Revised logarithmic Sobolev inequalities of fractional order

IF 1.3 3区 数学 Q2 MATHEMATICS, APPLIED
Marianna Chatzakou , Michael Ruzhansky
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引用次数: 0

Abstract

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on Rn with an explicit expression for the constant. Namely, we show that for all 0<s<n2 and a>0 we have the inequalityRn|f(x)|2log(|f(x)|2fL2(Rn)2)dx+ns(1+loga)fL2(Rn)2C(n,s,a)(Δ)s/2fL2(Rn)2 with an explicit C(n,s,a) depending on a, the order s, and the dimension n, and investigate the behaviour of C(n,s,a) for large n. Notably, for large n and when s=1, the constant C(n,1,a) is asymptotically the same as the sharp constant of Lieb and Loss that was computed in [13]. Moreover, we prove a similar type inequality for functions fLq(Rn)W1,p(Rn) whenever 1<p<n and p<qp(n1)np.
修订的分数阶对数索波列夫不等式
在这篇短文中,我们证明了 Rn 上带有分数阶导数的对数 Sobolev 不等式,并给出了常数的明确表达式。也就是说,我们证明对于所有 0<s<n2 和 a>;0,我们有不等式∫Rn|f(x)|2log(|f(x)|2‖f‖L2(Rn)2)dx+ns(1+lga)‖f‖L2(Rn)2≤C(n,s,a)‖(-Δ)s/2f‖L2(Rn)2,其中有一个显式 C(n、s,a) 取决于 a、阶 s 和维数 n,并研究 C(n,s,a) 在大 n 时的表现。值得注意的是,对于大 n 且 s=1 时,常数 C(n,1,a) 与 [13] 中计算的 Lieb 和 Loss 的尖锐常数渐近相同。此外,当 1<p<n 和 p<q≤p(n-1)n-p 时,我们证明了函数 f∈Lq(Rn)∩W1,p(Rn) 的类似不等式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.90
自引率
7.70%
发文量
71
审稿时长
6-12 weeks
期刊介绍: Founded in 1870, by Gaston Darboux, the Bulletin publishes original articles covering all branches of pure mathematics.
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