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An extension of the Forelli–Rudin projection theorem

Published online by Cambridge University Press:  20 January 2009

M. Mateljević  and
M. Pavlović
M. Mateljević
Affiliation:
Mathematički Fakultet, Studentski trg 16, 11000 Beograd, Yugoslavia
M. Pavlović
Affiliation:
Mathematički Fakultet, Studentski trg 16, 11000 Beograd, Yugoslavia
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Abstract

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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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 3 , October 1993 , pp. 375 - 389
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

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