Marie-Christine Chevalier

Peter Matthews

Gaëtan Hanen

Patrick Jumin

Deborah Stubbs

Bernard Roure

Evaluation of a procedure for positioning a crash test dummy in a railway

Keywords: rail, passive safety, crash test dummy, H-point, physical testing, virtual modelling

This technical note focuses on the evaluation of a procedure for positioning crash test dummy in rail environment before physical testing. Through physical implementation, a simple approach consisting in adjusting the location of the dummy on the seat was evaluated and validated. The data collected was used to accurately align a virtual dummy with real life information demonstrating that the procedure, coupled with virtual modelling, provides a powerful research tool by which to improve the passive safety of passengers travelling in rail vehicles.

BACKGROUND AND OBJECTIVES

In case of a crash, railway occupants may be subjected to sudden displacements of large amplitude leading to violent secondary contacts with various surfaces. The resulting injury risk is usually evaluated by reproducing the event in test facility: environment of the railway passengers is reconstructed on a test sled on which appropriate crash-test dummies are seated and submitted to a deceleration corresponding to the crash scenario. The mechanical response of the dummies allows an estimation of the injury risks.

For research purposes, the dummies have to be installed in a precise, recordable, repeatable and reproducible manner. RSSB developed a protocol to physically position dummies in a seated position prior to dynamic testing [START_REF] Matthews | Protocol for the Installation of ATD's for Crash Testing[END_REF]. The procedure was Manuscript implemented during the European SAFEINTERIORS project -focusing on the link between interior design for railway vehicles and the passengers' safety -in order to:

-evaluate, improve and validate the protocol; -give additional guidance regarding information to be measured and recorded; -transfer relevant data to a virtual modelling environment and assist validation allowing in-depth study while reducing number and costs of physical testing

TEST METHOD AND MATERIAL

Procedure

The procedure describes how to define a reference position of the occupant on a rail seat.

At first the position of the H-point (the midpoint between the hips) for a fiftieth percentile occupant relatively to the considered seat is defined. The devoted tool, an Hpoint machine (figure 1), is installed on the seat as required in the SAE J826 automotive procedure [START_REF]SAE Human Accommodation and Design Devices Standards Committee, Devices for Use in Defining and Measuring Vehicle Seating Accommodation[END_REF]. The position of this H-point is measured in a reference frame by using physical points of the seat. This data can be used for test repeatability and transferability to the virtual environment.

The second step consists in installing the crash test dummy so that its H-point matches, within a predefined tolerance, the location obtained here above. For this, the dummy is installed with its median line corresponding to the midline of the seat. Its legs are extended horizontally and pushed so that the dummy makes contact with the seat back.

In the absence of a footrest, the heels are placed on the floor; otherwise the insteps of the shoes are placed on the footrest. Afterwards, a flat rigid surface is pressed horizontally against the dummy's torso at a load of 250N. Then, the elbows are placed on the armrest with hands resting on the thighs whilst the feet are adjusted so that they have the same longitudinal coordinates. Finally, the knees are separated by 170mm, keeping the knees and ankles in the same vertical plane.

Then, the positioning of the dummy has to be validated: its H-point position is measured and compared to the value obtained in the first step. The procedure is repeated if deviation is outside a tolerance envelope of +/-20mm in the longitudinal and vertical directions. If not possible, the dummy is seated as close as possible to this envelope and its position recorded.

As a final step, all information relative to the dummy's posture and test arrangement (seat geometry, seat and table juxtaposition…) may be recorded with coordinate measurement equipment. Additional information may also be derived by data streaming this equipment over the required surfaces

Physical implementation

The procedure was implemented at Ifsttar by installing a regulatory frontal crash-test dummy (Hybrid III) in a SIRIUS Compin train seat. Measurements were taken using a FARO arm. The reference frame was the anchorage plate of the seat on the floor. The fixing bolt location of the armrest was also used, as it is easily identifiable in physical and virtual environments, and its position recorded. Each procedure (SAE J826 and positioning of the dummy with the feet on the floor) was repeated twice.

Numerical implementation

The seat was numerically reproduced at SNCF through a calibrated finite-element model (Altair HyperCrash® environment). A numerical model of the Hybrid III dummy was provided by Altair. Numerical implementation was carried out by adjusting the position of the seat and the posture of the dummy using the reference points and scanned envelopes recorded during physical positioning.

RESULTS

The H-point machine positioned on the seat is shown in figure 2.

Measurements with H-point manikin and Hybrid III are given in Table 1. Positioning of the numerical dummy with reference points and scanned lines is shown in Figure 4.

DISCUSSION

The feasibility of the procedure for installing a physical dummy on a seat was validated. This was very repeatable and precise in terms of H-point position: a difference less than 1 mm in both longitudinal and vertical directions was found when the procedure was applied twice. The H-point position of the dummy compared to the reference value (measured with the H-point machine) was shown to be within the tolerance box +/-20mm.

The arm positions has a great influence on the final kinematics and hence on injuries. Keeping the elbow on the armrest with the lower arm on the thigh was considered to be the most realistic position.

It was found useful to reference a set of common physical datum points on the seat in order to enable transferability between physical and virtual environments.

The data collected validates the initial positioning of the numerical dummy, improving correlation between virtual predictions and physical tests. However particular care is required on contact interfaces to avoid initial penetrations, edge intersections and numerical errors, especially between thighbones and pelvis foam. Thigh positioning could be achieved by a pre-calculation activating contact interfaces, including distortion and stress of the pelvis foam.

Limitations of this study concern the low flexibility of the dummy and access to it on the seat with coordinate measurement equipment. In trains, seats are often installed sideby-side or facing other seats or tables, complicating set-up of the dummy and associated measurements. A careful sequential assembly of the test scenario should aid the application of this procedure. This paper focuses on the Hybrid III dummy, the positioning procedure can also be applied to the RSSB developed Hybrid III RS dummy, specifically built for testing in the rail environment [START_REF] Parent | Evaluating Abdominal Injury in Impacts with Workstation Tables[END_REF].

The proposed protocol increases repeatability, reproducibility and precision of crash testing, making easier comparison of test results between test houses and going beyond physical testing to numerical simulation. It could be relevant for designers of rail transportation or be used as an input for future recommendations in railway standards relative to the interior passive safety.

Figure 3

 3 Figure3represents the final posture of the dummy after positioning with the feet on the floor. Positions of specific points and envelopes of the head, torso, upper and lower leg were recorded.

Table 1 : Comparison of H-point position measured after SAE J826 procedure and positioning of the dummy H-Point Procedure Longitudinal (X) mm

 1

	Vertical (Z) (mm)	Mean X	Mean Z

ACKNOWLEDGMENTS

Our most sincere thanks go to our partners of the SAFEINTERIORS project and to the UNEX staff at Ifsttar for their valuable contribution.

FUNDING

This work was supported by the European Community in the framework of the SAFEINTERIORS project [contract No FP6-031260] with the partners UNewcastle, Alstom, ATOC, BT, CIDAUT, DB, Ifsttar, IST, MIRA, RSSB, Siemens, SNCF, UNIFE and VUKV.

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Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Gaspard Jankowiak
email: jankowiak@ceremade.dauphine.fr

Keywords: Sobolev spaces, Sobolev inequality, Hardy-Littlewood-Sobolev inequality, logarithmic Hardy-Littlewood-Sobolev inequality, Onofri's inequality, Caffarelli-Kohn-Nirenberg inequalities, extremal functions, duality, best constants, stereographic projection, fast diffusion equation 2010 MSC: 26D10, 46E35, 35K55

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy-Littlewood-Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin-Talenti functions.

1. Introduction E. Carlen, J.A. Carrillo and M. Loss noticed in [START_REF] Carlen | Hardy-Littlewood-Sobolev inequalities via fast diffusion flows[END_REF] that Hardy-Littlewood-Sobolev inequalities in dimension d ≥ 3 can be deduced from some special Gagliardo-Nirenberg inequalities using a fast diffusion equation. Sobolev's inequalities and Hardy-Littlewood-Sobolev inequalities are dual. A fundamental reference for this issue is E.H. Lieb's paper [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF]. This duality has also been investigated using a fast diffusion flow in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF]. Although [START_REF] Carlen | Hardy-Littlewood-Sobolev inequalities via fast diffusion flows[END_REF] has motivated [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF], the two approaches are so far unrelated. Actually [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] is closely connected with the approach by Legendre's duality developed in [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF]. We shall take advantage of this fact in the present paper and also use of the flow introduced in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF].

For any d ≥ 3, the space D 1,2 (R d ) is defined as the completion of smooth solutions with compact support w.r.t. the norm

w → w := ∇w 2 L 2 (R d ) + w 2 L 2 * (R d ) 1/2
, where 2 * := 2 d d-2 . The Sobolev inequality in R d is

S d ∇u 2 L 2 (R d ) -u 2 L 2 * (R d ) ≥ 0 ∀ u ∈ D 1,2 (R d ) , (1)
where the best constant, or Aubin-Talenti constant, is given by

S d = 1 π d (d -2) Γ(d) Γ( d 2 ) 2 d
(see Appendix A for details). The optimal Hardy-Littlewood-Sobolev inequality

S d v 2 L 2 d d+2 (R d ) - R d v (-∆) -1 v dx ≥ 0 ∀ v ∈ L 2 d d+2 (R d ) (2)
involves the same best constant S d , as a result of the duality method of [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF]. When d ≥ 5, using a well chosen flow, it has been established in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] that the l.h.s. in [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF] is actually bounded from below by the l.h.s. in [START_REF] Almgren | Symmetric rearrangement is sometimes continuous[END_REF], multiplied by some positive proportionality constant. In our first result, we will remove the technical restriction d ≥ 5 and cover all dimensions d ≥ 3. An elementary use of the duality method -in fact a simple completion of the square methodprovides a simple upper bound on the optimal proportionality constant in any dimension.

Theorem 1. For any d ≥ 3, if q = d+2 d-2 the inequality

S d u q 2 L 2 d d+2 (R d ) - R d u q (-∆) -1 u q dx ≤ C d u
Inequality (3) is obtained with C d replaced by S d by expanding a well chosen square in Section 2. The lower bound on C d follows from an expansion of both sides of the inequality around the Aubin-Talenti functions, which are optimal for Sobolev and Hardy-Littlewood-Sobolev inequalities (see Section 2 for more details), and spectral estimates that will be studied in Section 3: see Corollary 6. The computation based on the flow as was done in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] can be optimized to get an improved inequality compared to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF], far from the Aubin-Talenti functions: see Theorem 9 in Section 4. As a consequence, we also prove the strict inequality C d < S d .

