Estimates for singular integrals on homogeneous groups

TL;DR

This paper establishes new $L^p$ boundedness and weighted inequalities for singular integral operators with rough kernels on homogeneous groups, using extrapolation techniques under sharp kernel conditions.

Contribution

It provides novel $L^p$ estimates and weighted inequalities for singular integrals on homogeneous groups, extending classical results to rough kernels.

Findings

Proved $L^p$ boundedness of singular integrals with rough kernels.

Established weighted $L^p$ inequalities for these operators.

Derived sharp conditions for kernel estimates.

Abstract

We consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups. We prove certain estimates for the operators that imply $L^p$ boundedness of them by an extrapolation argument under a sharp condition for the kernels. Also, we prove some weighted $L^p$ inequalities for the operators.