Continuous-time sparse domination

TL;DR

This paper extends sparse domination techniques to continuous-time martingales, providing sharp weighted inequalities for maximal operators and addressing longstanding questions in the theory of stochastic processes with jumps.

Contribution

It introduces a novel continuous-time sparse domination framework for martingales, including those with jumps, and proves sharp weighted inequalities for their maximal operators.

Findings

Established a sharp weighted L^p estimate for the maximal operator of martingales.

Provided a simple proof for the case where Y equals X.

Addressed a question from the 1970s regarding martingales with jumps.

Abstract

We develop the self similarity argument known as sparse domination in an abstract martingale setting, using a continuous time parameter. With this method, we prove a sharp weighted L^p estimate for the maximal operator Y^* of Y with respect to X. Here Y and X are uniformly integrable c\`adll\`ag Hilbert space valued martingales and Y differentially subordinate to X via the square bracket process. We also present a second, very simple proof of the special case Y=X. In this generality, notably including processes with jumps, the special case Y = X addresses a question raised in the late 70s by Bonami--L\'epingle.