  For a function $f$ on the hypercube $\{0,1\}^n$ with Fourier expansion
$f=\sum_{S\subseteq[n]}\hat f(S)\chi_S$, the hypercontractive inequality for
the noise operator allows bounding norms of $T_\rho f=\sum_S\rho^{|S|}\hat
f(S)\chi_S$ in terms of norms of $f$. If $f$ is Boolean-valued, the level-$d$
inequality allows bounding the norm of $f^{=d}=\sum_{|S|=d}\hat f(S)\chi_S$ in
terms of $E[f]$. These two inequalities play a central role in analysis of
Boolean functions and its applications.
  While both inequalities hold in a sharp form when the hypercube is endowed
with the uniform measure, they do not hold for more general discrete product
spaces. Finding a `natural' generalization was a long-standing open problem. In
[P. Keevash et al., Global hypercontractivity and its applications, J. Amer.
Math. Soc., to appear], a hypercontractive inequality for this setting was
presented, that holds for functions which are `global' -- namely, are not
significantly affected by a restriction of a small set of coordinates. This
hypercontractive inequality is not sharp, which precludes applications to the
symmetric group $S_n$ and to other settings where sharpness of the bound is
crucial. Also, no level-$d$ inequality for global functions over general
discrete product spaces is known.
  We obtain sharp versions of the hypercontractive inequality and of the
level-$d$ inequality for this setting. Our inequalities open the way for
diverse applications to extremal set theory and to group theory. We demonstrate
this by proving quantitative bounds on the size of intersecting families of
sets and vectors under weak symmetry conditions and by describing numerous
applications to the study of functions on $S_n$ -- including hypercontractivity
and level-$d$ inequalities, character bounds, variants of Roth's theorem and of
Bogolyubov's lemma, and diameter bounds, that were obtained using our
techniques.
