Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators

Abstract

Let be the high-order Schrödinger operator ${(-\Delta )}^2+V^2,$ where V is a non-negative potential satisfying the reverse Hölder inequality ($R{H}_q$), with $q>n/2$ and $n\ge 5$. In this paper, we prove that when $0<\alpha \le 2-n/q$, the adapted Lipschitz spaces ${\Lambda }_{\alpha /4}^{\mathcal{L}}$ we considered are equivalent to the Lipschitz space $C_L^{\alpha }$ associated to the Schrödinger operator $L=-\Delta +V$. In order to obtain this characterization, we should make use of some of the results associated to ${(-\Delta )}^2$. We also prove the regularity properties of fractional powers (positive and negative) of the operator $ℒ$, Schrödinger Riesz transforms, Bessel potentials and multipliers of the Laplace transforms type associated to the high-order Schrödinger operators.

1. Introduction and Statement of the Main Result

The classical Lipschitz spaces on ${\mathbb{R}}^n$, ${\Lambda }^{\alpha },$ and $\alpha >0$ play an important role in function theory, harmonic analysis and partial differential equations. When $0<\alpha <1$, they are defined as the set of functions $\phi$ such that $|\phi (x+z)-\phi \left(x\right)|\le {C\left|z\right|}^{\alpha }$$x,z\in {\mathbb{R}}^n$. We can say that these classes are between the continuous functions $C^0$ and derivable functions with continuous derivative, $C^1.$ However, Zygmund [1] argued that for applications in harmonic analysis, when $\alpha \to 1$, the space corresponds with a space bigger than $C^1$, the set of continuous functions $\phi$ such that

$$\left|\phi \right(x+z)+\phi (x-z)-2\phi (x\left)\right|\le C\left|z\right|,$$

$x,z\in {\mathbb{R}}^n$. This space ${\Lambda }^1$ is also known as the Zygmund space. For $\alpha >1$, ${\Lambda }^{\alpha }$ is defined as the class of smooth functions such that their first-order derivatives belong to ${\Lambda }^{\alpha -1}.$

To find some characterizations of these spaces, which are more suitable for some applications, is a recurrent object of research, such as expressions with finite differences, approximation properties, semigroup language, etc.; see [2,3,4,5,6] for instance. Stein and Taibleson [3,4,5,6] gave the characterizations of bounded Lipschitz functions via the Poisson semigroup, $e^{-t\sqrt{-\Delta }},$ and the Gauss semigroup, $e^{t\Delta }$, and deserve a special mention. The semigroup language allows us to obtain regularity results in these spaces. In particular, we can prove the boundedness of some fractional operators, such as fractional Laplacians, fractional integrals, Riesz transforms and Bessel potentials, in a much more simple way than using the classical definition of the spaces.

Taibleson and Stein raised the question of analyzing some Lipschitz spaces adapted to different “Laplacians”, and to find pointwise and semigroup estimate characterizations. In the case of the Ornstein–Ulhenbeck operator $\mathcal{O}=-\frac{1}{2}\Delta +x·\nabla$, in [7], some Lipschitz classes were defined by means of its Poisson semigroup, $e^{-t\sqrt{\mathcal{O}}}$, and in [8] a pointwise characterization was obtained for $0<\alpha <1$. Sometimes, “Lipschitz classes” are also known as “Hölder classes”. For the case of Hermite operator $\mathcal{H}=-\Delta +{\left|x\right|}^2$, adapted Hölder classes were defined pointwisely in [9]. These last classes were characterized in [10], also in the parabolic case, by using semigroups. The classical parabolic case was treated in [11]. In [12], they proved the characterization of Lipschitz spaces associated to the Schrödinger operators $L=-\Delta +V$, where V is a non-negative potential satisfying a reverse Hölder inequality.

In this paper, we consider the high-order Schrödinger operators in ${\mathbb{R}}^n$ with $n\ge 5$, that is, $\mathcal{L}={(-\Delta )}^2+V^2$, where V is a non-negative potential satisfying the reverse Hölder inequality:

$${\left(\frac{1}{\left|B\right|}{\int }_{B}V{\left(y\right)}^{q}dy\right)}^{1/q}\le \frac{C}{\left|B\right|}{\int }_{B}V\left(y\right)dy,$$

with an exponent $q>n/2$ for every ball B. In [13], the author proved the fundamental solution estimates and $L^p$-estimates for some Schrödinger-type operators. In [14], the authors proved that the Hardy spaces related with $ℒ$ is equivalent to the Hardy spaces related with L, which is defined in [15,16]. Based on the corrigendum of [14], throughout this paper, we always assume that the heat kernel, $e^{-tℒ}(x,y)$, satisfies the following inequality:

$$|e^{-tℒ}(x,y)|\le C{t}^{-n/4}exp\left\{-c\frac{{\left|x-y\right|}^{4/3}}{t^{1/3}}\right\}$$

for any $t\in (0,\infty )$ and $x,y\in {\mathbb{R}}^n,$ see [14,17]. Inspiring the results in [12], we can consider the Lipschitz spaces associated with $ℒ$ as in the Schrödinger setting.

In [18], the authors introduced the space $Li{p}_L^{\alpha }$, $0<\alpha <1$, as the set of functions which satisfy

$${\parallel \rho (·)}^{-\alpha }{f(·)\parallel }_{\infty }<\infty \mathrm{and}\underset{\left|y\right|>0}{sup}\frac{{\parallel f(·-y)-f(·)\parallel }_{\infty }}{{\left|y\right|}^{\alpha }}<\infty ,$$

(1)

where $\rho \left(x\right)$ is the critical radius associated to the potential V, see (7). Moreover, in [12] (Definition 1.1) and for $0<\alpha <2$, the authors consider the class of functions $C_L^{\alpha }$, as the set of measurable functions satisfying

$$\left\{\begin{array}{c}{\displaystyle M_{\alpha }^L\left[f\right]:={\parallel \rho {(·)}^{-\alpha }f\parallel }_{\infty }<\infty \mathrm{and}}\hfill \\ {\displaystyle N_{\alpha }\left[f\right]:=\underset{\left|y\right|>0}{sup}\frac{{\parallel f(·+y)+f(·-y)-2f(·)\parallel }_{\infty }}{{\left|y\right|}^{\alpha }}<\infty .}\hfill \end{array}\right.$$

(2)

This space is endowed with the norm

$${\parallel f\parallel }_{C_L^{\alpha }}:=M_{\alpha }^L\left[f\right]+N_{\alpha }\left[f\right].$$

They also proved that this space $C_L^{\alpha }$ can be described either as the set of functions such that

$$\left\{\begin{array}{c}{\displaystyle M_{\alpha }^L\left[f\right]<\infty \mathrm{and}}\hfill \\ {\displaystyle \parallel {\partial }_t^k{e}^{-tL}f{\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /2},k=[\alpha /2]+1,t>0,}\hfill \end{array}\right.$$

(3)

or as the set of functions such that

$$\left\{\begin{array}{c}{\displaystyle M_{\alpha }^L\left[f\right]<\infty ,\mathrm{and}}\hfill \\ {\displaystyle \parallel {\partial }_t^k{e}^{-tL}f{\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /2},k=[\alpha /2]+1,t>0,}\hfill \end{array}\right.$$

(4)

with the obvious norms; see Theorem 1.3 in [12].

One of the main results of this paper is to show that, the above coincidence of smooth class of functions defined either in (2), in (3), or in (4) also holds when considering the heat semigroup associated to the operator $ℒ$. In order to state the main theorem, we shall present our definitions.

Denote $W_t^{ℒ}(x,y)$ as the integral kernel of the heat semigroup of $e^{-tℒ}$ generated by $ℒ$. That means that, for suitable function f,

$$e^{-tℒ}f\left(x\right)=W_t^{ℒ}f\left(x\right)={\int }_{{\mathbb{R}}^n}{W}_t^{ℒ}(x,y)f\left(y\right)dy,x\in {\mathbb{R}}^n.$$

Definition 1.

Let $\alpha >0.$ We shall denote by ${\Lambda }_{\alpha /4}^{ℒ}$ the set of functions f such that

$$M_{\alpha }^{ℒ}\left[f\right]:={\parallel \rho {(·)}^{-\alpha }f\parallel }_{\infty }<\infty <\infty$$

and

$$\parallel {\partial }_t^k{W}_t^{ℒ}f{\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /4},\mathrm{with}k=[\alpha /4]+1,t>0.$$

(5)

The norm of this space is endowed with

$${\parallel f\parallel }_{{\Lambda }_{\alpha /4}^{ℒ}}:=S_{\alpha }^{ℒ}\left[f\right]+M_{\alpha }^{ℒ}\left[f\right],$$

where $S_{\alpha }^{ℒ}\left[f\right]$ is the infimum of the constants $C_{\alpha }$ appearing in (5).

Then, we obtain the following theorem.

Theorem 1.

Let $0<\alpha \le 2-n/q.$ Then

$$C_L^{\alpha }={\Lambda }_{\alpha /4}^{ℒ},$$

with equivalence of norms.

The proof of this theorem will follow some ideas developed in [12] partly. In particular, we shall need some new Lipschitz spaces associated to the operator ${(-\Delta )}^2$. These classes are introduced and discussed in Section 2.

As an application of Theorem 1, we shall prove the regularity properties of some operators associated to $ℒ$. We list here the operators which we are going to deal with. Their definitions are motivated by the Gamma formulas (see [19]).

• The Bessel potential of order $\beta >0$,

  $${(Id+ℒ)}^{-\beta /4}f\left(x\right)=\frac{1}{\Gamma (\beta /4)}{\int }_0^{\infty }{e}^{-s}{e}^{-sℒ}f\left(x\right)s^{\beta /4}\frac{ds}{s}.$$
• The fractional integral of order $\beta >0$,

  $$ℒ{}^{-\beta /4}f\left(x\right)=\frac{1}{\Gamma (\beta /4)}{\int }_0^{\infty }{e}^{-sℒ}f\left(x\right)s^{\beta /4}\frac{ds}{s}.$$
• The fractional “Laplacian” of order $\beta /4>0$,

  $$ℒ{}^{\beta /4}f\left(x\right)=\frac{1}{c_{\beta }}{\int }_0^{\infty }{(e^{-sℒ}-Id)}^{[\beta /4]+1}f\left(x\right)\frac{ds}{s^{1+\beta /4}}.$$
• The first-order Riesz transforms defined by

  $${\mathcal{R}}_i={\partial }_{x_i}\left({ℒ}^{-1/4}\right),\mathrm{and}{R}_i={ℒ}^{-1/4}\left({\partial }_{x_i}\right),i=1,\cdots ,n.$$

Then we have the following results.

Theorem 2.

Let $\alpha ,\beta >0$ and ${\mathcal{T}}_{\beta }$ denote the Bessel potential or the fractional integral of order β. Then,

(i) 

$\parallel {\mathcal{T}}_{\beta }{f\parallel }_{{\Lambda }_{\frac{\alpha +\beta }{4}}^{ℒ}}\le C{\parallel f\parallel }_{{\Lambda }_{\alpha /4}^{ℒ}},$

(ii) 

$\parallel {\mathcal{T}}_{\beta }{f\parallel }_{{\Lambda }_{\beta /4}^{ℒ}}\le C{\parallel f\parallel }_{\infty }.$

Theorem 3

(Hölder estimates). If $0<\beta <\alpha$ and $f\in {\Lambda }_{\alpha /4}^{ℒ},$ then,

$$\parallel {\mathcal{L}}^{\beta /4}{f\parallel }_{{\Lambda }_{\frac{\alpha +\beta }{4}}^{ℒ}}\le C{\parallel f\parallel }_{{\Lambda }_{\alpha /4}^{ℒ}}.$$

Theorem 4.

