# Rough pseudodifferential operators on Hardy spaces for Fourier integral
operators

Jan Rozendaal  Institute of Mathematics, Polish Academy of Sciences
ul. Śniadeckich 8
00-656 Warsaw
Poland
and Mathematical Sciences Institute
Australian National University
Acton ACT 2601
Australia<EMAIL_ADDRESS>
###### Abstract.

We prove mapping properties of pseudodifferential operators with rough symbols
on Hardy spaces for Fourier integral operators. The symbols $a(x,\eta)$ are
elements of $C^{r}_{*}S^{m}_{1,\delta}$ classes that have limited regularity
in the $x$ variable. We show that the associated pseudodifferential operator
$a(x,D)$ maps between Sobolev spaces
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ and
$\mathcal{H}^{t,p}_{FIO}({\mathbb{R}^{n}})$ over the Hardy space for Fourier
integral operator $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$. Our main result
implies that for $m=0$, $\delta=1/2$ and $r>n-1$, $a(x,D)$ acts boundedly on
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ for all $p\in(1,\infty)$.

###### Key words and phrases:

Rough pseudodifferential operators, Hardy spaces, Fourier integral operators,
paradifferential operators

###### 2020 Mathematics Subject Classification:

Primary 35S05. Secondary 42B35, 35S30, 35S50

This research was supported by ARC grant DP160100941, and partially supported
by NCN grants UMO2017/27/B/ST1/00078 and UMO2020/37/K/ST1/02765

##  1\. Introduction

###  1.1. Setting

Pseudodifferential operators have long been a powerful tool for the analysis
of partial differential equations. For example, they appear as parametrices
for various equations. They also arise naturally through paradifferential
calculus, where one collects the frequencies of a differential operator into
groups. In fact, the combination of these two tools, parametrices and
paradifferential calculus, has shown itself to be particularly potent when
studying equations with rough coefficients. Most relevant for the present
article are wave equations with rough coefficients, which arise for example
from Laplace–Beltrami operators on manifolds with a rough metric, or by
linearizing nonlinear wave equations with rough initial data.

Given a differential operator $A$ with rough coefficients, one can apply a
paradifferential smoothing procedure that goes back to Bony [1] (see also e.g.
Meyer [14, 15] and Taylor [30, 31]) to decompose $A$ as a sum of a smooth
pseudodifferential operator $A_{1}$ and a rough pseudodifferential operator
$A_{2}$ of lower order. Microlocal parametrices can then be constructed for
the pseudodifferential equation $(\partial_{t}^{2}-A_{1})u(t)=0$, and by
obtaining suitable mapping properties for $A_{2}$ one can use Duhamel’s
principle to obtain a parametrix for the rough wave equation
$(\partial_{t}^{2}-A)u(t)=0$ as well. The resulting parametrix then contains a
lot of information about the rough equation, while also being more tractable
than the bona fide solution operators to the equation. A specific instance of
this paradigm was developed by Smith in [22], and subsequently applied by both
Smith and Tataru to obtain powerful results for wave equations with rough
coefficients, such as Strichartz estimates [22, 27, 28, 29], propagation of
singularities [24], the related spectral cluster estimates [23], and well-
posedness of nonlinear wave equations with rough initial data [25].

In these applications, it suffices to obtain mapping properties between
Sobolev spaces over $L^{2}({\mathbb{R}^{n}})$ for the rough pseudodifferential
term. Such mapping properties are classical, even in the more general setting
of $L^{p}({\mathbb{R}^{n}})$-based Sobolev spaces [2, 13, 30]. However, when
considering the fixed-time $L^{p}$ regularity of wave equations with rough
coefficients, it no longer suffices to obtain mapping properties in the
$L^{p}({\mathbb{R}^{n}})$ scale, as will be explained next.

It has long been known that even smooth wave equations do not preserve the
$L^{p}$ regularity of initial data. In fact, for the classical wave propagator
one has

(1.1) $e^{it\sqrt{-\Delta}}:W^{2s(p),p}({\mathbb{R}^{n}})\to
L^{p}({\mathbb{R}^{n}})$

for $1<p<\infty$, $t\in{\mathbb{R}}$ and
$s(p):=\frac{n-1}{2}|\frac{1}{p}-\frac{1}{2}|$, as was shown by Peral [17] and
Miyachi [16], and the exponent $2s(p)$ cannot be improved. General wave
equations with smooth coefficients have the same fixed-time $L^{p}$
regularity, as is a consequence of the work of Seeger, Sogge and Stein from
1991 [20] on the $L^{p}$ regularity of Fourier integral operators. Moreover,
for $p=1$ and $p=\infty$ one obtains corresponding estimates upon replacing
$L^{p}({\mathbb{R}^{n}})$ by the local Hardy space
$\mathcal{H}^{1}({\mathbb{R}^{n}})$ and ${\mathrm{bmo}}({\mathbb{R}^{n}})$,
respectively. To the author’s best knowledge, since then, essentially no
progress has been made on determining the optimal $L^{p}$ regularity for a
general class of wave equations with rough coefficients.

One reason for the dearth of results on the fixed-time $L^{p}$ regularity for
wave equations with rough coefficients might be that, in general, iterative
approximation arguments are not useful for wave equations on the
$L^{p}({\mathbb{R}^{n}})$ scale for $p\neq 2$. Indeed, due to the
unboundedness of Fourier integral operators on $L^{p}({\mathbb{R}^{n}})$,
approximation arguments that involve iterative constructions typically lead to
a loss of regularity in each iteration step, and such constructions only work
for infinitely smooth initial data. However, in [21] Smith introduced a
powerful replacement of the local Hardy space
$\mathcal{H}^{1}({\mathbb{R}^{n}})$ that is adapted to Fourier integral
operators. More precisely, his space $\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$
is invariant under suitable Fourier integral operators of order zero, and in
particular under smooth wave propagators, and it satisfies Sobolev embeddings
that allow one to recover the results of Seeger, Sogge and Stein.

With an eye towards the fixed-time $L^{p}$ regularity of wave equations with
rough coefficients, in [11] Hassell, Portal and the author extended Smith’s
construction to a scale $(\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}}))_{1\leq
p\leq\infty}$ of Hardy spaces for Fourier integral operators. These spaces are
invariant under compactly supported Fourier integral operators of order zero
associated with a canonical transformation, and they satisfy the Sobolev
embeddings

(1.2)
$W^{s(p),p}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\subseteq
W^{-s(p),p}({\mathbb{R}^{n}})$

for $1<p<\infty$, with appropriate modifications for $p=1$ and $p=\infty$. By
considering the Sobolev space
$\mathcal{H}^{s(p),p}_{FIO}({\mathbb{R}^{n}})={\langle}D\rangle^{-s(p)}\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
over $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$, one directly recovers (1.1)
and the optimal $L^{p}$ regularity of Fourier integral operators.

However, beyond merely recovering existing results, the Hardy spaces for
Fourier integral operators allow for iterative constructions that were not
available before. In particular, one can use the combination of
paradifferential calculus and microlocal parametrices to prove that wave
equations with rough coefficients are solvable over
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$. Via (1.2), one then obtains as a
corollary the optimal $L^{p}$ regularity for such equations. For this strategy
to work, one has to construct a parametrix for a smooth pseudodifferential
equation on $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$, and determine the
mapping properties between Sobolev spaces over
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ of rough pseudodifferential
operators. In the present article we consider the latter problem. The results
from this article were subsequently used by Hassell and the author in the
companion paper [12] to obtain the first results on the optimal fixed-time
$L^{p}$ regularity for a general class of wave equations with rough
coefficients. We also note that in [6], invariant spaces for Fourier integral
operators were used in a different manner to obtain results for wave equations
with specific rough coefficients.

###  1.2. Main results

Our results are formulated in terms of a class $C^{r}_{*}S^{m}_{1,\delta}$ of
rough symbols. Symbols $a(x,\eta)$ in $C^{r}_{*}S^{m}_{1,\delta}$ behave like
elements of Hörmander’s $S^{m}_{1,\delta}$ class, except that they have
limited regularity in the $x$ variable (see Definition 2.6). For
$r\notin{{\mathbb{N}}}$ one requires $C^{r}({\mathbb{R}^{n}})$ regularity,
while for $r\in{{\mathbb{N}}}$ slightly less is assumed. One associates with
such a symbol $a$ the pseudodifferential operator $a(x,D)$ given by

$a(x,D)f(x)=\int_{{\mathbb{R}^{n}}}e^{ix\cdot\eta}a(x,\eta)\widehat{f}(\eta)\mathrm{d}\eta$

for suitable $f$. A special case of our main result is then as follows.

######  Theorem 1.1.

Let $r>n-1$ and $a\in C^{r}_{*}S^{0}_{1,1/2}$. Then

$a(x,D):\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$

is bounded for all $p\in(1,\infty)$.

Theorem 1.1 is a consequence of Theorem 4.1, which also deals with the case
where $r\leq n-1$, and which more generally considers mapping properties
between Sobolev spaces $\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ and
$\mathcal{H}^{t,p}_{FIO}({\mathbb{R}^{n}})$ over
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$, for suitable $s$ and $t$. From this
theorem one can in turn derive mapping properties on
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ for $C^{r}_{*}S^{0}_{1,\delta}$
symbols for general $\delta\in[0,1/2]$, as is shown in Corollary 4.5.

It should be noted that we consider $C^{r}_{*}S^{m}_{1,\delta}$ symbols with
$\delta>0$ not merely to formulate more general statements. In fact, when
applying the paradifferential smoothing procedure to a differential operator
$A$, the resulting smooth pseudodifferential operator $A_{1}$ and rough
pseudodifferential operator $A_{2}$ have $S^{m}_{1,\delta}$ and
$C^{r}_{*}S^{m}_{1,\delta}$ symbols, respectively, and only $\delta>0$ is of
interest here. Moreover, $\delta=1/2$ appears to be a critical value for wave
equations, as it is for the more general theory of Fourier integral operators.
We stress that Theorem 1.1 and the more general Theorem 4.1 are crucial for
the results on the $L^{p}$ regularity of wave equations with rough
coefficients in [12], since they provide one of the two pieces that are
required for the paradifferential approach to such equations.

However, the results in this article are of independent interest. For example,
for $\delta=0$ they apply to what are seemingly the simplest
pseudodifferential operators: multiplication operators. That
$L^{p}({\mathbb{R}^{n}})$ is invariant under multiplication by bounded
measurable functions is trivial, but it is nontrivial to determine the mapping
properties of multiplication operators on the Hardy spaces for Fourier
integral operators. This difficulty arises from the definition of the
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ norm as

(1.3)
$\|f\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}=\|q(D)f\|_{L^{p}({\mathbb{R}^{n}})}+\Big{(}\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)f\|_{L^{p}({\mathbb{R}^{n}})}^{p}\mathrm{d}{\omega}\Big{)}^{1/p}$

for $1<p<\infty$, with a suitable modification for $p=1$ (see Definition 2.1).
Here $q\in C^{\infty}_{c}({\mathbb{R}^{n}})$ is a low-frequency cutoff, and
the Fourier multipliers ${\varphi}_{{\omega}}(D)$ localize in frequency to a
paraboloid in the direction of ${\omega}\in S^{n-1}$. Since Fourier
multipliers and multiplication operators do not commute, it is not clear from
(1.3) how multiplication operators act on
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$, and the situation becomes even more
complicated when multiplying with a function of limited regularity.
Nonetheless, Corollaries 3.5 and 4.5 show that for each $r>0$, multiplication
with an element of $C^{r}_{*}({\mathbb{R}^{n}})$ is a bounded operator on
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ for all $p$ in an interval around
$2$ and for suitable $s$, both depending on $r$ and the dimension $n$.

###  1.3. Overview of the proof

To prove our results we use various techniques. Firstly, mapping properties of
rough pseudodifferential operators between Sobolev spaces over
$L^{p}({\mathbb{R}^{n}})$ are classical [2, 13, 30, 31], and by combining
these with the Sobolev embeddings in (1.2), one obtains some first results.
Moreover, it was shown in [11] that pseudodifferential operators with
$S^{0}_{1,1/2}$ symbols are bounded on
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ for all $p\in[1,\infty]$. For
$\delta<1/2$ one can then apply the paradifferential smoothing procedure, this
time to the rough symbol $a$ itself. As a result one obtains that, if $a\in
C^{r}_{*}S^{0}_{1,\delta}$ for some $r>0$ and $\delta<1/2$, then $a(x,D)$ acts
boundedly on $\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ for all $p$ in an
interval around $2$ and for suitable $s$, both depending on $r$, $n$ and
$\delta$.

However, for the critical case where $\delta=1/2$, such techniques do not
suffice. Instead, to prove Theorem 1.1, and the more general Theorem 4.1, we
first note that (1.3) allows us to apply Littlewood–Paley theory to the Hardy
spaces for Fourier integral operators, by applying it to
${\varphi}_{{\omega}}(D)f$ for each ${\omega}\in S^{n-1}$. In fact, it should
be noted that (1.3) is not how the $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
norm was originally defined in [21, 11]. It was shown by the author in [19]
that we may equivalently define the $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
norm in this manner for $1<p<\infty$, and the corresponding result for $p=1$
was obtained by Fan, Liu, Song and the author in [5].

