2.1 Reconstructing the total nodal length from local patches

The starting point of the proofs is the following observation: because the nodal volume is a local quantity (i.e., it is additive w.r.t. (disjoint) subsets of M), one may asymptotically reconstruct it based on averaging the local nodal volume of $f_{T}$ restricted to small balls w.r.t. their centres. That is,

(2.1) $$ \begin{align} \mathcal{V}(f_T)= T (1+o(1)) \int_{M} \mathcal{V}(F_x) d\operatorname{\mathrm{\operatorname{Vol}}}_g(x), \end{align} $$

where $F_x$ is a scaled local version

$$ \begin{align*}F_{x,T}(y)= F_x(y)= f_T(\exp_x(y/T))\end{align*} $$

of $f_{T}$ in the vicinity of $x\in M$ , defined on the unit Euclidean ball $y\in B_0(1)$ . Thus, to evaluate $\mathcal {V}(f)$ , it is sufficient to understand the nodal volume $\mathcal {V}(F_x)$ on average w.r.t. $x\in M$ .

With this notion in mind, rather than working with $f_{T}$ as a random field defined on a probability space $\Omega $ (where the random variables $a_{i}$ are defined), we may think of $F_x(\cdot )$ as a random field indexed by $B_0(1)$ , defined on the probability space $M\times \Omega $ . Thus, the local nodal volume $\mathcal {V}(F_x)$ is, in this sense, a random variable on the product space $M\times \Omega $ w.r.t. the normalized probability measure $\frac {d\operatorname {\mathrm {\operatorname {Vol}}}_g}{\operatorname {\mathrm {\operatorname {Vol}}}(M)}\otimes d\mathbb {P}$ . In light of the above, to use the observation (2.1), the proofs of Theorem 1.1 and Theorem 1.3 will borrow from two important preliminary steps: local asymptotic Gaussianity of $f_{T}$ in Proposition 4.1 (regarding the growing energy windows case) and Proposition 9.1 (regarding the constant energy windows case) and an anti-concentration estimate for $\mathcal {V}(F_x)$ in Proposition 6.1. We now explicate the meaning of these preliminary steps and give a sketch of their proofs.

2.2 Asymptotic Gaussianity

Because, under the assumptions on $\rho $ of either Theorem 1.1 or Theorem 1.3, the number of the summands within (1.2) is growing and the $a_{i}$ are i.i.d., the scaled version $F_{x,T}$ of $f_{T}$ should asymptotically behave like a Gaussian random field with correlations given by (1.1). It will be rigorously proved for either the growing window regime as in Theorem 1.1 or the constant width regime $\rho \equiv \rho _0$ , though for the former case, our arguments are significantly simplified.

Let us first explain the proof under the assumptions of Theorem 1.1. In this case, the asymptotic behavior of the correlation function of $F_{x,T}$ , postulated in (1.5), is given by the local Weyl’s law of Hörmander; see Section 4.2 below. It also follows that all the summands in (1.2) have size $o(v(T))$ . Therefore, an application of Linderberg’s Central Limit theorem, together with the Continuous Mapping Theorem, implies

$$ \begin{align*} \mathcal{V}(F_x) \overset{d}{\longrightarrow} \mathcal{V}(F_{\mu}) \hspace{10mm}T\rightarrow \infty, \end{align*} $$

where the convergence is in distribution uniformly w.r.t. $x\in M$ (that is, for all continuous and bounded functions $h:\mathbb {R}\rightarrow \mathbb {R}$ , $\mathbb {E}[h(\mathcal {V}(F_x))] \rightarrow \mathbb {E}[h(\mathcal {V}(F_{\mu }))]$ uniformly for all $x\in M$ ). This is the content of Proposition 4.1 below. Thus, in this case, for the asymptotic Gaussianity, there is no need to make use of the extra averaging with respect to $x\in M$ (but this will be required for the rest of the proof). Due to such a simplification, we will present the proof of Proposition 4.1 first in Section 4 so that the probabilistic arguments are easier to describe and can be separated from the more precise microlocal analysis techniques required in the constant energy window case, which we are going to discuss next.

In the case of constant energy windows $\rho \equiv \rho _0$ , there are at least two obstacles to the described approach:

1. (i) Around some ‘bad’ points $x\in M$ , the asymptotic behavior of the covariance kernel of $F_x$ may not coincide with (1.1).

2. (ii) Around some other ‘bad’ points, some of the summands in (1.2) could be as large, by order of magnitude, as $v(T)^{1/2}$ , occurring in reality, for example, on the sphere. Around these points the Central Limit Theorem is not applicable.

To overcome obstacle (i), the spectral projector operator

$$ \begin{align*}L^2(M)\rightarrow \operatorname{Sp}\{\phi_{i}\}_{\lambda_{i} \in [T-\rho, T]}\end{align*} $$

is carefully studied in Section 8. We show that, under the assumptions of Theorem 1.3 on the self-focal points of M, the asymptotics (1.5) hold outsideFootnote ¹ a set of points $x\in M$ of small measure. Sogge’s bound [Reference Sogge54] is used to prove that all the summands in (1.2) are of size $o(v(T)^{1/2})$ for x outside of another set of small measure, though crucially depending on T. Since, other than the vanishing measure of the bad sets, no other useful property of the family of bad sets is established (it would be useful if, for example, this family would be monotone decreasing with T growing), the Central Limit Theorem is not applicable with any fixed $x\in M$ .

Instead, a ‘triangular’ version of the Central Limit Theorem, allowing for the random variables to depend on a parameter, is applied with x varying with T; as it was explained above, the elegant way to express the outcome of its application as a single consolidated result is by thinking of x random uniform on the good set and, a forteriori, using the asymptotic vanishing of the excised set for x random uniform on M. Hence, the convergence of $F_{x}(\cdot )$ to the limit monochromatic random field is as a random field w.r.t. the probability measure $\frac {d\operatorname {\mathrm {\operatorname {Vol}}}_g}{\operatorname {\mathrm {\operatorname {Vol}}}(M)}\otimes d\mathbb {P}$ on $M\times \Omega $ (and the convergence of $F_{x}(\cdot )$ w.r.t. $d\mathbb {P}$ on $\Omega $ is not asserted for any given $x\in M$ ).

To the best of our knowledge, this aspect of our proofs, inspired by the de-randomization techniques [Reference Bourgain12, Reference Buckley and Wigman15], is novel in the context of the study of the geometry of random fields, and different from the rest of the literature on the subject, where the Central Limit Theorem is normally applied for every $x\in M$ fixed. The convergence, in distribution, of the random variables $\mathcal {V}(F_x)$ , also w.r.t. the probability measure $\frac {d\operatorname {\mathrm {\operatorname {Vol}}}_g}{\operatorname {\mathrm {\operatorname {Vol}}}(M)}\otimes d\mathbb {P}$ on $M\times \Omega $ , to $\mathcal {V}(F_{\mu })$ , follows directly from the convergence of the random fields $F_{x}$ to the limit random field $F_{\mu }$ , via the Continuous Mapping Theorem. This is the content of Proposition 9.1.

2.3 Anti-concentration

Since the outcome of Proposition 4.1 and Proposition 9.1 are valid outside a set of small probability (and outside a set of $x\in M$ of small volume), it is essential to demonstrate that the contribution of the exceptional set to (2.1) is negligible. In other words, we need to show that it is unlikely that a large proportion of the nodal set concentrates in a small portion of space, hence the term ‘anti-concentration’. This is precisely the purpose of the anti-concentration Proposition 6.1, whose proofs will be now discussed.

The required anti-concentration result is the existence of some function $h:\mathbb {R}_{>0}\rightarrow \mathbb {R}_{>0}$ , so that

$$ \begin{align*}\frac{h(t)}{t}\rightarrow \infty,\end{align*} $$

and that satisfies the estimate

(2.2) $$ \begin{align} \int_{M}\mathbb{E}[h(\mathcal{V}(F_x))]d\operatorname{\mathrm{\operatorname{Vol}}}_g(x)<C \end{align} $$

for some constant $C=C(M,g)>1$ , independent of T.

In Proposition 6.1, we will show that (2.2) holds true with $h(t)= t\cdot \log t$ provided that the energy window satisfies

$$ \begin{align*}\rho=\rho(T)\geq \begin{cases} T^{\vartheta_{n}}(\log T)^2 &2\le n\le 4 \\ \rho_0 & n\ge 5 \end{cases},\end{align*} $$

with $\vartheta _{n}$ given by (1.8).

Following the approach of Donnelly-Fefferman [Reference Donnelly and Fefferman21], Lin [Reference Lin35] and Jerison-Lebeau [Reference Christ and Kenig41, Section 14], the nodal volume of $f(\cdot )=f_{T}(\cdot )$ can be controlled via the doubling index (of the ‘harmonic lift’ of f). The doubling index is defined for any function

$$ \begin{align*}h: 3B= B(x,3r)\subseteq M\rightarrow \mathbb{R}\end{align*} $$

as

$$ \begin{align*}\mathcal{N}_h(x,r)=\mathcal{N}_h(B)= \log\frac{\sup_{2B}|h|}{\sup_{B}|h|}.\end{align*} $$

In Section 5, we will show, appealing to the analyticity of M, that the nodal volume of f in a ball of radius $r>0$ can be bounded as

$$ \begin{align*}\mathcal{V}(f, B_r)\lesssim r^{n-1} \mathcal{N}_{f^H}(\widetilde{B}_{8r})= r^{n-1}\mathcal{N}(\widetilde{B}_{8r}),\end{align*} $$

where $f^H$ is the harmonic lift of f to the manifold $M\times \mathbb {R}$ (at this stage it is instructive, although slightly imprecise, to think of the harmonic lift as $f^H(x,t)= f(x)\cdot \exp (T\cdot t)$ ), and $\widetilde {B}_{8r}$ is the ‘ball’

$$ \begin{align*}\widetilde{B}_{8r}=B_{8r}\times [-8r,8r].\end{align*} $$

The well-known bounds on the growth rate of eigenfunctions, as in [Reference Donnelly and Fefferman21], give

$$ \begin{align*}\mathcal{N}_{f^H}((x,0),c)=:\mathcal{N}(x,c)\lesssim T\end{align*} $$

for some constant $c=c(M,g)>0$ . The monotonicity of the doubling index (w.r.t. the radius $r>0$ ) for harmonic function [Reference Lin35] implies

$$ \begin{align*}\mathcal{N}(x,8/T) \lesssim \mathcal{N}(x, c)\lesssim T.\end{align*} $$

Thus, the statement (2.2) of Proposition 6.1 is equivalent, in essence, to an estimate of the type

(2.3) $$ \begin{align} \left(\operatorname{\mathrm{\operatorname{Vol}}} \otimes \mathbb{P}\right) \left(\left\{(x,\omega): \mathcal{V}(F_x)>H\right\}\right)\leq \frac{1}{H(\log H)^{2+\varepsilon}}, \end{align} $$

for all $1<H\lesssim T$ . The asymptotic estimate (2.1), together with the global bound $\mathcal {V}(f_T)\lesssim T$ , give

$$ \begin{align*}\operatorname{\mathrm{\operatorname{Vol}}}\left( x \in M : \mathcal{V}(F_x)>H\right)\lesssim H^{-1}.\end{align*} $$

Therefore, the aspired bound (2.3), holding with high probability w.r.t. the product space, is a logarithmic gain only over a bound holding deterministically for every band-limited function. This is the most delicate and, to our best knowledge, novel aspect of the proof of Proposition 6.1, described immediately below.

As discussed above, a large value of $\mathcal {V}(F_x)$ also implies a large value of the doubling index on $\tilde {B}=B(x,8/T) \times [-8/T,8/T]$ , which, in turn, may only happen if either the function has a large value on $2\tilde {B}$ or has a small value on $\tilde {B}$ , roughly speaking. The former case can be dealt with via the second moment method. However, controlling the small values of f (or rather, of $f^H$ ) is more delicate. Quantifying the Gaussian convergence obtained in Section 4, it is possible to control the probability of $f(x)$ being of ‘small’ depending on the $L^3$ -norm (cubed) of the eigenfunctions. After some computations, this leads to the bound

(2.4) $$ \begin{align} \sup_{x\in M} \mathbb{P} (\mathcal{N}(x,8/T)>H)\lesssim \exp(-H) + \sup_{\lambda_{i}\in [T-\rho,T]}||\phi_{i}||_{L^3}^3 v(T)^{-1/2}. \end{align} $$

Therefore, in order to obtain (2.3), it would be sufficient to control the second term on the r.h.s. of (2.4). Unfortunately, appealing to Sogge’s bound [Reference Sogge54] turns out to be not quite sufficient to yield Proposition 6.1. Thus, we will use one last ‘trick’, and by using the Gaussian convergence at various scales and the monotonicity of the doubling index, we will show that

$$ \begin{align*}\mathbb{P} (\sup_x \mathcal{N}(x,3/T)\leq T^{1-c})\geq 1 + O((\log T)^{-1})\end{align*} $$

for some constant $c=c(n)>0$ . This will reduce the range of H in (2.3), and thus the bound (2.4) will suffice to prove Propostion 6.1 in the appropriate range of $\rho $ specified above. This concludes the sketch of the proofs.

4.1 Notation and goal of Section 4

First, we will define the rescaled version of $f_T$ , as in (1.2), in geodesic balls of radius $T^{-1}$ . Let $x\in M$ and let $F_x$ be $f_T$ rescaled to the ball $B_g(x,1/T)$ in normal coordinates. More precisely, we define:

(4.1) $$ \begin{align} F_{T,x}(y)=F_x(y)= f(\exp_x(y/T)) \end{align} $$

for $y\in B(0,1)=:B_0\subseteq \mathbb {R}^n$ , where $\exp _x: \mathbb {R}^n\cong T_xM \rightarrow M$ is the exponential map. Notice that, in the definition of the exponential map, we have tacitly identified $\mathbb {R}^n$ with $T_xM$ via an Euclidean isometry. Moreover, we observe that since $(M,g)$ is analytic, the injectivity radius of M is uniformly bounded from below [Reference Cheeger20]; thus, from now on, we assume that $1/T$ is smaller than the injectivity radius so that the exponential map is a diffeomorphism. Furthermore, thanks to [Reference Nazarov and Sodin43, Section 8.1.2] due to Nazarov and Sodin (see also [Reference Rozenshein49, Section 2]), the map

$$ \begin{align*}(x,\omega)\in M\times \Omega \rightarrow F_x(\omega, \cdot)\in C^{\infty}(B_0)\end{align*} $$

is measurable.