(1) 

For $0<\alpha \le 1-n/q$, then $\parallel {\mathcal{R}}_i{f\parallel }_{{\Lambda }_{\alpha /4}^{ℒ}}\le C{\parallel f\parallel }_{{\Lambda }_{\alpha /4}^{ℒ}}$, $i=1,\cdots ,n$.

(2) 

For $1<\alpha \le 2-n/q$, then $\parallel R_i{f\parallel }_{{\Lambda }_{\alpha /4}^{ℒ}}\le C{\parallel f\parallel }_{{\Lambda }_{\alpha /4}^{ℒ}}$, $i=1,\cdots ,n$.

Theorem 5.

Let g be a measurable bounded function on $[0,\infty )$ and consider

$$m\left(\lambda \right)=\lambda {\int }_0^{\infty }{e}^{-s\lambda }g\left(s\right)ds,\lambda >0.$$

Then, for every $\alpha >0$, the multiplier operator of the Laplace transform type $m(ℒ)$ is bounded from ${\Lambda }_{\alpha /4}^{ℒ}$ into itself.

We organize the paper as follows. In Section 2, we prove the characterization results but related to ${(-\Delta )}^2.$ In Section 3, we give the proof of our characterization theorem listed above in a high-order Schrödinger setting. In the proof, we use the comparison of Lipschitz spaces associated to ${(-\Delta )}^2$ and $ℒ$. In Section 4, we prove the boundedness of the Bessel potentials, fractional integrals, fractional “Laplacian” and the first-order Riesz transforms related with high-order Schrödinger operators $ℒ$ on the adapted Lipschitz spaces.

Along this paper, C denotes a constant that may not be the same in each appearance. We will write the constant with subindexes if we need to emphasize the dependence on some parameters.

2. Lipschitz Spaces Related to the Biharmonic Operator ${(-\Delta )}^{\mathbf{2}}$

We shall define a new Lipschitz space associated to the biharmonic operator ${(-\Delta )}^2$, which will be called “the adapted Lipschitz space” later, in this section. In fact, we have double motivations. Firstly, we can use the definition to prove some regularity estimates for operators defined through ${(-\Delta )}^2$. Secondly, as we have said, we shall need them as a tool to prove results for high-order Schrödinger operators of the type $\mathcal{L}={(-\Delta )}^2+V^2.$

Consider the following Cauchy problem for the biharmonic heat equation

$$\left\{\begin{array}{c}({\partial }_t+{(-\Delta )}^2)u(x,t)=0\mathrm{in}{\mathbb{R}}_{+}^{n+1}\hfill \\ u(x,0)=f\left(x\right)\mathrm{in}{\mathbb{R}}^n.\hfill \end{array}\right.$$

(6)

The solution of this partial differential equation is given by

$$u(x,t)=W_{t}f\left(x\right)={\int }_{{\mathbb{R}}^n}{W}_t(x-y)f\left(y\right)dy,$$

where

$$W_t\left(x\right)={\mathfrak{F}}^{-1}\left(e^{-{\left|\xi \right|}^{4}t}\right)=t^{-n/4}g\left(\frac{x}{t^{1/4}}\right)$$

with

$$g\left(\xi \right)={\left(2\pi \right)}^{-n/2}{\int }_{{\mathbb{R}}^n}{e}^{i\xi \eta -{\left|\eta \right|}^4}d\eta ={\alpha }_n{\left|\xi \right|}^{1-n}{\int }_0^{\infty }{e}^{-s^4}{\left(\right|\xi \left|s\right)}^{n/2}{J}_{(n-2)/2}\left(\right|\xi \left|s\right)ds,\xi \in {\mathbb{R}}^n,$$

and ${\mathfrak{F}}^{-1}$ being the inverse Fourier transform. $J_v$ denotes that the v-th Bessel function and ${\alpha }_n>0$ is a normalization constant such that

$${\int }_{{\mathbb{R}}^n}g\left(\xi \right)d\xi =1.$$

For more details, see [20,21]. The following several results can be obtained by classical analysis (for details, see [22]):

(1)

If $f\in L^p\left({\mathbb{R}}^n\right)$, $1\le p\le \infty ,$ then

$${\displaystyle \underset{t\to 0}{lim}u(x,t)=f\left(x\right)a.e.x\in {\mathbb{R}}^n,}$$

and

$${∥u(·,t)∥}_{L^p\left({\mathbb{R}}^n\right)}\le C{∥f∥}_{L^p\left({\mathbb{R}}^n\right)}.$$

(2)

If $1\le p<\infty ,$ then

$${∥u(·,t)-f∥}_{L^p\left({\mathbb{R}}^n\right)}\to 0,\mathrm{when}t\to 0^{+}.$$

For the heat kernel $W_t\left(x\right)$ of the semigroup $e^{-t{(-\Delta )}^2}$, the following estimates are known.

Lemma 1

(See [23] and ([20], Lemma 2.4)). For $x\in {\mathbb{R}}^n$ and $t>0$, the following estimates hold:

(i) 

$$|W_t\left(x\right)|\le C{t}^{-n/4}{e}^{-c\frac{{\left|x\right|}^{4/3}}{t^{1/3}}},c=\frac{3·2^{1/3}}{16},$$

(ii) 

$$|{\partial }_t^l{\nabla }^k{W}_t\left(x\right)|\le C{\left(t^{\frac{1}{4}}+\left|x\right|\right)}^{-n-k-4l}{e}^{-c^{\prime }\frac{{\left|x\right|}^{4/3}}{t^{1/3}}},\forall k,l\ge 1,c^{\prime }<c,$$

(iii) 

$${∥{\partial }_t^l{\nabla }^k{W}_t(·)∥}_{L^1\left({\mathbb{R}}^n\right)}\le C{t}^{-l-\frac{k}{4}},\forall k,l\ge 1,$$

(iv) 

There exist $C,C^{\prime }>0$ such that for $0\le j\le 4,$

$$|{\nabla }^j{W}_t\left(x\right)|\le C{e}^{-C^{\prime }\left|x\right|},\forall (x,t)\in {\mathbb{R}}^n\times (0,1)\backslash \left(B_2\times (0,\frac{1}{2})\right).$$

2.1. Lipschitz Spaces Associated to ${(-\Delta )}^2$

We defined the spaces ${\Lambda }_{\alpha /4}^{{\Delta }^2}$ in [24].

Definition 2

(See ([24], Definition 1.1)). Let $\alpha >0.$ The spaces ${\Lambda }_{\alpha /4}^{{\Delta }^2}$ are defined as

$$\begin{array}{cc}\hfill {\Lambda }_{\alpha /4}^{{\Delta }^2}=& \left\{f:f\in L^{\infty }\left({\mathbb{R}}^n\right)\mathit{and}{∥{\partial }_t^k{W}_{t}f∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /4},k=[\alpha /4]+1\right\}.\hfill \end{array}$$

The norm is endowed as

$${\parallel f\parallel }_{{\Lambda }_{\alpha /4}^{{\Delta }^2}}:={∥f∥}_{\infty }+S_{\alpha }\left[f\right],$$

where $S_{\alpha }\left[f\right]$ denotes the infimum of the constants $C_{\alpha }$ appearing above.

Remark 1.

Because of the estimate(iii)in Lemma 1, in Definition 2, we can assume $t<1$.

The proposition in the following means that we can use any bigger integer than $[\alpha /4]+1$ in Definition 2.

Proposition 1

(See ([24], Proposition 2.3)). Let $\alpha >0$. A function $f\in {\Lambda }_{\alpha /4}^{{\Delta }^2}$ if, and only if, for $m\ge [\alpha /4]+1$, we have $\parallel {\partial }_t^m{W}_t{f\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_m{t}^{-m+\alpha /4}$ and $f\in L^{\infty }\left({\mathbb{R}}^n\right).$

In [24], the following theorem was proved.

Theorem 6

(See ([24], Theorem 1.2)). Let $0<\alpha <2$. Then the following three statements are equivalent:

(1) 

$f\in {\Lambda }_{\alpha /4}^{{\Delta }^2}$.

(2) 

$f\in \left\{f\in L^{\infty }\left({\mathbb{R}}^n\right):{∥{\partial }_t^k{e}^{-t\sqrt{-\Delta }}f∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha },k=\left[\alpha \right]+1\right\}$, where $e^{-t\sqrt{-\Delta }}$ is the classical Poisson kernel.

(3) 

${\displaystyle f\in \{f\in L^{\infty }\left({\mathbb{R}}^n\right):N_{\alpha }\left[f\right]:=\underset{\left|y\right|>0}{sup}\frac{{\parallel f(·+y)+f(·-y)-2f(·)\parallel }_{\infty }}{{\left|y\right|}^{\alpha }}<\infty \}}$.

Additionally,

$${∥f∥}_{{\Lambda }_{\alpha /4}^{{\Delta }^2}}\sim {∥f∥}_{L^{\infty }}+{\tilde{S}}_{\alpha }\left[f\right]\sim {∥f∥}_{L^{\infty }}+N_{\alpha }\left[f\right],$$

where ${\tilde{S}}_{\alpha }\left[f\right]$ denotes the infimum of the constants $C_{\alpha }$ appearing in $\left(2\right)$.

Remark 2.

In [3], E. M. Stein proved that $\left(2\right)$ and $\left(3\right)$ are equivalent.

In the following, we prove a parallel result to that of [3] (Proposition 9 in p. 147), which gives the relationship between the Lipschitz function and its derivatives. First, we give a lemma and a theorem which will be used later.

Lemma 2

(See ([24], Lemma 3.2)). Let $\alpha >0$ and $k=[\alpha /4]+1$. If ${∥{\partial }_t^k{W}_{t}f∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /4}$, then for every $j,m\in \mathbb{N}$ such that $m/4+j\ge k$, there exists a $C_{m,j}>0$ such that

$${∥{\partial }_{x_i}^m{\partial }_t^j{W}_{t}f∥}_{\infty }\le C_{m,j,\alpha }{t}^{-(m/4+j)+\alpha /4},\mathit{for}\mathit{every}i=1\cdots ,n.$$

Theorem 7

(See ([24], Theorem 1.6)). Suppose that $\alpha >1$ and $f\in L^{\infty }\left({\mathbb{R}}^n\right).$ Then,

$$f\in {\Lambda }_{\alpha /4}^{{\Delta }^2}\mathit{if}\mathit{and}\mathit{only}\mathit{if}{\partial }_{x_i}f\in {\Lambda }_{(\alpha -1)/4}^{{\Delta }^2},i=1,\cdots ,n.$$

In this case, the following equivalence holds

$${∥f∥}_{{\Lambda }_{\alpha /4}^{{\Delta }^2}}\sim \sum _{i=1}^n{∥{\partial }_{x_i}f∥}_{{\Lambda }_{(\alpha -1)/4}^{{\Delta }^2}}.$$

2.2. Adapted Lipschitz Spaces Associated to ${(-\Delta )}^2$

In this subsection, we should define a new Lipschitz class to prove our main results. For our results about the spaces adapted to the high order Schrödinger operator $ℒ$, it will be an auxiliary class. The crucial difference is that the functions do not need to be bounded.

Definition 3.

Let $\alpha >0.$ We define the spaces ${\Lambda }_{\alpha /4}^W$ as

$$\begin{array}{cc}\hfill {\Lambda }_{\alpha /4}^W=& \left\{f:{(1+|·\left|\right)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)\mathrm{and}{∥{\partial }_t^k{W}_{t}f∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /4},k=[\alpha /4]+1\right\}.\hfill \end{array}$$

We endow this class with the norm

$${\parallel f\parallel }_{{\Lambda }_{\alpha /4}^W}:=M_{\alpha }\left[f\right]+S_{\alpha }\left[f\right],$$

where $M_{\alpha }\left[f\right]:={\parallel (1+|·\left|\right)}^{-\alpha }{f\parallel }_{\infty }$.