In dimension d = 2, consider the probability measure dµ defined by dµ(x) := µ(x) dx with µ(x)

:= 1 π (1 + |x| 2 ) 2 ∀ x ∈ R 2 .
The Euclidean version of Onofri's inequality [START_REF] Onofri | On the positivity of the effective action in a theory of random surfaces[END_REF]

1 16 π R 2 |∇f | 2 dx -log R 2 e f dµ + R 2 f dµ ≥ 0 ∀ f ∈ D(R 2 ) (4)
plays the role of Sobolev's inequality in higher dimensions. Here the inequality is written for smooth and compactly supported functions in D(R 2 ), but can be extended to the appropriate Orlicz space which corresponds to functions such that both sides of the inequality are finite. This inequality is dual of the logarithmic Hardy-Littlewood-Sobolev inequality that can be written as follows: for any g ∈ L 1 + (R 2 ) with M = R 2 g dx, such that g log g, (1 + log |x| 2 ) g ∈ L 1 (R 2 ), we have

R 2 g log g M dx - 4 π M R 2 g (-∆) -1 g dx + M (1 + log π) ≥ 0 (5) with R 2
g (-∆) -1 g dx = -1 2 π R 2 ×R 2 g(x) g(y) log |x -y| dx dy .

Then, in dimension d = 2, we have an analogue of Theorem 1, which goes as follows.

Theorem 2. The inequality

R 2 g log g M dx - 4 π M R 2 g (-∆) -1 g dx + M (1 + log π) ≤ M 1 16 π ∇f 2 L 2 (R 2 ) + R 2 f dµ -log M (6)
holds for any function f ∈ D(R 2 ) such that M = R 2 e f dµ and g = e f µ.

Using for instance [START_REF] Almgren | Symmetric rearrangement is sometimes continuous[END_REF] or [START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S n[END_REF]Lemma 2] (also see [39, chapter 3-4]), it is known that optimality is achieved in (1), ( 2), ( 4) or ( 5) when the problem is reduced to radially symmetric functions. However, no such result applies when considering a difference of the terms in two such inequalities, like in (3) or [START_REF] Bianchi | A note on the Sobolev inequality[END_REF]. Optimality therefore requires a special treatment. In Section 2, we shall use the completion of the square method to establish the inequalities (without optimality) under an assumption of radial symmetry in case of Theorem 2. For radial functions, Theorem 1 can indeed be written with d > 2 considered as a real parameter and Theorem 2 corresponds, in this setting, to the limit case as d → 2 + . To handle the general case (without radial symmetry assumption), a more general setting is required. In Section 5, we extend the results established for Sobolev inequalities to weighted spaces and obtain an improved version of the Caffarelli-Kohn-Nirenberg inequalities (see Theorem 15). Playing with weights is equivalent to varying d or taking limits with respect to d, except that no symmetry assumption is required. This allows to complete the proof of Theorem 2.

Technical results regarding the computation of the constants, a weighted Poincaré inequality and the stereographic projection, the extension of the flow method of [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] to the case of the dimensions d = 3 and d = 4, and symmetry results for Caffarelli-Kohn-Nirenberg inequalities have been collected in various appendices.

At this point, we emphasize that Theorems 15 and 16, which are used as intermediate steps in the proof of Theorem 2 are slightly more general than, respectively, Theorems 1 and 2, except for the issue of the optimal value of the proportionality constant, which has not been studied. It is likely that the method used for Sobolev's inequality can be adapted, but since weights break the translation invariance, some care should be given to this question, which is of independent interest and known to raise a number of difficulties of its own (see for instance [START_REF] Dolbeault | About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF]). The question of a lower estimate of the proportionality constant in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] in connection with a larger family of Onofri type inequalities is currently being studied, see [START_REF] Jankowiak | Logarithmic Hardy-Littlewood-Sobolev and Onofri inequalities[END_REF].

Let us conclude this introduction by a brief review of the literature. To establish the inequalities, our approach is based on a completion of the square method which accounts for duality issues. Linearization (spectral estimates) and estimates based on a nonlinear flow are used for optimality issues. Although some of these methods have been widely used in the literature, for instance in the context of Hardy inequalities (see [START_REF] Bosi | Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators[END_REF] and references therein), it seems that they have not been fully exploited yet in the case of the functional inequalities considered in this paper. The main tool in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] is a flow of fast diffusion type, which has been considered earlier in [START_REF] Del Pino | On the extinction profile for solutions of u t = ∆u (N -2)/(N +2)[END_REF]. In dimension d = 2, we may refer to various papers (see for instance [START_REF] Daskalopoulos | On the Cauchy problem for u t = ∆ log u in higher dimensions[END_REF][START_REF] Daskalopoulos | On a singular diffusion equation[END_REF][START_REF] Daskalopoulos | Type II extinction profile of maximal solutions to the Ricci flow in R 2[END_REF]) in connection with Ricci's flow for properties of the solutions of the corresponding evolution equation.

Many papers have been devoted to the asymptotic behaviour near extinction of the solutions of nonlinear flows, in bounded domains (see for instance [START_REF] Berryman | Stability of the separable solution for fast diffusion[END_REF][START_REF] Galaktionov | Fast diffusion equation with critical Sobolev exponent in a ball[END_REF][START_REF] Savaré | The asymptotic profile of solutions of a class of doubly nonlinear equations[END_REF][START_REF] Bonforte | Behaviour near extinction for the Fast Diffusion Equation on bounded domains[END_REF]) or in the whole space (see [START_REF] King | Self-similar behaviour for the equation of fast nonlinear diffusion[END_REF][START_REF] Peletier | Self-similar solutions of a fast diffusion equation that do not conserve mass[END_REF][START_REF] Galaktionov | Asymptotic behaviour near finite-time extinction for the fast diffusion equation[END_REF] and references therein). In particular, the Cauchy-Schwarz inequality has been repeatedly used, for instance in [START_REF] Berryman | Stability of the separable solution for fast diffusion[END_REF][START_REF] Savaré | The asymptotic profile of solutions of a class of doubly nonlinear equations[END_REF], and turns out to be a key tool in the main result of [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF], as well as the solution with separation of variables, which is related to the Aubin-Talenti optimal function for [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF].

Getting improved versions of Sobolev's inequality is a question which has attracted lots of attention. See [START_REF] Brezis | Sobolev inequalities with remainder terms[END_REF] in the bounded domain case and [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical sobolev exponents[END_REF] for an earlier related paper. However, in [START_REF] Brezis | Sobolev inequalities with remainder terms[END_REF], H. Brezis and E. Lieb also raised the question of measuring the distance to the manifold of optimal functions in the case of the Euclidean space. A few years later, G. Bianchi and H. Egnell gave an answer in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] using the concentration-compactness method, with no explicit value of the constant. Since then, considerable efforts have been devoted to obtain quantitative improvements of Sobolev's inequality. On the whole Euclidean space, nice estimates based on rearrangements have been obtained in [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF] and we refer to [START_REF] Cianchi | Quantitative Sobolev and Hardy inequalities, and related symmetrization principles[END_REF] for an interesting review of various related results. The method there is in some sense constructive, but it hard to figure what is the practical value of the constant. As in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] our approach involves much weaker notions of distances to optimal functions, but on the other hand offers clear-cut estimates. Moreover, it provides an interesting way of obtaining global estimates based on a linearization around Aubin-Talenti optimal functions.

A completion of the square and consequences

Before proving the main results of this paper, let us explain in which sense Sobolev's inequality and the Hardy-Littlewood-Sobolev inequality, or Onofri's inequality and the logarithmic Hardy-Littlewood-Sobolev inequality, for instance, are dual inequalities.

To a convex functional F , we may associate the functional F * defined by Legendre's duality as

F * [v] := sup R d u v dx -F [u] . For instance, to F 1 [u] = 1 2 u 2 L p (R d ) defined on L p (R d ), we henceforth as- sociate F * 1 [v] = 1 2 v 2 L q (R d ) on L q (R d )
where p and q are Hölder conjugate exponents: 1/p + 1/q = 1. The supremum can be taken for instance on all functions in L p (R d ), or, by density, on the smaller space of the functions

u ∈ L p (R d ) such that ∇u ∈ L 2 (R d ). Similarly, to F 2 [u] = 1 2 S d ∇u 2 L 2 (R d ) , we associate F * 2 [v] = 1 2 S -1 d R d v (-∆) -1 v dx where (-∆) -1 v = G d * v with G d (x) = 1 d-2 |S d-1 | -1 |x| 2-d , when d ≥ 3
, and G 2 (x) = -1 2π log |x|. As a straightforward consequence of Legendre's duality, if we have a functional inequality of the form F 1 [u] ≤ F 2 [u], then we have the dual inequality

F * 1 [v] ≥ F * 2 [v].
In this sense, (1) and ( 2) are dual of each other, as it has been noticed in [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF]. Also notice that Inequality (2) is a consequence of Inequality [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF].

In this paper, we go one step further and establish that

F * 1 [u] -F * 2 [u] ≤ C (F 2 [u] -F 1 [u]) (7)
for some positive constant C, at least under some normalization condition (or up to a multiplicative term which is required for simple homogeneity reasons). Such an inequality has been established in [22, Theorem 1.2] when d ≥ 5. Here we extend it to any d ≥ 3 and get and improved value for the constant C. It turns out that the proof can be reduced to the completion of a square. Let us explain how the method applies in case of Theorem 1, and how Theorem 2 can be seen as a limit of Theorem 1 in case of radial functions.

Proof of Theorem 1, part 1: the completion of a square. Integrations by parts show that

R d |∇(-∆) -1 v| 2 dx = R d v (-∆) -1 v dx and, if v = u q with q = d+2 d-2 , R d ∇u • ∇(-∆) -1 v dx = R d u v dx = R d u 2 * dx .
Hence the expansion of the square

0 ≤ R d S d u 4 d-2 L 2 * (R d ) ∇u -∇(-∆) -1 v 2 dx shows that 0 ≤ S d u 8 d-2 L 2 * (R d ) S d ∇u 2 L 2 (R d ) -u 2 L 2 * (R d ) -S d u q 2 L 2 d d+2 (R d ) - R d u q (-∆) -1 u q dx .
Equality is achieved if and only if

S d u 4 d-2 L 2 * (R d ) u = (-∆) -1 v = (-∆) -1 u q , that is, if and only if u solves -∆u = 1 S d u -4 d-2 L 2 * (R d ) u q ,
which means that u is an Aubin-Talenti function, optimal for (1). This completes the proof of Theorem 1, up to the optimality of the proportionality constant, for which we know that

C d = C S d with C ≤ 1 . (8)
Incidentally, this also proves that v is optimal for (2).

As a first step towards the proof of Theorem 2, let us start with a result for radial functions. If d is a positive integer, we can define

s d := S d |S d-1 | 2 d
and get

s d = 4 d (d -2) Γ d+1 2 √ π Γ d 2 2 d . ( 9
)
Using this last expression allows us to consider d as a real parameter.

