Continuous Sobolev functions with singularity on arbitrary real-analytic sets

Yifei Pan and Yuan Zhang

Vol. 332 (2024), No. 2, 261–272

DOI: 10.2140/pjm.2024.332.261

Abstract

Near every point of a real-analytic set in ${ℝ}^n$, we make use of Hironaka’s resolution of singularities theorem to construct a family of continuous functions in $W_{\mathrm{loc}<!--FUNCTION\; APPLICATION-->}^{1,1}$ such that their weak derivatives have (removable) singularities precisely on that set.

Keywords

logarithms, real-analyticity, Sobolev spaces, resolution of singularities

Mathematical Subject Classification

Primary: 32C07

Secondary: 14E15, 46E35

Milestones

Received: 6 June 2024

Revised: 18 October 2024

Accepted: 15 November 2024

Published: 6 December 2024