We now define the universal scaling limit for the nodal length of $F_x$ . We denote $F_{\mu }$ to be the monochromatic isotropic Gaussian field on $B_0\subseteq \mathbb {R}^n$ with spectral measure $\mu $ , the (normalised) Lebesgue measure on the $n-1$ dimensional sphere $\mathbb {S}^{n-1}$ . Equivalently, $F_{\mu }$ has the covariance function

(4.2) $$ \begin{align} \mathbb{E}[ F_{\mu}(y)\cdot F_{\mu}(y')]= \int_{|\xi|=1}\exp\left(i\langle y-y', \xi\rangle\right)d\mu(\xi)=(2\pi)^{\Lambda}\frac{J_{\Lambda}(|y-y'|)}{|y-y'|^{\Lambda}}, \end{align} $$

with $\Lambda =(n-2)/2$ and where $J_{\Lambda }(\cdot )$ is the usual Bessel J function of order $\Lambda $ . In what follows we will use the shorthands

$$ \begin{align*} \mathcal{V}(F_x):=\mathcal{V}\left(F_x, \frac{1}{2}B_0\right) \quad\quad\quad\text{and} \quad\quad\quad \mathcal{V}(F_\mu):= \mathcal{V}\left(F_{\mu}, \frac{1}{2}B_0\right). \end{align*} $$

The aim of this section is to prove the following proposition:

Proposition 4.1. Let $F_x$ be as in (4.1), $F_{\mu }$ be as above. Then, under the assumptions of Theorem 1.1 on the energy window width $\rho =\rho (T)$ , uniformly for all $x\in M$ , we have

$$ \begin{align*} \mathcal{V}(F_x) \overset{d}{\longrightarrow} \mathcal{V}(F_{\mu}) \hspace{10mm}T\rightarrow \infty\nonumber \end{align*} $$

convergence in distribution.

Observe that in Proposition 4.1, the convergence to the Gaussian random field is claimed for fixed $x\in M$ , stronger than the average statement w.r.t. $x\in M$ , required for the proof of Theorem 1.1 (cf. Proposition 9.1 that is used for the proof of Theorem 1.3). This is where the growing energy window assumption of Theorem 1.1 is also used. In particular, Proposition 4.1 implies that

(4.3) $$ \begin{align} \mathcal{V}(F_x) \overset{d}{\longrightarrow} \mathcal{V}(F_{\mu}) \hspace{10mm}T\rightarrow \infty \end{align} $$

converges in distribution as a random variable on $(M\times \Omega , d\sigma )$ , where

$$ \begin{align*}d\sigma = (\operatorname{\mathrm{\operatorname{Vol}}}_g(M))^{-1}d\operatorname{\mathrm{\operatorname{Vol}}}_g \otimes d\mathbb{P}\end{align*} $$

(cf. Proposition 9.1).

4.2 Hörmander’s local Weyl’s law

In order to study $F_x$ as in (4.1), we will need the well-known local Weyl’s law of Hörmander [Reference Hörmander28, Theorem 4.4], which we do not present in its full generality but in a form convenient for our purposes. In particular, there are no restrictions on the width of the spectral windows $\rho $ in the following result (for $\rho $ too small, the error term dominates):

Proposition 4.3. Let $(M,g)$ be a compact, real analytic manifold with empty boundary. Let $x\in M$ and consider a (sufficiently small) coordinate patch $\Omega _x$ around x in normal coordinates. Then

$$ \begin{align*} \sup_{y,y'\in \Omega_x}\left|\sum_{\lambda_i\in[T- \rho, T]} \phi_{i}(y)\phi_{i}(y')- c_M T^{n} \mathcal{J}_{\Upsilon(T)}(Td_g(y,y')) \right| = O_{M,g}(T^{n-1}), \end{align*} $$

where $d_g(y,y')$ is the geodesic distance between $y,y'$ , $c_M>0$ is given in (1.3), $\Upsilon (T)=1-\frac {\rho }{T}$ and

(4.4) $$ \begin{align} \mathcal{J}_{\Upsilon(T)}(w)= \int_{\Upsilon(T) \leq |\xi|\leq 1}\exp(i\langle w,\xi \rangle)d\xi. \end{align} $$

Moreover, we can also differentiate both sides of (8.1) an arbitrary finite number of times; that is,

$$ \begin{align*} \sup_{y,y'\in \Omega_x}\frac{\left|\sum_{\lambda_i\in[T- \rho, T]} D^{\alpha}_{y}\phi_{i}(y)D^{\alpha'}_{y'}\phi_{i}(y')- \frac{c_M T^{n} D^{\alpha}_{y}D^{\alpha'}_{y'}\mathcal{J}_{\Upsilon(T)}(Td_g(y,y'))}{(2\pi)^{n}} \right|}{T^{|\alpha|+|\alpha'|}} = O_{M,g,\alpha}(T^{n-1}), \end{align*} $$

where $\alpha ,\alpha '$ are multi-indices and $\xi ^{\alpha }=(\xi _1^{\alpha _1},...,\xi _n^{\alpha _n})$ and the derivatives are understood after taking normal coordinates around the point x.

The following corollary will be quite useful later:

Corollary 4.4. Let $(M,g)$ be a compact, real analytic manifold with empty boundary. There exists some $\rho _0= \rho _0(M,g)$ such that for all $\rho \geq \rho _0$ , the following holds. Let $v(T)$ be as in (1.3). Then

$$ \begin{align*}\sum_{\lambda_i\in[T- \rho, T]} |\phi_{i}(x)|^2\asymp v(T),\end{align*} $$

where $A\asymp B$ means that there exist two constant $0<c<C$ , depending only on $(M,g),$ such that $cA\leq B\leq CA$ .

Although $\rho _{0}$ does not appear in the proof of Proposition 4.1 (and Theorem 1.1), the value of $\rho _0$ is fixed from now until the end of the article, for the results building up to Theorem 1.3. As a direct consequence of Proposition 4.3 and a straightforward calculation, we also have the following result:

Lemma 4.5. Let $(M,g)$ be a real analytic compact manifold with empty boundary of dimension n, let $f_T(\cdot )$ be as in (1.2), let $\rho (T)$ be the width of the energy window and let $c_M$ and $v(T)$ be as in (1.3). Then, under the assumption of Theorem 1.1 on the width of the spectral window $\rho (T)$ , we have

$$ \begin{align*}\mathbb{E}\left[|f_T(x)|^2\right]= \frac{1}{v(T)}\sum_{\lambda_i\in [T-\rho,T]} |\phi_{i}(x)|^2= (1+o(1)),\end{align*} $$

where the error term is uniform for all $x\in M$ . Moreover, for $F_x$ as in (4.1), we have

$$ \begin{align*} \sup_{\substack{x\in M\\ y,y'\in B_0}}\left|\mathbb{E}[F_x(y)\cdot F_x(y')]- (2\pi)^{\Lambda}\frac{J_{\Lambda}(|y-y'|)}{|y-y'|^{\Lambda}}\right|\rightarrow 0 & T\rightarrow \infty, \end{align*} $$

with $\Lambda =(n-2)/2$ and $ J_{\Lambda } (\cdot )$ the $\Lambda $ -th Bessel function. Further, one can differentiate both sides an arbitrary finite number of times; that is,

$$ \begin{align*} \mathbb{E}[D^{\alpha}F_x(y)\cdot D^{\alpha'}F_x(y')]=(-1)^{|\alpha'|}i^{|\alpha|+|\alpha'|} \int_{|\xi|=1}\xi^{\alpha+\alpha'}\exp\left(i\langle y-y', \xi\rangle\right)d\mu(\xi) +o_{T\rightarrow \infty}(1), \end{align*} $$

valid uniformly for all $x\in M,\,y,y'\in B_0$ , where $\alpha ,\alpha '$ are fixed multi-indices and $\xi ^{\alpha }=(\xi _1^{\alpha _1},...,\xi _n^{\alpha _n})$ .

Proof. By the first claim in Proposition 4.3 and the compactness of M, we have

$$ \begin{align*}\sum_{\lambda_{i} \in [T-\rho, T]}|\phi_i(x)|^2= c_M\rho(T) T^{n-1} +O(T^{n-1}),\end{align*} $$

where the error term is uniform for all $x\in M$ . Thus, the first claim in Lemma 4.5 follows by dividing both sides by $v(T)$ . In order to see the second claim in Lemma 4.5, let us take $y,y'\in B_0$ with $y\neq y'$ and let us rewrite the integral in (4.4) in the spherical coordinates and use the identity

$$ \begin{align*}\int_{S^{n-1}}\exp\left(i\langle u,\xi\rangle \right)d\mu(\xi)= (2\pi)^{\Lambda} \frac{J_{\Lambda}(|u|)}{|u|^{\Lambda}},\end{align*} $$

to obtain

(4.5) $$ \begin{align} \sum_{\lambda_i\in[T- \rho, T]} \phi_{i}(y')\phi_{i}(y)&= c_M\rho (T)T^{n-1} \int_{|\xi|=1} \exp(i\langle Td_g(y',y), \xi\rangle)d\mu(\xi) \nonumber \\& \quad + O\left(\rho(T)T^{n-1}d_g(y',y)\right) +O(T^{n-1}) \nonumber \\&= c_M \rho T^{n-1}(2\pi)^{\Lambda}\frac{J_{\Lambda}(|T d_g(y',y)|)}{|T d_g(y',y)|^{\Lambda}} \nonumber\\& \quad +O\left( \rho(T)T^{n-2}\right) + O(T^{n-1}), \end{align} $$

where $\Lambda = (n-2)/2$ . Thus, the second claim in Lemma 4.5 follows by dividing both sides of (4.5) by $v(T)$ and compactness of M. The third claim in Lemma 4.5 follows by the second claim in Proposition 4.3 and similar computation to (4.5) (and again the compactness of M).

4.3 Convergence of finite-dimensional distributions

In this section, we state and prove the following lemma about the convergence of finite-dimensional distributions of $F_x$ to the finite-dimensional distributions of $F_{\mu }$ .

Lemma 4.6 (Convergence of finite-dimensional distributions).

Let m be some positive integer, $B_0=B(0,1)$ , let $F_x$ be as in (4.1) and let $F_{\mu }$ be the random monochromatic wave as in (4.2). Then, under the assumptions of Theorem 1.1 on $\rho (T)$ , for every $y_1,...y_m\in B_0\subseteq \mathbb {R}^n$ , we have

$$ \begin{align*}&(F_x(y_1),..., F_x(y_m))\overset{d}{\longrightarrow}(F_{\mu}(y_1),..., F_{\mu}(y_m)) &T\rightarrow \infty, \end{align*} $$

where the convergence is in distribution uniformly for all $x\in M$ . Moreover, for every $\alpha =(\alpha _1,...,\alpha _n)$ , with $|\alpha |\leq 2$ , one has

$$ \begin{align*}& (D^{\alpha} F_x(y_1),..., D^{\alpha}F_x(y_m))\overset{d}{\longrightarrow}( D^{\alpha}F_{\mu}(y_1),..., D^{\alpha}F_{\mu}(y_m)) &T\rightarrow \infty. \end{align*} $$

In order to prove Lemma 4.6, we will need a simple (not sharp) bound on the maximum value of an eigenfunction in terms of its eigenvalue.

Claim 4.7. Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ and let $\phi _{\lambda }$ be a solution to the eigenvalue problem

$$ \begin{align*}\Delta_{g}\phi_{\lambda} +\lambda^2 \phi_{\lambda}=0.\end{align*} $$

Then, we have

$$ \begin{align*}\sup_{x\in M}|\phi_{\lambda}|^2\lesssim \lambda^{n-1} \log \lambda\end{align*} $$

and

$$ \begin{align*}\sup_{x\in M}\lambda^{-2\alpha}|D^{\alpha}\phi_{\lambda}|^2 \lesssim \lambda^{n-1} \log \lambda\end{align*} $$

for all multi indices $|\alpha |\leq 2$ .

Proof. Observe that, by the first part of Lemma 4.5, we have

$$ \begin{align*} \sup_{x \in M} |\phi_{\lambda}|^2 \leq \sum_{\lambda_{j} \in [\lambda-\log\lambda , \lambda]}|\phi_j|^2= c_M (\log \lambda) \lambda^{n-1} + O(\lambda^{n-1}) \end{align*} $$

and the bound on the supremum of $\phi _{\lambda }$ follows. The bound on the derivatives can be obtained similarly using the second part of Lemma 4.5.

We also recall, for the convenience of the reader, the following multidimensional version of Lindeberg-CLT (see, for example, [Reference Flasche and Kabluchko24, Proposition 6.2] and [Reference Billingsley10, Theorem 27.2]):

Lemma 4.8 (CLT).

Let $d>0$ be a positive integer and let $\{V_{n,k}\}_{n,k}$ be a triangular array of $\mathbb {R}^d$ -valued random variables, so that the random vectors lying on each of its rows are independent and of zero mean. That is, for any $n,k$ , $V_{n,k}= (V^{i}_{n,k})_{i=1}^d$ is a d-dimensional random vector with zero mean, and for every n fixed and every $k_1\neq k_2$ , the vectors $V_{n,k_1}$ and $V_{n,k_2}$ are independent. The random variables $V^i_{n,k}$ are normalized by setting

$$ \begin{align*}(s^i_{n})^2= \sum_{k} \mathbb{E}[(V^i_{n,k})^2]\end{align*} $$

and

$$ \begin{align*}\tilde{V}^i_{n,k}= (s^i_{n})^{-1}V^i_{n,k}.\end{align*} $$

We make the following two assumptions:

1. (1) The covariance matrices

   $$ \begin{align*}\left(\Sigma_{n,k}\right))_{ij}= \mathbb{E}[\tilde{V}^i_{n,k}\tilde{V}^j_{n,k}]\end{align*} $$
   of the k-th vector of $\{\tilde {V}_{n,k}\}_{n,k}$ satisfy
   $$ \begin{align*}\lim\limits_{n\rightarrow \infty} \sum_k \Sigma_{n,k}= \Sigma_0\end{align*} $$
   for some positive definite $d\times d$ -positive matrix.

2. (2) One has

   
   for any positive $\varepsilon>0$ , where is the indicator function.

