New pointwise bounds by Riesz potential type operators

Abstract

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