Lack of natural weighted estimates for some singular integral operators

Abstract:

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 \[ \int _{\mathbb {R}^n} |Tf(x)|^p w(x) dx \le C \int _{\mathbb {R}^n} M f(x)^p w(x) dx, \] 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: \[ \int _{\mathbb {R}^n} |Tf(x)|^p w(x) dx \le C \int _{\mathbb {R}^n} Mf(x)^p Mw(x) dx, \] 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.