Ioann Vasilyev
email: ioann.vasilyev@u-pec.fr

François Vigneron
email: francois.vigneron@u-pec.fr

Variation on a theme by Kiselev and Nazarov: Hölder estimates for non-local transport-diffusion, along a non-divergence-free BMO field

Keywords: Transport-diffusion, Hölder regularity, fractional diffusion equation, non-local operator, BMO drift, dual equation, conservation law, functional analysis, atoms. MSC primary: 35B65. MSC secondary: 35R11, 35Q35

 and is based on the dual evolution technique. The idea is to propagate an atom property (i.e. localization and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.

In this article, we are interested in the following transport-diffusion equation:

∂ t θ + (-∆) α/2 θ = (v • ∇)θ θ(0, x) = θ 0 (x). (1)
We consider this Cauchy problem where θ : [0, T ] × R d → R is unknown. The vector field v is given and is of bounded mean oscillation, i.e. it belongs to the space BMO(R d ). In what follows, we will not assume that div v = 0. We will also consider the periodic problem on T d . We are interested in the whole range of parameters 0 < α < 2 and in particular in the critical non-local diffusion (i.e. α = 1) where diffusion and transport are of similar order. We will, in passing, consider the classical local case α = 2; however, the local case is much better understood and we refer the reader, for exemple, to the recent monograph [START_REF] Bris | Parabolic equations with irregular data and related issues[END_REF] and the references therein.

For α > 1, the equation is called sub-critical because the drift is then of lower order than the diffusion, which means that the diffusion will be stronger at the smallest scales. On the contrary, for α < 1, the drift will be stronger than the diffusion at smaller scales and the equation is called supercritical. However, as the diffusion operator remains invariant under Galilean transforms, one can still expect a mild smoothing effect, as long as α > 0, because putative advected singularities cannot ride along a "defect" of the diffusion operator, simply because the homogeneous and isotropic operator does not have any.

The fractional derivative (-∆) α/2 is the Fourier multiplier by |ξ| α . It admits the following kernel expansion on R d (see e.g. [8, §2]):

(-∆) α/2 θ(x) = -c d,α pv R d θ(y) -θ(x) |y -x| d+α dy (2)
with c d,α > 0. As (-∆) α/2 is a non-local derivative of order α that does not induce phase-shift, it performs a sort of "graphical" interpolation between the graph of θ and that of -∆θ.

The goal is to establish uniform Hölder regularity estimates of the solution of (1) at a later time t > 0. As demonstrated by A. Kiselev and F. Nazarov [START_REF] Kiselev | Variation on a theme of Caffarelli and Vasseur[END_REF] and M. Dabkowski [START_REF] Dabkowski | Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation[END_REF] for α = 1, an elegant but powerful technique consists in using the atomic characterization of Hölder classes (see below). Multiplying the equation by a test-function ψ(t -s) where ψ solves the dual evolution equation, one can then exchange an atomic estimate of θ(t)ψ 0 against an atomic estimate of θ 0 ψ(t). Provided that the dual evolution propagates the atomic property in a controlled way, one then gets the desired Hölder regularity.

Another powerful approach, used e.g. by L.A. Caffarelli and A. Vasseur [START_REF] Caffarelli | Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation[END_REF] and L. Silvestre [START_REF] Silvestre | Hölder estimates for advection fractional-diffusion equations[END_REF], consists in using the parabolic De Giorgi method. The idea is to first establish a set of energy estimates. Then, one uses the rigidity given by the equation to deduce scalable uniform bounds. More precisely, one computes the energy cost of one oscillation of the solution between its maximum and minimum value. Given that the total amount of energy available is finite, this limits the size of the oscillations that the solution can perform at a given scale, which ultimately translates as regularity in the Hölder classes.

Finally, the method of the modulus of continuity was successfully introduced by A. Kiselev, F. Nazarov and A. Volberg [START_REF] Kiselev | Global well-posedness for the critical 2D dissipative quasi-geostrophic equation[END_REF] for SQG and later used by L. Silvestre, V. Vicol [START_REF] Silvestre | Hölder continuity for a drift-diffusion equation with pressure[END_REF] to study related PDEs.

The case of a divergence-free transport field in (1) is of particular interest as it closely relates to the quasi-geostrophic equation where d = 2 and u = ∇ ⊥ (-∆) -1/2 θ; see [START_REF] Constantin | On the critical dissipative quasi-geostrophic equation[END_REF], [START_REF] Caffarelli | Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation[END_REF], [START_REF] Kiselev | Global well-posedness for the critical 2D dissipative quasi-geostrophic equation[END_REF]. At a technical level, the computations of [START_REF] Kiselev | Variation on a theme of Caffarelli and Vasseur[END_REF], [START_REF] Dabkowski | Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation[END_REF] rely in a crucial way on the assumption that div v = 0. First, as they use the non-conservative form for the dual evolution equation

∂ s ψ + (-∆) α/2 ψ = -(v(t -s) • ∇)ψ
each integration by part requires a divergence-free field. Second, they invoque the maximum-principle and the decay of the L p -norms for this kind of equation, which was established by A. Cordoba, D. Cordoba [START_REF] Cordoba | A maximum principle applied to quasi-geostrophic equations[END_REF], but either for the particular vector field of SQG, or for a divergence-free transport field.

The key of our article is that we allow div v = 0. The sign of div v then becomes crucial because convergent characteristics will tend to create shocks and break down the regularity while divergent characteristics (rarefaction waves) have a natural smoothing effect and tend to reduce steep gradients. The presence of the diffusion operator fixes the issues of either non-existence or non-uniqueness that the pure transport would induce. In that context, the pertinent question is thus to establish quantitative estimates of the regularity norms.

Our main result is the following statement that provides uniform Hölder bounds for solutions of (1).

Theorem 1 We consider either R d or T d as the ambiant space, which is of dimension d > α, and an advection field

v ∈ C 1-α if 0 < α < 1, v ∈ BMO if 1 ≤ α ≤ 2. ( 3
)
In both cases, one requires:

(div v) -L d/α ≤ S α/2 = sup f ∈ Ḣα/2     |(-∆) α/4 f | 2 f 2 L 2d/(d-α)     (4)
where (div v) -is the negative part of the divergence (see (8) below) and Ḣα/2 denotes the homogeneous Sobolev space on R d , or the average-free space on T d . There exist β > 0 and C > 0 that depend solely on d, α and respectively either on v BMO or v C 1-α such that, for any θ 0 ∈ L q with 2 ≤ q ≤ ∞, the corresponding solution of (1) satisfies:

∀t ∈ (0, 1], θ(t, •) C β ≤ Ct -(β+ d q )/α θ 0 L q . ( 5
)
Moreover, if θ 0 ∈ C β , then: ∀t ≥ 0, θ(t, •) C β ≤ C ′ θ 0 C β (6)
for some constant C ′ .

Among the results quoted in this introduction, that of L. Silvestre [START_REF] Silvestre | Hölder estimates for advection fractional-diffusion equations[END_REF] is the only one that allows for non-divergence-free fields, so we will focus on this one. For α ≥ 1, he assumes that v ∈ L ∞ (R d ) but requires no other size constraint on v; for α < 1, he assumes that v ∈ C 1-α in order to compensate for the super-criticality of the equation. In both cases, L. Silvestre proves the Hölder continuity of the solution of (1) at positive times, namely that:

θ(t) C β (R d ) ≤ Ct -β/α θ 0 L ∞ (R d )
and he also establishes a similar C β/α -regularity estimate time-wise.

Our result is complimentary and does not quite compare to that of L. Silvestre when α ≥ 1 because, in that case, we do not assume that v is bounded. Instead, we assume that v belongs to BMO and we require the negative part of its divergence to be small. However, we do not constrain the size of (div v) + and, in particular any BMO divergence-free field is admissible in our result.

It is likely that a "universal" result holds true for a general advection field v = v 1 + v 2 with v 1 ∈ L ∞ and v 2 ∈ BMO with (div v 2 ) -small. Results in this spirit are known for general diffusion operators of order α = 2 (see [START_REF] Bris | Parabolic equations with irregular data and related issues[END_REF]). However, mixing the DeGiorgi and atomic methods is not obvious.

Remark 2

The critical value in (4) can be slightly relaxed but the constants β and C will then also depend on the size of (div v) -; see remark 26. When α = 2, the dimension d = 2 is excluded from our statement (and similarly the case α = d = 1) because the corresponding Sobolev Space H α/2 (R d ) fails to be included in L ∞ .

Our proof is inspired by A. Kiselev and F. Nazarov [START_REF] Kiselev | Variation on a theme of Caffarelli and Vasseur[END_REF] and we use the dual evolution technique. Following M. Dabkowski [START_REF] Dabkowski | Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation[END_REF], we also use L p -based atoms for more flexibility. In order to deal with non-divergence-free transport fields, the key is to replace the dual evolution equation by a transportdiffusion equation expressed in a conservative form, i.e. as a conservation law. The presence of a general exponent of diffusion α = 1 also brings about some technicalities that need to be addressed (see the end of §2).

There are many other takes on the question of regularity for transport-diffusion equations. For example, D. Bresch, P.-E. Jabin [START_REF] Bresch | Quantitative regularity estimates for compressible transport equations[END_REF] have studied the regularity of weak solutions to the advection equation set in conservative form. In particular, they investigated the case where both v and div v belong to some Lebesgue space, which allows point-wise unbounded variations of the field and of its compressibility.

Another related study, albeit slightly more distant from [START_REF] Ambrosio | Lecture notes on the DiPerna-Lions theory in abstract measure spaces[END_REF], is that of the kinetic Fokker-Planck equation. On that mater, we refer the reader to C. Mouhot [START_REF] Mouhot | Hörmander theories: new interplay[END_REF] and the references therein, and, e.g., to the regularity result by F. Golse, A. Vasseur [START_REF] Golse | Hölder Regularity for Hypoelliptic Kinetic Equations with Rough Diffusion Coefficients[END_REF].

