Plurisubharmonic functions and subellipticity of the dbar-Neumann problem on nonsmooth domains

We show subellipticity of the d-bar Neumann problem on domains with Lipschitz boundary in the presence of plurisubharmonic functions with Hessians of algebraic growth. In particular, a subelliptic estimate holds near a point where the boundary is piecewise smooth of finite type

Compactness of the ∂‾\overline{\partial}-Neumann operator on the intersection of two domains

Assume that $\Omega_{1}$ and $\Omega_{2}$ are two smooth bounded pseudoconvex domains in $\mathbb{C}^{2}$ that intersect (real) transversely, and that $\Omega_{1} \cap \Omega_{2}$ is a domain (i.e. is connected). If the $\overline{\partial}$-Neumann operators on $\Omega_{1}$ and on $\Omega_{2}$ are compact, then so is the $\overline{\partial}$-Neumann operator on $\Omega_{1} \cap \Omega_{2}$. The corresponding result holds for the $\overline{\partial}$-Neumann operators on $(0,n-1)$-forms on domains in $\mathbb{C}^{n}$

Estimates for the complex Green operator: symmetry, percolation, and interpolation

Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. We show that Sobolev estimates for the complex Green operator hold simultaneously for forms of symmetric bidegrees, that is, they hold for $(p,q)$--forms if and only if they hold for $(m-p,m-1-q)$--forms. Here $m$ equals the CR dimension of $M$ plus one. Symmetries of this type are known to hold for compactness estimates. We further show that with the usual microlocalization, compactness estimates for the positive part percolate up the complex, i.e. if they hold for $(p,q)$--forms, they also hold for $(p,q+1)$--forms. Similarly, compactness estimates for the negative part percolate down the complex. As a result, if the complex Green operator is compact on $(p,q_{1})$--forms and on $(p,q_{2})$--forms ($q_{1}\leq q_{2}$), then it is compact on $(p,q)$--forms for $q_{1}\leq q\leq q_{2}$. It is interesting to contrast this behavior of the complex Green operator with that of the $\overline{\partial}$--Neumann operator on a pseudoconvex domain.Comment: Added a reference to related work, removed a reference that was not quoted. To appear in Transactions of the American Mathematical Societ

Observations regarding compactness in the ∂‾\overline{\partial}-Neumann problem

We show that compactness of the $\overline{\partial}$-Neumann operator is independent of the metric, and we give a new proof of this independence for subellipticity. We define an abstract obstruction to compactness, namely the common zero set of all the compactness multipliers, and we identify this subset of the boundary for convex domains in $\mathbb{C}^{n}$ and for complete Hartogs domains in $\mathbb{C}^{2}$

Geometric sufficient conditions for compactness of the complex Green operator

We establish compactness estimates for $\bar{\partial}_{M}$ on a compact pseudoconvex CR-submanifold $M$ of $\mathbb{C}^{n}$ of hypersurface type that satisfies the (analogue of the) geometric sufficient conditions for compactness of the $\bar{\partial}$-Neumann operator given by the authors earlier. These conditions are formulated in terms of certain short time flows in complex tangential directions.Comment: 19 page

Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with d-bar

Boas and Straube proved a general sufficient condition for global regularity of the d-bar Neumann problem in terms of families of vector fields that commute approximately with d-bar. In this paper, we study the existence of these vector fields on a compact subset of the boundary whose interior is foliated by complex manifolds. This question turns out to be closely related to properties of interest from the point of view of foliation theory

Plurisubharmonic defining functions, good vector fields, and exactness of a certain one form

We show that the approaches to global regularity of the d-bar Neumann problem via the methods listed in the title are equivalent when the conditions involved are suitably modified. These modified conditions are also equivalent to one that is relevant in the context of Stein neighborhood bases and Mergelyan type approximation

Sobolev estimates for the complex Green operator on CR submanifolds of hypersurface type

Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. Our main result says that when a certain $1$-form on $M$ is exact on the null space of the Levi form, then the complex Green operator on $M$ satisfies Sobolev estimates. This happens in particular when $M$ admits a set of plurisubharmonic defining functions or when $M$ is strictly pseudoconvex except for the points on a simply connected complex submanifold

Compactness of the Complex Green Operator

Let \Omega\subset\C^n be a bounded smooth pseudoconvex domain. We show that compactness of the complex Green operator $G_{q}$ on $(0,q)$-forms on $b\Omega$ implies compactness of the $\bar{\partial}$-Neumann operator $N_{q}$ on $\Omega$. We prove that if $1 \leq q \leq n-2$ and $b\Omega$ satisfies $(P_q)$ and $(P_{n-q-1})$, then $G_{q}$ is a compact operator (and so is $G_{n-1-q}$). Our method relies on a jump type formula to represent forms on the boundary, and we prove an auxiliary compactness result for an `annulus' between two pseudoconvex domains. Our results, combined with the known characterization of compactness in the $\bar{\partial}$-Neumann problem on locally convexifiable domains, yield the corresponding characterization of compactness of the complex Green operator(s) on these domains.Comment: 17 pages. We added an appendix, fixed the proof of a main theorem, and revised the statement of another theorem. Also, we fixed some other typo

The Bergman kernel function: explicit formulas and zeroes

We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in C^3 defined by the inequality |z_1|+|z_2|+|z_3|<1, have zeroes