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).     

Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator may be defined as

where is an appropriate distance function and is the inverse Fourier transform. The behavior of on is described for , a rough distance function. We conjecture that this operator is bounded on when and , and unbounded when . This conjecture is verified for large ranges of .

     

Conjectures and problems on Bochner-Riesz means

The aim of this paper is to state some conjectures and problems on Bochner-Riesz means in multiple Fourier series and integrals. The progress on these conjectures and problems are also mentioned.     

Weighted inequalities for the Bochner-Riesz means related to the Fourier-Bessel expansions

We prove weighted inequalities for the Bochner-Riesz means for Fourier-Bessel series with more general weights w(x) than previously considered power weights. These estimates are given by using the local Ap theory and Hardy's inequalities with weights. Moreover, we also obtain weighted weak type (1,1) inequalities. The case when w(x)=xa is sketched and follows as a corollary of the main result.     

Almost Everywhere Convergence of Bochner-Riesz Means On The Heisenberg Group and Fractional Integration On The Dual

Let L denote the sub-Laplacian on the Heisenberg group H_n and the corresponding Bochner-Riesz operator. Let Q denote the homogeneous dimension and D the Euclideandimension of H_n. We prove convergence a.e. of the Bochner-Rieszmeans as r 0 for > 0and for all f L^p(H_n), provided that . Our proof is based on explicit formulas for the operators with a C, defined on the dual ofH_n by , which may be of independent interest. Here is given by for all (z,u) H_n. 2000 Mathematical Subject Classification: 22E30, 43A80.     

Boundedness for commutators of Bochner-Riesz operators below a critical index

In this paper, the authors obtain a necessary and sufficient condition for the LP bound-edness of commutators generated by Bochner-Riesz operators below a critical index and Lipschitz functions in two dimensional case. The authors also discuss a similar problem for higher dimensions.     

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<∞.     

Bochner—Riesz算子及其高阶交换子L^p有界的必要条件

通过选择适当的L^p函数并应用连续分解方法,给出了低于临界阶的Bochner—Riesz算子在L^p空间有界的新的证明,同时得到了该算子和Lipschitz函数构成的高阶交换子L^p有界性的必要条件．     

On Approximation Methods Generated by Bochner-Riesz Kernels

Means and families of operators generated by Bochner-Riesz kernels are studied. Some sharp results on their convergence are achieved. The equivalence of the approximation errors of these methods to smoothness quantities related to the Laplacian is proved.      

The convergence of the Bochner-Riesz means at the critical index

In this paper, we study the pointwise convergence of the Bochner-Riesz means at the critical index on the space We weaken the hypothesis, is a Lebesgue point", which is required on some research results by instead considering the convergence of averages of the function over balls when the radials of the balls approach to 0.

     

A note on commutators of Bochner-Riesz operator

In terms of continuous decomposition and choosing an appropriate BMO function, the authors obtain a sharp necessary condition for L ^p boundedness of the commutators generated by Bochner-Riesz operators below the critical index and BMO functions.      

On the focus order of planar polynomial differential equations

This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where n+1 is a prime p or a power of a prime pk, and show that these systems can have a focus order n²−n; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n²−1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too.     

紧Lie群上Hp（0＜P＜1）函数的临界阶Bochner－Riesz平均算子的有界性

本文利用尺度‖·‖_H(p,∞)研究了一般紧Ｌｉｅ群上Ｈ^p函数的临界阶Ｂｏｃｈｎｅｒ－Ｒｉｅｓｚ平均算子σ_R^δ:f→σ_R^δf的有界性，得到了如下结果：σ_R^δ是（Ｈ^p，Ｈ（ｐ，∞））型的，并且‖σ_R^δf‖_H(p,∞)≤C‖f‖_H^p     

Weighted norm inequalities of Bochner-Riesz means

Let w be a Muckenhoupt weight and be the weighted Hardy spaces. We use the atomic decomposition of and their molecular characters to show that the Bochner-Riesz means are bounded on for 0<p?1 and δ>max{n/p−(n+1)/2,[n/p]rw⁻¹(rw−1)−(n+1)/2}, where rw is the critical index of w for the reverse Hölder condition. We also prove the boundedness of the maximal Bochner-Riesz means for 0<p?1 and δ>n/p−(n+1)/2.     

A family of Bochner-Riesz type multipliers on the two-dimensional disk

Let s=(ν,μ)∈R² and define on R². Given p∈[1,+∞[, we prove some necessary and sufficient conditions on s such that ms be a Fourier multiplier for Lp. We employ two different techniques, according to ν=0 or ν≠0. In the latter case, the results obtained are optimal.     

准径向Bochner-Riesz算子极大交换子的有界性

建立了准径向Bochner-Riesz算子与Lipschitz函数生成的极大交换子的(L~p,Λ_(β-n/p)和(L~(n/β),BMO)有界性.     

Bochner-Riesz算子的多线性交换子的有界性

In this paper,the boundedness for the multilinear commutators of Bochner-Riesz operator is considered.We prove that the multilinear commutators generated by Bochner-Riesz operator and Lipschitz function are bounded from Lp（Rn）into ∧˙（β-np）（Rn）and from Lnβ（Rn）into BMO（Rn）.     

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.     

Bochner-Riesz算子的极大交换子的Hardy型估计

证明了Bochner-Riesz算子的极大交换子是一个从局部Hardy空间h~1(R~n)到空间h_q~1(R~n)=h~1(R~n)+L~q(R~n)(q>1)上的有界算子.     

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.