Let X be a Hermitian complex space of pure dimension n. We show that the d-bar-Neumann operator on (p,q)-forms is compact at isolated singularities of X if q>0 and p+q is not equal to n-1 or n. The main step is the construction of compact solution operators for the d-bar-equation on such spaces which is based on a general characterization of compactness in function spaces on singular spaces, and that leads also to a criterion for compactness of more general Green operators on singular spaces.
As an application of a new characterization of compactness of the $\bar\partial $-Neumann operator we derive a sufficient condition for compactness of the $\bar\partial $- Neumann operator on $(0,q)$-forms in weighted $L^2$-spaces on $\mathbb{C}^n.$
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 $\mathbb{C}^{2}$.
We prove that on smooth bounded pseudoconvex Hartogs domains in $\mathbb{C}^2$ compactness of the $\bar{\partial}$-Neumann operator is equivalent to compactness of all Hankel operators with symbols smooth on the closure of the domain.
We show that Property $(P)$ of $\partialΩ$, compactness of the $\bar{\partial}$-Neumann operators $N_1$, and compactness of Hankel operator on a smooth bounded pseudoconvex Hartogs domain $Ω={\{(z, w_1, w_2,\dots, w_n) \in \mathbb{C}^{n+1} \mid\sum_{k=1}^{n} |w_k|^2 < e^{-2φ(z)}, z\in\mathit{D}\}}$ are equivalent, where $D$ is a smooth bounded connected open set in $\mathbb{C}$.
The d-bar-Neumann operator on (0,q)-forms ($1\le q \le n$) on a bounded convex domain Omega in C^n is compact if and only if the boundary of Omega contains no complex analytic (equivalently: affine) variety of dimension greater than or equal to q.
By using a variant Property $(P_q)$ of Catlin, we discuss the relation of small set of weakly pseudoconvex points on the boundary of pseudoconvex domain and compactness of the $\overline{\partial}$-Neumann operator. In particular, we show that if the Hausdorff $(2n-2)$-dimensional measure of the weakly pseudoconvex points on the boundary of a smooth bounded pseudoconvex domain is zero, then the $\overline{\partial}$-Neumann operator $N_{n-1}$ is compact on $(0,n-1)$-level $L^2$-integrable forms.
Assume that $Ω_{1}$ and $Ω_{2}$ are two smooth bounded pseudoconvex domains in $\mathbb{C}^{2}$ that intersect (real) transversely, and that $Ω_{1} \cap Ω_{2}$ is a domain (i.e. is connected). If the $\overline{\partial}$-Neumann operators on $Ω_{1}$ and on $Ω_{2}$ are compact, then so is the $\overline{\partial}$-Neumann operator on $Ω_{1} \cap Ω_{2}$. The corresponding result holds for the $\overline{\partial}$-Neumann operators on $(0,n-1)$-forms on domains in $\mathbb{C}^{n}$.
Let $Ω$ be a $C^4$-smooth bounded pseudoconvex domain in $\mathbb{C}^2$. We show that if the $\overline{\partial}$-Neumann operator $N_1$ is compact on $L^2_{(0,1)}(Ω)$ then the embedding operator $\mathcal{J}:Dom(\overline{\partial})\cap Dom(\overline{\partial}^*) \to L^2_{(0,1)}(Ω)$ is $L^p$-regular for all $2\leq p<\infty$.
This paper surveys the theory of compactness of the d-bar-Neumann problem. It also contains several results which improve upon what was previously known.
We discuss compactness of the d-bar-Neumann operator in the setting of weighted L^2-spaces on C^n.$ For this purpose we use a description of relatively compact subsets of L^2- spaces. We also point out how to use this method to show that property (P) implies compactness for the d-bar-Neumann operator on a smoothly bounded pseudoconvex domain and mention an abstract functional analysis characterization of compactness of the d-bar-Neumann operator.
For smooth bounded pseudoconvex domains in $mathbb{C}^{2}$, we provide geometric conditions on (the points of infinite type in) the boundary which imply compactness of the $\bar{\partial}$-Neumann operator. It is noteworthy that the proof of compactness does \emph{not} proceed via verifying the known potential theoretic sufficient conditions.
We discuss compactness of the d-bar-Neumann operator in the setting of weighted L^2-spaces on b C^n. In addition we describe an approach to obtain the compactness estimates for the d-bar-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich - Kondrachov lemma.
