Dual Properties of Triebel–Lizorkin-Type Spaces and their Applications

Abstract

Let $s\in R$, $p\in (1,\infty )$, $\tau \in [0,\frac{1}{p}]$ and ${\mathcal{S}}_{\infty }(R^n)$ be the set of all Schwartz functions $\phi$ whose Fourier transforms $\hat{\phi }$ satisfy that ${\partial }^{\gamma }\hat{\phi }(0)=0$ for all $\gamma \in (N\cup \{0\}{)}^n$. Denote by ${}_V{\dot{F}}_{p,p}^{s,\tau }((R^n)$ the closure of ${\mathcal{S}}_{\infty }(R^n))$ in the Triebel–Lizorkin-type space ${\dot{F}}_{p,p}^{s,\tau }((R^n)$. In this paper, the authors prove that the dual space of ${}_V{\dot{F}}_{p,p}^{s,\tau }((R^n)$ is the Triebel–Lizorkin–Hausdorff space $F{\dot{H}}_{p^{\prime },p^{\prime }}^{-s,\tau }(R^n)$ via their $\phi$-transform characterizations together with the atomic decomposition characterization of the tent space $F{\dot{T}}_{p^{\prime },p^{\prime }}^{-s,\tau }(R_{\mathbb{Z}}^{n_{+}1})$, where $t^{\prime }$ denotes the conjugate index of $t\in [1,\infty ]$. This gives a generalization of the well-known duality that $(\mathrm{CMO}(R^n){)}^{\ast }=H^1(R^n)$ by taking $s=0$, $p=2$ and $\tau =\frac{1}{2}$ . As applications, the authors obtain the Sobolev-type embedding property, the smooth atomic and molecular decomposition characterizations, boundednesses of both pseudo-differential operators and the trace operators on $F{\dot{H}}_{p,p}^{s,\tau }(R^n)$; all of these results improve the existing conclusions.