Optimal domains and integral representations of Lp(G)-valued convolution operators via measures

S. Okada, W. J. Ricker*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    10 Citations (Scopus)

    Abstract

    Given 1 ≤ p < ∞, a compact abelian group G and a measure μ ∈ M (G), we investigate the optimal domain of the convolution operator Cμ(p) : f → f * μ (as an operator from Lp (G) to itself). This is the largest Köthe function space with order continuous norm into which Lp (G) is embedded and to which Cμ(p) has a continuous extension, still with values in Lp(G). Of course, the optimal domain depends on p and μ. Whereas Cμ(p) is compact precisely when μ ∈ M 0(G), this is not always so for the extension of C μ(p) to its optimal domain (which is always genuinely larger than Lp(G) whenever μ ∈ M0 (G)). Several characterizations of precisely when the extension is compact are presented.

    Original languageEnglish
    Pages (from-to)423-436
    Number of pages14
    JournalMathematische Nachrichten
    Volume280
    Issue number4
    DOIs
    Publication statusPublished - 2007

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