Jean Dolbeault

Maria J Esteban

Hardy-Littlewood-Sobolev and related inequalities: stability

Keywords: 2020 Mathematics Subject Classification. Primary 49J40, Secondary 26D10, 49K30, 46E35, 35J20, 49K20 Sobolev inequality, Hardy-Littlewood-Sobolev inequality, Gagliardo-Nirenberg-Sobolev inequalities, Bianchi-Egnell method, concentration-compactness method, spectral gap, Legendre duality, stability, fast diffusion equation

The purpose of this chapter is twofold. We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. We also contribute to the topic with some observations on constructive stability estimates for (HLS).

It is with great pleasure that we dedicate this paper to Elliott Lieb on the occasion of his 90th birthday.

A short review of some functional inequalities

 for the Hardy-Littlewood-Sobolev inequality which can be written as ∬ R 𝑛 ×R 𝑛 𝑓 (𝑥) |𝑥 -𝑦| -𝜆 𝑔(𝑦) 𝑑𝑥 𝑑𝑦 ≤ 𝐶 𝑝,𝜆,𝑛 𝑓 𝐿 𝑝 (R 𝑛 ) 𝑔 𝐿 𝑡 (R 𝑛 ) , (HLS 𝜆 )

for all 𝑓 ∈ 𝐿 𝑝 (R 𝑛 ), 𝑔 ∈ 𝐿 𝑡 (R 𝑛 ), with 1 < 𝑝, 𝑡 < +∞ such that 1/𝑝 + 1/𝑡 + 𝜆/𝑛 = 2 and 0 < 𝜆 < 𝑛. By duality he also obtained a new and simple proof of the fractional Sobolev inequality, which goes as follows. If 𝛼 ∈ (0, 𝑛/2) and 𝑞 = 2 𝑛/(𝑛 -2𝛼), then

(-Δ) 𝛼/2 𝑓 2 𝐿 2 (R 𝑛 ) ≥ 𝑆 𝑛, 𝛼 𝑓 2 𝐿 𝑞 (R 𝑛 ) (S 𝛼 )
for any smooth and compactly supported function 𝑓 . Refer to [START_REF] Lieb | Analysis[END_REF]Chapter 5] for considerations on the domain D 𝛼 of (-Δ) 𝛼/2 based on Fourier transforms: it is the space of all tempered distributions 𝑢 which vanishes at infinity and such that (-Δ) 𝛼/2 𝑢 is in 𝐿 2 (R 𝑛 ). According to [73, 5.9(1) and 5.10(2)], we have

(-Δ) -𝛼/2 𝑓 2 𝐿 2 (R 𝑛 ) = 𝑐 𝑛-2𝛼 𝑐 2𝛼 ∬ R 𝑛 ×R 𝑛
𝑓 (𝑥) 𝑓 (𝑦) |𝑥 -𝑦| 𝑛-2𝛼 𝑑𝑥 𝑑𝑦 with 𝑐 𝛼 = Γ(𝛼)/𝜋 𝛼/2 for any 𝑓 ∈ 𝐿 𝑝 (R 𝑛 ), so that

∬ R 𝑛 ×R 𝑛 𝑓 (𝑥) 𝑓 (𝑦) |𝑥 -𝑦| 𝑛-2𝛼 𝑑𝑥 𝑑𝑦 = 𝑐 2𝛼 𝑐 𝑛-2𝛼 ∫ R 𝑛 𝑓 (-Δ) -𝛼 𝑓 𝑑𝑥 .
Taking the Legendre transform of both sides of (HLS 𝜆 ) written for 𝑝 = 2 𝑛/(𝑛 + 2𝛼) and 𝜆 = 𝑛 -2𝛼, we realize that inequalities (HLS 𝜆 ) and (S 𝛼 ) are equivalent because 𝑞 = 2 𝑛/(𝑛 -2𝛼) is the Hölder conjugate of 𝑝. Moreover, the optimal constants are such that 𝑆 𝑛, 𝛼 = 𝑐 2𝛼 𝑐 𝑛-2𝛼 1 𝐶 𝑝,𝜆,𝑛 with 𝜆 = 𝑛 -2𝛼 and 𝑝 = 2 𝑛 𝑛 + 2𝛼

.

See [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF] and [START_REF] Carlen | Duality and stability for functional inequalities[END_REF] for further details on duality issues. Both (S 𝛼 ) and (HLS 𝜆 ) inequalities are invariant by scaling and conformally invariant. The classical case of Sobolev's inequality corresponds to (S 𝛼 ) with 𝛼 = 1. If 𝑛 ≥ 3, the inequality can be written with 𝑆 𝑛 = 𝑆 𝑛,1 as

∇ 𝑓 2 𝐿 2 (R 𝑛 ) ≥ 𝑆 𝑛 𝑓 2 𝐿 2𝑛 𝑛-2 (R 𝑛 ) (S)
for all functions 𝑓 ∈ 𝐿 2𝑛 𝑛-2 (R 𝑛 ) such that ∇ 𝑓 ∈ 𝐿 2 (R 𝑛 ). Inequality (S) has a long history. An early computation of the best constant and the optimizers for the Sobolev inequality among radial functions can be found in [START_REF] Bliss | An integral inequality[END_REF] (also see [START_REF] Rosen | Minimum value for 𝑐 in the Sobolev inequality 𝜙 3 ≤ 𝑐 ∇𝜙 3[END_REF] for the linear stability of the radial optimizers). Without symmetry, the inequality is proved in [START_REF] Soboleff | Sur un théorème d'analyse fonctionnelle[END_REF][START_REF] Sobolev | On a theorem of functional analysis[END_REF] and the computation of the constant and the optimal functions is classically attributed to T. Aubin [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF] and G. Talenti [START_REF] Talenti | Best constant in Sobolev inequality[END_REF], even if it seems that a first complete proof goes back to E. Rodemich's seminar [START_REF] Rodemich | The Sobolev inequalities with best possible constants[END_REF] (see [61, p. 158] for a quote). The value of the best constant 𝑆 𝑛 was found to be

𝑆 𝑛 = 𝜋 𝑛 (𝑛 -2) Γ(𝑛/2) Γ(𝑛)
and equality is achieved in (S) if and only if 𝑓 is in the (𝑑 + 2)-dimensional manifold M S of the Aubin-Talenti functions ℎ 𝜇, 𝑥 0 , 𝜎 (𝑥)

:= 𝜇 1 + 𝜎 |𝑥 -𝑥 0 | 2 -(𝑛-2)/2 ∀ 𝑥 ∈ R 𝑑
parametrised by (𝜇, 𝑥 0 , 𝜎) ∈ R × R 𝑑 × (0, +∞). In terms of duality, it is straightforward to check that the (HLS 𝜆 ) inequality corresponding to (S) is given by the particular choice 𝜆 = 𝑛 -2 and 𝑝 = 𝑡 of the parameters, and can be written as

∫ R 𝑛 𝑓 (-Δ) -1 𝑓 𝑑𝑥 ≤ C 𝑛 𝑓 2 𝐿 2𝑛 𝑛+2 (R 𝑛 ) (1.1)
with

C 𝑛 = 1 𝑆 𝑛 = 1 𝜋 𝑛 (𝑛 -2) Γ(𝑛) Γ(𝑛/2) 2 𝑛 . ( HLS
)
This case is important as it encodes the 𝐿 𝑝 smoothing properties of (-Δ) -1 on R 𝑛 . Inequality (HLS 𝜆 ) was introduced by G.H. Hardy and J.E. Littlewood [START_REF] Hardy | Some properties of fractional integrals[END_REF][START_REF] Hardy | Notes on the theory of series (XII): on certain inequalities connected with the calculus of variations[END_REF][START_REF] Hardy | Some properties of fractional integrals[END_REF] on R and generalised by S.L. Sobolev [START_REF] Soboleff | Sur un théorème d'analyse fonctionnelle[END_REF][START_REF] Sobolev | On a theorem of functional analysis[END_REF] to R 𝑁 . The proof of the sharp (HLS 𝜆 ) inequality is due to E.H. Lieb in [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF] with a more detailed proof in [START_REF] Lieb | Analysis[END_REF]. This proof uses rearrangement techniques. The first proof of existence of an optimal function without symmetrisation is due to P.-L. Lions in [74, Theorem 2.1] and relies on the concentration-compactness method. This proof is useful for generalizations involving non-symmetric convolution kernels, but does not allow to identify the optimal functions. Also see [START_REF] Struwe | Variational methods[END_REF]Section II.4] for the application to sharp Sobolev inequalities. The concentration-compactness method is currently used, see for instance [START_REF] Bonder | The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional brezis-nirenberg problem[END_REF][START_REF] De Pablo | Extremals in Hardy-Littlewood-Sobolev inequalities for stable processes[END_REF][START_REF] Palatucci | Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces[END_REF][START_REF] Zhang | Extremal problems of Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds[END_REF], although most papers on the topic focus on the fractional Sobolev inequality rather than on the corresponding (HLS 𝜆 ) inequalities. A new rearrangement-free proof was provided by R.L. Frank and E.H. Lieb in [START_REF] Frank | Spherical reflection positivity and the Hardy-Littlewood-Sobolev inequality[END_REF][START_REF] Frank | A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality[END_REF][START_REF] Frank | Sharp constants in several inequalities on the Heisenberg group[END_REF].

For the sake of simplicity, we restrict this presentation to few results and give sketches of the proofs only in the case of (S) and (1.1). Without pretending to any exhaustivity, we also list some results for (S 𝛼 ), (HLS 𝜆 ) and some related inequalities.

Gagliardo-Nirenberg-Sobolev inequalities refer to the two famous papers [START_REF] Gagliardo | Proprietà di alcune classi di funzioni in più variabili[END_REF][START_REF] Nirenberg | On elliptic partial differential equations[END_REF]. Within this class of inequalities, the inequality

𝑢 𝐿 𝑞 (R 𝑛 ) ≤ 𝐶 𝑢 1-𝜃 𝐿 𝑝 (R 𝑛 ) ∇𝑢 𝜃 𝐿 2 (R 𝑛 ) (GNS)
holds for all smooth compactly supported functions, and by density, for all functions

𝑢 ∈ 𝐿 𝑝 (R 𝑛 ) such that ∇𝑢 ∈ 𝐿 2 (R 𝑛 ).
We shall assume here that the parameters 𝑝 and 𝑞 are such that 2 ≤ 𝑝 < 𝑞 < 2 * where 2 * = +∞ if 𝑛 = 1 or 2, and 2

* = 2 𝑛/(𝑛 -2) if 𝑛 ≥ 3. The exponent 𝜃 = 2 𝑛 (𝑞-𝑝) 𝑞 (2 𝑑-𝑝 (𝑑-2))
is uniquely determined by the scaling properties of the inequalities. Optimality in inequality (GNS) is achieved among radial functions but there are only few cases for which the best constants and the minimizers are known, for instance if 𝑞 = 2 ( 𝑝 -1) (see [START_REF] Cordero-Erausquin | A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities[END_REF][START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF][START_REF] Gunson | Inequalities[END_REF]) or if 𝑑 = 1.

Optimality results for the logarithmic Sobolev (log-Sobolev) inequality, the logarithmic Hardy-Littlewood-Sobolev (log-HLS) inequality, the Onofri inequality or the Caffarelli-Kohn-Nirenberg inequalities, among many other functional inequalities, have attracted a lot of attention over the years and we may refer for instance to [START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S 𝑛[END_REF][START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF][START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] for some key papers and to [12, Chapter 1, bibliographical comments] for a short review. Functional inequalities in bounded domains or on manifolds will not be discussed here, except for a few results on the sphere S 𝑛 which are related to inequalities on R 𝑛 by the stereographic projection.

Once optimal constants are known and the set of optimising functions has been characterised, the next question is to understand stability: which kind of distance is measured by the deficit, that is, the difference of the two terms in the functional inequality, written with the optimal constant. A variety of answers has been obtained during the last 30 years and this is what we are now going to review, mostly in the case of (S) and (1.1).

