# zbMATH — the first resource for mathematics

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
##### MSC:
 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