Necessary conditions for two-weight inequalities for singular integral operators
We prove necessary conditions on pairs $(\mu ,\nu )$ of measures for a singular integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\leq p \lt \infty $, provided the kernel of $T$ satisfies a weak non-degeneracy condition first introduced by Stein (1993), and the measure $\mu $ satisfies a weak doubling condition related to the non-degeneracy of the kernel. We also show similar results for pairs $(\mu ,\sigma )$ of measures for the operator $T_\sigma f = T(fd\sigma )$, which has come to play an important role in the study of weighted norm inequalities. Our major tool is a careful analysis of the strong type inequalities for averaging operators; these results are of interest in their own right. Finally, as an application of our techniques, we show that in general a singular operator does not satisfy the endpoint strong type inequality $T : L^1(\nu ) \rightarrow L^1(\mu )$. Our results unify and extend a number of known results.