The role of fractional diffusion operators in physical models is growing (see e.g. J. Vaźquez's book [START_REF] Vázquez | The mathematical theories of diffusion[END_REF]). As for our motivations, we are interested in studying variants of a non-local Burgers equation introduced by C. Imbert, R. Shvydkoy and one of the present authors in [START_REF] Imbert | Global Well-Posedness of a Non-local Burgers Equation: the periodic case[END_REF]. The regularity of the solution for unsigned data remains an open problem that seems to share some nontrivial similarities with hydrodynamic turbulence.

The present article is structured as follows. In §1, we recall how Hölder classes can be characterized in terms of atoms. We draft the general ideas on how regularity can be obtained by studying the dual conservation law in §2. A weak version (i.e. in Lebesgue spaces) of the maximum principle is established in §3. It is then put to use in §4 to quantify how the atomic property is propagated. The proof of theorem 1 and, in particular, how the choices of structural constants should be made, is detailed in §5.

Notations. In this article, one uses the following common notations for a, b, x ∈ R:

a ∧ b = min{a, b}, a ∨ b = max{a, b} (7)
x + = a ∨ 0,

x -= (-x) + , x = x + -x -, |x| = x + + x -. (8)
Balls are denoted by B(x

0 , r) = x ∈ R d ; |x -x 0 | < r where | • | denotes the Euclidian distance on R d .

Atomic characterization of Hölder classes

Throughout the article, the constant A ≫ 1 is fixed, but is chosen arbitrarily large and ω ∈ (0, 1). One can check that none of the final estimates actually depend on the particular values of A or ω.

Definition 3 For r > 0 and p ∈ (1, ∞], the atomic class A p r (R d ) is defined as a subset of C ∞ (R d ) by the following three conditions:

R d ϕ(x)dx = 0, (9)
ϕ L 1 ≤ 1 and ϕ L p ≤ Ar -d(1-1 p ) , (10)
∃x 0 ∈ R d , R d |ϕ(x)|Ω(x -x 0 )dx ≤ r ω (11)
where

Ω(z) = |z| ω ∧ 1 ∈ L ∞ (R d ). If λ -1 ϕ(x) ∈ A p r (R d ) for some λ > 0, then one says that ϕ ∈ λ • A p r (R d ). ( 12
)
Remark 4 A typical example of an atom A p r of radius r < 1 is the function ϕ = ϕ r * ρ ǫ where ρ ǫ is an standard mollifier supported in B(0, ǫ) and

ϕ r (x) =      -Cr -d if |x -x 0 | ≤ r2 -1/d +Cr -d if r2 -1/d < |x -x 0 | ≤ r 0 if |x -x 0 | > r
for small enough constants C < min{|B(0, 1)| -1 ; A|B(0, 1)| -1/p ; |B(0, 1)|/(d + ω)} and ǫ > 0. The volume of the unit ball is |B(0, 1)| = π d/2 /Γ d 2 + 1 .

One can control the L q -norm of atoms for q ∈ [1, p] by a real interpolation estimate.

Proposition 5 If ϕ ∈ A p r (R d ), then for any 1 ≤ q ≤ p:

ϕ L q ≤ A 1-1/q 1-1/p r -d(1-1 q ) . (13)
Proof. Apply the interpolation inequality

f L q ≤ f 1-θ L 1 f θ L p with θ = (1 -1 q )/(1 -1 p ) ∈ [0, 1].
Atoms are the "poor man's wavelet" and offer a very comfortable characterization of Hölder's classes.

Proposition 6 For 0 < β < 1, a bounded function f belongs to C β (R d ) if and only if sup 0<r≤1 ϕ∈A p r (R d ) r -β R d f (x)ϕ(x)dx < ∞ ( 14
)
for some p ∈ (1, ∞]. Moreover, the left-hand side of ( 14) is equivalent to the usual semi-norm on

C β (R d ).
Proof. The proof is a classical exercise. See [START_REF] Kiselev | Variation on a theme of Caffarelli and Vasseur[END_REF], [START_REF] Dabkowski | Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation[END_REF] (resp. for p = ∞ and p < ∞) for a short proof that relies on the Littlewood-Paley theory [START_REF] Stein | Harmonic Analysis[END_REF]. The key is the equivalent control of sup j∈N 2 βj ∆ j f L ∞ where ∆ j is a smooth projection on the frequency scale of order 2 j .

Regularity through the dual conservation law

As explained in the previous section, in order to obtain estimates of the Hölder regularity of the solution of (1) at a time t > 0, one needs to control

r -β R d θ(t, x)ψ 0 (x)dx
where ψ 0 ∈ A p r (R d ) and this control needs to be uniform in 0 < r ≤ 1. Let us consider the following test function that solves the dual evolution problem, set in a conservative form:

∂ s ψ(s) + (-∆) α/2 ψ(s) = -div (v(t -s)ψ(s)) ψ(0, x) = ψ 0 (x). ( 15
)
One can then immediately check the following result.

Proposition 7 If θ is a smooth solution of (1) and ψ is a smooth solution of (15), then one has:

∀t ≥ 0, R d θ(t, x)ψ 0 (x)dx = R d θ 0 (x)ψ(t, x)dx. (16)
Proof. Let us indeed use ψ(t -s) as a test function for [START_REF] Ambrosio | Lecture notes on the DiPerna-Lions theory in abstract measure spaces[END_REF]. One gets:

∂ s θ(s) • ψ(t -s) + ((-∆) α/2 θ(s)) • ψ(t -s) = (v(s) • ∇)θ(s) • ψ(t -s).
Here and below, all double integrals are computed for (s, x) ∈ [0, t] × R d unless stated otherwise and the x-variable is not made explicit unless it is absolutely necessary. Integrating by part time-wise in the first integral and space-wise in the other two gives:

θ(s) • (∂ s ψ)(t -s) + R d θ(s)ψ(t -s) t 0 + θ(s) • ((-∆) α/2 ψ)(t -s) = - θ(s) • div(v(s)ψ(t -s)).
Thanks to [START_REF] Kiselev | Global well-posedness for the critical 2D dissipative quasi-geostrophic equation[END_REF], the double integrals cancel each other out and one is left with [START_REF] Kružkov | First order quasilinear equations in several independent variables[END_REF].

Remark 8 In [START_REF] Kiselev | Variation on a theme of Caffarelli and Vasseur[END_REF], the authors used the non-conservative dual form -v(t -s) • ∇ψ(s). This choice was harmless and perfectly adapted to their purpose because they assumed v to be divergence-free. Here, on the contrary, it is crucial that the right-hand side of (15) takes the form of a conservation law.

For the sake of the argument, let us assume that one will be able to show subsequently (which is indeed the case if α = 1) the following infinitesimal propagation property for (15):

ψ 0 ∈ A p r =⇒ ∀s ∈ [0, γr], ψ(s) ∈ (1 -h(r)s)A p r+Ks for a given value of p ∈ (1, ∞],
with universal constants γ, K that do not depend on r nor ψ 0 and a universal function h. One can then immediately infer a global propagation property:

ψ 0 ∈ A p r =⇒ ∀s ∈ 0, 1 -r K , ψ(s) ∈ f r (s)A p r+Ks with f ′ r (s) ≥ -h(r + Ks)f r (s). Let us introduce a function H such that H ′ (z) = h(z). Then f r (s) = exp H(r) -H(r + Ks) K
is an acceptable bound for the global propagation property. Coupled with [START_REF] Kružkov | First order quasilinear equations in several independent variables[END_REF], this means the following: for any solution of (1) and ψ 0 ∈ A p r , one has

R d θ(t, x)ψ 0 (x)dx = R d θ 0 (x)ψ(t, x)dx ≤ θ 0 C β (r + Kt) β f r (t).
One is thus able to propagate

C β (R d ) bounds of θ if f r (s) ≤ C( r r+Ks ) β
. This is the case if, for example, h(r) = δ/r with δ = Kβ. The regularization estimate is obtained in the same fashion.

Dealing with a general exponent α requires a slightly more careful computation. The fundamental idea remains that the dual conservation law propagates atoms and that a small gain on the amplitude of the atoms can be obtained as a tradeoff with a slight increase in the size of the atoms' radii.

The main technical difficulty is that the radii now grow as (r α + Ks) 1/α , which is not linear in s anymore, at least not when s ≫ r α . This non-linear region invades any neighborhood of r = s = 0 and the corresponding correction of amplitude will be h(r) = δ/r α . We found that the simplest workaround is to forfeit the ODE point of view presented here for α = 1 and to use direct estimates on the corresponding rate of change during a finite increment of an Euler scheme (see §4, estimate (51)).

Weak maximum principle for the dual conservation law

In this section, we establish the weak maximum principle i.e. the decay of the L p -norms for a non-local transport-diffusion equation written in a conservative form. In this section, one considers thus the following general problem for 0 < α ≤ 2:

∂ s ψ(s) + (-∆) α/2 ψ(s) = -div (v(s)ψ(s)) ψ(0, x) = ψ 0 (x) (17)
and we will assume, when necessary, that R d ψ 0 = 0. Subsequently, the results of this section will be applied to (15) at a given time t > 0 by choosing v(s) = v(t -s) ∈ BMO(R d ).

