Sharp Two-weight, weak-type norm inequalities for singular integral operators

We give a sufficient condition for singular integral operators and, more generally, Calderón-Zygmund operators to satisfy the weak $(p,p)$ inequality \[ u(\{ x\in \R^n : |Tf(x)| >t \}) \leq \frac{C}{t^p}\int_\subRn |f|^pv\,dx, \quad 1<p <\infty. \] Our condition is an $A_p$-type condition in the scale of Orlicz spaces: \[ \|u\|_{L(\log L)^{p-1+\delta},Q} \left(\frac{1}{|Q|}\int_Q v^{-p'/p}\,dx\right)^{p/p'} \leq K <\infty, \quad \delta >0.\] This conditions is stronger than the $A_p$ condition and is sharp since it fails when $\delta=0$.

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Published 1 January 1999