Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II

By Michael T. Lacey. 2012

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A conjecture of Nazarov--Treil--Volberg on the two weight inequality for the Hilbert transform is verified. Given two non-negative Borel measures u and w on the real line, the Hilbert transform $H_u$ maps $L^2(u)$ to $L^2(w)$ if and only if the pair of measures of satisfy a Poisson $A_2$ condition, and dual collections of testing conditions, uniformly over all intervals. This strengthens a prior characterization of Lacey-Sawyer-Shen-Uriate-Tuero arxiv:1201.4319. The latter paper includes a `Global to Local' reduction. This article solves the Local problem.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4dab78186f85deaf44d776dac2700e356b08981d

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Citations in the corpus

The two‐weight inequality for the Hilbert transform with general measures (2018, methodology): “We only need to remark that this result, by inspection of Lacey’s argument, remains valid with the definition of admissibility as formulated above; the third property imposed by Lacey [6] is only used when estimating the quantity size(Q).”
The A_2 theorem: Remarks and complements (2012, methodology): “The proof below follows the main lines of Treil’s argument [28], with one key difference: rather than splitting the summation over Q ∈ D in the expansion of 〈T (fσ), gω〉 into parts according to an ad hoc criterion such as σ(Q)〈f〉pQ ≥ ω(Q)〈g〉 ′ Q , I simply apply the “parallel corona” decomposition f”
Commutators in the two-weight setting (2015, result): “ has been obtained by Sawyer in [37,38]. For operators with a more singular nature we point to the work of Nazarov, Treil and Volberg in [33], Lacey, Sawyer, Shen, and Uriarte-Tuero in [24], Lacey in [20], Lacey, Sawyer, Shen, Uriarte-Tuero and Wick in [25] and Lacey and Wick COMMUTATORS IN THE TWO-”
Two weight norm inequalities for fractional integral operators and commutators (2014, background): “sucient for singular integral operators, beginning with the Hilbert transform. (Even this case is an extremely dicult problem which was only recently solved by Lacey, Sawyer, Shen and Uriarte-Tuero [50,53].) They also pointed out (see [101]) the close connection between testing conditions and the Da”