Singular Berezin Transforms

We give examples of pseudoconvex Reinhardt domains where the Berezin transform has integral kernel with singularities and, hence, fails to be a smoothing map. On the other hand, we show that this can never happen for a plane domain – in fact, then the Bergman kernel is always either identically zero or strictly positive everywhere on the diagonal – and also prove that, in contrast to the example by Wiegerinck from 1984, on any pseudoconvex Reinhardt domain the Bergman space can be finite-dimensional only if it reduces to the constant zero. Received: February 02, 2007. Accepted: May 28, 2007.     

Convergence of total variation of curvature measures

It is known that the curvature measures of parallel ɛ-neighbourhoods of a set with positive reach or a polyconvex set converge vaguely if ɛ tends to zero to the curvature measures of the set itself. We show that in the case of a set with positive reach, the total variations of the curvature measures converge as well, whereas in the case of a polyconvex set this is no more true in general. Supported by MSM 113200007 and GAČR 201/06/0302. Author’s address: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic     

On the Notion of the Bochner–Martinelli Integral for Domains with Rectifiable Boundary

In this paper we discuss the notion of the Bochner–Martinelli kernel for domains with rectifiable boundary in , by expressing the kernel in terms of the exterior normal due to Federer (see [17,18]). We shall use the above mentioned kernel in order to prove both Sokhotski–Plemelj and Plemelj–Privalov theorems for the corresponding Bochner–Martinelli integral, as well as a criterion of the holomorphic extendibility in terms of the representation with Bochner–Martinelli kernel of a continuous function of two complex variables. Explicit formula for the square of the Bochner–Martinelli integral is rediscovered for more general surfaces of integration extending the formula established first by Vasilevski and Shapiro in 1989. The proofs of all these facts are based on an intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis. Submitted: September 6, 2006. Accepted: November 1, 2006.     

Classification of locally homogeneous affine connections with arbitrary torsion on 2-dimensional manifolds

The aim of this paper is to classify (locally) all locally homogeneous affine connections with arbitrary torsion on two-dimensional manifolds. Herewith, we generalize the result given by B. Opozda for torsion-less case in [(2004) Classification of locally homogeneous connections on 2-dimensional manifolds. Diff Geom Appl 21: 173–198]. Authors’ addresses: Teresa Arias-Marco, Department of Geometry and Topology, University of Valencia, Vicente Andrés Estellés 1, 46100 Burjassot, Valencia, Spain; Oldřich Kowalski, Faculty of Mathematics and Physics of the Charles University, Sokolovská 83, 18600 Praha 8, Czech Republic     

Asymptotic Tian�CYau�CZelditch expansions on singular Riemann surfaces

In this paper, we give both lower and upper bound estimates of the Bergman kernel for a degeneration of Riemann surfaces with constant curvature −1. As a result, we give a geometric proof of the Riemann–Rock theorem for a singular Riemann surface.     

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.     

Maximal multihomogeneity of algebraic hypersurface singularities

From the degree zero part of the logarithmic vector fields along analgebraic hypersurface singularity we identify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one–to–one correspondence to maximal tori in the linear jet of the embedded automorphism group. These results are motivated by Kyoji Saito’s characterization of quasihomogeneity for isolated hypersurface singularities [Saito in Invent. Math. 14, 123–142 (1971)] and extend previous work with Granger and Schulze [Compos. Math. 142(3), 765–778 (2006), Theorem 5.4] and of Hauser and Müller [Nagoya Math. J. 113, 181–186 (1989), Theorem 4].     

Non-analytic Bergman and Szegö kernels for weakly pseudoconvex tube domains in $\mathbb C^2$

For any weakly pseudoconvex tube domain in with real analytic boundary, there exist points on the boundary off the diagonal where the Bergman kernel and the Szeg? kernel fail to be real analytic. Received April 6, 1999; in final form August 28, 1999 / Published online December 8, 2000     

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     

On the role of N.N. Bogolyubov in the development of the theory of nonlinear oscillations

We present a review of Bogolyubov’s works in the theory of nonlinear oscillations. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1014–1035, August, 1999.     

