We show that the weighted Bergman spaces of M-harmonic functions (functions annihilated by the invariant Laplacian on the unit ball of the complex n-space), as well as their analytic continuation (in the spirit of Rossi and Vergne), coincide with the certain Besov-type spaces, which were studied by Folland. Characterizations in terms of tangential derivatives are given, and for appropriate values of the weight parameter, these spaces are also shown to coincide with the subspaces of all M-harmonic fucntions in the Sobolev space of order t on the ball, 0≤ t≤ n. Unlike the holomorphic case, the last result is shown to fail in general for other values of t. The main tool in the proofs are asymptotic estimates for certain integrals of squared hypergeometric functions, which seem to be of interest in their own right and may find other applications.

We~describe the Dirichlet space of $M$-harmonic functions, i.e.~functions annihilated by the invariant Laplacian on~the unit ball of the complex $n$-space, as~the limit of the analytic continuation (in~the spirit of Rossi and Vergne) of the corresponding weighted Bergman spaces. Characterizations in terms of tangential derivatives are given, and the associated inner product is shown to be Moebius invariant. The pluriharmonic and harmonic cases are also briefly treated.

We prove uniqueness of the Moebius invariant semi-inner product on hyperbolic-harmonic functions on the unit ball of the real n-space, i.e. on functions annihilated by the hyperbolic Laplacian on the ball.

It has been an open problem - at least since M. Stoll's book "Harmonic and subharmonic function theory on the hyperbolic ball" (Cambridge University Press, 2016) - whether there exists a Moebius invariant Hilbert space of hyperbolic-harmonic functions on the unit ball of the real nspace, i.e. of functions annihilated by the hyperbolic Laplacian on the ball. We give an answer by describing a Dirichlettype space of hyperbolic-harmonic functions, as the analytic continuation (in the spirit of Rossi and Vergne) of the corresponding weighted Bergman spaces. Characterizations in terms of derivatives are given, and the associated semi- inner product is shown to be Moebius invariant. We also give a formula for the corresponding reproducing kernel. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http:// creativecommons.org/licenses/by-nc/4.0/).

Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we obtain expansions for the Szegö and the weighted Bergman kernels of M-harmonic functions, i.e. functions annihilated by the invariant Laplacian on the unit ball of the complex n-space. This yields, among others, an explicit formula for the M-harmonic Szegö kernel in terms of multivariable as well as single-variable hypergeometric functions, and also shows that most likely there is no explicit (“closed”) formula for the corresponding weighted Bergman kernels.

We give a description of the boundary singularity of the Szego kernel of upperM M-harmonic functions, i.e. functions annihilated by the invariant Laplacian, on the unit ball of the complexnn-space, in terms of the Gauss hypergeometric functions.

We identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in the sense of Shimura, on bounded symmetric domains. This also yields a description of the analogous kernels for spaces of “invariantly-polyanalytic” functions — a generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In both cases, the kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.

We consider a complex domain D x V in the space C-m x C-n and a family of weighted Bergman spaces on V defined by a weight e(-k phi(z , w)) for a pluri-subharmonic function phi(z, w) with a quantization parameter k. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain D. We consider the natural covariant differentiation del(z) on the sections, namely the unitary Chern connections preserving the Bergman norm. We prove a Dixmier trace formula for the curvature of the unitary connection and we find the asymptotic expansion for the curvatures R-(k)(Z,Z) for large k and for the induced connection [del((k))(Z), T-f((k))] on Toeplitz operators T-f. In the special case when the domain D is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for [del((k))(Z), T-f((k))] as Toeplitz operators. This generalizes earlier work of J.E. Andersen in Comm. Math. Phys. 255 (2005), 727-745. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D x V replaced by a general strictly pseudoconvex domain V subset of C-m x C-n fibered over a domain D subset of C-m. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.

We show that the usual Poincare metric is the only radial balanced metric on the disc with not too wild boundary behaviour. Additionally, we identify explicitly all radial metrics with such boundary behaviour which satisfy the balanced condition as far as germs at the boundary are concerned. Related results for the annulus and the punctured disc are also established.

We extend the classical theory of Qp-spaces of Aulaskari, Xiao and Zhao in two directions: first, by considering more general construction of such spaces; and second, by considering invariance with respect to weighted Möbius actions. In addition, related results for the associated Bloch-type spaces are also established.

