New sparse domination and weighted estimates for fractional operators beyond Calderón-Zygmund theory

Let L be a closed, densely defined operator on $L^2\left(R^n\right)$ 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.