# Upper bounds on the size of nodal sets for Gevrey and quasianalytic
Riemannian manifolds

Hamid Hezari  Department of Mathematics, UC Irvine, Irvine, CA 92617, USA
<EMAIL_ADDRESS>
###### Abstract.

We find new polynomial upper bounds for the size of nodal sets of
eigenfunctions when the Riemannian manifold has a Gevrey or quasianalytic
regularity.

##  1\. Introduction

Let $(M,g)$ be a smooth compact connected boundaryless Riemannian manifold of
dimension $n$. Suppose $-\Delta_{g}$ is the positive Laplace-Beltrami operator
on $(M,g)$ and $\psi_{\lambda}$ is an eigenfunction of $-\Delta_{g}$ with
eigenvalue $\lambda$. We are interested in upper bounds on the
$(n-1)$-dimensional Hausdorff measure $\mathcal{H}^{n-1}(Z_{\psi_{\lambda}})$
of the nodal set $Z_{\psi_{\lambda}}=\\{x\in M;\;\psi_{\lambda}(x)=0\\}$ of
$\psi_{\lambda}$. When $(M,g)$ belongs to the analytic class $C^{\omega}$,
Donnelly and Fefferman [DoFe88] proved the upper bound $C\sqrt{\lambda}$. In
the $C^{\infty}$ case and for $n\geq 3$, a result of Logunov [Lo18a] gives a
polynomial upper bound of the form $C\lambda^{\alpha}$ for some non-explicit
but apparently large $\alpha>\frac{1}{2}$ depending only on $n$. For $n=2$,
Logunov and Malinnikova [LoMa18b] find upper bounds of the form
$C\lambda^{\frac{3}{4}-\beta}$ for some small universal
$\beta\in(0,\frac{1}{4})$. In this paper, we are interested in the
intermediate cases of regularity, namely ultradifferentiable Riemannian
manifolds, also called Denjoy-Carleman (or Roumieu) classes $C^{\mathcal{M}}$
that lie between the analytic and smooth classes, i.e. $C^{\omega}\subset
C^{\mathcal{M}}\subset C^{\infty}$. We are mainly interested in Gevrey classes
and quasianalytic classes of Riemannian manifolds and obtain new upper bounds
for the size of nodal sets for these cases.

Our first result in this direction is:

######  Theorem 1.1.

Let $(M,g)$ be a compact connected boundaryless Riemannian manifold that
belongs to the Gevrey class $G^{s}$, $s\geq 1$. Let $\psi_{\lambda}$ be an
eigenfunction of $-\Delta_{g}$ with eigenvalue $\lambda\geq 0$. Then

$\mathcal{H}^{n-1}(Z_{\psi_{\lambda}})\leq C{\lambda}^{\frac{s}{2}},$

where the constant $C=C(M,g,s)$ depends only on $(M,g)$ and $s$.

For $s>1$ sufficiently small, Theorem 1.1 improves the polynomial upper bounds
of [Lo18a] and [LoMa18b], of course only in the Gevrey case. The proof uses
Crofton’s formula, hypoelliptic estimates in the Gevrey setting, and also a
fascinating and powerful result of [LoMa18a] on the measure of sublevel sets
of eigenfunctions. Our method also applies to the more general (Denjoy-
Carleman) classes of Riemannian manifolds. In particular, when $(M,g)$ belongs
to the quasianalytic class $C^{\mathcal{M}}$,
$\mathcal{M}_{k}=\left(\log(k+e)\right)^{k}$, we prove:

######  Theorem 1.2.

$\mathcal{H}^{n-1}(Z_{\psi_{\lambda}})\leq C\sqrt{\lambda}\log(\lambda+e)$.

This quasianalytic case is interesting because up to a factor
$\log(\lambda+e)$, the upper bound agrees with Yau’s upper bound conjecture
[Ya82].

More generally, if $(M,g)\in C^{\mathcal{M}}$, where $\mathcal{M}$ is a
regular sequence with _moderate growth_ (see the next section for
definitions), then we have:

######  Theorem 1.3.

$\mathcal{H}^{n-1}(Z_{\psi_{\lambda}})\leq
C\sqrt{\lambda}\;\left(\mathcal{M}_{\lfloor\sqrt{\lambda}\rfloor}\right)^{\frac{1}{\lfloor\sqrt{\lambda}\rfloor}}$.

When $\mathcal{M}_{k}=k!^{s-1}$, $s\geq 1$, we have $C^{\mathcal{M}}=G^{s}$
and thus Theorem 1.1 is in fact just a particular case of Theorem 1.3. We can
generate more results by considering more examples of weight sequences such as
$M_{k}=\log^{sk}(k+e)$, $s\geq 0$, which would provide the upper bounds

$\sqrt{\lambda}\log^{s}(\lambda+e).$

Although Theorem 1.3 is more general, but to keep the presentation smooth, we
focus mainly on the Gevrey case and only highlight the changes needed in the
notationally bothersome and less known case of Denjoy-Carleman classes.

It is worth mentioning that one can probably use the Bang degree (see
[NaSoVo04]), combined with the method of this paper, to obtain the above
estimates. It would also be interesting to study the case of manifolds with
boundary (see [LaMaNaNa21]) and Steklov eigenfunctions (see [Ze15]) in the
Gevrey setting.

##  2\. Hypoellipticity of $\Delta_{g}$ in the ultradifferentiable setting

We start this section by reviewing the ultradifferentiable classes that we are
interested in.

###  2.1. The Gevrey class and Denjoy-Carleman classes

Let $U\subset{\mathbb{R}}^{n}$ be an open set and let $s\geq 1$. The Gevrey
class $G^{s}(U)$ is defined as the space of smooth functions $f$ on $U$ such
that for each compact $K\subset U$ there exist $C$ and $A$ positive such that
for all multiindices $\alpha\in\mathbb{N}^{n}$, one has

$\sup_{K}|\partial^{\alpha}f(x)|\leq CA^{|\alpha|}|\alpha|!^{s},$

where $|\alpha|=\alpha_{1}+\dots+\alpha_{n}$. We refer to such functions as
$s$-Gevrey functions. When $s=1$ we have $G^{1}(U)=C^{\omega}(U)$, i.e. the
class of analytic functions on $U$. For $s\leq s^{\prime}$ we have
$G^{s}\subset G^{s^{\prime}}$. The space $G^{s}(U)$ is closed under addition
and multiplication.

A map $f=(f_{1},\dots,f_{n}):U\subset{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is
called $s$-Gevrey if each $f_{j}$ is $s$-Gevrey. The composition of two
$s$-Gevrey maps is $s$-Gevrey if the composition is well-defined.

A smooth manifold $M$ belongs to the Gevrey class $G^{s}$ if it has an atlas
$\\{(U_{\beta},\varphi_{\beta});\beta\in\mathcal{A}\\}$ whose transition maps
$\varphi_{\beta_{1}}\circ\varphi_{\beta_{2}}^{-1}$ are in $G^{s}$. An
$s$-Gevrey structure on $M$ is a maximal atlas of $s$-Gevrey charts. We say a
Riemannian manifold $(M,g)$ belongs to $G^{s}$ if in addition to $M$, the
metric $g$ also belongs to the Gevrey class of degree $s$, which is equivalent
to $g_{ij}\in G^{s}$ on every Gevrey chart $(U,\varphi)$ belonging to the
Gevrey structure. As examples of $G^{s}$ Riemannian manifolds, consider any
real analytic boundaryless manifold $M$ (such as a sphere or a torus), then
there is an infinite dimensional space of $s$-Gevrey metrics $g$ that $M$ can
be equipped with. This can be done first by defining Gevrey metrics locally on
Gevrey charts, then use an appropriate partition of unity to obtain a global
Gevrey metric. In the construction of the partition of unity, one needs to use
the bump functions,

(1) $g_{\gamma}(x)=\begin{cases}e^{-\frac{1}{(x(1-x))^{\gamma}}}&x\in(0,1),\\\
0&x\notin(0,1).\end{cases}$

For each $\gamma>0$, the function $g_{\gamma}$ belongs to
$G^{1+\frac{1}{\gamma}}$. The standard partition of unity theorem uses $g_{1}$
which is $2$-Gevrey and is not suitable for localizing functions within the
class $G^{s}$ if $s<2$.

