#  $L^{2}$ to $L^{p}$ bounds for spectral projectors on thin intervals in
Riemannian manifolds

Pierre Germain  Pierre Germain, Department of Mathematics, Huxley Building,
South Kensington Campus, Imperial College London, London SW7 2AZ, United
Kingdom<EMAIL_ADDRESS>
###### Abstract.

Given a Riemannian manifold with Laplace-Beltrami operator $\Delta$, consider
the projector on a thin interval $[\lambda-\delta,\lambda+\delta]$ associated
to the self-adjoint operator $\sqrt{-\Delta}$. Viewed as an operator from
$L^{2}(M)$ to $L^{p}(M)$, what is its operator norm? For $\delta=1$, this
question is fully answered by a celebrated theorem of Sogge, which applies to
any manifold. For $\delta<1$, the global geometry of the manifold comes into
play, and connections to a number of mathematical fields (such as Differential
Geometry, Combinatorics, Number Theory) appear, but the problem remains mostly
open. The aim of this article is to review known results, focusing on cases
exhibiting symmetry (where the regime where $\delta$ is polynomially small in
$\lambda$ becomes approachable) and emphasizing harmonic analytic rather than
microlocal methods.

###### Key words and phrases:

Riemannian manifolds, $L^{p}$ estimates, spectral projectors, eigenfunctions,
quasimodes

###### 2000 Mathematics Subject Classification:

…

###### Contents

  1. 1 Introduction
  2. 2 Sogge’s universal estimates for $\delta=1$
  3. 3 The Euclidean space and its quotients
  4. 4 The hyperbolic space and its quotients
  5. 5 Classical estimates implying spectral projector bounds
  6. 6 $L^{2}\to L^{p}$ projector bounds beyond complete smooth Riemannian manifolds
  7. 7 Some interesting questions

##  1\. Introduction

Consider a (smooth, complete) Riemannian manifold $M$, of dimension $d$, with
Laplace-Beltrami operator $\Delta$. Define the spectral projector
$P_{\lambda,\delta}$ through functional calculus by

$P_{\lambda,\delta}=\mathbf{1}_{[\lambda-\delta,\lambda+\delta]}\left(\sqrt{-\Delta}\right).$

It is the orthogonal projector on the (generalized) eigenmodes of
$\sqrt{-\Delta}$ with eigenvalue in $[\lambda-\delta,\lambda+\delta]$. We ask
the following question:

What is the operator norm of $P_{\lambda,\delta}$ from $L^{2}(M)$ to
$L^{p}(M)$?

A more classical question is to ask for $L^{p}$ bounds of $L^{2}$-normalized
eigenfunctions on compact manifolds, which would correspond to $\delta=0$.
Similarly, the question above could be formulated in terms of $L^{p}$ bounds
for $L^{2}$-normalized almost-eigenfunctions.

Why is it meaningful to consider the case $\delta>0$? First, this has the
advantage of applying to any manifold, be it compact or not, with discrete or
continuous spectrum, and this more general viewpoint is helpful. Second, in
the case of compact manifolds, the case $\delta=0$ seems in general completely
out of reach, while the case $\delta=1$ is fully understood, as we will see;
thus, interpolating between these two situations seems natural. Finally,
considering a window of positive width encodes information on the distribution
of the spectrum, together with the size of the eigenfunctions.

Before reviewing the theory, let us observe that the question of estimating
$\|P_{\lambda,\delta}\|_{L^{2}\to L^{p}}$ can be asked in different ways.
First, an immediate $TT^{*}$ argument shows that

(1.1) $\|P_{\lambda,\delta}\|_{L^{2}\to
L^{p}}^{2}=\|P_{\lambda,\delta}\|_{L^{p^{\prime}}\to L^{p}}.$

Second, it is essentially equivalent to estimate the operator norm of
$P_{\lambda,\delta}$, or that of the smoothed out version

$P_{\lambda,\delta}^{\flat}=\chi\left(\frac{\sqrt{-\Delta}-\lambda}{\delta}\right),$

where $\chi$ is a cutoff function; we will use indiscriminately all these
formulations111Up to logarithmic factors, it is also equivalent to consider
boundedness of the resolvent operator $(\Delta-z)^{-1}$, $\mathfrak{Im}z\neq
0$..

A beautiful textbook presentation of the theory for $\delta=1$ can be found in
[92], see also [91, 105, 106]. In this review note, we will try to give an
overview of progress on the case $\delta<1$, in particular in the case where
$\delta$ is polynomially small in $\lambda$.

A very closely connected topic is that of Quantum Unique Ergodicity for
negatively curved manifolds, or for general manifolds the description of
possible weak-$L^{2}$ limits of eigenfunctions. This is in many ways an
$L^{2}$ version of the $L^{p}$-problems which will occupy us in the present
review, with much of the same underlying phenomenology. We refer to [2, 81]
for surveys on Quantum Unique Ergodicity.

##  2\. Sogge’s universal estimates for $\delta=1$

###  2.1. Sogge’s theorem

In this subsection, we consider general compact manifolds $M$, without any
further restriction. Denoting by $(\lambda_{j})$ and $(\varphi_{j})$ the
eigenvalues and eigenvectors of the Laplacian on $M$, Hörmander’s bound [55],
see also [5, 74] states that

(2.1)
$\|\varphi_{j}\|_{L^{\infty}}\lesssim\lambda_{j}^{\frac{d-1}{2}}\|\varphi_{j}\|_{L^{2}}$

(this estimate arose in connection with the Weyl law, to which we will come
back in Section 5.1).

This classical bound was subsumed in the following much stronger result of
Sogge: it covers any $p>2$, allows for spectral projectors instead of
eigenfunctions, and it is not only providing an upper bound, but also a lower
bound. It will be convenient to denote

(2.2)
$\gamma(p)=\max\left[\frac{d-1}{2}-\frac{d}{p}\,,\,\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)\right],$

since this exponent will appear throughout the present review.

######  Theorem 2.1 (Sogge).

If $M$ is a complete Riemannian manifold with bounded geometry, then

(2.3) $\|P_{\lambda,1}\|_{L^{2}\to
L^{p}}\lesssim\lambda^{\gamma(p)}\qquad\mbox{if $2\leq p\leq\infty$}.$

Furthermore, there exists a constant $R_{0}$ such that: for any $\lambda_{0}$,
there exists $\lambda$ with $|\lambda-\lambda_{0}|<R_{0}$ and

$\|P_{\lambda,1}\|_{L^{2}\to L^{p}}\gtrsim\lambda^{\gamma(p)}\qquad\mbox{if
$2\leq p\leq\infty$}.$

The exponent $\gamma(p)$ is defined as a maximum in (2.2), and which of the
two terms in this maximum is dominant depends on how $p$ compares compares to
the Stein-Tomas exponent defined by

$p_{ST}=\frac{2(d+1)}{d-1}.$

More precisely

  * •

For $p\geq p_{ST}$, $P_{\lambda,1}$ is in the point-focusing regime: the term
$\lambda^{\frac{d-1}{2}-\frac{d}{p}}$ is dominant, and examples of functions
which focus at a point can be shown to achieve this value (up to a bounded
factor) for $\frac{\|P_{\lambda,1}f\|_{L^{p}}}{\|f\|_{L^{2}}}$.

  * •

For $2\leq p\leq p_{ST}$, $P_{\lambda,1}$ is in the geodesic-focusing regime:
the term $\lambda^{\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}$ is
dominant. Functions can be constructed which focus on a geodesic and achieve
this value (up to a bounded factor) for
$\frac{\|P_{\lambda,1}f\|_{L^{p}}}{\|f\|_{L^{2}}}$.

Figure 1. Consider a coordinate patch on a Riemannian manifold. One can then
contruct spectrally localized functions ($P_{\lambda,\delta}f=f$) which
concentrate their $L^{p}$ mass in a neighborhood of a given point (red), or a
given geodesic (blue). These functions saturate the value
$\lambda^{\gamma(p)}$ for the operator norm in (2.3) up to a bounded factor in
the regimes $p>p_{ST}$ and $p<p_{ST}$, respectively.

###### Proof of Theorem 2.3.

A detailed proof can be found in [88, 92] in the compact case, see also [AGL]
for the simple extension to the complete case. A sketch of the upper bound is
as follows.