When $f\in L^{\infty }\left({\mathbb{R}}^n\right)$, by the $L^{\infty }$-boundedness of the heat semigroup $e^{-t{\Delta }^2}$, we know that, for every $\ell \in \mathbb{N}$, ${\partial }_t^{\ell }{W}_{t}f$ is well defined. For functions $f\in {\Lambda }_{\alpha /4}^W$, the following lemma says that the definition above has sense.

Lemma 3.

Let f be a function with $M_{\alpha }\left[f\right]<\infty$. Then, for every $\ell \in \mathbb{N}$, ${\partial }_t^{\ell }{W}_{t}f$ is well defined. In addition, ${\displaystyle \underset{t\to 0}{lim}{W}_{t}f\left(x\right)=f\left(x\right),a.e.x\in {\mathbb{R}}^n.}$ Moreover, $W_{t}f\left(x\right)$ belongs to $C^{\infty }((0,+\infty )\times {\mathbb{R}}^n)$.

Proof. 

Observe that

$$\begin{array}{c}\hfill |W_{t}f\left(x\right)|\le C{\int }_{{\mathbb{R}}^n}\frac{exp\left(-c\frac{{|x-y|}^{4/3}}{t^{1/3}}\right)}{t^{n/4}}f\left(y\right)dy\le {\int }_{{\mathbb{R}}^n}\frac{exp\left(-c\frac{{|x-y|}^{4/3}}{t^{1/3}}\right)}{t^{n/4}}{(1+|y\left|\right)}^{\alpha }dy.\end{array}$$

If $\left|y\right|>2\left|x\right|$, then the above integral is less than

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}\frac{exp\left(-c\frac{{\left|y\right|}^{4/3}}{t^{1/3}}\right)}{t^{n/4}}{(1+|y\left|\right)}^{\alpha }dy\le C(1+t^{\alpha /4}).\end{array}$$

If $\left|y\right|\le 2\left|x\right|$, the last integral is convergent and bounded by $C(1+t^{\alpha /4}+|x{|}^{\alpha })$. The same arguments can be used for the derivatives ${\partial }_t^{\ell }{W}_{t}f$, $\ell \in \mathbb{N}\backslash \left\{0\right\}$.

Moreover, if $m/4+\ell \ge [\alpha /4]+1$, then ${\displaystyle \underset{t\to \infty }{lim}{\partial }_{x_i}^m{\partial }_t^{\ell }{W}_{t}f\left(x\right)=0}$, for every $x\in {\mathbb{R}}^n.$ Indeed, observe that

$$\begin{array}{c}\hfill |{\partial }_{x_i}^m{\partial }_t^{\ell }{W}_{t}f\left(x\right)|\le C{\int }_{{\mathbb{R}}^n}\frac{exp\left(-c\frac{{|x-y|}^{4/3}}{t^{1/3}}\right)\left|f\left(y\right)\right|}{t^{n/4+m/4+\ell }}dy\le C{\int }_{{\mathbb{R}}^n}\frac{exp\left(-c\frac{{|x-y|}^{4/3}}{t^{1/3}}\right)}{t^{n/4+m/4+\ell }}{(1+|y\left|\right)}^{\alpha }dy.\end{array}$$

If $\left|y\right|<2\left|x\right|$, the last integral is less than $C(1+t^{\alpha /4}+{\left|x\right|}^{\alpha })t^{-m/4-\ell }$. In the case $\left|y\right|>2\left|x\right|$, the integral is less than $C(1+t^{\alpha /4})t^{-m/4-\ell }.$

The proof of the second part of this lemma is similar to the proof of Lemma 2.15 in [12], but with some notation differences. We sketch it here. □

Now we will prove a pointwise description of the adapted Lipschitz spaces associated to ${(-\Delta )}^2$. First, we can obtain the proposition as follows.

Proposition 2.

Let $\alpha >0$. A function $f\in {\Lambda }_{\alpha /4}^W$ if, and only if, for all $m\ge [\alpha /4]+1$, we have $\parallel {\partial }_t^m{W}_t{f\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_m{t}^{-m+\alpha /4}$ and $M_{\alpha }\left[f\right]<\infty .$

Proof. 

Let $m\ge [\alpha /4]+1=k$. Using Lemma 1, ${\displaystyle |{\int }_{{\mathbb{R}}^n}{\partial }_t^k{W}_t(x,y)dy|\le \frac{C}{t^k}.}$ Then,

$$\begin{array}{c}\hfill \left|{\partial }_t^m{W}_{t}f\left(x\right)\right|=C\left|{\partial }_t^{m-k}{W}_{t/2}\left({\partial }_u^k{W}_{u}f\left(x\right){|}_{u=t/2}\right)\right|\le C_{\alpha }^{\prime }\frac{1}{t^{m-k}}{t}^{-k+\alpha /4}=C_m{t}^{-m+\alpha /4}.\end{array}$$

In addition, $M_{\alpha }\left[f\right]\le {∥f∥}_{{\Lambda }_{\alpha /4}^W}<\infty$ is trivial.

For the converse, since $|{\partial }_t^m{W}_{t}f\left(x\right)|\to 0\mathrm{as}t\to \infty ,$ we can integrate on t as many times as we need to obtain $\parallel {\partial }_t^k{W}_t{f\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /4}.$ □

We also need the following lemma in later proofs.

Lemma 4.

Let $\alpha >0$ and $k=[\alpha /4]+1$. If ${∥{\partial }_t^k{W}_{t}f∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /4}$, then for every $j,m\in \mathbb{N}$ such that $m/4+j\ge k$, there exists a $C_{m,j}>0$ such that

$${∥{\partial }_{x_i}^m{\partial }_t^j{W}_{t}f∥}_{\infty }\le C_{m,j,\alpha }{t}^{-(m/4+j)+\alpha /4},\mathit{for}\mathit{every}i=1\cdots ,n.$$

Proof. 

If $j\ge k$, we have

$$\begin{array}{cc}\hfill \left|{\partial }_{x_i}^m{\partial }_t^j{W}_{t}f\left(x\right)\right|& =C\left|{\int }_{{\mathbb{R}}^n}{\partial }_{x_i}^m{\partial }_v^{j-k}{W}_v(x-z){|}_{v=t/2}{\partial }_u^k{W}_{u}f\left(z\right){|}_{u=t/2}dz\right|\hfill \\ \hfill & \le \frac{C_{m,j,\alpha }{\parallel {\partial }_u^k{W}_{u}f{|}_{u=t/2}\parallel }_{\infty }}{t^{m/4+j-k}}{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}}{t^{n/4}}dy\hfill \\ \hfill & \le C_{m,j,\alpha }{t}^{-(m/4+j)+\alpha /4},x\in {\mathbb{R}}^n.\hfill \end{array}$$

If $j<k$, by proceeding as before, we obtain that $\left|{\partial }_{x_i}^m{\partial }_t^k{W}_{t}f\left(x\right)\right|\le C{t}^{-(m/4+k)+\alpha /4}$, $x\in {\mathbb{R}}^n$, and we obtain the result by integrating the previous estimate $k-j$ times, since $|{\partial }_{x_i}^m{\partial }_t^{\ell }{W}_{t}f\left(x\right)|\to 0$ as $t\to \infty$ as far as $m/4+\ell \ge k$. □

Theorem 8.

Let $0<\alpha <2.$ The following two statements are equivalent.

(1) 

$f\in {\Lambda }_{\alpha /4}^W$.

(2) 

${\displaystyle f\in \{f:M_{\alpha }\left[f\right]<\infty \mathit{and}{N}_{\alpha }\left[f\right]<\infty \}}$.

Moreover,

$${\parallel f\parallel }_{{\Lambda }_{\alpha /4}^W}\sim N_{\alpha }\left[f\right]+M_{\alpha }\left[f\right].$$

Proof. 

For any $x\in {\mathbb{R}}^n$ and $f\in {\Lambda }_{\alpha /4}^W$, and, for every $t>0$, $y\in {\mathbb{R}}^n$, we have

$$\begin{array}{cc}\hfill \left|f\right(x+y)& +f(x-y)-2f\left(x\right)|\le |W_{t}f(x+y)-f(x+y)|+|W_{t}f(x-y)-f(x-y)|\hfill \\ & +2|W_{t}f\left(x\right)-f\left(x\right)|+|W_{t}f(x+y)-W_{t}f\left(x\right)+W_{t}f(x-y)-W_{t}f\left(x\right)|.\hfill \end{array}$$

By using Lemma 3 we have

$$|W_{t}f\left(x\right)-f\left(x\right)|=\left|{\int }_0^t{\partial }_u{W}_{u}f\left(x\right)du\right|\le C{S}_{\alpha }\left[f\right]{\int }_0^t{u}^{-1+\alpha /4}du=C{S}_{\alpha }\left[f\right]t^{\alpha /4}.$$

In a parallel way, we can handle the two first summands. For the last summand, by using the chain rule, Lemmas 1 and 4, we have

$$\begin{array}{cc}\hfill & |W_{t}f(x+y)-W_{t}f\left(x\right)+W_{t}f(x-y)-W_{t}f\left(x\right)|\hfill \\ \hfill & =\left|{\int }_0^1{\partial }_{\theta }\left(W_{t}f(x+\theta y)+W_{t}f(x-\theta y)\right)d\theta \right|\hfill \\ \hfill & =\left|{\int }_0^1\left({\nabla }_u{W}_{t}f\left(u\right){|}_{u=x+\theta y}·y-{\nabla }_v{W}_{t}f\left(v\right){|}_{v=x-\theta y}·y\right)d\theta \right|\hfill \\ \hfill & =\left|{\int }_0^1{\int }_{-1}^1{\partial }_{\lambda }{\nabla }_u{W}_{t}f\left(u\right){|}_{u=x+\lambda \theta y}·yd\lambda d\theta \right|\hfill \\ & =\left|{\int }_0^1{\int }_{-1}^1{\nabla }_u^2{W}_{t}f\left(u\right){|}_{u=x+\lambda \theta y}·\theta {\left|y\right|}^{2}d\lambda d\theta \right|\hfill \\ & \le C{S}_{\alpha }\left[f\right]t^{-1/2+\alpha /4}{\left|y\right|}^2.\hfill \end{array}$$

Thus, by choosing $t={\left|y\right|}^4$, we obtain what we wanted.

For the converse, we assume that $N_{\alpha }\left[f\right]<\infty$ and $M_{\alpha }\left[f\right]<\infty$. Since

$${\displaystyle {\int }_{{\mathbb{R}}^n}{\partial }_y{W}_t\left(y\right)f(x+y)dy={\int }_{{\mathbb{R}}^n}{\partial }_t{W}_t(-y)f(x-y)dy={\int }_{{\mathbb{R}}^n}{\partial }_t{W}_t\left(y\right)f(x-y)dy}$$

and

$${\displaystyle {\int }_{{\mathbb{R}}^n}{\partial }_t{W}_t\left(y\right)dy=0,}$$

we have

$$\begin{array}{cc}\hfill |{\partial }_t{W}_{t}f\left(x\right)|& =\left|\frac{1}{2}{\int }_{{\mathbb{R}}^n}{\partial }_t{W}_t\left(y\right)(f(x-y)+f(x+y)-2f\left(x\right))dy\right|\hfill \\ & \le C{N}_{\alpha }\left[f\right]{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{\left|y\right|}^{4/3}}{t^{1/3}}}{\left|y\right|}^{\alpha }}{t^{n/4+1}}dy\le C{N}_{\alpha }\left[f\right]t^{-1+\alpha /4}.\hfill \end{array}$$

□

By the theorem above and Proposition 3.28 in [12], we know that, when $0<\alpha <1$, if $f\in {\Lambda }_{\alpha /4}^W$, then ${\displaystyle \underset{\left|y\right|>0}{sup}\frac{{\parallel f(·-y)-f(·)\parallel }_{\infty }}{{\left|y\right|}^{\alpha }}<\infty .}$

3. Lipschitz Spaces Associated to ${(-\Delta )}^{\mathbf{2}}+{\mathbf{V}}^{\mathbf{2}}$

In this section, we prove the characterization theorem in a high-order Schrödinger setting, i.e., the proof of Theorem 1.