In fact there are more general classes of smooth functions associated to
increasing sequences $\mathcal{M}=\\{\mathcal{M}_{k}\\}_{k\in{\mathbb{N}}}$ of
positive real numbers, called Denjoy-Carleman or Roumieu functions, and
denoted by $C^{\mathcal{M}}$. Given $U\subset{\mathbb{R}}^{n}$ open, we define
$C^{\mathcal{M}}(U)$ to be the class of functions $f$ on $U$ such that for all
$K\subset U$ compact, there exist $C$ and $A$ such that for all
$\alpha\in{\mathbb{N}}^{n}$,

$\sup_{K}|\partial^{\alpha}f(x)|\leq
CA^{|\alpha|}|\alpha|!\;\mathcal{M}_{|\alpha|}.$

The sequence $\mathcal{M}=\\{\mathcal{M}_{k}\\}_{k\in{\mathbb{N}}}$ is called
the weight sequence. By this definition, the $s$-Gevrey class corresponds to
the weight sequence $\mathcal{M}_{k}=k!^{s-1}$. Following [Fe20], we say
$\mathcal{M}$ is a _regular wight sequence_ if

$\displaystyle(i)\;\;$ $\displaystyle\mathcal{M}_{0}=1,$
$\displaystyle(ii)\;\;$ $\displaystyle\mathcal{M}_{k}\;\text{is increasing},$
$\displaystyle(iii)\;\;$
$\displaystyle\mathcal{M}_{k}^{2}\leq\mathcal{M}_{k-1}\mathcal{M}_{k+1},\quad\forall
k\in{\mathbb{N}}\quad\text{(logarithmic convexity)}$ $\displaystyle(iv)\;\;$
$\displaystyle\sup_{k\in{\mathbb{N}}^{>0}}\left(\frac{\mathcal{M}_{k+1}}{\mathcal{M}_{k}}\right)^{1/k}<\infty.$

In addition, a sequence $\mathcal{M}$ has moderate growth if

(2) $\displaystyle(v)\;\;$
$\displaystyle\sup_{k,m\in{\mathbb{N}}^{>0}}\left(\frac{\mathcal{M}_{k+m}}{\mathcal{M}_{k}\mathcal{M}_{m}}\right)^{1/(k+m)}<\infty.\qquad\qquad\qquad\qquad\qquad$

Conditions $(i)-(iii)$ imply that the class $C^{\mathcal{M}}$ is closed under
addition, multiplication, and composition. Condition (iv) guarantees that the
class is closed under derivation (see for example the paper [KrMiRa09] for a
broad review). The moderate growth condition $(v)$, in addition to regularity,
would provide $C^{\mathcal{M}}$-hypoellipticity for elliptic partial
differential operators with coefficients in $C^{\mathcal{M}}$. Note that
condition $(v)$ implies $(iv)$ and it also forces $C^{\mathcal{M}}\subset
G^{s}$ for some $s$ (see for example Lemma 3.2.1 of [Th03]). By a result of
Denjoy and Carleman, the class $C^{\mathcal{M}}$ with $\mathcal{M}$ satisfying
$(i)-(iii)$, is quasianalytic, meaning, functions which are flat at a point
are identically zero, if and only if

$\sum_{k=0}^{\infty}\frac{\mathcal{M}_{k}}{(k+1)\mathcal{M}_{k+1}}=\infty.$

For example the weight sequence $M_{k}=\log^{sk}(k+e)$ produces a
quasianalytic class if $0<s\leq 1$, and a non-quasianalytic class if $s>1$. A
non-quasianalytic class always accepts a partition of unity. Similar to the
notion of Gevrey Riemannian manifolds, we can define Denjoy-Carleman
Riemannian manifolds (see [Fe20]). Examples of compact quasianalytic
Riemannian manifolds can not be provided via a partition of unity. Instead,
one can use the implicit function theorem for $C^{\mathcal{M}}$-mappings (see
[Ko79]), or for example take a compact analytic manifold that is embedded in
the euclidean space, and for the metric consider a conformal factor that is in
a desired quasianalytic class.

###  2.2. Gevrey hypoellipticity of Laplacian and Gevrey regularity of
eigenfunctions

A partial differential operator $P$ with smooth coefficients on an open set
$U\subset{\mathbb{R}}^{n}$ is called hypoelliptic if $Pu\in C^{\infty}(U)$
implies that $u\in C^{\infty}(U)$. It is known that elliptic partial
differential operators (in fact more generally, elliptic pseudodifferential
operators) are hypoelliptic. Similarly, there is a notion of Gevrey
hypoellipticity, requiring that whenever $Pu\in G^{s}$ then $u\in G^{s}$. When
$s=1$, this property is the same as analytic hypoellipticity. Boutet de Monvel
and Kreé [BoKr67] proved that any elliptic partial differential operator with
$s$-Gevrey coefficients is $G^{s}$ hypoellliptic (the case $s=1$ was already
known). Therefore, an eigenfunction of the Laplacian of a Gevrey Riemannian
manifold must be Gevrey. However, in this paper we need the following
quantitative estimates.

######  Proposition 2.1.

Let $(M,g)$ ba compact connected boundaryless Riemannian manifold. Suppose
$(M,g)$ belongs to the class $G^{s}$. Then every eigenfunction
$\psi_{\lambda}$ of $\Delta_{g}$ also belongs to the class $G^{s}$. Moreover,
for every $s$-Gevrey chart $(U,\varphi)$ and every compact $K\subset U$, there
exist $C$ and $A$ independent of $\lambda$ such that

(3) $\sup_{K}|\partial^{\alpha}\psi_{\lambda}|\leq
Ce^{\sqrt{\lambda}}A^{|\alpha|}\alpha!^{s}\sup_{U}{|\psi_{\lambda}}|.$

######  Remark 2.2.

The factor $e^{\sqrt{\lambda}}$ can be replaced with any
$e^{\varepsilon\sqrt{\lambda}}$ at the cost of allowing the constants $C$ and
$A$ to be dependent on $\varepsilon$ (certainly $A\geq c/\varepsilon$).

######  Remark 2.3.

One could follow the argument of [DoFe88] (see proof of Lemma 7.1) and rescale
balls of radius $\lambda^{-1/2}$ to balls of radius one and then apply a
hypoellipticity argument as in [Ho69, Page 176] to obtain estimates of the
form:

(4) $\sup_{K}|\partial^{\alpha}\psi_{\lambda}|\leq
C(\sqrt{\lambda}A)^{|\alpha|}\alpha!^{s}\sup_{U}{|\psi_{\lambda}}|.$

Although estimates (4) are better than (3) for small values of $|\alpha|$, but
they are much worse when $|\alpha|\geq\sqrt{\lambda}$. For our purposes
estimates (3) play a key role.

###### Proof of Proposition 2.1.

We consider the function,

(5) $h(x,t)=e^{\sqrt{\lambda}t}\psi_{\lambda}(x),$

on the product manifold $\widetilde{M}=M\times{\mathbb{R}}$. Note that if we
denote $\Delta_{\tilde{g}}=\partial_{t}^{2}+\Delta_{g}$ to be the Laplace-
Beltrami operator associated to the metric $\tilde{g}$, defined as the product
of $g$ and the Euclidean metric $e_{0}$ on ${\mathbb{R}}$, then
$\Delta_{\tilde{g}}h=0$. In other words, $h$ is a harmonic extension of
$\psi_{\lambda}(x)$ into $\widetilde{M}=M\times{\mathbb{R}}$.

