Continuous Sobolev functions with singularity on arbitrary real-analytic sets

TL;DR

This paper constructs continuous Sobolev functions with singularities exactly on arbitrary real-analytic sets using Hironaka's resolution of singularity theorem, advancing understanding of singularities in Sobolev spaces.

Contribution

It introduces a method to explicitly construct Sobolev functions with prescribed singularities on any real-analytic set, leveraging resolution of singularities.

Findings

Functions have singularities precisely on the given real-analytic set.

The constructed functions are in the local Sobolev space $W^{1,1}_{loc}$.

The approach applies Hironaka's theorem to control singularities.

Abstract

Near every point of a real-analytic set in $\mathbb R^n$, we make use of Hironaka's resolution of singularity theorem to construct a family of continuous functions in $W^{1, 1}_{loc}$ such that their weak derivatives have (removable) singularity precisely on that set.