Quantitative stability results

In the celebrated paper [START_REF] Brezis | Sobolev inequalities with remainder terms[END_REF], H. Brezis and E.H. Lieb raise the question of the quantitative stability for Sobolev inequalities and gave an answer in bounded domains. A related result is proved by H. Egnell, F. Pacella, and M. Tricarico in [START_REF] Egnell | Some remarks on Sobolev inequalities[END_REF]. In the case of the Sobolev inequality in the whole Euclidean space, the first quantitative stability result is obtained by G. Bianchi and H. Egnell in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF]. They prove that

∇ 𝑓 2 𝐿 2 (R 𝑛 ) -𝑆 𝑛 𝑓 2 𝐿 2𝑛 𝑛-2 (R 𝑛 ) ≥ 𝜅 S inf ℎ ∈M S ∇ 𝑓 -∇ℎ 2 𝐿 2 (R 𝑛 ) (2.1)
for some positive constant 𝜅 S whose value is not known, as they argue by compactness using the concentration-compactness method and by contradiction. Stability results are of course not limited to (S). In recent years, the problem of proving stability for various sharp inequalities related to (S) in analysis and geometry, such as the isoperimetric inequality, the Brunn-Minkowski inequality, the logarithmic Sobolev inequality, etc., has been widely studied: see for instance [START_REF] Figalli | A sharp stability result for the relative isoperimetric inequality inside convex cones[END_REF][START_REF] Figalli | Quantitative stability for the Brunn-Minkowski inequality[END_REF][START_REF] Figalli | A mass transportation approach to quantitative isoperimetric inequalities[END_REF][START_REF] Figalli | Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation[END_REF]. The stability of the Sobolev inequality in the case of an 𝐿 𝑞 norm of the gradient with 𝑞 ≠ 2 is proved by A. Cianchi, N. Fusco, F. Maggi and A. Pratelli in [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF] using mass transportation, and improved recently by A. Figalli, R. Neumayer and Y. R.-Y. Zhang in [START_REF] Figalli | Gradient stability for the Sobolev inequality: the case 𝑝 ≥ 2[END_REF][START_REF] Figalli | Sharp gradient stability for the Sobolev inequality[END_REF][START_REF] Neumayer | A note on strong-form stability for the Sobolev inequality[END_REF]. The proof in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] uses the Hilbertian nature of 𝐻 1 (R 𝑑 ). When the Hilbertian nature of the underlying spaces is lost, as this is the case for the Gagliardo-Nirenberg-Sobolev inequalities, the same argument cannot be used. To circumvent this difficulty two approaches are used for (GNS). The first approach relies on a trick due to D. Bakry, that can be found in [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] (see a detailed computation in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]Section 1.3.1.2]): the (GNS) inequality with 𝑞 = 2 ( 𝑝 -1), whose optimal functions are generated by

u 𝑞 (𝑥) := 1 + |𝑥| 2 -1/(𝑞-1) ∀ 𝑥 ∈ R 𝑑 , (2.2)
is rewritten as a Sobolev inequality in a higher, artificial, dimension such that the optimal functions appear as Aubin-Talenti functions. This idea is used first by E. Carlen and A. Figalli in [START_REF] Carlen | Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation[END_REF] and later expanded by F. Seuffert in [START_REF] Seuffert | An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions[END_REF] and V.H. Nguyen in [START_REF] Nguyen | The sharp Gagliardo-Nirenberg-Sobolev inequality in quantitative form[END_REF]. Constants are anyway non-constructive as they rely on [START_REF] Bianchi | A note on the Sobolev inequality[END_REF]. Starting with [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF], another approach (which will not be reviewed here) has been developed for studying the stability of the critical points of (S 𝛼 ) in 𝐻 -1 : see [START_REF] Aryan | Stability of Hardy-Littlewood-Sobolev inequality under bubbling[END_REF][START_REF] Ciraolo | A quantitative analysis of metrics on R 𝑛 with almost constant positive scalar curvature, with applications to fast diffusion flows[END_REF][START_REF] Deng | Sharp quantitative estimates of Struwe's decomposition[END_REF][START_REF] Figalli | On the sharp stability of critical points of the Sobolev inequality[END_REF][START_REF] Frank | Reverse conformally invariant Sobolev inequalities on the sphere[END_REF][START_REF] Wei | Stability of Caffarelli-Kohn-Nirenberg inequality[END_REF]. Results are again non-constructive and we are not aware of any counterpart in the case of (HLS 𝜆 ).

There is a different line of thought which relies on entropy methods: see [12, Chapter 1] for results based on the calculus of variations. Notice here that stability is measured in terms of a relative Fisher information, which is not the same notion of distance as in (2.1) but via the Pinsker-Csiszár-Kullback inequality it controls a distance equivalent to

| 𝑓 | 𝑝 -|ℎ| 𝑝 𝐿 1 (R 𝑛 )
. Up to minor restrictions, these estimates can be made constructive and will be listed in the next section.

The Bianchi-Egnell method is not limited to the case 𝛼 = 1 of (S 𝛼 ). In [22, Theorem 1] S. Chen, R.L. Frank and T. Weth prove a stability result for (S 𝛼 ): for all 𝑓 ∈ D 𝛼 ⊂ 𝐿 𝑞 (R 𝑛 ) with 𝑞 = 2𝑛/(𝑛 -2𝛼) and 𝛼 ∈ (0, 𝑛/2),

(-Δ) 𝛼/2 𝑓 2 𝐿 2 (R 𝑛 ) -𝑆 𝑛, 𝛼 𝑓 2 𝐿 𝑞 (R 𝑛 ) ≥ 𝜅 S, 𝛼 inf ℎ ∈M S, 𝛼 (-Δ) 𝛼/2 ( 𝑓 -ℎ) 2 𝐿 2 (R 𝑛 ) (S 𝛼 )
where the manifold M S, 𝛼 of optimal functions is generated from

ℎ(𝑥) = 1 + |𝑥| 2 2𝛼-𝑛 2
by multiplications by a constant, translations and scalings. The computation of the spectrum of the linearized problem uses a reformulation on the sphere which will be illustrated in Section 5. The dual counterpart for (HLS 𝜆 ) inequalities is [16, Theorem 1.5] due to E. Carlen, which goes as follows.

Theorem 2.1. Let 𝑛 ≥ 2. For all 𝛼 ∈ (0, 𝑛/2), there is a constant 𝜅 HLS, 𝛼 > 0 depending only on 𝑛 and 𝛼 such that for all 𝑓 ∈ 𝐿 2𝑛/(𝑛+2𝛼) (R 𝑛 ),

𝑓 2 𝐿 2 𝑛 𝑛+2𝛼 (R 𝑛 ) -𝑆 𝑛, 𝛼 (-Δ) -𝛼/2 𝑓 2 𝐿 2 (R 𝑛 ) ≥ 𝜅 HLS, 𝛼 inf ℎ ∈M HLS, 𝛼 𝑓 -ℎ 2 𝐿 2 𝑛 𝑛+2𝛼 (R 𝑛 )
.

(2.3)

Here M HLS, 𝛼 is the manifold of all optimal functions, which is generated from

ℎ(𝑥) = 1 + |𝑥| 2 -𝑛+2𝛼
by multiplication by a constant, translations and scalings. Using a duality argument to relate (2.1) with (2.3) is not as straightforward as proving the equivalence of (S 𝛼 ) with (HLS 𝜆 ) when 𝑝 = 𝑡 and 𝜆 = 𝑛 -2𝛼. The interplay between stability bounds and quantitative convexity estimates is crucial. This is analysed in [START_REF] Carlen | Duality and stability for functional inequalities[END_REF], with the additional motivation of giving theoretical grounds for the stability bound associated with the Keller-Lieb-Thirring inequality. This inequality estimates the fundamental eigenvalue 𝜆(𝑉) of a Schrödinger operator -Δ + 𝑉 (𝑥) with 𝑉 𝐿 𝑡 (R 𝑛 ) for an appropriate 𝑡, and can be seen as the Legendre transform of (GNS) written with 𝑝 = 2: see [START_REF] Carlen | Stability estimates for the lowest eigenvalue of a Schrödinger operator[END_REF] for further details. Coming back to Theorem 2.1, an interesting consequence of the method is that 𝜅 HLS, 𝛼 is explicitly computable in terms of 𝜅 S . However, none of the two constants is known nor has been given an explicit estimate for them. Notice that a scheme of a direct proof based on a result of M. Christ in [START_REF] Christ | A sharpened Hausdorff-Young inequality[END_REF] has been proposed in [START_REF] Liu | Remainder terms for several inequalities on some groups of Heisenberg-type[END_REF] by H. Liu and A. Zhang in the case of the Heisenberg group. We refer to [START_REF] Figalli | Stability in geometric and functional inequalities[END_REF] for consequences of stability results. The remainder term in (2.1) is quadratic and as R. Frank points out in [START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF] (see also [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] for a comment on this), the power two is optimal since it is not possible to bound it from below with terms like

∇ 𝑓 2-𝛼 𝐿 2 (R 𝑛 ) ∇ 𝑓 -∇ℎ 𝛼 𝐿 2 (R 𝑛 )
for some 𝛼 < 2. Similar issues are addressed for instance in [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF][START_REF] Figalli | Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation[END_REF][START_REF] Figalli | Gradient stability for the Sobolev inequality: the case 𝑝 ≥ 2[END_REF][START_REF] Figalli | Sharp gradient stability for the Sobolev inequality[END_REF][START_REF] Neumayer | A note on strong-form stability for the Sobolev inequality[END_REF]. In the same spirit, on the sphere S 𝑛 with uniform probability measure 𝑑𝜇, R.L. Frank shows in [55, Theorem 2] that the optimal value for which there exists some 𝜅 > 0 for which

∇ 𝑓 2 𝐿 2 (S 𝑛 ) -𝑑 𝑞-2 𝑓 2 𝐿 𝑞 (S 𝑛 ) -𝑓 2 𝐿 2 (S 𝑛 ) ≥ 𝜅 ∇ 𝑓 2 𝐿 2 (S 𝑛 ) -𝑑 𝑞-2 𝑓 2 𝐿 2 (S 𝑛 ) -f 2 𝛼 2 ∇ 𝑓 2 𝐿 2 (S 𝑛 ) + 𝑑 𝑞-2 𝑓 2 𝐿 2 (S 𝑛 ) 𝛼-2 2 with f := ∫ S 𝑛 𝑓 𝑑𝜇, is 𝛼 = 4.
Here we use the notation

𝑓 𝐿 𝑞 (S 𝑛 ) = ∫ S 𝑛 | 𝑓 | 𝑞 𝑑𝜇 2 𝑞
.

Optimal remainder terms are also found to be quartic on the non-trivial manifold

S 1 (1/ √ 𝑛-2) × S 𝑛-1
as the minimizer is degenerate in the sense that there is a zero mode of the Hessian of the minimisation problem that does not come from symmetries of the set of minimizers. Notice that other results can be obtained for different choices of the distance on the sphere: see, e.g., [START_REF] Dolbeault | Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF].

Summarizing, in quantitative stability results obtained from methods à la Bianchi and Egnell, the exponent in the distance term is at least partially understood. Essentially nothing is known on the value of the constant 𝜅 S and similar constants in related stability inequalities, or even how to get such constructive estimates for strong notions of distances. There are some results for weaker notions of distance and this is what we shall expose in the next sections.

3 Stability results for (1.1) by parabolic methods E. Carlen, J.A. Carrillo and M. Loss prove in [START_REF] Carlen | Hardy-Littlewood-Sobolev inequalities via fast diffusion flows[END_REF] that

C 𝑛 𝑓 2 𝐿 2𝑛 𝑛+2 (R 𝑛 ) - ∫ R 𝑛 𝑓 (𝑥)(-Δ) -1 𝑓 (𝑥) 𝑑𝑥 = 8 𝑛 + 2 ∫ +∞ 0 𝑒 𝛽𝑡 D 𝑢 𝑛-1 𝑛+2 • , 𝑒 𝛽𝑡 𝑑𝑡 ,
where 𝑢 = 𝑢(𝑡, 𝑥) is the solution of the fast diffusion equation

𝜕𝑢 𝜕𝑡 = Δ 𝑢 𝑚 in R 𝑛 , 𝑡 ≥ 0 ,
with exponent 𝑚 = 𝑛/(𝑛 + 2) and initial datum 𝑢(0, 𝑥) = 𝑓 (𝑥). The deficit is measured by

D [𝑔] := C 𝑛 𝑛 (𝑛 -2) (𝑛 -1) 2 𝑔 2 𝐿 2 (R 𝑛 ) 𝑔 2( 𝑝-1) 𝐿 𝑝+1 (R 𝑛 ) -𝑔 2 𝑝 𝐿 2 𝑝 (R 𝑛 )
for any function 𝑔 ∈ 𝐻 1 (R 𝑛 ), where 𝑝 = (𝑛 + 1)/(𝑛 -1). By [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], it turns out that D [𝑔] ≥ 0 is a sharp form of (GNS) inequalities, with equality if and only if 𝑔(𝑥) = 1 + |𝑥| 2 1/( 𝑝-1) up to a multiplication by a constant, a translation and a scaling. The proof of the equality case can be achieved either by symmetrisation, variational methods and ODE techniques, or using the carré du champ method adapted to the fast diffusion equation (see [START_REF] Jüngel | Entropy methods for diffusive partial differential equations[END_REF] for detailed justifications and references). This result provides us with a proof of the sharp (HLS 𝜆 ) inequality, identifies the optimal function in terms of a Barenblatt profile and provides an integral deficit term. Moreover, the carré du champ method allows to bypass symmetrisation techniques. Another fast diffusion equation, now with exponent 𝑚 = (𝑛 -2)/(𝑛 + 2), provides us with similar results. If 𝑛 ≥ 3, a formal computation shows that

𝑑 𝑑𝑡 C 𝑛 𝑓 2 𝐿 2𝑛 𝑛+2 (R 𝑛 ) - ∫ R 𝑛 𝑓 (𝑥)(-Δ) -1 𝑓 (𝑥) 𝑑𝑥 = 2 𝑓 4 𝑛+2 𝐿 2𝑛 𝑛+2 (R 𝑛 ) D [𝑣]
where 𝑣 = 𝑢 𝑚 and, for any

𝑔 ∈ 𝐿 2𝑛 𝑛-2 (R 𝑛 ) such that ∇𝑔 ∈ 𝐿 2 (R 𝑛 ), D [𝑔] := ∇𝑔 2 𝐿 2 (R 𝑛 ) -𝑆 𝑛 𝑔 2 𝐿 2𝑛 𝑛-2 (R 𝑛 )
.

