Yannick Deleuze

A mathematical model of mast cell response to acupuncture needling

We introduce a new model of mast cell response to acupuncture needling based on the Keller-Segel model for chemotaxis. The needle manipulation induces the release of a chemoattractant by the mast cells. We show, in a simplified case, that blow-up of the solution occurs in finite time for large initial data concentrated around the acupoint. In those conditions, blow-up is the result of aggregation of cells and could indicate the efficiency of the acupuncture manipulation of the needle at one acupoint.

Introduction

During acupuncture treatment hair-thin needles are inserted in the skin via manual lifting and thrusting or rotating at specific points on the body, called acupoints [START_REF] Cheng | Chinese acupuncture and moxibustion[END_REF]. The needle interacts with the subcutaneous . This work was partially supported by Fondation Sciences Mathématiques de Paris. . Many thanks to Benoît Perthame for fruitful discussions about this work. Email address: yannick.deleuze@ljll.math.upmc.fr (Yannick Deleuze). loose connective tissue [START_REF] Langevin | Mechanical signaling through connective tissue: a mechanism for the therapeutic effect of acupuncture[END_REF]. This manipulation is shown to cause the wrapping of collagen fibers and send a mechanical signal to the extracellular matrix cells (ECM) [START_REF] Langevin | Evidence of connective tissue involvement in acupuncture[END_REF]. There is evidence that needle manipulation in acupuncture causes degranulation of mast calls directly through mechanical stress [START_REF] Yu | Role of Collagen Fibers in Acupuncture Analgesia Therapy on Rats[END_REF] [START_REF] Zhang | Role of Mast Cells in Acupuncture Effect: A Pilot Study[END_REF]. Mast cells are resident cells in connective tissues, in particular in tissues close to the external environment [START_REF] Metcalfe | Mast cells[END_REF]. They contain many granules rich in chemicals [START_REF] Thiriet | Biology and Mechanics of Blood Flows: Part I: Biology[END_REF]. Mast cells are well known for their role in inflammatory process where they accumulate at the site of inflammation in response to a chemical mediator [START_REF] Nilsson | C3a and C5a Are Chemotaxins for Human Mast Cells and Act Through Distinct Receptors Via a Pertussis Toxin-Sensitive Signal Transduction Pathway[END_REF]. However, they also appear to have a protective role [START_REF] Urb | The Role of Mast Cells in the Defence against Pathogens[END_REF].

To build our model, we make the assumption that mast cells, when mechanically stressed, release two main kinds of chemicals substances into the ECM : stimulants and chemoattractants. Some stimulants aim at triggering action potential to nearby nerve endings, that can lead for example to liberate opioids and analgesic in the brain [START_REF] Hsiao | A Neurovascular Transmission Model for Acupuncture-induced Nitric Oxide[END_REF]. Some stimulants increase the blood vessel lumen as well as its permeability and increase blood flow rate after reaching the heart [START_REF] Hsiu | Assessing the effects of acupuncture by comparing needling the hegu acupoint and needling nearby nonacupoints by spectral analysis of microcirculatory laser Doppler signals[END_REF] [START_REF] Kuo | Blood Flow Effect of Acupuncture on the Human Meridian[END_REF]. Then, to maintain a high level of the acupuncture response of mast cells, the chemoattractant participates in cell recruiting of neighbouring mast cells. Recruited mast cells, in turn, degranulate creating a positive feedback process and thus a sustained reaction to acupuncture needling.

Acupuncture as a whole is a complex system. We propose a mathematical model, related to the well known Keller-Segel system for chemotaxis [START_REF] Keller | Model for chemotaxis[END_REF], of mast cell response to acupuncture needling close to one acupoint.

Biological model

We consider the density n g (t, x) of granulated mast cells and the density n d (t, x) of degranulated mast cells around the needle insertion point. Granulated mast cells are stressed by a mechanical stimulus induced by the needle inserted at the acupoint. The function Φ(x) represents the stimulus signal. The function Φ is a smooth function of compact support from R 2 to [0, 1].

When excited, granulated mast cells release into the extracellular environment a chemoattractant, c(t, x), a neural stimulant, s n (t, x), and a endocrine stimulant, s e (t, x). Then, degranulated mast cells can regenerate their chemical mediators to become granulated mast cells again. We consider the release of chemical mediators as quasi-instantaneous and we neglect the transport by convection due to Stokes flow of the matrix fluid created by the motion of the needle.

The following system of partial differential equations in a domain Ω is a model to describe the mast cell response to acupuncture needling close to one acupoint :

                             ∂n g ∂t -D m ∇ 2 n g + ∇. (S n g ∇c) = -AΦn g + k r n d , t > 0, x ∈ Ω ∂c ∂t -D c ∇ 2 c = κ c AΦn g -δ c c, ∂n d ∂t -D m ∇ 2 n d = AΦn g -k r n d , ∂s n ∂t -D sn ∇ 2 s n = κ n AΦn g -δ sn s n , ∂s e ∂t -D se ∇ 2 s e = κ e AΦn g -δ se s e , (1)
with initial conditions

n g | t=0 = n 0 g , n d | t=0 = n 0 d , c| t=0 = c 0 , s n | t=0 = s 0 n , s e | t=0 = s 0 e .
All the parameters are constants. S > 0 is the sensitivity of the mast cells to the chemoattractant. A is the activation rate i.e., the rate of mass cells subjected to the mechanical stress Φ that will degranulate. D m , D c , D sn and D se are the diffusion coefficients. δ c , δ sn and δ se are the degradation rate. κ c , κ n and κ e are the release quantity coefficients of chemoattractant and stimulants. k r is the regeneration coefficient of degranulated mast cells.

This model differs from another chemotaxis model since the chemoattractant is only emitted close to the needle where the cells are mechanically stressed.

