Let $\Delta$ be the Laplace operator, that is $\Delta f = f''$. It is well known (see, e.g. this lecture notes, Chapter 2) that $\Delta\colon D(\Delta) \to L^1(0,1)$ with domain $D(\Delta) = W^{2,1}(0,1)$ is sectorial operator in $L^1(0,1)$.
My question is, if $\Delta\colon D(\Delta) \to W^{2,1}(0,1)$ with domain $D(\Delta) = \{ f \in W^{2,1}\colon \Delta f \in W^{2,1}(0,1)\}$ is still sectorial operator in $W^{2,1}(0,1)$? Is it known theory?