Shapes and shades of Analysis: in depth and beyond / Formes et nuances de l'analyse moderne

Collection Shapes and shades of Analysis: in depth and beyond / Formes et nuances de l'analyse moderne

Organisateur(s) Abakumov, Evgeny ; Charpentier, Stéphane ; Kupin, Stanislas ; Tomilov, Yuri ; Zarouf, Rachid
Date(s) 29/04/2024 - 03/05/2024
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00:00:00 / 00:00:00
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The matrix $A_2$ conjecture fails, or $3 / 2>1$

De Serguei Treil

The matrix $A_2$ condition on the matrix weight $W$$$[W]_{A_2}:=\sup _I\left|\langle W\rangle_I^{1 / 2}\left\langle W^{-1}\right\rangle_I^{1 / 2}\right|^2<\infty$$where supremum is taken over all intervals $I \subset \mathbb{R}$, and$$\langle W\rangle_I:=|I|^{-1} \int_I W(s) \mathrm{d} s,$$is necessary and sufficient for the Hilbert transform $T$ to be bounded in the weighted space $L^2(W)$.It was well known since early 90 s that $|T|_{L^2(W)} \gtrsim[W]_{A_2}^{1 / 2}$ for all weights, and that for some weights $|T|_{L^2(W)} \gtrsim[W]_{A_2}$. The famous $A_2$ conjecture (first stated for scalar weights) claims that the second bound is sharp, i.e.$$|T|_{L^2(W)} \lesssim[W]_{A_2}$$for all weights. After some significant developments (and some prizes obtained in the process) the scalar $A_2$ conjecture was finally proved: first by J. Wittwer for Haar multipliers, then by S. Petermichl for Hilbert Transform and for the Riesz transforms, and finally by T. Hytönen for general Calderón-Zygmund operators. However, while it was a general consensus that the $A_2$ conjecture is true in the matrix case as well, the best known estimate, obtained by Nazarov-Petermichl-Treil-Volberg (for all Calderón-Zygmund operators) was only $\lesssim[W]_{A_2}^{3 / 2}$. But this upper bound turned out to be sharp! In a recent joint work with K. Domelevo, S. Petermichl and A. Volberg we constructed weights $W$ such that$$|T|_{L^2(W)} \gtrsim[W]_{A_2}^{3 / 2},$$so the above exponent $3 / 2$ is a correct one. In the talk I'll explain motivations, history of the problem, and outline the main ideas of the construction. The construction is quite complicated, but it is an "almost a theorem" that no simple example is possible. This is joint work with K. Domelevo, S. Petermichl and A. Volberg.

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Données de citation

  • DOI 10.24350/CIRM.V.20168903
  • Citer cette vidéo Treil, Serguei (30/04/2024). The matrix $A_2$ conjecture fails, or $3 / 2>1$. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20168903
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