Blow-up in the case of only one state for mast cells

To verify our model (1), we show that blow-up of the density of mast cells occurs when the initial density of mast cells is higher at acupoint than those at nonacupoint locations like what Yu et al. observed in [START_REF] Yu | Role of Collagen Fibers in Acupuncture Analgesia Therapy on Rats[END_REF]. Blow-up corresponds to the aggregation of the mast cells close to the treated acupoint. To analyse mathematically the behaviour of our model, we first make some simplifications. From (1), we now consider the evolution of the granulated mast cell density n and the instantaneous diffusion of the chemoattractant c. We avoid the effects of boundary conditions and consider the system (2) in the full space R 2 . We also consider the stress function Φ to be equally distributed in a region close to the needle, in other words, for |x| ≤ λ, Φ(x) = 1. Those assumptions lead to the following system:

         ∂ t n -D m ∇ 2 n + ∇. (S n∇c) = -AΦ(x)n, t > 0, x ∈ R 2 , -∇ 2 c = κ c AΦ(x)n, n| t=0 = n 0 ≥ 0. (2)
Since the initial condition n 0 is nonnegative, the parabolic equation on n in (2) gives nonnegative solutions. Then, we quantify the spatial distribution of the cells by considering both of the total number of cells and the second moment

m 0 (t) := R 2 n(t, x)dx m 2 (t) := R 2 |x| 2 2 n(t, x)dx. ( 3
) Theorem 3.1 In R 2 , let n 0 ∈ L 1 + (R 2 , (1 + |x| 2 )dx).
Let n be a non-negative smooth solution of (2) and let [0, T * ) be the maximal interval of existence. Then, if the initial mass m 0 (0) is large enough and the second momentum m 2 (0) is small enough (to be precised in the proof ), the solution blows-up as t → T * .

Proof. The proof follows an argument previously introduced by Nagai in [START_REF] Nagai | Global Existence and Blow-up of Radial Solutions to a Parabolic-Elliptic System of Chemotaxis[END_REF], and used in [START_REF] Perthame | Transport Equations in Biology[END_REF] and [START_REF] Blanchet | Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions[END_REF]. First step: total mass. Taking the time derivative on (3) and using (2), we get

d dt m 0 (t) = R 2 -AΦ(x)n(t, x)dx ≥ -Am 0 (t). (4)
Then,

∀t ≤ T * , m 0 (0) ≥ m 0 (t) ≥ m 0 (0) e -AT * (5)
Second step: second moment. Taking the time derivative on (3) and using (2), we get

d dt m 2 (t) = R 2 |x| 2 2 D m ∇ 2 n(t, x) -∇. (Sn(t, x)∇c(t, x)) -AΦ(x)n(t, x) dx.
The second equation in [START_REF] Hecht | New Development in FreeFem++[END_REF] gives us ∇c(t, x) that we substitute back in the equation. From (5), we get the inequality

d dt m 2 (t) ≤ 2D m m 0 (t) 1 - ASκ c 8π m 0 (t) + 1 2π ASκ c R 2 R 2 x x -y |x -y| 2 (1 -Φ(y)) n(t, y)n(t, x)dydx.
We can assume that Φ in a k Φ -Lipchitz function and given |x| ≤ λ 2 and |y| > λ, then |x -y| ≥ λ 2 i.e.

d dt m 2 (t) ≤ 2D m m 0 (0) 1 - ASκ c 8π m 0 (0) e -AT * + 3 √ 2 2π ASκ c λ (m 2 (t)) 1 2 (m 0 (0)) 3 2 + 1 π ASκ c λ k Φ m 2 (t)m 0 (0).
Finally, following the arguments from [START_REF] Calvez | The Parabolic-Parabolic Keller-Segel Model in R2[END_REF], we have the following inequation

m 2 (t) ≤ m 2 (0) + t 0 f (m 2 (s))ds, (6)
where

f (ξ) = 2D m m 0 (0) 1 -ASκc 8π m 0 (0) e -LT * + 3 √ 2 2π ASκc λ ξ 1 2 m 0 (0) 3 2 + 1 π ASκc λ k Φ m 0 (0)ξ. f is a strictly increasing function and if m 0 (0) > 8π
ASκc e LT * , it exist a unique ξ * such that f (ξ * ) = 0. So, if m 2 (0) < ξ * , then f (m 2 (0)) < 0. Therefore, t 0 f (m 2 (s))ds < 0 and m 2 (t) ≤ m 2 (0) + tf (m 2 (0)) provided that (6) holds true. The second moment becomes nonpositive for T * ≥ -m2(0) f (m2(0)) which is impossible since n is nonnegative. Therefore, a singularity appears before that time and the solution n blows-up at the singularity point. This can only be appearance of a singular measure because these manipulations hold for correctly defined L 1 solutions (see [START_REF] Perthame | Transport Equations in Biology[END_REF]).

Existence in the case of only one state for mast cells

Considering the simplified model (2), we prove the following L p a priori estimate.

Theorem 4.1 In R 2 , let p > 1 and assume that n 0 ∈ L 1 + (R 2 , (1 + |x| 2 )dx).
There exists a constant α such that when the initial data satisfies m 0 (0) < 4α p A S κc , there exists a weak solution to (2) in L p (R 2 , dx) for all times.

Proof. We derive estimates based on the Sobolev inequalities following the argument in [START_REF] Jager | On Explosions of Solutions to a System of Partial Differential Equations Modelling Chemotaxis[END_REF].

Multiplying (2) by n p-1 and integrating, we get

1 p R 2 dn p dt = - R 2 4(p -1) p 2 |∇n p/2 | 2 + p -1 p R 2 A S κ c Φ(x)n p+1 - R 2 AΦ(x)n p .
To estimate the integral with power p + 1, we use the following Gagliardo-Nirenberg-Sobolev inequality

R 2 |n| p+1 ≤ C GN S R 2 |n| R 2 |∇n p/2 | 2 . (7)
Recalling that Φ(x) ≤ 1 and m 0 (t) ≤ m 0 (0),

1 p R 2 dn p dt ≤ p -1 p - 4 p + C GN S A S κ c m 0 (0) R 2 |∇n p/2 | 2 - R 2 AΦ(x)n p . (8)
Then, if m 0 (0) < 4 pC GN S A S κc , R 2 n p dx decays in time. From this a priori estimate, we may conclude the existence as done in [START_REF] Blanchet | Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions[END_REF].

