Weighted Bergman-Dirichlet and Bargmann-Dirichlet spaces of order $m$ in high dimensions

We introduce and study a generalization of the classical weighted Bergman and Dirichlet spaces on the unit ball in high dimension, the Bergman-Dirichlet spaces. Their counterparts on the whole $n$-complex space, the Bargmann-Dirichlet spaces, are also introduced and studied. Mainly, we give a complete description of the considered spaces, including orthonormal basis and the explicit formulas for their reproducing kernel functions. Moreover, we investigate their asymptotic behavior when the curvature goes to $0$.