# Improved spectral projection estimates

Matthew D. Blair  Department of Mathematics, University of New Mexico,
Albuquerque, NM 87131<EMAIL_ADDRESS>,  Xiaoqi Huang  Department of
Mathematics, University of Maryland, College Park. MD 20742<EMAIL_ADDRESS>and  Christopher D. Sogge  Department of Mathematics, Johns Hopkins
University, Baltimore, MD 21218<EMAIL_ADDRESS>
###### Abstract.

We obtain new improved spectral projection estimates on manifolds of non-
positive curvature, including sharp ones for relatively large spectral windows
for general tori. Our results are stronger than those in an earlier work of
the first and third authors [6], and the arguments have been greatly
simplified. We more directly make use of pointwise estimates that are implicit
in the work of Bérard [2] and avoid the use of weak-type spaces that were used
in the previous works [6] and [22]. We also simplify and strengthen the
bilinear arguments by exploiting the use of microlocal $L^{2}\to L^{q_{c}}$
Kakeya-Nikodym estimates and avoiding the of $L^{2}\to L^{2}$ ones as in
earlier results. This allows us to prove new results for manifolds of negative
curvature and some new sharp estimates for tori. We also have new and improved
techniques in two dimensions for general manifolds of non-positive curvature.

###### Key words and phrases:

Eigenfunctions, spectral cluster estimates

###### 2010 Mathematics Subject Classification:

58J50, 35P15

The second author was supported in part by an AMS-Simons travel grant. The
third author was supported in part by the NSF (DMS-1665373).

In memoriam: Steve Zelditch (1953-2022)

##  1\. Introduction

The purpose of this paper is to improve upon the spectral projection estimates
on compact manifolds of two of us [6], while at the same time simplifying the
arguments. As in the previous work we shall focus on obtaining improved
estimates for the critical exponent

(1.1) $q_{c}=\tfrac{2(n+1)}{n-1}$

on $n$-dimensional compact boundaryless Riemannian manioflds $(M,g)$ all of
whose sectional curvatures are non-positive. If we obtain improved
$L^{q_{c}}$-estimates compared to the universal bounds of one of us [19],
then, by interpolation and dyadic Sobolev estimates, we obtain improvements
for all other exponents $q>2$, including the difficult range of subcritical
exponents considered in earlier papers by Bourgain [8], Sogge [20], Sogge and
Zelditch [24] and Blair and Sogge [3], [4], [5] and [6].

We are able to obtain our improvements over the earlier results in [6] by
simplifying and strengthening the two main steps in the proof of the spectral
projection estimates there. As in [6] and its predecessor [22], we shall adapt
an argument of Bourgain [7] for Fourier restriction problems to obtain our
results. This allows us to split our estimate into two “heights”. The greater
“height” (which is easier to handle) corresponds to the size of
$L^{2}$-normalized zonal spherical harmonics on round spheres $S^{n}$ which
saturate the universal $L^{q}$-estimates of one of us [19] for exponents
$q\geq q_{c}$, while the lesser “height” (which is much harder to handle)
corresponds to the size of the highest weight spherical harmonics on $S^{n}$
which saturate the universal bounds for $2<q\leq q_{c}$. See also [18].

We are able to obtain the needed estimates for the larger heights in a much
more efficient manner than in the earlier works [6] and [22]. In these works
we first obtained weak-type $L^{q_{c}}$-estimates using the argument from [7]
along with pointwise estimates for smoothed out spectral projection kernels
that were obtained by the techniques of Bérard [2]. We then “upgraded” the
weak-type $L^{q_{c}}$ estimates in the end to $L^{q_{c}}$-estimates using the
Lorentz space improvements of Bak and Seeger [1] for “local” spectral
projection operators related to the “global” ones we were considering. We now
have a much more direct and simple argument based on Bérard-type estimates
that completely avoids the use of weak-type estimates and the corresponding
necessary interpolation argument involving the local estimates in [1] that
leads to unnecessary losses. Also, avoiding the use of weak-type norms in the
other main step which treats the “lower heights” simplifies the arguments
there.

As in [6], the second main step in the proof of our improved spectral
projection estimates utilizes bilinear techniques of Lee [17], Tao [25], Tao,
Vargas and Vega [26], Wolff [27] and others that were used to study Fourier
restriction problems in Euclidean space. We are able to make better use of the
bilinear techniques here by another more direct argument, which, like in the
other main step in the proof, is based on a single basic estimate and
straightforward arguments using “local operators”. The estimate that we use
for this is a “microlocal Kakeya-Nikodym” estimate going from $L^{2}(M)$ to
$L^{q_{c}}(M)$ that involves a microlocalized version of the smoothed out
spectral projection operators that we ultimately want to control. These
microlocal Kakeya-Nikodym $L^{2}\to L^{q_{c}}$ estimates were also used by two
of us in [6]; however, they were also paired with an additional superfluous
step that involved $L^{2}\to L^{2}$ Kakeya-Nikodym estimates (such as in [5])
that arose in all of the earlier works and was an artifact of the earliest
works in this thread by Bourgain [8] and Sogge [20]. Naturally, removing this
unnecessary step involving $L^{2}$–Kakeya-Nikodym bounds leads to improvements
over earlier results. We are able to obtain further improvements in two-
dimensions by tightening the bilinear estimates in this case versus the ones
in this dimension in [6] and earlier works.

Additionally, in some settings we are able to exploit specific geometric
features to obtain improved $L^{2}\to L^{q_{c}}$ Kakeya-Nikodym estimates
compared to the ones in [6] for general manifolds of non-positive curvatures.
These lead to further improvements in certain settings.

When $M$ is a torus, we are able to exploit favorable commutator properties
for operators that arise in the second step of the proof to obtain new
spectral projection estimates that we show to be sharp. These involve
relatively large spectral projection windows compared to those in the earlier
works of Bourgain and Demeter [9], Germain and Rydin Myerson [14] and Hickman
[16] that allowed very thin windows of width about the spectral parameter
$\lambda$ that go all the way to the wavelengths $\lambda^{-1}$. To be able to
obtain near sharp estimates for these very narrow spectral windows these
authors used decoupling which resulted in modest $\lambda^{\varepsilon}$
losses, while, as we mentioned, our new estimates are best possible for the
relatively large spectral scales we are able to handle with bilinear
techniques.

Also, for manifolds of strictly negative sectional curvatures, we are able to
exploit the rapid dispersive properties of wave kernels to obtain improvements
of the microlocal $L^{2}\to L^{q_{c}}$ Kakeya-Nikodym estimates compared to
the ones we can obtain for the general case of manifolds of non-positive
curvature. By exploiting these, we are able, in the case of manifolds of
negative curvature to obtain estimates for spectral windows of width
$(\log\lambda)^{-1}$ about $\lambda$ which are stronger in two-dimensions than
the (sharp) estimates that we obtain for ${\mathbb{T}}^{2}$ and, in
3-dimensions, that match up with the (sharp) estimates for ${\mathbb{T}}^{3}$
for this spectral scale. Unfortunately, unlike in the latter setting, we are
unable to show that these natural estimates are optimal and we discuss a
problem concerning geodesic concentration of log-quasimodes on manifolds
manifolds of negative curvature that seems to be related to recent work of
Brooks [11] and Eswarathasan and Nonnenmacher [13].

Let us now state our main results. We shall consider smoothed out spectral
projection operators of the form

(1.2) $\rho\bigl{(}T(\lambda-P)\bigr{)},\quad T=T(\lambda),$

with $P=\sqrt{-\Delta_{g}}$ where $\rho$ is a real-valued function satisfying

(1.3) $\rho\in{\mathcal{S}}({\mathbb{R}}),\quad\rho(0)=1,\quad\text{and
}\text{supp }\hat{\rho}\subset\\{|t|\in(\delta/2),\delta)\\},$

where $0<\delta<1$ will be specified later. For the most part, we shall take

(1.4) $T=c_{0}\log\lambda,$

with $c_{0}>0$ ultimately chosen sufficiently small depending on $(M,g)$.

Our main result which improves on estimates in [6] then is the following

######  Theorem 1.1.

Assume that $(M,g)$ is an $n$-dimensional compact manifold with non-positive
sectional curvatures. Then, for $T$ as above,

(1.5)
$\|\rho(T(\lambda-P))f\|_{L^{q_{c}}(M)}\lesssim\lambda^{1/q_{c}}\,\bigl{(}\log\lambda\bigr{)}^{-\sigma_{n}}\|f\|_{2},$

where $\sigma_{n}=\tfrac{2}{(n+1)q_{c}}$.

Let us note that if $\varepsilon(\lambda)=(\log\lambda)^{-1}$ then (1.5)
immediately implies bounds for the spectral projection operators

(1.6) $\chi_{[\lambda-\varepsilon(\lambda),\lambda+\varepsilon(\lambda)]}={\bf
1}_{[\lambda-\varepsilon(\lambda),\,\lambda+\varepsilon(\lambda)]}(P),$

with ${\bf 1}_{I}$ denoting the indicator function of the interval $I$. Indeed
since $\rho(0)=1$, by a simple orthogonality argument, Theorem 1.1 yields the
following.

######  Corollary 1.2.

If $\chi_{[\lambda-\varepsilon(\lambda),\lambda+\varepsilon(\lambda)]}$ is as
in (1.6) and $(M,g)$ as above has non-positive curvatures then for
$\lambda\geq 1$ and $\sigma_{n}=2/(n+1)q_{c}$

(1.7)
$\bigl{\|}\chi_{[\lambda-(\log\lambda)^{-1},\,\lambda+(\log\lambda)^{-1}]}f\bigr{\|}_{L^{q_{c}}(M)}\leq
C\lambda^{\frac{1}{q_{c}}}(\log\lambda)^{-\sigma_{n}}\|f\|_{L^{2}(M)}.$

Recall that the universal bounds of one of us [19] say that

(1.8) $\|\chi_{[\lambda-1,\,\lambda+1]}\|_{L^{2}(M)\to
L^{q}(M)}\lesssim\lambda^{\mu(q)},\quad 2<q\leq\infty,$

with

(1.9)
$\mu(q)=\max\Bigl{(}n(\tfrac{1}{2}-\tfrac{1}{q})-\tfrac{1}{2},\,\tfrac{n-1}{2}(\tfrac{1}{2}-\tfrac{1}{q})\Bigr{)}.$

Since $\mu(q_{c})=1/q_{c}$, (1.7) is a log-power improvement of the universal
bounds on manifolds of non-positive curvature. We need to use shrinking
spectral windows in order to improve the bounds (1.8) which are saturated on
any manifold (see [23]); however, for $S^{n}$ there is no improvement of (1.7)
even if one uses $(\log\lambda)^{-1}$-spectral windows as in (1.7). So the
assumption of non-positive curvature is needed to obtain the improved spectral
projection estimates in (1.7).

We also note that, by interpolating with the trivial $L^{2}\to L^{2}$
estimates for the spectral projection operators, (1.7) yields

(1.10)
$\bigl{\|}\chi_{[\lambda-(\log\lambda)^{-1},\,\lambda+(\log\lambda)^{-1}]}f\bigr{\|}_{L^{q}(M)}\leq\Bigl{(}\,(\log\lambda)^{-\frac{2}{n+1}}\lambda\Bigr{)}^{\mu(q)}\,\|f\|_{L^{2}(M)},\quad
2<q\leq q_{c},$

with $\mu(q)=\tfrac{n-1}{2}(\tfrac{1}{2}-\tfrac{1}{q})$ as in (1.9). By using
dyadic Sobolev estimates one can also obtain log-power improvements for
$q>q_{c}$; however, these would not be as strong as the $(\log\lambda)^{-1/2}$
improvements over the universal bounds for supercritical exponents of Bérard
[2] and Hassell and Tacy [15].

After we prove Theorem 1.1, we shall see that stronger estimates hold if one
assumes all of the sectional curvatures are negative, and that, in this case
the ones obtained for three-dimensions have numerology that is related to
recent toral spectral projection bounds of Hickman [16] and Germain and Rydin
Meyerson [14], while in two-dimensions are more favorable. We shall also
obtain in all dimensions $n\geq 2$ optimal spectral projection estimates on
$n$-dimensional tori for $\varepsilon(\lambda)$-windows with
$\varepsilon(\lambda)$ larger than a fixed negative power of $\lambda$ which
depends on the dimension.

Next, to motivate the two main steps in our proofs, let us recall the
eigenfunctions on $S^{n}$ that saturate the universal estimates (1.7). If we
wish to obtain improved bounds, such as those in (1.5) or (1.7), we need to
develop techniques that will be able to show that the types of extreme
eigenfunctions that exist on standard spheres cannot exist on manifolds of
non-positive curvature.

The eigenfunctions that saturate (1.8) on $S^{n}$ with eigenvalue
$\lambda=\sqrt{k(k+n-1)}$ are the zonal spherical harmonics of degree $k$,
$Z_{k}$. If they are $L^{2}$-normalized and if $B_{\pm}(c\lambda^{-1})$ are
geodesic balls of radius $c\lambda^{-1}$ about the two poles on $S^{n}$ then

(1.11) $|Z_{k}(x)|\approx\lambda^{\frac{n-1}{2}},\quad x\in
B_{\pm}(c\lambda^{-1}),$

for some sufficiently small uniform constant $c>0$. (See, e.g., [18]). As a
result, an easy calculation shows that for some $c_{0}>0$ we have the lower
bounds

$\|Z_{k}\|_{L^{q}(S^{n})}\geq
c_{0}\lambda^{n(\frac{1}{2}-\frac{1}{q})-\frac{1}{2}},\quad\text{if
}\,\,\lambda=\sqrt{k(k+n-1)}.$

This implies that (1.8) cannot be improved when $M=S^{n}$ for the range of
exponents for which $\mu(q)=n(\tfrac{1}{2}-\tfrac{1}{q})-\tfrac{1}{2}$, which
is $q\geq q_{c}$, where $q_{c}$ is as in (1.1).

The other extreme eigenfunctions for the above values of $\lambda$ are the
highest weight spherical harmonics of degree $k$, $Q_{k}$. If $\gamma\subset
S^{n}$ is the equator, and if ${\mathcal{T}}_{\lambda^{-1/2}}(\gamma)$ is a
$\lambda^{-1/2}$-tubular neighborhood of $\gamma$, then

$\|Q_{k}\|_{L^{2}({\mathcal{T}}_{\lambda^{-1/2}}\,(\gamma))}\approx
1\quad\text{if }\,\lambda=\sqrt{k(k+n-1)}\quad\text{and
}\,\,\|Q_{k}\|_{L^{2}(S^{n})}=1.$

(See, e.g. [18].) Therefore, as we shall show in §7, it is a simple exercise
using Hölder’s inequality to verify that for some uniform $c_{0}>0$

(1.12) $\|Q_{k}\|_{L^{q}(S^{n})}\geq
c_{0}\,\lambda^{\frac{n-1}{2}(\frac{1}{2}-\frac{1}{q})},\quad 2<q\leq\infty.$

As a result, (1.8) cannot be improved for $2<q\leq q_{c}$ since
$\mu(q)=\tfrac{n-1}{2}(\tfrac{1}{2}-\tfrac{1}{q})$ for such exponents. To
further motivate the splitting that we shall use in order to prove Theorem
1.1, we recall (see [18]) that the lower bounds in (1.12) are actually upper
bounds, and, in particular

(1.13)
$\|Q_{k}\|_{L^{\infty}(S^{n})}\approx\lambda^{\frac{n-1}{4}},\quad\text{if
}\,\,\lambda=\sqrt{k(k+n-1)}.$

We note that both the zonal spherical harmonics, $Z_{k}$, and the highest
weight spherical harmonics, $Q_{k}$, saturate the universal bounds (1.8) when
$q$ equals the critical exponent $q_{c}$ in (1.1). So estimates involving this
exponent are sensitive to both “point concentration” and “geodesic
concentration”. The two main steps in the proof of Theorem 1.1 will rule out
these types of extreme concentration under our curvature assumptions, and,
based on the arguments from §7 and (1.13), the “height” at which the
transition between these types of concentration can occur should be
$\lambda^{\frac{n-1}{4}}$.

Acknowledgement: We are deeply indebted to Steve Zelditch for the
encouragement and generous advice that he has given us over the years on this
project and its predecessors. In particular, his insistence that we should use
microlocal estimates rather than the original $L^{2}$ spatial Kakeya-Nikodym
estimates originally used by one of [20] was crucial for the progress that we
have made on these problems over the years. Moreover, the use of microlocal
estimates like (4.8) here has strengthened our bounds and greatly simplified
the arguments.

##  2\. The height splitting

As in some of our earlier works, we shall employ bilinear techniques that
require us to compose the “global operators”

(2.1) $\rho_{\lambda}=\rho(T(\lambda-P))$

in Theorem 1.1 with the “local operators”

(2.2) $\sigma_{\lambda}=\rho(\lambda-P).$

Also, as in earlier works, it will be convenient to localize a bit more. To
this end, let us write

(2.3) $I=\sum_{j=1}^{N}B_{j}(x,D),$

where each $B_{j}\in S^{0}_{1,0}(M)$ is a standard zero order pseudo-
differential with symbol supported in a small conic neighborhood of some
$(x_{j},\xi_{j})\in S^{*}M$. The size of the support will be described in a
bit; however, we note now that these operators will not depend on our spectral
parameter $\lambda\gg 1$. Also note that if $\beta\in C^{\infty}_{0}((1/2,2))$
is a Littlewood-Paley bump function which equals one near $t=1$ then the
dyadic operators

(2.4) $B=B_{j,\lambda}=B_{j}\circ\beta(P/\lambda)$

are uniformly bounded on $L^{p}$, i.e.,

(2.5) $\|B\|_{L^{p}(M)\to L^{p}(M)}=O(1)\quad\text{for }\,\,1\leq
p\leq\infty.$

We then shall further localize $\sigma_{\lambda}$ by setting

(2.6) $\tilde{\sigma}_{\lambda}=B\circ\sigma_{\lambda},$

where $B$ is one of the $N$ operators in (2.4). We also define the “semi-
global” operators

(2.7) $\tilde{\rho}_{\lambda}=\tilde{\sigma}_{\lambda}\circ\rho_{\lambda}.$

We note that the $\sigma_{\lambda}$ can be thought of as “smoothed out”
versions of the operators in (1.8). They enjoy the same operator norms and the
two sets of estimates are equivalent. Similarly, it is an easy exercise using
orthogonality to see that we have the uniform bounds

(2.8) $\|(I-\sigma_{\lambda})\circ\rho_{\lambda}\|_{L^{2}(M)\to
L^{q_{c}}(M)}\leq CT^{-1}\lambda^{\frac{1}{q_{c}}},\,\,\text{if }\,T\geq 1,\\\
\text{and
}\,\,\,\|\sigma_{\lambda}-\beta(P/\lambda)\circ\sigma_{\lambda}\|_{L^{2}(M)\to
L^{q_{c}}(M)}=O(\lambda^{-N}),\,\,\,\forall\,N.$

Therefore by (2.3) and (2.8), in order to prove (1.5), it suffices to show
that

(2.9)
$\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(M)}\lesssim\lambda^{\frac{1}{q_{c}}}(\log\lambda)^{-\sigma_{n}}\|f\|_{L^{2}(M)},$

if $T$ is as in (1.4).