Clearly, since $g\in G^{s}$, we have that $\tilde{g}\in G^{s}$ and
$\Delta_{\tilde{g}}$ is an elliptic differential operator with coefficients in
$G^{s}$. By Corollary 2.14 of [BoKr67], we have $h\in G^{s}$, which implies
that $\psi_{\lambda}\in G^{s}$. To obtain the estimates (3), we shall use
another result of [BoKr67, Prop 2.13] which implies that if $P$ is an elliptic
differential operator of order $2$ in $\Omega$ with coefficients in $G^{s}$,
then there exists an elliptic pseudodifferential operator $Q$ of order $-2$
such that $R=I-QP$ is a smoothing operator whose distributional kernel
$R(x,y)$ belong to the Gevrey class $G^{s}(\Omega\times\Omega)$. Now let
$(U,\varphi)$ be an arbitrary Gevrey chart of $M$ and consider
$P=\Delta_{\tilde{g}}$ on $\tilde{U}=U\times(-2,2)$. Since
$Ph=\Delta_{\tilde{g}}h=0$, we get $Rh=h$. For an arbitrary compact subset
$K\subset U$, we define the compact subset $\tilde{K}=K\times[-1,1]$ of
$\tilde{U}$. Then from $h=Rh=\int R(x,y)h(y)dy$ and the properties of $R$, we
obtain

$\sup_{\tilde{K}}|\partial^{\alpha}h|\leq
CA^{|\alpha|}\alpha!^{s}\sup_{\tilde{U}}{|h|}.$

Estimate (3) follows from this and the definition (5) of $h$ in terms of
$\psi_{\lambda}$.

∎

###  2.3. Denjoy-Carleman hypoellipticity

As mentioned earlier, if we add the moderate growth assumption $(v)$ to a
regular sequence $\mathcal{M}$, then any elliptic partial differential
operators with coefficients in $C^{\mathcal{M}}$ is
$C^{\mathcal{M}}$-hypoelliptic. This result follows from Theorem 4.1 of
Albanese, Jornet, and Oliaro [AlJoOl10] (see also Theorem 6.1 of [Fe20] for
systems), which states that if $P$ is a linear differential operator (not
necessarily elliptic) on $U\subset{\mathbb{R}}^{n}$ with coefficients in
$C^{\mathcal{M}}(U)$, then

$\text{WF}_{\mathcal{M}}(u)\subset\text{WF}_{\mathcal{M}}(Pu)\cup\Sigma,$

where $\Sigma$ is the characteristic set of $P$ and $\text{WF}_{\mathcal{M}}$
denotes the wavefront set associated to the class $C^{\mathcal{M}}$.111A
result of this type was first proved by Hörmander for operators with analytic
coefficients. If $Pu\in C^{\mathcal{M}}$ and $P$ is elliptic, then we obtain
that $\text{WF}_{\mathcal{M}}(u)=\emptyset$, thus $u\in C^{\mathcal{M}}(U)$.
In particular, this implies that an eigenfunction of a
$C^{\mathcal{M}}$-Riemannian manifold $(M,g)$ must be $C^{\mathcal{M}}$
regular. In fact, we have:

######  Proposition 2.4.

Let $(M,g)$ ba compact connected boundaryless Riemannian manifold that belongs
to the class $C^{\mathcal{M}}$, where $\mathcal{M}$ is a regular sequence with
moderate growth. Then every eigenfunction $\psi_{\lambda}$ of $\Delta_{g}$
belongs to the class $C^{\mathcal{M}}$. More precisely, for every
$C^{\mathcal{M}}$ chart $(U,\varphi)$ and every compact $K\subset U$, there
exist $C$ and $A$ independent of $\lambda$ such that

(6) $\sup_{K}|\partial^{\alpha}\psi_{\lambda}|\leq
Ce^{\sqrt{\lambda}}A^{|\alpha|}\alpha!\;\mathcal{M}_{|\alpha|}\;\sup_{U}{|\psi_{\lambda}}|.$

The proof is very similar to the one of Prop 2.1. Namely, we only need to show
that the harmonic extension $h$ of $\psi_{\lambda}$ to $M\times{\mathbb{R}}$,
satisfies $C^{\mathcal{M}}$-hypoellipticity estimates. However, this follows
from the proof of Theorem 4.1 of [AlJoOl10], from which equivalent estimates
can be driven for the Fourier transform of $h$.

##  3\. The measure of sublevel sets of eigenfunctions

In the article [LoMa18a], in the course of the proof of a propagation of
smallness result, the authors find the following lower bounds on the supremum
of $|\psi_{\lambda}|$ over an arbitrary measurable subset $E\subset M$:

(7)
$\sup_{E}|\psi_{\lambda}|\geq\frac{1}{K}\sup_{M}|\psi_{\lambda}|\left(\frac{|E|}{K|M|}\right)^{K\sqrt{\lambda}}.$

Here, $|\cdot|$ refer to the Riemannian volume measure associated to $g$ and
the constant $K>1$ is independent of $(\lambda,\psi_{\lambda})$ and $E$.

The estimates (7) resemble the Remez inequalities for polynomials: let
$p_{n}(x)$ be a polynomial of degree $n$ on an interval
$I\subset{\mathbb{R}}$. Then for any measurable subset $E\subset I$,

$\sup_{E}|p_{n}|\geq\sup_{I}|p_{n}|\left(\frac{|E|}{4|I|}\right)^{n}.$

The lower bounds (7) also strengthen _the tunneling estimates_ by allowing $E$
to be an arbitrary measurable set that can depend on $\lambda$, instead of an
open set independent of $\lambda$, which is assumed in standard tunneling
estimates in semiclassical analysis.

If in (7), we let $E$ to be the sublevel set $\\{|\psi_{\lambda}|\leq t\\}$,
and if we assume that $\psi_{\lambda}$ is $L^{\infty}(M)$ normalized, then we
obtain (see also Lemma 4.2 and Remark 4.3 of [LoMa18a]):

(8) $|\\{|\psi_{\lambda}|\leq t\\}|\leq K_{1}t^{\frac{1}{K\sqrt{\lambda}}},$

where $K_{1}$ and $K$ are independent of $\lambda$ and $t$. Interestingly, the
bounds (8) are analogous to Pólya’s inequality on the size of the sublevel
sets of polynomails with leading coefficient equal to one:

$|\\{x\in{\mathbb{R}};\,|p_{n}(x)|\leq t\\}|\leq
4\left(\frac{t}{2}\right)^{\frac{1}{n}}.$

######  Remark 3.1.

The bounds (8) can also be obtained directly from Lemma 4.2 of [LoMa18a],
which concerns the measure of sublevel sets of harmonic functions in terms of
their doubling indices.

##  4\. Proof in the Gervey case

In this section we prove Theorem 1.1 which concerns the Gevrey case. Since the
arguments for the qusi-analytic, or more generally the Denjoy-Carleman cases,
are notationally complicated and less intuitive, we postpone their
presentation to the last section of the article.

###  4.1. Reduction to a local problem

We cover $M$ with finitely many (independent of $\lambda$) $s$-Gevrey charts
$\\{(U_{\alpha},\varphi_{\alpha})\\}_{\alpha\in\mathcal{F}}$ such that each
$U_{\alpha}$ is identified (under $\varphi_{\alpha}$) with the $n$-dimensional
Euclidean ball

$B_{2}^{n}=\\{x\in{\mathbb{R}}^{n};\;|x|<2\\},$

and in addition we require that
$\\{\varphi_{\alpha}^{-1}(B^{n}_{1})\\}_{\alpha\in\mathcal{F}}$ covers $M$. To
prove our main theorem it suffices to prove that for each
$\alpha\in\mathcal{F}$, there exists $C_{\alpha}$ such that

$\mathcal{H}^{n-1}(Z_{\psi_{\lambda}}\cap\varphi_{\alpha}^{-1}(B^{n}_{1}))\leq
C_{\alpha}{\lambda}^{\frac{s}{2}},$

as this would imply that, with $C=\sum_{\alpha\in\mathcal{F}}C_{\alpha}$,

$\mathcal{H}^{n-1}(Z_{\psi_{\lambda}})\leq\sum_{\alpha\in\mathcal{F}}\mathcal{H}^{n-1}(Z_{\psi_{\lambda}}\cap\varphi_{\alpha}^{-1}(B^{n}_{1}))\leq
C{\lambda}^{\frac{s}{2}}.$

Fix a chart $(U,\varphi)$ in the covering specified above. From here on, we
consider the local eigenfunction
$\tilde{\psi}_{\lambda}=\psi_{\lambda}\circ\varphi^{-1}$ on the ball
$B^{n}_{2}$ and estimate the size of its nodal in the ball $B^{n}_{1}$. By
abuse of notation we will call this local eigenfunction $\psi_{\lambda}$
instead of $\tilde{\psi}_{\lambda}$.

