# Integrals of geometric random variables involving geodesics filling a pair of pants

Victor Le Guilloux

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Victor Le Guilloux. Integrals of geometric random variables involving geodesics filling a pair of pants. 2025. hal-05383831

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Distributed under a Creative Commons Attribution 4.0 International License
# INTEGRALS OF GEOMETRIC RANDOM VARIABLES INVOLVING GEODESICS
FILLING A PAIR OF PANTS

## VICTOR LE GUILLOUX

ABSTRACT. In this article we provide an integration formula for geometric random variables defined over the moduli space which involve the length of any closed geodesic filling a pair of pants. This generalizes the results of our previous paper, where such formula was only obtained for the particular case of a loop with exactly one self-intersection. As a consequence, when $\gamma$ is such a geodesic, we obtain an improvement of Mirzakhani's asymptotic equivalent of the Weil-Petersson expectation $\mathbb{E}[N_\gamma(a)]$, when $a \to \infty$, of the number $N_\gamma(a)$ of closed geodesics of length at most $a$ in the mapping class group orbit of $\gamma$.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Length function of a loop filling a pair of pants</td><td>5</td></tr><tr><td>3. Pseudo-convolution of polynomials with respect to length-type functions</td><td>15</td></tr><tr><td>4. Conclusion</td><td>26</td></tr><tr><td>References</td><td>27</td></tr></table>

## 1. INTRODUCTION

1.1. Main results. Let $S_{g,n}$ be a fixed surface of genus $g$ with $n$ labelled boundaries $\alpha_1, \dots, \alpha_n$. Assuming that $2g - 2 + n > 0$, we can consider the moduli space $\mathcal{M}_{g,n}(L)$ of hyperbolic metrics on $S_{g,n}$ with geodesic boundaries of lengths given by $L = (L_1, \dots, L_n) \in \mathbb{R}_{\ge 0}^n$ (a boundary of length 0 is a cusp). The aim of this article is to extend the results we proved in [Le 25] to arbitrary loops filling a pair of pants. Namely we study random variables on $\mathcal{M}_{g,n}(L)$ whose expression involves the lengths of closed geodesics filling a pair of pants. In order to consider functions defined on $\mathcal{M}_{g,n}(L)$ as random variables, we equip the moduli space with the Weil-Petersson probability measure, and we denote by $\mathbb{E}_{g,n}^L$ the expectation for this measure.

Given a closed loop $\gamma$ on $S_{g,n}$, we denote by $\mathcal{O}_\gamma$ the mapping class group orbit of the free homotopy class $[\gamma]$ of $\gamma$, and if $f: S_{g,n} \to X$ is an orientation-preserving homeomorphism from $S_{g,n}$ to a hyperbolic surface $X$, we denote for any test function $F: \mathbb{R} \to \mathbb{C}$ (e.g. compactly supported):

$$F^{\gamma}(X) = \sum_{[\gamma'] \in \mathcal{O}_{\gamma}} F(\ell_{f(\gamma')}(X)),$$

where $\ell_{f(\gamma')}$ is the hyperbolic length of the unique geodesic of $X$ freely homotopic to $f(\gamma')$. With this definition, $F^\gamma(X)$ does not depend on the choice of the homeomorphism $f$. As a consequence, $F^\gamma$ is a well-defined random variable on $\mathcal{M}_{g,n}(L)$.

Our main result is the following.
**Theorem 1.1.** Let $\gamma$ be a closed loop on $S_{g,n}$ filling a pair of pants. There exists a locally integrable function $V_{\mathcal{O}_{\gamma}}(\cdot, L): \mathbf{R} \to \mathbf{R}_{\ge 0}$, depending on the orbit of $\gamma$, such that for every test function $F: \mathbf{R} \to \mathbf{C}$, we have:

$$ \mathbb{E}_{g,n}^{L}[F^{\gamma}] = \int_{0}^{\infty} F(\ell) V_{\mathcal{O}_{\gamma}}(\ell, L) \frac{d\ell}{V_{g,n}(L)}. $$

Moreover there exist a polynomial $P_{\gamma}^{L}$ of degree $6g - 7 + 2n$ and constants $c_0, c > 0$ and $0 < \lambda_{\gamma} < 1$ such that for every $m \ge 1$ we have:

$$ \int_{0}^{m} e^{\ell} |V_{\mathcal{O}_{\gamma}}(\ell, L) - P_{\gamma}^{L}(\ell)| d\ell \le c_0 (1+m)^c e^{(1-\lambda_{\gamma})m}. $$

As a consequence, we obtain:

**Corollary 1.2.** Let $\gamma$ be a closed loop on $S_{g,n}$ filling a pair of pants. Given $a \ge 0$ and a hyperbolic metric $X \in \mathcal{M}_{g,n}(L)$, we denote by $N_{\gamma}(a)(X)$ the number of geodesics of $X$ in the same mapping class group orbit as $\gamma$ with length at most $a$. There exist a polynomial $Q_{\gamma}^{L}$ of degree $6g - 6 + 2n$ and constants $c > 0$ and $0 < \lambda_{\gamma} < 1$ such that:

$$ \mathbb{E}_{g,n}^{L}[N_{\gamma}(a)] = Q_{\gamma}^{L}(a) + \mathcal{O}_{a \to \infty}((1+a)^c e^{-\lambda_{\gamma}a}). $$

*Remark.* The coefficient $\lambda_{\gamma}$ provided by Theorem 1.1 is not necessarily optimal. Indeed, if we apply the method of this article to an eight-shaped loop $\gamma$, depending of the orbit of $\gamma$ we could obtain $\lambda_{\gamma} = \frac{1}{4}$ instead of $\lambda_{\gamma} = \frac{1}{2}$ we computed in [Le 25, Paragraph 4.2.2]

In what follows, we will work with fixed values of $g, n$ and $L$, hence we shall only denote by $\mathbb{E}$ the Weil-Petersson expectation on $\mathcal{M}_{g,n}(L)$.

**1.2. The Weil-Petersson model for random hyperbolic surfaces.** A marked hyperbolic surface is a couple $(X, f)$, where $X$ is a hyperbolic surface with labelled geodesic boundaries $\beta_1, \dots, \beta_n$ of lengths $L_1, \dots, L_n$ respectively and $f: S_{g,n} \to X$ is a homeomorphism which preserves the orientation and the labelling of the boundaries. Two marked hyperbolic surfaces $(X, f)$ and $(Y, g)$ are said to be equivalent is $g \circ f^{-1}$ is isotopic to an isometry. The Teichmüller space $\mathcal{T}_{g,n}(L)$ is the set of equivalence classes of marked hyperbolic surfaces. This space is a manifold of real dimension $6g - 6 + 2n$ and carries a natural symplectic form $\omega_{WP}$ called the *Weil-Petersson form*.

The Teichmüller space is additionally equipped with an action of the mapping class group, which is defined by:

$$ \mathrm{MCG}_{g,n} = \mathrm{Homeo}^{+}(S_{g,n}, \partial S_{g,n}) / \mathrm{Homeo}_{\circ}^{+}(S_{g,n}, \partial S_{g,n}), $$

where $\mathrm{Homeo}^{+}(S_{g,n}, \partial S_{g,n})$ is the group of orientation preserving homeomorphisms that restrict to the identity on $\partial S_{g,n}$, and $\mathrm{Homeo}_{\circ}^{+}(S_{g,n}, \partial S_{g,n})$ is its neutral component, i.e. the subgroup of homeomorphisms isotopic to the identity relatively to $\partial S_{g,n}$. If we consider a surface $S$ whose genus and number of boundaries or punctures is not specified, we denote by $\mathrm{MCG}(S)$ its mapping class group. The *moduli space* is the quotient space:

$$ \mathcal{M}_{g,n}(L) = \mathcal{T}_{g,n}(L) / \mathrm{MCG}_{g,n}. $$

Unlike the Teichmüller space, $\mathcal{M}_{g,n}(L)$ is not a manifold, but an *orbifold*. However the action of $\mathrm{MCG}_{g,n}$ on $\mathcal{T}_{g,n}(L)$ is symplectic, hence the Liouville measure associated to the Weil-Petersson symplectic form $\omega_{WP}$ induces a well-defined measure $\mu_{WP}$ on $\mathcal{M}_{g,n}(L)$, called the *Weil-Petersson measure*. The moduli space $\mathcal{M}_{g,n}(L)$ has a finite volume $V_{g,n}(L)$. More precisely we have:

**Theorem 1.3 ([Mir07a]).** The Weil-Petersson volume $V_{g,n}(L)$ is a polynomial in the variables $L_1, \dots, L_n$ of total degree $6g - 6 + 2n$, and we can write:

$$ V_{g,n}(L) = \sum_{i_1 + \cdots + i_n \le 3g - 3 + n} C_{i_1 \cdots i_n} L_1^{2i_1} \cdots L_n^{2i_n}, $$
where $C_{i_1 \dots i_n} \in \pi^{6g-6+2n-2(i_1+\dots+i_n)} \mathbb{Q}_{>0}$.

When equipped with the renormalized measure $\frac{1}{\mathcal{V}_{g,n}(L)}\mu_{WP}$, the moduli space is a probability space.

**1.3. Integration of geometric random variables and average geodesic counting.** A loop is a continuous map $\gamma$ from $[0, 1]$ to a surface $X$ such that $\gamma(0) = \gamma(1)$. Two loops $\gamma_1, \gamma_2: [0, 1] \to X$ are freely homotopic if there exists a continuous map $h: [0, 1]^2 \to X$ such that $h_{|\{0\} \times [0, 1]} = \gamma_1$ and $h_{|\{1\} \times [0, 1]} = \gamma_2$ and such that for all $t \in [0, 1]$, $h_{|\{t\} \times [0, 1]}$ is a loop. We then write $\gamma_1 \simeq \gamma_2$. The free homotopy class of a loop $\gamma$ will be denoted by $[\gamma]$. A loop $\gamma$ is said to be primitive if we cannot write $\gamma \simeq \gamma'^m$ for some other loop $\gamma'$ and $m \ge 2$.

Let $\gamma$ be a loop on a surface $X$. We say that $\gamma$ is in minimal position if it is an immersion, if its self-intersection points are double points, and if $\gamma$ has the minimal amount of self-intersections among all such loops in the free homotopy class of $\gamma$. The loop $\gamma$ is essential if it is non-contractible and not homotopic to any of the boundaries and cusps of $X$.

A multi-loop is a family of loops $\Gamma = (\gamma_1, \dots, \gamma_k)$. Given a multi-loop $\Gamma = (\gamma_1, \dots, \gamma_k)$ made of essential loops on a hyperbolic surface $X$, there exists a unique family $\Gamma' = (\gamma'_1, \dots, \gamma'_k)$ of geodesics on $X$ such that $\Gamma$ is freely homotopic to $\Gamma'$ (i.e. such that $\gamma_i \simeq \gamma'_i$ for all $i \in \{1, \dots, k\}$). We denote by $\ell_{\Gamma}(X)$ the vector $(\ell_{\gamma_1}(X), \dots, \ell_{\gamma_k}(X))$ made of the lengths of the geodesic representatives of the free homotopy classes $[\gamma_1], \dots, [\gamma_k]$. This definition of $\ell_{\Gamma}(X)$ holds even if some of the loops $\gamma_i$ are not essential (with the convention that the length of a cusp or a contractible curve is 0). A simple loop, i.e. a loop with no self-intersection, will also be called a *curve*. A *multi-curve* is a family of curves $\Gamma = (\gamma_1, \dots, \gamma_k)$ where $\gamma_1, \dots, \gamma_k$ are disjoint essential curves, such that for any $i \neq j$, $\gamma_i$ is not homotopic to $\gamma_j$ or $\gamma_j^{-1}$.

Given a multi-loop $\Gamma = (\gamma_1, \dots, \gamma_k)$ on the base surface $S_{g,n}$, we define the length function associated to $\Gamma$, denoted $\ell_{\Gamma}: \mathcal{T}_{g,n}(L) \to \mathbb{R}^k$, the following way: given a marked hyperbolic surface, $\ell_{\Gamma}(X, f)$ is the length vector of the geodesic representative of $f(\Gamma)$ on $X$.

The mapping class group also acts on the set of free homotopy classes of multi-loops on $S_{g,n}$ by $[\varphi] \cdot [\Gamma] = [\varphi \circ \Gamma]$, and if $\Gamma$ is a multi-loop, we denote by $\mathcal{O}_{\Gamma}$ the orbit of the free homotopy class $[\Gamma]$, and $\text{Stab}([\Gamma])$ the stabilizer of $[\Gamma]$, i.e. the subgroup of $\text{MCG}_{g,n}$ made of the elements $[\varphi]$ such that $\varphi \circ \gamma$ is homotopic to $\gamma$.

In this article we will be interested in random variables on $M_{g,n}(L)$ defined the following way. Given a hyperbolic surface $X$, each compact set of $\mathbb{R}$ contains only a finite number of lengths of closed geodesics on $X$. Hence if $F: \mathbb{R}^k \to \mathbb{C}$ is a test function (e.g. compactly supported), we can consider the following sum:

$$ \sum_{[\Gamma'] \in \mathcal{O}_{\Gamma}} F(\ell_{\Gamma'}(X, f)), $$

where $(X, f)$ is a marked hyperbolic surface. Since the sum is made over the whole orbit of $\Gamma$, it does not depend on the choice of the marking homeomorphism $f$. This sum therefore induces a well-defined random variable on $M_{g,n}(L)$, which will be denoted by:

$$ F^{\gamma}: X \mapsto \sum_{[\Gamma'] \in \mathcal{O}_{\Gamma}} F(\ell_{\Gamma'}(X)), \qquad (1) $$

i.e. we omit the marking homeomorphism $f$.

**Remarks.**

* The notations in the sum (1) are abusive. Indeed $\Gamma'$ is not a multi-loop on $X$, hence we would only be able to consider $\ell_{\Gamma'}(X, f)$, for some marking homeomorphism $f: S_{g,n} \to X$, but as we said before, since the sum is made on the whole orbit, the value of $F^{\gamma}(X)$ does not depend on $f$.

* Let $\gamma$ be a closed loop on $S_{g,n}$. If we take $F = 1_{[0,a]}$ in (1), where $a \ge 0$, we retrieve the counting random variable $N_{\gamma}(a)$.
Mirzakhani provided an integration formula which enables us to integrate random variables of the form $F^\Gamma$, when $\Gamma$ is a multi-curve.

**Theorem 1.4** ([Mir07b]). Let $\Gamma = (\gamma_1, \dots, \gamma_k)$ be a multi-curve on $S_{g,n}$. If $F: \mathbf{R}^k \to \mathbf{C}$ is a test function, we have:

$$ \mathbb{E}[F^\gamma] = \int_{\mathbf{R}^k \subset 0} F(x_1, \dots, x_k) x_1 \cdots x_k V_\Gamma(x_1, \dots, x_k, L) \frac{dx_1 \cdots dx_k}{V_{g,n}(L)}, $$

where $V_\Gamma(x_1, \dots, x_k, L)$ is the volume of the moduli space of the complement<fnref n="1" /> $S_{g,n} \setminus \Gamma$ of the multi-curve $\Gamma$, where the two boundary components of $S_{g,n} \setminus \Gamma$ corresponding to the curve $\gamma_i$ have length $x_i$. If $S_{g,n} \setminus \Gamma$ is disconnected, then $V_\Gamma(x_1, \dots, x_k, L)^2$ is the product of the volumes of the moduli spaces of the connected components. In particular $V_\Gamma$ is a polynomial.

