# SPECTRAL ESTIMATE FOR THE LAPLACE-BELTRAMI OPERATOR ON THE HYPERBOLIC HALF-PLANE

MARC ROUVEYROL

**ABSTRACT.** The purpose of this note is to investigate the concentration properties of spectral projectors on manifolds. This question has been intensively studied (by Logvinenko-Sereda, Nazarov, Jerison-Lebeau, Kovrizhkin, Egidi-Seelmann-Veselić, Burq-Moyano, among others) in connection with the uncertainty principle. We provide the first high-frequency results in a geometric setting which is neither Euclidean nor a perturbation of Euclidean. Namely, we prove the natural (and optimal) uncertainty principle for the spectral projector on the hyperbolic half-plane.

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. Assumption and main result</td><td>2</td></tr><tr><td>1.2. Some bibliography on spectral estimates</td><td>3</td></tr><tr><td>1.3. Scheme of the proof</td><td>4</td></tr><tr><td>2. Proof of Theorem 1</td><td>5</td></tr><tr><td>2.1. Proof of the spectral estimate</td><td>5</td></tr><tr><td>2.2. Proof of Lemma 2.1</td><td>8</td></tr><tr><td>2.3. Necessary condition</td><td>10</td></tr><tr><td>Appendix A. Logunov and Malinnikova's propagation of smallness inequality</td><td>13</td></tr><tr><td>Appendix B. Spectral estimate implies observability</td><td>14</td></tr><tr><td>Acknowledgements</td><td>16</td></tr><tr><td>References</td><td>16</td></tr></table>

<footnote>*Key words and phrases.* spectral estimate, propagation of smallness, hyperbolic half-plane, heat equation.</footnote>
# 1. INTRODUCTION

## 1.1. Assumption and main result.
In the present note, we prove spectral estimates from thick sets on the hyperbolic half-plane $\mathbb{H}^2 = \{(x, y) \in \mathbb{R}^2, y > 0\}$, endowed with its usual metric $g = \frac{1}{y^2}(dx^2 + dy^2)$. The Laplace-Beltrami operator on this manifold is $\Delta_g = y^2(\partial_x^2 + \partial_y^2)$, and the Riemannian measure corresponding to the metric $g$ is $\frac{dxdy}{y^2}$. Recall that the operator $-\Delta_g$ on $L^2(\mathbb{H}^2, \frac{dxdy}{y^2})$ with domain $H_g^2(\mathbb{R}^d)$ is self-adjoint with spectrum $[\frac{1}{4}, +\infty)$ (we refer to [Heb96, Gri09] for definitions and properties of Sobolev spaces on Riemannian manifolds and to [Ber11], [Bor16, Chapter 4] for spectral theory of the hyperbolic Laplacian). Thus, by the spectral theorem, it is possible to write

$$u = \int_{\frac{1}{4}}^{+\infty} dm_{\lambda}u, \quad \Pi_{\Lambda}u = \mathbb{1}_{\sqrt{-\Delta_g} \le \Lambda} u = \int_{\frac{1}{4}}^{\Lambda} dm_{\lambda}u,$$

where $dm_\lambda$ denotes the spectral measure of the operator $\sqrt{-\Delta_g}$ defined by functional calculus. For $\phi, \varphi$ two bounded, real-valued functions over $\mathbb{R}$, one has

$$\phi(\sqrt{-\Delta_g})u = \int_{\frac{1}{4}}^{+\infty} \phi(\lambda)dm_{\lambda}u, \\ \langle \phi(\sqrt{-\Delta_g})u, \varphi(\sqrt{-\Delta_g})u \rangle_{L^2(\mathbb{H}^2; \frac{dxdy}{y^2})} = \int_{\frac{1}{4}}^{+\infty} \phi(\lambda)\varphi(\lambda)(dm_{\lambda}u, u)$$

so that

$$(1.1) \qquad \|\phi(\sqrt{-\Delta_g})\Pi_\Lambda u\|_{L^2(\mathbb{H}^2; \frac{dxdy}{y^2})} \le \sup_{\lambda \in [\frac{1}{4}, \Lambda]} |\phi(\lambda)| \|\Pi_\Lambda u\|_{L^2(\mathbb{H}^2; \frac{dxdy}{y^2})}.$$

Geodesic balls for the metric $g$ are given by

$$(1.2) \qquad \forall (x, y) \in \mathbb{H}^2, r > 0, \quad B_r^g(x, y) = B_{y \sinh(r)}(x, y \cosh(r))$$

where $B_r^g$ denotes a geodesic ball of center $(x, y)$ and radius $r$ and $B_{y \sinh(r)}$ a Euclidean ball. Throughout the paper, we will consider a subset $\omega$ of $\mathbb{H}^2$ which satisfies the following assumption:

**Assumption 1.** — There exist $\delta, R > 0$ such that for any $(x, y) \in \mathbb{H}^2$, there holds

$$\mathrm{vol}_g(\omega \cap B_R^g(x, y)) = \int_{\omega \cap B_R^g(x, y)} \frac{dxdy}{y^2} \ge \delta.$$

It essentially states that the measure of the set $\omega$ is large enough and sufficiently spread out so as to be uniformly bounded from below in every geodesic ball, i.e. that $\omega$ is a thick set for the Riemannian metric and density (see [LS74, Kov01]). Note that in the literature, this lower usually concerns the ratio $\frac{\mathrm{vol}_g(\omega \cap B_R^g(x, y))}{\mathrm{vol}_g(B_R^g(x, y))}$. Since all balls of radius $R$ have equal volume in our setting, we omit this divider by including it in $\delta$.

Our main result is that this assumption is equivalent to a spectral estimate from set $\omega$, as follows.

**Theorem 1.** — Consider some Borel set $\omega \subset \mathbb{H}^2$ and define the spectral projector $\Pi_\Lambda$ as above, then the following conditions are equivalent:

(1) $\omega$ satisfies Assumption 1 for some $\delta, R > 0$;
$$ (2) \textit{ there exists some constant } C(\delta, R) > 0 \textit{ such that for any } \Lambda \ge \frac{1}{4}, u \in L^2\left(\mathbb{H}^2, \frac{dx dy}{y^2}\right), $$

$$ (1.3) \qquad \|\Pi_{\Lambda} u\|_{L^2\left(\mathbb{H}^2; \frac{dx dy}{y^2}\right)} \le C e^{C\Lambda} \|\Pi_{\Lambda} u\|_{L^2\left(\omega; \frac{dx dy}{y^2}\right)}. $$

## 1.2. Some bibliography on spectral estimates.
The first equivalence between a spectral estimate (formulated as an uncertainty principle for the Fourier transform) and a thickness condition on the observability set was obtained by Logvinenko and Sereda using complex analysis methods [LS74] (see also Zygmund [Zyg02, pp.202-208], Nazarov [Naz93] and Kovrizhkin [Kov01, Kov03]). The idea of using propagation of smallness inequalities for harmonic functions to prove these estimates can be traced back to Jerison and Lebeau [JL99], with applications to bounds on the Hausdorff measure of nodal sets.

These spectral estimates have had fruitful applications to controllability (or equivalently observability) of the heat equation and of parabolic systems since the works of Lebeau, Robbiano and Zuazua [LR95, LZ98]. In [Mil10], Miller provided an abstract framework for proving observability inequalities at optimal cost for parabolic systems. Phung and G. Wang generalized these observability estimates to any positive-measure time set using a telescoping series method [PW13], and Apraiz, Escauriaza, G. Wang and C. Zhang [AE13, AEWZ14] proved observability from any positive measure time-space set for the heat equation on bounded, star-shaped domains.

