Scaling limit of Modulation Spaces and Their Applications

TL;DR

This paper investigates the scaling properties of modulation spaces, introduces a generalized class that includes previous versions, and applies these to establish well-posedness results for nonlinear Schrödinger equations with rough initial data.

Contribution

It introduces a new class of generalized modulation spaces that unify previous versions and explores their applications in PDE well-posedness analysis.

Findings

Established local and global well-posedness for NLS in generalized modulation spaces.

Extended well-posedness results to rougher initial data than previous works.

Connected the scaling properties of modulation spaces with wavelet basis analysis.

Abstract

Modulation spaces $M^s_{p,q}$ were introduced by Feichtinger \cite{Fei83} in 1983. By resorting to the wavelet basis, B\'{e}nyi and Oh \cite{BeOh20} 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 the wavelet basis, we will introduce a class of generalized modulation spaces, which contain both Feichtinger's and B\'{e}nyi and Oh's modulation spaces. As their applications, we will give a local well-posedness and a (small data) global well-posedness results for NLS in some rougher generalized modulation spaces, which generalize the well posedness results of \cite{BeOk09} and \cite{WaHud07}, and certain super-critical initial data in $H^s$ or in $L^p$ are involved in these spaces.