# Fractional Fourier transforms, harmonic oscillator propagators and
Strichartz estimates on Pilipović and modulation spaces

Joachim Toft  Department of Mathematics, Linnæus University, Växjö, Sweden
<EMAIL_ADDRESS>,  Divyang G. Bhimani  Department of Mathematics, Indian
Institute of Science Education and Research, Pune, India
<EMAIL_ADDRESS>and  Ramesh Manna  School of Mathematical
Sciences, National Institute of Science Education and Research, Bhubaneswar,
An OCC of Homi Bhabha National Institute, Jatni 752050, India.
<EMAIL_ADDRESS>
###### Abstract.

We give a proof of that harmonic oscillator propagators and fractional Fourier
transforms are essentially the same. We deduce continuity properties and fix
time estimates for such operators on modulation spaces, and apply the results
to prove Strichartz estimates for such propagators when acting on Pilipović
and modulation spaces. Especially we extend some results by Balhara, Cordero,
Nicola, Rodino and Thangavelu. We also show that general forms of fractional
harmonic oscillator propagators are continuous on suitable Pilipović spaces.

###### Key words and phrases:

Pilopović spaces, Modulation spaces, Wiener amalgam, Bargmann transform,
Harmonic oscillator, propagators, Strichartz estimates

###### 2010 Mathematics Subject Classification:

primary 46F05; 44A15; 32A25; 35Q41; secondary 32A36

##  0\. Introduction

In the paper we investigate mapping properties for powers of harmonic
oscillators, their propagators and fractional Fourier transforms on Pilipović
spaces and modulation spaces. Especially we link fractional Fourier transforms
with harmonic oscillator propagators, which opens up for full transitions of
various properties between these operators.

We also deduce certain continuity properties for fractional Fourier transforms
on modulation spaces (including Wiener amalgam spaces). By using the link
between the fractional Fourier transform and harmonic oscillator propagators
we extend at the same time certain continuity properties of harmonic
oscillator propagators on modulation spaces, given in [8, 9, 15, 16]. These
investigations are also related to [13], where among others, $L^{p}$ and
Hausdorff-Young estimates for fractional Fourier transforms are established.
Thereafter we apply such continuity results to extend certain Strichartz
estimates for harmonic oscillator propagators in [16] when acting on
modulation spaces. In the end we apply our results to certain general classes
of time-dependent equations, similar to Schrödinger and heat equations. We
prove that several of these equations are ill-posed in the framework of
Schwartz space, Gelfand-Shilov spaces and their distribution spaces, but well-
posed in the framework of suitable Pilipović spaces.

Harmonic oscillators and their propagators are important in quantum mechanics,
e. g. when investigating free particles in quantum systems. An important
question concerns continuity for such operators. For the (standard) harmonic
oscillator

$H_{x}=|x|^{2}-\Delta_{x},\qquad x\in\mathbf{R}^{d},$

the corresponding propagator is given by

$P_{\varrho}=e^{-i\varrho H_{x}},$ (0.1)

which can also be formulated as

$P_{\varrho}=e^{-iH_{x,\varrho}},\qquad
H_{x,\varrho}=\varrho(|x|^{2}-\Delta_{x}).$ (0.1)′

Here $\varrho\in\mathbf{R}$. (See Sections 1–4 for more general operators of
such forms.) It is well-known that these operators are homeomorphisms on the
Schwartz space $\mathscr{S}(\mathbf{R}^{d})$ and its dual
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$, the set of tempered distributions.
(See [35] and Section 1 for notations.) Similar continuity properties hold
true with Pilipović spaces $\mathcal{H}_{0,s}(\mathbf{R}^{d})$ and
$\mathcal{H}_{s}(\mathbf{R}^{d})$, and their distribution spaces
$\mathcal{H}_{0,s}^{\prime}(\mathbf{R}^{d})$ and
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$, in place of
$\mathscr{S}(\mathbf{R}^{d})$ and $\mathscr{S}^{\prime}(\mathbf{R}^{d})$. We
recall that Fourier invariant (standard) Gelfand-Shilov are special cases of
Pilipović spaces. More precisely we have

$\displaystyle\mathcal{H}_{s}(\mathbf{R}^{d})$
$\displaystyle=\mathcal{S}_{s}(\mathbf{R}^{d})\neq\\{0\\},$
$\displaystyle\quad s$ $\displaystyle\geq\frac{1}{2},$
$\displaystyle\qquad\mathcal{H}_{s}(\mathbf{R}^{d})$
$\displaystyle\neq\mathcal{S}_{s}(\mathbf{R}^{d})=\\{0\\},$
$\displaystyle\quad s$ $\displaystyle<\frac{1}{2},$
$\displaystyle\mathcal{H}_{0,s}(\mathbf{R}^{d})$
$\displaystyle=\Sigma_{s}(\mathbf{R}^{d})\neq\\{0\\},$ $\displaystyle\quad s$
$\displaystyle>\frac{1}{2},$
$\displaystyle\qquad\mathcal{H}_{0,s}(\mathbf{R}^{d})$
$\displaystyle\neq\Sigma_{s}(\mathbf{R}^{d})=\\{0\\},$ $\displaystyle\quad s$
$\displaystyle\leq\frac{1}{2}$

(see [48, 53]).

Harmonic oscillators and their propagators also possess convenient mapping
properties in the background of suitable modulation spaces, a family of
(quasi-)Banach spaces which were introduced in [23] by H. Feichtinger and
further developed in [24, 25, 26, 28, 32]. For example, in [37, 38, 39, 40],
several continuity results for Schrödinger propagators including potential
terms when acting on modulation spaces are deduced. Harmonic oscillator
propagators are then obtained by choosing the potentials as $c|x|^{2}$ for
some positive constant $c$. For example, it is proved in [37, 38, 39] that for
the Fourier invariant modulation space $M^{p}(\mathbf{R}^{d})$, the map

$e^{iH_{x,\varrho}}\,:\,M^{p}(\mathbf{R}^{d})\to M^{p}(\mathbf{R}^{d})$ (0.2)

is continuous. See also [8, 9, 15, 16] for other results concerning mapping
properties of propagators on modulation spaces.

In Section 3 we give a strict proof, based on the Bargmann transform, of that
the propagator in (0.1) can be identified with fractional Fourier transforms
by the formula

$e^{-i\frac{\pi}{4}H_{x,\varrho}}=e^{-i\frac{\pi\varrho
d}{4}}\mathscr{F}_{\\!\varrho},$ (0.3)

for every $\varrho\in\mathbf{R}$. Here recall that the fractional Fourier
transform $\mathscr{F}_{\\!\varrho}$, acting on $\mathscr{S}(\mathbf{R}^{d})$
is given by

$\mathscr{F}_{\\!\varrho}f(\xi)=\langle
K_{d,\varrho}(\xi,\,\cdot\,),f\rangle,$

where $K_{d,\varrho}(\xi,x)$ is the distribution kernel, given by

$K_{d,\varrho}=\bigotimes_{j=1}^{d}K_{\varrho},$ (0.4)

with

$K_{\varrho}(\xi,x)=\begin{cases}\left(\frac{1-i\cot(\frac{\pi\varrho}{2})}{2\pi}\right)^{\frac{1}{2}}\exp\left(i\cdot\frac{(x^{2}+\xi^{2})\cos({\frac{\pi\varrho}{2}})-2{\xi
x}}{2\sin(\frac{\pi\varrho}{2})}\right),&\\!\varrho\in\mathbf{R}\setminus
2\mathbf{Z},\\\\[4.30554pt] \delta(\xi-x),&\\!\varrho\in
4\mathbf{Z},\\\\[4.30554pt] \delta(\xi+x),&\\!\varrho\in
2+4\mathbf{Z},\end{cases}$ (0.5)

Here $x,\xi\in\mathbf{R}$. (See e. g. [3, 45] and the references therein. See
also Section 1 for more details on fractional Fourier transform.) We observe
that

$\displaystyle(\mathscr{F}_{1}f)(\xi)$
$\displaystyle=\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{\mathbf{R}^{d}}f(x)e^{-i\langle
x,\xi\rangle}\,dx$ and $\displaystyle(\mathscr{F}_{3}f)(\xi)$
$\displaystyle=\frac{1}{(2\pi)^{\frac{d}{2}}}\int_{\mathbf{R}^{d}}f(x)e^{i\langle
x,\xi\rangle}\,dx$

are the (ordinary) Fourier transform and the inverse Fourier transform,
respectively.

Ideas on fractional Fourier transforms goes back to at least 1929 (cf.
historical notes and references in [12]). On the other hand, the first
rigorous approach seems to be given at first 1939 by Kober in [41]. Since then
numerous applications of fractional Fourier transforms have appeared. For
example, they were explicitly introduced in quantum mechanics around 1980 and
thereafter applied in optics (see e. g. [22, 44] and the references therein).
In quantum mechanics, the formulae (0.4) and (0.5) appear naturally by
considering certain rotations in the phase space and their induced actions on
quantum observables (see e. g. [15, 16, 18, 19, 20] and Remark 1.10 in Section
1). There are also several applications in signal analysis e. g. when
discussing rotation properties for time-frequency representations in time-
frequency analysis, phase retrieval, optics and pattern recognition. (See e.
g. [2, 3, 22, 45] and the references therein.) Here we also remark that in
some aspects, the theory of the fractional Fourier transform is merely a
special case of the metaplectic representation (see [20]).

In terms of the Bargmann transform $\mathfrak{V}_{d}$, fractional Fourier
transforms take the convenient form

$(\mathfrak{V}_{d}(\mathscr{F}_{\\!\varrho}f))(z)=(\mathfrak{V}_{d}f)(e^{-i\frac{\pi\varrho}{2}}z),\qquad
z\in\mathbf{C}^{d}.$ (0.6)

(See e. g. [5, 53].) The relation (0.6) was obtained for $\varrho=1$ already
in [5] by Bargmann.

A motivation of the identity (0.3) is given in e. g. [42], where several links
between the harmonic oscillator and fractional Fourier transforms are
established. We also remark that F. G. Mehler established a formula
(afterwards named _Mehler’s formula_) for the operator $e^{-H_{x,\varrho}}$,
$\varrho>0$, already in [43]. Observe that $e^{-H_{x,\varrho}}$ is the
canonical density operator in statistical physics, see [49, Section 3.4].

Analytic extensions of Mehler’s formula lead to that the kernel of
$e^{-i\frac{\pi}{4}H_{x,\varrho}}$ is essentially the same as the kernel of
the fractional Fourier transform (0.4) and (0.5).

In the aftermath of (0.3) one may link several results in e. g. [52, 53]
concerning mapping properties of fractional Fourier transforms on modulation
spaces with analogous results in e. g. [8, 9, 15, 16, 37, 38, 39, 40]
concerning mapping properties of the propagators on modulation spaces. Some
explicit demonstration of such transfers are given in Section 3. Here it is
addressed that $e^{iH_{x,\varrho}}$ is homeomorphic on
$M^{q}_{(\omega)}(\mathbf{R}^{d})$ for suitable weight $\omega$, because the
fractional Fourier transforms possess the same continuity properties in view
of [53, Proposition 7.1]. (See Propositions 3.4 and 3.5 in Section 3.) More
generally, in Section 3 we show that for $\varrho\in\mathbf{R}\setminus
2\mathbf{Z}$ and $q\leq p$, then $\mathscr{F}_{\\!\varrho}$ is continuous from
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ to $M^{q,p}_{(\omega)}(\mathbf{R}^{d})$,
and from $W^{q,p}_{(\omega)}(\mathbf{R}^{d})$ to
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$, for suitable weights $\omega$. Again, by
using the link (0.3) we transfer these continuity properties for fractional
Fourier transforms to harmonic oscillator propagators. In fact, the following
proposition is obtained by letting the weights in Theorems 3.6 and 3.7 in
Section 3, be trivially equal to one. (See also Proposition 0.1′ in Section
3.)

######  Proposition 0.1.

Let $\varrho\in\mathbf{R}$ and $p,q\in(0,\infty]$ be such that $q\leq p$. Then
the following is true:

  1. (1)

the map

$\displaystyle\mathscr{F}_{\\!\varrho}=e^{-i\frac{\pi}{4}H_{x,\varrho}}\,$
$\displaystyle:\,M^{p,q}(\mathbf{R}^{d})+W^{q,p}(\mathbf{R}^{d})$
$\displaystyle\to M^{q,p}(\mathbf{R}^{d})+W^{p,q}(\mathbf{R}^{d})$

is continuous;

  2. (2)

if in addition $\varrho\notin\mathbf{Z}$, then the map

$\displaystyle\mathscr{F}_{\\!\varrho}=e^{-i\frac{\pi}{4}H_{x,\varrho}}\,$
$\displaystyle:\,M^{p,q}(\mathbf{R}^{d})+W^{q,p}(\mathbf{R}^{d})$
$\displaystyle\to
M^{q,p}(\mathbf{R}^{d}){\textstyle\bigcap}W^{p,q}(\mathbf{R}^{d})$

is continuous.

There are several types of estimates behind the conclusions in the previous
proposition, which are expected to be applicable in non-linear partial
differential equations. Examples on such estimates are

$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{M^{q,p}}$
$\displaystyle=\|e^{-i\frac{\pi}{4}H_{x,\varrho}}f\|_{M^{q,p}}\lesssim|\sin(\textstyle{\frac{\pi\varrho}{2}})|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{M^{p,q}},$
$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{M^{q,p}}$
$\displaystyle=\|e^{-i\frac{\pi}{4}H_{x,\varrho}}f\|_{M^{q,p}}\lesssim|\cos(\textstyle{\frac{\pi\varrho}{2}})|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{W^{q,p}},$
$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{W^{p,q}}$
$\displaystyle=\|e^{-i\frac{\pi}{4}H_{x,\varrho}}f\|_{W^{p,q}}\lesssim|\cos(\textstyle{\frac{\pi\varrho}{2}})|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{M^{p,q}},$
and $\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{W^{p,q}}$
$\displaystyle=\|e^{-i\frac{\pi}{4}H_{x,\varrho}}f\|_{W^{p,q}}\lesssim|\sin(\textstyle{\frac{\pi\varrho}{2}})|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{W^{q,p}},$

still with $q\leq p$ (see Theorems 3.6 and 3.7).

Some of our investigations include more general propagators and fractional
Fourier transforms, where $\varrho$ in (0.1)′ and (0.3) is allowed to be any
complex number. If $\operatorname{Im}(\zeta)>0$, then $P_{\varrho}$ does
neither make sense as a continuous operator on $\mathscr{S}(\mathbf{R}^{d})$,
nor on any Fourier invariant Gelfand-Shilov space and their duals.
Consequently, in order to investigate such extended class of propagators, or,
even more generally, powers of $H_{x,\varrho}$ and their propagators, i. e.
operators of the forms

$H_{x,\varrho}^{r}\quad\text{and}\quad
P_{\varrho,r}=e^{-iH_{x,\varrho}^{r}},\qquad\varrho\in\mathbf{C},\
r\in\mathbf{R},$ (0.7)

other families of function and distribution spaces are needed. It turns out
that such continuity discussions can be performed in the framework of certain
Pilipović spaces and their distribution spaces.

In order to shed some further lights, we present the following propositions,
which are immediate consequences of our investigations. For the fractional
Fourier transforms, these conclusions also follows from the analysis in [53].

######  Proposition 0.2.

Let $\varrho\in\mathbf{C}$ and $s\in\overline{\mathbf{R}}_{\flat}$. Then the
following is true:

  1. (1)

if $s<\frac{1}{2}$, then $\mathscr{F}_{\\!\varrho}$ and
$e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are homeomorphisms on
$\mathcal{H}_{s}(\mathbf{R}^{d})$ and on
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$;

  2. (2)

if $\operatorname{Im}(\varrho)<0$ and $s\geq\frac{1}{2}$, then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are
continuous injections but not surjections on
$\mathcal{H}_{s}(\mathbf{R}^{d})$, $\mathscr{S}(\mathbf{R}^{d})$,
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and on
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$;

  3. (3)

if $\operatorname{Im}(\varrho)=0$, then $\mathscr{F}_{\\!\varrho}$ and
$e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are homeomorphisms on
$\mathcal{H}_{s}(\mathbf{R}^{d})$, $\mathscr{S}(\mathbf{R}^{d})$,
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and on
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$;

  4. (4)

if $\operatorname{Im}(\varrho)>0$ and $s\geq\frac{1}{2}$, then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are
discontinuous on $\mathcal{H}_{s}(\mathbf{R}^{d})$,
$\mathscr{S}(\mathbf{R}^{d})$, $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and on
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$.

The same holds true with $s>\frac{1}{2}$, $s\leq\frac{1}{2}$ and
$\mathcal{H}_{0,s}$ in place of $s\geq\frac{1}{2}$, $s<\frac{1}{2}$ and
$\mathcal{H}_{s}$ at each occurrence.

######  Proposition 0.3.

Let $\varrho\in\mathbf{C}$. Then the following is true:

  1. (1)

if $\operatorname{Im}(\varrho)<0$, then $\mathscr{F}_{\\!\varrho}$ and
$e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are continuous from
$\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$ to
$\mathcal{S}_{1/2}(\mathbf{R}^{d})$, and

$\mathscr{F}_{\\!\varrho}(\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d}))=e^{-i\frac{\pi}{4}H_{x,\varrho}}(\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d}))\subsetneq\mathcal{S}_{1/2}(\mathbf{R}^{d})\text{;}$

  2. (2)

if $\operatorname{Im}(\varrho)>0$, then $\mathscr{F}_{\\!\varrho}$ and
$e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are discontinuous from
$\mathcal{S}_{1/2}(\mathbf{R}^{d})$ to
$\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$, and

$\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})\subsetneq\mathscr{F}_{\\!\varrho}(\mathcal{S}_{1/2}(\mathbf{R}^{d}))=e^{-i\frac{\pi}{4}H_{x,\varrho}}(\mathcal{S}_{1/2}(\mathbf{R}^{d}))\subsetneq\mathcal{H}_{0,1/2}^{\prime}(\mathbf{R}^{d}).$

By usual inclusion relations for Pilipović spaces, it follows that Proposition
0.3 is a refinement of (2) and (4) in Proposition 0.2. In fact, consider the
inclusions

$\displaystyle\mathcal{S}_{s}(\mathbf{R}^{d})$
$\displaystyle\subsetneq\Sigma_{t}(\mathbf{R}^{d})\subsetneq\mathcal{S}_{t}(\mathbf{R}^{d})\subsetneq\mathscr{S}(\mathbf{R}^{d})$
$\displaystyle\subsetneq\mathscr{S}^{\prime}(\mathbf{R}^{d})\subsetneq\mathcal{S}_{t}(\mathbf{R}^{d})\subsetneq\Sigma_{t}^{\prime}(\mathbf{R}^{d})\subsetneq\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d}),\quad\frac{1}{2}\leq
s<t,$ (0.8)

between the Schwartz space, its distribution space, and all (standard) Fourier
invariant Gelfand-Shilov spaces of functions and ultra-distributions. Then
Proposition 0.3 (1) shows that if $\operatorname{Im}(\varrho)<0$, then the
images of the spaces in (0.8) under $\mathscr{F}_{\\!\varrho}$ and
$e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are strict subspaces of
$\mathcal{S}_{1/2}(\mathbf{R}^{d})$, the smallest space in (0.8). If instead
$\operatorname{Im}(\varrho)>0$, then Proposition 0.3 (2) shows that the image
of this smallest space $\mathcal{S}_{1/2}(\mathbf{R}^{d})$ under
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho}}$ is a
superspace of $\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$, the largest space
in (0.8).

This implies, roughly speaking, that the (standard) spaces in (0.8) are
disqualified when performing detailed continuity investigations of the
canonical density operator $e^{-H_{x,\varrho}}$ in statistical physics, and
that these spaces can not be used in continuity investigations of the inverse
$e^{H_{x,\varrho}}$ of that operator. On the other hand, Proposition 0.2 (1)
shows that Pilipović spaces and their distribution spaces, which are not
Gelfand-Shilov spaces of functions and distributions, are suitable for
continuity investigations of $e^{-H_{x,\varrho}}$ and $e^{H_{x,\varrho}}$.

In the most general case we allow $\varrho\in\mathbf{C}^{d}$, in which case
$H_{x,\varrho}$ is defined as

$H_{x,\varrho}=\sum_{j=1}^{d}\rho_{j}(x_{j}^{2}-\partial_{j}^{2}),$

and $\mathscr{F}_{\\!\varrho}$ is a fractional Fourier transform of the
multiple order $\varrho$ (see e. g. [20]). In Section 3 we show that (0.3) and
(0.6) still hold true for such general $\varrho$.