Due to the bilinear arguments that we shall employ, we need to make the height
decomposition using the semi-global operators $\tilde{\rho}_{\lambda}$. We
shall always, as we may, assume that the function $f$ in (2.9) is
$L^{2}$-normalized, i.e.,

(2.10) $\|f\|_{L^{2}(M)}=1.$

We shall then split our task (2.9) into estimating the $L^{q_{c}}$-norm of
$\tilde{\rho}_{\lambda}f$ over the two regions,

(2.11) $A_{+}=\\{x\in
M:\,|\tilde{\rho}_{\lambda}f(x)|\geq\lambda^{\frac{n-1}{4}+\frac{1}{8}}\,\\}$

and

(2.12) $A_{-}=\\{x\in
M:\,|\tilde{\rho}_{\lambda}f(x)|<\lambda^{\frac{n-1}{4}+\frac{1}{8}}\,\\},$

which basically correspond to the height (1.13) of the heighest weight
spherical harmonics. There is nothing special about the exponent $1/8$ in
(2.11) and (2.12). Just as in [6], it could be replaced by any sufficiently
small $\delta>0$ in what follows.

##  3\. Large height estimates

Let us describe here how to estimate the $L^{q_{c}}(A_{+})$ norm of
$\tilde{\rho}_{\lambda}f$. If $a\in C^{\infty}_{0}((-1,1))$ equals one on
$(-1/2,1/2)$, then we can do this using the “global estimate”

(3.1)
$G_{\lambda}(x,y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\bigl{(}1-a(t)\bigr{)}\,T^{-1}\hat{\Psi}(t/T)e^{it\lambda}\,\bigl{(}e^{-itP})(x,y)\,dt\\\
=O\bigl{(}\lambda^{\frac{n-1}{2}}\,\exp(C_{0}T)\Bigr{)},\,\,\,\Psi=\rho^{2}\,\,\,1\leq
T\lesssim\log\lambda.$

This estimate is valid when $(M,g)$ has nonpositive sectional curvatures (see
e.g., [2], [15], [22], [21]). One proves (3.1) by standard arguments after
lifting the calculation up to the universal cover. Since $\hat{\Psi}$ is
compactly supported, the number of terms in the sum that arises grows
exponentially in $T$ if $(M,g)$ has negative curvature, and this accounts for
the $\exp(C_{0}T)$ factor in (3.1). (See e.g., [21, §3.6] and the related
arguments in §5 below). Later in §6 we shall be able to obtain more favorable
estimates for tori than those in Theorem 1.1 in part by using the fact that in
this setting, the number of terms grows polynomially. Due to this, for tori we
shall use a slightly different splitting into the two heights compared to
(2.11) and (2.12).

Let us now see that if (3.1) is valid then we have the following.

######  Proposition 3.1.

Let $f$ and $A_{+}$ be as in (2.10) and (2.11) and $B$ be as in (2.4). Then
for $\lambda\gg 1$, if $T=c_{0}\log\lambda$ with $c_{0}>0$ sufficiently small,

(3.2) $\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{+})}\leq
C\lambda^{\frac{1}{q_{c}}}\,(\log\lambda)^{-1/2}\,\|f\|_{L^{2}(M)}.$

###### Proof.

We first note that, by (2.5) and (2.8), we have

$\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{+})}\leq\|B\rho_{\lambda}f\|_{L^{q_{c}}(A_{+})}+C\lambda^{\frac{1}{q_{c}}}/\log\lambda,$

since we are assuming that $f$ is $L^{2}$-normalized. Thus, we would have
(3.2) if we could show that

(3.3) $\|B\rho_{\lambda}f\|_{L^{q_{c}}(A_{+})}\leq
C\lambda^{\frac{1}{q_{c}}}(\log\lambda)^{-1/2}+\tfrac{1}{2}\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{+})}.$

To prove this we shall adapt an argument of Bourgain [7] and simplify related
ones in [22] and [6]. As mentioned before, we shall avoid using arguments that
utilize the Lorentz space estimates of Bak and Seeger [1], unlike in [22] and
[6].

To this end, choose $g$ such that

$\|g\|_{L^{q^{\prime}_{c}}(A_{+})}=1,\quad\text{and
}\,\,\,\|B\rho_{\lambda}f\|_{L^{q_{c}}(A_{+})}=\int
B\rho_{\lambda}f\,\cdot\bigl{(}{\bf 1}_{A_{+}}\cdot g\bigr{)}\,dx.$

Then since $\rho^{*}_{\lambda}=\rho_{\lambda}$ and
$\Psi(T(\lambda-P))=\rho_{\lambda}\circ\rho_{\lambda}$ for $\Psi$ as in (3.1),
by the Schwarz inequality, since we are assuming that $\|f\|_{2}=1$ we have

(3.4) $\displaystyle\|B\rho_{\lambda}f\|^{2}_{L^{q_{c}}(A_{+})}$
$\displaystyle=\Bigl{(}\,\int f\cdot\bigl{(}\rho_{\lambda}B^{*}\bigr{)}({\bf
1}_{A_{+}}\cdot g)(x)\,dx\,\Bigr{)}^{2}$
$\displaystyle\leq\int\bigl{|}\,\rho_{\lambda}B^{*}\bigl{(}{\bf
1}_{A_{+}}\cdot g\bigr{)}(x)\,\bigr{|}^{2}\,dx$
$\displaystyle=\int\bigl{(}B\circ\Psi(T(\lambda-P))\circ B^{*}\bigr{)}({\bf
1}_{A_{+}}\cdot g)(x)\,\cdot\,\overline{{\bf 1}_{A_{+}}(x)\,g(x)}\,dx$
$\displaystyle=\int\bigl{(}B\circ L_{\lambda}\circ B^{*}\bigr{)}({\bf
1}_{A_{+}}\cdot g)(x)\cdot\overline{{\bf 1}_{A_{+}}(x)\,g(x)}\,dx$
$\displaystyle\qquad\qquad+\int\bigl{(}B\circ G_{\lambda}\circ
B^{*}\bigr{)}({\bf 1}_{A_{+}}\cdot g)(x)\cdot\overline{{\bf
1}_{A_{+}}(x)\,g(x)}\,dx$ $\displaystyle=I+II.$

Here, $G_{\lambda}$ is the operator whose kernel is as in (3.1), while
$L_{\lambda}$ is the “local” operator

$L_{\lambda}=(2\pi T)^{-1}\int a(t)\hat{\Psi}(t/T)e^{it\lambda}e^{-itP}\,dt.$

Thus, $L_{\lambda}h=T^{-1}\sum_{j}m(\lambda;\lambda_{j})E_{j}h$, where $E_{j}$
denotes the projection onto the $j$th eigenspace of $P$ and the spectral
multiplier satisfies

$m(\lambda;\lambda_{j})=O((1+|\lambda-\lambda_{j}|)^{-N})$

for any $N$. Consequently, by the universal bounds in [19], we have

$\|L_{\lambda}\|_{L^{q_{c}^{\prime}}(M)\to L^{q_{c}}(M)}\lesssim
T^{-1}\lambda^{2/q_{c}}.$

Since $T=c_{0}\log\lambda$, if we use Hölder’s inequality and (2.5) we find
that

(3.5) $\displaystyle|I|$
$\displaystyle\leq\bigl{\|}BL_{\lambda}B^{*}\bigr{(}{\bf 1}_{A_{+}}\cdot
g\bigr{)}\bigr{\|}_{q_{c}}\cdot\|{\bf 1}_{A_{+}}\cdot g\|_{q_{c}^{\prime}}$
$\displaystyle\lesssim\bigl{\|}L_{\lambda}B^{*}\bigr{(}{\bf 1}_{A_{+}}\cdot
g\bigr{)}\bigr{\|}_{q_{c}}\cdot\|{\bf 1}_{A_{+}}\cdot g\|_{q_{c}^{\prime}}$
$\displaystyle\lesssim\lambda^{2/q_{c}}(\log\lambda)^{-1}\bigl{\|}B^{*}\bigr{(}{\bf
1}_{A_{+}}\cdot g\bigr{)}\bigr{\|}_{q_{c}^{\prime}}\cdot\|{\bf 1}_{A_{+}}\cdot
g\|_{q_{c}^{\prime}}$
$\displaystyle\lesssim\lambda^{2/q_{c}}(\log\lambda)^{-1}\|g\|^{2}_{L^{q_{c}^{\prime}}(A_{+})}$
$\displaystyle=\lambda^{2/q_{c}}(\log\lambda)^{-1}.$

To estimate $II$, we choose $c_{0}>0$ small enough so that if $C_{0}$ is the
constant in (3.1) then

$\exp(C_{0}T)\leq\lambda^{1/8}\quad\text{if
}\,\,T=c_{0}\log\lambda\quad\text{and }\,\,\lambda\gg 1.$

Consequently, (3.1) yields

$\|G_{\lambda}\|_{L^{1}(M)\to
L^{\infty}(M)}\leq\lambda^{\frac{n-1}{2}+\frac{1}{8}}.$

As a result, since the dyadic operators $B$ are uniformly bounded on $L^{1}$
and $L^{\infty}$, we can repeat the argument that we used to estimate $I$ to
see that

$|II|\leq C\lambda^{\frac{n-1}{2}}\lambda^{\frac{1}{8}}\,\bigl{\|}{\bf
1}_{A_{+}}\cdot g\bigr{\|}_{1}^{2}\leq
C\lambda^{\frac{n-1}{2}}\lambda^{\frac{1}{8}}\,\|g\|^{2}_{L^{q_{c}^{\prime}}(A_{+})}\,\|{\bf
1}_{A_{+}}\|_{q_{c}}^{2}=C\lambda^{\frac{n-1}{2}}\lambda^{\frac{1}{8}}\,\|{\bf
1}_{A_{+}}\|_{q_{c}}^{2}.$

If we recall the definition (2.11) of $A_{+}$, we can estimate the last
factor:

$\|{\bf
1}_{A_{+}}\|_{q_{c}}^{2}\leq\bigl{(}\lambda^{\frac{n-1}{4}+\frac{1}{8}}\bigr{)}^{-2}\|\tilde{\rho}_{\lambda}f\|^{2}_{L^{q_{c}}(A_{+})}.$

Therefore,

$|II|\lesssim\lambda^{-1/8}\|\tilde{\rho}_{\lambda}f\|^{2}_{L^{q_{c}}(A_{+})}\leq\bigl{(}\tfrac{1}{2}\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{+})}\bigr{)}^{2},$

assuming, as we may, that $\lambda$ is large enough.

If we combine this bound with the earlier one, (3.5), for $I$ we conclude that
(3.3) is valid which of course yields (3.2) and completes the proof of
Proposition 3.1. ∎

##  4\. Controlling small heights using microlocal Kakeya-Nikodym estimates

Note that, under our curvature assumption, Proposition 3.1 rules out the
existence of eigenfunctions like the zonal spherical harmonics $Z_{k}$ on
$S^{n}$ that maximally concentrate near points. On the other hand, given
(1.13) it does not rule out the existence of eigenfunctions like the highest
weight spherical harmonics since the $Q_{k}$ vanish on $A_{+}$. To complete
the proof of Theorem 1.1 we also need the following result for the
complementary set, $A_{-}$, which does rule out the existence of these types
of eigenfunctions with maximal concentration near geodesics.

######  Proposition 4.1.

Let $M$ have nonpositive curvature. Then

(4.1) $\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{-})}\leq
C\lambda^{\frac{1}{q_{c}}}\,(\log\lambda)^{-\sigma_{n}}\,\|f\|_{L^{2}(M)},$

where $\sigma_{n}$ is as in Theorem 1.1.

Let us collect the tools from [6] that we shall need to prove (4.1). We first
recall that the symbol $B(x,\xi)$ of $B$ in (2.4) is supported in a conic
neighborhood of some $(x_{0},\xi_{0})\in S^{*}M$. We may assume that its
symbol has small enough support so that we may work in a coordinate chart
$\Omega$ so that $x_{0}=0$, $\xi_{0}=(0,0,\dots,0,1)$ and
$g_{jk}(0)=\delta^{j}_{k}$ in the local coordinates. So we shall assume that
the $x$-support of $B(x,\xi)$ is contained in $\Omega$. We also may assume
that $B(x,\xi)$ vanishes when $\xi$ is outside of a small neighborhood of
$(0,\dots,0,1)$. These reductions and those that follow will contribute to the
number of summands in (2.3); however, it will be clear that the $N$ there will
be independent of $\lambda\gg 1$.

The bilinear arguments we shall need to use will involve microlocal cutoffs
corresponding to angular sectors of aperture $\approx\lambda^{-1/8}$. To
construct them write $x=(x^{\prime},x_{n})$ so that
$x^{\prime}=(x_{1},\dots,x_{n-1})$ denotes coordinates on the hypersurface
where $x_{n}=0$ in our coordinates. We shall want these cutoffs to commute
with the unit-speed geodesic flow $\chi_{t}:\,S^{*}M\to S^{*}M$ on the support
of $B(x,\xi)$ if $|t|\leq\delta$, where $\delta$ is as in (1.3). Recall that
$\chi_{t}$ is the flow generated by the Hamilton vector field associated with
the principal symbol

$p(x,\xi)=\bigl{(}\,\sum_{j,k}g^{jk}(x)\xi_{j}\xi_{k}\,\bigr{)}^{1/2}$

of $P=\sqrt{-\Delta_{g}}$. Here $g^{jk}(x)=\bigl{(}g_{jk}(x)\bigr{)}^{-1}$ is
the cometric and so $S^{*}M=\\{(x,\xi):\,p(x,\xi)=1\\}$.

If $|t|\leq 2\delta$ with $\delta>0$ small enough then the map

(4.2) $(t,x^{\prime},\eta)\to\chi_{t}(x^{\prime},0,\eta)\in
S^{*}M,\quad(x^{\prime},0,\eta)\in S^{*}M$

is a diffeomorphism from a neighborhood of $x^{\prime}=0$, $t=0$ and
$\eta=(0,\dots,0,1)$ to a neighborhood of
$(x_{0},\xi_{0})=(0,(0,\dots,0,1))\in S^{*}M$. Indeed since we are assuming
that $g_{jk}(0)=Identity$ the Jacobian of the map is the identity at
$(x_{0},\xi_{0})$. After possibly shrinking the support, we may assume that
these properties are valid on a neighborhood of $\text{supp }B(x,\xi)$.

Write the inverse of (4.2) as

(4.3)
$S^{*}M\ni(x,\omega)\to\bigl{(}\tau(x,\omega),\Phi(x,\omega),\Theta(x,\omega)\bigr{)}\in(-\delta,\delta)\times\\{y^{\prime}\in{\mathbb{R}}^{n-1}\\}\times
S^{*}_{(\Phi(x,\omega),0)}M.$

Thus, the unit speed geodesic passing through $(x,\omega)\in S^{*}\Omega$
arrives at the plane in our local coordinates where $y_{n}=0$ at
$\Phi(x,\omega)$, has covector $\Theta(x,\omega)\in
S^{*}_{(\Phi(x,\omega),0)}M$ there, and
$\tau(x,\omega)=d_{g}(x,(\Phi(x,\omega),0))$ is the geodesic distance between
$x$ and the point $(\Phi(x,\omega),0)$ on this plane.

We can now define the microlocal cutoffs that we shall use. First, we let
$\nu$ denote a $\lambda^{-1/8}$–separated set in $S^{n-1}$ lying in our conic
neighborhood of $\xi_{0}=(0,\dots,0,1)$. We then choose a partition of unity
$\sum\beta_{\nu}(\xi)=1$ on this set which includes a neighborhood of the
$\xi$-support of $B(x,\xi)$ so that each $\beta_{\nu}$ is supported in a
$2\lambda^{-1/8}$ cap about $\nu\in S^{n-1}$, and so that if
$\beta_{\nu}(\xi)$ is the homogeneous of degree zero extension of
$\beta_{\nu}$ to ${\mathbb{R}}^{n}\backslash 0$ we have

$|D^{\alpha}\beta_{\nu}(\xi)|\leq C_{\alpha}\lambda^{|\alpha|/8}\quad\text{if
}\,\,|\xi|=1.$

Finally, if $\psi\in C^{\infty}_{0}(\Omega)$ equals one in a neighborhood of
the $x$-support of $B(x,\xi)$, and if $\tilde{\beta}\in
C^{\infty}_{0}((0,\infty))$ equals one in a neighborhood of the support of the
Littlewood-Paley bump function in (2.4) we define

(4.4)
$q_{\nu}(x,\xi)=\psi(x)\,\beta_{\nu}\big{(}\Theta(x,\xi/p(x,\xi)\bigr{)}\,\tilde{\beta}\bigl{(}p(x,\xi)/\lambda\bigr{)}.$

It then follows that the pseudo-differential operators $Q_{\nu}$ with these
symbols belong to a bounded subset of $S^{0}_{7/8,1/8}(M)$. We have
constructed these operators so that

$q_{\nu}(x,\xi)=q_{\nu}\bigl{(}\chi_{t}(x,\xi)\bigr{)}\quad\text{on \, supp
}B(x,\xi)\,\,\text{if }\,\,|t|\leq 2\delta.$

Since

$\sigma_{\lambda}=\frac{1}{2\pi}\int e^{it\lambda}e^{-itP}\,\hat{\rho}(t)\,dt$

and $\hat{\rho}(t)=0$, $|t|\geq\delta$, as noted in [6] one can use Egorov’s
theorem and the universal estimates in [19] to see that

(4.5)
$\bigl{\|}\sigma_{\lambda}Q_{\nu}-Q_{\nu}\sigma_{\lambda}\bigr{\|}_{L^{2}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{\frac{1}{q_{c}}-\frac{1}{4}}.$

If $\delta>0$ in (1.3) is small enough we also have

(4.6)
$\tilde{\sigma}_{\lambda}-\sum_{\nu}\tilde{\sigma}_{\lambda}Q_{\nu}=R_{\lambda}\quad\text{where
}\,\|R_{\lambda}\|_{L^{p}\to L^{p}}=O(\lambda^{-N})\quad\forall\,N\,\,\text{if
}\,\,1\leq p\leq\infty.$

Finally, the support properties of the symbols imply that we have the uniform
bounds

(4.7) $\sum_{\nu}\|Q_{\nu}h\|^{2}_{L^{2}(M)}\leq C\|h\|^{2}_{L^{2}(M)}.$

Next, the global estimates that we shall use to prove the small heights bounds
in Proposition 4.1 are the following “global” “$L^{q_{c}}$-microlocal Kakeya-
Nikodym estimates”:

(4.8) $\sup_{\nu}\|Q_{\nu}\rho_{\lambda}\|_{L^{2}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{\frac{1}{q_{c}}}\cdot\bigl{(}\log\lambda\bigr{)}^{-\frac{1}{q_{c}}}.$

This, like (3.1), is valid when $(M,g)$ has non-positive sectional curvatures,
although in the next section we shall see that stronger bounds hold if the
sectional curvatures all assumed to be negative. The inequality (4.8) is a
slight improvement over the corresponding estimate (2.18) in [6].