3.1. Technical Results

Consider the critical radius function defined by

$$\rho \left(x\right)=sup\left\{r>0:\frac{1}{r^{n-2}}{\int }_{B(x,r)}V\left(y\right)dy\le 1\right\}.$$

(7)

First, we recall some properties of the auxiliary function $\rho \left(x\right)$, which will be used later.

Lemma 5

([25]). Let $V\in R{H}_{n/2}\left({\mathbb{R}}^n\right)$. Then there exist $C>0$ and $k_0>1$, such that for all $x,y\in {\mathbb{R}}^n$,

$$\frac{1}{C}\rho \left(x\right){\left(1+\frac{|x-y|}{\rho \left(x\right)}\right)}^{-k_0}\le \rho \left(y\right)\le C\rho \left(x\right){\left(1+\frac{|x-y|}{\rho \left(x\right)}\right)}^{\frac{k_0}{k_0+1}}.$$

In particular, $\rho \left(x\right)\sim \rho \left(y\right)$ if $|x-y|<C\rho \left(x\right)$.

We can prove the following kernel estimates of $e^{-t\mathcal{L}}$.

Lemma 6.

For every $N\in \mathbb{N}$, there exist positive constants C and c such that for all $x,y\in {\mathbb{R}}^n$ and $0<t<\infty$,

(i) 

${\displaystyle |W_t^{ℒ}(x,y\left)\right|\le C{t}^{-\frac{n}{4}}{\left(1+\frac{\sqrt{t}}{\rho {\left(x\right)}^2}+\frac{\sqrt{t}}{\rho {\left(y\right)}^2}\right)}^{-N}{e}^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}}$,

(ii) 

${\displaystyle |{\partial }_t^k{W}_t^{ℒ}(x,y\left)\right|\le C{t}^{-\frac{n+4k}{4}}{\left(1+\frac{\sqrt{t}}{\rho {\left(x\right)}^2}+\frac{\sqrt{t}}{\rho {\left(y\right)}^2}\right)}^{-N}{e}^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}.}$

Proof. 

For $\left(i\right)$, see Theorem 2.5 of [14].

Now we give the proof of $\left(ii\right)$. As $\mathcal{L}={(-\Delta )}^2+V^2$ is a non-negative self-adjoint operator, by the analytic semigroups theory developed in [26] (Section 2 in pp. 67–71), we can extend the semigroup ${\displaystyle {\left\{e^{-t\mathcal{L}}\right\}}_{t>0}}$ to a holomorphic semigroup ${\left\{T_{\xi }\right\}}_{\xi \in {\Delta }_{\pi /4}}$ uniquely. By a similar argument as in [16] (Corollary 6.4), the kernel $W_{\xi }^{ℒ}(x,y)$ of $T_{\xi }$ satisfies

$$|W_{\xi }^{ℒ}(x,y)|\le C_N{(\Re \xi )}^{-n/4}{\left(1+\frac{\sqrt{\Re \xi }}{\rho {\left(x\right)}^2}+\frac{\sqrt{\Re \xi }}{\rho {\left(y\right)}^2}\right)}^{-N}{e}^{-c\frac{{|x-y|}^{4/3}}{{(\Re \xi )}^{1/3}}}.$$

(8)

The Cauchy integral formula combined with (8) gives

$$\left|{\partial }_t^k{W}_t^{ℒ}(x,y)\right|=\left|\frac{k!}{2\pi i}{\int }_{\left|\xi -t\right|=t/10}\frac{W_{\xi }^{ℒ}(x,y)}{{(\xi -t)}^{k+1}}d\xi \right|\le \frac{C_{k,N}}{t^{n/4+k}}{\left(1+\frac{\sqrt{t}}{\rho {\left(x\right)}^2}+\frac{\sqrt{t}}{\rho {\left(y\right)}^2}\right)}^{-N}{e}^{-c\frac{{\left|x-y\right|}^{4/3}}{t^{1/3}}}.$$

Then, we complete the proof. □

Remark 3.

A consequence of Lemma 6 is that ${\displaystyle |{\int }_{{\mathbb{R}}^n}{\partial }_t^k{W}_t^{ℒ}(x,y)dy|\le \frac{C}{t^k}.}$

Lemma 7

(See ([14], Theorem 2.8)). There exist positive constants C and c such that

$$|W_t^{ℒ}(x,y)-W_t(x-y)|\le C{\left(\frac{\sqrt{t}}{\rho {\left(x\right)}^2}\right)}^{2-\frac{n}{q}}{t}^{-n/4}{e}^{-c\frac{{\left|x-y\right|}^{4/3}}{t^{1/3}}},$$

$x,y\in {\mathbb{R}}^n,t>0.$

Remark 4.

Observe that Lemma 5 for $x=0$, i.e.

$$\rho \left(y\right)\le C\rho \left(0\right){\left(1+\frac{\left|y\right|}{\rho \left(0\right)}\right)}^{\lambda },forsome0<\lambda <1,$$

implies that if $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$, then ${(1+|·\left|\right)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$. Therefore, for $0<\alpha <1$, Theorem 8 and the comment in the last part of Section 2 imply that $C_L^{\alpha }$ coincides with the space $Li{p}_L^{\alpha }$(see (1)).

3.2. Controlled Growth at Infinity

In this subsection, we mainly prove the following controlled growth at infinity of the heat semigroup ${\left\{e^{-tℒ}f\right\}}_{t>0}$. From this theorem, we can see that Definition 1 is equivalent to the natural definition of adapted Lipschitz spaces in the Schrödinger setting (see Definition 1.3 in [12]).

Theorem 9.

Let $\alpha >0$ and f be a function such that

$$\parallel {\partial }_t^k{W}_t^{\mathcal{L}}f{\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /4},\mathrm{with}k=[\alpha /4]+1,t>0.$$

Then, ${\displaystyle {\int }_{{\mathbb{R}}^n}{e}^{-\frac{{\left|x\right|}^{4/3}}{t^{1/3}}}\left|f\left(x\right)\right|dx<\infty }$ for every $t>0$ and ${\displaystyle \underset{t\to \infty }{lim}{\partial }_t^{\ell }{W}_t^{ℒ}f\left(x\right)=0}$ for every $\ell \in \mathbb{N}$ if and only if $M_{\alpha }^{ℒ}\left[f\right]<\infty$.

To prove Theorem 9, we need some lemmas and propositions. The proof of these lemmas and propositions are similar to the results in [12], only with some index difference. We leave the proofs of them to the interested reader.

Lemma 8.

If f is a measurable function such that there exists $t_0>0$ for which ${\displaystyle {\int }_{{\mathbb{R}}^n}{e}^{-\frac{{\left|x\right|}^{4/3}}{t_0^{1/3}}}f\left(x\right)dx<\infty .}$ Then, ${\displaystyle \underset{t\to 0}{lim}{W}_t^{\mathcal{L}}f\left(x\right)=f\left(x\right),a.e.x\in {\mathbb{R}}^n.}$ Moreover, $W_t^{ℒ}f\left(x\right)$ belongs to $C^{\infty }((0,+\infty )\times {\mathbb{R}}^n)$.

Proposition 3.

Let $\alpha >0$, $k=[\alpha /4]+1$ and f be a function such that

$${\displaystyle {\int }_{{\mathbb{R}}^n}{e}^{-\frac{{\left|x\right|}^{4/3}}{t^{1/3}}}\left|f\left(x\right)\right|dx<\infty }$$

for every $t>0$ and ${\displaystyle \underset{t\to \infty }{lim}{\partial }_t^{\ell }{W}_t^{ℒ}f\left(x\right)=0}$ for every $\ell \in \mathbb{N}$. Then,

$$\parallel {\partial }_t^k{W}_t^{ℒ}f{\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_{\alpha }{t}^{-k+\alpha /4}$$

if, and only if

$$\parallel {\partial }_t^m{W}_t^{ℒ}{f\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}\le C_m{t}^{-m+\alpha /4},\mathit{for}m\ge k.$$

Moreover, for each m, $C_m$ and $C_{\alpha }$ are comparable.

Lemma 9.

If f is a function such that $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$, for some $\alpha >0$, then, for every $\ell \in \mathbb{N}$ and $M>0$,

$$|{\partial }_t^{\ell }{W}_t^{ℒ}f\left(x\right)|\le C_{\ell }{M}_{\alpha }^{ℒ}\left[f\right]\frac{\rho {\left(x\right)}^{\alpha }}{t^{\ell }}{\left(1+\frac{\sqrt{t}}{\rho {\left(x\right)}^2}\right)}^{-M},$$

$x\in {\mathbb{R}}^n$, $t>0$.

As a direct consequence of Lemma 9, we can obtain the following proposition. Moreover, it corresponds with the “ if ” part of Theorem 9.

Proposition 4.

Let $\alpha >0$ and f be a measurable function. If $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$, then ${\displaystyle {\int }_{{\mathbb{R}}^n}{e}^{-\frac{{\left|x\right|}^{4/3}}{t^{1/3}}}\left|f\left(x\right)\right|dx<\infty }$ for every $t>0$ and ${\displaystyle \underset{t\to \infty }{lim}{\partial }_t^{\ell }{W}_t^{ℒ}f\left(x\right)=0}$ for every $\ell \in \mathbb{N}.$

Then, in order to complete the proof of Theorem 9, we need the following lemma.

Lemma 10.

Let $\alpha >0$ and $k=[\alpha /4]+1$. If $f\in {\Lambda }_{\alpha /4}^{ℒ}$, then for every $j,m\in \mathbb{N}$ such that ${\displaystyle \frac{m}{4}+j\ge k}$, there exists a $C_{m,j}>0$ such that

$${∥\frac{{\partial }_t^j{W}_t^{ℒ}f}{\rho {(·)}^m}∥}_{\infty }\le C_m{S}_{\alpha }^{ℒ}\left[f\right]t^{-(\frac{m}{4}+j)+\frac{\alpha }{4}}.$$

The following proposition corresponds with the “only if” part of Theorem 9. For completeness, we give the proof of this proposition by following the proof in ([12], Proposition 2.21).

Proposition 5.

Let $\alpha >0$ and f be a function such that ${\displaystyle {\int }_{{\mathbb{R}}^n}{e}^{-\frac{{\left|x\right|}^{4/3}}{t^{1/3}}}\left|f\left(x\right)\right|dx<\infty }$ for every $t>0$, ${\displaystyle \underset{t\to \infty }{lim}{\partial }_t^{\ell }{W}_t^{ℒ}f\left(x\right)=0}$ for every $\ell \in \mathbb{N}$, and (5). Then $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$.

Proof. 

