Martin Kolar, Ilya Kossovskiy, Dmitri Zaitsev
One of effective ways to solve the equivalence problem and describe moduli spaces for real submanifolds in complex space is the normal form approach. In this survey, we outline some normal form constructions in CR-geometry and formulate a number of open problems.
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We construct a Cauchy type formula on open subdomains of Riemann surfaces
We are interested in $L^p$-theory for the tangential Cauchy-Riemann operator in locally embeddable, $s$-concave, generic CR manifolds. We study the Dolbeault isomorphism and develop the Andreotti-Grauert theory in that setting. Using Serre duality, we solve the tangential Cauchy-Riemann equation with exact support and $L^p$-estimates.
We prove that every compact, pseudoconvex, orientable, CR manifold of $\C^n$, bounds a complex manifold in the $C^\infty$ sense. In particular, the tangential Cauchy-Riemann system has closed range.
In this paper we propose a systematic study of the Cauchy-Riemann operator in the $L^p$-setting in complex manifolds. We first consider $L^p_{loc}$-theory and then we develop an $L^p$ Andreotti-Grauert theory. Finally we consider Serre duality and its applications to the solvability of the Cauchy-Riemann equation with exact support in $L^p$-spaces.
We establish the $L^2$ theory for the Cauchy-Riemann equations on product domains provided that the Cauchy-Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on $(p,1)$-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.
The main aim of this paper is to describe the most adequate generalization of the Cauchy-Riemann system fixing properties of classical functions in octonionic case. An octonionic generalization of the Laplace transform is introduced. Octonionic generalizations of the inversion transformation, the gamma function and the Riemann zeta-function are given.
The Borel-Pompieu formula involving proportional fractional $ψ$-Cauchy-Riemann operators
本研究利用关于另一函数的Riemann-Liouville导数，证明了比例型分数阶$ψ$-Cauchy-Riemann算子框架下的四元数Borel-Pompieu公式的对应结果，拓展了相关算子的积分公式理论。
In this paper we first recall the definition of an octonion algebra and its algebraic properties. We derive the so called $e_4$-calculus and using it we obtain the list of generalized Cauchy-Riemann systems in octonionic monogenic functions. We define some bilinear forms and derive the corresponding symmetry groups.
The nonhomogeneous Cauchy-Riemann equation on families of open Riemann surfaces
本文通过非齐次Beltrami方程解决非齐次Cauchy-Riemann方程问题，并应用该结果得到复线丛在开放Riemann表面族上的Oka-Grauert原理。
We prove a general local existence theorem for nonlinear systems of Cauchy-Riemann operator of any order in one complex variable with initial values at a given point, which is a counterpart of local existence of ODE.
Using Hörmander $L^2$ method for Cauchy-Riemann equations from complex analysis, we study a simple differential operator $\bar{\partial}^k+a$ of any order (densely defined and closed) in weighted Hilbert space $L^2(\mathbb{C},e^{-|z|^2})$ and prove the existence of a right inverse that is bounded.
We show sufficient and necessary conditions, in terms of some partial differential equations with variable coefficients, for a quaternionic function to admit a continuous derivative in a open set in the sense of C. Schwartz.
We give a very simple argument to the effect that most germs of generic real analytic Cauchy-Riemann manifolds of positive CR dimension are not holomorphically embeddable into any generic real algebraic CR manifold of the same real codimension in a finite dimensional space. In particular, most such germs are not holomorphically equivalent to a germ of a generic real algebraic CR manifold.
A bicomplex proportional fractional $(\vartheta,φ)-$weighted Cauchy-Riemann operator using Riemann-Liouville derivatives with respect to an hyperbolic-valued function
本文首次定义了基于双曲值函数 Riemann-Liouville 导数的双复数比例分数加权 Cauchy-Riemann 算子，并证明了其分数 Borel-Pompeiu 公式。
In this note it is worked out a new set of Laplace-Like equations for quaternions through Riemann-Cauchy hypercomplex relations otained earlier \cite{BorgesZeMarcio}. As in the theory of functions of a complex variable, it is expected that this new set of Laplace-Like equations might be applied to a large number of Physical problems, providing new insights in the Classical Theory Fields.
In one complex variable, the existence of a compactly supported solution to the Cauchy-Riemann equation is related to the vanishing of certain integrals of the data; trying to generalize this approach, we find an explicit construction, via convolution, for a compactly supported solution in $\mathbb{C}^n$, which allows to estimate the $L^p$ norm of the solution. We also investigate the possible generalizations of this method to domains of the form $P\setminus Z$, where $P$ is a polydisc and $Z$ is the zero locus of some holomorphic function.