One obtains (4.8) via a simple interpolation argument and the following
pointwise microlocalized estimates

(4.9)
$K_{\lambda,\mu}(x,y)=T^{-1}\int\hat{\Psi}(t/T)\beta(t/\mu)\,e^{it\lambda}\Bigl{(}Q_{\nu}\circ\cos(tP)\circ
Q_{\nu}^{*}\Bigr{)}(x,y)\,dt\\\
=O\bigl{(}T^{-1}\,\mu^{1-\frac{n-1}{2}}\lambda^{\frac{n-1}{2}}\bigr{)},\,\,\Psi=(\rho)^{2},\,\,\quad
1\leq\mu\leq T=4c_{0}\log\lambda,$

if $\beta\in C^{\infty}_{0}((1/2,2))$ is a Littlewood-Paley bump function
satisfying $1=\sum_{-\infty}^{\infty}\beta(t/2^{j})$ for $t>0$, and $c_{0}>0$
is sufficiently small. To prove this one uses the Hadamard parametrix after
lifting the calculation up to the universal cover. The argument is almost
identical to the proof of (3.8) in [5] or the proof of Theorem 2.1 in [6]. The
pointwise bounds in (4.9) are more favorable than those in (3.1) since the
result of the microlocalization in (4.9) is that number of significant terms
in the calculation only grows linearly in $T$, unlike in the proof of (3.1)
where there is exponential growth. For more details, see the remark at the end
of §5.

By (4.9), the integral operator with kernel $K_{\lambda,\mu}$ maps
$L^{1}(M)\to L^{\infty}(M)$ with norm
$O\bigl{(}T^{-1}\,\mu^{1-\frac{n-1}{2}}\lambda^{\frac{n-1}{2}}\bigr{)}$. By
orthogonality it also maps $L^{2}\to L^{2}$ with norm $O(T^{-1}\mu)$. So by
interpolation, it maps $L^{q_{c}^{\prime}}\to L^{q_{c}}$ with norm
$O(T^{-1}\mu^{\frac{2}{n+1}})$. By taking $\mu=2^{j}\lesssim T$ and adding up
these dyadic estimates we find that if $a$ is as in (3.1) we have that the
operator with kernel

$T^{-1}\int\hat{\Psi}(t/T)(1-a(t))\,e^{it\lambda}\Bigl{(}Q_{\nu}\circ\cos(tP)\circ
Q_{\nu}^{*}\Bigr{)}(x,y)\,dt$

maps $L^{q_{c}^{\prime}}\to L^{q_{c}}$ with norm
$O((\lambda/T)^{\frac{2}{q_{c}}})$. Finally, since the local estimates from
[19] show that when we replace $(1-a(t))$ above by $a(t)$ the resulting
integral operator maps $L^{q_{c}^{\prime}}\to L^{q_{c}}$ with a better norm
norm $O(T^{-1}\lambda^{\frac{2}{q_{c}}})$, we must have

(4.10)
$\sup_{\nu}\|Q_{\nu}\Psi(T(\lambda-P))Q_{\nu}^{*}\|_{L^{q_{c}^{\prime}}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{\frac{2}{q_{c}}}\cdot\bigl{(}\log\lambda\bigr{)}^{-\frac{2}{q_{c}}},$

which implies (4.8) by a simple “$TT^{*}$” argument.

Bilinear decomposition

We also require the (local) bilinear harmonic arguments from [6] that take
advantage of the fact that the norm in (4.1) is taken over $A_{-}$. These
techniques go back to the bilinear Fourier restriction arguments from Tao
[25], Tao, Vargas and Vega [26], Lee [17], Wolff [27] and others.

We shall write using (4.6)

(4.11)
$\bigl{(}\tilde{\rho}_{\lambda}f\bigr{)}^{2}=\sum_{\nu,\tilde{\nu}}(\tilde{\sigma}_{\lambda}Q_{\nu}h)\cdot(\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}}h)+O(\lambda^{-N}),\quad
h=\rho_{\lambda}f.$

We shall organize the pairs of directions $(\nu,\nu^{\prime})\in S^{n-1}\times
S^{n-1}$ just as in [26]. So, let us write $\nu=(\nu^{\prime},\nu_{n})$ where
$\nu^{\prime}\in{\mathbb{R}}^{n-1}$ is near the origin since $\nu$ is close to
$(0,\dots,0,1)$.

To this end, consider the collection of dyadic cubes
$\\{\tau^{j}_{\mu^{\prime}}\\}$ in ${\mathbb{R}}^{n-1}$ of side length
$2^{j}$, where $\tau^{j}_{\mu^{\prime}}$ denotes the translation of
$[0,2^{j})^{n-1}$ by $\mu^{\prime}\in 2^{j}{\mathbb{Z}}^{n-1}$. Two such cubes
are said to be “close” if they are not adjacent but have adjacent parents of
sidelength $2^{j+1}$. If this is the case, we write
$\tau^{j}_{\mu^{\prime}}\sim\tau^{j}_{\tilde{\mu}^{\prime}}$. Close cubes are
separated by a distance which is comparable to $2^{j}$. As noted in [26], any
distinct $\nu^{\prime},\tilde{\nu}^{\prime}\in{\mathbb{R}}^{n-1}$ lie in a
unique pair of close cubes. Thus, there is a unique triple
$j,\mu^{\prime}\tilde{\mu}^{\prime}$ such that
$(\nu^{\prime},\tilde{\nu}^{\prime})\in\tau^{j}_{\mu^{\prime}}\times\tau^{j}_{\tilde{\mu}^{\prime}}$
and $\tau^{j}_{\mu^{\prime}}\sim\tau^{j}_{\tilde{\mu}^{\prime}}$. Since
$\nu^{\prime}$ is close to the origin for our $\nu\in S^{n-1}$, we need only
to consider integers $j\leq 0$.

If we then let $J\ll 0$ be the integer satisfying $2^{J-1}<8\lambda^{-1/8}\leq
2^{J}$, it follows that the sum in (4.11) can be organized as

(4.12) $\left(\sum_{J+1\leq j\leq
0}\,\,\sum_{(\nu^{\prime},\tilde{\nu}^{\prime})\in\\{\tau^{j}_{\mu^{\prime}}\times\tau^{j}_{\tilde{\mu}^{\prime}}:\,\tau^{j}_{\mu^{\prime}}\sim\tau^{j}_{\tilde{\mu}^{\prime}}\\}}+\sum_{(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}}\right)\bigl{(}\tilde{\sigma}_{\lambda}Q_{\nu}h\bigr{)}\cdot\bigl{(}\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}}h\bigr{)},$

where $\Xi_{J}$ denotes the remaining pairs not included in the first sum.
These include diagonal pairs where $\nu^{\prime}=\tilde{\nu}^{\prime}$, and
all $(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}$ satisfy
$|\nu^{\prime}-\tilde{\nu}^{\prime}|\lesssim\lambda^{-1/8}$. Thus, for each
fixed $\nu^{\prime}$, we have
$\\#\\{\tilde{\nu}^{\prime}:\,(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}\\}=O(1)$.

As in [6], write

(4.13)
$\Upsilon^{\text{diag}}(h)=\sum_{(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}}\bigl{(}\tilde{\sigma}_{\lambda}Q_{\nu}h\bigr{)}\cdot\bigl{(}\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}}h\bigr{)},\\\
\text{if }\,\,\lambda^{-1/8}\in(2^{J-4},2^{J-3}],\,\,\text{and }\quad
h=\rho_{\lambda}f.$

Then if we consider the first sum in the left of (4.12) over
$(\nu^{\prime},\tilde{\nu}^{\prime})\notin\Xi_{J}$ along with the error term
in (4.11),

(4.14) $\Upsilon^{\text{far}}(h)=\sum_{J+1\leq j\leq
0}\sum_{(\nu^{\prime},\tilde{\nu}^{\prime})\in\\{\tau^{j}_{\mu^{\prime}}\times\tau^{j}_{\tilde{\mu}^{\prime}}:\,\tau^{j}_{\mu^{\prime}}\sim\tau^{j}_{\tilde{\mu}^{\prime}}\\}}\bigl{(}\tilde{\sigma}_{\lambda}Q_{\nu}h\bigr{)}\cdot\bigl{(}\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}}h\bigr{)}+O(\lambda^{-N}),$

we have the favorable bilinear estimates

(4.15) $\int|\Upsilon^{\text{far}}(h)|^{q/2}\,dx\leq
C\lambda\,\bigl{(}\lambda^{\frac{7}{8}}\bigr{)}^{\frac{n-1}{2}(q-q_{c})}\|h\|_{L^{2}(M)}^{q},\,\,\text{if
}\,\,q\in(\tfrac{2(n+2)}{n},q_{c}).$

For, in the notation of [6], $\Upsilon^{\text{far}}(h)$ here equals
$\Upsilon^{\text{off}}(h)+\Upsilon^{\text{smooth}}(h)$ together with the error
term in (4.11), and so (4.15) follows from (4.7) in [6] and the fact that,
like the error term in (4.11),
$\|\Upsilon^{\text{smooth}}(h)\|_{q_{c}/2}\lesssim\lambda^{-N}\|h\|^{2}_{2}$
for all $N$, due to (4.5) in [6]. Based on this, one obtains (4.15). We note
that the estimates in [6] for the non-trivial part, $\Upsilon^{\text{off}}$ of
$\Upsilon^{\text{far}}$ are based on the bilinear oscillatory integral
estimates of Lee [17]. The terms $\Upsilon^{\text{smooth}}$ in [6] were
trivial microlocal error terms to allow us to use those estimates via
parabolic scaling. We have opted to lump together these two terms into (4.14)
to simplify the notation and argument a tiny bit in what follows.

The significance of (4.15) is that the second power of $\lambda$ in the right
is negative as $q<q_{c}$ there. On the other hand, this makes it somewhat
awkward to take advantage of this power-improvement to prove the $L^{q_{c}}$
bound in our remaining inequality (4.1). However, as in [6], we shall be able
to use (4.15) since the norm in the left hand side of (4.1) is taken over
$A_{-}$. We also note that, we have rewritten the left side of (4.11) as
follows

(4.16)
$(\tilde{\rho}_{\lambda}f)^{2}=(\tilde{\sigma}_{\lambda}h)^{2}=\Upsilon^{\text{diag}}(h)+\Upsilon^{\text{far}}(h),\quad
h=\rho_{\lambda}f.$

Next, let us use the fact that we can use (4.15) and (4.16) to prove a
slightly stronger version of estimates in [6] that will allow us to prove
(4.1). Specifically, as we shall show in the next subsection, we can use
(4.15) to obtain the following

(4.17) $\|\tilde{\sigma}_{\lambda}h\|_{L^{q_{c}}(A_{-})}\leq
C\lambda^{\frac{1}{q_{c}}}\lambda^{-\delta_{n}}\|h\|_{L^{2}(M)}+C\Bigl{(}\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}h\|_{L^{q_{c}}(M)}^{q_{c}}\Bigr{)}^{1/q_{c}},\,\,h=\rho_{\lambda}f,$

for certain $\delta_{n}>0$.

Let us now see how we can use (4.17) along with the microlocal Kakeya-Nikodym
estimate (4.8) to prove Proposition 4.1.

###### Proof of Proposition 4.1.

In this proof, let us express the microlocal log-power gain in (4.8) as

(4.18) $\varepsilon(\lambda)=(\log\lambda)^{-\frac{1}{q_{c}}}.$

With $h=\rho_{\lambda}f$, we have from (4.17) that for $n\geq 2$

(4.19) $\|\tilde{\rho}_{\lambda}f||_{L^{q_{c}}(A_{-})}\leq
C\lambda^{\frac{1}{q_{c}}}(\log\lambda)^{-1}\|f\|_{L^{2}(M)}+C\Bigl{(}\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}\rho_{\lambda}f\|_{L^{q_{c}}(M)}^{q_{c}}\Bigr{)}^{1/q_{c}}.$

Note that the universal (local) spectral projection bounds from [19] and (2.5)
yield

$\|\tilde{\sigma}_{\lambda}\|_{L^{2}\to
L^{q_{c}}}=O(\lambda^{\frac{1}{q_{c}}}).$

Using this and the fact that the $Q_{\nu}$ are almost orthogonal, we find that

(4.20)
$\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}\rho_{\lambda}f\|_{L^{q_{c}}(M)}^{2}\lesssim\lambda^{\frac{2}{q_{c}}}\sum_{\nu}\|Q_{\nu}\rho_{\lambda}f\|_{2}^{2}\lesssim\lambda^{\frac{2}{q_{c}}}\|\rho_{\lambda}f\|_{2}^{2}\lesssim\lambda^{\frac{2}{q_{c}}}\|f\|_{2}^{2}.$

Next, recall that $\tilde{\sigma}_{\lambda}=B\circ\sigma_{\lambda}$ where $B$
satisfies (2.5). Since the dyadic microlocal cutoffs $Q_{\nu}$ are also
uniformly bounded on all $L^{p}$ spaces, if we use (4.5), the first part of
(2.8) and our Kakeya-Nikodym bounds (4.8), we obtain the following bounds
which are uniform in $\nu$:

(4.21) $\displaystyle\|\tilde{\sigma}_{\lambda}Q_{\nu}\rho_{\lambda}$
$\displaystyle
f\|_{L^{q_{c}}(M)}\lesssim\|\sigma_{\lambda}Q_{\nu}\rho_{\lambda}f\|_{L^{q_{c}}(M)}$
$\displaystyle\lesssim\|Q_{\nu}\sigma_{\lambda}\rho_{\lambda}f\|_{L^{q_{c}}(M)}+\lambda^{\frac{1}{q_{c}}-\frac{1}{4}}\|f\|_{L^{2}(M)}$
$\displaystyle\lesssim\|Q_{\nu}\rho_{\lambda}f\|_{L^{q_{c}}(M)}+\|Q_{\nu}\circ(I-\sigma_{\lambda})\rho_{\lambda}f\|_{L^{q_{c}}(M)}+\lambda^{\frac{1}{q_{c}}-\frac{1}{4}}\|f\|_{L^{2}(M)}$
$\displaystyle\lesssim\|Q_{\nu}\rho_{\lambda}f\|_{L^{q_{c}}(M)}+\lambda^{\frac{1}{q_{c}}}(\log\lambda)^{-1}\|f\|_{L^{2}(M)}$
$\displaystyle\lesssim\lambda^{\frac{1}{q_{c}}}\,\varepsilon(\lambda)\,\|f\|_{L^{2}(M)},$

with, $\varepsilon(\lambda)$ as in (4.18).

If we combine (4.19), (4.20) and (4.21), we obtain

$\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{+})}\lesssim\lambda^{\frac{1}{q_{c}}}\,\bigl{(}\varepsilon(\lambda)\bigr{)}^{\frac{q_{c}-2}{q_{c}}}\|f\|_{2}=\lambda^{\frac{1}{q_{c}}}\bigl{(}\varepsilon(\lambda)\bigr{)}^{\frac{2}{n+1}}\,\|f\|_{2}.$

This combined with (3.2) yields

(4.22)
$\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(M)}\lesssim\lambda^{\frac{1}{q_{c}}}(\varepsilon(\lambda))^{\frac{2}{n+1}}\,\|f\|_{L^{2}(M)},$

since $(\varepsilon(\lambda))^{\frac{2}{n+1}}>(\log\lambda)^{-1/2}$. This
gives the bounds (1.5) in Theorem 1.1 if one recalls (2.9). ∎

Proving the bilinear estimates

Let us prove (4.17) by using estimates from [6] along with (4.15).

As in these inequalities $\tilde{\rho}_{\lambda}f=\tilde{\sigma}_{\lambda}h$,
with $h=\rho_{\lambda}f$. So, we need to control

$\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{-})}=\Bigl{(}\,\int_{\\{x:\,|\tilde{\sigma}_{\lambda}h(x)|>\lambda^{\frac{n-1}{4}+\frac{1}{8}}\\}}\bigl{|}(\tilde{\sigma}_{\lambda}h)^{2}\bigr{|}^{q_{c}/2}\,dx\,\Bigr{)}^{1/q_{c}}.$

If we recall (4.16) then of course we have for $q\in(\tfrac{2(n+2)}{n},q_{c})$
as in (4.15)

$|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h|^{q_{c}/2}\leq
2^{q/2}\,|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h|^{\frac{q_{c}-q}{2}}\,\bigl{(}\,|\Upsilon^{\text{diag}}(h)|^{q/2}+|\Upsilon^{\text{far}}(h)|^{q/2}\,\bigr{)}.$

Thus,

(4.23) $\displaystyle\|\tilde{\rho}_{\lambda}$ $\displaystyle
f\|_{L^{q_{c}}(A_{-})}^{q_{c}}=\int_{A_{-}}|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h|^{q_{c}/2}\,dx$
$\displaystyle\lesssim\int_{A_{-}}|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h|^{\frac{q_{c}-q}{2}}\,|\Upsilon^{\text{diag}}(h)|^{q/2}+\int_{A_{-}}|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h|^{\frac{q_{c}-q}{2}}|\,\Upsilon^{\text{far}}(h)|^{q/2}$
$\displaystyle=I+II.$

To estimate the second term we use (4.15), the ceiling for $A_{-}$ and the
fact that $\|f\|_{2}=1$ to see that

$II\lesssim\|\tilde{\sigma}_{\lambda}h\|_{L^{\infty}(A_{-})}^{q_{c}-q}\,\lambda^{-(q_{c}-q)(\frac{7}{8}\frac{n-1}{2})}\cdot\lambda\leq\bigl{(}\,\lambda^{\frac{n-1}{4}+\frac{1}{8}}\,\bigr{)}^{(q_{c}-q)}\cdot\lambda^{-(q_{c}-q)(\frac{7}{8}\frac{n-1}{2})}\cdot\lambda\\\
=\lambda^{-(q_{c}-q)[\frac{3(n-1)}{16}-\frac{1}{8}]}\cdot\lambda=\lambda^{1-\delta_{n}},\quad\delta_{n}>0,$

since $(q_{c}-q)\cdot(\tfrac{3(n-1)}{16}-\tfrac{1}{8})>0$.

To estimate $I$, we use the fact that by Lemma 4.2 in [6],

$\|\Upsilon^{\text{diag}}\|_{L^{{q_{c}}/2}(M)}\lesssim\Bigl{(}\,\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}h\|_{L^{q_{c}}(M)}^{2q^{*}}\,\Bigr{)}^{\frac{1}{q^{*}}}\,+\,\lambda^{-N},$

where $q^{*}=q/2$ if $n\geq 3$ and $q^{*}=3/2$ if $n=2$. Thus, by Hölder’s
inequality followed by Young’s inequality

$\displaystyle I$
$\displaystyle\leq\|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h\|_{L^{q_{c}/2}(A_{-})}^{\frac{q_{c}-q}{2}}\cdot\|\Upsilon^{\text{diag}}\|_{L^{q_{c}/2}(M)}^{q/2}$
$\displaystyle\leq\tfrac{q_{c}-q}{q_{c}}\|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h\|_{L^{q_{c}/2}(A_{-})}^{q_{c}/2}+\tfrac{q}{q_{c}}\|\Upsilon^{\text{diag}}\|_{L^{q_{c}/2}(M)}^{q_{c}/2}$
$\displaystyle\leq\tfrac{q_{c}-q}{q_{c}}\|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h\|_{L^{q_{c}/2}(A_{-})}^{q_{c}/2}+\Bigl{(}\,\sum_{\nu}\,\|\tilde{\sigma}_{\lambda}Q_{\nu}h\|_{L^{q_{c}}(M)}^{2q^{*}}\,\Bigr{)}^{\frac{1}{q^{*}}\frac{q_{c}}{2}}\,+\,\lambda^{-N}.$

Since $\tfrac{q_{c}-q}{q_{c}}<1$, the first term in the right can be absorbed
in the left side of (4.23).

