Luca Baracco, Tran Vu Khanh, Giuseppe Zampieri
We prove that CR lines in an exponentially degenerate boundary are propagators of holomorphic extension. This explains, in the context of the CR geometry, why in this situation the induced Kohn-Laplacian $\Box_b$ is not hypoelliptic according to recent work by Christ.
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We study the spectrum of the Kohn Laplacian $\square_b^t$ on the Rossi example $(\mathbb{S}^3, \mathcal{L}_t)$. In particular we show that $0$ is in the essential spectrum of $\square_b^t$, which yields another proof of the global non-embeddability of the Rossi example.
We study the Kohn-Laplacian and its fundamental solution on some model domains in $\mathbb C^{n+1}$, and further discuss the explicit kernel of the Cauchy-Szegö projections on these model domains using the real analysis method. We further show that these Cauchy-Szegö kernels are Calderón-Zygmund kernels under the suitable quasi-metric.
We give sharp and explicit upper bounds for the first positive eigenvalue $λ_1(\Box_b)$ of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^{n+1}$ in terms of their defining functions. As an application, we show that in the family of real ellipsoids, $λ_1(\Box_b)$ has a unique maximum value at the CR sphere.
A bounded domain in several complex variables with connected Lipschitz boundary is pseudoconvex if and only if the bottom of the (essential) spectrum of the Kohn Laplacian is positive on all (0, q)-forms with square-integrable coefficients.
We present an explicit formula for the leading coefficient in the asymptotic expansion of the eigenvalue counting function of the Kohn Laplacian on the unit sphere $\mathbb{S}^{2n-1}$.
Let $M^{(k)}$, $k=1,2,\ldots, n$, be the boundary of an unbounded polynomial domain $Ω^{(k)}$ of finite type in $\mathbb C ^2$, and let $\Box_b^{(k)}$ be the Kohn Laplacian on $M^{(k)}$. In this paper, we study multivariable spectral multipliers $m(\Box_b^{(1)},\ldots, \Box_b^{(n)})$ acting on the Shilov boundary $\widetilde{M}=M^{(1)} \times\cdots\times M^{(n)}$ of the product domain $Ω^{(1)}\times\cdots\times Ω^{(n)}$. We show that if a function $F(λ_1, \ldots ,λ_n)$ satisfies a Marcinkiewicz-type differential condition, then the spectral multiplier operator $m(\Box_b^{(1)}, \ldots, \Box_b^{…
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 compute the leading coefficient in the asymptotic expansion of the eigenvalue counting function for the Kohn Laplacian on the spheres. We express the coefficient as an infinite sum and as an integral.
In this note, we study the spectrum of the Kohn Laplacian on the unit spheres in $\mathbb{C}^n$ and revisit Folland's classical eigenvalue computation. We also look at the growth rate of the eigenvalue counting function in this context. Finally, we consider the growth rate of the eigenvalues of the perturbed Kohn Laplacian on the Rossi sphere in $\mathbb{C}^2$.
Let $(M,θ)$ be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue $λ_1$ of the Kohn-Laplacian $\Box_b$ on $(M,θ)$. In the present paper, we give a sharp upper bound for $λ_1$, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when $M$ is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webst…
The purpose of this paper is to establish local regularity of the solution operator to the Kohn-Laplace equation, called the complex Green operator, on abstract CR manifolds of hypersurface type. For a cut-off function $σ$, we introduce the $σ$-superlogarithmic property, a potential theoretical condition on CR manifolds. We prove that if the given datum is smooth on an open set containing the support of $σ$ then the solution is smooth on the interior of $\{x\in M:σ(x)=1\}$. Furthermore, we also study the smoothness of the integral kernel of the complex Green operator.
A real hypersurface in $\mathbb{C}^2$ is said to be Reinhardt if it is invariant under the standard $\mathbb{T}^2$-action on $\mathbb{C}^2$. Its CR geometry can be described in terms of the curvature function of its ``generating curve'', i.e., the logarithmic image of the hypersurface in the plane $\mathbb{R}^2$. We give a sharp upper bound for the first positive eigenvalue of the Kohn Laplacian associated to a natural pseudohermitian structure on a compact and strictly pseudoconvex Reinhardt real hypersurface having closed generating curve (which amounts to the $\mathbb{T}^2$-action being fre…
We prove rigidity for the Lichnerowicz-type eigenvalue estimate for the Kohn Laplacian on strictly pseudoconvex three-manifolds with nonnegative CR Paneitz operator and positive Webster curvature.
