Bertrand Lods

Mustapha Mokhtar-Kharroubi

Convergence Rate To Equilibrium For

Keywords: primary 82C40, secondary 35F15, 47D06 Kinetic equation, Boundary operators, Convergence to equilibrium, Inverse Laplace transform. C

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We consider here the time asymptotics for collisionless kinetic equations of the form

∂ t f (x, v, t) + v • ∇ x f (x, v, t) = 0, (x, v) ∈ Ω × V, t 0 (1.1a) with initial data f (x, v, 0) = f 0 (x, v), (x, v) ∈ Ω × V, (1.1b
) under di use boundary

f |Γ -= H(f |Γ + ), (1.1c)
where Ω is a bounded open subset of R d and V is a given closed subset of R d (see Assumptions 1.1 for major details), Γ ± = {(x, v) ∈ ∂Ω × V ; ±v • n(x) > 0} (n(x) being the outward unit normal at x ∈ ∂Ω) and H is a linear boundary operator relating the outgoing and incoming uxes f |Γ + and f |Γ -and is bounded on the trace spaces

L 1 ± = L 1 (Γ ± ; |v • n(x)|π(dx) ⊗ m(dv)) = L 1 (Γ ± , dµ ± (x, v))
where π denotes the Lebesgue surface measure on ∂Ω and m is a Borel measure on the set of velocities (see Assumptions 1.1 hereafter). The boundary operator

H : L 1 + → L 1
is nonnegative and stochastic, i.e.

Γ -

Hψ dµ -= Γ + ψ dµ + , ∀ψ ∈ L 1 (Γ + , dµ + ) (1.2)
so that (1.1) is governed by a stochastic C 0 -semigroup (U H (t)) t 0 on L 1 (Ω × V , dx ⊗ m(dv)) with generator T H . In a previous contribution [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF], a systematic study of (1.1) for general partly di use boundary operators H have been performed providing a general theory on the existence of an invariant density and its asymptotic stability (i.e. convergence to equilibrium), see also earlier one-dimensional results [START_REF]On asymptotic stability and sweeping of collisionless kinetic equations[END_REF]. However, the question of the rate of convergence to equilibrium has been left open by our contribution [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF] and is the main concern of the present paper.

1.1. Our contribution in a nutshell. The main question addressed in this paper is then the following: Question A. Determine a general class C ⊂ L 1 (Ω × V ) of initial datum f and a general rate function r : R + → R + such that

U H (t)f -̺ f Ψ H L 1 (Ω×V ) = O(r(t))
as t → 0 + for any f ∈ C (1.3) where Ψ H is the unique invariant density of T H with unit mass (see the subsequent Theorem 1.4),

̺ f = Ω×V f (x, v)dx ⊗ m(dv), f ∈ L 1 (Ω × V )
and lim t→0 r(t) = 0.

We answer this question here by considering only di use boundary operators for which, typically,

Hψ(x, v) = v ′ •n(x)>0 k(x, v, v ′ )ψ(x, v ′ ) |v ′ • n(x)|m(dv ′ ), (x, v) ∈ Γ - (1.4)
where,

v•n(x)<0 k(x, v, v ′ )|v • n(x)|m(dv) = 1, (x, v ′ ) ∈ Γ + . (1.5)
We do not consider the case where the velocities are bounded away from zero which deserves a separate analysis, mainly because in this case (U H (t)) t 0 exhibits a spectral gap and the convergence to equilibrium is exponential [START_REF]On eventual compactness of collisionless kinetic semigroups with non zero velocities[END_REF]. Let us describe more precisely our mathematical framework and the set of assumptions we adopt throughout the paper. First, the general assumptions on the phase space are the following Assumption 1.1. The phase space Ω × V is such that (1)

Ω ⊂ R d (d 2
) is an open and bounded subset with C 1 boundary ∂Ω.

(2) V ⊂ R d is the support of a nonnegative Borel measure m which is orthogonally invariant (i.e. invariant under the action of the orthogonal group of matrices in R d ).

(3) 0 ∈ V , m({0}) = 0 and m (V ∩ B(0, r)) > 0 for any r > 0 where B(0, r) = {v ∈ R d , |v| < r}. We denote by X 0 := L 1 (Ω × V , dx ⊗ m(dv)) endowed with its usual norm • X 0 . More generally, for any s 0, we set

X s := L 1 (Ω × V , max(1, |v| -s )dx ⊗ m(dv))
with norm • Xs .

Notice that the above Assumption (3) is necessary to ensure that the transport operator T H has at least the whole imaginary axis in its spectrum (see Theorem 4.6 for a precise statement).

With respect to our previous contribution [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF], as already mentioned, we do not consider abstract and general boundary operators here but focus our attention on the speci c case of a di use boundary operator satisfying the following assumption where we de ne, for λ ∈ C, Reλ 0 the following bounded operator

M λ : L 1 --→ L 1 + u -→ M λ u(x, v) = e -λ τ -(x,v) u(x -τ -(x, v)v, v), (x, v) ∈ Γ + ;
where τ -(x, v) := inf{ s > 0 ; x + sv / ∈ Ω} for any (x, v) ∈ Γ + (see Section 2.2 for more details on the travel time).

Assumption 1.2. The boundary operator H :

L 1 + → L 1
-is a bounded and stochastic operator of the form (1.4) which satis es the following 1) There exists some n ∈ N (where N is the set of nonnegative integers N = {0, 1, . . .}) such that

H ∈ B(L 1 + , Y - n+1 )
where, for any s 0, we de ne

Y ± s := {g ∈ L 1 ± ; Γ ± max(1, |v| -s )|g(x, v)|dµ + (x, v) < ∞}.
We will set

N H := sup{k ∈ N ; H ∈ B(L 1 + , Y - k+1 )}. (1.6)
and assume that N H < ∞ in all the paper. 2) The operator HM 0 H ∈ B(L 1 + , L 1 -) is weakly compact. 3) M 0 H is irreducible. 4) There exist p ∈ N and C > 0 such that

R (M ε+iη H) p B(L 1 + ) dη C ∀ε 0. (1.7)
We will give later in Section 6 practical criteria ensuring this set of assumptions to be met resorting notably to our previous contribution [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF] (see also Section 1.3 in this Introduction for some earlier considerations about Assumptions 1.2).

By means of a new and robust tauberian approach, we can answer Question A. The main contribution of this work is summarized in the following Theorem 1.3. Under Assumptions 1.1-1.2, for any f ∈ X N H +1 there exist a constant C f > 0 and

Θ f ∈ C 0 (R, X 0 ) ∩ L 1 (R, X 0 ) such that U H (t)f -̺ f Ψ H X 0 C f (1 + t) N H ε(t) ∀t 0 (1.8)
where

ε(t) = 1 1 + t + ∞ -∞ exp (iη t) Θ f (η)dη X 0
∀t 0 is such that lim t→∞ ε(t) = 0. If we assume moreover that there is C(p) > 0 and β > 0 such that

|η|>R (M iη H) p B(L 1 + ) dη C(p) R β , ∀R > 0 (1.9)
then, there is some positive constant K > 0 depending only on β and C(p) such that

∞ -∞ exp (iη t) Θ f (η)dη X 0 K ω f π t β β+1 ∀t 1 (1.10)
where ω f : R + → R + denotes the minimal modulus of continuity of the uniformly continuous mapping Θ f .

Our main Theorem provides therefore an explicit rate of convergence of the type

U H (t)f -̺ f Ψ H X 0 = O(t -N H ). (1.11)
It is important to point out that this rate of convergence is therefore prescribed by the maximal gain of integrability H is able to provide (corresponding to the parameter N H ) and therefore is governed by the action of the boundary operator H on small velocities. This important feature of collisionless transport equation is fully exploited in the companion paper [START_REF]On eventual compactness of collisionless kinetic semigroups with non zero velocities[END_REF] where, in the case of velocity bounded away from zero, the rate of convergence turns out to be exponential. The above explicit rate (1.11) can actually be strengthened into the semi-explicit

U H (t)f -̺ f Ψ H X 0 = o(t -N H ).
We wish to insist here on the fact that the rate of convergence given in (1.8) is optimal in the sense that the error function ε(t) is the exact correction to the rate (1 + t) -N H . Of course, this correction is only semi-explicit and to derive a more explicit rate, one needs to precisely determinate the rate of convergence to zero (granted by Riemann-Lebesgue Theorem) of the Fourier transform

t 0 -→ ∞ -∞ exp (iη t) Θ f (η)dη ∈ X 0
where Θ f (•) is de ned as the N H -derivative of some suitable boundary function (see (1.17) for a precise de nition). The second part of the Theorem, stated as (1.10) is a rst step towards this direction. The additional assumption (1.9) is easy to check in practice (see Section 6 where actually Assumption 1.2 4) is deduced from (1.9)). We point out also the following: a) We assumed for simplicity that N H is nite but, of course, if N H = ∞, then the above result remains valid and the rate of convergence we obtain then is of the type

U H (t)f -̺ f Ψ H X 0 = O (1 + t) -k ∀f ∈ X k+1 ,
for any k 0. It is an interesting open problem to determine whether this convergence can be upgraded to some (stretched) exponential convergence if f ∈ k X k (for instance, if the support of f is away from 0). b) If the decay at in nity of the mapping λ → (M λ H) p ∈ B(L 1 + ) is such that (1.9) is valid for any β > 0 large enough (up to change p), then the overall decay of ε(t) is as close as desired from the one of t → ω f π t .

This is the case for instance when H is associated to the Maxwellian boundary condition of Example 1.6 hereafter. c) Providing explicit estimates of the modulus of continuity ω f is an open problem. Some reasonable conjecture about this is given at the end of the paper (see Section 5.3).

The existence and uniqueness of the equilibrium density Ψ H as well as some qualitative convergence result has been obtained in a systematic way in [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]. Namely, under Assumption 1.2, one can deduce directly the following from [26, Theorem 6.5 and Section 7] and [START_REF]Quantitative tauberian approach to collisionless transport equations with di use boundary operators[END_REF]: Theorem 1.4. Under Assumption 1.2, the operator (T H , D(T H )) de ned by

D(T H ) = f ∈ X 0 ; v • ∇ x ψ ∈ X 0 ; f |Γ ± ∈ L 1 ± H f |Γ + = f |Γ -, T H f = -v • ∇ x f, f ∈ D(T H )
is the generator of a stochastic C 0 -semigroup (U H (t)) t 0 . Moreover, (U H (t)) t 0 is irreducible and has a unique invariant density Ψ H ∈ D(T H ) with

Ψ H (x, v) > 0 for a. e. (x, v) ∈ Ω × R d , Ψ H X 0 = 1
and Ker(T H ) = Span(Ψ H ). Moreover,

lim t→∞ U H (t)f -Pf X 0 = 0, ∀f ∈ X 0 (1.12)
where P denotes the ergodic projection

Pf = ̺ f Ψ H , with ̺ f = Ω×R d f (x, v)dx ⊗ m(dv), f ∈ X 0 .
Besides partial results in [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]Section 7], the strong convergence (1.12) has been obtained recently in our previous (unpublished) contribution [START_REF]Quantitative tauberian approach to collisionless transport equations with di use boundary operators[END_REF]. In that paper, we rst proved a general qualitative (without rate) convergence to equilibrium for (1.1) under Assumptions 1.1-1.2 via a Tauberian argument using Ingham's theorem. Moreover, we also addressed Question A and notably derived suboptimal rates of convergence to equilibrium for solutions to (1.1) under mild assumptions on the initial datum f thanks to recent quanti ed versions of Ingham's theorem [START_REF]Quanti ed versions of Ingham's theorem[END_REF]. Typically, with respect to Theorem 1.3, the rate obtained in [START_REF]Quantitative tauberian approach to collisionless transport equations with di use boundary operators[END_REF] were of the form

U H (t)f -̺ f Ψ H X 0 = O t -N H 2 for f ∈ X N H +1 .
The present paper is a signi cant improvement of the results of [START_REF]Quantitative tauberian approach to collisionless transport equations with di use boundary operators[END_REF] which do not longer use quanti ed versions of Ingham's theorem and strenghten in an almost optimal way the rate of convergence. We anticipate already that the tool which allows us to get rid of Ingham's theorem is the use of a suitable representation of the solutions to (1.1) combined with a tauberian approach. As far as we know, our construction is new and appears here for the rst time.

1.2. Related literature. Besides its fundamental role in the study of the Boltzmann equation with boundary conditions [START_REF]Non-isothermal boundary in the Boltzmann theory and Fourier law[END_REF][START_REF]Decay and continuity of the Boltzmann equation in bounded domains[END_REF][START_REF]Asymptotic stability of the Boltzmann equation with Maxwell boundary conditions[END_REF], the mathematical interest towards relaxation to equilibrium for collisionless equations is relatively recent in kinetic theory starting maybe with numerical evidences obtained in [START_REF]Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall[END_REF]. A precise description of the relevance of the question as well as very interesting results have been obtained then in [START_REF] Mokhtar-Kharroubi | On the speed of approach to equilibrium for a collisionless gas[END_REF]. We mention also the important contributions [START_REF]Equilibrating e ect of Maxwell-type boundary condition in highly rare ed gas[END_REF][START_REF]Free molecular ow with boundary e ect[END_REF] which obtain seemingly optimal rate of convergence when the spatial domain is a ball. For transport equations in a slab geometry, a Tauberian approach based upon Ingham's theorem has been introduced in [START_REF]Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry[END_REF]. Such an approach was then generalized to more general geometry (in higher dimension) and improved in our aforementioned unpublished manuscript [START_REF]Quantitative tauberian approach to collisionless transport equations with di use boundary operators[END_REF]. The two very recent works [START_REF]A semigroup approach to the convergence rate of a collisionless gas[END_REF][START_REF]A coupling approach for the convergence to equilibrium for a collisionless gas[END_REF] provide (nearly optimal) convergence rate for general domains Ω in a L 1 -setting. All these works are dealing with partially di use boundary operator of Maxwell-type for which

Hϕ(x, v) = α(x)ϕ(x, v -(v • n(x))n(x)) + (1 -α(x)) γ(x) M θ(x) (v) v ′ •n(x)>0 ϕ(x, v ′ )|v ′ • n(x)|m(dv ′ ) (1.13)
where, as above M θ(x) is a Maxwellian distribution given (see example ) for which the temperature θ(x) depends (continuously) on x ∈ ∂Ω and γ(x) is a normalization factor ensuring H to be stochastic. A nearly optimal rate of convergence for the boundary condition (1.13) in dimension d = 2, 3 has been obtained recently in [START_REF]A semigroup approach to the convergence rate of a collisionless gas[END_REF] thanks to a clever use of Harris's subgeometrical convergence theorem for Markov processes. A related probabilistic approach, based on coupling, has been addressed in [START_REF]A coupling approach for the convergence to equilibrium for a collisionless gas[END_REF] in dimension d 2 whenever θ(x) = θ is constant and in both these works, the rate of convergence is nearly optimal and given by

O (log(1 + t)) d+1 t d as t → ∞.
For such a model, with the notations of our Theorem 1.3,

N H = d -1
which suggests that our optimal correction ε(t) is at least of the order ε(t) = (log(1+t)) d+1 t . Let us nally mention the very recent contribution [START_REF]Damping of kinetic transport equation with di use boundary condition[END_REF] which closely follows the approach of [START_REF]A semigroup approach to the convergence rate of a collisionless gas[END_REF] and provides a L1 -L ∞ framework for solutions to (1.1) with exponential moments and obtain a rate of convergence similar to that of [START_REF]A semigroup approach to the convergence rate of a collisionless gas[END_REF] in the case of di use boundary operator of Maxwell-type as considered here.

Let us point out here that, even if the rate obtained in [START_REF]A semigroup approach to the convergence rate of a collisionless gas[END_REF][START_REF]A coupling approach for the convergence to equilibrium for a collisionless gas[END_REF][START_REF]Damping of kinetic transport equation with di use boundary condition[END_REF] are slightly better than the one obtained here, our contribution is not really comparable to those (1) First, we deal here with di erent kind of boundary conditions (in any dimension d 1)

and, even if we restrict ourselves to di use boundary condition, the structure of the kernel k(x, v, v ′ ) is much more general than the Maxwellian case (1.13). Notice in particular that our assumptions on the boundary operator H are relatively easy to check and quite general. On this aspect, our result can be seen as a systematic treatment of Question A for general di usive boundary conditions. Let us mention here that, even though the approach of [START_REF]A semigroup approach to the convergence rate of a collisionless gas[END_REF][START_REF]A coupling approach for the convergence to equilibrium for a collisionless gas[END_REF] is robust enough to be applied to more general boundary conditions than the Maxwellian one, such an approach requires the construction of some pointwise lower bounds for the solution to (1.1) which would di er from one boundary operator to another and, as such, some speci c work has to be done for each given boundary condition. (2) Second, the mathematical tools used in the present paper are completely new and different from those of the associated literature. In particular, since the analysis of [START_REF]A semigroup approach to the convergence rate of a collisionless gas[END_REF][START_REF]A coupling approach for the convergence to equilibrium for a collisionless gas[END_REF] is based upon a clever modi cation of Harris's convergence theorem, it is speci cally tailored to deal with the L 1 (or measure) functional framework 1 . Our approach on the contrary, though it uses in some places some speci c features of the L 1 setting, is denitely robust enough to be applied to a more general functional framework. In particular, it can be suitably modi ed to be applied (under suitable ad hoc assumptions on the boundary operator H) to some (weighted) L 2 or L ∞ settings which are particularly relevant for the study of the linearized Boltzmann equation with Maxwell-like boundary condition. (3) Our theory fully exploits the structure of equation (1.1), but we point out that our construction can be adapted to the study of collisional equations (see [START_REF]About the rate of convergence for collisional linear kinetic equations on the torus, a tauberian approach[END_REF] for neutron transport equation on the torus or [START_REF]A tauberian approach to the long-time behaviour of the linear spatially homogeneous Boltzmann equation[END_REF] for the spatially homogeneous linear Boltzmann equation with soft potentials) and we believe that it is virtually adaptable to other perturbative contexts using the representation of solution to linear collisional kinetic equation as a Dyson-Phillips series.

We insist here again on the fact that the rate derived in (1.8) is exact. Even if we derived, up to now, only suboptimal explicit convergence rate, the only restriction to get an optimal rate lies in the di culty we encountered in estimating accurately the decay to zero of the mapping

t 0 → ∞ -∞ exp (iη t) Θ f (η)dη ∈ X 0 . 1.3. Practical examples.
A few remarks are in order about our set of Assumptions:

• First, we gave in our previous contribution [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]Theorem 5.1] a precise de nition of a general class of boundary operator for which HM 0 H is weakly-compact. This class of operators was de ned in [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF] as the class of regular di use boundary operators and we will simply say here that H is di use. • Moreover, a practical criterion ensuring the above property 3) to occur is also given in [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]. In practice, as observed already, the typical operator we have in mind is given by (1.4). Under some strong positivity assumption on k(•, •, •), one can show that M 0 H is irreducible (see [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]Section 4]). • We believe that Assumption 4) is met for any regular di use boundary operators. We have been able to prove the result with ℓ = 2 for a slightly more restrictive class of boundary operators (see Proposition 6.12) whenever m(dv) is absolutely continuous with respect to the Lebesgue measure on R d .

We refer to Section 6 for more details on this set of assumptions and only provide here a brief list of examples covered by those assumptions and which are particularly relevant as models of boundary interactions in the kinetic theory of gas (see [START_REF]The mathematical theory of dilute gases[END_REF]).

Example 1.5 (Generalized Maxwell-type). The most typical example corresponds to a generalized Maxwell-type di use operator for which

k(x, v, v ′ ) = γ -1 (x)G(x, v)
where G : ∂Ω × V → R + is a measurable and nonnegative mapping such that

(i) G(x, •) is radially symmetric for π-almost every x ∈ ∂Ω; (ii) G(•, v) ∈ L ∞ (∂Ω) for almost every v ∈ V ;
(iii) The mapping x ∈ ∂Ω → γ(x) is continuous and bounded away from zero where

γ(x) := Γ -(x) G(x, v)|v • n(x)|m(dv) ∀x ∈ ∂Ω, (1.14)
i.e. there exist γ 0 > 0 such that γ(x) γ 0 for π-almost every x ∈ ∂Ω.

Example 1.6 (Maxwell di use boundary condition). A particularly relevant example is a special case of the previous one for which, m(dv) = dv and G is a given Maxwellian with temperature θ(x), i.e.

G(x, v) = M θ(x) (v), M θ (v) = (2πθ) -d/2 exp - |v| 2 2θ , x ∈ ∂Ω, v ∈ R d .
Then,

γ(x) = κ d θ(x) R d |w|M 1 (w)dw, x ∈ ∂Ω
for some positive constant κ d depending only on the dimension. The above assumption (iii) asserts that the temperature mapping x ∈ ∂Ω → θ(x) is bounded away from zero and continuous.

1.4. Method of proof. For the sake of clarity and in order to help the reading of the paper, we give here an idea of the main steps of the proof of Theorem 1.3. The main ideas behind the proof can be summarized in the next three steps:

Step 1) We exploit an explicit representation of the semigroup (U H (t)) t 0 obtained recently in [START_REF]Explicit transport semigroup associated to abstract boundary conditions[END_REF]. With this representation, similar to the Dyson-Phillips series for additive perturbative semigroup theory, the semigroup is expressed as a suitable strongly convergence series

U H (t)f = ∞ k=0 U k (t)f, f ∈ X 0 , t 0 (1.15)
where the family of operators {U k (t)} k 0 is de ned inductively. Typically, (U 0 (t)) t 0 denotes the semigroup generated by T 0 (corresponding to absorbing boundary condition H ≡ 0) whereas, for any k ∈ N, U k (t)f denotes the solution to (1.1) after having experienced k rebounds with the boundary.

Step 2) Precise estimates of the decay of each terms U k (t)f (for given k ∈ N) are obtained by suitably investigating the in uence of the boundary operator for small and large velocities. We in particular show that, if

f ∈ X N H +1 then n k=0 U k (t)f X 0 = O(t -N H -1 ) as t → ∞
for any n ∈ N. See Lemma 3.7 and Proposition 3.8 for more precise statements.