Thus, our bounds imply that for $n\geq 3$

(4.24)
$\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{-})}\lesssim\Bigl{(}\,\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}h\|_{L^{q_{c}}(M)}^{2q^{*}}\,\Bigr{)}^{\frac{1}{2q^{*}}}+\lambda^{\frac{1}{q_{c}}-\delta_{n}/q_{c}}\|f\|_{L^{2}(M)},$

and for $n=2$

(4.25)
$\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{-})}\lesssim\Bigl{(}\,\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}h\|_{L^{q_{c}}(M)}^{3}\,\Bigr{)}^{\frac{1}{3}}+\lambda^{\frac{1}{q_{c}}-\delta_{n}/q_{c}}\|f\|_{L^{2}(M)}.$

Note that the first term on the right side of (4.25) is larger than the one
appearing in (4.17) due to the inclusion $\ell^{6}\subset\ell^{3}$. In the
next subsection we shall see how we can modify the arguments we used for term
$I$ to get an improved bilinear type inequality rather than (4.25) when $n=2$.

Modified bilinear arguments for $n=2$

To be able to have a variant of (4.25) which involves an $\ell^{6}$ sum
instead of $\ell^{3}$, we need to change the definition of $Q_{\nu}$ a bit.
When $n=2$, instead of $\frac{1}{8}$, we fix a small $\varepsilon_{0}$, which
will be specified later, and let $\nu$ denote a
$\lambda^{-\varepsilon_{0}}$–separated set in $S^{1}$ lying in our conic
neighborhood of $\xi_{0}=(0,\dots,0,1)$. We then choose a partition of unity
$\sum\beta_{\nu}(\xi)=1$ on this set which includes a neighborhood of the
$\xi$-support of $B(x,\xi)$ so that each $\beta_{\nu}$ is supported in a
$2\lambda^{-\varepsilon_{0}}$ cap about $\nu\in S^{1}$, and so that if
$\beta_{\nu}(\xi)$ is the homogeneous of degree zero extension of
$\beta_{\nu}$ to ${\mathbb{R}}^{n}\backslash 0$ we have

$|D^{\alpha}\beta_{\nu}(\xi)|\leq
C_{\alpha}\lambda^{|\alpha|\varepsilon_{0}}\quad\text{if }\,\,|\xi|=1.$

If we let $q_{\nu}(x,\xi)$ then be defined as in (4.4), it follows that the
pseudo-differential operators $Q_{\nu}$ with symbols belong to a bounded
subset of $S^{0}_{1-\varepsilon_{0},\varepsilon_{0}}(M)$. And similar to
(4.5), we can use the arguments in [6], which involve applications of Egorov’s
theoerem and the universal bound in [19] to see that

(4.26)
$\bigl{\|}\sigma_{\lambda}Q_{\nu}-Q_{\nu}\sigma_{\lambda}\bigr{\|}_{L^{2}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{\frac{1}{q_{c}}-\frac{1}{2}+2\varepsilon_{0}},$

which is better than the estimate in (4.5) if $\varepsilon_{0}$ is small.

It is straightforward to check that (4.6) and (4.7) hold for the $Q_{\nu}$
operators defined as above, and by repeating the previous arguments, we also
have the analog of (4.8), as long as we let $T=c_{0}\log\lambda$ with
$c_{0}\ll\varepsilon_{0}$. These estimates ensure that the arguments
(4.18)-(4.22), which is the proof of Proposition 4.1, still work in our
current setting.

We are choosing the $\lambda^{-\varepsilon_{0}}$ scale instead of
$\lambda^{-\frac{1}{8}}$, since in our later arguments, we shall make explicit
use of the fact that the number of choices of $\nu$ is small. See the remark
below (4.29) for more details.

Write

$(\tilde{\rho}_{\lambda}h)^{2}=\,\Upsilon^{\text{diag}}(h)+\Upsilon^{\text{far}}(h),$

where $\Upsilon^{\text{far}}(h)$ is defined as in (4.14), and

(4.27)
$\Upsilon^{\text{diag}}(h)=\sum_{(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}}\bigl{(}\tilde{\sigma}_{\lambda}Q_{\nu}h\bigr{)}\cdot\bigl{(}\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}}h\bigr{)},\\\
\text{if }\,\,\lambda^{-\varepsilon_{0}}\in(2^{J-4},2^{J-3}],\,\,\text{and
}\quad h=\rho_{\lambda}f.$

Here as before, $\nu=(\nu^{\prime},\nu_{n})$, and the definition of $\Xi_{J}$
is the same as the one in (4.12), which include diagonal pairs where
$\nu^{\prime}=\tilde{\nu}^{\prime}$, and all
$(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}$ satisfy
$|\nu^{\prime}-\tilde{\nu}^{\prime}|\leq 32\lambda^{-\varepsilon_{0}}$. Thus,
for each fixed $\nu^{\prime}$, we have
$\\#\\{\tilde{\nu}^{\prime}:\,(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}\\}=O(1)$.

By (4.23), to prove improved version of (4.25), it suffices to control the
term $I$ involving $\Upsilon^{\text{diag}}(h)$. The other term $II$ involving
$\Upsilon^{\text{far}}(h)$ can be handled like before and it actually
satisfies better bound since we have improved bilinear estimates with the
wider angular separation for the off-diagonal terms here. To proceed, let us
define

$T_{\nu}h=\sum_{\tilde{\nu}^{\prime}:\,(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}}(\tilde{\sigma}_{\lambda}Q_{\nu}h)(\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}}h),$

and write

(4.28) $\displaystyle(\Upsilon^{\text{diag}}(h))^{2}$
$\displaystyle=\big{(}\sum_{\nu}T_{\nu}h\big{)}^{2}$
$\displaystyle=\sum_{|\nu_{1}^{\prime}-\nu_{2}^{\prime}|\geq
128\lambda^{-\varepsilon_{0}}}T_{\nu_{1}}hT_{\nu_{2}}h+\sum_{|\nu_{1}^{\prime}-\nu_{2}^{\prime}|\leq
128\lambda^{-\varepsilon_{0}}}T_{\nu_{1}}hT_{\nu_{2}}h$ $\displaystyle=A+B.$

Then for $q\in(4,6)$, we have

(4.29) $\displaystyle|\,\Upsilon^{\text{diag}}(h)|^{q/2}$
$\displaystyle\lesssim\big{|}\sum_{|\nu_{1}^{\prime}-\nu_{2}^{\prime}|\geq
128\lambda^{-\varepsilon_{0}}}T_{\nu_{1}}hT_{\nu_{2}}h\big{|}^{\frac{q}{4}}$
$\displaystyle\qquad\qquad+\big{|}\sum_{|\nu_{1}^{\prime}-\nu_{2}^{\prime}|\leq
128\lambda^{-\varepsilon_{0}}}T_{\nu_{1}}hT_{\nu_{2}}h\big{|}^{\frac{q}{4}}$
$\displaystyle=|A|^{\frac{q}{4}}+|B|^{\frac{q}{4}}.$

By splitting $(\Upsilon^{\text{diag}}(h))^{2}$ as a sum of $A$ and $B$ as
above, we essentially want to make use of the bilinear estimates for a second
time, where $A$ corresponds to the off-diagonal terms and $B$ corresponds to
the diagonal ones. However, for the off-diagonal terms, here we are not using
the classical Whitney-type decomposition as in (4.11)-(4.12) since the
$T_{\nu}$ operators themselves are defined as product of the $Q_{\nu}$
operators, which makes it difficult to define the $\sum_{\nu}T_{\nu}$
operators for certain collection of nearby $\nu$ indexes. Instead, we assume
that the number of choices of $\nu$ is small, which can be achieved by
choosing $\varepsilon_{0}$ to be small enough, so that after applying bilinear
estimates for each of the single term in $A$, we can sum them up in a rather
crude but still sufficient way. We shall also use the fact that when $n=2$,
$q_{c}=6$ and thus $\frac{6}{4}\in(1,2)$, which allows us to follow the
strategies in [6] to control the diagonal term $B$ using an almost orthogonal
type inequality(see (4.33) below).

More explicitly, to estimate the term involving $A$ in $I$, let us first
assume that $\nu_{1},\nu_{2}$ are fixed, with
$|\nu_{1}^{\prime}-\nu_{2}^{\prime}|\geq 128\lambda^{-\varepsilon_{0}}$. By
the Cauchy-Schwarz inequality and the remarks below (4.27), it is easy to see
that

$|T_{\nu_{1}}hT_{\nu_{2}}h|\leq
C\big{(}\sum_{\tilde{\nu}_{1}^{\prime}:\,|\tilde{\nu}_{1}^{\prime}-\nu_{1}^{\prime}|\leq
32\lambda^{-\varepsilon_{0}}}|\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{1}}h|^{2}\big{)}\big{(}\sum_{\tilde{\nu}_{2}^{\prime}:\,|\tilde{\nu}_{2}^{\prime}-\nu_{2}^{\prime}|\leq
32\lambda^{-\varepsilon_{0}}}|\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{2}}h|^{2}\big{)}.$

Since the number of terms in both summations are finite, it suffices to
control a single term involving
$|\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{1}}h\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{2}}h|^{2},$
where $|\tilde{\nu}_{1}^{\prime}-\tilde{\nu}_{2}^{\prime}|\approx
2^{j}\in(32\lambda^{-\varepsilon_{0}},1)$. To proceed, let
$\mu_{1},\mu_{2}\in{S}^{1}$ satisfy $|\mu_{1}-\mu_{2}|\approx 2^{j}$, and each
be associated with an operator $Q_{\mu_{1}}$, $Q_{\mu_{2}}$, where
$Q_{\mu_{1}}$ is the pseudo-differential operator with symbol
$q_{\mu_{1}}(x,\xi)$, and we assume $q_{\mu_{1}}(x,\xi)$ is supported in a
$2^{j}$ conical neighborhood of $\mu$, with $q_{\mu_{1}}(x,\xi)\equiv 1$ in a
neighborhood of the support of $Q_{\tilde{\nu}_{1}}$. It is not hard to see
that

$\|(I-Q_{\mu_{1}})\circ Q_{\tilde{\nu}_{1}}\|_{L^{2}\rightarrow
L^{2}}\lesssim_{N}\lambda^{-N},\,\,\forall\,\,\,N>0.$

Thus, if we we define $Q_{\mu_{2}}$ similarly, we have

$|\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{1}}h\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{2}}h|^{2}=|\tilde{\sigma}_{\lambda}Q_{\mu_{1}}Q_{\tilde{\nu}_{1}}h\tilde{\sigma}_{\lambda}Q_{\mu_{2}}Q_{\tilde{\nu}_{2}}h|^{2}+O(\lambda^{-N}).$

Now if we apply bilinear estimates at the scale $2^{j}$ in
$L^{\frac{q}{2}}$(see e.g., (4.12) in [6]), along with the fact that
$\tilde{\sigma}_{\lambda}h$ is bounded by
$\lambda^{\frac{n-1}{4}+\frac{1}{8}}$ (with $n=2$) on the set $A_{-}$, we
have, similar to before,

(4.30)
$\displaystyle\int_{A_{-}}|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h|^{\frac{6-q}{2}}|(\tilde{\sigma}_{\lambda}Q_{\mu_{1}}Q_{\tilde{\nu}_{1}}h)^{2}(\tilde{\sigma}_{\lambda}Q_{\mu_{2}}Q_{\tilde{\nu}_{2}}h)^{2}|^{q/4}\leq\lambda^{1-\delta_{2}},$

for some fixed $\delta_{2}>0$. Here the $\lambda^{-\delta_{2}}$ gain is
independent of $\varepsilon_{0}$ and $j$, whenever we have
$2^{j}\geq\lambda^{-\frac{1}{8}}$.

Since the number of choices of $\nu_{1}^{\prime},\nu_{2}^{\prime}$ in the sum
in $A$ is bounded by $\lambda^{2\varepsilon_{0}}$, (4.30) implies

(4.31)
$\displaystyle\int_{A_{-}}|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h|^{\frac{6-q}{2}}|A|^{q/4}\lesssim\lambda^{1+3\varepsilon_{0}-\delta_{2}}+\lambda^{-N}.$

By choosing $\varepsilon_{0}$ small enough so that
$3\varepsilon_{0}\leq\delta_{2}/2$, this gives us the desired bound.

To control the term involving $B$, let us define

(4.32) $\displaystyle S_{\nu_{1}}h$
$\displaystyle=\sum_{\nu_{2}^{\prime}:|\nu_{1}^{\prime}-\nu_{2}^{\prime}|\leq
128\lambda^{-\varepsilon}}T_{\nu_{1}}hT_{\nu_{2}}h$
$\displaystyle=\sum_{\tilde{\nu}_{1}^{\prime},\nu_{2}^{\prime},\tilde{\nu}_{2}^{\prime}}(\tilde{\sigma}_{\lambda}Q_{\nu_{1}}h)(\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{1}}h)(\tilde{\sigma}_{\lambda}Q_{\nu_{2}}h)(\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{2}}h),$

where $|\tilde{\nu}_{1}^{\prime}-\nu_{1}^{\prime}|\leq
32\lambda^{-\varepsilon_{0}}$,
$|\tilde{\nu}_{2}^{\prime}-\nu_{2}^{\prime}|\leq
32\lambda^{-\varepsilon_{0}}$, $|\nu_{1}^{\prime}-\nu_{2}^{\prime}|\leq
128\lambda^{-\varepsilon_{0}}$, and thus
$|\tilde{\nu}_{2}^{\prime}-\nu_{1}^{\prime}|\leq
160\lambda^{-\varepsilon_{0}}$. Overall, the number of terms in the second
line is finite, and we have $B=\sum_{\nu_{1}}S_{\nu_{1}}h$. If we use a
similar argument as in the proof of Lemma 4.2 in [6], it is not hard to show
the following almost orthogonal type inequality

(4.33)
$\displaystyle\|\sum_{\nu_{1}}S_{\nu_{1}}h\|_{L^{{3}/2}(M)}\lesssim\Bigl{(}\,\sum_{\nu_{1}}\|S_{\nu_{1}}h\|^{\frac{3}{2}}_{L^{{3}/2}(M)}\,\Bigr{)}^{\frac{2}{3}}\,+\,\lambda^{-N},$

which implies that

(4.34) $\displaystyle\|\sum_{\nu_{1}}$ $\displaystyle
S_{\nu_{1}}h\|_{L^{{3}/2}(M)}$
$\displaystyle\lesssim\Bigl{(}\,\sum_{\nu_{1}}\|\sum_{\tilde{\nu}_{1}^{\prime},\nu_{2}^{\prime},\tilde{\nu}_{2}^{\prime}}(\tilde{\sigma}_{\lambda}Q_{\nu_{1}}h)(\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{1}}h)(\tilde{\sigma}_{\lambda}Q_{\nu_{2}}h)(\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}_{2}}h)\|_{L^{{3}/2}(M)}^{\frac{3}{2}}\,\Bigr{)}^{\frac{2}{3}}\,+\,\lambda^{-N}$
$\displaystyle\lesssim\Bigl{(}\,\sum_{\nu_{1}}\|\sigma_{\lambda}Q_{\nu_{1}}h\|_{L^{6}(M)}^{6}\,\Bigr{)}^{\frac{2}{3}}\,+\,\lambda^{-N},$

where in the second inequality we essentially used the fact that for fixed
$\nu_{1}$, the number of terms in the sum is finite.

Thus, by Hölder’s inequality followed by Young’s inequality

$\displaystyle\int_{A_{-}}|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h|^{\frac{6-q}{2}}|B|^{q/4}$
$\displaystyle\leq\|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h\|_{L^{3}(A_{-})}^{\frac{6-q}{2}}\cdot\|B\|_{L^{3/2}(M)}^{q/4}$
$\displaystyle\leq\tfrac{6-q}{6}\|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h\|_{L^{3}(A_{-})}^{3}+\tfrac{q}{6}\|B\|_{L^{3/2}(M)}^{3/2}$
$\displaystyle\leq\tfrac{6-q}{6}\|\tilde{\sigma}_{\lambda}h\,\tilde{\sigma}_{\lambda}h\|_{L^{3}(A_{-})}^{3}+\sum_{\nu_{1}}\,\|\tilde{\sigma}_{\lambda}Q_{\nu_{1}}h\|_{L^{6}(M)}^{6}\,\,+\,\lambda^{-N}.$

Since $\tfrac{6-q}{6}<1$, the first term in the right can be absorbed in the
left side of (4.23).

Thus, the proof of (4.17) for the case $n=2$ is complete.

##  5\. Further improvements for negatively curved manifolds

As we shall see in the next subsection, if we assume that all of the sectional
curvatures of $(M,g)$ are negative then, as we shall show below, we can
improve the second global estimate, (4.8), that we used to

(5.1) $\sup_{\nu}\|Q_{\nu}\rho_{\lambda}\|_{L^{2}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{\frac{1}{q_{c}}}\cdot\varepsilon(\lambda),\quad\text{with
}\,\,\varepsilon(\lambda)=(\log\lambda)^{-1/2}.$

If one repeats the above arguments, one finds that this implies that we can
therefore improve Theorem 1.1 under this curvature assumption as follows.

######  Theorem 5.1.

Assume that $(M,g)$ is an $n$-dimensional compact manifold with all sectional
curvatures being negative. Then, if $T=c_{0}\log\lambda$ with $c_{0}>0$
sufficiently small, we have

(5.2)
$\|\rho(T(\lambda-P))f\|_{L^{q_{c}}(M)}\lesssim\lambda^{1/q_{c}}\,\bigl{(}\log\lambda\bigr{)}^{-\alpha_{n}}\|f\|_{2},$

where $\alpha_{n}=\tfrac{1}{(n+1)}$.

Note that when $n=2$, $q_{c}=6$, and so (5.2) says that

(5.3)
$\|\rho_{\lambda}f\|_{L^{6}(M)}\lesssim\lambda^{\frac{1}{6}}\,(\log\lambda)^{-\frac{1}{3}}\,\|f\|_{L^{2}(M)},\quad
n=2,$

while when $n=3$, $q_{c}=4$, and so (5.2) says that

(5.4)
$\|\rho_{\lambda}f\|_{L^{4}(M)}\lesssim\lambda^{\frac{1}{4}}(\log\lambda)^{-\frac{1}{4}}\,\|f\|_{L^{2}(M)},\quad
n=3.$

Estimates (5.3) and (5.4) should be compared to bounds in Hickman [16] and
Germain and Myerson [14] who extended the toral eigenfunction estimates of
Bourgain and Demeter [10]. As we shall see in §7, the three-dimensional
estimates (5.4) would be optimal if $M$ were a product manifold with $S^{1}$
as a factor; however, in this case, $M$ cannot have all sectional curvatures
negative, since some would have to equal zero. Perhaps not unexpectedly, in
two-dimensions, the bounds in (5.3) are stronger than the sharp estimates that
we have have obtained for two-dimensional tori for spectral windows of width
$(\log\lambda)^{-1}$. It would be interesting to determine whether or not this
phenomena persists in higher dimensions.

Proof of $L^{2}\to L^{q_{c}}$ Kakeya-Nikodym estimates assuming negative
curvature

The main step in the proof of (5.1) will be to establish certain pointwise
estimates using an argument that is almost identical one in [5]. Basically,
the only difference is that, here, we shall exploit the fact that, if all the
sectional curvatures of $(M,g)$ are negative, then the leading term in the
Hadamard parametrix for the solution of the wave equation in the universal
cover of $(M,g)$ decays exponentially in terms of the time parameter. This
fact was not used by two of us in [5] or its predecessor, Sogge and Zelditch
[24], since we were interested in results for general manifolds of non-
positive curvature and the techniques in these papers or [6] only provided
somewhat modest improvements for spectral projection estimates.

