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Geometric and Functional Analysis
We study the notion of reverse hypercontractivity. We show that reverse hypercontractive inequalities are implied by standard hypercontractive inequalities as well as by the modified log-Sobolev inequality. Our proof is based on a new comparison lemma for Dirichlet forms and an extension of the Stroock–Varopoulos inequality.
A consequence of our analysis is that all simple operators L=Id−E as well as their tensors satisfy uniform reverse hypercontractive inequalities. That is, for all q < p < 1 and every positive valued function f for t ≥ log(1−q)/(1−p) we have ∥e−tLf∥q ≥ ∥f∥p. This should be contrasted with the case of hypercontractive inequalities for simple operators where t is known to depend not only on p and q but also on the underlying space.
The new reverse hypercontractive inequalities established here imply new mixing and isoperimetric results for short random walks in product spaces, for certain card-shufflings, for Glauber dynamics in high-temperatures spin systems as well as for queueing processes. The inequalities further imply a quantitative Arrow impossibility theorem for general product distributions and inverse polynomial bounds in the number of players for the non-interactive correlation distillation problem with m-sided dice.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00039-013-0229-4.
Mossel, E., Oleszkiewicz, K., & Sen, A. (2013). On Reverse Hypercontractivity. Geometric and Functional Analysis, 23 (3), 1062-1097. http://dx.doi.org/10.1007/s00039-013-0229-4
Date Posted: 27 November 2017
This document has been peer reviewed.