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Noncommutative sharp dual Doob inequalities

Fedor Sukochev, Dejian Zhou

TL;DR

This work addresses sharp noncommutative dual Doob inequalities in a finite von Neumann algebra setting with an increasing sequence of subalgebras and conditional expectations. It proves two sharp inequalities with best constants: for $0<p\le1$, $\| \sum_{k=1}^n x_k\|_{L_p(\mathcal{M})} \le \frac{1}{p} \| \sum_{k=1}^n \mathcal{E}_k(x_k)\|_{L_p(\mathcal{M})}$, and for $1\le p\le 2$, $\| \sum_{k=1}^n \mathcal{E}_k(x_k)\|_{L_p(\mathcal{M})} \le p \| \sum_{k=1}^n x_k\|_{L_p(\mathcal{M})}$, with the constants being optimal. The proofs deploy two key lemmas, a telescoping argument on $A_n=\sum_{k=1}^n x_k$ and $B_n=\sum_{k=1}^n \mathcal{E}_k(x_k)$, and noncommutative Hölder and Young inequalities to establish the sharp bounds, offering a different route from prior approaches. The results yield sharp noncommutative martingale square-function inequalities, Burkholder-Gundy-type bounds, and a refined noncommutative Stein inequality, while recovering the classical commutative sharp estimates in Wa1991. Overall, the paper advances precise constant estimates and alternative proofs in noncommutative martingale theory with potential broader applicability.

Abstract

Let $(x_k)_{k=1}^n$ be positive elements in the noncommutative Lebesgue space $L_p(\mathcal{M})$, and let $(\mathcal{E}_k)_{k=1}^n$ be a sequence of conditional expectations with respect to an increasing subalgebras $(\mathcal{M}_n)_{k\geq1}$ of the finite von Neumann algebra $\mathcal{M}$. We establish the following sharp noncommutative dual Doob inequalities: \begin{equation*} \Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})}\leq \frac{1}{p} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})},\quad 0<p\leq 1, \end{equation*} and \begin{equation*} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})}\leq p\Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})},\quad 1\leq p\leq 2. \end{equation*} As applications, we obtain several noncommutative martingale inequalities with better constants.

Noncommutative sharp dual Doob inequalities

TL;DR

This work addresses sharp noncommutative dual Doob inequalities in a finite von Neumann algebra setting with an increasing sequence of subalgebras and conditional expectations. It proves two sharp inequalities with best constants: for , , and for , , with the constants being optimal. The proofs deploy two key lemmas, a telescoping argument on and , and noncommutative Hölder and Young inequalities to establish the sharp bounds, offering a different route from prior approaches. The results yield sharp noncommutative martingale square-function inequalities, Burkholder-Gundy-type bounds, and a refined noncommutative Stein inequality, while recovering the classical commutative sharp estimates in Wa1991. Overall, the paper advances precise constant estimates and alternative proofs in noncommutative martingale theory with potential broader applicability.

Abstract

Let be positive elements in the noncommutative Lebesgue space , and let be a sequence of conditional expectations with respect to an increasing subalgebras of the finite von Neumann algebra . We establish the following sharp noncommutative dual Doob inequalities: \begin{equation*} \Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})}\leq \frac{1}{p} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})},\quad 0<p\leq 1, \end{equation*} and \begin{equation*} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})}\leq p\Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})},\quad 1\leq p\leq 2. \end{equation*} As applications, we obtain several noncommutative martingale inequalities with better constants.
Paper Structure (2 sections, 10 theorems, 45 equations)

This paper contains 2 sections, 10 theorems, 45 equations.

Table of Contents

  1. Introduction and Main results
  2. Proof of the main results

Key Result

Theorem 1

Let $0<p\leq 1$. Then for any sequence of positive operators $(x_k)_{k=1}^n$ in $\mathcal{M}$, the following holds: The constant $1/p$ is best possible.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • proof
  • Corollary 6
  • proof
  • Lemma 7
  • proof
  • ...and 11 more