We show that on bounded Lipschitz pseudoconvex domains that admit good weight functions the $\overline{\partial}$-Neumann operators $N_q, \overline{\partial}^* N_{q}$, and $\overline{\partial} N_{q}$ are bounded on $L^p$ spaces for some values of $p$ greater than 2.
We observe that the continuity assumption on $f$ for the uniform estimates of the canonical solution to $\bar\partial u = f$ on products of $C^2$ bounded planar domains in \cite{DPZ} can be reduced to the boundedness assumption. This completely answers the original question raised by Kerzman in 1971. Moreover, the $L^p$ estimates of $\bar\partial$ is obtained for all $p \in [1, \infty]$.
An atomic representation for Hardy classes of solutions to nonhomogeneous Cauchy-Riemann equations
本文为非齐次 Cauchy–Riemann 方程解的 Hardy 类提出第二类原子表示，用于刻画分布边值、证明 Hilbert 变换连续性并构造 Schwarz 型边值问题的解。
A joint generalization of real smooth as well of complex manifolds are the Cauchy-Riemann manifolds. The main objective of the paper is to inroduce a class of symmetric CR manifolds containing both classes of Riemannian and Hermitian symmetric spaces. It turns out that the classical requirement of isolated fixed points for the symmetries is no longer adequate, because it would imply the Levi-flatness. Among all symmetric CR-manifolds we distinguish a large subclass consisting of all Shilov boundaries of bounded symmetric domains. For this class we calculate the polynomial and the rational conv…
We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly "$q$-pseudoconvex" higher codimensional submanifold $M$ of $\C^n$. Let us point out that our hypotheses do not suffice to guarantee subelliptic estimates, in general. Even more: hypoellipticity of the tangential C-R system is not in question (as shows the example by Kohn in case of a Levi-flat hypersurface). However our estimates suffice for existence of smooth solutions to the inhomogeneous C-R equations in certain degree. The main ingredients in our proofs are the weighted $L^2$ estimates by Hörmander a…
The Heisenberg group $\mathbb{H}^1$ is known to be conformally equivalent to the CR sphere $\mathbb{S}^3$ minus a point. We use this fact, together with the knowledge of the tangential Cauchy-Riemann operator on the compact CR manifold $\mathbb{S}^3$, to solve the corresponding operator on $\mathbb{H}^1$.
A historical review of the Cauchy-Riemann equations and the Cauchy Theorem
本文从历史角度系统梳理柯西-黎曼方程与柯西积分路径无关定理的关系，重点更新研究达朗贝尔和欧拉的贡献，并分析柯西论证中的不清晰之处及格林公式的演变。
Covariance property for a quaternionic fractional Cauchy-Riemann operator under affine transformation
本文针对四元数分数阶 Cauchy-Riemann 算子，引入适应仿射变换的分数阶超全纯函数类，证明其协变性，并推广了 Stokes、Borel-Pompeiu 和 Cauchy 公式。
We prove some formulas relating Cauchy-Riemann operators defined on hypercomplex subspaces of an alternative *-algebra to a differential operator associated with the concept of slice-regularity and to the spherical Dirac operator. These results in particular allow to introduce a definition of locally slice-regular function and open the path for local slice analysis. Since Cauchy-Riemann operators factor the corresponding Laplacian operators, the proven formulas let us also obtain several results about the harmonicity and polyharmonicity properties of slice-regular functions.
In this paper, we prove the infinite dimensionality of some local and global cohomology groups on abstract Cauchy-Riemann manifolds.
We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the solution set can be explicitly calculated. The methods also give local parametric formulas for pseudoholomorphic curves with respect to some continuous almost complex structures.
We compute the spectra of the Tanaka type Laplacians on the Rumin complex, a quotient of the tangential Cauchy-Riemann complex on the unit sphere in $C^n$. We prove that Szegö map is a unitary operator from a subspace of $(p, q-1)$-forms on the sphere defined by the Tanaka operators and the normal vector field onto the space of $L^2$-harmonic $(p, q)$-forms on the unit ball. Our results generalize earlier result of Folland.