On unbounded domains, Egidi and Veselić [EV18] and C. Wang, M. Wang, C. Zhang and Y. Zhang [WWZZ19] simultaneously proved the equivalence between observability of the (constant-coefficient) heat equation and thickness of the observation set on the whole space. The former used Kovrizhkin's formulation of the Logvinenko-Sereda theorem [Kov00, Kov01], while the latter's proof is based on the equivalence with Jerison-Lebeau type estimates for spectral projectors and the telescopic series method of [PW13]. The result of [WWZZ19] can be seen as the flat-Laplacian equivalent of Theorem 1. We refer to the works of Beauchard, Jaming, Martin and Pravda-Starov [BJPS21, MPS23], Egidi, Seelmann and their collaborators [ES21, ENS+20] and the references therein for further spectral estimate and parabolic controllability results from thick sets.

Studies about the interaction between spectral estimates for the Laplacian and the ambient geometry are more scarce in the literature. In [LM19], Lebeau and Moyano proved spectral estimates for Schrödinger operators with analytic potentials, when the metric is an analytic perturbation of the identity. More recently, Burq and Moyano proved spectral estimates for Laplace operators on compact, $W^{2,\infty}$ manifolds with Lipschitz metric and density [BM23], combining ideas of Jerison and Lebeau [JL99] with propagation of smallness results of Logunov and Malinnikova [LM18]. They then showed that the thickness condition for subsets of $\mathbb{R}^d$ implies spectral projector estimates for uniformly elliptic Laplacians with Lipschitz metric and density [BM21].

The aforementioned propagation of smallness result [LM18, Theorem 5.1] is a three-sphere type estimate for the gradients of solutions of homogeneous elliptic equations in divergence form. It plays a central role in the present article. A remarkable feature of this result is that the smaller set in the estimate is of Hausdorff dimension smaller than $d-1$, where $d$ denotes the dimension of the full space. We refer to [Log18a, Log18b, LM18] for details concerning the spirit and proofs of this result. For the convenience of the reader, a statement of the estimate is given in Appendix A.
To the best of our knowledge, the first spectral estimate results on non-compact manifolds with a geometry that is not a perturbation of the Euclidean one are those of Rose and Tautenhahn [RT24] on manifolds with Ricci curvature bounded from below. The main difference with Theorem 1 is that their estimate holds for values of $\Lambda$ below a certain threshold given by the geometry.

The main contribution of the present work is to give the first (again to the best of our knowledge) example of a high-frequency spectral estimate from thick sets on a non-compact manifold with a geometry that is not a perturbation of the Euclidean one. The proof is inspired by the works of Burq–Moyano [BM23, BM21], in which the link between the Logunov–Malinnikova inequality and spectral estimates was established, and the article [WWZZ19], where the authors use an observability inequality to recover a thickness condition on the observation set.

### 1.3. Scheme of the proof.
The rest of the paper is dedicated to proving Theorem 1. In the proof that Assumption 1 implies the spectral estimate, it is more natural to consider Euclidean rectangles than geodesic balls, hence the following remark:

**Remark** Consider $R'$ given by the one-to-one identity

$$ \tanh(R) = \min \left( 1 - 2^{-R'}, \left(\frac{3}{2}\right)^{R'} - 1 \right), $$

where $R$ is given by Assumption 1. Then, every rectangle

$$ \mathcal{R}_{j,k}(R') := (2^{R'j}k - 2^{R'j}, 2^{R'j}k + 2^{R'j})_x \times (2^{R'(j-1)}, 3^{R'} \times 2^{R'(j-1)})_y, $$

contains some geodesic ball of diameter $R$. Precisely, (1.2) gives

$$ (1.4) \qquad B_R^g \left( 2^{R'j}k, \frac{2^{R'j}}{\cosh(R)} \right) \subset \mathcal{R}_{j,k}(R'). $$

By (1.4), Assumption 1 implies the following assumption, which is the one used in the proof:

**Assumption 2.** — There exists some $\delta > 0$ such that for any $(j, k) \in \mathbb{Z}^2$, there holds

$$ \int_{\omega \cap \mathcal{R}_{j,k}(R')} \frac{dxdy}{y^2} \geq \delta. $$

It is easily checked that

$$ (1.5) \qquad \mathbb{H}^2 \subset \bigcup_{(j,k) \in \mathbb{Z}^2} \mathcal{R}_{j,k} $$

and that for any $x \in \mathbb{H}^2$, there holds

$$ \#\{(j,k), x \in \mathcal{R}_{j,k}\} \leq N $$

for some $N(R')$ independent of $x$. Thus,

$$ (1.6) \qquad \|\cdot\|_{L^2(\mathbb{H}^2; \frac{dxdy}{y^2})}^2 \leq \sum_{j,k} \|\cdot\|_{L^2(\mathcal{R}_{j,k}; \frac{dxdy}{y^2})}^2 \leq N \|\cdot\|_{L^2(\mathbb{H}^2; \frac{dxdy}{y^2})}^2. $$

The strategy of proof of the sufficient condition (namely that Assumption 1 implies the spectral estimate) adapts Burq and Moyano's [BM23] on the whole space: we apply Logunov and Malinnikova's propagation of smallness inequality to a well-chosen harmonic function $v_\Lambda(x, y, t)$, with $t$ an auxiliary variable and $(x, y)$ localized inside each $\mathcal{R}_{j,k}$. We then use a
Sobolev injection-type lemma to recover the spectral projector inequality (1.3). The main novelty arises from the fact that the operator $-\Delta_g = y^2(\partial_x^2 + \partial_y^2)$ is not uniformly elliptic over $\mathbb{H}^2$, hence the covering $(\mathcal{R}_{j,k})$ of the half-plane is chosen so that the metric $g$ does not vary too much over each subset. Besides, the multiplicative constant in Logunov and Malinnikova's estimate depends on the ellipticity and Lipschitz constants of the operator considered. To avoid dealing with this dependence, we perform a change of coordinates over every subset $\mathcal{R}_{j,k}$ to obtain constants uniform with respect to $(j, k)$.

The proof of the necessary condition is inspired by similar work by G. Wang, M. Wang, C. Zhang and Y. Zhang for the flat Laplacian on $\mathbb{R}^d$. In [WWZZ19], they deduce thickness of the set $\omega$ from a final-time observability estimate that has been known to derive from spectral projector estimates since the works of Miller [Mil10] and Phung and G. Wang [PW13]. We adapt this strategy to our framework using upper and lower Gaussian (in terms of the geodesic distance) bounds for the hyperbolic heat kernel.

The next section is split into three parts. In the first two, we prove that Assumption 2 implies the spectral inequality (1.3). In Subsection 2.1, we apply the Logunov–Malinnikova and Sobolev-type estimates to prove (1.3). Subsection 2.2 is dedicated to proving the Sobolev injection lemma. In subsection 2.3, we show that the spectral inequality (1.3) (or rather its corollary, observability inequality (2.17)) implies Assumption 1. The appendices do not contain any new mathematical content and are given for the sake of self-containedness. In Appendix A, we state the Logunov–Malinnikova inequality which is used in subsection 2.1. In Appendix B, we reproduce a proof that the spectral estimate (1.3) implies observability inequality (2.17), following [WWZZ19].

We conclude this introduction with a few notations. Throughout the rest of the article, we denote $z = (x, y)$ a generic element in the hyperbolic half-plane $\mathbb{H}^2$, $d_g(z, z')$ the geodesic distance between two points $z, z' \in \mathbb{H}^2$ and $d \operatorname{vol}_g := d \operatorname{vol}_g(z) = \frac{dxdy}{y^2}$ the Riemannian measure corresponding to the metric $g$. We denote by $\|\cdot\|_{L^2}$ norms involving the measure $d \operatorname{vol}_g(z)$ (or the measure $dtd \operatorname{vol}_g(z)$ when also integrating over the auxiliary variable $t$). Norms denoted $\|\cdot\|_{L^2}$ without the index $g$ are integrated using Lebesgue measure. The symbol $\lesssim$ means that the left-hand-side is bounded by the right-hand-side up to an implicit multiplicative constant which is independent of relevant quantities. The dependence of implicit constants is specified when deemed necessary. $A \simeq B$ means that both $A \lesssim B$ and $B \lesssim A$ hold.