Finally, in Section 4 we apply our results to deduce certain types of
Strichartz estimates. We recall that Strichartz estimates appears when finding
properties on solutions to Cauchy problems like the Schrödinger equation

$\begin{cases}i\partial_{t}u-H_{x}u=F,\\\\[4.30554pt]
u(0,x)=u_{0}(x),\qquad(t,x)\in I\times\mathbf{R}^{d}.\end{cases}$ (0.9)

Here $I=[0,\infty)$ or $I=[0,T]$ for some $T>0$, $F$ is a suitable function or
(ultra-)distribution on $I\times\mathbf{R}^{d}$ and $u_{0}$ is a suitable
function or (ultra-)distribution on $\mathbf{R}^{d}$. It follows that
continuity properties of the propagator

$\displaystyle(Ef)(t,x)$ $\displaystyle\equiv(e^{-itH_{x}}u_{0})(x),$
$\displaystyle\quad(t,x)$ $\displaystyle\in I\times\mathbf{R}^{d},$ (0.10)
$\displaystyle(S_{1}F)(t,x)$
$\displaystyle=\int_{0}^{t}(e^{-i(t-s)H_{x}}F(s,\,\cdot\,))(x)\,ds,$
$\displaystyle\qquad(t,x)$ $\displaystyle\in I\times\mathbf{R}^{d},$ (0.11)
and for $\displaystyle(S_{2}F)(t,x)$
$\displaystyle=\int_{I}(e^{-i(t-s)H_{x}}F(s,\,\cdot\,))(x)\,ds,$
$\displaystyle\quad(t,x)$ $\displaystyle\in I\times\mathbf{R}^{d},$ (0.12)

are essential when finding estimates for solutions to (0.9) (see [16, 30,
50]). Such estimates are called _Strichartz estimates_ (see also Subsection
1.8 for more details).

In Section 4 we deduce Strichartz estimates of the operators $E$, $S_{1}$ and
$S_{2}$ when acting on modulation spaces or Lebesgue spaces with values in
modulation spaces. For example by straight-forward applications of Proposition
0.1, we get the following result, which is a special case of Theorem 4.3 in
Section 4.

######  Theorem 0.4.

Let $p,q,r_{0}\in(0,\infty]$ be such that $q\leq p$ and

$\frac{1}{r_{0}}>d\left(\frac{1}{q}-\frac{1}{p}\right).$

Then $E$ is uniquely extendable to a continuous map

$E:M^{p,q}(\mathbf{R}^{d})+W^{q,p}(\mathbf{R}^{d})\to
L^{r_{0}}([0,T];M^{q,p}(\mathbf{R}^{d}))\bigcap
L^{r_{0}}([0,T];W^{p,q}(\mathbf{R}^{d})).$

Another application of Proposition 0.1 in combination of the Hardy-Littlewood-
Sobolev inequality leads to the following special case of Theorem 4.1 in
Section 4.

######  Theorem 0.5.

Let $p,p_{0},q\in(1,\infty]$ and $r_{0}\in(0,\infty)$ be such that

$0\leq d\left(\frac{1}{q}-\frac{1}{p}\right)<1,\quad
d\left(\frac{1}{q}-\frac{1}{p}\right)\leq 1+\frac{1}{r_{0}}-\frac{1}{p_{0}}.$

Then $S_{1}$ and $S_{2}$ from $C([0,T];M^{1}(\mathbf{R}^{d}))$ to
$L^{\infty}([0,T];M^{1}(\mathbf{R}^{d}))$ are uniquely extendable to
continuous mappings

$\displaystyle S_{j}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0}}([0,T];M^{p,q}(\mathbf{R}^{d})+W^{q,p}(\mathbf{R}^{d}))$
$\displaystyle\to L^{r_{0}}([0,T];M^{q,p}(\mathbf{R}^{d})\bigcap
W^{p,q}(\mathbf{R}^{d})),$

$j=1,2$.

In Section 4 we also discuss well-posed properties for more general equations,
where $H_{x}$ in (0.9) is replaced by $\zeta H_{x,\varrho}^{r}$ for some
$\zeta\in\mathbf{C}$, $\varrho\in\mathbf{C}^{d}$ and $r>0$ which satisfy
$\operatorname{Im}(\zeta\varrho_{j})>0$ for some $j$. Here we show that such
equations are ill-posed, not only in the framework of Schwartz functions and
tempered distributions, but also for Gelfand-Shilov functions and
distributions, while the equation is well-posed for suitable Pilipović spaces
and their distributions.

## Acknowledgement

The authors are grateful to Marianna Chatzakou, Maurice de Gosson and Alberto
Parmeggiani for valuable comments.

The first author was supported by Vetenskapsrådet (Swedish Science Council),
within the project 2019-04890. The second author is thankful for the research
grants (DST/INSPIRE/04/2016/001507) and the third author is thankful for the
research grants (DST/INSPIRE/04/2019/001914).

##  1\. Preliminaries

In this section we recall some basic facts. We start by discussing Gelfand-
Shilov spaces, then modulation spaces and thereafter Pilipović spaces and some
of their properties. Then we recall the Bargmann transform and some of its
properties, and introduce suitable classes of power series expansions and
entire functions on $\mathbf{C}^{d}$. Finally, in Subsection 1.8 we recall
some facts on Strichartz estimates.

###  1.1. Gelfand-Shilov spaces and their distribution spaces

We start by recalling definitions of Fourier invariant (standard) Gelfand-
Shilov spaces and their distribution spaces (cf. e. g. [29, 47]). Let $s\geq
0$ and $h\in\mathbf{R}_{+}$ be fixed. Then $\mathcal{S}_{s,h}(\mathbf{R}^{d})$
is the set of all $f\in C^{\infty}(\mathbf{R}^{d})$ such that

$\|f\|_{\mathcal{S}_{s,h}}\equiv\sup\frac{|x^{\beta}\partial^{\alpha}f(x)|}{h^{|\alpha|+|\beta|}(\alpha!\,\beta!)^{s}}$

is finite. Here the supremum is taken over all $\alpha,\beta\in\mathbf{N}^{d}$
and $x\in\mathbf{R}^{d}$.

Obviously $\mathcal{S}_{s,h}\subseteq\mathscr{S}$ is a Banach space which
increases with $h$ and $s$.

The _Gelfand-Shilov space_ $\mathcal{S}_{s}(\mathbf{R}^{d})$
($\Sigma_{s}(\mathbf{R}^{d})$) of Roumieu type (Beurling type) is the
projective limit (projective limit) of $\mathcal{S}_{s,h}(\mathbf{R}^{d})$
with respect to $h$. This implies that

$\mathcal{S}_{s}(\mathbf{R}^{d})=\bigcup_{h>0}\mathcal{S}_{s,h}(\mathbf{R}^{d})\quad\text{and}\quad\Sigma_{s}(\mathbf{R}^{d})=\bigcap_{h>0}\mathcal{S}_{s,h}(\mathbf{R}^{d})$
(1.1)

is a so-called LB-space and Fréchet space, respectively, with semi norms
$\|\,\cdot\,\|_{\mathcal{S}_{s,h}}$, $h>0$.

Let $\mathcal{S}_{s,h}^{\prime}(\mathbf{R}^{d})$ be the ($L^{2}$-)dual of
$\mathcal{S}_{s,h}(\mathbf{R}^{d})$. If $s\geq\frac{1}{2}$ ($s>\frac{1}{2}$),
then the _Gelfand-Shilov distribution space_
$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$
($\Sigma_{s}^{\prime}(\mathbf{R}^{d})$) is the projective limit (inductive
limit) of $\mathcal{S}_{s,h}^{\prime}(\mathbf{R}^{d})$ with respect to $h>0$.
Hence

$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})=\bigcap_{h>0}\mathcal{S}_{s,h}^{\prime}(\mathbf{R}^{d})\quad\text{and}\quad\Sigma_{s}^{\prime}(\mathbf{R}^{d})=\bigcup_{h>0}\mathcal{S}_{s,h}^{\prime}(\mathbf{R}^{d}).$
(1.1)′

We remark that (0.8) is true with dense embeddings, and that
$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ and
$\Sigma_{t}^{\prime}(\mathbf{R}^{d})$ are the (strong) duals of
$\mathcal{S}_{s}(\mathbf{R}^{d})$ and $\Sigma_{t}(\mathbf{R}^{d})$. On the
other hand, if $s<t\leq\frac{1}{2}$, then $\mathcal{S}_{s}(\mathbf{R}^{d})$
and $\Sigma_{t}(\mathbf{R}^{d})$ are trivially equal to $\\{0\\}$ (cf. [29,
46, 47]).

From now on we let $\mathscr{F}$ be the Fourier transform, given by

$(\mathscr{F}f)(\xi)=\widehat{f}(\xi)\equiv(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}f(x)e^{-i\langle
x,\xi\rangle}\,dx$

when $f\in L^{1}(\mathbf{R}^{d})$. Here $\langle\,\cdot\,,\,\cdot\,\rangle$
denotes the usual scalar product on $\mathbf{R}^{d}$. The map $\mathscr{F}$
extends uniquely to homeomorphisms on $\mathscr{S}^{\prime}(\mathbf{R}^{d})$,
$\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$ and
$\Sigma_{s}^{\prime}(\mathbf{R}^{d})$, and restricts to homeomorphisms on
$\mathscr{S}(\mathbf{R}^{d})$, $\mathcal{S}_{s}(\mathbf{R}^{d})$ and
$\Sigma_{s}(\mathbf{R}^{d})$, and to a unitary operator on
$L^{2}(\mathbf{R}^{d})$.

Next we recall some mapping properties of Gelfand-Shilov spaces under short-
time Fourier transforms. Let $\phi\in\mathscr{S}(\mathbf{R}^{d})$ be fixed.
For every $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$, the _short-time Fourier
transform_ $V_{\phi}f$ is the distribution on $\mathbf{R}^{2d}$ defined by the
formula

$(V_{\phi}f)(x,\xi)=\mathscr{F}(f\,\overline{\phi(\,\cdot\,-x)})(\xi)=(f,\phi(\,\cdot\,-x)e^{i\langle\,\cdot\,,\xi\rangle}).$
(1.2)

We recall that if $T(f,\phi)\equiv V_{\phi}f$ when
$f,\phi\in\mathcal{S}_{s}(\mathbf{R}^{d})$, then $T$ is uniquely extendable to
sequentially continuous mappings

$\displaystyle T\,$ $\displaystyle:\,$
$\displaystyle\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})\times\mathcal{S}_{s}(\mathbf{R}^{d})$
$\displaystyle\to\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d})\bigcap
C^{\infty}(\mathbf{R}^{2d}),$ $\displaystyle T\,$ $\displaystyle:\,$
$\displaystyle\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})\times\mathcal{S}_{s}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\to\mathcal{S}_{s}^{\prime}(\mathbf{R}^{2d}),$

and similarly with $\Sigma_{s}$ in place of $\mathcal{S}_{s}$ at each
occurrence (cf. [17, 53]). We also note that $V_{\phi}f$ takes the form

$V_{\phi}f(x,\xi)=(2\pi)^{-\frac{d}{2}}\int_{\mathbf{R}^{d}}f(y)\overline{\phi(y-x)}e^{-i\langle
y,\xi\rangle}\,dy$ (1.2)′

for admissible $f$.

There are several characterizations of Gelfand-Shilov spaces and their
distribution spaces, e. g. by suitable estimates of their Fourier and Short-
time Fourier transforms (cf. [14, 34, 53]).

###  1.2. Spaces of sequences

The definitions of Pilipović spaces and spaces of power series expansions are
based on certain spaces of sequences on $\mathbf{N}^{d}$, indexed by the
extended set

$\mathbf{R}_{\flat}=\mathbf{R}_{+}\bigcup\\{\,\flat_{\sigma}\,;\,\sigma\in\mathbf{R}_{+}\,\\},$

of $\mathbf{R}_{+}$. We extend the ordering relation on $\mathbf{R}_{+}$ to
the set $\mathbf{R}_{\flat}$, by letting

$s_{1}<\flat_{\sigma}<s_{2}\quad\text{and}\quad\flat_{\sigma_{1}}<\flat_{\sigma_{2}}$

when $s_{1},s_{2},\sigma_{1},\sigma_{2}\in\mathbf{R}_{+}$, satisfy
$s_{1}<\frac{1}{2}\leq s_{2}$ and $\sigma_{1}<\sigma_{2}$. (Cf. [53].)

######  Definition 1.1.

Let $s\in\mathbf{R}_{\flat}$ and $r,\sigma\in\mathbf{R}_{+}$.

  1. (1)

The set $\ell_{0}^{\prime}(\mathbf{N}^{d})$ consists of all formal sequences
$a=\\{a(\alpha)\\}_{\alpha\in\mathbf{N}^{d}}\subseteq\mathbf{C}$, and
$\ell_{0}(\mathbf{N}^{d})$ is the set of all
$a\in\ell_{0}^{\prime}(\mathbf{N}^{d})$ such that $a(\alpha)\neq 0$ for at
most finite numbers of $\alpha\in\mathbf{N}^{d}$;

  2. (2)

The Banach spaces $\ell_{s;r}^{\infty}(\mathbf{N}^{d})$ and
$\ell_{s;r}^{\infty,*}(\mathbf{N}^{d})$ consist of all
$a\in\ell_{0}^{\prime}(\mathbf{N}^{d})$ such that their corresponding norms

$\displaystyle\|a\|_{\ell_{s;r}^{\infty}}$
$\displaystyle=\begin{cases}\underset{\alpha\in\mathbf{N}^{d}}{\sup}|a(\alpha)e^{r|\alpha|^{\frac{1}{2s}}}|,&s\in\mathbf{R}_{+},\\\\[4.30554pt]
\underset{\alpha\in\mathbf{N}^{d}}{\sup}|a(\alpha)r^{-|\alpha|}\alpha!^{\frac{1}{2\sigma}}|,&s=\flat_{\sigma},\end{cases}$
and $\displaystyle\|a\|_{\ell_{s;r}^{\infty,*}}$
$\displaystyle=\begin{cases}\underset{\alpha\in\mathbf{N}^{d}}{\sup}|a(\alpha)e^{-r|\alpha|^{\frac{1}{2s}}}|,&s\in\mathbf{R}_{+},\\\\[4.30554pt]
\underset{\alpha\in\mathbf{N}^{d}}{\sup}|a(\alpha)r^{-|\alpha|}\alpha!^{-\frac{1}{2\sigma}}|,&s=\flat_{\sigma}.\end{cases}$

respectively, are finite;

  3. (3)

The space $\ell_{s}(\mathbf{N}^{d})$ ($\ell_{0,s}(\mathbf{N}^{d})$) is the
inductive limit (projective limit) of $\ell_{s;r}^{\infty}(\mathbf{N}^{d})$
with respect to $r>0$, and $\ell_{s}^{\prime}(\mathbf{N}^{d})$
($\ell_{0,s}^{\prime}(\mathbf{N}^{d})$) is the projective limit (inductive
limit) of $\ell_{s;r}^{\infty,*}(\mathbf{N}^{d})$ with respect to $r>0$.

We also let $\|\,\cdot\,\|_{\ell_{0;N}}$ be the semi-norm

$\|a\|_{\ell_{0;N}}\equiv\sup_{|\alpha|\leq N}|a(\alpha)|,\qquad
a\in\ell_{0}^{\prime}(\mathbf{N}^{d}),$ (1.3)

and $\ell_{0;N}(\mathbf{N}^{d})$ be the Banach space with norm (1.3) and
consisting of all $a\in\ell_{0}^{\prime}(\mathbf{N}^{d})$ such that
$a(\alpha)=0$ when $|\alpha|\geq N$. Then $\ell_{0}(\mathbf{N}^{d})$ is the
inductive limit of $\ell_{0;N}(\mathbf{N}^{d})$ with respect to $N$, and
$\ell_{0}^{\prime}(\mathbf{N}^{d})$ is a Fréchet space under the semi-norms
(Cf. [53].)

In what follows, $(\,\cdot\,,\,\cdot\,)_{\mathscr{H}}$ denotes the scalar
product in the Hilbert space $\mathscr{H}$.

######  Remark 1.2.

Let $s\in\overline{\mathbf{R}_{\flat}}$. Then the duals of

$\ell_{0,s}(\mathbf{N}^{d}),\quad\ell_{s}(\mathbf{N}^{d}),\quad\ell_{s}^{\prime}(\mathbf{N}^{d})\quad\text{and}\quad\ell_{0,s}^{\prime}(\mathbf{N}^{d})$
(1.4)

are given by

$\ell_{0,s}^{\prime}(\mathbf{N}^{d}),\quad\ell_{s}^{\prime}(\mathbf{N}^{d}),\quad\ell_{s}(\mathbf{N}^{d})\quad\text{and}\quad\ell_{0,s}(\mathbf{N}^{d}),$

respectively, with respect to unique extensions of the form
$(\,\cdot\,,\,\cdot\,)_{\ell^{2}(\mathbf{N}^{d})}$ on
$\ell_{0}(\mathbf{N}^{d})\times\ell_{0}(\mathbf{N}^{d})$. If $s>0$, then
$\ell_{0}(\mathbf{N}^{d})$ is dense in $\ell_{0}^{\prime}(\mathbf{N}^{d})$ and
the spaces in (1.4). (See e. g. [53])

###  1.3. Pilipović spaces and spaces of power series expansions on
$\mathbf{C}^{d}$

We recall that the Hermite function of order $\alpha\in\mathbf{N}^{d}$ is
defined by

$h_{\alpha}(x)=\pi^{-\frac{d}{4}}(-1)^{|\alpha|}(2^{|\alpha|}\alpha!)^{-\frac{1}{2}}e^{\frac{1}{2}\cdot{|x|^{2}}}(\partial^{\alpha}e^{-|x|^{2}}).$

It follows that

$h_{\alpha}(x)=((2\pi)^{\frac{d}{2}}\alpha!)^{-1}e^{-\frac{1}{2}\cdot{|x|^{2}}}p_{\alpha}(x),$

for some polynomial $p_{\alpha}$ of order $\alpha$ on $\mathbf{R}^{d}$, called
the Hermite polynomial of order $\alpha$. The Hermite functions are
eigenfunctions to the Fourier transform, and to the Harmonic oscillator

$H_{x,c}\equiv H_{x}+c,\qquad H_{x}\equiv|x|^{2}-\Delta_{x},\qquad
x\in\mathbf{R}^{d},$ (1.5)

which acts on functions and (ultra-)distributions defined on $\mathbf{R}^{d}$.
Here $c\in\mathbf{C}$ is fixed. More precisely, we have

$H_{x,c}h_{\alpha}=(2|\alpha|+d+c)h_{\alpha}.$ (1.6)

More generally, for any $c\in\mathbf{C}$ and
$\varrho=(\varrho_{1},\dots,\varrho_{d})\in\mathbf{C}^{d}$, we let

$H_{x,\varrho,c}\equiv\left(\sum_{j=1}^{d}\varrho_{j}(x_{j}^{2}-\partial_{x_{j}}^{2})\right)+c=\left(\sum_{j=1}^{d}\varrho_{j}H_{x_{j}}\right)+c,\qquad
x\in\mathbf{R}^{d}.$ (1.7)

Evidently, $H_{x,\varrho,c}$ is positive definite when

$\varrho\in\mathbf{R}^{d}_{+}\quad\text{and}\quad
c>-\sum_{j=1}^{d}\varrho_{j}.$

For conveniency we put $H_{x,\varrho,c}=H_{x,\varrho_{0},c}$ when
$\varrho=(\varrho_{0},\dots,\varrho_{0})\in\mathbf{C}^{d}$, and observe that

$H_{x,\varrho,c}=\varrho H_{x}+c\quad\text{when}\quad\varrho\in\mathbf{C}.$

It is well-known that the set of Hermite functions is a basis for
$\mathscr{S}(\mathbf{R}^{d})$ and an orthonormal basis for
$L^{2}(\mathbf{R}^{d})$ (cf. [48]). In particular, if $f,g\in
L^{2}(\mathbf{R}^{d})$, then

$\|f\|_{L^{2}(\mathbf{R}^{d})}^{2}=\sum_{\alpha\in\mathbf{N}^{d}}|c_{h}(f,\alpha)|^{2}\quad\text{and}\quad(f,g)_{L^{2}(\mathbf{R}^{d})}=\sum_{\alpha\in\mathbf{N}^{d}}c_{h}(f,\alpha)\overline{c_{h}(g,\alpha)},$

where

$\displaystyle f(x)$
$\displaystyle=\sum_{\alpha\in\mathbf{N}^{d}}c_{h}(f,\alpha)h_{\alpha}(x)$
(1.8) is the Hermite seriers expansion of $f$, and $\displaystyle
c_{h}(f,\alpha)$ $\displaystyle=(f,h_{\alpha})_{L^{2}(\mathbf{R}^{d})}$ (1.9)

is the Hermite coefficient of $f$ of order $\alpha\in\mathbf{R}^{d}$.

