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In the recent paper of Sawyer-Wick, a characterization of the two-weight$L^p$ norm inequality inequality for Calder\'on-Zygmund operators on pairs ofdoubling measures was given, which involves three conditions: a scalar testingcondition, a quadratic, or vector-valued, weak boundedness property and aquadratic, or vector-valued, Muckenhoupt condition $A_{p} ^{\ell^2,\operatorname{local}}$. In this paper we show that the quadratic weakboundedness property in Sawyer-Wick can be dispensed with, and in addition showthat the remaining quadratic Muckenhoupt condition in Sawyer-Wick cannot bedispensed with. For the latter point, we show more precisely that for $p \in\left (1, \infty \right ) \setminus \left \{ 2 \right \}$, the scalar $T1$theorem for doubling measures on $L^p$ fails for a Riesz transform in$\mathbb{R}^2$, despite holding for $p=2$: namely, we construct a pair ofdoubling measures $(\sigma, \omega)$ that satisfy the scalar $A_p$ conditionand the scalar $L^p$-testing conditions for an individual Riesz transform$R_j$, and yet $\left ( R_j \right )_{\sigma} : L^p (\sigma) \not \to L^p(\omega)$.

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