Weak type estimates for some maximal operators on the weighted Hardy spaces

Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spacesH _ω ^p (0 <p < 1, ω ∈A ₁) (0<p<1, ω∞A₁); in particular, weighted weak type endpoint estimates are obtained for the maximal operators arising from the Bochner-Riesz means and the spherical means onH _ω ^p .     

Some weighted estimates for Littlewood-Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces H_w^p (0<p?1) and on the weighted L^p spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means.     

Convolution with oscillating kernels: Bochner-Riesz and spherical partial sums operators

The paper is an up to date survey of results on theL ^p boundedness, 1 ≤p ≤ ∞, of convolution operators with oscillating kernels, mainly Bochner-Riesz and the spherical partial sums operators in ℝ ⁿ ,n ≥ 2 and their maximal operators. Consequences are norm convergence and almost everywhere convergence of Fourier series. Il lavoro è una rassegna di risultati, fino ai più recenti, sulla limitatezzaL ^p , 1 ≤p ≤ ∞, di operatori di convoluzione con nuclei oscillanti, con particolare riguardo agli operatori di Bochner-Riesz e agli operatori somme parziali sferiche. Come applicazione si hanno teoremi di convergenza in norma e quasi ovunque per serie di Fourier.

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(Conferenza tenuta il 27 giugno 1988)     

Spectra of Bochner-Riesz Means on Lp

In this article, the Bochner-Riesz means are shown to have either the unit interval [0,1] or the whole complex plane as their spectra on L^p,1≤p<∞.     

M ‐elliptic pseudo‐differential operators on L p (ℝn )

We construct the minimal and maximal extensions in L ^p (?ⁿ ), 1 < p < ∞, for M ‐elliptic pseudo‐differential operators initiated by Garello and Morando. We prove that they are equal and determine the domains of the minimal, and hence maximal, extensions of M ‐elliptic pseudo‐differential operators. For M ‐elliptic pseudodifferential operators with constant coefficients, the spectra and essential spectra are computed. An application to quantization is given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Group invariance and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounded operators

The Hilbert and Riesz transforms can be characterized up to scalar as the translation invariant operators that satisfy additionally certain relative invariance of conformal transformation groups. In this article, we initiate a systematic study of translation invariant operators from group theoretic viewpoints, and formalize a geometric condition that characterizes specific multiplier operators uniquely up to scalar by means of relative invariance of affine subgroups. After providing some interesting examples of multiplier operators having “large symmetry”, we classify which of these examples can be extended to continuous operators on L ^p (R ⁿ ) (1 < p < ∞). T. Kobayashi was partially supported by Grant-in-Aid for Scientific Research 18340037, Japan Society for the Promotion of Science. A. Nilsson was partially supported by Japan Society for the Promotion of Science.     

Approximation of Bochner-Riesz means of conjugate Fourier integrals below the critical index

In this paper we study the approximation on set of full measure for functions in Sobolev spaces L _m ¹ (R ⁿ) (m∈ℕ) by Bochner-Riesz means of conjugate Fourier integrals below the critical index. A theorem concerning the precise approximation orders with relation to the number m of space L _m ¹ (R ⁿ) and the index of Bochner-Riesz means is obtained. Supported by NNSFC.     

Maximal operators of commutators of Bochner-Riesz means with Lipschitz functions

This paper presents a weighted L ² estimate with power weights for the maximal operator of commutators generated by compactly supported multipliers and Lipschitz functions. As an application, we study the almost convergence of the commutators, which is generated by the Bochner-Riesz means under the critical index and Lipschitz functions, for functions in L ^p (p ⩾ 2).     

极大多线性Bochner-Riesz算子Triebel-Lizorkin空间估计

设BδA,*是由Bochner-Riesz算子生成的极大多线性Bochner-Riesz算子,其中DγA∈Λ.β(γ=m).我们得到了在一定条件下,极大多线性Bochner-Riesz算子在Triebel-Lizorkin空间中的有界性.     

极大多线性Bochner—Riesz算子Herz型估计

设Bδ^A,是由Bochner—Riesz算子生成的极大多线性Bochner—Riesz算子,其中D^γA∈Aβ（｜γ｜=m）．获得这个算子及其变形在中心Campanato空间的连续性．     

Applications of the purely imaginary powers of operators in Hilbert spaces

In C. R. Acad. Sci. Paris299 (1984), 173–176, we discussed purely imaginary powers A^iy(−∞ < y < + ∞) of linear operators A in Hilbert spaces. Here we utilize the results to consider the various problems: generation of cosine families in Hilbert spaces, coincidence of the definition domains of the fractional powers of operators, differentiability of the functions of the form A(·)⁰ (0 < θ < 1) where A(·) is an operator valued function defined on an interval [0, T], and so forth.     

