equations, and new well-posedness results for nonlinear wave equations. In
particular, we will prove Theorems 1.1 and 1.2.

Throughout, let $(M,g)$ be a compact Riemannian manifold of dimension
$n\in{{\mathbb{N}}}$, with (non-positive) Laplace–Beltrami operator
$\Delta_{g}$.

###  6.1. Linear wave equations

In this subsection we consider the Cauchy problem (1.1) on
$M\times\mathbb{R}$:

$\begin{cases}(\partial_{t}^{2}-\Delta_{g})u(x,t)=0,\\\ u(x,0)=u_{0}(x),\
\partial_{t}u(x,0)=u_{1}(x).\end{cases}$

Recall that $\Delta_{g}$ acts on functions in local coordinates as in (4.2).
Then $\Delta_{g}$ also acts on half densities, using the identification
$u\mapsto\rho_{g}^{-1/2}u$ between half densities and functions on $M$, where
$\rho_{g}$ is the Riemannian density on $M$. Moreover, $\Delta_{g}$ is a non-
positive operator on $L^{2}(M,\Omega_{1/2})$. For $t\in\mathbb{R}$, the
operators $\cos(t\sqrt{-\Delta_{g}})$ and
$\sin(t\sqrt{-\Delta_{g}})/\sqrt{-\Delta_{g}}$, initially defined through the
spectral calculus for $\Delta_{g}$ on $L^{2}(M,\Omega_{1/2})$, extend to
$\mathcal{D}^{\prime}(M,\Omega_{1/2})$, and the function
$u:\mathbb{R}\to\mathcal{D}^{\prime}(M,\Omega_{1/2})$,

(6.1)
$u(t):=\cos(t\sqrt{-\Delta_{g}})u_{0}+\frac{\sin(t\sqrt{-\Delta_{g}})}{\sqrt{-\Delta_{g}}}u_{1}$

for $t\in\mathbb{R}$, solves (1.1) for all
$u_{0},u_{1}\in\mathcal{D}^{\prime}(M,\Omega_{1/2})$.

Here we will be interested in the regularity of solutions for initial data in
the Hardy spaces for FIOs. To this end, we recall that (see e.g. [45, Sections
4.1 and 8.1]) $\cos(t\sqrt{-\Delta_{g}})$ and
$\sin(t\sqrt{-\Delta_{g}})/\sqrt{-\Delta_{g}}$ are compactly supported Fourier
integral operators of order $0$ and $-1$, respectively, associated with a
local canonical graph, for each $t\in\mathbb{R}$. Moreover, as space-time
solution operators, they are elements of
$I^{-1/4}(M,M\times\mathbb{R},{\mathcal{C}})$ and
$I^{-5/4}(M,M\times\mathbb{R},{\mathcal{C}})$ for ${\mathcal{C}}$ satisfying
the cinematic curvature condition. Hence Theorems 4.16 and 5.7, combined with
the uniform boundedness principle, immediately yield regularity results for
(1.1). However, in this case we have a distinguished time variable, and it is
natural to consider a version of these theorems which takes this into account.

######  Theorem 6.1.

Let $p\in[1,\infty]$, $s\in\mathbb{R}$ and $t_{0}>0$. Then there exists a
$C\geq 0$ such that, for all $u_{0}\in\mathcal{H}^{s,p}_{FIO}(M)$ and
$u_{1}\in\mathcal{H}^{s-1,p}_{FIO}(M)$, the function $u$ in (6.1) satisfies

(6.2) $\|u(t)\|_{\mathcal{H}^{s,p}_{FIO}(M)}\leq
C\big{(}\|u_{0}\|_{\mathcal{H}^{s,p}_{FIO}(M)}+\|u_{1}\|_{\mathcal{H}^{s-1,p}_{FIO}(M)}\big{)}$

for all $t\in[-t_{0},t_{0}]$. Moreover, if $n\geq 2$ and $p\in(2,\infty)$,
then for each $\varepsilon>0$ there exists a $C_{\varepsilon}\geq 0$ such that

(6.3)
$\Big{(}\int_{-t_{0}}^{t_{0}}\|u(t)\|_{W^{s-d(p)+s(p)-\varepsilon,p}(M)}^{p}\mathrm{d}t\Big{)}^{1/p}\leq
C_{\varepsilon}\big{(}\|u_{0}\|_{\mathcal{H}^{s,p}_{FIO}(M)}+\|u_{1}\|_{\mathcal{H}^{s-1,p}_{FIO}(M)}\big{)}.$

###### Proof.

For (6.2), Theorem 4.16 immediately yields norm bounds for each fixed
$t\in\mathbb{R}$, but we also need the bounds to be locally uniform in $t$.
Moreover, Theorem 5.7 yields (6.3) for $s-d(p)+s(p)-\varepsilon\geq 0$, but
for $s<d(p)-s(p)+\varepsilon$ we want to take advantage of the specific
structure of the underlying canonical relation, which yields integrability in
$t$ and (6.3). To do so, we will slightly modify the proof of Theorem 5.7.

Let $\phi\in C^{\infty}_{c}(\mathbb{R})$ be such that $\phi(t)=1$ for all
$t\in[-t_{0},t_{0}]$. Then it suffices to prove (6.2) and (6.3) with $u$
replaced by $\phi u$. Now, $\phi u=T_{0}+T_{1}$ for compactly supported
$T_{0}\in I^{-1/4}(M,M\times\mathbb{R},{\mathcal{C}})$ and $T_{1}\in
I^{-5/4}(M,M\times\mathbb{R},{\mathcal{C}})$, where ${\mathcal{C}}$ satisfies
the cinematic curvature condition.

Next, as in the proofs of Theorems 4.16 and 5.7, one can reduce to working in
local coordinates, and (due to the product structure of $M\times\mathbb{R}$)
one can choose geodesic normal coordinates independently for $M$ and
$\mathbb{R}$. This expresses each $T_{k}$ as a finite sum of compactly
supported elements of
$I^{-k-1/4}(\mathbb{R}^{n},\mathbb{R}^{n+1},{\mathcal{C}}^{\prime})$, where

${\mathcal{C}}^{\prime}=\\{(x,t,\xi,\tau,y,\eta)\mid(x,\xi)=\chi_{t}(y,\eta),\tau=p(x,\xi)\\}$

for $\chi_{t}$ a canonical transformation on $T^{*}(\mathbb{R}^{n})\setminus
o$ for each $t$ (see e.g. [45, pp. 129-130]). It then suffices to prove
mapping properties for these FIOs in local coordinates.

At this point one can reduce to the case where $s=d(p)-s(p)+\varepsilon$,
using a similar reduction as in the proof of Theorem 5.7. However, here one
has to replace the derivative $D_{z}=-i\partial_{z}$ with respect to $z=(x,t)$
in (5.13) by the derivative $D_{x}=-i\partial_{x}$ with respect to $x$. The
resulting compactly supported FIOs are still associated with
${\mathcal{C}}^{\prime}$, since ${\langle}D_{x}\rangle^{s-r}$ and
${\langle}D_{y}\rangle^{-(s-r)}$ are pseudodifferential operators.

Finally, an application of Lemma A.2 and Remark A.3 expresses each of the
resulting FIOs as a sum of operators in standard form. One can then apply
Corollary 2.4 at each time $t$ to obtain (6.2), with the desired uniformity in
$t$ coming from Remark 2.6. Moreover, Theorem 5.7 with
$s=d(p)-s(p)+\varepsilon$ yields (6.3). ∎

######  Remark 6.2.

One can reformulate (1.1) as an abstract Cauchy problem on
$X:=\mathcal{H}^{s,p}_{FIO}(M)\times\mathcal{H}^{s-1,p}_{FIO}(M)$. That is,
one wants to find a $U:\mathbb{R}\to X$ such that

(6.4)
$\frac{\mathrm{d}}{\mathrm{d}t}U(t)=\left(\\!\begin{array}[]{cc}\\!0&I\\!\\\
\\!\Delta_{g}&0\\!\end{array}\\!\right)U(t)\text{ and
}U(0)=\left(\\!\begin{array}[]{c}\\!u_{0}\\!\\\
\\!u_{1}\\!\end{array}\\!\right).$

Now, the function $u$ in (6.1) is strongly continuous on
$\mathcal{D}(M,\Omega_{1/2})$, as follows from its strong continuity on
$L^{2}(M,\Omega_{1/2})$ combined with Sobolev embeddings. Then Theorem 6.1,
combined with the density of $\mathcal{D}(M,\Omega_{1/2})$ from Proposition
4.9, implies that (6.4) is well posed for $p<\infty$. In particular, $u$ is
the unique strongly continuous solution to (1.1) as in Theorem 6.1. For
$p=\infty$, $u$ is weak-star continuous, cf. Proposition 4.13.

###  6.2. Nonlinear wave equations

We consider the cubic nonlinear wave equation from (1.4) on
$M\times\mathbb{R}$:

(6.5)
$\begin{cases}(\partial_{t}^{2}-\Delta_{g})u(x,t)=\pm|u(x,t)|^{2}u(x,t),\\\
u(x,0)=f_{1}(x),\ \partial_{t}u(x,0)=f_{2}(x).\end{cases}$

We will assume throughout that $M$ has dimension $n=2$, although our
techniques extend to higher dimensions and to other nonlinearities (see e.g.
Proposition 6.5 below).

Our notion of well-posedness of (6.5) is taken from [3]. Consider the abstract
evolution equation

(6.6) $u=L(f_{1},f_{2})+N(u,u,u),$

where $L:X\to S$ is a densely defined linear operator, and $N:S\times S\times
S\to S$ is a densely defined operator which is either linear or antilinear in
each of its variables. We say that (6.6) is _quantitatively well posed_ (with
initial data space $X$ and solution space $S$) if there exists a $C\geq 0$
such that

(6.7) $\displaystyle\|L(f_{1},f_{2})\|_{S}$ $\displaystyle\leq
C\|(f_{1},f_{2})\|_{X},$ (6.8) $\displaystyle\|N(u_{1},u_{2},u_{3})\|_{S}$
$\displaystyle\leq C\prod_{j=1}^{3}\|u_{j}\|_{S},$

for all $(f_{1},f_{2})\in X$ and $u_{1},u_{2},u_{3}\in S$. If (6.6) is
quantitatively well posed, then a fixed-point argument yields
$C_{0},\delta_{0}>0$ such that, for all

$(f_{1},f_{2})\in B_{X}(0,\delta_{0}):=\\{(g_{1},g_{2})\in
X\mid\|(g_{1},g_{2})\|_{X}<\delta_{0}\\},$

there exists a unique solution $u[f_{1},f_{2}]\in B_{S}(0,C_{0}\delta_{0})$ to
(6.6). Moreover, the map $(f_{1},f_{2})\mapsto u[f_{1},f_{2}]$ is Lipschitz
continuous from $B_{X}(0,\delta_{0})$ to $B_{S}(0,C_{0}\delta_{0})$, and one
can expand $u[f_{1},f_{2}]$ in terms of its Picard iterates. For more on this
see [3, Theorem 3].

In our setting, $S=S_{t_{0}}$ is a space-time function space on the time
interval $[0,t_{0}]$, for $t_{0}>0$. Then (6.6) arises by rewriting (6.5)
using Duhamel’s formula, with

$\begin{split}L(f_{1},f_{2})(t)&:=\cos(t\sqrt{-\Delta_{g}})f_{1}+\frac{\sin(t\sqrt{-\Delta_{g}})}{\sqrt{-\Delta_{g}}}f_{2},\\\
N(u_{1},u_{2},u_{3})(t)&:=\pm\int_{0}^{t}\frac{\sin((t-s)\sqrt{-\Delta_{g}})}{\sqrt{-\Delta_{g}}}u_{1}(s)\overline{u_{2}(s)}u_{3}(s)\mathrm{d}s,\end{split}$

for $t\in[0,t_{0}]$. In fact, we will improve (6.8) by showing that

(6.9) $\|N(u_{1},u_{2},u_{3})\|_{S}\leq
Ct_{0}^{\gamma}\prod_{i=1}^{3}\|u_{i}\|_{S}$

for all $t_{0}\in(0,1]$ and $u_{1},u_{2},u_{3}\in S=S_{t_{0}}$, where $C$ and
$\gamma>0$ are independent of $t_{0}\in(0,1]$. Then the fixed-point argument
also yields well-posedness for large data: for all $(f_{1},f_{2})\in X$ there
exists a $t_{0}>0$ such that the power series for $u[f_{1},f_{2}]$ converges
in $S_{t_{0}}$.

The main result of this section deals with two critical values of $p$, namely
$p=6=2(n+1)/(n-1)$ and $p=4=2n/(n-1)$. It implies in particular Theorem 1.2.

######  Theorem 6.3.

Let $\varepsilon,t_{0}>0$. Then (6.5) is quantitatively well posed with
initial data space

$X:=(\mathcal{H}^{\varepsilon,6}_{FIO}(M)+W^{1/2,2}(M))\times(\mathcal{H}^{\varepsilon-1,6}_{FIO}(M)+W^{-1/2,2}(M))$

and solution space

$S_{t_{0}}:=L^{4}\big{(}[0,t_{0}];L^{6}(M)\big{)}\cap
C\big{(}[0,t_{0}];\mathcal{H}^{\varepsilon,6}_{FIO}(M)+W^{1/2,2}(M)\big{)}.$

Moreover, (6.5) is quantitatively well posed with initial data space

(6.10)
$Y:=(\mathcal{H}^{\varepsilon,4}_{FIO}(M)+W^{3/8,2}(M))\times(\mathcal{H}^{\varepsilon-1,4}_{FIO}(M)+W^{-5/8,2}(M))$

and solution space

$T_{t_{0}}:=L^{24/7}\big{(}[0,t_{0}];L^{4}(M)\big{)}\cap
C\big{(}[0,t_{0}];\mathcal{H}^{\varepsilon,4}_{FIO}(M)+W^{3/8,2}(M)\big{)}.$

###### Proof.

The proof follows the same template as that of [38, Theorem 1.2]. We will only
consider $p=6$, with the other case being analogous. The spaces $W^{s,2}(M)$
arise due to the use of Strichartz estimates, to deal with the nonlinear term.

We have to prove (6.7) and (6.8). In the process we will also prove (6.9) for
$t_{0}\in(0,1]$. In fact, we will obtain estimates for the solution spaces
$L^{4}([0,t_{0}];L^{6}(M))$ and
$C([0,t_{0}];\mathcal{H}^{\varepsilon,6}_{FIO}(M)+W^{1/2,2}(M))$ separately.

Firstly, for a given $(f_{1},f_{2})\in X$, write
$(f_{1},f_{2})=(f_{11},f_{21})+(f_{12},f_{22})$ with

$(f_{11},f_{21})\in\mathcal{H}^{\varepsilon,6}_{FIO}(M)\times\mathcal{H}^{\varepsilon-1,6}_{FIO}(M)\text{
and }(f_{12},f_{22})\in W^{1/2,2}(M)\times W^{-1/2,2}(M).$

Then Hölder’s inequality, the second part of Theorem 6.1, and Strichartz
estimates on compact manifolds (cf. [26, 27]) yield

$\displaystyle\|L(f_{1},f_{2})\|_{L^{4}([0,t_{0}];L^{6}(M))}\leq
t_{0}^{1/12}\|L(f_{1},f_{2})\|_{L^{6}([0,t_{0}];L^{6}(M))}$
$\displaystyle\lesssim\|L(f_{11},f_{21})\|_{L^{6}([0,t_{0}];L^{6}(M))}+\|L(f_{12},f_{22})\|_{L^{6}([0,t_{0}];L^{6}(M))}$
$\displaystyle\lesssim\|(f_{11},f_{21})\|_{\mathcal{H}^{\varepsilon,6}_{FIO}(M)\times\mathcal{H}^{\varepsilon-1,6}_{FIO}(M)}+\|(f_{12},f_{22})\|_{W^{1/2,2}(M)\times
W^{-1/2,2}(M)}.$

Taking the infimum over all such decompositions, one obtains

$\|L(f_{1},f_{2})\|_{L^{4}([0,t_{0}];L^{6}(M))}\lesssim\|(f_{1},f_{2})\|_{X}.$

On the other hand, since $W^{1/2,2}(M)=\mathcal{H}^{1/2,2}_{FIO}(M)$ (see e.g.
Theorem 4.15 and Corollary 4.21), the first part of Theorem 6.1, with $p=6$
and $p=2$, yields

$\|L(f_{1},f_{2})\|_{L^{\infty}([0,t_{0}];\mathcal{H}^{\varepsilon,6}_{FIO}(M)+W^{1/2,2}(M))}\lesssim\|(f_{1},f_{2})\|_{X}.$

Moreover, applying Remark 6.2 with $p=6$ and $p=2$, we obtain

$L(f_{1},f_{2})\in
C\big{(}[0,t_{0}];\mathcal{H}^{\varepsilon,6}_{FIO}(M)+W^{1/2,2}(M)\big{)},$

which completes the proof of (6.7).

