Stability of Weighted Norm Inequalities

TL;DR

This paper investigates the stability of weighted norm inequalities for Riesz transforms, revealing that they are stable under biLipschitz changes for certain weights but unstable under rotations for others, highlighting fundamental differences between weight classes.

Contribution

It demonstrates the instability of Riesz transforms under rotations for doubling weights and establishes the necessity of testing conditions for characterizing inequalities.

Findings

Riesz transforms are stable under biLipschitz changes for $A_{ abla}$ weights.

Riesz transforms are unstable under rotations for doubling weights.

Iterated Riesz transforms of odd order are rotationally unstable on doubling weights.

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 for doubling measures, as opposed to the typically stable 'bump' conditions.