Then we have

$$ \begin{align*}W_{n}:=\sum_k\tilde{V}_{n,k} \overset{d}{\longrightarrow } N(0,\Sigma_0) \hspace{10mm} n\rightarrow \infty,\end{align*} $$

where the convergence is in distribution, and the rate of convergence depends on the rates of convergence in (1) and (2) only. That is, for every $h:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}$ bounded continuous,

$$ \begin{align*}\mathbb{E}[h(W_{n})] \rightarrow\mathbb{E}[h(Z)],\end{align*} $$

where $Z\sim N(0,\Sigma _0)$ , with the rate of convergence depending on h and the rate of convergence in (1) and (2).

We are now ready to prove Lemma 4.6.

Proof of Lemma 4.6.

First, we need a piece of notation that we will use throughout the proof. Let $\phi _{i,x}$ be the scaled restriction of $\phi _{i}$ to $B_g(x,4/T)$ via the exponential map; that is,

$$ \begin{align*}\phi_{i,x}(y)= \phi_i(\exp_x(y/T)),\end{align*} $$

for $y\in B(0,4)$ (here we tacitly assume that T is sufficiently large so that $4/T$ is less than the injectivity radius). Before embarking on the proof of Lemma 4.6, we also observe that by Claim 4.7, we have

(4.6) $$ \begin{align} \max_{\lambda_{i}\in [T-\rho, T]} \sup_{x \in M} \sup_{B_0} |\phi_i|^2\lesssim T^{n-1}\log T. \end{align} $$

Similarly, given a multi-index $|\alpha |\leq 2$ , we also have

(4.7) $$ \begin{align} \max_{\lambda_i\in[T-\rho, T]} \sup_{x\in M} \sup_{B_0} |D^{\alpha} \phi_{i,x}|^2 \lesssim T^{n-1}\log T. \end{align} $$

We are going to first consider the distribution of the vector $(F_x(y_1),...,F_x(y_m))$ for $x\in M$ . Thanks to Lemma 4.5, we have

(4.8) $$ \begin{align} &\sup_{\substack{i,j\in \{1,...,m\}\\ x\in M}}\left|\mathbb{E}[F_x(y_i)\cdot F_x(y_j)]- \mathbb{E}[F_{\mu}(y_i)\cdot F_{\mu}(y_j)] \right|\rightarrow 0 &T\rightarrow \infty. \end{align} $$

Therefore, by the multidimensional version of Lindeberg’s Central Limit Theorem (Lemma 4.8), and upon using (4.8), it suffices to prove that for every $\varepsilon>0$ , we have

(4.9)

where is the indicator function and $v(T)= c_M\rho T^{n-1}(1+o(1))$ .

Now we prove (4.9). Thanks to Lemma 4.5, we have

Therefore, to prove (4.9), it is sufficient to show that

(4.10)

where the supremum is over all $\lambda _{i}\in [T-\rho , T]$ , all $y\in B_0$ and all $x\in M$ . Thanks to (4.6) and the fact that $v(T)\gtrsim T^{n-1}(\log T)^2$ , we have

Hence, since the $a_i$ are i.i.d. with common distribution $a_0$ (say), we have

where we used Fubini and $\mathbb {E}[|a_{0}|^2]=1$ to switch the order of the limits. This concludes the proof of (4.10) and thus of (4.9).

In order to prove the convergence of the derivative vector and upon recalling the second part of Proposition 4.3, again by the multidimensional version of Lindeberg’s Central Limit Theorem, it is sufficient to prove that for any $\varepsilon>0$ and $|\alpha |\leq 2$ , we have

(4.11)

Similar to the above argument, (4.7) implies (4.11) if $|\alpha |\leq 2$ , thus concluding the proof of Lemma 4.6.

4.4 Tightness

The aim of this section is to show that Lemma 4.6 implies that $F_x$ converges as a random function to $F_{\mu }$ . To formally state the results of this section, let us first introduce some notation. Let $V=\overline {B_0}$ and let $\nu _{T}$ be the sequence of probability measures on $C^2(V)$ induced by the pushforward measure of $F_x$ (recall that, since the law of $f_{T}$ is locally constant, we may assume that T varies along a sequence); that is, for an open set $H\subseteq C^2(V)$ , we set

(4.12) $$ \begin{align} \nu_{T}(H):= (F_x)_{*}\mathbb{P}(H)= \mathbb{P}(F_x(\omega,\cdot)\in H). \end{align} $$

Lemma 4.6 says that there exists a subsequence $T_k$ such that $v_{T_k}$ converges to $\nu _{\infty }$ , the pushforward of $F_{\mu }$ onto $C^2(V)$ . Thus, to obtain the convergence of the whole sequence, it is enough to show that the sequence $v_T$ is tight.

A sequence of probability measures $\{\nu _k\}_{k=0}^{\infty }$ on some topological space X is tight if for every $\epsilon>0$ , there exists a compact set $K=K(\epsilon )\subseteq X$ such that

$$ \begin{align*}\nu_k(X\backslash K)\leq \epsilon,\end{align*} $$

uniformly for all $k\geq 0$ . We will need the following lemma, borrowed from [Reference Prigarin47, Lemma 1] (see also [Reference Billingsley11, Chapter 6 and 7]), which characterises the tightness in the space of continuously twice differentiable functions:

Proof. For condition $(1)$ of Lemma 4.9, we observe that Lemma 4.5 implies the bound

$$ \begin{align*}\mathbb{E}\left[|D^{\alpha}F_x(0)|^2\right]\lesssim 1\end{align*} $$

for $|\alpha |\leq 2$ and uniformly for all $x\in M$ . Thus, Chebyshev’s inequality yields

$$ \begin{align*}\mathbb{P}\left(|D^{\alpha}F_x(0)|>K \right)\lesssim K^{-2},\end{align*} $$

and condition (1) follows by taking $K= \epsilon ^{-1/2}$ .

To check condition $(2)$ of Lemma 4.9, we note that since $F_x$ is almost surely analytic, we have

(4.13) $$ \begin{align}\sup_{|y-y'|\leq \delta} |D^{\alpha}F_x(y)- D^{\alpha}F_x(y')|\lesssim \sup_{B_0}|\nabla D^{\alpha}F_x|\delta. \end{align} $$

Therefore, it is sufficient to prove the following claim:

(4.14) $$ \begin{align} \mathbb{P}( \sup_{B_0}|\nabla D^{\alpha}F_x|>K)\lesssim K^{-2} \end{align} $$

uniformly for all $x\in M$ . Indeed, as above, (4.14) together with (4.13) imply condition (2) by choosing $K=\epsilon \delta ^{-1}$ .

We are now going to prove (4.14). By Sobolev embedding, there exists some $t=t(n)>1$ , sufficently large depending on n only, so that $C^3(B_0)$ embeds in $H^t(B_0)$ , the Sobolev space. Thus, using Lemma 4.5, uniformly for all $x\in M$ , we have

(4.15) $$ \begin{align} \mathbb{E}\left[ ||F_x||_{C^3(B_0)}^2\right] \lesssim_n \mathbb{E}\left[ ||F_x||_{H^t(B_0)}^2\right]\lesssim 1, \end{align} $$

where the constant implied in the ‘ $\lesssim $ ’-notation is independent of T (and $x\in M$ ). Now, inequality (4.15) together with Chebyshev’s inequality implies (4.14), and this concludes the proof of Lemma 4.10.

As mentioned above, combining Lemma 4.6 and Lemma 4.9, we proved the following lemma; see, for example, [Reference Billingsley11, Theorem 7.1]:

4.5 Concluding the proof of Proposition 4.1

To conclude the proof of Proposition 4.1, we just need the following Lemma (see, for example, [Reference Romaniega and Sartori48, Lemma 6.2]), which shows that $\mathcal {V}(\cdot )$ – that is, the nodal volume – is a continuous map on the appropriate space of functions:

Lemma 4.12. Let $B\subseteq \mathbb {R}^n$ be a ball and define the (open) set

$$ \begin{align*}C^2_{*}(2B)=\{h\in C^2(2B): |h|+|\nabla h|>0 \}.\end{align*} $$

Then $\mathcal {V}(\cdot , B)$ is a continuous functional on $C^2_{*}(2B)$ .

We are now in the position to prove Proposition 4.1.

Proof of Proposition 4.1.

An application of Bulinskaya’s lemma (see, for example, [Reference Nazarov and Sodin43, Lemma 6]) on $F_{\mu }$ restricted to $V=\overline {B_0}$ yields that $F_{\mu }\in C^2_{*}(V)$ almost surely. Therefore, Lemma 4.11 and the Continuous Mapping Theorem [Reference Billingsley11, Theorem 2.7] imply

$$ \begin{align*}\mathcal{V}(F_x)\overset{d}{\rightarrow} \mathcal{V}(F_{\mu})\quad T\rightarrow \infty, \end{align*} $$

as required.

5.1 Bounding the nodal volume of sums of eigenfunctions

For a start, we introduce a few notions. Following [Reference Donnelly and Fefferman21] and [Reference Logunov37, Reference Logunov36], the doubling index of a function $h:M\rightarrow \mathbb {R}$ on a ball $B=B_g(x,r)\subseteq M$ is defined as

(5.1) $$ \begin{align} \mathcal{N}_h(B)=\mathcal{N}(x,r):=\log \frac{\sup_{2B} |h|}{\sup_B |h|}. \end{align} $$

The harmonic lift of $f_T$ in (1.2) is defined [Reference Christ and Kenig41, Page 231] (see also [Reference Lin35, Section 4]) as the unique solution $f^H: M\times \mathbb {R} \rightarrow \mathbb {R}$ of

(5.2) $$ \begin{align} (\Delta+ \partial_t^2)f^H(x,t)=0 \hspace{10mm} f^H(x,0)=0 \hspace{10mm}\partial_t f^H(x,0)= f_T. \end{align} $$

One may express $f^H$ explicitly as

(5.3) $$ \begin{align} f^H(x,t)=v^{-1/2}(T)\sum_{\lambda_i \in [T-\rho,T]} a_{i}\frac{\sinh(\lambda_i t)}{\lambda_i}\phi_{i}(x) , \end{align} $$

where $a_{i}$ and $\phi _{i}$ are as in (1.2) and $v(T)$ is as in (1.3). We also introduce the following piece of notation that we will use throughout this section:

$$ \begin{align*}\tilde{B}(x, r):= B_g(x,r)\times [-r,r] \subseteq M\times \mathbb{R}\end{align*} $$

will stand for the ‘ball’ of radius $r>0$ centred at a point $x\in M \cong M\times \{0\}$ , and the doubling index of $f^H$ on $\tilde {B}$ is defined via (5.1) as above, with $\tilde {B}$ in place of B. Finally, we recall that for any $s>0$ ,

$$ \begin{align*}s\tilde{B}:= B_g(x,s r)\times [-sr,sr]\end{align*} $$

is the radius- $sr$ ball centred at the same point as B.

The aim of this section is to prove the following result:

Proposition 5.1. Let $f_T$ and $f^H$ be as in (1.2) and (5.3), respectively. Then there exists some $\eta =\eta (M,g)>0$ with the following property: for every ball

$$ \begin{align*}\tilde{B}_r:=B_g(x,r)\times [-r,r] \subseteq M\times \mathbb{R}\end{align*} $$

centred at a point $x\in M \cong M\times \{0\}$ of radius $0<r<\eta /10$ , we have

$$ \begin{align*}\mathcal{V}\left(f_T, \tilde{B}_{r/2} \cap M\right)\cdot r^{-n+1}\lesssim \mathcal{N}(f^H, \tilde{B}_{8r}) ,\end{align*} $$

where the constant implied in the $\lesssim $ notation depends only on $(M,g)$ .

Before embarking on the proof of Proposition 5.1, we will recall some standard properties of the doubling index, which will be used throughout the rest of the paper.

5.2 Monotonicity of the doubling index and a few consequences

The fundamental property of the doubling index of an harmonic function, shown in [Reference Garofalo and Lin25], is that $\mathcal {N}(\cdot )$ is an almost monotonic function of the radial variable in the sense that

$$ \begin{align*}\mathcal{N}(\cdot, r_1) -C\leq (1+\varepsilon)\cdot\mathcal{N} (\cdot, r_2)+C\end{align*} $$

for $r_2\geq 2r_1$ and some $C=C(M,g)\geq 1 $ . Formally, we have the following (see [Reference Logunov36, Lemma 1.3]):

Lemma 5.2. Let $(\tilde {M},g)$ be a smooth manifold. For any $0<\varepsilon <1$ and any point $O\in \tilde {M}$ , there exists some $C=C(\tilde {M},g,O,\varepsilon )>0$ and $r_0= r_0(\tilde {M},g,O,\varepsilon )>0$ such that

$$ \begin{align*}t^{\mathcal{N}(x,r)(1-\varepsilon) - C} \leq \frac{\sup_{B_g(x,tr)}|u|}{\sup_{B_g(x,r)}|u|}\leq t^{\mathcal{N}(x,tr)(1+\varepsilon) + C}\end{align*} $$

uniformly for all harmonic functions $u: \tilde {M}\rightarrow \mathbb {R}$ , for all $x\in \tilde {M}$ and numbers $r>0$ , $t>2$ satisfying $B_g(x,tr)\subseteq B(O,r_0)$ .

We apply Lemma 5.2 in the following convenient settings. Fix $\varepsilon =1/2$ and $\tilde {M}=M \times [-10,10]$ in Lemma 5.2, covering $M\times [-10,10]$ by balls of radius $r_0$ . Upon using the compactness of $\tilde {M}$ , we obtain the following:

Corollary 5.3. Let $f^H: M\times [-10,10]\rightarrow \mathbb {R}$ be as in (5.2). There exists some $C=C(M,g)>0$ , independent of $f^H$ , such that

$$ \begin{align*}t^{\mathcal{N}(x,r)/2 - C} \leq \frac{\sup\limits_{\tilde{B}(x,tr)}|f^H|}{\sup\limits_{\tilde{B}(x,r)}|f^H|}\leq t^{2\mathcal{N}(x,tr) + C}\end{align*} $$

for all $x\in M\times [-10,10]$ and numbers $r>0$ , $t>2$ satisfying

$$ \begin{align*}B_g(x,tr)\times[-tr,tr]\subseteq M\times [-10,10] .\end{align*} $$

We conclude this section with a useful consequence of the monotonicity formula for the doubling index.

