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.[-]

Given $T>0$ and Hilbert spaces $V$ and $H$ where $V \hookrightarrow H$, a mapping $a:[0, T] \times V \times V \rightarrow \mathbb{K}$ is called non-autonomous form if $a(\cdot, v, w):[0, T] \rightarrow \mathbb{K}$ is measurable for all $v, w \in V$ and$$|a(t, v, w)| \leq M\|v\|_{V}\|w\|_{V} \quad \text { for all } t \in[0, T] \text { and } v, w \in V$$for some $M \geq 0$. The form is said to be coercive if there exists $\alpha>0$ with$$\operatorname{Re} a(t, v, v) \geq \alpha\|v\|_{V}^{2} \quad \text { for all } t \in[0, T] \text { and } v \in V$$An elegant result of Lions shows well-posedness of the problem$$\begin{cases}u^{\prime}(t)+\mathscr{A}(t) u(t) & =f(t) \tag{P}\\ u(0) & =u_{0}\end{cases}$$where $f \in L^{2}\left(0, T ; V^{\prime}\right)$ and $u_0 \in H$. Here, we consider the usual embedding $H \hookrightarrow V^{\prime}$ and the family of operators $\mathscr{A}(t) \in$ $\mathscr{L}\left(V, V^{\prime}\right)$ given by$$\langle\mathscr{A}(t) u, v\rangle=a(t, u, v) \quad \text { for all } t \in[0, T] \text { and } v, w \in V .$$In fact, one has maximal regularity in $V^{\prime}$, i.e.$$u \in H^1\left(0, T ; V^{\prime}\right) \cap L^2(0, T ; V) .$$Particularly, all the terms $u^{\prime}, \mathscr{A}(\cdot) u(\cdot)$ and $f$ belong to $L^{2}\left(0, T ; V^{\prime}\right)$. Frequently however, the part $A(t)$ of $\mathscr{A}(t)$ in $H$ given by$$D(A(t))=\{v \in V: \mathscr{A}(t) v \in H\}, \quad A(t) v=\mathscr{A}(t) v$$is more important because this operator incorporates the boundary conditions. Thus, an important problem is the following Lions' Problem (1961). If $f \in L^{2}(0, T ; H)$ and $u_{0} \in V$, does this imply $u \in H^{1}(0, T ; H)$ ?
The answer is "No", even if $u_{0}=0$. A first counterexample has been given by Dominik Dier (2014). It is based on the counterexample of McIntosh showing that $V \neq D\left(A^{\frac{1}{2}}\right)$ is possible.
On the other hand, if the form $a$ is sufficiently regular in time, then positive results hold by results of D. Dier, S. Fackler, E.M. Ouhabaz, C. Spina and others.

Organization of the project:
The project is organized in the following parts.

1. Consider the Gelfand triple $V \hookrightarrow H \hookrightarrow V^{\prime}$ and let $\mathscr{A}: V \rightarrow V^{\prime}$ be the operator associated to an autonomous, coercive form $a$ on $V$ and let $A$ be the part of $\mathscr{A}$ in $H$. Moreover, denote by $(T(t))_{t}$ the contractive, holomorphic $C_{0}$-semigroup on $H$ generated by $-A$, cf. [AVV19, Theorem 5.8]. The goal is then to prove that
$$\begin{equation*}
T(\cdot) x \in H^{1}(0, T ; H) \quad \text { if and only if } \quad x \in D\left(A^{\frac{1}{2}}\right) \tag{2.5}
\end{equation*}$$
The main steps in the proof are outlined in [ADF17, Section 4]. One of the main ingredients and the main focus of this talk lies in understanding that $D\left(A^{\frac{1}{2}}\right)=[H, D(A)]_{\frac{1}{2}}$. That is, the domain of the square root is an interpolation space! The proof of this fact is a special case of [Haa06, Theorem 6.6.9].

2. Lions' theorem on maximal regularity in $V^{\prime}$, cf. [AVV19, Theorem 17.15] and the above introduction. A key argument in the proof involves Lions' representation theorem, cf. [AVV19, Theorem 17.11].

3. Dier's counterexample, cf. [ADF17, Example 5.1] and [Die14].

4. A positive result: maximal regularity in $H$ for Lipschitz continuous forms. More precisely, we suppose that the nonautonomous form $a:[0, \tau] \times V \times V \rightarrow \mathbb{K}$ can be written as $a=a_{1}+b$ where $a_{1}$ and $b$ are bounded non-autonomous forms on $V$ with the following requirements:
(i) $a_{1}$ is symmetric, i.e. $a_{1}(t, x, y)=\overline{a_{1}(t, y, x)}$ for $x, y \in V$ and $0 \leq t \leq \tau$;
(ii) $a_{1}$ is coercive, i.e. there exists $\alpha>0$ with $a_{1}(t, x, x) \geq \alpha\|x\|_{V}^{2}$ for all $x \in V, 0 \leq t \leq \tau$;
(iii) $a_{1}$ is Lipschitz continuous, i.e. there exist $M_{1}^{\prime} \geq 0$ with
$$\left|a_{1}(t, x, y)-a_{1}(s, x, y)\right| \leq M_{1}^{\prime}|t-x|\|x\|_{V}\|y\|_{V}$$
for all $0 \leq t \leq \tau$ and all $x, y \in V$;
(iv) There exists $M_{b} \geq 0$ with $|b(t, x, y)| \leq M_{b}\|x\|_{V}\|y\|_{H}$ for all $0 \leq t \leq \tau$ and $x, y \in V$.
Then the statement of Lions' Problem as above holds true. For reference, cf. [ADLO14] and [AVV19, Theorem 18.2].
This talk's goal is giving a proof, possibly under somewhat stronger regularity assumptions on the form. For instance, if one even assumes $C^{1}$ regularity instead of Lipschitz regularity in time, then many technicalities become easier to handle.[-]

