# Characterizations of the Hardy space $\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$
for Fourier Integral Operators

Zhijie Fan, Naijia Liu, Jan Rozendaal and Liang Song  Zhijie Fan, Department
of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P.R. China
<EMAIL_ADDRESS>Naijia Liu, Department of Mathematics, Sun Yat-sen
University, Guangzhou, 510275, P.R. China<EMAIL_ADDRESS>Jan
Rozendaal, Mathematical Sciences Institute, Australian National University,
Acton ACT 2601, Australia<EMAIL_ADDRESS>Liang Song, Department
of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

The Hardy spaces for Fourier integral operators
$\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$, for $1\leq p\leq\infty$, were
introduced by Smith in [18] and Hassell et al. in [12]. In this article, we
give several equivalent characterizations of
$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$, for example in terms of
Littlewood–Paley g functions and maximal functions. This answers a question
from [15]. We also give several applications of the characterizations.

2010 Mathematical Subject Classification. 42B35, 35S30, 42B30

Key words and phrases: Fourier integral operators, Hardy spaces,
Littlewood–Paley g function, Maximal function

##  1\. Introduction

###  1.1. Overview

Hardy spaces have long been of great importance in harmonic analysis and
related fields. For example, the classical Hardy space
$H^{1}({\mathbb{R}^{n}})$ is the natural harmonic analytic substitute of the
Lebesgue space $L^{1}({\mathbb{R}^{n}})$ for the study of singular integral
operators (see [20, 11]). And in recent years, the theory of adapted Hardy
spaces has played a major role in the analysis of parabolic and elliptic
partial differential equations with rough coefficients. These adapted Hardy
spaces are the natural substitutes of $L^{p}({\mathbb{R}^{n}})$ when the
equation under consideration is not well behaved on $L^{p}({\mathbb{R}^{n}})$
for certain $1\leq p\leq\infty$ (see [2]). In turn, there are many
characterizations of Hardy spaces, such as in terms of area functionals,
Littlewood–Paley g functions and maximal functions. These characterizations
are powerful harmonic analytic tools, as they allow for different methods of
tackling a given problem.

Although singular integral operators are bounded on $H^{1}(\mathbb{R}^{n})$,
and thus bounded from $H^{1}({\mathbb{R}^{n}})$ to $L^{1}({\mathbb{R}^{n}})$,
the situation is quite different for oscillatory integral operators. Indeed,
Fourier integral operators (FIOs) of order zero are generally not bounded from
$H^{1}(\mathbb{R}^{n})$ to $L^{1}({\mathbb{R}^{n}})$ unless $n=1$. Fourier
integral operators are typical examples of oscillatory integrals, and they
arise naturally in classical analysis and partial differential equations, for
example as the solution operators to wave equations (for more on FIOs see [13,
8, 19]). As shown by Seeger, Sogge and Stein in [17], an FIO $T$ of order
zero, associated with a local canonical graph and having a compactly supported
Schwartz kernel, satisfies
$T:{\langle}D\rangle^{-\frac{n-1}{2}}H^{1}(\mathbb{R}^{n})\to
L^{1}(\mathbb{R}^{n})$, and the exponent $\frac{n-1}{2}$ cannot be improved in
general. Here ${\langle}D\rangle^{-\frac{n-1}{2}}$ is the Fourier multiplier
with symbol
$\xi\mapsto{\langle}\xi\rangle^{-\frac{n-1}{2}}=(1+|\xi|^{2})^{-\frac{n-1}{4}}$.
This result is often summarized by saying that FIOs “lose” $(n-1)/2$
derivatives on $H^{1}(\mathbb{R}^{n})$. Using duality and interpolation, one
in turn obtains optimal results about the $L^{p}$-boundedness of FIOs.

Although the loss of derivatives for FIOs on $L^{p}(\mathbb{R}^{n})$ is
unavoidable for $p\neq 2$ and $n>1$, it turns out that $L^{p}$-spaces are in
fact not the right function spaces for the analysis of FIOs. Indeed, in [18]
Hart Smith introduced a Hardy space, denoted
$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$, that is invariant under suitable FIOs
of order $0$, and this space is large enough to allow one to recover the
results in [17]. Recently, building on Smith’s work, in [12] Hassell, Portal
and the third author of this article introduced a full scale of Hardy spaces
$\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$, $1\leq p\leq\infty$, for Fourier
integral operators. These spaces are invariant under FIOs of order $0$, and
they satisfy Sobolev embeddings that allow one to directly recover the optimal
results about about $L^{p}$-boundedness of FIOs. This development should be
compared with the introduction of adapted Hardy spaces to replace
$L^{p}({\mathbb{R}^{n}})$ when considering parabolic and elliptic equations
with rough coefficients.

Apart from the intrinsic interest in determining the natural function spaces
for FIOs, the Hardy spaces for FIOs were introduced with applications to wave
equations with rough coefficients and nonlinear wave equations in mind.
Indeed, a common method for solving rough or nonlinear equations is to use
iterative constructions to build a solution, and if one loses derivatives in
each iteration step then such a process only works for infinitely smooth
initial data. On the other hand, since
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ is invariant under suitable FIOs of
order zero for all $1\leq p\leq\infty$, one can use iterative constructions to
build a solution on $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$, and then
afterwards use the Sobolev embeddings for
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ to deduce optimal results about
$L^{p}({\mathbb{R}^{n}})$ regularity.

In [12] (and implicitly already in [18]), the definition of
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ follows a template from the theory
of adapted Hardy spaces, using embeddings into tent spaces (for more on these
spaces see [7, 1]). However, an intrinsic difference between the theory of
parabolic and elliptic equations, for which adapted Hardy spaces have
classically been used, and that of hyperbolic equations is that the latter
exhibit propagation of singularities, the phenomenon whereby singularities of
the initial data are moved around by the solution operators. Propagation of
singularities takes place on phase space, i.e. on the cotangent bundle
$T^{*}(\mathbb{R}^{n})=\mathbb{R}^{n}\times{\mathbb{R}^{n}}$ of
${\mathbb{R}^{n}}$, so to obtain function spaces that are invariant under FIOs
one needs to move from ${\mathbb{R}^{n}}$ to phase space. For
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$, this is done by using wave packet
transforms to embed function spaces over ${\mathbb{R}^{n}}$ into tent spaces
over the cosphere bundle $S^{*}({\mathbb{R}^{n}})={\mathbb{R}^{n}}\times
S^{n-1}$ of ${\mathbb{R}^{n}}$. One can then use the established theory of
tent spaces to study the Hardy spaces for FIOs, and one can even prescribe
suitable kernel bounds that guarantee boundedness on the Hardy spaces for
FIOs. This opens the door to applications to wave equations with rough
coefficients.

Although the procedure used to define
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ in [12] leads to a robust theory
that builds on tools that have been successful in other areas of harmonic
analysis, this definition has several drawbacks. The resulting function space
norm involves a conical square function over $S^{*}({\mathbb{R}^{n}})$ that is
somewhat involved and not particularly amenable to direct calculations (see
(1.1) below). It is natural to wonder whether there are descriptions of
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ that are easier to work with
directly. Moreover, the classical Hardy space $H^{1}({\mathbb{R}^{n}})$ can be
characterized in a variety of ways, and one might ask whether similar
characterizations hold for the Hardy spaces for FIOs. Finally, although the
definition of $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ in [12] comes with
kernel conditions that guarantee boundedness on
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$, for applications to rough wave
equations one would also like to have access to powerful tools from
Littlewood–Paley theory, and these tools are not available in the theory of
tent spaces.

To address these issues, in [15] the third author of this paper proved several
characterizations of $\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$ for $1<p<\infty$.
In [15], $\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$ is first characterized in
terms of $L^{p}(\mathbb{R}^{n})$-norms of parabolic frequency localizations.
Then, as a corollary, any characterization of $L^{p}(\mathbb{R}^{n})$ yields a
corresponding version of $\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$. In
particular, in this manner one obtains characterizations of
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ in terms of Littlewood–Paley $g$
functions and in terms of maximal functions. These characterizations are more
amenable to direct calculations, and they allow one to incorporate powerful
tools from Littlewood–Paley theory. However, the methods of [15] do not work
for $p=1$ or $p=\infty$, and it was left as an open question whether similar
characterizations hold for $\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$ and
$\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$.

In the present article, we answer the question in [15] regarding
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$, by obtaining several equivalent
characterizations of $\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$, for example in
terms of Littlewood–Paley g functions and maximal functions. These
characterizations are similar to those in [15], and they allow one to
incorporate powerful tools from other parts of harmonic analysis for the study
of Fourier integral operators. We illustrate one such application, and we also
show how one can use the characterizations to perform explicit calculations
with the $\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$-norm.

###  1.2. Statement of results

To make our results precise, we first recall the definition of
$\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$ for $1\leq p<\infty$. Throughout, fix
$n\geq 2$. The results in this article go through for $n=1$ but reduce to
classical statements about the local Hardy space
$\mathcal{H}^{1}(\mathbb{R})$. Let
$S^{*}(\mathbb{R}^{n})={\mathbb{R}^{n}}\times S^{n-1}$ be the cosphere bundle
over $\mathbb{R}^{n}$, endowed with the standard measure
$\mathrm{d}x\mathrm{d}\omega$ and with a metric $d$ that arises from contact
geometry (see Section 2.2 for details). We note that
$(S^{*}(\mathbb{R}^{n}),d,\mathrm{d}x\mathrm{d}\omega)$ is a doubling metric
measure space. For $\sigma>0$ and $(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$,
we let $B_{\sqrt{\sigma}}(x,\omega):=\big{\\{}(y,\nu)\in
S^{*}(\mathbb{R}^{n})\mid d(y,\nu;x,\omega)<\sqrt{\sigma}\big{\\}}$ be the
ball around $(x,\omega)$ of radius $\sqrt{\sigma}$ with respect to the metric
$d$. Throughout, fix a $q\in C_{c}^{\infty}(\mathbb{R}^{n})$ such that
$q(\xi)=1$ for $|\xi|\leq 2$, and let $q(D)$ be the corresponding Fourier
multiplier operator. Also, for $0<\sigma<1$ we let $\theta_{\nu,\sigma}\in
C^{\infty}_{c}({\mathbb{R}^{n}})$ be a smooth function localized to the high
frequency region
$\big{\\{}\xi\in\mathbb{R}^{n}\mid|\xi|\eqsim\sigma^{-1},\big{|}\frac{\xi}{|\xi|}-\nu\big{|}\eqsim\sigma^{\frac{1}{2}}\big{\\}}$
(see (2.3) for the exact definition of $\theta_{\nu,\sigma}$). For
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ and $(x,{\omega})\in
S^{*}({\mathbb{R}^{n}})$, set

(1.1)
$S(f)(x,\omega):=\bigg{(}\int_{0}^{1}\fint_{B_{\sqrt{\sigma}}(x,\omega)}|\theta_{\nu,\sigma}(D)f(y)|^{2}\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}.$

We can now define $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ for $1\leq
p<\infty$.

######  Definition 1.1.

For $p\in[1,\infty)$, let $\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$ consist of
all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that $S(f)\in
L^{p}(S^{*}({\mathbb{R}^{n}}))$ and $q(D)f\in L^{p}({\mathbb{R}^{n}})$,
endowed with the norm

$\|f\|_{\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})}:=\bigg{(}\int_{S^{*}(\mathbb{R}^{n})}\big{(}S(f)(x,\omega)\big{)}^{p}\
\mathrm{d}x\mathrm{d}\omega\bigg{)}^{1/p}+\|q(D)f\|_{L^{p}(\mathbb{R}^{n})}.$

We note that this is not the original definition of
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ from [12]. However, it follows from
[15, Corollary 3.8] that Definition 1.1 is equivalent to the original
definition.

For the definition of $\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$ one has
to replace the conical square function in (1.1) by a Carleson measure
condition (see [12, Section 6]). However,
$\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$ will not play a significant
role in this article, and for our purposes it suffices to define
$\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$ as the dual of
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$ (see [12, Proposition 6.8]):

(1.2)
$\mathcal{H}^{\infty}({\mathbb{R}^{n}})=(\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}}))^{*}.$

Here the duality pairing is the standard duality pairing
${\langle}f,g\rangle_{{\mathbb{R}^{n}}}$ for
$f\in\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{S}^{\prime}({\mathbb{R}^{n}})$
and
$g\in\mathcal{S}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$.

Next, we define the Littlewood-Paley g function for Fourier integral operators
as follows: for $f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ and
$(x,\omega)\in S^{*}(\mathbb{R}^{n})$, set

(1.3)
$G(f)(x,\omega):=\bigg{(}\int_{0}^{1}|\theta_{\omega,\sigma}(D)f(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}.$

We can then introduce the second function space of interest in this article.

######  Definition 1.2.

Let $\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})$ consist of all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that $G(f)\in
L^{1}(S^{*}(\mathbb{R}^{n}))$ and $q(D)f\in L^{1}(\mathbb{R}^{n})$, endowed
with the norm

$\|f\|_{\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})}:=\|G(f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}+\|q(D)f\|_{L^{1}(\mathbb{R}^{n})}.$

By [4, Proposition 2.1 and Remark 2.2], the following continuous inclusion
holds:
$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})\subseteq\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})$.
However, until now it was not clear whether one also has
$\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})\subseteq\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$.
In this article we show that this inclusion also holds, so that
$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})=\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})$.

We will give two additional characterizations of
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$. To state these, let $\alpha>0$ and,
for $f\in\mathcal{S^{\prime}}(\mathbb{R}^{n})$ and $(x,\omega)\in
S^{*}({\mathbb{R}^{n}})$, set

$\mathcal{G}_{\alpha}^{*}(f)(x,\omega):=\bigg{(}\int_{0}^{1}\int_{S^{*}(\mathbb{R}^{n})}\frac{|\theta_{\nu,\sigma}(D)f(y)|^{2}}{\sigma^{n}(1+\sigma^{-1}d(x,\omega;y,\nu)^{2})^{n\alpha}}\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}.$

Also, let $\Phi\in\mathcal{S}({\mathbb{R}^{n}})$ be a Schwartz function such
that $\Phi(0)=1$, and for $\sigma>0$ and $\xi\in{\mathbb{R}^{n}}$ let
$\Phi_{\sigma}(\xi):=\Phi(\sigma\xi)$. The function $\varphi_{\omega}\in
C^{\infty}({\mathbb{R}^{n}})$ which occurs below is supported on a paraboloid
in the direction of ${\omega}\in S^{n-1}$, and it is defined in Section 2.3.
We recall that a tempered distribution
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ is a _bounded distribution_ if
$f\ast g\in L^{\infty}({\mathbb{R}^{n}})$ for all
$g\in\mathcal{S}({\mathbb{R}^{n}})$.

######  Definition 1.3.

Let $\mathcal{H}_{FIO,\max}^{1}(\mathbb{R}^{n})$ consist of all
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that $\varphi_{\omega}(D)f$ is
a bounded distribution for almost all $\omega\in S^{n-1}$,
$\int_{S^{*}(\mathbb{R}^{n})}\sup\limits_{\sigma>0}|\Phi_{\sigma}(D)\varphi_{\omega}(D)f(x)|\,\mathrm{d}x\mathrm{d}\omega<\infty$,
and $q(D)f\in L^{1}(\mathbb{R}^{n})$, endowed with the norm

$\|f\|_{\mathcal{H}_{FIO,\max}^{1}(\mathbb{R}^{n})}:=\int_{S^{*}(\mathbb{R}^{n})}\sup\limits_{\sigma>0}|\Phi_{\sigma}(D)\varphi_{\omega}(D)f(x)|\,\mathrm{d}x\mathrm{d}\omega+\|q(D)f\|_{L^{1}(\mathbb{R}^{n})}.$

Let $\mathcal{H}_{FIO,\mathcal{G}_{\alpha}^{*}}^{1}(\mathbb{R}^{n})$ consist
of all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that
$\mathcal{G}_{\alpha}^{*}(f)\in L^{1}(S^{*}(\mathbb{R}^{n}))$ and $q(D)f\in
L^{1}(\mathbb{R}^{n})$, endowed with the norm

$\|f\|_{\mathcal{H}_{FIO,\mathcal{G}_{\alpha}^{*}}^{1}(\mathbb{R}^{n})}:=\|\mathcal{G}_{\alpha}^{*}(f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}+\|q(D)f\|_{L^{1}(\mathbb{R}^{n})}.$

The following theorem is our main result.

