It is a well-known (and very powerful) fact that functions on Gaussian space become smoother under the Ornstein-Uhlenbeck semigroup. For instance, Nelson's hypercontractive inequality shows that if p > 1, then L^p functions are sent to L^q functions for some q > p. In 1989, Talagrand conjectured that quantitative smoothing is achieved even for functions in L^1, in the sense that under the...