Estimates for the Generalized Tricomi Equation

Abstract

We establish global-in-time, time-weighted Strichartz estimates for the linear inhomogeneous generalized Tricomi equation in spatial dimensions two and higher. At the conformally invariant integrability exponent, we obtain weighted spacetime estimates for an enlarged range of time-weight parameters, including the endpoint. Compared with previously established results, our estimates significantly extend the admissible range of time weights. In particular, in two spatial dimensions, the admissible upper bound reaches twice the degeneracy parameter. We also elucidate the connection between the endpoint estimate in three spatial dimensions and semilinear wave equations on spatially flat Friedmann–Lemaître–Robertson–Walker spacetimes in the decelerating expansion regime. Our results provide a useful analytical tool for investigating the long-time behavior of nonlinear wave equations on such cosmological backgrounds.

1. Introduction

Generalized Tricomi equations arise naturally in the study of wave propagation with time-dependent propagation speed and are also closely related to wave equations on time-dependent curved spacetimes. In particular, after a suitable change of the time variable and a rescaling of the unknown function, certain nonlinear wave equations on spatially homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes can be transformed into semilinear generalized Tricomi-type equations. In the decelerating regime, this leads naturally to equations whose principal part is of the form

$${\partial }_t^2-t^m\Delta .$$

(1)

Moreover, the transformation of the nonlinear term may introduce a time-dependent weight, leading to nonlinearities of the form $t^{\alpha }{\left|u\right|}^p$. Motivated by these connections, we consider the semilinear generalized Tricomi equation

$$\begin{array}{c}\hfill {\partial }_t^{2}u-t^m\Delta u=t^{\alpha }{\left|u\right|}^p,\end{array}$$

(2)

where $m>0$ and $\alpha \in \mathbb{R}$. The fundamental solution associated with the linear generalized Tricomi operator in (1) was explicitly constructed by Barros-Neto and Gelfand [1,2,3] and, independently, by Yagdjian [4]. For the semilinear problem (2) in dimensions $n\ge 2$ with $\alpha >-1$, the first local-existence result was established by Yagdjian in the pioneering work [5], under the assumptions

$$\left\{\begin{array}{cc}\hfill 1<p& \le 1+{\displaystyle \frac{2m}{(m+2)n+2}},\hfill \\ \hfill p& <1+{\displaystyle \frac{4(\alpha +2)}{(m+2)n-2(\alpha +2)}}.\hfill \end{array}\right.$$

The same work also addressed the long-time behavior of solutions. More precisely, ref. [5] (Theorem 1.3) shows that, if

$$1<p<1+{\displaystyle \frac{4}{(m+2)n-2}},$$

then (2) admits no global solution

$$u\in C([0,\infty ),L^{p+1}\left({\mathbb{R}}^n\right)).$$

By contrast, ref. [5] (Theorem 1.2) establishes the existence of a global small-data solution

$$u\in C([0,\infty ),L^{p+1}\left({\mathbb{R}}^n\right))\cap C^1([0,\infty ),{\mathcal{D}}^{\prime }\left({\mathbb{R}}^n\right))$$

provided that

$$\left\{\begin{array}{cc}\hfill p& \le 1+{\displaystyle \frac{2m}{(m+2)n+2}},\hfill \\ \hfill {\displaystyle \frac{(m+2)np}{p+1}}& \ge {\displaystyle \frac{2(p+1)+\max \{\alpha ,\alpha p\}}{p-1}}.\hfill \end{array}\right.$$

(3)

Taken together, however, these two results do not yield a complete characterization of the global dynamics: a gap remains between the range of p for which finite-time blow-up is known and that for which global small-data existence can be established. Moreover, the global-existence result itself requires an upper restriction on p.

The one-dimensional counterpart was subsequently studied by Galstian [6] for $\alpha >-1$ and $m>0$. In this setting, finite-time blow-up was proved for

$$1<p<1+{\displaystyle \frac{4}{m}}$$

whereas global existence for small initial data was obtained under condition (3).

A sharper description became available in the unweighted case $\alpha =0$. Witt, Yin, and the second author of the present paper investigated the initial-value problem for (2), with the data prescribed at a positive time $t_0$, and identified a critical exponent $p_{\mathrm{crit}}(n,m)>1$ separating the global-existence and blow-up regimes. More precisely, small-data solutions exist globally when $p>p_{\mathrm{crit}}(n,m)$, whereas finite-time blow-up may occur when $1<p\le p_{\mathrm{crit}}(n,m)$. For $n\ge 2$ and $m>0$, the exponent $p_{\mathrm{crit}}(n,m)$ is the positive root of

$$\left((m+2){\displaystyle \frac{n}{2}}-1\right)p^2+\left((m+2)(1-{\displaystyle \frac{n}{2}})-3\right)p-(m+2)=0,$$

while

$$p_{\mathrm{crit}}(1,m)=1+{\displaystyle \frac{4}{m}}$$

in the one-dimensional case (see [7,8,9,10]). For the semilinear Tricomi equation with mixed nonlinearities, a blow-up result and a lifespan estimate were established in [11].

We next turn to the Equation (2) and study the global behavior of solutions to the Cauchy problem

$$\left\{\begin{array}{c}{\partial }_t^{2}u-t^m\Delta u=t^{\alpha }{\left|u\right|}^p,\hfill \\ u(1,x)=\epsilon f\left(x\right),{\partial }_{t}u(1,x)=\epsilon g\left(x\right),\hfill \end{array}\right.$$

(4)

where $m>0$, $\alpha \in \mathbb{R}$, $p>1$, and $f,g\in C_c^{\infty }\left({\mathbb{R}}^n\right)$ with $n\ge 2$ and $suppf,suppg\subset B(0,M)$ for some $M>0$. For $\alpha >-2$, Palmieri and Reissig [12] (Theorem 1) proved the existence of a critical exponent $p_{\mathrm{crit}}(n,m,\alpha )>1$ such that, for suitable initial data, under the assumptions of (4), finite-time blow-up occurs for

$$1<p\le p_{\mathrm{crit}}(n,m,\alpha ).$$

This critical exponent is given by

$$p_{\mathrm{crit}}(n,m,\alpha )=\max \left\{p_1(n,m,\alpha ),p_2(n,m,\alpha )\right\}$$

with

$$p_1(n,m,\alpha )=1+{\displaystyle \frac{2(2+\alpha )}{(m+2)n-2}}$$

and $p_2(n,m,\alpha )$ denotes the positive root of

$$\left({\displaystyle \frac{m}{2(m+2)}}+{\displaystyle \frac{n-1}{2}}\right)p^2-\left({\displaystyle \frac{n+1}{2}}-{\displaystyle \frac{3m}{2(m+2)}}+{\displaystyle \frac{2\alpha }{m+2}}\right)p-1=0.$$

In order to investigate global existence of small-data solutions when $p>p_{\mathrm{crit}}(n,m,\alpha )$, we next turn to another distinguished exponent that plays a fundamental role in the global-existence theory, namely, the conformally invariant exponent. To motivate its counterpart for the generalized Tricomi equation, we first recall the corresponding picture for the classical semilinear wave equation

$${\partial }_t^{2}u-\Delta u={\left|u\right|}^p.$$

In this classical setting, the conformally invariant exponent is given by

$$p_{\mathrm{conf}}\left(n\right)={\displaystyle \frac{n+3}{n-1}}.$$

Strichartz, in his pioneering work [13], considered the linear inhomogeneous wave equation

$$\left\{\begin{array}{c}{\partial }_t^{2}u-\Delta u=F,\hfill \\ u(0,x)=0,{\partial }_{t}u(0,x)=0\hfill \end{array}\right.$$

(5)

and established the following Strichartz estimate

$${∥u∥}_{L^q([0,\infty )\times {\mathbb{R}}^n)}\le C{∥F∥}_{L^{{\displaystyle \frac{q}{q-1}}}([0,\infty )\times {\mathbb{R}}^n)}$$

with

$$q={\displaystyle \frac{2(n+1)}{n-1}}=p_{\mathrm{conf}}\left(n\right)+1.$$

Later, the significance of this exponent in the small-data global-existence theory is further illustrated by the seminal works of Lindblad and Sogge [14] and Georgiev, Lindblad, and Sogge [15]. Their arguments naturally distinguish between the two regimes

$$p_{\mathrm{crit}}\left(n\right)<p\le p_{\mathrm{conf}}\left(n\right)$$

and

$$p>p_{\mathrm{conf}}\left(n\right).$$

A key ingredient in treating these regimes, particularly at the conformal threshold, is provided by Strichartz estimates for the linear inhomogeneous wave Equation (5). This observation suggests that, in order to develop an analogous global-existence framework for the generalized Tricomi equation, one should first identify the appropriate conformal exponent and then establish the corresponding inhomogeneous Strichartz estimate. Guided by this classical picture, we define the conformally invariant exponent associated with (2) by

$$p_{\mathrm{conf}}(n,m,\alpha )={\displaystyle \frac{(m+2)n+4\alpha +6}{(m+2)n-2}},$$

(6)

which is derived from Noether’s theorem. For completeness, the derivation of (6) is presented in the Appendix A. Notice that the classical conformally invariant exponent is recovered when $m=\alpha =0$, since

$$p_{\mathrm{conf}}(n,0,0)=p_{\mathrm{conf}}\left(n\right).$$

The preceding discussion indicates that the linear inhomogeneous problem corresponding to the principal part of (2) should play a central role. We are therefore led to consider

$$\left\{\begin{array}{cc}\hfill & {\partial }_t^{2}w-t^m▵w=F(t,x),\hfill \\ \hfill & w(0,x)=0,{\partial }_{t}w(0,x)=0.\hfill \end{array}\right.$$

(7)

At the conformally invariant exponent, the natural spacetime integrability index is

$$q_0=p_{\mathrm{conf}}(n,m,\alpha )+1={\displaystyle \frac{2\left(\right(m+2)n+2+2\alpha )}{(m+2)n-2}}.$$

We now state our main weighted Strichartz estimate at this exponent.

Theorem 1.

Let $n\ge 2$ and $m>0$. If

$$0<\alpha \le {\displaystyle \frac{n}{n-1}}·m,$$

(8)

then for $F(t,x)$ satisfying $t^{-{\displaystyle \frac{\alpha }{q_0}}}F\in L^{{\displaystyle \frac{q_0}{q_0-1}}}([0,\infty )\times {\mathbb{R}}^n)$, the solution of (7) satisfies

$$\begin{array}{c}\hfill {∥t^{{\displaystyle \frac{\alpha }{q_0}}}w∥}_{L^{q_0}([0,\infty )\times {\mathbb{R}}^n)}\le C{∥t^{-{\displaystyle \frac{\alpha }{q_0}}}F∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}([0,\infty )\times {\mathbb{R}}^n)}.\end{array}$$

(9)

Remark 1.

