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

Let s ∈ R, τ ∈ [0,∞), p ∈ (1,∞) and q ∈ (1,∞]. In this paper, we introduce a new class of function spaces ˙ F s,τ p,q (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 F ˙ T s,τ p,q (Rn+1 + ) and determine their dual spaces F ˙W −s,τ/q p ,q (Rn), where s ∈ R, 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 F ˙H s,τ p,q (Rn) and prove that the dual space of F ˙H s,τ p,q (Rn) is just ˙ F −s,τ/q p ,q (Rn) when p, q ∈ (1,∞). © 2008 Elsevier Inc. All rights reserved.Let s ∈ R, τ ∈ [0,∞), p ∈ (1,∞) and q ∈ (1,∞]. In this paper, we introduce a new class of function spaces ˙ F s,τ p,q (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 F ˙ T s,τ p,q (Rn+1 + ) and determine their dual spaces F ˙W −s,τ/q p ,q (Rn), where s ∈ R, 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 F ˙H s,τ p,q (Rn) and prove that the dual space of F ˙H s,τ p,q (Rn) is just ˙ F −s,τ/q p ,q (Rn) when p, q ∈ (1,∞). © 2008 Elsevier Inc. All rights reserved.