Conical Fourier-Borel transformations for harmonic functionals on the Lie ball

Abstract

Let L(z) be the Lie norm on Ẽ = Cn+1 and L∗(z) the dual Lie norm. We denote by O∆(B(R)) the space of complex harmonic functions on the open Lie ball B(R) and by Exp∆(Ẽ; (A,L ∗)) the space of entire harmonic functions of exponential type (A,L∗). A continuous linear functional on these spaces will be called a harmonic functional or an entire harmonic functional. We shall study the conical Fourier–Borel transformations on the spaces of harmonic functionals or entire harmonic functionals.