###  4.2. Crofton’s formula and the proof of Theorem 1.1

The nodal set $Z_{\varphi_{\lambda}}$ is not necessarily a smooth hypersurface
but the $(n-1)$-Haussdorf measure of its singular set is zero. Thus Crofton’s
formula can be applied to calculate $\mathcal{H}^{n-1}(Z_{\psi_{\lambda}}\cap
B^{n}_{1})$ in terms of the number of intersection points of the nodal set
with lines going through $x\in B^{n}_{1}$ in the direction $\xi\in S^{n-1}$.
In fact, we have the following upper bound ([Fe20], [Ze15], [DoFe88])

(9) $\mathcal{H}^{n-1}(Z_{\psi_{\lambda}}\cap B^{n}_{1})\leq
C_{1}\iint_{B^{n}_{1}\times S^{n-1}}{n}(x,\xi)\,dxd\xi,$

where for each $(x,\xi)\in B^{n}_{1}\times S^{n-1}$, $n(x,\xi)$ is the number
of intersection points of the line $x+{\mathbb{R}}\xi$ with the nodal set
$Z_{\psi_{\lambda}}$, in the ball $B^{n}_{1}$. The constant $C_{1}$ depends
only on $n$. Note that $dxd\xi$ is the Liouville measure $d\mu$ on
$S^{*}B^{n}_{1}=B^{n}_{1}\times S^{n-1}$. The function $n(x,\xi)$ can take the
value $+\infty$ but only on a set of measure zero. In fact $n\in
L^{1}(S^{*}B^{n}_{1},d\mu)$. Now for each $m\in\mathbb{N}$, we denote

(10) $A_{m}=\\{(x,\xi)\in S^{*}B^{n}_{1};\;(m\sqrt{\lambda})^{s}\leq
n(x,\xi)<((m+1)\sqrt{\lambda})^{s}\\}.$

Because $n(x+t\xi,\xi)=n(x,\xi)$, the measurable subsets $A_{m}$ are unions of
lines.

The following lemma on the size of $A_{m}$ is crucial in the proof of Theorem
1.1.

######  Lemma 4.1.

There exist $c$ and $C_{2}$ both positive, dependent only on $(M,g)$ and $s$,
such that for all $m\in\mathbb{N}$ and $\lambda\geq 1$, we have,

$\mu(A_{m})=\iint_{A_{m}}dxd\xi\leq C_{2}e^{-c{m}}.$

Before we get into the proof of this lemma let us present how one can apply
this lemma and easily obtain a proof for Theorem 1.1.

###### Proof of Theorem 1.1.

Using (9) and the definition of $A_{m}$, we have

$\displaystyle\mathcal{H}^{n-1}(Z_{\psi_{\lambda}}\cap B^{n}_{1})$
$\displaystyle\leq C_{1}\iint_{S^{*}B^{n}_{1}}n(x,\xi)\,dxd\xi$
$\displaystyle=C_{1}\sum_{m=0}^{\infty}\iint_{A_{m}}n(x,\xi)\,dxd\xi$
$\displaystyle\leq C_{1}\sum_{m=0}^{\infty}\mu(A_{m})\sup_{(x,\xi)\in
A_{m}}n(x,\xi)$ $\displaystyle\leq
C_{1}\sum_{m=0}^{\infty}\mu(A_{m})((m+1)\sqrt{\lambda})^{s}$
$\displaystyle\leq
C_{1}C_{2}\lambda^{\frac{s}{2}}\sum_{m=0}^{\infty}(m+1)^{s}e^{-cm}\leq
C\lambda^{\frac{s}{2}}.$

∎

It remains to prove the main lemma.

##  5\. Proof of estimates on the size of $A_{m}$

In this section we prove Lemma 4.1. As we shall see, this lemma would follow
from another key lemma which we now state but prove later. Throughout this
section we assume that $\lambda\geq 1$ and

$\sup_{B^{n}_{2}}|\psi_{\lambda}|=1.$

Recall that

$A_{m}=\\{(x,\xi)\in S^{*}B^{n}_{1};\;(m\sqrt{\lambda})^{s}\leq
n(x,\xi)<((m+1)\sqrt{\lambda})^{s}\\}.$

######  Lemma 5.1.

For each $(x,\xi)\in A_{m}$, consider the line segment

$I_{x,\xi}:=(x+{\mathbb{R}}\xi)\cap B^{n}_{1},$

equipped with the $1$-dimensional Lebesgue measure $\mathcal{L}^{1}$ induced
from ${\mathbb{R}}$. Then, there exist $\ell\in{\mathbb{N}}^{>0}$ and an open
subset $I^{*}\subset I_{x,\xi}$ with
$\mathcal{L}^{1}(I^{*})\geq\frac{\mathcal{L}^{1}(I_{x,\xi})}{4^{\ell}}$, such
that

$\sup_{I^{*}}|\psi_{\lambda}|\leq C_{3}e^{(1-c_{0}m\ell)\sqrt{\lambda}}.$

The constants $c_{0}$ and $C_{3}$ are positive and only depend on $(M,g)$ and
$s$.

######  Remark 5.2.

We shall use $\mathcal{L}^{n}$ for the $n$-dimensional Lebesgue measure,
however, sometimes we simply use $|\cdot|$.

###### Proof of Lemma 4.1.

We only need the result of Lemma 5.1. In fact, the takeaway of that lemma is
that for each $(x,\xi)\in A_{m}$, there exists $\ell\in{\mathbb{N}}^{>0}$,
such that

(11) $\mathcal{L}^{1}(I_{x,\xi})\leq
4^{\ell}\,\mathcal{L}^{1}\left(I_{x,\xi}\cap\left\\{z\in
B^{n}_{1};\;|\psi_{\lambda}(z)|\leq
C_{3}e^{(1-c_{0}m\ell)\sqrt{\lambda}}\right\\}\right).$

Before giving the proof let us set some notations. For each $m\in{\mathbb{N}}$
and $\ell\in{\mathbb{N}}^{>0}$, we denote the $\mu$-measurable sets

$C_{m,\ell}=\\{(x,\xi)\in A_{m};\quad\text{\eqref{cor} holds}\\}.$

By Lemma 5.1,

$A_{m}=\bigcup_{\ell=1}^{\infty}C_{m,\ell}.$

We also denote

$L_{m,\ell}=\left\\{z\in B^{n}_{1};\;|\psi_{\lambda}(z)|\leq
C_{3}e^{(1-c_{0}m\ell)\sqrt{\lambda}}\right\\},$

and

$B_{m,\ell}=C_{m,\ell}\cap(L_{m,\ell}\times S^{n-1}).$

Note that $B_{m,\ell}$ is a $\mu$-measurable subset of $C_{m,\ell}$. Since by
the powerful estimate (8), one has

$\mathcal{L}^{n}(L_{m,\ell})\leq C_{4}e^{-\frac{c_{0}m\ell}{K}},$

by letting $c_{1}=\frac{c_{0}}{K}$, we get

$\mu(B_{m,\ell})\leq C_{5}e^{-c_{1}m\ell}.$

We claim that

(12) $\mu(C_{m,\ell})\leq 4^{\ell}\mu(B_{m,\ell}).$

If we know this estimate, then

$\displaystyle\mu(A_{m})\leq\sum_{\ell=1}^{\infty}\mu(C_{m,\ell})$
$\displaystyle\leq\sum_{\ell=1}^{\infty}4^{\ell}\mu(B_{m,\ell})\leq
C_{5}\sum_{\ell=1}^{\infty}4^{\ell}e^{-c_{1}m\ell}\leq
C_{5}\sum_{\ell=1}^{\infty}e^{(2-c_{1}m)\ell}.$

Therefore, for $m\geq\frac{4}{c_{1}}$, by the geometric sum we get

$\mu(A_{m})\leq C_{6}e^{2-c_{1}m}\leq C_{6}e^{-\frac{c_{1}}{2}m},$

which provides the estimate of Lemma 4.1 with $c=\frac{c_{1}}{2}$. For
$m<\frac{4}{c_{1}}$, Lemma 4.1 is trivial.

