Global Behavior of the 2D Dirac–Klein–Gordon System with a Class of Large Initial Data

Abstract

In this paper, we derive the global dynamic properties of the Dirac–Klein–Gordon system in ${\mathbb{R}}^{1+2}$ with a class of large initial data. We consider the case of a massless Dirac field coupled with a massive Klein–Gordon field. The initial data are bounded in certain weighted Sobolev spaces, where the Dirac field is small while the Klein–Gordon field can be arbitrarily large. We establish global existence and characterize the asymptotic behavior of the solutions, including sharp time decay and linear scattering.

1. Introduction

1.1. Model Problems

The Dirac–Klein–Gordon (DKG) system arises in particle physics as a model for Yukawa interactions between a Dirac field and a scalar field, see [1] for more details. In this paper, we consider the DKG system in ${\mathbb{R}}^{1+2}$ for a massless Dirac field coupled with a massive scalar field, which is mathematically formulated as follows:

$$\{\begin{array}{c}-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\psi =v\psi ,(t,x)\in [0,\infty )\times {\mathbb{R}}^2,\hfill \\ -\square v+v={\psi }^{*}{\gamma }^0\psi ,(t,x)\in [0,\infty )\times {\mathbb{R}}^2.\hfill \end{array}$$

(1)

The initial data are prescribed on the time slice $t=0$:

$$(\psi ,v,{\mathsf{\partial }}_{t}v){|}_{t=0}=({\psi }_0,v_0,v_1).$$

(2)

In the above expression, $i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }=i{\gamma }^0{\mathsf{\partial }}_0+i{\gamma }^1{\mathsf{\partial }}_1+i{\gamma }^2{\mathsf{\partial }}_2$ denotes the Dirac operator, with ${\mathsf{\partial }}_0={\mathsf{\partial }}_t$ and ${\mathsf{\partial }}_a={\mathsf{\partial }}_{x_a}$ for $a\in \{1,2\}$; $\psi =\psi (t,x):{\mathbb{R}}^{1+2}\to {\mathbb{C}}^2$ and $v=v(t,x):{\mathbb{R}}^{1+2}\to \mathbb{R}$ denote the Dirac field and the Klein–Gordon field, respectively. ${\psi }^{*}$ represents the complex conjugate transpose of the vector $\psi$, and ${\gamma }^{\mu }$ are the Dirac matrices. Einstein summation convention over repeated indices is adopted throughout the article. The Dirac matrices are defined by the following identities:

$${\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=-2g^{\mu \nu }\mathit{I},{\left({\gamma }^{\mu }\right)}^{*}=-g_{\mu \nu }{\gamma }^{\nu },$$

where $\mu ,\nu \in \{0,1,2\}$, $g=\left(g_{\mu \nu }\right)=\mathrm{diag}(-1,1,1)$ denotes the Minkowski metric in ${\mathbb{R}}^{1+2}$, $\left(g^{\mu \nu }\right)$ is the inverse matrix of $\left(g_{\mu \nu }\right)$, $\mathit{I}={\mathit{I}}_2$ is the $2\times 2$ identity matrix, and $A^{*}$ denotes the conjugate transpose of a matrix A. The wave operator is denoted by $\square =g^{\mu \nu }{\mathsf{\partial }}_{\mu }{\mathsf{\partial }}_{\nu }=-{\mathsf{\partial }}_t^2+{\mathsf{\partial }}_1^2+{\mathsf{\partial }}_2^2$. Particular representations of the Dirac matrices are given in Appendix A.

The DKG system in three spatial dimensions can be viewed as an effective field-theoretic model describing the interaction between a spin-${\displaystyle \frac{1}{2}}$ fermionic field and a spin-0 scalar field. It represents a simplified Yukawa-type coupling, where the scalar field mediates interactions between fermions, and captures certain features of fermion–boson interactions that are analogous to those in the electroweak sector of the Standard Model, such as fermion–Higgs coupling. In addition, the DKG system is relevant in lower-dimensional effective theories arising in condensed matter physics. In particular, two-dimensional Dirac-type equations appear in the description of low-energy excitations in graphene and surface states of topological insulators, where fermionic quasiparticles obey relativistic dispersion relations, see [2,3]. These considerations motivate the study of coupled Dirac–scalar field systems in lower dimensions.

The well-posedness and long-time behavior of the DKG system have been extensively studied. We first mention several works closely related to our study; with further discussion provided in Section 1.2. Grünrock-Pecher [4] demonstrated the global well-posedness for the DKG system with low-regularity, large data in two space dimensions, for all combinations of field masses. Dong-Wyatt [5] derived the global behavior (sharp time decay and linear scattering) of solutions to (1) with small, smooth, compactly supported initial data. The compactness assumption in [5] was subsequently removed in [6]. On the other hand, Candy-Herr [7] obtained conditional results on the global existence and scattering for large solutions of the massive DKG system in critical spaces in ${\mathbb{R}}^{1+3}$. In a recent work [8], the authors established the global dynamics of the DKG system with a small massless Dirac field and a large massive Klein–Gordon field in ${\mathbb{R}}^{1+3}$, which corresponds to the 3D version of (1). The aim of this paper is to demonstrate the global existence and asymptotic behavior of solutions to (1) with large initial data. This is motivated by the aforementioned works and large-data existence results for other wave-type equations, including those by Klainerman-Wang-Yang [9], Fang-Wang-Yang [10], and Yang-Yu [11], among others.

In this paper, we prove global existence, sharp time decay and linear scattering of the solutions to (1) and (2) under the condition that only the initial data for the Dirac field is small. Our assumption allows the Klein–Gordon field to have a finite weighted Sobolev norm with no restriction on its size. The smallness of the initial data for the Dirac field depends inversely polynomially on the size of the initial data for the Klein–Gordon field. These results demonstrate the global nonlinear stability of a large Klein–Gordon field when perturbed by a massless Dirac spinor field. In particular, we extend the results of [5,6,8] in several ways: $\left(\mathrm{i}\right)$ by addressing the long-range effects of an arbitrarily large Klein–Gordon field (thereby extending [5,6] to the case of large data); $\left(\mathrm{ii}\right)$ by considering the case of lower spatial dimensions, in contrast to [8].

The analysis of global existence and asymptotic dynamics for the DKG system with large initial data remains largely open, especially in lower spatial dimensions, where nonlinear interactions are stronger and dispersion is weaker. The main contribution of this paper is the establishment of the long-time dynamics of this system for a class of large initial data in two spatial dimensions. Our technical novelties include exploiting a vanishing structure in the Dirac energy, performing nonlinear transformations that generate rapidly decaying nonlinearities, and uncovering additional hidden structures in the nonlinear terms.

In the sequel, we use C to denote a universal constant whose value may change from line to line. As usual, $A\lesssim B$ means that $A\le CB$ for some constant C, and similar for $A\gtrsim B$. Given a vector or a scalar w, we use the Japanese bracket to denote $\langle w\rangle :={(1+|w|}^2{)}^{1/2}$.

1.2. Main Results and Discussion of the Related Literature

In this subsection, we will deal with the global dynamics of the solution to (1) and (2) around a nontrivial large solution.

For $l\in \mathbb{N}$, we denote by $H^l=H^l\left({\mathbb{R}}^2\right)$ the inhomogeneous Sobolev spaces and set $\nabla :=({\mathsf{\partial }}_1,{\mathsf{\partial }}_2)$. For $1\le p\le \infty$, we also denote $L^p=L^p\left({\mathbb{R}}^2\right)$. The main results of this paper are presented below.

Theorem 1.

Consider the Dirac–Klein–Gordon system (1) and let $N\ge 13$ be an integer. For any $K_{*}\ge 1$, there exists an ${ϵ}_0>0$, depending on $K_{*}$, such that for all initial data $({\psi }_0,v_0,v_1)$ satisfying

$$\begin{array}{cc}\hfill & \parallel {\psi }_0{\parallel }_{L^1}+\sum _{0\le k\le N}{\parallel \langle |x|\rangle }^{2+N}{\nabla }^k{\psi }_0{\parallel }_{L^2}\le ϵ\le {ϵ}_0,\hfill \\ \hfill & \sum _{0\le k\le N+1}\parallel \log (2+{|x|)\langle |x|\rangle }^{2+N}{\nabla }^k{v}_0{\parallel }_{L^2}+\sum _{0\le k\le N}\parallel \log (2+{|x|)\langle |x|\rangle }^{2+N}{\nabla }^k{v}_1{\parallel }_{L^2}\le K_{*},\hfill \end{array}$$

(3)

the Cauchy problem (1) and (2) admits a global solution $(\psi ,v)$ which decays as

$$|\psi (t,x)|\lesssim {ϵ}^{1/2}{\langle t+|x|\rangle }^{-1/2},|v(t,x)|\lesssim K_{*}^{2N}{\langle t+|x|\rangle }^{-1}.$$

(4)

Moreover, the solution $(\psi ,v)$ scatters linearly, i.e., there exists $({\psi }_0^{+},v_0^{+},v_1^{+})\in {\mathcal{X}}_{N-1}:=H^{N-1}\times H^N\times H^{N-1}$ such that

$$\underset{t\to +\infty }{\lim }{\parallel (\psi ,v,{\mathsf{\partial }}_{t}v)\left(t\right)-({\psi }^{+},v^{+},{\mathsf{\partial }}_t{v}^{+})\left(t\right)\parallel }_{{\mathcal{X}}_{N-1}}=0,$$

where $({\psi }^{+},v^{+})$ satisfies

$$\{\begin{array}{c}-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }{\psi }^{+}=0,(t,x)\in [0,\infty )\times {\mathbb{R}}^2,\hfill \\ -\square v^{+}+v^{+}=0,(t,x)\in [0,\infty )\times {\mathbb{R}}^2\hfill \end{array}$$

with the initial data $({\psi }^{+},v^{+},{\mathsf{\partial }}_t{v}^{+}){|}_{t=0}=({\psi }_0^{+},v_0^{+},v_1^{+})$.

Remark 1.

In Theorem 1, the constant ${ϵ}_0$ is related to $K_{*}$ by an inverse polynomial relationship, i.e.,

$$0<{ϵ}_0\le c{K}_{*}^{-4N(N+11)}.$$

Here, $c\in (0,1)$ is a small constant independent of $K_{*}$. Additionally, the implied constants in the relations in (4) are independent of ${ϵ}_0,ϵ$ and $K_{*}$.

Remark 2.

Theorem 1 can be interpreted as the global nonlinear stability of a large massive Klein–Gordon field under small perturbation of a massless Dirac field. Specifically, the problem (1) and (2) can be expressed as follows:

$$\left\{\begin{array}{c}-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\psi =(v^f+v^i)\psi ,(t,x)\in [0,\infty )\times {\mathbb{R}}^2,\hfill \\ -\square v^f+v^f=0,-\square v^i+v^i={\psi }^{*}{\gamma }^0\psi ,(t,x)\in [0,\infty )\times {\mathbb{R}}^2,\hfill \\ (\psi ,v^f,{\mathsf{\partial }}_t{v}^f,v^i,{\mathsf{\partial }}_t{v}^i){|}_{t=0}=({\psi }_0,v_0,v_1,0,0).\hfill \end{array}\right.$$

It is clear that $(\psi ,v)=(0,v^f)$ satisfies (1) with the initial data $(\psi ,v,{\mathsf{\partial }}_{t}v){|}_{t=0}=(0,v_0,v_1)$. Theorem 1 describes the global dynamics of solutions arising as small perturbations of this solution.

Over recent decades, significant progress has been made in studying the long-time behavior of solutions to wave-type equations. Here, we focus primarily on the works most relevant to our results.

For small initial data, the long-time dynamics of the Dirac–Klein–Gordon (DKG) system in ${\mathbb{R}}^{1+3}$ are well-established. Choquet-Bruhat [12] demonstrated global existence and scattering for the massless DKG system. For small, smooth initial data, Bachelot [13] showed asymptotic stability for the DKG system with a massive Dirac field and a massless Klein–Gordon field. For the combinations of field masses considered in [12,13], the uniform boundedness of the total energy of the solution was further obtained in [14]. In the low regularity setting, Wang [15] and Bejenaru-Herr [16] established global well-posedness and scattering for the DKG system under a non-resonance condition on the masses. We also refer to the work by D’Ancona-Foschi-Selberg [17] for an almost optimal local well-posedness result.

In the case of two space dimensions, the massive DKG system, which can be reduced to coupled Klein–Gordon equations, was studied by Simon-Taflin [18], Ozawa-Tsutaya-Tsutsumi [19] and Delort-Fang-Xue [20]. For small, smooth initial data, the asymptotic behavior of solutions to (1) was exploited by Dong-Wyatt [5] and Dong-Li-Ma-Yuan [6], see also the work of the author [21].

For large initial data, Grünrock-Pecher [4] demonstrated global existence for the DKG system with low-regularity, large data in ${\mathbb{R}}^{1+2}$, applicable to all combinations of field masses. Candy-Herr [7] established conditional large initial data scattering for the massive DKG system in ${\mathbb{R}}^{1+3}$. In a recent work [8], the authors proved global existence and asymptotic behavior for the DKG system in ${\mathbb{R}}^{1+3}$ with a large massive Klein–Gordon field. The same paper also considered the Klein–Gordon–Zakharov system in ${\mathbb{R}}^{1+3}$ with a large massless field.

We now discuss related research on large-data existence results for wave-type equations. Klainerman-Wang-Yang [9] and Fang-Wang-Yang [10] established global existence and asymptotic behavior of solutions to the massive Maxwell–Klein–Gordon equations in ${\mathbb{R}}^{1+3}$, assuming that the scalar field is small while the Maxwell field can be large. The corresponding results for the massless Maxwell–Klein–Gordon equations were proved by Yang [22] and Yang-Yu [11], see also Wei-Yang-Yu [23] on Yang-Mills-Higgs equations. Miao-Pei-Yu [24] demonstrated the global behavior of smooth solutions to the semilinear wave equation in ${\mathbb{R}}^{1+3}$ under the null condition with short pulse data introduced by Christodoulou [25]. Luk-Oh-Yang [26] investigated the Einstein scalar-field equations with spherically symmetric initial data and constructed global solutions with arbitrarily large (even infinitely large) initial bounded variation norms. Furthermore, Li [27] considered the Skyrme model in three space dimensions and obtained the global existence of hedgehog solutions with large data using a non-local energy bootstrap strategy.

This article is organized as follows. In Section 2, we first introduce the notations and key facts, followed by weighted energy estimates for both fields and Sobolev type inequalities. We also present some technical lemmas related to $L^2$ and $L^{\infty }$ estimates, as well as linear scattering for the fields. Section 3 and Section 4 are dedicated to the complete proof of Theorem 1. Specifically, in Section 3.1, we introduce a bootstrap argument for the solution to (1) and (2). The additional decay properties for the Klein–Gordon field and the gradient of the Dirac field are provided in Section 3.2. In Section 3.3, we demonstrate that the control of top-order energy and $L^2$-type norms for both fields can be maintained. Refined lower-order energy and pointwise bounds for the Dirac field and the Klein–Gordon field are established in Section 4.1 and Section 4.2, respectively. Finally, the proof of the linear scattering result in Theorem 1 is provided in Section 4.3. In Section 5, we summarize the main conclusions of this paper and present some perspectives. Several technical details are deferred to Appendix A, Appendix B and Appendix C.

2. Preliminaries

2.1. Notations

In this subsection, we introduce some notation that will be used in the proof of the main results.

We work in $(1+2)$ dimensional spacetime ${\mathbb{R}}^{1+2}$ with the Minkowski metric $g=\mathrm{diag}(-1,1,1)$, which is used to raise or lower indices. Spatial indices are represented by Roman letters $\{a,b,c,\dots \}$, ranging over $\{1,2\}$, while spacetime indices are denoted by Greek letters $\{\alpha ,\beta ,\gamma ,\dots \}$, ranging over $\{0,1,2\}$. Throughout the paper, we adopt the Einstein summation convention for repeated upper and lower indices. A point in ${\mathbb{R}}^{1+2}$ is denoted by $(t,x)=(x_0,x_1,x_2)$, where $t=x_0,x=(x_1,x_2)$, with $x^a=x_a$ for $a=1,2$. The spatial radius is denoted by $r:=|x|=\sqrt{x_1^2+x_2^2}$. Following Klainerman [28], we introduce the following vector fields:

(i)

Translations: ${\mathsf{\partial }}_{\alpha }:={\mathsf{\partial }}_{x_{\alpha }}$, for $\alpha =0,1,2$.

(ii)

Lorentz boosts: $L_a:=x_a{\mathsf{\partial }}_t+t{\mathsf{\partial }}_a$, for $a=1,2$.

(iii)

Rotation: $\mathsf{\Omega }:=x_1{\mathsf{\partial }}_2-x_2{\mathsf{\partial }}_1$.

(iv)

Scaling: $L_0=t{\mathsf{\partial }}_t+x^a{\mathsf{\partial }}_a$.

We also use the modified Lorentz boosts and rotation introduced by Bachelot [13],

$${\widehat{L}}_a:=L_a-{\displaystyle \frac{1}{2}}{\gamma }^0{\gamma }^a,\widehat{\mathsf{\Omega }}:=\mathsf{\Omega }-{\displaystyle \frac{1}{2}}{\gamma }^1{\gamma }^2,$$

which satisfy the following commutative property, namely:

$$[{\widehat{L}}_a,i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }]=0,\mathrm{for}a=1,2,[\widehat{\mathsf{\Omega }},i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }]=0.$$

Here the commutator $[A,B]$ is defined as

$$[A,B]:=AB-BA.$$

The good derivatives are denoted by

$$G_a={\mathsf{\partial }}_a+{\displaystyle \frac{x_a}{r}}{\mathsf{\partial }}_t,\mathrm{for}a=1,2.$$

(5)

Additionally, we define the ordered sets

$$\begin{array}{cc}\hfill {\left\{{\mathsf{\Gamma }}_k\right\}}_{k=1}^6:& =\{{\left({\mathsf{\partial }}_{\alpha }\right)}_{0\le \alpha \le 2},{\left(L_a\right)}_{1\le a\le 2},\mathsf{\Omega }\},\hfill \\ \hfill {\{{\widehat{\mathsf{\Gamma }}}_k\}}_{k=1}^6:& =\{{({\mathsf{\partial }}_{\alpha })}_{0\le \alpha \le 2},{({\widehat{L}}_a)}_{1\le a\le 2},\widehat{\mathsf{\Omega }}\},\hfill \end{array}$$

For any multi-index $I=(i_1,i_2,\dots ,i_6)\in {\mathbb{N}}^6$ of length $|I|={\sum }_{k=1}^6{i}_k$, we denote

$${\mathsf{\Gamma }}^I={\displaystyle \prod _{k=1}^6}{\mathsf{\Gamma }}_k^{i_k},{\widehat{\mathsf{\Gamma }}}^I={\displaystyle \prod _{k=1}^6}{\widehat{\mathsf{\Gamma }}}_k^{i_k}.$$

Furthermore, we also denote

$$({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)=({\mathsf{\partial }}_r,\mathsf{\Omega })\mathrm{and}{\mathsf{\Lambda }}^I={\mathsf{\Lambda }}_1^{i_1}{\mathsf{\Lambda }}_2^{i_2},\mathrm{for}I=(i_1,i_2)\in {\mathbb{N}}^2.$$

(6)

Given an index set $\mathsf{\Lambda }$, a finite set $\{P_{\lambda }:\lambda \in \mathsf{\Lambda }\}$ of linear operators and a linear operator Q, we write

$$Q=\underset{\lambda \in \mathsf{\Lambda }}{{\sum }^{\prime }}{P}_{\lambda }$$

if there exist some constants $c_{\lambda }$ such that we have

$$Q=\sum _{\lambda \in \mathsf{\Lambda }}{c}_{\lambda }{P}_{\lambda }.$$

When there are two parameters, $\lambda$ and $\theta$, by writing $Q_{\lambda }=\underset{\theta \in \mathsf{\Theta }}{{\sum }^{\prime }}{P}_{\lambda ,\theta }$, we mean that, for each $\lambda$, there exist the constants $c_{\lambda ,\theta }$ such that $Q_{\lambda }={\sum }_{\theta \in \mathsf{\Theta }}{c}_{\lambda ,\theta }{P}_{\lambda ,\theta }$. Especially, the expression

$$Q=\sum _{\lambda \in \mathsf{\Lambda }}\underset{\theta \in \mathsf{\Theta }}{{\sum }^{\prime }}{P}_{\lambda ,\theta }$$

means that there are the constants $c_{\lambda ,\theta }$ such that

$$Q=\sum _{\lambda \in \mathsf{\Lambda }}\sum _{\theta \in \mathsf{\Theta }}{c}_{\lambda ,\theta }{P}_{\lambda ,\theta }.$$

For any sufficiently regular $\mathbb{C}$-valued functions $f=f(t,x),g=g(t,x)$ defined on ${\mathbb{R}}^{1+2}$, we denote the standard null forms by

$$Q_0(f,g)={\mathsf{\partial }}_{\alpha }f{\mathsf{\partial }}^{\alpha }g.$$

In the case that f is ${\mathbb{C}}^2$-valued and g is $\mathbb{C}$-valued, $Q_0(f,g)$ is defined by each of its components. We fix a smooth cutoff function $\chi :\mathbb{R}\to \mathbb{R}$ such that

$${\chi }^{\prime }\ge 0,\chi \left(s\right)=\{\begin{array}{ccc}\hfill 0,& & \mathrm{for}s\le 1,\hfill \\ \hfill 1,& & \mathrm{for}s\ge 2.\hfill \end{array}$$

(7)

Let $1\le p<\infty$. For any sufficiently smooth function $f=f(t,x)$ defined on ${\mathbb{R}}^{1+2}$, we denote

$${\parallel f\parallel }_{L_x^p}={\parallel f\parallel }_{L_x^p\left({\mathbb{R}}^2\right)}:={\left({\int }_{{\mathbb{R}}^2}{|f(t,x)|}^p\mathrm{d}x\right)}^{1/p}{,\parallel f\parallel }_{L_x^{\infty }}={\parallel f\parallel }_{L_x^{\infty }\left({\mathbb{R}}^2\right)}:=\underset{x\in {\mathbb{R}}^2}{\sup }|f(t,x)|.$$

2.2. Estimates on Vector Fields and Null Forms

In this subsection, we present several key facts and estimates concerning vector fields and null forms.

Let $\mathit{I}$ denote the $2\times 2$ identity matrix and define the Hermitian matrices

$$T_{-}:=\mathit{I}-{\displaystyle \frac{x_a}{r}}{\gamma }^0{\gamma }^a,T_{+}:=\mathit{I}+{\displaystyle \frac{x_a}{r}}{\gamma }^0{\gamma }^a.$$

(8)

Following Dong-Wyatt [5], we introduce the following notation to illustrate the underlying structure in the nonlinear term ${\psi }^{*}{\gamma }^0\psi$ in (1):

$${[\phi ]}_{-}:=T_{-}\phi =\phi -{\displaystyle \frac{x_a}{r}}{\gamma }^0{\gamma }^a\phi ,{[\phi ]}_{+}:=T_{+}\phi =\phi +{\displaystyle \frac{x_a}{r}}{\gamma }^0{\gamma }^a\phi$$

(9)

for any function $\phi =\phi (t,x):{\mathbb{R}}^{1+2}\to {\mathbb{C}}^2$.

Proposition 1.

The following statements hold:

(i) 

Let $\phi =\phi (t,x):{\mathbb{R}}^{1+2}\to {\mathbb{C}}^2$ be a sufficiently smooth function and $I,J\in {\mathbb{N}}^6$. Then

$${\widehat{\mathsf{\Gamma }}}^I\phi ={\mathsf{\Gamma }}^I\phi +\sum _{|I^{\prime }|<|I|}{c}_{I^{\prime }}{\mathsf{\Gamma }}^{I^{\prime }}\phi ,{\mathsf{\Gamma }}^J\phi ={\widehat{\mathsf{\Gamma }}}^J\phi +\sum _{|J^{\prime }|<|J|}{d}_{J^{\prime }}{\widehat{\mathsf{\Gamma }}}^{J^{\prime }}\phi$$

for some constant matrices $c_{I^{\prime }},d_{J^{\prime }}$.

