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 ∫ ℝn |T f(x)≤ p w(x) dx ≤ C ∫ ℝn M f(x) p w(x) dx, where 0 < p < ∞, M is the Hardy-Littlewood maximal operator, w is any A ∞ weight and C is a constant depending upon p and the A ∞ 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: ∫ ℝn |T f(x)| p w(x) dx ≤ C ∫ ℝn M f(x) p Mw(x) dx, where 0 < p < 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 ∞ weights.