We study the weighted differential operators \(\begin{aligned} \Delta _\alpha = (1-|x|^2)\Bigl (\frac{1-|x|^2}{4}\Delta + \frac{\alpha }{2}\,x\cdot \nabla + \frac{\alpha (n-2-\alpha )}{4}\Bigr ) \end{aligned}\) on the unit ball \(\mathbb {B}^n\subset \mathbb {R}^n\) . These operators generalize the classical Laplacian and, for certain values of \(\alpha \) , recover the Laplace–Beltrami operator. Our main contributions are twofold. First, we derive sharp pointwise estimates for \(\alpha \) -harmonic functions (solutions to \(\Delta _\alpha u=0\) ) 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 \(\Delta _\alpha \) in terms of Gauss hypergeometric functions, showing an explicit relationship between radial eigenfunctions and the powers of the generalized \(\alpha \) -Poisson kernel. This unified framework links Euclidean and hyperbolic harmonic analysis.
