Moulay Youssef Barkatou
In this paper a new explicit integral formula is derived for solutions of the tangential Cauchy-Riemann equations on CR q-concave manifolds and optimal estimates in the Lipschitz norms are obtained.
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We prove smooth and analytic versions of the classical Schwarz reflection principle for transversal CR mappings between two CR manifolds of hypersurface type.
This paper first studies the regularity of conformal homeomorphisms on smooth locally embeddable strongly pseudoconvex CR manifolds. Then moduli of curve families are used to estimate the maximal dilatations of quasiconformal homeomorphisms. On certain CR 3-manifolds, namely, CR circle bundles over flat tori, extremal quasiconformal homeomorphisms in some homotopy classes are constructed. These extremal mappings have similar behaviors to Teichmüller mappings on Riemann surfaces.
Based on a new generalization of Cauchy-Riemann system presented in this paper, we introduce a class of quaternion-valued functions of a quaternionic variable, which are called algebraic regular functions. The set of algebraic regular functions is not only a real associative algebra, but also respect the composition of functions. Using algebraic regular functions as transition maps, we introduce a class of four-manifolds called the regular four-manifolds.
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.
Sobolev Regularity for the Bergman Projection on Relatively Compact Domains in Hermitian manifolds
本文推广了已有结果，为 Hermitian 流形中具有 Lipschitz 边界的相对紧区域上的 Bergman 投影提供了 $L^2$ Sobolev 正则性的充分条件，并通过 Hopf 流形实例验证了方法的有效性。
In this paper we study Hölder continuity of the pluricomplex Green function with logarithmic growth at infinity of a smooth generic submanifold of $\C^n$. In particular we prove that the pluricomplex Green function of any $C^2$-smooth generic compact submanifold of $\C^n$ (without boundary) is Lipschitz continuous in $\C^n$.
In this article the notion of ultradifferentiable CR manifold is introduced and an ultradifferentiable regularity result for finitely nondegenerate CR mappings is proven. Here ultradifferentiable means with respect to Denjoy-Carleman classes defined by weight sequences. Furthermore the regularity of infinitesimal CR automorphisms on ultradifferentiable abstract CR manifolds is investigated.
In this paper, we prove the $C^{1, 1}$-regularity of the plurisubharmonic envelope of a $C^{1,1}$ function on a compact Hermitian manifold. We also present examples to show this regularity is sharp.
Suppose that a smooth holomorphic curve $V$ has order of contact $η$ at a point $w_0$ in the boundary of a pseudoconvex domain $Ω$ in $\mathbb{C}^3.$ We show that the maximal gain in Hölder regularity for solutions of the $\bar{\partial}$-equation is at most $\frac{1}η.$
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.
The note concerns the $\bar\partial$ problem on product domains in $\mathbb C^2$. We show that there exists a bounded solution operator from $C^{k, α}$ into itself, $k\in \mathbb Z^+\cup \{0\}, 0<α< 1$. The regularity result is optimal in view of an example of Stein-Kerzman.
We show there is a solution operator to $\bar{\partial}$ which is bounded as a map $W^{s}_{(0,1)}(Ω)\cap\mbox{ker }\bar{\partial}\rightarrow W^{s}(Ω)$ for all $s\ge 0$.
Let $Ω$ be a product domain in $\mathbb C^n, n\ge 2$, where each slice has smooth boundary. We observe that the canonical solution operator for the $\bar\partial$ equation on $Ω$ is bounded in $W^{k,p}(Ω)$, $k\in \mathbb Z^+, 1<p<\infty$. This Sobolev regularity is sharp in view of Kerzman-type examples.
We reduce the problem of constructing a linear solution operator to the $\bar{\partial}$-equation on smoothly bounded weakly pseudoconvex domains, $Ω$, in $\mathbb{C}^2$ to the problem of the boundary $\bar{\partial}_b$-equation. We show there is a solution operator to $\bar{\partial}$ which is bounded as a map $W^{s}_{(0,1)}(Ω)\cap{ker} \bar{\partial} \rightarrow W^{s}(Ω)$ for all $s\ge 0$ if there is a corresponding solution operator to the $\bar{\partial}_b$-problem with analogous regularity properties.
