Triebel–Lizorkin space estimates for evolution equations with structure dissipation

Abstract

In this paper, we are concerned with the generalized wave equations in the homogeneous Triebel–Lizorkin spaces. The long time decay estimates are obtained by the decomposition of the unit, duality property and the multiplier theorems, extending the known results for the generalized wave equations with structure dissipation in the real Hardy spaces.

Appendix

Firstly, let us recall the characterization of $H^{p}(\mathbb {R}^{n})$ using the Riesz transforms. Let $R_{J},J=(j_{1},\ldots ,j_{s})\in \{0,1,\ldots ,n\}^{s},$ be the generalized Riesz transform of order s,  i.e. the Fourier multiplier transformation $T_{m}$ with

$$\begin{aligned} m(\xi )=m_{J}(\xi )=\left( -i\frac{\xi _{j_{1}}}{|\xi |}\right) \cdots \left( -i\frac{\xi _{j_{s}}}{|\xi |}\right) \ \ \ \xi \in \mathbb {R}^{n}, \end{aligned}$$

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where the factor $(-i\xi _{j_{1}}/|\xi |)$ shall be replaced by 1 if $j_i=0,$ where $i\in \{0,1,\ldots ,n\}$ [11].

With this assumption, we have the following theorem.

Theorem 5

([11]) Let $p > (n-1)/(n-1+s).$ Then $f\in L^2(\mathbb {R} ^{n}) \bigcap H^p(\mathbb {R}^{n})$ if and only if $R_Jf\in L^2(\mathbb {R} ^{n}) \bigcap L^p(\mathbb {R}^{n})$ for all $J\in \{0, 1, \ldots , n\}^s;$ and there exist constants C and $C^{^{\prime }}$ depending only on p, n,  and s such that

$$\begin{aligned} C \sum _J \Vert R_J f \Vert _{L^p} \le \Vert f \Vert _{H^p} \le C^{^{\prime }} \Vert R_J f \Vert _{L^p}, \ \ \ \ f\in L^2(\mathbb {R}^{n}) \bigcap H^p(\mathbb {R}^{n}). \end{aligned}$$

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Next, we recall a multiplier theorem of $H^{p}(\mathbb {R}^{n})$ in [11], where $\mathcal {M}(H^{p}(\mathbb {R}^{n}))$ denotes the set of all Fourier multipliers on $H^{p}(\mathbb {R}^{n}).$

Theorem 6

([11]) Let $c\ge 0, d\ge 0, 0< p_0 < 2, nd(1/p_0 - 1/2) = c,$ and $k=\max \{[n(1/p_0 - 1/2)] + 1, [n/2] + 1\}.$ Suppose that $m \in C^k(\mathbb {R}^n\setminus \{0\}), m(\xi )=0$ if $|\xi | \ge 1,$ and

$$\begin{aligned} \Big |\big (\frac{\partial }{\partial \xi }\big )^{\alpha }m(\xi )\Big | \le |\xi |^{c}(A|\xi |^{-1-d})^{|\alpha |}, \ \ \ |\alpha | \le k, \end{aligned}$$

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with some constant $A \ge 1.$ Then $m \in \mathcal {M}(H^p(\mathbb {R}^n))$ and

$$\begin{aligned} \Vert m\Vert _{\mathcal {M}(H^p(\mathbb {R}^n))} \le C A^{n(1/p - 1/2)}. \end{aligned}$$

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for $2 \ge p \ge p_0,$ where C is a constant independent of A.