## 2. PROOF OF THEOREM 1

### 2.1. Proof of the spectral estimate.
We prove inequality (1.3) under Assumption 2. For the sake of simplicity, the proof is written for $R' = 1$. It can be adapted easily to the case $R' > 0$ by adding appropriate $R'$ exponents in the bounds of the rectangles considered.

Take $u \in L^2(\mathbb{H}^2)$ and define

$$v_{\Lambda}(t, x, y) = \int_{\lambda=0}^{\Lambda} \frac{\sinh(\lambda t)}{\lambda} dm_{\lambda} u.$$

Then $-\Delta_g v_{\Lambda} = \int_0^{\Lambda} \frac{\sinh(\lambda t)}{\lambda} \lambda^2 dm_{\lambda} u$ so that

$$(\partial_t^2 + \Delta_g) v_{\Lambda} = 0.$$

Recalling that $\mathcal{R}_{j,k} = \{(x, y) \in \mathbb{H}^2, |x - 2^j k| < 2^j, 2^{j-1} < y < 3 \times 2^{j-1}\}$ for any $(j, k) \in \mathbb{Z}^2$ and $\mathbb{H}^2 \subset \cup_{j,k} \mathcal{R}_{j,k}$, let $1 = \sum_{(j,k) \in \mathbb{Z}^2} \chi_{j,k}$ be a partition of unity associated to this covering.
Over each rectangle $\mathcal{R}_{j,k}$, we perform the change of variables

$$ (X, Y) = \phi_{j,k}(x, y) = (2^{-j}x - k, 2^{-j}y) $$

so that $(X, Y)$ spans the rectangle $\mathcal{R} = (-1, 1)_X \times (\frac{1}{2}, \frac{3}{2})_Y$ when $(x, y)$ spans $\mathcal{R}_{j,k}$. This change of coordinates preserves the hyperbolic Laplacian:

$$ \Delta_g = y^2(\partial_x^2 + \partial_y^2) = 2^{2j}Y^2(2^{-2j}\partial_X^2 + 2^{-2j}\partial_Y^2) = Y^2(\partial_X^2 + \partial_Y^2). $$

We then define $V_{\Lambda}^{j,k}(t, X, Y) = v_{\Lambda}(t, \phi_{j,k}^{-1}(X, Y))$, which satisfies

$$ (\partial_t^2 + Y^2 \Delta_{X,Y}) V_{\Lambda}^{j,k}(t, X, Y) = (\partial_t^2 + \Delta_g) v_{\Lambda}(t, x, y) = 0, \quad (X, Y) \in \mathcal{R}. $$

Multiplying by $\frac{1}{Y^2}$, we get

$$ (\partial_t \frac{1}{Y^2} \partial_t + \Delta_{X,Y}) V_{\Lambda}^{j,k} = 0 \quad \text{over } \mathcal{R}, $$

which in turn can be written div($A\nabla_{t,X,Y}V_{\Lambda}^{j,k}$) = 0 for

$$ A(t, X, Y) = \begin{pmatrix} \frac{1}{Y^2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}. $$

It is clear that for any $T > 0$, $A$ is uniformly elliptic with Lipschitz coefficients over the box $(-T, T)_t \times \mathcal{R} = (-T, T)_t \times (-1, 1)_X \times (\frac{1}{2}, \frac{3}{2})_Y$, and its ellipticity and Lipschitz constants are independent of $T, j, k$.

Fix $0 < T_1 < T_2$ and $\mathcal{R}_2 = (-2, 2)_X \times (\frac{1}{4}, \frac{7}{4})_Y$. We denote

$$ (2.1) \quad \mathcal{K} = (-T_1, T_1) \times \mathcal{R}, \Omega = (-T_2, T_2) \times \mathcal{R}_2, F_{j,k} = \phi_{j,k}(\omega) \cap \mathcal{R}, E_{j,k} = \{0\} \times F_{j,k}. $$

Then by Assumption 2

$$ \int_{\omega \cap \mathcal{R}_{j,k}} \frac{dxdy}{y^2} = \int_{\phi_{j,k}(\omega) \cap \mathcal{R}} \frac{dXdY}{Y^2} \geq \delta. $$

Thus,

$$ (2.2) \qquad \mathrm{vol}_{\mathrm{euc}}(\phi_{j,k}(\omega) \cap \mathcal{R}) \geq \frac{\delta}{4} $$

uniformly with respect to $j, k$. By Logunov and Malinnikova's theorem [LM18, Theorem 5.1], the statement of which is recalled in Appendix A, we get

$$ (2.3) \qquad \sup_{\mathcal{K}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}| \leq C \sup_{E_{j,k}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^{\alpha} \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^{1-\alpha}. $$

The constants $C > 0, 0 < \alpha < 1$ depend on $\delta$ but do not depend on $u, \Lambda, j$ or $k$.

We change the $L^\infty$ norms to $L^2$ norms using a trick taken from [BM21, Section 2]. Assume that $\nabla V_{\Lambda}^{j,k}|_{\mathcal{K}}$ is not identically zero and let

$$ a = \left( \epsilon \frac{\sup_{\mathcal{K}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|}{\sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^{1-\alpha}} \right)^{\frac{1}{\alpha}}, \quad F'_{j,k} = \{(X, Y) \in F_{j,k}, |\nabla_{t,X,Y} V_{\Lambda}^{j,k}(t = 0, X, Y)| \leq a\}. $$
Let $E'_{j,k} = \{0\} \times F'_{j,k}$. One necessarily has $\text{vol}_{\text{euc}}(F'_{j,k}) \le \frac{\delta}{8}$. Indeed, otherwise, the Logunov-Malinnikova estimate (2.3) holds with identical constants but replacing $E_{j,k}$ with $E'_{j,k}$. This gives

$$ (2.4) \quad \sup_{\mathcal{K}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}| \le C a^{\alpha} \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^{1-\alpha} \le C \epsilon \sup_{\mathcal{K}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|. $$

For $C\epsilon < 1$, we obtain that $\nabla_{t,X,Y} V_{\Lambda}^{j,k}|_{\mathcal{K}}$ is identically zero, hence a contradiction. As a consequence,

$$ (2.5) \quad \int_{F_{j,k}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}(t=0, X, Y)|^2 dX dY \ge \int_{F_{j,k} \setminus F'_{j,k}} |\nabla V_{\Lambda}^{j,k}(t=0, X, Y)|^2 dX dY \\ \ge a^2 \frac{\delta}{8} \ge \frac{\delta}{8} \left( \epsilon \frac{\sup_{\mathcal{K}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|}{\sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^{1-\alpha}} \right)^{\frac{2}{\alpha}}. $$

Since $\nabla_{t,X,Y} V_{\Lambda}^{j,k}(t=0) = \begin{pmatrix} \Pi_{\Lambda} u \circ \phi_{j,k}^{-1} \\ 0 \\ \vdots \\ 0 \end{pmatrix}$, denoting $U_{\Lambda}^{j,k} = \Pi_{\Lambda} u \circ \phi_{j,k}^{-1}$, we get

$$ \int_{\mathcal{R}} |U_{\Lambda}^{j,k}(X, Y)|^2 dX dY \le \sup_{t \in (-T_1, T_1)} \int_{\mathcal{R}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^2 dX dY \\ \lesssim \sup_{\mathcal{K}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^2 \\ \lesssim \left( \int_{F_{j,k}} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}(t=0, X, Y)|^2 dX dY \right)^{\alpha} \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^{2-2\alpha} \\ \lesssim \left( \int_{F_{j,k}} |U_{\Lambda}^{j,k}(X, Y)|^2 dX dY \right)^{\alpha} \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^{2-2\alpha}. $$

The implicit multiplicative constant in the last inequality depends only on $\delta, \epsilon, T_1, T_2$ and numerical constants, and does not depend on $\Lambda, u, j, k$. Note that in the case where $R'$ (see Assumption 2 and the preliminary remark of this section) is fixed and different from 1, this constant also depends on $R'$ through the lower bound in (2.2), the definitions of $\mathcal{R}$ and $\mathcal{R}_2$, and the constants $\Lambda_1, \Lambda_2, \rho$ of Theorem 2.