In the spirit of the carré du champ method, it can be shown that the inequality D [𝑔] is monotone non-increasing with limit 0, so that D [𝑔] ≥ 0 is in fact the standard Sobolev inequality, with optimal constant 𝑆 𝑛 = 1/C 𝑛 . In practice the solution of the fast diffusion equation with 𝑚 = (𝑛 -2)/(𝑛 + 2) vanishes for some finite time 𝑇 > 0 and justifications require some additional care. For instance, it is convenient to use the inverse stereographic projection and rewrite the evolution equation on the unit sphere: see [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF][START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities[END_REF][START_REF] Jin | A fractional Yamabe flow and some applications[END_REF] for details. Altogether, we may conclude that

C 𝑛 𝑓 2 𝐿 2𝑛 𝑛+2 (R 𝑛 ) - ∫ R 𝑛 𝑓 (𝑥)(-Δ) -1 𝑓 (𝑥) 𝑑𝑥 = 2 ∫ 𝑇 0 D 𝑢 𝑚 (𝑡, •) 𝑑𝑡 .
In both cases associated with 𝑚 = 𝑛/(𝑛 + 2) and 𝑚 = (𝑛 -2)/(𝑛 + 2), it is not known how to express the deficit of the inequality in terms of the initial datum of the evolution equation.

Constructive stability results

Few constructive stability results are known so far. In [START_REF] Figalli | Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation[END_REF], A. Figalli, F. Maggi and A. Pratelli use mass transportation and rearrangements to prove constructive stability results for the 1-Sobolev anisotropic inequalities for functions of bounded variation. The initial idea to prove those results comes from a previous work of N. Fusco, F. Maggi and A. Pratelli, who prove a non-constructive stability result for the isotropic 1-Sobolev inequality for functions of bounded variation.

As a straightforward consequence of the duality approach, an explicit stability bound for (S 𝛼 ) is easily obtained where the distance to the Aubin-Talenti manifold M S, 𝛼 is measured in terms of the deficit in (HLS 𝜆 ) with 𝜆 = 𝑛 -2𝛼. Let us give some details. We consider the case of (HLS 𝜆 ) with the particular choice 𝜆 = 𝑛 -2𝛼 and 𝑝 = 𝑡 = 2 𝑛/(𝑛 + 2𝛼) of the parameters. The optimal function for (HLS 𝜆 ) is, up to a multiplication by a constant, a translation and a scaling, given by

𝑔 ★ (𝑥) := 1 + |𝑥| 2 𝜆 2 -𝑛 ∀ 𝑥 ∈ R 𝑑 .
We know from [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF] and [16, Theorem 1.2] that the optimal functions for (S 𝛼 ) is 𝑔 𝑟 ★ with 𝑟 = 𝑛-2𝛼 𝑛+2𝛼 , up to multiplications by a constant, translations and scalings. By a standard trick used for instance in [31, Theorem 1.2], [START_REF] Jin | A fractional Yamabe flow and some applications[END_REF]Theorem 1.4] and [68, Theorem 1, (i)], the expansion of the square in

0 ≤ ∫ R 𝑛 𝑓 4𝛼 𝑛-2𝛼 𝐿 𝑞 (R 𝑛 ) ∇(-Δ) 𝛼-1 2 𝑓 -𝑆 𝑛, 𝛼 ∇(-Δ) -1+𝛼 2 𝑔 2 𝑑𝑥 , with 𝑔 𝑟 = 𝑓 so that 𝑓 2 𝐿 𝑞 (R 𝑛 ) = 𝑔 2𝑟 𝐿 𝑝 (R 𝑛 ) if 𝑞 = 2 𝑛/(𝑛 -2𝛼), shows that 𝑆 𝑛, 𝛼 𝑔 2 𝐿 𝑝 (R 𝑛 ) -𝑆 𝑛, 𝛼 (-Δ) -𝛼/2 𝑔 2 𝐿 2 (R 𝑛 ) ≤ 𝑓 8𝛼 𝑛-2𝛼 𝐿 𝑞 (R 𝑛 ) (-Δ) 𝛼/2 𝑓 2 𝐿 2 (R 𝑛 ) -𝑆 𝑛, 𝛼 𝑓 2 𝐿 𝑞 (R 𝑛 ) .
Slightly better results are obtained using flow methods as in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF][START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities[END_REF][START_REF] Jankowiak | Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities[END_REF] which have not been detailed in Section 3. Notice that the expansion of the square applies to logarithmic inequalities corresponding to the limit cases as 𝛼 → 𝑛/2. See [38, Theorem 10] for further considerations in the case 𝛼 = 1 and 𝑛 = 2. The main drawback is that, in all these approaches, the stability is controlled only in a weaker norm.

The situation is slightly better for subcritical (GNS) inequalities. Constructive results are obtained by M. Bonforte et al. in [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] in a very small neighborhood of the manifold of the Aubin-Talenti functions and were later improved in [START_REF] Dolbeault | Fast diffusion equations: matching large time asymptotics by relative entropy methods[END_REF][START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF]. The global result of [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF] is explicit but sub-optimal as the remainder term is of the order of the square of the entropy while one expects a linear dependence in view of [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF][START_REF] Dolbeault | Fast diffusion equations: matching large time asymptotics by relative entropy methods[END_REF]. It is true also in the logarithmic case as explained in [START_REF] Dolbeault | Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities[END_REF] through scaling properties. Notice that stability results in logarithmic Sobolev inequalities is a widely studied question, with various results in 𝐿 1 and in Wasserstein distances: see [START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF][START_REF] Feo | Some remarks on the stability of the Log-Sobolev inequality for the Gaussian measure[END_REF][START_REF] Indrei | Deficit estimates for the logarithmic Sobolev inequality[END_REF][START_REF] Kim | Instability results for the logarithmic Sobolev inequality and its application to related inequalities[END_REF] and references therein. The proof of [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF] is based on the use of the remainder terms in the carré du champ method and has an already long history in the framework of entropy methods for linear diffusion equations: see [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF][START_REF] Arnold | Refined convex Sobolev inequalities[END_REF]. A remarkable property of the carré du champ method is that it also applies to nonlinear flows of fast diffusion type [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF][START_REF] Carrillo | Fine asymptotics for fast diffusion equations[END_REF][START_REF] Dolbeault | Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization[END_REF][START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF][START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF], eventually with a non-trivial metric (see [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] and references therein), on the Euclidean space or on manifolds: see for instance [START_REF] Demange | Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF][START_REF] Dolbeault | Nonlinear flows and rigidity results on compact manifolds[END_REF]. As a consequence, let us give some examples on the sphere, which are taken from [START_REF] Dolbeault | Improved interpolation inequalities and stability[END_REF][START_REF] Dolbeault | Improved interpolation inequalities on the sphere[END_REF][START_REF] Dolbeault | Interpolation inequalities on the sphere: linear vs. nonlinear flows (inégalités d'interpolation sur la sphère : flots non-linéaires vs. flots linéaires)[END_REF]. Using the uniform probability measure 𝑑𝜇 as in Section 2, for any 𝑞 ∈ (1, 2) ∪ (2, 2 * ) and any 𝑢 ∈ 𝐻 1 (S 𝑑 ),

∇𝑢 2 𝐿 2 (S 𝑛 ) -𝑑 𝑞-2 𝑢 2 𝐿 𝑞 (S 𝑛 ) -𝑢 2 𝐿 2 (S 𝑛 ) ≥ 𝑑 𝜓 𝑢 2 𝐿 𝑞 (S 𝑛 ) -𝑢 2 𝐿 2 (S 𝑛 ) (𝑞-2) 𝑢 2 𝐿 𝑞 (S 𝑛 ) 𝑢 2 𝐿 𝑞 (S 𝑛 )
holds for an explicit, strictly convex 𝐶 2 function 𝜓 such that 𝜓(0) = 𝜓 (0) = 0 (see [START_REF] Dolbeault | Improved interpolation inequalities and stability[END_REF] for details). Here equality is achieved only by constant functions, but optimality constants are determined by perturbations involving the spherical harmonics associated with the first eigenvalue of the Laplace-Beltrami operator on S 𝑛 . By requesting additional orthogonality constraints, which discard these spherical harmonics, the constant 𝑑 in the left-hand side can be improved (with 𝜓 = 0). See [37, Section 5] and [START_REF] Dolbeault | Improved interpolation inequalities and stability[END_REF]. The corresponding improved entropy -entropy production inequality can be reinterpreted as a stability result. The constant 𝜅 in front of the distance term is obtained through a classical minimisation problem. The value of the minimum is not explicitly known, except in the limit case of the logarithmic Sobolev inequality as 𝑞 → 2 or if additional symmetry assumptions are imposed.

Recent work by M. Bonforte et al. provides the first quantitative and constructive stability result for (GNS) and Sobolev inequalities on the Euclidean space, under some constraints. Using entropy methods and fast diffusion flows, the authors prove in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF] (also see [START_REF] Bonforte | Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations[END_REF] for a scheme of the proof and further considerations on the method) that stability for initial data can be deduced from improved decay rates of relative entropies in rescaled variables when the rescaling is chosen in order to match the solution with the best matching Barenblatt profile among all Barenblatt self-similar solutions: the stability is then measured by a relative Fisher information and takes the form inf

𝜑 ∈M GNS ∫ R 𝑛 (𝑞 -1) ∇𝑢 + 𝑢 𝑞 ∇𝜑 1-𝑞 2 𝑑𝑥 ,
where M GNS is the manifold generated from u 𝑞 defined in (2.2) by multiplications by a constant, translations and scalings. The result holds for any of the inequalities (GNS) with 𝑞 = 2 (𝑞 -1) ∈ (1, 2 * ) and also for the Sobolev inequality (S) when 𝑞 = 2 * , 𝑑 ≥ 3. The method relies on regularisation properties of the fast diffusion flows which introduce an integral decay condition on the initial data. Regularity and properties of the entropy are then used to extend the asymptotic stability results into the initial time layer, where nonlinear evolution takes place and the evolution of the entropy is controlled using the nonlinear carré du champ method by backward in time estimates. The core of the method is a constructive Harnack inequality based on J. Moser's methods and a fully quantitative global Harnack Principle for the nonlinear flow. This quantifies the threshold time after which the solution is in the asymptotic regime and the convergence is governed by an improved Hardy-Poincaré inequality, based on a spectral analysis. Improved decay rates are then extended to the initial time layer, thus proving an improved entropy -entropy production inequality for the solution, which is also valid for the initial data. The improved inequality is rephrased as a stability result while Barenblatt profiles are transformed into the Aubin-Talenti type functions u 𝑞 . The whole method relies on entropies, which requires to work with functions with a finite second moment, and on a global Harnack Principle, which provides us with a uniform threshold time if and only if the tails of the initial data have sufficient decay. These limitations are the price to pay in order to get a constructive stability estimate with an explicit constant.

Establishing constructive stability results for (1.1) is so far an open question.

