Mathematical Research Letters

Volume 6 (1999)

Number 4

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

Pages: 417 – 427



D. Cruz-Uribe (Trinity College)

C. Pérez (Universidad Autónoma de Madrid)


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 $\delta=0$.

Published 1 January 1999