Next, we prove (6.8) and (6.9) for $u_{1},u_{2},u_{3}\in S$. Firstly, Hölder’s
inequality and Strichartz estimates yield

(6.11) $\displaystyle\|N(u_{1},u_{2},u_{3})\|_{L^{4}([0,t_{0}];L^{6}(M))}\leq
t_{0}^{1/12}\|N(u_{1},u_{2},u_{3})\|_{L^{6}([0,t_{0}];L^{6}(M))}$
$\displaystyle\lesssim
t_{0}^{1/12}\|u_{1}u_{2}u_{3}\|_{L^{1}([0,t_{0}];L^{2}(M))}\leq
t_{0}^{1/3}\|u_{1}u_{2}u_{3}\|_{L^{4/3}([0,t_{0}];L^{2}(M))}$
$\displaystyle\leq
t_{0}^{1/3}\prod_{j=1}^{3}\|u_{j}\|_{L^{4}([0,t_{0}];L^{6}(M))},$

where the implicit constant is independent of $t_{0}$ if $t_{0}\in(0,1]$. On
the other hand, Minkowski’s inequality, the spectral theorem and Hölder’s
inequality yield

$\displaystyle\|N(u_{1},u_{2},u_{3})\|_{L^{\infty}([0,t_{0}];\mathcal{H}^{\varepsilon,6}_{FIO}(M)+W^{1/2,2}(M))}\leq\|N(u_{1},u_{2},u_{3})\|_{L^{\infty}([0,t_{0}];W^{1/2,2}(M))}$
$\displaystyle\leq\int_{0}^{t_{0}}\sup_{t\in[0,t_{0}]}\Big{\|}\frac{\sin((t-s)\sqrt{-\Delta_{g}})}{\sqrt{-\Delta_{g}}}u_{1}(s)\overline{u_{2}(s)}u_{3}(s)\Big{\|}_{W^{1/2,2}(M)}\mathrm{d}s$
$\displaystyle\lesssim\|u_{1}\overline{u_{2}}u_{3}\|_{L^{1}([0,t_{0}];L^{2}(M))}\leq
t_{0}^{1/4}\|u_{1}u_{2}u_{3}\|_{L^{4/3}([0,t_{0}];L^{2}(M))}$
$\displaystyle\leq
t_{0}^{1/4}\prod_{j=1}^{3}\|u_{j}\|_{L^{4}([0,t_{0}];L^{6}(M))}$

for an implicit constant independent of $t_{0}$. Using also the strong
continuity of $t\mapsto\sin(t\sqrt{-\Delta_{g}})$, cf. Remark 6.2, this
concludes the proof. ∎

######  Remark 6.4.

One can combine the same proof strategy with local smoothing estimates that
use classical Sobolev spaces as spaces of initial data. For $p=6$, this would
yield a weaker statement than that in Theorem 6.3, given that Theorem 6.1
improves upon the local smoothing conjecture for $p\geq 2(n+1)/(n-1)$. On the
other hand, for $p=4$, the main result of [14] allows one to show that (6.5)
is quantitatively well posed with initial data space

$Y:=(W^{\varepsilon,4}(M)+W^{3/8,2}(M))\times(W^{\varepsilon-1,4}(M)+W^{-5/8,2}(M))$

and solution space

$T_{t_{0}}:=L^{24/7}\big{(}[0,t_{0}];L^{4}(M)\big{)}.$

Due to the sharpness of the embeddings in (1.3), this initial data space
neither contains the one in (6.10), nor vice versa. Note, moreover, that in
this case there is no pointwise regularity statement as in Theorem 6.3.

In (6.11) we did not make full use of the regularizing effect of the Duhamel
term. The additional regularization can be used to solve the quintic nonlinear
wave equation

(6.12)
$\begin{cases}(\partial_{t}^{2}-\Delta_{g})u(x,t)=\pm|u(x,t)|^{4}u(x,t),\\\
u(x,0)=f_{1}(x),\ \partial_{t}u(x,0)=f_{2}(x),\end{cases}$

for small initial data.

######  Proposition 6.5.

Let $\varepsilon,t_{0}>0$. Then (6.12) is quantitatively well posed with
initial data space

$X:=(\mathcal{H}^{\varepsilon,6}_{FIO}(M)+W^{1/2,2}(M))\times(\mathcal{H}^{\varepsilon-1,6}_{FIO}(M)+W^{-1/2,2}(M))$

and solution space

$S_{t_{0}}=L^{6}\big{(}[0,t_{0}];L^{6}(M)\big{)}\cap
C\big{(}[0,t_{0}];\mathcal{H}^{\varepsilon,6}_{FIO}(M)+W^{1/2,2}(M)\big{)}$

The notion of quantitative well-posedness of (6.12) is the natural analog of
that of (6.5).

###### Proof.

The linear estimate (6.7) again follows from Strichartz estimates and from
Theorem 6.1. For the natural analog of (6.8) we also use Strichartz estimates:

$\displaystyle\Big{\|}\int_{0}^{t}\frac{\sin((t-s)\sqrt{-\Delta})}{\sqrt{-\Delta}}\prod_{j=1}^{5}u_{j}(s)ds\Big{\|}_{L^{6}([0,T],L^{6}(M))\cap
L^{\infty}([0,T];W^{1/2,2}(M))}$
$\displaystyle\lesssim\Big{\|}\prod_{j=1}^{5}u_{j}\Big{\|}_{L_{t}^{6/5}([0,T],L^{6/5}(M))}\lesssim\prod_{j=1}^{5}\|u_{j}\|_{L_{t}^{6}([0,T],L^{6}(M))}.\qed$

######  Remark 6.6.

Note that we did not show that (6.9) holds, so we do not obtain well-posedness
for large initial data.

Finally, global existence holds for the defocusing cubic nonlinear wave
equation:

(6.13)
$\begin{cases}(\partial_{t}^{2}-\Delta_{g})u(x,t)=-|u(x,t)|^{2}u(x,t),\\\
u(x,0)=f_{1}(x),\ \partial_{t}u(x,0)=f_{2}(x).\end{cases}$

The following proposition is a version on compact surfaces of [38, Theorem
5.8]. The proof, which involves a blow-up alternative, Sobolev embeddings and
an application of Grönwall’s inequality, is exactly as in the Euclidean case
(see also [9, 39]).

######  Proposition 6.7.

Let $s>1/2$ and

$(f_{1},f_{2})\in(\mathcal{H}^{s,6}_{FIO}(M)+W^{1/2,2}(M))\times(\mathcal{H}^{s-1,6}_{FIO}(M)+W^{-1/2,2}(M)).$

Let $u$ be the solution to (6.13) given by Theorem 6.3. Then $u\in
L^{4}([0,t_{0}];L^{6}(M))$ for all $t_{0}>0$.

##  7\. Sharpness of the results

In this section we show that Theorems 1.1 and 6.1 are sharp, and Theorems 1.3
and 5.7 are sharp under mild assumptions. To this end, we first prove that the
index $d(p)-s(p)$ in Corollary 5.4 cannot be improved, from which we then
derive the same conclusion in the setting of Theorem 5.7.

Throughout, fix a dimension $n\geq 2$.

###  7.1. Sharpness for operators in standard form

Throughout this subsection, fix $m\in\mathbb{R}$, $a\in
S^{m}(\mathbb{R}^{n+1}\times\mathbb{R}^{n})$ and a real-valued $\Phi\in
C^{\infty}(\mathbb{R}^{n+1}\times(\mathbb{R}^{n}\setminus\\{0\\}))$,
positively homogeneous of degree one in the fiber variable, such that
$\mathrm{rank}\,\partial_{z\eta}^{2}\Phi(z_{0},\eta_{0})=n$ for all
$(z_{0},\eta_{0})\in\mathrm{supp}(a)$ with $\eta_{0}\neq 0$. Set

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

for $f\in\mathcal{S}(\mathbb{R}^{n})$ and $z\in\mathbb{R}^{n+1}$. Using
terminology from Section A.3, we say that $T$ is a Fourier integral operator
in standard form.

To prove sharpness, we will proceed in a similar way as in [37, Section 5],
which dealt with the case where $Tf(x,t)=e^{it\phi(D)}f(x)$ for $\phi\in
C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ positively homogeneous of degree
one. A key step in that proof was to approximate $T$ by translations of the
form $x\mapsto x+t\partial_{\eta}\phi(\nu)$, which arise through propagation
of singularities from the canonical relation

$\\{(x,t,\eta,\phi(\eta),x+t\partial_{\eta}\phi(\eta),\eta)\\}.$

In the present variable-coefficient case, the relevant canonical relation is

$\\{(z,\partial_{z}\Phi(z,\eta),\partial_{\eta}\Phi(z,\eta),\eta)\\},$

and we will adjust the flow. However, we do not need to take the momentum
variable into account. This leads to the following version of [37, Lemma 5.1].

######  Lemma 7.1.

Suppose that $m=0$ and that there exists a non-empty conic
$V\subseteq\mathbb{R}^{n}\setminus\\{0\\}$ such that $a(0,\eta)=1$ and
$\Phi(0,\eta)=0$ for all $\eta\in V$ with $|\eta|\geq 1/2$. Suppose also that
$|(1-r)\nu+r\omega|\geq 1/2$ for all $\nu,\omega\in V\cap S^{n-1}$ and
$r\in[0,1]$. Then there exists a $C\geq 0$ such that

$|Th(z)-(2\pi)^{n}h(\partial_{\eta}\Phi(z,\nu))|\leq
C\|\widehat{h}\|_{L^{1}(\mathbb{R}^{n})}|z|(1+\gamma(h,\nu))$

for all $h\in\mathcal{S}(\mathbb{R}^{n})$ with
$\mathrm{supp}(\widehat{h})\subseteq
V\cap\\{\eta\in\mathbb{R}^{n}\mid|\eta|\geq 1/2\\}$ compact, all $\nu\in
S^{n-1}\cap V$ and all $z\in\mathbb{R}^{n+1}$ with $|z|\leq 1$. Here

$\gamma(h,\nu):=\sup\\{|\hat{\eta}-\nu||\eta|\mid\eta\in\mathrm{supp}(\widehat{h})\\}.$

###### Proof.

Fix $\nu\in S^{n-1}\cap V$ and $z\in\mathbb{R}^{n+1}$, and write

$\displaystyle
Th(z)-(2\pi)^{n}h(\partial_{\eta}\Phi(z,\nu))=\int_{\mathbb{R}^{n}}\big{(}e^{i\Phi(z,\eta)}a(z,\eta)-e^{i\partial_{\eta}\Phi(z,\nu)\cdot\eta}\big{)}\widehat{h}(\eta)\mathrm{d}\eta$
$\displaystyle=\int_{\mathbb{R}^{n}}e^{i\Phi(z,\eta)}(a(z,\eta)-1)\widehat{h}(\eta)\mathrm{d}\eta+\int_{\mathbb{R}^{n}}\big{(}e^{i\Phi(z,\eta)}-e^{i\partial_{\eta}\Phi(z,\nu)\cdot\eta}\big{)}\widehat{h}(\eta)\mathrm{d}\eta.$

We will bound each of the terms on the second line separately. The first term
is easy:

$\Big{|}\int_{\mathbb{R}^{n}}e^{i\Phi(z,\eta)}(a(z,\eta)-1)\widehat{h}(\eta)\mathrm{d}\eta\Big{|}\lesssim\|\widehat{h}\|_{L^{1}(\mathbb{R}^{n})}|z|,$

where we used Taylor approximation and the assumptions that $m=0$ and that
$a(0,\eta)=1$ for $\eta\in V$.

To deal with the second term, we first use Taylor approximation for the
exponential:

$\displaystyle\Big{|}\int_{\mathbb{R}^{n}}\big{(}e^{i\Phi(z,\eta)}-e^{i\partial_{\eta}\Phi(z,\nu)\cdot\eta}\big{)}\widehat{h}(\eta)\mathrm{d}\eta\Big{|}$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\big{|}e^{i(\Phi(z,\eta)-\partial_{\eta}\Phi(z,\nu)\cdot\eta)}-1\big{|}\,|\widehat{h}(\eta)|\mathrm{d}\eta$
$\displaystyle\leq\|\widehat{h}\|_{L^{1}(\mathbb{R}^{n})}\delta(h,\nu,z),$

where

$\delta(h,\nu,z):=\sup\\{|(\partial_{\eta}\Phi(z,\eta)-\partial_{\eta}\Phi(z,\nu))\cdot\eta|\mid\eta\in\mathrm{supp}(\widehat{h})\\}.$

It thus suffices to show that $\delta(h,\nu,z)\lesssim|z|\gamma(h,\nu)$. To do
so, for $\eta\in V$, we can use the homogeneity of $\Phi$ and another Taylor
expansion to write

$\displaystyle(\partial_{\eta}\Phi(z,\eta)-\partial_{\eta}\Phi(z,\nu))\cdot\eta$
$\displaystyle=(\partial_{\eta}\Phi(z,\hat{\eta})-\partial_{\eta}\Phi(z,\nu))\cdot\eta$
$\displaystyle=\Big{(}\int_{0}^{1}\partial_{\eta\eta}^{2}\Phi(z,(1-r)\nu+r\hat{\eta})(\hat{\eta}-\nu)\mathrm{d}r\Big{)}\cdot\eta.$

Finally, note that $\partial_{\eta\eta}^{2}\Phi(0,(1-r)\nu+r\hat{\eta})=0$ for
all $r\in[0,1]$, since $\Phi(0,\eta^{\prime})=0$ for $\eta^{\prime}\in V$ with
$|\eta^{\prime}|\geq 1/2$, and because $|(1-r)\nu+r\hat{\eta}|\geq 1/2$, by
assumption. Hence, for each $r$ we can use another Taylor approximation, this
time with respect to $z$, to obtain

$|\Phi(z,\eta)-\partial_{\eta}\Phi(z,\nu)\cdot\eta|\lesssim|z|\,|\hat{\eta}-\nu|\,|\eta|,$

as required. ∎

To prove sharpness for $2<p<2(n+1)/(n-1)$, we will use a version of [37, Lemma
5.2] which relates the Hardy spaces for FIOs to square functions as in [15,
14]. The proof in [37], which relies on Khintchine’s inequality, extends
directly to this setting. We use notation as in the decoupling norm in (5.3).

######  Lemma 7.2.

Let $p\in[1,\infty)$ and $s\in\mathbb{R}$ be such that
$T:\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})\to L^{p}(\mathbb{R}^{n+1})$ is
bounded. Then there exists a $C\geq 0$ such that

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

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

We are now ready to prove that the index $d(p)-s(p)$ in Corollary 5.4 is
sharp. Note, however, that our assumptions on $T$ are much weaker than those
in Section 5.1. In particular, we do not assume that the canonical relation
${\mathcal{C}}$ associated with $T$ satisfies the cinematic curvature
condition, nor even that the projection condition holds which guarantees
sharpness of the fixed-time $L^{p}$ estimates (see [40]).