Numerical tests

In this section, we will consider k r = 0 and thus focus on the two first equation of the system (1). Numerical simulations are carried out using the FreeFem++ software [START_REF] Hecht | New Development in FreeFem++[END_REF]. In the present study, we use one-degree polynomial (P 1 ) Lagrangian finite element type in space with a fully implicit Euler scheme in time. Let V h be the corresponding space of piecewise linear continuous functions. The formulation of the problem is the following: Find n m+1 g ∈ V h and c m+1 ∈ V h such that for all w 1 , w

2 ∈ V h Ω 1 dt n m+1 g -n m g w 1 + Ω D m ∇n m+1 g ∇w 1 - Ω S n m+1 g ∇c m ∇w 1 = - Ω AΦ(x)n m+1 g w 1 , Ω 1 dt c m+1 -c m w 2 + Ω D c ∇c m+1 ∇w 2 = Ω κ c AΦ(x)n m+1 g w 2 - Ω δ c cw 2 ,
At each step m, we have a non linear problem to solve. We use the fixed-point technique. To gain time, mesh adaptation (a subroutine of FreeFem++) fits the initial condition, i.e., a given cell distribution within the domain of interest, as the solution evolves locally. We apply Neumann boundary conditions (no cell and chemoattractant fluxes) at the border of the computational domain. According to the initial distribution of the mast cell distribution [Fig. 1 ] on the one hand and to the needle position with respect to the location of the peak cell density on the other, i.e., whether the

Conclusion

With the mathematical analysis of a simplified model we have shown a mechanism for blow-up in the chemotactic mechanism involved during acupuncture. A small second moment of the density of mast cells convey the idea that the cells are concentrated at the acupuncture points. Blow-up will occur when the initial number of mast cells is high enough and that they are sufficiently concentrated around the acupuncture points. Our model behaviour corresponds to the behaviour expected for the response of mast cells to acupuncture needling : a response of the mast cells exists when their density is low, for instance on a non-acupoint, but is greater when the initial density of mast cells is higher on a acupoint. In that case, the aggregation of mast cells could reveals the efficiency of the needle manipulation. We provide a numerical validation of the mathematical model of mast cell response to acupuncture needling.

1 |x-y| ≤ 2 λ

 12 . Moreover, (1 -Φ(y)) is bounded from above by 1 and (1 -Φ(y)) = 0 for |y| ≤ λ. This properties of the function Φ lead to

Figure 1 .

 1 Figure 1. Initial mast cell Gaussian distribution in an acupoint (concentrated distribution) and non-acupoint (dispersed distribution) with the same cell number.

Figure 2 .Figure 3 .

 23 Figure 2. Final distribution of mast cells at acupoint (left) and in a non-acupoint mast cell pool (right). Expected blow-up solution is only achieved at acupoint. Th0.01_case_1_n0_0.1_s_0_adapt_1_err_0.001_hmin_0.01_hmax_1_N_120.msh nnd0.01_case_1_n0_0.1_s_0_adapt_1_err_0.001_hmin_0.01_hmax_1_N_120.sol


Hajer Bahouri

Albert Cohen

Refined Sobolev inequalities in Lorentz spaces

We establish refined Sobolev inequalities between the Lorentz spaces and homogeneous Besov spaces. The sharpness of these inequalities is illustrated on several examples, in particular based on non-uniformly oscillating functions known as chirps. These results are also used to derive refined Hardy inequalities.

Introduction

The Sobolev inequality states that for any 0 < s < d 2 there exists a constant C > 0 such that for any function f defined on R d , f L p ≤ C f Ḣs , with 1 p := 1 2 -s d (therefore p is strictly between 2 and ∞). A refined version of this inequality was proved in [START_REF] Gérard | Inégalités de Sobolev précisées[END_REF] :

f L p ≤ C f 1-2s d Ḣs f 2s d Ḃs-d 2 ∞,∞ , where Ḃs-d 2
∞,∞ is the homogeneous Besov space. Similar results also hold when the smoothness of the function is measured in L q for 1 ≤ q < ∞: for any 0 < s < d q ,

f L p ≤ C f 1-qs d Ḃs q,q f qs d Ḃs-d q ∞,∞ , (1.1)
with 1 p := 1 q -s d , or the same kind of inequalities with Ẇ s,q in place of Ḃs q,q . We also refer to [START_REF] Cohen | Improved Sobolev inequalities[END_REF][START_REF] Cohen | Harmonic analysis of the space BV[END_REF][START_REF] Ledoux | On improved Sobolev embedding theorems[END_REF][START_REF] Martin | Sharp Gagliardo-Nirenberg inequalities via Symmetrization[END_REF][START_REF] Meyer | Oscillating Patterns in some Nonlinear Evolution Equations[END_REF] for similar results, involving in particular the space BV .

These inequalities are generalizations of the usual Sobolev embedding in Lebesgue spaces, their additional feature being that they are invariant under oscillations and fractal transforms see [START_REF] Bahouri | Refined Hardy inequalities[END_REF]. A significant application is the description of defect of compactness in Sobolev embedding (see [START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF] in the L 2 framework and [START_REF] Jaffard | Analysis of the lack of compactness in the critical Sobolev inbeddings[END_REF] in L p framework).

On the other hand, it is also known that Sobolev and Besov spaces also embed in Lorentz spaces L p,q (R d ) : with 1 p = 1 q -s d , one has f L p,q ≤ C f Ḃs q,q .

(1.2)

This may be derived from the standard inequalities in L p spaces by a real interpolation argument. This justifies looking for refined inequalities of the type (1.1) with L p,q (R d ) in place of L p (R d ).

