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

Nathan Keller

Published 2010-01-07, updated 2010-11-24Version 2

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.