A decomposition by non-negative functions in the Sobolev space $W^{k, 1}$

TL;DR

This paper demonstrates a method to decompose functions in the Sobolev space W^{k,1} into differences of two non-negative functions within the same space, using capacitary inequalities to control their norms.

Contribution

It introduces a novel decomposition technique in Sobolev spaces leveraging capacitary inequalities, providing norm control for the non-negative components.

Findings

Decomposition of W^{k,1} functions into non-negative parts.

Use of capacitary inequalities for norm control.

Applicable to functions in Sobolev spaces.

Abstract

We show how a strong capacitary inequality can be used to give a decomposition of any function in the Sobolev space $W^{k,1}(\mathbb{R}^d)$ as the difference of two non-negative functions in the same space with control of their norms.