It remains to prove the claim (12). We shall only use (11) and Fubini’s
Theorem. For simplicity, we drop the sub-indices $(m,\ell)$ from (12) and
denote $C=C_{m,\ell}$ and $B=B_{m,\ell}$. Let $\mathds{1}_{C}(x,\xi)$ be the
indicator function of $C$. Since $C$ is $\mu$-measurable, by Fubini’s Theorem,
for almost every $\xi\in S^{n-1}$ the function $\mathds{1}_{C}(\cdot,\xi)$ is
Lebesgue integrable on $B^{n}_{1}$ and,

$\mu(C)=\iint_{B^{n}_{1}\times
S^{n-1}}\mathds{1}_{C}(x,\xi)d\mu=\int_{S^{n-1}}\left(\int_{B^{n}_{1}}\mathds{1}_{C}(x,\xi)dx\right)\,d\xi=\int_{S^{n-1}}\mathcal{L}^{n}\left({(C)_{\xi}}\right)\,d\xi.$

Here, $(C)_{\xi}=\\{x\in B^{n}_{1};(x,\xi)\in C\\}$, which is measurable for
almost every $\xi$. We can write the same identity for $\mu(B)$, hence to
prove (11), it suffices to prove that

$\mathcal{L}^{n}((C)_{\xi})\leq
4^{\ell}\,\mathcal{L}^{n}((B)_{\xi}),\quad\text{almost every $\xi\in
S^{n-1}$}.$

Note that $(C)_{\xi}$ is a union of lines parallel to $\xi$ because by the
definition (10) of $A_{m}$, whenever $(x,\xi)\in A_{m}$, one has
$(x+{\mathbb{R}}\xi)\cap B^{n}_{1}\subset A_{m}$ and thus this property would
also hold for $C=C_{m,\ell}$, i.e., $(x+{\mathbb{R}}\xi)\cap B^{n}_{1}\subset
C$ whenever $(x,\xi)\in C$. We also note that if
$\mathcal{L}^{n}((C)_{\xi})=0$, there is nothing to prove because then
$\mathcal{L}^{n}((B)_{\xi})=0$ and the above estimate would be trivial.

Now, given $\xi$ with $\mathcal{L}^{n}((C)_{\xi})\neq 0$, we change
coordinates in $B^{n}_{1}$ by a rotation $x=R(y)$ so that $\xi=R(e_{1})$.
Then,

$\mathcal{L}^{n}((C)_{\xi})=\int_{B^{n}_{1}}\mathds{1}_{(C)_{\xi}}(x)dx=\int_{B^{n}_{1}}\mathds{1}_{R^{-1}((C)_{\xi})}(y)dy.$

Obviously, the set $R^{-1}((C)_{\xi})$ is measurable for almost all $\xi$ and
is a union of lines parallel to the $y_{1}$-axis. Using Fubini’s Theorem
again, we can write the last integral as

$\mathcal{L}^{n}((C)_{\xi})=\int_{B^{n}_{1}}\mathds{1}_{R^{-1}((C)_{\xi})}(y)dy=\int_{B^{n-1}_{1}}\mathcal{L}^{1}\left({(R^{-1}((C)_{\xi}))_{y^{\prime}}}\right)dy^{\prime},$

where
$(R^{-1}((C)_{\xi}))_{y^{\prime}}=\\{y_{1}\in[-1,1];\;y=(y_{1},y^{\prime})\in
R^{-1}((C)_{\xi})\\}$ is Lebesgue measurable in $[-1,1]$ for almost every
$y^{\prime}\in B^{n-1}_{1}$. Again, a similar identity holds for
$\mathcal{L}^{n}((B)_{\xi})$. Therefore, in order to prove (12), we only need
to show that

$\mathcal{L}^{1}\left({(R^{-1}((C)_{\xi}))_{y^{\prime}}}\right)\leq
4^{\ell}\mathcal{L}^{1}\left({(R^{-1}((B)_{\xi}))_{y^{\prime}}}\right).$

However, this is precisely (11). ∎

In the next section we prove Lemma 5.1, which is the technical part of the
paper.

##  6\. Proof of smallness of $\psi_{\lambda}$ on line segments containing
large number of zeros

We will use the following simple but important lemma.

######  Lemma 6.1.

Let $u$ be a smooth real-valued function on an interval
$I\subset{\mathbb{R}}$. Assume $u$ has at least $p$ many zeros (counting
multiplicities) in $I$. Then,

$\sup_{I}|u|\leq\frac{|I|^{p}}{p!}\sup_{I}|u^{(p)}|,$

where $|I|$ denotes the length of the interval $I$.

###### Proof.

A simple proof using Rolle’s theorem can be found in [Hi50][Lemma 4.a]. ∎

######  Remark 6.2.

This inequality was also proved and used via Hermite interpolation theorem by
[IgKu10][Theorem 2.4] in studying zeros of $1D$ parabolic partial differential
equations with Gevrey coefficients.

The following corollary of the lemma will be used many times in the course of
the proof of Lemma 5.1.

######  Proposition 6.3.

Let $(M,g)$ be a compact Riemannian manifold with no boundary, that belongs to
the class $G^{s}$, $s\geq 1$. Let $\psi_{\lambda}$ be an eigenunction of
$-\Delta_{g}$ with eigenvalue $\lambda\geq 1$. Assume
$\sup_{B^{n}_{2}}|\psi_{\lambda}|=1$. Consider $(x,\xi)\in S^{*}B^{n}_{1}$ and
assume $\psi_{\lambda}$ has at least $p$ zeros on a subinterval $I$ of
$I_{x,\xi}=(x+{\mathbb{R}}\xi)\cap B^{n}_{1}$. Then

$\sup_{I}|\psi_{\lambda}|\leq C_{7}(A|I|)^{p}\,p!^{s-1}e^{\sqrt{\lambda}},$

where $A$ and $C_{7}$ depend only on $(M,g)$ and $s$.

######  Remark 6.4.

This estimate is only useful when $|I|$ is very small in terms of $p$. We also
note that when $(M,g)$ belongs to a Denjoy-Carleman class $C^{\mathcal{M}}$
with $\mathcal{M}$ being a regular weight sequence having moderate growth, in
the above estimate we need to replace $p!^{s-1}$ with $\mathcal{M}_{|p|}$.