We obtain an analog of Weyl's law for the Kohn Laplacian on lens spaces. We also show that two 3-dimensional lens spaces with fundamental groups of equal prime order are isospectral with respect to the Kohn Laplacian if and only if they are CR isometric.
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.
Distribution of Eigenvalues of the Kohn Laplacian on Sphere Quotients
本文证明了球面商上 Kohn Laplacian 的 Weyl 型定理，指出在三维情况下可从其谱确定基本群，并给出了复 Green 算子的 Sobolev 估计。
Isospectral CR manifolds with respect to the Kohn Laplacian
该研究证明Kohn拉普拉斯算子的谱不能决定CR流形的等价类，构造了多对等谱但不等价的奇数维椭圆流形，并给出透镜空间及非循环基本群椭圆流形中的多种构造方式。
Let $X$ be a complex manifold and $M\subset X$ a compact, smooth, pseudoconvex CR manifold of dimension $2n-1$. (Here $n\ge 3$ or, in case $n=2$, it is made the extra assumption that $\dib_b$ has closed range on functions.) Assume that there exists a strictly CR-plurisubharmonic function in a neighborhood of $M$ in $X$. In this situation, there are here proved \begin{enumerate} \item[(i)] The global existence of $C^\infty$ solutions to the tangential Cauchy-Riemann operator $\dib_b$. \item[(ii)] The global hypoellipticity of the Kohn-Laplacian $\Box_b$, under the additional condition of "weak …
Let $X$ be an abstract orientable not necessarily compact CR manifold of dimension $2n+1$, $n\geq1$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Suppose that condition $Y(q)$ holds at each point of $X$, we establish asymptotics of the heat kernel of Kohn Laplacian with values in $L^k$. As an application, we give a heat kernel proof of Morse inequalities on compact CR manifolds. When $X$ admits a transversal CR $\mathbb R$-action, we also establish asymptotics of the $\mathbb R$-equivariant heat kernel of Kohn Laplacian with values in $L^k$. As an applicat…
Let $M= Γ\setminus \mathbb{H}_d$ be a compact quotient of the $d$-dimensional Heisenberg group $\mathbb{H}_d$ by a lattice subgroup $Γ$. We show that the eigenvalue counting function $N(λ)$ for any fixed element of a family of second order differential operators $\left\{\mathcal{L}_α\right\}$ on $M$ has asymptotic behavior $N\left(λ\right) \sim C_{d,α} \operatorname{vol}\left(M\right) λ^{d + 1}$, where $C_{d,α}$ is a constant that only depends on the dimension $d$ and the parameter $α$. As a consequence, we obtain an analog of Weyl's law (both on functions and forms) for the Kohn Laplacian on …
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.
Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds
本文研究了闭嵌入严格伪凸 CR 流形上 Kohn Laplacian 特征值作为伪埃尔米特结构泛函的连续性与可微性，定义了广义临界伪埃尔米特结构，给出了其充要条件并提供了显式例子。
Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstract CR manifold of hypersurface type, $n\geq 2$. We prove the following: (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, we also establish continuity for $G_0$ and $G_n$. Additionally, the $\bar\partial_b$-equation on $M$ can be solved in $C^\infty(M)$. (ii) If $M$ satisfies "a weak compactness property" on $(0,q_0)$-forms, then $G_q$ is a continuous opera…
Let $(\hat{X}, T^{1,0} \hat{X})$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0} \hat{X}$ is a CR structure on $\hat{X}$. Fix a point $p \in \hat{X}$ and take a global contact form $\hatθ$ so that $\hatθ$ is asymptotically flat near $p$. Then $(\hat{X}, T^{1,0} \hat{X}, \hatθ)$ is a pseudohermitian $3$-manifold. Let $G_p \in C^{\infty} (\hat{X} \setminus \{p\})$, $G_p > 0$, with $G_p(x) \sim \vartheta(x,p)^{-2}$ near $p$, where $\vartheta(x,y)$ denotes the natural pseudohermitian distance on $\hat{X}$. Consider the new pseudohermitian $…
The weighted Kohn Laplacian $\Box_φ$ is a natural second order elliptic operator associated to a weight $φ:\mathbb{C}^n\rightarrow\mathbb{R}$ and acting on $(0,1)$-forms, which plays a key role in several questions of complex analysis. We consider here the case of model monomial weights in $\mathbb{C}^2$, i.e., $ φ(z,w):=\sum_{(α,β)\inΓ}|z^αw^β|^2, $ where $Γ\subseteq \mathbb{N}^2$ is finite. Our goal is to prove coercivity estimates of the form $\Box_φ\geq μ^2$, where $μ:\mathbb{C}^n\rightarrow\mathbb{R}$ acts by pointwise multiplication on $(0,1)$-forms, and the inequality is in the sense of…