By Lemma 8, we have

$$\begin{array}{cc}\hfill \left|f\right(x\left)\right|& \le \underset{0<t<\rho {\left(x\right)}^4}{sup}\left|W_t^{ℒ}f\left(x\right)\right|\hfill \\ & \le \underset{0<t<\rho {\left(x\right)}^4}{sup}|W_t^{ℒ}f\left(x\right)-W_{\rho {\left(x\right)}^4}^{ℒ}f\left(x\right)|+|W_{\rho {\left(x\right)}^4}^{ℒ}f\left(x\right)|\hfill \\ & =I+II.\hfill \end{array}$$

Let $k=[\alpha /4]+1$. If $\alpha$ is not divisible by 4, by Lemma 10 with $j=1$ and $m=4(k-1)$, we have

$$\begin{array}{cc}\hfill I& \le \rho {\left(x\right)}^{4(k-1)}\underset{0<t<\rho {\left(x\right)}^4}{sup}{\int }_t^{\rho {\left(x\right)}^4}\left|\frac{{\partial }_s{W}_s^{ℒ}f\left(x\right)}{\rho {\left(x\right)}^{4(k-1)}}\right|ds\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{4(k-1)}\underset{0<t<\rho {\left(x\right)}^4}{sup}{\int }_t^{\rho {\left(x\right)}^4}{s}^{-k+\alpha /4}ds\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{4(k-1)}\underset{0<t<\rho {\left(x\right)}^4}{sup}({\left(\rho {\left(x\right)}^4\right)}^{-(k-1)+\alpha /4}-t^{-(k-1)+\alpha /4})\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }.\hfill \end{array}$$

When $\alpha$ is divisible by 4, we write

$$\begin{array}{cc}\hfill I& =\underset{0<t<\rho {\left(x\right)}^4}{sup}\left|{\int }_t^{\rho {\left(x\right)}^4}{\partial }_s{W}_s^{ℒ}f\left(x\right)ds\right|\hfill \\ & =\underset{0<t<\rho {\left(x\right)}^4}{sup}\left|{\int }_t^{\rho {\left(x\right)}^4}\left(-{\int }_s^{\rho {\left(x\right)}^4}{\partial }_u^2{W}_u^{ℒ}f\left(x\right)du+{\partial }_v{W}_v^{ℒ}f\left(x\right){|}_{v=\rho {\left(x\right)}^4}\right)ds\right|.\hfill \end{array}$$

By Lemma 10 with $j=4$ and $m=4(k-2)$, since $k=\alpha /4+1$, we get

$$\begin{array}{cc}\hfill |& {\int }_t^{\rho {\left(x\right)}^4}{\int }_s^{\rho {\left(x\right)}^4}{\partial }_u^2{W}_u^{ℒ}f\left(x\right)dudz|=\rho {\left(x\right)}^{4(k-2)}\left|{\int }_t^{\rho {\left(x\right)}^4}{\int }_s^{\rho {\left(x\right)}^4}\frac{{\partial }_u^2{W}_u^{ℒ}f\left(x\right)}{\rho {\left(x\right)}^{4(k-2)}}duds\right|\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha -4}{\int }_t^{\rho {\left(x\right)}^4}{\int }_s^{\rho {\left(x\right)}^4}{u}^{-1}duds\hfill \\ \hfill & =C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha -4}{\int }_t^{\rho {\left(x\right)}^4}(log\left(\rho {\left(x\right)}^4\right)-logs)ds\hfill \\ \hfill & =C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha -4}\left[log\left(\rho {\left(x\right)}^4\right)(\rho {\left(x\right)}^4-t)-(\rho {\left(x\right)}^{4}log\left(\rho {\left(x\right)}^4\right)-\rho {\left(x\right)}^4-tlogt+t)\right]\hfill \\ \hfill & =C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha -4}\left[tlog\left(\frac{t}{\rho {\left(x\right)}^4}\right)+\rho {\left(x\right)}^4-t\right]\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }.\hfill \end{array}$$

For the second summand of I, by Lemma 10 (with $j=1$ and $m=4(k-1)$), we have

$$\begin{array}{cc}\hfill & \underset{0<t<\rho {\left(x\right)}^4}{sup}(\rho {\left(x\right)}^4-t)\left|{\partial }_v{W}_v^{ℒ}f\left(x\right){|}_{v=\rho {\left(x\right)}^4}\right|\hfill \\ \hfill & =\underset{0<t<\rho {\left(x\right)}^4}{sup}(\rho {\left(x\right)}^4-t)\rho {\left(x\right)}^{4(k-1)}\frac{|{\partial }_v{W}_v^{ℒ}f\left(x\right){|}_{v=\rho {\left(x\right)}^4}|}{\rho {\left(x\right)}^{4(k-1)}}\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\underset{0<t<\rho {\left(x\right)}^4}{sup}(\rho {\left(x\right)}^4-t)\rho {\left(x\right)}^{\alpha }{\left(\rho {\left(x\right)}^4\right)}^{-1}\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }.\hfill \end{array}$$

For $II$, by Lemma 10 with $j=0$ and $m=4k$, we get

$$\begin{array}{c}\hfill II=|W_{\rho {\left(x\right)}^4}^{\mathcal{L}}f\left(x\right)|=\left|\frac{W_{\rho {\left(x\right)}^4}^{\mathcal{L}}f\left(x\right)}{\rho {\left(x\right)}^{4k}}\right|\rho {\left(x\right)}^{4k}\le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\left(\rho {\left(x\right)}^4\right)}^{-k+\alpha /4}\rho {\left(x\right)}^{4k}=C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }.\end{array}$$

□

Proposition 6.

Let $1<\alpha <2$, $f\in {\Lambda }_{\alpha /4}^W$ and for a certain ρ associated to a high order Schrödinger operator $ℒ$, we have $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$. For every $i=1,\cdots ,n$, ${\partial }_{x_i}f\in {\Lambda }_{\frac{\alpha -1}{4}}^W$ and $\rho {(·)}^{-(\alpha -1)}{\partial }_{x_i}f\in L^{\infty }\left({\mathbb{R}}^n\right)$. Moreover,

$$\parallel {\partial }_{x_i}{f\parallel }_{{\Lambda }_{\frac{\alpha -1}{4}}^W}\le C\left(S_{\alpha }\left[f\right]+M_{\alpha }^{\mathcal{L}}\left[f\right]\right).$$

3.3. Comparison of Lipschitz Spaces Associated to ${(-\Delta )}^2$ versus $\mathcal{L}$

In this subsection, we shall need the following lemma that can be found in [14].

Lemma 11

(See ([14], Lemma 2.7)). Let $V\in R{H}_q\left({\mathbb{R}}^n\right)$ with $q\in (n/2,\infty )$ and $n\ge 5$. Then there exists a positive constant C such that for all $x,y\in {\mathbb{R}}^n$ and $t\in (0,{\rho }^4\left(x\right)]$,

$${\int }_{{\mathbb{R}}^n}\frac{V^2\left(y\right)}{t^{n/4}}{e}^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}dy\le C{t}^{-1}{\left(\frac{\sqrt{t}}{\rho {\left(x\right)}^2}\right)}^{2-n/q}.$$

Then, we can obtain the following theorem.

Theorem 10.

Let $0<\alpha \le 4-2n/q$, and f be a function such that $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$. Then

$$\parallel {\partial }_t{W}_t^{ℒ}f-{\partial }_t{W}_t{f\parallel }_{\infty }\le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]t^{-1+\alpha /4}.$$

Proof. 

The existence of the derivatives ${\partial }_t{W}_{t}f\left(x\right)$ and ${\partial }_t{W}_t^{\mathcal{L}}f\left(x\right)$ follows from Lemmas 3 and 9. We analyze first the case $t\le \rho {\left(x\right)}^4$. By the properties of semigroups, we have (for details, see [14], (2.7))

$$W_{t}f\left(x\right)-W_t^{\mathcal{L}}f\left(x\right)={\int }_0^t{W}_{t-s}{V}^2{W}_s^{\mathcal{L}}f\left(x\right)ds.$$

Then we obtain

$$\begin{array}{cc}\hfill {\partial }_t(W_{t}f-W_t^{\mathcal{L}}f)& ={\int }_0^{t/2}\partial t{W}_{t-s}{V}^2{W}_s^{\mathcal{L}}fds+{\int }_{t/2}^t{W}_{t-s}{V}^2\partial s{W}_s^{\mathcal{L}}fds+W_{t/2}{V}^2{W}_{t/2}^{\mathcal{L}}f\hfill \\ \hfill & =:A+B+E.\hfill \end{array}$$

On the one hand, we have

$$\begin{array}{cc}\hfill A& ={\int }_0^{t/2}{\int }_{{\mathbb{R}}^n}\partial t{W}_{t-s}(x-z)V^2\left(z\right){\int }_{{\mathbb{R}}^n}{W}_s^{\mathcal{L}}(z,y)f\left(y\right)dydzds\hfill \\ \hfill & ={\int }_0^{t/2}{\int }_{{\mathbb{R}}^n}\partial t{W}_{t-s}(x-z)V^2\left(z\right)\left({\int }_{|y-z|\le \rho \left(z\right)}+\sum _{j=1}^{\infty }{\int }_{2^{j-1}\rho \left(z\right)<|y-z|\le 2^j\rho \left(z\right)}\right)\hfill \\ \hfill & W_s^{\mathcal{L}}(z,y)f\left(y\right)dydzds\hfill \\ \hfill & =:A_0+\sum _{j=1}^{\infty }{A}_j.\hfill \end{array}$$

By using Lemmas 5 and 6 and the fact $t-s\le t\le \rho {\left(x\right)}^4,$ for some $0<\lambda <1$ we have

$$\begin{array}{cc}\hfill |A_0|& \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{t/2}\frac{1}{t-s}{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{{(t-s)}^{1/3}}}}{{(t-s)}^{n/4}}{V}^2\left(z\right){\int }_{|y-z|\le \rho \left(z\right)}\frac{e^{-c\frac{{|z-y|}^{4/3}}{s^{1/3}}}}{s^{n/4}}\rho {\left(z\right)}^{\alpha }dydzds\hfill \\ & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{t/2}\frac{1}{t-s}{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{{(t-s)}^{1/3}}}}{{(t-s)}^{n/4}}{V}^2\left(z\right)\rho {\left(x\right)}^{\alpha }{\left(1+\frac{|x-z|}{\rho \left(x\right)}\right)}^{\lambda \alpha }dzds\hfill \\ \hfill & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{t/2}\frac{1}{t-s}{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{{(t-s)}^{1/3}}}}{{(t-s)}^{n/4}}{e}^{-c\frac{{|x-z|}^{4/3}}{\rho {\left(x\right)}^{4/3}}}{V}^2\left(z\right)\rho {\left(x\right)}^{\alpha }{\left(1+\frac{|x-z|}{\rho \left(x\right)}\right)}^{\lambda \alpha }dzds\hfill \\ \hfill & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }{\int }_0^{t/2}\frac{1}{t-s}{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{{(t-s)}^{1/3}}}}{{(t-s)}^{n/4}}{V}^2\left(z\right)dzds\hfill \\ & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }{\int }_0^{t/2}\frac{1}{{(t-s)}^2}{\left(\frac{\sqrt{t-s}}{\rho {\left(x\right)}^2}\right)}^{\alpha /2}ds\hfill \\ \hfill & \le C{M}_{\alpha }^{}\left[f\right]t^{-1+\alpha /4}.\hfill \end{array}$$

