Borderline Weak Type Estimates for Singular Integrals and Square Functions

Abstract

For any Calderón-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, Pérez, and Hytönen-Pérez, on the $L(\log L{)}^{\epsilon }$ scale. Also, for square functions $Sf$, and weights $w\in A_p$, the norm of $S$ from $L^p(w)$ to weak-$L^p(w)$, $2\le p<\infty$, is bounded by $[w{]}_{A_p}^{1/2}(1+\log [w{]}_{A_{\infty }}{)}^{1/2}$, which is a sharp estimate.

13 pages, 1 figure: V2 A new title, a new result on the square function, and a new author

References in corpus (2)

• An elementary proof of the $A_2$ Bound
• Weighted Weak Type Estimates for Square Functions

Cited by in corpus (3)

• Weighted norm inequalities for rough singular integral operators
• Borderline weighted estimates for commutators of singular integrals
• On the embedding of $A_1$ into $A_{\infty }$