Propagation of anisotropic Gabor singularities for Schrödinger type equations

Abstract

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity $a(\lambda x, \lambda ^\sigma \xi ) = \lambda ^{1+\sigma } a(x,\xi )$ for $\lambda > 0$ where $\sigma > 0$ is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin–Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.

1 Introduction

We prove results on propagation of anisotropic phase space singularities for the initial value Cauchy problem for evolution equations of the form

$$\begin{aligned} \left\{ \begin{array}{rl} \partial _t u(t,x) + i a^w(x,D_x) u (t,x) &{} = 0, \quad x \in \textbf{R}^{d}, \quad t \in [-T,T] \setminus \{ 0\} , \\ u(0,\cdot ) &{} = u_0. \end{array} \right. \end{aligned}$$

(1.1)

Here $T > 0$, $a^w(x,D_x)$ is a Weyl pseudodifferential operator and $u_0 \in \mathscr {S}'(\textbf{R}^{d})$ is a tempered distribution.

The Hamiltonian $a^w(x,D_x)$ is assumed to have real-valued principal symbol $a_0$. Following the fundamental idea of Hörmander we show that the singularities at time $t \in [-T,T]$ are the singularities of the initial datum $u_0$ transported by the Hamilton flow $\chi _t$ of the principal symbol $a_0$. The Hamilton flow $( x(t), \xi (t) ) = \chi _t(x,\xi )$ is the solution to Hamilton’s equation with initial datum $(x,\xi ) \in T^* \textbf{R}^{d} {\setminus } \{ (0,0) \}$, that is the solution to the system of ordinary differential equations

$$\begin{aligned} \left\{ \begin{array}{l} x' (t) = \nabla _{\xi } a_0 \left( x(t), \xi (t) \right) , \\ \xi ' (t) = -\nabla _{x} a_0 \left( x(t), \xi (t) \right) , \\ x(0) = x, \\ \xi (0) = \xi . \end{array} \right. \end{aligned}$$

The concept of phase space singularities that we use is the anisotropic Gabor wave front set, which is determined by an anisotropy parameter $\sigma > 0$. For $u \in \mathscr {S}'(\textbf{R}^{d})$ the anisotropic Gabor wave front set $\mathrm {WF_{g}^{ \sigma }}(u)$ is a $\sigma$-conic closed subset of $T^* \textbf{R}^{d} \setminus 0$. A $\sigma$-conic subset of $T^* \textbf{R}^{d} \setminus 0$ contains anisotropic phase space curves of the form

$$\begin{aligned} \lambda \mapsto ( \lambda x, \lambda ^\sigma \xi ) \in T^* \textbf{R}^{d} \setminus 0, \quad \lambda > 0, \end{aligned}$$

(1.2)

if one point of the curve belongs to the subset.

The anisotropic Gabor wave front set $\mathrm {WF_{g}^{ \sigma }}(u)$ is defined by means of the short-time Fourier transform $V_\varphi u (x, \xi ) = \mathscr {F}\left( u \, \overline{\varphi (\cdot -x) }\right) (\xi )$ where $\varphi \in \mathscr {S}(\textbf{R}^{d}) {\setminus } \{ 0 \}$ is a window function. To wit $z_0 = (x_0,\xi _0) \in T^* \textbf{R}^{d} {\setminus } 0$ satisfies $z_0 \notin \mathrm {WF_{g}^{ \sigma }}( u )$ if there exists an open set $U \subseteq T^* \textbf{R}^{d}$ such that $z_0 \in U$ and

$$\begin{aligned} \sup _{(x,\xi ) \in U, \ \lambda > 0} \lambda ^N |V_\varphi u (\lambda x, \lambda ^\sigma \xi )| < + \infty \quad \forall N \geqslant 0. \end{aligned}$$

This means that the short-time Fourier transform, which a priori is polynomially upper bounded, decays superpolynomially along curves of the form (1.2) in a neighborhood of $z_0$. For $u \in \mathscr {S}'(\textbf{R}^{d})$ we have $\mathrm {WF_{g}^{ \sigma }}(u) = \emptyset$ if and only if $u \in \mathscr {S}(\textbf{R}^{d})$ so $\mathrm {WF_{g}^{ \sigma }}(u)$ measures globally singular behavior in the sense of lack of smoothness or decay at infinity comprehensively.

We impose the condition that the Hamiltonian $a^w(x,D)$ has a real-valued principal symbol $a_0$ which satisfies the anisotropic homogeneity

$$\begin{aligned} a_0 (\lambda x, \lambda ^\sigma \xi ) = \lambda ^{1 + \sigma } a_0(x,\xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \quad \lambda > 0. \end{aligned}$$

(1.3)

This condition turns out to have several beneficial consequences for the problem we study.

First it implies that the Hamilton flow $\chi _t$ of $a_0$ commutes with the anisotropic scaling map

$$\begin{aligned} T^* \textbf{R}^{d} \setminus 0 \ni (x,\xi ) \mapsto ( \lambda x, \lambda ^\sigma \xi ) \in T^* \textbf{R}^{d} \setminus 0 \end{aligned}$$

for each $\lambda > 0$. This is a natural requirement for propagation results of the form $\mathrm {WF_{g}^{ \sigma }}(\mathscr {K}_t u_0) \subseteq \chi _t \mathrm {WF_{g}^{ \sigma }}(u_0)$, where $\mathscr {K}_t u_0 = e^{- i t a^w(x,D)} u_0$ denotes the solution operator (propagator) for (1.1), that we aim for, since $\mathrm {WF_{g}^{ \sigma }}(u)$ is $\sigma$-conic for all $u \in \mathscr {S}'(\textbf{R}^{d})$.

Secondly if $\sigma > 0$ is rational then condition (1.3) on the principal symbol allows us to prove the main result of this paper, that is the propagation of singularities

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}(\mathscr {K}_t u_0) = \chi _t ( \mathrm {WF_{g}^{ \sigma }}(u_0) ), \quad t \in [-T,T], \quad u_0 \in \mathscr {S}'(\textbf{R}^{d}), \end{aligned}$$

(1.4)

where $T > 0$.

The term “principal symbol” refers here to the pseudodifferential calculus of anisotropic Shubin symbols [7, 27, 32]. The symbols exhibit anisotropic behavior on phase space according to the assumed estimates

$$\begin{aligned} |\partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }a(x,\xi )| \lesssim ( 1 + |x| + |\xi |^{\frac{1}{\sigma }} )^{m - |\alpha | - \sigma |\beta |}, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad \alpha , \beta \in \textbf{N}^{d}, \end{aligned}$$

where again $\sigma > 0$ is a given anisotropy parameter, and $m \in \textbf{R}$ is the order. These symbol classes are denoted $G^{m,\sigma }$.

In the main result Theorem 8.3, we show (1.4) under the following assumptions. Suppose $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$, and let $a \in G^{1 + \sigma ,\sigma }$, $a \sim \sum _{j = 0}^{\infty } a_j$, where $a_0 \in C^\infty (\textbf{R}^{2d} {\setminus } 0)$ is real-valued and satisfies (1.3), whereas the lower order terms satisfy $a_j \in G^{(1+\sigma ) (1-2 j ), \sigma }$ for $j \geqslant 1$. An example of a symbol that satisfies the criteria is

$$\begin{aligned} a(x,\xi ) = c \psi (x,\xi ) \left( |x|^{2k} + |\xi |^{2m} \right) ^{\frac{1}{2} \left( \frac{1}{k} + \frac{1}{m} \right) } \end{aligned}$$

where $c \in \textbf{R}\setminus 0$ and $\psi$ is a smooth function vanishing in a small ball around the origin in $T^* \textbf{R}^{d}$.

We show that the solution operator is continuous on anisotropic Shubin–Sobolev spaces. This is of independent interest but also a tool for the proof of Theorem 8.3. The proof of the main result is based on ideas from [16]. More precisely our result is an anisotropic version of [21, Theorem 4.2] which treats propagation of the (isotropic) Gabor wave front set when the principal symbol is real-valued and homogeneous of order two on $T^* \textbf{R}^{d}$. With $\sigma = 1$ our result implies a weaker form of [21, Theorem 4.2].

The proof ideas for Theorem 8.3 and [21, Theorem 4.2] are based on Hörmander’s proof of [16, Theorem 23.1.4]. This result concerns Hamiltonians with first-order Hörmander type symbols, the continuity concerns classical Sobolev spaces, and the singularities are the classical smooth wave front set. The proof techniques rely on energy estimates, functional analysis and pseudodifferential calculus. Our proofs in this paper are worked out in detail as opposed to the rather brief arguments in [16, Chapter 23.1] and [21].

We also prove the propagation (1.4) for a different type of Hamiltonian of the form $a^w(x,D) = p(D) + \langle v,x \rangle$ where $p \in C^\infty (\textbf{R}^{d})$ is a sum of polynomials of each variable in $\textbf{R}^{d}$, with real coefficients, of order $m \geqslant 2$, $v \in \textbf{R}^{d}$ is a vector each of whose coordinate is nonzero, and $\sigma = \frac{1}{m-1}$. Since this setup includes the Airy operator $\frac{\textrm{d}^2}{\textrm{d}x^2} - x$ when $d =1$ we say that the corresponding equation (1.1) is of Airy–Schrödinger type. Using results from [31] we also formulate a version of (1.4) in the Gelfand–Shilov space functional framework and corresponding anisotropic wave front sets [26].

Denoting by $P_m$ the principal part of p, we show (1.4) where $\chi _t$ is the Hamilton flow of $P_m(\xi )$. This generalizes a particular case of [32, Theorem 5.1] where $v = 0$. Since $P_m(\xi )$ does not depend on x, the Hamiltonian flow for $P_m$ is trivial in the sense that it is constant in time with respect to the dual coordinates as $\chi _t (x, \xi ) = (x + t \nabla P_{m} (\xi ), \xi )$. This contrasts to the Hamilton flow in the main result Theorem 8.3 where both space and dual coordinates may depend on time. The techniques we use for Airy–Schrödinger equations are an explicit formula for the Schwartz kernel of the propagator and general results on propagation of singularities from [26, 27, 31, 32].

Our results in this paper fit in a project to investigate globally anisotropic pseudodifferential operators [3, 5, 7, 19, 26, 27] and propagation of global singularities for evolution equations [24, 31, 32]. The techniques are inspired from those of pseudodifferential operators defined by symbols that are anisotropic in the dual variables for fixed space coordinates. These ideas have been investigated e.g. in [11, 18, 22].

A major new feature of our main result Theorem 8.3 as opposed to earlier propagation results [32], is that it admits Hamiltonians that give rise to flows that are non-trivial in the sense that the dynamics involve all phase space coordinates.

Concerning the organization of the paper, Sect. 2 contains notations, background concepts and conventions, and Sect. 3 recalls material on anisotropic Shubin pseudodifferential calculus. Section 4 is devoted to Shubin–Sobolev modulation spaces in the anisotropic context, a recollection of localization operators, and an inequality of sharp Gårding type which is essential. In Sect. 5 we deduce propagation results for Airy–Schrödinger equations. Section 6 treats Hamiltonians that are anisotropically homogeneous as in (1.3) and their Hamilton flows, and in Sect. 7 we show existence and uniqueness of solutions to an inhomogeneous form of (1.1) in anisotropic Shubin–Sobolev spaces for Hamiltonian symbols in $G^{1+\sigma ,\sigma }$ with bounded imaginary part and $\sigma > 0$ rational. Then Sect. 8 is dedicated to the main result on propagation of singularities, and finally Sect. 9 consists of a very short discussion of examples.

2 Preliminaries

The unit sphere in $\textbf{R}^{d}$ is denoted $\textbf{S}^{d-1} \subseteq \textbf{R}^{d}$. An open ball of radius $r > 0$ centered in $x \in \textbf{R}^{d}$ is denoted ${\text {B}}_r (x)$, and ${\text {B}}_r(0) = {\text {B}}_r$. The transpose of a matrix $A \in \textbf{R}^{d \times d}$ is denoted $A^T$ and the inverse transpose of $A \in {\text {GL}}(d,\textbf{R})$ is $A^{-T}$. We write $f (x) \lesssim g (x)$ provided there exists $C>0$ such that $f (x) \leqslant C \, g(x)$ for all x in the domain of f and of g. If $f (x) \lesssim g (x) \lesssim f(x)$ then we write $f \asymp g$. We use the partial derivative $D_j = - i \partial _j$, $1 \leqslant j \leqslant d$, acting on functions and distributions on $\textbf{R}^{d}$, with extension to multi-indices. The bracket $\langle x\rangle = (1 + |x|^2)^{\frac{1}{2}}$ for $x \in \textbf{R}^{d}$ satisfies Peetre’s inequality with optimal constant [26, Lemma 2.1], that is

$$\begin{aligned} \langle x+y\rangle ^s \leqslant \left( \frac{2}{\sqrt{3}} \right) ^{|s|} \langle x\rangle ^s\langle y\rangle ^{|s|}\qquad x,y \in \textbf{R}^{d}, \quad s \in \textbf{R}. \end{aligned}$$

(2.1)

We use the normalization of the Fourier transform

$$\begin{aligned} \mathscr {F}f (\xi )= {{\widehat{f}}}(\xi ) = (2\pi )^{-\frac{d}{2}} \int _{\textbf{R}^{d}} f(x)e^{-i\langle x,\xi \rangle } \, \textrm{d}x, \qquad \xi \in \textbf{R}^{d}, \end{aligned}$$

for $f\in \mathscr {S}(\textbf{R}^{d})$ (the Schwartz space), where $\langle \, \cdot \, ,\, \cdot \, \rangle$ denotes the scalar product on $\textbf{R}^{d}$. The conjugate linear action of a distribution u on a test function $\phi$ is written $(u,\phi )$, consistent with the $L^2$ inner product $(\, \cdot \, ,\, \cdot \, ) = (\, \cdot \, ,\, \cdot \, )_{L^2}$ which is conjugate linear in the second argument.

Denote translation by $T_x f(y) = f( y-x )$ and modulation by $M_\xi f(y) = e^{i \langle y,\xi \rangle } f(y)$ for $x,y,\xi \in \textbf{R}^{d}$ where f is a function or distribution defined on $\textbf{R}^{d}$. If $\varphi \in \mathscr {S}(\textbf{R}^{d}) {\setminus } \{0\}$ then the short-time Fourier transform (STFT) of a tempered distribution $u \in \mathscr {S}'(\textbf{R}^{d})$ is defined by

$$\begin{aligned} V_\varphi u (x,\xi ) = (2\pi )^{-\frac{d}{2}} (u, M_\xi T_x \varphi ) = \mathscr {F}(u T_x {\overline{\varphi }})(\xi ), \quad x,\xi \in \textbf{R}^{d}. \end{aligned}$$

The function $V_\varphi u$ is smooth and polynomially bounded [13, Theorem 11.2.3], that is there exists $k \geqslant 0$ such that

$$\begin{aligned} |V_\varphi u (x,\xi )| \lesssim \langle (x,\xi )\rangle ^{k}, \quad (x,\xi ) \in T^* \textbf{R}^{d}. \end{aligned}$$

(2.2)

We have $u \in \mathscr {S}(\textbf{R}^{d})$ if and only if

$$\begin{aligned} |V_\varphi u (x,\xi )| \lesssim \langle (x,\xi )\rangle ^{-N}, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad \forall N \geqslant 0. \end{aligned}$$

(2.3)

The transform inverse to the STFT is given by

$$\begin{aligned} u = (2\pi )^{-\frac{d}{2}} \iint _{\textbf{R}^{2d}} V_\varphi u (x,\xi ) M_\xi T_x \varphi \, \textrm{d}x \, \textrm{d}\xi \end{aligned}$$

(2.4)

provided $\Vert \varphi \Vert _{L^2} = 1$, with action under the integral understood, that is

$$\begin{aligned} (u, f) = (V_\varphi u, V_\varphi f)_{L^2(\textbf{R}^{2d})} = (V_\varphi ^* V_\varphi u, f) \end{aligned}$$

(2.5)

for $u \in \mathscr {S}'(\textbf{R}^{d})$ and $f \in \mathscr {S}(\textbf{R}^{d})$, cf. [13, Theorem 11.2.5].

According to [13, Corollary 11.2.6] the topology for $\mathscr {S}(\textbf{R}^{d})$ can be defined by the collection of seminorms

$$\begin{aligned} \mathscr {S}(\textbf{R}^{d}) \ni \psi \mapsto \Vert \psi \Vert _m:= \sup _{z \in \textbf{R}^{2d}} \langle z\rangle ^m |V_\varphi \psi (z)|, \quad m \in \textbf{N}, \end{aligned}$$

(2.6)

for any $\varphi \in \mathscr {S}(\textbf{R}^{d}) \setminus 0$.

The Beurling type Gelfand–Shilov space $\Sigma _\nu ^\mu (\textbf{R}^{d})$ is for $\nu ,\mu ,h > 0$ is defined as the topological projective limit

$$\begin{aligned} \Sigma _\nu ^\mu (\textbf{R}^{d}) = \bigcap _{h>0} {\mathcal {S}}_{\nu ,h}^\mu (\textbf{R}^{d}) \end{aligned}$$

where ${\mathcal {S}}_{\nu ,h}^\mu (\textbf{R}^{d})$ is the Banach space of smooth functions that have finite

$$\begin{aligned} \Vert f\Vert _{{\mathcal {S}}_{\nu ,h}^\mu }\equiv \sup _{x \in \textbf{R}^{d}, \, \alpha ,\beta \in \textbf{N}^{d}} \frac{|x^\alpha D^\beta f(x)|}{h^{|\alpha + \beta |} \alpha !^\nu \, \beta !^\mu } \end{aligned}$$

norm [12]. The space $\Sigma _\nu ^\mu (\textbf{R}^{d})$ is a Fréchet space with respect to the seminorms $\Vert \cdot \Vert _{\mathcal S_{\nu ,h}^\mu }$ for $h > 0$, and $\Sigma _\nu ^\mu (\textbf{R}^{d})\ne \{ 0\}$ if and only if $\nu + \mu > 1$ [23].

If $\nu + \mu > 1$ the topological dual of $\Sigma _\nu ^\mu (\textbf{R}^{d})$ is the space of (Beurling type) Gelfand–Shilov ultradistributions [12, Section I.4.3]

$$\begin{aligned} (\Sigma _\nu ^\mu )'(\textbf{R}^{d}) =\bigcup _{h>0} (\mathcal S_{\nu ,h}^\mu )'(\textbf{R}^{d}). \end{aligned}$$

The space of ultradistributions $(\Sigma _\nu ^\mu )'(\textbf{R}^{d})$ may be equipped with several possibly different topologies [31]. In this paper we use exclusively the $\hbox {weak}^*$ topology.

The Gelfand–Shilov (ultradistribution) spaces enjoy invariance properties, with respect to translation, dilation, tensorization, coordinate transformation and (partial) Fourier transformation. The Fourier transform extends uniquely to homeomorphisms on $\mathscr {S}'(\textbf{R}^{d})$, from $(\Sigma _\nu ^\mu )'(\textbf{R}^{d})$ to $(\Sigma _\mu ^\nu )'(\textbf{R}^{d})$, and restricts to homeomorphisms on $\mathscr {S}(\textbf{R}^{d})$, from $\Sigma _\nu ^\mu (\textbf{R}^{d})$ to $\Sigma _\mu ^\nu (\textbf{R}^{d})$, and to a unitary operator on $L^2(\textbf{R}^{d})$.

3 Anisotropic Shubin pseudodifferential calculus

In this section we retrieve some essential facts from pseudodifferential calculus of anisotropic Shubin symbols [27, 32].

Let $\sigma > 0$. We use the weight function on $(x,\xi ) \in T^* \textbf{R}^{d}$

$$\begin{aligned} \theta _\sigma (x,\xi ) = 1 + |x| + |\xi |^{\frac{1}{\sigma }}. \end{aligned}$$

(3.1)

For this weight we have the following inequality of Peetre type [27]. If $s \in \textbf{R}$ then

$$\begin{aligned} \theta _\sigma ( x+y, \xi + \eta )^s \leqslant C_{\sigma ,s} \theta _\sigma ( x, \xi )^{|s|} \theta _\sigma ( y, \eta )^s, \quad x, y, \xi , \eta \in \textbf{R}^{d}. \end{aligned}$$

(3.2)

When $\sigma$ is rational, $\sigma = \frac{k}{m}$, $k,m \in \textbf{N}{\setminus } 0$, an alternative weight is

$$\begin{aligned} w_{k,m} (x, \xi ) = \left( 1 + | x |^{2k} + | \xi |^{2 m} \right) ^{\frac{1}{2}}. \end{aligned}$$

(3.3)

Note that

$$\begin{aligned} w_{k,m} \asymp \theta _{\sigma }^{k}. \end{aligned}$$

(3.4)

The motivation for using $w_{k,m}$ instead of $\theta _{\sigma }^{k}$ is that the former is smooth as opposed to the latter.

By [27, Eq. (3.4)] we have for $\sigma > 0$

$$\begin{aligned} \langle (x,\xi )\rangle ^{\min \left( 1, \frac{1}{\sigma }\right) } \lesssim \theta _\sigma (x,\xi ) \lesssim \langle (x,\xi )\rangle ^{\max \left( 1, \frac{1}{\sigma }\right) }, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

(3.5)

and for $k,m \in \textbf{N}\setminus 0$

$$\begin{aligned} \langle (x,\xi )\rangle ^{\min (k,m)} \lesssim w_{k,m} (x,\xi ) \lesssim \langle (x,\xi )\rangle ^{\max (k,m)}, \quad (x,\xi ) \in T^* \textbf{R}^{d}. \end{aligned}$$

(3.6)

The anisotropic Shubin symbols are defined as follows.

Definition 3.1

Let $\sigma > 0$ be real and $m \in \textbf{R}$. The space of ($\sigma$-)anisotropic Shubin symbols $G^{m,\sigma }$ of order m consists of functions $a \in C^\infty (\textbf{R}^{2d})$ that satisfy the estimates

$$\begin{aligned} |\partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }a(x,\xi )| \lesssim ( 1 + |x| + |\xi |^{\frac{1}{\sigma }} )^{m - |\alpha | - \sigma |\beta |}, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad \alpha , \beta \in \textbf{N}^{d}.\nonumber \\ \end{aligned}$$

(3.7)

The space $G^{m,\sigma }$ is a Fréchet space with respect to the seminorms on $a \in G^{m,\sigma }$ indexed by $j \in \textbf{N}$

$$\begin{aligned} \Vert a \Vert _j = \max _{|\alpha + \beta | \leqslant j} \sup _{(x,\xi ) \in \textbf{R}^{2d} } \theta _\sigma (x,\xi )^{-m + |\alpha | + \sigma |\beta |} \left| \partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }a(x,\xi ) \right| . \end{aligned}$$

If $\sigma = 1$ then $G^{m,\sigma }$ is the space of isotropic Shubin symbols with parameter $\rho = 1$ [20, 28]. Recall that the isotropic Shubin symbol of order m and parameter $0 \leqslant \rho \leqslant 1$, denoted $a\in G_\rho ^m$, satisfies

$$\begin{aligned} |\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )| \lesssim \langle (x,\xi )\rangle ^{m - \rho |\alpha + \beta |}, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad \alpha , \beta \in \textbf{N}^{d}. \end{aligned}$$

We have $G^{m,\sigma } \subseteq G_\rho ^{m_0}$, where $m_0 = \max (m, m/\sigma )$ and $\rho = \min (\sigma , 1/\sigma )$, and

$$\begin{aligned} \bigcap _{m \in \textbf{R}} G^{m,\sigma } = \mathscr {S}(\textbf{R}^{2d}). \end{aligned}$$

(3.8)

The following lemma is a tool for verification of membership in $G^{m,\sigma }$.