Step 3) We these two rst points, to investigate the decay of U H (t)(I -P), we only need to understand that of some suitable remainder of the series (1.15), say

S n (t)f = ∞ k=n+1 U k (t)f.
Notice that investigating S n (t) (I -P) f amounts to study carefully S n (t)f for some function f with zero mean, i.e. such that

̺ f := Ω×V f (x, v)dx ⊗ m(dv) = 0.
This is the most technical part of the paper. It requires a careful study of the spectral properties of T H and M 0 H and some tools from Fourier-Laplace analysis. More precisely, while the two previous points can be seen as a semigroup approach to Question A, this third step is rather a resolvent approach since we deduce the properties of the remainder

∞ k=n+1 U k (t)f
from the careful study of its Laplace transform which is related to the resolvent of T H . This is Let us describe with more details this third step. The resolvent of T H can be written, for λ ∈ C + as

R(λ, T H )f = R(λ, T 0 )f + ∞ k=0 Ξ λ H (M λ H) k G λ f
for some suitable operators Ξ λ , M λ and G λ described in Section 2.4. Then, for a given n ∈ N, we can show that the remainder S n (t) admits, as a Laplace transform,

∞ 0 exp (-λt) S n (t)f dt = ∞ k=n Ξ λ H (M λ H) k G λ f = Υ(λ)f, Reλ > 0.
Using properties of the inverse Laplace transform [START_REF]Vector-valued Laplace transforms and Cauchy problems[END_REF], we can describe then entirely S n (t)f in terms of Υ n (λ)f , namely

S n (t)f = exp(εt) 2π lim ℓ→∞ ℓ -ℓ exp (iηt) Υ n (ε + iη)f dη, ∀f ∈ X 0
for any t > 0, ε > 0. Of course, to hope deducing a decay of S n (t)f for large t, the positive exponential is a dramatic obstacle. This enforces to deduce a second representation formula for S n (t)f where the inverse Laplace transform is derived on the imaginary axis, i.e. for λ = iη, η ∈ R.

A rst mathematical di culty occurs here since Υ n (ε + iη)f is not even de ned for ε = 0. We need therefore to build, for suitable class of functions f , the boundary trace of Υ n (λ)f along the imaginary axis. This is the most technical part of the present work which will result in the following Theorem 1.7. Let f ∈ X N H +1 be such that

̺ f = Ω×V f (x, v)dx ⊗ m(dv) = 0. (1.16)
Then, for any n 0 the limit lim

ε→0 + Υ n (ε + iη)f,
exists in C N H 0 (R, X 0 ). Its limit is denoted Ψ n (η)f . Moreover, for n large enough (explicit), the mappings

η ∈ R -→ d k dη k Ψ n (η)f ∈ X 0 , 0 k N H are integrable.
Here above, for any Banach space (X, • X ) and any k ∈ N, we set

C k 0 (R, X) = h : R → X ; of class C k over R
and such that lim

|η|→∞ d j dη j h(η) X = 0 ∀j k and we endow C k 0 (R, X) with the norm h L ∞ (X) := sup η max 0 j k d j
dη j h(η) X which makes it a Banach space.

Remark 1.8. We point out here that we cannot expect a higher regularity for the mapping Ψ n (η)f ; the mapping

Υ n (ε + iη)f is not di erentiable N H + 1 times because M ε+iη H is di erentiable k- times if and only if H ∈ B(L 1 + , Y - k+1 ). Therefore, by de nition of N H , η → M ε+iη H ∈ B(L 1 + , X 0 ) is di erentiable exactly N H times and so is η → Υ n (ε + iη)f ∈ X 0 .
With this boundary function, one can prove the second representation formula (see Theorem 5.8)

S n (t)f = 1 2π ∞ -∞ exp (iη t) Ψ n (η)f dη , ∀t 0
for any f ∈ X N H +1 satisfying (5.10).

Recall that the condition ̺ f = 0 is equivalent to the condition Pf = 0, i.e. the above is somehow a representation formula for S n (t)(I -P)f . With this last representation formula, one easily get convinced that, if one can prove that the mapping ), then by integration by parts, we can expect

η ∈ R -→ Ψ n (η)f ∈ X 0 belongs to C N H 0 (R
S n (t)f = i t N H ∞ -∞ exp (iη t) d N H dη N H Ψ n (η)f dη 2π , t 0.
This provides at least a decay like S n (t)f = O(t -N H ). This decay is actually strengthened into

S n (t)f = o(t -N H )
by a simple use of Riemann-Lebesgue Theorem using that the mapping

η ∈ R -→ d N H dη N H Ψ n (η)f is integrable over R.
One sees here that this is exactly the contents of Theorem 1.3 where clearly

Θ f (η) = d N H dη N H Ψ n (η)f (1.17)
for a suitably large choice of the parameter n (related to p in (1.7)). We point out here that we cannot expect a higher regularity for the mapping Ψ n (η)f . Let us comment very brie y on the technical points behind the proof of Theorem 1.7. To study Υ n (λ) along the imaginary axis, we observe rst that

r σ (M λ H) < 1 Reλ > 0 but 1 is a (simple) eigenvalue of M 0 H. It turns out that λ → M λ H ∈ B(L
Υ n (ε + iη)f exists (η = 0)
and the convergence is locally uniform in η = 0. Because r σ (M 0 H) = 1, the treatment of the case η = 0 is very involved and the various technical results of Section 5 are devoted to this delicate point. In particular, by exploiting the fact that near λ = 0, the eigenvalue of M λ H of maximum modulus is algebraically simple (converging to 1 as λ → 0, see Proposition 4.7), and analyzing the corresponding spectral projection, we can show the existence of the boundary function for all the various operators involved in Υ n (ε + iη)f under the assumption (5.10) (see Lemma 4.10) where the convergence is uniform with respect to η ∈ R.

To prove then the smoothness of the boundary function Ψ n (•)f , we need an alternative representation of Υ n (λ) since it appears out of reach to compute the derivatives of ∞ k=n (M λ H) k G λ f at λ = 0 still because of the fact that r σ (M 0 H) = 1. It is easy to see that

Υ n (λ) = R(λ, T H ) -R(λ, T 0 ) - n k=0 Ξ λ (M λ H) k G λ
and the smoothness is then obtained through the well-known identity for derivatives of the resolvent

d k dλ k R(λ, T H ) = (-1) k k! R(λ, T H ) k+1 , Reλ > 0.
Finally, the last technical point is to show that the N H -th derivative of the boundary function is integrable. This point is the crucial one where Assumption 1.2 (4) is fully exploited. See Theorem 5.8 for details. All these results yield then to the decay rate (1.8). Then, the more precise investigation of the correction ε(t) and its link with the modulus of continuity of Θ f is deduced by a simple adaptation of considerations linking the fractional regularity of functions with the decay of its Fourier coe cients [START_REF]Classical Fourier analysis[END_REF]. This gives then (1.10) under the additional assumption (1.9).

As mentioned earlier, proving that Assumption 1.2 4) is met for a large class of di use boundary operators is a highly technical task and we devote Section 6 to this. The results of Section 6 are also related to a general change of variable formula transferring integrals in velocities into integrals over ∂Ω which has its own interest. It clari es several computations scattered in the literature [START_REF]Non-isothermal boundary in the Boltzmann theory and Fourier law[END_REF][START_REF]Decay and continuity of the Boltzmann equation in bounded domains[END_REF] and will be a fundamental tool for the analysis in the companion paper [START_REF]On eventual compactness of collisionless kinetic semigroups with non zero velocities[END_REF]. 1.5. Organization of the paper. In Section 2, we introduce the functional setting and notations used in the rest of the paper and recall several known results mainly from our previous contribution [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]. The representation of the semigroup (U H (t)) t 0 is discussed in Section 3.1 together with the decay rate of the iterates U k (t), settling the above Steps 1) and 2) of our approach. Section 4 is devoted to the ne analysis of the resolvent R(1, M λ H) which is well-de ned for Reλ > 0 but needs to be carefully extended to the imaginary axis λ = iη, η ∈ R. This section is the most technical one of the paper and we have to deal separately with the case η = 0 and η = 0.S uch an extension is a cornerstone in the construction of the boundary function lim ε→0 + Υ n (ε + iη, T H )f (for suitable f ) which is performed in Section 5. In this Section, we also establish the suitable representation formula for S n (t) thanks to the inverse Laplace transform (Section 5.1) and we provide the full proof of Theorem 1.3. In Section 6, we provide practical criteria ensuring Assumptions 1.2 to be met. It contains several results we believe to be of independent interest. We also illustrate how the examples described here above in Section 1.3 are covered by our results.

R

2.1. Functional setting. We introduce the partial Sobolev space

W 1 = {ψ ∈ X 0 ; v • ∇ x ψ ∈ X 0 }.
It is known [START_REF]Théorèmes de traces Lp pour les espaces de fonctions de la neutronique[END_REF][START_REF]Théorèmes de traces pour les espaces de fonctions de la neutronique[END_REF] that any ψ ∈ W 1 admits traces ψ |Γ ± on Γ ± such that

ψ |Γ ± ∈ L 1 loc (Γ ± ; dµ ± (x, v)) where dµ ± (x, v) = |v • n(x)|π(dx)
⊗ m(dv), denotes the "natural" measure on Γ ± . Notice that, since dµ + and dµ -share the same expression, we will often simply denote it by

dµ(x, v) = |v • n(x)|π(dx) ⊗ m(dv),
the fact that it acts on Γ -or Γ + being clear from the context. Note that [START_REF]Théorèmes de traces Lp pour les espaces de fonctions de la neutronique[END_REF][START_REF]Théorèmes de traces pour les espaces de fonctions de la neutronique[END_REF] that

∂Ω × V := Γ -∪ Γ + ∪ Γ 0 , where Γ 0 := {(x, v) ∈ ∂Ω × V ; v • n(x) = 0}. We introduce the set W = ψ ∈ W 1 ; ψ |Γ ± ∈ L 1 ± . One can show
W = ψ ∈ W 1 ; ψ |Γ + ∈ L 1 + = ψ ∈ W 1 ; ψ |Γ -∈ L 1 -.
Then, the trace operators B ± :

B ± : W 1 ⊂ X 0 → L 1 loc (Γ ± ; dµ ± ) ψ -→ B ± ψ = ψ |Γ ± , are such that B ± (W ) ⊆ L 1
± . Let us de ne the maximal transport operator T max as follows:

T max : D(T max ) ⊂ X 0 → X 0 ψ -→ T max ψ(x, v) = -v • ∇ x ψ(x, v), with domain D(T max ) = W 1 . Now, for any bounded boundary operator H ∈ B(L 1 + , L 1 -), de ne T H as T H ϕ = T max ϕ for any ϕ ∈ D(T H ), where D(T H ) = {ψ ∈ W ; ψ |Γ -= H(ψ |Γ + )}.
In particular, the transport operator with absorbing conditions (i.e. corresponding to H = 0) will be denoted by T 0 .

De nition 2.1. For any s 0, we de ne the function spaces

Y ± s = L 1 (Γ ± , max(1, |v| -s )dµ ± ) with the norm u Y ± s = Γ ± |u(x, v)| max(1, |v| -s )dµ ± (x, v).
In the same way, for any s 0, we introduce

X s = L 1 (Ω × V , max(1, |v| -s )dx ⊗ m(dv)) with norm f Xs := max(1, |v| -s )f X 0 , f ∈ X s .
Remark 2.2. Of course, for any s 0, Y ± s is continuously and densely embedded in L 1 ± . In the same way, X s is continuously and densely embedded in X 0 . Introduce, for any s ∈ N, the function

̟ s (v) = max(1, |v| -s ), v ∈ V.
One will identify, without ambiguity, ̟ s with the multiplication operator acting on L 1 ± or on X 0 , e.g.

̟ s : X 0 -→ X 0 f -→ ̟ s f (x, v) = ̟ s (v)f, (x, v) ∈ Ω × V.
Then, one sees that

Y ± s = {f ∈ L 1 ± ; ̟ s f ∈ L 1 ± }, X s = {f ∈ X 0 ; ̟ s f ∈ X 0 }.

2.2.

Travel time and integration formula. Let us now introduce the travel time of particles in Ω (with the notations of [START_REF]A new approach to transport equations associated to a regular eld: trace results and well-posedness[END_REF]), de ned as:

De nition 2.3. For any (x, v) ∈ Ω × V, de ne t ± (x, v) = inf{ s > 0 ; x ± sv / ∈ Ω}.
To avoid confusion, we will set τ ± (x, v)

:= t ± (x, v) if (x, v) ∈ ∂Ω × V.
With the notations of [START_REF]Decay and continuity of the Boltzmann equation in bounded domains[END_REF], t -is the backward exit time t b . From a heuristic perspective, t -(x, v) is the time needed by a particle having the position x ∈ Ω and the velocity -v ∈ V to reach the boundary ∂Ω. One can prove [START_REF]Functional analytic treatment of the initial boundary value problem for collisionless gases[END_REF]Lemma 1.5]

that t ± (•, •) is measurable on Ω × V . Moreover τ ± (x, v) = 0 for any (x, v) ∈ Γ ± whereas τ ∓ (x, v) > 0 on Γ ± . It holds (x, v) ∈ Γ ± ⇐⇒ ∃y ∈ Ω with t ± (y, v) < ∞ and x = y ± t ± (y, v)v.
In that case, τ ∓ (x, v) = t + (y, v) + t -(y, v). Notice also that,

t ± (x, v)|v| = t ± (x, ω) , ∀(x, v) ∈ Ω × V, v = 0, ω = |v| -1 v ∈ S d-1 . (2.1)
We have the following integration formulae from [START_REF]A new approach to transport equations associated to a regular eld: trace results and well-posedness[END_REF].

Proposition 2.4. For any h ∈ X 0 , it holds

Ω×V h(x, v)dx ⊗ m(dv) = Γ ± dµ ± (z, v) τ ∓ (z,v) 0 h (z ∓ sv, v) ds, (2.2)
and for any ψ ∈ L 1 (Γ -, dµ -),

Γ - ψ(z, v)dµ -(z, v) = Γ + ψ(x -τ -(x, v)v, v)dµ + (x, v). (2.3)
Remark 2.5. Notice that, because µ -(Γ 0 ) = µ + (Γ 0 ) = 0, we can extend the above identity (2.3) as follows: for any ψ ∈ L 1 (Γ -∪ Γ 0 , dµ -) it holds

Γ -∪Γ 0 ψ(z, v)dµ -(z, v) = Γ + ∪Γ 0 ψ(x -τ -(x, v)v, v)dµ + (x, v).
(2.4) 2.3. Decay of the semigroup assiociated to absorbing boundary conditions. We end this section with a decay property of the semigroup (U 0 (t)) t 0 on the hierarchy of spaces X k (k ∈ N):

Lemma 2.6. Given k ∈ N and f ∈ X k , one has

U 0 (t)f X 0 D k t k f X k , ∀t > 0.
Proof. Let f ∈ X k and t > 0 be xed. For simplicity, we introduce g(x, v) = |v| -k |f (x, v)|, (x, v) ∈ Ω × V and denote by g the extension by zero of g to R d × V. One has then

U 0 (t)f X 0 = Ω×V |v| k g(x -tv, v)1 t<t -(x,v) dxm(dv)
For a given x ∈ Ω, one has

V |v| k g(x -tv, v)1 t<t -(x,v) m(dv) V |v| k g(x -tv, v)m(dv).
Recalling that t > 0, x ∈ Ω are xed, we denote by m #,(t,x) (dy) the image measure of m(dv)

through the transform v ∈ V → y = x -tv ∈ R d and deduce V |v| k g(x -tv, v)1 t<t -(x,v) m(dv) R d |x -y| t k g y, x -y t m #,(t,x) (dy) D k t k R d g y, x -y t m #,(t,x) (dy).
Therefore,

U 0 (t)f X 0 D k t k R d dx R d g y, x -y t m #,(t,x) (dy).
By de nition of m #,(t,x) (dy), we can performe back the change of variable y → v = x-y t to get

U 0 (t)f X 0 D k t k V m(dv) R d g(x -tv, v)dx. Now, given v ∈ V , we perform the change of variable x ∈ R d → z = x -tv ∈ R d to get U 0 (t)f X 0 D k t k R d ×V g(z, v)dzm(dv) = D k t k g X 0
which gives the result.

We complement the above with the following technical property Lemma 2.7. For any k 0 and f ∈ X k+1 ,

∞ 0 U 0 (t)f X k dt D f X k+1 and ∞ 0 t k U 0 (t)f X 0 dt (2D) k+1 k + 1 f X k+1 .
Proof. We prove the result for k = 0. Assume f ∈ X 1 . Using (2.2) one computes for any t 0,

U 0 (t)f X 0 = Γ - dµ -(z, v) τ + (z,v) 0 |[U 0 (t)f ] (z + sv, v)| ds = Γ - dµ -(z, v) τ + (z,v) 0 1 [0,t -(z+sv,v)] (t) |f (z + (s -t)v, v)| ds Since t -(z + sv, v) = s for any (z, v) ∈ Γ -, we deduce that |U 0 (t)f X 0 = Γ - dµ -(z, v) τ + (z,v) 0 1 [0,s] (t) |f (z -(t -s)v, v)| ds. (2.5)
Integrating this identity and using Fubini's Theorem repeatedly yields

∞ 0 U 0 (t)f X 0 dt = Γ - dµ -(z, v) τ + (z,v) 0 ds s 0 |f (z -(t -s)v, v)| dt = Γ - dµ -(z, v) τ + (z,v) 0 ds s 0 |f (z -τ v, v)|dτ = Γ - dµ -(z, v) τ + (z,v) 0 |f (z -τ v, v)|dτ τ -(z,v) t ds i.e. ∞ 0 U 0 (t)f X 0 dt Γ - τ + (z, v)dµ -(z, v) τ + (z,v) 0 |f (z -τ v, v)|dτ. Because τ + (z, v) D/|v|, we see using again (2.2) that ∞ 0 U 0 (t)f X 0 dt D Ω×V |v| -1 |f (x, v)|dxm(dv) D f X 1 .
This proves the result for k = 0. Given now k 1, because the multiplication operator ̟ k and U 0 (t) commute (t 0), one has

U 0 (t)f X k = U 0 (t) (̟ k f ) X 0
which allows to apply the result obtained so far to ̟ k f ∈ X 1 and to conclude in the general case.

Let us now focus on the second estimate. One sees also from (2.5) that

∞ 0 t k U 0 (f )f X 0 dt = ∞ 0 t k dt Γ - dµ -(z, v) τ + (z,v) 0 1 [0,s] (t) |f (z -(t -s)v, v)| ds = Γ - dµ -(z, v) τ + (z,v) 0 |f (z -τ v, v)|dτ τ -(z,v) τ (s + τ ) k ds 1 k + 1 Γ - dµ -(z, v) τ + (z,v) 0 (τ + (z, v) + τ ) k+1 |f (z -τ v, v)|dτ 2 k+1 k + 1 Γ - τ + (z, v) k+1 dµ -(z, v) τ + 0 |f (z -τ v, v)|dτ
and one concludes as previously since τ + (z, v) D/|v|.

About the resolvent of T

H . For any λ ∈ C such that Reλ > 0, de ne      M λ : L 1 --→ L 1 + M λ u(x, v) = u(x -τ -(x, v)v, v)e -λτ -(x,v) , (x, v) ∈ Γ + u ∈ L 1 -;    G λ : X 0 -→ L 1 + G λ ϕ(x, v) = τ -(x,v) 0 ϕ(x -sv, v)e -λs ds, (x, v) ∈ Γ + ϕ ∈ X 0    R λ : X 0 -→ X 0 R λ ϕ(x, v) = t -(x,v) 0 ϕ(x -tv, v)e -λt dt, (x, v) ∈ Ω × V ; ϕ ∈ X 0 and      Ξ λ : L 1 --→ X 0 Ξ λ u(x, v) = u(x -t -(x, v)v, v)e -λt -(x,v) 1 {t -(x,v)<∞} , (x, v) ∈ Ω × V u ∈ L 1 -;
where 1 E denotes the characteristic function of the measurable set E. The interest of these operators is related to the resolution of the boundary value problem:

(λ -T max )f = g, B -f = u, (2.6)
where λ > 0, g ∈ X 0 and u is a given function over Γ -. Such a boundary value problem, with

u ∈ L 1 -can be uniquely solved (see [5, Theorem 2.1]) Theorem 2.8. Given λ > 0, u ∈ L 1 -and g ∈ X 0 , the function f = R λ g + Ξ λ u is the unique solution f ∈ D(T max ) of the boundary value problem (2.6). Moreover, B + f ∈ L 1 + and B + f L 1 + + λ f X 0 u L 1 -+ g X 0 .
Remark 2.9. Notice that Ξ λ is a lifting operator which, to a given u ∈ L 1 -, associates a function f = Ξ λ u ∈ D(T max ) whose trace on Γ -is exactly u. More precisely,

T max Ξ λ u = λΞ λ u, B -Ξ λ u = u, B + Ξ λ u = M λ u, ∀u ∈ L 1 -. (2.7)
Moreover, for any λ > 0, one sees with the choice u = 0 that R λ coincides with R(λ, T 0 ). The above theorem also shows that, for any λ > 0

Ξ λ B(L 1 -, X 0 ) λ -1 and R λ B(X 0 ) λ -1 . (2.8)
Moreover, one has the obvious estimates

M λ B(L 1 -,L 1 + ) 1, G λ B(X 0 ,L 1 + ) 1
for any λ > 0.

We can complement the above result with the following whose proof can be extracted from [26, Proposition 2.6]:

Proposition 2.10. For any λ ∈ C + it holds R(λ, T H ) = R λ + Ξ λ HR(1, M λ H)G λ = R(λ, T 0 ) + ∞ n=0 Ξ λ H (M λ H) n G λ (2.9)
where the series converges in B(X 0 ).

We make the following basic observation which will turn useful in the sequel

Proposition 2.11. If f ∈ X k+1 , 0 k N H , then g λ := R(λ, T H )f ∈ X k , ∀λ ∈ C + . Moreover, if ̺ f = 0 then ̺ g λ = 0 for all λ ∈ C + .
Proof. Assume that ̺ f = 0. The equation λ g λ -T H g λ = f implies, after integration, that

λ Ω×V g λ (x, v)dx ⊗ m(dv) = Ω×V f (x, v)dx ⊗ m(dv) = 0.
Because λ = 0, one sees that ̺ g λ = 0.