The estimate in Theorem 1 extends the admissible range of the time-weight parameter appearing in the existing global-existence theory. More precisely, Li, Wang and Yin [16] established global existence for (4) with small initial data when $n=2$, $m>0$, $-1<\alpha \le m$, and $p_{\mathrm{crit}}(2,m,\alpha )<p<p_{\mathrm{conf}}(2,m,\alpha )$. On the other hand, [17] obtained global small-data existence for $n=2$, $m>0$, $\alpha >0$, and

$$p\ge \max \{p_{conf}(2,m,\alpha ),{\displaystyle \frac{4\alpha }{m}}-1\},$$

as well as for $n=2$, $m>0$, $-2<\alpha <0$ and $p\ge p_{conf}(2,m,\alpha )$. The restriction $p\ge {\displaystyle \frac{4\alpha }{m}}-1$ in the argument of [17] is because the estimate corresponding to (9) is available only when

$$0<\alpha <{\displaystyle \frac{m+3}{m+2}}·m,$$

By comparison, condition (8) of Theorem 1 allows, when $n=2$,

$$0<\alpha \le 2m.$$

Thus, for every $m>0$, Theorem 1 enlarges the admissible range of α for the weighted Strichartz estimate.

Remark 2.

In [7], He, Witt, and Yin investigated the semilinear generalized Tricomi Equation (2) with $\alpha =0$. Their analysis used an unweighted Strichartz estimate for the corresponding inhomogeneous linear equation: for

$$q_0={\displaystyle \frac{2\left(\right(m+2)n+2)}{(m+2)n-2}},$$

the zero-data solution satisfies

$${\parallel w\parallel }_{L^{q_0}\left({\mathbb{R}}_{+}^{n+1}\right)}\le C{\parallel F\parallel }_{L^{{\displaystyle \frac{q_0}{q_0-1}}}\left({\mathbb{R}}_{+}^{n+1}\right)}.$$

Theorem 1 extends this estimate to a family of time-weighted bounds for $0<\alpha \le {\displaystyle \frac{nm}{n-1}}$, including the endpoint. In particular, when $n=2$ this range becomes $0<\alpha \le 2m$.

Remark 3.

The two time weights in (9) are chosen to be compatible with the pure-power nonlinearity in (2) at the conformal exponent. Indeed, if $p=p_{\mathrm{conf}}=q_0-1$, then

$$\parallel t^{-{\displaystyle \frac{\alpha }{q_0}}}{t}^{\alpha }{\left|u\right|}^{q_0-1}{\parallel }_{L^{{\displaystyle \frac{q_0}{q_0-1}}}}=\parallel t^{{\displaystyle \frac{\alpha }{q_0}}}u{\parallel }_{L^{q_0}}^{q_0-1}.$$

Consequently, the inhomogeneous estimate established here is naturally suited to a fixed-point argument for the associated semilinear problem. However, obtaining a global small-data existence theorem also requires a suitable homogeneous Strichartz estimate for nonzero initial data, together with a contraction argument in an appropriate solution space. These additional steps are beyond the scope of the present paper. The restriction $\alpha \le {\displaystyle \frac{nm}{n-1}}$ arises from the estimates used in Case IIIb in Section 3. We do not claim that it is necessary for the validity of the weighted Strichartz estimate; determining the optimal weight range remains an open question.

Remark 4.

The proof is tailored to the conformal integrability exponent $q_0$. In particular, the cancellation of dyadic frequency factors and the Hardy–Littlewood–Sobolev estimate in time use the precise relation between $q_0$ and α. Therefore, estimates at other integrability exponents do not follow automatically from the present argument. Developing complementary endpoint estimates and interpolating them, when compatible weights and function spaces are available, is a possible direction for future work.

To place Theorem 1 in its physical context, we now explain how the weighted generalized Tricomi equation considered above arises naturally from wave propagation on cosmological spacetimes. In general relativity, spatially homogeneous and isotropic cosmological models are described by Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes [18]. We focus on the spatially flat case with spatial topology ${\mathbb{R}}^3$, for which the Lorentzian metric on the four-dimensional manifold $M={(0,\infty )}_t\times {\mathbb{R}}^3$ takes the form

$$g=-d{t}^2+a^2\left(t\right)\left({\left(d{x}^1\right)}^2+{\left(d{x}^2\right)}^2+{\left(d{x}^3\right)}^2\right).$$

Wave propagation on such expanding backgrounds has been studied from several perspectives. Costa, Natário, and Oliveira [19] established decay estimates for solutions of the corresponding linear wave equation, while Galstian and Yagdjian [20] investigated semilinear Klein–Gordon equations and obtained global-existence results on FLRW spacetimes.

We specialize to the power-law scale factor

$$a\left(t\right)=t^q,$$

which describes the expansion of the spatial slices and becomes singular as $t\to 0^{+}$. Our interest lies in the future-time behavior of scalar waves on this background. In the coordinates $(t,x^1,x^2,x^3)$, the associated covariant wave equation takes the form

$${\square }_g\psi :={\partial }_t^2\psi +{\displaystyle \frac{3q}{t}}{\partial }_t\psi -{\displaystyle \frac{1}{t^{2q}}}{\Delta }_{{\mathbb{R}}^3}\psi =0,$$

and its semilinear counterpart is

$${\partial }_t^2\psi -{\displaystyle \frac{1}{t^{2q}}}\Delta \psi +{\displaystyle \frac{3q}{t}}{\partial }_t\psi ={\left|\psi \right|}^p.$$

(10)

The nonlinear dynamics of (10) depend strongly on the expansion rate. Tsutaya and Wakasugi [21] established finite-time blow-up results in an appropriate range of p, whereas Costa, Franzen [22] proved small-data global existence in the accelerated expansion regime $q>1$. In contrast, the corresponding global-existence problem for the full decelerating range $0<q<1$ remains open in general.

The power nonlinearity ${\left|\psi \right|}^p$ in (10) is adopted as a representative semilinear source term; it is not uniquely prescribed by the FLRW geometry. The metric determines the coefficients of the linear wave operator, whereas the choice of the nonlinear source is an additional modeling assumption. We focus on ${\left|\psi \right|}^p$ because it is a standard model for the study of blow-up, small-data global existence, and critical exponents, and because it leads to a generalized Tricomi equation with precisely the time weight relevant to our main theorem.

For $0<q<{\displaystyle \frac{1}{3}}$, put $c=1-3q$ and introduce

$$s=t^c,\hspace{1em}y=cx,\hspace{1em}U(s,y)=\psi (s^{{\displaystyle \frac{1}{c}}},{\displaystyle \frac{y}{c}}),m={\displaystyle \frac{4q}{c}}.$$

A direct calculation shows that

$$U_{ss}-s^m{\Delta }_{y}U=c_m{s}^{{\displaystyle \frac{3m}{2}}}{\left|U\right|}^p,c_m=c^{-2}={\displaystyle \frac{{(3m+4)}^2}{16}}.$$

Equivalently, with $v=c_m^{{\displaystyle \frac{1}{p-1}}}U$, this equation takes the normalized form

$$v_{ss}-s^m{\Delta }_{y}v=s^{{\displaystyle \frac{3m}{2}}}{\left|v\right|}^p.$$

If initial data are prescribed at $t=1$, the transformed initial surface is $s=1$. The spatial rescaling also gives

$${\parallel U(s,·)\parallel }_{L_y^r}=c^{{\displaystyle \frac{3}{r}}}{\parallel \psi (t,·)\parallel }_{L_x^r}.$$

When $n=3$ and $\alpha ={\displaystyle \frac{3m}{2}}$, we have $p_{\mathrm{conf}}=3$ and $q_0=4$. Hence, the conclusion of Theorem 1 specializes to

$$\parallel s^{{\displaystyle \frac{3m}{8}}}{w\parallel }_{L^4((0,\infty )\times {\mathbb{R}}^3)}\le C{\parallel s^{-{\displaystyle \frac{3m}{8}}}F\parallel }_{L^{{\displaystyle \frac{4}{3}}}((0,\infty )\times {\mathbb{R}}^3)}.$$

This is precisely the inhomogeneous estimate compatible with the cubic source $s^{{\displaystyle \frac{3m}{2}}}{\left|v\right|}^3$. A global small-data existence result for initial data prescribed at $s=1$ would additionally require a homogeneous estimate on $[1,\infty )$ in an appropriate initial-data norm and a contraction argument. Such a nonlinear theorem is not established in the present paper.

We conclude the introduction by outlining the main ideas underlying the proof of Theorem 1. After taking the spatial Fourier transform of (7), we obtain a microlocal representation of its solution. Combined with the Littlewood–Paley decomposition, this representation allows us to reduce the global weighted Strichartz estimate in Theorem 1 to a family of frequency-localized estimates. The main difficulty arises from the time-dependent weight $t^{{\displaystyle \frac{\alpha }{q_0}}}$, whose growth at large times becomes increasingly significant as $\alpha$ increases. To handle this difficulty, we divide the admissible range of ${\displaystyle \frac{\alpha }{q_0}}$ into several regimes and employ estimates adapted to the distinct behavior in each regime. Combining these estimates yields the required frequency-localized bound, from which the weighted Strichartz estimate in Theorem 1 follows.

The remainder of the paper is organized as follows. Section 2 collects the preliminary results required for the proof of Theorem 1, including a Littlewood–Paley estimate and a microlocal representation formula for the solution of (7). Section 3 is devoted to the proof of Theorem 1. The Appendix A provides the derivation of the conformally invariant exponent, well-posedness and initial traces of (7) and the details underlying the microlocal representation used in Section 2.

2. Preliminary

We first specify the solution class in which the zero initial conditions at the degenerate time $t=0$ are understood. For smooth forcing terms compactly supported away from $t=0$, Yagdjian [4] (Theorem 3.4) proved that the Cauchy problem (7) has a classical solution. The following lemma extends this construction to the weighted forcing class appearing in Theorem 1 and identifies the corresponding initial traces.

Lemma 1.

Let the assumptions of Theorem 1 hold, and let F satisfy $t^{-{\displaystyle \frac{\alpha }{q_0}}}F\in L^{{\displaystyle \frac{q_0}{q_0-1}}}([0,\infty )\times {\mathbb{R}}^n)$. For every

$$\gamma >n\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q_0}}\right),$$

(11)

there exists a unique solution w of (7) in the class

$$w\in C_{\mathrm{loc}}([0,\infty );H^{-\gamma }\left({\mathbb{R}}^n\right))\cap C_{\mathrm{loc}}^1([0,\infty );H^{-(\gamma +{\displaystyle \frac{m}{m+2}})}\left({\mathbb{R}}^n\right)),$$

(12)

where the equation holds in ${\mathcal{D}}^{\prime }((0,\infty )\times {\mathbb{R}}^n)$ and the initial conditions are interpreted in the indicated Sobolev spaces. More precisely, for every $T>0$ and $0\le t\le T$,

$$\begin{array}{cc}\hfill & {∥w\left(t\right)∥}_{H^{-\gamma }}+{∥{\partial }_{t}w\left(t\right)∥}_{H^{-(\gamma +{\displaystyle \frac{m}{m+2}})}}\le C_T{t}^{{\displaystyle \frac{\alpha +1}{q_0}}}{∥s^{-\frac{\alpha }{q_0}}F(s,x)∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}((0,t)\times {\mathbb{R}}^n)}.\hfill \end{array}$$

Consequently,

$$\underset{t\downarrow 0}{\lim }w\left(t\right)=0\mathrm{in}{H}^{-\gamma }\left({\mathbb{R}}^n\right),\underset{t\downarrow 0}{\lim }{\partial }_{t}w\left(t\right)=0\mathrm{in}{H}^{-(\gamma +{\displaystyle \frac{m}{m+2}})}\left({\mathbb{R}}^n\right).$$

(13)

We put the proof of Lemma 1 in the Appendix A.

