Compact pseudodifferential and Fourier integral operators via localization
Cody B. Stockdale, Cody Waters
TL;DR
This work develops a general framework for localized operators whose Gabor-frame matrix coefficients concentrate along the diagonal, proving boundedness on modulation spaces and compactness via a weak-compactness criterion. The authors introduce pullback modulation spaces $M^{p,q}_{m,\chi}$ and an abstract compactness theorem for $(\nu,\chi,g_1,g_2)$-localized operators, then apply it to Fourier integral operators, $\tau$-pseudodifferential operators, and three-parameter pseudodifferential operators, obtaining boundedness and compactness results under symbol-vanishing conditions such as $\lim_{(z,\zeta)\to\infty} V_{\varphi}\sigma(z,\zeta)=0$ or $\lim_{z\to\infty} \sigma(\cdot-z_1,\cdot-z_1,\cdot-z_2)=0$ in $\mathcal{S}'$. A key technical device is an atomic decomposition for weighted Sjöstrand classes enabling transfer from compact Fourier support to general symbols, and the framework recovers known $p=q$ results for $L^2$-based modulation spaces while extending to off-diagonal settings. The results offer a unified, robust approach to compactness in time-frequency analysis with potential impact on PDEs and signal processing. Key outcomes include new compactness criteria for three-parameter symbols and systematic treatment across FIOs and PDOs via a single abstract theorem.
Abstract
We present a general framework of localized operators, i.e., operators whose matrix coefficients with respect to the Gabor frame are concentrated on the diagonal. We show that localized operators are bounded between modulation spaces, and we deduce their compactness from an easily verifiable weak compactness condition. We apply this abstract formalism to unify and extend existing theorems for pseudodifferential and Fourier integral operators, and to obtain new results for three-parameter pseudodifferential operators.