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A reprise of the NTV conjecture for the Hilbert transform

Eric T. Sawyer

TL;DR

This work presents a modified, self-contained proof of the two-weight Hilbert transform inequality (the NTV conjecture) for measures $\sigma$ and $\omega$ with hole-based $A_2$ control. It refines the Haar-based decomposition and corona constructions, replacing Poisson-energy arguments with a Sawyer–Wick functional-energy framework and employing an $\mathcal{F}$-Corona/NTV reach structure to control all pieces of the bilinear form: disjoint, comparable, neighbor, diagonal, far, paraproduct, and stopping forms. Central ingredients include a CZ–PE corona with a refined functional-energy bound, the Intertwining Proposition for the far form, the Stopping Child Lemma, and a dual-tree decomposition to handle stopping forms, all tying back to the testing and hole $A_2$ characteristics. Consequently, the Hilbert transform is bounded from $L^2(\sigma)$ to $L^2(\omega)$ with operator norm controlled by $\mathfrak{T}_H^{loc}(\sigma,\omega)$, $\mathcal{A}_2^{hole}(\sigma,\omega)$, and $\mathcal{E}_2(\sigma,\omega)$, plus related constants, matching the two-weight theory in this setting and highlighting a streamlined approach distinct from prior Poisson-based methods.

Abstract

We give a slightly different proof of the NTV conjecture for the Hilbert transform that was proved by T. Hytönen, 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.

A reprise of the NTV conjecture for the Hilbert transform

TL;DR

This work presents a modified, self-contained proof of the two-weight Hilbert transform inequality (the NTV conjecture) for measures and with hole-based control. It refines the Haar-based decomposition and corona constructions, replacing Poisson-energy arguments with a Sawyer–Wick functional-energy framework and employing an -Corona/NTV reach structure to control all pieces of the bilinear form: disjoint, comparable, neighbor, diagonal, far, paraproduct, and stopping forms. Central ingredients include a CZ–PE corona with a refined functional-energy bound, the Intertwining Proposition for the far form, the Stopping Child Lemma, and a dual-tree decomposition to handle stopping forms, all tying back to the testing and hole characteristics. Consequently, the Hilbert transform is bounded from to with operator norm controlled by , , and , plus related constants, matching the two-weight theory in this setting and highlighting a streamlined approach distinct from prior Poisson-based methods.

Abstract

We give a slightly different proof of the NTV conjecture for the Hilbert transform that was proved by T. Hytönen, 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.
Paper Structure (16 sections, 6 theorems, 153 equations)

This paper contains 16 sections, 6 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Reprise of the proof of the $NTV$ conjecture
  3. First steps and outline of the decomposition of forms
  4. Control of the disjoint and comparable forms by testing and offset Muckenhoupt characteristics
  5. Control of the neighbour form by the offset Muckenhoupt characteristic
  6. Reduction of the home form by coronas
  7. Control of the far form by way of functional energy
  8. Reduction of the diagonal form by the NTV reach
  9. Control of the paraproduct form by the testing characteristic
  10. Control of the stopping form by the Poisson-Energy characteristic
  11. The Stopping Child Lemma
  12. Dual tree decomposition
  13. Completion of the proof
  14. Appendix
  15. Local forward testing
  16. ...and 1 more sections

Key Result

Theorem 2

Let $\sigma$ and $\omega$ be positive locally finite Borel measures on $\mathbb{R}$. The Hilbert transform $H_{\sigma}$ is bounded from $L^{2}\left( \sigma\right)$ to $L^{2}\left( \omega\right)$if and only if the two tailed Muckenhoupt characteristic with holes, and both of the testing characteris

Theorems & Definitions (16)

  • Conjecture 1
  • Theorem 2
  • Lemma 4: Monotonicity and Energy Lemma
  • Lemma 5: Poisson Decay Lemma
  • Definition 6
  • Definition 7
  • Definition 8
  • Proposition 9: The Intertwining Proposition
  • proof
  • Lemma 10: Stopping Child Lemma
  • ...and 6 more