  The hypercontractive inequality on the discrete cube plays a crucial role in
many fundamental results in the Analysis of Boolean functions, such as the KKL
theorem, Friedgut's junta theorem and the invariance principle. In these
results the cube is equipped with the uniform measure, but it is desirable,
particularly for applications to the theory of sharp thresholds, to also obtain
such results for general $p$-biased measures. However, simple examples show
that when $p = o(1)$, there is no hypercontractive inequality that is strong
enough.
  In this paper, we establish an effective hypercontractive inequality for
general $p$ that applies to `global functions', i.e. functions that are not
significantly affected by a restriction of a small set of coordinates. This
class of functions appears naturally, e.g. in Bourgain's sharp threshold
theorem, which states that such functions exhibit a sharp threshold. We
demonstrate the power of our tool by strengthening Bourgain's theorem, thereby
making progress on a conjecture of Kahn and Kalai and by establishing a
$p$-biased analog of the invariance principle.
  Our results have significant applications in Extremal Combinatorics. Here we
obtain new results on the Tur\'an number of any bounded degree uniform
hypergraph obtained as the expansion of a hypergraph of bounded uniformity.
These are asymptotically sharp over an essentially optimal regime for both the
uniformity and the number of edges and solve a number of open problems in the
area. In particular, we give general conditions under which the crosscut
parameter asymptotically determines the Tur\'an number, answering a question of
Mubayi and Verstra\"ete. We also apply the Junta Method to refine our
asymptotic results and obtain several exact results, including proofs of the
Huang--Loh--Sudakov conjecture on cross matchings and the
F\"uredi--Jiang--Seiver conjecture on path expansions.
