On the Fourier-Borel transformations of analytic functionals on the complex sphere

Abstract

Introduction.Let έ? (C d+1 ) and Exp(C d+1 ) be the spaces of entire functions on C d+1 and entire functions of exponential type, respectively.έ?\C d+1 ) and Exp'(C d+1 ) are the spaces dual to έ?(C d+1 ) and Exp(C d+1 ), respectively.For T e Exp'(C d+1 ) the Fourier-Borel transformation P λ is defined bywhere λeC, λ^O, is a fixed constant (Hashizume, Kowata, Minemura and Okamoto [2]).Martineau [4] determined the images of Exp'(C d+1 ) and some functional spaces on C d+1 by the Fourier-Borel transformation P λ .Let S = S d be the unit sphere in R d+1 and S denote the complex sphere in C d+1 .We put S(r) = {z e S; L(z) < r} and S[r] = {zeS; L(z) ^ r}, where L(z) is the Lie norm on C d+1 .^(S), ^(S(r)) and έ? (S[r]) denote the spaces of holomorphic functions on S, S(r), and S[r], respectively.Exp(S) denotes the restriction of Exp(C d+1 ) to S. Exp'(S), ^'(S), and <P\S[r\) are the spaces dual to Exp(S), ^(S), <£?(S(r)) and respectively.Exp'(S) can be regarded as a subspace of Exp'(C d+1 ).Morimoto [7] determined the images of Exp'(S) and έ?'(S) by the Fourier-Borel transformation P λ (Theorem 1.2).In this paper we will determine the images of έ?'(S(r)) and ^'(S[r]) by the Fourier-Borel transformation P λ .The images are characterized explicitly in terms of the dual Lie norm (Theorem 3.1).Consider a complex cone M = {z e C d+1 ; Σiίί z) = 0, z Φ 0}, which can be identified with the cotangent bundle of S minus its zero section.We define for /' e Exp'(S) <Λ.expte-*)> (zeM).Ff is the restriction of P_«/' to M. Ii [3] determined the images of H ntd by F, where H ntd is the space of spherical harmonics of degree n in dimension d + 1. Moreover if d is even, Ii [3] characterized the image of L\S) under this mapping F. In this paper we determine the image of L\S) for odd d (Theorem 2.4).We also determine the images of