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The method of alternating projections is used to examine how regularity of operators associated to the $${{\bar{\partial }}}$$ -Neumann problem percolates up the $${{\bar{\partial }}}$$ -complex. The approach revolves around operator identities—rather than estimates—that hold on any Lipschitz domain in $${{\mathbb {C}}}^n$$ , not necessarily bounded or pseudoconvex. We show that a geometric rate of convergence in von Neumann’s alternating projection algorithm, applied to two basic projection operators, is equivalent to $${{\bar{\partial }}}$$ having closed range. This implies that compactness of the $${{\bar{\partial }}}$$ -Neumann operator percolates up the $${{\bar{\partial }}}$$ -complex whenever $${{\bar{\partial }}}$$ has closed range at the corresponding form levels.

An integral solution operator for $${\bar{\partial }}$$ is constructed on product domains that include the punctured bidisc. This operator is shown to satisfy $$L^p$$ estimates for all $$1\le p

A sufficient condition for to have closed range is given for pseudoconvex, unbounded domains in .

A class of pseudoconvex domains in $\mathbb{C}^{n}$ generalizing the Hartogs triangle is considered. The $L^p$ boundedness of the Bergman projection associated to these domains is established, for a restricted range of $p$ depending on the "fatness" of domains. This range of $p$ is shown to be sharp.

Let Ω ⋐ C n \Omega \Subset \mathbb {C}^{n} be a domain with smooth boundary, whose Bergman projection B B maps the Sobolev space H k 1 ( Ω ) H^{k_{1}}(\Omega ) (continuously) into H k 2 ( Ω ) H^{k_{2}}(\Omega ) . We establish two smoothing results: (i) the full Sobolev norm ‖ B f ‖ k 2 \|Bf\|_{k_{2}} is controlled by L 2 L^2 derivatives of f f taken along a single, distinguished direction (of order ≤ k 1 \leq k_{1} ), and (ii) the projection of a conjugate holomorphic function in L 2 ( Ω ) L^{2}(\Omega ) is automatically in H k 2 ( Ω ) H^{k_{2}}(\Omega ) . There are obvious corollaries for when B B is globally regular.

Let B be the Bergman projection associated to a domain Ω on which the \({\bar\partial}\) -Neumann operator is compact. We show that arbitrary L2 derivatives of Bf are controlled by derivatives of f taken in a single, distinguished direction. As a consequence, functions not contained in \({C^{\infty}(\overline{\Omega})}\) that are mapped by B to \({C^{\infty}(\overline{\Omega})}\) are explicitly described.

We give three proofs of the fact that a smoothly bounded, convex domain in ℝn has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.

We give an example of a permutation of integers which preserves all convergent Fourier series and makes some divergent Fourier series converge.

Let $$\Omega \subset \subset \mathbb{C}^{n}$$ be a smoothly bounded domain. Suppose Ω has a defining function, such that the sum of any q eigenvalues of its complex Hessian is non-negative. We show that this implies global regularity of the Bergman projection, B j-1, and the $$\bar{\partial}$$ -Neumann operator, N j , acting on (0,j)-forms, for $$j\in\{q,\ldots,n\}$$ .

We establish a vanishing theorem on the square-integrable cohomology associated to the Cauchy-Riemann complex on some complete Kaehler manifolds. The hypothesis needed for this result is a growth condition on a primitive of the Kaehler form.

We show that the existence of a certain family of plurisubharmonic functions on a bounded domain in Cn implies that the ∂-Neumann operator associated to the domain is compact. Our condition generalizes previous work of Catlin on compactness of the ∂-Neumann operator.

First published in Proceedings of the American Mathematical Society in volume 130 and issue 1, published by the American Mathematical Society.

A new proof of Oka's lemma is given for smoothly bounded, pseudoconvex domains $D\subset\mathbb{C}^n$. The method of proof is then also applied to other convexity-like hypotheses on the boundary of $D$.