Lemma 5.4. Let $f^H: M\times [-10,10]\rightarrow \mathbb {R}$ be as in (5.2). There exist constants $C_1,C_2\geq 1$ depending only on $M,g$ , such that

$$ \begin{align*}\sup_{d_g(x,y)\leq r/8}\mathcal{N}(y, r/4)\leq C_1\cdot \mathcal{N}(x, r) + C_2,\end{align*} $$

where $y\in M\cong M\times \{0\}$ , uniformly for all $x\in M\times [-10,10]$ with

$$ \begin{align*}\tilde{B}(x,2r)\subseteq M\times[-10,10].\end{align*} $$

Proof. Since, in the relevant range, $d_g(x,y)\leq r/8$ , and by Corollary 5.3 applied with $t=8$ (say), we have

$$ \begin{align*}\sup_{\tilde{B}(y,r/2)}\big| f^H \big|\leq \sup_{\tilde{B}(x,r)}\left|f^H\right| \leq 8^{2\mathcal{N}(x,r) + C} \sup_{\tilde{B}(x,r/8)}\left|f^H\right| \leq \exp \left(C_1 \mathcal{N}(x,r) + C_2 \right)\cdot \sup_{\tilde{B}(y,r/4)}\left| f^H\right|,\end{align*} $$

as required.

5.3 Complexification of f

Since $(M, g)$ is real analytic and compact, by the Bruhat-Whitney Theorem [Reference Whitney and Bruhat55], there exists a complex manifold $M^{\mathbb {C}}$ where M embeds as a totally real manifold. Moreover, it is possible to analytically continue any Laplace eigenfunction $\phi _{i}$ to a holomorphic function $\phi _{i}^{\mathbb {C}}$ defined on a maximal uniform Grauert tube; that is, there exists some $\eta _0=\eta _0(M,g)>0$ such that $\phi _{i}^{\mathbb {C}}$ is an holomorphic function on

$$ \begin{align*} M_{\eta_0}^{\mathbb{C}}:=\{\zeta \in M^{\mathbb{C}}: \sqrt{\gamma}(\zeta) < \eta \}, \end{align*} $$

where $\sqrt {\gamma }(\cdot )$ is the Grauert tube function; see [Reference Zelditch60, Chapter 14] for details. For notational brevity, and in light of the fact that the precise value of $\eta _0$ will be unimportant, from now on we write $M^{\mathbb {C}}$ in place of $M_{\eta _0}^{\mathbb {C}}$ and let $f^{\mathbb {C}}$ , defined on $M^{\mathbb {C}}$ , be the complexification of f. The nodal volume of f can be controlled via the order of growth of $f^{\mathbb {C}}$ using the following classical fact, borrowed from [Reference Christ and Kenig41, Theorem 14.7] and [Reference Donnelly and Fefferman21, Proposition 6.7]:

Lemma 5.5. Let $B^{\mathbb {C}}\subseteq \mathbb {C}^n$ be a ball of radius $1$ and let H be a holomorphic function on $3B^{\mathbb {C}}$ . If, for some $N>1$ ,

$$ \begin{align*} |H|_{L^{\infty}(2B^{\mathbb{C}})}\leq e^{N}\cdot |H|_{L^{\infty}(B^{\mathbb{C}}\cap \mathbb{R}^n)}, \end{align*} $$

then

$$ \begin{align*} \mathcal{H}^{n-1}\left(\{H=0\} \cap \frac{1}{2}B^{\mathbb{C}} \cap \mathbb{R}^n\right)\lesssim_n N. \nonumber \end{align*} $$

5.4 Growth of $f^H$ and $f^{\mathbb {C}}$

In this section, we wish to quantify the growth of $f^{\mathbb {C}}$ in terms of the doubling index of $f^H$ . Our proof will proceed by using $f^H$ to control the derivatives of f so that we can bound the growth of $f^{\mathbb {C}}$ by bounding each term in its power series. Unfortunately, this approach requires introducing an extra (small) constant $c=c(M,g)>0$ in the next result in order to control the radius of convergence of the power series. We will then get rid of this extra technicality in the proof of Proposition 5.1 via a covering argument. All in all, the aim of this section is to prove the following result:

Lemma 5.6. There exist some (small) numbers $\eta _1=\eta _1(M,g)>0$ and $c=c(M,g)>0$ such that the following holds. Let f be as in (1.2), $f^H$ be as in (5.2) and $f^{\mathbb {C}}$ be the complexification of f. Moreover, let $\tilde {B}\subseteq M \times \mathbb {R}$ be a ball centred at a point lying on $M\cong M \times \{0\}$ of radius less than $\eta _1/10$ . Suppose that, for some (large) $N>1$ , one has

(5.4) $$ \begin{align} ||f^H||_{L^{\infty}(\tilde{B})}\leq e^N \cdot ||f^H||_{L^{\infty}(\frac{c}{2}\tilde{B})}. \end{align} $$

Then we have

$$ \begin{align*} \nonumber ||f^{\mathbb{C}}||_{L^{\infty}((2c\tilde{B}\cap M)^{\mathbb{C}})}\leq C' e^{CN} \cdot ||f||_{L^{\infty}(c\tilde{B}\cap M)} \end{align*} $$

for some constants $C,C'>1$ depending on $M,g$ only.

To prove Lemma 5.6, we first need the following result on the unique continuation of $f^H$ , borrowed from [Reference Christ and Kenig41, Page 231]; see also [Reference Lin35, Lemma 4.3].

Lemma 5.7. Let $x\in M$ . There exist constants $r_0=r_0(M,g,x)>0$ , $C_0=C_0(M,g,x, r_0)>0$ and $0< \beta =\beta (M,g,x, r_0)<1$ so that the following holds. Recall that the harmonic lift $f^H$ is the function defined in (5.2). Then one has

$$ \begin{align*} \left\| f^H\right\|_{L^{\infty}(\tilde{B}^{+})}\leq C_0 \left\| r\cdot \partial_t f^H\right\|^{\beta}_{L^{\infty}(2\tilde{B}\cap M)}\cdot \left\| f^H\right\|^{1-\beta}_{L^{\infty}(2\tilde{B}^{+})} \nonumber \end{align*} $$

uniformly w.r.t. balls $\tilde {B}$ of radius $r>0$ and centred at a point lying in $M\cong M\times \{0\}$ such that

$$ \begin{align*}\tilde{B}\subseteq B_g(x,r_0/4)\times [-r_0/4,r_0/4]\subseteq M \times \mathbb{R} ,\end{align*} $$

where $\tilde {B}^{+}= \tilde {B}\cap (M \times [0,\infty ))$ .

Note that, although not explicated in [Reference Christ and Kenig41], the constant $\beta $ in Lemma 5.7 depends only on a particular coordinate patch around the point $x\in M$ , provided this is sufficiently small. Therefore, $\beta $ is uniform with respect to all the balls contained in the said coordinate patch and well-separated from the boundaries, as stated in Lemma 5.7. We refer the reader to [Reference Alessandrini, Rondi, Rosset and Vessella1, Theorem 1.7] for the details (in a much more general scenario). We are now ready to prove Lemma 5.6.

Proof of Lemma 5.6.

First, given $x\in M$ , let $r_0=r_0(M,g,x)$ be given by Lemma 5.7. Covering M by balls of radius $r_0$ and using the compactness of M, we find that there exists some $\eta _1>0$ , depending only on M and g, such that the conclusion of Lemma 5.7 is applicable on every ball

$$ \begin{align*}B_g(x,\eta_1/2)\times [-\eta_1/2,\eta_1/2].\end{align*} $$

Moreover, we may assume that $\eta _1\leq \eta _0/2$ , with $\eta _0$ as constructed in Section 5.3. Now observe that, appealing to the compactness of M again, it is sufficient to prove Lemma 5.6 in every coordinate patch of radius $\eta _1$ . That is, it is sufficient to prove that, for every $x\in M$ , there exists some $c>0$ and some $C',C\geq 1$ , depending on $M,g,x,\eta _1$ , such that if (5.4) is satisfied, then one has

$$ \begin{align*} ||f^{\mathbb{C}}||_{L^{\infty}(2(c\tilde{B}\cap M)^{\mathbb{C}})}\leq C' e^{CN} \cdot||f||_{L^{\infty}(c\tilde{B}\cap M)} \end{align*} $$

uniformly w.r.t. balls $\tilde {B}$ of radius $r>0$ and centred at a point lying in $M\cong M\times \{0\}$ , such that

$$ \begin{align*}4\tilde{B}\subseteq B_g(x,\eta_1/2)\times [-\eta_1/2,\eta_1/2].\end{align*} $$

In what follows, this claim is established.

Since the supremum norm is scale invariant, we may rescale the metric and assume that $\tilde {B}$ has radius $r=1$ . Since $f^H$ satisfies

$$ \begin{align*}(\partial_t^2 +\Delta)f^H=0,\end{align*} $$

the elliptic estimates for the operator $\partial _t^2 +\Delta $ (see, for example, [Reference Hörmander27, Lemma 7.5.1 and equation (4.4.1)] or [Reference Evans23, Page 330]) imply that there exists some constants $C_1,C_2=C_1,C_2(M,g,\eta _1, x)$ such that for any $k>0$ , one has

$$ \begin{align*}\left\|f^H\right\|_{C^k(\frac{1}{2}\tilde{B})} \leq C_1^k k!\cdot \left\|f^H\right\|_{L^2(\frac{3}{4}\tilde{B})} \leq C_2^k k! \cdot \left\|f^H\right\|_{L^{\infty}(\tilde{B})}.\end{align*} $$

Moreover, by the definition (5.2) of $f^H$ , for any multi-index $\alpha $ so that $|\alpha |= k$ , we have

$$ \begin{align*}\sup_{\frac{1}{2}(\tilde{B} \cap M)}|D^{\alpha} f| \leq 2 \cdot \|f^H\|_{C^{k}(\tilde{B})}.\end{align*} $$

Therefore, we obtain the bound

(5.5) $$ \begin{align} \sup_{\frac{1}{2}(\tilde{B}\cap M)} \frac{|D^{\alpha} f|}{|\alpha|!} \leq 2 C_2^{k} \cdot \left\|f^H\right\|_{L^{\infty}(\tilde{B})}. \end{align} $$

Now we are going to bound the r.h.s. of (5.5) using the assumed doubling property (5.4). First, observe that since $\sinh (\cdot )$ is an odd function, we have

$$ \begin{align*}\left\|f^H\right\|_{L^{\infty}(\tilde{B}^{+})} = \left\|f^H\right\|_{L^{\infty}(\tilde{B})}.\end{align*} $$

Thus, using the assumption (5.4) on the doubling of $f^H$ (for some $c>0$ to be chosen later), the assumption $r= 1$ , and the equality

$$ \begin{align*}\left\| \partial_t f^H\right\|_{L^{\infty}(r'\tilde{B}\cap M)}= \| f\|_{L^{\infty}(r'\tilde{B} \cap M)}\end{align*} $$

that follows from (5.2) for any $r'>0$ , Lemma 5.7 implies

(5.6) $$ \begin{align} \left\| f^H\right\|_{L^{\infty}(\tilde{B}^{+})}= \left\| f^H\right\|_{L^{\infty}(\tilde{B})}\leq e^N \cdot \left\| f^H\right\|_{L^{\infty}(\frac{c}{2}\tilde{B})}= e^N \cdot \left\| f^H\right\|_{L^{\infty}(\frac{c}{2}\tilde{B}^{+})} \nonumber \\ \lesssim_M e^{C_3 N}\cdot \|f\|^{\beta}_{L^{\infty}(c\tilde{B} \cap M)}\cdot \left\| f^H\right\|^{1-\beta}_{L^{\infty}(cB^{+})} \leq e^{C_3N} \cdot \|f\|^{\beta}_{L^{\infty}(c\tilde{B} \cap M)}\cdot \left\| f^H\right\|^{1-\beta}_{L^{\infty}(\tilde{B}^{+})} \end{align} $$

for some $0<\beta =\beta (M,g,x, \eta _1)<1$ and some $C_3=C_3(M,g,x,\eta _1)>1$ . Since $f^H$ is an analytic function, we have $||f^H||_{L^{\infty }(\tilde {B}^{+})}\neq 0$ ; thus, (5.6) implies

(5.7) $$ \begin{align} ||f^H||_{L^{\infty}(\tilde{B}^{+})}\lesssim_{\beta} e^{C_4N} \cdot \| f\|_{L^{\infty}(c\tilde{B} \cap M)} \end{align} $$

for some $C_4=C_4(\beta ,M,g)>1$ . Therefore, combining (5.5) and (5.7), we obtain

(5.8) $$ \begin{align} \sup_{\frac{1}{2}(\tilde{B} \cap M)}\frac{|D ^{\alpha}f|}{|\alpha|!}\lesssim e^{C_5N +C_6|\alpha|}\sup_{c\tilde{B} \cap M} |f| \end{align} $$

for some $C_5,C_6>1$ depending only on $M,g, x, \eta _1$ . Since f is real analytic, we may expand

$$ \begin{align*}f^{\mathbb{C}}(z)= \sum_{\alpha} \frac{D^{\alpha} f(y)}{|\alpha| !} z^{|\alpha|}\end{align*} $$

into an absolutely convergent Taylor series in $(2c\tilde {B}\cap M)^{\mathbb {C}}$ for some sufficiently small $0<c=c(M,g,x, \eta _1)<1/2$ . Then, (5.8) gives

$$ \begin{align*}\sup_{(2c\tilde{B}\cap M)^{\mathbb{C}}} |f^{\mathbb{C}}|\leq C_7 \exp(C_8 N) \sup_{c\tilde{B} \cap M} |f|\end{align*} $$

for some constants $C_7,C_8>1$ depending only on $M,g, x, \eta _1$ , as required.