Lemma 3.2

If $m \in \textbf{R}$, $\sigma ,r > 0$ and $a \in C^\infty (\textbf{R}^{2d})$ satisfies

$$\begin{aligned}&{} \left| \partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }a (\lambda x, \lambda ^\sigma \xi ) \right| \lesssim \lambda ^{m-|\alpha | - \sigma |\beta |}, \ (x,\xi ) \in T^* {\textbf {R}}^{d}, \ \ |(x,\xi )| = r, \lambda \geqslant 1, \alpha , \beta \in {\textbf {N}}^{d}, \end{aligned}$$

(3.9)

then $a \in G^{m,\sigma }$.

Proof

Let $(y,\eta ) \in \textbf{R}^{2d} \setminus {\text {B}}_{r}$. By [27, Section 3] $(y,\eta ) = (\lambda x, \lambda ^\sigma \xi )$ for a unique $(x,\xi ) \in \textbf{R}^{2d}$ such that $|(x,\xi )| = r$ and $\lambda \geqslant 1$. Combining

$$\begin{aligned} 1 + |y| + |\eta |^{\frac{1}{\sigma }} = 1 + \lambda ( |x| + |\xi |^{\frac{1}{\sigma }} ) \asymp 1+ \lambda \end{aligned}$$

with (3.9) we obtain for any $\alpha , \beta \in \textbf{N}^{d}$

$$\begin{aligned} \left| \partial _{y} ^{\alpha }\partial _{\eta } ^{\beta }a (y, \eta ) \right| \lesssim (1+\lambda )^{m -|\alpha | - \sigma |\beta |} \lesssim ( 1 + |y| + |\eta |^{\frac{1}{\sigma }} )^{m -|\alpha | - \sigma |\beta |}. \end{aligned}$$

The same estimate is trivial for $(y,\eta ) \in {\text {B}}_{r}$ so referring to (3.7) we may conclude that $a \in G^{m,\sigma }$. $\square$

Corollary 3.3

If $\sigma > 0$, $m \geqslant 0$ and $a \in C^\infty (\textbf{R}^{2d})$ is anisotropically homogeneous as

$$\begin{aligned} a( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{m} a(x,\xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad \lambda > 0, \end{aligned}$$

(3.10)

then $a \in G^{m, \sigma }$.

For $a \in G^{m,\sigma }$ and $\tau \in \textbf{R}$ a pseudodifferential operator in the $\tau$-quantization is defined by

$$\begin{aligned} a_\tau (x,D) f(x){}&{} = (2\pi )^{-d} \int _{{\textbf {R}}^{2d}} e^{i \langle x-y, \xi \rangle } a ( (1-\tau ) x + \tau y,\xi ) \, f(y) \, \text {d}y \, \text {d} \xi , \end{aligned}$$

(3.11)

for $f \in \mathscr {S}({\textbf {R}}^{d})$ when $m < - d \sigma$. The definition extends to $m \in \textbf{R}$ if the integral is viewed as an oscillatory integral. If $\tau = 0$ we get the Kohn–Nirenberg quantization $a_0(x,D) = a(x,D)$ and if $\tau = \frac{1}{2}$ we have the Weyl quantization $a_{1/2}(x,D) = a^w(x,D)$. The Weyl quantization enjoys a simple formal adjoint relation: $a^w(x,D)^* = {\overline{a}}^w(x,D)$. We will use exclusively the Weyl quantization in this paper. By [27, Proposition 3.3 (i)] the symbol classes $G^{m,\sigma }$ are homeomorphically invariant under change of quantization parameter $\tau \in \textbf{R}$, for any $\sigma > 0$ and $m \in \textbf{R}$. If $a \in G^{m,\sigma }$ then the operator $a^w(x,D)$ acts continuously on $\mathscr {S}(\textbf{R}^{d})$ and extends uniquely by duality to a continuous operator on $\mathscr {S}'(\textbf{R}^{d})$ [27, 28]. If $a \in \mathscr {S}'(\textbf{R}^{2d})$ then $a^w(x,D)$ extends to a continuous operator $a^w(x,D): \mathscr {S}(\textbf{R}^{d}) \rightarrow \mathscr {S}'(\textbf{R}^{d})$. If $a \in \mathscr {S}(\textbf{R}^{2d})$ then $a^w(x,D)$ is regularizing, in the sense that it is continuous $a^w(x,D): \mathscr {S}'(\textbf{R}^{d}) \rightarrow \mathscr {S}(\textbf{R}^{d})$ with $\mathscr {S}'(\textbf{R}^{d})$ equipped with the strong topology [6].

If $a \in \mathscr {S}'(\textbf{R}^{2d})$ then

$$\begin{aligned} (a^w(x,D) f, g) = (2 \pi )^{-d} (a, W(g,f) ), \quad f, g \in \mathscr {S}(\textbf{R}^{d}), \end{aligned}$$

(3.12)

where the cross-Wigner distribution [10, 13] is defined as

$$\begin{aligned} W(g,f) (x,\xi ) = \int _{\textbf{R}^{d}} g (x+y/2) \overline{f(x-y/2)} e^{- i \langle y, \xi \rangle } \textrm{d}y, \quad (x,\xi ) \in \textbf{R}^{2d}. \end{aligned}$$

If $f,g \in \mathscr {S}(\textbf{R}^{d})$ then $W(g,f) \in \mathscr {S}(\textbf{R}^{2d})$.

Given a sequence of symbols $a_j \in G^{m_j,\sigma }$, $j=1,2,\dots$, such that $m_j \rightarrow - \infty$ as $j \rightarrow \infty$ we write

$$\begin{aligned} a \sim \sum _{j = 1}^\infty a_j \end{aligned}$$

provided that for any $n \geqslant 2$

$$\begin{aligned} a - \sum _{j = 1}^{n-1} a_j \in G^{\mu _n,\sigma } \end{aligned}$$

where $\mu _n = \max _{j \geqslant n} m_j$. By [27, Lemma 3.2] there exists a symbol $a \in G^{m,\sigma }$ where $m = \max _{j \geqslant 1} m_j$ such that $a \sim \sum _{j = 1}^\infty a_j$ under the stated circumstances. The symbol a is unique modulo $\mathscr {S}(\textbf{R}^{2d})$.

The bilinear Weyl product $a {{\#}}b$ of two symbols $a \in G^{m,\sigma }$ and $b \in G^{n,\sigma }$ is defined as the product of symbols corresponding to operator composition: $( a {{\#}}b)^w(x,D) = a^w(x,D) b^w (x,D)$. By [27, Proposition 3.3 (ii)] the Weyl product is continuous ${{\#}}: G^{m,\sigma } \times G^{n,\sigma } \rightarrow G^{m+n,\sigma }$. The asymptotic expansion formula for the Weyl product [16, 28] is

$$\begin{aligned} a {{\#}}b(x,\xi ) \sim \sum _{\alpha , \beta \geqslant 0} \frac{(-1)^{|\beta |}}{\alpha ! \beta !} \ 2^{-|\alpha +\beta |} D_x^\beta \partial _{\xi } ^{\alpha }a(x,\xi ) \, D_x^\alpha \partial _{\xi } ^{\beta }b(x,\xi ). \end{aligned}$$

(3.13)

If $a \in G^{m,\sigma }$ and $b \in G^{n,\sigma }$ then each term in the sum belongs to $G^{m+n -(1+\sigma )|\alpha +\beta |,\sigma }$.

For $\sigma > 0$ a $\sigma$-conic subset $\Gamma \subseteq T^* \textbf{R}^{d} {\setminus } 0$ is closed under the operation $T^* \textbf{R}^{d} {\setminus } 0 \ni (x,\xi ) \mapsto ( \lambda x, \lambda ^\sigma \xi )$ for all $\lambda > 0$. By [27, Definition 3.4 and Lemma 3.5] (cf. also [32, Remark 3.4]) it is possible to construct $\sigma$-conic open subsets of given points in $T^* \textbf{R}^{d} {\setminus } 0$, and corresponding cutoff functions.

A symbol $a \in G^{m,\sigma }$ is said to be non-characteristic at $z_0 \in T^* \textbf{R}^{d} \setminus 0$, if

$$\begin{aligned} |a( x, \xi )| \geqslant C \theta _\sigma (x,\xi )^{m}, \quad (x,\xi ) \in \Gamma , \quad |(x,\xi )| \geqslant R \end{aligned}$$

(3.14)

for $C,R > 0$, where $\Gamma \subseteq T^* \textbf{R}^{d} {\setminus } 0$ is an open $\sigma$-conic subset containing $z_0$. The complement in $T^* \textbf{R}^{d} {\setminus } 0$ of the non-characteristic points is called the characteristic set ${\text {char}}_\sigma (a) \subseteq T^* \textbf{R}^{d} {\setminus } 0$. It is a closed and $\sigma$-conic subset of $T^* \textbf{R}^{d} {\setminus } 0$. This is a particular case of [27, Definition 3.8].

In most respects the anisotropic Shubin pseudodifferential calculus for the symbol classes $G^{m,\sigma }$ with $m \in \textbf{R}$ and $\sigma > 0$ works as the isotropic calculus in [20, 28]. In fact [27, Section 3] contains the basics of the anisotropic pseudodifferential calculus, and by [27, Lemma 6.3] and its proof it is possible to construct parametrices for elliptic symbols. Thus if $\sigma > 0$ and $a \in G^{m,\sigma }$ is elliptic in the sense of ${\text {char}}_\sigma (a) = \emptyset$, that is,

$$\begin{aligned} |a( x, \xi )| \geqslant C \theta _\sigma (x,\xi )^{m}, \quad (x,\xi ) \in \textbf{R}^{2d} \setminus {\text {B}}_R, \end{aligned}$$

(3.15)

for $C,R > 0$, then there exists an elliptic symbol $b \in G^{-m,\sigma }$ such that

$$\begin{aligned} a {{\#}}b = 1 + r_1, \quad b {{\#}}a = 1 + r_2, \end{aligned}$$

where $r_1, r_2 \in \mathscr {S}(\textbf{R}^{2d})$.

The following definition concerns the anisotropic Gabor wave front set $\mathrm {WF_{g}^{ \sigma }}( u ) \subseteq T^* \textbf{R}^{d} \setminus 0$ of $u \in \mathscr {S}'(\textbf{R}^{d})$ [27, Definition 4.1] which is important in this paper. It is a closed and $\sigma$-conic subset of $T^* \textbf{R}^{d} \setminus 0$ well adapted to the anisotropic Shubin calculus.

Definition 3.4

Suppose $u \in \mathscr {S}'(\textbf{R}^{d})$, $\varphi \in \mathscr {S}(\textbf{R}^{d}) \setminus 0$, and $\sigma > 0$. Then $z_0 = (x_0,\xi _0) \in T^* \textbf{R}^{d} \setminus 0$ satisfies $z_0 \notin \mathrm {WF_{g}^{ \sigma }}( u )$ if there exists an open set $U \subseteq T^* \textbf{R}^{d}$ such that $z_0 \in U$ and

$$\begin{aligned} \sup _{(x,\xi ) \in U, \ \lambda > 0} \lambda ^N |V_\varphi u (\lambda x, \lambda ^\sigma \xi )| < + \infty \quad \forall N \geqslant 0. \end{aligned}$$

(3.16)

If $\sigma = 1$ then $\mathrm {WF_{g}^{ \sigma }}( u ) = \mathrm {WF_\textrm{g}}(u)$ that denotes the usual Gabor wave front set [17, 25], which is isotropic in phase space. The $\sigma$-conic sets are then ordinary cones in $T^* \textbf{R}^{d} \setminus 0$, that is sets closed under multiplication with a positive parameter.

The definition of $\mathrm {WF_{g}^{ \sigma }}( u )$ does not depend on $\varphi \in \mathscr {S}(\textbf{R}^{d}) {\setminus } 0$ [27, Proposition 4.2], and [27, Proposition 4.3 (i)] says that

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}(\widehat{u}) = \mathcal {J}\textrm{WF}_{\textrm{g}}^{\frac{1}{\sigma }} (u), \quad u \in \mathscr {S}'(\textbf{R}^{d}), \end{aligned}$$

(3.17)

where

$$\begin{aligned} \mathcal {J}= \left( \begin{array}{cc} 0 &{} I_d \\ -I_d &{} 0 \end{array} \right) \in \textbf{R}^{2d \times 2d} \end{aligned}$$

(3.18)

is the matrix that defines the symplectic group [10].

By [32, Proposition 3.5] we may express the anisotropic Gabor wave front set of $u \in \mathscr {S}'(\textbf{R}^{d})$ as

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}(u) = \bigcap _{a \in G^{m,\sigma }: \ a^w(x,D) u \in \mathscr {S}} {\text {char}}_{\sigma } (a) \end{aligned}$$

(3.19)

for any $m \in \textbf{R}$. For $\sigma > 0$, $m \in \textbf{R}$, $a \in G^{m,\sigma }$ and $u \in \mathscr {S}'(\textbf{R}^{d})$ we have the microlocal and microelliptic inclusions

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( a^w(x,D) u ) \subseteq \mathrm {WF_{g}^{ \sigma }}(u) \subseteq \mathrm {WF_{g}^{ \sigma }}( a^w(x,D) u ) \bigcup {\text {char}}_\sigma (a) \end{aligned}$$

(cf. [27, Proposition 5.1 and Theorem 6.4] which are stated slightly more generally).

At a few occasions we will use anisotropic wave front sets in the Gelfand–Shilov functional framework. The Gelfand–Shilov wave front set of $u \in (\Sigma _\nu ^\mu )' (\textbf{R}^{d})$ with $\nu + \mu > 1$ is based on the following facts. If $\varphi \in \Sigma _\nu ^\mu (\textbf{R}^{d}) {\setminus } 0$ then

$$\begin{aligned} | V_\varphi u (x,\xi )| \lesssim e^{r (|x|^{\frac{1}{\nu }} + |\xi |^{\frac{1}{\mu }})} \end{aligned}$$

for some $r > 0$, and $u \in \Sigma _\nu ^\mu (\textbf{R}^{d})$ if and only if

$$\begin{aligned} | V_\varphi u (x,\xi )| \lesssim e^{-r (|x|^{\frac{1}{\nu }} + |\xi |^{\frac{1}{\mu }})} \end{aligned}$$

for all $r > 0$. See e.g., [30, Theorems 2.4 and 2.5]. The $\nu ,\mu$-Gelfand–Shilov wave front set $\textrm{WF}^{\nu ,\mu } (u) \subseteq T^* \textbf{R}^{d} {\setminus } 0$ is defined as follows.

Definition 3.5

Let $\nu ,\mu > 0$ satisfy $\nu + \mu > 1$, and suppose $\varphi \in \Sigma _\nu ^\mu (\textbf{R}^{d}) {\setminus } 0$ and $u \in (\Sigma _\nu ^\mu )'(\textbf{R}^{d})$. Then $(x_0,\xi _0) \in T^* \textbf{R}^{d} {\setminus } 0$ satisfies $(x_0,\xi _0) \notin \textrm{WF}^{\nu ,\mu } (u)$ if there exists an open set $U \subseteq T^*\textbf{R}^{d} \setminus 0$ containing $(x_0,\xi _0)$ such that

$$\begin{aligned} \sup _{\lambda> 0, \ (x,\xi ) \in U} e^{r \lambda } |V_\varphi u(\lambda ^\nu x, \lambda ^\mu \xi )| < \infty , \quad \forall r > 0. \end{aligned}$$

The requested decay is thus exponential rather than superpolynomial as for $\mathrm {WF_{g}^{ \sigma }}$. The $\nu ,\mu$-Gelfand–Shilov wave front set is a closed and $\mu /\nu$-conic subset of $T^* \textbf{R}^{d} \setminus 0$ [26].

The next result identifies powers of the weight $w_{k,m}$ defined in (3.3) as anisotropic Shubin symbols.

Lemma 3.6

Let $k,m \in \textbf{N}\setminus 0$ and $\sigma = \frac{k}{m}$. If $n \in \textbf{R}$ then $w_{k,m}^n \in G^{ n k, \sigma }$.

Proof

To simplify notation we write $w = w_{k,m}$. It is clear that $w \in C^{\infty } (\textbf{R}^{2d})$ and that w is positive everywhere. We claim that for $\alpha , \beta \in \textbf{N}^{d}$ we can write

$$\begin{aligned} \partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }w^n (x, \xi ) = \left( w (x, \xi )^2 \right) ^{\frac{n}{2} - |\alpha + \beta |} \, p_{\alpha ,\beta } (w,x,\xi ) \end{aligned}$$

(3.20)

where $p_{\alpha ,\beta }$ are polynomials of the form

$$\begin{aligned} p_{\alpha ,\beta } (w,x,\xi ) = \sum _{2 j + \frac{|\gamma |}{k} + \frac{|\kappa |}{m} \leqslant \left( 2 - \frac{1}{k} \right) |\alpha | + \left( 2 - \frac{1}{m} \right) |\beta |} c_{j,\gamma ,\kappa } w^{2j} x^\gamma \xi ^\kappa \end{aligned}$$

(3.21)

with real coefficients $c_{j,\gamma ,\kappa }$ for $(j,\gamma ,\kappa ) \in \textbf{N}\times \textbf{N}^{d} \times \textbf{N}^{d}$.

In fact the claim follows from an induction argument with respect to $|\alpha + \beta |$, starting with

$$\begin{aligned} \partial _{x_\ell } \left( \left( w (x, \xi )^2 \right) ^{\frac{n}{2}} \right) = n k |x|^{2 (k-1)} x_\ell \, \left( w (x, \xi )^2 \right) ^{\frac{n}{2} - 1} \end{aligned}$$

and

$$\begin{aligned} \partial _{\xi _\ell } \left( \left( w (x, \xi )^2 \right) ^{\frac{n}{2}} \right) = n m |\xi |^{2 (m-1)} \xi _\ell \, \left( w (x, \xi )^2 \right) ^{\frac{n}{2} - 1} \end{aligned}$$

for $1 \leqslant \ell \leqslant d$.

Next we estimate a generic monomial in (3.21), using $2 j + \frac{|\gamma |}{k} + \frac{|\kappa |}{m} \leqslant \left( 2 - \frac{1}{k} \right) |\alpha | + \left( 2 - \frac{1}{m} \right) |\beta |$, as

$$\begin{aligned} \left| w^{2 j} x^\gamma \xi ^\kappa \right| \leqslant w(x,\xi ) ^{2 j + \frac{|\gamma |}{k} + \frac{| \kappa |}{m}} \leqslant w(x,\xi ) ^{\left( 2 - \frac{1}{k} \right) |\alpha | + \left( 2 - \frac{1}{m} \right) |\beta |}. \end{aligned}$$

Inserting into (3.20) and exploiting (3.4) finally give for any $\alpha ,\beta \in \textbf{N}^{d}$ the estimate

$$\begin{aligned} \left| \partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }w^n (x, \xi ) \right|&= w(x, \xi )^{n - 2 |\alpha + \beta |} \, \left| p_{\alpha ,\beta } (w,x,\xi ) \right| \\&\lesssim w(x, \xi )^{n - \frac{1}{k} |\alpha | - \frac{1}{m} |\beta |} \\&\asymp \theta _\sigma (x,\xi )^{n k - |\alpha | - \sigma |\beta | }. \end{aligned}$$

$\square$

Suppose $\sigma > 0$ is rational, that is $\sigma = \frac{k}{m}$ with $k,m \in \textbf{N}\setminus 0$. In [7, Proposition 4.2] the authors identify the symbol class $a \in G^{n,\sigma }$ with $n \in \textbf{R}$ as the Weyl–Hörmander symbol class [16, Chapter 18.4]

$$\begin{aligned} G^{n,\sigma } = S( h_g^{-\frac{n}{1 + \sigma }}, g) \end{aligned}$$

(3.22)

defined by the metric

$$\begin{aligned} g = \frac{\textrm{d}x^2}{\left( 1 + |x|^{2k} + |\xi |^{2m} \right) ^{\frac{1}{k}}} + \frac{\textrm{d}\xi ^2}{ \left( 1 + |x|^{2k} + |\xi |^{2m} \right) ^{\frac{1}{m}} } \end{aligned}$$

and the weight $h_g^{-\frac{n}{1 + \sigma }}$. Here $h_g$ is the so called Planck function associated to g [7, 16]. The Planck function is according to [7, Remark 2.4]

$$\begin{aligned} h_g (x,\xi ) = \left( 1 + |x|^{2k} + |\xi |^{2m} \right) ^{- \frac{1}{2} \left( \frac{1}{k} + \frac{1}{m} \right) } = \left( w_{k,m} (x,\xi ) \right) ^{- \left( \frac{1}{k} + \frac{1}{m} \right) }. \end{aligned}$$

(3.23)

From this two conclusions follows: First we observe that $h_g$ satisfies the so-called uncertainty principle

$$\begin{aligned} h_g (x,\xi ) \leqslant 1 \quad \forall (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

(3.24)

and secondly by Lemma 3.6 we have $h_g \in G^{-1-\sigma ,\sigma }$.

4 Globally anisotropic Shubin–Sobolev spaces, localization operators and a sharp Gårding inequality

In this paper we will often use the following parametrized family of Hilbert modulation spaces. These spaces also have an independent interest. Proposition 4.2 complements the anisotropic Shubin pseudodifferential calculus in [27].

Definition 4.1

Let $\varphi \in \mathscr {S}(\textbf{R}^{d}) \setminus 0$. The anisotropic Shubin–Sobolev modulation space $M_{\sigma ,s} (\textbf{R}^{d})$ with anisotropy parameter $\sigma > 0$ and order $s \in \textbf{R}$ is the Hilbert subspace of $\mathscr {S}'(\textbf{R}^{d})$ defined by the norm

$$\begin{aligned} \Vert u \Vert _{M_{\sigma ,s}} = \left( \iint _{\textbf{R}^{2 d}} |V_\varphi u (x,\xi )|^2 \, \theta _\sigma (x,\xi )^{2 s} \, \textrm{d}x \, \textrm{d}\xi \right) ^{\frac{1}{2}}. \end{aligned}$$

(4.1)

For any $\sigma > 0$ we have $M_{\sigma ,0} (\textbf{R}^{d}) = L^2(\textbf{R}^{d})$ [13], and $M_{\sigma ,s_1} (\textbf{R}^{d}) \subseteq M_{\sigma ,s_2}(\textbf{R}^{d})$ is a continuous inclusion when $s_1 \geqslant s_2$. It holds

$$\begin{aligned} \mathscr {S}(\textbf{R}^{d}) = \bigcap _{s \in \textbf{R}} M_{\sigma ,s} (\textbf{R}^{d}), \quad \mathscr {S}'(\textbf{R}^{d}) = \bigcup _{s \in \textbf{R}} M_{\sigma ,s} (\textbf{R}^{d}), \end{aligned}$$

(4.2)

and $\{ \Vert \cdot \Vert _{M_{\sigma ,s}}, s \geqslant 0\}$ is a family of seminorms that defines the Fréchet space topology on $\mathscr {S}(\textbf{R}^{d})$ [13].

The next continuity result is a natural generalization of the isotropic Shubin calculus. More precisely it generalizes [20, Proposition 1.5.5] and [28, Theorem 25.2].

Proposition 4.2

Let $\sigma > 0$ and $m,s \in \textbf{R}$. If $a \in G^{m,\sigma }$ then

$$\begin{aligned} a^w(x,D): M_{\sigma ,s + m} (\textbf{R}^{d}) \rightarrow M_{\sigma ,s} (\textbf{R}^{d}) \end{aligned}$$

(4.3)

is continuous.