2.5. Some auxiliary operators. For λ = 0, we can extend the de nition of these operators in an obvious way but not all the resulting operators are bounded in their respective spaces. However, we see from the above integration formula (2.3), that

M 0 ∈ B(L 1 -, L 1 + ) with M 0 u L 1 + = u L 1 -, ∀u ∈ L 1 -.
In the same way, one deduces from (2.2) that for any nonnegative ϕ ∈ X 0 :

Γ + G 0 ϕ(x, v)dµ + (x, v) = Γ + dµ + (x, v) τ -(x,v) 0 ϕ(x -sv, v)ds = Ω×V ϕ(x, v)dx ⊗ m(dv) (2.10) which proves that G 0 ∈ B(X 0 , L 1 + ) with G 0 ϕ L 1 + = ϕ X 0 , ∀ϕ ∈ X 0 .
Notice that, more generally, for any η ∈ R

G iη ∈ B(X 0 , L 1 + ), M iη ∈ B(L 1 -, L 1 + ) with G iη B(X 0 ,L 1 + ) 1, M iη B(L 1 -,L 1 + ) 1.
To be able to provide a rigorous de nition of the operators Ξ 0 and R 0 we need the following The interest of the above boundary spaces lies in the following (see [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]Lemma 2.8] where (2.12) is proven for k = 1 but readily extends to k ∈ N):

Lemma 2.12. For any u ∈ Y - 1 one has Ξ 0 u ∈ X 0 with Ξ 0 u X 0 = Γ - u(x, v)τ + (x, v)dµ + (x, v) D u Y - 1 , ∀u ∈ Y - 1 (2.11)
where we recall that D is the diameter of Ω. Moreover, given k [START_REF]Linear modulus of linear operator[END_REF]. We wish to emphasise here that, if H satis es assumptions 1.2 1), then

1, if u ∈ Y - k then M 0 u ∈ Y + k and Ξ 0 u ∈ X k-1 with M 0 u Y + k = u Y - k and Ξ 0 u X k-1 D u Y - k (2.12) If f ∈ X 1 then G 0 f ∈ Y + 1 and R 0 f ∈ D(T 0 ) ⊂ X and T 0 R 0 f = -f . Remark 2.
Ψ H ∈ X n ∀n N H Indeed, recall from [26, Proposition 4.2], that Ψ H = Ξ 0 H φ where φ ∈ L 1 + is such that M 0 H φ = φ.
From Assumption 1.2 1), H φ ∈ Y - n+1 and from (2.12), φ ∈ Y - n+1 and Ψ H ∈ X n . An immediate but fundamental consequence of the above Lemma is the following which will be used repeatedly in the sequel: Corollary 2.14. It holds

M 0 H ∈ B(L 1 + , Y + N H +1 ), Ξ 0 H ∈ B(L 1 + , X N H ) (2.13)
In particular, for any f ∈ L 1 + and any ε 0, the mapping

η ∈ R -→ Ξ ε+iη Hf ∈ X 0 is of class C N H with d k dη k Ξ ε+iη Hf X 0 D k Ξ 0 H B(L 1 + ,X k ) f L 1 + k ∈ {0, . . . , N H }.
Proof. The proof is a direct consequence of (2.12) which implies

M 0 ∈ B(Y - k , Y + k ) while Ξ 0 ∈ B(Y - k , X k-1 ) for any k 1. Since H ∈ B(L 1 + , Y - N H +1
), we get (2.13). One checks then in a straightforward way that, for any k N H ,

d k dη k Ξ ε+iη Hf = (-it -) k Ξ ε+iη Hf where (-it -) k is the multiplication operator by the measure mapping (x, v) ∈ Ω × V → (-it -(x, v)) k . This gives directly the result since (-it -(x, v)) k D k ̟ k (v) for any (x, v) ∈ Ω × V and Ξ ε+iη H B(L 1 + ,X k ) M 0 H B(L 1 + ,X k ) Ξ 0 H B(L 1 + ,X N H ) ∀ε 0, η ∈ R as soon as k N H .
Remark 2.15. As observed in the Introduction, the fact that the maximal gain of integrability for H is measured by N H is what make the above C N H the maximal regularity of the mapping η → Ξ ε+iη H ∈ B(L 1 + , X 0 ). One may wonder here if some additional assumption like

H ∈ B(L 1 + , Y - N H +1+α
), for some α ∈ (0, 1) would induce some additional fractional derivative that could be exploited.

One has the following

Lemma 2.16. For any ε 0, the mapping

η ∈ R -→ M ε+iη H ∈ B(L 1 + ) is uniformly continuous on R. Consequently, lim |η|→∞ (M ε+iη H) p B(L 1 + ) = 0 ∀ε > 0 (2.14)
where p is de ned through (1.7).

Proof. Let ε 0 be xed. Given η 1 , η 2 ∈ R, one has

M ε+iη 1 H -M ε+iη 2 H = (exp (-iη 1 τ -) -exp (-iη 2 τ -)) M ε H so that, because the mapping t → e it ∈ C is 2-Lipschitz, M ε+iη 1 Hϕ -M ε+iη 2 H B(L 1 + ) 2|η 1 -η 2 | τ -M ε H B(L 1 + ) 2|η 1 -η 2 | τ -M 0 H B(L 1 + ) . Now, because H ∈ B(L 1 + , Y - 1 ) and τ -(x, v) D/|v|, one sees that τ -M 0 H B(L 1 + ) D M 0 H B(L 1 + ,Y + 1 ) D M 0 B(Y - 1 ,Y + 1 ) H B(L 1 + ,Y - 1 ) . Thus, M ε+iη 1 Hϕ -M ε+iη 2 H B(L 1 + ) 2D|η 1 -η 2 | H B(L 1 + ,Y - 1 )
which proves that the mapping

η ∈ R → M ε+iη H ∈ B(L 1 +
) is uniformly continuous on R. We deduce then (2.14) from the uniform continuity and the integrability at in nity ensured by (1.7).

A D P

3.1. First de nitions. We recall here a useful representation of the semigroup U H (t) as a kind of Dyson-Phillips expansion series introduced in [START_REF]Explicit transport semigroup associated to abstract boundary conditions[END_REF]. We recall the de nition of the C 0 -semigroup generated by T 0 :

U 0 (t)f (x, v) = f (x -tv, v)1 {t<t -(x,v)} , f ∈ X 0 , t 0.
We begin with the following de nition where

D 0 = {f ∈ D(T max ) ; B -f = 0 = B + f } and U 0 (t) = U 0 (t) (t 0): De nition 3.1. Let t 0, k 1 and f ∈ D 0 be given. For (x, v) ∈ Ω × V with t -(x, v) < t
, there exists a unique y ∈ ∂Ω with (y, v) ∈ Γ -and a unique 0 < s < min(t, τ + (y, v)) such that x = y + sv and then one sets

[U k (t)f ](x, v) = HB + U k-1 (t -s)f (y, v), We set [U k (t)f ](x, v) = 0 if t -(x, v) t and U k (0)f = 0. Remark 3.2. Clearly, for (x, v) ∈ Ω × V with t -(x, v) < t, the unique (y, v) ∈ Γ -and s ∈ (0, min(t, τ + (y, v)) such that x = y + sv are y = x -t -(x, v)v, s = t -(x, v)
so that the above de nition reads

[U k (t)f ](x, v) = H(B + U k-1 (t -s)f (x -sv, v) s=t -(x,v)
.

For a di use boundary operator H the expression of U n (t) is fully explicit, namely Lemma 3.3. If H is given by (1.4), then, for any n ∈ N and any

(x, v) ∈ Ω × V , it holds [U n (t)f ] (x, v) = Γ + (y 0 ) k(y 0 , v, v 0 )|v 0 • n(y 0 )|m(dv 0 ) Γ + (y 1 ) k(y 1 , v 0 , v 1 )|v 1 • n(y 1 )|m(dv 1 ) . . . Γ + (y n-2 ) k(y n-2 , v n-3 , v n-2 ) |v n-2 • n(y n-2 )|m(dv n-2 )× × At(x,v,v 0 ,...,v n-2 ) k(y n-1 , v n-2 , v n-1 | |v n-1 • n(y n-1 )| × f y n-1 -t -t -(x, v) - n-2 k=0 τ -(y k , v k ) v n-1 , v n-1 m(dv n-1 ) , (3.1)
for any f ∈ D 0 where

y 0 = x -t -(x, v)v, y k+1 = y k -τ -(y k , v k )v k , k = 0, . . . , n -2.
and,

A t (x, v, v 0 , . . . , v n-2 ) = v n-1 ∈ Γ + (y n-1 ) ; n-2 k=0 τ -(y k , v k ) < t -t -(x, v) < n-1 k=0 τ -(y k , v k ) .
Proof. The proof is by direct inspection. For instance, it is easy to see that, given f ∈ D 0 and t > t -(x, v), one has

[U 1 (t)f ] (x, v) = Γ + (y 0 ) k(y 0 , v, v 0 )f (y 0 -(t -t -(x, v))v 0 , v 0 ) × 1 {t -(x,v)<t<t -(x,v)+τ -(y 0 ,v 0 )} |v 0 • n(y 0 )|m(dv 0 ), and
[U 2 (t)f ] (x, v) = Γ + (y 0 ) k(y 0 , v, v 0 )|v 0 • n(y 0 )|m(dv 0 ) Γ + (y 1 ) k(y 1 , v 0 , v 1 )f (y 1 -(t -t -(x, v) -τ -(y 0 , v 0 ))v 1 , v 1 ) × 1 {t -(x,v)+τ -(y 0 ,v 0 )<t<t -(x,v)+τ -(y 0 ,v 0 )+τ -(y 1 ,v 1 )} |v 1 • n(y 1 )|m(dv 1 ).
The proof for n 3 is then easily deduced by induction.

One has then the following proven in [START_REF]Explicit transport semigroup associated to abstract boundary conditions[END_REF] (see also [4, Appendix A and Theorem 3.8]):

Theorem 3.4. For any k 1, f ∈ D 0 one has U k (t)f ∈ X 0 for any t 0 with U k (t)f X 0 f X 0 .
In particular, U k (t) can be extended to be a bounded linear operator, still denoted

U k (t) ∈ B(X 0 ) with U k (t) B(X 0 ) 1 ∀t 0, k 1. (3.2)
Moreover, the following holds for any k 1 (1) (U k (t)) t 0 is a strongly continuous family of B(X 0 ).

(2) For any f ∈ X 0 and any t, s 0, it holds

U k (t + s)f = k j=0 U j (t)U k-j (s)f. (3) For any f ∈ D 0 , the mapping t 0 → U k (t)f is di erentiable with d dt U k (t)f = U k (t)T max f ∀t 0. (4) For any f ∈ D 0 , one has U k (t)f ∈ D(T max ) for all t 0 with T max U k (t)f = U k (t)T max f = U k (t)T 0 f.
(5) For any f ∈ D 0 and any t 0, the traces

B ± U k (t)f ∈ L 1 ± and the mappings t 0 → B ± U k (t)f ∈ L 1
± are continuous. Moreover, for all f ∈ X and t > 0, one has

B ± t 0 U k (s)f ds ∈ L 1 ± with B - t 0 U k (s)f ds = HB + t 0 U k-1 (s)f ds. (6) For any f ∈ D 0 , it holds t 0 B + U k (s)f L 1 + ds t 0 B + U k-1 (s)f L 1 + ds, ∀t 0.
(7) For any f ∈ X 0 and λ > 0, setting

L k (λ)f = ∞ 0 exp(-λt)U k (t)f dt one has, for k 1, L k (λ)f ∈ D(T max ) with T max L k (λ)f = λ L k (λ)f and B ± L k (λ)f ∈ L 1 ± with B -L k (λ)f = HB + L k-1 (λ)f B + L k (λ)f = (M λ H) k G λ f. (8) For any f ∈ X 0 , the series ∞ k=0 U k (t)f is strongly convergent and it holds U H (t)f = ∞ k=0 U k (t)f
Remark 3.5. One sees from the point [START_REF]A coupling approach for the convergence to equilibrium for a collisionless gas[END_REF] together with [26, Theorem 2.4] that, for any k 1,

L k (λ)f = Ξ λ HB + L k-1 (λ)f. Since L 0 (λ)f = R λ f we deduce that L 1 (λ) = Ξ λ HB + R λ = Ξ λ HG λ ,
and, since B + Ξ λ = M λ , one gets by induction that, for any k 1,

L k (λ) = Ξ λ H (M λ H) k-1 G λ .
In particular, one sees that, in the representation series (2.9) that, for any n 0

Ξ λ H (M λ H) n G λ f = ∞ 0 exp(-λ t)U n+1 (t)f dt (3.3)
for any λ > 0 which is of course coherent with the above point (4).

3.2.

Decay of the iterates. We extend the decay of the semigroup (U 0 (t))) t 0 obtained in Lemma 2.6 to the iterates (U k (t)) t 0 . To do so, we rst observe that Assumption 1.2 1) implies a nice behaviour of H for small velocities. More precisely, introducing for any δ > 0, the operator

H (δ) ∈ B(L 1 + , L 1 -)
given by

H (δ) ψ(x, v) = 1 |v| δ Hψ(x, v) ∀ψ ∈ L 1 + , (x, v) ∈ Γ - (3.4)
Then, for any ψ ∈ L 1 + , one has

H (δ) ψ L 1 -= Γ - 1 |v| δ |Hψ(x, v)|dµ -(x, v) δ n+1 Γ - 1 |v| δ |Hψ(x, v)||v| -(n+1) dµ -(x, v)
from which we deduce the estimate

H (δ) B(L 1 H B(L 1 + ,Y - n+1 ) δ n+1 ∀n N H . (3.5)
Having such a property in mind, we can deduce the decay of U k (t) as t → ∞ for any k ∈ N. For the clarity of exposition, we give full details for the decay of U 1 (t).

Lemma 3.6. Let k ∈ N and f ∈ X k . Then, there exists some universal constant C k > 0 (depending only on H, k and D but not on f ) such that

U 1 (t)f X 0 C k t -(N H +1) + t -k f X k , ∀t > 0 where N H is de ned in Assumption 1.2.
Proof. The proof is based upon the decomposition of H for small and large velocities. Namely, we introduce, for some δ > 0 to be determined, the splitting

H = H (δ) + H (δ)
where H (δ) is de ned in (3.4) and

H (δ) ψ(x, v) = 1 |v|>δ Hψ(x, v), ψ ∈ L 1 + , (x, v) ∈ Γ -. With such a splitting, one has of course, U 1 (t) = U (δ) 1 (t) + U (δ) 1 (t), t > 0, δ > 0 where U (δ) 1 (t), U (δ)
1 (t) are constructed as in De nition 3.1 with H replaced respectively by H (δ) and H (δ) . Let now x k 1, f ∈ X k , t > 0. One has

U 1 (t)f X 0 U (δ) 1 (t)f X 0 + U (δ) 1 (t)f X 0 U (δ) 1 (t)f X 0 + H (δ) B(L 1 + ,L 1
-) f X 0 where we used (A.9). Using now (3.5), there is C (depending only on H) such that

U 1 (t)f X 0 Cδ N H +1 f X 0 + U (δ) 1 (t)f X 0 ∀δ > 0. (3.6)
Let us focus then on the estimate for

U (δ) 1 (t)f . The crucial point is of course that U (δ) 1 (t)f is supported on Ω × {v ∈ V ; |v| > δ} and, on this set, τ -(•, •) is uniformly bounded since t -(x, v) D |v| D δ ∀(x, v) ∈ Ω × V, |v| > δ.
Let us then consider x ∈ Ω and |v| > δ and t > D δ > t -(x, v). We recall (see Lemma 3.3) that

U (δ) 1 (t)f (x, v) = 1 |v|>δ Γ + (y 0 ) k(y 0 , v, v 0 )f (y 0 -(t -t -(x, v))v 0 , v 0 ) × 1 {t -(x,v)<t<t -(x,v)+τ -(y 0 ,v 0 )} |v 0 • n(y 0 )|m(dv 0 ).
One sees that, for t > t -(x, v) and |v| > δ,

1 {t -(x,v)<t<t -(x,v)+τ -(y 0 ,v 0 )} = 0 ⇐⇒ τ -(y 0 , v 0 ) > t -t -(x, v) t - D δ .
Since moreover τ -(y 0 , v 0 )

D |v 0 | one deduces that 1 {t -(x,v)<t<t -(x,v)+τ -(y 0 ,v 0 )} = 0 =⇒ D |v 0 | > t - D δ .
Therefore,

U (δ) 1 (t)f (x, v) 1 |v|>δ Γ + (y 0 ) 1 |v 0 | D t-D δ k(y 0 , v, v 0 ) × 1 {t -(x,v)<t<t -(x,v)+τ -(y 0 ,v 0 )} |f (y 0 -(t -t -(x, v))v 0 , v 0 )| |v 0 • n(y 0 )|m(dv 0 ). Introducing g(z, w) = |w| -k |f (z, w)| , ∀(z, w) ∈ Ω × V one deduces easily that, for any t > D δ , U (δ) 1 (t)f (x, v) D j t -D δ j U (δ) 1 (t)g (x, v)
Therefore,

U (δ) 1 (t)f X 0 D j t -D δ j U (δ) 1 (t)g X 0 D j t -D δ j g X 0 , ∀t > D δ
where we used (3.2). Combining this with (3.6), and since max( f X 0 , g X 0 ) f X k , one deduces that

U 1 (t)f X 0 Cε N H +1 + D j t -D δ j f X k , ∀t > D δ .
Choosing δ such that t = 2D δ , we get the result. We generalise this to the other iterates Lemma 3.7. Let k ∈ N and f ∈ X k . Then, there exists some universal constant C k > 0 (depending only on H, k and D but not on f ) such that, for any n 1,

U n (t)f X 0 C k (2 n -1) n + 1 t (N H +1) + n + 1 t k f X k , ∀t > 0
where N H is de ned in Assumption 1.2.

Proof. The proof uses the same ideas introduced in the proof for n = 1. Given ε > 0, we still introduce the splitting H = H (δ) + H (δ) . With the representation of H given by (1.4), it is clear that

           H (δ) ψ(x, v) = v ′ •n(x)>0 k (δ) (x, v, v ′ )ψ(x, v ′ ) |v ′ • n(x)|m(dv ′ ), H (δ) ψ(x, v) = v ′ •n(x)>0 k (δ) (x, v, v ′ )ψ(x, v ′ ) |v ′ • n(x)|m(dv ′ ), ψ ∈ L 1 + , (x, v) ∈ Γ -
where

k (δ) (x, v, v ′ ) = 1 |v| δ k(x, v, v ′ ), k (δ) (x, v, v ′ ) = 1 |v|>δ k(x, v, v ′ ),
and

(x, v) ∈ Γ -, v ′ ∈ Γ + (x). Since k(•, •, •) = k (δ) (•, •, •) + k (δ) (•,
•, •), using the representation formula (3.1) one sees by a simple combinatorial argument that, for n 1, one can write

U n (t) = U (δ) n (t) + U (δ) n (t)
where

U (δ) n (t) is given by (3.1) with all kernels k(•, •, •) replaced with k (δ) (•, •, •) whereas the reminder term U (δ) n (t) is the some of 2 n -1 operators U (δ) n (t) = 2 n -1 j=1 V (j) n (t)
where, for any j ∈ {1, . . . ,

2 n -1}, V (j)
n (t) is de ned by (3.1) with at least one kernel k (δ) (•, •, •). Alternatively, this means that V (j) n (t) is de ned as in De nition A.3 for a family of boundary operators (H 1 , . . . , H n ) where there is at least one i ∈ {1, . . . , n} such that H i = H (δ) (the other ones being indi erently H (δ) or H (δ) ). Using Proposition A.6, one has then

V (j) n (t) B(X 0 ) H (δ) B(L 1 + ,L 1 -) . Therefore U (δ) n (t)f X 0 (2 n -1) H (δ) B(L 1 + ,L 1 -) f X 0 and U n (t)f X 0 U (δ) n (t)f X 0 + C(2 n -1) δ N H +1 f X 0 , t > 0, ε > 0 (3.7)
where we used (3.5). We focus now on the expression of U (δ)

n (t)f . As before, U (δ) n (t)f is sup- ported on Ω × {v ∈ V ; |v| > δ}. We have, from (3.1), U (δ) n (t)f (x, v) = 1 |v|>δ Γ δ + (y 0 ) k(y 0 , v, v 0 )|v 0 • n(y 0 )|m(dv 0 ) Γ δ + (y 1 ) k(y 1 , v 0 , v 1 )|v 1 • n(y 1 )|m(dv 1 ) . . . Γ δ + (y n-2 ) k(y n-2 , v n-3 , v n-2 ) |v n-2 • n(y n-2 )|m(dv n-2 )× × A δ t (x,v,v 0 ,...,v n-2 ) k(y n-1 , v n-2 , v n-1 | |v n-1 • n(y n-1 )| × f y n-1 -t -t -(x, v) - n-2 k=0 τ -(y k , v k ) v n-1 , v n-1 m(dv n-1 ) , (3.8)
where

y 0 = x -t -(x, v)v, y k+1 = y k -τ -(y k , v k )v k (k = 0, . . . , n -2), Γ δ + (y k ) = Γ + (y j ) ∩ {v ∈ V ; |v| > δ}, k = 0, . . . , n -2
and,

A δ t (x, v, v 0 , . . . , v n-2 ) = v n-1 ∈ Γ δ + (y n-1 ) ; n-2 k=0 τ -(y k , v k ) < t -t -(x, v) < n-1 k=0 τ -(y k , v k ) .
Notice that, since |v| > δ and |v k | > δ for any k = 0, . . . , n -2, one has

t -(x, v) + n-2 k=0 τ -(y k , v k ) nD δ .
Let us consider then t > nD δ . One sees then that if

v n-1 ∈ A δ t (x, v, v 0 , . . . , v n-2 ) then D |v n-1 | τ -(y n-1 , v n-1 ) t -t -(x, v) + n-2 k=0 τ -(y k , v k ) t - nD δ which implies |v n-1 | < D t -nD δ . (3.9)
As in the proof of Lemma 3.6, one introduce now

g(z, w) = |w| -k |f (z, w)| , (z, w) ∈ Ω × V
and sees then from (3.7),

U (δ) n (t)f (x, v) 1 |v|>δ Γ δ + (y 0 ) k(y 0 , v, v 0 )|v 0 • n(y 0 )|m(dv 0 ) Γ δ + (y 1 ) k(y 1 , v 0 , v 1 )|v 1 • n(y 1 )|m(dv 1 ) . . . Γ δ + (y n-2 ) k(y n-2 , v n-3 , v n-2 ) |v n-2 • n(y n-2 )|m(dv n-2 )× × A δ t (x,v,v 0 ,...,v n-2 ) k(y n-1 , v n-2 , v n-1 | |v n-1 • n(y n-1 )| × |v n-1 | k g y n-1 -t -t -(x, v) - n-2 k=0 τ -(y k , v k ) v n-1 , v n-1 m(dv n-1 ) .
Therefore, from (3.9),

U (δ) n f (x, v) D t -nD δ k U (δ) n (t)g (x, v) t > nD δ .
Consequently

U (δ) n (t)f X 0 D k t -nD δ k U (δ) n (t)g X 0 D k t -nD δ k g X 0 , t > nD δ
thanks to (3.2). Combining this with (3.7) we nally obtain

U n (t)f X 0 D k t -nD δ k g X 0 + C(2 n -1) δ N H +1 f X 0 , t > nD δ .
Picking now δ > 0 such that t = (n+1)D δ we get the result since both g X 0 and f X 0 are smaller than f X k .