We study a complex valued version of the Sobolev inequalities and its relationship to compactness of the d-bar-Neumann operator. For this purpose we use an abstract characterization of compactness derived from a general description of precompact subsets in L^2-spaces. Finally we remark that the d-bar-Neumann operator can be continuously extended provided a subelliptic estimate holds.
On a smooth, bounded pseudoconvex domain $Ω$ in $\mathbb{C}^n$, to verify that Catlin's Property ($P$) holds for $bΩ$, it suffices to check that it holds on the set of D'Angelo infinite type boundary points. In this note, we consider the support of the Levi core, $S_{\mathfrak{C}(\mathcal{N})}$, a subset of the infinite type points, and show that Property ($P$) holds for $bΩ$ if and only if it holds for $S_{\mathfrak{C}(\mathcal{N})}$. Consequently, if Property ($P$) holds on $S_{\mathfrak{C}(\mathcal{N})}$, then the $\overline{\partial}$-Neumann operator $N_1$ is compact on $Ω$.
In this paper we discuss compactness estimates for the $\bar \partial $-Neumann problem in the setting of weighted $L^2$-spaces on $\mathbb{C}^n.$ For this purpose we use a version of the Rellich - Lemma for weighted Sobolev spaces.
By establishing a unified estimate of the twisted Kohn-Morrey-Hörmander estimate and the $q$-pseudoconvex Ahn-Zampieri estimate, we discuss variants of Property $(P_q)$ of Catlin and Property $(\widetilde{P_q})$ of McNeal on the boundary of a smooth pseudoconvex domain in $\mathbb{C}^n$ for certain high level forms. These variant conditions on the one side, imply $L^2$-compactness of the $\overline{\partial}$-Neumann operator on the associated domain, on the other side, are different from the classical Property $(P_q)$ and Property $(\widetilde{P_q})$. As an application of our result, we show …
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.
We study the asymptotic behavior, in a ``semi-classical limit'', of the first eigenvalues (i.e. the groundstate energies) of a class of Schrödinger operators with magnetic fields and the relationship of this behavior with compactness in the $\bar\partial$-Neumann problem on Hartogs domains in $\C^2$
We provide geometric conditions on the set of boundary points of infinite type of a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$ which imply that the $\bar{\partial}$-Neumann operator is compact. These conditions are formulated in terms of certain short time flows in suitable complex tangential directions. It is noteworthy that compactness is \emph{not} established via the known potential theoretic sufficient conditions. Our results generalize to $\mathbb{C}^{n}$ the corresponding $\mathbb C^{2}$ results due to the second author.
Let D be a bounded pseudoconvex domain in $C^n, n\geq 2, 0\leq p\leq n,$ and $1\leq q\leq n-1.$ We show that compactness of the dbar-Neumann operator, $N_{p,q+1},$ on square integrable (p,q+1)-forms is equivalent to compactness of the commutators $[P_{p,q}, \bar{z}_j]$ on square integrable dbar-closed (p,q)-forms for $1\leq j\leq n$ where $P_{p,q}$ is the Bergman projection on (p,q)-forms. We also show that compactness of the commutator of the Bergman projection with functions continuous on the closure percolates up in the dbar-complex on dbar-closed forms and square integrable holomorphic for…
We derive a necessary condition for compactness of the weighted $\overline\partial$-Neumann operator on the space $L^2(\mathbb C^n,e^{-φ})$, under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension. Moreover, we compute the essential spectrum of the complex Laplacian for decoupled weights, $φ(z) = φ_1(z_1) + \dotsb + φ_n(z_n)$, and investigate (non-) compactness of the $\overline\partial$-Neumann operator in this case. More can be said if every $Δφ_j$ defines a nontrivial doubling measure.
Let $\D=\D_1\setminus \Dc_2$, where $\D_1$ and $\D_2$ are two smooth bounded pseudoconvex domains in $\C^n, n\geq 3,$ such that $\Dc_2\subset \D_1.$ Assume that the $\dbar$-Neumann operator of $\D_1$ is compact and the interior of the Levi-flat points in the boundary of $\D_2$ is not empty (in the relative topology). Then we show that the Hankel operator on $\D$ with symbol $ϕ, H^{\D}_ϕ,$ is compact for every $ϕ\in C(\Dc)$ but the $\dbar$-Neumann operator on $\D$ is not compact.
