# Asymptotic estimates of solution to damped fractional wave equation

## Abstract

It is known that the damped fractional wave equation has the diffusive structure as $$t\rightarrow \infty$$. Let $$u(t,x)=e^{-t}\cosh (t\sqrt{L})f(x)+e^{-t} \frac{\sinh (t\sqrt{L})}{\sqrt{L}}(f(x)+g(x))$$ be the solution of the Cauchy problem for the damped fractional wave equation, where $$\sqrt{L}$$ involves the fractional Laplacian $$(-\triangle )^{\alpha}$$ on the space variable. We can study the decay estimate of the solution $$u(t,x)$$ over the time t by means of the Cauchy problem for the parabolic equation. In this paper, we consider, for $$0<\alpha <1$$, the Cauchy problem in the two- and three-dimensional spaces for the damped fractional wave equation and the corresponding parabolic equation and obtain the Triebel–Lizorkin space estimate of the difference of solutions. At the same time, we also consider, for $$\alpha =1$$, the case of the Cauchy problem in the four-dimensional space and obtain a Triebel–Lizorkin space estimate.

## 1 Introduction

We consider the Cauchy problem of the damped fractional wave equation (DFWE)

$${(\widetilde{A} )}\quad \quad \left \{ \textstyle\begin{array}{ll} \partial _{tt}u+2u_{t}+(-\Delta )^{\alpha }u=0,~~~\alpha >0, & \\ u(0,x)=f(x),~u_{t}(0,x)=g(x), & \end{array}\displaystyle \right .$$

where $$(t,x)\in (0,\infty )\times \mathbb{R} ^{n}$$, and $$(-\triangle )^{\alpha}$$ is the fractional Laplacian defined by

$$\widehat{(-\triangle )^{\alpha}f}(\xi )=\left \vert \xi \right \vert ^{2 \alpha}\widehat{f}(\xi ).$$

Via the Fourier transform, we easily find the solution of the Cauchy problem:

\begin{aligned} u(t,x)=e^{-t}\cosh (t\sqrt{L})f(x)+e^{-t} \frac{\sinh (t\sqrt{L})}{\sqrt{L}}(f(x)+g(x)), \end{aligned}
(1.1)

where L is the Fourier multiplier with symbol $$1-\left \vert \xi \right \vert ^{2\alpha }$$, that is, for a function G, we define the operator $$G(L)$$ via the Fourier transform $$(\widehat{G(L)f})(\xi )=G(1-\left \vert \xi \right \vert ^{2\alpha}) \widehat{f}(\xi )$$.

When the initial position $$f(x)=0$$, we obtain the solution

\begin{aligned} u(t,x)=e^{-t}\frac{\sinh (t\sqrt{L})}{\sqrt{L}}(g(x)). \end{aligned}

In the case $$\alpha =1$$, the damped fractional wave equation is reduced to the damped wave equation (DWE), which and its relative topics are studied by many authors (see [3, 6, 8, 1114, 23, 25]). It is also well known that the damped fractional wave equation is an important mathematical model in studying many physics and applied mathematics problems. Hence studies of both DWE and DFWE have attracted many authors. Checking the existing literature, we can easily find many papers addressing various research problems on this topic. For instance, we refer the reader to [3, 4, 7, 8, 12, 14, 15, 20] and references therein for researches on the local and global well-posedness of the Cauchy problem, space-time estimates, asymptotic estimates, etc. The asymptotic properties of $$u(t,x)$$ as $$t\rightarrow \infty$$ were studied by many authors (see [3, 9, 10, 12, 14, 15, 17, 19, 20]). As for dissipative equations, one of interesting problems is to study the decay estimate of the solution $$u(t,x)$$ over the time t.

To study this problem, we further introduce the Cauchy problem for the parabolic equation

$${(\widetilde{B})}\quad \quad \left \{ \textstyle\begin{array}{ll} \phi _{t}(t,x)+\frac{(-\triangle )^{\alpha}}{2}\phi (t,x)=0, & \\ \phi (0,x)=f(x)+g(x) & \end{array}\displaystyle \right .$$

and the Cauchy problem for the wave equation

$$(\widetilde{C})\quad \quad \left \{ \textstyle\begin{array}{ll} \upsilon _{tt}(t,x)+(-\triangle )^{\alpha}\upsilon (t,x)=0, & \\ \upsilon (0,x)=f(x), & \\ \upsilon _{t}(0,x)=g(x). \end{array}\displaystyle \right .$$

It is well known that the solution $$\phi \left ( t,x\right )$$ of () is given by

$$\phi (t,x)=G_{t}(g+f)(x),$$

where

$$G_{t}(g)(x)=e^{-\frac{t}{2}(-\triangle )^{\alpha}}(g).$$
(1.2)

We use the notation $$G_{t}$$, since its kernel is the Gauss function.

Also, we know that the solution of $$(\widetilde{C})$$ is

$$\upsilon (t,x)=\cos t\left \vert \triangle \right \vert ^{ \frac{\alpha}{2}}f(x)+ \frac{\sin t\left \vert \triangle \right \vert ^{\frac{\alpha}{2}}}{\left \vert \triangle \right \vert ^{\frac{\alpha}{2}}}g(x).$$

We denote by $$W_{t,\alpha}$$ the wave operator involving the fraction Laplacian $$\left \vert \triangle \right \vert ^{\alpha}$$,

$$W_{t,\alpha}(g)(x)= \frac{\sin t\left \vert \triangle \right \vert ^{\frac{\alpha}{2}}}{\left \vert \triangle \right \vert ^{\frac{\alpha}{2}}}g(x),$$
(1.3)

and particularly for $$\alpha =1$$,

$$W_{t}(g)(x)=W_{t,1}(g)(x).$$

In [19, Theorem 1.1], we find the following result.

### Theorem 1.1

([19])

Let $$n=3$$, let u be the solution in $$(\widetilde{A})$$, and let ϕ be the solution in $$(\widetilde{B})$$. Let $$\alpha =1$$. Then

$$\left \Vert \left ( u-\phi \right ) \left ( t,\cdot \right ) -e^{-t} \mathbf{W}_{0}\left ( t,f,g\right ) \right \Vert _{L^{p}\left ( \mathbb{R} ^{3} \right ) }\preceq t^{-\frac{3}{2}\left ( 1/q-1/p\right ) -1}\left \Vert f,g\right \Vert _{L^{q}\left ( \mathbb{R} ^{3}\right ) }$$

for $$t>t_{0}>0$$ and $$1\leq q\leq p\leq \infty$$, where

$$\mathbf{W}_{0}\left ( t,f,g\right ) =\left ( 1/2+t/8\right ) W_{t} \left ( f\right ) +\partial _{t}\left ( W_{t}\left ( f\right ) \right ) +W_{t}\left ( g\right )$$

and

$$\left \Vert f,g\right \Vert _{L^{q}\left ( \mathbb{R} ^{n}\right ) }= \left \Vert f\right \Vert _{L^{q}\left ( \mathbb{R} ^{n}\right ) }+ \left \Vert g\right \Vert _{L^{q}\left ( \mathbb{R} ^{n}\right ) }.$$

Here we adopt the notation $$\mathbf{W}_{0}$$ from [19].

In Theorem 1.1, taking $$f(x)=0$$, we obtain the following weak result of Theorem 1.1.

### Theorem 1.2

Let n = 3, let u be the solution of $$(\widetilde{A})$$, and let ϕ be the solution of $$(\widetilde{B})$$. Let $$\alpha =1$$. Then

$$\left \Vert \left ( u-\phi \right ) \left ( t,\cdot \right ) -e^{-t}W_{t} \left ( g\right ) \right \Vert _{L^{p}\left ( \mathbb{R} ^{3}\right ) } \preceq t^{-\frac{3}{2}\left ( 1/q-1/p\right ) -1}\left \Vert g \right \Vert _{L^{q}\left ( \mathbb{R} ^{3}\right ) }$$

for $$t>t_{0}>0$$ and $$1\leq q\leq p\leq \infty$$.

Motivated by Theorems 1.1 and 1.2, the aim of the paper is multifold. First, we extend Theorem 1.2 to the case $$0<\alpha <1$$. Second, we establish the asymptotic theorem in more general function spaces, the Triebel–Lizorkin spaces $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})\,(\gamma \in \mathbb{R},\,p, \,q\geq 1 )$$, which put functions into a wider frame, and $$\dot{F}_{p}^{0,2}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$$, $$1< p<\infty$$. Third and more important, we will use a method different from the proof of Theorem 1.1. We recall that the proof of Theorem 1.1 is based on the kernel estimate, which is complicated in its computation, and details can be found in [19]. We will use a much simpler method of directly using the multiplier theorem on the space $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$, recently established in [5].

We establish the following three theorems. In the following, $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ denote the Triebel–Lizorkin spaces.

### Theorem 1.3

Let $$n=3$$, let u be the solution of $$(\widetilde{A})$$, and let ϕ be the solution of $$(\widetilde{B})$$. Let $$6/7\leq \alpha <1$$. Then

$$\left \Vert \left ( u-\frac{1}{2}\phi \right ) \left ( t,\cdot \right ) -e^{-t}W_{t,\alpha }\left ( g\right ) \right \Vert _{\dot{F}_{p}^{ \gamma ,q}\left ( \mathbb{R} ^{3}\right ) }\preceq t^{- \frac{3}{2\alpha }\left ( 1/r-1/p\right ) -1}\left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}\left ( \mathbb{R} ^{3}\right ) }$$

for $$t>t_{0}>0$$ and $$1\leq r\leq p\leq \infty$$.

### Theorem 1.4

Let $$n=2$$, let u be the solution of $$(\widetilde{A})$$, and let ϕ be the solution of $$(\widetilde{B})$$. Let $$2/3\leq \alpha <1$$. Then

$$\left \Vert \left ( u-\frac{1}{2}\phi \right ) \left ( t,\cdot \right ) -e^{-t}W_{t,\alpha }\left ( g\right ) \right \Vert _{\dot{F}_{p}^{ \gamma ,q}\left ( \mathbb{R} ^{2}\right ) }\preceq t^{- \frac{1}{\alpha }\left ( 1/r-1/p\right ) -1}\left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}\left ( \mathbb{R} ^{2}\right ) }$$

for $$t>t_{0}>0$$ and $$1\leq r\leq p\leq \infty$$.

We also consider the case of $$\alpha =1$$ and $$n=4$$.

### Theorem 1.5

Let $$n=4$$, let u be the solution of $$(\widetilde{A})$$, and let ϕ be the solution in $$(\widetilde{B})$$. Let $$\alpha =1$$. Then

$$\left \Vert \left ( u-\frac{1}{2}\phi \right ) \left ( t,\cdot \right ) -e^{-t}W_{t}\left ( g\right ) \right \Vert _{\dot{F}_{p}^{ \gamma ,q}\left ( \mathbb{R} ^{4}\right ) }\preceq t^{-2\left ( 1/r-1/p \right ) -1}\left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}\left ( \mathbb{R}^{4}\right ) }$$

for $$t>t_{0}>0$$ and $$1< r\leq p<\infty$$, provided that

$$1/r-1/p< 1/2.$$

### Remark 1

We must point out that the reason we only extend Theorem 1.2 by considering $$f(x)=0$$ is merely for simplicity and illustration of our method. Our method easily works in the case $$f(x)\neq 0$$, an extension of Theorem 1.1.

