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

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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 M¹, ᵖ(Rⁿ) via a grand Littlewood-Paley function.