We provide an essentially complete dictionary of all implications among the basic and fundamental conditions in weighted theory such as the doubling, one weight $A_p(w)$, $A_\infty$ and $C_p$ conditions as well as the two weight $\A_p(\o,\s)$ and the ``buffer" Energy and Pivotal conditions. The most notable implication is that in the case of $A_\infty$ weights the two weight $\A_p$ condition implies the $p-$Pivotal condition hence giving an elegant and short proof of the known NTV-conjecture with $p=2$ for $A_\infty$ weights in terms of existing T1 theory. We also provide a quite technical construction inspired by \cite{GaKS} proving that we can have doubling weights satisfying the $C_p$ condition which are not in $A_\infty$. We obtain a local two weight $Tb$ theorem with an energy side condition for higher dimensional fractional Calder\'{o}n-Zygmund operators. The proof follows the general outline of the proof for the corresponding one-dimensional $Tb$ theorem in \cite{SaShUr12}, but encountering a number of new challenges, including several arising from the failure in higher dimensions of T. Hyt\"{o}nen's one-dimensional two weight $A_{2}$ inequality. Hyt\"{o}nen used this inequality 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.We show that it is not the case even in the presence of the energy conditions used in one dimension \cite{SaShUr12}. Thus, in order to obtain a local $T_b$ theorem in higher dimensions, it was 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.
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