New Sparse Domination and Weighted Estimates for Fractional Operators Beyond Calder\'on-Zygmund Theory

TL;DR

This paper develops new sparse domination techniques to establish weighted estimates for fractional operators associated with a broad class of differential operators, extending beyond classical Calderón-Zygmund theory.

Contribution

It introduces minimal assumptions for sparse domination of fractional operators and provides a new criterion applicable to various differential operators and fractional integrals.

Findings

Established two-weight and Bloom weighted estimates for fractional operators.

Developed a new sparse domination criterion for fractional operators.

Extended the applicability of weighted estimates beyond classical Calderón-Zygmund operators.

Abstract

Let $L$ be a closed, densely defined operator on $L^2(\mathbb{R}^n)$ satisfying suitable $L^p-L^q$ off-diagonal estimates of order $\kappa > 0$. This paper aims to investigate the two-weight estimate and the Bloom weighted estimate for the fractional operator $L^{-\alpha/\kappa}$ with $0 < \alpha < n$ through the method of sparse domination. Our assumptions on the operators are minimal, and our result applies to a wide range of differential operators. As a byproduct, we also establish a new sparse domination criterion for a general class of fractional operators, including the classical fractional integral.