Analytic and algebraic geometers often study the same geometric structures but bring different methods to bear on them. While this dual approach has been spectacularly successful at solving problems, the language differences between algebra and analysis also represent a difficulty for students and researchers in geometry, particularly complex geometry. The PCMI program was designed to partially address this language gulf, by presenting some of the active developments in algebraic and analytic geometry in a form suitable for students on the ""other side"" of the analysis-algebra language divide. One focal point of the summer school was multiplier ideals, a subject of wide current interest in both subjects. The present volume is based on a series of lectures at the PCMI summer school on analytic and algebraic geometry. The series is designed to give a high-level introduction to the advanced techniques behind some recent developments in algebraic and analytic geometry. The lectures contain many illustrative examples, detailed computations, and new perspectives on the topics presented, in order to enhance access of this material to non-specialists. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute.|Analytic and algebraic geometers often study the same geometric structures but bring different methods to bear on them. While this dual approach has been spectacularly successful at solving problems, the language differences between algebra and analysis also represent a difficulty for students and researchers in geometry, particularly complex geometry. The PCMI program was designed to partially address this language gulf, by presenting some of the active developments in algebraic and analytic geometry in a form suitable for students on the ""other side"" of the analysis-algebra language divide. One focal point of the summer school was multiplier ideals, a subject of wide current interest in both subjects. The present volume is based on a series of lectures at the PCMI summer school on analytic and algebraic geometry. The series is designed to give a high-level introduction to the advanced techniques behind some recent developments in algebraic and analytic geometry. The lectures contain many illustrative examples, detailed computations, and new perspectives on the topics presented, in order to enhance access of this material to non-specialists. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute.

Proofs of two results about a monomial ideal -- describing membership in auxiliary ideals associated to the monomial ideal -- are given which do not invoke resolution of singularities. The AM--GM inequality is used as a substitute for taking a log resolution of the monomial ideal.

We construct bounded, plurisubharmonic functions with maximally large Hessians near the boundary of a smoothly bounded convex domain in Cn. As a corollary, the equality of the order of contact of the boundary with complex analytic varieties (the D'Angelo type) and the order of contact with complex lines is demonstrated.

We present several generalizations of the Riemann-Lebesgue lemma. Our approach highlights the role of cancellation in the Riemann—Lebesgue lemma.

Toeplitz operators on strongly pseudoconvex domains in $\mathbb{C}^n$, constructed from the Bergman projection and with symbol equal to a positive power of the distance to the boundary, are considered. The mapping properties of these operators on $L^p$, as the power of the distance varies, are established.

We establish new results on weighted $L^2$ extension of holomorphic top forms with values in a holomorphic line bundle, from a smooth hypersurface cut out by a holomorphic function. The weights we use are determined by certain functions that we call denominators. We give a collection of examples of these denominators related to the divisor defined by the submanifold., Comment: To Appear in Ann. Inst. Fourier

This volume is the proceedings of a conference held at Ohio State University in May of 1999. Over sixty mathematicians from around the world participated in this conference and principal lectures were given by some of the most distinguished experts in the field. The proceedings volume contains fully refereed research articles from some of the principal speakers, including: Salah Baouendi (UCSD), David Barrett (Univ. Michigan), Bo Berndtsson (Goteborg), David Catlin (Purdue Univ.), Micheal Christ (Berkeley), John D'Angelo (Univ. Illinois), Xiaojun Huang (Rutgers), J. J. Kohn (Princeton), Y.-T. Siu (Harvard), and Emil Straube (Texas A & M).

We construct holomorphic support functions on a smoothly bounded, convex domain of finite type in ℂ n which satisfy sharp, nonisotropic estimates near the fixed boundary point where the functions vanish.

For any smoothly bounded, strongly pseudoconvex domain Ω in ℂ n , the boundary of a suitable ball in ℂ n locally approximates b Ω near p ∈ b Ω, i.e., their defining functions agree up to 3-jets near p. This approximation is often the first step taken when analyzing various domain dependent functions and operators on Ω (e.g., the Bergman and Szego kernels, the ∂-Neumann operator, Poisson kernels), as it allows certain results on the behavior of the operators (or functions) associated to the ball and various tools of classical analysis to be brought to bear on questions about the behavior of the operators on Ω. See, for instance, [F],[F-S],[P]. When the domain Ω is only weakly pseudoconvex, there is no single model domain which locally approximates b Ω, in any reasonable sense, near some z 0 ∈ b Ω. Moreover, the description of an appropriate approximating domain in terms of analytic information on the defining function is not known in general, even when the Levi form degenerates to finite order. Additionally, even when an approximating domain for b Ω near z 0 ∈ b Ω can be found, its shape may vary greatly from the shape of an approximating domain for b Ω near another z 1 ∈ bΩ, even when z 1 is close to z 0.

The holomorphic sectional curvatures in the Bergman metric of a smooth bounded pseudoconvex domain in C 2 {{\mathbf {C}}^2} are shown to be bounded in absolute value near a poinit of finite type in the boundary

We present the theory of twisted \(L^2\) estimates for the Cauchy–Riemann operator and give a number of recent applications of these estimates. Among the applications: extension theorem of Ohsawa–Takegoshi type, size estimates on the Bergman kernel, quantitative information on the classical invariant metrics of Kobayshi, Caratheodory, and Bergman, and sub elliptic estimates on the \(\bar{\partial }\)-Neumann problem. We endeavor to explain the flexibility inherent to the twisted method, through examples and new computations, in order to suggest further applications.