We consider the problem of estimating small ball probabilities 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 -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, arises as an extremal case in this study. The proofs exploit the convexity and hypercontractivity properties of the Gaussian measure.
The first author was supported by the NSF Grant DMS-1812240.
The second author was supported in part by the Simons foundation.
The third author was supported by the NSF Grant DMS-1612936 and by Simons Foundation grant 638224.
Part of this work was conducted while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, supported by NSF Grant DMS-1440140. The hospitality of MSRI and of the organizers of the program on Geometric Functional Analysis is gratefully acknowledged. The authors are also grateful to an anonymous referee whose comments improved the style of this exposition.
"Hypercontractivity and lower deviation estimates in normed spaces." Ann. Probab. 50 (2) 688 - 734, March 2022. https://doi.org/10.1214/21-AOP1543