# Local smoothing and Hardy spaces for Fourier integral operators

Jan Rozendaal  Institute of Mathematics, Polish Academy of Sciences
ul. Śniadeckich 8
00-656 Warsaw
Poland<EMAIL_ADDRESS>
###### Abstract.

We show that the Hardy spaces for Fourier integral operators form natural
spaces of initial data when applying $\ell^{p}$-decoupling inequalities to
local smoothing for the wave equation. This yields new local smoothing
estimates which, in a quantified manner, improve the bounds in the local
smoothing conjecture on $\mathbb{R}^{n}$ for $p\geq 2(n+1)/(n-1)$, and
complement them for $2<p<2(n+1)/(n-1)$. These estimates are invariant under
application of Fourier integral operators, and they are essentially sharp.

###### Key words and phrases:

Local smoothing, wave equation, Hardy space for Fourier integral operators

###### 2020 Mathematics Subject Classification:

Primary 42B37. Secondary 35L05, 42B35, 35S30.

This research was supported by grant DP160100941 of the Australian Research
Council, and by NCN grant UMO2017/27/B/ST1/00078. The research leading to
these results has received funding from the Norwegian Financial Mechanism
2014-2021, grant 2020/37/K/ST1/02765.

##  1\. Introduction

This article concerns the local smoothing conjecture for the Euclidean wave
equation on $\mathbb{R}^{n}$. The phenomenon of local smoothing revolves
around determining, for each $1<p<\infty$, the minimal $s\in\mathbb{R}$ for
which there exists a $C\geq 0$ such that

(1.1)
$\Big{(}\int_{0}^{1}\|e^{it\sqrt{-\Delta}}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\leq
C\|f\|_{W^{s,p}(\mathbb{R}^{n})}$

for all $f\in W^{s,p}(\mathbb{R}^{n})$. The optimal fixed-time estimates for
the half-wave propagator $e^{it\sqrt{-\Delta}}$ from [16, 15] imply that one
may choose $s=2s(p)$, where we write

$s(p):=\frac{n-1}{2}\Big{|}\frac{1}{2}-\frac{1}{p}\Big{|}$

throughout. This estimate is sharp for $1<p\leq 2$, but it can be improved for
$p>2$ and $n\geq 2$. In fact, the local smoothing conjecture, as originally
formulated by Sogge in [24], stipulates that (1.1) should hold with
$s=\sigma(p)+\varepsilon$ for each $\varepsilon>0$, where $\sigma(p)=0$ for
$2<p\leq 2n/(n-1)$, and $\sigma(p)=2s(p)-1/p$ for $p>2n/(n-1)$.

Although the local smoothing conjecture is still open in full generality,
there are many partial results available, and the conjecture was proved by
Guth, Wang and Zhang [9] for $n=2$. For the purposes of the present article,
however, it is relevant to highlight the article [3] by Bourgain and Demeter.
Building on work by Wolff [27], they showed that (1.1) holds with
$s=\sigma(p)+\varepsilon$ for $p\geq 2(n+1)/(n-1)$, by proving the
$\ell^{p}$-decoupling inequality. More precisely, one has

(1.2)
$\Big{(}\int_{0}^{1}\|e^{it\sqrt{-\Delta}}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\leq
C\|f\|_{W^{d(p)+\varepsilon,p}(\mathbb{R}^{n})},$

where

$d(p):=\begin{cases}2s(p)-\frac{1}{p}&\text{if }p\geq\frac{2(n+1)}{n-1},\\\
s(p)&\text{if }2\leq p<\frac{2(n+1)}{n-1}.\end{cases}$

### Main results

The local smoothing conjecture is sharp, in the sense that (1.1) does not hold
for $s<\sigma(p)$ (see e.g. [2, Remark 2.6]). Nonetheless, in this article we
obtain improved local smoothing estimates for $p\geq 2(n+1)/(n-1)$, by working
with a different space of initial data in (1.2). Our main result, formulated
in terms of the Sobolev spaces
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})={\langle}D\rangle^{-s}\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$
over the Hardy spaces for Fourier integral operators
$\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ (see Definition 2.1), is as follows.

######  Theorem 1.1.

Let $p\in(2,\infty)$ and $\varepsilon>0$. Then there exists a $C>0$ such that

(1.3)
$\Big{(}\int_{0}^{1}\|e^{it\sqrt{-\Delta}}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\leq
C\|f\|_{\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})}$

for all $f\in\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})$.

Theorem 1.1 is a special case of Theorem 4.4, which considers the more general
class of operators of the form $e^{it\phi(D)}$, where $\phi\in
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ is positively homogeneous of
degree $1$ and such that $\mathrm{rank}(\partial_{\xi\xi}^{2}\phi)\equiv n-1$.
Moreover, the exponent $d(p)-s(p)$ in (1.3) is sharp, as is shown in Theorem
5.3.

The Hardy space $\mathcal{H}^{1}_{FIO}(\mathbb{R}^{n})$ for Fourier integral
operators (FIOs) was introduced by Smith in [23], and decades later his
construction was extended to a scale
$(\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n}))_{1\leq p\leq\infty}$ of invariant
spaces for FIOs by Hassell, Portal and the author in [10]. More precisely,
$\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ is invariant under FIOs of order zero
which have a compactly supported Schwartz kernel and are associated with a
local canonical graph, and one has

(1.4)
$W^{s(p),p}(\mathbb{R}^{n})\subseteq\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})\subseteq
W^{-s(p),p}(\mathbb{R}^{n})$

for all $1<p<\infty$ and $s\in\mathbb{R}$, with the natural modifications
involving the local Hardy space $\mathcal{H}^{1}(\mathbb{R}^{n})$ for $p=1$,
and $\mathrm{bmo}(\mathbb{R}^{n})$ for $p=\infty$. In particular,

$W^{d(p)+\varepsilon,p}(\mathbb{R}^{n})\subseteq\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})\subseteq
W^{d(p)-2s(p)+\varepsilon,p}(\mathbb{R}^{n})$

for $2<p<\infty$, and (1.3) recovers (1.2). However, since the exponents in
(1.4) are sharp, (1.3) is in fact a strict improvement of (1.2). And since
$d(p)=\sigma(p)$ for $p\geq 2(n+1)/(n-1)$, Theorem 1.1 improves upon the local
smoothing conjecture for such $p$.

Note that $d(p)-s(p)=\max(0,s(p)-1/p)$,

$d(p)-s(p)=0=\sigma(p)\quad\text{for}\quad 2<p\leq 2\frac{n}{n-1},$

and

$d(p)-s(p)=0<2s(p)-\frac{1}{p}=\sigma(p)<d(p)\quad\text{for}\quad
2\frac{n}{n-1}<p<2\frac{n+1}{n-1}.$

Hence the sharpness of the embeddings in (1.4) implies that, for
$2<p<2(n+1)/(n-1)$, (1.3) neither follows from the local smoothing conjecture
nor implies it.

It was shown in [12] that the $\varepsilon$ loss in (1.2) can be removed for
$n\geq 4$ and $p>2+\frac{4}{n-3}>2\frac{n+1}{n-1}$. It is not clear whether
one may also let $\varepsilon=0$ in (1.3) for certain $p$. For
$p>2+\frac{4}{n-3}$, by the sharpness of the exponents in (1.4), Theorem 1.1
neither follows from the local smoothing result in [12] nor implies it.

### Applications

The invariance of $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ under FIOs and
related operators shows that (1.3) and (1.2) are invariant under application
of such operators to the initial data. Corollary 4.5 collects a few instances
of this paradigm.

In particular, the Cauchy problem associated to the half-wave equation
$\partial_{t}u(t)=i\sqrt{-\Delta}u(t)$ is well posed on
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$. Hence in Theorem 1.1 one in fact
has

$[t\mapsto e^{it\sqrt{-\Delta}}f]\in
C(\mathbb{R};\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n}))\cap
L^{p}_{{\mathrm{loc}}}(\mathbb{R};L^{p}(\mathbb{R}^{n}))$

for all $f\in\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})$.
Here it is relevant to note that, since
$(e^{it\sqrt{-\Delta}})_{t\in\mathbb{R}}$ is a group, one has
$e^{it\sqrt{-\Delta}}f\in\mathcal{H}^{r,p}_{FIO}(\mathbb{R}^{n})$ for some
$t\in\mathbb{R}$ if and only if this holds for all $t\in\mathbb{R}$, and no
smoothing can occur. Instead, the half-wave group has a smoothing effect on
initial data with respect to the Sobolev embeddings in (1.4).

Local smoothing estimates have applications to various problems in harmonic
and geometric analysis, such as spherical maximal theorems (see [2, 25]). It
would be of interest to examine applications of the results in this article to
these areas.

In [21], Schippa obtained local smoothing estimates for the Schrödinger
equation using invariant spaces for Schrödinger propagators, the modulation
spaces. He then used those estimates to prove new local and global well-
posedness results for the nonlinear Schrödinger equation. Very recently, in
[20], Schippa and the author applied this approach to function spaces related
to the Hardy spaces for FIOs, to prove new local and global well-posedness
results for nonlinear wave equations.