5.5 Concluding the proof of Proposition 5.1

We are finally in a position to prove Proposition 5.1:

Proof. First, we take $\eta =c\min \{\eta _0,\eta _1, r_0(M)\}$ , where $\eta _0$ is given at the beginning of Section 5.3, $\eta _1$ is given by Lemma 5.6 and $r_0(M)$ is the injectivity radius of M and $c=c(M,g)>0$ is as in Lemma 5.6. Next, denote

$$ \begin{align*}\tilde{B}_r=B_g(x,r)\times [-r,r]\end{align*} $$

to be the ball as in the statement of Proposition 5.1 and let $c_1=c_1(M,g)= c/8$ with $c=c(M,g)$ given as in Lemma 5.6. We cover the ball $\frac {1}{2}\tilde {B}_r$ by balls $\tilde {B}^i$ of radius $c_1r/2 $ and centre $x^i$ so that

$$ \begin{align*}\mathcal{V}\left(f, \tilde{B}_{r/2} \cap M \right)\lesssim \max_i \mathcal{V}\left(f, \tilde{B}^i \cap M \right),\end{align*} $$

where the constant implied in the $\lesssim $ notation depends only on $(M,g)$ . Now let $f^i_{c_1r}$ be a version of f rescaled by a factor of $c_1r$ in the ball $\tilde {B}^i$ in the normal coordinates; that is,

$$ \begin{align*}f^i_{c_1r}= f (\exp_{x^i}(c_1r y))\end{align*} $$

for $y\in B_0\subset \mathbb {R}^n$ , where $B_0$ is the unit ball. The scaling property of the nodal volume gives

(5.9) $$ \begin{align} \mathcal{V}\left(f, \tilde{B}^{i} \cap M \right)\lesssim r^{n-1} \mathcal{V}\left(f^i_{c_1r}, \frac{1}{2}B_0 \right), \end{align} $$

where the constant implied in the $\lesssim $ notation depends only on $(M,g).$ Thus, Lemma 5.5 and invariance of the $L^{\infty }$ -norm w.r.t. scaling imply

(5.10) $$ \begin{align} \mathcal{V}\left(f^i_{c_1r}, \frac{1}{2}B_0 \right)\lesssim \log \frac{\sup_{2(B_0)^{\mathbb{C}}}|(f^i)_{c_1r}^{\mathbb{C}}|}{\sup_{B_0}|f^i_{c_1r}|}\lesssim \log \frac{\sup_{2(\tilde{B}^{i} \cap M)^{\mathbb{C}}}|f^{\mathbb{C}}|}{\sup_{(\tilde{B}^i \cap M)}|f|}, \end{align} $$

where $f^{\mathbb {C}}$ is the complexification of f. Then, denoting $N^i= \mathcal {N}_{f^H} (2c_1^{-1}\tilde {B}^i)$ with $f^H$ as in (5.2), Lemma 5.6, applied under the assumption $r<\eta /10$ , and Corollary 5.3 yield

(5.11) $$ \begin{align} \log \frac{\sup_{2(\tilde{B}^{i} \cap M)^{\mathbb{C}}}|f^{\mathbb{C}}|}{\sup_{(\tilde{B}^i \cap M)}|f|}\lesssim \log \frac{\sup_{c^{-1}\tilde{B}^i }|f^{H}|}{\sup_{\frac{1}{2}\tilde{B}^{i}_r}|f^H|} \lesssim N^i. \end{align} $$

Since $\tilde {B}^i$ have, by construction, radius $cr/8$ , we find that

$$ \begin{align*}N^i= \mathcal{N}(x^i, r/4)\lesssim \mathcal{N}_{f^H}( \tilde{B}_{8r}),\end{align*} $$

where the second inequality follows from Lemma 5.4. Hence, the statement of Proposition 5.1 follows by combining (5.9), (5.10) and (5.11).

5.6 Estimates for the local nodal volume

In this section, we deduce a bound on $\mathcal {V}(F_x)$ from Proposition 5.1. We begin with the following estimate (see [Reference Christ and Kenig41, page 231]):

Lemma 5.8. Let $f^H$ be as in (5.2) and let $\tilde {B} \subseteq M \times \mathbb {R}$ be a ball centred at some point on $M\cong M\times \{0\}$ and of any radius less than $\eta /10$ , where $\eta $ is given by Proposition 5.1. Then

(5.12) $$ \begin{align} ||f^H||_{L^{\infty}(2\tilde{B})}\lesssim e^{CT} ||f^H||_{L^{\infty}(\tilde{B})} \end{align} $$

with some $C=C(M,g)>1$ .

Although we do not wish to reproduce the proof of Lemma 5.8 in full detail, for the sake of completeness, we quickly indicate how Lemma 5.8 follows from Lemma 5.7. Indeed, applying Lemma 5.7 to $B= M$ with $L^2$ -norm instead of $L^{\infty }$ -norm (which is possible by elliptic estimates), upon observing that $||f^H||_{L^2(M\times [-a,a])}\asymp \exp (T) \sum _{i} |\phi _{i}|^2$ for $a=1,2$ , we obtain

$$ \begin{align*} ||f^H||_{L^{\infty}(2\tilde{B})}\lesssim e^{CT} ||f^H||_{L^{\infty}(\tilde{B})} \nonumber \end{align*} $$

at macroscopic scales. Now, Lemma 5.8 follows by Corollary 5.3. As a direct consequence of Lemma 5.8, we have the following bound:

Lemma 5.9. For $F_x$ as in (4.1), one has

$$ \begin{align*} \sup_{x\in M} \mathcal{V}(F_x)\lesssim T. \nonumber \end{align*} $$

Proof. Applying Proposition 5.1 on f as in (1.2) with

$$ \begin{align*}\tilde{B}= B_g(x,1/T)\times (-1/T,1/T)\end{align*} $$

(where we tacitly assume that T is sufficiently large so that $1/T\leq \eta /80$ with $\eta $ as in Proposition 5.1), we obtain

$$ \begin{align*} \mathcal{V}\left(f, \frac{1}{2}\tilde{B} \cap M\right) T^{n-1} \lesssim \mathcal{N}_{f^H}(8\tilde{B}).\end{align*} $$

Lemma 5.8 gives $\mathcal {N}_{f^H}(8\tilde {B})\lesssim T$ . Therefore, Lemma 5.9 follows upon noticing that the definition (4.1) of $F_x$ , being the scaled version of $f_{T}$ , implies that

$$ \begin{align*}\mathcal{V}(F_x) \lesssim T^{n-1} \mathcal{V}\left(f, \frac{1}{2}\tilde{B} \cap M\right) \lesssim \mathcal{N}_{f^H}(8\tilde{B})\lesssim T.\\[-41pt] \end{align*} $$

6.1 Weak $L^1$ estimate for the local nodal volume

We collect here a well-known result about the locality of the nodal volume, which will also be useful later in the proof, allowing for an $L^1$ -weak type estimate for the volume of $x\in M$ with $\mathcal {V}(F_x)$ large.

Lemma 6.2. Let $F_x$ and f be as in (4.1) and (1.2), respectively, and let $\omega _n$ be the unit n-ball volume. Then one has:

$$ \begin{align*} \mathcal{V}(f)= \frac{2^n T}{\omega_{n}}\left(1 +o_{T\rightarrow \infty}(1)\right)\cdot\int_M \mathcal{V}(F_x)d\operatorname{\mathrm{\operatorname{Vol}}}(x). \nonumber \end{align*} $$

Proof. First, we observe that we may write

(6.1)

where is the indicator function and $\mathcal {H}^{n-1}$ is the Hausdorff measure. Then, integrating both sides of (6.1) and using Fubini’s Theorem, we have

(6.2) $$ \begin{align} \int_M \mathcal{V}\left(f, B_g\left(x,\frac{1}{2T}\right)\right)\ d\operatorname{\mathrm{\operatorname{Vol}}}(x)= \int_{f^{-1}(0)} \operatorname{\mathrm{\operatorname{Vol}}}_g\left( B_g\left(y,\frac{1}{2T}\right)\right)\ d\mathcal{H}^{n-1}(y). \end{align} $$

Now, we observe that in light of the definition of $F_x$ in Section 4.1, since $1/T$ is smaller than the injectivity radius of M, we have

$$ \begin{align*}\mathcal{V} (F_x)= \mathcal{V}\left(f, B_g\left(x,\frac{1}{2T}\right)\right) \cdot T^{n-1}(1+o_{T\rightarrow \infty}(1)),\end{align*} $$

where we have used the scaling property of the nodal volume

$$ \begin{align*}\mathcal{V}(F_x)=\mathcal{V}\left(F_x(\cdot), \frac{1}{2}B_0\right)=T^{n-1} \mathcal{V} \left(F_x(T\cdot), B\left(\frac{1}{2T}\right)\right).\end{align*} $$

Thus, the l.h.s. of (6.2) is

(6.3) $$ \begin{align} \int_M \mathcal{V}\left(f, B_g\left(x,\frac{1}{2T}\right)\right)\ d\operatorname{\mathrm{\operatorname{Vol}}}(x)= \frac{1}{T^{n-1}} \left(1 +o_{T\rightarrow \infty}(1)\right)\int_M \mathcal{V}(F_x)d\operatorname{\mathrm{\operatorname{Vol}}}_g(x). \end{align} $$

Moreover, for all $y\in M$ , we also have

$$ \begin{align*}\operatorname{\mathrm{\operatorname{Vol}}}_g\left( B_g\left(y,\frac{1}{2T}\right)\right) = \operatorname{\mathrm{\operatorname{Vol}}}_{\mathbb{R}^n} \left(B\left(0,\frac{1}{2T}\right)\right)\left( 1 + O\left( T^{-1}\right)\right)= \frac{\omega_{n}}{(2T)^n}(1 +O(T^{-1})).\end{align*} $$

Thus, the r.h.s. of (6.2) is

(6.4) $$ \begin{align} \int_{f^{-1}(0)} \operatorname{\mathrm{\operatorname{Vol}}}_g\left( B_g\left(y,\frac{1}{2T}\right)\right)\ d\mathcal{H}^{n-1}(y)= \frac{\omega_n}{(2T)^n}\left(1 + O\left( T^{-1}\right)\right)\cdot \mathcal{V}(f) \end{align} $$

Hence, Lemma 6.2 follows upon inserting (6.3) and (6.4) into (6.2).

As a direct consequence of Lemma 6.2, we have the following result:

Corollary 6.3. Let $F_x$ be as in (4.1). Then, uniformly for all $t>0$ ,

$$ \begin{align*}\operatorname{\mathrm{\operatorname{Vol}}}_g\left(x: \mathcal{V}(F_x)>t\right)\lesssim t^{-1}.\end{align*} $$

Proof. We first aim to prove (1.6), that is claimed no novelty of, but was decided to be included for the sake of completeness. Let $\eta>0$ be as prescribed by Proposition 5.1. Then, by Proposition 5.1 and Lemma 5.8, we have

$$ \begin{align*}\mathcal{V}(f_T, B_{\eta})\lesssim T\end{align*} $$

for any ball $B_{\eta }\subseteq M$ of radius $\eta /20$ . Covering M by finitely many such balls, we obtain

(6.5) $$ \begin{align} \mathcal{V}(f_T)\lesssim T, \end{align} $$

which is (1.6). Inserting (6.5) into Lemma 6.2, we obtain

$$ \begin{align*}\int_M \mathcal{V}(F_x)d\operatorname{\mathrm{\operatorname{Vol}}}(x) \lesssim 1.\end{align*} $$

Hence, Corollary 6.3 follows from Markov’s inequality.

6.2 A probabilistic anti-concentration inequality for the doubling index

The aim of this section is to prove the following result that, unlike Proposition 6.1 for $n\le 4$ , will not require the growth of $\rho (T)$ :

Lemma 6.4. Let $f^H$ be as in (5.3), $\eta>0$ as in Proposition 5.1, $v(T)$ as in (1.3) and $\rho _0$ as in Corollary 4.4. Moreover, given a parameter $100/\eta \leq A\leq 50 T$ , let

$$ \begin{align*}\tilde{B}_A=B_g(x,A^{-1})\times \left[-\frac{1}{A},\frac{1}{A}\right]\subseteq M\times [-10,10]\end{align*} $$

be a ball centred at some $(x,0)\in M \times \{0\}$ . If $\rho (T)\geq \rho _0$ , then for all $Q\geq 100$ , one has

$$ \begin{align*} \mathbb{P}\left(\mathcal{N}_{f^H}(\tilde{B}_A)> \frac{Q \cdot T}{A}\right)\lesssim \exp\left(-\frac{Q\cdot T}{50A}\right) + E_A(x), \nonumber \end{align*} $$

where

$$ \begin{align*}E_A(x):=v(T)^{-3/2}\cdot\sum_{\lambda_{i}\in [T-\rho,T]}A^n\int_{B_g(x, (2A)^{-1})} |\phi_i(y)|^3d\operatorname{\mathrm{\operatorname{Vol}}}_g(y)\end{align*} $$

and the constant involved in the ‘ $\lesssim $ ’-notation may depend on $M,g$ , but not on $Q,A,T$ or x.

Before giving a proof to Lemma 6.4, we would like to describe the intuition behind its proof. By the definition (5.1) of the doubling index, a large doubling index of $f^H$ indicates a rapid growth of $f^H$ on concentric balls. This could happen in two possible scenarios: either $f^H$ is large on the larger ball or $f^H$ is small on the smaller ball. The probability of the former event can be controlled using some $L^2$ -bounds and Chebyshev’s inequality, whereas the probability of the latter one is controlled with the following lemma:

Lemma 6.5. Let $f^H(x)$ be as in (5.3), $v(T)$ as in (1.3), and $\rho _0$ as in Corollary 4.4. Denote

$$ \begin{align*}\Psi(t)= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^t \exp\left(-\frac{z^2}{2}\right)dz\end{align*} $$

to be the standard Gaussian cumulative distribution function, and, given $x\in M $ , denote $\tilde {x}= (x, (100 T)^{-1})\in M\times [-10,10]$ . Then if $\rho (T)\geq \rho _0$ , one has

$$ \begin{align*}\sup_{t\in \mathbb{R}}\left|\mathbb{P}\left(\frac{f^H(\tilde{x})}{\mathbb{E}[|f^H(\tilde{x})|^2]^{1/2}}\leq t\right) - \Psi(t)\right|\lesssim E(x) ,\end{align*} $$

where the constant implied in the ‘ $\lesssim $ ’-notation depends only on $(M,g)$ and

$$ \begin{align*}E(x):= v(T)^{-3/2}\sum_{\lambda_{i}\in [T-\rho,T]} |\phi_i(x)|^3.\end{align*} $$

Proof. For $\lambda _i \in [T-\rho , T]$ , let us write

$$ \begin{align*}X_i= X_i(\tilde{x}):= \frac{a_i}{\lambda_i^{-1}} \sinh \left( \frac{\lambda_i}{100T}\right)\cdot \phi_i(x),\end{align*} $$

and, on recalling that $\lambda _i/T= 1+o_{T\rightarrow \infty }(1)$ ,

(6.6) $$ \begin{align} \sigma_i = \sigma_i(\tilde{x}):= \mathbb{E}\left[|X_i|^2\right]^{1/2} = \lambda_i^{-1} \left|\sinh \left( \frac{\lambda_i}{100T}\right)\cdot\phi_i(x)\right| \asymp T^{-1}\cdot |\phi_i(x)|, \end{align} $$

where the constant in the ‘ $\asymp $ ’-notation is absolute. Moreover, let

$$ \begin{align*}\tau_i= \tau_i(\tilde{x}):=\mathbb{E}\left[|X_i(\tilde{x})|^3\right]\asymp \sigma_i^3(\tilde{x}).\end{align*} $$

