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

### Dachun Yang

Beijing Normal University, China### Wen Yuan

Beijing Normal University, China

## Abstract

Let $s∈R$, $p∈(1,∞)$, $τ∈[0,p1 ]$ and $S_{∞}(R_{n})$ be the set of all Schwartz functions $φ$ whose Fourier transforms $φ $ satisfy that $∂_{γ}φ (0)=0$ for all $γ∈(N∪{0})_{n}$. Denote by $_{V}F˙_{p,p}((R_{n})$ the closure of $S_{∞}(R_{n}))$ in the Triebel–Lizorkin-type space $F˙_{p,p}((R_{n})$. In this paper, the authors prove that the dual space of $_{V}F˙_{p,p}((R_{n})$ is the Triebel–Lizorkin–Hausdorff space $FH˙_{p_{′},p_{′}}(R_{n})$ via their $φ$-transform characterizations together with the atomic decomposition characterization of the tent space $FT˙_{p_{′},p_{′}}(R_{Z})$, where $t_{′}$ denotes the conjugate index of $t∈[1,∞]$. This gives a generalization of the well-known duality that $(CMO(R_{n}))_{∗}=H_{1}(R_{n})$ by taking $s=0$, $p=2$ and $τ=21 $ . 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 $FH˙_{p,p}(R_{n})$; all of these results improve the existing conclusions.

## Cite this article

Dachun Yang, Wen Yuan, Dual Properties of Triebel–Lizorkin-Type Spaces and their Applications. Z. Anal. Anwend. 30 (2011), no. 1, pp. 29–58

DOI 10.4171/ZAA/1422