Proof

By a small modification of the proof of [4, Proposition 3.2] it follows that $a \in G^{m,\sigma }$ if and only if

$$\begin{aligned} \left| \partial _{z_1} ^{\alpha }\partial _{z_2} ^{\beta }\mathcal {T}_\varphi a( z,\zeta ) \right| \lesssim \theta _\sigma (z)^{m - |\alpha | - \sigma |\beta |} \langle \zeta \rangle ^{-N}, \quad z, \zeta \in \textbf{R}^{2d}, \quad \alpha , \beta \in \textbf{N}^{d}, \quad N \geqslant 0,\nonumber \\ \end{aligned}$$

(4.4)

where $z = (z_1,z_2)$ with $z_1, z_2 \in \textbf{R}^{d}$, $g \in \mathscr {S}(\textbf{R}^{2d}) {\setminus } 0$, and where $\mathcal {T}_\varphi u$ is defined by

$$\begin{aligned} \mathcal {T}_\varphi u(x,\xi )=(2\pi )^{-\frac{d}{2}} (u,T_x M_{\xi } \varphi ) = e^{i \langle x, \xi \rangle } V_\varphi u(x,\xi ), \quad x, \xi \in \textbf{R}^{d}, \end{aligned}$$

for $u \in \mathscr {S}'(\textbf{R}^{d})$ and $\varphi \in \mathscr {S}(\textbf{R}^{d}) \setminus 0$. In fact in the original proof [4] we only have to replace the weight $\langle \cdot \rangle$ used there by $\theta _\sigma$, take into account the behavior with respect to derivatives of $a \in G^{m,\sigma }$ with respect to $z_1$ and $z_2$ respectively, and use (3.2).

Let $\varphi \in \mathscr {S}(\textbf{R}^{d}) \setminus 0$ and set $\Phi = W(\varphi , \varphi ) \in \mathscr {S}(\textbf{R}^{2d}) {\setminus } 0$. If $u \in \mathscr {S}'(\textbf{R}^{d})$ then by [27, Eq. (5.3)] we have

$$\begin{aligned} |V_\varphi (a^w(x,D) u) (z)| \lesssim \int _{\textbf{R}^{2d}} |V_\varphi u(z-w)| \, \left| V_\Phi a \left( z-\frac{w}{2}, \mathcal {J}w \right) \right| \, \textrm{d}w. \end{aligned}$$

(4.5)

We obtain from (4.4), (3.2) and (3.5) the estimates

$$\begin{aligned} \left| V_\Phi a \left( z - \frac{w}{2}, \mathcal {J}w \right) \right|&\lesssim \theta _\sigma ( z-w )^{m} \theta _\sigma (w)^{|m|} \langle w \rangle ^{-N} \\&\lesssim \theta _\sigma ( z-w )^{m} \langle w \rangle ^{- \left( N - |m| \max (1, \frac{1}{\sigma }) \right) }, \quad z, w \in \textbf{R}^{2d}, \quad N \geqslant 0. \end{aligned}$$

Combining this with (4.5), Minkowski’s inequality and again (3.2) yields

$$\begin{aligned} \Vert a^w (x,D) u \Vert _{M_{\sigma ,s}}&= \left\| V_\varphi (a^w (x,D) u) \, \theta _\sigma ^s \right\| _{L^2 (\textbf{R}^{2d})} \\&\lesssim \left\| \int _{\textbf{R}^{2d}} |V_\varphi u(\cdot -w)| \, \left| V_\Phi a \left( \cdot -\frac{w}{2}, \mathcal {J}w \right) \right| \, \textrm{d}w \, \theta _\sigma (\cdot )^s \right\| _{L^2 (\textbf{R}^{2d})} \\&\lesssim \left\| \int _{\textbf{R}^{2d}} |V_\varphi u(\cdot -w)| \, \theta _\sigma ( \cdot -w )^{m + s} \langle w \rangle ^{- \left( N - ( |m| + |s| ) \max (1, \frac{1}{\sigma }) \right) } \, \textrm{d}w \right\| _{L^2 (\textbf{R}^{2d})} \\&\lesssim \left\| V_\varphi u \, \theta _\sigma ^{m + s} \right\| _{L^2 (\textbf{R}^{2d})} \\&= \Vert u \Vert _{M_{\sigma ,m + s}} \end{aligned}$$

provided $N \geqslant 0$ is sufficiently large. $\square$

Let $\varphi \in \mathscr {S}(\textbf{R}^{d})$ satisfy $\Vert \varphi \Vert _{L^2} = 1$. A localization operator $A_a$ with symbol $a \in \mathscr {S}'(\textbf{R}^{2d})$ is defined as

$$\begin{aligned} (A_a f, g) = (a V_\varphi f, V_\varphi g) = (V_\varphi ^* a V_\varphi f, g), \quad f,g \in \mathscr {S}(\textbf{R}^{d}), \end{aligned}$$

(4.6)

that is $A_a = V_\varphi ^* a V_\varphi$. Then $A_a: \mathscr {S}(\textbf{R}^{d}) \rightarrow \mathscr {S}'(\textbf{R}^{d})$ is continuous. We will assume that $\varphi$ is a Gaussian.

By [14, Theorem 1.1] we have for any $\sigma > 0$ and $s \in \textbf{R}$

$$\begin{aligned} \Vert A_{\theta _\sigma ^s} u \Vert _{L^2(\textbf{R}^{d})} \asymp \Vert u \Vert _{M_{\sigma ,s}(\textbf{R}^{d})} \end{aligned}$$

(4.7)

which means that $A_{\theta _\sigma ^s}: M_{\sigma ,s}(\textbf{R}^{d}) \rightarrow L^2(\textbf{R}^{d})$ is an isometry.

If $a \in \mathscr {S}'(\textbf{R}^{2d})$ and $\varphi$ is a Gaussian on $\textbf{R}^{d}$ we have $A_a = b^w(x,D)$ where $b = a * \psi$ with $\psi$ a Gaussian on $\textbf{R}^{2d}$ [20, Proposition 1.7.9]. If $\sigma > 0$ and $a \in G^{m, \sigma }$ then also $b \in G^{m, \sigma }$, and a real-valued implies that also b is real-valued [20, Theorem 1.7.10].

In Sect. 7 we will need the following inequality of sharp Gårding type.

Lemma 4.3

Let $k,m \in \textbf{N}\setminus 0$ and $\sigma = \frac{k}{m}$. If $a \in G^{2\left( 1 + \sigma \right) ,\sigma }$ and $a \geqslant 0$ then there exists $c > 0$ such that

$$\begin{aligned} ( a^w(x,D) f, f) \geqslant - c \Vert f \Vert _{L^2}^2, \quad f \in \mathscr {S}(\textbf{R}^{d}). \end{aligned}$$

(4.8)

Proof

By (3.22) we have $G^{2\left( 1 + \sigma \right) ,\sigma } = S( h_g^{-2}, g)$, where the Planck function $h_g$ is defined by (3.23) and satisfies the uncertainty principle (3.24). The conclusion is now a consequence of the Fefferman–Phong inequality [16, Theorem 18.6.8]. $\square$

5 Propagation of anisotropic Gabor wave front sets for evolution equations of Airy–Schrödinger type

In this section we consider the evolution equation

$$\begin{aligned} \left\{ \begin{array}{rl} \partial _t u(t,x) + i \left( p(D_x) + \langle v, x \rangle \right) u (t,x) &{} = 0, \quad x \in \textbf{R}^{d}, \quad t \in \textbf{R}, \\ u(0,\cdot ) &{} = u_0. \end{array} \right. \end{aligned}$$

(5.1)

Here $v = (v_1, \dots , v_d) \in \textbf{R}^{d}$ is a vector with nonzero entries: $v_j \ne 0$, $1 \leqslant j \leqslant d$, and $p: \textbf{R}^{d} \rightarrow \textbf{R}$ is a polynomial with real coefficients of order $m \geqslant 2$ which is a sum of one variable polynomials, that is

$$\begin{aligned} p (\xi ) = \sum _{j=1}^d p_j(\xi _j), \quad \xi = ( \xi _1, \xi _2, \dots , \xi _d) \in \textbf{R}^{d}, \end{aligned}$$

(5.2)

where

$$\begin{aligned} p_j (\xi _j) = \sum _{k=0}^{m_j} c_{j,k} \xi _j^k, \quad c_{j,k} \in \textbf{R}, \quad c_{j,m_j} \ne 0, \end{aligned}$$

(5.3)

and $\max _{j=1}^d \deg p_j = \max _{j=1}^d m_j = m$. The principal part of p is

$$\begin{aligned} P_m (\xi ) = \sum _{j \in \{1, \dots ,d \}: \ m_j = m} c_{j,m} \xi _j^m. \end{aligned}$$

(5.4)

We say that the equation (5.1) is of Airy–Schrödinger type, since when $d = 1$ a particular case of the Hamiltonian is the operator

$$\begin{aligned} a(x,D) = \frac{\textrm{d}^2}{\textrm{d}x^2} - x \end{aligned}$$

which defines the Airy equation $a(x,D) f = 0$. This equation is satisfied by the Airy function [16, Chapter 7.6].

First we deduce the explicit solution $u(t,x) = \mathscr {K}_t u_0 (x)$ to (5.1) defined by the propagator $\mathscr {K}_t$, and in particular an expression for the Schwartz kernel $K_t$ of $\mathscr {K}_t$ for each $t \in \textbf{R}$. Let $q_j$ be primitive polynomials of $p_j$:

$$\begin{aligned} q_j' = p_j, \quad 1 \leqslant j \leqslant d. \end{aligned}$$

(5.5)

If $u_0 \in \mathscr {S}(\textbf{R}^{d})$ then the solution to (5.1) is given by

$$\begin{aligned} \begin{aligned} u(t,x)&= (2 \pi )^{-\frac{d}{2}} \int _{\textbf{R}^{d}} e^{i \left( \langle x,\xi - v t \rangle + \sum _{j=1}^d v_j^{-1} \left( q_j(\xi _j - t v_j ) - q_j( \xi _j) \right) \right) } \widehat{u}_0(\xi ) \textrm{d}\xi \\&= e^{- i t \langle x, v \rangle } \mathscr {F}^{-1} \left( e^{i \varphi _t } \widehat{u}_0 \right) (x) = M_{- t v} \mathscr {F}^{-1} \left( e^{i \varphi _t } \widehat{u}_0 \right) (x) \end{aligned} \end{aligned}$$

(5.6)

where

$$\begin{aligned} \varphi _t (\xi ) = \sum _{j=1}^d v_j^{-1} \left( q_j(\xi _j - t v_j) - q_j( \xi _j) \right) . \end{aligned}$$

(5.7)

This can be confirmed by insertion of (5.6) into (5.1).

The solution operator

$$\begin{aligned} \mathscr {K}_t f = M_{- t v} \mathscr {F}^{-1} \left( e^{i \varphi _t } \widehat{f} \right) \end{aligned}$$

(5.8)

is unitary on $L^2(\textbf{R}^{d})$, and since $\varphi _{-t}(\xi ) = - \varphi _t(\xi + t v )$ we obtain for $f,g \in \mathscr {S}(\textbf{R}^{d})$

$$\begin{aligned} ( \mathscr {K}_t f, g)&= \left( \widehat{f}, e^{- i \varphi _t} \mathscr {F}\left( M_{t v} g \right) \right) = \left( \widehat{f}, e^{- i \varphi _t} T_{t v} \widehat{g} \right) = \left( \widehat{f}, T_{t v} \left( e^{- i \varphi _t(\cdot + t v )} \widehat{g} \right) \right) \\&= \left( f, \mathscr {F}^{-1} \left( T_{t v} \left( e^{i \varphi _{-t}} \widehat{g} \right) \right) \right) = \left( f, M_{t v} \mathscr {F}^{-1} \left( e^{i \varphi _{-t}} \widehat{g} \right) \right) \ = (f, \mathscr {K}_{-t} g) \end{aligned}$$

so $\mathscr {K}_t^* = \mathscr {K}_{-t} = \mathscr {K}_t^{-1}$. If $t_1, t_2 \in \textbf{R}$ then

$$\begin{aligned} \varphi _{t_1}(\xi -t_2 v) + \varphi _{t_2} (\xi ) = \varphi _{t_1 + t_2} (\xi ) \end{aligned}$$

which gives

$$\begin{aligned} \mathscr {K}_{t_1} \mathscr {K}_{t_2} f&= M_{- t_1 v} \mathscr {F}^{-1} \left( e^{i \varphi _{t_1} } \mathscr {F}\left( M_{- t_2 v} \mathscr {F}^{-1} \left( e^{i \varphi _{t_2} } \widehat{f} \right) \right) \right) \\&= M_{- t_1 v} \mathscr {F}^{-1} \left( e^{i \varphi _{t_1} } T_{- t_2 v} \left( e^{i \varphi _{t_2} } \widehat{f} \right) \right) \\&= M_{- t_1 v} \mathscr {F}^{-1} \left( T_{- t_2 v} \left( e^{i \left( \varphi _{t_1}(\cdot -t_2 v) + \varphi _{t_2} \right) } \widehat{f} \right) \right) \\&= M_{- (t_1 + t_2) v} \mathscr {F}^{-1} \left( e^{i \varphi _{t_1 + t_2} } \widehat{f} \right) = \mathscr {K}_{t_1 + t_2} f \end{aligned}$$

so the map $\textbf{R}\ni t \mapsto \mathscr {K}_t$ is in fact a one-parameter group of unitary operators.

The Schwartz kernel of the solution operator $\mathscr {K}_t$ is

$$\begin{aligned} K_t (x,y)= & {} (2 \pi )^{-d} \int _{\textbf{R}^{d}} e^{i \left( \langle x,\xi - v t \rangle - \langle y, \xi \rangle + \varphi _t(\xi ) \right) } \textrm{d}\xi \nonumber \\= & {} (2 \pi )^{-\frac{d}{2}} e^{- i t \langle x, v \rangle } \mathscr {F}^{-1} \left( e^{ i \varphi _t }\right) (x-y) \nonumber \\= & {} (2 \pi )^{-\frac{d}{2}} e^{- i t \langle x, v \rangle } \left( 1 \otimes \mathscr {F}^{-1} e^{i \varphi _t} \right) \circ \kappa ^{-1}(x,y) \in \mathscr {S}'(\textbf{R}^{2d}) \end{aligned}$$

(5.9)

where $\kappa \in \textbf{R}^{2d \times {2d}}$ is the matrix defined by $\kappa (x,y) = (x+\frac{y}{2}, x - \frac{y}{2})$ for $x,y \in \textbf{R}^{d}$. We note that $\mathscr {K}_t$ acts continuously on $\mathscr {S}(\textbf{R}^{d})$ for any $t \in \textbf{R}$, and extends uniquely to a continuous linear operator on $\mathscr {S}' (\textbf{R}^{d})$ by

$$\begin{aligned} ( \mathscr {K}_t u, \varphi ):= (u, \mathscr {K}_{-t} \varphi ), \quad u \in \mathscr {S}'(\textbf{R}^{d}), \quad \varphi \in \mathscr {S}(\textbf{R}^{d}). \end{aligned}$$

For each $1 \leqslant j \leqslant d$ we have

$$\begin{aligned} q_j(\xi _j - t v_j) - q_j( \xi _j)&= \sum _{k=0}^{m_j} \frac{c_{j,k}}{k+1} \left( (\xi _j - t v_j)^{k+1} - \xi _j^{k+1} \right) \\&= \sum _{k=0}^{m_j} \left( - c_{j,k} t v_j \xi _j^k + \frac{c_{j,k}}{k+1} \sum _{n=2}^{k+1} \left( {\begin{array}{c}k+1\\ n\end{array}}\right) (-t v_j)^{n} \xi _j^{k+1-n} \right) . \end{aligned}$$

Hence the phase function $\varphi _t (\xi )$ is a polynomial of order m with highest order term

$$\begin{aligned} \varphi _{t,m} (\xi ) = - t \sum _{j \in \{1, \dots ,d \}: \ m_j = m} c_{j,m} \xi _j^m = - t P_m(\xi ). \end{aligned}$$

We may now give a result which generalizes a particular case of [32, Theorem 5.1]. More precisely, in the quoted result the polynomial p is arbitrary with real coefficients, whereas here we assume the particular “separable” form (5.2). On the other hand, in Theorem 5.1 below we allow a vector $v \in \textbf{R}^{d}$ with nonzero entries. The result uses the Hamilton flow corresponding to the principal part $P_m(\xi )$ of the polynomial $p(\xi )$, that is

$$\begin{aligned} \chi _t (x, \xi ) = (x + t \nabla P_{m} (\xi ), \xi ), \quad t \in \textbf{R}, \quad (x, \xi ) \in T^* \textbf{R}^{d} \setminus 0. \end{aligned}$$

(5.10)

Theorem 5.1

Let p be a polynomial with real coefficients defined by (5.2), (5.3), of order $m = \max _{j=1}^d \deg p_j \geqslant 2$, with principal part $P_m$ defined by (5.4). Denote the Hamilton flow of $P_m(\xi )$ as in (5.10). Suppose $\mathscr {K}_t: \mathscr {S}' (\textbf{R}^{d}) \rightarrow \mathscr {S}' (\textbf{R}^{d})$ is the solution operator for the evolution equation (5.1), with Schwartz kernel (5.9) where $\varphi _t$ is defined by (5.5) and (5.7). Then

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( \mathscr {K}_t u)&= \chi _t \left( \mathrm {WF_{g}^{ \sigma }}(u) \right) , \quad t \in \textbf{R}, \quad u \in \mathscr {S}'(\textbf{R}^{d}), \quad \sigma = \frac{1}{m-1}, \end{aligned}$$

(5.11)

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( \mathscr {K}_t u)&= \mathrm {WF_{g}^{ \sigma }}(u), \quad t \in \textbf{R}, \quad u \in \mathscr {S}'(\textbf{R}), \quad \sigma < \frac{1}{m-1}. \end{aligned}$$

(5.12)

Proof

By [27, Theorems 7.1 and 7.2] we have

$$\begin{aligned} \begin{aligned} \textrm{WF}_{\textrm{g}}^{m-1} \left( e^{i \varphi _t } \right)&\subseteq \{ \left( x, \nabla \varphi _{t,m} (x) \right) \in \textbf{R}^{2 d}: x \in \textbf{R}^{d} \setminus 0 \} \\&= \{ \left( x, - t \nabla P_{m} (x) \right) \in \textbf{R}^{2 d}: x \in \textbf{R}^{d} \setminus 0 \}, \\ \mathrm {WF_{g}^{ \sigma }}\left( e^{i \varphi _t } \right)&\subseteq \left( \textbf{R}^{d} \setminus 0 \right) \times \{0\}, \quad \sigma > m-1. \end{aligned} \end{aligned}$$

(5.13)

Combining (5.13) with [27, Eq. (4.6) and Proposition 4.3 (i)], cf. (3.17), gives

$$\begin{aligned} \begin{aligned} \textrm{WF}_{\textrm{g}}^{\frac{1}{m-1}} \left( \mathscr {F}^{-1} e^{i \varphi _t } \right)&\subseteq \{ \left( t \nabla P_m (x), x) \right) \in \textbf{R}^{2d}: x \in \textbf{R}^{d} \setminus 0 \}, \\ \mathrm {WF_{g}^{ \sigma }}\left( \mathscr {F}^{-1} e^{i \varphi _t } \right)&\subseteq \{0\} \times \left( \textbf{R}^{d} \setminus 0 \right) , \quad \sigma < \frac{1}{m-1}. \end{aligned} \end{aligned}$$

(5.14)

Now (5.9), [27, Corollary 5.2 and Proposition 4.3 (ii)], [32, Proposition 3.2], [27, Proposition 5.3 (iii)] and (5.14) yield if $\sigma = \frac{1}{m-1}$

$$\begin{aligned}&\mathrm {WF_{g}^{ \sigma }}(K_t) = \mathrm {WF_{g}^{ \sigma }}( \left( 1 \otimes \mathscr {F}^{-1} e^{i \varphi _t} \right) \circ \kappa ^{-1} ) \\&= \left( \begin{array}{cc} \kappa &{} 0 \\ 0 &{} \kappa ^{-T} \end{array} \right) \mathrm {WF_{g}^{ \sigma }}\left( 1 \otimes \mathscr {F}^{-1} e^{i \varphi _t} \right) \\&\subseteq \{ ( \kappa ( x_1, x_2), \kappa ^{-T}(\xi _1, \xi _2) ) \in T^* \textbf{R}^{2d}: \\&\qquad \qquad \qquad \qquad \qquad (x_1, \xi _1) \in \mathrm {WF_{g}^{ \sigma }}(1) \cup \{ 0 \}, \ (x_2, \xi _2) \in \mathrm {WF_{g}^{ \sigma }}( \mathscr {F}^{-1} e^{i \varphi _t} ) \cup \{ 0 \} \} \setminus 0 \\&= \{ ( \kappa ( x_1, t \nabla P_{m} (x_2) ), \kappa ^{-T} (0, x_2) \in T^* \textbf{R}^{2d}: \ x_1, x_2 \in \textbf{R}^{d} \} \setminus 0 \\&= \left\{ \left( x_1 +t \frac{1}{2} \nabla P_{m} (x_2), x_1 - t \frac{1}{2} \nabla P_{m} (x_2) , x_2, - x_2 \right) \in T^* \textbf{R}^{2d}: \ x_1, x_2 \in \textbf{R}^{d} \right\} \setminus 0 \\&= \left\{ \left( x_1 + t \nabla P_{m} (x_2), x_1, x_2, - x_2 \right) \in T^* \textbf{R}^{2d}: \ x_1, x_2 \in \textbf{R}^{d} \right\} \setminus 0. \end{aligned}$$

Since $m \geqslant 2$ we have $\nabla P_{m} (0) = 0$. Hence $\mathrm {WF_{g}^{ \sigma }}(K_t)$ does not contain points of the form $(x, 0, \xi , 0)$ nor of the form $(0, x, 0, -\xi )$ for any $(x,\xi ) \in T^* \textbf{R}^{d} {\setminus } 0$. We may therefore apply [32, Theorem 4.4] which gives for $u \in \mathscr {S}'({\textbf {R}}^d)$

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( \mathscr {K}_t u)\subseteq&{} \mathrm {WF_{g}^{ \sigma }}(K_t)' \circ \mathrm {WF_{g}^{ \sigma }}(u) \nonumber \\=&{} \{ (x,\xi ) \in T^* {\textbf {R}}^d: \ \exists (y,\eta ) \in \mathrm {WF_{g}^{ \sigma }}(u), \ (x,y, \xi , - \eta ) \in \mathrm {WF_{g}^{ \sigma }}(K_t) \} \nonumber \\\subseteq&{} \{ ( x_1 + t \nabla P_{m} (x_2), x_2) \in T^*{\textbf {R}}^d: \ (x_1,x_2) \in \mathrm {WF_{g}^{ \sigma }}(u) \} \nonumber \\=&{} \chi _t \left( \mathrm {WF_{g}^{ \sigma }}(u) \right) . \end{aligned}$$

(5.15)

Since $\mathscr {K}_t^{-1} = \mathscr {K}_{-t}$ and $\chi _t^{-1} = \chi _{-t}$ we may strengthen (5.15) into

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( \mathscr {K}_t u) = \chi _t \left( \mathrm {WF_{g}^{ \sigma }}(u) \right) , \quad t \in \textbf{R}, \quad u \in \mathscr {S}'(\textbf{R}^{d}), \quad \sigma = \frac{1}{m-1}. \end{aligned}$$

We have proved (5.11).

