# Hypercontractivity and Lower Deviation Estimates in Normed Spaces

## TL;DR

This paper develops dimension-dependent bounds for small ball probabilities of sub-additive, positively homogeneous functions under Gaussian measures, utilizing hypercontractivity and convexity properties to optimize estimates.

## Contribution

It introduces new bounds for small deviation probabilities that depend on analytic and statistical parameters, improving understanding of Gaussian measures in normed spaces.

## Key findings

- Bounds are tight up to constants.
- Extremal case is the maximum of Gaussian variables.
- Results apply to seminorms with optimized parameters.

## Abstract

We consider the problem of estimating small ball probabilities $\mathbb P\{f(G) \leqslant \delta \mathbb Ef(G)\}$ for sub-additive,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, $\|G\|_\infty = \max_{ i \leqslant n}|g_i|$ arises as an extremal case in this study. The proofs exploit the convexity and hypercontractivity properties of the Gaussian measure.