# The scalar $T1$ theorem for pairs of doubling measures fails for Riesz transforms when $p>2$ is an even integer

arxiv（2023）

Abstract

For $p \in \{ 2m ~:~ m \geq 2 \} \cup \{\frac{2m}{2m-1} ~:~ m \geq 2 \}$, we show that on $L^p$, the scalar $T1$ theorem for doubling measures fails for a Riesz transform in $\mathbb{R}^2$, despite holding for $p=2$, as shown by Alexis-Sawyer-Uriarte-Tuero. More precisely, we construct a pair of doubling measures $(\sigma, \omega)$ that satisfy the scalar $A_p$ condition and the 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 provide an improvement of the positive result of Sawyer-Wick for all $p \neq 2$ by removing the vector-valued, or quadratic, weak-boundedness property, showing that if a pair of doubling measures satisfies scalar testing and a vector-valued, or quadratic, Muckenhoupt condition $A_{p} ^{\ell^2, \operatorname{local}}$, then the norm inequality holds.

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