**Corollary 1.5.** If $\gamma$ is a closed curve (i.e. $\gamma$ is a simple loop) on $S_{g,n}$, then $\mathbb{E}[N_\gamma(a)]$ is a polynomial of degree $6g - 6 + 2n$ in the variable $a$.

Mirzakhani also computed an asymptotic equivalent of $N_\gamma(a)$ and $\mathbb{E}[N_\gamma(a)]$ when $a \to \infty$, for any loop $\gamma$ on $S_{g,n}$.

**Theorem 1.6** ([Mir16]). Let $\gamma$ be a closed loop on $S_{g,n}$. There exists a constant $n_\gamma \in \mathbf{Q}$ and a smooth and integrable function $B: M_{g,n}(L) \to \mathbf{R}_{\ge 0}$ such that for all $X \in M_{g,n}(L)$, we have:

$$ N_\gamma(a)(X) \underset{a \to \infty}{\sim} n_\gamma \frac{B(X)}{b_{g,n}} a^{6g-6+2n}, $$

$$ \text{where } b_{g,n} = \int_{M_{g,n}(L)} B d\mu_{WP}. $$

Mirzakhani proved in the same paper that this result can be integrated in order to obtain the following.

**Corollary 1.7** ([Mir16]). Let $\gamma$ be a closed loop on $S_{g,n}$. We have:

$$ \mathbb{E}[N_\gamma(a)] \underset{a \to \infty}{\sim} \frac{n_\gamma}{V_{g,n}(L)} a^{6g-6+2n}. $$

Erlandsson and Souto obtained similar results for geodesics on a once-holed torus in [ES16] by using a different approach, and Erlandsson, Parlier and Souto in [EPS20], Erlandsson and Souto in [ES24] and Trin in [Tri24] proved similar results for more general notions of lengths of loops. Given $a > 0$ and an integer $k \ge 0$, Sapir computed in [Sap16a; Sap16b] bounds on the number of closed geodesics of length at most $a$ and with at most $k$ self-intersections on any hyperbolic surface.

We also obtained in [Le 25] a more precise asymptotic expansion of $\mathbb{E}[N_\gamma(a)]$ when $a \to \infty$ for eight-shaped loops $\gamma$. However the tools we used to study the figure eight do not work anymore for more complex loops. The aim of this article is therefore to develop a new method suited to the study of any loop filling a pair of pants, in order to generalize the results of [Le 25]. Our further goal will be to generalize these results to even more complex loops.

Similarly to the results we obtained in [Le 25], even though our Corollary 1.2 improves the asymptotic equivalent of Corollary 1.7, we do not know if it is possible to deduce a better pointwise estimate than Theorem 1.6 from our Theorem 1.1.

<footnote n="1">¹By moduli space of the complement, we mean the moduli space of hyperbolic structures on $S_{g,n} \setminus \Gamma$.</footnote>

<footnote n="2">²We refer to [Mir07b] for a more expanded expression of $V_\Gamma(x_1, \dots, x_k, L)$.</footnote>
1.4. **Links with the work of Anantharaman and Monk.** The lengths of closed geodesics on a hyperbolic surface X are linked to the eigenvalues of the Laplace operator on X by the Selberg trace formula [Sel56]. Therefore, having a better understanding of the behavior of random variables of the form $F^\Gamma$ could help us understanding the spectral geometry of random hyperbolic surfaces.

Anantharaman and Monk exploited this fact in [AM24; AM25] in order to compute the optimal spectral gap for random hyperbolic surfaces of large genus. Since they consider closed surfaces of growing genus g, the notion of MCG$_{g,n}$-orbits of loops is not well-suited to their study. To remedy this problem they defined the notion of *local topological type* of a loop. Given a local topological type T, and a test function F, the *T-average of F over surfaces of genus g* is defined to be:

$$ \langle F \rangle_g^T = \mathbb{E} \left[ \sum_{\substack{\gamma \sim T \\ \gamma \in \mathcal{G}(X)}} F(\ell_\gamma(X)) \right], $$

where $\mathcal{G}(X)$ is the set of closed geodesics on $X \in \mathcal{M}_g$ and $\gamma \sim T$ means that $\gamma$ has local topological type T. Anantharaman and Monk proved in [AM24] that there exists $V_g^T \in L_{loc}^1(\mathbf{R}_{\ge 0})$, called the *volume function associated with local topological type T on surfaces of genus g*, such that for any test function F, we have:

$$ \langle F \rangle_g^T = \int_0^\infty F(\ell) V_g^T(\ell) d\ell. $$

Contrary to Anantharaman and Monk, we fix the topology of the base surface $S_{g,n}$ (i.e. we fix g and n), hence we are more interested in studying quantities of the form $\mathbb{E}[F^\gamma]$ rather than $\langle F \rangle_g^T$, i.e. we sort the loops according to their MCG$_{g,n}$-orbit and not their local topological type. However we can adapt some of the tools developed in [AM24; AM25] to our framework. We can therefore obtain the existence of functions $V_{\mathcal{O}_\gamma}(\cdot, L)$ in Theorem 1.1 by using similar arguments as for the existence of the volume functions $V_g^T$. In this article, we will be studying the large $\ell$ behavior of $V_{\mathcal{O}_\gamma}(\ell, L)$. The proof of Theorem 1.1 is also inspired from [AM25], contrary to [Le 25], which resorted to more explicit computations. We hope that our estimates will help us in the future having a better understanding of the spectral behavior of random hyperbolic surfaces of fixed topology.

1.5. **Length-type functions and pseudo-convolution.** The density function $V_{\mathcal{O}_\gamma}(\cdot, L)$ in Theorem 1.1 is a linear combination of *pseudo-convolution of polynomials*, obtained by disintegrating the measure $x_1 \dots x_n V_\Gamma(x_1, \dots, x_n, L) dx_1 \dots dx_n$ along the level sets of the length function $h_\gamma$ of the loop $\gamma$. One of the contributions of this paper is that we define the class of *length-type functions* and prove that (up to a diffeomorphism $\mathbf{R} \to \mathbf{R}$ close to $\mathrm{Id}_{\mathbf{R}}$ in a $\mathcal{C}^1$ sense) $h_\gamma$ is a length-type function. Length-type functions behave similarly to piecewise linear forms up to a $\mathcal{C}^1$-small error term. We then study the properties of length-type functions and we compute the asymptotic behavior of the pseudo-convolution of polynomials with respect to length-type functions.

The process of pseudo-convolution was introduced by Anantharaman and Monk [AM25] to compute the behavior at high genus of the volume functions $V_g^T$, and we used it in [Le 25] to study eight-shaped loops. The usual convolution of locally integrable functions $f_1, \dots, f_n$ consists in disintegrating the measure $f_1 \cdots f_n dx_1 \cdots dx_n$ along the level sets of the linear form $(x_1, \dots, x_n) \mapsto x_1 + \cdots + x_n$, and in a similar manner, the *pseudo-convolution of* $f_1, \dots, f_n$ consists in disintegrating $f_1 \cdots f_n dx_1 \cdots dx_n$ along the level set of a more general function $h: \mathbf{R}^n \to \mathbf{R}$ which behaves similarly to a (piecewise) linear form.

## 2. LENGTH FUNCTION OF A LOOP FILLING A PAIR OF PANTS

2.1. **Resolutions of a non-simple loop.** In order to describe the length of a non-simple loop $\gamma$ in Fenchel-Nielsen coordinates, we shall open the self-intersections of $\gamma$. This process is called the *resolution of $\gamma$ at its self-intersecting points*. We recall from [RY14] the properties of the resolution process and its applications in hyperbolic geometry.
**Definition 2.1** ([RY14]). Let $\gamma$ be a closed loop self-intersecting transversally at a point $p$. The resolutions of $\gamma$ at $p$ are the loops obtained by opening the self-intersection at $p$ by going along $\gamma$ towards $p$ and turning left (respectively right) at $p$.

<figure><img src="image_3.png" /><figcaption>FIGURE 1. The resolutions of a loop at the self-intersection point $p$.</figcaption></figure>

Since the resolutions of a loop $\gamma$ at a self-intersecting point can be taken in an arbitrarily small neighborhood of $\gamma$, its resolutions are homotopic to loops on the surface filled by $\gamma$.

**Definition 2.2.** Let $\gamma$ be a closed loop in minimal position on a hyperbolic surface $X$. The surface filled by $\gamma$, denoted by $S(\gamma)$ is defined the following way:

(i) we consider $\varepsilon > 0$ small enough so the regular neighborhood $\mathcal{V}_{\varepsilon}(\gamma) = \{x \in X | d(x, \gamma) < \varepsilon\}$ retracts to $\gamma$;

(ii) denoting by $C_1, \dots, C_k$ the connected components of $X \setminus \mathcal{V}_{\varepsilon}(\gamma)$ homeomorphic to a disk, we define:

$$S(\gamma) = \mathcal{V}_{\varepsilon}(\gamma) \cup \bigcup_{i=1}^{k} C_i.$$

*Remark.* We need the surface $X$ to be endowed with a hyperbolic metric only to be able to consider the regular neighborhood $\mathcal{V}_{\varepsilon}(\gamma)$. But the surface filled by $\gamma$ is well-defined up to isotopy, and it does not depend on the choice of the metric. Hence we can consider the surface filled by a loop $\gamma$ on the base surface $S_{g,n}$.

*Example.* The surface filled by a figure-eight is a pair of pants.

**Lemma 2.3** ([RY14]). Let $\gamma$ be a closed loop self-intersecting transversally at $p$. One way of resolving the self-intersection at $p$ splits $\gamma$ into two loops $\alpha, \alpha'$ and the other way of resolving $\gamma$ gives a single loop $\beta$.

The loops $\alpha, \alpha'$ are called the separating resolutions of $\gamma$ at $p$ and $\beta$ is called the non-separating resolution of $\gamma$ at $p$.
<figure><img src="image_4.png" /><figcaption>FIGURE 2. The resolutions of an eight-shaped loop. The curves $\alpha$ and $\alpha'$ are the separating resolutions and $\beta$ is the non-separating one.</figcaption></figure>

**Definition 2.4 ([RY14]).** Let $\gamma$ be a loop, $p$ be a self-intersection point of $\gamma$ and let $t_1, t_2 \in (0,1)$ such that $\gamma(t_1) = \gamma(t_2) = p$. If the subloop $\gamma_{|[t_1,t_2]}$ is homotopically trivial, we call it a *curl* of $\gamma$ and $p$ is its *vertex*.

A *Reidemeister move I on $\gamma$ at $p$* consists in removing a curl of $\gamma$ of vertex $p$.

If $\gamma$ is homotopically non-trivial, the *curling number of $\gamma$*, denoted by $c(\gamma)$ is the maximal number of Reidemeister moves I possible on $\gamma$. If $\gamma$ is homotopically trivial, $c(\gamma)$ is defined by adding 1 to the former definition.

<figure><img src="image_5.png" /><figcaption>FIGURE 3. A Reidemeister move I removing a curl of vertex $p$.</figcaption></figure>

**2.2. Length functions in Fenchel-Nielsen coordinates.** The length of a non-simple geodesic on a hyperbolic surface is closely related to the length of its resolutions. We can therefore describe an algorithm consisting in successively resolving a closed geodesic at its self-intersecting points in order to find an expression of its length in terms of lengths of simple closed geodesics.

**Lemma 2.5 ([RY14]).** Let $\gamma$ be a closed geodesic on a hyperbolic surface.

(i) $c(\gamma) = 0$.

(ii) The *curling number of the non-separating resolution of $\gamma$ at a self-intersection $p$ is at most 1*, and it is 1 if and only if there exists another self-intersection $p'$ of $\gamma$ such that the domain bounded by $\gamma$ between $p$ and $p'$ is a contractible triangle. The only possible case for it to happen is shown on Figure 4.
<figure><img src="image_6.png" /><figcaption>FIGURE 4. The only possible case for the curling number of the non-separating resolution of $\gamma$ at $p$ to be 1.</figcaption></figure>

**Lemma 2.6 ([RY14]).** Let $\gamma$ be a non-simple closed geodesic on a hyperbolic surface $X$ and let us denote by $\alpha, \alpha'$ the separating resolutions of $\gamma$ at one of its self-intersecting points $p$ and $\beta$ its non-separating resolution at $p$.

(i) If $c(\beta) = 0$, then:

$$ \cosh\left(\frac{\ell_{\gamma}(X)}{2}\right) = 2\cosh\left(\frac{\ell_{\alpha}(X)}{2}\right)\cosh\left(\frac{\ell_{\alpha'}(X)}{2}\right) + \cosh\left(\frac{\ell_{\beta}(X)}{2}\right). $$

(ii) If $c(\beta) = 1$, then:

$$ \cosh\left(\frac{\ell_{\gamma}(X)}{2}\right) = 2\cosh\left(\frac{\ell_{\alpha}(X)}{2}\right)\cosh\left(\frac{\ell_{\alpha'}(X)}{2}\right) - \cosh\left(\frac{\ell_{\beta}(X)}{2}\right). $$

**2.3. Length-type functions.** By using iteratively Lemma 2.6 on the resolutions of a self-intersecting loop, we can compute the length function of a loop filling a pair of pants. When written in Fenchel-Nielsen coordinates, such length functions share common properties which we will exploit in order to prove Theorem 1.1. This is the reason why we shall define and study the class of length-type functions.

From now on, for convenience of notations, we shall use the bold font $\mathbf{x}_n$ to denote the vector $(x_1, \dots, x_n)$, and $d\mathbf{x}_n = dx_1 \cdots dx_n$.

**Definition 2.7.** A function $h: \mathbf{R}^n \ge 0 \to \mathbf{R}$ is said to be *length-type* if there exists a finite generating set $B$ of the space $(\mathbf{R}^n)^*$ of linear forms on $\mathbf{R}^n$ satisfying $B = -B$, and positive numbers $\{a_T\}_{T \in B}$ such that:

$$ h = \log \left[ \sum_{T \in B} a_T e^T \right]. $$

Let $C \subset \{0 \le x_1 \le \dots \le x_n\}$ be a convex cone defined by a finite number of linear inequations. A function $h: C \to \mathbf{R}$ is said to be a simplified length-type function if $h$ is real-analytic on the interior of $C$, if it extends to a continuous function on the closure of $C$ and if there exist a linear form $\Lambda \in (\mathbf{R}^n)^*$ and a function $\psi: C \to \mathbf{R}$ such that:

(i) $h = \Lambda + \psi$;

(ii) for all $\mathbf{x}_n \in C$, we have $\Lambda(\mathbf{x}_n) \ge \frac{x_n}{2} \ge 0$;

(iii) there exists a constant $c > 0$ such that for all $1 \le i \le n$ and for all $\mathbf{x}_n \in C$, $|\frac{\partial \psi}{\partial x_i}(\mathbf{x}_n)| \le c e^{-x_i}$.

**Remark.** The class of length-type functions is stable under permutation of the variables, i.e. if $h: \mathbf{R}^n \to \mathbf{R}$ is a length-type function and $M \in \text{GL}_n(\mathbf{R})$ is a permutation matrix, then $h \circ M$ is a length-type function.

