Zeitschrift für Analysis und ihre Anwendungen

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Volume 30, Issue 1, 2011, pp. 29–58
DOI: 10.4171/ZAA/1422

Published online: 2011-01-03

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

Dachun Yang[1] and Wen Yuan[2]

(1) Beijing Normal University, China
(2) Beijing Normal University, China

Let s ∈ ℝ,| p ∈ (1,∞), τ ∈ [0, 1/p] and S(ℝn) 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 VFp,p ((ℝn) the closure of S(ℝn)) in the Triebel–Lizorkin-type space Fp,p ((ℝn). In this paper, the authors prove that the dual space of VFp,p ((ℝn) is the Triebel–Lizorkin–Hausdorff space FHp',p' ((ℝn) via their φ -transform characterizations together with the atomic decomposition characterization of the tent space FTp',p' ((ℝn+1)Z), where t′ denotes the conjugate index of t ∈ [1,∞]. This gives a generalization of the well-known duality that (CMO(ℝn))* = H1(ℝn) 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 FHp,p ((ℝn); all of these results improve the existing conclusions.

Keywords: Hausdorff capacity, Besov space, Triebel–Lizorkin space, tent space, duality, atom, molecule, embedding, pseudo-differential operator, trace

Yang Dachun, Yuan Wen: Dual Properties of Triebel-Lizorkin-Type Spaces and their Applications. Z. Anal. Anwend. 30 (2011), 29-58. doi: 10.4171/ZAA/1422