A Characterization of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Grand Littlewood-Paley Functions

Abstract

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 $p\in (n/(n+1),\infty )$, we give a new characterization of the Hajłasz-Sobolev spaces ${\dot{M}}^{1,p}({\mathbb{R}}^n)$ via a grand Littlewood-Paley function.

J. Funct. Anal. (to appear)