Length-type functions and simplified length-type functions are closely linked, as will be evidenced by Theorem 2.11. But let us prove first that, by resolving enough times a non-simple geodesic $\gamma$ filling a pair of pants on a hyperbolic surface $X$, we can eventually write $2\log(2\cosh(\ell_{\gamma}(X)/2))$ as a length-type function in Fenchel-Nielsen coordinates.
**Proposition 2.8.** Let $\gamma$ be a closed geodesic on a hyperbolic surface such that $S(\gamma)$ is a pair of pants and let $\alpha_1, \alpha_2$ and $\alpha_3$ be the connected components of $\partial S(\gamma)$. There exists a function $h_\gamma: \mathbb{R}_{\ge 0}^3 \to \mathbb{R}$ such that $H_\gamma = 2 \log(2 \cosh(\frac{h_\gamma}{2}))$ is a length-type function and:

$$ \ell_{\gamma}(X) = h_{\gamma}(\ell_{\alpha_1}(X), \ell_{\alpha_2}(X), \ell_{\alpha_3}(X)). $$

Moreover if $\varphi: X \to X$ is a homeomorphism, we have $\ell_{\varphi \circ \gamma}(X) = h_{\gamma}(\ell_{\varphi \circ \alpha_1}(X), \ell_{\varphi \circ \alpha_2}(X), \ell_{\varphi \circ \alpha_3}(X))$.

*Proof.* We prove this result by induction on the number of self-intersections of $\gamma$.

If $\gamma$ has exactly one self-intersection, let us denote by $\alpha, \alpha'$ its separating resolutions at its self-intersecting point and $\beta$ its non separating resolution. The three loops $\alpha, \alpha'$ and $\beta$ are simple, hence there exists $\sigma \in \mathfrak{S}_3$ such that $[\alpha, \alpha', \beta] = [\alpha_{\sigma(1)}, \alpha_{\sigma(2)}, \alpha_{\sigma(3)}]$ and we have:

$$ \begin{aligned} \ell_{\gamma}(X) &= 2\operatorname{argcosh}\left(2\cosh\left(\frac{\ell_{\alpha}(X)}{2}\right)\cosh\left(\frac{\ell_{\alpha'}(X)}{2}\right) + \cosh\left(\frac{\ell_{\beta}(X)}{2}\right)\right) \\ &= 2\operatorname{argcosh}\left(\cosh\left(\frac{\ell_{\alpha}(X) + \ell_{\alpha'}(X)}{2}\right) + \cosh\left(\frac{\ell_{\alpha}(X) - \ell_{\alpha'}(X)}{2}\right) + \cosh\left(\frac{\ell_{\beta}(X)}{2}\right)\right). \end{aligned} $$

Since the function

$$ H: (x, y, z) \mapsto 2\log\left[2\cosh\left(\frac{x+y}{2}\right) + 2\cosh\left(\frac{x-y}{2}\right) + 2\cosh\left(\frac{z}{2}\right)\right] $$

is length-type, the result is true if $\gamma$ has exactly one self-intersection.

Let us now assume that the result is true for geodesics with at most $k > 0$ self-intersections and let $\gamma$ be a geodesic with $k + 1$ self-intersections. Let $p$ be a self-intersecting point of $\gamma$. Same as before, let us denote by $\alpha$ and $\alpha'$ the separating resolutions of $\gamma$ at $p$ and $\beta$ its non-separating resolution. Two scenarios may then occur.

* $c(\beta) = 0$;

* $c(\beta) = 1$.

The only possible way to have $c(\beta) = 1$ is displayed on Figure 4. In this case there exists another self-intersecting point $p'$ of $\gamma$ such that the curling number of the non-separating resolution of $\gamma$ at $p'$ is 0. By resolving at $p'$ instead of $p$ we can therefore always assume that $c(\beta) = 0$. This way, according to the induction hypothesis and Lemma 2.6, we can write:

$$ 2 \cosh \left( \frac{\ell_{\gamma}(X)}{2} \right) = \left( \sum_{T \in B_{\alpha}} a_T e^{T(\ell_1, \ell_2, \ell_3)} \right) \left( \sum_{T' \in B_{\alpha'}} a_{T'} e^{T'(\ell_1, \ell_2, \ell_3)} \right) + \sum_{S \in B_{\beta}} b_S e^{S(\ell_1, \ell_2, \ell_3)}, \quad (2) $$

where $\ell_i = \ell_{\alpha_i}(X)$ for all $i \in \{1, 2, 3\}$, $B_\alpha, B_{\alpha'}$ and $B_\beta$ are subsets of $(\mathbb{R}^3)^*$ stable under multiplication by -1, and $\{a_T\}_{T \in B_\alpha}, \{a_{T'}\}_{T' \in B_{\alpha'}}, \{b_S\}_{S \in B_\beta} \subset \mathbb{R}_{>0}$. However the sets $B_\alpha, B_{\alpha'}$ and $B_\beta$ may not be generating sets of $(\mathbb{R}^3)^*$. Indeed for instance if $\alpha$ is homotopic to a simple (possibly non-primitive) geodesic then $S(\alpha)$ is a cylinder, in which case we cannot apply the induction hypothesis to $\alpha$.

If the three loops $\alpha, \alpha'$ and $\beta$ are homotopic to simple closed geodesics, there exist integers $m_1, m_2, m_3 > 0$ and a permutation $\sigma \in \mathfrak{S}_3$ such that:

$$ \cosh\left(\frac{\ell_{\gamma}(X)}{2}\right) = 2\operatorname{argcosh}\left(\cosh\left(\frac{m_{\sigma(1)}\ell_{\sigma(1)} + m_{\sigma(2)}\ell_{\sigma(2)}}{2}\right) + \cosh\left(\frac{m_{\sigma(1)}\ell_{\sigma(1)} - m_{\sigma(2)}\ell_{\sigma(2)}}{2}\right) + \cosh\left(\frac{m_{\sigma(3)}\ell_{\sigma(3)}}{2}\right)\right). $$
Hence, writing:

$$H_{\gamma}(x_1, x_2, x_3) = 2 \log \left[ 2 \cosh \left( \frac{m_{\sigma(1)}x_{\sigma(1)} + m_{\sigma(2)}x_{\sigma(2)}}{2} \right) + 2 \cosh \left( \frac{m_{\sigma(1)}x_{\sigma(1)} - m_{\sigma(2)}x_{\sigma(2)}}{2} \right) \right. \\ \left. + 2 \cosh \left( \frac{m_{\sigma(3)}x_{\sigma(3)}}{2} \right) \right],$$

we can conclude similarly to the case where $\gamma$ has one self-intersection.

In any cases, Expression (2) enables us to write:

$$2 \log \left[ 2 \cosh \left( \frac{\ell_{\gamma}(X)}{2} \right) \right] = 2 \log \left[ \sum_{T \in B_{\gamma}} c_T e^{T(\ell_1, \ell_2, \ell_3)} \right], \quad (3)$$

with $B_{\gamma} = (B_{\alpha} + B_{\alpha'}) \cup B_{\beta}$ and $\{c_T\}_{T \in B_{\gamma}} \subset \mathbf{R}_{>0}$. Hence if $\beta$ is not homotopic to a simple loop, then $\beta$ fills a pair of pants and $S(\beta) = S(\gamma)$. According to the induction hypothesis $B_{\beta}$ is a generating subset of $(\mathbf{R}^3)^*$, so is $B_{\gamma} \supset B_{\beta}$. Finally, if at least one of the loops $\alpha, \alpha'$ is not homotopic to a simple curve, then the set $B_{\alpha} + B_{\alpha'}$ is a generating subset of $(\mathbf{R}^3)^*$. We can therefore conclude in every cases that $B_{\gamma}$ is a generating set, which enables us to write:

$$\ell_{\gamma}(X) = h_{\gamma}(\ell_1, \ell_2, \ell_3),$$

where $2 \log(2 \cosh(\frac{h_{\gamma}}{2}))$ is a length-type function.

Given $p$ a self-intersection point of $\gamma, \alpha, \alpha'$ the separating resolutions of $\gamma$ at $p$ and $\beta$ the non-separating resolution such that $c(\beta) = 0$, and $\varphi: X \to X$ a homeomorphism, by resolving $\varphi \circ \gamma$ at $\varphi(p)$, we obtain:

$$\cosh \left( \frac{\ell_{\varphi \circ \gamma}(X)}{2} \right) = 2 \cosh \left( \frac{\ell_{\varphi \circ \alpha}(X)}{2} \right) \cosh \left( \frac{\ell_{\varphi \circ \alpha'}(X)}{2} \right) + \cosh \left( \frac{\ell_{\varphi \circ \beta}(X)}{2} \right),$$

hence by iterating the same process on the resolutions of $\gamma$, we can eventually write:

$$\ell_{\varphi \circ \gamma}(X) = h_{\gamma}(\ell_{\varphi \circ \alpha_1}(X), \ell_{\varphi \circ \alpha_2}(X), \ell_{\varphi \circ \alpha_3}(X)).$$

□

If $\gamma$ is a closed loop filling a pair of pants and if $\alpha_1, \alpha_2, \alpha_3$ are the connected components of $\partial S(\gamma)$, then some of the curves $\alpha_i$ may be homotopic to one another in the ambient surface $S_{g,n}$, or may be non-essential. This happens when $S(\gamma)$ is a pair of pants embedded inside a once-holed torus contained in $S_{g,n}$, or if some of the boundaries of $S(\gamma)$ are homotopic to boundaries of $S_{g,n}$. If $S(\gamma)$ is embedded inside a holed torus, then $\ell_{\gamma}$ is given by the restriction of $h_{\gamma}$ on planes of the form $\{x = y\}$. If some boundaries of $S(\gamma)$ are homotopic to boundaries of $S_{g,n}$, then $\ell_{\gamma}$ is given by the restriction of $h_{\gamma}$ on (intersections of) planes of the form $\{x = L_i\}$ with $1 \le i \le n$. The following lemma ensures that in each case, $2 \log(2 \cosh(\frac{\ell_{\gamma}}{2}))$ is a length-type function in Fenchel-Nielsen coordinates.

**Lemma 2.9.** Let $h: \mathbf{R}_{\ge 0}^n \to \mathbf{R}$ be a length-type function.

(i) For any $L \ge 0$ and $1 \le i \le n$, the function $\mathbf{x}_{n-1} \mapsto h(x_1, \dots, x_{i-1}, L, x_i, \dots, x_{n-1})$ obtained from $h$ by freezing $i$-th variable to $L$ is a length-type function.

(ii) The function $\mathbf{x}_{n-1} \mapsto h(x_1, \mathbf{x}_{n-1})$ obtained by restricting $h$ on the hyperplane $\{x_1 = x_2\}$ is a length-type function.

*Proof.* Let us denote:

$$h = \log \left[ \sum_{T \in B} a_T e^T \right]$$

(i) If $T \in B$, we denote $\hat{T}_i: \mathbf{x}_{n-1} \mapsto T(x_1, \dots, x_{i-1}, 0, x_i, \dots, x_{n-1})$. Then the set $\hat{B}_i = \{\hat{T}_i | T \in B\}$ is a generating subset of $(\mathbf{R}^{n-1})^*$ such that $-\hat{B}_i = \hat{B}_i$. Moreover, if for any $S \in \hat{B}_i$ we denote $b_S =$
$\sum_{\hat{T}_i=S} a_T e^{T(0, \dots, 0, L, 0, \dots, 0)} > 0$, we have for all $\mathbf{x}_{n-1} \in \mathbf{R}_{\ge 0}^{n-1}$:

$$h(x_1, \dots, x_{i-1}, L, x_i, \dots, x_n) = \log \left[ \sum_{S \in \hat{B}_i} b_S e^{S(\mathbf{x}_{n-1})} \right].$$

Hence $\mathbf{x}_{n-1} \mapsto h(x_1, \dots, x_{i-1}, L, x_i, \dots, x_n)$ is a length-type function.

(ii) If $T \in B$, we denote $T_1^2: \mathbf{x}_{n-1} \mapsto T(x_1, \mathbf{x}_{n-1})$ and $B_1^2 = \{T_1^2|T \in B\}$. Similarly to what we did before, $B_1^2$ is a generating subset of $(\mathbf{R}^{n-1})^*$ such that $B_1^2 = -B_1^2$, and when denoting for any $S \in B_1^2$, $c_S = \sum_{T_1^2=S} a_T$, we have:

$$h(x_1, \mathbf{x}_{n-1}) = \log \left[ \sum_{S \in B_j^i} c_S e^{S(\mathbf{x}_{n-1})} \right],$$

for all $\mathbf{x}_{n-1} \in \mathbf{R}_{\ge 0}^{n-1}$. Hence this defines a length-type function. $\square$

**Remark.** Since length-type functions are stable under permutations of the variables, using 2.9(ii), the restriction of length-type function to any hyperplane of the form $\{x_i = x_j\}$ is a length-type function.

When putting together Proposition 2.8 and Lemma 2.9, we obtain the following conclusion.

**Corollary 2.10.** We note $u: \ell \mapsto 2 \log(2 \cosh(\ell/2))$. Let $\gamma$ be a loop filling a pair of pants on $S_{g,n}$ and let $\alpha_1, \alpha_2, \alpha_3$ be the connected components of $\partial S_{g,n}$. We extract a maximal multi-curve from $(\alpha_1, \alpha_2, \alpha_3)$ and complete it into a pants decomposition $\mathcal{P}$ of $S_{g,n}$. When written into Fenchel-Nielsen coordinates associated to $\mathcal{P}$, $u \circ \ell_{\gamma}$ is a length-type function.

The key idea of the proof of our main result, similarly to what we did in [Le 25], is to exploit the analogy between the usual convolution of polynomials on $\mathbf{R}_{\ge 0}^n$ and the disintegration of a measure with polynomial density with respect to the Lebesgue measure along the level sets of length-type functions. In order to be able to discuss the behavior of such a disintegration, given a length-type function $h$ we shall partition $\mathbf{R}_{\ge 0}^n$ into cones where $h$ can be written as a simplified length-type function (up to a linear change of variables).

**Theorem 2.11.** Let $h: \mathbf{R}_{\ge 0}^n \to \mathbf{R}$ be a length-type function. There exist a partition of $\mathbf{R}_{\ge 0}^n$ into cones $U_1, \dots, U_m$ defined by a finite number of linear (strict and non-strict) inequations, and linear isomorphisms $\Phi_1, \dots, \Phi_m \in \mathrm{GL}(\mathbf{R}^n)$ such that each $\Phi_i$ maps the cone $U_i$ to a cone $C_i \subset \{0 \le x_1 \le \dots \le x_n\}$, and each $h \circ \Phi_i^{-1}$ is a simplified length-type function.

**Proof.** Let us denote:

$$h = \log \left[ \sum_{i=1}^{k} a_i e^{T_i} \right].$$

If $i_0 \in \{1, \dots, k\}$, we consider the cone:

$$V_{i_0} = \left\{ \max_{1 \le i \le k} T_i = T_{i_0} \right\} \cap \mathbf{R}_{\ge 0}^n.$$

We allow ourselves to define $V_{i_0}$ by using strict or non strict inequations in order for the family $(V_j)_{1 \le j \le k}$ to realize a partition of $\mathbf{R}_{\ge 0}^n$. Inside $V_{i_0}$, we can write:

$$h = T_{i_0} + \log a_{i_0} + \log \left[ 1 + \frac{1}{a_{i_0}} \sum_{i \ne i_0} a_i e^{-(T_{i_0} - T_i)} \right].$$

Let us note $S_{i_0, i} = T_{i_0} - T_i$ for all $i \ne i_0$ and $c_{i_0} = \log a_{i_0}$. Let us now check all the different possible ways of ordering the linear forms $S_{i_0, i}$ and $T_{i_0}$ inside $V_{i_0}$. Let $O$ be such an order, i.e. if $O = (S'_1, \dots, S'_k)$ is a permutation of the family $(S_{i_0, i}, T_{i_0})_{i \ne i_0}$, we denote:

$$U_O = \{0 < S'_1 < \dots < S'_k\},$$
where each $<$ is either $<$ or $\le$, chosen so that the cones of the form $U_O$ realize a partition of $V_{i_0}$.

