Sobolev spaces with non-isotropic dilations and square functions of Marcinkiewicz type

TL;DR

This paper characterizes weighted Sobolev spaces associated with non-isotropic dilations using Marcinkiewicz-type square functions, including those with repeated averaging, advancing understanding of these function spaces.

Contribution

It introduces a new characterization of weighted Sobolev spaces with non-isotropic dilations via Marcinkiewicz-type square functions, including repeated averaging methods.

Findings

Characterization of Sobolev spaces using Marcinkiewicz square functions.

Extension to spaces defined with repeated averaging operations.

Provides tools for analyzing non-isotropic dilation effects in Sobolev spaces.

Abstract

We consider the weighted Sobolev spaces associated with non-isotropic dilations of Calder\'on-Torchinsky and characterize the spaces by the square functions of Marcinkiewicz type including those defined with repeated uses of averaging operation.