we pose

For $ 0< \alpha < 1/2$, almost surely, for all $ 1< r < 2$, and finitely supported functions $ f, g$, there is a sparse operator $ \Lambda _{1,r}$ so that $$\langle M _{\alpha } f , g \rangle \lesssim \Lambda _{1,r} (f,g).$$

We turn to the context for our paper. The concept of sparse operators arose from Lerner’s remarkable median inequality [@MR2721744]. It’s application to weighted inequalities was advanced by several authors, with a high point of this development being Lerner’s argument [@MR3085756] showing that the weighted norm of Calderón-Zygmund operators is comparable to that of the norms of sparse operators. This lead to the question of pointwise control, namely Theorem \[t:czo\]. First established by Conde-Alonso and Rey [@14094351], also see Lerner and Nazarov [@150805639], the author [@150105818] established Theorem \[t:czo\] with a stopping time argument. The latter argument was extended by Bernicot, Frey and Petermichl [@MR3531367] to a setting where the operators are generated by semigroups, including examples outside the scope of classical Calderón-Zygmund theory. For closely related developments see [@MR3484688; @151005789]. The sparse bounds for commutators [@160401334; @160405506] are remarkably powerful. Edging beyond the Calderón-Zygmund context, Benau, Bernicot and Frey [@160506401] have supplied sparse 