First, one can reduce matters to an operator $\widetilde{P}_{\lambda,1}$ which
is a slightly modified version of $P_{\lambda,1}$, simpler to estimate since
it is local. Through pseudodifferential calculus, one can then obtain a very
precise description of the kernel of $\widetilde{P}_{\lambda,1}$:

(2.4) $\widetilde{P}_{\lambda,1}f(x)=\int_{M}K_{\lambda,1}(x,y)f(y)\,dx,\qquad
K_{\lambda,1}(x,y)=\lambda^{\frac{d-1}{2}}a(x,y)e^{i\lambda\psi(x,y)}+O(\lambda^{-\infty}),$

where $a\in\mathcal{C}_{0}^{\infty}$. The desired result follows by
interpolation between the following Lebesgue exponents

  * •

For $p=2$, the bound $O(1)$ is an immediate consequence of the definition of
$\widetilde{P}_{\lambda,1}$.

  * •

For $p=\infty$, the bound $O(\lambda^{\frac{d-1}{2}})$ follows immediately
from the formula (2.4) and the Cauchy-Schwarz inequality.

  * •

Finally, $p=p_{ST}$ is the critical exponent. The properties of the phase
$\psi(x,y)$ now come into play: indeed, it satisfies the Carleson-Sjölin
condition, which, roughly speaking, requires that its Hessian has rank $n-1$.
The $L^{2}\to L^{p}$ bounds on Carleson-Sjölin operators (due in dimension 2
to Carleson-Sjölin [32] and Hörmander [56], and in higher dimension to Stein
[98]) then give the desired result. These bounds can be thought of as a
variable coefficient generalization of the Stein-Tomas theorem, which will
presented in Section 3.

∎

As observed in [3], a consequence of the optimality of the Sogge estimates is
the universal lower bound

(2.5) $\|P_{\lambda,\delta}\|_{L^{2}\to
L^{p}}\gtrsim\lambda^{\gamma(p)}\delta^{\frac{1}{2}}\qquad\mbox{if $2\leq
p\leq\infty$}$

(more precisely: for any $\lambda_{0},\delta$, there exists $\lambda$ such
that $|\lambda-\lambda_{0}|<R_{0}$, and such that the above holds).

###  2.2. The case of the sphere

As we saw, the universal Sogge estimate (2.3) corresponds to the case
$\delta=1$, and is sharp for any manifold. Is it possible to improve this
bound by taking $\delta$ smaller? The example of the sphere shows that the
answer is negative in general. Indeed, the spectrum of the Laplace-Beltrami
operator on $\mathbb{S}^{d}$ is given by $\\{k(k+d-1),\,k\in\mathbb{Z}\\}$, so
that

$P_{\lambda,\delta}=P_{\lambda,\frac{1}{2}}\qquad\mbox{if}\;\;\delta<\frac{1}{2},\;\;\lambda=\sqrt{k(k+d-1)},\;k\in\mathbb{Z}$

(and $\lambda$ sufficiently big). This implies that operator bounds for
$P_{\lambda,\delta}$ and $P_{\lambda,\frac{1}{2}}$ are equal and that the
universal Sogge estimate (2.3) will be saturated by eigenfunctions; the two
following examples appear in [87] and are fundamental.

  * •

In the point-focusing regime, zonal spherical harmonics saturate (2.3). By
definition, zonal functions can be written as functions of the geodesic
distance to a point on the sphere. Each eigenspace contains a zonal function,
unique up to the symmetry group $SO(d+1)$. Normalizing this function in
$L^{2}$, it reaches a size $\sim\lambda^{\frac{d-1}{2}}$ on a set of diameter
$\sim\lambda^{-1}$ around the pole, leading to the estimate
$\|f\|_{L^{p}}\sim\lambda^{\frac{d-1}{2}-\frac{d}{p}}\|f\|_{L^{2}}$ for
$p>p_{ST}$.

  * •

In the geodesic-focusing regime, highest weight spherical harmonics saturate
(2.3). Viewing the sphere $\mathbb{S}^{d-1}$ as embedded in
$\mathbb{R}^{d+1}$, they can be taken to be the trace on the sphere of
$(x_{1}+ix_{2})^{k}$. Normalizing this function in $L^{2}$, it reaches a size
$\sim\lambda^{\frac{d-1}{4}}$ on a neighbourhood of size
$\sim\lambda^{-\frac{1}{2}}$ around the great circle $x_{1}^{2}+x_{2}^{2}=1$,
leading to the estimate
$\|f\|_{L^{p}}\sim\lambda^{\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}$
for $p<p_{ST}$.

Figure 2. On the sphere, zonal spherical harmonics concentrate most of their
$L^{p}$ mass close to the North pole (in red), and highest weight spherical
harmonics concentrate most of their $L^{p}$ mass close to the equator (in
blue). These eigenfunctions of the Laplacian saturate the Sogge bound (2.3)
for $p>p_{ST}$ and $p<p_{ST}$ respectively. Of course, the North pole and the
equator do not play a distinguished role, since the Laplacian is invariant
under $O(d+1)$, and this figure can be rotated.

###  2.3. When are Sogge’s estimates optimal for eigenfunctions?

We saw that, in the case of the sphere, the Sogge bound (2.3) agrees with the
optimal bound for eigenfunctions, or in other words that the spectral
projectors for $\delta=0$ and $\delta=1$ have comparable operator norms. Is it
possible to characterize all manifolds for which this phenomenon occurs?

####  2.3.1. Point-focusing regime

A line of research initiated by Sogge and Zelditch [94, 95, 96, 90] and
pursued by Canzani and Galkowski [29, 31] seeks conditions on a compact
manifold $M$ under which there exists, or not, a sequence of eigenfunctions
$\varphi_{j}$ associated to eigenvalues $\lambda_{j}$ exhibiting maximum
eigenfunction growth, namely

(2.6)
$\|\varphi_{j}\|_{L^{\infty}}\sim\lambda_{j}^{\frac{d-1}{2}}\|\varphi_{j}\|_{L^{2}}.$

Of course, this implies that $\|P_{\lambda,\delta}\|_{L^{2}\to
L^{\infty}}\gtrsim\lambda^{\frac{d-1}{2}}$ for any $\delta>0$, along a
sequence of $\lambda$.

While very powerful and precise results have been proved along these lines, we
will give here a simple example of such a statement, which can be found in
[93]. For $x\in M$, define $\mathcal{L}_{x}\subset S_{x}^{*}M$ (unit cotangent
bundle) to be the set of directions $\xi\in T_{x}^{*}M$ such that the geodesic
$\gamma$ with initial data $\gamma(0)=x$, $\dot{\gamma}(0)=\xi$ returns to
$x$. Then there cannot exists a sequence of eigenfunctions with maximal
eigenfunction growth unless $\mathcal{L}_{x}$ has positive measure for some
$x\in M$.

This is a strong constraint, of course reminiscent of the case of the sphere;
but it is not stable under perturbation. Indeed, it was showed [94] that for
generic metrics,
$\|\varphi_{j}\|_{L^{\infty}}=o(\lambda_{j}^{\frac{d-1}{2}})$.

Finally, we mention the original point of view in [100], which relates
poinwise $L^{\infty}$ growth to correlations between eigenfunctions at
different points of the manifold.

####  2.3.2. Geodesic-focusing regime

We call a geodesic stable if its associated Poincaré map only has eigenvalues
of modulus 1. If there exists a stable closed geodesic, one can construct a
sequence $(\lambda_{j})$ going to infinity and associated quasimodes $(f_{j})$
such that

$\left\|\left[\Delta+\lambda_{j}^{2}\right]f_{j}\right\|_{L^{2}}\lesssim_{N}\lambda_{j}^{-N}\qquad\mbox{for
any $j,N\in\mathbb{N}$;}$

furthermore, these quasimodes concentrate within
$\frac{1}{\sqrt{\lambda_{j}}}$ of the geodesic, and are bounded pointwise by
$\lambda_{j}^{\frac{d-1}{4}}$. To the best of our knowledge, this exact
statement does not appear in the literature, but should follow from the
construction of quasimodes supported on stable geodesics [51, 77, 40, 1].

As a consequence of the existence of these quasimodes, the spectral projectors
can be bounded from below by

$\|P_{\lambda_{j},\delta}\|_{L^{2}\to
L^{p}}\gtrsim_{N}\lambda_{j}^{\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}\qquad\mbox{if
$\delta>\lambda_{j}^{-N}$}.$

Note that the right-hand side matches the universal bound of Sogge in the
regime $p<p_{ST}$.

