I wish to show the domain of a strongly continuous semigroup $S(t)$ is some sobolev space, for instance, for the heat semigroup, it is known that $$D(A)=W^{2,p}$$
I believe a general strategy to get above would be to show $$D(A)\subseteq W^{2,p},$$ and $$W^{2,p}\subseteq D(A) $$ However I am uncertain about how to achieve the above in detail, would be appreciate if anyone could outline me some steps to prove things like $D(A)=W^{2,p}.$