######  Theorem 7.3.

Let $p\in(2,\infty)$ and $s\in\mathbb{R}$ be such that
$T:\mathcal{H}^{s+m}_{FIO}(\mathbb{R}^{n})\to L^{p}(\mathbb{R}^{n+1})$ is
bounded. Suppose that there exist a $z_{0}\in\mathbb{R}^{n+1}$, a non-empty
open conic set $V\subseteq\mathbb{R}^{n}\setminus\\{0\\}$ and $c,C>0$ such
that $|a(z_{0},\eta)|\geq c|\eta|^{m}$ for all $\eta\in V$ with $|\eta|\geq
C$. Then $s\geq d(p)-s(p)$.

###### Proof.

The approach is similar to that in [37, Theorem 5.3], as are the examples that
show sharpness. However, we need additional work to reduce to a setting where
we can apply Lemma 7.1.

By translating and precomposing with ${\langle}D\rangle^{-m}$, we may assume
that $z_{0}=0$ and $m=0$, and by adding a smoothing operator which modifies
the low frequencies near $z_{0}$, we may assume that $C=1/2$. We may also
suppose, without loss of generality, that $|(1-r)\nu+r\omega|\geq 1/2$ for all
$\nu,\omega\in V\cap S^{n-1}$ and $r\in[0,1]$. Finally, we will assume for the
moment that $a(0,\eta)=1$ and $\Phi(0,\eta)=0$ for all $\eta\in V$ with
$|\eta|\geq 1/2$. This assumption will be removed at the end of the proof.

Now, we want to show that $s\geq d(p)-s(p)=\max(s(p)-1/p,0)$. Fix a standard
Littlewood–Paley decomposition $(\phi_{k})_{k=0}^{\infty}\subseteq
C^{\infty}_{c}(\mathbb{R}^{n})$, as in the proof of Corollary 5.4, and let
$k\in\mathbb{Z}_{+}$. Our construction will only rely on estimates for large
$k$. Given that $V$ is non-empty and open, we may thus choose $k$ large enough
such that the collection $V_{k}$ of $\nu\in\Theta_{k}$ with
$\mathrm{supp}(\chi_{\nu})\subseteq V$ is non-empty. Note that then $V_{k}$
has approximately $2^{k(n-1)/2}$ elements, because $V$ has a fixed nonzero
aperture.

We first prove that $s\geq s(p)-1/p$. In this case we will apply Lemma 7.1 to
functions whose Fourier support fills up a dyadic-parabolic piece.

We will construct a collection $(f_{\nu})_{\nu\in
V_{k}}\subseteq\mathcal{S}(\mathbb{R}^{n})$ with the following properties. For
each $\nu\in V_{k}$ one has $\phi_{k}(D)f_{\nu}=\chi_{\nu}(D)f_{\nu}=f_{\nu}$,
and $\operatorname{Re}(f_{\nu}(x))\gtrsim 1$ for all $x\in\mathbb{R}^{n}$ with
$|x|\leq 2^{-k}$. Moreover,
$\|\mathcal{F}f_{\nu}\|_{L^{1}(\mathbb{R}^{n})}\eqsim 1$ and
$\|f_{\nu}\|_{L^{p}(\mathbb{R}^{n})}\eqsim 2^{-k\frac{n+1}{2p}}$. Here all the
implicit constants are independent of $\nu$ and $k$. To construct such a
collection, one may fix a $\psi\in\mathcal{S}(\mathbb{R}^{n})$ such that
$\psi(x)\geq 1$ whenever $|x|\leq 1$, and $\widehat{\psi}(\xi)=0$ if
$|\xi|>c^{\prime}$, for some small $c^{\prime}>0$. Then set

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

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

Now set $f:=\sum_{\nu\in V_{k}}f_{\nu}$. Then
$\mathrm{supp}(\widehat{f}\,)\subseteq V$,
$f\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ and

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

by [37, Proposition 4.1] (see also [37, Equation (5.3)]).

Now, $\Phi$ and all of its derivatives are bounded on compact subsets of
$\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})$. Hence
$z\mapsto\partial_{\eta}\Phi(z,\nu)$ is a locally Lipschitz flow for each
$\nu\in V_{k}$, with Lipschitz constants independent of $\nu$. Moreover, by
assumption, $\partial_{\eta}\Phi(0,\nu)=0$. It thus follows that
$\operatorname{Re}(f_{\nu}(\partial_{\eta}\Phi(z,\nu)))\gtrsim 1$ whenever
$|z|\lesssim 2^{-k}$ for some implicit constant independent of $k$ and $\nu$.
On the other hand, because
$\mathrm{supp}(f_{\nu})\subseteq\mathrm{supp}(\phi_{k}\chi_{\nu})$, one has
$|\hat{\eta}-\nu|\,|\eta|\lesssim 2^{k/2}$ for all
$\eta\in\mathrm{supp}(f_{\nu})$. Hence Lemma 7.1 yields

$|Tf_{\nu}(z)-(2\pi)^{n}f_{\nu}(\partial_{\eta}\Phi(z,\nu))|\lesssim|z|2^{k/2}$

whenever $|z|\lesssim 2^{-k}$. By combining all this, we find

$\operatorname{Re}(Tf_{\nu}(z))\geq(2\pi)^{n}\operatorname{Re}(f_{\nu}(\partial_{\eta}\Phi(z,\nu)))-|Tf_{\nu}(z)-(2\pi)^{n}f_{\nu}(\partial_{\eta}\Phi(z,\nu))|\gtrsim
1$

if $|z|\leq c_{1}2^{-k}$ for some small $c_{1}>0$ independent of $k$ and
$\nu$. Now we can use the assumption on $T$ and (7.1):

$\displaystyle 2^{-k\frac{n+1}{p}}2^{k\frac{n-1}{2}}$
$\displaystyle\lesssim\Big{(}\int_{\mathbb{R}^{n+1}}\Big{|}\sum_{\nu\in
V_{k}}Tf_{\nu}(z)\Big{|}^{p}\mathrm{d}z\Big{)}^{1/p}=\|Tf\|_{L^{p}(\mathbb{R}^{n+1})}\lesssim\|f\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}$
$\displaystyle\eqsim
2^{k(s+\frac{n-1}{2}(\frac{1}{2}-\frac{1}{p})-\frac{1}{p})}.$

This implies that $s\geq s(p)-1/p$.

Next, we will show that $s\geq 0$, which is the required statement for
$2<p<2(n+1)/(n-1)$. Here we will apply Lemma 7.1 to functions with frequency
support of unit size in a dyadic-parabolic piece, and we will also use Lemma
7.2.

Let $\psi\in\mathcal{S}(\mathbb{R}^{n})$ be as before, and set
$g_{\nu}(x):=e^{i2^{k}\nu\cdot x}\psi(x)$ for $\nu\in V_{k}$ and
$x\in\mathbb{R}^{n}$. Since $\widehat{\psi}(\xi)=0$ if $|\xi|>c^{\prime}$, for
some small $c^{\prime}>0$, we may again suppose that
$\phi_{k}(D)g_{\nu}=\chi_{\nu}(D)g_{\nu}=g_{\nu}$. Also,
$\|g_{\nu}\|_{L^{p}(\mathbb{R}^{n})}\eqsim
1\eqsim\|\mathcal{F}(g_{\nu})\|_{L^{1}(\mathbb{R}^{n})}$. Set $g:=\sum_{\nu\in
V_{k}}g_{\nu}$. Then $\mathrm{supp}(\widehat{g})\subseteq V$,
$g\in\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ and

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

by [37, Proposition 4.1] (see also [37, Equation (5.5)]).

Let $\nu\in V_{k}$. Since $\psi(x)\geq 1$ for $|x|\leq 1$, we find in a
similar way as before that $|g_{\nu}(\partial_{\eta}\Phi(z,\nu))|\gtrsim 1$
whenever $|z|\lesssim 1$, for small implicit constants independent of $k$ and
$\nu$. Moreover, since
$\mathrm{supp}(g_{\nu})\subseteq\mathrm{supp}(\phi_{k}\chi_{\nu})$ is of unit
size, Lemma 7.1 yields

$|Tg_{\nu}(z)-(2\pi)^{n}g_{\nu}(\partial_{\eta}\Phi(z,\nu))|\lesssim|z|$

if $|z|\lesssim 1$. Combined, this implies that

$|Tg_{\nu}(z)|\geq(2\pi)^{n}|g_{\nu}(\partial_{\eta}\Phi(z,\nu))|-|Tg_{\nu}(z)-(2\pi)^{n}g_{\nu}(\partial_{\eta}\Phi(z,\nu))|\gtrsim
1$

whenever $|z|\leq c_{2}$ for some small $c_{2}>0$ independent of $k$ and
$\nu$. Now we can use the assumption on $T$, Lemma 7.2 and (7.2):

$\displaystyle 2^{k\frac{n-1}{4}}$
$\displaystyle\lesssim\Big{(}\int_{\mathbb{R}^{n+1}}\Big{(}\sum_{\nu\in
V_{k}}|Tg_{\nu}(z)|^{2}\Big{)}^{p/2}\mathrm{d}z\Big{)}^{1/p}$
$\displaystyle=\Big{(}\int_{\mathbb{R}^{n+1}}\Big{(}\sum_{\nu\in
V_{k}}|T\chi_{\nu}(D)g(z)|^{2}\Big{)}^{p/2}\mathrm{d}z\Big{)}^{1/p}\lesssim\|g\|_{\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})}\eqsim
2^{k(s+\frac{n-1}{4})},$

which shows that $s\geq 0$.

We have now proved that $s\geq d(p)-s(p)$, and it only remains to remove the
assumption that $a(0,\eta)=1$ and $\Phi(0,\eta)=0$ for all $\eta\in V$ with
$|\eta|\geq 1/2$. To do so, and in particular for the second part of this
assumption, we will use the invariance of
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ under FIOs.

First note that, so far, we have not used any information about $V$ and $c$,
beyond what is listed in the statement of the theorem, as well as the
assumption that $|(1-r)\nu+r\omega|\geq 1/2$ for all $\nu,\omega\in V\cap
S^{n-1}$ and $r\in[0,1]$, which can be made without loss of generality by
choosing $V$ small enough. Hence, by choosing slightly smaller $V$ and $c$, we
may suppose that the closure of $V$ in $\mathbb{R}^{n}\setminus\\{0\\}$ is a
strict subset of a closed cone $V_{0}\subseteq\mathbb{R}^{n}\setminus\\{0\\}$
such that $|a(z_{0},\eta)|\geq c$ for all $\eta\in V_{0}$ with $|\eta|\geq
c^{\prime}$, for some $c^{\prime}<1/2$.

Let $\rho\in C^{\infty}(\mathbb{R}^{n})$ be such that
$\mathrm{supp}(\rho)\subseteq V_{0}\cap\\{\eta\in\mathbb{R}^{n}\mid|\eta|\geq
c^{\prime}\\}$, and such that $\rho(\eta)=1$ for all $\eta\in
V\cap\\{\eta\in\mathbb{R}^{n}\mid|\eta|\geq 1/2\\}$. Set

$\theta(\eta):=e^{-i\Phi(0,\eta)}\frac{\rho(\eta)}{a(0,\eta)}$

for $\eta\in\mathbb{R}^{n}$. Then $\theta(D)$ is an FIO in standard form, with
phase function $x\cdot\eta-\Phi(0,\eta)$ and symbol $\rho(\eta)/a(0,\eta)$. In
fact, $\theta(D)\in
I^{0}(\mathbb{R}^{n},\mathbb{R}^{n};{\mathcal{C}}^{\prime})$ for

${\mathcal{C}}^{\prime}:=\\{(x,\eta,x-\partial_{\eta}\Phi(0,\eta),\eta)\mid(x,\eta)\in\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})\\},$

which is the graph of $(y,\eta)\mapsto(y+\partial_{\eta}\Phi(0,\eta),\eta)$ on
$T^{*}\mathbb{R}^{n}\setminus o$. It thus follows from [17, Theorem 6.10] (see
also [36, Remark 3.7]) that
$\theta(D):\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})\to\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$
is bounded.

By assumption on $T$, this in turn implies that
$T\theta(D):\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})\to
L^{p}(\mathbb{R}^{n+1})$ is bounded. Moreover, $T\theta(D)$ is an FIO in
standard form, with phase function given by $\Phi(z,\eta)-\Phi(0,\eta)$ for
$\eta\in V\cap\\{\eta\in\mathbb{R}^{n}\mid|\eta|\geq 1/2\\}$, and symbol given
by $a(z,\eta)/a(0,\eta)$ for such $\eta$. Now the first part of the proof
shows that $s\geq d(p)-s(p)$, thereby concluding the proof for general $T$ as
in the statement of the theorem. ∎

######  Remark 7.4.

Note that the sharpness examples are universal, in the sense that they can be
used for any FIO satisfying the conditions of Theorem 7.3. Also note that,
when combined with Proposition C.3, Theorem 7.3 shows that the exponent $d(p)$
in Theorem 5.1 is sharp whenever $T$ is non-characteristic somewhere.

###  7.2. Sharpness on manifolds

Throughout, let $(M,g)$ and $(N,g^{\prime})$ be complete Riemannian manifolds
of dimensions $n$ and $n+1$, respectively, with bounded geometry.

We will prove that the index $d(p)-s(p)$ in Theorem 5.7 is sharp. However, the
assumptions under which we show sharpness are, for the most part, much weaker
than those in Theorem 5.7, and in particular we do not assume that
${\mathcal{C}}$ satisfies the cinematic curvature condition.

######  Theorem 7.5.

Let $T\in I^{m-1/4}(M,N;{\mathcal{C}})$, for $m\in\mathbb{R}$ and
${\mathcal{C}}\subseteq(T^{*}N\setminus o)\times(T^{*}M\setminus o)$ a
homogeneous canonical relation such that the projections
$\Pi_{N}:{\mathcal{C}}\to N$ and $\Pi_{M}:{\mathcal{C}}\to M$ are submersions.
Let $p\in(2,\infty)$ and $s,r\in\mathbb{R}$. Suppose that

$T:\mathcal{H}^{s+m,p}_{FIO}(M)\to W^{r,p}(N)$

is bounded, and that $T$ is non-characteristic at some point
$c\in{\mathcal{C}}$. Then $r\leq s-d(p)+s(p)$.

Note that, if ${\mathcal{C}}$ satisfies the cinematic curvature condition,
then it also satisfies the projection conditions in the theorem, given that
the natural projection $\Pi_{M}:T^{*}M\setminus o\to M$ is a submersion.
Moreover, the space-time solution operator to the Cauchy problem (1.1)
satisfies the conditions of the theorem, given that the wave propagators are
invertible and therefore elliptic. Hence the index $d(p)-s(p)$ in Theorems 1.1
and 6.1 is sharp on every compact manifold.

