Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n
Robert Berman
备注：
v1: 11 pages v2: 19 pages. Substantial revision: regularity assumption on the weight weakened to C^1,1, setting of polynomials with a given Newton polytope considered, examples and a figure added
Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is differentiable and all of its first partial derivatives are locally Lipshitz continuous. The convergence is studied in the large k limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the Monge-Ampere measure of the weight function itself on a certain set. A setting of polynomials associated to a given Newton polytope, scaled by k, is also considered. These results apply directly to the study of the distribution of zeroes of random polynomials and of the eigenvalues of random normal matrices.
We introduce the notion of "virtual Bergman kernel" and apply it to the computation of the Bergman kernel of "domains inflated by Hermitian balls", in particular when the base domain is a bounded symmetric domain.
Herein, the theory of Bergman kernel is developed to the weighted case. A general form of weighted Bergman reproducing kernel is obtained, by which we can calculate concrete Bergman kernel functions for specific weights and domains.
We investigate Lp regularity of weighted Bergman projections and zeros of weighted Bergman kernels for the weights that are radially symmetric and comparable to 1 on the unit disc.
We investigate $L^p$ regularity of weighted Bergman projections on the unit disc and $L^p$ regularity of ordinary Bergman projections in higher dimensions.
We discuss topics related to zeroes of the Bergman kernels, and present a method for generating Bergman kernels with arbitrarily, but finitely, many zeroes. It is also shown that a Bergman kernel induced by a radial weight on the unit disk cannot have infinitely many zeroes. Similar questions for the Segal-Bargmann spaces of the complex plane are briefly discussed.
We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
The goal of this note is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in $L^2_{0,q}(Ω)\cap\ker\bar\partial_q$, fail for $(0,q)$-forms, $q \geq 1$. We do, however, provide a careful construction of the Bergman kernel and explicitly compute the Bergman kernel on $(0,n-1)$-forms. For the ball in $\mathbb{C}^2$, we also show that the size of the Bergman kernel on $(0,1)$-forms is not governed by the control metric, in stark contrast to Bergman kernel on functions.
In this paper we give sharp norm estimates for the Bergman operator acting from weighted mixed-norm spaces to weighted Hardy spaces in the ball, endowed with natural norms.
We solved the problem of the best rational approximation of the Bergman kernels on the unit circle of the complex plane in the quadratic and uniform metrics.
In this paper we consider the zeros of the Bergman kernel of the Fock-Bargmann-Hartogs domain $D_{n,m}$. We describe how the existence of zeros of the Bergman kernel depends on the integers $m$ and $n$ with the help of the interlacing property.
In this article, we consider Bergman kernels with respect to modules at boundary points, and obtain a log-subharmonicity property of the Bergman kernels, which deduces a concavity property related to the Bergman kernels. As applications, we reprove the sharp effectiveness result related to a conjecture posed by Jonsson-Mustcată and the effectiveness result of strong openness property of the modules at boundary points.
Let $AL^{2}_ϕ(\mathbb{D})$ denote the closed subspace of $L^{2}(\mathbb{D},e^{-2ϕ}dλ)$ consisting of holomorphic functions in the unit disc ${\mathbb D}$. For certain class of subharmonic funcions $ϕ: {\mathbb D}\rightarrow{\mathbb D}$, we prove upper pointwise estimate for the Bergman kernel for $AL^{2}_ϕ(\mathbb{D})$
We give a characterization of $L_h^2$-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.
We compute explicitly the Bergman kernels of all two dimensional monomial polyhedra, a class of domains including the Hartogs triangle and some of its generalizations. The kernel is computed from the representation of such domains as quotient of the product of a disc and punctured disc.
We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in C^3 defined by the inequality |z_1|+|z_2|+|z_3|<1, have zeroes.
In this work we provide an asymptotic expansion for the Szegő kernel associated to a suitably defined Hardy space on the the non-smooth worm domain $D'_β$. After describing the singularities of the kernel, we compare it with an asymptotic expansion of the Bergman kernel. In particular, we show that the Bergman kernel has the same singularities of the first derivative of the Szegő kernel with respect to any of the variables. On the side, we prove the boundedness of the Bergman projection operator on Sobolev spaces of integer order.
We use Stokes's theorem to establish an explicit and concrete connection between the Bergman and Szegő projections on the disc, the ball, and on strongly pseudoconvex domains.
We obtain the octonionic Bergman kernel for half space in the octonionic analysis setting by two different methods. As a consequence, we unify the kernel forms in both complex analysis and hyper-complex analysis.
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.
We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of conformal mappings in Hardy or weighted Bergman spaces by studying Euclidean areas. Applying these results, we prove several consequences for such mappings that extend known results for Hardy spaces to weighted Bergman spaces. Moreover, we introduce a number which is the analogue of the Hardy number for weighted Bergman spaces. We derive various expressions f…
We extend our work on nonseparated interpolating sequences, originally developed for Bergman spaces with weights of the form $(1 - |z|^2)^α$, to more general weights.
In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.
