On a partial differential operator associated with eigenvalue problems of invariant laplacian

Abstract

For $\lambda = 4m(n+m)$, a function $f(z)=(1-\mid{z}\mid^2)^mg(z)$ with $g\in{X}\lambda$, the eigenspace of the invariant Laplacian $\widetilde{\triangle}$ in the unit ball $B_n$ of $C^n$, satisfies an elliptic differential equation $\triangle_mf= 0$. We make a study of the operator $\triangle_m$ as another way to study $\widetilde(\triangle} - 4m(n + m)$. For example, if $Z_m$ denotes the class of all solutions f in $C^2(B_n)$ of $\triangle_mf = 0$, we obtain an $L^2$-growth condition for the projection of a function in $Z_m$ onto H(p,q), the space of all harmonic homogeneous polynomials on $C^n$ of degree p in z and of degree q in z, to be 0 unless either $p \leq m$ or $q \leq m$. This corresponds and gives another way to obtain the $L^2$-growth condition for a function in $X_\lambda$ to be in the M-subspace $Y_4$ of $X_\lambda$ in [1,3,7]. $Y_4$ is the space of pluriharmonic functions in case $\lambda = 0$.