The preceding discussion suggests that, for ”most” manifolds, the bounds on
$\|P_{\lambda,\delta}\|_{L^{2}\to L^{p}}$ should improve as $\delta$ decreases
from $\delta=1$. In the remainder of this review article, we will try and
characterize this improvement in some specific examples.

##  3\. The Euclidean space and its quotients

###  3.1. The Euclidean space

Consider the extension operator mapping functions on the sphere
$\mathbb{S}^{d-1}$ to functions on the whole space $\mathbb{R}^{d}$ through
the formula

$f\mapsto\mathcal{F}\left[f\,d\sigma_{|\mathbb{S}^{d-1}}\right],$

where $\mathcal{F}$ is the Fourier transform and $d\sigma_{|\mathbb{S}^{d-1}}$
is the surface measure on $\mathbb{S}^{d-1}$. The Stein-Tomas theorem
establishes the boundedness of this operator from $L^{2}(\mathbb{S}^{d-1})$ to
$L^{p}(\mathbb{R}^{d})$ if $p\geq p_{ST}$. Recently, several works have
investigated the existence and structure of functions which saturate the
operator norm [85, 42]. The Stein-Tomas theorem can be reformulated in terms
of our spectral projectors:

######  Theorem 3.1 (Stein-Tomas [99, 102]).

For any $\lambda>1$, $\delta<1$,

(3.1) $\|P_{\lambda,\delta}\|_{L^{2}\to
L^{p}}\sim\lambda^{\frac{d-1}{2}-\frac{d}{p}}\delta^{\frac{1}{2}}+\lambda^{\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}\delta^{\frac{d+1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}\qquad\mbox{if
$2\leq p\leq\infty$}.$

Before sketching the proof of this theorem, we remark that the Stein-Tomas
estimate above can be extended to asymptotically conic, non-trapping manifolds
[50]; extensions to non-elliptic, constant coefficients operators can also be
considered [68].

###### Proof.

By scaling, this theorem is a consequence of the universal Sogge estimate
(2.3); but it came first, and can be proved by simpler means. We sketch the
proof in dimension 3, where formulas are cleaner. The smoothed out operator
$P^{\chi}_{\lambda,\delta}$ defined as
$\chi\left(\frac{\sqrt{-\Delta}-\lambda}{\delta}\right)$ (where
$\chi\in\mathcal{C}_{0}^{\infty}$) can be expressed through a radial
convolution kernel

(3.2)
$P^{\chi}_{\lambda,\delta}f(x)=\int_{\mathbb{R}^{3}}K_{\lambda,\delta}(|x-y|)f(y)\,dy,\qquad\mbox{with}\qquad
K_{\lambda,\delta}(r)=\delta\widehat{\chi}(\delta r)\lambda\frac{\sin(\lambda
r)}{r}.$

The proof consists in splitting $K_{\lambda,\delta}$ into

$K_{\lambda,\delta}(r)=\sum_{j}K_{\lambda,\delta}^{j}(r),\qquad\mbox{and
accordingly}\qquad P_{\lambda,\delta}=\sum_{j}P_{\lambda,\delta}^{j},$

where each $K_{\lambda,\delta}^{j}$ is smoothly localized where $r\sim 2^{j}$.
Then the operator norm of $P_{\lambda,\delta}^{j}$ can be estimated in two
ways

  * •

$\|P_{\lambda,\delta}^{j}\|_{L^{1}\to
L^{\infty}}\lesssim\|K_{\lambda,\delta}^{j}\|_{\infty}$

  * •

$\|P_{\lambda,\delta}^{j}\|_{L^{2}\to
L^{2}}\lesssim\|m_{\lambda,\delta}^{j}\|_{\infty}$, where
$m_{\lambda,\delta}^{j}$ is the Fourier symbol of $P_{\lambda,\delta}^{j}$.

By interpolation, this gives a bound for
$\|P_{\lambda,\delta}^{j}\|_{L^{p^{\prime}}\to L^{p}}$, and, after filling out
the estimates and summing over $j$, the desired bound for
$\|P_{\lambda,\delta}\|_{L^{p^{\prime}}\to L^{p}}$ if $p\neq p_{ST}$. The
critical point $p_{ST}$ requires complex interpolation. ∎

Just like for Sogge’s estimate (2.3), one can distinguish two regimes in
(3.1):

  * •

In the point-focusing regime $p\geq p_{ST}$, the dominant term is
$\lambda^{\frac{d-1}{2}-\frac{d}{p}}\delta^{\frac{1}{2}}$. It is achieved up
to a bounded factor for the spherical example $f$ (the terminology coming from
the radial symmetry of $f$) given in Fourier by
$\widehat{f}(\xi)=\mathbf{1}_{[\lambda-\delta,\lambda+\delta]}(|x|)$, which
has a sharp peak at the origin.

  * •

In the geodesic-focusing regime $p\leq p_{ST}$, the dominant term is
$\lambda^{\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}\delta^{\frac{d+1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}$;
it can be achieved up to a bounded factor by the so-called Knapp example which
we now describe. It is the function defined, in Fourier space, by
$\widehat{f}=\mathbf{1}_{R}$, where $R$ is a rectangle of dimensions
$\sim(\sqrt{\lambda\delta})^{d-1}\times\delta$ contained in the annulus
$\\{\xi,\,\lambda-\delta<|\xi|<\lambda+\delta\\}$. In physical space, it is
mostly supported on the dual rectangle $\widehat{R}$, of dimensions
$((\lambda\delta)^{-\frac{1}{2}})^{d-1}\times\delta$.

These two examples are depicted in Figure 3, both in physical space and in
Fourier space.

Figure 3.  The spherical example (red) and the Knapp example (blue) for the
Euclidean space represented in frequency and physical space: the Fourier
transform exchanges the left and right columns. In frequency space, the
supports are contained within the annulus with inner and outer radii
$\lambda-\delta$ and $\lambda+\delta$, either the full annulus or a rectangle
contained in it. In physical space, the $L^{p}$ mass concentrates at a point
or on the dual rectangle, respectively.

###  3.2. Tori

We now turn to quotients of the Euclidean space, namely tori. As in the whole
space, we can rely on Fourier analysis (Fourier series rather than the Fourier
transform) to carry out the analysis, but as we shall see, it becomes much
more challenging.

Choosing a family of independent vectors $(e_{i})$ of $\mathbb{R}^{d}$
generating the lattice $\Lambda=\mathbb{Z}e_{1}+\dots+\mathbb{Z}e_{d}$, we
will denote $\mathbb{T}^{d}=\mathbb{T}^{d}_{\Lambda}$ for the quotient
$\mathbb{R}^{d}/(\mathbb{Z}e_{1}+\dots+\mathbb{Z}e_{d})$.

####  3.2.1. The case $p=\infty$

A basis of eigenfunctions of the Laplacian on $\mathbb{T}^{d}$ is provided by
complex exponentials of the type $e^{ik\cdot x}$, where $k$ belongs to the
dual lattice $\Lambda^{*}$. It is then easy to see that

$\|P_{\lambda,\delta}\|_{L^{1}\to
L^{\infty}}=\\#\\{k\in\Lambda^{*},\,\lambda-\delta<|k|<\lambda+\delta\\},$

which brings us to a lattice point counting problem. This is a variant of the
more classical problem of estimating the number of lattive points in a sphere
of radius $\lambda$:

$N(\lambda)=\\#\\{k\in\Lambda^{*},\,|k|<\lambda\\}.$

To leading order, $N(\lambda)$ equals $C_{\Lambda}\lambda^{d}$, and the
problem is to bound the error
$E(\lambda)=\left|P(\lambda)-C_{\Lambda}\lambda^{d}\right|$. A full
understanding of optimal bounds on $E(\lambda)$ is still missing in dimensions
2 and 3; we refer the reader to [49, 61, 73, 76] for entry points into the
literature.

Though bounds on $E(\lambda)$ imply bounds on
$\|P_{\lambda,\delta}\|_{L^{1}\to L^{\infty}}$ by the above characterization,
this cannot lead to an optimal answer, and the questions of estimating
$E(\lambda)$ and $\|P_{\lambda,\delta}\|_{L^{1}\to L^{\infty}}$ are not
equivalent.