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.[-]

In the ISem, we have encountered sectorial operators $A$ on a Hilbert space $H$. In Lecture 6 we have defined the exponential $\mathrm{e}^{-t A}$ for $t>0$ if the sectoriality angle of $A$ is smaller than $\frac{\pi}{2}$, the so-defined family $\left(\mathrm{e}^{-t A}\right)_{t>0}$ is called the semigroup associated with $A$. In Proposition 6.6 it was shown that the semigroup yields the solution to the abstract Cauchy problem$$\begin{aligned}\partial_{t} u(t)+A u(t) & =0, \quad(t>0) \\u(0+) & =u_{0}\end{aligned}$$by setting $u(t):=\mathrm{e}^{-t A} u_{0}$. In the same way, one can solve the equation$$\begin{align*}\partial_{t} u(t)+A u(t) & =f(t), \quad(t>0) \tag{2.1}\\u(0+) & =0\end{align*}$$by computing the convolution of $\mathrm{e}^{-t A}$ with $f$; that is,$$u(t):=\int_{0}^{t}e^{-(t-s)A}f(s)ds.$$One can now show that sectoriality of $A$ yields the maximal $L_{2}$-regularity of (2.1); that is, if $f\in L_{2}(0,\infty ;H)$ then the sodefined solution $u$ satisfies $u\in H^{1}(0,\infty ;H)$ or equivalently (due to (2.1)) $Au\in L_{2}(0,\infty ;H)$. It is the main object of this project to generalise this result to operators on Banach spaces $X$.
As we will see, sectoriality is not enough to ensure maximal regularity of (2.1). In fact, some stronger property is needed, namely $\mathscr{R}$-sectoriality, which in the Hilbert space case is equivalent to sectoriality. Moreover, the goal to prove such a result for all Banach spaces turns out to be too ambitious, so we will restrict our attention to so-called UMD spaces (sometimes also called $\mathscr{HT}$-spaces to reflect their relation to the Hilbert transform). This class of Banach spaces turns out to be suited for the application of techniques from Fourier analysis, which will be one of the main tools to prove our goal, which can be formulated as:

Maximal regularity of (2.1) in a UMD space is equivalent to $\mathscr{R}$-sectoriality of $A$.

The main source for this project will be [1], where our main result can be found in Theorem 4.4. Moreover, we will have a look at elliptic operators in divergence form, now on $L_{p}(\mathbb{R^{n}})$ and not on $L_{2}(\mathbb{R^{n}})$, and study the $\mathscr{R}$-sectoriality of those operators. If time permits, we can continue the study of elliptic operators, now on half-spaces and on domains.[-]

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 aim of this project is a deeper investigation of off-diagonal estimates. In the ISem lectures, in Theorem 11.16, it has already been shown that off-diagonal estimates in combination with Sobolev embeddings lead to $L^{p}$-extrapolation for the resolvents of an elliptic operator $L$ in divergence form on $\mathbb{R}^{n}$. More precisely, if $\left|\frac{1}{p}-\frac{1}{2}\right|<\frac{1}{n}$, then there exists $C>0$ such that $\left\|(1+t L)^{-1} u\right\|_{p} \leqslant C\|u\|_{p}$ for all $t>0, u \in L^{p} \cap L^{2}\left(\mathbb{R}^{n}\right)$.
A related (more difficult!) question is for what range of $p \in(1, \infty)$ the norm equivalence $\|\sqrt{L} u\|_{2} \simeq\|\nabla u\|_{2}$ from Theorem 12.1 (the Kato square root property for $L$ !) extrapolates to $L^{p}\left(\mathbb{R}^{n}\right)$. It turns out that there are different ranges of $p$ for the two estimates $\|\sqrt{L} u\|_{p} \lesssim\|\nabla u\|_{p}$ and $\|\nabla u\|_{p} \lesssim\|\sqrt{L} u\|_{p}$. The latter estimate is generally known as $L^{p}$-boundedness of the Riesz transform, and this is what shall be the core of the project.
Starting point of the project is the AMS memoir [1], which starts with an excellent introduction into the topic; you can find a preprint version of the memoir on the arXiv (with different numbering of theorems than in the published version, unfortunately). An important abstract $L^{p}$-extrapolation result is Theorem 1.1 in [1], the application Riesz transforms on $L^{p}$ can be found in Section 4.1. This approach is due to Blunck and Kunstmann [2, 3]. If time permits, we can also study the approach of Shen [4] to Riesz transforms. The precise selection of topics will be decided among the participants of the project.[-]