The Szego Kernel for Non-Pseudoconvex Domains in C²

<p>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 <em>Bergman projection</em> <strong><em>B</em></strong> 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 <em>Szeg� projection</em> <strong><em>S</em></strong> 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</p> <p><strong><em>B</em></strong>[f](z) = ∫_Ω,f(w)<strong><em>B</em></strong>(z,w)dw, <strong><em>S</em></strong>[f](z) = ∫_{∂Ω,}f(w)<strong><em>S</em></strong>(z,w)do(w).</p> <p>The distributions <strong><em>B</em></strong> and <strong><em>S</em></strong> are known respectively as the Bergman and Szeg� kernels. In an attempt to prove that <strong><em>B</em></strong> and <strong><em>S</em></strong> are bounded operators on L^p, 1 < p < ∞, many authors have obtained size estimates for the kernels B and S for <em>pseudoconvex</em> domains in â„‚ⁿ.</p> <p>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 <em>and off</em> the diagonal for a specific non-smooth, <em>{non-convex</em> 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.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Grantor dc:publisher

University of Montana

Year

2011

Author dc:creator

• Gilliam, Michael

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

Repository record dc:identifier

https://scholarworks.umt.edu/etd/1093

OAI identifier oai:identifier

oai:scholarworks.umt.edu:etd-2112