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Note on the Wiener compactification and the \(H^ p\)-space of harmonic functions. (English) Zbl 0537.30030

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
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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
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