We let $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ be the set of all formal
Hermite series expansions in (1.8), and
$\mathcal{A}_{0}^{\prime}(\mathbf{C}^{d})$ be the set of all formal power
series expansions

$F(z)=\sum_{\alpha\in\mathbf{N}^{d}}c(F,\alpha)e_{\alpha}(z),\qquad
e_{\alpha}(z)=\frac{z^{\alpha}}{\sqrt{\alpha!}},\ \alpha\in\mathbf{N}^{d},$
(1.10)

on $\mathbf{C}^{d}$. Then the map

$\displaystyle T_{\mathcal{H}}\,$ $\displaystyle:$
$\displaystyle\,\\{c(\alpha)\\}_{\alpha\in\mathbf{N}^{d}}$
$\displaystyle\mapsto\sum_{\alpha\in\mathbf{N}^{d}}c(\alpha)h_{\alpha}$ (1.11)
is bijective from $\ell_{0}^{\prime}(\mathbf{N}^{d})$ to
$\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$, and $\displaystyle
T_{\mathcal{A}}\,$ $\displaystyle:$
$\displaystyle\,\\{c(\alpha)\\}_{\alpha\in\mathbf{N}^{d}}$
$\displaystyle\mapsto\sum_{\alpha\in\mathbf{N}^{d}}c(\alpha)e_{\alpha}$ (1.12)

is bijective from $\ell_{0}^{\prime}(\mathbf{N}^{d})$ to
$\mathcal{A}_{0}^{\prime}(\mathbf{C}^{d})$. We let the topologies of
$\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ and
$\mathcal{A}_{0}^{\prime}(\mathbf{C}^{d})$ be inherited from
$\ell_{0}^{\prime}(\mathbf{N}^{d})$ through the mappings $T_{\mathcal{H}}$ and
$T_{\mathcal{A}}$, respectively.

######  Definition 1.3.

Let $s\in\overline{\mathbf{R}_{\flat}}$.

  1. (1)

The spaces

$\mathcal{H}_{0,s}(\mathbf{R}^{d}),\quad\mathcal{H}_{s}(\mathbf{R}^{d}),\quad\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})\quad\text{and}\quad\mathcal{H}_{0,s}^{\prime}(\mathbf{R}^{d}),$
(1.13)

and their topologies, are the images under the map $T_{\mathcal{H}}$ of the
spaces and their topologies in (1.4), respectively.

  2. (2)

The spaces

$\mathcal{A}_{0,s}(\mathbf{C}^{d}),\quad\mathcal{A}_{s}(\mathbf{C}^{d}),\quad\mathcal{A}_{s}^{\prime}(\mathbf{C}^{d})\quad\text{and}\quad\mathcal{A}_{0,s}^{\prime}(\mathbf{C}^{d}),$
(1.14)

and their topologies, are the images under the map $T_{\mathcal{A}}$ of the
spaces and their topologies in (1.4), respectively.

The spaces $\mathcal{H}_{s}(\mathbf{R}^{d})$ and
$\mathcal{H}_{0,s}(\mathbf{R}^{d})$ in Definition 1.3 are called _Pilipović
spaces_ of _Roumieu_ respectively _Beurling types_ of order $s$, and
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$ and
$\mathcal{H}_{0,s}^{\prime}(\mathbf{R}^{d})$ are called _Pilipović
distribution spaces_ of _Roumieu_ respectively _Beurling types_ of order $s$.

There are several characterizations of Pilipović spaces, e. g. in terms of
estimates of powers of the harmonic oscillator on the involved functions (see
e. g. [4, 27, 53]).

######  Remark 1.4.

Let $s_{1},s_{2}\in\overline{\mathbf{R}}_{\flat}$. For future references we
recall that

$\displaystyle\mathcal{H}_{s_{1}}(\mathbf{R}^{d})$
$\displaystyle=\mathcal{S}_{s_{1}}(\mathbf{R}^{d}),$
$\displaystyle\quad\mathcal{H}_{s_{1}}^{\prime}(\mathbf{R}^{d})$
$\displaystyle=\mathcal{S}_{s_{1}}^{\prime}(\mathbf{R}^{d}),$
$\displaystyle\quad s_{1}$ $\displaystyle\geq\frac{1}{2},$ (1.15)
$\displaystyle\mathcal{H}_{0,s_{2}}(\mathbf{R}^{d})$
$\displaystyle=\Sigma_{s_{2}}(\mathbf{R}^{d}),$
$\displaystyle\quad\mathcal{H}_{0,s_{2}}^{\prime}(\mathbf{R}^{d})$
$\displaystyle=\Sigma_{s_{2}}^{\prime}(\mathbf{R}^{d}),$ $\displaystyle\quad
s_{2}$ $\displaystyle>\frac{1}{2},$

while for the other choices of $s_{1},s_{2}$ we have

$\displaystyle\mathcal{H}_{s_{1}}(\mathbf{R}^{d})$
$\displaystyle\neq\mathcal{S}_{s_{1}}(\mathbf{R}^{d})=\\{0\\},$
$\displaystyle\qquad s_{1}$ $\displaystyle<\frac{1}{2},$ (1.16)
$\displaystyle\mathcal{H}_{0,s_{2}}(\mathbf{R}^{d})$
$\displaystyle\neq\Sigma_{s_{2}}(\mathbf{R}^{d})=\\{0\\},$
$\displaystyle\qquad 0<s_{2}$ $\displaystyle\leq\frac{1}{2},$

and that $\mathcal{H}_{s_{1}}(\mathbf{R}^{d})$ and
$\mathcal{H}_{0,s_{2}}(\mathbf{R}^{d})$ in (1.15) and (1.16) are dense in
$\mathscr{S}(\mathbf{R}^{d})$. (See e. g. [47, 53].)

Hence, any non-trivial Gelfand-Shilov space and its distribution space, agree
with corresponding Pilipović space and its distribution space. In particular,
Gelfand-Shilov spaces and their distribution spaces can be characterized in
similar ways as Pilipović spaces and their distribution spaces in terms of
estimates of their coefficients in their Hermite function expansions.

In this context we also recall that the Schwartz space and the set of tempered
distributions can be characterized as

$\displaystyle f$ $\displaystyle\in\mathscr{S}(\mathbf{R}^{d})$
$\displaystyle\Leftrightarrow$ $\displaystyle\quad|c_{h}(f,\alpha)|$
$\displaystyle\lesssim\langle\alpha\rangle^{-N}$ $\displaystyle\text{for
every}\ N\geq 0$ (1.17) and $\displaystyle f$
$\displaystyle\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$
$\displaystyle\Leftrightarrow$ $\displaystyle\quad|c_{h}(f,\alpha)|$
$\displaystyle\lesssim\langle\alpha\rangle^{N}$ $\displaystyle\text{for some}\
N\geq 0.$ (1.18)

(See e. g. [48]). Here and in what follows we let

$\langle x\rangle=(1+|x|^{2})^{\frac{1}{2}}.$

###  1.4. Weight functions

Next we recall some facts on weight functions. A _weight_ on $\mathbf{R}^{d}$
is a positive function $\omega\in L^{\infty}_{loc}(\mathbf{R}^{d})$ such that
$1/\omega\in L^{\infty}_{loc}(\mathbf{R}^{d})$. The set of weights on
$\mathbf{R}^{d}$ is denoted by $\mathscr{P}_{\\!A}(\mathbf{R}^{d})$. In the
sequel we usually assume that $\omega$ is _moderate_ , or _$v$ -moderate_ for
some positive function $v\in L^{\infty}_{loc}(\mathbf{R}^{d})$. This means
that

$\omega(x+y)\lesssim\omega(x)v(y),\qquad x,y\in\mathbf{R}^{d}.$ (1.19)

Here $A\lesssim B$ means that $A\leq cB$ for a suitable constant $c>0$, and we
write $A\asymp B$ when $A\lesssim B$ and $B\lesssim A$. We note that (1.19)
implies that $\omega$ fulfills the estimates

$v(-x)^{-1}\lesssim\omega(x)\lesssim v(x),\quad x\in\mathbf{R}^{d}.$ (1.20)

We let $\mathscr{P}_{E}(\mathbf{R}^{d})$ be the sets of all moderate weights
on $\mathbf{R}^{d}$.

In several situations we also deal with weights which are radial symmetric in
each phase space variable $(x_{j},\xi_{j})$. The set of such weights is
denoted by $\mathscr{P}_{\\!A,r}(\mathbf{R}^{2d})$. That is,
$\mathscr{P}_{\\!A,r}(\mathbf{R}^{2d})$ consists of all
$\omega\in\mathscr{P}_{\\!A}(\mathbf{R}^{2d})$ such that
$\omega(x,\xi)=\omega_{0}(\rho)$ for some
$\omega_{0}\in\mathscr{P}_{\\!A}(\mathbf{R}^{d})$, where
$\rho_{j}=x_{j}^{2}+\xi_{j}^{2}$.

It can be proved that if $\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$, then
$\omega$ is $v$-moderate for some $v(x)=e^{r|x|}$, provided the positive
constant $r$ is chosen large enough (cf. [33]). In particular, (1.20) shows
that for any $\omega\in\mathscr{P}_{E}(\mathbf{R}^{d})$, there is a constant
$r>0$ such that

$e^{-r|x|}\lesssim\omega(x)\lesssim e^{r|x|},\quad x\in\mathbf{R}^{d}.$ (1.21)

We also let $\mathscr{P}(\mathbf{R}^{d})$ be the set of all weights $\omega$
on $\mathbf{R}^{d}$ such that $\omega$ is moderated by
$v(x,\xi)=(1+|x|+|\xi|)^{r}$, for some $r\geq 0$. Evidently,
$\mathscr{P}(\mathbf{R}^{d})\subseteq\mathscr{P}_{E}(\mathbf{R}^{d})$.

We say that $v$ is _submultiplicative_ if $v$ is _even_ and (1.19) holds with
$\omega=v$. In the sequel, $v$ always stand for a submultiplicative weight if
nothing else is stated.

###  1.5. Modulation spaces and Wiener amalgam spaces

Before defining modulation spaces we first address some notions on mixed norm
spaces of Lebesgue types. Let $p,q\in(0.\infty]$ and $r=\min(p,q)$. For any
$f\in L^{r}_{\operatorname{loc}}(\mathbf{R}^{2d})$, let

$\displaystyle\|f\|_{L^{p,q}}=\|f\|_{L^{p,q}(\mathbf{R}^{2d})}$
$\displaystyle\equiv\|g_{1,f,p}\|_{L^{q}(\mathbf{R}^{d})},$ where
$\displaystyle\quad g_{1,f,p}(\xi)$
$\displaystyle\equiv\|f(\,\cdot\,,\xi)\|_{L^{p}(\mathbf{R}^{d})}$ and
$\displaystyle\|f\|_{L^{p,q}_{*}}=\|f\|_{L^{p,q}_{*}(\mathbf{R}^{2d})}$
$\displaystyle\equiv\|g_{2,f,q}\|_{L^{p}(\mathbf{R}^{d})},$ where
$\displaystyle\quad g_{2,f,q}(x)$
$\displaystyle\equiv\|f(x,\,\cdot\,)\|_{L^{q}(\mathbf{R}^{d})}.$

We also let $L^{p,q}(\mathbf{R}^{2d})$ and $L^{p,q}_{*}(\mathbf{R}^{2d})$ be
the quasi-Banach spaces which consist of all $f\in
L^{r}_{\operatorname{loc}}(\mathbf{R}^{2d})$ such that $\|f\|_{L^{p,q}}$ and
$\|f\|_{L^{p,q}_{*}}$ are finite, respectively.

Let $\phi\in\Sigma_{1}(\mathbf{R}^{d})\setminus 0$, $p,q\in(0.\infty]$ and
$\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$. Then the modulation spaces
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and $W^{p,q}_{(\omega)}(\mathbf{R}^{d})$
consist of all $f\in\Sigma_{1}^{\prime}(\mathbf{R}^{d})$ such that
$V_{\phi}f\cdot\omega$ belongs to $L^{p,q}(\mathbf{R}^{2d})$ respectively
$L^{p,q}_{*}(\mathbf{R}^{2d})$. We equip $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$
and $W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ with the quasi-norms

$f\mapsto\|f\|_{M^{p,q}_{(\omega)}}\equiv\|V_{\phi}f\cdot\omega\|_{L^{p,q}}\quad\text{and}\quad
f\mapsto\|f\|_{W^{p,q}_{(\omega)}}\equiv\|V_{\phi}f\cdot\omega\|_{L^{p,q}_{*}},$
(1.22)

respectively. For conveniency we also set
$M^{p}_{(\omega)}=M^{p,p}_{(\omega)}$, and remark that
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is one of the most common types of
modulation spaces. We also set $M^{p,q}=M^{p,q}_{(\omega)},$ and
$M^{p}=M^{p}_{(\omega)}$ when $\omega=1$, and similarly for
$W^{p,q}_{(\omega)}$ spaces.

Modulation spaces with $\omega\in\mathscr{P}(\mathbf{R}^{2d})$ were introduced
by Feichtinger in [23]. The theory was thereafter extended and generalized in
several ways (see e. g. [24, 25, 26, 28]). In the following proposition we
list some basic properties for modulation spaces. We refer to [23, 25, 28, 32,
51] for the proof.

######  Proposition 1.5.

Let $r\in(0,1]$, $p,p_{j},q_{j}\in(0,\infty]$ and
$\omega,\omega_{j},v\in\mathscr{P}_{E}(\mathbf{R}^{2d})$, $j=1,2$, be such
that $r\leq p,q$, $p_{1}\leq p_{2}$, $q_{1}\leq q_{2}$,
$\omega_{2}\lesssim\omega_{1}$, and let $\omega$ be $v$-moderate. Then the
following is true:

  1. (1)

$\Sigma_{1}(\mathbf{R}^{d})\subseteq
M^{p,q}_{(\omega)}(\mathbf{R}^{d}),W^{p,q}_{(\omega)}(\mathbf{R}^{d})\subseteq\Sigma_{1}^{\prime}(\mathbf{R}^{d})$
with continuous inclusions. If in addition $p,q<\infty$, then
$\Sigma_{1}(\mathbf{R}^{d})$ is dense in $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$
and $W^{p,q}_{(\omega)}(\mathbf{R}^{d})$. If, more restricted,
$\omega\in\mathscr{P}(\mathbf{R}^{2d})$, then similar facts hold true with
$\mathscr{S}$ in place of $\Sigma_{1}$ at each occurrence;

  2. (2)

if $\phi\in M^{r}_{(v)}(\mathbf{R}^{d})\setminus 0$, then $f\in
M^{p,q}_{(\omega)}(\mathbf{R}^{d})$, if and only if
$\|V_{\phi}f\cdot\omega\|_{L^{p,q}}$ is finite, and $f\in
W^{p,q}_{(\omega)}(\mathbf{R}^{d})$, if and only if
$\|V_{\phi}f\cdot\omega\|_{L^{p,q}_{*}}$ is finite. In particular,
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and $W^{p,q}_{(\omega)}(\mathbf{R}^{d})$
are independent of the choice of $\phi\in M^{r}_{(v)}(\mathbf{R}^{d})\setminus
0$. Moreover, different choices of $\phi$ in (1.22) give rise to equivalent
quasi-norms;

  3. (3)

$M^{p_{1},q_{1}}_{(\omega_{1})}(\mathbf{R}^{d})\subseteq
M^{p_{2},q_{2}}_{(\omega_{2})}(\mathbf{R}^{d})$ and
$W^{p_{1},q_{1}}_{(\omega_{1})}(\mathbf{R}^{d})\subseteq
W^{p_{2},q_{2}}_{(\omega_{2})}(\mathbf{R}^{d})$;

  4. (4)

if $\omega_{0}(\xi,x)=\omega(-x,\xi)$, then $\mathscr{F}$ is a homeomorphism
from $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ to
$W^{q,p}_{(\omega_{0})}(\mathbf{R}^{d})$.

######  Remark 1.6.

In the framework of Pilipović distribution spaces, the definition of
modulation spaces are extended in [53] to include more general weights (which
are not necessary moderate). In these approaches the window function $\phi$ is
fixed and equal to the Gaussian
$\phi(x)=\pi^{-\frac{d}{4}}e^{-\frac{1}{2}|x|^{2}}$. For any
$\omega\in\mathscr{P}_{\\!A}(\mathbf{R}^{2d})$ and $p,q\in(0,\infty]$, the
modulation spaces $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and
$W^{p,q}_{(\omega)}(\mathbf{R}^{d})$ consist of all
$f\in\mathcal{H}_{\flat_{1}}^{\prime}(\mathbf{R}^{d})$ such that corresponding
quasi-norms in (1.22) are finite. It is proved in [53] that
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ and $W^{p,q}_{(\omega)}(\mathbf{R}^{d})$
are _quasi-Banach spaces_. If in addition $p,q\geq 1$, then these spaces are
_Banach spaces_.

###  1.6. Spaces of entire functions and the Bargmann transform

Let $\Omega\subseteq\mathbf{C}^{d}$ be open. Then $A(\Omega)$ denotes the set
of all analytic functions in $\Omega$.

Next we recall some properties of the Bargmann transform (cf. [5, 6]). We set

$\displaystyle\langle
z,w\rangle=\sum_{j=1}^{d}z_{j}w_{j}\quad\text{and}\quad(z,w)=\langle
z,\overline{w}\rangle,\quad\text{when}$ $\displaystyle
z=(z_{1},\dots,z_{d})\in\mathbf{C}^{d}\quad\text{and}\quad
w=(w_{1},\dots,w_{d})\in\mathbf{C}^{d},$

and otherwise $\langle\,\cdot\,,\,\cdot\,\rangle$ denotes the duality between
test function spaces and their corresponding duals. The Bargmann transform
$\mathfrak{V}_{d}f$ of $f\in L^{2}(\mathbf{R}^{d})$ is defined by the formula

$(\mathfrak{V}_{d}f)(z)=\pi^{-\frac{d}{4}}\int_{\mathbf{R}^{d}}\exp\Big{(}-\frac{1}{2}(\langle
z,z\rangle+|y|^{2})+2^{\frac{1}{2}}\langle z,y\rangle\Big{)}f(y)\,dy$ (1.23)

(cf. [5]). We note that if $f\in L^{2}(\mathbf{R}^{d})$, then the Bargmann
transform $\mathfrak{V}_{d}f$ of $f$ is the entire function on
$\mathbf{C}^{d}$, given by

$(\mathfrak{V}_{d}f)(z)=\int_{\mathbf{R}^{d}}\mathfrak{A}_{d}(z,y)f(y)\,dy,$

or

$(\mathfrak{V}_{d}f)(z)=\langle f,\mathfrak{A}_{d}(z,\,\cdot\,)\rangle,$
(1.24)

where the Bargmann kernel $\mathfrak{A}_{d}$ is given by

$\mathfrak{A}_{d}(z,y)=\pi^{-\frac{d}{4}}\exp\Big{(}-\frac{1}{2}(\langle
z,z\rangle+|y|^{2})+2^{\frac{1}{2}}\langle z,y\rangle\Big{)}.$

Evidently, the right-hand side in (1.24) makes sense when
$f\in\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$ and defines an element in
$A(\mathbf{C}^{d})$, since $y\mapsto\mathfrak{A}_{d}(z,y)$ can be interpreted
as an element in $\mathcal{S}_{1/2}(\mathbf{R}^{d})$ with values in
$A(\mathbf{C}^{d})$.

It was proved in [5] that $f\mapsto\mathfrak{V}_{d}f$ is a bijective and
isometric map from $L^{2}(\mathbf{R}^{d})$ to the Hilbert space
$A^{2}(\mathbf{C}^{d})\equiv B^{2}(\mathbf{C}^{d})\cap A(\mathbf{C}^{d})$,
where $B^{2}(\mathbf{C}^{d})$ consists of all measurable functions $F$ on
$\mathbf{C}^{d}$ such that

$\|F\|_{B^{2}}\equiv\Big{(}\int_{\mathbf{C}^{d}}|F(z)|^{2}d\mu(z)\Big{)}^{\frac{1}{2}}<\infty.$
(1.25)

Here $d\mu(z)=\pi^{-d}e^{-|z|^{2}}\,d\lambda(z)$, where $d\lambda(z)$ is the
Lebesgue measure on $\mathbf{C}^{d}$. We recall that $A^{2}(\mathbf{C}^{d})$
and $B^{2}(\mathbf{C}^{d})$ are Hilbert spaces, where the scalar product are
given by

$(F,G)_{B^{2}}\equiv\int_{\mathbf{C}^{d}}F(z)\overline{G(z)}\,d\mu(z),\quad
F,G\in B^{2}(\mathbf{C}^{d}).$ (1.26)

If $F,G\in A^{2}(\mathbf{C}^{d})$, then we set $\|F\|_{A^{2}}=\|F\|_{B^{2}}$
and $(F,G)_{A^{2}}=(F,G)_{B^{2}}$.

In [5] it is also proved that

$\mathfrak{V}_{d}h_{\alpha}=e_{\alpha},\quad\text{where}\quad
e_{\alpha}(z)\equiv\frac{z^{\alpha}}{\sqrt{\alpha!}},\quad
z\in\mathbf{C}^{d}.$ (1.27)

In particular, the Bargmann transform maps the orthonormal basis
$\\{h_{\alpha}\\}_{\alpha\in\mathbf{N}^{d}}$ in $L^{2}(\mathbf{R}^{d})$
bijectively into the orthonormal basis
$\\{e_{\alpha}\\}_{\alpha\in\mathbf{N}^{d}}$ of monomials in
$A^{2}(\mathbf{C}^{d})$.

