Contents Online

# Mathematical Research Letters

## Volume 6 (1999)

### Number 4

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

Pages: 417 – 427

DOI: https://dx.doi.org/10.4310/MRL.1999.v6.n4.a4

#### Authors

#### Abstract

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