HYPERCONTRACTIVITY AND LOWER DEVIATION ESTIMATES IN NORMED SPACES

• 成果类型: Article
• 署名作者: Paouris, Grigoris; Tikhomirov, Konstantin; Valettas, Petros
• 署名单位: Texas A&M University System; Texas A&M University College Station; University System of Georgia; Georgia Institute of Technology; University of Missouri System; University of Missouri Columbia; University of Missouri System; University of Missouri Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1543
• 发表日期: 2022
• 页码: 688-734
• 关键词: small ball probability; dvoretzkys theorem; random version; INEQUALITY; sections

摘要

We consider the problem of estimating small ball probabilities P[f (G) <= delta E f (G)} for subadditive, positively homogeneous functions f with respect to the Gaussian measure. We establish estimates that depend on global parameters of the underlying function, which take into account analytic and statistical measures, such as the variance and the L-1-norms of its partial derivatives. This leads to dimension-dependent bounds for small ball and lower small deviation estimates for seminorms when the linear structure is appropriately chosen to optimize the aforementioned parameters. Our bounds are best possible up to numerical constants. In all regimes, parallel to G parallel to(infinity) = max(i <= n)vertical bar g(i)vertical bar arises as an extremal case in this study. The proofs exploit the convexity and hypercontractivity properties of the Gaussian measure.