5 Two local stability results for (1.1)

A local stability result in the norm of relative uniform convergence

In the spirit of [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities[END_REF], we first establish a constructive stability result of (1.1) in a neighbourhood of the optimal functions, with respect to the very strong topology of relative uniform convergence. See [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF] for an illustration of the interest of this framework. The price to pay is that the result applies in a neighbourhood of M := M HLS 1 in a topology stronger than the natural topology. Equality in (1.1) is achieved by the function

𝑢 ★ (𝑥) = 1 + |𝑥| 2 -(𝑛+2)/2 ∀ 𝑥 ∈ R 𝑑 .
Let us define the functions

𝑓 0 := 𝑢 𝑛-2 𝑛+2 ★ = 1 + |𝑥| 2 -𝑛-2 2 , 𝑓 𝑖 := 𝑥 𝑖 1 + |𝑥| 2 𝑓 0 , 𝑓 𝑛+1 := 1 -|𝑥| 2 1 + |𝑥| 2 𝑓 0 ,
and the norm

𝑣 2 ★ := ∫ R 𝑛 𝑢 -4 𝑛+2 ★ 𝑣(𝑥) 2 𝑑𝑥 = ∫ R 𝑛 𝑣(𝑥) 2 1 + |𝑥| 2 2 𝑑𝑥 . (5.1)
For perturbations

𝑢 ★ + 𝜀 𝑢 4 𝑛+2 ★ 𝑔 of 𝑢 ★ such that 𝑢 -𝑛-2 𝑛+2 ★ 𝑔 𝐿 ∞ (R 𝑛 ) ≤ 1 , (5.2)
there is a stability result for (1.1) which is uniform in 𝜀 > 0 small enough. Notice that (5.2) can be rewritten as

(1 -𝜀) 𝑢 ★ ≤ 𝑢 𝜀 := 𝑢 ★ + 𝜀 𝑢 4 𝑛+2 ★ 𝑔 ≤ (1 + 𝜀) 𝑢 ★ .
This means that 𝑢 𝜀 /𝑢 ★ is 𝜀-close to 1, i.e., 𝑢 𝜀 is close to 𝑢 ★ in the topology of relative uniform convergence (see [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities[END_REF]) associated with the norm 𝑣 ↦ → 𝑣/𝑢 ★ 𝐿 ∞ (R 𝑛 ) . Theorem 5.1. Let 𝑛 ≥ 3. If 𝑔 fullfils (5.2) and the orthogonality conditions

∫ R 𝑛 𝑔 𝑓 𝑖 1 + |𝑥| 2 2 𝑑𝑥 = 0 ∀ 𝑖 = 0, 1, . . . , 𝑛 + 1 , (5.3)
then for any 𝜀 ∈ (0, 1), the function

𝑢 𝜀 = 𝑢 ★ + 𝜀 𝑢 4 𝑛+2 ★ 𝑔 satisfies C 𝑛 𝑢 2 𝐿 2 𝑛 𝑛+2 (R 𝑛 ) - ∫ R 𝑛 𝑢 (-Δ) -1 𝑢 𝑑𝑥 ≥ 𝜅 𝑛 𝑢 -𝑢 ★ 2 ★ with 𝜅 𝑛 := 8 (𝑛+1) 3 𝑛 (𝑛+2) 2 (𝑛+4) .
Proof. Let 𝑝 = 2 𝑛/(𝑛 + 2). By using the fact that 𝑢 ★ is optimal, inequality (1.1) can be written as H [𝑢] ≥ 0 with

H [𝑢] := C 𝑛 𝑢 2 𝐿 𝑝 (R 𝑛 ) -𝑢 ★ 2 𝐿 𝑝 (R 𝑛 ) -2 𝑢 ★ 4 𝑛+2 𝐿 𝑝 (R 𝑛 ) ∫ R 𝑛 𝑢 𝑛-2 𝑛+2 ★ (𝑢 -𝑢 ★ ) 𝑑𝑥 - ∫ R 𝑛 (𝑢 -𝑢 ★ ) (-Δ) -1 (𝑢 -𝑢 ★ ) 𝑑𝑥 .
Assume that 𝑔 satisfies (5.2) and (5.3). We notice that

∫ R 𝑛 𝑢 𝑛-2 𝑛+2 ★ (𝑢 -𝑢 ★ ) 𝑑𝑥 = ∫ R 𝑛 𝑓 0 (𝑢 -𝑢 ★ ) 𝑑𝑥 = 0 ,
and as a consequence

H [𝑢] = C 𝑛 𝑢 2 𝐿 𝑝 (R 𝑛 ) -𝑢 ★ 2 𝐿 𝑝 (R 𝑛 ) - ∫ R 𝑛 (𝑢 -𝑢 ★ ) (-Δ) -1 (𝑢 -𝑢 ★ ) 𝑑𝑥 .
By a Taylor-Lagrange expansion, we have

|𝑢| 𝑝 -𝑢 𝑝 ★ -𝑝 𝑢 𝑝-1 ★ (𝑢 -𝑢 ★ ) -1 2 𝑝 ( 𝑝 -1) 𝑢 𝑝-2 ★ (𝑢 -𝑢 ★ ) 2 = 1 6 𝑝 ( 𝑝 -1) ( 𝑝 -2) |𝜉 | 𝑝-4 𝜉 (𝑢 -𝑢 ★ ) 3
for some intermediate value 𝜉 between 𝑢 ★ and 𝑢. With 𝑝 = 2 𝑛/(𝑛 + 2) ∈ (1, 2), 𝑢 ★ > 0 and |𝑢 -𝑢 ★ | < 𝜀 𝑢 ★ for some 𝜀 ∈ (0, 1), we know that 𝜉 is positive. In that case, either 𝑢 < 𝑢 ★ and the right-hand side is positive, or 𝑢 ≥ 𝑢 ★ and

|𝜉 | 𝑝-4 𝜉 (𝑢 -𝑢 ★ ) 3 ≤ 𝑢 𝑝-3 ★ (𝑢 -𝑢 ★ ) 3 .

By using |𝑔| ≤ 𝑢

𝑛-2 𝑛+2

★ by (5.2), the expansion applied with 𝑢 = 𝑢 𝜀 shows that

1 6 𝑝 ( 𝑝 -1) ( 𝑝 -2) |𝜉 | 𝑝-4 𝜉 (𝑢 -𝑢 ★ ) 3 ≥ - 4 𝑛 (𝑛 -2) 3 (𝑛 + 2) 3 𝜀 3 𝑢 4 𝑛+2 ★ 𝑔 2 .
After taking into account (5.3) and 𝑢

4 𝑛+2 ★ (𝑥) = 1 + |𝑥| 2 -2 , we obtain ∫ R 𝑛 |𝑢 𝜀 | 𝑝 𝑑𝑥 ≥ ∫ R 𝑛 𝑢 𝑝 ★ 𝑑𝑥 + 𝑛 (𝑛 -2) (𝑛 + 2) 2 𝜀 2 1 - 4 𝜀 3 (𝑛 + 2) ∫ R 𝑛 𝑢 4 𝑛+2 ★ 𝑔 2 𝑑𝑥 .
𝑢 𝜀 2 𝐿 𝑝 (R 𝑛 ) -𝑢 ★ 2 𝐿 𝑝 (R 𝑛 ) ≥ 1 - 4 𝜀 3 (𝑛 + 2) 𝜀 2 M [𝑔]
where

M [𝑔] := 𝑛 -2 𝑛 + 2 𝑢 ★ 4 𝑛+2 𝐿 𝑝 (R 𝑛 ) ∫ R 𝑛 𝑔(𝑥) 2 1 + |𝑥| 2 2 𝑑𝑥 . With D [𝑔] := ∫ R 𝑛 𝑔(𝑥) 1 + |𝑥| 2 2 (-Δ) -1 𝑔(𝑥) 1 + |𝑥| 2 2 𝑑𝑥 ,
we know that

H [𝑢 𝜀 ] ≥ C 𝑛 M [𝑔] 𝜀 2 1 - 4 𝜀 3 (𝑛 + 2) - D [𝑔] C 𝑛 M [𝑔]
.

Finally, we study the quotient D [𝑔]/M [𝑔] among functions which satisfy the orthogonality conditions (5.3). To this end, let us consider the orthonormal basis of functions of 𝐿 2 R 𝑛 , (1 + |𝑥| 2 ) -2 𝑑𝑥 made of the spherical harmonics on S 𝑛 mapped into R 𝑛 via the stereographic projection. They satisfy the equations

-Δ𝑔 𝑘 = 𝜇 𝑘 𝑔 𝑘 1 + |𝑥| 2 2 , 𝜇 𝑘 = 4 𝑘 (𝑘 + 𝑛 -1) + 𝑛 (𝑛 -2)
and as a consequence, we have

D [𝑔 𝑘 ] = 1 𝜇 𝑘 ∫ R 𝑛 |𝑔 𝑘 (𝑥)| 2 1 + |𝑥| 2 2 𝑑𝑥 . Under condition (5.3), 𝑔 = 𝑘 ≥2 𝑎 𝑘 𝑔 𝑘 is such that ∫ R 𝑛 𝑔(𝑥) 2 1 + |𝑥| 2 2 𝑑𝑥 = ∑ 𝑘 ≥2 |𝑎 𝑘 | 2 and D [𝑔] = ∑ 𝑘 ≥2 |𝑎 𝑘 | 2 𝜇 𝑘 .
Since (𝜇 𝑘 ) 𝑘 ∈N is an increasing sequence, it follows that

D [𝑔] ∫ R 𝑛 𝑔 ( 𝑥) 2 1+ | 𝑥 | 2 2 𝑑𝑥 ≤ 1 𝜇 2 = 1 (𝑛 + 2) (𝑛 + 4) (5.4)
Taking into account the fact that

𝑢 ★ 𝑝 𝐿 𝑝 (R 𝑛 ) = 2 1-𝑛 𝜋 𝑛+1 2 Γ( 𝑛+1 2
) and the expression of C 𝑛 in (1.1), we obtain

H [𝑢 𝜀 ] ≥ 4 𝜀 2 𝑛 (𝑛 + 2) 1 𝑛 + 4 - 𝜀 3 (𝑛 + 2) ≥ 𝜅 𝑛 𝜀 2 ∫ R 𝑛 𝑔(𝑥) 2 1 + |𝑥| 2 2 𝑑𝑥 ,
because 𝜀 < 1. This completes the proof of Theorem 5.1.

A stability result similar to Theorem 5.1 can be established for (HLS 𝜆 ) with the particular choice 𝜆 = 𝑛 -2𝛼 and 𝑝 = 𝑡 = 2 𝑛/(𝑛 -2𝛼) of the parameters, 𝛼 ∈ (0, 𝑛/2), by adapting the computations of [START_REF] Jankowiak | Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities[END_REF]Section 4].

Let us consider on 𝐻

-1 (R 𝑛 ) the norm 𝑓 = ∇(-Δ) -1 𝑓 𝐿 2 (R 𝑛 ) such that 𝑓 2 = ∫ R 𝑛 𝑓 (-Δ) -1 𝑓 𝑑𝑥 = 𝜋 2-𝑛 2 Γ(𝑛 -2) ∬ R 𝑛 ×R 𝑛 𝑓 (𝑥) 𝑓 (𝑦) |𝑥 -𝑦| 𝑛-2 𝑑𝑥 𝑑𝑦 .
Here we consider again the case 𝜆 = 𝑛 -2 of (HLS 𝜆 ), with 𝑝 = 𝑡 = 2 𝑛/(𝑛 + 2), and recall that 𝐿 𝑝 (R 𝑛 ) ⊂ 𝐻 -1 (R 𝑛 ). For any 𝑓 ∈ 𝐿 𝑝 (R 𝑛 ) and any ℎ ∈ M := M HLS,1 , let us define the quotient

Q[ 𝑓 , ℎ] := 𝑓 2 𝐿 𝑝 (R 𝑛 ) -𝑆 𝑛 (-Δ) -1/2 𝑓 2 𝐿 2 (R 𝑛 ) 𝑓 -ℎ 2 .
Our goal is to prove that for some 𝜅 > 0

inf ℎ ∈M Q[ 𝑓 , ℎ] ≥ 𝜅 (5.5)
for all functions in a small neighbourhood of M , to be defined. Theorem 5.1 is already a result of this type: if

𝑔 = 1 𝜀 ( 𝑓 -𝑢 ★ ) 𝑢 -4 𝑛+2 ★ satisfies (5.
2) for some 𝜀 ∈ (0, 1), using (5.4), we obtain a first local stability result with 𝜅 = (𝑛 + 2) (𝑛 + 4) 𝜅 𝑛 . An important drawback is that the topology of the relative uniform convergence is very strong so that the eligible functions 𝑓 is a small set in the natural function space.