Recall that since $y$ is roughly of size $2^j$ over $\mathcal{R}_{j,k}$, we have for any function $f \in L^2(\mathcal{R})$,

$$ \int_{\mathcal{R}} |f(X, Y)|^2 dX dY = \int_{\mathcal{R}_{j,k}} 2^{-2j} |f \circ \phi_{j,k}(x, y)|^2 dxdy \approx \int_{\mathcal{R}_{j,k}} |f \circ \phi_{j,k}(x, y)|^2 \frac{dxdy}{y^2}. $$

Applying this to $f = U_{\Lambda}^{j,k}$ and $f = U_{\Lambda}^{j,k} \mathbb{1}_{\phi_{j,k}(\omega)}$, we get

$$ (2.6) \quad \|U_{\Lambda}^{j,k}\|_{L^2(\mathcal{R})} \simeq \|\Pi_{\Lambda} u\|_{L_g^2(\mathcal{R}_{j,k})}, \quad \|U_{\Lambda}^{j,k}\|_{L^2(F_{j,k})} \simeq \|\Pi_{\Lambda} u\|_{L_g^2(\mathcal{R}_{j,k} \cap \omega)}. $$

Hence,

$$ (2.7) \quad \|\Pi_{\Lambda} u\|_{L_g^2(\mathcal{R}_{j,k})}^2 \le C \|\Pi_{\Lambda} u\|_{L^2(\mathcal{R}_{j,k} \cap \omega)}^{2\alpha} \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^{2-2\alpha}. $$

We now state the following Sobolev injection lemma, the proof of which is postponed to the next subsection.
**Lemma 2.1.** — There exist constants $C, K > 0$ independent of $u$ and $\Lambda$ such that for any $u \in L_g^2(\mathbb{H}^2)$,

$$ \sum_{j,k} \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^2 \le Ce^{K\Lambda} \|\Pi_{\Lambda} u\|_{L_g^2(\mathbb{H}^2)}^2. $$

By Young's inequality,

$$ a^{2-2\alpha}b^{2\alpha} \le C(a^2 + b^2). $$

Thus, by (2.7), for any $D > 0$,

$$ (2.8) \quad \|\Pi_{\Lambda} u\|_{L_g^2(\mathcal{R}_{j,k})}^2 \le Ce^{-D\Lambda} \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^2 + Ce^{\frac{2-2\alpha}{2\alpha} D\Lambda} \|\Pi_{\Lambda} u\|_{L_g^2(\omega \cap \mathcal{R}_{j,k})}^2. $$

Summing over $(j, k)$ and applying Lemma 2.1 then gives

$$ \begin{aligned} \|\Pi_{\Lambda} u\|_{L_g^2(\mathbb{H}^2)}^2 &\le Ce^{-D\Lambda} \sum_{j,k} \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^2 + Ce^{\frac{2-2\alpha}{2\alpha} D\Lambda} \|\Pi_{\Lambda} u\|_{L_g^2(\omega \cap \mathcal{R}_{j,k})}^2 \\ &\le Ce^{-D\Lambda} e^{K\Lambda} \|\Pi_{\Lambda} u\|_{L_g^2(\mathbb{H}^2)}^2 + Ce^{\frac{2-2\alpha}{2\alpha} D\Lambda} \|\Pi_{\Lambda} u\|_{L_g^2(\omega)}^2. \end{aligned} $$

Fixing $D > K$ gives the result for $\Lambda$ greater than some $\Lambda_0$ satisfying $Ce^{(K-D)\Lambda_0} < 1$. For $\Lambda < \Lambda_0$, the result is proved by applying Theorem 1 to $\Pi_{\Lambda} u$.

**2.2. Proof of Lemma 2.1.** We now prove Lemma 2.1. Recall that $\Omega$ denotes the set $(-T_2, T_2)_t \times (-2, 2)_X \times (\frac{1}{4}, \frac{7}{4})_y$ and $V_{\Lambda}^{j,k} = v_{\Lambda} \circ \phi_{j,k}$. By the Sobolev injection in dimension 3, we have

$$ \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^2 \le \|\chi \nabla_{t,X,Y} V_{\Lambda}^{j,k}\|_{H^2(\Omega^{(1)})}^2 $$

with $\chi$ a smooth cutoff function supported in

$$ \Omega^{(1)} = (-T_3, T_3) \times (-2 - \eta, 2 + \eta) \times \left(\frac{1}{4} - \eta, \frac{7}{4} + \eta\right) $$

and equal to 1 over $\Omega$. $\eta$ denotes a positive, small enough constant and $T_3 > T_2$. We can then apply elliptic regularity (see [Eva22, Section 6.3]) to the operator $-(\partial_t^2 + Y^2\Delta_{X,Y})$, which is uniformly elliptic over $\Omega^{(1)}$, to obtain

$$ \begin{aligned} \|\chi \nabla_{t,X,Y} V_{\Lambda}^{j,k}\|_{H^2(\Omega^{(1)})}^2 &\lesssim \|(1 - \partial_t^2 - Y^2\Delta_{X,Y})\chi \nabla_{t,X,Y} V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 \\ &\lesssim \|(1 - \partial_t^2 - Y^2\Delta_{X,Y})\chi \partial_t V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 + \|(1 - \partial_t^2 - Y^2\Delta_{X,Y})\chi \nabla_{X,Y} V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2. \end{aligned} $$

The first term is bounded up to some multiplicative constant by

$$ \begin{aligned} \|\partial_t V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 + \|\partial_t^2 V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 + \|\partial_t^3 V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 \\ + \|\nabla_{X,Y} \partial_t V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 + \|(Y^2\Delta_{X,Y}) \partial_t V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 \end{aligned} $$

and the second term by

$$ \begin{aligned} \|\nabla_{X,Y} V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 + \|\nabla_{X,Y} \partial_t V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 + \|\nabla_{X,Y} \partial_t^2 V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 \\ + \|Y^2 \nabla_{X,Y} V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 + \|Y \Delta_{X,Y} V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 + \|\nabla_{X,Y}(Y^2 \Delta_{X,Y} V_{\Lambda}^{j,k})\|_{L^2(\Omega^{(1)})}^2 \end{aligned} $$
where for the last four terms we used the identity

$$Y^2\Delta(\chi\nabla f) = Y^2(\Delta\chi)\nabla f + Y^2\nabla^2 f(\nabla\chi) - \chi\Delta f\left(\binom{0}{2Y}\right) + \chi\nabla(Y^2\Delta f)$$

and $\nabla^2 f$ denotes the Hessian matrix of a function $f$. We again swell $\Omega^{(1)}$ by $\eta$ in the $(X, Y)$ variables to bound the Hessian of $V_{\Lambda}^{j,k}$ by its Laplacian and lower order terms:

$$\|\nabla_{X,Y}^2 V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(1)})}^2 \lesssim \|V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(2)})}^2 + \|\nabla_{X,Y} V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(2)})}^2 + \|\Delta_{X,Y} V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(2)})}^2$$

with $\Omega^{(2)} = (-T_3, T_3) \times (-2 - 2\eta, 2 + 2\eta) \times (\frac{1}{4} - 2\eta, \frac{7}{4} + 2\eta)$ and an implicit multiplicative constant independent of $(j, k, \Lambda)$.