The paper is organized as follows. In Sect. 2, we review the definition of the Triebel–Lizorkin spaces and some known results. The proofs of the main theorems can be found in Sect. 3.

Throughout this paper, we write $$A\preceq B$$ if there is a constant $$C>0$$, independent of all essential values and variables, such that $$A\leq CB$$. We write $$A\simeq B$$ if there exists a positive constant C, independent of all essential values and variables, such that $$C^{-1}B\leq A\leq CB$$.

## 2 Preliminaries

In this section, we give some basic properties of the Triebel–Lizorkin spaces, which can also be referred to [1].

### 2.1 Definition [24]

Fix a function $$\varphi \in C^{\infty}(\mathbb{R}^{n})$$ satisfying $$supp(\varphi )\subset \{ \xi :2<\left \vert \xi \right \vert \leq 4 \}$$, $$0\leq \varphi (\xi )\leq 1$$, and $$\varphi (\xi )>c>0$$ for $$\frac{5}{2}\leq \left \vert \xi \right \vert \leq \frac{7}{2}$$. Denote $$\varphi _{j}(\xi )=\varphi (2^{-j}\xi )$$. By an easy normalization we may also require that φ satisfies

$$\sum _{j=-\infty}^{\infty}\varphi _{j}(\xi )=1 \quad \text{for all} \quad \xi \in \mathbb{R}^{n}\backslash \{0\}.$$

Denote the functions $$\phi _{j}(x)$$ by $$\widehat{\phi _{j}}(\xi )=\varphi _{j}(\xi )$$ for $$j\in \mathbb{Z}$$. For $$\gamma \in \mathbb{R}$$ and $$0< p$$, $$q<\infty$$, the homogeneous Triebel–Lizorkin space $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ is the set of all distributions f satisfying

$$\left \Vert f\right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}= \left \Vert (\sum _{j\in{\mathbb{Z}}}\left \vert 2^{\gamma j}(\phi _{j} \ast f)\right \vert ^{q})^{\frac{1}{q}}\right \Vert _{L^{p}( \mathbb{R}^{n})}< \infty .$$

If we change the sum $$\sum _{j\in \mathbb{Z}}$$ to $$\sum _{j\geq 0}$$ in the above expression and add an extra term $$\left \Vert \psi \ast f\right \Vert _{L^{p}}$$ to the sum for a suitable function ψ, then we obtain the norm of the inhomogeneous Triebel–Lizorkin space $$F_{p}^{\gamma ,q}$$. The definitions of $$\dot{F}_{p}^{\gamma ,q}$$ and $$F_{p}^{\gamma ,q}$$ are independent of the choice of functions ϕ and ψ.

### 2.2 Imbedding and lifting [24]

The space $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ has the Sobolev imbedding relationship

$$\dot{F}_{p}^{\gamma ,q_{1}}(\mathbb{R}^{n})\subset \dot{F}_{p}^{ \gamma ,q_{2}}(\mathbb{R}^{n})$$

for $$q_{1}\leq q_{2}$$. For any $$1\leq q_{1}$$, $$q_{2}\leq \infty$$ and $$-\infty <\gamma _{2}<\gamma _{1}<\infty$$,

$$\dot{F}_{p_{1}}^{\gamma _{1},q_{1}}(\mathbb{R}^{n})\subset \dot{F}_{p_{2}}^{ \gamma _{2},q_{2}}(\mathbb{R}^{n}),$$

provided that

$$\gamma _{1}-\frac{n}{p_{1}}=\gamma _{2}-\frac{n}{p_{2}}.$$

The space $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ also has the lifting property

$$\left \Vert f\right \Vert _{\dot{F}_{p}^{\gamma ,q}}\simeq \left \Vert R_{\gamma}f\right \Vert _{\dot{F}_{p}^{0,q}},$$

where $$R_{\gamma}$$ is the Riesz potential defined by

$$\widehat{R_{\gamma }f}(\xi )=\left \vert \xi \right \vert ^{\gamma} \widehat{f}(\xi ).$$

### 2.3 Complex interpolation of Triebel–Lizorkin space

Suppose $$-\infty < s_{0}$$, $$s_{1}<\infty$$, $$0<\theta <1$$, and $$0< p_{0},\,p_{1},\,q_{0}$$, $$q_{1}\leq \infty$$. Then

$$(\dot{F}_{p_{0}}^{s_{0},q_{0}},\,\dot{F}_{p_{1}}^{s_{1},q_{1}})_{ \theta}=\dot{F}_{p}^{s,q}$$

for $$s=(1-\theta )s_{0}+\theta s_{1}$$, $$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$$, and $$\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}}$$.

### 2.4 Multiplier theorem of the Triebel–Lizorkin space

Given a positive integer l and $$s\in{R}$$, we assume that $$m\in C^{l}(\mathbb{R}^{n}\backslash \{0\})$$ satisfies the following condition:

$$\sup _{R>0}\{R^{-n+2s+2\left \vert \sigma \right \vert}\int _{R< \left \vert \xi \right \vert \leq 2R}\left \vert \partial ^{\sigma}m( \xi )\right \vert ^{2} d\xi \}\leq A_{\sigma}.$$
(2.1)

This condition can be replaced by the simple condition

$$\left \vert \partial ^{\sigma}m(\xi )\right \vert \preceq \left \vert \xi \right \vert ^{-\left \vert \sigma \right \vert}.$$
(2.2)

### Proposition 2.1

(The multiplier theorem of $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$; see [5])

Let $$\gamma \in{\mathbb{R}}$$, $$0< p<\infty$$, and $$0< q\leq \infty$$. Given a positive integer l, we assume that $$m\in{C^{l}(\mathbb{R}^{n}\backslash \left \{ 0\right \})}$$ satisfies condition (2.2) for all multiindices σ with $$\left \vert \sigma \right \vert \leq l$$, where

$$l>\max (\frac{n}{p},\frac{n}{q})+\frac{n}{2}.$$

Then

$$\left \Vert T_{m}(f)\right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})} \preceq \left \Vert f\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}.$$

If $$l>\lambda +\frac{n}{2}$$ for a sufficiently large λ, then

$$\left \Vert T_{m}(f)\right \Vert _{\dot{F}_{\infty}^{\gamma ,q}( \mathbb{R}^{n})}\preceq \left \Vert f\right \Vert _{\dot{F}_{\infty}^{ \gamma ,q}(\mathbb{R}^{n})},$$

where

$$T_{m}(f)(x)=\int _{R^{n}}m(\xi )\widehat{f}(\xi )e^{i< \xi ,x>}d\xi ,$$

and we say that m is an $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ multiplier if $$T_{m}$$ is bounded on $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$.

To prove main theorems, we introduce an easy lemma.

### Lemma 2.2

Let $$\{T_{t}\}$$ be a family of convolution operators satisfying

$$\left \Vert T_{t}\left ( f\right ) \right \Vert _{L^{\infty }\left ( \mathbb{R} ^{n}\right ) }\preceq \left ( B+t\right ) ^{-N}\left \Vert f\right \Vert _{L^{1}\left ( \mathbb{R} ^{n}\right ) }.$$

Then for all $$1\leq q\leq \infty$$, we have

$$\left \Vert T_{t}\left ( f\right ) \right \Vert _{\dot{F}_{\infty }^{ \gamma ,q}(\mathbf{\mathbb{R} }^{n})}\preceq \left ( B+t\right ) ^{-N}\left \Vert f\right \Vert _{ \dot{F}_{1}^{\gamma ,q}(\mathbf{\mathbb{R} }^{n})}.$$

Proof The Triebel–Lizorkin space $$\dot{F}_{p}^{\gamma ,q}(\mathbf{\mathbb{R} }^{n})$$ is the set of all distributions f satisfying

$$\left \Vert f\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbf{\mathbb{R} }^{n})}=\left \Vert \left ( \sum _{j\in \mathbb{Z}}\left \vert 2^{ \gamma j}(\phi _{j}\ast f)\right \vert ^{q}\right ) ^{1/q}\right \Vert _{L^{p}(\mathbf{\mathbb{R} }^{n})}< \infty .$$

Hence

\begin{aligned} \left \Vert T_{t}\left ( f\right ) \right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbf{\mathbb{R} }^{n})} =&\left \Vert \left ( \sum _{j\in \mathbb{Z}}\left \vert 2^{ \gamma j}(\phi _{j}\ast T_{t}\left ( f\right ) )\right \vert ^{q} \right ) ^{1/q}\right \Vert _{L^{p}(\mathbf{\mathbb{R} }^{n})} \\ =&\left \Vert \left ( \sum _{j\in \mathbb{Z}}\left \vert 2_{{}}^{ \gamma j}T_{t}(\phi _{j}\ast f)\right \vert ^{q}\right ) ^{1/q} \right \Vert _{L^{p}(\mathbf{\mathbb{R} }^{n})}. \end{aligned}

To prove the lemma, with the lifting property of $$\dot{F}_{p}^{\gamma ,q}$$, we may assume that $$\gamma =0$$.

Now by assumption

\begin{aligned} \left \Vert T_{t}\left ( f\right ) \right \Vert _{\dot{F}_{\infty }^{0, \infty }(\mathbf{\mathbb{R} }^{n})} =&\sup _{x}\sup _{j}\left \vert T_{t}\left ( \phi _{j}\ast f \right ) \left ( x\right ) \right \vert \\ \preceq &\sup _{j}\sup _{x}\left \vert T_{t}\left ( \phi _{j}\ast f \right ) \left ( x\right ) \right \vert \\ \preceq &\left ( B+t\right ) ^{-N}\sup _{j}\left \Vert \phi _{j} \ast f\right \Vert _{L^{1}\left ( \mathbb{R} ^{n}\right ) } \\ \preceq &\left ( B+t\right ) ^{-N}\left \Vert f\right \Vert _{ \dot{F}_{1}^{0,\infty }(\mathbf{\mathbb{R} }^{n})}. \end{aligned}

Also,

\begin{aligned} \left \Vert T_{t}\left ( f\right ) \right \Vert _{\dot{F}_{\infty }^{0,1}(\mathbf{\mathbb{R} }^{n})} =&\sup _{x}\sum _{j\in \mathbb{Z}}\left \vert T_{t}\left ( \phi _{j}\ast f\right ) \left ( x\right ) \right \vert \\ \preceq &\sum _{j\in \mathbb{Z}}\sup _{x}\left \vert T_{t}\left ( \phi _{j}\ast f\right ) \left ( x\right ) \right \vert \\ \preceq &\left ( B+t\right ) ^{-N}\sum _{j\in \mathbb{Z}}\left \Vert \left ( \phi _{j}\ast f\right ) \right \Vert _{L^{1}\left ( \mathbb{R} ^{n}\right ) } \\ =&\left ( B+t\right ) ^{-N}\left \Vert f\right \Vert _{\dot{F}_{1}^{0,1}(\mathbf{\mathbb{R} }^{n})}. \end{aligned}

An interpolation yields that

$$\left \Vert T_{t}\left ( f\right ) \right \Vert _{\dot{F}_{\infty }^{0,q}(\mathbf{\mathbb{R} }^{n})}\preceq \left ( B+t\right ) ^{-N}\left \Vert f\right \Vert _{ \dot{F}_{1}^{0,q}(\mathbf{\mathbb{R} }^{n})}$$

for any $$1\leq q\leq \infty$$. The lemma is proved.

