Singular extension of critical Sobolev mappings under an exponential weak-type estimate

TL;DR

This paper constructs extensions of critical Sobolev maps into manifolds with exponential weak-type estimates, advancing understanding of Sobolev extension problems in geometric analysis.

Contribution

It introduces a method to extend Sobolev maps with critical regularity into manifolds, satisfying exponential weak-type estimates, applicable to various geometric settings.

Findings

Constructed Sobolev extensions with exponential weak-type estimates

Extended results to half-spaces and hyperbolic spaces

Provided new tools for geometric Sobolev extension problems

Abstract

Given $m \in \mathbb{N} \setminus \{0\}$ and a compact Riemannian manifold $\mathcal{N}$, we construct for every map $u$ in the critical Sobolev space $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$, a map $U : \mathbb{B}^{m + 1} \to \mathcal{N}$ whose trace is $u$ and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity.