# From spectral cluster to uniform resolvent estimates on compact manifolds

Jean-Claude Cuenin  Department of Mathematical Sciences, Loughborough
University, Loughborough, Leicestershire, LE11 3TU United Kingdom
<EMAIL_ADDRESS>
(Date: November 12, 2020)

###### Abstract.

It is well known that uniform resolvent estimates imply spectral cluster
estimates. We show that the converse is also true in some cases. In
particular, the universal spectral cluster estimates of Sogge [48] for the
Laplace–Beltrami operator on compact Riemannian manifolds without boundary
directly imply the uniform Sobolev inequality of Dos Santos Ferreira, Kenig
and Salo [15], without any reference to parametrices. This observation also
yields new resolvent estimates for manifolds with boundary or with nonsmooth
metrics, based on spectral cluster bounds of Smith–Sogge [47] and Smith, Koch
and Tataru [29], respectively. We also convert the recent spectral cluster
bounds of Canzani and Galkowski [8] to improved resolvent bounds. Moreover, we
show that the resolvent estimates are stable under perturbations and use this
to establish uniform Sobolev and spectral cluster inequalities for Schrödinger
operators with singular potentials.

###### Key words and phrases:

Spectral cluster estimates, uniform resolvent estimates

###### 2020 Mathematics Subject Classification:

58J50, 35P15

##  1\. Introduction

Let $\Delta_{g}$ be the Laplace-Beltrami operator on a compact boundaryless
Riemannian manifold $(M,g)$ of dimension $n\geq 3$. Dos Santos Ferreira, Kenig
and Salo [15] proved the uniform Sobolev inequality

(1)
$\displaystyle\|u\|_{L^{\frac{2n}{n-2}}(M)}\lesssim\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})u\|_{L^{\frac{2n}{n+2}}(M)}.$

This generalizes a celebrated result of Kenig, Ruiz and Sogge [27] in the
Euclidean case to the setting of compact manifolds without boundary. For the
torus the inequality (1) is due to Shen [39]. Bourgain, Shao, Sogge and Yao
[6] obtained improved bounds on negatively curved manifolds and on the torus.
Using similar techniques, Krupchyk and Uhlmann [35] generalized (1) to higher
order elliptic differential operators satisfying a suitable curvature
condition. Inequality (1) is equivalent to a $L^{q^{\prime}}\to L^{q}$
estimate for the resolvent operator
$(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}$, with $q=2n/(n-2)$. Frank and
Schimmer [17], and, independently, Burq, Dos Santos Ferreira and Krupchyk [7],
proved the endpoint version

(2)
$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2}))^{-1}\|_{L^{q^{\prime}}(M)\to
L^{q}(M)}\lesssim\langle\lambda\rangle^{2\sigma(q)-1},$

where $\langle\lambda\rangle=2+|\lambda|$, $q=q_{n}:=2(n+1)/(n-1)$ is the
critical exponent, and

(3)
$\displaystyle\sigma(q)=\min\bigl{(}\,n(\tfrac{1}{2}-\tfrac{1}{q})-\tfrac{1}{2},\,\tfrac{n-1}{2}(\tfrac{1}{2}-\tfrac{1}{q})\,\bigr{)}.$

In fact, all other estimates (2) for $2\leq q\leq 2n/(n-2)$ follow from the
endpoint case by Sobolev embedding and by interpolation with the trivial bound
for $q=2$. In particular, this yields (1) since $2\sigma(q)-1=0$ for
$q=2n/(n-2)$.

It is also well known (see e.g. [17, Lemma 10]) that (2) implies Sogge’s [48]
spectral cluster estimates

(4) $\displaystyle\|\Pi_{[\lambda,\lambda+1]}\|_{L^{2}(M)\to
L^{q}(M)}\lesssim\langle\lambda\rangle^{\sigma(q)}$

for $2\leq q\leq\infty$. Here
$\Pi_{[\lambda,\lambda+1]}:=\mathbf{1}_{[\lambda,\lambda+1]}(\sqrt{-\Delta_{g}})$,
defined by the functional calculus, is the orthogonal projection onto the
subspace of functions in $L^{2}(M)$ that are spectrally localized to
frequencies in the unit length window $[\lambda,\lambda+1]$. The $q=\infty$
bound is related to the sharp counting function remainder estimate of
Avakumović, Levitan and Hörmander [1, 36, 22]. On general manifolds with
boundary, Grieser [20] was the the first to prove $L^{\infty}$ estimates for
eigenfunctions (but not spectral clusters) without convexity or concavity
assumptions.

It does not seem to be known whether the converse implication $(3)\implies(2)$
(or even $(3)\implies(1)$) is true in full generality. To paraphrase the
authors of [15]: “It seems that such estimates cannot be obtained as a direct
consequence of the spectral cluster estimates obtained by Sogge”. This forced
them to redo the Hadamard parametrix construction of [48]. The paper [15]
provides a fairly complete study of the mapping properties of the Hadamard
parametrix for the resolvent and the corresponding remainder term, using
oscillatory integral theorems. Only a special case of these estimates is
needed to prove (1). Frank and Schimmer [17] avoided the explicit construction
of the Hadamard parametrix, but used some of its off-diagonal (non self-dual)
estimates established in [15]. The approach of [6] avoids oscillatory integral
theorems and only uses stationary phase to prove (1). Sogge’s [48] universal
spectral cluster bounds are used to handle a “global term”. However, the proof
still uses the short-time Hadamard parametrix for the wave propagator
$\cos(t\sqrt{-\Delta_{g}})$ to handle a “local term”. The bounds for the
latter are independent of the global geometry of the manifold. One of our main
results, Theorem 2.1 (or rather its generalization, Theorem 4.2), is close to
[6, Theorem 1.3], which shows that improvements of the $L^{q^{\prime}}\to
L^{q}$ resolvent bounds for $(\Delta_{g}+(\lambda+\mathrm{i}\mu)^{2})^{-1}$ to
a larger region $|\mu|\geq\epsilon(\lambda)$, with $\epsilon(\lambda)\leq 1$
and $\lambda\geq 1$, are equivalent to $L^{2}\to L^{q}$ spectral cluster
bounds in shrinking frequency windows $[\lambda,\lambda+\epsilon(\lambda)]$.
The first difference between this and our result is that we are in a sense
concerned with the opposite of improved bounds. Namely, we consider metrics of
low regularity and seek bounds on the resolvent without losses compared to the
smooth case. In fact, we will show how the two approaches can be combined to
get improved resolvent bounds from improved spectral cluster bounds, without
any a priori assumption on the smoothness of the metric. Of course, it is
unreasonable to expect improved estimates for metrics of such low regularity
as considered here, but the point is that all the geometric information is now
encoded in the spectral cluster bound. The second difference is that we
completely avoid parametrix estimates and, again, solely rely on the spectral
cluster bounds (or the equivalent quasimode bounds). In the case of improved
estimates, we replace the step in [6] involving the Hadamard parametrix by a
more abstract argument. As an application, we convert the recent spectral
cluster bounds of Canzani and Galkowski [8] to improved resolvent bounds under
pure dynamical assumptions on the manifold. The Hadamard parametrix
construction breaks down for metrics with limited smoothness. Smith [40, 42]
and Tataru [53, 54, 55] used wave packet techniques to construct parametrices
for regularized wave operators and established Strichartz and square function
estimates for the variable-coefficient wave equation with rough coefficients.
In contrast to the smooth case, one cannot hope for asymptotic expansions of
the fundamental solution modulo smoothing operators since the errors in the
parametrix are too large. However, for the sake of proving estimates, they can
be considered as forcing terms, and the equation can be solved iteratively.
The wave packet parametrices require roughly two derivatives on the
coefficients as this guarantees uniqueness of the bicharacteristic flow. For
metrics with $C^{1,1}$ coefficients one could prove the parametrix bounds for
the resolvent estimates concurrently with those for the spectral cluster
bounds. This is essentially the approach of Koch and Tataru [31, Theorem 2.5].
The difference between the $TT^{*}$ version of the spectral cluster bound (or
quasimode bound) and the resolvent is merely a causal factor
$\mathbf{1}_{t\geq 0}$, where $t$ denotes one of the spatial coordinates (see
the proof of Lemma 4.3). The dispersive ($L^{1}\to L^{\infty}$) estimate [31,
Proposition 7.4] is insensitive to this factor, and the spectral cluster and
resolvent estimates both follow from this. However, this strategy fails to
prove resolvent analogues of the spectral cluster bounds of Smith and Sogge
[47] or Smith, Koch and Tataru [29]. In those cases, the metric is only
Lipschitz, and good parametrix bounds hold only on small spatial scales.
Highly nontrivial energy propagation estimates are needed to prove estimates
on the unit scale. Following the arguments of these papers carefully, one
could perhaps prove resolvent estimates directly. However, the preceding
discussion should make it clear that this is not completely straightforward.
Theorem 2.1 has the advantage that it treats the spectral cluster estimates as
a black box and thus yields several new resolvent estimates in one stroke.

A simple argument involving the Phragmén–Lindelöf maximum principle (see e.g.
[7]) shows that (2) is equivalent to the bound

(5) $\displaystyle\|(\Delta_{g}+z)^{-1}\|_{L^{q^{\prime}}(M)\to
L^{q}(M)}\lesssim_{\delta}|z|^{\sigma(q)-1/2}$

for arbitrary $\delta>0$ and all $z\in{\mathbb{C}}\setminus[0,\infty)$ with
$\mathrm{Im}\,\sqrt{z}\geq\delta$. Here we take the branch of the square root
with $\mathrm{Im}\,\sqrt{\cdot}>0$. After possibly multiplying the metric by
an innocuous factor, we may always assume that $\delta=1$, in accordance with
(2). Because $\sigma(q)$ is decreasing in $1/q$, the exponent of $|z|$ is
always nonpositive. Hence these estimates are uniform in the region111This
corresponds to the outside of a parabola, see [15, 6].
$\mathrm{Im}\,\sqrt{z}\geq 1$, which is why they are called _uniform_
resolvent estimates. We will also call the resolvent version of (1) a “uniform
Sobolev inequality”. Spectral cluster estimates (4) with frequency windows of
unit length are called “universal” since they hold on any compact boundaryless
Riemannian manifold. These are best possible in terms of the exponent
$\sigma(q)$, see [49, Chapter 5]. However, they are rarely optimal for
individual eigenfunctions (see e.g. [50, 5]). If $M$ is the sphere or, more
generally, a Zoll manifold, then no improvements are possible, due to the high
multiplicity of eigenvalues. Similarly, (5) is best possible in terms of the
lower bound $\mathrm{Im}\,\sqrt{z}\geq 1$ for these manifolds [6, 17]. In the
low-regularity setting that we will consider here, $g\in C^{s}$, there may be
losses in the exponents of (4) and (5), i.e. $\sigma(q)$ may have to be
replaced by $\gamma(q)>\sigma(q)$. We will refer to spectral cluster estimates
with exponent as in the smooth case, i.e. $\gamma(q)=\sigma(q)$, as “perfect”
estimates.222This terminology is borrowed from a seminar talk of Daniel
Tataru. For the resolvent, we distinguish two types of perfect estimates. The
first we call estimates with “perfect exponents”. This means that the exponent
is the same as in (5) (and hence the same as for elliptic estimates), but the
region $\Omega\subset{\mathbb{C}}$ for which the estimate holds may be smaller
than the “perfect region” $\Omega=\\{\mathrm{Im}\,\sqrt{z}\geq 1\\}$. In the
second type the region $\Omega$ is perfect, but the exponent may be larger
than in (5). If both the exponent and the region is perfect, we simply say
that we have perfect resolvent estimates. Note that, by Sobolev embedding, if
we have perfect exponents for $q_{1}$, then the same is true for all
$q_{2}\geq q_{1}$. We will consider regions of the form
$\Omega_{\rho}=\\{z=(\lambda+\mathrm{i}\mu)^{2}:\,\lambda\geq
1,\,|\mu|\lesssim\lambda^{\rho}\\}$, disregarding the elliptic region (the
outside of the union of a disk around the origin and a cone in the direction
of the positive $z$-axis) in which (5) trivially holds. Hence the relevant
range for $\rho$ is $0\leq\rho\leq 1$, and the perfect region corresponds to
$\rho=0$.

