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For any operator $T$ whose bilinear form can be dominated by a sparsebilinear 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 maximalfunction defined by directly using the local $A_\infty$ characteristic of theweight (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$M_{L({log log} L) ({log log log} L) ({log log log log} L)^{1+\epsilon}}$.

Borderline Weak--Type Estimates for Sparse Bilinear Forms Involving $A_\infty$ Maximal Functions