Optimal domains and integral representations of convolution operators in LP(G)

Susumu Okada, W.J. Ricker

    Research output: Contribution to journalArticlepeer-review

    12 Citations (Scopus)

    Abstract

    Given 1 ≤ p <∞, a compact abelian group G and a function g ∈ L1 (G), we identify the maximal (i.e. optimal) domain of the convolution operator Cg (p) : f ↦ f * g (as an operator from Lp(G) to itself). This is the largest Danach function space (with order continuous norm) into which Lp(G) is embedded and to which Cg (p) has a continuous extension, still with values in Lp(G). Of course, the optimal domain depends on p and g. Whereas Cg (p) is compact, this is not always so for the extension of Cg (p) to its optimal domain. Several characterizations of precisely when this is the case are presented.
    Original languageEnglish
    Pages (from-to)525-546
    Number of pages22
    JournalIntegral Equations and Operator Theory
    Volume48
    Issue number4
    DOIs
    Publication statusPublished - 2004

    Keywords

    • Convolution operator
    • Optimal domain
    • Vector measures in Lp(G)

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