Poincare inequalities and compact embeddings from Sobolev type spaces into weighted Lq spaces on metric spaces

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into L-q spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincare type inequalities. We show how to construct such suitable coverings and Poincare inequalities. For locally doubling measures, we prove a self-improvement property for two weighted Poincare inequalities, which applies also to lower dimensional measures.& nbsp;We simultaneously treat various Sobolev spaces, such as the Newtonian, fractional Hajlasz and Poincare type spaces, for rather general measures and sets, including fractals and domains with fractal boundaries. By considering lower dimensional measures on the boundaries of such domains, we obtain trace embeddings for the above spaces. In the case of Newtonian spaces we exactly characterize when embeddings into L-q spaces with respect to another measure are compact. Our tools are illustrated by concrete examples. For measures satisfying suitable dimension conditions, we recover several classical embedding theorems on domains and fractal sets in R-n. (C) 2022 The Author(s). Published by Elsevier Inc.& nbsp;

Automated summary

This article studies the compactness and boundedness of embeddings from Sobolev-type spaces on metric spaces into L-q spaces with respect to another measure. The Sobolev spaces considered can be of fractional order and some statements also allow non-doubling measures. The results are formulated in a general form, using sequences of covering families and local Poincare-type inequalities. Various Sobolev spaces are simultaneously treated, including the Newtonian, fractional Hajlasz, and Poincare-type spaces. For locally doubling measures, a self-improvement property for two weighted Poincare inequalities is proven.