### 2.5 Known results of an operator

We will further consider the multiplier operator $$K_{\alpha ,\beta}\ast f$$ with

$$K_{\alpha ,\beta}(x)=\int _{\mathbb{R}^{n}}\Psi (\left \vert \xi \right \vert ) \frac{\sin \left \vert \xi \right \vert ^{\alpha}}{\left \vert \xi \right \vert ^{\beta}}e^{ix \cdot \xi}d\xi$$

or

$$K_{\alpha ,\beta}(x)=\int _{\mathbb{R}^{n}}\Psi (\left \vert \xi \right \vert ) \frac{\cos \left \vert \xi \right \vert ^{\alpha}}{\left \vert \xi \right \vert ^{\beta}}e^{ix \cdot \xi}d\xi ,$$

where $$\Psi \in C^{\infty}$$ is supported on the set $$\{\left \vert \xi \right \vert ,\left \vert \xi \right \vert \geq C \}$$ for some $$C\geq 1$$. The following two results are well known.

### Theorem 2.3

([1])

Let $$0<\alpha <1$$. Then

$$\left \Vert K_{\alpha ,\beta}\ast g\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}\preceq \left \Vert g\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}$$

if

$$\beta \geq \alpha n\left \vert \frac{1}{p}-\frac{1}{2}\right \vert .$$

Moreover,

$$\left \Vert K_{\alpha ,\beta}\ast g\right \Vert _{L^{1}(\mathbb{R}^{n})} \preceq \left \Vert g\right \Vert _{L^{1}(\mathbb{R}^{n})}$$

if and only if

$$\beta >\frac{\alpha n}{2}.$$

### Theorem 2.4

([2])

Let $$\alpha =1$$. Then

$$\left \Vert K_{1,\beta}\ast g\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq \left \Vert g\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}$$

if

$$\beta \geq (n-1)\left \vert \frac{1}{p}-\frac{1}{2}\right \vert .$$

Moreover,

$$\left \Vert K_{1,\beta}\ast g\right \Vert _{L^{1}(\mathbb{R}^{n})} \preceq \left \Vert g\right \Vert _{L^{1}(\mathbb{R}^{n})}$$

if and only if

$$\beta >\frac{n-1}{2}.$$

## 3 Proofs of main theorems

### 3.1 Discussion

Let $$\Phi (\xi )$$ be a $$C^{\infty }$$ radial function with values in $$[0,1]$$, supported in the set $$\{\xi :\left \vert \xi \right \vert \leq 1/2\}$$, and satisfying $$\Phi (\xi )\equiv 1$$ in the set $$\{\xi :\left \vert \xi \right \vert \leq 7/16\}$$.

Set

$$\Psi (\xi )=1-\Phi (\xi ).$$

Then $$\Psi (\xi )$$ has support in

$$\{\xi :\left \vert \xi \right \vert >7/16\}.$$

Denote

$$S(t)=e^{-t}\frac{\sinh \left ( t\sqrt{L}\right ) }{\sqrt{L}}.$$

Then we have

$$S(t)=\Psi (D)S(t)+\Phi (D)S(t).$$

Here $$\Phi (D)S(t)$$ is the low-frequency part of $$S(t)$$ defined by

$$\Phi (D)S(t)g(x)=e^{-t}\int _{\mathbf{\mathbb{R} }^{n}}\Phi (\xi ) \frac{\sinh (t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }})}{\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}}\widehat{g}( \xi )e^{i< x,\xi >}d\xi ,$$

and $$\Psi (D)S(t)$$ is the high-frequency part of $$S(t)$$ defined by

\begin{aligned} \Psi (D)S(t)g(x) =&e^{-t}\int _{\mathbf{\mathbb{R} }^{n}}\Psi (\xi ) \frac{\sinh (t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }})}{\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}}\widehat{g}( \xi )e^{i< x,\xi >}d\xi . \end{aligned}

Similarly, we may write

\begin{aligned} \phi \left ( t,x\right ) =&\int _{\mathbf{\mathbb{R} }^{n}}\Phi (\xi )e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2 \alpha }}\widehat{g}(\xi )e^{i< x,\xi >}d\xi \\ &+\int _{\mathbf{\mathbb{R} }^{n}}\Psi (\xi )e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2 \alpha }}\widehat{g}(\xi )e^{i< x,\xi >}d\xi \\ =&\Phi (D)G_{t}\left ( g\right ) (x)+\Psi (D)G_{t}\left ( g\right ) (x), \end{aligned}

recalling that

$$\phi \left ( t,x\right ) =G_{t}\left ( g\right ) (x).$$

By these definitions

\begin{aligned} &\left \Vert \left ( u-\frac{1}{2}\phi \right ) \left ( t,\cdot \right ) -e^{-t}W_{t}\left ( g\right ) \right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbf{\mathbb{R} }^{n})} \\ \leq &\left \Vert \Phi (D)S(t)g-\frac{1}{2}\Phi (D)G_{t}\left ( g \right ) \right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbf{\mathbb{R} }^{n})}+\frac{1}{2}\left \Vert \Psi (D)G_{t}\left ( g\right ) \right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbf{\mathbb{R} }^{n})} \\ &+\left \Vert \Psi (D)S(t)g-e^{-t}\Psi (D)W_{t,\alpha }\left ( g \right ) \right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbf{\mathbb{R} }^{n})}+\left \Vert e^{-t}\Phi (D)W_{t,\alpha }\left ( g\right ) \right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbf{\mathbb{R} }^{n})}. \end{aligned}

Clearly, to prove Theorems 1.3 and 1.4, it suffices to show the following three propositions.

### Proposition 3.1

For any $$1\leq r\leq p\leq \infty$$ and $$\alpha ,\,q>0$$,

\begin{aligned}& \left \Vert \Phi (D)S(t)g-\frac{1}{2}\Phi (D)G_{t}(g)\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}( \frac{1}{r}-\frac{1}{p})-1}\left \Vert g\right \Vert _{\dot{F}_{r}^{ \gamma ,q}(\mathbb{R}^{n})}, \end{aligned}
(3.1)
\begin{aligned}& \left \Vert \Psi (D)G_{t}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{n})}, \end{aligned}
(3.2)
\begin{aligned}& \left \Vert e^{-t}\Phi (D)W_{t,\alpha}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}(\frac{1}{r}- \frac{1}{p})-1}\left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}( \mathbb{R}^{n})}. \end{aligned}
(3.3)

### Proposition 3.2

For any $$1\leq r\leq p\leq \infty$$ and $$q>0$$,

$$\left \Vert \Psi (D)S(t)g-e^{-t}\Psi (D)W_{t,\alpha}(g)\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{3})}\preceq t^{-\frac{3}{2\alpha}( \frac{1}{r}-\frac{1}{p})-1}\left \Vert g\right \Vert _{\dot{F}_{r}^{ \gamma ,q}(\mathbb{R}^{3})},$$
(3.4)

provided that

$$1>\alpha \geq \frac{6}{7}.$$

### Proposition 3.3

For any $$1\leq r\leq p\leq \infty$$ and $$q>0$$,

$$\left \Vert \Psi (D)S(t)g-e^{-t}\Psi (D)W_{t,\alpha}(g)\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{2})}\preceq t^{-\frac{1}{\alpha}( \frac{1}{r}-\frac{1}{p})-1}\left \Vert g\right \Vert _{\dot{F}_{r}^{ \gamma ,q}(\mathbb{R}^{2})},$$
(3.5)

provided that

$$1>\alpha \geq \frac{2}{3}.$$

### 3.2 Proof of Proposition 3.1

We first show (3.2). By the definition we have

$$\Psi (D)G_{t}\left ( g\right ) \left ( x\right ) =\int _{ \mathbf{\mathbb{R} }^{n}}\Psi (\xi )e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2 \alpha }}\widehat{g}(\xi )e^{i< x,\xi >}d\xi .$$

Thus

\begin{aligned} \left \Vert \Psi (D)G_{t}(g)\right \Vert _{L^{\infty}( \mathbb{R}^{n})}&\preceq \int _{\mathbb{R}^{n}}\left \vert \Psi (\xi ) \right \vert e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha}} \left \Vert \widehat{g}\right \Vert _{L^{\infty}(\mathbb{R}^{n})}d \xi \\ &\preceq \left \Vert g\right \Vert _{L^{1}(\mathbb{R}^{n})} \frac{1}{t^{N}}\int _{\mathbb{R}^{n}}\left \vert \Psi (\xi )\right \vert \left \vert \xi \right \vert ^{-2\alpha N}d\xi \\ &\preceq \frac{1}{t^{N}}\left \Vert g\right \Vert _{L^{1}(\mathbb{R}^{n})} \end{aligned}

for any

$$N>\frac{n}{2\alpha}.$$

By Lemma 2.2 we now have that for $$N>\frac{n}{2\alpha }$$,

\begin{aligned} \left \Vert \Psi (D)G_{t}(g)\right \Vert _{\dot{F}_{ \infty}^{\gamma ,q}(\mathbb{R}^{n})}\preceq \frac{1}{t^{N}}\left \Vert g\right \Vert _{\dot{F}_{1}^{\gamma ,q}(\mathbb{R}^{n})}. \end{aligned}
(3.6)

On the other hand, the Fourier multiplier of $$\Psi (D)G_{t}$$ is $$\Psi (\xi )e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha }}$$. Thus it is easy to see that for any multi-index σ,

$$\left \vert (\frac{\partial}{\partial \xi})^{\sigma}\Psi (\xi )e^{- \frac{t}{2}\left \vert \xi \right \vert ^{2\alpha}}\right \vert \preceq t^{-N}\left \vert \xi \right \vert ^{-\sigma},$$

which, by the multiplier theorem of space $$\dot{F}_{p}^{\gamma ,q}$$ (Proposition 2.1), yields that

$$\left \Vert \Psi (D)G_{t}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq t^{-N}\left \Vert g\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}$$
(3.7)

for any $$1< p\leq \infty$$. We obtain (3.2) by an easy interpolation between (3.6) and (3.7).

