Embedding and compact embedding for weighted and abstract sobolev spaces

Abstract

Let $\Omega$ be an open set in a metric space $H$, $1\le p_0,p\le q<\infty$, $a\ge 0,b,\gamma\in \R$. Suppose $\sigma,\mu,w$ are Borel measures. Combining results of \cite{Ch09,CW11,CRW}, we study embedding and compact embedding theorems of sets $\CS\subset L^1_{\sigma,loc}(\Omega)\times L^p_w(\Omega)$ to $L^q_\mu(\Omega)$ (projection to the first component) where $\CS$ (abstract Sobolev space) satisfies a Poincar\'e type inequality, $\sigma$ satisfies certain weak doubling property and $\mu$ is absolutely continuous w.r.t. $\sigma$. In particular, when $H=\R^n$, $w,\mu,\rho$ are weights so that $\rho$ is essentially constant on each ball deep inside in $\Omega\setminus F$, $F$ is a finite collection of points and hyperplanes, with the help of a simple observation, we apply our result to the studies of embedding and compact embedding of $L^{p_0}_{\rho^\gamma}(\Omega)\cap E^p_{w\rho^b}(\Omega)$ and weighted fractional Sobolev spaces to $L^q_{\mu\rho^a}(\Omega)$ where $E^p_{w\rho^b}(\Omega)$ is the space of locally integrable functions in $\Omega$ such that their weak derivatives are in $L^p_{w\rho^b}(\Omega)$. In $\R^n$, our assumptions are mostly sharp. Besides extending numerous results in the literature, we also extend a result of Bourgain et al \cite{BBM02} on cubes to John domains.