Next we recall a Littlewood–Paley estimate that will be used in the proof of Theorem 1, see [14] (Lemma 3.8).

Lemma 2.

Let

$$\rho \in C_0^{\infty }\left(\left({\displaystyle \frac{1}{2}},2\right)\right)$$

(14)

satisfy

$$\sum _{j\in \mathbb{Z}}\rho \left(2^{-j}\tau \right)=1,\tau >0.$$

For $j\in \mathbb{Z}$, let $G_j$ denote the Littlewood–Paley piece of G defined by

$$G_j(t,x)={\left(2\pi \right)}^{-n}{\int }_{{\mathbb{R}}^n}{e}^{ix·\xi }\rho \left(2^{-j}\left|\xi \right|\right)\widehat{G}(t,\xi )d\xi ,j\in Z.$$

Then

$$\begin{array}{cc}\hfill {\parallel G\parallel }_{L_t^s{L}_x^q}\le C{\left({\displaystyle \sum _{j=-\infty }^{\infty }}{\parallel G_j\parallel }_{L_t^s{L}_x^q}^2\right)}^{{\displaystyle \frac{1}{2}}},& 2\le q<\infty ,2\le s\le \infty ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\left({\displaystyle \sum _{j=-\infty }^{\infty }}{\parallel G_j\parallel }_{L_t^r{L}_x^p}^2\right)}^{{\displaystyle \frac{1}{2}}}\le C{\parallel G\parallel }_{L_t^r{L}_x^p},& 1<p\le 2,1\le r\le 2.\hfill \end{array}$$

In order to establish Theorem 1, we still need to give a suitable representation of the solution w to (7). Here and below denote

$${\varphi }_m\left(t\right)={\displaystyle \frac{2}{m+2}}{t}^{{\displaystyle \frac{m+2}{2}}}.$$

We then obtain the following representation lemma.

Lemma 3.

$$\begin{array}{cc}\hfill w(t,x)& =\sum _{\kappa ,{\kappa }^{\prime }\in \{+,-\}}{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )}{a}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau ,\hfill \end{array}$$

(15)

where the phase functions ${\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}$ are defined as

$${\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )=x·\xi +\kappa {\varphi }_m\left(t\right)\left|\xi \right|+{\kappa }^{\prime }{\varphi }_m\left(\tau \right)\left|\xi \right|$$

and the amplitude functions $a_{\pm ,\pm }(t,\tau ,\xi )$ satisfy

$$|{\partial }_{\xi }^{\kappa }{a}_{\pm ,\pm }(t,\tau ,\xi )|\lesssim (1+{\varphi }_m{\left(t\right)\left|\xi \right|)}^{-{\displaystyle \frac{m}{2(m+2)}}}(1+{\varphi }_m{\left(\tau \right)\left|\xi \right|)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\left|\xi \right|}^{-{\displaystyle \frac{2}{m+2}}-\left|\kappa \right|}.$$

(16)

The derivation of (15) and (16) is similar to the proof procedure for Lemma 3.4 of Section 3 in [7]. For the reader’s convenience, the details are included in Appendix A.

3. Proof of Theorem 1

To prove (9), set

$$\beta ={\displaystyle \frac{\alpha }{q_0}}.$$

In view of the definition of $q_0$, we have

$$\begin{array}{cc}\hfill q_0& ={\displaystyle \frac{2\left(\right(m+2)n+2\alpha +2)}{(m+2)n-2}}\hfill \\ \hfill & ={\displaystyle \frac{2\left(\right(m+2)n+2)}{(m+2)n-2-4\frac{\alpha }{q_0}}}\hfill \\ \hfill & ={\displaystyle \frac{2\left(\right(m+2)n+2)}{(m+2)n-2-4\beta }}.\hfill \end{array}$$

Moreover, the condition

$$0<\alpha \le {\displaystyle \frac{n}{n-1}}·m$$

is equivalent to

$$0<\beta \le {\displaystyle \frac{mn}{2(n+1)}}.$$

Using the representation formula (15), we may write

$$\begin{array}{cc}\hfill t^{\beta }w& =\sum _{\kappa ,{\kappa }^{\prime }\in \{+,-\}}{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )}{t}^{\beta }{a}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & =\sum _{\kappa ,{\kappa }^{\prime }\in \{+,-\}}{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )}{t}^{\beta }{\tau }^{\beta }{a}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,\xi )\widehat{{\tau }^{-\beta }F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & :=\sum _{\kappa ,{\kappa }^{\prime }\in \{+,-\}}{A}_{\kappa ,{\kappa }^{\prime }}\left({\tau }^{-\beta }F\right).\hfill \end{array}$$

We next localize $A_{\kappa ,{\kappa }^{\prime }}$ to dyadic spatial frequencies. For $\lambda >0$, let

$$a_{\kappa ,{\kappa }^{\prime },\lambda }(t,\tau ,\xi )=\rho \left({\displaystyle \frac{\left|\xi \right|}{\lambda }}\right)t^{\beta }{\tau }^{\beta }{a}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,\xi )$$

where $\rho$ is the cutoff function introduced in (14), and define

$$A_{\kappa ,{\kappa }^{\prime },\lambda }G={\int }_{[0,t)\times {\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{i\{{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )-y·\xi \}}{a}_{\kappa ,{\kappa }^{\prime },\lambda }(t,\tau ,\xi )G(\tau ,y)d\xi d\tau dy.$$

(17)

By Lemma 2 and the finite overlap of dyadic projections, it suffices to establish

$${∥A_{\kappa ,{\kappa }^{\prime },\lambda }G∥}_{L^{q_0}([0,\infty )\times {\mathbb{R}}^n)}\le C{∥G∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}([0,\infty )\times {\mathbb{R}}^n)}.$$

(18)

where the constant C is independent of $\lambda$.

Indeed, once (18) is available, Lemma 2, together with

$${\displaystyle \frac{q_0}{q_0-1}}={\displaystyle \frac{2\left(\right(m+2)n+2)}{(m+2)n+6+4\beta }}<2,$$

yields

$$\begin{array}{cc}\hfill \parallel A_{\kappa ,{\kappa }^{\prime }}{G\parallel }_{L^{q_0}}^2& \le C\sum _{j\in \mathbb{Z}}{\parallel A_{\kappa ,{\kappa }^{\prime },2^j}G\parallel }_{L^{q_0}}^2\hfill \\ \hfill & \le C\sum _{j\in \mathbb{Z}}\sum _{k:|j-k|\le C_0}{\parallel A_{\kappa ,{\kappa }^{\prime },2^j}{G}_k\parallel }_{L^{q_0}}^2\hfill \\ \hfill & \le C\sum _{j\in \mathbb{Z}}\sum _{k:|j-k|\le C_0}{\parallel G_k\parallel }_{L^{{\displaystyle \frac{q_0}{q_0-1}}}}^2\hfill \\ \hfill & \le {C\parallel G\parallel }_{L^{{\displaystyle \frac{q_0}{q_0-1}}}([0,\infty )\times {\mathbb{R}}^n)}^2,\hfill \end{array}$$

(19)

where

$${\widehat{G}}_k(\tau ,\xi )=\rho (2^{-k}\left|\xi \right|)\widehat{G}(\tau ,\xi ).$$

Taking $G={\tau }^{-\beta }F$ in (19) therefore gives (9).

Thus, it remains to establish (18). The argument depends on the size of $\beta$, and we divide the proof into three regimes. Throughout the rest of the proof, let

$$p_0={\displaystyle \frac{q_0}{q_0-1}}.$$

Case I $\mathbf{0}<\beta \le {\displaystyle \frac{\mathbf{m}}{(\mathbf{m}+\mathbf{2})(\mathbf{n}-\mathbf{1})+\mathbf{4}}}$

In this range, the symbol estimate (16) allows us to absorb an appropriate power of t into the amplitude. More precisely,

$$\left|{\partial }_{\xi }^{\kappa }(t^{{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}-\beta }{a}_{\kappa ,{\kappa }^{\prime },\lambda }(t,\tau ,\xi ))\right|\le {\left|\xi \right|}^{-{\displaystyle \frac{2(n+1)(\beta +1)}{(m+2)n+2}}-\left|\kappa \right|}.$$

Indeed, by (16)

$$\begin{array}{cc}\hfill & \left|{\partial }_{\xi }^{\kappa }\left(t^{{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}-\beta }{a}_{\kappa ,{\kappa }^{\prime },\lambda }(t,\tau ,\xi )\right)\right|\hfill \\ \hfill & \le t^{{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}-\beta }{t}^{\beta }{\left(1+{\varphi }_m\left(t\right)\left|\xi \right|\right)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\tau }^{\beta }{\left(1+{\varphi }_m\left(\tau \right)\left|\xi \right|\right)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\left|\xi \right|}^{-\left|\kappa \right|-{\displaystyle \frac{2}{m+2}}}\hfill \\ \hfill & \le {\left|\xi \right|}^{-{\displaystyle \frac{2m(\beta +1)}{(m+2)\left(\right(m+2)n+2)}}-{\displaystyle \frac{2\beta }{m+2}}-{\displaystyle \frac{2}{m+2}}-\left|\kappa \right|}\hfill \\ \hfill & ={\left|\xi \right|}^{-{\displaystyle \frac{2(n+1)(\beta +1)}{(m+2)n+2}}}\hfill \end{array}$$

Consequently, if we introduce the normalized amplitude

$$b(t,\tau ,\xi )={\lambda }^{{\displaystyle \frac{2(n+1)(\beta +1)}{(m+2)n+2}}}{t}^{{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}-\beta }{a}_{\kappa ,{\kappa }^{\prime },\lambda }(t,\tau ,\xi ).$$

then

$$\left|{\partial }_{\xi }^{\kappa }b(t,\tau ,\xi )\right|\le {\left|\xi \right|}^{-\left|\kappa \right|}.$$

Accordingly, $A_{\lambda }$ can be written as

$$\begin{array}{cc}\hfill A_{\kappa ,{\kappa }^{\prime },\lambda }G=& {\lambda }^{-{\displaystyle \frac{2(n+1)(\beta +1)}{(m+2)n+2}}}\hfill \\ \hfill & \times {\int }_{[0,\infty )\times {\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{i\left\{{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )-y·\xi \right\}}{t}^{\beta -{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}}b(t,\tau ,\xi )G(\tau ,y)d\xi dyd\tau .\hfill \end{array}$$

For fixed t and $\tau$, denote the corresponding spatial oscillatory integral operator by

$$T_{t,\tau }f\left(x\right)={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{i\left\{{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )-y·\xi \right\}}{t}^{\beta -{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}}b(t,\tau ,\xi )f\left(y\right)d\xi dy.$$

Then, we have

$$A_{\kappa ,{\kappa }^{\prime },\lambda }G={\int }_0^t{\lambda }^{-{\displaystyle \frac{2(n+1)(\beta +1)}{(m+2)n+2}}}{T}_{t,\tau }Gd\tau .$$