We deduce directly from Lemmas 2.6 and 3.7 the following

Proposition 3.8. Assume that f ∈ X N H +1
then, for any n 1, there exists

C n > 0 such that n k=0 U k (t)f X 0 C n t -(N H +1) f X N H +1 ∀t > 0.
(3.10) 3.3. Representation formulae for remainder terms. Introducing

S n (t) := U H (t) - n k=0 U k (t), n 1, t > 0
one has the following Proposition 3.9. For any n ∈ N, n p where p de ned through (1.7). Then, for any f ∈ X 0 , one has

S n (t)f = exp(εt) 2π lim ℓ→∞ ℓ -ℓ exp (iηt) Υ n (ε + iη)f dη, ∀f ∈ X 0 (3.11)
for any t > 0, ε > 0.

Proof. One notices that, for any n 0 and any f ∈ X 0 , it holds

∞ 0 exp (-λt) S n (t)f dt = ∞ k=n ∞ 0 exp (-λt) U k+1 (t)f dt = ∞ k=n Ξ λ H (M λ H) k G λ f = Υ n (λ)f, Reλ > 0
where we used (3.3) together with the fact that, for Reλ > 0, M λ H B(L 1 + ) < 1. Since moreover, for any f ∈ X 0 , the mapping t 0 → S n (t)f is continuous and bounded, with S n (0)f = 0, one applies the complex Laplace inversion formula [2, Theorem 4.2.21] to deduce

S n (t)f = exp(εt) 2π lim L→∞ 1 2L L -L dℓ ℓ -ℓ exp (iηt) Υ n (ε + iη)f dη, ∀f ∈ X 0 (3.12)
for any t > 0, ε > 0, i.e. S n (t)f is the Cesarò limit of the family

ℓ -ℓ exp (iηt) Υ n (ε + iη)f dη ℓ .
Let us prove it is actually a classical limit. Fix ε > 0 and f ∈ X 0 . Arguing as in (5.17),

Υ n (ε + iη) B(X 0 ,X k ) Ξ 0 B(Y - k ,X k ) H B(L 1 + ,Y - k ) (M ε+iη H) n B(L 1 + ) R(1, M ε+iη H) B(L 1 + ) G ε+iη B(X 0 ,L 1 + )
Since, for any ε > 0,

R(1, M ε+iη H) B(L 1 + ) R(1, M ε H) B(L 1 + ) while sup η G ε+iη B(X 0 ,L 1 + )
1 we deduce that there exists C ε > 0 such that

Υ n (ε + iη) B(X 0 ,X k ) C ε (M ε+iη H) n B(L 1 + ) , ∀η ∈ R.
For n p, one has

(M ε+iη H) n B(L 1 + ) (M ε+iη H) p B(L 1
+ ) , we deduce from (1.7) that there is

M ε > 0 such that ∞ -∞ Υ n (ε + iη) B(X 0 ,X k ) dη M ε , ∀ε > 0.
This of course implies that

∞ -∞ exp ((ε + iη)t) Υ n (ε + iη) B(X 0 ,X k ) dη M ε exp(εt), ∀ε > 0.
In particular, for any f ∈ X 0 , the limit

lim ℓ→∞ 1 2π ℓ -ℓ exp ((ε + iη)t) Υ n (ε + iη)f dη
exists in X k . Since its Cesarò limit is S n (t)f , we deduce the result.

Remark 3.10. It appears that the convergence actually holds in operator norm, i.e. the integral

exp(εt) 2π lim ℓ→∞ ℓ -ℓ exp (iηt) Υ n (ε + iη)dη converges to S n (t) in B(X 0 , X k ) as ℓ → ∞.
As a consequence, under Assumption 1.2, one sees that S n (t) is a compact operator for n large enough.

It is clear that, from the above representation formula, no decay of S n (t) can be expected because of the growing function exp(εt). As said already in the Introduction, we will need therefore to derive a second representation formula showing that (3.12) actually holds for ε = 0. Of course, to do so, we need rst to suitably de ne the trace of Υ n (λ) on the imaginary axis, i.e. to de ne properly the limit as ε → 0 + of Υ(ε + iη)f for suitable f . This will require several preliminary de nitions and regularity estimates of the various operators de ning Υ n (ε + iη).

G

This section is devoted to the core technical results which will allow to de ne the boundary function of the mapping λ ∈ C + -→ Υ n (λ) ∈ B(X 0 ).

We rst start with the convergence of the various operators M λ , Ξ λ and G λ . The proof of this result is postponed to the Appendix A Proposition 4.1. For any f ∈ X 0 , the limit

lim ε→0 + G ε+iη f -G iη f L 1 + = 0 (4.1)
uniformly with respect to η ∈ R with

lim |η|→∞ sup ε∈[0,1] G ε+iη f L 1 + = 0. (4.2)
Moreover, for any η ∈ R and any k ∈ N,

M δ+iη -M iη B(Y - k+1 ,Y k ) δ D, Ξ ε+iη -Ξ iη B(Y - k+1 ,X k ) ε D (4.3)
where D is the diameter of Ω. Consequently, for any k N H where N H is de ned in (1.6)

M δ+iη H -M iη H B(L 1 + ,Y + k ) δ D H B(L 1 + ,Y - k+1 )
and

Ξ ε+iη H -Ξ iη H B(L 1 + ,X k ) ε D H B(L 1 + ,Y - k+1 ) ∀η ∈ R. (4.4)
Moreover, for any j ∈ N, one has

lim δ→0 (M ε+iη H) j -(M iη H) j B(L 1 + ) = 0 (4.5)
uniformly with respect to η ∈ R.

Remark 4.2. An important consequence of the above Proposition 4.1 is that the holomorphic func-

tions λ ∈ C + → M λ H ∈ B(L 1 + ) and λ ∈ C + → Ξ λ H ∈ B(L 1
+ , X 0 ) can be extended to continuous functions on C + . Moreover, one easily deduces that

∞ -∞ (M iη H) p B(L 1 + ) dη < ∞
where p > 0 is de ned through (1.7).

4.1. Spectral properties of M λ H along the imaginary axis. We study here more carefully the properties of M iη H for η ∈ R. Proof. We give the proof only for Reλ = 0, the case Reλ > 0 being similar. Since

|M iη Hψ(x, v)| = |M 0 Hψ(x, v)| for any ψ ∈ L 1 + , (x, v) ∈ Γ + , η ∈ R, one sees that M iη H ∈ B(L 1 + ) with |M iη H| M 0 H
where |M iη H| denotes the absolute value operator of M iη H (see [START_REF]Linear modulus of linear operator[END_REF]). The operator M 0 H being power compact, the same holds for |M iη H| by a domination argument so that

r σ (|M iη H|) < r σ (M 0 H) = 1
which is a contradiction. Therefore, r σ (|M iη H|) < 1 and, since r σ (M iη H) r σ (|M iη H|), the conclusion holds true.

We deduce the following Corollary 4.4. For any

η 0 ∈ R \ {0}, there is 0 < δ < 1 2 |η 0 | such that lim ε→0 + sup |η-η 0 |<δ R(1, M ε+iη H) -R(1, M iη H) B(L 1 + ) = 0. Proof. Notice that, if 0 < δ < |η 0 | 2 then, η = 0 whenever |η -η 0 | < δ.
Without loss of generality, we can assume η 0 > 0. From Proposition 4.3, there is ̺ ∈ (0, 1) such that r σ (M iη 0 H) < ̺ < 1.

In particular, there is ℓ ∈ N such that

(M iη 0 H) ℓ 1 ℓ B(L 1 + ) < ̺ < 1. Since M iη H converges to M iη 0 H in operator norm as η → η 0 , there is δ < η 0 2 such that (M iη H) ℓ B(L 1 + ) < ̺ ℓ ∀η ∈ (η 0 -δ, η 0 + δ).
Because of Eq. (4.5), there is ε 0 > 0 small enough, such that, for any 0 < ε < ε 0 we also have

(M ε+iη H) ℓ B(L 1 + ) < ̺ ℓ ∀η ∈ (η 0 -δ, η 0 + δ).
One has then

R(1, M ε+iη H) = ∞ n=0 (M ε+iη H) n = ∞ k=0 ℓ-1 j=0 (M ε+iη H) kℓ+j
and, similarly

R(1, M iη H) = ∞ k=0 ℓ-1 j=0 (M iη H) kℓ+j ∀η ∈ (η 0 -δ 0 , η 0 + δ 0 ). Therefore, R(1, M ε+iη H) -R(1, M iη H) = ∞ k=0 ℓ-1 j=0 (M ε+iη H) kℓ+j -(M iη H) kℓ+j
for any ε ∈ (0, ε 0 ), η ∈ (η 0 -δ 0 , η 0 + δ 0 ). Using again (4.5), each term of the series converges to 0 as η → 0 uniformly with respect to η ∈ (η 0 -δ 0 , η 0 + δ 0 ). To prove the result, it is enough therefore to show that the remainder of the series can be made arbitrarily small in operator norm uniformly on (η 0 -δ 0 , η 0 + δ 0 ). Since, for any k, j 0

(M ε+iη H) kℓ+j B(L 1 + ) (M ε+iη H) kℓ B(L 1 + ) (M ε+iη H) ℓ k B(L 1 + ) ̺ kℓ
for any ε ∈ [0, ε 0 ), we get that, for any n 0

sup |η-η 0 |<δ 0 sup ε∈(0,ε 0 ) ∞ k=n ℓ-1 j=0 (M ε+iη H) kℓ+j -(M iη H) kℓ+j B(L 1 + ) 2ℓ ∞ k=n ̺ ℓ k
which tends to 0 as n → ∞. This combined with the term-by-term convergence of the series as ε → 0 gives the result. We rst deduce from this property of T 0 the following:

Theorem 4.6. If H ∈ B(L 1 + , L 1
-) satis es Assumption 1.2 then iR ⊂ S(T H ). Proof. According to Theorem 4.5, it holds

lim ε→0 + R(ε + iη, T 0 ) B(X 0 ) = +∞, ∀η ∈ R. ( 4
Ξ ε+iη H B(L 1 + ,X 0 ) < ∞.
Moreover sup ε>0,η∈R G ε+iη B(X 0 ,L 1 + ) < ∞ we get that, for any η ∈ R, η = 0, it holds: lim sup

ε→0 + Ξ ε+iη HR(1, M ε+iη H)G ε+iη B(X 0 ) < ∞.
This, together with (4.6) and (2.9) proves that, for any η ∈ R, η = 0, it holds lim sup

ε→0 + R(ε + iη, T H ) B(X 0 ) = ∞,
whence iη ∈ S(T H ) for any η = 0. Recalling that 0 ∈ S p (T H ) we get the conclusion. 4.3. Spectral properties of M λ H in the vicinity of λ = 0. We recall that, being M 0 H stochastic and irreducible, the spectral radius r σ (M 0 H) = 1 is an algebraically simple and isolated eigenvalue of M 0 H and there exists 0 < r < 1 such that

S(M 0 H) \ {1} ⊂ {z ∈ C ; |z| < r}
and there is a normalised and positive eigenfunction ϕ 0 such that

M 0 H ϕ 0 = 1, Γ + ϕ 0 dµ + = 1.
Because M 0 H is stochastic, the dual operator (M 0 H) ⋆ (in L ∞ (Γ + , dµ + )) admits the eigenfunction ϕ ⋆ 0 = 1 Γ + associated to the algebraically simple eigenvalue 1. The spectral projection of M 0 H associated to the eigenvalue 1 is then de ned as

P(0) = 1 2iπ {|z-1|=r 0 } R(z, M 0 H)dz
where r 0 > 0 is chosen so that {z ∈ C ; |z -1| = r 0 } ⊂ {z ∈ C ; |z| > r}. Such a spectral structure is somehow inherited by M λ H for λ small enough:

Proposition 4.7. For any λ ∈ C + the spectrum of M λ H is given by

S(M λ H) = {0} ∪ {ν j (λ) ; j ∈ N λ ⊂ N}
where, N λ is a (possibly nite) subset of N and, for each j ∈ N λ , ν j (λ) is an isolated eigenvalue of M λ H of nite algebraic multiplicities and 0 being the only possible accumulation point of the sequence {ν j (λ)} j∈N λ . Moreover,

|ν j (λ)| < 1 for any j ∈ N λ , λ = 0.
Finally, there exists δ 0 > 0 such that, for any |λ| δ 0 , λ ∈ C + ,

S(M λ H) ∩ {z ∈ C ; |z -1| < ε} = {ν(λ)}
where ν(λ) is an algebraically simple eigenvalue of M λ H such that

lim λ→0 ν(λ) = 1
and there exist an eigenfunction ϕ λ of M λ H and an eigenfunction ϕ ⋆ λ of (M λ H) ⋆ associated to ν(λ) such that lim

λ→0 ϕ(λ) -ϕ 0 L 1 + = 0, lim λ→0 ϕ ⋆ λ -ϕ ⋆ 0 L ∞ (Γ + ,dµ + ) = 0. Proof. Since (M λ H) 2 (M 0 H) 2 , one has that (M λ H) 2 is
P(λ) = 1 2iπ {|z-1|=r 0 } R(z, M λ H)dz, (4.7)
is converging in operator norm to P(0) as λ → 0 (Reλ 0) so that, in particular, up to reduce again δ 0 ,

dim(Range(P(λ))) = dim(Range(P(0))) = 1, |λ| < δ 0 , Reλ 0.
This shows that

S in (M λ H) = S(M λ H) ∩ {z ∈ C ; |z -1| < ε} = {ν(λ)}, |λ| < δ 0 , Reλ 0,
where ν(λ) is a algebraically simple eigenvalue of M λ H. Notice that, clearly

lim λ→0 ν(λ) = 1 (Reλ 0).
In the same way, de ning

P(λ) ⋆ = 1 2iπ {|z-1|=r 0 } R(z, (M λ H) ⋆ )dz, |λ| δ 0 , Reλ 0 it holds that lim λ→0 P(λ) ⋆ -P(0) ⋆ B(L ∞ (Γ + ,dµ + )) = 0.
Set ϕ λ := P(λ)ϕ 0 , λ ∈ C + . Since ϕ λ converges to P(0)ϕ 0 = ϕ 0 = 0, we get that ϕ λ = 0 for λ small enough and, since ν(λ) is algebraically simple, ϕ(λ) is an eigenfunction of M λ H for |λ| small enough. In the same way, for |λ| small enough, ϕ ⋆ λ := P(λ) ⋆ ϕ ⋆ 0 -→ P(0) ⋆ ϕ ⋆ 0 = 1 as λ → 0 and ϕ ⋆ λ is an eigenfunction of (M λ H) ⋆ associated to the eigenvalue ν(λ). From now, we de ne δ > 0 small enough, so that the rectangle

C δ := {λ ∈ C ; 0 Reλ δ , |Imλ| δ} ⊂ {λ ∈ C ; |λ| < δ 0 } ,
where δ 0 is introduced in the previous Proposition 4.7.

Lemma 4.8. The mapping

λ ∈ C δ -→ P(λ) ∈ B(L 1 + ) is di erentiable with P ′ (0) = - 1 2iπ {|z-1|=r 0 } R(z, M 0 H)(τ -M 0 H)R(z, M 0 H)dz.
More generally, for any η ∈ (-δ, δ),

d dη P(iη) = - 1 2iπ {|z-1|=r 0 } R(z, M iη H) d dη M iη H R(z, M iη H)dz.
Proof. The only di culty is to prove the di erentiability on the imaginary axis. As soon as z / ∈ S (M λ H) for any λ ∈ C δ , one has

d dλ R(z, M λ H) = -R(z, M λ H) d dλ M λ H R(z, M λ H), so that d dλ P(λ) = - 1 2iπ {|z-1|=r 0 } R(z, M λ H) d dλ M λ H R(z, M λ H)dz ∀λ ∈ C δ and, since lim λ→0 d dλ (M λ H) = -lim λ→0 (τ -M λ H) = -τ -M 0 H
we easily get the di erentiability in 0. The same computations also give

d dη P(ε+iη) = - 1 2iπ {|z-1|=r 0 } R(z, M ε+iη ) d dη M ε+iη H R(z, M ε+iη H)dz, ∀η ∈ R\{0}.
Using now Prop. 5.4 which asserts that d dη M ε+iη H converges to d dη M iη H as ε → 0 + uniformly with respect to η, we deduce the second part of the Lemma.

We can complement the above result with the following: Lemma 4.9. With the notations of Proposition 4.7, the function λ ∈ C δ → ν(λ) ∈ C is di erentiable with derivative ν ′ (λ) such that the limit

ν ′ (0) = lim λ→0 ν ′ (λ)
exists and is given by

ν ′ (0) = - Γ + τ -(x, v)ϕ 0 (x, v)dµ + (x, v) < 0.
Proof. Recall that we introduced in the proof of Proposition 4.7 the functions

ϕ λ = P(λ)ϕ 0 , ϕ ⋆ λ = P(λ) ⋆ ϕ ⋆ 0 , λ ∈ C δ
which are such that lim λ→0 ϕ λ = ϕ 0 and lim λ→0 ϕ ⋆ λ = ϕ ⋆ 0 = 1 Γ + . Introducing the duality bracket •, • between L 1 + and its dual (L 1 + ) ⋆ = L ∞ (Γ + , dµ + ), we have in particular

lim λ→0 ϕ λ , ϕ ⋆ λ = ϕ 0 , ϕ ⋆ 0 = Γ + ϕ 0 dµ + = 1.
Moreover, the mappings λ

∈ C δ → ϕ λ ∈ L 1 + and λ ∈ C δ → ϕ ⋆ λ ∈ (L 1 + ) ⋆ are di erentiable with d dλ ϕ λ = d dλ P(λ)ϕ 0 , d dλ ϕ ⋆ λ = d dλ P(λ) ⋆ ϕ ⋆ 0 . Since M λ Hϕ λ = ν(λ)ϕ λ so that M λ Hϕ λ , ϕ ⋆ λ = ν(λ) ϕ λ , ϕ ⋆ λ , we deduce rst that λ ∈ C δ → ν(λ)
is di erentiable, and di erentiating the above identity yields

d dλ (M λ Hϕ λ ) = ν ′ (λ)ϕ λ + ν(λ) d dλ ϕ λ .
Computing the derivatives and multiplying with ϕ ⋆ λ and integrating over Γ + we get

d dλ M λ H ϕ λ + M λ H d dλ ϕ λ , ϕ ⋆ λ = ν ′ (λ) ϕ λ , ϕ ⋆ λ + ν(λ) d dλ ϕ λ , ϕ ⋆ λ . Using that d dλ M λ H = -τ -M λ H whereas M λ H d dλ ϕ λ , ϕ ⋆ λ = d dλ ϕ λ , (M λ H) ⋆ ϕ ⋆ λ = ν(λ) d dλ ϕ λ , ϕ ⋆ λ we obtain -τ -M λ Hϕ λ , ϕ ⋆ λ + ν(λ) d dλ ϕ λ , ϕ ⋆ λ = ν ′ (λ) ϕ λ , ϕ ⋆ λ + ν(λ) d dλ ϕ λ , ϕ ⋆ λ . Thus -τ -M λ Hϕ λ , ϕ ⋆ λ = ν ′ (λ) ϕ λ , ϕ ⋆ λ , ∀λ ∈ C δ .
Letting λ → 0, we get that lim λ→0 ν ′ (λ) = -τ -M 0 Hϕ 0 , ϕ ⋆ 0 which is the desired result since M 0 Hϕ 0 = ϕ 0 .

4.4. Boundary functions for R(λ, T 0 ) and R(λ, T H ). We have now all the tools at hands to de ne the traces of the functions λ

∈ C + → R(λ, T 0 )f ∈ X 0 and λ ∈ C + → R(λ, T H )f
along the imaginary axis. We will distinguish between the two cases η = 0 and η = 0. For the latter case, the technical di culty is tremendously increased due to the fact that 1 lies in the spectrum of M 0 H and we will resort to the careful study of the spectral properties of M λ H for λ ∈ C + with |λ| small. To handle this case, we will need the additional assumption that f has zero mean ̺ f = 0 (see (5.10)) together with a slight additional integrability f ∈ X 1 . Notice that the constraint (5.10) exactly means that Pf = 0 (where P is the spectral projection associated to the (dominant) zero eigenvalue of T H ) or equivalently, f = (I -P)f . For such a case, the assumption (5.10) that f has zero mean will be fully exploited. Related to this assumption (5.10), we introduce

X 0 k := {f ∈ X k ; ̺ f = 0}, k ∈ N.
which is a closed subspace of X k . Notice that, endowed with the X k -norm, X 0 k is a Banach space. Since

Ω×V U H (t)f dx ⊗ m(dv) = Ω×V f dx ⊗ m(dv), ∀t 0, f ∈ X 0 one has Ω×V R(λ, T H )f dx ⊗ m(dv) = Ω×V ∞ 0 e -λ t U H (t)f dt dx ⊗ m(dv) = 1 λ Ω×V f dx ⊗ m(dv), ∀λ ∈ C +
and therefore the resolvent and all its iterates R(λ, T H ) k leave X 0 0 invariant (k 0). An important consequence of the spectral result stated in Prop. 4.7 is the following Lemma 4.10. For any f ∈ X 0 1 the limit lim

ε→0 + R(1, M ε+iη H)G ε+iη f = Φ(η)f exists in X 0 where Φ(η)f := R(1, M iη H)G iη f if η = 0 R (1, M 0 H (I -P(0))) G 0 f -1 ν ′ (0) [P ′ (0)G 0 f + P(0)G ′ 0 f )] if η = 0 (4.8)
Moreover, the convergence is uniform on any compact subset of R.

