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Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces

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Abstract

Let \(\tilde \nabla \) and τ denote the invariant gradient and invariant measure on the unit ball B of ℂn, respectively. Assume that f is a holomorphic function on B and ϕ ∈ C2(ℝ) is a nonnegative, nondecreasing, convex function. Then f belongs to the Hardy-Orlicz space H ϕ(B>) if and only if

$$\int\limits_B {\varphi ''(\log \left| {f(z)} \right|)\frac{{\left| {\tilde \nabla f(z)} \right|^2 }}{{\left| {f(z)} \right|^2 }}} (1 - \left| z \right|^2 )^n d\tau (z) < \infty .$$

Analogous characterizations of Bergman-Orlicz spaces are obtained. Bibliography: 9 titles.

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Authors and Affiliations

  1. St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia

    E. S. Dubtsov

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  1. E. S. Dubtsov
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 43–53.

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Dubtsov, E.S. Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces. J Math Sci 141, 1531–1537 (2007). https://doi.org/10.1007/s10958-007-0059-8

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  • DOI: https://doi.org/10.1007/s10958-007-0059-8

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