The sharp weighted maximal inequalities for noncommutative martingales

TL;DR

This paper develops optimal weighted maximal inequalities for noncommutative martingales in operator algebras, with applications to noncommutative harmonic analysis and singular integrals.

Contribution

It introduces the first sharp weighted inequalities for noncommutative martingales, improving understanding of weighted bounds in noncommutative harmonic analysis.

Findings

Established optimal dependence of constants on weight characteristics.

Derived weighted estimates for noncommutative Hardy-Littlewood maximal operator.

Provided weighted bounds for noncommutative maximal truncations of singular integrals.

Abstract

The purpose of the paper is to establish weighted maximal $L_p$-inequalities in the context of operator-valued martingales on semifinite von Neumann algebras. The main emphasis is put on the optimal dependence of the $L_p$ constants on the characteristic of the weight involved. As applications, we establish weighted estimates for the noncommutative version of Hardy-Littlewood maximal operator and weighted bounds for noncommutative maximal truncations of a wide class of singular integrals.