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Invariant metrics and the boundary behavior of holomorphic functions on domains in \({\mathbb{C}}^ n\). (English) Zbl 0728.32002
The author develops a version of \(H^ p\) space theory which begins with the observation that the optimal approach regions for the existence of boundary values depend on geometrical properties of the domain. These properties are in turn reflected in the behaviour of invariant metrics. The theory begins with a distance \(\rho\) and a volume V on a smoothly bounded domain \(\Omega\) which are compatible, i.e. such that the sub- mean-value property holds for plurisubharmonic functions on metric balls. Approach regions with vertex at a point \(P\in \partial \Omega\) are defined in terms of the distance to the interior normal line segment at P, namely \[ L_{\alpha}(P)=\{z\in \Omega | \quad \rho (z,N_ P)<\alpha \}. \] Balls on the boundary are defined by projecting these approach regions onto \(\partial \Omega\). A measure on \(\partial \Omega\) is then obtained from the Carathéodory construction. A distance is also defined on \(\partial \Omega\) using the boundary balls, and the assumption is made that this distance is directionally limited, so that a covering theorem of Federer can be applied. This leads to an estimate on the distribution function of the maximal function \[ \limsup_{L_{\alpha}(P)\ni z\to P}| f(z)| \] associated to a function \(f\in H^ p(\Omega)\). The main result is that a function \(f\in H^ p(\Omega)\) has boundary values within the constructed approach regions at almost all boundary points.
The reader should compare the study of admissible boundary values in E. M. Stein, Boundary behavior of holomorphic functions of several complex variables (1972; Zbl 0242.32005). The present paper is basically a more invariant version of \(H^ p\) space theory.
Reviewer: I.Graham (Toronto)

MSC:
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32F45 Invariant metrics and pseudodistances in several complex variables
32A40 Boundary behavior of holomorphic functions of several complex variables
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