Scaling limit of modulation spaces and their applications

Abstract

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.

Introduction

Let $\mathcal{S}:=\mathcal{S}({\mathbb{R}}^d)$ be the Schwartz space and ${\mathcal{S}}^{\prime }$ be its dual space. Feichtinger [11] in 1983 introduced the notion of modulation spaces $M_{p,q}^s$ by using the short time Fourier transform (STFT) which is now a basic tool in time-frequency analysis. Recall that the STFT of a function f with respect to a window function $g\in \mathcal{S}$ is defined as (see [11], [16])

$$V_{g}f(x,\xi )=\underset{{\mathbb{R}}^d}{\int }{e}^{-\mathrm{i}t\cdot \xi }\overline{g(t-x)}f(t)dt=〈f,M_{\xi }{T}_{x}g〉,$$

where $T_x:g\to g(\cdot -x)$ and $M_{\xi }:g\to e^{\mathrm{i}\cdot \xi }g$ denote the translation and modulation operators, respectively. The STFT is closely related to the wave packet transform of Córdoba and Fefferman [8] and the Wigner transform [16], which is now a basic tool in time frequency analysis theory. For any $1⩽p,q⩽\infty$, $s\in \mathbb{R}$, we denote

$${\Vert f\Vert }_{M_{p,q}^s}={\Vert {〈\xi 〉}^s{\Vert V_{g}f(x,\xi )\Vert }_{L^p({\mathbb{R}}_x^d)}\Vert }_{L^q({\mathbb{R}}_{\xi }^d)}.$$

Modulation spaces $M_{p,q}^s$ are defined as the space of all tempered distributions $f\in {\mathcal{S}}^{\prime }$ for which ${\Vert f\Vert }_{M_{p,q}^s}$ is finite (cf. [11]). Modulation spaces $M_{p,q}^s$ are also said to be Feichtinger spaces.

It is well-known that translation $T_x$, modulation $M_{\xi }$ and scaling $D_{\lambda }:f\to f(\lambda \cdot )$ ($\lambda >0$) are fundamental operators in Banach function spaces. Feichtinger spaces $M_{p,q}^0$ are invariant under translation and modulation operators. However, $M_{p,q}^0$ have variant scalings, different functions in $M_{p,q}^0$ may have different scalings and the sharp scaling property was obtained in [21] (see also [17] for α-modulation spaces). Recently, a modified version of Feichtinger spaces with invariant scalings were introduced by Bényi and Oh [2]. Applying the scaling of the frequency-uniform partition like ${\{{\psi }_{j,k}:=\psi (2^{-j}\cdot -k)\}}_{j\in \mathbb{Z},k\in {\mathbb{Z}}^d}$, they introduced the following function spaces ${\mathfrak{M}}_{p,q,r}^{\mathbf{w}}$ with $p,q,r\in [1,\infty )$. Denote

$${\Vert f\Vert }_{M_{p,q}^{[j]}}={\Vert {\Vert {\psi }_{j,k}(D)f\Vert }_{L^p({\mathbb{R}}^d)}\Vert }_{{\ell }_k^q({\mathbb{Z}}^d)},$$

where

$${\psi }_{j,k}(D)f={\mathcal{F}}^{-1}{\psi }_{j,k}\mathcal{F}f.$$

By carefully choosing the weight $\mathbf{w}={\{w_j\}}_{j\in \mathbb{Z}}$, say

$$w_j\lesssim \{\begin{array}{cc}{2}^{-\epsilon j},\hfill & j⩾0,\hfill \\ 2^{d(1/p+1/q-1+\epsilon )j},\hfill & j<0,\hfill \end{array}$$

one can define the following (cf. [2])

$${\Vert f\Vert }_{{\mathfrak{M}}_{p,q,r}^{\mathbf{w}}}={\Vert w_j{\Vert f\Vert }_{M_{p,q}^{[j]}}\Vert }_{{\ell }_j^r(\mathbb{Z})}.$$

For some special weights w, Bényi and Oh [2] showed that ${\mathfrak{M}}_{p,q,r}^{\mathbf{w}}$ has invariant scalings ${\Vert f(\lambda \cdot )\Vert }_{{\mathfrak{M}}_{p,q,r}^{\mathbf{w}}}\sim {\lambda }^a{\Vert f\Vert }_{{\mathfrak{M}}_{p,q,r}^{\mathbf{w}}}$ for all $f\in {\mathfrak{M}}_{p,q,r}^{\mathbf{w}}$, where $a\in \mathbb{R}$ is independent of $f\in {\mathfrak{M}}_{p,q,r}^{\mathbf{w}}$. However, ${\mathfrak{M}}_{p,q,r}^{\mathbf{w}}$ cannot cover Feichtinger spaces $M_{p,q}^s$, which can be regarded as a modified version of Feichtinger spaces with invariant scalings.

Since ${\mathfrak{M}}_{p,q,r}^{\mathbf{w}}$ contains two indices $q,r$ to control the growth of the regularity arising from translations and scalings in frequency spaces, the regularity of ${\mathfrak{M}}_{p,q,r}^{\mathbf{w}}$ depends on both translation index k and scaling index $2^j$. It follows that k and j are not completely independent of each other, which leads to that the dual space of ${\mathfrak{M}}_{p,q,r}^{\mathbf{w}}$ is not ${\mathfrak{M}}_{p^{\prime },q^{\prime },r^{\prime }}^{1/\mathbf{w}}$ (see Section 4).