### Idea behind the proof of Theorem 1.1

The proof exploits the full strength of the $\ell^{p}$-decoupling inequality,
a corollary of the $\ell^{2}$-decoupling inequality from [3]. This inequality
bounds, for a suitable $g\in\mathcal{S}(\mathbb{R})$ and for initial data $f$
with frequency support in a dyadic annulus, the $L^{p}(\mathbb{R}^{n+1})$ norm
of $g(t)e^{it\sqrt{-\Delta}}f(x)$ by an $\ell^{p}$ sum involving a dyadic-
parabolic decomposition of the frequency support of $f$. This dyadic-parabolic
decomposition goes back to [8] and plays a key role in the proof of the
optimal $L^{p}$ regularity of Fourier integral operators [22]. On each dyadic
annulus one makes a different discrete decomposition, and the right-hand side
of the decoupling inequality involves a different _decoupling norm_ on each
annulus.

By contrast, the Hardy spaces for FIOs involve a continuous dyadic-parabolic
decomposition which is the same on each frequency annulus, and one observation
in this article is that the discrete and continuous decompositions are
equivalent on dyadic annuli (see Proposition 4.1). It then follows from the
fact that the half-wave group is polynomially bounded on the Hardy spaces for
FIOs that, when applied to the function $g(t)e^{it\sqrt{-\Delta}}f(x)$ in the
setting of Theorem 1.1, the right-hand side of the $\ell^{p}$-decoupling
inequality is equivalent to the
$\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})$ norm of $f$. In
fact, Theorem 1.1 is a consequence of Theorem 4.4, which is proved in the same
manner but relies on a more general class of $\ell^{p}$-decoupling
inequalities, from [4]. As is shown in Corollary 4.2, the Hardy spaces for
FIOs are adapted to each of these decoupling inequalities.

Hence, at least when restricted to dyadic frequency annuli, the Hardy spaces
for FIOs form the _largest_ space of initial data for which one can obtain
local smoothing estimates when applying $\ell^{p}$-decoupling inequalities in
the manner in which they are typically used. Moreover, since the exponent
$d(p)-s(p)$ in (1.3) is sharp, for $2<p<2(n+1)/(n-1)$ one cannot improve upon
the local smoothing conjecture by using the Hardy spaces for FIOs as spaces of
initial data. Here it is relevant to note that the example which shows that
the exponent $d(p)-s(p)$ is sharp involves, for $2<p<2(n+1)/(n-1)$, a square
function which was used in [9] to prove the full local smoothing conjecture
for $n=2$ (see Lemma 5.2). On the other hand, while the local smoothing
conjecture is still open for $n\geq 3$, this article contains the essentially
optimal local smoothing estimates with the Hardy spaces for FIOs as spaces of
initial data.

### Organization

Section 2 contains background on the Hardy spaces for FIOs, and Section 3
concerns basic properties of FIOs. In Section 4 we prove our main result, and
in Section 5 we show that this result is essentially sharp.

### Notation

The natural numbers are ${{\mathbb{N}}}=\\{1,2,\ldots\\}$, and
${\mathbb{Z}}_{+}:={{\mathbb{N}}}\cup\\{0\\}$. Throughout this article we fix
$n\in{{\mathbb{N}}}$ with $n\geq 2$.

For $\xi\in\mathbb{R}^{n}$ we write ${\langle}\xi\rangle=(1+|\xi|^{2})^{1/2}$,
and $\hat{\xi}=\xi/|\xi|$ if $\xi\neq 0$. We use multi-index notation, where
$\partial_{\xi}=(\partial_{\xi_{1}},\ldots,\partial_{\xi_{n}})$ and
$\partial^{\alpha}_{\xi}=\partial^{\alpha_{1}}_{\xi_{1}}\ldots\partial^{\alpha_{n}}_{\xi_{n}}$
for $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in{\mathbb{Z}}_{+}^{n}$. Moreover,
$\partial_{x\eta}^{2}\Phi$ is the mixed Hessian of a function $\Phi$ of the
variables $x$ and $\eta$. The Fourier transform of
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ is denoted by $\mathcal{F}f$ or
$\widehat{f}$, and the Fourier multiplier with symbol
$\varphi\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ is denoted by $\varphi(D)$.

We write $f(s)\lesssim g(s)$ to indicate that $f(s)\leq Cg(s)$ for all $s$ and
a constant $C>0$ independent of $s$, and similarly for $f(s)\gtrsim g(s)$ and
$g(s)\eqsim f(s)$.

##  2\. Hardy spaces for Fourier integral operators

In this section we first introduce the Hardy spaces for Fourier integral
operators, and then we collect a few of their basic properties.

###  2.1. Definitions

Fix a non-negative radial $\varphi\in C^{\infty}_{c}(\mathbb{R}^{n})$ such
that $\varphi(\xi)=0$ for $|\xi|>1$, and $\varphi\equiv 1$ in a neighborhood
of zero. For $\omega\in S^{n-1}$, $\sigma>0$ and
$\xi\in\mathbb{R}^{n}\setminus\\{0\\}$, set
$\varphi_{\omega,\sigma}(\xi):=c_{\sigma}\varphi(\tfrac{\hat{\xi}-\omega}{\sqrt{\sigma}})$,
where
$c_{\sigma}:=(\int_{S^{n-1}}\varphi(\frac{e_{1}-\nu}{\sqrt{\sigma}})^{2}\mathrm{d}\nu)^{-1/2}$
for $e_{1}=(1,0,\ldots,0)$ the first basis vector of $\mathbb{R}^{n}$ (this
choice is irrelevant). Here and throughout, $\mathrm{d}\nu$ is the normalized
surface measure on the unit sphere $S^{n-1}$. Also set
$\varphi_{\omega,\sigma}(0):=0$. Let $\Psi\in C^{\infty}_{c}(\mathbb{R}^{n})$
be non-negative, radial, such that $\Psi(\xi)=0$ if $|\xi|\notin[1/2,2]$, and
such that
$\int_{0}^{\infty}\Psi(\sigma\xi)^{2}\frac{\mathrm{d}\sigma}{\sigma}=1$ for
all $\xi\neq 0$. Now, for $\omega\in S^{n-1}$ and $\xi\in\mathbb{R}^{n}$, set

$\varphi_{\omega}(\xi):=\int_{0}^{4}\Psi(\sigma\xi)\varphi_{\omega,\sigma}(\xi)\frac{\mathrm{d}\sigma}{\sigma}.$

Some properties of these functions are as follows (see [18, Remark 3.3]):

  1. (1)

For all $\omega\in S^{n-1}$ and $\xi\neq 0$ one has

(2.1) $\varphi_{\omega}(\xi)=0\text{ if }|\xi|<\tfrac{1}{8}\text{ or
}|\hat{\xi}-\omega|>2|\xi|^{-1/2}.$

  2. (2)

For all $\alpha\in{\mathbb{Z}}_{+}^{n}$ and $\beta\in{\mathbb{Z}}_{+}$ there
exists a $C_{\alpha,\beta}\geq 0$ such that

$|(\omega\cdot\partial_{\xi})^{\beta}\partial^{\alpha}_{\xi}\varphi_{\omega}(\xi)|\leq
C_{\alpha,\beta}|\xi|^{\frac{n-1}{4}-\frac{|\alpha|}{2}-\beta}$

for all $\omega\in S^{n-1}$ and $\xi\neq 0$.

  3. (3)

For all $\alpha\in{\mathbb{Z}}_{+}^{n}$ there exists a $C_{\alpha}\geq 0$ such
that

$\Big{|}\partial_{\xi}^{\alpha}\Big{(}\int_{S^{n-1}}\varphi_{\omega}(\xi)\mathrm{d}\omega\Big{)}^{-1}\Big{|}\leq
C_{\alpha}|\xi|^{\frac{n-1}{4}-|\alpha|}$

for all $\xi\in\mathbb{R}^{n}$ with $|\xi|\geq 1/2$. Hence there exists a
radial $m\in S^{(n-1)/4}(\mathbb{R}^{n})$ such that if
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ satisfies
$\mathrm{supp}(\widehat{f}\,)\subseteq\\{\xi\in\mathbb{R}^{n}\mid|\xi|\geq
1/2\\}$, then

$f=\int_{S^{n-1}}m(D)\varphi_{\nu}(D)f\mathrm{d}\nu.$

In (3), $S^{(n-1)/4}(\mathbb{R}^{n})$ is the standard class of symbols $a$ of
order $(n-1)/4$. That is, $a\in C^{\infty}(\mathbb{R}^{n})$, and for all
$\alpha\in{\mathbb{Z}}_{+}^{n}$ there exists a $C_{\alpha}\geq 0$ such that

$|\partial_{\eta}^{\alpha}a(\eta)|\leq
C_{\alpha}{\langle}\eta\rangle^{(n-1)/4-|\alpha|}$

for all $\eta\in\mathbb{R}^{n}$.

Throughout, for simplicity of notation, we write
$\mathcal{H}^{p}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$ for $1<p<\infty$, and
$\mathcal{H}^{\infty}(\mathbb{R}^{n})=\mathrm{bmo}(\mathbb{R}^{n})$. Moreover,
$\mathcal{H}^{1}(\mathbb{R}^{n})$ is the classical local Hardy space. We also
fix a $q\in C^{\infty}_{c}(\mathbb{R}^{n})$ such that $q(\xi)=1$ for
$|\xi|\leq 2$.