When proving (2) in a coordinate patch of the manifold, the crucial region in
phase space is $|x|\lesssim 1$, $|\xi|\sim\lambda$. Away from this region, one
has stronger elliptic estimates, and the inequality follows by Sobolev
embedding. We will omit the arguments concerning elliptic estimates and refer
to the proof of [32, Corollary 5] or that of [41, Theorem 7]. In the crucial
phase space region, the difference between spectral cluster and resolvent
estimates is at most logarithmic in the frequency $\lambda$. In fact, this
holds in a rather abstract setting, see Proposition 3.3. Similar abstract
considerations (see e.g. [38, Theorem XIII.25] or [18, Corollary 3.3]) show
that the spectral cluster bound (4) is equivalent to the analogue of (2) for
the imaginary part of the resolvent. Hence, any difference between (2) and (4)
can only come from the real part. The logarithmic loss incurred by the
abstract bounds hints at an inefficiency in detecting cancellations. This is
most clearly seen in the constant coefficient case, where the distributional
formula

$\displaystyle(|\xi|^{2}-(\lambda+\mathrm{i}0)^{2})^{-1}=\operatorname{p.v.}(|\xi|^{2}-\lambda^{2})^{-1}+\mathrm{i}\pi\delta(|\xi|^{2}-\lambda^{2})$

reveals cancellations of the real part. More generally, in the case where the
operator has only absolutely continuous spectrum, Guillarmou and Hassell [18,
Lemma 4.1] showed that the cancellation effects can be exploited by an
integration by parts argument. This requires stronger pointwise bounds on the
spectral measure and all its derivatives of order $\leq n/2$. The cancellation
is related to the causal structure of the resolvent mentioned before.

Most modern proofs of spectral cluster and resolvent estimates rely on wave
equation techniques. One popular approach is via square function estimates for
the wave equation [37, 42, 2, 9, 43], another one is via Strichartz estimates
for a half wave equation [31, 7, 34, 30]. The latter is purely stationary,
i.e. all the variables are spatial. Locally in phase space, one can single out
a distinguished direction that plays the role of a fictional time. In fact,
the papers listed under the first approach also use this phase space
localization. While square function estimates are of independent interest, the
second approach seems more natural for a purely stationary problem. More
importantly, the homogeneous Strichartz estimates (corresponding to spectral
cluster bounds) can easily be upgraded to inhomogeneous estimates
(corresponding to resolvent bounds) by invoking the Christ–Kiselev lemma [10,
46, 52]. For square function estimates, the passage from homogeneous to
inhomogeneous estimates is more involved, as the recent work of Smith [43]
shows.

##  2\. Main results

###  2.1. From spectral cluster to resolvent bounds

######  Theorem 2.1.

Let $(M,g)$ be a compact Riemannian manifold without boundary and with
Lipschitz metric $g$. Assume that $2\leq q\leq 2n/(n-2)$ if $n\geq 3$ and
$2\leq q<\infty$ if $n=2$, and that the spectral cluster estimate

(6) $\displaystyle\|\Pi_{[\lambda,\lambda+1]}\|_{L^{2}(M)\to
L^{q}(M)}\lesssim\langle\lambda\rangle^{\gamma(q)}$

holds for some $\gamma(q)\in[\sigma(q),\sigma(q)+1/2]$. Then we have the
uniform resolvent estimate

(7)
$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}\|_{L^{q^{\prime}}(M)\to
L^{q}(M)}\lesssim\langle\lambda\rangle^{2\gamma(q)-1}.$

######  Remark 2.2.

The restrictions on $\gamma(q)$ are natural since, by Sobolev embedding, the
bounds trivially hold for $\gamma(q)=\sigma(q)+1/2$. On the other hand, we
cannot have better bounds than in the smooth case.

######  Remark 2.3.

We actually prove a more general result (Theorem 4.2) that allows us to obtain
resolvent estimates with perfect exponents in a an intermediate region
$\Omega_{\rho}\subset{\mathbb{C}}$, even when perfect spectral cluster
estimates are not available.

######  Remark 2.4.

We do not touch here on operator-theoretic considerations related to the
definition of $\Delta_{g}$ and domain issues. In the context of low-regularity
metrics, it is advantageous to define the operator by quadratic form methods,
see e.g. [13, Chapter 6]. Thus, we implicitly assume that $\Delta_{g}$ is in
divergence form and that the resolvent equation
$\Delta_{g}u+(\lambda+\mathrm{i})^{2}u=f$ is interpreted in a weak sense.

###  2.2. Stability under perturbations

######  Theorem 2.5.

Assume that the conditions of Theorem 2.1 are satisfied with perfect exponent
$\gamma(q)=\sigma(q)$ and that $V\in L^{p}(M)$ with $1/p=1/q^{\prime}-1/q$.
Then there exists $\Lambda=\Lambda(M,q,V)\geq 1$ such that

(8)
$\displaystyle\|(\Delta_{g}+V+(\lambda+\mathrm{i})^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\langle\lambda\rangle^{2\sigma(q)-1}$

for $\lambda\geq\Lambda$. Moreover, if $n=2$ or $n\geq 3$ and $q<2n/(n-2)$,
then $\Lambda$ depends on $V$ only though $\|V\|_{L^{p}}$.

######  Remark 2.6.

The above also follows from recent results by Blair, Huang, Sire and Sogge
[3]. However, for the exponents $2\leq q\leq 2n/(n-2)$ and the self-dual
estimates considered here, we can relax the assumptions in the abstract
Theorem 2.1 of [3]. Instead of assuming the quasimode _and_ the uniform
Sobolev estimate for the unperturbed operator, (2.2) and (2.6) in [3], we only
need to assume one of them (since they are equivalent).

######  Remark 2.7.

Since $M$ is compact, Hölder’s inequalities implies that $L^{p_{2}}(M)\subset
L^{p_{1}}(M)$ for $p_{2}\geq p_{1}$. We will subsequently consider critically
singular potentials $V\in L^{n/2}(M)$. For fixed $V$, the restriction
$\lambda\geq\Lambda$ can easily be removed since the inequality for
$\lambda\leq\Lambda$ is trivial.

######  Remark 2.8.

The stability holds in greater generality, not just for potential
perturbations, see Proposition 4.8. Similar results may be found in [31, 32].

######  Corollary 2.9.

Perfect spectral cluster estimates (i.e. (6) with $\gamma(q)=\sigma(q)$) are
stable under perturbations by real-valued potentials $V\in L^{n/2}(M)$.

###  2.3. Manifolds with nonsmooth metrics

######  Theorem 2.10.

Let $\Delta_{g}$ be the Laplace–Beltrami operator on an $n$-dimensional
compact boundaryless Riemannian manifold with $C^{s}$ metric333Lipschitz if
$s=1$ and $C^{1,1}$ if $s=2$., $0\leq s\leq 2$. Assume that $2\leq q\leq
2n/(n-2)$ if $n\geq 3$ and $2\leq q<\infty$ if $n=2$.

  * (i)

If $s=2$, then we have the perfect estimates

(9)
$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\langle\lambda\rangle^{2\sigma(q)-1}.$

  * (ii)

If $s=1$ and $n=2$, then (9) holds for $q>8$, and

(10)
$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}\|_{L^{\frac{8}{7}}\to
L^{8}}\lesssim\langle\lambda\rangle^{2\sigma(q)-1}\ln^{3}\langle\lambda\rangle.$

  * (iii)

For any $0\leq s\leq 2$ and with $\rho=\frac{2-s}{2+s}$, we have

(11)
$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i}\lambda^{\rho})^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\langle\lambda\rangle^{2\sigma(q)-1}.$

These estimates are stable under perturbations by real-valued potentials $V\in
L^{n/2}(M)$.

######  Remark 2.11.

$(i)$ follows from the perfect spectral cluster estimates for $C^{1,1}$
metrics, due to Smith [42]. The resolvent estimates in this case are implicit
in the work of Koch and Tataru [31, Theorem 2.5].

######  Remark 2.12.

Counterexamples of Smith and Sogge [44] show that, in the Hölder scale,
$C^{1,1}$ is best possible for perfect spectral cluster estimates in the range
$2\leq q\leq q_{n}$. This also implies that this regularity is optimal for the
endpoint $L^{q_{n}^{\prime}}\to L^{q_{n}}$ resolvent estimate. It is an
interesting question what the minimal regularity for the uniform Sobolev
inequality (1) is.

######  Remark 2.13.

Recently, Smith [43], extending results of Chen and Smith [9], obtained
uniform $L^{p}\to L^{q}$ resolvent estimates for certain (not necessarily
dual) exponents obeying the Sobolev gap condition $1/p-1/q=2/n$ (thus,
including (1), but not (2)) for manifolds of bounded sectional curvature. This
setting is more general than that of $C^{1,1}$ metrics since the latter give
rise to bounded curvature. We should emphasize here that our proof only works
for self-dual estimates. Using the spectral cluster bounds of [9], we could
also obtain the results of $(i)$ under the bounded curvature condition.

######  Remark 2.14.

For Lipschitz metrics, Koch, Smith and Tataru [30] proved sharp spectral
cluster estimates in two dimensions. Perfect estimates hold for $q>8$ (with a
logarithmic loss at the endpoint $q=8$). This accounts for the resolvent
estimate in $(ii)$. The higher dimensional bounds of [30] do not yield uniform
resolvent bounds. The relevant issue is whether one has perfect estimates for
the Sobolev exponent $q=2n/(n-2)$. The results in [30] only yield perfect
estimates for larger exponents $q>(6n-2)/(n-1)$. It is conjectured that
perfect estimates should hold for $q\geq 2(n+2)/(n-1)$. If true, this would
imply the uniform Sobolev inequality (1) in dimensions $n=3,4$.

######  Remark 2.15.

All our estimates have perfect exponents, and in $(i),(ii)$ we also have
perfect regions. In $(iii)$, we obtain resolvent estimates in an intermediate
(between the perfect and the trivial) region
$\Omega_{\rho}\subset{\mathbb{C}}$. This is sharp at the critical exponent
$q=q_{n}$ for all $s\in[0,2]$ since the spectral cluster estimates in this
case are sharp [41]. It is also sharp in the trivial case $s=0$ as the
spectral cluster bounds are no better than what can be obtained by Sobolev
embedding [12]. In all other cases, $(iii)$ is presumably not sharp. The proof
uses results of Smith [41] for $1\leq s\leq 2$ and of Koch, Smith and Tataru
[28] for $0\leq s\leq 1$. A minor obstruction in the latter case is that
Theorem 2.1 requires at least Lipschitz regularity. We will overcome this by
localizing to a smaller scale.

###  2.4. Manifolds with boundary

######  Theorem 2.16.

Let $\Delta_{g}$ be the Laplace–Beltrami operator on a $n$-dimensional compact
Riemannian manifold with boundary, subject to either Dirichlet or Neumann
boundary conditions. Assume that $2\leq q\leq 2n/(n-2)$ if $n\geq 3$ and
$2\leq q<\infty$ if $n=2$.

  * (i)

We have the estimates

$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i}\lambda^{1/3})^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\langle\lambda\rangle^{2\sigma(q)-1}.$

  * (ii)

If $n=2,3,4$ and $q\geq 2(n+2)/(n-1)$, then

$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\langle\lambda\rangle^{2\sigma(q)-1}.$

  * (iii)

If $M$ has strictly concave boundary and the boundary conditions are
Dirichlet, then

$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\langle\lambda\rangle^{2\sigma(q)-1}\ln\langle\lambda\rangle.$

These estimates are stable under perturbations by real-valued potentials $V\in
L^{n/2}(M)$.

######  Remark 2.17.

$(ii)$ implies that the uniform Sobolev inequality (1) holds for manifolds
with boundary in dimensions $n=2,3,4$. The proof is based on a deep result by
Smith and Sogge [47], who proved that perfect spectral cluster estimates hold
in the stated range. For $n=2$ their result is best possible. The conjectured
range for perfect spectral cluster estimates in higher dimensions is
$q\geq(6n+4)/(3n-4)$. If true, this would imply the uniform Sobolev inequality
in all dimensions $n\geq 3$. The higher dimensional estimates of [47] are
still good enough to obtain this in dimensions $n=3,4$. Their results also
yield spectral cluster bounds for $n\geq 5$, $q\geq 4$, but these do not imply
uniform Sobolev estimates since $2n/(n-2)<4$.

######  Remark 2.18.

The estimates for manifolds with boundary are closely related to those for
manifolds with Lipschitz metrics. The reason is that one can double the
manifold and reflect the metric across the boundary in an odd (for Dirichlet
boundary conditions) or even (for Neumann boundary conditions) way. This
produces a manifold without boundary, but with Lipschitz metric, which
accounts for the simple bound in $(i)$.

######  Remark 2.19.

The fact that the bounds in [47] were proved by reducing to a manifold without
boundary is important here since Theorem 2.1 does not directly apply to
manifolds with boundary. Such a generalization would imply perfect resolvent
estimates for manifolds with strictly concave boundaries, based on the
corresponding spectral cluster bounds of Grieser [19] (in two dimensions) or
those of Smith and Sogge [45] (in any dimension). Microlocally, the proof of
Theorem 2.1 does not work at “glancing” points of $T^{*}(\partial M)$, see
[23, Chapter 24]. It would be interesting to prove a version of Theorem 2.1
for manifolds with boundary. However, the abstract bound (31) holds
irrespective of boundaries and yield a weaker bound with a logarithmic loss in
$(iii)$.