Next, we prove (3.3). By the definition we see that

$$e^{-t}\Phi (D)W_{t,\alpha}(g)(x)=e^{-t}\int _{\mathbb{R}^{n}}\Phi ( \xi ) \frac{\sin (t\left \vert \xi \right \vert ^{\alpha})}{\left \vert \xi \right \vert ^{\alpha}} \widehat{g}(\xi )e^{i< x,\xi >}d\xi .$$

Clearly,

$$\left \Vert e^{-t}\Phi (D)W_{t,\alpha}(g)\right \Vert _{L^{\infty}( \mathbb{R}^{n})}\preceq (\int _{\mathbb{R}^{n}}\left \vert \Phi (\xi ) \right \vert \frac{\left \vert \sin (t\left \vert \xi \right \vert ^{\alpha})\right \vert}{\left \vert \xi \right \vert ^{\alpha}}d \xi )e^{-t}\left \Vert g\right \Vert _{L^{1}(\mathbb{R}^{n})}.$$

Here

$$\int _{\mathbb{R}^{n}}\left \vert \Phi (\xi )\right \vert \frac{\left \vert \sin (t\left \vert \xi \right \vert ^{\alpha})\right \vert}{\left \vert \xi \right \vert ^{\alpha}}d \xi \leq t\int _{\left \vert \xi \right \vert < \frac{1}{t^{\frac{1}{\alpha}}}}\left \vert \Phi (\xi )\right \vert d \xi +\int _{\left \vert \xi \right \vert \geq \frac{1}{t^{\frac{1}{\alpha}}}}\left \vert \Phi (\xi )\right \vert \frac{d\xi}{\left \vert \xi \right \vert ^{\alpha}} \leq t^{1-\frac{n}{\alpha}}+C$$

for some positive constant C. Thus from Lemma 2.2 we obtain

$$\left \Vert e^{-t}\Phi (D)W_{t,\alpha}(g)\right \Vert _{\dot{F}_{ \infty}^{\gamma ,q}(\mathbb{R}^{n})}\preceq e^{-t}(t^{1- \frac{n}{\alpha}}+C)\left \Vert g\right \Vert _{\dot{F}_{1}^{\gamma ,q}( \mathbb{R}^{n})}.$$
(3.8)

On the other hand, the Fourier multiplier of $$e^{-t}\Phi (D)W_{t,\alpha}$$ is $$e^{-t}\Phi (\xi ) \frac{\sin (t\left \vert \xi \right \vert ^{\alpha})}{\left \vert \xi \right \vert ^{\alpha}}$$. Thus it is easy to see that for any multi-index σ,

$$\left \vert (\frac{\partial}{\partial \xi})^{\sigma}e^{-t}\Phi (\xi ) \frac{\sin (t\left \vert \xi \right \vert ^{\alpha})}{\left \vert \xi \right \vert ^{\alpha}} \right \vert \preceq e^{-t}t^{\left \vert \sigma \right \vert}\left \vert \xi \right \vert ^{-\left \vert \sigma \right \vert},$$

which, by the multiplier theorem of the space $$\dot{F}_{p}^{\gamma ,q}$$, yields

$$\left \Vert e^{-t} \Phi (D)W_{t,\alpha}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}\preceq e^{-t}t^{\left \vert \sigma \right \vert}\left \Vert f\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}.$$
(3.9)

An easy interpolation from (3.8) and (3.9) now gives (3.3).

Finally, we show (3.1). By the definition we have that

\begin{aligned} &\Phi (D)S(t)g\left ( x\right ) -\frac{1}{2}\Phi (D)G_{t}\left ( g \right ) \left ( x\right ) \\ =&\int _{\mathbf{\mathbb{R} }^{n}}\Phi (\xi )\left ( \frac{e^{-t}\sinh (t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }})}{\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}}-\frac{e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha }}}{2} \right ) \widehat{g}(\xi )e^{i< x,\xi >}d\xi , \end{aligned}

where we write the multiplier as a sum of multipliers

$$\Phi (\xi )\left ( \frac{e^{-t}\sinh (t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }})}{\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}}- \frac{e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha }}}{2}\right )$$
\begin{aligned} =& \frac{-\Phi (\xi )e^{-t}e^{-t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}}}{2\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}}+ \frac{\Phi (\xi )e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha }}}{2\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}} \left ( e^{(t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}-t+ \frac{t\left \vert \xi \right \vert ^{2\alpha }}{2})}-1\right ) \\ &+\left \vert \xi \right \vert ^{2\alpha }e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha }}\cdot \frac{\left ( 1-\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}\right ) }{2\left \vert \xi \right \vert ^{2\alpha }\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}}\Phi (\xi ) \\ =&m_{1}\left ( t,\xi \right ) +m_{2}\left ( t,\xi \right ) +m_{3} \left ( t,\xi \right ) m_{4}\left ( \xi \right ) \text{\ }. \end{aligned}

Clearly, since the support of $$\Phi (\xi )$$ implies $$\left \vert \xi \right \vert \leq 2/3$$, using the same proof as that of (3.2), we easily check that for all $$1\leq r\leq p\leq \infty$$,

$$\left \Vert T_{m_{1}}g\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbf{\mathbb{R} }^{n})}\preceq t^{-\frac{n}{2\alpha }\left ( 1/r-1/p\right ) -1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}( \mathbf{\mathbb{R} }^{n})}.$$

For the multiplier $$m_{2}\left ( t,\xi \right )$$, we see that the corresponding operator satisfies

\begin{aligned} \left \Vert T_{m_{2}\left ( t,\cdot \right ) }\left ( g\right ) \right \Vert _{L^{\infty }\left ( \mathbb{R} ^{n}\right ) } \preceq & \left \Vert g\right \Vert _{L^{1}\left ( \mathbb{R} ^{n}\right ) } \int _{\mathbb{R} ^{n}}\left \vert \frac{\Phi (\xi )e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha }}}{2\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}}\left ( e^{(t \sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}-t+ \frac{t\left \vert \xi \right \vert ^{2\alpha }}{2})}-1\right ) \right \vert d\xi \\ \preceq &\left \Vert g\right \Vert _{L^{1}\left ( \mathbb{R} ^{n} \right ) }\int _{\mathbb{R} ^{n}}\left \vert \Phi (\xi )e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha }}\left ( e^{tO(\left \vert \xi \right \vert ^{4\alpha })}-1\right ) \right \vert d\xi \\ \preceq &\left \Vert g\right \Vert _{L^{1}\left ( \mathbb{R} ^{n} \right ) }t\int _{\mathbb{R} ^{n}}\Phi (\xi )e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha }}\left \vert \xi \right \vert ^{4\alpha }d\xi \\ \preceq &\left \Vert g\right \Vert _{L^{1}\left ( \mathbb{R} ^{n} \right ) }t^{-1}t^{-\frac{n}{2\alpha }}\int _{\mathbb{R} ^{n}}e^{-\left \vert \xi \right \vert ^{2\alpha }}\left \vert \xi \right \vert ^{4\alpha }d\xi \\ \preceq &t^{-\frac{n}{2\alpha }-1}\left \Vert g\right \Vert _{L^{1} \left ( \mathbb{R} ^{n}\right ) }. \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{aligned}

Thus by Lemma 2.2 we obtain

$$\left \Vert T_{m_{2}(t,\cdot )}(g)\right \Vert _{\dot{F}_{\infty}^{ \gamma ,q}(\mathbb{R}^{n})}\preceq t^{-1}t^{-\frac{n}{2\alpha}}\left \Vert g\right \Vert _{\dot{F}_{1}^{\gamma ,q}(\mathbb{R}^{n})}.$$
(3.10)

On the other hand, in the multiplier

$$m_{2}(t,\xi )= \frac{\Phi (\xi )e^{-\frac{t}{2}\left \vert \xi \right \vert ^{2\alpha}}}{2\sqrt{1-\left \vert \xi \right \vert ^{2\alpha}}}(e^{t \sqrt{1-\left \vert \xi \right \vert ^{2\alpha}}-t+ \frac{t\left \vert \xi \right \vert ^{2\alpha}}{2}}-1),$$

we note that in the support of $$\Phi (\xi )$$, we have that

$$(e^{(t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha}}-t+ \frac{t\left \vert \xi \right \vert ^{2\alpha}}{2})}-1)=e^{-t\varphi ( \xi )}-1,$$

where $$\varphi \in C^{\infty}{(\mathbb{R}^{n}\backslash \{0\}),}$$ and

$$\left \vert (\frac{\partial}{\partial \xi})^{\sigma}\varphi (\xi ) \right \vert \preceq \left \vert \xi \right \vert ^{4\alpha -\left \vert \sigma \right \vert}$$

in the support of Φ.

It is easy to see, by the multiplier theorem on $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$, that for any $$1\leq p\leq \infty$$,

$$\left \Vert T_{m_{2}(t,\cdot )}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq t^{-1}\left \Vert g\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}.$$

Interpolation between the above inequality and (3.10) now yields

$$\left \Vert T_{m_{2}}g\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{n})}$$
(3.11)

for any $$1\leq r\leq p\leq \infty$$ and $$1\leq q\leq \infty$$.

Also, it is easy to check that the multiplier

$$m_{4}(\xi )= \frac{(1-\sqrt{1-\left \vert \xi \right \vert ^{2\alpha}})}{2\left \vert \xi \right \vert ^{2\alpha}\sqrt{1-\left \vert \xi \right \vert ^{2\alpha}}} \Phi (\xi )$$

is an $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ multiplier for any $$1\leq p$$, $$q\leq \infty$$.

On the other hand,

$$m_{3}(t,\xi )=\left \vert \xi \right \vert ^{2\alpha}e^{-\frac{t}{2} \left \vert \xi \right \vert ^{2\alpha}}$$

is an $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ multiplier for any $$1\leq p$$, $$q\leq \infty$$ with the bound $$\preceq t^{-1}$$.

In addition, by Lemma 2.2 this implies

$$\left \Vert T_{m_{3}(t,\cdot )}(g)\right \Vert _{\dot{F}_{\infty}^{ \gamma ,q}(\mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}-1}\left \Vert g\right \Vert _{\dot{F}_{1}^{\gamma ,q}(\mathbb{R}^{n})}.$$

Now by interpolation we obtain that

$$\left \Vert T_{m_{3}(t,\cdot )}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{n})}.$$

Finally, this becomes

$$\left \Vert T_{m_{3}m_{4}}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq \left \Vert T_{m_{3}}(g)\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}( \frac{1}{r}-\frac{1}{p})-1}\left \Vert g\right \Vert _{\dot{F}_{r}^{ \gamma ,q}(\mathbb{R}^{n})}$$
(3.12)

for any $$1\leq r\leq p\leq \infty$$ and $$1\leq q\leq \infty$$.

The proof of Proposition 3.1 is completed.

### 3.3 Proofs of Propositions 3.2 and 3.3

The proofs of Propositions 3.2 and 3.3 are the same, so we will show Proposition 3.2. Precisely, we will show the following formula:

$$\left \Vert \Psi (D)S(t)g-e^{-t}\Psi (D)W_{t,\alpha}(g)\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{3})}\preceq t^{-\frac{3}{2\alpha}( \frac{1}{r}-\frac{1}{p})-1}\left \Vert g\right \Vert _{\dot{F}_{r}^{ \gamma ,q}(\mathbb{R}^{3})}$$

for any $$1\leq r\leq p\leq \infty$$ and $$1\leq q\leq \infty$$.

First, we need the following lemma.

