an:04006656
Zbl 0621.32012
Wang, Jian; Fang, Xiang
On Baernstein's theorem on the upper half-space and polydiscs
ZH
Acta Math. Sin. 29, 393-398 (1986).
00151814
1986
j
32A35 32A30 30D50 31B05 42B30
upper half-space; BMOA; bounded mean oscillation; harmonic functions; Poisson integral; polydiscs; characteristic boundary
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)\(\equiv t\). \textit{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 \(\Delta^ n\).
S.H.Tung
Zbl 0492.30026