On $\Phi$-inequalities for martingale fractional integration and their Bellman functions

TL;DR

This paper develops new $\

Contribution

It introduces novel Bellman functions to prove $\

Findings

Established $\

Demonstrated $\

Provided a martingale model for differential operator inequalities.

Abstract

Inspired by a conjecture of Vladimir Maz'ya on $\Phi$-inequalities in the spirit of Bourgain and Brezis, we establish some $\Phi$-inequalities for fractional martingale transforms. These inequalities may be thought of as martingale models of $\Phi$-inequalities for differential operators. The proofs rest on new simple Bellman functions.