BiLipschitz instability of weighted norm inequalities for singular integral operators

We introduce singular integral operators — including the Hilbert transform and the Riesz transforms — and discuss characterizations of their boundedness between weighted \(L^2\) spaces. In particular: under what circumstances is boundedness stable under a biLipschitz change of variables? We go through examples of measure pairs for which biLipschitz stability fails — one a basic example involving the Hilbert transform, the other a pair of doubling measures for which the Riesz transform \(R_1\) in the plane is unstable.