(ii) 

For any multi-indices $I,J\in {\mathbb{N}}^6$ and $0\le \alpha \le 2$, we have

$${\mathsf{\Gamma }}^I{\mathsf{\Gamma }}^J={\mathsf{\Gamma }}^{I+J}+\underset{|K|<|I|+|J|}{{\sum }^{\prime }}{\mathsf{\Gamma }}^K,[{\mathsf{\Gamma }}^I,{\mathsf{\partial }}_{\alpha }]=\sum _{0\le \beta \le 2}\underset{|K|<|I|}{{\sum }^{\prime }}{\mathsf{\partial }}_{\beta }{\mathsf{\Gamma }}^K=\sum _{0\le \beta \le 2}\underset{|K|<|I|}{{\sum }^{\prime }}{\mathsf{\Gamma }}^K{\mathsf{\partial }}_{\beta }.$$

(iii) 

Let $u=u(t,x):{\mathbb{R}}^{1+2}\to \mathbb{R}$, $\phi =\phi (t,x),\mathsf{\Phi }=\mathsf{\Phi }(t,x):{\mathbb{R}}^{1+2}\to {\mathbb{C}}^2$ be sufficiently smooth functions and $I\in {\mathbb{N}}^6$. Then, we have

$${\widehat{\mathsf{\Gamma }}}^I\left(u\phi \right)=\sum _{I_1+I_2=I}({\mathsf{\Gamma }}^{I_1}u)({\widehat{\mathsf{\Gamma }}}^{I_2}\phi ),$$

(10)

$${\mathsf{\Gamma }}^I\left({\phi }^{*}{\gamma }^0\mathsf{\Phi }\right)=\sum _{I_1+I_2=I}{({\widehat{\mathsf{\Gamma }}}^{I_1}\phi )}^{*}{\gamma }^0({\widehat{\mathsf{\Gamma }}}^{I_2}\mathsf{\Phi }).$$

(11)

(iv) 

For any functions φ and $\mathsf{\Phi }:{\mathbb{R}}^{1+2}\to {\mathbb{C}}^2$, we have

$${\phi }^{*}{\gamma }^0\mathsf{\Phi }={\displaystyle \frac{1}{4}}\left\{{[\phi ]}_{-}^{*}{\gamma }^0{[\mathsf{\Phi }]}_{+}+{[\phi ]}_{+}^{*}{\gamma }^0{[\mathsf{\Phi }]}_{-}\right\}.$$

(12)

In addition, for any sufficiently smooth function $\mathsf{\Phi }:{\mathbb{R}}^{1+2}\to {\mathbb{C}}^2$, let $\phi :=i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\mathsf{\Phi }$. Then

$${[\phi ]}_{-}=i\left(\mathit{I}-{\displaystyle \frac{x_b}{r}}{\gamma }^0{\gamma }^b\right){\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\mathsf{\Phi }=i\left(\mathit{I}-{\displaystyle \frac{x_b}{r}}{\gamma }^0{\gamma }^b\right){\gamma }^a{G}_a\mathsf{\Phi }.$$

(v) 

For any sufficiently smooth function u defined on ${\mathbb{R}}^{1+2}$, it holds that

$$\langle t-r\rangle |\mathsf{\partial }u|+\langle t+r\rangle \sum _{1\le a\le 2}|G_{a}u|\lesssim |L_{0}u|+\sum _{|I|=1}|{\mathsf{\Gamma }}^{I}u|.$$

(vi) 

For any smooth $\mathbb{C}$-valued or ${\mathbb{C}}^2$-valued function f and $\mathbb{C}$-valued function g defined on ${\mathbb{R}}^{1+2}$ and any multi-index $I\in {\mathbb{N}}^6$, we have

$$\begin{array}{cc}\hfill |Q_0(f,g)|& \lesssim {\displaystyle \frac{1}{\langle t+r\rangle }}\left\{|L_{0}f|·|\mathsf{\partial }g|+\sum _{|J_1|,|J_2|\le 1}|{\mathsf{\Gamma }}^{J_1}f|·|{\mathsf{\Gamma }}^{J_2}g|\right\},\hfill \\ \hfill |{\mathsf{\Gamma }}^I{Q}_0(f,g)|& \lesssim \sum _{|I_1|+|I_2|\le |I|}|Q_0({\mathsf{\Gamma }}^{I_1}f,{\mathsf{\Gamma }}^{I_2}g)|,\hfill \\ \hfill |{\mathsf{\Gamma }}^I{Q}_0(f,g)|& \lesssim \sum _{|I_1|+|I_2|\le |I|}\left\{{\displaystyle \frac{1}{\langle t+r\rangle }}\sum _{|J_1|,|J_2|\le 1}|{\mathsf{\Gamma }}^{J_1}{\mathsf{\Gamma }}^{I_1}f|·|{\mathsf{\Gamma }}^{J_2}{\mathsf{\Gamma }}^{I_2}g|+{\displaystyle \frac{\langle t-r\rangle }{\langle t+r\rangle }}|\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}f|·|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}g|\right\}.\hfill \end{array}$$

Proof. 

The proofs of $\left(i\right)$–$\left(vi\right)$ can be found in [5,6,29]. For completeness, we provide a brief sketch of these proofs below. $\left(i\right)$ follows directly from the definitions of the vector fields presented in Section 2.1. $\left(ii\right)$ can be derived from the following identities:

$$\begin{array}{cc}& [{\mathsf{\partial }}_{\alpha },L_a]=\underset{0\le \beta \le 2}{{\sum }^{\prime }}{\mathsf{\partial }}_{\beta },[{\mathsf{\partial }}_{\alpha },\mathsf{\Omega }]=\underset{0\le \beta \le 2}{{\sum }^{\prime }}{\mathsf{\partial }}_{\beta },[{\mathsf{\partial }}_{\alpha },L_0]={\mathsf{\partial }}_{\alpha },0\le \alpha \le 2,1\le a\le 2,\hfill \\ & [Z_k,Z_l]=\underset{Z_j\in V}{{\sum }^{\prime }}{Z}_j,Z_k,Z_l\in V:=\{{\left(L_a\right)}_{1\le a\le 2},\mathsf{\Omega }\},\hfill \\ & [L_0,\mathsf{\Omega }]=[L_0,L_a]=0,1\le a\le 2.\hfill \end{array}$$

For $\left(iii\right)$, a straightforward calculation shows that

$$\begin{array}{cc}\hfill {\widehat{L}}_a\left(u\phi \right)& =(L_{a}u)\phi +u({\widehat{L}}_a\phi ),\mathrm{for}1\le a\le 2,\hfill \\ \hfill L_a\left({\phi }^{*}{\gamma }^0\mathsf{\Phi }\right)& ={(L_a\phi )}^{*}{\gamma }^0\mathsf{\Phi }+{\phi }^{*}{\gamma }^0{L}_a\mathsf{\Phi }={({\widehat{L}}_a\phi )}^{*}{\gamma }^0\mathsf{\Phi }+{\displaystyle \frac{1}{2}}{\phi }^{*}{\gamma }^0{\gamma }^a{\gamma }^0\mathsf{\Phi }+{\phi }^{*}{\gamma }^0{\widehat{L}}_a\mathsf{\Phi }+{\displaystyle \frac{1}{2}}{\phi }^{*}{\gamma }^0{\gamma }^0{\gamma }^a\mathsf{\Phi }\hfill \\ & ={({\widehat{L}}_a\phi )}^{*}{\gamma }^0\mathsf{\Phi }+{\phi }^{*}{\gamma }^0{\widehat{L}}_a\mathsf{\Phi },\mathrm{for}1\le a\le 2,\hfill \end{array}$$

and similarly for $\widehat{\mathsf{\Omega }}$. Hence, $\left(iii\right)$ follows from these identities and the induction argument. For $\left(iv\right)$, let $T_{+}$ and $T_{-}$ be as in (8). We write

$$\phi ={\displaystyle \frac{T_{-}\phi +T_{+}\phi }{2}},\mathsf{\Phi }={\displaystyle \frac{T_{-}\mathsf{\Phi }+T_{+}\mathsf{\Phi }}{2}}.$$

By direct calculation,

$$T_{-}{\gamma }^0{T}_{-}=T_{+}{\gamma }^0{T}_{+}=0.$$

(13)

Furthermore, by using that ${\mathsf{\partial }}_a=G_a-{\displaystyle \frac{x_a}{r}}{\mathsf{\partial }}_t$, we can express

$${\gamma }^0{\mathsf{\partial }}_t+{\gamma }^a{\mathsf{\partial }}_a={\gamma }^0\left(\mathit{I}-{\displaystyle \frac{x_a}{r}}{\gamma }^0{\gamma }^a\right){\mathsf{\partial }}_t+{\gamma }^a{G}_a.$$

(14)

Then, the relation (13) implies

$$\left(\mathit{I}-{\displaystyle \frac{x_b}{r}}{\gamma }^0{\gamma }^b\right){\gamma }^0\left(\mathit{I}-{\displaystyle \frac{x_a}{r}}{\gamma }^0{\gamma }^a\right)=T_{-}{\gamma }^0{T}_{-}=0.$$

(15)

Hence, $\left(iv\right)$ follows from (13)–(15). For the proofs of $\left(v\right)$ and the first two inequalities in $\left(vi\right)$, we refer to [29] (page 39 and Proposition 1.1, page 58 and Lemma 3.3), and use the identity

$$G_{a}v={\displaystyle \frac{1}{r}}\left(L_{a}v+(r-t){\mathsf{\partial }}_{a}v\right)={\displaystyle \frac{1}{t}}\left(L_{a}v-{\displaystyle \frac{x_a}{r}}(r-t){\mathsf{\partial }}_{t}v\right).$$

The proof of the last inequality in $\left(vi\right)$ follows directly from the identity

$$L_0=(t-r){\mathsf{\partial }}_t+(r-t){\mathsf{\partial }}_r+(x^a/r)L_a$$

and the first two inequalities in $\left(vi\right)$. □

2.3. Energy and $L^2$-Type Estimates

In this subsection, we establish weighted energy estimates and $L^2$-type estimates for wave-type equations.

Given $\delta \ge 0$ and $\phi =\phi (t,x):{\mathbb{R}}^{1+2}\to {\mathbb{C}}^2$, we denote

$${\mathcal{E}}_D^{\delta }(t,\phi ):={\int }_{{\mathbb{R}}^2}{\langle t\rangle }^{-\delta }{|\phi (t,x)|}^2\mathrm{d}x+{\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-\delta }{\displaystyle \frac{{|[\phi ]}_{-}{(s,x)|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s,$$

(16)

where we refer to (9) for the definition of ${[\phi ]}_{-}$. For simplicity, we denote ${\mathcal{E}}_D(t,\phi ):={\mathcal{E}}_D^0(t,\phi )$, which is the ghost energy adapted to the Dirac equation; the ghost energy for the wave equation was originally introduced by Alinhac [30]. The following lemma provides weighted energy inequalities for the Dirac field. Using this lemma, we find that when a vector field ${\widehat{\mathsf{\Gamma }}}^I$ is applied to both sides of the first equation in (1), the term $v{\widehat{\mathsf{\Gamma }}}^I\psi$ in the nonlinearity vanishes from the energy inequality. This will help address the challenge of closing the top-order energy estimate for $\psi$, which arises from the presence of the large Klein–Gordon field.

Lemma 1.

Let $\delta \ge 0$. Suppose $\phi =\phi (t,x)$ is the solution to the Cauchy problem

$$-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }{\phi +u\phi =G\mathrm{with}\phi |}_{t=0}={\phi }_0,$$

(17)

where $u=u(t,x):{\mathbb{R}}^{1+2}\to \mathbb{R}$ is a real-valued function, ${\phi }_0={\phi }_0\left(x\right):{\mathbb{R}}^2\to {\mathbb{C}}^2$ and $G=G(t,x):{\mathbb{R}}^{1+2}\to {\mathbb{C}}^2$ are sufficiently nice functions such that the terms on the right hand sides of the estimates below are finite. Then, we have

$${\mathcal{E}}_D^{\delta }(t,\phi )\lesssim {\mathcal{E}}_D^{\delta }(0,\phi )+{\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-\delta }|{\phi }^{*}(s,x){\gamma }^{0}G(s,x)|\mathrm{d}x\mathrm{d}s,$$

(18)

$${[{\mathcal{E}}_D^{\delta }(t,\phi )]}^{{\displaystyle \frac{1}{2}}}\lesssim {[{\mathcal{E}}_D^{\delta }(0,\phi )]}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\parallel {\langle s\rangle }^{-{\displaystyle \frac{\delta }{2}}}G(s,x)\parallel }_{L_x^2}\mathrm{d}s,$$

(19)

$${\parallel \langle r-t\rangle \chi (r-2t)\phi (t,x)\parallel }_{L_x^2}\lesssim \parallel \langle r\rangle {\phi }_0{\parallel }_{L_x^2}+{\int }_0^t{\parallel \langle r-s\rangle \chi (r-2s)G(s,x)\parallel }_{L_x^2}\mathrm{d}s,$$

(20)

where we recall (7) for the definition of χ.

Proof. 

Let ω be a weight. Multiplying $i\omega {\phi }^{*}{\gamma }^0$ to both sides of (17), we obtain

$$\omega {\phi }^{*}{\mathsf{\partial }}_t\phi +\omega {\phi }^{*}{\gamma }^0{\gamma }^a{\mathsf{\partial }}_a\phi +iu\omega {\phi }^{*}{\gamma }^0\phi =i\omega {\phi }^{*}{\gamma }^{0}G.$$

Taking the complex conjugate of the last equality, we have

$$\omega {\mathsf{\partial }}_t{\phi }^{*}\phi +\omega {\mathsf{\partial }}_a{\phi }^{*}{\gamma }^0{\gamma }^a\phi -iu\omega {\phi }^{*}{\gamma }^0\phi =-i\omega G^{*}{\gamma }^0\phi .$$

Adding the last two equalities, we get

$$\omega {\mathsf{\partial }}_t\left({\phi }^{*}\phi \right)+\omega {\mathsf{\partial }}_a\left({\phi }^{*}{\gamma }^0{\gamma }^a\phi \right)=i\omega \left({\phi }^{*}{\gamma }^{0}G-G^{*}{\gamma }^0\phi \right),$$

which yields

$${\mathsf{\partial }}_t\left(\omega {\phi }^{*}\phi \right)+{\mathsf{\partial }}_a\left(\omega {\phi }^{*}{\gamma }^0{\gamma }^a\phi \right)-{\mathsf{\partial }}_t\omega {\phi }^{*}\phi -{\mathsf{\partial }}_a\omega {\phi }^{*}{\gamma }^0{\gamma }^a\phi =i\omega \left({\phi }^{*}{\gamma }^{0}G-G^{*}{\gamma }^0\phi \right).$$

(21)

Integrating (21) over $[0,t]\times {\mathbb{R}}^2$, we see that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}\omega (t,x){|\phi |}^2(t,x)\mathrm{d}x+{\int }_0^t{\int }_{{\mathbb{R}}^2}\left\{-{\mathsf{\partial }}_t\omega {\phi }^{*}\phi -{\mathsf{\partial }}_a\omega {\phi }^{*}{\gamma }^0{\gamma }^a\phi \right\}\mathrm{d}x\mathrm{d}\tau \hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\omega (0,x){|\phi |}^2(0,x)\mathrm{d}x+i{\int }_0^t{\int }_{{\mathbb{R}}^2}\omega \left({\phi }^{*}{\gamma }^{0}G-G^{*}{\gamma }^0\phi \right)\mathrm{d}x\mathrm{d}\tau .\hfill \end{array}$$

(22)

Let ${\omega }_1:={\langle t\rangle }^{-\delta }{e}^{\tilde{q}(r-t)}$ and ${\omega }_2:={\langle r-t\rangle }^2{\chi }^2(r-2t)$, where

$$\tilde{q}\left(s\right):={\int }_{-\infty }^s{\displaystyle \frac{1}{{\langle s^{\prime }\rangle }^{\frac{6}{5}}}}\mathrm{d}{s}^{\prime }.$$

(23)

By elementary calculation, we find

$$\begin{array}{cc}\hfill -{\mathsf{\partial }}_t{\omega }_1{\phi }^{*}\phi -{\mathsf{\partial }}_a{\omega }_1{\phi }^{*}{\gamma }^0{\gamma }^a\phi & =\delta {\langle t\rangle }^{-\delta -2}t{e}^{\tilde{q}(r-t)}{\phi }^{*}\phi +{\displaystyle \frac{1}{2}}{\langle t\rangle }^{-\delta }{e}^{\tilde{q}(r-t)}{\langle r-t\rangle }^{-6/5}{|{[\phi ]}_{-}|}^2\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2}}{\langle t\rangle }^{-\delta }{e}^{\tilde{q}(r-t)}{\langle r-t\rangle }^{-6/5}{|{[\phi ]}_{-}|}^2,\hfill \\ \hfill -{\mathsf{\partial }}_t{\omega }_2{\phi }^{*}\phi -{\mathsf{\partial }}_a{\omega }_2{\phi }^{*}{\gamma }^0{\gamma }^a\phi & =2{\langle r-t\rangle }^2\chi (r-2t){\chi }^{\prime }(r-2t){\phi }^{*}\phi \hfill \\ \hfill & +\left\{(r-t){\chi }^2(r-2t)+{\langle r-t\rangle }^2\chi (r-2t){\chi }^{\prime }(r-2t)\right\}{|{[\phi ]}_{-}|}^2\hfill \\ \hfill & \ge 0,\hfill \end{array}$$

(24)

where we use that

$${\phi }^{*}\phi -{\displaystyle \frac{x_a}{r}}{\phi }^{*}{\gamma }^0{\gamma }^a\phi ={\displaystyle \frac{1}{2}}{|{[\phi ]}_{-}|}^2.$$

First, we substitute $\omega ={\omega }_1$ into (22). Using (24) and the fact that $e^{\tilde{q}(r-t)}\sim 1$, we obtain (18). Next, by differentiating (22) with respect to t, and applying (24) along with the Hölder inequality, we derive (19). Finally, setting $\omega ={\omega }_2$ in (22), differentiating (22) with respect to t and using (24) and the Hölder inequality, we obtain (20). □

Let $m=0$ or 1. We introduce the standard energy and Alinhac’s ghost weight energy for the $2D$ linear wave and Klein–Gordon equations as follows:

$$\begin{array}{cc}\hfill {\mathcal{E}}_m(t,u):& ={\int }_{{\mathbb{R}}^2}\left({|\mathsf{\partial }u(t,x)|}^2+m^2{|u(t,x)|}^2\right)\mathrm{d}x,\hfill \\ \hfill {\mathcal{G}}_m(t,u):& ={\mathcal{E}}_m(t,u)+{\int }_0^t{\int }_{{\mathbb{R}}^2}{\displaystyle \frac{{|Gu(s,x)|}^2+m^2{|u(s,x)|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}\mathrm{s},\hfill \end{array}$$

(25)

where

$$|\mathsf{\partial }u|={\left(\sum _{0\le \alpha \le 2}{|{\mathsf{\partial }}_{\alpha }u|}^2\right)}^{1/2},|Gu|={\left(\sum _{1\le a\le 2}{|G_{a}u|}^2\right)}^{1/2}.$$

We denote for simplicity $\mathcal{E}(t,u):={\mathcal{E}}_0(t,u)$ and $\mathcal{G}(t,u):={\mathcal{G}}_0(t,u)$. Given $\delta \ge 0$, we also denote

$${\mathcal{G}}_m^{\delta }(t,u):={\int }_{{\mathbb{R}}^2}{\langle t\rangle }^{-\delta }\left({|\mathsf{\partial }u(t,x)|}^2+m^2{|u(t,x)|}^2\right)\mathrm{d}x+{\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-\delta }{\displaystyle \frac{{|Gu(s,x)|}^2+m^2{|u(s,x)|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}\mathrm{s}.$$

It is obvious that ${\mathcal{G}}_m^0(t,u)={\mathcal{G}}_m(t,u)$.

Lemma 2.

Let $m=0$ or 1 and $\delta \ge 0$. Suppose u is the solution to the Cauchy problem

$$-\square u+m^{2}u=F\mathrm{with}(u,{\mathsf{\partial }}_{t}u){|}_{t=0}=(u_0,u_1),$$

(26)

where $u_0=u_0\left(x\right)$ and $u_1=u_1\left(x\right):{\mathbb{R}}^2\to \mathbb{R}$ and $F=F(t,x):{\mathbb{R}}^{1+2}\to \mathbb{R}$ are sufficiently nice functions such that the terms on the right hand sides of the estimates below are finite. Then

$${[{\mathcal{E}}_m(t,u)]}^{{\displaystyle \frac{1}{2}}}\lesssim {[{\mathcal{E}}_m(0,u)]}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\parallel F(s,x)\parallel }_{L_x^2}\mathrm{d}s,$$

(27)

$${\mathcal{G}}_m^{\delta }(t,u)\lesssim {\mathcal{E}}_m(0,u)+{\int }_0^t{\langle s\rangle }^{-\delta }{\parallel F(s,x)\parallel }_{L_x^2}·{\parallel {\mathsf{\partial }}_{t}u(s,x)\parallel }_{L_x^2}\mathrm{d}s,$$

(28)

$${[{\mathcal{G}}_m^{\delta }(t,u)]}^{{\displaystyle \frac{1}{2}}}\lesssim {[{\mathcal{E}}_m(0,u)]}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{\delta }{2}}}{\parallel F(s,x)\parallel }_{L_x^2}\mathrm{d}s,$$

(29)

$$\begin{array}{cc}\hfill & \parallel \langle r-t\rangle \chi (r-2t)\left(|\mathsf{\partial }u|+m|u|\right){(t,x)\parallel }_{L_x^2}\lesssim {\parallel \langle r\rangle \left(|\mathsf{\partial }u|+m|u|\right)(0,x)\parallel }_{L_x^2}\hfill \\ \hfill & +{\int }_0^t{\parallel \langle r-s\rangle \chi (r-2s)F(s,x)\parallel }_{L_x^2}\mathrm{d}s.\hfill \end{array}$$

(30)

Proof. 

