Sobolev Differentiability Properties of Logarithmic Modulus of Real Analytic Functions

Abstract

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

Final version, 15 pages, published in J. Funct. Anal