Sharp weighted norm estimates beyond Calder\'on-Zygmund theory

Abstract

We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are Riesz transforms / multipliers or paraproducts associated with a second order elliptic operator. It also applies to such operators whose unweighted continuity is restricted to Lebesgue spaces with certain ranges of exponents $(p_0,q_0)$ where $1\le p_0<2<q_0\le \infty$. The norm estimates obtained are powers $\alpha$ of the characteristic used by Auscher and Martell. The critical exponent in this case is $\mathfrak{p}=1+\frac{p_0}{q'_0}$. We prove $\alpha=\frac1{p-p_0}$ when $p_0<p\le \mathfrak{p}$ and $\alpha= \frac{q_0-1}{q_0-p}$ when $\mathfrak{p}\le p<q_0$. In particular, we are able to obtain the sharp $A_2$ estimates for non-integral singular operators which do not fit into the class of Calder\'on-Zygmund operators. These results are new even in the Euclidean space and are the first ones for operators whose kernel does not satisfy any regularity estimate.