调制空间的标度极限及其应用
Scaling limit of Modulation Spaces and Their Applications
摘要：
调制空间$M^s_{p,q}$由Feichtinger \cite{Fei83}在1983年引入。Bényi和Oh \cite{BeOh20}借助于小波基定义了Feichtinger调制空间的一个修正版本，强调了其在偏微分方程中的可能应用。通过仔细研究调制空间的标度性质及其与小波基的联系，我们将引入一类广义调制空间，它包含Feichtinger和Bényi和Oh的调制空间。作为它们的应用，我们将在一些更粗糙的广义调制空间中给出NLS的一个局部适定性和一个（小数据）整体适定性结果，这些结果推广了\cite{BeOk09}和\cite{WaHud07}的适定性结果，并且在这些空间中涉及到$H^s$或$L^p$中的某些超临界初始数据.
英文摘要：
Modulation spaces $M^s_{p,q}$ were introduced by Feichtinger \cite{Fei83} in 1983. By resorting to the wavelet basis, Bényi 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 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 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.
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