(20)

Since $t\ge |t-\tau |$ and

$$\beta \le {\displaystyle \frac{m}{(m+2)(n-1)+4}},$$

the $L^2$-bound takes the form

$$\parallel T_{t,\tau }{f\parallel }_{L^2\left({\mathbb{R}}^n\right)}\le C{\left|t-\tau \right|}^{\beta -{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}}{∥f∥}_{L^2\left({\mathbb{R}}^n\right)}.$$

(21)

On the other hand, the stationary-phase estimate yields

$$\begin{array}{cc}\hfill \parallel T_{t,\tau }{f\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}& \le {\lambda }^n{t}^{\beta -{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}}{\left(1+\lambda |\kappa {\varphi }_m\left(t\right)+{\kappa }^{\prime }{\varphi }_m\left(\tau \right)|\right)}^{-{\displaystyle \frac{n-1}{2}}}{∥f∥}_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C{\lambda }^{{\displaystyle \frac{n+1}{2}}}{t}^{\beta -{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}}{\left|{\varphi }_m\left(t\right)-{\varphi }_m\left(\tau \right)\right|}^{-{\displaystyle \frac{n-1}{2}}}{∥f∥}_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C{\lambda }^{{\displaystyle \frac{n+1}{2}}}{\left|t-\tau \right|}^{\beta -{\displaystyle \frac{(\beta +1)m}{(m+2)n+2}}}{\left|t-\tau \right|}^{-{\displaystyle \frac{n-1}{2}}·{\displaystyle \frac{m+2}{2}}}{∥f∥}_{L^1\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(22)

Interpolating between (21) and (22), we obtain

$$\parallel T_{t,\tau }{f\parallel }_{L^{q_0}\left({\mathbb{R}}^n\right)}\le C{\lambda }^{{\displaystyle \frac{2(\beta +1)(n+1)}{(m+2)n+2}}}{\left|t-\tau \right|}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}}{∥f∥}_{L^{p_0}\left({\mathbb{R}}^n\right)}.$$

(23)

By (20) and the estimate of $T_{t,\tau }$ in (23) we obtain

$$\begin{array}{cc}\hfill & \parallel A_{\kappa ,{\kappa }^{\prime },\lambda }{G\parallel }_{L^{q_0}([0,\infty )\times {\mathbb{R}}^n)}\hfill \\ \hfill & ={∥{\int }_0^t{\lambda }^{-{\displaystyle \frac{2(n+1)(\beta +1)}{(m+2)n+2}}}{T}_{t,\tau }Gd\tau ∥}_{L^{q_0}([0,\infty )\times {\mathbb{R}}^n)}\hfill \\ \hfill & \le {∥{\int }_0^t{\lambda }^{-{\displaystyle \frac{2(n+1)(\beta +1)}{(m+2)n+2}}}{∥T_{t,\tau }G∥}_{L^{q_0}\left({\mathbb{R}}^n\right)}d\tau ∥}_{L^{q_0}\left([0,\infty )\right)}\hfill \\ \hfill & \le C{∥{\int }_0^{\infty }{\lambda }^{-{\displaystyle \frac{2(n+1)(\beta +1)}{(m+2)n+2}}}{\lambda }^{{\displaystyle \frac{2(n+1)(\beta +1)}{(m+2)n+2}}}{|t-\tau |}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}}{∥G(\tau ,·)∥}_{L^{p_0}\left({\mathbb{R}}^n\right)}d\tau ∥}_{L^{q_0}\left(\mathbb{R}\right)}\hfill \\ \hfill & =C{∥{\int }_0^{\infty }{|t-\tau |}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}}{∥G(\tau ,·)∥}_{L^{p_0}\left({\mathbb{R}}^n\right)}d\tau ∥}_{L^{q_0}\left(\mathbb{R}\right)}.\hfill \end{array}$$

Finally, since

$$1-({\displaystyle \frac{1}{p_0}}-{\displaystyle \frac{1}{q_0}})={\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}},$$

the Hardy–Littlewood–Sobolev inequality in the time variable yields

$$\begin{array}{cc}\hfill {\parallel {\int }_0^{\infty }{|t-\tau |}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}}{∥G(\tau ,·)∥}_{L^{p_0}\left({\mathbb{R}}^n\right)}d\tau \parallel }_{L^{q_0}\left(\mathbb{R}\right)}& \le C{∥G∥}_{L^{p_0}([0,\infty )\times {\mathbb{R}}^n)}.\hfill \end{array}$$

This proves (18) in Case I.

• Case II ${\displaystyle \frac{\mathbf{m}}{(\mathbf{m}+\mathbf{2})(\mathbf{n}-\mathbf{1})+\mathbf{4}}}<\beta \le {\displaystyle \frac{\mathbf{m}}{\mathbf{4}}}$

In this intermediate range, a direct use of (16) gives

$$\begin{array}{cc}\hfill & \left|{\partial }_{\xi }^{\kappa }{a}_{\kappa ,{\kappa }^{\prime },\lambda }(t,\tau ,\xi )\right|\hfill \\ \hfill & \le t^{\beta }(1+{\varphi }_m{\left(t\right)\left|\xi \right|)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\tau }^{\beta }(1+{\varphi }_m{\left(\tau \right)\left|\xi \right|)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\left|\xi \right|}^{-{\displaystyle \frac{2}{m+2}}}{\left|\xi \right|}^{-\left|\kappa \right|}\hfill \\ \hfill & \le {\left|\xi \right|}^{-{\displaystyle \frac{2(2\beta +1)}{m+2}}-\left|\kappa \right|}.\hfill \end{array}$$

We therefore normalize the amplitude by setting

$$b(t,\tau ,\xi )={\lambda }^{{\displaystyle \frac{2(2\beta +1)}{m+2}}}{a}_{\kappa ,{\kappa }^{\prime },\lambda }(t,\tau ,\xi ).$$

With this choice,

$$A_{\kappa ,{\kappa }^{\prime },\lambda }G={\lambda }^{-{\displaystyle \frac{2(2\beta +1)}{m+2}}}{\int }_{[0,\infty )\times {\mathbb{R}}^n}{\int }_{{\mathcal{R}}^n}{e}^{i\left\{{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )-y·\xi \right\}}b(t,\tau ,\xi )G(\tau ,y)d\xi dyd\tau .$$

For fixed t and $\tau$, let

$$T_{t,\tau }f\left(x\right)={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{i\left\{{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )-y·\xi \right\}}b(t,\tau ,\xi )f\left(y\right)d\xi dy,$$

then $A_{\kappa ,{\kappa }^{\prime },\lambda }G$ can be represented as

$$A_{\kappa ,{\kappa }^{\prime },\lambda }G={\int }_0^t{\lambda }^{-{\displaystyle \frac{2(2\beta +1)}{m+2}}}{T}_{t,\tau }G\mathrm{d}\tau .$$

(24)

The normalized symbol satisfies the standard $L^2$ bound

$$\parallel T_{t,\tau }{f\parallel }_{L^2\left({\mathbb{R}}^n\right)}\le C{∥f∥}_{L^2\left({\mathbb{R}}^n\right)}.$$

(25)

Moreover, the stationary-phase estimate, together with

$$\beta >{\displaystyle \frac{m}{(m+2)(n-1)+4}},$$

gives

$$\begin{array}{cc}\hfill \parallel T_{t,\tau }{f\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}& \le C{\lambda }^n{(1+\lambda \left|\kappa {\varphi }_m\left(t\right)+{\kappa }^{\prime }{\varphi }_m\left(\tau \right)\right|)}^{-{\displaystyle \frac{n-1}{2}}}{∥f∥}_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C{\lambda }^n{(1+\lambda \left|{\varphi }_m\left(t\right)-{\varphi }_m\left(\tau \right)\right|)}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{2(\beta +1)(m+2)}}}{∥f∥}_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C{\lambda }^{n-{\displaystyle \frac{(m+2)n-2-4\beta }{2(\beta +1)(m+2)}}}{\left|t-\tau \right|}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{4(\beta +1)}}}{∥f∥}_{L^1\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(26)

Interpolation between (25) and (26) then yields

$$\parallel T_{t,\tau }{f\parallel }_{L^{q_0}\left({\mathbb{R}}^n\right)}\le C{\lambda }^{{\displaystyle \frac{4\beta +2}{m+2}}}{\left|t-\tau \right|}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}}{∥f∥}_{L^{p_0}\left({\mathbb{R}}^n\right)}.$$

(27)

Using the identity

$$1-({\displaystyle \frac{1}{p_0}}-{\displaystyle \frac{1}{q_0}})={\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}$$

we apply the Hardy–Littlewood–Sobolev inequality in time and obtain by (24) and (27) that

$$\begin{array}{cc}\hfill & \parallel A_{\kappa ,{\kappa }^{\prime },\lambda }{G\parallel }_{L^{q_0}([0,\infty )\times {\mathbb{R}}^n)}\hfill \\ \hfill & ={∥{\int }_0^t{\lambda }^{-{\displaystyle \frac{2(2\beta +1)}{m+2}}}{T}_{t,\tau }Gd\tau ∥}_{L^{q_0}([0,\infty )\times {\mathbb{R}}^n)}\hfill \\ \hfill & \le {∥{\int }_0^t{\lambda }^{-{\displaystyle \frac{2(2\beta +1)}{m+2}}}{∥T_{t,\tau }G∥}_{L^{q_0}\left({\mathbb{R}}^n\right)}d\tau ∥}_{L^{q_0}([0,\infty )\times {\mathbb{R}}^n)}\hfill \\ \hfill & \le C{∥{\int }_0^{\infty }{\lambda }^{-{\displaystyle \frac{2(2\beta +1)}{m+2}}}{\lambda }^{{\displaystyle \frac{4\beta +2}{m+2}}}{|t-\tau |}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}}{∥G(\tau ,·)∥}_{L^{p_0}\left({\mathbb{R}}^n\right)}d\tau ∥}_{L^{q_0}[0,\infty )}\hfill \\ \hfill & \le C{∥G∥}_{L^{p_0}([0,\infty )\times {\mathbb{R}}^n)}.\hfill \end{array}$$

Hence, (18) also holds throughout Case II.

