Hardy classes and Riemann surfaces of Parreau-Widom type

4.5.

HARDY CLASSES AND RIEMANN SURFACES OF PARREAU--WIDOM TYPE*

The theory of Hardy classes on the unit disc and its abstract generalization have received considerable attention in recent years [I-3]. The case of compact bordered surfaces has also been studied in detail. But our knowledge for infinitely connected surfaces seems to be relatively scarce. Our basic question is this: For which class of Riemann surfaces can one get a fruitful extension of the Hardy class theory on the disc? We propose here the class of Riemann surfaces of Parreau--Widom type as a most promising candidate. Definition. L e t R be a hyperbolic Riemann surface, G(a, z) the Green function for R with pole at a point a, a ~ R, and B(a, a) the first Betti number of the subdomain R(a, a) = {z e

R:G(a, z) > a} with a > 0.

We say that R is of Parreau--Widom type if I~+~(~,~)~<+~.

We first describe some relevant results which show that such surfaces are nice. following, R denotes a surface of Parreau--Widom type, unless stated otherwise. (I) Parreau every Green line R for any hounded the boundary data

In the

[4]: (a) Every positive harmonic function on R has a limit along almost issuing from any fixed origin. (b) The Diriehlet problem on Green lines on measurable boundary function has a unique solution, which converges to along almost all Green lines.

(2) Widom [5]: For a hyperbolic Riemann surface R, H~(R, ~) for any complex flat unitary line bundle ~ over R has a nonzero element if and only if R is of Parreau-Widom type, where H~(R, ~) denotes the set of all bounded holomorphic sections of ~. (3) Hasumi [6]: (a) Almost every Green line I on R issuing from any fixed origin 0 converges to a point b in the Martin boundary A of R and the correspondence 1 -+ b is measurepreserving with respect to the Green measure and the harmonic measure on A for the point 0. (b) The usual solution of the Dirichlet problem on R for any bounded measurable boundary function on A converges to the boundary data along almost all Green lines. (c) The inverse Cauchy theorem holds. (4) Under some further restrictions, Neville [7] proved the direct and the inverse C a u c h y theorems as well as the invariant subspace theorems of Beurling type for the Hardy classes HP(R) on R and, independently, Hasumi [6] showed the direct Cauchy theorem and the invariant subspace theorems for LP on A. (5) Further results may be seen in [8-11]. (6) In a paper still in preparation Hayashi shows that the restrictions mentioned (4) are redundant and that the correspondence 1 § b in (3) (a) is almost one-to-one.

in

Although the surfaces of Parreau-Widom type occupy a rather restricted place in the category of Riemann surfaces, we believe that they probably form the widest class of "nice" domains as far as the Hardy class theory is concerned. We may try to weaken our hypothesis on R. For instance, one may think of the condition (S): the set H~(R) of bounded holomorphic functions on R separates points of R. This looks unlikely, however, for there exists a Riemann surface (even a plane domain) R satisfying the condition (S), for which H~(R) is not dense in H2(R) (of. [12]). We ask, quite vaguely, which of the basic results in the unit disc case have natural extensions to our surfaces. Certain results can of course be true under conditions weaker than the Parreau--Widom. It would be interesting to find the weakest possible assumption for each basic result in the Hardy class theory. Finally we mention a few concrete questions without claiming that these are typical. In Hayashi's study of invariant subspaces (el. [6]), unicity of the expression for simply invariant subspaces remains unsettled and he asks: (i) Does H~(R, 5) for any ~ have only constant common inner factors? He also asks, in connection with his study [10], the following question: (ii) Is a generalized F. and M. Riesz theorem true for measures on Wiener's harmonic boundary, which are orthogonal to H~? Another problem: (iii) Characterize those *MORISUKE HASUMI.

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Ibaraki University, Department of Mathematics, Mito, Ibaraki, 310, Japan.

surfaces R for which H~(R, ~) for every ~ has an element wiLhout zero. This was once communicated from Widom and seems to be still open. On the other hand, plane domains of Parreau--Widom type are not very well known. (iv) Characterize closed subsets E of the Riemann sphere S for which S\ E is of Parreau-Widom type (cf. [13, 7]). LITERATURE CITED I. 2. 3. 4. 5 6 7 8 9 10 11. 12. 13.

K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall (1962). H. Helson, Lectures on Invariant Subspaces, Academic Press, New York (1964). T. Gamelin, uniform Algebras, Prentice-Hall (1969). M. Parreau, "Th~orSme de Fatou et problhme de Dirichlet pour les lignes de Green de certaines surfaces de Riemann," Ann. Acad. Sci. Fenn. Ser. At, No. 250/25 (1958). H. Widom, "Hp sections of vector bundles over Riemann surfaces," Ann. Math., __94, 304324 (1971). M. Hasumi, "Invariant subspaces on open Riemann surfaces," Ann. Inst. Fourier, Grenoble, 24, No. 4, 241-286 (1974); II, ibid., 26, No. 2, 273-299 (1976). C--~ Neville, "Invariant subspaces of Hardy classes on infinitely connected open surfaces," Mem. Am. Math. Soc., No. 160 (1975). Ch. Pommerenke, "On the Green's function of Fuchsian groups," Ann. Acad. Sci. Fenn., Ser. AI, No. 2, 408-427 (1976). C. Stanton, "Bounded analytic functions on a class of open Riemann surfaces," Pacific J. Math., 59, 557-565 (1975). M. Hayashi, "Linear extremal problems on Riemann surfaces," preprint (1977). M. Hasumi, "The Fatou theorem for open Riemann surfaces," Duke Math. J., 43, 731-746 (1976). M. Hasumi, "Hardy classes on plane domains," Inst. Mittag--Leffler Rep., No. 2 (1977). M. Voichiek, "Extreme points of bounded analytic functions on infinitely connected regions," Proc. Am. Math. Soc., 17, 1366-1369 (1966).

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