Hypercontractivity on HDX II: Symmetrization and q-Norms

Bourgain’s symmetrization theorem is a powerful technique reducing boolean analysis on product spaces to the cube. It states that for any product Ωi⊗ d, function f: Ωi⊗ d → , and q > 1: ||T_1/2f(x)||_q ≤||f(r,x)||_q ≤||T_c_qf(x)||_q where Tρf = ∑ρSf=S is the noise operator and f(r,x) = ∑rSf=S(x) ‘symmetrizes’ f by convolving its Fourier components f=SS ⊆ [d] with a random boolean string r ∈ ± 1d. In this work, we extend the symmetrization theorem to high dimensional expanders (HDX). Building on (O’Donnell and Zhao 2021), we show this implies nearly-sharp (2→q)-hypercontractivity for partite HDX. This resolves the main open question of (Gur, Lifshitz, and Liu STOC 2022) and gives the first fully hypercontractive subsets X ⊂ [n]d of support n·exp((d)), an exponential improvement over Bafna, Hopkins, Kaufman, and Lovett’s n·exp(exp(d)) bound (BHKL STOC 2022). Adapting (Bourgain JAMS 1999), we also give the first booster theorem for HDX, resolving a main open question of BHKL. Our proof is based on two elementary new ideas in the theory of high dimensional expansion. First we introduce ‘q-norm HDX’, generalizing standard spectral notions to higher moments, and observe every spectral HDX is a q-norm HDX. Second, we introduce a simple method of coordinate-wise analysis on HDX which breaks high dimensional random walks into coordinate-wise components and allows each component to be analyzed as a 1-dimensional operator locally within X. This allows for application of standard tricks such as the replacement method, greatly simplifying prior analytic techniques. This is an extended abstract. The full paper may be found at https://arxiv.org/abs/2408.16687