Keller and Kindler recently established a quantitative version of the famous
Benjamini–Kalai–Schramm theorem on the noise sensitivity of Boolean functions.
Their result was extended to the continuous Gaussian setting by Keller, Mossel
and Sen by means of a Central Limit Theorem argument. In this work we present a
unified approach to these results, in both discrete and continuous settings. The
proof relies on semigroup decompositions together with a suitable cut-off
argument, allowing for the efficient use of the classical hypercontractivity
tool behind these results. It extends to further models of interest such as
families of log-concave measures and Cayley and Schreier graphs. In particular
we obtain a quantitative version of the Benjamini–Kalai–Schramm theorem for the
slices of the Boolean cube.