New pointwise bounds by Riesz potential type operators

We investigate new pointwise bounds for a class of rough integral operators, $T_{\mathrm{\Omega },\alpha }$, for a parameter $0<\alpha <n$ 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 $\mathrm{\Omega }$. As a result of our pointwise inequalities, we obtain several new Sobolev mappings of the form $T_{\mathrm{\Omega },\alpha }:{\dot{W}}^{1,p}\to L^q$