Local real analysis in locally homogeneous spaces

TL;DR

This paper develops local real analysis tools in locally homogeneous spaces, establishing key estimates for singular and fractional integrals and their commutators, motivated by subelliptic PDEs.

Contribution

It introduces the concept of locally homogeneous spaces and proves L^p and Holder estimates for integrals and commutators within this framework.

Findings

Established L^p and Holder estimates for singular and fractional integrals.

Proved boundedness of commutators with BMO and VMO functions.

Motivated by applications to subelliptic equations.

Abstract

We introduce the concept of locally homogeneous space, and prove in this context L^p and Holder estimates for singular and fractional integrals, as well as L^p estimates on the commutator of a singular or fractional integral with a BMO or VMO function. These results are motivated by local a-priori estimates for subelliptic equations.