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Discrete weighted Hardy inequalities with different kinds of boundary conditions

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Abstract

This paper studies the weighted Hardy inequalities on the discrete intervals with four different kinds of boundary conditions. The main result is the uniform expression of the basic estimate of the optimal constant with the corresponding boundary condition. Firstly, one-side boundary condition is considered, which means that the sequences vanish at the right endpoint (ND-case). Based on the dual method, it can be translated into the case vanishing at left endpoint (DN-case). Secondly, the condition is the case that the sequences vanish at two endpoints (DD-case). The third type of condition is the generality of the mean zero condition (NN-case), which is motivated from probability theory. To deal with the second and the third kinds of inequalities, the splitting technique is presented. Finally, as typical applications, some examples are included.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China

    Zhong Wei Liao

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  1. Zhong Wei Liao
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Correspondence to Zhong Wei Liao.

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Supported by NSFC (Grant No. 11131003) and by the “985” project from the Ministry of Education in China

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Liao, Z.W. Discrete weighted Hardy inequalities with different kinds of boundary conditions. Acta. Math. Sin.-English Ser. 32, 993–1013 (2016). https://doi.org/10.1007/s10114-016-5675-0

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  • DOI: https://doi.org/10.1007/s10114-016-5675-0

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