A Sobolev Space Property of Logarithm of Lipschitz functions

TL;DR

This paper investigates the Sobolev space properties of the logarithm of Lipschitz functions, focusing on the integrability of their gradient quotients, which has implications for understanding function regularity.

Contribution

It establishes new conditions under which the logarithm of Lipschitz functions belongs to certain Sobolev spaces, advancing the theoretical understanding of function regularity.

Findings

Identifies integrability conditions for the quotient | abla f|/|f|

Provides new Sobolev space membership results for log of Lipschitz functions

Enhances understanding of regularity properties of Lipschitz functions

Abstract

For a Lipschitz function $f$ on an open set in $\mathbb{R}^n$, we consider the $L^{n}$ integrability of the quotient $\frac{|\nabla f|}{|f|}$ over its natural domain of definition.