Vol. 303, No. 2, 2019

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Embedding and compact embedding for weighted and abstract Sobolev spaces

Seng-Kee Chua

Vol. 303 (2019), No. 2, 519–568
Abstract

Let Ω be an open set in a metric space H, 1 p0,p q < , a,b,γ , a 0. Suppose σ,μ,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 S Lσ,loc1(Ω) × Lwp(Ω) to Lμq(Ω) (projection to the first component) where S (abstract Sobolev space) satisfies a Poincaré-type inequality, σ satisfies certain weak doubling property and μ is absolutely continuous with respect to σ. In particular, when H = n, w,μ,ρ are weights so that ρ is essentially constant on each ball deep inside in Ω 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 Lργp0 (Ω) Ewρbp(Ω) and weighted fractional Sobolev spaces to Lμρaq(Ω), where Ewρbp(Ω) is the space of locally integrable functions in Ω such that their weak derivatives are in Lwρbp(Ω). In 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
Authors
Seng-Kee Chua
Department of Mathematics
National University of Singapore
Singapore