  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
Strook-Varapolos inequality.
  A consequence of our analysis is that {\em 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 $t \geq
\log \frac{1-q}{1-p}$ we have $\| e^{-tL}f\|_{q} \geq \| f\|_{p}$. 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.
