Year: 2025
Type: article
Abstract: 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 \mathrm{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 \lVert \mathbf{1}_{E} f \rVert_{L^p (\mathbb{R}^d;X)}\geq C \lVert f \rVert_{L^p (\mathbb{R}^d;X)}. The constant is explicitly known independent 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.
DOI: 10.4171/zaa/1794
Authors: Clemens Bombach, Martin Tautenhahn
Institutions: Chemnitz University of Technology, Leipzig University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4408335216