######  Theorem 1.4.

Let $\alpha>2$. Then

$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})=\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})=\mathcal{H}_{FIO,\max}^{1}(\mathbb{R}^{n})=\mathcal{H}_{FIO,\mathcal{G}_{\alpha}^{*}}^{1}(\mathbb{R}^{n}),$

with equivalence of norms.

Theorem 1.4 is proved in the main text as Theorems 3.8, 4.1 and 5.2. In
Section 6 we give two applications of this result:

  1. (1)

Theorem 6.1, which shows that a large class of operators that are bounded on
$L^{p}({\mathbb{R}^{n}})$ for $1<p<\infty$ are also bounded on
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ for all $1\leq p\leq\infty$;

  2. (2)

Proposition 6.4, which determines in a relatively explicit manner the
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$-norm of functions with suitably
restricted frequency support.

Our goal with this last section is to indicate how the equivalent
characterizations of $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ can be used to
incorporate techniques from other parts of harmonic analysis and to perform
calculations involving the $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$-norm.
Other applications, to wave equations with rough coefficients, make use of
Littlewood–Paley theory and will follow in future work.

###  1.3. Comparison to previous work

In [15] it is shown, for $1<p<\infty$, that an
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ satisfies
$f\in{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}$ if and only if
${\varphi}_{{\omega}}(D)\in L^{p}({\mathbb{R}^{n}})$ for almost all
${\omega}\in S^{n-1}$,
$\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)f\|_{L^{p}({\mathbb{R}^{n}})}^{p}\mathrm{d}{\omega}<\infty$,
and $q(D)f\in L^{p}({\mathbb{R}^{n}})$. Moreover, in this case one has

(1.4)
$\|f\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}\eqsim\bigg{(}\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)f\|_{L^{p}({\mathbb{R}^{n}})}^{p}\mathrm{d}{\omega}\bigg{)}^{1/p}+\|q(D)f\|_{L^{p}({\mathbb{R}^{n}})}$

for an implicit constant independent of $f$. Using classical characterizations
of $L^{p}({\mathbb{R}^{n}})$ in terms of Littlewood–Paley g functions and
maximal functions, one obtains from this similar characterizations of
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ for $1<p<\infty$ as are given in
Theorem 1.4 for $p=1$. In fact, Theorem 1.4 answers an open question from [15]
whether such characterizations also hold for $p=1$. Indeed, in [15, Remark
4.3] the question is posed whether, if
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ is such that
${\varphi}_{{\omega}}(D)\in H^{1}({\mathbb{R}^{n}})$ for almost all
${\omega}\in S^{n-1}$,
$\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)f\|_{H^{1}({\mathbb{R}^{n}})}\mathrm{d}{\omega}<\infty$,
and $q(D)f\in L^{1}({\mathbb{R}^{n}})$, then
$f\in\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$ and

$\|f\|_{\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})}\lesssim\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{H^{1}(\mathbb{R}^{n})}\,\mathrm{d}\omega+\|q(D)f\|_{L^{1}(\mathbb{R}^{n})}$

for an implicit constant of $f$. Using classical characterizations of
$H^{1}({\mathbb{R}^{n}})$ in terms of Littlewood–Paley g functions, it is
straightforward to show (see Proposition 3.1) that Theorem 1.4 gives an
affirmative answer to this question. We leave as an open problem the question
whether a similar characterization also holds for $p=\infty$ (see Remark 4.2).

It should be noted that the techniques used in this article to prove Theorem
1.4 are quite different from those employed in [15] to deal with
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ for $1<p<\infty$, although we do use
the parabolic frequency localizations that played a key role in [15]. More
precisely, the characterizations of $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
for $1<p<\infty$ are proved in [15] by showing that each
$f\in\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ satisfies
${\varphi}_{{\omega}}(D)\in L^{p}({\mathbb{R}^{n}})$ for almost all
${\omega}\in S^{n-1}$,
$\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)f\|_{L^{p}({\mathbb{R}^{n}})}^{p}\mathrm{d}{\omega}<\infty$,
and $q(D)f\in L^{p}({\mathbb{R}^{n}})$, with

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

After that one uses duality to obtain the reverse inequality. In the
terminology of the present article, this amounts to showing that
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{p}_{FIO,G}({\mathbb{R}^{n}})$,
where $\mathcal{H}^{p}_{FIO,G}({\mathbb{R}^{n}})$ is defined in an analogous
manner as in Definition 1.2, and then using duality to obtain the reverse
inclusion. For $p=1$, where we are interested in the inclusion
$\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$,
such an approach does not appear to work. This is because
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$ is not the dual of
$\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$, and also because the norm of
$\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$ is of a different nature than
that of $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ for $p<\infty$, so that the
techniques in [15] do not apply there. Instead, we prove the inclusion
$\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$
directly, using e.g. pointwise inequalities for a maximal function of Peetre
type, as well as boundedness of the vector-valued Hardy–Littlewood maximal
function. Our proof is motivated in part by arguments from [6, 5, 14].

This article is organized as follows. In Section 2, we recall some notation
and background on the metric $d$ and the wave packets that are used to define
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$. In Section 3 we then show that
$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})=\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$,
and in Section 4 we derive from this that
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})=\mathcal{H}^{1}_{FIO,\max}({\mathbb{R}^{n}})$.
Finally, in Section 5 we show that
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})=\mathcal{H}^{1}_{FIO,{\mathcal{G}}_{\alpha}^{*}}({\mathbb{R}^{n}})$,
thereby completing the proof of Theorem 1.4. We conclude with Section 6, which
contains two applications of our main result.

##  2\. Notation and preliminaries

###  2.1. Notation

The natural numbers are $\mathbb{N}=\\{1,2,\ldots\\}$, and
${{\mathbb{Z}}}_{+}:={{\mathbb{N}}}\cup\\{0\\}$. Throughout,
$n\in{{\mathbb{N}}}$ with $n\geq 2$ is fixed. For
$\xi,\eta\in{\mathbb{R}^{n}}$ we write
${\langle}\xi\rangle:=(1+|\xi|^{2})^{1/2}$ and
${\langle}\xi,\eta\rangle:=\xi\cdot\eta$, and for $\xi\neq 0$ we set
$\hat{\xi}:=\xi/|\xi|$. We use multi-index notation, where
$\partial^{\alpha}_{\xi}=\partial^{\alpha_{1}}_{\xi_{1}}\ldots\partial^{\alpha_{n}}_{\xi_{n}}$
for $\xi=(\xi_{1},\ldots,\xi_{n})\in{\mathbb{R}^{n}}$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in{{\mathbb{Z}}}_{+}^{n}$.

The Schwartz class and the class of tempered distributions on
${\mathbb{R}^{n}}$ are denoted by $\mathcal{S}(\mathbb{R}^{n})$ and
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$, respectively. The Fourier transform of
an $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ is denoted by $\mathcal{F}f$,
and for $f\in L^{1}({\mathbb{R}^{n}})$ it is normalized as follows:

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

For $m:{\mathbb{R}^{n}}\to{\mathbb{C}}$ a measurable function of temperate
growth, $m(D)$ is the Fourier multiplier with symbol $m$.

The volume of a measurable subset $B$ of a measure space is denoted by $V(B)$.
If $V(B)<\infty$, then for an integrable function $f:B\to{\mathbb{C}}$ we
write

$\displaystyle\fint_{B}f(x)\,\mathrm{d}x:=\frac{1}{V(B)}\int_{B}f(x)\,\mathrm{d}x.$

The indicator function of a set $E$ is denoted by $\mathbf{1}_{E}$. For
$(X,\mu)$ a measure space and $p,q\in[1,\infty)$, we denote by
$L^{p}(X;\ell^{q})$ the space of all sequences
$\\{f_{j}\\}_{j\in{{\mathbb{N}}}}$ of measurable functions
$f_{j}:X\to{\mathbb{C}}$, $j\in{{\mathbb{N}}}$, such that

$\displaystyle\|\\{f_{j}\\}_{j\in{{\mathbb{N}}}}\|_{L^{p}(X;\ell^{q})}:=\bigg{(}\int_{X}\|\\{f_{j}(x)\\}_{j\in{{\mathbb{N}}}}\|_{\ell^{q}}^{p}\mathrm{d}\mu(x)\bigg{)}^{1/p}<\infty.$

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

###  2.2. A metric on the cosphere bundle

In this subsection, we collect some background on the underlying metric
measure space which will be considered throughout. The relevant metric arises
from contact geometry, but for this article we will only need a few basic
facts about this metric. For more details on the material presented here, see
[12, Section 2.1].

Throughout, we denote elements of the sphere $S^{n-1}$ by ${\omega}$ or $\nu$,
and we let $g_{S^{n-1}}$ be the standard Riemannian metric on $S^{n-1}$. Let
$S^{*}({\mathbb{R}^{n}}):={\mathbb{R}^{n}}\times S^{n-1}$ be the cosphere
bundle of $\mathbb{R}^{n}$, endowed with the standard measure
$\mathrm{d}x\mathrm{d}{\omega}$. The $1$-form
$\alpha_{S^{n-1}}:=\hat{\xi}\cdot dx$ on $S^{*}({\mathbb{R}^{n}})$ determines
a contact structure on $S^{*}({\mathbb{R}^{n}})$, the smooth distribution of
codimension $1$ hypersurfaces of $T(S^{*}({\mathbb{R}^{n}}))$ given by the
kernel of $\alpha_{S^{n-1}}$. Then
$(S^{*}({\mathbb{R}^{n}}),\alpha_{S^{n-1}})$ is a contact manifold. Together,
the product metric $dx^{2}+g_{S^{n-1}}$ and the contact form determine a sub-
Riemannian metric $d$ on $S^{*}(\mathbb{R}^{n})$:

(2.1) $\displaystyle
d(x,\omega;y,v):={\rm\inf\limits_{\gamma}}\int_{0}^{1}|\gamma^{\prime}(s)|\,\mathrm{d}s.$

for $(x,\omega),(y,\nu)\in S^{*}(\mathbb{R}^{n})$. Here the infimum is taken
over all piecewise $C^{1}$ curves $\gamma:[0,1]\rightarrow
S^{*}(\mathbb{R}^{n})$ such that $\gamma(0)=(x,\omega)$, $\gamma(1)=(y,\nu)$
and $\alpha_{S^{n-1}}(\gamma^{\prime}(s))=0$ for almost all $s\in[0,1]$.
Moreover, $|\gamma^{\prime}(s)|$ is the length of the vector
$\gamma^{\prime}(s)$ with respect to $dx^{2}+dg_{S^{n-1}}$.

It is shown in [12, Lemma 2.1] that

$\displaystyle
d(x,\omega;y,\nu)\eqsim\big{(}|\langle\omega,x-y\rangle|+|x-y|^{2}+|\omega-\nu|^{2}\big{)}^{1/2}$

for an implicit constant independent of $(x,{\omega}),(y,\nu)\in
S^{*}({\mathbb{R}^{n}})$. The following is [12, Lemma 2.3].

######  Lemma 2.1.

There exists a constant $C>0$ such that, for all $(x,\omega)\in
S^{*}(\mathbb{R}^{n})$, one has

$\displaystyle\frac{1}{C}\tau^{2n}\leq V(B_{\tau}(x,\omega))\leq C\tau^{2n}$

if $\tau\in(0,1)$ and

$\displaystyle\frac{1}{C}\tau^{n}\leq V(B_{\tau}(x,\omega))\leq C\tau^{n}$

if $\tau\geq 1$. In particular,

$\displaystyle V(B_{\lambda\tau}(x,\omega))\leq
C\lambda^{2n}V(B_{\tau}(x,\omega))$

for all $\tau>0$ and $\lambda\geq 1$, and
$(S^{*}(\mathbb{R}^{n}),d,\mathrm{d}x\mathrm{d}\omega)$ is a doubling metric
measure space.

###  2.3. Wave packets

In this subsection we introduce the wave packets that are used to define the
Hardy spaces for Fourier integral operators. For more on this, see [12,
Section 4] and [15, Section 3].

Fix a non-negative radial $\varphi\in C_{c}^{\infty}(\mathbb{R}^{n})$ such
that $\varphi\equiv 1$ in a neighborhood of zero and $\varphi(\xi)=0$ for
$|\xi|>1$. For $\sigma>0$, ${\omega}\in S^{n-1}$ and
$\xi\in{\mathbb{R}^{n}}\setminus\\{0\\}$ set
$c_{\sigma}:=\big{(}\int_{S^{n-1}}\varphi(\frac{e_{1}-\nu}{\sqrt{\sigma}})^{2}\mathrm{d}\nu\big{)}^{-1/2}$,
where $e_{1}$ is the first basis vector of ${\mathbb{R}^{n}}$ (this particular
choice is irrelevant), and
$\varphi_{\omega,\sigma}(\xi):=c_{\sigma}\varphi\big{(}\frac{\hat{\xi}-\omega}{\sqrt{\sigma}}\big{)}$.
Also let $\varphi_{\omega,\sigma}(0):=0$. Next, let
$\Psi\in\mathcal{S}(\mathbb{R}^{n})$ be a non-negative radial function, with
$\Psi(\xi)=0$ if $|\xi|\notin[\frac{1}{2},2]$, $\Psi(\xi)=c>0$ if
$|\xi|\in[\frac{3}{4},\frac{3}{2}]$, and

(2.2)
$\int_{0}^{\infty}\Psi(\sigma\xi)^{2}\frac{\mathrm{d}\sigma}{\sigma}=1\quad(\xi\neq
0).$

For $\sigma>0$ and $\xi\in{\mathbb{R}^{n}}$ set
$\Psi_{\sigma}(\xi):=\Psi(\sigma\xi)$. Now, for $\omega\in S^{n-1}$, write

$\varphi_{\omega}(\xi):=\int_{0}^{4}\Psi_{\tau}(\xi){\varphi}_{{\omega},\tau}(\xi)\frac{\mathrm{d}\tau}{\tau}$

and, if $\sigma\in(0,1)$,

(2.3) $\theta_{\omega,\sigma}(\xi):=\Psi_{\sigma}(\xi)\varphi_{\omega}(\xi).$

These wave packets were introduced in [15], and in this article they have
already appeared in Definitions 1.1, 1.2 and 1.3.

We also introduce some new wave packets. Set

(2.4)
$\eta(\xi):=\begin{cases}\frac{\Psi(\xi)}{\sum_{j\in\mathbb{Z}}\Psi(2^{-j}\xi)^{2}}&\text{for
}\xi\neq 0,\\\ 0&\text{for }\xi=0,\end{cases}$

and, for ${\omega}\in S^{n-1}$ and $0<\sigma<1$,

(2.5)
$\chi_{\omega,\sigma}(\xi):=\begin{cases}\frac{\eta(\sigma\xi)\varphi_{\omega}(\xi)}{\int_{S^{n-1}}\varphi_{\nu}(\xi)^{2}\mathrm{d}\nu}&\text{
for }\xi\in{\rm supp}(\theta_{\omega,\sigma}),\\\
0&\text{otherwise}.\end{cases}$

We collect some properties of these wave packets in the following lemma.