With these equivalent characterizations and Littlewood–Paley theory in hand,
we first remove the lowest and the highest frequencies of $a$. This reduces to
the case where ${\mathcal{F}}a(\cdot,\eta)(\xi)=0$ unless
$|\eta|^{1/2}\lesssim|\xi|\lesssim(1+|\eta|)^{\gamma}$, for some
$\gamma\in[1/2,1)$. The highest frequencies are dealt with using the Sobolev
embeddings in (1.2) and results about rough pseudodifferential operators on
$W^{s,p}({\mathbb{R}^{n}})$ from [13]. On the other hand, the term with the
lowest frequencies is an $S^{0}_{1,1/2}$ symbol, and the corresponding
pseudodifferential operator is bounded on
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ for all $p\in[1,\infty]$ and
$s\in{\mathbb{R}}$. The remaining frequencies, and specifically those $\xi$
for which $|\xi|\eqsim|\eta|^{1/2}$, are the most problematic, and dealing
with these is the heart of the proof.

Next, we use a symbol decomposition from [4] to reduce to the case where
$a(x,\eta)=\sum_{k=0}a_{k}(x)\psi_{k}(\eta)$ for suitable multiplication
operators $a_{k}$ and dyadic localizations $\psi_{k}$. We then proceed to
group together the frequencies of each $a_{k}$, as well as those of the
function $f$ to which $a(x,D)$ is applied. Such a paradifferential approach
has been used in e.g. [2, 13, 30, 31] to study rough pseudodifferential
operators on $L^{p}({\mathbb{R}^{n}})$. However, in those cases the
frequencies are grouped together in dyadic annuli, whereas in our setting it
is necessary to apply a further decomposition to the frequencies of $f$, the
so-called dyadic-parabolic or second dyadic decomposition. The proof of [7,
Proposition 7.4] is somewhat similar to this part of our argument, although
there the authors obtain $L^{2}$ bounds.

We then use an anisotropic version of the Mikhlin multiplier theorem, Lemma
2.4, to estimate away the Fourier multipliers ${\varphi}_{{\omega}}(D)$,
uniformly in ${\omega}\in S^{n-1}$, after which we can use supremum norm
estimates on $L^{p}({\mathbb{R}^{n}})$ to deal with the multiplication
operators $a_{k}$. In the process we pick up terms that blow up if $r\leq
n-1$, and this is where the restriction $r>n-1$ in Theorem 1.1 comes in. To
conclude the proof we use the boundedness of the Hardy–Littlewood maximal
function on $L^{p}(S^{n-1};\ell^{2})$. We note that the use of the anisotropic
multiplier theorem and the boundedness of the Hardy–Littlewood maximal
function are what restricts our proof to $p\in(1,\infty)$ (see Remark 4.3). We
will not deal with the cases $p=1$ and $p=\infty$ in Theorem 4.1 in the
present article, although the results in Section 3 hold for all
$p\in[1,\infty]$.

###  1.4. Organization of this article

This article is organized as follows. In Section 2 we collect some background
for the rest of the article. In Section 2.1 we first introduce the Hardy
spaces for Fourier integral operators. Then we collect some results from
Littlewood–Paley theory that will be used for the proof of Theorem 4.1, such
as the anisotropic multiplier theorem in Lemma 2.4. In Section 2.2 we
introduce the rough symbol classes, as well as the paradifferential smoothing
procedure. In Section 3 we then derive some first results about the
boundedness of rough pseudodifferential operators on Hardy spaces for Fourier
integral operators. These results will suffice for many purposes, but not for
$C^{r}_{*}S^{m}_{1,1/2}$ symbols. That critical case is dealt with in Section
4, where in particular Theorem 1.1 is proved.

###  1.5. Notation

The natural numbers are ${{\mathbb{N}}}=\\{1,2,\ldots\\}$, and the nonnegative
integers are ${{\mathbb{Z}}}_{+}={{\mathbb{N}}}\cup\\{0\\}$. Throughout this
article, we fix $n\in{{\mathbb{N}}}$ with $n\geq 2$. Our techniques can also
be applied for $n=1$, but in that case the results are classical, by
Proposition 3.1 and because
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}})=\mathcal{H}^{s,p}({\mathbb{R}})$ for
all $p\in[1,\infty]$ and $s\in{\mathbb{R}}$, cf. (2.1) and (2.9).

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
$\partial^{\alpha}_{\xi}=\partial^{\alpha_{1}}_{\xi_{1}}\ldots\partial^{\alpha_{n}}_{\xi_{n}}$
and $\xi^{\alpha}=\xi_{1}^{\alpha_{1}}\ldots\xi_{n}^{\alpha_{n}}$ for
$\xi=(\xi_{1},\ldots,\xi_{n})\in{\mathbb{R}^{n}}$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in{{\mathbb{Z}}}_{+}^{n}$.

The spaces of Schwartz functions and tempered distributions are
$\mathcal{S}({\mathbb{R}^{n}})$ and $\mathcal{S}^{\prime}({\mathbb{R}^{n}})$,
respectively. The Fourier transform of
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ is denoted by ${\mathcal{F}}f$ or
$\widehat{f}$. If $f\in L^{1}({\mathbb{R}^{n}})$ then

$\displaystyle{\mathcal{F}}f(\xi)=\int_{{\mathbb{R}^{n}}}e^{-ix\xi}f(x)\mathrm{d}x\quad(\xi\in{\mathbb{R}^{n}}).$

The Fourier multiplier with symbol
${\varphi}\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ is denoted by
${\varphi}(D)$.

The Hölder conjugate of $p\in[1,\infty]$ is denoted by $p^{\prime}$. The
volume of a measurable subset $B$ of a measure space $(\Omega,\mu)$ is $|B|$.
For an integrable $F:B\to{\mathbb{C}}$, we write

$\fint_{B}F(x)\mathrm{d}\mu(x)=\frac{1}{|B|}\int_{B}F(x)\mathrm{d}\mu(x)$

if $|B|<\infty$. The space of bounded linear operators between Banach spaces
$X$ and $Y$ is ${\mathcal{L}}(X,Y)$, and
${\mathcal{L}}(X):={\mathcal{L}}(X,X)$. We will often simply write
$\|\cdot\|_{F}$ for the norm of a function space $F({\mathbb{R}^{n}})$ over
${\mathbb{R}^{n}}$.

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\. Preliminaries

In this section we collect the required background on the Hardy spaces for
Fourier integral operators, rough symbol classes, and on the paradifferential
smoothing procedure.

###  2.1. Hardy spaces for Fourier integral operators

In this subsection we introduce the Hardy spaces for Fourier integral
operators, and we collect some of their basic properties.

Throughout, to simplify various statements, for $s\in{\mathbb{R}}$ and
$p\in[1,\infty]$ we write

(2.1)
$\mathcal{H}^{s,p}({\mathbb{R}^{n}}):=\begin{cases}W^{s,p}({\mathbb{R}^{n}})&\text{if
}p\in(1,\infty),\\\
{\langle}D\rangle^{-s}\mathcal{H}^{1}({\mathbb{R}^{n}})&\text{if }p=1,\\\
{\langle}D\rangle^{-s}{\mathrm{bmo}}({\mathbb{R}^{n}})&\text{if
}p=\infty.\end{cases}$

Here $\mathcal{H}^{1}({\mathbb{R}^{n}})$ is the local Hardy space from [9].
Recall that $\mathcal{H}^{1}({\mathbb{R}^{n}})$ consists of all
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ such that $q(D)f\in
L^{1}({\mathbb{R}^{n}})$ and $(1-q)(D)f\in H^{1}({\mathbb{R}^{n}})$, for
$H^{1}({\mathbb{R}^{n}})$ the classical Hardy space, endowed with the norm

(2.2)
$\|f\|_{\mathcal{H}^{1}({\mathbb{R}^{n}})}:=\|q(D)f\|_{L^{1}({\mathbb{R}^{n}})}+\|(1-q)(D)f\|_{H^{1}({\mathbb{R}^{n}})}.$

Throughout, $q\in C^{\infty}_{c}({\mathbb{R}^{n}})$ is such that $q(\zeta)=1$
for $|\zeta|\leq 2$. Moreover, ${\mathrm{bmo}}({\mathbb{R}^{n}})$ consists of
all $f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ such that $q(D)f\in
L^{\infty}({\mathbb{R}^{n}})$ and
$(1-q)(D)f\in{\mathrm{BMO}}({\mathbb{R}^{n}})$, with norm

(2.3)
$\|f\|_{{\mathrm{bmo}}({\mathbb{R}^{n}})}:=\|q(D)f\|_{L^{\infty}({\mathbb{R}^{n}})}+\|(1-q)(D)f\|_{{\mathrm{BMO}}({\mathbb{R}^{n}})}.$

Also recall that ${\mathrm{bmo}}({\mathbb{R}^{n}})$ is the dual of
$\mathcal{H}^{1}({\mathbb{R}^{n}})$.

Fix a non-negative radial ${\varphi}\in C^{\infty}_{c}({\mathbb{R}^{n}})$ such
that ${\varphi}(\zeta)=0$ for $|\zeta|>1$, and ${\varphi}\equiv 1$ in a
neighborhood of zero. For ${\omega}\in S^{n-1}$, $\sigma>0$ and
$\zeta\in{\mathbb{R}^{n}}\setminus\\{0\\}$, set
${\varphi}_{{\omega},\sigma}(\zeta):=c_{\sigma}{\varphi}\big{(}\tfrac{\hat{\zeta}-{\omega}}{\sqrt{\sigma}}\big{)}$,
where
$c_{\sigma}:=\big{(}\int_{S^{n-1}}{\varphi}\big{(}\tfrac{e_{1}-\nu}{\sqrt{\sigma}}\big{)}^{2}\mathrm{d}\nu\big{)}^{-1/2}$
for $e_{1}=(1,0,\ldots,0)$ the first basis vector of ${\mathbb{R}^{n}}$ (this
choice is irrelevant). Also let ${\varphi}_{{\omega},\sigma}(0):=0$. Next, fix
a non-negative radial $\Psi\in C^{\infty}_{c}({\mathbb{R}^{n}})$ such that
$\Psi(\zeta)=0$ if $|\zeta|\notin[1/2,2]$, and

$\int_{0}^{\infty}\Psi(\sigma\zeta)^{2}\frac{\mathrm{d}\sigma}{\sigma}=1\quad(\zeta\neq
0).$

For ${\omega}\in S^{n-1}$ and $\zeta\in{\mathbb{R}^{n}}$, set

${\varphi}_{{\omega}}(\zeta):=\int_{0}^{4}\Psi(\tau\zeta){\varphi}_{{\omega},\tau}(\zeta)\frac{\mathrm{d}\tau}{\tau}.$

We will use the following properties of these functions, cf. [19, Remark 3.3]
and [5, Lemma 2.2]:

  1. (1)

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

(2.4) ${\varphi}_{{\omega}}(\zeta)=0\text{ if }|\zeta|<\tfrac{1}{8}\text{ or
}|\hat{\zeta}-{\omega}|>2|\zeta|^{-1/2};$

  2. (2)

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

(2.5)
$|({\omega}\cdot\nabla_{\zeta})^{\beta}\partial^{\alpha}_{\zeta}{\varphi}_{{\omega}}(\zeta)|\leq
C_{\alpha,\beta}|\zeta|^{\frac{n-1}{4}-\frac{|\alpha|}{2}-\beta}$

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

  3. (3)

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

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

for all $\zeta\in{\mathbb{R}^{n}}$ with $|\zeta|\geq 1/2$. 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
${\mathrm{supp}}(\widehat{f}\,)\subseteq\\{\zeta\in{\mathbb{R}^{n}}\mid|\zeta|\geq
1/2\\}$, then

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

In (3) and throughout, for $\gamma\in{\mathbb{R}}$ we denote by
$S^{\gamma}({\mathbb{R}^{n}})$ the class of $m\in
C^{\infty}({\mathbb{R}^{n}})$ such that for each
$\alpha\in{{\mathbb{Z}}}_{n}^{+}$ there exists a $C_{\alpha}^{\prime}\geq 0$
such that

$|\partial_{\zeta}^{\alpha}m(\zeta)|\leq
C_{\alpha}^{\prime}|\zeta|^{\gamma-|\alpha|}$

for all $\zeta\in{\mathbb{R}^{n}}$. Each $m\in S^{\gamma}({\mathbb{R}^{n}})$
yields a bounded operator
$m(D):\mathcal{H}^{s+\gamma,p}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}({\mathbb{R}^{n}})$
for all $s\in{\mathbb{R}}$ (see Lemma 3.1).

Recall that $q\in C^{\infty}_{c}({\mathbb{R}^{n}})$ is such that $q(\zeta)=1$
for $|\zeta|\leq 2$. We can now define the Hardy spaces for Fourier integral
operators.

######  Definition 2.1.

For $p\in[1,\infty)$ we let $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ consists
of all $f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ 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

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

We also set
$\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}}):=(\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}}))^{*}$.
Moreover, for $p\in[1,\infty]$ and $s\in{\mathbb{R}}$ we let
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ consist 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

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

It is straightforward to see that, for $1\leq p<\infty$ and
$s\in{\mathbb{R}}$, one has

(2.7)
$\|f\|_{\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})}\eqsim\|q(D)f\|_{L^{p}({\mathbb{R}^{n}})}+\Big{(}\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)f\|_{\mathcal{H}^{s,p}({\mathbb{R}^{n}})}^{p}\mathrm{d}{\omega}\Big{)}^{1/p}$

for all $f\in\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$, with implicit
constants independent of $f$.

