Tanaka, Hiroshi; Schiff, Joel L. Note on the Wiener compactification and the \(H^ p\)-space of harmonic functions. (English) Zbl 0537.30030 Proc. Japan Acad., Ser. A 59, 231-233 (1983). Let R be a hyperbolic Riemann surface, and denote by HB(R) (resp. HB’(R)) the class of all bounded harmonic (resp. quasibounded harmonic) functions on R. For \(1<p<\infty\) denote by \(H^ p(R)\) the Hardy space of all harmonic functions u on R such that \(| u|^ p\) has a harmonic majorant. It is well known that \(HB(R)\subseteq H^ p(R)\subseteq HB'(R),\) and that if \(\dim HB(R)<\infty,\) then \(HB(R)=H^ p(R)=HB'(R).\) In the present paper, it is shown that if any two of the classes HB(R), \(H^ p(R)\), and HB’(R) coincide, then necessarily \(\dim HB(R)<\infty,\) for \(1<p<\infty.\) MSC: 30F15 Harmonic functions on Riemann surfaces Keywords:class of bounded harmonic functions; class of quasi-bounded harmonic functions; hyperbolic Riemann surface; Hardy space of all harmonic functions PDFBibTeX XMLCite \textit{H. Tanaka} and \textit{J. L. Schiff}, Proc. Japan Acad., Ser. A 59, 231--233 (1983; Zbl 0537.30030) Full Text: DOI References: [1] Lumer-Naim, L.: #P-spaces of harmonic functions. Ann. Inst. Fourier, 17, 425-469 (1967). · Zbl 0153.43102 [2] Sario, L., and M. Nakai: Classification Theory of Riemann Surfaces. Berlin-Heidelberg-New York, Springer (1970). · Zbl 0199.40603 [3] Schiff, J. L.: H^-spaces of harmonic functions and the Wiener compactification. Math. Z., 132, 135-140 (1973). · Zbl 0256.31001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.