###### Proof.

Firstly, we may work in a sufficiently small conic neighborhood of $c$, to be
chosen later. More precisely, let $S_{1}$ and $S_{2}$ be compactly supported
pseudodifferential operators of order zero on $N$ and $M$, whose kernels are
smooth away from small conic neighborhoods of
$(z_{0},\zeta_{0},z_{0},\zeta_{0})$ and $(y_{0},\eta_{0},y_{0},\eta_{0})$, and
whose principal symbols are equal to one near $(z_{0},\zeta_{0})$ and
$(y_{0},\eta_{0})$, respectively. Then, by the composition theorem for FIOs
and because the canonical relation of a pseudodifferential operator is the
diagonal, $S_{1}TS_{2}\in I^{m-1/4}(M,N;{\mathcal{C}}^{\prime})$ is non-
characteristic at $c$, and ${\mathcal{C}}^{\prime}$ is a small neighborhood of
$c$ in ${\mathcal{C}}$. Moreover, $S_{2}$ is bounded on
$\mathcal{H}^{s+m,p}_{FIO}(M)$, by Theorem 4.16, and $S_{1}:W^{r,p}(N)\to
W^{r,p}(N)$ is bounded as well, by Corollary 4.21 and because
pseudodifferential operators of order zero are bounded on
$W^{r,p}(\mathbb{R}^{n+1})$. Hence
$S_{1}TS_{2}:\mathcal{H}^{s+m,p}_{FIO}(M)\to W^{r,p}(N)$ is bounded, and we
may replace $T$ by $S_{1}TS_{2}$ in the remainder. For simplicity of notation,
we will continue working with $T$ and simply assume that $T\in
I^{m-1/4}(M,N;{\mathcal{C}})$ for ${\mathcal{C}}$ a small neighborhood of $c$.

Next, we reduce to working in local coordinates. Let uniformly locally finite
families $K\subseteq{\mathcal{A}}$, $L\subseteq{\mathcal{B}}$,
$(\psi_{\kappa})_{\kappa\in K}\subseteq C^{\infty}_{c}(M)$ and
$(\theta_{\lambda})_{\lambda\in L}\subseteq C^{\infty}_{c}(N)$ as in Lemma A.1
be given, where ${\mathcal{A}}$ and ${\mathcal{B}}$ are the smooth structures
on $M$ and $N$, respectively. Write $c=(z_{0},\zeta_{0},y_{0},\eta_{0})$, and
fix $\kappa\in K$ and $\lambda\in L$ such that $\psi_{\kappa}(y_{0})\neq 0$
and $\theta_{\lambda}(z_{0})\neq 0$. Set
$\tilde{T}f:=\lambda_{*}(\theta_{\lambda}T\kappa^{*}(\tilde{\psi}_{\kappa}f))$
for $f\in\mathcal{D}(\mathbb{R}^{n})$. Then, by the composition theorem for
FIOs, $\tilde{T}\in
I^{m}(\mathbb{R}^{n},\mathbb{R}^{n+1};\tilde{{\mathcal{C}}})$ for

$\tilde{{\mathcal{C}}}:=\\{(\lambda(z),(\partial\lambda(z))_{*}\zeta,\kappa(y),(\partial\kappa(y))_{*}\eta)\mid(z,\zeta,y,\eta)\in{\mathcal{C}},z\in
U_{\lambda},y\in U_{\kappa}\\}.$

Note that the natural projections
$\Pi_{\mathbb{R}^{n+1}}:\tilde{{\mathcal{C}}}\to\mathbb{R}^{n+1}$ and
$\Pi_{\mathbb{R}^{n}}:\tilde{{\mathcal{C}}}\to\mathbb{R}^{n}$ are submersions,
by the assumption on ${\mathcal{C}}$. Moreover, the principal symbol of
$\tilde{T}$ is just the principal symbol of $T$, expressed in local
coordinates near $z_{0}$ and $y_{0}$. In particular, $\tilde{T}$ is non-
characteristic at

$\tilde{c}:=(\lambda(z_{0}),(\partial\lambda(z_{0}))_{*}\zeta_{0},\kappa(y_{0}),(\partial\kappa(y_{0}))_{*}\eta_{0})\in\tilde{{\mathcal{C}}},$

given that $\theta_{\lambda}(z_{0})\neq 0\neq\psi_{\kappa}(y_{0})$. Also, by
Definition 4.2 and Corollary 4.21, one has
$\lambda_{*}(\theta_{\lambda}T):\mathcal{H}^{s+m,p}_{FIO}(M)\to
W^{r,p}(\mathbb{R}^{n+1})$, and $f\mapsto\kappa^{*}(\tilde{\psi}_{\kappa}f)$
is bounded from $\mathcal{H}^{s+m,p}_{FIO}(\mathbb{R}^{n})$ to
$\mathcal{H}^{s+m,p}_{FIO}(M)$ by Theorem 4.16. Hence
$\tilde{T}:\mathcal{H}^{s+m,p}_{FIO}(\mathbb{R}^{n})\to
W^{r,p}(\mathbb{R}^{n+1})$ is bounded. We may thus work with $\tilde{T}$ in
the remainder. Note also that $\tilde{T}$ is compactly supported.

Next, we want to reduce to the case where $r=0$. To this end, note that
${\langle}D\rangle^{r}\tilde{T}{\langle}D\rangle^{-r}:\mathcal{H}^{s-r+m,p}_{FIO}(\mathbb{R}^{n})\to
L^{p}(\mathbb{R}^{n+1})$ is bounded. Since $\tilde{T}$ is compactly supported,
we can multiply $\tilde{T}$ by smooth cutoffs $\rho_{1}$ and $\rho_{2}$, and
apply Lemma A.5 twice, to write

${\langle}D\rangle^{r}\tilde{T}{\langle}D\rangle^{-r}={\langle}D\rangle^{r}\rho_{1}\tilde{T}\rho_{2}{\langle}D\rangle^{-r}=\tilde{S}_{1}\tilde{T}\tilde{S}_{2}+R,$

for compactly supported pseudodifferential operators $\tilde{S}_{1}$ and
$\tilde{S}_{2}$ of order $r$ and $-r$, respectively, and a smoothing operator
$R:\mathcal{S}^{\prime}(\mathbb{R}^{n})\to\mathcal{S}(\mathbb{R}^{n+1})$. Then
$\tilde{S}_{1}\tilde{T}\tilde{S}_{2}:\mathcal{H}^{s-r+m,p}_{FIO}(\mathbb{R}^{n})\to
L^{p}(\mathbb{R}^{n+1})$ is bounded, and the composition theorem for FIOs
implies that $\tilde{S}_{1}\tilde{T}\tilde{S}_{2}\in
I^{m-1/4}(\mathbb{R}^{n},\mathbb{R}^{n+1};\tilde{C})$. Moreover, since
${\langle}D\rangle^{r}$ and ${\langle}D\rangle^{-r}$ are elliptic and because
the smoothing term $R$ does not change the principal symbol,
$\tilde{S}_{1}\tilde{T}\tilde{S}_{2}$ is a compactly supported element of
$I^{m-1/4}(\mathbb{R}^{n},\mathbb{R}^{n+1};\tilde{C})$ that is non-
characteristic at $\tilde{c}$.

It only remains to express $\tilde{S}_{1}\tilde{T}\tilde{S}_{2}$ in standard
form. This is where the projection conditions on $\tilde{C}$ come in. Namely,
these allow us to apply Lemma A.2 to write
$\tilde{S}_{1}\tilde{T}\tilde{S}_{2}$ as a sum of FIOs of order $m-1/4$ as in
Section 7.1, composed with changes of coordinates, plus a smoothing term.
Moreover, at the start of the proof we observed that we are allowed to work in
a sufficiently small neighborhood of $c$ in ${\mathcal{C}}$ (and therefore
also of $\tilde{c}$ in $\tilde{C}$). Hence Remark A.4 in fact yields a
decomposition
$\tilde{S}_{1}\tilde{T}\tilde{S}_{2}=T^{\prime}S^{\prime}+R^{\prime}$, for
$T^{\prime}$ an FIO of order $m-1/4$ as in Section 7.1, $S^{\prime}$ a change
of coordinates on $\mathbb{R}^{n}$, and
$R^{\prime}:\mathcal{S}^{\prime}(\mathbb{R}^{n})\to\mathcal{S}(\mathbb{R}^{n+1})$
a smoothing operator. Then $T^{\prime}S^{\prime}$ is non-characteristic at
$\tilde{c}$ and bounded from $\mathcal{H}^{s-r+m,p}_{FIO}(\mathbb{R}^{n})$ to
$L^{p}(\mathbb{R}^{n+1})$, given that $R^{\prime}$ does not affect these
properties. And, since $\tilde{S}_{1}\tilde{T}\tilde{S}_{2}$ is compactly
supported and we are allowed to work in a sufficiently small neighborhood of
$\tilde{c}$, the change of coordinates $S^{\prime}$ does not affect the
mapping properties of $\tilde{S}_{1}\tilde{T}\tilde{S}_{2}$ either.

To conclude, the operator $T^{\prime}$ is as in Section 7.1, is non-
characteristic at some point $c^{\prime}$ (obtained from $\tilde{c}$ in terms
of the change of coordinates $S^{\prime}$), and is bounded from
$\mathcal{H}^{s-r+m,p}_{FIO}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n+1})$.
Thus Theorem 7.3 yields $s-r\geq d(p)-s(p)$, as required. ∎

## Acknowledgments

The authors would like to thank the referee for their careful reading of the
manuscript and for many helpful suggestions. The second author would also like
to thank Po–Lam Yung for many inspiring conversations about the phenomenon of
local smoothing.

##  Appendix A Geometry and Fourier integral operators

In this appendix we collect some basics on manifolds with bounded geometry and
on the theory of Fourier integral operators on manifolds. We first discuss the
main geometric assumptions on the underlying manifolds. We then recall some
basics on distributional densities on manifolds, which will allow us to
introduce Fourier integral operators on manifolds.

###  A.1. Manifolds with bounded geometry

In this subsection we recall some basics from Riemannian geometry. The
underlying geometric assumptions correspond to those used in [50, 51, 48] to
define the classical local Hardy spaces on manifolds.

Throughout this subsection, we let $(M,g)$ be a complete Riemannian manifold
of dimension $n\in{{\mathbb{N}}}$. Denote by ${\mathcal{A}}$ the smooth
structure on $M$. That is, ${\mathcal{A}}$ is a maximal smooth atlas of
homeomorphisms
$\kappa:U_{\kappa}\to\tilde{U}_{\kappa}\subseteq\mathbb{R}^{n}$, where the
$U_{\kappa}\subseteq M$ are open sets that cover $M$, and

(A.1) $\mu_{\lambda\kappa}:=\lambda\circ\kappa^{-1}:\kappa(U_{\kappa}\cap
U_{\lambda})\to\lambda(U_{\kappa}\cap U_{\lambda})$

is a diffeomorphism for all $\kappa,\lambda\in{\mathcal{A}}$ such that
$U_{\kappa}\cap U_{\lambda}\neq\emptyset$.

We assume throughout that $(M,g)$ has _bounded geometry_ , by which we mean
that the injectivity radius of $(M,g)$ is positive, and that all covariant
derivatives $\nabla^{k}R$, $k\in\mathbb{Z}_{+}$, of the Riemannian curvature
tensor $R$ are bounded. Recall that the injectivity radius
$\mathrm{inj}(M)\in[0,\infty]$ is the infimumum over $p\in M$ of the supremum
over all $r>0$ such that the exponential map

$\exp_{p}:B_{r}(0)\to U_{r}(p):=\exp_{p}(B_{r}(0))\subseteq M$

at $p$ is a diffeomorphism. Here
$B_{r}(x):=\\{y\in\mathbb{R}^{n}\mid|x-y|<r\\}\subseteq\mathbb{R}^{n}$ for all
$x\in\mathbb{R}^{n}$, where we identify $T_{p}M$ with $\mathbb{R}^{n}$. Then
$\kappa=\exp^{-1}_{p}:U_{r}(p)\to B_{r}(0)$ is an element of ${\mathcal{A}}$,
and the corresponding local coordinates are _geodesic normal coordinates_. We
will often alternate between denoting the domain of $\kappa$ by $U_{\kappa}$
and by $U_{r}(p)$.

One can reformulate the assumption that all covariant derivatives of the
curvature tensor are bounded using the matrix expression
$(g_{ij})_{i,j=1}^{n}$ of the metric tensor in local coordinate charts (see
[11]). That is, we assume that for all $0<r<\mathrm{inj}(M)$ there exists a
$c>0$ and, for every $\alpha\in\mathbb{Z}_{+}^{n}$, a $C_{\alpha}\geq 0$ such
that

(A.2) $|\det(g_{ij}(y))_{i,j=1}^{n}|\geq
c\quad\text{and}\quad|\partial^{\alpha}_{y}g_{ij}(y)|\leq C_{\alpha}\text{ for
}1\leq i,j\leq n,$

in every normal geodesic coordinate chart $\exp^{-1}_{p}:U_{r}(p)\to
B_{r}(0)$, $p\in M$, and for all $y\in B_{r}(0)$. This implies similar uniform
bounds for the inverse matrix $(g^{ij})_{i,j=1}^{n}$.

There is yet another equivalent formulation of the assumption of bounded
geometry; this one, in terms of the changes of coordinates in (A.1), will be
used frequently by us. Namely, we assume that, for all $0<r<\mathrm{inj}(M)$
and $\alpha\in\mathbb{Z}_{+}^{n}$, there exists a $C_{\alpha}\geq 0$ such that

(A.3) $|\partial^{\alpha}_{y}\mu_{\lambda\kappa}(y)|\leq C_{\alpha}$

for all normal geodesic charts $\kappa=\exp_{p}^{-1}:U_{r}(p)\to B_{r}(0)$ and
$\lambda=\exp_{q}^{-1}:U_{r}(q)\to B_{r}(0)$ with $U_{r}(p)\cap
U_{r}(q)\neq\emptyset$, $p,q\in M$, and all $y\in\kappa(U_{r}(p)\cap
U_{r}(q))$.

To avoid confusion, we note that some authors use the terminology “bounded
geometry” for the weaker assumption that $M$ has positive injectivity radius
and Ricci tensor bounded from below (see e.g. [31, 30]).

For us, the main reason to make the assumption of bounded geometry is that it
implies the existence of a _uniformly locally finite_ cover by geodesic balls,
in a sense which is specified by the following lemma.

######  Lemma A.1.

For every $\delta\in(0,\mathrm{inj}(M)/3]$, there exist countable collections
$K\subseteq{\mathcal{A}}$ and $(\psi_{\kappa})_{\kappa\in K}\subseteq
C^{\infty}_{c}(M)$ with the following properties.

  1. (1)

Each $\kappa=\exp_{p_{\kappa}}^{-1}:U_{\delta}(p_{\kappa})\to B_{\delta}(0)$,
for $\kappa\in K$, is a geodesic normal coordinate chart for some
$p_{\kappa}\in M$, and $\cup_{\kappa\in K}U_{\delta}(p_{\kappa})=M$.

  2. (2)

There exists an $N\in{{\mathbb{N}}}$ such that, for every $\kappa\in K$, the
collection of $\lambda\in K$ with $U_{\mathrm{inj}(M)/3}(p_{\kappa})\cap
U_{\mathrm{inj}(M)/3}(p_{\lambda})\neq\emptyset$ contains at most $N$
elements.

  3. (3)

For every $\kappa\in K$ one has $\mathrm{supp}(\psi_{\kappa})\subseteq
U_{\delta}(p_{\kappa})$, and

$\sum_{\kappa\in K}\psi_{\kappa}(x)^{2}=1\quad(x\in M).$

  4. (4)

Writing $\tilde{\psi}_{\kappa}:=\psi_{\kappa}\circ\kappa^{-1}$ for $\kappa\in
K$, for each $\alpha\in\mathbb{Z}_{+}^{n}$ one has

$\sup\\{|\partial_{y}^{\alpha}\tilde{\psi}_{\kappa}(y)|\mid\kappa\in K,y\in
B_{\delta}(0)\\}<\infty.$

Note that the family $(\psi_{\kappa})_{\kappa\in K}$ is also uniformly locally
finite, in the sense that, for each $x\in M$, one has $\psi_{\kappa}(x)\neq 0$
for at most $N$ different $\kappa\in K$. In fact, we typically choose
$\delta\leq\mathrm{inj}(M)/9$, which allows us to treat intersecting geodesic
balls within the same coordinate chart. This is in turn useful for the proof
of the crucial Corollary 3.4.