• Case III ${\displaystyle \frac{\mathbf{m}}{\mathbf{4}}}<\beta \le {\displaystyle \frac{\mathbf{m}}{\mathbf{2}}}·{\displaystyle \frac{\mathbf{n}}{\mathbf{n}+\mathbf{1}}}$

In the remaining range, the time weights carried by the amplitude require a more refined analysis. Set

$$b(t,\tau ,\xi )=\lambda a_{\kappa ,{\kappa }^{\prime },\lambda }(t,\tau ,\xi ).$$

It follows from (16) that

$$\left|{\partial }_{\xi }^{\kappa }b(t,\tau ,\xi )\right|\lesssim t^{\beta -{\displaystyle \frac{m}{4}}}{\tau }^{\beta -{\displaystyle \frac{m}{4}}}{\lambda }^{-\left|\kappa \right|}.$$

Consequently,

$$A_{\kappa ,{\kappa }^{\prime },\lambda }G={\lambda }^{-1}{\int }_{[0,t)\times {\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{i\{{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )-y·\xi \}}b(t,\tau ,\xi )G(\tau ,y)d\xi dyd\tau .$$

Set

$$T_{t,\tau }f\left(x\right)={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{i\{{\mathrm{\Phi }}_{\kappa ,{\kappa }^{\prime }}(t,\tau ,x,\xi )-y·\xi \}}b(t,\tau ,\xi )f\left(y\right)d\xi dy,$$

then $A_{\kappa ,{\kappa }^{\prime },\lambda }G$ can be written as

$$A_{\kappa ,{\kappa }^{\prime },\lambda }G={\int }_0^t{\lambda }^{-1}{T}_{t,\tau }G\mathrm{d}\tau .$$

(28)

For $T_{t,\tau }$ we first obtain the $L^2$-estimate

$${∥T_{t,\tau }f(·)∥}_{L^2\left({\mathbb{R}}^n\right)}\le C{t}^{\beta -{\displaystyle \frac{m}{4}}}{\tau }^{\beta -{\displaystyle \frac{m}{4}}}{∥f(·)∥}_{L^2\left({\mathbb{R}}^n\right)}.$$

(29)

The stationary-phase method also yields

$$\begin{array}{cc}\hfill \parallel T_{t,\tau }f(·){\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}& \lesssim {\lambda }^n{\left(1+\lambda \left|\kappa {\varphi }_m\left(t\right)+{\kappa }^{\prime }{\varphi }_m\left(\tau \right)\right|\right)}^{-{\displaystyle \frac{n-1}{2}}}{t}^{\beta -{\displaystyle \frac{m}{4}}}{\tau }^{\beta -{\displaystyle \frac{m}{4}}}{∥f(·)∥}_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \lesssim {\lambda }^n{(1+\lambda \left|{\varphi }_m\left(t\right)-{\varphi }_m\left(\tau \right)\right|)}^{-\left(n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\right)}{t}^{\beta -{\displaystyle \frac{m}{4}}}{\tau }^{\beta -{\displaystyle \frac{m}{4}}}{∥f(·)∥}_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \lesssim {\lambda }^{{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}}{\left|{\varphi }_m\left(t\right)-{\varphi }_m\left(\tau \right)\right|}^{-\left(n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\right)}{t}^{\beta -{\displaystyle \frac{m}{4}}}{\tau }^{\beta -{\displaystyle \frac{m}{4}}}{∥f(·)∥}_{L^1\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(30)

Here the assumption

$${\displaystyle \frac{m}{4}}<\beta \le {\displaystyle \frac{m}{2}}·{\displaystyle \frac{n}{n+1}}$$

guarantees that

$$0<n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\le {\displaystyle \frac{n-1}{2}}.$$

Unlike in the previous two cases, however, the remaining factors involving t and $\tau$ cannot be controlled uniformly by the same argument. We therefore distinguish between the the separated-time region region $t\ge 2\tau$ and the comparable-time region $t<2\tau$.

• Case IIIa $\mathbf{t}\ge \mathbf{2}\tau$

In this region, $t-\tau \ge {\displaystyle \frac{t}{2}}$. Since $\beta >{\displaystyle \frac{m}{4}}$, it follows that

$$t^{\beta -{\displaystyle \frac{m}{4}}}{\tau }^{\beta -{\displaystyle \frac{m}{4}}}\lesssim {|t-\tau |}^{2\beta -{\displaystyle \frac{m}{2}}}.$$

Combining this observation with (29), we obtain

$${∥T_{t,\tau }f(·)∥}_{L^2\left({\mathbb{R}}^n\right)}\le C{|t-\tau |}^{2\beta -{\displaystyle \frac{m}{2}}}{∥f(·)∥}_{L^2\left({\mathbb{R}}^n\right)},$$

(31)

Similarly, (30) implies

$${∥T_{t,\tau }f(·)∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C{\lambda }^{{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}}{\left|t-\tau \right|}^{-\left(n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\right){\displaystyle \frac{m+2}{2}}}{|t-\tau |}^{2\beta -{\displaystyle \frac{m}{2}}}{∥f(·)∥}_{L^1\left({\mathbb{R}}^n\right)}.$$

(32)

Interpolating between (31) and (32) gives the desired bound

$${∥T_{t,\tau }f(·)∥}_{L^{q_0}\left({\mathbb{R}}^n\right)}\le C\lambda {\left|t-\tau \right|}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}}{∥f(·)∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}\left({\mathbb{R}}^n\right)}.$$

(33)

Case IIIb $\mathbf{2}\tau >\mathbf{t}$

In this region, t and $\tau$ are comparable. By the mean value theorem,

$${\varphi }_m\left(t\right)-{\varphi }_m\left(\tau \right)={\eta }^{{\displaystyle \frac{m}{2}}}(t-\tau ),$$

for some $\eta \in [\tau ,t]$. Since

$$\eta \ge \tau >{\displaystyle \frac{t}{2}},$$

estimate (30) can be rewritten as

$$\begin{array}{cc}\hfill & {∥T_{t,\tau }f(·)∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C{\lambda }^{{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}}{\eta }^{-{\displaystyle \frac{m}{2}}\left(n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\right)}{\left|t-\tau \right|}^{-\left(n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\right)}{t}^{\beta -{\displaystyle \frac{m}{4}}}{\tau }^{\beta -{\displaystyle \frac{m}{4}}}{∥f(·)∥}_{L^1\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C{\lambda }^{{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}}{t}^{-{\displaystyle \frac{m}{2}}\left(n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\right)}{\left|t-\tau \right|}^{-\left(n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\right)}{t}^{2\beta -{\displaystyle \frac{m}{2}}}{∥f(·)∥}_{L^1\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(34)

Meanwhile, (29) reduces to

$${∥T_{t,\tau }f(·)∥}_{L^2\left({\mathbb{R}}^n\right)}\le C{t}^{2\beta -{\displaystyle \frac{m}{2}}}{∥f(·)∥}_{L^2\left({\mathbb{R}}^n\right)}.$$

(35)

Interpolating (35) and (34), we find

$$\begin{array}{cc}\hfill & {∥T_{t,\tau }f(·)∥}_{L^{q_0}\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C\lambda t^{2\beta -{\displaystyle \frac{m}{2}}-{\displaystyle \frac{m}{2}}\left(n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\right){\displaystyle \frac{4(\beta +1)}{(m+2)n+2}}}{\left|t-\tau \right|}^{1-{\displaystyle \frac{4n(\beta +1)}{(m+2)n+2}}}{∥f(·)∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(36)

The upper bound

$$\beta \le {\displaystyle \frac{mn}{2(n+1)}}$$

implies

$$\begin{array}{cc}\hfill & 2\beta -{\displaystyle \frac{m}{2}}-{\displaystyle \frac{m}{2}}\left(n-{\displaystyle \frac{(m+2)n+2}{4(\beta +1)}}\right){\displaystyle \frac{4(\beta +1)}{(m+2)n+2}}\hfill \\ \hfill & =2\beta -{\displaystyle \frac{2mn(\beta +1)}{(m+2)n+2}}\hfill \\ \hfill & \le 0,\hfill \end{array}$$

Hence, (36) yields

$$\begin{array}{cc}\hfill {∥T_{t,\tau }f(·)∥}_{L^{q_0}\left({\mathbb{R}}^n\right)}& \le C\lambda {\left|t-\tau \right|}^{2\beta -{\displaystyle \frac{2mn(\beta +1)}{(m+2)n+2}}}{\left|t-\tau \right|}^{1-{\displaystyle \frac{4n(\beta +1)}{(m+2)n+2}}}{∥f(·)∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le C\lambda {\left|t-\tau \right|}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}}{∥f(·)∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(37)

Combining (33) and (37), we obtain the same time-decay estimate in both subregions. Since

$$1-({\displaystyle \frac{q_0-1}{q_0}}-{\displaystyle \frac{1}{q_0}})={\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}},$$

(28), together with the Hardy–Littlewood–Sobolev inequality in the time variable, gives

$$\begin{array}{cc}\hfill \parallel A_{\kappa ,{\kappa }^{\prime },\lambda }{G\parallel }_{L^{q_0}([0,\infty )\times {\mathbb{R}}^n)}& ={∥{\int }_0^t{\lambda }^{-1}{T}_{t,\tau }Gd\tau ∥}_{L^{q_0}([0,\infty )\times {\mathbb{R}}^n)}\hfill \\ \hfill & \le {∥{\int }_0^t{\lambda }^{-1}{∥T_{t,\tau }G∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}\left({\mathbb{R}}^n\right)}d\tau ∥}_{L^{q_0}\left([0,\infty )\right)}\hfill \\ \hfill & \le C{∥{\int }_0^{\infty }{\lambda }^{-1}\lambda {|t-\tau |}^{-{\displaystyle \frac{(m+2)n-2-4\beta }{(m+2)n+2}}}{∥G(\tau ,·)∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}\left({\mathbb{R}}^n\right)}d\tau ∥}_{L^{q_0}\left([0,\infty )\right)}\hfill \\ \hfill & \le C{∥G∥}_{L^{{\displaystyle \frac{q_0}{q_0-1}}}([0,\infty )\times {\mathbb{R}}^n)}.\hfill \end{array}$$

Thus, (18) holds in Case III as well. Together with Cases I and II, this proves (18) for the entire range

$$0<\beta \le {\displaystyle \frac{mn}{2(n+1)}}.$$

As explained above, (18), Lemma 2, and the choice $G={\tau }^{-\beta }F$ imply (9). This completes the proof of Theorem 1.

Appendix A.1. Derivation of the Conformally Invariant Exponent

Lemma A1.

The conformally invariant exponent associated with (4) is

$$p_{conf}(n,m,\alpha )={\displaystyle \frac{(m+2)n+4\alpha +6}{(m+2)n-2}}.$$

Proof. 

Let

$$u^{\prime }=({\partial }_{t}u,{\nabla }_{x}u)=({\partial }_{0}u,{\partial }_{1}u,\dots ,{\partial }_{n}u).$$

Associated with (4), we consider the Lagrangian density

$$L(u,u^{\prime })={\displaystyle \frac{1}{2}}|{\partial }_t{u|}^2-{\displaystyle \frac{1}{2}}{t}^m|{\nabla }_x{u|}^2+{\displaystyle \frac{t^{\alpha }}{p+1}}{\left|u\right|}^{p}u.$$

Note that for any $u\in \mathbb{R}$,

$${\displaystyle \frac{d}{du}}\left({\displaystyle \frac{{\left|u\right|}^{p}u}{p+1}}\right)={\left|u\right|}^p,$$

and

$${\displaystyle \frac{\partial L}{\partial u}}(u,\partial u)=t^{\alpha }{\left|u\right|}^p,{\displaystyle \frac{\partial L}{\partial u_t}}(u,\partial u)=u_t,{\displaystyle \frac{\partial L}{\partial u_j}}(u,\partial u)=-t^m{u}_j,j=1,2,\dots ,n.$$

Hence, we obtain from the corresponding Euler–Lagrange equation

$${\partial }_t^{2}u-t^m\Delta u=t^{\alpha }{\left|u\right|}^p,$$

which is just the equation in (4), and it is valid for sign-changing solutions.