Lemma 3. Assume that d ∈ R and d > 2. Then 0 ≤ s d ∞ 0 u 2 d d-2 r d-1 dr 1+ 2 d - ∞ 0 u d+2 d-2 (-∆) -1 u d+2 d-2 r d-1 dr ≤ c d ∞ 0 u 2 d d-2 r d-1 dr 4 d s d ∞ 0 |u ′ | 2 r d-1 dr - ∞ 0 u 2 d d-2 r d-1 dr d-2 d holds for any radial function u ∈ D 1,2 (R d ) with optimal constant c d ≤ s d .
Here we use the notation (-∆) -1 v = w to express the fact that w is the solution to

w ′′ + d-1 r w ′ + v = 0, that is, (-∆) -1 v (r) = ∞ r s 1-d s 0 v(t) t d-1 dt ds ∀ r > 0 . ( 10
)
Proof. In the case of a radially symmetric function u, and with the standard abuse of notations that amounts to identify u(x) with u(r), r = |x|, Inequality (1) can be written as

s d ∞ 0 |u ′ | 2 r d-1 dr ≥ ∞ 0 |u| 2 d d-2 r d-1 dr 1-2 d . (11)
However, if u is considered as a function of one real variable r, then the inequality also holds for any real parameter d ∈ (2, ∞) and is equivalent to the one-dimensional Gagliardo-Nirenberg inequality

s d R |w ′ | 2 dt + 1 4 (d -2) 2 R |w| 2 dt ≥ R |w| 2 d d-2 dt 1-2 d
as can be shown using the Emden-Fowler transformation

u(r) = (2 r) -d-2 2 w(t) , t = -log r . (12)
The corresponding optimal function is, up to a multiplication by a constant, given by w

⋆ (t) = (cosh t) -d-2 2 ∀ t ∈ R ,
which solves the Euler-Lagrange equation

-(p -2) 2 w ′′ + 4 w -2 p |w| p-2 w = 0 .
for any real number d > 2 and the optimal function for ( 11) is

u ⋆ (r) = (2 r) -d-2 2 w ⋆ (-log r) = 1 + r 2 -d-2 2
up to translations, multiplication by a constant and scalings. This establishes [START_REF] Brezis | Sobolev inequalities with remainder terms[END_REF]. See Appendix A for details on the computation of s d . The reader is in particular invited to check that the expression of s d is consistent with the one of S d given in the introduction.

Next we apply Legendre's transform to [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] and get a Hardy-Littlewood-Sobolev inequality that reads

∞ 0 v (-∆) -1 v r d-1 dr ≤ s d ∞ 0 v 2 d d+2 r d-1 dr 1+ d 2 (13)
for any d > 2. Inequality (13) holds on the functional space which is obtained by completion of the space of smooth compactly supported radial functions with respect to the norm defined by the r.h.s. in [START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S n[END_REF]. Inequality [START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S n[END_REF] is the first inequality of Lemma 3.

Finally, we apply the completion of the square method. By expanding

0 ≤ ∞ 0 a u ′ -(-∆) -1 v ′ 2 r d-1 dr with a = s d ∞ 0 u 2 d d-2 r d-1 dr 2 d and v = u d-2
d+2 , we establish the second inequality of Lemma 3 (with optimal constant c d ≤ s d ). Now let us turn our attention to the case d = 2 and to Theorem 2. Using the fact that d in Lemma 3 is a real parameter, we can simply consider the limit of the inequalities as d → 2 + .

Corollary 4. For any function

f ∈ L 1 (R + ; r dr) such that f ′ ∈ L 2 (R + ; r dr) and M = ∞ 0 e f (1 + r 2 ) -2 2 r dr, we have the inequality 0 ≤ ∞ 0 e f log e f M (1 + r 2 ) 2 2 r dr (1 + r 2 ) 2 - 2 M ∞ 0 e f (1 + r 2 ) 2 (-∆) -1 e f (1 + r 2 ) 2 2 r dr + M ≤ M 1 8 ∞ 0 |f ′ | 2 r dr + ∞ 0 f 2 r dr (1 + r 2 ) 2 -log ∞ 0 e f 2 r dr (1 + r 2 ) 2 . ( 14
)
Here again (-∆) -1 is defined by [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical sobolev exponents[END_REF], but it coincides with the inverse of -∆ acting on radial functions.

Proof. We may pass to the limit in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] written in terms of

u(r) = u ⋆ (r) 1 + d-2 2 d f
to get the radial version of Onofri's inequality for f . By expanding the expression of |u ′ | 2 we get

u ′2 = u ′2 ⋆ + d -2 d u ′ ⋆ (u ⋆ f ) ′ + d -2 2 d 2 (u ′ ⋆ f + u ⋆ f ′ ) 2 .
Using the fact that lim d→2 + (d -2) s d = 1,

s d = 1 d -2 + 1 2 - 1 2 log 2 + o(1) as d → 2 + ,
and lim

d→2 + 1 d -2 ∞ 0 |u ′ ⋆ | 2 r d-1 dr = 1 , 1 d -2 ∞ 0 |u ′ ⋆ | 2 r d-1 dr -1 ∼ - 1 2 (d -2) , lim d→2 + 1 d -2 ∞ 0 u ′ ⋆ (u ⋆ f ) ′ r d-1 dr = ∞ 0 f 2 r dr (1 + r 2 ) 2 , lim d→2 + 1 4 d 2 ∞ 0 |f ′ | 2 u 2 ⋆ r d-1 dr = 1 16 ∞ 0 |f ′ | 2 r dr ,
and finally

lim d→2 + ∞ 0 |u ⋆ (1 + d-2 2 d f )| 2 d d-2 r d-1 dr = ∞ 0 e f r dr (1 + r 2 ) 2 , so that, as d → 2 + , ∞ 0 |u ⋆ (1 + d-2 2 d f )| 2 d d-2 r d-1 dr d-2 d -1 ∼ d -2 2 log ∞ 0 e f r dr (1 + r 2 ) 2 .
By keeping only the highest order terms, which are of the order of (d -2), and passing to the limit as d → 2 + in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF], we obtain that

1 8 ∞ 0 |f ′ | 2 r dr + ∞ 0 f 2 r dr (1 + r 2 ) 2 ≥ log ∞ 0 e f 2 r dr (1 + r 2 ) 2 ,
which is Onofri's inequality written for radial functions.

Similarly, we can pass to the limit as d → 2 + in [START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S n[END_REF]. Let v be a compactly supported smooth radial function, considered as a function of r ∈ [0, ∞) and let us compute the limit as

d → 2 + of h(d) := ∞ 0 v 2 d d+2 r d-1 dr 1+ 2 d - 1 s d ∞ 0 v k d [v] r d-1 dr
where

k d [v] := (-∆) -1 v is given by (10) for any d ≥ 2. If d > 2, since (2 -d) ∞ 0 v(r) k d [v](r) r d-1 dr = (2 -d) ∞ 0 v(r) r d-1 ∞ r s 1-d s 0 v(t) t d-1 dt ds dr = (2 -d) ∞ 0 r 1-d r 0 v(t) t d-1 dt 2 dr = -2 ∞ 0 r v(r) r 0 v(t) t d-1 dt dr we see that lim d→2 + h(d) = 0 since 2 ∞ 0 r v(r) r 0 v(t) t dt dr = ∞ 0 r v(r) dr 2 .
Let us compute the O(d -2) term. With the above expression, it is now easy to check that lim

d→2 + h(d) d -2 = 1 2 ∞ 0 v r dr ∞ 0 v log v ∞ 0 v r dr r dr - log 2 -1 2 ∞ 0 r v(r) dr 2 + 2 ∞ 0 v r dr ∞ 0 v(r) r log r dr -2 ∞ 0 r v(r) r 0 v(t) t log t dt dr = 1 2 ∞ 0 v r dr ∞ 0 v log v ∞ 0 v r dr r dr - log 2 -1 2 ∞ 0 r v(r) dr 2 + 2 ∞ 0 v r dr ∞ r v(t) t log t dt since 1 (d-2) s d ∼ 1 + d-2
2 (log 2 -1). A computation corresponding to d = 2 similar to the one done above for d > 2 shows that, when d = 2,

∞ 0 v k 2 [v] r dr = ∞ 0 v(r) r ∞ r 1 s s 0 v(t) t dt ds dr = ∞ 0 1 r r 0 v(t) t dt 2 dr = -2 ∞ 0 r log r v(r) r 0 v(t) t dt dr , thus proving that lim d→2 + h(d) d -2 = 1 2 ∞ 0 v r dr ∞ 0 v log v ∞ 0 v r dr r dr - ∞ 0 v k 2 [v] r d-1 dr - 1 2 (log 2 -1) ∞ 0 r v(r) dr 2 .
Now let us consider as above the limit

u d+2 d-2 = (1 + r 2 ) -d+2 2 (1 + d-2 2 d f ) d+2 d-2 → (1 + r 2 ) -2 e f =: g as d → 2.
This concludes the proof of Corollary 4 by passing to the limit in the inequalities of Lemma 3 and taking v = g.

Proof of Theorem 2: a passage to the limit in the radial case. If we consider g as a function on R 2 ∋ x with r = |x|, this means that lim

d→2 + h(d) d -2 = 1 2 R 2 g dx R 2 g log g R 2 g dx dx -2 π R 2 g (-∆) -1 g dx + 1 2 (1 + log π) R 2
g dx 2 which precisely corresponds to the terms involved in ( 5), up to a factor

1 2 M = 1 2 R 2 g dx.
The proof in the non-radial case will be provided at the end of Section 5.

Linearization

In the previous section, we have proved that the optimal constant C d in (3) is such that C d ≤ S d . Let us prove that C d ≥ d d+4 S d using a special sequence of test functions. Let F and G be the positive integral quantities associated with, respectively, the Sobolev and Hardy-Littlewood-Sobolev inequalities:

F [u] := S d ∇u 2 L 2 (R d ) -u 2 L 2 * (R d ) , G[v] := S d v 2 L 2 d d+2 (R d ) - R d v (-∆) -1 v dx .
Since that, for the Aubin-Talenti extremal function u ⋆ , we have

F [u ⋆ ] = G[u q ⋆ ]
= 0, so that u ⋆ gives a case of equality for (3), a natural question to ask is whether the infimum of F [u]/G[u q ], under an appropriate normalization of u L 2 * (R d ) , is achieved as a perturbation of the u ⋆ .

Recall that u ⋆ is the Aubin-Talenti extremal function

u ⋆ (x) := (1 + |x| 2 ) -d-2 2 ∀ x ∈ R d .
With a slight abuse of notations, we use the same notation as in Section 2.