######  Definition 2.1.

For $p\in[1,\infty)$, $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ consists of
those $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that $q(D)f\in
L^{p}(\mathbb{R}^{n})$,
$\varphi_{\omega}(D)f\in\mathcal{H}^{p}(\mathbb{R}^{n})$ for almost all
$\omega\in S^{n-1}$, and

$\Big{(}\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{\mathcal{H}^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}\omega\Big{)}^{1/p}<\infty,$

endowed with the norm

$\|f\|_{\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})}:=\|q(D)f\|_{L^{p}(\mathbb{R}^{n})}+\Big{(}\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{\mathcal{H}^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}\omega\Big{)}^{1/p}.$

Moreover,
$\mathcal{H}^{\infty}_{FIO}(\mathbb{R}^{n}):=(\mathcal{H}^{1}_{FIO}(\mathbb{R}^{n}))^{*}$.
For $p\in[1,\infty]$ and $s\in\mathbb{R}$,
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ consists of all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that
${\langle}D\rangle^{s}f\in\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, endowed with
the norm

$\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}:=\|{\langle}D\rangle^{s}f\|_{\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})}.$

In fact, this is not how the Hardy spaces for FIOs were originally defined in
[23] and [10], namely in terms of conical square functions and tent spaces
over the cosphere bundle. That these definitions are equivalent was shown in
[7, 18].

###  2.2. Properties

First recall the Sobolev embeddings from (1.4). By [10, Theorem 7.4], these in
fact hold for all $p\in[1,\infty]$, upon replacing $W^{s,p}(\mathbb{R}^{n})$
by ${\langle}D\rangle^{-s}\mathcal{H}^{p}(\mathbb{R}^{n})$.

The Hardy spaces for FIOs form a complex interpolation scale. That is, let
$p_{0},p_{1},p\in[1,\infty]$, $s_{0},s_{1},s\in\mathbb{R}$ and
$\theta\in[0,1]$ be such that
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$ and
$s=(1-\theta)s_{0}+\theta s_{1}$. Then, as follows from [10, Proposition 6.7],
one has

$[\mathcal{H}^{s_{0},p_{0}}_{FIO}(\mathbb{R}^{n}),\mathcal{H}^{s_{1},p_{1}}_{FIO}(\mathbb{R}^{n})]_{\theta}=\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n}).$

Moreover, by [10, Proposition 6.8], for all $p\in[1,\infty)$ and
$s\in\mathbb{R}$ one has

$(\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n}))^{*}=\mathcal{H}^{-s,p^{\prime}}_{FIO}(\mathbb{R}^{n}).$

Here the duality pairing is the standard distributional pairing
${\langle}f,g\rangle$ for
$f\in\mathcal{S}(\mathbb{R}^{n})\subseteq\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$
and
$g\in\mathcal{H}^{-s,p^{\prime}}_{FIO}(\mathbb{R}^{n})\subseteq\mathcal{S}^{\prime}(\mathbb{R}^{n})$.
By [10, Proposition 6.6], the Schwartz functions lie dense in
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$, so this uniquely determines the
duality pairing.

The following lemma concerns a key feature of the Hardy spaces for FIOs: the
relationship between the $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ norm and the
$\mathcal{H}^{p}(\mathbb{R}^{n})$ norm of a function with frequency support in
a dyadic-parabolic region.

######  Lemma 2.2.

Let $p\in[1,\infty]$ and $s\in\mathbb{R}$. Then there exists a $C>0$ such that
the following statements hold for all
$f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ such that

(2.2)
$\mathrm{supp}(\widehat{f}\,)\subseteq\\{\xi\in\mathbb{R}^{n}\mid|\xi|\in[R/2,2R],|\hat{\xi}-\nu|\leq
2R^{-1/2}\\}$

for some $R\geq 1$ and $\nu\in S^{n-1}$:

  1. (1)

If $p\leq 2$, then

$\frac{1}{C}R^{s-s(p)}\|f\|_{L^{p}(\mathbb{R}^{n})}\leq\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\leq
CR^{s-s(p)}\|f\|_{L^{p}(\mathbb{R}^{n})}.$

  2. (2)

If $p>2$, then

$\frac{1}{C}R^{s+s(p)}\|f\|_{L^{p}(\mathbb{R}^{n})}\leq\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\leq
CR^{s+s(p)}\|f\|_{L^{p}(\mathbb{R}^{n})}.$

###### Proof.

Since $f$ has frequency support in a dyadic annulus, one has

$\|{\langle}D\rangle^{s}f\|_{\mathcal{H}^{p}(\mathbb{R}^{n})}\eqsim
R^{s}\|f\|_{\mathcal{H}^{p}(\mathbb{R}^{n})}\eqsim
R^{s}\|f\|_{L^{p}(\mathbb{R}^{n})}.$

Hence the statement is a direct consequence of [7, Proposition 6.4]. ∎

##  3\. Fourier integral operators

For the general theory of Fourier integral operators, and the associated
notions from symplectic geometry, we refer to [13, 6, 25]. To prove Theorem
1.1 we will only use Corollary 3.6, concerning the operators $e^{it\phi(D)}$.
However, these operators are part of a larger class of Fourier integral
operators, introduced in Definition 3.1, which are treated by the same
methods. One of the key properties of the Hardy spaces for FIOs is their
invariance under such operators.

###  3.1. Definitions

For $m\in\mathbb{R}$, recall that $S^{m}_{1,0}$ consists of all $a\in
C^{\infty}(\mathbb{R}^{2n})$ such that

(3.1)
$\|a\|_{\alpha,\beta}:=\sup_{(x,\eta)\in\mathbb{R}^{2n}}{\langle}\eta\rangle^{-m+|\alpha|}|\partial_{x}^{\beta}\partial_{\eta}^{\alpha}a(x,\eta)|<\infty$

for all $\alpha,\beta\in{\mathbb{Z}}_{+}^{n}$. In [10] a larger symbol class
was considered, the elements of which behave in an anisotropic manner which
matches the dyadic-parabolic decomposition of
$\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$. This class, denoted by
$S^{m}_{1/2,1/2,1}$, contains Hörmander’s $S^{m}_{1,1/2}$ class but is
strictly contained in $S^{m}_{1/2,1/2}$. The statements in Proposition 3.3 (2)
and Lemma 3.5 below in fact extend to this larger symbol class.

In the following definition, a special case of [10, Definition 2.11], and
throughout, $o:=\mathbb{R}^{n}\times\\{0\\}$ denotes the zero section.

######  Definition 3.1.

Let $m\in\mathbb{R}$, $a\in S^{m}_{1,0}$ and $\Phi\in
C^{\infty}(\mathbb{R}^{2n}\setminus o)$, and set

(3.2)
$Tf(x):=\int_{\mathbb{R}^{n}}e^{i\Phi(x,\eta)}a(x,\eta)\widehat{f}(\eta)\mathrm{d}\eta$

for $f\in\mathcal{S}(\mathbb{R}^{n})$ and $x\in\mathbb{R}^{n}$. Then $T$ is a
Fourier integral operator of order $m$ in _standard form_ , associated with a
_global canonical graph_ , if:

  1. (1)

$\Phi$ is real-valued and positively homogeneous of degree $1$ in the
$\eta$-variable;

  2. (2)

$\sup_{(x,\eta)\in\mathbb{R}^{2n}\setminus
o}|\partial_{x}^{\beta}\partial_{\eta}^{\alpha}\Phi(x,\hat{\eta})|<\infty$ for
all $\alpha,\beta\in{\mathbb{Z}}_{+}^{n}$ with $|\alpha|+|\beta|\geq 2$;

  3. (3)

$\inf_{(x,\eta)\in\mathbb{R}^{2n}\setminus
o}|\det\partial^{2}_{x\eta}\Phi(x,\eta)|>0$;

  4. (4)

For each $x\in\mathbb{R}^{n}$, $\eta\mapsto\partial_{x}\Phi(x,\eta)$ is a
bijection on $\mathbb{R}^{n}\setminus\\{0\\}$.

######  Remark 3.2.

By Hadamard’s global inverse function theorem [14, Theorem 6.2.8], condition
(4) is superfluous for $n\geq 3$. Moreover, (4) holds if
$\Phi(x,\eta)=x\cdot\eta+\phi(\eta)$ for some $\phi\in
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ which is positively homogeneous of
degree $1$.

If (4) holds, then it follows from the global inverse function theorem that
the map
$(\partial_{\eta}\Phi(x,\eta),\eta)\mapsto(x,\partial_{x}\Phi(x,\eta))$ is a
homogeneous canonical transformation on $\mathbb{R}^{2n}\setminus o$, and the
canonical relation of $T$ is the graph of this transformation.

Recall that a Fourier integral operator of order $m$, associated with a local
canonical graph and having a compact Schwartz kernel, can, modulo an operator
with a Schwartz kernel which is a Schwartz function, be expressed as a finite
sum of operators which in appropriate coordinate systems are as in (3.2),
where the symbol $a$ has compact support in the $x$-variable (see e.g. [25,
Proposition 6.2.4]). In this case, (2) is automatically satisfied, (3) holds
on the support of $a$, and the map in (4) is a locally defined homogeneous
canonical transformation. By contrast, in Definition 3.1, the symbols are not
required to have compact spatial support, but the conditions on the phase
function are required to hold on all of $\mathbb{R}^{2n}\setminus o$.