Turning to the proof of (5.1), just as in [5], we shall want to make use of
the fact that if $Q_{\nu}(x,y)$ denotes the kernels of our microlocal cutoffs
there, then we have the uniform bounds

(5.5) $\sup_{x}\int|Q_{\nu}(x,y)|\,dy\leq C.$

This implies that $Q_{\nu}:\,L^{\infty}(M)\to L^{\infty}(M)$ uniformly, and,
since they are also uniformly bounded on $L^{2}(M)$, by interpolation and
duality, we conclude that

(5.6) $\|Q_{\nu}\|_{L^{p}(M)\to L^{p}(M)}=O(1),\quad 2\leq p\leq\infty\\\
\text{and }\,\,\|Q^{*}_{\nu}\|_{L^{p}(M)\to L^{p}(M)}=O(1),\quad 1\leq p\leq
2.$

Also, of course if, as in (4.9), $\Psi=\rho^{2}$, then (5.1) is equivalent to
the estimate

(5.7)
$\bigl{\|}\,Q_{\nu}\Psi(T(\lambda-P))Q^{*}_{\nu}\,\bigr{\|}_{L^{q_{c}^{\prime}}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{2/q_{c}}\,(\log\lambda)^{-1},$

if, as before,

(5.8) $T=c_{0}\log\lambda,$

with $c_{0}>0$ sufficiently small.

By (5.6), this would be a consequence of

(5.9)
$\bigl{\|}\,Q_{\nu}\Psi(T(\lambda-P))\,\bigr{\|}_{L^{q_{c}^{\prime}}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{2/q_{c}}\,(\log\lambda)^{-1}.$

The proof of this will closely follow the related $L^{2}\to L^{2}$ Kakeya-
Nikodym estimate (3.6) in [5].

First, as was done in the proof of Proposition 3.1, let us split our smoothed
out spectral projection operators $\Psi(T(\lambda-P))$ into the sum of a
“local” and “global” part. This time, since we shall want to dyadically
decompose the “global” piece, let us assume, as before, that our Littlewood-
Paley function $\beta\in C^{\infty}_{0}((1/2,2))$ satisfies
$\sum_{j=-\infty}^{\infty}\beta(t/2^{j})=1$, $t>0$. We then let

$\beta_{0}(t)=1-\sum_{j=0}^{\infty}\beta(j/2^{j})\in
C^{\infty}_{0}({\mathbb{R}}_{+}),$

and note that $\beta_{0}(t)\equiv 1$ for $t>0$ near the origin. The “local”
operator then is

$L_{\lambda}=\frac{1}{2\pi T}\int e^{i\lambda
t}e^{-itP}\beta_{0}(|t|)\,\hat{\Psi}(t/T)\,dt,$

and we have

$\|Q_{\nu}L_{\lambda}\|_{L^{q^{\prime}_{c}}(M)\to
L^{q_{c}}(M)}=O(\lambda^{2/q_{c}}\,(\log\lambda)^{-1})$

using (5.6), (5.7) and the universal local spectral projection estimates in
[19] just as we did in the proof of Proposition 3.1.

As a result of these local bounds, we would have (5.9) and be done if we could
show that

(5.10) $\|Q_{\nu}G_{\lambda}\|_{L^{q^{\prime}_{c}}(M)\to
L^{q_{c}}(M)}=O(\lambda^{2/q_{c}}\,(\log\lambda)^{-1}),$

if

(5.11) $G_{\lambda}=\frac{1}{2\pi T}\int e^{i\lambda
t}e^{-itP}\,\bigl{(}1-\beta_{0}(|t|)\bigr{)}\,\hat{\Psi}(t/T)\,dt.$

Note that if we replace $e^{-itP}$ by $e^{itP}$ here, then the resulting
operator has a kernel which is $O(\lambda^{-N})$ and so trivially enjoys the
bounds in (5.10).

Consequently, by Euler’s formula, it suffices to show that if

(5.12) $\tilde{G}_{\lambda}=\frac{1}{\pi T}\int e^{i\lambda
t}\cos(t\sqrt{-\Delta_{g}})\,\bigl{(}1-\beta_{0}(|t|)\bigr{)}\,\hat{\Psi}(t/T)\,dt,$

then

(5.13) $\|Q_{\nu}\tilde{G}_{\lambda}\|_{L^{q^{\prime}_{c}}(M)\to
L^{q_{c}}(M)}=O(\lambda^{2/q_{c}}\,(\log\lambda)^{-1}).$

If we let for $\mu=2^{j}\geq 1$

(5.14) $\tilde{G}_{\lambda,\mu}=\frac{1}{\pi T}\int e^{i\lambda
t}\cos(t\sqrt{-\Delta_{g}})\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dt,$

then since

$\tilde{G}_{\lambda}=\sum_{\\{\mu=2^{j}:\,j\geq 0\\}}\tilde{G}_{\lambda,\mu},$

in order to prove (5.13), it suffices to show that

(5.15) $\|Q_{\nu}\tilde{G}_{\lambda,\mu}\|_{L^{q^{\prime}_{c}}(M)\to
L^{q_{c}}(M)}\leq C\lambda^{2/q_{c}}\,(\log\lambda)^{-1}\,\mu^{-1}.$

We note that since $\hat{\Psi}$ is compactly supported, by Huygens’ principle,
$\tilde{G}_{\lambda,\mu}=0$ if $\mu$ is larger than a fixed multiple of $T$ as
in (5.8).

As before, we shall prove (5.15) using interpolation. To this end, we first
note that since the Fourier transform of

$t\to T^{-1}\beta(|t|/\mu)\hat{\Psi}(t/T)$

is $O(\mu/T)$, by (5.6) and orthogonality, we have

(5.16) $\|Q_{\nu}\tilde{G}_{\lambda}\|_{L^{2}(M)\to L^{2}(M)}=O(\mu/T).$

The other estimate that we shall use, which requires our curvature assumption,
is that

(5.17) $\|Q_{\nu}\tilde{G}_{\lambda}\|_{L^{1}(M)\to L^{\infty}(M)}\leq
C_{K,N}T^{-1}\lambda^{\frac{n-1}{2}}\mu^{-N},\quad N=1,2,\dots,$

if all of the sectional curvatures of $(M,g)$ are assumed to be
$\leq-K^{2}<0$. This is analogous to (3.8′) in [5] which did not include the
rapid decay $\mu^{-N}$ since the curvatures there were merely assumed to be
non-positive. Also, there, we did not break things up dyadically as have done
so here.

By a simple interpolation argument (as in the proof of (4.8) above), (5.16)
and (5.17) imply (5.15). So, we have reduced matters to establishing (5.17).

Just as in [5] and other works, we have switched from the half-wave operator
in (5.11) to the use of $\cos t\sqrt{-\Delta_{g}}$ so that we can use the
Hadamard parametrix and the Cartan-Hadamard theorem to lift the calculations
that will be needed for (5.17) up to the universal cover
$({\mathbb{R}}^{n},\tilde{g})$ of $(M,g)$. This approach was used earlier in
[5] and [24].

To this end, let $\\{\alpha\\}=\Gamma$ denote the group of deck transformation
preserving the associated the associated covering map
$\kappa:\,{\mathbb{R}}^{n}\to M$ coming from the exponential map at the point
in $M$ with coordinates $0$ in $\Omega$ above. The metric $\tilde{g}$ on
${\mathbb{R}}^{n}$ that we mentioned above is the pullback of $g$ via
$\kappa$. Choose a Dirichlet domain $D\simeq M$ for $M$ centered at the lift
of the point with coordinates $0$.

Following [2], [24] and others, we recall that if $\tilde{x}$ denotes the lift
of $x\in M$ to $D$, then we have the following formula

$\bigl{(}\cos
t\sqrt{-\Delta_{g}}\bigr{)}(x,y)=\sum_{\alpha\in\Gamma}\bigl{(}\cos
t\sqrt{-\Delta_{\tilde{g}}}\bigr{)}(\tilde{x},\alpha(\tilde{y})),$

which is a cousin of the classical Poisson summation formula. As a result, for
later use, if $K_{\lambda,\mu}(x,y)$ denotes the kernel of
$\tilde{G}_{\lambda,\mu}$ then we have the formula

(5.18)
$K_{\lambda,\mu}(x,y)=\sum_{\alpha\in\Gamma}K_{\lambda,\mu}(\tilde{x},\alpha(\tilde{y})),\\\
\text{if }\,\,K_{\lambda,\mu}(\tilde{x},\tilde{y})=\frac{1}{\pi T}\int
e^{i\lambda t}\,\bigl{(}\cos
t\sqrt{-\Delta_{\tilde{g}}}\bigr{)}(\tilde{x},\tilde{y})\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dt.$

The key estimate for us, which is an improvement of (3.8) in [5] and requires
our curvature assumption is that

(5.19) $|K_{\lambda,\mu}(\tilde{x},\tilde{y})|\leq
C_{K,N}T^{-1}\,\lambda^{\frac{n-1}{2}}\,\bigl{(}d_{\tilde{g}}(\tilde{x},\tilde{y})\bigr{)}^{-\frac{n-1}{2}}\mu^{-N},\quad
N=1,2,\dots.$

To prove this, we note that this kernel vanishes if
$d_{\tilde{g}}(\tilde{x},\tilde{y})$ is larger than $2\mu$ by the Huygens
principle since $\beta(|t|/\mu)=0$ for $|t|\geq 2\mu$. Also, since this
function also vanishes for $|t|\leq\mu/2$, it is straightforward to use the
Hadamard parametrix to see that
$K_{\lambda,\mu}(\tilde{x},\tilde{y})=O(\lambda^{-N})$ for every $N$ if
$d_{\tilde{g}}(\tilde{x},\tilde{y})\leq\mu/4$, which is better than the bounds
in (5.19) as $\mu\lesssim\log\lambda\ll\lambda$.

Due to these simple facts, to prove (5.19), we may assume that
$d_{\tilde{g}}(\tilde{x},\tilde{y})\approx\mu$. We shall make use of the
Hadamard parametrix for $\cos t\sqrt{-\Delta_{\tilde{g}}}$. As is well known
and described, for instance, in [2], [21] and [24], the leading term in this
parametrix is of the form

(5.20)
$w_{0}(\tilde{x},\tilde{y})\cdot(2\pi)^{-n}\int_{{\mathbb{R}}^{n}}e^{id_{\tilde{g}}(\tilde{x},\tilde{y})\xi_{1}}\,\cos(t|\xi|)\,d\xi,$

where in geodesic normal coordinates about $\tilde{x}$ this leading
coefficient is given by

(5.21) $w_{0}(\tilde{x},\tilde{y})=\bigl{(}\,\text{det
}\tilde{g}_{{}_{ij}}(\tilde{y})\,\bigr{)}^{-1/4}.$

Thus, if in geodesic polar coordinates the volume element is given by

$dV_{\tilde{g}}(\tilde{y})=\bigl{(}\,{\cal
A}(t,\omega)\,\bigr{)}^{n-1}\,dtd\omega,\,\,\,t=d_{\tilde{g}}(\tilde{x},\tilde{y}),$

then

$w_{0}(\tilde{x},\tilde{y})=\bigl{(}\,t/{\mathcal{A}}(t,\omega)\,\bigr{)}^{\frac{n-1}{2}}.$

By the classical Günther comparison theorem from Riemannian geometry (see [12,
§III.4])

(5.22) ${\mathcal{A}}(t,\omega)\geq\frac{1}{K}\,\sinh(Kt),$

if, as we are assuming all the sectional curvatures are $\leq-K^{2}<0$. Thus,

(5.23) $w_{0}(\tilde{x},\tilde{y})\leq C_{K,N}\mu^{-N}\quad\text{if
}\,\,\,d_{\tilde{g}}(\tilde{x},\tilde{y})\approx\mu.$

Also, a routine stationary phase calculation yields

(5.24) $T^{-1}\iint e^{i\lambda
t}e^{id_{\tilde{g}}(\tilde{x},\tilde{y})\xi_{1}}\,\cos(t|\xi|)\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dtd\xi\\\
=O(T^{-1}\lambda^{\frac{n-1}{2}}\,(d_{\tilde{g}}(\tilde{x},\tilde{y}))^{-\frac{n-1}{2}}),\quad\text{if
}\,\,d_{\tilde{g}}(\tilde{x},\tilde{y})\approx\mu\geq 1.$

Due to (5.23) and (5.24), if we replace $(\cos
t\sqrt{-\Delta_{\tilde{g}}})(\tilde{x},\tilde{y})$ in (5.18) with the leading
term (5.20) in its Hadamard parametrix, the resulting expression will satisfy
the bounds in (5.19). Since one can also use stationary phase to see that the
lower order terms in the Hadamard parametrix (see, e.g., [21, Theorem 2.4.1])
will each make contributions which are $O(\lambda^{\frac{n-1}{2}-1})$ which is
better than the bounds posited in (5.19), we obtain this bound.

Unfortunately, although (5.19) is sufficient for obtaining bounds like (3.1),
it, by itself, can not be sufficient for proving (5.17) since there are
$\exp(c\mu)$ terms in the sum in (5.18) with
$d_{\tilde{g}}(\tilde{x},\tilde{y})\approx\mu$. To prove (5.17), as in [5], we
need to use the fact that (5.17) includes the microlocal cutoff which means
that the kernel of $Q_{\nu}\tilde{G}_{\lambda,\mu}$,

(5.25) $\sum_{\alpha\in\Gamma}\frac{1}{\pi T}\int e^{i\lambda
t}\,\bigl{(}Q_{\nu}\cos
t\sqrt{-\Delta_{\tilde{g}}}\bigr{)}(\tilde{x},\alpha(\tilde{y}))\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dt,$

only involves $O(\mu)$ non-trivial terms. In the above $Q_{\nu}$ denotes the
pullback of the operator $Q_{\nu}$ to the fundamental domain $D$.

To see this we shall argue exactly as in [5]. We first recall that through the
point $x\in\Omega$ there is a unique geodesic $\gamma_{x,\nu}$ which passes
through the plane $z_{n}=0$ with covector $\xi_{\nu}$ at this point in the
plane. After modifying the coordinates in $\Omega$, we may assume that
$z^{\prime}=0$ at the intersection point in the plane. Then, as in [5], we let
$\tilde{\gamma}(t)$, $t\in{\mathbb{R}}$ denote the lift of the projection of
the geodesic $\gamma$ in $\Omega$ to the universal cover and

${\mathcal{T}}_{R}(\tilde{\gamma})=\\{\tilde{z}:\,d_{\tilde{g}}(\tilde{\gamma},\tilde{z})\leq
R\\}$

denote an $R$-tube about $\tilde{\gamma}$. Then, just as in [5], if $R$ is
fixed sufficiently large and
$\alpha(D)\cap{\mathcal{T}}_{R}(\tilde{\gamma})=\emptyset$, the corresponding
summand in (5.25) is $O(\lambda^{-1})$ by Toponogov’s theorem and microlocal
arguments. Indeed, this is exactly how (3.9) in [5] was proved and one can
simply repeat the arguments there. Since there are $O(\lambda^{1/2})$ non-zero
terms in (5.25) if $c_{0}$ in (5.8) is fixed small enough, we obtain in this
case

$\sum_{\\{\alpha:\,\alpha(D)\cap{\mathcal{T}}_{R}(\tilde{\gamma})=\emptyset\\}}\frac{1}{\pi
T}\int e^{i\lambda t}\,\bigl{(}Q_{\nu}\cos
t\sqrt{-\Delta_{\tilde{g}}}\bigr{)}(\tilde{x},\alpha(\tilde{y}))\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dt=O(\lambda^{-1/2}).$

Therefore, the operator with this kernel maps $L^{1}(M)$ to $L^{\infty}(M)$
with norm $O(\lambda^{-1/2})$, which is much better than the bounds in (5.17)
as $\mu\leq\log\lambda$.

Since (5.17) is equivalent to the statement that the kernel of
$Q_{\nu}\tilde{G}_{\lambda,\mu}$ has sup-norm bounded by the right side of
(5.17), it suffices to show that the remaining part of the sum in (5.25) where
$\alpha(D)\cap{\mathcal{T}}_{R}(\tilde{\gamma})\neq\emptyset$ can be bounded
by the right side of (5.17). As above, the terms where
$d_{\tilde{g}}(\tilde{x},\alpha(\tilde{y}))$ is not comparable to $\mu$ yield
$O(\lambda^{-N})$ contributions and so can be ignored. Thus, since, by (5.6),
$Q_{\nu}:\,L^{\infty}\to L^{\infty}$, our task now would be complete if we
could show that

(5.26)
$\sum_{\\{\alpha:\,\alpha(D)\cap{\mathcal{T}}_{R}(\tilde{\gamma})\neq\emptyset,\,\text{and
}\,d_{\tilde{g}}(\tilde{x},\alpha(\tilde{y}))\approx\mu\\}}\frac{1}{T}\Bigl{|}\,\int
e^{i\lambda t}\,\bigl{(}\cos
t\sqrt{-\Delta_{\tilde{g}}}\bigr{)}(\tilde{x},\alpha(\tilde{y}))\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dt\,\Bigr{|}\\\
\leq C_{K,N}T^{-1}\,\lambda^{\frac{n-1}{2}}\,\mu^{-N},\quad N=1,2,\dots.$

It is easy to see directly or by arguments in [5], this sum only involves
$O(\mu)$ terms. By (5.19), each is of these is
$O(T^{-1}\lambda^{\frac{n-1}{2}}\mu^{-N})$ for any $N$, which yields (5.26)
and completes the proof.

Remark: This argument also yields the microlocal Kakeya-Nikodym estimate (4.8)
in which we were merely assuming that $(M,g)$ has non-positive curvature. In
this case, the leading coefficient of the Hadamard parametrix is no longer
rapidly decreasing. So, we cannot use (5.23). However, we do have that $w_{0}$
is bounded also by the Günther comparison theorem. If we repeat the above
arguments, we would see that, up to an
$O(T^{-1}\lambda^{\frac{n-1}{2}}\mu^{-N})$ error, the kernels
$K_{\lambda,\mu}$ in (4.9) would be dominated by the left side of (5.26).
Since $w_{0}$ is bounded, by (5.24), each summand is
$O(T^{-1}\lambda^{\frac{n-1}{2}}\mu^{-\frac{n-1}{2}})$ since
$d_{\tilde{g}}(\tilde{x},\alpha(\tilde{y}))\approx\mu$ in the sum. Since there
are $O(\mu)$ terms, we conclude that the left side of (5.26) under the
assumption of non-positive curvature is
$O(T^{-1}\lambda^{\frac{n-1}{2}}\mu^{1-\frac{n-1}{2}})$, and so we obtain
(4.9).

##  6\. Some sharp estimates on tori

In this section, we shall see how we can modify the previous arguments to get
the following sharp improved spectral projection bounds on tori.

######  Theorem 6.1.

Let $n\geq 2$,
${\mathbb{T}}^{n}={\mathbb{R}}/(\ell_{1}\mathbb{Z})\times\dots\times{\mathbb{R}}/(\ell_{n}\mathbb{Z})$
denote the rectangular torus with $\ell_{i}\geq 1$. Then,

(6.1)
$\|\rho(T(\lambda-P))f\|_{L^{q_{c}}(\mathbb{T}^{n})}\lesssim\lambda^{1/q_{c}}\,T^{-1/q_{c}}\|f\|_{2},\,\,\,\,\,\forall\,\,1\leq
T\leq\lambda^{\frac{1}{n+3}-\varepsilon}.$

Here $\varepsilon$ is an arbitrary small constant, and for convenience we are
assuming $\ell_{i}\geq 1$, which implies that the injectivity radius of the
torus is bounded below by $\frac{1}{2}$. The implicit constants in (6.1) may
depend on $\ell_{i}$ and $\varepsilon$, but do not depend on $T$ and
$\lambda$. As we shall see later in Proposition 7.1, the $T^{-1/q_{c}}$ gain
in the above estimate is sharp.

