Counterexample to the off-testing condition in two dimensions

Abstract

In proving the local $T_b$ Theorem for two weights in one dimension [SaShUT] Sawyer, Shen and Uriarte-Tuero used a basic theorem of Hyt\"{o}nen [Hy] to deal with estimates for measures living in adjacent intervals. Hyt\"{o}nen's theorem states that the off-testing condition for the Hilbert transform is controlled by the Muckenhoupt's $A_2$ and $A^*_2$ conditions. 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 Hyt\"{o}nen's theorem holds that permits analogous control of terms involving measures that live on adjacent cubes. In this paper we show that it is not the case even in the presence of the energy conditions used in one dimension [SaShUT]. 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 Hyt\"{o}nen's off-testing condition for the two weight fractional integral and the Riesz transform inequalities is not controlled by Muckenhoupt's $A_2^\alpha$ and $A_2^{\alpha,*}$ conditions and energy conditions.