Likewise if $\sigma < \frac{1}{m-1}$ then again (5.9), [27, Corollary 5.2 and Proposition 4.3 (ii)], [32, Proposition 3.2], [27, Proposition 5.3 (iii)] and (5.14) yield

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}(K_t)&\subseteq \{ ( \kappa ( x_1, x_2), \kappa ^{-T}(\xi _1, \xi _2) ) \in T^* \textbf{R}^{2 d}: \\&\qquad \qquad \qquad \qquad (x_1, \xi _1) \in \mathrm {WF_{g}^{ \sigma }}(1) \cup \{ 0 \}, \ (x_2, \xi _2) \in \mathrm {WF_{g}^{ \sigma }}( \mathscr {F}^{-1} e^{i \varphi _t} ) \cup \{ 0 \} \} \setminus 0 \\&\subseteq \{ ( \kappa ( x_1, 0 ), \kappa ^{-T}(0, x_2) \in T^* \textbf{R}^{2 d}: \ x_1, x_2 \in \textbf{R}^{d} \} \setminus 0 \\&= \left\{ \left( x_1 , x_1, x_2, - x_2 \right) \in T^* \textbf{R}^{2d}: \ x_1, x_2 \in \textbf{R}^{d} \right\} \setminus 0. \end{aligned}$$

Again [32, Theorem 4.4] gives

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( \mathscr {K}_t u) = \mathrm {WF_{g}^{ \sigma }}(u), \quad t \in \textbf{R}, \quad u \in \mathscr {S}'(\textbf{R}^{d}), \quad \sigma < \frac{1}{m-1}, \end{aligned}$$

which proves (5.12). $\square$

Remark 5.2

The conclusion from (5.11) and (5.12) is that the propagation of singularities for the equation (5.1) works exactly as when $v = 0$, as described in [32, Theorem 5.1]. The Hamiltonian in (5.1) is $a(x,\xi ) = p(\xi ) + \langle v, x \rangle$, but the propagation of singularities follows the Hamiltonian flow of $P_m(\xi )$. Note that $a_0(x,\xi ) = P_m(\xi )$ satisfies the anisotropic homogeneity

$$\begin{aligned} a_0( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{1 + \sigma } a_0(x,\xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad \lambda > 0, \end{aligned}$$

if $\sigma = \frac{1}{m-1}$, so $a_0 \in G^{1 + \sigma ,\sigma }$ according to Corollary 3.3.

If we decompose the polynomial p as

$$\begin{aligned} p(\xi ) = P_m(\xi ) + \sum _{j=0}^{m-1} P_j (\xi ) \end{aligned}$$

where each term $P_j (\xi )$ is homogeneous of degree j for $0 \leqslant j \leqslant m$, then each term $P_j$ satisfies

$$\begin{aligned} P_j( \lambda ^\sigma \xi ) = \lambda ^{\frac{j}{m-1}} P_j(\xi ), \quad \xi \in \textbf{R}^{d}, \quad \lambda > 0, \quad 0 \leqslant j \leqslant m. \end{aligned}$$

Thus $a-a_0 = \sum _{j=0}^{m-1} P_j + \langle v, x \rangle = \sum _{j=0}^{m-1} b_j$ with terms $b_j$, considered as functions on $(x,\xi ) \in T^* \textbf{R}^{d}$, of homogeneities

$$\begin{aligned} b_j( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{\frac{j}{m-1}} b_j(x,\xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad \lambda > 0, \quad 0 \leqslant j \leqslant m-1.\nonumber \\ \end{aligned}$$

(5.16)

The terms $b_j$ have all smaller order $\frac{j}{m-1} = j \sigma$ of anisotropic homogeneity than the principal part $a_0 = P_m$ which has order $1 + \sigma = m \sigma$, and which governs the propagation of singularities. Thus one may see $a - a_0$ as lower order perturbations of the Hamiltonian that do not affect propagation of singularities. Note that the Hamiltonian term $\langle v, x \rangle$ satisfies (5.16) with $j = m-1$.

As a complementary result we formulate a version of Theorem 5.1 in the framework of Beurling type Gelfand–Shilov spaces $\Sigma _\nu ^\mu (\textbf{R}^{d})$ for $\nu + \mu > 1$ and their dual ultradistribution spaces $\left( \Sigma _\nu ^\mu \right) '(\textbf{R}^{d})$.

Theorem 5.3

Let p be a polynomial with real coefficients defined by (5.2), (5.3), of order $m = \max _{j=1}^d \deg p_j \geqslant 2$, with principal part $P_m$ defined by (5.4). Denote the Hamilton flow of $P_m(\xi )$ as in (5.10). Suppose $\mathscr {K}_t: \mathscr {S}(\textbf{R}^{d}) \rightarrow \mathscr {S}(\textbf{R}^{d})$ is the solution operator for the evolution equation (5.1), with Schwartz kernel (5.9) where $\varphi _t$ is defined by (5.5) and (5.7).

If $\nu \geqslant \mu (m-1) > 1$ then $\mathscr {K}_t$ is continuous on $\Sigma _\nu ^\mu (\textbf{R}^{d})$, extends uniquely to a continuous linear operator on $\left( \Sigma _\nu ^\mu \right) '(\textbf{R}^{d})$, and

$$\begin{aligned} \textrm{WF}^{\nu ,\mu } ( \mathscr {K}_t u)&= \chi _t \left( \textrm{WF}^{\nu ,\mu } (u) \right) , \quad t \in \textbf{R}, \quad u \in \left( \Sigma _\nu ^\mu \right) '(\textbf{R}^{d}), \quad \nu = \mu (m-1) > 1, \end{aligned}$$

(5.17)

$$\begin{aligned} \textrm{WF}^{\nu ,\mu } ( \mathscr {K}_t u)&= \textrm{WF}^{\nu ,\mu } (u), \quad t \in \textbf{R}, \quad u \in \left( \Sigma _\nu ^\mu \right) '(\textbf{R}^{d}), \quad \nu> \mu (m-1) > 1. \end{aligned}$$

(5.18)

Proof

In the Gelfand–Shilov functional framework we have, similar to (5.13), by [31, Theorems 6.1 and 6.2] if $\nu > \frac{1}{m-1}$

$$\begin{aligned} \begin{aligned} \textrm{WF}^{\nu ,\nu (m-1)} \left( e^{i \varphi _t } \right)&\subseteq \{ \left( x, - t \nabla P_{m}(x) \right) \in \textbf{R}^{2d}: x \in \textbf{R}^{d} \setminus 0 \}, \\ \textrm{WF}^{\nu ,\mu } \left( e^{i \varphi _t } \right)&\subseteq \left( \textbf{R}^{d} \setminus 0 \right) \times \{0\}, \quad \mu > \nu (m-1). \end{aligned} \end{aligned}$$

(5.19)

As before [26, Eq. (3.8) and Proposition 3.6 (i)] give

$$\begin{aligned} \begin{aligned} \textrm{WF}^{\nu (m-1),\nu } \left( \mathscr {F}^{-1} e^{i \varphi _t } \right)&\subseteq \{ \left( t \nabla P_{m}(x), x \right) \in \textbf{R}^{2d}: x \in \textbf{R}^{d} \setminus 0 \}, \\ \textrm{WF}^{\mu ,\nu } \left( \mathscr {F}^{-1} e^{i \varphi _t } \right)&\subseteq \{0\} \times \left( \textbf{R}^{d} \setminus 0 \right) , \quad \mu > \nu (m-1). \end{aligned} \end{aligned}$$

(5.20)

From [31, Proposition 4.5] and [26, Proposition 3.6 (ii), Corollary 6.4 and Proposition 7.1 (iii)] we obtain if $\nu = \mu (m-1) > 1$

$$\begin{aligned} \textrm{WF}^{\nu ,\mu } (K_t) \subseteq \left\{ \left( x_1 + t \nabla P_{m}(x_2), x_1, x_2, - x_2 \right) \in T^* \textbf{R}^{2d}: \ x_1, x_2 \in \textbf{R}^{d} \right\} \setminus 0, \end{aligned}$$

and if $\nu> \mu (m-1) > 1$

$$\begin{aligned} \textrm{WF}^{\nu ,\mu } (K_t) \subseteq \left\{ \left( x_1, x_1, x_2, - x_2 \right) \in T^* \textbf{R}^{2d}: \ x_1, x_2 \in \textbf{R}^{d} \right\} \setminus 0. \end{aligned}$$

At this point [31, Theorem 5.5] yields the following two final conclusions: If $\nu \geqslant \mu (m-1) > 1$ then $\mathscr {K}_t$ is continuous on $\Sigma _\nu ^\mu (\textbf{R}^{d})$ and extends uniquely to a continuous linear operator on $\left( \Sigma _\nu ^\mu \right) '(\textbf{R}^{d})$, and the propagation of singularities follows (5.17) and (5.18). $\square$

Again the overall conclusion is that propagation of singularities works as if $v = 0$.

5.1 Fourier transformation of the evolution equation

Next we take the Fourier transform $\mathscr {F}u(t,\cdot )$. If we denote this Fourier transform for simplicity still by $u(t,\cdot )$, then we obtain from (5.1) the evolution equation

$$\begin{aligned} \left\{ \begin{array}{rl} \partial _t u(t,x) + i \left( - \langle v, D_x \rangle + p(x) \right) u (t,x) &{} = 0, \quad x \in \textbf{R}^{d}, \quad t \in \textbf{R}, \\ u(0,\cdot ) &{} = u_0. \end{array} \right. \end{aligned}$$

(5.21)

where again $v \in \textbf{R}^{d}$ is a vector with nonzero entries: $v_j \ne 0$, $1 \leqslant j \leqslant d$.

Referring to (5.7) and (5.8) the solution is now for $u_0 \in \mathscr {S}(\textbf{R}^{d})$

$$\begin{aligned} \begin{aligned} u(t,x)&= \widetilde{\mathscr {K}}_t u_0 = \mathscr {F}\mathscr {K}_t \mathscr {F}^{-1} u_0&= T_{-t v} \left( e^{i \varphi _t } u_0 \right) (x) \\&= e^{i \varphi _t (x + t v)} u_0 (x + t v) \\&= e^{ - i \varphi _{-t} (x)} u_0 (x + t v). \end{aligned} \end{aligned}$$

(5.22)

The solution operator $\widetilde{\mathscr {K}}_t$ is continuous on $\mathscr {S}(\textbf{R}^{d})$ and extends to a continuous operator on $\mathscr {S}'(\textbf{R}^{d})$. Now (5.11) combined with [27, Proposition 4.3 (i)] give

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( {\widetilde{\mathscr {K}}}_t u_0 ) = \widetilde{\chi }_t \left( \mathrm {WF_{g}^{ \sigma }}(u_0) \right) , \quad t \in \textbf{R}, \quad u_0 \in \mathscr {S}'(\textbf{R}^{d}), \quad \sigma = m-1, \end{aligned}$$

(5.23)

where

$$\begin{aligned} \widetilde{\chi }_t (x, \xi ) = \mathcal {J}\chi _t (-\mathcal {J})(x,\xi ) = (x, \xi - t \nabla P_{m} (x)), \quad t \in \textbf{R}, \quad (x, \xi ) \in T^* \textbf{R}^{d} \setminus 0,\nonumber \\ \end{aligned}$$

(5.24)

and $\chi _t$ is defined by (5.10). This is the Hamilton flow corresponding to the principal part $P_m(x)$ of the polynomial p(x). We also obtain

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( {\widetilde{\mathscr {K}}}_t u_0 ) = \mathrm {WF_{g}^{ \sigma }}(u_0), \quad t \in \textbf{R}, \quad u \in \mathscr {S}'(\textbf{R}^{d}), \quad \sigma > m-1. \end{aligned}$$

(5.25)

These considerations, combined with a similar discussion in the Gelfand–Shilov framework, may be summarized as follows.

Theorem 5.4

Let p be a polynomial with real coefficients defined by (5.2), (5.3), of order $m = \max _{j=1}^d \deg p_j \geqslant 2$, with principal part $P_m$ defined by (5.4). Denote the Hamilton flow of $P_m(x)$ as in (5.24). Suppose ${\widetilde{\mathscr {K}}}_t: \mathscr {S}(\textbf{R}^{d}) \rightarrow \mathscr {S}(\textbf{R}^{d})$ is the solution operator (5.22), where $\varphi _t$ is defined by (5.5) and (5.7), for the evolution equation (5.21). Then

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( {\widetilde{\mathscr {K}}}_t u )&= \widetilde{\chi }_t \left( \mathrm {WF_{g}^{ \sigma }}(u) \right) , \quad t \in \textbf{R}, \quad u \in \mathscr {S}'(\textbf{R}^{d}), \quad \sigma = m-1, \\ \mathrm {WF_{g}^{ \sigma }}( {\widetilde{\mathscr {K}}}_t u )&= \mathrm {WF_{g}^{ \sigma }}(u), \quad t \in \textbf{R}, \quad u \in \mathscr {S}'(\textbf{R}^{d}), \quad \sigma > m-1. \end{aligned}$$

If $\nu \geqslant \mu (m-1) > 1$ then ${\widetilde{\mathscr {K}}}_t$ is continuous on $\Sigma _\mu ^\nu (\textbf{R}^{d})$, extends uniquely to a continuous linear operator on $\left( \Sigma _\mu ^\nu \right) '(\textbf{R}^{d})$, and

$$\begin{aligned} \textrm{WF}^{\mu ,\nu } ( {\widetilde{\mathscr {K}}}_t u)&= \widetilde{\chi }_t \left( \textrm{WF}^{\mu ,\nu } (u) \right) , \quad t \in \textbf{R}, \quad u \in \left( \Sigma _\mu ^\nu \right) '(\textbf{R}^{d}), \quad \nu = \mu (m-1)> 1, \\ \textrm{WF}^{\mu ,\nu } ( {\widetilde{\mathscr {K}}}_t u)&= \textrm{WF}^{\mu ,\nu } (u), \quad t \in \textbf{R}, \quad u \in \left( \Sigma _\mu ^\nu \right) '(\textbf{R}^{d}), \quad \nu> \mu (m-1) > 1. \end{aligned}$$

The conclusion from Theorem 5.4 is that the propagation of singularities for (5.21) works again exactly as when $v = 0$, in both the tempered Schwartz distribution and the Gelfand–Shilov ultradistribution frameworks, respectively.

Remark 5.5

Consider the Hamiltonian $a(x,\xi ) = p(x) - \langle v, \xi \rangle$ in the equation (5.21), with $\deg p = m$. The propagation of $\mathrm {WF_{g}^{ \sigma }}$ with $\sigma = m-1$ is governed by $a_0(x,\xi ) = P_m(x)$ which satisfies

$$\begin{aligned} a_0( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{m} P_m(x) = \lambda ^{1+\sigma } a_0(x,\xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad \lambda > 0. \end{aligned}$$

Thus $a_0 \in G^{1 + \sigma ,\sigma } = G^{m,m-1}$ according to Corollary 3.3. This is similar to Remark 5.2. In Sect. 6 we will study more general Hamiltonians that satisfy this type of anisotropic homogeneity.

When the Hamiltonian Weyl symbol $a(x,\xi ) = p(\xi )$ is a polynomial in $\xi$ of the form (5.2) then the Hamilton flow is as in (5.10) that is

$$\begin{aligned} \chi _t (x, \xi ) = (x + t \nabla P_{m} (\xi ), \xi ), \quad t \in \textbf{R}, \quad (x, \xi ) \in T^* \textbf{R}^{d} \setminus 0, \end{aligned}$$

(5.26)

where $P_m$ is the principal part of p. When instead the Weyl symbol depends on x, $a(x,\xi ) = p(x)$, with the same assumptions on p, we obtain the Hamilton flow (5.24) that is

$$\begin{aligned} \chi _t (x, \xi ) = (x, \xi - t \nabla P_{m} (x)), \quad t \in \textbf{R}, \quad (x, \xi ) \in T^* \textbf{R}^{d} \setminus 0. \end{aligned}$$

(5.27)

Define for $\sigma > 0$ the anisotropic scaling map $\Lambda _\sigma (\lambda ): T^* \textbf{R}^{d} \rightarrow T^* \textbf{R}^{d}$ as

$$\begin{aligned} \Lambda _\sigma (\lambda ) (x,\xi ) = (\lambda x, \lambda ^\sigma \xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad \lambda > 0. \end{aligned}$$

(5.28)

For suitable $\sigma > 0$ the Hamilton flows (5.26) and (5.27) commute with $\Lambda _\sigma (\lambda )$ for all $\lambda > 0$. In fact, if $\chi _t$ is defined by (5.26) and $\sigma = \frac{1}{m-1}$ then for $\lambda > 0$

$$\begin{aligned} \chi _t ( \lambda x, \lambda ^\sigma \xi ) = (\lambda x + t \nabla P_{m} (\lambda ^\sigma \xi ), \lambda ^\sigma \xi ) = ( \lambda (x + t \nabla P_{m} (\xi )), \lambda ^\sigma \xi ) = \Lambda _\sigma (\lambda ) \chi _t(x,\xi ). \end{aligned}$$

Likewise if $\chi _t$ is defined by (5.27) and $\sigma = m-1$ then for $\lambda > 0$

$$\begin{aligned} \chi _t ( \lambda x, \lambda ^\sigma \xi ) = (\lambda x, \lambda ^\sigma \xi - t \nabla P_{m} ( \lambda x)) = ( \lambda x, \lambda ^\sigma (\xi - t \nabla P_{m} ( x) ) ) = \Lambda _\sigma (\lambda ) \chi _t(x,\xi ). \end{aligned}$$

Thus in both cases the Hamilton flow $\chi _t$ commutes with anisotropic scaling $\Lambda _\sigma$

$$\begin{aligned} \chi _t \Lambda _\sigma (\lambda ) = \Lambda _\sigma (\lambda ) \chi _t, \quad \lambda > 0, \quad t \in \textbf{R}, \end{aligned}$$

(5.29)

suppressing the variables $(x,\xi ) \in T^* \textbf{R}^{d} \setminus 0$.

Remark 5.6

The commutativity (5.29) means that the considered Hamilton flows are consistent with the propagation inclusion that we aim for, namely

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( \mathscr {K}_t u) \subseteq \chi _t \left( \mathrm {WF_{g}^{ \sigma }}(u) \right) , \quad t \in \textbf{R}, \quad u \in \mathscr {S}'(\textbf{R}^{d}), \end{aligned}$$

(5.30)

for the solution operator (propagator) $\mathscr {K}_t$ of a Schrödinger type evolution equation.

Indeed the inclusion (5.30) requires that the image of $\chi _t$ of any $\mathrm {WF_{g}^{ \sigma }}(u) \subseteq T^* \textbf{R}^{d} {\setminus } 0$ for $u \in \mathscr {S}'(\textbf{R}^{d})$ contains a closed $\sigma$-conic subset of $T^* \textbf{R}^{d} \setminus 0$. It is not known if for any closed $\sigma$-conic subset of $\Gamma \subseteq T^* \textbf{R}^{d} {\setminus } 0$ there exists $u \in \mathscr {S}'(\textbf{R}^{d})$ such that $\mathrm {WF_{g}^{ \sigma }}(u) = \Gamma$ except when $\sigma = 1$. In fact if $\sigma = 1$ then [29, Theorem 6.1] answers the question affirmatively. Nevertheless it seems reasonable to ask that the image of $\chi _t$ of any closed $\sigma$-conic subset of $T^* \textbf{R}^{d} \setminus 0$ contains a closed $\sigma$-conic subset of $T^* \textbf{R}^{d} \setminus 0$. Then in particular a $\sigma$-conic curve of the form

$$\begin{aligned} R_{x,\xi } = \{ (\lambda x, \lambda ^\sigma \xi ) \in T^* \textbf{R}^{d} \setminus 0, \ \lambda > 0 \} \end{aligned}$$

where $(x,\xi ) \in \textbf{S}^{2d-1}$, must be mapped into another such curve, that is, $\chi _t R_{x,\xi } = R_{z}$ where $z \in \textbf{S}^{2d-1}$.

6 Anisotropically homogeneous Hamiltonians and their flows

Given a Hamiltonian $a: \textbf{R}^{2d} \setminus 0 \rightarrow \textbf{R}$ of class $C^\infty$, Hamilton’s system of equations is

$$\begin{aligned} \left\{ \begin{array}{l} x' (t) = \nabla _{\xi } a \left( x(t), \xi (t) \right) , \\ \xi ' (t) = -\nabla _{x} a \left( x(t), \xi (t) \right) , \\ x(0) = x, \\ \xi (0) = \xi , \end{array} \right. \end{aligned}$$

(6.1)

for initial datum $(x,\xi ) \in T^* \textbf{R}^{d} {\setminus } 0$ and $t \in (-T,T)$ with $T > 0$. By the Picard–Lindelöf theorem there is a unique solution $( x(t), \xi (t) ) = \chi _t(x,\xi )$, $\chi _t: \textbf{R}^{2d} {\setminus } 0 \rightarrow \textbf{R}^{2d} {\setminus } 0$, $t \in (-T,T)$, for some $T > 0$. It is called the Hamiltonian flow. In general the maximal T depends on $(x,\xi )$. The map $(-T,T) \ni t \rightarrow \chi _t$ satisfies $\chi _{t_1+t_2} = \chi _{t_1} \chi _{t_2}$ and $\chi _t^{-1} = \chi _{-t}$ [1]. The solution $\chi _t$ is a symplectomorphism on $T^* \textbf{R}^{d}$ for fixed $t \in (-T,T)$ [8], $C^1$ with respect to t, and hence a $C^1$ diffeomorphism on $T^* \textbf{R}^{d}$. If the level sets of a are compact then the solution $\chi _t(x,\xi )$ extends to all $t \in \textbf{R}$ [1]. Using the matrix (3.18) we may write the differential equation in (6.1) as

$$\begin{aligned} \begin{aligned} \left( \begin{array}{l} x' (t) \\ \xi '(t) \end{array} \right)&= \mathcal {J}\nabla _{x,\xi } a \left( x(t), \xi (t) \right) . \end{aligned} \end{aligned}$$

(6.2)

Suppose the solution $\chi _t: \textbf{R}^{2d} {\setminus } 0 \rightarrow \textbf{R}^{2d} {\setminus } 0$ is well defined for $t \in (-T,T)$ with the parameter $T > 0$ valid for all initial data $(x,\xi ) \in T^* \textbf{R}^{d} {\setminus } 0$. The assumption $a \in C^\infty (\textbf{R}^{2d} {\setminus } 0)$ and [15, Theorem V.4.1] imply that

$$\begin{aligned}{} & {} (-T,T) \times \textbf{R}^{2d} \setminus 0 \ni (t, x,\xi ) \mapsto \partial _{x} ^{\alpha } \partial _{\xi } ^{\beta } \chi _t (x,\xi ) \in \textbf{R}^{2d} \setminus 0 \in C ((-T,T) \times \textbf{R}^{2d} \setminus 0) \nonumber \\{} & {} \quad \forall \alpha , \beta \in \textbf{N}^{d}, \end{aligned}$$

(6.3)

and in particular $\chi _t \in C^\infty ( \textbf{R}^{2d} {\setminus } 0, \textbf{R}^{2d} {\setminus } 0)$ for each $t \in (-T,T)$.

The next lemma will be used in the proofs of Proposition 6.2 and its converse Proposition 6.4.

Lemma 6.1

If $\sigma > 0$ and $a \in C^\infty (\textbf{R}^{2d} \setminus 0)$ is real-valued then

$$\begin{aligned} a( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{\sigma +1} a(x,\xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \quad \lambda > 0, \end{aligned}$$

(6.4)

holds if and only if

$$\begin{aligned} \lim _{(x,\xi ) \rightarrow (0,0)} a(x,\xi )&= 0, \end{aligned}$$

(6.5)

$$\begin{aligned} \nabla _x a ( \lambda x, \lambda ^\sigma \xi )&= \lambda ^{\sigma } \nabla _x a ( x, \xi ), \quad (x,\xi ) \in T^* {\textbf {R}}^{d} \setminus 0, \quad \lambda > 0, \quad \text { and } \end{aligned}$$

(6.6)

$$\begin{aligned} \nabla _\xi a ( \lambda x, \lambda ^\sigma \xi )&= \lambda \nabla _\xi a ( x, \xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \quad \lambda > 0, \end{aligned}$$

(6.7)

hold.