A local stability result in a weighted norm

Here we want to work in the more natural framework of the norm defined by (5.1). We recall that 𝑝 = 2 𝑛/(𝑛 + 2). By Hölder's inequality, we can write

𝑓 2 𝐿 𝑝 (R 𝑛 ) ≤ 𝑢 ★ 4 𝑛+2 𝐿 𝑝 (R 𝑛 ) 𝑓 2 ★ , (5.6)
which means that 𝑓 2 ★ is a stronger norm than 𝑓 2 𝐿 𝑝 (R 𝑛 ) . Equality holds in (5.6) for 𝑓 > 0 such that 𝑓 𝐿 𝑝 (R 𝑛 ) = 𝑢 ★ 𝐿 𝑝 (R 𝑛 ) if and only if 𝑓 = 𝑢 ★ so that

𝑓 2 𝐿 𝑝 (R 𝑛 ) -𝑢 ★ 4 𝑛+2 𝐿 𝑝 (R 𝑛 ) 𝑓 2 ★
measures the distance of 𝑓 to 𝑢 ★ . This can be made precise using (1 + 𝑠) 2/ 𝑝 ≥ 1 + 2 𝑠/𝑝 for any 𝑠 ≥ 0 and, according to the generalised Pinsker-Csiszár-Kullback inequality (see e.g., [6, Proposition 1.1]) as follows: for any function 𝑓 ∈ 𝐿 𝑝 (R 𝑛 ),

𝑓 2 𝐿 𝑝 (R 𝑛 ) -𝑢 ★ 2 𝐿 𝑝 (R 𝑛 ) ≥ 2 𝑝 𝑢 ★ 2-𝑝 𝐿 𝑝 (R 𝑛 ) ∫ R 𝑛 | 𝑓 | 𝑝 𝑑𝑥 - ∫ R 𝑛 𝑢 𝑝 ★ 𝑑𝑥 ≥ 2 1-2 𝑝 ( 𝑝 -1) | 𝑓 | -𝑢 ★ 2 𝐿 𝑝 (R 𝑛 ) .
For simplicity, we shall assume again that 𝛼 = 1, so that 𝜆 = 𝑛 -2 as in Section 5.1. The extension to 𝛼 ≠ 1 is left to the reader. Since 𝑢 ★ is a critical point of H [ 𝑓 ], we can write that

1 + Q[ 𝑓 , 𝑢 ★ ] = C 𝑛 𝑓 2 𝐿 𝑝 (R 𝑛 ) -𝑢 ★ 2 𝐿 𝑝 (R 𝑛 ) -2 𝑢 ★ 4 𝑛+2 𝐿 𝑝 (R 𝑛 ) ∫ R 𝑛 𝑢 𝑛-2 𝑛+2 ★ ( 𝑓 -𝑢 ★ ) 𝑑𝑥 𝑓 -𝑢 ★ 2
Theorem 5.2. Let 𝑛 ≥ 3. Let 𝑓 ∈ 𝐿 𝑝 (R 𝑛 ) be a nonnegative function such that the orthogonality conditions (5.3) are fullfiled by

𝑔 = ( 𝑓 -𝑢 ★ ) 𝑢 -4 𝑛+2 ★ . If one has 𝑛 + 4 3 (𝑛 + 2) (1 + 𝜂) ∫ R 𝑛 𝑢 𝑝-3 ★ | 𝑓 -𝑢 ★ | 3 𝑑𝑥 ≤ ∫ R 𝑛 𝑢 𝑝-2 ★ | 𝑓 -𝑢 ★ | 2 𝑑𝑥 (5.7)
for some 𝜂 > 0, then (5.5) holds with 𝜅 = 4 𝜂 𝑛 (1+𝜂) . Proof. Let us consider ℎ such that 𝑓 -𝑢 ★ = 𝜀 ℎ with 𝜀 := 𝑓 -𝑢 ★ and define By (5.8), we have X ≥ (𝑛 + 2) (𝑛 + 4), which concludes the proof.

X := ℎ 2 ★ = ∫ R 𝑛 𝑢 𝑝-2 ★ ℎ 2 𝑑𝑥 and Y := ∫ R 𝑛 𝑢 𝑝-3 ★ |ℎ| 3 𝑑𝑥 . A Taylor-Lagrange expansion shows that ∫ R 𝑛 | 𝑓 | 𝑝 𝑑𝑥 ≥ ∫ R 𝑛 𝑢 𝑝 ★ 𝑑𝑥 + 𝑛 (𝑛 -2) (𝑛 + 2) 2 𝜀 2 X - 4 𝑛 (𝑛 -2) 3 (𝑛 + 2) 3 𝜀 3 Y . Using (1 + 𝑠) 2/ 𝑝 ≥ 1 + 2 𝑠 𝑝 for any 𝑠 ≥ 0, we obtain 𝑓 2 𝐿 𝑝 (R 𝑛 ) -𝑢 ★ 2 𝐿 𝑝 (R 𝑛 ) ≥ 𝐾 0 𝜀 2 𝑛 (𝑛 -2) (𝑛 + 2) 2 X - 4
With the notations of the proof of Theorem 5.2, Condition (5.7) can be rewritten as X -K Y ≥ 0 with K = 𝑛+4 3 (𝑛+2) 𝜀 (1 + 𝜂). This somewhat unusual condition has two implications:

(1) ℎ ∈ 𝐿 𝑝 (R 𝑛 ) is bounded in the stronger norm ∫ R 𝑛 𝑢 𝑝-3 ★ |ℎ| 3 𝑑𝑥 1/3 , which is however a much weaker condition than (5.2),

(2) the function ℎ is limited to an explicit neighbourhood of 0, in strong norms.

A detailed statement goes as follows. The stability assumption (5.9) means that

0 ≤ X -K Y ≤ X - K a X 3-𝑝 2-𝑝 = X 1 - K a X 1 2-𝑝 , 0 ≤ X -K Y ≤ (a Y) 2-𝑝 3-𝑝 -K Y = Y 2-𝑝 3-𝑝 a 2-𝑝 3-𝑝 -K Y 1 3-𝑝 ,
which proves X ≤ a K -1 2-𝑝 and Y ≤ K -1 a K -1 2-𝑝 , i.e., (5.10a) and (5.10b). By Hölder's inequality, we obtain

H 2 𝐿 1 𝑛 2 𝑛 + 2 𝑛 - 2 𝜀 2 = 4 𝜀 2 𝑛

 21222242 On the other hand, Inequality(5.4) applied to 𝑔 = 1 + |𝑥| 2 2 ( 𝑓 -𝑢 ★ ) shows that ∫ R 𝑛 ( 𝑓 -𝑢 ★ ) (-Δ) -1 ( 𝑓 -𝑢 ★ ) 𝑑𝑥 ≤ 𝑓 -[ 𝑓 ] = C 𝑛 𝑓 2 𝐿 𝑝 (R 𝑛 ) -𝑢 ★ 𝑝 (R 𝑛 ) -∫ R 𝑛 ( 𝑓 -𝑢 ★ ) (-Δ) -1 ( 𝑓 -𝑢 ★ ) 𝑑𝑥 ≥ 𝑛 (𝑛 -2) (𝑛 + 2) 2 X -4 𝑛 (𝑛 -2) 3 (𝑛 + 2) 3 𝜀 Y -𝜀 2 X (𝑛 + 2) (𝑛 + 4)

Proposition 5 . 3 . 3 ★|ℎ| 3 9 ) 2 ★ℎ 2 3 ★|ℎ| 3 2 - 2 ★ ℎ 2 3 ★ |ℎ| 3

 53339223322233 If ℎ is a function in 𝐿 𝑝 (R 𝑛 ) such that ∫ R 𝑛 𝑢 𝑝-𝑑𝑥 < +∞, which satisfies ∫for some K > 0, then ℎ also has the following properties:∫ R 𝑛 𝑢 𝑝-𝑑𝑥 ≤ K -1 ℎ 𝑝 2-𝑝 𝐿 𝑝 (R 𝑛 ) 𝑑𝑥 ≤ K -1 K -1 ℎ 𝑝 2-𝑝 𝐿 𝑝 (R 𝑛 ) 𝐿 𝑝 (R 𝑛 ) ≤ K -1 𝑢 ★ 𝐿 𝑝 (R 𝑛 ) With a = ∫ R 𝑛 |ℎ| 𝑝 𝑑𝑥 1 𝑝 , X := ∫ R 𝑛 𝑢 𝑝-𝑑𝑥, Y := ∫ R 𝑛 𝑢 𝑝-𝑑𝑥as in the proof of Theorem 5.2, the inequality ∫

2 with

 2 b := 𝑢 ★ 𝐿 𝑝 (R 𝑛 ) . After taking into account (5.10a), this amounts to a ≥ b K 𝑝 2-𝑝 , which concludes the proof of (5.10c).

  By a Taylor-Lagrange expansion again, we know that𝑠 𝑞 ≥ 𝑠 𝑞 ★ + 𝑞 𝑠 ★ 𝑞-1 (𝑠 -𝑠 ★ ) if 𝑞 > 1 and 𝑠 > 𝑠 ★ > 0. Applied with 𝑞 = 1 + 2/𝑛, 𝑠 = ∫ R 𝑛 |𝑢 𝜀 | 𝑝 𝑑𝑥 with 𝜀 ∈ (0,

		1) so
	that 4 𝜀 3 (𝑛+2) < 1, and 𝑠 ★ =	∫ R 𝑛 𝑢 𝑝 ★ 𝑑𝑥, we infer that

Funding. This research has been partially supported by the projects EFI ANR-17-CE40-0030 (J.D.) and molQED (M.J.E.) of the French National Research Agency (ANR).


Paolo Zappalà

Amal Benhamiche

Matthieu Chardy

Francesco De Pellegrini

Rosa Figueiredo

A timing game approach for the roll-out of new mobile technologies

Keywords: technology adoption, timing games, two-player extensive form games

When adopting a novel mobile technology, a mobile network operator faces the dilemma of determining which is the best time to start the installation of next generation equipment onto the existing infrastructure. In a strategic context, the best possible time for deployment is also the best response to competitors' actions, subject to normative and material constraints and to the customer's adoption curve. We formulate in this paper a finite discrete-time game which captures the main features of the problem for a two-player game played over a prescribed finite horizon. Our numerical results provide insights on the possible optimal tradeoffs for an operator between fixed costs and installation strategies.

I. INTRODUCTION

In the telecommunication industry, the roll-out of a new mobile communication technology is a core challenge faced periodically by mobile operators, requiring to upgrade their infrastructure to keep up with ever-increasing traffic demand and key technology upgrades. Indeed, while standardization efforts are already taking off for next generation 6G mobile networks, it is only very recently that 5G technology was introduced into the mobile communication market [START_REF] Oughton | The cost, coverage and rollout implications of 5G infrastructure in Britain[END_REF]. What happened is that the current ever-increasing traffic demand has rendered the previous generation of 4G mobile networks inadequate. This has forced most telecommunication operators in the last months to install the 5G technology on their sites in order to meet a satisfactory user-experience to their customers.

A key success element for an operator, in this context, is the timing of the upgrades of their infrastructure and the most efficient timing available to them is defined by a competitive setting. Indeed, operators may want to defer the deployment costs for an economic advantage. But an untimely delay with respect to their competitors may represent a risk of future loss, mainly because this might induce their customers to switch to other operators' network. Furthermore, different types of sites have different return on investment and a different customer base, depending on multiple factors, including operators' positioning on the mobile communication market and geographical specifics.

Motivated by the recurring pattern of innovation in the mobile telecommunication industry, in this paper we propose a strategic framework to model the introduction of new wireless technologies and the main tradeoffs therein. Solution concepts and tools from game theory let us identify optimal technology ⋆ Orange Innovation, Chatillon, France; † LIA, Avignon University, Avignon, France.

deployment plans. We focus on the case of two competing operators performing their technology upgrades on a given set of sites.

The problem is formulated in the form of a discrete-time timing game [START_REF] Arcidiacono | Estimation of dynamic discrete choice models in continuous time[END_REF], where actions sets available to each player are defined by logistic and normative constraints. For instance, all operators who joined the auction for the allocation of the 5G band spectrum have to follow certain deployment constraints set by the regulator [START_REF] Oughton | The cost, coverage and rollout implications of 5G infrastructure in Britain[END_REF]. In turn, the operators' utility is a function of the customers' technology adoption and their distribution over the operator sites, of costs incurred for the technology upgrades and how effective it is the promotion and marketing for the launch of the new technology. Quality of service offered by operators is assumed a piece of public information, as it is constantly monitored by independent authorities (see [3] for France). A player, i.e., an operator, is hence able to track the evolution of the competitors' 5G deployment. On the other hand, upon choosing a 5G service provider, customers are assumed to bind to this operator until the end of the time horizon. To this aim, as a new technology is introduced, the operators use financial subsidies to accelerate its adoption e.g. using discounted offers or advertisement. We assume that such subsidies are budgeted strategically for the whole time horizon and simultaneously adopted by the operators at the beginning the deployment phase.