###### Proof.

Observe that $I$ must be of the form $I=x+J\xi$ for some closed interval
$J\in{\mathbb{R}}$. Obviously, $|I|=|J|$. Now we denote

$u_{\lambda}(t):=\psi_{\lambda}(x+t\xi),\quad t\in J.$

By assumption, the function $u_{\lambda}$ has at least $p$ zeros in $J$. Thus
by Lemma 6.1, we have

$\sup_{J}|u_{\lambda}|\leq\frac{|J|^{p}}{p!}\sup_{J}|u^{(p)}_{\lambda}|.$

Calculating $u_{\lambda}^{(p)}$ in terms of the partial derivatives of
$\psi_{\lambda}$, we get

$\sup_{J}|u^{(p)}_{\lambda}|\leq
n^{p}\sup_{|\alpha|=p}\sup_{B^{n}_{1}}|\partial^{\alpha}\psi_{\lambda}|.$

Therefore, by the quantitative hypoellipticity estimates (3) and the
normalization $\|\psi_{\lambda}\|_{L^{\infty}(B_{2}^{n})}=1$, we obtain:

$\sup_{I}|\psi_{\lambda}|\leq C_{7}(A|I|)^{p}p!^{s-1}e^{\sqrt{\lambda}}.$

∎

###### Proof of Lemma 5.1.

Throughout we fix $(x,\xi)\in A_{m}$. This means that

$(m\sqrt{\lambda})^{s}\leq n(x,\xi)<((m+1)\sqrt{\lambda})^{s}.$

Recall that, $n(x,\xi)=|I_{x,\xi}\cap Z_{\psi_{\lambda}}|$, i.e. the number of
zeros of $\psi_{\lambda}$ on the line segment
$I_{x,\xi}=(x+{\mathbb{R}}\xi)\cap B^{n}_{1}$. We may assume during the proof
that $m\geq 32$ and $\lambda\geq 1$, because if $m<32$, then the estimate in
Lemma 6.3 is trivial with the choice of $\ell=1$ and $c_{0}\leq\frac{1}{32}$.

Let

$n_{0}=\lfloor(m\sqrt{\lambda})^{s}\rfloor,\qquad N_{0}=\lceil
C_{0}(m\sqrt{\lambda})^{s-1}\rceil,$

and

$p_{0}=\frac{n_{0}}{N_{0}}.$

The constant $C_{0}\geq 1$ will be specified later, independently from $m$,
$\lambda$, and $(x,\xi)$. In the first step of our argument, we partition the
segment $I_{x,\xi}$ into $N_{0}$ many equal length subintervals
$I_{1},\dots,I_{N_{0}}$, each, of course, of length
$\frac{|I_{x,\xi}|}{N_{0}}<\frac{2}{N_{0}}$. We then define

$S_{0}=\\{1\leq j\leq N_{0}\\},$

and

(13) $S_{1}=\\{j\in S_{0};\quad|I_{j}\cap Z_{\psi_{\lambda}}|\geq\kappa
p_{0}\\},$

where $\kappa\in[\frac{1}{4},1]$ is a universal constant that will be
specified later. Now there are two cases:

  * (1a)

$|S_{1}|>\frac{1}{4}N_{0}$;

  * (1b)

$|S_{0}\setminus S_{1}|>\frac{3}{4}N_{4}$.

If case (1a) happens, then by Prop 6.3 with $p=\lfloor\kappa p_{0}\ \rfloor$,
we have for each $j\in S_{1}$:

$\sup_{I_{j}}|\psi_{\lambda}|\leq
C_{7}(A|I_{j}|)^{p}p!^{s-1}e^{\sqrt{\lambda}}.$

Next, using that $p!\leq p^{p}$, $\kappa\in[\frac{1}{4},1]$, and
$m\sqrt{\lambda}\geq 32$, we get for $j\in S_{1}$:

(14) $\begin{array}[]{r@{}l}\sup_{I_{j}}|\psi_{\lambda}|&\leq
C_{7}\left(2A\frac{p^{s-1}}{N_{0}}\right)^{p}e^{\sqrt{\lambda}}\\\ &\leq
C_{7}\left(4A\frac{n_{0}^{s-1}}{N_{0}^{s}}\right)^{\kappa
p_{0}-1}e^{\sqrt{\lambda}}\\\ &\leq
C_{7}\left(\frac{4A}{C_{0}^{s}}\right)^{\frac{m\sqrt{\lambda}}{16C_{0}}}e^{\sqrt{\lambda}}.\end{array}$

If we choose $C_{0}$ so that $\frac{4A}{C_{0}^{s}}=\frac{1}{16e}$, then Lemma
5.1 follows with the choice of $\ell=1$, $I^{*}=\cup_{j\in S_{1}}I_{j}$, and
$c_{0}=\min\\{\frac{1}{16C_{0}},\frac{1}{32}\\}$. Note that

$|I^{*}|=\sum_{j\in
S_{1}}|I_{j}|\geq\frac{N_{0}}{4}\cdot\frac{|I_{x,\xi}|}{N_{0}}=\frac{|I_{x,\xi}|}{4},$

which confirms the condition on $I^{*}$ in the lemma.

The case (1b), i.e. $|S_{0}\setminus S_{1}|>\frac{3}{4}N_{0}$, is more
complicated and splits into two subcases:

  * ($1b^{\prime}$)

$\sum_{j\in S_{0}\setminus S_{1}}|I_{j}\cap
Z_{\psi_{\lambda}}|\geq\frac{n_{0}}{2}$;

  * ($1b^{\prime\prime}$)

$\sum_{j\in S_{0}\setminus S_{1}}|I_{j}\cap
Z_{\psi_{\lambda}}|<\frac{n_{0}}{2}$.

Assume ($1b^{\prime}$) holds. We claim that with the choice of
$\kappa=\frac{39}{60}$ in (13), we have

(15) $\left|\\{j\in S_{0}\setminus S_{1};\quad|I_{j}\cap
Z_{\psi_{\lambda}}|\geq\frac{p_{0}}{4}\\}\right|\geq\frac{3}{8}N_{0}.$

To prove this, first note that

$\sum_{j\in S_{0}\setminus S_{1}}\kappa p_{0}-|I_{j}\cap
Z_{\psi_{\lambda}}|=\kappa p_{0}|S_{0}\setminus S_{1}|-\sum_{j\in
S_{0}\setminus S_{1}}|I_{j}\cap Z_{\psi_{\lambda}}|\leq\kappa
p_{0}N_{0}-\frac{n_{0}}{2}=\left(\kappa-\frac{1}{2}\right)n_{0}.$

Thus,

(16) $\frac{1}{|S_{0}\setminus S_{1}|}\sum_{j\in S_{0}\setminus S_{1}}\kappa
p_{0}-|I_{j}\cap
Z_{\psi_{\lambda}}|\leq\left(\kappa-\frac{1}{2}\right)\frac{n_{0}}{|S_{0}\setminus
S_{1}|}\leq\frac{4}{3}\left(\kappa-\frac{1}{2}\right)p_{0}.$

Now we implement the following simple lemma for the above average estimate.

######  Lemma 6.5.

Let $\\{z_{j}\\}_{j=1}^{d}$ be a set of non-negative real numbers such that

$\frac{z_{1}+\dots+z_{d}}{d}\leq T.$

Then the subset $S=\\{j;\;z_{j}\leq 2T\\}$ has at least half density in
$\\{1,2,\dots,d\\}$, i.e. $\frac{|S|}{d}\geq\frac{1}{2}$.

