Abstract
This paper studies the Hardy-type inequalities on the intervals (may be infinite) with two weights, either vanishing at two endpoints of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new in the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.
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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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Chen, M.F. Bilateral Hardy-type inequalities. Acta. Math. Sin.-English Ser. 29, 1–32 (2013). https://doi.org/10.1007/s10114-012-2316-0
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DOI: https://doi.org/10.1007/s10114-012-2316-0
Keywords
- Hardy-type inequality
- vanishing at two endpoints
- mean zero
- splitting technique
- normed linear space
- Nash inequality
- logarithmic Sobolev inequality