Geometric characterizations of embedding theorems

for Sobolev, Besov, and Triebel–Lizorkin spaces on spaces of homogeneous type - via orthonormal wavelets

Yanchang Han, Yongsheng Han, Ziyi He, Ji Li, Cristina Pereyra*

*Corresponding author for this work

    Research output: Contribution to journalArticle

    2 Citations (Scopus)

    Abstract

    It was well known that geometric considerations enter in a decisive way in many questions. The embedding theorem arises in several problems from partial differential equations, analysis, and geometry. The purpose of this paper is to provide a deep understanding of analysis and geometry with a particular focus on embedding theorems for spaces of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure μ satisfies the doubling property only. We prove that embedding theorems hold on spaces of homogeneous type if and only if geometric conditions, namely the measures of all balls have lower bounds, hold. We make no additional geometric assumptions on the quasi-metric or the doubling measure, and thus, the results of this paper extend to the full generality of all related previous ones, in which the extra geometric assumptions were made on both the quasi-metric d and the measure μ. As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov, and Triebel–Lizorkin spaces. The crucial tool used in this paper is the remarkable orthonormal wavelet basis constructed recently by Auscher–Hytönen on spaces of homogeneous type in the sense of Coifman and Weiss.

    Original languageEnglish
    Number of pages32
    JournalJournal of Geometric Analysis
    Early online date24 Oct 2020
    DOIs
    Publication statusE-pub ahead of print - 24 Oct 2020

    Keywords

    • Besov space
    • Distributions
    • Embedding
    • Orthonormal wavelet
    • Sobolev space
    • Spaces of homogeneous type
    • Test function space
    • Triebel–Lizorkin space

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