Sobolev embedding for M^1,p spaces is equivalent to a lower bound of the measure

2019/03/14 by Ryan Alvarado, Alvarado, Ryan, Przemysław Górka +3 · 2 citations
Mathematics · #42B37 #43A85 #46E35 #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Primary 30L99 #Secondary 31E05

paper · pdf · doi:10.48550/arxiv.1903.05793

openalex publication_date 2019/03/14 · openalex created_date 2019/03/22 · openalex updated_date 2026/07/28

Abstract

It has been known since 1996 that a lower bound for the measure, μ(B(x,r))≥ br^s, implies Sobolev embedding theorems for Sobolev spaces M^1,p defined on metric-measure spaces. We prove that, in fact Sobolev embeddings for M^1,p spaces are equivalent to the lower bound of the measure.

Citations

• Sobolev embeddings, extensions and measure density condition
• Quasiconformal maps in metric spaces with controlled geometry
• The Poincaré inequality is an open ended condition
• Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

Cited by

• On the dimension distortion under fractionally smooth mappings
• A priori Hölder estimates for equations degenerating on nodal sets

Related

• Sobolev embedding implies regularity of measure in metric measure spaces
• Lower bound of measure and embeddings of Sobolev, Besov and Triebel-Lizorkin spaces
• Metric Sobolev spaces I: equivalence of definitions
• On mappings generating embedding operators in Sobolev classes on metric measure spaces
• Measure density and Embeddings of Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces