A reprise of the NTV conjecture for the Hilbert transform
Abstract
We give a slightly different 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. Uriarte-Tuero, building on previous work of F. Nazarov, S. Treil and A. Volberg. After modifying the decomposition of the main bilinear form, we give a new proof of control of functional energy that is based on the potential Theorem 1 of [Saw3], rather than the Poisson Theorem 2 that is used in all other proofs in the literature. This approach was pioneered in the first version of Sawyer and Wick [SaWi] on the ArXiv. Then we alter the bottom-up corona construction, the size functional, the straddling lemmas, and the use of recursion of admissible collections of pairs of intervals, from M. Lacey [Lac]. However, the essence of control of the stopping form remains as in the fundamental work of Lacey.
Keywords
Cite
@article{arxiv.2302.13920,
title = {A reprise of the NTV conjecture for the Hilbert transform},
author = {Eric T. Sawyer},
journal= {arXiv preprint arXiv:2302.13920},
year = {2025}
}
Comments
27 pages, the new control of functional energy corrected in this version, which also includes an appendix on the alternate approach of Sawyer and Wick to controlling functional energy. Thanks to Brett Wick for continuing fruitful discussions