Proof

It is immediate to see that (6.4) implies (6.6) and (6.7). Since any $(y,\eta ) \in T^* \textbf{R}^{d} \setminus 0$ can be written as $(y,\eta ) = (\lambda x, \lambda ^\sigma \xi )$ for a unique $\lambda > 0$ and a unique $(x,\xi ) \in \textbf{S}^{2d-1}$ [27, Section 3], also (6.5) follows from (6.4).

Assume on the other hand (6.5), (6.6) and (6.7). Let $(x,\xi ) \in T^* \textbf{R}^{d} \setminus 0$ and define the function $f(t) = a (t x, t \xi )$ for $t > 0$. Then we have for $0< \varepsilon < 1$

$$\begin{aligned} a(x,\xi ) = f(1) = \int _{\varepsilon }^1 f'(t) \, \textrm{d}t + f (\varepsilon ) = \int _{\varepsilon }^1 \langle \nabla _{x,\xi } a(t (x,\xi )), (x,\xi ) \rangle \, \textrm{d}t + a ( \varepsilon ( x, \xi )) \end{aligned}$$

which gives for $\lambda > 0$, using (6.6) and (6.7),

$$\begin{aligned} a( \lambda x, \lambda ^\sigma \xi )&= \int _{\varepsilon }^1 \langle \nabla _{x,\xi } a(t ( \lambda x, \lambda ^\sigma \xi )), ( \lambda x, \lambda ^\sigma \xi ) \rangle \, \textrm{d}t + a ( \varepsilon ( \lambda x, \lambda ^\sigma \xi )) \\&= \lambda ^{\sigma +1} \int _{\varepsilon }^1 \langle \nabla _{x,\xi } a(t (x, \xi )), ( x, \xi ) \rangle \, \textrm{d}t + a ( \varepsilon ( \lambda x, \lambda ^\sigma \xi )) \\&= \lambda ^{\sigma +1} \Big ( a(x,\xi ) - a ( \varepsilon ( x, \xi )) \Big ) + a ( \varepsilon ( \lambda x, \lambda ^\sigma \xi )). \end{aligned}$$

The claim (6.4) now follows from the limit as $\varepsilon \rightarrow 0^+$ using the assumption (6.5). $\square$

In the following result we show that the Hamilton flow commutes with anisotropic scaling for Hamiltonians with the anisotropic homogeneity (6.4).

Proposition 6.2

Let $\sigma > 0$, and suppose $a \in C^\infty (\textbf{R}^{2d} \setminus 0)$ is real-valued and satisfies

$$\begin{aligned} a( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{\sigma +1} a(x,\xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \quad \lambda > 0. \end{aligned}$$

(6.8)

Then there exists $T > 0$ such that the Hamilton flow $\chi _t(x,\xi )$ defined by the function a is well defined for $t \in [-T,T]$ uniformly for all $(x,\xi ) \in T^* \textbf{R}^{d} \setminus 0$, and $\chi _t$ satisfies

$$\begin{aligned} \chi _t( \Lambda _\sigma (\lambda ) (x,\xi ) ) = \Lambda _\sigma (\lambda ) \chi _t (x,\xi ), \quad \lambda > 0, \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \quad t \in [-T,T],\nonumber \\ \end{aligned}$$

(6.9)

where $\Lambda _\sigma (\lambda ): T^* \textbf{R}^{d} \rightarrow T^* \textbf{R}^{d}$ is defined in (5.28).

Proof

The assumption (6.8) and Lemma 6.1 give the anisotropic homogeneities

$$\begin{aligned} \begin{aligned} \nabla _x a ( \lambda x, \lambda ^\sigma \xi )&= \lambda ^{\sigma } \nabla _x a ( x, \xi ), \\ \nabla _\xi a ( \lambda x, \lambda ^\sigma \xi )&= \lambda \nabla _\xi a ( x, \xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \quad \lambda > 0, \end{aligned} \end{aligned}$$

which can be written as

$$\begin{aligned} \nabla _{x,\xi } a ( \lambda x, \lambda ^\sigma \xi ) = \Lambda _{\frac{1}{\sigma }} (\lambda ^{\sigma }) \nabla _{x,\xi } a ( x, \xi ). \end{aligned}$$

(6.10)

This gives $\lim _{(x,\xi ) \rightarrow (0,0)} \nabla _{x,\xi } a(x,\xi ) = 0$. Set

$$\begin{aligned} M = \sup _{0< |(x,\xi )| \leqslant \frac{3}{2}} | \nabla _{x,\xi } a(x,\xi )| < + \infty . \end{aligned}$$

If $(x,\xi ) \in \textbf{S}^{2d-1}$ then by the Picard–Lindelöf theorem [15, Theorem II.1.1] the Hamilton flow stays in the ball $\chi _t(x,\xi ) \in \overline{{\text {B}}_{\frac{1}{2}}(x,\xi )}$ if $-T \leqslant t \leqslant T$ and $T = \frac{1}{2\,M}$. Thus there exists $T > 0$ such that the Hamilton flow $\chi _t: \textbf{S}^{2d-1} \rightarrow \textbf{R}^{2d} \setminus 0$ is well defined for $-T \leqslant t \leqslant T$ uniformly over $\textbf{S}^{2d-1}$.

Let $(x,\xi ) \in T^* \textbf{R}^{d} \setminus 0$ and set $(x(t), \xi (t) ) = \chi _t(x,\xi )$. For $T_0 > 0$ sufficiently small we have $( x(t), \xi (t) ) \in T^* \textbf{R}^{d} {\setminus } 0$ for $t \in [-T_0,T_0]$. From (6.2), (6.10) and

$$\begin{aligned} \mathcal {J}\Lambda _{\frac{1}{\sigma }} (\lambda ^{-\sigma }) = \Lambda _{\sigma } (\lambda ^{-1}) \mathcal {J}\end{aligned}$$

(6.11)

we obtain

$$\begin{aligned} \frac{d}{dt} \chi _t(x,\xi ) = \left( \begin{array}{l} x' (t) \\ \xi '(t) \end{array} \right)&= \mathcal {J}\nabla _{x,\xi } a \left( x(t), \xi (t) \right) \\&= \mathcal {J}\Lambda _{\frac{1}{\sigma }} (\lambda ^{-\sigma }) \nabla _{x,\xi } a \left( \lambda x(t), \lambda ^\sigma \xi (t) \right) \\&= \Lambda _{\sigma } (\lambda ^{-1}) \mathcal {J}\nabla _{x,\xi } a \left( \lambda x(t), \lambda ^\sigma \xi (t) \right) \end{aligned}$$

which may be written

$$\begin{aligned} ( \lambda x'(t), \lambda ^\sigma \xi '(t) ) = \mathcal {J}\nabla _{x,\xi } a \left( \lambda x(t), \lambda ^\sigma \xi (t) \right) . \end{aligned}$$

Thus $( \lambda x(t), \lambda ^\sigma \xi (t) )$ solves (6.1) for $t \in [-T_0,T_0]$ with initial datum $(\lambda x, \lambda ^\sigma \xi )$, for any $\lambda > 0$. If we choose $\lambda > 0$ such that $|(\lambda x, \lambda ^\sigma \xi )| = 1$ then the solution is well defined for $t \in [-T,T]$ by the first part of the proof. The solution $( \lambda x(t), \lambda ^\sigma \xi (t) )$ hence extends to $t \in [-T,T]$ for all $\lambda > 0$. By the uniqueness of the solution we have $\chi _t (\lambda x, \lambda ^\sigma \xi ) = ( \lambda x(t), \lambda ^\sigma \xi (t) )$. It follows that the Hamilton flow $\chi _t: \textbf{R}^{2d} {\setminus } 0 \rightarrow \textbf{R}^{2d} {\setminus } 0$ is well defined in the interval $t \in [-T,T]$ uniformly over the phase space $\textbf{R}^{2d} \setminus 0$. In conclusion we have

$$\begin{aligned} \Lambda _\sigma (\lambda ) \chi _t(x,\xi ) = ( \lambda x(t), \lambda ^\sigma \xi (t) ) = \chi _t( \lambda x, \lambda ^\sigma \xi ) = \chi _t ( \Lambda _\sigma (\lambda ) (x,\xi ) ) \end{aligned}$$

for $(x,\xi ) \in T^* \textbf{R}^{d} \setminus 0$, $\lambda > 0$ and $t \in [-T,T]$. $\square$

Remark 6.3

With the assumptions of Proposition 6.2, for any $t \in [-T,T]$ we have

$$\begin{aligned} \lim _{(x,\xi ) \rightarrow (0,0)} \chi _t (x,\xi ) = 0. \end{aligned}$$

In fact this is an immediate consequence of (6.9). So defining $\chi _t(0,0) = (0,0)$ we could extend the Hamilton flow as a continuous bijection $\chi _t: \textbf{R}^{2d} \rightarrow \textbf{R}^{2d}$ for $t \in [-T,T]$. By [15, Theorem V.4.1] we know that $\chi _t \in C^\infty ( \textbf{R}^{2d} {\setminus } 0, \textbf{R}^{2d} {\setminus } 0)$ but we cannot extend the smoothness to the new domain point (0, 0).

Next we show a converse of Proposition 6.2.

Proposition 6.4

Let $a \in C^\infty (\textbf{R}^{2d} \setminus 0)$ be real-valued and suppose

$$\begin{aligned} \lim _{(x,\xi ) \rightarrow (0,0)} a(x,\xi ) = 0. \end{aligned}$$

Suppose the solution $\chi _t(x,\xi )$ to (6.1) is well defined for $t \in [-T,T]$ for some $T > 0$ for all $(x,\xi ) \in T^* \textbf{R}^{d} {\setminus } 0$. If $\sigma > 0$ and (6.9) holds true then a satisfies the homogeneity

$$\begin{aligned} a( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{\sigma +1} a( x, \xi ), \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \quad \lambda > 0. \end{aligned}$$

(6.12)

Proof

For $(x,\xi ) \in T^* \textbf{R}^{d} \setminus 0$ we denote $( x(t), \xi (t) ) = \chi _t(x,\xi )$. Formula (6.9) means that the solution to (6.1) with $(x,\xi )$ replaced by $(\lambda x, \lambda ^\sigma \xi )$ for $\lambda > 0$ is $\Lambda _\sigma (\lambda ) \chi _t( x, \xi ) = (\lambda x(t), \lambda ^\sigma \xi (t) )$.

Let $(x,\xi ) \in T^* \textbf{R}^{d} \setminus 0$. From (6.2) and (6.9) we obtain for any $\lambda > 0$

$$\begin{aligned} \mathcal {J}\nabla _{x,\xi } a \left( x(t), \xi (t) \right)&= \frac{d}{dt} \chi _t(x,\xi ) = \frac{d}{dt} \left( \Lambda _\sigma (\lambda ^{-1}) \chi _t \left( \Lambda _\sigma (\lambda ) (x,\xi ) \right) \right) \\&= \Lambda _\sigma (\lambda ^{-1}) \frac{d}{dt} \left( \chi _t \left( \Lambda _\sigma (\lambda ) (x,\xi ) \right) \right) \\&= \Lambda _\sigma (\lambda ^{-1}) \mathcal {J}\nabla _{x,\xi } a \left( \lambda x(t), \lambda ^\sigma \xi (t) \right) . \end{aligned}$$

With aid of (6.11) and $\mathcal {J}^{-1} = -\mathcal {J}$ this gives

$$\begin{aligned} \nabla _{x,\xi } a \left( x(t), \xi (t) \right)&= - \mathcal {J}\Lambda _\sigma (\lambda ^{-1}) \mathcal {J}\nabla _{x,\xi } a \left( \lambda x(t), \lambda ^\sigma \xi (t) \right) \\&= \Lambda _{\frac{1}{\sigma }} (\lambda ^{-\sigma }) \nabla _{x,\xi } a \left( \lambda x(t), \lambda ^\sigma \xi (t) \right) . \end{aligned}$$

For $t=0$ we get

$$\begin{aligned} \nabla _x a ( \lambda x, \lambda ^\sigma \xi )&= \lambda ^\sigma \nabla _x a ( x, \xi ), \\ \nabla _\xi a ( \lambda x, \lambda ^\sigma \xi )&= \lambda \nabla _\xi a ( x, \xi ) \end{aligned}$$

which together with the assumption $\lim _{(x,\xi ) \rightarrow (0,0)} a(x,\xi ) = 0$ is equivalent to (6.12) by Lemma 6.1. $\square$

We note that a function a that satisfies (6.12) is determined by its values on the unit sphere $\textbf{S}^{2d-1}$, and

$$\begin{aligned} a( x, \xi ) = \lambda _\sigma ^{\sigma +1} (x,\xi ) \, a( p_\sigma (x,\xi ) ), \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \end{aligned}$$

where $\lambda _\sigma : \textbf{R}^{2d} \setminus 0 \rightarrow \textbf{R}_+$ and $p_\sigma : \textbf{R}^{2d} \setminus 0 \rightarrow \textbf{S}^{2d-1}$ are smooth functions defined in [27, Section 3].

Examples of Hamiltonians that satisfy $a \in C^\infty (\textbf{R}^{2d} {\setminus } 0)$ and (6.12) are the homogeneous polynomials that depend on either x or $\xi$ (but not both) studied in Sect. 5 (cf. Remarks 5.2 and 5.5), that is

$$\begin{aligned} a(x,\xi )= & {} P_m(x), \quad \sigma = m-1, \\ a(x,\xi )= & {} P_m(\xi ), \quad \sigma = \frac{1}{m-1}, \end{aligned}$$

where $m \in \textbf{N}$ and $m \geqslant 2$. Other examples are

$$\begin{aligned} a(x,\xi ) = c \left( |x| + |\xi |^{\frac{1}{\sigma }} \right) ^{\sigma +1}, \end{aligned}$$

where $\sigma > 0$ and $c \in \textbf{R}\setminus 0$, and

$$\begin{aligned} a(x,\xi ) = c \left( |x|^{2k} + |\xi |^{2m} \right) ^{ \frac{1}{2} \left( \frac{1}{k} + \frac{1}{m} \right) }, \end{aligned}$$

with $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$ and $c \in \textbf{R}{\setminus } 0$.

Note that the Hamiltonians

$$\begin{aligned} a(x,\xi ) = c_1 |x|^{\sigma +1} + c_2 |\xi |^{1 + \frac{1}{\sigma }} \end{aligned}$$

with $c_1, c_2 \in \textbf{R}$ and $\sigma > 0$,

$$\begin{aligned} a(x,\xi ) = c_1 |x|^{2k} + c_2 |\xi |^{1 + \frac{1}{2k-1}} \end{aligned}$$

with $k \in \textbf{N}{\setminus } 0$, $\sigma = 2k-1$ and $c_1, c_2 \in \textbf{R}$, and

$$\begin{aligned} a(x,\xi ) = c_1 |x|^{1 + \frac{1}{2k-1}} + c_2 |\xi |^{2k} \end{aligned}$$

with $\sigma = \frac{1}{2k-1}$ and $c_1, c_2 \in \textbf{R}$, all satisfy (6.12). But none of them satisfy $a \in C^\infty (\textbf{R}^{2d} {\setminus } 0)$.

The final result in this section will be useful in Sect. 8. It says that the $G^{m,\sigma }$ property of a symbol is preserved under composition with a Hamiltonian flow that satisfies the anisotropic scaling commutativity (6.9). We need a cutoff function $\psi _\delta (x,\xi ) = \varphi (|x|^2 + |\xi |^2) \in C^\infty (\textbf{R}^{2d})$ where $\varphi \in C^{\infty }(\textbf{R})$, $0 \leqslant \varphi \leqslant 1$, $\varphi (t) = 0$ for $t \leqslant \frac{\delta ^2}{4}$ and $\varphi (t) = 1$ for $t \geqslant \delta ^2$ for a given $\delta > 0$. Thus $\psi _\delta \big |_{{\text {B}}_{\frac{\delta }{2}}} \equiv 0$ and $\psi _\delta \big |_{\textbf{R}^{2d} {\setminus } {\text {B}}_\delta } \equiv 1$.

Proposition 6.5

Let $\sigma , \delta , T > 0$, and suppose $\chi _t \in C^\infty ( \textbf{R}^{2d} {\setminus } 0, \textbf{R}^{2d} {\setminus } 0)$ for $-T \leqslant t \leqslant T$ is a Hamiltonian flow that satisfies the anisotropic scaling commutativity (6.9). If $a \in G^{m,\sigma }$ then $b_t = \psi _\delta (a \circ \chi _t) \in G^{m,\sigma }$ uniformly for all $-T \leqslant t \leqslant T$.

Proof

Let $(x,\xi ) \in T^* \textbf{R}^{d}$ satisfy $|(x,\xi )| \geqslant \delta$ and let $\lambda \geqslant 1$. From (6.9) we obtain

$$\begin{aligned} b_t( \lambda x, \lambda ^\sigma \xi ) = a \left( \chi _t ( \lambda x, \lambda ^\sigma \xi ) \right) = a \left( \Lambda _\sigma (\lambda ) \chi _t ( x, \xi ) \right) = a \left( \lambda \chi _{t,1} ( x, \xi ), \lambda ^\sigma \chi _{t,2} ( x, \xi ) \right) \end{aligned}$$

decomposing $\chi _t = ( \chi _{t,1}, \chi _{t,2} )$ into its two $\textbf{R}^{d}$ component functions. For $1 \leqslant k \leqslant d$ we denote by $\chi _{t,j,k}$ the component with index k of $\chi _{t,j}$ for $j = 1,2$.

We claim that for $|(x,\xi )| > \delta$, $\lambda \geqslant 1$, and $\alpha , \beta \in \textbf{N}^{d}$ we have

$$\begin{aligned} \begin{aligned}&\partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }\Big ( a \left( \lambda \chi _{t,1} ( x, \xi ), \lambda ^\sigma \chi _{t,2} ( x, \xi )\right) \Big ) \\&\quad = \sum _{|\gamma + \kappa | \leqslant |\alpha +\beta |} \lambda ^{|\gamma | + \sigma |\kappa |} \left( \partial _{x} ^{\gamma }\partial _{\xi } ^{\kappa }a \right) \Big ( \Lambda _\sigma (\lambda ) \chi _t ( x, \xi ) \Big ) f_{\gamma ,\kappa }(x,\xi ) \end{aligned} \end{aligned}$$

(6.13)

where $f_{\gamma ,\kappa } \in C^\infty ( \textbf{R}^{2d} \setminus 0)$ are smooth functions. In fact the claim follows by induction with respect to $|\alpha +\beta |$, starting with $|\alpha +\beta | = 1$, as follows. With $e_k \in \textbf{N}^{d}$ denoting the standard basis vector, $1 \leqslant k \leqslant d$, we may write $\partial _{x_k} a(x,\xi ) = \partial _{x} ^{e_k} a(x,\xi )$ and $\partial _{\xi _k} a(x,\xi ) = \partial _{\xi } ^{e_k} a(x,\xi )$. If $|\alpha +\beta | = 1$ we have either

$$\begin{aligned}&\partial _{x_j} \Big ( a \left( \lambda \chi _{t,1} ( x, \xi ) , \lambda ^\sigma \chi _{t,2} ( x, \xi )\right) \Big ) \\&\quad = \sum _{k=1}^d \left( \lambda \left( \partial _{x} ^{e_k} a \right) \Big ( \Lambda _\sigma (\lambda ) \chi _t ( x, \xi ) \Big ) \frac{\partial \chi _{t,1,k}}{\partial x_j} (x,\xi ) + \lambda ^\sigma \left( \partial _{\xi } ^{e_k} a \right) \Big ( \Lambda _\sigma (\lambda ) \chi _t ( x, \xi ) \Big ) \frac{\partial \chi _{t,2,k}}{\partial x_j} (x,\xi ) \right) \end{aligned}$$

or

$$\begin{aligned}&\partial _{\xi _j} \Big ( a \left( \lambda \chi _{t,1} ( x, \xi ) , \lambda ^\sigma \chi _{t,2} ( x, \xi )\right) \Big ) \\&\quad = \sum _{k=1}^d \left( \lambda \left( \partial _{x} ^{e_k} a \right) \Big ( \Lambda _\sigma (\lambda ) \chi _t ( x, \xi ) \Big ) \frac{\partial \chi _{t,1,k}}{\partial \xi _j} (x,\xi ) + \lambda ^\sigma \left( \partial _{\xi } ^{e_k} a \right) \Big ( \Lambda _\sigma (\lambda ) \chi _t ( x, \xi ) \Big ) \frac{\partial \chi _{t,2,k}}{\partial \xi _j} (x,\xi ) \right) \end{aligned}$$

for $1 \leqslant j \leqslant d$. Thus (6.13) holds when $|\alpha +\beta | = 1$. The induction step follows straight-forwardly. It follows that (6.13) holds for all $\alpha , \beta \in \textbf{N}^{d}$, $|(x,\xi )| > \delta$, and $\lambda \geqslant 1$, as claimed.

We fix $r > \delta$ and consider any $(x,\xi ) \in T^* \textbf{R}^{d}$ such that $|(x,\xi )| = r$. Using (6.13), the assumption $a \in G^{m,\sigma }$ and

$$\begin{aligned} \inf _{\begin{array}{c} |t| \leqslant T \\ |(x,\xi )| = r \end{array}} |\chi _t (x,\xi )| > 0, \quad \sup _{\begin{array}{c} |t| \leqslant T \\ |(x,\xi )| = r \end{array}} |\chi _t (x,\xi )| < \infty , \end{aligned}$$

we estimate for $\alpha , \beta \in \textbf{N}^{d}$

$$\begin{aligned} \lambda ^{|\alpha | + \sigma |\beta |} \left| \Big ( \partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }b_t \Big ) ( \lambda x, \lambda ^\sigma \xi ) \right|&= \left| \partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }\Big ( b_t ( \lambda x, \lambda ^\sigma \xi ) \Big ) \right| \\&= \left| \partial _{x} ^{\alpha }\partial _{\xi } ^{\beta }\Big ( a \left( \Lambda _\sigma (\lambda ) \chi _t ( x, \xi ) \right) \Big ) \right| \\&\lesssim \sum _{|\gamma + \kappa | \leqslant |\alpha +\beta |} \lambda ^{|\gamma | + \sigma |\kappa |} \left| \left( \partial _{x} ^{\gamma }\partial _{\xi } ^{\kappa }a \right) \Big ( \Lambda _\sigma (\lambda ) \chi _t ( x, \xi ) \Big ) \right| \\&\lesssim \sum _{|\gamma + \kappa | \leqslant |\alpha +\beta |} \lambda ^{|\gamma | + \sigma |\kappa |} \theta _\sigma \left( \lambda \chi _{t,1} ( x, \xi ) , \lambda ^\sigma \chi _{t,2} ( x, \xi ) \right) ^{m - |\gamma | - \sigma |\kappa |} \\&\lesssim \lambda ^m \end{aligned}$$

for all $\lambda \geqslant 1$. The conclusion $b_t \in G^{m,\sigma }$ uniformly for all $-T \leqslant t \leqslant T$ is now a consequence of Lemma 3.2. $\square$

7 Solutions to a class of Schrödinger type equations with anisotropic Hamiltonians

In the sequel we use $k,m \in \textbf{N}\setminus 0$ and $\sigma = \frac{k}{m}$. We consider in this section first the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{rl} \partial _t u(t,x) + i a^w(x,D) u (t,x) &{} = f(t,x), \quad x \in \textbf{R}^{d}, \quad 0 < t \leqslant T, \\ u(0,\cdot ) &{} = u_0, \end{array} \right. \end{aligned}$$

(7.1)

where $T > 0$ and $a \in G^{1 + \sigma ,\sigma }$. Later we will extend the time domain to $[-T,T]$.

Simplifying notation we set $M_s = M_{\sigma ,s}(\textbf{R}^{d})$ for $s \in \textbf{R}$ and $a^w = a^w(x,D)$. The main purpose of the section is to show existence and uniqueness of solutions to (7.1) considering $u(t,\cdot )$ as a function of t with values in $M_s$ spaces.