Proof. Let f ∈ X 1 be xed with ̺ f = 0. For any ε > 0, η ∈ R, one writes

R(1, M ε+iη H)G ε+iη f = R(1, M ε+iη H(I -P(ε + iη)))G ε+iη f + R(1, M ε+iη HP(ε + iη))G
I -P(ε + iη)) = 1 2iπ {|z|=r ′ } R(z, M ε+iη H)dz so that lim ε→0 + I -P(ε + iη)) = I -P(iη)) in B(L 1 + ) uniformly with respect to |η| < δ. Consequently, lim ε→0 + sup |η| δ R(1, M ε+iη H(I -P(ε + iη)))G iη f -R(1, M iη H(I -P(iη)))G iη f L 1 + = 0. (4.9)
On the other hand,

R(1, M ε+iη HP(ε + iη))G ε+iη f = 1 1 -ν(ε + iη) P(ε + iη)G ε+iη f so that lim ε→0 + R(1, M ε+iη HP(ε + iη))G ε+iη f = 1 1 -ν(iη) P(iη)G iη f, ∀η = 0
where the limit is meant in L 1 + and we used the continuity of λ ∈ C δ → ν(λ). Whenever η = 0, we have

R(1, M ε HP(ε))G ε f = 1 1 -ν(ε) P(ε)G ε f = ε 1 -ν(ε) P(ε)G ε f -P(0)G 0 f ε
where we used the fact that, since ̺ f = 0 and G 0 is stochastic, one has

Γ + G 0 f dµ + = 0 so P(0)G 0 f = 0.
As already seen, the derivative G ′ (0)f exists since f ∈ X 1 and therefore, by virtue of Lemma 4.8,

lim ε→0 + P(ε)G ε f -P(0)G 0 f ε = P ′ (0)G 0 f + P(0)G ′ 0 f.
According to Lemma 4.9,

lim ε→0 + ε 1 -ν(ε) = - 1 ν ′ (0) > 0 so that lim ε→0 + R(1, M ε HP(ε))G ε f = - 1 ν ′ (0) P ′ (0)G 0 f + P(0)G ′ 0 f .
Finally, we obtain that

lim ε→0 + R(1, M ε H)G ε f exists in L 1
+ and is given by

R(1, M 0 H(I -P(0)))G 0 f - 1 ν ′ (0) P ′ (0)G 0 f + P(0)G ′ 0 f .
This proves the convergence. Let us prove that the convergence is uniform with respect to |η| δ. According to (4.9), we only need to prove that the convergence

lim ε→0 + R(1, M ε+iη HP(ε + iη))G ε+iη f towards F (η) = R(1, M iη HP(iη))G iη f if η = 0 -1 ν ′ (0) [P ′ (0)G 0 f + P(0)G ′ 0 f )] if η = 0
. is uniform with respect to |η| < δ. We argue by contradiction, assuming that there exist c > 0, a sequence (ε n ) n ⊂ (0, ∞) converging to 0 and a sequence

(η n ) n ⊂ (-δ, δ) such that R(1, M εn+iηn HP(ε n + iη n ))G εn+iηn f -F (η n ) L 1 + c > 0. (4.10)
Up to considering a subsequence, if necessary, we can assume without loss of generality that lim n η n = η 0 with |η 0 | δ. First, one sees that then η 0 = 0 since the convergence of R(1, M ε+iη HP(ε+ iη)G ε+iη f to F (η) is actually uniform in any neighbourhood around η 0 = 0 (see (??)). Because η 0 = 0, de ning

λ n := ε n + iη n , n ∈ N, the sequence (λ n ) n ⊂ C δ is converging to 0. Now, as before, R(1, M λn HP λn )G λn f = λ n 1 -ν(λ n ) P(λ n )G λn f -P(0)G 0 f λ n , n ∈ N with lim n→∞ λ n 1 -ν(λ n ) = - 1 ν ′ (0) , lim n→∞ P(λ n )G λn f -P(0)G 0 f λ n = P ′ (0)G 0 f + P(0)G ′ 0 f . Therefore, lim n→∞ R(1, M λn HP(λ n ))G λn f = - 1 ν ′ (0) P ′ (0)G 0 f + P(0)G ′ 0 f .
One also has

F (η n ) = R(1, M iηn HP(iη n ))G iηn f = iη n 1 -ν(iη n ) P(iη n )G iηn f -P(0)G 0 f iη n , n ∈ N so that F (iη n ) has the same limit -1 ν ′ (0) [P ′ (0)G 0 f + P(0)G ′ 0 f ] as n → ∞.
This contradicts (4.10).

One deduce from this the following Proposition 4.11. For any f ∈ X 0 0 ,

Ξ ε+iη HR(1, M ε+iη H)G iη f ∈ X k ∀k ∈ {0, . . . , N H } and, for any k N H , lim ε→0 + Ξ ε+iη HR(1, M ε+iη H)G iη f = Ξ iη HΦ(η)f (4.11)
where the convergence is meant in C 0 (R, X k ).

Proof. The rst part of the result is clear from the regularising properties of Ξ ε+iη H. Let us focus on the proof of (4.11) and let us x f ∈ X 0 satisfying (5.

Ξ ε+iη HR(1, M ε+iη H)G iη f -Ξ iη HΦ(η)f X k εD H B(L 1 + ,Y - k+1 ) sup η∈[a,b] R(1, M ε+iη H)G iη f X k + Ξ 0 B(Y - k+1 ,X k ) H B(L 1 + ,Y - k+1 ) sup η∈[a,b] R(1, M ε+iη H)G iη f -Ξ iη HΦ(η)f X k (4.12)
where we used that

sup η∈R Ξ iη H B(L 1 + ,X k ) Ξ 0 B(Y - k+1 ,X k ) H B(L 1 + ,Y - k+1 )
. From Corollary 4.10 one concludes that sup ε∈(0,1) sup η∈[a,b] R(1, M ε+iη H)G iη f X k is nite while the last term in (4.12) converges to 0 as ε → 0 + . This shows that

lim ε→0 + sup η∈[a,b] Ξ ε+iη HR(1, M ε+iη H)G iη f -Ξ iη HΦ(η)f X k = 0.
Let us then focus on |η| > R, R > 0 arbitrary. We already saw that

sup ε∈[0,1] sup |η|>R Ξ ε+iη HR(1, M ε+iη H) B(L 1 + ,X k ) < ∞,
Ξ ε+iη HR(1, M ε+iη H)G ε+iη f X k = 0
and this implies clearly the result.

Regarding the behaviour of R(ε + iη, T 0 )f , one has Proposition 4.12. For any f ∈ X 0 and ε > 0, the mapping

η ∈ R -→ R(ε + iη, T 0 )f ∈ X 0 belongs to C k 0 (R, X 0 ) for any k ∈ N. Moreover, given k ∈ N, for any f ∈ X k+1 , lim ε→0 + R(ε + iη, T 0 )f exists in C 0 (R, X k ). Its limit is denoted R(iη, T 0 )f.
Proof. We begin with the rst part of the Proposition. Recalling that

R(ε + iη, T 0 )f = ∞ 0 e -iηt ε -εt U 0 (t)f dt with t ∈ R → e -εt U 0 (t)f ∈ X 0 Bochner integrable one deduces from Riemann-Lebesgue Theorem that lim |η|→∞ R(ε + iη, T 0 )f X 0 = 0. Given k ∈ N, because d k dη k R(ε + iη, T 0 )f = (-i) k ∞ 0 t k e -iηt e -εt U 0 (t)f dt
the exact same argument shows that

lim |η|→∞ d k dη k R(ε + iη, T 0 )f X 0 = 0 which proves that η ∈ R → R(ε + iη, T 0 )f belongs to C k 0 (R, X 0 ).
Let us focus now on the limit for ε → 0 + . Given f ∈ X k+1 , we deduce from Lemma 2.7 and the dominated convergence theorem that

lim ε→0 + R(ε + iη, T 0 )f = lim ε→0 + ∞ 0 e -iηt e -εt U 0 (t)f dt = ∞ 0 e -iηt U 0 (f )f dt exists in X k . The limit is of course denoted R(iη, T 0 ) and one has R(ε + iη, T 0 )f -R(iη, T 0 )f X k ∞ 0 e -εt -1 U 0 (t)f X k dt ∀η ∈ R, ε > 0.
Thus lim

ε→0 + sup η∈R R(ε + iη, T 0 )f -R(iη, T 0 )f X k = 0 (4.13)
still using the fact that t → U 0 (t)f X k is integrable over [0, ∞) and the dominated convergence theorem.

Remark 4.13. One deduces from the above Proposition and Banach-Steinhaus Theorem [8, Theorem 2.2, p. 32] that

C k := sup R(ε + iη, T 0 ) B(X k+1 ,X k ) ; ε ∈ (0, 1] ; η ∈ R < ∞ ∀k ∈ N. (4.14) Corollary 4.14. Given k ∈ N, if g : λ ∈ C + -→ g(λ) ∈ X k+1
is a continuous mapping such that

lim |η|→∞ g(ε + iη) X k+1 = 0 ∀ε > 0 while the limit g(η) := lim ε→0 + g(ε + iη)
exists in X k+1 uniformly with respect to η ∈ R, then

lim ε→0 + R(ε + iη, T 0 )g(ε + iη) = R(iη, T 0 ) g(η) in C 0 (R, X k ).
Proof. Since the convergence (4.13) holds for any f ∈ X k+1 , the convergence is of course uniform on any compact subset of X k+1 . Since by assumption the mapping g :

η ∈ R → g(η) ∈ X 0 belongs to C 0 (R, X k+1 ), the set { g(η) ; η ∈ R} is a compact subset of X k+1 .
Thus sup

ε→0 + sup η∈R R(ε + iη, T 0 ) g(η) -R(iη, T 0 ) g(η) X k = 0
Now, noticing that, for any ε ∈ (0, 1], it holds

R(ε + iη, T 0 )g(ε + iη) -R(iη, T 0 ) g(η) X k R(ε + iη, T 0 ) (g(ε + iη) -g(η)) X k + R(ε + iη, T 0 ) g(η) -R(iη, T 0 ) g(η) X k C k g(ε + iη) -g(η) X k+1 + R(ε + iη, T 0 ) g(η) -R(iη, T 0 ) g(η) X k
where C k is de ned in (4.14), we deduce easily that

lim ε→0 + sup η∈R R(ε + iη, T 0 )g(ε + iη) -R(iη, T 0 ) g(η) X k = 0
which proves the result.

The convergence established in Prop. 4.12 extends to derivatives of R(ε

+ iη, T 0 )f Lemma 4.15. Given k ∈ N and f ∈ X k+1 . It holds lim ε→0 + sup η∈R d k dη k R(ε + iη, T 0 )f - d k dη k R(iη, T 0 )f X 0 = 0.
Consequently, the mapping

η ∈ R -→ R(iη, T 0 )f ∈ X 0 belongs to C k 0 (R, X 0 ). Proof. As already established d k dη k R(ε + iη, T 0 )f = (-i) k ∞ 0 e -iηt e -εt U 0 (t)f dt and, since R(iη, T 0 )f = ∞ 0 e -iηt U 0 (t)f dt one sees easily that, if f ∈ X k+1 , d k dη k R(iη, T 0 )f = (-i) k ∞ 0 e -iηt t k U 0 (t)f dt
is well-de ned in X 0 thanks to Lemma 2.7. One concludes then exactly as in Prop. 4.12.

Recalling that

R(ε + iη, T H ) = R(ε + iη, T 0 ) + Ξ ε+iη HR(1, M ε+iη H)G ε+iη
the previous results allows to prove the following Proposition 4.16. For any k ∈ N, k N H and any f ∈ X 0 k+1 the limit lim

ε→0 + R(ε + iη, T H )f exists in C 0 (R, X k ).
We denote by R(iη, T H )f the limit.

Proof. Using the above splitting R(ε

+ iη, T H ) = R(ε + iη, T 0 ) + Ξ ε+iη HR(1, M ε+iη H)G ε+iη , Prop. 4.
12 shows the convergence of the rst term and (4.11) gives the one of the second one. This proves the result with

R(iη, T H )f = R(iη, T 0 )f + Ξ iη HΦ(η)f and of course the mapping η ∈ R → R(iη, T H )f belongs to C 0 (R, X k ).
Remark 4.17. Notice that

R(ε + iη, T H ) ∈ B(X k+1 , X k ) ∀ε > 0, η ∈ R
and one sees from Prop. 4.16 that R(iη, T H ) ∈ B(X 0 k+1 , X k ) with, thanks to Banach-Steinhaus Theorem,

sup R(ε + iη, T H ) B(X 0 k+1 ,X k ) ; ε ∈ (0, 1] ; η ∈ R := C k < ∞ for any k ∈ {0, . . . , N H } . Notice also that one cannot hope to go beyond the threshold value k = N H since Ξ iη H maps L 1 + in X N H but not in X N H +1 .
As it was the case for R(ε+iη, T 0 ), the above convergence extends to derivatives. The crucial observation is the following general property of the resolvent

d k dλ k R(λ, T H ) = (-1) k k!R(λ, T H ) k+1 , λ ∈ C + .
One has then the key technical result Proposition 4.18. Assume that H satis es assumptions 1.2 and let f ∈ X 0 N H +1 . Then,

lim ε→0 + [R(ε + iη, T H )] k f := [R(iη, T H ] k f in C 0 R, X N H -(k-1)
holds for any k ∈ {0, . . . , N H + 1}.

Proof. The proof is made by induction over k ∈ N, k N H + 1 For k = 1, the result holds true by Proposition 4.16. Let k ∈ N, k N H and assume the result to be true for any j ∈ {1, . . . , k}.

Let us prove the result is still true for k + 1. One recalls that

R(ε + iη, T H ) = R(ε + iη, T 0 ) + Υ 0 (ε + iη) where Υ 0 (λ) = Ξ λ HR(1, M λ H)G λ . One has [R(ε + iη, T H )] k+1 f = R(ε + iη, T 0 ) [R(ε + iη, T H )] k f + Υ 0 (ε + iη) [R(ε + iη, T H )] k f. (4.15)
On the one hand, our induction hypothesis implies that lim ε→0

+ [R(ε + iη, T H ] k f = [R(iη, T H )] k f in C 0 (R, X N H +1-k ).
Thanks to Corollary 4.14, we deduce that

lim ε→0 + R(ε + iη, T 0 ) [R(ε + iη, T H )] k f = R(iη, T 0 ) [R(iη, T H )] k f in C 0 (R, X N H -k ).
(4.16) On the other hand, from the induction hypothesis

lim |η|→∞ [R(iη, T H ] k f X N H +1-k = 0 which implies in particular that lim |η|→∞ [R(iη, T H ] k f X 0 = 0. Since moreover [R(iη, T H ] k f ∈ X 0 0 and sup η∈R Ξ iη HΦ(η) B(X 0 0 ,X N H -k ) < ∞
where Φ(η)f is de ned in Lemma 4.10, one sees that

lim |η|→∞ Ξ iη HΦ(η) [R(iη, T H ] k f X N H -k = 0.
Now, to prove that

lim ε→0 Υ 0 (ε + iη) [R(ε + iη, T H )] k f = Ξ iη HΦ(η) [R(iη, T H )] k f in C 0 (R, X N H -k
), one argues along the exact same lines as those used to prove (4.16) since, according to (4.11),

lim ε→0 sup η∈R Υ 0 (ε + iη)g -Ξ iη HΦ(η)g X N H -k = 0, ∀g ∈ X 0 0
one can resume the argument of Corollary 4.14 to deduce that, for any continuous mapping

g : λ ∈ C + -→ g(λ) ∈ X 0 0 such that lim |η|→∞ g(ε + iη) X 0 = 0 ∀ε > 0 and g(η) := lim ε→0 + g(ε + iη)
exists in X 0 0 uniformly with respect to η ∈ R, it holds lim

ε→0 sup η∈R Υ 0 (ε + iη)g(ε + iη) -Ξ iη HΦ(η) g(η) X N H -k = 0.
Applying this with g(ε

+ iη) = [R(ε + iη, T H )] k f and g(iη) = [R(iη, T H )] k f we deduce that lim ε→0 Υ 0 (ε + iη) [R(ε + iη, T H )] k f = Ξ iη HΦ(η) [R(iη, T H ] k f in C 0 (R, X N H -k )
which, combined with (4.16) and (4.15) achieves the induction.

A fundamental consequence of the previous Proposition is Corollary 4.19. Assume that H satis es assumptions 1.2. For any f ∈ X 0 N H +1 , the mapping η ∈ R -→ R(iη, T H )f de ned in Proposition 4.16 belongs to C N H 0 (R, X 0 ) and the convergence lim

ε→0 + R(ε + iη, T H )f = R(iη, T H f ) holds in C N H 0 (R, X 0 ). Proof. Let f ∈ X 0 N H +1 be xed. Since, for any k ∈ {1, . . . , N H } and any ε > 0, η ∈ R, d k dη k R(ε + iη, T H )f = (-i) k k! [R(ε + iη, T H )] k+1
f the result follows directly from Proposition 4.18 where the derivatives of R(iη, T H )f are de ned by

d k dη k R(iη, T H )f = (-i) k k! [R(iη, T H )] k+1 f for any k ∈ {0, . . . , N H }. 5. D Υ n (λ)
This section is devoted to the construction of the trace along the imaginary axis, that is when

λ = iη, η ∈ R, of Υ n (λ)f = Ξ λ H (M λ H) n R(1, M λ H)G λ f, λ ∈ C, Reλ 0, n ∈ N
for a suitable class of function f. The crucial observation here is the following alternative representation of Υ n (λ)f which can also be written as

Υ n (λ) = R(λ, T H ) -R(λ, T 0 ) - n k=0 Ξ λ H (M λ H) k G λ , λ ∈ C + (5.1)
as can easily be seen from the fact that

R(λ, T H ) = R(λ, T 0 ) + ∞ k=0 Ξ λ H (M λ H) k G λ .
We already investigated the existence and regularity of the traces on the imaginary axis of the rst two terms in (5.1) so we just need to focus on the properties of the nite sum

s n (λ) := n p=0 Ξ λ H (M λ H) p G λ , λ ∈ C + . (5.2)
The di erentiability of the various involved (single) operators is summarized in the following whose proof is easy and postponed to Appendix A. In the sequel, the notion of di erentiability of functions h : λ ∈ C + → h(λ) ∈ Y (where Y is a given Banach space) is the usual one but we have to emphasize the fact that limits are always meant in C + 2 2 This means for instance that, if λ0 ∈ C+, h is di erentiable means that it is holomorphic in a neighborhoud of λ0 whereas, for λ0 = iη0, η0 ∈ R, the di erentiability at λ0 of h at means that there exists h ′ (λ0) ∈ Y such that

lim λ→λ 0 λ∈C + h(λ) -h(λ0) λ -λ0 -h ′ (λ0) Y = 0
Proposition 5.1. We have the following general di erentiability properties:

(1) For any k 1 and any f ∈ X k , the limit

lim λ→0 d k dλ k G λ f = (-1) k G 0 (t k + f ) exists in L 1 + and sup λ∈C + d j dλ j G λ f L 1 + D j f X j D j f X k , ∀j ∈ {0, . . . , k}. (5.3) (2) For any k N H d k dη k M ε+iη H - d k dη k M iη H B(L 1 + ) ε D H B(L 1 + ,Y - k+1 ) ∀η ∈ R, ε > 0.
(5.4)

Remark 5.2. Notice that the above expression of

d j dλ j G λ f shows that, for f ∈ X j , d j dλ j G λ f = (-1) j G λ t j + f (5.5)
where t j + denotes the multiplication operator by the function

(x, v) → t j + (x, v). As a consequence Corollary 5.3. For any k ∈ N such that H ∈ B(L 1 + , Y - k+1 ) (i.e. k N H ), the function λ = ε + iη ∈ C + -→ d j dη j M λ H ∈ B(L 1 + ), 0 j k
can be extended to a continuous functions on C + . In particular, the mapping

η ∈ R -→ M iη H ∈ B(L 1 + ) is of class C N H (5.6)
with bounded derivatives up to order N H . In the same way, the function

λ = ε + iη ∈ C + -→ d j dη j Ξ λ H ∈ B(L 1 + , X 0 ), 0 j k
can be extended to a continuous functions on C + and the mapping

η ∈ R -→ Ξ iη H ∈ B(L 1 + ) is of class C N H (5.7)
with bounded derivatives up to order N H .

Moreover, for any ϕ ∈ Y - 1 , the limit lim λ→0

d dλ M λ ϕ exists in L 1 + . In particular, lim λ→0 d dλ M λ H = -τ -M 0 H exists in B(L 1 + )
where, as before, we use the same symbol τ -for the measurable function τ -(•, •) and the multiplication operator by that function and we recall that limits are meant in C + , i.e. lim We refer to Appendix A for the full proof of Proposition 5.1 as well as that of Corollary 5.3. All the above results allow to prove the regularity the nite sum s n (λ) de ned by (5.2) Proposition 5.4. For any f ∈ X N H +1 , the mapping

η ∈ R -→ s n (ε + iη)f ∈ X 0 belongs to C N H 0 (R, X 0 ) for any ε 0 with lim |η|→∞ sup ε∈(0,1] d k dη k s n (ε + iη)f X 0 = 0 (5.8)
for any k ∈ {0, . . . , N H }. In particular

lim ε→0 + sup η∈R s n (ε + iη)f -s n (iη)f X 0 = 0.
Proof. Let n ∈ N be xed. For simplicity of notations, for any p ∈ {0, . . . , n}, we de ne

L p (λ) = (M λ H) p , λ ∈ C +
and denotes its derivatives of order j by L

p (λ). Computing derivatives with Leibniz rule we get, for any k ∈ N

d k dλ k s n (λ)f = n p=0 k ℓ=0 k ℓ d k-ℓ dλ k-ℓ Ξ λ H d ℓ dλ ℓ [L p (λ)G λ f ] = n p=0 k ℓ=0 ℓ j=0 ℓ j k ℓ d k-ℓ dλ k-ℓ Ξ λ H L (j) p (λ) d ℓ-j dλ ℓ-j [G λ f ]
Now, as observed (see the proofs of Proposition 5.1 and Corollary 5.3), for any j

d j dλ j Ξ λ = (-1) j t j -Ξ λ , d j dλ j G λ f = (-1) j G λ t j + f
where t j ± denote here the multiplication operator by t ± (x, v) j . Therefore

d k dλ k s n (λ)f = n p=0 k ℓ=0 ℓ j=0 ℓ j k ℓ (-1) k-j t k-ℓ -Ξ λ HL (j) p (λ)G λ t ℓ-j + f . (5.9) If f ∈ X k , then t ℓ-j + f ∈ X 0 and L (j) p (λ)G λ t ℓ-j + f ∈ L 1
-so that (see Eqs. (5.6)-(5.7))

HL (j) p (λ)G λ t ℓ-j + f ∈ Y - N H +1 and Ξ λ HL (j) p (λ)G λ t ℓ-j + f ∈ X N H . Then, if k N H , t ℓ-j -Ξ λ HL (j) p (λ)G λ t ℓ-j
+ f ∈ X 0 for all ℓ ∈ {0, . . . , k}, j ∈ {0, . . . , ℓ}. This easily proves that the mapping

λ ∈ C + -→ s n (λ)f ∈ X 0 is of class C N H with sup λ∈C + d k dλ k s n (λ)f X 0 C k f X N H +1
for some positive C k > 0 depending only on k ∈ {0, . . . , N H }. Let us now prove (5.8) which will also prove the fact that the mapping η → s n (ε + iη)f belongs to C N H 0 (R, X 0 ). The proof of (5.8) follows exactly the lines of Proposition 4.11. Indeed, for any k N H , p ∈ {0, . . . , n}, ℓ ∈ {0, . . . , k} and j ∈ {0, . . . , ℓ}, one easily see that, for any R > 0, sup

ε∈[0,1] sup |η|>R t k-ℓ -Ξ ε+iη HL (j) p (ε + iη) B(L 1 + ,X 0 ) sup ε∈[0,1] sup |η|>R Ξ ε+iη HL (j) p (ε + iη) B(L 1 + ,X k ) Ξ 0 B(Y - k+1 ,X k ) H B(L 1 + ,Y - k+1 ) sup |η|>R L (j) p (iη) B(L 1 + ,X k ) .
Now, one can prove that that there is

C k > 0 such that sup |η|>R L (j) p (iη) B(L 1 + ,X k ) C k < ∞ (see Lemma A.2 in Appendix A) from which sup ε∈[0,1] sup |η|>R t k-ℓ -Ξ ε+iη HL (j) p (ε + iη) B(L 1 + ,X 0 ) < ∞.
Combining this with (4.2) in Prop. 4.1 and the representation (5.9) proves (5.8). The fact that s n (ε+

iη)f converges to s n (iη)f in C N H 0 (R, X 0
) is then deduced from (5.9) and the limits established in Proposition 4.1.

