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

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.