On recalling (5.3), observe that we have

$$ \begin{align*}\mathbb{E}\left[|f^H(\tilde{x})|^2\right] = \sum_{\lambda_i\in [T-\rho, T]} \sigma_i^2(\tilde{x}).\end{align*} $$

By the well-known Berry-Esseen Theorem [Reference Berry8, Reference Esseen22] applied to the sum of the $X_i$ ’s, we have

(6.7) $$ \begin{align} \sup_{t\in \mathbb{R}}\left|\mathbb{P}\left(\frac{f^H(\tilde{x})}{\mathbb{E}[f^H(\tilde{x})]^{1/2}} < t\right) - \Psi(t)\right|\lesssim \left(\sum_{\lambda_{i} \in [T-\rho, T]}|\sigma_{i}(\tilde{x})|^2\right)^{-3/2} \sum_{\lambda_{i} \in [T-\rho, T]} \tau_i(\tilde{x}) \nonumber \\ \lesssim \left(\sum_{\lambda_{i} \in [T-\rho, T]}|\sigma_{i}|^2\right)^{-3/2}\sum_{\lambda_{i} \in [T-\rho, T]} |\sigma_{i}|^3,\end{align} $$

where the constant implied in the ‘ $\lesssim $ ’-notation is absolute. Using (6.6), the r.h.s of (6.7) may be bounded as

$$ \begin{align*}\left(\sum_{\lambda_{i} \in [T-\rho, T]}|\sigma_{i}(\tilde{x})|^2\right)^{-3/2}\sum_{\lambda_{i} \in [T-\rho, T] } |\sigma_{i}(\tilde{x})|^3 \lesssim \left(\sum_{\lambda_{i} \in [T-\rho, T]}|\phi_{i}(x)|^2\right)^{-3/2}\sum_{\lambda_{i} \in [T-\rho, T]} |\phi_{i}(x)|^3.\end{align*} $$

Hence, Lemma 6.5 follows from Corollary 4.4; that is,

$$ \begin{align*}\sum_{\lambda_{i} \in [T-\rho, T]}|\phi_{i}(x)|^2\asymp v(T).\\[-49pt] \end{align*} $$

Lemma 6.5 gives the following bound on the supremum of $f^H$ of the ball $\tilde {B}_A$ as in Lemma 6.7:

Corollary 6.6. Let $f^H$ be as in (5.3), $\eta>0$ as in Proposition 5.1, and $\rho _0$ as in Corollary 4.4. Given a parameter $100/\eta \leq A\leq 50 T$ , let

$$ \begin{align*}\tilde{B}_A=B_g(x,A^{-1})\times \left[-\frac{1}{A},\frac{1}{A}\right]\subseteq M\times [-10,10]\end{align*} $$

be a ball centred at some $(x,0)\in M \times \{0\}$ . Moreover, let us write $c(\tilde {x}):= \mathbb {E}[|f^H(\tilde {x})|^2]$ , where $\tilde {x}:=(x, (100T)^{-1})$ . Suppose that $\rho (T)\geq \rho _0$ . Then, for all $\tau>0$ (which may depend on A), we have

$$ \begin{align*}\mathbb{P}\left(\sup_{\tilde{B}_A}\left|\frac{f^H}{c(\tilde{x})^{1/2}}\right|\leq \tau\right)\lesssim \tau + E_A(x),\end{align*} $$

where

$$ \begin{align*}E_A(x):=v(T)^{-3/2}\cdot\sum_{\lambda_{i}\in [T-\rho,T]}A^n\int_{B_g(x, (2A)^{-1})} |\phi_i(y)|^3d\operatorname{\mathrm{\operatorname{Vol}}}_g(y),\end{align*} $$

and the constant involved in the ‘ $\lesssim $ ’-notation may depend on $M,g$ but not on $A,T$ or x.

Proof. Since for every $\tilde {y}:= (y, (100T)^{-1})$ with $y\in B_g(x, A^{-1})$ , we have

$$ \begin{align*}\mathbb{P}\left(\sup_{\tilde{B}_A}\left|\frac{f^H}{c(\tilde{x})^{1/2}}\right|\leq \tau\right) \leq \mathbb{P} \left( \left|\frac{f^H(\tilde{y})}{c(\tilde{x})^{1/2}}\right|\leq \tau\right),\end{align*} $$

we have the bound

$$ \begin{align*}\mathbb{P}\left(\sup_{\tilde{B}_A}\left|\frac{f^H}{c(\tilde{x})^{1/2}}\right|\leq \tau\right) \leq \inf_{y\in B_g(x, A^{-1})} \mathbb{P} \left( \left|\frac{f^H(\tilde{y})}{c(\tilde{x})^{1/2}}\right|\leq \tau\right).\end{align*} $$

Bounding the infimum by the average (over, say, a slightly smaller ball), we obtain

$$ \begin{align*} \mathbb{P}\left(\sup_{\tilde{B}_A}\left|\frac{f^H}{c(\tilde{x})^{1/2}}\right|\leq \tau\right) \lesssim A^n \int_{B_g(x, (2A)^{-1})} \mathbb{P} \left(\left| \frac{f^H(\tilde{y})}{c(\tilde{x})^{1/2}}\right|\leq\tau \right) d\operatorname{\mathrm{\operatorname{Vol}}}_g(y) \\ \lesssim A^n \int_{B_g(x, (2A)^{-1})} \mathbb{P} \left(\left| \frac{f^H(\tilde{y})}{c(\tilde{y})^{1/2}}\right|\leq \frac{c(\tilde{x})^{1/2}}{c(\tilde{y})^{1/2}}\tau \right) d\operatorname{\mathrm{\operatorname{Vol}}}_g(y). \end{align*} $$

Therefore, Lemma 6.5 yields

(6.8) $$ \begin{align} \mathbb{P}\left(\sup_{\tilde{B}_A}\left|\frac{f^H}{c(\tilde{x})^{1/2}}\right|\leq \tau\right) \lesssim c(\tilde{x})^{1/2}\tau A^n \int_{B_g(x, (2A)^{-1})} \frac{1}{c(\tilde{y})^{1/2}} d\operatorname{\mathrm{\operatorname{Vol}}}_g(y) + E_A(x). \end{align} $$

Now, using Lemma 4.5, for all $y\in B_g(x, A^{-1})$ (and so in particular for x), we have

$$ \begin{align*} c(\tilde{y})= \mathbb{E}\left[|f^H(y, (100T)^{-1})|^2\right]= \sum_{\lambda_{i} \in [T-\rho, T]} \left| \lambda_i^{-1}\sinh\left(\frac{\lambda_{i}}{100 T}\right) \phi_i(x)\right|{}^2 \nonumber \\ \asymp T^{-2} \sum_{\lambda_{i} \in [T-\rho, T]} |\phi_i(x)|^2\asymp T^{-2}. \nonumber \end{align*} $$

Thus, the first term on the r.h.s. of (6.8) is

$$ \begin{align*}c(\tilde{x})^{1/2}\tau A^n \int_{B_g(x, (2A)^{-1})} \frac{1}{c(\tilde{y})^{1/2}}\asymp \tau ,\end{align*} $$

and this concludes the proof of Corollary 6.6.

We are finally ready to prove Lemma 6.4:

Proof of Lemma 6.4.

To simplify notation, we will use the following shorthand: $\tilde {x}= (x,(100T)^{-1})$ and $\tilde {B}=\tilde {B}_A$ . First, we may renormalize $f^H$ by dividing it by the nonvanishing number $c(\tilde {x}):=\mathbb {E}[|f^H(x,(100T)^{-1})|^2]$ ; that is, by a slight abuse of notation, we write $f^{H}$ in place of

(6.9) $$ \begin{align} \frac{f^H}{\mathbb{E}[|f^H(x,(100T)^{-1})|^2]^{1/2}}= \frac{f^H}{c(\tilde{x})^{1/2}}= \frac{1}{c(\tilde{x})^{1/2} v(T)^{1/2}} \sum_{\lambda_{i} \in [T-\rho, T]} a_{i}\frac{\sinh(\lambda_i t)}{\lambda_i}\phi_i(x), \end{align} $$

throughout the proof of Lemma 6.4. We are now in a position to commence the proof of Lemma 6.4.

To bound the probability that the doubling index us large, we note that it could occur under two possible scenarios: either $\sup \limits _{\tilde {B}}|f^H|$ is small or $\sup \limits _{2\tilde {B}}|f^H|$ is large. Given some $\tau>0$ to be determined later, we write

(6.10) $$ \begin{align} \mathbb{P}\left( \mathcal{N} (\tilde{x}, A^{-1})> \frac{Q \cdot T}{A} \right) &= \mathbb{P}\left( \mathcal{N} (\tilde{x}, A^{-1}) > \frac{Q \cdot T}{A} \hspace{3mm}\text{and} \hspace{3mm}\sup_{\tilde{B}}|f^H|<\tau \right)\nonumber\\ & \quad + \mathbb{P}\left(\mathcal{N} (\tilde{x}, A^{-1})> \frac{Q \cdot T}{A} \hspace{3mm}\text{and} \hspace{3mm} \sup_{\tilde{B}}|f^H|\geq \tau\right). \end{align} $$

The first term on the r.h.s. of (6.10) can be bounded as

$$ \begin{align*} \mathbb{P}\left( \mathcal{N} (\tilde{x}, A^{-1})> \frac{Q \cdot T}{A} \hspace{3mm}\text{and} \hspace{3mm}\sup_{\tilde{B}}|f^H|<\tau \right) \leq \mathbb{P} ( \sup_{\tilde{B}} |f^H|\leq \tau ). \nonumber \end{align*} $$

Thus, Corollary 6.6 gives

(6.11) $$ \begin{align} \mathbb{P}\left( \mathcal{N} (\tilde{x}, A^{-1})> \frac{Q \cdot T}{A} \hspace{3mm}\text{and} \hspace{3mm}\sup_{\tilde{B}}|f^H|<\tau \right) \lesssim \tau + E_A(x). \end{align} $$

Now we bound the second term on the r.h.s. of (6.10). To this end, we use the definition (5.1) of the doubling index, and since, under the relevant event, $\sup _{\tilde {B}}|f^H|\ge \tau $ , we may write under the same event

$$ \begin{align*}\frac{Q\cdot T}{A}<\mathcal{N}(\tilde{x}, A^{-1}) \leq \log \frac{||f^H||_{L^{\infty}(2\tilde{B})}}{\tau}.\end{align*} $$

Thus, we obtain

(6.12) $$ \begin{align} ||f^H||_{L^{\infty}(2\tilde{B})}\geq \exp \left(\frac{Q\cdot T}{A}\right)\tau. \end{align} $$

Now we claim the following:

(6.13) $$ \begin{align} \mathbb{E}\left[||f^H||^2_{L^{\infty}(2\tilde{B})}\right] \lesssim \exp \left( 8 \frac{T}{A}\right), \end{align} $$

where the constant implied in the ‘ $\lesssim $ ’-notation may depend on $(M,g)$ only. Upon using the elliptic regularity [Reference Evans23, Page 330], we have

$$ \begin{align*}||f^H||^2_{L^{\infty}(2\tilde{B})}\lesssim A^{n+1}||f^H||^2_{L^2(4\tilde{B})},\end{align*} $$

where the constant implies in the ‘ $\lesssim $ ’-notation depends only on $(M,g)$ . Therefore, using the formula (5.3), exchanging the order of the expectation and the summation, and upon bearing in mind that $f^{H}$ is normalized via (6.9), we have

$$ \begin{align*} \mathbb{E}\left[||f^H||^2_{L^{\infty}(2\tilde{B})}\right]&\lesssim A^{n+1} \mathbb{E}\left[||f^H||^2_{L^2(4\tilde{B})}\right] \nonumber \\ &\lesssim c(\tilde{x})^{-1} v(T)^{-1}\sum_{\lambda_i} \frac{\sinh (8 \lambda_i/A)}{\lambda_{i}^2} A^{n} \int_{B_g(x,4/A)} |\phi_{i}(x)|^2 d\operatorname{\mathrm{\operatorname{Vol}}}_g(x), \nonumber \end{align*} $$

where the constant implied in the ‘ $\lesssim $ ’-notation may depend on $(M,g)$ only. Switching the sum with the integral, using Lemma 4.5, the obvious bound $\sinh (\cdot )\leq \exp (\cdot )$ , and, again, $\lambda _i/T = 1+o_{T\rightarrow \infty }(1)$ , we obtain

(6.14) $$ \begin{align} \mathbb{E}\left[||f^H||^2_{L^{\infty}(2\tilde{B})}\right]^2\lesssim c(\tilde{x})^{-1} T^{-2} \exp \left( 8 \frac{T}{A}\right), \end{align} $$

where, again, the constant implied in the ‘ $\lesssim $ ’-notation may depend on $(M,g)$ only. Since $c(\tilde {x})^{-1}\asymp T^{-2}$ , (6.13) follows from (6.14).

Using (6.13) together with Chebyshev’s inequality, (6.10), (6.11) and (6.12), we obtain

$$ \begin{align*} \nonumber \mathbb{P}\left(\mathcal{N} (\tilde{x}, A^{-1})> \frac{Q \cdot T}{A}\right)\lesssim \tau + \exp\left( 8 \frac{T}{A} - \frac{2Q\cdot T}{A}\right) \tau^{-2} + E_A(x). \end{align*} $$

Hence, Lemma 6.4 follows by taking $\tau = \exp (-Q T/(50A))$ and $Q\geq 100$ (say).

6.3 Sogge’s bound and the decay of the doubling index

The aim of this section is to prove the following lemma, which shows that outside an event of small probability, the doubling index decreases uniformly for all $x\in M$ . In accordance with the results in the previous section except Proposition 6.1 for $n\le 4$ , the following lemma is stated for $\rho (T)\geq \rho _0$ without the growth assumption of Theorem 1.1.

