There are many operators associated with a domain O ⊂ Cn with smooth boundary ∂O. There are two closely related projections that are of particular interest. The Bergman projection B is the orthogonal projection of L²(O) onto the closed subspace L²(O) ∩ O (O), where O (O) is the space of all holomorphic functions on O. The Szego projection S is the orthogonal projection of L²(∂O) onto the space H²(O) of boundary values of elements of O (O). On O, these projection operators have integral representations Bf z=W fwB z,wdw,S fz =6Wf wSz,w dsw. The distributions B and S are known respectively as the Bergman and Szego kernels. In an attempt to prove that B and S are bounded operators on Lp, 1 p infinity, many authors have obtained size estimates for the kernels B and S for pseudoconvex domains in Cn .;In this thesis, we restrict our attention to the Szego kernel for a large class of domains of the form O = {(z, w) ∈ C2 : Im[w] > b ( Re[z])}. 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 Szego kernel S is absolutely convergent off the diagonal only. Carracino proves that the Szego 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 Szego kernel for non-pseudoconvex domains O. I take b to be an arbitrary even-degree polynomial with positive leading coefficient and identify the set in C2xC2 on which the Szego kernel is absolutely convergent. For a polynomial b, we will see that the Szego 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 Lp(∂O), 1 p infinity, for a large class of non-pseudoconvex domains O.
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