Borderline Weak--Type Estimates for Sparse Bilinear Forms Involving $A_\infty$ 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 $L^1(\widetilde{M}w)$ into weak--$L^1(w)$. Our main innovation is that $\widetilde{M}$ is a maximal function defined by directly using the local $A_\infty$ characteristic of the weight (rather than Orlicz norms). Prior results are due to Coifman\&Fefferman, P\'{e}rez, Hyt\"onen\&P\'erez, and Domingo-Salazar\&Lacey\&Rey. As we discuss, but do not prove, the maximal functions we use seem to be on the order of $M_{L({log log} L) ({log log log} L) ({log log log log} L)^{1+\epsilon}}$.