Auteurs : Treil, Serguei (Auteur de la Conférence)
CIRM (Editeur )
Résumé :
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.
Keywords : Matrix weights; martingale transform; Hilbert transform
Codes MSC :
42B20
- Singular and oscillatory integrals, several variables
42B35
- Function spaces arising in harmonic analysis
47A30
- Norms (inequalities, more than one norm, etc.)
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Informations sur la Rencontre
Nom de la rencontre : Shapes and shades of Analysis: in depth and beyond / Formes et nuances de l'analyse moderne Organisateurs de la rencontre : Abakumov, Evgeny ; Charpentier, Stéphane ; Kupin, Stanislas ; Tomilov, Yuri ; Zarouf, Rachid Dates : 29/04/2024 - 03/05/2024
Année de la rencontre : 2024
URL Congrès : https://conferences.cirm-math.fr/3004.html
DOI : 10.24350/CIRM.V.20168903
Citer cette vidéo:
Treil, Serguei (2024). The matrix $A_2$ conjecture fails, or $3 / 2>1$. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20168903
URI : http://dx.doi.org/10.24350/CIRM.V.20168903
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