A brief note on the well-posedness theory

For smooth v(s, x), the well-posedness theory of the scalar conservation law

∂ s ψ(s) = -div (v(s)ψ(s))
was established by S.N. Kružkov [START_REF] Kružkov | First order quasilinear equations in several independent variables[END_REF], in the setting of entropy solutions. The theory was refined and generalized to the non-conservative convective form by R.J. DiPerna and P.L. Lions [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]; their theory ensures that assuming a transport field

v ∈ L 1 ([0, T ]; W 1,1 ) with (div v) -∈ L 1 ([0, T ]; L ∞
) is enough to guarantee the existence, uniqueness and stability in the proper function spaces. The key idea is a celebrated commutation lemma:

ρ ε * (v • ∇)ψ -v • ∇(ρ ε * ψ) → 0 in L 1 ([0, T ]; L β loc ).
On R d , an additional assumption of mild growth at infinity is required, e.g.

v ∈ (1 + |x|) • (L 1 + L ∞ ).
To handle unbounded data, the idea is to use renormalization, i.e. to consider Φ(ψ) for suitable smooth and bounded Φ. For a review of the fundamental ideas and the last developments of the theory, we refer the reader to the monograph [START_REF] Bris | Parabolic equations with irregular data and related issues[END_REF] by C. Le Bris and P.L. Lions, and the references therein. See also the lecture notes by L. Ambrosio and D. Trevisan [START_REF] Ambrosio | Lecture notes on the DiPerna-Lions theory in abstract measure spaces[END_REF] or those of C. De Lellis [START_REF] Lellis | Notes on hyperbolic systems of conservation laws and transport equations[END_REF].

Adding the coercive diffusion term (-∆) α/2 ψ in (17) with 0 < α ≤ 2 obviously does not alter these results. On the contrary, the assumptions on the transport field can even be relaxed. For example, for α = 2 and even with a fully general second-order elliptic operator, one can accept a field v ∈

L 2 + W 1,1 with (div v) -∈ L ∞ , as mentioned in [3, §3.2].
The local well-posedness of ( 17) is thus classical; see e.g. [START_REF] Bresch | Quantitative regularity estimates for compressible transport equations[END_REF].

Remark 9 If the transport term takes the conservative form, the equation is called a conservation law; if not, it is referred to as a general convection. When the Laplace operator has variable coefficients, then the term conservative is preferred to describe the equation with the operator written in divergence form -∂ i (a ij ∂ j ), regardless of whether the transport part is a convection or a conservation law. In our present case, however, the fractional power (-∆) α is obviously a conservative operator so our use of the adjective conservative concerns only the form of the advection term.

Propagation of positivity

The classical positivity result for α = 2 can be generalized for fractional diffusions.

Proposition 10 If ψ is a solution to [START_REF] Mouhot | Hörmander theories: new interplay[END_REF], stemming from ψ 0 ≥ 0, then ψ(s) ≥ 0 for any s ≥ 0.

Proof. Let us sketch the argument first. If a solution of ( 17) is smooth and positive, then at a first contact point with zero, say (s 0 , x 0 ), it reaches a global minimum. One thus has ψ(s 0 , x 0 ) = 0 and ∇ψ(s 0 , x 0 ) = 0, and therefore:

div (vψ) = (div v)ψ(s 0 , x 0 ) + (v • ∇)ψ(s 0 , x 0 ) = 0.
Moreover, for 0 < α < 2, by ( 2), there exists a positive kernel K d,α such that:

(-∆) α/2 ψ(s 0 , x 0 ) = - R d (ψ(s 0 , y) -ψ(s 0 , x 0 )) K d,α (y -x 0 )dy ≤ 0
and the inequality is strict, unless ψ(s 0 , •) ≡ 0. The equation ensures that ∂ s ψ(s 0 , x 0 ) ≥ 0 and, in particular, the solution remains positive forever. To make the proof fully rigorous, one proceeds e.g. as in [13, §2.1]: for given T, R > 0 and ψ 0 > 0, one considers the approximation ψ R where the kernel K d,α is restricted to B(0, R) and

s 0 = inf {s ∈ (0, T ) ; ∃x 0 ∈ B(0, R), ψ R (s, x 0 ) = 0} .
By compactness, s 0 is attained and s 0 > 0. As ψ R (s, •) is not identically zero, the previous computation ensures that ∂ s ψ R (s 0 , x 0 ) > 0 and thus ψ R had to be negative in the neighborhood of x 0 a short time before s 0 , which is contradictory. For a general ψ 0 ≥ 0, the data can be approximated by a strictly positive mollification, whose strict positivity propagates downstream. Passing to the limit at a later time s > 0 ensures therefore that ψ(s) ≥ 0.

Propagation of the L 1 norm and conservation of momentum

The simple structure of ( 17) inherited from the underlying conservation law plays in our favor.

Proposition 11 Let ψ be a solution to [START_REF] Mouhot | Hörmander theories: new interplay[END_REF]. Then

ψ(s, •) L 1 ≤ ψ 0 L 1 ( 18
)
and

R d ψ(s, x)dx = R d ψ 0 (x)dx. ( 19
)
Proof. For the first statement, let us decompose ψ 0 = ψ + 0 -ψ - 0 where both ψ + 0 and ψ - 0 are positive and have disjoint supports. Let ψ + and ψ -be the solutions to the equation with initial data ψ + 0 and ψ - 0 correspondingly. Then, by linearity, ψ(s) = ψ + (s) -ψ -(s) and therefore

ψ(s, •) L 1 ≤ ψ + (s, •) L 1 + ψ -(s, •) L 1 .
Equation ( 17) and an integration by part ensure that:

d ds R d ψ ± (s, x)dx = - R d (-∆) α/2 ψ ± - R d div(vψ ± )dx = 0.
As the propagation of positivity yields that ψ ± ≥ 0, one gets ψ ± (s, [START_REF] Silvestre | Hölder estimates for advection fractional-diffusion equations[END_REF]. The identity [START_REF] Silvestre | Hölder continuity for a drift-diffusion equation with pressure[END_REF] is immediate.

•) L 1 = ψ ± 0 L 1 and finally ψ(s, •) L 1 ≤ ψ + 0 L 1 + ψ - 0 L 1 = ψ 0 L 1 , hence
Remark 12 Note that, because of the diffusion, the functions ψ ± of the previous proof will not coincide, in general, with the positive and negative parts ψ ± of ψ.

Estimate of the L p norm

For h < d/2 let us introduce the constant in the Sobolev embedding Ḣh (R d ) ⊂ L 2d/(d-2h) (see e.g. [START_REF] Talenti | Best Constant in Sobolev Inequality[END_REF]):

S h (R d ) -1 = sup f 2 L 2d/(d-2h) ; f ∈ Ḣh (R d ), R d |(-∆) h/2 f | 2 = 1 > 0. ( 20
)
The idea is to relax the uniform control given by the maximum principle for ( 17) into a weaker one in the scale of Lebesgue spaces.

Proposition 13 For any α ∈ (0, 2] and any dimension d > α, if the transport field satisfies

(p -1) (div v) -L ∞ t L d/α x ≤ S α/2 (R d ) ( 21
)
for some p ≥ 2 (eventually restricted to p = 2 n with n ∈ N if α < 2), then any solution of (17) satisfies

ψ(s) p L p + S α/2 (R d ) s 0 ψ(τ ) p L σ dτ ≤ ψ 0 p L p with σ = dp d -α (22)
and in particular ∀q ∈

[1, p], ∀s ≥ 0, ψ(s) L q ≤ ψ 0 L q (23)
for any ψ 0 ∈ L 1 ∩ L p . In particular, when div(v) ≥ 0, the estimate (23) holds for 1 ≤ q ≤ ∞.

Remark 14

The following proof also establishes that all solutions of (17) satisfy:

ψ(s) L p ≤ ψ 0 L p e t(1-1 p ) (div v) -L ∞ t,x (24)
regardless of the diffusion term and independently of [START_REF] Talenti | Best Constant in Sobolev Inequality[END_REF]. For what follows, we are however interested in getting a better (i.e. non-increasing) control of the L p -norm as given by ( 22)-(23).

Proof. Using p|ψ| p-2 ψ as a multiplier for the equation leads to:

d ds R d |ψ(s, x)| p dx + p R d |ψ| p-2 ψ • (-∆) α/2 ψ = -p R d div (vψ) |ψ| p-2 ψ.
For the integral on the right-hand side, an integration by part gives:

R d div (vψ) |ψ| p-2 ψ = -(p -1) R d |ψ| p-2 ψ(v • ∇)ψ = -(p -1) R d div (vψ) |ψ| p-2 ψ + (p -1) R d |ψ| p div v.
One thus has this identity:

R d div (vψ) |ψ| p-2 ψ = 1 - 1 p R d |ψ| p div v (25)
and thus

d ds R d |ψ(s, x)| p dx + p R d |ψ| p-2 ψ • (-∆) α/2 ψ = -(p -1) R d |ψ| p div v. ( 26
)
For the integral on the left-hand side of (26) and when α = 2, the following identity holds:

p R d |ψ| p-2 ψ • (-∆)ψ = p(p -1) R d |ψ| p-2 |∇ψ| 2 = 4 1 - 1 p R d |∇(|ψ| p/2 )| 2 ≥ 0. ( 27a
)
For 0 < α < 2, one needs to replace (27a) because the Leibniz formula is no longer valid; instead, one follows the ideas of [START_REF] Cordoba | A maximum principle applied to quasi-geostrophic equations[END_REF]. The key is the point-wise inequality [8, Prop. 2.3]:

2ψ • (-∆) α/2 ψ ≥ (-∆) α/2 (|ψ| 2 )
which follows immediately from the kernel representation (2) of (-∆) α/2 . Applied recursively n -1 times when p = 2 n and n ≥ 1 is an integer, it provides for 1 ≤ k ≤ n -1 (or without intermediary k if n = 1):

p R d |ψ| p-2 ψ • (-∆) α/2 ψ ≥ p 2 k R d |ψ| p-2 k (-∆) α/2 (|ψ| 2 k ) ≥ 2 R d |(-∆) α/4 (ψ p/2 )| 2 . ( 27b
)
Compared to [8, Lemma 2.4], the inequality (27b) is improved by a factor of 2. Overall, for p ≥ 2 (restricted to exact powers of 2 if 0 < α < 2), the evolution of the L p -norm of smooth solutions of ( 17) obeys the following inequality:

d ds ψ p L p + 2 R d |(-∆) α/4 (ψ p/2 )| 2 ≤ -(p -1) R d |ψ| p div v. (28)
Obviously, only the focusing zones (i.e. regions where div v < 0) of the transport field can contribute to an increase of the L p norm; the other just tends to spread ψ out. Using the notation (8) for the negative part, one thus has the following estimate:

d ds ψ p L p + 2 R d |(-∆) α/4 (ψ p/2 )| 2 ≤ (p -1) R d |ψ| p (div v) -. (29)
In dimension d ≥ 2 and for 0 < α < 2, one uses the Sobolev embedding [START_REF] Stein | Harmonic Analysis[END_REF]. For σ = dp/(d -α), one thus has:

ψ p L σ = ψ p/2 2 L 2d/(d-α) ≤ 1 S α/2 (R d ) R d |(-∆) α/4 ψ p/2 | 2 .
The estimate (29) becomes:

d ds ψ p L p + 2S α/2 (R d ) • ψ p L σ ≤ (p -1) R d |ψ| p (div v) -. (30)
Finally, as the conjugate exponent of q = d/α > 1 satisfies pq ′ = σ, one splits the right-hand side in the following way:

R d |ψ| p (div v) -≤ ψ p L σ (div v) -L d/α .
Thanks to the smallness assumption [START_REF] Talenti | Best Constant in Sobolev Inequality[END_REF], it is then possible to bootstrap the Lebesgue norm into the left-hand side. In that case, (30) ensures that d ds ψ L p + S α/2 (R d ) • ψ p L σ ≤ 0, which gives [START_REF] Vázquez | The mathematical theories of diffusion[END_REF]. One can then interpolate with [START_REF] Silvestre | Hölder estimates for advection fractional-diffusion equations[END_REF] to control all L q norms for 1 ≤ q ≤ p.

Remark 15 If C α,p (v) = 2S α/2 (R d ) -(p -1) (div v) -L ∞ t L d/α x > 0 then (22) still holds, but with the constant S α/2 (R d ) replaced by C α,p (v), which is not uniform in v anymore.
Remark 16 An improved version of (27b) valid for average-free functions is found in [START_REF] Constantin | Unique ergodicity for fractionally dissipated, stochastically forced 2d-Euler equations[END_REF] or [7, Prop. 2.4]:

R d |ψ| p-2 ψ • (-∆) α/2 ψ ≥ 1 p (-∆) α/2 (ψ p/2 ) 2 L 2 + C ψ p L p .
However, in our case, using a Sobolev embedding for ψ p/2 provides some additional integrability and in particular a control of the L σ -norm with σ > p. This gain will be crucial in what follows. It also allows us to put a restriction on the L d/α -norm of (div v) -, instead of requiring smallness in L ∞ .

Note that on T d , the Sobolev embedding Ḣh (T d ) ⊂ L 2d/(d-2h) (20) is only valid for average-free functions. However, ψ p/2 is not average-free in general (i.e. p = 2), even if ψ is so. For the periodic case, one will use the following simpler result, whose proof is also contained above.

Proposition 17 If ψ is an average-free solution of (17) on T d with α ∈ (0, 2] and d > α and

(div v) -L ∞ t L d/α x ≤ S α/2 (T d ), then ψ(s) 2 L 2 + S α/2 (T d ) s 0 ψ(τ ) 2 L σ dτ ≤ ψ 0 2 L 2 with σ = 2d d -α •

Propagation of the atom property by the dual conservation law

As long as the advection field has mildly convergent characteristics (expressed precisely by ( 57)), the weak maximum principle implies that the (non-local) diffusion propagates the properties of atoms. It is possible to trade a slow increase in each atomic radius to gain some decay in amplitude.

Local propagation

Proposition 18 Let us assume that 1 ≤ α ≤ 2 and d > α and that the velocity field v ∈ BMO satisfies

(p -1) (div v) -L ∞ t L d/α x ≤ S α/2 (R d ) ( 31
)
for some p ≥ 2 (eventually restricted to p = 2 n with n ∈ N if α < 2) such that p > d d -(α -ω) with 0 < ω < α ∧ 1.
Then there exist constants δ, K and γ, depending only on d, p, α and v BMO , such that for all r ∈ (0, 1], the following implication holds:

ψ 0 ∈ A p r =⇒ ∀s ∈ [0, γr α ], ψ(s, •) ∈ 1 - δs r α A p (r α +Ks) 1/α (32)
where ψ denotes the solution of the Cauchy problem [START_REF] Mouhot | Hörmander theories: new interplay[END_REF]. The constant A, which is implicit in the definition of A p r , has to be chosen large enough. The admissible threshold for A, which also depends only on d, p, α and v BMO , is specified in remark 21.

Remark 19

The proposition holds with p = 2 if d > 2(α -ω), which is always possible if one chooses ω such that α -1 < ω < 1 when α < 2 (and ω > 1/2 when α = 2 and d ≥ 3); in this case (57) is also the least restrictive. Thanks to proposition 17, the result then also holds, mutatis mutandis, on T d .

Proof. The proof of proposition 18 is inspired by [START_REF] Kiselev | Variation on a theme of Caffarelli and Vasseur[END_REF] and [START_REF] Dabkowski | Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation[END_REF], though the fractional derivative requires some additional care. Thanks to [START_REF] Silvestre | Hölder continuity for a drift-diffusion equation with pressure[END_REF], the zero-average property of atoms is obviously propagated by [START_REF] Mouhot | Hörmander theories: new interplay[END_REF]. Let x(s) be the solution to the following ODE, which tracks the average flow on a ball of size r. It is obviously well defined for v ∈ L 1 loc (R + × R d ):

x ′ (s) = vB(x(s),r)

x(0) = x 0 where vB(x,r) (s) = 1 |B(x, r)| B(x,r) v(s, y)dy. ( 33
)
Step 1. Strict decay of the L 1 -norm. One introduces S = ψ(s) -1 ({0}) and D ± = supp ψ ± (s). Arguing as in [14, §4] and taking advantage of the conservative form of ( 17):

d ds ψ(s) L 1 = - D ± ψ |ψ| (-∆) α/2 ψ + S |(-∆) α/2 ψ|.
The kernel formula (2) allows us to improve [START_REF] Silvestre | Hölder estimates for advection fractional-diffusion equations[END_REF] and gives

d ds ψ(s) L 1 ≤ - c d,α 2
D 2 ± ψ(y) |ψ(y)| - ψ(x) |ψ(x)| ψ(y) -ψ(x) |y -x| d+α dydx (34) because (if S is a set of strictly positive measure) S |(-∆) α/2 ψ(y)|dy ≤ c d,α x∈D ± y∈S |ψ(x)| |y -x| d+α dydx.
The right-hand side of (34) is negative:

d ds ψ(s) L 1 ≤ -c d,α D + D - dy |x -y| d+α ψ + (s, x)dx + D - D + dy |x -y| d+α ψ -(s, x)dx .
Obviously, the domains of integration D ± can be reduced to D ± ∩ B(x(s), 100r). Provided that most of the L 1 -mass of ψ is localized in B(x(s), 100r), which, as explained in [START_REF] Kiselev | Variation on a theme of Caffarelli and Vasseur[END_REF], is ensured by the 3rd part of the proof, it ends up giving:

d ds ψ(s) L 1 -r -α
i.e. for δ and γ > 0 small enough

∀s ∈ [0, γr α ], ψ(s) L 1 ≤ 1 - δs r α • (35)
Step 2. Strict decay of the L p norm. We have already proven that, under the smallness assumption (57), the right-hand side of (30) can be resorbed within the elliptic term, i.e.

d ds ψ p L p ≤ -S α/2 (R d ) • ψ p L σ < 0.
Next, as σ = dp d-α > p, one can use an elegant idea of [8, p. 517], which is to combine the interpolation inequality ψ L p ≤ ψ 1-θ L 1 ψ θ L σ for θ = (p-1)d (p-1)d+α with the propagation of the L 1 -norm [START_REF] Silvestre | Hölder estimates for advection fractional-diffusion equations[END_REF]. As ψ 0 is an atom A p r , it ensures that:

d ds ψ p L p ≤ -S α/2 (R d ) • ψ 0 -αp (p-1)d L 1 ψ p/θ L p ≤ -S α/2 (R d ) • ψ p/θ L p . ( 36
)
This is a Riccati-type ODE that can be solved explicitly:

ψ(s) p L p ≤ ψ 0 -αp (p-1)d L p + αS α/2 (R d ) (p-1)d • s -(p-1)d α .
Using the atom property ψ 0 L p ≤ Ar -d(1-1 p ) and rearranging the terms, one gets:

ψ(s) L p ≤ Ar -d(1-1 p ) 1 + αS α/2 (R d ) (p-1)d • A αp (p-1)d r -α s -d α (1-1 p ) i.e. ψ(s) L p ≤ Ar(s) -d(1-1 p ) with r(s) = (r α + C d,p,α A µ s) 1/α and µ = α d(1 -1 p ) • (37)
One chooses δ > 0 small enough, then

0 < K < C d,p,α A µ - 2δ d α 1 -1 p • (38)
Thanks to the reversed Bernoulli inequality (1

-x) -1/β ≤ 1 + 2x/β for β > 1 and x ∈ [0, 1/2], this choice ensures that ∀t ∈ [0, γ], 1 + C d,p,α A µ t 1 + Kt ≥ 1 + 2δt d α (1 -1 p ) ≥ (1 -δt) -1/[ d α (1-1 p )] with γ = d α (1-1 p ) 2δ C d,p,α A µ -2δ/[ d α (1-1 p )] K
-1 and thus, after substituting t = s/r α :

ψ(s) L p ≤ A 1 - δs r α (r α + Ks) -d α (1-1 p ) (39)
for s ∈ [0, γr α ].

