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, …