We obtain higher order estimates for a parabolic flow on a compact Hermitian manifold. As an application, we prove that a bounded $\hatω$-plurisubharmonic solution of an elliptic complex Monge-Ampère equation is smooth under an assumption on the background Hermitian metric $\hatω$. This generalizes a result of Székelyhidi and Tosatti on Kähler manifolds.
On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds
本文研究复流形中具有C²边界的a_q域上∂-方程解的正则性，在特定Levi特征值条件下证明了解在闭包上存在并获得1/2阶导数增益。
In this paper, we prove weighted $L^p$ estimates for the canonical solutions on product domains. As an application, we show that if $p\in [4, \infty)$, the $\bar\partial$ equation on the Hartogs triangle with $L^p$ data admits $L^p$ solutions with the desired estimates. For any $ε>0$, by constructing an example with $L^p$ data but having no $L^{p+ε}$ solutions, we verify the sharpness of the $L^p$ regularity on the Hartogs triangle.
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…
Combining V. Koziarz's observation about the regularity of some modified section related to the initial extension with J. McNeal--D. Varolin's regularity argument, we generalize two theorems of McNeal--Varolin for the $L^2$ extension of $\bar\partial$-closed high-degree forms on a Stein manifold to the weakly pseudoconvex Kähler case under mixed positivity conditions.
We characterize infinitesimal generators on complete hyperbolic complex manifolds without any regularity assumption on the Kobayashi distance. This allows to prove a general Loewner type equation with regularity of any order $d\in [1,+\infty ]$. Finally, based on these results, we focus on some open problems naturally arising.
We prove regularity of solutions of the $\bar\partial$-problem in the Hölder-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains which is of independent interest, and on techniques that were recently developed by the first-named author to prove estimates for the $\bar\partial$-problem on strongly pseudoconvex domains of class $C^2$.
We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a minimal manifold into such a foliated target is either generically smooth or geometrically highly constrained, and to show generic smoothness of sufficiently regular CR transversal CR maps between pseudoconvex hypersurfaces. As an application, we discuss boundary regularity of proper holomorphic maps into bounded symmetric domains.
Hölder regularity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds
研究了紧致Hermitian流形上极值函数的Hölder连续性，证明其局部性质及加权极值函数的Hölder连续性，扩展了Nguyen在紧致Kähler流形上的结果。
The regularity of the $\bar{\partial}$-problem on the domain $\{|{z_1}|<|{z_2}|<1\}$ in $\mathbb{C}^2$ is studied using $L^2$ methods. Estimates are obtained for the canonical solution in weighted $L^2$-Sobolev spaces with a weight that is singular at the point $(0,0)$. The canonical solution for $\dbar$ with weights is exact regular in the weighted Sobolev spaces away from the singularity $(0,0)$. In particular, the singularity of the Bergman projection for the Hartogs triangle is contained at the singular point and it does not propagate.
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…
We introduce a class of embedded CR manifolds satisfying a geometric condition that we call weak $Y(q)$. For such manifolds, we show that dbar-b has closed range on $L^2$ and that the complex Green operator is continuous on $L^2$. Our methods involves building a weighted norm from a microlocal decomposition. We also prove that at any Sobolev level there is a weight such that the complex Green operator inverting the weighted Kohn Laplacian is continuous. Thus, we can solve the dbar-b equation in $C^\infty$.
