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

TL;DR

This paper characterizes Haj{}asz-Sobolev and Triebel-Lizorkin spaces using grand Littlewood-Paley functions, providing new equivalences on Euclidean and certain metric spaces, and offers a novel characterization for specific Haj{}asz-Sobolev spaces.

Contribution

It establishes the equivalence between classical and grand Triebel-Lizorkin spaces on Euclidean and metric spaces, and introduces a new characterization of Haj{}asz-Sobolev spaces via grand Littlewood-Paley functions.

Findings

Equivalence between Haj{}asz-Sobolev and grand Triebel-Lizorkin spaces.

New characterization of Haj{}asz-Sobolev spaces for p in (n/(n+1), ) using grand Littlewood-Paley functions.

Applicability on Euclidean spaces and doubling, reverse doubling metric spaces.

Abstract

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 $p\in(n/(n+1),\infty)$, 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.