Some classical problems of the theory of analytic functions in domains of Parreau-Widom type.

*(Russian)*Zbl 0761.30020Continuing his study of function theory on general plane domains [Mat. Sb., Nov. Ser. 101(143), 189–203 (1976; Zbl 0358.46035); ibid. 111(153), 557–578 (1980; Zbl 0434.30023); ibid. 135(177), No. 4, 497–513 (1988; Zbl 0663.30028)], the author now gives a very interesting observation concerning Hardy classes on domains of Parreau-Widom type. Let \(D\) be a domain of hyperbolic type contained in the extended complex plane \(\hat{\mathbb C}\). A function \(f\) on \(D\) is said to be multiplicative if it is multiple-valued and analytic on \(D\) such that \(| f|\) is single valued. Such a function defines a character \(\Gamma_f\) of the fundamental group \(\pi_1(D)\) for \(D\), which is called the character of \(f\). Given any character \(\Gamma\) of \(\pi_1(D)\), we denote by \(H^\infty(D,\Gamma)\) the totality of bounded multiplicative functions \(f\) on \(D\) with \(\Gamma_ f=\Gamma\). A domain \(D\) is said to be of Parreau-Widom type if \(H^\infty(D,\Gamma)\neq\{0\}\) for any character \(\Gamma\) of \(\pi_1(D)\). Basic results concerning such domains may be found in the reviewer’s book [Hardy classes on infinitely connected Riemann surfaces. Berlin etc.: Springer-Verlag (1983; Zbl 0523.30028)].

In the book mentioned above the main tools for studying surfaces of Parreau-Widom type were Martin compactification and Green lines. On the other hand, in the case of compact bordered surfaces, F. Forelli [Ill. J. Math. 10, 367–380 (1966; Zbl 0141.31401)] employed the conditional expectation operators to study invariant subspaces of Hardy classes. This method was further exploited via Poincaré’s theta series by C. J. Earle and A. Marden [Ill. J. Math. 13, 202–219 (1969; Zbl 0169.10202)]. In a paper by Ch. Pommerenke [Ann. Acad. Sci. Fenn., Ser. A 2, 409–427 (1976; Zbl 0363.30029)] Fuchsian groups, appearing as the cover transformation groups of surfaces of Parreau-Widom type, were characterized and Earle and Marden’s result was extended to such groups. This means that Forelli’s conditional expectation operator method may be used not only for finitely connected surfaces but also for infinitely connected surfaces of Parreau-Widom type. This is done explicitly on the paper under review.

Let \(D\) be a hyperbolic domain, \(\pi:\Delta\to D\) be the universal covering map from the unit disk \(\Delta\) to \(D\) and \({\mathcal G}=\{\gamma\}\) the group of cover transformations for \(\pi\). The quotient space \(\Delta/{\mathcal G}\) is isomorphic with \(D\) and the Hardy class \(H^p(D)\) is identified with the subspace \(H^p(\Delta,{\mathcal G})\) of \({\mathcal G}\)- invariant functions in \(H^p(\Delta)\). Let \(g(z)=\prod\{e^{- \arg\gamma(0)}\gamma(z)\): \(\gamma\in{\mathcal G}\}\) be the Blaschke product corresponding to the group \({\mathcal G}\). Pommerenke loc. cit. showed that \(D\) (or \({\mathcal G}\)) is of Parreau-Widom type if and only if \(g'(z)\) is of bounded characteristic.

