The height function of a sparse collection: a Bellman function approach

Sparse operators have emerged as a powerful method to extract sharp constants in harmonic analysis inequalities, for example in the context of bounding singular integral operators. We investigate the level sets of height functions for sparse collections, or, in other words, weak-type (1,1) inequalities for sparse operators applied to constant functions. We use another notable method from dyadic harmonic analysis, also famous for its ability to produce sharp constants, the Bellman function method. Specifically, we find the exact Bellman function maximizing level sets of $\mathcal{A}_\alpha 1\!\!1$, where $\mathcal{A}_\alpha$ is the (localized) sparse operator associated with a binary Carleson sequence.