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We prove hypercontractive inequalities on high dimensional expanders. As inthe settings of the p-biased hypercube, the symmetric group, and the Grassmannscheme, our inequalities are effective for global functions, which arefunctions that are not significantly affected by a restriction of a small setof coordinates. As applications, we obtain Fourier concentration, small-setexpansion, and Kruskal-Katona theorems for high dimensional expanders. Ourtechniques rely on a new approximate Efron-Stein decomposition for highdimensional link expanders.

Hypercontractivity on High Dimensional Expanders: Approximate Efron-Stein Decompositions for $\varepsilon$-Product Spaces