###  3.2. Properties

The following proposition lists some classes of operators which are bounded on
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ but in general not on
$W^{s,p}(\mathbb{R}^{n})$.

######  Proposition 3.3.

Let $p\in[1,\infty]$, $s\in\mathbb{R}$ and $T$ satisfy one of the following
conditions:

  1. (1)

$T$ is a Fourier integral operator of order zero, associated with a local
canonical graph, with a compactly supported Schwartz kernel;

  2. (2)

$T$ is a Fourier integral operator of order zero in standard form, associated
with a global canonical graph;

  3. (3)

$T=\cos(t_{0}\sqrt{L})$, for $t_{0}\in\mathbb{R}$ and $L$ an elliptic
divergence-form operator with bounded real-valued $C^{1,1}(\mathbb{R}^{n})$
coefficients. Moreover, one has $1<p<\infty$, $2s(p)<1$, and $-2+s(p)\leq
s\leq 2-s(p)$.

Then
$T:\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})\to\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$
is bounded.

###### Proof.

(1): This follows in the same way as the case $s=0$ from [10]; we will use the
same terminology as there. By [10, Proposition 2.14], there exist an
$m\in{{\mathbb{N}}}$, normal oscillatory integral operators
$(T_{j,1})_{j=1}^{m}$ and $(T_{j,2})_{j=1}^{m}$, and a smoothing operator $R$,
such that $T=\sum_{j=1}^{m}T_{j,1}T_{j,2}+R$. As in the proof of [10,
Corollary 5.4], when conjugated with a suitable isometric wave packet
transform $W:L^{2}(\mathbb{R}^{n})\to L^{2}(T^{*}\mathbb{R}^{n})$, the
$T_{j,1}$ and $T_{j,2}$ can each be written as a sum of an operator which
satisfies off-singularity bounds and a residual operator. The smoothing
operator $R$ yields a residual operator $WRW^{*}$ as well, by [10, Proposition
5.3]. All these operators are bounded on tent spaces over the cosphere bundle,
by [10, Theorem 3.7 and Proposition 3.6], but for $s\neq 0$ it is important to
note that they are also bounded on the weighted tent spaces
$T^{p}_{s}(S^{*}\mathbb{R}^{n})$ used in [11]. This is shown in the same way
as [11, Proposition 2.4], where the canonical relations under consideration
are canonical graphs on all of $\mathbb{R}^{2n}\setminus o$, which plays no
role in that proof. Moreover, [11, Proposition 3.4] says that
$\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\eqsim\|Wf\|_{T^{p}_{s}(S^{*}\mathbb{R}^{n})}$.
It follows from this that the $T_{j,1}$, $T_{j,2}$ and $R$ are all bounded on
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$, and therefore so is $T$.

(2): Let $q^{\prime}\in C^{\infty}_{c}(\mathbb{R}^{n})$ equal $1$ near zero,
and set $T_{1}:=T(1-q^{\prime})(D)$ and $T_{2}:=Tq^{\prime}(D)$. As in (1),
$WT_{1}W^{*}$ satisfies off-singularity bounds, by [10, Corollary 5.2]. Then
the same arguments as above show that $T_{1}$ is bounded on
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$.

On the other hand, for the low-frequency term $T_{2}$, one can integrate by
parts and apply Young’s inequality as in the proof of [5, Theorem 1.18] to
show that $\partial_{x}^{\alpha}T_{2}$ is bounded on $L^{p}(\mathbb{R}^{n})$
for each $\alpha\in{\mathbb{Z}}_{+}^{n}$. In particular,
$T_{2}:L^{p}(\mathbb{R}^{n})\to W^{N,p}(\mathbb{R}^{n})$ for each
$N\in{{\mathbb{N}}}$. Now choose $N>s+s(p)$, write
$q^{\prime}=q^{\prime}q^{\prime\prime}$ for some $q^{\prime\prime}\in
C^{\infty}_{c}(\mathbb{R}^{n})$, so that
$q^{\prime\prime}(D):\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})\to
L^{p}(\mathbb{R}^{n})$, and use (1.4), to obtain

$T_{2}=T_{2}q^{\prime\prime}(D):\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})\to
W^{N,p}(\mathbb{R}^{n})\subseteq\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n}).$

(3): This is contained in [11, Corollary 5.7]. ∎

######  Remark 3.4.

If the coefficients of the operator $L$ in (3) have additional regularity,
then the conditions on $p$ and $s$ can be weakened, cf. [11]. Moreover,
similar statements hold for the other solution operator to the Cauchy problem
associated with the wave equation $(\partial_{t}^{2}+L)u(t,x)=0$, and by [17,
19] pseudodifferential operators with rough coefficients are also bounded on
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ for suitable $p$ and $s$.

In fact, we will need a more precise statement about the operator norm of
families of Fourier integral operators as in (2).

######  Lemma 3.5.

For each $t\in\mathbb{R}$, let $T_{t}$ be a Fourier integral operator of order
zero in standard form, associated with a global canonical graph, with symbol
$a_{t}$ and phase function $\Phi_{t}$. Suppose that, for all
$\alpha,\beta\in{\mathbb{Z}}_{+}^{n}$, there exist $\varepsilon,\kappa,M>0$
such that the following conditions hold for all $t\in\mathbb{R}$:

  * •

$\|a_{t}\|_{\alpha,\beta}\leq\kappa(1+|t|)^{M}$;

  * •

$\sup_{(x,\eta)\in\mathbb{R}^{2n}\setminus
o}|\partial_{x}^{\beta}\partial_{\eta}^{\alpha}\Phi_{t}(x,\hat{\eta})|\leq\kappa(1+|t|)^{M}$
if $|\alpha|+|\beta|\geq 2$;

  * •

$\inf_{(x,\eta)\in\mathbb{R}^{2n}\setminus
o}|\det\partial^{2}_{x\eta}\Phi(x,\eta)|\geq\varepsilon$.

Let $p\in[1,\infty]$ and $s\in\mathbb{R}$. Then there exist $C,N>0$ such that

$\|T_{t}\|_{{\mathcal{L}}(\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n}))}\leq
C(1+|t|)^{N}$

for all $t\in\mathbb{R}$.

###### Proof.

This follows by keeping track of the constants in the proof of Proposition 3.3
(2). See also [10, Remark 5.6] and the remark after [10, Theorem 6.10]. ∎

We will apply this lemma to Fourier integral operators as in Remark 3.2.

######  Corollary 3.6.

Let $p\in[1,\infty]$ and $s\in\mathbb{R}$, and let $\phi\in
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be positively homogeneous of
degree $1$. Then there exist $C,N>0$ such that

$\|e^{it\phi(D)}\|_{{\mathcal{L}}(\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n}))}\leq
C(1+|t|)^{N}$

for all $t\in\mathbb{R}$.

######  Remark 3.7.

In the proof of our main result, we will only need bounds for $e^{it\phi(D)}$
when restricted to dyadic frequency shells. In this case one can rely directly
on kernel estimates, and one does not need bounds on all of
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$. On the other hand, the proof of
Lemma 3.5 also relies on these kernel estimates (see also [18, Remark 3.7] for
a simplification for the operators $e^{it\phi(D)}$).

##  4\. Main results

Throughout this section, for each $k\in{\mathbb{Z}}_{+}$, fix a maximal
collection $\Theta_{k}\subseteq S^{n-1}$ of unit vectors such that
$|\nu-\nu^{\prime}|\geq 2^{-k/2}$ for all $\nu,\nu^{\prime}\in\Theta_{k}$.
Note that $\Theta_{k}$ has approximately $2^{k(n-1)/2}$ elements. Let
$(\chi_{\nu})_{\nu\in\Theta_{k}}\subseteq
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be an associated partition of
unity. That is, each $\chi_{\nu}$ is positively homogeneous of degree $0$ and
satisfies $0\leq\chi_{\nu}\leq 1$ and

$\mathrm{supp}(\chi_{\nu})\subseteq\\{\xi\in\mathbb{R}^{n}\setminus\\{0\\}\mid|\hat{\xi}-\nu|\leq
2^{-k/2+1}\\}.$

Moreover, $\sum_{\nu\in\Theta_{k}}\chi_{\nu}(\xi)=1$ for all $\xi\neq 0$, and
for all $\alpha\in{\mathbb{Z}}_{+}^{n}$ and $\beta\in{\mathbb{Z}}_{+}$ there
exists a $C_{\alpha,\beta}\geq 0$ independent of $k$ such that, if
$2^{k-1}\leq|\xi|\leq 2^{k+1}$, then

$|(\hat{\xi}\cdot\partial_{\xi})^{\beta}\partial_{\xi}^{\alpha}\chi_{\nu}(\xi)|\leq
C_{\alpha,\beta}2^{-k(|\alpha|/2+\beta)}$

for all $\nu\in\Theta_{k}$. Such a collection is straightforward to construct,
in a similar manner as the wave packets in Section 2 (see [26, Section IX.4]).
Note also that the collection $\\{\mathcal{F}^{-1}(\chi_{\nu})\mid
k\in{\mathbb{Z}}_{+},\nu\in\Theta_{k}\\}$ is uniformly bounded in
$L^{1}(\mathbb{R}^{n})$.

