# A reprise of the NTV conjecture for the Hilbert transform

## TL;DR

This paper presents a new proof of the NTV conjecture for the Hilbert transform, utilizing a different decomposition and control approach based on Sawyer's potential theorem, building on prior foundational work.

## Contribution

It introduces a novel proof technique for the NTV conjecture, modifying key components like the bilinear form decomposition and energy control methods.

## Key findings

- New proof of the NTV conjecture for the Hilbert transform
- Modified decomposition of the bilinear form
- Alternative control of functional energy using Sawyer's potential theorem

## 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.