On the Characterization of Triebel–Lizorkin Type Spaces of Analytic Functions

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    Abstract

    We consider different characterizations of Triebel–Lizorkin type spaces of analytic functions on the unit disc. Even though our results appear in the folklore, detailed descriptions are hard to find, and in fact we are unable to discuss the full range of parameters. Without additional effort we work with vector-valued analytic functions, and also consider a generalized scale of function spaces, including for example so-called Q-spaces. The primary aim of this note is to generalize, and clarify, a remarkable result by Cohn and Verbitsky, on factorization of Triebel–Lizorkin spaces. Their result remains valid for functions taking values in an arbitrary Banach space, provided that the vector-valuedness “sits in the right factor”. On the other hand, if we impose vector-valuedness on the “wrong” factor, then the factorization theorem fails even for functions taking values in a separable Hilbert space.

    Original languageEnglish
    Pages (from-to)1491-1517
    JournalJournal of Fourier Analysis and Applications
    Volume24
    Issue number6
    Early online date2017 Dec 4
    DOIs
    Publication statusPublished - 2018 Dec

    Subject classification (UKÄ)

    • Mathematical Analysis

    Keywords

    • Factorization
    • Q-spaces
    • Triebel–Lizorkin spaces
    • Vector-valued analytic functions

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