A Cauchy--Riemann-Type CK Extension for Monogenic Functions
本文提出与双轴分解相关的广义柯西-科瓦列夫斯卡娅延拓，通过单演层分解构造算子CK_{z,k}，刻画双轴单演函数类，并研究其在齐次多项式上的作用，得到双轴单演平面波及其偏微分方程组。
This paper aims at studying a functional $K$-transformation $w\left( z \right)\to \widetilde{w}\left( z \right)=w\left( z \right)K\left( z \right)$ that is made to reconsider the complex differentiability for a given complex function $w$ and subsequently we obtain structural holomorphic to judge a complex function to be complex structural differentiable. Since $K\left( z \right)$ can be chosen arbitrarily, thus it has greatly generalized the applied practicability. And we particularly consider $K \left( z \right)= 1+κ\left( z \right)$, then we found an unique Carleman-Bers-Vekua equations whic…
In our recent work [25] we showed that $C^\infty$ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in $\mathbb C^{2}$ are not analytic in general. This result raised again the question on the nature of CR-maps of real-analytic hypersurfaces. In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the {\em sectorial analyticity property}. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeare…
The $k$-Cauchy-Fueter operators, $k=0,1,\ldots$, are quaternionic counterparts of the Cauchy-Riemann operator in the theory of several complex variables. The weighted $L^2$ method to solve Cauchy-Riemann equation is applied to find the canonical solution to the non-homogeneous $k$-Cauchy-Fueter equation in a weighted $L^2$-space, by establishing the weighted $L^2$ estimate. The weighted $k$-Bergman space is the space of weighted $L^2$ integrable functions annihilated by the $k$-Cauchy-Fueter operator, as the counterpart of the Fock space of weighted $L^2$-holomorphic functions on $\mathbb{C}^n…
We introduce the Nonlinear Cauchy-Riemann equations as Bäcklund transformations for several nonlinear and linear partial differential equations. From these equations we treat in details the Laplace and the Liouville equations by deriving general solution for the nonlinear Liouville equation. By Möbius transformation we relate solutions for the Poincare model of hyperbolic geometry, the Klein model in half-plane and the pseudo-sphere. Conformal form of the constant curvature metrics in these geometries, stereographic projections and special solutions are discussed. Then we introduce the hyperbo…
For certain real hypersurfaces in the projective space, of signature (1,n), we study the filling problem for small deformations of the CR structure (the other signatures being well understood). We characterize the deformations which are fillable, and prove that they have infinite codimension in the set of all CR structures. This result generalizes the cases of the 3-sphere and of signature (1,1) to higher dimension.
Let $M$ be a compact abstract $CR$ manifold of arbitrary $CR$ codimension. Under certain conditions on the Levi form we prove the infinite dimensionality of some global cohomology groups of $M$.
We have obtained the explicit versions and precisions for the Hodge-Riemann decomposition of formes on affine algebraic curve V. The main application consists in the construction of Faddeev-Green function for Laplacian on V. Basing on this [HM](arXiv:0804.3951 and J.Geom.Anal., 2008,18), we extended from the case X in C to the case of bordered Riemann surface X in V the R.Novikov (1988) scheme for the effective reconstruction of conductivity function sigma on X through Dirichlet-to-Neumann mapping on bX for solutions of d(sigma d^cU)=0. In Sec.4 we give a correction of the paper [HM].
On the surjectivity of the Cauchy-Riemann and Laplace operators on weighted spaces of smooth functions
本文研究了在特定加权空间上Cauchy-Riemann和拉普拉斯算子的满射性，并通过增长条件确定其满射条件。
Classic complex analysis is built on structural function $K=1$ only associated with Cauchy-Riemann equations, subsequently various generalizations of Cauchy-Riemann equations start to break this situation. The goal of this article is to show that only structural function $K=Const$ such that Liouville's theorem is held, otherwise, it's not valid any more on complex domain based on structural holomorphic, the correction should be $w=Φ{{e}^{-K}}$, where $Φ=Const$. Those theories in complex analysis which keep constant are unable to be held as constant in the framework of structural holomorphic. S…
We prove that if two real-analytic hypersurfaces in $\mathbb C^2$ are equivalent formally, then they are also $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$ are algebraic (in particular are convergent). The result is obtained by using the recent {\em CR - DS technique}, connecting degenerate CR-manifolds and Dynamical Systems, and employing subsequently the {\em multisummability theory} of divergent power series used in the Dynamical Systems theory.
Let $X$ be a CR manifold with transversal, proper CR $G$-action. We show that $X/G$ is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorises uniquely over a holomorphic map on $X/G$. We then use this result and complex geometry to proof an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal $G \rtimes S^1$-action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.
In this article we provide a generalized version of the result of L.H. Son and W. Tutschke \cite{tut} on the solvability of first order systems on the plane whose initial functions are arbitrary holomorphic functions. This is achieved by considering the more general concept of holomorphicity with respect to the structure polynomial $X^2 + βX + α$. It is shown that the Son-Tutschke lemma on the construction of complex linear operators associated to the Cauchy-Riemann operator remains valid when interpreted for a large class of real parameters $α$ and $β$ including the elliptic case but also cas…
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 article, we consider metrically thin singularities A of the tangential Cauchy-Riemann operator on smoothly embedded Cauchy-Riemann manifolds M. The main result states removability within the space of locally integrable functions on M under the hypothesis that the (dim M-2)-dimensional Hausdorff volume of A is zero and that the CR-orbits of M and M-A are comparable.