Step 3. Propagation of the concentration. With x(s) defined by (33), one considers

χ(s) = R d ψ(s, x)Ω(x -x(s))dx. (40)
Using the equation ( 17) and the fact that (-∆) α/2 is self-adjoint, the derivative of χ satisfies:

χ ′ (s) = R d (v -vB(x(s),r) ) • ∇Ω(x -x(s))ψ(s, x)dx - R d ψ(s, x) • (-∆) α/2 Ω(x -x(s))dx. (41)
Let us collect obvious estimates for the derivatives of Ω:

|∇Ω(z)| |z| -(1-ω) • 1 B(0,2) (z), (42a)
|(-∆) α/2 Ω(z)| |z| -(α-ω) + • 1 B(0,2) (z) + |z| -2-α • 1 B(0,2) c (z). (42b)
They follow easily from the scaling properties of the Fourier transform (and thus of (-∆) α/2 ) and from the kernel representation [START_REF] Bresch | Quantitative regularity estimates for compressible transport equations[END_REF]. Recall that we assume ω < min{α, 1} throughout the proof.

3a. Transport term in χ ′ . Let us introduce E k (s) := {x ∈ R d : |x-x(s)| ∈ [2 k-1 r, 2 k r)} to estimate I 1 = T d (v -vB(x(s),r) ) • ∇Ω(x -x(s))ψ(s, x)dx .
One has

I 1 J 0 + ∞ k=1 J k with J 0 = B(x(s),r) |v -vB(x(s),r) ||x -x(s)| -(1-ω) |ψ| and J k = E k (s) |v -vB(x(s),r) ||ψ| r -(1-ω) 2 -k(1-ω) .
For J 0 , we use the Hölder inequality with a -1 + b -1 + c -1 = 1 and the BMO property

J 0 ≤ v -vB(x(s),r) L a (B(x(s),r)) ψ 0 L b |x -x(s)| -(1-ω) L c (B(x(s),r)) v BMO • r d/a • A b * r -d(1-1 b ) • r d c -(1-ω) r -(1-ω) A b * v BMO with b * = 1 -1 b 1 -1 p = p ′ b ′ • (43)
Here, one should comment on the choice of the powers a, b, c. Obviously, we have to take c < d/(1 -ω) for local integrability reasons. Second, as we used the decay of the L b norm of ψ given by proposition 13 followed by proposition 5 on ψ 0 , one needs

p ≥ b > d/(d -(1 -ω))
. Since a can be chosen arbitrary large, it is always possible to find a proper triplet (a, b, c) as soon as

p > d d -(1 -ω) • (44)
For J k with k ≥ 1, we apply the Hölder inequality with a pair of conjugate powers q 1 and q ′ 1 , with q 1 > d/(1 -ω). Thanks to (44), one thus has q ′ 1 < d d-(1-ω) < p, which ensures again that we have propagation of the L q ′ 1 norm of ψ and that proposition 5 may be used liberally on ψ 0 . One also uses that for BMO functions, the averages of adjacent dyadic balls are comparable and that

ψ L 1 (E k ) ≤ 2 kd/q 1 ψ L q ′
1 (E k ) uniformly for r ∈ (0, 1]. One thus gets:

E k (s) |v -vB(x(s),r) ||ψ| ≤ E k (s) |v -vB(x(s),2 k r) ||ψ| + E k (s) |v B(x(s),2 k r) -vB(x(s),r) ||ψ| v -vB(x(s),2 k r) L q 1 (B(x(s),2 k r)) ψ 0 L q ′ 1 + k v BMO ψ L 1 (E k ) (1 + k)2 kd/q 1 A p ′ /q 1 v BMO .
As we choose d/q 1 < 1 -ω, the geometric series in k ≥ 1 is convergent and p ′ /q 1 < b * , and thus:

I 1 r -(1-ω) A b * v BMO . (45)
Let us observe that, due to the admissible range for b, the value of b * can be chosen arbitrarily within the interval

1 -ω d(1 -1 p ) < b * ≤ 1.
In the next part of this proof, we will chose b * to be as close as possible to the lowest bound.

Remark 20 As supp ∇Ω ⊂ B(0, 2) the series of J k terms is limited to k | log r|. However, this upper bound becomes arbitrarily large when r → 0. Our previous estimate is uniform for r ∈ (0, 1].

3b. Nonlocal viscous term in χ ′ . Let us now consider the second term of (41):

I 2 = R d ψ(s, x) • (-∆) α/2 Ω(x -x(s))dx .
Recall that we assume α > ω. Thanks to (42b), for any 0 < ρ ≤ r < 1, one has:

I 2 B(x(s),ρ) |x -x(s)| -(α-ω) |ψ(s, x)|dx + ρ -(α-ω) ψ 0 L 1 .
We apply the Hölder inequality with another pair of conjugate powers q 2 and q ′ 2 , with 1 ≤ q 2 < d α-ω , which is always possible. One also needs q ′ 2 ≤ p to ensure the propagation of the L q ′ 2 norm by proposition 13, i.e. 1 q 2 ≤ 1 -1 p . Such a choice is possible if

p > d d -(α -ω) • (46)
As α ≥ 1, this restriction on p supersedes (44). One gets

I 2 ≤ |x -x(s)| -(α-ω) L q 2 (B(x(s),ρ)) ψ 0 L q ′ 2 + ρ -(α-ω) ψ 0 L 1 i.e. I 2 ρ d q 2 -(α-ω) A p ′ /q 2 r -d/q 2 + ρ -(α-ω) .
The optimal choice for ρ is given by ρ = rA -p ′ /d , which belongs indeed to (0, r] as A ≫ 1. Substituting this value in the previous estimate of I 2 gives:

I 2 r -(α-ω) A µ * with µ * = α -ω d(1 -1 p ) • (47)
3c. Conclusion. Putting together (41) with ( 45) and (47), one gets:

|χ ′ (s)| ≤ I 1 + I 2 r -(1-ω) A b * v BMO + r -(α-ω) A µ * .
After integration and considering that χ(0) ≤ r ω and 1 ≤ r -1 ≤ r -α because α ≥ 1:

χ(s) ≤ r ω 1 + C ′ d,p,α A b * v BMO + A µ * s r α .
Provided A is large enough, one may amend the previous choice (38) of δ and K to ensure that

K ≥ α ω δ + C ′ d,p,α A b * v BMO + A µ * , (48)
which, in turn, ensures that

∀t ∈ [0, γ], 1 + C ′ d,p,α A b * v BMO + A µ * t ≤ (1 -δt)(1 + Kt) ω/α
i.e. , with t = s/r α :

∀s ∈ [0, γr α ], χ(s) ≤ 1 - δs r α (r α + Ks) ω/α . ( 49
)
This concludes the proof of ψ(s) ∈ 1 -δs r α A p (r α +Ks) In turn, this condition is satisfied by choosing b * as small as possible and because α > ω ∨ (1 -ω).

Remark 22 It could be tempting to discard the L 1 -property from the atom definition and use proposition 28 from the appendix to control this norm a-posteriori. In this case, instead of (38) and (48), one is led to choose δ and K such that

α ω δ + C ′ d,p,α A b v BMO + A μ ≤ K < C d,p,α A μ - 2δ d α 1 -1 p with b = ω+d(1-1 b ) ω+d(1-1 p ) ∈ (μ/α, 1] and μ = α ω+d(1-1 p )
. As both sides are order A μ, it is not clear anymore that the choice can be resolved for some large value of A. This alternate path is thus a subtle deadlock.

Global propagation

Proposition 23 In the conditions of Proposition 18, the constants δ, K are such that

ψ 0 ∈ A p r =⇒ ∀s > 0, ψ(s, •) ∈ r α r α + Ks δ/K A (r α +Ks) 1/α ( 50
)
where ψ denotes the solution of the Cauchy problem [START_REF] Mouhot | Hörmander theories: new interplay[END_REF].

Proof. We keep the assumptions and notations of proposition 18. Let us split the time-line in consecutive intervals [ℓγr α , (ℓ + 1)γr α ] with ℓ ∈ N. For ℓ = 0, one has

∀s ∈ [0, γr α ], 1 - δs r α ≤ 1 + Ks r α -δ/K = r α r α + Ks δ/K .
Let us assume that, for some integer ℓ ∈ N, one has:

ψ(ℓγr α , •) ∈ (1 + Kℓγ) -δ/K A r(1+Kℓγ) 1/α .
Then for any s ∈ [ℓγr α , (ℓ + 1)γr α ], proposition 18 gives:

ψ(s, •) ∈ (1 + Kℓγ) -δ/K 1 + KS R α -δ/K A (R α +KS) 1/α
with S = s -ℓγr α and R = r(1 + Kℓγ) 1/α . The new radius is an exact match:

(R α + KS) 1/α = (r α + Ks) 1/α . (51)
Similarly, the amplitude satisfies:

(1 + Kℓγ) 1 + KS R α = 1 + Ks r α •
The proposition thus follows by induction on ℓ ∈ N.

4.3 Modifications in the case 0 < α < 1

Throughout §4, the assumption α ≥ 1 has only been used in the third step of the proof of proposition 18, i.e. to ascertain the propagation of the concentration. Let us investigate in this subsection how to deal with the case 0 < α < 1.