######  Remark 2.2.

We note that this is not how the Hardy spaces for Fourier integral operators
were originally defined in [21] and [11]. There these spaces were introduced
using a conical square function over the cosphere bundle, or equivalently
using wave packet transforms and tent spaces over the cosphere bundle. That
definition also includes an intrinsic description of
$\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$. It was shown in [19] for
$1<p<\infty$, and in [5] for $p=1$, that the spaces in Definition 2.1 coincide
with the original Hardy spaces for Fourier integral operators. The same then
holds for $p=\infty$, by duality (see [11, Proposition 6.8]).

For $p\in[1,\infty]$ set

(2.8) $s(p):=\tfrac{n-1}{2}|\tfrac{1}{p}-\tfrac{1}{2}|.$

With notation as in (2.1), by [11, Theorem 7.4] the following Sobolev
embeddings hold for all $p\in[1,\infty]$ and $s\in{\mathbb{R}}$:

(2.9)
$\mathcal{H}^{s+s(p),p}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s-s(p),p}({\mathbb{R}^{n}}),$

and the exponents in these embeddings are sharp (see [11, Remark 7.9] and [5,
Remark 6.5]).

A key idea in this article is to apply Littlewood-Paley theory to (2.7).
Throughout, $(\psi_{j})_{j=0}^{\infty}\subseteq
C^{\infty}_{c}({\mathbb{R}^{n}})$ is a fixed Littlewood–Paley decomposition.
That is,

(2.10) $\sum_{j=0}^{\infty}\psi_{j}(\xi)=1\quad(\xi\in{\mathbb{R}^{n}}),$

$\psi_{0}(\xi)=0$ for $|\xi|>1$, $\psi_{1}(\xi)=0$ if $|\xi|\notin[1/2,2]$,
and $\psi_{j}(\xi)=\psi_{1}(2^{-j+1}\xi)$ for all $j>1$ and
$\xi\in{\mathbb{R}^{n}}$. In fact, one can assume that $\psi_{1}(\xi)=0$ for
$|\xi|\notin[(1+\varepsilon)/2,2-\varepsilon]$ for some $\varepsilon>0$, an
additional assumption which is of minor convenience in the proof of Theorem
4.1 below. For notational simplicity, we also set $\psi_{j}:=0$ for $j<0$. We
will use the following standard lemma (see e.g. [32, Section 2.5]).

######  Lemma 2.3.

Let $p\in[1,\infty)$ and $s\in{\mathbb{R}}$. Let
$(\chi_{k})_{k=0}^{\infty}\subseteq C^{\infty}_{c}({\mathbb{R}^{n}})$ be such
that $\chi_{k}(\xi)=\chi_{1}(2^{-k+1}\xi)$ for all $k>1$ and
$\xi\in{\mathbb{R}^{n}}$, and $\chi_{1}(\xi)=0$ if $|\xi|\notin[1/4,4]$. Then
there exists a $C>0$ such that the following holds for all
$(f_{k})_{k=0}^{\infty}\subseteq L^{p}({\mathbb{R}^{n}})$ with

${\mathrm{supp}}(\widehat{f}_{k})\subseteq\\{\xi\in{\mathbb{R}^{n}}\mid|\xi|\in[2^{k-3},2^{k+1}]\\}$

for $k\geq 1$, and
${\mathrm{supp}}(\widehat{f}_{0})\subseteq\\{\xi\in{\mathbb{R}^{n}}\mid|\xi|\leq
2\\}$.

  1. (1)

If $(\sum_{k=0}^{\infty}4^{ks}|f_{k}|^{2})^{1/2}\in L^{p}({\mathbb{R}^{n}})$
then $\sum_{k=0}^{\infty}f_{k}\in\mathcal{H}^{s,p}({\mathbb{R}^{n}})$ and

$\Big{\|}\sum_{k=0}^{\infty}f_{k}\Big{\|}_{\mathcal{H}^{s,p}({\mathbb{R}^{n}})}\leq
C\Big{\|}\Big{(}\sum_{k=0}^{\infty}4^{ks}|f_{k}|^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}({\mathbb{R}^{n}})}.$

  2. (2)

If $f_{k}=\chi_{k}(D)f$ for all $k\geq 0$ and some
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$, and if
$\sum_{k=0}^{\infty}f_{k}\in\mathcal{H}^{s,p}({\mathbb{R}^{n}})$, then
$(\sum_{k=0}^{\infty}4^{ks}|f_{k}|^{2})^{1/2}\in L^{p}({\mathbb{R}^{n}})$ and

$\Big{\|}\Big{(}\sum_{k=0}^{\infty}4^{ks}|f_{k}|^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}({\mathbb{R}^{n}})}\leq
C\Big{\|}\sum_{k=0}^{\infty}f_{k}\Big{\|}_{\mathcal{H}^{s,p}({\mathbb{R}^{n}})}.$

We will combine Lemma 2.3 with the following result about multipliers on the
space $L^{p}({\mathbb{R}^{n}};\ell^{2})$ of $p$-integrable $\ell^{2}$-valued
functions. Recall that $e_{1}$ is the first basis vector of
${\mathbb{R}^{n}}$, although this choice is again immaterial.

######  Lemma 2.4.

Let $p\in(1,\infty)$. Then
${\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{{\omega}}(D)\in{\mathcal{L}}(L^{p}({\mathbb{R}^{n}};\ell^{2}))$
for all ${\omega}\in S^{n-1}$, and

$\|{\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{{\omega}}(D)\|_{{\mathcal{L}}(L^{p}({\mathbb{R}^{n}};\ell^{2}))}=\|{\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{e_{1}}(D)\|_{{\mathcal{L}}(L^{p}({\mathbb{R}^{n}};\ell^{2}))}.$

###### Proof.

By applying a rotation, using the definition of ${\varphi}_{{\omega}}$ and the
assumption that ${\varphi}$ is radial, it suffices to show that
${\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{e_{1}}(D)\in{\mathcal{L}}(L^{p}({\mathbb{R}^{n}};\ell^{2}))$.
To this end we claim that, for every $\alpha\in{{\mathbb{Z}}}_{+}^{n}$, one
has

(2.11)
$\sup_{\xi\in{\mathbb{R}^{n}}}\big{|}\xi^{\alpha}\partial_{\xi}^{\alpha}\big{(}{\langle}\xi\rangle^{-\frac{n-1}{4}}{\varphi}_{e_{1}}(\xi)\big{)}\big{|}<\infty.$

Taking the claim for granted momentarily, it then follows from the
Marcinkiewicz–Lizorkin multiplier theorem (see e.g. [33, Proposition 3] for a
version with values in a Banach space) that
${\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{e_{1}}(D)\in{\mathcal{L}}(L^{p}({\mathbb{R}^{n}};\ell^{2}))$.

We conclude by proving (2.11). Let
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in{{\mathbb{Z}}}_{+}^{n}$ and
$\xi=(\xi_{1},\ldots,\xi_{n})\in{\mathrm{supp}}({\varphi}_{e_{1}})$, and write
$\xi^{\prime}:=\xi-\xi_{1}e_{1}=(0,\xi_{2},\ldots,\xi_{n})$. Then, by (2.4),
one has

$|\xi^{\prime}|^{2}\leq(\xi_{1}-|\xi|)^{2}+\xi_{2}^{2}+\ldots+\xi_{n}^{2}=|\xi-|\xi|e_{1}|^{2}=|\xi|^{2}|\hat{\xi}-e_{1}|\lesssim|\xi|.$

Hence, with $\alpha^{\prime}:=(0,\alpha_{2},\ldots,\alpha_{n})$, (2.5) yields

$\displaystyle|\xi^{\alpha}\partial_{\xi}^{\alpha}{\varphi}_{e_{1}}(\xi)|=|\xi_{1}^{\alpha_{1}}(\xi^{\prime})^{\alpha^{\prime}}\partial_{\xi_{1}}^{\alpha_{1}}\partial_{\xi}^{\alpha^{\prime}}{\varphi}_{e_{1}}(\xi)|\lesssim|\xi|^{\alpha_{1}+\frac{\alpha^{\prime}}{2}}|\xi|^{\frac{n-1}{4}-\alpha_{1}-\frac{\alpha^{\prime}}{2}}=|\xi|^{\frac{n-1}{4}}.$

Now (2.11) follows from the Leibniz rule. ∎

######  Remark 2.5.

As was pointed out by the referee,
${\langle}D\rangle^{-(n-1)/4}{\varphi}_{e_{1}}(D)$ is in fact an anisotropic
Calderon–Zygmund operator, associated to the dilation
$\xi\mapsto(r^{2}\xi_{1},r\xi^{\prime})$ for $r>1$. Hence it also follows from
the general theory of anisotropic Calderon–Zygmund operators that
${\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{e_{1}}(D)\in{\mathcal{L}}(L^{p}({\mathbb{R}^{n}};\ell^{2}))$.
See, for example, [3] in the scalar-valued case, with the extension to
functions with values in a Hilbert space being standard (see [26, Chapter I]).

###  2.2. Rough symbols

We first introduce some classes of rough symbols, and then we describe a
smoothing procedure for such symbols.

Let $(\psi_{j})_{j=0}^{\infty}\subseteq C^{\infty}_{c}({\mathbb{R}^{n}})$ be
the Littlewood–Paley decomposition from (2.10). For $r>0$ we let the _Zygmund
space_ $C^{r}_{*}({\mathbb{R}^{n}})$ consist of all
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ such that $\psi_{j}(D)f\in
L^{\infty}({\mathbb{R}^{n}})$ for all $j\geq 0$, with

$\|f\|_{C^{r}_{*}({\mathbb{R}^{n}})}:=\sup_{j\in{{\mathbb{Z}}}_{+}}2^{jr}\|\psi_{j}(D)f\|_{L^{\infty}({\mathbb{R}^{n}})}<\infty.$

Then (see [32])

(2.12) $\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})\subsetneq
C^{r}_{*}({\mathbb{R}^{n}})=C^{r}({\mathbb{R}^{n}})\cap
L^{\infty}({\mathbb{R}^{n}})$

if $r\notin{{\mathbb{N}}}$, and

(2.13) $C^{r-1,1}({\mathbb{R}^{n}})\cap
L^{\infty}({\mathbb{R}^{n}})\subsetneq\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})\subsetneq
C^{r}_{*}({\mathbb{R}^{n}})$

if $r\in{{\mathbb{N}}}$. Here $\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$ is as
in (2.1). Moreover, for $r=l+s\notin{{\mathbb{N}}}$ with
$l\in{{\mathbb{Z}}}_{+}$ and $s\in(0,1)$, $C^{r}({\mathbb{R}^{n}})$ consists
of those $f\in C^{l}({\mathbb{R}^{n}})$ such that for each
$\alpha\in{{\mathbb{Z}}}_{+}^{n}$ with $|\alpha|=l$, the partial derivative
$\partial_{x}^{\alpha}f$ is Hölder continuous with parameter $s$. For
$r\in{{\mathbb{N}}}$, $C^{r-1,1}({\mathbb{R}^{n}})$ consists of those $f\in
C^{r-1}({\mathbb{R}^{n}})$ such that $\partial_{x}^{\alpha}f$ is Lipschitz for
all $\alpha\in{{\mathbb{Z}}}_{+}^{n}$ with $|\alpha|=r-1$. We also note in
passing that $C^{r}_{*}({\mathbb{R}^{n}})$ is equal to the Besov space
$B^{r}_{\infty,\infty}({\mathbb{R}^{n}})$ for all $r>0$.

Recall that, for $m\in{\mathbb{R}}$ and $\delta\in[0,1]$, the Hörmander symbol
class $S^{m}_{1,\delta}$ is the space of $a\in C^{\infty}({\mathbb{R}}^{2n})$
such that, for all $\alpha,\beta\in{{\mathbb{Z}}}_{+}^{n}$, there exists a
$C_{\alpha,\beta}\geq 0$ with

$|\partial_{x}^{\beta}\partial_{\eta}^{\alpha}a(x,\eta)|\leq
C_{\alpha,\beta}{\langle}\eta\rangle^{m-|\alpha|+\delta|\beta|}$

for all $x,\eta\in{\mathbb{R}^{n}}$. We will be interested in versions of
these symbols that are rough in the $x$ variable.