######  Remark 2.20.

The Lipschitz singularity produced by the doubling procedure is of a special
type, which is why the bounds of [47] yield uniform Sobolev inequalities in
higher dimensions, while those of [30] do not. Incidentally, the proof in [47]
yields the conjectured results for the special Lipschitz singularities
considered there. This explains the similarity of the conjectured estimates
for Lipschitz metrics and the actual estimates in $(ii)$ for manifolds with
boundary.

###  2.5. Improved estimates

Following Canzani–Galkowski [8], we denote by ${\Xi}$ the collection of
maximal unit speed geodesics for $(M,g)$ and define, for $x\in M$,

$\mathcal{C}_{{}_{x}}^{r,t}:=\big{\\{}\gamma(t):\,\gamma\in\Xi,\,\gamma(0)=x,\,\exists\,n-1\text{
conjugate points to }x\text{ in }\gamma(t-r,t+r)\big{\\}},$

where conjugate points are counted with multiplicity.

######  Theorem 2.21.

Let $\Delta_{g}$ be the Laplace–Beltrami operator on an $n$-dimensional
compact boundaryless Riemannian manifold with smooth metric. Assume that
$q_{n}<q\leq 2n/(n-2)$ if $n\geq 3$ and $q_{n}<q<\infty$ if $n=2$. Assume that
there exist $t_{0}>0$ and $a>0$ such that

$\inf_{x_{1},x_{2}\in M}d\big{(}x_{1},\mathcal{C}_{x_{2}}^{r_{t},t}\big{)}\geq
r_{t},\qquad\text{ for }t\geq t_{0},$

with $r_{t}=\frac{1}{a}e^{-at}.$ Then we have the improved resolvent estimate

$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i}\epsilon(\lambda))^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\langle\lambda\rangle^{2\sigma(q)-1},$

where $\epsilon(\lambda)=1/\ln\langle\lambda\rangle$. These estimates are
stable under perturbations by real-valued potentials $V\in L^{n/2}(M)$.

######  Remark 2.22.

For spectral clusters, Canzani-Galkowski [8, Theorem 2] prove a more general
result that does not require global geometric assumptions on $(M,g)$, but only
purely dynamical assumptions. In order to avoid lengthy definitions we stated
the resolvent estimates for the simpler version [8, Theorem 1], but the
corresponding result for the general version is also valid. It is interesting
to note that, similar to our method, the techniques of [8] do not require
long-time wave parametrices.

######  Remark 2.23.

Since [8, Theorem 1] includes as a special case manifolds without conjugate
points, it generalizes the results of Hassell and Tacy [21], where logarithmic
improvements for $q>q_{n}$ were proved for manifolds with non-positive
curvature. Blair, Huang, Sire and Sogge [3, Theorem 1.3] used these to obtain
uniform Sobolev inequalities for Schrödinger operators with singular
potentials on such manifolds. Theorem 2.21 generalizes the latter result (in a
limited range of exponents) in the same way.

######  Remark 2.24.

As before, one could interpolate with the trivial $L^{2}\to L^{2}$ bound to
get resolvent estimates for the full range of $q$. Then one would get a small
loss $\epsilon(\lambda)^{-\nu}$, for arbitrary $\nu>0$, at the critical
exponent $q_{n}$.

###  2.6. Fractional Schrödinger operators

######  Theorem 2.25.

Let $(M,g)$ be an $n$-dimensional compact boundaryless Riemannian manifold,
$n\geq 2$, with $C^{1,1}$ metric. Assume that $2n/(n+1)\leq\alpha\leq n$ and
$2\leq q\leq 2n/(n-\alpha)$ if $n>\alpha$ or $2\leq q<\infty$ if $n=\alpha$.
If $V\in L^{n/\alpha}(M)$, then

$\displaystyle\|((-\Delta_{g})^{\alpha/2}+V-(\lambda+\mathrm{i})^{\alpha})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\langle\lambda\rangle^{2\sigma(q)+1-\alpha}.$

######  Corollary 2.26.

If $H_{V}=(-\Delta_{g})^{\alpha/2}+V\geq 0$ and
$\Pi^{V}_{[\lambda,\lambda+1]}=\mathbf{1}_{[\lambda,\lambda+1]}(H_{V}^{1/\alpha})$,
then

$\displaystyle\|\Pi^{V}_{[\lambda,\lambda+1]}\|_{L^{2}\to
L^{q}}\lesssim\langle\lambda\rangle^{\sigma(q)}$

holds for $2\leq q\leq 2n/(n-\alpha)$ if $n>\alpha$ or $2\leq q<\infty$ if
$n=\alpha$.

######  Remark 2.27.

Note that, under the assumption of Theorem 2.25, $H_{V}$ is bounded from
below. Hence, after possibly adding a constant, we can always assume
$H_{V}\geq 0$. Here the operator $(-\Delta_{g})^{\alpha/2}$ is the so-called
spectral fractional Laplacian, defined by the functional calculus. Thus the
spectral cluster bounds for $V=0$ coincide with those for the Laplace-Beltrami
operator.

######  Remark 2.28.

Huang, Sire and Zhang [24] proved the spectral cluster estimate in Corollary
2.26 in the case of smooth metric for a larger range of exponents $q$. Here we
only use resolvent estimates as input, which limits the range to the stated
one. Estimates for higher exponents could be derived by using heat kernel
bounds as in [24]. For larger $q$ the condition $V\in L^{n/\alpha}(M)$ is not
enough, but the spectral cluster bounds in [24] hold under the additional
assumption that $V$ is in the Kato class of order $\alpha$, see also [4, 3,
16] for similar recent results involving critically singular potentials.

######  Remark 2.29.

Although Theorem 2.25 does not quite follow from Theorem 2.1, a slight
modification of its proof suffices. In fact, Theorem 2.1 could be stated for
pseudodifferential operators of real principal type. We will leave such
generalizations to a future investigation.

######  Remark 2.30.

Related to the previous remark, the strategy for the fractional Laplacian
could also be adapted to simplify and generalize result of Krupchyk and
Uhlmann [35] for higher order differential operators. Here one assumes that
the fibers of the unit cosphere bundle associated to the principal symbol are
strictly convex. Once more, this could also be deduced from the results of
Koch and Tataru [31, Theorem 2.5].

###  2.7. Overview of results

The following table summarizes the above results.

Operator | Regularity/boundary | Spectral cluster | Resolvent
---|---|---|---
$-\Delta_{g}$, | $g\in C^{\infty}$ | [48] |  [15], [6], [35], [7], [17]
| $g\in C^{s}$ | [42, 41, 28, 30] | Theorem 2.10
| $\partial M\neq\emptyset$ |  [47], [45] | Theorems 2.16
$-\Delta_{g}+V$ | $V\in L^{n/2}$ |  [4], [3] |  [3], Theorems 2.10, 2.16
$(-\Delta_{g})^{\alpha/2}+V$ | $V\in L^{n/\alpha}$ | [24] | Theorem 2.25

###  2.8. Outline of the paper

In Section 3 we prove, in a rather abstract setting, that sharp spectral
cluster estimates imply resolvent estimates that are sharp up to a logarithmic
loss in $\lambda$. In Section 4 we prove Theorem 2.1 and Theorem 2.5. In
Section 5 we then show how Theorems 2.10, 2.16 and 2.25 follow from a
combination of these results and known spectral cluster estimates. We will
also discuss simple proofs of known resolvent bounds on compact manifolds and
on Euclidean space. In the latter case, we show that the Kenig–Ruiz–Sogge
bounds directly from the Tomas–Stein theorem.

###  2.9. Notation

If not indicated otherwise, $L^{q}$ always stands for $L^{q}(M)$. The norm of
an operator $A:L^{p}\to L^{q}$, initially defined on $L^{2}\cap L^{p}$ and
extended by continuity, is denoted by $\|A\|_{L^{p}\to L^{q}}$. We write
$2^{*}$ for the Sobolev exponent $2n/(n-2)$ if $n\geq 3$ and any fixed,
sufficiently large number $2^{*}<\infty$ for $n=2$. We will call a bump
function adapted to a set if it is supported on that set and equal to one on a
slightly smaller dilate of the same set. $a\lesssim b$ means $a\leq Cb$ for
some universal constant $C$ (independent $\lambda$). If the constant is
assumed to be small, we write $a\ll b$. The Weyl quantization of a
pseudodifferential symbol $a$ is denoted by $a^{w}$.

##  3\. Abstract results

We consider the following setting: Let $X$ be a $\sigma$-finite measure space
and let $A$ be a self-adjoint (possibly unbounded) operator on $L^{2}(X)$. The
Lebesgue spaces $L^{p}(X)$ are _compatible_ as $p$ varies in $[1,\infty]$, in
the sense that $L^{p_{1}}(X)\cap L^{p_{2}}(X)$ is dense in both spaces in the
intersection and complete with respect to the norm $\|\cdot\|_{L^{p_{1}}\cap
L^{p_{2}}}=\|\cdot\|_{L^{p_{1}}}+\|\cdot\|_{L^{p_{2}}}$ (see [14, Problem
2.2.9]). We fix $q\in(2,\infty)$ and assume that the resolvent operators
$R(z,A)=(A-z)^{-1}$, $z\in{\mathbb{C}}\setminus{\mathbb{R}}$, have a
_consistent_ extension as bounded operators $L^{q^{\prime}}(X)\to L^{2}(X)$
and $L^{q^{\prime}}(X)\to L^{q}(X)$. Consistency means that $R(z,A)$ coincides
with the $L^{2}$-resolvent operator on the intersection $L^{2}(X)\cap
L^{q^{\prime}}(X)$ (see [14]). By the Stone-Weierstrass theorem this implies
consistency of $m(A)$ for arbitrary bounded Borel functions $m$ (see e.g. [56,
Lemma 4.4]). We will use the notation $\Pi_{[a,b]}:=\mathbf{1}_{[a,b]}(A)$ for
$a,b\in{\mathbb{R}}$, $a<b$.

We start with a spectral multiplier estimate.

######  Lemma 3.1.

Let $m_{1},m_{2}:[0,\infty)\to{\mathbb{C}}$ be bounded Borel functions and set
$m:=m_{1}m_{2}$. Further, let $2\leq q\leq\infty$, $J\subset{\mathbb{R}}$ an
interval, and let $(\tau_{k})_{k=0}^{N}$ be a partition of $J$. Assume that

(12) $\displaystyle\sup_{0\leq k\leq
N}\|\Pi_{[\tau_{k},\tau_{k+1}]}\|_{L^{q^{\prime}}\to L^{2}}<\infty,$ (13)
$\displaystyle
M_{j}^{2}:=\sum_{k=0}^{N}\sup_{\tau\in[\tau_{k},\tau_{k+1}]}|m_{j}(\tau)|^{2}\|\Pi_{[\tau_{k},\tau_{k+1}]}\|^{2}_{L^{q^{\prime}}\to
L^{2}}<\infty,$

for $j=1,2$. Then we have the spectral multiplier estimates

(14) $\displaystyle\|\mathbf{1}_{J}(A)m_{j}(A)\|_{L^{q^{\prime}}\to L^{2}}\leq
M_{j},\quad\|\mathbf{1}_{J}(A)m(A)\|_{L^{q^{\prime}}\to L^{q}}\leq
M_{1}M_{2}.$

######  Remark 3.2.

Lemma 3.1 is similar to [6, Lemma 2.3], but improves upon the latter when
applied to the spectrally localized resolvent. The reason is that it uses the
$TT^{*}$ argument in a more efficient way and avoids the use of the triangle
inequality.

###### Proof.

For the proof we set $\Pi_{k}:=\Pi_{[\tau_{k},\tau_{k+1}]}$. In the following
we only consider functions $u\in\mathop{\rm Ran}\mathbf{1}_{J}(A)$. For
$j=1,2$ and $\delta>0$ we define the spectral multipliers

$\displaystyle
S_{j,\delta}u:=\sum_{k=0}^{N}(\sup_{\tau\in[\tau_{k},\tau_{k+1}]}|m_{j}(\tau)|+\delta)^{-1}m_{j}(A)\Pi_{k}u.$

By the spectral theorem444We will sometimes use the phrase “by orthogonality”
or “by the functional calculus” for arguments relating to the spectral
theorem., we have $\|S_{j,\delta}\|_{L^{2}\to L^{2}}\leq 1$. By orthogonality
we obtain the bound

$\displaystyle\|m_{j}(A)u\|_{L^{2}}^{2}$
$\displaystyle=\sum_{k=0}^{N}\sup_{\tau\in[\tau_{k},\tau_{k+1}]}(|m_{j}(\tau)|+\delta)^{2}\|\Pi_{k}(S_{j,\delta}u)\|_{L^{2}}^{2}$
$\displaystyle\leq\sum_{k=0}^{N}\sup_{\tau\in[\tau_{k},\tau_{k+1}]}(|m_{j}(\tau)|+\delta)^{2}\|\Pi_{k}\|_{L^{q^{\prime}}\to
L^{2}}^{2}\|u\|_{L^{q^{\prime}}}^{2}$

for $u\in L^{2}\cap L^{q^{\prime}}$. Taking $\delta\to 0$, the first bound in
(14) follows by dominated convergence and density of $L^{2}\cap
L^{q^{\prime}}$ in $L^{q^{\prime}}$. Since the assumption (13) is invariant
under complex conjugation of $m_{j}$, we obtain the same bound for
$m_{j}(A)^{*}$. The Schwarz inequality then yields

$\displaystyle|\langle m(A)u,v\rangle|=|\langle
m_{2}(A)u,m_{1}(A)^{*}v\rangle|\leq
M_{1}M_{2}\|u\|_{L^{q^{\prime}}}\|v\|_{L^{q^{\prime}}}.$

Hence, the second bound in (14) follows from duality of $L^{q}(X)$ and
$L^{q^{\prime}}(X)$ (see e.g. [14, Theorem 2.1.10]). ∎

######  Proposition 3.3.

Let $0<\epsilon\leq\lambda$. Then the following estimates hold.

