Summary: 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 \[ \int_{\mathbb{R} ^n} | Tf(x)|^p\, w(x)\, dx \leq C\,\int_{\mathbb{R} ^n} M f(x)^p\, w(x)\,dx, \] 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: \[ \int_{\mathbb{R} ^n} | Tf(x)|^p\, w(x)\, dx \leq C\,\int_{\mathbb{R} ^n} Mf(x)^p\, Mw(x)\,dx, \] where \(0<p\leq 1\) 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.