The statement and the proof of Lemma A.1 are essentially contained in [41,
Lemmas A1.1.2 and A1.1.3]. See also [52, Section 7.1]) and [29, Lemma 2.1].

###  A.2. Distributional densities

In this subsection we collect some basics on distributional densities on
manifolds, cf. [10, 21]. We apply this theory to a Riemannian manifold $(M,g)$
with bounded geometry, but for the material presented here we may assume that
$M$ is a general $n$-dimensional smooth manifold.

Let $\gamma\in\mathbb{R}$. We write $\mathcal{D}(M,\Omega_{\gamma})$ for the
space of smooth compactly supported sections of the complex line bundle
$\Omega_{\gamma}(M)$. The fibers of this bundle are the one-dimensional vector
spaces $\Omega_{\gamma}(T_{x}M)$, $x\in M$, consisting of all functions
$\sigma:\Lambda^{n}T_{x}M\to{\mathbb{C}}$, on the exterior product
$\Lambda^{n}T_{x}M$, with the property that $\sigma(\lambda
v)=|\lambda|^{\gamma}\sigma(v)$ for all $v\in\Lambda^{n}T_{x}M$ and
$\lambda\in\mathbb{R}\setminus\\{0\\}$. If $\rho_{0}\in
C^{\infty}(M,\Omega_{1})$ is a fixed strictly positive density on $M$ (such as
the Riemannian density $\rho_{g}$), then

(A.4) $f\mapsto f\rho_{0}^{\gamma}\quad(f\in C^{\infty}_{c}(M))$

is a bijection between $C^{\infty}_{c}(M)$ and
$\mathcal{D}(M,\Omega_{\gamma})$.

The topology on $\mathcal{D}(M,\Omega_{\gamma})$ is defined in the same manner
as that of the test functions $C^{\infty}_{c}(M)$. In fact,
$\mathcal{D}(M,\Omega_{\gamma})$ inherits the topology of $C^{\infty}_{c}(M)$
via (A.4). Moreover, the resulting topology does not depend on the choice of
$\rho_{0}$, by definition of the inductive limit topology on
$C^{\infty}_{c}(M)$ and because the quotient of two strictly positive
densities is bounded from above and below on compact subsets of $M$.

A distributional density of order $\gamma$ on $M$ is a continuous linear
functional on $\mathcal{D}(M,\Omega_{1-\gamma})$. We denote by
$\mathcal{D}^{\prime}(M,\Omega_{\gamma})$ the space of distributional
densities of order $\gamma$, endowed with the weak topology, and
${\langle}u,\rho\rangle_{M}$ denotes the duality between
$u\in\mathcal{D}^{\prime}(M,\Omega_{\gamma})$ and
$\rho\in\mathcal{D}(M,\Omega_{1-\gamma})$. The space
$C^{\infty}(M,\Omega_{\gamma})$ of smooth sections of $\Omega_{\gamma}$ is
embedded in $\mathcal{D}^{\prime}(M,\Omega_{\gamma})$, via
${\langle}u,\rho\rangle_{M}=\int_{M}u(x)\rho(x)$ for $u\in
C^{\infty}(M,\Omega_{\gamma})$ and $\rho\in\mathcal{D}(M,\Omega_{1-\gamma})$.
This is well defined because $u\rho\in\mathcal{D}(M,\Omega_{1})$ is an
integrable density in the classical sense. Moreover,
$\mathcal{D}(M,\Omega_{\gamma})$ is dense in
$\mathcal{D}^{\prime}(M,\Omega_{\gamma})$, as follows by approximating a
$u\in\mathcal{D}^{\prime}(M,\Omega_{\gamma})$ in local coordinate charts by
test functions, and then lifting these back to $M$ (see e.g. (A.10)).

As is typical for the coordinate-invariant theory of Fourier integral
operators (see e.g. [10, 23]), we will mostly consider $\gamma=1/2$ in this
definition, in which case we refer to _half densities_. However, we note that
$u\mapsto\rho_{0}^{\gamma-1}u$ identifies functionals on $C^{\infty}_{c}(M)$
with distributional densities of a general111In particular, after making this
identification, our approach coincides with the notion of distributions on
manifolds from [52, Chapter 7] (where $\gamma=1$) and [4, 5] (where
$\gamma=0$). order $\gamma\in\mathbb{R}$, as in (A.4).

Let $\gamma\in\mathbb{R}$. Recall that if $A:E\to F$ is a linear map between
$n$-dimensional vector spaces, then the pullback
$A^{*}:\Omega_{\gamma}(F)\to\Omega_{\gamma}(E)$ is given by

(A.5) $(A^{*}\sigma)(v)=\sigma(Av_{1}\wedge\ldots\wedge Av_{n})$

for $\sigma\in\Omega_{\gamma}(F)$ and $v=v_{1}\wedge\ldots\wedge
v_{n}\in\Lambda^{n}E$. If $N$ is another $n$-dimensional manifold and
$\chi:M\to N$ is a diffeomorphism, then the pullback
$\chi^{*}_{\gamma}:C^{\infty}(N,\Omega_{\gamma})\to
C^{\infty}(M,\Omega_{\gamma})$ is

(A.6)
$\chi^{*}_{\gamma}\rho(x):=(\partial_{x}\chi(x))^{*}\big{(}\rho(\chi(x))\big{)}$

for $\rho\in C^{\infty}(N,\Omega_{\gamma})$ and $x\in M$. Finally, the
pullback $\chi_{\gamma}^{*}u\in\mathcal{D}^{\prime}(M,\Omega_{\gamma})$ of
$u\in\mathcal{D}^{\prime}(N,\Omega_{\gamma})$ is given by

(A.7)
${\langle}\chi_{\gamma}^{*}u,\rho\rangle_{M}:={\langle}u,(\chi^{-1})_{1-\gamma}^{*}\rho\rangle_{N}$

for $\rho\in\mathcal{D}(M,\Omega_{1-\gamma})$. We also set
$(\chi_{*})_{\gamma}:=(\chi^{-1})^{*}_{\gamma}$, acting on smooth or
distributional densities. We will almost exclusively consider $\gamma=1/2$, in
which case we write $\chi^{*}=\chi^{*}_{1/2}$ and
$\chi_{*}=(\chi^{-1})^{*}_{1/2}$, for simplicity of notation.

Let $(U_{\kappa})_{\kappa\in K}$ be a cover of $M$ by coordinate charts
$\kappa:U_{\kappa}\to\tilde{U}_{\kappa}\subseteq\mathbb{R}^{n}$, and set

(A.8) $u_{\kappa}:=(\kappa_{*})_{\gamma}u$

for $u\in\mathcal{D}^{\prime}(M,\Omega_{\gamma})$ and
$\kappa\in{\mathcal{A}}$. This gives rise to a collection
$(u_{\kappa})_{\kappa\in K}$ of distributions
$u_{\kappa}\in\mathcal{D}^{\prime}(\tilde{U}_{\kappa})=\mathcal{D}^{\prime}(\tilde{U}_{\kappa},\Omega_{\gamma})$
such that

(A.9) $u_{\kappa}=(\lambda\circ\kappa^{-1})^{*}_{\gamma}u_{\lambda}$

in $D^{\prime}(\kappa(U_{\lambda}\cap U_{\kappa}))$, for all
$\kappa,\lambda\in K$. Here and throughout, we identify distributions on
$\mathbb{R}^{n}$ with distributional densities of order $\gamma$ using the
standard basis. Note that

${\langle}u_{\kappa},\rho_{\kappa}\rangle_{\mathbb{R}^{n}}={\langle}(\kappa_{*})_{\gamma}u,(\kappa_{*})_{1-\gamma}\rho\rangle_{\mathbb{R}^{n}}={\langle}u,\rho\rangle_{M}$

for all $u\in\mathcal{D}^{\prime}(M,\Omega_{\gamma})$,
$\rho\in\mathcal{D}(U_{\kappa},\Omega_{1-\gamma})$ and $\kappa\in K$.

Conversely, for any collection $(u_{\kappa})_{\kappa\in K}$ of
$u_{\kappa}\in\mathcal{D}^{\prime}(\tilde{U}_{\kappa})$,

(A.10) $u:=\sum_{\kappa\in
K}\psi_{\kappa}^{2}\kappa_{\gamma}^{*}(u_{\kappa})\in\mathcal{D}^{\prime}(M,\Omega_{\gamma})$

is a well-defined distributional density. Here $(\psi_{\kappa}^{2})_{\kappa\in
K}$ is some partition of unity. If we additionally restrict this
correspondence to those $(u_{\kappa})_{\kappa\in K}$ satisfying (A.9), then
(A.8) and (A.10) are each other’s inverses.

One can also express (A.9) without any reference to distributional densities.
Indeed, the pullback $\chi_{\gamma}^{*}u\in\mathcal{D}^{\prime}(V)$ of a
distribution $u\in\mathcal{D}^{\prime}(W)$ is given by

(A.11)
${\langle}\chi^{*}_{\gamma}u,\phi\rangle_{\mathbb{R}^{n}}={\langle}u,|\det\partial(\chi^{-1})|^{1-\gamma}\phi\circ\chi^{-1}\rangle_{\mathbb{R}^{n}}$

for $\phi\in\mathcal{D}(V)$. Here we used that
$A^{*}\sigma=|\det(A)|^{1-\gamma}\sigma$ in (A.5) when $E=F$. We note that
(A.11), with $\gamma=0$, corresponds to the pullback of a distribution on
$\mathbb{R}^{n}$ as it is typically defined. Now (A.11) shows that (A.9) is
equivalent to the identity

(A.12)
$u_{\kappa}=|\det\partial(\lambda\circ\kappa^{-1})|^{\gamma}(\lambda\circ\kappa^{-1})^{*}_{0}u_{\lambda}$

in $\mathcal{D}^{\prime}(\kappa(U_{\lambda}\cap U_{\kappa}))$, for all
$\kappa,\lambda\in K$.

The remarks above show that one could alternatively define distributional
densities of order $\gamma$ to be collections $(u_{\kappa})_{\kappa\in K}$ of
distributions $u_{\kappa}\in\mathcal{D}^{\prime}(\tilde{U}_{\kappa})$
satisfying (A.12). However, doing so would complicate notation and
computations in Sections 3 and 4. Instead, in those sections we use an
association of sequences of distributions to distributional densities which is
more symmetric with respect to the partition of unity
$(\psi_{\kappa}^{2})_{\kappa\in K}$ in (A.10).

###  A.3. Fourier integral operators

This subsection contains some background on the Fourier integral operators
under consideration in this article. We first collect the relevant
definitions, and then some basic results about these operators.

#### Definitions

We refer to [22, 23, 10] for the general theory of Fourier integral operators
on manifolds and the associated symplectic geometry. In particular, as in [23,
Section 25.1], Fourier integral operators are defined using the class
$I^{m}(X,\Lambda,\Omega_{1/2})\subseteq\mathcal{D}^{\prime}(X,\Omega_{1/2})$
of Lagrangian distributions of order $m\in\mathbb{R}$, for $X$ a smooth
manifold and $\Lambda\subseteq T^{*}X\setminus o$ a closed conic Lagrangian
submanifold. Recall that $o$ is the zero section. On the other hand, our main
results rely on estimates in local coordinates, and these distributions have
well-known concrete oscillatory integral representations in local coordinates
(see e.g. Lemma A.2 below).

For $m\in\mathbb{R}$ and $n_{1},n_{2}\in{{\mathbb{N}}}$, we will work with the
standard Kohn–Nirenberg symbol class
$S^{m}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})$, consisting of those $a\in
C^{\infty}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})$ such that

(A.13)
$\sup\\{{\langle}\eta\rangle^{|\beta|-m}|\partial_{z}^{\alpha}\partial_{\eta}^{\beta}a(z,\eta)|\mid
z\in\mathbb{R}^{n_{1}},\eta\in\mathbb{R}^{n_{2}}\\}<\infty$

for all $\alpha\in\mathbb{Z}_{+}^{n_{1}}$ and
$\beta\in\mathbb{Z}_{+}^{n_{2}}$. We write $S^{m}(\mathbb{R}^{n_{2}})$ for the
subspace of symbols which do not depend on the $z$ variable. As in [17], one
could consider symbols from more general symbol classes, but for simplicity we
will not do so here.

Let $M$ and $N$ be smooth manifolds. The Schwartz kernel theorem (see [10,
Theorem 1.1.1]) yields a bijection between continuous linear maps
$T:\mathcal{D}(M,\Omega_{1/2})\to\mathcal{D}^{\prime}(N,\Omega_{1/2})$ and
kernels $K\in\mathcal{D}^{\prime}(N\times M,\Omega_{1/2})$. A _homogeneous
canonical relation_ from $T^{*}M$ to $T^{*}N$ is a submanifold
${\mathcal{C}}\subseteq(T^{*}N\setminus o)\times(T^{*}M\setminus o)$ such that

${\mathcal{C}}^{\prime}:=\\{(x,y,\xi,-\eta)\mid(x,\xi,y,\eta)\in{\mathcal{C}}\\}$

is a closed conic Lagrangian submanifold of $T^{*}(N\times M)\setminus o$.
Then a _Fourier integral operator of order $m\in\mathbb{R}$ associated with
${\mathcal{C}}$_ is an operator
$T:\mathcal{D}(M,\Omega_{1/2})\to\mathcal{D}^{\prime}(N,\Omega_{1/2})$ such
that the Schwartz kernel of $T$ is an element of $I^{m}(N\times
M,{\mathcal{C}}^{\prime},\Omega_{1/2})$. The class of all such Fourier
integral operators will be denoted by $I^{m}(M,N;{\mathcal{C}})$.

An FIO $T\in I^{m}(M,N;{\mathcal{C}})$ is _compactly supported_ if its
Schwartz kernel $K$ is compactly supported in $N\times M$, and _properly
supported_ if both projections from $\mathrm{supp}(K)\subseteq N\times M$, to
$N$ and to $M$, are proper maps. We say that $T$ is _non-characteristic_ at a
point $c\in{\mathcal{C}}$ if the principal symbol $a\in
S^{m}({\mathcal{C}},L_{{\mathcal{C}}})$ of $T$ is of the form $a=s_{0}a_{0}$
for some $a_{0}\in S^{m}({\mathcal{C}},L_{{\mathcal{C}}})$ homogeneous of
degree $m$ and nowhere vanishing, and $|s_{0}|$ bounded away from zero in a
conic neighborhood of $c$. We refer to [10, 23] for more on the principal
symbol of an FIO, the Maslov line bundle $L_{{\mathcal{C}}}$ and the symbol
class $S^{m}({\mathcal{C}},L_{{\mathcal{C}}})$, but we note that these notions
reduce to more concrete concepts when working in local coordinates (see
Section 7 and (C.1)). One says that $T$ _elliptic_ if it is non-characteristic
at every point $c\in{\mathcal{C}}$.

The transpose
$T^{t}:\mathcal{D}(N,\Omega_{1/2})\to\mathcal{D}^{\prime}(M,\Omega_{1/2})$ of
a $T$, given by ${\langle}T^{t}v,u\rangle_{M}={\langle}Tu,v\rangle_{N}$ for
$v\in\mathcal{D}(N,\Omega_{1/2})$ and $u\in\mathcal{D}(M,\Omega_{1/2})$, is an
element of $I^{m}(N,M;{\mathcal{C}}^{-1})$, where

(A.14)
${\mathcal{C}}^{-1}:=\\{(y,\eta,x,\xi)\mid(x,\xi,y,\eta)\in{\mathcal{C}})\subseteq(T^{*}M\setminus
o)\times(T^{*}N\setminus o)$

is the inverse canonical relation.

