Taiwanese Journal of Mathematics

Weighted Bloch, Lipschitz, Zygmund, Bers, and Growth Spaces of the Ball: Bergman Projections and Characterizations

H. Turgay Kaptanoğlu and Serdar Tülü

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Abstract

We determine precise conditions for the boundedness of Bergman projections from Lebesgue classes onto the spaces in the title, which are members of the same one-parameter family of spaces. The projections provide integral representations for the functions in the spaces. We obtain many properties of the spaces as straightforward corollaries of the projections, integral representations, and isometries among the spaces. We solve the Gleason problem and an extremal problem for point evaluations in each space. We establish maximality of these spaces among those that exhibit Mobius-type invariances and possess decent functionals. We find new Hermitian non-Kahlerian metrics that characterize half of these spaces by Lipschitz-type inequalities.

Article information

Source
Taiwanese J. Math., Volume 15, Number 1 (2011), 101-127.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406164

Digital Object Identifier
doi:10.11650/twjm/1500406164

Mathematical Reviews number (MathSciNet)
MR2780274

Zentralblatt MATH identifier
1247.32012

Subjects
Primary: 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx] 32A18: Bloch functions, normal functions
Secondary: 30D45: Bloch functions, normal functions, normal families 26A16: Lipschitz (Hölder) classes 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.) 46E15: Banach spaces of continuous, differentiable or analytic functions 47B38: Operators on function spaces (general) 47B34: Kernel operators 32M99: None of the above, but in this section 32F45: Invariant metrics and pseudodistances

Keywords
Bergman projection Bloch Lipschitz Zygmund growth Bers Besov space isometry Gleason problem slice function boundary growth Taylor coefficient extremal point evaluation duality interpolation $\alpha$-Möbius invariance decent functional maximal space Hermitian metric Kähler metric geodesic completeness Laplace-Beltrami operator holomorphic sectional curvature

Citation

Kaptanoğlu, H. Turgay; Tülü, Serdar. Weighted Bloch, Lipschitz, Zygmund, Bers, and Growth Spaces of the Ball: Bergman Projections and Characterizations. Taiwanese J. Math. 15 (2011), no. 1, 101--127. doi:10.11650/twjm/1500406164. https://projecteuclid.org/euclid.twjm/1500406164


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