When dealing with the super-critical case, L. Silvestre [START_REF] Silvestre | Hölder estimates for advection fractional-diffusion equations[END_REF] assumes a higher regularity for the advection field. We will do the same here and assume respectively that v ∈ C 1-α (R d ) or C 1-α (T d ). The identity (41) still holds. To deal with the transport term, one uses

v(x) -vB(x(s),r) ≤ 1 |B(x(s), r)| B(x(s),r) |v(x) -v(y)|dy ≤ v C 1-α B(x(s),r) |x -y| 1-α dy, which gives v(x) -vB(x(s),r) ≤ (|x -x(s)| + r) 1-α v C 1-α . (52)
For J 0 , one takes a = ∞ and the same constraints for the exponents b and c, thus

J 0 r -(α-ω) A b * v C 1-α .
Similarly, for J k , one takes q 1 = ∞:

E k (s) |v -vB(x(s),r) ||ψ| 2 k(1-α) r 1-α v C 1-α thus J k 2 -k(α-ω) r -(α-ω) v C 1-
α and as ω < α, the geometric series in k is convergent. The estimate (45) can therefore be replaced by

I 1 r -(α-ω) A b * v C 1-α with b * = 1 -ω d(1 -1 p ) + ε, ε > 0. (53)
For the non-local viscous term, the estimate (47) remains valid. The sole difference is that now

d d -(1 -ω) > d d -(α -ω)
and, consequently, the requirement (44) trumps (46).

Putting together (41) with ( 53) and (47), one gets:

|χ ′ (s)| r -(α-ω) A b * v C 1-α + A µ * and χ(s) ≤ r ω 1 + C ′ d,p,α A b * v C 1-α + A µ * s r α .
The conclusion is identical, provided that A is large enough and that the choice of K and δ ensures

K ≥ α ω δ + C ′ d,p,α A b * v C 1-α + A µ * (54)
instead of (48). Note that to reconcile (54) with (38) for large A, one needs α > ω ∨ (1 -ω), which is always possible if α > 1/2. However, when α ≤ 1/2, one needs one final modification, which is to replace the average vB(x(s),r) by the point-wise value v(x(s)), where:

x ′ (s) = v(x(s)), x(0) = x 0 . (55)
In this case, estimate (52) is improved one step further into the following one:

|v(x) -v(x(s))| ≤ |x -x(s)| 1-α v C 1-α . (56)
This changes J 0 into

J 0 = B(x(s),r) |x -x(s)| -(α-ω) |ψ|,
which is then estimated in an identical manner to I 2 . Note that this modification also allows us to drop all requirements concerning b * and in particular (44), which is beneficial for any α ∈ (0, 1). Let us finally point out that, in the other parts of the proof, the requirement d > α now allows for any dimension d ≥ 1. We have thus established the following statement:

Proposition 24 Let us assume that 0 < α < 1 and d ≥ 1 and that the velocity field v ∈ C 1-α satisfies

(p -1) (div v) -L ∞ t L d/α x ≤ S α/2 (R d ) (57)
for some p = 2 n with n ∈ N such that

p > d d -(α -ω) with 0 < ω < α.
Then there exist constants δ, K and γ (and a lower threshold for A), depending only on d, p, α and v C 1-α , such that for all r ∈ (0, 1], the following implication holds:

ψ 0 ∈ A p r =⇒ ∀s > 0, ψ(s, •) ∈ 1 + Ks r α -δ/K A (r α +Ks) 1/α ( 58
)
where ψ denotes the solution of the Cauchy problem [START_REF] Mouhot | Hörmander theories: new interplay[END_REF].

Remark 25 Note that we can take p = 2 in the previous statement (and thus, using remark 19, claim a similar one in the case of

T d ) if d 2 > α -ω.
Such a choice is always possible.

Proof of Theorem 1

The proof of theorem 1 is now straightforward.

Given d ≥ 2 and 1 ≤ α ≤ 2 (with d ≥ 3 when α = 2), one chooses ω ∈ (0, 1) such that α -1 < ω < 1 if α < 2, or ω > 1/2 if α = 2. One checks immediately that ω < α < d and d > 2(α -ω). Let us now consider an advection vector field v ∈ BMO with

(div v) -L d/α ≤ S α/2 .
One takes p = 2. One chooses the constant A, which is implicit in the definition of atoms, according to the threshold mentioned in remark 21; this threshold depends solely on d, α and v BMO . One considers the constants γ, δ and K given by propositions 18 and 23 and sets

β = αδ/K.
The value of β depends on d, α and v BMO .

For d ≥ 1 and 0 < α < 1, one chooses ω such that (α -d 2 ) + < ω < α and p = 2. In this case, the BMO norm is replaced by the C 1-α -norm in all computations.

Remark 26 When (div v) -L d/α < 2S α/2 , one can still run the following proof. However, the choice of A and of all constants then depends not only on v BMO but also on C(v) = 2S α/2 -(div v) -L d/α > 0 and degenerates as C(v) → 0. See remark 15.

Propagation of the Hölder regularity

For any solution θ of (1) stemming from θ 0 ∈ C β and for ψ 0 ∈ A 2 r , identity [START_REF] Kružkov | First order quasilinear equations in several independent variables[END_REF] implies that: where ψ is solution of the dual equation [START_REF] Kiselev | Global well-posedness for the critical 2D dissipative quasi-geostrophic equation[END_REF], which, by proposition 23, is an atom of calibrated size. Using [START_REF] Kiselev | Variation on a theme of Caffarelli and Vasseur[END_REF] for θ 0 , one gets:

r -β R d θ(t, x)ψ 0 (x)dx r -β 1 + Kt r α -δ/K (r α + Kt) β/α θ 0 C β = θ 0 C β .
A second application of ( 14) then ensures that θ(t) ∈ C β and that

θ(t) C β ≤ C θ 0 C β .
The constant C is the implicit one in [START_REF] Kiselev | Variation on a theme of Caffarelli and Vasseur[END_REF].

Remark 27 The same argument also gives θ(t) C β ′ ≤ C θ 0 C β ′ for any 0 ≤ β ′ ≤ β.

Gain in Hölder regularity

One can use the Hölder inequality and proposition 5 with p = 2 to control R d θ 0 (x)ψ(t, x)dx ≤ θ 0 L q ψ(t) L q ′ ≤ A 2/q r β θ 0 L q (r α + Kt)

(β+ d q )/α r β t -(β+ d q )/α θ 0 L q
for any Lebesgue exponent q such that 2 ≤ q ≤ ∞. One thus gets a regularization estimate:

θ(t, •) C β ≃ sup 0<r≤1 ψ 0 ∈A 2 r r -β R d
θ(t, x)ψ 0 (x)dx ≤ Ct -(β+ d q )/α θ 0 L q with a constant C that depends on q and A and thus ultimately on d, α and v BMO .

A Appendix: on the L 1 -control of atoms

Even without the a-priori constraint ψ L 1 ≤ 1, one can control the L 1 -norm of atoms (or any L q norm for q ≤ p) by a real interpolation estimate.

and, more generally, for any 1 ≤ q ≤ p, one has ϕ L q ≤ C d,p,q A ω+d(1-1/q) ω+d(1-1/p) r -d(1-1 q ) . (60)

Remark 29 Compared to proposition 5, these estimates "lose" powers of A, which would provoke a critical collision of exponents in the previous proof (see remark 22). |B(0, 1)| 1-1 p + r ρ ω and (59) follows from choosing the optimal value ρ = r A|B(0, 1)| 1-1 p -1/(ω+d(1-1 p )) . For the second estimate, one proceeds similarly with τ ∈ [0, r]; for clarity, we do not track the constant related to |B(0, 1)|:

R d |ϕ| q ≤ B(x 0 ,τ ) |ϕ| q + R d \B(x 0 ,τ ) |ϕ| q R d |ϕ| p q p τ d(1-q p ) + τ -ω R d \B(x 0 ,τ ) |ϕ(x)|Ω(x -x 0 )dx p-q p-1 R d |ϕ| p q-1 p-1
, thanks to the Hölder inequality (with p/q ≥ 1) for the first term, and the interpolation inequality f L q ≤ f 1-θ L 1 f θ L p with θ = (1 -1 q )/(1 -1 p ) ∈ [0, 1] for the second. We now use the fact that ϕ ∈ A p r and deduce that

R d |ϕ| q Ar -d(1-1 p ) q τ d(1-q p ) + r τ ω( p-q p-1 )
(A p r -(p-1)d ) q-1 p-1 .

The optimal choice for τ is the one that balances the weight of both terms; it is τ = rA -p/(d(p-1)+ωp) . The computation then boils down to R d |ϕ| q r -(q-1)d A dp(q-1)+ωpq d(p-1)+ωp i.e. ϕ L q A ω+d(1-1/q) ω+d(1-1/p) r -d(1-1 q )

and the lemma is proven.

  x)ψ 0 (x)dx = R d θ 0 (x)ψ(t, x)dx

Proposition 28

 28 If ϕ satisfies ϕ L p ≤ Ar -d(1-1 p ) ,and∃x 0 ∈ R d , R d |ϕ(x)|Ω(x -x 0 )dx ≤ r ω for some 0 < r ≤ 1 and p ∈ (1, ∞], then ϕ L 1 ≤ C d,p A ω/(ω+d(1-1 p ))

Proof.

  For any ρ ∈ [0, r], one hasϕ L 1 ≤ B(x 0 ,ρ) |ϕ| + ρ -ω R d \B(x 0 ,ρ) |ϕ(x)|Ω(x -x 0 )dx ≤ A ρ r d(1-1 p )

  1/α . Remark 21 Our conditions (38) and (48) generalize respectively the conditions 4.6 -4.15 of [14]. Both conditions are compatible, provided A is chosen large enough, because µ > b * ∨ µ * .