To prove (6.1), if we define $\tilde{\rho}_{\lambda}$ as in (2.7), by arguing
as before, it suffices to show that

(6.2)
$\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}\leq\lambda^{1/q_{c}}T^{-1/q_{c}}\|f\|_{2}.$

######  Proposition 6.2.

Let

$A_{+}=\\{x\in{\mathbb{T}}^{n}:\,|\tilde{\rho}_{\lambda}f(x)|\geq
C(T\lambda)^{\frac{n-1}{4}}\,\\}\subset{\mathbb{T}}^{n},$

where $C$ is a large enough constant which may depend on $\ell_{i}$. Then,

(6.3) $\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{+})}\leq
C\lambda^{\frac{1}{q_{c}}}\,T^{-1/2}\,\|f\|_{L^{2}({\mathbb{T}}^{n})}.$

###### Proof.

We first note that, by (2.5) and (2.8), we have

$\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{+})}\leq\|B\rho_{\lambda}f\|_{L^{q_{c}}(A_{+})}+C\lambda^{\frac{1}{q_{c}}}/T,$

since we are assuming that $f$ is $L^{2}$-normalized. Thus, we would have
(6.3) if we could show that

(6.4) $\|B\rho_{\lambda}f\|_{L^{q_{c}}(A_{+})}\leq
C\lambda^{\frac{1}{q_{c}}}T^{-1/2}+\tfrac{1}{2}\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{+})}.$

If we repeat the argument in (3.4), by duality,

$\displaystyle\|B\rho_{\lambda}f\|^{2}_{L^{q_{c}}(A_{+})}$
$\displaystyle\leq\int\big{(}B\circ L_{\lambda}\circ B^{*}\big{)}\bigl{(}{\bf
1}_{A_{+}}\cdot g\bigr{)}(x)\cdot\overline{{\bf 1}_{A_{+}}(x)\,g(x)}\,dx$
$\displaystyle\qquad\qquad+\int\big{(}B\circ G_{\lambda}\circ
B^{*}\big{)}\bigl{(}{\bf 1}_{A_{+}}\cdot g\bigr{)}(x)\cdot\overline{{\bf
1}_{A_{+}}(x)\,g(x)}\,dx$ $\displaystyle=I+II.$

Here, $G_{\lambda}$ is the operator whose kernel is in (3.1), while
$L_{\lambda}$ is the “local” operator

$L_{\lambda}=(2\pi T)^{-1}\int a(t)\hat{\Psi}(t/T)e^{it\lambda}e^{-itP}\,dt.$

As before the local operator $L_{\lambda}$ satisfies

$\|L_{\lambda}\|_{L^{q_{c}^{\prime}}({\mathbb{T}}^{n})\to
L^{q_{c}}({\mathbb{T}}^{n})}\lesssim T^{-1}\lambda^{2/q_{c}}.$

By Hölder’s inequality, we have

(6.5) $\displaystyle|I|$
$\displaystyle\leq\bigl{\|}BL_{\lambda}B^{*}\bigr{(}{\bf 1}_{A_{+}}\cdot
g\bigr{)}\bigr{\|}_{q_{c}}\cdot\|{\bf 1}_{A_{+}}\cdot g\|_{q_{c}^{\prime}}$
$\displaystyle\lesssim\bigl{\|}L_{\lambda}B^{*}\bigr{(}{\bf 1}_{A_{+}}\cdot
g\bigr{)}\bigr{\|}_{q_{c}}\cdot\|{\bf 1}_{A_{+}}\cdot g\|_{q_{c}^{\prime}}$
$\displaystyle\lesssim\lambda^{2/q_{c}}T^{-1}\bigl{\|}B^{*}\bigr{(}{\bf
1}_{A_{+}}\cdot g\bigr{)}\bigr{\|}_{q_{c}^{\prime}}\cdot\|{\bf 1}_{A_{+}}\cdot
g\|_{q_{c}^{\prime}}$
$\displaystyle\lesssim\lambda^{2/q_{c}}T^{-1}\|g\|^{2}_{L^{q_{c}^{\prime}}(A_{+})}$
$\displaystyle=\lambda^{2/q_{c}}T^{-1}.$

To handle the second term, we shall use the fact that if $\beta\in
C^{\infty}_{0}((1/2,2))$ and

(6.6)
$G^{j}_{\lambda}(x,y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\bigl{(}1-a(t)\bigr{)}\beta(t/2^{j})\,T^{-1}\hat{\Psi}(t/T)e^{it\lambda}\,\bigl{(}e^{-itP})(x,y)\,dt,$

then the integral operator with this kernel maps $L^{1}\to L^{\infty}$ with
norm $O\bigl{(}T^{-1}\,2^{\frac{n+1}{2}j}\lambda^{\frac{n-1}{2}}\bigr{)}$. To
see this, note that if we replace $e^{-itP}$ above by $e^{itP}$, the kernel is
$O(\lambda^{-N})$ for all $N>0$. Thus by Euler’s formula, it suffices to show
that for any fixed $x,y$

$\big{|}\int_{-\infty}^{\infty}\bigl{(}1-a(t)\bigr{)}\beta(t/2^{j})\,T^{-1}\hat{\Psi}(t/T)e^{it\lambda}\,\bigl{(}\cos(tP))(x,y)\,dt\big{|}=O\bigl{(}T^{-1}\,2^{\frac{n+1}{2}j}\lambda^{\frac{n-1}{2}}\bigr{)}.$

We postpone the proof of this to (6.26)-(6.29) below.

As a result, since the dyadic operators $B$ are uniformly bounded on $L^{1}$
and $L^{\infty}$, we can repeat the argument that we used to estimate $I$ to
see that

(6.7) $\|G_{\lambda}\|_{L^{1}({\mathbb{T}}^{n})\to
L^{\infty}({\mathbb{T}}^{n})}\leq\sum_{1\leq 2^{j}\leq
T}\|G^{j}_{\lambda}\|_{L^{1}({\mathbb{T}}^{n})\to
L^{\infty}({\mathbb{T}}^{n})}\lesssim(T\lambda)^{\frac{n-1}{2}}.$

Then (6.7) yields

$|II|\leq C_{0}(T\lambda)^{\frac{n-1}{2}}\,\bigl{\|}{\bf 1}_{A_{+}}\cdot
g\bigr{\|}_{1}^{2}\leq
C_{0}(T\lambda)^{\frac{n-1}{2}}\,\|g\|^{2}_{q_{c}^{\prime}}\,\|{\bf
1}_{A_{+}}\|_{q_{c}}^{2}=C_{0}(T\lambda)^{\frac{n-1}{2}}\,\|{\bf
1}_{A_{+}}\|_{q_{c}}^{2}.$

If we recall the definition of $A_{+}$ and use Chebyshev’s inequality we can
estimate the last factor:

$\|{\bf
1}_{A_{+}}\|_{q_{c}}^{2}\leq\bigl{(}C(T\lambda)^{\frac{n-1}{4}}\bigr{)}^{-2}\|\tilde{\rho}_{\lambda}f\|^{2}_{L^{q_{c}}(A_{+})}.$

By choosing the constant $C$ large enough such that
$C^{-2}C_{0}\leq\frac{1}{4}$, we have,

$|II|\leq\tfrac{1}{4}\|\tilde{\rho}_{\lambda}f\|^{2}_{L^{q_{c}}(A_{+})}.$

If we combine this bound with the earlier one, (6.5), for $I$ we conclude that
(6.4) is valid. ∎

######  Proposition 6.3.

Let

(6.8) $A_{-}=\\{x\in{\mathbb{T}}^{n}:\,|\tilde{\rho}_{\lambda}f(x)|\leq
C(T\lambda)^{\frac{n-1}{4}}\,\\}\subset{\mathbb{T}}^{n}.$

Then, for any fixed $\varepsilon>0$ we have

(6.9) $\|\tilde{\rho}_{\lambda}f\|_{L^{q_{c}}(A_{-})}\leq
C\lambda^{\frac{1}{q_{c}}}\,T^{-1/q_{c}}\,\|f\|_{L^{2}({\mathbb{T}}^{n})},\,\,\,\,\,\forall\,\,1\leq
T\leq\lambda^{\frac{1}{n+3}-\varepsilon}.$

In order to make use of the bilinear estimates, we shall work in the
microlocalized modes. First, let $\nu$ denote a $T^{-1}$–separated set in
$S^{n-1}$ lying in a neighborhood of $\xi_{0}=(0,\dots,0,1)$. We then choose a
partition of unity $\sum\beta_{\nu}(\xi)=1$ on this set which includes a
neighborhood of the $\xi$-support of $B(x,\xi)$ so that each $\beta_{\nu}$ is
supported in a $2T^{-1}$ cap about $\nu\in S^{n-1}$, and so that if
$\beta_{\nu}(\xi)$ is the homogeneous of degree zero extension of
$\beta_{\nu}$ to ${\mathbb{R}}^{n}\backslash 0$ we have

$|D^{\alpha}\beta_{\nu}(\xi)|\leq C_{\alpha}T^{|\alpha|}\quad\text{if
}\,\,|\xi|=1.$

And as in (4.4), we define the symbols

(6.10)
$q_{\nu}(x,\xi)=\psi(x)\,\beta_{\nu}\big{(}\Theta(x,\xi/p(x,\xi)\bigr{)}\,\tilde{\beta}\bigl{(}p(x,\xi)/\lambda\bigr{)}.$

where $\psi\in C^{\infty}_{0}(\Omega)$ equals one in a neighborhood of the
$x$-support of $B(x,\xi)$, and $\tilde{\beta}\in C^{\infty}_{0}((0,\infty))$
equals one in a neighborhood of the support of the Littlewood-Paley bump
function in (2.4). In particular, in a local coordinate chart of
${\mathbb{T}}^{n}$, we always have $p(x,\xi)=|\xi|$, which is independent of
$x$. Also recall that as in (4.3), $\Theta(x,\xi/p(x,\xi)\bigr{)}$ is the
third component function of the inverse of the map

(6.11) $(t,x^{\prime},\eta)\to\chi_{t}(x^{\prime},0,\eta)\in
S^{*}M,\quad(x^{\prime},0,\eta)\in S^{*}M,$

in a neighborhood of $x^{\prime}=0$, $t=0$ and $\eta=(0,\dots,0,1)$. And on
the flat torus ${\mathbb{T}}^{n}$, we always have
$\chi_{t}(x^{\prime},0,\eta)=(x^{\prime}+t\eta^{\prime},t\eta_{n},\eta)$ for
$(t,x^{\prime},\eta)$ in this small neighborhood, thus
$\Theta(x,\xi/p(x,\xi)\bigr{)}=\xi/|\xi|$, which is also independent of $x$.
It then follows that, by letting $\xi=\lambda\xi$, the pseudo-differential
operators $Q_{\nu}$ with these symbols have Schwartz kernel

(6.12) $Q_{\nu}(x,y)=\frac{\lambda^{n}}{(2\pi)^{n}}\int e^{i\lambda\langle
x-y,\xi\rangle}\psi(x)\,\beta_{\nu}\big{(}\xi/|\xi|\bigr{)}\,\tilde{\beta}(|\xi|)d\xi,$

which satisfies

(6.13) $\sup_{x}\int|Q_{\nu}(x,y)|\,dy\leq C.$

This implies that $Q_{\nu}:\,L^{\infty}(M)\to L^{\infty}(M)$ uniformly, and,
since they are also uniformly bounded on $L^{2}(M)$, by interpolation and
duality, we conclude that

(6.14) $\|Q_{\nu}\|_{L^{p}(M)\to L^{p}(M)}=O(1),\quad 2\leq p\leq\infty\\\
\text{and }\,\,\|Q^{*}_{\nu}\|_{L^{p}(M)\to L^{p}(M)}=O(1),\quad 1\leq p\leq
2.$

For later use, we shall record several other properties the of $Q_{\nu}$
operators. First, as in (4.5) and (4.26), we can use the arguments in [6] to
see that

(6.15)
$\bigl{\|}\sigma_{\lambda}Q_{\nu}-Q_{\nu}\sigma_{\lambda}\bigr{\|}_{L^{2}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{\frac{1}{q_{c}}-\frac{1}{2}}T^{2},$

Also (4.6) and (4.7) hold for the $Q_{\nu}$ operators defined in (6.12), as
long as $\delta>0$ in (1.3) is small enough. That is

(6.16)
$\tilde{\sigma}_{\lambda}-\sum_{\nu}\tilde{\sigma}_{\lambda}Q_{\nu}=R_{\lambda}\quad\text{where
}\,\|R_{\lambda}\|_{L^{p}\to L^{p}}=O(\lambda^{-N})\quad\forall\,N\,\,\text{if
}\,\,1\leq p\leq\infty,$

and

(6.17) $\sum_{\nu}\|Q_{\nu}h\|^{2}_{L^{2}(M)}\leq C\|h\|^{2}_{L^{2}(M)}.$

Furthermore, as in (4.8), we have the following “global”
“$L^{q_{c}}$-microlocal Kakeya-Nikodym estimates”:

(6.18) $\sup_{\nu}\|Q_{\nu}\rho_{\lambda}\|_{L^{2}({\mathbb{T}}^{n})\to
L^{q_{c}}({\mathbb{T}}^{n})}\lesssim_{\delta_{0}}\lambda^{\frac{1}{q_{c}}}\cdot
T^{-\frac{1}{q_{c}}},\,\,\,\forall\,1\leq T\leq\lambda^{1-\delta_{0}}.$

Here $\delta_{0}$ is a fixed constant that can be arbitrarily small, the
condition $T\leq\lambda^{1-\delta_{0}}$ ensures that
$\lambda^{-1}T\leq\lambda^{-\delta_{0}}$, which implies that the $Q_{\nu}$
operator is rapidly decreasing outside a $\lambda^{-\delta_{0}}$ neighborhood
of the diagonal. The estimate (6.18) is analogous to (4.8) where
$T=c_{0}\log\lambda$ there, while here in the torus case, we can take $T$ to
be almost as large as $\lambda$.

If $\Psi=\rho^{2}$, then (6.18) is equivalent to the estimate

(6.19)
$\bigl{\|}\,Q_{\nu}\Psi(T(\lambda-P))Q^{*}_{\nu}\,\bigr{\|}_{L^{q_{c}^{\prime}}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{2/q_{c}}\,T^{-2/q_{c}},$

By (6.14), this would be a consequence of

(6.20)
$\bigl{\|}\,Q_{\nu}\Psi(T(\lambda-P))\,\bigr{\|}_{L^{q_{c}^{\prime}}(M)\to
L^{q_{c}}(M)}\lesssim\lambda^{2/q_{c}}\,T^{-2/q_{c}}.$

If we repeat the arguments in (5.9)–(5.15), in order to prove (6.20), it
suffices to show that

(6.21) $\|Q_{\nu}\tilde{G}_{\lambda,\mu}\|_{L^{q^{\prime}_{c}}(M)\to
L^{q_{c}}(M)}\leq C\lambda^{2/q_{c}}\,T^{-1}\,\mu^{\frac{2}{n+2}},$

where for $\mu=2^{j}\geq 1$

(6.22) $\tilde{G}_{\lambda,\mu}=\frac{1}{\pi T}\int e^{i\lambda
t}\cos(t\sqrt{-\Delta_{g}})\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dt,$

since, after adding up the dyadic estimates for $\mu=2^{j}\leq T$, this gives
us the right side of (6.20). And as before, the other operators involving
$L_{\lambda}$ and the difference $G_{\lambda}-\tilde{G}_{\lambda}$ satisfy
better estimates.

As in (5.16), by using (6.14), it is not hard to see that

(6.23) $\|Q_{\nu}\tilde{G}_{\lambda}\|_{L^{2}(M)\to L^{2}(M)}=O(\mu/T).$

Thus, by interpolation, (6.21) would be a consequence of

(6.24) $\|Q_{\nu}\tilde{G}_{\lambda}\|_{L^{1}(M)\to L^{\infty}(M)}\lesssim
T^{-1}\lambda^{\frac{n-1}{2}}\mu^{1-\frac{n-1}{2}}.$

As before, we prove (6.24) by lifting the calculations up to the universal
cover, where in our current case the universal cover is just
${\mathbb{R}}^{n}$ with the standard metric. More explicitly, we shall use the
fact that

(6.25) $\bigl{(}\cos
t\sqrt{-\Delta_{{\mathbb{T}}^{n}}}\bigr{)}(x,y)=\sum_{j\in\mathbb{Z}^{n}}\bigl{(}\cos
t\sqrt{-\Delta_{{\mathbb{R}}^{n}}}\bigr{)}(x-(y+j_{\ell})),$

if $j_{\ell}=(\ell_{1}j_{1},\ell_{2}j_{2},\dots,\ell_{n}j_{n})$ for
$j=(j_{1},j_{2},\dots,j_{n})\in\mathbb{Z}^{n}$. This also follows from the
classical Poisson summation formula. Here we are abusing notation a bit by
viewing $x\in{\mathbb{T}}^{n}$ as a point in ${\mathbb{R}}^{n}$, since the
torus can always be identified with the cube
$Q=[-\ell_{1}/2,\ell_{1}/2]\times\dots\times[-\ell_{n}/2,\ell_{n}/2]\in{\mathbb{R}}^{n}$,
which is a fundamental domain for the covering. As a result, if
$K_{\lambda,\mu}(x,y)$ denotes the kernel of $\tilde{G}_{\lambda,\mu}$ then we
have the formula

(6.26)
$K_{\lambda,\mu}(x,y)=\sum_{j\in\mathbb{Z}^{n}}\tilde{K}_{\lambda,\mu}(x,(y+j_{\ell})),$

if

(6.27) $\displaystyle\tilde{K}_{\lambda,\mu}(x,y)$ $\displaystyle=\frac{1}{\pi
T}\int e^{i\lambda t}\,\bigl{(}\cos
t\sqrt{-\Delta_{{\mathbb{R}}^{n}}}\bigr{)}(x,y)\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dt.$
$\displaystyle=\frac{1}{\pi
T}\cdot(2\pi)^{-n}\int\int_{{\mathbb{R}}^{n}}e^{i\lambda t}e^{i\langle
x-y,\xi\rangle}\,\bigl{(}\cos
t|\xi|\bigr{)}\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dtd\xi.$

Note that by finite speed of propagation, the above kernel vanishes if
$|x-y|\geq 2\mu$, since $\beta(|t|/\mu)=0$ for $|t|\geq 2\mu$. Also, since
this function also vanishes for $|t|\leq\mu/2$, by integrating by parts, it is
straightforward to check that $\tilde{K}_{\lambda,\mu}(x,y)=O(\lambda^{-N})$
for every $N$ if $|x-y|\leq\mu/4$. Thus, we may assume that $|x-y|\approx\mu$.
In this case, by using a stationary phase argument, we have

(6.28)
$\tilde{K}_{\lambda,\mu}(x,y)=\sum_{\pm}T^{-1}\lambda^{\frac{n-1}{2}}\mu^{-\frac{n-1}{2}}e^{\pm
i\lambda|x-y|}a_{\pm}(\lambda,|x-y|)+O(\lambda^{-N})\\\
\text{if}\,\,|x-y|\approx\mu\geq 1,$

where for $j\geq 0$, $|\partial_{r}^{j}a_{\pm}(\lambda,r)|\leq
C_{j}r^{-j},\,\,r\geq\lambda^{-1}.$

Since for fixed $x,y$ the number of choices of $j_{\ell}$ such that
$|x-(y+j_{\ell})|\approx\mu$ is $O(\mu^{n})$. By summing up all possible
choices of $j_{\ell}$, (6.28) also implies that

(6.29) $|K_{\lambda,\mu}(x,y)|\lesssim
T^{-1}\lambda^{\frac{n-1}{2}}\mu^{\frac{n+1}{2}},$

which is equivalent to $\|\tilde{G}_{\lambda,\mu}\|_{L^{1}\rightarrow
L^{\infty}}=O(T^{-1}\lambda^{\frac{n-1}{2}}\mu^{\frac{n+1}{2}})$.

