arXiv subjects
Publications and source records attributed to Tran Vu Khanh.
In this paper, we study supnorm and modified Hölder estimates for the integral solution of the di-bar-equation on a class of convex domains of general type in $\C^2$ that includes many infinite type examples.
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.
This note is aimed at simplifying current literature about compactness estimates for the Kohn-Laplacian on CR manifolds. The approach consists in a tangential basic estimate in the formulation given by the first author in \cite{Kh10} which refines former work by Nicoara \cite{N06}. It has been proved by Raich \cite{R10} that on a CR manifold of dimension $2n-1$ which is compact pseudoconvex of hypersurface type embedded in $\C^n$ and orientable, the property named "$(CR-P_q)$" for $1\leq q\leq \frac{n-1}2$, a generalization of the one introduced by Catlin in \cite{C84}, implies compactness estimates for the Kohn-Laplacian $\Box_b$ in degree $k$ for any $k$ satisfying $q\leq k\leq n-1-q$. The same result is stated by Straube in \cite{S10} without the assumption of orientability. We regain these results by a simplified method and extend the conclusions in two directions. First, the CR manifold is no longer required to be embedded. Second, when $(CR-P_q)$ holds for $q=1$ (and, in case $n=1$, under the additional hypothesis that $\dib_b$ has closed range on functions) we prove compactness also in the critical degrees $k=0$ and $k=n-1$.
We discuss, both for systems of complex vector fields and for sums of squares, the phenomenon discovered by Kohn of hypoellipticity with loss of derivatives.
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.
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 want to point out that under a suitable choice of $f$ and $\M$, $(f\T-\M)^k$ is the subelliptic, superlogarithmic, compactness and subelliptic multiplier estimate. The thesis also aims at exhibiting some relevant classes of domains which enjoy Property $(f\T-\M\T-P)^k$ and at discussing recent literature on the $\bar\partial$-Neumann problem in the framework of this property.
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.
For the $\bar\partial$-Neumann problem on a regular coordinate domain $Ω\subset \C^{n+1}$, we prove $ε$-subelliptic estimates for an index $ε$ which is in some cases better than $ε=\frac1{2m}$ ($m$ being the {\it multiplicity}) as it was previously proved by Catlin and Cho in \cite{CC08}. This also supplies a much simplified proof of the existing literature. Our approach is founded on the method by Catlin in \cite{C87} which consists in constructing a family of weights $\{ϕ^δ\}$ whose Levi form is bigger than $δ^{-2ε}$ on the $δ$-strip around $\partialΩ$.
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