  We consider a two weight $L^{p}(\mu) \to L^{q}(\nu)$-inequality for well
localized operators as defined and studied by F. Nazarov, S. Treil and A.
Volberg when $p=q=2$. A counterexample of F. Nazarov shows that the direct
analogue of these results fails for for $p=q\not=2$. Here a new square function
testing condition is introduced and applied to characterize the two weight norm
inequality. The use of the square function testing condition is also
demonstrated in connection with certain positive dyadic operators.

Classical Analysis and ODEs