A two weight inequality for the Hilbert transform assuming an Energy Hypothesis
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Subject to a range of side conditions, the two weight inequality for the
Hilbert transform is characterized in terms of (1) a Poisson A_2 condition on
the weights (2) A forward testing condition, in which the two weight inequality
is tested on intervals (3) and a backwards testing condition, dual to (2). A
critical new concept in the proof is an Energy Condition, which incorporates
information about the distribution of the weights in question inside intervals.
This condition is a consequence of the three conditions above. The Side
Conditions are termed 'Energy Hypotheses'. At one endpoint they are necessary
for the two weight inequality, and at the other, they are the Pivotal
Conditions of Nazarov-Treil-Volberg. This new concept is combined with a known
proof strategy devised by Nazarov-Treil-Volberg. A counterexample shows that
the Pivotal Condition are not necessary for the two weight inequality.
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