######  Definition 2.6.

Let $r>0$, $m\in{\mathbb{R}}$ and $\delta\in[0,1]$. Then
$C^{r}_{*}S^{m}_{1,\delta}$ is the collection of
$a:{\mathbb{R}}^{2n}\to{\mathbb{C}}$ such that for each
$\alpha\in{{\mathbb{Z}}}_{+}^{n}$ there exists a $C_{\alpha}\geq 0$ with the
following properties:

  1. (1)

For all $x,\eta\in{\mathbb{R}^{n}}$ one has $a(x,\cdot)\in
C^{\infty}({\mathbb{R}^{n}})$ and

(2.14) $|\partial_{\eta}^{\alpha}a(x,\eta)|\leq
C_{\alpha}{\langle}\eta\rangle^{m-|\alpha|}.$

  2. (2)

For all $\eta\in{\mathbb{R}^{n}}$ one has
$\partial_{\eta}^{\alpha}a(\cdot,\eta)\in C^{r}_{*}({\mathbb{R}^{n}})$ and

(2.15)
$\|\partial_{\eta}^{\alpha}a(\cdot,\eta)\|_{C^{r}_{*}({\mathbb{R}^{n}})}\leq
C_{\alpha}{\langle}\eta\rangle^{m-|\alpha|+r\delta}.$

Moreover, $\mathcal{H}^{r,\infty}S^{m}_{1,\delta}$ is the collection of $a\in
C^{r}_{*}S^{m}_{1,\delta}$ such that, in (2), for all
$\eta\in{\mathbb{R}^{n}}$ one has
$\partial_{\eta}^{\alpha}a(\cdot,\eta)\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$
and

$\|\partial_{\eta}^{\alpha}a(\cdot,\eta)\|_{\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})}\leq
C_{\alpha}{\langle}\eta\rangle^{m-|\alpha|+r\delta}.$

Note that
$S^{m}_{1,\delta}\subsetneq\mathcal{H}^{r,\infty}S^{m}_{1,\delta}\subsetneq
C^{r}_{*}S^{m}_{1,\delta}$ for all $r>0$, $m\in{\mathbb{R}}$ and
$\delta\in[0,1]$.

Given $a\in C^{r}_{*}S^{m}_{1,\delta}$ for some $r>0$, $m\in{\mathbb{R}}$ and
$\delta\in[0,1]$, we define the pseudodifferential operator
$a(x,D):\mathcal{S}({\mathbb{R}^{n}})\to\mathcal{S}^{\prime}({\mathbb{R}^{n}})$
with symbol $a$ by

(2.16)
$a(x,D)f(x):=\frac{1}{(2\pi)^{n}}\int_{{\mathbb{R}^{n}}}e^{ix\eta}a(x,\eta)\widehat{f}(\eta)\mathrm{d}\eta$

for $f\in\mathcal{S}({\mathbb{R}^{n}})$ and $x\in{\mathbb{R}^{n}}$.

######  Remark 2.7.

In (2.16) we only defined $a(x,D)f$ for $f\in\mathcal{S}({\mathbb{R}^{n}})$,
whereas we will consider the boundedness of $a(x,D)$ on
$\mathcal{H}^{s,p}({\mathbb{R}^{n}})$ and
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$, for $p\in[1,\infty]$ and
$s\in{\mathbb{R}}$. For $p<\infty$ the Schwartz functions lie dense in
$\mathcal{H}^{s,p}({\mathbb{R}^{n}})$ and
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ for all $s\in{\mathbb{R}}$ (see
[11, Proposition 6.6]), so by obtaining suitable norm bounds for
$f\in\mathcal{S}({\mathbb{R}^{n}})$ we can consider the unique extension of
$a(x,D)$ to these spaces. However, for $p=\infty$ the Schwartz functions are
not dense in $\mathcal{H}^{s,\infty}({\mathbb{R}^{n}})$ and
$\mathcal{H}^{s,\infty}_{FIO}({\mathbb{R}^{n}})$, and extending (2.16) by
adjoint action to $f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ is problematic
both due to the low regularity of $a$ and, even in the smooth case, for
$\delta=1$ and $a\in S^{m}_{1,1}$. For suitable $s\in{\mathbb{R}}$ one can
define $a(x,D)f$ for $f\in\mathcal{H}^{s,\infty}({\mathbb{R}^{n}})$ or
$f\in\mathcal{H}^{s,\infty}_{FIO}({\mathbb{R}^{n}})$ by duality, using also a
symbol decomposition as in the proof of Theorem 4.1 (see [13] or [31, Section
1.2]). However, because the case $p=\infty$ plays a very minor role in this
article, we will not concern ourselves with such matters. Instead, when we
write $a(x,D):X\to Y$ for function spaces $X$ and $Y$ over ${\mathbb{R}^{n}}$
that contain $\mathcal{S}({\mathbb{R}^{n}})$, we simply mean that
$\|a(x,D)f\|_{Y}\lesssim\|f\|_{X}$ for all
$f\in\mathcal{S}({\mathbb{R}^{n}})$.

A basic pseudodifferential operator is the multiplication operator $a(x,D)$
associated with a function
$a\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{r,\infty}S^{0}_{1,0}$.
Such multiplication operators in turn give rise to other pseudodifferential
operators, through paradifferential calculus. For example, for $r>0$ and
$b\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$, we define the high-high
paraproduct operator $R_{b}$ and the high-low paraproduct operator $\pi_{b}$
as follows. For $f\in\mathcal{S}({\mathbb{R}^{n}})$ and $x\in{\mathbb{R}^{n}}$
set

(2.17)
$R_{b}(f)(x):=\sum_{k=0}^{\infty}\sum_{j=k-5}^{k+5}(\psi_{j}(D)b)(x)(\psi_{k}(D)f)(x)$

and

(2.18)
$\pi_{b}(f)(x):=\sum_{k=0}^{\infty}\sum_{j=k+6}^{\infty}(\psi_{j}(D)b)(x)(\psi_{k}(D)f)(x).$

One can check (see also the proof of Lemma 2.9) that $R_{b}$ and $\pi_{b}$ are
pseudodifferential operators with symbols in $S^{-r}_{1,1}$ and
$\mathcal{H}^{r,\infty}S^{-r}_{1,1}$, respectively. A minor role will be
played in the proof of Lemma 3.1 by the following mapping property of these
operators, the proof of which can be skipping during a first reading.

######  Lemma 2.8.

Let $r>0$ and $b\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$. Then

$R_{b},\pi_{b}:\mathcal{H}^{-r,p}({\mathbb{R}^{n}})\to\mathcal{H}^{-\varepsilon,p}({\mathbb{R}^{n}})$

for all $p\in[1,\infty]$ and $\varepsilon>0$.

###### Proof.

By decomposing $b$ into a sum of finitely many functions with restricted
Fourier supports and then moving derivatives around as in the proofs of [30,
Proposition 3.5.B and Proposition 3.5.F], it suffices to show that for each
$c\in\mathcal{H}^{0,\infty}({\mathbb{R}^{n}})={\mathrm{bmo}}({\mathbb{R}^{n}})$
one has

(2.19)
$R_{c},\pi_{c}:\mathcal{H}^{0,p}({\mathbb{R}^{n}})\to\mathcal{H}^{-\varepsilon,p}({\mathbb{R}^{n}}).$

Since $R_{c}$ and $\pi_{c}$ are Calderon-Zygmund operators, as follows for
example from [30, Lemma 3.5.E], one then immediately obtains (2.19) for
$1<p<\infty$.

To deal with $p=1$ and $p=\infty$, we argue in different ways for $R_{c}$ and
$\pi_{c}$. We first consider $\pi_{c}$, and we give an argument which in fact
directly works for all $p\in[1,\infty]$. Let
$f\in\mathcal{S}({\mathbb{R}^{n}})$ and use the support properties of
$\psi_{k}$, as well as the fact that
$({\mathcal{F}}^{-1}(\psi_{k}))_{k=0}^{\infty}\subseteq
L^{1}({\mathbb{R}^{n}})$ is uniformly bounded, to find an
$M\in{{\mathbb{Z}}}_{+}$ such that

$\displaystyle 2^{-l\varepsilon/2}\|\psi_{l}(D)\pi_{c}(f)\|_{L^{p}}\leq
2^{-l\varepsilon/2}\sum_{k=0}^{\infty}\sum_{j=k+6}^{\infty}\big{\|}\psi_{l}(D)\big{(}(\psi_{j}(D)c)(\psi_{k}(D)f)\big{)}\big{\|}_{L^{p}}$
$\displaystyle=2^{-l\varepsilon/2}\sum_{j=l-M}^{l+M}\sum_{k=0}^{j-6}\big{\|}\psi_{l}(D)\big{(}(\psi_{j}(D)c)(\psi_{k}(D)f)\big{)}\big{\|}_{L^{p}}$
$\displaystyle\lesssim
2^{-l\varepsilon/2}\sum_{j=l-M}^{l+M}\sum_{k=0}^{j-6}\|(\psi_{j}(D)c)(\psi_{k}(D)f)\|_{L^{p}}$
$\displaystyle\lesssim\sum_{j=l-M}^{l+M}2^{-j\varepsilon/2}\sum_{k=0}^{j-6}\|\psi_{j}(D)c\|_{L^{\infty}}\|\psi_{k}(D)f\|_{L^{p}}\lesssim
j2^{-j\varepsilon/2}\|c\|_{{\mathrm{bmo}}}\sup_{k\geq
0}\|\psi_{k}(D)f\|_{L^{p}}$

for all $l\geq 0$. For the final inequality we used that
$\|\psi_{j}(D)c\|_{L^{\infty}}\eqsim\|\psi_{j}(D)c\|_{{\mathrm{bmo}}}\lesssim\|c\|_{{\mathrm{bmo}}}$,
a standard fact which follows for example from the Littlewood-Paley
decomposition of $H^{1}({\mathbb{R}^{n}})$ and duality, combined with the
uniform boundedness of $({\mathcal{F}}^{-1}(\psi_{k}))_{k=0}^{\infty}$ in
$L^{1}({\mathbb{R}^{n}})$. Now the Besov space embeddings

(2.20) $\mathcal{H}^{0,p}({\mathbb{R}^{n}})\subseteq
B^{0}_{p,\infty}({\mathbb{R}^{n}})\subseteq
B^{-\varepsilon}_{p,1}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{-\varepsilon,p}({\mathbb{R}^{n}})$

from [32, Sections 2.3.2, 2.5.7 and 2.5.8] conclude the proof for $\pi_{c}$.

Next, for $R_{c}$ we note that, since $R_{c}$ is a Calderon-Zygmund operator,
[10, Theorem 4.2.6] yields $R_{c}:H^{1}({\mathbb{R}^{n}})\to
L^{1}({\mathbb{R}^{n}})$. Combined with the embedding
$L^{1}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{-\varepsilon,1}({\mathbb{R}^{n}})$,
which in turn follows from (2.20) because $L^{1}({\mathbb{R}^{n}})\subseteq
B^{0}_{1,\infty}({\mathbb{R}^{n}})$, we can use (2.2) to see that

$R_{c}(1-q)(D):\mathcal{H}^{1}({\mathbb{R}^{n}})\to
L^{1}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{-\varepsilon,1}({\mathbb{R}^{n}}).$

Moreover, since $q\in C^{\infty}_{c}({\mathbb{R}^{n}})$, there exists an
$N\geq 0$ such that

$\displaystyle\|R_{c}(q(D)f)\|_{L^{1}}$
$\displaystyle\leq\sum_{k=0}^{N}\sum_{j=k-5}^{k+5}\|(\psi_{j}(D)c)(\psi_{k}(D)q(D)f)\|_{L^{1}}$
$\displaystyle\lesssim\sum_{k=0}^{N}\sum_{j=k-5}^{k+5}\|\psi_{j}(D)c\|_{L^{\infty}}\|\psi_{k}(D)q(D)f\|_{L^{1}}$
$\displaystyle\lesssim\sum_{k=0}^{N}\sum_{j=k-5}^{k+5}\|c\|_{{\mathrm{bmo}}}\|q(D)f\|_{L^{1}}\lesssim\|c\|_{{\mathrm{bmo}}}\|f\|_{\mathcal{H}^{1}}$

for all $f\in\mathcal{S}({\mathbb{R}^{n}})$. This proves (2.19) for $R_{c}$
and $p=1$.

Finally, for $p=\infty$ one can use [18, Lemma 3.3, Definition 3.11 and
Corollary 7.3], since $(c,f)\mapsto R_{c}(f)$ is a bilinear Coifman-Meyer
multiplier. ∎

We next describe a symbol smoothing procedure that decomposes a rough symbol
as a sum of a smooth part and a rough part with additional decay. Let $a\in
C^{r}_{*}S^{m}_{1,\delta}$ for $r>0$, $m\in{\mathbb{R}}$ and $\delta\in[0,1]$.
Let $\gamma\in[0,1]$ be given, and recall from Section 2.1 that ${\varphi}\in
C^{\infty}_{c}({\mathbb{R}^{n}})$ satisfies ${\varphi}\equiv 1$ near zero. Now
set, for $(x,\eta)\in{\mathbb{R}}^{2n}$,

$a^{\sharp}_{\gamma}(x,\eta):=\sum_{k=0}^{\infty}\big{(}{\varphi}(2^{-\gamma
k}D)a(\cdot,\eta)\big{)}(x)\psi_{k}(\eta)$

and

$a^{\flat}_{\gamma}(x,\eta):=a(x,\eta)-a^{\sharp}_{\gamma}(x,\eta)=\sum_{k=0}^{\infty}\big{(}(1-{\varphi})(2^{-\gamma
k}D)a(\cdot,\eta)\big{)}(x)\psi_{k}(\eta).$

For the final identity we used that $\sum_{k=0}^{\infty}\psi_{k}(\eta)=1$. The
decomposition $a=a^{\sharp}_{\gamma}+a^{\flat}_{\gamma}$ has the following
properties.

######  Lemma 2.9.

Let $r>0$, $m\in{\mathbb{R}}$ and $\delta,\gamma\in[0,1]$ with
$\gamma\geq\delta$. Then for each $a\in C^{r}_{*}S^{m}_{1,\delta}$ one has
$a^{\sharp}_{\gamma}\in S^{m}_{1,\gamma}$ and $a^{\flat}_{\gamma}\in
C^{r}_{*}S^{m-(\gamma-\delta)r}_{1,\gamma}$. If
$a\in\mathcal{H}^{r,\infty}S^{m}_{1,\delta}$ then additionally
$a^{\flat}_{\gamma}\in\mathcal{H}^{r,\infty}S^{m-(\gamma-\delta)r}_{1,\gamma}$.

