Strongly singular convolution operators on the weighted Herz-type Hardy spaces

Let 0<p≤1<q<0, andw ₁ ,w ₂ ∈ A ₁ (Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the homogeneous weighted Herz-type Hardy spacesH K^{α, p} _q(w₁; w₂) to the homogeneous weighted Herz spacesK ^{α, p} _q (w₁; w₂), provided α=n(1−1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spacesH K ^{α, p} _q (w ₁;w ₂) is also investigated. Supported by the National Natural Science Foundation of China     

具有粗糙核的多线性算子在Herz型空间的有界性

§ 1　Introduction and main resultsLet b∈BMO(Rn) and T be a standard Calderon-Zygmund singular integral operator.Define the commutator[b,T] as follows.[b,T] f(x) =b(x) Tf(x) -T(bf) (x) .In [3 ] ,the boundedness ofthe commutator[b,T] wasestablished on Lp(Rn) .There are thesimilar results in [1 ,2 ] when the commutator was substituted with the multilinearoperators generated by the singular integral operator T and a Taylor series A(see thedefinition below) .Recently,many mathematicians h…     

Isolation and Component Structure in Spaces of Composition Operators

We establish a condition that guarantees isolation in the space of composition operators acting between H^p(B_N) and H^q(B_N), for 0 < p ≤ ∞, 0 < q < ∞, and N ≥ 1. This result will allow us, in certain cases where 0 < q < p ≤ ∞, completely to characterize the component structure of this space of operators.     

含有p拉普拉斯算子方程的解的存在性研究

By using the perturbation theories on sums of ranges for nonlinear accretive mappings of Calvert and Gupta (1978), the abstract result on the existence of a solution u ∈ Lp (Ω) to nonlinear equations involving p-Laplacian operator △p, where 2N/N 1 ＜ p ＜ ∞ and N (≥ 1 )denotes the dimension of RN,is studied. The equation discussed and the methods shown in the paper are continuation and complement to the corresponding results of Li and Zhen's previous papers. To obtain the result ,some new techniques are used.     

Optimal query error of quantum approximation on some Sobolev classes

We study the approximation of the imbedding of functions from anisotropic and general-ized Sobolev classes into Lq([0,1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from LpN to LNq , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wpr ([0,1]d)) to Lq([0,1]d) space for all 1 q,p ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.     

L p approximation by Bernstein-Kantorovich quasi-interpolants

Bernstein-Kantorovich quasi-interpolants K^（2r-1）n（f, x） are considered and direct, inverse and equivalence theorems with Ditzian-Totik modulus of smoothness ω^2rφ（f, t）p （1 ≤ p ≤＋∞） are obtained.     

变量核分数次积分在Hardy空间的有界性

§ 1　 Introduction and main resultsLet Sn- 1 be the unitsphere in Rn(n≥ 2 ) equipped with normalized Lebesgue measure dσ= dσ(z′) .We say that a functionΩ(x,z) defined on Rn× Rnbelongs to L∞ (Rn)× Lr(Sn- 1 )(r≥ 1 ) ,ifΩ(x,z) satisfies the following two conditions,(i) for any x,z∈Rnandλ>0 ,there hasΩ(x,λz) =Ω(x,z) ;(ii)‖Ω‖L∞(Rn)× Lr(Sn- 1) :=supx∈ Rn∫Sn- 1|Ω(x,z′) | rdσ(z′) 1 / r<∞ .For 0 <α相似文献   

Toeplitz and composition operators on H2 (B n)

The composition operators with closed rangc on H²(B _n) are characterized, and the Fredholmness of products of Toeplitz and composition operators discussed. Moreover, using composition operators, the spectra of Toeplitz operators are studied. Project supported by the National Natural Sciencc Foundation of China and the Postdoctoral Science Foundation of China.     

Operators on Differential Form Spaces for Riemann Surfaces

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Predholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on ∧1 2, a（M） for Riemann surface M are defined and some properties of these operators are discussed.     

