  We prove a Logvinenke-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.
