The scalar T1 theorem for pairs of doubling measures fails for Riesz transforms when p not 2

Abstract We show that for an individual Riesz transform in the setting of doubling measures, the scalar theorem fails when : for each , we construct a pair of doubling measures on with doubling constant close to that of Lebesgue measure that also satisfy the scalar condition and the full scalar ‐testing conditions for an individual Riesz transform , and yet . On the other hand, we improve upon the quadratic , or vector‐valued, theorem of Sawyer and Wick [J. Geom. Anal. 35 (2025), 44] when on pairs of doubling measures: we dispense with their vector‐valued weak boundedness property to show that for pairs of doubling measures, the two‐weight norm inequality for the vector Riesz transform is characterized by a quadratic Muckenhoupt condition and a quadratic testing condition. Finally, in the Appendix, we use constructions of Kakaroumpas and Treil [Adv. Math. 376 (2021), 107450] to show that the two‐weight norm inequality for the maximal function cannot be characterized solely by the condition when the measures are doubling, contrary to reports in the literature.