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 $x \in X$ ![-]

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.[-]

One of my recent main interests has been the characterization of boundedness of (integral) operators between two $L^p$ spaces equipped with two different measures. Some recent developments have indicated a need of "Banach spaces and their applications" also in this area of Classical Analysis. For instance, while the theory of two-weight $L^2$ inequalities is already rich enough to deal with a number of singular operators (like the Hilbert transform), the $L^p$ theory has been essentially restricted to positive operators so far. In fact, a counterexample of $F$. Nazarov shows that the common "Sawyer testing" or "David-Journé $T(1)$" type characterization will fail, in general, in the two-weight $L^p$ world. What comes to rescue is what we so often need to save the $L^2$ results in an Lp setting: $R$-boundedness in place of boundedness! Even in the case of positive operators, it turns out that a version of "sequential boundedness" is useful to describe the boundedness of operators from $L^p$ to $L^q$ when $q < p$. - This is about my recent joint work with T. Hänninen and K. Li, as well as the work of my student E. Vuorinen.

two-weight inequalities - boundedness - singular operators[-]

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.[-]