A characterization of Hajlasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions

In this paper, we establish the equivalence between the Hajlasz-Sobolev spaces or classical Triebel-Lizorkin spaces and it class of grand Triebel-Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when p is an element of (n/(n + 1), infinity), we give a new characterization of the Hajlasz-Sobolev spaces (M) over dot(1),P (R-n) via a grand Littlewood-Paley function. (C) 2009 Elsevier Inc. All rights reserved.