I follow the notation of Salamon '87: Infinite Dimensional Linear Systems With Unbounded Control and Observation: A Functional Analytic Approach
Let
$$W\subset H \subset V$$
be Hilbert spaces with continuous, dense injections, and let $A\in \mathcal L(W, H)$ be the generator of a strongly continuous semigroup $S(t) \in \mathcal L (H)$, and consider the abstract differential equation $$ \dot x = Ax + f, \quad x(0)=x_0 \in H $$ for $f \in L^2(0,\infty; W)$.
Then, there are plenty regularity results on well posedness that guarantee that the solution operator $(f\mapsto x)$ is a bounded linear map from $L^2(0,\infty; V)$ into $L^2(0,\infty; H)$:
But what if I have additional regularity, namely $f \in L^2(0,\infty; H)$ and $x_0 \in W$. Can I get an estimate like, for any $T>0$,
$$ \|x(t)\|_{L^2(0,T;W)} \leq c \|u(t)\|_{L^2(0,T;H)}\quad (?) $$
Any hint or references? An example for $W\subset H \subset V$ is $H^1 \subset L^2 \subset H^{-1}$.