There are many operators associated with a domain Ω ⊂ [special characters omitted] with smooth boundary ∂Ω. There are two closely related projections that are of particular interest. The Bergman projection [special characters omitted] is the orthogonal projection of L²(Ω) onto the closed subspace L²(Ω) ∩ [special characters omitted](Ω), where [special characters omitted](Ω) is the space of all holomorphic functions on Ω. The Szegö projection [special characters omitted] is the orthogonal projection of L²(∂Ω) onto the space H²(Ω) of boundary values of elements of [special characters omitted](Ω). On Ω, 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 Szegö kernels. In an attempt to prove that [special characters omitted] and [special characters omitted] are bounded operators on L^p, 1 < p < ∞, many authors have obtained size estimates for the kernels B and S for pseudoconvex domains in [special characters omitted].

In this thesis, we restrict our attention to the Szegö kernel for a large class of domains of the form Ω = {(z, w) ∈ [special characters omitted] : Im[w] > b ( Re[z])}. Such a domain fails to be pseudoconvex precisely when b is not convex on all of [special characters omitted]. 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 Szegö kernel S is absolutely convergent off the diagonal only. Carracino proves that the Szegö 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 Szegö kernel for non-pseudoconvex domains Ω. I take b to be an arbitrary even-degree polynomial with positive leading coefficient and identify the set in [special characters omitted] on which the Szegö kernel is absolutely convergent. For a polynomial b, we will see that the Szegö 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 L^p(∂Ω), 1 < p < ∞, for a large class of non-pseudoconvex domains Ω.