The KKL inequality and Rademacher type 2

TL;DR

This paper extends the Kahn--Kalai--Linial inequality to Banach spaces of Rademacher type 2 and establishes new bounds involving functions with specific integrability conditions, advancing understanding in vector-valued inequalities.

Contribution

It proves a vector-valued KKL inequality in Banach spaces of Rademacher type 2 and derives a new inequality involving a nondecreasing function with integrability conditions.

Findings

Established the KKL inequality in Rademacher type 2 Banach spaces.

Derived a new inequality relating function deviations to derivatives with integrability conditions.

Provided bounds involving the type 2 constant and specific functions for vector-valued functions.

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 $h\geq 0$ with $0<\int_{1}^{\infty}\frac{h(t)}{t^{2}}\mathrm{dt}<\infty$ we have the inequality \begin{align*} \|f - \mathbb{E}f\|_2 \leq 12 \, T_{2}(X) \left(\int_{1}^{\infty}\frac{h(t)}{t^{2}} \mathrm{dt} \right)^{1/2} \, \left(\sum_{j=1}^n \frac{\|D_j f\|^{2}_2}{h\left( \log \frac{\|D_j f\|_2}{\|D_j f\|_1} \right)}\right)^{1/2} \end{align*} for all $f :\{-1,1\}^{n} \to X$ and all $n\geq 1$, where $X$ is a normed space and $T_{2}(X)$ is the associated type 2 constant.