### Lemma 3.4

In the proofs of propositions, we may assume that

$$supp(\Psi )\subset \{\xi :\left \vert \xi \right \vert \geq 100\}.$$

Proof: We explain why Lemma 3.4 is reasonable. In fact, we write the function $$\Theta (\xi )=1-\Psi (\xi )-\Phi (\xi )$$. Let $$\Omega (\xi )$$ be a $$C^{\infty }$$ radial function with values in the interval $$[0,1]$$, supported in the set $$\{\xi :3/4\leq \left \vert \xi \right \vert \leq 4/3\}$$, and satisfying $$\Omega (\xi )\equiv 1$$ in the set $$\{7/8\leq \left \vert \xi \right \vert \leq 8/7\}$$. Write

$$\Lambda (\xi )=\Theta (\xi )-\Omega (\xi ).$$

It is clear that

$$1=\Psi (\xi )+\Phi (\xi )+\Lambda (\xi )+\Omega (\xi ).$$

To prove Proposition 3.2, we also need to check that for any dimension n,

$$\left \Vert \Lambda (D)S(t)(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}+\left \Vert \Omega (D)S(t)(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}(\frac{1}{r}- \frac{1}{p})-1}\left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}( \mathbb{R}^{n})}$$
(3.13)

and

$$\left \Vert e^{-t}\Omega (D)W_{t,\alpha}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}+\left \Vert e^{-t}\Lambda (D)W_{t,\alpha}(g) \right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}\preceq t^{- \frac{n}{2\alpha}(\frac{1}{r}-\frac{1}{p})-1}\left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{n})}.$$
(3.14)

The proof of (3.14) is exactly the same as that of (3.3). We need to show (3.13).

First, we know that the multiplier of $$\Lambda (D)S(t)$$ is

$$e^{-t} \frac{\Lambda (\xi )\sin (t\sqrt{\left \vert \xi \right \vert ^{2\alpha}-1})}{\sqrt{\left \vert \xi \right \vert ^{2\alpha}-1}},$$

and we recall that the support of $$\Lambda (\xi )$$ is contained in the set

$$\{\frac{8}{7}\leq \left \vert \xi \right \vert \leq 150\}.$$

Clearly,

$$\Lambda (\xi )(\left \vert \xi \right \vert ^{2\alpha}-1)^{- \frac{1}{2}}\sin (t\sqrt{\left \vert \xi \right \vert ^{2\alpha}-1})$$

is a Schwartz function satisfying

$$\left \vert \partial _{\xi }^{\sigma }\left ( e^{-t} \frac{\Lambda (\xi )\sin (t\sqrt{\left \vert \xi \right \vert ^{2\alpha }-1})}{\sqrt{\left \vert \xi \right \vert ^{2\alpha }-1}} \right ) \right \vert \preceq e^{-t}\left ( 1+t\right ) ^{\left \vert \sigma \right \vert }.$$

Thus by the multiplier theorem it is trivial to see that

$$\left \Vert \Lambda (D)S(t)(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{n})}.$$
(3.15)

We know that the multiplier of $$\Omega (D)S(t)$$ is

$$\frac{e^{-t}\Omega (\xi )\sinh (t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha}})}{\sqrt{1-\left \vert \xi \right \vert ^{2\alpha}}}.$$

By the Taylor expansion, in the support of $$\Omega (\xi )$$, we see that

$$\Omega (\xi ) \frac{\sinh (t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }})}{\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}}=\Omega (\xi )\sum _{k=0}^{ \infty } \frac{t^{2k+1}\left ( 1-\left \vert \xi \right \vert ^{2\alpha }\right ) ^{k}}{\left ( 2k+1\right ) !}.$$

Hence it is easy to see that in the support of $$\Omega (\xi )$$, we have that for any multiindex σ,

\begin{aligned} &\left \vert \partial _{\xi }^{\sigma }\left ( \frac{\sinh (t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }})}{\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}} \right ) \right \vert \\ \preceq &\sum _{k=\left \vert \sigma \right \vert }^{\infty } \frac{t^{2k+1}k^{\left \vert \sigma \right \vert }\left ( 1/2\right ) ^{k}}{\left ( 2k+1\right ) !}\left \vert \xi \right \vert ^{-\left \vert \sigma \right \vert}. \end{aligned}

We note that for any N,

\begin{aligned} e^{-t} \frac{t^{2k+1}k^{\left \vert \sigma \right \vert }}{\left ( 2k+1\right ) !} =&t^{-N} \frac{e^{-t}t^{2k+N}k^{\left \vert \sigma \right \vert }\left ( 2k+N\right ) !}{\left ( 2k+N\right ) !\left ( 2k+1\right ) !} \\ \preceq &t^{-N} \frac{k^{\left \vert \sigma \right \vert }\left ( 2k+N\right ) !}{\left ( 2k+1\right ) !}, \end{aligned}

since

$$\frac{t^{2k+N}e^{-t}}{\left ( 2k+N\right ) !}\leq 1.$$

Thus we obtain that for any multi-index σ,

\begin{aligned} &\left \vert \partial _{\xi }^{\sigma } \frac{e^{-t}\Omega (\xi )\sinh (t\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }})}{\sqrt{1-\left \vert \xi \right \vert ^{2\alpha }}} \right \vert \\ \preceq &\left \vert \Omega _{0}(\xi )\right \vert t^{-N}\sum _{k= \left \vert \sigma \right \vert }^{\infty } \frac{\left ( 2k+N\right ) !k^{\left \vert \sigma \right \vert }\left ( 1/2\right ) ^{k}}{\left ( 2k+1\right ) !} \left \vert \xi \right \vert ^{-\sigma} \\ \preceq &\left \vert \Omega _{0}(\xi )\right \vert t^{-N}\left \vert \xi \right \vert ^{-\sigma} \end{aligned}

for any positive integer N, where $$\Omega _{0}(\xi )$$ is a Schwartz function supported in the set $$\{\xi :3/4\leq \left \vert \xi \right \vert \leq 4/3\}$$. Now it is easy to check that

$$\left \Vert \Omega (D)S(t)(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq t^{-\frac{n}{2\alpha}(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{n})}.$$
(3.16)

Finally, (3.13) follows from (3.15) and (3.16).

By Lemma 3.4 we may assume that suppΨ is contained in the set $$\left \{ \xi :\left \vert \xi \right \vert \geq 100\right \}$$. The boundedness of $$K_{\alpha ,\beta}\ast f$$ can be found in Theorems 2.3 and 2.4. Now we recall some well-known results.

### Theorem 3.5

([21])

For $$0<\alpha <1$$, the kernel $$K_{\alpha ,\beta}$$ has the estimates

$$K_{\alpha ,\beta}(x)=O(\left \vert x\right \vert ^{- \frac{\beta -n+\frac{n\alpha}{2}}{\alpha -1}}) \quad \textit{for} \quad \left \vert x\right \vert \leq 1$$

and

$$\left \vert K_{\alpha ,\beta}(x)\right \vert \preceq \left \vert x \right \vert ^{-N} \quad \textit{for} \quad \left \vert x\right \vert >1$$

for any positive integer N.

### Remarks 1

(1) In Theorem 3.5, let $$\beta =2\alpha$$ and $$n=3$$. Then

$$K_{\alpha ,\beta}(x)=O(1)\quad \text{for} \quad \left \vert x\right \vert \leq 1$$

if

$$\alpha \geq \frac{6}{7}.$$

Let $$\beta =2\alpha$$ and $$n=2$$. Then

$$K_{\alpha ,\beta}(x)=O(1) \quad \text{for}\quad \left \vert x\right \vert \leq 1$$

if

$$\alpha \geq \frac{2}{3}.$$

(2) If

$$0< \frac{\beta -n+\frac{n\alpha}{2}}{\alpha -1}< n,$$

then by Theorem 3.5 and the Sobolev imbedding theorem we know that

$$\left \Vert K_{\alpha ,\beta}\ast g\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}\leq \left \Vert g\right \Vert _{\dot{F}_{r}^{ \gamma ,q}(\mathbb{R}^{n})},$$

provided that

$$\frac{1}{r}-\frac{1}{p}\leq \frac{\beta -\frac{n\alpha}{2}}{n(1-\alpha )}.$$

We now turn to show (3.4). Write

\begin{aligned} &\Psi (D)S(t)g\left ( x\right ) -e^{-t}\Psi (D)W_{t,\alpha }\left ( g \right ) \left ( x\right ) \\ =&e^{-t}\int _{\mathbf{\mathbb{R} }^{n}}\Psi (\xi )\left ( \frac{\sin (t\left \vert \xi \right \vert ^{\alpha }\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }})}{\left \vert \xi \right \vert ^{\alpha }\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }}}-\frac{\sin t\left \vert \xi \right \vert ^{\alpha }}{\left \vert \xi \right \vert ^{\alpha }} \right ) \widehat{g}(\xi )e^{i< x,\xi >}d\xi \\ =&\int _{\mathbf{\mathbb{R} }^{n}}(m_{5}(t,\xi )\widehat{g}(\xi )e^{i< x,\xi >} +m_{6}(t,\xi ) \widehat{g}(\xi )e^{i< x,\xi >})d\xi , \end{aligned}

where

$$m_{5}(t,\xi )=e^{-t}\Psi (\xi )\left \{ \frac{\sin t\left \vert \xi \right \vert ^{\alpha }}{\left \vert \xi \right \vert ^{\alpha }} \left ( \frac{1-\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }}}{\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }}} \right ) \right \}$$
(3.17)

and

$$m_{6}(t,\xi )=e^{-t}\Psi (\xi )\left ( \frac{\sin (t\left \vert \xi \right \vert ^{\alpha }\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }})-\sin t\left \vert \xi \right \vert ^{\alpha }}{\left \vert \xi \right \vert ^{\alpha }\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }}}\right ) .$$
(3.18)

Without loss of generality, we may write

$$\Psi (\xi )=\Psi _{1}(\xi )\Psi _{2}(\xi ),$$

where $$\Psi _{1}(\xi )$$ and $$\Psi _{2}(\xi )$$ have the same smoothness and support as those of $$\Psi (\xi )$$. We further write

$$m_{5}(t,\xi )=m_{7}(t,\xi )m_{8}(t,\xi )$$

with

$$m_{7}(t,\xi )=e^{-t/2}\Psi _{1}(\xi ) \frac{\sin t\left \vert \xi \right \vert ^{\alpha }}{\left \vert \xi \right \vert ^{2\alpha }}$$

and

$$m_{8}(t,\xi )=e^{-t/2}\Psi _{2}(\xi ) \frac{\left ( 1-\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }}\right ) \left \vert \xi \right \vert ^{\alpha }}{\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }}}.$$

In the support of $$\Psi _{2}(\xi )$$, by the Taylor expansion,

\begin{aligned} \left ( 1-\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }}\right ) \left \vert \xi \right \vert ^{\alpha } =& \frac{1}{2\left \vert \xi \right \vert ^{\alpha }}-\frac{1}{8\left \vert \xi \right \vert ^{3\alpha }}+O(\left \vert \xi \right \vert ^{-3\alpha}) \\ =&\frac{1}{2\left \vert \xi \right \vert ^{\alpha }}+G(\left \vert \xi \right \vert ), \end{aligned}

where $$G(\left \vert \xi \right \vert )$$ is a $$C^{\infty }$$ function in the support of $$\Psi _{2}(\xi )$$ and satisfies

$$\left \vert \partial _{\xi }^{\sigma }G(\left \vert \xi \right \vert ) \right \vert \preceq \frac{1}{\left \vert \xi \right \vert ^{3\alpha +\left \vert \sigma \right \vert }}$$

for any multiindex σ. Thus it is easy to check that

$$\left \Vert T_{m_{8}(\cdot ,t)}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq t^{-N}\left \Vert g\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}.$$
(3.19)

