Subdyadic time-frequency analysis: Gabor frames, modulation spaces, and Miyachi multipliers

Abstract: We present a time-frequency framework adapted to dispersive phase functions via a subdyadic geometry in phase space. On top of this geometry we construct stable Gabor frames with quantitative control of overlap, almost orthogonality, and off-diagonal decay. Based on these frames we introduce modulation spaces consistent with the subdyadic scale and establish window and lattice independence, identifications in the Hilbertian case, duality, and natural inclusion relations. Within this setting we develop a theory for two-sided Miyachi multipliers, relying on discrete almost diagonalization and Wiener-Jaffard type results for well-localized matrices, and obtain boundedness on weighted modulation spaces. Finally, we define a Gabor-type wavefront set adapted to the subdyadic geometry and prove its invariance and ellipticity with respect to smooth order-zero pseudodifferential operators. Taken together, these results provide a unified tool both for global microlocal analysis and for the design of stable numerical schemes in high-frequency regimes.