From now on we chose an order $O$ such that the associated cone $U_O$ is non-empty. Since the set $B = \{T_1, \dots, T_k\}$ is a generating set of $(\mathbf{R}^n)^*$, we have $\langle S'_1, \dots, S'_k \rangle = (\mathbf{R}^n)^*$. Hence we can define:

$$ 
\begin{align*}
k_1 &= 1 \\
k_2 &= \min \{ j > k_1 | S'_j \notin \langle S'_1 \rangle \} \\
&\vdots \\
k_n &= \min \{ j > k_{n-1} | S'_j \notin \langle S'_1, \dots, S'_{k_{n-1}} \rangle \}.
\end{align*}
$$

The map $\Phi_O = (S'_{k_1}, \dots, S'_{k_n})$ is therefore a linear isomorphism. Moreover $\Phi_O$ maps the cone $U_O$ to a cone $C_O \subset \{0 \le x_1 \le \dots \le x_n\}$ and we have:

$$ \sum_{i \neq i_0} a_i e^{-S_{i_0, i}} = \sum_{p=1}^{n} \sum_{\substack{S'_i \in \langle S'_{k_1}, \dots, S'_{k_p} \rangle \\ S'_i \notin \langle S'_{k_1}, \dots, S'_{k_{p-1}} \rangle}} a'_i e^{-S'_i}, $$

where $(a'_i)$ is a well-chosen permutation of $(a_i)_{i \neq i_0}$. Let us denote:

$$ 
\begin{align*}
r_1 &= \frac{1}{a_{i_0}} \sum_{S'_i \in \langle S'_{k_1} \rangle} a'_i e^{-(S'_i - S'_{k_1})} \\
r_2 &= \frac{1}{a_{i_0}} \sum_{\substack{S'_i \in \langle S'_{k_1}, S'_{k_2} \rangle \\ S'_i \notin \langle S'_{k_1} \rangle}} a'_i e^{-(S'_i - S'_{k_2})} \\
&\vdots \\
r_n &= \frac{1}{a_{i_0}} \sum_{\substack{S'_i \in \langle S'_{k_1}, \dots, S'_{k_n} \rangle \\ S'_i \notin \langle S'_{k_1}, \dots, S'_{k_{n-1}} \rangle}} a'_i e^{-(S'_i - S'_{k_n})}.
\end{align*}
$$

Thus we have:

$$ h = T_{i_0} + c_{i_0} + \log \left[ 1 + \sum_{p=1}^{n} r_p e^{-S'_{k_p}} \right], $$

and for all $\mathbf{x}_n \in C_O$, we have:

$$ h \circ \Phi_O^{-1}(\mathbf{x}_n) = T_{i_0} \circ \Phi_O^{-1}(\mathbf{x}_n) + c_{i_0} + \log \left[ 1 + \sum_{p=1}^{n} r_p \circ \Phi_O^{-1}(\mathbf{x}_n) e^{-x_p} \right], \quad (4) $$

Let us denote $\Lambda_O = T_{i_0} \circ \Phi_O^{-1}$, $\psi_O = h \circ \Phi_O^{-1} - \Lambda_O$ and $\rho_p = r_p \circ \Phi_O^{-1}$ for all $1 \le p \le n$.

The function $h \circ \Phi_O^{-1}$ is analytic on the interior of $C_O$ and extends continuously to the closure of $C_O$. Moreover if $S'_{k_n} \neq T_{i_0}$, there exists $i \neq i_0$ such that $S'_{k_n} = T_{i_0} - T_i$. Hence we have $S'_{k_n} \le 2T_{i_0}$ on $U_O$, which implies that $\Lambda_O(\mathbf{x}_n) \ge \frac{x_n}{2}$ for all $\mathbf{x}_n \in C_O$. Since each function $\rho_p$ only depends on $\mathbf{x}_p$ and is a linear combination of exponential of negative linear forms on $C_O$, when derivating (4) we conclude that for every $1 \le p \le n$, there exists $c_p > 0$ such that for all $\mathbf{x}_n \in C_O$ we have $|\frac{\partial \psi_O}{\partial x_p}(\mathbf{x}_n)| \le c_p e^{-x_p}$.

Hence $h \circ \Phi_O^{-1}$ is a simplified length-type function. $\square$

<figure><img src="image_7.png" /></figure>
2.4. **An integration formula for geodesics filling a pair of pants.** Let $\gamma$ be a closed loop in minimal position such that $S(\gamma)$ is a pair of pants (see Definition 2.2 for the definition of the surface $S(\gamma)$ filled by $\gamma$), $\alpha_1, \alpha_2, \alpha_3$ be the connected components of $\partial S(\gamma)$, oriented so that $S(\gamma)$ is on the left of these curves, and $\Gamma = (\alpha_1, \alpha_2, \alpha_3)$. The components of $\Gamma$ are disjoint, however $\Gamma$ is not necessarily a multi-curve since some of its components may be non-essential or there may be $i \neq j$ such that $\alpha_i$ is homotopic to $\alpha_j^{-1}$ in $S_{g,n}$. To provide an integration formula for loops filling a pair of pants, our goal will be to use Mirzakhani's integration formula on a maximal multi-curve extracted from $\Gamma$. Given a test function $F: \mathbf{R} \to \mathbf{C}$, we first need to compare

$$F^{\gamma} = \sum_{[\gamma'] \in \mathcal{O}_{\gamma}} F \circ \ell_{\gamma'}$$

and

$$(F \circ h_{\gamma})^{\Gamma} = \sum_{[\alpha'_1, \alpha'_2, \alpha'_3] \in \mathcal{O}_{\Gamma}} F \circ h_{\gamma}(\ell_{\alpha'_1}, \ell_{\alpha'_2}, \ell_{\alpha'_3}).$$

Recall that $\mathcal{O}_{\gamma} = \{[\varphi \circ \gamma] | [\varphi] \in \text{MCG}_{g,n}\}$, $\text{Stab}([\gamma]) = \{[\varphi] \in \text{MCG}_{g,n} | [\varphi \circ \gamma] = [\gamma]\}$, and the (multi-)loops we consider are parametrized.

Let $G(\gamma)$ be the subgroup of $\text{MCG}_{g,n}$ generated by the Dehn twists around the connected components of $\partial S(\gamma)$ and by the mapping class group $\text{MCG}(S_{g,n} \setminus S(\gamma))$ of the complement of $S(\gamma)$. The group $G(\gamma)$ is a subgroup of the stabilizer $\text{Stab}([\gamma])$, we denote:

$$m(\gamma) = [\text{Stab}([\gamma]) : G(\gamma)].$$

**Lemma 2.12.** Let $\gamma$ be a loop such that $S(\gamma)$ is a pair of pants. The index $m(\gamma)$ is finite.

*Proof.* Let $\varphi: S_{g,n} \to S_{g,n}$ be an orientation-preserving homeomorphism such that $\varphi(\gamma)$ is homotopic to $\gamma$. Then (up to isotopy) $\varphi$ restricts to a homeomorphism of $S(\gamma)$, and therefore induces a permutation $\sigma_{\varphi} \in \mathfrak{S}_3$ of the connected components of $\partial S(\gamma)$. This induces a group morphism from $\text{Stab}([\gamma]) \subset \text{MCG}_{g,n}$ to $\mathfrak{S}_3$, whose kernel is $G(\gamma)$.

Indeed, if $\sigma_{\varphi} = \text{Id}$, then by composing by an ambient isotopy of $S_{g,n}$ we can assume without loss of generality that $\varphi$ restricts to the identity on $\partial S(\gamma)$, and the homotopy class $[\varphi|_{S(\gamma)}]$ of the restriction $\varphi|_{S(\gamma)}$ relatively to $\partial S(\gamma)$ is therefore an element of the mapping class group $\text{MCG}(S(\gamma))$ of $S(\gamma)$. Moreover since $S(\gamma)$ is a pair of pants, the group $\text{MCG}(S(\gamma))$ is generated by the Dehn twists around $\partial S(\gamma)$. Thus $[\varphi] \in G(\gamma)$. The inclusion of $G(\gamma)$ into the kernel of $[\varphi] \mapsto \sigma_{\varphi}$ is clear.

This proves that $[\text{Stab}([\gamma]) : G(\gamma)] \leq |\mathfrak{S}_3| = 6$. $\square$

**Corollary 2.13.** Let $\gamma$ be a closed loop filling a pair of pants and denote by $\alpha_1, \alpha_2, \alpha_3$ the connected components of $S(\gamma)$, oriented so that $S(\gamma)$ is on the left of these curves. If $\Gamma = (\alpha_1, \alpha_2, \alpha_3)$, then $G(\gamma) = \text{Stab}([\Gamma])$ and for every test function $F: \mathbf{R} \to \mathbf{C}$, we have:

$$F^{\gamma} = \frac{1}{m(\gamma)} (F \circ h_{\gamma})^{\Gamma}.$$

*Proof.* Given complete systems of representatives $\mathcal{U}$ and $\mathcal{V}$ of $\text{MCG}_{g,n}/\text{Stab}([\gamma])$ and $\text{Stab}([\gamma])/G(\gamma)$ respectively, the set $\mathcal{U}\mathcal{V} = \{[\varphi\psi] | [\varphi] \in \mathcal{U}, [\psi] \in \mathcal{V}\}$ is a complete system of representatives of $\text{MCG}_{g,n}/G(\gamma)$. It follows, according to Proposition 2.8, that:

$$(F \circ h_{\gamma})^{\Gamma} = \sum_{[\varphi] \in \mathcal{U}} \sum_{[\psi] \in \mathcal{V}} F \circ h_{\gamma} \circ \ell_{\varphi \circ \psi \circ \Gamma} \\ = \sum_{[\varphi] \in \mathcal{U}} \sum_{[\psi] \in \mathcal{V}} F \circ \ell_{\varphi \circ \psi \circ \gamma},$$
but since $\mathcal{V} \subset \text{Stab}([\gamma])$, we have $\ell_{\varphi \circ \psi \circ \gamma} = \ell_{\varphi \circ \gamma}$ for all $[\psi] \in \mathcal{V}$, hence:

$$ (F \circ h_{\gamma})^{\Gamma} = m(\gamma) \sum_{[\varphi] \in \mathcal{U}} F \circ \ell_{\varphi \circ \gamma} = m(\gamma) \sum_{[\gamma'] \in \mathcal{O}_{\gamma}} F \circ \ell_{\gamma'} = m(\gamma) F^{\gamma}. $$

□

**Example.** If $[\alpha_1, \alpha_2, \alpha_3]$ and $[\alpha_2, \alpha_1, \alpha_3]$ belong to the same MCG$_{g,n}$-orbit (e.g. if both $\alpha_1$ and $\alpha_2$ separate a one-handle), then the loop displayed on Figure 5 is an example of a loop $\gamma$ such that $m(\gamma) = 2$. Indeed, due to the symmetries of $\gamma$, any $[\varphi] \in \text{Stab}([\gamma])$ such that $[\varphi]$ induces a non-trivial permutation of $\{[\alpha_1], [\alpha_2], [\alpha_3]\}$ only exchanges $[\alpha_1]$ and $[\alpha_2]$, hence $m(\gamma) \le 2$. Since we assumed that $[\alpha_1, \alpha_2, \alpha_3]$ and $[\alpha_2, \alpha_1, \alpha_3]$ belong to the same orbit, such a $[\varphi]$ exists and we have $m(\gamma) = 2$.

<figure><img src="image_8.png" /><figcaption>FIGURE 5. A loop $\gamma$ with $m(\gamma) = 2$.</figcaption></figure>

**Definition 2.14.** Let $\gamma$ be a closed loop on $S_{g,n}$ such that $S(\gamma)$ is a pair of pants. We say that $\gamma$ exactly fills a pair of pants if the connected components of $\partial S(\gamma)$ are pairwise non homotopic. Otherwise we say that $\gamma$ almost fills a holed torus.

If $\gamma$ exactly fills a pair of pants, we say that $\gamma$ is of type $(\mathcal{P}_i)$, with $i \in \{0, 1, 2\}$, if $\partial S(\gamma)$ has exactly $i$ non-essential connected components.

If $\gamma$ almost fills a torus, we say that $\gamma$ is of type $(\mathcal{T}_j)$, with $j \in \{0, 1\}$, if $S(\gamma)$ has exactly $j$ non-essential connected component.

**Remark.** If the closed loop $\gamma$ is of type $(\mathcal{T}_1)$, then we have $S_{g,n} = S_{1,1}$.

Depending on the type of $\gamma$, using Mirzakhani's integration formula the expectation $\mathbb{E}[F^{\gamma}]$ can be written an integral of one, two or three variables.

**Theorem 2.15.** Let $\gamma$ be a loop on $S_{g,n}$ such that $S(\gamma)$ is a pair of pants. There exists a polynomial $Q_{\gamma}^{g,n,L}$ of at most three variables, depending only on the orbit of $\gamma$, such that for every test function $F: \mathbf{R} \to \mathbf{C}$:

(i) if $\gamma$ is of type $(\mathcal{P}_0)$, then $Q_{\gamma}^{g,n,L}$ has three variables and:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^3} F(h_{\gamma}(x, y, z)) Q_{\gamma}^{g,n,L}(x, y, z) \frac{dxdydz}{V_{g,n}(L)}; $$

(ii) if $\gamma$ is of type $(\mathcal{P}_1)$, then $Q_{\gamma}^{g,n,L}$ has two variables and there exists $1 \le i \le n$ such that, after having possibly permuted the variables:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}^2} F(h_{\gamma}(x, y, L_i)) Q_{\gamma}^{g,n,L}(x, y) \frac{dxdy}{V_{g,n}(L)}; $$
(iii) if $\gamma$ is of type $(\mathcal{P}_2)$, then $Q_{\gamma}^{g,n,L}$ has one variable and there exist $1 \le i, j \le n$ such that, after having possibly permuted the variables:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}} F(h_{\gamma}(x, L_i, L_j)) Q_{\gamma}^{g,n,L}(x) \frac{dx}{V_{g,n}(L)}; $$

(iv) if $\gamma$ is of type $(\mathcal{T}_0)$, then $Q_{\gamma}^{g,n,L}$ has two variables and after having possibly permuted the variables, we have:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{\ge 0}^2} F(h_{\gamma}(x, x, y)) Q_{\gamma}^{g,n,L}(x, y) \frac{dxdy}{V_{g,n}(L)}; $$

(v) if $\gamma$ is of type $(\mathcal{T}_1)$, then $g = n = 1$ and $Q_{\gamma}^{1,1,L_1}(x) = x$, and after having possibly permuted the variables, we have:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{>0}} F(h_{\gamma}(x, x, L_1)) Q_{\gamma}^{1,1,L_1}(x) \frac{dx}{V_{1,1}(L_1)}. $$

*Proof.* Let us note by $\alpha_1, \alpha_2$ and $\alpha_3$ the connected components of $\partial S(\gamma)$, oriented so that $S(\gamma)$ is on the left of these curves. We denote by $\Gamma$ the multi-curve obtained from $(\alpha_1, \alpha_2, \alpha_3)$ by removing the non-essential curves (if there are any), and if there exist $i_1, i_2$ such that $\alpha_{i_1}$ is homotopic to $\alpha_{i_2}^{-1}$, by removing $\alpha_{i_2}$. Let $k \in \{1, 2, 3\}$ be the number of components of $\Gamma$. According to Corollary 2.13 and Mirzakhani's integration formula, there exists an integer $m(\gamma)$ such that:

$$ \mathbb{E}[F^{\gamma}] = \frac{1}{m(\gamma)} \int_{\mathbf{R}_{\ge 0}^k} F \circ \tilde{h}_{\gamma}(\mathbf{x}_k) V_{\Gamma}(\mathbf{x}_k, L) \frac{d\mathbf{x}_k}{V_{g,n}(L)}, $$

where $\tilde{h}_{\gamma}$ is the restriction of $h_{\gamma}$ on some subspace of the form $\{x = L_i\}, \{x = y\}, \{x = L_i, y = L_j\}$ or $\{x = y, z = L_i\}$. $\square$

### 3. PSEUDO-CONVOLUTION OF POLYNOMIALS WITH RESPECT TO LENGTH-TYPE FUNCTIONS

**3.1. Disintegration and pseudo-convolution.** Let $h: \mathbf{R}_{\ge 0}^n \to \mathbf{R}$ be a measurable function such that for each compact subset $K \subset \mathbf{R}$, the set $h^{-1}(K)$ has finite measure, and $f \in L_{\text{loc}}^{1}(\mathbf{R}_{\ge 0}^n)$. When integrating $f$ on the level set $h^{-1}(\ell)$ defined by the implicit relation $\ell = h(\mathbf{x}_n)$, we denote by $\frac{d\mathbf{x}_n}{d\ell}$ the disintegration of the Lebesgue measure on the level sets of $h$. In particular, if $\gamma$ is a closed loop filling a pair of pants and $F: \mathbf{R} \to \mathbf{C}$ is a test function, we can therefore write:

$$ \mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}} F(\ell) \left( \int_{\ell=h(\mathbf{x}_k)} Q_{\gamma}^{g,n,L}(\mathbf{x}_k) \frac{d\mathbf{x}_k}{d\ell} \right) d\ell, $$

where $2\log(2\cosh(\frac{h}{2}))$ is a length-type function. This is the reason why in this section we study the disintegrations of measures with polynomials density against Lebesgue measure, with respect to length-type functions.