######  Lemma 2.2.

For $\omega\in S^{n-1}$ and $0<\sigma<1$, let
$\gamma_{\omega,\sigma}\in\\{\theta_{\omega,\sigma},\chi_{\omega,\sigma}\\}$.
Then $\gamma_{\omega,\sigma}\in C_{c}^{\infty}(\mathbb{R}^{n})$, and

(2.6) ${\rm
supp}(\gamma_{\omega,\sigma})\subseteq\big{\\{}\xi\in{\mathbb{R}^{n}}\mid\tfrac{1}{2}\sigma^{-1}\leq|\xi|\leq
2\sigma^{-1},|\hat{\xi}-\omega|\leq 2\sqrt{\sigma}\big{\\}}.$

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

(2.7)
$\displaystyle|\langle\omega,\nabla_{\xi}\rangle^{\beta}\partial_{\xi}^{\alpha}\gamma_{\omega,\sigma}(\xi)|\leq
C_{\alpha,\beta}\sigma^{-\frac{n-1}{4}+\frac{|\alpha|}{2}+\beta}$

for all $(\xi,\omega,\sigma)\in S^{*}(\mathbb{R}^{n})\times(0,1)$. For each
$N\geq 0$, there exists a constant $C_{N}\geq 0$ such that

(2.8) $\displaystyle|\mathcal{F}^{-1}(\gamma_{\omega,\sigma})(x)|\leq
C_{N}\sigma^{-\frac{3n+1}{4}}(1+\sigma^{-1}|x|^{2}+\sigma^{-2}\langle\omega,x\rangle^{2})^{-N}$

for all $(x,\omega,\sigma)\in S^{*}(\mathbb{R}^{n})\times(0,1)$. Finally, for
all $\alpha\in{{\mathbb{Z}}}_{+}^{n}$ there exists a constant $C_{\alpha}\geq
0$ such that

(2.9)
$\Big{|}\partial_{\xi}^{\alpha}\bigg{(}\int_{S_{n-1}}{\varphi}_{\nu}(\xi)\mathrm{d}\nu\bigg{)}^{-1}\Big{|}\leq
C_{\alpha}|\xi|^{\frac{n-1}{4}-|\alpha|}$

for all $\xi\in{\mathbb{R}^{n}}$ with $|\xi|\geq 1/2$.

###### Proof.

For $\gamma_{\omega,\sigma}=\theta_{\omega,\sigma}$, the required statements
are contained in [15, Lemma 3.2]. It is also shown there (see [15, Remark 3.3]
and the arguments for (2.9) below) that, for all $\alpha\in\mathbb{Z}_{+}^{n}$
and $\beta\in\mathbb{Z}_{+}$, there exists constants
$C^{\prime}_{\alpha},C^{\prime}_{\alpha,\beta}\geq 0$ such that

(2.10)
$\Big{|}\partial_{\xi}^{\alpha}\bigg{(}\int_{S^{n-1}}\varphi_{\nu}(\xi)^{2}\mathrm{d}\nu\bigg{)}^{-1}\Big{|}\leq
C_{\alpha}^{\prime}\sigma^{|\alpha|}$

and

(2.11)
$|\langle\omega,\nabla_{\xi}\rangle^{\beta}\partial_{\xi}^{\alpha}\varphi_{\omega}(\xi)|\leq
C_{\alpha,\beta}^{\prime}\sigma^{-\frac{n-1}{4}+\frac{|\alpha|}{2}+\beta}$

for all ${\omega}\in S^{n-1}$, $0<\sigma<1$ and $\xi\in{\rm supp}{\hskip
1.42271pt}(\theta_{\omega,\sigma})$.

For $\gamma_{\omega,\sigma}=\chi_{\omega,\sigma}$, we first use the properties
of $\Psi$ to note that

$\sum_{j\in{{\mathbb{Z}}}}\Psi(2^{-j}\xi)^{2}\geq c$

for all $\xi\neq 0$, since there exists a $j\in{{\mathbb{Z}}}$ such that
$2^{-j}|\xi|\in[\frac{3}{4},\frac{3}{2}]$. In turn, this implies that $\eta$
is well defined, and it is straightforward to see that in fact $\eta\in
C^{\infty}_{c}({\mathbb{R}^{n}})$. It now follows that
$\chi_{{\omega},\sigma}\in C^{\infty}_{c}({\mathbb{R}^{n}})$ is well-defined
with ${\rm supp}{\hskip 1.42271pt}(\chi_{{\omega},\sigma})={\rm supp}{\hskip
1.42271pt}(\theta_{{\omega},\sigma})$. Moreover, clearly

$|\partial_{\xi}^{\alpha}\eta(\sigma\xi)|=\sigma^{|\alpha|}|(\partial_{\xi}^{\alpha}\eta)(\sigma\xi)|\lesssim\sigma^{|\alpha|}$

for all $\alpha\in{{\mathbb{Z}}}_{+}^{n}$, with an implicit constant
independent of $\sigma>0$ and $\xi\in{\mathbb{R}^{n}}$. By combining this with
(2.10) and (2.11), it follows that $\chi_{{\omega},\sigma}$ satisfies (2.7).
For (2.8) one now integrates by parts with respect to the operator

$\displaystyle
L:=\big{(}1+\sigma^{-1}|x|^{2}+\sigma^{-2}\langle\omega,x\rangle^{2}\big{)}^{-1}\big{(}1-\sigma^{-1}\Delta_{\xi}-\sigma^{-2}\langle\omega,\nabla_{\xi}\rangle^{2}\big{)}$

in the expression

$\displaystyle\mathcal{F}^{-1}(\chi_{\omega,\sigma})(x):=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}{\chi}_{\omega,\sigma}(\xi)\,\mathrm{d}\xi\quad(x\in\mathbb{R}^{n}),$

using (2.7) and the support properties of $\chi_{{\omega},\sigma}$. See [12,
Lemma 4.1] for more details.

Finally, (2.9) is obtained in the same manner as (2.10). More precisely, let
$c^{\prime}>0$ be such that ${\varphi}(\xi)=1$ for $|\xi|\leq c^{\prime}$, and
fix $\xi\in{\mathbb{R}^{n}}$ with $|\xi|\geq\frac{1}{2}$. Set
$E_{\xi}:=\\{\nu\in S^{n-1}\mid|\nu-\hat{\xi}|\leq c^{\prime}|2\xi|^{-1/2}\\}$
and $F_{\xi}:=\\{\nu\in
S^{n-1}\mid|\nu-\hat{\xi}|\leq\sqrt{2}|\xi|^{-1/2}\\}$. Then
$|E_{\xi}|\eqsim|F_{\xi}|\eqsim|\xi|^{-\frac{n-1}{2}}$. Moreover,

$c_{\tau}\Psi_{\tau}(\xi){\varphi}\big{(}\tfrac{\hat{\xi}-\nu}{\sqrt{\tau}}\big{)}=c_{\tau}\Psi_{\tau}(\xi)$

for $\nu\in E_{\xi}$ and $\tau\geq|2\xi|^{-1}$, and

$c_{\tau}\Psi_{\tau}(\xi){\varphi}\big{(}\tfrac{\hat{\xi}-\nu}{\sqrt{\tau}}\big{)}=0$

for all $\nu\notin F_{\xi}$ and $\tau>0$, where we used that
$\Psi_{\tau}(\xi)=0$ if $|\xi|\notin[\frac{1}{2}\tau^{-1},2\tau^{-1}]$. Since
$|\xi|\geq 1/2$, the latter observation yields

${\varphi}_{\nu}(\xi)=\int_{|2\zeta|^{-1}}^{\min(4,2|\zeta|^{-1})}c_{\tau}\Psi_{\tau}(\xi){\varphi}\big{(}\tfrac{\hat{\xi}-\nu}{\sqrt{\tau}}\big{)}\frac{\mathrm{d}\tau}{\tau}=\int_{|2\xi|^{-1}}^{2|\xi|^{-1}}c_{\tau}\Psi_{\tau}(\xi){\varphi}\big{(}\tfrac{\hat{\xi}-\nu}{\sqrt{\tau}}\big{)}\frac{\mathrm{d}\tau}{\tau}.$

We also note that $c_{\tau}\eqsim\tau^{-\frac{n-1}{4}}$ for implicit constants
independent of $\tau>0$, as is shown in [12, Lemma 4.1]. We now combine all of
this with (2.2) and (2.11) to obtain

$\displaystyle|\xi|^{-\frac{n-1}{4}}=|\xi|^{-\frac{n-1}{4}}\int_{0}^{\infty}\Psi_{\tau}(\xi)^{2}\frac{\mathrm{d}\tau}{\tau}\lesssim|\xi|^{-\frac{n-1}{4}}\int_{0}^{\infty}\Psi_{\tau}(\xi)\frac{\mathrm{d}\tau}{\tau}=|\xi|^{-\frac{n-1}{4}}\int_{|2\xi|^{-1}}^{2|\xi|^{-1}}\Psi_{\tau}(\xi)\frac{\mathrm{d}\tau}{\tau}$
$\displaystyle\eqsim\int_{E_{\xi}}\int_{|2\xi|^{-1}}^{2|\xi|^{-1}}c_{\tau}\Psi_{\tau}(\xi)\frac{\mathrm{d}\tau}{\tau}\mathrm{d}\nu=\int_{E_{\xi}}{\varphi}_{\nu}(\xi)\mathrm{d}\nu\leq\int_{S^{n-1}}{\varphi}_{\nu}(\xi)\mathrm{d}\nu=\int_{F_{\xi}}{\varphi}_{\nu}(\xi)\mathrm{d}\nu\lesssim\int_{F_{\xi}}|\xi|^{\frac{n-1}{4}}\mathrm{d}\nu\eqsim|\xi|^{-\frac{n-1}{4}}.$

In particular,
$\int_{S^{n-1}}{\varphi}_{\nu}(\xi)\mathrm{d}\nu\eqsim|\xi|^{-\frac{n-1}{4}}$.
By combining similar arguments with the bounds in (2.11) for the derivatives
of ${\varphi}_{\nu}$, one can show that for all $\alpha\in\mathbb{Z}_{+}^{n}$
and $\beta\in\mathbb{Z}_{+}$ one has

$\Big{|}{\langle}\hat{\xi},\nabla_{\xi}\rangle^{\beta}\partial_{\xi}^{\alpha}\bigg{(}\int_{S^{n-1}}{\varphi}_{\nu}(\xi)\mathrm{d}\nu\bigg{)}^{-1}\Big{|}\lesssim|\xi|^{\frac{n-1}{4}-\frac{|\alpha|}{2}-\beta},$

with an implicit constant dependent on $\alpha$ and $\beta$ but independent of
$\xi\in{\mathbb{R}^{n}}$ with $|\xi|\geq\frac{1}{2}$. However,
$\int_{S^{n-1}}{\varphi}_{\nu}(\xi)\mathrm{d}\nu$ is radial in $\xi$, so that
in fact the stronger bounds in (2.9) hold. ∎

We will also need the following corollary. The estimates in (2.12) were called
off-singularity bounds in [12], and they are useful for showing that an
operator is bounded on $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$, $1\leq
p\leq\infty$.

######  Corollary 2.3.

For $w,\nu\in S^{n-1}$ and $\sigma,\tau\in(0,1)$, let
$K_{\sigma,\tau}^{{\omega},\nu}$ be the integral kernel associated with the
operator

$f\mapsto\theta_{\omega,\sigma}(D)\chi_{\nu,\tau}(D)f$

on $\mathcal{S}({\mathbb{R}^{n}})$. Then for each $N\geq 0$ there exists a
$C_{N}\geq 0$, independent of ${\omega}$, $\nu$, $\sigma$ and $\tau$, such
that

(2.12) $|K_{\sigma,\tau}^{{\omega},\nu}(x,y)|\leq
C_{N}\min\Big{(}\frac{\sigma}{\tau},\frac{\tau}{\sigma}\Big{)}^{N}\rho^{-n}(1+\rho^{-1}d(x,\omega;y,\nu)^{2})^{-N}$

for all $x,y\in{\mathbb{R}^{n}}$, where $\rho=\min(\sigma,\tau)$.

###### Proof.

To obtain (2.12), it suffices to repeat the arguments in [15, Proposition 3.6]
(see also [15, Remark 3.7] and [12, Theorem 5.1]), which rely only on
integration by parts and on the properties of the wave packets in Lemma 2.2. ∎

##  3\. The Littlewood–Paley g function characterization

This section is devoted to showing that
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})=\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$.
By [4, Proposition 2.1 and Remark 2.2] (see also [12, Equation (2.9)]) one has
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$,
so it suffices to show that
$\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})\subseteq\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$.
To do so, we first collect some preliminary results that will be used to prove
the required embedding.

###  3.1. Preliminary results

In this subsection we first prove a useful equivalent characterization of
$\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$, from which we derive a Sobolev
embedding for $\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$. Then we prove a
technical lemma that will afterwards be used to obtain a pointwise inequality
for a maximal function of Peetre type. This maximal function will in turn play
a crucial role in proving the main result of this section.

######  Proposition 3.1.

An $f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ satisfies
$f\in\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})$ if and only if $q(D)f\in
L^{1}({\mathbb{R}^{n}})$, ${\varphi}_{{\omega}}(D)f\in
H^{1}({\mathbb{R}^{n}})$ for almost all ${\omega}\in S^{n-1}$, and

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

Moreover, in this case one has

$\|f\|_{\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})}\eqsim\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{H^{1}(\mathbb{R}^{n})}\,\mathrm{d}\omega+\|q(D)f\|_{L^{1}(\mathbb{R}^{n})}.$

###### Proof.

By the Littlewood–Paley g function characterization of
$H^{1}({\mathbb{R}^{n}})$ (see [20] or [21]), it suffices to show that an
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ satisfies
$f\in\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})$ if and only if $q(D)f\in
L^{1}({\mathbb{R}^{n}})$ and $G^{\prime}(f)\in
L^{1}(S^{*}({\mathbb{R}^{n}}))$, with

$\|f\|_{\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})}\eqsim\|G^{\prime}(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}+\|q(D)f\|_{L^{1}({\mathbb{R}^{n}})}.$

Here

$G^{\prime}(f)(x,{\omega}):=\bigg{(}\int_{0}^{\infty}|\theta_{{\omega},\sigma}(D)f(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}$

for $(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$. In turn, since $G(f)\leq
G^{\prime}(f)$ pointwise, it suffices to prove that each
$f\in\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$ satisfies $G^{\prime}(f)\in
L^{1}(S^{*}({\mathbb{R}^{n}}))$ and
$\|G^{\prime}(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}\lesssim\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}$.

