Area integrals and h_p norms of analytic functions

Abstract

Let R be a Riemann surface with Green's function g(z,a). Let a εR and p>0 be fixed. For a function f analytic on R, let aεR be the least harmonic majorant of |f(z)|^p on R and

For an f with f(a) = 0. we define

. where g ^* (z,a) denotes the harmonic conjugate of g(z,a). In this paper we show the equality

in the case for regular Riemann surfaces and the inequality

for general Riemann surfaces. We also consider a higher dimensional case.

AMS (MOS):

• 30DS5
• 31B05