We may notice that u ⋆ solves

-∆u ⋆ = d (d -2) u d+2 d-2
⋆ which allows to compute the optimal Sobolev constant as

S d = 1 d (d -2) R d u 2 * ⋆ dx -2 d ( 15
)
using [START_REF] Carlen | Hardy-Littlewood-Sobolev inequalities via fast diffusion flows[END_REF]. See Appendix A for details. This shows that

1 S d F [u] = ∇u 2 L 2 (R d ) -d (d -2) R d u 2 * dx 1-2 d R d u 2 * ⋆ dx 2 d
.

The goal of this section is to perform a linearization. By expanding

F [u ε ] with u ε = u ⋆ + ε f , for some f such that R d f u⋆ (1+|x| 2 ) 2 dx = 0 at order two in terms of ε, we get that 1 S d F [u ε ] = ε 2 F[f ] + o(ε 2 )
where

F[f ] := R d |∇f | 2 dx -d (d + 2) R d |f | 2 (1 + |x| 2 ) 2 dx .
According to Lemma 17 (see Appendix B), we know that

F[f ] ≥ 4 (d + 2) R d |f | 2 (1 + |x| 2 ) 2 dx for any f ∈ D 1,2 (R d ) such that R d f f i (1 + |x| 2 ) 2 dx = 0 ∀ i = 0 , 1 , 2 , . . . d + 1 , (16)
where

f 0 := u ⋆ , f i (x) = x i 1 + |x| 2 u ⋆ (x) and f d+1 (x) := 1 -|x| 2 1 + |x| 2 u ⋆ (x) .
Notice for later use that

-∆f 0 = d (d -2) f 0 (1 + |x| 2 ) 2

and

-

∆f i = d (d + 2) f i (1 + |x| 2 ) 2 ∀ i = 1 , 2 , . . . d + 1 . Also notice that R d f i f j (1 + |x| 2 ) 2 dx = 0 for any i, j = 0, 1, . . . d + 1, j = i.
Similarly, we can consider the functional G as given above, associated with the Hardy-Littlewood-Sobolev inequality, and whose minimum

G[v ⋆ ] = 0 is achieved by v ⋆ := u q ⋆ , q = d+2 d-2 . Consistently with the above computations, let v ε := (u ⋆ + ε f ) q = v ⋆ 1 + ε f u⋆ q where f is such that R d f f 0 (1+|x| 2 ) 2 dx = 0. By expanding G[v ε ] at order two in terms of ε, we get that G[v ε ] = ε 2 d + 2 d -2 2 G[f ] + o(ε 2 )
where

G[f ] := 1 d (d + 2) R d |f | 2 (1 + |x| 2 ) 2 dx - R d f (1 + |x| 2 ) 2 (-∆) -1 f (1 + |x| 2 ) 2 dx . Lemma 5. Ker(F) = Ker(G).
It is straightforward to check that the kernel is generated by f i with i = 1, 2, . . . d, d + 1. Details are left to the reader. Next, by Legendre duality we find that

1 2 R d |g| 2 (1 + |x| 2 ) 2 dx = sup f R d f g (1 + |x| 2 ) 2 dx - 1 2 R d |f | 2 (1 + |x| 2 ) 2 dx , 1 2 R d g (1 + |x| 2 ) 2 (-∆) -1 g (1 + |x| 2 ) 2 dx = sup f R d f g (1 + |x| 2 ) 2 dx - 1 2 R d |∇f | 2 dx .
Here the supremum is taken for all f satisfying the orthogonality conditions [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF]. It is then straightforward to see that duality holds if g is restricted to functions satisfying (16) as well. Consider indeed an optimal function f subject to [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF]. There are Lagrange multipliers

µ i ∈ R such that g -f - d+1 i=0 µ i f i = 0
and after multiplying by f (1 + |x| 2 ) -2 , an integration shows that

R d f g (1 + |x| 2 ) 2 dx = R d |f | 2 (1 + |x| 2 ) 2 dx
using the fact that f satisfies [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF]. On the other hand, if g satisfies ( 16), after multiplying by g (1 + |x| 2 ) -2 , an integration gives

R d |g| 2 (1 + |x| 2 ) 2 dx = R d f g (1 + |x| 2 ) 2 dx ,
which establishes the first identity of duality. As for the second identity, the optimal function satisfies the Euler-Lagrange equation

g (1 + |x| 2 ) 2 + ∆ f = d+1 i=0 µ i f i (1 + |x| 2 ) 2
for some Lagrange multipliers that we again denote by µ i . By multiplying by f and (-∆) -1 g (1 + |x| 2 ) -2 , we find that

R d f g (1 + |x| 2 ) 2 dx = R d |∇f | 2 dx R d g (1 + |x| 2 ) 2 (-∆) -1 g (1 + |x| 2 ) 2 dx = R d f g (1 + |x| 2 ) 2 dx
where we have used the fact that

R d f i (1 + |x| 2 ) 2 (-∆) -1 g (1 + |x| 2 ) 2 dx = R d g (1 + |x| 2 ) 2 (-∆) -1 f i (1 + |x| 2 ) 2 dx = 0 because (-∆) -1 f i (1 + |x| 2 ) -2 is proportional to f i .
As a straightforward consequence, the dual form of Lemma 17 then reads as follows.

Corollary 6. For any g satisfying the orthogonality conditions (16), we have

R d g (1 + |x| 2 ) 2 (-∆) -1 g (1 + |x| 2 ) 2 dx ≤ 1 (d + 2) (d + 4) R d g 2 (1 + |x| 2 ) 2 dx .
Moreover, if f obeys to (16), then we have

4 d (d + 2) (d + 4) R d f 2 (1 + |x| 2 ) 2 dx ≤ G[f ] ≤ 1 d (d + 2) 2 (d + 4) F[f ]
and equalities are achieved in L 2 (R d , (1 + |x| 2 ) -2 dx).

Proof. The first inequality follows from the above considerations on duality and the second one from the definition of G, using

4 d (d + 2) (d + 4) = 1 d (d + 2) - 1 (d + 2) (d + 4)
.

To establish the last inequality, we can decompose f on (f k ) k , the stereographic projection of the spherical harmonics associated to eigenvalues λ k = k (k+d-1) with k ≥ 2, so as to meet condition [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF]. See Appendix B for more details. The corresponding eigenvalues for the Laplacian operator on the Euclidean space are

µ k = 4 λ k + d (d -2), so that -∆f k = µ k f k (1 + |x| 2 ) -2 , with f k L 2 (R d , (1+|x| 2 ) -2 dx) = 1. By writing f = k≥2 a k f k we have F[f ] = k≥2 c k , with c k := a 2 k (µ k -µ 1 ) , G[f ] = k≥2 d k , with d k := a 2 k 1 µ 1 - 1 µ k , with c k = µ 1 µ k d k ≤ µ 1 µ 2 d k since (µ k ) k is increasing in k. This yields F[f ] G[f ] ≤ µ 1 µ 2 = d (d + 2) 2 (d + 4) , with equality for f = f 2 .
As a consequence of Corollary 6 and ( 15), we have found that

1 C := S d C d = inf G[u q ] =0 u 8 d-2 L 2 * (R d ) S d F [u] G[u q ] ≤ 1 d 2 (d + 2) 2 inf f F[f ] G[f ] = d + 4 d , (17)
where the last infimum is taken on the set of all non-trivial functions in L 2 (R d , (1+|x| 2 ) -2 dx) satisfying ( 16). This establishes the lower bound in (3).

Remark 7. One may hope to get a better estimate by considering the case f ∈ Ker(F) = Ker(G) and expanding F and G to the fourth order in ε but, interestingly, this yields exactly the same lower bound on C d as the linearization shown above.

Improved inequalities and nonlinear flows

In Section 3, the basic strategy was based on the completion of a square. The initial approach for the improvement of Sobolev inequalities in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] was based on a fast diffusion flow. Let us give some details and explain how even better results can be obtained using a combination of the two approaches.

Let us start with a summary of the method of [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF]. It will be convenient to define the functionals

J d [v] := R d v 2 d d+2 dx and H d [v] := R d v (-∆) -1 v dx -S d v 2 L 2 d d+2 (R d )
.

Consider a positive solution v of the fast diffusion equation

∂v ∂t = ∆v m t > 0 , x ∈ R d , m = d -2 d + 2 (18)
and define the functions

J(t) := J d [v(t, •)] and H(t) := H d [v(t, •)] .
We shall denote by J 0 and H 0 the corresponding initial values. Elementary computations show that

J ′ = -(m + 1) ∇v m 2 L 2 (R d ) ≤ - m + 1 S d J 1-2 d = - 2 d d + 2 1 S d J 1-2 d , (19)
where the inequality is a consequence of Sobolev's inequality. Hence v has a finite extinction time T > 0 and since

J(t) 2 d ≤ J 2 d 0 - 4 d + 2 t S d ,
we find that

T ≤ d + 2 4 S d J 2 d 0 .
We notice that H is nonpositive because of the Hardy-Littlewood-Sobolev inequality and by applying the flow of ( 18), we get that

1 2 J -2 d H ′ = S d ∇u 2 L 2 (R d ) -u 2 L 2 * (R d ) with u = v d-2 d+2 .
The right hand side is nonnegative because of Sobolev's inequality. One more derivation with respect to t gives that

H ′′ = J ′ J H ′ -4 m S d J 2 d K ( 20
)
where

K := R d v m-1 |(-∆)v m -Λ v| 2 dx and Λ := -d+2 2 d J ′
J . This identity makes sense in dimension d ≥ 5, because, close to the extinction time, v behaves like the Aubin-Talenti functions. The reader is invited to check that all terms are finite when expanding the square in K and can refer to [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] for more details. It turns out that the following estimate is also true if d = 3 or d = 4. Lemma 8. Assume that d ≥ 3. With above notations, we have

H ′′ H ′ ≤ J ′ J .
The main idea is that even if each of the above integrals is infinite, there are cancellations in low dimensions. To clarify this computation, it is much easier to get rid of the time-dependence corresponding to the solution with separation of variables and use the inverse stereographic projection to recast the problem on the sphere. The sketch of the proof of this lemma will be given in Appendix C.

