N_(ψ,φ)-type Quotient Modules over the Bidisk

Let H~2■ be the Hardy space over ~the bidisk ■, and let M_(ψ,φ)=[(ψ(z)-φ(w))~2] be the submodule generated by(ψ(z)-φ(w))~2, where ψ(z) and φ(w) are nonconstant inner functions.The related quotient module is denoted by ■. In this paper, we give a complete characterization for the essential normality of N_(ψ,φ). In particular, if ψ(z) = z, we simply write M_(ψ,φ)and N_(ψ,φ) as M_φ and N_φ respectively. This paper also studies compactness of evaluation operators L(0)|N_φand R(0)|N_φ, essential spectrum of compression operator S_z on N_φ, essential normality of compression operators S_z and S_w on N_φ.     

A Dominated Theorem on 5L(2, R) and Its Application

In this paper, we give the following dominated theorem: Let φ(g) ∈ L1(G//K),φε(t)=ε> 0, and the least radical decreasing dominatedfunction φ(t) = sup |φ(y)| ∈L1(G//K). If shtφ(t) is monotonically decreasingon (0, ∞), then for any f∈L1loc(G//K) , the following inequality holds:sup |φε * f(x)| ≤ Cmf(x),where mf(x) is the Hardy-Littlewood maximal function of f, and C = ||φ||1.An application of this dominated theorem is also given.     

ALGEBRAIC EXTENSION OF ＊-A OPERATOR

In this paper, we study various properties of algebraic extension of *-A operator.Specifically, we show that every algebraic extension of *-A operator has SVEP and is isoloid.And if T is an algebraic extension of *-A operator, then Weyl's theorem holds for f(T), where f is an analytic functions on some neighborhood of σ(T) and not constant on each of the components of its domain.     

DIFFERENTIAL OPERATORS OF INFINITE ORDER IN THE SPACE OF RAPIDLY DECREASING SEQUENCES

We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it.The object of this article is to study the equivalence of a differential operator of infinite order;that is φ(D) =sum from k=0 to ∞φ_κD~κ.φ_κ constant numbers an a power of D.D~n,meaning,is there a isomorphism X(from s onto s) such that X_φ(D) = D~nX?.We prove that if φ(D) is equivalent to D~n,then φ(D) is of finite order,in fact a polynomial of degree n.The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n=1.     

Hardy-Orlicz 空间到 Bloch-Orlicz 型空间上的复合算子

Let φ be an analytic self-map of D. The composition operator C_φ is the operator defined on H(D) by C_φ(f) = f ? φ. In this paper, we investigate the boundedness and compactness of the composition operator C_φ from Hardy-Orlicz spaces to Bloch-Orlicz type spaces.     

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R~n×R~m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R~n× R~m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R~n× R~m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R~n× R~m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R~n× R~m) to L~φ(R~n× R~m)and from H~φ_A(R~n×R~m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R~n× R~m and are new even for classical product Orlicz-Hardy spaces.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

Property(ω_1) and Single Valued Extension Property

In this note we study the property(ω1),a variant of Weyl's theorem by means of the single valued extension property,and establish for a bounded linear operator defined on a Banach space the necessary and sufficient condition for which property(ω1) holds.As a consequence of the main result,the stability of property(ω1) is discussed.     

Polycyclic groups admitting a regular automorphism of order four

Let G be a polycyclic group and α a regular automorphism of order four of G. If the map φ: G→ G defined by g~φ= [g, α] is surjective, then the second derived group of G is contained in the centre of G. Abandoning the condition on surjectivity, we prove that C_G(α~2) and G/[G, α~2] are both abelian-by-finite.     

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L~p(R)and the Hardy space H~1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H~p(R) with 0 p 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H~p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H~1(R).     

WEYL＇S TYPE THEOREMS AND HYPERCYCLIC OPERATORS

For a bounded operator T acting on an infinite dimensional separable Hilbert space H,we prove the following assertions: (i) If T or T* ∈ SC,then generalized aBrowder's theorem holds for f(T) for every ...     

Nonuniform sampling and approximation in Sobolev space from perturbation of the framelet system

The Sobolev space H~?(R~d), where ? d/2, is an important function space that has many applications in various areas of research. Attributed to the inertia of a measurement instrument, it is desirable in sampling theory to recover a function by its nonuniform sampling. In the present paper, based on dual framelet systems for the Sobolev space pair(H~s(R~d), H~(-s)(R~d)), where d/2 s ?, we investigate the problem of constructing the approximations to all the functions in H~?(R~d) by nonuniform sampling. We first establish the convergence rate of the framelet series in(H~s(R~d), H~(-s)(R~d)), and then construct the framelet approximation operator that acts on the entire space H~?(R~d). We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters, and obtain an estimate bound for the perturbation error. Our result shows that under the condition d/2 s ?, the approximation operator is robust to shift perturbations. Motivated by Hamm(2015)'s work on nonuniform sampling and approximation in the Sobolev space, we do not require the perturbation sequence to be in ?~α(Z~d). Our results allow us to establish the approximation for every function in H~?(R~d) by nonuniform sampling. In particular, the approximation error is robust to the jittering of the samples.     

