Elsevier

Journal of Functional Analysis

Volume 255, Issue 10, 15 November 2008, Pages 2760-2809
Journal of Functional Analysis

A new class of function spaces connecting Triebel–Lizorkin spaces and Q spaces

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Abstract

Let sR, τ[0,), p(1,) and q(1,]. In this paper, we introduce a new class of function spaces F˙p,qs,τ(Rn) which unify and generalize the Triebel–Lizorkin spaces with both p(1,) and p= and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel–Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and Qα(Rn), J. Funct. Anal. 208 (2004) 377–422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces FT˙p,qs,τ(R+n+1) and determine their dual spaces FW˙p,qs,τ/q(Rn), where sR, p,q[1,), max{p,q}>1, τ[0,q(max{p,q})], and t denotes the conjugate index of t(1,); as an application of this, we further introduce certain Hardy–Hausdorff spaces FH˙p,qs,τ(Rn) and prove that the dual space of FH˙p,qs,τ(Rn) is just F˙p,qs,τ/q(Rn) when p,q(1,).

Keywords

Triebel–Lizorkin space
Q space
Tent space
Calderón reproducing formula
Capacity
Dual space

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