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 $p\ne 2$: 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 ${\left(R_j\right)}_{\sigma }:L^p(\sigma )\nrightarrow L^p(\omega )$. On the other hand, we improve upon the quadratic, or vector-valued, $T1$ theorem of Sawyer-Wick when $p\ne 2$ 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 $A_p^{{\ell }^2,\mathrm{local}}$, 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.

43 pages, 6 figures. To appear in JLMS. Thanks to the thoughtful referee's comments that helped make the final version of this paper much clearer