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On Baernstein’s theorem on the upper half-space and polydiscs. (Chinese) Zbl 0621.32012
Let \(\phi\) (t) be a non-negative strictly increasing subadditive function on [0,\(\infty)\) with \(\phi\) (t)\(\to \infty\) as \(t\to \infty\). Let Q be a fixed cube in \({\mathbb{R}}^ n\) with sides parallel to the coordinate axes. Denote \(BMO_{\phi}(Q)\) for the set of all functions f such that \(\phi\) (\(| f(x)|)\) is locally integrable on Q, with \(\| f\|^{\phi}_{BMO}=\sup_{I\subseteq Q}\frac{1}{| I|}\int_{I}\phi (| f(x)-f(I)|) dx<\infty,\) where I is a subcube with sides parallel to sides of Q, \(| I|\) the Lebesgue measure of I and f(I) the average of f over I. \(BMO_ t(Q)\) is the usual BMO(Q) when \(\phi\) (t)\(\equiv t\). A. Baernstein introduced [Aspects of contemporary complex analysis, Proc. instr. Conf. Durham/Engl. 1979, 3-36 (1980; Zbl 0492.30026)] the set BMOA of functions with bounded mean oscillation over the unit circle T whose Poisson extensions to the unit disc \(\Delta\) are analytic, and proved a theorem which established the equivalence between the set of all normalized hyperbolic translates of a function analytic in \(\Delta\) to be bounded in the Nevanlinna class and the exponential decrease of the distribution of the function. In this paper the authors extend this theorem to the set \(BMOH_{\phi}({\mathbb{R}}_+^{n+1})\) of all harmonic functions on \({\mathbb{R}}_+^{n+1}\) from the Poisson integral of functions in \(BMO_{\phi}({\mathbb{R}}^ n)\) and the set \(BMOH_{\phi}(\Delta^ n)\) on polydiscs \(\Delta^ n\) derived from the functions in \(BMO_{\phi}(T^ n)\) over the characteristic boundary \(T^ n\) of \(\Delta^ n\).
Reviewer: S.H.Tung
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A30 Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30-XX)
30D50 Blaschke products, etc. (MSC2000)
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
42B30 \(H^p\)-spaces