  We prove hypercontractive inequalities on high dimensional expanders. As in
the settings of the p-biased hypercube, the symmetric group, and the Grassmann
scheme, our inequalities are effective for global functions, which are
functions that are not significantly affected by a restriction of a small set
of coordinates. As applications, we obtain Fourier concentration, small-set
expansion, and Kruskal-Katona theorems for high dimensional expanders. Our
techniques rely on a new approximate Efron-Stein decomposition for high
dimensional link expanders.
