Weighted Local Estimates for Singular Integral Operators

TL;DR

This paper develops new local weighted estimates for singular integral operators using median decompositions, extending classical bounds to broader weight classes and function spaces.

Contribution

It introduces a local median decomposition approach to control singular integrals with weights beyond A_infinity, enabling two-weight inequalities in generalized spaces.

Findings

Weighted local estimates for singular integrals established

Two-weight L^p-L^q bounds proven for broader weight classes

Results applicable to generalized Orlicz-Campanato and Morrey spaces

Abstract

A local median decomposition is used to prove that a weighted local mean of a function is controlled by a weighted local mean of its local sharp maximal function. Together with (a local version of) the estimate $M^{\sharp}_{0,s}(Tf)(x) \le c\,Mf(x)$ for Calder\'{o}n-Zygmund singular integral operators, this allows us to express the local weighted integral control of $Tf$ by $Mf$. Similar estimates hold for $T$ replaced by singular integrals with kernels satisfying H\"{o}rmander-type conditions or integral operators with homogeneous kernels, and $M$ replaced by an appropriate maximal function $M_T$. Using sharper bounds in the local median decomposition we prove two-weight, $L^p_v$-$L^q_w$ estimates for singular integral operators for $1<p\le q<\infty$. In all cases, the results include weights that are not necessarily $A_{\infty}$. The local nature of these estimates leads to results involving weighted generalized Orlicz-Campanato and Orlicz-Morrey spaces.