In the last two lines, we used Lemma 11 and the fact that ${\displaystyle 0<\alpha /2<\alpha \le 2-n/q}$. We shall deal with the summation of $A_j$. Observe that by using Lemma 5 with $2N>\lambda \alpha$ and Lemma 6, we have

$$\begin{array}{cc}\hfill & \left|{\int }_{2^{j-1}\rho \left(z\right)<|y-z|\le 2^j\rho \left(z\right)}{W}_s^{\mathcal{L}}(z,y)f\left(y\right)dy\right|\hfill \\ & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_{2^{j-1}\rho \left(z\right)<|y-z|\le 2^j\rho \left(z\right)}\frac{e^{-c\frac{{|z-y|}^{4/3}}{s^{1/3}}}}{s^{n/4}}{\left(1+\frac{\sqrt{s}}{\rho {\left(z\right)}^2}+\frac{\sqrt{s}}{\rho {\left(y\right)}^2}\right)}^{-N}\rho {\left(y\right)}^{\alpha }dy\hfill \\ & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_{2^{j-1}\rho \left(z\right)<|y-z|\le 2^j\rho \left(z\right)}\frac{e^{-c\frac{{|z-y|}^{4/3}}{s^{1/3}}}}{s^{n/4}}{\left(\frac{\sqrt{s}}{\rho {\left(z\right)}^2}\right)}^{-N}\rho {\left(z\right)}^{\alpha }{2}^{j\lambda \alpha }dy\hfill \\ & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_{2^{j-1}\rho \left(z\right)<|y-z|\le 2^j\rho \left(z\right)}{e}^{-c\frac{{\left(2^j\rho \left(z\right)\right)}^{4/3}}{s^{1/3}}}{\left(\frac{2^{2j}\rho {\left(z\right)}^2}{\sqrt{s}}\right)}^N{2}^{-2jN}\frac{e^{-c\frac{{|z-y|}^{4/3}}{s^{1/3}}}}{s^{n/4}}\rho {\left(z\right)}^{\alpha }{2}^{j\lambda \alpha }dy\hfill \\ & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(z\right)}^{\alpha }{2}^{-j(2N-\lambda \alpha )}.\hfill \end{array}$$

The rest of the computation is the same as in the case of $A_0$. Now we analyze B.

$$\begin{array}{cc}\hfill B& ={\int }_{t/2}^t{\int }_{{\mathbb{R}}^n}{W}_{t-s}(x-z)V^2\left(z\right){\int }_{{\mathbb{R}}^n}\partial s{W}_s^{\mathcal{L}}(z,y)f\left(y\right)dydzds\hfill \\ \hfill & ={\int }_{t/2}^t{\int }_{{\mathbb{R}}^n}{W}_{t-s}(x-z)V^2\left(z\right)\left({\int }_{|y-z|\le \rho \left(z\right)}+\sum _{j=1}^{\infty }{\int }_{2^{j-1}\rho \left(z\right)<|y-z|\le 2^j\rho \left(z\right)}\right)\hfill \\ \hfill & \partial s{W}_s^{\mathcal{L}}(z,y)f\left(y\right)dydzds\hfill \\ \hfill & =B_0+\sum _{j=1}^{\infty }{B}_j.\hfill \end{array}$$

Analogously to $A_0$, we have

$$\begin{array}{cc}\hfill |B_0|& \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_{t/2}^t{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{{(t-s)}^{1/3}}}}{{(t-s)}^{n/4}}{V}^2\left(z\right){\int }_{|y-z|\le \rho \left(z\right)}\frac{e^{-c\frac{{|z-y|}^{4/3}}{s^{1/3}}}}{s^{n/4+1}}\rho {\left(z\right)}^{\alpha }dydzds\hfill \\ \hfill & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_{t/2}^t{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{{(t-s)}^{1/3}}}}{{(t-s)}^{n/4+1}}{V}^2\left(z\right)\rho {\left(x\right)}^{\alpha }{\left(1+\frac{|x-z|}{\rho \left(x\right)}\right)}^{\lambda \alpha }dzds.\hfill \end{array}$$

We can continue as in the case of $A_0$. $B_j$ is parallel to the case $A_j$ with the obvious changes. Finally, we analyze E.

$$\begin{array}{cc}\hfill & E=C{\int }_{{\mathbb{R}}^n}{W}_{t/2}(x-z)V^2\left(z\right){\int }_{{\mathbb{R}}^n}{W}_{t/2}^{\mathcal{L}}(z,y)f\left(y\right)dydz\hfill \\ \hfill & =C{\int }_{{\mathbb{R}}^n}{W}_{t/2}(x-z)V^2\left(z\right)\left({\int }_{|y-z|\le \rho \left(z\right)}+\sum _{j=1}^{\infty }{\int }_{2^{j-1}\rho \left(z\right)<|y-z|\le 2^j\rho \left(z\right)}\right)W_{t/2}^{\mathcal{L}}(z,y)f\left(y\right)dydz\hfill \\ \hfill & =E_0+\sum _j{E}_j.\hfill \end{array}$$

Regarding $E_0$, we use Lemmas 5, 6 and 11 to obtain

$$\begin{array}{cc}\hfill |E_0|& \le C{M}_{\alpha }^{}\left[f\right]{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{t^{1/3}}}}{t^{n/4}}{V}^2\left(z\right){\int }_{|y-z|\le \rho \left(z\right)}\frac{e^{-c\frac{{|z-y|}^{4/3}}{t^{1/3}}}}{t^{n/4}}\rho {\left(z\right)}^{\alpha }dydz\hfill \\ \hfill & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{t^{1/3}}}}{t^{n/4}}{V}^2\left(z\right)\rho {\left(x\right)}^{\alpha }{\left(1+\frac{|x-z|}{\rho \left(x\right)}\right)}^{\lambda \alpha }dydz\hfill \\ & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{t^{1/3}}}}{t^{n/4}}{e}^{-c\frac{{|x-z|}^{4/3}}{\rho {\left(x\right)}^{4/3}}}{V}^2\left(z\right)\rho {\left(x\right)}^{\alpha }{\left(1+\frac{|x-z|}{\rho \left(x\right)}\right)}^{\lambda \alpha }dz\hfill \\ \hfill & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }{\int }_{{\mathbb{R}}^n}\frac{e^{-c\frac{{|x-z|}^{4/3}}{t^{1/3}}}}{t^{n/4}}{V}^2\left(z\right)dz\hfill \\ \hfill & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }\frac{1}{t}{\left(\frac{\sqrt{t}}{\rho {\left(x\right)}^2}\right)}^{\alpha /2}\hfill \\ \hfill & \le C{t}^{-1+\alpha /4}.\hfill \end{array}$$

$E_j$, $j=1,2,\cdots$, can be handled similarly to $A_j$ and $B_j$ with some obvious changes.

Now we consider the case $t\ge \rho {\left(x\right)}^4$. From Lemmas 5 and 6,

$$\begin{array}{c}|{\partial }_t{W}_{t}f\left(x\right)-{\partial }_t{W}_t^{\mathcal{L}}f\left(x\right)|\le 2C{\int }_{{\mathbb{R}}^n}\frac{1}{t}\frac{e^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}}{t^{n/4}}\left|f\left(y\right)\right|dy\hfill \\ \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_{{\mathbb{R}}^n}\frac{1}{t}\frac{e^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}}{t^{n/4}}\rho {\left(y\right)}^{\alpha }dy\hfill \\ =C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\frac{1}{t}{\int }_{|x-y|\le \rho \left(x\right)}\frac{e^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}}{t^{n/4}}\rho {\left(x\right)}^{\alpha }dy\hfill \\ \hspace{1em}+C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\frac{1}{t}\sum _j{\int }_{2^{j-1}\rho \left(x\right)<|x-y|\le 2^j\rho \left(x\right)}\frac{e^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}}{t^{n/4}}\rho {\left(y\right)}^{\alpha }dy\hfill \\ \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\frac{1}{t}\rho {\left(x\right)}^{\alpha }\hfill \\ \hspace{1em}+C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\frac{1}{t}\sum _j{\int }_{2^{j-1}\rho \left(x\right)<|x-y|\le 2^j\rho \left(x\right)}\frac{e^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}}{t^{n/4}}{e}^{-c\frac{{\left(2^j\rho \left(x\right)\right)}^{4/3}}{t^{1/3}}}\rho {\left(x\right)}^{\alpha }{\left(2^j\right)}^{\lambda \alpha }dy\hfill \\ \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]t^{-1+\alpha /4}\hfill \\ \hspace{1em}+C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\frac{1}{t}{t}^{\alpha /4}\sum _j{2}^{j(\lambda -1)\alpha }{\int }_{2^{j-1}\rho \left(x\right)<|x-y|\le 2^j\rho \left(x\right)}\frac{e^{-c\frac{{|x-y|}^{4/3}}{t^{1/3}}}}{t^{n/4}}dy\hfill \\ \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]t^{-1+\alpha /4}.\hfill \end{array}$$

□

As a consequence of the previous results, the following theorem can be obtained.

Theorem 11.

For $0<\alpha \le 2-n/q$, a measurable function $f\in {\Lambda }_{\alpha /4}^{\mathcal{L}}$ if, and only if, $f\in {\Lambda }_{\alpha /4}^W$ and $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right).$

By Theorems 8 and 11, we obtain the proof of Theorem 1.

4. Proof of Regularity Properties

In this section, we give the proof of Theorems 2–5. First, we prove a lemma in the following.

Lemma 12.

Let $\beta >0$ and ${\mathcal{T}}_{\beta }$ be either the operator ${(Id+\mathcal{L})}^{-\beta /4}$ or the operator ${\mathcal{L}}^{-\beta /4}$. If f is a function such that $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$ for some $\alpha >0$, then ${\mathcal{T}}_{\beta }f\left(x\right)$ is well defined and satisfies

$$M_{\alpha +\beta }^{\mathcal{L}}[{\mathcal{T}}_{\beta }f]\le C{M}_{\alpha }^{\mathcal{L}}\left[f\right].$$

Moreover, if $f\in L^{\infty }\left({\mathbb{R}}^n\right)$, then ${\mathcal{T}}_{\beta }f\left(x\right)$ is well defined and

$$M_{\beta }^{\mathcal{L}}[{\mathcal{T}}_{\beta }f]\le C{\parallel f\parallel }_{\infty }.$$

Proof. 

If $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$ for some $\alpha >0$, then by Lemma 9, we obtain

$$\begin{array}{cc}\hfill & |{(Id+\mathcal{L})}^{-\beta /4}f\left(x\right)|=|\frac{1}{\Gamma (\beta /4)}{\int }_0^{\infty }{e}^{-s}{W}_s^{\mathcal{L}}\left(x\right)s^{\beta /4}\frac{ds}{s}|\hfill \\ \hfill & \le C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{\rho {\left(x\right)}^4}\rho {\left(x\right)}^{\alpha }{s}^{\beta /4}\frac{ds}{s}+C{M}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_{\rho {\left(x\right)}^4}^{\infty }\rho {\left(x\right)}^{\alpha }{\left(\frac{\rho {\left(x\right)}^4}{s}\right)}^{\beta /4+1}{s}^{\beta /4}\frac{ds}{s}\hfill \\ & =C{M}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha +\beta },x\in {\mathbb{R}}^n.\hfill \end{array}$$

The same estimate works for ${\mathcal{L}}^{-\beta /4}f.$ The proof in the second case runs parallel since Lemma 9 has an obvious version for bounded functions. □

Now, we are in a position to prove Theorem 2.

Proof of Theorem 2.