We show that there is no radial balanced metric on the unit ball of the Kepler manifold with not too wild boundary behavior. Additionally, we identify explicitly the weights corresponding to radial metrics with such boundary behavior which satisfy the balanced condition as far as germs at the boundary are concerned. Related results for Poincare metrics are also established.

We study the complex geometry of generalized Kepler manifolds, defined in Jordan theoretic terms, introduce Hilbert spaces of holomorphic functions defined by radial measures, and find the complete asymptotic expansion of the corresponding reproducing kernels for Kähler potentials, both in the flat and bounded setting.

It was conjectured by the first author and Peetre that the higher Laplace–Beltrami operators generate the whole ring of invariant operators on bounded symmetric domains. We give a proof of the conjecture for domains of rank ≤ 6 by using a graph manipulation of Kähler curvature tensor. We also compute higher order terms in the asymptotic expansions of the Bergman kernels and the Berezin transform on bounded symmetric domain.

In this paper, we construct noncommutative coherent states using various families of unitary irreducible representations (UIRs) of G(nc), a connected, simply connected nilpotent Lie group, which was identified as the kinematical symmetry group of noncommutative quantum mechanics for a system of two degrees of freedom in an earlier paper. Similarly described are the degenerate noncommutative coherent states arising from the degenerate UIRs of Gnc. We then compute the reproducing kernels associated with both these families of coherent states and study the Berezin-Toeplitz quantization of the observables on the underlying 4-dimensional phase space, analyzing in particular the semi-classical asymptotics for both these cases.

We develop a theory of Toeplitz, and to some extent Hankel, operators on the kernels of powers of the boundary d-bar operator, suggested by Boutet de Monvel and Guillemin, and on their analogues, somewhat better from the point of view of complex analysis, defined using instead the covariant Cauchy-Riemann operators of Peetre and the second author. For the former, Dixmier class membership of these Hankel operators is also discussed. Our main tool are the generalized Toeplitz operators (with pseudodifferential symbols), in particular there appears naturally the problem of finding parametrices of matrices of such operators in situations when the principal symbol fails to be elliptic.

For a class of $O(n+1,R)$ invariant measures on the Kepler manifold possessing finite moments of all orders, we describe the reproducing kernels of the associated Bergman spaces, discuss the corresponding asymptotic expansions of Tian-Yau-Zelditch, and study the relevant Hankel operators with conjugate holomorphic symbols. Related reproducing kernels on the minimal ball are also discussed. Finally, we observe that the Kepler manifold either does not admit balanced metrics, or such metrics are not unique.

We give criteria for the membership of Hankel operators on the Hardy space on the disc in the Dixmier class, and establish estimates for their Dixmier trace. In contrast to the situation in the Bergman space setting, it turns out that there exist Dixmier-class Hankel operators that are not measurable (that is, their Dixmier trace depends on the choice of the underlying Banach limit), as well as Dixmier-class Hankel operators that do not belong to the Schatten-Lorentz ideal. A related question concerning logarithmic interpolation of Besov spaces is also discussed.

For weights ρ which are either radial on the unit ball or depend only on the vertical coordinate on the upper half-space, we describe the asymptotic behaviour of the corresponding weighted harmonic Bergman kernels with respect to ρα as α→+∞. This can be compared to the analogous situation for the holomorphic case, which is of importance in the Berezin quantization as well as in complex geometry.

We present an introduction to the Berezin and Berezin-Toeplitz quantizations, starting from their historical origins and relationships with other quantization methods, discussing various instructive examples like the Segal-Bargmann-Fock space, and culminating by highlights of proofs of the existence of these quantizations using both the Boutet de Monvel theory and the approach via Fefferman's expansion and Forelli-Rudin construction. The exposition strives to be reasonably self-contained and accessible to nonexperts.

A general theory of Berezin-Toeplitz quantization for symmetric spaces is presented, with emphasis on representation-theoretic asymptotic expansions, which applies to spaces of compact and non-compact type, both in the classical setting of hermitian symmetric spaces and also for their real forms. The Berezin (or Wick type) calculus and its opposite "anti-Wick" type calculus are treated on an equal footing.