For general $f\in\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ we now set

$\mathfrak{V}_{d}f\equiv(T_{\mathcal{A}}\circ T_{\mathcal{H}}^{-1})f,\qquad
f\in\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d}),$ (1.28)

where $T_{\mathcal{H}}$ and $T_{\mathcal{A}}$ are given by (1.11) and (1.12).
It follows from (1.27) that $\mathfrak{V}_{d}f$ in (1.28) agrees with
$\mathfrak{V}_{d}f$ in (1.23) when $f\in L^{2}(\mathbf{R}^{d})$, and that this
is the only way to extend the Bargmann transform continuously to a continuous
map from $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ to
$\mathcal{A}_{0}^{\prime}(\mathbf{C}^{d})$. From these observations and
definitions, we get the following. The details are left for the reader.

######  Proposition 1.7.

Let $s\in\overline{\mathbf{R}_{\flat}}$. Then $\mathfrak{V}_{d}$ is a
homeomorphism from $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ to
$\mathcal{A}_{0}^{\prime}(\mathbf{C}^{d})$, and restricts to homeomorphisms
from the spaces in (1.13) to the spaces in (1.14), respectively.

It follows that if $f,g\in L^{2}(\mathbf{R}^{d})$ and $F,G\in
A^{2}(\mathbf{C}^{d})$, then

$\displaystyle(f,g)_{L^{2}(\mathbf{R}^{d})}$
$\displaystyle=\sum_{\alpha\in\mathbf{N}^{d}}c_{h}(f,\alpha)\overline{c_{h}(g,\alpha)},$
(1.29) $\displaystyle(F,G)_{A^{2}(\mathbf{C}^{d})}$
$\displaystyle=\sum_{\alpha\in\mathbf{N}^{d}}c(F,\alpha)\overline{c(G,\alpha)}.$

By the definitions we get the following proposition on duality for Pilipović
spaces and their Bargmann images. The details are left for the reader.

######  Proposition 1.8.

Let $s_{1}\in\mathbf{R}_{\flat}$ and $s_{2}\in\overline{\mathbf{R}}_{\flat}$.
Then the form $(\,\cdot\,,\,\cdot\,)_{L^{2}(\mathbf{R}^{d})}$ on
$\mathcal{H}_{0}(\mathbf{R}^{d})\times\mathcal{H}_{0}(\mathbf{R}^{d})$ is
uniquely extendable to sesqui-linear forms on

$\displaystyle\mathcal{H}_{s_{2}}^{\prime}(\mathbf{R}^{d})\times\mathcal{H}_{s_{2}}(\mathbf{R}^{d}),$
$\displaystyle\mathcal{H}_{s_{2}}(\mathbf{R}^{d})\times\mathcal{H}_{s_{2}}^{\prime}(\mathbf{R}^{d}),$
$\displaystyle\mathcal{H}_{0,s_{1}}^{\prime}(\mathbf{R}^{d})\times\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d})$
and on
$\displaystyle\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d})\times\mathcal{H}_{0,s_{1}}^{\prime}(\mathbf{R}^{d}).$

The (strong) duals of $\mathcal{H}_{s_{2}}(\mathbf{R}^{d})$ and
$\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d})$ are equal to
$\mathcal{H}_{s_{2}}^{\prime}(\mathbf{R}^{d})$ and
$\mathcal{H}_{0,s_{1}}^{\prime}(\mathbf{R}^{d})$, respectively, through the
form $(\,\cdot\,,\,\cdot\,)_{L^{2}(\mathbf{R}^{d})}$.

The same holds true if the spaces in (1.13) and the form
$(\,\cdot\,,\,\cdot\,)_{L^{2}(\mathbf{R}^{d})}$ are replaced by corresponding
spaces in (1.14) and the form $(\,\cdot\,,\,\cdot\,)_{A^{2}(\mathbf{C}^{d})}$,
at each occurrence.

If $s\in\overline{\mathbf{R}_{\flat}}$, $f\in\mathcal{H}_{s}(\mathbf{R}^{d})$,
$g\in\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$,
$F\in\mathcal{A}_{s}(\mathbf{C}^{d})$ and
$G\in\mathcal{A}_{s}^{\prime}(\mathbf{C}^{d})$, then
$(f,g)_{L^{2}(\mathbf{R}^{d})}$ and $(F,G)_{A^{2}(\mathbf{C}^{d})}$ are
defined by the formula (1.29). It follows that

$c_{h}(f,\alpha)=c(F,\alpha)\quad\text{when}\quad F=\mathfrak{V}_{d}f,\
G=\mathfrak{V}_{d}g.$ (1.30)

holds for such choices of $f$ and $g$.

######  Remark 1.9.

In [27, 53], the spaces in (1.14), contained in
$\mathcal{A}_{0,\flat_{1}}^{\prime}(\mathbf{C}^{d})=A(\mathbf{C}^{d})$ are
identified as canonical spaces of analytic functions. For example it is here
shown that if $\sigma_{1}>0$ and $\sigma_{2}>1$, then

$\displaystyle\mathcal{A}_{\flat_{\sigma_{1}}}(\mathbf{C}^{d})$
$\displaystyle=\\{\,F\in A(\mathbf{C}^{d})\,;\,|F(z)|\lesssim
e^{r|z|^{\frac{2\sigma_{1}}{\sigma_{1}+1}}}\ \text{for some $r>0$}\,\\}$ and
$\displaystyle\mathcal{A}_{\flat_{\sigma_{2}}}^{\prime}(\mathbf{C}^{d})$
$\displaystyle=\\{\,F\in A(\mathbf{C}^{d})\,;\,|F(z)|\lesssim
e^{r|z|^{\frac{2\sigma_{2}}{\sigma_{2}-1}}}\ \text{for every $r>0$}\,\\}.$

###  1.7. Fractional Fourier transforms

We recall that (multiple ordered) fractional Fourier transform
$\mathscr{F}_{\\!\varrho}$ with respect to
$\varrho=(\varrho_{1},\dots,\varrho_{d})\in\mathbf{R}^{d}$ is the operator
with kernel given by $K_{d,\varrho}$ in (0.4) and (0.5) (see e. g. [20]).
Evidently,

$\mathscr{F}_{\\!\varrho}=\mathscr{F}_{\\!\varrho_{1}}\otimes\cdots\otimes\mathscr{F}_{\\!\varrho_{d}},\qquad\varrho=(\varrho_{1},\dots,\varrho_{d})\in\mathbf{R}^{d},$
(1.31)

and it follows that $\mathscr{F}_{\\!\varrho}$ makes sense as homeomorphisms
on

$\mathcal{H}_{0,s}(\mathbf{R}^{d}),\quad\mathcal{H}_{s}(\mathbf{R}^{d}),\quad\mathscr{S}(\mathbf{R}^{d}),\quad\mathscr{S}^{\prime}(\mathbf{R}^{d}),\quad\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})\quad\text{and}\quad\mathcal{H}_{0,s}^{\prime}(\mathbf{R}^{d}),$

and to a unitary operators on $L^{2}(\mathbf{R}^{d})$ (see e. g. [5, 53]). For
conveniency we put $\mathscr{F}_{\\!\varrho_{0}}=\mathscr{F}_{\\!\varrho}$
when $\varrho=(\varrho_{0},\dots,\varrho_{0})\in\mathbf{R}^{d}$.

######  Remark 1.10.

Apart from the cases when $\varrho$ in $\mathscr{F}_{\\!\varrho}$ are
(multiple) integers, the formula for the fractional Fourier transform might
not look like a visual eye candy, because the kernel $K_{\varrho}(\xi,x)$ is
rather complex. However the formula appear naturally by using suitable changes
of symplectic coordinates in quantum mechanics. In fact, the symplectic map
which rotates $(x,\xi)$ in the phase space $\mathbf{R}^{2d}$ with angle
$-\frac{\pi}{2}$ into $(\xi,-x)$, induces that the observables in quantum
mechanics (which are operators) should be conjugated by the Fourier transform
$\mathscr{F}=\mathscr{F}_{1}$ (see e. g. [15, 16, 18, 19, 20]). It might then
be natural to define the fractional Fourier transform
$\mathscr{F}_{\\!\varrho}$ to be the operator which should conjugate the
quantum observables, when the phase space is rotated with the angle
$-\varrho\frac{\pi}{2}$. That is,

_Symplectic map_ |  |  | _Conjugation of quantum observables_
---|---|---|---
Rotation with angle $-\frac{\pi}{2}$ |  |  | $\mathscr{F}_{1}$
Rotation with angle $-\varrho\frac{\pi}{2}$ |  |  | $\mathscr{F}_{\\!\varrho}$

This gives a unique definition of $\mathscr{F}_{\\!\varrho}$, and after some
computations, it follows that the kernel of $\mathscr{F}_{\\!\varrho}$ is
given by $K_{d,\varrho}$ in (0.4) and (0.5). An equivalent approach and which
leads to the same formulae, consist of using the metaplectic representation of
symplectic group and considering metaplectic operators. (See e. g. [15, 16,
18, 19, 20] for more facts on metaplectic representations and corresponding
operators.)

A slightly equivalent way to reach the fractional Fourier transform consists
of investigating mapping properties of the Bargmann transform,
$\mathfrak{V}_{d}$. It is proved already in [5] that the Bargmann image of
$\widehat{f}$ is given by

$(\mathfrak{V}_{d}(\mathscr{F}f))(z)=(\mathfrak{V}_{d}\widehat{f})(z)=(\mathfrak{V}_{d}\widehat{f})(-iz)=(\mathfrak{V}_{d}\widehat{f})(e^{-i\frac{\pi}{2}}z).$

That is, the Bargmann image of $\widehat{f}$ is obtained by retrieving
corresponding image of $f$ and then (again) rotating the argument with the
angle $-\frac{\pi}{2}$. In particular, the Fourier transform can be evaluated
as

$\mathscr{F}_{1}=\mathfrak{V}_{d}^{-1}\circ
U_{1}\circ\mathfrak{V}_{d},\qquad(U_{1}F)(z)=F(e^{-i\frac{\pi}{2}}z).$

It might then be natural to define the fractional Fourier transform as

$\mathscr{F}_{\\!\varrho}=\mathfrak{V}_{d}^{-1}\circ
U_{\varrho}\circ\mathfrak{V}_{d},\qquad(U_{\varrho}F)(z)=F(e^{-i\varrho\frac{\pi}{2}}z),$

and a straight-forward computations show that we attain the same formula of
the kernel $K_{d,\varrho}$ of $\mathscr{F}_{\\!\varrho}$ as before.

Due to the previous remark, the Bargmann image of $\mathscr{F}_{\\!\varrho}$
in (0.4), (0.5) and (1.31) takes the form

$(\mathfrak{V}_{d}(\mathscr{F}_{\\!\varrho}f))(z)=(\mathfrak{V}_{d}f)(e^{-\frac{i}{2}{\pi}\varrho_{1}}z_{1},\dots,e^{-\frac{i}{2}{\pi}\varrho_{d}}z_{d}),\qquad
f\in\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d}).$ (1.32)

Let $\phi(x)=\pi^{-\frac{d}{4}}e^{-\frac{1}{2}|x|^{2}}$. Then we recall that
the Bargmann transform and the short-time Fourier transform can be linked as

$\displaystyle V_{\phi}f(x,\xi)$
$\displaystyle=(2\pi)^{-\frac{d}{2}}e^{-\frac{1}{4}|z|^{2}}e^{-\frac{i}{2}\langle
x,\xi\rangle}(\mathfrak{V}_{d}f)(2^{-\frac{1}{2}}\overline{z}),$ (1.33)
$\displaystyle z$ $\displaystyle=x+i\xi,\ x,\xi\in\mathbf{R}^{d}$

(see (1.28) in [52]). A combination of (1.32) and (1.33) gives that if
$\varrho\in\mathbf{R}^{d}$,

$\displaystyle R_{1,\varrho_{j}}(x_{j},\xi_{j})$
$\displaystyle=(\cos\theta_{j})x_{j}+(\sin\theta_{j})\xi_{j},$ (1.34)
$\displaystyle R_{2,\varrho_{j}}(x_{j},\xi_{j})$
$\displaystyle=-(\sin\theta_{j})x_{j}+(\cos\theta_{j})\xi_{j},$
$\displaystyle\qquad\theta_{j}$
$\displaystyle=\textstyle{\frac{\pi\varrho_{j}}{2}}$ $\displaystyle
R_{k,\varrho}(x,\xi)$
$\displaystyle=\big{(}R_{k,\varrho_{1}}(x_{1},\xi_{1}),\dots,R_{k,\varrho_{d}}(x_{d},\xi_{d})\big{)},$
$\displaystyle\qquad k$ $\displaystyle=1,2,$ $\displaystyle
A_{d,\varrho}(x,\xi)$
$\displaystyle=(R_{1,\varrho}(x,\xi),R_{2,\varrho}(x,\xi)),$ $\displaystyle
U_{d,\varrho}(x,\xi)$
$\displaystyle=R_{1,\varrho}(x,\xi)+iR_{2,\varrho}(x,\xi),$

then

$\displaystyle(V_{\phi}(\mathscr{F}_{\\!\varrho}f))(x,\xi)$
$\displaystyle=(2\pi)^{-\frac{d}{2}}e^{-\frac{1}{4}|z|^{2}}e^{-\frac{i}{2}\langle
x,\xi\rangle}(\mathfrak{V}_{d}f)(2^{-\frac{1}{2}}\overline{U_{d,\varrho}(z)})$
(1.35)
$\displaystyle=e^{i\frac{1}{4}\Phi_{\varrho}(x,\xi)}V_{\phi}f(A_{d,\varrho}(x,\xi)),\quad
z=x+i\xi\in\mathbf{C}^{d}.$

For conveniency we set

$A_{d,\varrho_{0}}(x,\xi)=A_{d,\varrho}(x,\xi)\quad\text{and}\quad
U_{d,\varrho_{0}}(x,\xi)=U_{d,\varrho}(x,\xi)$

when

$\varrho=(\varrho_{0},\dots,\varrho_{0})\in\mathbf{R}^{d}\quad\text{and}\quad
x,\xi\in\mathbf{R}^{d}.$

It it then clear that (1.35) still hold true when $\mathscr{F}_{\\!\varrho}$
is the fractional Fourier transform on
$\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ of order $\varrho\in\mathbf{R}$.

As in [53] we observe that if, more generally, $\varrho\in\mathbf{C}^{d}$,
then the map

$f\mapsto\big{(}z\mapsto(\mathfrak{V}_{d}f(e^{-\frac{i}{2}\pi\varrho_{1}}z_{1},\dots,e^{-\frac{i}{2}\pi\varrho_{d}}z_{d})\big{)}$

makes sense as a homeomorphism from $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$
into $\mathcal{A}_{0}^{\prime}(\mathbf{C}^{d})$. In similar ways as in [53],
we define the fractional Fourier transform $\mathscr{F}_{\\!\varrho}$ by
(1.32) when $\varrho\in\mathbf{C}^{d}$. Then it follows that still
$\mathscr{F}_{\\!\varrho}$ is continuous on
$\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$.

###  1.8. Strichartz estimates

We recall that for a linear operator $R$ acting on suitable functions or
(ultra-)distributions on $\mathbf{R}^{d}$, Strichartz estimates appears when
finding properties on solutions to Cauchy problems like the generalized
inhomogeneous Schrödinger equation

$\begin{cases}i\partial_{t}u-Ru=F,\\\\[4.30554pt]
u(0,x)=u_{0}(x),\qquad(t,x)\in I\times\mathbf{R}^{d}.\end{cases}$ (1.36)

(See e. g. [50] and the references therein.) Here $I=[0,\infty)$ or $I=[0,T]$
for some $T>0$, $F$ is a suitable function or (ultra-)distribution on
$I\times\mathbf{R}^{d}$, $R$ is a linear operator acting on functions or
distributions on $\mathbf{R}^{d}$, and $u_{0}$ is a suitable function or
(ultra-)distribution on $\mathbf{R}^{d}$. The solution of (1.36) is formally
given by

$u(t,x)=(e^{-itR}u_{0})(x)-i\int_{0}^{t}(e^{i(t-s)R}F(t,\,\cdot\,))(x)\,ds.$
(1.37)

In particular it follows that continuity properties of the propagator

$\displaystyle(E_{R}f)(t,x)$ $\displaystyle\equiv(e^{-itR}u_{0})(x),$
$\displaystyle\quad(t,x)$ $\displaystyle\in I\times\mathbf{R}^{d}.$ (1.38) as
well as for the operator $\displaystyle(S_{1,R}F)(t,x)$
$\displaystyle=\int_{0}^{t}(e^{-i(t-s)R}F(s,\,\cdot\,))(x)\,ds,$
$\displaystyle\qquad(t,x)$ $\displaystyle\in I\times\mathbf{R}^{d},$ (1.39)
are essential for finding estimates for solutions to (1.36). We observe that
the $L^{2}(I\times\mathbf{R}^{d})$ adjoint $E_{R}^{*}$ of $E_{R}$ is given by
$\displaystyle(E_{R}^{*}F)(x)$
$\displaystyle=\int_{I}(e^{isR}F(s,\,\cdot\,))(x)\,ds,$ $\displaystyle\quad x$
$\displaystyle\in\mathbf{R}^{d},$ (1.40) and that the composition $E_{R}\circ
E_{R}^{*}$ of $E_{R}$ and $E_{R}^{*}$ is the operator $S_{2,R}$, similar to
$S_{1,R}$, and given by $\displaystyle(S_{2,R}F)(t,x)$
$\displaystyle=\int_{I}(e^{-i(t-s)R}F(s,\,\cdot\,))(x)\,ds,$
$\displaystyle\quad(t,x)$ $\displaystyle\in I\times\mathbf{R}^{d}.$ (1.41)

We recall that continuity properties of $E_{R}$ (or $E_{R}^{*}$) are strongly
linked to continuity properties for $S_{2,R}$ (see [30]). Estimates on the
operator $E_{R}$ in (1.38) is called _homogeneous Strichartz estimates_ ,
while estimates for $S_{1,R}$ in (1.39), or even for $S_{2,R}$ in (1.41), are
called _inhomogeneous Strichartz estimates_.

In our situation, the operator $R$ is given by the operator $H_{x,\varrho,c}$
for some $c\in\mathbf{C}$ and $\varrho\in\mathbf{R}^{d}$ or
$\varrho\in\mathbf{C}^{d}$, and for such choice of $R$, we put $E=E_{R}$ and
$S_{j}=S_{j,R}$, $j=1,2$. Hence

$\displaystyle(Ef)(t,x)$
$\displaystyle\equiv(e^{-itH_{x,\varrho,c}}u_{0})(x),$
$\displaystyle\quad(t,x)$ $\displaystyle\in I\times\mathbf{R}^{d},$ (1.38)′
$\displaystyle(E^{*}F)(x)$
$\displaystyle=\int_{I}(e^{isH_{x,\varrho,c}}F(s,\,\cdot\,))(x)\,ds,$
$\displaystyle\quad x$ $\displaystyle\in\mathbf{R}^{d}.$ (1.40)′
$\displaystyle(S_{1}F)(t,x)$
$\displaystyle=\int_{0}^{t}(e^{-i(t-s)H_{x,\varrho,c}}F(s,\,\cdot\,))(x)\,ds,$
$\displaystyle\quad(t,x)$ $\displaystyle\in I\times\mathbf{R}^{d},$ (1.39)′
and $\displaystyle(S_{2}F)(t,x)$
$\displaystyle=\int_{I}(e^{-i(t-s)H_{x,\varrho,c}}F(s,\,\cdot\,))(x)\,ds,$
$\displaystyle\quad(t,x)$ $\displaystyle\in I\times\mathbf{R}^{d}.$ (1.41)′

In Section 4 we deduce continuity properties for the operators $E$ when acting
on modulation spaces, and for $S_{1}$ and $S_{2}$ when acting on Lebesgue
spaces with values in modulation spaces.

##  2\. Powers of generalized harmonic oscillator propagators on Pilipović
spaces

In this section we show that powers of harmonic oscillators, $H_{x,c}$, or
more generally $H_{x,\varrho,c}$, are continuous on Pilipović spaces. If in
addition $H_{x,\varrho,c}$ is injective, then we show that powers of
$H_{x,\varrho,c}$ are in fact homeomorphisms on Pilipović spaces and their
distribution spaces. We also consider harmonic oscillator propagators and
deduce homeomorphism properties of such operators on Pilipović spaces.

In the last part we show that powers of harmonic oscillators are continuous on
Hilbert modulation spaces of the form
$M^{2,2}_{(\vartheta_{r})}(\mathbf{R}^{d})$, where
$\vartheta_{r}(x,\xi)=\langle(x,\xi)\rangle^{r}$.