State of the art. In the literature, several papers studied the investment optimization problem faced by operators aiming to mitigate installation costs of the 5G technology and yet satisfy increasing traffic demand. One approach considers the behaviour of the operator's customers [START_REF] Cambier | Optimizing the investments in mobile networks and subscriber migrations for a telecommunication operator[END_REF], [START_REF] Chen | Financial analysis of 4G network deployment[END_REF]; such models do not take into account the competition among operators. Another approach is to set a cooperative game which determines the social optimum and how investments should be allocated among the operators [START_REF] Gajić | Competition of wireless providers for atomic users: Equilibrium and social optimality[END_REF]- [START_REF] Cano | On optimal infrastructure sharing strategies in mobile radio networks[END_REF]. Some models do account for the fact that operators are in competition to serve potential customers [START_REF] Cano | A non-cooperative game approach for RAN and spectrum sharing in mobile radio networks[END_REF]- [START_REF] Datar | A mechanism for price differentiation and slicing in wireless networks[END_REF], but they neglect the temporal dimension. In the economic theory, the introduction of a new technology belongs to a specific class of models, called innovation timing games. This category of games concerns two players selecting the time at which they act [START_REF] Hoppe | Innovation timing games: a general framework with applications[END_REF]- [START_REF] Duan | Economic analysis of 4G network upgrade[END_REF]. This standard scenario is concerned, for instance, with the dynamics of an incumbent which defends from a possible entrant in a market [START_REF] Smirnov | Blocking in a timing game with asymmetric players[END_REF]. Results for more than two players are derived in [START_REF] Argenziano | Clustering in-Player Preemption Games[END_REF], [START_REF] Steg | On preemption in discrete and continuous time[END_REF] under special assumptions. In game theory, games where players pick a time when they act are called timing games and can be either in continuous [START_REF] Laraki | Continuous-time games of timing[END_REF] or discrete [START_REF] Arcidiacono | Estimation of dynamic discrete choice models in continuous time[END_REF] time. In the literature, discrete timing games have been used for marketing decisions [START_REF] Draganska | Discrete choice models of firms' strategic decisions[END_REF]. In this work we focus on a two-player discrete timing game, in which two operators decide when to start the roll-out. Our model also belongs to the class of sequential games [START_REF] Fudenberg | Subgame-Perfect Equilibria of Finite-and Infinite-Horizon Games[END_REF], in which players act in turns one after another. In the literature, results are only given for specific categories of timing games: stochastic games with one choice [START_REF] Riedel | Subgame-perfect equilibria in stochastic timing games[END_REF], Stackelberg games with random-ordered players [START_REF] Doraszelski | Dynamic stochastic games with sequential state-to-state transitions[END_REF] and games with small discrete time intervals [START_REF] Schmidt-Dengler | The timing of new technology adoption: The case of MRI[END_REF].

Main contribution. To the best of our knowledge, this is the first work that introduces a timing game model for the roll-out of mobile technologies. The model can factor in the customers' adoption dynamics, the operators' installation strategies on multiple classes of sites, and constraints related to logistics and regulation. The system model relies on an extensive form game solved with a tailored-made formulation based on a classic resolution method [START_REF] Stengel | Efficient computation of behavior strategies[END_REF]. Numerical simulations provide interesting insight into the strategic approach of the operators and the resulting market shares.

The paper is structured as follows. In Section II we introduce the model and in Section III we provide an example of application. In Section IV we introduce the methods used for model simulations. In Section V we report for numerical results and we interpret them in light of the possible strategies for the operators. Section VI ends the paper.

II. SYSTEM MODEL

Modern telecommunication markets have a few mobile operators in strong competition with one another. A significant fraction of their annual turnover is spent on the maintenance and the upgrade of their network. Thus, the main objective of our model is to study strategies to optimize the allocation of operators' resources. Strategies such as prioritising the time of installation on more profitable sites or performing offers and marketing investments have to be optimized in light of competitors' choices, expected returns and regulation constraints. We restrict to the case of two telecommunication operators where both act as rational players. Each player seeks the optimal strategy to maximize their own return. At the moment of introducing a new mobile technology on the market, both operators first invest a promotion and marketing budget, P&M budget hereafter, to boost the customer base, i.e., launching marketing campaigns and releasing a fixed number of promotion subscription offers. In our model each operator i ∈ N chooses independently their P&M budget from a discrete set s i ∈ S i . Second, they launch the field deployment campaign for the new technology, i.e., they schedule when and where to deploy investments on their own sites. A site is an area where both operators can install the new technology. Since operators often build sites close to the competitors', we consider perfect overlap of the sites. We assume that there exist few classes of sites, depending on their profitability: metropolitan sites, urban sites, and rural sites. Each of such classes has a different profitability depending, e.g., on the number of subscribers. Operators act on a discrete time horizon T = {1, ..., T }. We thus introduce the following parameters:

• N = {1, 2}, set of players; • S i , i.e., the P&M budget options for player i ∈ N (chosen at time t = 0); • T = {1, ..., T }, set of time-intervals over which operators act to install the new technology; • A , site classes.

The possible actions of a player are defined by the variables: • s i ∈ S i , the P&M budget chosen by player i at time 0; • A it ∈ P(A ), a subset of classes of sites (possibly empty) on which operator i ∈ N installs the new technology at time t ∈ T . After the installation, no other intervention is made. We thus denote by t ia ∈ T the time at which operator i installs the technology on class a ∈ A (in vectorial form

t i ∈ T |A | ).
Each operator's schedule is bounded by some constraints:

τ ≤t |A iτ | ≥ R t .
The strategy (s i , t i ) ∈ S i × T |A | is a choice made by player i ∈ N on the budget s i ∈ S and on the roll-out scheme t i ∈ T |A | . This choice is subject to the maximisation of the player's objective, which is modeled by her utility function

u i : S 1 × S 2 × (T |A | ) 2 → R.
The utility functions depend on multiple parameters, which involve costs, characteristics of the market and adoption dynamics. We defer such analysis to Section III, in which we provide an example of its model.

Strategic framework. Players have conflicting interests, seeking to gain the largest market share, while keeping costs under control. As such, an operator cannot compute their own solution independently from their competitors. We thus identify an equilibrium of the game, i.e., a couple of strategies such that players are satisfied with them if they are both played. We add a further assumption: players can observe, at time t, the history of actions taken by the other player for t ′ < t, and react accordingly. The choice of the opponent's P&M budget is observable by both players after time t = 0.

A convenient model to encode the above assumptions is that of a game in extensive form [START_REF]Contributions to the Theory of Games[END_REF], whose mathematical model is based on a game tree. In a game tree, arcs outgoing from a node represent the possible choices available to a player. The sequence of nodes and arcs represent the sequence of choices taken by the players and the moments at which players act. A leaf or terminal node h reached after both players have performed T actions has no outgoing arcs and represents an outcome of the game. We denote with H the set of terminal nodes h ∈ H; occasionally we will use the same notation to identify the unique path that leads to a terminal node. Every outcome h ∈ H is assigned a pair of values (u 1 (h), u 2 (h)) ∈ R 2 which correspond to the values assigned by each player to such outcome. The higher the value of u i (h), the higher the value a player assigns to the combination of actions that leads to the final node h. Every combination of strategies (s 1 , s 2 , t 1 , t 2 ) ∈ S 1 ×S 2 ×(T |A | ) 2 leads to a unique outcome h ∈ H. For our system the game tree is defined as the Service providers (SP) game (cf. Definition 1)

Definition 1 (Service providers (SP) game): The service providers game ⟨N, S 1 , S 2 , A , T, u⟩ is an extensive form game with two players N = {1, 2} competing over set of classes of sites A in which:

• at the root vertex both players choose at the same time t = 0 and independently the P&M budgets s 1 ∈ S 1 and s 2 ∈ S 2 ;

• the players act in sequence at every round t ⩾ 1, starting from player 1. At every step they can decide on which subset of class of sites A 1t ∈ P(A ) and A 2t ∈ P(A ) install the new technology, given the logistic and regulator constraints;

• after T rounds the game ends; the actions chosen at each round are evaluated by utility functions u i hereafter defined;

• players' utility function u = (u 1 , u 2 ), where u i :

S 1 × S 2 × (T |A | ) 2 → R.
Example. Figures 1 and2 show a representation of a game with the following properties: • A = {A, B}, two classes of sites.

• T = {1, 2}, horizon of two time-intervals.

• S 1 = {no, low, high} and S 2 = {no, low, mid, high} set of possible P&M budgets.

Overall, we can distinguish two phases of the game. In the first phase both players choose their P&M budget for the launch of the technology (details are given in Section II-A). In the second phase players act in sequence one after another and choose on which sites install the new technology (Section II-B).

A. First phase: P&M budget choice

Both players pick independently their P&M budget s i ∈ S i , i = 1, 2 (cf. Figure 1). Once budgets (s 1 , s 2 ) are revealed, the players proceed in choosing the timing of their investments. We denote this second part of the game Γ(s 1 , s 2 ), which is represented in Figure 2 for our example. In Section II-B we discuss in detail how to compute the timing of investments as a solution (t

1 , t 2 ) ∈ (T |A | ) 2 of game Γ(s 1 , s 2 ).
(s 1 , s 2 , t 1 , t 2 ) ∈ S 1 × S 2 × (T |A | ) 2 is evaluated by the utility function u(s 1 , s 2 , t 1 , t 2 ) ∈ R 2 .
We define map M : S 1 × S 2 ∈ R 2 which evaluates for every pair (s 1 , s 2 ) the solution of the game Γ(s 1 , s 2 ):

M (s 1 , s 2 ) := u(s 1 , s 2 , t 1 , t 2 ),
where (t 1 , t 2 ) ∈ (T |A | ) 2 solves Γ(s 1 , s 2 ). Let us suppose now to have already computed the solutions of the second phase M (s 1 , s 2 ) for every (s 1 , s 2 ) ∈ S 1 × S 2 with the method given in Section II-B. Table I shows such computation for the game of Figure 1. We would like to find a solution of the game, i.e., a Nash Equilibrium for the corresponding matrix game: a pair of strategies such that an operator, known the strategy of the other one, does not deviate unilaterally.

Definition 2 (SP game, solution first phase): Given a SP game ⟨N, S 1 , S 2 , A , T, u⟩ and its correspondent matrix 2 the optimal installation times chosen at times t ≥ 1, we say (s 1 , s 2 ) ∈ S 1 × S 2 is an equilibrium if for all s 1 ∈ S 1 and s 2 ∈ S 2 we have:

M = (M 1 , M 2 ) : (s 1 , s 2 ) → u(s 1 , s 2 , σ(s 1 , s 2 )), with (s 1 , s 2 ) chosen at time t = 0 and σ(s 1 , s 2 ) ∈ (T |A | )
M 1 (s 1 , s 2 ) ≥ M 1 (s 1 , s 2 ), M 2 (s 1 , s 2 ) ≥ M 2 (s 1 , s 2 ).

B. Second phase: timing of network investments

Let both operators have chosen their P&M budgets (s 1 , s 2 ) ∈ S 1 × S 2 . The second phase of the game is represented by the subtree Γ(s 1 , s 2 ) of Figure 2. At the root the first player chooses a subset of sites where to invest at time t = 1. The choices are represented by the outgoing arcs. At the following node the second player chooses their own site class for time t = 1. Their actions lead to a node at which the first player chooses the installations to be made at time t = 2, and so on until the planning for each time interval is assigned. The two operators know which choice of the P&M budget is made by the other operator and then act in sequence one after another. At every time t ∈ T each operator i ∈ N chooses a class of sites where to invest. Formally, the subtree Γ(s 1 , s 2 ) is an extensive form game, identified by the tuple Γ(s 1 , s 2 ) = ⟨N, H, u⟩ from now on, where H is the set of terminal nodes of the game tree. With some notations' abuse h ∈ H identifies either a terminal node or the path that leads to it from the root. Note that along a path a player can pick a site class only once (installation can be done only once).

In the example of Figure 2 path h = {{B}, {A}, {A}, {B}} is highlighted: at time t = 1 the first operator installs the new technology on class B, the second operator installs it on class A, while at time t = 2 the two operators do the opposite.