A straightforward consequence is the fact that

H ′′ H ′ ≤ -κ with κ := 2 d d + 2 J -2 d 0 S d
where the last inequality is a consequence of [START_REF] Daskalopoulos | Type II extinction profile of maximal solutions to the Ricci flow in R 2[END_REF]. Two integrations with respect to t show that

-H 0 ≤ 1 κ H ′ 0 (1 -e -κ T ) ≤ 1 2 C S d J 2 d 0 H ′ 0 with C = d + 2 d (1 -e -d/2 ) ,
which is the main result of [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF] (when d ≥ 5), namely

-H 0 ≤ C S d J 4 d 0 S d ∇u 0 2 L 2 (R d ) -u 0 2 L 2 * (R d ) with u 0 = v d-2 d+2 0 .
Since this inequality holds for any initial datum u 0 = u, we have indeed shown that

-H d [v] ≤ C S d J d [v] 4 d S d ∇u 2 L 2 (R d ) -u 2 L 2 * (R d ) ∀ u ∈ D 1,2 (R d ) , v = u d+2 d-2 .
It is straightforward to check that our result of Theorem 1 is an improvement, not only because the restriction d ≥ 5 is removed, but also because the inequality holds with

d d+4 ≤ C < 1 < d+2 d (1 -e -d/2
). In other words, the result of Theorem 1 is equivalent to

-H 0 ≤ 1 2 C S d J 2 d 0 H ′ 0 with C = d d + 4 . ( 21
)
Up to now, we have not established yet the fact that C < 1. This is what we are now going to do. Now let us reinject in the flow method described above our improved 20 inequality of Theorem 1, which can also be written as

C S d J 4 d d + 2 2 d S d J ′ + J 1-2 d -H ≤ 0 ( 22
)
if v is still a positive solution of [START_REF] Daskalopoulos | On a singular diffusion equation[END_REF]. From Lemma 8, we deduce that

H ′ ≤ κ 0 J with κ 0 := H ′ 0 J 0 .
Since t → J(t) is monotone decreasing, there exists a function Y such that

H(t) = -Y(J(t)) ∀ t ∈ [0, T ) .
Differentiating with respect to t, we find that

-Y ′ (J) J ′ = H ′ ≤ κ 0 J
and, by inserting this expression in [START_REF] Dolbeault | Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion[END_REF], we arrive at

C - d + 2 2 d κ 0 S 2 d J 1+ 4 d Y ′ + S d J 1+ 2 d + Y ≤ 0 .
Summarizing, we end up by considering the differential inequality

Y ′ C S d s 1+ 2 d + Y ≤ d + 2 2 d C κ 0 S 2 d s 1+ 4 d , Y(0) = 0 , Y(J 0 ) = -H 0 (23)
on the interval [0, J 0 ] ∋ s. It is then possible to obtain estimates as follows.

On the one hand we know that

Y ′ ≤ d + 2 2 d κ 0 S d s 2 d
and, hence,

Y(s) ≤ 1 2 κ 0 S d s 1+ 2 d ∀ s ∈ [0, J 0 ] .
On the other hand, after integrating by parts on the interval [0, J 0 ], we get

1 2 H 2 0 -C S d J 1+ 2 d 0 H 0 ≤ 1 4 C κ 0 S 2 d J 2+ 4 d 0 + d + 2 d C S d J 0 0 s 2 d Y(s) ds .
Using the above estimate, we find that

d + 2 d S d J 0 0 s 2 d Y(s) ds ≤ 1 4 J 2+ 4 d 0 ,
and finally 1 2

H 2 0 -C S d J 1+ 2 d 0 H 0 ≤ 1 2 Cκ 0 S 2 d J 2+ 4 d 0 .
This is a strict improvement of ( 21) when C = 1 since ( 21) is then equivalent to -S d J

1+ 2 d 0 H 0 ≤ 1 2 Cκ 0 S 2 d J 2+ 4 d 0 .
However, it is a strict improvement of (21) if C < 1 only when |H 0 | = -H 0 is large enough (we will come back to this point in Remarks 10 and 11). Altogether, we have shown an improved inequality that can be stated as follows.

Theorem 9. Assume that d ≥ 3. Then we have

0 ≤ H d [v] + S d J d [v] 1+ 2 d ϕ J d [v] 2 d -1 S d ∇u 2 L 2 (R d ) -u 2 L 2 * (R d ) ∀ u ∈ D 1,2 (R d ) , v = u d+2 d-2
where ϕ(x) := √ C 2 + 2 C x -C for any x ≥ 0.

Proof. We have shown that

y 2 + 2 C y-C κ 0 ≤ 0 with y = -H 0 /(S d J 1+ 2 d 0 ) ≥ 0. This proves that y ≤ √ C 2 + Cκ 0 -C, which proves that -H 0 ≤ S d J 1+ 2 d 0 C 2 + C κ 0 -C after recalling that 1 2 κ 0 = H ′ 0 J 0 = J d [v 0 ] 2 d -1 S d ∇u 0 2 L 2 (R d ) -u 0 2 L 2 * (R d ) .
Remark 10. We may observe that x → x -ϕ(x) is a convex nonnegative function which is equal to 0 if and only if x = 0. Moreover, we have

ϕ(x) ≤ x ∀ x ≥ 0
with equality if and only if x = 0. However, one can notice that

ϕ(x) ≤ C x ⇐⇒ x ≥ 2 1 -C C .
Remark 11. A more careful analysis of [START_REF] Dolbeault | Radial symmetry and symmetry breaking for some interpolation inequalities[END_REF] shows that

Y(s) ≤ 1 2 1 + 2 κ 0 C -1 C S d s 1+ 2 d ,
which shows that the inequality of Theorem 9 holds with the improved function

ϕ(x) := C 2 + C x + 1 2 C 2 1 + 4 x C -1 -C
but again the reader is invited to check that ϕ(x) ≤ x for any x ≥ 0 and lim x→0 + ϕ(x)/x = 1.

Corollary 12. With the above notations, we have C < 1.

Proof. Assume by contradiction that C = 1. With the notations of Section 3, let us consider a minimizing sequence (u n ) n∈N for the functional u

→ F [u] G[u q ]
but assume that J d [u q n ] = J d [u q ⋆ ] =: J ⋆ for any n ∈ N. This condition is not restrictive because of the homogeneity of the inequality. It implies that (G[u q n ]) n∈N is bounded. If lim n→∞ G[u q n ] > 0, then we also have L := lim n→∞ F [u n ] > 0, at least up to the extraction of a subsequence. As a consequence we find that

0 = lim n→∞ S d J 4 d ⋆ F [u n ] -G[u q n ] = S d lim n→∞ J 4 d ⋆ F [u n ] -J 1+ 2 d ⋆ ϕ J 2 d -1 ⋆ F [u n ] + lim n→∞ S d J 1+ 2 d ⋆ ϕ J 3.1],
we know that up to translations and dilations, u k converges to u ⋆ . Thus there exists f k such that u k = u ⋆ + f k with f k → 0, and then

1 C = S d C d = lim k→∞ 1 d 2 (d + 2) 2 F[f k ] G[f k ] ≥ d + 4 d .
This shows that C ≤ d d+4 , a contradiction. We may observe that C < 1 means C d < S d . This completes the the proof of Theorem 1.

Caffarelli-Kohn-Nirenberg inequalities and duality

Let 2 * := ∞ if d = 1 or 2, 2 * := 2 d/(d -2) if d ≥ 3 and a c := (d -2)/2. Consider the space D 1,2 a (R d ) obtained by completion of D(R d \ {0}) with respect to the norm u → |x| -a ∇u 2 L 2 (R d ) .
In this section, we shall consider the Caffarelli-Kohn-Nirenberg inequalities

R d |u| p |x| bp dx 2 p ≤ C a,b R d |∇u| 2 |x| 2a dx (24)
These inequalities generalize to D 1,2 a (R d ) the Sobolev inequality (1) and in particular the exponent p is given in terms of a and b by

p = 2 d d -2 + 2 (b -a)
as can be checked by a simple scaling argument. A precise statements on the range of validity of (24) goes as follows.

Lemma 13. [11] Let d ≥ 1. For any p ∈ [2, 2 * ] if d ≥ 3 or p ∈ [2, 2 * ) if d = 1 or 2, there exists a positive constant C a,b such that (24) holds if a, b and p are related by b = a -a c + d/p, with the restrictions a < a c , a ≤ b ≤ a + 1 if d ≥ 3, a < b ≤ a + 1 if d = 2 and a + 1/2 < b ≤ a + 1 if d = 1.
At least for radial solutions in R d , weights can be used to work as in Section 2 as if the dimension d was replaced by the dimension (d -2a). We will apply this heuristic idea to the case d = 2 and a < 0, a → 0 in order to prove Theorem 2. See Appendix D for symmetry results for optimal functions in [START_REF] Dolbeault | About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF].

On D 1,2 a (R d ), let us define the functionals

F 1 [u] := 1 2 R d |u| p |x| bp dx 2 p and F 2 [u] := 1 2 C a,b R d |∇u| 2
|x| 2a dx so that Inequality [START_REF] Dolbeault | About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF] amounts to

F 1 [u] ≤ F 2 [u]. Assume that •, • denotes the natural scalar product on L 2 R d , |x| -2a dx , that is, u, v := R d u v |x| 2a dx
and denote by u = u, u 1/2 the corresponding norm. Consider the operators

A a u := ∇u , A * a w := -∇ • w + 2a x |x| 2 • w and L a u := A * a A a u = -∆u + 2a x |x| 2 • ∇u defined for u and w respectively in L 2 R d , |x| -2a dx and L 2 R d , |x| -2a dx d .
Elementary integrations by parts show that

u, L a u = A a u, A a u = A a u 2 = R d |∇u| 2 |x| 2a dx .
If we define the Legendre dual of F i by

F * i [v] = sup u∈D 1,2 a (R d ) ( u, v -F i [u]
), then it is clear that we formally have the inequality

F * 2 [v] ≤ F * 1 [v] for any v ∈ L q (R d , |x| -(2a-b) q dx) ∩ L a (D 1,2 a (R d ))
, where q is Hölder's conjugate of p, i.e.

1 p + 1 q = 1 .
Using the invertibility of L a , we indeed observe that

F * 2 [v] = u, v -F 2 [u] with v = C a,b L a u ⇐⇒ u = 1 C a,b L -1 a v ,
hence proving that

F * 2 [v] = 1 2 C a,b v, L -1 a v .
Similarly, we get that

F * 1 [v] = u, v -F 1 [u] with |x| -2a v = κ 2-p |x| -bp u p-1 (25)
and

κ = R d |u| p |x| bp dx 1 p = u, v = R d |v| q |x| (2a-b) q dx 1 q
, that is

F * 1 [v] = 1 2 R d |v| q |x| (2a-b) q dx 2 q
. This proves the following result. Lemma 14. With the above notations and under the same assumptions as in Lemma 13, we have

1 C a,b v, L -1 a v ≤ R d |v| q |x| (2a-b) q dx 2 q ∀ v ∈ L q (R d , |x| -(2a-b) q dx) ∩ L a (D 1,2 a (R d )) .
The next step is based on the completion of the square. Let us compute

A a u -λ A a L -1 a v 2 = A a u 2 -2 λ A a u, A a L -1 a v + λ 2 A a L -1 a v, A a L -1 a v = A a u 2 -2 λ u, v + λ 2 v, L -1 a v .
With the choice λ = 1/C a,b and v given by ( 25), we have proved the following Theorem 15. Under the assumptions of Lemma 13 and with the above notations, for any

u ∈ D 1,2 a (R d ) and any v ∈ L q (R d , |x| -(2a-b) q dx) ∩ L a (D 1,2 a (R d )) we have 0 ≤ R d |v| q |x| (2a-b) q dx 2 q - 1 C a,b v, L -1 a v ≤ C a,b R d |∇u| 2 |x| 2a dx - R d |u| p |x| bp dx 2 p
if u and v are related by [START_REF]Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations[END_REF], if a, b and p are such that b = a -a c + d/p and verify the conditions of Lemma 13, and if q = p/(p -1).

