1,319 research outputs found
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 -Neumann operator on the intersection of two domains
Assume that and are two smooth bounded pseudoconvex
domains in that intersect (real) transversely, and that
is a domain (i.e. is connected). If the
-Neumann operators on and on are
compact, then so is the -Neumann operator on . The corresponding result holds for the
-Neumann operators on -forms on domains in
Estimates for the complex Green operator: symmetry, percolation, and interpolation
Let be a pseudoconvex, oriented, bounded and closed CR submanifold of
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 --forms if and only if they hold for
--forms. Here equals the CR dimension of 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 --forms,
they also hold for --forms. Similarly, compactness estimates for the
negative part percolate down the complex. As a result, if the complex Green
operator is compact on --forms and on --forms (), then it is compact on --forms for . It
is interesting to contrast this behavior of the complex Green operator with
that of the --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 -Neumann problem
We show that compactness of the -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 and for complete Hartogs
domains in
Geometric sufficient conditions for compactness of the complex Green operator
We establish compactness estimates for on a compact
pseudoconvex CR-submanifold of of hypersurface type that
satisfies the (analogue of the) geometric sufficient conditions for compactness
of the -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 be a pseudoconvex, oriented, bounded and closed CR submanifold of
of hypersurface type. Our main result says that when a certain
-form on is exact on the null space of the Levi form, then the complex
Green operator on satisfies Sobolev estimates. This happens in particular
when admits a set of plurisubharmonic defining functions or when 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 on -forms on
implies compactness of the -Neumann operator on
. We prove that if and satisfies
and , then is a compact operator (and so is ).
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 -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
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