  In this paper we derive tight bounds on the expected value of products of
{\em low influence} functions defined on correlated probability spaces. The
proofs are based on extending Fourier theory to an arbitrary number of
correlated probability spaces, on a generalization of an invariance principle
recently obtained with O'Donnell and Oleszkiewicz for multilinear polynomials
with low influences and bounded degree and on properties of multi-dimensional
Gaussian distributions. The results derived here have a number of applications
to the theory of social choice in economics, to hardness of approximation in
computer science and to additive combinatorics problems.