Let $f\in\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$ and note that

$\theta_{{\omega},\sigma}(\xi)=\Psi_{\sigma}(\xi){\varphi}_{{\omega}}(\xi)=\int_{0}^{4}\Psi_{\sigma}(\xi)\Psi_{\tau}(\xi){\varphi}_{{\omega},\tau}(\xi)\frac{\mathrm{d}\tau}{\tau}=0$

for all $\xi\in{\mathbb{R}^{n}}$ if $\sigma>16$, since for all $\tau>0$ one
has $\Psi_{\tau}(\xi)=0$ if $|\xi|\notin[\tau^{-1}/2,2\tau^{-1}]$. Hence one
in fact has

$G^{\prime}(f)(x,{\omega})=\bigg{(}\int_{0}^{16}|\theta_{{\omega},\sigma}(D)f(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\leq
G(f)(x,{\omega})+\bigg{(}\int_{1}^{16}|\theta_{{\omega},\sigma}(D)f(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}$

for all $(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$, and it suffices to show
that

$\int_{S^{*}({\mathbb{R}^{n}})}\bigg{(}\int_{1}^{16}|\theta_{{\omega},\sigma}(D)f(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}{\omega}\lesssim\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}.$

But this is proved by noting that $\theta_{{\omega},\sigma}(D)(1-q)(D)f=0$ for
$\sigma>1$, since $q(\xi)=1$ for $|\xi|\leq 2$, and then reasoning as follows:

$\displaystyle\int_{S^{*}({\mathbb{R}^{n}})}\bigg{(}\int_{1}^{16}|\theta_{{\omega},\sigma}(D)f(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}{\omega}=\int_{S^{*}({\mathbb{R}^{n}})}\bigg{(}\int_{1}^{16}|\theta_{{\omega},\sigma}(D)q(D)f(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}{\omega}$
$\displaystyle\lesssim\int_{\mathbb{R}^{n}}\sup_{\frac{1}{2}\leq\sigma\leq
16,\omega\in
S^{n-1}}|\theta_{{\omega},\sigma}(D)q(D)f(x)|\mathrm{d}x\lesssim\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|q(D)f(y)|}{(1+|x-y|)^{n+1}}\,\mathrm{d}y\mathrm{d}x$
$\displaystyle\lesssim\|q(D)f\|_{L^{1}(\mathbb{R}^{n})}\leq\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}.$

Note that the bounds for ${\mathcal{F}}^{-1}(\theta_{{\omega},\sigma})$ that
we used in the penultimate line are contained in (2.8). ∎

We can now derive a useful Sobolev embedding for
$\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$, which is formulated in terms of
the local real Hardy space $\mathcal{H}^{1}({\mathbb{R}^{n}})$ defined by
Goldberg [10]. For $r\in C^{\infty}_{c}({\mathbb{R}^{n}})$ fixed, this space
consists of all $f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ such that
$r(D)f\in L^{1}({\mathbb{R}^{n}})$ and $(1-r)(D)f\in H^{1}({\mathbb{R}^{n}})$,
with the norm

(3.13)
$\|f\|_{\mathcal{H}^{1}({\mathbb{R}^{n}})}\eqsim\|r(D)f\|_{L^{1}({\mathbb{R}^{n}})}+\|(1-r)(D)f\|_{H^{1}({\mathbb{R}^{n}})}.$

Up to norm equivalence, this definition does not depend on the specific choice
of $r$.

######  Proposition 3.2.

The map
${\langle}D\rangle^{-\frac{n-1}{4}}:\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})\to\mathcal{H}^{1}({\mathbb{R}^{n}})$
is bounded. Hence $\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})\subseteq
W^{-\frac{n-1}{4},1}({\mathbb{R}^{n}})$.

###### Proof.

For the first statement we let $r:=q$ and fix
$f\in\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$. Then
$q(D){\langle}D\rangle^{-\frac{n-1}{4}}f\in L^{1}({\mathbb{R}^{n}})$ with

$\|q(D){\langle}D\rangle^{-\frac{n-1}{4}}f\|_{L^{1}({\mathbb{R}^{n}})}\lesssim\|q(D)f\|_{L^{1}({\mathbb{R}^{n}})}\leq\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}.$

To show that $(1-q)(D){\langle}D\rangle^{-\frac{n-1}{4}}f\in
H^{1}({\mathbb{R}^{n}})$, define $m\in C^{\infty}({\mathbb{R}^{n}})$ by

(3.14)
$m(\xi):=\begin{cases}(1-q(\xi)){\langle}\xi\rangle^{-\frac{n-1}{4}}\big{(}\int_{S^{n-1}}{\varphi}_{\nu}(\xi)\mathrm{d}\nu\big{)}^{-1}&\text{if
}|\xi|\geq 1/2,\\\ 0&\text{otherwise}.\end{cases}$

It follows from (2.9) that $m(D):H^{1}({\mathbb{R}^{n}})\to
H^{1}({\mathbb{R}^{n}})$ is continuous, and one has

(3.15)
$(1-q)(D){\langle}D\rangle^{-\frac{n-1}{4}}f=\int_{S^{n-1}}m(D){\varphi}_{{\omega}}(D)f\mathrm{d}{\omega}$

since $q(\xi)=1$ if $|\xi|\geq 2$. Hence
$(1-q)(D){\langle}D\rangle^{-\frac{n-1}{4}}f\in H^{1}({\mathbb{R}^{n}})$ with

$\displaystyle\|(1-q)(D){\langle}D\rangle^{-\frac{n-1}{4}}f\|_{H^{1}({\mathbb{R}^{n}})}$
$\displaystyle=\Big{\|}\int_{S^{n-1}}m(D){\varphi}_{{\omega}}(D)f\mathrm{d}{\omega}\Big{\|}_{H^{1}({\mathbb{R}^{n}})}$
$\displaystyle\leq\int_{S^{n-1}}\|m(D){\varphi}_{{\omega}}(D)f\|_{H^{1}({\mathbb{R}^{n}})}\mathrm{d}{\omega}\lesssim\int_{S^{n-1}}\|{\varphi}_{{\omega}}(D)f\|_{H^{1}({\mathbb{R}^{n}})}\mathrm{d}{\omega}\lesssim\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})},$

where for the final inequality we used Proposition 3.1.

The second statement of the proposition now follows from the inclusion
$H^{1}({\mathbb{R}^{n}})\subseteq L^{1}({\mathbb{R}^{n}})$. ∎

######  Remark 3.3.

The same embedding as in Proposition 3.2 was obtained for
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$ in [12, Theorem 7.4], with a
somewhat similar proof. However, we cannot appeal to that result here since we
have not yet shown that
$\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$
(and in fact we will use Proposition 3.2 to prove this inclusion).

We will also need the following technical lemma.

######  Lemma 3.4.

Let $0<r\leq 1$, and let $\\{b_{l}\\}_{l=1}^{\infty}\subseteq[0,\infty]$ and
$\\{d_{l}\\}_{l=1}^{\infty}\subseteq[0,\infty)$ be two sequences. Assume that
there exist $C_{0},N_{0}>0$ such that

(3.16) $d_{l}\leq C_{0}2^{lN_{0}}\quad(l\in{{\mathbb{N}}}),$

and that for each $N>N_{0}$ there exists a $C_{N}>0$ such that

(3.17) $d_{l}\leq
C_{N}\sum_{j=1}^{\infty}2^{-|j-l|N}b_{j}d_{j}^{1-r}\quad(l\in{{\mathbb{N}}}).$

Then

$d_{l}^{r}\leq
C_{N}\sum_{j=1}^{\infty}2^{-|j-l|Nr}b_{j}\quad(l\in{{\mathbb{N}}}).$

###### Proof.

The proof of Lemma 3.4 is essentially contained in [16], but for the reader’s
convenience we give a simple proof here. Without loss of generality, we may
assume that $\\{d_{l}\\}_{l=1}^{\infty}$ is not the zero sequence, and then
(3.16) shows that
$D_{l,N}:=\sup_{k\in{{\mathbb{N}}}}2^{-|l-k|N}d_{k}\in(0,\infty)$ for all
$l\in{{\mathbb{N}}}$ and $N>N_{0}$. Now (3.17) yields

$\displaystyle D_{l,N}$
$\displaystyle\leq\sup_{k\in{{\mathbb{N}}}}2^{-|l-k|N}C_{N}\sum_{j=1}^{\infty}2^{-|j-k|N}b_{j}d_{j}^{1-r}\leq
C_{N}\sum_{j=1}^{\infty}2^{-|j-l|N}b_{j}d_{j}^{1-r}$ $\displaystyle\leq
C_{N}\sum_{j=1}^{\infty}2^{-|j-l|N}b_{j}2^{|j-l|N(1-r)}D_{l,N}^{1-r}=C_{N}\sum_{j=1}^{\infty}2^{-|j-l|Nr}b_{j}D_{l,N}^{1-r}$

for all $j\in{{\mathbb{N}}}$. Multiplying by $D_{l,N}^{r-1}$, we obtain from
this the required conclusion:

$d_{l}^{r}\leq D_{l,N}^{r}\leq C_{N}\sum_{j=1}^{\infty}2^{-|j-l|Nr}b_{j}.\qed$

For the main result of this section we will work with a Peetre type maximal
function. For $\alpha>0$, $f\in\mathcal{S^{\prime}}(\mathbb{R}^{n})$ and
$(x,\omega,\sigma)\in S^{*}(\mathbb{R}^{n})\times(0,\infty)$, set

$\displaystyle M_{\alpha}^{*}(f)(x,\omega,\sigma):=\sup_{(y,\nu)\in
S^{*}(\mathbb{R}^{n})}\frac{|\theta_{\nu,\sigma}(D)f(y)|}{(1+\sigma^{-1}d(x,\omega;y,\nu)^{2})^{\alpha}},$

where the metric $d$ on $S^{*}({\mathbb{R}^{n}})$ is as in Section 2.2. We
will apply Lemma 3.4 to a sequence arising from this maximal function, and in
the following lemma we show that the growth condition (3.16) is satisfied for
this sequence.

######  Lemma 3.5.

Let $\alpha>0$. Then there exists a $C_{\alpha}>0$ with the following
property. For all $f\in W^{-\frac{n-1}{4},1}(\mathbb{R}^{n})$,
$(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$, $l\in{{\mathbb{N}}}$ and
$\sigma\in(1,2)$, one has

$M_{\alpha}^{*}(f)(x,\omega,2^{-l}\sigma)\leq
C_{\alpha}2^{ln}\|f\|_{W^{-\frac{n-1}{4},1}(\mathbb{R}^{n})}.$

###### Proof.

Fix $f\in W^{-\frac{n-1}{4},1}(\mathbb{R}^{n})$, $(x,{\omega})\in
S^{*}({\mathbb{R}^{n}})$, $l\in{{\mathbb{N}}}$ and $\sigma\in(1,2)$. For
$\tau\in(0,1)$ and $\xi\in{\mathbb{R}^{n}}$, set
$\tilde{\theta}_{\omega,\tau}(\xi):=\tau^{\frac{n-1}{4}}\langle\xi\rangle^{\frac{n-1}{4}}\theta_{\omega,\tau}(\xi)$.
It is straightforward to see that $\tilde{\theta}_{{\omega},\tau}\in
C^{\infty}_{c}({\mathbb{R}^{n}})$, with the same support properties and upper
bounds as $\theta_{{\omega},\tau}$ from Lemma 2.2, with constants independent
of $\tau$. In particular, using (2.7), we obtain

$\displaystyle M_{\alpha}^{*}(f)(x,\omega,2^{-l}\sigma)$
$\displaystyle=\sup_{(y,\nu)\in
S^{*}(\mathbb{R}^{n})}\frac{|\theta_{\nu,2^{-l}\sigma}(D)f(y)|}{(1+2^{l}\sigma^{-1}d(x,\omega;y,\nu)^{2})^{\alpha}}\leq\sup_{(y,\nu)\in
S^{*}(\mathbb{R}^{n})}|\theta_{\nu,2^{-l}\sigma}(D)f(y)|$ $\displaystyle\leq
2^{\frac{n-1}{4}l}\sup_{(y,\nu)\in
S^{*}(\mathbb{R}^{n})}|\tilde{\theta}_{\nu,2^{-l}\sigma}(D)\langle
D\rangle^{-\frac{n-1}{4}}f(y)|$ $\displaystyle\leq
2^{\frac{n-1}{4}l}\sup_{(y,\nu)\in
S^{*}(\mathbb{R}^{n})}\int_{{\mathbb{R}^{n}}}|{\mathcal{F}}^{-1}(\tilde{\theta}_{\nu,2^{-l}\sigma})(y-z){\langle}D\rangle^{-\frac{n-1}{4}}f(z)|\,\mathrm{d}z$
$\displaystyle\lesssim 2^{nl}\int_{\mathbb{R}^{n}}|\langle
D\rangle^{-\frac{n-1}{4}}f(z)|\,\mathrm{d}z=2^{nl}\|f\|_{W^{-\frac{n-1}{4},1}(\mathbb{R}^{n})}.\qed$

Having verified the conditions of Lemma 3.4, we can now apply this lemma to
obtain a useful inequality for our maximal function.

######  Proposition 3.6.

Let $\alpha>0$ and $r\in(0,1)$. Then for each $N>0$ there exists a
$C_{\alpha,r,N}>0$ such that, for all $\sigma\in(1,2)$, $l\in{{\mathbb{N}}}$
and $f\in W^{-\frac{n-1}{4},1}(\mathbb{R}^{n})$ with ${\mathcal{F}}f(\xi)=0$
for $|\xi|\leq 2$, one has

$[M_{\alpha}^{*}(f)(x,\omega,2^{-l}\sigma)]^{r}\leq
C_{\alpha,r,N}\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(x,\omega;z,\mu)^{2})^{-\alpha
r}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|^{r}\mathrm{d}z\mathrm{d}\mu.$

Note that the Fourier transform ${\mathcal{F}}f$ of an $f\in
W^{-\frac{n-1}{4},1}({\mathbb{R}^{n}})$ is a function of at most polynomial
growth, so the pointwise condition ${\mathcal{F}}f(\xi)=0$ for $|\xi|\leq 2$
is well defined. We also note that the assumption $f\in
W^{-\frac{n-1}{4},1}({\mathbb{R}^{n}})$ can be extended to $f\in
W^{s,1}({\mathbb{R}^{n}})$ for some $s\in\mathbb{R}$, but we will not need
such generality in the remainder.

###### Proof.

Clearly we may consider $N\geq\alpha$. Fix $(x,{\omega})\in
S^{*}({\mathbb{R}^{n}})$, $\sigma\in(1,2)$, $l\in{{\mathbb{N}}}$ and $f\in
W^{-\frac{n-1}{4},1}({\mathbb{R}^{n}})$ with ${\mathcal{F}}f(\xi)=0$ for
$|\xi|\leq 2$. Recall from the proof of Lemma 2.2 that $\eta$, as defined in
(2.4), satisfies $\eta\in C^{\infty}_{c}({\mathbb{R}^{n}})$ and ${\rm
supp}{\hskip 1.42271pt}(\eta)={\rm supp}{\hskip
1.42271pt}(\Psi)\subseteq\\{\xi\in\mathbb{R}^{n}:\frac{1}{2}\leq|\xi|\leq
2\\}$. As we did for $\Psi$, write $\eta_{\tau}(\xi):=\eta(\tau\xi)$ for
$\tau>0$ and $\xi\in{\mathbb{R}^{n}}$. Then, by definition, the following
identity holds for $\xi\neq 0$:

$\sum_{j\in\mathbb{Z}}\eta_{2^{-j}\sigma}(\xi)\Psi_{2^{-j}\sigma}(\xi)=1.$

Now, by the assumption on the support of ${\mathcal{F}}f$ and because
$\sigma\in(1,2)$, one has $\Psi_{2^{-j}\sigma}(D)f=0$ for $j\leq 0$. Hence,
using the definition of $\chi_{\mu,2^{-j}\sigma}$ from (2.5), a direct
calculation yields

$\displaystyle\theta_{\nu,2^{-l}\sigma}(D)f(y)$
$\displaystyle=\sum_{j\in{{\mathbb{Z}}}}\theta_{\nu,2^{-l}\sigma}(D)\eta_{2^{-j}\sigma}(D)\Psi_{2^{-j}\sigma}(D)f(y)=\sum_{j=1}^{\infty}\theta_{\nu,2^{-l}\sigma}(D)\eta_{2^{-j}\sigma}(D)\Psi_{2^{-j}\sigma}(D)f(y)$
$\displaystyle=\sum_{j=1}^{\infty}\int_{S^{n-1}}\theta_{\nu,2^{-l}\sigma}(D)\chi_{\mu,2^{-j}\sigma}(D)\theta_{\mu,2^{-j}\sigma}(D)f(y)\,\mathrm{d}\mu$

for all $(y,\nu)\in S^{*}({\mathbb{R}^{n}})$. Now apply Corollary 2.3 to

$K_{2^{-l}\sigma,2^{-j}\sigma}^{\nu,\mu}(y,z)={\mathcal{F}}^{-1}(\theta_{\nu,2^{-l}\sigma}\chi_{\mu,2^{-j}\sigma})(y-z)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{i\langle
y-z,\xi\rangle}\theta_{\nu,2^{-l}\sigma}(\xi)\chi_{\mu,2^{-j}\sigma}(\xi)\,\mathrm{d}\xi,$

for $(z,\mu)\in S^{*}({\mathbb{R}^{n}})$ and $j\in{{\mathbb{N}}}$, to obtain

$\displaystyle|\theta_{\nu,2^{-l}\sigma}(D)f(y)|$
$\displaystyle\leq\sum_{j=1}^{\infty}\Big{|}\int_{S^{n-1}}\theta_{\nu,2^{-l}\sigma}(D)\chi_{\mu,2^{-j}\sigma}(D)\theta_{\mu,2^{-j}\sigma}(D)f(y)\,\mathrm{d}\mu\Big{|}$
$\displaystyle\leq\sum_{j=1}^{\infty}\int_{S^{n-1}}\int_{\mathbb{R}^{n}}|K_{2^{-l}\sigma,2^{-j}\sigma}^{\nu,\mu}(y,z)\theta_{\mu,2^{-j}\sigma}(D)f(z)|\,\mathrm{d}z\mathrm{d}\mu$
$\displaystyle\lesssim\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(y,\nu;z,\mu)^{2})^{-N}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|\,\mathrm{d}z\mathrm{d}\mu.$

