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Conjectures and problems on Bochner-Riesz means

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Frontiers of Mathematics in China 2013年第6期8卷 1237-1251页

作者： shanzhen lu School of Mathematical Sciences Beijing Normal University Beijing 100875 China

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.

关键词： Bochner-Riesz means Fourier series Fourier integral norm convergence almost everywhere convergence

A characterization of λ-central BMO space

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Frontiers of Mathematics in China 2013年第1期8卷 229-238页

作者： Fayou ZHAO shanzhen lu Department of Mathematics Shanghai University Shanghai 200444 China School of Mathematical Sciences Beijing Normal University Beijing 100875 China

We give a characterization of the λ-central BMO space via the boundedness of commutators of n-dimensional Hardy operators.

关键词： λ-central BMO space n-dimensional Hardy operator commutator

Toeplitz operators related to strongly singular Calderón-Zygmund operators

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Science China Mathematics 2006年第8期49卷 1048-1064页

作者： LIN Yan lu shanzhen School of Mathematical Sciences Beijing Normal University Beijing 100875 China School of Mathematical Sciences Beijing Normal University Beijing 100875 China

In this paper, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and Lipschitz function b∈Λβ0(Rn) is discussed from Lp(Rn) to Lq(Rn), 1/q=1/p-β0/n, and from Lp(Rn) to Triebel-Lizorkin space Fβ0,∞p. We also obtain the boundedness of generalized Toeplitz operatorθbα0 from LP(Rn) to Lq(Rn), 1/q =1/p-α0+β0/n. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and BMO function b is discussed on LP(Rn), 1 < p <∞.

关键词： Toeplitz operator strongly singular Calderon-Zygmund operator Lipschitz space Triebel-Lizorkin space BMO.

Boundedness for a class of fractional integrals with a rough kernel related to block spaces

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Frontiers of Mathematics in China 2016年第1期11卷 173-187页

作者： Xiao YU shanzhen lu Department of Mathematics Shangrao Normal University Shangrao 334001 China School of Mathematical Science Beijing Normal University Beijing 100875 China

We prove the boundedness of fractional integral with a rough kernel on Triebel-Lizorkin spaces, where the rough kernel belongs to the block space and does not need to satisfy any moment conditions on the unit sphere.

关键词： Fractional integral block space Triebel-Lizorkin space Fouriertransform interpolation

ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR (I)

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Chinese Annals of Mathematics,Series B 1997年第2期18卷 181-190页

作者： CHEN WENGU HU GUOEN lu shanzhen Department of Mathematics Beijing Normal UniversityBeijing 100875China.

The authors consider the multilinear oscillatory singular integral operator defined by T A 1,A 2,…,A k f(x)=∫ R n e iP(x,y) ∏k j=1 R m j (A j;x,y)Ω(x-y)|x-y| n+M f(y)dy,where ... 详细信息

The authors consider the multilinear oscillatory singular integral operator defined by T A 1,A 2,…,A k f(x)=∫ R n e iP(x,y) ∏k j=1 R m j (A j;x,y)Ω(x-y)|x-y| n+M f(y)dy,where P(x,y) is a real-valued polynomial on R n× R n , Ω is homogeneous of degree zero, R m j (A j;x,y) denotes the m j -th order Taylor series remainder of A j at x expanded about y , M=∑kj=1 m j . It is shown that if Ω belongs to the space L log +L(S n-1 ) and has vanishing moment up to order M , then‖T A 1,A 2,…,A k f‖ q C ∏kj=1∑|α|=mj‖D αA j‖ r j ‖f‖ p, provided that 1

关键词： Multilinear operator Oscillatory singularintegral Maximal operator

Multiple weighted estimates for maximal vector-valued commutator of multilinear Calderon-Zygmund singular integrals

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Frontiers of Mathematics in China 2017年第3期12卷 531-558页

作者： Dongxiang CHEN shanzhen lu Suzhen MAO Department of Mathematics Jiangxi Normal University Nanchang 330022 China School of Mathematical Science Beijing Normal University Beijing 100875 China

This paper concerns with multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. The Cotlar-type inequality of maximal vector-valued multilinear singular integrals operator is obtained. On the other hand, pointwise estimates for sharp maximal function of two kinds of maximal vector-valued multilinear singular integrals and maximal vector-valued commutators are also established. By the weighted estimates of a class of new variant maximal operator, Cotlar＇s inequality and the sharp maximal flmction estimates, multiple weighted strong estimates and weak estimates for maximal vector-valued singular integrals of multilinear operators and those for maximal vector-valued commutator of multilinear singular integrals are obtained.

关键词： Multilinear singular integrals vector-valued commutator multiple weight maximal functions

Endpoint estimates for n-dimensional Hardy operators and their commutators

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Science China Mathematics 2012年第10期55卷 1977-1990页

作者： ZHAO FaYou FU ZunWei lu shanzhen Department of Mathematics Shanghai UniversityShanghai 200444China Department of Mathematics Linyi UniversityLinyi 276005China School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China

In this paper,the sharp bound for the weak-type(1,1) inequality for the n-dimensional Hardy operator is ***,the precise norms of generalized Hardy operators on the type of Campanato spaces are worked *** applications,the corresponding norms of the Riemann-Liouville integral operator and the n-dimensional Hardy operator are *** is also proved that the n-dimensional Hardy operator maps from the Hardy space into the Lebesgue *** endpoint estimate for the commutator generated by the Hardy operator and the(central) BMO function is also discussed.

关键词： sharp bound generalized Hardy operator commutator Campanato space Hardy space

Commutators of n-dimensional rough Hardy operators

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Science China Mathematics 2011年第1期54卷 95-104页

作者： FU ZunWei lu shanzhen ZHAO FaYou School of Mathematical Sciences Beijing Normal University Beijing 100875 China Department of Mathematics Linyi University Linyi 276005 China

In this paper, we study central BMO estimates for commutators of n-dimensional rough Hardy operators. Furthermore, λ-central BMO estimates for commutators on central Morrey spaces are discussed.

关键词： n-dimensional rough Hardy operator commutator central BMO space central Morrey space

Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces

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Science China Mathematics 2012年第12期55卷 2469-2480页

作者： WANG ShiMo lu shanzhen YAN DunYan School of Mathematics Science Heilongjiang University Haerbin 150080 China School of Mathematics Beijing Normal University Beijing 100875 China School of Mathematics Graduate University Chinese Academy of Sciences Beijing 100049 China

In this paper, Hardy operator H on n-dimensional product spaces G = (0, ∞)n and its adjoint operator H* are investigated. We use novel methods to obtain two main results. One is that we characterize the sufficient and necessary conditions for the operators H and H* being bounded from Lp(G, xα) to Lq(G, xβ), and the bounds of the operators H and H* are explicitly worked out. The other is that when 1 < p = q < +∞, norms of the operators H and H* are obtained.

关键词： n-dimensional Hardy operator Hausdorff-Young inequality power weight