A simplified proof of an improved NTV conjecture for the Hilbert transform

Eric T. Sawyer

Published 2023-02-27Version 1

We give a simplified proof of the NTV conjecture for the Hilbert transform that was proved by T. Hyt\"onen, M. Lacey, E.T. Sawyer, C.-Y. Shen and I. Uriartre-Tuero, building on previous work of F. Nazarov, S. Treil and A. Volberg. Our proof shows a bit more, namely that the Hilbert transform is bounded from one weighted space to another if and only if the two local testing conditions hold and the classical offset Muckenhoupt condition holds. The proof avoids the use of functional energy, two weight inequalities for Poisson integrals, and recursion of admissible collections of pairs of intervals, but retains the bottom-up corona construction, and a variant of the straddling lemmas, from M. Lacey. Finally, the proof can be extended to certain more general operators on the real line, without assuming a classical energy condition.