Note that the above estimate is not sufficient for proving (6.24). To prove
(6.24), as in the negative curvature case, we need to use the fact that (6.24)
includes the microlocal cutoff which means that the kernel of
$Q_{\nu}\tilde{G}_{\lambda,\mu}$

(6.30) $\displaystyle\sum_{j\in\mathbb{Z}^{n}}$ $\displaystyle\frac{1}{\pi
T}\int e^{i\lambda t}\,\bigl{(}Q_{\nu}\cos
t\sqrt{-\Delta_{{\mathbb{R}}^{n}}}\bigr{)}(x,y+j_{\ell})\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dt$
$\displaystyle=\sum_{j\in\mathbb{Z}^{n}}$ $\displaystyle
K_{\lambda,\mu.\nu}(x,(y+j_{\ell}))$

only involves $O(\mu)$ non-trivial terms for $\mu\gg 1$. In the above
$Q_{\nu}$ denotes the pullback of the operator $Q_{\nu}$ to the fundamental
domain $Q$, which has the same Schwartz kernel as in (6.12).

More explicitly, for fixed $x,y$ and $\mu$, if we define

(6.31)
$D_{main}=\Big{\\{}j\in\mathbb{Z}^{n}:\Big{|}\pm\frac{x-(y+j_{\ell})}{|x-(y+j_{\ell})|}-\nu\Big{|}\leq
C\mu^{-1},\,\,\,|x-(y+j_{\ell})|\approx\mu\Big{\\}},$

and

(6.32)
$D_{error}=\\{j\in\mathbb{Z}^{n}:\Big{|}\pm\frac{x-(y+j_{\ell})}{|x-(y+j_{\ell})|}-\nu\Big{|}\geq
C\mu^{-1},\,\,\,|x-(y+j_{\ell})|\approx\mu\\},$

for some large enough constant $C$. Then the contributions from the terms
$j\in D_{error}$ in (6.30) is negligible. To see this, note that by (6.12) and
(6.28), modulo $O(\lambda^{-N})$ errors, the kernel of
$K_{\lambda,\mu.\nu}(x,y)$ is

(6.33) $\displaystyle C_{T,\lambda,\mu}\iint e^{i\lambda\langle
x-z,\xi\rangle\pm
i\lambda|z-y|}a_{\pm}(\lambda,|z-y|)\psi(x)\tilde{\psi}(z)\,\beta_{\nu}\big{(}\xi/|\xi|\bigr{)}\,\tilde{\beta}(|\xi|)d\xi
dz,$

where
$C_{T,\lambda,\mu}=(2\pi)^{-n}\lambda^{\frac{3n-1}{2}}T^{-1}\mu^{-\frac{n-1}{2}}$
and $\tilde{\psi}\in C^{\infty}_{0}(\Omega)$ equals one in a neighborhood of
the $x$-support of $\psi$. Here we introduced the extra the cut-off function
$\tilde{\psi}(z)$, which is allowed since as mentioned before, the $Q_{\nu}$
operator is rapidly decreasing outside a $\lambda^{-\delta_{0}}$ neighborhood
of the diagonal. And since for fixed $y$, as in (6.28), the integral is taken
over the region $|z-y|\approx u\gg 1$, we can also assume that $|x-y|\approx
u$.

Note that fixed $x$ and $y$ with $|x-y|\approx\mu$, if

(6.34) $\Big{|}\pm\frac{x-y}{|x-y|}-\nu\Big{|}\geq C\mu^{-1},$

for some large enough constant $C$, then it is not hard to check that for all
$z$ in the support of $\tilde{\psi}$ and $\xi$ in the support of $\beta_{\nu}$

(6.35) $\Big{|}\pm\frac{z-y}{|z-y|}-\xi\Big{|}\geq C(2\mu)^{-1}.$

Thus, integration by parts in $z$ in (6.33) yields

(6.36)
$|K_{\lambda,\mu.\nu}(x,y)|\lesssim_{N}(\lambda\mu^{-1})^{-N}\lambda^{\frac{3n-1}{2}}T^{-1}\mu^{-\frac{n-1}{2}},$

which implies that

(6.37) $\sum_{j\in
D_{error}}|K_{\lambda,\mu.\nu}(x,y+j_{\ell})|\lesssim_{N}(\lambda\mu^{-1})^{-N}\lambda^{\frac{3n-1}{2}}T^{-1}\mu^{\frac{n+1}{2}}.$

Since we are assuming $\mu\leq T\leq\lambda^{1-\delta_{0}}$, (6.37) is better
than desired when $N$ is large enough.

On the other hand, it is straightforward to check that $\\#D_{main}=O(\mu)$,
and for each fixed $j\in D_{main}$,

(6.38) $\displaystyle|K_{\lambda,\mu.\nu}(x,y+j_{\ell})|$
$\displaystyle=\Big{|}\frac{1}{\pi T}\int e^{i\lambda t}\,\bigl{(}Q_{\nu}\cos
t\sqrt{-\Delta_{{\mathbb{R}}^{n}}}\bigr{)}(x,y+j_{\ell})\,\beta(|t|/\mu)\,\hat{\Psi}(t/T)\,dt\Big{|}$
$\displaystyle=\Big{|}\int
Q_{\nu}(x,z)\tilde{K}_{\lambda,\mu}(z,y+j_{\ell})dz\Big{|}$
$\displaystyle\leq\sup_{x}\int|Q_{\nu}(x,z)|\,dz\cdot\sup_{z,y}|\tilde{K}_{\lambda,\mu}(z,y)|$
$\displaystyle=O(T^{-1}\lambda^{\frac{n-1}{2}}\mu^{-\frac{n-1}{2}}),$

where used (6.13) and (6.28) in the last line. This implies that

(6.39) $\sum_{j\in
D_{main}}|K_{\lambda,\mu.\nu}(x,y+j_{\ell})|\lesssim\lambda^{\frac{n-1}{2}}T^{-1}\mu^{1-\frac{n-1}{2}}.$

Thus the proof of (6.24) is complete.

Recall that the standard orthonormal basis for Laplacian on the rectangular
torus ${\mathbb{T}}^{n}$ is

$\\{e^{2\pi
i(\frac{j_{1}}{\ell_{1}}x_{1}+\dots+\frac{j_{n}}{\ell_{n}}x_{n})}(\ell_{1}\cdot\dots\ell_{n})^{\frac{1}{2}},\,\,\,j\in\mathbb{Z}^{n}\\}.$

For later use, let us also define

(6.40)
$\tilde{Q}_{\nu}f=\sum_{j\in\mathbb{Z}^{n}}\eta_{\nu}(j_{\ell^{\prime}})a_{j}e^{2\pi
i\langle j_{\ell^{\prime}},\cdot
x\rangle},\,\,\,\text{if}\,\,\,f=\sum_{j\in\mathbb{Z}^{n}}a_{j}e^{2\pi
i\langle j_{\ell^{\prime}},\cdot x\rangle},$

where
$j_{\ell^{\prime}}=(\frac{j_{1}}{\ell_{1}},\dots,\frac{j_{n}}{\ell_{n}})$ for
$j=(j_{1},\dots,j_{n})\in\mathbb{Z}^{n}$ and $\eta_{\nu}\in
C_{0}^{\infty}({\mathbb{R}}^{n})$ equals one in a neighborhood of the support
of $\beta_{\nu}$ and

$\text{supp
}(\eta_{\nu})=\\{\xi\in{\mathbb{R}}^{n}:\big{|}\frac{\xi}{|\xi|}-\nu\big{|}\leq
4T^{-1}\\},$

It is obvious that $\tilde{Q}_{\nu}$ commute with the operators
$\sigma_{\lambda}$ and $\rho_{\lambda}$, and

(6.41) $\sum_{\nu}\|\tilde{Q}_{\nu}f\|_{2}^{2}\lesssim\|f\|_{2}^{2},$

if $\nu$ is chosen from a $T^{-1}$ separated set on ${S}^{n-1}$.

On the other hand, we also have

(6.42)
$\|Q_{\nu}f-Q_{\nu}\tilde{Q}_{\nu}f\|_{L^{p}({\mathbb{T}}^{n})}\leq\lambda^{-N}\|f\|_{2},\,\,\,\forall\,p\geq
2,\,\,\,1\leq T\leq\lambda^{1-\delta_{0}}.$

To see this, since $Q_{\nu}$ is rapidly decreasing outside a
$\lambda^{-\delta_{0}}$ neighborhood of the diagonal, it suffices to show
(6.42) with $Q_{\nu}$ replaced by

(6.43) $Q_{\nu}(x,y)=\frac{\lambda^{n}}{(2\pi)^{n}}\int e^{i\lambda\langle
x-y,\xi\rangle}\psi(x)\tilde{\psi}(y)\,\beta_{\nu}\big{(}\xi/|\xi|\bigr{)}\,\tilde{\beta}(|\xi|)d\xi,$

where $\tilde{\psi}\in C^{\infty}_{0}(\Omega)$ equals one in a neighborhood of
the $x$-support of $\psi$. Now, if $f=\sum_{j\in\mathbb{Z}^{n}}a_{j}e^{2\pi
i\langle j_{\ell^{\prime}},\cdot x\rangle},$

(6.44) $Q_{\nu}f-Q_{\nu}\tilde{Q}_{\nu}f\\\
=\frac{\lambda^{n}}{(2\pi)^{n}}\sum_{j\in\mathbb{Z}^{n}}\int
e^{i\lambda\langle x-y,\xi\rangle+2\pi i\langle
j_{\ell^{\prime}},y\rangle}(1-\eta_{\nu}(j_{\ell}))\psi(x)\tilde{\psi}(y)\,\beta_{\nu}\big{(}\xi/|\xi|\bigr{)}\,\tilde{\beta}(|\xi|)d\xi
dy,$

integrating by parts in $y$ and using Sobolev give us (6.42).

We have collect all the necessary properties we need for $Q_{\nu}$. Now write

$(\tilde{\rho}_{\lambda}h)^{2}=\,\Upsilon^{\text{diag}}(h)+\Upsilon^{\text{far}}(h),$

where $\Upsilon^{\text{far}}(h)$ is defined as in (4.14), and

(6.45)
$\Upsilon^{\text{diag}}(h)=\sum_{(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}}\bigl{(}\tilde{\sigma}_{\lambda}Q_{\nu}h\bigr{)}\cdot\bigl{(}\tilde{\sigma}_{\lambda}Q_{\tilde{\nu}}h\bigr{)},\\\
\text{if }\,\,T^{-1}\in(2^{J-4},2^{J-3}],\,\,\text{and }\quad
h=\rho_{\lambda}f.$

Here as before, $v=(v^{\prime},v_{n})$, and the definition of $\Xi_{J}$ is the
same as the one in (4.12), which include diagonal pairs where
$\nu^{\prime}=\tilde{\nu}^{\prime}$, and all
$(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}$ satisfy
$|\nu^{\prime}-\tilde{\nu}^{\prime}|\lesssim T^{-1}$. Thus, for each fixed
$\nu^{\prime}$, we have
$\\#\\{\tilde{\nu}^{\prime}:\,(\nu^{\prime},\tilde{\nu}^{\prime})\in\Xi_{J}\\}=O(1)$.

For $\Upsilon^{\text{far}}(h)$, we have the favorable bilinear estimates

(6.46) $\int|\Upsilon^{\text{far}}(h)|^{q/2}\,dx\leq C\lambda\,\bigl{(}\lambda
T^{-1}\bigr{)}^{\frac{n-1}{2}(q-q_{c})}\|h\|_{L^{2}(M)}^{q},\,\,\text{if
}\,\,q\in(\tfrac{2(n+2)}{n},q_{c}).$

This is analogous to (4.15), where we replaced $\lambda^{7/8}$ in (4.15) by
$\lambda T^{-1}$ here. And as is pointed out below (4.15), (6.46) essentially
follows form (4.7) in [6] up to rapidly decaying terms.

If we combine (6.46) with the fact that $|\tilde{\rho}_{\lambda}f(x)|\leq
C(T\lambda)^{\frac{n-1}{4}}$ on $A_{-}$, we have

(6.47)
$\displaystyle\int_{A_{-}}|\tilde{\sigma}_{\lambda}h|^{\frac{q_{c}-q}{2}}|\,\Upsilon^{\text{far}}(h)|^{q/2}$
$\displaystyle\lesssim\|\tilde{\sigma}_{\lambda}h\|_{L^{\infty}(A_{-})}^{q_{c}-q}\,\bigl{(}\lambda
T^{-1}\bigr{)}^{\frac{n-1}{2}(q-q_{c})}\cdot\lambda$
$\displaystyle\leq\,\big{(}\lambda
T\big{)}^{\frac{n-1}{4}(q_{c}-q)}\cdot\bigl{(}\lambda
T^{-1}\bigr{)}^{\frac{n-1}{2}(q-q_{c})}\cdot\lambda$
$\displaystyle=\bigl{(}\lambda^{-\frac{n-1}{4}}T^{\frac{3(n-1)}{4}}\bigr{)}^{(q_{c}-q)}\cdot\lambda.$

Thus, if one repeat the arguments in (4.23)–(4.25) for the $Q_{\nu}$ operator
defined in this section, then for dimensions $n\geq 3$, we have

(6.48) $\|\tilde{\sigma}_{\lambda}h\|_{L^{q_{c}}(A_{-})}\leq
C\lambda^{\frac{1}{q_{c}}}\mu_{n}(T,\lambda)\|h\|_{L^{2}({\mathbb{T}}^{n})}+C\Bigl{(}\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}h\|_{L^{q_{c}}({\mathbb{T}}^{n})}^{q_{c}}\Bigr{)}^{1/q_{c}},\,\,h=\rho_{\lambda}f.$

And for $n=2$, where $q_{c}=6$, this is replaced by

(6.49) $\|\tilde{\sigma}_{\lambda}h\|_{L^{6}(A_{-})}\leq
C\lambda^{\frac{1}{6}}\mu_{2}(T,\lambda)\|h\|_{L^{2}({\mathbb{T}}^{2})}+C\Bigl{(}\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}h\|_{L^{6}({\mathbb{T}}^{2})}^{3}\Bigr{)}^{1/3},\quad
h=\rho_{\lambda}f.$

Here
$\mu_{n}(T,\lambda)=(\lambda^{-\frac{n-1}{4}}T^{\frac{3(n-1)}{4}})^{1-\frac{q}{q_{c}}}$,
for any $\tfrac{2(n+2)}{n}<q<q_{c}$. If we require $\mu_{n}(T,\lambda)\leq
T^{-\frac{1}{q_{c}}}$ and let $q\downarrow\tfrac{2(n+2)}{n}$, we get the
condition $T\leq\lambda^{\frac{1}{n+3}-\varepsilon}$ for arbitrary small
$\varepsilon$.

###### Proof of Proposition 6.3.

With $h=\rho_{\lambda}f$ in (6.48), we have for $n\geq 3$,

(6.50) $\|\tilde{\rho}_{\lambda}f||_{L^{q_{c}}(A_{-})}\leq
C\lambda^{\frac{1}{q_{c}}}T^{-\frac{1}{q_{c}}}\|f\|_{L^{2}({\mathbb{T}}^{n})}+C\Bigl{(}\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}\rho_{\lambda}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}^{q_{c}}\Bigr{)}^{1/q_{c}},$

if $1\leq T\leq\lambda^{\frac{1}{n+3}-\varepsilon}$. Note that the universal
(local) spectral projection bounds from [19] yield

$\|\tilde{\sigma}_{\lambda}\|_{L^{2}\to
L^{q_{c}}}=O(\lambda^{\frac{1}{q_{c}}}).$

Using this and (6.42), we have

(6.51) $\displaystyle\sum_{\nu}$
$\displaystyle\|\tilde{\sigma}_{\lambda}Q_{\nu}\rho_{\lambda}f-\tilde{\sigma}_{\lambda}Q_{\nu}\rho_{\lambda}\tilde{Q}_{\nu}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}^{2}$
$\displaystyle=\sum_{\nu}\|\tilde{\sigma}_{\lambda}(Q_{\nu}-Q_{\nu}\tilde{Q}_{\nu})\rho_{\lambda}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}^{2}$
$\displaystyle\lesssim\lambda^{\frac{2}{q_{c}}}\sum_{\nu}\|(Q_{\nu}-Q_{\nu}\tilde{Q}_{\nu})\rho_{\lambda}f\|_{2}^{2}$
$\displaystyle\lesssim\lambda^{\frac{2}{q_{c}}-N}\|\rho_{\lambda}f\|_{2}^{2}\lesssim\lambda^{\frac{2}{q_{c}}-N}\|f\|_{2}^{2}.$

Thus, it suffices to show that

(6.52)
$\Bigl{(}\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}\rho_{\lambda}\tilde{Q}_{\nu}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}^{q_{c}}\Bigr{)}^{1/q_{c}}\leq
C\lambda^{\frac{1}{q_{c}}}T^{-\frac{1}{q_{c}}}\|f\|_{2}.$

First, we show that for fixed $\nu$, Note that by (2.8), (6.14), (6.15) and
(6.18), we have the following bounds which are uniform in $\nu$:

(6.53)
$\displaystyle\|\tilde{\sigma}_{\lambda}Q_{\nu}\rho_{\lambda}\tilde{Q}_{\nu}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}\leq\|\sigma_{\lambda}Q_{\nu}\rho_{\lambda}\tilde{Q}_{\nu}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}$
$\displaystyle\leq\|Q_{\nu}\sigma_{\lambda}\rho_{\lambda}\tilde{Q}_{\nu}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}+C\lambda^{\frac{1}{q_{c}}-\frac{1}{2}}T^{2}\|\tilde{Q}_{\nu}f\|_{2}$
$\displaystyle\lesssim\|Q_{\nu}\rho_{\lambda}\tilde{Q}_{\nu}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}+\|Q_{\nu}\circ(I-\sigma_{\lambda})\rho_{\lambda}\tilde{Q}_{\nu}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}+C\lambda^{\frac{1}{q_{c}}-\frac{1}{2}}T^{2}\|\tilde{Q}_{\nu}f\|_{2}$
$\displaystyle\lesssim\|Q_{\nu}\rho_{\lambda}\tilde{Q}_{\nu}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}+\lambda^{\frac{1}{q_{c}}}T^{-1}\|\tilde{Q}_{\nu}f\|_{2}+\lambda^{\frac{1}{q_{c}}}\,T^{-\frac{1}{q_{c}}}\,\|\tilde{Q}_{\nu}f\|_{2}$
$\displaystyle\lesssim\lambda^{\frac{1}{q_{c}}}\,T^{-\frac{1}{q_{c}}}\,\|\tilde{Q}_{\nu}f\|_{2},$

where we used (2.5) in the first line, (6.15) in the second line, (2.8) in the
fourth line and (6.18) in the last line. We also we used the fact that
$\lambda^{\frac{1}{q_{c}}-\frac{1}{2}}T^{2}\leq\lambda^{\frac{1}{q_{c}}}T^{-\frac{1}{q_{c}}}$
for $1\leq T\leq\lambda^{\frac{1}{n+3}-\varepsilon}$ in the fourth line.

