Scaling limit of Modulation Spaces and Their Applications
Abstract
Modulation spaces Msp,q were introduced by Feichtinger Fei83 in 1983. By resorting to the wavelet basis, B\'enyi and Oh 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\'enyi 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 BeOk09 and WaHud07, and certain super-critical initial data in Hs or in Lp are involved in these spaces.