We will need the following lemma which says that $C^1([0,T], \mathscr {S})$ is dense in $C([0,T], M_\mu ) \cap C^1 ([0,T], M_\nu )$ for any $\mu , \nu \in \textbf{R}$.

Lemma 7.1

If $\mu , \nu \in \textbf{R}$ and $u \in C([0,T], M_\mu ) \cap C^1 ([0,T], M_\nu )$ then there exists a sequence $(u_n)_{n \geqslant 1} \subseteq C^1([0,T], \mathscr {S})$ such that

$$\begin{aligned}&\lim _{n \rightarrow +\infty } \sup _{0 \leqslant t \leqslant T} \Vert u_n(t,\cdot ) - u(t,\cdot ) \Vert _{M_\mu } = 0, \end{aligned}$$

(7.2)

$$\begin{aligned}&\lim _{n \rightarrow +\infty } \sup _{0 \leqslant t \leqslant T} \Vert \partial _t u_n(t,\cdot ) - \partial _t u(t,\cdot ) \Vert _{M_\nu } = 0. \end{aligned}$$

(7.3)

Proof

Let $\varphi \in \mathscr {S}(\textbf{R}^{d})$ satisfy $\Vert \varphi \Vert _{L^2} = 1$. We use the approximations (cf. [13])

$$\begin{aligned} u_n (t,\cdot ) = V_\varphi ^* \chi _n V_\varphi u(t,\cdot ) \in \mathscr {S}(\textbf{R}^{d}), \quad 0 \leqslant t \leqslant T, \end{aligned}$$

where $\chi _n$ is the indicator function of the ball ${\text {B}}_n \subseteq \textbf{R}^{2d}$, $n \in \textbf{N}{\setminus } 0$.

By [13, Eq. (11.29)] we have on the one hand

$$\begin{aligned} \left| V_\varphi \left( u_n (t,\cdot ) - u_n (\tau ,\cdot )\right) \right| \leqslant \left( \chi _n \left| V_\varphi \left( u (t,\cdot ) - u (\tau ,\cdot ) \right) \right| \right) * |V_\varphi \varphi | \end{aligned}$$

(7.4)

and on the other hand, using (2.5) in the form $V_\varphi ^* V_\varphi = \textrm{id}_{\mathscr {S}'}$,

$$\begin{aligned} \left| V_\varphi \left( u_n (t,\cdot ) - u (t,\cdot )\right) \right| \leqslant \left( ( 1 - \chi _n ) \left| V_\varphi u (t,\cdot ) \right| \right) * |V_\varphi \varphi |. \end{aligned}$$

(7.5)

With $m \geqslant 0$ we write using (3.2) and (3.5)

$$\begin{aligned} \langle z\rangle ^m&\lesssim \theta _\sigma (z)^{m \max (1,\sigma )} \lesssim \theta _\sigma (z-w)^{m \max (1,\sigma )} \theta _\sigma (w)^{m \max (1,\sigma )} \\&\leqslant \theta _\sigma (z-w)^{m \max (1,\sigma ) + |\mu | + \mu } \theta _\sigma (w)^{m \max (1,\sigma )}, \quad z,w \in \textbf{R}^{2d}, \end{aligned}$$

which inserted into (7.4) gives by means of the Cauchy–Schwarz inequality, again (3.5) and (2.3)

$$\begin{aligned}&\langle z\rangle ^m \left| V_\varphi \left( u_n (t,\cdot ) - u_n (\tau ,\cdot )\right) (z) \right| \\&\quad \lesssim \left( \chi _n \theta _\sigma ^{m \max (1,\sigma ) +|\mu |} \theta _\sigma ^\mu \left| V_\varphi \left( u (t,\cdot ) - u (\tau ,\cdot ) \right) \right| \right) * \left( \theta _\sigma ^{m \max (1,\sigma )} |V_\varphi \varphi | \right) (z) \\&\quad \leqslant \sup _{\textbf{R}^{2d}} \left( \chi _n \theta _\sigma ^{m\max (1,\sigma ) + |\mu |} \right) \left\| \theta _\sigma ^\mu \left| V_\varphi \left( u (t,\cdot ) - u (\tau ,\cdot ) \right) \right| \right\| _{L^2(\textbf{R}^{2d})} \left\| \theta _\sigma ^{m \max (1,\sigma )} |V_\varphi \varphi | \right\| _{L^2(\textbf{R}^{2d})} \\&\quad \lesssim \Vert u (t,\cdot ) - u (\tau ,\cdot ) \Vert _{M_\mu }. \end{aligned}$$

Referring to the assumption $u \in C([0,T], M_\mu )$ and to the seminorms (2.6) this shows that $u_n \in C([0,T], \mathscr {S})$, and $u_n \in C^1([0,T], \mathscr {S})$ follows similarly from $\partial _t u_n (t,\cdot ) = V_\varphi ^* \chi _n V_\varphi \partial _t u(t,\cdot )$, replacing $\mu$ with $\nu$ and using the assumption $u \in C^1 ([0,T], M_\nu )$.

From (7.5) and Young’s inequality we obtain, again using (3.2), (2.3) and (3.5),

$$\begin{aligned} \Vert u_n(t,\cdot ) - u(t,\cdot ) \Vert _{M_\mu }&= \left\| \theta _\sigma ^\mu \left| V_\varphi \left( u_n (t,\cdot ) - u (t,\cdot ) \right) \right| \right\| _{L^2(\textbf{R}^{2d})} \\&\lesssim \left\| \left( ( 1 - \chi _n ) \theta _\sigma ^\mu \left| V_\varphi u (t,\cdot ) \right| \right) * \left( \theta _\sigma ^{|\mu |} |V_\varphi \varphi | \right) \right\| _{L^2(\textbf{R}^{2d})} \\&\lesssim \left\| ( 1 - \chi _n ) \theta _\sigma ^\mu \, V_\varphi u (t,\cdot ) \right\| _{L^2(\textbf{R}^{2d})} \left\| \theta _\sigma ^{|\mu |} V_\varphi \varphi \right\| _{L^1(\textbf{R}^{2d})} \\&\lesssim \left\| ( 1 - \chi _n ) \theta _\sigma ^\mu \, V_\varphi u (t,\cdot ) \right\| _{L^2(\textbf{R}^{2d})} := f_n (t). \end{aligned}$$

Note the monotonicity $f_n (t) \geqslant f_{n+1} (t)$ for each $n \in \textbf{N}{\setminus } 0$, and by the assumption $u \in C([0,T], M_\mu )$ and dominated convergence we get $\lim _{n \rightarrow \infty } f_n (t) = 0$ for each $t \in [0,T]$. For each $n \in \textbf{N}\setminus 0$ we have $f_n \in C([0,T])$. In fact

$$\begin{aligned} | f_n (t) - f_n (\tau )|&= \left| \left\| ( 1 - \chi _n ) \theta _\sigma ^\mu V_\varphi u (t,\cdot ) \right\| _{{L^2(\textbf{R}^{2d})}} - \left\| (1 - \chi _n ) \theta _\sigma ^\mu V_\varphi u (\tau ,\cdot ) \right\| _{{L^2(\textbf{R}^{2d})}} \right| \\&\leqslant \left\| ( 1 - \chi _n ) \theta _\sigma ^\mu \left( V_\varphi u (t,\cdot ) - V_\varphi u (\tau ,\cdot ) \right) \right\| _{{L^2(\textbf{R}^{2d})}} \\&\leqslant \left\| \theta _\sigma ^\mu \left( V_\varphi u (t,\cdot ) - V_\varphi u (\tau ,\cdot ) \right) \right\| _{{L^2(\textbf{R}^{2d})}} \\&= \Vert u (t,\cdot ) - u (\tau ,\cdot ) \Vert _{M_\mu } \end{aligned}$$

so $f_n \in C([0,T])$ follows from the assumption $u \in C([0,T], M_\mu )$. Now it follows from Dini’s theorem that $f_n (t) \rightarrow 0$ uniformly for $t \in [0,T]$ as $n \rightarrow \infty$. This means that we have shown (7.2), and (7.3) follows in the same fashion. $\square$

Remark 7.2

We note that Lemma 7.1 is true also when we replace the interval [0, T] with $[-T,T]$.

By (3.4) we have $w_{k,m}^{1/k} \asymp \theta _\sigma$ when $\sigma = \frac{k}{m}$ and $k,m \in \textbf{N}{\setminus } 0$. Combining this with (4.1), (4.7) and [14, Theorem 1.1] it follows that if $s \in \textbf{R}$ then the symbol $\theta _\sigma ^s$ for the localization operator (4.7) that defines the isometry $M_s \rightarrow L^2$ can be replaced by $w_{k,m}^{s/k}$. We denote for simplicity this localization operator by $A_s = A_{w_{k,m}^{s/k}}$. We will need the following auxiliary result.

Lemma 7.3

Let $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$ and $a \in G^{1 + \sigma ,\sigma }$, and suppose that

$$\begin{aligned} \textrm{Im}\, a (x,\xi ) \leqslant C_1, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

for $C_1 > 0$. If $s \in \textbf{R}$ then $A_s \, a^w A_s^{-1} = b^w$ where $b \in G^{1 + \sigma ,\sigma }$ and

$$\begin{aligned} \textrm{Im}\, b (x,\xi ) \leqslant C_2, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

for some $C_2 > 0$.

Proof

By Lemma 3.6 we have $w_{k,m}^{s/k} \in G^{s,\sigma }$ for the symbol of $A_s$. From [20, Theorem 1.7.10] it follows that $A_s = a_1^w$ where $a_1 \in G^{s,\sigma }$ is real-valued, cf. Sect. 4.

The symbol $w_{k,m}^{s/k}$ for $A_s$ is positive everywhere and elliptic, cf. (3.15). By the proof of [2, Theorem 8.2] (cf. also [20, Proposition 1.7.12]), slightly modified to the anisotropic calculus, it follows that $A_s$ is invertible on $\mathscr {S}$, and $A_s^{-1} = c^w$ where $c \in G^{-s,\sigma }$. From

$$\begin{aligned} (A_s f, f) = \int _{\textbf{R}^{2d}} w_{k,m}^{s/k} (z) |V_\varphi f (z)|^2 \, \textrm{d}z > 0 \end{aligned}$$

for all $f \in \mathscr {S}\setminus 0$ it follows that $(c^w f, f) > 0$ for all $f \in \mathscr {S}\setminus 0$ which implies that c is a real-valued symbol. Indeed we have

$$\begin{aligned} 2i \, ( (\textrm{Im}\, c)^w f,f)&= ( c^w f,f) - ({{\overline{c}}}^w f, f)\\&= ( c^w f,f) - (f, c^w f) = ( c^w f,f) - \overline{(c^w f,f) } = 0 \end{aligned}$$

for all $f \in \mathscr {S}$, which by polarization yields

$$\begin{aligned} 4 ( (\textrm{Im}\, c)^w f,g )&= \left( (\textrm{Im}\, c)^w (f+g),f+g \right) - \left( (\textrm{Im}\, c)^w (f-g),f-g \right) \\&\quad + i \left( (\textrm{Im}\, c)^w (f+ig),f+ig \right) {-} i \left( (\textrm{Im}\, c)^w (f-ig),f{-}ig \right) = 0 \end{aligned}$$

for all $f,g \in \mathscr {S}$. This implies $\textrm{Im}\, c \equiv 0$.

Finally from $b^w = A_s \, a^w A_s^{-1} = a_1^w a^w c^w$ and (3.13) we obtain

$$\begin{aligned} b = a_1 {{\#}}a {{\#}}c = a_1 \, a \, c + b_1 \end{aligned}$$

where $b_1 \in G^{0, \sigma }$ is bounded. Thus since $a_1 \, c \in G^{0, \sigma }$ is also bounded we get

$$\begin{aligned} \textrm{Im}\, b = (\textrm{Im}\, a) \, a_1 \, c + \textrm{Im}\, b_1 \leqslant C_1 \sup _{\textbf{R}^{2d}} \left( a_1 \, c \right) + \sup _{\textbf{R}^{2d}} \textrm{Im}\, b_1 = C_2 < \infty \end{aligned}$$

for some $C_2 > 0$. $\square$

Remark 7.4

The proof of Lemma 7.3 shows that from the added assumption

$$\begin{aligned} |\textrm{Im}\, a (x,\xi )| \leqslant C_1, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

follows the stronger conclusion

$$\begin{aligned} |\textrm{Im}\, b (x,\xi ) | \leqslant C_2, \quad (x,\xi ) \in T^* \textbf{R}^{d}. \end{aligned}$$

The following two results Lemma 7.5 and Theorem 7.9 are detailed adaptations of [16, Lemma 23.1.1 and Theorem 23.1.2] from the calculus of Hörmander symbols to the anisotropic Shubin calculus.

Lemma 7.5

Let $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$, $a \in G^{1 + \sigma , \sigma }$, and suppose that

$$\begin{aligned} \textrm{Im}\, a (x,\xi ) \leqslant C, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

(7.6)

for $C > 0$. If $s \in \textbf{R}$, $u \in C([0,T], M_{s + 1 + \sigma } ) \cap C^1 ([0,T], M_s)$ then

$$\begin{aligned} f (t) = \partial _t u (t,\cdot ) + i a^w u(t,\cdot ) \in C([0,T], M_s ), \end{aligned}$$

(7.7)

and there exists $c > 0$ such that

$$\begin{aligned} \Vert u(t) \Vert _{M_s} \lesssim e^{c t} \Vert u(0) \Vert _{M_s} + \int _0^t e^{c (t-\tau )} \Vert f(\tau ) \Vert _{M_s} \textrm{d}\tau \end{aligned}$$

(7.8)

for $0 \leqslant t \leqslant T$.

Proof

First we prove the result for $s = 0$. The assumptions, $M_{1 + \sigma } \subseteq L^2$ and Proposition 4.2 imply

$$\begin{aligned} \left\{ \begin{array}{l} t \mapsto u (t,\cdot ) \in C([0,T], L^2 ), \\ t \mapsto \partial _t u (t,\cdot ) \in C([0,T], L^2 ), \\ t \mapsto a^w u (t,\cdot ) \in C([0,T], L^2 ) \end{array} \right. \Longrightarrow \quad t \mapsto f (t) \in C([0,T], L^2 ). \end{aligned}$$

The conclusion (7.7) follows.

By Lemma 7.1 we may replace $L^2$ by $\mathscr {S}$ above. The assumptions $a \in G^{1 + \sigma , \sigma }$ and $\textrm{Im}\, a (x,\xi ) \leqslant C$ make Lemma 4.3 applicable. Combining with (3.12) and the fact that W(g, g) is real-valued [13] we get for $g \in \mathscr {S}(\textbf{R}^{d})$

$$\begin{aligned} \textrm{Re}( i a^w g, g)&= - \textrm{Im}( a^w g, g) = - (2 \pi )^{-d} \textrm{Im}( a, W(g,g) ) = - (2 \pi )^{-d} ( \textrm{Im}\, a, W(g,g) ) \\&= - ( (\textrm{Im}\, a)^w g, g) = ( ( C - \textrm{Im}\, a)^w g,g) - C \Vert g \Vert _{L^2}^2 \geqslant - (b+C) \Vert g \Vert _{L^2}^2 \end{aligned}$$

where $b > 0$. If $0 \leqslant t \leqslant T$ and $\mu \in \textbf{R}$ this gives, writing $u(t) = u(t,\cdot )$ for brevity,

$$\begin{aligned} \partial _t \left( e^{- 2 \mu t} \Vert u(t) \Vert _{L^2}^2 \right)&= 2 e^{- 2 \mu t} \left( \textrm{Re}\left( \partial _t u(t), u(t) \right) - \mu \Vert u(t) \Vert _{L^2}^2 \right) \\&= 2 e^{- 2 \mu t} \left( \textrm{Re}\left( f(t), u(t) \right) - \textrm{Re}\big ( (i a + \mu )^w u(t), u(t) \big ) \right) \\&\leqslant 2 e^{- 2 \mu t} \textrm{Re}\left( f(t), u(t) \right) \end{aligned}$$

provided $\mu \geqslant b + C$.

Integration gives for any $0 \leqslant \nu \leqslant t$

$$\begin{aligned} e^{- 2 \mu \nu } \Vert u(\nu ) \Vert _{L^2}^2&\leqslant \Vert u(0) \Vert _{L^2}^2 + 2 \int _0^\nu e^{- 2 \mu \tau } \Vert f(\tau ) \Vert _{L^2} \, \Vert u( \tau ) \Vert _{L^2} \, \textrm{d}\tau \\&\leqslant \Vert u(0) \Vert _{L^2}^2 + 2 \int _0^t e^{- 2 \mu \tau } \Vert f(\tau ) \Vert _{L^2} \, \Vert u( \tau ) \Vert _{L^2} \, \textrm{d}\tau \\&\leqslant \Vert u(0) \Vert _{L^2}^2 + 2 M(t) \int _0^t e^{- \mu \tau } \Vert f(\tau ) \Vert _{L^2} \, \textrm{d}\tau \end{aligned}$$

with

$$\begin{aligned} M(t) = \sup _{0 \leqslant \tau \leqslant t} e^{- \mu \tau } \Vert u(\tau ) \Vert _{L^2}. \end{aligned}$$

Thus

$$\begin{aligned} \left( M(t) - \int _0^t e^{- \mu \tau } \Vert f(\tau ) \Vert _{L^2} \, \textrm{d}\tau \right) ^2 \leqslant \left( \Vert u(0) \Vert _{L^2} + \int _0^t e^{- \mu \tau } \Vert f(\tau ) \Vert _{L^2} \, \textrm{d}\tau \right) ^2 \end{aligned}$$

which yields

$$\begin{aligned} e^{- \mu t} \Vert u(t) \Vert _{L^2}&\leqslant M(t) \leqslant \Vert u(0) \Vert _{L^2} + 2 \int _0^t e^{- \mu \tau } \Vert f(\tau ) \Vert _{L^2} \, \textrm{d}\tau . \end{aligned}$$

We have now shown (7.8) for $s = 0$ and $c = \mu$.

Next let $s \in \textbf{R}$ and $u \in C([0,T], M_{s + 1 + \sigma } ) \cap C^1 ([0,T], M_s)$. Then $\partial _t u, a^w u \in C ([0,T], M_s)$, again appealing to Proposition 4.2, and the conclusion (7.7) follows. By the proof of Lemma 7.3 we know that $A_s = a_1^w$ with $a_1 \in G^{s,\sigma }$. Proposition 4.2 yields

$$\begin{aligned}&A_s u \in C ([0,T], M_{\sigma +1}) \cap C^1([0,T], L^2), \quad a^w A_s u \in C ([0,T], L^2) \\&\Longrightarrow \quad \partial _t A_s u + i a^w A_s u \in C ([0,T], L^2). \end{aligned}$$

By Lemma 7.3 the symbol $b \in G^{1 + \sigma , \sigma }$ defined by $b^w = A_s \, a^w A_s^{-1}$ satisfies $\textrm{Im}\, b \leqslant C_2$ for some $C_2 > 0$. The inequality (7.8) with $a=b$ and $s = 0$ thus gives

$$\begin{aligned} \Vert A_s u(t) \Vert _{L^2} \lesssim e^{c t} \Vert A_s u(0) \Vert _{L^2} + \int _0^t e^{c (t-\tau )} \Vert \partial _t A_s u(\tau ) + i b^w A_s u(\tau ) \Vert _{L^2} \textrm{d}\tau \end{aligned}$$

which finally yields

$$\begin{aligned} \Vert u(t) \Vert _{M_s} \asymp \Vert A_s u(t) \Vert _{L^2}&\lesssim e^{c t} \Vert A_s u(0) \Vert _{L^2}\\&\quad \ + \int _0^t e^{c (t-\tau )} \Vert A_s \left( \partial _t u(\tau ) + i A_s^{-1} b ^w A_s u(\tau ) \right) \Vert _{L^2} \textrm{d}\tau \\&\asymp e^{c t} \Vert u(0) \Vert _{M_s} + \int _0^t e^{c (t-\tau )} \Vert \partial _t u(\tau ) + i a^w u(\tau ) \Vert _{M_s} \textrm{d}\tau . \end{aligned}$$

$\square$

Remark 7.6

If we strengthen the assumption (7.6) with a lower bound as

$$\begin{aligned} - C \leqslant \textrm{Im}\, a (x,\xi ) \leqslant C, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

(7.9)

for $C > 0$, then the time direction may be reversed in Lemma 7.5. More precisely the lower bound in (7.9) yields the estimate

$$\begin{aligned} \textrm{Re}( i a^w g, g) \leqslant (b+C) \Vert g \Vert _{L^2}^2, \quad g \in \mathscr {S}. \end{aligned}$$

Straightforward modifications of the argument in the proof for the case $s = 0$ leads to the estimate

$$\begin{aligned} \Vert u(-t) \Vert _{L^2} \lesssim e^{c t} \Vert u(0) \Vert _{L^2} + \int _{-t}^0 e^{c (t+\tau ) } \Vert f(\tau ) \Vert _{L^2} \, \textrm{d}\tau \end{aligned}$$

for $c > 0$ and $0 \leqslant t \leqslant T$. Taking into account Remark 7.4 a statement replacing (7.8) can then be formulated as follows. If $s \in \textbf{R}$, $u \in C([-T,T], M_{s + 1 + \sigma } ) \cap C^1 ([-T,T], M_s)$ then

$$\begin{aligned} \Vert u(t) \Vert _{M_s} \lesssim e^{c |t|} \Vert u(0) \Vert _{M_s} + \int _{|\tau | \leqslant |t| } e^{c (|t| + |\tau |)} \Vert f(\tau ) \Vert _{M_s} \textrm{d}\tau \end{aligned}$$

for $-T \leqslant t \leqslant T$, where $c > 0$.

The final tool for the proof of existence and uniqueness of a solution to (7.1) we need is the following approximation lemma.

Lemma 7.7

Let $s \in \textbf{R}$. If $f \in L^1([0,T], M_s)$ then there exists a sequence $(f_n)_{n \geqslant 1} \subseteq C_c^\infty ( (0,T), \mathscr {S}( \textbf{R}^{d}) )$ such that

$$\begin{aligned} \lim _{n \rightarrow + \infty } \Vert f_n - f \Vert _{L^1([0,T], M_s)} = 0. \end{aligned}$$

(7.10)

Proof

Since $C([0,T], M_s) {\subseteq } L^1([0,T], M_s)$ is dense we may assume $f {\in } C([0,T], M_s)$, and by Lemma 7.1 we may assume $f \in C([0,T], \mathscr {S})$. Thus we have

$$\begin{aligned} \lim _{n \rightarrow +\infty } \sup _{ |\tau | \leqslant \frac{1}{2}} \int _{\frac{\tau +1}{n}}^{T + \frac{\tau -1}{n}} \left\| f \left( t - \frac{\tau }{n}\right) - f(t) \right\| _{M_s} \, \textrm{d}t = 0. \end{aligned}$$

(7.11)

We regularize f with respect to $t \in [0,T]$ as

$$\begin{aligned} f_n (t) = \psi _n * \left( f \chi _n \right) (t) \in C_c^\infty ( (0,T), \mathscr {S}(\textbf{R}^{d})) \end{aligned}$$

where $\chi _n \in C_c^\infty (\textbf{R})$ is the indicator function for the interval $[\frac{1}{n},T-\frac{1}{n}] \subseteq \textbf{R}$, $\psi \in C_c^\infty (\textbf{R})$, $\psi \geqslant 0$, ${\text {supp}}\psi \subseteq [-\frac{1}{2},\frac{1}{2}]$, $\int _\textbf{R}\psi (x) \, \textrm{d}x = 1$, and $\psi _n(x) = n \psi (n x)$.