####  3.2.2. The case $\delta=0$ (eigenfunctions)

The earliest $L^{p}$ bound on eigenfunctions of the Laplacian on the torus is
the famous result of Cooke-Zygmund [35, 107] that, on $\mathbb{T}^{2}$, an
eigenfunction $\varphi$ of the Laplacian satisfies

$\|\varphi\|_{L^{4}}\lesssim\|\varphi\|_{L^{2}}.$

Whether this estimate can be extended to any $p>4$ has remained an open
problem. If $\Lambda=\mathbb{Z}^{d}$, Bourgain [18] conjectured the general
bound for an eigenfunction $\varphi$ associated to the eigenvalue $\lambda$

(3.3)
$\|\varphi\|_{L^{p}}\lesssim\lambda^{\frac{d}{2}-1-\frac{d}{p}}\|\varphi\|_{L^{2}}\qquad\mbox{for
$p>p^{*}=\frac{2d}{d-2}$}$

This conjecture was the subject of a number of subsequent papers [19, 20, 21]
culminating in the $\ell^{2}$ decoupling estimate of Bourgain and Demeter
[22]. Though the $\ell^{2}$ decoupling theorem was a landmark result in
harmonic analysis, it only proved the conjecture in the range
$p\geq\frac{2(d-1)}{d-3}$.

####  3.2.3. General values of $p$ and $\delta$

The conjecture (3.3) gives $L^{p}$ bounds on eigenfunctions of the Laplacian
on the torus. This is meaningful if $\Lambda=\mathbb{Z}^{d}$, but this
question becomes trivial on generic tori, for which eigenspaces of the
Laplacian have uniformly bounded dimension. A natural generalization consists
in asking the question which is at the heart of the present review article,
namely that of estimating $\|P_{\lambda,\delta}\|_{L^{2}\to L^{p}}$. The
author and Myerson [45] proposed the following conjecture222Notice that it
implies the conjecture (3.3) by choosing $\Lambda=\mathbb{Z}^{d}$ and
$\delta\sim\lambda^{-1}$.

(3.4) $\|P_{\lambda,\delta}\|_{L^{2}\to
L^{p}}\lesssim\lambda^{\frac{d-1}{2}-\frac{d}{p}}\delta^{\frac{1}{2}}+(\lambda\delta)^{\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}\qquad\mbox{if
$2\leq p\leq\infty$ and $\delta>\lambda^{-1}$}.$

As in the case $\delta=1$, and in the case of the Euclidean space, the two
summands on the right-hand side are achieved up to a bounded factor for
eigenfunctions focusing at a point, or on a geodesic. However, the dominant
term does not depend on how $p$ compares to $p_{ST}$; it is rather a condition
depending on $p$, $\lambda$, and $\delta$.

  * •

In the point-focusing regime, the dominant term is
$\lambda^{\frac{d-1}{2}-\frac{d}{p}}\delta^{\frac{1}{2}}$. This operator norm
for $P_{\lambda,\delta}$ can be achieved up to a bounded factor for functions
which are evenly spread amongst all Fourier modes $k\in\Lambda^{*}$ such that
$\lambda-\delta<|k|<\lambda+\delta$: this is the torus analog of the spherical
example.

  * •

In the geodesic-focusing regime, the dominant term is
$(\lambda\delta)^{\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}$. This
can be achieved up to a bounded factor for functions which, in physical space,
are focusing on a closed geodesic of $\mathbb{T}^{d}$: this is the torus
analog of the Knapp example.

The spherical and Knapp examples for the torus are depicted in Figure 4, both
in physical and in frequency space.

Figure 4.  The spherical example (red) and the Knapp example (blue) for the
torus represented in frequency and physical space: the Fourier series exchange
the left and right columns. In frequency space, the supports are contained
within the annulus with inner and outer radii $\lambda-\delta$ and
$\lambda+\delta$, either the full annulus or a rectangle contained in it. In
physical space, the $L^{p}$ mass concentrates close at a point or on the dual
rectangle, respectively.

In the following theorem, we do not attempt to gather all the cases for which
the conjecture is known to be true, but rather record two relevant instances
where it holds. For the current state of the art, the interested reader can
consult [37, 46, 54], besides the papers which have already been mentioned.

######  Theorem 3.2.

The conjecture (3.4) is verified

  * (i)

for $\Lambda=\mathbb{Z}^{d}$, if $p\geq\frac{2(d-1)}{d-3}$ and $d\geq 4$

  * (ii)

for general $\Lambda$, if $p\leq p_{ST}$.

###### Proof.

The first assertion is due to Bourgain and Demeter [22]; its proof combines
the $\ell^{2}$ decoupling theorem with arithmetic properties of the lattice
$\mathbb{Z}^{d}$.

The second assertion turns out to be a rather direct consequence of $\ell^{2}$
decoupling. To simplify notations, we focus on the case $d=2$, for which
$p_{ST}=6$, and choose the lattice to be $\Lambda=\mathbb{Z}^{2}$. By
interpolation with the trivial case $p=2$, it suffices to bound the $L^{6}$
norm of a function $f\in L^{2}(\mathbb{T}^{2})$ which is Fourier supported on
an annulus of inner radius $\lambda-\delta$ and outer radius $\lambda+\delta$.
We write $f$ as the sum of its Fourier series $f=\sum a_{k}e^{2\pi ik\cdot x}$
and rescale variables to $X=\lambda x$ and $K=k/\lambda$ in order to apply the
$\ell^{2}$ decoupling theorem:

$\displaystyle\|f\|_{L^{6}(\mathbb{T}^{2})}$
$\displaystyle=\left\|\sum_{k\in\mathbb{Z}^{2}}a_{k}e^{2\pi ik\cdot
x}\right\|_{L^{6}(\mathbb{T}^{2})}\lesssim\left(\frac{\delta}{\lambda}\right)^{\frac{1}{3}}\left\|\phi\left(\frac{\delta
X}{\lambda}\right)\sum_{K\in\mathbb{Z}^{2}/\lambda}a_{\lambda K}e^{2\pi
iK\cdot X}\right\|_{L^{6}(\mathbb{R}^{2})},$

where the cutoff function $\phi$ can be chosen to have compactly supported
Fourier transform. As a result, the Fourier transform of the function on the
right-hand side is supported on a $\delta/\lambda$-neighborhood of
$\mathbb{S}^{d-1}$. It can be written as a sum of functions which are
supported in Fourier on caps $\theta$ with dimensions
$\sim\frac{\delta}{\lambda}\times\frac{\delta^{\frac{1}{2}}}{\lambda^{\frac{1}{2}}}$.
These caps are almost disjoint rectangles which cover the annulus, and to
which we associate a smooth partition of unity $(\chi_{\theta})$. By
$\ell^{2}$ decoupling, the $L^{6}$ norm above is bounded by

$\lesssim_{\epsilon}\left(\frac{\delta}{\lambda}\right)^{\frac{1}{3}-\epsilon}\left(\sum_{\theta}\left\|\phi\left(\frac{\delta
X}{\lambda}\right)\sum_{K\in\mathbb{Z}^{2}/\lambda}\chi_{\theta}(K)a_{\lambda
K}e^{2\pi iK\cdot
X}\right\|_{L^{6}(\mathbb{R}^{2})}^{2}\right)^{\frac{1}{2}}.$

At this point, we use the inequality
$\|g\|_{L^{p}(\mathbb{R}^{2})}\lesssim\|g\|_{L^{2}(\mathbb{R}^{2})}|\operatorname{Supp}\widehat{g}|^{\frac{1}{2}-\frac{1}{p}}$,
which is valid for $p\geq 2$ and follows by applying successively the
Hausdorff-Young, Hölder, and the Plancherel inequalities. We use this
inequality for $g_{\theta}(X)=\phi\left(\frac{\delta
X}{\lambda}\right)\sum_{K}\chi_{\theta}(K)a_{\lambda K}e^{2\pi iK\cdot X}$.
Since each cap contains at most $\lambda^{\frac{1}{2}}\delta^{\frac{1}{2}}$
rescaled lattice points, its Fourier transform is supported on the union of at
most $\lambda^{\frac{1}{2}}\delta^{\frac{1}{2}}$ balls of radius
$O(\delta/\lambda)$, giving
$|\operatorname{Supp}\widehat{f}|\lesssim\lambda^{-\frac{1}{2}}\delta^{\frac{3}{2}}$.
Thus the $L^{6}$ norm we are trying to bound is less than

$\displaystyle\lesssim\left(\frac{\delta}{\lambda}\right)^{\frac{1}{3}-\epsilon}(\lambda^{-\frac{1}{2}}\delta^{\frac{3}{2}})^{\frac{1}{3}}\left(\sum_{\theta}\left\|\phi\left(\frac{\delta
X}{\lambda}\right)\sum_{K\in\mathbb{Z}^{2}/\lambda}\chi_{\theta}(\lambda
K)a_{\lambda K}e^{2\pi iK\cdot
X}\right\|_{L^{2}(\mathbb{R}^{2})}^{2}\right)^{\frac{1}{2}}$

By almost orthogonality and periodicity of the Fourier series, this is bounded
by
$\lambda^{\epsilon}(\lambda\delta)^{\frac{1}{6}}\|f\|_{L^{2}(\mathbb{T}^{2})}$,
which is the desired estimate. ∎

Before completing this section on Euclidean tori, we mention two related
directions. First, the two-dimensional Euclidean cylinder is considered in
[44]; in this case, there are no difficulties related to number theoretical
questions, and the conjecture (3.4) could be proved. Second, Euclidean tori
are the simplest examples of manifolds with integrable geodesic flow; this
broader class was investigated in [17, 103].