If, instead of (25), we simply require that

|x| -2a v = |x| -bp u p-1 ,
then the inequality becomes

0 ≤ C a,b R d |v| q |x| (2a-b) q dx 2 q -v, L -1 a v ≤ C a,b R d |u| p |x| bp dx 2 p (p-2) C a,b R d |∇u| 2 |x| 2a dx - R d |u| p |x| bp dx 2 p
Hence Theorem 15 generalizes Theorem 1, which is recovered in the special case a = b = 0, d ≥ 3. Because of the positivity of the l.h.s. due to Lemma 14, the inequality in Theorem 15 is an improvement of the Caffarelli-Kohn-Nirenberg inequality [START_REF] Dolbeault | About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF]. It can also be seen as an interpolation result, namely 2

R d |v| q |x| (2a-b) q dx 2 q = 2 R d |u| p |x| bp dx 2 p ≤ C a,b R d |∇u| 2 |x| 2a dx + 1 C a,b v, L -1 a v
whenever u and v are related by [START_REF]Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations[END_REF]. The explicit value of C a,b is not known unless equality in [START_REF] Dolbeault | About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF] is achieved by radial functions, that is when symmetry holds. See Proposition 19 in Appendix D for some symmetry results. Now, as in [START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF], we may investigate the limit (a, b) → (0, 0) with b = α a/(1 + α) in order to investigate the Onofri limit case. A key observation is that optimality in ( 24) is achieved by radial functions for any α ∈ (-1, 0) and a < 0, |a| small enough. In that range C a,b is known and given by (D.1).

Proof of Theorem 2 (continued). Theorem 2 has been established for radial functions in Section 2. Now we investigate the general case. We shall restrict our purpose to the case of dimension d = 2. For any α ∈ (-1, 0), let us denote by dµ α the probability measure on R 2 defined by dµ α := µ α dx where

µ α := 1 + α π |x| 2 α (1 + |x| 2 (1+α) ) 2 .
It has been established in [START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF]

that log R 2 e u dµ α - R 2 u dµ α ≤ 1 16 π (1 + α) R 2 |∇u| 2 dx ∀ u ∈ D(R 2 ) , (26)
where D(R 2 ) is the space of smooth functions with compact support. By density with respect to the natural norm defined by each of the inequalities, the result also holds on the corresponding Orlicz space.

We adopt the strategy of [30, Section 2.3] to pass to the limit in ( 24) as (a, b) → (0, 0) with b = α α+1 a. Let

a ε = - ε 1 -ε (α + 1) , b ε = a ε + ε, p ε = 2 ε , and
u ε (x) = 1 + |x| 2 (α+1) -ε 1-ε ,
assuming that u ε is an optimal function for (24), define

κ ε = R 2 u ε |x| aε+ε 2/ε dx = R 2 |x| 2 α 1 + |x| 2 (1+α) 2 u 2 ε |x| 2aε dx = π α + 1 Γ 1 1-ε 2 Γ 2 1-ε , λ ε = R 2 |∇u ε | |x| a 2 dx = 4 a 2 ε R 2 |x| 2 (2 α+1-aε) 1 + |x| 2 (1+α) 2 1-ε dx = 4 π |a ε | 1 -ε Γ 1 1-ε 2 Γ 2 1-ε . Then w ε = (1 + 1 2 ε u) u ε is such that lim ε→0 + 1 κ ε R 2 |w ε | pε |x| bεpε dx = R 2 e u dµ α , lim ε→0 + 1 ε 1 λ ε R 2 |∇w ε | 2 |x| 2aε dx -1 = R 2 u dµ α + 1 16 (1 + α) π ∇u 2 L 2 (R 2 ) .
Hence we can recover (26) by passing to the limit in [START_REF] Dolbeault | About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF] as ε → 0 + . On the other hand, if we pass to the limit in the inequality stated in Theorem 15,

A standard computation shows that

S d |∇v| 2 dω + 1 4 d (d -2) S d |v| 2 dω = R d |∇u| 2 dx and S d |v| q dω = R d |u| q 2 1+|x| 2 d-(d-2) q 2 dx .
On S d , the kernel of the Laplace-Beltrami operator is generated by the constants and the lowest positive eigenvalue is λ 1 = d. The corresponding eigenspace is generated by v 0 (ω) = 1 and v i (ω) = ω i , i = 1, 2, . . . d + 1. All eigenvalues of the Laplace-Beltrami operator are given by the formula

λ k = k (k + d -1) ∀ k ∈ N
according to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF]. We still denote by u ⋆ the Aubin-Talenti extremal function

u ⋆ (x) := (1 + |x| 2 ) -d-2 2 ∀ x ∈ R d .
Using the inverse stereographic projection, the reader is invited to check that Sobolev's inequality is equivalent to the inequality

4 d (d -2) S d |∇v| 2 dω + S d |v| 2 dω ≥ |S d | 2 d S d |v| 2 d d-2 dω d-2 d
so that the Aubin-Talenti extremal function is transformed into a constant function on the sphere and incidentally this shows that

S d = 4 d (d -2) |S d | -2 d .
With these preliminaries on the Laplace-Beltrami operator and the stereographic projection in hand, we can now state the counterpart on R d of the Poincaré inequality on S d .

Lemma 17. For any function

f ∈ D 1,2 (R d ) such that R d f u ⋆ (1 + |x| 2 ) 2 dx = 0 , R d f (1 -|x| 2 ) u ⋆ (1 + |x| 2 ) 3 dx = 0 , and
R d f x i u ⋆ (1 + |x| 2 ) 3 dx = 0 ∀ i = 1 , 2 , . . . d the following inequality holds R d |∇f | 2 dx ≥ (d + 2) (d + 4) R d f 2 (1 + |x| 2 ) 2 dx .
Proof. On the sphere we know that 2 dσ d is monotone decreasing, and hence

S d |∇v| 2 dω + 1 4 d (d -2) S d v 2 dω ≥ λ 2 + 1 4 d (d -2) S d v 2 dω = 1 4 (d + 2)(d + 4) S d v 2 dω if v is orthogonal to v i for any i = 0,
H ′′ = J ′ J H ′ + 2 J S d Q ′ ≤ J ′ J H ′ .
This establishes the proof of Lemma 8 for any d ≥ 3.

Appendix D. Symmetry in Caffarelli-Kohn-Nirenberg inequalities

In this Appendix, we recall some known results concerning symmetry and symmetry breaking in the Caffarelli-Kohn-Nirenberg inequalities [START_REF] Dolbeault | About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF]. Proposition 19. Assume that d ≥ 2. There exists a continuous function α : (2, 2 * ) → (-∞, 0) such that lim p→2 * α(p) = 0 for which the equality case in [START_REF] Dolbeault | About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF] is not achieved among radial functions if a < α(p) while for a < α(p) equality is achieved by This result summarizes a list of partial results that have been obtained in various papers. Existence of optimal functions has been dealt with in [START_REF] Catrina | On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions[END_REF], while Condition (i) in Proposition 19 has been established in [START_REF] Felli | Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type[END_REF]. See [START_REF] Dolbeault | On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities[END_REF] for the existence of the curve p → α(p), [START_REF] Dolbeault | Radial symmetry and symmetry breaking for some interpolation inequalities[END_REF][START_REF] Dolbeault | About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities[END_REF] for various results on symmetry in a larger class of inequalities, and [START_REF] Dolbeault | Symmetry of extremals of functional inequalities via spectral estimates for linear operators[END_REF] for Property (ii) in Proposition 19. Numerical computations of the branches of non-radial optimal functions and formal asymptotic expansions at the bifurcation point have been collected in [START_REF]A scenario for symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF][START_REF] Dolbeault | Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations[END_REF]. The paper [START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF]

2 , 1 -r 2 1+r 2 ∈ 4

 2124 1, . . . d + 1. The conclusion follows from the stereographic projection. if v ⋆ is defined by (C.1). If v solves the fast diffusion equation (18) on R d , then we may use the inverse stereographic projection (see Appendix B) to define the function w on S d such that v(t, x) = e -where τ = -log(T -t), r = |x| and y = 2 x 1+r S d ⊂ R d × R. With no loss of generality, assume that c = λ = 1 and x 0 = 0. According to Lemma 18, w uniformly converges as τ → ∞ to 1 on S d . Let dσ d denote the measure induced on S d ⊂ R d+1 by Lebesgue's measure on R d+1 . We may then write J(t) = e -d with τ = -log(T -t), so that dτ dt = e τ . Hence w solves w τ = L w where L denotes the Laplace-Beltrami operator on the sphere S d , and d+2 dσ d .Using the Cauchy-Schwarz inequality, that is, by writing that

u 2 δ

 2 ⋆ (x) := 1 + |x| (ac-a) -δ ∀ x ∈ R d where δ = ac+b-a 1+a-b . Moreover the function α has the following properties (i) For any p ∈ (2, 2 * ), α(p) ≥ a c -2 d-1 p 2 -4 . (ii) For any p ∈ 2, 2 d 2 -d+1 d 2 -3 d+3 , α(p) ≤ a c -If d = 2, lim p→2 * β(p)/α(p) = 0 where β(p) := α(p) -a c + d/p.

  deals with the special case of dimension d = 2 and contains Property (iii) in Proposition 19, which can be rephrased as follows: the region of radial symmetry contains the region corresponding to a ≥ α(p) and b ≥ β(p), and the parametric curve p → (α(p), β(p)) converges to 0 as p → 2 * = ∞ tangentially to the axis b = 0. For completeness, let us mention that [5, Theorem 3.1] covers the case a > a c -d/p also we will not use it. Finally, let us observe that in the symmetric case, the expression of C a,b can be computed explicitly in terms of the Γ function as C a,b = |S d-1 | of the unit sphere is given by |S d-1 | = 2 π d 2 / Γ d 2 .

d -1 ⋆ F [u n ] -G[u q n ] ,a contradiction since the last term is nonnegative by Theorem 9 and, as observed in Remark 10, J4/d ⋆ F [u n ] -J 1+2/d ⋆ ϕ J 2/d-1 ⋆ F [u n ] is positive unless F [u n ] = 0.Hence we know that L = lim n→∞ F [u n ] = 0 and lim n→∞ G[u q n ] = 0. According to the caracterisation of minimizers of G by Lieb[START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF] Theorem

Acknowlegments. This work has been partially supported by the projects STAB, NoNAP and Kibord of the French National Research Agency (ANR). The authors warmfully thank Mr. Nguyen Van Hoang, who found a mistake in a previous version of this paper. c 2014 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes.

we arrive at the following result, for any α ∈ (-1, 0). Theorem 16. Let α ∈ (-1, 0]. With the above notations, we have

for any u ∈ D, where u and v are related by

.

The case α = 0 is achieved by taking the limit as α → 0 -. Since -∆ log µ α = 8 π (1 + α) µ α holds for any α ∈ (-1, 0], the proof of Theorem 2 is now completed, with µ = µ 0 .