Writing

$$\begin{aligned} f_n (t) - f (t) = \int _{-\frac{1}{2n}}^\frac{1}{2n} \psi _n(\tau ) \Big ( \big ( f (t - \tau ) - f(t) \big ) \chi _n (t - \tau ) {+} f(t) \big ( \chi _n (t - \tau ) - 1 \big ) \Big ) \, \textrm{d}\tau \end{aligned}$$

we may estimate

$$\begin{aligned} \int _0^T \Vert f_n (t) - f (t) \Vert _{M_s} \, \textrm{d}t \leqslant I_n + J_n \end{aligned}$$

where

$$\begin{aligned} I_n&= \int _{-\frac{1}{2n}}^\frac{1}{2n} \psi _n(\tau ) \int _{0}^T \Vert f (t-\tau ) - f(t) \Vert _{M_s} \chi _n (t-\tau ) \, \textrm{d}t \, \textrm{d}\tau \\&= \int _{-\frac{1}{2n}}^\frac{1}{2n} \psi _n(\tau ) \int _{\tau +\frac{1}{n}}^{T+\tau -\frac{1}{n}} \Vert f (t-\tau ) - f(t) \Vert _{M_s} \, \textrm{d}t \, \textrm{d}\tau \\&= \int _{-\frac{1}{2}}^\frac{1}{2} \psi (\tau ) \int _{\frac{\tau +1}{n}}^{T + \frac{\tau -1}{n}} \left\| f \left( t- \frac{\tau }{n} \right) - f(t) \right\| _{M_s} \, \textrm{d}t \, \textrm{d}\tau \\&\quad \longrightarrow 0, \quad n \rightarrow + \infty \end{aligned}$$

using (7.11). Finally

$$\begin{aligned} J_n&= \int _{-\frac{1}{2n}}^\frac{1}{2n} \psi _n(\tau ) \int _{0}^T \Vert f (t) \Vert _{M_s} \left( 1 - \chi _n (t-\tau ) \right) \, \textrm{d}t \, \textrm{d}\tau \\&= \int _{-\frac{1}{2}}^\frac{1}{2} \psi (\tau ) \int _{0}^T \Vert f(t) \Vert _{M_s} \left( 1 - \chi _n \left( t- \frac{\tau }{n} \right) \right) \, \textrm{d}t \, \textrm{d}\tau \\&= \int _{-\frac{1}{2}}^\frac{1}{2} \psi (\tau ) \left( \int _{0}^{\frac{\tau +1}{n}} + \int _{T + \frac{\tau -1}{n}}^T \right) \Vert f(t) \Vert _{M_s} \, \textrm{d}t \, \textrm{d}\tau \\&\leqslant \int _{-\frac{1}{2}}^\frac{1}{2} \psi (\tau ) \left( \int _{0}^{\frac{3}{2 n}} + \int _{T - \frac{3}{2 n}}^T \right) \Vert f(t) \Vert _{M_s} \, \textrm{d}t \, \textrm{d}\tau \\&= \left( \int _{0}^{\frac{3}{2 n}} + \int _{T - \frac{3}{2 n}}^T \right) \Vert f(t) \Vert _{M_s} \, \textrm{d}t \\&\quad \longrightarrow 0, \quad n \rightarrow + \infty . \end{aligned}$$

We have shown (7.10). $\square$

Remark 7.8

Again we note (cf. Remark 7.2) that Lemma 7.7 is true also when we replace the interval [0, T] with $[-T,T]$.

We have now finally arrived at a point where we may state and prove the existence and uniqueness of solutions to (7.1) that are continuous on the spaces $M_s$.

Theorem 7.9

Let $T > 0$, $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$, $a \in G^{1 + \sigma ,\sigma }$, suppose

$$\begin{aligned} \textrm{Im}\, a (x,\xi ) \leqslant C, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

for $C > 0$, and let $s \in \textbf{R}$. If $u_0 \in M_s$ and $f \in L^1([0,T], M_s)$, then the equation (7.1) has a unique solution $u \in C ([0,T], M_s)$.

Proof

First we assume $u_0 \in \mathscr {S}$ and $f \in C_c^\infty ( (0,T), \mathscr {S}( \textbf{R}^{d}) )$.

Let $\psi \in C_c^\infty ( (0,T) \times \textbf{R}^{d})$. With $\psi (t) = \psi (t,\cdot )$ we have $\psi (t), \partial _t \psi (t), {\overline{a}}^w \psi (t) \in C( (0,T), \mathscr {S})$. Let $\nu \in \textbf{R}$. Lemma 7.5 applied to $t \mapsto \psi (T-t,\cdot )$ and $- {{\overline{a}}}$ gives

$$\begin{aligned} \sup _{0 \leqslant t \leqslant T} \Vert \psi (t) \Vert _{M_{-\nu }}&\lesssim \int _0^T \Vert - \partial _t \psi (T-t,\cdot ) - i {{\overline{a}}}^w \psi (T-t,\cdot ) \Vert _{M_{-\nu }} \, \textrm{d}t \\&= \int _0^T \Vert \partial _t \psi (t,\cdot ) + i {{\overline{a}}}^w \psi (t,\cdot ) \Vert _{M_{-\nu }} \, \textrm{d}t. \end{aligned}$$

This implies

$$\begin{aligned} \left| \int _0^T ( f (t), \psi (t) ) \, \textrm{d}t \right|&\leqslant \Vert f \Vert _{L^1([0,T], M_{\nu })} \Vert \psi \Vert _{L^\infty ([0,T], M_{-\nu })} \lesssim \sup _{0 \leqslant t \leqslant T} \Vert \psi (t) \Vert _{M_{-\nu }} \\&\lesssim \int _0^T \Vert \partial _t \psi (t,\cdot ) + i {{\overline{a}}}^w \psi (t,\cdot ) \Vert _{-\nu } \, \textrm{d}t. \end{aligned}$$

Thus

$$\begin{aligned} L^1( (0,T], M_{-\nu }) \ni -\partial _t \psi (t,\cdot ) - i \overline{a}^w \psi (t,\cdot ) \mapsto \int _0^T ( f (t), \psi (t) ) \, \textrm{d}t \end{aligned}$$

is an anti-linear continuous functional. By the Hahn–Banach theorem it can be extended to a functional on $L^1( (0,T], M_{-\nu })$. From [9, Theorem IV.1 and Corollary IV.4] we know that the dual space of $L^1( (0,T], M_{-\nu })$ can be identified with $L^\infty ( (0,T], M_\nu )$ through the natural pairing. Hence there exists $u \in L^\infty ( (0,T], M_\nu ) \subseteq \mathscr {D}' ( (0,T) \times \textbf{R}^{d})$ such that

$$\begin{aligned} \int _0^T ( f (t), \psi (t)) \, \textrm{d}t&= \int _0^T (u (t), -\partial _t \psi (t,\cdot ) - i {{\overline{a}}}^w \psi (t,\cdot ) ) \, \textrm{d}t \\&= \int _0^T (\partial _t u (t) + i a^w u (t), \psi (t) ) \, \textrm{d}t. \end{aligned}$$

It follows from this argument that $\partial _t u + i a^w u = f$ in $\mathscr {D}'( (0,T) \times \textbf{R}^{d})$. From $u \in L^\infty ( (0,T], M_\nu )$, $a \in G^{1 + \sigma ,\sigma }$ and Proposition 4.2 it follows that $\partial _t u \in L^\infty ( (0,T], M_{\nu -(1+\sigma )})$. If we set $g(0) = u_0$ and

$$\begin{aligned} g(t) = \int _0^t \partial _t u(\tau ) \, \textrm{d}\tau + u_0, \quad 0 < t \leqslant T, \end{aligned}$$

then $g \in C([0,T], M_{\nu -(1+\sigma )})$, and it follows from Lebesgue’s differentiation theorem for Bochner integrals [9, Theorem II.2.9] that $g'(t) = \partial _t u(t)$ for almost all $t \in [0,T]$.

If $\psi \in C_c^\infty ( (0,T) \times \textbf{R}^{d})$ then we obtain from this

$$\begin{aligned} ( u, \partial _t \psi )&= - ( \partial _t u, \psi ) = - \int _{\textbf{R}^{d}} \int _0^T \partial _t g (t,x) \overline{\psi (t,x)} \, \textrm{d}t \ \textrm{d}x = (g, \partial _t \psi ) \end{aligned}$$

which shows that $u = g \in C( [0,T], M_{\nu -(1+\sigma )})$. Now $\partial _t u + i a^w u = f$ and $a \in G^{1 + \sigma ,\sigma }$ yields $u \in C^1( [0,T], M_{\nu -2(1+\sigma )})$. We may now apply Lemma 7.5 and conclude

$$\begin{aligned} \sup _{0 \leqslant t \leqslant T} \Vert u(t) \Vert _{M_{\nu - 2(1+\sigma )}} \lesssim \Vert u_0 \Vert _{M_{\nu - 2(1+\sigma )}} + \int _0^T \Vert f(t) \Vert _{M_{\nu - 2(1+\sigma )}} \, \textrm{d}t. \end{aligned}$$

Since $\nu \in \textbf{R}$ is arbitrary we get the following conclusion. If $u_0 \in \mathscr {S}$ and $f \in C_c^\infty ( (0,T), \mathscr {S}( \textbf{R}^{d}) )$ then for any $\nu \in \textbf{R}$ there exists a solution

$$\begin{aligned} u \in C ( [0,T], M_{\nu +1+\sigma }) \cap C^1 ( [0,T], M_{\nu }) \end{aligned}$$

to (7.1) such that

$$\begin{aligned} \sup _{0 \leqslant t \leqslant T} \Vert u(t) \Vert _{M_{\nu } } \lesssim \Vert u_0 \Vert _{M_{\nu }} + \int _0^T \Vert f(t) \Vert _{M_\nu } \, \textrm{d}t. \end{aligned}$$

(7.12)

If $u_0 \in M_s$ and $f \in L^1( [0,T], M_s)$ we take sequences $(u_n)_{n = 1}^{\infty } \subseteq \mathscr {S}$ and $( f_n)_{n = 1}^{\infty } \subseteq C_c^\infty ((0,T), \mathscr {S}( \textbf{R}^{d}) )$ such that $\Vert u_n - u_0 \Vert _{M_s} \rightarrow 0$ and $\Vert f_n - f \Vert _{L^1([0,T], M_s)} \rightarrow 0$ as $n \rightarrow +\infty$. The former is possible due to [13, Proposition 11.3.4], and the latter thanks to Lemma 7.7. By the first part of the proof there exists a sequence $(u_n(t))_{n = 1}^{+\infty } \subseteq C( [0,T], M_{s+1+\sigma })$ such that $\partial _t u_n (t) + i a^w u_n (t) = f_n (t)$ and $u_n(0) = u_n$ for each $n \geqslant 1$. By (7.12) with $\nu = s$ the sequence $(u_n (t) )_n$ is a Cauchy sequence in $C( [0,T], M_s)$. It follows that $( \partial _t u_n(t))_{n = 1}^{+\infty } \subseteq C( [0,T], M_{s})$ is a Cauchy sequence in $L^1( [0,T], M_{s-1-\sigma })$.

The sequence $(u_n (t) )_n$ converges in $C( [0,T], M_s)$ to $u (t) \in C( [0,T], M_s)$, and the sequence $(\partial _t u_n (t) )_n$ converges in $L^1( [0,T], M_{s-1-\sigma })$ to $v(t) \in L^1( [0,T], M_{s-1-\sigma })$, $v + i a^w u = f$ in $L^1( [0,T], M_{s-1-\sigma })$, and $u(0) = u_0$.

If $\psi \in C_c^\infty ( (0,T) \times \textbf{R}^{d})$ then

$$\begin{aligned} ( \partial _t u,\psi )&= - ( u, \partial _t \psi ) = - \lim _{n \rightarrow \infty } \int _0^T ( u_n (t) , \partial _t \psi (t,\cdot ) ) \, \textrm{d}t \\&= \lim _{n \rightarrow \infty } \int _0^T ( \partial _t u_n (t) , \psi (t,\cdot ) ) \, \textrm{d}t \\&= \int _0^T ( v (t) , \psi (t,\cdot ) ) \, \textrm{d}t = (v, \psi ) \end{aligned}$$

which shows that $v = \partial _t u$ in $L^1( [0,T], M_{s-1-\sigma })$. We conclude that $\partial _t u + i a^w u = f$ in $L^1( [0,T], M_{s-1-\sigma })$, $u(0) = u_0$, $u \in C( [0,T], M_{s})$, and

$$\begin{aligned} \sup _{0 \leqslant t \leqslant T} \Vert u(t) \Vert _{M_{s} } \lesssim \Vert u_0 \Vert _{M_{s}} + \int _0^T \Vert f(t) \Vert _{M_s} \, \textrm{d}t. \end{aligned}$$

It remains to prove the uniqueness of the solution. Suppose $u \in C( [0,T], M_s)$, $\partial _t u + i a^w u = 0$ and $u(0) = 0$. Then $u \in C^1( [0,T], M_{s-1-\sigma })$ by Proposition 4.2, and thus by Lemma 7.5 we have $u(t) = 0$ in $M_{s-1-\sigma }$, which implies $u(t) = 0$ in $M_{s}$, for each $t \in [0,T]$. $\square$

By Remarks 7.2, 7.6, 7.8 and straightforward modifications in the proof of Theorem 7.9 we may strengthen the assumption on $\textrm{Im}\,a$, reverse the time direction and obtain results for the equation

$$\begin{aligned} \left\{ \begin{array}{rl} \partial _t u(t,x) + i a^w(x,D) u (t,x) &{} = f(t,x), \quad x \in \textbf{R}^{d}, \quad t \in [-T,T] \setminus 0, \\ u(0,\cdot ) &{} = u_0. \end{array} \right. \end{aligned}$$

(7.13)

Corollary 7.10

Let $T > 0$, $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$, $a \in G^{1 + \sigma , \sigma }$, suppose

$$\begin{aligned} |\textrm{Im}\, a (x,\xi ) | \leqslant C, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

for $C > 0$, and let $s \in \textbf{R}$. If $u_0 \in M_s$ and $f \in L^1([-T,T], M_s)$, the equation (7.13) has a unique solution $u \in C ([-T,T], M_s)$.

Since

$$\begin{aligned}{} & {} L^1 ([-T,T], \mathscr {S}(\textbf{R}^{d})) = \bigcap _{s \in \textbf{R}} L^1 ([-T,T], M_s),\\{} & {} C ([-T,T], \mathscr {S}(\textbf{R}^{d})) = \bigcap _{s \in \textbf{R}} C ([-T,T], M_s) \end{aligned}$$

we get the following corollary taking into account (4.2).

Corollary 7.11

Let $T > 0$, $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$, $a \in G^{1 + \sigma , \sigma }$, and suppose

$$\begin{aligned} | \textrm{Im}\, a (x,\xi ) | \leqslant C, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

for $C > 0$. If $f \in L^1 ([-T,T], \mathscr {S}(\textbf{R}^{d}))$ and $u_0 \in \mathscr {S}(\textbf{R}^{d})$ then the unique solution to (7.13) satisfies $u \in C ([-T,T], \mathscr {S}(\textbf{R}^{d}))$.

Finally we state a result dual to Corollary 7.11.

Corollary 7.12

Let $T > 0$, $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$, $a \in G^{1 + \sigma , \sigma }$, and suppose

$$\begin{aligned} | \textrm{Im}\, a (x,\xi ) | \leqslant C, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \end{aligned}$$

for $C > 0$. If $u_0 \in \mathscr {S}'$ then by (4.2) there exists $s \in \textbf{R}$ such that $u_0 \in M_s$. If $f \in L^1 ([-T,T], M_s)$ then the unique solution to (7.13) satisfies $u \in C ([-T,T], M_s)$.

8 Propagation of anisotropic Gabor wave front sets for Schrödinger type equations

The following lemma is an anisotropic version of [6, Lemma 3.6]. Its proof is similar so we omit it (cf. also [27, Lemma 3.2]). The lemma will be used in the proof of Proposition 8.2 which is essential for our main result Theorem 8.3.

Lemma 8.1

Suppose $\sigma > 0$, $r_j(t) \in C([-T,T], G^{m_j,\sigma })$ for $j \geqslant 0$, where $(m_j)_{j=0}^\infty \subseteq \textbf{R}$ is decreasing, $[-T,T] \ni t \mapsto \partial _t r_j(t)(z)$ is continuous for each $z \in T^* \textbf{R}^{d}$, and $\partial _t r_j(t) \in L^\infty ([-T,T], G^{m_j,\sigma })$ for all $j \geqslant 0$. Then there exists $r(t) \in C([-T,T], G^{m_0,\sigma })$ such that for any $n \geqslant 1$

$$\begin{aligned} r(t) - \sum _{j=0}^{n-1} r_j(t) \in C([-T,T], G^{m_n,\sigma }). \end{aligned}$$

We write $r(t) \sim \sum _{j=0}^{\infty } r_j(t)$.

Note that r(t) is unique modulo an element in $C([-T,T], \mathscr {S}(\textbf{R}^{2d}))$.

If $r_j(t) \in L^\infty ([-T,T], G^{m_j,\sigma })$ for $j \geqslant 0$ we abuse the notation $r(t) \sim \sum _{j=0}^{\infty } r_j(t)$ to mean

$$\begin{aligned} r(t) - \sum _{j=0}^{n-1} r_j(t) \in L^\infty ([-T,T], G^{m_n,\sigma }) \end{aligned}$$

for $n \geqslant 1$. In this interpretation r(t) is unique modulo an element in $L^\infty ([-T,T], \mathscr {S}(\textbf{R}^{2d}))$. Thus in Lemma 8.1 it holds $\partial _t r(t) \sim \sum _{j=0}^{\infty } \partial _t r_j(t)$ in the latter sense.

In the next result we use the cutoff function $\psi _\delta$ introduced prior to Proposition 6.5.

Proposition 8.2

Let $\delta > 0$, $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$, and suppose that $a \in G^{1 + \sigma ,\sigma }$, $a \sim \sum _{j = 0}^{\infty } a_j$, where $a_0 \in C^\infty (\textbf{R}^{2d} {\setminus } 0)$ is real-valued,

$$\begin{aligned} a_0( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{1+\sigma } a_0(x,\xi ), \quad \lambda > 0, \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \end{aligned}$$

(8.1)

and $a_j \in G^{(1+\sigma ) (1- 2 j ), \sigma }$ for $j \geqslant 1$. The Hamiltonian flow $\chi _t: T^* \textbf{R}^{d} {\setminus } 0 \rightarrow T^* \textbf{R}^{d} {\setminus } 0$ corresponding to the Hamiltonian $a_0$ is then defined for $-T \leqslant t \leqslant T$ with $T > 0$.

If $q_0 \in G^{0,\sigma }$ then there exists a function $t \mapsto q(t)$ such that $q(0) = \psi _\delta q_0$,

$$\begin{aligned}&q(t) \in C([-T,T], G^{0,\sigma }), \end{aligned}$$

(8.2)

$$\begin{aligned}&q(t) \sim \sum _{j = 0}^{\infty } q_j (t), \quad q_j (t) \in C([-T,T], G^{- 2 j (1+\sigma ),\sigma }), \end{aligned}$$

(8.3)

$$\begin{aligned}&\partial _t q(t) \sim \sum _{j = 0}^{\infty } \partial _t q_j (t), \quad \partial _t q_j (t) \in L^\infty ([-T,T], G^{- 2 j (1+\sigma ),\sigma }), \end{aligned}$$

(8.4)

$$\begin{aligned}&q_0(t) (x,\xi ) = \psi _\delta (x,\xi ) q_0( \chi _{-t} (x,\xi )), \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad t \in [-T,T], \end{aligned}$$

(8.5)

and $r(t) \in L^\infty ([-T,T], \mathscr {S}(\textbf{R}^{2d}))$ where

$$\begin{aligned} r(t)^w = q(t)^w \left( \partial _t + i a^w \right) - \left( \partial _t +i a^w \right) q(t)^w. \end{aligned}$$

Proof

The claim that the Hamiltonian flow $\chi _t(x,\xi ) \in T^* \textbf{R}^{d} {\setminus } 0$ corresponding to the Hamiltonian $a_0$ is defined for $-T \leqslant t \leqslant T$ with the same parameter $T > 0$ for all initial data $(x,\xi ) \in T^* \textbf{R}^{d} \setminus 0$ is a consequence of Proposition 6.2.

We will design q(t) such that (8.2), (8.3), (8.4) and (8.5) are satisfied, and, noting that $\partial _t q(t)^w = q(t)^w \partial _t + \left( \partial _t q(t) \right) ^w$,

$$\begin{aligned} r(t) = i \left( q(t) {{\#}}a - a {{\#}}q(t) \right) - \partial _t q(t) \in L^\infty ([-T,T], \mathscr {S}(\textbf{R}^{2d}) ). \end{aligned}$$

(8.6)

By [16, Theorem 18.5.4] we have

$$\begin{aligned} \begin{aligned}&i \left( q(t) {{\#}}a - a {{\#}}q(t) \right) (x,\xi ) \\ {}&\sim \sum _{j=0}^{\infty } \frac{(-1)^{j+1}}{(2j + 1)! 2^{ 2 j}} \left( \langle \partial _x, \partial _\eta \rangle - \langle \partial _y, \partial _\xi \rangle \right) ^{2 j + 1} q(t)(x,\xi ) a (y,\eta ) \Big |_{(y,\eta ) = (x,\xi )} \\ {}&\sim \{ q(t), a \}(x,\xi ) \\ {}&\quad + \sum _{j=1}^{\infty } \frac{(-1)^{j+1}}{(2j + 1)! 2^{ 2 j }} \left( \langle \partial _x, \partial _\eta \rangle - \langle \partial _y, \partial _\xi \rangle \right) ^{2 j + 1} q(t)(x,\xi ) a (y,\eta ) \Big |_{(y,\eta ) = (x,\xi )} \end{aligned} \end{aligned}$$

(8.7)

where we use the Poisson bracket notation

$$\begin{aligned} \{ q (t),a \} = \langle \nabla _\xi q(t), \nabla _x a \rangle - \langle \nabla _x q(t), \nabla _\xi a \rangle = \langle \mathcal {J}\nabla _{x,\xi } q(t), \nabla _{x,\xi } a \rangle . \end{aligned}$$

If we introduce for $j \geqslant 0$ the bilinear differential operator

$$\begin{aligned} \{ f,g \}_{j} (x,\xi ) = (-1)^j \left( \langle \partial _x, \partial _\eta \rangle - \langle \partial _y, \partial _\xi \rangle \right) ^{j} f(x,\xi ) g (y,\eta ) \Big |_{(y,\eta ) = (x,\xi )} \end{aligned}$$

(8.8)

then $\{ f,g \}_{1} = \{ f,g\}$, so $\{ f,g \}_{j}$ extends the Poisson bracket to higher order differential operators. Note that for $j \geqslant 0$

$$\begin{aligned} a \in G^{m,\sigma }, \quad b \in G^{n,\sigma } \quad \Longrightarrow \quad \{ a, b \}_j \in G^{m + n - j(1+\sigma ),\sigma }. \end{aligned}$$

(8.9)

The notation (8.8) allows us to abbreviate (8.7) as

$$\begin{aligned} i \left( q(t) {{\#}}a - a {{\#}}q(t) \right) \sim \sum _{j=0}^{\infty } \frac{(-1)^{j}}{(2j + 1)! 2^{ 2 j }} \, \{ q(t),a \}_{2j+1}. \end{aligned}$$

Inserting $a \sim \sum _{j = 0}^{\infty } a_j$ and $q(t) \sim \sum _{j = 0}^{\infty } q_j (t)$, and collecting terms of order $j \geqslant 0$ gives

$$\begin{aligned} i \left( q(t) {{\#}}a - a {{\#}}q(t) \right) \sim \sum _{j=0}^{\infty } \sum _{k + n + m = j} \frac{(-1)^{m}}{(2m + 1)! 2^{ 2 m }} \, \{ q_k (t),a_n \}_{2m+1} \end{aligned}$$

(8.10)

since $\{ q_k (t),a_n \}_{2\,m+1} \in C([-T,T], G^{- 2 j (1+\sigma ),\sigma })$ when $k + n + m = j$.