We prove only (i), and estimate (ii) can be proved similarly. Let $f\in {\Lambda }_{\alpha /4}^{\mathcal{L}}$. Lemma 9 with $\ell =0$ together with Fubini’s theorem allows us to obtain

$${\displaystyle W_t^{\mathcal{L}}({(Id+)}^{-\beta /4}f)\left(x\right)=\frac{1}{\Gamma (\beta /4)}{\int }_0^{\infty }{e}^{-s}{W}_t^{\mathcal{L}}(W_s^{\mathcal{L}}f)\left(x\right)s^{\beta /4}\frac{ds}{s}.}$$

By Lemma 10 with $j=1$ and $m\in \mathbb{N}$ such that $[\alpha /4+\beta /4]+1\le 1+m/4$, we have

$$\begin{array}{c}{\int }_0^{\infty }\left|e^{-s}{\partial }_t{W}_t^{\mathcal{L}}\right(W_s^{\mathcal{L}}f\left)\left(x\right)\right|s^{\beta /4}\frac{ds}{s}={\int }_0^{\infty }\left|e^{-s}{\partial }_w{W}_w^{\mathcal{L}}f\left(x\right){|}_{w=t+s}\right|s^{\beta /4}\frac{ds}{s}\hfill \\ \hfill \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{\infty }{e}^{-s}\rho {\left(x\right)}^m{(t+s)}^{-(m/4+1)+\alpha /4}{s}^{\beta /4}\frac{ds}{s}.\end{array}$$

We can bound the function in the last integral by a uniform (in a neighborhood of t) integrable function (of s). This allows us to interchange the derivative with respect to t and the integral with respect to s in the above expression.

Let $\ell =[\alpha /4+\beta /4]+1$. By using the above arguments again and using the hypothesis, we have

$$\begin{array}{cc}\hfill & |{\partial }_t^{\ell }{W}_t^{\mathcal{L}}({(Id+\mathcal{L})}^{-\beta /4}f\left(x\right)\left)\right|=\left|\frac{1}{\Gamma (\beta /4)}{\int }_0^{\infty }{e}^{-s}{\partial }_t^{\ell }{W}_t^{\mathcal{L}}\left(W_s^{\mathcal{L}}f\right)\left(x\right)s^{\beta /4}\frac{ds}{s}\right|\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{\infty }{e}^{-s}\left|{\partial }_w^{\ell }{W}_w^{\mathcal{L}}f\left(x\right){|}_{w=t+s}\right|s^{\beta /4}\frac{ds}{s}\hfill \\ & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{\infty }{e}^{-s}{(t+s)}^{-\ell +\alpha /4}{s}^{\beta /4}\frac{ds}{s}\hfill \\ & \stackrel{\frac{s}{t}=u}{\le }C{S}_{\alpha }^{\mathcal{L}}\left[f\right]t^{\alpha /4+\beta /4-\ell }{\int }_0^{\infty }\frac{u^{\beta /4}{e}^{-tu}}{{(1+u)}^{\ell -\alpha /4}}\frac{du}{u}\hfill \\ & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]t^{\alpha /4+\beta /4-\ell }.\hfill \end{array}$$

When $f\in L^{\infty }\left({\mathbb{R}}^n\right)$, we apply Lemma 6 and we obtain for $\ell =[\beta /4]+1$ that ${\displaystyle |{\partial }_t^{\ell }{W}_t^{\mathcal{L}}{W}_s^{\mathcal{L}}f\left(x\right)|\le C\frac{{\parallel f\parallel }_{\infty }}{t^{\ell }}}$. Then we can proceed as before.

Together with Lemma 12, we end the proof of the theorem. □

In order to prove Theorem 3, we need a lemma as follows.

Lemma 13.

Let $0<\beta <\alpha$ and f be a function in the space ${\Lambda }_{\alpha /4}^{\mathcal{L}}$. Then ${\mathcal{L}}^{\beta /4}f$ is well defined and

$$M_{\alpha -\beta }^{\mathcal{L}}\left[{\mathcal{L}}^{\beta /4}f\right]\le C_{\alpha ,\beta }{\parallel f\parallel }_{{\Lambda }_{\alpha /4}^{\mathcal{L}}}.$$

Proof. 

We can write

$$\begin{array}{c}\hfill |{\mathcal{L}}^{\beta /4}f\left(x\right)|=\left|\frac{1}{c_{\beta }}\left({\int }_0^{\rho {\left(x\right)}^4}+{\int }_{\rho {\left(x\right)}^4}^{\infty }\right){(e^{-s\mathcal{L}}-Id)}^{[\beta /4]+1}f\left(x\right)\frac{ds}{s^{1+\beta /4}}\right|=|I+II|.\end{array}$$

As $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$, by Lemma 9 we have

$$\left|II\right|\le C{M}_{\alpha }^{\mathcal{L}}[f]{\int }_{\rho {\left(x\right)}^4}^{\infty }\rho {\left(x\right)}^{\alpha }\frac{ds}{s^{1+\beta /4}}=C{M}_{\alpha }^{\mathcal{L}}[f]\rho {\left(x\right)}^{\alpha -\beta }.$$

Now we shall estimate $\left|I\right|$. Let $\ell =[\beta /4]+1$. We have

$${\displaystyle {(e^{-s\mathcal{L}}-Id)}^{[\beta /4]+1}f\left(x\right)=\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^s\cdots {\int }_0^s}}{\partial }_{s_1}\cdots {\partial }_{s_{\ell }}{W}_{s_1+\cdots +s_{\ell }}^{\mathcal{L}}f\left(x\right)d{s}_1\cdots d{s}_{\ell }.}$$

If $\beta /4<\alpha /4<\ell$, then by Lemma 10 (with $k:=[\alpha /4]+1=\ell$, $j=l$ and $m=0$) we have

$$|{(e^{-s\mathcal{L}}-Id)}^{\ell }f\left(x\right)|\le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^s\cdots {\int }_0^s}}\frac{d{s}_{\ell }\cdots d{s}_1}{{(s_1+\cdots +s_{\ell })}^{\ell -\alpha /4}}\le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]s^{\alpha /4}.$$

Therefore,

$${\displaystyle \left|I\right|\le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{\rho {\left(x\right)}^4}{s}^{\alpha /4}\frac{ds}{s^{1+\beta /4}}=C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha -\beta }.}$$

If $\ell <\alpha /4$, then $k>\ell$ and by Lemma 10 we obtain, for $0<s\le \rho {\left(x\right)}^4$,

$$\begin{array}{cc}\hfill |{(e^{-s\mathcal{L}}-Id)}^{\ell }f\left(x\right)|& =|\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^s\cdots {\int }_0^s}}(-{\int }_{s_1+\cdots +s_{\ell }}^{\ell \rho {\left(x\right)}^4}\frac{{\partial }_u^{\ell +1}{W}_u^{\mathcal{L}}f\left(x\right)}{{\left(\rho {\left(x\right)}^4\right)}^{k-(\ell +1)}}du{\left(\rho {\left(x\right)}^4\right)}^{k-(\ell +1)}\hfill \\ \hfill & +\frac{{\partial }_{\nu }^{\ell }{W}_{\nu }^{\mathcal{L}}f\left(x\right){|}_{\nu =\ell \rho {\left(x\right)}^4}}{{\left(\rho {\left(x\right)}^4\right)}^{k-\ell }}{\left(\rho {\left(x\right)}^4\right)}^{k-\ell })d{s}_1\cdots d{s}_{\ell }|\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\left(\rho {\left(x\right)}^4\right)}^{k-(\ell +1)}\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^s\cdots {\int }_0^s}}{\int }_{s_1+\cdots +s_{\ell }}^{\ell {\left(\rho \left(x\right)\right)}^4}{u}^{-k+\alpha /4}dud{s}_1\cdots d{s}_{\ell }\hfill \\ & +C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{(\ell \rho {\left(x\right)}^4)}^{-k+\alpha /4}{\left(\rho {\left(x\right)}^4\right)}^{k-\ell }{s}^{\ell }.\hfill \end{array}$$

If $\alpha$ is not divisible by 4, then we have, for $0<s\le \rho {\left(x\right)}^4$,

$$\begin{array}{c}|{(e^{-s\mathcal{L}}-Id)}^{\ell }f\left(x\right)|\hfill \\ \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\left(\rho {\left(x\right)}^4\right)}^{k-(\ell +1)}\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^s\cdots {\int }_0^s}}({(s_1+\cdots +s_{\ell })}^{-k+\alpha /4+1}\hfill \\ +{(\ell \rho {\left(x\right)}^4)}^{-k+\alpha /4+1})d{s}_1\cdots d{s}_{\ell }+C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\left(\rho {\left(x\right)}^4\right)}^{\alpha /4-\ell }{s}^{\ell }\hfill \\ \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\left({\left(\rho {\left(x\right)}^4\right)}^{k-(\ell +1)}{s}^{-k+\alpha /4+\ell +1}+{\left(\rho {\left(x\right)}^4\right)}^{-\ell +\alpha /4}{s}^{\ell }\right).\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill {\displaystyle \left|I\right|}& \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\left({\left(\rho {\left(x\right)}^4\right)}^{k-\ell -1}{\int }_0^{\rho {\left(x\right)}^4}{s}^{-k+\alpha /4+\ell -\beta /4}ds+{\left(\rho {\left(x\right)}^4\right)}^{-\ell +\alpha /4}{\int }_0^{\rho {\left(x\right)}^4}{s}^{\ell -\beta /4-1}ds\right)\hfill \\ & =C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha -\beta }.\hfill \end{array}$$

If $\alpha$ is divisible by 4, then $k=\alpha /4+1$ and, for $0<s\le \rho {\left(x\right)}^4$,

$$\begin{array}{cc}\hfill \left|\right(e^{-s\mathcal{L}}& {-Id)}^{\ell }f\left(x\right)|\hfill \\ \hfill & \le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\left(\rho {\left(x\right)}^4\right)}^{\alpha /4-\ell }\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^s\cdots {\int }_0^s}}(log(\ell {\left(\rho \left(x\right)\right)}^4)-log(s_1+\cdots +s_{\ell }))d{s}_1\cdots d{s}_{\ell }\hfill \\ & +C{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\left(\rho {\left(x\right)}^4\right)}^{\alpha /4-\ell }{s}^{\ell }.\hfill \end{array}$$

In order to solve the last integral, by changing variables ${\tilde{s}}_1=s_1,{\tilde{s}}_2=s_2,\cdots ,{\tilde{s}}_{\ell -1}=s_{\ell -1},\tilde{s}=s_1+\cdots +s_{\ell }$. Then we can proceed as in the proof of Proposition 5. We obtain, in this case,

$${\int }_0^{\rho {\left(x\right)}^4}\frac{{(e^{-s\mathcal{L}}-Id)}^{\ell }f\left(x\right)}{s^{1+\beta /4}}ds\le C{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha -\beta }.$$

Combining the estimates of I and $II$, we can finish the proof of Lemma 13. □

Proof of Theorem 3.