Let ω be a weight. By multiplying both sides of (26) by $2\omega {\mathsf{\partial }}_{t}u$ and performing elementary calculation, we obtain

$${\mathsf{\partial }}_t\left(\omega (|\mathsf{\partial }u{|}^2+m^2{u}^2)\right)-2{\mathsf{\partial }}_a\left(\omega {\mathsf{\partial }}_{t}u{\mathsf{\partial }}^{a}u\right)-{\mathsf{\partial }}_t{\omega (|\mathsf{\partial }u|}^2+m^2{u}^2)+2{\mathsf{\partial }}_a\omega {\mathsf{\partial }}_{t}u{\mathsf{\partial }}^{a}u=2\omega {\mathsf{\partial }}_{t}uF.$$

(31)

Integrating (31) over $[0,t]\times {\mathbb{R}}^2$, we get

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}\omega (t,x)\left({|\mathsf{\partial }u(t,x)|}^2+m^2{|u(t,x)|}^2\right)\mathrm{d}x\hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^2}\left\{-{\mathsf{\partial }}_t{\omega (|\mathsf{\partial }u|}^2+m^2{u}^2)+2{\mathsf{\partial }}_a\omega {\mathsf{\partial }}_{t}u{\mathsf{\partial }}^{a}u\right\}\mathrm{d}x\mathrm{d}\tau \hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\omega (0,x)\left({|\mathsf{\partial }u(0,x)|}^2+m^2{|u(0,x)|}^2\right)\mathrm{d}x+2{\int }_0^t{\int }_{{\mathbb{R}}^2}\omega {\mathsf{\partial }}_{t}uF\mathrm{d}x\mathrm{d}\tau .\hfill \end{array}$$

(32)

Let ${\omega }_1:={\langle t\rangle }^{-\delta }{e}^{\tilde{q}(r-t)}$ and ${\omega }_2:={\langle r-t\rangle }^2{\chi }^2(r-2t)$, where $\tilde{q}$ is defined in (23). By direct calculation, we find

$$\begin{array}{cc}\hfill & -{\mathsf{\partial }}_t{\omega }_1{(|\mathsf{\partial }u|}^2+m^2{u}^2)+2{\mathsf{\partial }}_a{\omega }_1{\mathsf{\partial }}_{t}u{\mathsf{\partial }}^{a}u\hfill \\ \hfill & =\delta {\langle t\rangle }^{-\delta -2}t{e}^{\tilde{q}(r-t)}{(|\mathsf{\partial }u|}^2+m^2{u}^2{)+\langle t\rangle }^{-\delta }{e}^{\tilde{q}(r-t)}{\langle r-t\rangle }^{-6/5}\left\{{|Gu|}^2+m^2{u}^2\right\}\hfill \\ \hfill & \ge {\langle t\rangle }^{-\delta }{e}^{\tilde{q}(r-t)}{\langle r-t\rangle }^{-6/5}\left\{{|Gu|}^2+m^2{u}^2\right\},\hfill \\ \hfill & -{\mathsf{\partial }}_t{\omega }_2{(|\mathsf{\partial }u|}^2+m^2{u}^2)+2{\mathsf{\partial }}_a{\omega }_2{\mathsf{\partial }}_{t}u{\mathsf{\partial }}^{a}u\hfill \\ \hfill & =2{\langle r-t\rangle }^2\chi (r-2t){\chi }^{\prime }{(r-2t)(|\mathsf{\partial }u|}^2+m^2{u}^2)\hfill \\ \hfill & +\left\{2(r-t){\chi }^2(r-2t)+2{\langle r-t\rangle }^2\chi (r-2t){\chi }^{\prime }(r-2t)\right\}\left\{{|Gu|}^2+m^2{u}^2\right\}\ge 0,\hfill \end{array}$$

(33)

where we use that

$${|\mathsf{\partial }u|}^2+m^2{u}^2+2(x_a/r){\mathsf{\partial }}_{t}u{\mathsf{\partial }}^{a}u={|Gu|}^2+m^2{u}^2.$$

The estimate (27) follows from (29) (by setting $\delta =0$). Let $\omega ={\omega }_1$ in (32). Using (33) and the observation that $e^{\tilde{q}(r-t)}\sim 1$, we obtain (28). By differentiating (32) with respect to t, and using (33) along with the Hölder inequality, the estimate (29) follows. Next, setting $\omega ={\omega }_2$ in (32), differentiating (32) with respect to t, and using (33) and the Hölder inequality, we derive (30). □

We denote the conformal energy for the 2D linear wave equation by

$${\mathcal{E}}_{\mathrm{con}}(t,u):={\int }_{{\mathbb{R}}^2}\left(|L_0{u+u|}^2+{|Lu|}^2+{|\mathsf{\Omega }u|}^2\right)(t,x)\mathrm{d}x,$$

where

$$|Lu|:={\left(\sum _{1\le a\le 2}{|L_{a}u|}^2\right)}^{1/2}.$$

The lemma below provides both conformal energy and $L^2$ estimates for the 2D linear wave equation. The proof of these results can be found in [31] and [32] (Theorems 4.3.1 and 4.6.1), respectively.

Lemma 3

(See [31,32]). Let u be the solution to the Cauchy problem

$$-\square u=F\mathrm{with}(u,{\mathsf{\partial }}_{t}u){|}_{t=0}=(u_0,u_1),$$

where $u_0=u_0\left(x\right)$ and $u_1=u_1\left(x\right):{\mathbb{R}}^2\to \mathbb{R}$ and $F=F(t,x):{\mathbb{R}}^{1+2}\to \mathbb{R}$ are sufficiently nice functions such that the terms on the right hand sides of the estimates below are finite. Then the following estimates hold:

$$\begin{array}{cc}\hfill {[{\mathcal{E}}_{\mathrm{con}}(t,u)]}^{{\displaystyle \frac{1}{2}}}& \lesssim {[{\mathcal{E}}_{\mathrm{con}}(0,u)]}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\parallel \langle s+r\rangle F(s,x)\parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\parallel u(t,x)\parallel }_{L_x^2}& \lesssim \parallel u_0{\parallel }_{L_x^2}+{\log }^{{\displaystyle \frac{1}{2}}}(2+t)\left(\parallel u_1{\parallel }_{L_x^1}+{\parallel u_1\parallel }_{L_x^2}\right)\hfill \\ & +{\log }^{{\displaystyle \frac{1}{2}}}(2+t){\int }_0^t\left({\parallel F(s,x)\parallel }_{L_x^1}+{\parallel F(s,x)\parallel }_{L_x^2}\right)\mathrm{d}s.\hfill \end{array}$$

2.4. Sobolev Type Inequalities and Decay Estimates

In this subsection, we state Sobolev-type inequalities and derive decay properties for the Dirac and Klein–Gordon fields.

The following two global Sobolev inequalities, established by Klainerman [33] and Georgiev [34], are crucial to our study, particularly since the scaling vector field $L_0$ is excluded.

Lemma 4.

For all sufficiently regular functions $f=f(t,x)$ defined in ${\mathbb{R}}^{1+2}$, we have

$$\begin{array}{cc}\hfill |f(t,x)|& \lesssim {\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}}\sum _{|I|\le 3}{\parallel {\mathsf{\Gamma }}^{I}f(t,x)\parallel }_{L_x^2},\hfill \\ \hfill |f(t,x)|& \lesssim {\langle r\rangle }^{-{\displaystyle \frac{1}{2}}}\sum _{\begin{array}{c}|I|\le 2\\ I=(i_1,i_2)\ne (2,0)\end{array}}{\parallel {\mathsf{\Lambda }}^{I}f(t,x)\parallel }_{L_x^2},\hfill \end{array}$$

where we recall (6) for the definition of ${\mathsf{\Lambda }}^I$.

Proof. 

The first inequality was proved by Georgiev [34], see [34] (Lemma 2.4). The second inequality follows from the standard Sobolev inequality on ${\mathbb{S}}^1$, see Klainerman [33] (Proposition 1) for the proof. □

We denote

$$\begin{array}{cc}\hfill {\mathcal{C}}^{\mathrm{in}}:& =\{(t,x)\in [0,\infty )\times {\mathbb{R}}^2:|x|\le 3t+3\},\hfill \\ \hfill {\mathcal{C}}^{\mathrm{ex}}:& =\{(t,x)\in [0,\infty )\times {\mathbb{R}}^2:|x|\ge 2t+3\}.\hfill \end{array}$$

(34)

Next, we present the extra decay for the gradient of a Dirac field within a cone, see [5] (Lemma 3.5).

Lemma 5

(See [5]). Let φ be the solution to

$$-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }{\phi =G\mathrm{with}\phi |}_{t=0}={\phi }_0.$$

Then, it holds that

$$|\mathsf{\partial }\phi |\lesssim {\displaystyle \frac{1}{\langle t-r\rangle }}\sum _{|I|\le 1}|{\widehat{\mathsf{\Gamma }}}^I\phi |+{\displaystyle \frac{t}{\langle t-r\rangle }}|G|\mathrm{in}{\mathcal{C}}^{\mathrm{in}}.$$

The lemma below addresses the additional decay property for the linear Klein–Gordon equation, which was first observed by Klainerman [35].

Lemma 6

(See [6,35,36]). Let u be the solution to

$$-\square u+u=F\mathrm{with}(u,{\mathsf{\partial }}_{t}u){|}_{t=0}=(u_0,u_1).$$

Then, we have the following:

$$|u|\lesssim {\displaystyle \frac{\langle t-r\rangle }{\langle t+r\rangle }}\sum _{|I|\le 1}|\mathsf{\partial }{\mathsf{\Gamma }}^{I}u|+|F|.$$

Let ${\left\{{\phi }_j\right\}}_{j=0}^{\infty }$ be a Littlewood–Paley partition of unity, i.e.,

$$\begin{array}{cc}& 1={\displaystyle \sum _{j=0}^{\infty }}{\phi }_j\left(s\right),s\ge 0,{\phi }_j\in C_0^{\infty }\left(\mathbb{R}\right),{\phi }_j\ge 0\mathrm{for}j\ge 0,\hfill \\ & \mathrm{supp}{\phi }_0\cap [0,\infty )=[0,2],\mathrm{supp}{\phi }_j\subset [2^{j-1},2^{j+1}]\mathrm{for}j\ge 1.\hfill \end{array}$$

Next, we state the decay result for the linear Klein–Gordon equation as established by Georgiev [34].

Theorem 2

(See [34]). Assume u is the solution to the Cauchy problem

$$-\square u+u=F\mathrm{with}(u,{\mathsf{\partial }}_{t}u){|}_{t=0}=(u_0,u_1),$$

where $u_0=u_0\left(x\right)$ and $u_1=u_1\left(x\right):{\mathbb{R}}^2\to \mathbb{R}$ and $F=F(t,x):{\mathbb{R}}^{1+2}\to \mathbb{R}$ are sufficiently nice functions such that the terms on the right hand side of the estimate below are finite. Then, the following holds:

$$\begin{array}{cc}\hfill \langle t+r\rangle |u(t,x)|& \lesssim {\displaystyle \sum _{j=0}^{\infty }}\sum _{|I|\le 4}\underset{0\le s\le t}{\max }{\phi }_j\left(s\right)\parallel \langle s+|y|\rangle {\mathsf{\Gamma }}^I{F(s,y)\parallel }_{L_y^2}\hfill \\ & +{\displaystyle \sum _{j=0}^{\infty }}\sum _{|I|\le 5}\parallel \langle |y|\rangle {\phi }_j(|y|){\mathsf{\Gamma }}^I{u(0,y)\parallel }_{L_y^2}.\hfill \end{array}$$

As a result, we obtain the following simplified form of Theorem 2.

Corollary 1

(See [6]). Under the assumptions of Theorem 2, suppose that

$$\sum _{|I|\le 4}\underset{0\le s\le t}{\max }{\langle s\rangle }^{{\delta }_1}\parallel \langle s+|y|\rangle {\mathsf{\Gamma }}^I{F(s,y)\parallel }_{L_y^2}\le \tilde{C}$$

for some constant ${\delta }_1>0$. Then, we have

$$\langle t+r\rangle |u(t,x)|\lesssim \tilde{C}+\sum _{|I|\le 5}\parallel \log (2+|y|)\langle |y|\rangle {\mathsf{\Gamma }}^I{u(0,y)\parallel }_{L_y^2}.$$

2.5. Nonlinear Transformations

In this subsection, we recall several nonlinear transformations and present a lemma concerning the linear scattering of both fields.

In the lemma below, we introduce some nonlinear transformations from [5], which are used to remove the slowly decaying nonlinear terms in (1). These transformations are in the spirit of Shatah’s normal form method [37], and additional relevant techniques can be found in [38].

To obtain refined estimates for the Dirac field $\psi$, we adopt an idea from [39] (Lemma 3). Specifically, we introduce $\mathsf{\Psi }$, which satisfies $-\square \mathsf{\Psi }=-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\psi$ with initial data $(0,-i{\gamma }^0{\psi }_0)$. The pointwise estimate for $\psi$ then follows from that of $\mathsf{\Psi }$ and the relation $\psi =i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\mathsf{\Psi }$.

Lemma 7

(See [5,6]). Let $(\psi ,v)$ be the solution to (1) and (2) and let Ψ be the solution to

$$-\square \mathsf{\Psi }=-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\psi =v\psi \mathrm{with}(\mathsf{\Psi },{\mathsf{\partial }}_t\mathsf{\Psi }){|}_{t=0}=(0,-i{\gamma }^0{\psi }_0).$$

Then, the following statements hold:

(i) 

$$\psi =i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }{\mathsf{\Psi }\mathrm{and}|[\psi ]}_{-}|\lesssim \sum _{1\le a\le 2}|G_a\mathsf{\Psi }|.$$

(ii) 

Let $\tilde{\psi }:=\psi +i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)$, $\tilde{v}:=v-{\psi }^{*}{\gamma }^0\psi$ and $\tilde{\mathsf{\Psi }}:=\mathsf{\Psi }+v\psi$. Then

$$\begin{array}{cc}\hfill -i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\tilde{\psi }& =F_{\tilde{\psi }}:=-2{\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi +iv{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)+\left({\psi }^{*}{\gamma }^0\psi \right)\psi ,\hfill \\ \hfill -\square \tilde{v}+\tilde{v}& =F_{\tilde{v}}:=i{\mathsf{\partial }}_{\mu }\left(v{\psi }^{*}\right){\gamma }^0{\gamma }^{\mu }\psi -i{\psi }^{*}{\gamma }^0{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)+2{\mathsf{\partial }}_{\alpha }{\psi }^{*}{\gamma }^0{\mathsf{\partial }}^{\alpha }\psi ,\hfill \\ \hfill -\square \tilde{\mathsf{\Psi }}& =F_{\tilde{\psi }}=-2{\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi +iv{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)+\left({\psi }^{*}{\gamma }^0\psi \right)\psi .\hfill \end{array}$$

Let $M=0$ or 1, and let $\phi =\phi (t,x):{\mathbb{R}}^{1+2}\to {\mathbb{C}}^2$, $u=u(t,x):{\mathbb{R}}^{1+2}\to \mathbb{R}$ satisfy

$$-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }{\phi +M\phi =G\mathrm{with}\phi |}_{t=0}={\phi }_0,$$

(35)

$$-\square u+u=F\mathrm{with}(u,{\mathsf{\partial }}_{t}u){|}_{t=0}=(u_0,u_1),$$

(36)

respectively. For $l\in \mathbb{N}$, we denote by $H^l=H^l\left({\mathbb{R}}^2\right)$ the inhomogeneous Sobolev spaces. The lemma below provides sufficient conditions for the linear scattering of the solutions to (35) and (36). The proof can be found in [5] (Lemma 2.8).

Lemma 8

(See [5]). Let $u_0\in H^{l+1}$ and $u_1,{\phi }_0\in H^l$ for some $l\in \mathbb{N}$, and let φ and u be the global solutions to (35) and (36), respectively. Then the following statements hold:

(i) 

Suppose

$${\int }_0^{+\infty }{\parallel G\left(\tau \right)\parallel }_{H^l}\mathrm{d}\tau <\infty ,$$

then φ scatters to a free solution in $H^l$, i.e., there exists ${\phi }_0^{+}\in H^l$ such that

$$\underset{t\to +\infty }{\lim }{\parallel \phi \left(t\right)-{\phi }^{+}\left(t\right)\parallel }_{H^l}=0,$$

where ${\phi }^{+}$ solves

$$-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }{\phi }^{+}+M{\phi }^{+}=0\mathrm{with}{\phi }^{+}{|}_{t=0}={\phi }_0^{+}.$$

(ii) 

If

$${\int }_0^{+\infty }{\parallel F\left(\tau \right)\parallel }_{H^l}\mathrm{d}\tau <\infty ,$$

then u scatters to a free solution in ${\mathcal{H}}_l:=H^{l+1}\times H^l$, i.e., there exists $(u_0^{+},u_1^{+})\in {\mathcal{H}}_l$ such that

$$\underset{t\to +\infty }{\lim }{\parallel (u,{\mathsf{\partial }}_{t}u)\left(t\right)-(u^{+},{\mathsf{\partial }}_t{u}^{+})\left(t\right)\parallel }_{{\mathcal{H}}_l}=0,$$

where $u^{+}$ solves

$$-\square u^{+}+u^{+}=0\mathrm{with}(u^{+},{\mathsf{\partial }}_t{u}^{+}){|}_{t=0}=(u_0^{+},u_1^{+}).$$

3. Proof of Theorem 1

3.1. Bootstrap Setting

In this subsection, we introduce a bootstrap framework for the solutions to (1) and (2).

Let $N\ge 13$ be an integer, $0<\delta \ll 1$, and define $K:=K_{*}^N$, where $K_{*}\ge 1$ is the large constant given in Theorem 1. Let $C_1\gg 1$ and $0<ϵ<1$ be two constants to be determined later. To prove Theorem 1, let $(\psi ,u)$ be the solution to (1) and (2). We assume that the following estimates hold for $t\in [2,T]$:

$${[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\psi )]}^{{\displaystyle \frac{1}{2}}}\le C_1{ϵ}^{1-\delta |I|}{\langle t\rangle }^{\delta },\mathrm{for}|I|\le N,$$

(37)

$${\left({\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-3\delta }{\displaystyle \frac{|{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\le C_1{ϵ}^{1-\delta |I|},\mathrm{for}|I|\le N,$$

(38)

$$\parallel \langle r-t\rangle \chi (r-2t){\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}\le C_1{ϵ}^{1-\delta |I|}{\langle t\rangle }^{\delta },\mathrm{for}|I|\le N,$$

(39)

$${[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\psi )]}^{{\displaystyle \frac{1}{2}}}\le C_1{ϵ}^{1-\delta (|I|+3/2)},\mathrm{for}|I|\le N-1,$$

(40)

$$\parallel \langle t+r\rangle {[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{\parallel }_{L_x^2}\le C_1{ϵ}^{1-\delta (|I|+3/2)}{\langle t\rangle }^{2\delta },\mathrm{for}|I|\le N-2,$$

(41)

$${\langle t+r\rangle }^{{\displaystyle \frac{3}{2}}-2\delta }|{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}|+{\langle t+r\rangle }^{{\displaystyle \frac{1}{2}}-2\delta }\langle t-r\rangle |{\widehat{\mathsf{\Gamma }}}^I\psi |\le C_1{ϵ}^{1-\delta (|I|+9/2)},\mathrm{for}|I|\le N-5,$$

(42)

$${[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}\le C_{1}K{\langle t\rangle }^{3\delta },\mathrm{for}|I|\le N,$$

(43)

$${\left({\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-7\delta }{\displaystyle \frac{|{\mathsf{\Gamma }}^I{v|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}\mathrm{s}\right)}^{{\displaystyle \frac{1}{2}}}\le C_{1}K,\mathrm{for}|I|\le N,$$

(44)

$$\parallel \langle r-t\rangle \chi (r-2t){\mathsf{\Gamma }}^I{v\parallel }_{L_x^2}\le C_{1}K,\mathrm{for}|I|\le N,$$

(45)

$${[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}\le C_{1}K,\mathrm{for}|I|\le N-1,$$

(46)

$$\langle t+r\rangle |{\mathsf{\Gamma }}^{I}v|\le C_{1}K,\mathrm{for}|I|\le N-6.$$

(47)

For all initial data $({\psi }_0,v_0,v_1)$ satisfying (3), we denote ${\overrightarrow{v}}_0:=(v_0,v_1)$ and set

$$T_{*}=T_{*}({\psi }_0,{\overrightarrow{v}}_0):=\sup \{T\in [0,+\infty ):\mathrm{The}\mathrm{estimates}\left(37\right)–\left(47\right)\mathrm{hold}\mathrm{for}t\in [2,T]\}.$$

Theorem 1 follows from the proposition below.

Proposition 2.

For any initial data $({\psi }_0,{\overrightarrow{v}}_0)$ satisfying (3) in Theorem 1, we have $T_{*}=+\infty$.

The remaining part of Section 3 and all of Section 4 are devoted to the proof of Proposition 2, which is accomplished by refining the estimates (37)–(47). In the following, the implied constants in ≲ do not depend on $C_1,ϵ$ and K. Using (40), (46) and Lemma 4, we obtain

$$|{\widehat{\mathsf{\Gamma }}}^I\psi |\lesssim C_1{ϵ}^{1-\delta (|I|+9/2)}{\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}},\mathrm{for}|I|\le N-4,$$

(48)

$$|{\mathsf{\Gamma }}^{I}v|\lesssim C_{1}K{\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}},\mathrm{for}|I|\le N-4.$$

(49)

Therefore, the boundedness of the low-order energy yields decay properties of the solution, reflecting the dispersive nature of the evolution.

3.2. Extra Decay for $\mathsf{\partial }\psi$ and v

The following lemma, which establishes extra decay estimates for v and the gradient of $\psi$, will be frequently used in the proof of Proposition 2.

Lemma 9.

Under the assumptions (37)–(47), the following statements hold:

$\left(i\right)$

In the region ${\mathcal{C}}^{\mathrm{in}}$ (as defined in (34)), we have

$$\begin{array}{cc}\hfill |\mathsf{\partial }{\widehat{\mathsf{\Gamma }}}^I\psi |& \lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+11/2)}{\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }{\langle t-r\rangle }^{-2},\mathrm{for}|I|\le N-6,\hfill \\ \hfill |\mathsf{\partial }{\widehat{\mathsf{\Gamma }}}^I\psi |& \lesssim {\displaystyle \frac{C_{1}K}{\langle t-r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi |+C_1{ϵ}^{1-\delta (|I|+1/2)}{\displaystyle \frac{{\langle t\rangle }^{\frac{1}{2}+2\delta }}{{\langle t-r\rangle }^2}}\sum _{|I^{\prime }|\le |I|}|{\mathsf{\Gamma }}^{I^{\prime }}v|,\mathrm{for}|I|\le N-1.\hfill \end{array}$$

On the other hand, in the region ${\mathcal{C}}^{\mathrm{ex}}$, the following holds:

$$|{\widehat{\mathsf{\Gamma }}}^I\psi |\lesssim C_1{ϵ}^{1-\delta (|I|+2)}{\langle t+r\rangle }^{-{\displaystyle \frac{3}{2}}+\delta },\mathrm{for}|I|\le N-2.$$

$\left(ii\right)$

In $[0,\infty )\times {\mathbb{R}}^2$, we have

$$\begin{array}{cc}\hfill |{\mathsf{\Gamma }}^{I}v|& \lesssim C_{1}K{\langle t+r\rangle }^{-2}\langle t-r\rangle +C_1^2{ϵ}^{2-\delta (|I|+9)}{\langle t+r\rangle }^{-2+2\delta },\mathrm{for}|I|\le N-8,\hfill \\ \hfill |{\mathsf{\Gamma }}^{I}v|& \lesssim {\displaystyle \frac{\langle t-r\rangle }{\langle t+r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|\mathsf{\partial }{\mathsf{\Gamma }}^{I^{\prime }}v|\hfill \\ & +C_1{ϵ}^{1-\delta N}\left\{{\langle t+r\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\sum _{|I^{\prime }|\le |I|}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}{\psi |+\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}}\sum _{|I^{\prime }|\le |I|}|{[{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi ]}_{-}|\right\},\mathrm{for}|I|\le N-1.\hfill \end{array}$$

Additionally, in the region ${\mathcal{C}}^{\mathrm{ex}}$, we have

$$|{\mathsf{\Gamma }}^{I}v|\lesssim C_{1}K{\langle t+r\rangle }^{-{\displaystyle \frac{3}{2}}},\mathrm{for}|I|\le N-2.$$

Proof. 

