New Sparse Domination and Weighted Estimates for Fractional Operators Beyond Calderón-Zygmund Theory

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 $\hat{I}º>0$. This paper aims to investigate the two-weight estimate and the Bloom weighted estimate for the fractional operator $L^{-\hat{I}\pm /\hat{I}º}$ with $0<\hat{I}\pm <n$ 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.