Appendix A. Some useful formulae

We recall that

for any q > 0. An integration by parts shows that f (q + 2) = q q+1 f (q). The following formulae are reproduced with no change from [START_REF] Del Pino | A logarithmic Hardy inequality[END_REF] (also see [START_REF] Dolbeault | Symmetry of extremals of functional inequalities via spectral estimates for linear operators[END_REF][START_REF]Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations[END_REF]). The function w(t) := (cosh t)

and we can define

29

Using the function f , we can compute

+ 2 and get the relations

.

In particular, this establishes (9), namely

for any d > 2. The expression of the optimal constant in Sobolev's inequality (1):

denotes the volume of the unit sphere, for any integer d ≥ 3, follows from the duplication formula

according for instance to [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]. See [START_REF] Dolbeault | Spectral estimates on the sphere[END_REF]Appendix B.4] for further details.

Appendix B. Poincaré inequality and stereographic projection

On

, and define the stereographic projection Σ :

The North Pole N corresponds to z = 1 (and is formally sent at infinity) while the equator (corresponding to z = 0) is sent onto the unit sphere S d-1 ⊂ R d . Now we can transform any function v on S d into a function u on R d using

Appendix C. Flow on the sphere and consequences

We recall that Equation ( 18) admits special solutions with separation of variables given by

where u ⋆ (x) := (1 + |x| 2 ) -(d-2)/2 is the Aubin-Talenti extremal function, x ∈ R d and 0 < t < T . Such a solution is generic near the extinction time T , in the following sense. Lemma 18. [START_REF] Del Pino | On the extinction profile for solutions of u t = ∆u (N -2)/(N +2)[END_REF][START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF]. For any solution v of (18) with nonnegative, not identically zero initial datum v 0 ∈ L 2d/(d+2) (R d ), there exists T > 0, λ > 0, c > 0 and x 0 ∈ R d such that v(t, •) ≡ 0 for any t ∈ (0, T ) and


Adrien Barbaresi
email: adrien.barbaresi@ens-lyon.fr

Finding viable seed URLs for web corpora: a scouting approach and comparative study of available sources

The conventional tools of the "web as corpus" framework rely heavily on URLs obtained from search engines. Recently, the corresponding querying process became much slower or impossible to perform on a low budget. I try to find acceptable substitutes, i.e. viable link sources for web corpus construction. To this end, I perform a study of possible alternatives, including social networks as well as the Open Directory Project and Wikipedia. Four different languages (Dutch, French, Indonesian and Swedish) taken as examples show that complementary approaches are needed. My scouting approach using open-source software leads to a URL directory enriched with metadata which may be used to start a web crawl. This is more than a drop-in replacement for existing tools since said metadata enables researchers to filter and select URLs that fit particular needs, as they are classified according to their language, their length and a few other indicators such as host-and markup-based data.

Introduction

1.1 The "web as corpus" paradigm and its URL seeds problem

The state of the art tools of the "web as corpus" framework rely heavily on URLs obtained from search engines. The BootCaT method [START_REF] Baroni | BootCaT: Bootstrapping corpora and terms from the web[END_REF] consists in repeated search engine queries using several word seeds that are randomly combined, first coming from an initial list and later from unigram extraction over the corpus itself. As a result, so-called "seed URLs" are gathered which are used as a starting point for web crawlers. This approach is not limited to English: it has been successfully used by [START_REF] Baroni | The WaCky Wide Web: A collection of very large linguistically processed web-crawled corpora[END_REF] and [START_REF] Kilgarriff | A Corpus Factory for Many Languages[END_REF] for major world languages.

Until recently, the BootCaT method could be used in free web corpus building approaches. To my best knowledge it is now passé because of increasing limitations on the search engines' APIs, which make the querying process on a low budget much slower or impossible. Other technical difficulties include diverse and partly unknown search biases due in part to search engine optimization tricks as well as undocumented PageRank adjustments. All in all, the APIs may be too expensive and/or too unstable to support large-scale corpus building projects.

API changes are combined with an evolving web document structure and a slow but inescapable shift from "web as corpus" to "web for corpus" due to the increasing number of web pages and the necessity of using sampling methods at some stage. This is what I call the post-BootCaT world in web corpus construction. 1Moreover, the question whether the method used so far, i.e. randomizing keywords, provides a good overview of a language is still open. It now seems reasonable to look for alternatives, so that research material does not depend on a single data source, as this kind of black box effect combined with paid queries really impedes reproducibility of research. Using diverse sources of URL seeds could at least ensure that there is not a single bias, but several.

Additionally, the lack of interest and project financing when dealing with certain less-resourced languages makes it necessary to use light-weight approaches where costs are lowered as much as possible [START_REF] Scannell | The Crúbadán Project: Corpus building for under-resourced languages[END_REF]. In this perspective, a preliminary light scouting approach and a fullfledged focused crawler like those used by the Spiderling [START_REF] Suchomel | Efficient Webcrawling for large text corpora[END_REF] or the COW [START_REF] Schäfer | Building large corpora from the web using a new efficient tool chain[END_REF] projects are complementary. A "web for corpus" crawling method using a seed set enriched with metadata as described in this article may yield better results, e.g. ensure a more diverse and less skewed sample distribution in a population of web documents, and/or reach faster a given quantitative goal.

Looking for alternatives, what issues do

we face?

Search engines have not been taken as a source simply because they were convenient. They actually yield good results in terms of linguistic quality. The main advantage was to outsource operations such as web crawling and website quality filtering, which are considered to be too costly or too complicated to deal with while the main purpose is actually to build a corpus.

In fact, it is not possible to start a web crawl from scratch, so the main issue to tackle can be put this way: where may we find web pages which are bound to be interesting for corpus linguists and which in turn contain many links to other interesting web pages?

Researchers in the machine translation field have started another attempt to outsource competence and computing power, making use of data gathered by the CommonCrawl project2 to find parallel corpora [START_REF] Smith | Dirt Cheap Web-Scale Parallel Text from the Common Crawl[END_REF]. Nonetheless, the quality of the links may not live up to their expectations. First, purely URL-based approaches are a trade-off in favor of speed which sacrifices precision, and language identification tasks are a good example of this phenomenon [START_REF] Baykan | Web Page Language Identification Based on URLs[END_REF]. Second, machine-translated content is a major issue, so is text quality in general, especially when it comes to web texts [START_REF] Arase | Machine Translation Detection from Monolingual Web-Text[END_REF]. Third, mixed-language documents slow down text gathering processes [START_REF] King | Labeling the Languages of Words in Mixed-Language Documents using Weakly Supervised Methods[END_REF]. Fourth, link diversity is a also problem, which in my opinion has not got the attention it deserves. Last, the resource is constantly moving. There are not only fast URL changes and ubiquitous redirections. Following the "web 2.0" paradigm, much web content is being injected from other sources, so that many web pages are now expected to change any time.3 Regular exploration and re-analysis could be the way to go to ensure the durability of the resource.

In the remainder of this paper, I introduce a scouting approach which considers the first issue, touches on the second one, provides tools and metrics to address the third and fourth, and adapts to the last. In the following section I describe my methodology, then I show in detail which metrics I decided to use, and last I discuss the results.

Method

2.1 Languages studied I chose four different languages in order to see if my approach generalizes well: Dutch, French, Indonesian and Swedish. It enables me to compare several language-dependent web spaces which ought to have different if not incompatible characteristics. In fact, the "speaker to website quantity" ratio is probably extremely different when it comes to Swedish and Indonesian. I showed in a previous study that this affects greatly link discovery and corpus construction processes (Barbaresi, 2013a).

French is spoken on several continents and Dutch is spoken in several countries (Afrikaans was not part of this study). Indonesian offers an interesting point of comparison, as the chances to find web pages in this language during a crawl at random are scarce. For this very reason, I explicitly chose not to study English or Chinese because they are clearly the most prominently represented languages on the web.

Data sources

I use two reference points, the first one being the existing method depending on search engine queries, upon which I hope to cast a new light with this study. The comparison grounds on URLs retrieved using the BootCaT seed method on the meta-engine E-Tools4 at the end of 2012. The second reference point consists of social networks, to whose linguistic structure I already dedicated a study (Barbaresi, 2013b) where the method used to find the URLs is described in detail. I chose to adopt a different perspective, to re-examine the URLs I gathered and to add relevant metadata in order to see how they compared to the other sources studied here.

I chose to focus on three different networks: FriendFeed, an aggregator that offers a broader spectrum of retrieved information; identi.ca, a microblogging service similar to Twitter; and Reddit, a social bookmarking and microblogging platform. Perhaps not surprisingly, these data sources display the issues linked to API instability mentioned above. The example of identi.ca is telling: until March 2013, when the API was closed after the company was bought, it was a social microblogging service built on open source tools and open standards, the advantages compared to Twitter include the Creative Commons license of the content, and the absence of limitations on the total number of pages seen.

Another data source is the Open Directory Project (DMOZ5 ), where a selection of links is curated according to their language and/or topic. The language classification is expected to be adequate, but the amount of viable links is an open question, as well as the content.

Last, the free encyclopedia Wikipedia is another spam-resilient data source in which the quality of links is expected to be high. It is acknowledged that the encyclopedia in a given language edition is a useful resource, the open question resides in the links pointing to the outside world, as it is hard to get an idea of their characteristics due to the large number of articles, which is rapidly increasing even for an under-resourced language such as Indonesian.

Processing pipeline

The following sketch describes how the results below were obtained:

1. URL harvesting: queries or archive/dump traversal, filtering of obvious spam and nontext documents.

2. Operations on the URL queue: redirection checks, sampling by domain name.

3. Download of the web documents and analysis: collection of host-and markup-based data, HTML code stripping, document validity check, language identification.

Links pointing to media documents were excluded from this study, as its final purpose is to enable construction of a text corpus. The URL checker removes non-http protocols, images, PDFs, audio and video files, ad banners, feeds and unwanted hostnames like twitter.com, google.com, youtube.com or flickr.com. Additionally, a proper spam filtering is performed on the whole URL (using basic regular expressions) as well as at domain name level using a list of blacklisted domains comparable to those used by e-mail services to filter spam. As a page is downloaded or a query is executed, links are filtered on-the-fly using a series of heuristics described below, and finally the rest of the links are stored.

There are two other major filtering operations to be aware of. The first concerns the URLs, which are sampled prior to the download. The main goal of this operation is strongly related to my scouting approach. Since I set my tools on an exploration course, this allows for a faster execution and provides us with a more realistic image of what awaits a potential exhaustive crawler. Because of the sampling approach, the "big picture" cannot easily be distorted by a single website. This also avoids "hammering" a particular server unduly and facilitates compliance with robots.txt as well as other ethical rules. The second filter deals with the downloaded content: web pages are discarded if they are too short. Web documents which are more than a few megabytes long are also discarded.

