Gabor Orthonormal Bases with Maximal Localization and Gabor Frame Operator on Local Fields

We provide an explicit construction of a Gabor orthonormal bases for a local field $K$ that provides maximal localization in both time and frequency. Such a localization is not true in case of $\mathbb{R}$ due to the uncertainty principle. In particular, we construct examples of functions $f \in L^2(K)$ such that the support of the ambiguity function of $f$ is of minimum measure. Moreover, we establish a quantitative uncertainty principle for local fields, which follows as a consequence of Lieb's inequalities for general locally compact abelian group. In addition, we develop fundamental operator representations for Gabor systems defined over local fields.