Growth of Generalized Harmonic Functions and Radial Eigenfunctions

We study the weighted differential operators (Formula presented.) on the unit ball Bn⊂Rn. These operators generalize the classical Laplacian and, for certain values of α, recover the Laplace–Beltrami operator. Our main contributions are twofold. First, we derive sharp pointwise estimates for α-harmonic functions (solutions to Δαu=0) using a Poisson-type integral representation. This extends prior results on harmonic, hyperbolic harmonic functions, and weighted planar harmonic functions to higher dimensions. Second, we characterize radial eigenfunctions of Δα in terms of Gauss hypergeometric functions, showing an explicit relationship between radial eigenfunctions and the powers of the generalized α-Poisson kernel. This unified framework links Euclidean and hyperbolic harmonic analysis.