L^p（R^n） Boundedness for the Maximal Operator of Multilinear Singular Integrals with Calderon-Zygmund Kernels

In this paper, the authors consider a class of maximal multilinear singular integral operators and maximal multilinear oscillatory singular integral operators with standard Calderón–Zygmund kernels, and obtain their boundedness on L ^p (ℝ ⁿ ) for 1 < p < ∞. Research supported by Professor Xu Yuesheng’s Research Grant in the program of "One hundred Distinguished Young Scientists" of the Chinese Academy of Sciences     

极大多线性Bochner-Riesz算子Herz型估计

设BδA,*是由Bochner-Riesz算子生成的极大多线性Bochner-Riesz算子,其中DγA∈Λ.β(|γ|=m).获得这个算子及其变形在中心Campanato空间的连续性.     

Strong Convergence Theorems for Two Parameter Vilenkin-Fourier Series

As main result we prove that certain means of the partial sums of two-parameter Vilenkin-Fourier series are uniformly bounded operators from H ^P to L ^p (0 < p ≦ 1). The Hardy space H ^p (0 < p ≦ 1) will be defined by means of a diagonal maximal function. As a consequence we obtain a so-called strong convergence theorem for the Vilenkin-Fourier partial sums. Some dual inequalities are also verified for BMO spaces.     

Some estimates for Bochner-Riesz operators on the weighted Herz-type Hardy spaces

In this paper, by using the atomic decomposition and molecular characterization of the homogeneous and non-homogeneous weighted Herz-type Hardy spaces , we obtain some weighted boundedness properties of the Bochner-Riesz operator and the maximal Bochner-Riesz operator on these spaces for α=n(1/p−1/q), 0<p?1 and 1<q<∞.     

On Multiple Singular Integrals along Polynomial Curves with Rough Kernels

This paper is devoted to the study of a class of singular integral operators defined by polynomial mappings on product domains. Some rather weak size conditions, which imply the Lp boundedness of these singular integral operators as well as the corresponding maximal truncated singular integral operators for some fixed 1〈p〈 ∞,are given.     

Spectral analysis of certain compact factors for Gaussian dynamical systems

For factors of a Gaussian automorphismT determined by compact subgroups of the group of unitary operators acting onL ² of the spectral measure ofT, we prove that the maximal spectral multiplicity is either 1 or infinity. As an application, we show that the maximal multiplicity of those factors an allL ^p, 1<p<+∞, is the same.     

ON LAGRANGE INTERPOLATION TO ｜x｜^α（1 〈α 〈 2） WITH EQUALLY SPACED NODES

S.M. Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [−1,1], In 2000, M. Rever generalized S.M. Lozinskii’s result to |x|^α(0≤α≤1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|^α(1<α<2).     

Existence and multiplicity of solutions for fourth-order boundary value problems with parameters

In this paper we study the existence and multiplicity of the solutions for the fourth-order boundary value problem (BVP) u(4)(t)+ηu^″(t)−ζu(t)=λf(t,u(t)), 0<t<1, u(0)=u(1)=u^″(0)=u^″(1)=0, where is continuous, ζ,η∈R and λ∈R⁺ are parameters. By means of the idea of the decomposition of operators shown by Chen [W.Y. Chen, A decomposition problem for operators, Xuebao of Dongbei Renmin University 1 (1957) 95-98], see also [M. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Gostehizdat, Moscow, 1956], and the critical point theory, we obtain that if the pair (η,ζ) is on the curve ζ=−η²/4 satisfying η<2π², then the above BVP has at least one, two, three, and infinitely many solutions for λ being in different interval, respectively.     

A family of cohen-macaulay modules over singularities of type xt + y3

Let K be a field with char K ≠ 3 and it two positive integers such that 1 ≤i <t/2,t ≠ 3i. The classification problem for maximal Cohen-Macaulay modules over K[[X,Y]]/(X^t+Y³ ) is complicated if t≥ 6, because there exist parameter families of non-isomorphic maximal Cohen-Macaulay modules [Sc], or [GK], [Yo, Ch.9] and [DG]). Here we describe parameter families of such modules N, such that N/YN is a direct sum of copies of K[[X]]/(X ⁱ)K[[X]]/(X^t-i ).