Hypercontractivity and comparison of moments of iterated maxima and   minima of independent random variables
  We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.'s. It turns out that the idea of hypercontractivity for minima is closely related to small ball probabilities and Gaussian correlation inequalities. 
P. Hitczenko, Stanis{\l}aw Kwapie\'n, Wenbo V. Li, Gideon Schechtman,   Thomas Schlumprecht, Joel Zinn