We consider two types of FIOs, by making geometric assumptions on their
canonical relations. Firstly, we deal with $T\in I^{m}(M,N;{\mathcal{C}})$ for
${\mathcal{C}}$ a canonical relation which is a _local canonical graph_. This
means that the natural projections

$\Pi_{T^{*}N}:{\mathcal{C}}\to T^{*}N\setminus o\ \ \text{and}\ \
\Pi_{T^{*}M}:C\to T^{*}M\setminus o$

are local diffeomorphisms, or equivalently that ${\mathcal{C}}$ is locally the
graph of a symplectomorphism $\chi$ between open subsets of $T^{*}M\setminus
o$ and $T^{*}N\setminus o$. This condition implies in particular that $M$ and
$N$ have the same dimension, and ${\mathcal{C}}^{-1}$ is then a local
canonical graph as well. If $T$ is properly supported, then
$T:\mathcal{D}(M,\Omega_{1/2})\to\mathcal{D}(N,\Omega_{1/2})$ continuously.
Then $T^{t}$ has the same properties with $M$ and $N$ reversed, so $T$ extends
uniquely to a continuous map
$T:\mathcal{D}^{\prime}(M,\Omega_{1/2})\to\mathcal{D}^{\prime}(N,\Omega_{1/2})$
by setting ${\langle}Tu,v\rangle_{N}:={\langle}u,T^{t}v\rangle_{M}$ for
$u\in\mathcal{D}^{\prime}(M,\Omega_{1/2})$ and
$v\in\mathcal{D}(N,\Omega_{1/2})$.

Secondly, we consider the case where $M$ has dimension $n\geq 2$ and $N$ has
dimension $n+1$. Then, cf. [34], we say that ${\mathcal{C}}$ satisfies the
_cinematic curvature condition_ if the natural projections

$\Pi_{T^{*}M}:{\mathcal{C}}\to T^{*}M\setminus o\ \ \text{and}\ \
\Pi_{N}:{\mathcal{C}}\to N$

are submersions, and if for each $z\in\Pi_{N}{\mathcal{C}}$ the conic
hypersurface $\Pi_{T^{*}_{z}N}{\mathcal{C}}\subseteq
T^{*}_{z}N\setminus\\{0\\}$ has everywhere $n-1$ non-vanishing principal
curvatures. Here $\Pi_{T_{z}^{*}N}:{\mathcal{C}}\to
T^{*}_{z}N\setminus\\{0\\}$ is the natural projection onto the fiber
$T^{*}_{z}N$ of $z$. This condition is invariant under changes of coordinates
on $M$ and $N$.

#### Results

We now collect some basic results about Fourier integral operators.

Let ${\mathcal{C}}$ be a homogeneous canonical relation from
$T^{*}\mathbb{R}^{n}$ to $T^{*}\mathbb{R}^{n+k}$, for $n\in{{\mathbb{N}}}$ and
$k\in\mathbb{Z}_{+}$, and let $m\in\mathbb{R}$ and $a\in
S^{m}(\mathbb{R}^{n+k}\times\mathbb{R}^{n})$. Also let
$V\subseteq\mathbb{R}^{n+k}\times(\mathbb{R}^{n}\setminus\\{0\\})$ be open,
conic and such that $(z,\eta)\in V$ whenever $(z,\eta)\in\mathrm{supp}(a)$ and
$\eta\neq 0$. Let $\Phi\in C^{\infty}(V)$ be real-valued and positively
homogenous of degree $1$ in the fiber variable $\eta$, and let $T\in
I^{m-k/4}(\mathbb{R}^{n},\mathbb{R}^{n+k};{\mathcal{C}})$ be such that

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

for all $f\in\mathcal{D}(\mathbb{R}^{n})$ and $z\in\mathbb{R}^{n+k}$. We then
say that $T$ is a Fourier integral operator _in standard form_ , with phase
function $\Phi$ and symbol $a$. Moreover, we call an operator
$S:\mathcal{D}^{\prime}(\mathbb{R}^{n})\to\mathcal{D}^{\prime}(\mathbb{R}^{n})$
a _change of coordinates_ if there exists a diffeomorphism $\chi:U\to
U^{\prime}$ between open subsets $U,U^{\prime}\subseteq\mathbb{R}^{n}$, and a
$\psi\in C_{c}(U^{\prime})$, such that $Sf=\chi^{*}_{0}(\psi f)$ for all
$f\in\mathcal{D}^{\prime}(\mathbb{R}^{n})$. Finally, an operator
$R:\mathcal{S}^{\prime}(\mathbb{R}^{n_{1}})\to\mathcal{S}(\mathbb{R}^{n_{2}})$
is a _smoothing operator_ if the Schwartz kernel of $R$ is an element of
$\mathcal{S}(\mathbb{R}^{n_{2}}\times\mathbb{R}^{n_{1}})$.

The following lemma reduces the analysis of relatively general Fourier
integral operators to operators in standard form.

######  Lemma A.2.

Let $n\in{{\mathbb{N}}}$ and $k\in\mathbb{Z}_{+}$, and let ${\mathcal{C}}$ be
a homogeneous canonical relation from $\mathbb{R}^{n}$ to $\mathbb{R}^{n+k}$
such that the projections

$\Pi_{\mathbb{R}^{n+k}}:{\mathcal{C}}\to\mathbb{R}^{n+k}\ \ \text{and}\ \
\Pi_{T^{*}\mathbb{R}^{n}}:{\mathcal{C}}\to T^{*}\mathbb{R}^{n}\setminus o$

are submersions. Let $T\in
I^{m-k/4}(\mathbb{R}^{n},\mathbb{R}^{n+k};{\mathcal{C}})$, for
$m\in\mathbb{R}$, be compactly supported. Then there exist an
$l\in{{\mathbb{N}}}$, homogeneous canonical relations
$({\mathcal{C}}_{j})_{j=1}^{l}$ from $\mathbb{R}^{n}$ to $\mathbb{R}^{n+k}$,
Fourier integral operators $(T_{j})_{j=1}^{l}$ in standard form satisfying
$T_{j}\in I^{m-k/4}(\mathbb{R}^{n},\mathbb{R}^{n+k};{\mathcal{C}}_{j})$ for
each $j\in\\{1,\ldots,l\\}$, changes of coordinates $(S_{j,1})_{j=1}^{l}$ and
$(S_{j,2})_{j=1}^{l}$, and a smoothing operator
$R:\mathcal{S}^{\prime}(\mathbb{R}^{n})\to\mathcal{S}(\mathbb{R}^{n+k})$, with
the following properties:

  1. (1)

$T=\sum_{j=1}^{l}S_{j,1}T_{j}S_{j,2}+R$;

  2. (2)

For each $j\in\\{1,\ldots,l\\}$, the phase function $\Phi_{j}\in
C^{\infty}(V_{j})$ of $T_{j}$ satisfies
$\mathrm{rank}\,\partial^{2}_{x\eta}\Phi_{j}(z,\eta)=n$ for all $(z,\eta)\in
V_{j}$, where $x=(z_{1},\ldots,z_{n})$, and the symbol $a_{j}\in
S^{m}(\mathbb{R}^{n+k}\times\mathbb{R}^{n})$ has compact support in
$\mathbb{R}^{n+k}$.

Moreover, suppose that $k=1$ and that ${\mathcal{C}}$ satisfies the cinematic
curvature condition. Then one may additionally assume that the following holds
for all $j\in\\{1,\ldots,l\\}$ and $(z_{0},\eta_{0})\in V_{j}$. One has
$\Pi_{T^{*}_{z_{0}}\mathbb{R}^{n+1}}{\mathcal{C}}_{j}=\\{\partial_{z}\Phi_{j}(z_{0},\eta)\mid(z_{0},\eta)\in
V_{j}\\}$, and the unit normal vectors $\pm\theta$ to
$\Pi_{T^{*}_{z_{0}}\mathbb{R}^{n+1}}{\mathcal{C}}_{j}$ at
$\partial_{z}\Phi_{j}(z_{0},\eta_{0})$ satisfy

(A.15)
$\mathrm{rank}\,\partial^{2}_{\eta\eta}(\partial_{z}\Phi_{j}(z_{0},\eta)\cdot\theta)|_{\eta=\eta_{0}}=n-1.$

One may express (A.15) as in (5.1):

$\mathrm{rank}\,\partial_{\eta\eta}^{2}(\partial_{z}\Phi_{j}(z_{0},\eta)\cdot
G_{j}(z_{0},\eta_{0}))|_{\eta=\eta_{0}}=n-1,$

where
$G_{j}(z_{0},\eta_{0}):=G_{j,0}(z_{0},\eta_{0})/|G_{j,0}(z_{0},\eta_{0})|$ is
the generalized Gauss map.

A statement similar to the first one in the lemma can be found in [23, pp.
29-30], under more general assumptions on the canonical relation. Note that
the assumptions of the lemma hold with $k=0$ if and only if ${\mathcal{C}}$ is
a local canonical graph, since ${\mathcal{C}}$ then has dimension $2n$. In
this case, the statement is essentially contained in [23, pp. 26-27], [45,
Proposition 6.2.4], [17, Proposition 2.14] and [10, p. 108], and the changes
of coordinates $(S_{j,1})_{j=1}^{l}$ can be omitted. The proof for general $k$
is similar to that for $k=0$. The second statement can in turn be found in
[45, Section 8.1]. One can also find a more detailed proof of both statements
in [29, Lemma 2.2].

######  Remark A.3.

In Section 6, we apply this lemma to canonical relations ${\mathcal{C}}$ from
$\mathbb{R}^{n}$ to $\mathbb{R}^{n+1}$ of the form

${\mathcal{C}}=\\{(x,t,\xi,\tau,y,\eta)\mid(x,\xi)=\chi_{t}(y,\eta),\tau=q(x,t,\xi)\\},$

where each $\chi_{t}$ is a canonical transformation on
$T^{*}(\mathbb{R}^{n})\setminus o$. In this case the changes $S_{j,1}$ of $z$
coordinates are not needed to guarantee that
$\partial_{x\eta}^{2}\Phi_{j}(x,t,\eta)=n$. It follows from the inverse
function theorem that, in suitable local coordinates, any canonical relation
satisfying the cinematic curvature condition has this form (see e.g. [45,
Section 8.1]).

######  Remark A.4.

It is relevant for the proof of Theorem 7.5 to note that, if ${\mathcal{C}}$
is contained in a sufficiently small neighborhood of a given point
$\tilde{c}\in{\mathcal{C}}$, then the decomposition in (1) simplifies and the
summation and one of the changes of coordinates are not necessary. More
precisely, one may write $T=T^{\prime}S^{\prime}+R$ for $T^{\prime}\in
I^{m-k/4}(\mathbb{R}^{n},\mathbb{R}^{n+k};{\mathcal{C}}^{\prime})$ a Fourier
integral operator in standard form associated with a homogeneous canonical
relation ${\mathcal{C}}^{\prime}$, a change of coordinates $S^{\prime}$, and a
smoothing operator $R$.

Finally, for our main results we use the following lemma, a slight variation
of [22, Proposition 18.1.22]. It allows us to apply the composition theorem
for Fourier integral operators (see e.g. [23, Theorem 25.2.3] or [45, Theorem
6.2.2]), which deals with properly supported operators. The proof of the lemma
is similar to that of [22, Proposition 18.1.22] and can also be found in [29,
Lemma 2.4].

######  Lemma A.5.

Let $T_{1}$ and $T_{2}$ be pseudodifferential operators on $\mathbb{R}^{n}$,
of orders $m_{1}\in\mathbb{R}$ and $m_{2}\in\mathbb{R}$, respectively. Suppose
that either $T_{1}$ or $T_{2}$ is compactly supported. Then there exist a
compactly supported pseudodifferential $S$ of order $m_{1}+m_{2}$, and a
smoothing operator $R$, such that $T_{1}T_{2}=S+R$.

##  Appendix B Atomic decomposition

This appendix contains the atomic decomposition of
$\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$. This atomic decomposition was
proved for $s=(n-1)/4$ in [43]. In the proof of Theorem 3.6, we use that every
$\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$-atom has compact support, and this
support condition is not preserved when directly applying the fractional
derivative ${\langle}D\rangle^{-s-(n-1)/4}$ to an
$\mathcal{H}^{(n-1)/4,1}_{FIO}(\mathbb{R}^{n})$-atom. Moreover, we believe
that it is of independent interest to include an atomic decomposition of
$\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$ for general $s\in\mathbb{R}$.

Throughout this section, fix $s\in\mathbb{R}$. An $f\in
W^{s,2}(\mathbb{R}^{n})$ is an
$\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$-_atom_ , associated with a ball
$B_{\sqrt{\tau}}(y,\nu)\subseteq S^{*}\mathbb{R}^{n}$, for $(y,\nu)\in
S^{*}\mathbb{R}^{n}$ and $\tau>0$, if

(B.1)
$\mathrm{supp}(f)\subseteq\\{x\in\mathbb{R}^{n}\mid(|\nu\cdot(x-y)|+|x-y|^{2})^{1/2}\leq\sqrt{\tau}\\}$

and $\|M_{\tau}^{\nu}(D)f\|_{W^{s,2}(\mathbb{R}^{n})}\leq\tau^{-n/2}$. Here,
for $\xi\in\mathbb{R}^{n}\setminus\\{0\\}$,

$M^{\nu}_{\tau}(\xi):=\big{(}1+\tau^{-2}\big{(}{\langle}\xi\rangle^{-1}+\min(|\hat{\xi}-\nu|^{2},|\hat{\xi}+\nu|^{2})\big{)}^{2}\big{)}^{n/2}.$

Then $M^{\nu}_{\tau}$ is a weight which is concentrated in part of the union
of two cones, in the direction of $\nu$ and $-\nu$. In this terminology,
$B_{\sqrt{\tau}}(y,\nu)$ is a ball of radius $\sqrt{\tau}$ around $(y,\nu)$
with respect to a sub-Riemannian metric on $S^{*}\mathbb{R}^{n}$ which arises
from contact geometry, and $f$ is supported in approximately the projection of
such a ball onto $\mathbb{R}^{n}$. However, for the present article these
subtleties play no role whatsoever, given that we will only use the compact
support condition (B.1) and Proposition B.1. Instead, we refer the interested
reader to [43, 17] for more on the nomenclature.

The following proposition contains the atomic decomposition of
$\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$. Fix $r\in
C^{\infty}_{c}(\mathbb{R}^{n})$ such that $r(\xi)=1$ if $|\xi|\leq 1$, and
$r(\xi)=0$ if $|\xi|\geq 2$.

######  Proposition B.1.

The collection of $\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$-atoms of radius at
most $1$ is a uniformly bounded subset of
$\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$. Moreover, there exists a $C\geq 0$
with the following property. For each
$f\in\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$ there exists a sequence
$(f_{k})_{k=1}^{\infty}$ of $\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$-atoms,
associated with balls of radius at most $1$, and an
$(\alpha_{k})_{k=1}^{\infty}\in\ell^{1}$, such that

(B.2)
$f=\sum_{k=1}^{\infty}\alpha_{k}f_{k}+r(D)f\quad\text{and}\quad\sum_{k=1}^{\infty}|\alpha_{k}|\leq
C\|f\|_{\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})}.$

###### Proof.

The proof in [43] for $s=(n-1)/4$, which in turn is a modification of the
proof of the atomic decomposition of $\mathcal{H}^{1}(\mathbb{R}^{n})$,
extends almost directly to general $s\in\mathbb{R}$. Most notably, to obtain
the decomposition in (B.2), one may replace the factor
${\langle}D\rangle^{(n-1)/2}$ in [43, Lemma 3.10] by
${\langle}D\rangle^{s+(n-1)/4}$, after which the remainder of the proof is
identical to that of [43, Theorem 3.5].

