Weighted Bergman–Dirichlet and Bargmann–Dirichlet Spaces in High Dimension

Abstract

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 C n , 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.