(15) $\displaystyle\|\Pi_{[\lambda,\lambda+\epsilon]}\|_{L^{q^{\prime}}\to
L^{2}}\lesssim\epsilon\lambda\|\Pi_{[0,2\lambda]}\mathrm{Im}\,(A^{2}-(\lambda+\mathrm{i}\epsilon)^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{2}},$

(16)
$\displaystyle\|\Pi_{[0,2\lambda]}(A^{2}-(\lambda+\mathrm{i}\epsilon)^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{2}}\lesssim(\epsilon\lambda)^{-1}\sup_{k}\|\Pi_{[\epsilon
k,\epsilon(k+1)]}\|_{L^{q^{\prime}}\to L^{2}},$

(17) $\displaystyle\|\Pi_{[\lambda,\lambda+\epsilon]}\|^{2}_{L^{q^{\prime}}\to
L^{2}}\lesssim\epsilon\lambda\|\Pi_{[0,2\lambda]}\mathrm{Im}\,(A^{2}-(\lambda+\mathrm{i}\epsilon)^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}},$

(18)
$\displaystyle\|\Pi_{[0,2\lambda]}(A^{2}-(\lambda+\mathrm{i}\epsilon)^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim(\epsilon\lambda)^{-1}\ln\langle\lambda/\epsilon\rangle\sup_{k}\|\Pi_{[\epsilon
k,\epsilon(k+1)]}\|^{2}_{L^{q^{\prime}}\to L^{2}},$

For $\mu\gtrsim\epsilon$ and $\beta>1$ we also have

(19)
$\displaystyle\|\Pi_{[0,2\lambda]}(A^{2}-(\lambda+\mathrm{i}\mu)^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim(\epsilon\lambda)^{-1}\langle\mu/\lambda\rangle^{-1}\ln\langle\lambda/\mu\rangle\sup_{k}\|\Pi_{[\epsilon
k,\epsilon(k+1)]}\|^{2}_{L^{q^{\prime}}\to L^{2}},$

(20)
$\displaystyle\|\Pi_{[0,2\lambda]}(A^{2}-(\lambda+\mathrm{i}\mu)^{2})^{-\beta}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim_{\beta}(\epsilon\lambda)^{-\beta}(\epsilon/\mu)^{\beta-1}\langle\mu/\lambda\rangle^{-\beta}\sup_{k}\|\Pi_{[\epsilon
k,\epsilon(k+1)]}\|^{2}_{L^{q^{\prime}}\to L^{2}}.$

The suprema are taken over $k\in{\mathbb{Z}}^{+}$ with $k\leq
2\lambda/\epsilon$.

######  Remark 3.4.

The relationship between $L^{q^{\prime}}\to L^{2}$ spectral cluster and
$L^{q^{\prime}}\to L^{2}$ resolvent estimates is well known. The bound (17) is
contained in [17, Lemma 10]. Estimates similar to (20) may be found e.g. in
[33] for the Hermite operator. The bounds (18)–(19) seem to be new.

###### Proof.

To prove (15) we use that

(21)
$\displaystyle\mathrm{Im}\,(\tau^{2}-(\lambda+\mathrm{i}\epsilon)^{2})^{-1}\gtrsim(\epsilon\lambda)^{-1}\mathbf{1}_{[\lambda,\lambda+\epsilon]}(\tau).$

By the functional calculus it follows that, for $u\in L^{2}\cap
L^{q^{\prime}}$

(22)
$\displaystyle\|\Pi_{[\lambda,\lambda+\epsilon]}u\|_{L^{2}}\lesssim\epsilon\lambda\|\Pi_{[0,2\lambda]}\mathrm{Im}\,(A^{2}-(\lambda+\mathrm{i}\epsilon)^{2})^{-1}u\|_{L^{2}}.$

Hence, (15) again follows by density of $L^{2}\cap L^{q^{\prime}}$ in
$L^{q^{\prime}}$. The same argument, together with the fact that
$\Pi_{[\lambda,\lambda+\epsilon]}^{2}=\Pi_{[\lambda,\lambda+\epsilon]}$,
yields

$\displaystyle\|\Pi_{[\lambda,\lambda+\epsilon]}u\|^{2}_{L^{2}}=\langle\Pi_{[\lambda,\lambda+\epsilon]}u,u\rangle\lesssim\epsilon\lambda\langle\mathrm{Im}\,(A^{2}-(\lambda+\mathrm{i}\epsilon)^{2})^{-1}u,u\rangle,$

and this implies (17). Inequality (18) is a special case of (19) for
$\mu=\epsilon$. To prove (16), (19), (20) we use Lemma 3.1 with
$J=[0,2\lambda]$, $N=\lceil 2\lambda/\epsilon\rceil$, $\tau_{k}=\epsilon k$,
and with

(23) $\displaystyle
m_{j}(\tau)=(\tau^{2}-(\lambda+\mathrm{i}\mu)^{2}))^{-\alpha},\quad\alpha\in\\{1,1/2,\beta/2\\}$

for $j=1,2$. We will show that $m_{j}$ is smooth on the $\epsilon$-scale and
bound the sum (13) in the definition of $M_{j}$ by a Darboux (Riemann)
integral. The required smoothness is captured by the bound

(24) $\displaystyle\sup_{\tau\in[\epsilon
k,\epsilon(k+1)]}|m_{j}(\tau)|^{2}\lesssim\inf_{\tau\in[\epsilon
k,\epsilon(k+1)]}|m_{j}(\tau)|^{2},$

with an implied constant that is uniform in $\lambda,\epsilon,\mu$. This holds
since, for $\tau\in[\epsilon k,\epsilon(k+1)]$, we have
$|\tau+(\lambda+\mathrm{i}\mu)|\approx\lambda$ and
$|\tau-(\lambda+\mathrm{i}\mu)|\approx|\epsilon k-\lambda|+\mu$. The first
bound is obvious, and so is the upper bound for the second. We also have the
lower bound $||\epsilon k-\lambda|-\epsilon|+\mu$ for the second. We only need
to consider the case $|\epsilon k-\lambda|\geq C\mu$ since the other case is
again obvious. If $\mu\geq C^{\prime}\epsilon$, we get the lower bound
$|\epsilon k-\lambda|(1-1/(CC^{\prime}))$. Note that $C^{\prime}$ is fixed,
and we may choose $C=1/(2C^{\prime})$, establishing the desired lower bound.

Consider the partition $P$ of $[0,(N+1)\epsilon]\supset[0,2\lambda]$ given by
$\tau_{0}<\tau_{1}\ldots<\tau_{N+1}$ and define the Darboux sums

$\displaystyle s(P)=\epsilon\sum_{k=0}^{N}\inf_{\tau\in[\epsilon
k,\epsilon(k+1)]}|m_{j}(\tau)|^{2},\quad
S(P)=\epsilon\sum_{k=0}^{N}\sup_{\tau\in[\epsilon
k,\epsilon(k+1)]}|m_{j}(\tau)|^{2}.$

Obviously, we have

(25) $\displaystyle s(P)\leq\int_{0}^{(N+1)\epsilon}|m_{j}(\tau)|^{2}{\rm
d}\tau\leq S(P).$

On the other hand, by (24), we also have the converse inequalities, up to
constants. Hence,

$\displaystyle
S(P)\lesssim\int_{0}^{4\lambda}|\tau^{2}-(\lambda+\mathrm{i}\mu)^{2})|^{-2\alpha}{\rm
d}\tau\lesssim(\lambda+\mu)^{-2\alpha}\mu^{1-2\alpha}\ln^{\nu}\langle\lambda/\mu\rangle,$

with $\nu=1$ if $\alpha=1/2$ and $\nu=0$ if $\alpha>1/2$. Since, by definition
of $M_{j}$ and $S(P)$,

(26) $\displaystyle M_{j}^{2}\leq\epsilon^{-1}S(P)\sup_{1\leq k\leq
N}\|\Pi_{k}\|^{2}_{L^{q^{\prime}}\to L^{2}},$

the claims follows from (14). ∎

In most applications, the spectral localization can easily be removed by
Sobolev embedding and elliptic estimates. Moreover, the spectral cluster
bounds usually take the form (27). For convenience we record the following
corollary.

######  Corollary 3.5.

Let $\epsilon=\epsilon(\lambda)$ and $\delta=\delta(\lambda)$ be such that
$0<\delta,\epsilon\leq\lambda$. The following bounds are equivalent.

  * (a)

The spectral cluster bound:

(27) $\displaystyle\|\Pi_{[\lambda,\lambda+\epsilon]}\|_{L^{2}\to
L^{q}}\lesssim\delta\langle\lambda\rangle^{\gamma(q)}.$

  * (b)

The spectrally localized $L^{2}\to L^{q}$ resolvent bound:

(28) $\displaystyle\|\Pi_{\leq
2\lambda}(A^{2}-(\lambda+\mathrm{i}\epsilon)^{2})^{-1}\|_{L^{2}\to
L^{q}}\lesssim\epsilon^{-1}\delta\langle\lambda\rangle^{\gamma(q)-1}.$

  * (c)

The quasimode bound:

(29)
$\displaystyle\|u\|_{L^{q}}\lesssim\delta\langle\lambda\rangle^{\gamma(q)}\|u\|_{L^{2}}+\epsilon^{-1}\delta\langle\lambda\rangle^{\gamma(q)-1}\|(A^{2}-\lambda^{2})u\|_{L^{2}},$

for all functions $u$ spectrally localized in $[0,2\lambda]$.

Moreover, if $\mu=\mu(\lambda)$, $\mu\gtrsim\epsilon$, these bounds imply the
spectral cluster estimate

(30) $\displaystyle\|\Pi_{[\lambda,\lambda+\mu]}\|_{L^{2}\to
L^{q}}\lesssim(\mu/\epsilon)^{1/2}\delta\langle\lambda\rangle^{\gamma(q)}$

and the spectrally localized $L^{q^{\prime}}\to L^{q}$ resolvent estimate

(31) $\displaystyle\|\Pi_{\leq
2\Lambda}(A^{2}-(\lambda+\mathrm{i}\mu)^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\epsilon^{-1}\langle\mu/\lambda\rangle^{-1}\ln\langle\lambda/\mu\rangle\delta^{2}\lambda^{2\gamma(q)-1}.$

###### Proof.

The equivalence of $(a)$, $(b)$ and (31) follow from Proposition 3.3. The
implication $(b)\implies(c)$ follows from the triangle inequality, and
$(c)\implies(a)$ follows from

$\displaystyle\|(A^{2}-\lambda^{2})\Pi_{[\lambda,\lambda+\epsilon]}u\|_{L^{2}}\lesssim\epsilon\lambda\|u\|_{L^{2}}.$

The spectral cluster estimate (30) follows from (27) and orthogonality. ∎

######  Remark 3.6.

The bounds (17), (31) show that if the spectral cluster estimate (27) is
sharp, then the resolvent estimate (31) with $\mu=\epsilon$ will be sharp up
to a logarithm. It also shows that there is a separation of scales at
$\mu=\lambda$, at which point the logarithm becomes bounded. This corresponds
to an elliptic estimate outside a cone. It is slightly less obvious that (31)
is almost sharp for intermediate values $\epsilon\lesssim\mu\lesssim\lambda$.
To see this, one uses orthogonality and the assumed sharpness of (27) to get a
lower bound

$\displaystyle\|\Pi_{[\lambda,\lambda+\mu]}\|_{L^{2}\to
L^{q}}\gtrsim(\mu/\epsilon)^{1/2}\lambda^{\gamma(q)}.$

Then the same argument leading to (22) with $\mu$ in place of $\epsilon$
yields the conclusion.