**Definition 3.1.** Let $C \subset \mathbf{R}_{\ge 0}^n$ be a cone defined by a finite number of linear inequations such that $C$ has a non-empty interior. If $\bar{1} \le i \le n$, we denote:

$$ \hat{C}_i = \{(x_j)_{j \neq i} \in \mathbf{R}_{\ge 0}^{n-1} | \exists x_i \in \mathbf{R}_{\ge 0}, (x_1, \dots, x_n) \in C\}. $$

Let $h: \hat{C} \to \mathbf{R}$ be a proper map. We say that $h$ satisfies Assumption $(\mathbf{H}_n^i)$ if $h$ is $\mathscr{C}^1$ and if for any $(x_j)_{j \neq i} \in \hat{C}_i$, the function:

$$ h_i = h_i^{(x_j)_{j \neq i}}: x_i \mapsto h(\mathbf{x}_n) $$

is a $\mathscr{C}^1$-diffeomorphism from an interval of the form $(X_0((x_j)_{j \neq i}), X_1((x_j)_{j \neq i}))$ onto its image, where $X_1((x_j)_{j \neq i})$ may be infinite. Its inverse will then be denoted by:

$$ \ell \mapsto h_i^{-1}(\ell, (x_j)_{j \neq i}). $$
If $h$ satisfies Assumption $(\mathbf{H}_n^i)$ for all $1 \le i \le n$, we say that $h$ satisfies Assumption $(\mathbf{H}_n)$.

**Proposition 3.2.** Let $h: \mathbf{R}_{>0}^n \to \mathbf{R}$ be a function satisfying Assumption $(\mathbf{H}_n^i)$. For any $\ell$ belonging in the image of $h$, the function $(x_j)_{j \ne i} \mapsto h_i^{-1}(\ell, (x_j)_{j \ne i})$ is defined on:

$$U_{\ell}^{i} = \{(x_j)_{j \ne i} \in \mathbf{R}_{>0}^{n-1} | h_{i}^{(x_j)_{j \ne i}}(X_0((x_j)_{j \ne i})) < \ell < h_{i}^{(x_j)_{j \ne i}}(X_1((x_j)_{j \ne i}))\}$$

if $h$ is an increasing function of $x_i$ and:

$$U_{\ell}^{i} = \{(x_j)_{j \ne i} \in \mathbf{R}_{>0}^{n-1} | h_{i}^{(x_j)_{j \ne i}}(X_1((x_j)_{j \ne i})) < \ell < h_{i}^{(x_j)_{j \ne i}}(X_0((x_j)_{j \ne i}))\}$$

otherwise. Moreover if $f \in L_{\text{loc}}^1(\mathbf{R}^n)$, we have for almost every $\ell \in \mathbf{R}$:

$$\int_{\ell=h(\mathbf{x}_n)} f(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} = \int_{U_{\ell}^i} f(x_1, \dots, x_{i-1}, h_i^{-1}(\ell, (x_j)_{j \ne i}), x_{i+1}, \dots, x_n) \left| \frac{\partial h_i^{-1}}{\partial \ell}(\ell, (x_j)_{j \ne i}) \right| \prod_{j \ne i} dx_j. \quad (5)$$

**Lemma 3.3** (Change of variables). Let $C \subset \mathbf{R}^n$ be an open set, $h: C \to \mathbf{R}$ be a proper and measurable function and $f \in L_{\text{loc}}^1(\mathbf{R}^n)$.

(i) Let $\Phi: U \to C$ be a $\mathcal{C}^1$-diffeomorphism, we have:

$$\int_{\ell=h(\mathbf{y}_n)} f(\mathbf{y}_n) \mathbf{1}_C(\mathbf{y}_n) \frac{d\mathbf{y}_n}{d\ell} = \int_{\ell=h \circ \Phi(\mathbf{x}_n)} f \circ \Phi(\mathbf{x}_n) |J_{\Phi}(\mathbf{x}_n)| \mathbf{1}_U(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell},$$

where $J_{\Phi}$ is the determinant of the Jacobian matrix of $\Phi$.

(ii) Let $u: \mathbf{R} \to \mathbf{R}$ be a $\mathcal{C}^1$-diffeomorphism. Let us denote $H = u \circ h$ and:

$$F(\ell) = \int_{\ell=h(\mathbf{x}_n)} f(\mathbf{x}_n) \mathbf{1}_C(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell},$$

and

$$G(\ell) = \int_{\ell=H(\mathbf{x}_n)} f(\mathbf{x}_n) \mathbf{1}_C(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell}.$$

We have $F = |u'|G \circ u$.

*Proof.* The point (i) immediately comes from the change of variables formula in $\mathbf{R}^n$. To prove the point (ii), let us consider a test function $\varphi: \mathbf{R} \to \mathbf{R}$. We have on a one hand, by definition:

$$\int_C \varphi(h(\mathbf{x}_n)) f(\mathbf{x}_n) d\mathbf{x}_n = \int_{\mathbf{R}} \varphi(\ell) F(\ell) d\ell,$$

and on the other hand, we have:

$$\begin{align*}
\int_C \varphi(h(\mathbf{x}_n)) f(\mathbf{x}_n) d\mathbf{x}_n &= \int_C \varphi \circ u^{-1}(H(\mathbf{x}_n)) f(\mathbf{x}_n) d\mathbf{x}_n \\
&= \int_{\mathbf{R}} \varphi \circ u^{-1}(\ell) G(\ell) d\ell \\
&= \int_{\mathbf{R}} \varphi(L) |u'(L)| G \circ u(L) dL,
\end{align*}$$

which makes us able to conclude. $\square$

We recall the definition of the pseudo-convolution from [AM25], which generalizes the usual convolution.

**Definition 3.4.** [AM25] Let $\varphi: \mathbf{R}^n \to \mathbf{R}$ be a measurable and locally bounded function and $h$ a measurable function defined on the support of $\varphi$ such that for every compact subset $K \subset \mathbf{R}$, the set $h^{-1}(K) \cap \{\varphi \ne 0\}$ has finite measure.
Let $f_1, \dots, f_n \in L_{\text{loc}}^1(\mathbf{R})$. The $(h, \varphi)$-convolution of $f_1, \dots, f_n$ is the pushforward measure $\mu$ of $\varphi \times \bigotimes_{i=1}^n f_i \, dx_i$ by the map $h$, i.e. for every test function $g$:

$$ \int_{\mathbf{R}} g d\mu = \int_{\mathbf{R}^n} g(h(x_1, \dots, x_n)) \varphi(x_1, \dots, x_n) f_1(x_1) \cdots f_n(x_n) dx_1 \cdots dx_n. $$

When the measure $\mu$ admits a density with respect to the Lebesgue measure, we will denote for almost every $\ell \in \mathbf{R}$:

$$ \frac{d\mu}{d\ell}(\ell) = f_1 \star \cdots \star f_n(\ell) \Big|_{\varphi}^h. $$

**3.2. Friedman-Ramanujan and sub-Friedman-Ramanujan functions.** The definition of Friedman-Ramanujan functions given by Anantharaman and Monk in [AM24] is not completely suited to our framework. Indeed in the following computations we shall see that it may be possible to find loops such that the exponential remainder term is not decaying like $(1+\ell)^c e^{-\frac{\ell}{2}}$, but like $(1+\ell)^c e^{-\nu\ell}$ for some $\nu \in (0, 1)$. This is the reason why we need to generalize the definition of Friedman-Ramanujan functions.

**Definition 3.5.** Let $\lambda \in (0, 1)$. A continuous function $f: \mathbf{R} \to \mathbf{C}$ belongs to the class $\mathcal{R}(\lambda)$ if there exist constants $c, c_1 > 0$ such that:

$$ \forall \ell \ge 0, |f(\ell)| \le c_1 (1+\ell)^c e^{\lambda \ell}. $$

We denote by $\mathcal{R}_w(\lambda)$ the space of locally integrable functions $f: \mathbf{R} \to \mathbf{C}$ such that for some constants $c, c_1 > 0$:

$$ \forall m \ge 1, \int_0^m |f(\ell)| d\ell \le c_1 (1+m)^c e^{\lambda m}. $$

The space of $\lambda$-Friedman-Ramanujan functions is the space $\mathcal{F}(\lambda) = e^\ell \mathbf{C}[\ell] \oplus \mathcal{R}(\lambda)$. The degree of a Friedman-Ramanujan function is the degree of its polynomial part.

We similarly define the space of weak $\lambda$-Friedman-Ramanujan functions to be the space $\mathcal{F}_w(\lambda) = e^\ell \mathbf{C}[\ell] \oplus \mathcal{R}_w(\lambda)$.

If $m \ge 0$, we denote $\mathcal{F}^m(\lambda) = e^\ell \mathbf{C}_{m-1}[\ell] \oplus \mathcal{R}(\lambda)$ and $\mathcal{F}_w^m(\lambda) = \mathbf{C}_{m-1}[\ell] \oplus \mathcal{R}_w(\lambda)$.

A function $f$ is said to be sub-$\lambda$-Friedman-Ramanujan if $\ell \mapsto e^\ell f(\ell)$ belongs to $\mathcal{F}(1-\lambda)$, and we denote $\mathcal{S}\mathcal{F}(\lambda) = e^{-\ell}\mathcal{F}(1-\lambda)$ and $\mathcal{S}\mathcal{F}^m(\lambda) = e^{-\ell}\mathcal{F}^{m+1}(1-\lambda)$. We similarly define the spaces $\mathcal{S}\mathcal{F}_w(\lambda)$ and $\mathcal{S}\mathcal{F}_w^m(\lambda)$ of weak sub-$\lambda$-Friedman-Ramanujan functions.

We finally denote $\mathcal{S}\mathcal{R}(\lambda) = e^{-\ell}\mathcal{R}(1-\lambda)$ and $\mathcal{S}\mathcal{R}_w(\lambda) = e^{-\ell}\mathcal{R}_w(1-\lambda)$.

**Remark.** With this definition, a $\lambda$-Friedman Ramanujan function is an $(e+1)$-Ramanujan function of order $e^\lambda$, as defined by Friedman in [Fri03], but with continuous variable.

**Definition 3.6.** We denote by $\mathcal{P}$ the operator consisting in taking the primitive vanishing at 0, and $\mathcal{L} = \text{Id} - \mathcal{P}$. When considering functions of several variables, we shall denote by $\mathcal{P}_{x_i}$ and $\mathcal{L}_{x_i}$ when we apply these operators with respect to the variable $x_i$ if there is an ambiguity on the variable.

If $f$ is defined on an interval of the form $[x_0, +\infty)$ with $x_0 > 0$, we extend it to be 0 on $[0, x_0)$, and $\mathcal{P}f$ is the primitive vanishing at 0 of this extension of $f$.

**Lemma 3.7 (Characterization of Friedman-Ramanujan functions).** Let $\lambda \in (0, 1)$ and $m \ge 0$. The function $f$ belongs to $\mathcal{F}^m(\lambda)$ (respectively $\mathcal{F}_w^m(\lambda)$) if and only if $\mathcal{L}^m f \in \mathcal{R}(\lambda)$ (respectively $\mathcal{R}_w(\lambda)$).

### 3.3. Convolution with respect to simplified length-type functions.

**3.3.1. Computations in one variable.** We assume in this paragraph that $x \mapsto h(x)$ is a $\mathcal{C}^1$-diffeomorphism from $(0, +\infty)$ to $(h_0, +\infty)$ and we denote by $h \mapsto x(h)$ its inverse.

**Lemma 3.8.** Let $0 < x_0 < x_1 \le \infty$ and let $f: \mathbf{R} \to \mathbf{R}$ be a $\mathcal{C}^1$ function.
(i) If $h$ is an increasing function, we have:

$$ \mathcal{L}_h \left[ e^h f \circ x(h) \mathbf{1}_{(x_0, x_1)} \circ x(h) \right] = e^{h(x_0)} f(x_0) \mathbf{1}_{(h(x_0), +\infty)}(h) - e^{h(x_1)} f(x_1) \mathbf{1}_{(h(x_1), +\infty)}(h) \\ + \mathcal{P}_h \left[ e^h f' \circ x(h) \frac{\partial x}{\partial h}(h) \mathbf{1}_{(x_0, x_1)} \circ x(h) \right]. $$

(ii) If $h$ is a decreasing function, we have:

$$ \mathcal{L}_h \left[ e^h f \circ x(h) \mathbf{1}_{(x_0, x_1)} \circ x(h) \right] = -e^{h(x_0)} f(x_0) \mathbf{1}_{(h(x_0), +\infty)}(h) + e^{h(x_1)} f(x_1) \mathbf{1}_{(h(x_1), +\infty)}(h) \\ + \mathcal{P}_h \left[ e^h f' \circ x(h) \frac{\partial x}{\partial h}(h) \mathbf{1}_{(x_0, x_1)} \circ x(h) \right]. $$

*Proof.* Let us assume that $h$ is increasing (the computations are very similar if $h$ is decreasing). We have:

$$ \begin{aligned} \mathcal{P}_h \left[ e^h f \circ x(h) \mathbf{1}_{(x_0, x_1)} \circ x(h) \right] &= \mathbf{1}_{(x_0, x_1)} \circ x(h) \int_{h(x_0)}^{h} e^t f \circ x(t) dt + \mathbf{1}_{(x_1, +\infty)} \circ x(h) \int_{h(x_0)}^{h(x_1)} e^t f(t) dt \\ &= \mathbf{1}_{(x_0, x_1)} \circ x(h) \left( e^h f \circ (h) - e^{h(x_0)} f(x_0) - \int_{h(x_0)}^{h} e^t f' \circ x(t) \frac{\partial x}{\partial h}(t) dt \right) \\ &\quad + \mathbf{1}_{(x_1, +\infty)} \circ x(h) \left( e^{h(x_1)} f(x_1) - e^{h(x_0)} f(x_0) - \int_{h(x_0)}^{h(x_1)} e^t f' \circ x(t) \frac{\partial x}{\partial h}(t) dt \right) \\ &= \mathbf{1}_{(x_0, x_1)} \circ x(h) e^h f \circ x(h) - \mathbf{1}_{(x_0, +\infty)} \circ x(h) e^{h(x_0)} f(x_0) \\ &\quad + \mathbf{1}_{(x_1, +\infty)} \circ x(h) e^{h(x_1)} f(x_1) - \mathcal{P}_h \left[ \mathbf{1}_{(x_0, x_1)} \circ x(h) e^h f' \circ x(h) \frac{\partial x}{\partial h}(h) \right]. \end{aligned} $$

