Eigenspaces of the invariant laplacian in the unit ball of $C^n$

Abstract

The differential operator $L_{pq}$ is defined by $(L_{pq}y)(t) = 4(1-t) (t(1-t)y^```` + (p+q+n-(p+q+1)t]y^``-pqy), p, q=0,1,2, …. The eigenfunctions of $L_{pq}$ constitute the radial parts of the base functions of -subspaces of the eigenspaces of the invariant Laplacian $\overline{Δ}$. For certain eigenvalue (4m(m+n), m=0,1,2, …), p and q, we find the eigenfunctions of $L_{pq}$ explicitly and for n=2, we find a base functions of the space of all homogeneous harmonic polynomial of bidegree(p,q). These results applied to get a base functions of certain M-spaces. We also give some relations between some M-spaces in the eigenspaces of the invariant Laplacian and certain spaces of potentials.