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 $T1$ theorem fails when $p \neq 2$: for each $ p \in (1, \infty) \setminus \{2\}$, we construct a pair of doubling measures $(\sigma, \omega)$ on $\mathbb{R}^2$ with doubling constant close to that of Lebesgue measure that also satisfy the scalar $\mathcal{A}_p$ condition and the full scalar $L^p$-testing conditions for an individual Riesz transform $R_j$, and yet $\left ( R_j \right )_{\sigma} : L^p (\sigma) \not \to L^p (\omega)$. On the other hand, we improve upon the quadratic, or vector-valued, $T1$ theorem of Sawyer-Wick when $p \neq 2$ 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 $L^p$ norm inequality for the vector Riesz transform is characterized by a quadratic Muckenhoupt condition $A_{p} ^{\ell^2, \operatorname{local}}$, and a quadratic testing condition. Finally, in the appendix, we use constructions of Kakaroumpas-Treil to show that the two-weight norm inequality for the maximal function cannot be characterized solely by the $A_p$ condition when the measures are doubling, contrary to reports in the literature.