###  4.1. A discrete description of the
$\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ norm

Using the decomposition of unity from above, we can give a discrete
characterization of the $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ norm of a
function with frequency support in a dyadic annulus.

######  Proposition 4.1.

Let $p\in[1,\infty]$ and $s\in\mathbb{R}$. Then there exists a $C>0$ such that
the following holds. Let $f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ be such
that $\mathrm{supp}(\widehat{f}\,)\subseteq\\{\xi\in\mathbb{R}^{n}\mid
2^{k-1}\leq|\xi|\leq 2^{k+1}\\}$ for some $k\in{\mathbb{Z}}_{+}$. Then

$\frac{1}{C}\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\leq
2^{k(s+\frac{n-1}{2}(\frac{1}{2}-\frac{1}{p}))}\Big{(}\sum_{\nu\in\Theta_{k}}\|\chi_{\nu}(D)f\|_{L^{p}(\mathbb{R}^{n})}^{p}\Big{)}^{1/p}\leq
C\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$

for $p<\infty$, while for $p=\infty$ one has

$\frac{1}{C}\|f\|_{\mathcal{H}^{s,\infty}_{FIO}(\mathbb{R}^{n})}\leq
2^{k(s+\frac{n-1}{4})}\max_{\nu\in\Theta_{k}}\|\chi_{\nu}(D)f\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C\|f\|_{\mathcal{H}^{s,\infty}_{FIO}(\mathbb{R}^{n})}.$

###### Proof.

By a standard argument, using a Littlewood–Paley decomposition of
$L^{p}(\mathbb{R}^{n})$ (and thereby also of
$\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$), we may suppose that
$s=-\frac{n-1}{2}(\frac{1}{2}-\frac{1}{p})$.

Note that the statement is true for $p=2$. Indeed,
$\mathcal{H}^{2}_{FIO}(\mathbb{R}^{n})=L^{2}(\mathbb{R}^{n})$ and

$\Big{(}\sum_{\nu}\|\chi_{\nu}(D)f\|_{L^{2}(\mathbb{R}^{n})}^{2}\Big{)}^{1/2}=\Big{(}\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}\sum_{\nu}\chi_{\nu}(\xi)^{2}\mathrm{d}\xi\Big{)}^{1/2}\eqsim\|f\|_{L^{2}(\mathbb{R}^{n})}.$

In the final step we used that
$1\lesssim\sum_{\nu}\chi_{\nu}(\xi)^{2}\leq(\sum_{\nu}\chi_{\nu}(\xi))^{2}=1$
for all $\xi\neq 0$. More precisely, since there exists a
$c_{n}\in{{\mathbb{N}}}$, independent of $k$ and $\Theta_{k}$, such that there
are at most $c_{n}$ elements $\nu\in\Theta_{k}$ with $\chi_{\nu}(\xi)\neq 0$,
one has $\sum_{\nu}\chi_{\nu}(\xi)^{2}\geq\max_{\nu}\chi_{\nu}(\xi)^{2}\geq
1/c_{n}^{2}$.

Hence, by interpolation and duality, it suffices to prove the statement for
$p=1$. We may also choose $k$ large enough so that $q(D)f=0$, as follows e.g.
from the Sobolev embeddings in (1.4). Note that Lemma 2.2 yields

(4.1)
$\|f\|_{\mathcal{H}^{(n-1)/4,1}_{FIO}(\mathbb{R}^{n})}\leq\sum_{\nu}\|\chi_{\nu}(D)f\|_{\mathcal{H}^{(n-1)/4,1}_{FIO}(\mathbb{R}^{n})}\eqsim\sum_{\nu}\|\chi_{\nu}(D)f\|_{L^{1}(\mathbb{R}^{n})}.$

On the other hand,

$\sum_{\nu}\|\chi_{\nu}(D)f\|_{\mathcal{H}^{(n-1)/4,1}_{FIO}(\mathbb{R}^{n})}=\sum_{\nu}\int_{E_{\nu}}\|\chi_{\nu}(D){\langle}D\rangle^{(n-1)/4}\varphi_{\omega}(D)f\|_{\mathcal{H}^{1}(\mathbb{R}^{n})}\mathrm{d}\omega$

for $E_{\nu}:=\\{\omega\in S^{n-1}\mid|\omega-\nu|\leq 2^{-k/2+3}\\}$. Since
the $\chi_{\nu}(D)$ have kernels that are uniformly in
$L^{1}(\mathbb{R}^{n})$, and because the $\nu\in\Theta_{k}$ are $2^{-k/2}$
separated, one now has

$\displaystyle\sum_{\nu}\|\chi_{\nu}(D)f\|_{\mathcal{H}^{(n-1)/4,1}_{FIO}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\sum_{\nu}\int_{E_{\nu}}\|{\langle}D\rangle^{(n-1)/4}\varphi_{\omega}(D)f\|_{\mathcal{H}^{1}(\mathbb{R}^{n})}\mathrm{d}\omega$
$\displaystyle\lesssim\|f\|_{\mathcal{H}^{(n-1)/4,1}_{FIO}(\mathbb{R}^{n})}.$

Together with (4.1), this concludes the proof. ∎

By combining this proposition with Corollary 3.6, we can derive the following
corollary, which will in turn play a key role in the proof of our main result.

######  Corollary 4.2.

Let $p\in[1,\infty]$, $s\in\mathbb{R}$ and $0\neq
g\in\mathcal{S}(\mathbb{R})$. Let $\phi\in
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be positively homogeneous of
degree $1$. Then there exists a $C>0$ such that the following holds. Let
$f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ be such that
$\mathrm{supp}(\widehat{f}\,)\subseteq\\{\xi\in\mathbb{R}^{n}\mid
2^{k-1}\leq|\xi|\leq 2^{k+1}\\}$ for some $k\in{\mathbb{Z}}_{+}$. Then

$\displaystyle\frac{1}{C}\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$
$\displaystyle\leq
2^{k(s+\frac{n-1}{2}(\frac{1}{2}-\frac{1}{p}))}\Big{(}\sum_{\nu\in\Theta_{k}}\int_{\mathbb{R}}\|\chi_{\nu}(D)g(t)e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\leq C\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$

for $p<\infty$, while for $p=\infty$ one has

$\frac{1}{C}\|f\|_{\mathcal{H}^{s,\infty}_{FIO}(\mathbb{R}^{n})}\leq
2^{k(s+\frac{n-1}{4})}\max_{\nu\in\Theta_{k}}\sup_{t\in\mathbb{R}}\|\chi_{\nu}(D)g(t)e^{it\phi(D)}f\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C\|f\|_{\mathcal{H}^{s,\infty}_{FIO}(\mathbb{R}^{n})}.$

###### Proof.

We consider $p<\infty$. The argument for $p=\infty$ is identical, up to a
change in notation. By Proposition 4.1, one has

$\displaystyle
2^{k(s+\frac{n-1}{2}(\frac{1}{2}-\frac{1}{p}))}\Big{(}\sum_{\nu\in\Theta_{k}}\int_{\mathbb{R}}\|\chi_{\nu}(D)g(t)e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\eqsim\Big{(}\int_{\mathbb{R}}\|g(t)e^{it\phi(D)}f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}.$

Now Corollary 3.6 yields

$\displaystyle\Big{(}\int_{\mathbb{R}}\|g(t)e^{it\phi(D)}f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\lesssim\sup_{t\in\mathbb{R}}\|(1+|t|)^{2}g(t)e^{it\phi(D)}f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}.$

On the other hand,

$\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}=\|e^{-it\phi(D)}e^{it\phi(D)}f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\lesssim\|g(t)e^{it\phi(D)}f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$

on any interval $I\subseteq\mathbb{R}$ such that $|g(t)|\gtrsim 1$ for all
$t\in I$. It follows that

$\Big{(}\int_{\mathbb{R}}\|g(t)e^{it\phi(D)}f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\eqsim\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}.\qed$

###  4.2. Main result

Our main theorem concerns more general constant-coefficient operators than the
Euclidean half-wave group $(e^{it\sqrt{-\Delta}})_{t\in\mathbb{R}}$. The
generality of our result is restricted by the validity of the following
$\ell^{p}$-decoupling inequality.

######  Proposition 4.3.

Let $p\in(2,\infty)$, and let $\phi\in
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be positively homogeneous of
degree $1$ and such that $\mathrm{rank}(\partial_{\xi\xi}^{2}\phi)\equiv n-1$.
Let $g\in\mathcal{S}(\mathbb{R})$ be such that $|g(t)|\geq 1$ for $t\in[0,1]$,
and $\widehat{g}(\tau)=0$ for $\tau\notin[-1,1]$. Then, for each
$\varepsilon>0$, there exists a $C>0$ such that

$\Big{(}\int_{0}^{1}\|e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{\frac{1}{p}}\leq
C2^{k(d(p)+\varepsilon)}\Big{(}\sum_{\nu\in\Theta_{k}}\int_{\mathbb{R}}\|\chi_{\nu}(D)g(t)e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{\frac{1}{p}}$

for every $f\in L^{p}(\mathbb{R}^{n})$ with
$\mathrm{supp}(\widehat{f}\,)\subseteq\\{\xi\in\mathbb{R}^{n}\mid
2^{k-1}\leq|\xi|\leq 2^{k+1}\\}$ for some $k\in{\mathbb{Z}}_{+}$.