##  4\. The hyperbolic space and its quotients

###  4.1. Eigenfunctions of compact manifolds of nonpositive curvature

Manifolds of nonpositive curvature form a much broader class than the
hyperbolic space and its quotients (it even includes Euclidean tori), and a
natural context in which improvements to the Hörmander bound (2.1) are
expected. A long line of research [8, 53, 52, 11, 12, 13] focused on this
question through microlocal analysis, ultimately resulting in the following
theorem.

######  Theorem 4.1.

If the manifold $M$ is compact and has nonnegative curvature, then, denoting
its eigenfunctions and eigenvalues by $(\varphi_{j},\lambda_{j})$, there holds

$\|\varphi_{j}\|_{L^{p}}\lesssim\lambda_{j}^{\gamma(p)}(\log\lambda_{j})^{-\kappa},$

where $\kappa=\kappa(p)=\frac{1}{2}$ for $p>p_{ST}$, and $\kappa>0$ for
$p\in(2,p_{ST}]$.

The above theorem is fundamental in the analysis of the spectral resolution of
the Laplacian: it deals with the hardest case, that of compact manifolds, and
it does so without any symmetry assumptions, under the single geometric
condition of nonpositive curvature. Similar improvements can actually be
obtained in even more general contexts, see [31, 28, 30].

However, stronger bounds are expected: under chaoticity assumptions for the
geodesic flow, the random wave model of Berry [9] leads to predicting a bound
$\sqrt{\log\lambda_{j}}$ (see for instance [100] for the heuristic argument,
and [4] for numerics which appear to support this conjecture).

Such a bound seems utterly out of reach of current analytic methods. However,
in the hope of achieving some progress, we will now shift the focus and
consider a more restricted class of examples, namely the hyperbolic space and
its quotients, allowing also for infinite volume.

###  4.2. The hyperbolic space

The question of estimating $\|P_{\lambda,\delta}\|_{L^{2}\to L^{p}}$ can be
fully answered for the hyperbolic space. After partial results were obtained
in [57], this was done in [33], and the optimal dependence of the bounds on
$p$ was proved in [43].

######  Theorem 4.2.

If $M$ is the hyperbolic space $\mathbb{H}^{d}$,

$\|P_{\lambda,\delta}\|_{L^{2}\to
L^{p}}\lesssim[(p-2)^{-1}+1]\lambda^{\gamma(p)}\delta^{\frac{1}{2}}.$

The reader should by now not be surprised that the upper bound in this theorem
comprises two regimes, as reflected in the definition of $\gamma(p)$ in 2.2:
these are of course the point-focusing and the geodesic-focusing regime. Just
like in the Euclidean case, a spherical example and a Knapp example can be
constructed, which nearly saturate the bounds in the point-focusing and
geodesic-focusing regimes respectively. These examples are depicted in Figure
5, both in physical space (hyperbolic space) and in frequency space (the
appropriate analog of the Fourier transform being provided by the Helgason-
Fourier transform).

Figure 5.  The spherical example (red) and the Knapp example (blue) for the
hyperbolic space represented in frequency and physical space. On the physical
side, we use the Poincaré disk model to represent the hyperbolic space; and on
the frequency side, it is the Fourier-Helgason transform which provides the
dual picture. In frequency space, the supports are contained within the
annulus with inner and outer radii $\lambda-\delta$ and $\lambda+\delta$,
either the full annulus or a rectangle contained in it. In physical space, the
$L^{p}$ mass concentrates at a point, or along a geodesic; notice how the
instability of geodesics on the hyperbolic space leads to spreading at the
ends of the Knapp example.

Comparing this bound to the Euclidean case or the sphere, it appears that the
strongest bound is reached in the hyperbolic case, the weakest in the
spherical case, with the Euclidean case being a middle ground. This can be
explained, at least partially, in relation to the stability of geodesics: the
more stable geodesics are, the easier it is for almost-eigenfunctions to focus
on them, and therefore the weaker the bounds for
$\|P_{\lambda,\delta}\|_{L^{2}\to L^{p}}$.

###### Proof of Theorem 4.2.

We sketch the proof in dimension 3, where formulas are cleaner. The operator
$P_{\lambda,\delta}$ can then be expressed through a radial convolution kernel

$P_{\lambda,\delta}f(x)=\int_{\mathbb{H}^{3}}K_{\lambda,\delta}(\operatorname{dist}(x,y))f(y)\,dy,\qquad\mbox{with}\qquad
K_{\lambda,\delta}(r)=\delta\widehat{\chi}(\delta r)\lambda\frac{\sin(\lambda
r)}{\sinh r}.$

This is quite similar to the Euclidean case (3.2) over small distances, but
over large distances, the kernel decays exponentially. Nevertheless, it is
possible to follow closely the scheme of the Euclidean proof: split the kernel
dyadically, estimate all summands in the $L^{2}\to L^{p}$ topology, and then
conclude through real or complex interpolation. ∎

###  4.3. Infinite area quotients

Only the case of dimension 2 has been explored so far, with the following
theorem, which essentially asserts that the bounds are identical to the case
of the hyperbolic space, up to possible supbolynomial factors.

######  Theorem 4.3 ([3]).

Consider a geometrically finite hyperbolic quotient
$M=\Gamma\backslash\mathbb{H}^{2}$ of the hyperbolic space by a Fuchsian
subgroup. For any $\epsilon,N>0$,

$\|P_{\lambda,\delta}\|_{L^{2}\to
L^{p}}\lesssim_{p,\epsilon}\lambda^{\gamma(p)+\epsilon}\delta^{\frac{1}{2}}\quad\mbox{if
$\delta>\lambda^{-N}$}.$

Furthermore, $\epsilon$ and $N$ can be taken equal to $0$ and $\infty$
respectively if the exponent of convergence of $N$ is $<\frac{1}{2}$.

###### Proof.

Decomposition into compact and unbounded parts. By a structure theorem for
hyperbolic surfaces [14], $M$ can be written as the union of a compact part,
hyperbolic ends (identical to hyperbolic cylinders) and cusps. It is easy to
see that cusps prevent boundedness of the projectors, and thus we only
consider the case when they are absent of the decomposition. We now consider a
partition of unity

$\chi_{c}(x)+\sum_{j=1}^{N}\chi_{hf}^{j}(x)=1\qquad\mbox{for all $x\in M$},$

where $\chi_{c}$ is compactly supported, the supports of the $\chi_{hf}^{j}$
are disjoint, and $M$ reduces to a hyperbolic funnel (halves of hyperbolic
cylinders) on the support of each $\chi_{hf}^{j}$.