For $\left(i\right)$, applying the vector field ${\widehat{\mathsf{\Gamma }}}^I$ with $|I|\le N$ to both sides of the first equation in (1) and using Proposition 1 (see (10)), we find

$$-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }{\widehat{\mathsf{\Gamma }}}^I\psi ={\widehat{\mathsf{\Gamma }}}^I\left(v\psi \right)=\sum _{I_1+I_2=I}{\mathsf{\Gamma }}^{I_1}v{\widehat{\mathsf{\Gamma }}}^{I_2}\psi .$$

By Lemma 5, along with (42) and (47), for $|I|\le N-6$, we have

$$\begin{array}{cc}\hfill |\mathsf{\partial }{\widehat{\mathsf{\Gamma }}}^I\psi |{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}& \lesssim {\displaystyle \frac{1}{\langle t-r\rangle }}\sum _{|J|\le 1}|{\widehat{\mathsf{\Gamma }}}^J{\widehat{\mathsf{\Gamma }}}^I\psi |+{\displaystyle \frac{t}{\langle t-r\rangle }}\sum _{|I_1|+|I_2|\le |I|}|{\mathsf{\Gamma }}^{I_1}v|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\hfill \\ & \lesssim C_1{ϵ}^{1-\delta (|I|+11/2)}{\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }{\langle t-r\rangle }^{-2}+C_1^{2}K\sum _{|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+9/2)}{\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }{\langle t-r\rangle }^{-2}\hfill \\ & \lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+11/2)}{\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }{\langle t-r\rangle }^{-2},\hfill \end{array}$$

where ${\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}$ denotes the characteristic function of ${\mathcal{C}}^{\mathrm{in}}$. Next, we prove the second inequality in $\left(i\right)$. If $4\le |I|\le N-1$, then for any $|I_1|+|I_2|\le |I|$, it holds that $|I_2|\le |I|-4$ or $|I_1|\le 3\le N-6$. Hence, using Lemma 5, (42) and (47) again, we see that

$$\begin{array}{cc}\hfill |\mathsf{\partial }{\widehat{\mathsf{\Gamma }}}^I\psi |{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}& \lesssim {\displaystyle \frac{1}{\langle t-r\rangle }}\sum _{|J|\le 1}|{\widehat{\mathsf{\Gamma }}}^J{\widehat{\mathsf{\Gamma }}}^I\psi |+{\displaystyle \frac{t}{\langle t-r\rangle }}\sum _{|I_1|+|I_2|\le |I|}|{\mathsf{\Gamma }}^{I_1}v|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\hfill \\ & \lesssim {\displaystyle \frac{1}{\langle t-r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi |+{\displaystyle \frac{t}{\langle t-r\rangle }}\sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|\end{array}}|{\mathsf{\Gamma }}^{I_1}v|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |+{\displaystyle \frac{t}{\langle t-r\rangle }}\sum _{\begin{array}{c}|I_2|\le |I|-4\\ |I_1|\le |I|\end{array}}|{\mathsf{\Gamma }}^{I_1}v|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\hfill \\ & \lesssim {\displaystyle \frac{1}{\langle t-r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi |+{\displaystyle \frac{C_{1}K}{\langle t-r\rangle }}\sum _{|I_2|\le |I|}|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |+{\displaystyle \frac{{\langle t\rangle }^{\frac{1}{2}+2\delta }}{{\langle t-r\rangle }^2}}\sum _{\begin{array}{c}|I_2|\le |I|-4\\ |I_1|\le |I|\end{array}}{C}_1{ϵ}^{1-\delta (|I_2|+9/2)}|{\mathsf{\Gamma }}^{I_1}v|\hfill \\ & \lesssim C_{1}K{\displaystyle \frac{1}{\langle t-r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi |+C_1{ϵ}^{1-\delta (|I|+1/2)}{\displaystyle \frac{{\langle t\rangle }^{\frac{1}{2}+2\delta }}{{\langle t-r\rangle }^2}}\sum _{|I^{\prime }|\le |I|}|{\mathsf{\Gamma }}^{I^{\prime }}v|.\hfill \end{array}$$

On the other hand, for $|I|\le 3\le N-6$, the last estimate also holds because

$$\begin{array}{cc}\hfill |\mathsf{\partial }{\widehat{\mathsf{\Gamma }}}^I\psi |{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}& \lesssim {\displaystyle \frac{1}{\langle t-r\rangle }}\sum _{|J|\le 1}|{\widehat{\mathsf{\Gamma }}}^J{\widehat{\mathsf{\Gamma }}}^I\psi |+{\displaystyle \frac{t}{\langle t-r\rangle }}\sum _{|I_1|+|I_2|\le |I|}|{\mathsf{\Gamma }}^{I_1}v|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\hfill \\ & \lesssim {\displaystyle \frac{1}{\langle t-r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi |+{\displaystyle \frac{t}{\langle t-r\rangle }}\sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|\end{array}}|{\mathsf{\Gamma }}^{I_1}v|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\lesssim C_{1}K{\displaystyle \frac{1}{\langle t-r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi |.\hfill \end{array}$$

For the last inequality in $\left(i\right)$, we claim that

$$|{\mathsf{\Lambda }}^J\left(\langle r-t\rangle \chi (r-2t-1)\right)|\lesssim \langle r-t\rangle \chi (r-2t),\mathrm{for}\mathrm{any}|J|\le 2.$$

(50)

Indeed, by direct calculation,

$$\begin{array}{cc}& {\mathsf{\partial }}_r\left(\langle r-t\rangle \right)={\langle r-t\rangle }^{-1}(r-t),\mathsf{\Omega }\left(\langle r-t\rangle \right)=0,\hfill \\ & {\mathsf{\partial }}_r\left(\chi (r-2t-1)\right)={\chi }^{\prime }(r-2t-1),\mathsf{\Omega }\left(\chi (r-2t-1)\right)=0.\hfill \end{array}$$

We observe that, on the support of $\chi (r-2t-1)$, we have $r-2t\ge 2$ and $\chi (r-2t)=1$. Hence, the claim follows. By Lemma 4, (50) and (39), for $|I|\le N-2$, it holds that

$$\begin{array}{cc}\hfill {\langle r\rangle }^{{\displaystyle \frac{1}{2}}}\langle r-t\rangle \chi (r-2t-1)|{\mathsf{\Gamma }}^I\psi (t,x)|& \lesssim \sum _{\begin{array}{c}|J|\le 2\\ J=(j_1,j_2)\ne (2,0)\end{array}}{\parallel {\mathsf{\Lambda }}^J\left(\langle r-t\rangle \chi (r-2t-1){\mathsf{\Gamma }}^I\psi \right)\parallel }_{L_x^2}\hfill \\ & \lesssim \sum _{|I^{\prime }|\le |I|+2}{\parallel \langle r-t\rangle \chi (r-2t){\mathsf{\Gamma }}^{I^{\prime }}\psi \parallel }_{L_x^2}\lesssim C_1{ϵ}^{1-\delta (|I|+2)}{\langle t\rangle }^{\delta }.\hfill \end{array}$$

In the region ${\mathcal{C}}^{\mathrm{ex}}$, we have $r-2t-1\ge 2$, which gives $\chi (r-2t-1)=1$ and $\langle r\rangle \sim \langle t+r\rangle \sim \langle r-t\rangle$. It follows that

$$|{\mathsf{\Gamma }}^I\psi (t,x)|\lesssim C_1{ϵ}^{1-\delta (|I|+2)}{\langle t+r\rangle }^{-{\displaystyle \frac{3}{2}}+\delta },\mathrm{for}|I|\le N-2\mathrm{and}(t,x)\in {\mathcal{C}}^{\mathrm{ex}}.$$

For $\left(ii\right)$, using Lemma 6 and Proposition 1 (see (11) and (12)), along with (42), (47) and (48), for $|I|\le N-8$, we obtain that

$$\begin{array}{cc}\hfill |{\mathsf{\Gamma }}^{I}v|& \lesssim {\displaystyle \frac{\langle t-r\rangle }{\langle t+r\rangle }}\sum _{|J|\le 1}|\mathsf{\partial }{\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^{I}v|+|{\mathsf{\Gamma }}^I\left({\psi }^{*}{\gamma }^0\psi \right)|\hfill \\ & \lesssim {\displaystyle \frac{\langle t-r\rangle }{\langle t+r\rangle }}\sum _{|I^{\prime }|\le |I|+2}|{\mathsf{\Gamma }}^{I^{\prime }}v|+\sum _{|I_1|+|I_2|\le |I|}|{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\hfill \\ & \lesssim C_{1}K{\langle t+r\rangle }^{-2}\langle t-r\rangle +C_1^2\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_1|+9/2)}{ϵ}^{1-\delta (|I_2|+9/2)}{\langle t+r\rangle }^{-2+2\delta }\hfill \\ & \lesssim C_{1}K{\langle t+r\rangle }^{-2}\langle t-r\rangle +C_1^2{ϵ}^{2-\delta (|I|+9)}{\langle t+r\rangle }^{-2+2\delta },\hfill \end{array}$$

and for $|I|\le N-1$, we derive

$$\begin{array}{cc}\hfill |{\mathsf{\Gamma }}^{I}v|& \lesssim {\displaystyle \frac{\langle t-r\rangle }{\langle t+r\rangle }}\sum _{|J|\le 1}|\mathsf{\partial }{\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^{I}v|+|{\mathsf{\Gamma }}^I\left({\psi }^{*}{\gamma }^0\psi \right)|\hfill \\ & \lesssim {\displaystyle \frac{\langle t-r\rangle }{\langle t+r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|\mathsf{\partial }{\mathsf{\Gamma }}^{I^{\prime }}v|+\sum _{\begin{array}{c}|I_1|\le N-5\\ |I_1|+|I_2|\le |I|\end{array}}|{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |+\sum _{\begin{array}{c}|I_2|\le N-5\\ |I_1|+|I_2|\le |I|\end{array}}|{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\hfill \\ & \lesssim {\displaystyle \frac{\langle t-r\rangle }{\langle t+r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|\mathsf{\partial }{\mathsf{\Gamma }}^{I^{\prime }}v|+C_1{\langle t+r\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\sum _{\begin{array}{c}|I_1|\le N-5\\ |I_2|\le |I|\end{array}}{ϵ}^{1-\delta (|I_1|+9/2)}|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\hfill \\ & +C_1{\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}}\sum _{\begin{array}{c}|I_2|\le N-5\\ |I_1|\le |I|\end{array}}{ϵ}^{1-\delta (|I_2|+9/2)}|{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}|\hfill \\ & \lesssim {\displaystyle \frac{\langle t-r\rangle }{\langle t+r\rangle }}\sum _{|I^{\prime }|\le |I|+1}|\mathsf{\partial }{\mathsf{\Gamma }}^{I^{\prime }}v|+C_1{ϵ}^{1-\delta N}\left\{{\langle t+r\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\sum _{|I^{\prime }|\le |I|}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}{\psi |+\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}}\sum _{|I^{\prime }|\le |I|}|{[{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi ]}_{-}|\right\}.\hfill \end{array}$$

Finally, using Lemma 4, together with (50) and (45), for $|I|\le N-2$, we find

$$\begin{array}{cc}\hfill {\langle r\rangle }^{{\displaystyle \frac{1}{2}}}\langle r-t\rangle \chi (r-2t-1)|{\mathsf{\Gamma }}^{I}v(t,x)|& \lesssim \sum _{\begin{array}{c}|J|\le 2\\ J=(j_1,j_2)\ne (2,0)\end{array}}{\parallel {\mathsf{\Lambda }}^J\left(\langle r-t\rangle \chi (r-2t-1){\mathsf{\Gamma }}^{I}v\right)\parallel }_{L_x^2}\hfill \\ & \lesssim \sum _{|I^{\prime }|\le |I|+2}{\parallel \langle r-t\rangle \chi (r-2t){\mathsf{\Gamma }}^{I^{\prime }}v\parallel }_{L_x^2}\lesssim C_{1}K,\hfill \end{array}$$

hence we deduce that

$$|{\mathsf{\Gamma }}^{I}v(t,x)|\lesssim C_{1}K{\langle t+r\rangle }^{-{\displaystyle \frac{3}{2}}},\mathrm{for}|I|\le N-2\mathrm{and}(t,x)\in {\mathcal{C}}^{\mathrm{ex}}.$$

The proof is complete. □

3.3. Top-Order Energy and $L^2$-Type Estimates for $\psi$ and v

In this subsection, we establish refined estimates for the top-order energy and $L^2$-type norms of both fields. The following proposition establishes energy estimates, spacetime $L^2$ estimates, and weighted $L_x^2$ estimates for the Dirac field in the exterior region. In particular, it provides refined bounds for the bootstrap assumptions (37)–(39).

Proposition 3.

Under the assumptions (37)–(47), the following estimates hold for $|I|\le N$:

$$\begin{array}{cc}& {[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\psi )]}^{{\displaystyle \frac{1}{2}}}\lesssim Kϵ+C_1^{{\displaystyle \frac{3}{2}}}{K}^{{\displaystyle \frac{1}{2}}}{ϵ}^{1-\delta (|I|-1/4)}{\langle t\rangle }^{\delta },\hfill \\ & {\left({\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-3\delta }{\displaystyle \frac{|{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\lesssim Kϵ+C_1^{{\displaystyle \frac{3}{2}}}{K}^{{\displaystyle \frac{1}{2}}}{ϵ}^{1-\delta (|I|-1/4)},\hfill \\ & \parallel \langle r-t\rangle \chi (r-2t){\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}\lesssim Kϵ+C_1^{2}K{ϵ}^{1-\delta (|I|-1/2)}{\langle t\rangle }^{\delta }.\hfill \end{array}$$

Proof. 

The proof is divided into the following three steps.

• $\mathbf{Step}\mathbf{1}.$ The top-order energy estimate for ψ: Applying the vector field ${\widehat{\mathsf{\Gamma }}}^I$ with $|I|\le N$ to both sides of the first equation in (1) and using Proposition 1 (see (10)), we obtain

  $$-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }{\widehat{\mathsf{\Gamma }}}^I\psi ={\widehat{\mathsf{\Gamma }}}^I\left(v\psi \right)=v{\widehat{\mathsf{\Gamma }}}^I\psi +\underset{\begin{array}{c}|I_1|+|I_2|\le |I|\\ |I_2|\le |I|-1\end{array}}{{\sum }^{\prime }}{\mathsf{\Gamma }}^{I_1}v{\widehat{\mathsf{\Gamma }}}^{I_2}\psi .$$

  (51)

  Here, we assume without loss of generality that $|I|\ge 1$; otherwise the sum ${\sum }^{\prime }$ on the right hand side of (51) vanishes. By Lemma 1 (see (18)) and (12), we deduce

  $$\begin{array}{cc}\hfill {\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\psi )& \lesssim {\mathcal{E}}_D(0,{\widehat{\mathsf{\Gamma }}}^I\psi )+\sum _{\begin{array}{c}|I_1|+|I_2|\le |I|\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\int }_{{\mathbb{R}}^2}|{({\widehat{\mathsf{\Gamma }}}^I\psi )}^{*}(s,x){\gamma }^0({\mathsf{\Gamma }}^{I_1}v{\widehat{\mathsf{\Gamma }}}^{I_2}\psi )(s,x)|\mathrm{d}x\mathrm{d}s\hfill \\ \hfill & \lesssim {\left(Kϵ\right)}^2+{\mathcal{R}}_1+{\mathcal{R}}_2,\hfill \end{array}$$

  (52)

  where we recall that $K:=K_{*}^N$ (see Section 3.1), and the terms ${\mathcal{R}}_j,j=1,2$ are defined as follows:

  $$\begin{array}{cc}\hfill {\mathcal{R}}_1:& =\sum _{\begin{array}{c}|I_1|+|I_2|\le |I|\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\int }_{{\mathbb{R}}^2}|{\widehat{\mathsf{\Gamma }}}^I\psi |·|{\mathsf{\Gamma }}^{I_1}v|·|{[{\widehat{\mathsf{\Gamma }}}^{I_2}\psi ]}_{-}|\mathrm{d}x\mathrm{d}s,\hfill \\ \hfill {\mathcal{R}}_2:& =\sum _{\begin{array}{c}|I_1|+|I_2|\le |I|\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\int }_{{\mathbb{R}}^2}|{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}|·|{\mathsf{\Gamma }}^{I_1}v|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\mathrm{d}x\mathrm{d}s.\hfill \end{array}$$

  If $5\le |I|\le N$, then for any $|I_1|+|I_2|\le |I|$, we have either $|I_2|\le |I|-5$ or $|I_1|\le 4\le N-6$ (here we recall that $N\ge 13$). It follows that

  $${\mathcal{R}}_1\lesssim {\mathcal{R}}_1^1+{\mathcal{R}}_1^2,{\mathcal{R}}_2\lesssim {\mathcal{R}}_2^1+{\mathcal{R}}_2^2,$$

  where

  $$\begin{array}{cc}\hfill {\mathcal{R}}_1^1& =\sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|-1\end{array}}{\int }_0^t\parallel {\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·{\parallel {[{\widehat{\mathsf{\Gamma }}}^{I_2}\psi ]}_{-}\parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\mathcal{R}}_1^2& =\sum _{\begin{array}{c}|I_2|\le |I|-5\\ |I_1|\le |I|\end{array}}{\int }_0^t\parallel {\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^2}·{\parallel {[{\widehat{\mathsf{\Gamma }}}^{I_2}\psi ]}_{-}\parallel }_{L_x^{\infty }}\mathrm{d}s,\hfill \\ \hfill {\mathcal{R}}_2^1& =\sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|-1\end{array}}{\int }_0^t\parallel {[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{\parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\mathcal{R}}_2^2& =\sum _{\begin{array}{c}|I_2|\le |I|-5\\ |I_1|\le |I|\end{array}}{\int }_0^t{\parallel \langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{\parallel }_{L_x^2}·\parallel {\langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^2}·{\parallel {\langle r-s\rangle }^{{\displaystyle \frac{6}{5}}}{\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

  (53)

  Using (37)–(38), (42)–(44) and (47), we derive

  $$\begin{array}{cc}\hfill & {\mathcal{R}}_1^1+{\mathcal{R}}_2^1\lesssim \sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|-1\end{array}}{\int }_0^t\parallel {\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim C_1^{3}K{ϵ}^{1-\delta |I|}\sum _{|I_2|\le |I|-1}{ϵ}^{1-\delta |I_2|}{\int }_0^t{\langle s\rangle }^{-1+2\delta }\mathrm{d}s\lesssim C_1^{3}K{ϵ}^{2-\delta (2|I|-1)}{\langle t\rangle }^{2\delta },\hfill \end{array}$$

  (54)

  $$\begin{array}{cc}\hfill & {\mathcal{R}}_1^2\lesssim C_1^{3}K{ϵ}^{1-\delta |I|}\sum _{|I_2|\le |I|-5}{ϵ}^{1-\delta (|I_2|+9/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+6\delta }\mathrm{d}s\lesssim C_1^{3}K{ϵ}^{2-\delta (2|I|-1/2)},\hfill \\ \hfill & {\mathcal{R}}_2^2\lesssim C_1\sum _{\begin{array}{c}|I_2|\le |I|-5\\ |I_1|\le |I|\end{array}}{ϵ}^{1-\delta (|I_2|+9/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{10}}+2\delta }{\parallel \langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{\parallel }_{L_x^2}·{\parallel {\langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{\mathsf{\Gamma }}^{I_1}v\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim C_1{ϵ}^{1-\delta (|I|-1/2)}\sum _{|I_1|\le |I|}{\left({\int }_0^t\parallel {\langle s\rangle }^{-{\displaystyle \frac{3}{2}}\delta }{\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_0^t\parallel {\langle s\rangle }^{-{\displaystyle \frac{7}{2}}\delta }{\displaystyle \frac{{\mathsf{\Gamma }}^{I_1}v}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \lesssim C_1^{3}K{ϵ}^{2-\delta (2|I|-1/2)}.\hfill \end{array}$$

  (55)

  Combining the above estimates, for $5\le |I|\le N$, we obtain

  $${\mathcal{R}}_1+{\mathcal{R}}_2\lesssim C_1^{3}K{ϵ}^{2-\delta (2|I|-1/2)}{\langle t\rangle }^{2\delta }.$$

  (56)

  On the other hand, if $|I|\le 4\le N-6$, we claim that (56) also holds. Indeed, in this case, we have ${\mathcal{R}}_1\lesssim {\mathcal{R}}_1^1$ and ${\mathcal{R}}_2\lesssim {\mathcal{R}}_2^1$. Hence, the estimate (56) follows from (54). Combining (52) and (56), we deduce

  $${[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\psi )]}^{{\displaystyle \frac{1}{2}}}\lesssim Kϵ+C_1^{{\displaystyle \frac{3}{2}}}{K}^{{\displaystyle \frac{1}{2}}}{ϵ}^{1-\delta (|I|-1/4)}{\langle t\rangle }^{\delta },\mathrm{for}|I|\le N.$$

  (57)

  This estimate shows that the top-order energy of the Dirac field exhibits polynomial growth in time. It suggests that the nonlinear effects accumulate slowly and do not destroy the dominant dispersive behavior of the field.
• $\mathbf{Step}\mathbf{2}.$ The weighted spacetime $L^2$ estimate for ψ: For $|I|\le N$, by Lemma 1 (see (18)) and (51), we see that

  $$\begin{array}{cc}\hfill & {\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-3\delta }{\displaystyle \frac{|{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s\hfill \\ \hfill & \lesssim {\mathcal{E}}_D(0,{\widehat{\mathsf{\Gamma }}}^I\psi )+\sum _{\begin{array}{c}|I_1|+|I_2|\le |I|\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-3\delta }|{({\widehat{\mathsf{\Gamma }}}^I\psi )}^{*}{\gamma }^0({\mathsf{\Gamma }}^{I_1}v{\widehat{\mathsf{\Gamma }}}^{I_2}\psi )|\mathrm{d}x\mathrm{d}s\hfill \\ \hfill & \lesssim {\left(Kϵ\right)}^2+{\tilde{\mathcal{R}}}_1+{\tilde{\mathcal{R}}}_2,\hfill \end{array}$$

  (58)

  where

  $$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_1:& =\sum _{\begin{array}{c}|I_1|+|I_2|\le |I|\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-3\delta }|{\widehat{\mathsf{\Gamma }}}^I\psi |·|{\mathsf{\Gamma }}^{I_1}v|·|{[{\widehat{\mathsf{\Gamma }}}^{I_2}\psi ]}_{-}|\mathrm{d}x\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{R}}}_2:& =\sum _{\begin{array}{c}|I_1|+|I_2|\le |I|\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-3\delta }|{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}|·|{\mathsf{\Gamma }}^{I_1}v|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |\mathrm{d}x\mathrm{d}s.\hfill \end{array}$$

  If $5\le |I|\le N$, then for any $|I_1|+|I_2|\le |I|$, we have either $|I_2|\le |I|-5$ or $|I_1|\le 4\le N-6$. Then,

  $${\tilde{\mathcal{R}}}_1\lesssim {\tilde{\mathcal{R}}}_1^1+{\tilde{\mathcal{R}}}_1^2,{\tilde{\mathcal{R}}}_2\lesssim {\tilde{\mathcal{R}}}_2^1+{\tilde{\mathcal{R}}}_2^2,$$

  where

  $$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_1^1& =\sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\langle s\rangle }^{-3\delta }\parallel {\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·{\parallel {[{\widehat{\mathsf{\Gamma }}}^{I_2}\psi ]}_{-}\parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{R}}}_1^2& =\sum _{\begin{array}{c}|I_2|\le |I|-5\\ |I_1|\le |I|\end{array}}{\int }_0^t{\langle s\rangle }^{-3\delta }\parallel {\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^2}·{\parallel {[{\widehat{\mathsf{\Gamma }}}^{I_2}\psi ]}_{-}\parallel }_{L_x^{\infty }}\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{R}}}_2^1& =\sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\langle s\rangle }^{-3\delta }\parallel {[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{\parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{R}}}_2^2& =\sum _{\begin{array}{c}|I_2|\le |I|-5\\ |I_1|\le |I|\end{array}}{\int }_0^t{\langle s\rangle }^{-3\delta }{\parallel \langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{\parallel }_{L_x^2}·\parallel {\langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^2}·{\parallel {\langle r-s\rangle }^{{\displaystyle \frac{6}{5}}}{\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

