Bellman function method for general operators on martingales

TL;DR

This paper demonstrates how the Bellman function method can be used to analyze the $L^p$-norms of a broad class of operators on martingales, including those not directly related to martingale transforms.

Contribution

It introduces a unified Bellman function approach to study the $L^p$-boundedness of general martingale operators, extending previous methods to a wider class of operators.

Findings

Bellman function method applies to various martingale operators.

Unified approach encodes $L^p$-boundedness for almost all operators.

Examples include Haar transforms and Walsh system operators.

Abstract

It is shown that the Bellman function method can be applied to study the $L^p$-norms of general operators on martingales, i.e., of operators that are not necessarily martingale transforms. Informally, we provide a single Bellman-type function that "encodes" the $L^p$-boundedness of "almost all" operators from Gundy's extrapolation theorem. As examples of such operators, we consider the Haar transforms and the operator whose $L^p$-boundedness underlies Rubio de Francia's inequality for the Walsh system.