###### Proof.

The proof is elementary. Let $S^{c}$ be the complement of $S$. Then

$\frac{2T|S^{c}|}{d}<\frac{\sum_{j\in
S^{c}}z_{j}}{d}\leq\frac{\sum_{j=1}^{d}z_{j}}{d}\leq T,$

which implies that $\frac{|S^{c}|}{d}<\frac{1}{2}$. ∎

Applying this lemma to the set $\left\\{\kappa p_{0}-|I_{j}\cap
Z_{\psi_{\lambda}}|;\quad j\in S_{0}\setminus S_{1}\right\\}$, we obtain:

$\left|j\in S_{0}\setminus S_{1};\quad\kappa p_{0}-|I_{j}\cap
Z_{\psi_{\lambda}}|\leq\frac{8}{3}\left(\kappa-\frac{1}{2}\right)p_{0}\right|\geq\frac{|S_{0}\setminus
S_{1}|}{2}\geq\frac{3}{8}N_{0}.$

However,

$\kappa p_{0}-|I_{j}\cap
Z_{\psi_{\lambda}}|\leq\frac{8}{3}\left(\kappa-\frac{1}{2}\right)p_{0}\implies|I_{j}\cap
Z_{\psi_{\lambda}}|\geq\left(\kappa-\frac{8}{3}\left(\kappa-\frac{1}{2}\right)\right)p_{0}.$

The claim (15) follows since for $\kappa=\frac{39}{60}$, we have
$\kappa-\frac{8}{3}\left(\kappa-\frac{1}{2}\right)=\frac{1}{4}$. Given (15),
we use Prop 6.3 again, with $p=\lfloor\frac{p_{0}}{4}\rfloor$, and follow the
same array of estimates as in (14). In fact we get exactly the same estimate
with the same $C_{0}$, $c_{0}$, and $\ell$. Even the length of the obtained
union of subintervals is bounded below by the same quantity:

$|I|\geq\frac{3N_{0}}{8}\cdot\frac{|I_{x,\xi}|}{N_{0}}=\frac{3|I_{x,\xi}|}{8}>\frac{I_{x,\xi}}{4}.$

Next, we need to deal with the case ($1b^{\prime\prime}$), which turns out to
be more involved.

So let us assume $|S_{0}\setminus S_{1}|>\frac{3}{4}N_{0}$ and $\sum_{j\in
S_{0}\setminus S_{1}}|I_{j}\cap Z_{\psi_{\lambda}}|<\frac{n_{0}}{2}$.
Equivalently, we have:

$|S_{1}|\leq\frac{1}{4}N_{0}\quad\text{and}\quad\sum_{j\in S_{1}}|I_{j}\cap
Z_{\psi_{\lambda}}|\geq\frac{n_{0}}{2}.$

In the second step of our argument, we let

$N_{1}=|S_{1}|,\quad n_{1}=\frac{n_{0}}{2}\quad\text{and}\quad
p_{1}=\frac{n_{1}}{N_{1}}.$

We also denote

$S_{2}=\\{j\in S_{1};\ \quad|I_{j}\cap Z_{\psi_{\lambda}}|\geq\kappa
p_{1}\\}.$

We repeat the same procedure we previously applied to $S_{1}$, to $S_{2}$. So
we get cases ($2a$), ($2b^{\prime}$), and ($2b^{\prime\prime}$) and everywhere
we replace $p_{0}$ with $p_{1}$, $n_{0}$ with $n_{1}$, and $N_{0}$ with
$N_{1}$. In fact, by continuing this procedure, we obtain for each
$k\in{\mathbb{N}}^{>0}$, subsets $S_{k}$ defined inductively by

$S_{k}=\\{j\in S_{k-1};\quad|I_{j}\cap Z_{\psi_{\lambda}}|\geq\kappa
p_{k-1}\\}.$

We also have for each $k$:

$N_{k}=|S_{k}|,\quad n_{k}=\frac{n_{0}}{2^{k}}\quad\text{and}\quad
p_{k}=\frac{n_{k}}{N_{k}}.$

Furthermore, we have

$|S_{k}|\leq\frac{1}{4}N_{k-1}\leq\frac{1}{4^{k}}N_{0}\quad\text{and}\quad\sum_{j\in
S_{k}}|I_{j}\cap
Z_{\psi_{\lambda}}|\geq\frac{n_{k-1}}{2}=\frac{n_{0}}{2^{k}}.$

This procedure is not infinite, because as soon as $\frac{1}{4^{k}}N_{0}<1$ we
get $S_{k}=\emptyset$, and the procedure stops. This corresponds to
$k=\lceil\frac{\log N_{0}}{\log 4}\rceil+1$. To prove Lemma 6.3, we need to
show that for each of the intermediate cases ($ka$) and ($kb^{\prime}$), the
lemma follows. So let us fix $k\in{\mathbb{N}}^{>0}$. Cases ($ka$) and
($kb^{\prime}$) are:

  * ($ka$)

$|S_{k}|>\frac{1}{4}N_{k-1}$

  * ($kb^{\prime}$)

$|S_{k-1}\setminus S_{k}|>\frac{3}{4}N_{k-1}$ and $\sum_{j\in S_{k-1}\setminus
S_{k}}|I_{j}\cap Z_{\psi_{\lambda}}|<\frac{n_{k-1}}{2}$.

Suppose case ($ka$) occurs. By the definition of $S_{k}$, for each $j\in
S_{k}$, we have at least $\kappa p_{k-1}$ zeros of $\psi_{\lambda}$ on
$I_{j}$. Now we partition each interval $I_{j}$, $j\in S_{k}$, into
$\lfloor\frac{p_{k-1}}{p_{0}}\rfloor$ many equal length intervals. Hence there
must exist $J_{j}\subset I_{j}$ of length
$\frac{|I_{j}|}{\lfloor\frac{p_{k-1}}{p_{0}}\rfloor}$ on which
$\psi_{\lambda}$ has at least

$\frac{\kappa p_{k-1}}{\lfloor\frac{p_{k-1}}{p_{0}}\rfloor}\geq\kappa p_{0}$

many zeros. So using Prop 6.3, with $I=J_{j}$ and $p=\lfloor\kappa
p_{0}\rfloor$, we obtain

(17) $\begin{array}[]{r@{}l}\sup_{J_{j}}|\psi_{\lambda}|&\leq
C_{7}(A|J_{j}|)^{p}p!^{s-1}e^{\sqrt{\lambda}}\\\ &\leq
C_{7}\left(4A\frac{p^{s-1}}{N_{0}}\cdot\frac{p_{0}}{p_{k-1}}\right)^{p}e^{\sqrt{\lambda}}\\\
&\leq
C_{7}\left(4A\frac{n_{0}^{s-1}}{N_{0}^{s}}\cdot\frac{p_{0}}{p_{k-1}}\right)^{\kappa
p_{0}-1}e^{\sqrt{\lambda}}\\\ &\leq
C_{7}\left(\frac{4A}{C_{0}^{s}}\cdot\frac{p_{0}}{p_{k-1}}\right)^{\frac{m\sqrt{\lambda}}{16C_{0}}}e^{\sqrt{\lambda}}.\end{array}$

Since in (14), we have already chosen $C_{0}$ and $c_{0}$ to satisfy
$\frac{4A}{C_{0}^{s}}=\frac{1}{16e}$ and
$c_{0}=\min\\{\frac{1}{16C_{0}},\frac{1}{32}\\}$, we obtain

(18) $\sup_{J_{j}}|\psi_{\lambda}|\leq
C_{7}\left(\frac{p_{0}}{16ep_{k-1}}\right)^{c_{0}m\sqrt{\lambda}}e^{\sqrt{\lambda}}.$

The total interval on which we have this estimate is $I^{*}=\cup_{j\in
S_{k}}J_{j}$, which has length

$\displaystyle|I^{*}|=|S_{k}||J_{j}|$
$\displaystyle\geq\frac{1}{4}N_{k-1}\cdot\frac{|I_{j}|}{\lfloor\frac{p_{k-1}}{p_{0}}\rfloor}=\frac{N_{k-1}}{4N_{0}}\cdot\frac{|I_{x,\xi}|}{\lfloor\frac{p_{k-1}}{p_{0}}\rfloor}$
$\displaystyle\geq\frac{n_{k-1}}{8n_{0}}\cdot\left(\frac{p_{0}}{p_{k-1}}\right)^{2}{|I_{x,\xi}|}$
$\displaystyle=\frac{1}{2^{k+2}}\cdot\left(\frac{p_{0}}{p_{k-1}}\right)^{2}{|I_{x,\xi}|}.$

Let $\ell\in N^{>0}$ be the unique integer that satisfies

(19)
$\frac{1}{4^{\ell}}\leq\frac{1}{2^{k+2}}\cdot\left(\frac{p_{0}}{p_{k-1}}\right)^{2}<\frac{1}{4^{\ell-1}}.$