Lemma 6.7. Let $f^H$ be as in (5.3), and $\rho _0$ as in Corollary 4.4. If $\rho (T)\geq \rho _0$ , then there exists some constant $C=C(M,g)>1$ such that

$$ \begin{align*}\mathbb{P}\left(\sup_{x\in M} \mathcal{N}_{f^H}\left((x,0), A^{-1}\right)\geq C \frac{T}{A} \right)\lesssim (\log T)^{-1},\end{align*} $$

where

$$ \begin{align*} \nonumber A=A(T)= \begin{cases} T^{\frac{n-1}{4n}}\rho(T)^{\frac{1}{2n}}\cdot \frac{1}{(\log T)^{ \frac{1}{n}}} & n\leq 4 \\ T^{\frac{1}{n}}\rho(T)^{\frac{1}{2n}} \cdot \frac{1}{(\log T)^{\frac{1}{n}}} & n\geq 5. \end{cases}. \end{align*} $$

To prove Lemma 6.7, we use Lemma 6.4 together with the following bound on the $L^p$ -norm of eigenfunctions due to Sogge [Reference Sogge54]. Let $\phi _{i}$ be an eigenfunction with eigenvalue $\lambda _i^2$ . Then we have the following estimate on the $L^p$ norms of $\phi _{i}$ (see also [Reference Zelditch60, Theorem 10.1]):

(6.15) $$ \begin{align} ||\phi_{i}||_{L^p(M)}\lesssim \lambda_i ^{\sigma(p)} ||\phi_{i}||_{L^2(M)}, \end{align} $$

where

$$ \begin{align*} \sigma(p)=\begin{cases} \frac{n-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right) & 2< p \leq \frac{2(n+1)}{n-1} \\ n\left(\frac{1}{2}-\frac{1}{p}\right)-1/2 & p\geq \frac{2(n+1)}{n-1}. \end{cases} \end{align*} $$

We are now in a position to prove Lemma 6.7.

Proof of Lemma 6.7.

Let $A=A(T)$ be some parameter to be chosen later and apply Lemma 6.4 on the ball

$$ \begin{align*}\tilde{B}_{A}=B_g(x,A^{-1})\times [-1/A,1/A]\end{align*} $$

with $Q= 100$ (say) to yield

$$ \begin{align*}\mathbb{P}\left( \mathcal{N}_{f^H}(\tilde{x}, A^{-1})\geq \frac{100T}{A}\right)\lesssim \exp \left(- \frac{2T}{A}\right) +E_A(x),\end{align*} $$

where $\tilde {x}:=(x,0)$ . Using the monotonicity of the doubling index of Lemma 5.4 with $r=A$ , we deduce that there exists some (large) constant $C_1=C_1(M,g)\geq 1$ such that

$$ \begin{align*}\mathbb{P}\left( \sup_{ y\in B_g(x , (10A)^{-1} )}\mathcal{N}_{f^H}(y, (4A)^{-1})\geq \frac{C_1T}{A}\right)\lesssim \exp \left(- \frac{2T}{A}\right) +E_A(x),\end{align*} $$

where we have tacitly assumed that $T/A$ is sufficiently large depending on $M,g$ only. Taking the union bound over at most $O(A^n)$ balls $B(x_j, (10A)^{-1})$ , we obtain

(6.16) $$ \begin{align} \mathbb{P}\left( \sup_{ x\in M} \mathcal{N}_{f^H}(\tilde{x}, (4A)^{-1})\geq \frac{C_1T}{A} \right)\lesssim A^n\exp \left(- \frac{2T}{A}\right) +\sum_j E_A(x_j). \end{align} $$

Assuming that A is sufficiently large so that $A^{-1}$ is smaller than the injectivity radius, each ball $B(x_j, (10A)^{-1})$ intersects finitely many (depending on n only) other balls in the collection; therefore,

$$ \begin{align*}v(T)^{3/2}\sum_j E_A(x_j)= A^n \sum_{\lambda_{i}\in [T,T-\rho]} \sum_j \int_{B(x_j, (10A)^{-1})}|\phi_{i}|^3 d\operatorname{\mathrm{\operatorname{Vol}}} \lesssim A^n \sum_{\lambda_{i}\in [T,T-\rho]} ||\phi_i||_{L^3(M)}^3.\end{align*} $$

Using Sogge’s bound (6.15), we conclude that

(6.17) $$ \begin{align} \sum_j E_A(x_j)\lesssim \sum_{\lambda_{i}\in [T,T-\rho]} \frac{A^n T^{3\sigma (3)}}{v(T)^{3/2}}, \end{align} $$

with $\sigma (3)$ as in (6.15).

Finally, inserting (6.17) into (6.16) and summing over i, which gives a contribution of $v(T)$ , we obtain

$$ \begin{align*} \mathbb{P}\left( \sup_{ x\in M} \mathcal{N}_{f^H}(\tilde{x}, 4^{-1}A^{-1})\geq \frac{C_1T}{A} \right)\lesssim A^n\exp \left(- \frac{2T}{A}\right) +A^n T^{3\sigma(3)} v(T)^{-1/2}. \nonumber \end{align*} $$

Hence, Lemma 6.7 follows by observing that

$$ \begin{align*} 3\sigma(3)=\begin{cases} \frac{n-1}{4} & n\leq 4 \\ \frac{n}{2}-\frac{3}{2} & n\geq 5 \end{cases}, \nonumber \end{align*} $$

$$ \begin{align*}v(T)^{1/2}\asymp T^{\frac{n-1}{2}}\rho(T)^{1/2},\end{align*} $$

and taking for $n\leq 4$ ,

$$ \begin{align*}A= T^{\frac{n-1}{4n}}\rho(T)^{\frac{1}{2n}}\cdot \frac{1}{4(\log T)^{\frac{1}{n}}}\end{align*} $$

and for $n>5$ ,

$$ \begin{align*}A= T^{\frac{1}{n}}\rho(T)^{\frac{1}{2n}} \cdot \frac{1}{4(\log T)^{\frac{1}{n}}}.\\[-46pt] \end{align*} $$

6.4 Concluding the proof of Proposition 6.1

Proof of Proposition 6.1.

Since the proof of the Proposition 6.1 is somewhat long, we break it up into a series of steps:

Step 1: Controlling the distribution of $\mathcal {V}(F_x)$ .

Recall that $d\sigma = \frac {d\operatorname {\mathrm {\operatorname {Vol}}}_g}{\operatorname {\mathrm {\operatorname {Vol}}}(M)} \otimes d\mathbb {P}$ . The aim of this step is to obtain some bounds on $\sigma \left ( \mathcal {V}(F_x)>t\right )$ for all $t\geq C_0$ for some $C_0=C_0(M,g)\geq 1$ . First, by Proposition 5.1, bearing in mind the rescaling factor, we have

$$ \begin{align*}\mathcal{V}(F_x)\leq C_1 \mathcal{N}_{f^H}((x,0), 8T^{-1}) \hspace{5mm} \text{and}\hspace{5mm} \mathcal{N}_{f^H}((x,0), 8T^{-1}):= \mathcal{N}_T(x)=\mathcal{N}(x)\end{align*} $$

for some $C_1=C(M,g)\geq 1$ . Therefore, Lemma 6.4, applied with $A= 4T$ and $Q= c_0t:= C_1^{-1}t$ (which is larger than $100$ taking $C_0$ sufficiently large in terms of $C_1$ ), gives

$$ \begin{align*} \nonumber \mathbb{P}(\mathcal{V}(F_x)\geq t)\leq \mathbb{P}(\mathcal{N}(x)\geq c_0t)\lesssim \exp\left(- \frac{c_0t}{10}\right) +E_T(x), \end{align*} $$

where $E_T$ is as in Lemma 6.4 (and we write $E_T$ in place of $E_{T/8}$ as shorthand). Thus, we have

(6.18) $$ \begin{align} \sigma \left( \mathcal{V}(F_x)>t\right)= \frac{1}{\operatorname{\mathrm{\operatorname{Vol}}}(M)} \int_M \mathbb{P}(\mathcal{V}(F_x)\geq t) d\operatorname{\mathrm{\operatorname{Vol}}}_g \nonumber \\ \lesssim \exp\left(- \frac{c_0t}{10}\right) +\int_M E_T(x)d\operatorname{\mathrm{\operatorname{Vol}}}_g. \end{align} $$

We are now going to bound the second term on the r.h.s. of (6.18). By Sogge’s bound, we have

$$ \begin{align*}\int_M |\phi_i(x)|^3 d\operatorname{\mathrm{\operatorname{Vol}}}_g \lesssim T^{3\sigma(3)},\end{align*} $$

with $\sigma (3)$ as in (6.15). Therefore, in light of the fact that the sum over i in the definition of $E_T(x)$ in Lemma 6.5 has $v(T)$ -terms, $v(T)\asymp \rho (T)T^{n-1}$ and exchanging the integrals, we have

(6.19) $$ \begin{align} \int_M E_T(x)d\operatorname{\mathrm{\operatorname{Vol}}}\lesssim T^{\alpha(n)} \rho(T)^{-1/2}, \end{align} $$

with

$$ \begin{align*} \nonumber \alpha(n):= 3\sigma(3)- \frac{n-1}{2}= \begin{cases} -\frac{n-1}{4} & n\leq 4 \\ -1 & n\geq 5 \end{cases}. \end{align*} $$

Inserting (6.19) into (6.18), we see that

(6.20) $$ \begin{align} \sigma \left( \mathcal{V}(F_x)>t\right)\lesssim \exp\left(- \frac{c_0t}{10}\right) + T^{\alpha(n)}\rho(T)^{-1/2}. \end{align} $$

Step 2: Sharpening the upper bound

By Lemma 5.9, we have

$$ \begin{align*}\sup\limits_{x\in M}\mathcal{V}(F_x)\lesssim T.\end{align*} $$

Our task in this step is to obtain a better upper bound, outside an event of small probability, using Lemma 6.7. Let $A=A(T)$ be as in Lemma 6.7. Since the monotonicity of the doubling index of Lemma 5.4 implies that

$$ \begin{align*}\mathcal{N}(x)\lesssim \mathcal{N}_{f^H}\left((x,0), A^{-1}\right) +C_3\end{align*} $$

for some $C_3=C_3(M,g)>1$ , an application of Lemma 6.7 gives

(6.21) $$ \begin{align} \sup_{x\in M} \mathcal{V}(F_x) \leq C_4 \frac{T}{A}=: p(T) \end{align} $$

for some $C_4=C_4(M,g)>0$ , outside an event $\Omega _1$ with $\mathbb {P}(\Omega _1)\lesssim (\log T)^{-1}$ .

We now show that the event $\Omega _1$ does not positively contribute to the integral of Proposition 6.1. Indeed, we write

(6.22)

Since $h(t)= t\log t$ and $\mathcal {V}(F_x)\lesssim T$ , the second term on the r.h.s of (6.22) can be bounded by

$$ \begin{align*}(\log T)^{-1} \sup_{\omega\in \Omega}\int_{M} h(\mathcal{V}(F_x))d\operatorname{\mathrm{\operatorname{Vol}}}(x) \lesssim \sup_{\omega \in \Omega} \int_{M}\mathcal{V}(F_x) d\operatorname{\mathrm{\operatorname{Vol}}}(x) =O(1),\end{align*} $$

where, in the last inequality, we have used Lemma 6.2, which is deterministic, in the form

$$ \begin{align*}\sup_{\omega \in \Omega} \int_{M}\mathcal{V}(F_x) d\operatorname{\mathrm{\operatorname{Vol}}}(x)= O(1).\end{align*} $$

Thus, we have shown that

(6.23)

with $p(T)$ as in (6.21). This concludes step 2.

Step 3: Collecting the estimates.

We begin by rewriting the integral on the r.h.s of (6.23) as

Integrating by parts, we obtain

(6.24)

with $C_0$ as in Step 1. Now, recall that in Step 2, we had

$$ \begin{align*} \nonumber A=A(T)= \begin{cases} T^{\frac{n-1}{4n}}\rho(T)^{\frac{1}{2n}}\cdot \frac{1}{(\log T)^{ \frac{1}{n}}} & n\leq 4 \\ T^{\frac{1}{n}}\rho(T)^{\frac{1}{2n}} \cdot \frac{1}{(\log T)^{\frac{1}{n}}} & n\geq 5, \end{cases} \end{align*} $$

and in Step 1, we had

$$ \begin{align*} \nonumber \alpha(n):= 3\sigma(3)- \frac{n-1}{2}= \begin{cases} -\frac{n-1}{4} & n\leq 4 \\ -1 & n\geq 5 \end{cases}. \end{align*} $$

Since $h(t)=t\log t$ , we have $h'(t)= \log t +1\leq 2\log t$ . Thus, using Step 1, namely (6.20), Step 2, (6.24) and the definition of $A(T)$ above, we have

(6.25)

where the constant implied in the ‘ $\lesssim $ ’-notation depends only on $(M,g)$ and

$$ \begin{align*} q(T):= \begin{cases} T \cdot T^{-\frac{n-1}{4}\left(1+\frac{1}{n}\right)} & n\leq 4 \\ T^{-\frac{1}{n}} &n\geq 5. \end{cases} \nonumber \end{align*} $$

Hence, taking

$$ \begin{align*} \rho(T)\geq \begin{cases} T^{\frac{-n^2+4n+1}{2(n+1)}}(\log T)^2 & n\leq 4 \\ 1 &n\geq 5 \end{cases}, \nonumber \end{align*} $$

we see that the r.h.s. of (6.25) is bounded, as required.

8.1 Preliminaries: Geodesic flow and the spectrum of $\sqrt {-\Delta }$

For a reference to the facts contained in this section, we suggest the exposition in [Reference Zelditch60]. Let $T^{*}M$ and $S^{*}M$ be the co-tangent and the co-sphere bundle on M, respectively. The geodesic flow

$$ \begin{align*} G^t: T^{*}M \rightarrow T^{*}M \end{align*} $$

is the Hamiltonian flow of the metric norm function

$$ \begin{align*} \nonumber& H: T^{*}M \rightarrow \mathbb{R} &H(x,\xi)= \sum_{i,j=1}^n g^{ij}\xi_i\xi_j, \end{align*} $$

where $g=g_{ij}$ is the metric on M and $g^{ij}$ is its inverse. Since $G^t$ is homogeneous, from now on, we will consider only its restriction to $S^{*}M$ . We will need the following simple lemma (see also [Reference Safarov and Vassiliev51, Lemma 1.3.8]):

Proof. Since $(M,g)$ is real analytic, the geodesic flow $G^{t}(\cdot ,\cdot )$ is a real analytic function on $S^{\star }M$ . Therefore, for fixed $t>0$ , solutions to

$$ \begin{align*}G^t(x,\xi)= (x,\xi)\end{align*} $$

consist of the zero set of an analytic function. This must have codimension at least $1$ or be trivial.

