Analytic functions and analytic functionals on some balls in the complex Euclidean spaces
by
Keiko Fujita
Saga University, Japan

In this talk, we shall consider analytic functions and analytic functionals on the N_p-balls [B\tilde]_p(r) defined as follows:

For p >= 1, we define the following N_p-ball [B\tilde]_p (r) of radius r with center at 0 in the complex Euclidean space [(E)\tilde]=Cⁿ⁺¹;

~ B   p  (r) = ì í î z in ~ E   ; é ë  1 2 ì í î L(z)p+ æ è  |z2| L(z) ö ø p   ü ý þ ù û 1/p   < r ü ý þ ,

where L(z) is the Lie norm and z²=z₁²+z₂²+ ... + z_n+1².

Note that [B\tilde]₂(r)={ z in [(E)\tilde]; ||z|| ² = |z₁|²+ ... +|z_n+1|² < r²} is the complex Euclidean ball, [B\tilde]₁(r) is the dual Lie ball and [B\tilde](r)= \cap _{p > 0}[B\tilde]_p(r) = { z in [(E)\tilde]; L(z) < r} is the Lie ball.

It is well-known that a holomorphic function f in [(E)\tilde] can locally be expanded into the double series

f(z) = \infty å k=0  [k/2] å l=0  (z2)l fk, k-2l (z),

where f_{k, k-2l} is the homogeneous harmonic polynomials of degree k-2l.

For holomorphic functions on [B\tilde]_p(r) we have the following theorem:

Theorem Let f(z) = \sum_k=0^\infty \sum_l=0^[k/2] (z²)^l f_{k, k-2l} (z). Then we have

f in O( ~ B   p  (r)) <===> limsup k --> \infty  é ë æ è  2kl!(k-l)! k! ö ø 1/p   || fk, k-2l|| C(S1) ù û 1/k   <= 1/r,

where O([B\tilde]_p(r)) denotes the space of holomorphic functions on [B\tilde]_p(r) and || f||_C(S1) = sup_{z in S1} |f(z)| is the supremum norm on the unit real sphere S₁.

Furthermore, for analytic functionals on [B\tilde]_p(r) we also have similar results to the above theorem.

We shall treat such a kind of theorem and related topics.

Date received: August 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-74.