Sparse domination and the strong maximal function (Q1721936)

A family \(\mathcal S\) of sets is called \(\eta\)-sparse if for all \(Q\in \mathcal S\) there exists a subset \(E(Q)\subset Q\) with \(|E(Q)|\geq \eta |Q|\) and \(\{E(Q)\}\) is pairwise disjoint. Many operators \(T\) in Harmonic Analysis are dominated by sparse forms, meaning that \(|\langle Tf, g\rangle |\leq \sum_{Q\in \mathcal S} \langle |f|\rangle_Q \langle |g|\rangle_Q |Q|\) for nice functions, where \(\langle |f|\rangle_Q \langle=\frac{1}{|Q|}\int_Q |f|\). This is a helpful tool to get boundedness and sharp constants for the norm of the operator acting on function spaces. The authors study the problem of sparsely dominating the bi-parametric analogue of the dyadic maximal function, meaning \(\mathcal M_S f(x)=\sup _{x\in R} \langle |f|\rangle_R \) where the supremum is taken over all dyadic rectangles containing \(x\) with sides parallel to the axes. Their main result establishes that for every \(C>0\) and \(0<\eta<1\) there exists a pair of compactly supported functions such that \(|\langle \mathcal M_S f, g\rangle |\geq C\sum_{R\in \mathcal S} \langle |f|\rangle_R \langle |g|\rangle_R |R|\) for all \(\eta\)-sparse collections \(\mathcal S\) of dyadic rectangles with sides parallel to the axes. The proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance.