Extending external rays throughout the Julia sets of rational maps

For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of the form Fλ(z) = z ⁿ + λ/z ⁿ where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology of the Julia set.     

Carleson measures and the BMO space on the p ‐adic vector space

For a prime number p, let Q _p be the p ‐adic field and let Q _p ^d denote a vector space over Q _p which consists of all d ‐tuples of Q _p . Then we study the p ‐adic version of the Calderón–Zygmund decomposition, Carleson measures on the vector space Q _p ^{d +1} and the space BMO ( Q _p ^d ) of functions of bounded mean oscillation on Q _p ^d . In particular, it turns out that the operator norms of various oncoming operators are independent of the dimension d and the prime number p, which is one of the big differences from that of the Euclidean case. Interestingly, the independence of the dimension d and p makes it possible to develop Harmonic Analysis on the infinite dimensional p ‐adic vector space as the importance had already been pointed out in the Euclidean case (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sufficient conditions for the projective freeness of Banach algebras

Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H^∞(X) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.     

Existence of Solutions and Star-shapedness in Minty Variational Inequalities

Minty Variational Inequalities (for short, Minty VI) have proved to characterize a kind of equilibrium more qualified than Stampacchia Variational Inequalities (for short, Stampacchia VI). This conclusion leads to argue that, when a Minty VI admits a solution and the operator F admits a primitive f (that is F= f′), then f has some regularity property, e.g. convexity or generalized convexity. In this paper we put in terms of the lower Dini directional derivative a problem, referred to as Minty VI(f′_,K), which can be considered a nonlinear extension of the Minty VI with F=f′ (K denotes a subset of ℝⁿ). We investigate, in the case that K is star-shaped, the existence of a solution of Minty VI(f’_,K) and increasing along rays starting at x^* property of (for short, F ɛIAR (K,x^*)). We prove that Minty VI(f’_,K) with a radially lower semicontinuous function fhas a solution x^* ɛker K if and only if FɛIAR(K, x^*). Furthermore we investigate, with regard to optimization problems, some properties of increasing along rays functions, which can be considered as extensions of analogous properties holding for convex functions. In particular we show that functions belonging to the class IAR(K,x^*) enjoy some well-posedness properties.     

Harmonic Analysis on a Galois Field and Its Subfields

Complex functions χ(m) where m belongs to a Galois field GF(p ^ℓ ), are considered. Fourier transforms, displacements in the GF(p ^ℓ )×GF(p ^ℓ ) phase space and symplectic transforms of these functions are studied. It is shown that the formalism inherits many features from the theory of Galois fields. For example, Frobenius transformations and Galois groups are introduced in the present context. The relationship between harmonic analysis on GF(p ^ℓ ) and harmonic analysis on its subfields, is studied.      

Group Classification of Eikonal Equations for the Wave Equation in Inhomogeneous Media

We present a group classification of the family of equations of the form (∇ψ)² = 1/v ²(x, y, z), called the eikonal equations, which are equations of characteristics for equations describing acoustic and electromagnetic waves in inhomogeneous media. For some cases of equations with linear and quadratic functions v(x, y, z), having the sense of the speed of waves in a medium, we present explicit solutions (eikonals of point sources) and completely describe the geometry of rays.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

On the second conjugate of several convex functions in general normed vector spaces

When dealing with convex functions defined on a normed vector space X the biconjugate is usually considered with respect to the dual system (X, X ^*), that is, as a function defined on the initial space X. However, it is of interest to consider also the biconjugate as a function defined on the bidual X ^**. It is the aim of this note to calculate the biconjugate of the functions obtained by several operations which preserve convexity. In particular we recover the result of Fitzpatrick and Simons on the biconjugate of the maximum of two convex functions with a much simpler proof.      

General Littlewood–Paley functions and singular integral operators on product spaces

This paper is devoted to the study on the L^p ‐mapping properties for certain singular integral operators with rough kernels and related Littlewood–Paley functions along “polynomial curves” on product spaces ?^m × ?ⁿ (m ≥ 2, n ≥ 2). By means of the method of block decomposition for kernel functions and some delicate estimates on Fourier transforms, the author proves that the singular integral operators and Littlewood–Paley functions are bounded on L^p (?^m × ?ⁿ ), p ∈ (1, ∞), and the bounds are independent of the coefficients of the polynomials. These results essentially improve or extend some well‐known results. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Product of functions in BMO and <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> in non-homogeneous spaces

Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition and may not be doubling, we define the product of functions in the regular BMO and the atomic block H ¹ in the sense of distribution, and show that this product may be split into two parts, one in L ¹ and the other in some Hardy-Orlicz space.     

Entropy,Approximation Quantities and the Asymptotics of the Modulus of Continuity

The paper deals with the approximation of bounded real functions f on a compact metric space (X, d) by so-called controllable step functions in continuation of [Ri/Ste]. These step functions are connected with controllable coverings, that are finite coverings of compact metric spaces by subsets whose sizes fulfil a uniformity condition depending on the entropy numbers ε_n(X) of the space X. We show that a strong form of local finiteness holds for these coverings on compact metric subspaces of IR^m and S^m. This leads to a Bernstein type theorem if the space is of finite convex information. In this case the corresponding approximation numbers ε_n(f) have the same asymptotics its ω(f, ε_n(X)) for f ε C(X). Finally, the results concerning functions f ε M(X) and f ε C(X) are transferred to operators with values in M(X) and C(X), respectively.     