Also, by Theorem 2.3 we obtain that

$$\left \Vert T_{m_{7}(t,\cdot )}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{n})}\preceq t^{-N}\left \Vert g\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})},$$
(3.20)

provided that

$$n\leq \frac{2}{\left \vert \frac{1}{p}-\frac{1}{2}\right \vert}.$$

Particularly, Remark (1) gives that if $$\alpha \geq 6/7$$, then

$$\left \Vert T_{m_{7}(t,\cdot )}(g)\right \Vert _{L^{\infty}( \mathbb{R}^{3})}\preceq e^{-\frac{t}{2}}t^{N_{0}}\left \Vert g\right \Vert _{L^{1}(\mathbb{R}^{3})},$$

and if $$\alpha \geq 2/3$$, then

$$\left \Vert T_{m_{7}(t,\cdot )}(g)\right \Vert _{L^{\infty}( \mathbb{R}^{2})}\preceq e^{-\frac{t}{2}}t^{N_{0}}\left \Vert g\right \Vert _{L^{1}(\mathbb{R}^{2})},$$

where $$N_{0}$$ is a large integer. Hence by Lemma 2.2 we have

\begin{aligned}& \left \Vert T_{m_{7}(t,\cdot )}(g)\right \Vert _{\dot{F}_{\infty}^{ \gamma ,q}(\mathbb{R}^{3})}\preceq e^{-\frac{t}{2}}t^{N_{0}}\left \Vert g\right \Vert _{\dot{F}_{1}^{\gamma ,q}(\mathbb{R}^{3})} \quad \text{for}\quad \alpha \geq \frac{6}{7}, \end{aligned}
(3.21)
\begin{aligned}& \left \Vert T_{m_{7}(t,\cdot )}(g)\right \Vert _{\dot{F}_{\infty}^{ \gamma ,q}(\mathbb{R}^{2})}\preceq e^{-\frac{t}{2}}t^{N_{0}}\left \Vert g\right \Vert _{\dot{F}_{1}^{\gamma ,q}(\mathbb{R}^{2})} \quad \text{for} \quad \alpha \geq \frac{2}{3}. \end{aligned}
(3.22)

Clearly, by interpolating between (3.20) and (3.21) we obtain that for $$1\leq r\leq p\leq \infty$$,

$$\left \Vert T_{m_{7}(t,\cdot )}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{3})}\preceq t^{-\frac{3}{2\alpha}(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{3})},$$
(3.23)

provided that $$\alpha \geq \frac{6}{7}$$.

Similarly, by an interpolation between (3.20) and (3.22) we obtain that for $$1\leq r\leq p\leq \infty$$,

$$\left \Vert T_{m_{7}(t,\cdot )}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{2})}\preceq t^{-\frac{1}{\alpha}(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{2})},$$
(3.24)

provided that $$\alpha \geq \frac{2}{3}$$.

Since $$m_{5}(t,\xi )=m_{7}(t,\xi )m_{8}(t,\xi )$$, by (3.19) and (3.23) we obtain

$$\left \Vert T_{m_{5}(t,\cdot )}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{3})}\preceq t^{-\frac{3}{2\alpha}(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{3})},$$

provided that $$\alpha \geq \frac{6}{7}$$.

Similarly, by (3.19) and (3.24) we have

$$\left \Vert T_{m_{5}(t,\cdot )}(g)\right \Vert _{\dot{F}_{p}^{\gamma ,q}( \mathbb{R}^{2})}\preceq t^{-\frac{1}{\alpha}(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{2})},$$

provided that $$\alpha \geq \frac{2}{3}$$.

Finally, for the multiplier $$m_{6}(t,\xi )$$, we write

\begin{aligned} m_{6}(t,\xi ) =&e^{-t}\Psi (\xi )\left ( \frac{\sin (t\left \vert \xi \right \vert ^{\alpha }\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }})-\sin t\left \vert \xi \right \vert ^{\alpha }}{\left \vert \xi \right \vert ^{\alpha }\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }}}\right ) \\ =&e^{-t}\Psi _{1}(\xi )\left ( \frac{\sin (t\left \vert \xi \right \vert ^{\alpha }\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }})-\sin t\left \vert \xi \right \vert ^{\alpha }}{\left \vert \xi \right \vert ^{\alpha }} \right ) \frac{\Psi _{1}(\xi )}{\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }}} \\ =&\mu _{1}(t,\xi )\cdot \mu _{2}\left ( t,\xi \right ) , \end{aligned}

where, without loss of generality, we assume that

$$\Psi (\xi )=\Psi _{1}(\xi )\Psi _{1}(\xi )$$

with $$\Psi _{1}$$ and Ψ having the same properties.

Again, by the multiplier theorem of $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ it is easy to see that $$\mu _{2}(\xi ,t)$$ is an $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ multiplier for any $$0< p$$, $$q<\infty$$ and

$$\left \Vert T_{\mu _{2}(\cdot ,t)}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}\preceq \left \Vert g\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}.$$

It remains to estimate $$T_{\mu _{1}}$$.

Recall that in

$$\mu _{1}(t,\xi )=e^{-t}\Psi _{1}(\xi )( \frac{\sin (t\left \vert \xi \right \vert ^{\alpha}\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha}})-\sin t\left \vert \xi \right \vert ^{\alpha}}{\left \vert \xi \right \vert ^{\alpha}}),$$

we can write

$$t\left \vert \xi \right \vert ^{\alpha}\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha}}=t\left \vert \xi \right \vert ^{\alpha}+tH( \left \vert \xi \right \vert )$$

where $$H\in C^{\infty}({\mathbb{R}^{n}\backslash \{0\}})$$ and

$$\left \vert \partial _{\xi}^{\sigma}H(\left \vert \xi \right \vert ) \right \vert \preceq \frac{1}{\left \vert \xi \right \vert ^{\alpha +\left \vert \sigma \right \vert}}$$

for any multiindex σ in the support of $$\Psi _{1}$$.

We have

\begin{aligned} &\sin (t\left \vert \xi \right \vert ^{\alpha }\sqrt{1-\left \vert \xi \right \vert ^{-2\alpha }})-\sin t\left \vert \xi \right \vert ^{ \alpha } \\ =&\sin (t\left \vert \xi \right \vert ^{\alpha })\cos \left ( tH( \left \vert \xi \right \vert )\right ) +\cos (t\left \vert \xi \right \vert ^{\alpha })\sin \left ( tH(\left \vert \xi \right \vert ) \right ) -\sin t\left \vert \xi \right \vert ^{\alpha } \\ =&\sin (t\left \vert \xi \right \vert ^{\alpha })\left \{ \cos \left ( tH(\left \vert \xi \right \vert )\right ) -1\right \} +\cos (t \left \vert \xi \right \vert ^{\alpha })\sin \left ( tH(\left \vert \xi \right \vert )\right ) . \end{aligned}

Hence

\begin{aligned} \mu _{1}(t,\xi ) =&e^{-t}\Psi _{1}(\xi ) \frac{\sin (t\left \vert \xi \right \vert ^{\alpha })\left \{ \cos \left ( tH(\left \vert \xi \right \vert )\right ) -1\right \} }{\left \vert \xi \right \vert ^{\alpha }}+e^{-t} \Psi _{1}(\xi ) \frac{\cos (t\left \vert \xi \right \vert ^{\alpha })\sin \left ( tH(\left \vert \xi \right \vert )\right ) }{\left \vert \xi \right \vert ^{\alpha }} \\ =&e^{-t}\Psi _{1}(\xi ) \frac{\sin (t\left \vert \xi \right \vert ^{\alpha })}{\left \vert \xi \right \vert ^{2\alpha }}\cdot \left \{ \cos \left ( tH( \left \vert \xi \right \vert )\right ) -1\right \} \left \vert \xi \right \vert ^{\alpha } \\ &+e^{-t}\Psi _{1}(\xi ) \frac{\cos (t\left \vert \xi \right \vert ^{\alpha })}{\left \vert \xi \right \vert ^{2\alpha }}\cdot \sin \left ( tH(\left \vert \xi \right \vert )\right ) \left \vert \xi \right \vert ^{ \alpha }. \end{aligned}

We may again assume that

$$\Psi _{1}(\xi )=\Psi _{2}(\xi )\Psi _{2}(\xi ),$$

where $$\Psi _{2}$$ has the same properties as Ψ. We have that

$$\mu _{1}(t,\xi )=\mu _{3}(t,\xi )\mu _{4}(t,\xi )+\mu _{5}(t,\xi ) \mu _{6}(t,\xi ),$$

where

\begin{aligned} \mu _{3}(t,\xi ) =&e^{-t/2}\Psi _{2}(\xi ) \frac{\sin (t\left \vert \xi \right \vert ^{\alpha })}{\left \vert \xi \right \vert ^{2\alpha }}, \\ \mu _{4}(t,\xi ) =&e^{-t/2}\Psi _{2}(\xi )\left \{ \cos \left ( tH( \left \vert \xi \right \vert )\right ) -1\right \} \left \vert \xi \right \vert ^{\alpha }, \\ \mu _{5}(t,\xi ) =&e^{-t/2}\Psi _{2}(\xi ) \frac{\cos (t\left \vert \xi \right \vert ^{\alpha })}{\left \vert \xi \right \vert ^{2\alpha }}, \\ \mu _{6}(t,\xi ) =&e^{-t/2}\Psi _{2}(\xi )\sin \left ( tH(\left \vert \xi \right \vert )\right ) \left \vert \xi \right \vert ^{ \alpha }. \end{aligned}

By the multiplier theorem for $$\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})$$ it is easy to check that

\begin{aligned}& \left \Vert T_{\mu _{4}(\cdot ,t)}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}\preceq (1+t)^{-N}\left \Vert g\right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})},\\& \left \Vert T_{\mu _{6}(\cdot ,t)}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{n})}\preceq (1+t)^{-N}\left \Vert g\right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}. \end{aligned}

Now by Theorem 3.5 and Theorem 2.3, together with (1) in Remarks, we finally obtain

$$\left \Vert T_{\mu _{4}(\cdot ,t)}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{3})}+\left \Vert T_{\mu _{6}(\cdot ,t)}(g) \right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{3})}\preceq (1+t)^{-N} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{3})},$$

provided that $$\alpha \geq \frac{6}{7}$$, and

$$\left \Vert T_{\mu _{4}(\cdot ,t)}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{2})}+\left \Vert T_{\mu _{6}(\cdot ,t)}(g) \right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{2})}\preceq (1+t)^{-N} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{2})},$$

provided that $$\alpha \geq \frac{2}{3}$$.

The proposition is proved.