###### Proof.

First note that

$\Big{(}\int_{0}^{1}\|e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\leq\Big{(}\int_{\mathbb{R}}\|g(t)e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$

and that the Fourier transform (in $n+1$ variables) of $g(t)e^{it\phi(D)}f(x)$
is supported in $\\{(\xi,\tau)\mid|\tau-\phi(\xi)|\leq 1\\}$. Moreover, the
condition $\mathrm{rank}(\partial_{\xi\xi}^{2}\phi)\equiv n-1$ is equivalent
to the condition that the cone $\\{(\xi,\phi(\xi))\mid\xi\neq 0\\}$ has
everywhere $n-1$ non-vanishing principal curvatures. Since
$e^{it\phi(D)}f=\sum_{\nu\in\Theta_{k}}\chi_{\nu}(D)e^{it\phi(D)}f$, after
rescaling to unit frequencies, the proposition follows from the
$\ell^{p}$-decoupling inequality in [4], for hypersurfaces with non-vanishing
Gaussian curvature, in a similar manner as how the $\ell^{2}$-decoupling
inequality for the cone follows from the $\ell^{2}$-decoupling inequality for
the paraboloid in [3]. For more on this see [2, Theorem 5.3] and [1]. ∎

We can now state and prove our main result, of which Theorem 1.1 is a
corollary.

######  Theorem 4.4.

Let $p\in(2,\infty)$ and $s\in\mathbb{R}$, and let $\phi\in
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be positively homogeneous of
degree $1$ and such that $\mathrm{rank}(\partial_{\xi\xi}^{2}\phi)\equiv n-1$.
Then, for each $\varepsilon>0$, there exists a $C>0$ such that

$\Big{(}\int_{0}^{1}\|e^{it\phi(D)}f\|_{W^{s,p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\leq
C\|f\|_{\mathcal{H}^{s+d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})}$

for all $f\in\mathcal{H}^{s+d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})$.

###### Proof.

We may consider the case where $s=0$. We also claim that we may suppose that
$\mathrm{supp}(\widehat{f}\,)\subseteq\\{\xi\in\mathbb{R}^{n}\mid|\xi|\in[2^{k-1},2^{k+1}]\\}$
for some $k\in{\mathbb{Z}}_{+}$, and show that

(4.2)
$\Big{(}\int_{0}^{1}\|e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\lesssim\|f\|_{\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})}.$

Indeed, suppose that we have proved this, and let a general
$f\in\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})$ be given.
Let $q^{\prime}\in C^{\infty}_{c}(\mathbb{R}^{n})$ and $\kappa>0$ be such that
$q(\xi)=0$ for $|\xi|>\kappa$, and $q^{\prime}(\xi)=1$ for $|\xi|\leq\kappa$.
For each $t\in(0,1)$, write $f=q^{\prime}(tD)q^{\prime}(D)q(D)f+(1-q)(D)f$.
The kernels of $q^{\prime}(D)e^{i\phi(D)}$ and
$q^{\prime}(D){\langle}D\rangle^{-d(p)+s(p)-\varepsilon}$ are integrable, so

$\displaystyle\Big{(}\int_{0}^{1}\|e^{it\phi(D)}q^{\prime}(tD)q^{\prime}(D)q(D)f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\leq\sup_{t\in(0,1)}\|e^{it\phi(D)}q^{\prime}(tD)q^{\prime}(D)q(D)f\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|q^{\prime}(D)q(D)f\|_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\|{\langle}D\rangle^{d(p)-s(p)+\varepsilon}q(D)f\|_{L^{p}(\mathbb{R}^{n})}\leq\|f\|_{\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})}.$

On the other hand, since $q(\xi)=1$ for $|\xi|\leq 2$, we can use a standard
Littlewood–Paley decomposition to write
$(1-q)(D)f=\sum_{k=3}^{\infty}\psi(2^{-k+1}D)(1-q)(D)f$. Here $\psi\in
C^{\infty}_{c}(\mathbb{R}^{n})$ satisfies
$\sum_{k=-\infty}^{\infty}\psi(2^{-k+1}\xi)=1$ for all $\xi\neq 0$, and
$\psi(\xi)=0$ if $|\xi|\notin[1/2,2]$. The convolution kernels of the
$\psi(2^{-k+1}D)(1-q)(D)$ form a uniformly bounded subset of
$L^{1}(\mathbb{R}^{n})$, so

$\|\psi(2^{-k+1}D)(1-q)(D){\langle}D\rangle^{d(p)-s(p)+\varepsilon}f\|_{\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})}\lesssim\|{\langle}D\rangle^{d(p)-s(p)+\varepsilon}f\|_{\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})}$

for an implicit constant independent of $k\in{\mathbb{Z}}_{+}$ and $f$ (see
[7, Theorem 6.1]). Moreover, for all $k\in{\mathbb{Z}}_{+}$, $r\in\mathbb{R}$
and $h\in\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, one has

$\|\psi(2^{-k+1}D)h\|_{\mathcal{H}^{r,p}_{FIO}(\mathbb{R}^{n})}\eqsim
2^{kr}\|\psi(2^{-k+1}D)h\|_{\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})},$

as follows from the definition of $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ and
from a standard fact about the $L^{p}(\mathbb{R}^{n})$ norm of functions with
frequency support in a dyadic annulus. Hence, using the assumption with
$\varepsilon$ replaced by $\varepsilon/2$, we obtain

$\displaystyle\Big{(}\int_{0}^{1}\|e^{it\phi(D)}(1-q)(D)f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\leq\sum_{k=3}^{\infty}\Big{(}\int_{0}^{1}\|e^{it\phi(D)}\psi(2^{-k+1}D)(1-q)(D)f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\lesssim\sum_{k=3}^{\infty}\|\psi(2^{-k+1}D)(1-q)(D)f\|_{\mathcal{H}^{d(p)-s(p)+\varepsilon/2,p}_{FIO}(\mathbb{R}^{n})}$
$\displaystyle\eqsim\sum_{k=3}^{\infty}2^{-k\varepsilon/2}\|\psi(2^{-k+1}D)(1-q)(D)f\|_{\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\sum_{k=3}^{\infty}2^{-k\varepsilon/2}\|f\|_{\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})}\lesssim\|f\|_{\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})}.$

This proves the claim, and it remains to establish (4.2) whenever
$\mathrm{supp}(\widehat{f}\,)\subseteq\\{\xi\in\mathbb{R}^{n}\mid|\xi|\in[2^{k-1},2^{k+1}]\\}$
for some $k\in{\mathbb{Z}}_{+}$.

Let the partition of unity $(\chi_{\nu})_{\nu\in\Theta_{k}}$ be as above. Also
let $g\in\mathcal{S}(\mathbb{R})$ be such that $|g(t)|\geq 1$ for $t\in[0,1]$,
and $\widehat{g}(\tau)=0$ for $\tau\notin[-1,1]$. Then Proposition 4.3 yields

$\displaystyle\Big{(}\int_{0}^{1}\|e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\leq\Big{(}\int_{\mathbb{R}}\Big{\|}\sum_{\nu\in\Theta_{k}}g(t)\chi_{\nu}(D)e^{it\phi(D)}f\Big{\|}_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\lesssim
2^{k(d(p)+\varepsilon)}\Big{(}\\!\sum_{\nu\in\Theta_{k}}\int_{\mathbb{R}}\|\chi_{\nu}(D)g(t)e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$

for implicit constants independent of $f$ and $k$. By Corollary 4.2, the final
quantity is equivalent to
$\|f\|_{\mathcal{H}^{d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})}$, as
required. ∎

###  4.3. Invariance of estimates

As an immediate corollary of Theorem 1.1, we obtain the invariance of local
smoothing estimates under FIOs and related operators. Indeed, under the
assumptions of Theorem 4.4, if $T$ is a bounded operator on
$\mathcal{H}^{s+d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})$, then Theorem
1.1 yields a $C^{\prime}\geq 0$ such that

(4.3)
$\Big{(}\int_{0}^{1}\|e^{it\phi(D)}Tf\|_{W^{s,p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\leq
C^{\prime}\|f\|_{\mathcal{H}^{s+d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})}$

for all $f\in\mathcal{H}^{s+d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})$.
Now Proposition 3.3 yields the following result.

######  Corollary 4.5.

Let $p\in(2,\infty)$ and $s\in\mathbb{R}$, and let $\phi\in
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be positively homogeneous of
degree $1$ and such that $\mathrm{rank}(\partial_{\xi\xi}^{2}\phi)\equiv n-1$.
Let $T$ be an operator as in Proposition 3.3, where in (3) we suppose instead
that $-2+2s(p)\leq s+d(p)+\varepsilon\leq 2$. Then, for each $\varepsilon>0$,
there exists a $C^{\prime}>0$ such that (4.3) holds for all
$f\in\mathcal{H}^{s+d(p)-s(p)+\varepsilon,p}_{FIO}(\mathbb{R}^{n})$.

