Microlocal analysis, time-frequency transforms and local uncertainty principles

In this paper we propose an uncertainty principle of local type involving the short-time Fourier transform in the frame of ultradifferentiable spaces. The classical local uncertainty principle for the Fourier transform can be seen as a refinement of the Heisenberg inequality; the results we prove in this paper assert that the $L^1$ content of a function $f$ in a measurable $E$ must be small as the measure of $E$ is small and/or the $\xi$-dispersion of the short-time Fourier transform of $f$ is small, giving then a new result related with classical Fourier analysis and time-frequency representations.