### 3.4 Proof of Theorem 1.5

Let

$$\Omega (x,t)=\lim _{\sigma \rightarrow 0^{+}}\int _{\mathbb{R}^{4}} \frac{e^{-\sigma \left \vert \xi \right \vert} e^{it\left \vert \xi \right \vert}}{\left \vert \xi \right \vert ^{2}} \Psi (\xi )e^{i< x,\xi >}d\xi ,$$

and let

$$T_{t}(f)(x)=\Omega (\cdot ,\,t)\ast f(x).$$

By Theorem 2.4 (see also [16]) it is easy to check the following:

### Proposition 3.6

$$T_{1}(f)(x)$$ is bounded on $$H^{p}(\mathbb{R}^{4})$$ if and only if

$$\left \vert \frac{1}{p}-\frac{1}{2}\right \vert \leq \frac{2}{3},$$

and $$T_{1}(f)(x)$$ is bounded on $$L^{1}(\mathbb{R}^{4})$$.

We also need the following lemma.

### Lemma 3.7

Let $$n=4$$, and let

$$\Omega (x,t)=\lim _{\sigma \rightarrow 0^{+}}\int _{\mathbb{R}^{4}} \frac{e^{-\sigma \left \vert \xi \right \vert }e^{it\left \vert \xi \right \vert}}{\left \vert \xi \right \vert ^{2}} \Psi (\xi )e^{i< x,\xi >}d\xi .$$

Fixing $$t_{0}$$, for $$t>t_{0}$$ and any $$N>0$$, we have

\begin{aligned}& \left \vert \Omega (x,t)\right \vert \preceq \frac{1}{\left \vert x\right \vert ^{2}} \quad \textit{for} \quad \left \vert x\right \vert < \frac{t}{2},\\& \left \vert \Omega (x,t)\right \vert \preceq \left \vert x\right \vert ^{-N} \quad \textit{for} \quad \left \vert x\right \vert \geq 2t,\\& \left \vert \Omega (x,t)\right \vert \preceq \left \vert t-\left \vert x\right \vert \right \vert ^{-\frac{1}{2}}t^{-\frac{3}{2}} \quad \textit{for} \quad \frac{t}{2}\leq \left \vert x\right \vert < 2t. \end{aligned}

Proof: Without loss of generality, we may assume $$t_{0}=1$$. Let

$$V_{\nu }\left ( r\right ) = \frac{J_{\nu }\left ( r\right ) }{r^{\nu }},$$

where $$J_{\nu }\left ( r\right )$$ is the Bessel function of order ν. From [22] we have

\begin{aligned} \Omega \left ( x,t\right ) =&\lim _{\sigma \rightarrow 0+}\int _{0}^{ \infty }\frac{e^{-\sigma r}e^{itr}}{r^{-1}}\Psi (r)V_{1}\left ( r\left \vert x \right \vert \right ) dr \\ \approx &\frac{1}{\left \vert x\right \vert }\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty }e^{-\sigma r}e^{itr}\Psi (r)J_{1} \left ( r\left \vert x\right \vert \right ) dr, \end{aligned}

where the Bessel function has the following asymptotic expansion on the interval $$[1,\infty )$$:

$$J_{\nu }(r)=\sqrt{\frac{2}{\pi r}}cos(r-\frac{\pi \nu }{2}- \frac{\pi }{4})+\sum _{j=1}^{L}a_{j}e^{ir}r^{-\frac{1}{2}-j}+\sum _{j=1}^{L}b_{j}e^{-ir}r^{-\frac{1}{2}-j}+\frac{e^{\pm ir}}{r^{\frac{3}{2}+L}}E_{L}(r),$$

where $$a_{j}$$, $$b_{j}$$ are constants for all j, and $$E_{L}$$ is a $$C^{\infty }$$ function satisfying

$$\frac{d^{k}}{dr^{k}}E_{L}(r)=O(r^{-k})\quad \text{ as }r\rightarrow \infty$$

for $$k=0,\,1,\,2,\dots$$ and any positive integer L.

Using the asymptotic expansion of $$J_{1}$$, for $$\left \vert x\right \vert >1$$, we have

\begin{aligned} \Omega \left ( x,t\right ) =& \frac{a_{0}}{\left \vert x\right \vert ^{\frac{3}{2}}}\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{1/2}}\Psi (r)e^{ir\left \vert x\right \vert }dr \\ &+\frac{b_{0}}{\left \vert x\right \vert ^{\frac{3}{2}}}\lim _{ \sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{1/2}}\Psi (r)e^{-ir\left \vert x \right \vert }dr \\ &+\sum _{j=1}^{L} \frac{a_{j}}{\left \vert x\right \vert ^{\frac{3}{2}+j}}\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{1/2+j}}\Psi (r)e^{ir\left \vert x\right \vert }dr \\ &+\sum _{j=1}^{L} \frac{b_{j}}{\left \vert x\right \vert ^{\frac{3}{2}+j}}\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{1/2+j}}\Psi (r)e^{-ir\left \vert x\right \vert }dr \\ &+\lim _{\sigma \rightarrow 0+} \frac{1}{\left \vert x\right \vert ^{\frac{3}{2}+1+L}}\int _{0}^{\infty }\frac{e^{-\sigma r}e^{itr}}{r^{1/2+1+L}}E_{L}(r) \Psi (r)e^{\pm ir\left \vert x\right \vert }dr, \end{aligned}

where

$$a_{0}=\frac{1}{2}e^{-i3\pi /4},\text{ \ }b_{0}=\frac{1}{2}e^{i3\pi /4}.$$

We will only check the second term:

$$\frac{b_{0}}{\left \vert x\right \vert ^{\frac{3}{2}}}\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{1/2}}\Psi (r)e^{-ir\left \vert x \right \vert }dr= \frac{b_{0}}{\left \vert x\right \vert ^{\frac{3}{2}}}\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{ir\left ( t-\left \vert x\right \vert \right ) }}{r^{1/2}} \Psi (r)dr,$$

since we can obtain the same estimates for all other terms.

Invoking integration by parts, we easily see that for $$\left \vert x\right \vert >2t$$,

$$\left \vert \frac{b_{0}}{\left \vert x\right \vert ^{\frac{3}{2}}} \lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{ir\left ( t-\left \vert x\right \vert \right ) }}{r^{1/2}} \Psi (r)dr\right \vert =O\left ( \left \vert x\right \vert ^{-N} \right )$$

for any $$N>0$$.

We estimate $$\Omega \left ( x,t\right )$$ for $$\left \vert x\right \vert \leq \frac{t}{2}$$.

Write

\begin{aligned} \Omega \left ( x,t\right ) =&\lim _{\sigma \rightarrow 0+}\int _{0}^{ \infty }\frac{e^{-\sigma r}e^{itr}}{r^{-1}}\Phi \left ( r\left \vert x\right \vert \right ) \Psi (r)V_{1}\left ( r\left \vert x\right \vert \right ) dr \\ &+\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{-1}}\Pi \left ( r\left \vert x\right \vert \right ) \Psi (r)V_{1} \left ( r\left \vert x\right \vert \right ) dr, \end{aligned}

where Φ is the function defined as before and is supported in the set $$\{\xi :\left \vert \xi \right \vert \leq 3/4\}$$, and

$$\Pi \left ( r\left \vert x\right \vert \right ) =1-\Phi \left ( r \left \vert x\right \vert \right ) .$$

Using the support condition of Φ and the property of the Bessel function

$$V_{\gamma }\left ( r\right ) =O(1),$$

we then have that

\begin{aligned} &\lim _{\sigma \rightarrow 0+}\left \vert \int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{-1}}\Phi \left ( r\left \vert x\right \vert \right ) \Psi (r)V_{1}\left ( r\left \vert x\right \vert \right ) dr\right \vert \\ \preceq &\left \vert \int _{2}^{1/\left \vert x\right \vert }rdr \right \vert \preceq \left \vert x\right \vert ^{-2}. \end{aligned}

In another integral

\begin{aligned} \lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{-1}}\Pi \left ( r\left \vert x\right \vert \right ) \Psi (r)V_{1} \left ( r\left \vert x\right \vert \right ) dr, \end{aligned}

we notice that supp$$\Psi (r)\subset \left \{ r:r>100\right \}$$ and supp$$\Pi \left ( r\left \vert x\right \vert \right ) \subset \left \{ r:r \geq \frac{1}{3\left \vert x\right \vert }\right \}$$. We may write

\begin{aligned} &\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{-1}}\Pi \left ( r\left \vert x\right \vert \right ) \Psi (r)V_{1} \left ( r\left \vert x\right \vert \right ) dr \\ =&\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{-1}}\Pi \left ( r\left \vert x\right \vert \right ) V_{1}\left ( r \left \vert x\right \vert \right ) dr. \end{aligned}

By the asymptotic expansion of Bessel function we have that

$$\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{itr}}{r^{-1}}\Pi \left ( r\left \vert x\right \vert \right ) V_{1}\left ( r \left \vert x\right \vert \right ) dr$$
\begin{aligned} =&\sum _{j=0}^{L}\frac{a_{j}}{\left \vert x\right \vert ^{3/2+j}} \lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{ir\left ( t+\left \vert x\right \vert \right ) }}{r^{1/2+j}} \Pi \left ( r\left \vert x\right \vert \right ) dr \\ &+\sum _{j=0}^{L}\frac{b_{j}}{\left \vert x\right \vert ^{3/2+j}} \lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{ir\left ( t-\left \vert x\right \vert \right ) }}{r^{1/2+j}} \Pi \left ( r\left \vert x\right \vert \right ) dr \\ &+\lim _{\sigma \rightarrow 0+} \frac{1}{\left \vert x\right \vert ^{\frac{5}{2}+L}}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{ir\left ( t+\left \vert x\right \vert \right ) }}{r^{\frac{3}{2}+L}} \Pi \left ( r\left \vert x\right \vert \right ) dr. \end{aligned}

Estimates of all above terms are the same, so we will only work on the leading term

$$\frac{b_{0}}{\left \vert x\right \vert ^{3/2}}\lim _{\sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{ir\left ( t-\left \vert x\right \vert \right ) }}{r^{1/2}} \Pi \left ( r\left \vert x\right \vert \right ) \Psi (r)dr.$$

An integration by parts yields that

\begin{aligned} &\left \vert \frac{b_{0}}{\left \vert x\right \vert ^{3/2}}\lim _{ \sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{ir\left ( t-\left \vert x\right \vert \right ) }}{r^{1/2}} \Pi \left ( r\left \vert x\right \vert \right ) \Psi (r)dr\right \vert \\ \preceq & \left \vert x\right \vert ^{-2}\text{ \ \ \ \ for }\left \vert x\right \vert < t/2, \end{aligned}

and for $$t/2\leq \left \vert x\right \vert <2t$$,

\begin{aligned} \left \vert \frac{b_{0}}{\left \vert x\right \vert ^{3/2}}\lim _{ \sigma \rightarrow 0+}\int _{0}^{\infty } \frac{e^{-\sigma r}e^{ir\left ( t-\left \vert x\right \vert \right ) }}{r^{1/2}} \Pi \left ( r\left \vert x\right \vert \right ) \Psi (r)dr\right \vert \preceq &\left \vert x\right \vert ^{-3/2}\left \vert t-\left \vert x\right \vert \right \vert ^{-1/2}. \end{aligned}

Lemma 3.7 is proved.

