A characterization of $\dot W^{1,p}(\mathbb R^d)$

TL;DR

This paper provides a new characterization of the homogeneous Sobolev space ot W^{1,p}(bR^d) for 1<p<, based on the oscillations of functions over balls, linking to operator theory and derivative-free Sobolev descriptions.

Contribution

It introduces a novel oscillation-based characterization of ot W^{1,p}(bR^d), expanding understanding of Sobolev spaces without derivatives.

Findings

Characterization in terms of oscillations on balls of varying sizes and centers.

Connections to trace ideal properties of commutators with singular integrals.

Relevance to derivative-free Sobolev space descriptions.

Abstract

For $1<p<\infty$ we give a characterization of the Sobolev space $\dot W^{1,p}(\mathbb R^d)$ in terms of the oscillations of a function on balls of varying centers and radii. Our work is motivated both by the study of trace ideal properties of commutators with singular integral operators and by work of Nguyen and by Brezis, Van Schaftingen and Yung on derivative-free characterizations of Sobolev spaces.