\({\mathcal M}\)-subspaces of \(X_ \lambda\) (Q1320947)

For \(\lambda\in\mathbb{C}\), \(X_ \lambda\) denotes the space of all \(f\in C^ 2(B)\) for which \(\widetilde{\Delta} f=\lambda f\) (\(B\) denotes the unit ball in \(\mathbb{C}^ n\), \(\widetilde{\Delta}\) is the invariant Laplacian). \(L^ 2\)-growth condition for a function in \(X_ \lambda\) to be in \(Y_ 4\) is obtained; this result extends corresponding results for \(X_ 0\) of [\textit{W. Rudin}, A smoothness condition that implies pluriharmonicity (preprint); \textit{P. Ahern} and \textit{W. Rudin}, Indag. Math., New Ser. 2, No. 2, 141-147 (1991; Zbl 0762.32001)]. A necessary and sufficient condition for a function \(g\in X_ \lambda\) to be represented by \(P^ \alpha [G]\) for some \(G\in L^ 2(\partial B)\) \((\alpha> 1/2)\) is given. Further a description of \(Y_ 3\) in terms of \({\mathcal M}_ \alpha\) in the case \(\lambda= 4m(m+n)\) \((m=0,1,2,\dots)\) is given.