The extension property for domains with one singular point

TL;DR

This paper demonstrates that arbitrary outward cuspidal domains can be transformed into Lipschitz ones, enabling the extension of Sobolev functions and broadening the scope of previous extension results.

Contribution

It introduces a bi-Lipschitz transformation for arbitrary outward cuspidal domains, extending Sobolev extension results to more general domains.

Findings

Arbitrary outward cuspidal domains are bi-Lipschitz equivalent to Lipschitz ones.

Extension results for Sobolev functions are extended to all outward cuspidal domains.

A limit case of extension results is established.

Abstract

An arbitrary outward cuspidal domain is shown to be bi-Lipschitz equivalent to a Lipschitz outward cuspidal domain via a global transformation. This allows us to extend earlier Sobolev extension results on Lipschitz outward cuspidal domains from the work of Maz'ya and Poborchi to arbitrary outward cuspidal domains. We also establish a limit case of extension results on outward cuspidal domains.