Now we are in a position to prove Theorem 1.5. Since Proposition 3.1 holds for all dimensions n and all $$\alpha >0$$, it suffices to show that for $$\alpha =1$$,

$$\left \Vert \Psi (D)S(t)g-e^{-t}\Psi (D)W_{t}(g)\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{4})}\preceq t^{-2(\frac{1}{r}- \frac{1}{p})-1}\left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}( \mathbb{R}^{4})}$$

for $$t>t_{0}>0$$ and $$1\leq r\leq p<\infty$$, provided that

$$\frac{1}{r}-\frac{1}{p}< \frac{1}{2}.$$

Recall that (see the above argument on $$0<\alpha <1$$)

\begin{aligned} &\Psi (D)S(t)g(x)-e^{-t}\Psi (D)W_{t}(g)(x) \\ &=e^{-t}\int _{\mathbb{R}^{n}}\Psi (\xi )( \frac{\sin (t\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}})}{\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}}}- \frac{\sin t\left \vert \xi \right \vert}{\left \vert \xi \right \vert}) \widehat{g}(\xi )e^{i< x,\xi >}d\xi \\ &=\int _{\mathbb{R}^{n}}(\widetilde{m_{5}}(t,\xi )+\widetilde{m_{6}}(t, \xi ))\widehat{g}(\xi )e^{i< x,\xi >}d\xi , \end{aligned}

where

$$\widetilde{m_{5}}(t,\xi )=e^{-t}\Psi (\xi ) \frac{\sin t\left \vert \xi \right \vert}{\left \vert \xi \right \vert}( \frac{1-\sqrt{1-\left \vert \xi \right \vert ^{-2}}}{\sqrt{1-\left \vert \xi \right \vert ^{-2}}})$$

and

$$\widetilde{m_{6}}(t,\xi )=e^{-t}\Psi (\xi )( \frac{\sin (t\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}})-\sin t\left \vert \xi \right \vert}{\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}}}).$$

Similar to the case of $$0<\alpha <1$$, we can get

$$\left \Vert T_{\widetilde{m_{5}}(t,\cdot )}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{4})}\preceq t^{-2(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{4})}.$$
(3.25)

Finally, for the multiplier $$\widetilde{m_{6}}(t,\xi )$$, we write

\begin{aligned} \widetilde{m_{6}}(t,\xi ) & =e^{-t}\Psi (\xi )( \frac{\sin (t\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}}-\sin t\left \vert \xi \right \vert )}{\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}}}) \\ & =e^{-t}\Psi _{1}(\xi )( \frac{\sin (t\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}})-\sin t\left \vert \xi \right \vert}{\left \vert \xi \right \vert}) \frac{\Psi _{1}(\xi )}{\sqrt{1-\left \vert \xi \right \vert ^{-2}}} \\ &= \widetilde{\mu _{1}}(t,\xi )\cdot \widetilde{\mu _{2}}(t,\xi ), \end{aligned}

where we assume that

$$\Psi (\xi )=\Psi _{1}(\xi )\Psi _{1}(\xi )$$

with $$\Psi _{1}$$ and Ψ having the same properties.

Again, by the multiplier theorem it is easy to see that

$$\left \Vert T_{\widetilde{\mu _{2}}(\cdot ,t)}(g)\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}\preceq \left \Vert g\right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}.$$

Recall that in

$$\widetilde{\mu _{1}}(t,\xi )=e^{-t}\Psi _{1}(\xi )\left ( \frac{\sin (t\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}})-\sin t\left \vert \xi \right \vert }{\left \vert \xi \right \vert } \right ) ,$$

we can write

$$t\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}}=t \left \vert \xi \right \vert +th(\left \vert \xi \right \vert ),$$

where $$h\in C^{\infty}(\mathbb{R}^{n}\backslash \{0\})$$, and

$$\left \vert \partial _{\xi}^{\sigma}h(\left \vert \xi \right \vert ) \right \vert \preceq \frac{1}{\left \vert \xi \right \vert ^{1+\left \vert \sigma \right \vert}}$$

for any multiindex σ in the support of $$\Psi _{1}$$.

We may write

\begin{aligned} &\sin (t\left \vert \xi \right \vert \sqrt{1-\left \vert \xi \right \vert ^{-2}})-\sin t\left \vert \xi \right \vert \\ =&\sin (t\left \vert \xi \right \vert )\left \{ \cos \left ( th( \left \vert \xi \right \vert )\right ) -1\right \} +\cos (t\left \vert \xi \right \vert )\sin \left ( th(\left \vert \xi \right \vert ) \right ) . \end{aligned}

Hence

\begin{aligned} \widetilde{\mu _{1}}(t,\xi ) =&e^{-t}\Psi _{1}(\xi ) \frac{\sin (t\left \vert \xi \right \vert )}{\left \vert \xi \right \vert ^{2}} \cdot \left \{ \cos \left ( th(\left \vert \xi \right \vert )\right ) -1 \right \} \left \vert \xi \right \vert \\ &+e^{-t}\Psi _{1}(\xi ) \frac{\cos (t\left \vert \xi \right \vert )}{\left \vert \xi \right \vert ^{2}}\cdot \sin \left ( th(\left \vert \xi \right \vert )\right ) \left \vert \xi \right \vert . \end{aligned}

We may again assume that

$$\Psi _{1}(\xi )=\Psi _{2}(\xi )\Psi _{2}(\xi ),$$

where $$\Psi _{2}$$ has the same properties as Ψ.

We have that

$$\widetilde{\mu _{1}}(t,\xi )=\widetilde{\mu _{3}}(t,\xi ) \widetilde{\mu _{4}}(t,\xi )+\widetilde{\mu _{5}}(t,\xi ) \widetilde{\mu _{6}}(t,\xi ),$$

where

\begin{aligned} \widetilde{\mu _{3}}(t,\xi ) =&e^{-t/2}\Psi _{2}(\xi ) \frac{\sin (t\left \vert \xi \right \vert )}{\left \vert \xi \right \vert ^{2}}, \\ \widetilde{\mu _{4}}(t,\xi ) =&e^{-t/2}\Psi _{2}(\xi )\left \{ \cos \left ( th(\left \vert \xi \right \vert )\right ) -1\right \} \left \vert \xi \right \vert , \\ \widetilde{\mu _{5}}(t,\xi ) =&e^{-t/2}\Psi _{2}(\xi ) \frac{\cos (t\left \vert \xi \right \vert )}{\left \vert \xi \right \vert ^{2}}, \\ \widetilde{\mu _{6}}(t,\xi ) =&e^{-t/2}\Psi _{2}(\xi )\sin \left ( th( \left \vert \xi \right \vert )\right ) \left \vert \xi \right \vert . \end{aligned}

By the multiplier theorem it is easy to check that

\begin{aligned}& \left \Vert T_{\widetilde{\mu _{4}}(\cdot ,t)}(g)\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}\preceq (1+t)^{-N}\left \Vert g\right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})},\\& \left \Vert T_{\widetilde{\mu _{6}}(\cdot ,t)}(g)\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}\preceq (1+t)^{-N}\left \Vert g\right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{n})}. \end{aligned}

Next, we only work on

$$T_{\widetilde{\mu _{3}}(\cdot ,t)}(g)(x)=e^{-\frac{t}{2}}\int _{ \mathbb{R}^{n}}\Psi _{2}(\xi ) \frac{\sin (t\left \vert \xi \right \vert )}{\left \vert \xi \right \vert ^{2}} \widehat{g}(\xi )e^{i< x,\xi >}d\xi ,$$

since the estimate of $$T_{\widetilde{\mu _{5}}}$$ is the same.

By Lemma 3.7 and generalized Young’s inequality we know that

\begin{aligned} \left \Vert \int _{\mathbb{R}^{n}}\Psi _{2}(\xi ) \frac{\sin (t\left \vert \xi \right \vert )}{\left \vert \xi \right \vert ^{2}} \widehat{g}(\xi )e^{i< \cdot ,\xi >}\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{4})} &\approx \left \Vert \Omega (\cdot ,t) \ast g\right \Vert _{\dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{4})} \\ &\preceq \left \Vert \Omega (\cdot ,t)\right \Vert _{L^{s} ( \mathbb{R}^{4})}\left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}( \mathbb{R}^{4})}, \end{aligned}

where

$$\frac{1}{r}-\frac{1}{p}=1-\frac{1}{s}$$

for s arbitrarily close to 2 and less than 2. We write

$$s=2-\epsilon ,$$

where ε is an arbitrarily small positive number. Now we have that by Lemma 3.7

$$\left \Vert \Omega (\cdot ,t)\right \Vert _{L^{s}(\mathbb{R}^{4})}^{s}=O(1).$$

Therefore we obtain that

$$\left \Vert e^{-\frac{t}{2}} \Omega (\cdot ,t)\ast g\right \Vert _{ \dot{F}_{p}^{\gamma ,q}(\mathbb{R}^{4})}\preceq t^{-N}\left \Vert g \right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{4})}$$

for any

$$0\leq \frac{1}{r}-\frac{1}{p}< \frac{1}{2}$$

and $$N>0$$.

Furthermore, by combining all estimates we obtain that

$$\left \Vert T_{\widetilde{m_{6}}(t,\cdot )}(g)\right \Vert _{\dot{F}_{p}^{ \gamma ,q}(\mathbb{R}^{4})}\preceq t^{-2(\frac{1}{r}-\frac{1}{p})-1} \left \Vert g\right \Vert _{\dot{F}_{r}^{\gamma ,q}(\mathbb{R}^{4})}$$

whenever

$$0\leq \frac{1}{r}-\frac{1}{p}< \frac{1}{2}.$$

Theorem 1.5 is proved.

The damped wave equation has the diffusive structures as $$t\rightarrow \infty$$. The estimates in the paper indicate the decay behavior of the diffusion. Problem $$(\widetilde{A})$$ with $$n=1$$ related to the asymptotic behavior of solutions to the system of the compressible flow through porous media. Also, the asymptotic estimates we obtained may potentially study the Cauchy problem for the semilinear damped fractional wave equation

$${(\widetilde{D} )}\quad \quad \left \{ \textstyle\begin{array}{ll} \partial _{tt}u+2u_{t}+(-\Delta )^{\alpha }u=\left \vert u\right \vert ^{\alpha}u,~~~\alpha >0, & \\ u(0,x)=f(x),~u_{t}(0,x)=g(x). & \end{array}\displaystyle \right .$$

See [1719].

## Availability of data and materials

This is not applicable in our parper.

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## Acknowledgements

Discussion with professor Weichao Guo led us to clarifying the generalized Young inequality. We thank him for his help.

## Funding

This work was supported by the National Key Research and Development Program of China (No. 2022YFA1005700); the Natural Science Foundation of Guangdong Province (No.2023A1515012034) and the National Natural Science Foundation of China (Nos. 12371105, 11971295).

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Correspondence to Meizhong Wang.

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Wang, M., Fan, D. Asymptotic estimates of solution to damped fractional wave equation. J Inequal Appl 2024, 100 (2024). https://doi.org/10.1186/s13660-024-03181-7