Stability of weighted norm inequalities

Abstract

We show that while individual Riesz transforms are two-weight norm stable under biLipschitz change of variables on $A_{\infty }$ weights, they are two-weight norm unstable under even rotational change of variables on doubling weights. More precisely, we show that individual Riesz transforms are unstable under a set of rotations having full measure, which includes rotations arbitrarily close to the identity. This provides an operator theoretic distinction between $A_{\infty }$ weights and doubling weights. More generally, all iterated Riesz transforms of odd order are rotationally unstable on pairs of doubling weights, thus demonstrating the need for characterizations of iterated Riesz transform inequalities using testing conditions as appearing in the work of Nazarov, Treil and Volberg, and other works by subsets of the authors Alexis, Lacey, Sawyer, Shen, Uriarte-Tuero and Wick, as opposed to the typically stable ’bump’ conditions.