Approximate identities in variable L p spaces

We give conditions for the convergence of approximate identities, both pointwise and in norm, in variable L ^p spaces. We unify and extend results due to Diening [8], Samko [18] and Sharapudinov [19]. As applications, we give criteria for smooth functions to be dense in the variable Sobolev spaces, and we give solutions of the Laplace equation and the heat equation with boundary values in the variable L ^p spaces. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

Boundedness of Commutators with Lipschitz Functions in Non-homogeneous Spaces

Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition, the authors prove that for a class of commutators with Lipschitz functions which include commutators generated by Calderon-Zygmund operators and Lipschitz functions as examples, their boundedness in Lebesgue spaces or the Hardy space H1 (μ) is equivalent to some endpoint estimates satisfied by them. This result is new even when the underlying measureμis the d-dimensional Lebesgue measure.     

Spectral expansion on the entire axis of the green function for a three-layer medium in fundamental functions of a nonself-adjoint Sturm-Liouville operator

We give a new representation of the Green function in the space R ² for the Helmholtz equation with coefficient that is a complex-valued piecewise constant function depending on a single variable and taking three values. This representation has the form of an expansion in fundamental functions, i.e., bounded (on the entire line R ¹) solutions of the Sturm-Liouville equation with a complex coefficient. The spectrum consists of two rays parallel to the real axis in the complex plane of the spectral parameter. The origin of the rays is determined by constants characterizing the coefficient in the equation.     

Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations

This article is concerned with some properties of Stepanov-like almost automorphic (S ^p -a.a.) functions. We establish a composition theorem about S ^p -a.a. functions, and with its help, study the existence and uniqueness of almost automorphic solutions for semilinear evolution equations in Banach spaces. Moreover, integration and differentiation of S ^p -a.a. functions are discussed. Some theorems extend earlier results.     

Carleson measure characterization of Bloch functions

We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized Carleson measure;f B ₀ if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.     

Absolutely continuous Hardy–Sobolev functions in the unit disc

Let f(z) be an analytic function defined in the unit disc whose fractional derivative of order belongs to H^p, 0<p1. We show that as a consequence of a monotonicity condition on the decay of the Taylor coefficients, it is possible to improve the usual radial boundary growth estimate for H^p functions by a logarithmic factor. As a consequence we show that under certain regularity conditions imposed on the decay and oscillations of the absolute values of the function's Taylor coefficients, it is possible to estimate the function's modulus of continuity and modulus of absolute continuity and that a consequence of this is that as p→0, these functions will be generally smoother. Examples are also given of Hardy–Sobolev functions having modulus of absolute continuity different than modulus of continuity.     

Some notes on densely continuous forms

We consider a special space of set-valued functions (multifunctions), the space of densely continuous forms D(X, Y) between Hausdorff spaces X and Y, defined in [HAMMER, S. T.—McCOY, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266] and investigated also in [HOLá, L’.: Spaces of densely continuous forms, USCO and minimal USCO maps, Set-Valued Anal. 11 (2003), 133–151]. We show some of its properties, completing the results from the papers [HOLY, D.—VADOVIČ, P.: Densely continuous forms, pointwise topology and cardinal functions, Czechoslovak Math. J. 58(133) (2008), 79–92] and [HOLY, D.—VADOVIČ, P.: Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps, Acta Math. Hungar. 116 (2007), 133–144], in particular concerning the structure of the space of real-valued locally bounded densely continuous forms D _p ^*(X) equipped with the topology of pointwise convergence in the product space of all nonempty-compact-valued multifunctions. The paper also contains a comparison of cardinal functions on D _p ^*(X) and on real-valued continuous functions C _p (X) and a generalization of a sufficient condition for the countable cellularity of D _p ^*(X). This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904 and by the Eco-Net (EGIDE) programme of the Laboratoire de Mathématiques de l’Université de Saint-Etienne (LaMUSE), France.     

Boundedness of Marcinkiewicz integrals and their commutators in H1 (?n × ?m)

The authors establish the boundedness of Marcinkiewicz integrals from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ¹(ℝ ⁿ × ℝ ^m ) and their commutators with Lipschitz functions from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ^q (ℝ ⁿ × ℝ ^m ) for some q > 1.     

The eigen functions of the complex projective space

In the complexn-dimensional projective spaceCP ⁿ , let λ _p (=4p(p+n)) be the eigen value of the Laplace-Beltrami operator andH _p be the space of all eigen functions of eigen value λ _p . The reproducing kernelh _p (z, w) ofH _p is constructed explicitly in this paper, and a system of complete orthogohal functions ofH _p is constructed fromh _p (z,w)(p=1,2, …). Partially supported by NSF of China