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

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