  We note that ${\tilde{\mathcal{R}}}_1^2\lesssim {\mathcal{R}}_1^2$ and ${\tilde{\mathcal{R}}}_2^2\lesssim {\mathcal{R}}_2^2$, where ${\mathcal{R}}_1^2$ and ${\mathcal{R}}_2^2$ are as in (53). Hence, by (55), we obtain

  $${\tilde{\mathcal{R}}}_1^2+{\tilde{\mathcal{R}}}_2^2\lesssim C_1^{3}K{ϵ}^{2-\delta (2|I|-1/2)}.$$

  Additionally, the estimate (54) yields

  $$\begin{array}{cc}& {\tilde{\mathcal{R}}}_1^1+{\tilde{\mathcal{R}}}_2^1\lesssim \sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\langle s\rangle }^{-3\delta }\parallel {\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s\hfill \\ & \lesssim C_1^{3}K{ϵ}^{1-\delta |I|}\sum _{|I_2|\le |I|-1}{ϵ}^{1-\delta |I_2|}{\int }_0^t{\langle s\rangle }^{-1-\delta }\mathrm{d}s\lesssim C_1^{3}K{ϵ}^{2-\delta (2|I|-1)}.\hfill \end{array}$$

  It follows that for $5\le |I|\le N$,

  $${\tilde{\mathcal{R}}}_1+{\tilde{\mathcal{R}}}_2\lesssim C_1^{3}K{ϵ}^{2-\delta (2|I|-1/2)}.$$

  (59)

  On the other hand, if $|I|\le 4\le N-6$, we can also obtain (59), since ${\tilde{\mathcal{R}}}_1+{\tilde{\mathcal{R}}}_2$ is bounded by ${\tilde{\mathcal{R}}}_1^1+{\tilde{\mathcal{R}}}_2^1$ above. Combining (58) and (59), we derive

  $$\begin{array}{c}\hfill {\left({\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-3\delta }{\displaystyle \frac{|{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\lesssim Kϵ+C_1^{{\displaystyle \frac{3}{2}}}{K}^{{\displaystyle \frac{1}{2}}}{ϵ}^{1-\delta (|I|-1/4)},\mathrm{for}|I|\le N.\end{array}$$

  (60)

  Hence, we obtain the boundedness of the weighted spacetime $L_{t,x}^2$ norm of the Dirac field. In particular, the cumulative effect of nonlinear interactions remains globally controllable over long times, thereby preventing persistent concentration or instability formation.
• $\mathbf{Step}\mathbf{3}.$ $L_x^2$ estimate for ψ in the exterior region: For $|I|\le N$, by Lemma 1 (see (20)) and (51), we find

  $$\parallel \langle r-t\rangle \chi (r-2t){\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}\lesssim {\parallel \langle r\rangle {\widehat{\mathsf{\Gamma }}}^I\psi (0,x)\parallel }_{L_x^2}+\widehat{\mathcal{R}}\lesssim Kϵ+\widehat{\mathcal{R}},$$

  (61)

  where

  $$\widehat{\mathcal{R}}:=\sum _{\begin{array}{c}|I_1|+|I_2|\le |I|\\ |I_2|\le |I|-1\end{array}}{\int }_0^t{\parallel \langle r-s\rangle \chi (r-2s){\mathsf{\Gamma }}^{I_1}v{\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s.$$

  If $5\le |I|\le N$, then for any $|I_1|+|I_2|\le |I|$, we have $|I_2|\le |I|-5$ or $|I_1|\le 4\le N-6$. Hence,

  $$\widehat{\mathcal{R}}\lesssim {\widehat{\mathcal{R}}}_1+{\widehat{\mathcal{R}}}_2,$$

  where

  $$\begin{array}{cc}\hfill {\widehat{\mathcal{R}}}_1:& =\sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|-1\end{array}}{\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·{\parallel \langle r-s\rangle \chi (r-2s){\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\widehat{\mathcal{R}}}_2:& =\sum _{\begin{array}{c}|I_2|\le |I|-5\\ |I_1|\le |I|\end{array}}{\int }_0^t\parallel \langle r-s\rangle \chi (r-2s){\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^2}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi {\mathbf{1}}_{\{r\ge 2s+1\}}\parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

  Here, ${\mathbf{1}}_{\{r\ge 2s+1\}}$ denotes the characteristic function of the set $\{(s,x)\in [0,\infty )\times {\mathbb{R}}^2:|x|\ge 2s+1\}$. Using (39), (42), (45) and (47), we obtain

  $$\begin{array}{cc}\hfill {\widehat{\mathcal{R}}}_1& \lesssim C_1^{2}K\sum _{|I_2|\le |I|-1}{ϵ}^{1-\delta |I_2|}{\int }_0^t{\langle s\rangle }^{-1+\delta }\mathrm{d}s\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|-1)}{\langle t\rangle }^{\delta },\hfill \\ \hfill {\widehat{\mathcal{R}}}_2& \lesssim C_1^{2}K\sum _{|I_2|\le |I|-5}{ϵ}^{1-\delta (|I_2|+9/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\mathrm{d}s\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|-1/2)},\hfill \end{array}$$

  where we use the fact that $\langle s+r\rangle \sim \langle s-r\rangle \sim \langle r\rangle \gtrsim \langle s\rangle$ in the region $\{(s,x)\in [0,\infty )\times {\mathbb{R}}^2:|x|\ge 2s+1\}$. Hence, for $5\le |I|\le N$, we deduce

  $$\widehat{\mathcal{R}}\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|-1/2)}{\langle t\rangle }^{\delta }.$$

  (62)

  On the other hand, for $|I|\le 4\le N-6$, we can also obtain (62) by using the fact that $\widehat{\mathcal{R}}\lesssim {\widehat{\mathcal{R}}}_1$ and the estimate of ${\widehat{\mathcal{R}}}_1$ above. Combining (61) and (62), we conclude that

  $$\parallel \langle r-t\rangle \chi (r-2t){\widehat{\mathsf{\Gamma }}}^I{\psi \parallel }_{L_x^2}\lesssim Kϵ+C_1^{2}K{ϵ}^{1-\delta (|I|-1/2)}{\langle t\rangle }^{\delta },\mathrm{for}|I|\le N.$$

  (63)

  We see that the weighted $L_x^2$ norm of the Dirac field in the exterior region exhibits mild polynomial growth in time. In other words, the outward propagation of the Dirac field remains predominantly dispersive.

The conclusion of the proposition follows from (57), (60) and (63). □

Next, we establish refined bounds for the bootstrap assumptions (43)–(45) by deriving energy and spacetime $L^2$ estimates for the Klein–Gordon field, together with weighted $L_x^2$ estimates in the exterior region.

Proposition 4.

Under the assumptions (37)–(47), we obtain the following estimates for $|I|\le N$:

$$\begin{array}{cc}& {[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}\lesssim K+C_1^2{ϵ}^{2-\delta (N+9/2)}{\langle t\rangle }^{3\delta },\hfill \\ & {\left({\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-7\delta }{\displaystyle \frac{|{\mathsf{\Gamma }}^I{v|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\lesssim K+C_1^{{\displaystyle \frac{3}{2}}}{K}^{{\displaystyle \frac{1}{2}}}{ϵ}^{1-\delta (N/2+9/4)},\hfill \\ & \parallel \langle r-t\rangle \chi (r-2t)\left(|\mathsf{\partial }{\mathsf{\Gamma }}^{I}v|+|{\mathsf{\Gamma }}^{I}v|\right){\parallel }_{L_x^2}\lesssim K+C_1^2{ϵ}^{2-\delta (N+9/2)}.\hfill \end{array}$$

Proof. 

The proof is organized into three steps below.

• $\mathbf{Step}\mathbf{1}.$ The top-order energy estimate for v: Applying the vector field ${\mathsf{\Gamma }}^I$ with $|I|\le N$ to both sides of the second equation in (1), we derive

  $$-\square {\mathsf{\Gamma }}^{I}v+{\mathsf{\Gamma }}^{I}v={\mathsf{\Gamma }}^I\left({\psi }^{*}{\gamma }^0\psi \right).$$

  (64)

  Then, by Lemma 2 (see (29)) and Proposition 1 (see (11) and (12)), we obtain

  $${[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}\lesssim {[{\mathcal{E}}_1(0,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left({\psi }^{*}{\gamma }^0\psi \right)\parallel }_{L_x^2}\mathrm{d}s\lesssim K+\mathcal{T},$$

  (65)

  where we recall that $K=K_{*}^N$, and the term $\mathcal{T}$ is defined as

  $$\mathcal{T}:=\sum _{|I_1|+|I_2|\le |I|}{\int }_0^t\parallel |{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi {|\parallel }_{L_x^2}\mathrm{d}s.$$

  We recall that $N\ge 13$. Hence,

  $$\mathcal{T}\le {\mathcal{T}}_1+{\mathcal{T}}_2,$$

  where

  $$\begin{array}{cc}\hfill {\mathcal{T}}_1:& =\sum _{\begin{array}{c}|I_1|\le N-5\\ |I_1|+|I_2|\le |I|\end{array}}{\int }_0^t\parallel {[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\mathcal{T}}_2:& =\sum _{\begin{array}{c}|I_2|\le N-5\\ |I_1|+|I_2|\le |I|\end{array}}{\int }_0^t{\parallel \langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^2}·{\parallel {\langle r-s\rangle }^{{\displaystyle \frac{3}{5}}}{\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

  Using (37) and (42), we deduce

  $$\begin{array}{cc}\hfill {\mathcal{T}}_1& \lesssim C_1^2\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_1|+9/2)}{ϵ}^{1-\delta |I_2|}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+3\delta }\mathrm{d}s\lesssim C_1^2{ϵ}^{2-\delta (|I|+9/2)},\hfill \\ \hfill {\mathcal{T}}_2& \lesssim C_1\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+9/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }{\parallel {\langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim C_1\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+9/2)}{\left({\int }_0^t{\langle s\rangle }^{-1+4\delta }\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_0^t{\parallel {\langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \lesssim C_1^2\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta |I_1|}{ϵ}^{1-\delta (|I_2|+9/2)}{\langle t\rangle }^{3\delta }\lesssim C_1^2{ϵ}^{2-\delta (|I|+9/2)}{\langle t\rangle }^{3\delta },\hfill \end{array}$$

  (66)

  which leads to

  $$\mathcal{T}\lesssim C_1^2{ϵ}^{2-\delta (|I|+9/2)}{\langle t\rangle }^{3\delta }.$$

  (67)

  Combining (65) and (67), we find

  $${[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}\lesssim K+C_1^2{ϵ}^{2-\delta (|I|+9/2)}{\langle t\rangle }^{3\delta },\mathrm{for}|I|\le N.$$

  (68)

  This characterizes the polynomial-in-time growth of the top-order energy of the Klein–Gordon field, showing that nonlinear interactions generate only controlled long-time growth in the high-frequency regime while preserving the dispersive nature of the field.
• $\mathbf{Step}\mathbf{2}.$ The weighted spacetime $L^2$ estimate for v: Using Lemma 2 (see (28)), (64) and (43), for $|I|\le N$, we see that

  $$\begin{array}{cc}\hfill {\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-7\delta }{\displaystyle \frac{|{\mathsf{\Gamma }}^I{v|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s& \lesssim {\mathcal{E}}_1(0,{\mathsf{\Gamma }}^{I}v)+{\int }_0^t{\langle s\rangle }^{-7\delta }\parallel {\mathsf{\Gamma }}^I\left({\psi }^{*}{\gamma }^0\psi \right){\parallel }_{L_x^2}·{\parallel {\mathsf{\partial }}_t{\mathsf{\Gamma }}^{I}v\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim K^2+\tilde{\mathcal{T}},\hfill \end{array}$$

  (69)

  where

  $$\tilde{\mathcal{T}}:=C_{1}K{\int }_0^t{\langle s\rangle }^{-4\delta }{\parallel {\mathsf{\Gamma }}^I\left({\psi }^{*}{\gamma }^0\psi \right)\parallel }_{L_x^2}\mathrm{d}s.$$

  We have

  $$\tilde{\mathcal{T}}\lesssim {\tilde{\mathcal{T}}}_1+{\tilde{\mathcal{T}}}_2$$

  with

  $$\begin{array}{cc}\hfill {\tilde{\mathcal{T}}}_1:& =C_{1}K\sum _{\begin{array}{c}|I_1|\le N-5\\ |I_1|+|I_2|\le |I|\end{array}}{\int }_0^t{\langle s\rangle }^{-4\delta }\parallel {[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{T}}}_2:& =C_{1}K\sum _{\begin{array}{c}|I_2|\le N-5\\ |I_1|+|I_2|\le |I|\end{array}}{\int }_0^t{\langle s\rangle }^{-4\delta }{\parallel \langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^2}·{\parallel {\langle r-s\rangle }^{{\displaystyle \frac{3}{5}}}{\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

  From (66) and (38), we obtain

  $$\begin{array}{cc}\hfill {\tilde{\mathcal{T}}}_1& \lesssim C_{1}K·{\mathcal{T}}_1\lesssim C_1^{3}K{ϵ}^{2-\delta (|I|+9/2)},\hfill \\ \hfill {\tilde{\mathcal{T}}}_2& \lesssim C_1^{2}K\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+9/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}-2\delta }{\parallel {\langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}\parallel }_{L_x^2}\mathrm{d}s\hfill \\ & \lesssim C_1^{2}K\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+9/2)}{\left({\int }_0^t{\langle s\rangle }^{-1-\delta }\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_0^t{\langle s\rangle }^{-3\delta }{\parallel {\langle r-s\rangle }^{-{\displaystyle \frac{3}{5}}}{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \lesssim C_1^{3}K\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta |I_1|}{ϵ}^{1-\delta (|I_2|+9/2)}\lesssim C_1^{3}K{ϵ}^{2-\delta (|I|+9/2)},\hfill \end{array}$$

  which yields

  $$\tilde{\mathcal{T}}\lesssim C_1^{3}K{ϵ}^{2-\delta (|I|+9/2)}.$$

  (70)

  Combining (69) and (70), we conclude

  $${\left({\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-7\delta }{\displaystyle \frac{|{\mathsf{\Gamma }}^I{v|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\lesssim K+C_1^{{\displaystyle \frac{3}{2}}}{K}^{{\displaystyle \frac{1}{2}}}{ϵ}^{1-\delta (N/2+9/4)},\mathrm{for}|I|\le N.$$

  (71)

  Such a weighted spacetime $L^2$-type estimate captures the propagation behavior of the Klein–Gordon field near the light cone.
• $\mathbf{Step}\mathbf{3}.$ $L_x^2$ estimate for v in the exterior region: By Lemma 2 (see (30)), (64), (39) and (42), for $|I|\le N$, we find

  $$\begin{array}{cc}\hfill & \parallel \langle r-t\rangle \chi (r-2t)\left(|\mathsf{\partial }{\mathsf{\Gamma }}^{I}v|+|{\mathsf{\Gamma }}^{I}v|\right){(t,x)\parallel }_{L_x^2}\hfill \\ \hfill & \lesssim \parallel \langle r\rangle \left(|\mathsf{\partial }{\mathsf{\Gamma }}^{I}v|+|{\mathsf{\Gamma }}^{I}v|\right){(0,x)\parallel }_{L_x^2}+{\int }_0^t{\parallel \langle r-s\rangle \chi (r-2s){\mathsf{\Gamma }}^I\left({\psi }^{*}{\gamma }^0\psi \right)(s,x)\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim K+\sum _{|I_1|+|I_2|\le |I|}{\int }_0^t\parallel \langle r-s\rangle \chi (r-2s)|{\mathsf{\Gamma }}^{I_1}\psi |·|{\mathsf{\Gamma }}^{I_2}\psi {|\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim K+\sum _{\begin{array}{c}|I_1|\le N-5\\ |I_1|+|I_2|\le |I|\end{array}}{\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}\psi {\mathbf{1}}_{\{r\ge 2s+1\}}{\parallel }_{L_x^{\infty }}·{\parallel \langle r-s\rangle \chi (r-2s){\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim K+C_1^2\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_1|+9/2)}{ϵ}^{1-\delta |I_2|}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+3\delta }\mathrm{d}s\lesssim K+C_1^2{ϵ}^{2-\delta (|I|+9/2)},\hfill \end{array}$$

  (72)

  where ${\mathbf{1}}_{\{r\ge 2s+1\}}$ denotes the characteristic function of $\{(s,x)\in [0,\infty )\times {\mathbb{R}}^2:|x|\ge 2s+1\}$. This yields weighted $L^2$ estimates for the Klein–Gordon field in the exterior region and reflects the stability of the field near the outgoing light cone in the far-field regime.

The conclusion of the proposition follows from (68), (71) and (72). □

• $\mathbf{Refined}\mathbf{estimates}\mathbf{of}\mathbf{top}\mathbf{order}:$ By Propositions 3 and 4, for $|I|\le N$, the following improved estimates for (37)–(39) and (43)–(45) hold:

  $$\begin{array}{cc}\hfill & \max \{{[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\psi )]}^{{\displaystyle \frac{1}{2}}},{\parallel \langle r-t\rangle \chi (r-2t){\widehat{\mathsf{\Gamma }}}^I\psi \parallel }_{L_x^2}\}\le {\displaystyle \frac{1}{2}}{C}_1{ϵ}^{1-\delta |I|}{\langle t\rangle }^{\delta },\hfill \\ \hfill & {\left({\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-3\delta }{\displaystyle \frac{|{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\le {\displaystyle \frac{1}{2}}{C}_1{ϵ}^{1-\delta |I|},\hfill \\ \hfill & {[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}\le {\displaystyle \frac{1}{2}}{C}_{1}K{\langle t\rangle }^{3\delta },\hfill \\ \hfill & \max \left\{{\left({\int }_0^t{\int }_{{\mathbb{R}}^2}{\langle s\rangle }^{-7\delta }{\displaystyle \frac{|{\mathsf{\Gamma }}^I{v|}^2}{{\langle r-s\rangle }^{\frac{6}{5}}}}\mathrm{d}x\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}},{\parallel \langle r-t\rangle \chi (r-2t)\left(|\mathsf{\partial }{\mathsf{\Gamma }}^{I}v|+|{\mathsf{\Gamma }}^{I}v|\right)\parallel }_{L_x^2}\right\}\le {\displaystyle \frac{1}{2}}{C}_{1}K,\hfill \end{array}$$

  (73)

  provided that

  $$0<\delta \le {\displaystyle \frac{2}{N+6}},C_1\ge CK,0<ϵ\le {\left(C{C}_{1}K\right)}^{-2/\delta }$$

  (74)

  for some large constant $C>0$ (independent of $C_1,ϵ$ and K).

4. Proof of Theorem 1—Continued

4.1. Lower-Order Energy and Pointwise Estimates for $\psi$

In this subsection, we prove refined lower-order energy estimates and pointwise bounds for $\psi$. In particular, we derive improved versions of the estimates (40)–(42).

Let $\tilde{\psi }:=\psi +i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)$ and $\tilde{\mathsf{\Psi }}:=\mathsf{\Psi }+v\psi$. Then, by Lemma 7, for $|I|\le N-1$, we see that

$$-i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }{\widehat{\mathsf{\Gamma }}}^I\tilde{\psi }={\widehat{\mathsf{\Gamma }}}^I{F}_{\tilde{\psi }},-\square {\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }}={\mathsf{\Gamma }}^I{F}_{\tilde{\psi }},$$

(75)

where

$$F_{\tilde{\psi }}:=-2{\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi +iv{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)+\left({\psi }^{*}{\gamma }^0\psi \right)\psi .$$

(76)

The following propositions, Propositions 5–7, provide estimates for the first term in the expression of $F_{\tilde{\psi }}$.

Proposition 5.

Under the assumptions (37)–(47), the following estimates hold for $|I|\le N-1$:

$${\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left({\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi \right)\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}.$$

Proof. 

The proof is similar to (and simpler than) that of Proposition 6 below and is therefore deferred to Appendix B. □

Proposition 6.

Under the assumptions (37)–(47), for $|I|\le N-2$, we have

$${\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left({\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi \right)\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}.$$

Proof. 

The proof is based on the following main ideas. First, we decompose spacetime into interior and exterior regions. Second, we exploit the additional decay of both fields in these regions. Finally, we combine these ingredients to obtain the desired conclusion. We now present the details of the proof.

For $|I|\le N-2$, we have

$${\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left({\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi \right)\parallel }_{L_x^2}\mathrm{d}s\lesssim {\tilde{\mathcal{R}}}^{\mathrm{in}}+{\tilde{\mathcal{R}}}^{\mathrm{ex}},$$

(77)

where

$${\tilde{\mathcal{R}}}^{\mathrm{in}}:={\int }_0^t\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left({\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi \right){\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}{\parallel }_{L_x^2}\mathrm{d}s,{\tilde{\mathcal{R}}}^{\mathrm{ex}}:={\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left({\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi \right){\mathbf{1}}_{{\mathcal{C}}^{\mathrm{ex}}}\parallel }_{L_x^2}\mathrm{d}s.$$

By $\left(vi\right)$ in Proposition 1, for $|I|\le N-2$, we find

$${\tilde{\mathcal{R}}}^{\mathrm{in}}\lesssim {\tilde{\mathcal{R}}}_1^{\mathrm{in}}+{\tilde{\mathcal{R}}}_2^{\mathrm{in}}$$

(78)

with

$$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_1^{\mathrm{in}}:& =\sum _{\begin{array}{c}|I_1|+|I_2|\le |I|\\ |J_1|,|J_2|\le 1\end{array}}{\int }_0^t\parallel |{\mathsf{\Gamma }}^{J_1}{\mathsf{\Gamma }}^{I_1}v|·|{\mathsf{\Gamma }}^{J_2}{\mathsf{\Gamma }}^{I_2}\psi {|{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}\parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{R}}}_2^{\mathrm{in}}:& =\sum _{|I_1|+|I_2|\le |I|}{\int }_0^t\parallel \langle s-r\rangle |\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v|·|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi {|{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}\parallel }_{L_x^2}\mathrm{d}s.\hfill \end{array}$$

If $5\le |I|\le N-2$, then for any $|I_1|+|I_2|\le |I|$, we have either $|I_2|\le |I|-5$ or $|I_1|\le 4\le N-7$. Then,

$${\tilde{\mathcal{R}}}_1^{\mathrm{in}}\lesssim {\tilde{\mathcal{R}}}_{1,1}^{\mathrm{in}}+{\tilde{\mathcal{R}}}_{1,2}^{\mathrm{in}},$$

(79)

where

$$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_{1,1}^{\mathrm{in}}:& =\sum _{\begin{array}{c}|I_1|\le N-7,|I_2|\le |I|\\ |J_1|,|J_2|\le 1\end{array}}{\int }_0^t\parallel {\mathsf{\Gamma }}^{J_1}{\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{J_2}{\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{R}}}_{1,2}^{\mathrm{in}}:& =\sum _{\begin{array}{c}|I_2|\le |I|-5,|I_1|+|I_2|\le |I|\\ |J_1|,|J_2|\le 1\end{array}}{\int }_0^t\parallel {\displaystyle \frac{{\mathsf{\Gamma }}^{J_1}{\mathsf{\Gamma }}^{I_1}v}{\langle r-s\rangle }}{\parallel }_{L_x^2}·{\parallel \langle r-s\rangle {\mathsf{\Gamma }}^{J_2}{\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

Using (37) and (47), we deduce

$${\tilde{\mathcal{R}}}_{1,1}^{\mathrm{in}}\lesssim C_1^{2}K\sum _{|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+1)}{\int }_0^t{\langle s\rangle }^{-1+\delta }\mathrm{d}s\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }.$$

(80)

We turn to the estimate for ${\tilde{\mathcal{R}}}_{1,2}^{\mathrm{in}}$. For any $|I_1|\le |I|\le N-2,|J_1|\le 1$ and $s\in [0,t]$, using Lemma 9, along with (37) and (43), we see that

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{{\mathsf{\Gamma }}^{J_1}{\mathsf{\Gamma }}^{I_1}v}{\langle r-s\rangle }}{\parallel }_{L_x^2}& \lesssim {\langle s\rangle }^{-1}\sum _{|J|\le |I_1|+2}{\parallel \mathsf{\partial }{\mathsf{\Gamma }}^{J}v\parallel }_{L_x^2}\hfill \\ & +C_1{ϵ}^{1-\delta N}\left\{{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\sum _{|J|\le |I_1|+1}{\parallel {\widehat{\mathsf{\Gamma }}}^J\psi \parallel }_{L_x^2}+{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}}\sum _{|J|\le |I_1|+1}\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}}{\langle r-s\rangle }}{\parallel }_{L_x^2}\right\}\hfill \\ & \lesssim C_{1}K{\langle s\rangle }^{-1+3\delta }+C_1^2{ϵ}^{1-\delta N}{ϵ}^{1-\delta (|I_1|+1)}{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+3\delta }+C_1{ϵ}^{1-\delta N}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}}\sum _{|J|\le |I_1|+1}\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}}{\langle r-s\rangle }}{\parallel }_{L_x^2}.\hfill \end{array}$$

This together with (42) yields

$$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_{1,2}^{\mathrm{in}}& \lesssim C_1^{2}K\sum _{|I_2|\le |I|-5}{ϵ}^{1-\delta (|I_2|+11/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+5\delta }\mathrm{d}s\hfill \\ \hfill & +C_1^3{ϵ}^{1-\delta N}\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_1|+1)}{ϵ}^{1-\delta (|I_2|+11/2)}{\int }_0^t{\langle s\rangle }^{-2+5\delta }\mathrm{d}s\hfill \\ \hfill & +C_1^2{ϵ}^{1-\delta N}\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+11/2)}\sum _{|J|\le |I_1|+1}{\int }_0^t{\langle s\rangle }^{-1+2\delta }·\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}}{\langle r-s\rangle }}{\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1/2)}+C_1^3{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}+C_1^3{ϵ}^{1-\delta N}\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{2-\delta (|I_1|+|I_2|+13/2)}\hfill \\ \hfill & \lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1/2)}+C_1^3{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)},\hfill \end{array}$$

(81)

where we use (38) in the second inequality to obtain that

$${\int }_0^t{\langle s\rangle }^{-1+2\delta }·\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}}{\langle r-s\rangle }}{\parallel }_{L_x^2}\mathrm{d}s\lesssim {\left({\int }_0^t{\langle s\rangle }^{-2+7\delta }\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}·{\left({\int }_0^t{\langle s\rangle }^{-3\delta }\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}}{\langle r-s\rangle }}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\lesssim C_1{ϵ}^{1-\delta |J|}.$$

Combining (79)–(81), for $5\le |I|\le N-2$, we deduce

$${\tilde{\mathcal{R}}}_1^{\mathrm{in}}\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^3{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}.$$

(82)

On the other hand, for $|I|\le 4\le N-7$, ${\tilde{\mathcal{R}}}_1^{\mathrm{in}}$ is bounded by ${\tilde{\mathcal{R}}}_{1,1}^{\mathrm{in}}$, and hence (82) holds as well.

