On the Bergman metric of pseudoconvex domains in a complex projective space

We prove a localization principle of the Bergman kernel form and metric for pseudoconvex domains in the complex projective space. An estimate of the Bergman distance is also given.

     

On the existence of polyanalytic functions

We give a sharp estimate for the codimension of the poly‐Bergman space in the poly‐Bergman space over the punctured domain. It is established the behaviour at the infinity point of polyanalytic Bergman functions on the complement of closed disks. In the main result of the paper, we prove that for and the j‐polyanalytic Bergman space over the domain U is trivial precisely when the complement of U has at most one point and at most two points or three points lying in a circle, respectively. We point out the differences between the domains over which the Bergman space and the non‐analytic poly‐Bergman space are trivial.     

Compact composition operators on the Bloch space in polydiscs

Let Uⁿ be the unit polydisc of ℂⁿ and φ=(φ₁, ⋯, φ _n ) a holomorphic self-map of Uⁿ. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Uⁿ) if and only if for every ε > 0, there exists a δ > 0, such that

Boundary behavior of the Bergman kernel function on pseudoconvex domains with comparable Levi form

Let Ω be a smoothly bounded pseudoconvex domain in and let be a point of finite type. We also assume that the Levi form of bΩ is comparable in a neighborhood of z₀. Then we get a quantity which bounds from above and below the Bergman kernel function in a small constant and large constant sense.     

Bergman Kernels for Almost Spherical Domains with Boundaries of the Class H σ

This paper is devoted to the behavior of the Bergman kernels for almost spherical strictly pseudoconvex domains in . We find several first terms of the asymptotics for the Bergman kernel of a strictly pseudoconvex domain with H -smooth boundary, 4 < < 6. New terms in comparison with the case of C ⁶-smooth boundary appear but the growth rate of the remainder remains the same. Bibliography: 16 titles.     

The commutant of analytic Toeplitz operators on Bergman space

In this paper, using the matrix skills and operator theory techniques we characterize the commutant of analytic Toeplitz operators on Bergman space. For f（z） = z^ng（z） （n ≥1）, g（z） = b0 ＋ b1z^p1 ＋b2z^p2 ＋.. , bk ≠ 0 （k = 0, 1, 2,...）, our main result is =A′（Mf） = A′（Mzn）∩A′（Mg） = A′（Mz^s）, where s = g.c.d.（n,p1,p2,...）. In the last section, we study the relation between strongly irreducible curve and the winding number W（f,f（α））, α ∈ D.     

Unique continuation theorems for the\bar \partial - Operator and applications

We formulate a unique continuation principle for the inhomogeneous Cauchy-Riemann equations near a boundary pointz ₀ of a smooth domain in complex euclidean space. The principle implies that the Bergman projection of a function supported away fromz ₀ cannot vanish to infinite order atz ₀ unless it vanishes identically. We prove that the principle holds in planar domains and in domains where the problem is known to be analytic hypoelliptic. We also demonstrate the relevance of such questions to mapping problems in several complex variables. The last section of the paper deals with unique continuation properties of the Szegő projection and kernel in planar domains. Research supported by NSF Grant DMS-8922810.     

Lp ‐estimates for the Bergman projection on strictly pseudoconvex nonsmooth domains

We consider the Bergman projection on Henkin–Leiterer domains, bounded strictly pseudoconvex domains which have defining functions whose gradient is allowed to vanish. Our result describes the mapping properties of the Bergman projection between weighted L^p spaces, with the weights being powers of the gradient of the defining function. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact regularity of the Bergman and Szegö projections on domains with partially transverse symmetries

IfD is a smooth bounded pseudoconvex domain in C ⁿ that has symmetries transverse on the complement of a compact subset of the boundary consisting of points of finite type, then the Bergman projection forD maps the Sobolev spaceW ^r (D) continuously into itself and the Szegö projection maps the Sobolev spaceWsur(bD) continuously into itself. IfD has symmetries, coming from a group of rotations, that are transverse on the complement of aB-regular subset of the boundary, then the Bergman projection, the Szegö projection, and the -Neumann operator on (0, 1)-forms all exactly preserve differentiability measured in Sobolev norms. The results hold, in particular, for all smooth bounded strictly complete pseudoconvex Hartogs domains in C², as well as for Sibony's counterexample domain that fails to have sup-norm estimates for solutions of the -equation.     

