A ratio inequality for nonnegative martingales and their differential subordinates

Adam Osękowski

doi:10.1214/20-ecp301

Abstract

We prove a sharp inequality for the ratio of the maximal functions of nonnegative martingales and their differential subordinates. An application in the theory of weighted inequalities is given.

Highlights

Suppose that (Ω, F, P) is a complete probability space, filtered with a nondecreasing right-continuous family (Ft)t≥0 of sub-σ-fields of F, such that F0 contains all the events of probability 0
We prove a sharp inequality for the ratio of the maximal functions of nonnegative martingales and their differential subordinates
Let X, Y be two adapted martingales, taking values in a certain separable Hilbert space H with norm | · | and scalar product denoted by the dot ·

Summary

Introduction

Suppose that (Ω, F , P) is a complete probability space, filtered with a nondecreasing right-continuous family (Ft)t≥0 of sub-σ-fields of F , such that F0 contains all the events of probability 0. Throughout the paper we will assume that the process Y is dominated by X in the sense of the so-called differential subordination: following Bañuelos and Wang [1] and Wang [17], we say that Y is differentially subordinate to X, if the process ([X, X]t − [Y, Y ]t)t≥0 is nondecreasing and nonnegative as a function of t The origins of this domination principle go back to the works of Burkholder [3] in the discrete-time case: a martingale g = (gn)n≥0 is differentially subordinate to f = (fn)n≥0, if for any n ≥ 0 we have |dgn| ≤ |dfn| almost surely. Observe that the inequalities of Theorems 1.1 and 1.2 can be regarded as comparisons of sizes of X and Y measured separately in terms of strong or weak Lp norms and, as such, they do not say anything about the joint behavior of X and Y From this point of view, a very natural functional to study is the ratio |Y |/|X|. The final part of the paper is devoted to some applications of the estimate (1.2) in the theory of weighted inequalities

A special function and its properties

An application