A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions

In this paper, we establish the equivalence between the Hajłasz–Sobolev spaces or classical Triebel– Lizorkin spaces and a 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 ∈ (n/(n + 1),∞), we give a new characterization of the Hajłasz–Sobolev spaces M˙ 1,p(Rn) via a grand Littlewood–Paley function. © 2009 Elsevier Inc. All rights reserved