Siqi Fu, Chunhui Qiu, Weixia Zhu
In this paper we study spectral stability of the $\bar\partial$-Neumann Laplacian under the Kohn-Nirenberg elliptic regularization. We obtain quantitative estimates for stability of the spectrum of the $\bar\partial$-Neumann Laplacian when either the operator or the underlying domain is perturbed.
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The Kohn-Nireberg domains are unbounded domains in the complex Euclidean space of dimension 2 upon which many outstanding questions are yet to be explored. The primary aim of this article is to demonstrate that the Bergman and Caratheodory metrics of any Kohn-Nirenberg domains are positive and complete.
We give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two. It is expressed in terms of a generalized model, which captures local geometry of the hypersurface both in the complex tangential and nontangential directions. As an application we obtain a new class of nonconvexifiable pseudoconvex hypersurfaces with convex models.
For $1\leq q \leq n-2$, we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on $L^2$ $(0,q)$-forms on the unit ball in $(2n+1)$-dimensional Heisenberg space.
This article gives a rapid introduction to the complex Neumann problem and Kohn's algorithm for generating subelliptic estimate.
The purpose of this note is threefold. (i) To explain the effective Kohn algorithm for multipliers in the complex Neumann problem and its difference with the full-real-radical Kohn algorithm, especially in the context of an example of Catlin-D'Angelo concerning the ineffectivness of the latter. (ii) To extend the techniques of multiplier ideal sheaves for the complex Neumann problem to general systems of partial differential equations. (iii) To present a new procedure of generation of multipliers in the complex Neumann problem as a special case of the multiplier ideal sheaves techniques for ge…
A theory of global regularity of the d-bar-Neumann operator is developed which unifies the two principal approaches to date, namely the one via compactness due to Kohn-Nirenberg and Catlin, and the one via plurisubharmonic defining functions and/or vector fields that commute approximately with d-bar due to Boas and the author.
We describe along the guidelines of Kohn "Quantitative estimates..." (1999), the constant E_s which is needed to control the commutator of a totally real vector field T with di-bar* in order to have Sobolev s-regularity of the Bergman projection in any degree of forms, on a smooth pseudoconvex domain D of the complex space. This statement, not explicit in Kohn's paper, yields Straube's Theorem in "A sufficient condition..." (2008). Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function r of D, the operators (T^+)^{-delta/2} and (-r)^{delta/2…
We prove local hypoellipticity of the complex Laplacian $\Box$ and of the Kohn Laplacian $\Box_b$ in a pseudoconvex boundary when, for a system of cut-off $η$, the gradient $\partial_bη$ and the Levi form $\partial_b\bar\partial_bη^2$ are subelliptic multipliers in the sense of Kohn.
This article chronicles a development that started around 1990 with \cite{BoasStraube91}, where the authors showed that if a smooth bounded pseudoconvex domain $Ω$ in $\mathbb{C}^{n}$ admits a defining function that is plurisubharmonic at points of the boundary, then the $\overline{\partial}$--Neumann operators on $Ω$ preserve the Sobolev spaces $W^{s}_{(0,q)}(Ω)$, $s\geq 0$. The same authors then proved a further regularity result and made explicit the role of D'Angelo forms for regularity (\cite{BoasStraube93}). A few years later, Kohn (\cite{Kohn99}) initiated a quantitative study of the re…
We prove that the $\bar\partial$-Neumann solution operator is locally regular in a domain which has compactness estimates, is of finite type outside a curve transversal to the CR directions and for which the holomorphic tangential derivatives of a defining function are subelliptic multipliers in the sense of Kohn.
We study the $\bar{\partial}_b$-Neumann problem for domains $Ω$ contained in a strictly pseudoconvex manifold M^{2n+1} whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts of a single CR function w. When the Kohn Laplacian is a priori known to have closed range in L^2, we prove sharp regularity and estimates for solutions. We establish a condition on the boundary which is sufficient for the Kohn Laplacian to be Fredholm on $L^2_{(0,q)}(Ω)$ and show that this condition always holds when M is embedded as a hypersurface in C^{n+1}. We…
This paper presents a class of domains on which the Kohn Laplacian and the dib-bar-Neumann problem are hypoelliptic but not superlogarithmic and which, moreover, have a set of points of Levi-degeneracy with positive CR dimension.
We establish sharp regularity and Fredholm theorems for the \bar{\partial}_b-Neumann problem on domains satisfying some non-generic geometric conditions. We use these domains to construct explicit examples of bad behaviour of the Kohn Laplacian: it is not always hypoelliptic up to the boundary, its partial inverse is not compact and it is not globally subelliptic.
We adapt the results of Part 1 to include the unit ball in the Heisenberg group, the model domain with characteristic boundary points. In particular, we construct function spaces on which the Kohn Laplacian with the \bar{\partial}_b-Neumann boundary conditions is an isomorphism. As an application, we establish sharp regularity for a canonical solution to the inhomogenous \bar{\partial}_b equation on the unit ball.
This thesis starts from a review on current research on the local hypoellipticity of the $\bar\partial$-Neumann problem. It presents the classical method of regularity from estimates of the energy: subelliptic as well as superlogarithmic. More recent material is included in which the regularity of the solution is obtained from the geometry of the singularities of the Levi form. The new contribution to this discussion consists in a general weighted Kohn-Hörmander-Morrey formula twisted by a pseudodifferential operator. As an application, a new class of domains for which the $\bar\partial$-Neuma…
Sharp subelliptic estimates in the $\bar\partial$-Neumann problem via an uncertainty principle
本文通过引入$\bar\partial$-不确定性原理，确定了环境维数不超过5时Kohn特殊域在原点处次椭圆估计的尖锐阶数，解决了该问题中一个长期开放的关键方面。
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