Stability of global wave front sets by perturbations of frames
Chiara Boiti, David Jornet, Alessandro Oliaro
TL;DR
The paper addresses the stability of global Gabor wave front sets for ultradifferentiable Beurling classes under perturbations of the time-frequency frame. Using the Gabor transform and modulation space techniques, it proves two main invariance results: (i) the Gabor $\omega$-wave front set $\mathrm{WF}^G_\omega(u)$ remains unchanged under Christensen-type $\varepsilon$-perturbations of a Gabor frame, even when the perturbed family is not a frame for all of $L^2(\\mathbb{R}^d)$; (ii) the same WF is stable under nonstationary Gabor frames, provided the nonstationarity is controlled so that the canonical duals stay in $\mathcal{S}_\omega(\mathbb{R}^d)$ (achieved by showing $1/G\in\mathcal{O}_{M,\omega}(\mathbb{R}^d)$). The results extend to distributions in $\mathcal{S}'_\omega(\mathbb{R}^d)$ and highlight the robustness of microlocal regularity notions in time-frequency contexts, with implications for global propagation of singularities in pseudodifferential operators within ultradifferentiable frameworks.
Abstract
In this paper we consider the Gabor wave front set of ultradistributions in the frame of ultradifferentiable functions. We prove that such a wave front set, defined through a Gabor frame on a regular lattice, is not affected by perturbations of the frame, in two different cases: when we consider $\varepsilon$-perturbations of Christensen type, and when we consider nonstationary Gabor frames.