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.[-]

For an open set $\Omega \subset \mathbb{R}^{d}$ with an Ahlfors regular boundary, solvability of the Dirichlet problem for Laplaces equation, with boundary data in $L^{p}$ for some $p<\infty$, is equivalent to quantitative, scale invariant absolute continuity (more precisely, the weak- $A_{\infty}$ property) of harmonic measure with respect to surface measure on $\partial \Omega$. A similar statement is true in the caloric setting. Thus, it is of interest to find geometric criteria which characterize the open sets for which such absolute continuity (hence also solvability) holds. Recently, this has been done in the harmonic case. In this talk, we shall discuss recent progress in the caloric setting, in which we show that quantitative absolute continuity of caloric measure, with respect to surface measure on the parabolic Ahlfors regular (lateral) boundary $\Sigma$, implies parabolic uniform rectifiability of $\Sigma$. We observe that this result may be viewed as the solution of a certain 1-phase free boundary problem. This is joint work with S. Bortz, J. M. Martell and K. Nyström.[-]

To solve the Kato conjecture in the lectures, we first reformulated the Kato property as a square function estimate. One of the main characters in this reformulation was McIntosh's theorem, which states that a sectorial operator $L$ on a Hilbert space $H$ has a bounded $H^{\infty}$-calculus if and only if for some (equivalently all) nonzero $f \in H_{0}^{\infty}\left(S_{\varphi}\right)$ the quadratic estimate$$\begin{equation*}\left(\int_{0}^{\infty}\|f(t L) u\|_{H}^{2} \frac{\mathrm{d} t}{t}\right)^{1 / 2} \approx\|u\|_{H}, \quad u \in H \tag{2.3}\end{equation*}$$holds. Since neither the definition of the $H^{\infty}$-calculus, nor the statement of McIntosh's theorem explicitly use the Hilbert space structure of $H$, one may wonder if this theorem is also true for Banach spaces. This would, for example, be a useful tool in the study of the Kato property in $L^{p}(\Omega)$ with $p \neq 2$.In [1], it was shown that for a sectorial operator $L$ on $L^{p}(\Omega)$ the quadratic estimates need to be adapted, taking the form$$\begin{equation*}\left\|\left(\int_{0}^{\infty}|f(t L) u|^{2} \frac{\mathrm{d} t}{t}\right)^{1 / 2}\right\|_{L^{p}(\Omega)} \approx\|u\|_{L^{p}(\Omega)}, \quad u \in L^{p}(\Omega) \tag{2.4}\end{equation*}$$Note that (2.3) and (2.4) coincide for $p=2$ by Fubini's theorem.The connection between $H^{\infty}$-calculus and quadratic estimates in [1] is not yet as clean as the statement we know in the Hilbert space setting. Only after introducing randomness, through a notion called $\mathscr{R}$-sectoriality, we arrive at a formulation in $L^{p}(\Omega)$ fully analogous to McIntosh's theorem [3]. In this project, we will explore the intricacies of McIntosh theorem in $L^{p}(\Omega)$. Moreover, we will discuss what happens in a general Banach space $X$ [2]. Note that (2.4) does not have an obvious interpretation in this case, as $|x|^{2}$ has no meaning for $x \in X$ ![-]

In this talk we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded $H^\infty$-calculus on weighted $L^p$-spaces for power weights which fall outside the classical class of $A_p$-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with very rough inhomogeneous boundary data.
The talk is based on joint work with Nick Lindemulder.[-]