###### Proof.

The result is essentially contained in [30, Section 1.3] (see also [31,
Section 1.3]), but for the convenience of the reader we sketch the argument
here. Let $\alpha,\beta\in{{\mathbb{Z}}}_{+}^{n}$ and
$\eta\in{\mathbb{R}^{n}}$. Since ${\varphi}\in
C^{\infty}_{c}({\mathbb{R}^{n}})$, we can use an $L^{1}$-estimate for
$\partial_{x}^{\beta}{\mathcal{F}}^{-1}({\varphi})$, as well as (2.14), to
obtain

$\big{\|}\partial_{x}^{\beta}\partial_{\eta}^{\alpha}\big{(}{\varphi}(2^{-\gamma
k}D)a(\cdot,\eta)\big{)}\big{\|}_{L^{\infty}}\lesssim
2^{k\gamma|\beta|}\|\partial_{\eta}^{\alpha}a(\cdot,\eta)\|_{L^{\infty}}\lesssim
2^{k\gamma|\beta|}{\langle}\eta\rangle^{m-|\alpha|}$

for all $k\geq 0$. Noting that ${\langle}\eta\rangle\eqsim 2^{k}$ if
$\eta\in{\mathrm{supp}}(\psi_{k})$, one can derive from this that
$a^{\sharp}_{\gamma}\in S^{m}_{1,\gamma}$.

On the other hand, for all $\xi\in{\mathbb{R}^{n}}$ and $j\geq 0$ one has
$\psi_{j}(\xi)=0$ if $|\xi|\notin[2^{j-2},2^{j}]$, and there exists a $c>0$
such that ${\varphi}(\xi)=1$ if $|\xi|\leq c$. Hence, for some $M\geq 0$,
(2.15) yields

$\displaystyle\|\partial_{\eta}^{\alpha}(1-{\varphi})(2^{-\gamma
k}D)a(\cdot,\eta)\|_{L^{\infty}}\leq\sum_{j\geq\gamma
k-M}\|(1-{\varphi})(2^{-\gamma
k}D)\psi_{j}(D)\partial_{\eta}^{\alpha}a(\cdot,\eta)\|_{L^{\infty}}$
$\displaystyle\lesssim\sum_{j\geq\gamma
k-M}\|\psi_{j}(D)\partial_{\eta}^{\alpha}a(\cdot,\eta)\|_{L^{\infty}}\leq\sum_{j\geq\gamma
k-M}2^{-jr}\|\partial_{\eta}^{\alpha}a(\cdot,\eta)\|_{C^{r}_{*}}\lesssim
2^{-kr\gamma}{\langle}\eta\rangle^{m-|\alpha|+r\delta}$

for all $k\geq 0$. In the same manner as before, this implies that
$\|\partial^{\alpha}_{\eta}a^{\flat}_{\gamma}(\cdot,\eta)\|_{L^{\infty}}\lesssim{\langle}\eta\rangle^{m-(\gamma-\delta)r-|\alpha|}$.
Finally, by (2.15) and because $C^{r}_{*}({\mathbb{R}^{n}})$ is preserved
under convolution with elements of $L^{1}({\mathbb{R}^{n}})$, one has

(2.21) $\|\partial_{\eta}^{\alpha}(1-{\varphi})(2^{-\gamma
k}D)a(\cdot,\eta)\|_{C^{r}_{*}}\lesssim\|\partial_{\eta}^{\alpha}a(\cdot,\eta)\|_{C^{r}_{*}}\lesssim
2^{m-|\alpha|+\delta r},$

which in turn implies that $a^{\flat}_{\gamma}\in
C^{r}_{*}S^{m-(\gamma-\delta)r}_{1,\gamma}$. If
$a\in\mathcal{H}^{r,\infty}S^{m}_{1,\delta}$ then (2.21) holds with
$C^{r}_{*}$ replaced by $\mathcal{H}^{r,\infty}$, so that
$a^{\flat}_{\gamma}\in\mathcal{H}^{r,\infty}S^{m-(\gamma-\delta)r}_{1,\gamma}$.
∎

##  3\. The subcritical case

In this section we prove results about rough pseudodifferential operators
acting on Hardy spaces for Fourier integral operators, in the subcritical
case. The results in this section suffice for many purposes, except that they
do not yield boundedness on $\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ for a
pseudodifferential operator with symbol $a\in C^{r}_{*}S^{0}_{1,1/2}$ unless
$p=2$.

We first state an extension of a result from [2] about rough
pseudodifferential operators acting on the classical function spaces
$\mathcal{H}^{s,p}({\mathbb{R}^{n}})$ from (2.1).

######  Lemma 3.1.

Let $r>0$, $m\in{\mathbb{R}}$, $\delta\in[0,1]$, $p\in[1,\infty]$ and $a\in
C^{r}_{*}S^{m}_{1,\delta}$. Then

(3.1)
$a(x,D):\mathcal{H}^{s+m,p}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}({\mathbb{R}^{n}})$

for $-(1-\delta)r<s<r$. If $\delta<1$ and
$a\in\mathcal{H}^{r,\infty}S^{m}_{1,\delta}$, then (3.1) also holds for $s=r$.
If additionally $a=b^{\flat}_{\delta}$ for some
$b\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$, then (3.1) also holds for
$s=-(1-\delta)r$ and $m=-\delta r$.

###### Proof.

For the first statement see [13, Theorems 2.3 and 2.4]. The second statement,
concerning $s=r$ in (3.1), is contained in [13, Theorem 2.2]. For the final
statement, let $R_{b}$ and $\pi_{b}$ be as in (2.17) and (2.18), respectively,
and write
$b^{\flat}_{\delta}(x,D)=R_{b}+\pi_{b}+b^{\flat}_{\delta}(x,D)-R_{b}-\pi_{b}$.
Lemma 2.8 shows that

$R_{b}+\pi_{b}:\mathcal{H}^{-r,p}({\mathbb{R}^{n}})\to\mathcal{H}^{-(1-\delta)r,p}({\mathbb{R}^{n}}).$

Moreover, the symbol $c$ of $b^{\flat}_{\delta}(x,D)-R_{b}-\pi_{b}$ is given
by $c=c_{1}+c_{2}$, where

$c_{1}(x,\eta):=\sum_{k=1}^{\infty}\Big{(}-{\varphi}(2^{-\delta
k}D)+\sum_{j=0}^{k-6}\psi_{j}(D)\Big{)}b(\cdot,\eta)(x)\psi_{k}(\eta)$

and

$c_{2}(x,\eta):=-{\varphi}(D)b(\cdot,\eta)(x)\psi_{0}(\eta)$

for $x,\eta\in{\mathbb{R}^{n}}$. For ${\varphi}$ supported sufficiently close
to zero, one has ${\mathcal{F}}c_{1}(\cdot,\eta)(\xi)=0$ if
$|\xi|\geq|\eta|/10$. Moreover, as in the proof of Lemma 2.9 one can show that
for all $\alpha,\beta\in{{\mathbb{Z}}}_{+}^{n}$ there exists a
$C_{\alpha,\beta}\geq 0$ such that

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}c_{1}(x,\eta)|\leq
C_{\alpha,\beta}{\langle}\eta\rangle^{|\beta|-|\alpha|}$

for all $x,\eta\in{\mathbb{R}^{n}}$. Now [30, Proposition 3.4.F] yields

$c_{1}(x,D):\mathcal{H}^{t-\delta
r,p}({\mathbb{R}^{n}})\to\mathcal{H}^{t,p}({\mathbb{R}^{n}})$

for all $t\in{\mathbb{R}}$, and specifically for $t=s$. One can check directly
that the same mapping property holds for $c_{2}(x,D)$. This concludes the
proof. ∎

We next state a result about pseudodifferential operators with smooth symbols
on $\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$. This lemma reduces to a
theorem in [11], but we note that it can also be proved in a similar manner as
Theorem 4.1 below (see Remark 4.4).

######  Lemma 3.2.

Let $m\in{\mathbb{R}}$ and $a\in S^{m}_{1,1/2}$. Then

$a(x,D):\mathcal{H}^{s+m,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for all $p\in[1,\infty]$ and $s\in{\mathbb{R}}$.

###### Proof.

In the case where $m=s=0$, the statement is contained in [11, Theorem 6.10].
For general $m$ and $s$, it suffices to show that

${\langle}D\rangle^{s}a(x,D){\langle}D\rangle^{-m-s}:\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$

is bounded. But ${\langle}D\rangle^{s}a(x,D){\langle}D\rangle^{-m-s}$ has a
symbol in $S^{0}_{1,1/2}$, cf. [30, Proposition 0.3.C], so the general case is
reduced to what we have already shown. ∎

We now prove a proposition that yields, in particular, boundedness of
pseudodifferential operators with $C^{r}_{*}S^{-\varepsilon}_{1,\delta}$
symbols on $\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$, for any
$r,\varepsilon>0$ and $p$ close enough to $2$. Recall from (2.8) that, for
$p\in[1,\infty]$, $s(p)=\frac{n-1}{2}|\frac{1}{2}-\frac{1}{p}|$.

######  Proposition 3.3.

Let $r>0$, $m\in{\mathbb{R}}$, $\delta\in[0,1]$, $p\in[1,\infty]$ and $a\in
C^{r}_{*}S^{m}_{1,\delta}$. Then

(3.2)
$a(x,D):\mathcal{H}^{s+2s(p)+m,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for $-(1-\delta)r<s+s(p)<r$. If $\delta<1$ and
$a\in\mathcal{H}^{r,\infty}S^{m}_{1,\delta}$, then (3.2) also holds for
$s+s(p)=r$. If additionally $a=b^{\flat}_{\delta}$ for some
$b\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$, then (3.2) also holds for
$s+s(p)=-(1-\delta)r$ and $m=-\delta r$.

###### Proof.

The first statement follows by combining Lemma 3.1 and the Sobolev embeddings
for $\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ from (2.9):

$a(x,D):\mathcal{H}^{s+2s(p)+m,p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s+s(p)+m,p}({\mathbb{R}^{n}})\to\mathcal{H}^{s+s(p),p}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}}).$

The remaining statements follow in the same manner from Lemma 3.1 and (2.9). ∎

######  Remark 3.4.

For the proof of Theorem 4.1 it is convenient to note that (3.2) also holds
for $s+s(p)=-(1-\delta)r$ and $m=-\delta r$ when
$a=((b^{\flat}_{\delta^{\prime}})^{\flat}_{\delta^{\prime}})^{\flat}_{\delta}$
for some $b\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$ if
$\delta^{\prime}\in[0,\delta]$. This follows either from a minor modification
of the proof of Lemma 3.1, or by noting that

$((b^{\flat}_{\delta^{\prime}})^{\flat}_{\delta^{\prime}})^{\flat}_{\delta}(x,D)-b^{\flat}_{\delta}(x,D):\mathcal{H}^{t-\delta
r,p}({\mathbb{R}^{n}})\to\mathcal{H}^{t,p}({\mathbb{R}^{n}})$

is bounded for all $t\in{\mathbb{R}}$, as in the proof of Lemma 3.1.

By combining Proposition 3.3 with the symbol smoothing procedure from Section
2.2, we obtain the following corollary. Much of this result will in fact be
improved in Corollary 4.5. However, the range of Sobolev indices in the latter
result is smaller than what we will obtain here. Moreover, the proof of
Corollary 4.5, via Theorem 4.1, is relatively involved, and we believe that
there is value in demonstrating how a simpler proof yields a boundedness
statement that already suffices for many purposes.

######  Corollary 3.5.

Let $r>0$, $m\in{\mathbb{R}}$, $\delta\in[0,1/2]$, $p\in[1,\infty]$ and $a\in
C^{r}_{*}S^{m}_{1,\delta}$. Set $\sigma:=\max(0,2s(p)-(\frac{1}{2}-\delta)r)$.
Then

(3.3)
$a(x,D):\mathcal{H}^{s+m+\sigma,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for $-r/2<s+s(p)<r$. If $a\in\mathcal{H}^{r,\infty}S^{m}_{1,\delta}$, then
(3.3) also holds for $s+s(p)=r$. If
$a\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$ and $p\in(1,\infty)$, then
(3.3) holds for all $-r/2\leq s+s(p)\leq r$, with $m=\delta=0$.

