An extension of the Forelli–Rudin projection theorem

Abstract

For a measurable function f on the unit ball B in ℂn we define (M1f)(w), |w|<1, to be the mean modulus of f over a hyperbolic ball with center at w and of a fixed radius. The space , 0<p<∞, is defined by the requirement that M1f belongs to the Lebesgue space Lp. It is shown that the subspace of Lp spanned by holomorphic functions coincides with the corresponding subspace of . It is proved that if s>(n+1)(p−1−1), 0<p<1, then this subspace is complemented in by the projection whose reproducing kernel is . As corollaries we get an extension of the Forelli–Rudin projection theorem and we show that a holomorphic function f is Lp-integrable, 0<p<∞, over the unit ball B iff u = Ref is Lp-integrable over B. Finally, we sketch an alternative proof of the main result of this paper in the case 0<p<1.