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.[-]