Counterexample to the off-testing condition in two dimensions
Volume 167 / 2022
Abstract
In proving the local $T_b$ theorem for two weights in one dimension, Sawyer, Shen and Uriarte-Tuero used a basic theorem of Hytönen to deal with estimates for measures living in adjacent intervals. Hytönen’s theorem states that the off-testing constant for the Hilbert transform is controlled by Muckenhoupt’s $\mathcal {A}_2$ and $\mathcal {A}^*_2$ constants. So in attempting to extend the two-weight $T_b$ theorem to higher dimensions, it is natural to ask if a higher-dimensional analogue of Hytönen’s theorem holds that permits analogous control of terms involving measures that live on adjacent cubes. In this paper, we show that this is not the case even in the presence of the energy conditions used in one dimension. Thus, in order to obtain a local $T_b$ theorem in higher dimensions, it will be necessary to find some substantially new arguments to control the notoriously difficult “nearby form”. More precisely, we show that Hytönen’s off-testing constant for the two-weight fractional integral and the Riesz transform inequalities is not controlled by Muckenhoupt’s $\mathcal {A}_2^\alpha $ and $\mathcal {A}_2^{\alpha ,*}$ constants and energy constants.