For ${\tilde{\mathcal{R}}}_2^{\mathrm{in}}$, we first consider $5\le |I|\le N-2$. In this case, for any $|I_1|+|I_2|\le |I|$, we have either $|I_2|\le |I|-5$ or $|I_1|\le 4\le N-9$. Hence,

$${\tilde{\mathcal{R}}}_2^{\mathrm{in}}\lesssim {\tilde{\mathcal{R}}}_{2,1}^{\mathrm{in}}+{\tilde{\mathcal{R}}}_{2,2}^{\mathrm{in}},$$

(83)

where

$$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_{2,1}^{\mathrm{in}}:& =\sum _{\begin{array}{c}|I_1|\le N-9\\ |I_1|+|I_2|\le |I|\end{array}}{\int }_0^t\parallel \langle s-r\rangle |\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v|·|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi {|{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}\parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{R}}}_{2,2}^{\mathrm{in}}:& =\sum _{\begin{array}{c}|I_2|\le |I|-5\\ |I_1|+|I_2|\le |I|\end{array}}{\int }_0^t\parallel \langle s-r\rangle |\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v|·|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi {|{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}\parallel }_{L_x^2}\mathrm{d}s.\hfill \end{array}$$

By Lemma 9 and (47), for any $|I_1|+|I_2|\le |I|$ with $|I_1|\le N-9$, we find

$$\begin{array}{cc}& \langle s-r\rangle |\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v|·|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi |{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}\hfill \\ & \lesssim |\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v|·\left\{C_{1}K\sum _{|J|\le |I_2|+1}|{\widehat{\mathsf{\Gamma }}}^J\psi |+C_1{ϵ}^{1-\delta (|I_2|+1/2)}{\displaystyle \frac{{\langle s\rangle }^{\frac{1}{2}+2\delta }}{\langle s-r\rangle }}\sum _{|J|\le |I_2|}|{\mathsf{\Gamma }}^{J}v|\right\}\hfill \\ & \lesssim {\left(C_{1}K\right)}^2{\langle s\rangle }^{-1}\sum _{|J|\le |I_2|+1}|{\widehat{\mathsf{\Gamma }}}^J\psi |\hfill \\ & +C_1{ϵ}^{1-\delta (|I_2|+1/2)}{\langle s\rangle }^{{\displaystyle \frac{1}{2}}+2\delta }\left\{C_{1}K{\langle s\rangle }^{-2}+C_1^2{ϵ}^{2-\delta (|I_1|+10)}{\langle s\rangle }^{-2+2\delta }\right\}\sum _{|J|\le |I_2|}|{\mathsf{\Gamma }}^{J}v|,\hfill \end{array}$$

from which it follows that

$$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_{2,1}^{\mathrm{in}}& \lesssim C_1^3{K}^2\sum _{|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+1)}{\int }_0^t{\langle s\rangle }^{-1+\delta }\mathrm{d}s+C_1^3{K}^2\sum _{|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+1/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\mathrm{d}s\hfill \\ \hfill & +C_1^{4}K\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+1/2)}{ϵ}^{2-\delta (|I_1|+10)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+4\delta }\mathrm{d}s\hfill \\ \hfill & \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{3-\delta (|I|+11)},\hfill \end{array}$$

(84)

where we use (37) and (46). For any $|I_1|+|I_2|\le |I|$ with $|I_2|\le |I|-5\le N-7$, Lemma 9 implies that

$$\begin{array}{cc}& \langle s-r\rangle |\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v|·|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi |{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}\hfill \\ & \lesssim \left\{{\displaystyle \frac{\langle s-r\rangle }{\langle s+r\rangle }}\sum _{|J|\le |I_1|+2}|\mathsf{\partial }{\mathsf{\Gamma }}^{J}v|+C_1{ϵ}^{1-\delta N}\left[{\langle s+r\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\sum _{|J|\le |I_1|+1}|{\widehat{\mathsf{\Gamma }}}^J{\psi |+\langle s+r\rangle }^{-{\displaystyle \frac{1}{2}}}\sum _{|J|\le |I_1|+1}|{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}|\right]\right\}\hfill \\ & ·\langle s-r\rangle |\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi |{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}\hfill \\ & \lesssim {\mathcal{J}}_1+{\mathcal{J}}_2+{\mathcal{J}}_3,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathcal{J}}_1:& ={\langle s\rangle }^{-1}{\langle s-r\rangle }^2|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi |{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}·\sum _{|J|\le |I_1|+2}|\mathsf{\partial }{\mathsf{\Gamma }}^{J}v|,\hfill \\ \hfill {\mathcal{J}}_2:& =C_1{ϵ}^{1-\delta N}{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\langle s-r\rangle |\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi |·\sum _{|J|\le |I_1|+1}|{\widehat{\mathsf{\Gamma }}}^J\psi |,\hfill \\ \hfill {\mathcal{J}}_3:& =C_1{ϵ}^{1-\delta N}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}}{\langle s-r\rangle }^2|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi |{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{in}}}·\sum _{|J|\le |I_1|+1}\left|{\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}}{\langle s-r\rangle }}\right|.\hfill \end{array}$$

Using Lemma 9 along with (42), we derive

$$\begin{array}{cc}\hfill {\mathcal{J}}_1& \lesssim {\langle s\rangle }^{-1}{C}_1^{2}K{ϵ}^{1-\delta (|I_2|+11/2)}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }·\sum _{|J|\le |I_1|+2}|\mathsf{\partial }{\mathsf{\Gamma }}^{J}v|\hfill \\ & \lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1/2)}{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }·\sum _{|J|\le |I_1|+2}|\mathsf{\partial }{\mathsf{\Gamma }}^{J}v|,\hfill \\ \hfill {\mathcal{J}}_2& \lesssim C_1{ϵ}^{1-\delta N}{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }{C}_1{ϵ}^{1-\delta (|I_2|+11/2)}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }·\sum _{|J|\le |I_1|+1}|{\widehat{\mathsf{\Gamma }}}^J\psi |\hfill \\ & \lesssim C_1^2{ϵ}^{1-\delta N}{ϵ}^{1-\delta (|I_2|+11/2)}{\langle s\rangle }^{-2+4\delta }·\sum _{|J|\le |I_1|+1}|{\widehat{\mathsf{\Gamma }}}^J\psi |,\hfill \\ \hfill {\mathcal{J}}_3& \lesssim C_1{ϵ}^{1-\delta N}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}}{C}_1^{2}K{ϵ}^{1-\delta (|I_2|+11/2)}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }·\sum _{|J|\le |I_1|+1}\left|{\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}}{\langle s-r\rangle }}\right|\hfill \\ & \lesssim C_1^{3}K{ϵ}^{1-\delta N}{ϵ}^{1-\delta (|I_2|+11/2)}{\langle s\rangle }^{-1+2\delta }·\sum _{|J|\le |I_1|+1}\left|{\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}}{\langle s-r\rangle }}\right|.\hfill \end{array}$$

Then, from (37), (38) and (43), we see that

$$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_{2,2}^{\mathrm{in}}& \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+5\delta }\mathrm{d}s\hfill \\ \hfill & +C_1^3{ϵ}^{1-\delta N}\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_1|+1)}{ϵ}^{1-\delta (|I_2|+11/2)}{\int }_0^t{\langle s\rangle }^{-2+5\delta }\mathrm{d}s\hfill \\ \hfill & +C_1^{3}K{ϵ}^{1-\delta N}\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+11/2)}\sum _{|J|\le |I_1|+1}{\int }_0^t{\langle s\rangle }^{-1+2\delta }\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^J\psi ]}_{-}}{\langle s-r\rangle }}{\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1/2)}+C_1^3{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}\hfill \\ \hfill & +C_1^{4}K{ϵ}^{1-\delta N}\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_1|+1)}{ϵ}^{1-\delta (|I_2|+11/2)}\hfill \\ \hfill & \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1/2)}+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}.\hfill \end{array}$$

(85)

Combining (83)–(85), for $5\le |I|\le N-2$, we conclude

$${\tilde{\mathcal{R}}}_2^{\mathrm{in}}\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}.$$

(86)

On the other hand, for $|I|\le 4\le N-9$, we can bound ${\tilde{\mathcal{R}}}_2^{\mathrm{in}}$ by ${\tilde{\mathcal{R}}}_{2,1}^{\mathrm{in}}$, hence (86) also holds. Combining (78), (82) and (86), we obtain

$${\tilde{\mathcal{R}}}^{\mathrm{in}}\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}.$$

(87)

We turn to the estimate for ${\tilde{\mathcal{R}}}^{\mathrm{ex}}$. If $3\le |I|\le N-2$, then for any $|I_1|+|I_2|\le |I|$, we have $|I_2|\le |I|-3$ or $|I_1|\le 2\le N-3$. It follows that

$${\tilde{\mathcal{R}}}^{\mathrm{ex}}\lesssim \sum _{|I_1|+|I_2|\le |I|}{\int }_0^t\parallel \langle s+r\rangle |\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v|·|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi {|{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{ex}}}\parallel }_{L_x^2}\mathrm{d}s\lesssim {\tilde{\mathcal{R}}}_1^{\mathrm{ex}}+{\tilde{\mathcal{R}}}_2^{\mathrm{ex}},$$

where

$$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_1^{\mathrm{ex}}:& =\sum _{\begin{array}{c}|I_1|\le N-3\\ |I_2|\le |I|\end{array}}{\int }_0^t\parallel \mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{ex}}}{\parallel }_{L_x^{\infty }}·{\parallel \langle s+r\rangle \mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi {\mathbf{1}}_{{\mathcal{C}}^{\mathrm{ex}}}\parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{R}}}_2^{\mathrm{ex}}:& =\sum _{\begin{array}{c}|I_2|\le |I|-3\\ |I_1|\le |I|\end{array}}{\int }_0^t\parallel \langle s+r\rangle \mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{ex}}}{\parallel }_{L_x^2}·{\parallel \mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi {\mathbf{1}}_{{\mathcal{C}}^{\mathrm{ex}}}\parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

We observe that for $(s,x)\in {\mathcal{C}}^{\mathrm{ex}}$, it holds that $\langle r+s\rangle \sim \langle r\rangle \sim \langle r-s\rangle$ and $\chi (r-2s)=1$. Then, by (39), (45) and Lemma 9, we deduce

$$\begin{array}{cc}\hfill {\tilde{\mathcal{R}}}_1^{\mathrm{ex}}& \lesssim \sum _{\begin{array}{c}|I_1|\le N-3\\ |I_2|\le |I|\end{array}}{\int }_0^t\parallel \mathsf{\partial }{\mathsf{\Gamma }}^{I_1}v{\mathbf{1}}_{{\mathcal{C}}^{\mathrm{ex}}}{\parallel }_{L_x^{\infty }}·{\parallel \langle r-s\rangle \chi (r-2s)\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s\hfill \\ & \lesssim C_1^{2}K\sum _{|I_2|\le |I|}{ϵ}^{1-\delta (|I_2|+1)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+\delta }\mathrm{d}s\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)},\hfill \\ \hfill {\tilde{\mathcal{R}}}_2^{\mathrm{ex}}& \lesssim \sum _{\begin{array}{c}|I_2|\le |I|-3\\ |I_1|\le |I|\end{array}}{\int }_0^t\parallel \langle r-s\rangle \chi (r-2s)\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^2}·{\parallel \mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi {\mathbf{1}}_{{\mathcal{C}}^{\mathrm{ex}}}\parallel }_{L_x^{\infty }}\hfill \\ & \lesssim C_1^{2}K\sum _{|I_2|\le |I|-3}{ϵ}^{1-\delta (|I_2|+3)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+\delta }\mathrm{d}s\lesssim C_1^{2}K{ϵ}^{1-\delta |I|},\hfill \end{array}$$

hence,

$${\tilde{\mathcal{R}}}^{\mathrm{ex}}\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)}.$$

(88)

On the other hand, for $|I|\le 2\le N-3$, we have ${\tilde{\mathcal{R}}}^{\mathrm{ex}}\lesssim {\tilde{\mathcal{R}}}_1^{\mathrm{ex}}$. Hence we can obtain (88) as well. Combining (77), (87) and (88), for $|I|\le N-2$, we conclude

$${\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left({\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi \right)\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}.$$

The proof is complete. □

Proposition 7.

Under the assumptions (37)–(47), for $|I|\le N-2$, we have

$${\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left({\mathsf{\partial }}_{\alpha }v{\mathsf{\partial }}^{\alpha }\psi \right)\parallel }_{L_x^1}\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+3/2)}.$$

Proof. 

The proof is similar to that of Proposition 6 and is deferred to Appendix C. □

Next, we show the estimates for the second term appearing in the expression of $F_{\tilde{\psi }}$ (see (76)).

Proposition 8.

Under the assumptions (37)–(47), for $|I|\le N-1$, we obtain

$${\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)},$$

and for $|I|\le N-2$, we have

$$\begin{array}{cc}& {\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{3-\delta (|I|+16)},\hfill \\ & {\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^1}\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }.\hfill \end{array}$$

Proof. 

First, we present the estimate for ${\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^2}\mathrm{d}s$, where $|I|\le N-1$.

If $6\le |I|\le N-1$, then for any $|I_1|+|I_2|+|I_3|\le |I|$, one of the following cases occurs: (i) $|I_2|,|I_3|\le |I|-6$; (ii) $|I_1|,|I_3|\le |I|-6$; (iii) $|I_1|,|I_2|\le 5\le N-7$. Then, using (37), (43), (47) and (48), we find

$$\begin{array}{cc}\hfill & {\int }_0^t\parallel {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right){\parallel }_{L_x^2}\mathrm{d}s\lesssim \sum _{\begin{array}{c}|I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel |{\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}v|·|{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}v|·|{\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi {|\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim \sum _{\begin{array}{c}|I_1|,|I_2|\le N-7,|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·\parallel {\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}{v\parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi \parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & +\sum _{\begin{array}{c}|I_2|,|I_3|\le |I|-6,|I_1|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}{v\parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}{v\parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s\hfill \\ \hfill & \lesssim C_1^3{K}^2\sum _{|I_3|\le |I|}{ϵ}^{1-\delta (|I_3|+1)}{\int }_0^t{\langle s\rangle }^{-2+\delta }\mathrm{d}s+C_1^3{K}^2\sum _{|I_3|\le |I|-6}{ϵ}^{1-\delta (|I_3|+11/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+3\delta }\mathrm{d}s\hfill \\ \hfill & \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}.\hfill \end{array}$$

(89)

On the other hand, for $|I|\le 5\le N-7$, the estimate (89) remains valid, as ${\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^2}\mathrm{d}s$ is bounded by the sum in the second line of (89).

We now provide the estimate for ${\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^2}\mathrm{d}s$ with $|I|\le N-2$.

If $7\le |I|\le N-2$, then for any $|I_1|+|I_2|+|I_3|\le |I|$, one of the following cases happens: (i) $|I_2|,|I_3|\le |I|-7$; (ii) $|I_1|,|I_3|\le |I|-7$; (iii) $|I_1|,|I_2|\le 6\le N-7$. Hence,

$$\begin{array}{cc}& {\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^2}\mathrm{d}s\hfill \\ & \lesssim \sum _{\begin{array}{c}|I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel \langle s+r\rangle |{\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}v|·|{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}v|·|{\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi {|\parallel }_{L_x^2}\mathrm{d}s\lesssim {\mathcal{J}}_1+{\mathcal{J}}_2,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathcal{J}}_1& :=\sum _{\begin{array}{c}|I_1|,|I_2|\le N-7,|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel \langle s+r\rangle {\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·\parallel {\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}{v\parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\mathcal{J}}_2& :=\sum _{\begin{array}{c}|I_2|,|I_3|\le |I|-7\\ |I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}{v\parallel }_{L_x^2}·\parallel {\displaystyle \frac{\langle s+r\rangle }{\langle s-r\rangle }}{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}v{\parallel }_{L_x^{\infty }}·{\parallel \langle s-r\rangle {\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

Then, using Lemma 9, (37), (42), (46) and (47), we see that

$$\begin{array}{cc}\hfill {\mathcal{J}}_1& \lesssim C_1^3{K}^2\sum _{|I_3|\le |I|}{ϵ}^{1-\delta (|I_3|+1)}{\int }_0^t{\langle s\rangle }^{-1+\delta }\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta },\hfill \\ \hfill {\mathcal{J}}_2& \lesssim C_1^{2}K\sum _{\begin{array}{c}|I_3|\le |I|-7\\ |I_2|+|I_3|\le |I|\end{array}}{ϵ}^{1-\delta (|I_3|+11/2)}{\int }_0^t\left\{C_{1}K{\langle s\rangle }^{-1}+C_1^2{ϵ}^{2-\delta (|I_2|+10)}{\langle s\rangle }^{-1+2\delta }\right\}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }\mathrm{d}s\hfill \\ & \lesssim C_1^3{K}^2{ϵ}^{1-\delta |I|}+C_1^{4}K{ϵ}^{3-\delta (|I|+16)},\hfill \end{array}$$

and it follows that

$${\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{3-\delta (|I|+16)}.$$

(90)

On the other hand, for $|I|\le 6\le N-7$, we can also obtain (90), as ${\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^2}\mathrm{d}s$ is bounded by ${\mathcal{J}}_1$ above.

It remains to address the estimate for ${\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^1}\mathrm{d}s$, where $|I|\le N-2$.

For any $|I_1|+|I_2|+|I_3|\le |I|\le N-2$, at least two of $|I_j|,j=1,2,3$ are no greater than $N-7$. It follows that

$$\begin{array}{cc}\hfill & {\int }_0^t\parallel {\mathsf{\Gamma }}^I\left(v{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right){\parallel }_{L_x^1}\mathrm{d}s\lesssim \sum _{\begin{array}{c}|I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel |{\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}v|·|{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}v|·|{\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi {|\parallel }_{L_x^1}\mathrm{d}s\hfill \\ \hfill & \lesssim \sum _{\begin{array}{c}|I_1|,|I_2|\le N-7,|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}{v\parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi \parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & +\sum _{\begin{array}{c}|I_2|,|I_3|\le N-7\\ |I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}{v\parallel }_{L_x^2}·\parallel {\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}{v\parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi \parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim C_1^3{K}^2\sum _{|I_3|\le |I|}{ϵ}^{1-\delta (|I_3|+1)}{\int }_0^t{\langle s\rangle }^{-1+\delta }\mathrm{d}s\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta },\hfill \end{array}$$

(91)

where we use (37), (46) and (47). Combining (89)–(91), we obtain the conclusion of the proposition. □

For the last term in the definition of $F_{\tilde{\psi }}$ in (76), we have the following estimates:

Proposition 9.

Under the assumptions (37)–(47), the estimate

$${\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right)\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+9)}$$

is valid for $|I|\le N-1$. Additionally, for $|I|\le N-2$, the following holds:

$$\begin{array}{cc}& {\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right)\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+11)}{\langle t\rangle }^{2\delta },\hfill \\ & {\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right)\parallel }_{L_x^1}\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+8)}.\hfill \end{array}$$

Proof. 