In turn, we can use that $d$ is a metric and that $N\geq\alpha$ to derive from
this that

$\displaystyle
M_{\alpha}^{*}(f)(x,\omega,2^{-l}\sigma)\lesssim\sup_{(y,\nu)\in
S^{*}(\mathbb{R}^{n})}\frac{|\theta_{\nu,2^{-l}\sigma}(D)f(y)|}{(1+2^{l}d(x,\omega;y,\nu)^{2})^{\alpha}}$
$\displaystyle\lesssim\sup_{(y,\nu)\in
S^{*}(\mathbb{R}^{n})}\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(y,\nu;z,\mu)^{2})^{-N}(1+2^{l}d(x,\omega;y,\nu)^{2})^{-\alpha}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|\,\mathrm{d}z\mathrm{d}\mu$
$\displaystyle\leq\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(x,\omega;z,\mu)^{2})^{-\alpha}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|\,\mathrm{d}z\mathrm{d}\mu$
$\displaystyle=\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(x,\omega;z,\mu)^{2})^{-\alpha}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|^{r}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|^{1-r}\mathrm{d}z\mathrm{d}\mu$
$\displaystyle\leq\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}\frac{(1+2^{j}d(x,\omega;z,\mu)^{2})^{\alpha(1-r)}}{(1+2^{l}d(x,\omega;z,\mu)^{2})^{\alpha}}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|^{r}\mathrm{d}z\mathrm{d}\mu(M_{\alpha}^{*}(f)(x,\omega,2^{-j}\sigma))^{1-r}.$

Moreover, since one has

$1+2^{j}d(x,\omega;z,\mu)^{2}\leq(1+2^{l}d(x,\omega;z,\mu)^{2})2^{|l-j|}$

for all $j\in{{\mathbb{N}}}$ and $(z,\mu)\in S^{*}({\mathbb{R}^{n}})$, we can
write

$\displaystyle M_{\alpha}^{*}(f)(x,\omega,2^{-l}\sigma)$
$\displaystyle\lesssim\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}\frac{(1+2^{j}d(x,\omega;z,\mu)^{2})^{\alpha(1-r)}}{(1+2^{l}d(x,\omega;z,\mu)^{2})^{\alpha}}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|^{r}\mathrm{d}z\mathrm{d}\mu(M_{\alpha}^{*}(f)(x,\omega,2^{-j}\sigma))^{1-r}$
$\displaystyle\leq\sum_{j=1}^{\infty}2^{-|j-l|(N-\alpha)}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(x,\omega;z,\mu)^{2})^{-\alpha
r}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|^{r}\mathrm{d}z\mathrm{d}\mu\big{(}M_{\alpha}^{*}(f)(x,\omega,2^{-j}\sigma)\big{)}^{1-r}.$

Finally, we can apply Lemmas 3.4 and 3.5 to this estimate to obtain

$(M_{\alpha}^{*}(f)(x,\omega,2^{-l}\sigma))^{r}\lesssim\sum_{j=1}^{\infty}2^{-|j-l|(N-\alpha)r}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(x,\omega;z,\mu)^{2})^{-\alpha
r}|\theta_{\mu,2^{-j}\sigma}(D)f(z)|^{r}\mathrm{d}z\mathrm{d}\mu.\qed$

To conclude this subsection we collect the following result from [16].

######  Lemma 3.7.

Let $(X,\tilde{d},\mu)$ be a metric measure space, where $\tilde{d}$ is a
metric and $\mu$ is a nonnegative, doubling, Borel measure. Let
$p,q\in[1,\infty)$ and $N>0$, and let $\\{g_{j}\\}_{j\in{{\mathbb{Z}}}}$ be a
sequence of nonnegative measurable functions on $X$. For each
$l\in{{\mathbb{Z}}}$ set

$\displaystyle h_{l}:=\sum_{j=-\infty}^{\infty}2^{-|j-l|N}g_{j}.$

Then there exists a $C=C(p,q,N)>0$ such that

$\displaystyle||\\{h_{l}\\}_{l\in{{\mathbb{Z}}}}||_{L^{p}(X;\ell^{q})}\leq
C||\\{g_{j}\\}_{j\in{{\mathbb{Z}}}}||_{L^{p}(X;\ell^{q})}.$

###  3.2. The main embedding

After this preliminary work, we are ready to prove the main result of this
section.

######  Theorem 3.8.

One has

$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})=\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})$

with equivalent norms.

###### Proof.

As already noted, it follows from [4, Proposition 2.1 and Remark 2.2] that
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})$
continuously. More precisely, it is shown in [4, Proposition 2.1 and Remark
2.2] that the following inequality holds for the square functions $S$ and $G$
from (1.1) and (1.3), respectively:

(3.18)
$\|G(g)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}\lesssim\|S(g)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}\quad(g\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})).$

So it remains to prove that
$\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})\subseteq\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$.

Fix $f\in\mathcal{H}_{FIO,G}^{1}(\mathbb{R}^{n})$. First note that one
trivially has $q(D)f\in L^{1}({\mathbb{R}^{n}})$ and

$\|q(D)f\|_{L^{1}({\mathbb{R}^{n}})}\leq\|G(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}+\|q(D)f\|_{L^{1}({\mathbb{R}^{n}})}=\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}.$

So it suffices to show that $S(f)\in L^{1}(S^{*}({\mathbb{R}^{n}}))$ and

$\|S(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}\lesssim\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}.$

To this end, we decompose $f$ into its low-frequency and high-frequency
components:

$\displaystyle\|S(f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}\leq\|S(q(D)f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}+\|S((1-q)(D)f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}.$

For the low-frequency part we use that
${\langle}D\rangle^{-\frac{n-1}{4}}:\mathcal{H}^{1}({\mathbb{R}^{n}})\to\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$
is continuous (see [12, Theorem 7.4]), where
$\mathcal{H}^{1}({\mathbb{R}^{n}})$ is as defined in (3.13). Choosing $r$ in
the definition of $\mathcal{H}^{1}({\mathbb{R}^{n}})$ such that $r\equiv 1$ on
${\rm supp}{\hskip 1.42271pt}(q)$, we obtain

(3.19) $\displaystyle\|S(q(D)f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}$
$\displaystyle\leq\|q(D)f\|_{\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})}\lesssim\|\langle
D\rangle^{\frac{n-1}{4}}q(D)f\|_{\mathcal{H}^{1}(\mathbb{R}^{n})}$
$\displaystyle\eqsim\|\langle
D\rangle^{\frac{n-1}{4}}q(D)f\|_{L^{1}(\mathbb{R}^{n})}\lesssim\|q(D)f\|_{L^{1}(\mathbb{R}^{n})}\leq\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})},$

where in the penultimate inequality we used that $q\in
C^{\infty}_{c}({\mathbb{R}^{n}})$.

Next, we consider the high-frequency component $h:=(1-q)(D)f$. We fix
$\alpha>n$ and claim that it suffices to prove the following two inequalities:

(3.20)
$S(h)(x,\omega)\lesssim\bigg{(}\int_{0}^{1}[M_{\alpha}^{*}(h)(x,\omega,\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}$

for all $(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$, and

(3.21)
$\int_{S^{*}({\mathbb{R}^{n}})}\bigg{(}\int_{0}^{1}[M_{\alpha}^{*}(h)(x,\omega,\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}{\omega}\lesssim\|G(h)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}.$

Indeed, by combining these inequalities with (3.19) and (3.18), we obtain

$\displaystyle\|S(f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}$
$\displaystyle\leq\|S(q(D)f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}+\|S(h)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}\lesssim\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}+\|G(h)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}$
$\displaystyle\leq\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}+\|G(q(D)f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}+\|G(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}$
$\displaystyle\lesssim\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}+\|S(q(D)f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}+\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}\lesssim\|f\|_{\mathcal{H}^{1}_{FIO,G}({\mathbb{R}^{n}})}.$

Hence in the remainder we will focus on proving (3.20) and (3.21).

#### Estimate (3.20)

This estimate follows from a straightforward calculation. For all
$(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$, $\sigma>0$ and $(y,\nu)\in
B_{\sqrt{\sigma}}(x,\omega)$ one has $1\leq
1+\sigma^{-1}d(x,\omega;y,\nu)^{2}\leq 2$. Hence

$\displaystyle S(h)(x,\omega)$
$\displaystyle=\bigg{(}\int_{0}^{1}\fint_{B_{\sqrt{\sigma}}(x,\omega)}|\theta_{\nu,\sigma}(D)h(y)|^{2}\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}$
$\displaystyle\leq\bigg{(}\int_{0}^{1}\sup\limits_{(y,\nu)\in
B_{\sqrt{\sigma}}(x,\omega)}|\theta_{\nu,\sigma}(D)h(y)|^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}$
$\displaystyle\lesssim\bigg{(}\int_{0}^{1}\sup\limits_{(y,\nu)\in
B_{\sqrt{\sigma}}(x,\omega)}\frac{|\theta_{\nu,\sigma}(D)f(y)|^{2}}{(1+\sigma^{-1}d(x,\omega;y,\nu)^{2})^{2\alpha}}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}$
$\displaystyle\leq\bigg{(}\int_{0}^{1}\sup\limits_{(y,\nu)\in
S^{*}(\mathbb{R}^{n})}\frac{|\theta_{\nu,\sigma}(D)f(y)|^{2}}{(1+\sigma^{-1}d(x,\omega;y,\nu)^{2})^{2\alpha}}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}=\bigg{(}\int_{0}^{1}[M_{\alpha}^{*}(h)(x,\omega,\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}.$

#### Estimate (3.21)

The idea of the proof is to write

$\displaystyle\int_{S^{*}({\mathbb{R}^{n}})}\bigg{(}\int_{0}^{1}[M_{\alpha}^{*}(h)(x,\omega,\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}{\omega}$
$\displaystyle=\int_{S^{*}({\mathbb{R}^{n}})}\bigg{(}\sum_{l=1}^{\infty}\int_{1}^{2}[M_{\alpha}^{*}(h)(x,\omega,2^{-l}\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}{\omega}$
$\displaystyle=\bigg{\|}\bigg{\\{}\bigg{(}\int_{1}^{2}[M_{\alpha}^{*}(h)(\cdot,\cdot,2^{-l}\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}\bigg{\\}}_{l=1}^{\infty}\bigg{\|}_{L^{1/r}(S^{*}(\mathbb{R}^{n});\ell^{2/r})}^{1/r}$

for a suitably chosen $r\in(0,1)$. We will bound the sequence in the final
term by a suitable expression involving the Hardy–Littlewood maximal function,
and then we combine boundedness properties of this maximal function with Lemma
3.7 to obtain (3.21).

For the moment, fix $(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$ and
$l\in{{\mathbb{N}}}$. We will use the pointwise estimate in Proposition 3.6
for $M_{\alpha}^{*}(h)$. Note that Proposition 3.6 indeed applies to $h$,
given that Proposition 3.2 shows that $f\in
W^{-\frac{n-1}{4},1}({\mathbb{R}^{n}})$ and therefore $h=(1-q)(D)f\in
W^{-\frac{n-1}{4},1}({\mathbb{R}^{n}})$ as well. And one has

${\mathcal{F}}h(\xi)=(1-q(\xi)){\mathcal{F}}f(\xi)=0$

for $|\xi|\leq 2$ because $q(\xi)=1$ for such $\xi$. Now, since $\alpha>n$ we
can choose $r\in(n/\alpha,1)$ and $N>0$ and apply Proposition 3.6 to obtain

$\displaystyle[M_{\alpha}^{*}(h)(x,\omega,2^{-l}\sigma)]^{r}\lesssim\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(x,\omega;y,v)^{2})^{-\alpha
r}|\theta_{\nu,2^{-j}\sigma}(D)h(y)|^{r}\,\mathrm{d}y\mathrm{d}\nu$

for all $l\in{{\mathbb{N}}}$ and $\sigma\in(1,2)$. Hence the triangle
inequality and Minkowski’s inequality yield

(3.22)
$\displaystyle\bigg{(}\int_{1}^{2}[M_{\alpha}^{*}(h)(x,\omega,2^{-l}\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}$
$\displaystyle\lesssim\bigg{(}\int_{1}^{2}\bigg{(}\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(x,\omega;y,v)^{2})^{-\alpha
r}|\theta_{\nu,2^{-j}\sigma}(D)h(y)|^{r}\,\mathrm{d}y\mathrm{d}\nu\bigg{)}^{2/r}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}$
$\displaystyle\leq\sum_{j=1}^{\infty}2^{-|j-l|N}\int_{S^{*}(\mathbb{R}^{n})}2^{ln}(1+2^{l}d(x,\omega;y,v)^{2})^{-\alpha
r}\bigg{(}\int_{1}^{2}|\theta_{\nu,2^{-j}\sigma}(D)h(y)|^{2}\frac{d\sigma}{\sigma}\bigg{)}^{r/2}\mathrm{d}y\mathrm{d}\nu.$

Next, we will bound each of the terms is this series separately.