Benjamin Girault

Antonio Ortega

Shrikanth S Narayayan

GRAPH VERTEX SAMPLING WITH ARBITRARY GRAPH SIGNAL HILBERT SPACES

Keywords: Graph signal processing, vertex sampling, graph signal Hilbert space

Graph vertex sampling set selection aims at selecting a set of vertices of a graph such that the space of graph signals that can be reconstructed exactly from those samples alone is maximal. In this context, we propose to extend sampling set selection based on spectral proxies to arbitrary Hilbert spaces of graph signals. Enabling arbitrary inner product of graph signals allows then to better account for vertex importance on the graph for a sampling adapted to the application. We first state how the change of inner product impacts sampling set selection and reconstruction, and then apply it in the context of geometric graphs to highlight how choosing an alternative inner product matrix can help sampling set selection and reconstruction.

INTRODUCTION

Over the past few years, the field of graph signal processing (GSP) has been building up a framework to analyze, process and transform graph supported data with the goal of understanding data supported by arbitrary discrete structures [START_REF] Shuman | The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains[END_REF][START_REF] Ortega | Graph Signal Processing: Overview, Challenges, and Applications[END_REF]. In this paper, we are interested in the question of sampling set selection: Where on the graph do we perform measurements so as the dimension of the space of smooth signals we can reconstruct exactly is as high as possible? While there have been a number of contributions studying this problem in a GSP context in the past few years [START_REF] Shomorony | Sampling large data on graphs[END_REF][START_REF] Chen | Discrete Signal Processing on Graphs: Sampling Theory[END_REF][START_REF] Anis | Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies[END_REF], in all of them the space of graph signals is assumed to be equipped with the canonical inner product: the dot product.

Recently, we introduced a generalization of the graph Fourier transform to arbitrary Hilbert spaces of graph signals, allowing to equip that space with any inner product [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF]. We showed that this additional parameter (i.e., the inner product) allows better control over the importance of the nodes of the graph, with their relative importance influencing the design of a suitable graph Fourier transform. For example, the random walk Laplacian approach based on the inner product whose matrix is the degree matrix D, is shown to yield the most interesting results in spectral clustering [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF][START_REF] Von | A tutorial on spectral clustering[END_REF]. In image processing, bilateral filtering and non-local means are based on the inner product I + D and perform well in image denoising [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF][START_REF] Milanfar | A Tour of Modern Image Filtering: New Insights and Methods, Both Practical and Theoretical[END_REF]. Still in image processing, a recent work shows that the inner product matrix can be used towards perceptually better compression by integrat-

The research is based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA Contract No 2017-17042800005. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.

Work partially funded by NSF under grant CCF-1527874.

ing the importance of each pixel towards the Structural SIMilarity (SSIM) [START_REF] Lu | Perceptually Inspired Weighted MSE Optimization using Irregularity-Aware Graph Fourier Transform[END_REF]. Finally, in point cloud attribute compression, a regionadaptive approach can be seen as a hierarchical graph Fourier transform with specific graphs and inner products at each level [START_REF] De Queiroz | Compression of 3D Point Clouds Using a Region-Adaptive Hierarchical Transform[END_REF][START_REF] Chou | A Volumetric Approach to Point Cloud Compression-Part I: Attribute Compression[END_REF][START_REF] Pavez | Region Adaptive Graph Fourier Transform for 3d Point Clouds[END_REF].

Our contribution is twofold. First, in Sec. 2, we show how the sampling approach of [START_REF] Anis | Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies[END_REF] can be generalized to any inner product. In particular, our generalization describes a method that can applied to many other GSP tools to extend them to arbitrary Hilbert spaces, not just sampling set selection. Our second contribution, in Sec. 3, is to experimentally study the impact of the inner product on sampling and reconstruction for geometric graphs and several inner product. Doing so, we show that it has an impact on both theoretical worse case scenario error bounds and mean squared reconstruction error, and we give evidence that the Voronoi cell area inner product proposed in [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF] is a good candidate.

SAMPLING WITH SPECTRAL PROXIES

In this section, we describe how the graph vertex sampling set selection approach of [START_REF] Anis | Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies[END_REF] can be extended to any arbitrary Hilbert space of graph signals. To that end, we use the graph Fourier transform adapted to arbitrary Hilbert spaces that we proposed in [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF] to obtain the required notion of bandlimitedness.

Background: Graph Fourier Transform in Arbitrary Hilbert Spaces

Let G = (V, E, W) be a graph with vertex set |V| = N , edge set E, and weight matrix W such that wij is the weight of the edge from vertex i to vertex j. G may be directed or undirected. A graph signal x maps any vertex i to a scalar value xi ∈ C. We assume that there exists a non-negative graph signal variation operator ∆(x) ≥ 0 that depends on the edge set E and weight matrix W. In [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF], we proposed a generalization of the graph Fourier transform (GFT) for any Hilbert space of graph signals using the variation operator ∆. We denote Q the matrix of the graph signal Hilbert space inner product verifying x, y Q = y * Qx with . * the conjugate transpose operator. Above, and in the rest of this paper, we color the inner product matrix Q in blue to highlight where it impacts the sampling set selection approach of [START_REF] Anis | Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies[END_REF]. In this paper, we further assume that the variation operator verifies ∆(x) = x * Mx with M a non-negative Hermitian matrix. 1 We denote this graph Fourier transform the (M, Q)-GFT.

More precisely, the (M, Q)-GFT projects a graph signal onto an orthonormal2 basis of graph Fourier modes {u l } l defined as solution to the following iterative minimizations:

min u L :∀l<L, u L ,u l Q =0 ∆(uL) subj. to uL Q = 1.
Since ∆(x) = x * Mx, the set of graph Fourier modes that is solution to the minimizations above is exactly the set of eigenvectors of the matrix Theorem 1]. The analysis and synthesis formula for the (M, Q)-GFT are then given by:

Z = Q -1 M [6,
x l = [Fx] l = x, u l Q = [U * Qx] l x = F -1 x = U x = l x l u l = l x, u l Q u l ,
with U = [u0 . . . uN-1] the matrix whose columns are the graph Fourier modes. Moreover, the graph frequencies are naturally defined as the graph variations of the graph Fourier modes:

λ l = ∆(u l ) = u * l
Mu l , and are equal to the eigenvalues of Z.

In this setting, a graph signal is ω-bandlimited if:

∀λ l > ω, x l = 0.

Consistent Reconstruction in Hilbert Spaces

Similar to [START_REF] Anis | Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies[END_REF], our goal is to sample vertices of a graph such that the error made during reconstruction of an ω-bandlimited graph signal is minimal. In other words, given a signal x of which we only know the noisy samples yS = xS + n on vertices in S ⊆ V, we would like to recover x from yS . We denote x the estimated signal. Before looking at the question of sampling set selection, we study the problem of signal reconstruction given a sampling set S. One approach called consistent reconstruction is to perform a least squares estimate of the spectral coefficients of the signal and apply the inverse Fourier transform, thus leveraging the bandlimited hypothesis of the signal x [13]:

x = UVR argmin x R yS -USR xR 2 Q S ,
where R = {1, . . . , r} and λr is the largest frequency smaller than ω. Under mild conditions, this estimate verifies xS = yS . Notice above the use of the inner product . Q S of the Hilbert subspace of graph signals supported by the vertex sampling subspace S: This is necessary since ys lives in the Hilbert space of graph signals whose inner product matrix is QS , not the usual dot product. We obtain the following close-form solution:

3 x = UVR U * SR QS USR -1 U * SR QS yS . (1)

Effect of Noise

Here we study the impact of noisy measurements on reconstruction. More precisely, we look at the error covariance matrix when the noise distribution is a QS -white noise, i.e., n ∼ N (0, Q -1 S ) [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF]. Considering this type of noise is necessary to have a noise with flat power spectrum, i.e., with E[ n n * ] = I. Let e = xx be this error. In the Hilbert space of graph signal with inner product matrix Q, the error covariance matrix is then:4

E = E [ee * Q] = UVR [U * SR QS USR] -1 U * VR Q.
The A-optimal sampling design minimizing the Q-Mean Squared Error (Q-MSE) [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF] is then obtained as:

S A-opt = argmin |S|=m tr(E) = argmin |S|=m tr [U * SR QS USR] -1 ,
while the E-optimal sampling design minimizing the maximum eigenvalue of E is given by:

S E-opt = argmin |S|=m λmax(E) = argmax |S|=m σmin Q 1 2 S USR . (2)
In particular, a larger value of σmin Q

1 2
S USR leads to a smaller reconstruction error under noisy observations.

Effect of Model Mismatch

Now, we assume that the signal we wish to reconstruct is not exactly bandlimited and can be written as x = x + x ⊥ , where x is the orthogonal projection of x on the space of ω-bandlimited signals, and the non-zero x ⊥ correspond to the high-pass components of the signal. Using the E-optimal sampling design, the reconstruction error can be bounded by [START_REF] Eldar | Sampling with Arbitrary Sampling and Reconstruction Spaces and Oblique Dual Frame Vectors[END_REF]:

x -x Q σmin Q 1 2 S USR -1 x ⊥ Q . ( 3
)
In other words, a choice of M and Q that leads to a larger minimum singular value of Q

1/2
S USR leads to a smaller influence of a model mismatch on the reconstruction error.

Noticeably, the influence of both the noise and model mismatch are minimized when σmin Q 1/2 S USR is maximized. We will use this to experimentally study the performance of various choices of the matrix Q in Sec. 3.

