Compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces

2024/06/26 by Ryan Alvarado, Alvarado, Ryan, Przemysław Górka +3
Mathematics · Medicine · #Advanced Mathematical Physics Problems #Advanced Harmonic Analysis Research #Soft tissue tumor case studies

paper · pdf · doi:10.48550/arxiv.2406.18527

Abstract

We establish necessary and sufficient conditions guaranteeing compactness of embeddings of fractional Sobolev spaces, Besov spaces, and Triebel-Lizorkin spaces, in the general context of quasi-metric-measure spaces. Although stated in the setting of quasi-metric spaces, the main results in this article are new, even in the metric setting. Moreover, by considering the more general category of quasi-metric spaces we are able to obtain these characterizations for optimal ranges of exponents that depend (quantitatively) on the geometric makeup of the underlying space.

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