Compactness of the d-bar Neumann operator is studied for weakly pseudoconvex bounded Hartogs domains in two dimensions. A nonsmooth example is constructed in which condition (P) fails to hold, yet the Neumann operator is compact. The main result, in contrast, is that for smoothly bounded Hartogs domains, condition (P) of Catlin and Sibony is equivalent to compactness. The analyses of both compactness and condition (P) boil down to properties of the lowest eigenvalues of certain sequences of Schrodinger operators, with and without magnetic fields, parametrized by a Fourier variable resulting fr…
By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, $\mathbb{C}$-convex domains, and homogeneous domains. For this class of domains, we show that compactness of the $\bar{\partial}$-Neumann operator on $(0,q)$-forms is equivalent to the boundary not containing any $q$-dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of d…
We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain in C^n is an obstruction to compactness of the d-bar-Neumann operator on the domain, provided that at some point of M, the Levi form has the maximal possible rank n-1-dim(M) (i.e. the domain is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue. We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmon…
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.
We give a sufficient condition for subelliptic estimates for the d-bar-Neumann operator on smoothly bounded, pseudoconvex domains in $\mathbb{C}^n$. This condition is a quantified version of McNeal's condition ($\tilde{P}$) for compactness of the d-bar-Neumann operator, and it extends Catlin's condition for subellipticity as it is less stringent.
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 establish a connection between compactness of Hankel operators and geometry of the underlying domain through compactness multipliers for the $\overline{\partial}$-Neumann operator. In particular, we prove that any compactness multiplier induces a compact Hankel operator. We also generalize the notion of compactness multipliers to vector fields and matrices and then we use this generalization to generate compact Hankel operators.
Let $Ω\subset\C^n$ be a bounded smooth pseudoconvex domain. We show that compactness of the complex Green operator $G_{q}$ on $(0,q)$-forms on $bΩ$ implies compactness of the $\bar{\partial}$-Neumann operator $N_{q}$ on $Ω$. We prove that if $1 \leq q \leq n-2$ and $bΩ$ 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 co…
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.
The $\bar{\partial}$-Neumann problem is the fundamental boundary value problem in several complex variables. It features an elliptic operator coupled with non-coercive boundary conditions. The problem is globally regular on many, but not all, pseudoconvex domains. We discuss several recent developments in the $L^{2}$-Sobolev theory of the $\bar{\partial}$-Neumann problem that concern compactness and global regularity.
The spectrum of the d-bar-Neumann Laplacian on the Fock space $L^2(\mathbb{C}^n, e^{-|z|^2})$ is explicitly computed. It turns out that it consists of positive integer eigenvalues each of which is of infinite multiplicity. Spectral analysis of the d-bar-Neumann Laplacian on the Fock space is closely related to Schrödinger operators with magnetic field and to the complex Witten-Laplacian.
A twisted $\bar{\partial}_f$-Neumann problem associated to a singularity $(\mathscr{O}_n,f)$ is established. By constructing the connection to the Koszul complex for toeplitz $n$-tuples $(f_1,\cdots,f_n)$ on Bergman spaces $B^0(D)$, we can solve this $\bar{\partial}_f$-Neumann problem. Moreover, we can compute the cohomology of the $L^2$ holomorphic Koszul complex $(B^*(D),\partial f\wedge)$ explicitly
The $\bar{\partial}$-Neumann operator (the inverse of the complex Laplacian) is shown to be noncompact on certain domains in complex Euclidean space. These domains are either higher-dimensional analogs of the Hartogs triangle, or have such a generalized Hartogs triangle imbedded appropriately in them.
We obtain weighted estimates for the $\bar{\partial}$-Neumann operator on intersections of two smooth strictly pseudoconvex domains in $\mathbb{C}^2$. The regularity estimates are described with the use of Sobolev norms with weights which are powers of the defining functions of the two domains.
For a domain $D$ of $\mathbb{C}^n$ which is weakly $q$-pseudoconvex or $q$-pseudoconcave we give a sufficient condition for subelliptic estimates for the $\bar{\partial}$-Neumann problem. The paper extends to domains which are not necessarily pseudoconvex, the results and the techniques of Catlin. MSC: 32D10, 32U05, 32V25
We introduce a new integral representation formula in the d-bar Neumann Theory on weakly pseudoconvex domains which satisfies certain estimates analogous to the basic L^2 estimate. It is expected that more complete estimates can be obtained in case the boundary is of finite type.