First, we handle the estimate for ${\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right)\parallel }_{L_x^2}\mathrm{d}s$ with $|I|\le N-1$. By $\left(i\right)$, $\left(iii\right)$ and $\left(iv\right)$ (see (10)–(12)) in Proposition 1, we deduce

$${\int }_0^t\parallel {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right){\parallel }_{L_x^2}\mathrm{d}s\lesssim \sum _{|I_1|+|I_2|+|I_3|\le |I|}{\int }_0^t\parallel |{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |·|{\widehat{\mathsf{\Gamma }}}^{I_3}\psi {|\parallel }_{L_x^2}\mathrm{d}s\lesssim {\mathcal{I}}_1+{\mathcal{I}}_2,$$

where

$$\begin{array}{cc}\hfill {\mathcal{I}}_1:& =\sum _{\begin{array}{c}|I_1|,|I_2|\le N-5\\ |I_1|+|I_2|+|I_3|\le |I|\end{array}}{\int }_0^t\parallel {[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^{\infty }}·\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}{\psi \parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_3}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\mathcal{I}}_2:& =\sum _{\begin{array}{c}|I_2|,|I_3|\le N-5\\ |I_1|+|I_2|+|I_3|\le |I|\end{array}}{\int }_0^t\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}}{\langle s-r\rangle }}{\parallel }_{L_x^2}·\parallel \langle s-r\rangle {\widehat{\mathsf{\Gamma }}}^{I_2}{\psi \parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_3}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

Then, from (37), (38) and (42), we find

$$\begin{array}{cc}\hfill {\mathcal{I}}_1& \lesssim C_1^3\sum _{|I_1|+|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_1|+9/2)}{ϵ}^{1-\delta (|I_2|+9/2)}{ϵ}^{1-\delta |I_3|}{\int }_0^t{\langle s\rangle }^{-2+5\delta }\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+9)},\hfill \\ \hfill {\mathcal{I}}_2& \lesssim C_1^2\sum _{|I_1|+|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_2|+9/2)}{ϵ}^{1-\delta (|I_3|+9/2)}{\int }_0^t{\langle s\rangle }^{-1+4\delta }\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}}{\langle s-r\rangle }}{\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+9)},\hfill \end{array}$$

which yields

$${\int }_0^t\parallel {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right){\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+9)},\mathrm{for}|I|\le N-1.$$

(92)

We now proceed to estimate ${\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right)\parallel }_{L_x^2}\mathrm{d}s$, for $|I|\le N-2$. We have

$${\int }_0^t\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right){\parallel }_{L_x^2}\mathrm{d}s\lesssim \sum _{|I_1|+|I_2|+|I_3|\le |I|}{\int }_0^t\parallel \langle s+r\rangle |{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |·|{\widehat{\mathsf{\Gamma }}}^{I_3}\psi {|\parallel }_{L_x^2}\mathrm{d}s\lesssim {\tilde{\mathcal{I}}}_1+{\tilde{\mathcal{I}}}_2,$$

where

$$\begin{array}{cc}\hfill {\tilde{\mathcal{I}}}_1:& =\sum _{\begin{array}{c}|I_1|,|I_2|\le N-5\\ |I_1|+|I_2|+|I_3|\le |I|\end{array}}{\int }_0^t\parallel \langle s+r\rangle {[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^{\infty }}·\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}{\psi \parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_3}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\tilde{\mathcal{I}}}_2:& =\sum _{\begin{array}{c}|I_2|,|I_3|\le N-5\\ |I_1|+|I_2|+|I_3|\le |I|\end{array}}{\int }_0^t\parallel \langle s+r\rangle {[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^2}·\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}{\psi \parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_3}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

Then, using (40)–(42) and (48), we derive

$$\begin{array}{cc}\hfill {\tilde{\mathcal{I}}}_1& \lesssim C_1^3\sum _{|I_1|+|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_1|+9/2)}{ϵ}^{1-\delta (|I_2|+9/2)}{ϵ}^{1-\delta (|I_3|+3/2)}{\int }_0^t{\langle s\rangle }^{-1+2\delta }\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+11)}{\langle t\rangle }^{2\delta },\hfill \\ \hfill {\tilde{\mathcal{I}}}_2& \lesssim C_1^3\sum _{|I_1|+|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_1|+3/2)}{ϵ}^{1-\delta (|I_2|+9/2)}{ϵ}^{1-\delta (|I_3|+9/2)}{\int }_0^t{\langle s\rangle }^{-1+2\delta }\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+11)}{\langle t\rangle }^{2\delta },\hfill \end{array}$$

which gives

$${\int }_0^t\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right){\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+11)}{\langle t\rangle }^{2\delta },\mathrm{for}|I|\le N-2.$$

(93)

Finally, we address the estimate for ${\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right)\parallel }_{L_x^1}\mathrm{d}s$, where $|I|\le N-2$. Clearly,

$${\int }_0^t\parallel {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right){\parallel }_{L_x^1}\mathrm{d}s\lesssim \sum _{|I_1|+|I_2|+|I_3|\le |I|}{\int }_0^t\parallel |{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |·|{\widehat{\mathsf{\Gamma }}}^{I_3}\psi {|\parallel }_{L_x^1}\mathrm{d}s\lesssim {\widehat{\mathcal{I}}}_1+{\widehat{\mathcal{I}}}_2,$$

where

$$\begin{array}{cc}\hfill {\widehat{\mathcal{I}}}_1:& =\sum _{\begin{array}{c}|I_1|,|I_2|\le N-5\\ |I_1|+|I_2|+|I_3|\le |I|\end{array}}{\int }_0^t\parallel {[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^{\infty }}·\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}{\psi \parallel }_{L_x^2}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_3}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\widehat{\mathcal{I}}}_2:& =\sum _{\begin{array}{c}|I_2|,|I_3|\le N-5\\ |I_1|+|I_2|+|I_3|\le |I|\end{array}}{\int }_0^t{\langle s\rangle }^{-1}\parallel \langle s+r\rangle {[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^2}·\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}{\psi \parallel }_{L_x^2}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_3}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

From (40)–(42) and (48), we see that

$$\begin{array}{cc}\hfill {\widehat{\mathcal{I}}}_1& \lesssim C_1^3\sum _{|I_1|+|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_1|+9/2)}{ϵ}^{1-\delta (|I_2|+3/2)}{ϵ}^{1-\delta (|I_3|+3/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+8)},\hfill \\ \hfill {\widehat{\mathcal{I}}}_2& \lesssim C_1^3\sum _{|I_1|+|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_1|+3/2)}{ϵ}^{1-\delta (|I_2|+3/2)}{ϵ}^{1-\delta (|I_3|+9/2)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+2\delta }\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+8)},\hfill \end{array}$$

which yields

$${\int }_0^t\parallel {\mathsf{\Gamma }}^I\left(\left({\psi }^{*}{\gamma }^0\psi \right)\psi \right){\parallel }_{L_x^1}\mathrm{d}s\lesssim C_1^3{ϵ}^{3-\delta (|I|+8)},\mathrm{for}|I|\le N-2.$$

(94)

Combining (92)–(94), the proof is complete. □

Combining Propositions 5–9, we derive refined bounds for the lower-order energy, $L^2$ and $L^{\infty }$ type norms of $\psi$, as stated below.

Proposition 10.

Under the assumptions (37)–(47), for $|I|\le N-1$, we have

$${[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\psi )]}^{{\displaystyle \frac{1}{2}}}\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}+C_1^3{ϵ}^{3-\delta (|I|+9)},$$

and for $|I|\le N-2$, we obtain

$$\parallel \langle t+r\rangle {[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{\parallel }_{L_x^2}\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }{\log }^{{\displaystyle \frac{1}{2}}}(2+t)+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}{\langle t\rangle }^{2\delta }.$$

In addition, the following holds for $|I|\le N-5$:

$$\langle t+r\rangle |{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}|+\langle t-r\rangle |{\widehat{\mathsf{\Gamma }}}^I\psi |\lesssim {\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}}\{C_1^3{K}^2{ϵ}^{1-\delta (|I|+4)}{\langle t\rangle }^{\delta }{\log }^{{\displaystyle \frac{1}{2}}}(2+t)+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+10)}{\langle t\rangle }^{2\delta }\}.$$

Proof. 

The idea is to derive energy and $L^2$-type estimates for the functions $\tilde{\psi }$ and $\tilde{\mathsf{\Psi }}$. These, together with Sobolev-type inequalities, yield the corresponding estimates for ψ. The details follow.

By (75) (see Lemma 7), (19) and Propositions 5, 8 and 9, for $|I|\le N-1$, we see that

$$\begin{array}{cc}\hfill & {[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\tilde{\psi })]}^{{\displaystyle \frac{1}{2}}}\lesssim {[{\mathcal{E}}_D(0,{\widehat{\mathsf{\Gamma }}}^I\tilde{\psi })]}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\parallel {\widehat{\mathsf{\Gamma }}}^I{F}_{\tilde{\psi }}\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim Kϵ+C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}+C_1^3{ϵ}^{3-\delta (|I|+9)}\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}+C_1^3{ϵ}^{3-\delta (|I|+9)}.\hfill \end{array}$$

(95)

Let $\tilde{\mathsf{\Psi }}:=\mathsf{\Psi }+v\psi$. For $|I|\le N-2$, it follows from (75), Lemmas 2 and 3 and Propositions 6, 8 and 9 that

$$\begin{array}{cc}\hfill {[\mathcal{E}(t,{\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }})+{\mathcal{E}}_{\mathrm{con}}(t,{\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }})]}^{{\displaystyle \frac{1}{2}}}& \lesssim {[\mathcal{E}(0,{\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }})+{\mathcal{E}}_{\mathrm{con}}(0,{\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }})]}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\parallel \langle s+r\rangle {\mathsf{\Gamma }}^I{F}_{\tilde{\psi }}\parallel }_{L_x^2}\mathrm{d}s\hfill \\ & \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}{\langle t\rangle }^{2\delta }.\hfill \end{array}$$

In addition, using Lemma 3 and Propositions 5 and 7–9, for $|I|\le N-2$, we deduce

$$\begin{array}{cc}\hfill \parallel {\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }}{(t,x)\parallel }_{L_x^2}& \lesssim \parallel {\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }}{(0,x)\parallel }_{L_x^2}+{\log }^{{\displaystyle \frac{1}{2}}}(2+t)\left(\parallel {\mathsf{\partial }}_t{\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }}{(0,x)\parallel }_{L_x^1}+{\parallel {\mathsf{\partial }}_t{\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }}(0,x)\parallel }_{L_x^2}\right)\hfill \\ & +{\log }^{{\displaystyle \frac{1}{2}}}(2+t){\int }_0^t\left(\parallel {\mathsf{\Gamma }}^I{F}_{\tilde{\psi }}{(s,x)\parallel }_{L_x^1}+{\parallel {\mathsf{\Gamma }}^I{F}_{\tilde{\psi }}(s,x)\parallel }_{L_x^2}\right)\mathrm{d}s\hfill \\ & \lesssim Kϵ{\log }^{{\displaystyle \frac{1}{2}}}(2+t)+\left\{C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}+C_1^3{ϵ}^{3-\delta (|I|+9)}\right\}{\log }^{{\displaystyle \frac{1}{2}}}(2+t)\hfill \\ & +\left\{C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+3/2)}\right\}{\log }^{{\displaystyle \frac{1}{2}}}(2+t)\hfill \\ & \lesssim \left\{C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+3/2)}\right\}{\log }^{{\displaystyle \frac{1}{2}}}(2+t).\hfill \end{array}$$

For $|I|\le N-2$, the last two estimates imply

$$\begin{array}{cc}\hfill \parallel L_0{\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }}{\parallel }_{L_x^2}+\sum _{|J|\le 1}{\parallel {\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^I\tilde{\mathsf{\Psi }}\parallel }_{L_x^2}& \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }{\log }^{{\displaystyle \frac{1}{2}}}(2+t)\hfill \\ \hfill & +C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}{\langle t\rangle }^{2\delta }.\hfill \end{array}$$

(96)

We recall that $\tilde{\psi }=\psi +i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)$. For $|I|\le N-1$, we see that

$$\begin{array}{cc}\hfill & {[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\left(i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right))]}^{{\displaystyle \frac{1}{2}}}\lesssim {\parallel {\widehat{\mathsf{\Gamma }}}^I\left(i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)\parallel }_{L_x^2}+{\left({\int }_0^t\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^I\left(i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)]}_{-}}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \lesssim \sum _{|I^{\prime }|\le |I|}\left\{\parallel {\mathsf{\Gamma }}^{I^{\prime }}\left(i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right){\parallel }_{L_x^2}+{\left({\int }_0^t\parallel {\displaystyle \frac{{\mathsf{\Gamma }}^{I^{\prime }}\left(i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)\right)}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\right\}\hfill \\ \hfill & \lesssim \sum _{|I_1|+|I_2|\le |I|+1}\left\{\parallel |{\mathsf{\Gamma }}^{I_1}v|·|{\mathsf{\Gamma }}^{I_2}\psi {|\parallel }_{L_x^2}+{\left({\int }_0^t\parallel {\displaystyle \frac{|{\mathsf{\Gamma }}^{I_1}v|·|{\mathsf{\Gamma }}^{I_2}\psi |}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\right\}.\hfill \end{array}$$

(97)

If $4\le |I|\le N-1$, then for any $|I_1|+|I_2|\le |I|+1$, we have $|I_2|\le |I|-4$ or $|I_1|\le 4\le N-4$. Hence,

$$\begin{array}{cc}\hfill & \sum _{|I_1|+|I_2|\le |I|+1}\parallel |{\mathsf{\Gamma }}^{I_1}v|·|{\mathsf{\Gamma }}^{I_2}\psi {|\parallel }_{L_x^2}\hfill \\ \hfill & \lesssim \sum _{\begin{array}{c}|I_1|\le N-4\\ |I_2|\le |I|+1\end{array}}\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·\parallel {\mathsf{\Gamma }}^{I_2}{\psi \parallel }_{L_x^2}+\sum _{\begin{array}{c}|I_2|\le |I|-4\\ |I_1|\le |I|+1\end{array}}\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^2}·{\parallel {\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^{\infty }}\hfill \\ \hfill & \lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{-{\displaystyle \frac{1}{2}}+\delta }+C_1^{2}K\sum _{|I_2|\le |I|-4}{ϵ}^{1-\delta (|I_2|+9/2)}{\langle t\rangle }^{-{\displaystyle \frac{1}{2}}+3\delta }\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)},\hfill \end{array}$$

(98)

where we use (37), (43), (48) and (49). On the other hand, for $|I|\le 3\le N-5$, the last estimate remains true, since

$$\sum _{|I_1|+|I_2|\le |I|+1}\parallel |{\mathsf{\Gamma }}^{I_1}v|·|{\mathsf{\Gamma }}^{I_2}{\psi |\parallel }_{L_x^2}\lesssim \sum _{\begin{array}{c}|I_1|\le N-4\\ |I_2|\le |I|+1\end{array}}\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^2}\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)}.$$

For $4\le |I|\le N-1$ and any $|I_1|+|I_2|\le |I|+1$, we have $|I_2|\le |I|-4$ or $|I_1|\le 4\le N-6$. Then, by (37), (44), (47) and (48), we derive

$$\begin{array}{cc}\hfill & \sum _{|I_1|+|I_2|\le |I|+1}{\left({\int }_0^t\parallel {\displaystyle \frac{|{\mathsf{\Gamma }}^{I_1}v|·|{\mathsf{\Gamma }}^{I_2}\psi |}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \lesssim \sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|+1\end{array}}{\left({\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}^2·{\parallel {\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}+\sum _{\begin{array}{c}|I_2|\le |I|-4\\ |I_1|\le |I|+1\end{array}}{\left({\int }_0^t\parallel {\displaystyle \frac{{\mathsf{\Gamma }}^{I_1}v}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2·{\parallel {\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^{\infty }}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \lesssim C_1^{2}K\sum _{|I_2|\le |I|+1}{ϵ}^{1-\delta |I_2|}{\left({\int }_0^t{\langle s\rangle }^{-2+2\delta }\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +C_1\sum _{\begin{array}{c}|I_2|\le |I|-4\\ |I_1|\le |I|+1\end{array}}{ϵ}^{1-\delta (|I_2|+9/2)}{\left({\int }_0^t{\langle s\rangle }^{-1}\parallel {\displaystyle \frac{{\mathsf{\Gamma }}^{I_1}v}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)}.\hfill \end{array}$$

(99)

On the other hand, for $|I|\le 3\le N-7$, we can also obtain (99) by using that

$$\sum _{|I_1|+|I_2|\le |I|+1}{\left({\int }_0^t\parallel {\displaystyle \frac{|{\mathsf{\Gamma }}^{I_1}v|·|{\mathsf{\Gamma }}^{I_2}\psi |}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\lesssim \sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|+1\end{array}}{\left({\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}^2·{\parallel {\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)}.$$

Combining (95) and (97)–(99), we conclude that

$${[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\psi )]}^{{\displaystyle \frac{1}{2}}}\lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}+C_1^3{ϵ}^{3-\delta (|I|+9)},\mathrm{for}|I|\le N-1.$$

(100)

Therefore, we derive the uniform boundedness of the low-order energy of the Dirac field, which indicates that the field remains globally stable throughout the evolution.

We recall that $\tilde{\mathsf{\Psi }}=\mathsf{\Psi }+v\psi$ and observe that

$$L_0=(t-r){\mathsf{\partial }}_t+(r-t){\mathsf{\partial }}_r+(x^a/r)L_a.$$

If $4\le |I|\le N-2$, then for any $|I_1|+|I_2|\le |I|+1$, we have $|I_2|\le |I|-4$ or $|I_1|\le 4\le N-6$. Then,

$$\begin{array}{cc}\hfill & \parallel L_0{\mathsf{\Gamma }}^I{\left(v\psi \right)\parallel }_{L_x^2}+\sum _{|J|\le 1}\parallel {\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^I{\left(v\psi \right)\parallel }_{L_x^2}\lesssim \sum _{|I^{\prime }|\le |I|+1}{\parallel \langle t-r\rangle {\mathsf{\Gamma }}^{I^{\prime }}\left(v\psi \right)\parallel }_{L_x^2}\hfill \\ \hfill & \lesssim \sum _{|I_1|+|I_2|\le |I|+1}\parallel \langle t-r\rangle |{\mathsf{\Gamma }}^{I_1}v|·|{\mathsf{\Gamma }}^{I_2}\psi {|\parallel }_{L_x^2}\hfill \\ \hfill & \lesssim \sum _{\begin{array}{c}|I_1|\le N-6\\ |I_2|\le |I|+1\end{array}}\parallel \langle t-r\rangle {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^{\infty }}·\parallel {\mathsf{\Gamma }}^{I_2}{\psi \parallel }_{L_x^2}+\sum _{\begin{array}{c}|I_2|\le |I|-4\\ |I_1|\le |I|+1\end{array}}\parallel {\mathsf{\Gamma }}^{I_1}{v\parallel }_{L_x^2}·{\parallel \langle t-r\rangle {\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^{\infty }}\hfill \\ \hfill & \lesssim C_1^{2}K\sum _{|I_2|\le |I|+1}{ϵ}^{1-\delta |I_2|}{\langle t\rangle }^{\delta }+C_1^{2}K\sum _{|I_2|\le |I|-4}{ϵ}^{1-\delta (|I_2|+9/2)}{\langle t\rangle }^{-{\displaystyle \frac{1}{2}}+2\delta }\hfill \\ \hfill & \lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta },\hfill \end{array}$$

(101)

where we use (37), (42), (46) and (47). This also holds for $|I|\le 3\le N-7$. Combining (96) and (101), for $|I|\le N-2$, we obtain

$$\begin{array}{cc}\hfill \parallel L_0{\mathsf{\Gamma }}^I{\mathsf{\Psi }\parallel }_{L_x^2}+\sum _{|J|\le 1}{\parallel {\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^I\mathsf{\Psi }\parallel }_{L_x^2}& \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }{\log }^{{\displaystyle \frac{1}{2}}}(2+t)\hfill \\ \hfill & +C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}{\langle t\rangle }^{2\delta }.\hfill \end{array}$$

(102)

For $|I|\le N-5$, Lemma 4 and (102) yield

$$\begin{array}{cc}\hfill & |L_0{\mathsf{\Gamma }}^I\mathsf{\Psi }|+\sum _{|J|\le 1}|{\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^I\mathsf{\Psi }|\lesssim {\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}}\left\{\sum _{|J^{\prime }|\le 3}\parallel {\mathsf{\Gamma }}^{J^{\prime }}{L}_0{\mathsf{\Gamma }}^I{\mathsf{\Psi }\parallel }_{L_x^2}+\sum _{|J|\le 1,|J^{\prime }|\le 3}{\parallel {\mathsf{\Gamma }}^{J^{\prime }}{\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^I\mathsf{\Psi }\parallel }_{L_x^2}\right\}\hfill \\ \hfill & \lesssim {\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}}\left\{\sum _{|I^{\prime }|\le |I|+3}\parallel L_0{\mathsf{\Gamma }}^{I^{\prime }}{\mathsf{\Psi }\parallel }_{L_x^2}+\sum _{|J|\le 1,|I^{\prime }|\le |I|+3}{\parallel {\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^{I^{\prime }}\mathsf{\Psi }\parallel }_{L_x^2}\right\}\hfill \\ \hfill & \lesssim {\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}}\left\{C_1^3{K}^2{ϵ}^{1-\delta (|I|+4)}{\langle t\rangle }^{\delta }{\log }^{{\displaystyle \frac{1}{2}}}(2+t)+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+10)}{\langle t\rangle }^{2\delta }\right\}.\hfill \end{array}$$

(103)

By $\left(i\right)$ in Lemma 7, we have $\psi =i{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\mathsf{\Psi }$. Then, by (102) and (103) and $\left(iv\right)$ and $\left(v\right)$ in Proposition 1, for $|I|\le N-2$, we find

$$\begin{array}{cc}\hfill \parallel \langle t+r\rangle {[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{\parallel }_{L_x^2}& \lesssim \sum _{|I^{\prime }|\le |I|}\left\{\parallel L_0{\mathsf{\Gamma }}^{I^{\prime }}{\mathsf{\Psi }\parallel }_{L_x^2}+\sum _{|J|\le 1}{\parallel {\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^{I^{\prime }}\mathsf{\Psi }\parallel }_{L_x^2}\right\}\hfill \\ \hfill & \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{\delta }{\log }^{{\displaystyle \frac{1}{2}}}(2+t)+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+13/2)}{\langle t\rangle }^{2\delta }.\hfill \end{array}$$

(104)

Additionally, for $|I|\le N-5$, we have

$$\begin{array}{cc}\hfill & \langle t+r\rangle |{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}|+\langle t-r\rangle |{\widehat{\mathsf{\Gamma }}}^I\psi |\lesssim \sum _{|I^{\prime }|\le |I|}\left\{|L_0{\mathsf{\Gamma }}^{I^{\prime }}\mathsf{\Psi }|+\sum _{|J|\le 1}|{\mathsf{\Gamma }}^J{\mathsf{\Gamma }}^{I^{\prime }}\mathsf{\Psi }|\right\}\hfill \\ \hfill & \lesssim {\langle t+r\rangle }^{-{\displaystyle \frac{1}{2}}}\{C_1^3{K}^2{ϵ}^{1-\delta (|I|+4)}{\langle t\rangle }^{\delta }{\log }^{{\displaystyle \frac{1}{2}}}(2+t)+C_1^{4}K{ϵ}^{1-\delta N}{ϵ}^{2-\delta (|I|+10)}{\langle t\rangle }^{2\delta }\}.\hfill \end{array}$$

(105)

The last two estimates demonstrate the time decay of the $L^2$ and $L^{\infty }$ norms of the “good” component of the Dirac field. The enhanced decay associated with this good component reveals that the nonlinear interactions are effectively weakened by the underlying structural cancellations. The proof is completed by combining (100), (104) and (105). □

4.2. Lower-Order Energy and Pointwise Estimates for v

In this subsection, we establish improved lower-order energy estimates and pointwise bounds for v.

Let $\tilde{v}:=v-{\psi }^{*}{\gamma }^0\psi$. We recall from Lemma 7 that

$$-\square \tilde{v}+\tilde{v}=F_{\tilde{v}}:=i{\mathsf{\partial }}_{\mu }\left(v{\psi }^{*}\right){\gamma }^0{\gamma }^{\mu }\psi -i{\psi }^{*}{\gamma }^0{\gamma }^{\mu }{\mathsf{\partial }}_{\mu }\left(v\psi \right)+2{\mathsf{\partial }}_{\alpha }{\psi }^{*}{\gamma }^0{\mathsf{\partial }}^{\alpha }\psi .$$

(106)

In the following two propositions, we show improved estimates for the lower-order energy and pointwise bounds of v.

Proposition 11.

Under the assumptions (37)–(47), the following holds:

$${[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}\lesssim K+C_1^{3}K{ϵ}^{2-\delta (N+10)},\mathrm{for}|I|\le N-1.$$

Proof. 

The proof proceeds by first establishing lower-order energy estimates for $\tilde{v}$, from which the corresponding estimates for v are obtained, since their difference consists of quadratic terms. The details are provided below.

