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.[-]

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 discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates.
We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local equilibrium, a Gaussian with the same macroscopic parameters. Due to the absence of dissipation w.r.t. the spatial direction, convergence to the global equilibrium is only possible thanks to the transport term that mixes various positions. Hence, such models are hypocoercive.
We shall prove exponential convergence towards the equilibrium with explicit rates for several linear, space periodic BGK-models in dimension 1 and 2. Their BGK-operators differ by the number of conserved macroscopic quantities (like mass, momentum, energy), and hence their hypocoercivity index. Our discussion includes also discrete velocity models, and the local exponential stability of a nonlinear BGK-model.
The third part of the talk is concerned with the entropy method for (non)symmetric Fokker-Planck equations, which is a powerful tool to analyze the rate of convergence to the equilibrium (in relative entropy and hence in L1). The essence of the method is to first derive a differential inequality between the first and second time derivative of the relative entropy, and then between the entropy dissipation and the entropy. For hypocoercive Fokker-Planck equations, i.e. degenerate parabolic equations (with drift terms that are linear in the spatial variable) we modify the classical entropy method by introducing an auxiliary functional (of entropy dissipation type) to prove exponential decay of the solution towards the steady state in relative entropy. The obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck equation (with nonquadratic potential).[-]

In this talk we study for $p\in \left ( 1,\infty \right )$ the $L^{p}$-realization of the vector-valued Schrödinger operator $\mathcal{L}u:= div\left ( Q\triangledown u \right )+Vu$. Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, and a perturbation theorem by Okazawa, we prove that $L^{p}$, the $L^{p}$-realization of $\mathcal{L}$, defined on the intersection of the natural domains of the differential and multiplication operators which form $\mathcal{L}$, generates a strongly continuous contraction semigroup on $L^{p}\left ( \mathbb{R}^{d} ;\mathbb{C}^{m}\right )$. We also study additional properties of the semigroup such as positivity, ultracontractivity, Gaussian estimates and compactness of the resolvent. We end the talk by giving some generalizations obtained recently and several examples.[-]