Borderline Weak Type Estimates for Singular Integrals and Square Functions

For any Calder'on-Zygmund operator $T$, any weight $w$, and $\alpha >1$, the operator $T$ is bounded as a map from $ L ^{1} (M _{ L \log\log L (\log\log\log L) ^{\alpha } } w )$intoweak-$L^1(w)$. The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman-Fefferman, P'erez, and Hyt"onen-P'erez, on the $L(\log L{)}^{ϵ}$ scale. Also, for square functions $Sf$, and weights $w\in A_p$, the norm of $S$ from $L^p(w)$ to weak-$L^p(w)$, $2\le p<\mathrm{\infty }$, is bounded by $ [w] _{A_p}^{1/2} (1+\log [w] {A \infty }) ^{1/2} $, which is a sharp estimate.