Szego Projections Research Articles

Norm Estimates for the Bergman and Cauchy–Szegö Projections Over the Siegel Upper Half-Space

We obtain several estimates for the $$L^p$$ operator norms of the Bergman and Cauchy–Szego projections over the the Siegel upper half-space. As a by-product, we also determine the precise value of the $$L^p$$ operator norm of a family of integral operators over the Siegel upper half-space.

Bergman versus Szego via conformal mapping

We compare the Bergman and Szego projections associated to bounded, simply connected planar domains with Holder continuous boundary. It is shown that their difference gains a derivative in an appropriate range of Sobolev or Lipschitz norms.

An Analogue of the Kerzman-Stein Formula for the Bergman and Szegö Projections

We show that the difference between the Bergman and Szego projections on a smooth, bounded planar domain gains a derivative in the Lp-Sobolev and Lipschitz spaces.

Comparing the Bergman and Szegö projections on domains with subelliptic boundary Laplacian

We prove that the difference between the Bergman and Szego projections on a bounded, pseudoconvex domain (with C ∞ boundary) is smoothing whenever the boundary Laplacian is subelliptic. An equivalent statement is that the Bergman projection can be represented as a composition of the Szego and harmonic Bergman projections (along with the restriction and Poisson extension operators) modulo an error that is smoothing. We give several applications to the study of optimal mapping properties for these projections and their difference.

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

We study regularity of Bergman and Szego 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.

Estimates for the Bergman and Szego projections for pseudoconvex domains of finite type with locally diagonalizable Levi Form

In this paper, we give precise isotropic and non-isotropic estimates for the Bergman and Szego projections of a bounded pseudoconvex domain whose boundary points are all of finite type and with locally diagonalizable Levi form. Additional local results on estimates of invariant metrics are also given.

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj-Calderon-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szego projections and the corresponding Lp-decompositions. Our approach relies on an extension of the classical Calderon-Zygmund theory of singular integral operators which allow one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.

On the structure of Bergman and poly-Bergman spaces

Let II be the upper half-plane inℂ, consider the Bergman space $$\mathcal{A}^2 (II)$$ , the subspace of all analytic functions fromL 2(II). The complete decomposition ofL 2(II) onto Bergman and Bergman type spaces of poly-analytic and poly-anti-analytic functions is obtained. The orthogonal Bergman type projections onto each of these subspaces are described. Connections with the Hardy spaces and the Szego projections are established.

A characterization of ellipsoids and balls in ${\bf C}\sp n$

The following fact is proved: If D is a smooth bounded domain in C for which Bergman and Szego projections are equal on smooth harmonic functions then D is a ball.