Definition 3 (SP game, second phase): Given a SP game ⟨N, S 1 , S 2 , A , T, u⟩ and a couple of P&M budget choices (s 1 , s 2 ) ∈ S 1 × S 2 we consider the extensive form game Γ(s 1 , s 2 ) = ⟨N, H, u⟩, whose paths h = (h it ) i∈N,t∈T ∈ H have the following properties:

• for all (i, t) ∈ N × T we have h it ∈ P(A );

• for all i ∈ N and for all t, t ′ ∈ T with t ̸ = t ′ we have

h it ∩ h it ′ = ∅.
Later in the section we introduce the solution of a game, which is a specific terminal node h ∈ H. We represent such node with a vector (t ia ) i∈N,a∈A , which indicates the time t ia ∈ T at which operator i installs the new technology on class a.

In the tree representing the game (see Figure 2) a step is represented by a node and the actions that follows by the outgoing arcs. A subgame describes the part of the game that follows a given step: it corresponds to the part of the tree that follows a node. Formally, we identify the node with the vector of actions that leads to it. Such vector is a prefix h of the vector that corresponds to a path h ∈ H. A subgame is the collection of the paths that share the same prefix. Given path h ∈ H, we write h = h + h ′ to show that h is a prefix and h ′ is the rest of the path.

Definition 4 (subgame): Given a game Γ(s 1 , s 2 ) = ⟨N, H, u⟩ and a path h ∈ H, a subgame is a game Γ = ⟨N, H, u⟩ such that:

• H is the set of paths in H sharing prefix h: H = {h ′ : ∃h ∈ H, h = h + h ′ };
• The utility function in Γ corresponds to the ones in Γ, i.e. for all h ′ ∈ H we have that:

u(h ′ ) = u(h + h ′ ).
As defined before, we would like to identify a solution of the game under the assumption that every operator at time t can observe the actions taken by the other player at time t ′ < t. This corresponds to require the solution to be an equilibrium for every subgame, i.e., to be a subgame perfect equilibrium (SPE) [START_REF] Fudenberg | Subgame-Perfect Equilibria of Finite-and Infinite-Horizon Games[END_REF]. Operators can forecast future actions, starting from the bottom of the tree, since they both know that they choose the actions that maximise their utility. The algorithm that finds recursively the best action is called backward induction algorithm [START_REF] Fudenberg | Subgame-Perfect Equilibria of Finite-and Infinite-Horizon Games[END_REF]. If an operator is on a subgame corresponding to a leaf of the game tree, they pick an action that maximises their utility. In recursive manner, in any parent subgame an operator can identify the actions played subsequently and also identify the action that leads to the best outcome for them. The solution identified by this algorithm corresponds to a SPE. A generic finite extensive form game with perfect information has a unique SPE equilibrium [START_REF] Fudenberg | Subgame-Perfect Equilibria of Finite-and Infinite-Horizon Games[END_REF] and it is generated by the backward induction algorithm [START_REF]Contributions to the Theory of Games[END_REF].

The choice of the order of the operators can be arbitrary. However, while in reality they act simultaneously, when both players can quickly adjust their strategy accordingly, the outcome of the game undergoes a small perturbation, as we see in simulations.

III. SUBSCRIBER DYNAMICS AND OPERATOR UTILITY

So far we have not discussed the operators' utility function

u i : S 1 × S 2 × (T |A | ) 2 → R.
As described next, it highly depends on the adoption dynamics, i.e., how many customers switch to the new mobile technology at each time unit. To characterize the utility function we fix some assumptions: 1) following [START_REF] Cambier | Optimizing the investments in mobile networks and subscriber migrations for a telecommunication operator[END_REF], every customer decides to switch to the new technology at a given time t and sticks to the choice till the end of the time horizon; 2) the quantity of customers switching at every time depends on a given dynamics, which is known a priori by both operators; 3) every customer has a preference over the two operators; if their preferred operator does not offer the new technology at time t they subscribe to the other operator if it offers it, otherwise they wait for one of them to offer it; 4) an operator can acquire only a limited amount of new customers per time interval; 5) once a customer subscribes for an operator, it sticks to it. Operators choose a pair of P&M budgets (s 1 , s 2 ) ∈ S 1 ×S 2 ; such choices influence the potential market of customers willing to switch to the new technology.

We consider first the scenario where all sites belong to the same class, and we then extend the analysis to the case where sites belong to different ones.

A. Single-class model

For every t ∈ T , the percentage of customers adopting the new mobile technology is identified by {y t } t∈T , subject to the condition that t∈T y t = 1 because all customers eventually switch to the new technology. Hence, {y t } can be seen as the time-step increment y t := y adopt (t + 1) -y adopt (t) of the adoption curve y adopt (t), which is assumed to be nondecreasing concave and y(∞) = 1. A model often used in the literature [START_REF] Oughton | Towards 5G: Scenario-based assessment of the future supply and demand for mobile telecommunications infrastructure[END_REF] for adoption curves is y adopt (t) = 1 -e -λt , where the parameter λ > 0 determines the adoption speed.

Preferences over the offers proposed by operators 1 and 2 are a function of the P&M budgets. They are distributed with same proportion p 1 (s 1 , s 2 ) and p 2 (s 1 , s 2 ) = 1 -p i (s 1 , s 2 ) for all t ∈ T . We fix p

1 (s 1 , s 2 ) = p01+k•s1 1+k•(s1+s2) and p 2 (s 1 , s 2 ) = p02+k•s2 1+k•(s1+s2)
, where p 01 , p 02 ∈ [0, 1] with p 01 + p 02 = 1, the preference of a customer for operator 1 or 2, respectively, if

s 1 = s 2 = 0.
Here, k is a parameter that weights the influence of the P&M budget on the customers. We recall that y t is the potential market at time t. In order to capture these customers the operators have to install the new technology; once their sites are equipped with the new technology, we assume that mobile operators can acquire at most τ > 0 share of new customers per time slot. This constraint models the fact that potential customers resolve for the technology little by little and not all at once. Let α i (t) := α i (t; s 1 , s 2 , t 1 , t 2 ) the revenue for player i which can be ascribed to customers acquired up to time t ∈ T , respectively, under a given multistrategy (s 1 , s 2 , t 1 , t 2 ) (in case of a single site (s 1 , s 2 , t 1 , t 2 )). Once acquired, they are retained until the end of the horizon: the revenue generated by the final market share is α i (T ).

On the other hand, operators face the costs related to the installation of the new technology and the subsidies. P&M budgets s i ∈ S i are constant fixed costs. Conversely, c i (t) > 0 is the installation cost at time step t ∈ T : it is discounted to account for the depreciation since installation time t and lower maintenance costs over the period. Player i installing at time t = 1 incurs in cost c i (1) = c i,0 > 0, whereas we assume c(T

) = 0. A linear discounted cost model sets c i (t) = c i,0 • T -t T -1 .
Finally, the utility for player i writes

u i (s 1 , s 2 , t 1 , t 2 ) = α i (T ) -c i (t i ) -s i .
Summing up, the game is determined by parameters:

• λ > 0: adoption speed;

• τ > 0: maximal share of customers that can be acquired in a time interval;

• S 1 , S 2 ⊂ [0, +∞) with |S 1 | < ∞ and |S 2 | < ∞: discrete sets of choices for the P&M budget; • p 0i ∈ [0, 1], i = 1
, 2: fraction of customers preferring operator i when no P&M budget is deployed s 1 = s 2 = 0;

• k > 0, weight of the P&M budget;

• c 0 > 0, the installation cost at time t = 1.

We now describe how the functions α 1 (t) and α 2 (t) depend on the strategies chosen by the players among the states t ∈ T .

The system state at time t is also described by the following variables: 1) y t , the percentage of customers who decide to switch to the new technology at time slot t; 2) t i , time at which operator i installs the technology; 3) d t , the demand of customers who want to switch to the new technology at time t and who are not served before time t; 4) d it , the fraction of d t who prefer the operator i (with d i0 = 0); 5) r it , customers that operator i can accept at time t. In particular, if an operator i has installed the technology at time t i , they can accept up to τ customers per interval of time:

r it = τ • 1{t ≥ t i }.
The demand at time t is updated with the new costumers. All the customers that are added to the demand at time t are y t . Given p i := p i (s 1 , s 2 ) the fraction of customers who prefer operator i, we have that the number of customers who prefer i at time t are p i • y t , i.e. ∀i ∈ N :

d it ← d i,t-1 + p i • y t .
Up on their demand and on their supply, the operators add their customers. The operator i can absorb the unfulfilled demand of its competitor j. The dynamics of customers acquired by operator i is thus governed by equation

α i (t + 1) = α i (t)+min(d it , r it ) + [min(d jt -r jt , r it -d it )] +
where i ̸ = j. Finally, the demand at time t+1 is set at d i,t+1 ← [d it -r it -[r jt -d jt ] + ] + since costumers acquired at time t by operator i are removed from their demand at time t + 1.

B. Multi-class model

Let us consider a model for more than one class of sites |A | > 1. We assume further that every customer is served on sites belonging to same class, and thus introduce the parameter ω a ∈ (0, 1), the percentage of users that are served on sites of class a ∈ A . Accordingly, we assign to every parameter an index referring to the relative class. The potential market y a,t for class a ∈ A is subject to the condition t∈T y a,t = ω a • y t . The utility function for the multi-class case differs from the single-class by the fact that it sums the customers acquired on every class of sites and the costs of the respective installations. For every player i ∈ N the utility writes

u i (s 1 , s 2 , t 1 , t 2 ) = a∈A α i,a (T ) - a∈A c i (t i,a ) -s i
where the first summation is the revenue calculated across all sites, and α i,a and c i (t i,a ) are respectively the utility and the cost for player i for sites of class a ∈ A .

IV. UPPER BOUNDS

As described in Section II, the SPE determined by backward induction requires the explicit construction of the game tree, whose size is polynomial in the time horizon but exponential in the number of site classes. An efficient method for the solution of the game is the one introduced by Von Stengel [START_REF] Stengel | Efficient computation of behavior strategies[END_REF], solving the equilibrium of extensive-form games via a bilevel optimization problem. Compared to the SPE proposed before, the solution found provides an upper bound to any Nash equilibrium of the timing game, and so to the SPE as well. The method is defined around the concept of sequence. A sequence is a vector of consecutive actions played by the same player. Let us consider a game with T = 2 and a path h = (h 11 , h 21 , h 12 , h 22 ) ∈ H, as for instance h = {{B}, {A}, {A}, {B}} of Figure 2. The actions h 11 and h 12 are played by the first player, while h 21 and h 22 are played by the second player. A sequence of the first player is seq 1 = (h 11 , h 12 ) and a sequence of the second player is seq 2 = (h 21 , h 22 ). Formally, for every player i ∈ N we consider the set of their sequences

Σ i = {seq i = (h kτ ), k = i, τ ≤ t, t ∈ T }. Let x ∈ {0, 1} |Σ1| and y ∈ {0, 1} |Σ2|
the vectors which define the probability for a sequence to be played. We confine to pure strategies, i.e., either x seq1 = 1 if sequence seq 1 is played or x seq1 = 0 if sequence seq 1 is not played. Every path h that leads to a leaf and the two corresponding sequences (seq 1h , seq 2h ) ∈ Σ 1 × Σ 2 are evaluated by the utility function u(h). We define thus the matrices U 1 , U 2 : Σ 1 × Σ 2 → R that map couples of sequences to their utilities: U 1 seq 1h ,seq 2h = u 1 (h) and U 2 seq 1h ,seq 2h = u 2 (h). The utilities of the players can be thus written in the form x T U 1 y and x T U 2 y. Note that the players' sequences are constrained according to the SP game. Indeed, let us suppose that after the path h = ({B}, {A}) the first player has already played seq 1 = ({B}) and thus can choose among the actions {∅, {A}}. We can write the corresponding sequences seq ′ 1 = ({B}, ∅) and seq ′′ 1 = ({B}, {A}). If any of these sequences is chosen, then the sequence seq 1 is also chosen. We thus write seq 1 = seq ′ 1 + seq ′′ 1 . All such causal constraints Ex = e and F y = f can be built according to the same principle. Finally, the Von Stengel solution for the SP game is a solution of the following bilevel problem (VS).

u V S 1 = max x x T U 1 y (VS) s.t. Ex = e x ∈ {0, 1} |Σ1| y = max y x T U 2 y s.t. F y = f y ∈ {0, 1} |Σ2|
This bilevel optimization problem is linear in the size of the game [START_REF] Stengel | Efficient computation of behavior strategies[END_REF]. The following result, whose proof is omitted for the sake of space, ensures that Problem (VS) provides an upper bound to the utility of any Nash equilibrium of the game. Theorem 1: Given an extensive-form game ⟨N, H, u⟩, the solution u V S 1 ∈ R of the corresponding optimization problem (VS) and the outcome of a Nash equilibrium h N E ∈ H, we have:

u V S 1 ≥ u 1 (h N E ).
We apply Theorem 1 to get an upper bound of the utility of the first player in a subgame perfect equilibrium. Indeed, since the subgame perfect equilibrium is a Nash equilibrium, we have that (VS) provides such upper bound. In Section V the method is deployed to approximate the value of the utility of the games.