Kohn-Rossi cohomology of spherical CR manifolds
本文利用Patterson-Sullivan测度定义的典范接触形式和Kohn Laplacian的Weitzenböck型公式，证明了某些球面CR流形的Kohn-Rossi上同调的消没定理，并证明了部分结果的最优性。
In this article, we follow the arguments in a paper of Y-T. Siu to study the effective termination of Kohn's algorithm for special domains in $\mathbb{C}^{3}$. We make explicit the effective constants and generic conditions that appear there, and we obtain an explicit expression for the regularity of the Dolbeault laplacian for the $\overline{\partial}$-Neumann problem. Specifically, on a local peudoconvex domain of the special shape \[ Ω:= \bigg\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}:\ 2\text{Re}\ z_{3}+ \sum_{i=1}^{N}|F_{i}(z_{1},z_{2})|^{2}<0 \bigg\} \] with holomorphic function germs $F_{1}…
Let $Ω$ be a pseudoconvex domain in $\mathbb C^n$ with smooth boundary $bΩ$. We define general estimates $(f\text{-}\mathcal M)^k_Ω$ and $(f\text{-}\mathcal M)^k_{bΩ}$ on $k$-forms for the complex Laplacian $\Box$ on $Ω$ and the Kohn-Laplacian $\Box_b$ on $bΩ$. For $1\le k\le n-2$, we show that $(f\text{-}\mathcal M)^k_{bΩ}$ holds if and only if $(f\text{-}\mathcal M)^k_Ω$ and $(f\text{-}\mathcal M)^{n-k-1}_Ω$ hold. Our proof relies on Kohn's method in [Ann. of Math. (2), 156(1):213--248, 2002].
We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solve $\square_b$ works whenever $\overline{\partial}_b$ on the compact CR manifold has closed range in $L^2$; in particular, as in Beals and Greiner, it does not require the CR manifold to be the boundary of a strongly pseudoconvex domain in $\mathbb{C}^2$. Our result provides in tu…
For a pseudoconvex domain in complex space, we prove the equivalence of the local hypoellipticity of the system (di-bar, di-bar*) with the system (di-bar_b,di-bar*_b) induced in the boundary. This develops a result of ours which used the theory of the "harmonic" extension by Kohn. This technique is inadequate for the purpose of the present paper and must be replaced by the "holomorphic" extension introduced by the authors in former work.
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 new pointwise bounds for weighted Bergman kernels in $\mathbb{C}^n$, whenever a coercivity condition is satisfied by the associated weighted Kohn Laplacian on $(0,1)$-forms. Our results extend the ones obtained in $\mathbb{C}$ by Christ. Our main idea is to develop a version of Agmon theory (originally introduced to deal with Schrödinger operators) for weighted Kohn Laplacians on $(0,1)$-forms, inspired by the fact that these are unitarily equivalent to certain generalized Schrödinger operators.
It is here proved that if a pseudoconvex CR manifold $M$ of hypersurface type has a certain "type", that we quantify by a vanishing rate $F$ at a submanifold of CR dimension $0$, then $\Box_b$ "gains $f^2$ derivatives" where $f$ is defined by inversion of $F$. Indeed the estimate is more accurate and it involves the Levi form of $M$ and of additional weights, instead of $\Box_b$. Next a general tangential estimate, "twisted" by a pseudodifferential operator $Ψ$ is established. The combination of the two yields a general "$f$-estimate" twisted by $Ψ$. We apply the twisted estimate for $Ψ$ which…
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.
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…
We study the regularity of Radon measures $μ$ which satisfy that there exists a function $h_μ$ in $H^1(Ω)$, stationary harmonic such that $Δh_μ=μ$ in $Ω$ (here $Ω$ is an open set of $\mathbb{R}^2$). Such conditions appear in physical contexts such as the study of a limiting vorticity measure associated to a family $(u_\varepsilon)_\varepsilon$ of solutions of the Ginzburg-Landau system without magnetic field. Under these conditions we prove that locally there exists a harmonic function $H$ such that the support of the measure is contained in the set of zeros of $H$. Using the local structure o…
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