SOME ENDPOINT ESTIMATES FOR COMMUTATORS OF θ-TYPE CALDERON-ZYGMUND OPERATORS

In this note, the authors prove that the commutator generated by θ-type Calderón-Zygmund operator T and a Lipschitz function is bounded from L ^p (R ⁿ ) into (R ⁿ ) and also maps from (R ⁿ ) into BMO (R ⁿ ). Supported by NSFC(10571014), NSFC(10571156), the Doctor Foundation of Jxnu (2443), the Natural Science Foundation of Jiangxi province(2008GZS0051).     

Two-side estimates for sums of absolute values of Fourier coefficients of functions from Hω(Tm)

The class of functions H ^ω is considered, where ω(t) is a continuity modulus monotone in the sense of Hardy and satisfying some condition C. The behavior of the value is obtained, where is the sum of absolute values of Fourier coefficients of a function f ∈ L(T ^m ) in pth power. Original Russian Text ? D.M. D’yachenko, 2008, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2008, Vol. 63, No. 3, pp. 19–26.     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

Some characterizations of weighted Herz-type Hardy spaces and their applications

In this paper, the authors first establish some new real-variable characterizations of Herz-type Hardy spaces and , where ω₁,ω₃ ∈ A₁-weight, 1<q>∞,n(1−1/q)≤α<∞ and 0<p<∞. Then, using these new characterizations, they investigate the convergence of a bounded set in these spaces, and study the boundedness of some potential operators on these spaces. Supported by the NNSF of China     

Vive la différence III

We show that, consistently, there is an ultrafilter on ω such that if N _n^ℓ = (P _n^ℓ ∪ Q _n^ℓ, P _n^ℓ, Q _n^ℓ, R _n^ℓ) (for ℓ = 1, 2, n < ω), P _n^ℓ ∪ Q _n^ℓ ⊆ ω, and are models of the canonical theory t ^ind of the strong independence property, then every isomorphism from onto is a product isomorphism. The first version of this work done in 93; First typed: May 1993. This research was partially supported by the United States-Israel Binational Science Foundation. Publication 509     

The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from L P( · )(Γ)

We consider the Riemann–Hilbert problem in the following setting: find a function whose boundary values ϕ⁺(t) satisfy the condition a.e. on Γ. Here D is a simply connected domain bounded by a simple closed curve Γ, and K ^{p( · )}(D;ω) is the set of functions ϕ(z) representable in the form , where ω(z) is a weight function and (K _Γφ )(z) is a Cauchy type integral whose density φ is integrable with a variable exponent p(t). It is assumed that Γ is a piecewise-Lyapunov curve without zero angles, ω(z) is an arbitrary power function and p(t) satisfies the Log-H?lder condition. The solvability conditions are established and solutions are constructed. These solutions largely depend on the coefficients a, b, c, the weight ω, on the values of p(t) at the angular points of Γ and on the values of angles at these points. Submitted: May 13, 2007. Revised: August 8, 2007 and August 28, 2007. Accepted: November 8, 2007.     

Borel hierarchies in infinite products of Polish spaces

Let H be a product of countably infinite number of copies of an uncountable Polish space X. Let Σ_ξ be the class of Borel sets of additive class ξ for the product of copies of the discrete topology on X (the Polish topology on X), and let . We prove in the Lévy-Solovay model that

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> boundedness for multilinear fractional integral on product spaces

For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by

The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π＇ be automorphic irreducible cuspidal representations of GLm（QA） and GLm′ （QA）, respectively, and L（s, π×π′） be the Rankin-Selberg L-function attached to π and π＇. Without assuming the Generalized Ramanujan Conjecture （GRC）, the author gives the generalized prime number theorem for L（s, π × π′） when π =π＇. The result generalizes the corresponding result of Liu and Ye in 2007.     

Fine Properties of the Integrated Density of States and a Quantitative Separation Property of the Dirichlet Eigenvalues

We consider one-dimensional difference Schr?dinger equations with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V ₀(x) of degree k ₀, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval , does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k ₀ of them fall into any disk of radius exp. In addition, we obtain the lower bound (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies. Received: February 2006, Accepted: December 2007     

Roots of C*-algebra elements under numerical range condition

Let be a C*-algebra with unit 1. For each a ∈ , the C*-algebra numerical range is defined by V(a) = {φ(a): φ ∈ , φ ≥ 0,φ(1) = 1}. In a 2003 paper Li, Rodman and Spitkovsky have found the ω-th roots of elements in C*-algebra under a numerical range condition, when ω ∈ [1,∞). In this paper, we will give a short proof of the above result in the case of ω is a positive integer number. We also give a simple proof for ω-th root of an element a ∈ , when ω ∈ [1,∞) and V(a)∩ {z ∈ ℂ: z ≤ 0} = . The first author was supported by the Shiraz university Research Council Grant No. 86-GRSC-32.     

