Criteria for Optimal Global Integrability of Haj{\l}asz-Sobolev Functions

TL;DR

This paper establishes geometric criteria for domains in Euclidean space to support specific Haj{ extl}asz-Sobolev and Trudinger imbeddings, advancing understanding of function space integrability conditions.

Contribution

It introduces new geometric conditions that determine when domains support certain Haj{ extl}asz-Sobolev-Poincaré and Trudinger imbeddings, extending previous theoretical frameworks.

Findings

Derived criteria for Haj{ extl}asz-Sobolev-Poincaré imbeddings.

Established conditions for Haj{ extl}asz-Trudinger imbeddings.

Extended the theory of function space embeddings in geometric domains.

Abstract

The author establishes some geometric criteria for a domain of ${\mathbb R}^n$ with $n\ge2$ to support a $(pn/(n-ps),\,p)_s$-Haj{\l}asz-Sobolev-Poincar\'e imbedding with $s\in(0,\,1]$ and $p\in(n/(n+s),\,n/s)$ or an $s$-Haj{\l}asz-Trudinger imbedding with $s\in(0,\,1]$.