For $(j, k) \in \mathbb{Z}^2$, we denote $\Omega_{j,k}^{(2)} = \phi_{j,k}^{-1}(\Omega^{(2)})$. Switching back to coordinates $(x, y)$ and using (2.6), we deduce the following estimates:

$$\|Y^l \partial_t^m V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(2)})}^2 = \|\partial_t^m v_{\Lambda}\|_{L_g^2(\Omega_{j,k}^{(2)})}^2, \quad (l, m) \in \{0, 1, 2\}^2,$$

$$\|Y^l \nabla_{X,Y} \partial_t^m V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(2)})}^2 \approx \|y \nabla_{x,y} \partial_t^m v_{\Lambda}\|_{L_g^2(\Omega_{j,k}^{(2)})}^2, \quad (l, m) \in \{0, 1, 2\}^2,$$

$$\|Y^l \Delta_{X,Y} V_{\Lambda}^{j,k}\|_{L^2(\Omega^{(2)})}^2 \approx \|y^2 \Delta_{x,y} v_{\Lambda}\|_{L_g^2(\Omega_{j,k}^{(2)})}^2, \quad l \in \{1, 2\},$$

$$\|\nabla_{X,Y}(Y^2\Delta_{X,Y}V_{\Lambda}^{j,k})\|_{L^2(\Omega^{(2)})}^2 \approx \|y\nabla_{x,y}(y^2\Delta_{x,y}v_{\Lambda})\|_{L_g^2(\Omega_{j,k}^{(2)})}^2.$$

Combining (2.10), (2.11) and (2.12) and summing over $(j, k) \in \mathbb{Z}^2$ then yields

$$(2.13) \qquad \sum_{j,k} \sup_{\Omega} |\nabla_{t,X,Y} V_{\Lambda}^{j,k}|^2 \lesssim \left( \sum_{m=1}^3 \|\partial_t^m v_{\Lambda}\|_{L_g^2((-T_2,T_2)\times\mathbb{H}^2)}^2 \right) + \left( \sum_{m=0}^2 \|y\nabla_{x,y} \partial_t^m v_{\Lambda}\|_{L_g^2((-T_2,T_2)\times\mathbb{H}^2)}^2 \right) \\ + \|\Delta_g v_{\Lambda}\|_{L_g^2((-T_2,T_2)\times\mathbb{H}^2)}^2 + \|\Delta_g \partial_t v_{\Lambda}\|_{L_g^2((-T_2,T_2)\times\mathbb{H}^2)}^2 + \|y\nabla_{x,y}\Delta_g v_{\Lambda}\|_{L_g^2((-T_2,T_2)\times\mathbb{H}^2)}^2.$$

Recall that

$$(2.14) \qquad \partial_t^p ((-\Delta_g)^{q/2} v_{\Lambda}) = \int_0^{\Lambda} \lambda^{p+q-1} \sinh^{(p)}(\lambda t) dm_{\Lambda} u, \quad p, q \in \mathbb{N}$$

so by (1.1) the first, third and fourth terms in (2.13) can be bounded by

$$Ce^{C\Lambda} \|\Pi_{\Lambda} u\|_{L_g^2(\mathbb{H}^2)}.$$

There remains to bound the terms involving a gradient, using integration by parts in the $(X, Y)$ variables:

**Lemma 2.2.** — Call any of the functions $v_{\Lambda}, \partial_t v_{\Lambda}, \partial_t^2 v_{\Lambda}, \Delta_g v_{\Lambda}$, then

$$\|y \nabla_{x,y} h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)}^2 \lesssim \|h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)}^2 + \|\Delta_g h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)}^2.$$

*Proof of Lemma 2.2.* Recall that

$$\|y \nabla_{x,y} h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)}^2 \le \sum_{j,k} \|\nabla_{X,Y} (h \circ \phi_{j,k}^{-1})\|_{L^2(\Omega^{(2)})}^2.$$

Let $\tilde{\chi}(X, Y)$ be a cutoff function equal to 1 over $(-2 - 2\eta, 2 + 2\eta)_X \times (\frac{1}{4} - 2\eta, \frac{7}{4} + 2\eta)_Y$, supported in $(-2 - 3\eta, 2 + 3\eta)_X \times (\frac{1}{4} - 3\eta, \frac{7}{4} + 3\eta)_Y$. Then by integration by parts and the
Cauchy-Schwarz inequality, denoting $\Omega^{(3)} = (-T_3, T_3) \times (-2-3\eta, 2+3\eta)_X \times (\frac{1}{4}-3\eta, \frac{7}{4}+3\eta)_Y$ and $H_{j,k} = h \circ \phi_{j,k}^{-1}$, we get

$$ \begin{aligned} \|\nabla_{X,Y} H_{j,k}\|_{L^2(\Omega^{(2)})}^2 &\le \|\nabla_{X,Y}(\tilde{\chi} H_{j,k})\|_{L^2(\Omega^{(3)})}^2 \\ &\le \|\tilde{\chi} H_{j,k}\|_{L^2(\Omega^{(3)})} \|\Delta_{X,Y}(\tilde{\chi} H_{j,k})\|_{L^2(\Omega^{(3)})} \\ &\le C \|H_{j,k}\|_{L^2(\Omega^{(3)})} \\ &\times \left( \|H_{j,k}\|_{L^2(\Omega^{(3)})} + \|\nabla_{X,Y} H_{j,k}\|_{L^2(\Omega^{(3)})} + \|\Delta_{X,Y} H_{j,k}\|_{L^2(\Omega^{(3)})} \right) \end{aligned} $$

Changing variables and summing over $(j, k)$ gives

$$ (2.15) \quad \|y \nabla_{x,y} h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)}^2 \le C \left( \|h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)}^2 \right. \\ \qquad \left. \qquad + \|y \nabla_{x,y} h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)} \|h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)} + \|\Delta_g h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)}^2 \right). $$

The result then comes from the inequality $ab \le \frac{1}{2}(\varepsilon a^2 + \frac{1}{\varepsilon} b^2)$ with $\varepsilon = \frac{1}{2C}$, applied to $a = \|y \nabla_{x,y} h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)}$ and $b = \|h\|_{L_g^2((-T_3,T_3) \times \mathbb{H}^2)}$. $\square$

Combining this lemma with (2.13), we can replace all terms in the right-hand-side of (2.13) by squared $L_g^2$-norms of terms of the form (2.14) with $0 \le p + q \le 4$. We then conclude the proof of Lemma 2.1 using estimate (1.1) and the inequality

$$ \sup_{0 \le m \le 4, 0 \le \lambda \le \Lambda, |t| \le T_3} \lambda^m (\cosh(\lambda t) + |\sinh(\lambda t)|) \le C e^{K\Lambda} $$

for some constants $C, K > 0$ independent of $\Lambda$ but allowed to depend on $T_3$. This completes the proof that Assumption 1 implies the spectral estimate (1.3).

2.3. **Necessary condition.** We now prove that Assumption 1 is necessary for the spectral estimate to hold, using the heat kernel approach developed in [WWZZ19] for the flat Laplacian. The first step is the following observability estimate:

**Lemma 2.3** (Spectral estimate implies observability). — Consider $\omega \subset \mathbb{H}^2$ a measurable set such that the spectral estimate (1.3) holds for $\omega$. For any solution $u$ of the heat equation on $\mathbb{H}^2$

$$ (2.16) \quad \partial_t u - \Delta_g u = 0 \quad \text{in } (0, +\infty)_t \times \mathbb{H}_z^2, \quad u(0, \cdot) = u_0 \in L_g^2(\mathbb{H}^2), $$

for any $T > 0$, there exists a positive constant $C_{\text{obs}}(T, \omega)$ such that

$$ (2.17) \quad \int_{\mathbb{H}^2} |u(T, z)|^2 d\text{vol}_g(z) \le C_{\text{obs}} \int_0^T \int_\omega |u(s, z)|^2 d\text{vol}_g(z) ds. $$

We provide a proof and references concerning this classical result in Appendix B for the sake of completeness.

