A two weight inequality for the Hilbert transform assuming an Energy Hypothesis

Let σ and ω be locally finite positive Borel measures on R. Subject to the pair of weights satisfying a side condition, we characterize boundedness of the Hilbert transform H from L2(σ) to L2(ω) in terms of the A2 condition[∫I(|I||I|+|x−xI|)2dω(x)∫I(|I||I|+|x−xI|)2dσ(x)]12⩽C|I|, and the two testing conditions: For all intervals I in R∫IH(1Iσ)(x)2dω(x)⩽C∫Idσ(x),∫IH(1Iω)(x)2dσ(x)⩽C∫Idω(x). The proof uses the beautiful Corona argument of Nazarov, Treil and Volberg. There is a range of side conditions, termed Energy Conditions; at one endpoint, the Energy Conditions are also a consequence of the testing conditions above, and at the other endpoint they are the Pivotal Conditions of Nazarov, Treil and Volberg. We detail an example which shows that the Pivotal Conditions are not necessary for boundedness of the Hilbert transform.