In particular, by combining (4.3) with (1.4), one sees that the local
smoothing estimates in (1.2) are invariant under application of the operators
in Corollary 4.5.

######  Remark 4.6.

A variable-coefficient version of (1.2) holds on compact manifolds [1]. This
implies in particular that (1.2) is invariant under application of FIOs
associated with a local canonical graph which have a compactly supported
Schwartz kernel, as in Proposition 3.3 (1). However, by itself this does not
imply the stronger estimate (4.3) for such operators. In this article we will
not explore variable-coefficient versions of Theorem 4.4.

##  5\. Sharpness of the results

In this section we show that the exponent $d(p)-s(p)$ in Theorems 1.1 and 4.4
is sharp. We also prove a proposition which deals with the case where
$p\in[1,2]\cup\\{\infty\\}$.

We need two lemmas. The first one relates certain FIOs to translation
operators, and will be applied to functions with frequency support in a
dyadic-parabolic piece. A similar statement holds for more general FIOs, using
bicharacteristic flows on phase space (see [11]), but we will not need such
generality here.

######  Lemma 5.1.

Let $\phi\in C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be positively
homogeneous of degree $1$, and let $h\in\mathcal{S}(\mathbb{R}^{n})$ be such
that $\mathrm{supp}(\widehat{h})\subseteq\mathbb{R}^{n}\setminus\\{0\\}$ is
compact. Let $\nu\in S^{n-1}$ and set

$\kappa(\phi,h,\nu):=\sup\\{|(\partial_{\xi}\phi(\hat{\xi})-\partial_{\xi}\phi(\nu))\cdot\xi|\mid\xi\in\mathrm{supp}(\widehat{h})\\}.$

Then

$|e^{it\phi(D)}h(x)-h(x+t\partial_{\xi}\phi(\nu))|\leq\|\widehat{h}\|_{L^{1}(\mathbb{R}^{n})}\kappa(\phi,h,\nu)|t|$

for all $x\in\mathbb{R}^{n}$ and $t\in\mathbb{R}$ such that
$\kappa(\phi,h,\nu)|t|\leq 1$.

###### Proof.

We use the homogeneity of $\phi$ to write

$\displaystyle(2\pi)^{n}\big{(}e^{it\phi(D)}h(x)-h(x+t\partial_{\xi}\phi(\nu))\big{)}=\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}(e^{it\phi(\xi)}-e^{it\partial_{\xi}\phi(\nu)\cdot\xi})\widehat{h}(\xi)\mathrm{d}\xi$
$\displaystyle=\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}e^{it\partial_{\xi}\phi(\nu)\cdot\xi}(e^{it(\partial_{\xi}\phi(\hat{\xi})-\partial_{\xi}\phi(\nu))\cdot\xi}-1)\widehat{h}(\xi)\mathrm{d}\xi.$

Now simply note that, in the last line, the term in brackets is bounded in
absolute value by $|t|\kappa(\phi,h,\nu)e^{|t|\kappa(\phi,h,\nu)}$. ∎

Next, to prove sharpness for $2<p<2(n+1)/(n-1)$, we will use the following
lemma, which relates the Hardy spaces for FIOs to square functions similar to
those in [9]. Throughout, we use notation as in Section 4.

######  Lemma 5.2.

Let $\phi\in C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be positively
homogeneous of degree $1$, and let $p\in[1,\infty)$, $s\in\mathbb{R}$ and
$C\geq 0$ be such that

(5.1)
$\Big{(}\int_{0}^{1}\|e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\leq
C\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$

for all $f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$. Then there exists a
$C^{\prime}\geq 0$ such that

$\Big{(}\int_{0}^{1}\int_{\mathbb{R}^{n}}\Big{(}\sum_{\nu\in\Theta_{k}}|\chi_{\nu}(D)e^{it\phi(D)}f(x)|^{2}\Big{)}^{p/2}\mathrm{d}x\mathrm{d}t\Big{)}^{1/p}\leq
C^{\prime}\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$

for all $f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ such that
$\mathrm{supp}(\widehat{f}\,)\subseteq\\{\xi\in\mathbb{R}^{n}\mid
2^{k-1}\leq|\xi|\leq 2^{k+1}\\}$ for some $k\in{\mathbb{Z}}_{+}$.

###### Proof.

Let $(\varepsilon_{\nu})_{\nu\in\Theta_{k}}$ be an independent sequence of
Rademacher random variables, and for $x\in\mathbb{R}^{n}$ write

$f_{\varepsilon}(x):=\sum_{\nu\in\Theta_{k}}\varepsilon_{\nu}\chi_{\nu}(D)f(x).$

Then Proposition 4.1 shows that
$f_{\varepsilon}\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ with
$\|f_{\varepsilon}\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\eqsim\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$.
Now one simply combines (5.1) with Khintchine’s inequality. ∎

We can now state the main result of this section, the proof of which relies on
examples that are typically used to show sharpness of decoupling inequalities
(see e.g. [9, 27]). These examples were communicated to us by Po–Lam Yung.

######  Theorem 5.3.

Let $\phi\in C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be positively
homogeneous of degree $1$. Let $p\in(2,\infty)$, $s\in\mathbb{R}$ and $C\geq
0$ be such that

(5.2)
$\Big{(}\int_{0}^{1}\|e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\leq
C\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$

for all $f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$. Then $s\geq d(p)-s(p)$.

###### Proof.

We want to prove $s\geq d(p)-s(p)=\max(s(p)-\frac{1}{p},0)$. We first show
that $s\geq s(p)-\frac{1}{p}$. To this end we apply Lemma 5.1 in the case
where the frequency support of the relevant function fills up a complete
dyadic-parabolic piece.

More precisely, for $k\in{\mathbb{Z}}_{+}$, let
$(f_{\nu})_{\nu\in\Theta_{k}}\subseteq\mathcal{S}(\mathbb{R}^{n})$ be a
collection with the following properties. For each $\nu\in\Theta_{k}$ one has
$\chi_{\nu}(D)f_{\nu}=f_{\nu}$, and $f_{\nu}(x)\geq 1$ for all
$x\in\mathbb{R}^{n}$ with $|x|\leq 2^{-k}$. Moreover,
$\|\mathcal{F}(f_{\nu})\|_{L^{1}(\mathbb{R}^{n})}\eqsim 1$ and
$\|f_{\nu}\|_{L^{p}(\mathbb{R}^{n})}\eqsim 2^{-k\frac{n+1}{2p}}$ for implicit
constants independent of $\nu$ and $k$. To construct such a collection, one
may start off with a function $\psi\in\mathcal{S}(\mathbb{R}^{n})$ such that
$\psi(x)\geq 1$ whenever $|x|\leq 1$, and $\widehat{\psi}(\xi)=0$ if
$|\xi|>c$, for some small $c>0$. Then set

$f_{\nu}(x):=e^{i2^{k}\nu\cdot x}\psi(2^{k}\nu\cdot
x+2^{k/2}\Pi_{\nu}^{\perp}x)$

where $\Pi^{\perp}_{\nu}$ is the orthogonal projection onto the complement of
the span of $\nu$.

Next, let $f:=\sum_{\nu\in\Theta_{k}}f_{\nu}$. Then Proposition 4.1 yields
$f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ and

(5.3) $\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\eqsim
2^{k(s+\frac{n-1}{2}(\frac{1}{2}-\frac{1}{p}))}\Big{(}\sum_{\nu\in\Theta_{k}}\|f_{\nu}\|_{L^{p}(\mathbb{R}^{n})}^{p}\Big{)}^{1/p}\eqsim
2^{k(s+\frac{n-1}{2}(\frac{1}{2}-\frac{1}{p})-\frac{1}{p})}.$

Moreover, Lemma 5.1 and the assumptions on $f_{\nu}$ yield

(5.4) $|e^{it\phi(D)}f_{\nu}(x)-f_{\nu}(x+t\partial_{\xi}\phi(\nu))|\lesssim
t2^{-k/2}$

whenever $0\leq t\lesssim 2^{-k/2}$, for implicit constants independent of
$\nu$ and $k$. Also, $f_{\nu}(x+t\partial_{\xi}\phi(\nu))\geq 1$ if $t\leq
2^{-k}$ and $|x|\leq c^{\prime}2^{-k}$ for some small $c^{\prime}>0$, since
$f_{\nu}(x)\geq 1$ for $|x|\leq 2^{-k}$ and because $\partial_{\xi}\phi$ is
uniformly bounded on $S^{n-1}$. By combining this with (5.4), we obtain

$e^{it\phi(D)}f_{\nu}(x)\geq
f_{\nu}(x+t\partial_{\xi}\phi(\nu))-|e^{it\phi(D)}f_{\nu}(x)-f_{\nu}(x+t\partial_{\xi}\phi(\nu))|\gtrsim
1$

for $|x|\leq c^{\prime}2^{-k}$ and $0\leq t\leq c^{\prime\prime}2^{-k}$ for
some small enough $c^{\prime\prime}>0$. Now (5.2) and (5.3) yield

$\displaystyle 2^{-k\frac{n+1}{p}}2^{k\frac{n-1}{2}}$
$\displaystyle\lesssim\Big{(}\int_{0}^{1}\Big{\|}\sum_{\nu\in\Theta_{k}}e^{it\phi(D)}f_{\nu}\Big{\|}_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}=\Big{(}\int_{0}^{1}\|e^{it\phi(D)}f\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\lesssim\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\eqsim
2^{k(s+\frac{n-1}{2}(\frac{1}{2}-\frac{1}{p})-\frac{1}{p})},$

which implies that $s\geq s(p)-\frac{1}{p}$.

