Global hypercontractivity and its applications

Understand

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.