We now show that the observability estimate (2.17) implies Assumption 1. Consider the heat kernel on the hyperbolic half-plane:

$$ (2.18) \quad H(t, z, z') = \frac{\sqrt{2}}{(4\pi t)^{\frac{3}{2}}} e^{-\frac{t}{4}} \int_{d_g(z, z')}^{\infty} \frac{se^{-\frac{s^2}{4t}} ds}{(\cosh(s) - \cosh(d_g(z, z'))^{1/2}}, \quad \forall z, z' \in \mathbb{H}^2, t > 0. $$
Any solution $u$ of the heat equation (2.16) then satisfies

$$ (2.19) \qquad u(t, z) = \int_{\mathbb{H}^2} H(t, z, z')u_0(z')d\operatorname{vol}_g(z'), $$

and by the semi-group property for $e^{t\Delta_g}$, one has

$$ (2.20) \qquad H(t, z_1, z_2) = \int_{\mathbb{H}^2} H(s, z_1, z)H(t-s, z, z_2)d\operatorname{vol}_g(z), \quad \forall z_1, z_2 \in \mathbb{H}^2, 0 < s < t. $$

We refer to [McK70, GN98] for a proof of the expression of the heat kernel on the hyperbolic plane and to [CLY19, Gri09, Chapters 4, 7] and the references therein for a more detailed study of the heat kernel on manifolds.

We now fix some $z_0 \in \mathbb{H}^2$ and show that there exist some $R, \delta > 0$ independent of $z_0$ such that $\operatorname{vol}_g(B_R^g(z_0) \cap \omega) \ge \delta$. Let us apply Lemma 2.3 with the initial condition

$$ u_0(z) = H(1, z, z_0). $$

By (2.20), the corresponding solution $u(t, z)$ of the heat equation is

$$ (2.21) \qquad u(t, z) = H(t + 1, z, z_0), \quad t > 0, z \in \mathbb{H}^2. $$

Inserting (2.21) in the observability inequality (2.17) for time $T=1$ yields

$$ (2.22) \qquad \int_{\mathbb{H}^2} |H(2, z, z_0)|^2 d\operatorname{vol}_g(z) \le C_{\operatorname{obs}}(1) \int_0^1 \int_\omega |H(s + 1, z, z_0)|^2 d\operatorname{vol}_g(z) ds. $$

The next lemma allows to conclude:

**Lemma 2.4 (Gaussian bounds).** — The heat kernel $H$ given by (2.18) admits Gaussian (as functions of $d_g(z, z')$) lower and upper bounds. More precisely, there exist some constants $\alpha, \beta > 0$ such that the lower bound

$$ (2.23) \qquad H(2, z, z_0) \gtrsim e^{-\beta d_g(z, z_0)^2} $$

and the upper bound

$$ (2.24) \qquad H(t, z, z_0) \lesssim \frac{\sqrt{\gamma t}}{f(\gamma t)} e^{-\frac{\alpha}{t} d_g(z, z_0)^2}, \quad t > 0, $$

hold for some $\gamma > 0$ and $f(t) = e^{\frac{t}{4}t^{\frac{3}{2}}}.$

We deduce the necessary condition of Theorem 1 from the lemma, which will be proved afterwards. The l.h.s. of (2.22) can be bounded from below as follows, with $C$ a positive constant

$$ \int_{\mathbb{H}^2} |H(2, z, z_0)|^2 d\operatorname{vol}_g(z) \ge C \int_{\mathbb{H}^2} e^{-2\beta d_g(z, z_0)^2} d\operatorname{vol}_g(z). $$

The integral in the right-hand-side is independent of $z_0$, as shown by the change of variables

$$ z' = y_0^{-1}(z - (x_0, 0)). $$

Indeed, the hyperbolic distance and the measure $d\operatorname{vol}_g$ are invariant by scaling and horizontal translation (see for example [Bor16, (2.6) and (2.8)]), so that $d_g(z, z_0) = d_g(z', (0, 1))$ and $d\operatorname{vol}_g(z) = d\operatorname{vol}_g(z')$.
We now bound the r.h.s. of (2.22) from above using the upper bound (2.24). We have

$$ 
\begin{aligned}
\int_0^1 \int_{\omega} |H(s+1, z, z_0)|^2 d\mathrm{vol}_g(z) ds &\le C' \int_0^1 \int_{\omega} \frac{(s+1)\gamma}{f((s+1)\gamma)^2} e^{-2\frac{\alpha}{s+1}d_g(z,z_0)^2} d\mathrm{vol}_g(z) ds \\
&\le \frac{C'\gamma}{f(\gamma)^2} \int_{\omega} e^{-\alpha d_g(z,z_0)^2} d\mathrm{vol}_g(z).
\end{aligned}
$$

We introduce a parameter $L > 0$ and split the domain of integration into $B_L^g(z_0)$ and $B_L^g(z_0)^c$:

$$ 
\begin{aligned}
&\int_0^1 \int_{\omega} |H(s+1, z, z_0)|^2 d\mathrm{vol}_g(z) \\
&\le \frac{C'\gamma}{f(\gamma)^2} \left( \int_{\omega \cap B_L^g(z_0)} e^{-\alpha d_g(z,z_0)^2} d\mathrm{vol}_g(z) + \int_{\omega \cap B_L^g(z_0)^c} e^{-\alpha d_g(z,z_0)^2} d\mathrm{vol}_g(z) \right) \\
&\le \frac{C'\gamma}{f(\gamma)^2} \left( \mathrm{vol}_g(\omega \cap B_L^g(z_0)) + e^{-\frac{\alpha}{2}L^2} \int_{\mathbb{H}^2} e^{-\frac{\alpha}{2}d_g(z,z_0)^2} d\mathrm{vol}_g(z) \right).
\end{aligned}
$$

The first term is precisely the volume we wish to bound from below to prove Assumption 1. The integral in the second term is independent from $z_0$ for the same reason as before.

Plugging the lower and upper bounds together gives

$$ (2.25) \qquad C'' \le \mathrm{vol}_g(\omega \cap B_L^g(z_0)) + e^{-\frac{\alpha}{2}L^2} \int_{\mathbb{H}^2} e^{-\frac{\alpha}{2}d_g(z',(0,1))^2} d\mathrm{vol}_g(z') $$

where $C'' = \frac{C f(\gamma)^2}{C'\gamma} \int_{\mathbb{H}^2} e^{-2\beta d_g(z',(0,1))^2} d\mathrm{vol}_g(z')$ does not depend on $z_0$. We fix $L$ large enough for the second term of the r.h.s. to be smaller than $\frac{C''}{2}$. Assumption 1 then holds for $R = L$ and $\delta = \frac{C''}{2}$.

