Polynomially bounded solutions to the Loewner differential equation in several complex variables

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z,t)=etAz+?, where A∈L(Cn,Cn) has the property m(A)>0. Here m(A)=min{R〈A(z),z〉:‖z‖=1}. We also give sufficient conditions for g(z,t)=L(f(z,t)) to be polynomially bounded, where f(z,t) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation.     

Solution of a Loewner chain equation in several complex variables

We find a solution to the Loewner chain equation in the case when the infinitesimal generator satisfies h(0,t)=0, Dh(0,t)=A for any A∈L(Cn,Cn) with m(A)>0. We also study the related classes of spirallike mappings, mappings with parametric representation and asymptotically spirallike mappings.     

Loewner chains and parametric representation in several complex variables

Let B be the unit ball of with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f(z,t) and the transition mapping v(z,s,t) associated to f(z,t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈B. Moreover, we prove that a mapping f∈H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e^−tf(z,t)}_t?0 is a normal family on B and f(z)=f(z,0) for z∈B. Also we show that univalent solutions f(z,t) of the generalized Loewner differential equation in higher dimensions are unique when {e^−tf(z,t)}_t?0 is a normal family on B. Finally we show that the set S⁰(B) of mappings which have parametric representation on B is compact.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

多复变数的二阶微分从属

在文[1－2]的基础上讨论了多复变数中的二阶微分从属,取得了一些结果。这些结果是文[1-5]中的一些结果的推广。     

Growth and approximation of generalized biaxially symmetric potentials in several complex variables

The paper deals with the aspect of the growth of entire functions of several complex variables. The growth of the entire functions with respect to each of the variables separately has been studied by defining partial order and partial type. Finally, we have studied the growth and polynomial approximation of entire function generalized biaxially symmetric potential with respect to each of the variables separately.     

Sine functional equation in several variables

Making use of some transparent convolutional approach we consider the sine functional equation in several variables. Received: 8 April 2005     

On approximation and generalized type of analytic functions of several complex variables

In the present paper, we study the polynomial approximation of analytic functions of several complex variables. The characterizations of generalized type of analytic functions of several complex variables have been obtained in terms of approximation and interpolation errors.     

Interval expansion method for nonlinear equation in several variables

An interval expansion method is presented to find the solution of nonlinear equation in several variables with guaranteed accuracy. In this method, the existence theorem for the solution of nonlinear equation is obtained by using two newly defined operators. A new algorithm for solving nonlinear equation in several variables is given. Furthermore, we prove that the new method is global convergent under mild conditions.     

Evolution families and the Loewner equation II: complex hyperbolic manifolds

We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds. M. D. Contreras and S. Díaz-Madrigal were partially supported by the Ministerio de Ciencia e Innovación and the European Union (FEDER), project MTM2006-14449-C02-01, by La Consejería de Educación y Ciencia de la Junta de Andalucía, and by the European Science Foundation Research Networking Programme HCAA.     

Toeplitz operators in several complex variables

Let S be the unit sphere in Cⁿ. We investigate the properties of Toeplitz operators on S, i.e., operators of the form Tφf = P(φf) where φ?L^∞(S) and P denotes the projection of L²(S) onto H²(S). The aim of this paper is to determine how far the extensive one-variable theory remains valid in higher dimensions. We establish the spectral inclusion theorem, that the spectrum of T_φ contains the essential range of φ, and obtain a characterization of the Toeplitz operators among operators on H²(S) by an operator equation. Particular attention is paid to the case where φ ? H^∞(S) + C(S) where C(S) denotes the algebra of continuous functions on S. Finally we describe a class of Toeplitz operators useful for providing counterexamples—in particular, Widom's theorem on the connectedness of the spectrum fails when n > 1.     

A generalized Stefan problem in several space variables

A multidimensional, multiphase problem of Stefan type, involving quasilinear parabolic equations and nonlinear boundary conditions is considered. Regularization techniques and monotonicity methods are exploited. Existence and uniqueness of a weak solution to the problem, as well as continuous and monotone dependence of the solution upon data are shown.     

Solutions for a generalized fractional anomalous diffusion equation

In this paper, we investigate the solutions for a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time-dependent diffusion coefficient and an external force into account, which subjects to the natural boundaries and the generic initial condition. We obtain explicit analytical expressions for the probability distribution and study the relation between our solutions and those obtained within the maximum entropy principle by using the Tsallis entropy.     

Parabolic starlike mappings in several complex variables

Let B ⁿ be the unit ball in with respect to an arbitrary norm. We obtain sharp growth and covering theorems for parabolic starlike mappings on B ⁿ as well as coefficient estimates, and consider various examples.     

Asymptotically spirallike mappings in several complex variables

In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space ℂ ⁿ . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric representation, where A ∈ L(ℂ ⁿ , ℂ ⁿ ), and prove that if dt < ∞ (this integral is convergent if k ₊(A) < 2m(A)), then a mapping f ∈ S(B ⁿ ) is A-asymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f(z, t) such that D f(0, t) = e ^At , {e ^−At f(·, t)} _t≥0 is a normal family on B ⁿ and f = f(·, 0). In particular, a spirallike mapping with respect to A ∈ L(ℂ ⁿ , ℂ ⁿ ) with dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A* = 2I _n , then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings. Partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221. Partially supported by Grant-in-Aid for Scientific Research (C) no. 19540205 from Japan Society for the Promotion of Science, 2007. Partially supported by Romanian Ministry of Education and Research, CEEX Program, Project 2-CEx06-11-10/2006.     

The Bloch theorem in several complex variables

The background theory for the Bloch theorem is generalized to several complex variables. This work involves study of the Bergman kernel functions in order to extend work of Landau and Bonk. The main conclusion is an estimate for Bloch’s constant for mappings of domains of the first classical type. In the special case of then-dimensional ball, the estimate of Bloch’s constant coincides with that of Liu.