We have all the tools to prove the rst part of Theorem 1.7 in the Introduction.

Theorem 5.5. Let f ∈ X N H +1 be such that

̺ f = Ω×V f (x, v)dx ⊗ m(dv) = 0.
(5.10)

Then, for any n 0 the limit lim

ε→0 + Υ n (ε + iη)f, exists in C N H 0 (R, X 0 ). Its limit is denoted Ψ n (η)f .
Proof. We know from Corollary 4.19 that

lim ε→0 + R(ε + iη, T H )f = R(iη, T H )f holds in C N H 0 (R, X 0 ).
In the same way, Lemma 4.15 shows that lim

ε→0 + R(ε + iη, T 0 )f = R(iη, T 0 )f holds in C N H 0 (R, X 0 ).
Since one sees easily from Prop. 5.4 that lim

ε→0 + s n (ε + iη)f = s n (iη)f in C N H 0 (R, X 0 )
we get the result from the representation (5.1).

Remark 5.6. Notice that, using the representation

Υ n (ε + iη)f = Ξ ε+iη H (M ε+iη H) n R(1, M ε+iη H)G ε+iη , ε > 0, η ∈ R
together with Lemma 4.10, it is easy to see, by uniqueness of the limit, that

Ψ n (η)f = Ξ iη H (M iη H) n Φ(η)f, ∀η ∈ R. (5.11)
where Φ(η)f is de ned in (4.8).

In the following, we show also that, if n is large enough, the boundary function is also integrable.

Lemma 5.7. Assume that n 2 N H p (with p de ned in (1.7)) and f ∈ X 0 N H +1 . Then, the derivatives of the trace function

η ∈ R → Ψ n (η)f ∈ X 0 are integrable with moreover R d k dη k Ψ n (η)f X 0 dη < ∞ ∀k ∈ {0, . . . , N H }.
Proof. We use here the representation of the boundary function Ψ n (η)f in (5.11). We recall that,

for |η| > R, it holds Φ n (η)f = R(1, M iη H)G iη f and Ψ n (η)f = Υ n (iη)f = Ξ iη H (M iη H) n R(1, M iη H)G iη f
where we can write

R(1, M iη H) = ∞ m=0 p-1 r=0 (M iη H) mp+r where R > 0 is chosen such that (M iη H) p B(L 1 + ) 1 2 for |η| > R.
Introducing, as in the proof of Proposition 5.4,

L k (λ) = (M λ H) k , λ ∈ C + ; k ∈ N one has then Ψ n (η)f = ∞ m=0 p-1 r=0 Ξ iη HL mp+r+n (iη)G iη f
Exactly as in Proposition 5.4, Eq. (5.9), we have,

d k dη k Ψ n (η)f = ∞ m=0 p-1 r=0 k ℓ=0 ℓ j=0 ℓ j k ℓ (-i) k-j t k-ℓ -Ξ iη HL (j) mp+r+n (iη)G iη t ℓ-j + f Recall now that, for any ℓ -j k N H , sup |η|>R G iη t ℓ-j + f L 1 + f X N H +1 and t k-ℓ -Ξ iη H B(L 1 + ) Ξ 0 B(Y - k ,X k ) H B(L 1 + ,Y - k )
from which

d k dη k Ψ n (η)f X 0 C k f X N H +1 ∞ m=0 p-1 r=0 k j=0 L (j) mp+r+n (iη) B(L 1 + )
, ∀|η| > R. (5.12)

We chose now n 2 k p and use inequality (A.4) in Lemma A.2 to deduce that, for any m 0, r 0 and j ∈ {0, . . . , k}, there is

C k > 0 such that L (j) mp+r+n (iη) B(L 1 + ) C k ((m + 1)p + n) k L ⌊ mp+r+n 2 k ⌋ (iη) B(L 1 + )
.

Then, since n 2 k p, one has ⌊ mp+r+n

2 k ⌋ p + ⌊ m 2 k ⌋p so that L (j) mp+r+n (iη) B(L 1 + ) C k L p (iη) B(L 1 + ) ((m + 1)p + n) k L ⌊ m 2 k ⌋p (iη) B(L 1 + ) C k L p (iη) B(L 1 + ) ((m + 1)p + n) k 2 -⌊ m 2 k ⌋
where we recall that L p (iη)

B(L 1 + ) 1 2 for any |η| > R. Since ∞ m=0 p-1 r=0 k j=0 ((m + 1)p + n) k 2 -⌊ m 2 k ⌋ p(k + 1) ∞ m=0 ((m + 1)p + n) k 2 -⌊ m 2 k ⌋ < ∞
we deduce from (5.12) that there is a positive constant β k > 0 such that

d k dη k Ψ n (η)f X 0 β k f X N H +1 L p (iη) B(L 1 + ) ∀|η| > R. (5.13)
Using (1.7) once again, we deduce the result.

5.1.

A second representation of the remainder terms. Theorem 5.5 allows to establish the following new representation of S n (t) where we recall that S n (t) = U H (t) -n k=0 U k (t) has been de ned in Proposition 3.8. Theorem 5.8. For any n p where p is de ned through (1.7) and any f ∈ X N H +1 satisfying (5.10), one has

S n (t)f = lim ℓ→∞ 1 2π ℓ -ℓ exp (iηt) Ψ n (η)f dη = 1 2π ∞ -∞ exp (iηt) Ψ n (η)f dη, ∀t > 0
(5.14) where the convergence holds in X 0 . Consequently, for any n 2 N H p and any f ∈ X 0

N H +1 , S n (t)f = - i t N H ∞ -∞ exp (iηt) d N H dη N H Ψ n (η)f dη 2π (5.15)
holds true for any t 0 where the convergence of the integral holds in X 0 .

Proof. We rst deduce from the uniform convergence obtained in Theorem 5.5 that, for any ℓ > 0 and any f ∈ X N H +1 and any t 0

lim ε→0 + 1 2π ℓ -ℓ exp ((ε + iη)t) Υ n (ε + iη)f dη = 1 2π ℓ -ℓ exp (iηt) Ψ n (η)f dη (5.16)
where the convergence occurs in X 0 .

For any η = 0, one has

Ψ n (η)f = Ξ iη H (M iη H) n R(1, M iη H)G iη f so that Ψ n (η)f X 0 Ξ 0 B(Y - 1 ,X 0 ) H B(L 1 + ,Y - 1 ) (M iη H) n B(L 1 + ) R(1, M iη H) B(L 1 + ) G iη B(X 1 ,L 1 + ) f X 1 Ξ 0 B(Y - 1 ,X 0 ) H B(L 1 + ,Y - 1 ) (M iη H) n B(L 1 + ) R(1, M iη H) B(L 1 + ) f X 1 (5.17) where we used that |Ξ iη | Ξ 0 , G iη B(X 0 ,L 1 + ) G 0 B(X 0 ,L 1 + ) 1 and M iη H B(L 1 + )
1. Recall (see Lemma 2.16) that we can nd R > 0 large enough such that

(M iη H) p B(L 1 + ) 1 2 , ∀|η| > R.
Arguing exactly as in Corollary 4.4, one proves easily that, for |η| > R,

R(1, M iη H) = ∞ m=0 p-1 j=0 (M iη H) mp+j from which R(1, M iη H) B(L 1 + ) p ∞ m=0 (M iη H) p m B(L 1 + ) = p 1 -(M iη H) p B(L 1 + ) 2p
Combining this with (5.17), one gets

Ψ n (η)f X 0 2p Ξ 0 B(Y - 1 ,X 0 ) H B(L 1 + ,Y - 1 ) (M iη H) n B(L 1 + ) f X 1 , |η| > R.
Moreover, for n p, (M iη H) n

B(L 1 + ) (M iη H) p B(L 1 + ) so that, there exists C > 0 such that Ψ n (η)f X 0 C (M iη H) p B(L 1 + ) f X 1 |η| > R, n p.
The same exact reasoning shows that, actually, sup

+ iη)t) Υ n (ε + iη)f X 0 C exp(t) (M iη H) p B(L 1 + ) f X 1 ε∈(0,1) exp ((ε
for any |η| > R and any n p. Since the mapping

|η| > R -→ (M iη H) p B(L 1
+ ) is integrable thanks to (1.7), we deduce, from the dominated convergence theorem that

lim ε→0 + |η|>R exp ((ε + iη)t) Υ n (ε + iη)f dη = |η|>R exp(iηt)Ψ n (η)f dη
where the convergence holds in X 0 . This proves (5.14) according to the representation formula (3.12). The proof of (5.15) is then deduced easily after N H integration by parts, using Lemma 5.7.

5.2.

Decay rate: proof of Theorem 1.3. We can prove our main result stated in the Introduction Proof of Theorem 1.3. Let us x f ∈ X N H +1 . To prove the result, we can assume without loss of generality that

̺ f = 0, i.e f ∈ X 0 N H +1 . Of course, the term Θ f (•) in Theorem 1.3 is given by Θ f (η) = d N H dη N H Ψ n (η)f ∈ X 0 , η ∈ R
for some suitable choice of n ∈ N. Recall rst that, for any n ∈ N and any t 0

U H (t)f = ∞ k=0 U k (t)f = n k=0 U k (t) + S n (t)f
where, according to Proposition 3.8,

n k=0 U k (t)f X 0 C n (1 + t) -N H -1 , ∀t 0
for some positive constant C n depending on n and f (but not on t). Choosing now n 2 N H p and using (5.15), one obtains

U H (t)f X 0 C n (1 + t) -N H -1 + t -N H F n (t)
where

F n (t) = (-i) N H 2π ∞ -∞ exp (iηt) d N H dη N H Ψ n (η)f dη X 0
is such that lim t→∞ F n (t) = 0 according to Riemann-Lebesgue Theorem (recall the mapping Lemma 5.7). This proves the rst part of the result. Let us now prove the second part of it and assume that (1.9) holds true, i.e.

η → d N H dη N H Ψ n (η)f ∈ X 0 is integrable over R according to
|η|>R (M iη H) p B(L 1 + ) dη C(p) R β , ∀R > 0
for some C(p) > 0. We already observed that the mapping

Θ f : η ∈ R -→ d N H dη N H Ψ n (η)f ∈ X 0 belongs to C 0 (R, X 0
) so is uniformly continuous. This allows to de ne a (minimal) modulus of continuity

ω f (s) := sup Θ f (η 1 ) -Θ f (η 2 ) X 0 ; η 1 , η 2 ∈ R, |η 1 -η 2 | s , s 0.
The estimate then comes from some standard reasoning about Fourier transform (see for instance [16, Theorem 3.3.9 (b), p. 196] for similar considerations for the decay of Fourier coe cients of Hölder function). Namely, introducing the Fourier transform of the (Bochner integrable) function

Θ f as Θ f (t) = R exp(iηt)Θ f (η)dη ∈ X 0 , t 0
one has then, since e iπ = -1 = exp(iπt/t), t > 0,

Θ f (t) = - R exp iηt + i π t t Θ f (η)dη = - R exp (iyt) Θ f y - π t dy
which gives, taking the mean of both the expressions of Θ f (t),

Θ f (t) = 1 2 R exp (iηt) Θ f (η) -Θ f η - π t dη.
Consequently, if one assumes that R > 2π,

Θ f (t) X 0 1 2 |η| R Θ f (η) -Θ f η - π t X 0 dη + |η|> R 2 Θ f (η) X 0 dη
where we used that {η ∈ R ; |η

+ π t | > R} ⊂ {η ∈ R ; |η| > R -π} ⊂ {η ∈ R ; |η| > R 2 } since t 1, R -π > R 2 .
Therefore, using the modulus of continuity ω f and the additional assumption (1.9), we deduce that

Θ f (t) X 0 2Rω f π t + |η|> R 2 Θ f (η) X 0 dη 2Rω f π t + 2 β C(p)R -β f X N H +1 , ∀R > 2π, t 1.
(5.18)

Optimising then the parameter R, i.e. choosing

R = 2 β-1 β C(p) f X N H +1 ω f π t 1 β+1
(up to work with t t 0 1 to ensure that R > 2π), we obtain the desired estimate.

Remark 5.9. Notice that, if (1.9) holds true for any β > 0 large enough, then one sees that the decay rate of Θ f (t) can be estimated as

Θ f (t) X 0 = O ω f π t α
for any α ∈ (0, 1). This means that the decay rate of Θ f (t) X 0 can be made as close as possible (without reaching it) to that of t → ω f π t . 5.3. Comments and conjecture. On the basis of the second part of Theorem 1.3, a careful study of the modulus of continuity ω f (•) of the mapping

η ∈ R -→ Θ f (η) = d N H dη N H Ψ n (η)f ∈ X 0 where Ψ n (η)f = Ξ iη H [M iη H] n Φ(η)f (with Φ(η)f de ned in (4. 8
)) would be fundamental to make more explicit our main rate of convergence

U H (t)f -̺ f Ψ H X 0 C f (1 + t) N H ε(t) ∀t 0.
We actually are able to evaluate the modulus of continuity of the bounded and uniformly continuous mapping

χ : η ∈ R -→ d N H dη N H Ξ iη H ∈ B(L 1 + , X 0 ) (5.19)
(see Remark 5.10) and we conjecture that it also the modulus of continuity of Θ f (•). Namely,

ω f (s) C 0 ω(s) + C 1 s ∀s 0 (5.20)
for some positive C 0 , C 1 > 0 where ω(•) is the (minimal) modulus of continuity of χ.

We point out that this is indeed the case on R \ (-δ, δ), δ > 0. However, the general strategy of this paper seems to be such that we cannot deal with the modulus of continuity in the neighborhoud of the origin of a derivative of order N H .

Remark 5.10. The interest of the above conjecture is of course that investigating the modulus of continuity of χ is technically much simpler. In particular, under the additional assumption that there exists α ∈ (0, 1) such that

H ∈ B(L 1 + , Y - N H +1+α ) (5.21)
then, one can prove that the modulus of continuity ω(•) of χ is such that

ω(s) c 1 |s| k H 1-α+k H , ∀s 0
for some c 1 > 0 and where k H is de ned as k H := max{s 0 ; lim sup

ε→0 + ε -s H (ε) B(L 1 + ,L 1 -)
< ∞} with H (ε) = 1 |v|<ε H de ned in (3.5). In this case, if the conjecture (5.20) is correct, the rate of convergence in Theorem 1.3 would be upgraded in

U H (t)f -̺ f Ψ H X 0 = O t -N H - βk H (1+β)(1-α+k H ) , ∀f ∈ X N H +1 (5.22)
In particular, for the model described in Example 1.6, since β can be chosen arbitrarily and (5.21) holds for any α ∈ (0, 1), the rate of convergence would be

U H (t)f -̺ f Ψ H X 0 = O t -d+δ , ∀f ∈ X d
for any δ > 0 (see Section 6.3 for details).

6. A A 1.2
We recall here that, in our previous contribution [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF], we introduced a general class of di use boundary operators that we called regular and proved in [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]Theorem 5.1], that, if H is a regular and stochastic di use boundary operator, then

HM 0 H ∈ B(L 1 + , L 1
-) is weakly-compact, i.e. Assumption 1.2 2) is met. We also provided in [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF] practical criteria ensuring that M 0 H is irreducible, satisfying then Assumption 1.2 3). Point 1) of Assumption 1.2 is of course something that has to be computed for each speci c boundary operator H. We focus on this Section on some practical criteria yielding to Assumption 1.2 4).

6.1. Some useful change of variables. We establish now a fundamental change of variable formula which has its own interest. A systematic use of such a change of variable will be made in the companion paper [START_REF]On eventual compactness of collisionless kinetic semigroups with non zero velocities[END_REF]. We begin with the following technical lemma: Lemma 6.1. Let x ∈ ∂Ω be xed. We denote by B d-1 the closed unit ball of R d-1 and de ne

p : z ∈ B d-1 -→ p(z) = x -τ -(x, σ(z))σ(z) ∈ ∂Ω where σ(z) = (σ 1 (z), . . . , σ d (z)) = z 1 , . . . , z d-1 , 1 -|z| 2 ∈ S d-1 , z ∈ B d-1 .

De ning

O x := {z ∈ B d-1 ; σ(z) • n(x) > 0 ; σ(z) • n(p(z)) < 0} , it holds that p is di erentiable on O x and det ∂p(z) ∂z i , ∂p(z) ∂z j 1 i,j d-1 = τ -(x, σ(z)) d-1 (σ(z) • n(p(z))) σ d (z) 2 ∀z ∈ O x .
Proof. The fact that p(•) is di erentiable on O x is a consequence of [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]Lemma A.4]. We recall in particular here that the set

Γ + (x) = {ω ∈ S d-1 ; (x, ω) ∈ Γ + and ξ(x, ω) = (x - τ -(x, ω)ω, ω) ∈ Γ -} is an open subset of {ω ∈ S d-1 ; (x, ω) ∈ Γ + } and τ -(x, •) is di eren- tiable on Γ + (x) with ∂ ω τ -(x, ω) = τ -(x, ω) w • n(ξ(x, ω)) n(ξ s (x, ω)), ω ∈ Γ + (x), (6.1)
where ξ s (x, ω) = x -τ -(x, ω)ω. Since, for any z ∈ O x , p(z) = ξ s (x, σ(z)), this translates in a straightforward way to the di erentiability of p. Moreover, one deduces from (6.1) that

∂ i p(z) = -(∇ ω τ -(x, σ(z)) • ∂ i σ(z)) σ(z) -τ -(x, σ(z))∂ i σ(z) = - τ -(x, σ(z)) σ(z) • n(p(z)) [(n(p(z)) • ∂ i σ(z)) σ(z) + (n(p(z)) • σ(z)) ∂ i σ(z)] ,
where we denote for simplicity ∂ i = ∂ ∂z i . We therefore get

∂ i p(z) ; ∂ j p(z) = τ -(x, σ(z)) σ(z) • n(p(z)) 2 (n(p(z)) • ∂ i σ(z)) (n(p(z)) • ∂ j σ(z)) + (n(p(z)) • σ(z)) 2 (∂ i σ(z) • ∂ j σ(z)) .
Let us x z ∈ O x . We denote by P(z) the matrix whose entries are

P ij (z) = ∂ i p(z) ; ∂ j p(z) , 1 i, j d -1. Using that ∂ i σ(z) • ∂ j σ(z) = δ ij + z i z j σ 2 d (z)
, where σ d (z) is the last component of σ(z), i.e. σ d (z) = 1 -|z| 2 , one sees that

P ij (z) = τ 2 -(x, σ(z)) δ ij + z i z j σ 2 d (z) + 1 (σ(z) • n(p(z))) 2 (n(p(z)) • ∂ i σ(z)) (n(p(z)) • ∂ j σ(z)) .
For simplicity, we will simply denote by n the unit vector n(p(z)) and σ = σ(z). Introducing the vectors u, p ∈ R d-1 whose components are

u i := n • ∂ i σ σ • n , p i := z i σ d , i = 1, . . . , d -1 we have P ij (z) = τ -(x, σ) 2 [δ ij + p i p j + u i u j ] so that det (P(z)) = (τ -(x, σ)) 2(d-1) det (I d-1 + p ⊗ p + u ⊗ u) .
Recall that, for any invertible matrix A and any vectors x, y ∈ R d-1 , then

det (A + x ⊗ y) = det(A) 1 + y, A -1 x . (6.2)
We apply this identity rst by considering A = I d-1 + p ⊗ p and get

det (P(z)) = (τ -(x, σ)) 2(d-1) det(A) 1 + u, A -1 u .
To compute det(A), one uses again (6.2) to deduce

det(A) = 1 + p, p = 1 + |z| 2 σ 2 d = σ -2 d .
One also can compute in a direct way the inverse of A given by A

-1 = I d-1 -1 1+|p| 2 p ⊗ p from which u, A -1 u = |u| 2 - p, u 2 1 + |p| 2 .
This results in

det (P(z)) = (τ -(x, σ)) 2(d-1) 1 + |u| 2 - p, u 2 1 + |p| 2 . (6.3)
Let us make this more explicit. One easily checks that

u i = 1 σ • n n i - n d σ d z i and p, u = 1 (σ • n)σ d d-1 i=1 n i z i - n d σ d z 2 i . Noticing that d-1 i=1 n i z i = (σ • n) -σ d n d , it holds p, u = 1 (σ • n)σ d σ • n -σ d n d - n d σ d |z| 2 = 1 σ 2 d σ d - n d σ • n ,
where we use again that σ 2 d + |z| 2 = 1. Since one also has

|u| 2 = 1 (σ • n) 2 d-1 i=1 n i - n d σ d z i 2 = 1 σ 2 d (σ • n) 2 d-1 i=1 (σ d n i -z i n d ) 2 ,
we get easily after expanding the square and using that d-1 i=1

n 2 i = 1 -n 2 d , |u| 2 = 1 σ 2 d (σ • n) 2 σ 2 d + n 2 d -2(σ • n)σ d n d .
One nally obtains, using (6.3),

det (P(z)) = τ -(x, σ) 2(d-1) σ 2 d 1 + 1 σ 2 d (σ • n) 2 σ 2 d + n 2 d -2(σ • n)σ d n d - 1 σ 2 d σ d - n d σ • n 2
and little algebra gives

det (P(z)) = τ -(x, σ) 2(d-1) σ 2 d (σ • n) 2 which is the desired result.
We complement the above with the following Lemma 6.2. For any x ∈ ∂Ω, introduce