At this point, it is convenient to replace $P_{\lambda,\delta}$ by a smoothed-
out version of this projector (which is indifferent as far as $L^{2}\to L^{p}$
bounds are concerned), and to express it through the Schrödinger group

(4.1)
$P_{\lambda,\delta}^{\flat}=\chi_{0}\left(\frac{\Delta+\lambda^{2}}{\lambda\delta}\right)=\frac{\lambda\delta}{\sqrt{2\pi}}\int\widehat{\chi_{0}}(\lambda\delta
t)e^{i\lambda^{2}t}e^{it\Delta}\,dt,$

where $\chi_{0}$ is a smooth cutoff function. Combining this formula with the
partition of unity defined earlier, we will rely on the decomposition
(simplifying slightly the formulas)

$P_{\lambda,\delta}^{\flat}=P_{\lambda,1}\chi_{c}(x)P_{\lambda,\delta}^{\flat}+P_{\lambda,1}\chi_{hf}(x)P_{\lambda,\delta}^{\flat}.$

Bounding the compact part. The crucial ingredient is the local $L^{2}$
smoothing estimate, which follows from Bourgain-Dyatlov [23]. As explained in
Subsection 5.2 below, this results in the local smoothing bound

$\|\chi_{c}(x)e^{it\Delta}f\|_{L^{2}_{t}L^{2}_{x}}\lesssim\lambda^{-\frac{1}{2}+\epsilon}\|f\|_{L^{2}_{x}}$

(for any $\epsilon>0$). Combining this bound with formula (4.1), and the
Cauchy-Schwarz inequality, we see that

$\|\chi_{c}(x)P_{\lambda,\delta}^{\flat}\|_{L^{2}}\lesssim\lambda\delta\left\|\chi_{c}(x)\int\widehat{\chi_{0}}(\lambda\delta
t)e^{i\lambda^{2}t}P_{\lambda}e^{it\Delta}\,dt\right\|_{L^{2}}\lesssim(\lambda\delta)^{\frac{1}{2}}\|\chi_{c}(x)e^{it\Delta}f\|_{L^{2}_{t}L^{2}_{x}}\lesssim\lambda^{\epsilon}\delta^{\frac{1}{2}}\|f\|_{L^{2}_{x}}.$

Using in addition the Sogge theorem, we can bound

$\left\|P_{\lambda,1}\chi_{c}(x)P_{\lambda,\delta}^{\flat}f\right\|_{L^{p}}\lesssim\lambda^{\gamma(p)}\left\|\chi_{c}(x)P_{\lambda,\delta}^{\flat}f\right\|_{L^{2}}\lesssim\lambda^{\gamma(p)+\epsilon}\delta^{\frac{1}{2}}.$

Bounding the hyperbolic part. To treat the contribution of the hyperbolic
ends, the idea is to write

$\chi_{hf}(x)\chi_{0}\left(\frac{\Delta+\lambda^{2}}{\lambda\delta}\right)=\frac{\lambda\delta}{\sqrt{2\pi}}\int\widehat{\chi_{0}}(\lambda\delta
t)e^{i\lambda^{2}t}P_{\lambda}\chi_{hf}(x)e^{it\Delta}\,dt.$

Then, the function $u=\chi_{hf}^{j}(x)e^{it\Delta}$ is supported on the $j$-th
funnel where it satisfies the Schrödinger equation

$i\partial_{t}u-\Delta u=[\Delta,\chi_{hf}^{j}]e^{it\Delta}f.$

The source term (on the right) is controlled by the local smoothing estimates;
in order to control $u$, there remains to derive sufficiently sharp
Strichartz, smoothing, and dispersive estimates on the funnel. ∎

###  4.4. Finite area quotients

In the case where $\Gamma\backslash\mathbb{H}^{2}$ has finite area, Iwaniec
and Sarnak [63], see also the review [81], conjectured the following: any
eigenfunction $\phi_{\lambda_{j}}$ associated to the eigenvalue $\lambda_{j}$
satisfies, for $K$ compact and any $\epsilon>0$

$\|\varphi_{\lambda_{j}}\|_{L^{\infty}(K)}\lesssim\lambda_{j}^{\epsilon}\|\phi_{\lambda}\|_{L^{2}}$

(which reduces to
$\|\varphi_{\lambda}\|_{L^{\infty}(X)}\lesssim\lambda^{\epsilon}$ on a compact
surface, for a normalized eigenfunction).

This conjecture should be regarded as very hard, since, for $p=\infty$ and in
the particular case of the modular surface, it implies the Lindelöf hypothesis
[80]. We refer to [27, 59, 60] for progress on this problem in the case of
arithmetic manifolds.

Note that a markedly different behavior can occur in dimension $d=3$, as
evidenced by the result in [79].

##  5\. Classical estimates implying spectral projector bounds

In this section, we examine how estimates for the spectrum of the Schrödinger
operator (Weyl law), or the decay of the Schrödinger group (smoothing
estimates, dispersive estimates, Strichartz estimates) imply bounds for the
spectral projectors $P_{\lambda,\delta}$ in various regimes.

###  5.1. Weyl law

If $M$ is compact, let $(\lambda_{j}^{2})$ denote the eigenvalues and
$(\varphi_{j})$ the normalized eigenfunctions of the Laplace-Beltrami
operator. The counting function enjoys the following asymptotics [62, 91]

$N(\lambda)=\\#\\{\lambda_{j}^{2}\;\mbox{eigenvalue of $-\Delta$
with}\;\lambda_{j}<\lambda\\}=C\lambda^{d}+O(\lambda^{d-1})$

(for a constant $C$ corresponding to the volume of the unit cotangent bundle).
This estimate is sharp for spheres and Zoll manifolds, but the error term is
expected to be smaller than $O(\lambda^{d-1})$ for ”most” manifolds. Known
improvements are as follows: $o(\lambda^{d-1})$ if the set of closed geodesics
has measure zero [41], $O(\lambda^{d-1}(\log\lambda)^{-1})$ if $M$ has
negative curvature [8], and $O(\lambda^{d-1-\delta})$ (for some $\delta>0$
depending on $M$) if $M$ is integrable [36].

Local Weyl laws refer to (pointwise) estimates on the function

$N_{x}(\lambda)=\sum_{\lambda_{j}<\lambda}|\varphi_{j}(x)|^{2},\qquad\mbox{with
$x\in M$ and $\lambda>0$}.$

For any compact manifold [55, 94],

(5.1) $N_{x}(\lambda)=C\lambda^{d}+O(\lambda^{d-1})\qquad\mbox{(uniformly in
$x$)}.$

To see the connection to operator norms of $P_{\lambda,\delta}$, observe that
the kernel of $P_{\lambda,\delta}$ is given by
$K_{\lambda,\delta}(x,y)=\sum_{\lambda-\delta<\lambda_{j}<\lambda+\delta}\varphi_{j}(x)\varphi_{j}(y)$.
By the Cauchy-Schwarz inequality,

$|K_{\lambda,\delta}(x,y)|\leq\sqrt{N_{x}(\lambda+\delta)-N_{x}(\lambda-\delta)}\sqrt{N_{y}(\lambda+\delta)-N_{y}(\lambda-\delta)}.$

Using in addition the bound (5.1) gives the theorem of Sogge (2.3) for
$p=\infty$, namely

$\|P_{\lambda,1}\|_{L^{1}\to L^{\infty}}\lesssim\lambda^{d-1}.$

Any improvement of the error term in (5.1) leads to an improved estimate for
$\|P_{\lambda,\delta}\|_{L^{1}\to L^{\infty}}$.

###  5.2. Local smoothing estimates

Assume that the resolvent $R(\lambda)=(\Delta-\lambda^{2}-i0)^{-1}$ enjoys the
local smoothing bound

$\|\chi R(\lambda)\chi\|_{L^{2}\to L^{2}}\lesssim\lambda^{-1+\epsilon};$

here, $\epsilon\geq 0$ and $\chi$ is a localized function which can be
compactly supported or decay at a given rate depending on the context. It
follows from Kato’s theorem that the Schrödinger group also exhibits local
smoothing:

(5.2) $\left\|\chi
P_{\lambda,1}e^{-it\Delta}f\right\|_{L^{2}_{t}L^{2}(M)}\lesssim\lambda^{-\frac{1}{2}+\epsilon}\|f\|_{L^{2}}.$

Bounds of this type were proved for various infinite volume manifolds, in
particular asymptotically Euclidean or hyperbolic, basic references include
[34, 38, 39, 66].

It is then possible to follow ideas similar to those presented in Subsection
4.3.

###  5.3. Dispersive estimates

Assume that dispersive estimates hold on $M$ with the same decay rate as on
Euclidean space

(5.3) $\|e^{it\Delta}\|_{L^{p^{\prime}}\to
L^{p}}\lesssim|t|^{\frac{d}{p}-\frac{d}{2}},\qquad p\geq 2$

(see the review [82] and also [83, 84]). Expressing again the smoothed-out
operator $P_{\lambda,\delta}^{\flat}$ in terms of the Schrödinger group, it
can be decomposed into

$P_{\lambda,\delta}^{\flat}=\frac{\lambda\delta}{\sqrt{2\pi}}\int\widehat{\chi}(\lambda\delta
t)e^{i\lambda^{2}t}e^{it\Delta}\,dt\sim\frac{\lambda\delta}{\sqrt{2\pi}}\left[\int_{|t|\lesssim\frac{1}{\lambda}}+\int_{\frac{1}{\lambda}\lesssim|t|\lesssim\frac{1}{\lambda\delta}}\right]e^{i\lambda^{2}t}e^{it\Delta}\,dt=P_{small}+P_{large}$

(here, integration bounds such as $|t|\lesssim\frac{1}{\lambda}$ should be
understood in terms of smooth cutoffs).

