Sobolev Differentiability Properties of Logarithmic Modulus of Real Analytic Functions

Ziming Shi, Ruixiang Zhang

Published 2022-05-04Version 1

Let $f$ be a real analytic function at a point $p$ in $\mathbb{R}^n $ for $n \geq 2$, and suppose the zero set of $f$ has codimension at least $2$ at $p$. We show that $\log |f|$ is $W^{1,1}_{\operatorname{loc}}$ at $p$. In particular, this implies the differential inequality $|\nabla f |\leq V |f|$, where $V \in L^1_{\operatorname{loc}}$. As an application, we derive an inequality relating the {\L}ojasiewicz exponent and the singularity exponent for such functions.