Mediterr. J. Math. (2017) 14:166
published online July 15, 2017
Springer International Publishing AG 2017
Weighted Bergman–Dirichlet and
Bargmann–Dirichlet Spaces in High
Aymane El Fardi, Allal Ghanmi,
Ahmed Intissar and Mohammed Ziyat
Abstract. In this paper, we consider and study the n-dimensional exten-
sion of the Bergman–Dirichlet and Bargmann–Dirichlet spaces intro-
duced recently in El Hamyani et al. (Ann Glob Anal Geom 49(1):59–72,
2016). We give a complete description of the considered spaces, includ-
ing the explicit closed formulas for their reproducing kernel functions.
Moreover, we investigate their asymptotic behavior when the curvature
goes to 0.
Mathematics Subject Classiﬁcation. Primary 32A10, 31C25, 32A36;
Secondary 32A17, 32K99.
Keywords. Weighted Bergman–Dirichlet spaces, weighted Bargmann–
Dirichlet spaces, reproducing kernel functions, hypergeometric functions.
The Bargmann–Fock space on the n-complex space C
and the so-called
weighted Bergman and Dirichlet spaces, on the open unit ball B
examples of functional spaces in the theory of analytic functions. Such spaces
play important roles in function and operator theories, as well as in modern
analysis, probability and statistical analysis. For a nice introduction and sur-
veys of these analytic spaces we refer the reader to [1,2,8,9,12–14] and the
Recently, two new classes of analytic function spaces of Sobolev type,
labeled by a nonnegative integer m, have been introduced and studied in
. The ﬁrst one is the Bergman–Dirichlet space generalizing the weighted
Bergman and Dirichlet spaces on the disk D(0,R) in the complex plane C.
The second is the Bargmann–Dirichlet space generalizing the Bargmann–
Fock space on the complex plane C = D(0, +∞). They are reproducing ker-
nel Hilbert spaces. Their reproducing kernel functions have been calculated
explicitly and expressed in terms of the hypergeometric functions.