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