In this paper, such inequalities will be established. The "price to pay" will be a slight modification in one index of the Besov space of negative smoothness, namely

f L p,q ≤ C f 1-qs d Ḃs q,q f qs d Ḃs-d q ∞,q , (1.3)
for the same range of indices (s, p, q) as in (1.1). It will be shown that this change is unavoidable and that such inequalities are sharp. The structure of the paper is the following. First, in §2, we briefly recall some facts on Lorentz and Besov spaces, we establish estimates for Besov spaces which are used in the sequel. In §3 we give the 1 proof of the refined Sobolev inequalities (1.3), which follows from the same line of reasoning as in [START_REF] Gérard | Inégalités de Sobolev précisées[END_REF] or [START_REF] Ledoux | On improved Sobolev embedding theorems[END_REF]. The sharpness of the inequality is discussed in §4. As with the improved Sobolev inequality (1.1), these inequalities remain sharp for oscillatory functions, but more interestingly the necessity of the above mentioned change of index can be proved by considering non-uniformly oscillating functions known as chirps. Such functions have been deeply studied by Meyer and Jaffard, see [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF][START_REF] Meyer | Oscillating Patterns in some Nonlinear Evolution Equations[END_REF] and arise naturally in signal processing. Finally, as a by-product of (1.3) we obtain in §5 a refined version of the Hardy inequalities in the L q framework, namely

|f (x)| q |x| sq 1 q ≤ C f 1-qs d Ḃs q,q f qs d Ḃs-d q ∞,q
.

(1.4)

A similar inequality was derived in [START_REF] Bahouri | Refined Hardy inequalities[END_REF] in the case q = 2. However, our method of proof is significantly simpler and applies to arbitrary values of q. Let us stress that Hardy inequalities are of important use in analysis: among other applications, we can mention blow-up methods or the study of pseudo-differentional operators with singular coefficients.

Acknowledgement: the authors are grateful to the anonymous reviewers for their constructive suggestions in improving the paper.

Lorentz spaces and Besov spaces

For any measurable function f on R d , we define its distribution and rearrangement functions

µ f (t) := |{x ; |f (x)| ≥ t} and f * (s) := inf{t ; µ f (t) ≤ s}.
For 1 ≤ p < ∞ and 1 ≤ q ≤ ∞, the Lorentz space L p,q (R d ) is defined as the set of all measurable functions f such that the function s 1 p f * (s) belongs to L q (R + , ds s ). The L p,q quasi-norm is defined by

f L p,q := +∞ 0 (s 1 p f * (s)) q ds s 1 q , (2.5)
with the standard modification when q = ∞ which corresponds to the weak L p space wL p = L p,∞ . It is well known that L p,p = L p and that Lorentz spaces can be derived from L p spaces by the real interpolation method. In particular, when 1 < p < ∞ we have L p,q = [L 1 , L ∞ ] θ,q with 1 p = 1 -θ. We refer to [START_REF] Bergh | Interpolation soaces. An introduction[END_REF][START_REF] Bennett | Interpolation of operators[END_REF] for these classical results.

In this paper, we shall use the expression of the L p,q quasi-norm in terms of the distribution function µ f , namely

f L p,q = p +∞ 0 (tµ f (t) 1 p ) q dt t 1 q . (2.6)
To derive this expression, one first writes

f * (s) q = q f * (s) 0 t q-1 dt, so that f q L p,q = +∞ 0 s q p -1 f * (s) q ds = q +∞ 0 s q p -1 [ f * (s) 0 t q-1 dt]ds = q +∞ 0 t q-1 [ µ f * (t) 0 s q p -1 ds]dt = p +∞ 0 t q-1 µ f * (t) q p dt.
Observing that µ f = µ f * we obtain (2.6).

There exists several equivalent definitions of Besov spaces either based on Littlewood-Paley decompositions, moduli of smoothness, approximation procedures, wavelet decompositions. We refer to [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF] and [START_REF] Devore | Constructive Approximation[END_REF] for a general treatment, and only recall here the definition based on Littlewood-Paley theory.

Let ϕ be a function of d variables such that its Fourier transform φ is non-negative, C ∞ , supported in the ball {|ω| ≤ M } for some M > 1 and such that φ(ω) = 1 if |ω| ≤ 1. For j ∈ Z we define the operator S j acting on tempered distributions of d variables by

S j f = 2 dj ϕ(2 j •) * f and ∆ j = S j+1 -S j or equivalently ∆ j f = 2 dj ψ(2 j •) * f, with ψ(ω) = φ( ω 2 ) -φ(ω).
For s < d p and q ≥ 1, we define the homogeneous Besov space Ḃs p,q (R d ) as the space of tempered distributions f such that J j=-J ∆ j f converges towards f in S ′ as J → +∞ and such that f Ḃs

p,q := (2 sj ∆ j f L p ) j∈Z ℓ q
is finite. It is known and not difficult to check that this definition is independent of the initial choice of the function ϕ with the above mentioned property, the resulting norms being equivalent. The condition that s < d p is necessary to ensure the fact that we have defined a proper Banach space (otherwise convergence should be interpreted in S ′ /Π n where Π n is the space of polynomials of total degree n = ⌊s -d p ⌋, for more details see [8] and the discussion therein p. 153-155).

We shall consider Besov spaces of regularity index s which is either positive or negative. Using discrete Hardy inequalities, it is also possible to characterize these spaces using only the operator S j : for s > 0, we have

f Ḃs p,q ∼ (2 sj (I -S j )f L p ) j∈Z ℓ q ,
and for s < 0, f Ḃs p,q ∼ (2 sj S j f L p ) j∈Z ℓ q . Equivalent norms for Besov spaces can also be obtained with the discrete frequencies 2 j replaced by a continuous one: introducing for any λ > 0 the operators

S λ f = λ d ϕ(λ•) * f and ∆ λ f = S 2λ f -S λ f = λ d ψ(λ•) * f,
we then find that f Ḃs p,q is equivalent to the norm of the function

F (λ) := λ s ∆ λ f L p (2.7)
in L q (R + , dt t ). By similar arguments as when working with discrete frequency, we can equivalently choose

F (λ) := λ s (I -S λ )f L p (2.8) for s > 0 or F (λ) := λ s S λ f L p , (2.9)
for s < 0. For the purpose of our analysis, we shall also need estimates on the derivative F ′ (λ) = dF dλ , as expressed by the following result.

