A Characterization of Haj{\l}asz-Sobolev and Triebel-Lizorkin Spaces via Grand Littlewood-Paley Functions

In this paper, we establish the equivalence between the Haj{\l}asz-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 $p\in (n\mathrm{/}(n+1),\mathrm{\infty })$, we give a new characterization of the Haj{\l}asz-Sobolev spaces $\dot M^{1, p}({\mathbb R}^n)$ via a grand Littlewood-Paley function.