This paper treats subelliptic estimates for the $\bar{\partial}$-Neumann problem on a class of domains known as regular coordinate domains. Our main result is that the largest subelliptic gain for a regular coordinate domain is bounded below by a purely algebraic number, the inverse of twice the multiplicity of the ideal associated to a given boundary point.
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 study the solution of the d-bar-Neumann problem on (0,1)-forms on the product of two half-planes in C^2. In, particular, we show the solution can be decomposed into functions smooth up to the boundary and functions which are singular at the singular points of the boundary. Furthermore, we show the singular functions are log and arctan terms.
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 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…
Let $Ω$ be an unbounded, pseudoconvex domain in $\Bbb C^n$ and let $φ$ be a $\mathcal C^2$-weight function plurisubharmonic on $Ω$. We show both necessary and sufficient conditions for existence and compactness of a weighted $\bar\partial$-Neumann operator $N_φ$ on the space $L^2_{(0,1)}(Ω,e^{-φ})$ in terms of the eigenvalues of the complex Hessian $(\partial ^2φ/\partial z_j\partial\bar z_k)_{j,k}$ of the weight. We also give some applications to the unweighted $\bar\partial$-Neumann problem on unbounded domains.
It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $\bar\partial$-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for $(0,1)$-forms) and Ben Moussa (for $(0,q)$-forms when $q\geq 1$). In this paper, we explore necessary conditions and sufficient conditions for maxima…
In this paper we discuss compactness of the canonical solution operator to d-bar on weigthed L^2- spaces on C^n. For this purpose we apply ideas which were used for the Witten Laplacian in the real case and various methods of spectral theory of these operators. We also point out connections to the theory of Dirac and Pauli operators.
This thesis deals with Partial Differential Equations in Several Complex Variables and especially focuses on a general estimate for the $\bar\partial$-Neumann problem on a domain which is $q$-pseudoconvex or $q$-pseudoconcave at a boundary point $z_0$. Generalizing Property ($P$) by \cite{C84}, we define Property $(f\T-\M\T-P)^k$ at $z_0$. This property yields the estimate {(f\T-\M)^k} \qquad \no{f(Λ)\mathcal M u}^2\le c(\no{\bar\partial u}^2+\no{\bar\partial^*u}^2+\no{u}^2)+C_\M\no{u}^2_{-1} for any $u\in C^\infty_c(U\cap \barΩ)^k\cap \T{Dom}(\dib^*)$ where $U$ is a neighborhood of $z_0$. We …
Let $Γ(\cdot,λ)$ be smooth, i.e.\, $\mathcal C^\infty$, embeddings from $\barΩ$ onto $\bar{Ω^λ}$, where $Ω$ and $Ω^λ$ are bounded domains with smooth boundary in the complex plane and $λ$ varies in $I=[0,1]$. Suppose that $Γ$ is smooth on $\barΩ\times I$ and $f$ is a smooth function on $\partialΩ\times I$. Let $u(\cdot,λ)$ be the harmonic functions on $Ω^λ$ with boundary values $f(\cdot,λ)$. We show that $u(Γ(z,λ),λ)$ is smooth on $\barΩ\times I$. Our main result is proved for suitable Hölder spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solu…
In this note, we find an equivalent boundary integral equation to the classical $\bar{\partial}$-Neumann problem. The new equation contains an equivalent regularity to the global regularity of the $\bar{\partial}$-Neumann problem. We also use the integral equations to observe a new-found rigidity property of the $\bar{\partial}$-Neumann problem on the unit ball in $\mathbb{C}^2$.
We study the $\bar\partial$-Neumann Laplacian from spectral theoretic perspectives. In particular, we show how pseudoconvexity of a bounded domain is characterized by positivity of the $\bar\partial$-Neumann Laplacian.
We show, under an assumption on the weakly pseudoconvex points, the trivial Diederich--Fornæss index directly implies the global regularities of the $\bar{\partial}$-Neumann operators.
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.
We show that if the closure of a smooth pseudoconvex domain admits a "sufficiently nice" Stein neighborhood basis, then the d-bar Neumann problem on the domain is globally regular.
The spectrum of the d-bar-Neumann Laplacian on a polydisc in several complex variables is explicitly computed. The calculation exhibits that the spectrum consists of eigenvalues, some of which, in particular the smallest ones, are of infinite multiplicity.
A smooth bounded pseudoconvex domain in two complex variables is of finite type if and only if the number of eigenvalues of the d-bar-Neumann Laplacian that are less than or equal to $λ$ has at most polynomial growth as $λ$ goes to infinity.
