Besov-Bergman spaces of $M$-harmonic functions
Petr Blaschke, Miroslav Engliš
TL;DR
This work identifies the analytic-continuation of weighted $M$-harmonic Bergman spaces $\mathcal{M}_s$ on the unit ball with Besov-type and Sobolev spaces, using only tangential derivatives. Central to the approach is a decomposition of $M$-harmonic functions into bidegree $(p,q)$ pieces via bigraded spherical harmonics, and the action of the tangential operator square $\square$ on these components, enabling componentwise definitions like $(I+\square)^t f$ and explicit norm equivalences. A key technical advance is a uniform bound for the coefficients $c_{pq}(s)$, ensuring robust equivalences among Bergman, Besov, and Hardy-tangential norms, and enabling the identification of $M$-harmonic Sobolev spaces with tangential data for $0\le t\le n$. The results illuminate how complex tangential derivatives control Sobolev regularity in the invariant-harmonic setting and extend classical holomorphic Besov–Sobolev correspondences to $M$-harmonic functions, with potential broader applicability through the hypergeometric-estimate techniques developed.
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.