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The matrix $A_2$ conjecture fails, i.e. $3/2>1$

Komla Domelevo, Stefanie Petermichl, Sergei Treil, Alexander Volberg

Abstract

We show that the famous matrix $A_2$ conjecture is false: the norm of the Hilbert Transform in the space $L^2(W)$ with matrix weight $W$ is estimated below by $C[W]_{{A}_2}^{3/2}$.

The matrix $A_2$ conjecture fails, i.e. $3/2>1$

Abstract

We show that the famous matrix conjecture is false: the norm of the Hilbert Transform in the space with matrix weight is estimated below by .
Paper Structure (41 sections, 22 theorems, 252 equations, 5 figures)

This paper contains 41 sections, 22 theorems, 252 equations, 5 figures.

Table of Contents

  1. Definitions and main result
  2. Introduction
  3. Motivation
  4. Some scalar history
  5. Vector history
  6. The idea of our counterexample
  7. Preliminaries
  8. Some definitions
  9. Properties of A2 weights
  10. Representation of weights as martingales
  11. Construction of the dyadic weight
  12. Families of weights
  13. Starting point
  14. About initial eigenvalues
  15. Rotations
  16. ...and 26 more sections

Key Result

Theorem 1.1

There exists a constant $c>0$ such that for all sufficiently large $\boldsymbol{\mathcal{Q}}$ there exist a $2\times 2$ matrix weight $W=W{\newline}_{\!\!\boldsymbol{\mathcal{Q}}}$(with real entries), $[W]{\newline}_{\mathbf{A}_2}\le \boldsymbol{\mathcal{Q}}$ and a function $g\in L^2(W)$, $g:\mathbb

Figures (5)

  • Figure 1: Rotation: The picture shows the rotation step. The blue ellipses are the images of the unit ball under operators $\langle W \rangle _I$ (the top line) and $\langle W \rangle _{I_\pm}$ (bottom line). The red ellipses are the corresponding images under the averages $\langle W^{-1} \rangle _I$ and $\langle W^{-1} \rangle _{I_\pm}$ respectively.
  • Figure 2: Stretching $(x,y) \mapsto (x_+,y_+)$
  • Figure 3: Dyadic tree: the picture shows two dyadic time steps: a rotation followed by stretching.
  • Figure 4: The first intervals on which periodisation is applied are $\operatorname{Start}_1=\{ I^0_-,I^0_+\}$. The function $f|_{I^0_-}$ is copied $2^{N_1}=8$ times onto the $8$ intervals in $\mathcal{D}_4\cap{I^0_-}$. The grandchildren $\mathcal{D}_6\cap{I^0_-}$ of those $8$ intervals constitute the new starting intervals $\operatorname{Start}_2\cap{I^0_-}$ below $I^0_-$. Each of these grandchildren carries a copy of one of the grandchildren of $I^0_-$. For example the interval $J\in\mathcal{D_3}$ marked in red is the second grandchild (counting from the left) of $I^0_-$. Correspondingly the intervals $I\in\operatorname{Start}_2\cap{I^0_-}$ marked in red are the second grandchildren of intervals in $\mathcal{D}_4\cap{I^0_-}$. As such, the second difference $\Delta^2_I f$ on those intervals is a compressed version of the second difference $\Delta^2_J f$ through the map $\Psi_{J,I}$, with $J=\mathcal{F}(I)$.
  • Figure 5: The first intervals on which quasi-periodisation is applied are again $\operatorname{Start}_1=\{ I^0_-,I^0_+\}$. The function $f|_{I^0_-}$ is copied $2^{N_1}-2=6$ times on the $6$ regular stopping intervals $\mathcal{R}(I^0_-)$ in $\mathcal{D}_4\cap{I^0_-}$ that are not touching the boundary of $I^0_-$. The two blue intervals from $\mathcal{D}_4\cap{I^0_-}$ that are touching the boundary of $I^0_-$ are the two exceptional stopping intervals $\mathcal{E}(I^0_-)$ below $I^0_-$, the red intervals are $\mathcal{E}(I^0_+)$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • Remark 3.5
  • proof : Proof of Lemma \ref{['l:stop']}
  • Remark 3.6
  • ...and 44 more