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, Friedguts junta theorem and the invariance principle of Mossel, ODonnell 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 Bourgains sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgains 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, ODonnell, and Oleszkiewicz. In a companion paper, we give applications to the solution of two open problems in Extremal Combinatorics.
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