If we combine (6.41) and (6.53), we have

(6.54)
$\Bigl{(}\sum_{\nu}\|\tilde{\sigma}_{\lambda}Q_{\nu}\rho_{\lambda}\tilde{Q}_{\nu}f\|_{L^{q_{c}}({\mathbb{T}}^{n})}^{q_{c}}\Bigr{)}^{1/q_{c}}\leq
C\lambda^{\frac{1}{q_{c}}}T^{-\frac{1}{q_{c}}}(\sum_{\nu}\|\tilde{Q}_{\nu}f\|^{2}_{2})^{\frac{1}{2}}\leq
C\lambda^{\frac{1}{q_{c}}}T^{-\frac{1}{q_{c}}}\|f\|_{2},$

which is desired. The proof for the case $n=2$ is similar. If we combine
Proposition 6.2 and Proposition 6.3, we get (6.2). ∎

Remark: In the next section, we shall show that the results in Theorem 6.1 are
optimal in terms of how the bounds depend on $T$. It would be interesting to
see whether the optimal results in Theorem 6.1 are valid for
$T\lesssim\lambda^{1/2-}$.

##  7\. Geodesic concentration of quasimodes and lower bounds of
$L^{q}(M)$-norms

In this section we shall show that the bounds in Theorem 6.1 are sharp in
terms of their dependence on $T$ even though we expect the bounds to hold for
a larger range of $T$, at least $T\in[1,\lambda^{1/2-}]$. Bourgain and Demeter
[9] and Hickman [16] (see also Germain and Rydin Myerson [14]) obtained near
optimal bounds with $T$ is larger than a fixed power of $\lambda$; however,
since their arguments used decoupling they involved
$\lambda^{\varepsilon}$-losses for arbitrary $\varepsilon>0$.

We shall consider here relationships between the problem of obtaining lower
bounds for the $L^{q}(M)$ norms of quasimodes $\Psi_{\lambda}$ of frequency
$\lambda\gg 1$ on $M$ and the problem of establishing their nontrivial
$L^{2}$-mass concentration near a fixed closed geodesic $\gamma\in M$. Here,
we are assuming that the spectrum associated with $P=\sqrt{-\Delta_{g}}$ of
the quasimodes $\Psi_{\lambda}$ belongs to intervals of the form
$[\lambda-\varepsilon(\lambda),\lambda+\varepsilon(\lambda)]$, where, as
$\lambda$ goes to infinity through a sequence $\lambda_{j}\to+\infty$ we have
$\varepsilon(\lambda)\searrow 0$. We shall also assume that these quasimodes
are $L^{2}$-normalized, i.e.,

(7.1) $\|\Psi_{\lambda}\|_{L^{2}(M)}=1.$

Additionally, to get lower bounds for $L^{q}$-norms with $q>2$, if

(7.2)
${\mathcal{T}}_{\lambda,N}={\mathcal{T}}_{N(\lambda)\cdot\lambda^{-1/2}}(\gamma)=\\{y\in
M:\,d_{g}(y,\gamma)\leq N(\lambda)\cdot\lambda^{-1/2}\\},\,\,\,N=N(\lambda),$

is a tube of width $N(\lambda)\cdot\lambda^{-1/2}$ about our closed geodesic
$\gamma$, we shall assume that we have the following uniform lower bounds for
the $L^{2}({\mathcal{T}}_{\lambda,N})$ mass of our quasimodes

(7.3) $\|\Psi_{\lambda}\|_{L^{2}({\mathcal{T}}_{\lambda,N})}\geq\delta_{0}>0.$

Typically, we shall have $N(\lambda)\to+\infty$ as $\lambda\to+\infty$, and
(7.3) with $N(\lambda)\equiv 1$ corresponds to the concentration of Gaussian
beam quasimodes, which is expected to be the tightest possible concentration
about geodesics. It is also not natural to consider tubes which are wider than
the injectivity radius of $M$. So, by (7.2), it is natural to assume, as we
shall, that

(7.4) $1\leq N(\lambda)=O(\lambda^{1/2-}).$

Note that if we assume (7.3), then by Hölder’s inequality we have for $q>2$

$\delta_{0}\leq\|\Psi_{\lambda}\|_{L^{2}({\mathcal{T}}_{\lambda,N})}\leq|{\mathcal{T}}_{\lambda,N}|^{\frac{1}{2}-\frac{1}{q}}\,\|\Psi_{\lambda}\|_{L^{q}(M)}.$

Since the Riemannian volume of the tubes satisfies
$|{\mathcal{T}}_{\lambda,N}|\approx\bigl{(}N(\lambda)\cdot\lambda^{-1/2}\bigr{)}^{(n-1)}$,
this gives the lower bounds

(7.5)
$\bigl{(}N(\lambda)\bigr{)}^{-(n-1)(\frac{1}{2}-\frac{1}{q})}\,\lambda^{\frac{n-1}{2}(\frac{1}{2}-\frac{1}{q})}\lesssim\|\Psi_{\lambda}\|_{L^{q}(M)}$

for the lower bounds of our quasimodes satisfying (7.3). Tighter concentration
of $L^{2}$-mass near the geodesic (i.e. smaller $N(\lambda)$) implies stronger
lower bounds.

Note that for the standard round sphere, $S^{n}$, the $L^{2}$-normalized
highest weight spherical harmonics $Q_{\lambda}$ are eigenfunctions satisfying
(7.3) with $N(\lambda)\equiv 1$, and the lower bounds in (7.5) with
$\Psi_{\lambda}=Q_{\lambda}$ are sharp since
$\|Q_{\lambda}\|_{L^{q}(S^{n})}\approx\lambda^{\frac{n-1}{2}(\frac{1}{2}-\frac{1}{q})}$
if $q>2$, and this fact shows that the universal bounds in [19] are best
possible for $2<q\leq q_{c}$. (See [18].)

Let us also see here that we can also obtain natural lower bounds for $L^{q}$
estimates for this range of $q$ if $M$ is a product manifold with $S^{1}$ as
one of its factors. So let us assume for now that

(7.6) $M=S^{1}\times X^{n-1}$

is a Cartesian product of $S^{1}$ with an $(n-1)$-dimensional compact manifold
$X^{n-1}$ equipped with the product metric. If $x_{0}\in X^{n-1}$ is fixed,
let us consider the geodesic $\gamma=S^{1}\times\\{x_{0}\\}$ in $M$. Also, for
$\lambda=1,2,3,\dots$ consider the function on $S^{1}\times X^{n-1}$ defined
by

(7.7)
$\Psi_{\lambda}(\theta,x)=(\lambda^{1/2}/N)^{-(n-1)/2}e^{i\lambda\theta}\,\beta\bigl{(}P_{X}/(\lambda^{1/2}/N)\bigr{)}(x_{0},x),\quad
N=N(\lambda),$

where $0\leq\beta\in C^{\infty}_{0}((1/2,2))$ is a Littlewood-Paley bump
function and $P_{X}=\sqrt{-\Delta_{X^{n-1}}}$ is the first order operator on
$X^{n-1}$ coming from its metric Laplacian. Here $\beta(P_{X}/\mu)(x,y)$ is
the kernel of the operator $\beta(P_{X}/\mu)$ on $X=X^{n-1}$ defined by the
spectral theorem.

Let us now prove the following simple result on such product manifolds.

######  Proposition 7.1.

Let $M=S^{1}\times X^{n-1}$ be as in (7.6) with $n\geq 2$. Then if
$\varepsilon(\lambda)\searrow 0$ and (7.4) is valid with
$N(\lambda)=(\varepsilon(\lambda))^{-1/2}$, there is a geodesic $\gamma\in M$
so that we have (7.3). Consequently, we have

(7.8)
$\|\chi_{[\lambda-\varepsilon(\lambda),\lambda+\varepsilon(\lambda)]}\|_{L^{2}(S^{1}\times
X^{n-1})\to L^{q}(S^{1}\times X^{n-1})}\geq
c_{0}\,\bigl{(}\varepsilon(\lambda)\cdot\lambda)^{\frac{n-1}{2}(\frac{1}{2}-\frac{1}{q})},\\\
q\in(2,q_{c}],\quad q_{c}=\tfrac{2(n+1)}{n-1}.$

######  Remark 7.2.

Note that the universal bounds in [19] imply that the operators in (7.8) map
$L^{2}\to L^{q}$ with norm
$O\bigl{(}\lambda^{\frac{n-1}{2}(\frac{1}{2}-\frac{1}{q})}\bigr{)}$, and so
(7.8) says that for manifolds as in (7.6) one can, at best, improve these
bounds by a factor of
$\varepsilon(\lambda))^{\frac{n-1}{2}(\frac{1}{2}-\frac{1}{q})}$.
Consequently, since tori are product manifolds involving $S^{1}$ (perhaps
scaled) as a factor, the bounds in Theorem 6.1 are sharp for the range
$\varepsilon(\lambda)\in[\lambda^{-\frac{1}{n+3}+},1]$ there.

As we pointed out before, Hickman [16] showed that for tori if one includes
additional arbitrary $\lambda^{\varepsilon}$ factors in the right side of
(7.8) then this controls the bounds of the projection operators there. Further
results were obtained by Germain and Rydin Myerson [14] as well as the lower
bound (7.8) for tori using a slightly different and more direct method.

###### Proof.

We shall take $\gamma=S^{1}\times\\{x_{0}\\}$ as mentioned above. We then
notice that, if $\\{\mu_{k}\\}$ are the eigenvalues of $P_{X}$ and
$\\{e_{k}\\}$ is the associated orthonormal basis, then

$\displaystyle\bigl{\|}\beta\bigl{(}P_{X}/(\lambda^{1/2}/N)\bigr{)}(x_{0},\,\cdot\,)\|_{L^{2}(X^{n-1})}^{2}$
$\displaystyle=\sum_{k}\beta^{2}(\mu_{k}/(\lambda^{1/2}/N))\,|e_{k}(x_{0})|^{2}$
$\displaystyle\approx\\#\\{\mu_{k}:\,\mu_{k}\approx(\lambda^{1/2}/N)\\}\approx(\lambda^{1/2}/N)^{n-1}.$

Therefore, the functions in (7.7) have $L^{2}(S^{1}\times X^{n-1})$ norms
which are comparable to one.

Consequently, by (7.5) with $N(\lambda)=(\varepsilon(\lambda))^{-1/2}$ we
would have the lower-bound (7.8) if we could show that for large
$\lambda\in{\mathbb{N}}$

(7.9) $\text{Spec }\Psi_{\lambda}\subset[\lambda-
C_{0}\varepsilon(\lambda),\lambda+C_{0}\varepsilon(\lambda)],$

for some fixed constant, as well as

(7.10) $\|\Psi_{\lambda}\|_{L^{2}(\\{y\in S^{1}\times
X^{n-1}:\,d_{g}(y,\gamma)\leq(\varepsilon(\lambda)\lambda)^{-1/2}\\})}\geq\delta_{0},\quad\gamma=S^{1}\times\\{x_{0}\\},\,\,x_{0}\in
X^{n-1},$

for some fixed $\delta_{0}>0$.

To prove (7.9) we note that $\Psi_{\lambda}(\theta,x)$ is a linear combination
of eigenfunctions on $S^{1}\times X^{n-1}$ of the form
$e^{i\lambda\theta}e_{k}(x)$ where $\lambda\in{\mathbb{N}}$ is fixed and
$\mu_{k}\approx\lambda^{1/2}/N(\lambda)=\sqrt{\varepsilon(\lambda)\lambda}$.
Since

$\sqrt{-\bigl{(}\tfrac{\partial^{2}}{\partial\theta^{2}}+\Delta_{X}\bigr{)}}\bigl{(}e^{i\lambda\theta}e_{k}(x)\bigr{)}=\sqrt{\lambda^{2}+\mu_{k}^{2}}\,\bigl{(}e^{i\lambda\theta}e_{k}(x)\bigr{)}$

and

$\sqrt{\lambda^{2}+\mu_{k}^{2}}-\lambda=O(\varepsilon(\lambda))\quad\text{if
}\,\,\mu_{k}\leq C\sqrt{\varepsilon(\lambda)\lambda},$

we conclude that (7.9) is valid.

To prove (7.10) it suffices to see that if $B(x_{0},r)$ is the geodesic ball
of radius $r\ll 1$ about $x_{0}$ in $X^{n-1}$ then we have the lower bound

(7.11)
$\bigl{(}\varepsilon(\lambda)\lambda\bigr{)}^{-\frac{n-1}{4}}\,\bigl{\|}\beta\bigl{(}P_{X}/(\varepsilon(\lambda)\lambda)^{1/2}\bigr{)}(x_{0},\,\cdot\,)\|_{L^{2}(B(x_{0},\,(\varepsilon(\lambda)\lambda)^{-1/2}))}\geq\delta_{0}>0.$

This is standard calculation. Since
$\varepsilon(\lambda)\lambda\geq\lambda^{\sigma}$ for some $\sigma>0$ because
of our assumption (7.4) one can argue as in [23, §4.3] to see that for large
$\lambda$ we have

$\beta\bigl{(}P_{X}/(\varepsilon(\lambda)\lambda)^{1/2}\bigr{)}(x_{0},y)\approx(\sqrt{\varepsilon(\lambda)\lambda})^{n-1}\quad\text{if
}\,\,\,y\in B(x_{0},\,c_{0}(\varepsilon(\lambda)\lambda)^{-1/2})$

for some fixed $c_{0}>0$, which yields (7.11) and completes the proof. ∎

Concentration problems for manifolds of negative curvature

As we pointed out before the analog of (7.8) with $\varepsilon(\lambda)\equiv
1$ is valid on $S^{n}$ since there are $L^{2}$-normalized eigenfunctions on
$S^{n}$ for which (7.3) is valid with $N(\lambda)\equiv 1$, which means they
have a fixed fraction of their $L^{2}$-mass in $\lambda^{-1/2}$ tubes about a
geodesic. The proof of Proposition 7.1 showed that on tori one can find
$\varepsilon(\lambda)$-quasimodes with fixed $L^{2}$-mass in
$(\varepsilon(\lambda))^{-1/2}\lambda^{-1/2}$ tubes around a closed geodesic
whenever $\varepsilon(\lambda)\in[\lambda^{-1+},1]$. So we have natural
concentration results for quasimodes near geodesics in manifolds of positive
and zero curvature.

We wonder if there are any analogous results for manifolds of negative
curvature. Recently there has be somewhat related work by Brooks [11],
Eswarathasan and Nonnenmacher [13] and others who showed that there is strong
scarring of logarithmic quasimodes on compact quotients of ${\mathbb{H}}^{n}$.
These say that, on such manifolds, if $\gamma$ is a closed geodesic then there
is a sequence of $L^{2}$-normalized quasimodes $\Psi_{\lambda}$ satisfying
(7.9) (or a related $L^{2}$-quasimode condition) so that the quantum
probability measures $|\Psi_{\lambda}|^{2}\,dVol_{g}$ exhibit strong scarring
along $\gamma$. By this we mean that weak limits of these measures must
include a positive multiple of the Dirac mass on $\gamma$, and so the $\Psi$
are tightly concentrated on this geodesic.

It would be interesting if one could find more quantitative versions of these
results that might lead to nontrivial lower bounds for the $L^{2}\to
L^{q_{c}}$ operator norms of the operators in (7.8) with
$\varepsilon(\lambda)=(\log\lambda)^{-1}$ (or perhaps larger). So it would be
interesting to see if for certain $N(\lambda)\nearrow$ we could find
logarithmic quasimodes $\Psi_{\lambda}$ (so
$\varepsilon(\lambda)\approx(\log\lambda)^{-1}$) on hyperbolic quotients for
which we have (7.2). It is routine to see that on any manifold one cannot have
for $\varepsilon(\lambda)\searrow 0$

(7.12)
$\bigl{\|}\chi_{[\lambda-\varepsilon(\lambda),\lambda+\varepsilon(\lambda)]}\bigr{\|}_{L^{2}\to
L^{q_{c}}}=o(\sqrt{\varepsilon(\lambda)}\lambda^{\frac{1}{q_{c}}}),$

since these bounds would imply that the operators map $L^{2}\to L^{\infty}$
with norm $o(\sqrt{\varepsilon(\lambda)}\lambda^{\frac{n-1}{2}})$, and this
would violate the Weyl counting formula.

Even though Hassell and Tacy [15] showed that there is $(\log\lambda)^{-1/2}$
improvements over the universal bounds in [19] on the $L^{2}\to L^{q}$ norms
of these operators for $q>q_{c}$ when $M$ has negative curvature and
$\varepsilon(\lambda)=(\log\lambda)^{-1}$, given Proposition 7.1 one might
expect such an improvement to not hold when $q$ is the critical exponent
$q=q_{c}$. This would follow from showing that there is a closed geodesic
$\gamma$ and $L^{2}$-normalized logarithmic quasimodes as described above so
that one has the non-trivial lower bounds (7.3) with
$N(\lambda)=o((\log\lambda)^{q_{c}/4})=o((\log\lambda)^{\frac{n+1}{2(n-1)}})$.
So, for instance, when $n=2$, if one could find $\Psi_{\lambda}$ as above with
fixed lower bounds of $L^{2}$-mass in
$o((\log\lambda)^{3/2}\lambda^{-\frac{1}{2}})$ neighborhoods of a closed
geodesic $\gamma$, then the $\sqrt{\varepsilon(\lambda)}=(\log\lambda)^{-1/2}$
improvement of Hassell and Tacy [15] for $q>q_{c}$ could not hold for
$q=q_{c}$.

Also, it would be interesting to know whether our estimates in (5.3) for log-
quasimodes on surfaces of negative curvature are sharp. If one could construct
a sequence of such quasimodes for which
$\|\Psi_{\lambda}\|_{L^{2}({\mathcal{T}})}\geq\delta_{0}$ for tubes
${\mathcal{T}}={\mathcal{T}}_{\lambda}(\gamma)$ of width
$\log\lambda\cdot\lambda^{-1/2}$ about a periodic geodesic $\gamma$, then
(5.3) would be saturated. Note that these are much wider tubes than the ones
we used for $S^{1}\times X^{n-1}$, which is to be expected due to the much
faster divergence of geodesics on manifolds of negative curvature.

As was kindly pointed out to us by Eswarathasan, due to the role of the
Ehrenfest time in the constructions, it does not seem likely that the
techniques in [11] or [13] could be used to prove this. It would also be
interesting to see whether such a result might hold for log-log quasimodes, in
which case one would replace the above $\log\lambda$ terms with
$\log\log\lambda$. So $\varepsilon(\lambda)=(\log\log\lambda)^{-1}$ and
$N(\lambda)=o((\log\log\lambda)^{\frac{n+1}{2(n-1)}})$. If we then had (7.3)
this would imply that the $\sqrt{\varepsilon(\lambda)}$ improvements for the
$L^{2}\to L^{q}$ norms of the operators in (7.8) of Hassell and Tacy for
$q>q_{c}$ could not hold for $q=q_{c}$ with
$\varepsilon(\lambda)=(\log\log\lambda)^{-1}$.

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