A dual approach to Burkholder's $L^p$ estimates

TL;DR

This paper presents an alternative proof of Burkholder's $L^p$ estimates for differentially subordinate martingales, connecting the estimates to a new boundary value problem.

Contribution

It introduces a novel boundary value problem approach to establish $L^p$ estimates, offering a different perspective from classical proofs.

Findings

Proof of $L^p$ estimates via boundary value problem

New connection between martingale inequalities and boundary problems

Potential for generalized applications in stochastic analysis

Abstract

The paper contains an alternative proof of the celebrated $L^p$ estimates for differentially subordinate martingales established by Burkholder and Wang in the eighties and nineties. The approach links the validity of the estimate to the existence of a lower solution to a novel boundary value problem.