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A note on certain square functions on product spaces

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Science china mathematics 2006年第1期49卷 98-108页

作者： WANG Meng, chen jiecheng2 & fan dashan department of mathematics, zhejiang university (at Yuquan campus), hangzhou 310027, china department of mathematics, zhejiang university (at Xixi campus), hangzhou 310028, china department of mathematics, university of Wisconsin-Milwaukee, Milwaukee, WI53201, USA department of mathematics Zhejiang University (at Yuquan campus) Hangzhou China department of mathematics Zhejiang University (at Xixi campus) Hangzhou China department of mathematics University of Wisconsin-Milwaukee Milwaukee USA

We study certain square functions on product spaces Rn × Rm, whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log+ L) away from Rn × {0} and {0} × Rm by means of polynomial distortions in the radial variable. As a model case, we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn × Rm)(P > 1) for Ω∈ e Llog+ L(Sn-1 × Sm-1) satisfying the cancellation condition.

关键词： square function, Marcinkiewicz integral, product space.

Hausdorff Operators on Function Spaces

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Chinese Annals of mathematics,Series B 2012年第4期33卷 537-556页

作者： jiecheng chen dashan fan Jun LI department of mathematics Zt~ejiang Normal University Jinhua 321004 China department of mathematics Zhejiang University Hangzhou 310027 China department of mathematics University of Wisconsin-Milwaukee Milwaukee WI 53201 USA. department of mathematics Zhejiang University Hangzhou 310027 China.

The authors mainly study the Hausdorff operators on Euclidean space Rn. They establish boundedness of the Hausdorff operators in various function spaces, such as Lebesgue spaces, Hardy spaces, local Hardy spaces and Herz type spaces. The results reveal that the Hausdorff operators have better performance on the Herz type Hardy spaces HKP（Rn） than their performance on the Hardy spaces Hv（Rn） when 0 〈 p 〈 1. Also, the authors obtain some new results and reprove or generalize some known results for the high dimensional Hardy operator and adjoint Hardy operator.

关键词： Hausdorff operator Hardy operator Hardy space Herz space

Estimates for wave and Klein-Gordon equations on modulation spaces

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Science china mathematics 2012年第10期55卷 2109-2123页

作者： chen jiecheng fan dashan department of mathematics Zhejiang Normal University Jinhua 321004 China department of mathematics University of Wisconsin-Milwaukee Milwaukee WI 53201 USA

We prove that the fundamental semi-group e^it（m^2│△│）^1/2 （m≠ 0） of the Klein-Gordon equation is bounded on the modulation space M^8p,q（R^n） for all 0 〈 p, q ≤∞ and s ∈ R. Similarly, we prove that the wave semi-group e^it│△│^1/2 is bounded on the Hardy type modulation spaces μ^εp,q（R^n） for all 0 〈 p, q ≤ ∞, and s ∈R. All the bounds have an asymptotic factor t^n│1/p-1/2│ as t goes to the infinity. These results extend some known results for the case of p ≥ 1. Also, some applications for the Cauchy problems related to the semi-group eit（m^2I＋│△│）1/2 are obtained. Finally we discuss the optimum of the factor t^n│1/p-1/2│ and raise some unsolved problems.

关键词： Klein-Gordon equation wave equation modulation space

Transference on Some Non-convolution Operators from Euclidean Spaces to Torus

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Chinese Annals of mathematics,Series B 2011年第1期32卷 59-68页

作者： Yandan ZHANG dashan fan jiecheng chen department of mathematics Zhejiang University of Science and Technology Hangzhou 310023 China department of mathematics University of Wisconsin-Milwaukee Milwaukee 53711 USA Corresponding author. department of mathematics Zhejiang University Hangzhou 310017 China

The authors prove the certain de Leeuw type theorems on some non-convolution operators,and give some applications on certain known results.

关键词： n-Torus de Leeuw's theorem Commutator

Oscillatory hyper Hilbert transforms along variable curves

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Frontiers of mathematics in china 2019年第4期14卷 673-692页

作者： jiecheng chen dashan fan Meng WANG College of Mat hematics and Computer Science Zhejiang Normal University Jinhua 321001 China department of mathematics University of Wisconsin-Milwaukee Milwaukee WI 53301 USA School of Mathematical Science Zhejiang University Hangzhou 310027 China

For n = 2 or 3 and x ∈ Rn, we study the oscillatory hyper Hilbert transform Tα,β f(x)=∫Rf(x-Г(t,x))e^-i|t|-β|t|^-1-α dt along an appropriate variable curveГ(t, x) in R^n (namely,Г(t, x) is a curve in R^n for each fixed x), where α〉β〉0. We obtain some LP boundedness theorems of Tα,β, under some suitable conditions on α and β. These results are extensions of some earlier theorems. However, Tα,β f(x) is not a convolution in general. Thus, we only can partially employ the Plancherel theorem, and we mainly use the orthogonality principle to prove our main theorems.

关键词： Hyper Hilbert transform variable curve

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

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Science china mathematics 2018年第9期61卷 1647-1664页

作者： jiecheng chen Jiawei Dai dashan fan Xiangrong Zhu department of mathematics Zhejiang Normal University department of mathematics University of Wisconsin-Milwaukee

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).

关键词： Hausdorff operator Hardy space atomic characterization molecule