Momentarily fix $j\in{{\mathbb{N}}}$, and write

$F(y,\nu):=\bigg{(}\int_{1}^{2}|\theta_{\nu,2^{-j}\sigma}(D)h(y)|^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}$

for $(y,\nu)\in S^{*}({\mathbb{R}^{n}})$. Also let $\mathcal{M}$ be the
centered Hardy-Littlewood operator on
$(S^{*}(\mathbb{R}^{n}),d,\mathrm{d}x\mathrm{d}\omega)$ given by

$\displaystyle\mathcal{M}(f)(x,\omega):=\sup_{(x,\omega)\in
B}\frac{1}{V(B)}\int_{B}|f(y,\nu)|\,\mathrm{d}y\mathrm{d}\nu$

for $f\in L^{1}_{\mathrm{loc}}(S^{*}({\mathbb{R}^{n}}))$, where the supremum
is taken over all balls $B\subseteq S^{*}({\mathbb{R}^{n}})$ with center
$(x,{\omega})$. Then

$\int_{S^{*}(\mathbb{R}^{n})}2^{ln}\Big{(}1+2^{l}d(x,\omega;y,v)^{2}\Big{)}^{-\alpha
r}F(y,\nu)\,\mathrm{d}y\mathrm{d}\nu=\sum_{k=0}^{\infty}\int_{C_{k}}\frac{1}{2^{-ln}}\Big{(}1+\frac{d(x,\omega;y,v)^{2}}{2^{-l}}\Big{)}^{-\alpha
r}F(y,\nu)\,\mathrm{d}y\mathrm{d}\nu,$

where

$C_{0}=\\{(y,\nu)\in S^{*}({\mathbb{R}^{n}}):d(x,\omega;y,v)\leq\sqrt{2}\
2^{-l/2}\\}=B_{\sqrt{2}\,2^{-l/2}}(x,{\omega})$

and

$C_{k}=\\{(y,\nu)\in
S^{*}({\mathbb{R}^{n}}):2^{k/2}2^{-l/2}<d(x,\omega;y,v)\leq
2^{(k+1)/2}2^{-l/2}\\}$

for $k\in{{\mathbb{N}}}$. We bound each of the terms in this series
separately, recalling from Lemma 2.1 that
$V(B_{\tau}(x,\omega))\lesssim\tau^{2n}$ for all $\tau>0$. We obtain

$\displaystyle\int_{C_{0}}\frac{1}{2^{-ln}}\Big{(}1+\frac{d(x,\omega;y,v)^{2}}{2^{-l}}\Big{)}^{-\alpha
r}F(y,\nu)\,\mathrm{d}y\mathrm{d}\nu\leq\int_{C_{0}}2^{ln}F(y,\nu)\,\mathrm{d}y\mathrm{d}\nu$
$\displaystyle=V(B_{\sqrt{2}\ 2^{-l/2}}(x,\omega))2^{ln}\fint_{B_{\sqrt{2}\
2^{-l/2}}(x,\omega)}F(y,\nu)\,\mathrm{d}y\mathrm{d}\nu\lesssim
2^{n}\mathcal{M}(F)(x,\omega)$

and, for $k\in{{\mathbb{N}}}$,

$\displaystyle\int_{C_{k}}\frac{1}{2^{-ln}}\Big{(}1+\frac{d(x,\omega;y,v)^{2}}{2^{-l}}\Big{)}^{-\alpha
r}F(y,\nu)\,\mathrm{d}y\mathrm{d}\nu\leq\frac{1}{2^{k\alpha
r}}\frac{1}{2^{-ln}}\int_{C_{k}}F(y,\nu)\,\mathrm{d}y\mathrm{d}\nu$
$\displaystyle\leq\frac{1}{2^{k\alpha
r}}\frac{1}{2^{-ln}}V(B_{2^{(k+1)/2}2^{-l/2}}(x,\omega))\fint_{B_{2^{(k+1)/2}2^{-l/2}}(x,\omega)}F(y,\nu)\,\mathrm{d}y\mathrm{d}\nu\lesssim\frac{1}{2^{k(\alpha
r-n)}}2^{n}\mathcal{M}(F)(x,\omega).$

Since $r>\frac{n}{\alpha}$, the series converges and we obtain

$\int_{S^{*}(\mathbb{R}^{n})}2^{ln}\Big{(}1+2^{l}d(x,\omega;y,v)^{2}\Big{)}^{-\alpha
r}F(y,\nu)\,\mathrm{d}y\mathrm{d}\nu\lesssim\mathcal{M}(F)(x,\omega)=\mathcal{M}\bigg{[}\bigg{(}\int_{1}^{2}|\theta_{\cdot,2^{-j}\sigma}(D)h(\cdot)|^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}\bigg{]}(x,\omega).$

Now (3.22) yields

(3.23)
$\bigg{(}\int_{1}^{2}[M_{\alpha}^{*}(h)(x,\omega,2^{-l}\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}\lesssim\sum_{j=1}^{\infty}2^{-|j-l|N}\mathcal{M}\bigg{[}\bigg{(}\int_{1}^{2}|\theta_{\cdot,2^{-j}\sigma}(D)h(\cdot)|^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}\bigg{]}(x,\omega).$

We have now obtained suitable bounds for each of the terms in our original
sequence, and we will use these bounds to complete the proof of (3.21).

For $(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$ and $j\in{{\mathbb{N}}}$, write

$g_{j}(x,\omega):=\mathcal{M}\bigg{[}\bigg{(}\int_{1}^{2}|\theta_{\cdot,2^{-j}\sigma}(D)h(\cdot)|^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}\bigg{]}(x,\omega),$

and for $l\in{{\mathbb{N}}}$ set

$h_{l}(x,\omega):=\sum_{j=1}^{\infty}2^{-|j-l|N}g_{j}(x,\omega).$

Then we can combine (3.23) with Lemma 3.7, as well as the boundedness of
$\mathcal{M}$ on $L^{1/r}(S^{*}({\mathbb{R}^{n}});\ell^{2/r})$ (see [9,
Section 6.6]), to obtain

$\displaystyle\int_{S^{*}({\mathbb{R}^{n}})}\bigg{(}\int_{0}^{1}[M_{\alpha}^{*}(h)(x,\omega,\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}{\omega}=\bigg{\|}\bigg{\\{}\bigg{(}\int_{1}^{2}[M_{\alpha}^{*}(h)(\cdot,\cdot,2^{-l}\sigma)]^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}\bigg{\\}}_{l=1}^{\infty}\bigg{\|}_{L^{1/r}(S^{*}(\mathbb{R}^{n});\ell^{2/r})}^{1/r}$
$\displaystyle\lesssim\|\\{h_{l}\\}_{l=1}^{\infty}\|^{1/r}_{L^{1/r}(S^{*}(\mathbb{R}^{n});\ell^{2/r})}\lesssim\|\\{g_{j}\\}_{j=1}^{\infty}\|^{1/r}_{L^{1/r}(S^{*}(\mathbb{R}^{n});\ell^{2/r})}=\bigg{\|}\bigg{\\{}\mathcal{M}\bigg{[}\bigg{(}\int_{1}^{2}|\theta_{\cdot,2^{-j}\sigma}(D)h(\cdot)|^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}\bigg{]}\bigg{\\}}_{j=1}^{\infty}\bigg{\|}_{L^{1/r}(S^{*}(\mathbb{R}^{n});\ell^{2/r})}^{1/r}$
$\displaystyle\lesssim\bigg{\|}\bigg{\\{}\bigg{(}\int_{1}^{2}|\theta_{\cdot,2^{-j}\sigma}(D)h(\cdot)|^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{r/2}\bigg{\\}}_{j=1}^{\infty}\bigg{\|}_{L^{1/r}(S^{*}(\mathbb{R}^{n});\ell^{2/r})}^{1/r}=\bigg{\|}\bigg{(}\int_{0}^{1}|\theta_{\cdot,\sigma}(D)h(\cdot)|^{2}\,\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\bigg{\|}_{L^{1}(S^{*}(\mathbb{R}^{n}))}$
$\displaystyle=\|G(h)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}.$

This concludes the proof of (3.21) and thereby of the theorem. ∎

##  4\. Maximal function characterization

In [15] a maximal function characterization of
$\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$ was obtained for $1<p<\infty$. As an
immediate corollary of what we have already shown, we can extend this
characterization to $\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$, by showing that
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})=\mathcal{H}^{1}_{FIO,\max}({\mathbb{R}^{n}})$.

######  Theorem 4.1.

One has

$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})=\mathcal{H}_{FIO,\max}^{1}(\mathbb{R}^{n})$

with equivalence of norms.

###### Proof.

From the maximal function characterization of $H^{1}({\mathbb{R}^{n}})$ (see
[11, Theorem 2.1.4]), we know that an
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ satisfies
$f\in\mathcal{H}_{FIO,\max}^{1}(\mathbb{R}^{n})$ if and only if $q(D)f\in
L^{1}({\mathbb{R}^{n}})$, ${\varphi}_{{\omega}}(D)f\in
H^{1}({\mathbb{R}^{n}})$ for almost all ${\omega}\in S^{n-1}$, and

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

Moreover, in this case one has

$\|f\|_{\mathcal{H}_{FIO,\max}^{1}(\mathbb{R}^{n})}\eqsim\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{H^{1}(\mathbb{R}^{n})}\,\mathrm{d}\omega+\|q(D)f\|_{L^{1}(\mathbb{R}^{n})}.$

Hence the required statement follows from Proposition 3.1 and Theorem 3.8. ∎

######  Remark 4.2.

The proof of Theorem 4.1 relies on the following characterization of
$\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$, obtained by combining Proposition
3.1 and Theorem 3.8: an $f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ satisfies
$f\in\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$ if and only if $q(D)f\in
L^{1}({\mathbb{R}^{n}})$, ${\varphi}_{{\omega}}(D)f\in
H^{1}({\mathbb{R}^{n}})$ for almost all ${\omega}\in S^{n-1}$, and
$\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{H^{1}(\mathbb{R}^{n})}\,\mathrm{d}\omega<\infty$.
In this case one has

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

A similar characterization of $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ was
obtained in [15] for $1<p<\infty$, but it is not clear whether one can also
characterize $\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$ in this manner.
More precisely, a natural question is whether an
$f\in\mathcal{S}^{\prime}({\mathbb{R}^{n}})$ satisfies
$f\in\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$ if and only if $q(D)f\in
L^{\infty}({\mathbb{R}^{n}})$,
${\varphi}_{{\omega}}(D)f\in{\mathrm{BMO}}({\mathbb{R}^{n}})$ for almost all
${\omega}\in S^{n-1}$, and $\operatorname*{ess\,sup}_{{\omega}\in
S^{n-1}}\|\varphi_{\omega}(D)f\|_{{\mathrm{BMO}}(\mathbb{R}^{n})}<\infty$, and
whether in this case

$\|f\|_{\mathcal{H}_{FIO}^{\infty}(\mathbb{R}^{n})}\eqsim\operatorname*{ess\,sup}_{{\omega}\in
S^{n-1}}\|\varphi_{\omega}(D)f\|_{{\mathrm{BMO}}(\mathbb{R}^{n})}+\|q(D)f\|_{L^{\infty}(\mathbb{R}^{n})}.$

One can use duality to show that if $q(D)f\in L^{\infty}({\mathbb{R}^{n}})$,
${\varphi}_{{\omega}}(D)f\in{\mathrm{BMO}}({\mathbb{R}^{n}})$ for almost all
${\omega}\in S^{n-1}$, and $\operatorname*{ess\,sup}_{{\omega}\in
S^{n-1}}\|\varphi_{\omega}(D)f\|_{{\mathrm{BMO}}(\mathbb{R}^{n})}<\infty$,
then $f\in\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$ with

$\|f\|_{\mathcal{H}_{FIO}^{\infty}(\mathbb{R}^{n})}\lesssim\operatorname*{ess\,sup}_{{\omega}\in
S^{n-1}}\|\varphi_{\omega}(D)f\|_{{\mathrm{BMO}}(\mathbb{R}^{n})}+\|q(D)f\|_{L^{\infty}(\mathbb{R}^{n})}.$

However, it is not clear whether the reverse inequality also holds. We leave
this as an open problem.

##  5\. ${\mathcal{G}}_{\alpha}^{*}$ characterization

In this section, we will prove that
$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})=\mathcal{H}_{FIO,\mathcal{G}_{\alpha}^{*}}^{1}(\mathbb{R}^{n})$
for $\alpha>2$. To do so, we will need the following quantitative change of
aperture formula from [15, Lemma 2.2] (see also [3]).

######  Lemma 5.1.

There exists a $C\geq 0$ such that, for all $\lambda\geq 1$ and $F\in
L^{2}_{\mathrm{loc}}(S^{*}(\mathbb{R}^{n})\times(0,\infty))$, one has

$\displaystyle\int_{S^{*}(\mathbb{R}^{n})}\bigg{(}\int_{0}^{\infty}\fint_{B_{\lambda\sqrt{\sigma}}(x,\omega)}|F(y,\nu,\sigma)|^{2}\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}\omega$
$\displaystyle\leq
C\lambda^{n}\int_{S^{*}(\mathbb{R}^{n})}\bigg{(}\int_{0}^{\infty}\fint_{B_{\sqrt{\sigma}}(x,\omega)}|F(y,\nu,\sigma)|^{2}\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}\omega$

whenever the second term is finite.

For the next theorem, recall that
$\mathcal{H}_{FIO,\mathcal{G}_{\alpha}^{*}}^{1}(\mathbb{R}^{n})$ consists of
all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that
$\mathcal{G}_{\alpha}^{*}(f)\in L^{1}(S^{*}(\mathbb{R}^{n}))$ and $q(D)f\in
L^{1}(\mathbb{R}^{n})$, endowed with the norm

$\|f\|_{\mathcal{H}_{FIO,\mathcal{G}_{\alpha}^{*}}^{1}(\mathbb{R}^{n})}=\|\mathcal{G}_{\alpha}^{*}(f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}+\|q(D)f\|_{L^{1}(\mathbb{R}^{n})}.$

Here

$\mathcal{G}_{\alpha}^{*}(f)(x,\omega)=\bigg{(}\int_{0}^{1}\int_{S^{*}(\mathbb{R}^{n})}\frac{|\theta_{\nu,\sigma}(D)f(y)|^{2}}{\sigma^{n}(1+\sigma^{-1}d(x,\omega;y,\nu)^{2})^{n\alpha}}\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}$

for $(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$.

######  Theorem 5.2.

Let $\alpha>2$. Then

$\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})=\mathcal{H}_{FIO,\mathcal{G}_{\alpha}^{*}}^{1}(\mathbb{R}^{n})$

with equivalent norms.

###### Proof.

We first show that
$\mathcal{H}_{FIO,\mathcal{G}_{\alpha}^{*}}^{1}(\mathbb{R}^{n})\subseteq\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$.
Let $f\in\mathcal{H}_{FIO,\mathcal{G}_{\alpha}^{*}}^{1}(\mathbb{R}^{n})$. It
suffices to prove that $S(f)\in L^{1}(S^{*}({\mathbb{R}^{n}}))$ with
$\|S(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}\lesssim\|\mathcal{G}_{\alpha}^{*}(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}$.
To this end, observe that for all $(x,{\omega})\in S^{*}({\mathbb{R}^{n}})$,
$\sigma>0$ and $(y,\nu)\in B_{\sqrt{\sigma}}(x,{\omega})$, one has
$1+\sigma^{-1}d(x,\omega;y,v)^{2}\leq 2$. Moreover, Lemma 2.1 yields that
$V(B_{\sqrt{\sigma}}(x,{\omega}))\eqsim\sigma^{n}$ for all $\sigma\in(0,1)$.
Hence

$\displaystyle S(f)(x,\omega)$
$\displaystyle=\bigg{(}\int_{0}^{1}\fint_{B_{\sqrt{\sigma}}(x,\omega)}|\theta_{\nu,\sigma}(D)f(y)|^{2}\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}$
$\displaystyle\lesssim\bigg{(}\int_{0}^{1}\int_{B_{\sqrt{\sigma}}(x,\omega)}\frac{|\theta_{\nu,\sigma}(D)f(y)|^{2}}{V(B_{\sqrt{\sigma}}(x,{\omega}))(1+\sigma^{-1}d(x,\omega;y,\nu)^{2})^{n\alpha}}\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}$
$\displaystyle\lesssim\mathcal{G}_{\alpha}^{*}(f)(x,\omega).$

Thus $S(f)\in L^{1}(S^{*}({\mathbb{R}^{n}}))$ with
$\|S(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}\lesssim\|\mathcal{G}_{\alpha}^{*}(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}$.