###  2.1. Continuity of powers of $H_{x,c}$ and their propagators

For any $c\in\mathbf{C}$, it follows that $H_{x,c}$ is continuous on
$\mathcal{H}_{0}(\mathbf{R}^{d})$, and that

$H_{x,c}f(x)=\sum_{\alpha\in\mathbf{N}^{d}}\left(2|\alpha|+d+c\right)c_{h}(f,\alpha)h_{\alpha}(x),$
(2.1)

when $f\in\mathcal{H}_{0}(\mathbf{R}^{d})$ is given by (1.8). By duality it
follows that $H_{x,c}$ on $\mathcal{H}_{0}(\mathbf{R}^{d})$ is uniquely
extendable to a continuous map on $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$,
and that (2.1) still holds true when
$f\in\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ is given by (1.8).

In the same way it follows that if $r\geq 0$ is real, then

$\displaystyle H_{x,c}^{r}\,$ $\displaystyle:$
$\displaystyle\sum_{\alpha\in\mathbf{N}^{d}}c_{h}(f,\alpha)h_{\alpha}$
$\displaystyle\mapsto$
$\displaystyle\sum_{\alpha\in\mathbf{N}^{d}}(2|\alpha|+d+c)^{r}c_{h}(f,\alpha)h_{\alpha}$
(2.2) and $\displaystyle e^{\zeta H_{x,c}^{r}}\,$ $\displaystyle:$
$\displaystyle\sum_{\alpha\in\mathbf{N}^{d}}c_{h}(f,\alpha)h_{\alpha}$
$\displaystyle\mapsto$
$\displaystyle\sum_{\alpha\in\mathbf{N}^{d}}e^{(2|\alpha|+d+c)^{r}}c_{h}(f,\alpha)h_{\alpha}$
(2.3)

are continuous on $\mathcal{H}_{0}(\mathbf{R}^{d})$, and uniquely extendable
to continuous mappings on $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$. More
generally, if $r\in\mathbf{R}$, and

$\displaystyle c$
$\displaystyle\in\mathbf{C}\setminus\\{\,-2n-d\,;\,n\in\mathbf{N}\,\\}$ and
$\displaystyle\quad r$ $\displaystyle\in\mathbf{R}$ (2.4) or $\displaystyle c$
$\displaystyle\in\mathbf{C}$ and $\displaystyle\quad r$
$\displaystyle\in\overline{\mathbf{R}}_{+},$ (2.5)

then the operators (2.2) and (2.3) are continuous on
$\mathcal{H}_{0}(\mathbf{R}^{d})$ and on
$\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$. The following result extends these
continuity properties to other Pilipović spaces.

######  Proposition 2.1.

Let $\zeta\in\mathbf{C}$, $r\in\mathbf{R}$ and $c\in\mathbf{C}$ be as in (2.4)
or as in (2.5), and let $s,s_{1},s_{2}\in\overline{\mathbf{R}_{\flat}}$ be
such that $0<s_{1}\leq\frac{1}{2r}$ and $s_{2}<\frac{1}{2r}$. Then the
following is true:

  1. (1)

the map (2.2) on $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ restricts to
continuous mappings on $\mathscr{S}(\mathbf{R}^{d})$,
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and on the spaces in (1.13). If (2.4)
holds true, then these mappings are homeomorphisms;

  2. (2)

the map (2.3) on $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ restricts to
homeomorphisms on

$\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d}),\quad\mathcal{H}_{s_{2}}(\mathbf{R}^{d}),\quad\mathcal{H}_{s_{2}}^{\prime}(\mathbf{R}^{d})\quad\text{and
on}\quad\mathcal{H}_{0,s_{1}}^{\prime}(\mathbf{R}^{d}).$ (2.6)

If in addition $\zeta\in i\mathbf{R}$, then the map (2.3) is homeomorphic on
$\mathscr{S}(\mathbf{R}^{d})$, $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and the
spaces in (1.13)

###### Proof.

The assertion (1) follows from (2.2), the definitions of the spaces in (1.13)
and the fact that

$\\{c(\alpha)\\}_{\mathbf{N}^{d}}\mapsto\\{(2|\alpha|+d+c)^{r}c(\alpha)\\}_{\mathbf{N}^{d}}$
(2.7)

is continuous on the sequence spaces in Definition 1.1 (3). The homeomorphism
property in the case $c\neq-d$ follows from the fact that the map (2.7) is
then a continuous bijection.

In the same way, (2) follows from (2.3) and the fact that

$\\{c(\alpha)\\}_{\mathbf{N}^{d}}\mapsto\\{e^{\zeta(2|\alpha|+d+c)^{r}}c(\alpha)\\}_{\mathbf{N}^{d}}$
(2.8)

is a homeomorphism on the sequence spaces which correspond to the spaces in
(2.6). ∎

We also have the following negative result concerning continuity for the
operator in (2.3).

######  Proposition 2.2.

Let $\zeta\in\mathbf{C}$ be such that $\operatorname{Re}(\zeta)>0$, $r>0$,
$c\in\mathbf{C}$, and let $s,s_{1},s_{2}\in\overline{\mathbf{R}_{\flat}}$ be
such that $s_{1}>\frac{1}{2r}$ and $s_{2}\geq\frac{1}{2r}$. Then the following
is true:

  1. (1)

the map (2.3) is discontinuous from $\mathscr{S}(\mathbf{R}^{d})$ to
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$;

  2. (2)

the map (2.3) is discontinuous from $\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d})$ to
$\mathcal{H}_{0,s_{1}}^{\prime}(\mathbf{R}^{d})$;

  3. (3)

the map (2.3) is discontinuous from $\mathcal{H}_{s_{2}}(\mathbf{R}^{d})$ to
$\mathcal{H}_{s_{2}}^{\prime}(\mathbf{R}^{d})$.

###### Proof.

We only prove (1) and (2), and then in the case when $s_{1}\in\mathbf{R}$ and
$c=0$. The other cases follow by similar arguments and are left for the
reader.

Since $e^{itH_{x}^{r}}$ is a homeomorphism on all involved spaces when $t$ is
real, we may assume that $\zeta>0$ is real. Let

$f=\sum_{\alpha\in\mathbf{N}^{d}}e^{-(\log(1+|\alpha|))^{2}}h_{\alpha},$

i. e. the Hermite coefficients for $f$ are given by
$c_{h}(f,\alpha)=e^{-(\log(1+|\alpha|))^{2}}$. Since

$e^{-(\log(1+|\alpha|))^{2}}\lesssim\langle\alpha\rangle^{-N}$

for every $N\geq 1$, it follows that $f\in\mathscr{S}(\mathbf{R}^{d})$ in view
of (1.17).

On the other hand, by (2.3) we get

$c_{h}(e^{\zeta
H_{x}^{r}}f,\alpha)=e^{\zeta(2|\alpha|+d)^{r}}e^{-(\log(1+|\alpha|))^{2}}\gtrsim
e^{\zeta(|\alpha|+d)^{r}}\gtrsim\langle\alpha\rangle^{N},$

for every $N\geq 1$. Hence (1.18) shows that $e^{\zeta
H_{x}^{r}}f\notin\mathscr{S}^{\prime}(\mathbf{R}^{d})$, while
$f\in\mathscr{S}(\mathbf{R}^{d})$. This gives (1).

In order to prove (2), let $s\in\mathbf{R}$ be such that
$\frac{1}{2r}<s<s_{1}$, and let

$f=\sum_{\alpha\in\mathbf{N}^{d}}e^{-(1+|\alpha|)^{\frac{1}{2s}}}h_{\alpha},$

i. e. the Hermite coefficients for $f$ are given by
$c_{h}(f,\alpha)=e^{-(1+|\alpha|)^{\frac{1}{2s}}}$. Since

$e^{-(1+|\alpha|)^{\frac{1}{2s}}}\lesssim
e^{-r(1+|\alpha|)^{\frac{1}{2s_{1}}}}$

for every $r>0$, it follows that $f\in\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d})$
in view of the definitions of $\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d})$.

On the other hand, by (2.3) we get

$\displaystyle c_{h}(e^{\zeta H_{x}^{r}}f,\alpha)$
$\displaystyle=e^{\zeta(2|\alpha|+d)^{r}}e^{-(1+|\alpha|)^{\frac{1}{2s}}}$
$\displaystyle\gtrsim e^{\zeta(|\alpha|+d)^{r}}\gtrsim
e^{r(1+|\alpha|)^{\frac{1}{2s}}}\gtrsim e^{r(1+|\alpha|)^{\frac{1}{2s_{1}}}},$

for every $r>0$. Hence $e^{\zeta
H_{x}^{r}}f\notin\mathcal{H}_{0,s_{1}}^{\prime}(\mathbf{R}^{d})$, due to the
definitions, while $f\in\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d})$. This gives
(2), and the result follows. ∎

###  2.2. Extensions to powers of $H_{x,\varrho,c}$ and their propagators

The previous results can be extended to allow $H_{x,\varrho,c}$ in place of
$H_{x,c}$, for more general choices of $\varrho\in\mathbf{C}^{d}$. We notice
that

$\displaystyle H_{x,\varrho,c}^{r}\,$ $\displaystyle:$
$\displaystyle\sum_{\alpha\in\mathbf{N}^{d}}c_{h}(f,\alpha)h_{\alpha}$
$\displaystyle\mapsto$
$\displaystyle\sum_{\alpha\in\mathbf{N}^{d}}(2\langle\alpha,\varrho\rangle+\operatorname{sum}(\varrho)+c)^{r}c_{h}(f,\alpha)h_{\alpha}$
(2.2)′ and $\displaystyle e^{\zeta H_{x,\varrho,c}^{r}}\,$ $\displaystyle:$
$\displaystyle\sum_{\alpha\in\mathbf{N}^{d}}c_{h}(f,\alpha)h_{\alpha}$
$\displaystyle\mapsto$
$\displaystyle\sum_{\alpha\in\mathbf{N}^{d}}e^{\zeta(2\langle\alpha,\varrho\rangle+\operatorname{sum}(\varrho)+c)^{r}}c_{h}(f,\alpha)h_{\alpha}$
(2.3)′

when $f\in\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$, and

$\displaystyle c$
$\displaystyle\in\mathbf{C}\setminus\\{\,-2\langle\alpha,\varrho\rangle-\operatorname{sum}(\varrho)\,;\,\alpha\in\mathbf{N}^{d}\,\\}$
and $\displaystyle\quad r$ $\displaystyle\in\mathbf{R}$ (2.4)′ or
$\displaystyle c$ $\displaystyle\in\mathbf{C}$ and $\displaystyle\quad r$
$\displaystyle\in\overline{\mathbf{R}}_{+},$ (2.5)′

Here and in what follows we let

$\operatorname{sum}(\varrho)=\sum_{j=1}^{d}\varrho_{j},\quad\text{when}\quad\varrho=(\varrho_{1},\dots,\varrho_{d})\in\mathbf{C}^{d}.$

By similar arguments as in the proof of Proposition 2.1 we get the following
extension. The details are left for the reader.

######  Proposition 2.1${}^{\prime}\\!\\!$.

Let $\zeta\in\mathbf{C}$,$r\in\mathbf{R}$ $\varrho\in\mathbf{C}^{d}$ and
$c\in\mathbf{C}$ be as in (2.4)′ or as in (2.5)′, and let
$s,s_{1},s_{2}\in\overline{\mathbf{R}_{\flat}}$ be such that
$0<s_{1}\leq\frac{1}{2r}$ and $s_{2}<\frac{1}{2r}$. Then the following is
true:

  1. (1)

the map (2.2)′ on $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ restricts to
continuous mappings on $\mathscr{S}(\mathbf{R}^{d})$,
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and on the spaces in (1.13). If in
addition (2.4)′ holds true, then these mappings are homeomorphisms;

  2. (2)

the map (2.3)′ on $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ restricts to
homeomorphisms on

$\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d}),\quad\mathcal{H}_{s_{2}}(\mathbf{R}^{d}),\quad\mathcal{H}_{s_{2}}^{\prime}(\mathbf{R}^{d})\quad\text{and
on}\quad\mathcal{H}_{0,s_{1}}^{\prime}(\mathbf{R}^{d}).$ (2.6)′

If in addition $\zeta\varrho_{j}^{r}\in i\mathbf{R}$ for every $j$, then the
map (2.3)′ is homeomorphic on $\mathscr{S}(\mathbf{R}^{d})$,
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and the spaces in (1.13)

In the same way, similar arguments as in the proof of Proposition 2.2 gives
the following extension. The details are left for the reader.

######  Proposition 2.2′.

Let $r>0$, $\varrho\in\mathbf{C}^{d}$ be such that
$\operatorname{Re}(\zeta\varrho_{j}^{r})>0$ for every $j=1,\dots,d$,
$c\in\mathbf{C}$, and let $s,s_{1},s_{2}\in\overline{\mathbf{R}_{\flat}}$ be
such that $s_{1}>\frac{1}{2r}$ and $s_{2}\geq\frac{1}{2r}$. Then the following
is true:

  1. (1)

the map (2.3) is discontinuous from $\mathscr{S}(\mathbf{R}^{d})$ to
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$;

  2. (2)

the map (2.3) is discontinuous from $\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d})$ to
$\mathcal{H}_{0,s_{1}}^{\prime}(\mathbf{R}^{d})$;

  3. (3)

the map (2.3) is discontinuous from $\mathcal{H}_{s_{2}}(\mathbf{R}^{d})$ to
$\mathcal{H}_{s_{2}}^{\prime}(\mathbf{R}^{d})$.

###  2.3. Powers of harmonic oscillator on Hilbert modulations spaces

Let $r\in\mathbf{R}$, $p,q\in(0,\infty]$, $s>1$,
$\omega\in\mathscr{P}_{E,s}(\mathbf{R}^{2d})$ and
$\vartheta_{r}(x,\xi)=\langle(x,\xi)\rangle^{r}$. Then it follows from [1,
Proposition 1.34′] that $H_{x}^{N}$ is homeomorphic from
$M^{p,q}_{(\omega\vartheta_{N})}(\mathbf{R}^{d})$ to
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ when $N$ is an integer. So far, we are
not able to prove any such result when the integer $N$ is replaced by a
general real number for such general modulation spaces. On the other hand we
have the following for certain Hilbert modulation spaces. See also [11, 21]
for similar results. Here we restrict ourself to weights which are rotational
invariant with respect to each phase space, or complex variable. That is, we
assume that

$\omega(x,\xi)=\omega_{0}(\rho),\qquad\rho_{j}=x_{j}^{2}+\xi_{j}^{2},$ (2.9)

for some positive function $\omega_{0}$ on ${\overline{\mathbf{R}}}_{+}^{d}$.

######  Theorem 2.3.

Let $r,t\in\mathbf{R}$, $\vartheta_{r}(x,\xi)=\langle(x,\xi)\rangle^{r}$ and
suppose that $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ satisfies (2.9) for
some positive function $\omega_{0}$ on ${\overline{\mathbf{R}}}_{+}^{d}$. Then
the following is true:

  1. (1)

$H_{x}^{r}$ on $\mathcal{H}_{0}(\mathbf{R}^{d})$ is uniquely extendable to a
homeomorphism from $M^{2}_{(\omega)}(\mathbf{R}^{d})$ to
$M^{2}_{(\omega/\vartheta_{r})}(\mathbf{R}^{d})$, with bound of $H_{x}^{r}$
which is independent of $r$;

  2. (2)

$e^{itH_{x}^{r}}$ on $\mathcal{H}_{0}(\mathbf{R}^{d})$ is uniquely extendable
to a homeomorphism on $M^{2}_{(\omega)}(\mathbf{R}^{d})$, with bound of
$e^{itH_{x}^{r}}$ which is independent of $t$ and $r$.

The following lemma shows that $M^{2}_{(\omega)}(\mathbf{R}^{d})$ with weights
of the form (2.9) can be expressed as norm estimates of Hermite series
expansions of the involved functions and distributions. Here we extract the
weight function

$\nu_{\omega}:\mathbf{N}^{d}\to\mathbf{R}_{+},$

from $\omega$ by the formula

$\displaystyle\nu_{\omega}(\alpha)$
$\displaystyle\equiv\left(\alpha!^{-1}\int_{\mathbf{R}^{d}_{+}}r^{\alpha}\omega_{0}(r)^{2}e^{-(r_{1}+\cdots+r_{d})}\,dr\right)^{\frac{1}{2}},\qquad\alpha\in\mathbf{N}^{d}.$
(2.10)

######  Lemma 2.4.

Let $\omega\in\mathscr{P}_{E}(\mathbf{R}^{2d})$ be such that (2.9) holds for
some positive function $\omega_{0}$ on ${\overline{\mathbf{R}}}_{+}^{d}$. Also
let $\nu_{\omega}$ be given by (2.10). Then $M^{2}_{(\omega)}(\mathbf{R}^{d})$
consists of all $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$ such that

$\|f\|_{[\omega]}\equiv\left(\sum_{\alpha\in\mathbf{N}^{d}}|c_{h}(f,\alpha)\nu_{\omega}(\alpha)|^{2}\right)^{\frac{1}{2}}$
(2.11)

is finite. The Hilbert norm $\|\,\cdot\,\|_{[\omega]}$ is equivalent to
$\|\,\cdot\,\|_{M^{2}_{(\omega)}}$.

###### Proof.

The result follows by straight-forward applications of [53, Theorem 3.5], and
that the Bargmann image of $M^{2}_{(\omega)}(\mathbf{R}^{d})$ is equal to
$A^{2}_{(\omega)}(\mathbf{C}^{d})$. The details are left for the reader. ∎

###### Proof of Theorem 2.3.

Let $\|\,\cdot\,\|_{[\omega]}$ be the norm given in (2.11). Then (2.2) shows
that

$\displaystyle c_{h}(H_{x}^{r}f,\alpha)$
$\displaystyle=(2|\alpha|+d)^{r}c(f,\alpha)$ and $\displaystyle
c_{h}(e^{itH_{x}^{r}}f,\alpha)$
$\displaystyle=e^{it(2|\alpha|+d)^{r}}c(f,\alpha).$

Since $|e^{it(2|\alpha|+d)^{r}}|=1$ and that $\omega$ satisfies (2.9), the
assertion follows from Lemma 2.4. ∎

######  Remark 2.5.

So far we are not able to extend Theorem 2.3 to more general modulation spaces
$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$, without the assumption (2.9) on the
weight $\omega$. Suppose that $\omega$ is the same as in Theorem 2.3. Then
using standard embeddings for modulation spaces like

$\displaystyle M^{p,q}_{(\omega\vartheta_{\tau_{1}})}(\mathbf{R}^{d})$
$\displaystyle\hookrightarrow M^{2}_{(\omega)}(\mathbf{R}^{d})\hookrightarrow
M^{p,q}_{(\omega/\vartheta_{\tau_{2}})}(\mathbf{R}^{d}),$ (2.12)
$\displaystyle\tau_{1}$ $\displaystyle\geq
d\max\Big{(}0,\frac{1}{2}-\frac{1}{p},\frac{1}{2}-\frac{1}{q}\Big{)},$
$\displaystyle\tau_{2}$ $\displaystyle\geq
d\max\Big{(}0,\frac{1}{p}-\frac{1}{2},\frac{1}{q}-\frac{1}{2}\Big{)},$

with strict inequalities if $p\neq 2$ or $q\neq 2$, it follows that

$H^{r_{0}}e^{itH^{r}}:M^{p,q}_{(\omega)}(\mathbf{R}^{d})\to
M^{p,q}_{(\omega/\vartheta_{\theta})}(\mathbf{R}^{d}),\quad\theta=r_{0}+\tau_{1}+\tau_{2}$
(2.13)

is continuous, with bounds which are independent of $r$. In fact, by Theorem
2.3 and (2.12) we have

$\displaystyle H^{r_{0}}e^{itH^{r}}M^{p,q}_{(\omega)}(\mathbf{R}^{d})$
$\displaystyle\hookrightarrow
H^{r_{0}}e^{itH^{r}}M^{2}_{(\omega/\vartheta_{\tau_{1}})}(\mathbf{R}^{d})$
$\displaystyle\hookrightarrow
M^{2}_{(\omega/\vartheta_{r_{0}+\tau_{1}})}(\mathbf{R}^{d})\hookrightarrow
M^{p,q}_{(\omega/\vartheta_{\theta})}(\mathbf{R}^{d}).$

##  3\. General harmonic oscillator propagators and fractional Fourier
transforms

In this section we prove that generalized harmonic oscillator propagators of
the forms $e^{irH_{x,\varrho,c}}$ are essentially fractional Fourier
transforms of multiple orders. We use such identities to link some results in
[8] with results in [53], especially when such operators act on (weighted)
modulation spaces.