###### Proof.

As in Lemma 2.9, we write $a=a^{\sharp}_{1/2}+a^{\flat}_{1/2}$ with
$a^{\sharp}_{1/2}\in S^{m}_{1,1/2}$ and $a^{\flat}_{1/2}\in
C^{r}_{*}S^{m-(1/2-\delta)r}_{1,1/2}$. Then Lemma 3.2 yields

$a^{\sharp}_{1/2}(x,D):\mathcal{H}^{s+m+\sigma,p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s+m,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for all $s\in{\mathbb{R}}$. On the other hand, Proposition 3.3 shows that

(3.4)
$a^{\flat}_{1/2}(x,D):\mathcal{H}^{s+m+\sigma,p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s+2s(p)+m-(1/2-\delta)r,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for $-r/2<s+s(p)<r$. This proves the first statement. For the second and third
statement one only has to extend (3.4) to the cases where $s+s(p)=r$ and
$s+s(p)=-r/2$, respectively. But this follows from Proposition 3.3 as well,
noting that if $a\in\mathcal{H}^{r,\infty}S^{m}_{1,\delta}$ then
$a^{\flat}_{1/2}\in\mathcal{H}^{r,\infty}S^{m-(1/2-\delta)r}_{1,1/2}$, by
Lemma 2.9. ∎

######  Remark 3.6.

For $a\in C^{r}_{*}S^{m}_{1,\delta}$ with $\delta<1/2$, one can obtain results
for $s$ in a slightly larger range than that in Corollary 3.5. To this end one
chooses $\delta^{\prime}\in(\delta,1/2)$ and applies the symbol smoothing
procedure to write
$a=a^{\sharp}_{\delta^{\prime}}+a^{\flat}_{\delta^{\prime}}$ with
$a^{\sharp}_{\delta^{\prime}}\in S^{m}_{1,\delta^{\prime}}$ and
$a^{\flat}_{\delta^{\prime}}\in
C^{r}_{*}S^{m-(\delta^{\prime}-\delta)r}_{1,\delta^{\prime}}$. Then Lemma 3.2
and Proposition 3.3 combine to yield

(3.5)
$a(x,D):\mathcal{H}^{s+m+\sigma^{\prime},p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for $-(1-\delta^{\prime})r<s+s(p)<r$ and
$\sigma^{\prime}:=\max(0,2s(p)-(\delta^{\prime}-\delta)r)$. Note that
$\sigma^{\prime}\geq\sigma$, so that (3.3) is stronger than (3.5). However,
for $p$ sufficiently close to $2$ one has $\sigma=\sigma^{\prime}=0$, and then
(3.5) is as strong as (3.3) and holds for a bigger range of $s$.

######  Remark 3.7.

In Definition 2.6 one can consider symbols
$a\in\mathcal{H}^{r,q}S^{m}_{1,\delta}$ with coefficients in Sobolev spaces
$\mathcal{H}^{r,q}({\mathbb{R}^{n}})$ for $q<\infty$, by replacing
$C^{r}_{*}({\mathbb{R}^{n}})$ in (2.15) by
$\mathcal{H}^{r,q}({\mathbb{R}^{n}})$. The Sobolev embedding
$\mathcal{H}^{r,q}({\mathbb{R}^{n}})\subseteq C^{r-n/q}_{*}({\mathbb{R}^{n}})$
allows one to directly derive mapping properties for pseudodifferential
operators with $\mathcal{H}^{r,q}S^{m}_{1,\delta}$ symbols from the results in
this article. However, slightly better results can be obtained in an
alternative manner. Mapping properties for pseudodifferential operators with
$\mathcal{H}^{r,q}S^{m}_{1,\delta}$ symbols on
$\mathcal{H}^{s,p}({\mathbb{R}^{n}})$ were obtained in [13]. As in Proposition
3.3, one can combine such results with the Sobolev embeddings for
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ in (2.9). Moreover, the symbol
decomposition in Lemma 2.9 also applies to $\mathcal{H}^{r,q}S^{m}_{1,\delta}$
symbols (see [31, Proposition 1.8.2]). Just as in Corollary 3.5, one can then
derive mapping properties of operators with
$\mathcal{H}^{r,q}S^{m}_{1,\delta}$ symbols on
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$. This applies in particular to
multiplication operators associated with functions
$g\in\mathcal{H}^{r,q}({\mathbb{R}^{n}})$.

##  4\. The critical case

In Corollary 3.5 we proved boundedness of pseudodifferential operators with
$C^{r}_{*}S^{0}_{1,\delta}$ symbols on
$\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ for any $r>0$ and $\delta<1/2$, at
least for $p$ sufficiently close to $2$ and $s$ sufficiently close to $-s(p)$.
However, for $\delta=1/2$, Corollary 3.5 only yields boundedness for $p=2$.
Lemma 3.2 shows that a pseudodifferential operator with an $S^{0}_{1,1/2}$
symbol is bounded on $\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ for all
$p\in[1,\infty]$ and $s\in{\mathbb{R}}$, but that leaves open the question
whether operators with $C^{r}_{*}S^{0}_{1,1/2}$ symbols, for reasonably small
$r$, are bounded on $\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ for $p\neq 2$.
In this section we show that this is indeed the case.

The main result of this section is as follows. It shows in particular that
pseudodifferential operators with $C^{r}_{*}S^{0}_{1,1/2}$ symbols are bounded
on $\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$ for all $1<p<\infty$ and
$-r/2+2s(p)<s+s(p)<r$, as long as $r>n-1$. Theorem 1.1 is a special case of
this.

######  Theorem 4.1.

Let $r>0$, $m\in{\mathbb{R}}$, $p\in(1,\infty)$ and $a\in
C^{r}_{*}S^{m}_{1,1/2}$. For $\varepsilon>0$, set

$\tau:=\begin{cases}0&\text{if }r>n-1,\\\ \varepsilon&\text{if }r=n-1,\\\
2s(p)\big{(}1-\frac{r}{n-1}\big{)}&\text{if }r<n-1,\end{cases}$

and

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

Then

(4.1)
$a(x,D):\mathcal{H}^{s+m+\tau,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for $-(1-\gamma)r<s+s(p)<r$. If $a\in\mathcal{H}^{r,\infty}S^{m}_{1,1/2}$,
then (4.1) also holds for $s+s(p)=r$. If $a=b^{\flat}_{1/2}$ for some
$b\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$, then (4.1) holds for all
$-(1-\gamma)r\leq s+s(p)\leq r$, with $m=-r/2$.

###### Proof.

By replacing $a(x,D)$ by $a(x,D){\langle}D\rangle^{-m}$, we may assume that
$m=0$. For the final statement this requires replacing $a(x,D)$ by
$b^{\flat}_{1/2}(x,D){\langle}D\rangle^{r/2}$.

#### Symbol smoothing

We first use the symbol smoothing procedure and the results from the previous
section to remove some of the frequencies of $a$. More precisely, we claim
that it suffices to prove the following statement. If $a\in
C^{r}_{*}S^{0}_{1,1/2}$ has the additional property that, for some $c>0$ and
all $\eta\in{\mathbb{R}^{n}}$, one has

(4.2)
${\mathrm{supp}}({\mathcal{F}}a(\cdot,\eta))\subseteq\\{\xi\in{\mathbb{R}^{n}}\mid
c|\eta|^{1/2}\leq|\xi|\leq\tfrac{1}{16}(1+|\eta|)^{\gamma}\\},$

then (4.1) holds for all $s\in{\mathbb{R}}$.

To prove this claim, let a general $a\in C^{r}_{*}S^{0}_{1,1/2}$ be given and
note that

$\tfrac{1}{2}\leq\gamma\leq\tfrac{1}{2}+\tfrac{2s(p)}{n-1}<\tfrac{1}{2}+\tfrac{n-1}{2(n-1)}=1$

and

$\tau\geq 2s(p)-(\gamma-\tfrac{1}{2})r.$

We apply the symbol smoothing procedure in Lemma 2.9 twice, once to $a$ and
once to $a^{\flat}_{1/2}$, to write

(4.3)
$a=a^{\sharp}_{1/2}+a^{\flat}_{1/2}=a^{\sharp}_{1/2}+(a^{\flat}_{1/2})^{\sharp}_{\gamma}+(a^{\flat}_{1/2})^{\flat}_{\gamma}.$

Then $a^{\sharp}_{1/2}\in S^{0}_{1,1/2}$ and
$(a^{\flat}_{1/2})^{\flat}_{\gamma}\in
C^{r}_{*}S^{-(\gamma-1/2)r}_{1,\gamma}$, by Lemma 2.9. Hence

(4.4)
$a^{\sharp}_{1/2}(x,D):\mathcal{H}^{s+\tau,p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for all $s\in{\mathbb{R}}$, by Lemma 3.2, and

(4.5)
$(a^{\flat}_{1/2})^{\flat}_{\gamma}(x,D):\mathcal{H}^{s+\tau,p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s+2s(p)-(\gamma-1/2)r,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for $-(1-\gamma)r<s+s(p)<r$, by Proposition 3.3. Moreover, if
$a\in\mathcal{H}^{r,\infty}S^{0}_{1,1/2}$ then
$(a^{\flat}_{1/2})^{\flat}_{\gamma}\in\mathcal{H}^{r,\infty}S^{-(\gamma-1/2)r}_{1,\gamma}$,
again by Lemma 2.9, and then Proposition 3.3 shows that (4.5) also holds for
$s+s(p)=r$. Finally, if $a(x,D)=b_{1/2}^{\flat}(x,D){\langle}D\rangle^{r/2}$
for some $b\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$, then Remark 3.4 shows
that (4.5) also holds for $s+s(p)=-(1-\gamma)r$.

It thus follows from (4.3), (4.4) and (4.5) that it suffices to prove (4.1)
for all $s\in{\mathbb{R}}$ with $a(x,D)$ replaced by
$(a^{\flat}_{1/2})^{\sharp}_{\gamma}(x,D)$. It is straightforward to check
that, for ${\varphi}$ supported sufficiently close to zero,
$(a^{\flat}_{1/2})^{\sharp}_{\gamma}$ has the property in (4.2) for some $c>0$
and all $\eta\in{\mathbb{R}^{n}}$:

${\mathrm{supp}}\big{(}{\mathcal{F}}\big{(}(a^{\flat}_{1/2})^{\sharp}_{\gamma}(\cdot,\eta)\big{)}\big{)}\subseteq\\{\xi\in{\mathbb{R}^{n}}\mid
c|\eta|^{1/2}\leq|\xi|\leq\tfrac{1}{16}(1+|\eta|)^{\gamma}\\}.$

Moreover, $a^{\flat}_{1/2}\in C^{r}_{*}S^{0}_{1,1/2}$ and
$(a^{\flat}_{1/2})^{\flat}_{\gamma}\in
C^{r}_{*}S^{-(\gamma-1/2)r}_{1,\gamma}\subseteq C^{r}_{*}S^{0}_{1,1/2}$, so
$(a^{\flat}_{1/2})^{\sharp}_{\gamma}\in C^{r}_{*}S^{0}_{1,1/2}$ as well, by
(4.3). This proves the claim, and the remainder of the proof is dedicated to
showing (4.1) for all $s\in{\mathbb{R}}$ and $a\in C^{r}_{*}S^{0}_{1,1/2}$
satisfying (4.2).

#### A symbol decomposition

Next, we will use a symbol decomposition as in [4, Chapter 2] (see also [2,
13, 31]). Let $(\psi_{k})_{k=0}^{\infty}$ be the Littlewood-Paley
decomposition from (2.10), and suppose that $\psi_{1}(\eta)=0$ for
$|\eta|\notin[(1+\varepsilon)/2,2-\varepsilon]$ for some $\varepsilon>0$. Let
$(\widetilde{\psi}_{k})_{k=0}^{\infty}\subseteq
C^{\infty}_{c}({\mathbb{R}^{n}})$ be such that $\widetilde{\psi}_{k}\equiv 1$
on ${\mathrm{supp}}(\psi_{k})$ for each $k\geq 0$,
$\widetilde{\psi}_{0}(\eta)=0$ for $|\eta|>2$, and such that for all $k\geq 1$
and $\eta\in{\mathbb{R}^{n}}$ one has $\widetilde{\psi}_{1}(\eta)=0$ if
$|\eta|\notin[1/2,2]$, and
$\widetilde{\psi}_{k}(\eta)=\widetilde{\psi}_{1}(2^{-k+1}\eta)$. By
decomposing the function

$\zeta\mapsto a(x,2^{k+1}\pi\zeta)\psi_{k}(2^{k+1}\pi\zeta)$

on $[-1/2,1/2]^{n}$ into its Fourier modes, we can write

$a(x,\eta)\psi_{k}(\eta)=\sum_{\beta\in{{\mathbb{Z}}}^{n}}c_{k,\beta}(x)e^{i\beta
2^{-k}\eta}\widetilde{\psi}_{k}(\eta)$

for all $k\geq 0$ and $x,\eta\in{\mathbb{R}^{n}}$, where

(4.6) $c_{k,\beta}(x)=\int_{[-1/2,1/2]^{n}}e^{-2\pi
i\beta\zeta}a(x,2^{k+1}\pi\zeta)\psi_{k}(2^{k+1}\pi\zeta)\mathrm{d}\zeta.$

Then

(4.7)
$a(x,\eta)=\sum_{k=0}^{\infty}a(x,\eta)\psi_{k}(\eta)=\sum_{\beta\in{{\mathbb{Z}}}^{n}}\sum_{k=0}^{\infty}c_{k,\beta}(x)e^{i\beta
2^{-k}\eta}\widetilde{\psi}_{k}(\eta).$

Here the $c_{k,\beta}$ decay rapidly as $|\beta|\to\infty$, as can be seen by
integrating by parts in (4.6) and using that $a\in C^{r}_{*}S^{0}_{1,1/2}$ and
that $\psi_{k}(\eta)=\psi_{1}(2^{-k+1}\eta)=0$ if
$|\eta|\notin[2^{k-2},2^{k}]$ and $k\geq 1$. Moreover, for each $k\geq 0$ one
has ${\mathcal{F}}c_{k,\beta}(\xi)=0$ if $|\xi|<c2^{(k-2)/2}$ or
$|\xi|>2^{k\gamma-3}\geq\tfrac{1}{16}(1+2^{k})^{\gamma}$, as follows from
assumption (4.2) and the support condition on $\psi_{k}$.

