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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.

Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.

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

Regularity and irregularity of the Bergman projection on $L^p$ spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable $\gamma$. A surprising consequence of the analysis is that, whenever $\gamma$ is irrational, the Bergman projection is bounded only for $p=2$., Comment: 21 pages, 2 figures

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.

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 $$j\in\{q,\ldots,n\}$$ .