Logarithmic Sobolev inequalities for harmonic measures on spheres
Logarithmic Sobolev inequalities for harmonic measures on spheres
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DOI：
10.1016/j.matpur.2013.11.008
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发表时间：
2014-07
期刊：
影响因子：
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通讯作者：
F. Barthe;Yutao Ma;Zheng-liang Zhang
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文献类型：
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作者：
F. Barthe;Yutao Ma;Zheng-liang Zhang
本文考虑调和测度族μ xn（指标为x∈ Rn，|X| < 1）在Rn中的单位球面Sn − 1上，其中n <$3。本文研究了对数Sobolev不等式和对数Poincaré不等式的最优常数，分别记为CLS（μ xn）和CP（μ xn）.我们证明庞加莱常数本质上只取决于维数，如1/（n− 1）<$CP（μ xn）<$2/（n− 2）。C LS（μ x n）的行为更加复杂，取决于位置和维度之间的相互作用。
In this paper, we consider the family of harmonic measures μ x n (indexed by x∈ R n with| x|< 1) on the unit sphere S n− 1 in R n, for n⩾ 3. We study the corresponding optimal constants of logarithmic Sobolev and Poincaré inequalities, denoted respectively by C LS (μ x n) and C P (μ x n). We show that the Poincaré constant essentially depends on the dimension only, as 1/(n− 1)⩽ C P (μ x n)⩽ 2/(n− 2). The behavior of C LS (μ x n) is more intricate and depends on the interplay between position and dimension.