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

Abstract

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 L1(M˜w) into weak–L1(w). Our main innovation is that M˜ is a maximal function defined by directly using the local A∞ characteristic of the weight (rather than Orlicz norms). Prior results are due to Coifman&Fefferman, Pérez, Hytönen&Pérez, and Domingo-Salazar&Lacey&Rey.