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.[-]