Hence we conclude by using the fact that $\mathcal{L}_h = \text{Id} - \mathcal{P}_h$. $\square$

### 3.3.2. Convolution in $(A, +\infty)^n$.

**Theorem 3.9.** Let $C \subset \{0 \le x_1 \le \dots \le x_n\}$ be a cone defined by a finite amount of linear inequations and $h: C \to \mathbf{R}$ be a simplified length-type function. Let $A \ge 0$ and $i \in \{1, \dots, n\}$ such that $h$ satisfies $(\mathbf{H}_n^i)$ on $C^A = C \cap (A, +\infty)^n$. There exist $(n-1)$-dimensional cones $C_1, \dots, C_p \subset \{0 \le x_1 \le \dots \le x_{n-1}\}$ defined by a finite number of linear inequations, simplified length-type functions $h_1, \dots, h_p$, linear forms $Y^1, \dots, Y^p \in (\mathbf{R}^{n-1})^*$ and $1 \le q \le p$ such that for any $\eta \in \mathbf{R}^*$ and any $\mathcal{C}^1$-functions $f_1, \dots, f_n$ on $\mathbf{R}$, we have for all $\ell \ge 0$:

$$ \mathcal{L}_{\ell} \left[ e^{\ell} \left( f_1 \star \cdots \star f_n(\ell) \right) \Big|_{\mathbf{1}_{C_A}}^{h} \right] = \frac{1}{\eta} \left[ \xi_i(\ell) + \chi_{\eta}(\ell) + \sum_{m=1}^{q} \theta_m(\ell) - \sum_{m=q+1}^{p} \theta_m(\ell) \right], $$

where:

$$ \theta_m(\ell) = \mathcal{P}_\ell \left[ e^\ell \int_{\ell = h_m(\mathbf{x}_{n-1})} f_i\left(Y^m(\mathbf{x}_{n-1})\right) \prod_{j \neq i} f_j(x_j) \mathbf{1}_{C_m \cap (A, +\infty)^{n-1}}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell} \right], $$

$$ \xi_i(\ell) = \mathcal{P}_\ell \left[ e^\ell \left( f_1 \star \cdots \star f_{i-1} \star f'_i \star f_{i+1} \star \cdots \star f_n(\ell) \right) \Big|_{\mathbf{1}_{C_A}}^h \right], $$

$$ \chi_{\eta}(\ell) = \mathcal{L}_{\ell} \left[ e^{\ell} \left( f_1 \star \cdots \star f_n(\ell) \right) \Big|_{\left(\eta - \frac{\partial h}{\partial x_i}\right) \mathbf{1}_{C_A}}^{h} \right]. $$

*Remark.* The constant $\eta$ will later be chosen as a well-suited coefficient of the linear part of $h$.

*Proof.* Let us denote $h = \Lambda + \psi$, where $\Lambda$ is a linear form on $\mathbf{R}^n$, and $c > 0$ such that:

$$ \forall 1 \le j \le n, \forall \mathbf{x}_n \in C, \left| \frac{\partial \psi}{\partial x_j}(\mathbf{x}_n) \right| \le c e^{-x_j} $$
For convenience of notations we assume that $i = n$. Let $T_1, \dots, T_k \in (\mathbf{R}^n)^*$ such that:

$$C = \{\mathbf{x}_n \in \mathbf{R}_{\ge 0}^n | \forall 1 \le j \le k, T_j(\mathbf{x}_n) > 0\}.$$

Here $>=$ is either $\ge$ or $>$ depending on the inequality used to define $C$. Let us denote $T_j: \mathbf{x}_n \mapsto a_{j1}x_1 + \dots + a_{jn}x_n$ for all $1 \le j \le k$ and:

$$I = \{j | a_{jn} = 0\}$$

$$J_0 = \{j | a_{jn} > 0\}$$

$$J_1 = \{j | a_{jn} < 0\},$$

and:

$$X_0: \mathbf{x}_{n-1} \mapsto \max_{j \in J_0} \left\{ \frac{-a_{j1}x_1 - \dots - a_{j,n-1}x_{n-1}}{a_{jn}} \right\}$$

$$X_1: \mathbf{x}_{n-1} \mapsto \min_{j \in J_1} \left\{ \frac{-a_{j1}x_1 - \dots - a_{j,n-1}x_{n-1}}{a_{jn}} \right\}.$$

If $J_1 = \emptyset$ we let $X_1 = +\infty$. We finally note:

$$C_0 = \left( \bigcap_{j \in I} \{\mathbf{x}_{n-1} | T_j(\mathbf{x}_{n-1}) > 0\} \right) \cap \{\mathbf{x}_{n-1} | 0 < X_0(\mathbf{x}_{n-1}) < X_1(\mathbf{x}_{n-1})\} \subset \mathbf{R}_{\ge 0}^{n-1}.$$

Here we use the notation $T_j(\mathbf{x}_{n-1})$ even if $T_j$ is a linear form on $\mathbf{R}^n$, but we do it only for linear forms $T_j$ with $j \in I$, i.e. such that $a_{jn} = 0$, thus these linear forms do not depend on $x_n$.

The functions $X_0$ and $X_1$ are piecewise linear. Let $C_1, \dots, C_q \subset C_0$ be the subcones on which $X_0$ is linear and $C_{q+1}, \dots, C_p \subset C_0$ the subcones on which $X_1$ is linear.

With these notations, we have:

$$\mathbf{x}_n \in C \iff \mathbf{x}_{n-1} \in C_0 \text{ and } X_0(\mathbf{x}_{n-1}) < x_n < X_1(\mathbf{x}_n).$$

We also denote $h_m: \mathbf{x}_{n-1} \in C_m \mapsto h(\mathbf{x}_{n-1}, X_0(\mathbf{x}_{n-1}))$ for $1 \le m \le q$ and $h_m: \mathbf{x}_{n-1} \in C_m \mapsto h(\mathbf{x}_{n-1}, X_1(\mathbf{x}_{n-1}))$ for $q+1 \le m \le p$.

Since $C \subset \{0 \le x_1 \le \dots \le x_n\}$, we can assume without loss of generality that $\mathbf{x}_n \mapsto x_1$ and $\mathbf{x}_n \mapsto x_{j+1} - x_j$ belong to $\{T_1, \dots, T_k\}$ for all $1 \le j \le n-1$. In particular the linear form $\mathbf{x}_n \mapsto x_n - x_{n-1}$ is one of $T_1, \dots, T_k$, which implies that $X_0(\mathbf{x}_{n-1}) \ge x_{n-1}$ (and therefore $X_1(\mathbf{x}_{n-1}) \ge x_{n-1}$). This implies in turn that the $h_m$'s are simplified length-type functions. Indeed, if $m \le q$ the linear part of $h_m$ is $\mathbf{x}_{n-1} \mapsto \Lambda(\mathbf{x}_{n-1}, X_0(\mathbf{x}_{n-1})), \text{ and for all } \mathbf{x}_{n-1} \in C_m \text{ we have:}$

$$\Lambda(\mathbf{x}_{n-1}, X_0(\mathbf{x}_{n-1})) \ge \frac{X_0(\mathbf{x}_{n-1})}{2} \ge \frac{x_{n-1}}{2},$$

and for all $1 \le j < n$:

$$\frac{\partial}{\partial x_j} \psi(\mathbf{x}_{n-1}, X_0(\mathbf{x}_{n-1})) = \frac{\partial \psi}{\partial x_j}(\mathbf{x}_{n-1}, X_0(\mathbf{x}_{n-1})) + \frac{\partial X_0}{\partial x_j} \frac{\partial \psi}{\partial x_n}(\mathbf{x}_{n-1}, X_0(\mathbf{x}_{n-1})),$$

hence:

$$\left| \frac{\partial}{\partial x_j} \psi(\mathbf{x}_{n-1}, X_0(\mathbf{x}_{n-1})) \right| \le c e^{-x_j} + c \left| \frac{\partial X_0}{\partial x_j} \right| e^{-X_0(\mathbf{x}_{n-1})} \le c \left( 1 + \left| \frac{\partial X_0}{\partial x_j} \right| \right) e^{-x_j}.$$

The same goes on if $m > q$ by replacing $X_0$ by $X_1$.

With this in mind, let us prove the result.

According to Proposition 3.2, we can write:

$$f_1 \star \cdots \star f_n(\ell)\Big|_{1_{C^A}}^h = \int_{U_\ell^n} \prod_{j=1}^{n-1} f_j(x_j) f_n(h_n^{-1}(\ell, \mathbf{x}_{n-1})) 1_{C^A}(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1})) \left| \frac{\partial h_n^{-1}}{\partial \ell}(\ell, \mathbf{x}_{n-1}) \right| d\mathbf{x}_{n-1}.$$
The function $h$ satisfies $(\mathbf{H}_n^n)$, hence $\frac{\partial h_n^{-1}}{\partial \ell} \neq 0$. Let us assume for example that $\frac{\partial h_n^{-1}}{\partial \ell} > 0$. For any $\eta \neq 0$, we have:

$$ \frac{\partial h_n^{-1}}{\partial \ell}(\ell, \mathbf{x}_{n-1}) = \frac{1}{\eta} \left( 1 + \frac{\partial h_n^{-1}}{\partial \ell}(\ell, \mathbf{x}_{n-1}) \left( \eta - \frac{\partial h}{\partial x_n}(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1})) \right) \right), $$

hence:

$$ e^{\ell} \left( f_1 \star \cdots \star f_n(\ell) \right)_{1^{C^A}}^{h} = \frac{e^{\ell}}{\eta} \int_{U_{\ell}^n} \prod_{j=1}^{n-1} f_j(x_j) f_n(h_n^{-1}(\ell, \mathbf{x}_{n-1})) 1_{C^A}(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1})) d\mathbf{x}_{n-1} \quad (6) $$

$$ + \frac{e^{\ell}}{\eta} \int_{\ell=h(\mathbf{x}_n)} \prod_{j=1}^{n} f_j(x_j) \left( \eta - \frac{\partial h}{\partial x_n}(\mathbf{x}_n) \right) 1_{C^A}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell}. \quad (7) $$

When applying the operator $\mathcal{L}_\ell$, the term (7) gives the term $\frac{1}{\eta}\chi_\eta(\ell)$. Let us now focus on the term (6). Let $C_0^A = C_0 \cap (A, +\infty)^{n-1}$. When applying Lemma 3.8, we obtain:

$$
\begin{align}
\mathcal{L}_{\ell} \left[ \int_{U_{\ell}^{n}} e^{\ell} \prod_{j=1}^{n-1} f_j(x_j) f_n(h_n^{-1}(\ell, \mathbf{x}_{n-1})) 1_{C^A}(\mathbf{x}_{n-1}, h_n^{-1}(\ell, \mathbf{x}_{n-1})) d\mathbf{x}_{n-1} \right] \nonumber \\
&= \int_{U_{\ell}^{n}} \prod_{j=1}^{n-1} f_j(x_j) 1_{C_0^A}(\mathbf{x}_{n-1}) \mathcal{L}_{\ell} \left[ e^{\ell} f_n \circ h_n^{-1}(\ell, \mathbf{x}_{n-1}) 1_{(X_0(\mathbf{x}_{n-1}), X_1(\mathbf{x}_{n-1}))} \circ h_n^{-1}(\ell, \mathbf{x}_{n-1}) \right] d\mathbf{x}_{n-1} \nonumber \\
&= \int_{U_{\ell}^{n}} \prod_{j=1}^{n-1} f_j(x_j) 1_{C_0^A}(\mathbf{x}_{n-1}) e^{h(\mathbf{x}_{n-1}, X_0(\mathbf{x}_{n-1}))} f_n(X_0(\mathbf{x}_{n-1})) 1_{\{h(\mathbf{x}_{n-1}, X_0(\mathbf{x}_{n-1})) < \ell\}} d\mathbf{x}_{n-1} \tag{8} \\
&\quad - \int_{U_{\ell}^{n}} \prod_{j=1}^{n-1} f_j(x_j) 1_{C_0^A}(\mathbf{x}_{n-1}) e^{h(\mathbf{x}_{n-1}, X_1(\mathbf{x}_{n-1}))} f_n(X_1(\mathbf{x}_{n-1})) 1_{\{h(\mathbf{x}_{n-1}, X_1(\mathbf{x}_{n-1})) < \ell\}} d\mathbf{x}_{n-1} \tag{9} \\
&\quad + \int_{U_{\ell}^{n}} \prod_{j=1}^{n-1} f_j(x_j) 1_{C_0^A}(\mathbf{x}_{n-1}) \mathcal{P}_{\ell} \left[ e^{\ell} f'_n \circ h_n^{-1} \frac{\partial h_n^{-1}}{\partial \ell}(\ell, \mathbf{x}_{n-1}) 1_{(X_0(\mathbf{x}_{n-1}), X_1(\mathbf{x}_{n-1}))} \circ h_n^{-1}(\ell, \mathbf{x}_{n-1}) \right] d\mathbf{x}_{n-1}. \tag{10}
\end{align}
$$

Hence, by disintegrating along the level sets of the functions $h_1, \dots, h_p$, the terms (8) and (9) respectively give:

$$ \sum_{m=1}^{q} \int_{0}^{\ell} e^{t} \left( \int_{t=h_{m}(\mathbf{x}_{n-1})} f_{n}(X_{0}(\mathbf{x}_{n-1})) \prod_{j=1}^{n-1} f_{j}(x_{j}) 1_{C_{m}^{A}}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{dt} \right) dt, \quad (11) $$

and:

$$ - \sum_{m=q+1}^{p} \int_{0}^{\ell} e^{t} \left( \int_{t=h_{m}(\mathbf{x}_{n-1})} f_{n}(X_{1}(\mathbf{x}_{n-1})) \prod_{j=1}^{n-1} f_{j}(x_{j}) 1_{C_{m}^{A}}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{dt} \right) dt. \quad (12) $$

The terms (11) and (12) therefore give the following term:

$$ \mathcal{P}_{\ell} \left[ \sum_{m=1}^{q} e^{\ell} \int_{\ell=h_m(\mathbf{x}_{n-1})} f_n(X_0(\mathbf{x}_{n-1})) \prod_{j=1}^{n-1} f_j(x_j) 1_{C_m \cap (A, +\infty)^{n-1}}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell} \right. \\ \left. - \sum_{m=q+1}^{p} e^{\ell} \int_{\ell=h_m(\mathbf{x}_{n-1})} f_n(X_1(\mathbf{x}_{n-1})) \prod_{j=1}^{n-1} f_j(x_j) 1_{C_m \cap (A, +\infty)^{n-1}}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell} \right]. $$

Finally, the term (10) gives:

$$ \mathcal{P}_{\ell} \left[ e^{\ell} \int_{\ell=h(\mathbf{x}_n)} f_1(x_1) \cdots f_{n-1}(x_{n-1}) f'_n(x_n) 1_{C^A}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right]. $$
Putting these terms together we obtain:

$$ \mathcal{L}_{\ell} \left[ e^{\ell} \left( f_1 \star \cdots \star f_n (\ell) \right) \Big|_{1_{CA}}^{h} \right] = \frac{1}{\eta} \left[ \xi_n(\ell) + \chi_{\eta}(\ell) + \sum_{m=1}^{q} \theta_m(\ell) - \sum_{m=q+1}^{p} \theta_m(\ell) \right]. $$

