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March 2022 Hypercontractivity and lower deviation estimates in normed spaces
Grigoris Paouris, Konstantin Tikhomirov, Petros Valettas
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Ann. Probab. 50(2): 688-734 (March 2022). DOI: 10.1214/21-AOP1543

Abstract

We consider the problem of estimating small ball probabilities P{f(G)δEf(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 L1-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=maxin|gi| arises as an extremal case in this study. The proofs exploit the convexity and hypercontractivity properties of the Gaussian measure.

Funding Statement

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.

Acknowledgments

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.

Citation

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Grigoris Paouris. Konstantin Tikhomirov. Petros Valettas. "Hypercontractivity and lower deviation estimates in normed spaces." Ann. Probab. 50 (2) 688 - 734, March 2022. https://doi.org/10.1214/21-AOP1543

Information

Received: 1 January 2020; Revised: 1 July 2021; Published: March 2022
First available in Project Euclid: 24 March 2022

Digital Object Identifier: 10.1214/21-AOP1543

Subjects:
Primary: 46B09
Secondary: 52A21

Keywords: Alon–Milman theorem , Gaussian convexity , hypercontractvity , Ornstein–Uhlenbeck semigroup , superconcentration , Talagrand’s L1−L2 bound

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 2 • March 2022
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