Next, we show that $s\geq 0$. Here we apply Lemma 5.1 to functions with
frequency support of unit size in a dyadic-parabolic piece, and we also rely
on Lemma 5.2. Let $\psi\in\mathcal{S}(\mathbb{R}^{n})$ be as before, and for
$k\in{\mathbb{Z}}_{+}$, $\nu\in\Theta_{k}$ and $x\in\mathbb{R}^{n}$ set

$g_{\nu}(x):=e^{i2^{k}\nu\cdot x}\psi(x).$

Since $\widehat{\psi}(\xi)=0$ if $|\xi|>c$, for some small $c>0$, we may again
assume that $\chi_{\nu}(D)g_{\nu}=g_{\nu}$. Also,
$\|g_{\nu}\|_{L^{p}(\mathbb{R}^{n})}\eqsim
1\eqsim\|\mathcal{F}(g_{\nu})\|_{L^{1}(\mathbb{R}^{n})}$.

Write $g:=\sum_{\nu\in\Theta_{k}}g_{\nu}$. Then
$g\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ with

(5.5) $\|g\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\eqsim
2^{k(s+\frac{n-1}{2}(\frac{1}{2}-\frac{1}{p}))}\Big{(}\sum_{\nu\in\Theta_{k}}\|g_{\nu}\|_{L^{p}(\mathbb{R}^{n})}^{p}\Big{)}^{1/p}\eqsim
2^{k(s+\frac{n-1}{4})},$

by Proposition 4.1. Moreover, since
$\mathrm{supp}(\mathcal{F}g_{\nu})\subseteq\mathrm{supp}(\chi_{\nu})$ is of
unit size, Lemma 5.1 yields

$\big{|}e^{it\phi(D)}g_{\nu}(x)-g_{\nu}(x+t\partial_{\xi}\phi(\nu))\big{|}\lesssim
t$

whenever $0\leq t\leq\kappa$, for some small $\kappa>0$ independent of $k$ and
$\nu$. As before, one also has $g_{\nu}(x+t\partial_{\xi}\phi(\nu))\geq 1$
whenever $|x|\lesssim 1$ and $0\leq t\lesssim 1$ are small enough. Hence

$e^{it\phi(D)}g_{\nu}(x)\geq
g_{\nu}(x+t\partial_{\xi}\phi(\nu))-\big{|}e^{it\phi(D)}g_{\nu}(x)-g_{\nu}(x+t\partial_{\xi}\phi(\nu))\big{|}\gtrsim
1$

for $|x|\leq\kappa^{\prime}$ and $0\leq t\leq\kappa^{\prime\prime}$, for small
enough $\kappa^{\prime},\kappa^{\prime\prime}>0$. Now combine this with (5.2),
Lemma 5.2 and (5.5) to obtain

$\displaystyle 2^{k\frac{n-1}{4}}$
$\displaystyle\lesssim\Big{(}\int_{0}^{1}\int_{\mathbb{R}^{n}}\Big{(}\sum_{\nu\in\Theta_{k}}|e^{it\phi(D)}g_{\nu}(x)|^{2}\Big{)}^{p/2}\mathrm{d}x\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle=\Big{(}\int_{0}^{1}\int_{\mathbb{R}^{n}}\Big{(}\sum_{\nu\in\Theta_{k}}|\chi_{\nu}(D)e^{it\phi(D)}g(x)|^{2}\Big{)}^{p/2}\mathrm{d}x\mathrm{d}t\Big{)}^{1/p}\lesssim\|g\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$
$\displaystyle\eqsim 2^{k(s+\frac{n-1}{4})},$

which implies that $s\geq 0$. ∎

######  Remark 5.4.

Recall that Theorem 4.4 relied on the $\ell^{p}$-decoupling inequality from
[4], for the hypersurface associated with $(e^{it\phi(D)})_{t\in\mathbb{R}}$,
in Proposition 4.3. As a byproduct, Theorem 5.3 shows that each of these
decoupling inequalities is sharp.

######  Remark 5.5.

To prove the inequality $s\geq s(p)-\frac{1}{p}$ in Theorem 5.3, i.e. deal
with the case where $p\geq 2(n+1)/(n-1)$, we cannot simply rely on the fact
that the local smoothing conjecture is sharp. Indeed, while such an approach
suffices for $(e^{it\sqrt{-\Delta}})_{t\in\mathbb{R}}$, or more generally
under the conditions of Theorem 4.4, Theorem 5.3 does not make any assumptions
on the rank of $\partial_{\xi\xi}^{2}\phi$, and the $L^{p}$-mapping properties
of $e^{it\phi(D)}$ improve when $\mathrm{rank}(\partial_{\xi\xi}^{2}\phi)$ is
small (see [22]). See also Remark 5.7.

For completeness we also include the following proposition, for
$p\notin(2,\infty)$.

######  Proposition 5.6.

Let $\phi\in C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be positively
homogeneous of degree $1$. Let $p\in[1,2]\cup\\{\infty\\}$ and
$s\in\mathbb{R}$. Then the following are equivalent:

  1. (1)

One has $[t\mapsto e^{it\phi(D)}f]\in
L^{p}([0,1];\mathcal{H}^{p}(\mathbb{R}^{n}))$ for all
$f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$;

  2. (2)

$s\geq s(p)$.

###### Proof.

By Corollary 4.2 and the Sobolev embeddings in (1.4), one has

(5.6) $[t\mapsto e^{it\phi(D)}]\in
C(\mathbb{R};\mathcal{H}^{s(p),p}_{FIO}(\mathbb{R}^{n}))\subseteq
L^{p}([0,1];\mathcal{H}^{p}(\mathbb{R}^{n}))$

for all $1\leq p\leq\infty$ and
$f\in\mathcal{H}^{s(p),p}_{FIO}(\mathbb{R}^{n})$, which proves the implication
(2)$\Rightarrow$(1).

For $p=\infty$, the converse implication follows directly from the first part
of the proof of Theorem 5.3, up to a change of notation. For $p\leq 2$, let
$f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ be such that

$\mathrm{supp}(\widehat{f}\,)\subseteq\\{\xi\in\mathbb{R}^{n}\mid
2^{k-1}\leq|\xi|\leq 2^{k+1},|\hat{\xi}-\omega|\leq 2^{-k/2+1}\\}$

for some $k\in{\mathbb{Z}}_{+}$ and $\omega\in S^{n-1}$. Then Corollary 4.2
and Lemma 2.2 yield

$2^{k(s(p)-s)}\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\eqsim
2^{k(s(p)-s)}\|e^{i\phi(D)}f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\eqsim\|e^{it\phi(D)}f\|_{\mathcal{H}^{p}(\mathbb{R}^{n})},$

for any $t\in[0,1]$. Now integrate this in time and apply (1) to obtain

$2^{k(s(p)-s)}\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\eqsim\Big{(}\int_{0}^{1}\|e^{it\phi(D)}f\|_{\mathcal{H}^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}\lesssim\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})},$

so that $s\geq s(p)$. ∎

As in Remark 5.5, Proposition 5.6 does not follow in full generality from the
fact that no local smoothing can occur for $p\notin(2,\infty)$.

######  Remark 5.7.

We briefly summarize what we have shown. Let $1\leq p\leq\infty$ and
$s\in\mathbb{R}$, and let $\phi\in C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$
be positively homogeneous of degree $1$. By (5.6), one has

(5.7) $[t\mapsto e^{it\phi(D)}f]\in
L^{p}([0,1];\mathcal{H}^{p}(\mathbb{R}^{n}))\text{ for all
}f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$

whenever $s\geq s(p)$. Because $(e^{it\phi(D)})_{t\in\mathbb{R}}$ is a group,
there can be no improvement of the index in the first inclusion in (5.6), not
even at a single time. Moreover, the condition $s\geq s(p)$ is also necessary
for (5.7) in some situations, for example when $p\in[1,2]\cup\\{\infty\\}$ or
when $\phi\equiv 0$, by Proposition 5.6 and because the Sobolev embeddings in
(1.4) are sharp. On the other hand, for $p\in(2,\infty)$, the condition $s\geq
d(p)-s(p)$ is necessary for any $\phi$, while $s>d(p)-s(p)$ suffices when
$\mathrm{rank}(\partial_{\xi\xi}^{2}\phi)$ is maximal everywhere.

## Acknowledgments

This work arose from conversations with Po-Lam Yung, who made various
suggestions that helped improve it significantly. The author is very grateful
to him for sharing his ideas, without which this article would not have been
written. The author would also like to thank Robert Schippa and the anonymous
referee for useful comments.

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