  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.
