Variable step mollifiers and applications

TL;DR

This paper introduces a variable step mollifier that preserves boundary values and demonstrates its boundedness and approximation properties across various function spaces, with applications to convex intersection density theory.

Contribution

The paper develops a novel variable step mollifier that maintains boundary values and extends existing theories in convex analysis.

Findings

Proves boundedness in Lebesgue, Sobolev, and BV spaces.

Establishes approximation results for the mollifier.

Applies the mollifier to extend convex intersection density theory.

Abstract

We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections.