H\"older-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation

TL;DR

This paper demonstrates that the extreme spectral values of pseudodifferential operators and the frame bounds of Gabor systems exhibit H"older continuity with respect to certain parameters, extending previous Lipschitz results.

Contribution

It establishes H"older continuity of spectral values and Gabor frame bounds for broader symbol classes and window functions, with explicit H"older exponents.

Findings

Spectral values depend H"older continuously on parameters for larger symbol classes.

Gabor frame bounds are continuous or H"older continuous depending on the window function.

Explicit H"older exponents are derived for Gabor frame bounds.

Abstract

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 $g\in M^1_s(\mathbb{R}^d)$, $0<s\leq 2$, with the H\"older exponent explicitly given.