A Simple Proof of Dieudonn-Manin Classification Theorem

The Dieudonn-Manin classification theorem on φ-modules (φ-isocrystals) over a perfect field plays a very important role in p-adic Hodge theory. In this note, in a more general setting we give a new proof of this result, and in the course of the proof, we also give an explicit construction of the Harder-Narasimhan filtration of a φ-module.     

Hardy-type inequalities on strong and weak Orlicz-Lorentz spaces

In the present paper, the characterization of strong-type modular inequality ∫ 0 ∞φ(Sf (t))w(t)dt≤∫0∞ φ(Cf (t))w(t)dt, f↓ is given, where φ∈Δ’ and S is a Hardy operator. Furthermore, the equivalent conditions of modular inequalities and norm inequalities related to weak Orlicz-Lorentz spaces are researched. We also explore the conditions for Orlicz-Lorentz spaces and weak Orlicz-Lorentz spaces to be normable. Finally, the weak boundedness of certain Hardy-type operators on Orlicz-Lorentz spaces is studied.     

CALDER6N-ZYGMUND OPERAORS ON THE HARDY SPACES OF WEIGHTED HERZ TYPE

Applying the decomposition theorems in [1] and [2] , we obtain the boundedness theorem of Calderbn-Zygmund operator of type 6 on the Hardy spaces of weighted Herz type and establish interpolation theorem of linear operators on the weighted Herz spaces. -     

Some limit results on supremum of Shepp statistics for fractional Brownian motion

Define the incremental fractional Brownian field Z_H(τ,s)=B_H(s+τ)-B_H(s),where B_H(s) is a standard fractional Brownian motion with Hurst parameter H ∈(0,1).In this paper,we first derive an exact asymptotic of distribution of the maximum M_H(T_u)=sup_τ∈[0,1],s∈[0,xT_u]Z_H(τ,s),which holds uniformly for x ∈[A,B]with A,B two positive constants.We apply the findings to analyse the tail asymptotic and limit theorem of MH(τ) with a random index τ.In the end,we also prove an ahnost sure limit theorem for the maximum M_(1/2)(T) with non-random index T.     

$(\omega_1)$性质与单值延拓性质

In this note we study the property（ω1）,a variant of Weyl＇s theorem by means of the single valued extension property,and establish for a bounded linear operator defined on a Banach space the necessary and sufficient condition for which property（ω1） holds.As a consequence of the main result,the stability of property（ω1） is discussed.     

Integral Operators Between Fock Spaces∗

In this paper, the authors study the integral operator Sφf(z) = Z C φ(z, w)f(w)dλα(w) induced by a kernel function φ(z, ·) ∈ F ∞α between Fock spaces. For 1 ≤ p ≤ ∞, they prove that Sφ : F 1 α → F p α is bounded if and only if sup a∈C kSφkakp,α < ∞, (?) where ka is the normalized reproducing kernel of F 2 α; and, Sφ : F 1 α → F p α is compact if and only if lim |a|→∞ kSφkakp,α = 0. When 1 < q ≤ ∞, it is also proved that the condition (?) is not sufficient for boundedness of Sφ : F q α → F p α . In the particular case φ(z, w) = eαzw?(z ? w) with ? ∈ F 2 α, for 1 ≤ q < p < ∞, they show that Sφ : F p α → F q α is bounded if and only if ? = 0; for 1 < p ≤ q < ∞, they give sufficient conditions for the boundedness or compactness of the operator Sφ : F p α → F q α.     

Sz\`{a}sz-Mirakjan 算子拟中插式的点态逼近等价定理

Recently some classical operator quasi-interpolants were introduced to obtain much faster convergence. A.T. Diallo investigated some approximation properties of Szasz-Mirakjan Quasi-Interpolants, but he obtained only direct theorem with Ditzian-Totik modulus wφ^2r （f, t）. In this paper, we extend Diallo＇s result and solve completely the characterization on the rate of approximation by the method of quasi-interpolants to functions f ∈ CB[0, ∞） by making use of the unified modulus wφ^2r（f, t） （0≤λ≤ 1）.     

Operators on the orthogonal complement of the Dirichlet space(Ⅱ)

In this paper we first prove that a dual Hankel operator R φ on the orthogonal complement of the Dirichlet space is compact for φ∈ W 1,∞(D),and then that a semicommutator of two Toeplitz operators on the Dirichlet space or two dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space is compact.We also prove that a dual Hankel operator R φ with φ∈ W 1,∞(D) is of finite rank if and only if B φ is orthogonal to the Dirichlet space for some finite Blaschke product B,and give ...