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 via a grand Littlewood–Paley function.