The corresponding action functional is stationary with respect to compactly supported variations. More precisely, for every

$$\psi \in C_0^{\infty }\left({\mathbb{R}}^{n+1}\right),$$

the Lagrangian variational principle gives

$${\displaystyle \frac{d}{d\epsilon }}\int \int L(u+\epsilon \psi ,{(u+\epsilon \psi )}^{\prime })dtdx{|}_{\epsilon =0}=0.$$

Consequently, u satisfies the associated Euler–Lagrange equation

$${\displaystyle \frac{\partial L}{\partial u}}(u,\partial u)-{\displaystyle \sum _{j=0}^n}{\partial }_j\left({\displaystyle \frac{\partial L}{\partial u_j}}(u,\partial u)\right)=0,$$

(A1)

where $u_j={\partial }_{j}u$ for $j=0,1,...,n$.

We next examine the scaling symmetry of (4), which will be used to identify the exponent for which the corresponding action possesses the appropriate variational symmetry. The equation is invariant under the scaling

$$u(t,x)⟶{\lambda }^{{\displaystyle \frac{\alpha +2}{p-1}}}u(\lambda t,{\lambda }^{{\displaystyle \frac{m+2}{2}}}x).$$

Accordingly, define

$$u_{\lambda }={\lambda }^{{\displaystyle \frac{\alpha +2}{p-1}}}u(\lambda t,{\lambda }^{{\displaystyle \frac{m+2}{2}}}x).$$

Differentiating the Lagrangian density along this one-parameter family gives

$${\partial }_{\lambda }L(u_{\lambda },u_{\lambda }^{\prime })={\displaystyle \frac{\partial L}{\partial u}}(u_{\lambda },u_{\lambda }^{\prime }){\partial }_{\lambda }{u}_{\lambda }+{\displaystyle \sum _{j=0}^n}{\displaystyle \frac{\partial L}{\partial u_j}}(u_{\lambda },u_{\lambda }^{\prime }){\partial }_j{\partial }_{\lambda }{u}_{\lambda }.$$

(A2)

Using the Euler–Lagrange Equation (A1), the right-hand side of (A2) can be expressed in divergence form:

$${\partial }_{\lambda }L(u_{\lambda },u_{\lambda }^{\prime })={\displaystyle \sum _{j=0}^n}{\partial }_j\left({\displaystyle \frac{\partial L}{\partial u_j}}(u_{\lambda },u_{\lambda }^{\prime }){\partial }_{\lambda }{u}_{\lambda }\right).$$

On the other hand, differentiation of the scaling family at $\lambda =1$ yields

$${\partial }_{\lambda }{u}_{\lambda }{|}_{\lambda =1}={\displaystyle \frac{\alpha +2}{p-1}}u+t{\partial }_{t}u+{\displaystyle \frac{m+2}{2}}\sum _{j=1}{x}_j{\partial }_{j}u.$$

Substitution into the preceding identity therefore gives

$${\partial }_{\lambda }L(u_{\lambda },u_{\lambda }^{\prime }){|}_{\lambda =1}={\displaystyle \sum _{j=0}^n}{\partial }_j\left({\displaystyle \frac{\partial L}{\partial u_j}}(u,u^{\prime })\left({\displaystyle \frac{\alpha +2}{p-1}}u+t{\partial }_{t}u+{\displaystyle \frac{m+2}{2}}\sum _{j=1}{x}_j{\partial }_{j}u\right)\right).$$

(A3)

The same derivative can also be computed directly from the scaling behavior of the Lagrangian density. Indeed,

$$L(u_{\lambda },u_{\lambda }^{\prime })={\lambda }^{{\displaystyle \frac{2(\alpha +2)}{p-1}}+2}\left(L(u,u^{\prime })\right)(\lambda t,{\lambda }^{{\displaystyle \frac{m+2}{2}}}x)$$

and hence

$${\partial }_{\lambda }L(u_{\lambda },u_{\lambda }^{\prime }){|}_{\lambda =1}=\left({\displaystyle \frac{2(\alpha +2)}{p-1}}+2\right)L(u,u^{\prime })+t{\partial }_{t}L(u,u^{\prime })+{\displaystyle \frac{m+2}{2}}{\displaystyle \sum _{j=1}^n}{x}_j{\partial }_{j}L(u,u^{\prime }).$$

Equating this expression with (A3) and rearranging the resulting terms, we obtain

$$\begin{array}{cc}\hfill & {\partial }_t\left({\displaystyle \frac{\partial L}{\partial u_0}}(u,u^{\prime })\left({\displaystyle \frac{\alpha +2}{p-1}}u+t{\partial }_{t}u+{\displaystyle \frac{m+2}{2}}\sum _{j=1}{x}_j{\partial }_{j}u\right)-tL(u,u^{\prime })\right)\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^n}{\partial }_j\left({\displaystyle \frac{\partial L}{\partial u_j}}(u,u^{\prime })\left({\displaystyle \frac{\alpha +2}{p-1}}u+t{\partial }_{t}u+{\displaystyle \frac{m+2}{2}}\sum _{j=1}{x}_j{\partial }_{j}u\right)-{\displaystyle \frac{m+2}{2}}{x}_{j}L(u,u^{\prime })\right)\hfill \\ \hfill & =\left({\displaystyle \frac{2(\alpha +2)}{p-1}}+1-n{\displaystyle \frac{m+2}{2}}\right)L(u,u^{\prime }).\hfill \end{array}$$

(A4)

The left-hand side of (A4) is a space time divergence, as displayed below

$${\mathrm{div}}_{t,x}(Q_0,Q_1,\dots ,Q_n).$$

Since $suppu(t,·)$ has compact support, integration of (A4) over ${\mathbb{R}}^n$ eliminates the spatial-divergence terms and gives

$${\displaystyle \frac{d}{dt}}{\int }_{{\mathbb{R}}^n}{Q}_{0}dx=\left({\displaystyle \frac{2(\alpha +2)}{p-1}}+1-n{\displaystyle \frac{m+2}{2}}\right){\int }_{{\mathbb{R}}^n}L(u,u^{\prime })dx.$$

(A5)

For the scaling transformation to generate a conservation law through Noether’s theorem, the coefficient on the right-hand side of (A5) must vanish. We are therefore led to the condition

$${\displaystyle \frac{2(\alpha +2)}{p-1}}+1-n{\displaystyle \frac{m+2}{2}}=0,$$

Solving this equation for p yields

$$p_{\mathrm{conf}}(n,m,\alpha )={\displaystyle \frac{(m+2)n+4\alpha +6}{(m+2)n-2}}.$$

This proves the assertion. □

Appendix A.2. Well-Posedness and Initial Traces of (7)

We show the proof of Lemma 1 in Section 2. Throughout, we use the parameters of Theorem 1:

$$\begin{array}{c}n\ge 2,m>0,0<\alpha \le {\displaystyle \frac{nm}{n-1}},\\ q_0={\displaystyle \frac{2\left(\right(m+2)n+2+2\alpha )}{(m+2)n-2}},r={\displaystyle \frac{q_0}{q_0-1}},\beta ={\displaystyle \frac{\alpha }{q_0}}.\end{array}$$

In particular, $q_0>2$, $1<r<2$, and $r^{\prime }=q_0$. Define

$$\begin{array}{cc}\hfill Y_{\beta }& =\left\{F:t^{-\beta }F\in L^r((0,\infty )\times {\mathbb{R}}^n)\right\},\hfill \\ \hfill X_{\beta }& =\left\{w:t^{\beta }w\in L^{q_0}((0,\infty )\times {\mathbb{R}}^n)\right\},\hfill \end{array}$$

with their natural weighted norms.

To proceed further, we divide the argument into four steps.

• Step 1: Local time integrability of the forcing.

Since $1<r<2$ and (11) holds, Sobolev embedding and duality imply

$$H^{\gamma }\left({\mathbb{R}}^n\right)\hookrightarrow L^{r^{\prime }}\left({\mathbb{R}}^n\right),{∥f∥}_{H^{-\gamma }}\le C{∥f∥}_{L^r},r^{\prime }=q_0.$$

Put $G(s,x)=s^{-\beta }F(s,x)$. For $0<t\le T$, Hölder’s inequality gives

$$\begin{array}{cc}\hfill {\int }_0^t{∥F\left(s\right)∥}_{H^{-\gamma }}ds& \le C{\int }_0^t{s}^{\beta }{∥G\left(s\right)∥}_{L_x^r}ds\hfill \\ \hfill & \le C{\left({\int }_0^t{s}^{\beta q_0}ds\right)}^{1/q_0}{∥G∥}_{L^r((0,t)\times {\mathbb{R}}^n)}\hfill \\ \hfill & =C{(\alpha +1)}^{-1/q_0}{t}^{(\alpha +1)/q_0}{∥G∥}_{L^r((0,t)\times {\mathbb{R}}^n)},\hfill \end{array}$$

(A6)

since $\beta q_0=\alpha >0$. Thus $F\in L^1((0,T);H^{-\gamma }\left({\mathbb{R}}^n\right))$ for every finite T.

• Step 2: Uniform estimates for the Fourier fundamental solution.

For $\rho \ge 0$ and $0\le s\le t$, let $K(t,s,\rho )$ solve

$${\partial }_t^{2}K+t^m{\rho }^{2}K=0,K(s,s,\rho )=0,{\partial }_{t}K(s,s,\rho )=1.$$

(A7)

We claim that, for every $T>0$,

$$\underset{0\le s\le t\le T}{\sup }\left|K(t,s,\rho )\right|\le C_T,\underset{0\le s\le t\le T}{\sup }\left|{\partial }_{t}K(t,s,\rho )\right|\le C_T{(1+\rho )}^{\sigma }.$$

(A8)

For $0\le \rho \le 1$, these estimates follow directly by applying Gronwall’s inequality to the first-order system associated with (A7); its coefficients are uniformly bounded on $[0,T]$.

Now suppose $\rho \ge 1$ and set

$$\delta ={\rho }^{-2/(m+2)}\in (0,1],{\rho }^2{\delta }^{m+2}=1.$$

For $s\le t\le \delta$, the Volterra form of (A7) reads

$${\partial }_{t}K(t,s,\rho )=1-{\rho }^2{\int }_s^t{u}^{m}K(u,s,\rho )du.$$

Let $b=\min \{\delta ,T\}$ and, when $s\le b$, put $M={\sup }_{s\le u\le b}\left|{\partial }_{u}K(u,s,\rho )\right|$. As $\left|K\right(u,s,\rho \left)\right|\le (u-s)M$, we obtain, for $s\le t\le b$,

$$\begin{array}{cc}\hfill |{\partial }_{t}K(t,s,\rho )|& \le 1+{\rho }^{2}M{\int }_s^t{u}^m(u-s)du\hfill \\ \hfill & \le 1+{\rho }^{2}M{\int }_0^{\delta }{u}^{m+1}du=1+{\displaystyle \frac{M}{m+2}}.\hfill \end{array}$$

Consequently,

$$M\le {\displaystyle \frac{m+2}{m+1}}<2,\left|K(t,s,\rho )\right|\le 2\delta (s\le t\le \min \{\delta ,T\}).$$

(A9)

If $s>\min \{\delta ,T\}$, the preceding short-time case is not used.