□

**Lemma 3.10.** Let $C \subset \{0 \le x_1 \le \cdots \le x_n\}$ be a cone defined by a finite number of linear inequations and $h: C \to \mathbf{R}$ be a simplified length-type function. There exist $1 \le i \le n$, $A \ge 0$, $\eta \ne 0$ and $0 < \lambda < 1$ such that $h$ satisfies Assumption $(\mathbf{H}_n^i)$ on $C^A$ and for every polynomial functions $f_1, \dots, f_n$, we have:

$$ f_1 \star \cdots \star f_n \Big|_{(\eta - \frac{\partial h}{\partial x_i}) 1_{CA}}^h \in \mathcal{SR}_w(\lambda). $$

*Proof.* Let us denote $h = \Lambda + \psi$, where $\Lambda \in (\mathbf{R}^n)^*$, and:

$$ i = \max \left\{ j \in \{1, \dots, n\} \middle| \frac{\partial \Lambda}{\partial x_j} \neq 0 \right\}, $$

and $\eta = \frac{\partial \Lambda}{\partial x_i}$ (which is not necessary positive). The reason why we chose this value of $i$ is because $x_i$ is the largest variable involved in the expression of $\Lambda$ in the cone $C$, which will make us able to have a majoration of $\Lambda(\mathbf{x}_n) - x_i$. For $A \ge 0$ large enough, $h$ therefore satisfies Assumption $(\mathbf{H}_n^i)$. Moreover on the one hand there exists a constant $c_0 > 0$ such that:

$$ \Lambda - c_0 \le h \le \Lambda + c_0, $$

and on the other hand, since $h$ is a simplified length-type function, there exists a constant $c > 0$ such that for all $\mathbf{x}_n \in C$, we have:

$$ \left| \frac{\partial \psi}{\partial x_i}(\mathbf{x}_n) \right| \le c e^{-x_i}. $$

Hence if $a \ge 0$, we have:

$$ \begin{aligned} & \int_0^a e^\ell \left| \int_{\ell=h(\mathbf{x}_n)} \prod_{i=1}^n f_j(x_j) \left( \eta - \frac{\partial h}{\partial x_i}(\mathbf{x}_n) \right) 1_{CA}(\mathbf{x}_n) \frac{d\mathbf{x}_n}{d\ell} \right| d\ell \\ & \le e^{c_0} \int_{CA} 1_{\{h(\mathbf{x}_n) \le a\}} e^{\Lambda(\mathbf{x}_n)} \left| \prod_{j=1}^n f_j(x_j) \frac{\partial \psi}{\partial x_i}(\mathbf{x}_n) \right| d\mathbf{x}_n \\ & \le c e^{c_0} \int_{CA} 1_{\{h(\mathbf{x}_n) \le a\}} e^{\Lambda(\mathbf{x}_n) - x_i} \left| \prod_{j=1}^n f_j(x_j) \right| d\mathbf{x}_n. \end{aligned} $$

Let:

$$ M = \max_{1 \le j \le n} \frac{\partial \Lambda}{\partial x_j}. $$

Since $\Lambda$ is positive on $C$, we have $M > 0$, and for all $\mathbf{x}_n \in C$, we have:

$$ \Lambda(\mathbf{x}_n) \le M(x_1 + \cdots + x_i) \le iMx_i. $$

If $\frac{1}{iM} < 1$, we denote $\nu = 1 - \frac{1}{iM}$, and if $\frac{1}{iM} \ge 1$, we let $\nu$ be any number in $(0, 1)$. In each case, we have for all $\mathbf{x}_n \in C$:

$$ \Lambda(\mathbf{x}_n) - x_i \le \nu \Lambda(\mathbf{x}_n). $$
This, in addition to the fact that $\Lambda(\mathbf{x}_n) \ge \frac{x_n}{2}$, yields:

$$ \int_{C^A} \mathbb{1}_{\{h(\mathbf{x}_n) \le a\}} e^{\Lambda(\mathbf{x}_n) - x_i} \left| \prod_{j=1}^{n} f_j(x_j) \right| d\mathbf{x}_n \le \int_{C^A} \mathbb{1}_{\{h(\mathbf{x}_n) \le a\}} e^{\nu\Lambda(\mathbf{x}_n)} \left| \prod_{j=1}^{n} f_j(x_j) \right| d\mathbf{x}_n \\ \le e^{\nu(c_0+a)} \int_{C^A} \mathbb{1}_{\{\frac{x_n}{2} \le a+c_0\}} \left| \prod_{j=1}^{n} f_j(x_j) \right| d\mathbf{x}_n. $$

Thus there exist constants $K, c' > 0$ such that:

$$ \int_0^a e^\ell \left| f_1 \star \cdots \star f_n(\ell) \right|^{\frac{h}{(\eta - \frac{\partial h}{\partial x_i}) \mathbb{1}_{C^A}}} d\ell \le K(1+a)^{c'} e^{\nu a}. $$

We can therefore conclude that $f_1 \star \cdots \star f_n|_{(\eta - \frac{\partial h}{\partial x_i}) \mathbb{1}_{C^A}}^{\frac{h}{(\eta - \frac{\partial h}{\partial x_i}) \mathbb{1}_{C^A}}} \in \mathcal{SR}_w(\lambda)$ with $\lambda = 1 - \nu$. $\square$

**Lemma 3.11.** Let $C \subset \{0 \le x_1 \le \cdots \le x_n\}$ be a cone defined by a finite number of linear inequations and $h: C \to \mathbf{R}$ be a simplified length-type function. Let $1 \le i \le n$, $\eta \ne 0$ and $A \ge 0$ be chosen as in Lemma 3.10, and $C_1, \dots, C_p, h_1, \dots, h_p, Y^1, \dots, Y^p$ and $1 \le q \le p$ be chosen as in Theorem 3.9. If $f_1, \dots, f_n$ are polynomial functions, we denote:

$$ \Theta_i^k(\ell) = \sum_{m=1}^q e^\ell \int_{\ell=h_m(\mathbf{x}_{n-1})} f_i^{(k)}(Y^m(\mathbf{x}_{n-1})) \prod_{j \ne i} f_j(x_j) \mathbb{1}_{C_m^A}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell} \\ - \sum_{m=q+1}^p e^\ell \int_{\ell=h_m(\mathbf{x}_{n-1})} f_i^{(k)}(Y^m(\mathbf{x}_{n-1})) \prod_{j \ne i} f_j(x_j) \mathbb{1}_{C_m^A}(\mathbf{x}_{n-1}) \frac{d\mathbf{x}_{n-1}}{d\ell}. $$

If $K_i = \deg f_i$, there exist $0 < \lambda < 1$ which depends only on $h$, and $r \in \mathcal{R}_w(\lambda)$ such that:

$$ \mathcal{L}_{\ell}^{K_i+1} \left[ e^{\ell} \left( f_1 \star \cdots \star f_n(\ell) \right) \bigg|_{\mathbb{1}_{C^A}}^{h} \right] = \sum_{k=0}^{K_i} \frac{1}{\eta^{k+1}} \mathcal{L}_{\ell}^{K_i-k} \mathcal{P}_{\ell}^{k+1} \Theta_i^k(\ell) + r(\ell). $$

*Proof.* We prove by induction that there exists $0 < \lambda < 1$ such that for any $K \ge 0$, there exists $r_K \in \mathcal{R}_w(\lambda)$ such that we have:

$$ \mathcal{L}_{\ell}^{K+1} \left[ e^{\ell} \left( f_1 \star \cdots \star f_n(\ell) \right) \bigg|_{\mathbb{1}_{C^A}}^{h} \right] = \frac{1}{\eta^{K+1}} \mathcal{P}_{\ell}^{K+1} \left[ e^{\ell} \left( f_1 \star \cdots \star f_{i-1} \star f_i^{(K+1)} \star f_{i+1} \star \cdots \star f_n(\ell) \right) \bigg|_{\mathbb{1}_{C^A}}^{h} \right] \\ + \sum_{k=0}^{K} \frac{1}{\eta^{k+1}} \mathcal{L}_{\ell}^{K-k} \mathcal{P}_{\ell}^{k+1} \Theta_i^k(\ell) + r_K(\ell). \qquad (13) $$
According to Theorem 3.9 and Lemma 3.10, the result is true when $K = 0$. Now if we assume it to be true for some $K \ge 0$, by applying the induction hypothesis and Theorem 3.9 we have:

$$
\begin{align*}
\mathcal{L}_{\ell}^{K+2} \left[ e^{\ell} \left( f_1 \star \cdots \star f_n(\ell) \right) \Big|_{1_{CA}}^{h} \right] &= \frac{1}{\eta^{K+1}} \mathcal{P}_{\ell}^{K+1} \mathcal{L}_{\ell} \left[ e^{\ell} \left( f_1 \star \cdots \star f_{i-1} \star f_i^{(K+1)} \star f_{i+1} \star \cdots \star f_n(\ell) \right) \Big|_{1_{CA}}^{h} \right] \\
&\quad + \sum_{k=0}^{K} \frac{1}{\eta^{k+1}} \mathcal{L}_{\ell}^{K+1-k} \mathcal{P}_{\ell}^{k+1} \Theta_i^k(\ell) + \mathcal{L}_{\ell} r_K(\ell) \\
&= \frac{1}{\eta^{K+2}} \mathcal{P}_{\ell}^{K+2} \left[ e^{\ell} \left( f_1 \star \cdots \star f_{i-1} \star f_i^{(K+2)} \star f_{i+1} \star \cdots \star f_n(\ell) \right) \Big|_{1_{CA}}^{h} \right] \\
&\quad + e^{\ell} \left( f_1 \star \cdots \star f_{i-1} \star f_i^{(K+1)} \star f_{i+1} \star \cdots \star f_n(\ell) \right) \Big|_{\left(\eta - \frac{\partial h}{\partial x_i}\right)_{1_{CA}}}^{h} \right] \\
&\quad + \frac{1}{\eta^{K+2}} \mathcal{P}_{\ell}^{K+2} \Theta_i^{K+1}(\ell) + \sum_{k=0}^{K} \frac{1}{\eta^{k+1}} \mathcal{L}_{\ell}^{K+1-k} \mathcal{P}_{\ell}^{k+1} \Theta_i^k(\ell) + \mathcal{L}_{\ell} r_K(\ell)
\end{align*}
$$

According to Lemma 3.10, we have:

$$
\ell \mapsto e^{\ell} \left( f_1 \star \cdots \star f_{i-1} \star f_i^{(K+1)} \star f_{i+1} \star \cdots \star f_n(\ell) \right) \Big|_{\left(\eta - \frac{\partial h}{\partial x_i}\right)_{1_{CA}}}^{h} \in \mathcal{R}_w(\lambda),
$$

hence if we denote:

$$
r_{K+1}: \ell \mapsto \mathcal{L}_{\ell} r_K(\ell) + \frac{1}{\eta^{K+2}} \mathcal{P}_{\ell}^{K+2} \left[ e^{\ell} \left( f_1 \star \cdots \star f_{i-1} \star f_i^{(K+1)} \star f_{i+1} \star \cdots \star f_n(\ell) \right) \Big|_{\left(\eta - \frac{\partial h}{\partial x_i}\right)_{1_{CA}}}^{h} \right],
$$

we can conclude by induction that (13) is true for all $K \ge 0$. We complete the proof by taking $K = K_i$. $\square$

**Corollary 3.12.** Let $C \subset \{0 \le x_1 \le \cdots \le x_n\}$ be a cone defined by a finite number of linear inequations and $h: C \to \mathbf{R}$ be a simplified length-type function. There exists $0 < \lambda < 1$ such that for every polynomial functions $f_1, \dots, f_n$ of degree $K_1, \dots, K_n$ respectively, we have:

$$
f_1 \star \cdots \star f_n \Big|_{1_{CA}}^h \in S \mathcal{F}_w^{K_1+\cdots+K_n+n-1}(\lambda).
$$

*Proof.* We prove the result by induction on $n$. Let us first assume that $n = 1$. In this case we either have $C = \mathbf{R}_{\ge 0}$ or $C = \mathbf{R}_{> 0}$ depending on the type of inequality used to define $C$. Since $h$ is a simplified length-type function, there exist constants $\lambda \ge \frac{1}{2}$ and $c, \ell_0 \ge 0$ such that for all $\ell > \ell_0$ we have:

$$
h^{-1}(\ell) = \frac{\ell - c}{\lambda} + O\left(e^{-\frac{\ell}{\lambda}}\right).
$$

If $f_1$ is a polynomial, we have:

$$
f_1 \Big|_{1_{(A, +\infty)}}^h = f_1 \circ h^{-1} \cdot (h^{-1})' 1_{(\ell_0, +\infty)}.
$$

Hence $f_1|_{1_{(A, +\infty)}}^h$ is a sub-Friedman-Ramanujan function with same degree as $f_1$.

Let us now assume that the result is true for all rank $k \in \{1, \dots, n-1\}$ and let $f_1, \dots, f_n$ be polynomials of degree $K_1, \dots, K_n$ respectively. According to Lemma 3.11, there exists $i \in \{1, \dots, n\}$ such that we can write $\mathcal{L}_{\ell}^{K_i+1} [e^{\ell} (f_1 \star \cdots \star f_n(\ell))|_{1_{CA}}^h]$ as a linear combination of terms of the form:

$$
\mathcal{P}_{\ell}^{k+1} \mathcal{L}_{\ell}^{K_i-k} \left[ e^{\ell} \int_{\ell=\tilde{h}(\mathbf{x}_n)} f_i^{(k)}(Y(\mathbf{x}_{n-1})) \prod_{j \neq i} f_j(x_j) \mathbf{1}_{\tilde{C}^A}(\mathbf{x}_n) \frac{d\mathbf{x}_{n-1}}{d\ell} \right],
$$

where $0 \le k \le K_i$ and $\tilde{C} \subset \{0 \le x_1 \le \cdots \le x_{n-1}\}$ is a cone defined by a finite number of linear
inequations, $Y$ is a linear form and $\tilde{h}: \tilde{C} \to \mathbf{R}$ is a simplified length-type function. The function
$\mathbf{x}_{n-1} \mapsto f_i^{(k)}(Y(\mathbf{x}_{n-1})) \prod_{j \neq i} f_j(x_j)$ is therefore a polynomial function of the variable $\mathbf{x}_{n-1}$ of total degree $K_1 + \cdots + K_n - k$. Hence, by linearity and by using our induction hypothesis, we obtain $0 < \lambda < 1$ such that:

$$ \ell \mapsto e^{\ell} \int_{\ell = \tilde{h}(\mathbf{x}_n)} f_i^{(k)}(Y(\mathbf{x}_{n-1})) \prod_{j \neq i} f_j(x_j) \mathbf{1}_{\tilde{C}^A}(\mathbf{x}_n) \frac{d\mathbf{x}_{n-1}}{d\ell} \in \mathcal{F}_w^{K_1+\cdots+K_n-k+n-2}(1-\lambda). $$

We can finally conclude that $f_1 \star \cdots \star f_n|_{\mathbf{1}_{C^A}}^h \in \mathcal{S}\mathcal{F}_w^{K_1+\cdots+K_n+n-1}(\lambda)$. $\square$

### 3.3.3. Convolution in $[0, A]^j \times (A, +\infty)^{n-j}$.

**Lemma 3.13.** Let $C \subset \{0 \le x_1 \le \cdots \le x_n\}$ be a cone defined by a finite number of linear inequations and $h: C \to \mathbf{R}$ be a simplified length-type function. If $A \ge 0$ is large enough, for any $1 \le j \le n$ and any $f_1, \dots, f_n \in L_{\text{loc}}^1(\mathbf{R})$, the convolution $f_1 \star \cdots \star f_n|_{\mathbf{1}_{C \cap [0, A]^j \times (A, +\infty)^{n-j}}}^h$ exists.

