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

Let s epsilon R, tau epsilon [0, infinity), p epsilon (1, infinity) and q epsilon (1, infinity]. In this paper, we introduce a new class of function spaces <(F)over dot>(s,tau)(p,q)(R-n) which unify and generalize the Triebel-Lizorkin spaces with both p epsilon (1, infinity) and p = infinity and Q spaces. By establishing the Carleson measure charactetization 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 Q(alpha) (R-n), J. Funct. Anal. 208 (2004) 377-422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces F<(T)over dot>(s,tau)(p,q)(R-+(n+1)) and determine their dual spaces F<(W)over dot>(-s,tau/q)(p',q') (R-n), where s epsilon R, p,q epsilon [1, infinity), max{p,q} > 1, tau epsilon [0, q/(max{p,q})'], and t' denotes the conjugate index of t epsilon (1, infinity); as an application of this, we further introduce certain Hardy-Hausdorff spaces F<(H)over dot>(s,tau)(p,q)(R-n) and prove that the dual space of F<(H)over dot>(s,tau)(p,q) (R-n) is just <(F)over dot>(-s,tau/q)(p',q')(R-n) when p, q epsilon (1, infinity). (C) 2008 Elsevier Inc. All fights reserved.