G x := {ω ∈ S d-1 ; (x, ω) ∈ Γ + ; ω • n(x -τ -(x, ω)ω) = 0}. Then, |G x | = 0,
where here | • | denotes the Lebesgue surface measure over S d-1 . Moreover, with the notations of Lemma 6.1, the set

G x = z ∈ B d-1 ; σ(z) = (z, 1 -|z| 2 ) ∈ G x has zero Lebesgue measure (in R d-1 ).
Proof. The proof is based on Sard's Theorem. Let x ∈ ∂Ω be xed. Introducing the function

Ψ : y ∈ ∂Ω \ {x} -→ Ψ(y) = x -y |x -y| , it holds that Ψ is a C 1 function. For any ω ∈ S d-1 setting y ω = x -τ -(x, ω)ω ∈ ∂Ω, one has ω = Ψ(y ω ).
Let us prove that G x is included in the set of critical values of Ψ. To do this, we show that, if ω ∈ G x , then y ω is a critical point of Ψ, i.e. the di erential dΨ(y ω ) is not injective. Since actually Ψ is de ned and smooth on R d \ {x}, its di erential on ∂Ω \ {x} is nothing but the restriction of its di erential on R d \ {x} on the tangent hyperplane to ∂Ω, i.e., for any y ∈ ∂Ω \ {x}, one has

dΨ(y) : h ∈ T y -→ - 1 |x -y| P zy (h),
where T y is the tangent space of ∂Ω at y ∈ ∂Ω \ {x}, z y = x-y |x-y| and, for any z ∈ R d , P z denotes the orthogonal projection onto the hyperplane orthogonal to z,

P z h = h ⊥ z := h -h, z z, z = z |z| ∈ S d-1 , h ∈ R d .
Now, one notices that

ω ∈ G x =⇒ ω • n(y ω ) = 0, ω = Ψ(y ω ).
In particular, one has ω ∈ T yω and P zy ω (ω) = 0 since z yω = x-yω |x-yω| = ω. In particular, dΨ(y ω ) is not injective (ω = 0 belongs to its kernel). We proved that, if ω ∈ G x , then it is a critical value of Ψ and Sard's Theorem implies in particular that the measure of G x is zero. Now, to prove that G x is also of zero measure, one simply notices that G x is the image of G x through the smooth function

P : ω ∈ S d-1 \ {w d = 0} -→ P(ω) = (ω 1 , . . . , ω d-1 ) ∈ B d-1 .
In particular, from the rst part of the lemma, G x is included in the set of critical values of the smooth function P • Ψ, and we conclude again with Sard's Theorem. Lemma 6.3. Assume that ∂Ω satis es Assumption 1.1. For any x ∈ ∂Ω, we set

S + (x) = σ ∈ S d-1 ; σ • n(x) > 0 .
Then, for any nonnegative measurable mapping g : S d-1 → R, one has,

S + (x) g(σ) |σ • n(x)|dσ = ∂Ω g
x -y |x -y| J (x, y)π(dy), and

J (x, y) = 1 Σ + (x) (y) |(x -y) • n(x)| |x -y| d+1 |(x -y) • n(y)|, ∀y ∈ Σ + (x) (6.4) with Σ + (x) = {y ∈ ∂Ω : ]x, y[ ⊂ Ω ; (x -y) • n(x) > 0 ; n(x -y) • n(y) < 0}
where ]x, y[ = {tx + (1 -t)y ; 0 < t < 1} is the open segment joining x and y 3 . In particular, for any nonnegative measurable G : ∂Ω → R and any measurable ϕ : R + → C, we have

S + (x) |σ • n(x)| ϕ(τ -(x, σ)) G(x -τ -(x, σ)σ)dσ = Σ + (x)
G(y)ϕ(|x -y|)J (x, y)π(dy). Proof. Let now x ∈ ∂Ω be given. We can assume without generality that the system of coordinates in R d is such that n(x) = (0, . . . , 1). For a given f : S + (x) → R + , it holds

S + (x) f (σ)dσ = B d-1 f (z, 1 -|z| 2 ) dz 1 -|z| 2 ,
where B d-1 = {z ∈ R d-1 ; |z| < 1}. Moreover, according to Lemma 6.2,

S + (x) f (σ)dσ = S + (x)\Gx f (σ)dσ, while B d-1 f (z, 1 -|z| 2 ) dz 1 -|z| 2 = B d-1 \Gx f (z, 1 -|z| 2 ) dz 1 -|z| 2 .
Then, for the special choice of f (σ) = g(σ) |σ • n(x)|, we get

S + (x) |σ • n(x)| g(σ)dσ = B d-1 g(σ(z)))1 B d-1 \Gx (z)dz (6.6) with σ(z) = (z 1 , . . . , z d-1 , 1 -|z| 2 ) for |z| < 1.
Notice that, with the notations of Lemma 6.1, one has B d-1 \ G x = O x . Still using the notations of Lemma 6.1, we introduce the mapping

p : z ∈ O x → y = p(z) = x -τ -(x, σ(z))σ(z) which is such that p(O x ) = Σ + (x). With this change of variable, notice that τ -(x, σ(z)) = |x -y|, since σ(z) ∈ S d-1 , and therefore σ(z) = x -p(z) τ -(x, σ(z)) = x -y |x -y| .
According to [30,

B d-1 1 B d-1 \Gx (z)g(σ(z))dz = ∂Ω g x -y |x -y| J (x, y)π(dy), where J (x, y) = 1 det(P(z)) 1 Ox (z)
has to be expressed in terms of x and y. Using Lemma 6.1, one has

J (x, y) = (σ(z) • n(p(z)))σ d (z) τ -(x, σ(z)) d-1 1 Ox (z)
with, as mentioned, τ -(x, σ(z)) = |x -y|, p(z) = y and σ(z) = x-y |x-y| . Notice that

σ d (z) = σ(z) • n(z) = (x -y) |x -y| • n(x)
which gives the desired expression (6.4) of J (x, y). Now, if g(σ) = ϕ(τ -(x, σ))G(x-τ -(x, σ)σ), we get (6.5).

We end this section with a useful technical result. Proof. First, one notices that the straightforward estimate

J (x, y) |x -y| 1-d (6.7)
is not strong enough to derive the result (see the subsequent Lemma 6.5 for more details on this point). We need to proceed in a di erent way. Observe that, thanks to Remark ??, for any y ∈ ∂Ω, it holds

|x-y| δ J (x, y)π(dx) = |x-y| δ J (y, x)π(dx) = S + (y) 1 (0,δ] (τ -(y, σ))dσ,
where we used Lemma 6.3 with the functions ϕ(r) = 1 [0,δ] (r) and G ≡ 1. Clearly, for any xed y ∈ ∂Ω

lim δ→0 + S + (y) 1 (0,δ] (τ -(y, σ))dσ = 0 (6.8)
according to the dominated convergence theorem, so one needs to check that the convergence (6.8) is uniform with respect to y ∈ ∂Ω. Assume it is not the case so that there exist c > 0, a sequence {y n } n ⊂ ∂Ω and a sequence (δ n ) n ⊂ (0, ∞) converging to 0 such that S + (yn)

1 (0,δn] (τ -(y n , σ))dσ c ∀n ∈ N.
First, one deduces from Fatou's lemma that

0 < c lim sup n S d-1 1 S + (yn) (σ)1 (0,δn ] (τ -(y n , σ))dσ S d-1 lim sup n 1 S + (yn) (σ)1 (0,δn ] (τ -(y n , σ))dσ. (6.9)
Of course, there is no loss of generality in assuming that {y n } n converges to some y ∈ ∂Ω. Now, ∂Ω being of class C 1 , it holds that lim n n(y n ) = n(y) and therefore there is n 0 ∈ N such that

S + (y) ⊂ S + (y n ) ∀n n 0 .
Moreover, for σ ∈ S + (y), τ + (y, σ) > 0, and, since τ + is lower-semicontinuous on ∂Ω × V (see [START_REF]Functional analytic treatment of the initial boundary value problem for collisionless gases[END_REF]Lemma 1.5]), it holds that

lim inf n→∞ τ -(y n , σ) τ -(y, σ) > 0 ∀σ ∈ S + (y).
As a consequence, one has

lim sup n→∞ 1 (0,δ] (τ -(y n , σ)) = 0 ∀σ ∈ S + (y).
Since {σ ∈ S d-1 ; σ • n(x) = 0} is a subset of S d-1 of zero Lebesgue measure, we see that

lim sup n 1 S + (yn) (σ)1 (0,δ] (τ -(y n , σ)) = 0
for almost every σ ∈ S d-1 which contradicts (6.9). This proves the result.

Whenever the boundary ∂Ω is more regular than merely C 1 one can strengthen the estimate (6.7). Namely, one has the following result (see [START_REF]Decay and continuity of the Boltzmann equation in bounded domains[END_REF]Lemma 2] for a similar result for ∂Ω of class C 2 ) Lemma 6.5. Assume that ∂Ω is of class C 1,α , α ∈ (0, 1) then, there exists a positive constant

C Ω > 0 such that |(x -y) • n(x)| C Ω |x -y| 1+α , ∀x, y ∈ ∂Ω.
Consequently, with the notations of Lemma 6.3, there is a positive constant C > 0 such that

J (x, y) C |x -y| d-1-2α , ∀x, y ∈ ∂Ω, x = y.
Proof. The intuition behind the estimate is that, from the smoothness of ∂Ω, for any x = y ∈ ∂Ω, if e x (y) = x-y |x-y| denotes the unit vector with direction x -y, then lim

x) = 1 1 + |∇Φ(u 0 )| 2 (∇Φ(u 0 ), -1) , so that (x -y) • n(x) = 1 1 + |∇Φ(u 0 )| 2 u -u 0 ; ∇Φ(u 0 ) d-1 -(Φ(u) -Φ(u 0 )) , where •, • d-1 is the inner product in R d-1 . Since, Φ(u) -Φ(u 0 ) = 1 0 u -u 0 , ∇Φ(tu + (1 -t)u 0 ) d-1 dt = u -u 0 ; ∇Φ(u 0 ) d-1 + u -u 0 , 1 0 (∇Φ(tu + (1 -t)u 0 ) -∇Φ(u 0 )) dt d-1 , (
we see that

|Φ(u) -Φ(u 0 -u -u 0 ; ∇Φ(u 0 ) d-1 | |u -u 0 | 1 0 |∇Φ(tu + (1 -t)u 0 ) -∇Φ(u 0 )| dt .
Since ∇Φ ∈ C 0,α , denoting by

C Φ = sup u 1 ,u 2 ∈U |∇Φ(u 1 ) -∇Φ(u 2 )| |u 1 -u 2 | α
the Hölder semi-norm of ∇Φ, we get

|Φ(u) -Φ(u 0 -u -u 0 ; ∇Φ(u 0 ) d-1 | C Φ |u-u 0 | 1+α 1 0 t α dt = C Φ α + 1 |u-u 0 | 1+α , u ∈ U.
We deduce then that

|(x -y) • n(x)| C Φ α + 1 |u -u 0 | 1+α C Φ α + 1 |x -y| 1+α , for u ≃ u 0 .
Since ∂Ω is compact and Ω bounded, this easily yields the conclusion. Now, from (6.4), we get

J (x, y) C 2 Ω |x -y| d-1-2α ∀y ∈ Γ + (x),
which achieves the proof.

Remark 6.6. The above result is still true for ∂Ω of class C 2 which would correspond to α = 1. In such a case, one has

|(x -y) • n(x)| C Ω |x -y| 2 , x, y ∈ ∂Ω and J (x, y) C 2 Ω |x -y| d-3 , ∀y ∈ Γ + (x).
Notice in particular, that, in such a case one has J (x, y) C 2 Ω D 3-d for d = 2, 3, i.e. J is bounded. In such a case, applying Lemma 6.3 with ϕ = 1, one sees that in dimension d = 2, 3, the boundedness of J (x, y) implies the existence of a positive constant C > 0 such that

S + (x) G(x -τ -(x, σ)σ) |σ • n(x)|dσ C Σ + (x) G(y)π(dy) ∀G 0.
This easily allows us to recover [15, Lemma 2.3, Eq. (2.6)].

6.2. Practical criterion ensuring Assumption 1.2 4). We provide here some practical Assumptions under which (1.7) will hold. We recall rst the following generalization of the polar decomposition theorem (see [33,

L 1 (R d , m) it holds R d ψ(v)m(dv) = 1 |S d-1 | ∞ 0 m 0 (d̺) S d-1 ψ(̺ σ)dσ
where dσ denotes the Lebesgue measure on S d-1 with surface |S d-1 |.

We can deduce from the above change of variables the following useful expression for HM λ H. Recall that H is assumed to be given by (1.4) Proposition 6.8. For any λ ∈ C + , it holds

HM λ Hϕ(x, v) = Γ + J λ (x, v, y, w)ϕ(y, w) |w • n(y)|m(dw)π(dy) (6.10)
where

J λ (x, v, y, w) = J (x, y) ∞ 0 ̺ k x, v, ̺ x -y |x -y| × × k y, ̺ x -y |x -y| , |w| exp -λ |x -y| ̺ m 0 (d̺) |S d-1 | (6.11)
for any (x, v) ∈ Γ -, (y, w) ∈ Γ + .

Proof. The proof follows by direct inspection. Indeed, for any ϕ ∈ L 1 + and (x, v) ∈ Γ -:

HM λ Hϕ(x, v) = Γ + (x) exp(-λ τ -(x, v ′ ))k(x, v, v ′ )|v ′ • n(x)|m(dv ′ ) w•n(x-τ -(x,v ′ )v ′ )>0 k(x-τ -(x, v ′ )v ′ , v ′ , w)ϕ(x-τ -(x, v ′ )v ′ , w)|w•n(x-τ -(x, v ′ )v ′ )|m(dw).
Then, using polar coordinates v ′ = ̺ σ and the fact that τ -(x, v ′ ) = ̺ -1 τ -(x, σ), we can use Proposition 6.3, Eq. (6.5) (with h(s) = exp(-λ ̺ -1 s)) and Lemma 6.7 to get

HM λ Hϕ(x, v) = 1 |S d-1 | Σ + (x) J (x, y)π(dy) Γ + (y) ϕ(y, w)|w • n(y)|m(dw)× × ∞ 0 ̺ k x, v, ̺ x -y |x -y| exp -λ |x -y| ̺ π(dy)k y, ̺ x -y |x -y| , |w| m 0 (d̺)
which gives the result.

Remark 6.9. In the special case of Example 1.5, one checks readily that

J λ (x, v, y, w) = γ(x) -1 G(x, v)J (x, y) × γ -1 (y) ∞ 0 ̺ G(y, ̺) exp -λ |x -y| ̺ m 0 (d̺) |S d-1 |
for any (x, v) ∈ Γ -, y ∈ ∂Ω. In particular, J λ (x, v, y, w) does not depend on w.

Thanks to this representation of HM λ H, we can make the following set of assumptions ensuring (1.7) to hold true. Assumption 6.10. Let H ∈ B(L 1 + , L 1 -) be given by (1.4) where the kernel k(x, v, v ′ ) is nonnegative, measurable and satis es (1.5). Assume that m 0 is given by 4 where we adopted the notations

m 0 (d̺) = |S d-1 |̺ d-1 ̟(̺)d̺ for some positive ̟(̺) > 0 with lim ̺→∞ ̺ d+2 k y, v, ̺ x -y |x -y| k y, ̺ x -y |x -y| , w ̟(̺) = 0, ∀(x, v) ∈ Γ -, (y, w) ∈ Γ + ; (6.12) sup (y,w)∈Γ + σ∈S d-1 ∞ 0 ̺ d+1 ̺ ̟(̺) |∇ 2 k(y, ̺σ, w)| + k(y, ̺σ, w) ̺ ̟ ′ (̺) + ̟(̺) d̺ < ∞; (6.13) and sup x∈∂Ω sup (y,w)∈Γ + σ∈S d-1 ∞ 0 ̺ d+2 ̟(̺)k(y, ̺σ, w)d̺ Γ -(x) |∇ 3 k(x, v, ̺σ)| µ x (dv) < ∞. ( 6
∇ 2 k(x, v, w) = ∇ v k(x, v, w), ∇ 3 k(x, v, w) = ∇ w k(x, v, w), for (x, v) ∈ Γ -, w ∈ Γ + (x).
We can then prove the following: Lemma 6.11. Under Assumption 6.10 and if ∂Ω is of class C 1,α with α > 1 2 , then for any λ

∈ C + , λ = 0, it holds sup (y,w)∈Γ + Γ + |J λ (x, v, y, w)| dµ -(x, v) C |λ|
for some positive C > 0 depending only on k and ∂Ω.

Proof. From (6.11) and Lemma 6.5, one has for all (x, v) ∈ Γ -, (y, w)

∈ Γ + |J λ (x, v, y, w)| C Ω |x -y| d-1-2α ∞ 0 ̺ d k x, v, ̺ x -y |x -y| × × k y, ̺ x -y |x -y| , w exp -λ |x -y| ̺ ̟(̺)d̺ .
for some positive constant C Ω . We observe that the last integral can be written as:

1 λ|x -y| ∞ 0 ̺ d+2 k x, v, ̺ x -y |x -y| k y, ̺ x -y |x -y| , w λ|x -y| ̺ 2 exp -λ |x -y| ̺ = d d̺ exp(-λ|x-y|̺ -1 )

̟(̺)d̺

which, after integration by parts and using (6.12), is equal to

- 1 λ|x -y| ∞ 0 exp -λ|x -y|̺ -1 d d̺ ̺ d+2 k x, v, ̺ x -y |x -y| k y, ̺ x -y |x -y| , |w| d̺.
This results in the following estimate for the kernel J λ (x, v, y, w):

|J λ (x, v, y, w)| C Ω |λ| |x -y| d-2α |I(λ, x, y, v, w)| with I(λ, x, v, y, w) = ∞ 0 exp -λ|x -y|̺ -1 d d̺ ̺ d+2 k x, v, ̺ x -y |x -y| k y, ̺ x -y |x -y| , |w| d̺
for any λ = 0, (x, v) ∈ Γ -, (y, w) ∈ Γ + . Distributing the derivative with respect to ̺ thanks to Leibniz rule, one writes Combining all these estimates, we nally obtain that there exists some positive constant C such that The above, combined with Proposition 6.8 yields the following Proposition 6.12. Assume that Assumption 6.10 are in force and ∂Ω is of class C

I(λ, x, v, y, w) = 4 j=1 I j (λ, x, v, y, w) with                                    I 1 (λ, x, v, y, w) = ∞ 0 σ x,y • ∇ 2 k (y, ̺σ x,y , w) ̺ d+2 ̟(̺)k (x, v, ̺σ x,y ) exp -λ|x -y|̺ -1 d̺ I 2 (λ, x, v, y, w) = ∞ 0 ̺ d+2 ̟(̺)σ x,y • ∇ 3 k (x, v, ̺σ x,y ) k (y, ̺σ x,y , w) exp -λ|x -y|̺ -1 d̺ I 3 (λ, x, v, y, w) = ∞ 0 ̺ d+2 ̟ ′ (̺)k(x, v, ̺σ x,y ) k(y, ̺σ x,y ,

6.3.

Examples. We revisit here the examples of practical application in the kinetic theory of gases introduced in the Introduction. We focus here on the case on which

V = R d , m(dv) = dv
for simplicity but of course the case of measure m absolutely continuous with respect to the Lebesgue measure m(dv) = ̟(|v|)dv is easily deduced from our analysis. We give full details for the Example 1.6 in the Introduction which is the most studied model in the framework we are dealing with here (see [START_REF] Mokhtar-Kharroubi | On the speed of approach to equilibrium for a collisionless gas[END_REF][START_REF]A semigroup approach to the convergence rate of a collisionless gas[END_REF][START_REF]Damping of kinetic transport equation with di use boundary condition[END_REF].

We recall that, here k(x, v, v ′ ) = γ -1 (x)M θ(x) (v) with In this case, one sees that γ(x) c d √ θ 0 for some explicit c d > 0 and k(x, v, v ′ ) c -1 d M θ 0 (v). With this, it is easy to deduce that the boundary operator H associated to k(•, •, •) is dominated by a rank-one operator on L 1 + and as such is a regular di use boundary operator (see [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]Remark 3.6]). In particular, from [START_REF]Invariant density and time asymptotics for collisionless kinetic equations with partly di use boundary operators[END_REF]Theorem 5.1],

HM 0 H ∈ B(L 1 + , L
m 0 (d̺) = |S d-1 |̺ d-1 d̺
we see easily that k(x, v, v ′ ) satisfy Assumptions 6.10 (with ̟ ≡ 1). Indeed, notice that k(x, v, v ′ ) is independent of v ′ and depends only on |v| and we denote simply

k(x, v, v ′ ) = G(x, |v|).
Then, (6.12) simply reads since ∇ 2 k(y, v, w) = -v θ(y)γ(y) M θ(y) (v) = -v θ(y) G(y, |v|). Since γ(y) and θ(•) are bounded from below, the result follows easily. Finally, (6.14) is obviously satis ed since ∇ 3 k(x, v, w) = 0. Therefore, Assumption 6.10 is met and one deduces from Proposition 6.12 that, if ∂Ω is of class C 1,α with α > 1 2 , then (1.7) holds true with p = 3 and Assumption 1.2 4) is met. Let us now determine N H for which Assumption 1.2 is met. Direct computations show that, for any integer k,

H ∈ B(L 1 + , Y - k+1 ) ⇐⇒ ∞ 0 |v| -k M θ(x) (v)dv < ∞ ∀x ∈ ∂Ω ⇐⇒ k < d, (6.15)
which means that N H = d -1 (since N H needs to be an integer). Therefore, for this model, we can reformulate Theorem 1.3 as follows, if ∂Ω is of class C 1,α with α > 1 2 , then the following holds: for any f ∈ X d

U H (t)f -̺ f Ψ H X 0 = o (1 + t) d-1 .
Of course, one can be more explicit about the form of o (1 + t) d-1 and, in this case, one sees also that (1.9) holds true for any choice of p and β > 0 since, according to Proposition 6.12, for any n > 2

(M λ H) n B(L 1 + ) C n |λ| n 2 ∀|λ| > 1.
so that

sup η∈R G ε+iη f -G iη f L 1 + Γ + dµ + (x, v) τ -(x,v) 0
1 -e -ε t |f (x -tv, v)|dt.

