New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces

Lixin Yan, Dachun Yang*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    2 Citations (Scopus)

    Abstract

    In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajłasz-Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajłasz-Sobolev spaces on metric measure spaces.

    Original languageEnglish
    Pages (from-to)133-159
    Number of pages27
    JournalMathematische Zeitschrift
    Volume255
    Issue number1
    DOIs
    Publication statusPublished - Jan 2007

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