Failure of necessity of the energy condition
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abstract

We give an example of a pair of weights (u,v) on the line, and an elliptic
convolution singular integral operator H on the line, such that H_u is bounded
from L^2(u) to L^2(v), yet the measure pair (u,v) fails to satisfy the backward
energy condition. The key to the construction is that the kernel K of H has
flat spots where d/dx K(x) = 0. Conversely, we show that if H is gradient
elliptic, i.e. d/dx K(x) =< c < 0, then the energy conditions are necessary for
boundedness of H, and by our theorem in arXiv:1603.04332v2, the T1 theorem
holds for H.