A Marcinkiewicz integral type characterization of the Sobolev space

TL;DR

This paper introduces a new Marcinkiewicz integral-based characterization of Sobolev spaces, simplifies existing proofs, and extends the results to weighted Sobolev spaces with Muckenhoupt weights.

Contribution

It provides a novel higher-dimensional characterization of Sobolev spaces and generalizes the results to weighted cases, improving understanding and proof simplicity.

Findings

New characterization of Sobolev space $W^{1,p}$ using Marcinkiewicz integral

Simplified proof of a recent Sobolev space result

Extension to weighted Sobolev spaces with Muckenhoupt weights

Abstract

In this paper we present a new characterization of the Sobolev space $W^{1,p}$, $1<p<\infty$ which is a higher dimensional version of a result of Waterman. We also provide a new and simplified proof of a recent result of Alabern, Mateu and Verdera. Finally, we generalize the results to the case of weighted Sobolev spaces with respect to a Muckenhoupt weight.