###  3.1. Identifications between fractional Fourier transforms and harmonic
oscillator type propagators

Let $\varrho\in\mathbf{C}^{d}$ and $c\in\mathbf{C}$. Then the operator
$H_{x,\varrho,c}$ is transformed into the operator

$\displaystyle 2H_{\varrho,c,\mathfrak{V}}$
$\displaystyle=2H_{\varrho,\mathfrak{V}}+\operatorname{sum}(\varrho)+c,\quad\text{where}\quad
H_{\varrho,\mathfrak{V}}=\sum_{j=1}^{d}\varrho_{j}z_{j}\partial_{z_{j}}.$ That
is, $\displaystyle\mathfrak{V}_{d}\circ H_{x,\varrho,c}$
$\displaystyle=(2H_{\varrho,\mathfrak{V}}+\operatorname{sum}(\varrho)+c)\circ\mathfrak{V}_{d},\quad
H_{\varrho,\mathfrak{V}}=\sum_{j=1}^{d}\varrho_{j}z_{j}\partial_{z_{j}},$
(3.1) or equivalently, $\displaystyle\mathfrak{V}_{d}H_{x,\varrho,c}f$
$\displaystyle=(2H_{\varrho,\mathfrak{V}}+\operatorname{sum}(\varrho)+c)F,\qquad
F=\mathfrak{V}_{d}f.$

For convenience we put

$H_{\varrho_{0},c,\mathfrak{V}}=H_{\varrho,c,\mathfrak{V}}\quad\text{and}\quad
H_{\varrho_{0},\mathfrak{V}}=H_{\varrho,\mathfrak{V}},$

when $\varrho=(\varrho_{0},\dots,\varrho_{0})\in\mathbf{C}^{d}$, and we put

$H_{\mathfrak{V}}=H_{1,\mathfrak{V}}.$

In particular, it follows from (3.1) that $\frac{1}{2}H_{x,-d}$ is transformed
into $H_{\mathfrak{V}}$. It also follows from (3.1) that if
$r,\zeta\in\mathbf{C}$ satisfy $\operatorname{Re}(r)\geq 0$, then

$\mathfrak{V}_{d}\circ e^{\zeta
H_{x,\varrho,c}^{r}}=e^{\zeta(2H_{\varrho,\mathfrak{V}}+\operatorname{sum}(\varrho)+c)^{r}}\circ\mathfrak{V}_{d},$
(3.2)

as continuous operators from $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$ to
$\mathcal{A}_{0}^{\prime}(\mathbf{C}^{d})$.

By straight-forward computations we get

$H_{\varrho,\mathfrak{V}}e_{\alpha}(z)=\langle\varrho,\alpha\rangle
e_{\alpha}(z)$

(see e. g. [5]). This implies that

$\displaystyle e^{\zeta H_{\varrho,\mathfrak{V}}}e_{\alpha}(z)$
$\displaystyle=e^{\zeta\langle\varrho,\alpha\rangle}e_{\alpha}(z)=e_{\alpha}(e^{\zeta\varrho_{1}}z_{1},\dots,e^{\zeta\varrho_{d}}z_{d}),$
(3.3) for every $\zeta\in\mathbf{C}$, which gives $\displaystyle(e^{\zeta
H_{\varrho,\mathfrak{V}}}F)(z)$
$\displaystyle=F(e^{\zeta\varrho_{1}}z_{1},\dots,e^{\zeta\varrho_{d}}z_{d}),\qquad
F\in\mathcal{A}_{0}^{\prime}(\mathbf{C}^{d}).$ (3.4)

######  Remark 3.1.

We observe that the map

$F(z)\mapsto(e^{\zeta
H_{\varrho,\mathfrak{V}}}F)(z)=F(e^{\zeta\varrho_{1}}z_{1},\dots,e^{\zeta\varrho_{d}}z_{d})$

is a continuous bijection on the spaces

$\mathcal{A}_{s_{1}}(\mathbf{C}^{d}),\quad\mathcal{A}_{0,s_{2}}(\mathbf{C}^{d}),\quad\mathcal{A}_{s_{1}}^{\prime}(\mathbf{C}^{d}),\quad\mathcal{A}_{0,s_{2}}^{\prime}(\mathbf{C}^{d}),$
(3.5)

when $s_{1}<\frac{1}{2}$ and $s_{2}\leq\frac{1}{2}$. See [53] for definition
and some characterizations of the spaces in (3.5). This is also a consequence
of Proposition 2.1.

We recall that the fractional Fourier transform $\mathscr{F}_{\\!\varrho}$ of
(multiple) order $\varrho\in\mathbf{C}^{d}$ satisfies

$(\mathfrak{V}_{d}(\mathscr{F}_{\\!\varrho}f))(z)=(\mathfrak{V}_{d}f)(e^{-i\frac{\pi\varrho_{1}}{2}}z_{1},\dots,e^{-i\frac{\pi\varrho_{d}}{2}}z_{d}),\qquad
f\in\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d}).$ (3.6)

Hence, a combination of Proposition 2.1′, (3.1), (3.2) and (3.4) with

$e^{-iH_{x,\varrho,c_{1}}}=e^{i(c_{2}-c_{1})}e^{-iH_{x,\varrho,c_{2}}},\qquad
c_{1},c_{2}\in\mathbf{C}$

gives the following extention of results given in [42, p. 161].

######  Theorem 3.2.

Let $\varrho\in\mathbf{C}^{d}$, $c\in\mathbf{C}$ and
$s,s_{1},s_{2}\in\overline{\mathbf{R}_{\flat}}$ be such that
$0<s_{1}\leq\frac{1}{2}$ and $s_{2}<\frac{1}{2}$. Then

$e^{-i\frac{\pi}{4}H_{x,\varrho,c}}=e^{-i\frac{\pi}{4}(\operatorname{sum}(\varrho)+c)}\mathscr{F}_{\\!\varrho}$
(3.7)

as operators on $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$. The operators in
(3.7) restrict to homeomorphisms on the spaces in (2.6).

Evidently, (3.7) is the same as

$\mathscr{F}_{\\!\varrho}=e^{i\frac{\pi}{4}(\operatorname{sum}(\varrho)+c)}e^{-i\frac{\pi}{4}H_{x,\varrho,c}},$
(3.7)′

and can be used to transfer properties between fractional Fourier transform
and harmonic oscillator propagators.

By combining with Theorem 3.2, we get the following extensions of Propositions
0.2 and 0.3 from the introduction. The details are left for the reader.

######  Proposition 0.2${}^{\prime}\\!\\!$.

Let $I_{d}=\\{1,\dots,d\\}$, $\varrho\in\mathbf{C}^{d}$ and
$s\in\overline{\mathbf{R}}_{\flat}$. Then the following is true:

  1. (1)

if $s<\frac{1}{2}$, then $\mathscr{F}_{\\!\varrho}$ and
$e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are homeomorphisms on
$\mathcal{H}_{s}(\mathbf{R}^{d})$ and on
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$;

  2. (2)

if $\operatorname{Im}(\varrho_{j})\leq 0$ for every $j\in I_{d}$,
$\operatorname{Im}(\varrho_{j_{0}})<0$ for some $j_{0}\in I_{d}$ and
$s\geq\frac{1}{2}$, then $\mathscr{F}_{\\!\varrho}$ and
$e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are continuous injections but not
surjections on $\mathcal{H}_{s}(\mathbf{R}^{d})$,
$\mathscr{S}(\mathbf{R}^{d})$, $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and on
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$;

  3. (3)

if $\operatorname{Im}(\varrho_{j})=0$ for every $j\in I_{d}$, then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are
homeomorphisms on $\mathcal{H}_{s}(\mathbf{R}^{d})$,
$\mathscr{S}(\mathbf{R}^{d})$, $\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and on
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$;

  4. (4)

if $\operatorname{Im}(\varrho_{j})>0$ for some $j\in I_{d}$ and
$s\geq\frac{1}{2}$, then $\mathscr{F}_{\\!\varrho}$ and
$e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are discontinuous on
$\mathcal{H}_{s}(\mathbf{R}^{d})$, $\mathscr{S}(\mathbf{R}^{d})$,
$\mathscr{S}^{\prime}(\mathbf{R}^{d})$ and on
$\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d})$.

The same holds true with $s>\frac{1}{2}$, $s\leq\frac{1}{2}$ and
$\mathcal{H}_{0,s}$ in place of $s\geq\frac{1}{2}$, $s<\frac{1}{2}$ and
$\mathcal{H}_{s}$ at each occurrence.

######  Proposition 0.3${}^{\prime}\\!\\!$.

Let $I_{d}=\\{1,\dots,d\\}$ and $\varrho\in\mathbf{C}^{d}$. Then the following
is true:

  1. (1)

if $\operatorname{Im}(\varrho_{j})<0$ for every $j\in I_{d}$, then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are
continuous from $\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$ to
$\mathcal{S}_{1/2}(\mathbf{R}^{d})$, and

$\mathscr{F}_{\\!\varrho}(\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d}))=e^{-i\frac{\pi}{4}H_{x,\varrho}}(\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d}))\subsetneq\mathcal{S}_{1/2}(\mathbf{R}^{d})\text{;}$

  2. (2)

if $\operatorname{Im}(\varrho_{j})>0$ for every $j\in I_{d}$, then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are
discontinuous from $\mathcal{S}_{1/2}(\mathbf{R}^{d})$ to
$\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$, and

$\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})\subsetneq\mathscr{F}_{\\!\varrho}(\mathcal{S}_{1/2}(\mathbf{R}^{d}))=e^{-i\frac{\pi}{4}H_{x,\varrho}}(\mathcal{S}_{1/2}(\mathbf{R}^{d}))\subsetneq\mathcal{H}_{0,1/2}^{\prime}(\mathbf{R}^{d})\text{;}$

  3. (3)

if $\operatorname{Im}(\varrho_{j})>0$ for some $j\in I_{d}$, then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho}}$ are
discontinuous from $\mathcal{S}_{1/2}(\mathbf{R}^{d})$ to
$\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$, and

$\mathscr{F}_{\\!\varrho}f\in\mathcal{H}_{0,1/2}^{\prime}(\mathbf{R}^{d})\setminus\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})\quad\text{and}\quad
e^{-i\frac{\pi}{4}H_{x,\varrho}}f\in\mathcal{H}_{0,1/2}^{\prime}(\mathbf{R}^{d})\setminus\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$

for some $f\in\mathcal{S}_{1/2}(\mathbf{R}^{d})$.

In the following proposition we point out some auxiliary group properties for
fractional Fourier transforms of complex orders, which extends similar results
in [42] for fractional Fourier transforms of real orders. The result follows
by straight-forward applications of (3.6) and the fact that the Bargmann
transform is injective. The details are left for the reader.

######  Proposition 3.3.

For any $\varrho\in\mathbf{C}^{d}$, let $\mathscr{F}_{\\!\varrho}$ be acting
on $\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})$. Then
$\\{\mathscr{F}_{\\!\varrho}\\}_{\varrho\in\mathbf{C}^{d}}$ is a commutative
group under composition, with identity element
$\mathscr{F}_{0}=\operatorname{Id}_{\mathcal{H}_{0}^{\prime}(\mathbf{R}^{d})}$,
and

$\displaystyle\mathscr{F}_{\\!\varrho_{1}}\circ\mathscr{F}_{\\!\varrho_{2}}=\mathscr{F}_{\\!\varrho_{1}+\varrho_{2}},\qquad\mathscr{F}_{\\!\varrho}^{-1}=\mathscr{F}_{\\!-\varrho},\qquad\mathscr{F}_{\\!\varrho+\varrho_{0}}$
$\displaystyle=\mathscr{F}_{\\!\varrho},$ (3.8)
$\displaystyle\varrho,\varrho_{1},\varrho_{2}$
$\displaystyle\in\mathbf{C}^{d},\ \varrho_{0}\in 4\mathbf{Z}^{d}.$

###  3.2. Continuity for one-parameters fractional Fourier transforms and
harmonic oscillator propagators on modulation spaces

As an example we shall next transfer mapping properties of
$\mathscr{F}_{\\!\varrho}$, $\varrho\in\mathbf{R}$, when acting on certain
classes of modulation spaces into analogous properties for harmonic oscillator
propagators.

For fractional Fourier transforms on modulation spaces we recall the following
special case of [53, Proposition 7.1]. We refer to Subsection 1.4 for
notations on weight classes.

######  Proposition 3.4.

Let $\varrho\in\mathbf{R}$, $p\in(0,\infty]$ and
$\omega\in\mathscr{P}_{\\!A,r}(\mathbf{R}^{2d})$. Then
$\mathscr{F}_{\\!\varrho}$ is an isometric homeomorphism on
$M^{p}_{(\omega)}(\mathbf{R}^{d})$.

A combination of Theorem 3.2 and Proposition 3.4 now gives the following. (See
[8, Theorem 2.6] and [9, Theorem 1.7] in the case when $p\geq 1$ and
$\omega=1$. See also [15] in the case when $p\geq 1$ and $\omega$ is
moderatated by polynomially bounded weights.)

######  Proposition 3.5.

Let $c\in\mathbf{C}$, $\varrho\in\mathbf{R}$, $p\in(0,\infty]$ and
$\omega\in\mathscr{P}_{\\!A,r}(\mathbf{R}^{2d})$. Then
$e^{-i\frac{\pi}{4}H_{x,\varrho,c}}$ is a homeomorphism on
$M^{p}_{(\omega)}(\mathbf{R}^{d})$.

We also have the following extensions of [16, Proposition 5.1] and the
previous propositions. Here the involved weights are allowed to belong to the
general class $\mathscr{P}_{\\!A}(\mathbf{R}^{2d})$, and are linked as

$\displaystyle\omega_{\varrho}(x,\xi)$
$\displaystyle=\omega(A_{d,\varrho}(x,\xi))$ (3.9)

(see Subsections 1.4, 1.7 and Remark 1.6)

######  Theorem 3.6.

Let $\varrho\in\mathbf{R}\setminus 2\mathbf{Z}$, $c\in\mathbf{C}$,
$\omega,\omega_{\varrho}\in\mathscr{P}_{\\!A}(\mathbf{R}^{2d})$ and
$p,q\in(0,\infty]$ be such that $q\leq p$ and (3.9) hold. Then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho,c}}$ on
$\mathcal{H}_{\flat_{1}}^{\prime}(\mathbf{R}^{d})$ restrict to continuous
mappings from $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ to
$M^{q,p}_{(\omega_{\varrho})}(\mathbf{R}^{d})$, and from
$W^{q,p}_{(\omega)}(\mathbf{R}^{d})$ to
$W^{p,q}_{(\omega_{\varrho})}(\mathbf{R}^{d})$, and

$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{M^{q,p}_{(\omega_{\varrho})}}$
$\displaystyle=e^{-\frac{\pi}{4}\cdot\operatorname{Im}(c)}\|e^{-i\frac{\pi}{4}H_{x,\varrho,c}}f\|_{M^{q,p}_{(\omega_{\varrho})}}$
$\displaystyle\lesssim|\sin(\textstyle{\frac{\pi\varrho}{2}})|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{M^{p,q}_{(\omega)}},$
$\displaystyle\qquad f$ $\displaystyle\in M^{p,q}_{(\omega)}(\mathbf{R}^{d})$
(3.10) and
$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{W^{p,q}_{(\omega_{\varrho})}}$
$\displaystyle=e^{-\frac{\pi}{4}\cdot\operatorname{Im}(c)}\|e^{-i\frac{\pi}{4}H_{x,\varrho,c}}f\|_{W^{p,q}_{(\omega_{t})}}$
$\displaystyle\lesssim|\sin(\textstyle{\frac{\pi\varrho}{2}})|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{W^{q,p}_{(\omega)}},$
$\displaystyle\qquad f$ $\displaystyle\in W^{q,p}_{(\omega)}(\mathbf{R}^{d}).$
(3.11)

######  Theorem 3.7.

Let $\varrho\in\mathbf{R}\setminus(2\mathbf{Z}+1)$,
$\omega,\omega_{\varrho}\in\mathscr{P}_{\\!A}(\mathbf{R}^{2d})$ and
$p,q\in(0,\infty]$ be such that $q\leq p$ and (3.9) hold. Then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho,c}}$ on
$\mathcal{H}_{\flat_{1}}^{\prime}(\mathbf{R}^{d})$ restrict to continuous
mappings from $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ to
$W^{p,q}_{(\omega_{\varrho})}(\mathbf{R}^{d})$, and from
$W^{q,p}_{(\omega)}(\mathbf{R}^{d})$ to
$M^{q,p}_{(\omega_{\varrho})}(\mathbf{R}^{d})$, and

$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{W^{p,q}_{(\omega_{\varrho})}}$
$\displaystyle=e^{-\frac{\pi}{4}\cdot\operatorname{Im}(c)}\|e^{-i\frac{\pi}{4}H_{x,\varrho,c}}f\|_{W^{p,q}_{(\omega_{\varrho})}}$
$\displaystyle\lesssim|\cos(\textstyle{\frac{\pi\varrho}{2}})|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{M^{p,q}_{(\omega)}},$
$\displaystyle\qquad f$ $\displaystyle\in M^{p,q}_{(\omega)}(\mathbf{R}^{d})$
(3.12) and
$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{M^{q,p}_{(\omega_{\varrho})}}$
$\displaystyle=e^{-\frac{\pi}{4}\cdot\operatorname{Im}(c)}\|e^{-i\frac{\pi}{4}H_{x,\varrho,c}}f\|_{M^{q,p}_{(\omega_{t})}}$
$\displaystyle\lesssim|\cos(\textstyle{\frac{\pi\varrho}{2}})|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{W^{q,p}_{(\omega)}},$
$\displaystyle\qquad f$ $\displaystyle\in W^{q,p}_{(\omega)}(\mathbf{R}^{d}).$
(3.13)

For the proofs of Theorems 3.6 and 3.7 we need the following version of
Minkowski’s inequality.

######  Lemma 3.8.

Let $\varrho\in\mathbf{R}$, $p,q\in(0,\infty]$ be such that $q\leq p$,
$A_{d,\varrho}$ be as in Subsection 1.7 and let $T_{\varrho}$ from
$\Sigma_{1}(\mathbf{R}^{2d})$ to $\Sigma_{1}(\mathbf{R}^{2d})$ be given by

$(T_{\varrho}f)(x,\xi)=f(A_{d,\varrho}(x,\xi)),\qquad f\in
L^{q}_{loc}(\mathbf{R}^{2d}).$

Then the following is true:

  1. (1)

if $\varrho\neq 2\mathbf{Z}$, then $T_{\varrho}$ extends uniquely to a
continuous map from $L^{p,q}(\mathbf{R}^{2d})$ to $L^{q,p}(\mathbf{R}^{2d})$
and from $L^{q,p}_{*}(\mathbf{R}^{2d})$ $L^{p,q}_{*}(\mathbf{R}^{2d})$, and

$\displaystyle\|T_{\varrho}f\|_{L^{q,p}}$
$\displaystyle\leq|\sin{\textstyle{(\frac{\pi\varrho}{2})}}|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^{p,q}},$
$\displaystyle\qquad f$ $\displaystyle\in
L^{q}_{loc}(\mathbf{R}^{2d})\text{;}$ (3.14)

  2. (2)

if $\varrho\neq 2\mathbf{Z}+1$, then $T_{\varrho}$ extends uniquely to a
continuous map from $L^{p,q}(\mathbf{R}^{2d})$ to
$L^{p,q}_{*}(\mathbf{R}^{2d})$ and from $L^{q,p}_{*}(\mathbf{R}^{2d})$ to
$L^{q,p}(\mathbf{R}^{2d})$, and

$\displaystyle\|T_{\varrho}f\|_{L^{p,q}_{*}}$
$\displaystyle\leq|\cos{\textstyle{(\frac{\pi\varrho}{2})}}|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^{p,q}},$
$\displaystyle\qquad f$ $\displaystyle\in
L^{q}_{loc}(\mathbf{R}^{2d})\text{.}$ (3.15)

The same holds true with $L^{p,q}_{*}$ and $L^{q,p}_{*}$ in place of $L^{q,p}$
and $L^{p,q}$, respectively, at each occurrence.

###### Proof.

We only prove (1). The assertion (2) follows by similar arguments and is left
for the reader.