Lemma 2.1 There exists a constant K which depends only on s, d, p, q and the choice of ϕ such that for all f ∈ Ḃs p,q , the function

G(λ) := λF ′ (λ) belongs to L q (R + , dt t ) with G L q (R+, dt t ) ≤ K f Ḃs p,q , (2.10)
where F is defined by (2.7), or by (2.8) when s > 0, or by (2.9) when s < 0.

Proof: we only consider the case where F is defined by (2.7), the two others being treated in the same way. We first estimate the derivative of the function g(λ) := ∆ λ f L p . Clearly, we have

|g ′ (λ)| ≤ d dλ (λ d ψ(λ•)) * f L p = λ d-1 ψ(λ•) * f + dλ d-1 ψ(λ•) * f L p ,
with ψ(x) = x • ∇ψ(x). Since we have

F ′ (λ) = sλ s-1 g(λ) + λ s g ′ (λ), it follows that |G(λ)| ≤ (s + d)F (λ) + λ s λ d ψ(λ•) * f L p := (s + d)F (λ) + F (λ)
The function ψ satisfies ψ(ω

) = φ( ω 2 ) -φ(ω) with φ(ω) := φ(ω) + ω • ∇ φ(ω).
Since φ is C ∞ , compactly supported in the ball {|ω| ≤ M }, and such that φ(ω) = 1 if |ω| ≤ 1, we obtain from the definition of Besov spaces that

G L q (R+, dt t ) ≤ (s + d) F L q (R+, dt t ) + F L q (R+, dt t ) ≤ K f Ḃs p,q ,
for some K which depends on (s, d, p, q, ϕ). ⋄

Proof of the refined inequality

We first observe that we may prove (1.3) in the particulary case where f is a C ∞ compactly supported function, and derive by density the general validity of the inequality.

We decompose f into low and high frequencies according to

f = S λ f + (I -S λ )f
where S λ is the convolution operator defined in the previous section. From the characterization of Besov spaces which was recalled in §2.2, we have

∞ 0 [F (λ)] q dλ λ ≤ C 0 f q Ḃs-d q ∞,q
where C 0 is an absolute constant and

F (λ) := λ s-d q S λ f L ∞ .
We also have

+∞ 0 λ qs (I -S λ )f q L q dλ λ ≤ C 1 f q Ḃs q,q
, where C 1 is also an absolute constant. We now consider the function λ → t(λ) defined by

t(λ) = 2λ d p F (λ) = 2λ s+ d p -d q S λ f L ∞ = 2 S λ f L ∞ .
This function may not be monotone but it tends towards 0 as λ → 0 and towards

M := 2 f L ∞ ≥ f L ∞ as λ → +∞.
We may thus use it as a change of variable in the expression (2.6) of the L p,q quasi-norm, which yields

f q L p,q = p +∞ 0 t q µ f (t) q p dt = p 2M 0 t q µ f (t) q p dt = p +∞ 0 t(λ) q-1 t ′ (λ) µ f (t(λ)) q p dλ. (3.11)
From the triangle inequality we have

x ; |f (x)| > t ⊂ x ; |S λ f (x)| > t 2 ∪ x ; |(I -S λ )f (x)| > t 2 Since we have |S λ f (x)| ≤ t(λ) 2 for all x, it follows that {x ; |f (x)| > t(λ)} ⊂ {x ; |(I -S λ )f | > t(λ) 2 }, i.e. µ f (t(λ)) ≤ µ (I-S λ )f ( t(λ)
2 ). Combining this observation together with (3.11), we obtain

f q L p,q ≤ p +∞ 0 t(λ) q-1 t ′ (λ)[µ (I-S λ )f ( t(λ) 2 )] q p dλ = 2 q d +∞ 0 [λ d p F (λ)] q-1 (λ d p -1 F (λ))[µ (I-S λ )f ( t(λ)
2 )]

q p dλ + 2 q p +∞ 0 [λ d p F (λ)] q-1 (λ d p F ′ (λ))[µ (I-S λ )f ( t(λ) 2 )] q p dλ = 2 q d +∞ 0 λ dq p -1 F (λ) q [µ (I-S λ )f ( t(λ)
2 )]

q p dλ + 2 q p +∞ 0 λ dq p F (λ) q-1 F ′ (λ)[µ (I-S λ )f ( t(λ)
2 )] q p dλ := T 0 + T 1

In order to estimate further the two above terms, we remark that

µ (I-S λ )f ( t(λ) 2 ) ≤ ( t(λ) 2 ) -q (I -S λ )f q L q = (λ d p F (λ)) -q (I -S λ )f q L q .
For T 0 , this gives

T 0 ≤ 2 q d +∞ 0 λ dq p -1 F (λ) q [(λ d p F (λ)) -q (I -S λ )f q L q ] q p dλ = 2 q d +∞ 0 F (λ) q(1-q p ) [λ d-dq p (I -S λ )f q L q ] q p dλ λ = 2 q d +∞ 0 F (λ) q(1-q p ) [λ qs (I -S λ )f q L q ] q p dλ λ .
Applying Hölder inequality, we thus obtain

