Generalized Hankel operators and the generalized solution operator to $ \bar \partial $ on the Fock space and on the Bergman space of the unit disc

In this paper we consider Hankel operators = (Id – P ₁) from A ²(?, |z |²) to A ^2,1(?, |z |²)^⊥. Here A ²(?, |z |²) denotes the Fock space A ²(?, |z |²) = {f: f is entire and ‖f ‖² = ∫_? |f (z)|² exp (–|z |²) dλ (z) < ∞}. Furthermore A ^2,1(?, |z |²) denotes the closure of the linear span of the monomials { z ⁿ : n, l ∈ ?, l ≤ 1} and the corresponding orthogonal projection is denoted by P ₁. Note that we call these operators generalized Hankel operators because the projection P ₁ is not the usual Bergman projection. In the introduction we give a motivation for replacing the Bergman projection by P ₁. The paper analyzes boundedness and compactness of the mentioned operators. On the Fock space we show that is bounded, but not compact, and for k ≥ 3 that is not bounded. Afterwards we also consider the same situation on the Bergman space of the unit disc. Here a completely different situation appears: we have compactness for all k ≥ 1. Finally we will also consider an analogous situation in the case of several complex variables. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compactness of the Canonical Solution Operator of $ \bar \partial $ on Bounded Pseudoconvex Domains

On bounded pseudoconvex domains Ω the orthogonal projection P_q : L²_(p,q) (Ω) → ker _q is given by P_q = Id – S_{q+1 q} = Id – ^*_q+1N_{q+1 q}, where S_q is the canonical solution operator of the ‐equation and N_q is the ‐Neumann operator. We prove a formula for the solution operator S_q restricted on (0, q)‐forms with holomorphic coefficients. And as an application we get a characterization of compactness of the solution operator restricted on (0, q)‐forms with holomorphic coefficients. On general (0, q)‐forms we show that this condition is necessary for compactness of the solution operator.     

Strongly Exposed Points in the Ball of the Bergman Space

We investigate which boundary points in the closed unit ball of the Bergman space A¹ are strongly exposed. This requires study of the Bergman projection and its kernel, the annihilator of Bergman space. We show that all polynomials in the boundary of the unit ball are strongly exposed.     

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.     

Characterizations for Besov spaces and applications. Part I

The main theorem of this paper gives a characterization for holomorphic Besov space B^p(D) over a large class of bounded domains D in , which states that there is a bounded linear operator so that PV_D=I on B^p(D), where P is the Bergman projection, and is the biholomorphic invariant measure with K(z,z) being Bergman kernel function for D. Moreover, some application for characterizing Schatter von Neumann p-class small Hankel operation is given as a direct consequence of this theorem.     

Global boundary regularity for the $ \bar \partial $‐equation on q‐pseudoconvex domains

We introduce a notion of q ‐pseudoconvex domain of new type for a bounded domain of ?ⁿ and prove that for given a ‐closed (p, r)‐form, r ≥ q, that is smooth up to the boundary, there exists a (p, r – 1)‐form smooth up to the boundary which is a solution of ‐equation on a bounded q ‐pseudoconvex domain. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the growth of the Bergman kernel near an infinite-type point

We study diagonal estimates for the Bergman kernels of certain model domains in \mathbbC²{\mathbb{C}^{2}} near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range—roughly speaking—from being “mildly infinite-type” to very flat at the infinite-type points.     

On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let be a domain with smooth boundary and let α be a C ²- diffeomorphism on satisfying the Carleman condition .We denote by the C*-algebra generated by the Bergman projection of G, all multiplication operators aI and the operator where is the Jacobian of α. A symbol algebra of is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator is isomorphic and isometric to . Submitted: February 11, 2001?Revised: January 27, 2002     

Sharp Norm Estimates for Weighted Bergman Projections in the Mixed Norm Spaces

In this paper, we show that the norm of the Bergman projection on L^p,q-spaces in the upper half-plane is comparable to csc(π/q). Then we extend this result to a more general class of domains, known as the homogeneous Siegel domains of type II.     

Forelli-Rudin type theorem in pluriharmonic Bergman spaces with small index

It is proved that the Bergman type operatorT, is a bounded projection from the pluriharmonic Bergman spaceL ^p (B)∩h(B) onto Bergman spaceL ^p (B) ∩ H(B) for 0p 1 ands (p¹-1)(n+1). As an application it is shown that the Gleason’s problem can be solved in Bergman space L^P(B)∩H(B) for 0p 1. Project supported by the National Natural Science Foundation of China (Grant No. 19871081) and the Doctoral Program Foundation of the State Education Commission of China.     

L p,q -Boundedness of Bergman Projections in Homogeneous Siegel Domains of Type II

In this paper, we generalize to homogeneous Siegel domains of second kind the L ^p -continuity properties of the Bergman projection. Precisely, we give an improvement of the index p using Fourier analysis as in the case of convex homogeneous tube type domains (Nana and Trojan in Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) X:477–511, 2011).     

Operators on harmonic Bergman spaces

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL ^p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

Schatten class hankel operators on the Bergman space

In this paper we characterize Hankel operatorsH _f andH _f on the Bergman spaces of bounded symmetric domains which are in the Schatten p-class for 2p< and f inL ² using a Jordan algebra characterization of bounded symmetric domains and properties of the Bergman metric.     

Continuity of Bergman and Szegö projections on weighted-sup function spaces on pseudoconvex domains

We study regularity of Bergman and Szeg? projections on Sobolev type weighted-sup spaces. The paper covers the case of strongly pseudoconvex domains with C⁴ boundary and, partially, domains of finite type in the sense of D’Angelo. Received: 6 October 2005     

Hankel-Toeplitz type operators onL a 1 (Ω)

This paper studies Hankel and Toeplitz operators on the Bergman spaceL _a ¹ () of bounded symmetric domains. These operators are defined in terms of a certain bounded projection onL ¹(,dV). The main results of the paper include several characterizations for the boundedness and (weak-star) compactness of these Hankel-Toeplitz type operators. When the symbol is conjugate holomorphic, our results here are similar to those obtained by Békollé, Berger, Coburn, and Zhu [2] in theL ²-Bergman space context.Research partially supported by the National Science Foundation     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA ²(C ^N),N≧1.     

A smoothing property of the Bergman projection

Let B be the Bergman projection associated to a domain Ω on which the -Neumann operator is compact. We show that arbitrary L ² derivatives of Bf are controlled by derivatives of f taken in a single, distinguished direction. As a consequence, functions not contained in that are mapped by B to are explicitly described.     

Weighted generalization of the Ramadanov’S theorem and further considerations

We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space ?^N, 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. Skwarczyński (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.     

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.     

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 \(p=2\).