To treat $t\ge \delta$, introduce

$$E\left(t\right)=|{\partial }_t{K(t,s,\rho )|}^2+t^m{\rho }^2{\left|K(t,s,\rho )\right|}^2.$$

Direct differentiation and (A7) give

$$E^{\prime }\left(t\right)=m{t}^{m-1}{\rho }^2{\left|K(t,s,\rho )\right|}^2\le {\displaystyle \frac{m}{t}}E\left(t\right),t>0.$$

Thus, for $0<a\le t$ with $a\ge s$,

$$E\left(t\right)\le {(t/a)}^{m}E\left(a\right).$$

(A10)

If $s\ge \delta$, take $a=s$. Since $E\left(s\right)=1$,

$$|{\partial }_t{K(t,s,\rho )|\le (t/s)}^{m/2}\le T^{m/2}{\delta }^{-m/2},\left|K(t,s,\rho )\right|\le {\displaystyle \frac{1}{\rho s^{m/2}}}\le \delta .$$

(A11)

If $s<\delta \le t$, estimate (A9) at $t=\delta$ and ${\rho }^2{\delta }^{m+2}=1$ imply

$$E\left(\delta \right)\le 4+4{\rho }^2{\delta }^{m+2}=8.$$

Now (A10) yields

$$|{\partial }_{t}K(t,s,\rho )|\le \sqrt{8}{T}^{m/2}{\delta }^{-m/2},\left|K(t,s,\rho )\right|\le {\displaystyle \frac{\sqrt{8}}{\rho {\delta }^{m/2}}}=\sqrt{8}\delta .$$

(A12)

Since

$${\delta }^{-m/2}={\rho }^{m/(m+2)}={\rho }^{\sigma },{\left(\rho {\delta }^{m/2}\right)}^{-1}={\rho }^{-2/(m+2)}=\delta ,$$

(A9), (A11), and (A12) prove (A8). In fact, for $\rho \ge 1$ the stronger bound $\left|K(t,s,\rho )\right|\le C{\rho }^{-2/(m+2)}$ holds. Crucially, all constants are independent of how close s is to zero.

• Step 3: Definition of the Duhamel solution and its traces.

Let $\mathcal{K}(t,s)$ be the spatial Fourier multiplier with symbol $K(t,s,|\xi \left|\right)$. By (A8) and Plancherel’s theorem,

$$\begin{array}{cc}\hfill {∥\mathcal{K}(t,s)f∥}_{H^{-\gamma }}& \le C_T{∥f∥}_{H^{-\gamma }},\hfill \\ \hfill {∥{\partial }_t\mathcal{K}(t,s)f∥}_{H^{-(\gamma +\sigma )}}& \le C_T{∥f∥}_{H^{-\gamma }}\hfill \end{array}(0\le s\le t\le T).$$

(A13)

The multiplier symbols are pointwise continuous in their time variables, and the bounds in (A13) imply strong continuity on the indicated Sobolev spaces by dominated convergence in Fourier space. Therefore, by Step 1, the Bochner integral

$$w\left(t\right)={\int }_0^t\mathcal{K}(t,s)F\left(s\right)ds$$

(A14)

is well-defined in $H^{-\gamma }$. Since $K(t,t,\rho )=0$, the usual Leibniz rule, justified first for step functions of s and then by (A13) and $L^1$ approximation, yields

$${\partial }_{t}w\left(t\right)={\int }_0^t{\partial }_t\mathcal{K}(t,s)F\left(s\right)ds\mathrm{in}{H}^{-(\gamma +\sigma )}.$$

Consequently,

$$w\in C([0,T];H^{-\gamma })\cap C^1([0,T];H^{-(\gamma +\sigma )}).$$

(A15)

From (A13) and (A6),

$$\begin{array}{cc}\hfill & {∥w\left(t\right)∥}_{H^{-\gamma }}+{∥{\partial }_{t}w\left(t\right)∥}_{H^{-(\gamma +\sigma )}}\hfill \\ \hfill & \le C_T{\int }_0^t{∥F\left(s\right)∥}_{H^{-\gamma }}ds\hfill \\ \hfill & \le C_T{t}^{(\alpha +1)/q_0}{∥s^{-\beta }F∥}_{L^r((0,t)\times {\mathbb{R}}^n)},\hfill \end{array}$$

(A16)

where the factor ${(\alpha +1)}^{-1/q_0}$ is absorbed into $C_T$. Since $(\alpha +1)/q_0>0$, (A16) proves both limits in (13). It also shows that the traces exist strongly in the claimed negative-order Sobolev spaces, rather than merely in ${\mathcal{D}}^{\prime }\left({\mathbb{R}}^n\right)$.

• Step 4: Approximation and uniqueness.

Choose $G_j\in C_c^{\infty }((0,\infty )\times {\mathbb{R}}^n)$ such that

$$G_j\to G=s^{-\beta }F\mathrm{in}{L}^r((0,\infty )\times {\mathbb{R}}^n),$$

and put $F_j(s,x)=s^{\beta }{G}_j(s,x)$. Every $F_j$ is smooth and compactly supported away from $s=0$, and

$${∥F_j-F∥}_{Y_{\beta }}={∥G_j-G∥}_{L^r}⟶0.$$

Let $w_j$ be the classical zero-data solution given by (A14) with $F_j$ in place of F. The difference estimate obtained from (A13) and (A6) gives

$$\begin{array}{cc}\hfill & \underset{0\le t\le T}{\sup }\left({∥w_j\left(t\right)-w_k\left(t\right)∥}_{H^{-\gamma }}+{∥{\partial }_t{w}_j\left(t\right)-{\partial }_t{w}_k\left(t\right)∥}_{H^{-(\gamma +\sigma )}}\right)\hfill \\ \hfill & \le C_T{∥F_j-F_k∥}_{Y_{\beta }}⟶0.\hfill \end{array}$$

(A17)

Hence, $w_j\to w$ in the class (A15); in particular, the zero initial traces pass to the limit. Also, $F_j\to F$ in $L^1((0,T);H^{-\gamma })$, by (A6). Passing to the limit in ${\partial }_t^2{w}_j-t^m\Delta w_j=F_j$ gives

$${\partial }_t^{2}w-t^m\Delta w=F\mathrm{in}{\mathcal{D}}^{\prime }((0,T)\times {\mathbb{R}}^n).$$

(A18)

The constants in (A17) do not depend on the distance of the supports of $F_j$ and $F_k$ from $s=0$.

For completeness, uniqueness holds within the class (12) for the prescribed data, and not just among solutions obtained by approximation. Indeed, let z solve the homogeneous equation with both zero initial traces in that class. For $\mathbb{R}>0$, let $P_{\mathbb{R}}=\chi (\left|D\right|/\mathbb{R})$, where $\chi \in C_c^{\infty }\left({\mathbb{R}}^n\right)$ equals one near the origin. Then $z_{\mathbb{R}}=P_{\mathbb{R}}z\in C^1([0,T];L^2)$ satisfies

$${\partial }_t^2{z}_{\mathbb{R}}+t^m{\left|D\right|}^2{z}_{\mathbb{R}}=0,z_{\mathbb{R}}\left(0\right)={\partial }_t{z}_{\mathbb{R}}\left(0\right)=0,$$

(A19)

in the $L^2$-valued distributional sense. On the range of $P_{\mathbb{R}}$, ${\left|D\right|}^2$ is a bounded operator in $L^2$. Thus, (A19) is an ordinary differential equation in the Banach space $L^2$, with continuous coefficients; its zero-data solution is uniquely zero, by the integral equation and Gronwall’s inequality. Hence, $P_{\mathbb{R}}z=0$ for every R, and letting $\mathbb{R}\to \infty$ in ${\mathcal{D}}^{\prime }\left({\mathbb{R}}^n\right)$ gives $z=0$.

Appendix A.3. Microlocal Representation of the Solution

Proof of Lemma 3.

Let w be the solution of (7). We prove the representation formula (15) holds with an amplitude satisfying the symbol estimate (16).

We begin with the representation of the solution operator obtained in [5]. Namely,

$$w(t,x)={\int }_0^t\left(V_2(t,D_x)V_1(\tau ,D_x)-V_1(t,D_x)V_2(\tau ,D_x)\right)F(\tau ,x)d\tau ,$$

(A20)

Here, $V_j(t,D_x)$ and ($j=1,2$) are Fourier multiplier operators with symbols $V_j(t,\xi )=V_j(t,|\xi \left|\right)$ given by

$$\begin{array}{cc}\hfill & V_1(t,\xi )\hfill \\ \hfill & ={\displaystyle \frac{\Gamma \left(\frac{m}{m+2}\right)}{\Gamma \left(\frac{m}{2(m+2)}\right)}}\left\{e^{{\displaystyle \frac{z}{2}}}{H}_{+}\left({\displaystyle \frac{m}{2(m+2)}},{\displaystyle \frac{m}{m+2}};z\right)+e^{-{\displaystyle \frac{z}{2}}}{H}_{-}\left({\displaystyle \frac{m}{2(m+2)}},{\displaystyle \frac{m}{m+2}};z\right)\right\}\hfill \end{array}$$

(A21)

and

$$\begin{array}{cc}\hfill & V_2(t,\xi )\hfill \\ \hfill & =t{\displaystyle \frac{\Gamma \left(\frac{m+4}{m+2}\right)}{\Gamma \left(\frac{m+4}{2(m+2)}\right)}}\left\{e^{{\displaystyle \frac{z}{2}}}{H}_{+}\left({\displaystyle \frac{m+4}{2(m+2)}},{\displaystyle \frac{m+4}{m+2}};z\right)+e^{-{\displaystyle \frac{z}{2}}}{H}_{-}\left({\displaystyle \frac{m+4}{2(m+2)}},{\displaystyle \frac{m+4}{m+2}};z\right)\right\},\hfill \end{array}$$

(A22)

where $z=2i\varphi \left(t\right)\left|\xi \right|$. For $\alpha ,\nu \in \mathbb{R}$, $\omega \in \mathbb{C}$, we have

$$\begin{array}{cc}\hfill H_{+}(\alpha ,\nu ;\omega )& ={\displaystyle \frac{e^{-i\pi (\nu -\alpha )}}{e^{i\pi (\nu -\alpha )}-e^{-i\pi (\nu -\alpha )}}}{\displaystyle \frac{1}{\Gamma (\nu -\alpha )}}{\omega }^{\alpha -\nu }{\int }_{\infty }^{(0+)}{e}^{-\theta }{\theta }^{\nu -\alpha -1}{\left(1-{\displaystyle \frac{\theta }{\omega }}\right)}^{\alpha -1}d\theta ,\hfill \\ \hfill H_{-}(\alpha ,\nu ;\omega )& ={\displaystyle \frac{1}{e^{i\pi \alpha }-e^{-i\pi \alpha }}}{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\omega }^{-\alpha }{\int }_{\infty }^{(0+)}{e}^{-\theta }{\theta }^{\alpha -1}{\left(1+{\displaystyle \frac{\theta }{\omega }}\right)}^{\nu -\alpha -1}d\theta .\hfill \end{array}$$