######  Remark 3.7.

In most applications $\delta(\lambda)$ will be a power of $\lambda$ and hence
could be omitted by redefining $\gamma(q)$. However, in some cases
$\delta(\lambda)=\ln^{\nu}\langle\lambda\rangle$ for some
$\nu\in{\mathbb{R}}$, see e.g. (10). For the Laplace–Beltrami operator on a
compact boundaryless manifold, corresponding to $A=\sqrt{-\Delta_{g}}$, the
relevant ranges for smooth metrics are $\epsilon\in[\lambda^{-1},1]$ and
$\delta\in[\epsilon(\lambda)^{1/2},1]$, with
$\delta(\lambda),\epsilon(\lambda)$ monotonically decreasing [24]. The
universal estimates of Sogge [48] correspond to $\delta,\epsilon=1$ and
$\gamma(q)=\sigma(q)$. In the Euclidean case, the Tomas–Stein bound (53)
corresponds to $\delta=\epsilon^{1/2}$ and $\epsilon>0$.

##  4\. From spectral cluster to resolvent bounds

We now consider a setting where the logarithmic loss in (31) can be removed.
We assume that $X$ is a smooth compact Riemannian manifold $(M,g)$ without
boundary and $A=\sqrt{-\Delta_{g}}$. The methods we use require Lipschitz
regularity of the metric. This condition is natural for the paradifferential
reductions to a first order frequency localized equation (see [41, 31] and
especially Step 1 in the proof of [30, Theorem 2.1]).

Since we will be concerned mainly with rough metrics here we focus on
universal bounds ($\epsilon=1$); in fact, it will be advantageous to allow
$\epsilon\geq 1$. For smooth metrics, one can follow the proof of Bourgain,
Shao, Sogge and Yao [6, Theorem 1.3] to show that improved spectral cluster
bounds ($\epsilon=o(1)$ as $\lambda\to\infty$) imply improved resolvent
bounds; this was done for $q=2^{*}$ in [6]. The proof of [6, Theorem 1.3] uses
the Hadamard parametrix for the wave equation, which is not available under
our limited smoothness assumptions. We therefore include the argument for
improved bounds, circumventing the Hadamard parametrix and extending the
result of [6] to all $2\leq q\leq 2^{*}$.

###  4.1. Symbol classes

We will work with the following symbol classes (c.f. [30]).

######  Definition 4.1.

Let $a(t,x,\xi,\lambda)$ be a family of smooth functions in $(x,\xi)\in
T^{*}{\mathbb{R}}^{d}$, depending on parameters $t\in{\mathbb{R}}$ and
$\lambda\geq 1$. For $\kappa\in[0,1]$, we say $a(t,x,\xi,\lambda)\in
S_{\lambda,\lambda^{\kappa}}$ if, for all multi-indices $\alpha,\beta$,

(32)
$\displaystyle|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(t,x,\xi,\lambda)|$
$\displaystyle\lesssim_{\alpha,\beta}\lambda^{-|\beta|+\kappa|\alpha|}.$

We say $a(t,x,\xi,\lambda)\in C^{k}S_{\lambda,\lambda^{\kappa}}$ if the
stronger bound

(33)
$\displaystyle|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(t,x,\xi,\lambda)|$
$\displaystyle\lesssim_{\alpha,\beta}\lambda^{-|\beta|+\kappa(|\alpha|-k)_{+}}$

holds, where $x_{+}=\max(x,0)$.

In the proof of Theorem 2.1 we will only consider the case $k=1$, $\kappa=1$.
We will also set $d=n-1$ and consider $t$ as one of the spatial coordinates.
Our symbols will always be localized to $|x|\lesssim 1$ and
$|\xi|\lesssim\lambda$. Henceforth, we will suppress the dependence on
$\lambda$. As remarked in [30], since $a\in S_{\lambda,\lambda}$ is simply a
bounded family of zero order symbols in the Hörmander class $S^{0}_{0,0}$,
rescaled by $(t,x,\xi)\mapsto(\lambda t,\lambda x,\xi/\lambda)$, the Kohn-
Nirenberg quantization $a(t,x,D_{x})$ and the Weyl quantization
$a^{w}(t,x,D_{x})$ are $L^{2}$ bounded (with uniform bounds in $t$ and
$\lambda$) [26]. We will use the Weyl quantization here. The asymptotic laws
for composition and adjoint for symbols $a,b\in C^{1}S_{\lambda,\lambda}$ hold
to first order, viz.

(34) $\displaystyle
a^{w}(t,x,D_{x})b^{w}(t,x,D_{x})=(ab)^{w}(t,x,D_{x})+\lambda^{-1}R_{1},$ (35)
$\displaystyle
a^{w}(t,x,D_{x})^{*}=(\overline{a})^{w}(t,x,D_{x})+\lambda^{-1}R_{2},$

where the errors $R_{1},R_{2}$ are quantizations of symbols in
$S_{\lambda,\lambda}$ and hence bounded in $L^{2}({\mathbb{R}}^{d})$.

###  4.2. A generalization of Theorem 2.1

######  Theorem 4.2.

Let $(M,g)$ be a compact Riemannian manifold without boundary and with
Lipschitz metric $g$. Assume that $2\leq q\leq 2^{*}$, and that the spectral
cluster estimate (27) holds for some $\gamma(q)\in[\sigma(q),\sigma(q)+1/2]$
and $0<\delta,\epsilon\leq\lambda$. Then

(36)
$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i}\epsilon)^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\epsilon^{-1}\delta^{2}\lambda^{2\gamma(q)-1}.$

###### Proof.

We first consider the case $\epsilon\geq 1$, and use Corollary 3.5 to see that
the assumed spectral cluster bound (6) implies the $L^{2}\to L^{q}$ bound
(28). By the dual of this, it is enough to show that, for $u$ a solution to
the Helmholtz equation555We thus consider the resolvent equation
$(\Delta_{g}+(\lambda+\mathrm{i}\epsilon)^{2})u=\lambda f$. We multiplied the
forcing term by $\lambda$ so that, after localizing in frequency, (37) turns
into a first order equation with a forcing term of unit size.

(37) $\displaystyle(\Delta_{g}+\lambda^{2})u=\lambda
f-2\mathrm{i}\epsilon\lambda u+\epsilon^{2}u,$

we have

(38)
$\displaystyle\|u\|_{L^{q}}\lesssim\delta\lambda^{\gamma(q)}(\|u\|_{L^{2}}+\epsilon^{-1}\delta\lambda^{\gamma(q)}\|f\|_{L^{q^{\prime}}}).$

Since $M$ is compact, by a partition of unity subordinate to suitable local
coordinates, we may assume without loss of generality that $u$ is supported in
a small coordinate patch. Note that the $L^{2}$ norm of the localization error
$[-\Delta_{g},\phi]u$ (where $\phi$ is a localizing function and
$[\cdot,\cdot]$ is the commutator) is of order $\lambda\|u\|_{L^{2}}$ and can
thus be absorbed into the forcing term (the right hand side of (37)). It is
this step that is problematic for $\epsilon<1$ since we would need the better
error bound $\epsilon\lambda\|u\|_{L^{2}}$. We choose coordinate patches in
such a way that, in local coordinates, $g$ is close to the Euclidean metric in
Lipschitz norm after extending $g$ to all of ${\mathbb{R}}^{n}$. From now on
we will thus consider (37) as an equation on ${\mathbb{R}}^{n}$ and assume
that $u,f$ are supported in a unit cube.

The characteristic set of the Helmholtz operator $\Delta_{g}+\lambda^{2}$ is
contained in the phase space region $B_{\lambda}=\\{|x|\lesssim
1,\,|\xi|\lesssim\lambda\\}$. As already pointed out in the introduction,
since stronger elliptic estimates hold in the noncharacteristic region (see
e.g. [41, Corollary 5]), we assume from now on that $u$ is microlocalized to
$B_{\lambda}$. This means that, for some bump function $\chi$ adapted to
$B_{\lambda}$, we have $u=\chi^{w}(x,D)u$ modulo negligible errors. More
precisely, this means that
$u=\chi^{w}(x,D)u+\mathcal{O}_{L^{2}}(\lambda^{-N}\|u\|_{L^{2}})$ for any
$N>0$. By a conical partition of unity, we may actually assume that $u$ is
Fourier localized to the conical region $|\xi_{1}-\lambda|\lesssim\lambda$,
$|\xi^{\prime}|\ll\lambda$. Then (37) turns into a first order equation

(39)
$\displaystyle(D_{x_{1}}-a_{\lambda}^{w}(x_{1},x^{\prime},D_{x^{\prime}}))u=\widetilde{\chi}^{w}(x,D)f+\epsilon
Ru,$

where $a_{\lambda}$ is a real-valued (positive) symbol, $\widetilde{\chi}\in
S_{\lambda,\lambda}$ is supported in $B_{\lambda}$, and $R$ denotes a generic
operator with $L^{2}\to L^{2}$ norm of unit size. The reduction to (39) is
standard (see e.g. [41, 30]) but we provide a few details for clarity of
exposition. First observe that a conical localization $\Gamma(D_{x})$ in
Fourier space produces an error that may again be absorbed into the forcing
term since $[\Gamma(D_{x}),g^{ij}]=\mathcal{O}_{L^{2}\to
L^{2}}(\lambda^{-1})$. Indeed, from the formula

$\displaystyle[\Gamma(D_{x}),g^{ij}]u(x)=\int\Gamma^{\vee}(x-y)(g^{ij}(x)-g^{ij}(y))u(y){\rm
d}y,$

where $\Gamma^{\vee}$ denotes the inverse Fourier transform of $\Gamma$, using
Hölder’s inequality, Fubini’s and Plancherel’s theorem and the fact that
$|\Gamma^{\vee}(x-y)(x_{i}-y_{i})|^{2}=|(\partial_{\xi_{i}}\Gamma)^{\vee}(x-y)|^{2}$,
it follows that

$\displaystyle\|[\Gamma(D_{x}),g^{ij}]u\|_{L^{2}}^{2}\lesssim\|\partial_{\xi}\Gamma\|_{L^{2}}^{2}\|u\|_{L^{2}}^{2}.$

The implicit constant depends on the Lipschitz norm of $g$, and the
$y$-integral disappeared (yields only a constant factor) since we are
assuming, as we may, that $g$ coincides with the Euclidean metric outside a
compact set. The next step replaces $g$ by a smoothed metric $g_{\lambda}$,
which is obtained from $g$ by smoothly truncating $\widehat{g}(\xi)$ to
frequencies $|\xi|\ll\lambda$. Since

$\displaystyle\|g-g_{\lambda}\|_{L^{\infty}({\mathbb{R}}^{n})}\lesssim\lambda^{-1},\quad\|\nabla(g-g_{\lambda})\|_{L^{\infty}({\mathbb{R}}^{n})}\lesssim
1,$

the smoothing error may again be absorbed into the forcing term. Since $g$ is
pointwise close to the flat metric we have $g_{\lambda}^{11}(x)>0$, and we may
factor

(40)
$\displaystyle\lambda^{2}-\sum_{i,j=1}^{n}g_{\lambda}^{ij}(x)\xi_{i}\xi_{j}=g_{\lambda}^{11}(x)(\xi_{1}+\widetilde{a}(x,\xi^{\prime},\lambda))(\xi_{1}-a(x,\xi^{\prime},\lambda)),$

where $\widetilde{a},a>0$ are symbols in $\lambda C^{1}S_{\lambda,\lambda}$.
This holds since each derivative of $g_{\lambda}$ beyond the first costs a
factor of $\lambda$. Since the left hand side of (40) equals the principal
symbol of $\lambda^{2}+\Delta_{g}$, the first order calculus (34) yields

$\displaystyle\lambda^{2}+\Delta_{g}=g_{\lambda}^{11}(x)(D_{x_{1}}+\widetilde{a}(x,D_{x^{\prime}},\lambda))(D_{x_{1}}-a(x,D_{x^{\prime}},\lambda))+r(x,D_{x})$

for some $r\in\lambda S_{\lambda,\lambda}$. Owing to the conical frequency
localization, $(D_{x_{1}}+\widetilde{a}(x,D_{x^{\prime}},\lambda))$ admits a
parametrix $Q$ with symbol in $q\in\lambda^{-1}S_{\lambda,\lambda}$. Then we
obtain

$\displaystyle(D_{x_{1}}-a^{w}(x_{,}D_{x^{\prime}},\lambda))u=\lambda
Qf+\epsilon Ru,$

which is already close to (39). We now apply $\chi^{w}(x,D)$ to both sides and
absorb the commutator error into the second term on the right. This gives us
(39) with $\widetilde{\chi}=\lambda\chi q$, which is in $S_{\lambda,\lambda}$
and supported on $B_{\lambda}$, as desired. By the same argument as before we
may replace $a(x,\xi^{\prime},\lambda)$ by a smoothed symbol
$a_{\lambda}(x,\xi^{\prime})$ whose $x$-Fourier transform is supported at
frequencies $|\xi|\ll\lambda$, and $a_{\lambda}\in\lambda
C^{1}S_{\lambda,\lambda}$; the error can be adsorbed into the second term on
the right of (39). We may also extend $a_{\lambda}$ globally so that
$a_{\lambda}(x,\xi^{\prime})=\lambda$ for $|\xi|\gg\lambda$, while preserving
the frequency localization in $x$. This completes the reduction of (37) to the
first order equation (39). In addition, we have shown that we may consider
(39) as an equation on ${\mathbb{R}}^{n}$, with $a_{\lambda}$ having the
properties just mentioned.