The paper is divided into two parts. In the first part, the author examines in detail the property (DCT) (= “Direct Cauchy Theorem”), which is crucial in the study of invariant subspaces. Let \(D\) be a domain of Parreau-Widom type. We define the operator \(E\) by setting \(E(H)(z)=g(z)/g'(z)\sum_{\gamma\in{\mathcal G}} H(\gamma(z))\gamma'(z)/\gamma(z)\) for any analytic function \(H(z)\) on \(\Delta\). Then, \(H\) is \({\mathcal G}\)-invariant if and only if \(E(H)=H\). For a function \(H(z)\) on \(\partial\Delta\) we set \(E(H)(z)=\rho^{-1} \sum_{\gamma\in{\mathcal G}} H(\gamma(z))|\gamma'(z)|\) for \(z\in\partial\Delta\), where \(\rho(z)=\sum_{\gamma\in{\mathcal G}}|\gamma'(z)|\). Then, \(E\) defines a projection of the norm from \(L^p(d\theta)\) onto \(L^p(d\theta,{\mathcal G})\). Let \(g^*(z)\) be the inner factor of \(g'(z)\) and \(\Gamma_ *\) denotes the character of \({\mathcal G}\) defined by \(g^*\), namely, \(g^*(\gamma(z))=\Gamma_ *(\gamma)g^*(z)\) for any \(\gamma\in{\mathcal G}\). By use of the operator \(E\) the author first proves that \(E(H^p(\Delta))\subset H^p(\Delta,\Gamma_*)/g^*\) and that \(\| g^*E(f)\|_ p\leq\| f\|_ p\). This fact is then used to prove among others the following:

Theorem 2: \(\overline{[E(H_0^p(\Delta))]}^ \perp =H^q(\Delta,{\mathcal G})\) and \(\overline{[E(g^* H^p(\Delta))]}^\perp= \frac{1}{g^*} H_0^q(\Delta,\Gamma_*)\) for any \(p\geq 1\), where \([\cdot]^\perp\) denotes the annihilator in the space \(L^q(d\theta,\mathcal G)\) and \(H_0^q(\Delta,\Gamma_ *)\) is the set of functions \(f\in H_0^q(\Delta)\) for which \(f\circ\gamma=\Gamma_*(\gamma)\circ f\) for any \(\gamma\in{\mathcal G}\).

Theorem 3: For any \(D\) of Parreau-Widom type the property (DCT) holds for the subspace \(\overline{E(H^1(\Delta))}\), i.e. \(f(0)=\int_{\partial\Delta} f \,d\theta/2\pi\) for any \(f\in\overline{E(H^1(\Delta))}\).

He further obtains a characterization of domains for which (DCT) holds.

In the second part, by use of this earlier work on harmonic domains, orthogonal measures are studied for domains of Parreau-Widom type satisfying (DCT). Let \(D\) be such a domain and let \(A(D)\) be the space of functions, bounded on \(D\) and continuous on \(\bar D\). Theorems 9 and 10: Every measure \(\mu\) on the boundary \(\partial D\), orthogonal to \(A(D)\), is absolutely continuous with respect to the harmonic measure on \(\partial D\) if and only if \(A(D)\) is pointwise boundedly dense in \(H^\infty(D)\). For other results including a generalization of the F. and M. Riesz theorem and the Rudin-Carleson theorem we refer the reader to the original paper.

In the book mentioned above the main tools for studying surfaces of Parreau-Widom type were Martin compactification and Green lines. On the other hand, in the case of compact bordered surfaces, F. Forelli [Ill. J. Math. 10, 367–380 (1966; Zbl 0141.31401)] employed the conditional expectation operators to study invariant subspaces of Hardy classes. This method was further exploited via Poincaré’s theta series by C. J. Earle and A. Marden [Ill. J. Math. 13, 202–219 (1969; Zbl 0169.10202)]. In a paper by Ch. Pommerenke [Ann. Acad. Sci. Fenn., Ser. A 2, 409–427 (1976; Zbl 0363.30029)] Fuchsian groups, appearing as the cover transformation groups of surfaces of Parreau-Widom type, were characterized and Earle and Marden’s result was extended to such groups. This means that Forelli’s conditional expectation operator method may be used not only for finitely connected surfaces but also for infinitely connected surfaces of Parreau-Widom type. This is done explicitly on the paper under review.

