On the Poisson Transform on the bounded domain of type IV.

Let D be the Lie ball in C² and let A′(S) be the space of all hyperfunctions over the Shilov boundary S of D. The aim of this paper is to give a necessary and sufficient condition on the Poisson transform P_λf of an element f in the space A′(S) for f to be in L²(S). More precisely, we establish for any λ … R\{0} that: (i) Let F = P_λf, f … L²(S). Then we have ||F||²_* = sup _t>0 1/t² ∫₀^t (∫^r1₀ ∫_{S(O(2)×O(2))} |F(ka_R.0)|² sinh (r₁ - r₂) sinh (r₁ + r₂) dkdr₂)dr₁ < ∞. (ii) Let f be a hyperfunction on S such that its image F = P_λf satisfies the growth condition ||F||_* < ∞, then necessarily such f is in L²(S). [ABSTRACT FROM AUTHOR]