A natural relation between the scaling of frequency-uniform partition and the dilations of Feichtinger spaces is the following dilation identity

$${\Vert f\Vert }_{M_{p,q}^{[j]}}={\lambda }^{d/p}{\Vert f(\lambda \cdot )\Vert }_{M_{p,q}^0},\lambda =2^{-j}.$$

By resorting to this identity and the scaling properties of Feichtinger spaces, we will consider the scaling limit spaces of $M_{p,q}^0$, which contain both Feichtinger's and Bényi and Oh's modulation spaces. Also, we will characterize the dual spaces of ${\mathfrak{M}}_{p,q,r}^{\mathbf{w}}$, by a class of new function spaces which are rougher than ${\mathfrak{M}}_{p^{\prime },q^{\prime },r^{\prime }}^{1/\mathbf{w}}$. Moreover, we will study nonlinear Schrödinger equation (NLS) in those scaling limit spaces of $M_{p,q}^0$ and obtain some local and (small data) global well-posedness results, which generalize the results of [3] and [23] and certain $L^p$ or $H^s$ super-critical initial data are involved in our results.

Throughout this paper, we will use the following notations. $C⩾1,c\le 1$ will denote constants which can be different at different places, we will use $A\lesssim B$ to denote $A⩽CB$; $A\sim B$ means that $A\lesssim B$ and $B\lesssim A$. We write $a\vee b=\max (a,b)$ and $a\wedge b=\min (a,b)$. ${\mathbb{Z}}_{+}=\{j\in \mathbb{Z}:j⩾0\}$ and ${\mathbb{Z}}_{-}=\{j\in \mathbb{Z}:j⩽0\}$. Let Λ be a set with finitely many elements, #Λ stands for the number of the elements contained in Λ. We denote $|x{|}_{\infty }={\max }_{i=1,...,d}|x_i|$, $|x|=|x_1|+...+|x_d|$ and $〈x〉={(1+x_1^2+...+x_d^2)}^{1/2}$ for any $x\in {\mathbb{R}}^d$. $Q(x_0,\delta )$ denotes the cube in ${\mathbb{R}}^d$ with center at $x_0$ and side length 2δ. We denote by $\mathcal{F}f$ or $\hat{f}$ the Fourier transform of f, and by ${\mathcal{F}}^{-1}f$ the inverse Fourier transform of f. For any $1⩽p⩽\infty$, we denote by $p^{\prime }$ the dual number of p, i.e., $1/p+1/p^{\prime }=1$. $L^p=L^p({\mathbb{R}}^d)$ (${\ell }^p$) stands for the (sequence) Lebesgue space for which the norm is written as ${\Vert \cdot \Vert }_p$ (${\Vert \cdot \Vert }_{{\ell }^p}$). $H^s={〈\mathrm{\nabla }〉}^{-s}{L}^2$ with ${〈\mathrm{\nabla }〉}^{-s}={\mathcal{F}}^{-1}{〈\xi 〉}^{-s}\mathcal{F}$ for any $s\in \mathbb{R}$. The following two inequalities will be frequently used in this paper (cf. [4]).

Proposition 1.1

(Multiplier estimate) Let $1⩽r⩽\infty ,L⩾[d/2]+1$ and $\rho \in H^L$. Then we have

$${\Vert {\mathcal{F}}^{-1}\rho \mathcal{F}f\Vert }_r\lesssim {\Vert \rho \Vert }_2^{1-d/2L}{\left(\sum _{i=1}^d{\Vert {\partial }_{x_i}^L\rho \Vert }_2\right)}^{d/2L}{\Vert f\Vert }_r.$$

Proposition 1.2 Proposition 1.3.2 of [22]

Let $1⩽p⩽q⩽\infty$, $b>0$, ${\xi }_0\in {\mathbb{R}}^d$. Denote $L_{B({\xi }_0,b)}^p=\{f\in L^p:\mathrm{supp}\hat{f}\subset B({\xi }_0,b)\}$. Then there exists $C>0$ such that

$${\Vert f\Vert }_q⩽C{b}^{d(1/p-1/q)}{\Vert f\Vert }_p$$

holds for all $f\in L_{B({\xi }_0,b)}^p$ and C is independent of $b>0$ and ${\xi }_0\in {\mathbb{R}}^d$.

The paper is organized as follows. In Section 2 we introduce two kinds of scaling limit spaces of $M_{p,q}^0$ and show that they are dual spaces in Section 3. In Section 4 we consider a generalized version of the scaling limit spaces of $M_{p,q}^0$ which contain both Feichtinger's and Benyi and Oh's modulation spaces. The dilation property of the scaling limit spaces of $M_{p,q}^0$ will be considered in Section 5. The algebraic structure is of importance in applications to PDE and we will obtain an algebraic property of the scaling limit spaces of $M_{p,q}^0$ in Section 6. As applications to NLS, we will obtain a local well-posedness result and a (small data) global well-posedness result in Sections 7 and 8, respectively.