Regularity of the Siciak-Zaharjuta extremal function on compact Kähler manifolds
本文证明紧Kähler流形上极值函数的正则性为局部性质，将连续性与Hölder连续性等价于多重位势理论中的经典概念，并据此给出紧集$(\Cc^\al, \Cc^{\al'})$-正则性的有效刻画，证明$\bR^n$中所有紧的胖次解析子集均满足该正则性。
Regularity of solutions to Monge--Ampère equations on compact Hermitian manifolds
研究紧致赫尔德流形上退化复monge-ampere方程解的稳定性与霍尔德连续性，建立扭曲陈-里齐流的统一直径界。
Let (X,L) be a polarized compact manifold, i.e. L is an ample line bundle over X and denote by H the infinite dimensional space of all positively curved Hermitian metrics on L equipped with the Mabuchi metric. In this short note we show, using Bedford-Taylor type envelope techniques developed in the authors previous work [ber2], that Chen's weak geodesic connecting any two elements in H are C^{1,1}-smooth, i.e. the real Hessian is bounded, for any fixed time t, thus improving the original bound on the Laplacins due to Chen. This also gives a partial generalization of Blocki's refinement of Che…
We prove stability of solutions of the complex Monge-Ampère equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be Hölder continuous. As an application we extend a recent result of Székelyhidi and Tosatti on Kähler-Einstein equation from Kähler to Hermitian manifolds.
Let $(X,ω)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Ampère equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are Hölder continuous with the same exponent as in the Kähler case \cite{DDGKPZ14}. Our techniques also apply to the setting of big cohomology classes on compact Kähler manifolds.
Let M of real dimension 2n-1 be a compact, orientable, weakly pseudoconvex manifold of dimension at least five, embedded in C^N (n less than or equal to N), of codimension one or more in C^N, and endowed with the induced CR structure. We show the tangential Cauchy-Riemann operator has closed range on such a manifold M, hence we get global existence and regularity results for the \bar\partial_b problem. We also show the middle (i.e. corresponding to (p,q) forms for q between 1 and n-2) \bar\partial_b cohomology groups of M with respect to L^2, Sobolev s, and smooth coefficients are finite and i…
In this paper, we show that for each $k\in \mathbb Z^+, p>4$, there exists a solution operator $\mathcal T_k$ to the $\bar\partial$ problem on the Hartogs triangle that maintains the same $W^{k, p}$ regularity as that of the data. According to a Kerzman-type example, this operator provides solutions with the optimal Sobolev regularity.
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 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.
This note seeks to prove the existence of a canonical solution operator to the $\bar\partial$-equation that preserves Hölder regularity on product domains. It is a well known fact that such solution operators do not in general gain Hölder regularity, and as such, our solution operator is optimal in this regard.
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.
Hölder regularity of the $\bar\partial-$equation on the polydisc
本文研究了 polydisc 上 $\bar\partial-$ 方程的解算子，证明其保持 Hölder 正则性，并指出该算子在正则性方面是最佳的。
We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact Kähler manifolds have the optimal $C^{1,1}$ regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green's functions on Kähler manifolds. We then go on to prove the same regularity for envelopes when the manifold is assumed to have boundary. As an application, we answer affirmatively a question of Ross--Witt-Nyström concerning the Hele-Shaw flow on an arbitrary Reimann surface.
We introduce a notion of super-potential (canonical function) associated to positive closed (p,p)-currents on compact Kaehler manifolds and we develop a calculus on such currents. One of the key points in our study is the use of deformations in the space of currents. As an application, we obtain several results on the dynamics of holomorphic automorphisms: regularity and uniqueness of the Green currents. We also get the regularity, the entropy, the ergodicity and the hyperbolicity of the equilibrium measures.
As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of $\mathbb{R}^r\times \mathbb{C}^n$ (for some $r$ and $n$) in such a way that the structure is locally the span of $\frac{\partial}{\partial t_1},\ldots, \frac{\partial}{\partial t_r},\frac{\partial}{\partial \overline{z}_1},\ldots, \frac{\partial}{\partial \overline{z}_n}$; where $\mathbb{R}^r\times \mathbb{C}^n$ has coordinates $(t_1,\ldots, t_r, z_1,\ldots, z_n)$. In this paper, we give optimal regularit…
Let $f$ be a meromorphic correspondence on a compact Kähler manifold $X$ of dimension $k$. Assume that its topological degree is larger than the dynamical degree of order $k-1$. We obtain a quantitative regularity of the equilibrium measure of $f$ in terms of its super-potentials.
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