  The classical hypercontractive inequality for the noise operator on the
discrete cube plays a crucial role in many of the fundamental results in the
Analysis of Boolean functions, such as the KKL (Kahn-Kalai-Linial) theorem,
Friedgut's junta theorem and the invariance principle of Mossel, O'Donnell and
Oleszkiewicz. In these results the cube is equipped with the uniform
($1/2$-biased) 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 $p$ is small there
is no hypercontractive inequality that is strong enough for such applications.
  In this paper, we establish an effective hypercontractivity 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. An additional application of
our hypercontractivity theorem, is a $p$-biased analog of the seminal
invariance principle of Mossel, O'Donnell, and Oleszkiewicz. In a companion
paper, we give applications to the solution of two open problems in Extremal
Combinatorics.
