Lack of natural weighted estimates for some singular integral operators

We show that the classical Hörmander condition, or analogously the $L^r$-Hörmander condition, for singular integral operators $T$ is not sufficient to derive Coifman's inequality

where $0<p<\infty$, $M$ is the Hardy-Littlewood maximal operator, $w$ is any $A_{\infty}$ weight and $C$ is a constant depending upon $p$ and the $A_{\infty}$ constant of $w$. This estimate is well known to hold when $T$ is a Calderón-Zygmund operator.

As a consequence we deduce that the following estimate does not hold:

where $0<p\le 1$ and where $w$ is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever $T$ is a Calderón-Zygmund operator.

One of the main ingredients of the proof is a very general extrapolation theorem for $A_\infty$ weights.