The hardy and bellman operators in spaces connected with H( \mathbb{T} ) and BMO( \mathbb{T} )

Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series $ \sum\limits_{n = 1}^\infty {a_n } Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series cos nx. According to one result of G.H. Hardy, the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. But if 1 < p ≤ ∞ and f ∈ L ^p [0, π], then the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. Similar assertions are true for sine series. This allows one to define the Hardy operator on L ^p (), 1 ≤ p < ∞, and to define the Bellman operator on L ^p (), 1 < p ≤ ∞. In this paper we prove that the Bellman operator boundedly acts in VMO(), and the Hardy operator also maps a certain subspace C() onto VMO(). We also prove the invariance of certain classes of functions with given majorants of modules of continuity or best approximations in the spaces H(), L(), VMO() with respect to the Hardy and Bellman operators. Original Russian Text ? S.S. Volosivets and B.I. Golubov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 4–13.     

Deterministic extractors for affine sources over large fields

An (n,k)-affine source over a finite field is a random variable X = (X ₁,..., X _n ) ∈ , which is uniformly distributed over an (unknown) k-dimensional affine subspace of . We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n ^c (where c is a large enough constant). Our main results are as follows:

Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Let be a separable Hilbert space, an open convex subset, and f: a smooth map. Let Ω be an open convex set in with , where denotes the closure of Ω in . We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ , where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L ²-space under the flow of a class of kinetic transport equations so called BGK model.      

On potentially nilpotent double star sign patterns

A matrix whose entries come from the set {+, −, 0} is called a sign pattern matrix, or sign pattern. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, the characterization problem for some potentially nilpotent double star sign patterns is discussed. A class of double star sign patterns, denoted by (m, 2), is introduced. We determine all potentially nilpotent sign patterns in (3, 2) and (5, 2), and prove that one sign pattern in (3, 2) is potentially stable. Supported by youth scientific funds of the Education Department of Jiangxi Province (No. GJJ09460), Natural Science Foundation of Jiangxi Normal University (No.2058), National Natural Science Foundation of China (No.10601001).     

On some new criteria for infinite differentiability of periodic functions

We study the set of infinitely differentiable periodic functions in terms of generalized -derivatives defined by a pair of sequences ψ ₁ and ψ ₂. It is shown that every function ƒ from the set has at least one derivative whose parameters ψ ₁ and ψ ₂ decrease faster than any power function and, at the same time, for any function ƒ ∈ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ ₁ and ψ ₂ have the same rate of decrease and for which the -derivative no longer exists. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1399–1409, October, 2007.     

The \bar \partial -Neumann operator and commutators of the Bergman projection and multiplication operators

We prove that compactness of the canonical solution operator to restricted to (0, 1)-forms with holomorphic coefficients is equivalent to compactness of the commutator defined on the whole L _(0,1)²(Ω), where is the multiplication by and is the orthogonal projection of L _(0,1)²(Ω) to the subspace of (0, 1) forms with holomorphic coefficients. Further we derive a formula for the -Neumann operator restricted to (0, 1) forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by z and . Partially supported by the FWF grant P19147-N13.     

Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ₀(x) satisfies |d ϕ₀(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ² (resp. ℝ²×T) The limit,v(t,x), of the corresponding velocity fields {v ^ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω₀(x). Moreover, (ℤ²)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.     

Algebraic Polynomials with Non-identical Random Coefficients

The asymptotic estimate for the expected number of real zeros of a random algebraic polynomial is known. The identical random coefficients a_j(ω) are normally distributed defined on a probability space , ω ∈Ω. The estimate for the expected number of zeros of the derivative of the above polynomial with respect to x is also known, which gives the expected number of maxima and minima of Q_n(x, ω). In this paper we provide the asymptotic value for the expected number of zeros of the integration of Q_n(x,ω) with respect to x. We give the geometric interpretation of our results and discuss the difficulties which arise when we consider a similar problem for the case of .     

Duals of holomorphic Besov spaces on the polydisks and diagonal mappings

Let U ⁿ be the unit polydisk in C ⁿ and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω ₁, ..., ω _n ), ω _j ∈ S(1 ≤ j ≤ n) and f ∈ H(U ⁿ ). The function f is said to be in holomorphic Besov space B _p (ω) if