Adjusting the cutoff function, the operator $P_{small}$ can be written as
$\delta P_{\lambda,1}^{\flat}$, to which Sogge’s theorem applies:
$\|P_{small}\|_{L^{p^{\prime}}\to
L^{p}}\lesssim\lambda^{d-1-\frac{2d}{p}}\delta$ if $p\geq p_{ST}$. Turning to
$P_{large}$, it can be bounded through the dispersive estimates (5.3) to
obtain, if $p\geq p^{*}=\frac{2d}{d-2}$,

$\left\|\int_{\frac{1}{\lambda}\lesssim|t|\lesssim\frac{1}{\lambda\delta}}\lambda\delta\widehat{\chi}(\lambda\delta
t)e^{i\lambda^{2}t}e^{it\Delta}\,dt\right\|_{L^{p^{\prime}}\to
L^{p}}\lesssim\lambda\delta\int_{\frac{1}{\lambda}\lesssim|t|\lesssim\frac{1}{\lambda\delta}}|t|^{\frac{d}{p}-\frac{d}{2}}\,dt\lesssim\lambda^{\frac{d}{2}-\frac{d}{p}}\delta\lesssim\lambda^{d-1-\frac{2d}{p}}\delta.$

Overall, we found that, if the dispersive estimates (5.3) hold, the $L^{2}\to
L^{p}$ norm of $P_{\lambda,\delta}$ enjoys the optimal bound
$\lambda^{\frac{d-1}{2}-\frac{d}{p}}\delta^{\frac{1}{2}}$ as soon as $p\geq
p^{*}=\frac{2d}{d-2}$.

###  5.4. Strichartz estimates

Strichartz estimates on $\mathbb{R}^{d}$ are given by

$\|e^{it\Delta}f\|_{L^{p}_{t}L^{q}_{x}}\lesssim\|f\|_{L^{2}},\quad\mbox{if
$p,q\geq 2$, \;\; $\frac{2}{p}+\frac{d}{q}=\frac{d}{2}$}.$

This estimate was first proved [47, 48, 104] for $p>2$, and the endpoint
result for $p=p^{*}=\frac{2d}{d-2}$

$\|e^{it\Delta}f\|_{L^{2}_{t}L^{p^{*}}_{x}}\lesssim\|f\|_{L^{2}}$

(which will be of particular relevance for us) was then obtained by Keel and
Tao [67].

Strichartz estimates were subsequently extended to perturbations of the
Euclidean space: either short range perturbations yielding a global in time
estimate [97, 78], or long range perturbations yielding local in time
estimates [25, 15]

Assuming that this endpoint estimate holds on $M$, we can apply it together
with the Cauchy-Schwarz inequality to the mollified spectral projector
expressed through the Schrödinger group (4.1) and obtain

$\|P_{\lambda,\delta}^{\flat}f\|_{L^{p^{*}}}\lesssim\lambda\delta(\lambda\delta)^{-\frac{1}{2}}\|e^{it\Delta}f\|_{L^{2}([-\frac{1}{\lambda\delta},\frac{1}{\lambda\delta}],L^{p^{*}}(M))}\lesssim\lambda^{\frac{1}{2}}\delta^{\frac{1}{2}}\|f\|_{L^{2}};$

(neglecting tails of $\widehat{\chi_{0}}$). This shows that the endpoint
Strichartz estimate up to time $\sim\frac{1}{\lambda\delta}$ implies the
optimal $L^{2}\to L^{p^{*}}$ estimate for $P_{\lambda,\delta}$.

##  6\. $L^{2}\to L^{p}$ projector bounds beyond complete smooth Riemannian
manifolds

While the present review focuses on bounds for spectral projectors on smooth
complete Riemannian manifolds, many interesting variants can be considered. In
the list below, we try to summarize all generalizations which have been
investigated, with the relevant references.

  * •

Non-smooth metric [69, 70]

  * •

Manifold with boundary [89, 86]

  * •

Sub-Riemannian manifolds [75, 6]

  * •

Pseudo-differential operators [72]

  * •

Schrödinger operators [65, 71, 64, 10, 58]. In the asymptotically Euclidean
case, a related question is the boundedness of wave operators [82], which can
be combined with the Stein-Tomas theorem to prove a natural analog.

  * •

Restriction of eigenfunctions to lower-dimensional submanifolds [26, 24, 32,
101]

  * •

Finite graphs (size $N$), for which one should substitute the infinite size
limit $N\to\infty$ to the high frequency limit $\lambda\to\infty$ which is of
interest for manifolds. Nearly optimal pointwise bounds for eigenvectors could
be proved for random graphs [7].

##  7\. Some interesting questions

We conclude with some interesting questions

###  7.1. Geometry of maximizers.

In all known examples, the functions at which the operator norm of
$P_{\lambda,\delta}$ is nearly achieved concentrate asymptotically (as
$\lambda\to\infty$) either at a point, or along a geodesic. Could other
possibilities arise?

###  7.2. Infinite volume manifolds

. In all known examples, $\|P_{\lambda,\delta}\|_{L^{2}\to
L^{p}}\lesssim\lambda^{\frac{d-1}{2}-\frac{d}{p}}\delta^{\frac{1}{2}}+\lambda^{\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}$,
where the first term comes from point focusing, and the second from geodesic
focusing on a stable closed geodesic. Is this bound indeed universal? At least
for all manifolds with a simple behavior at infinity (e.g. flat, conic,
hyperbolic)?

For $p<p_{ST}$, very different behaviors can occur for
$\|P_{\lambda,\delta}\|_{L^{2}\to L^{p}}$. One finds that this is
$\sim\lambda^{\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)}\delta^{\alpha}$,
with $\alpha=0$ if there exists a stable closed geodesic,
$\alpha=\frac{d+1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)$ on the Euclidean
space, $\alpha=\frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)$ on Euclidean
cylinders, and $\alpha=\frac{1}{2}$ on the hyperbolic space (these values of
$\alpha$ can be interpreted in relation to the stability of geodesics). Are
other values of $\alpha$ possible?

###  7.3. Equidistribution of eigenfunctions and of the spectrum

. In the compact case, is it possible to establish a relation between
equidistribution of eigenfunctions on the manifold, and equidistribution of
the spectrum over the reals? For instance, denoting $\eta$ for a quantity
related to the scale over which eigenvalues are equidistributed, could there
hold an estimate such as
$\|\varphi\|_{L^{\infty}}\lesssim\lambda^{\frac{d-1}{2}-\frac{d}{p}}\eta^{\frac{1}{2}}$
for any eigenfunction $\varphi$ with eigenvalue $\lambda$, and for $p$
sufficiently large? On the one hand, such an estimate seems to fail for
surfaces of revolution, whose spectrum can be equidistributed, but which admit
exceptional eigenfunctions concentrating at the poles. On the other hand, such
an estimate is verified on spheres and rational or irrational tori. An analogy
can also be made with random matrices, for which a connection between quantum
unique ergodicy and GUE statistics for the spectrum can be established in some
cases [16].

###  7.4. Ignoring stable closed geodesics

Stable closed geodesics are the obstacle to an improvement over the sphere
case for $p$ close to $2$. Would an improvement become possible by localizing
away from stable geodesics?

###  7.5. Link with the Fourier restriction problem

. The question of estimating $\|P_{\lambda,\delta}\|_{L^{2}\to L^{p}}$
provided a generalization of the Fourier restriction problem to arbitrary
Riemannian manifolds, when the density in Fourier space is $L^{2}$. Is it
possible (and interesting) to obtain such a generalization for the analog of
$L^{p}$ density in Fourier space?

