Weighted Bergman Kernels Research Articles

The weighted Bergman kernels and projections on a class of unbounded Hartogs domains

In this paper, we study the weighted Bergman kernels and projections on a class of unbounded Hartogs domains. As applications of our main results, we can obtain the explicit formulas for the weighted Bergman kernels on a wide class of unbounded Hartogs domains with different types of weight functions, and show that the associated weighted Bergman projections are bounded on the weighted [Formula: see text] spaces if and only if [Formula: see text].

Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains

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.

M-harmonic reproducing kernels on the ball

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.

The affine ensemble: determinantal point processes associated with the $ax + b$ group

We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane $\mathbb{C}^{+}$ associated with the $ax + b$ (affine) group, depending on an admissible Hardy function $\psi$. We obtain the asymptotic behavior of the variance, the exact value of the asymptotic constant, and non-asymptotic upper and lower bounds for the variance on a compact set $\Omega \subset \mathbb{C}^{+}$. As a special case one recovers the DPP related to the weighted Bergman kernel. When $\psi$ is chosen within a finite family whose Fourier transform are Laguerre functions, we obtain the DPP associated to hyperbolic Landau levels, the eigenspaces of the finite spectrum of the Maass Laplacian with a magnetic field.

A L2 estimate for the minimal solution of [formula omitted] on the unit ball

We extend the result of Schuster and Varolin on L2-estimates for the minimal solution of ∂‾ on the unit ball B in Cn. Estimates of weighted Bergman kernels are also considered.

Toeplitz operators and skew Carleson measures for weighted Bergman spaces on strongly pseudoconvex domains

In this paper we study properties of Toeplitz operators on weighted Bergman spaces of bounded strongly pseudoconvex domains. We prove that a Toeplitz operator built using a weighted Bergman kernel of weight β and integrating against a measure μ maps continuously a weighted Bergman space Ap1α1(D) into Ap2α2(D) if and only if μ is a (λ,γ)-skew Carleson measure, where λ=1+1p1−1p2 and γ=1λ(β+α1p1−α2p2). This generalizes results obtained by Pau and Zhao on the unit ball, and by Abate, Raissy and Saracco on a smaller class of Toeplitz operators on strongly pseudoconvex domains.

The weighted Bergman space on a sector and a degenerate parabolic equation

In this work we solve a degenerate parabolic equation for the half line with Dirichlet boundary data, and use some results from the theory of Reproducing Kernel Hilbert Spaces to show that the null reachable space of this degenerate parabolic equation is a RKHS of analytic functions on a sector, whose reproducing kernel can be written in terms of the weighted Bergman kernel on the half plane C+.

Bergman Spaces Under Maps of Monomial Type

For appropriate domains $$\Omega _{1}, \Omega _{2}$$ , we consider mappings $$\Phi _{\mathbf {A}}:\Omega _{1}\rightarrow \Omega _{2}$$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $${\mathcal {A}}^{2}(\Omega _{1})$$ into finitely many closed subspaces indexed by characters of a finite abelian group associated to the mapping $$\Phi _{\mathbf {A}}$$ . We then show that each subspace is isomorphic to a weighted Bergman space on $$\Omega _{2}$$ . This leads to a formula for the Bergman kernel on $$\Omega _{1}$$ as a sum of weighted Bergman kernels on $$\Omega _{2}$$ .

General fractional derivatives and the Bergman projection

In this note we study some basic properties of general fractional derivatives induced by weighted Bergman kernels. As an application we demonstrate a method for generating pre-images of analytic functions under weighted Bergman projections. This approach is useful for proving the surjectivity of weighted Bergman projections in cases when the target space is not a subspace of the domain space (such situations arise often when dealing with Bloch and Besov spaces). We also discuss a fractional Littlewood-Paley formula.

Weighted Bergman Kernels and Mathematical Physics

We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains Ω ⊂ C n but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is a complex manifold, and turn out to be an efficient computational tool that is useful for the calculation of transition probability amplitudes from a classical state (identified to a coherent state) to another. We review the weighted version (for weights of the form γ = | φ | m on strictly pseudoconvex domains Ω = { φ < 0 } ⊂ C n ) of Fefferman’s asymptotic expansion of the Bergman kernel and discuss its possible extensions (to more general classes of weights) and implications, e.g., such as related to the construction and use of Fefferman’s metric (a Lorentzian metric on ∂ Ω × S 1 ). Several open problems are indicated throughout the survey.

REGULARITY OF THE WEIGHTED BERGMAN PROJECTION ON THE FOCK–BARGMANN–HARTOGS DOMAIN

The Fock–Bargmann–Hartogs domain $D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$, where $\unicode[STIX]{x1D707}>0$, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of $D_{n,m}(\,\unicode[STIX]{x1D707})$ with respect to the weight $(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$, where $\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$ and $\unicode[STIX]{x1D6FC}>-1$. Then, for $p\in [1,\infty ),$ we show that the corresponding weighted Bergman projection $P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$ is unbounded on $L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$, except for the trivial case $p=2$. This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^{p}$ irregular when $p\in [1,\infty )\setminus \{2\}$, in contrast to the well-known positive $L^{p}$ regularity result on a bounded strongly pseudoconvex domain.

