Borderline weak-type estimates for sparse bilinear forms involving A∞ maximal functions

For any operator T whose bilinear form can be dominated by a sparse bilinear form, we prove that T is bounded as a map from L-1((M) over tildew) into weak-L-1(w). Our main innovation is that (M) over tilde is a maximal function defined by directly using the local A(infinity) characteristic of the weight (rather than Orlicz norms). Prior results are due to Coifman&Fefferman, Perez, Hytiinen&Perez, and Domingo-Salazar&Lacey&Rey. (C) 2021 Elsevier Inc. All rights reserved.