Since 1 -e -ε t 1 for any ε > 0, t 0, the dominated convergence theorem combined with (2.10) gives the result. Let now consider (A.2) and (A.3). We give the proof for M iη , the proof for Ξ iη being exactly the same. Let η ∈ R be xed. Let ϕ ∈ Y - 1 be given and ε > 0. One has

M
G ε+iη f L 1 + = 0.
The proof resorts from Riemann-Lebesgue Theorem, the only slightly delicate point being to make the Riemann-Lebesgue argument uniform with respect to ε. We write Notice that, since

G ε+iη f (x, v) =
Γ + dµ + (x, v) R h ε x,v (s) ds
|G ε+iη f (x, v)| 1 2 h ε x,v (•) -h ε x,v • - πη |η| 2 L 1 (R)
. Now, writing h ε x,v (s) = e -εs h 0 x,v (s) we see that

h ε x,v (•) -h ε x,v • -
|G ε+iη f (x, v)| h 0 x,v (•) L 1 (R)
and since t + (x -tv, v) = t. In particular, G 0 (t + f ) ∈ L 1 + since f ∈ X 1 and we can invoke the dominated convergence theorem to get the conclusion. The result for higher-order derivatives proceed along the same lines. Let us now prove (5.3). For f ∈ X k , it holds for µ-a. e. (x, v) ∈ Γ + d j dλ j G λ f (x, v) = (-1) j τ -(x,v) 0 s j f (x -sv, v) exp(-λs)ds.

Γ + h 0 x,v (•) L 1 (R) dµ + (x, v) = f X 0 < ∞,
Introducing ϕ(x, v) = |f (x, v)| t + (x, v) j , (x, v) ∈ Ω × R d , we get easily that

d j dλ j G λ f (x, v) τ -(x,v) 0 ϕ(x -sv, v)ds = G 0 ϕ(x, v).
Then, according to (2.10),

d j dλ j G λ f L 1 + G 0 ϕ L 1 + ϕ X 0
For j k, it is clear that ϕ X 0 D j f X j D j f X k and the conclusion follows.

(2) For ϕ ∈ Y - k+1 , ε > 0, η ∈ R one checks easily that

d k dη k M ε+iη ϕ(x, v) - d k dη k M iη ϕ(x, v) = (-i) k τ -(x, v) k (exp (-(ε + iη)τ -(x, v)) -exp (-iητ -(x, v))) M 0 ϕ(x, v),
for any (x, v) ∈ Γ -. Therefore

d k dη k M ε+iη ϕ(x, v) - d k dη k M iη ϕ(x, v) = τ -(x, v) k |M 0 ϕ(x, v)| |exp(-ε(τ -(x, v)) -1|
and, reasoning as in Lemma A.1, we get

d k dη k M ε+iη ϕ(x, v) - d k dη k M iη ϕ(x, v) ε D j+1 |v| -k-1 |M 0 ϕ(x, v)|
and the result follows.

We give the proof of Corollary 5.3

H 2 . 5 .

 25 Some auxiliary operators 3. About Dyson-Phillips representation 3.1. First de nitions 3.2. Decay of the iterates 3.3. Representation formulae for remainder terms 4. General regularity and spectral results 4.1. Spectral properties of M λ H along the imaginary axis 4.2. First consequence on the spectrum of T H 4.3. Spectral properties of M λ H in the vicinity of λ = 0. 4.4. Boundary functions for R(λ, T 0 ) and R(λ, T H ) 1 1. I

Proposition 4 . 3 .

 43 For any λ ∈ C \ {0} with Reλ 0,r σ (M λ H) < 1.

r

  ess (|M iη H|) = 0 where r ess (•) denotes the essential spectral radius. We prove that r σ (|M iη H|) < 1 by contradiction: assume, on the contrary, r σ (|M iη H|) = 1 > r ess (|M iη H|) = 0, then r σ (|M iη H|) is an isolated eigenvalue of |M iη H| with nite algebraic multiplicity and also an eigenvalue of the dual operator, associated to a nonnegative eigenfunction. From the fact that |M iη H| M 0 H with |M iη H| = M 0 H, one can invoke[START_REF]Frobenius theory of positive operators: Comparison theorems and applications[END_REF] Theorem 4.3] to get that

4. 2 .Theorem 4 . 5 .

 245 First consequence on the spectrum of T H . We recall the following result (see[32, Theorem 1.1.(c)]): Under Assumption 1.1 3), S(T 0 ) = {λ ∈ C ; Reλ 0}.

  weakly compact and the structure of S(M λ H) follows. The fact that all eigenvalues have modulus less than one comes from Proposition 4.3. This gives the rst part of the Proposition. For the second part, because M λ H converges in operator norm towards M 0 H as λ → 0 (λ ∈ C + ), it follows from general results about the separation of the spectrum [19, Theorem 3.16, p.212] that, for |λ| < δ 0 small enough, the curve {z ∈ C ; |z -1| = r 0 } is separating the spectrum S(M λ H) into two disjoint parts, say S(M λ H) = S in (M λ H) ∪ S ext (M λ H) where S in (M λ H) ⊂ {z ∈ C ; |z -1| < r 0 } and S ext (M λ H) ⊂ {z ∈ C ; |z -1| > r 0 }. Moreover, the spectral projection of M λ H associated to S in (M λ H), de ned as,

  ε+iη f since P(λ) commutes with M λ H. Notice that, with the notations of Proposition 4.7, S (M ε+iη H [I -P(ε + iη)]) ⊂ {z ∈ C ; |z| < r} so that r σ (M ε+iη H(I -P(ε + iη))) r < 1. One has then, for r < r ′ < r 0

  10) and k ∈ {0, . . . , N H }. Let [a, b] be a compact subset of R. Using (4.4) in Proposition 4.1 we see that sup η∈[a,b]

  λ→0 {. . .} = lim λ→0 λ∈C + {. . .}. where • Y is the norm on Y .

( 6 . 5 ) 3

 653 Observe that, if Ω is convex, thenΣ+(x) = {y ∈ ∂Ω ; (xy) • n(x) > 0 and (xy) • n(y) < 0}and ∂Ω = x∈∂Ω Σ+(x) whereas, if Ω is not convex x∈∂Ω Σ+(x) = ∂Ω.

Lemma 6 . 4 .

 64 Assume that ∂Ω satis es Assumption 1.1. Then lim δ→0 + sup y∈∂Ω |x-y| δ J (x, y)π(dx) = 0.

  y→x e x (y) • n(x) = 0, since e x (y) tends to be tangent to ∂Ω. Then (x -y) • n(x) is of the order |x -y| 1+α for x ≃ y. Let us make this rigorous. For a given x ∈ ∂Ω, one can nd a local parametrization of a neighbourhood O x ⊂ ∂Ω, containing x as O x = {(u, Φ(u)) ; u ∈ U } , where U is an open subset of R d-1 and Φ : U → O x is a C 1,α -di eomorphism. Denoting by | • | the euclidian norm of R d-1 and by ∇Φ the gradient of Φ (in R d-1 ), we get, with x = (u 0 , Φ(u 0 )) ∈ ∂Ω, n

Lemma 6 . 7 .

 67 Lemma 6.13, p.113]): Let m 0 be the image of the measure m under the transformation v ∈ R d → |v| ∈ [0, ∞), i.e. m 0 (I) = m {v ∈ R d ; |v| ∈ I} for any Borel subset I ⊂ R + . Then, for any ψ ∈

.14) 4

 4 Notice that this amounts to a measure m which is absolutely continuous with respect to the Lebesgue measure over R d , namely m(dv) = ̟(|v|)dv.

0 ̺ 0 ̺ 0 ̺

 000 w) exp -λ|x -y|̺ -1 d̺I 4 (λ, x, v, y, w) = (d + 2) ∞ d+1 ̟(̺)k(x, v, ̺σ x,y ) k(y, ̺σ x,y , w) exp -λ|x -y|̺ -1 d̺where we adopt the short-hand notation σ x,y = x-y |x-y| , (x = y). Using the normalisation condition (1.5), one hasΓ -(x) |I 1 (λ, x, v, y, w)| |v • n(x)|m(dv) ∞ d+2 ̟(̺) |σ x,y • ∇ 2 k (y, ̺σ x,y , w)| d̺ ∞ d+2 ̟(̺) |∇ 2 k (y, ̺σ x,y , w)| d̺.Thus, assumption (6.13) yields sup (y,w)∈Γ + Γ -(x) |I 1 (λ, x, v, y, w)| |v • n(x)|m(dv)| C. In the same way, one sees easily that (6.13) implies that sup (y,w)∈Γ + Γ -(x) (|I 3 (λ, x, v, y, w)| + |I 4 (λ, x, v, y, w)|) |v • n(x)|m(dv) C . Finally, one checks easily that (6.14) implies sup x∈∂Ω sup (y,w)∈Γ + Γ -(x) |I 2 (λ, x, v, y, w)| |v • n(x)|m(dv)| C .

Γ

  -(x) |J λ (x, v, y, w)| |v • n(x)|m(dv) C |λ||x -y| d-2α ∀x ∈ ∂Ω, ∀(y, w) ∈ Γ + . |x -y| d-2α < ∞we get the desired result.

M

  θ (v) = (2πθ) -d/2 exp -|v| 2 2θ , x ∈ ∂Ω, v ∈ R d . and γ(x) = κ d θ(x) R d |w|M 1 (w)dw, x ∈ ∂Ωfor some positive constant κ d depending only on the dimension ensuring in particular (1.5). We assume here that the temperature mapping x ∈ ∂Ω → θ(x) is bounded away from zero and continuous and denote θ 0 := inf x∈∂Ω θ(x) > 0.

̺ 1 γ 2 exp -̺ 2 1 ∞ 0 ̺

 12210 d+2 G (y, ̺) = 0, ∀(x, v) ∈ Γ -, (y, w) ∈ Γ +which obviously hold true since G(y, ̺) = (y)(2πθ(y)) d 2θ(y) . In the same way, (6.13) reads simply sup (y,w)∈Γ + σ∈S d-d+1 ̺ 2 θ(y) G(y, ̺) + G(y, ̺) d̺ < ∞;

= 1 . 1 +|v| - 1 1 + 1 +ε D ϕ Y - 1 whichL 1 +BM 1 + ) 1 .

 111111111 ε+iη ϕ -M iη ϕ L 1 + = Γ + e -(ε+iη)τ -(x,v) -e -iητ -(x,v) |M 0 ϕ(x, v)|dµ + (x, v) = Γ + |exp(-ετ -(x, v)) -1| |M 0 ϕ(x, v)|dµ + (x, v) C 0 ε Γ + τ -(x, v) |M 0 ϕ(x, v)|dµ + (x, v) where C 0 := sup s>0 | exp(-s)-1| s Now, because τ -(x, v) D|v| -1 we get M ε+iη ϕ -M iη ϕ L |M 0 ϕ(x, v)|dµ + (x, v) = D C 0 ε M 0 ψ L where ψ(x, v) = |v| -1 ϕ(x, v). Because M 0 ψ L 1 + = ψ L 1 -= ϕ Y -1 we obtain M ε+iη ϕ -M iη ϕ L proves (A.2). Now, since Range(H) ⊂ Y - 1 , one deduces (A.3) directly from (A.2). Proof of Proposition 4.1. Inequalities (4.3) and (4.4) are true for k = 0 (recall then that Y + 0 =thanks to Lemma A.1. The proof for general k is exactly the same and is omitted here. Let us focus on (4.5). The proof is done by induction on j ∈ N. For j = 1, the result is true, see(A.3). Noticing that, for any j ∈ N(M ε+iη H) j+1 -(M iη H) j+1 B(L 1 + ) (M ε+iη H) j -(M iη H) j ε+iη H -M iη H B(L 1 + )we easily get the result since M iη H B(L It remains only to prove (4.2), i.e.

Re

  -iηs h ε x,v (s)ds, h ε x,v (s) = 1 [0,t -(x,v)] (s)f (x -sv, v)e -εs , s ∈ R, (x, v) ∈ Γ + .

-iη s+ πη |η| 2

 2 Γ + dµ + (x, v) R h 0x,v (s) ds = f X 0 < ∞ we deduce from Fubini's Theorem that, for any ε 0 and µ + -a. e. (x, v) ∈ Γ + , the mappings ∈ R -→ h ε x,v(s)belongs to L 1 (R). Thus, according to Riemman-Lebesgue Theorem,lim |η|→∞ G ε+iη f (x, v) = 0 for µ + -a. e. (x, v) ∈ Γ +Actually, this convergence can be made uniform with respect to ε. Indeed, recalling the classical proof of the Riemann-Lebesgue Theorem, we writeR e -iηs h ε x,v (s)ds = -R e h ε x,v (s)ds = -R e -iηs h ε x,v s -πη |η| 2 dswhere we used that e -iπ = -1. Thus, R e -iηs h ε x,v (s)ds = 1 2 R e -iηs h ε x,v (s) -h ε x,v s -π η |η| 2 ds and

πη |η| 2 L 1 1 ]

 211 (R) R e -εs h 0 x,v (s) -h 0 x,v s -L 1 (R) .The rst term is independent of ε and goes to zero as |η| → ∞ owing to the continuity of translation. For the second term, there isC R > 0 such that sup |G ε+iη f (x, v)| = 0 for µ + -a. e. (x, v) ∈ Γ + .

2 . 1 )

 21 we deduce the result from the dominated convergence theorem.A.Di erentiability properties of M λ H, Ξ λ H and G λ . We give here the full proof of Proposition 5.1Proof of Proposition 5.1. We prove the various points of the Proposition.(For λ ∈ C + and f ∈ X 1 , one has d dλ G λ f (x, v) = -τ -(x,v) 0 tf (x -tv, v)e -λt dt, for a. e.(x, v) ∈ Γ + .Notice that, for any f ∈ X 1 and any(x, v) ∈ Γ + τ -(x,v) 0 tf (x -tv, v)dt = τ -(x,v) 0 t + (x -tv, v)f (x -tv, v)dt = G 0 (t + f )(x, v)

  Decay of the semigroup assiociated to absorbing boundary conditions 2.4. About the resolvent of T

	1. Introduction
	1.1. Our contribution in a nutshell
	1.2. Related literature
	1.3. Practical examples
	1.4. Method of proof
	1.5. Organization of the paper
	Acknowledgements
	2. Reminders of known results
	2.1. Functional setting
	2.2. Travel time and integration formula
	2.3.

  Since, under (1.6), the range of H is included in Y - 1 and Ξ 0 u X 0 u Y -

			.6)
	From Proposition 4.3 and Banach-Steinhaus Theorem [8, Theorem 2.2, p. 32], for any η = 0,
		lim sup
	Lemma 2.12), one has	ε→0 1	1 (see for any u ∈ Y -
		sup
		ε>0,η∈R

+ R(1, M ε+iη H) B(L 1 + ) < ∞.

  1,α with α > 1 2 . There exists a positive constant C such that (M λ H) 2 ∈ C + , λ = 0. In particular, (1.7) holds true with p = 4.Proof. It is clear from Proposition 6.8 that, for any ψ ∈ L1 + ,(M λ H) 2 ψ L 1 |J λ (x, v, y, w)| dµ -(x, v) so that, using that M λ B(L 1 |J λ (x, v, y, w)| dµ -(x, v)and we conclude then with Lemma 6.11. Since then, for any ε 0 and η > 0, (M ε+iη H)4

					C
					B(L 1 + )	|λ|
	holds for any λ +	M λ B(L 1 -,L 1 + ) HM λ Hψ L 1 -
		M λ B(L 1 -,L 1 + )	Γ +	|ψ(y, w)|dµ + (y, w)	Γ -
		-,L 1 + )	1 we get
		(M λ H) 2 ψ L 1 +	sup (y,w)∈Γ + Γ -
					C 2
					B(L 1 + )	|ε + iη| 2

  1 -) is weakly compact which means that Assumption 1.2 2) is met. Moreover, since k(x, v, v ′ ) > 0 for any (x, v) ∈ Γ + and any v ′ ∈ Γ -(x), we deduce from [26, Remark 4.5] that M 0 H is irreducible, i.e. Assumption 1.2 1) is met. For this model, recalling here that the measure m 0 over R + is given by

  Email address: mustapha.mokhtar-kharroubi@univ-fcomte.fr

	U		S	T	C	C	A	, D	E	S	,
	C	U	S	, 218/ , 10134 T	, I	.
	Email address: bertrand.lods@unito.it
	U		B	F	C	, E	M		, CNRS UMR 6623, 16,	G	,
	25030 B	C	, F

or more generally, to abstract state spaces where the norm is additive on the positive cone

Acknowledgements. BL gratefully acknowledges the nancial support from the Italian Ministry of Education, University and Research (MIUR), "Dipartimenti di Eccellenza" grant 2018-2022. Part of this research was performed while the second author was visiting the "Laboratoire de Mathématiques CNRS UMR 6623" at Université de Franche-Comté in February 2020. He wishes to express his gratitude for the nancial support and warm hospitality o ered by this Institution.

Therefore, with the notations of Theorem 1.3, the decay of ∞ -∞ exp (iη t) Θ f (η)dη X 0 can be made as close as desired from ω f π t

where ω f : R + → R + denotes the minimal modulus of continuity of the uniformly continuous mapping Θ f . Since moreover

thanks to (6.15), one sees that, if the conjecture (5.20) holds true, then one would have

thanks to Remark 5.10.

We see that, here above, we fully exploit the fact that the model in Example 1.6, the kernel was radially symmetric with respect to v and did not depend on w. Such properties are still shared by the Example 1.5 given in the Introduction and henceforth, it can be treated exactly along the same lines as those described here (under ad hoc explicit condition ensuring Assumption 6.10 to hold true). For such a model, one sees also that N H is the maximal n ∈ N for which γ(n, d) < ∞ where, for all s 0,

Clearly, the precise value of N H depends on the explicit expression of G.

A

A. P

A.1. Fine properties of G λ , M λ H and Ξ λ H. We collect here all the technical details useful for the proof of Proposition 4.1. The main step is the following which corresponds to Prop. 4.1 for k = 0:

Lemma A.1. For any f ∈ X 0 , the limit

uniformly with respect to η ∈ R. For any η ∈ R, it holds

where D is the diameter of Ω. Consequently,

) is continuous for any 0 j k. Thanks to (5.4), we can let ε → 0 and conclude that the derivatives exist and are continuous on R. Now, for any λ ∈ C + , one has

so that, by the dominated convergence theorem, lim λ→0

The conclusion follows easily. The proof for Ξ λ H proceeds along the same line. For ϕ ∈ Y - k+1 , ε > 0, η ∈ R one checks easily that

for any (x, v) ∈ Ω × V. So, as in (5.4),

We deduce the result as in the previous point.

With the notations of the above proof, we have also the following technical Lemma regarding derivatives of L N (iη) which was used in the proof of Proposition 5.7.

Lemma A.2. For any j ∈ {1, . . . , N H }, there exists Cj > 0 such that

Proof. The proof is based upon elementary but tedious computations. For simplicity of notations, we will simply here denote • for the norm • B(L 1 + ) . We notice rst that, since L N (iη) = (L 1 (iη)) N , one has for the rst derivative:

We also denote Since L 1 (iη) 1 and

which results in

(iη) and proves the result for j = 1. Now, for j = 2, one has

L (1) r (iη)L

(1)

N -r (iη).

One has again, L

1 (iη)

so that, as before

The last sum is bounded like in the previous step while, for the rst sum, we apply (A.5) to N -r so that

and

This clearly gives the rough estimate (using

and proves the result for j = 2. By a tedious but simple induction argument, we deduce then the result for any j ∈ {0, . . . , N H -1}. Recall that, for j ∈ N,

+ ,Y + j ) := C j which is nite as long as j N H .

A.3. Additional properties of the Dyson-Phillips iterates.

For technical reasons we need to introduce a slightly di erent expression for the iterates U k (t) de ned in Section 3.1 where we allow various boundary operator to enter the construction. More precisely, let us consider a sequence (H n ) n∈N of boundary operators

We mimic then the construction of [START_REF]Explicit transport semigroup associated to abstract boundary conditions[END_REF] and de ne, for any t 0, V 0 (t) = U 0 (t) = U 0 (t) and

there exists a unique y ∈ ∂Ω with (y, v) ∈ Γ -and a unique 0 < s < min(t, τ + (y, v)) such that x = y + sv and then one sets

We establish here the properties of such a sequence of Dyson-Phillips operators. We rst recall the following, taken from [3, Proposition 3, Corollary 2] (see also [START_REF]An L p -approach to the well-posedness of transport equations associated to a regular eld: Part II[END_REF]Proposition 3.6]):

One then proves by induction, exactly as in [3, Theorem 3.2] (see also [4, Theorem 3.9]), the following Theorem A.5. For any k 1, f ∈ D 0 one has U k (t)f ∈ X 0 for any t 0 with

In particular, U k (t) can be extended to be a bounded linear operator, still denoted

Moreover, the following holds for any k 1 (1) (V k (t)) t 0 is a strongly continuous family of B(X 0 ).

(2) For any f ∈ D 0 , one has V k (t)f ∈ D(T max ) for all t 0 with

(3) For any f ∈ D 0 and any t 0, the traces

One can actually sharpen estimate (A.7)

Proposition A.6. For any n 1, f ∈ D 0 , one has V n (t)f ∈ X 0 for any t 0 with

In particular, V n (t) can be extended to a bounded linear operator, still denoted V n (t) ∈ B(X 0 ) with

Proof. The proof is made by induction. Let f ∈ D 0 and t 0 be xed. For n = 1, one deduce from (2.2) that

From the de nition of V 1 (t), for µ --a. e. (z, v) ∈ Γ -and s ∈ (0, τ + (z, v)), one has

-) ( f X 0 -U 0 (t)f X 0 ) . This proves (A.9) for n = 1. Assume then the result to be true for n 1 and let us prove for n + 1. Using (2.2) one has, a before,

Using then (A.8), we deduce that

and the result follows.