*Proof.* Let us denote $h = \Lambda + \psi$, where $\Lambda$ is a linear form on $\mathbf{R}^n$.

If $0 < j < n$ and if $\frac{\partial \Lambda}{\partial x_i} \neq 0$ for some $i > j$, then we can disintegrate with respect to the $i$-th variable if $A$ is large enough. Else, let $1 \le i \le j$, for any $\hat{x}_i = (x_k)_{k \neq i} \in [0, A]^{j-1} \times (A, +\infty)^{n-j}$ we denote by $[a(\hat{x}_i), b(\hat{x}_i)]$ the interval on which $h_i = h_{i}^{\hat{x}_i}$: $x_i \mapsto h(\mathbf{x}_n)$ is defined and $t_1(\hat{x}_i), \dots, t_{m(\hat{x}_i)}(\hat{x}_i)$ the points on which $h_{i}^{\hat{x}_i}$ admits a local extremum. Since $h$ is analytic on the interior of $C$, $m(\hat{x}_i)$ is finite. We denote $t_0(\hat{x}_i) = a(\hat{x}_i), t_{m(\hat{x}_i)+1} = b(\hat{x}_i)$ and for all $1 \le s \le m(\hat{x}_i)$, we denote by $\ell \mapsto h_{i,s}^{-1}(\ell, \hat{x}_i)$ the inverse of $h_i$ on $[t_s(\hat{x}_i), t_{s+1}(\hat{x}_i)]$.

Similarly to what we did in the proof of Theorem 3.9, there exists a cone $C_0 \subset \mathbf{R}_{\ge 0}^{n-1}$ defined by a finite number of linear inequations such that:

$$ \mathbf{x}_n \in C \iff \hat{x}_i \in C_0 \text{ and } a(\hat{x}_i) < x_i < b(\hat{x}_i), $$

where each $<$ is either $\le$ or $<$, depending on the fact that the inequalities used to define $C$ are strict or not.

If $\varphi: \mathbf{R} \to \mathbf{R}$ is a compactly supported $\mathcal{C}^{\infty}$ function, we have:

$$ \begin{aligned} & \int_{C \cap [0, A]^j \times (A, +\infty)^{n-j}} \varphi(h(\mathbf{x}_n)) \prod_{k=1}^{n} f_k(x_k) d\mathbf{x}_n \\ &= \int_{C_0} \prod_{k \neq i} f_k(x_k) \left( \sum_{s=0}^{m(\hat{x}_i)} \int_{t_s(\hat{x}_i)}^{t_{s+1}(\hat{x}_i)} \varphi(h(\mathbf{x}_n)) f_i(x_i) dx_i \right) \prod_{k \neq i} dx_k \\ &= \int_{\mathbf{R}} \varphi(\ell) \left( \int_{C_0} \prod_{k \neq i} f_k(x_k) \sum_{s=0}^{m(\hat{x}_i)} f_i(h_{i,s}^{-1}(\ell, \hat{x}_i)) \left| \frac{\partial h_{i,s}^{-1}}{\partial \ell}(\ell, \hat{x}_i) \right| \mathbf{1}_{\{t_s(\hat{x}_i) < \ell < t_{s+1}(\hat{x}_i)\}} \prod_{k \neq i} dx_k \right) d\ell. \end{aligned} $$

This yields that for all $\ell \in \mathbf{R}$:

$$ \begin{aligned} & f_1 \star \cdots \star f_n(\ell)|_{\mathbf{1}_{C \cap [0, A]^j \times (A, +\infty)^{n-j}}}^h \\ &= \int_{C_0} \prod_{k \neq i} f_k(x_k) \sum_{s=0}^{m(\hat{x}_i)} f_i(h_{i,s}^{-1}(\ell, \hat{x}_i)) \left| \frac{\partial h_{i,s}^{-1}}{\partial \ell}(\ell, \hat{x}_i) \right| \mathbf{1}_{\{t_s(\hat{x}_i) < \ell < t_{s+1}(\hat{x}_i)\}} \prod_{k \neq i} dx_k. \end{aligned} $$

$\square$

**Lemma 3.14.** Let $C \subset \{0 \le x_1 \le \cdots \le x_n\}$ be a cone defined by a finite number of linear inequations and $h: C \to \mathbf{R}$ be a simplified length-type function. There exist $0 < \lambda < 1$ and $A \ge 0$ such that for any
0 ≤ j ≤ n and any polynomial functions f₁, ..., fₙ of degree K₁, ..., Kₙ respectively, we have:

$$f_1 \star \cdots \star f_n \Big|_{\mathbb{I}_{C \cap [0, A]^j \times (A, +\infty)^{n-j}}}^h \in \mathcal{S} \mathcal{F}_w^{K_{j+1}+\cdots+K_n+n-j-1}(\lambda).$$

*Proof.* Let us denote h = Λ + ψ with Λ ∈ (Rⁿ)\* and:

$$i = \max \left\{ k \in \{1, \dots, n\} \middle| \frac{\partial \Lambda}{\partial x_k} \neq 0 \right\}.$$

If j ≥ i, then the convolution $f_1 \star \cdots \star f_n|_{\mathbb{I}_{C \cap [0, A]^j \times (A, +\infty)^{n-j}}}^h$ has compact support. Indeed, Λ only depends on the variables x₁, ..., xᵢ, hence Λ is bounded on $C \cap [0, A]^j \times (A, +\infty)^{n-j}$. Thus h is bounded on $C \cap [0, A]^j \times (A, +\infty)^{n-j}$, which implies that the convolution is compactly supported.

It is therefore sufficient to consider the convolution with j < i. If we take A ≥ 0 large enough, h satisfies Assumption (Hᵢ) and we can apply Theorem 3.9 with respect to this variable. Moreover the proofs of Lemmas 3.10 and 3.11 hold for this choice of i. Thus we can write $\mathcal{L}_\ell^{K_i+1}[e^\ell(f_1 \star \cdots \star f_n|_{\mathbb{I}_{C \cap [0, A]^j \times (A, +\infty)^{n-j}}}^h)]$ as a linear combination of elements of $\mathcal{R}_w(\lambda)$ (for some 0 < λ < 1) and of images under powers of $\mathcal{L}$ and $\mathcal{P}$ of pseudo-convolutions of the form:

$$e^\ell(\tilde{f}_1 \star \cdots \star \tilde{f}_{n-1}(\ell)|_{\mathbb{I}_{\tilde{C} \cap [0, A]^j \times (A, +\infty)^{n-1-j}}}^{\tilde{h}}), \quad (14)$$

where $\tilde{C} \subset \{0 \le x_i \le \cdots \le x_{n-1}\}$ is an $(n-1)$-dimensional cone defined by a finite amount of linear inequations, $\tilde{h}: \tilde{C} \to \mathbb{R}$ is a simplified length-type function and $\tilde{f}_1, \dots, \tilde{f}_n$ are polynomials.

If the linear part of $\tilde{h}$ only depends on its j first variables, the convolution (14) has compact support and therefore belongs to $\mathcal{R}_w(\lambda)$. If the linear part of $\tilde{h}$ depends on one of the variables $x_{j+1}, \dots, x_n$, we repeat the same process as above. At each step the number of variables that do not stay in a compact decreases, which implies that after applying the operator $\mathcal{L}$ on $\ell \mapsto e^\ell(f_1 \star \cdots \star f_n(\ell)|_{\mathbb{I}_{C \cap [0, A]^j \times (A, +\infty)^{n-j}}}^h)$ at most $K_{j+1} + \cdots + K_n + n - j$ times, we obtain a function with compact support.

This yields:

$$\ell \mapsto e^\ell \left( f_1 \star \cdots \star f_n(\ell) \Big|_{\mathbb{I}_{C \cap [0, A]^j \times (A, +\infty)^{n-j}}}^h \right) \in \mathcal{F}_w^{K_{j+1}+\cdots+K_n+n-j}(\lambda),$$

which in turn enables us to conclude.

<figure><img src="image_9.png" /></figure>

We immediately deduce the following:

**Corollary 3.15.** Let $C \subset \{0 \le x_1 \le \cdots \le x_n\}$ be a cone defined by a finite number of linear inequations and h: $C \to \mathbb{R}$ be a simplified length-type function. There exists 0 < λ < 1 such that for any polynomial functions $f_1, \dots, f_n$ of degree $K_1, \dots, K_n$ respectively, we have:

$$f_1 \star \cdots \star f_n \Big|_{\mathbb{I}_C}^h \in \mathcal{S} \mathcal{F}_w^{K_1+\cdots+K_n+n-1}(\lambda).$$

*Proof.* Since $C \subset \{0 \le x_1 \le \cdots \le x_n\}$, we have:

$$f_1 \star \cdots \star f_n \Big|_{\mathbb{I}_C}^h = \sum_{j=0}^{n} f_1 \star \cdots \star f_n \Big|_{\mathbb{I}_{C \cap [0, A]^j \times (A, +\infty)^{n-j}}}^h.$$

We conclude by using Corollary 3.12 and Lemma 3.14.

<figure><img src="image_10.png" /></figure>
**3.4. Convolution with respect to length-type functions.** We are now able to compute the asymptotic behavior of the convolution of polynomials with respect to length-type functions.

**Theorem 3.16.** Let $h: \mathbf{R}_{\ge 0}^n \to \mathbf{R}$ be a length-type function. There exists $0 < \lambda < 1$ such that for every polynomial functions $f_1, \dots, f_n$ of degree $K_1, \dots, K_n$ respectively, we have:

$$f_1 \star \cdots \star f_n \Big|_{\mathbf{1}_{\mathbf{R}_{\ge 0}^n}}^h \in \mathcal{S}\mathcal{F}_w^{K_1+\cdots+K_n+n-1}(\lambda)$$

*Proof.* It is sufficient to prove the result with $f_i: x \mapsto x^{K_i}$ for all $1 \le i \le n$.

According to Theorem 2.11, we partition $\mathbf{R}_{\ge 0}^n$ into cones $U_1, \dots, U_m$ defined by a finite number of linear inequations, and we consider linear isomorphisms $\Phi_1, \dots, \Phi_m \in \mathrm{GL}(\mathbf{R}^n)$ such that each $h_m = h \circ \Phi_m^{-1}$ is a simplified length-type function defined over $C_i = \Phi_i(U_i)$. We therefore have:

$$x_1^{K_1} \star \cdots \star x_n^{K_n} \Big|_{\mathbf{1}_{\mathbf{R}_{\ge 0}^n}}^h = \sum_{k=1}^m x_1^{K_1} \star \cdots \star x_n^{K_n} \Big|_{\mathbf{1}_{U_k}}^h.$$

For any $1 \le k \le m$, let us denote $\Phi_k = (S_{k1}, \dots, S_{kn})$. The functions $S_{ki}$ are linear forms and the Jacobian determinant $J_{\Phi_k}$ is constant. According to Lemma 3.3, we have for all $\ell \in \mathbf{R}$:

$$x_1^{K_1} \star \cdots \star x_n^{K_n}(\ell) \Big|_{\mathbf{1}_{U_k}}^h = \frac{1}{|J_{\Phi_k}|} \int_{\ell=h_k(\mathbf{y}_n)} \prod_{i=1}^n S_{ki}(\mathbf{y}_n)^{K_i} \mathbf{1}_{C_k}(\mathbf{y}_n) \frac{d\mathbf{y}_n}{d\ell}.$$

Thus we can write $x_1^{K_1} \star \cdots \star x_n^{K_n} \Big|_{\mathbf{1}_{U_k}}^h$ as a linear combination of terms of the form $y_1^{N_1} \star \cdots \star y_n^{N_n} \Big|_{\mathbf{1}_{C_k}}^{h_k}$, where $N_1 + \cdots + N_n = K_1 + \cdots + K_n$, which enables us to conclude. $\square$

**4. CONCLUSION**

**Theorem 4.1.** Let $\gamma$ be a closed loop on $S_{g,n}$ such that $S(\gamma)$ is a pair of pants. There exist $0 < \lambda_\gamma < 1$ and a function $V_{\mathcal{O}_\gamma}(\cdot, L) \in \mathcal{S}\mathcal{F}_w^{6g-7+2n}(\lambda_\gamma)$ such that for every test function $F: \mathbf{R} \to \mathbf{C}$, we have:

$$\mathbb{E}[F^{\gamma}] = \int_{0}^{\infty} F(\ell) V_{\mathcal{O}_{\gamma}}(\ell, L) \frac{d\ell}{V_{g,n}(L)}.$$

*Proof.* Let us denote $u: \ell \mapsto 2\log(2\cosh(\ell/2))$. According to Theorem 2.15, we can write:

$$\mathbb{E}[F^{\gamma}] = \int_{\mathbf{R}_{\ge 0}^k} F(h(\mathbf{x}_k)) Q(\mathbf{x}_k) \frac{d\mathbf{x}_k}{V_{g,n}(L)},$$

where $Q$ is a polynomial of total degree $6g-6+2n-k$ and $k \in \{1, 2, 3\}$. Moreover, according to Lemmas 2.8 and 2.9, the function $H = u \circ h$ is a length-type function. Hence, on a one hand by using Theorem 3.16 we can conclude that the function:

$$W: \ell \mapsto \int_{\ell=H(\mathbf{x}_k)} Q(\mathbf{x}_k) \mathbf{1}_{\mathbf{R}_{\ge 0}^k}(\mathbf{x}_k) \frac{d\mathbf{x}_k}{d\ell}$$

belongs to $\mathcal{S}\mathcal{F}_w^{6g-7+2n}(\lambda_\gamma)$ for some $0 < \lambda_\gamma < 1$.

On the other hand, when applying Lemma 3.3, we have for almost every $\ell \in \mathbf{R}$:

$$V_{\mathcal{O}_\gamma}(\ell, L) = u'(\ell)W(u(\ell)),$$

and since $u'(\ell) = 1+O(e^{-\ell})$ and $u(\ell) = \ell+O(e^{-\ell})$, we can conclude that $V_{\mathcal{O}_\gamma}(\cdot, L) \in \mathcal{S}\mathcal{F}_w^{6g-7+2n}(\lambda_\gamma)$. $\square$

**Corollary 4.2.** Under the same assumptions we have $a \mapsto \mathbb{E}[N_\gamma(a)] \in \mathcal{S}\mathcal{F}_w^{6g-6+2n}(\lambda_\gamma)$.
Proof. Let us denote $V_{O_γ}(.,L) = P_γ^L + r_γ^L$, where $P_γ^L$ is a polynomial of degree $6g - 7 + 2n$. Since $V_{O_γ}(.,L) \in SF_w(\lambda_γ)$, we have $r_γ^L \in L^1(\mathbf{R})$ and there exists $c > 0$ such that:

$$ \int_a^\infty |r(\ell)| d\ell = \mathcal{O} \left( (1+a)^c e^{-\lambda_\gamma a} \right). $$

Thus:

$$ \mathbb{E}[N_γ(a)] = \int_0^a P_γ^L(\ell) d\ell + \int_0^\infty r_γ^L(\ell) d\ell - \int_a^\infty r(\ell) d\ell, $$

which makes us able to conclude. □

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