A Two Weight Inequality for the Hilbert transform Assuming an Energy Hypothesis

By Michael T. Lacey, Eric T. Sawyer, Ignacio Uriarte-Tuero. 2010

52 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

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In Applied Mathematics

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Citations in the corpus

Systems of dyadic cubes in a doubling metric space (2010, background): “These random dyadic systems have been instrumental in the development of the non-doubling theory of singular integrals [23, 24] and its applications to analytic capacity [27, 28] as well as sharp one-weight [9, 13] and two-weight [16, 25] inequalities for classical singular integrals.”
The two‐weight inequality for the Hilbert transform with general measures (2018, unknown): “1] and so-called ‘pivotal’ [16] and ‘energy’ [10] hypotheses.”
Sharp weighted estimates for classical operators (2010, methodology): “The exponent in Theorem 1.8 was first conjectured by Lerner [31] for the continuous square function; he also showed it was the best possible.”
A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure (2008, methodology): “Corresponding results for even the Hilbert transform have only recently been obtained [Nazarov et al. 2010; Lacey et al. 2011] and even then only for p = 2; evidently these are much harder to obtain.”