Optimal estimates for hyperbolic Poisson integrals of functions in [formula omitted] with [formula omitted] and radial eigenfunctions of the hyperbolic Laplacian

Abstract

Assume that p∈(1,∞] and u=Ph[ϕ], where ϕ∈Lp(Sn−1,Rn). Then for any x∈Bn, we obtain the sharp inequalities |u(x)|≤Cq1q(x)(1−|x|2)n−1p‖ϕ‖Lpand|u(x)|≤Cq1q(1−|x|2)n−1p‖ϕ‖Lp for some function Cq(x) and constant Cq in terms of Gauss hypergeometric and Gamma functions, where q is the conjugate of p. This result generalizes and extends some known results from harmonic mapping theory (Kalaj and Marković (2012, Theorems 1.1 and 1.2) and Axler et al. (1992, Proposition 6.16)). The proofs are mainly based on certain characterizations of the radial eigenfunctions of the hyperbolic Laplacian Δh, which are of independent interest.