Borderline Weak Type Estimates for Singular Integrals and Square Functions
For any Calderón-Zygmund operator $ T$, any weight $ w$, and $ α>1$, the operator $ T$ is bounded as a map from $ L ^{1} (M _{ L \log\log L (\log\log\log L) ^{α} } 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, Pérez, and Hytönen-Pérez, on the $ L (\log L) ^{ε}$ 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.
math.CA↗