Finite-Term Relations for Planar Orthogonal Polynomials

We prove by elementary means that, if the Bergman orthogonal polynomials of a bounded simply-connected planar domain, with sufficiently regular boundary, satisfy a finite-term relation, then the domain is algebraic and characterized by the fact that Dirichlet’s problem with boundary polynomial data has a polynomial solution. This, and an additional compactness assumption, is known to imply that the domain is an ellipse. In particular, we show that if the Bergman orthogonal polynomials satisfy a three-term relation then the domain is an ellipse. This completes an inquiry started forty years ago by Peter Duren. To Peter Duren on the occasion of his seventieth birthday The first author was partially supported by the National Science Foundation Grant DMS- 0350911. Received: October 15, 2006. Revised: January 22, 2007.     

Characterisation of homogeneous polynomials which are constant on complex-tangential curves in the boundary of the unit ball of

We prove that a complex-tangential curve γ in the boundary of the unit ball of having the property that there exists a homogeneous polynomial P such that P=1 on γ has constant curvature. This implies that a homogeneous polynomial P having the property that there exists a closed complex-tangential curve γ (respectively a totally real 2-dimensional submanifold) in the boundary of the unit ball of such that P=1 on γ (respectively |P|=1 on γ) reduces to a monomial by a unitary chage of variables. These results represent a positive answer to conjectures of H. O. Kim.     

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     

Local embeddability of CR manifolds into spheres

In this paper we solve local CR embeddability problem of smooth CR manifolds into spheres under a certain nondegeneracy condition on the Chern–Moser’s curvature tensor. We state necessary and sufficient conditions for the existence of CR embeddings as finite number of equations and rank conditions on the Chern–Moser’s curvature tensors and their derivatives. We also discuss the rigidity of those embeddings. J.-W. Oh was partially supported by BK21-Yonsei University.     

Green function estimate for censored stable processes

 Sharp two-sided estimates for Green functions of censored α-stable process Y in a bounded C ^1,1 open set D are obtained, where α  (1, 2). It is shown that the Martin boundary and minimal Martin boundary of Y can all be identified with the Euclidean boundary of D. Sharp two-sided estimates for the Martin kernel of Y are also derived. Received: 27 January 2002 / Revised version: 10 June 2002 / Published online: 24 October 2002 This research is supported in part by NSF Grant DMS-0071486. Mathematics Subject Classification (2002): Primary: 60J45, 31C35; Secondary: 60G52, 31C15 Keywords or phrases: Censored stable process – Green function – Capacity – Martin boundary – Martin kernel – Harmonic function     

Superconvergence in the generalized finite element method

In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945–981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727–758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289–314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct. I. Babuška’s research was partially supported by NSF Grant # DMS-0341982 and ONR Grant # N00014-99-1-0724. U. Banerjee’s research was partially supported by NSF Grant # DMS-0341899. J. E. Osborn’s research was supported by NSF Grant # DMS-0341982.     

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.     

Geodesics for the capacity metric in doubly connected domains

We give the qualitative behavior of geodesics of the capacity metric defined on the annulus and study the variation of its Gaussian curvature. We make explicit the relation between the capacities, the Bergman kernel and the reduced Bergman kernel on doubly connected domains and give some applications.     

Asymptotic of the heat kernel in general Benedicks domains

 Using a new inequality relating the heat kernel and the probability of survival, we prove asymptotic ratio limit theorems for the heat kernel (and survival probability) in general Benedicks domains. In particular, the dimension of the cone of positive harmonic measures with Dirichlet boundary conditions can be derived from the rate of convergence to zero of the heat kernel (or the survival probability). Received: 31 March 2002 / Revised version: 12 August 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 60J65, 31B05 Key words or phrases: Positive harmonic functions – Ratio limit theorems – Survival probability