On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions
On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions
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关于扩散半群和极值函数超收缩性的积分准则
DOI：
10.1016/0022-1236(92)90084-v
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发表时间：
1992
影响因子：
1.7
通讯作者：
M. Ledoux
中科院分区：
文献类型：
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作者：
M. Ledoux
在对D. Bakry和M.金刚砂（数学讲义，第1123卷，第1123页。175-206，Springer-Verlag，纽约/柏林1985）和O. S.罗索斯斯（J. Funct. 42（1981）102-109;J. Funct. Anal.65（1986）358-367），研究了D. Bakry和M. Emery（见前参考文献）及其在扩散半群的超压缩性中的应用。特别是，一个简短的证明的Ornstein-Uhlenbeck半群的超压缩性的性质，我们的论述统一在一个简单的方式几个以前的结果，插值顺利从谱间隙不等式对数Sobolev不等式，甚至真正的Sobolev不等式。同时研究了球面上Ornstein-Uhlenbeck半群和热半群的超压缩性和对数Sobolev不等式的极函数。
In the line of investigation of the works by D. Bakry and M. Emery (Lecture Notes in Math., Vol. 1123, pp. 175–206, Springer-Verlag, New York/Berlin 1985) and O. S. Rothaus (J. Funct. Anal.42(1981) 102–109;J. Funct. Anal.65(1986) 358–367), we study an integral inequality behind the “Γ2criterion” of D. Bakry and M. Emery (see previous reference) and its applications to hypercontractivity of diffusion semigroups. With, in particular, a short proof of the hypercontractivity property of the Ornstein-Uhlenbeck semigroup, our exposition unifies in a simple way several previous results, interpolating smoothly from the spectral gap inequalities to logarithmic Sobolev inequalities and even true Sobolev inequalities. We examine simultaneously the extremal functions for hypercontractivity and logarithmic Sobolev inequalities of the Ornstein-Uhlenbeck semigroup and heat semigroup on spheres.