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 $T(1)$ theorem of David and Journé is one of the most remarkable theorems in harmonic analysis. The theorem reduces the study of $L^{p}$ boundedness of a singular integral operator, $T$ to testing a 'testing condition', that is, verifying $T(1)$ is in the space $B M O$. A simplistic view of these theorems is that they shift the task of verifying boundedness for all functions (globally) to that of verifying a condition on all cubes. More general testing conditions, e.g. 'local $T(b)$' conditions, allow one to adapt the testing function to the cube and/or weaken conditions on the operator. These 'local $T(b)$ theorems' are an important ingredient to the initial solution to the Kato problem.
The project will introduce the concepts of $T(1) / T(b)$ theory for singular integrals, Littlewood-Paley theory, Carleson measures and stopping time arguments. The goal is to present the 'original' proof of the Kato problem and, possibly, look at more recent developments.[-]

We investigate the Dirichlet problem for a non-divergence form elliptic operator $L=a^{i j}(x) D_{i j}+b^{i}(x) D_{i}-c(x)$ in a bounded domain of $\mathbb{R}^{d}$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's function in a ball and derive two-sided pointwise estimates for it. Utilizing these results, we demonstrate the equivalence of regular points for $L$ and those for the Laplace operator, characterized via the Wiener test. This equivalence facilitates the unique solvability of the Dirichlet problem with continuous boundary data in regular domains. Furthermore, we construct the Green's function for $L$ in regular domains and establish pointwise bounds for it.[-]

It is known that in 3D exterior domains Ω with the compact smooth boundary $\partial \Omega$, two spaces $X^{r}_{har}\left ( \Omega \right )$ and $V^{r}_{har}\left ( \Omega \right )$ of $L^{r}$-harmonic vector fields $h$ with $h\cdot v\mid _{\partial \Omega }= 0$ and $h\times v\mid _{\partial \Omega }= 0$ are both of finite dimensions, where $v$ denotes the unit outward normal to $\partial \Omega$. We prove that for every $L^{r}$-vector field $u$, there exist $h\in X^{r}_{har}\left ( \Omega \right )$, $w\in H^{1,r}\left ( \Omega \right )^{3}$ with div $w= 0$ and $p\in H^{1,r}\left ( \Omega \right )$ such that $u$ is uniquely decomposed as $u= h$ + rot $w$ + $\bigtriangledown p$.
On the other hand, if for the given $L^{r}$-vector field $u$ we choose its harmonic part $h$ from $V^{r}_{har}\left ( \Omega \right )$, then we have a similar decomposition to above, while the unique expression of $u$ holds only for $1< r< 3$. Furthermore, the choice of $p$ in $H^{1,r}\left ( \Omega \right )$ is determined in accordance with the threshold $r= 3/2$.
Our result is based on the joint work with Matthias Hieber, Anton Seyferd (TU Darmstadt), Senjo Shimizu (Kyoto Univ.) and Taku Yanagisawa (Nara Women Univ.).[-]