Sampling Set Selection using Spectral Proxies

In [START_REF] Anis | Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies[END_REF], the authors propose to select a sampling set using:

S opt k = argmax |S|=m Ω k (S), (4)
where Ω k (S) is the k th order approximate cutoff frequency of the sampling set S, i.e., the minimum bandwidth of a graph signal verifying xS = 0. This cutoff frequency is in turn defined using the k th spectral proxy of the cutoff frequency of a given graph signal:

ω k (x) =   Q -1 M k x Q x Q   1/k
, leading to:

Ω k (S) = min x:x S =0 ω k (x) (5)
= min

x S c xSc * H k (S c ) * H k (S c ) xSc xSc * QSc xSc 1 2k = [σmin (H k (S c ))] 1 k ,
where . c is the set complement operation, with associated singular vector φ * k , and with:

H k (S c ) = Q 1 2 Q -1 M k VS c Q -1 2 S c .
The optimization problem of (4) being combinatorial, a greedy heuristic is used to select the vertices of the sampling set one after the other. More precisely, the cost of adding vertex is obtained using:

ω α k (x, t) = ω k (x) + α x * diag(t)x x * x λ α k (t) = min x ω α k (x, t)
such that adding vertex i to the sampling set S leads to:

∂λ α k ∂ti (1S ) = ∂ω α k ∂ti (φ * k , 1S ) = α [φ * k ] i φ * k 2 .
Notice above how Q does not appear in the norm: The relaxation

x * diag(1 S )x x * x
(when t = 1S ) of the constraint xS = 0 in (5) does not involve Q. Indeed, the constraint is enforcing that x is zero on the set S, which we translate into most of the nodes of S having values close to zero, rather than the QS -norm of xS being close to zero compared to x Q . Note that choosing the Q-norm can be shown to be equivalent to using the (Q -1 2 MQ -1 2 , I)-GFT using the change of variable y = Q 1 2 x in λ α k which would defeat the purpose of an alternative Hilbert space of graph signals.

We then adopt the same greedy approach that selects the next vertex i that maximizes | [φ * k ] i |, i.e., that increases the most the cutoff frequency thus optimizing (5).

Additional Background: Reconstruction with POCS

Although (1) can be used to reconstruct a graph signal, it requires computing the (M, Q)-GFT matrix, i.e., computing an eigendecomposition. To avoid performing this eigendecomposition, we adopt the Projection On Convex Sets (POCS) approach of [START_REF] Sunil K Narang | Localized iterative methods for interpolation in graph structured data[END_REF][START_REF] Gadde | Active Semi-Supervised Learning using Sampling Theory for Graph Signals[END_REF]. To reconstruct the signal, we iteratively project it onto (PS ) the set of signals verifying xS = yS (recovering the observed samples) and then onto (Pω) the set of ω-bandlimited graph signals until the loop converges or exceeds a maximum number of iterations.

Implementing the projection on (Pω) can be done using an ideal low-pass graph filter of cutoff frequency ω [START_REF] Gadde | Active Semi-Supervised Learning using Sampling Theory for Graph Signals[END_REF]. In practice an ideal low-pass graph filter still needs an eigendecomposition of Z, so instead, we adopt the same strategy of approximating it with a Chebyshev polynomial approximation of the following graph filter frequency response:

h(λ) = 1 1 + exp (α(λ -ω))
, with α controlling how sharp the transition is around the cutoff frequency. We denote h ω m,α (λ) this polynomial approximation of order m, and the projection on (Pω) of the signal x is approximated by

h ω m,α (Q -1 M)x.

EXPERIMENTS WITH GEOMETRIC GRAPHS

One of the challenges of comparing a spectral method efficacy depending on the graph Fourier transform is to actually measure this efficacy and synthesize ground truth data that is independent of the graph Fourier transforms being compared. To that end, geometric graphs provide an appealing setting where the underlying Euclidean geometry holds the ground truth. More precisely, let us consider a portion of a two dimensional space, e.g. a square, where a spatially continuous phenomena can be measured. Such a setting can model a variety of real world examples such as weather readings, geologic earth of the ground, or heat propagation in a metal sheet. In this section, we experiment with this setting of geometric graphs, and study the application of sampling set seletion and signal reconstruction described in Sec. 2.

There are however a number of parameters that we need to choose before performing vertex sampling: i) the edge set E, ii) the weights of each edge wij, iii) the graph signal variation operator ∆(x), and iv) the graph signal inner product Q. Here we use the classical approach of a complete graph with edge weights given by a Gaussian kernel of the distance between vertices. Let dist(i, j) be the Euclidean distance between any two vertices (locations), wij = expdist(i, j) 2 /(2σ 2 ) be a Gaussian kernel of the distance, and D = diag(W1) be the degree matrix associated to the weight matrix W. Then, using the combinatorial Laplacian M = L = D -W, we choose the classical graph signal variation ∆(x) = x * Lx.

Using this setting, we compare three (L, Q)-GFTs: Q = I, Q = D, and Q = C where C is the diagonal matrix with Voronoi cell areas of each vertex [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF]. In each experiment, multiple realizations of geometric graphs of 100 vertices are obtained by uniformly drawing vertices in a 10 × 10 square.

In all of our experiments, we use k = 3 for the spectral proxies ω k (x).

Error Bound

In (3), we showed that the reconstruction error is upper bounded by a quantity that depends on the minimum singular value of Q 1/2 S USR, and showed that this minimum singular value should be large to have a smaller worse case reconstruction error. In Fig. 1, we show the resulting average minimum singular value for 5,000 realizations of geometric graphs. Q = C outperforms the other two choices of inner product matrix Q. Interestingly, the random walk Laplacian is the less performing of all three approaches and sees even a large drop of performance for sampling sets larger than 40% of the vertex set. This may suggest that this approach is better at capturing global trends on the graph (smooth graph signals easily described by a small subset of vertices) than finer grained details. However, the reconstruction error also depends on the model mismatch energy (second term in the r.h.s. of (3)). To better quantify this reconstruction error, we study it on concrete signals next.

Although this only shows that the worse case reconstruction sce- nario is less impacted by noise with Q = C, this highlights the property that choosing a quantity that more finely describes the underlying structure can actually help in making the right sampling set selection.

Synthetic Graph Signals

In this section, we are interested in simulating sampling and reconstruction of graph signals, and studying the average error made by all three choices of matrix Q. Noticeably, the usual approach of choosing a bandlimited ground truth graph signal and perturbing it with additive Gaussian noise is flawed in our setting: There are multiple definitions of bandlimitedness that each depend on a different (L, Q)-GFT. To lift this difficulty, and to be true to our motivations, we consider a smooth ground truth continuous signal. More precisely, we consider four pure horizontal sinewaves sω(x, y) = sin(2πωx) and choose ω = ω k such that there are exactly k ∈ {2, 3, 4, 5} horizontal oscillations in the underlying continuous space. To study the mean squared error, we also perturb the measurements of sω at each vertex with a Gaussian noise ni ∼ N (0, σ 2 ), independently for each vertex. Note that this additive noise corresponds to measurement errors. Even though it is modeled differently depending on Q, choosing one Q over another is not about filtering out uniformly the noise, but filtering it out where it is important to do so. In other words, using an approach that removes uniformly the noise (Q = I) may be detrimental if the noised removed is not important, while selectively removing the noise can help attain our goal (here, having an accurate representation of the underlying continuous signal).

Additionally, we do not plot the 2-norm of the error, but the C-norm, based on the Voronoi cell area since we showed previously that it corresponds to the error made in the underlying continuous space if we were interpolating the measurement in the continuous domain [START_REF] Girault | Irregularity-Aware Graph Fourier Transforms[END_REF]. It is therefore a more realistic measure of how accu-rate our reconstruction is with respect to the underlying continuous space.

Fig. 2 reports these mean squared errors for each of the 12 cases (four signals, and three noise levels with standard deviations σ ∈ {0.1, 0.2, 0.4}). Noticeably, the (L, D)-GFT based approach slightly outperforms the other two for the smoothest signal, but quickly becomes less effective as signal smoothness decreases. For sω 3 and sω 4 we remark that the MSE of the (L, D)-GFT approach actually decreases for a larger sampling set size than the other two methods. As stated in [START_REF] Anis | Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies[END_REF], this suggests that the cutoff frequency of the signal being considered is actually higher with respect to the (L, D)-GFT than the other two methods.

On the other hand, (L, C) performed the best for sω 3 , sω 4 and sω 5 with barely larger error for smaller sampling sets, and then outperforming the other two methods, thus showing the importance of correctly choosing Q.

CONCLUSIONS AND PERSPECTIVES

In this paper, we showed how to extend the sampling set selection through spectral proxies method of [START_REF] Anis | Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies[END_REF] to any Hilbert space of graph signals. Doing so, we highlighted how the inner product matrix Q impacts the method, and the assumptions made to derive close-form solutions. We also showed on synthetic geometric graphs experiments how the choice of Q alters reconstruction error bounds and reconstruction errors for some smooth graph signals.

This work opens up several interesting perspectives. In particular, the Voronoi cell area inner product is shown to be helpful, but can we do better? Can we find a better inner product for geometric graphs, or an alternative choice of edge weights to better match the inner product? This also opens up many possibilities for other types of graphs, such as social network graphs for which we know already that the heavy-tail degree distribution is an obstacle to graph signal processing.

Fig. 1 : 1 S

 11 Fig. 1: Average value of

Fig. 2 :

 2 Fig. 2: Mean C-norm of the reconstruction error for various ground truth graph signals sω k (columns), and various amount of additive Gaussian noise with standard deviation σ (rows), depending on the size of the sampling set |S|.

Extending our current contribution to any ∆ by considering the fundamental matrix of the graph Z instead of Q -1 M is straightforward, but we choose not to do it here to better highlight how Q changes the setting of[START_REF] Anis | Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies[END_REF].

[START_REF] Ortega | Graph Signal Processing: Overview, Challenges, and Applications[END_REF] Orthonormality is with respect to the Q inner product.

Note that Q S is itself a proper inner product matrix on the graph signals on S [14, Observation 7.1.2]

We use the generalization of covariance matrix K to arbitrary Hilbert spaces verifying Kx, y = Ez [ z, y x, z ]. Notice also how the covariance matrix of n is actually the identity matrix in the corresponding Hilbert space:E[nn * Q S ] = Q -1 S Q S = I.