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 $L ^{1} (M _{ L \log\log L (\log\log\log L) ^{\alpha } } w )$ into weak-$L^1(w)$. 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 $L (\log L) ^{\epsilon }$ scale. Also, for square functions $S f$, and weights $w \in A_p$, the norm of $S$ from $L ^p (w)$ to weak-$L^p (w)$, $2\leq p < \infty$, is bounded by $[w] _{A_p}^{1/2} (1+\log [w] _{A_ \infty }) ^{1/2}$, which is a sharp estimate.