Date of Award
Doctor of Philosophy (PhD)
This thesis is a study of various weighted estimates for the Bergman and Szegö projections on domains in several complex variables. The starting point of our analysis is a bounded, pseudoconvex domain D ⊂ Cn. The Bergman and Szegö projections are both orthogonal projections onto spaces of holomorphic functions associated with D. While it is immediate that both of these operators are bounded on L2, it has been a topic of substantial interest to determine their mapping properties on Lp, where the boundary geometry of D plays amajor role. Given a linear operator T acting on measurable functions that is bounded on L2 or Lp with respect to Lebesgue measure dµ, it is of substantial interest in harmonic analysis to determine the absolutely continuous measures σdµ such that T is also bounded on Lp(σdµ). In the case that T is a Calderón-Zygmund operator, it is known that the correct sufficient condition for boundedness is the Ap condition. A closely related weight class, called the Bp class, is known to characterize the weighted Lp boundedness of the Bergman projection on the unit ball. The goal of this thesis is to establish weighted estimates for the two projection operators for weights in the Ap or (generalized) Bp classes in a much more general context. We especially focus on strongly pseudoconvex domains with minimal or near-minimal boundary smoothness. In Chapter 1, we introduce the spaces of holomorphic functions and related projection operators, define the relevant weight classes, and discuss the history of unweighted and weighted Lp estimates for the projection operators on various classes of domains. We also establish common notation and state the main results in the thesis. In Chapter 2, we establish weighted Lp estimates for the Bergman projection when 1 < p < ∞ and σ ∈ Bp on several classes of smoothly bounded, pseudoconvex domains where explicit size and smoothness on the Bergman kernel are known. In the case of strongly pseudoconvex domains, the necessity of the Bp condition is also proved. Next, in Chapter 3 we shift our focus to strongly pseudoconvex domains with a lower level of boundary regularity. For such domains, explicit estimates on the kernel functions are not known, so one must instead use an operator-theoretic technique pioneered by Kerzman and Stein. This technique involves relating the Bergman or Szegö projection to a non-orthogonal projection which has a non-canonical, yet explicitly constructed kernel. In this chapter, we specifically prove weighted Lp estimates for the Szegö projection for 1 < p < ∞ and σ ∈ Ap on strongly pseudoconvex domains with C2 boundary. In Chapter 4, we establish weighted Lp estimates for the Bergman projection for 1 < p < ∞ and σ ∈ Bp on strongly pseudoconvex domains with C4 boundary. These estimates are obtained using very similar techniques to Chapter 3. At the end of the chapter, we prove weighted estimates in the minimally smooth (C2) case for the Bergman projection for a special class of weights that are a power of the distance to the boundary. We also provide an application of the proof techniques to mapping properties of Toeplitz operators. Finally, in Chapter 5 we establish some endpoint estimates for both projection operators. In particular, we prove that on a strongly pseudoconvex domain with C4 boundary, if σ belongs to B1, the Bergman projection is weighted weak-type (1,1). We also establish that on a strongly pseudoconvex domain with C3 boundary, if σ belongs to A1, the Szegö projection is also weighted weak-type (1,1). Finally, we provide some other endpoint estimates, including weighted Kolmogorov and Zygmund inequalities, as well as an estimate for the Bergman projection for p = ∞ in terms of the Bloch norm.
Chair and Committee
Wagner, Nathan Andrew, "Weighted Estimates for the Bergman and Szegö Projections" (2022). Arts & Sciences Electronic Theses and Dissertations. 2663.