To show that the collection of $\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$-atoms
of radius at most $1$ is uniformly bounded, one can modify the notion of a
$\phi$-molecule in [43, Definition 3.3], again by changing the factor
${\langle}D\rangle^{(n-1)/2}$ to ${\langle}D\rangle^{s+(n-1)/4}$. Then one can
show as in [43, Lemma 3.4(b)] that an
$\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$-atom is the sum of two such
$\phi$-molecules, using that [43, Lemma 2.13(b)] is already adapted to general
$s$. In turn, that the collection of these $\phi$-molecules form a uniformly
bounded subset of $\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$ follows
immediately from the case where $s=(n-1)/4$, given that $\phi$-molecules need
not be compactly supported and are well behaved under application of
${\langle}D\rangle^{s-(n-1)/4}$. ∎

We note in passing, and to prove the localization principle for
$\mathcal{H}^{s,p}(\mathbb{R}^{n})$ in (3.11) in the same way as that for
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ in Theorem 3.6, that the atomic
decomposition of $\mathcal{H}^{s,1}(\mathbb{R}^{n})$ also extends from $s=0$
to general $s\in\mathbb{R}$, by a similar modification of the proof. An $f\in
W^{s,2}(\mathbb{R}^{n})$ is an $\mathcal{H}^{s,1}(\mathbb{R}^{n})$-_atom_ ,
associated with a ball $B_{\tau}(y)\subseteq\mathbb{R}^{n}$, for
$y\in\mathbb{R}^{n}$ and $\tau>0$, if $\mathrm{supp}(f)\subseteq B_{\tau}(y)$
and $\|f\|_{W^{s,2}(\mathbb{R}^{n})}\leq\tau^{-n/2}$.

######  Lemma B.2.

The collection of $\mathcal{H}^{s,1}(\mathbb{R}^{n})$-atoms is a uniformly
bounded subset of $\mathcal{H}^{s,1}(\mathbb{R}^{n})$. Moreover, there exists
a $C\geq 0$ with the following property. For each
$f\in\mathcal{H}^{s,1}(\mathbb{R}^{n})$ there exists a sequence
$(f_{k})_{k=1}^{\infty}$ of $\mathcal{H}^{s,1}(\mathbb{R}^{n})$-atoms,
associated with balls of radius at most $1$, and an
$(\alpha_{k})_{k=1}^{\infty}\in\ell^{1}$, such that

$f=\sum_{k=1}^{\infty}\alpha_{k}f_{k}+r(D)f\quad\text{and}\quad\sum_{k=1}^{\infty}|\alpha_{k}|\leq
C\|f\|_{\mathcal{H}^{s,1}(\mathbb{R}^{n})}.$

##  Appendix C More on variable-coefficient Wolff-type inequalities

In this appendix we will prove a converse statement to Proposition 5.3. Such a
converse statement is contained in [37, Corollary 4.2] for the Euclidean half-
wave equation, with a relatively simple proof that uses the invertibility of
the Euclidean half-wave propagators $e^{it\sqrt{-\Delta}}$, for
$t\in\mathbb{R}$. By contrast, a general FIO need not be invertible, and in
fact the operators in Proposition 5.3 are never invertible at fixed times, due
to the assumption that their symbol has compact spatial support. Instead, we
may merely hope for microlocal invertibility wherever the operator is non-
characteristic. This makes the converse statement to Proposition 5.3 less
elegant than that in [37, Corollary 4.2]. Nonetheless, the main result of this
appendix shows that, microlocally near any point where the operator $T$ is
non-characteristic, one may equivalently replace the decoupling norm in (5.4)
by the $\mathcal{H}^{m-s(p),p}_{FIO}(\mathbb{R}^{n})$ norm of the initial
data.

Throughout, we consider an operator $T\in
I^{m-1/4}(\mathbb{R}^{n},\mathbb{R}^{n+1};{\mathcal{C}})$ as in Section 5.1.
Additionally, we will assume that ${\mathcal{C}}$ is as in Remark A.3:

${\mathcal{C}}=\\{(x,t,\xi,\tau,y,\eta)\mid(x,t,\eta)\in\mathrm{supp}(a),\eta\neq
0,(x,\xi)=\chi_{t}(y,\eta),\tau=q(x,t,\xi)\\},$

where each $\chi_{t}$ is a homogeneous canonical transformation between open
subsets of $T^{*}(\mathbb{R}^{n})\setminus o$. This assumption always holds
locally in appropriate coordinate systems and, in this appendix, it only
serves to simplify the statement of the main result. In particular, it implies
the following. Set $T_{t}f(x):=Tf(x,t)$ for $f\in\mathcal{S}(\mathbb{R}^{n})$,
$x\in\mathbb{R}^{n}$ and $t\in\mathbb{R}$. Then $T_{t}\in
I^{m}(\mathbb{R}^{n},\mathbb{R}^{n};{\mathcal{C}}_{t})$, where

${\mathcal{C}}_{t}:=\\{(x,\xi,y,\eta)\mid\exists\,\tau\text{ such that
}(x,t,\xi,\tau,y,\eta)\in{\mathcal{C}}\\}$

is the graph of $\chi_{t}$ whenever this set is non-empty. The inverse
canonical relation ${\mathcal{C}}_{t}^{-1}$, from (A.14), is then the graph of
$\chi_{t}^{-1}$.

We note that $T$ is non-characteristic at a point
$c=(z_{0},\zeta_{0},y_{0},\eta_{0})\in{\mathcal{C}}$ if and only if there
exists a conic neighborhood
$V_{0}\subseteq\mathbb{R}^{n+1}\times(\mathbb{R}^{n}\setminus\\{0\\})$ of
$(z_{0},\eta_{0})$ and a $c>0$ such that

(C.1) $|a(z,\eta)|\geq c{\langle}\eta\rangle^{m}\text{ for all }(z,\eta)\in
V_{0}.$

Indeed, the principal symbol of each $T_{t}$ is (see e.g. [23, p. 27])

(C.2) $a(x,t,\eta)|\\!\det\partial_{x\eta}^{2}\Phi(x,t,\eta)|^{-1/2},$

and $T$ is non-characteristic at $c$ if and only if this quantity is bounded
in absolute value, both from above and below, by multiples of
${\langle}\eta\rangle^{m}$, uniformly near $z_{0}=(x_{0},t_{0})$. Since
$\det\partial_{x\eta}^{2}\Phi(x,t,\eta)\neq 0$, this reduces to (C.1).

Also recall that the _microsupport_
$\operatorname{WF}^{\prime}(R)\subseteq\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})$
of a pseudodifferential operator
$R:\mathcal{S}^{\prime}(\mathbb{R}^{n})\to\mathcal{S}^{\prime}(\mathbb{R}^{n})$
with symbol $r$ is the complement of the set of
$(x_{0},\eta_{0})\in\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})$ such
that, in some conic neighborhood of $(x_{0},\eta_{0})$, one has
$r(x,\eta)=O({\langle}\eta\rangle^{-l})$ for each $l\in\mathbb{R}$.

For our main result we need a lemma concerning a microlocal parametrix for
$T$. Let $(\chi_{\nu})_{\nu\in\Theta_{k}}$ be as in the decoupling norm in
(5.3), and let $(\phi_{k})_{k=0}^{\infty}$ be a standard Littlewood–Paley
decomposition, as in the proof of Corollary 5.4. Fix real-valued
$(\tilde{\chi}_{\nu})_{\nu\in\Theta_{k}}$ and
$(\tilde{\phi}_{k})_{k=0}^{\infty}\subseteq C^{\infty}_{c}(\mathbb{R}^{n})$
with similar properties. More precisely, each $\tilde{\chi}_{\nu}$ is
positively homogeneous of degree $0$, satisfies

$\mathrm{supp}(\tilde{\chi}_{\nu})\subseteq\\{\eta\in\mathbb{R}^{n}\setminus\\{0\\}\mid|\hat{\eta}-\nu|\leq
2^{-k/2+2}\\}$

and $\tilde{\chi}_{\nu}\chi_{\nu}=\chi_{\nu}$, and for all
$\alpha\in\mathbb{Z}_{+}^{n}$ and $\beta\in\mathbb{Z}_{+}$ there exists a
$\tilde{C}_{\alpha,\beta}\geq 0$ independent of $k$ such that, if
$2^{k-2}\leq|\eta|\leq 2^{k+2}$, then

$|(\hat{\eta}\cdot\partial_{\eta})^{\beta}\partial_{\eta}^{\alpha}\tilde{\chi}_{\nu}(\eta)|\leq\tilde{C}_{\alpha,\beta}2^{-k(|\alpha|/2+\beta)}$

for all $\nu\in\Theta_{k}$. The $(\tilde{\phi}_{k})_{k=0}^{\infty}$ have
inverse Fourier transforms uniformly in $L^{1}(\mathbb{R}^{n})$ and satisfy
$\mathrm{supp}(\tilde{\phi}_{k})\subseteq\\{\eta\in\mathbb{R}^{n}\mid
2^{k-2}\leq|\eta|\leq 2^{k+2}\\}$ and $\tilde{\phi}_{k}\phi_{k}=\phi_{k}$ for
each $k$.

######  Lemma C.1.

Suppose that $T$ is non-characteristic at a point
$c=(x_{0},t_{0},\zeta_{0},y_{0},\eta_{0})\in{\mathcal{C}}$. Then there exists
a conic neighborhood
$V\subseteq\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})$ of
$(y_{0},\eta_{0})$, an interval $J\subseteq\mathbb{R}$ containing $t_{0}$,
and, for each $t\in J$, a compactly supported $S_{t}\in
I^{-m}(\mathbb{R}^{n},\mathbb{R}^{n};{\mathcal{C}}_{t}^{-1})$ with the
following properties.

  1. (1)

For all $p\in[1,\infty]$ and $s\in\mathbb{R}$ one has

$\sup\\{\|\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)S_{t}\|_{{\mathcal{L}}(W^{s,p}(\mathbb{R}^{n}),W^{s+m,p}(\mathbb{R}^{n}))}\mid
t\in J,k\in\mathbb{Z}_{+},\nu\in\Theta_{k}\\}<\infty;$

  2. (2)

The pseudodifferential operator
$R:\mathcal{S}^{\prime}(\mathbb{R}^{n})\to\mathcal{S}^{\prime}(\mathbb{R}^{n})$,
given by

$Rf(x):=f(x)-\int_{J}S_{t}T_{t}f(x)\mathrm{d}t$

for $f\in\mathcal{S}(\mathbb{R}^{n})$ and $x\in\mathbb{R}^{n}$, satisfies
$\operatorname{WF}^{\prime}(R)\cap V=\emptyset$.

###### Proof.

The construction is analogous to that of a parametrix for a single FIO
associated with a canonical graph, as in [10, Theorem 4.2.5], but we will
sketch the details since our setting is slightly different.

Let $V_{0}$ be as in (C.1), and let $J\subseteq\mathbb{R}$ and
$V^{\prime}\subseteq\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})$ be
the projections of $V_{0}$ with respect to the $t$ variable and the $(y,\eta)$
variable, respectively. We may assume that the reciprocal of the symbol in
(C.2) is bounded, uniformly in $t\in J$, by a multiple of
${\langle}\eta\rangle^{-m}$. Then, as in [10, Theorem 4.2.5], one can cut off
this reciprocal away from a small neighborhood of $c$ to construct, for each
$t\in J$ and each conic subset $V\subseteq V^{\prime}$ which is closed in
$\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})$, a properly supported
$\tilde{S}_{t}\in
I^{-m}(\mathbb{R}^{n},\mathbb{R}^{n};{\mathcal{C}}_{t}^{-1})$ such that
$\operatorname{WF}^{\prime}(I-\tilde{S}_{t}T_{t})\cap V=\emptyset$. The
relevant bounds for $\tilde{S}_{t}$ and the pseudodifferential operator
$R_{t}:=I-\tilde{S}_{t}T_{t}$ are uniform in $t$. Moreover, since $a$ has
compact spatial support and because each $\tilde{S}_{t}$ is properly
supported, we may assume in addition that each $\tilde{S}_{t}$ is compactly
supported. By assumption, ${\mathcal{C}}_{t}$ is a canonical graph, and
therefore so is ${\mathcal{C}}_{t}^{-1}$.

Now set $S_{t}:=|J|^{-1}\tilde{S}_{t}$ for each $t\in J$. Then (2) holds by
construction. Moreover, (1) is the dual of a standard estimate from the
$L^{p}$ theory of FIOs (see (5.6)):

$\|S_{t}^{*}\tilde{\chi}_{\nu}(D)\tilde{\phi}_{k}(D)\|_{{\mathcal{L}}(W^{s-m,p}(\mathbb{R}^{n}),W^{s,p}(\mathbb{R}^{n})}\lesssim
1,$

with an implicit constant which is uniform in $k$, $\nu$ and $t\in J$, due to
Remark 2.6. ∎

######  Remark C.2.

Note that we did not use anywhere in the proof that ${\mathcal{C}}$ satisfies
the cinematic curvature condition. Indeed, we only used the specific form of
the operator $T$ in (5.2), and the assumption that the canonical relations
${\mathcal{C}}_{t}$ are canonical graphs. In fact, the results in this
appendix hold under these more general assumptions. The product structure of
the symbol in Section 5.1 also plays no role here, nor do the low frequencies.
In fact, by the very statement of Proposition C.3, it suffices to only require
(C.1) when $|\eta|\geq 1/2$.

We can now prove a converse to Proposition 5.3, a version for general FIOs of
[37, Corollary 4.2]. Recall that the wavefront set
$\operatorname{WF}(f)\subseteq\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})$
of an $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ is the complement of the set
of $(y_{0},\eta_{0})$ such that there exists a pseudodifferential operator
$S$, elliptic at $(y_{0},\eta_{0})$, with $Sf\in\mathcal{S}(\mathbb{R}^{n})$.

######  Proposition C.3.

Suppose that $T$ is non-characteristic at a point
$(z_{0},\zeta_{0},y_{0},\eta_{0})\in{\mathcal{C}}$. Then there exists a conic
neighborhood $V$ of $(y_{0},\eta_{0})$ and, for all $p\in(2,\infty)$ and
$L\geq 0$, a $C\geq 0$ such that

(C.3) $\|\phi_{k}(D)f\|_{\mathcal{H}^{m-s(p),p}_{FIO}(\mathbb{R}^{n})}\leq
C\big{(}\|T\phi_{k}(D)f\|_{L^{p,k}_{\mathrm{dec}}(\mathbb{R}^{n+1})}+2^{-kL}\|\phi_{k}(D)f\|_{L^{2}(\mathbb{R}^{n})}\big{)}$

for all $k\in\mathbb{Z}_{+}$ and $f\in L^{2}(\mathbb{R}^{n})$ with
$\operatorname{WF}(f)\subseteq V$.

###### Proof.

By replacing $T$ by $T{\langle}D\rangle^{-m}$, we may suppose that $m=0$.

Let $V$, $J$ and $R$ be as in Lemma C.1, and write
$f_{k}:=\phi_{k}(D)f=\tilde{\phi}_{k}(D)\phi_{k}(D)f$. Then

(C.4)
$f_{k}=\sum_{\nu\in\Theta_{k}}\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)R\chi_{\nu}(D)f_{k}+\sum_{\nu\in\Theta_{k}}\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)(1-R)\chi_{\nu}(D)f_{k},$

and we will bound the $\mathcal{H}^{-s(p),p}_{FIO}(\mathbb{R}^{n})$ norm of
each of the terms in (C.4) separately.