V. NUMERICAL RESULTS

In this section we characterize a scenario for two operators N = {1, 2} and three classes of sites A = {A, B, C}, which correspond to metropolitan (A), urban (B) and rural sites (C). The roll-out occurs over a period of twelve time intervals T = {1, . . . , 12}, corresponding to twelve quarters, i.e., three years. Due to logistics constraints, operators can install only on a single class of sites per interval of time, i.e., for all i ∈ N and all t ∈ T , |A it | ≤ 1. Moreover, each of them is forced by the regulator to install the new technology on at least one class of sites before the end of the first year (t ≤ 4), i.e. R 4 = 1, and to deploy it everywhere before the end of the game (t ≤ 12), i.e. R 12 = 3. The first class has a larger customer base than the other two, which are instead of comparable size: ω A = 0.5, ω B = 0.3, and ω C = 0.2. Also, we set λ = 0.1 and τ = 1 6 . We use baseline preferences (p 01 , p 02 ) = (0.6, 0.4) with k = 3, that is operator 1 is the incumbent and operator 2 is the entrant. The initial investment cost c 0 = 0.5.

To identify the Nash equilibria of the game, we should compute M (s 1 , s 2 ) for every value of the P&M budget s 1 ≥ 0 and s 2 ≥ 0. We recall that M (s 1 , s 2 ) is the utility of the unique SPE of the game in extensive form Γ(s 1 , s 2 ). Since we can perform such computation only for a finite number of games, we make a suitable discrete selection of the values s 1 ∈ S 1 and s 2 ∈ S 2 . The choice of S 1 = {0, 0.1, 0.2} and S 2 = {0, 0.1, 0.2, 0.3} is justified hereafter. We refer to them also as respectively {no, low, mid} and {no, low, mid, high} budget for clarity.

We suppose that large values of s 1 and s 2 are not chosen by the operators. Indeed, if the operators invest too much in the P&M budget, they cannot pay off their investment. We would like thus to exclude those strategies that are not played by the operators. Formally, a strategy for the first player s 1 is said to be dominated if there exists another strategy s ′ 1 that provides better utility, no matter what the second player chooses as their strategy.

We first explore the best responses for different budget choices and under different customers' baseline preferences over the two operators. Figure 3a) shows the best response of the entrant operator, where s 2 ∈ [0, 0.3], when the incumbent operator does not spend any P&M budget for the launch of the new technology, i.e., s 1 = 0. The utilities of the two players evolve, given different choices of the budget of the second operator. The best response for the entrant operator is to play s 2 = 0.03 because it maximises the utility of the second player M 2 (s 1 , s 2 ) = 0.25. We can observe that the utility of the incumbent M 1 (s 1 , s 2 ) is generically decreasing in the P&M budget of the entrant s 2 .

We will now prove that some strategies are dominated, so that they can be excluded to compute Nash equilibria. Figure 3b) shows how the utility of the first player evolves when they choose s 1 = 0 and the second player chooses s 2 ≥ 0. Strategies dominated by s 1 = 0 can be computationally expensive to calculate for many values s 1 > 0. However, we can provide an upper bound u V S 1 (s 1 , s 2 ; s 1 ) of function u 1 (s 1 , s 2 ; s 1 ) by using the method introduced in Section IV. The solution of (VS) provides an upper bound to the utility of the first player at the SPE. We compute it for different values of s 1 ≥ 0. Figure 3b) shows the graphs of u V S 1 (s 1 , s 2 ; s 1 ) for s 1 = {0.1, 0.2, 0.3, 0.4, 0.5}. We find that the upper bound of the utility of the first player given s 1 ≥ 0.3 is always lower than the utility given by s 1 = 0, i.e. u V S 1 (s 1 , s 2 ; s 1 ≥ 0.3) < u 1 (s 1 , s 2 ; s 1 = 0) for all s 2 ∈ S 2 . Since u V S 1 (s 1 , s 2 ; s 1 ≥ 0.3) ≥ u 1 (s 1 , s 2 ; s 1 ≥ 0.3), we conclude that any strategy s 1 ≥ 0.3 does not provide a better utility to the first player than s 1 = 0.0, no matter what the second player chooses. We thus exclude any value of s 1 ≥ 0.3 for the first player. By performing a similar analysis on the second player we can thus limit the choice of the discrete sets to S 1 = {0, 0.1, 0.2} and S 2 = {0, 0.1, 0.2, 0.3}.

The choice of the P&M budget changes the customers' baseline preference for operators. Figure 4 shows how the market shares are distributed over the classes of sites as a function of the resulting customers' preference. As we observe there, in some cases the investment can become profitable only if an operator obtains the monopoly over a class of sites. We recall that the regulator's constraint forces both operators to install early on at least one class of sites. Where to invest early determines the possibility to create a monopoly on a certain class of sites and/or prevent the opponent to get one. For instance, if the incumbent has almost the monopoly, i.e., p 1 ≥ 0.9, they find it profitable to install early on all the classes. If the incumbent is strongly dominant, i.e. 0.7 ≤ p 1 < 0.9, they do not find it profitable to invest on class C, which is left as a monopoly to Operator 2. If Operator 1 is weakly dominant, i.e. 0.5 ≤ p 1 < 0.7, they get the monopoly on class B, but they are also forced to share the customer base on class A and even leave the monopoly on class C to Operator 2.

The solutions of the game are reported in Figure 5. More precisely Figure 5a) reports on the solutions of the timing game for each P&M budget pair, whereas Table 5b) reports on the corresponding attained utility.

From Figure 5a) we observe how different P&M budgetsby influencing the preferences of customers -lead to different investment plans. Also, they induce different shares of the customer base over different classes of sites as observed by comparing the possible scenarios of Figure 4.

For instance, if both operators do not invest any P&M budget for the launch of the new technology, namely (s 1 , s 2 ) = (no, no), they both deploy from the start (t = 1) in rural areas (class C). Afterwards, the first operator (the incumbent) installs in urban areas (B) and finally both operators install in metropolitan areas (A). Operators roughly split evenly customers in metropolitan areas (A) and in rural areas (C), while the incumbent obtains the monopoly in urban areas (B), i.e., where they install first.

We consider now the case when the incumbent increases the P&M budget: (s 1 , s 2 ) = (low, no). In this case they install immediately in metropolitan areas (A), i.e., at time t = 1. They thus obtain a monopoly over this class of sites. The entrant reacts by installing in rural areas (C) at time t = 2 and takes over those sites by getting a monopoly there. At time t = 3 the incumbent operator installs in urban areas (B) and gets all the market share for those sites. Finally, in such case metropolitan (A) and urban areas (B) are almost completely assigned to the first player, while rural areas (C) are monopoly of the second player.

While the existence of a Nash equilibrium in mixed strategies is guaranteed by theory [START_REF] Nash | Equilibrium points in n-person games[END_REF], Figure 5b) indicates that there is only one pure Nash equilibrium (low, mid). We notice that strategy s 2 = mid is the best response of the entrant operator to any strategy of the incumbent. Since the incumbent operator is a rational player, they know that the entrant operator plays s 2 = mid and thus chooses s 1 = low because they increase users' preference to them which maximises their utility. The equilibrium (low, mid) is not Pareto optimal: indeed, both operators attain larger utility by playing (no, low). However, outcome (no, low) is not an equilibrium: if the incumbent does not invest any P&M budget, the entrant gets larger utility by further increasing it. The incumbent thus invests a small P&M budget to avoid the entrant to get an excessive share of the market.

VI. CONCLUSION

Inspired by the ongoing roll-out of 5G mobile networks we introduced the Service providers (SP) game, a timing-game model to determine the optimal strategy for the installation of new mobile technologies. We consider a scenario where two operators have to perform investments over classes of sites with different profitability profiles. The model permits to analyse the dynamics of the roll-out strategies used by the operators accounting for promotion and marketing investments and regulatory constraints. We have provided a solution concept in the form of an equilibrium for the game and a method to compute it. Numerical results show that different strategic patterns depend on the balance between the baseline market share of incumbent and entrant operator. P&M investments 4, the operators split the market in classes A and C, while in class B operator 1 has the monopoly. b) Results of the simulation: utility, displayed in the following order (u 1 , u 2 ). Nash equilibrium is given by (low, mid).

may change the share presence of the operators, but higher investments may even result in lesser profits for both. Discussion and future works. We have observed that the timing of the investments may vary significantly depending on the customers' preferences. A P&M budget able to steer the customers' preferences represents thus a key component of operators' strategic behavior (cf. Figure 4). To this respect, since we are interested in pure equilibria, the range of the P&M budget can be bounded by exclusion of dominated strategies. This provides a computational advantage, because comparing the utility appearing in the matrix game M is expensive, since it requires to compute the SPE of multiple timing games. We have showed that the technique introduced in Section IV provides an effective mean to bound the utility that a player attains at the SPE and by computing the upper bound of multiple timing games we could exclude several dominated strategies.

The same model proposed in this paper can be extended to describe roll-out strategies for individual sites. Indeed, an improved granularity of the model would allow a better understanding of the impact of logistic and regulatory constraints at the local level. However, the complexity of the computation of the equilibrium of the game grows exponentially with the number of sites. In future works we will explore efficient methods to both reduce the space search and to categorize best responses at the equilibrium.

Fig. 1 .

 1 Fig. 1. The P&M budgets are picked first. Both players choose a low budget, i.e. (s 1 , s 2 ) = (low, low), leading thus to game F = Γ(s 1 , s 2 ) = Γ(low, low).

Fig. 2 .

 2 Fig. 2. After the P&M budgets are decided, players choose sequentially to add or not the new technology to sites of class A and B at each round t ∈ T = {1, 2, . . . , T }.

  Fig. 3. a) Evolution of M (s 1 , s 2 ) with s 1 = 0.0 and s 2 ∈ [0.0, 0.3]. b) Utility of the first player M 1 (s 1 , s 2 ) for s 1 = 0.0 and s 2 ∈ [0.0, 0.3], and its upper bound u V S 1 (s 1 , s 2 ) with s 1 = {0.1, 0.2, 0.3, 0.4, 0.5} and s 2 ≥ 0.0. The strategies s 1 ∈ {0.3, 0.4, 0.5} are dominated by the strategy s 1 = 0.0.

Fig. 4 .

 4 Fig. 4. Different share distribution for different values of p 1 ∈ {0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5}. The three columns represent the three classes of sites {A, B, C}.

  • Logistic constraints: the operator i can invest on a limited number Z i of classes of sites at each time t ≥ 1. Thus for every player it holds |A it | ≤ Z i ; • Regulator constraints: operators must cover a sufficient number of sites within the deadlines fixed by the regulator; before every time t at least R t classes of sites have to support the new technology. Thus for all players and for all t ≥ 1 it holds

TABLE I MATRIX

 I REPRESENTATION OF THE FIRST PHASE.

  Fig.5. a) Optimal timing: displayed on two lines as (t 1A , t 1B , t 1C ) p 1 and (t 2A , t 2B , t 2C ) p 2 . The baseline preferences (p 1 , p 2 ) influence the strategy for investment timings. For instance, if operators choose (s 1 , s 2 ) = (no, no) the baseline preferences are (p 1 , p 2 ) = (0.6, 0.4). As we can see from column p 1 = 0.6 of Figure

	a)			Operator 2		b)
	Operator 1	no low mid	no (9, 5, 1) 0.60 (8, 12, 1) 0.40 (1, 3, 12) 0.69 (11, 12, 2) 0.31 (2, 4, 12) 0.75	low (10, 4, 11) 0.46 (10, 4, 9) 0.54 (3, 2, 1) 0.56 (2, 11, 12) 0.44 (1, 3, 12) 0.63	mid (12, 9, 4) 0.37 (2, 9, 12) 0.63 (10, 4, 11) 0.47 (10, 4, 9) 0.53 (9, 3, 8) 0.55	high (12, 11, 3) 0.32 (2, 8, 12) 0.68 (9, 12, 4) 0.41 (8, 3, 12) 0.59 (10, 4, 11) 0.48	no low mid	no (0.30, 0.12) (0.36, 0.02) (0.30, 0.03)	Operator 2 (0.26, 0.22) low (0.15, 0.24) mid (0.16, -0.01) (0.16, 0.12) (0.11, -0.07) (0.05, 0.08)	high (0.11, 0.18) (0.11, 0.07) (0.07, 0.02)
			(11, 12, 3) 0.25	(2, 12, 1) 0.37	(1, 12, 11) 0.45	(10, 4, 9) 0.52