To proceed, we set $x_{1}=t$ and use coordinates
$(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{n-1}$ instead of
$(x_{1},x^{\prime})$. The dual coordinates are denoted by $(\tau,\xi)$.
Assuming, as we are, that the spectral cluster estimate (27), and hence the
quasimode estimate (29), hold, we can reverse the steps in the above
reduction666In fact, (29) with $A=\sqrt{-\Delta_{g}}$ and (41) are equivalent
and differ only by an elliptic estimate, see [31, Lemma 3.5] (which also holds
for the symbols considered here). to obtain the microlocalized quasimode
estimate

(41)
$\displaystyle\|u\|_{L_{t,x}^{q}}\lesssim\delta\lambda^{\gamma(q)}(\|u\|_{L_{t,x}^{2}}+\epsilon^{-1}\|(D_{t}-a_{\lambda}^{w}(t,x,D_{x}))u\|_{L^{2}_{t,x}}).$

Then (38) is a consequence of Lemma 4.3 below. There we will assume that
$\widetilde{\chi}$ is not just in $S_{\lambda,\lambda}$ but is a bump
function. To justify this assumption we need that operators with
$S_{\lambda,\lambda}$ symbols are $L^{p}$ bounded for $1<p<\infty$. Since we
already know that they are $L^{2}$ bounded, this follows from Calderon–Zygmund
theory since their Schwartz kernels satisfy standard estimates. In fact, after
rescaling $(t,x,\xi)\mapsto(\lambda t,\lambda x,\xi/\lambda)$, it suffices
estimate the kernel $K$ of an operator corresponding to an $S_{0,0}$ symbol,
localized to frequencies $|\xi|\lesssim 1$; this satisfies the trivial bound
$|K(x,y)|+|\nabla_{x,y}K(x,y)|\lesssim 1$.

We now turn to the case $\epsilon<1$. To this end, we partially follow the
strategy of [6] and represent the resolvent operator as a cosine transform,

$\displaystyle(\Delta_{g}+(\lambda+\mathrm{i}\epsilon)^{2})^{-1}f=\frac{1}{\mathrm{i}(\lambda+\mathrm{i}\epsilon)}\int_{0}^{\infty}{\rm
e}^{\mathrm{i}\ell\lambda-\ell\epsilon}\cos(\ell\sqrt{-\Delta_{g}})f{\rm
d}\ell.$

We will also need the localized version

$\displaystyle\mathfrak{S}_{\rm
loc}f=\frac{1}{\mathrm{i}(\lambda+\mathrm{i}\epsilon)}\int_{0}^{\infty}\rho(\ell){\rm
e}^{\mathrm{i}\ell\lambda-\ell\epsilon}\cos(\ell\sqrt{-\Delta_{g}})f{\rm
d}\ell,$

where $\rho$ is a bump function adapted to $[-\delta_{0},\delta_{0}]$, with
$\delta_{0}\ll 1$ smaller than half the injectivity radius of $(M,g)$. If $u$
satisfies the Helmholtz equation (37), we set $u_{\rm
loc}=\lambda\mathfrak{S}_{\rm loc}f$. An easy integration by parts argument
then shows that

$\displaystyle(\Delta_{g}+(\lambda+\mathrm{i}\epsilon)^{2})u_{\rm loc}=\lambda
f+\lambda\,r_{\rm loc},$

where

$\displaystyle r_{\rm
loc}=\int_{0}^{\infty}(\rho^{\prime}(\ell)-\mathrm{i}(\lambda+\mathrm{i}\epsilon)^{-1}\rho^{\prime\prime}(\ell)){\rm
e}^{\mathrm{i}\ell\lambda-\ell\epsilon}\cos(\ell\sqrt{-\Delta_{g}})f{\rm
d}\ell$

is readily estimated in $L^{2}$ norm by $\|f\|_{L^{2}}$. Again, by Corollary
3.5, we have the following quasimode estimate for $u_{\rm loc}$,

$\displaystyle\|u_{\rm
loc}\|_{L^{q}}\lesssim\epsilon^{-1}\delta\langle\lambda\rangle^{\gamma(q)}\|u_{\rm
loc}\|_{L^{2}}.$

If we assume, as we may, that $f$ is supported in a suitably small coordinate
patch, then the finite propagation speed of $\cos(\ell\sqrt{-\Delta_{g}})$
localizes $u_{\rm loc}$ to the same patch. We have thus avoided the
localization error for $u_{\rm loc}$ and can repeat the previous arguments to
get the correct bound

$\displaystyle\|u_{\rm
loc}\|_{L^{q}}\lesssim\epsilon^{-1}\delta^{2}\lambda^{2\gamma(q)}\|f\|_{L^{q^{\prime}}}.$

Note that by the support properties of $u_{\rm loc}$ we are allowed to apply
Lemma 4.3 for arbitrarily long time intervals. The remaining part $u-u_{\rm
loc}$ can be estimated exactly as in [6]. Since the argument there is only
given for $q=2n/(n-2)$, $n\geq 3$, we provide some details and simplifications
for the case considered here. By definition, $u-u_{\rm
loc}=m(\sqrt{-\Delta_{g}})f$, where $m$ is the multiplier

$\displaystyle
m(\tau)=\frac{\lambda}{\mathrm{i}(\lambda+\mathrm{i}\epsilon)}\int_{0}^{\infty}(1-\rho(\ell)){\rm
e}^{\mathrm{i}\ell\lambda-\ell\epsilon}\cos(\ell\tau){\rm d}\ell.$

The main observation is that, since $1-\rho(\ell)$ vanishes near the origin,
there are no boundary terms and we can integrate by parts as many times as we
please. Using Euler’s formula for the cosine function, we deduce that for
arbitrary $N\in{\mathbb{N}}$, we have

$\displaystyle|m(\tau)|\lesssim_{N}\langle\tau-\lambda\rangle^{-N},$

where we used that derivatives are uniform in $\epsilon<1$. We now use Lemma
3.1 with $\tau_{k},J,N$ as in the proof of Proposition 3.3 and with $m_{1}=m$,
$m_{2}=1$. By (25)–(26), we have

$\displaystyle
M_{1}\lesssim\epsilon^{-1}\delta^{2}\langle\lambda\rangle^{2\gamma(q)-1},\quad
M_{2}\lesssim\epsilon^{-1}\delta^{2}\langle\lambda\rangle^{2\gamma(q)+1},$

which implies that

$\displaystyle\|u-u_{\rm
loc}\|_{L^{q}}\lesssim\epsilon^{-1}\delta^{2}\lambda^{2\gamma(q)}\|f\|_{L^{q^{\prime}}}.$

This completes the proof in the case $\epsilon<1$. ∎

######  Lemma 4.3.

Assume that $a_{\lambda}\in C^{1}S_{\lambda,\lambda}$ and $\epsilon>0$. Let
$\chi,\widetilde{\chi}$ be bump functions adapted to $|x|\lesssim 1$,
$|\xi|\lesssim\lambda$, and assume that $u$ solves

(42)
$\displaystyle(D_{t}-a_{\lambda}^{w}(t,x,D_{x}))u=f_{1}+\widetilde{\chi}^{w}(x,D_{x})f_{2},\quad
u(0)=u_{0}$

in $[0,\epsilon^{-1}]\times{\mathbb{R}}^{n-1}$. Then (41) implies

(43)
$\displaystyle\|\chi^{w}(x,D_{x})u\|_{L_{t,x}^{q}}\lesssim\epsilon^{-1/2}\delta\lambda^{\gamma(q)}(\|u_{0}\|_{L_{x}^{2}}+\|f_{1}\|_{L_{t}^{1}L_{x}^{2}}+\epsilon^{-1/2}\delta\lambda^{\gamma(q)}\|f_{2}\|_{L_{t,x}^{q^{\prime}}}),$

with norms over $(t,x)\in[0,\epsilon^{-1}]\times{\mathbb{R}}^{n-1}$.

###### Proof.

Let $S(t,s):L^{2}({\mathbb{R}}^{n-1})\to L^{2}({\mathbb{R}}^{n-1})$ be the
unitary propagators associated to $a_{\lambda}^{w}$. Note that, due to the
frequency localization, we can arrange that $|a_{\lambda}|\lesssim\lambda$.
Therefore, $a_{\lambda}^{w}$ is a bounded self-adjoint operator. We then have
Duhamel’s formula

$\displaystyle u(t)=S(t,0)u_{0}+\mathrm{i}\int_{0}^{t}S(t,s)f_{1}(s){\rm
d}s+\mathrm{i}\int_{0}^{t}S(t,s)\widetilde{\chi}^{w}(x,D_{x})f_{2}(s){\rm
d}s.$

We apply (41) to $u(t,x)=\psi(\epsilon t)S(t,0)\chi^{w}(x,D_{x})u_{0}(x)$,
where $\psi$ is adapted to $[0,1]$. This is permitted since the flow $S(t,0)$
preserves the frequency localization, due to the truncation of the symbol to
$x$-frequencies $\ll\lambda$; see the comment after (2.4) in [30]. The result
is that

(44)
$\displaystyle\|S(t,0)\chi(x,D_{x})u_{0}\|_{L_{t,x}^{q}}\lesssim\epsilon^{-1/2}\delta\lambda^{\gamma(q)}\|u_{0}\|_{L_{x}^{2}}.$

Here we used energy conservation
$\|S(t,0)u_{0}\|_{L^{\infty}_{t}L^{2}_{x}}\leq\|u_{0}\|_{L^{2}_{x}}$ and
Hölder’s inequality, as well as the fact that the $L^{2}$ norm of
$\psi(\epsilon\cdot)$ is of order $\epsilon^{-1/2}$. This yields (43) in the
case $f_{2}=0$. By a standard $TT^{*}$ argument and the group property of the
propagator the latter implies

(45)
$\displaystyle\big{\|}\int_{0}^{\epsilon^{-1}}\chi(x,D_{x})S(t,s)\widetilde{\chi}(x,D_{x})F(s){\rm
d}s\big{\|}_{L_{t,x}^{q}}\lesssim\epsilon^{-1}\delta^{2}\lambda^{2\gamma(q)}\|F\|_{L_{t,x}^{q^{\prime}}}.$

An application of the Christ–Kiselev lemma [10], as formulated e.g. in [46,
52], yields the same bound for the forward propagator $\mathbf{1}_{t\geq
s}S(t,s)$. Then (43) follows from Duhamel’s formula. ∎

######  Remark 4.4.

Lemma 4.3 (with $\delta,\epsilon=1$) is similar to [31, Proposition 4.8]. The
first difference is that we omitted the curvature condition (A3) in [31] and
replaced it by the black box assumption that the spectral cluster bound (6)
holds. In contrast, the real principal type condition (A2) in [31] is crucial
for the reduction to the first order equation (39). In the context of Theorem
2.1, the condition (A3) is of course also satisfied: The fibers of the
cosphere bundle $|\xi|_{g}=1$ have everywhere nonzero Gaussian curvature. But
this information is now encoded in the spectral cluster bound. The second
difference is that [31, Proposition 4.8] applies only to symbols
$a_{\lambda^{1/2}}$ in the smaller class $C^{2}S_{\lambda,\lambda^{1/2}}$, at
which one would arrive, after similar reductions as in the proof of Theorem
2.1, if one had started with a $C^{1,1}$ metric. Lipschitz functions are well
approximated by $C^{1,1}$ functions on the $\lambda^{-1/3}$ scale. On this
scale the wave packet parametrix constructions work777Koch and Tataru [31] use
Gaussian coherent states (the FBI transform), while Smith [42] uses states
with compact Fourier support., and one can prove the spectral cluster
(homogeneous Stricharz) and resolvent (inhomogeneous Strichartz) bounds
concurrently. These follow from dispersive ($L^{1}\to L^{\infty}$) estimates
on the regularized flow corresponding to $a_{\lambda^{1/2}}$. For such
estimates, the causal factor $\mathbf{1}_{t\geq s}$ is innocuous. On the other
hand, the flow of $a_{\lambda}$ in Lemma 4.3 is not regular enough for the
dispersive estimates to hold.

###  4.3. Stability under perturbations

We will show that the resolvent estimates are stable under perturbations
$V=V(\lambda)$ with the mapping properties

(46) $\displaystyle V:W_{\lambda}^{\frac{1}{2},2}\cap W_{\lambda}^{s(q),q}\to
W_{\lambda}^{-\frac{1}{2},2}+W_{\lambda}^{-s(q),q^{\prime}},$

Here, $1-s(q)=n(1/2-1/q)$ and $W_{\lambda}^{s,p}$ are Sobolev spaces flattened
at frequency $\lambda$,

$\displaystyle\|u\|_{W_{\lambda}^{s,p}}=\|(\lambda^{2}-\Delta_{g})^{s/2}u\|_{L^{p}}.$

We assume that we have perfect estimates for the exponent $q$ under
consideration, i.e.

