An unified approach to the Junta theorem for discrete and continuous models

In a recent paper, T. Austin has proved an analogous theorem for the continuous torus of the original Junta theorem proved by Friedgut in the case of the Boolean cube. Analogous statements have been established recently in discrete cases such as the discrete Tori by Ellis et.al., and in the case of slices of the Boolean cube by Wimmer and Filmus. In the continuous case, through the notion of geometric influences, a statement has also been established by Keller, Mossel and Sen for Boltzmann probability measures. In this article, we broaden the scope of the arguments of T. Austin to get an unified proof of these results, removing the restriction to Boolean functions. Indeed, the proof of T. Austin relies on semigroup arguments and can be performed in a general framework that covers both Cayley or Schreier graphs or product of log-concave probability measures.