Let T be the noise operator acting on functions on the boolean cube 0 1 n . Let f be a nonnegative function on 0 1 n and let q 1 . We upper bound the q norm of T f by the average q norm of conditional expectations of f , given sets of roughly (1 − 2 ) r ( q ) n variables, where r is an explicitly defined function of q .

We describe some applications for error-correcting codes and for matroids. In particular, we derive an upper bound on the weight distribution of duals of BEC-capacity achieving binary linear codes. This improves the known bounds on the linear-weight components of the weight distribution of constant rate binary Reed-Muller codes for almost all rates.