On the optimal $L_p$-$L_4$ Khintchine inequality

TL;DR

The paper determines the optimal dimension-free constant in the Khintchine inequality between the $p$-th and the 4th moments for $p\ge 4$, proving $C_{p,4}=\frac{\gamma_p}{\gamma_4}$. It provides two complementary proofs: a direct moment-comparison method valid for $4\le p\le 8$, and a reduction-based approach for $p\ge 5$ that relies on an auxiliary extremal-projection result. As an application, it derives a stability bound for the $p$-th vs second-moment Khintchine inequality with deficit $\frac{1}{6}\sum_i a_i^4$, and establishes optimal dimension-specific behavior. The work combines geometric analysis on spheres, a change of variables to a constrained hyperplane, and Nazarov–Podkorytov-type comparison principles to identify extremal configurations.

Abstract

We derive optimal dimension independent constants in the classical Khintchine inequality between the $p$th and fourth moment for $p\ge 4$. As an application we deduce stability estimates for the Khintchine inequality between the $p$th and second moment for $p \geq 4$.

Figures (1)

• Figure 1: The sets $A_\gamma$ for $\gamma \in(\frac{1}{3},\frac{1}{2})$, $\gamma=\frac{1}{2}$, and $\gamma \in (\frac{1}{2},1)$.

Theorems & Definitions (30)

• Theorem 1
• Proposition 2
• Corollary 1
• Proposition 3
• Lemma 4
• proof
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['thm:cp4']} for $4 \leq p \leq 8$
• Lemma 6