A Logvinenko-Sereda theorem for vector-valued functions and application to control theory

We prove a Logvinenko-Sereda Theorem for vector valued functions. That is, for an arbitrary Banach space $X$, all $p \in [1,\infty]$, all $\lambda \in (0,\infty)^d$, all $f \in L^p (\mathbb{R}^d ; X)$ with $\operatorname{supp} \mathcal{F} f \in \times_{i=1}^d (-\lambda_i/2 , \lambda_i /2)$, and all thick sets $E \subseteq \mathbb{R}^d$ we have \begin{equation*} \lVert \mathbf{1}_E f \rVert_{L^p (\mathbb{R}^d)} \geq C \lVert f \rVert_{L^p (\mathbb{R}^d)} . \end{equation*} The constant is explicitly known in dependence of the geometric parameters of the thick set and the parameter $\lambda$. As an application, we study control theory for normally elliptic operators on Banach spaces whose coefficients of their symbol are given by bounded linear operators. This includes systems of coupled parabolic equations or problems depending on a parameter.