Pointwise estimates for rough operators with applications to Sobolev inequalities

TL;DR

This paper establishes pointwise bounds for rough operators using Riesz potentials, leading to new Sobolev inequalities that extend classical results to more general operators.

Contribution

It introduces novel pointwise estimates for rough operators and derives new Sobolev inequalities involving these operators.

Findings

Operators satisfy pointwise bounds by Riesz potentials

New Sobolev-type inequalities are derived

Applications to analysis of rough operators

Abstract

We investigate Sobolev inequalities for several rough operators. We prove that several operators satisfy a pointwise bound by the Riesz potential applied to the gradient. From this inequality, we derive several new Sobolev-type inequalities with an operator on the left-hand side.