On an inclusion between operator ideals

Let 1 ⩽ q < p < ∞ and 1/r:= 1/p max(q/2, 1). We prove that L _r,p ^(c), the ideal of operators of Gel’fand type l _r,p , is contained in the ideal Π _p,q of (p, q)-absolutely summing operators. For q > 2 this generalizes a result of G. Bennett given for operators on a Hilbert space.     

A Note on the Mean Value of Numbersof the Solutions of x^α≡ 1（modn）

Let T = T(p, q, α) be the number of solutions of the congruence x^α ≡ 1 (mod p^ηq^θ). Let A and B be sets of primes satisfying x₁ < p ≤ x₂ and y₁ < q ≤ y₂, respectively. A mean value estimation of is given. Supported by National Natural Science Foundation of China (No. 19971024) and Zhejiang Provincial Natural Science Foundation of China (No. 199047)     

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝⁿ are bounded from HK ^α1,p1 _q1 (ℝⁿ) ×···×HK ^αk,pk _qk (ℝⁿ) into HK ^α,p _q (ℝⁿ) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders. Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999     

The Characterization of Finite Simple Groups, L3（3^2m-1）（m≥2）, by Their Element Orders

In this paper the following theorem is proved： Every group L3（q） for q = 3^（2m-1）（m≥2） is characterized by its set of element orders.     

Global Existence and Nonexistence for a Strongly Coupled Parabolic System with Nonlinear Boundary Conditions

This paper deals with the strongly coupled parabolic system ut = v^m△u, vt = u^n△v, （x, t） ∈Ω × （0,T） subject to nonlinear boundary conditions 偏du/偏dη = u^αv^p, 偏du/偏dη= u^qv^β, （x, t） ∈ 偏dΩ × （0, T）, where Ω 包含 RN is a bounded domain, m, n are positive constants and α,β, p, q are nonnegative constants. Global existence and nonexistence of the positive solution of the above problem are studied and a new criterion is established. It is proved that the positive solution of the above problem exists globally if and only if α 〈 1,β 〈 1 and （m ＋p）（n ＋ q） ≤ （1 - α）（1 -β）.     

Linear fractional composition operators on the Dirichlet space in the unit ball

We investigate the adjoints of linear fractional composition operators Cφ acting on classical Dirichlet space D(BN ) in the unit ball BN of CN , and characterize the normality and essential normality of Cφ on D(BN ) and the Dirichlet space modulo constant function D0(BN ), where φ is a linear fractional map of BN . In addition, we also show that for any non-elliptic linear fractional map φ of BN , the composition maps σ ο φ and φ ο σ are elliptic or parabolic linear fractional maps of BN .     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

Boundedness Properties of Pseudo-Differential and Calderón-Zygmund Operators on Modulation Spaces

In this article, we study the boundedness of pseudo-differential operators with symbols in S _ρ,δ ^m on the modulation spaces M ^p,q . We discuss the order m for the boundedness Op(S _ρ,δ ^m )⊂ℒ(M ^p,q ) to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space M ^p,q with q≠2. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on M ^p,q whose symbols are of the class S _1,δ ⁰ with 0<δ<1.      

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

On Approximation by Reciprocals of Spherical Harmonics in L p Norm

Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.     

Compactness properties of certain integral operators related to fractional integration

Suppose 1≤p,q≤∞ and α > (1/p−1/q)₊. Then we investigate compactness properties of the integral operator when regarded as operator from L_p[0,1] into L_q[0,1]. We prove that its Kolmogorov numbers tend to zero faster than exp(−c_αn^1/2). This extends former results of Laptev in the case p=q=2 and of the authors for p=2 and q=∞. As application we investigate compactness properties of related integral operators as, for example, of the difference between the fractional integration operators of Riemann–Liouville and Weyl type. It is shown that both types of fractional integration operators possess the same degree of compactness. In some cases this allows to determine the strong asymptotic behavior of the Kolmogorov numbers of Riemann–Liouville operators. In memoria of Eduard (University of the West Indies) who passed away in October 2004.