From Zygmund space to Bergman–Zygmund space

Hong Rae Cho, Hyungwoon Koo, Young Joo Lee

    Research output: Contribution to journalArticlepeer-review

    Abstract

    Let 0<p<∞,α>-1, and β,γ∈R. Let μ be a finite positive Borel measure on the unit disk D. The Zygmund space Lp,β(dμ) consists of all measurable functions f on D such that |f|plogβ(e+|f|)∈L1(dμ) and the Bergman–Zygmund space Aαp,β is the set of all analytic functions in Lp,β(dAα), where dAα=cα(1-|z|2)αdA. We prove an interpolation theorem for the Zygmund space assuming the weak type estimates on the Zygmund spaces themselves at the end points rather than the weak Lp-Lq type estimates at the end points. We show that the Bergman–Zygmund space is equal to the logβ(e/(1-|z|))dAα(z) weighted Bergman space as a set and characterize the bounded and compact Carleson measure μ from Aαp,β into Ap,γ(dμ), respectively. The Carleson measure characterizations are of the same type for any pairs of (β,γ) whether β<γ or γ≤β.

    Original languageEnglish
    Article number58
    JournalBanach Journal of Mathematical Analysis
    Volume18
    Issue number3
    DOIs
    Publication statusPublished - 2024 Jul

    Bibliographical note

    Publisher Copyright:
    © Tusi Mathematical Research Group (TMRG) 2024.

    Keywords

    • 30H20
    • 46B70
    • 46E30
    • Bergman–Zygmund space
    • Carleson measure
    • Interpolation
    • Logarithmic weighted Bergman space
    • Quasinormed Fréchet space
    • Zygmund space

    ASJC Scopus subject areas

    • Analysis
    • Algebra and Number Theory

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