For $|I|\le N-1$, we see that

$$\begin{array}{cc}& {\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left(i{\mathsf{\partial }}_{\mu }\left(v{\psi }^{*}\right){\gamma }^0{\gamma }^{\mu }\psi \right)\parallel }_{L_x^2}\mathrm{d}s\hfill \\ & \lesssim \sum _{\begin{array}{c}|I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel |{\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}v|·|{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}\psi |·|{\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi {|\parallel }_{L_x^2}\mathrm{d}s\lesssim {\mathcal{I}}_1+{\mathcal{I}}_2,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathcal{I}}_1:& =\sum _{\begin{array}{c}|I_1|,|I_2|\le N-7\\ |I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel {\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}{v\parallel }_{L_x^{\infty }}·\parallel {\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}{\psi \parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi \parallel }_{L_x^2}\mathrm{d}s,\hfill \\ \hfill {\mathcal{I}}_2:& =\sum _{\begin{array}{c}|I_2|,|I_3|\le N-6\\ |I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}{\int }_0^t\parallel {\displaystyle \frac{{\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}v}{\langle r-s\rangle }}{\parallel }_{L_x^2}·\parallel \langle r-s\rangle {\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}{\psi \parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi \parallel }_{L_x^{\infty }}\mathrm{d}s.\hfill \end{array}$$

By (37), (42), (44), (47) and (48), we find

$$\begin{array}{cc}\hfill {\mathcal{I}}_1& \lesssim C_1^{3}K\sum _{|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_2|+11/2)}{ϵ}^{1-\delta (|I_3|+1)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+\delta }\mathrm{d}s\lesssim C_1^{3}K{ϵ}^{2-\delta (|I|+13/2)},\hfill \\ \hfill {\mathcal{I}}_2& \lesssim C_1^2\sum _{|I_1|+|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_2|+11/2)}{ϵ}^{1-\delta (|I_3|+11/2)}{\int }_0^t{\langle s\rangle }^{-1+2\delta }\parallel {\displaystyle \frac{{\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}v}{\langle r-s\rangle }}{\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^{3}K{ϵ}^{2-\delta (|I|+11)},\hfill \end{array}$$

which implies

$${\int }_0^t\parallel {\mathsf{\Gamma }}^I\left(i{\mathsf{\partial }}_{\mu }\left(v{\psi }^{*}\right){\gamma }^0{\gamma }^{\mu }\psi \right){\parallel }_{L_x^2}\mathrm{d}s\lesssim C_1^{3}K{ϵ}^{2-\delta (|I|+11)},\mathrm{for}|I|\le N-1.$$

(107)

Similarly, from $\left(vi\right)$ in Proposition 1, along with (37), (42) and (48), for $|I|\le N-1$, we derive

$$\begin{array}{cc}\hfill & {\int }_0^t{\parallel {\mathsf{\Gamma }}^I\left({\mathsf{\partial }}_{\alpha }{\psi }^{*}{\gamma }^0{\mathsf{\partial }}^{\alpha }\psi \right)\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim \sum _{|I_1|+|I_2|\le |I|}{\int }_0^t\parallel {\langle s\rangle }^{-1}\sum _{|J_1|,|J_2|\le 1}|{\mathsf{\Gamma }}^{J_1}{\mathsf{\Gamma }}^{I_1}\psi |·|{\mathsf{\Gamma }}^{J_2}{\mathsf{\Gamma }}^{I_2}\psi |+{\displaystyle \frac{\langle s-r\rangle }{\langle s+r\rangle }}|\mathsf{\partial }{\mathsf{\Gamma }}^{I_1}\psi |·|\mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi |{\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim \sum _{\begin{array}{c}|I_1|\le N-5,|I_1|+|I_2|\le |I|\\ |J_1|,|J_2|\le 1\end{array}}{\int }_0^t{\langle s\rangle }^{-1}\parallel {\mathsf{\Gamma }}^{J_1}{\mathsf{\Gamma }}^{I_1}{\psi \parallel }_{L_x^{\infty }}·{\parallel {\mathsf{\Gamma }}^{J_2}{\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & +\sum _{\begin{array}{c}|I_1|\le N-6,|I_1|+|I_2|\le |I|\\ |J_1|,|J_2|\le 1\end{array}}{\int }_0^t{\langle s\rangle }^{-1}\parallel \langle s-r\rangle \mathsf{\partial }{\mathsf{\Gamma }}^{I_1}{\psi \parallel }_{L_x^{\infty }}·{\parallel \mathsf{\partial }{\mathsf{\Gamma }}^{I_2}\psi \parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim C_1^2\sum _{|I_1|+|I_2|\le |I|}{ϵ}^{1-\delta (|I_1|+11/2)}{ϵ}^{1-\delta (|I_2|+1)}{\int }_0^t{\langle s\rangle }^{-{\displaystyle \frac{3}{2}}+3\delta }\mathrm{d}s\lesssim C_1^2{ϵ}^{2-\delta (|I|+13/2)}.\hfill \end{array}$$

(108)

Then, using Lemma 2 (see (29)), (106)–(108), we find

$$\begin{array}{cc}\hfill {[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^I\tilde{v})]}^{{\displaystyle \frac{1}{2}}}& \lesssim {[{\mathcal{E}}_1(0,{\mathsf{\Gamma }}^I\tilde{v})]}^{{\displaystyle \frac{1}{2}}}+{\int }_0^t{\parallel {\mathsf{\Gamma }}^I{F}_{\tilde{v}}(s,x)\parallel }_{L_x^2}\mathrm{d}s\hfill \\ \hfill & \lesssim K+C_1^{3}K{ϵ}^{2-\delta (|I|+11)},\mathrm{for}|I|\le N-1.\hfill \end{array}$$

(109)

We recall that $\tilde{v}=v-{\psi }^{*}{\gamma }^0\psi$. For $|I|\le N-1$, it is clear that

$${[{\mathcal{G}}_1\left(t,{\mathsf{\Gamma }}^I\left({\psi }^{*}{\gamma }^0\psi \right)\right)]}^{{\displaystyle \frac{1}{2}}}\lesssim {\mathcal{J}}_1+{\mathcal{J}}_2,$$

where

$${\mathcal{J}}_1=\sum _{|I^{\prime }|\le |I|+1}{\parallel {\mathsf{\Gamma }}^{I^{\prime }}\left({\psi }^{*}{\gamma }^0\psi \right)\parallel }_{L_x^2},{\mathcal{J}}_2=\sum _{|I^{\prime }|\le |I|+1}{\left({\int }_0^t\parallel {\displaystyle \frac{{\mathsf{\Gamma }}^{I^{\prime }}\left({\psi }^{*}{\gamma }^0\psi \right)}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}.$$

By (37), (38), (42) and (48), we see that

$$\begin{array}{cc}\hfill {\mathcal{J}}_1& \lesssim \sum _{|I_1|+|I_2|\le |I|+1}\parallel |{\widehat{\mathsf{\Gamma }}}^{I_1}\psi |·|{\widehat{\mathsf{\Gamma }}}^{I_2}{\psi |\parallel }_{L_x^2}\lesssim \sum _{\begin{array}{c}|I_1|\le N-4\\ |I_1|+|I_2|\le |I|+1\end{array}}\parallel {\widehat{\mathsf{\Gamma }}}^{I_1}{\psi \parallel }_{L_x^{\infty }}·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}\hfill \\ \hfill & \lesssim C_1^2\sum _{|I_1|+|I_2|\le |I|+1}{ϵ}^{1-\delta (|I_1|+9/2)}{ϵ}^{1-\delta |I_2|}{\langle t\rangle }^{-{\displaystyle \frac{1}{2}}+\delta }\lesssim C_1^2{ϵ}^{2-\delta (|I|+11/2)},\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill {\mathcal{J}}_2& \lesssim \sum _{|I_1|+|I_2|\le |I|+1}{\left({\int }_0^t\parallel {\displaystyle \frac{|{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}|·|{\widehat{\mathsf{\Gamma }}}^{I_2}\psi |}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \lesssim \sum _{\begin{array}{c}|I_1|\le N-5\\ |I_1|+|I_2|\le |I|+1\end{array}}{\left({\int }_0^t\parallel {[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}{\parallel }_{L_x^{\infty }}^2·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}+\sum _{\begin{array}{c}|I_2|\le N-4\\ |I_1|+|I_2|\le |I|+1\end{array}}{\left({\int }_0^t\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2·{\parallel {\widehat{\mathsf{\Gamma }}}^{I_2}\psi \parallel }_{L_x^{\infty }}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \lesssim C_1^2\sum _{|I_1|+|I_2|\le |I|+1}{ϵ}^{1-\delta (|I_1|+9/2)}{ϵ}^{1-\delta |I_2|}{\left({\int }_0^t{\langle s\rangle }^{-3+6\delta }\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & +C_1\sum _{|I_1|+|I_2|\le |I|+1}{ϵ}^{1-\delta (|I_2|+9/2)}{\left({\int }_0^t{\langle s\rangle }^{-1}\parallel {\displaystyle \frac{{[{\widehat{\mathsf{\Gamma }}}^{I_1}\psi ]}_{-}}{{\langle r-s\rangle }^{\frac{3}{5}}}}{\parallel }_{L_x^2}^2\mathrm{d}s\right)}^{{\displaystyle \frac{1}{2}}}\lesssim C_1^2{ϵ}^{2-\delta (|I|+11/2)}.\hfill \end{array}$$

Combining the above estimates, we obtain

$${[{\mathcal{G}}_1\left(t,{\mathsf{\Gamma }}^I\left({\psi }^{*}{\gamma }^0\psi \right)\right)]}^{{\displaystyle \frac{1}{2}}}\lesssim C_1^2{ϵ}^{2-\delta (|I|+11/2)},\mathrm{for}|I|\le N-1.$$

This together with (109) yields

$${[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}\lesssim K+C_1^{3}K{ϵ}^{2-\delta (|I|+11)},\mathrm{for}|I|\le N-1.$$

Hence, the uniform boundedness of the low-order energy of the Klein–Gordon field is proved, showing that nonlinear interactions are controlled and dispersive effects dominate the large-time dynamics. The proof is done. □

Proposition 12.

Under the assumptions (37)–(47), the following estimate holds:

$$\langle t+r\rangle |{\mathsf{\Gamma }}^{I}v|\lesssim K+C_1^{3}K{ϵ}^{2-\delta (N+9)}+C_1^4{ϵ}^{1-\delta N}{ϵ}^{3-\delta (N+10)},\mathrm{for}|I|\le N-6.$$

Proof. 

For $|I|\le N-2$ and $s\in [0,t]$, we have

$$\parallel \langle s+|y|\rangle {\mathsf{\Gamma }}^I\left(i{\mathsf{\partial }}_{\mu }\left(v{\psi }^{*}\right){\gamma }^0{\gamma }^{\mu }\psi \right){(s,y)\parallel }_{L_y^2}\lesssim {\widehat{\mathcal{T}}}_1+{\widehat{\mathcal{T}}}_2,$$

where

$$\begin{array}{cc}\hfill {\widehat{\mathcal{T}}}_1:& =\sum _{\begin{array}{c}|I_1|,|I_2|\le N-7\\ |I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}\parallel \langle s+|y|\rangle {\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}{v\parallel }_{L_y^{\infty }}·\parallel {\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}{\psi \parallel }_{L_y^{\infty }}·{\parallel {\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi \parallel }_{L_y^2},\hfill \\ \hfill {\widehat{\mathcal{T}}}_2:& =\sum _{\begin{array}{c}|I_2|,|I_3|\le N-6\\ |I_1|+|I_2|+|I_3|\le |I|\\ |J_1|+|J_2|+|J_3|\le 1\end{array}}\parallel \langle s+|y|\rangle |{\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}v|·|{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}\psi |·|{\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi {|\parallel }_{L_y^2}.\hfill \end{array}$$

Then, from (37), (47) and (48), we see that

$${\widehat{\mathcal{T}}}_1\lesssim C_1^{3}K\sum _{|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_2|+11/2)}{ϵ}^{1-\delta (|I_3|+1)}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+\delta }\lesssim C_1^{3}K{ϵ}^{2-\delta (|I|+13/2)}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+\delta }.$$

For any $|I_1|+|I_2|+|I_3|\le |I|$ with $|I_2|,|I_3|\le N-6$ and any $|J_1|+|J_2|+|J_3|\le 1$, using Lemma 9, (42) and (48), we deduce

$$\begin{array}{cc}& \langle s+|y|\rangle |{\mathsf{\Gamma }}^{I_1}{\mathsf{\Gamma }}^{J_1}v|·|{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}\psi |·|{\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi |\hfill \\ & \lesssim \left\{\langle s-|y|\rangle \sum _{|I^{\prime }|\le |I_1|+2}|\mathsf{\partial }{\mathsf{\Gamma }}^{I^{\prime }}v|+C_1{ϵ}^{1-\delta N}{\langle s+|y|\rangle }^{{\displaystyle \frac{1}{2}}}\sum _{|I^{\prime }|\le |I_1|+1}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi |\right\}·|{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}\psi |·|{\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi |\hfill \\ & \lesssim \langle s-|y|\rangle |{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}\psi |·|{\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi |\sum _{|I^{\prime }|\le |I_1|+2}|\mathsf{\partial }{\mathsf{\Gamma }}^{I^{\prime }}v|\hfill \\ & +C_1{ϵ}^{1-\delta N}{\langle s+|y|\rangle }^{{\displaystyle \frac{1}{2}}}|{\mathsf{\Gamma }}^{I_2}{\mathsf{\Gamma }}^{J_2}\psi |·|{\mathsf{\Gamma }}^{I_3}{\mathsf{\Gamma }}^{J_3}\psi |\sum _{|I^{\prime }|\le |I_1|+1}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi |\hfill \\ & \lesssim C_1^2{ϵ}^{1-\delta (|I_2|+11/2)}{ϵ}^{1-\delta (|I_3|+11/2)}{\langle s\rangle }^{-1+2\delta }\sum _{|I^{\prime }|\le |I_1|+2}|\mathsf{\partial }{\mathsf{\Gamma }}^{I^{\prime }}v|\hfill \\ & +C_1^3{ϵ}^{1-\delta N}{ϵ}^{1-\delta (|I_2|+11/2)}{ϵ}^{1-\delta (|I_3|+11/2)}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}}\sum _{|I^{\prime }|\le |I_1|+1}|{\widehat{\mathsf{\Gamma }}}^{I^{\prime }}\psi |.\hfill \end{array}$$

This together with (37) and (43) gives

$$\begin{array}{cc}\hfill {\widehat{\mathcal{T}}}_2& \lesssim C_1^{3}K\sum _{|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_2|+11/2)}{ϵ}^{1-\delta (|I_3|+11/2)}{\langle s\rangle }^{-1+5\delta }\hfill \\ & +C_1^4{ϵ}^{1-\delta N}\sum _{|I_1|+|I_2|+|I_3|\le |I|}{ϵ}^{1-\delta (|I_1|+1)}{ϵ}^{1-\delta (|I_2|+11/2)}{ϵ}^{1-\delta (|I_3|+11/2)}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+\delta }\hfill \\ & \lesssim \left\{C_1^{3}K{ϵ}^{2-\delta (|I|+11)}+C_1^4{ϵ}^{1-\delta N}{ϵ}^{3-\delta (|I|+12)}\right\}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+\delta }.\hfill \end{array}$$

Combining the above estimates, for $|I|\le N-2$ and $s\in [0,t]$, we obtain

$$\parallel \langle s+|y|\rangle {\mathsf{\Gamma }}^I\left(i{\mathsf{\partial }}_{\mu }\left(v{\psi }^{*}\right){\gamma }^0{\gamma }^{\mu }\psi \right){(s,y)\parallel }_{L_y^2}\lesssim \left\{C_1^{3}K{ϵ}^{2-\delta (|I|+11)}+C_1^4{ϵ}^{1-\delta N}{ϵ}^{3-\delta (|I|+12)}\right\}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+\delta }.$$

(111)

By the proof of (108), we also derive

$$\parallel \langle s+|y|\rangle {\mathsf{\Gamma }}^I\left({\mathsf{\partial }}_{\alpha }{\psi }^{*}{\gamma }^0{\mathsf{\partial }}^{\alpha }\psi \right){(s,y)\parallel }_{L_y^2}\lesssim C_1^2{ϵ}^{2-\delta (|I|+13/2)}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+3\delta },\mathrm{for}|I|\le N-2,s\in [0,t].$$

(112)

From (106), (111) and (112), for $|I|\le N-2$ and $s\in [0,t]$, we find

$$\parallel \langle s+|y|\rangle {\mathsf{\Gamma }}^I{F}_{\tilde{v}}{(s,y)\parallel }_{L_y^2}\lesssim \left\{C_1^{3}K{ϵ}^{2-\delta (|I|+11)}+C_1^4{ϵ}^{1-\delta N}{ϵ}^{3-\delta (|I|+12)}\right\}{\langle s\rangle }^{-{\displaystyle \frac{1}{2}}+3\delta }.$$

(113)

Then, Corollary 1 and (113) imply

$$\langle t+r\rangle |{\mathsf{\Gamma }}^I\tilde{v}|\lesssim K+C_1^{3}K{ϵ}^{2-\delta (|I|+15)}+C_1^4{ϵ}^{1-\delta N}{ϵ}^{3-\delta (|I|+16)},\mathrm{for}|I|\le N-6.$$

(114)

We recall that $\tilde{v}=v-{\psi }^{*}{\gamma }^0\psi$. Hence, from (48) and (114), we obtain that

$$\langle t+r\rangle |{\mathsf{\Gamma }}^{I}v|\lesssim K+C_1^{3}K{ϵ}^{2-\delta (|I|+15)}+C_1^4{ϵ}^{1-\delta N}{ϵ}^{3-\delta (|I|+16)},\mathrm{for}|I|\le N-6.$$

This establishes the optimal time decay for the Klein–Gordon field and reflects that the nonlinear evolution preserves the underlying linear dispersive propagation mechanism. The proof is complete. □

• $\mathbf{Refined}\mathbf{bounds}\mathbf{of}\mathbf{lower}\mathbf{order}:$ By Propositions 10–12, we have the following refined estimates for (40)–(42) and (46) and (47):

  $$\begin{array}{cc}\hfill & {[{\mathcal{E}}_D(t,{\widehat{\mathsf{\Gamma }}}^I\psi )]}^{{\displaystyle \frac{1}{2}}}\le {\displaystyle \frac{1}{2}}{C}_1{ϵ}^{1-\delta (|I|+3/2)},\mathrm{for}|I|\le N-1,\hfill \\ \hfill & \parallel \langle t+r\rangle {[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}{\parallel }_{L_x^2}\le {\displaystyle \frac{1}{2}}{C}_1{ϵ}^{1-\delta (|I|+3/2)}{\langle t\rangle }^{2\delta },\mathrm{for}|I|\le N-2,\hfill \\ \hfill & {\langle t+r\rangle }^{{\displaystyle \frac{3}{2}}-2\delta }|{[{\widehat{\mathsf{\Gamma }}}^I\psi ]}_{-}|+{\langle t+r\rangle }^{{\displaystyle \frac{1}{2}}-2\delta }\langle t-r\rangle |{\widehat{\mathsf{\Gamma }}}^I\psi |\le {\displaystyle \frac{1}{2}}{C}_1{ϵ}^{1-\delta (|I|+9/2)},\mathrm{for}|I|\le N-5,\hfill \\ \hfill & {[{\mathcal{G}}_1(t,{\mathsf{\Gamma }}^{I}v)]}^{{\displaystyle \frac{1}{2}}}\le {\displaystyle \frac{1}{2}}{C}_{1}K,\mathrm{for}|I|\le N-1,\hfill \\ \hfill & \langle t+r\rangle |{\mathsf{\Gamma }}^{I}v|\le {\displaystyle \frac{1}{2}}{C}_{1}K,\mathrm{for}|I|\le N-6,\hfill \end{array}$$

  (115)

  provided that

  $$0<\delta \le {\displaystyle \frac{2}{N+11}},C_1\ge CK,0<ϵ\le {\left(C{C}_1^{3}K\right)}^{-2/\delta }$$

  (116)

  for some large constant $C>0$ (independent of $C_1,ϵ$ and K). Combining (73) and (74) and (115) and (116), the estimates (37)–(47) have been refined by choosing $\delta ,C_1$ and $ϵ$ as in (116). The proof of Proposition 2 is complete, which implies the global existence and pointwise decay results in Theorem 1.

4.3. Proof of the Scattering Result

By (75), (106), Propositions 5, 8 and 9, (107) and (108), for $|I|\le N-1$, we find

$$\begin{array}{cc}\hfill {\int }_0^{+\infty }{\parallel {\mathsf{\Gamma }}^I{F}_{\tilde{\psi }}\parallel }_{L_x^2}\mathrm{d}s& \lesssim C_1^3{K}^2{ϵ}^{1-\delta (|I|+1)}+C_1^3{ϵ}^{3-\delta (|I|+9)}<\infty ,\hfill \\ \hfill {\int }_0^{+\infty }{\parallel {\mathsf{\Gamma }}^I{F}_{\tilde{v}}\parallel }_{L_x^2}\mathrm{d}s& \lesssim C_1^{3}K{ϵ}^{2-\delta (|I|+11)}<\infty ,\hfill \end{array}$$

Then, by Lemma 8, we conclude that $(\tilde{\psi },\tilde{v})$ scatters to a free solution in $H^{N-1}\times {\mathcal{H}}_{N-1}$, where ${\mathcal{H}}_{N-1}:=H^N\times H^{N-1}$. In addition, by the proofs of (97), (98) and (110), we see that

$$\begin{array}{cc}\hfill \parallel {\mathsf{\Gamma }}^I(\psi -\tilde{\psi }){\parallel }_{L_x^2}& \lesssim C_1^{2}K{ϵ}^{1-\delta (|I|+1)}{\langle t\rangle }^{-{\displaystyle \frac{1}{2}}+3\delta },\mathrm{for}|I|\le N-1,\hfill \\ \hfill \parallel {\mathsf{\Gamma }}^I(v-\tilde{v}){\parallel }_{L_x^2}& \lesssim C_1^2{ϵ}^{2-\delta (|I|+11/2)}{\langle t\rangle }^{-{\displaystyle \frac{1}{2}}+\delta },\mathrm{for}|I|\le N,\hfill \end{array}$$

therefore,

$$\underset{t\to +\infty }{\lim }\left\{\parallel \psi \left(t\right)-\tilde{\psi }{\left(t\right)\parallel }_{H^{N-1}}+\parallel v\left(t\right)-\tilde{v}{\left(t\right)\parallel }_{H^N}+{\parallel {\mathsf{\partial }}_{t}v\left(t\right)-{\mathsf{\partial }}_t\tilde{v}\left(t\right)\parallel }_{H^{N-1}}\right\}=0.$$

As a consequence, $(\psi ,v)$ scatters to a free solution in $H^{N-1}\times {\mathcal{H}}_{N-1}$.

5. Conclusions and Outlook

In this paper, we establish the global asymptotic dynamics of the Dirac–Klein–Gordon (DKG) system in ${\mathbb{R}}^{1+2}$, where a massless Dirac field is coupled to a massive Klein–Gordon field. We consider a class of large initial data in which the Dirac component is small, while the Klein–Gordon component may be arbitrarily large. For such data, we prove global existence, sharp time decay, and linear scattering of solutions. To the best of our knowledge, this is the first result concerning the asymptotic behavior of solutions to the two-dimensional DKG system with large data.

During the proof of Theorem 1, several natural questions remain open:

•

Other combinations of field masses.

In this work, we consider the case of a massless Dirac field coupled to a massive Klein–Gordon field. It would be interesting to investigate whether similar results can be established for other mass combinations, such as the massive DKG system (both fields are massive) or the massless case (both fields are massless).

•

Large data without smallness conditions.

In Theorem 1, we study the case in which a small Dirac field is coupled with a large Klein–Gordon field, where the smallness parameter ${ϵ}_0$ depends polynomially on the size $K_{*}$ of the initial Klein–Gordon data. A natural question is whether this dependence can be removed, or, more generally, whether global asymptotic behavior can be established for general large initial data without imposing any smallness condition.