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Scaling limit of modulation spaces and their applications
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-01-20, DOI: 10.1016/j.acha.2021.01.002
Mitsuru Sugimoto, Baoxiang Wang

Modulation spaces ${M}_{p,q}^{s}$ were introduced by Feichtinger [11] in 1983. Bényi and Oh [2] defined a modified version to Feichtinger's modulation spaces for which the symmetry scalings are emphasized for its possible applications in PDE. By carefully investigating the scaling properties of modulation spaces and their connections with Bényi and Oh's modulation spaces, we introduce the scaling limit versions of modulation spaces, which contains both Feichtinger's and Bényi and Oh's modulation spaces. As their applications, we will give a local well-posedness and a (small data) global well-posedness results for nonlinear Schrödinger equation in some scaling limit of modulation spaces, which generalize the well posedness results of [3] and [23], and certain super-critical initial data in ${H}^{s}$ or in ${L}^{p}$ are involved in these spaces.