Surjectivity of real-linear Cauchy--Riemann operators: from the minimal Harder--Narasimhan slope to automatic transversality
本文针对实线性柯西-黎曼算子建立共形不变的L²满射性判据，利用极小Harder–Narasimhan斜率得到渐近判据，提出滤过原理简化满射性问题，为伪全纯曲线提供高阶自动横截性判据并应用于标准近凯勒6-球面。
Generalized Cauchy-Riemann equations and relevant PDE
本文综述 Beltrami 方程理论向广义 Cauchy-Riemann 方程的推广，并梳理其与矩阵值系数 $A$-调和方程及经典边值问题解理论的关系。
In [8] it was shown that the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we demonstrate a fundamental property of this operator: it transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a "complicated" Vekua equation from positive formal powers and a generating sequence of "simpler" Vekua equation. Similar results is obta…
Cauchy formulae and Hardy spaces in discrete octonionic analysis
该研究推进离散八元数分析基础构建，建立离散八元数版Borel-Pompeiu公式与柯西积分公式，构造离散单演八元数柯西变换，并借此定义上下半空间离散八元数哈代空间与普莱姆利投影公式。
This paper extends approach of recent author's paper devoted to special classes of exact solutions of the static Maxwell system in inhomogeneous isotropic media and new generalizations of the Cauchy-Riemann system in $\mathbb R^3$. Two families of generalizations of the Cauchy-Riemann system with variable coefficients in $\mathbb R^4$ are presented in the context of non-Euclidean geometry. Analytic models of a wide range of potential meridional vector fields in $\mathbb R^4$ are characterized using a family of Vekua type systems in cylindrical coordinates. The specifics of meridional fields al…
The $k$-Cauchy-Fueter operators and complexes are quaternionic counterparts of the Cauchy-Riemann operator and the Dolbeault complex in the theory of several complex variables. To develop the function theory of several quaternionic variables, we need to solve the non-homogeneous $k$-Cauchy-Fueter equation over a domain under the compatibility condition, which naturally leads to a Neumann problem. The method of solving the $\overline{\partial}$-Neumann problem in the theory of several complex variables is applied to this Neumann problem. We introduce notions of $k$-plurisubharmonic functions an…
This article is concerned with the geometry of germs of real analytic surfaces in $(\mathbb{C}^2,0)$ having an isolated Cauchy-Riemann (CR) singularity at the origin. These are perturbations of {\it Bishop quadrics}. There are two kinds of CR singularities stable under perturbation~: {\it elliptic} and {\it hyperbolic}. Elliptic case was studied by Moser-Webster \cite{moser-webster} who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to {\it normal form} from which lots of geometric features can be read off. In this article we focus on perturbations o…
Fourier coefficients of normalized Cauchy transforms
本文研究规范化 Cauchy 变换 Fourier 系数的唯一性问题，通过同时逼近与循环内函数存在性分析，给出 Kahane–Katzenelson 定理的新非概率证明，并证明 de Branges–Rovnyak 空间中不存在一致容许控制函数。
We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy-Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of critical…
We study the behavior of the Riemann zeta function on the critical line when the imaginary part of the argument is sampled by the Cauchy random walk. We develop a complete second order theory for the corresponding system of random variables and show that it behaves almost like a system of non-correlated variables. Exploiting this fact in relation with known criteria for almost sure convergence allows to investigate its almost sure asymptotic behavior.
On the pointwise existence of Cauchy $\rm{P.V.}$ integrals
本文扩展了标量 Riemann-Hilbert 理论中的 Sokhotski-Plemelj 公式，在比 Dini 连续性更弱的逐点连续性与 $L^1$ 条件下证明了该公式，并给出了 Cauchy $\ m{P.V.}$ 积分存在的充要条件。
A Riemann--Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals
本文提出一种基于 Riemann–Hilbert 表示的 O(N) 数值方法，用于计算不相交区间上正交多项式的递推系数、逐点值及 Cauchy 变换，并展示其在可积系统与逼近论中的应用。
Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian are bounded below by a positive constant provided the CR Paneitz operator is non-negative and the Webster curvature is positive. Our lower bound for the non-zero eigenvalues is sharp and is attained on S^3. A consequence of our lower bound is that all compact CR 3-manifolds with non-negative CR Paneitz operator and positive CR Yamabe constant are embeddable. N…
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