Gleason’s problem in weighted Bergman space on egg domains

It is shown that on the egg domains:

Logarithmic growth of the Bergman Kernel for weakly pseudoconvex domains in ℂ3 of finite type

If G???ⁿis pseudoconvex with smooth boundary and z_o∈bG is a point_of finite type, one can ask the question: Does the Bergman kernel K_G(z,¯z) grow like a rational power of dist(z,bG) when z approaches z_o nontangentially? This question is suggested by observations for the domains U_pg defined below. But the answer to this question is in general negative as is shown by a counterexample in ?³.     

Sharpness of certain Campbell and Pommerenke estimates

The paper is concerned with the sharpness of some well-known estimates in universal linear-invariant families of regular functions. It is shown that the estimate of | arg ƒ′(z)|,z ∈ Δ = {z: |z| < 1} obtained by Pommerenke in 1964 is sharp; the extremal function is found. A lower estimate for the Schwarzian derivative in is obtained. For , a sharp estimate of order of the function ƒr(z)=ƒ(rz)/r withr ∈ (0, 1) is found; this estimate is applied to solve other problems. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 665–672, May, 1998.     

The Bergman kernel: Explicit formulas,deflation, Lu Qi-Keng problem and Jacobi polynomials

We investigate the Bergman kernel function for the intersection of two complex ellipsoids {(z,w ₁,w ₂) ∈ C ⁿ⁺²: |z ₁|²+...+|z _n |²+|w ₁| ^q < 1, |z ₁|²+...+|z _n |²+|w ₂| ^r < 1}. We also compute the kernel function for {(z ₁,w ₁,w ₂) ∈ C³: |z ₁|^2/n + |w ₁| ^q < 1, |z ₁|^2/n + |w ₂| ^r < 1} and show deflation type identity between these two domains. Moreover in the case that q = r = 2 we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.     

Hankel operators on the Bergman spaces of strongly pseudoconvex domains

We characterize functions fL ²(D) such that the Hankel operators H_f are, respectively, bounded and compact on the Bergman spaces of bounded strongly pseudoconvex domains.Research partially supported by a grant of the National Science Foundation.     

Comparison of the Bergman and Szegö kernels

The quotient of the Szegö and Bergman kernels for a smooth bounded pseudoconvex domains in Cn is bounded from above by a constant multiple of for any p>n, where δ is the distance to the boundary. For a class of domains that includes those of D?Angelo finite type and those with plurisubharmonic defining functions, the quotient is also bounded from below by a constant multiple of for any p<−1. Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of δ.     

Bergman Subspaces and Subkernels: Degenerate $$L^p$$ Mapping and Zeroes

Regularity and irregularity of the Bergman projection on \(L^p\) spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable \(\gamma \). A surprising consequence of the analysis is that, whenever \(\gamma \) is irrational, the Bergman projection is bounded only for \(p=2\).     

Local analyticity for the [`(?)]\overline \partial -Neumann problem in completely decoupled pseudoconvex domains-Neumann problem in completely decoupled pseudoconvex domains

In this paper we prove local analyticity of solutions to the -Neumann problem up to the boundary of rigid, completely decoupled pseudoconvex domains with real-analytic boundary. These are domains that are locally of the form Imw > Σ |h _k (z _k )|² with eachh _k holomorphic and vanishing only at 0. As in those earlier papers, we use purelyL ² methods and must construct a special holomorphic vector fieldM and then use carefully balanced polynomials inM to localize high powers ofT = ∂/∂t effectively, wheret = Rew.