A generalized Poincaré inequality for Gaussian measures
A generalized Poincaré inequality for Gaussian measures
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DOI：
10.1090/s0002-9939-1989-0954373-7
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发表时间：
1989-02
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影响因子：
--
通讯作者：
W. Beckner
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文献类型：
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作者：
W. Beckner
对高斯测度和球面测度，得到了Poincaré不等式和对数Sobolev不等式之间的一个新的不等式，它们是一个尖锐的插值不等式.经典的Poincaré不等式给出了零化常数的正自伴算子的第一非平凡特征值的估计。对于高斯测度dp = T\k（2n）~（12）Xkdxk和Ornstein-Uhlenbeck过程的生成元N = -A + x · V，Poincaré不等式简单地表示为估计（1）\f\2dp-[j fdp）<j\Vf\2dp.这是埃尔米特多项式民间传说中的一个古老的不等式，可能在20世纪30年代的数学家和物理学家在一维中与特征值问题有关（[9]，[10]）。它在包括偏微分方程[7]和统计学[4]在内的各种学科中都是一个有用的工具。这里的目的是注意到这个估计的一个推广形式，它以一种尖锐的方式插在Gross [5]得到的高斯测度的Poincaré不等式和对数Sobolev不等式之间。定理1.对于f ∈ L2（dp），1 < p < 2，且e-1 = sjp 1，（2）J1/12 dp J | e-tNf 12 dp<（2-p）j| V/l2 dp和（3）j l/l2 dp（/ l/l”dp）“<（2 p）j| V/l2 dp。这样的估计与维数无关，因此可以将高斯测度空间视为无限维。这两者之间的关系编辑于1988年8月1日收到。1980年数学学科分类（1985年修订）。小学42 B 99;中学60 D 05。这项工作得到了国家科学基金会的部分支持。© 1989美国数学学会0002-9939/89 $1.00+ $.25每页
New inequalities are obtained which interpolate in a sharp way between the Poincaré inequality and the logarithmic Sobolev inequality for both Gaussian measure and spherical surface measure. The classical Poincaré inequality provides an estimate for the first nontrivial eigenvalue of a positive self-adjoint operator that annihilates constants. For the Gaussian measure dp = T\k(2n)~{'2e~({l2)Xkdxk and the generator of the Ornstein-Uhlenbeck process N = -A + x • V, the Poincaré inequality is simply the estimate (1) ¡\f\2dp-[j fdp) <j\Vf\2dp. This is an old inequality in the folklore of Hermite polynomials and probably was known in one dimension to both mathematicians and physicists in the 1930's in relation to eigenvalue problems ([9], [10]). It has been a useful tool in diverse subjects including partial differential equations [7] and statistics [4]. The purpose here is to note a generalized form of this estimate which interpolates in a sharp way between the Poincaré inequality and the logarithmic Sobolev inequality for Gaussian measures obtained by Gross [5]. Theorem 1. For f e L2(dp), 1 < p < 2 and e~' = sjp 1, (2) J l/l2 dp J \e~tNf\2 dp<(2-p)j |V/l2 dp and (3) j l/l2 dp (/ l/l" dp) " < (2 p) j |V/l2 dp. Such estimates are independent of dimension so one can regard the Gaussian measure space as infinite dimensional. The relation between these two Received by the editors August 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 42B99; Secondary 60D05. This work was partially supported by the National Science Foundation. © 1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page