Note on \(H^ p\) on Riemann surfaces (Q1087670)

The following result is obtained: For an arbitrary holomorphic function f on an arbitrary open Riemann surface R and any positive real number p, there exist two subregions \(S_ 1\) and \(S_ 2\) of R with \(S_ 1\cup S_ 2=R\) such that \(f| S_ i\) belongs to the Hardy class \(H^ p(S_ i)\) \((i=1,2)\). The result is originally obtained by \textit{R. Bañuelos} and \textit{T. Wolff} [Proc. Am. Math. Soc. 95, 217-218 (1985; Zbl 0582.30025)] when R is the unit disk. The proof of the above result given in this note does not use the analyticity of f or subharmonicity of \(| f| ^ p\) or even the continuity of f but only uses the local boundedness of f on R to deduce that \(| f| ^ p\) admits a harmonic majorant on \(S_ i\) \((i=1,2)\). Thus in reality a much more general result than stated is obtained in this note.