For the other inclusion we let $f\in\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$
and show that
$\|\mathcal{G}_{\alpha}^{*}(f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}\lesssim\|S(f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}$.
Note that

$\displaystyle(1+s)^{-n\alpha}\leq{\mathbf{1}}_{[0,1]}(s)+2\sum_{k=1}^{\infty}2^{-k}\mathbf{1}_{{}_{[0,1]}}\bigg{(}\frac{s}{2^{k/(n\alpha)}}\bigg{)}=:g(s)$

for all $s\geq 0$, as can be seen for $s>1$ by letting
$k_{0}\in{{\mathbb{N}}}$ be such that $2^{(k_{0}-1)/n\alpha}<s\leq
2^{k_{0}/n\alpha}$. Now apply Lemma 2.1 to obtain, for all $(x,{\omega})\in
S^{*}({\mathbb{R}^{n}})$,

$\displaystyle\mathcal{G}_{\alpha}^{*}(f)(x,\omega)\leq\bigg{(}\int_{0}^{1}\int_{S^{*}(\mathbb{R}^{n})}\sigma^{-n}|\theta_{\nu,\sigma}(D)f(y)|^{2}g(\sigma^{-1}d(x,\omega;y,\nu)^{2})\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\eqsim\bigg{(}\int_{0}^{1}\int_{B_{\sqrt{\sigma}}(x,\omega)}\sigma^{-n}|\theta_{\nu,\sigma}(D)f(y)|^{2}\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}+\sum_{k=1}^{\infty}2^{-k}\int_{0}^{1}\int_{B_{2^{k/(2n\alpha)}\sqrt{\sigma}}(x,\omega)}\sigma^{-n}|\theta_{\nu,\sigma}(D)f(y)|^{2}\,\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\lesssim\bigg{(}\int_{0}^{1}\fint_{B_{\sqrt{\sigma}}(x,\omega)}|\theta_{\nu,\sigma}(D)f(y)|^{2}\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}+\sum_{k=1}^{\infty}2^{-k+k/\alpha}\int_{0}^{1}\fint_{B_{2^{k/(2n\alpha)}\sqrt{\sigma}}(x,\omega)}|\theta_{\nu,\sigma}(D)f(y)|^{2}\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\lesssim\bigg{(}\int_{0}^{1}\fint_{B_{\sqrt{\sigma}}(x,\omega)}|\theta_{\nu,\sigma}(D)f(y)|^{2}\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{\frac{1}{2}}+\sum_{k=1}^{\infty}2^{(-k+k/\alpha)/2}\bigg{(}\int_{0}^{1}\fint_{B_{2^{k/(2n\alpha)}\sqrt{\sigma}}(x,\omega)}|\theta_{\nu,\sigma}(D)f(y)|^{2}\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{\frac{1}{2}}.$

We can then conclude the proof using Lemma 5.1, with
$F(y,\nu,\sigma)=\theta_{\nu,\sigma}(D)f(y)$:

$\displaystyle\|\mathcal{G}_{\alpha}^{*}(f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))}$
$\displaystyle\lesssim\|S(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}+\sum_{k=1}^{\infty}2^{-k/2+k/(2\alpha)}\int_{S^{*}(\mathbb{R}^{n})}\bigg{(}\int_{0}^{1}\fint_{B_{2^{k/(2n\alpha)}\sqrt{\sigma}}(x,\omega)}|\theta_{\nu,\sigma}(D)f(y)|^{2}\mathrm{d}y\mathrm{d}\nu\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{1/2}\mathrm{d}x\mathrm{d}\omega$
$\displaystyle\lesssim\|S(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}+\sum_{k=1}^{\infty}2^{-k/2+k/\alpha}\|S(f)\|_{L^{1}(S^{*}({\mathbb{R}^{n}}))}\lesssim\|S(f)\|_{L^{1}(S^{*}(\mathbb{R}^{n}))},$

where for the final inequality we used that $\alpha>2$. ∎

##  6\. Applications

In this section we give two applications of the results in the previous
sections. The aim here is to show how the characterizations in this article
can be used to incorporate techniques from other parts of harmonic analyis,
and to demonstrate that the characterizations are amenable to direct
calculations. Other applications of these characterizations, to operators with
rough coefficients, will follow in future work.

We first prove that a large class of singular integral operators which are
bounded on $L^{p}({\mathbb{R}^{n}})$ for $1<p<\infty$ are also bounded on
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ for all $1\leq p\leq\infty$. Recall
the definition of the local Hardy space $\mathcal{H}^{1}({\mathbb{R}^{n}})$
from (3.13).

######  Theorem 6.1.

Let $m\in L^{\infty}({\mathbb{R}^{n}})$ be such that
$m(D):\mathcal{H}^{1}({\mathbb{R}^{n}})\to L^{1}({\mathbb{R}^{n}})$ is
bounded. Then
$m(D):\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
is bounded for all $p\in[1,\infty]$.

###### Proof.

We first consider the case where $p=1$. It follows from the inclusion
$H^{1}({\mathbb{R}^{n}})\subseteq L^{1}({\mathbb{R}^{n}})$ that
$H^{1}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{1}({\mathbb{R}^{n}})$, and
therefore $m(D):H^{1}({\mathbb{R}^{n}})\to L^{1}({\mathbb{R}^{n}})$ is
bounded. Now, for $j\in\\{1,\ldots,n\\}$, let $R_{j}(D)$, where
$R_{j}(\xi):=-i\xi_{j}/|\xi|$ for
$\xi=(\xi_{1},\ldots,\xi_{n})\in{\mathbb{R}^{n}}\setminus\\{0\\}$, be the
$j$-th Riesz transform. Then the Riesz transform characterization of
$H^{1}({\mathbb{R}^{n}})$ (see [20, Section III.4.3]) shows that
$m(D):H^{1}({\mathbb{R}^{n}})\to H^{1}({\mathbb{R}^{n}})$ with

$\displaystyle\|m(D)f\|_{H^{1}({\mathbb{R}^{n}})}$
$\displaystyle\eqsim\|m(D)f\|_{L^{1}({\mathbb{R}^{n}})}+\sum_{j=1}^{n}\|R_{j}(D)m(D)f\|_{L^{1}({\mathbb{R}^{n}})}\lesssim\|f\|_{H^{1}({\mathbb{R}^{n}})}+\sum_{j=1}^{n}\|m(D)R_{j}(D)f\|_{L^{1}({\mathbb{R}^{n}})}$
$\displaystyle\lesssim\|f\|_{H^{1}({\mathbb{R}^{n}})}+\sum_{j=1}^{n}\|R_{j}(D)f\|_{H^{1}({\mathbb{R}^{n}})}\lesssim\|f\|_{H^{1}({\mathbb{R}^{n}})}$

for all $f\in H^{1}({\mathbb{R}^{n}})$, where we also used that the Riesz
transforms are bounded on $H^{1}({\mathbb{R}^{n}})$.

Now let $f\in\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$. By Proposition 3.1 and
Theorem 3.8, it suffices to show that $q(D)m(D)f\in L^{1}({\mathbb{R}^{n}})$,
${\varphi}_{{\omega}}(D)m(D)f\in H^{1}({\mathbb{R}^{n}})$ for almost all
${\omega}\in S^{n-1}$, and

$\int_{S^{n-1}}\|\varphi_{\omega}(D)m(D)f\|_{H^{1}(\mathbb{R}^{n})}\,\mathrm{d}\omega+\|q(D)m(D)f\|_{L^{1}({\mathbb{R}^{n}})}\lesssim\|f\|_{\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})}.$

But this follows from the boundedness of $m(D)$ on $H^{1}({\mathbb{R}^{n}})$
and from $\mathcal{H}^{1}({\mathbb{R}^{n}})$ to $L^{1}({\mathbb{R}^{n}})$, if
one takes $r\in C^{\infty}_{c}({\mathbb{R}^{n}})$ in the definition of
$\mathcal{H}^{1}({\mathbb{R}^{n}})$ such that $r\equiv 1$ on ${\rm
supp}{\hskip 1.42271pt}(q)$:

$\displaystyle\int_{S^{n-1}}\|\varphi_{\omega}(D)m(D)f\|_{H^{1}(\mathbb{R}^{n})}\,\mathrm{d}\omega+\|q(D)m(D)f\|_{L^{1}({\mathbb{R}^{n}})}$
$\displaystyle=\int_{S^{n-1}}\|m(D)\varphi_{\omega}(D)f\|_{H^{1}(\mathbb{R}^{n})}\,\mathrm{d}\omega+\|m(D)q(D)f\|_{L^{1}({\mathbb{R}^{n}})}\lesssim\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{H^{1}(\mathbb{R}^{n})}\,\mathrm{d}\omega+\|q(D)f\|_{\mathcal{H}^{1}({\mathbb{R}^{n}})}$
$\displaystyle=\int_{S^{n-1}}\|\varphi_{\omega}(D)f\|_{H^{1}(\mathbb{R}^{n})}\,\mathrm{d}\omega+\|q(D)f\|_{L^{1}({\mathbb{R}^{n}})}\eqsim\|f\|_{\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})},$

where we again used Proposition 3.1 and Theorem 3.8 for the final equivalence.

Next, we consider the case $p=\infty$. It is straightforward to check that

$\overline{m(D)^{*}g(x)}=\overline{\overline{m}(D)g(x)}=m(D)\tilde{g}(-x)$

for all $g\in\mathcal{S}({\mathbb{R}^{n}})$ and $x\in{\mathbb{R}^{n}}$, where
$\tilde{g}(y):=\overline{g(-y)}$ for $y\in{\mathbb{R}^{n}}$. Hence

$\|m(D)^{*}g\|_{L^{1}({\mathbb{R}^{n}})}=\big{\|}\overline{m(D)^{*}g}\big{\|}_{L^{1}({\mathbb{R}^{n}})}=\|m(D)\tilde{g}\|_{L^{1}({\mathbb{R}^{n}})}\lesssim\|\tilde{g}\|_{\mathcal{H}^{1}({\mathbb{R}^{n}})}=\|g\|_{\mathcal{H}^{1}({\mathbb{R}^{n}})},$

so that $m(D)^{*}:\mathcal{H}^{1}({\mathbb{R}^{n}})\to
L^{1}({\mathbb{R}^{n}})$. It now follows from what we have shown for $p=1$
that
$m(D)^{*}:\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$
is continuous. Hence (1.2) and the density of $\mathcal{S}({\mathbb{R}^{n}})$
in $\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})$ (see [12, Proposition 6.6]) imply
that for all $f\in\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})$ one has

$\displaystyle\|m(D)f\|_{\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})}$
$\displaystyle\eqsim\sup|{\langle}m(D)f,g\rangle_{{\mathbb{R}^{n}}}|=\sup|{\langle}f,m(D)^{*}g\rangle_{{\mathbb{R}^{n}}}|\lesssim\sup\|f\|_{\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})}\|m(D)^{*}g\|_{\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})}$
$\displaystyle\lesssim\sup\|f\|_{\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})}\|g\|_{\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})}=\|f\|_{\mathcal{H}^{\infty}_{FIO}({\mathbb{R}^{n}})},$

where the supremum is taken over all $g\in\mathcal{S}({\mathbb{R}^{n}})$ such
that $\|g\|_{\mathcal{H}^{1}_{FIO}({\mathbb{R}^{n}})}\leq 1$. This proves the
required statement for $p=\infty$.

Finally, for $1<p<\infty$ one can use complex interpolation, by [12,
Proposition 6.7]. ∎

######  Remark 6.2.

For sufficiently smooth $m$ the conclusion of Theorem 6.1 was already obtained
in [12, Theorem 6.10]. This is the case, for example, if $m\in
C^{\infty}({\mathbb{R}^{n}})$ satisfies standard symbol estimates of the form

(6.24) $|\partial^{\alpha}_{\xi}m(\xi)|\leq
C_{\alpha}{\langle}\xi\rangle^{-|\alpha|}\quad(\xi\in{\mathbb{R}^{n}})$

for all $\alpha\in{{\mathbb{Z}}}_{+}^{n}$, and such estimates hold e.g. for
the local Riesz transforms. However, the techniques used to prove [12, Theorem
6.10] involve repeated integration by parts and require more regularity than
the Mikhlin multiplier theorem. Hence Theorem 6.1 allows one to incorporate
results from other parts of harmonic analysis that are not accessible without
the characterizations in this article.

In [12, Theorem 6.10] it is shown that
$m(D):\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
for all $1\leq p\leq\infty$ if $m\in C^{\infty}({\mathbb{R}^{n}})$ is such
that for all $\alpha\in{{\mathbb{Z}}}_{+}^{n}$ and
$\beta\in{{\mathbb{Z}}}_{+}$ there exists a $C_{\alpha,\beta}\geq 0$ with

$|{\langle}\hat{\xi},\nabla_{\xi}\rangle^{\beta}\partial_{\xi}^{\alpha}m(\xi)|\leq
C_{\alpha,\beta}{\langle}\xi\rangle^{-\frac{|\alpha|}{2}-\beta}\quad(\xi\in{\mathbb{R}^{n}}\setminus\\{0\\}).$

It is not clear whether one also has
$m(D):\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
for some $p\neq 2$ under the weaker assumption that $m\in
S^{0}_{1/2}({\mathbb{R}^{n}})$. Here $S^{\gamma}_{1/2}({\mathbb{R}^{n}})$, for
$\gamma\in\mathbb{R}$, consists of all $m\in C^{\infty}({\mathbb{R}^{n}})$
such that for all $\alpha\in{{\mathbb{Z}}}_{+}^{n}$ there exists a
$C_{\alpha}\geq 0$ with

$|\partial_{\xi}^{\alpha}m(\xi)|\leq
C_{\alpha,\beta}{\langle}\xi\rangle^{\gamma-\frac{|\alpha|}{2}}\quad(\xi\in{\mathbb{R}^{n}}).$

Using the alternative characterizations of
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ we can easily obtain a slightly
weaker result.

######  Corollary 6.3.

Let $\gamma\in[0,n/4]$. Then each $m\in S^{-\gamma}_{1/2}({\mathbb{R}^{n}})$
satisfies
$m(D):\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
for all $p\in[1,\infty]$ with $\big{|}\frac{1}{2}-\frac{1}{p}\big{|}\leq
2\gamma/n$.

###### Proof.

For $\gamma=n/4$ one has $m(D):\mathcal{H}^{1}({\mathbb{R}^{n}})\to
L^{1}({\mathbb{R}^{n}})$ by [20, Section VII.5.12], and then Theorem 6.1
concludes the proof. For $\gamma=0$ the result follows from Plancherel’s
theorem, given that
$\mathcal{H}^{2}_{FIO}({\mathbb{R}^{n}})=L^{2}({\mathbb{R}^{n}})$. Stein
interpolation then yields the required result for $0<\gamma<n/4$.
Alternatively, for $0<\gamma<n/4$ one can directly combine the
characterization of $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ from (1.4) with
$L^{p}({\mathbb{R}^{n}})$-bounds for $m(D)$ from [20, Section VII.5.12]. ∎

Next, we determine in a relatively explicit manner the
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ norm, for $1\leq p\leq\infty$, of
functions with frequency support in one of the dyadic-parabolic regions in
(2.6). For simplicity of notation we write
$\mathcal{H}^{p}({\mathbb{R}^{n}})=L^{p}({\mathbb{R}^{n}})$ for $1<p<\infty$,
and
$\mathcal{H}^{\infty}({\mathbb{R}^{n}})=\text{bmo}({\mathbb{R}^{n}})=(\mathcal{H}^{1}({\mathbb{R}^{n}}))^{*}$.

######  Proposition 6.4.

Let $p\in[1,\infty]$ and set
$s_{p}:=\frac{n-1}{2}\big{|}\frac{1}{2}-\frac{1}{p}\big{|}$. Then for each
$A>1$ there exists a $C>0$ such that the following statements hold for all
$f\in\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$. Suppose that there exist
$\tau>0$ and $\nu\in S^{n-1}$ with

(6.25) ${\rm supp}{\hskip
1.42271pt}({\mathcal{F}}(f))\subseteq\\{\xi\in{\mathbb{R}^{n}}\mid|\xi|\in[\tau^{-1}/A,A\tau^{-1}],|\hat{\xi}-\nu|\leq
A\sqrt{\tau}\\}.$

Then the following assertions hold.

  1. (1)

If $p\leq 2$, then

(6.26)
$\frac{1}{C}\|{\langle}D\rangle^{-s_{p}}f\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}\leq\|f\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}\leq
C\|{\langle}D\rangle^{-s_{p}}f\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}.$

Hence there does not exist an $s<s_{p}$ such that
${\langle}D\rangle^{-s}:\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{p}({\mathbb{R}^{n}})$
is bounded.

  2. (2)

If $p>2$, then

(6.27)
$\frac{1}{C}\|{\langle}D\rangle^{s_{p}}f\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}\leq\|f\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}\leq
C\|{\langle}D\rangle^{s_{p}}f\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}.$

Hence there does not exist an $s<s_{p}$ such that
${\langle}D\rangle^{-s}:\mathcal{H}^{p}({\mathbb{R}^{n}})\to\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
is bounded.

