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

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 μ 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.