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On First Order interpolation Inequalities with Weights on the Heisenberg Group

Acta Mathematica Sinica,English Series 2011年第12期27卷 2493-2506页

作者： Ya Zhou HAN Peng Cheng NIU Shu Tao ZHANG Department of Mathematics College of Science China diliang University Hangzhou 310018 P. R. China Department of Applied Mathematics Northwestern Polytechnical University Xi＇an 710072 P. R. China Department of Mathematics College of Science China Jiliang University Hangzhou 310018 P. R. China

In this paper, sufficient and necessary conditions for the first order interpolation inequalities with weights on the Heisenberg group are given. The necessity is discussed by polar coordinates changes of the Heisenberg group. Establishing a class of Hardy type inequalities via a new representation formula for functions and Hardy-Sobolev type inequalities by interpolation, we derive the sufficiency. Finally, sharp constants for Hardy type inequalities are determined.

关键词： interpolation inequality Hardy-Sobolev type inequality Hardy type inequality Heisenberg group

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CLASSIFICATION OF SOLUTIONS FOR A CLASS OF SINGULAR INTEGRAL SYSTEM

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Acta Mathematica Scientia 2011年第4期31卷 1449-1456页

作者：许建开谭忠 College of Sciences Hunan Agriculture University School of Mathematical Sciences Xiamen University

In this paper, we consider the following integral system： u（x） = R n v q （y）｜ x y ｜ nα dy, v（x） = R n u p （y）｜ x y ｜ nμ dy, （0.1） where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes... 详细信息

In this paper, we consider the following integral system： u（x） = R n v q （y）｜ x y ｜ nα dy, v（x） = R n u p （y）｜ x y ｜ nμ dy, （0.1） where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of （0.1） are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].

关键词： Hardy-Littlewood-Sobolev inquality integral equation moving plane interpolation inequality radial symmetry

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interpolation INEQUALITIES AND DECAY ESTIMATES FOR DERIVATIVES

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Analysis in Theory and Applications 2000年第3期16卷 10-31页

作者： Lu Xuguang (Tsinghua University , China) Department of Mathematics and Science Tsinghua University Beijing China

Under the only assumption of the cone property for a given domain Ω?R n, it is proved that interpolation inequalities for intermediate derivatives of functions in the Sobolev spaces Wm,p (Ω) or even in some weighted Sobolve spaces W w m,p (Ω) still hold. That is, the usual additional restrictions that Ω is bounded or has the uniform cone property are both removed. The main tools used are polynomial inequalities, by which it is also obtained pointwise version interpolation inequalities for smooth and analytic functions. Such pointwise version inequalities give explicit decay estimates for derivatives at infinity in unbounded domains which have the cone property. As an application of the decay estimates, a previous result on radial basis function approximation of smooth functions is extended to the derivative-simultaneous approximation.

关键词： Orthogonal Matrix Decay Estimate Weighted Sobolev Space interpolation inequality Cone Versus

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The Proof of Sobolev Embedding inequality

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Chinese Quarterly Journal of Mathematics 2009年第2期24卷 298-302页

作者： ZHANG Yuan-zhang ZHANG Yong-ping GUO Xiu-lan College of Mathematics North China Institute of Water Conservancy and Hydroelectric Power Zhengzhou 450011 China Henan Women＇s Administrative Training School Zhengzhou 450002 China College of Science Henan University of Technology Zhengzhou 450001 China

We consider the problem about the space embedded by the space and the embedding inequality. With the HSlder inequality and interpolation inequality, we give the proof of the space embedding theorem and the space holde... 详细信息

关键词： Sobolev space HSlder inequality interpolation inequality embedding theorem embedding inequality

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