Inspired by a conjecture of Vladimir Maz’ya on Φ-inequalities in the spirit of Bourgain and Brezis, we establish some Φ-inequalities for fractional martingale transforms. These inequalities may be thought of as martingale models of Φ-inequalities for differential operators. The proofs rest on new simple Bellman functions.
"On Maz’ya’s Φ-Inequalities for Martingale Fractional Integration and Their Bellman Functions." Michigan Math. J. Advance Publication 1 - 18, 2023. https://doi.org/10.1307/mmj/20216116