Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation

We build on our recent results on the Lipschitz dependence of the extreme spectral values of one-parameter families of pseudodifferential operators with symbols in a weighted Sj\"ostrand class. We prove that larger symbol classes lead to H\"older continuity with respect to the parameter. This result is then used to investigate the behavior of frame bounds of families of Gabor systems $\mathcal{G}(g,\alpha\Lambda)$ with respect to the parameter $\alpha>0$, where $\Lambda$ is a set of non-uniform, relatively separated time-frequency shifts, and $g\in M^1_s(\mathbb{R}^d)$, $0\leq s\leq 2$. In particular, we show that the frame bounds depend continuously on $\alpha$ if $g\in M^1(\mathbb{R}^d)$, and are H\"older continuous if $g\in M^1_s(\mathbb{R}^d)$, $0