A simple reduction from a biased measure on the discrete cube to the uniform measure

TL;DR

This paper introduces a straightforward method to translate results from the uniform measure to biased measures on the discrete cube, simplifying the analysis of Fourier-Walsh expansions and related inequalities.

Contribution

It provides a simple reduction technique that generalizes key inequalities like Bonami-Beckner and Talagrand's bounds to biased measures, maintaining tightness up to constants.

Findings

Reduction from biased to uniform measure for Fourier analysis

Generalized Bonami-Beckner hypercontractivity to biased measures

Extended Talagrand's boundary size bounds to biased measures

Abstract

We show that certain statements related to the Fourier-Walsh expansion of functions with respect to a biased measure on the discrete cube can be deduced from the respective results for the uniform measure by a simple reduction. In particular, we present simple generalizations to the biased measure $\mu_p$ of the Bonami-Beckner hypercontractive inequality, and of Talagrand's lower bound on the size of the boundary of subsets of the discrete cube. Our generalizations are tight up to constant factors.