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 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-biased measures. However, simple examples show that when 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 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, making progress on two conjectures of Kahn and Kalai (both these conjectures were open when we arXived this paper in 2019; one of them was solved in 2022; the other is still open), and proving a-biased analogue of the seminal invariance principle of Mossel, O’Donnell, and Oleszkiewicz. In this 2023 version of our paper we will also survey many further applications of our results that have been obtained by various authors since we arXived the first version in 2019.
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