The KKL inequality and Rademacher type 2

2022/04/11 by Ivanisvili, Paata, Stone, Yonathan · 1 citation
#46B07 #46B09 #60E15 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.2204.05195

Abstract

We show that a vector-valued Kahn--Kalai--Linial inequality holds in every Banach space of Rademacher type 2. We also show that for any nondecreasing function h≥ 0 with 0<∫₁^∞\frach(t)t²dt<∞ we have the inequality ‖f - 𝔼f‖₂ ≤ 12 T₂(X) (∫₁^∞\frach(t)t² dt )^1/2 (∑_j=1ⁿ \frac‖D_j f‖²₂h( log (‖D_j f‖₂)/(‖D_j f‖₁) ))^1/2 for all f :\-1,1\ⁿ → X and all n≥ 1, where X is a normed space and T₂(X) is the associated type 2 constant.

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