Weighted Bergman Kernel, Directional Lelong Number and John–Nirenberg Exponent

Let $\psi$ be a plurisubharmonic function on the closed unit ball and $K_{t\psi}(z)$ the Bergman kernel on the unit ball with respect to the weight $t\psi$. We show that the boundary behavior of $K_{t\psi}(z)$ is determined by certain directional Lelong number of $\psi$ for all $t$ smaller than the John-Nirenberg exponent of $\psi$ associated to certain family of nonisotropic balls, which is always positive.

Weighted Sub-Bergman Hilbert spaces in the unit ball of ℂ n

Abstract In this note, we study defect operators in the case of holomorphic functions of the unit ball of ℂn. These operators are built from weighted Bergman kernel with a holomorphic vector. We obtain a description of sub-Hilbert spaces and we give a sufficient condition so that theses spaces are the same.

English

We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space ℂN, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given by M. Skwarczynski (1980), highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover, we show that convergence of weighted Bergman kernels implies this property, which will give a characterization of the domains, for which the inverse of the Ramadanov’s theorem holds.

The hyperbolic-type point process

In this paper, we introduce a two-parameters determinantal point process in the Poincaré disc and compute the asymptotics of the variance of its number of particles inside a disc centered at the origin and of radius $r$ as $r \rightarrow 1^-$. Our computations rely on simple geometrical arguments whose analogues in the Euclidean setting provide a shorter proof of Shirai's result for the Ginibre-type point process. In the special instance corresponding to the weighted Bergman kernel, we mimic the computations of Peres and Virag in order to describe the distribution of the number of particles inside the disc.

Convergence speed of weighted Bergman kernels towards extremal functions

We construct inner products by the Bernstein-Markov inequality on spaces of holomorphic sections of high powers of a line bundle. The corresponding weighted Bergman kernel functions converge to an extremal function. We obtain a uniform convergence speed.

On the localization of the minimum integral related to the weighted Bergman kernel and its application

In this note, under an additional condition, we present an alternative proof of a stability theorem for the boundary asymptotics of the Bergman kernel due to T. Ohsawa. Our method relies on the localization of the minimum integral related to the weighted Bergman kernel.

Weighted Bergman kernel functions associated to meromorphic functions

We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with modulus squared weight of a meromorphic function in the case that the meromorphic function has a finite number of zeros on the domain and a concrete formula for the unweighted kernel is known. We apply this theory to the study of the Lu Qi-keng problem.

Coercivity of weighted Kohn Laplacians: The case of model monomial weights in ℂ²

The weighted Kohn Laplacian ◻ φ \Box _\varphi is a natural second order elliptic operator associated to a weight φ : C n → R \varphi :\mathbb {C}^n\rightarrow \mathbb {R} and acting on ( 0 , 1 ) (0,1) -forms, which plays a key role in several questions of complex analysis. We consider here the case of model monomial weights in C 2 \mathbb {C}^2 , i.e., \[ φ ( z , w ) := ∑ ( α , β ) ∈ Γ | z α w β | 2 , \varphi (z,w):=\sum _{(\alpha ,\beta )\in \Gamma }|z^\alpha w^\beta |^2, \] where Γ ⊆ N 2 \Gamma \subseteq \mathbb {N}^2 is finite. Our goal is to prove coercivity estimates of the form \[ ( ∗ ) ◻ φ ≥ μ 2 , (*)\hspace {10pc} \Box _\varphi \geq \mu ^2,\hspace {10pc} \] where μ : C n → R \mu :\mathbb {C}^n\rightarrow \mathbb {R} acts by pointwise multiplication on ( 0 , 1 ) (0,1) -forms, and the inequality is in the sense of self-adjoint operators. We proved in 2015 how to derive from ( ∗ ) (*) new pointwise bounds for the weighted Bergman kernel associated to φ \varphi . Here we introduce a technique to establish ( ∗ ) (*) with \[ μ ( z , w ) = c ( 1 + | z | a + | w | b ) ( a , b ≥ 0 ) , \mu (z,w)=c(1+|z|^a+|w|^b) \qquad (a,b\geq 0), \] where a , b ≥ 0 a,b\geq 0 depend on (and are easily computable from) Γ \Gamma . As a corollary we also prove that, for a wide class of model monomial weights, the spectrum of ◻ φ \Box _\varphi is discrete if and only if the weight is not decoupled, i.e., Γ \Gamma contains at least a point ( α , β ) (\alpha ,\beta ) with α ≠ 0 ≠ β \alpha \neq 0\neq \beta . Our methods comprise a new holomorphic uncertainty principle and linear optimization arguments.

The Weighted Bergman Kernel and the Green\u2019s Function

We study the connection between the weighted Bergman kernel and the Green’s function on a domain Wsubset mathbb {C} for which the Green’s function exists.