Let \(D\) be a hyperbolic domain, \(\pi:\Delta\to D\) be the universal covering map from the unit disk \(\Delta\) to \(D\) and \({\mathcal G}=\{\gamma\}\) the group of cover transformations for \(\pi\). The quotient space \(\Delta/{\mathcal G}\) is isomorphic with \(D\) and the Hardy class \(H^p(D)\) is identified with the subspace \(H^p(\Delta,{\mathcal G})\) of \({\mathcal G}\)- invariant functions in \(H^p(\Delta)\). Let \(g(z)=\prod\{e^{- \arg\gamma(0)}\gamma(z)\): \(\gamma\in{\mathcal G}\}\) be the Blaschke product corresponding to the group \({\mathcal G}\). Pommerenke loc. cit. showed that \(D\) (or \({\mathcal G}\)) is of Parreau-Widom type if and only if \(g'(z)\) is of bounded characteristic.

The paper is divided into two parts. In the first part, the author examines in detail the property (DCT) (= “Direct Cauchy Theorem”), which is crucial in the study of invariant subspaces. Let \(D\) be a domain of Parreau-Widom type. We define the operator \(E\) by setting \(E(H)(z)=g(z)/g'(z)\sum_{\gamma\in{\mathcal G}} H(\gamma(z))\gamma'(z)/\gamma(z)\) for any analytic function \(H(z)\) on \(\Delta\). Then, \(H\) is \({\mathcal G}\)-invariant if and only if \(E(H)=H\). For a function \(H(z)\) on \(\partial\Delta\) we set \(E(H)(z)=\rho^{-1} \sum_{\gamma\in{\mathcal G}} H(\gamma(z))|\gamma'(z)|\) for \(z\in\partial\Delta\), where \(\rho(z)=\sum_{\gamma\in{\mathcal G}}|\gamma'(z)|\). Then, \(E\) defines a projection of the norm from \(L^p(d\theta)\) onto \(L^p(d\theta,{\mathcal G})\). Let \(g^*(z)\) be the inner factor of \(g'(z)\) and \(\Gamma_ *\) denotes the character of \({\mathcal G}\) defined by \(g^*\), namely, \(g^*(\gamma(z))=\Gamma_ *(\gamma)g^*(z)\) for any \(\gamma\in{\mathcal G}\). By use of the operator \(E\) the author first proves that \(E(H^p(\Delta))\subset H^p(\Delta,\Gamma_*)/g^*\) and that \(\| g^*E(f)\|_ p\leq\| f\|_ p\). This fact is then used to prove among others the following:

Theorem 2: \(\overline{[E(H_0^p(\Delta))]}^ \perp =H^q(\Delta,{\mathcal G})\) and \(\overline{[E(g^* H^p(\Delta))]}^\perp= \frac{1}{g^*} H_0^q(\Delta,\Gamma_*)\) for any \(p\geq 1\), where \([\cdot]^\perp\) denotes the annihilator in the space \(L^q(d\theta,\mathcal G)\) and \(H_0^q(\Delta,\Gamma_ *)\) is the set of functions \(f\in H_0^q(\Delta)\) for which \(f\circ\gamma=\Gamma_*(\gamma)\circ f\) for any \(\gamma\in{\mathcal G}\).

Theorem 3: For any \(D\) of Parreau-Widom type the property (DCT) holds for the subspace \(\overline{E(H^1(\Delta))}\), i.e. \(f(0)=\int_{\partial\Delta} f \,d\theta/2\pi\) for any \(f\in\overline{E(H^1(\Delta))}\).

He further obtains a characterization of domains for which (DCT) holds.

In the second part, by use of this earlier work on harmonic domains, orthogonal measures are studied for domains of Parreau-Widom type satisfying (DCT). Let \(D\) be such a domain and let \(A(D)\) be the space of functions, bounded on \(D\) and continuous on \(\bar D\). Theorems 9 and 10: Every measure \(\mu\) on the boundary \(\partial D\), orthogonal to \(A(D)\), is absolutely continuous with respect to the harmonic measure on \(\partial D\) if and only if \(A(D)\) is pointwise boundedly dense in \(H^\infty(D)\). For other results including a generalization of the F. and M. Riesz theorem and the Rudin-Carleson theorem we refer the reader to the original paper.

Reviewer: M. Hasumi (Mito / Ibaraki)

##### MSC:

30F99 | Riemann surfaces |

30D55 | \(H^p\)-classes (MSC2000) |