Borderline Weak--Type Estimates for Sparse Bilinear Forms Involving $A_{\mathrm{\infty }}$ 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 $L^1(\tilde{M}w)$ into weak–$L^1(w)$. Our main innovation is that $\tilde{M}$ is a maximal function defined by directly using the local $A_{\mathrm{\infty }}$ 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 $M_{L(loglogL)(logloglogL)(loglogloglogL{)}^{1+ϵ}}$.