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Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces

Authors:

E S Dubtsov

Institution:

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

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. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 43–53.

Keywords:

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