First suppose that $p<\infty$. Then $q<\infty$. Let
$\theta=\frac{\pi\varrho}{2}$, $f\in\mathscr{S}(\mathbf{R}^{d})$,
$p_{0}=p/q\geq 1$, $h\in\mathscr{S}(\mathbf{R}^{d})$ be such that
$\|h\|_{L^{p_{0}^{\prime}}}\leq 1$,

$f_{\varrho}(x,\xi)=f(A_{d,\varrho}(x,\xi))\quad\text{and}\quad
g_{\varrho}(\xi)=\int_{\mathbf{R}^{d}}|f_{\varrho}(x,\xi)|^{q}\,dx.$

Since $f_{0}=f$ and

$(y,\eta)=A_{d,\varrho}(x,\xi)\quad\Leftrightarrow\quad(x,\xi)=A_{d,-\varrho}(y,\eta),$

we get

$\displaystyle|(g_{\varrho},h)_{L^{2}}|$
$\displaystyle=\left|\iint_{\mathbf{R}^{2d}}|f_{0}(A_{d,\varrho}(x,\xi))|^{q}h(\xi)\,dxd\xi\right|$
$\displaystyle=\left|\iint_{\mathbf{R}^{2d}}|f_{0}(y,\eta)|^{q}h((\sin\theta)y+(\cos\theta)\eta)\,dxd\xi\right|$
$\displaystyle\leq\int_{\mathbf{R}^{d}}\||f_{0}(\,\cdot\,,\eta)|^{q}\|_{L^{p_{0}}}\|h((\sin\theta)\,\cdot\,+(\cos\theta)\eta)\|_{L^{p_{0}^{\prime}}}\,d\xi$
$\displaystyle=|\sin\theta|^{-\frac{d}{p_{0}^{\prime}}}\|h\|_{L^{p_{0}^{\prime}}}\int_{\mathbf{R}^{d}}\||f_{0}(\,\cdot\,,\eta)|^{q}\|_{L^{p_{0}}}\,d\xi\leq|\sin\theta|^{-\frac{d}{p_{0}^{\prime}}}\|f_{0}\|_{L^{p,q}}^{q}.$

By taking the supremum over all possible $h$ with
$\|h\|_{L^{p_{0}^{\prime}}}\leq 1$ we obtain

$\|g_{\varrho}\|_{L^{p_{0}}}\leq|\sin\theta|^{-\frac{d}{p_{0}^{\prime}}}\|f_{0}\|_{L^{p,q}}^{q}=|\sin\theta|^{-\frac{d}{p_{0}^{\prime}}}\|f\|_{L^{p,q}}^{q},$

which is the same as (3.14) when $f\in\mathscr{S}(\mathbf{R}^{2d})$. Since
$\mathscr{S}(\mathbf{R}^{2d})$ is dense in $L^{p,q}(\mathbf{R}^{2d})$ when
$p,q<\infty$, (3.14) follows when $p<\infty$.

Next suppose that $p=\infty$. The result is obviously true when $q=\infty$.
Therefore suppose that $q<\infty$, and let $f\in L^{q}_{loc}(\mathbf{R}^{2d})$
and

$g(\xi)=\|f(\,\cdot\,,\xi)\|_{L^{p}}.$

Then

$\int_{\mathbf{R}^{d}}|f_{\varrho}(x,\xi)|^{q}\,dx\leq\int_{\mathbf{R}^{d}}|g((-\sin\theta)x+(\cos\theta)\xi)|^{q}\,dx=|\sin\theta|^{-d}\|g\|_{L^{\infty,q}}^{q}$

If we take the supremum over $\xi\in\mathbf{R}^{d}$, then we obtain (3.14) for
$p=\infty$, and we have proved (3.14) for any $p\in(0,\infty]$. This gives
(1), and the result follows. ∎

###### Proof of Theorems 3.6 and 3.7.

By

$\mathscr{F}(M^{p,q}_{(\omega_{1})}(\mathbf{R}^{d}))=W^{q,p}_{(\omega_{2})}(\mathbf{R}^{d}),\quad\text{when}\quad\omega_{1}(x,\xi)=\omega_{2}(\xi,-x)$

and that $|\cos(\frac{\pi(\varrho+1)}{2})|=|\sin(\frac{\pi\varrho}{2})|$, it
follows that Theorem 3.7 is the Fourier version of Theorem 3.6. Hence it
suffices to prove Theorem 3.6. By (3.7) it also follows that it suffices to
prove the norm estimates for $\mathscr{F}_{\\!\varrho}f$ in (3.10) and (3.11).

Let $\phi(x)=\pi^{-\frac{d}{4}}e^{-\frac{1}{2}|x|^{2}}$. Then (1.35) gives

$\displaystyle F_{\varrho}(x,\xi)$
$\displaystyle=|(V_{\phi}(\mathscr{F}_{\\!\varrho}f))(x,\xi)\omega_{\varrho}(x,\xi)|=|(V_{\phi}f)(A_{d,\varrho}(x,\xi))\omega(A_{d,\varrho}(x,\xi))|.$
We also have $\displaystyle F_{0}(x,\xi)$
$\displaystyle=|(V_{\phi}f)(x,\xi)\omega(x,\xi)|\quad\text{and}\quad
F_{\varrho}(x,\xi)=F_{0}(A_{d,\varrho}(x,\xi)).$

By Lemma 3.8 and the identities above we obtain

$\|\mathscr{F}_{\\!\varrho}f\|_{M^{q,p}_{(\omega_{\varrho})}}\asymp\|F_{\varrho}\|_{L^{q,p}}\leq|\sin{\textstyle{(\frac{\pi\varrho}{2})}}|^{d(\frac{1}{p}-\frac{1}{q})}\|F_{0}\|_{L^{p,q}}\asymp|\sin{\textstyle{(\frac{\pi\varrho}{2})}}|^{d(\frac{1}{p}-\frac{1}{q})}\|f\|_{M^{p,q}_{(\omega)}},$

and (3.10) follows. In similar ways one obtains (3.11). The details are left
for the reader, and the result follows. ∎

######  Remark 3.9.

By choosing $\omega=1$ and $p=q^{\prime}\geq 2$, Theorem 3.7 agrees with [16,
Proposition 5.1] by Cordero and Nicola.

It is evident that Theorems 3.6 and 3.7 implies the following weighted version
of Proposition 0.1 in the introduction.

######  Proposition 0.1′.

Let $\varrho\in\mathbf{R}$, $c\in\mathbf{C}$,
$\omega,\omega_{\varrho}\in\mathscr{P}_{\\!A}(\mathbf{R}^{2d})$ and
$p,q\in(0,\infty]$ be such that $q\leq p$ and (3.9) holds. Then the following
is true:

  1. (1)

the map

$\displaystyle\mathscr{F}_{\\!\varrho}=e^{-i\frac{\pi}{4}H_{x,\varrho,c}}\,$
$\displaystyle:\,M^{p,q}_{(\omega)}(\mathbf{R}^{d})+W^{q,p}_{(\omega)}(\mathbf{R}^{d})$
$\displaystyle\to
M^{q,p}_{(\omega_{\varrho})}(\mathbf{R}^{d})+W^{p,q}_{(\omega_{\varrho})}(\mathbf{R}^{d})$

is continuous;

  2. (2)

if in addition $\varrho\notin\mathbf{Z}$, then the map

$\displaystyle\mathscr{F}_{\\!\varrho}=e^{-i\pi H_{x,\varrho,c}}\,$
$\displaystyle:\,M^{p,q}_{(\omega)}(\mathbf{R}^{d})+W^{q,p}_{(\omega)}(\mathbf{R}^{d})$
$\displaystyle\to
M^{q,p}_{(\omega_{\varrho})}(\mathbf{R}^{d}){\textstyle\bigcap}W^{p,q}_{(\omega_{\varrho})}(\mathbf{R}^{d})$

is continuous.

By choosing $p=q$ in (1) in previous proposition, we get Propositions 3.4 and
3.5.

###  3.3. Extensions to multiple ordered fractional Fourier transforms

By using similar arguments as in the proofs of Theorems 3.6 and 3.7, it
follows that that the following extensions hold true. Again we refer to
Subsections 1.4 and 1.7 for notations. The details are left for the reader.

######  Theorem 3.6′.

Let $\varrho\in\mathbf{R}^{d}\setminus 2\mathbf{Z}^{d}$, $c\in\mathbf{C}$,
$\omega,\omega_{\varrho}\in\mathscr{P}_{\\!A}(\mathbf{R}^{2d})$ and
$p,q\in(0,\infty]$ be such that $q\leq p$ and (3.9) hold. Then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho,c}}$ on
$\mathcal{H}_{\flat_{1}}^{\prime}(\mathbf{R}^{d})$ restrict to continuous
mappings from $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ to
$M^{q,p}_{(\omega_{\varrho})}(\mathbf{R}^{d})$, and from
$W^{q,p}_{(\omega)}(\mathbf{R}^{d})$ to
$W^{p,q}_{(\omega_{t})}(\mathbf{R}^{\varrho})$, and

$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{M^{q,p}_{(\omega_{\varrho})}}$
$\displaystyle=e^{-\frac{\pi}{4}\cdot\operatorname{Im}(c)}\|e^{-i\frac{\pi}{4}H_{x,\varrho,c}}f\|_{M^{q,p}_{(\omega_{\varrho})}}$
$\displaystyle\lesssim\big{(}{\textstyle{\prod_{j=1}^{d}}}|\sin(\textstyle{\frac{\pi\varrho_{j}}{2}})|^{\frac{1}{p}-\frac{1}{q}}\big{)}\|f\|_{M^{p,q}_{(\omega)}},$
$\displaystyle\qquad f$ $\displaystyle\in M^{p,q}_{(\omega)}(\mathbf{R}^{d})$
(3.10)′ and
$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{W^{p,q}_{(\omega_{\varrho})}}$
$\displaystyle=e^{-\frac{\pi}{4}\cdot\operatorname{Im}(c)}\|e^{-i\frac{\pi}{4}H_{x,\varrho,c}}f\|_{W^{p,q}_{(\omega_{\varrho})}}$
$\displaystyle\lesssim\big{(}{\textstyle{\prod_{j=1}^{d}}}|\sin(\textstyle{\frac{\pi\varrho_{j}}{2}})|^{\frac{1}{p}-\frac{1}{q}}\big{)}\|f\|_{W^{q,p}_{(\omega)}},$
$\displaystyle\qquad f$ $\displaystyle\in W^{q,p}_{(\omega)}(\mathbf{R}^{d}).$
(3.11)′

######  Theorem 3.7′.

Let $\varrho\in\mathbf{R}^{d}\setminus(2\mathbf{Z}d+1)^{d}$,
$\omega,\omega_{\varrho}\in\mathscr{P}_{\\!A}(\mathbf{R}^{2d})$ and
$p,q\in(0,\infty]$ be such that $q\leq p$ and (3.9) hold. Then
$\mathscr{F}_{\\!\varrho}$ and $e^{-i\frac{\pi}{4}H_{x,\varrho,c}}$ on
$\mathcal{H}_{\flat_{1}}^{\prime}(\mathbf{R}^{d})$ restrict to continuous
mappings from $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ to
$W^{p,q}_{(\omega_{\varrho})}(\mathbf{R}^{d})$, and from
$W^{q,p}_{(\omega)}(\mathbf{R}^{d})$ to
$M^{q,p}_{(\omega_{\varrho})}(\mathbf{R}^{d})$, and

$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{W^{p,q}_{(\omega_{\varrho})}}$
$\displaystyle=e^{-\frac{\pi}{4}\cdot\operatorname{Im}(c)}\|e^{-i\frac{\pi}{4}H_{x,\varrho,c}}f\|_{W^{p,q}_{(\omega_{\varrho})}}$
$\displaystyle\lesssim\big{(}{\textstyle{\prod_{j=1}^{d}}}|\cos(\textstyle{\frac{\pi\varrho_{j}}{2}})|^{\frac{1}{p}-\frac{1}{q}}\big{)}\|f\|_{M^{p,q}_{(\omega)}},$
$\displaystyle\qquad f$ $\displaystyle\in M^{p,q}_{(\omega)}(\mathbf{R}^{d})$
(3.12)′ and
$\displaystyle\|\mathscr{F}_{\\!\varrho}f\|_{M^{q,p}_{(\omega_{\varrho})}}$
$\displaystyle=e^{-\frac{\pi}{4}\cdot\operatorname{Im}(c)}\|e^{-i\frac{\pi}{4}H_{x,\varrho,c}}f\|_{M^{q,p}_{(\omega_{\varrho})}}$
$\displaystyle\lesssim\big{(}{\textstyle{\prod_{j=1}^{d}}}|\cos(\textstyle{\frac{\pi\varrho_{j}}{2}})|^{\frac{1}{p}-\frac{1}{q}}\big{)}\|f\|_{W^{q,p}_{(\omega)}},$
$\displaystyle\qquad f$ $\displaystyle\in W^{q,p}_{(\omega)}(\mathbf{R}^{d}).$
(3.13)′

##  4\. Applications to Strichartz estimates, and some further continuity
properties for certain partial differential equations

In this section we apply results from the previous sections to extend certain
Strichartz estimates in [16], with initial data in suitable Wiener amalgam
spaces. Thereafter we deduce further continuity properties for a family of
equations involving certain Schrödinger equations and heat equations.

###  4.1. Strichartz estimates for certain Schrödinger equations

We shall deduce Strichartz estimates in the framework of the operator $E$,
$S_{1}$ and $S_{2}$ in Subsection 1.8 (see (1.38)′–(1.41)′). If $T>0$, then it
follows by straight-forward estimates that $S_{1}$ and $S_{2}$ are continuous
from $C([0,T];M^{1}(\mathbf{R}^{d}))$ to
$L^{\infty}([0,T];M^{1}(\mathbf{R}^{d}))$.

By Proposition 3.5 it follows that $E$, $S_{1}$ and $S_{2}$ are uniquely
defined and continuous. In the following result we extend the operator $S_{j}$
to act between spaces of the form $L^{r}([0,T];W^{p,q}(\mathbf{R}^{d}))$.

######  Theorem 4.1.

Let $p,p_{0},q\in[1,\infty]$ and $r_{0}\in(0,\infty]$ be such that

$0\leq d\left(\frac{1}{q}-\frac{1}{p}\right)<1,\quad
d\left(\frac{1}{q}-\frac{1}{p}\right)\leq 1+\frac{1}{r_{0}}-\frac{1}{p_{0}},$
(4.1)

with strict inequalities when $q<p$ and $p_{0}=1$, or when $q<p$ and
$r_{0}=\infty$. Also let $S_{1}$ and $S_{2}$ from
$C([0,T];M^{1}(\mathbf{R}^{d}))$ to $L^{\infty}([0,T];M^{1}(\mathbf{R}^{d}))$
be given by (1.39)′ and (1.41)′, and let
$\omega\in\mathscr{P}_{\\!A,r}(\mathbf{R}^{2d})$. Then $S_{1}$ and $S_{2}$ is
uniquely extendable to a continuous mappings

$\displaystyle S_{j}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0}}([0,T];M^{p,q}_{(\omega)}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{r_{0}}([0,T];M^{q,p}_{(\omega)}(\mathbf{R}^{d})),$ (4.2) $\displaystyle
S_{j}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0}}([0,T];M^{p,q}_{(\omega)}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{r_{0}}([0,T];W^{p,q}_{(\omega)}(\mathbf{R}^{d})),$ (4.3) $\displaystyle
S_{j}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0}}([0,T];W^{q,p}_{(\omega)}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{r_{0}}([0,T];M^{q,p}_{(\omega)}(\mathbf{R}^{d})),$ (4.4) and $\displaystyle
S_{j}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0}}([0,T];W^{q,p}_{(\omega)}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{r_{0}}([0,T];W^{p,q}_{(\omega)}(\mathbf{R}^{d})),$ (4.5)

$j=1,2$.

Theorem 4.1 can also be formulated in the following way (cf. Theorem 0.5 in
the introduction).

######  Theorem 0.5′.

Let $p,p_{0},q\in[1,\infty]$ and $r_{0}\in(0,\infty]$ be such that

$0\leq d\left(\frac{1}{q}-\frac{1}{p}\right)<1,\quad
d\left(\frac{1}{q}-\frac{1}{p}\right)\leq 1+\frac{1}{r_{0}}-\frac{1}{p_{0}},$

with strict inequalities when $q<p$ and $p_{0}=1$, or when $q<p$ and
$r_{0}=\infty$. Also let $\omega\in\mathscr{P}_{\\!A,r}(\mathbf{R}^{2d})$.
Then $S_{1}$ and $S_{2}$ from $C([0,T];M^{1}(\mathbf{R}^{d}))$ to
$L^{\infty}([0,T];M^{1}(\mathbf{R}^{d}))$ are uniquely extendable to
continuous mappings

$\displaystyle S_{j}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0}}([0,T];M^{p,q}_{(\omega)}(\mathbf{R}^{d})+W^{q,p}_{(\omega)}(\mathbf{R}^{d}))$
$\displaystyle\to L^{r_{0}}([0,T];M^{q,p}_{(\omega)}(\mathbf{R}^{d})\bigcap
W^{p,q}_{(\omega)}(\mathbf{R}^{d})),$

$j=1,2$, and

$\|S_{j}F\|_{L^{r_{0}}([0,T];M^{q,p}_{(\omega)}(\mathbf{R}^{d}))}+\|S_{j}F\|_{L^{r_{0}}([0,T];W^{p,q}_{(\omega)}(\mathbf{R}^{d}))}\\\\[4.30554pt]
\lesssim\min\left(\|F\|_{L^{p_{0}}([0,T];M^{p,q}_{(\omega)}(\mathbf{R}^{d}))},\|F\|_{L^{p_{0}}([0,T];W^{q,p}_{(\omega)}(\mathbf{R}^{d}))}\right),\quad
j=1,2.$ (4.6)

We need some preparations for the proof of Theorem 4.1 and start with the
following.

######  Lemma 4.2.

Suppose that $p_{0},q_{0},r_{0}\in[1,\infty]$ satisfy

$\frac{1}{p_{0}}+\frac{1}{q_{0}}\leq 1+\frac{1}{r_{0}}\quad\text{and}\quad
q_{0}>1,$ (4.7)

with strict inequality when $q_{0}<\infty$ and $p_{0}=1$, or when
$q_{0}<\infty$ and $r_{0}=\infty$. Let $T>0$ and $\phi_{q_{0}}(t)=|\sin
t|^{-\frac{1}{q_{0}}}$ or $\phi_{q_{0}}(t)=|\cos t|^{-\frac{1}{q_{0}}}$,
$t\in\mathbf{R}$. Also let $T_{1}$ and $T_{2}$ be the mappings from $C[0,T]$
to $C[0,T]$, given by

$(T_{1}h)(t)\equiv\int_{0}^{T}\phi_{q_{0}}(t-s)h(s)\,ds\quad\text{and}\quad(T_{2}h)(t)\equiv\int_{0}^{t}\phi_{q_{0}}(t-s)h(s)\,ds,$

when $0\leq t\leq T$. Then $T_{1}$ and $T_{2}$ are uniquely extendable to
continuous mappings from $L^{p_{0}}[0,T]$ to $L^{r_{0}}[0,T]$.

###### Proof.

We only prove the assertions for $T_{1}$. The assertions for $T_{2}$ follows
by similar arguments and is left for the reader.

First suppose that $1<p_{0}$ and $r_{0}<\infty$, and let $\psi$ be a
measurable complex-valued function on $\mathbf{R}$ such that

$|\psi(t)|\lesssim|t|^{-\frac{1}{q_{0}}}.$

We observe that if $k$ is an integer, then

$\displaystyle|\phi_{q_{0}}(t)|$
$\displaystyle\lesssim|t-k\pi|^{-\frac{1}{q_{0}}}$ when
$\displaystyle\quad|t-k\pi|$ $\displaystyle\leq\frac{\pi}{2},$ or
$\displaystyle|\phi_{q_{0}}(t)|$
$\displaystyle\lesssim|t-(k+{\textstyle{\frac{1}{2}}})\pi|^{-\frac{1}{q_{0}}}$
when $\displaystyle\quad|t-(k+{\textstyle{\frac{1}{2}}})\pi|$
$\displaystyle\leq\frac{\pi}{2}.$

Since $L^{p}[0,T]$ decreases with $p$, we may assume that equality is attained
in the first inequality in (4.7). Then it follows from Lebesgue’s theorem and
Hardy-Littlewood-Sobolev inequality in [35, Theorem 4.5.3] that
$f\mapsto\psi*f$ from $C_{0}(\mathbf{R})$ to $C(\mathbf{R})$ is uniquely
extendable to a continuous map from $L^{p_{0}}(\mathbf{R})$ to
$L^{r_{0}}(\mathbf{R})$.

We may now divide $T_{1}$ into a finite sum

$T_{1}=\sum_{j=1}^{N}T_{1,N},$

where

$(T_{1,j}f)(t)=\int_{a_{j}}^{b_{j}}f(t-s)\psi_{j}(s)\,ds,$

with $\psi_{j}$ being measurable functions on $\mathbf{R}$ which satisfy

$|\psi_{j}(t)|\lesssim|t-a_{j}|^{-\frac{1}{q_{0}}}\quad\text{or}\quad|\psi_{j}(t)|\lesssim|t-b_{j}|^{-\frac{1}{q_{0}}},$

for some $a_{j}\leq b_{j}$, $j=1,\dots,N$. Since each $T_{1,j}$ is uniquely
extendable to a continuous map from $L^{p_{0}}(\mathbf{R})$ to
$L^{r_{0}}(\mathbf{R})$, in view of the previous part of the proof, the
asserted continuity assertions for $T_{1}$ follows in the case $1<p_{0}$ and
$r_{0}<\infty$.