(47)
$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\lambda^{2\sigma(q)-1},$

where we recall that $\sigma(q)=n(1/2-1/q)-1/2$ for $q\geq q_{n}$. This makes
the the estimates compatible with Sobolev embedding. Note that the spaces in
(46) are isomorphic as $\lambda$ varies, but we ask for uniform bounds in
$\lambda$. More precisely, we will assume that

(48) $\displaystyle
M(\Lambda):=\sup_{\lambda\geq\Lambda}\|V(\lambda)\|_{X(\lambda)\to
X^{\prime}(\lambda)}<\infty$

for some (and hence all) $\Lambda\geq 1$. Here we introduced the spaces

$\displaystyle X(\lambda)=W_{\lambda}^{\frac{1}{2},2}\cap
W_{\lambda}^{s(q),q},\quad
X^{\prime}(\lambda)=W_{\lambda}^{-\frac{1}{2},2}+W_{\lambda}^{-s(q),q^{\prime}}.$

To prove the stability we first add in the $L^{2}$ estimates in the
$L^{q^{\prime}}\to L^{q}$ resolvent bound.

######  Lemma 4.5.

Assume that (47) holds. Then we have

$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}u\|_{\lambda^{-1/2}L^{2}\cap\lambda^{\sigma(q)-1/2}L^{q}}\lesssim\|u\|_{\lambda^{1/2}L^{2}+\lambda^{-\sigma(q)+1/2}L^{q^{\prime}}}.$

###### Proof.

The $L^{q^{\prime}}\to L^{q}$ bound holds by assumption. The $L^{2}\to L^{q}$
bound is equivalent to the spectral cluster bound (Corollary 3.5) and hence
follows from the $L^{q^{\prime}}\to L^{q}$ bound (Proposition 3.3). The
$L^{q}\to L^{2}$ bound then follows by duality. The $L^{2}\to L^{2}$ bound
follows from the spectral theorem. ∎

Next we add in the elliptic $W_{\lambda}^{-1,2}\to W_{\lambda}^{1,2}$
estimates, similar to [32, Theorem 7], to obtain an even stronger version.

######  Lemma 4.6.

Assume that (47) holds. Then there exists $C_{0}>0$ such that

(49)
$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}\|_{X^{\prime}(\lambda)\to
X(\lambda)}\leq C_{0}.$

###### Proof.

This follows from the previous lemma together with the Sobolev embeddings

(50) $\displaystyle W_{\lambda}^{1,2}\subset W_{\lambda}^{s(q),q},\quad
W_{\lambda}^{-s(q),q^{\prime}}\subset W_{\lambda}^{-1,2},$

and the observation that for functions localized to frequency
$\lesssim\lambda$, we have

$\displaystyle\|u\|_{W_{\lambda}^{s(q),q}}\lesssim\|u\|_{\lambda^{-s(q)}L^{q}},\quad\|u\|_{\lambda^{s(q)}L^{q^{\prime}}}\lesssim\|u\|_{W_{\lambda}^{-s(q),q^{\prime}}}.$

Note again that $s(q)=-\sigma(q)+1/2$. ∎

######  Remark 4.7.

The $L^{2}$ based bounds allow one to handle gradient perturbations or
perturbations of the metric, similarly to [32], [25], [11].

We will assume that for some $c\in(0,1)$,

(51) $\displaystyle\Lambda_{0}=\inf\\{\Lambda\geq 1:\,M(\Lambda)\leq
c\,C_{0}^{-1}\\}<\infty.$

######  Proposition 4.8.

Assume that (49) holds and that $V=V(\lambda)$ is a family of perturbations
satisfying (48), (51). Then

(52)
$\displaystyle\|(\Delta_{g}+V+(\lambda+\mathrm{i})^{2})^{-1}\|_{X^{\prime}(\lambda)\to
X(\lambda)}\leq C_{0}(1-c)^{-1}$

for all $\lambda\geq\Lambda_{0}$.

###### Proof.

This is an immediate consequence of Lemma 4.6, the second resolvent formula

$\displaystyle(\Delta_{g}+V+(\lambda+\mathrm{i})^{2})^{-1}=(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}(\mathbf{1}+V(\Delta_{g}+V+(\lambda+\mathrm{i})^{2})^{-1})$

and a Neumann series argument. ∎

The estimate (52) is effective in the sense that it is valid for a whole
family of perturbations satisfying appropriate bounds. We are now turning our
attention to non-effective bounds, i.e. we fix a perturbation $V$ and allow
the constants in the estimates depend on $V$ itself. We assume that (46) holds
and take norms for fixed $\lambda$, say $\lambda=1$. For simplicity we only
consider zero order perturbations (potentials). That is, we assume that $V\in
L^{p}(M)$ with $1/p=1/q^{\prime}-1/q$. Since $M$ is compact, we have the
inclusion $L^{p_{2}}(M)\subset L^{p_{1}}(M)$ for $p_{2}\geq p_{1}$.

###### Proof of Theorem 2.1.

If $n=2$ or $n=3$ and $q<2^{*}$, then by Lemma 4.5 we have

$\displaystyle M(\Lambda)\leq\Lambda^{2\sigma(q)-1}\|V\|_{L^{p}}.$

If we fix $c=1/2$ in the definition of $\Lambda_{0}$, as we do now, this
implies that

$\displaystyle\Lambda_{0}\leq(2C_{0}\|V\|_{L^{p}})^{\frac{1}{1-2\sigma(q)}}.$

Note that $1-2\sigma(q)>0$ here. The claim then follows from Proposition 4.8.

Now consider the case $n\geq 3$ and $q=2^{*}$, so that $2\sigma(q)-1=0$. For
$\alpha_{0}>0$ we set

$\displaystyle V_{>\alpha_{0}}=V\mathbf{1}_{\\{x\in
M:\,|V(x)|>\alpha_{0}\\}},\quad V_{\leq\alpha_{0}}=V-V_{>\alpha_{0}}.$

Since $V\in L^{n/2}$ we may choose $\alpha_{0}$ so large that
$\|V_{>\alpha_{0}}\|_{L^{n/2}}\leq C_{0}^{-1}/4$. Since
$\|V_{\leq\alpha_{0}}\|_{L^{\infty}}\leq\alpha_{0}$ we have

$\displaystyle\|V\|_{X(\lambda)\to
X^{\prime}(\lambda)}\leq\|V_{>\alpha_{0}}\|_{L^{q^{\prime}}\to
L^{q}}+\lambda^{-1}\|V_{\leq\alpha_{0}}\|_{L^{2}\to
L^{2}}\leq\tfrac{1}{4}C_{0}^{-1}+\lambda^{-1}\alpha_{0},$

which yields

$\displaystyle
M(\Lambda)\leq\tfrac{1}{4}C_{0}^{-1}+\Lambda^{-1}\alpha_{0},\quad\Lambda_{0}\leq
4\alpha_{0}C_{0}.$

Thus, the claim again follows from Proposition 4.8. ∎

##  5\. Applications

###  5.1. From Tomas–Stein to Kenig–Ruiz–Sogge

In the remainder of this section we will consider applications of Theorem 2.1.
Ultimately, we will obtain new uniform resolvent estimates for compact
manifolds with boundary or nonsmooth metrics. As a warm-up, we start by re-
proving several known resolvent estimates, using the corresponding spectral
cluster bound.

The prototype of a spectral cluster bound is the Tomas–Stein inequality, which
can be stated as a spectral cluster bound for the Euclidean Laplacian
$-\Delta$ (see [49, Chapter 5]),

(53)
$\displaystyle\|\Pi_{[\lambda,\lambda+\epsilon]}\|_{L^{2}({\mathbb{R}}^{n})\to
L^{q_{n}}({\mathbb{R}}^{n})}\lesssim\epsilon^{1/2}\lambda^{1/q_{n}}$

for $\lambda,\epsilon>0$. This also follows from the Euclidean version of (4)
by scaling. We will show that (53) implies the resolvent bound

(54)
$\displaystyle\|(\Delta+(\lambda+\mathrm{i}0)^{2})^{-1}u\|_{L^{q_{n}}({\mathbb{R}}^{n})}\lesssim\lambda^{-\frac{2}{n+1}}\|u\|_{L^{q_{n}^{\prime}}({\mathbb{R}}^{n})},$

for $\lambda\geq 1$. This is the Kenig–Ruiz–Sogge bound888The condition
$\lambda\geq 1$ can be removed by scaling. Moreover, all other bounds $2\leq
q\leq 2^{*}$ follow from (54) by Sobolev embedding. in the special case of
self-dual exponents ([27, Theorem 2.3]). By Littlewood–Paley theory999It is
not strictly necessary to invoke Littlewood–Paley theory. We could just use
the two regions $|\xi|\leq 2\lambda$ and its complement, as before. and
Minkowski’s inequality for integrals this would be a consequence of the same
estimate for $P_{k}u$, where $P_{k}$ is the Littlewood–Paley projection onto
frequencies $2^{k-1}\leq|\xi|\leq 2^{k+1}$. In the elliptic region
$2^{k}\ll\lambda$ or $2^{k}\gg\lambda$, Bernstein inequalities (or Sobolev
embedding) and the Mikhlin multiplier theorem yield the bound

$\displaystyle\|(\Delta+(\lambda+\mathrm{i}0)^{2})^{-1}P_{k}u\|_{L^{q_{n}}({\mathbb{R}}^{n})}\lesssim\frac{2^{k\frac{2n}{n+1}}}{(2^{k}+\lambda)^{2}}\|P_{k}u\|_{L^{q_{n}^{\prime}}({\mathbb{R}}^{n})}.$

The factor on the right is dominated by $\lambda^{-\frac{2}{n+1}}$ in the
region considered. It remains to treat the case $2^{k}\approx\lambda$. The
Tomas–Stein estimate (53) is of the form of the abstract spectral cluster
bound (27) with $\delta=\epsilon^{1/2}$ and
$\gamma(q_{n})=\sigma(q_{n})=1/q_{n}$. Also, note that the exponent in (54)
$2/(n+1)=2\gamma(q_{n})-1$. We thus aim to prove that, if (27) holds for
$A=\sqrt{-\Delta}$ and $\delta,\gamma(q_{n})$ as above, then (36) holds for
$\epsilon\in(0,1]$ and with norms over
$[-\epsilon^{-1},\epsilon^{-1}]\times{\mathbb{R}}^{n-1}$. Since the right hand
side is independent of $\epsilon$ and equals $\lambda^{2\gamma(q_{n})-1}$,
taking $\epsilon\to 0$ we would then get (54). As in the proof of Theorem 2.1,
this would follow from the bound (38) with
$(\Delta+(\lambda+\mathrm{i}\epsilon)^{2})u=\lambda f$, whereas (53) implies
(41) and thus (43), by Lemma (4.3). As before, we switch to $(t,x)$
coordinates. Here we have the explicit expressions

$\displaystyle
a_{\lambda}(\xi)=\sqrt{\lambda^{2}-|\xi|^{2}},\quad\xi\in{\mathbb{R}}^{n-1}$

and $S(t,s)={\rm e}^{\mathrm{i}(t-s)a_{\lambda}(D_{x})}$. Moreover, the cutoff
$\chi$ is independent of $(t,x)$. We apply (43) to $\psi(\epsilon t)u(t,x)$,
with $\psi$ adapted to $[0,1]$ and $\psi(0)=0$, to get (38) with norms over
$(t,x)\in[0,\epsilon^{-1}]\times{\mathbb{R}}^{n-1}$. Again, we are using that
$\psi(\epsilon\cdot)$ has $L^{2}$-norm $\epsilon^{-1/2}$. Similarly, solving
the equation backwards in time, we get (38) with norms over
$(t,x)\in[-\epsilon^{-1},0]\times{\mathbb{R}}^{n-1}$. This completes the proof
of (38) and thus of the Kenig–Ruiz–Sogge bound (54).

###  5.2. Consequences of Sogge’s bounds

As explained in the introduction, combining Theorem (2.1) with Sogge’s bounds
(4), we recover the result of Frank–Schimmer [17] and Burq–Dos Santos
Ferreira–Krupchyk [7], and thus also that of Dos Santos Ferreira–Kenig–Salo
[15].

###  5.3. Nonsmooth metrics

As already discussed in Section 2, part (i) of Theorem 2.10 can be deduced
from several references [42, 31, 9, 43], either with or without the help of
Theorem 2.1. The easiest route is to combine Smith’s result [42] with Theorem
2.1. Alternatively, one can apply [31, Theorem 2.5] to the Weyl quantization
of the symbol $p(x,\xi)=\lambda^{-1}(\lambda^{2}-g^{ij}(x)\xi_{i}\xi_{j})$,
localized to frequency $\lambda$. This is in $\lambda
S_{\lambda,\lambda^{1/2}}$ after suitable global extension and satisfies (A2),
(A3) in [31] with $k=0$. This yields (43) with $\epsilon=1$ and
$\gamma(q)=\sigma(q)$. The remainder of the proof is identical to that of
Theorem 2.1. The use of the Weyl quantization is immaterial since the error
compared to $\Delta_{g}+\lambda^{2}$ can be absorbed into the forcing term
$f_{1}$ in (43).

