Sobolev Differentiability Properties of Logarithmic Modulus of Real Analytic Functions

TL;DR

This paper proves that for real analytic functions with zero sets of codimension at least two, the logarithm of their absolute value is locally integrable in the Sobolev space W^{1,1}, implying certain differential inequalities.

Contribution

It establishes the Sobolev differentiability of the logarithmic modulus of real analytic functions near their zeros under codimension conditions.

Findings

log |f| is W^{1,1}_{loc} near the origin

Differential inequality | abla f| leq V |f| holds with V in L^1_{loc}

Zero set codimension condition is crucial for results

Abstract

Let $f$ be the germ of a real analytic function at the origin in $\mathbb{R}^n $ for $n \geq 2$, and suppose the codimension of the zero set of $f$ at $\mathbf{0}$ is at least $2$. We show that $\log |f|$ is $W^{1,1}_{\operatorname{loc}}$ near $\mathbf{0}$. In particular, this implies the differential inequality $|\nabla f |\leq V |f|$ holds with $V \in L^1_{\operatorname{loc}}$.