Note that (6.25) holds in particular for the wave packets
${\mathcal{F}}^{-1}(\theta_{\nu,\tau})$ and
${\mathcal{F}}^{-1}(\chi_{\nu,\tau})$, by Lemma 2.2.

###### Proof.

We first deal with the low frequencies of $f$. Let $r,r^{\prime}\in
C^{\infty}_{c}({\mathbb{R}^{n}})$ be such that $r\equiv 1$ on ${\rm
supp}{\hskip 1.42271pt}(q)$, and $r^{\prime}\equiv 1$ on ${\rm supp}{\hskip
1.42271pt}(r)$. Then
${\langle}D\rangle^{s}r^{\prime}(D):\mathcal{H}^{p}({\mathbb{R}^{n}})\to\mathcal{H}^{p}({\mathbb{R}^{n}})$
is bounded for all $s\in\mathbb{R}$, so the Sobolev embeddings for
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ from [12, Theorem 7.4] yield

$\displaystyle\|{\langle}D\rangle^{s_{p}}r(D)f\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}$
$\displaystyle=\|{\langle}D\rangle^{2s_{p}}r^{\prime}(D){\langle}D\rangle^{-s_{p}}r(D)f\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}\lesssim\|{\langle}D\rangle^{-s_{p}}r(D)f\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}$
$\displaystyle\lesssim\|r(D)f\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}\lesssim\|{\langle}D\rangle^{s_{p}}r(D)f\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}$
$\displaystyle=\|{\langle}D\rangle^{2s_{p}}r^{\prime}(D){\langle}D\rangle^{-s_{p}}r(D)f\|_{\mathcal{H}^{1}({\mathbb{R}^{n}})}\lesssim\|{\langle}D\rangle^{-s_{p}}r(D)f\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}.$

Hence all the norms of $r(D)f$ under consideration are equivalent, and it
suffices to prove (6.26) and (6.27) with $f$ replaced by $g:=(1-r)(D)f$. Note
that $q(D)g=0$.

(1): By the Sobolev embeddings for $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
one has
$\|{\langle}D\rangle^{-s_{p}}g\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}\lesssim\|g\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}$,
so it remains to show that
$\|g\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}\lesssim\|{\langle}D\rangle^{-s_{p}}g\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}$.
To this end, first note that ${\varphi}_{{\omega}}(\xi)=0$ if $|\xi|<1/8$ or
$|\hat{\xi}-{\omega}|>2|\xi|^{-1/2}$ (see e.g. [15, Remark 3.3]). It is then
easy to check, using the support properties of ${\mathcal{F}}(g)$, that
$\theta_{{\omega},\sigma}(D)g=0$ if $|{\omega}-\nu|>3A\sqrt{\tau}$ or
$\sigma\notin[\tau/(2A),2A\tau]$. Let $E_{\nu,\tau}:=\\{{\omega}\in
S^{n-1}\mid|{\omega}-\nu|\leq 3A\sqrt{\tau}\\}$, and note that
$|E_{\nu,\tau}|\eqsim\tau^{\frac{n-1}{2}}$ for implicit constants independent
of $\nu$ and $\tau$. We now use the characterization in Theorem 3.8 for $p=1$,
and the corresponding one in [15, Corollary 4.5] for $1<p\leq 2$. By combining
this with the bounds for ${\mathcal{F}}^{-1}(\theta_{{\omega},\sigma})$ from
(2.8), we obtain

$\displaystyle\|g\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}^{p}$
$\displaystyle\eqsim\int_{S^{*}({\mathbb{R}^{n}})}\bigg{(}\int_{0}^{1}|\theta_{{\omega},\sigma}(D)g(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{p/2}\mathrm{d}x\mathrm{d}{\omega}=\int_{{\mathbb{R}^{n}}}\int_{E_{\nu,\tau}}\bigg{(}\int_{\tau/4}^{4\tau}|\theta_{{\omega},\sigma}(D)g(x)|^{2}\frac{\mathrm{d}\sigma}{\sigma}\bigg{)}^{p/2}\mathrm{d}{\omega}\mathrm{d}x$
$\displaystyle\lesssim\int_{E_{\nu,\tau}}\int_{{\mathbb{R}^{n}}}\sup_{\sigma\in[\tau/(2A),2A\tau]}|\theta_{{\omega},\sigma}(D)g(x)|^{p}\mathrm{d}x\mathrm{d}{\omega}$
$\displaystyle\lesssim\int_{E_{\nu,\tau}}\int_{{\mathbb{R}^{n}}}\sup_{\sigma\in[\tau/(2A),2A\tau]}\bigg{(}\int_{{\mathbb{R}^{n}}}\sigma^{-\frac{3n+1}{4}}(1+\sigma^{-1}|x-y|^{2}+\sigma^{-2}{\langle}{\omega},x-y\rangle^{2})^{-(n+1)}|g(y)|\mathrm{d}y\bigg{)}^{p}\mathrm{d}x\mathrm{d}{\omega}$
$\displaystyle\eqsim\tau^{-p\frac{n-1}{4}}\int_{E_{\nu,\tau}}\int_{{\mathbb{R}^{n}}}\bigg{(}\int_{{\mathbb{R}^{n}}}\tau^{-\frac{n+1}{2}}(1+\tau^{-1}|x-y|^{2}+\tau^{-2}{\langle}{\omega},x-y\rangle^{2})^{-(n+1)}|g(y)|\mathrm{d}y\bigg{)}^{p}\mathrm{d}x\mathrm{d}{\omega}.$

Now, an anisotropic substitution shows that
$\int_{{\mathbb{R}^{n}}}\tau^{-\frac{n+1}{2}}(1+\tau^{-1}|z|^{2}+\tau^{-2}{\langle}{\omega},z\rangle^{2})^{-(n+1)}\mathrm{d}z\lesssim
1$ for all ${\omega}\in{S}^{n-1}$. Using this twice, in conjunction with
Hölder’s inequality, we obtain

$\displaystyle\int_{{\mathbb{R}^{n}}}\bigg{(}\int_{{\mathbb{R}^{n}}}\tau^{-\frac{n+1}{2}}(1+\tau^{-1}|x-y|^{2}+\tau^{-2}{\langle}{\omega},x-y\rangle^{2})^{-(n+1)}|g(y)|\mathrm{d}y\bigg{)}^{p}\mathrm{d}x$
$\displaystyle\lesssim\int_{{\mathbb{R}^{n}}}\int_{{\mathbb{R}^{n}}}\tau^{-\frac{n+1}{2}}(1+\tau^{-1}|x-y|^{2}+\tau^{-2}{\langle}{\omega},x-y\rangle^{2})^{-(n+1)}|g(y)|^{p}\mathrm{d}y\mathrm{d}x\lesssim\|g\|_{L^{p}({\mathbb{R}^{n}})}^{p}$

for each ${\omega}\in{S}^{n-1}$. It follows that

$\displaystyle\|g\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}^{p}\lesssim\tau^{-p\frac{n-1}{4}}\int_{E_{\nu,\tau}}\|g\|_{L^{p}({\mathbb{R}^{n}})}^{p}\mathrm{d}{\omega}\eqsim\tau^{ps_{p}}\|g\|_{L^{p}({\mathbb{R}^{n}})}^{p}\eqsim\|{\langle}D\rangle^{-s_{p}}g\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}^{p}.$

The very last equivalence of norms is derived in a standard manner from the
support properties of ${\mathcal{F}}(f)$, using for example a Littlewood–Paley
description of the $\mathcal{H}^{p}({\mathbb{R}^{n}})$-norm and a change of
square functions. This proves (6.26).

To conclude the proof of (1), we will apply (6.26) to
${\mathcal{F}}^{-1}(\theta_{\nu,\tau})$ for $\tau\in(0,1)$ and a given $\nu\in
S^{n-1}$. Let $\Psi^{\prime}\in C^{\infty}_{c}({\mathbb{R}^{n}})$ be such that
$\Psi^{\prime}(\xi)=0$ for $|\xi|\notin[1/4,4]$ and such that
$\Psi^{\prime}\equiv 1$ on ${\rm supp}{\hskip 1.42271pt}(\Psi)$. Then for all
$s\in\mathbb{R}$ and $\xi\in{\rm supp}{\hskip 1.42271pt}(\theta_{\nu,\tau})$
one has

$\displaystyle{\langle}\xi\rangle^{-s_{p}}\theta_{\nu,\tau}(\xi)=\tau^{s_{p}-s}m_{\tau}(\xi){\langle}\xi\rangle^{-s}\theta_{\nu,\tau}(\xi),$

where
$m_{\tau}(\xi):={\langle}\xi\rangle^{s-s_{p}}\tau^{s-s_{p}}\Psi^{\prime}(\tau\xi)$
for $\xi\neq 0$. Note that $m_{\tau}$ satisfies standard symbol estimates as
in (6.24), with constants independent of $\tau$. Hence
$m_{\tau}(D):\mathcal{H}^{p}({\mathbb{R}^{n}})\to\mathcal{H}^{p}({\mathbb{R}^{n}})$
is bounded, uniformly in $\tau\in(0,1)$. Then, by (6.26),

$\displaystyle\|{\mathcal{F}}^{-1}(\theta_{\nu,\tau})\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}$
$\displaystyle\eqsim\|{\langle}D\rangle^{-s_{p}}{\mathcal{F}}^{-1}(\theta_{\nu,\tau})\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}=\tau^{s_{p}-s}\|m_{\tau}(D){\langle}D\rangle^{-s}{\mathcal{F}}^{-1}(\theta_{\nu,\tau})\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}$
$\displaystyle\lesssim\tau^{s_{p}-s}\|{\langle}D\rangle^{-s}{\mathcal{F}}^{-1}(\theta_{\nu,\tau})\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}.$

For $s<s_{p}$ the right-hand side tends to zero as $\tau\to 0$, and it follows
that in this case one does not have
$\|{\langle}D\rangle^{-s}{\mathcal{F}}^{-1}(\theta_{\nu,\tau})\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}\lesssim\|{\mathcal{F}}^{-1}(\theta_{\nu,\tau})\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}$.
That is,
${\langle}D\rangle^{-s}:\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\to\mathcal{H}^{p}({\mathbb{R}^{n}})$
is not bounded.

(2): By the Sobolev embeddings for $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$
from [12, Theorem 7.4], one has
$\|g\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}\lesssim\|{\langle}D\rangle^{s_{p}}g\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}$.
For $2<p<\infty$ this also follows from the arguments used above to derive the
corresponding inequality for $p\leq 2$. We will use duality to show that
$\|{\langle}D\rangle^{s_{p}}g\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}\lesssim\|g\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}$.

First, for $B>1$ write

$F_{\nu,\tau,B}:=\\{\xi\in{\mathbb{R}^{n}}\mid|\xi|\in[\tau^{-1}/B,B\tau^{-1}],|\hat{\xi}-\nu|\leq
B\sqrt{\tau}\\},$

and let $\rho\in C^{\infty}_{c}({\mathbb{R}^{n}})$ be such that $\rho\equiv 1$
on $F_{\nu,\tau,A}$ and $\rho\equiv 0$ outside $F_{\nu,\tau,2A}$. Then
$\rho(D)h\in\mathcal{S}({\mathbb{R}^{n}})$ with ${\rm supp}{\hskip
1.42271pt}({\mathcal{F}}(\rho(D)h))\subseteq F_{\nu,\tau,2A}$ for every
$h\in\mathcal{S}({\mathbb{R}^{n}})$. Moreover, by taking Fourier transforms in
the standard duality pairing ${\langle}g,h\rangle_{{\mathbb{R}^{n}}}$ between
$g$ and $h$, one obtains

(6.28)
${\langle}g,h\rangle_{{\mathbb{R}^{n}}}={\langle}g,\rho(D)h\rangle_{{\mathbb{R}^{n}}},$

where we used that ${\rm supp}{\hskip 1.42271pt}({\mathcal{F}}g)\subseteq
F_{\nu,\tau,A}$, by assumption. Next, by what we have shown in part (1) with
$A$ replaced by $2A$, one has

(6.29) $\displaystyle I_{1}$
$\displaystyle:=\\{h\in\mathcal{S}({\mathbb{R}^{n}})\mid{\rm supp}{\hskip
1.42271pt}({\mathcal{F}}{h})\subseteq
F_{\nu,\tau,2A},\|{\langle}D\rangle^{-s_{p}}h\|_{\mathcal{H}^{p^{\prime}}({\mathbb{R}^{n}})}\leq
c\\}$ $\displaystyle\subseteq I_{2}$
$\displaystyle:=\\{h\in\mathcal{S}({\mathbb{R}^{n}})\mid{\rm supp}{\hskip
1.42271pt}({\mathcal{F}}{h})\subseteq
F_{\nu,\tau,2A},\|h\|_{\mathcal{H}^{p^{\prime}}_{FIO}({\mathbb{R}^{n}})}\leq
1\\}$

for some $c>0$ independent of $g$, $\nu$ and $\tau$. Since
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})=(\mathcal{H}^{p^{\prime}}_{FIO}({\mathbb{R}^{n}}))^{*}$,
where the duality pairing is the standard duality pairing between
$f_{1}\in\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\subseteq\mathcal{S}^{\prime}({\mathbb{R}^{n}})$
and
$f_{2}\in\mathcal{S}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{p^{\prime}}_{FIO}({\mathbb{R}^{n}})$
(see [12, Proposition 6.8]), and because
$\mathcal{S}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{p^{\prime}}_{FIO}({\mathbb{R}^{n}})$
is dense (cf. [12, Proposition 6.6]), we can combine (6.28) and (6.29) to
obtain

$\displaystyle\|g\|_{\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})}\eqsim\sup_{g\in
I_{2}}|{\langle}g,h\rangle_{{\mathbb{R}^{n}}}|\geq\sup_{h\in
I_{1}}|{\langle}g,h\rangle_{{\mathbb{R}^{n}}}|\eqsim\|{\langle}D\rangle^{s_{p}}g\|_{\mathcal{H}^{p}({\mathbb{R}^{n}})}.$

This proves (6.27). For the final statement in (2) one argues in a similar
manner as for $p\leq 2$. ∎

######  Remark 6.5.

Proposition 6.4 shows that the $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$-norm
behaves differently depending on whether $p<2$ or $p>2$. Recall from [12,
Theorem 7.4] that

(6.30)
$W^{s_{p},p}({\mathbb{R}^{n}})\subseteq\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})\subseteq
W^{-s_{p},p}({\mathbb{R}^{n}})$

for $1<p<\infty$, with suitable modifications for $p=1$ and $p=\infty$. For
$1<p<2$ functions with frequency support in a dyadic-parabolic region have
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$-norm comparable to the
$W^{-s_{p},p}({\mathbb{R}^{n}})$-norm that appears on the right-hand side of
(6.30). Informally speaking, such functions have a “small” norm on the
$L^{p}$-scale. On the other hand, for $2<p<\infty$ the same functions have
$\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$-norm comparable to the
$W^{s_{p},p}({\mathbb{R}^{n}})$-norm on the left-hand side of (6.30); here the
norm is “large” on the $L^{p}$-scale.

The fact that the Sobolev embeddings in (6.30) are sharp was already observed
in [12, Remark 7.9], as a consequence of the optimal $L^{p}$-regularity of
FIOs and the fact that $\mathcal{H}^{p}_{FIO}({\mathbb{R}^{n}})$ is invariant
under suitable FIOs of order zero. On the other hand, Proposition 6.4 gives an
explicit class of examples that also shows that one of the Sobolev embeddings
is optimal.

#### Acknowledgments

The authors would like to thank Lixin Yan for helpful discussions. Z.J. Fan is
supported by International Program for Ph.D. Candidates, Sun Yat-sen
University. J. Rozendaal is supported by grant DP160100941 of the Australian
Research Council. L. Song is supported by NNSF of China (No. 11622113) and NSF
for distinguished Young Scholar of Guangdong Province (No. 2016A030306040).

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