By this choice of $\ell$, $|I^{*}|$ certainly satisfies the required lower
bound $\frac{|I_{x,\xi}|}{4^{\ell}}$. Applying this to (18), we get

(20) $\sup_{J_{j}}|\psi_{\lambda}|\leq
C_{7}\left(2^{k}4^{-\ell}\right)^{\frac{c_{0}}{2}m\sqrt{\lambda}}e^{\sqrt{\lambda}}.$

On the other hand, by the definition of $p_{k-1}$, we have

$\frac{p_{k-1}}{p_{0}}=\frac{\frac{n_{k-1}}{N_{k-1}}}{p_{0}}\geq\frac{\frac{n_{0}/2^{k-1}}{N_{0}/4^{k-1}}}{2p_{0}}\geq{2^{k-1}}.$

Using this and (19), we must have that $4^{\ell}\geq 8^{k}$, or equivalently
$k\leq\frac{2}{3}\ell$, therefore (20) becomes

(21) $\sup_{J_{j}}|\psi_{\lambda}|\leq
C_{7}\left(2^{-4/3\ell}\right)^{\frac{c_{0}}{2}m\sqrt{\lambda}}e^{\sqrt{\lambda}}=C_{7}e^{(1-\frac{2}{3}(\log
2){c_{0}}m\ell)\sqrt{\lambda}}\leq C_{7}e^{(1-{c_{0}}m\ell)\sqrt{\lambda}}.$

This proves the lemma in the case ($ka$). Finally assume case ($kb^{\prime}$)
occurs. Then as in the case ($1b^{\prime}$), we claim and prove that for
$\kappa=\frac{39}{60}$,

(22) $\left|\\{j\in S_{k-1}\setminus S_{k};\quad|I_{j}\cap
Z_{\psi_{\lambda}}|\geq\frac{p_{k-1}}{4}\\}\right|\geq\frac{3}{8}N_{k-1}.$

Up to a change of indices, the proof of (22) is identical to the proof of
claim (15). We then follow as in the case ($ka$), meaning, for each $j\in
S_{k-1}\setminus S_{k}$ that satisfies the lower bound $|I_{j}\cap
Z_{\psi_{\lambda}}|\geq\frac{p_{k-1}}{4}$, we find a subinterval $J_{j}\subset
I_{j}$ with length $\frac{|I_{j}|}{\lfloor\frac{p_{k-1}}{p_{0}}\rfloor}$ on
which $\psi_{\lambda}$ has at least $\lfloor\frac{p_{0}}{4}\rfloor$ many
zeros. We then argue exactly as in the case ($ka$) presented above. ∎

##  7\. Proof in the case of Denjoy-Carleman Riemannian manifolds

In this section, we sketch the proof of Theorem 1.3. A main tool is
$C^{\mathcal{M}}$ hypoellipticity estimates for eigenfunctions which we
already presented in (6). The proof of the upper bounds on the size of nodal
sets follows the same arguments as in the Gevrey case but with some
modifications that we now present. Assume that $\mathcal{M}_{k}$ is a regular
sequence with moderate growth and let

$J_{k}=(k!\mathcal{M}_{k})^{1/k}.$

It is known that $J_{k}$ is an increasing sequence. Then we change the
definition of the set $A_{m}$ in (10), to

$A_{m}=\left\\{(x,\xi)\in
S^{*}B^{n}_{1};\;J_{m\lfloor\sqrt{\lambda}\rfloor}\leq
n(x,\xi)<J_{(m+1)\lfloor\sqrt{\lambda}\rfloor}\right\\}.$

As we will see, we still get the same estimates $\mu(A_{m})\leq Ce^{-cm}$ as
in Lemma 4.1 on the size of $A_{m}$. Following the same steps as we gave for
the proof of Theorem 1.1, we get

(23) $\displaystyle\mathcal{H}^{n-1}(Z_{\psi_{\lambda}}\cap B^{n}_{1})$
$\displaystyle\leq
C\sum_{m=0}^{\infty}e^{-cm}J_{(m+1)\lfloor\sqrt{\lambda}\rfloor}.$

To obtain bounds on the last sum, we utilize the following lemma:

######  Lemma 7.1.

Let $\mathcal{M}_{k}$ be a regular sequence with moderate growth. Then there
exists a constant $\gamma>0$ such that for all $m,j\in{\mathbb{N}}^{>0}$, we
have

$\left(\mathcal{M}_{mj}\right)^{1/mj}\leq
e^{2\gamma}\left(\mathcal{M}_{j}\right)^{1/j}m^{\gamma}.$

###### Proof.

Let
$A=\sup_{j,k\in{\mathbb{N}}^{>0}}\left(\frac{\mathcal{M}_{j+k}}{\mathcal{M}_{j}\mathcal{M}_{k}}\right)^{1/(j+k)}$.
By the definition of moderate growth, we have $A<\infty$. Then using the
convexity of $\mathcal{M}$ and the moderate growth assumption, one can show
that (see [Th03][equations (6) and (9)]):

$\mathcal{M}_{j+1}\leq A^{2}\mathcal{M}_{j}^{1+\frac{1}{j}}.$

Using this inequality repeatedly, we obtain for all $j,k\in{\mathbb{N}}^{>0}$
that

$(\mathcal{M}_{j+k})^{1/(j+k)}\leq\mathcal{M}_{j}^{1/j}A^{2+2\sum_{i=0}^{k-1}\frac{1}{j+i}}\leq\mathcal{M}_{j}^{1/j}A^{4+2\log\left(\frac{j+k}{j}\right)}.$

If we put $k=(m-1)j$ into this inequality, then the lemma follows with
$\gamma=2\log A$. ∎

###### Proof of Theorem 1.3.

Let us continue on (23). The proof is straightforward because by the above
lemma and the definition of $J_{k}$:

$\displaystyle\mathcal{H}^{n-1}(Z_{\psi_{\lambda}}\cap B^{n}_{1})$
$\displaystyle\leq
C\sum_{m=0}^{\infty}e^{-cm}J_{(m+1)\lfloor\sqrt{\lambda}\rfloor}$
$\displaystyle\leq
C_{1}\sum_{m=0}^{\infty}e^{-cm}(m+1)^{1+\gamma}\sqrt{\lambda}\left(\mathcal{M}_{\lfloor\sqrt{\lambda}\rfloor}\right)^{1/\lfloor\sqrt{\lambda}\rfloor}$
$\displaystyle\leq
C_{2}\sqrt{\lambda}\left(\mathcal{M}_{\lfloor\sqrt{\lambda}\rfloor}\right)^{1/\lfloor\sqrt{\lambda}\rfloor}.$

∎

To prove the exponential upper bound $\mu(A_{m})\leq Ce^{-cm}$, we only need
to know that Lemma 5.1 holds in the Denjoy-Carleman setting. In fact there
will be no change in the statement of Lemma 5.1. As we pointed out in Remark
6.4, in the Denjoy-Carleman case there is a change in the estimates of Prop
6.3:

$\sup_{I}|\psi_{\lambda}|\leq
C_{7}(A|I|)^{p}\,\mathcal{M}_{|p|}e^{\sqrt{\lambda}}.$

However, this will not change the result of Lemma 5.1 as the proof reveals.
The only changes required are in the initial quantities used in the first step
of the proof:

$n_{0}=\lfloor J_{m\lfloor\sqrt{\lambda}\rfloor}\rfloor,\quad
N_{0}=\left\lfloor\frac{n_{0}}{m\sqrt{\lambda}}\right\rfloor,\quad
p_{0}=\frac{n_{0}}{N_{0}}.$

Correspondingly, these will change the quantities $n_{k}$, $N_{k}$, and
$p_{k}$. The proof is otherwise identical to the proof in the Gevrey case.

### Acknowledgement

The author would like to thank Steve Zelditch for their comments on the first
draft of this article and for bringing to attention the works of [Hi50] and
[NaSoVo04]. The research of the author is supported by the Simons
Collaborations Grants for Mathematicians 638398.

## References

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