By [5] (Section 3), for $\varphi \left(t\right)\left|\xi \right|\ge 1$,

$$\begin{array}{cc}\hfill \left|{\partial }_{\xi }^{\beta }{H}_{+}\left(\alpha ,\gamma ;2i{\varphi }_m\left(t\right)\left|\xi \right|\right)\right|& \le C{\left({\varphi }_m\left(t\right)\left|\xi \right|\right)}^{\alpha -\gamma }{\left(1+{\left|\xi \right|}^2\right)}^{-{\displaystyle \frac{\left|\beta \right|}{2}}},\hfill \end{array}$$

(A23)

$$\begin{array}{cc}\hfill \left|{\partial }_{\xi }^{\beta }{H}_{-}\left(\alpha ,\gamma ;2i{\varphi }_m\left(t\right)\left|\xi \right|\right)\right|& \le C{\left({\varphi }_m\left(t\right)\left|\xi \right|\right)}^{-\alpha }{\left(1+{\left|\xi \right|}^2\right)}^{-{\displaystyle \frac{\left|\beta \right|}{2}}}.\hfill \end{array}$$

(A24)

To exploit these asymptotic estimates while retaining control in the region where ${\varphi }_m\left(t\right)\left|\xi \right|$ is small, we decompose the symbol into high- and low-frequency components relative to the scale ${\varphi }_m{\left(t\right)}^{-1}$. Choose $\chi \in C^{\infty }\left({\mathbb{R}}_{+}\right)$ such that

$$\chi \left(s\right)=\left\{\begin{array}{cc}1,\hfill & s\ge 2,\hfill \\ 0,\hfill & s\le 1.\hfill \end{array}\right.$$

Then, for any test function f

$$\begin{array}{cc}\hfill V_1(t,|\xi \left|\right)\widehat{f}\left(\xi \right)& =\chi ({\varphi }_m\left(t\right)\left|\xi \right|)V_1(t,|\xi \left|\right)\widehat{f}\left(\xi \right)+(1-\chi ({\varphi }_m\left(t\right)\left|\xi \right|\left)\right)V_1(t,|\xi \left|\right)\widehat{f}\left(\xi \right)\hfill \\ \hfill & :={\widehat{v}}_1(t,\xi )+{\widehat{v}}_2(t,\xi ).\hfill \end{array}$$

(A25)

Applying (A21), (A23), and (A24) to the high-frequency component in (A25), we obtain

$$v_1(t,x)=C_m\left({\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +{\varphi }_m\left(t\right)\left|\xi \right|\right)}{a}_{11}(t,\xi )\widehat{f}\left(\xi \right)d\xi +{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -{\varphi }_m\left(t\right)\left|\xi \right|\right)}{a}_{12}(t,\xi )\widehat{f}\left(\xi \right)d\xi \right),$$

where $C_m>0$ depends only on m, and the amplitudes $a_{1l}$ and $l=1,2$ satisfy

$$\left|{\partial }_{\xi }^{\beta }{a}_{1l}(t,\xi )\right|\le C_{l\beta }{\left|\xi \right|}^{-\left|\beta \right|}{\left({\varphi }_m\left(t\right)\left|\xi \right|\right)}^{-{\displaystyle \frac{m}{2(m+2)}}}.$$

It remains to control the complementary low-frequency component. By [23],

$$V_1(t,|\xi \left|\right)=e^{-{\displaystyle \frac{z}{2}}}\mathrm{\Phi }\left({\displaystyle \frac{m}{2(m+2)}},{\displaystyle \frac{m}{m+2}};z\right),$$

where $\mathrm{\Phi }$ is the confluent hypergeometric function, analytic in $z=2i{\varphi }_m\left(t\right)\left|\xi \right|$. This analyticity, together with the support properties of $1-\chi$, gives

$$\left|{\partial }_{\xi }^{\beta }\left((1-\chi ({\varphi }_m\left(t\right)\left|\xi \right|\left)\right)V_1(t,|\xi \left|\right)\right)\right|\le C{\left(1+{\varphi }_m\left(t\right)\left|\xi \right|\right)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\left|\xi \right|}^{-\left|\beta \right|}.$$

Hence, the low-frequency contribution admits the representation

$$v_2(t,x)=C_m\left({\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +{\varphi }_m\left(t\right)\left|\xi \right|\right)}{a}_{21}(t,\xi )\widehat{f}\left(\xi \right)d\xi +{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -{\varphi }_m\left(t\right)\left|\xi \right|\right)}{a}_{22}(t,\xi )\widehat{f}\left(\xi \right)d\xi \right),$$

with the corresponding amplitudes satisfying

$$\left|{\partial }_{\xi }^{\beta }{a}_{2l}(t,\xi )\right|\le C_{l\beta }{\left(1+{\varphi }_m\left(t\right)\left|\xi \right|\right)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\left|\xi \right|}^{-\left|\beta \right|}.$$

Combining the high- and low-frequency components, we conclude that

$$V_1(t,D_x)f\left(x\right)=C_m\left({\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +{\varphi }_m\left(t\right)\left|\xi \right|\right)}{a}_1(t,\xi )\widehat{f}\left(\xi \right)d\xi +{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -{\varphi }_m\left(t\right)\left|\xi \right|\right)}{a}_2(t,\xi )\widehat{f}\left(\xi \right)d\xi \right),$$

(A26)

where for $l=1,2$

$$\left|{\partial }_{\xi }^{\beta }{a}_l(t,\xi )\right|\le C_{l\beta }{\left(1+{\varphi }_m\left(t\right)\left|\xi \right|\right)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\left|\xi \right|}^{-\left|\beta \right|}.$$

(A27)

he same argument yields the corresponding representation and symbol bounds for $V_2(t,D_x)$:

$$V_2(t,D_x)f\left(x\right)=C_m\left({\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +{\varphi }_m\left(t\right)\left|\xi \right|\right)}{a}_3(t,\xi )\widehat{f}\left(\xi \right)d\xi +{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -{\varphi }_m\left(t\right)\left|\xi \right|\right)}{a}_4(t,\xi )\widehat{f}\left(\xi \right)d\xi \right),$$

(A28)

where for $l=3,4$

$$\left|{\partial }_{\xi }^{\beta }{a}_l(t,\xi )\right|\le C_{l\beta }{\left(1+{\varphi }_m\left(t\right)\left|\xi \right|\right)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\left|\xi \right|}^{-{\displaystyle \frac{2}{m+2}}-\left|\beta \right|}.$$

(A29)

We now return to the representation (A20). We first handle

$${\int }_0^t{V}_2(t,D_x)V_1(\tau ,D_x)F(\tau ,x)d\tau .$$

By (A26) and (A28) we can write

$$\begin{array}{cc}\hfill & {\int }_0^t{V}_2(t,D_x)V_1(\tau ,D_x)F(\tau ,x)d\tau \hfill \\ \hfill & ={\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +({\varphi }_m\left(t\right)+{\varphi }_m\left(\tau \right))\left|\xi \right|\right)}{a}_3(t,\xi )a_1(\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +({\varphi }_m\left(t\right)-{\varphi }_m\left(\tau \right))\left|\xi \right|\right)}{a}_3(t,\xi )a_2(\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -({\varphi }_m\left(t\right)-{\varphi }_m\left(\tau \right))\left|\xi \right|\right)}{a}_4(t,\xi )a_1(\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -({\varphi }_m\left(t\right)+{\varphi }_m\left(\tau \right))\left|\xi \right|\right)}{a}_4(t,\xi )a_2(\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau ,\hfill \end{array}$$

while for the second term in (A20),

$${\int }_0^t{V}_1(t,D_x)V_2(\tau ,D_x)F(\tau ,x)d\tau$$

we have

$$\begin{array}{cc}\hfill & {\int }_0^t{V}_1(t,D_x)V_2(\tau ,D_x)F(\tau ,x)d\tau \hfill \\ \hfill & ={\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +({\varphi }_m\left(t\right)+{\varphi }_m\left(\tau \right))\left|\xi \right|\right)}{a}_1(t,\xi )a_3(\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +({\varphi }_m\left(t\right)-{\varphi }_m\left(\tau \right))\left|\xi \right|\right)}{a}_1(t,\xi )a_4(\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -({\varphi }_m\left(t\right)-{\varphi }_m\left(\tau \right))\left|\xi \right|\right)}{a}_2(t,\xi )a_3(\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -({\varphi }_m\left(t\right)+{\varphi }_m\left(\tau \right))\left|\xi \right|\right)}{a}_2(t,\xi )a_4(\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau ,\hfill \end{array}$$

Define the four amplitudes by

$$a_{+,+}(t,\tau ,\xi )=a_3(t,\xi )a_1(\tau ,\xi )-a_1(t,\xi )a_3(\tau ,\xi ),$$

$$a_{+,-}(t,\tau ,\xi )=a_3(t,\xi )a_2(\tau ,\xi )-a_1(t,\xi )a_4(\tau ,\xi ),$$

$$a_{-,+}(t,\tau ,\xi )=a_4(t,\xi )a_1(\tau ,\xi )-a_2(t,\xi )a_3(\tau ,\xi ),$$

$$a_{-,-}(t,\tau ,\xi )=a_4(t,\xi )a_2(\tau ,\xi )-a_2(t,\xi )a_4(\tau ,\xi ).$$

Then, we have

$$\begin{array}{cc}\hfill w(t,x)& ={\int }_0^t\left(V_2(t,D_x)V_1(\tau ,D_x)-V_1(t,D_x)V_2(\tau ,D_x)\right)F(\tau ,x)d\tau \hfill \\ \hfill & ={\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +{\varphi }_m\left(t\right)\left|\xi \right|+{\varphi }_m\left(\tau \right)\left|\xi \right|\right)}{a}_{+,+}(t,\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi +{\varphi }_m\left(t\right)\left|\xi \right|-{\varphi }_m\left(\tau \right)\left|\xi \right|\right)}{a}_{+,-}(t,\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -{\varphi }_m\left(t\right)\left|\xi \right|+{\varphi }_m\left(\tau \right)\left|\xi \right|\right)}{a}_{-,+}(t,\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau \hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^n}{e}^{i\left(x·\xi -{\varphi }_m\left(t\right)\left|\xi \right|-{\varphi }_m\left(\tau \right)\left|\xi \right|\right)}{a}_{-,-}(t,\tau ,\xi )\widehat{F}(\tau ,\xi )d\xi d\tau .\hfill \end{array}$$

(A30)

Furthermore, (A27) and (A29) yield uniform estimates for all $a_{\pm ,\pm }(t,\tau ,\xi )$

$$\left|{\partial }_{\xi }^{\beta }{a}_{\pm ,\pm }(t,\tau ,\xi )\right|\le C{\left(1+{\varphi }_m\left(t\right)\left|\xi \right|\right)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\left(1+{\varphi }_m\left(\tau \right)\left|\xi \right|\right)}^{-{\displaystyle \frac{m}{2(m+2)}}}{\left|\xi \right|}^{-{\displaystyle \frac{2}{m+2}}-\left|\beta \right|}.$$

(A31)

Thus, we have established the representation formula (15) together with the amplitude bound (16). □