Lemma 8.3 implies that the geodesic flow on a real analytic manifold is either aperiodic if the set of closed geodesics has measure zero or periodic with (minimal) period $H>0$ if $G^H=id$ . For the former case, the two-term Weyl’s law of Duistermaat-Guilleimin(-Ivrii) states

$$ \begin{align*}|\{i>0: \lambda_{i}\leq T\}|= c_MT^{n} + o(T^{n-1}).\end{align*} $$

For the latter case, the spectrum of $\sqrt {\Delta }$ is a union of clusters of the form

$$ \begin{align*}C_k:=\left\{\frac{2\pi}{H}\left( k+ \frac{\beta}{4}\right) + \mu_{k_i} \hspace{4mm} \text{for} \hspace{4mm} i=1,...,d_k\right\} \hspace{5mm} k=1,2...,\end{align*} $$

where $\mu _{k_i}= O(k^{-1})$ uniformly for all i, $d_k$ is a polynomial in k of degree $n-1$ and $\beta $ is the common Morse index of the closed geodesics of M.

8.2 Local Weyl’s law revisited

The aim of this section is to prove Proposition 8.2. As we will see below, Proposition 8.2 is a direct consequence of Egorov’s Theorem and the following:

Proposition 8.4. Let $(M,g)$ be a compact, real analytic manifold with empty boundary, $\rho _0$ as in Corollary 4.4, and suppose that either the geodesic flow on M is periodic or $x\in M$ is not a self-focal point (Definition 1.2). Then

(8.1) $$ \begin{align} \sup_{y,y'\in B_g(x,10/T)}\left|\sum_{\lambda_i\in[T- \rho_0, T]} \phi_{i}(y)\phi_{i}(y')- c_M T^{n} \mathcal{J}_{\Upsilon(T)}(Td_g(y,y')) \right| = o_x(T^{n-1}), \end{align} $$

where $d_g(y,y')$ is the geodesic distance between $y,y'$ , $c_M>0$ is given in (1.3), $\Upsilon (T)=1-\frac {\rho _0}{T}$ and

$$ \begin{align*} \mathcal{J}_{\Upsilon(T)}(w)= \int_{\Upsilon(T) \leq |\xi|\leq 1}\exp(i\langle w,\xi \rangle)d\xi. \end{align*} $$

Moreover, we can also differentiate both sides of (8.1) an arbitrary finite number of times; that is,

$$ \begin{align*} \sup_{y,y'\in B_g(x,10/T)}\frac{\left|\sum_{\lambda_i\in[T- \rho_0, T]} D^{\alpha}_{y}\phi_{i}(y)D^{\alpha'}_{y'}\phi_{i}(y')- \frac{c_M T^{n} D^{\alpha}_{y}D^{\alpha'}_{y'}\mathcal{J}_{\Upsilon(T)}(Td_g(y,y'))}{(2\pi)^{n}} \right|}{T^{|\alpha|+|\alpha'|}} = o_x(T^{n-1}), \end{align*} $$

where $\alpha ,\alpha '$ are multi-indices, and $\xi ^{\alpha }=(\xi _1^{\alpha _1},...,\xi _n^{\alpha _n})$ and the derivatives are understood after taking normal coordinates around the point x.

The proof of Proposition 8.4 follows directly from the following two lemmas. In the periodic case, we have a full asymptotic expansion for the spectral projector kernel [Reference Zelditch58, Theorem 2]; see also [Reference Zelditch59]. In particular, we have the following:

The second lemma is borrowed from Canzani-Hanin [Reference Canzani and Hanin16, Reference Canzani and Hanin17]; see also the preceding work of Safarov [Reference Safarov52]:

We are finally ready to prove Proposition 8.2:

Proof of Proposition 8.2.

First we observe that, under the assumptions of Theorem 1.3, (8.1) and its term-wise differentiation hold for almost all $x\in M$ , that is outside a set of measure zero. Indeed, thanks to Lemma 8.3, the geodesic flow on M is either aperiodic or periodic. In the latter case, the conclusion of Proposition 8.4 holds for all $x\in M$ . In the former case, Proposition 8.4 holds for almost all $x\in M$ . Thus, it remains to show that (8.1) and its term-wise differentiation, holding for almost all $x\in M$ implies the conclusion of Proposition 8.2.

Following along identical lines to the proof of Proposition 4.3 (which we do not reproduce here for the sake of brevity) shows that the function

$$ \begin{align*}h(x):= \sup_{y,y'\in B_0} \left|\mathbb{E}[F_x(y)\cdot F_x(y')] - (2\pi)^{\Lambda}\frac{J_{\Lambda}(|y-y'|)}{|y-y'|^{\Lambda}}\right|\end{align*} $$

converges point-wise to $0$ for almost all $x\in M$ . Therefore, Egorov’s Theorem implies that there exists a (measurable) set $\mathcal {A}_1=\mathcal {A}_1(T)\subseteq M$ of volume $\operatorname {\mathrm {\operatorname {Vol}}}(\mathcal {A}_1)=o(1)$ such that h converges to zero uniformly for all $x\in M\backslash \mathcal {A}_1$ . This concludes the proof of the first claim of Proposition 8.2 (recall that the set of relevant T is a discrete subset of ${\mathbb {R}}$ ). The proof of the second claim is similar and therefore omitted.

9.1 Sogge’s bound and large values of the eigenfunctions

In addition to the possible failure of the local Weyl’s law around self-focal points, another difficulty of the constant energy window regime is the possibility of

$$ \begin{align*}\sup_x|\phi_i(x)|\asymp v(T)^{1/2}\end{align*} $$

for some $\phi _i$ in the summation (1.2) (for example, it might occur for the sphere $\mathbb {S}^n$ ). This problem was not present in the growing energy window case thanks to Claim 4.7. Thus, we would not be able to apply Lindeberg’s CLT as in the proof of Lemma 4.6. In order to circumvent this difficulty, we show that $|\phi _i(x)|= o(T)$ for all $\lambda _{i}\in [T-\rho _0, T]$ and for all $x\in M$ outside a set of small measure. That is, the main result of this section is to prove the following consequence of (6.15):

In order to state a preliminary result towards the proof of Lemma 9.2, we recall some notation. Given a Laplace eigenfunction $\phi _{i}$ , we denote by $\phi _{i,x}$ the scaled restriction of $\phi _{i}$ to $B_g(x,4/T)$ via the exponential map; that is,

$$ \begin{align*}\phi_{i,x}(y)= \phi_i(\exp_x(y/T))\end{align*} $$

for $y\in B(0,4)$ (here we tacitly assume that T is sufficiently large so that $4/T$ is less than the injectivity radius). With this notation in mind, we prove the following consequence of elliptic regularity for harmonic functions:

Before embarking on the proof of Lemmas 9.2 and 9.3, we would like to briefly discuss their statements. Lemmas 9.2 and 9.3 are stated in the precise form that will be used in Section 4.3. However, in the literature, the conclusions of Lemmas 9.2 and 9.3 are often stated as

(9.2) $$ \begin{align} & \sup_{B_g(x,c/\lambda_i)}|\phi_i| \lesssim K^{-1}v(T)^{1/2} &\sup_{B_g(x,c/\lambda_{i})}|\phi_i|^2 \lesssim \int_{B(0,2c)} |\phi_{i,x}(y)|^2dy \end{align} $$

for some small $c=c(M)$ with an analogous statement for the bounds on the derivatives. Since $\rho _0=O_M(1)$ , and for $\lambda _{i}\in [T-\rho _0,T]$ ,

$$ \begin{align*}\lambda_{i}^{-1}= T^{-1}(1+o(1)),\end{align*} $$

(9.2) is equivalent to Lemmas 9.2 and 9.3 up to the constant $2$ and a simple covering argument.

Proof of Lemma 9.3.

Given $\phi _{i}$ , let us consider the function $h(x,t)= \phi _{i}(x)e^{\lambda _i t}$ defined on $M\times [-2,2]$ and let us write $h_{T}(\cdot )= h (T^{-1}\cdot )$ (where the rescaling is to be understood in normal coordinates). Then, since the supremum norm is scale invariant, we have

(9.3) $$ \begin{align} \sup_{B_g(x,2/T)}|\phi_i|\lesssim \sup_{B_g(x,2/T)\times [-2/T,2/T]}|h| \lesssim ||h_T||_{L^{\infty}(\tilde{B})} \end{align} $$

(9.4) $$ \begin{align} \sup_{B_g(x,2/T)}|D^{\alpha}\phi_i| \lesssim \sup_{B_g(x,2/T)\times [-2/T,2/T]}|D^{\alpha} h| \lesssim T^{\alpha} ||h_T||_{C^1(\tilde{B})}, \end{align} $$

where $\tilde {B}= B_g(x,2)\times [-2,2]$ and $|\alpha |\leq 2$ is a multi-index. Since h is a harmonic function ( $\Delta h=0$ ) and $\tilde {B}$ has radius $4$ , for any $k\geq 0$ , elliptic regularity [Reference Evans23, Page 330] gives

(9.5) $$ \begin{align} ||h_T||_{C^k(\tilde{B})}\lesssim_k ||h_T||_{L^2(2\tilde{B})}. \end{align} $$

The constant implied in the notation is independent of $x\in M$ . Thus, Lemma 9.3 follows by inserting (9.5) into (9.3) and (9.4) and noticing that $||h_T||_{L^2(2\tilde {B})} \lesssim || \phi _{i,x}||_{L^2(B(0,4))}$ .

Proof of Lemma 9.2.

First, we observe that given $p\geq 2$ , the function $x\rightarrow x^{p/2}$ is convex for $x\geq 0$ . Therefore, applying Jensen’s inequality to part (1) of Lemma 9.3, we obtain

(9.6) $$ \begin{align} \left( \sup_{B_g(x,2/T)} |\phi_{i}|\right)^p \lesssim_p \left( \int_{B(0,4)} |\phi_{i,x}(y)|^2dy\right)^{p/2} \lesssim_p\int_{B(0,4)} |\phi_{i,x}(y)|^p dy \end{align} $$

and similarly,

(9.7) $$ \begin{align} \left(\sup_{B_g(x,2/T)}|T^{-\alpha}D^{\alpha}\phi_i|\right)^p \lesssim_p \int_{B(0,4)} |\phi_{i,x}(y)|^p dy, \end{align} $$

where $|\alpha |\leq 2$ is a multi-index. We are now going to prove part (1) of Lemma 9.2. By Sogge’s bound (6.15) with $p\leq 2(n+1)/(n-1)$ , bearing in mind that $||\phi _i||_{L^2}=1$ , we have

$$ \begin{align*} \left(\int_{M} |\phi_{i}(x)|^pd\operatorname{\mathrm{\operatorname{Vol}}}_g(x) \right)^{1/p}\lesssim T^{\frac{n-1}{2}\left(\frac{1}{2}- \frac{1}{p}\right)}=: \tilde{T} \nonumber \end{align*} $$

for all $\lambda _i \leq T$ . Thus, integrating both sides of (9.6) with respect to $x\in M$ and exchanging the order of the integrals, we obtain

$$ \begin{align*}\int_M\left( \sup_{B_g(x,2/T)} |\phi_{i}|\right)^p d\operatorname{\mathrm{\operatorname{Vol}}}_g(x) \lesssim \int_{B(0,4)} \int_M|\phi_{i,x}(y)|^p d\operatorname{\mathrm{\operatorname{Vol}}}_g(x) dy\lesssim \tilde{T}^p.\end{align*} $$

Therefore, by Chebyshev’s bound, for any $K_1>0$ , we have

$$ \begin{align*} \operatorname{\mathrm{\operatorname{Vol}}}_g\left( \left\{x\in M: \sup_{B_g(x,2/T)}|\phi_{i}|\geq K_1\right\} \right) \lesssim K_1^{-p} \tilde{T}^p,\nonumber \end{align*} $$

and taking the union bound over the $O(v(T))$ choices for i, we deduce

(9.8) $$ \begin{align} \operatorname{\mathrm{\operatorname{Vol}}}_g\left( \left\{x\in M: \max_{\lambda_i \in [T-\rho_0,T]}\sup_{B_g(x,2/T)}|\phi_{i}|\geq K_1\right\} \right) \lesssim K_1^{-p} v(T) \cdot \tilde{T}^p. \end{align} $$

Thus, taking $K_1=K^{-1}v(T)^{1/2}\gtrsim K^{-1}( T^{n-1})^{1/2}$ in (9.8) and recalling that $\tilde {T}=T^{\frac {n-1}{2}\left (\frac {1}{2}- \frac {1}{p}\right )}$ , we have

(9.9) $$ \begin{align}\operatorname{\mathrm{\operatorname{Vol}}}_g\left( \left\{x\in M: \max_{\lambda_i \in [T-\rho_0,T]}\sup_{B_g(x,2/T)}|\phi_{i}|\geq K^{-1} v(T)^{1/2}\right\} \right)\lesssim_p K^p T^{\nu(n,p)}, \end{align} $$

where

$$ \begin{align*} \nu(n,p):&= -p\frac{n-1}{2} + n-1 + \frac{n-1}{2}\left(\frac{p}{2}-1\right) \nonumber \\ &= \frac{n-1}{2}\left(1- \frac{p}{2}\right). \nonumber \end{align*} $$

Hence, taking $p=2(n+1)/(n-1)$ in (9.9), we have

$$ \begin{align*}\operatorname{\mathrm{\operatorname{Vol}}}_g\left( \left\{x\in M: \max_{\lambda_i \in [T-\rho_0,T]}\sup_{B_g(x,2/T)}|\phi_{i}|\geq K^{-1} v(T)^{1/2}\right\} \right)\lesssim K^p T^{-1},\end{align*} $$

as required. Thanks to (9.7), the proof of part (2) of Lemma 9.2 follows along the lines of the proof of its part (1).

9.2 Concluding the proof of Proposition 9.1

As we saw in the course of the proof of Proposition 4.1, to prove Proposition 9.1, it is sufficient to consider the convergence of finite-dimensional distributions. Indeed, Lemma 4.10 implies that the measure induced by $F_x$ (as a random variable on $(M\times \Omega , d\sigma )$ ) onto $C^2(B_0)$ is tight. Thus, to conclude the proof of Proposition 9.1, it is sufficient to prove the following result: