Fractional integration for irregular martingales

TL;DR

This paper introduces two versions of Hardy--Littlewood--Sobolev inequalities tailored for discrete-time martingales, addressing issues of irregular filtrations and extending applicability beyond martingale transforms.

Contribution

It proposes novel fractional integration inequalities for martingales, including one that works with irregular filtrations despite potential vanishing issues.

Findings

Two versions of inequalities are established

One version uses martingale transforms, which may vanish with irregular filtrations

The other version is analytically meaningful without the martingale property

Abstract

We suggest two versions of the Hardy--Littlewood--Sobolev inequality for discrete time martingales. In one version, the fractional integration operator is a martingale transform, however, it may vanish if the filtration is excessively irregular; the second version lacks the martingale property while being analytically meaningful for an arbitrary filtration.