To deal with the first term on the right-hand side of (C.4), let $S$ be a
pseudodifferential operator of order zero, the symbol of which is supported
away from $\operatorname{WF}^{\prime}(R)$ and is equal to $1$ on $V$. Then
$\operatorname{WF}^{\prime}(R)\cap\operatorname{WF}^{\prime}(S)=\emptyset$
and, for each $r\in\mathbb{R}$,

$\|(1-S)\chi_{\nu}(D)f_{k}\|_{W^{r,2}(\mathbb{R}^{n})}\lesssim\|\chi_{\nu}(D)f_{k}\|_{W^{-L-(n-1)/2,2}(\mathbb{R}^{n})}$

uniformly in $k$ and $\nu$. Hence (2.2) and a standard Sobolev embedding
yield, for some $r\in\mathbb{R}$,

$\displaystyle\Big{\|}\sum_{\nu\in\Theta_{k}}\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)R\chi_{\nu}(D)f_{k}\Big{\|}_{\mathcal{H}^{-s(p),p}_{FIO}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\sum_{\nu\in\Theta_{k}}\|\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)R\chi_{\nu}(D)f_{k}\|_{W^{r,2}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\sum_{\nu\in\Theta_{k}}\|RS\chi_{\nu}(D)f_{k}\|_{W^{r,2}(\mathbb{R}^{n})}+\sum_{\nu\in\Theta_{k}}\|R(1-S)\chi_{\nu}(D)f_{k}\|_{W^{r,2}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\sum_{\nu\in\Theta_{k}}\|\chi_{\nu}(D)f_{k}\|_{W^{-L-(n-1)/2,2}(\mathbb{R}^{n})}\lesssim
2^{k(n-1)/2}\|f_{k}\|_{W^{-L-(n-1)/2,2}(\mathbb{R}^{n})}$ $\displaystyle\eqsim
2^{-kL}\|f_{k}\|_{L^{2}(\mathbb{R}^{n})},$

where we also used that the $\tilde{\phi}_{k}(D)$ and $\tilde{\chi}_{\nu}(D)$
have kernels uniformly in $L^{1}(\mathbb{R}^{n})$.

On the other hand, there exists a constant $N\geq 0$, independent of $k$ and
$\mu\in\Theta_{k}$, such that $\chi_{\mu}\tilde{\chi}_{\nu}\neq 0$ for at most
$N$ elements $\nu\in\Theta_{k}$. Hence one can use [37, Proposition 4.1] and
Hölder’s inequality to write

$\displaystyle\Big{\|}\sum_{\nu\in\Theta_{k}}\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)(1-R)\chi_{\nu}(D)f_{k}\Big{\|}_{\mathcal{H}^{-s(p),p}_{FIO}(\mathbb{R}^{n})}$
$\displaystyle\eqsim\Big{(}\sum_{\mu\in\Theta_{k}}\Big{\|}\sum_{\nu\in\Theta_{k}}\chi_{\mu}(D)\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)(1-R)\chi_{\nu}(D)f_{k}\Big{\|}_{L^{p}(\mathbb{R}^{n})}^{p}\Big{)}^{1/p}$
$\displaystyle\leq\Big{(}\sum_{\mu\in\Theta_{k}}\Big{(}\sum_{\nu\in\Theta_{k}}\|\chi_{\mu}(D)\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)(1-R)\chi_{\nu}(D)f_{k}\|_{L^{p}(\mathbb{R}^{n})}\Big{)}^{p}\Big{)}^{1/p}$
$\displaystyle\lesssim\Big{(}\sum_{\mu\in\Theta_{k}}\sum_{\nu\in\Theta_{k}}\|\chi_{\mu}(D)\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)(1-R)\chi_{\nu}(D)f_{k}\|_{L^{p}(\mathbb{R}^{n})}^{p}\Big{)}^{1/p}$
$\displaystyle\lesssim\Big{(}\sum_{\nu\in\Theta_{k}}\|\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)(1-R)\chi_{\nu}(D)f_{k}\|_{L^{p}(\mathbb{R}^{n})}^{p}\Big{)}^{1/p}$
$\displaystyle=\Big{(}\sum_{\nu\in\Theta_{k}}\Big{\|}\int_{J}\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)S_{t}T_{t}\chi_{\nu}(D)f_{k}\mathrm{d}t\Big{\|}_{L^{p}(\mathbb{R}^{n})}^{p}\Big{)}^{1/p}.$

Now Lemma C.1 (1), together with Hölder’s inequality and the fact that $J$ is
a bounded interval, allows us to bound the quantity on the final line by

$\displaystyle\Big{(}\sum_{\nu\in\Theta_{k}}\Big{(}\int_{J}\|\tilde{\phi}_{k}(D)\tilde{\chi}_{\nu}(D)S_{t}T_{t}\chi_{\nu}(D)f_{k}\|_{L^{p}(\mathbb{R}^{n})}\mathrm{d}t\Big{)}^{p}\Big{)}^{1/p}$
$\displaystyle\lesssim\Big{(}\sum_{\nu\in\Theta_{k}}\Big{(}\int_{J}\|T_{t}\chi_{\nu}(D)f_{k}\|_{L^{p}(\mathbb{R}^{n})}\mathrm{d}t\Big{)}^{p}\Big{)}^{1/p}\lesssim\Big{(}\sum_{\nu\in\Theta_{k}}\int_{J}\|T_{t}\chi_{\nu}(D)f_{k}\|_{L^{p}(\mathbb{R}^{n})}^{p}\mathrm{d}t\Big{)}^{1/p}$
$\displaystyle\leq\|Tf_{k}\|_{L^{p,k}_{\mathrm{dec}}(\mathbb{R}^{n+1})},$

as required. ∎

######  Remark C.4.

The reason to formulate (C.3) using the $\phi_{k}(D)f$, instead of a general
$f$ with $\mathrm{supp}(\widehat{f}\,)\subseteq\\{\eta\in\mathbb{R}^{n}\mid
2^{k-1}\leq|\eta|\leq 2^{k+1}\\}$, is that a function with the latter property
is necessarily smooth. Hence, in order for the condition
$\operatorname{WF}(f)\subseteq V$ to be useful, we dyadically localize a
single function $f$ in frequency, and it is the uniformity in $k$ which is
relevant. Note also that, when deriving local smoothing estimates, Theorem 5.1
is applied to functions of the form $\phi_{k}(D)f$.

For the same reason, one could replace the a priori assumption that $f\in
L^{2}(\mathbb{R}^{n})$ by $f\in W^{s,2}(\mathbb{R}^{n})$ for any
$s\in\mathbb{R}$. On the other hand, we work with square integrable functions
to match the setting of Theorem 5.1. Doing so limits us to $p>2$, because of
the use of a standard Sobolev embedding. The same proof as above would work
for any $p\in[1,\infty]$ upon replacing $L^{2}(\mathbb{R}^{n})$ by
$L^{p}(\mathbb{R}^{n})$.

######  Remark C.5.

One could attempt to choose the set $V$ as large as possible, which would lead
to a stronger statement in Proposition C.3. Moreover, one could work with a
more refined notion of wavefront set, taking into account only microlocalized
$\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ regularity. For the sake of
simplicity, we will not do so here. Note also that, since the symbol $a$ has
compact spatial support, the operator $T$ always has characteristic points.

## References

  * [1] T. Aubin.  Nonlinear analysis on manifolds. Monge-Ampère equations, volume 252 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, New York, 1982.
  * [2] E. Bahuaud, C. Guenther, J. Isenberg, and R. Mazzeo.  Wellposedness of nonlinear flows on manifolds of bounded geometry.  Online at https://arxiv.org/abs/2210.15886, 2022.
  * [3] I. Bejenaru and T. Tao.  Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation.  J. Funct. Anal., 233(1):228–259, 2006.
  * [4] D. Beltran, J. Hickman, and C. Sogge.  Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds.  Anal. PDE, 13(2):403–433, 2020.
  * [5] D. Beltran, J. Hickman, and C. Sogge.  Sharp local smoothing estimates for Fourier integral operators.  In Geometric aspects of harmonic analysis, volume 45 of Springer INdAM Ser., pages 29–105. Springer, Cham, 2021.
  * [6] J. Bourgain and C. Demeter.  The proof of the $l^{2}$ decoupling conjecture.  Ann. of Math. (2), 182(1):351–389, 2015.
  * [7] J. Bourgain and C. Demeter.  Decouplings for curves and hypersurfaces with nonzero Gaussian curvature.  J. Anal. Math., 133:279–311, 2017.
  * [8] L. Carleson and P. Sjölin.  Oscillatory integrals and a multiplier problem for the disc.  Studia Math., 44:287–299. (errata insert), 1972.
  * [9] B. Dodson, A. Soffer, and T. Spencer.  Global well-posedness for the cubic nonlinear Schrödinger equation with initial data lying in $L^{p}$-based Sobolev spaces.  J. Math. Phys., 62(7):Paper No. 071507, 13, 2021.
  * [10] J. Duistermaat.  Fourier integral operators.  Modern Birkhäuser Classics. Birkhäuser/Springer, New York, 2011\.  Reprint of the 1996 edition, based on the original lecture notes published in 1973.
  * [11] J. Eichhorn.  The boundedness of connection coefficients and their derivatives.  Math. Nachr., 152:145–158, 1991.
  * [12] Z. Fan, N. Liu, J. Rozendaal, and L. Song.  Characterizations of the Hardy space $\mathcal{H}_{\rm FIO}^{1}(\mathbb{R}^{n})$ for Fourier integral operators.  Studia Math., 270(2):175–207, 2023.
  * [13] D. Frey and P. Portal.  ${L}^{p}$ estimates for wave equations with specific ${C}^{0,1}$ coefficients.  Preprint available at https://arxiv.org/abs/2010.08326, 2020.
  * [14] C. Gao, B. Liu, C. Miao, and Y. Xi.  Square function estimates and local smoothing for Fourier integral operators.  Proc. Lond. Math. Soc. (3), 126(6):1923–1960, 2023.
  * [15] L. Guth, H. Wang, and R. Zhang.  A sharp square function estimate for the cone in $\mathbb{R}^{3}$.  Ann. of Math. (2), 192(2):551–581, 2020.
  * [16] M. Haase.  The functional calculus for sectorial operators, volume 169 of Operator Theory: Advances and Applications.  Birkhäuser Verlag, Basel, 2006.
  * [17] A. Hassell, P. Portal, and J. Rozendaal.  Off-singularity bounds and Hardy spaces for Fourier integral operators.  Trans. Amer. Math. Soc., 373(8):5773–5832, 2020.
  * [18] A. Hassell and J. Rozendaal.  $L^{p}$ and $\mathcal{H}_{FIO}^{p}$ regularity for wave equations with rough coefficients.  Pure Appl. Anal., 5(3):541–599, 2023.
  * [19] Y. Heo, F. Nazarov, and A. Seeger.  Radial Fourier multipliers in high dimensions.  Acta Math., 206(1):55–92, 2011.
  * [20] L. Hörmander.  Oscillatory integrals and multipliers on $FL^{p}$.  Ark. Mat., 11:1–11, 1973.
  * [21] L. Hörmander.  The analysis of linear partial differential operators. I.  Classics in Mathematics. Springer-Verlag, Berlin, 2003.  Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)].
  * [22] L. Hörmander.  The analysis of linear partial differential operators. III.  Classics in Mathematics. Springer, Berlin, 2007.  Pseudo-differential operators, Reprint of the 1994 edition.
  * [23] L. Hörmander.  The analysis of linear partial differential operators. IV.  Classics in Mathematics. Springer-Verlag, Berlin, 2009.  Fourier integral operators, Reprint of the 1994 edition.
  * [24] T. Hytönen, J. van Neerven, M. Veraar, and L. Weis.  Analysis in Banach Spaces. Volume I: Martingales and Littlewood–Paley Theory, volume 63 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3).  Springer, 2016.
  * [25] A. Iosevich, B. Liu, and Y. Xi.  Microlocal decoupling inequalities and the distance problem on Riemannian manifolds.  Amer. J. Math., 144(6):1601–1639, 2022.
  * [26] L. V. Kapitanskiĭ.  Estimates for norms in Besov and Lizorkin-Triebel spaces for solutions of second-order linear hyperbolic equations.  Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 171(Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsiĭ. 20):106–162, 185–186, 1989.
  * [27] L. V. Kapitanskiĭ.  Some generalizations of the Strichartz-Brenner inequality.  Algebra i Analiz, 1(3):127–159, 1989.
  * [28] S. Lee and A. Seeger.  Lebesgue space estimates for a class of Fourier integral operators associated with wave propagation.  Math. Nachr., 286(7):743–755, 2013.
  * [29] N. Liu, J. Rozendaal, L. Song, and Yan. L. X.  Local smoothing and Hardy spaces for Fourier integral operators on manifolds.  Preprint version 1, available at https://arXiv:2211.10521v1, 2022\.
  * [30] A. Martini, S. Meda, and M. Vallarino.  Maximal characterisation of local Hardy spaces on locally doubling manifolds.  Math. Z., 300(2):1705–1739, 2022.
  * [31] S. Meda and G. Veronelli.  Local Riesz transform and local Hardy spaces on Riemannian manifolds with bounded geometry.  J. Geom. Anal., 32(2):Paper No. 55, 57, 2022.
  * [32] W. Minicozzi, II and C. Sogge.  Negative results for Nikodym maximal functions and related oscillatory integrals in curved space.  Math. Res. Lett., 4(2-3):221–237, 1997.
  * [33] A. Miyachi.  On some estimates for the wave equation in $L^{p}$ and $H^{p}$.  J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27(2):331–354, 1980.
  * [34] G. Mockenhaupt, A. Seeger, and C. Sogge.  Local smoothing of Fourier integral operators and Carleson-Sjölin estimates.  J. Amer. Math. Soc., 6(1):65–130, 1993.
  * [35] J. C. Peral.  $L^{p}$ estimates for the wave equation.  J. Funct. Anal., 36(1):114–145, 1980.
  * [36] J. Rozendaal.  Characterizations of Hardy spaces for Fourier integral operators.  Rev. Mat. Iberoam., 37(5):1717–1745, 2021.
  * [37] J. Rozendaal.  Local smoothing and Hardy spaces for Fourier integral operators.  J. Funct. Anal., 283(12):Paper No. 109721, 22, 2022.
  * [38] J. Rozendaal and R. Schippa.  Nonlinear wave equations with slowly decaying initial data.  J. Differential Equations, 350:152–188, 2023.
  * [39] R. Schippa.  On smoothing estimates in modulation spaces and the nonlinear Schrödinger equation with slowly decaying initial data.  J. Funct. Anal., 282(5):Paper No. 109352, 46, 2022.
  * [40] A. Seeger, C. D. Sogge, and E. M. Stein.  Regularity properties of Fourier integral operators.  Ann. of Math. (2), 134(2):231–251, 1991.
  * [41] M. A. Shubin.  Spectral theory of elliptic operators on noncompact manifolds.  Number 207, pages 5, 35–108. 1992.  Méthodes semi-classiques, Vol. 1 (Nantes, 1991).
  * [42] L. Skrzypczak.  Atomic decompositions on manifolds with bounded geometry.  Forum Math., 10(1):19–38, 1998.
  * [43] H. Smith.  A Hardy space for Fourier integral operators.  J. Geom. Anal., 8(4):629–653, 1998.
  * [44] C. Sogge.  Propagation of singularities and maximal functions in the plane.  Invent. Math., 104(2):349–376, 1991.
  * [45] C. Sogge.  Fourier integrals in classical analysis, volume 210 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, second edition, 2017.
  * [46] E. M. Stein.  Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 1993.  With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.
  * [47] R. Strichartz.  Analysis of the Laplacian on the complete Riemannian manifold.  J. Funct. Anal., 52(1):48–79, 1983.
  * [48] M. Taylor.  Hardy spaces and BMO on manifolds with bounded geometry.  J. Geom. Anal., 19(1):137–190, 2009.
  * [49] H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators, volume 18 of North-Holland Mathematical Library.  North-Holland Publishing Co., Amsterdam-New York, 1978.
  * [50] H. Triebel.  Spaces of Besov–Hardy–Sobolev type on complete Riemannian manifolds.  Ark. Mat., 24(2):299–337, 1986.
  * [51] H. Triebel.  Characterizations of function spaces on a complete Riemannian manifold with bounded geometry.  Math. Nachr., 130:321–346, 1987.
  * [52] H. Triebel.  Theory of Function Spaces. II, volume 84 of Monographs in Mathematics.  Birkhäuser Verlag, Basel, 1992.