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.[-]

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.[-]