The Szego Kernel for Non-Pseudoconvex Domains in C²

Author

Michael Gilliam, The University of Montana

Year of Award

2011

Document Type

Dissertation

Degree Type

Doctor of Philosophy (PhD)

Degree Name

Mathematics

Department or School/College

Department of Mathematical Sciences

Committee Chair

Jennifer Halfpap

Commitee Members

Eric Chesebro, Michael Schneider, Karel Stroethoff, Thomas Tonev

Keywords

Bergman Kernel, Non-Pseudoconvex Domain, Pseudoconvex Domain, Szego Kernel, Szego Projection

Abstract

There are many operators associated with a domain Ω ⊂ â„‚ⁿ with smooth boundary ∂Ω. There are two closely related projections that are of particular interest. The Bergman projection B is the orthogonal projection of L²(Ω) onto the closed subspace L²(Ω)∩O(Ω), where O(Ω)is the space of all holomorphic functions on Ω. The Szeg� projection S is the orthogonal projection of L²(∂Ω) onto the space H²(Ω) of boundary values of elements of O(Ω). These projection operators have integral representations

B[f](z) = ∫_Ω,f(w)B(z,w)dw, S[f](z) = ∫_{∂Ω,}f(w)S(z,w)do(w).

The distributions B and S are known respectively as the Bergman and Szeg� kernels. In an attempt to prove that B and S are bounded operators on L^p, 1 < p < ∞, many authors have obtained size estimates for the kernels B and S for pseudoconvex domains in â„‚ⁿ.

In this thesis, we restrict our attention to the Szeg� kernel for a large class of domains of the form 1 Such a domain fails to be pseudoconvex precisely when b is not convex on all of R. In an influential paper, Nagel, Rosay, Stein, and Wainger obtain size estimates for both kernels and sharp mapping properties for their respective operators in the convex setting. Consequently, if b is a convex polynomial, the Szeg� kernel S is absolutely convergent off the diagonal only. Carracino proves that the Szeg� kernel has singularities on and off the diagonal for a specific non-smooth, {non-convex piecewise defined quadratic b. Her results are novel since very little is known for the Szeg� kernel for non-pseudoconvex domains 2. I take b to be an arbitrary even-degree polynomial with positive leading coefficient and identify the set in 3 on which the Szeg� kernel is absolutely convergent. For a polynomial b, we will see that the Szeg� kernel is smooth off the diagonal if and only if b is convex. These results provide an incremental step toward proving the projection S is bounded on 4, for a large class of non-pseudoconvex domains Ω.

Recommended Citation

Gilliam, Michael, "The Szego Kernel for Non-Pseudoconvex Domains in C²" (2011). Graduate Student Theses, Dissertations, & Professional Papers. 1093.
https://scholarworks.umt.edu/etd/1093