Part (ii) follows from [30, Theorem 1.1] and Theorem 2.1. At the endpoint
$q=8$, the result could also be deduced (with the same logarithmic loss as for
the spectral cluster bound in [30]) without using Theorem 2.1 directly, but
instead applying the Christ–Kiselev lemma to the last formula in Step 1 in the
proof of [30, Theorem 2.1].

It remains to prove part (iii) of Theorem 2.10.

###### Proof of Theorem 2.10 (iii).

For $1\leq s\leq 2$ this is a consequence of Lemma 5.1 and Theorem 4.2. For
$0\leq s<1$ the estimate (56) is still true as long as $2\leq q\leq 2^{*}$,
but we cannot directly use Theorem 4.2. However, we can use its proof if we
first localize to the $\lambda^{-\rho}$ spatial scale. This works because the
estimates (38) and (41), with $\delta=\lambda^{\rho/2}$ and
$\epsilon=\lambda^{\rho}$, are localizable to this scale, and the coefficents
of the metric can be approximated by $C^{1,1}$ functions [41, Corollary 6].
After rescaling to the unit scale, the new frequency is
$\mu=\lambda^{1-\rho}$. We can then repeat the arguments of Theorem 4.2 for
the regularized operator (with the smoothed metric). By [41, Corollary 7] we
have the localized analogue of (41),

(55) $\displaystyle\|u\|_{L^{q_{n}}(B_{R})}\lesssim
R^{-1/2}\lambda^{1/q_{n}}(\|u\|_{L^{2}(B_{R}^{*})}+R\lambda^{-1}\|(\Delta_{g}+\lambda^{2})u\|_{L^{2}(B_{R}^{*})}),$

where $R=\lambda^{-\rho}$ and $B_{R}$ is the double of $B_{R}$; see also the
last estimate before (15) in [28]. Lemma 4.3 (again for the regularized
operator) then yields the localized analogue of (38),

$\displaystyle\|u\|_{L^{q_{n}}(B_{R})}\lesssim
R^{-1/2}\lambda^{1/q_{n}}(\|u\|_{L^{2}(B_{R}^{*})}+R^{1/2}\lambda^{1/q_{n}-1}\|(\Delta_{g}+\lambda^{2})u\|_{L^{q_{n}^{\prime}}(B_{R}^{*})}).$

Summing over a cover of $M$ by balls $B_{R}$ with bounded overlap and adding
in the $L^{2}\to L^{q}$ resolvent bound (28) as in the proof of Theorem 4.2
yields the claimed bound. ∎

######  Lemma 5.1.

If $1\leq s\leq 2$ and $\rho=(2-s)/(2+s)$, then the spectral cluster estimates

(56) $\displaystyle\|\Pi_{[\lambda,\lambda+\lambda^{\rho}]}\|_{L^{2}\to
L^{q}}\lesssim\lambda^{\sigma(q)+\rho/2}$

hold for all $2\leq q\leq\infty$.

###### Proof.

By interpolation with the $L^{2}\to L^{\infty}$ bound [41, (5)] we can reduce
this to the bound for $q=q_{n}$. Using the equivalence of spectral cluster and
quasimode estimates (see Corollary 3.5) and the same covering argument as
before, this follows from (55). ∎

######  Remark 5.2.

Lemma 5.1 shows that we have no loss estimates for spectral windows of length
$\lambda^{\rho}$. Indeed, orthogonality and optimality of the universal
estimates (4) show that (56) is best possible. Lemma 5.1 is in a sense a dual
version of [41, Theorem 2], which states no loss estimates for the spectral
windows of unit length hold on the $\lambda^{-\rho}$ spatial scale.

We conclude this subsection with resolvent estimates for $C^{s}$ metrics that
are perfect with respect to the spectral region, but not with respect to the
exponent. They rely on spectral cluster bounds of Smith [41]. These are sharp
at the critical exponent $q_{n}$, but improvements are expected for larger
$q$.

######  Theorem 5.3.

Let $\Delta_{g}$ be the Laplace–Beltrami operator on an $n$-dimensional
compact boundaryless Riemannian manifold with $C^{s}$ metric, $1\leq s\leq 2$.
If

(57) $\displaystyle 2\leq q\leq\tfrac{2n(s+2)+s-2}{(n-2)(s+2)}=:q_{s},$

and $q<\infty$ if $n=2$, then we have

(58)
$\displaystyle\|(\Delta_{g}+(\lambda+\mathrm{i})^{2})^{-1}\|_{L^{q^{\prime}}\to
L^{q}}\lesssim\langle\lambda\rangle^{2\gamma(q)-1},$

where $\gamma(q)=\sigma(q)+\tfrac{1}{q}\tfrac{2-s}{2+s}\leq\tfrac{1}{2}$.

###### Proof.

This follows from [41, Corollary 3] in combination with Theorem 2.1. ∎

######  Remark 5.4.

Even though Theorem 5.3 does not yield a uniform Sobolev inequality, i.e.
$q<2^{*}$ if $n\geq 3$, the value of $q_{s}$ is surprisingly close to the
Sobolev exponent. The table below depicts the ratio $q_{s}/2^{*}$ for various
values of $n$ and $s$.

| $n=3$ | $n=4$ | $n=5$
---|---|---|---
$s=1$ | $0.944444$ | $0.958333$ | $0.966667$
$s=1.3$ | $0.964646$ | $0.973485$ | $0.978788$
$s=1.6$ | $0.981481$ | $0.986111$ | $0.988889$

###  5.4. Manifolds with boundary

###### Proof of Theorem 2.16.

(i) follows from Theorem 2.10 (iii), see Remark 2.18.

(ii): Smith and Sogge proved the spectral cluster bound (6) with

$\displaystyle\gamma(q)=\begin{cases}\frac{4}{3}\sigma(q)&\quad\mbox{if}\quad\frac{2(n+2)}{n-1}\leq
q\leq q_{n},\\\
\sigma(q)+\frac{1}{3}\big{(}\varepsilon(q)-\frac{1}{q}\big{)}_{-}&\quad\mbox{if}\quad
q_{n}\leq q\leq 2^{*},\end{cases}$

where $x_{-}=\max(0,-x)$ and $\varepsilon(q)=(n-1)(1/2-1/q)-2/q\geq 0$. See
[47, Theorem 2.1] for $n=2$ and [47, Theorem 7.2] for $n\geq 3$. In fact, they
proved homogeneous square function estimates for solutions to the wave
equation, which imply the spectral cluster estimates (see e.g. [45], [42]).
Since Theorem 2.1 applies only to boundaryless manifolds, we have referred to
the corresponding results of [47]. These results apply to boundaryless
manifolds with special Lipschitz metrics (smooth, with a Lipschitz singularity
across a hypersurface). Thus, we are in the setting of Theorem 2.1, and the
claim follows.

(iii): Smith and Sogge [45] have also proved perfect spectral cluster
estimates for the Dirichlet realization of the Laplace-Beltrami operator on a
smooth manifold with strictly concave boundary, see Remark 2.19. The bound in
(iii) is an immediate consequence of these results and e.g. (31), with
$\delta,\epsilon=1$ there. ∎

###  5.5. Improved estimates

To prove Theorem 2.21 we only need to combine the spectral cluster bound [8,
Theorem 1] of Canzani and Galkowski with Theorem 4.2. Note that this is the
only instance where we need the version for improved bounds ($\epsilon<1$).
The result in [8, Theorem 1] is stated in a semiclassical framework, but it
straightforward to translate it to a bound of the form (29), or equivalently
(27). More precisely, [8, Theorem 1] asserts that

$\displaystyle\|u\|_{L^{q}}\lesssim
h^{-\sigma(q)}\left(\frac{\|u\|_{L^{2}}}{\sqrt{\log h^{-1}}}+\frac{\sqrt{\log
h^{-1}}}{h}\|(-h^{2}\Delta_{g}-1)u\|_{H_{\rm
scl}^{\frac{n-3}{2}-\frac{n}{q}}}\right)$

for $0<h\ll 1$. Here we considered the special case $U=M$ in the theorem of
Canzani–Galkowski and changed the notation to $p=q$, $\delta(p)=\sigma(q)$. We
also translate this from the semiclassical to the high frequency setting by
taking $h=1/\lambda$, obtaining

$\displaystyle\|u\|_{L^{q}}\lesssim\frac{\lambda^{\sigma(q)}}{\sqrt{\log\lambda}}\|u\|_{L^{2}}+\lambda^{\sigma(q)-1}\sqrt{\log\lambda}\|(\Delta_{g}+\lambda^{2})u\|_{L^{2}}.$

We used that the semiclassical Sobolev norms $H_{\rm scl}^{2}$ appearing in
the previous estimate are innocuous in the frequency-localized regime, i.e.
that

$\displaystyle\|(-h^{2}\Delta_{g}-1)u\|_{H_{\rm
scl}^{s}}\lesssim\|(-h^{2}\Delta_{g}-1)u\|_{L^{2}}$

for $u$ Fourier supported in $|\xi|\lesssim 1/h$. Now a direct comparison with
(29), with $\gamma(q)=\sigma(q)$, $\epsilon=1/\ln\langle\lambda\rangle$ and
$\delta=\sqrt{\epsilon}$ yields the claimed resolvent estimate.

###  5.6. Fractional Schrödinger operators

We give a brief argument how the proof of Theorem 2.1 (or that of Theorem 4.2)
can be modified in such a way that Theorem 2.25 follows from the spectral
cluster bounds of Smith [42]. This also gives a flavor of other possible
generalizations. The point of departure is the first order equation (39). This
is still valid if $u$ satisfies
$((-\Delta_{g})^{\alpha/2}-(\lambda+\mathrm{i})^{\alpha})u=\lambda^{\alpha-1}f$
(here we set $\delta,\epsilon=1$). The reason is that the symbol
$p(x,\xi)=|\xi|_{g}^{\alpha}-\lambda^{\alpha}$ is of real principal type, with
$|p_{\xi}|\gtrsim\lambda^{\alpha-1}$ on $\\{p=0\\}$ and at frequency
$\lambda$. Since $g$ is one order smoother that in the assumptions of Theorem
4.2, it would be sufficient to perform the symbol smoothing at frequency
$|\xi|\ll\sqrt{\lambda}$ as opposed to $\lambda$. Then, in a conic region and
in suitable coordinates,

$\displaystyle p(x,\xi)=e(x,\xi)(\xi_{1}-a(x,\xi^{\prime},\lambda)),$

where $a>0$ is a symbol in $\lambda C^{2}S_{\lambda,\sqrt{\lambda}}$, and
$e(x,\xi)$ is elliptic and of size $\lambda^{\alpha-1}$. Let $Q$ be a
parametrix for $e^{w}$, with a symbol in
$\lambda^{1-\alpha}S_{\lambda,\lambda}$. As in the proof of Theorem 4.2 first
order calculus yields

(59)
$\displaystyle(D_{x_{1}}-a_{\lambda}(x,D_{x^{\prime}}))u=\lambda^{\alpha-1}Qf+Ru$

where $R=\mathcal{O}_{L^{2}\to L^{2}}(1)$ and the $x$-frequencies of
$a_{\lambda}(x,\xi^{\prime})$ are truncated to $|\xi|\ll\sqrt{\lambda}$. We
proceed as in the proof of Theorem 4.2. Using the abstract results of
Corollary 3.5, we first rewrite Smith’s spectral cluster bound [42] as a
quasimode estimate

(60)
$\displaystyle\|u\|_{L^{q}}\lesssim\langle\lambda\rangle^{\sigma(q)}\|u\|_{L^{2}}+\langle\lambda\rangle^{\sigma(q)-1}\|(\Delta_{g}+\lambda^{2})u\|_{L^{2}},$

then we reverse the steps in the reduction to (59) to convert (60) to a
microlocalized estimate

$\displaystyle\|u\|_{L^{q}}\lesssim\lambda^{\sigma(q)}(\|u\|_{L^{2}}+\|(D_{x_{1}}-a_{\lambda}^{w}(x,D_{x^{\prime}}))u\|_{L^{2}}).$

This is exactly the same as (41) but with $\delta,\epsilon=1$. Thus the
claimed resolvent estimate follows from Lemma 4.3, just as in the proof of
Theorem 4.2.

### Acknowledgements

I would like to thank Rupert Frank for suggesting the topic of uniform
resolvent estimates on manifolds with boundary, from which the present work
emerged. I would also like to thank Herbert Koch, Hart Smith, Chris Sogge and
Daniel Tataru for illuminating discussion during various stages of the
project. I am grateful to Jeff Galkowski for discussions that lead to a
version of Theorem 4.2 that is applicable to improved estimates.

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