Harmonic Bergman functions as radial derivatives of Bergman functions

In the setting of the half-space of the euclidean -space, we show that every harmonic Bergman function is the radial derivative of a Bergman function with an appropriate norm bound.

     

Behavior of the Bergman kernel and metric near convex boundary points

The boundary behavior of the Bergman metric near a convex boundary point of a pseudoconvex domain is studied. It turns out that the Bergman metric at points in the direction of a fixed vector tends to infinity, when is approaching , if and only if the boundary of does not contain any analytic disc through in the direction of .

     

A remark on the Bergman stability

Let , be a sequence of bounded pseudoconvex domains that converges, in the sense of Boas, to a bounded domain . We show that if can be described locally as the graph of a continuous function in suitable coordinates for , then the Bergman kernel of converges to the Bergman kernel of uniformly on compact subsets of .     

The harmonic Bergman kernels for punctured domains

We obtain the explicit formulae for the harmonic Bergman kernels of Bn／{0} and Rn／Bn and study the connection between harmonic Bergman kernel and weighted harmonic Bergman kernel.We also get the explicit formula for the weighted harmonic Bergman kernel of Bn／{0} with the weight 1/｜x｜4.     

Bergman kernel and metric on non-smooth pseudoconvex domains

A stability theorem of the Bergman kernel and completeness of the Bergman metric have been proved on a type of non-smooth pseudoconvex domains defined in the following way:D = {z∈U|r(z)} <whereU is a neighbourhood of andr is a continuous plurisubharmonic function onU. A continuity principle of the Bergman Kernel for pseudoconvex domains with Lipschitz boundary is also given, which answers a problem of Boas.     

The Bergman metric on a Stein manifold with a bounded plurisubharmonic function

In this article, we use the pluricomplex Green function to give a sufficient condition for the existence and the completeness of the Bergman metric. As a consequence, we proved that a simply connected complete Kähler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature , which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such manifolds.     

On Beurling-type theorems in weighted and Bergman spaces

We prove that analytic operators satisfying certain series of operator inequalities possess the wandering subspace property. As a corollary, we obtain Beurling-type theorems for invariant subspaces in certain weighted and Bergman spaces.

     

Complemented invariant subspaces and interpolation sequences

It is shown that the invariant subspace of the Bergman space of the unit disc, generated by a finite union of Hardy interpolation sequences, is complemented in .

     

The method of variation of the domain for poly‐Bergman spaces

If , is an increasing sequence (well ordered by inclusion) of domains then the sequence of poly‐Bergman projections on the domains strongly converges to the poly‐Bergman projection on the limit domain. As a corollary some properties of the poly‐Bergman spaces on the half‐planes are deduced from the corresponding ones in the unit disk. We obtain explicit representation of the poly‐Bergman projections in terms of the two‐dimensional singular integral operators , likewise explicit formulas for the poly‐Bergman kernels. We prove that the poly‐Bergman projections on the sectors with a non‐smooth boundary do not admit the usual representations by the two‐dimensional singular integral operators. The variation of the domain and the latter peculiarity of the poly‐Bergman projections allow us to furnish a larger class of domains not admitting Dzhuraev's formulas.     

Equilibrium point of Green's function for the annulus and Eisenstein series

We study the motion of the equilibrium point of Green's function and give an explicit parametrization of the unique zero of the Bergman kernel of the annulus. This problem is reduced to solving the equation , where is the usual Eisenstein series.

     

Harmonic Bergman Functions on Half-Spaces

We study harmonic Bergman functions on the upper half-space of . Among our main results are: The Bergman projection is bounded for the range ; certain nonorthogonal projections are bounded for the range ; the dual space of the Bergman -space is the harmonic Bloch space modulo constants; harmonic conjugation is bounded on the Bergman spaces for the range ; the Bergman norm is equivalent to a ``normal derivative norm' as well as to a ``tangential derivative norm'.

     

Lu Qi-Keng's problem on some complex ellipsoids

In this note, we give an improvement on the Bergman kernel for the domain . As an application, we describe how the zeroes of the kernel depend on the defining parameters p,m,n. We also consider the domain .     

Fixed point formula for holomorphic functions

We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of , we make use of the Bergman kernel of this domain. The Lefschetz number is proved to be the sum of the usual contributions of fixed points of the map in and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points.

     

Explicit Formulae for the Wave Kernels for the Laplacians Δαβ in the Bergman Ball Bn,n≥1

For the perturbed Bergman Laplacians given by in the unit ball B ⁿ of C ⁿ we establish explicit formulae for the corresponding wave equations in B _n. The formulae obtained generalise, for arbitrary , the formulae given in [2] and [5] for the wave equation associated to the shifted Bergman Laplacian =₀₀ in B _n. Moreover, using an analytic continuation argument, we are able to give explicit formulae for the solutions of the wave equation associated to a two parameter family of Laplacians _, on C ⁿ which are natural deformations of the Fubini-Study Laplacian on the Projective space P ⁿ(C) , n 1, viewed as the dual space of the Bergman ball B _n.     

The canonical solution operator to restricted to Bergman spaces

We first show that the canonical solution operator to restricted to -forms with holomorphic coefficients can be expressed by an integral operator using the Bergman kernel. This result is used to prove that in the case of the unit disc in the canonical solution operator to restricted to -forms with holomorphic coefficients is a Hilbert-Schmidt operator. In the sequel we give a direct proof of the last statement using orthonormal bases and show that in the case of the polydisc and the unit ball in 1,$"> the corresponding operator fails to be a Hilbert-Schmidt operator. We also indicate a connection with the theory of Hankel operators.

     

Hypercyclic operators that commute with the Bergman backward shift

The backward shift on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: ``Which operators that commute with inherit its hypercyclicity?' We show that the problem reduces to the study of operators of the form where is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question of hypercyclicity for such an operator depends on how freely is allowed to approach the unit circle as .

     

Finite rank Toeplitz operators on the Bergman space

Given a complex Borel measure with compact support in the complex plane the sesquilinear form defined on analytic polynomials and by , determines an operator from the space of such polynomials to the space of linear functionals on . This operator is called the Toeplitz operator with symbol . We show that has finite rank if and only if is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.

     

Harmonic functions of maximal growth: invertibility and cyclicity in Bergman spaces

In the theory of commutative Banach algebras with unit, an element generates a dense ideal if and only if it is invertible, in which case its Gelfand transform has no zeros, and the ideal it generates is the whole algebra. With varying degrees of success, efforts have been made to extend the validity of this result beyond the context of Banach algebras. For instance, for the Hardy space on the unit disk, it is known that all invertible elements are cyclic (an element is cyclic if its polynomial multiples are dense), but cyclic elements need not be invertible. In this paper, we supply examples of functions in the Bergman and uniform Bergman spaces on the unit disk which are invertible, but not cyclic. This answers in the negative questions raised by Shapiro, Nikolskii, Shields, Korenblum, Brown, and Frankfurt.

     

The backward shift on Dirichlet-type spaces

We study the backward shift operator on Hilbert spaces (for ) which are norm equivalent to the Dirichlet-type spaces . Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an elementary formula expressing the inner product on in terms of a weighted superposition of backward shifts.