T 0 ≤ 2 q d +∞ 0 F (λ) q dλ λ 1-q p +∞ 0 λ qs (I -S λ )f q L q dλ λ q p ≤ 2 q d +∞ 0 F (λ) q dλ λ qs d +∞ 0 λ qs (I -S λ )f q L q dλ λ 1-qs d ≤ C f qs d Ḃs-d q ∞,q f 1-qs d Ḃs q,q q .
where C is an absolute constant. Proceeding in an exactly similar manner for T 1 , we obtain

T 1 ≤ 2 q p +∞ 0 F (λ) q-1 λF ′ (λ) dλ λ qs d +∞ 0 λ qs (I -S λ )f q L q dλ λ 1-qs d .
Using Lemma 2.1 together with a Hölder inequality, we see that the integral in the first factor can again be controlled by f q Ḃs-d q ∞,q

, and therefore we also obtain

T 1 ≤ C f qs d Ḃs-d q ∞,q f 1-qs d Ḃs q,q q ,
where C is an absolute constant. Combining our estimates for T 0 and T 1 we have proved that

f q L p,q ≤ C f qs d Ḃs-d q ∞,q f 1-qs d Ḃs q,q q ,
which is the desired result.

Remark 3.1 For our range of indices (s, p, q), we also have a continuous embedding of Ḃs q,r into L p,r for any 1 ≤ r ≤ ∞ (this can easily be proved from the standard Sobolev embedding by a real interpolation argument). This suggests a correponding improved inequality of the form

f L p,r ≤ C f 1-qs d Ḃs q,r f qs d Ḃs-d q ∞,r
, but the proof of this does not seem obvious to us, neither by interpolation arguments, nor by an adaptation of our current proof (that corresponds to the case r = q).

Sharpness of the inequality

We shall discuss the sharpness of (1.3) through two examples. The first one shows that the refined estimates are sharp with respect to oscillating functions: consider for all positive ω, the function

ϕ ω (x) = e iωx1 ϕ(x),
where ϕ is a fixed function in S(R d ). For such a function, ϕ ω L p,q is independent of ω since the L p,q quasi-norm only depends on the modulus. On the other hand for |ω| > 1,

ϕ ω Ḃs q,q ∼ ω s .
Therefore, the Sobolev inequality (1.2) is not sharp when ω is large. It is also easily checked that for

|ω| > 1 ϕ ω Ḃs-d q ∞,q ∼ ω s-d q ,
(see [START_REF] Bahouri | Refined Hardy inequalities[END_REF] for the proof of these equivalences) and therefore

ϕ ω 1-qs d Ḃs q,q ϕ ω qs d Ḃs-d/q ∞,q ∼ ω s(1-qs d )+(s-d q ) qs d = 1,
which shows that (1.3) is sharp for such oscillatory functions. The defect of this first example is that it does not make much distinction between (1.3) and the improved Sobolev inequality (1.1) in L p spaces : indeed, we also have that ϕ ω L p is independent of ω and that ϕ ω Ḃs-d q ∞,∞ ∼ ω s-d q . In particular, this example does not reveal the necessity of using Ḃs-d q ∞,q in place of Ḃs-d q ∞,∞ when we want to control the L p,q quasi-norm. A finer example is needed order to show that this cannot be avoided. This example should contain oscillations in a wide range of frequencies in order to make the distinction between Ḃs-d q ∞,q and Ḃs-d q ∞,∞ , whereas it should also contain a wide range of magnitudes in order to distinguish between L p and L p,q . Our example has roughly the form of a hyperbolic "chirp"

f (x) = x -α sin( 1 x ),
with α > 0. For the sake of simplicity, we have placed ourselves in the one-dimensional case d = 1 although this example can be generalized to arbitrary dimensions. In order to simplify our analysis, we shall define a variant of f in the form of a wavelet series. We recall that the Haar system is defined by the functions

ψ j,k (x) = 2 j 2 ψ(2 j x -k), j, k ∈ Z, where ψ := χ [0, 1 2 [ -χ [ 1 2 ,1]
. Besov spaces Ḃs p,q can be characterized as follows by wavelets (see e.g. [START_REF] Meyer | Ondelettes et opérateurs[END_REF] or [START_REF] Cohen | Numerical analysis of wavelet methods[END_REF]): if f = j,k d j,k ψ j,k , then

f Ḃs p,q ∼ 2 (s+ 1 2 -1 p )j (d j,k ) k∈Z ℓ p j∈Z ℓ q .
In the case of the Haar system, this characterisation holds if 1 p -1 < s < 1 p (due to the lack of smoothness of the functions ψ j,k ) which is sufficient for our purpose.

Let q ≥ 1 and 0 < s < 1 q , and let 1 p := 1 q -s. For J > 0, we consider the function

f J (x) = J j=1 2 ( 1 p - 1
2 )j ψ j,1 (x) = J j=1 2 j p ψ(2 j x -1). (4.12)

Note that the wavelets in the expansion of f J have disjoint supports. We first notice that we have

f J Ḃs q,q ∼ J j=1 2 (qs+ q 2 -1)j 2 ( q p -q 2 )j 1 q = J 1 q .
For the Besov norms of negative indices, we have

f J Ḃs-1 q ∞,∞ ∼ J sup j=1 2 (s-1 q + 1 2 )j 2 ( 1 p -1 2 )j = 1, and f J Ḃs-1 q ∞,q ∼ J j=1 [2 (s-1 q + 1 2 )j 2 ( 1 p -1 2 )j ] q 1 q = J 1 q .
Since the wavelets in the expansion of f J have disjoint supports, the L p norm is of the order

f J L p = J j=1 2 j p ψ(2 j • -1) p L p 1 p ∼ J 1/p .
Remarking that the rearrangement function of f J is simply given by

f * J (s) = |f J (s + 2 -J )|,
we can also easily evaluate its L p,q quasi-norm using (2.5) which gives

f J L p,q = J j=1 2 jq p 2 -j 0 (s + 2 -j -2 -J ) q p -1 ds 1 q ∼ J j=1 2 jq p 2 -j 2 -j( q p -1) 1 q = J 1 q .
From all these estimates, we obtain that

f J 1-qs Ḃs q,q f J qs Ḃs-1 q ∞,∞ ∼ J 1 q -s = J 1 p ∼ f J L p , and f J 1-qs Ḃs q,q f J qs Ḃs-1 q ∞,q ∼ J 1 q -s J s = J 1 q ∼ f J L p,q .
Again (1.3) and (1.1) are sharp, but this example also shows that the use of Ḃs-d q ∞,q in place of Ḃs-d q ∞,∞ is unavoidable for the validity of (1.3).

