On some boundary properties of bounded analytic functions and the maximum modulus principle in domains of arbitrary connectivity.

*(Russian)*Zbl 0663.30028The author continues his study on boundary behavior of bounded analytic functions defined on plane domains of infinite connectivity, which he calls harmonic domains. For his previous work see Mat. Sb., Nov. Ser. 101(143), 189-203 (1976; Zbl 0353.46035) and ibid. 111(153), 557-578 (1980; Zbl 0434.30023). Let D be a domain in the extended complex plane whose boundary \(\partial D\) is bounded. Let \(\pi\) : \(\Delta\) \(\mapsto D\) be a universal covering map from the open unit disk \(\Delta\) onto D and let G be the group of cover transformations associated with \(\pi\). Then D is called harmonic if, for any G-invariant positive measurble function u on \(\partial D\) which is bounded away from zero and satisfies
\[
\int^{2\pi}_{0}\log u(\theta)d\theta <\infty,
\]
there exists a G- invariant analytic function H of Smirnov class of \(\Delta\) such that \(| H(e^{i\theta})| =u(\theta)\) a.e. on \(\partial D.\)

The author first strengthens his characterization of harmonic domains given in his previous papers. He shows among others that D is harmonic if and only if the Shilov boundary of \(H^{\infty}(D)\) and the Choquet boundary of \(h^{\infty}(D)\) are naturally homeomorphic, where \(H^{\infty}(D)\) (resp. \(h^{\infty}(D))\) denotes the space of bounded analytic (resp. harmonic) functions on D. As a consequence, harmonicity of domains is seen to be characterized by the following generalized maximum principles. Namely, suppose that a set \(E\subset \partial D\) is given for each boundary point \(\zeta\in \partial D\) such that \(\zeta\in \bar E\). Let \(F\subset \partial D\). Then, the set family \(E_ F=\{E_{\zeta}:\) \(\zeta\in \partial D\setminus F\}\) is said to satisfy the general maximum principle (GMP) for \(H^{\infty}(D)\) (resp. \(h^{\infty}(D))\) if \[ \sup_{D}| f| =\sup_{\zeta \in \partial D\setminus F}\{\limsup \{| f(z)|: z\to \zeta,\quad z\in E_{\zeta}\}\} \] holds for any \(f\in H^{\infty}(D)\) (resp. \(h^{\infty}(D))\). Then, a domain D is harmonic if and only if the set family \(E_ F\) satisfies the GMP for \(h^{\infty}(D)\) whenever so does for \(H^{\infty}(D)\). This enables to strengthen his earlier result concerning the localization principle for harmonicity of domains as follows: suppose there exists a set \(E\subset \partial D\) of harmonic measure zero such that for each point \(\zeta\in \partial D\setminus E\) there exists a neighborhood \(U_{\zeta}\) of \(\zeta\) such that every component of \(D\cap U_{\zeta}\) is harmonic, then D is harmonic. He then gives positive as well as negative examples. Finally, he describes in detail the role played by the Shilov boundary in the theory of cluster values of bounded analytic functions on D.

The author first strengthens his characterization of harmonic domains given in his previous papers. He shows among others that D is harmonic if and only if the Shilov boundary of \(H^{\infty}(D)\) and the Choquet boundary of \(h^{\infty}(D)\) are naturally homeomorphic, where \(H^{\infty}(D)\) (resp. \(h^{\infty}(D))\) denotes the space of bounded analytic (resp. harmonic) functions on D. As a consequence, harmonicity of domains is seen to be characterized by the following generalized maximum principles. Namely, suppose that a set \(E\subset \partial D\) is given for each boundary point \(\zeta\in \partial D\) such that \(\zeta\in \bar E\). Let \(F\subset \partial D\). Then, the set family \(E_ F=\{E_{\zeta}:\) \(\zeta\in \partial D\setminus F\}\) is said to satisfy the general maximum principle (GMP) for \(H^{\infty}(D)\) (resp. \(h^{\infty}(D))\) if \[ \sup_{D}| f| =\sup_{\zeta \in \partial D\setminus F}\{\limsup \{| f(z)|: z\to \zeta,\quad z\in E_{\zeta}\}\} \] holds for any \(f\in H^{\infty}(D)\) (resp. \(h^{\infty}(D))\). Then, a domain D is harmonic if and only if the set family \(E_ F\) satisfies the GMP for \(h^{\infty}(D)\) whenever so does for \(H^{\infty}(D)\). This enables to strengthen his earlier result concerning the localization principle for harmonicity of domains as follows: suppose there exists a set \(E\subset \partial D\) of harmonic measure zero such that for each point \(\zeta\in \partial D\setminus E\) there exists a neighborhood \(U_{\zeta}\) of \(\zeta\) such that every component of \(D\cap U_{\zeta}\) is harmonic, then D is harmonic. He then gives positive as well as negative examples. Finally, he describes in detail the role played by the Shilov boundary in the theory of cluster values of bounded analytic functions on D.

Reviewer: M.Hasumi