An embedding theorem for weighted Sobolev classes on a John domain: case of weights that are functions of a distance to a certain h-set

TL;DR

This paper establishes embedding theorems for weighted Sobolev classes on John domains with weights depending on the distance to a specific h-set, expanding understanding of function space embeddings in geometric contexts.

Contribution

It provides a new embedding theorem for weighted Sobolev classes on John domains with weights related to the distance from an h-set, a novel geometric setting.

Findings

Embedding theorems for weighted Sobolev classes are proved.

Results depend on the geometry of John domains and h-sets.

Theorems extend classical embeddings to weighted, geometric contexts.

Abstract

Let $\Omega$ be a John domain, and let $\Gamma\subset \partial \Omega$ be an $h$-set. For some functions $h$ and some weight functions depending on distance from $\Gamma$, embedding theorems for a weighted Sobolev class is obtained.