Refined Hardy inequalities

Let as in the previous sections 1 < p < ∞ and s ∈]0, d q [, with 1 p = 1 q -s d . For such indices, we claim that the refined Hardy inequality (1.4) holds. It is actually a direct consequence of (1.3) combined with a generalized version of Hölder's inequality known as O'Neil inequalities : if 1 ≤ p 1 , p 2 , q 1 , q 2 ≤ ∞, then for any f ∈ L p1,q1 (R d ) and g ∈ L p2,q2 (R d ),

f g L p,q (R d ) ≤ C f L p 1 ,q 1 (R d ) g L p 2 ,q 2 (R d ) , (5.13)
where 1 p = 1 p1 + 1 p2 and 1 q = 1 q1 + 1 q2 and C = C(p 1 , p 2 , q 1 , q 2 ) (a first proof of these inequalities using rearrangements can be found in [START_REF] Neil | Convolution operators on L(p, q) spaces[END_REF], while a more modern proof would proceed by bilinear interpolation between the usual inequalities for Lebesgue spaces).

We now take g(x) = 1 |x| s and apply (5.13), in the specific form

f g L q,q (R d ) ≤ C f L p,q (R d ) g L r,∞ (R d )
where 1 q = 1 p + 1 r , i.e. r = d s . Since obviously g ∈ L r,∞ (R d ), we have |f (x)| q |x| sq 1 q ≤ C f L p,q (R d ) .

Combining this with (1.3), we obtain (1.4). Let us mention that this inequality is obtained in [START_REF] Bahouri | Refined Hardy inequalities[END_REF] for q = 2 by a different method which can not be generalized to the general L q framework. Concerning the sharpness of (1.4), we can make similar observations as for (1.3). In particular, this inequality is sharp for oscillatory functions of the type ϕ ω (x) = e iωx1 ϕ(x). It is also easily checked that the sequence f J defined by (4.12) satisfies |f J (x)| q |x| sq 1 q ∼ J 1 q ∼ f J L p,q (R d ) , making (1.4) sharp for such functions, and justifying the use of Ḃs-d q ∞,q in place of Ḃs-d q ∞,∞ .


Sophie David

Tge Adonis

Documenting some Uses of the Isidore Platform

The availability of millions of digital resources, and of the means for processing, analysing, signalling, and exposing them, should make it possible to facilitate the emergence of new research objects and new questions. After a brief description of Isidore, this paper proposes first results in observing the Isidore platform because it presents all the basic materials a researcher needs: publications, blogs, websites and conference calendars. Then we discuss how facets proposed by Isidore are currently being used and finally we propose some issues which are being studied.

1. What is Isidore?

Introduction

It is increasingly recognized that Digital Humanities will change researchers' practices (ThatCamp 2010[START_REF] Juanals | De l'émergence de nouvelles technologies intellectuelles[END_REF][START_REF] Vinck | La gestion de la recherche. Nouveaux problèmes[END_REF]. The availability of millions of digital resources, and of the means for processing, analysing, signalling, and exposing them, should make it possible to deal with old problems in new ways, or facilitate the emergence of new research objects and new questions, in particular because of interdisciplinary exchange and the creation of novel web-enabled links, etc. which were initially inconceivable (this is for instance the position of the Dariah project). Nevertheless, it remains difficult to precisely objectivise these expected benefits. It is true that it is still early days. It is also true that only a few surveys have documented and analysed these new or innovative usages [START_REF] Garron | Formes de lecture et usages du Web[END_REF][START_REF] Gallezo | De la numérisation des revues à l'expérimentation d'une édition de recherche processuelle[END_REF]. Documenting and analysing usages requires setting up methods of observation (Le Marec 2001). Several are possible, depending partially on the chosen "object", partially on the observation framework (evaluation, documentation, etc.), and above all, on how the objects to be evaluated are considered [START_REF] Amar | Classification Procedures for Software Evaluation[END_REF]. We are interested in observing the Isidore system (http://www.rechercheisidore.fr/apropos), because it presents, on a single platform, all the basic materials a researcher needs: publications (to determine the state of the art, to access documents, etc.); blogs (to follow research in the making, to track research developments, etc.); websites (to access results, to belong to a community, etc.); and conference calendars (to monitor the state of the art, to announce results, etc.). We propose to analyse different uses of the platform, basing the analysis on elements provided by Google Analytics. Three immediate interests concerning Isidore itself will be addressed: a) to provide some initial answers to a set of precise questions about the use of Isidore; b) to identify the elements which could lead to improvements in the platform; c) to identify the elements which could improve observation of how the platform is used. There is a fourth, additional interest: from the collected documentation, to begin to develop the problematics and methods to observe "composites"1 (Le Marec 2001), i.e. to carry out a survey of the usage of Isidore. We will first give the key elements for understanding the operating model of Isidore. Second, we formulate the sets of questions we seek to answer; we then briefly describe the corpus of statistics on which our analyses are based. Lastly, we present our results and analyses.