The formulation of this question depends on a choice of a distinguished basis
of (generalized) eigenfunctions. If the manifold is of infinite volume and for
instance asymptotically Euclidean, such a distinguished basis could be
provided by scattering theory. If the manifold is compact, then the
eigenspaces of the Laplacian are generically non-degenerate, and this provides
naturally an orthonormal basis of $L^{2}$, which will be denoted
$(\varphi_{k})$. It is tempting to ask about the $\ell^{q}(\mathbb{N})\to
L^{p}(M)$ boundedness properties of
$(c_{k})_{k\in\mathbb{N}}\mapsto\sum_{|\lambda_{k}-\lambda|<\delta}c_{k}\varphi_{k}$
(denoting $\varphi_{k}$ for the $k$-th eigenfunction). In the case of the
torus, it was noticed by Bourgain and Demeter [20] that the case of general
$q$ is an immediate consequence of $q=2$. Does this remain true for other
manifolds?

###  7.6. Nonsymmetric and nonintegrable compact manifolds

We conclude with this most ambitious question: if the manifold under
consideration is neither symmetric, nor integrable, there does not exist, to
the best of our knowledge, any result giving more than logarithmic
improvements over the universal Sogge estimate. Could that be obtained in a
single example?

### Acknowledgements

The author is grateful to Tristan Léger for his suggestions to improve a
preliminary version, and to Paul Bourgade, Isabelle Gallagher, Chris Sogge,
Stefan Steinerberger, Daniel Tataru, Laurent Thomann and Maciej Zworski for
enlightening conversations during the writing of this review article.

When working on this article, the author was supported by a start-up grant
from Imperial College and a Wolfson fellowship

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# A Comprehensive System for Secondary Structure Analysis of Protein Models

Vedh Ramalingam Kannan

(February 2024)

###### Abstract

In protein structure analysis, the accurate characterization of secondary
structure elements is crucial for understanding protein function and dynamics.
This paper presents a software system designed for the comprehensive analysis
of the secondary structure of protein models. Leveraging phi ($\phi$) and psi
($\psi$) torsion angles, the system utilises K-means clustering and outlier
detection techniques to identify and classify folding structures within
protein models. Through the visualisation of the Ramachandran plot, the
software enables the differentiation of various secondary structure motifs,
including alpha-helices, beta-sheets, and other structural elements. The
incorporation of customisable threshold values facilitates the identification
of outliers, providing insights into potential structural anomalies or
misalignments within protein models. Overall, this software system offers
researchers a powerful tool for comprehensive secondary structure analysis, to
aid in enhancing understanding of protein structures created using both
traditional methods such as X-Ray Diffraction and contemporary methods such as
Artificial Intelligence.

##  1 Background and Introduction

In protein structure analysis, the Ramachandran plot serves as a fundamental
tool for evaluating the quality and validity of protein models by visually
representing the distribution of phi ($\phi$) and psi ($\psi$) angles for
amino acid residues in the protein backbone [1]. The $\phi$ angle signifies
the rotation around the C-N bond, while the $\psi$ angle signifies the
rotation around the C-C bond.

Different folding structures, including alpha-helices, beta-sheets, and other
motifs, exhibit distinct patterns of torsion angles. For instance, the right-
handed alpha-helix is characterised by specific ranges of $\phi$ and $\psi$
angles, approximately $-57°$ to $-47°$ and $-57°$ to $-47°$, respectively.
Conversely, the left-handed alpha-helix displays similar ranges, albeit with
positive values for both $\phi$ and $\psi$ angles. The $3_{10}$ helix and the
pi helix also possess unique torsion angle preferences, with ranges of
approximately $-74°$ to $-4°$ and $-57°$ to $-70°$, respectively. In contrast,
beta-sheets are characterised by different torsion angle distributions. The
parallel beta-sheet exhibits $\phi$ and $\psi$ angles around -119° and +113°,
respectively, while the anti-parallel beta sheet is characterised by ideal
angles around $-139°$ and $+135°$ [2].

These regions, defined by the sterics of amino acid residues, are delineated
by the Ramachandran plot and a large number of deviations of residues from
these regions, commonly termed as Ramachandran outliers, may indicate
potential issues with the protein model and necessitate further examination.

This study has developed research software aimed at simplifying the
identification of various folding structures, offering a user-friendly
approach to gaining in-depth insights into this secondary structure of
proteins.

##  2 Methods

###  2.1 Data Preprocessing

The raw data for this analysis is protein models in the *.pdb file format. A
script in the software generates a *.tsv file of all these files to extract
$\phi$ and $\psi$ angles for each amino acid residue. The Bio.PDB module is
then used to parse the PDB files and extract the required structural
information, writing to file as shown below [3].

Figure 1: An example TSV file generated from a protein model in the RCSB
databank

The above file shows each amino acid chain (column 1), the $\phi$ angles
(column 2) and the $\psi$ angles (column 3) for each amino acid residue
(column 4) in the protein model. All following computational analyses are
performed on a file of this format.

###  2.2 K-Means Clustering

The core functionality of the software relies on the application of the
K-means clustering algorithm to group $\phi$ and $\psi$ angles extracted from
protein models into distinct clusters representing different folding
structures. K-means clustering is a partitioning algorithm that aims to divide
a dataset into K clusters, where each data point belongs to the cluster with
the nearest mean, or centroid, and iteratively updates the centroids to
minimise the within-cluster sum of squares. It is characterised by the
formula:

$\underset{\mathbf{C}}{\text{argmin}}\sum_{i=1}^{k}\sum_{\mathbf{x}\in
S_{i}}||\mathbf{x}-\mathbf{\mu}_{i}||^{2}$ (1)

  * •

$k$ is the number of clusters.

  * •

$S_{i}$ is the set of data points assigned to cluster $i$.

  * •

$\mathbf{\mu}_{i}$ is the centroid (mean) of cluster $i$

  * •

$\mathbf{C}$ represents the set of cluster centroids.

In the context of protein structure analysis, the number of clusters (K) is
determined based on the number of folding structures of interest. Each cluster
represents a distinct folding structure characterised by specific ranges of
$\phi$ and $\psi$ angles. For example, to identify right-handed alpha-helices,
the software initialises K centroids corresponding to the ideal $\phi$ and
$\psi$ angles of right-handed alpha-helices, as mentioned above section 1.
Similarly, centroids representing beta-sheets and other motifs are initialised
accordingly. The clustering algorithm then iteratively assigns data points to
the nearest centroids and updates the centroids to minimise the within-cluster
sum of squares, effectively grouping the $\phi$ and $\psi$ angles into
clusters corresponding to different folding structures. The outcome of this
program is a plot as shown below in figure 2

Figure 2: Ramachandran Plot of model 1rzt

This plot was generated using model 1rzt from the Protein Data Bank [4]. It is
a model of the crystal structure of DNA polymerase lambda complexed with a two
nucleotide gap DNA molecule.

###  2.3 Outliers

In addition to clustering, the software includes a feature for outlier
detection to identify residues that deviate significantly from the expected
torsion angle distributions. Outliers in the Ramachandran plot may indicate
potential issues with the protein model, such as structural anomalies or
misalignments. The software employs a distance-based approach to identify
outliers, where residues that fall beyond a certain threshold euclidean
distance from the cluster centroids are flagged as outliers with an ’x’ mark.
Notably, empirical observations have shown that setting the threshold within
the range of $85\leq x\leq 100$ yields results closely aligned with the
validation reports provided by the Protein Data Bank (PDB) for the
corresponding models. The outcome of introducing this into the program can be
seen in figure 3.

Figure 3: Ramachandran Plot of model 1nql

This plot was generated using model 1nql from the Protein Data Bank [5]. It is
a model of the structure of the extracellular domain of human epidermal growth
factor (EGF) receptor in an inactive (low pH) complex with EGF.

##  3 Conclusion

This study has presented a research software aimed at enhancing the accuracy
assessment of protein models through the analysis of phi ($\phi$) and psi
($\psi$) torsion angles. Through the utilisation of K-means clustering and
outlier detection techniques, the software successfully identified and
characterised folding structures within protein models, allowing for a more
comprehensive understanding of their secondary structure.

By leveraging the Ramachandran plot, the software facilitated the
visualization and differentiation of various folding phenomena, including
alpha-helices and beta-sheets, providing researchers with valuable insights
into protein conformation and stability. Additionally, the incorporation of
outlier detection mechanisms enabled the identification of residues deviating
significantly from expected torsion angle distributions, thereby pinpointing
potential structural anomalies or misalignments within protein models.

##  4 Data Availability

All protein models used for testing in this study were either found on the
Protein Data Bank (https://rcsb.org) or created using AlphaFold [6] [7]. All
software which was used can be found in the author’s Github repository [8].

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