Embedding and compact embedding for weighted and abstract Sobolev spaces

Seng-Kee Chua

Vol. 303 (2019), No. 2, 519–568

DOI: 10.2140/pjm.2019.303.519

Abstract

Let $\Omega$ be an open set in a metric space $H$, $1\le p_0,p\le q<\infty$, $a,b,\gamma \in ℝ$, $a\ge 0$. Suppose $\sigma ,\mu ,w$ are Borel measures. Combining results from earlier work (2009) with those obtained in work with Wheeden (2011) and with Rodney and Wheeden (2013), we study embedding and compact embedding theorems of sets $\mathfrak{S}\subset L_{\sigma ,loc}^1\left(\Omega \right)\times L_w^p\left(\Omega \right)$ to $L_{\mu }^q\left(\Omega \right)$ (projection to the first component) where $\mathfrak{S}$ (abstract Sobolev space) satisfies a Poincaré-type inequality, $\sigma$ satisfies certain weak doubling property and $\mu$ is absolutely continuous with respect to $\sigma$. In particular, when $H={ℝ}^n$, $w,\mu ,\rho$ are weights so that $\rho$ is essentially constant on each ball deep inside in $\Omega \setminus F$, and $F$ is a finite collection of points and hyperplanes. With the help of a simple observation, we apply our result to the study of embedding and compact embedding of $L_{{\rho }^{\gamma }}^{p_0}\left(\Omega \right)\cap E_{w{\rho }^b}^p\left(\Omega \right)$ and weighted fractional Sobolev spaces to $L_{\mu {\rho }^a}^q\left(\Omega \right)$, where $E_{w{\rho }^b}^p\left(\Omega \right)$ is the space of locally integrable functions in $\Omega$ such that their weak derivatives are in $L_{w{\rho }^b}^p\left(\Omega \right)$. In ${ℝ}^n$, our assumptions are mostly sharp. Besides extending numerous results in the literature, we also extend a result of Bourgain et al. (2002) on cubes to John domains.

Keywords

John domains, Hölmander's vector fields, $A_p$ weights, $\delta$-doubling, reverse-doubling, density theorems, Poincaré inequalities, fractional derivatives

Mathematical Subject Classification 2010

Primary: 26D10, 46E35

Milestones

Received: 7 December 2016

Revised: 10 January 2019

Accepted: 2 June 2019

Published: 4 January 2020