Trace inequalities for Sobolev martingales

TL;DR

This paper investigates trace inequalities for Sobolev martingales, employing the Bellman function method to establish conditions for these inequalities to hold, extending classical results to the martingale setting.

Contribution

It introduces a Bellman function approach to derive trace inequalities for Sobolev martingales, providing nearly necessary and sufficient conditions.

Findings

Established trace inequalities for Sobolev martingales.

Identified conditions on martingale spaces and transforms for inequality validity.

Extended classical trace inequality results to the martingale framework.

Abstract

We study limiting trace inequalities in the style of Maz'ya and Meyers--Ziemer for Sobolev martingales. We develop the Bellman function approach to such estimates, which allows to provide sufficient and almost necessary conditions on the martingale space and the martingale transform under which the trace inequalities hold true