# 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, 1/*p*] and *S_∞(ℝ_n*) be the set of all Schwartz functions φ whose Fourier transforms φ^ satisfy that ∂γφ^(0) = 0 for all γ ∈ (ℕ ∪ {0})*n*. Denote by *VF_sτ_p,p* ((ℝ_n_) the closure of *S_∞(ℝ_n*)) in the Triebel–Lizorkin-type space *F_sτ_p,p* ((ℝ_n_). In this paper, the authors prove that the dual space of *VF_sτ_p,p* ((ℝ_n_) is the Triebel–Lizorkin–Hausdorff space *FH_sτ_p',p'* ((ℝ_n_) via their φ -transform characterizations together with the atomic decomposition characterization of the tent space *FT_sτ_p',p'* ((ℝ_n_+1)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 = 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 *FH_sτ_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