Sobolev Differentiability Properties of Logarithmic Modulus of Real Analytic Functions

TL;DR

This work establishes that for a real analytic germ $f$ at the origin in $\mathbb{R}^n$ with $n\ge 2$ and codimension of its zero set at least 2, the logarithmic modulus $\log|f|$ belongs to $W^{1,1}_{\mathrm{loc}}$, which yields the differential inequality $|\nabla f| \le V|f|$ with $V\in L^1_{\mathrm{loc}}$. The authors provide a precise local and global formulation, prove the optimality of the integrability threshold, and relate the result to local invariants via the Łojasiewicz gradient inequality and the singularity exponent, showing $\alpha_0+\beta_0 \ge 1$. The analysis combines Weierstrass preparation, o-minimal tame geometry, and a removable singularity argument to pass differentiability across the zero set. Sharpness is demonstrated through explicit polynomial examples and a blow-up result for the critical exponent $n-d$, establishing the boundary between finite and infinite integrals of $|\nabla f|/|f|$ near the zero set. The results illuminate the structure of real-analytic zeros and provide a robust bridge between Sobolev regularity and algebraic-geometric properties of zero loci.

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 $V \in L^1_{\operatorname{loc}}$.

Theorems & Definitions (44)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5: Stratification of analytic sets
• Proposition 2.6