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.

翻译：在高维扩张器上,我们证明存在超强的不平等性。 正如在有偏向的超立方体、对称组和格拉斯曼体系的环境下一样,我们的不平等性对于全球功能是有效的,这些功能没有受到一组小坐标限制的显著影响。作为应用,我们获得了高维扩张器的Fourier浓度、小型扩张和Kruskal-Katona理论。我们的技术依靠一种新型的Efron-Stein分解法。