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 obtained regarding construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theory are considered and examples are given to illustrate the procedure.
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.
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.
A bicomplex proportional fractional $(\vartheta,φ)-$weighted Cauchy-Riemann operator using Riemann-Liouville derivatives with respect to an hyperbolic-valued function
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.
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 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.
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 $p \in [1, \infty]$.
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…
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.
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…
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…
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…
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…
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…