By inspecting the properties of each of the series in (4.7) for fixed
$\beta\in{{\mathbb{Z}}}_{+}^{n}$, using in particular the rapid decay of the
$c_{k,\beta}$ and that $a\in C^{r}_{*}S^{0}_{1,1/2}$, it follows that we may
suppose in the remainder that $a$ has the form

(4.8)
$a(x,\eta)=\sum_{k=0}^{\infty}a_{k}(x)\chi_{k}(\eta)\quad(x,\eta\in{\mathbb{R}^{n}}),$

where $(a_{k})_{k=0}^{\infty}\subseteq C^{r}_{*}({\mathbb{R}^{n}})$ satisfies

(4.9) $\sup_{k\geq
0}\|a_{k}\|_{L^{\infty}({\mathbb{R}^{n}})}+2^{-kr/2}\|a_{k}\|_{C^{r}_{*}({\mathbb{R}^{n}})}<\infty$

and, for all $k\geq 0$ and $\eta\in{\mathbb{R}^{n}}$,

(4.10)
${\mathrm{supp}}({\mathcal{F}}a_{k})\subseteq\\{\xi\in{\mathbb{R}^{n}}\mid
c2^{(k-2)/2}\leq|\xi|\leq 2^{k\gamma-3}\\}.$

The latter assumption implies that in fact $(a_{k})_{k=0}^{\infty}\subseteq
C^{\infty}({\mathbb{R}^{n}})$. Moreover, $(\chi_{k})_{k=0}^{\infty}\subseteq
C^{\infty}_{c}({\mathbb{R}^{n}})$ is such that $\chi_{0}(\eta)=0$ if
$|\eta|>1$, $\chi_{1}(\eta)=0$ if $|\eta|\notin[1/2,2]$, and
$\chi_{k}(\eta)=\chi_{1}(2^{-k+1}\eta)$ for all $k>1$ and
$\eta\in{\mathbb{R}^{n}}$.

#### The low-frequency components

After this preliminary work, we now get to the heart of the proof. Let
$f\in\mathcal{S}({\mathbb{R}^{n}})$. For $a$ given by (4.8), by (2.7) we need
to show that

(4.11)
$\|q(D)a(x,D)f\|_{L^{p}}+\Big{(}\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)a(x,D)f\|_{\mathcal{H}^{s,p}}^{p}\mathrm{d}{\omega}\Big{)}^{1/p}\lesssim\|f\|_{\mathcal{H}^{s+\tau,p}_{FIO}}$

for an implicit constant independent of $f$. Throughout, for $k\geq 0$ write
$f_{k}:=\chi_{k}(D)f$.

We first deal with the low-frequency component $q(D)a(x,D)f$. Here one can
directly combine the smoothing property of $q(D)$ with Lemma 3.1 and the
Sobolev embeddings in (2.9), but instead we make an observation which will
also be useful later on. For each $k\geq 1$ one has

(4.12) ${\mathcal{F}}(a_{k}f_{k})(\xi)=0\quad(|\xi|\notin[2^{k-3},2^{k+1}]),$

as follows from the support properties of ${\mathcal{F}}(a_{k})$ and
${\mathcal{F}}(f_{k})=\chi_{k}{\mathcal{F}}(f)$. Indeed,

(4.13)
${\mathrm{supp}}({\mathcal{F}}(a_{k}f_{k}))={\mathrm{supp}}({\mathcal{F}}(a_{k})\ast{\mathcal{F}}(f_{k}))\subseteq{\mathrm{supp}}({\mathcal{F}}(a_{k}))+{\mathrm{supp}}({\mathcal{F}}(f_{k})),$

and for $\zeta\in{\mathrm{supp}}({\mathcal{F}}(a_{k}))$ and
$\eta\in{\mathrm{supp}}({\mathcal{F}}(f_{k}))$ one has $|\zeta|\leq
2^{k\gamma-3}\leq 2^{k-3}$, $|\eta|\in[2^{k-2},2^{k}]$ and

(4.14) $2^{k-3}=2^{k-2}-2^{k-3}\leq|\zeta+\eta|\leq 2^{k}+2^{k-3}\leq
2^{k+1}.$

Since $q(\xi)=0$ for $|\xi|\geq 2$, it follows from (4.12) that

$q(D)a(x,D)f=\sum_{k=0}^{3}q(D)(a_{k}f_{k}).$

Now the fact that $q\in C^{\infty}_{c}({\mathbb{R}^{n}})$, the assumption that
$(a_{k})_{k=0}^{\infty}\subseteq C^{\infty}({\mathbb{R}^{n}})$ and
$(\chi_{k})_{k=0}^{\infty}\subseteq C^{\infty}_{c}({\mathbb{R}^{n}})$, and the
Sobolev embeddings from (2.9) yield

$\displaystyle\|q(D)a(x,D)f\|_{L^{p}}$
$\displaystyle\leq\sum_{k=0}^{3}\|q(D)(a_{k}f_{k})\|_{L^{p}}\lesssim\sum_{k=0}^{3}\|a_{k}f_{k}\|_{\mathcal{H}^{s+\tau-s(p),p}}$
$\displaystyle\lesssim\sum_{k=0}^{3}\|\chi_{k}(D)f\|_{\mathcal{H}^{s+\tau-s(p),p}}\lesssim\|f\|_{\mathcal{H}^{s+\tau-s(p),p}}\lesssim\|f\|_{\mathcal{H}^{s+\tau,p}_{FIO}}.$

This proves half of (4.11).

For the other half of (4.11) first note that, since $q\in
C^{\infty}_{c}({\mathbb{R}^{n}})$, there exists an $N\geq 0$ such that
$a(x,D)q(D)f=\sum_{k=0}^{N}a_{k}\chi_{k}(D)q(D)f$. By considering functions
with a single nonzero coordinate, it follows in particular from Lemma 2.4 that
${\varphi}_{{\omega}}(D):\mathcal{H}^{s+\frac{n-1}{4},p}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}({\mathbb{R}^{n}})$
uniformly in ${\omega}\in S^{n-1}$. Hence we obtain in the same manner as
before that

$\displaystyle\|{\varphi}_{{\omega}}(D)a(x,D)q(D)f\|_{\mathcal{H}^{s,p}}\leq\sum_{k=0}^{N}\|{\varphi}_{w}(D)(a_{k}q(D)\chi_{k}(D)f)\|_{\mathcal{H}^{s,p}}$
$\displaystyle\lesssim\sum_{k=0}^{N}\|a_{k}q(D)\chi_{k}(D)f\|_{\mathcal{H}^{s+(n-1)/4,p}}\lesssim\sum_{k=0}^{N}\|q(D)\chi_{k}(D)f\|_{\mathcal{H}^{s+(n-1)/4,p}}$
$\displaystyle\lesssim\sum_{k=0}^{N}\|f\|_{\mathcal{H}^{s+\tau-s(p),p}}\lesssim\|f\|_{\mathcal{H}^{s+\tau-s(p),p}}\lesssim\|f\|_{\mathcal{H}^{s+\tau,p}_{FIO}},$

for implicit constants independent of ${\omega}$ and $f$. Hence

$\Big{(}\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)a(x,D)q(D)f\|_{\mathcal{H}^{s,p}}^{p}\mathrm{d}{\omega}\Big{)}^{1/p}\lesssim\|f\|_{\mathcal{H}^{s+\tau,p}_{FIO}},$

and for (4.11) it remains to show that

$\Big{(}\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)a(x,D)(1-q)(D)f\|_{\mathcal{H}^{s,p}}^{p}\mathrm{d}{\omega}\Big{)}^{1/p}\lesssim\|f\|_{\mathcal{H}^{s+\tau,p}_{FIO}}.$

Moreover, since
$q(D):\mathcal{H}^{s+\tau,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s+\tau,p}_{FIO}({\mathbb{R}^{n}})$
is bounded, as follows e.g. from (2.9) or Lemma 3.2, it suffices to show that

$\Big{(}\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)a(x,D)(1-q)(D)f\|_{\mathcal{H}^{s,p}}^{p}\mathrm{d}{\omega}\Big{)}^{1/p}\lesssim\|(1-q)(D)f\|_{\mathcal{H}^{s+\tau,p}_{FIO}}.$

This is what we will focus on in the remainder of the proof. However, for
simplicity of notation and because the only property that we will use of
$(1-q)(D)f$ is that $\chi_{0}(D)(1-q)(D)f=0$, we will continue working with
$f$ and prove more generally that

(4.15)
$\Big{(}\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)a(x,D)f\|_{\mathcal{H}^{s,p}}^{p}\mathrm{d}{\omega}\Big{)}^{1/p}\lesssim\|f\|_{\mathcal{H}^{s+\tau,p}_{FIO}}$

under the additional assumption that $\chi_{0}(D)f=0$.

#### The high-frequency component

In the remainder of the proof we will show (4.15) under the additional
assumption that $f_{0}=\chi_{0}(D)f=0$. To this end, write
$a_{k,j}:=\psi_{j}(D)a_{k}$ for $k,j\geq 0$, and $a_{k,j}:=0$ for $j<0$. Note
that, by (4.10), there exists an $M\geq 0$ such that

$a(x,D)f=\sum_{k=1}^{\infty}\sum_{j=-M+k/2}^{k\gamma-2}a_{k,j}f_{k}.$

For convenience, we assume that $M\geq 3$. Let ${\omega}\in S^{n-1}$. Then
Lemma 2.3 and (4.12) yield

(4.16) $\displaystyle\|{\varphi}_{{\omega}}(D)a(x,D)f\|_{\mathcal{H}^{s,p}}$
$\displaystyle=\Big{\|}\sum_{k=1}^{\infty}{\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{{\omega}}(D)\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}a_{k,j}f_{k}\Big{)}\Big{\|}_{\mathcal{H}^{s+(n-1)/4,p}}$
$\displaystyle\lesssim\Big{\|}\Big{(}\sum_{k=1}^{\infty}4^{k(s+\frac{n-1}{4})}\Big{|}{\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{{\omega}}(D)\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}a_{k,j}f_{k}\Big{)}\Big{|}^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}}$

for an implicit constant independent of ${\omega}$. Next, as in (2.6), let
$m\in S^{(n-1)/4}({\mathbb{R}^{n}})$ be such that
$f_{k}=\int_{S^{n-1}}m(D){\varphi}_{\nu}(D)f_{k}\,\mathrm{d}\nu$ for all
$k\geq 1$, which is possible since $\widehat{f}_{k}(\zeta)=0$ for
$|\zeta|<1/2$. Write $f_{k,\nu}:=m(D){\varphi}_{\nu}(D)f_{k}$ for $\nu\in
S^{n-1}$. We claim that

(4.17)
${\varphi}_{{\omega}}(D)\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}a_{k,j}f_{k}\Big{)}={\varphi}_{{\omega}}(D)\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}a_{k,j}\int_{F_{{\omega},k,j}}f_{k,\nu}\,\mathrm{d}\nu\Big{)},$

where

(4.18) $F_{{\omega},k,j}:=\\{\nu\in S^{n-1}\mid|\nu-{\omega}|\leq
2^{3+M+j-k}\\}$

for $-M+k/2\leq j\leq k\gamma-2$. Note that $|F_{{\omega},k,j}|\eqsim
2^{(n-1)(j-k)}$.

To prove (4.17), let $k\geq 1$, $-M+k/2\leq j\leq k\gamma-2$, $\nu\in
S^{n-1}$, $\zeta\in{\mathrm{supp}}({\mathcal{F}}(a_{k,j}))$ and
$\eta\in{\mathrm{supp}}({\mathcal{F}}(f_{k,\nu}))$ be given. Then
$|\hat{\eta}-\nu|\leq 2^{1-k/2}$ and, using (4.14),

(4.19)
$\frac{\big{|}|\zeta+\eta|-|\eta|\big{|}}{|\zeta+\eta|}\leq\frac{|\zeta|}{|\zeta+\eta|}\leq\frac{2^{j}}{2^{k-3}}.$

Hence

$\Big{|}\frac{\eta}{|\zeta+\eta|}-\nu\Big{|}\leq|\hat{\eta}-\nu|+\Big{|}\frac{\eta}{|\zeta+\eta|}-\frac{\eta}{|\eta|}\Big{|}\leq
2^{1-k/2}+\frac{\big{|}|\zeta+\eta|-|\eta|\big{|}}{|\zeta+\eta|}\leq
2^{1-k/2}+2^{3+j-k}$

and, using (4.19) again,

$\displaystyle\Big{|}\frac{\zeta+\eta}{|\zeta+\eta|}-\nu\Big{|}$
$\displaystyle\leq\Big{|}\frac{\eta}{|\zeta+\eta|}-\nu\Big{|}+\frac{|\zeta|}{|\zeta+\eta|}\leq
2^{1-k/2}+2^{4+j-k}=2^{1-k+k/2}+2^{4+j-k}$ $\displaystyle\leq
2^{1+M+j-k}+2^{4+j-k}\leq 2^{2+M+j-k},$

where we also used that $M\geq 3$. If $\nu\notin F_{{\omega},k,j}$ then this
in turn implies that

$\displaystyle\Big{|}\frac{\zeta+\eta}{|\zeta+\eta|}-{\omega}\Big{|}$
$\displaystyle\geq|\nu-{\omega}|-\Big{|}\frac{\zeta+\eta}{|\zeta+\eta|}-\nu\Big{|}>2^{3+M+j-k}-2^{2+M+j-k}$
$\displaystyle=2^{2+M+j-k}\geq 2^{2-k+k/2}\geq
2^{1-k/2}\geq|\zeta+\eta|^{-1/2},$

so that $\zeta+\eta\notin{\mathrm{supp}}({\varphi}_{{\omega}})$, by (2.4). By
(4.13) with $a_{k}$ replaced by $a_{k,j}$ and $f_{k}$ replaced by $f_{k,\nu}$,
this proves (4.17).