*Proof of lemma 2.4.* There remains to prove Gaussian lower and upper bounds for the heat kernel $H(t, z, z')$. We start with the lower bound. The explicit expression (2.18) gives

$$ 
\begin{aligned}
H(2, z, z') &\gtrsim \int_{d_g(z, z')}^{\infty} \frac{se^{-\frac{s^2}{8}} ds}{(\cosh(s) - \cosh(d_g(z, z')))^{\frac{1}{2}}} ds \\
&\gtrsim \int_{d_g(z, z')}^{\infty} \frac{se^{-\frac{s^2}{8}} ds}{(\cosh(s) - 1)^{\frac{1}{2}}} ds \\
&\gtrsim \int_{d_g(z, z')}^{\infty} \frac{se^{-\frac{s^2}{8}} ds}{\sinh(s/2)}. 
\end{aligned}
$$

Then since $\sinh(\frac{s}{2}) \le \frac{1}{2}e^{\frac{s^2}{2}}$ and $\frac{s}{2} + \frac{s^2}{8} \le \frac{1+s^2}{2}$, we get the lower bound

$$ H(2, z, z') \gtrsim \int_{d_g(z, z')}^{\infty} se^{-\frac{s^2}{2}} ds \gtrsim e^{-\frac{1}{2}d_g(z, z')}^2. $$

To prove the upper bound, we employ the strategy developed by Grigor'yan in [Gri97, Gri99, Section 5.1], which consists in deducing off-diagonal estimates from diagonal estimates. H
satisfies the following diagonal estimate:

$$ (2.26) \qquad \begin{aligned} H(t, z, z) &= \frac{\sqrt{2}}{(4\pi)^{\frac{3}{2}}} \int_0^\infty \frac{se^{-\frac{s^2}{4t}}}{(\cosh(s) - 1)^{\frac{1}{2}}} ds \\ &\lesssim \frac{1}{f(t)} \int_0^\infty \frac{se^{-\frac{s^2}{4t}}}{\sinh(\frac{s}{2})} ds \\ &\lesssim \frac{1}{f(t)} \int_0^\infty e^{-\frac{s^2}{4t}} ds \quad \text{as } \sinh\left(\frac{s}{2}\right) \ge \frac{s}{2} \text{ for positive } s \\ &\lesssim \frac{t^{\frac{1}{2}}}{f(t)}. \end{aligned} $$

where $f(t)$ is the function introduced in Lemma 2.4. The estimate is independent of $z$.

The upper bound can then be derived from [Gri97, Theorem 1.1], the assumption of which is satisfied by $\frac{f(t)}{t^{\frac{1}{2}}} = te^{\frac{t}{4}}$ (see Section 2, Example 2.). Thus, there exist some positive constants $\alpha, K, \gamma$ such that for any $(z, z') \in (\mathbb{H}^2)^2$, one has

$$ H(t, z, z') \le \frac{K(\gamma t)^{\frac{1}{2}}}{f(\gamma t)} e^{-\frac{\alpha}{t} d_g(z, z')}^2. $$

This completes the proof of the lemma.

<figure><img src="image_1.png" /></figure>

## APPENDIX A. LOGUNOV AND MALINNIKOVA'S PROPAGATION OF SMALLNESS INEQUALITY

We recall the precise statement of the estimate by Logunov and Malinnikova which we use in the proof of Theorem 1, namely [LM18, Theorem 5.1]:

**Theorem 2 ([LM18]).** — Consider $\Omega$ a bounded domain in $\mathbb{R}^d$. There exists a constant $c \in (0, 1)$ that depends only on the dimension $d$ such that the following holds: take any $u$ solution of an elliptic equation in divergence form

$$ (A.1) \qquad \operatorname{div}(A\nabla u) = 0 \text{ over } \Omega $$

where $A = (a_{i,j}(x))_{i,j}$ is a symmetric, uniformly elliptic matrix with Lipschitz entries:

$$ \Lambda_1^{-1} |\zeta|^2 \le \langle A\zeta, \zeta \rangle \le \Lambda_1 |\zeta|^2, \quad |a_{i,j}(x) - a_{i,j}(y)| \le \Lambda_2 |x - y|. $$

Let $m, \delta, \rho$ be positive numbers, and assume sets $E, K \subset \Omega$ satisfy

$$ C_{\mathcal{H}}^{n-1-c+\delta}(E) > m, \quad \operatorname{dist}(E, \partial\Omega) > \rho, \operatorname{dist}(K, \partial\Omega) > \rho. $$

Then there exist $C, \alpha > 0$ depending only on $m, \delta, \rho, \Lambda_1, \Lambda_2, \Omega$ such that

$$ \sup_K |\nabla u| \le C \sup_E |\nabla u|^\alpha \sup_\Omega |\nabla u|^{1-\alpha}. $$

Taking $\delta = c$, the result holds if $E$ is a set of Hausdorff dimension $n - 1$.
## APPENDIX B. SPECTRAL ESTIMATE IMPLIES OBSERVABILITY

In this Appendix, we prove that on any set where the spectral estimate (1.3) holds, solutions of the heat equation (2.16) satisfy the observability inequality (2.17). The link between both was established by Miller [Mil10] while Phung and Wang established observability from any set of positive measure [PW13]. The present proof follows closely that of [WWZZ19, Lemmas 2.3, 2.4] and we present it for the sake of completeness.

The proof is divided into two parts. In the first one, we use the spectral inequality to prove the interpolation estimate (B.1). Differences with [WWZZ19] are that the Plancherel-based estimates for Fourier multipliers are replaced by inequality (1.1), and that the authors prove the inequality for any $\theta \in (0, 1)$ while we focus on the case $\theta = 2$. In the second part, observability is deduced from the interpolation inequality using a telescoping series method: there we follow [WWZZ19] in the simpler case where $F$ equals $(0, T)$ rather than any positive-measure subset.

*Proof of Lemma 2.3.* Assume that (1.3) holds for the set $\omega$ and consider a solution $u$ of the heat equation (2.16) on $\mathbb{H}^2$. Fix $T > 0$ and denote $u_0 = u(0, \cdot)$. Then,

$$u(T, z) = e^{T\Delta_g} u_0(z), \quad z \in \mathbb{H}^2.$$

We first prove that $u$ satisfies the following Hölder-type inequality for some constant $\tilde{C}$:
(B.1)

$$\int_{\mathbb{H}^2} |u(T, z)|^2 d\text{vol}_g \le \exp\left(\tilde{C}\left(1 + \frac{1}{T}\right)\right) \left(\int_{\omega} |u(T, z)|^2 d\text{vol}_g\right)^{\frac{1}{2}} \left(\int_{\mathbb{H}^2} |u_0(z)|^2 d\text{vol}_g\right)^{\frac{1}{2}}.$$

Denote $\Pi_{>\Lambda}$ the spectral projector defined by

$$\Pi_{>\Lambda} u = \int_{\Lambda}^{+\infty} dm_{\lambda} u, \quad u \in L_g^2(\mathbb{H}^2),$$

then $u_0 = \Pi_{>\Lambda} u_0 + \Pi_{>\Lambda} u_0$. By the spectral estimate (1.3), there exists some $K > 0$ such that

$$\begin{aligned} \int_{\mathbb{H}^2} |u(T, z)|^2 d\text{vol}_g &\le 2 \int_{\mathbb{H}^2} |e^{T\Delta_g} \Pi_{>\Lambda} u_0(z)|^2 d\text{vol}_g + 2 \int_{\mathbb{H}^2} |e^{T\Delta_g} \Pi_{>\Lambda} u_0(z)|^2 d\text{vol}_g \\ &\le 2Ke^{K\Lambda} \int_{\omega} |e^{T\Delta_g} \Pi_{>\Lambda} u_0(z)|^2 d\text{vol}_g + 2 \int_{\mathbb{H}^2} |e^{T\Delta_g} \Pi_{>\Lambda} u_0(z)|^2 d\text{vol}_g \\ &\le 4Ke^{K\Lambda} \int_{\omega} |e^{T\Delta_g} u_0(z)|^2 d\text{vol}_g + (2 + 4Ke^{K\Lambda}) \int_{\mathbb{H}^2} |e^{T\Delta_g} \Pi_{>\Lambda} u_0(z)|^2 d\text{vol}_g. \end{aligned}$$

Since $\int_{\mathbb{H}^2} |e^{T\Delta_g} \Pi_{>\Lambda} u_0(z)|^2 d\text{vol}_g \le e^{-T\Lambda^2} \|\Pi_{>\Lambda} u_0\|_{L_g^2(\mathbb{H}^2)}^2 \le e^{-T\Lambda^2} \|u_0\|_{L_g^2(\mathbb{H}^2)}^2$, it follows that

$$\begin{aligned} \int_{\mathbb{H}^2} |u(T, z)|^2 d\text{vol}_g &\le 4Ke^{K\Lambda} \int_{\omega} |e^{T\Delta_g} u_0(z)|^2 d\text{vol}_g + (2 + 4Ke^{K\Lambda})e^{-T\Lambda^2} \|u_0\|_{L_g^2(\mathbb{H}^2)}^2 \\ &\le 6K \left( e^{K\Lambda} \int_{\omega} |e^{T\Delta_g} u_0(z)|^2 d\text{vol}_g + e^{K\Lambda - T\Lambda^2} \|u_0\|_{L_g^2(\mathbb{H}^2)}^2 \right) \end{aligned} \quad (B.2)$$

