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.[-]

The goal of the talk is to present selected results in real harmonic analysis in the rational Dunkl setting. We shall start by deriving estimates for the generalized translations$$\tau_{\mathbf{x}} f(-\mathbf{y})=c_{k}^{-1} \int_{\mathbb{R}^{N}} E(\mathbf{x}, i \xi) E(\mathbf{y},-i \xi) \mathcal{F} f(\xi) d w(\xi)$$of certain radial and non-radial functions $f$ on $\mathbb{R}^{N}$, including estimates for the integral kernel of the heat Dunkl semigroup. Here $d w(\mathbf{x})=$ $\prod_{\alpha \in R}|\langle\alpha, \mathbf{x}\rangle|^{k(\alpha)} d \mathbf{x}$ denotes the associated measure, $E(\mathbf{x}, \mathbf{y})$ is the Dunkl kernel, and $\mathcal{F} f(\xi)=c_{k}^{-1} \int_{\mathbb{R}^{N}} f(\mathbf{x}) E(-i \xi, \mathbf{x}) f(\mathbf{x}) d w(\mathbf{x})$ is the Dunkl transform. The obtained estimates will be given by means of the distance $d(\mathbf{x}, \mathbf{y})$ of the orbit of $\mathbf{x}$ to the orbit of $\mathbf{y}$ under the action of the reflection group $G$, that is,$$d(\mathbf{x}, \mathbf{y})=\min _{\sigma \in G}\|\sigma(\mathbf{x})-\mathbf{y}\|$$the Euclidean distance $\|\mathbf{x}-\mathbf{y}\|$, and $d w$-volumes of the Euclidean balls and they will be in the spirit of estimates needed in real harmonic analysis on spaces of homogeneous type.Then, if time permits, we shall discuss selected results, parallel to classical ones, which are proved by utilizing the obtained estimates for the generalized translation. In particular, we will be interested in:- boundedness of maximal functions on various function spaces,- characterizations of the real Hardy space $H^{1}$ in the Dunkl setting- boundedness of the Dunkl transform multiplier operators,- boundedness of singular integral operators,- upper and lower bounds for Littlewood-Paley square functions. The results are joint works with Jean-Philippe Anker and Agnieszka Hejna.[-]

The boundedness (on $L^p$ spaces) of commutators $[b,T] = bT-Tb$ of pointwise multiplication $b$ and singular integral operators $T$ has been well studied for a long time. Curiously, the necessary conditions for this boundedness to happen are generally less understood than the sufficient conditions, for instance what comes to the assumptions on the operator $T$. I will discuss some new results in this direction, and show how this circle of ideas relates to the mapping properties of the Jacobian (the determinant of the derivative matrix) on first order Sobolev spaces. This is work in progress at the time of submitting the abstract, so I will hopefully be able to present some fairly fresh material.[-]