$\mathcal H$-Harmonic Bergman Projection on the Hyperbolic Ball

TL;DR

This paper characterizes the boundedness of the $eta$-weighted Bergman projection on Lebesgue and Bergman spaces of $ ext{H}$-harmonic functions on the hyperbolic ball, confirming a recent conjecture and providing kernel estimates.

Contribution

It precisely determines boundedness conditions for the $ ext{H}$-harmonic Bergman projection and verifies a conjecture, also deriving kernel estimates and analyzing the projection to the Bloch space.

Findings

Boundedness conditions for $P_eta$ from $L^p_ ext{alpha}$ to $ ext{Bergman spaces

Upper estimates for the reproducing kernel and its derivatives

Verification of a recent conjecture of M. Stoll

Abstract

We determine precisely when the Bergman projection $P_\beta$ is bound\-ed from Lebesgue spaces $L^p_\alpha$ to weighted Bergman spaces $\mathcal B^p_\alpha$ of $\mathcal H$-harmonic functions on the hyperbolic ball, and verify a recent conjecture of M. Stoll. We obtain upper estimates for the reproducing kernel of the $\mathcal H$-harmonic Bergman space $\mathcal B^2_\alpha$ and its partial derivatives. We also consider the projection from $L^\infty$ to the Bloch space $\mathcal B$ of $\mathcal H$-harmonic functions.