The remainder (8.6) can now be written $r(t) \sim \sum _{j=0}^\infty r_j (t)$ as an asymptotic sum in $L^\infty ([-T,T], G^{0,\sigma })$ with

$$\begin{aligned} r_j (t) = \sum _{k + n + m = j} \frac{(-1)^{m}}{(2m + 1)! 2^{ 2 m }} \, \{ q_k (t),a_n \}_{2m+1} - \partial _t q_j(t) \in L^\infty ([-T,T], G^{-2j(1+\sigma ),\sigma })\nonumber \\ \end{aligned}$$

(8.11)

for $j \geqslant 0$. In the proof we show how to pick $\{ q_j(t)\}_{j=0}^\infty$ with the stated properties so that $r(t) \in L^\infty ([-T,T], \mathscr {S}(\textbf{R}^{2d}) )$.

Set for $(x,\xi ) \in T^* \textbf{R}^{d}$ and $t \in [-T,T]$

$$\begin{aligned} q_0 (t)(x, \xi ) = \psi _\delta (x,\xi ) q_0 ( \chi _{-t} (x,\xi ) ) \end{aligned}$$

so that (8.5) is satisfied. The purpose of the factor $\psi _\delta$ is to make $q_0 (t)$ a well defined smooth function also around $(0,0) \in T^* \textbf{R}^{d}$ where $\chi _{-t}$ may not be smooth. For each $(x,\xi ) \in T^* \textbf{R}^{d}$, $t \mapsto q_0 (t)(x, \xi ) \in C([-T,T])$. By Proposition 6.5 we have $q_0 (t) \in G^{0,\sigma }$ uniformly for all $t \in [-T,T]$.

We write $q_0 (t) ( \chi _t (x, \xi ) ) = \psi _\delta ( \chi _t (x, \xi ) ) q_0 ( x,\xi )$. Then differentiation with respect to t, $\partial _t \chi _t(x,\xi ) = \mathcal {J}\nabla a_0 ( \chi _t(x,\xi ) )$ (cf. (6.2)) and the Chain Rule give for $(x,\xi ) \in T^* \textbf{R}^{d} {\setminus } 0$

$$\begin{aligned} \{ a_0, \psi _\delta \} ( \chi _{t} (x,\xi ) ) q_0 ( x,\xi )= & {} \langle \nabla \psi _\delta ( \chi _{t} (x,\xi ) ), \mathcal {J}\nabla a_0 ( \chi _t(x,\xi ) ) \rangle q_0 ( x,\xi ) \nonumber \\= & {} \langle \nabla \psi _\delta ( \chi _{t} (x,\xi ) ), \partial _t \chi _t(x,\xi ) \rangle q_0 ( x,\xi ) \nonumber \\= & {} \left( \partial _t q_0(t) \right) ( \chi _t (x, \xi ) ) + \langle \nabla q_0 (t) ( \chi _{t} (x,\xi ) ), \mathcal {J}\nabla a_0 ( \chi _{t} (x,\xi ) ) \rangle \nonumber \\= & {} \left( \partial _t q_0(t) \right) ( \chi _t (x, \xi ) ) - \{ q_0 (t), a_0 \} ( \chi _{t} (x,\xi ) ). \end{aligned}$$

(8.12)

Thus for all $(x,\xi ) \in T^* \textbf{R}^{d}$

$$\begin{aligned} \partial _t q_0 (t)(x, \xi ) = \{ q_0(t), a_0 \}(x,\xi ) - \{ \psi _\delta , a_0 \} (x,\xi ) q_0 ( \chi _{-t} (x,\xi ) ). \end{aligned}$$

Note that the right hand side is supported in $\textbf{R}^{2d} {\setminus } {\text {B}}_{\frac{\delta }{2}}$, and the second term is compactly supported in ${\overline{{\text {B}}}}_\delta \subseteq T^* \textbf{R}^{d}$.

The function $[-T,T] \ni t \mapsto \partial _t q_0 (t)(x, \xi )$ is continuous for each $(x,\xi ) \in T^* \textbf{R}^{d}$. Indeed the continuity of the first term

$$\begin{aligned}{}[-T,T] \ni t \mapsto \{ q_0 (t), a_0 \} (x,\xi ) = - \langle \nabla q_0 (t) (x,\xi ), \mathcal {J}\nabla a_0 ( x,\xi ) \rangle \end{aligned}$$

is a consequence of (6.3) and the chain rule, and the continuity of the second term has been verified above.

From $q_0 (t) \in G^{0,\sigma }$ and (8.9) it now follows that $\partial _t q_0(t) \in L^\infty ([-T,T], G^{0,\sigma })$, and then the continuity of $[-T,T] \ni t \mapsto \partial _t q_0 (t)(x, \xi )$ and the mean value theorem gives $q_0 (t) \in C( [-T,T],G^{0,\sigma })$. By (8.11) we have

$$\begin{aligned} r_0(t) = \{ q_0 (t), a_0 \} - \partial _t q_0(t) = \{ \psi _\delta , a_0 \} q_0 \circ \chi _{-t} \in L^\infty ( [-T,T], G^{0, \sigma }) \end{aligned}$$

which implies that ${\text {supp}}\,r_0(t) \subseteq {\overline{{\text {B}}}}_\delta \subseteq T^* \textbf{R}^{d}$ for all $t \in [-T,T]$ so in fact we have $r_0(t) \in L^\infty ( [-T,T], C_c^{\infty } )$. This means that the principal symbol of r(t) vanishes: $r_0(t) \sim 0$.

Next we eliminate the second highest order term in (8.11) $r_1(t) \in L^\infty ( [-T,T], G^{- 2(1+\sigma ), \sigma })$ by a proper choice of $q_1(t) \in C( [-T,T], G^{- 2(1+\sigma ), \sigma })$. The term in $C( [-T,T], G^{- 2(1+\sigma ), \sigma })$ in (8.10) is

$$\begin{aligned} \{ q_0(t), a_1 \} + \{ q_1(t), a_0 \} - \frac{1}{24} \{ q_0 (t),a_0 \}_{3}. \end{aligned}$$

Define

$$\begin{aligned} \rho _1(t) = \{ q_0(t), a_1 \} - \frac{1}{24} \{ q_0 (t),a_0 \}_{3} \in C( [-T,T], G^{- 2(1+\sigma ), \sigma }) \end{aligned}$$

(8.13)

so that $\rho _1(t) + \{ q_1(t), a_0 \}$ is the term in $C( [-T,T], G^{- 2(1+\sigma ), \sigma })$ in (8.10). Define

$$\begin{aligned} q_1(t) ( \chi _t (x,\xi ) ) = \int _0^t \rho _1 (\tau ) ( \chi _\tau (x,\xi ) ) \, \textrm{d}\tau \end{aligned}$$

(8.14)

or equivalently

$$\begin{aligned} q_1(t) ( x,\xi ) = \int _0^t \rho _1 (\tau ) ( \chi _{\tau - t} (x,\xi ) ) \, \textrm{d}\tau . \end{aligned}$$

From (8.13) and Proposition 6.5 it follows that $q_1 (t) \in G^{- 2(1+\sigma ),\sigma }$ uniformly for all $t \in [-T,T]$. We differentiate (8.14) with respect to t which gives if $(x,\xi ) \in T^* \textbf{R}^{d} \setminus 0$

$$\begin{aligned} \rho _1 (t) ( \chi _t (x,\xi ) )&= (\partial _t q_1(t)) ( \chi _t (x,\xi ) ) + \langle \nabla q_1 (t) ( \chi _t (x,\xi ) ), \partial _t \chi _{t} (x,\xi ) \rangle \\&= (\partial _t q_1(t)) ( \chi _t (x,\xi ) ) - \{ q_1 (t), a_0 \} ( \chi _{t} (x,\xi ) ). \end{aligned}$$

Thus $\partial _t q_1(t) - \{ q_1 (t), a_0 \} = \rho _1 (t)$ which implies $\partial _t q_1 (t)\in L^\infty ( [-T,T], G^{- 2(1+\sigma ),\sigma })$.

Let $(x,\xi ) \in T^* \textbf{R}^{d}$ be fixed. We know that $[-T,T] \ni t \mapsto \rho _1 (t) (x,\xi )$ is continuous, and the continuity of

$$\begin{aligned}{}[-T,T] \ni t \mapsto \{ q_1 (t), a_0 \} (x,\xi ) = - \langle \nabla q_1 (t) (x,\xi ), \mathcal {J}\nabla a_0 ( x,\xi ) \rangle \end{aligned}$$

is a consequence of the continuity of

$$\begin{aligned}{}[-T,T] \ni t \mapsto \partial _{z} q_1(t) ( x,\xi ) = \int _0^t \partial _{z} \left( \rho _1 (\tau ) ( \chi _{\tau - t} (x,\xi ) ) \right) \, \textrm{d}\tau \end{aligned}$$

for $z = x_j$ and $z = \xi _j$ for all $1 \leqslant j \leqslant d$. In turn, the latter is a consequence of

$$\begin{aligned} \partial _{z} \left( q_1(t + s) - q_1(t) \right) ( x,\xi )&= \int _t^{t+s} \partial _{z} \left( \rho _1 (\tau ) ( \chi _{\tau - t - s} (x,\xi ) ) \right) \, \textrm{d}\tau \\&+ \int _0^t \partial _{z} \Big ( \rho _1 (\tau ) ( \chi _{\tau - t - s} (x,\xi ) ) - \rho _1 (\tau ) ( \chi _{\tau - t } (x,\xi ) ) \Big ) \, \textrm{d}\tau , \end{aligned}$$

the chain rule, and again (6.3).

It follows that $[-T,T] \ni t \mapsto \partial _t q_1 (t) (x,\xi )$ is continuous for each $(x,\xi ) \in T^* \textbf{R}^{d}$. Combining this with $\partial _t q_1 (t)\in L^\infty ( [-T,T], G^{- 2(1+\sigma ),\sigma })$ we may conclude that $q_1(t) \in C( [-T,T], G^{- 2(1+\sigma ),\sigma })$. Referring to (8.11) this implies that $r_1 (t) \in L^\infty ( [-T,T], G^{- 2(1+\sigma ),\sigma })$ and

$$\begin{aligned} r_1(t)&= \{ q_1(t), a_0 \} + \{ q_0 (t), a_1 \} - \frac{1}{24} \{ q_0 (t),a_0 \}_{3} - \partial _t q_1(t) \\&= \{ q_1(t), a_0 \} + \rho _1 (t) - \partial _t q_1(t) = 0 \end{aligned}$$

which shows that the choice of $q_1(t)$ in (8.14) indeed eliminates $r_1(t) \in L^\infty ( [-T,T], G^{- 2(1+\sigma ),\sigma })$.

In a similar way we construct $q_j(t) \in C( [-T,T], G^{- 2 j (1+\sigma ),\sigma } )$ for $j \geqslant 2$ using $\{ q_k (t) \}_{k=0}^{j-1}$, by defining

$$\begin{aligned} \rho _j(t) = \sum _{k + n + m = j, \ k < j} \frac{(-1)^{m}}{(2m + 1)! 2^{ 2 m }} \, \{ q_k(t),a_n \}_{2 m +1} \in C( [-T,T], G^{- 2 j (1+\sigma ), \sigma } )\nonumber \\ \end{aligned}$$

(8.15)

and

$$\begin{aligned} q_j(t) ( \chi _t (x,\xi ) ) = \int _0^t \rho _j (\tau ) ( \chi _\tau (x,\xi ) ) \, \textrm{d}\tau . \end{aligned}$$

(8.16)

As before $\partial _t q_j(t) \in L^\infty ( [-T,T], G^{- 2 j (1+\sigma ), \sigma })$, $q_j(t) \in C( [-T,T], G^{- 2 j (1+\sigma ), \sigma })$, and $\partial _t q_j(t) - \{ q_j (t), a_0 \} = \rho _j (t)$, which yields $r_j (t) \in L^\infty ( [-T,T], G^{- 2j (1+\sigma ), \sigma })$ (cf. (8.11)) and

$$\begin{aligned} r_j(t)&= \sum _{k + n + m = j} \frac{(-1)^{m}}{(2m + 1)! 2^{ 2 m }} \, \{ q_k(t),a_n \}_{2 m +1} - \partial _t q_j(t) \\ {}&= \{ q_j(t), a_0 \} + \rho _j(t) - \partial _t q_j(t) = 0. \end{aligned}$$

So $r_j(t) \sim 0$ for all $j \geqslant 0$ which means that

$$\begin{aligned} r(t) \in \bigcap _{j \geqslant 0} L^\infty ( [-T,T], G^{- 2j (1+\sigma ),\sigma }) = L^\infty ( [-T,T], \mathscr {S}(\textbf{R}^{2d})). \end{aligned}$$

Finally defining q(t) by (8.3), Lemma 8.1 shows that (8.2) and (8.4) hold. The claim $q(0) = \psi _\delta q_0$ is a consequence of $q_0(0) = \psi _\delta q_0$ and $q_j(0) = 0$ for $j \geqslant 1$. $\square$

Combining Corollaries 7.10 and 7.11 with Proposition 8.2 we obtain our main result about propagation of anisotropic Gabor singularities for the evolution equation

$$\begin{aligned} \left\{ \begin{array}{rl} \partial _t u(t,x) + i a^w(x,D) u (t,x) &{} = 0, \quad x \in \textbf{R}^{d}, \quad t \in [-T,T] \setminus 0, \\ u(0,\cdot ) &{} = u_0. \end{array} \right. \end{aligned}$$

(8.17)

Theorem 8.3

Let $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$, and suppose that $a \in G^{1 + \sigma ,\sigma }$, $a \sim \sum _{j = 0}^{\infty } a_j$, where $a_0 \in C^\infty (\textbf{R}^{2d} {\setminus } 0)$ is real-valued,

$$\begin{aligned} a_0( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{1+\sigma } a_0(x,\xi ), \quad \lambda > 0, \quad (x,\xi ) \in T^* \textbf{R}^{d} \setminus 0, \end{aligned}$$

(8.18)

and $a_j \in G^{(1+\sigma ) (1-2 j ), \sigma }$ for $j \geqslant 1$. The Hamiltonian flow $\chi _t: T^* \textbf{R}^{d} {\setminus } 0 \rightarrow T^* \textbf{R}^{d} {\setminus } 0$ corresponding to the Hamiltonian $a_0$ is then defined for $-T \leqslant t \leqslant T$ with $T > 0$. If $u_0 \in \mathscr {S}'(\textbf{R}^{d})$ then (8.17) has a unique solution denoted $\mathscr {K}_t u_0$, and we have

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( \mathscr {K}_t u_0) = \chi _t \mathrm {WF_{g}^{ \sigma }}(u_0), \quad t \in [-T,T]. \end{aligned}$$

Proof

By Proposition 6.2 there exists $T > 0$ such that the Hamiltonian flow $\chi _t: T^* \textbf{R}^{d} {\setminus } 0 \rightarrow T^* \textbf{R}^{d} {\setminus } 0$ corresponding to the Hamiltonian $a_0$ is well defined for $-T \leqslant t \leqslant T$.

By (4.2) there exists $s \in \textbf{R}$ such that $u_0 \in M_s$. From Corollary 7.10 we obtain the existence of a unique solution $u(t) = \mathscr {K}_t u_0 \in C( [-T,T], M_s)$ to (8.17).

Let $z_0 \in T^* \textbf{R}^{d} {\setminus } \left( \mathrm {WF_{g}^{ \sigma }}(u_0) \cup \{ 0 \} \right)$. We may assume that $|z_0| = 1$. By (3.19) with $m = 0$ there exists $q_0 \in G^{0,\sigma }$ such that $q_0^w u_0 \in \mathscr {S}$ and $z_0 \notin {\text {char}}_\sigma (q_0)$. By (3.14) there exists a $\sigma$-conic neighborhood $\Gamma \subseteq T^* \textbf{R}^{d} \setminus 0$ such that $z_0 \in \Gamma$, and $|q_0(x,\xi ) | \geqslant C > 0$ when $(x,\xi ) \in \Gamma \setminus {\text {B}}_r$ for some $r > 0$.

Let $0 < \delta \leqslant |\chi _t (z_0)|$ for all $t \in [-T,T]$. By Proposition 8.2 there exists $q(t) \in C([-T,T], G^{0,\sigma })$ such that $q(t) \sim \sum _{j = 0}^{\infty } q_j (t)$ with $q_j (t) \in C([-T,T], G^{- 2j(1+\sigma ),\sigma })$ for $j \geqslant 0$, $q_0(t) (x,\xi ) = \psi _\delta (x,\xi ) q_0( \chi _{-t} (x,\xi ))$ and $r(t) \in L^\infty ([-T,T], \mathscr {S}(\textbf{R}^{2d}))$ where

$$\begin{aligned} r(t)^w = q(t)^w \left( \partial _t + i a^w \right) - \left( \partial _t +i a^w \right) q(t)^w. \end{aligned}$$

This gives

$$\begin{aligned} 0 = q(t)^w \left( \partial _t + i a^w \right) u(t) = \left( \partial _t +i a^w \right) q(t)^w u(t) + r(t)^w u(t) \end{aligned}$$

that is

$$\begin{aligned} \left( \partial _t +i a^w \right) q(t)^w u(t) = - r(t)^w u(t). \end{aligned}$$

Set $f (t) = - r(t)^w u(t)$. By (3.8) and Proposition 4.2 we have for any $m \in \textbf{R}$

$$\begin{aligned} \sup _{|t| \leqslant T} \Vert f(t) \Vert _{M_{m + s}} \lesssim \sup _{|t| \leqslant T} \Vert u(t) \Vert _{M_{s}} < \infty \end{aligned}$$

which by (4.2) implies that $f \in L^\infty ( [-T,T], \mathscr {S}(\textbf{R}^{d})) \subseteq L^1 ( [-T,T], \mathscr {S}(\textbf{R}^{d}))$.

Thus $q(t)^w u(t)$ solves the equation (7.13), and for the initial value we have

$$\begin{aligned} q(0)^w u(0) = q_0(0)^w u(0) = \left( \psi _\delta q_0 \right) ^w u_0 = q_0^w u_0 + \left( (\psi _\delta -1) q_0 \right) ^w u_0 \in \mathscr {S} \end{aligned}$$

due to $\psi _\delta -1 \in C_c^\infty (\textbf{R}^{2d})$. At this point we may apply Corollary 7.11 which gives $q(t)^w u(t) \in \mathscr {S}(\textbf{R}^{d})$ for all $t \in [-T,T]$. We note that $q_0 (t) (\chi _t(x,\xi )) = q_0(x,\xi )$ if $|\chi _t(x,\xi )| \geqslant \delta$, and $\chi _t \Gamma \subseteq T^* \textbf{R}^{d} {\setminus } 0$ is a $\sigma$-conic neighborhood of $\chi _t(z_0)$, which is a consequence of Proposition 6.2. This implies that $\chi _t(z_0) \notin {\text {char}}_\sigma ( q(t) )$, since the lower order terms $\{ q_j(t) \}_{j \geqslant 1}$ in q(t) decay on $T^* \textbf{R}^{d}$. By (3.19) this means that $\chi _t(z_0) \notin \mathrm {WF_{g}^{ \sigma }}( u(t) ) = \mathrm {WF_{g}^{ \sigma }}( \mathscr {K}_t u_0 )$. We have shown

$$\begin{aligned} \mathrm {WF_{g}^{ \sigma }}( \mathscr {K}_t u_0) \subseteq \chi _t \mathrm {WF_{g}^{ \sigma }}(u_0), \quad t \in [-T,T]. \end{aligned}$$

The opposite inclusion follows from $\mathscr {K}_t^{-1} = \mathscr {K}_{-t}$ and $\chi _t^{-1} = \chi _{-t}$. $\square$

Remark 8.4

In Theorem 5.1 the Hamiltonian has by Remark 5.2 the form

$$\begin{aligned} a = \sum _{j=0}^m a_j \end{aligned}$$

where $a_j$ is real-valued for all $0 \leqslant j \leqslant m$, $\sigma = \frac{1}{m-1}$, $a_0 \in G^{1+\sigma ,\sigma }$ satisfies (8.18), and $a_j \in G^{\sigma (m-j),\sigma } = G^{(1+\sigma ) \left( 1 - \frac{j}{m} \right) ,\sigma } \subseteq G^{1,\sigma }$ for $1 \leqslant j \leqslant m$.

In Theorem 8.3 on the other hand $\sigma = \frac{k}{m}$ and the Hamiltonian is $a \sim \sum _{j=0}^\infty a_j$, $a_0$ is again real-valued and satisfies (8.18), and $a_j \in G^{(1+\sigma ) (1-2 j ), \sigma } \subseteq G^{-(1+\sigma ), \sigma }$ for $j \geqslant 1$.

Comparing Theorem 5.1 and Theorem 8.3 we may conclude that the former is not a particular case of the latter, due to the different assumptions on the perturbation $a - a_0$ of the Hamiltonian.

9 Examples

Let again $\psi _\delta (x,\xi ) = \varphi (|x|^2 + |\xi |^2) \in C^\infty (\textbf{R}^{2d})$ where $\varphi \in C^{\infty }(\textbf{R})$, $0 \leqslant \varphi \leqslant 1$, $\varphi (t) = 0$ for $t \leqslant \frac{\delta ^2}{4}$ and $\varphi (t) = 1$ for $t \geqslant \delta ^2$ for a given $\delta > 0$. Thus $\psi _\delta \big |_{{\text {B}}_{\frac{\delta }{2}}} \equiv 0$ and $\psi _\delta \big |_{\textbf{R}^{2d} \setminus {\text {B}}_\delta } \equiv 1$.

Example 9.1

Let $\delta > 0$, $c \in \textbf{R}{\setminus } 0$, $k,m \in \textbf{N}{\setminus } 0$, $\sigma = \frac{k}{m}$, and set

$$\begin{aligned} a(x,\xi ) = c \psi _\delta (x,\xi ) \left( |x|^{2k} + |\xi |^{2m} \right) ^{\frac{1}{2} \left( \frac{1}{k} + \frac{1}{m} \right) }. \end{aligned}$$

Then

$$\begin{aligned} a( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{1+\sigma } a(x,\xi ), \quad \lambda \geqslant 1, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad | (x,\xi ) | \geqslant \delta , \end{aligned}$$

and $a \in G^{1+\sigma ,\sigma }$. Theorem 8.3 applies to this Hamiltonian.

Example 9.2

Let $c_1, c_2 \in \textbf{R}\setminus 0$, $k \in \textbf{N}\setminus 0$, and set

$$\begin{aligned} a(x,\xi ) = \psi _\delta (x,\xi ) \left( c_1 |x|^{\frac{2k}{2k-1}} + c_2 |\xi |^{2 k} \right) . \end{aligned}$$

With $\sigma = \frac{1}{2k-1}$ we have

$$\begin{aligned} a( \lambda x, \lambda ^\sigma \xi ) = \lambda ^{1+\sigma } a(x,\xi ), \quad \lambda \geqslant 1, \quad (x,\xi ) \in T^* \textbf{R}^{d}, \quad | (x,\xi ) | \geqslant \delta . \end{aligned}$$

However, we note that the singularity (non-smoothness) of the term $|x|^{\frac{2k}{2k-1}} = |x|^{1 + \sigma }$ at the origin is not annihilated by the cutoff function $\psi _\delta$ unless $k = 1$. For this purpose, we would need a cutoff function that depends on x only. But this type of cutoff function does not fit into the calculus with $G^{m,\sigma }$ symbols. So $a \notin G^{1+\sigma ,\sigma }$ and we cannot apply Theorem 8.3 to this Hamiltonian.