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Weighted Harmonic Bloch Spaces on the Ball

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Abstract

We study the family of weighted harmonic Bloch spaces \(b_\alpha , \alpha \in {\mathbb {R}}\), on the unit ball of \({\mathbb {R}}^n\). We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We consider a class of integral operators related to harmonic Bergman projection and determine precisely when they are bounded on \(L^\infty _\alpha \). We define projections from \(L^\infty _\alpha \) to \(b_\alpha \) and as a consequence obtain integral representations. We solve the Gleason problem and provide atomic decomposition for all \(b_\alpha , \alpha \in {\mathbb {R}}\). Finally we give an oscillatory characterization of \(b_\alpha \) when \(\alpha >-1\).

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Authors and Affiliations

  1. Department of Mathematics, Namik Kemal University, 59030, Tekirdağ, Turkey

    Ömer Faruk Doğan

  2. Department of Mathematics, Anadolu University, 26470, Eskisehir, Turkey

    A. Ersin Üreyen

Authors
  1. Ömer Faruk Doğan
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  2. A. Ersin Üreyen
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Correspondence to A. Ersin Üreyen.

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Communicated by H. Turgay Kaptanoglu.

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Doğan, Ö.F., Üreyen, A.E. Weighted Harmonic Bloch Spaces on the Ball. Complex Anal. Oper. Theory 12, 1143–1177 (2018). https://doi.org/10.1007/s11785-017-0645-9

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  • DOI: https://doi.org/10.1007/s11785-017-0645-9

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