If $\ell =\left[\beta /4\right]+1$ and $m=\left[(\alpha -\beta )/4\right]+1$, then

$$m+\ell =\left[(\alpha -\beta )/4\right]+1+[\beta /4]+1>\alpha /4-\beta /4+\beta /4=\alpha /4.$$

Since $m+\ell \in \mathbb{N}$, we have $m+\ell \ge [\alpha /4]+1.$

By the same argument in the proof of Lemma 13, we have

$$\begin{array}{cc}\hfill & |{\partial }_t^m{W}_t^{\mathcal{L}}(\beta /4f\left)\left(x\right)\right|\hfill \\ & =|c_{\beta }{\int }_0^{\infty }{\partial }_t^m{W}_t^{\mathcal{L}}\left(\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^s\cdots {\int }_0^s}}{\partial }_{\nu }^{\ell }{W}_{\nu }^{\mathcal{L}}{|}_{\nu =s_1+\cdots +s_{\ell }}f\left(x\right)d{s}_1\cdots d{s}_{\ell }\right)\frac{ds}{s^{1+\beta /4}}|\hfill \\ \hfill & =|c_{\beta }{\int }_0^{\infty }\left(\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^s\cdots {\int }_0^s}}{\partial }_{\nu }^{m+\ell }{W}_{\nu }^{\mathcal{L}}{|}_{\nu =t+s_1+\cdots +s_{\ell }}f\left(x\right)d{s}_1\cdots d{s}_{\ell }\right)\frac{ds}{s^{1+\beta /4}}|\hfill \\ & \le C_{\beta }{S}_{\alpha }^{\mathcal{L}}[f]{\int }_0^{\infty }\left(\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^s\cdots {\int }_0^s}}{(t+s_1+\cdots +s_{\ell })}^{-(m+\ell )+\alpha /4}d{s}_1\cdots d{s}_{\ell }\right)\frac{ds}{s^{1+\beta /4}}\hfill \\ & =C_{\beta }{S}_{\alpha }^{\mathcal{L}}[f]{\int }_0^t(\cdots )\frac{ds}{s^{1+\beta /4}}+C_{\beta }{S}_{\alpha }^{\mathcal{L}}[f]{\int }_t^{\infty }(\cdots )\frac{ds}{s^{1+\beta /4}}\hfill \\ \hfill & =:C_{\beta }{S}_{\alpha }^{\mathcal{L}}[f](A+B).\hfill \end{array}$$

Then, we will estimate A and B separately. By change of variables, we obtain

$$\begin{array}{cc}\hfill A& =C_{\beta }{t}^{-m+\alpha /4}{\int }_0^t\underset{\begin{array}{c}\ell \end{array}}{\underbrace{{\int }_0^{s/t}\cdots {\int }_0^{s/t}}}{(1+s_1+\cdots +s_{\ell })}^{-(m+\ell )+\alpha /4}d{s}_1\cdots d{s}_{\ell }\frac{ds}{s^{1+\beta /4}}\hfill \\ & \le C_{\beta }{t}^{-m+\alpha /4}{\int }_0^t{\left(\frac{s}{t}\right)}^{\ell }\frac{ds}{s^{1+\beta /4}}=C_{\beta }{t}^{-m+\alpha /4-\ell }{\int }_0^t\frac{ds}{s^{1+\beta /4-\ell }}=C_{\beta }{t}^{-m+(\alpha -\beta )/4}.\hfill \end{array}$$

On the other hand,

$$\begin{array}{cc}\hfill B& \le {\int }_t^{\infty }\sum _{j=0}^{\ell }\frac{C_j}{{(t+js)}^{m-\alpha /4}}\frac{ds}{s^{1+\beta /4}}=\sum _{j=0}^{\ell }{\int }_t^{\infty }\frac{C_j}{{(t+js)}^{m-\alpha /4}}\frac{ds}{s^{1+\beta /4}}\hfill \\ & \le \sum _{j=0}^{\ell }{C}_j{t}^{-m+(\alpha -\beta )/4}\le C{t}^{-m+(\alpha -\beta )/4}.\hfill \end{array}$$

The last inequalities can be obtained by observing that $t\le t+js\le (1+\ell )s$ inside the integrals together with the discussion about the sign of $m-\alpha /4$. Together with Lemma 13, we complete the proof of this theorem. □

Now, we will prove Theorem 4.

Proof of Theorem 4.

Let $0<\alpha \le 1-n/q$ and $f\in {\Lambda }_{\alpha /4}^{\mathcal{L}}$. We have ${\mathcal{L}}^{-1/4}f\in {\Lambda }_{\frac{\alpha +1}{4}}^{\mathcal{L}}$ by Theorem 2. By Theorem 11, this means that ${\mathcal{L}}^{-1/4}f\in {\Lambda }_{\frac{\alpha +1}{4}}^W$ and $\rho {(·)}^{-(\alpha +1)}{\mathcal{L}}^{-1/4}f\in L^{\infty }\left({\mathbb{R}}^n\right)$, where ${\Lambda }_{\frac{\alpha +1}{4}}^W$ denotes the adapted Lipschitz space associated to the biharmonic operator ${(-\Delta )}^2$ (see Definition 3). Therefore, by Proposition 6, we obtain ${\mathcal{R}}_{i}f={\partial }_{x_i}\left({\mathcal{L}}^{-1/4}f\right)\in {\Lambda }_{\alpha /4}^W$ and $\rho {(·)}^{-\alpha }{\mathcal{R}}_{i}f\in L^{\infty }\left({\mathbb{R}}^n\right)$. Thus, Theorem 11 gives the first statement of the theorem.

Suppose $1<\alpha \le 2-n/q$ and $f\in {\Lambda }_{\alpha /4}^{\mathcal{L}}$. By Theorem 11, this means that $f\in {\Lambda }_{\alpha /4}^W$ and $\rho {(·)}^{-\alpha }f\in L^{\infty }\left({\mathbb{R}}^n\right)$. By Proposition 6, we obtain ${\partial }_{x_i}f\in {\Lambda }_{\frac{\alpha -1}{4}}^W$ and $\rho {(·)}^{-(\alpha -1)}{\partial }_{x_i}f\in L^{\infty }\left({\mathbb{R}}^n\right)$. Again, by Theorem 11, this means that ${\partial }_{x_i}f\in {\Lambda }_{\frac{\alpha -1}{4}}^{\mathcal{L}}$ and by Theorem 2, we obtain $R_{i}f={\mathcal{L}}^{-1/4}\left({\partial }_{x_i}f\right)\in {\Lambda }_{\alpha /4}^{\mathcal{L}}$. □

At last, we complete the proof of Theorem 5.

Proof of Theorem 5.

Lemmas 9 and 10 guarantee the integrability of ${\partial }_s\left(W_s^{\mathcal{L}}f\left(x\right)\right)$ as a function of $s.$ Hence, we can write

$$\begin{array}{c}m\left(\mathcal{L}\right)f\left(x\right)={\int }_0^{\infty }(-{\partial }_s\left(W_s^{\mathcal{L}}f\left(x\right)\right)g\left(s\right)ds\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =\left({\int }_0^{\rho {\left(x\right)}^4}+{\int }_{\rho {\left(x\right)}^4}^{\infty }\right)-{\partial }_s\left(W_s^{\mathcal{L}}f\left(x\right)\right)g\left(s\right)ds=I+II.\end{array}$$

Using Lemma 9, we obtain

$$\begin{array}{cc}\hfill \left|II\right|& \le {C\parallel g\parallel }_{\infty }{M}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }{\int }_{\rho {\left(x\right)}^4}^{\infty }\frac{1}{s}{\left(1+\frac{s}{\rho {\left(x\right)}^4}\right)}^{-M}ds\hfill \\ & ={C\parallel g\parallel }_{\infty }{M}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }{\int }_1^{\infty }\frac{1}{u{(1+u)}^M}du\le C{\parallel g\parallel }_{\infty }{M}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }.\hfill \end{array}$$

Now we estimate I. Let $k=[\alpha /4]+1$. If $\alpha$ is not divisible by 4, by Lemma 10, we obtain

$$\begin{array}{c}\left|I\right|\le \rho {\left(x\right)}^{4(k-1)}{\int }_0^{\rho {\left(x\right)}^4}\frac{|{\partial }_s{W}_s^{\mathcal{L}}f\left(x\right)|}{\rho {\left(x\right)}^{4(k-1)}}\left|g\left(s\right)\right|ds\hfill \\ \hfill \le {C\parallel g\parallel }_{\infty }{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{4(k-1)}{\int }_0^{\rho {\left(x\right)}^4}{s}^{-k+\alpha /4}ds=C{\parallel g\parallel }_{\infty }{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }.\end{array}$$

If $\alpha$ is divisible by 4, by Lemma 10 we have

$$\begin{array}{cc}\hfill \left|I\right|& =\left|{\int }_0^{\rho {\left(x\right)}^4}\left({\int }_s^{\rho {\left(x\right)}^4}\frac{{\partial }_u^2{W}_u^{\mathcal{L}}f\left(x\right)}{\rho {\left(x\right)}^{4(k-2)}}du\rho {\left(x\right)}^{4(k-2)}-\frac{{\partial }_{\nu }{W}_{\nu }^{\mathcal{L}}f\left(x\right){|}_{\nu =\rho {\left(x\right)}^4}}{\rho {\left(x\right)}^{4(k-1)}}\rho {\left(x\right)}^{4(k-1)}\right)g\left(s\right)ds\right|\hfill \\ \hfill & \le {C\parallel g\parallel }_{\infty }{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{\rho {\left(x\right)}^4}\left({\int }_s^{\rho {\left(x\right)}^4}{u}^{-1}du\rho {\left(x\right)}^{\alpha -4}+\rho {\left(x\right)}^{\alpha -4}\right)ds\hfill \\ \hfill & ={C\parallel g\parallel }_{\infty }{S}_{\alpha }^{\mathcal{L}}\left[f\right]\left({\int }_0^{\rho {\left(x\right)}^4}(log\left(\rho {\left(x\right)}^4\right)-logs)\rho {\left(x\right)}^{\alpha -4}ds+\rho {\left(x\right)}^{\alpha }\right)\hfill \\ \hfill & \le {C\parallel g\parallel }_{\infty }{S}_{\alpha }^{\mathcal{L}}\left[f\right]\rho {\left(x\right)}^{\alpha }.\hfill \end{array}$$

Up to now, we have shown that

$$M_{\alpha }^{\mathcal{L}}\left[m\left(\mathcal{L}\right)\left(f\right)\right]\le C{\parallel f\parallel }_{{\Lambda }_{\alpha /4}^{\mathcal{L}}}.$$

Now we want to see that $\parallel {\partial }_t^k{W}_t^{\mathcal{L}}m{\left(\mathcal{L}\right)\left(f\right)\parallel }_{\infty }\le C{t}^{-k+\alpha /4}$. Fubini’s theorem together with Lemmas 9 and 10 allow us to interchange the integral with derivatives and kernels. Then,

$$\begin{array}{cc}\hfill & |{\partial }_t^k{W}_t^{\mathcal{L}}m\left(\mathcal{L}\right)\left(f\right)\left(x\right)|=\left|{\int }_0^{\infty }{\partial }_u^{k+1}{W}_u^{\mathcal{L}}f\left(x\right){|}_{u=t+s}a\left(s\right)ds\right|\hfill \\ \hfill & \le {C\parallel g\parallel }_{\infty }{S}_{\alpha }^{\mathcal{L}}\left[f\right]{\int }_0^{\infty }\frac{1}{{(t+s)}^{k+1-\alpha /4}}ds\hfill \\ \hfill & ={C\parallel g\parallel }_{\infty }{S}_{\alpha }^{\mathcal{L}}\left[f\right]t^{-(k+1)+\alpha /4}{\int }_0^{\infty }\frac{t}{{(1+r)}^{k+1-\alpha /4}}dr\hfill \\ \hfill & \le {C\parallel g\parallel }_{\infty }{S}_{\alpha }^{\mathcal{L}}\left[f\right]t^{-k+\alpha /4}.\hfill \end{array}$$

We complete the proof of the theorem. □

5. Conclusions

In this paper, we prove that, when $0<\alpha \le 2-n/q$, the adapted Lipschitz spaces ${\Lambda }_{\alpha /4}^{\mathcal{L}}$ associated to $\mathcal{L}={(-\Delta )}^2+V^2$ that we considered are equivalent to the Lipschitz space $C_L^{\alpha }$ associated to the Schrödinger operator $L=-\Delta +V$. We also establish the regularity properties of the fractional powers (positive and negative) of the operator $ℒ$, Schrödinger Riesz transforms, Bessel potentials and multipliers of Laplace transforms type associated to the high-order Schrödinger operators $ℒ$.