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 present examples where the inhomogeneous $\bar{\partial}_b$ equation can always be solved smoothly up to the boundary on (p,q)-forms with 0<q<n-1.
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.
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.
Let $Ω$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$. We prove that the $\bar\partial-Neumann operator $N$ exists for $(p,q)$-forms on $Ω$. Furthermore, there exists a $t_0>0$ such that the operators $N$, $\bar\partial^*N$, $\bar\partial N$ and the Bergman projection are regular in the Sobolev space $W^t (\barΩ) $ for $t<t_0$. The boundary estimates above have applications in complex geometry. We use the estimates to prove the nonexistence of $C^{2, α}$ real Levi-flat hypersurfaces in $\mathbb CP^n$. We also show that there exist no non-zero $L^2$-holomorphic $(p, 0)$-…
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 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.
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 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.
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 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.
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 $\mathbb{C}^2$.
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.
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 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 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.
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 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 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.
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 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.
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.
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.
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.
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 $\mathbb{C}$.
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.
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.
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.
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 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.
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 study spectral stability of the $\bar\partial$-Neumann Laplacian on a bounded domain in $\mathbb{C}^n$ when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the $\bar\partial$-Neumann Laplacian on bounded pseudoconvex domains in $\mathbb{C}^n$, lower semi-continuity properties on pseudoconvex domains that satisfy property ($P$), and quantitative estimates on smooth bounded pseudoconvex domains of finite D'Angelo type in $\mathbb{C}^n$.
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.
Let $M$ be a strongly pseudoconvex complex $G$-manifold with compact quotient $M/G$. We provide a simple condition on forms $α$ sufficient for the regular solvability of the equation $\square u=α$ and other problems related to the $\bar\partial$-Neumann problem on $M$.
We show there exists an L^p solution, for p>2, to the dbar-Neumann problem on an edge domain in C^2 for (0,1)-forms, and we explicitly compute the singularities, which are of complex logarithmic and arctangent type, along the edge, of the solution.
Let $Ω\subset\mathbb{C}^m$ be a bounded pseudoconvex domain with smooth boundary. For each $k\in\mathbb{N}$, we give a sufficient condition to estimate the $\bar\partial$-Neumann operator in the Sobolev space $W^k(Ω)$. The key feature of our results is a precise formula for $k$ in terms of the geometry of the boundary of $Ω$.
In the paper we give a lower estimate for the essential norm of the $\bar{\partial}$-Neumann operator on convex domains and worm domains of Diederich and Fornaess.
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…
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 prove heat kernel estimates for the $\bar\partial$-Neumann Laplacian acting in spaces of differential forms over noncompact, strongly pseudoconvex complex manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated Laplace-Beltrami operator on functions.
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.
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…
Let $M$ be a strongly pseudoconvex complex manifold which is also the total space of a principal $G$-bundle with $G$ a Lie group and compact orbit space $\bar M/G$. Here we investigate the $\bar\partial$-Neumann Laplacian on $M$. We show that it is essentially self-adjoint on its restriction to compactly supported smooth forms. Moreover we relate its spectrum to the existence of generalized eigenforms: an energy belongs to $σ(\square)$ if there is a subexponentially bounded generalized eigenform for this energy. Vice versa, there is an expansion in terms of these well-behaved eigenforms so tha…
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…
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…
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.
We study Hardy spaces $H^p_ν$ of the conjugate Beltrami equation $\bar{\partial} f=ν\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $ν\in W^{1,r}$ with $r>2$, in the range $r/(r-1)<p<\infty$. We develop a theory of conjugate functions and apply it to solve Dirichlet and Neumann problems for the conductivity equation $\nabla.(σ\nabla u)=0$ where $σ=(1-ν)/(1+ν)$. In particular situations, we also consider some density properties of traces of solutions together with boundary approximation issues.
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 give a simple proof of the fact that an "$f$-estimate" for the $\bar\partial$-Neumann problem implies a lower bound on the geomatric type of the boundary along any complex one dimensional variety. The proof uses the existence of peak functions which is in turn a consequence of the $f$-estimate.
Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ be the total space of a principal bundle $G\to M\to X$ so that $M$ is also a strongly pseudoconvex complex manifold. In this work, we show that if $G$ acts by holomorphic transformations in $M$, then the complex Laplacian $\square$ on $M$ has the following properties: The kernel of $\square$ restricted to the forms $Λ^{p,q}$ with $q$ positive is a closed, $G$-invariant subspace in $L^{2}(M,Λ^{p,q})$ of finite $G$-dimension. Secondly, we show that if $q$ is positive, then the image of $\square$ contains a closed, $…