Now we can combine (4.16), (4.17) and Lemma 2.4 to write

$\displaystyle\|{\varphi}_{{\omega}}(D)a(x,D)f\|_{\mathcal{H}^{s,p}}$
$\displaystyle\lesssim\Big{\|}\Big{(}\sum_{k=1}^{\infty}4^{k(s+\frac{n-1}{4})}\Big{|}{\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{{\omega}}(D)\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}a_{k,j}f_{k}\Big{)}\Big{|}^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}}$
$\displaystyle=\Big{\|}\Big{(}\sum_{k=1}^{\infty}4^{k(s+\frac{n-1}{4})}\Big{|}{\langle}D\rangle^{-\frac{n-1}{4}}{\varphi}_{{\omega}}(D)\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}a_{k,j}\int_{F_{{\omega},k,j}}f_{k,\nu}\,\mathrm{d}\nu\Big{)}\Big{|}^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}}$
$\displaystyle\lesssim\Big{\|}\Big{(}\sum_{k=1}^{\infty}4^{k(s+\frac{n-1}{4})}\Big{|}\sum_{j=-M+k/2}^{k\gamma-2}a_{k,j}\int_{F_{{\omega},k,j}}f_{k,\nu}\,\mathrm{d}\nu\Big{|}^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}}$
$\displaystyle\lesssim\Big{\|}\Big{(}\sum_{k=1}^{\infty}4^{k(s+\frac{n-1}{4})}\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}|a_{k,j}|2^{(n-1)(j-k)}\fint_{F_{{\omega},k,j}}|f_{k,\nu}|\mathrm{d}\nu\Big{)}^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}}.$

Next, by (4.9), one has

$\|a_{k,j}\|_{L^{\infty}}=\|\psi_{j}(D)a_{k}\|_{L^{\infty}}\leq
2^{-jr}\|a_{k}\|_{C^{r}_{*}}\lesssim 2^{(\frac{k}{2}-j)r}$

for all $k,j\geq 0$. Hence

$\displaystyle\|{\varphi}_{{\omega}}(D)a(x,D)f\|_{\mathcal{H}^{s,p}}$
$\displaystyle\lesssim\Big{\|}\Big{(}\sum_{k=1}^{\infty}4^{k(s+\frac{n-1}{4})}\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}2^{k(\frac{r}{2}+1-n)+j(n-1-r)}\fint_{F_{{\omega},k,j}}|f_{k,\nu}|\mathrm{d}\nu\Big{)}^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}}$
$\displaystyle=\Big{\|}\Big{(}\sum_{k=1}^{\infty}\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}2^{k(\frac{r}{2}-\tau-\frac{n-1}{2})+j(n-1-r)}\fint_{F_{{\omega},k,j}}2^{k(s+\tau-\frac{n-1}{4})}|f_{k,\nu}|\mathrm{d}\nu\Big{)}^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}}.$

Let $M$ be the centered Hardy–Littlewood maximal function on $S^{n-1}$, given
for $g\in L^{1}(S^{n-1})$ by

$Mg({\omega}):=\sup_{B}\fint_{B}|g(\nu)|\mathrm{d}\nu\quad({\omega}\in
S^{n-1}),$

where the supremum is taken over all balls $B\subseteq S^{n-1}$ with center
${\omega}$. Also, for $x\in{\mathbb{R}^{n}}$ and $k\geq 1$ set

$g_{k,x}(\nu):=2^{k(s+\tau-\frac{n-1}{4})}f_{k,\nu}(x)\quad(\nu\in S^{n-1}).$

Now, $\tau$ and $\gamma$ are chosen so that

$\sum_{j=-M+k/2}^{k\gamma-2}2^{k(\frac{r}{2}-\tau-\frac{n-1}{2})+j(n-1-r)}\lesssim
1,$

as is straightforward to check. We can combine all this with the boundedness
of $M$ on $L^{p}(S^{n-1};\ell^{2})$ (see [8, Section 6.6]) to obtain

$\displaystyle\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)a(x,D)f\|_{\mathcal{H}^{s,p}}^{p}\mathrm{d}{\omega}$
$\displaystyle\lesssim\int_{S^{n-1}}\int_{{\mathbb{R}^{n}}}\\!\Big{(}\sum_{k=1}^{\infty}\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}2^{k(\frac{r}{2}-\tau-\frac{n-1}{2})+j(n-1-r)}\fint_{F_{{\omega},k,j}}|g_{k,x}(\nu)|\mathrm{d}\nu\Big{)}^{2}\Big{)}^{p/2}\mathrm{d}x\mathrm{d}{\omega}$
$\displaystyle\leq\int_{S^{n-1}}\int_{{\mathbb{R}^{n}}}\\!\Big{(}\sum_{k=1}^{\infty}\Big{(}\sum_{j=-M+k/2}^{k\gamma-2}2^{k(\frac{r}{2}-\tau-\frac{n-1}{2})+j(n-1-r)}Mg_{k,x}({\omega})\Big{)}^{2}\Big{)}^{p/2}\mathrm{d}x\mathrm{d}{\omega}$
$\displaystyle\lesssim\int_{{\mathbb{R}^{n}}}\int_{S^{n-1}}\\!\Big{(}\sum_{k=1}^{\infty}|Mg_{k,x}({\omega})|^{2}\Big{)}^{p/2}\mathrm{d}{\omega}\mathrm{d}x\lesssim\int_{{\mathbb{R}^{n}}}\int_{S^{n-1}}\\!\Big{(}\sum_{k=1}^{\infty}|g_{k,x}(\nu)|^{2}\Big{)}^{p/2}\mathrm{d}\nu\mathrm{d}x$
$\displaystyle=\int_{S^{n-1}}\Big{\|}\Big{(}\sum_{k=1}^{\infty}4^{k(s+\tau-\frac{n-1}{4})}|\chi_{k}(D)m(D){\varphi}_{\nu}(D)f|^{2}\Big{)}^{1/2}\Big{\|}_{L^{p}}^{p}\mathrm{d}\nu.$

Now we apply Lemma 2.3 to this expression, and use that $m\in
S^{(n-1)/4}({\mathbb{R}^{n}})$ and that $\sum_{k=0}^{\infty}\chi_{k}\in
S^{0}({\mathbb{R}^{n}})$, as is straightforward to check, to obtain

$\displaystyle\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)a(x,D)f\|_{\mathcal{H}^{s,p}}^{p}\mathrm{d}{\omega}$
$\displaystyle\lesssim\int_{S^{n-1}}\Big{\|}\sum_{k=1}^{\infty}m(D)\chi_{k}(D){\varphi}_{\nu}(D)f\Big{\|}_{\mathcal{H}^{s+\tau-(n-1)/4,p}}^{p}\mathrm{d}\nu$
$\displaystyle\lesssim\int_{S^{n-1}}\|{\varphi}_{\nu}(D)f\|_{\mathcal{H}^{s+\tau,p}}^{p}\mathrm{d}\nu\lesssim\|f\|_{\mathcal{H}^{s+\tau,p}_{FIO}}^{p}.$

This proves the second half of (4.11) and concludes the proof. ∎

######  Remark 4.2.

It follows from the proof of Theorem 4.1 that the operator norm of $a(x,D)$
depends on $a$ through a finite number of the $C^{r}_{*}S^{m}_{1,\delta}$
seminorms of $a$. More precisely, for all $p\in(1,\infty)$ and
$-(1-\gamma)r<s+s(p)<r$ one has

$\|a(x,D)\|_{{\mathcal{L}}(\mathcal{H}^{s+m+\tau,p}_{FIO}({\mathbb{R}^{n}}),\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}}))}\lesssim\max_{|\alpha|\leq
N}C_{\alpha},$

with an implicit constant independent of $a\in C^{r}_{*}S^{m}_{1,\delta}$.
Here $C_{\alpha}$, for $\alpha\in{{\mathbb{Z}}}_{+}^{n}$, is such that (2.14)
and (2.15) hold. Similar statements apply for the extremal values
$s+s(p)=-(1-\gamma)r$ and $s+s(p)=r$ in Theorem 4.1, as well as to other
results in this article such as Proposition 3.3 and Corollary 3.5.

######  Remark 4.3.

For $p=1$ one can still rely on Littlewood-Paley theory as in Lemma 2.3.
However, there are two obstacles in extending our proof of Theorem 4.1 to the
case where $p=1$. Firstly, Lemma 2.4 played a crucial role in the proof, and
it is unclear to the author whether that lemma extends (in some form) to the
case where $p=1$. Secondly, we used the assumption that $1<p<\infty$ when
relying on the boundedness of the Hardy–Littlewood maximal function on
$L^{p}(S^{n-1};\ell^{2})$. We choose to not pursue this matter any further in
the present article.

######  Remark 4.4.

Theorem 4.1 recovers the statement of Lemma 3.2. One might note that we in
fact used Lemma 3.2 in the proof of Theorem 4.1 to deal with the low
frequencies of $a$, captured by $a^{\sharp}_{1/2}$. However, it is not too
difficult to adapt the proof of Theorem 4.1 to also deal with
$a^{\sharp}_{1/2}$ directly. This involves a modification of the definition of
$F_{{\omega},k,j}$ in (4.18), by setting $F_{{\omega},k,j}:=\\{\nu\in
S^{n-1}\mid|\nu-{\omega}|\leq\kappa 2^{-k/2}\\}$ for $j<-M+k/2$ and a suitable
$\kappa>0$. Moreover, one has to use the supremum bounds for
$(a_{k})_{k=0}^{\infty}$ in (4.9). Then similar arguments yield an alternative
proof of Lemma 3.2.

Through an application of Lemma 2.9, one obtains the following generalization
of Theorem 4.1. Note that (4.20) is an improvement of (3.3), although it holds
for a slightly smaller range of $s$.

######  Corollary 4.5.

Let $r>0$, $m\in{\mathbb{R}}$, $\delta\in[0,1/2]$, $p\in(1,\infty)$ and $a\in
C^{r}_{*}S^{m}_{1,\delta}$. For $\varepsilon>0$, set

$\rho:=\begin{cases}0&\text{if
}2s(p)(1-\frac{r}{n-1})\leq(\frac{1}{2}-\delta)r\text{ and
}\delta\neq\frac{1}{2},\\\ 0&\text{if }r>n-1\text{ and }\delta=\frac{1}{2},\\\
\varepsilon&\text{if }r=n-1\text{ and }\delta=\frac{1}{2},\\\
2s(p)(1-\frac{r}{n-1})-(\frac{1}{2}-\delta)r&\text{otherwise},\end{cases}$

and let $\gamma$ be as in Theorem 4.1. Then

(4.20)
$a(x,D):\mathcal{H}^{s+m+\rho,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for $-(1-\gamma)r<s+s(p)<r$. If $a\in\mathcal{H}^{r,\infty}S^{0}_{1,\delta}$,
then (4.20) also holds for $s+s(p)=r$. If
$a\in\mathcal{H}^{r,\infty}({\mathbb{R}^{n}})$, then (4.20) holds for all
$-(1-\gamma)r\leq s+s(p)\leq r$, with $m=\delta=0$.

Note that $\rho=\max(0,\tau-(\frac{1}{2}-\delta)r)$, where $\tau$ is as in
Theorem 4.1.

###### Proof.

Use Lemma 2.9 to write $a=a^{\sharp}_{1/2}+a^{\flat}_{1/2}$ with
$a^{\sharp}_{1/2}\in S^{m}_{1,1/2}$ and $a^{\flat}_{1/2}\in
C^{r}_{*}S^{m-(1/2-\delta)r}_{1,1/2}$. By Lemma 3.2, or Theorem 4.1, one has

$a^{\sharp}_{1/2}(x,D):\mathcal{H}^{s+m+\rho,p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s+m,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for all $s\in{\mathbb{R}}$. Moreover, by Theorem 4.1,

$a^{\flat}_{1/2}(x,D):\mathcal{H}^{s+m+\rho,p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{s+m-(1/2-\delta)r+\tau,p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{s,p}_{FIO}({\mathbb{R}^{n}})$

for $-(1-\gamma)r<s+s(p)<r$. The final two statements follow from the final
two statements in Theorem 4.1, applied to $a^{\flat}_{1/2}$. ∎

## Acknowledgements

The author would like to thank Andrew Hassell for many useful conversations
about the article, and both Andrew Hassell and Pierre Portal for valuable
advice. The author is also grateful to Dorothee Frey for a conversation
regarding the use of the anisotropic multiplier theorem in Lemma 2.4, and to
the anonymous referee for various helpful comments.

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