We obtain a quantitative estimate of Bergman distance when $Ω\subset \mathbb{C}^n$ is a bounded domain with log-hyperconvexity index $α_l(Ω)>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2(\log A)^q$-integrability of the Bergman kernel $K_Ω(\cdot, w)$ when $α_l(Ω)>0$.
Let $\{M_j\}$ be a sequence of complete Riemannian surfaces which converges in the sense of Cheeger-Gromov to a complete Riemannian surface $M$. We study the convergence of the Bergman kernel $K_{M_j}$ of $M_j$ by using isoperimetric inequalities.
Herein, the Laplace transform representations for functions of weighted holomorphic Bergman spaces on the tube domains are developed. Then a weighted version of the edge-of-the-wedge theorem is derived as a byproduct of the main results.
In this paper we obtain the full asymptotic expansion of the Bergman-Hodge kernel associated to a high power of a holomorphic line bundle with non-degenerate curvature. We also explore some relations with asymptotic holomorphic sections on symplectic manifolds
We give an elementary proof of the existence of an asymptotic expansion in powers of $k$ of the Bergman kernel associated to $L^k$, where $L$ is a positive line bundle. We also give an algorithm for computing the coefficients in the expansion.
We study weighted Bergman projections on the Hartogs triangle in $\mathbb{C}^2$. We show that projections corresponding to exponentially vanishing weights have degenerate $L^p$ mapping properties.
We prove a general version of \cite[Theorem 4.1]{Boas84} to obtain Sobolev estimates for weighted Bergman projections on convex Reinhardt domains by using the Prékopa-Leindler inequality.
We consider a certain Hartogs domain which is related to the Fock-Bargmann space. We give an explicit formula for the Bergman kernel of the domain in terms of the polylogarithm functions. Moreover we solve the Lu Qi-Keng problem of the domain in some cases.
This article is concerned with asymptotics of equivariant Bergman kernels and partial Bergman kernels for polarized projective Kahler manifolds invariant under a Hamiltonian holomorphic $S^1$ action. Asymptotics of partial Bergman kernel are obtained in the allowed region $\mathcal{A}$ resp. forbidden region $\mathcal{F}$, generalizing results of Shiffman-Zelditch, Shiffman-Tate-Zelditch and Pokorny-Singer for toric Kahler manifolds. The main result gives scaling asymptotics of equivariant Bergman kernels and partial Bergman kernels in the transition region around the interface $\partial \math…
In this paper, we investigate some properties of planar harmonic mappings. First, we generalize the main results in \cite{CPW3} and \cite{HT}, and then discuss the relationship between area integral means and harmonic Hardy spaces or harmonic weighted Bergman spaces. At the end, coefficient estimates of mappings in weighted Bergman spaces are obtained.
In this paper, we study the Bergman kernel $B_φ(x,y)$ of generalized Bargmann-Fock spaces in the setting of Clifford algebra. The versions of $L^2$-estimate method and weighted subharmonic inequality for Clifford algebra are established. Consequently we show the existence of $B_φ(x,y)$ and then give some estimates on and off the diagonal. As a by-product, we also obtain an upper estimate of the weighted harmonic Bergman kernel.
In this paper we investigate the Bergman kernel function for intersection of two complex ellipsoids $\{(z,w_1,w_2) \in \mathbb{C}^{n+2} : |z_1|^2 + \cdots + |z_n|^2 + |w_1|^q < 1, \quad |z_1|^2 + \cdots + |z_n|^2 + |w_2|^r < 1\}.$
By introducing a suitable new definition for the inner product on the octonionic Bergman space, we determine the explicit form of the octonionic Bergman kernel, in the framework of octonionic analysis which is non-commutative and non-associative.
We introduce a family of weighted BMO and VMO spaces for the unit ball and use them to characterize bounded and compact Hankel operators between different Bergman spaces. In particular, we resolve two problems left open by S. Janson in 1988 and R. Wallsten in 1990.
We show that weighted Bergman projections, corresponding to weights of the form $M(z)(1-|z|^2)^α$ where $α>-1$ and $M(z)$ is a radially symmetric, strictly positive and at least $C^2$ function on the unit disc, are $L^p$ regular.
We study the complex symmetric structure of weighted composition--differentiation operators of order $n $ on the weighted Bergman spaces $A_α^2$ with respect to some conjugations. Then we provide some examples of these operators.
Using the techniques developed in \cite{SunSun}, we give estimations of the Bergman kernel of the punctured disk with the standard complete Poincaré metric. As an application, we improve the result of \cite{AMM} on the Bergman kernels of punctured Riemann surfaces near singularities.
We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with weight the modulus squared 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.
A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szegö and the Bergman kernel on the symmetrized polydisc.
The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function vanishes on a portion of the plane, the off-spectral region. This type of behaviour is observed when the metric is negatively curved somewhere, or when we study partial Bergman kernels in the context of positively curved metrics. In this work, we cover these two situations in a unified way, for exponentially varying planar measures on the complex plane. We obtain u…
Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct image sheaf. In general, following Schumacher's idea, we use the Lie derivative to compute the variation. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
