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.     

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     

Hölder andL p -estimates for the\bar \partial -equation on non-smooth strictlyq-convex domainson non-smooth strictlyq-convex domains

A solution operator for the \(\bar \partial \) -equation on strictlyq-convex domains with nonsmooth boundary is constructed. It is proved that the solution satisfies optimal 1/2-Hölder andL ^p estimates.     

Szegö kernel asymptotics and Morse inequalities on CR manifolds

Let X be an abstract compact orientable CR manifold of dimension ${2n-1, n\,\geqslant\,2}$ , and let L ^k be the k-th tensor power of a CR complex line bundle L over X. We assume that condition Y(q) holds at each point of X. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in L ^k , for large k. After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convex–concave manifolds.     

Szegö theorem,Carathéodory domains,and boundedness of calculating functionals

Convex Chebyshev sets M in a linear space X with norm or nonsymmetric norm ‖ · ‖ which are contained in a subspace H of X are considered. It is proved that if | { |_{H, φ} is the nonsymmetric norm on H determined by the Minkowski functional of , where B is the unit ball of X and ‖φ‖, with respect to 0, then M is a Chebyshev set in for any φ. From this result sufficient and necessary conditions for the convexity of Chebyshev sets and bounded Chebyshev sets contained in a subspace H of X are derived.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 3–15.Original Russian Text Copyright © 2005 by A. R. Alimov.     

Szegö and Polymonogenic Bergman Kernels for Half-Space and Strip Domains,and Single-Periodic Functions in Clifford Analysis

In this paper, we consider half-space domains (semi-infinite in one of the dimensions) and strip domains (finite in one of the dimensions) in real Euclidean spaces of dimension at least 2. The Szegö reproducing kernel for the space of monogenic and square integrable functions on a strip domain is obtained in closed form as a monogenic single-periodic function, viz a monogenic cosecant. The relationship between the Szegö and Bergman kernel for monogenic functions in a strip domain is explicitated in the transversally Fourier transformed setting. This relationship is then generalised to the polymonogenic Bergman case. Finally, the half-space case is considered specifically and the simplifications are pointed out.     

Apollonian isometries of planar domains are Möbius mappings

The Apollonian metric is a generalization of the hyperbolic metric, defined in a much larger class of open sets. Beardon introduced the metric in 1998, and asked whether its isometries are just the Möbius mappings. In this article we show that this is the case in all open subsets of the plane with at least three boundary points.     

A Direct Connection Between the Bergman and Szegő Projections

We use Stokes’s theorem to establish an explicit and concrete connection between the Bergman and Szeg? projections on the disc, the ball, and on strongly pseudoconvex domains.     

The Szegö function on a non-rectifiable arc

Let Γ be a simple Jordan arc in the complex plane. The Szegö function, by definition, is a holomorphic in ? Γ function with a prescribed product of its boundary values on Γ. The problem of finding the Segö function in the case of piecewise smooth Γ was solved earlier. In this paper we study this problem for non-rectifiable arcs. The solution relies on properties of the Cauchy transform of certain distributions with the support on Γ.     

A Remark on a Theorem of Szegö

Let be a polynomial with complex coefficients and define, for , where ||P|| is the euclidean norm of the polynomial P. By a theorem of Szegö where is the Mahler measure of F. Recently, J. Dégot proved an effective version of this result. In this paper we sharpen Dégot's result, under the additional hypotheses that F is a square-free polynomial with integer coefficients and without reciprocal factors.     

On the asymptotics of the on-diagonal Szegö kernel of certain Reinhardt domains

We compute the leading and subleading terms in the asymptotic expansion of the Szegö kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.     

Gleason’s problem in weighted Bergman space on egg domains

It is shown that on the egg domains:

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory.     

Solutions of Tikhonov Functional Equations and Applications to Multiplication Operators on Szegö Spaces

We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear operator equations (when defined from reproducing kernel Hilbert spaces into general Hilbert spaces), by using the Hilbert–Schmidt property and tensor product of Hilbert spaces. As a concrete case, we shall consider generalized fractional functions formed by the quotient of Bergman functions by Szegö functions considered from the multiplication operators on the Szegö spaces.     

On Octonionic Regular Functions and the Szegö Projection on the Octonionic Heisenberg Group

We discuss the octonionic regular functions and the octonionic regular operator on the octonionic Heisenberg group. This is the octonionic version of CR function theory in the theory of several complex variables and regular function theory on the quaternionic Heisenberg group. By identifying the octonionic algebra with \(\mathbb{R }^{8}\) , we can write the octonionic regular operator and the associated Laplacian operator as real \((8\times 8)\) -matrix differential operators. Then we use the group Fourier transform on the octonionic Heisenberg group to analyze the associated Laplacian operator and to construct its kernel. This kernel is exactly the Szegö kernel of the orthonormal projection from the space of \(L^{2}\) functions to the space of \(L^{2}\) regular functions on the octonionic Heisenberg group.     

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szegö kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kähler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szegö kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Engli? (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.