Weak type estimates for functions of Marcinkiewicz type with fractional integrals of mixed homogeneity

TL;DR

This paper establishes endpoint weak type estimates for Marcinkiewicz-type square functions involving fractional integrals with mixed homogeneity, extending classical results to non-isotropic settings.

Contribution

It generalizes Fefferman's result by considering fractional integrals of mixed homogeneity instead of Euclidean Riesz potentials.

Findings

Proves endpoint weak type estimates for non-isotropic fractional integrals

Extends classical Marcinkiewicz theory to mixed homogeneity cases

Provides a framework for analyzing non-isotropic fractional integral operators

Abstract

We prove the endpoint weak type estimate for square functions of Marcinkiewicz type with fractional integrals associated with non-isotropic dilations. This generalizes a result of C. Fefferman on functions of Marcinkiewicz type by considering fractional integrals of mixed homogeneity in place of the Riesz potentials of Euclidean structure.