New pointwise bounds by Riesz potential type operators

Abstract

We investigate new pointwise bounds for a class of rough integral operators, $T_{\hat{I}\text{c◯},\hat{I}\pm }$, for a parameter $0<\hat{I}\pm <n$ that includes classical rough singular integrals of Calderón and Zygmund, rough hypersingular integrals, and rough fractional integral operators. We prove that the rough integral operators are bounded by a sparse potential operator that depends on the size of the symbol $\hat{I}\text{c◯}$. As a result of our pointwise inequalities, we obtain several new Sobolev mappings of the form $T_{\hat{I}\text{c◯},\hat{I}\pm }:{\dot{W}}^{1,p}\to L^q$

Version 3: typos corrected