The Isidore platform (Isidore 2011) is both a technical search engine and a new approach to designing a flexible workstation for targeted end-users, namely researchers, students, engineers, and librarians working in the Humanities. Isidore is based on radically different principles to Google, Bing, etc. First, the crawled sources were carefully chosen after a preliminary study of end-users' needs. Consequently, sources come from different categories of web pages: journals, blogs, RSS feeds and announcements of scientific events. Secondly, a panel of thesauri and list of references (produced by the scientific community) were selected in order to enhance the indexing process. Thirdly, metadata are the cornerstone of the whole process, from crawling to the user interfaces. Isidore is powered by different modules which exchange information, metadata and data, connected by the web of data and linked data methods, and stored in an RDF triple-store. Isidore, as a semantic web system, also offers a SPARQL endpoint. With regard to the question of user interfaces, results are displayed without any ranking order but several selection filters (or facets), around 10, are available. The choice to use facets rather others query refinement systems is based on several studies [START_REF] Yee | Faceted metadata for image search and browsing[END_REF][START_REF] Dakka | Automatic Extraction of Useful Facet Hierarchies from Text Database[END_REF] which have shown how they upgraded the relevance of the query. These facets enable the end-user to navigate through the results in order to find relevant sources. Furthermore, by analysing results and data linking (Isidore uses linked data methods) some suggestions are put forward.

Questions, Results and Analyses

Our aim is to answer four sets of questions (only three of which will be tackled here2 ): -Do users formulate the different types of queries which are available? -Do users use the facet system? To what extent? What is the distribution of facets? -Do users browse on thanks to the enrichment provided by hints? To what extent? Do they use generalist thesauri or specialized ones? The statistics used here come from Google Analytics. The period observed is the third week in July 20113 . We have analysed 2385 queries, which represents 3377 pages and 2702 unique visits. In quantitative terms, that represents a "weak" week of activity (when compared with the Piwik results).

Four Types of Use

Four specific uses can be identified: By 'direct query' (a) we mean a query where the user writes an expression in the dedicated box on the home page; by 'more options query' (b), a query where the user writes the query using the more options possibilities (one or several options): author, title, term, before, after; by 'facet exploration' (c), a query where the user explores the resources with the facets but does not write any expression; by 'hints query' (d), a query where the user, after a query of type (a) or (b), uses the enrichment to continue browsing. Almost 70% of users write expressions, which is not very surprising. Queries of type (c) and (d) account for more than 30%, equally divided, and are therefore significant from a quantitative point of view. It thus seems relevant to examine the facet system and the use of enrichment in greater detail.

Using Facets

Facets are used to explore the resources (cf. the facet exploration). But they can also be used to restrict or to shift a query scope. Overall, 26.2% of the queries are constrained by one or several facets (i.e. 630 queries out of 2385, which represents 941 facets; up to 5 different facets can be used in the same query). It is also interesting to cross-link these constraints with regard to the types of queries identified above: Very strong contrasts can be observed: facets are used a) because the query is not at first contextualized (cf. direct query versus more options query, in which users can type more items of information (author, title, etc.)), or b) because exploration has been chosen (facet exploration). A little more surprising is the low rate of hints queries (which should resemble the direct query rate, because a hints query is like a new query). No doubt the fact that the hints query is not immediately available, requiring formulation of a query, access to one specific resource, access to this service (which presupposes that the need is real and the service is well understood), could explain this result. However, another explanation (apart from the hypothesis of a bias, which is also possible) can also be given. The hints query is used as another way to restrict the number of results; if the meaning of the chosen term is sufficiently specified in the context, there would be no need to restrict the search even further4 . The distribution of the facets is the following: It can be seen that there is a great difference in the use of the two thesauri. Two explanations can be suggested: the version of Pactols used in Isidore contains approximately 60,000 terms, while Rameau contains approximately 200,000 terms. Rameau is much more heavily used simply because it contains far more terms. Second, Rameau is a generic thesaurus, while Pactols is archaeologyand history-oriented.

Conclusion

These different observations are the first step in the analysis of the use of Isidore. We can confirm that the four types of queries are used, as well as the facet system and the hints system. However, it appears that having access to the 'routes' chosen by the users would be a fruitful way to improve our analyses: we could produce a better analysis of the enrichment; we could have a better idea of

Direct query More options query Facet exploration Hints query Total Number

		922	732	372	359	2385
	%	38.7	30.7	15.6	15.1	100

Direct query More options query Facet exploration Hints query Constrained by at least 1 facet

		341	34	232	23
	% (number of queries = 2385)	14.3	1.43	9.73	0.96
	% (number of queries in each query type)	36.98	4.64	62.37	6.41

  Pactols and Geonames. At the moment, users can use hints query only with Rameau and Pactols. The results are displayed below:

		Type of documents	Discipline Century Language	Historical period	Categories	Collection Institution and	Others Total
	Number	270	214	185	78		64	50	54	26	941
	%	28.69	22.74	19.66	8.29		6,80	5,31	5,74	2.76	100
	2.3. Enrichment
	Three thesauri are used: Rameau, Rameau	Pactols	Total queries 5
			Number	345	15		359

Composites means "work situations, in which people mobilize material objects and representations […], carry out actions […], and operate norm systems or operating rules" (Le Marec

2001: 187). Field surveys are therefore favored.

It seems relevant to us to analyze the form of the queries from a linguistic point of view (simple terms versus complex terms, expressions, sentences, grammatical categories, proper names, etc.). We can then assess the possible influence of a Google search. This point is important, because it outlines a specific appropriation by users, and determines possible corrective actions, additional explanations, etc. We have used Piwik, an open source service, which was put in place on the beta version of Isidore; we have statistics for more than 6 months of use. Results will be given in the final version, if this paper is accepted.

In the final version, we will extend the period in order to get statistics on several weeks and to check the representativeness of our results.

Furthermore the user knows in advance how many resources are tagged with this term. Of course, with this method of observation, we can not know how many trials have been done before a click.

One user formulated a query using both thesauri. That is why the number of queries is 359 and not 360.

4

other uses of the facet system or the hints system, which should perhaps not be analysed only as restrictions. It would also enable us to answer other types of questions: are there exclusive uses of Isidore? For example are there users who are just interested in calendars, just in blogs, or just in PhDs? Or can types of query chaining be identified? In any case, we need better tools to answer more sophisticated questions.