Besov-Bergman spaces of $M$-harmonic functions

Abstract: We~show that the weighted Bergman spaces of M-harmonic functions (functions annihilated by the invariant Laplacian on the unit ball of the complex n-space), as~well as their analytic continuation (in~the spirit of Rossi and Vergne), coincide with the certain Besov-type spaces, which were studied by Folland. Characterizations in terms of tangential derivatives are given, and for appropriate values of the weight parameter, these spaces are also shown to coincide with the subspaces of all M-harmonic fucntions in the Sobolev space of order~$t$ on the~ball, $0\le t\le n$. Unlike the holomorphic case, the~last result is shown to fail in general for other values of~$t$. The~main tool in the proofs are asymptotic estimates for certain integrals of squared hypergeometric functions, which seem to be of interest in their own right and may find other applications.