Next suppose that $p_{0}=1$. Then $r_{0}<q_{0}$ and $q_{0}<\infty$, or
$r_{0}=q_{0}=\infty$. This implies that $\phi_{q_{0}}\in L^{r_{0}}[0,T]$. By
Minkowski’s inequality we obtain

$\|T_{1}h\|_{L^{r_{0}}[0,T]}\leq\int_{0}^{T}\|\phi_{q_{0}}(\,\cdot\,-s)\|_{L^{r_{0}}[0,T]}|h(s)|\,ds\leq
2\|\phi_{q_{0}}\|_{L^{r_{0}}[0,T]}\|h\|_{L^{1}[0,T]},$

and the result follows in the case $p_{0}=1$.

Finally suppose that $r_{0}=\infty$ and $q_{0}<\infty$. Then
$p_{0}^{\prime}<q_{0}$, giving that $\phi_{q_{0}}\in L^{p_{0}^{\prime}}[0,T]$.
Hence, Hölder’s inequality gives

$|(T_{1}h)(t)|\leq\|\phi_{q_{0}}(t-\,\cdot\,)\|_{L^{p_{0}^{\prime}}[0,T]}\|h\|_{L^{p_{0}}[0,T]}\leq
2\|\phi_{q_{0}}\|_{L^{p_{0}^{\prime}}[0,T]}\|h\|_{L^{p_{0}}[0,T]},$

and the result follows. ∎

###### Proof of Theorem 4.1.

We only prove the continuity for (4.2). The other cases follow by similar
arguments and are left for the reader.

Since $L^{r_{0}}[0,T]$ is decreasing with $r_{0}$ and that (4.1) is obviously
true for some $r_{0}\geq 1$, we may assume that $r_{0}\geq 1$. We also observe
that (4.1) implies that $q\leq p$, which makes it possible to apply Theorem
3.6.

Let $q_{0}\in\mathbf{R}\cup\\{\infty\\}$ be defined by

$\frac{1}{q_{0}}=d\left(\frac{1}{q}-\frac{1}{p}\right)$

and let $\phi_{q_{0}}$ be the same as in Lemma 4.2. Then $q_{0}\in(1,\infty]$.
A combination of Theorem 3.6 and Minkowski’s inequality gives

$\displaystyle\|(SF)(t,\,\cdot\,)\|_{M^{q,p}_{(\omega)}}$
$\displaystyle\leq\int_{0}^{T}\|(e^{-i\pi(t-s)H_{x,c}}F(s,\,\cdot\,))\|_{M^{q,p}_{(\omega)}}\,ds$
$\displaystyle\lesssim\int_{0}^{T}\phi_{q_{0}}(t-s)\|F(s,\,\cdot\,))\|_{M^{p,q}_{(\omega)}}\,ds.$

By applying the $L^{r_{0}}[0,T]$ norm on the last inequality and using Lemma
4.2 we get

$\|SF\|_{L^{r_{0}}([0,T];M^{q,p}_{(\omega)})}\lesssim\|SF\|_{L^{p_{0}}([0,T];M^{p,q}_{(\omega)})},$

and the result follows. ∎

Next we shall apply Theorems 3.6 and 4.1 to deduce continuity properties for
the operator $E$ in (1.40)′, and thereby obtain Strichartz estimates for the
harmonic oscillator propagator $e^{-itH_{x,c}}$. The first result is the
following and is a straight-forward consequence of Theorem 3.6. The details
are left for the reader. Here we let $L^{p}_{\text{w}}(\Omega)$ be the weak
$L^{p}(\Omega)$ space when $p\in(0,\infty]$, which is often denoted by
$L^{p,\infty}(\Omega)$ or $L^{p,*}(\Omega)$ in the literature (see e. g. [7]).

######  Theorem 4.3.

Let $c\in\mathbf{C}$, $\omega\in\mathscr{P}_{E}^{r}(\mathbf{R}^{2d})$ and
$p,q,r_{0}\in(0,\infty]$ be such that $q\leq p$ and

$\frac{1}{r_{0}}=d\left(\frac{1}{q}-\frac{1}{p}\right).$ (4.8)

Then $E$ in (1.40)′ is uniquely extendable to a continuous map

$E:M^{p,q}_{(\omega)}(\mathbf{R}^{d})+W^{q,p}_{(\omega)}(\mathbf{R}^{d})\to
L^{r_{0}}_{{\text{\rm{w}}}}([0,T];M^{q,p}_{(\omega)}(\mathbf{R}^{d}))\bigcap
L^{r_{0}}_{{\text{\rm{w}}}}([0,T];W^{p,q}_{(\omega)}(\mathbf{R}^{d})),$

and

$\|Ef\|_{L^{r_{0}}_{{\text{\rm{w}}}}([0,T];M^{q,p}_{(\omega)})}+\|Ef\|_{L^{r_{0}}_{{\text{\rm{w}}}}([0,T];W^{p,q}_{(\omega)})}\lesssim\|f\|_{M^{p,q}_{(\omega)}+W^{q,p}_{(\omega)}},\\\\[4.30554pt]
f\in M^{p,q}_{(\omega)}(\mathbf{R}^{d})+W^{q,p}_{(\omega)}(\mathbf{R}^{d}).$
(4.9)

Here recall that if $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are quasi-Banach
spaces, then $\mathcal{B}_{1}+\mathcal{B}_{2}$ is the quasi-Banach space

$\\{\,f_{1}+f_{2}\,;\,f_{1}\in\mathcal{B}_{1},\ f_{2}\in\mathcal{B}_{2}\,\\}$

equipped with the quasi-norm

$\|f\|_{\mathcal{B}_{1}+\mathcal{B}_{2}}\equiv\inf_{f=f_{1}+f_{2}}\left(\|f_{1}\|_{\mathcal{B}_{1}}+\|f_{2}\|_{\mathcal{B}_{2}}\right).$

######  Remark 4.4.

If $p=q$ in Theorem 4.3, then $r_{0}=\infty$ in (4.8). By Theorems 3.6 and 3.7
it follows that $E$ in (1.40)′ is uniquely extendable to a continuous map

$E:M^{p}_{(\omega)}(\mathbf{R}^{d})\to
L^{\infty}(\mathbf{R};M^{p}_{(\omega)}(\mathbf{R}^{d})),$

and

$\|Ef\|_{L^{\infty}(\mathbf{R};M^{p}_{(\omega)})}\lesssim\|f\|_{M^{p}_{(\omega)}},\qquad
f\in M^{p}_{(\omega)}(\mathbf{R}^{d}).$ (4.9)′

The next result follows by combining Theorem 4.1 with general techniques in
Section 2 in [30] for Strichartz estimates in order to deduce further
continuity properties for $E$ and $E^{*}$.

######  Theorem 4.5.

Let $p,p_{0}\in[1,\infty]$ be such that

$2\leq p<\frac{2d}{d-1}\quad\text{and}\quad
d\left(1-\frac{2}{p}\right)=\frac{2}{p_{0}^{\prime}}.$ (4.10)

Then $E^{*}$ in (1.40)′ is uniquely extendable to continuous mappings

$\displaystyle E^{*}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0}}([0,T];M^{p,p^{\prime}}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{2}(\mathbf{R}^{d}),$ (4.11) $\displaystyle E^{*}$ $\displaystyle:\,$
$\displaystyle L^{p_{0}}([0,T];W^{p^{\prime},p}(\mathbf{R}^{d}))$
$\displaystyle\to L^{2}(\mathbf{R}^{d}),$ (4.12)

and $E$ in (1.38)′ is uniquely extendable to continuous mappings

$\displaystyle E$ $\displaystyle:\,$ $\displaystyle L^{2}(\mathbf{R}^{d})$
$\displaystyle\to L^{p_{0}^{\prime}}([0,T];M^{p^{\prime},p}(\mathbf{R}^{d})),$
(4.13) $\displaystyle E$ $\displaystyle:\,$ $\displaystyle
L^{2}(\mathbf{R}^{d})$ $\displaystyle\to
L^{p_{0}^{\prime}}([0,T];W^{p,p^{\prime}}(\mathbf{R}^{d})),$ (4.14)

###### Proof.

By choosing $r_{0}=p_{0}^{\prime}$, $q=p^{\prime}$ and $\omega=1$, it follows
that (4.2) and (4.5) takes the forms

$\displaystyle S_{2}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0}}([0,T];M^{p,p^{\prime}}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{p_{0}^{\prime}}([0,T];M^{p^{\prime},p}(\mathbf{R}^{d}))$ (4.2)′ and
$\displaystyle S_{2}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0}}([0,T];W^{p^{\prime},p}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{p_{0}^{\prime}}([0,T];W^{p,p^{\prime}}(\mathbf{R}^{d})).$ (4.5)′

Since the ranks in (4.2)′ and (4.5)′ are the duals to their domains, the
result follows by straight-forward applications of the equivalences
(2.1)–(2.3) in [30]. ∎

By Theorem 4.5 and the fact that $S_{2}=E\circ E^{*}$, we get the following
(see also [30, Corollary 2.1]).

######  Theorem 4.6.

Let $p_{j},p_{0,j}\in[1,\infty]$ be such that

$2\leq p_{j}<\frac{2d}{d-1}\quad\text{and}\quad
d\left(1-\frac{2}{p_{j}}\right)=\frac{2}{p_{0,j}^{\prime}},$ (4.15)

$j=1,2$. Then $S_{2}$ in (1.41)′ is uniquely extendable to continuous mappings

$\displaystyle S_{2}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0,1}}([0,T];M^{p_{1},p_{1}^{\prime}}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{p_{0,2}^{\prime}}([0,T];M^{p_{2}^{\prime},p_{2}}(\mathbf{R}^{d})),$ (4.16)
$\displaystyle S_{2}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0,1}}([0,T];M^{p_{1},p_{1}^{\prime}}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{p_{0,2}^{\prime}}([0,T];W^{p_{2},p_{2}^{\prime}}(\mathbf{R}^{d})),$ (4.17)
$\displaystyle S_{2}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0,1}}([0,T];W^{p_{1}^{\prime},p_{1}}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{p_{0,2}^{\prime}}([0,T];M^{p_{2}^{\prime},p_{2}}(\mathbf{R}^{d})),$ (4.18)
and $\displaystyle S_{2}$ $\displaystyle:\,$ $\displaystyle
L^{p_{0,1}}([0,T];W^{p_{1}^{\prime},p_{1}}(\mathbf{R}^{d}))$ $\displaystyle\to
L^{p_{0,2}^{\prime}}([0,T];W^{p_{2},p_{2}^{\prime}}(\mathbf{R}^{d})).$ (4.19)

The same holds true with $S_{1}$ in place of $S$ at each occurrence.

######  Remark 4.7.

We observe that (4.19) is the same as (45) in [16].

######  Remark 4.8.

We observe that the conditions (4.10) and (4.15) imply that $p_{0}<2$ and
$p_{0,j}<2$.

###  4.2. Continuity properties for a family of equations related to
Schrödinger and heat equations

Next we consider more general equations, given by (1.36), where $u_{0}$ is a
suitable function or ultra-distribution on $\mathbf{R}^{d}$, $F$ is a suitable
function or ultra-distribution on $\mathbf{R}^{d+1}$ and
$R=H_{x,\varrho,c}^{r}$ for some $r\geq 0$, $\varrho\in\mathbf{C}^{d}$ and
$c\in\mathbf{C}$. That is, we consider

$\begin{cases}i\partial_{t}u-H_{x,\varrho,c}^{r}u=F,\\\\[4.30554pt]
u(0,x)=u_{0}(x),\qquad(t,x)\in[0,T]\times\mathbf{R}^{d},\end{cases}$ (4.20)

where $T>0$ is fixed. Here we observe that (4.20) is a partial differential
equation when $r$ is an integer. For general $r$, (4.20) becomes a pseudo-
differential equation.

By (1.37) it follows that the formal solution is given by

$u(t,x)=(e^{-itH_{x,\varrho,c}^{r}}u_{0})(x)-i\int_{0}^{t}(e^{i(t-s)H_{x,\varrho,c}^{r}}F(t,\,\cdot\,))(x)\,ds.$
(4.21)

Hence, questions on well-posed properties for the equation (4.20) rely
completely on continuity properties of the propagator

$\displaystyle(E_{\varrho,r,c}f)(t,x)$
$\displaystyle\equiv(e^{-itH_{x,\varrho,c}^{r}}u_{0})(x),\qquad(t,x)\in[0,T]\times\mathbf{R}^{d}.$
(4.22) as well as for the operators $\displaystyle(S_{1,\varrho,r,c}F)(t,x)$
$\displaystyle=\int_{0}^{t}(e^{-i(t-s)H_{x,\varrho,c}^{r}}F(s,\,\cdot\,))(x)\,ds,\qquad(t,x)\in[0,T]\times\mathbf{R}^{d},$
(4.23) and $\displaystyle(S_{2,\varrho,r,c}F)(t,x)$
$\displaystyle=\int_{0}^{T}(e^{-i(t-s)H_{x,\varrho,c}^{r}}F(s,\,\cdot\,))(x)\,ds,\qquad(t,x)\in[0,T]\times\mathbf{R}^{d}.$
(4.24)

Here we remark that there are different definitions of well-posed problems in
the literature. We say that the problem (4.20) is _well-posed_ if the solution
$u=u(t,x)$ depends continuously on the initial data $u_{0}$. If (4.20) fails
to be well-posed, then (4.20) is called _ill-posed_.

By Proposition 2.2 it follows that the following is true, which shows that the
operator (4.22) easily become discontinuous in the framework of classical
functions and (ultra-)distribution spaces. The details are left for the
reader.

######  Proposition 4.9.

Let $r,T>0$, $c\in\mathbf{C}$ and $\varrho\in\mathbf{C}^{d}$ be such that
$\operatorname{Im}(\varrho_{j}^{r})>0$ for some $j\in\\{1,\dots,d\\}$. Then
following is true:

  1. (1)

$E_{\varrho,r,c}$ in (4.22) is discontinuous from
$\mathscr{S}(\mathbf{R}^{d})$ to $\mathscr{S}^{\prime}(\mathbf{R}^{d})$;

  2. (2)

if in addition $r\geq 1$, then $E_{\varrho,r,c}$ in (4.22) is discontinuous
from $\mathcal{S}_{1/2}(\mathbf{R}^{d})$ to
$\mathcal{S}_{1/2}^{\prime}(\mathbf{R}^{d})$.

On the other hand, by Proposition 2.1, it follows that the mappings (4.22),
(4.23) and (4.24) are continuous on suitable Pilipović spaces, which is
explained in the following result. The details are left for the reader. Here
we let

$\displaystyle L^{1}([0,T];\mathcal{H}_{0,s}(\mathbf{R}^{d}))$
$\displaystyle=\underset{r>0}{\operatorname{proj\,lim\,}}L^{1}([0,T];\mathcal{H}_{s;r}(\mathbf{R}^{d})),$
$\displaystyle L^{1}([0,T];\mathcal{H}_{s}(\mathbf{R}^{d}))$
$\displaystyle=\underset{r>0}{\operatorname{ind\,lim\,}}L^{1}([0,T];\mathcal{H}_{s;r}(\mathbf{R}^{d})),$
$\displaystyle L^{1}([0,T];\mathcal{H}_{s}^{\prime}(\mathbf{R}^{d}))$
$\displaystyle=\underset{r>0}{\operatorname{proj\,lim\,}}L^{1}([0,T];\mathcal{H}_{s;r}^{\prime}(\mathbf{R}^{d})),$
and $\displaystyle L^{1}([0,T];\mathcal{H}_{0,s}^{\prime}(\mathbf{R}^{d}))$
$\displaystyle=\underset{r>0}{\operatorname{ind\,lim\,}}L^{1}([0,T];\mathcal{H}_{s;r}^{\prime}(\mathbf{R}^{d})),$

where $\mathcal{H}_{s;r}(\mathbf{R}^{d})$ and
$\mathcal{H}_{s;r}^{\prime}(\mathbf{R}^{d})$ are the images of
$\ell_{s;r}^{\infty}(\mathbf{N}^{d})$ and
$\ell_{s;r}^{\infty,*}(\mathbf{N}^{d})$, respectively under the map
$T_{\mathcal{H}}$ in (1.11), also in topological sense. (See also Definition
1.1.)

######  Theorem 4.10.

Let $r,T>0$, $c\in\mathbf{C}$, $\varrho\in\mathbf{C}^{d}$ and let
$s,s_{1},s_{2}\in\overline{\mathbf{R}_{\flat}}$ be such that
$0<s_{1}\leq\frac{1}{2r}$ and $s_{2}<\frac{1}{2r}$. Then following is true:

  1. (1)

$E_{\varrho,r,c}$ in (4.22) is a homeomorphism on the spaces in (2.6);

  2. (2)

$S_{j,\varrho,r,c}$ in (4.23) and (4.24) are homeomorphisms on the spaces

$\displaystyle L^{1}([0,T];\mathcal{H}_{0,s_{1}}(\mathbf{R}^{d})),$
$\displaystyle L^{1}([0,T];\mathcal{H}_{s_{2}}(\mathbf{R}^{d})),$ (4.25)
$\displaystyle L^{1}([0,T];\mathcal{H}_{s_{2}}^{\prime}(\mathbf{R}^{d}))$ and
$\displaystyle L^{1}([0,T];\mathcal{H}_{0,s_{1}}^{\prime}(\mathbf{R}^{d})).$

As consequences of Proposition 4.9 and Theorem 4.10 we get the following,
concerning well-posed properties of the equation (4.20).

######  Corollary 4.11.

Suppose that $r\geq 1$, $c\in\mathbf{C}$ and $\varrho\in\mathbf{C}^{d}$ are
such that $\operatorname{Im}(\varrho_{j}^{r})>0$ for some
$j\in\\{1,\dots,d\\}$. Then the following is true:

  1. (1)

the equation (4.20) is ill-posed in the framework of Schwartz functions,
Gelfand-Shilov spaces, and their dual spaces;

  2. (2)

the equation (4.20) is well-posed for the Pilipović spaces and their dual
spaces in (2.6).

######  Remark 4.12.

If $c=0$, $r=1$, $F=0$ and $\varrho_{j}=i$ for every $j$ in Corollary 4.11,
then (4.20) takes the form

$\begin{cases}\partial_{t}u=-\Delta_{x}+|x|^{2},\\\\[4.30554pt]
u(0,x)=u_{0}(x),\qquad(t,x)\in[0,T]\times\mathbf{R}^{d}.\end{cases}$ (4.20)′

That is we obtain a sort of heat equation, where the potential term $|x|^{2}$
is included. The minus sign in front of the Laplace operator $\Delta_{x}$
implies that we are searching for a solution when moving backwards in time.

Corollary 4.11 then shows that it is _not meaningful_ to investigate (4.20) in
the framework of Schwartz spaces, Gelfand-Shilov spaces and their distribution
spaces. On the other hand, it follows from the same corollary that it always
makes sense to investigate such problems in the framework of Pilipović spaces
which are not Gelfand-Shilov spaces, and their distributions.

###  4.3. Some further applications and remarks

A question which appears is whether our results are applicable to problems
like

$\begin{cases}\partial_{t}u-\operatorname{Op}^{w}(a)u=F,\\\\[4.30554pt]
u(0,x)=u_{0}(x),\qquad(t,x)\in[0,T]\times\mathbf{R}^{d},\end{cases}$ (4.26)

when $a$ is a positive definite quadratic form on $\mathbf{R}^{2d}$. Here
$\operatorname{Op}^{w}(a)$ is the _Weyl quantization_ of $a$, i. e. the
operator on $\mathscr{S}(\mathbf{R}^{d})$, given by

$\operatorname{Op}^{w}(a)f(x)=(2\pi)^{-d}\iint_{\mathbf{R}^{2d}}a({\textstyle{\frac{1}{2}}}(x+y),\xi)f(y)e^{i\langle
x-y,\xi\rangle}\,dyd\xi.$

The operator $\operatorname{Op}^{w}(a)$ is continuous on
$\mathscr{S}(\mathbf{R}^{d})$ and on any Pilipović space, which extends
uniquely to a continuous map on $\mathscr{S}^{\prime}(\mathbf{R}^{d})$, and to
any Pilipović distribution space. (See e. g. [35, 54].)

By introducing suitable new symplectic coordinates it follows that
$\operatorname{Op}^{w}(a)$ takes the form

$\operatorname{Op}^{w}(a)=\sum_{j=1}^{d}\varrho_{j}(x_{j}^{2}-\partial_{x_{j}}^{2})+c,$

for some $\varrho_{j}>0$, $j=1,\dots,d$, and some real constant $c$, in these
new coordinates. (See Section 18.6 in [35].)

By Proposition 4.9 and Theorem 4.10 it follows that (4.26) is ill-posed in the
framework of Schwartz spaces, Gelfand-Shilov spaces and their distribution
spaces, but well-posed for other types of Pilipović spaces of functions and
distributions. Here we notice that we need to keep staying in these new
symplectic coordinates, since Pilipović spaces which are not Gelfand-Shilov
spaces, are not invariant under general changes of symplectic coordinates.
(See e. g. [53].)

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