One of the many meaningful equivalent norms on BMO uses a Carleson-measure condition on the gradient of the Poisson extension. This is closely related to the Dirichlet problem for the Laplacian in the upper half-space with boundary data in BMO. The Poisson semigroup provides the unique solution in appropriate classes, and it is bounded on BMO, that is, it propagates the space boundary space in the transversal direction. If the tangential Laplacian is replaced by a general elliptic operator in divergence form, boundedness of the Poisson semigroup on BMO can fail in any dimension n ≥ 3. Somewhat unexpectedly, its gradient persists to give rise to a Carleson measure with norm equivalent to the BMO-norm at the boundary in dimensions n = 3, 4 and hence a unique solution to the corresponding Dirichlet problem. In my talk, I will try to explain the broader context behind this phenomenon and why we still do not know if the result is sharp.
Based on joint work with (of course) Pascal. It is Chapter 18 of our book but you will not have to read the seventeen preceding chapters to follow.[-]

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 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.[-]