Fix $\eta \in (0, 1)$ and $\Lambda$ such that $e^{K\Lambda - T\Lambda^2} = \eta$, then one has

$$\Lambda = \frac{K + \sqrt{K^2 + 4T \log \frac{1}{\eta}}}{2T} \le \frac{1}{T} \left( K + \sqrt{T \log \frac{1}{\eta}} \right).$$
Thus,

$$ 
\begin{aligned}
e^{K\Lambda} & \le \exp\left(\frac{K^2}{T}\right) \exp\left(\frac{K}{\sqrt{T}}\sqrt{\log\frac{1}{\eta}}\right) \\
& \le \exp\left(\frac{K^2}{T}\right) \exp\left(\frac{K^2}{2T} + \frac{1}{2}\log\frac{1}{\eta}\right) \\
& \le \exp\left(\frac{3K^2}{2T}\right) \eta^{-1}.
\end{aligned}
$$

Setting

$$ \eta = \left( \frac{\int_{\omega} |u(T, z)|^2 d\operatorname{vol}_g}{\int_{\mathbb{H}^2} |u_0(z)|^2 d\operatorname{vol}_g} \right)^{\frac{1}{2}} < 1 $$

yields estimate (B.1).

Then, by (B.1) and translation in time, one gets that for $0 < t_1 < t_2$ and $\varepsilon > 0$,

$$ (B.3) \quad \int_{\mathbb{H}^2} |u(t_2, z)|^2 d\operatorname{vol}_g \le \frac{1}{\varepsilon} e^{2\tilde{C}\left(1+\frac{1}{t_2-t_1}\right)} \int_{\omega} |u(t_2, z)|^2 d\operatorname{vol}_g + \varepsilon \int_{\mathbb{H}^2} |u(t_1, z)|^2 d\operatorname{vol}_g. $$

Now set $l_1 = T$, $\lambda \in (\frac{1}{\sqrt{2}}, 1)$ and $l_{m+1} = \lambda^m l_1$ for $m \ge 0$. Take an integer $m$ and some $s$ such that

$$ 0 < l_{m+2} < l_{m+1} \le s < l_m \le T, $$

then by (B.3),

$$ \int_{\mathbb{H}^2} |u(l_m, z)|^2 d\operatorname{vol}_g \le \frac{1}{\varepsilon} e^{2\tilde{C}\left(1+\frac{1}{l_{m+1}-l_{m+2}}\right)} \int_{\omega} |u(s, z)|^2 d\operatorname{vol}_g + \varepsilon \int_{\mathbb{H}^2} |u(l_{m+2}, z)|^2 d\operatorname{vol}_g. $$

Integrating over $s \in (l_{m+1}, l_m)$, we get

$$ (B.4) \quad \int_{\mathbb{H}^2} |u(l_m, z)|^2 d\operatorname{vol}_g \\
\le \varepsilon \int_{\mathbb{H}^2} |u(l_{m+2}, z)|^2 d\operatorname{vol}_g + \frac{1}{\varepsilon} \frac{1}{l_m - l_{m+1}} e^{2\tilde{C}\left(1+\frac{1}{l_{m+1}-l_{m+2}}\right)} \int_{l_{m+1}}^{l_m} \int_{\omega} |u(s, z)|^2 d\operatorname{vol}_g ds. $$

Using the inequality $l_m - l_{m+1} \ge e^{-\frac{1}{l_m - l_{m+1}}}$, and the identities

$$ l_m - l_{m+1} = \frac{1}{1 + \lambda}(l_m - l_{m+2}), \quad l_{m+1} - l_{m+2} = \frac{\lambda}{1 + \lambda}(l_m - l_{m+2}), $$

we obtain

$$ (B.5) \quad \int_{\mathbb{H}^2} |u(l_m, z)|^2 d\operatorname{vol}_g \\
\le \varepsilon \int_{\mathbb{H}^2} |u(l_{m+2}, z)|^2 d\operatorname{vol}_g + \frac{1}{\varepsilon} e^{2\tilde{C}} e^{\frac{C'}{l_m - l_{m+2}}} \int_{l_{m+1}}^{l_m} \int_{\omega} |u(s, z)|^2 d\operatorname{vol}_g ds $$

with $C' = 1 + \lambda + \frac{2\tilde{C}(1+\lambda)}{\lambda}$. (B.5) can be rewritten as

$$ (B.6) \quad \varepsilon e^{-\frac{C'}{l_m - l_{m+2}}} \int_{\mathbb{H}^2} |u(l_m, z)|^2 d\operatorname{vol}_g - \varepsilon^2 e^{-\frac{C'}{l_m - l_{m+2}}} \int_{\mathbb{H}^2} |u(l_{m+2}, z)|^2 d\operatorname{vol}_g \\
\le e^{2\tilde{C}} \int_{l_{m+1}}^{l_m} \int_{\omega} |u(s, z)|^2 d\operatorname{vol}_g ds. $$
Setting $\mu := \frac{1}{2-\lambda^{-2}} > 1$ and $\varepsilon = \exp\left[-\frac{(\mu-1)C'}{l_m-l_{m+2}}\right]$ yields

$$ (B.7) \quad \begin{aligned} & e^{-\frac{\mu C'}{l_m-l_{m+2}}} \int_{\mathbb{H}^2} |u(l_m, z)|^2 d\text{vol}_g - e^{-\frac{(2\mu-1)C'}{l_m-l_{m+2}}} \int_{\mathbb{H}^2} |u(l_{m+2}, z)|^2 d\text{vol}_g \\ & \le e^{2\tilde{C}} \int_{l_{m+1}}^{l_m} \int_{\omega} |u(s, z)|^2 d\text{vol}_g ds. \end{aligned} $$

Now

$$ \frac{2\mu - 1}{l_m - l_{m+2}} = \frac{\mu}{\lambda^2(l_m - l_{m+2})} = \frac{\mu}{l_{m+2} - l_{m+4}}, $$

hence

$$ (B.8) \quad \begin{aligned} & e^{-\frac{\mu C'}{l_m-l_{m+2}}} \int_{\mathbb{H}^2} |u(l_m, z)|^2 d\text{vol}_g - e^{-\frac{\mu C'}{l_{m+2}-l_{m+4}}} \int_{\mathbb{H}^2} |u(l_{m+2}, z)|^2 d\text{vol}_g \\ & \le e^{2\tilde{C}} \int_{l_{m+1}}^{l_m} \int_{\omega} |u(s, z)|^2 d\text{vol}_g ds. \end{aligned} $$

Summing this inequality over all odd integers $m$, we obtain

$$ e^{\frac{\mu C'}{l_1-l_3}} \int_{\mathbb{H}^2} |u(l_1, x)|^2 dx \le e^{2\tilde{C}} \int_0^1 \int_{\omega} |u(s, z)|^2 d\text{vol}_g ds, $$

which in turn gives the observability estimate (2.17). $\square$

## ACKNOWLEDGEMENTS

The author would like to thank Nicolas Burq and Alix Deleporte for insightful discussions about the topic of this article. This work was funded by a CDSN PhD grant from Ecole Normale Supérieure Paris-Saclay via the Hadamard Doctoral School of Mathematics (EDMH).

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UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405, ORSAY, FRANCE
*Email address: marc.rouveyrol@universite-paris-saclay.fr*