Regarding the web pages, the software fetches them from a list, strips the HTML code, sends raw text to a server instance of langid.py (description below) and retrieves the server response, on which it performs a basic heuristic tests.

Metadata

The metadata described in this section can be used in classificatory or graph-based approaches. I use some of them in the results below but did not exhaust all the possible combinations in this study. There are nine of them in total, which can be divided in three categories: corpus size metrics, which are related to word count measures, web science metrics, which ought to be given a higher importance in web corpus building, and finally the language identification, which is performed using an external tool.

Corpus size metrics

Web page length (in characters) was used as a discriminating factor. Web pages which were too short (less than 1,000 characters long after HTML stripping) were discarded in order to avoid documents containing just multimedia (pictures and/or videos) or microtext collections for example, as the purpose was to simulate the creation of a general-purpose text corpus.

The page length in characters after stripping was recorded, as well as the number of tokens, so that the total number of tokens of a web corpus built on this URL basis can be estimated. The page length distribution is not normal, with a majority of short web texts and a few incredibly long documents at the end of the spectrum, which is emphasized by the differences between mean and median values used in the results below and justifies the mention of both.

Web science metrics

Host sampling is a very important step because the number of web pages is drastically reduced, which makes the whole process more feasible and more well-balanced, i.e. less prone to host biases. IP-based statistics corroborate this hypothesis, as shown below.

The deduplication operation is elementary, it takes place at document level, using a hash function. The IP diversity is partly a relevant indicator, as it can be used to prove that not all domain names lead to the same server. Nonetheless, it cannot detect the duplication of the same document across many different servers with different IPs, which in turn the elementary deduplication is able to reveal.

Links that lead to pages within the same domain name and links which lead to other domains are extracted from the HTML markup. The first number can be used to find possible spam or irrelevant links, with the notable exception of websites like Amazon or Wikipedia, which are quite easy to list. The latter may be used to assess the richness (or at a given level the suspiciousness) of a website by the company it keeps. While this indicator is not perfect, it enables users to draw conclusions without fetching all the downstream URLs.

Moreover, even if I do not take advantage of this information in this study, the fetcher also records all the links it "sees" (as an origin-destination pair), which enables graph-based approaches such as visualization of the gathered network or the as-sessment of the "weight" of a website in the URL directory. Also, these metadata may very well be useful for finding promising start URLs.

Language identification

I consider the fact that a lot of web pages have characteristics which make it hard for "classical" NLP approaches like web page language identification based on URLs [START_REF] Baykan | Web Page Language Identification Based on URLs[END_REF] to predict the languages of the links with certainty. That is why mature NLP tools have to be used to qualify the incoming URLs and enable a language-based filtering based on actual facts.

The language identification tool I used is langid.py [START_REF] Lui | langid.py: An Off-the-shelf Language Identification Tool[END_REF]. It is opensource, it incorporates a pre-trained model and it covers 97 languages, which is ideal for tackling the diversity of the web. Its use as a web service makes it a fast solution enabling distant or distributed work.

As the software is still under active development, it can encounter difficulties with rare encodings. As a result, the text gets falsely classified as for example Russian or Chinese. The languages I studied are not affected by these issues. Still, language identification at document level raises a few problems regarding "parasite" languages [START_REF] Scannell | The Crúbadán Project: Corpus building for under-resourced languages[END_REF].

Using a language identification system has a few benefits: it enables finding "regular" texts in terms of statistical properties and excluding certain types of irregularities such as encoding problems. Web text collections are smoothed out in relation to the statistical model applied for each language target, which is a partly destructive but interesting feature.

There are cases where the confidence interval of the language identifier is highly relevant, for instance if the page is multi-lingual. Then there are two main effects: on one hand the confidence indicator gets a lower value, so that it is possible to isolate pages which are likely to be in the target language only. On the other hand, the language guessed is the one with the largest number of identifiable words: if a given web page contains 70 % Danish and 30 % English, then it will be classified as being written in Danish, with a low confidence interval: this information is part of the metadata I associate with each web page. Since nothing particular stood out in this respect I do not mention it further.

URLs

Characteristics of the BootCaT approach

First of all, I let my toolchain run on URLs obtained using the BootCaT approach, in order to get a glimpse of its characteristics. I let the URL extractor run for several weeks on Indonesian and Swedish and only a few days for Dutch and French, since I was limited by the constraints of this approach, which becomes exponentially slower as one adds target languages. 6 The results commented below are displayed in table 1. The domain name reduction has a substantial impact on the set of URLs, as about a quarter of the URLs at best (for French) have different domain names. This is a first hint at the lack of diversity of the URLs found using the BootCaT technique.

Unsurprisingly, the majority of links appear to be in the target language, although the language filters do not seem to perform very well. As the adequate matching of documents to the user's language is paramount for search engines, it is probably a bias of the querying methodology and its random tuples of tokens. In fact, it is not rare to find unexpected and undesirable documents such as word lists or search engine optimization traps.

The length of web documents is remarkable, it indicates that there are likely to contain long texts. Moreover, the median length seems to be quite constant across the three languages at about 8,000 tokens, whereas it is less than half that (3,600) for Dutch. All in all, it appears to be an advantage which clearly explains why this method has been considered to be successful. The potential corpus sizes are noteworthy, especially when enough URLs where gathered in the first place, which was already too impracticable in my case to be considered a sustainable option.

The number of different IPs, i.e. the diversity in terms of hosts, seems to get gradually lower as the URL list becomes larger. The fact that the same phenomenon happens for Indonesian and Swedish, with one host out of five being "new", indicates a strong tendency.

Social networks

Due to the mixed nature of the experimental setting, no conclusions can be drawn concerning the single components. The more than 700,000 URLs that were analyzed give an insight regarding the usefulness of these sources. About a tenth of it remained as responding websites with different domain names, which is the lowest ratio of this study. It may be explained by the fast-paced evolution of microblogs and also by the potential impurity of the source compared to the user-reviewed directories whose results I describe next.

As I did not target the studied languages during the URL collection process, there were merely a few hundred different domain names to be found, with the exception of French, which was a lot more prominent.

Table 2 provides an overview of the results. The mean and median lengths are clearly lower than in the search engine experiment. In the case of French, with a comparable number of remaining URLs, the corpus size estimate is about 2.5 times smaller. The host diversity is comparable, and does not seem to be an issue at this point.

All in all, social networks are probably a good candidate for web corpora, but they require a focused approach of microtext to target a particular community of speakers. ing the best ratio in this study between unfiltered and remaining URLs. The lack of web pages written in Indonesian is a problem for this source, whereas the other languages seem to be far better covered. The adequacy of the web pages with respect to their language is excellent, as shown in table 3. These results underline the quality of the resource.

DMOZ

On the other hand, document length is the biggest issue here. The mean and median values indicate that this characteristic is quite homogeneous the document collection. This may easily be explained by the fact that the URLs which are listed on DMOZ mostly lead to corporate homepages for example, which are clear and concise, the eventual "real" text content being somewhere else. What's more, the websites in question are not text reservoirs by nature. Nonetheless, the sheer quantity of listed URLs compensates for this fact. The corpus sizes for Dutch and French are quite reasonable if one bears in mind that the URLs were sampled.

The relative diversity of IPs compared to the number of domain names visited is another indicator that the Open Directory leads to a wide range of websites. The directory performs well compared to the sources mentioned above, it is also much easier to crawl. It did not cost us more than a few lines of code followed by a few minutes of runtime to gather the URLs.

Wikipedia

The characteristics of Wikipedia are quite similar, since the free encyclopedia also makes dumps available, which are easily combed through in order to gather start URLs. Wikipedia also compares favorably to search engines or social networks when it comes to the sampling operation and page availability. It is a major source of URLs, with numbers of gathered URLs in the millions for languages like French. As Wikipedia is not a URL directory by nature, it is interesting to see what are the characteristics of the pages it links to are. The results are shown in table 3.

First, the pages referenced in a particular language edition of Wikipedia often point to web pages written in a foreign language. According to my figures, this is a clear case, all the more since web pages in Indonesian are rare. Still, with a total of more than 4,000 retained web texts, it fares a lot better than DMOZ or social networks.

The web pages are longer than the ones from DMOZ, but shorter than the rest. This may also be related to the large number of concise homepages in the total. Nonetheless, the impressive number of URLs in the target language is decisive for corpus building purposes, with the second-biggest corpus size estimate obtained for French.

The IP-related indicator yields good results with respect to the number of URLs that were retrieved. Because to the high number of analyzed URLs the figures between 30 and 46% give an insight into the concentration of web hosting providers on the market.

Discussion

I also analyzed the results regarding the number of links that lead out of the page's domain name. For all sources, I found no consistent results across languages, with figures varying by a factor of three. Nonetheless, there seem to be a tendency towards a hierarchy in which the search engines are on top, followed by social networks, Wikipedia and DMOZ. This is one more hint at the heterogeneous nature of the data sources I examined with respect to the criteria I chose.

This hierarchy is also one more search engines queries are believed to be fast and reliable in terms of quantity. This method was fast, as the web pages are long and full of links, which enables to rapidly harvest a large number of web pages without having to worry about going round in circles. The researchers using the Boot-CaT method probably took advantage of the undocumented but efficient filtering operations which search engines perform in order to lead to reliable documents. Since this process takes place in a competitive sector where this kind of information can be sold, it may explain why the companies now try to avoid giving it away for free.

In the long run, several questions regarding URL quality remain open. As I show using a highcredibility source such as Wikipedia, the search engines results are probably closer to the maximum amount of text that is to be found on a given website than the other sources, all the more when the sampling procedure chooses a page at random without analyzing the rest of a website and thus without maximizing its potential in terms of tokens. Nonetheless, confrontation with the constantly increasing number of URLs to analyze and necessarily limited resources make a website sampling by domain name useful. This is part of my cost-efficient approach, where the relatively low performance of Wikipedia and DMOZ is compensated by the ease of URL extraction. Besides, the size of the potential corpora mentioned here could increase dramatically if one was to remove the domain name sampling process and if one was to select the web pages with the most out-domain links for the crawl.

What's more, DMOZ and Wikipedia are likely to improve over time concerning the number of URLs they reference. As diversity and costs (temporal or financial) are real issues, a combined approach could take the best of all worlds and provide a web crawler with distinct and distant starting points, between the terse web pages referenced in DMOZ and the expected "freshness" of social networks. This could be a track to consider, as they could provide a not inconsiderable amount of promising URLs.

Finally, from the output of the toolchain to a full-fledged web corpus, other fine-grained instruments as well as further decisions processes [START_REF] Schäfer | The Good, the Bad, and the Hazy: Design Decisions in Web Corpus Construction[END_REF] will be needed. The fact that web documents coming from several sources already differ by our criteria does not exclude further differences regarding text content. By way of consequence, future work could include a few more linguistically relevant text quality indicators in order to go further in bridging the gap between web data, NLP and corpus linguistics.

Conclusion

I evaluated several strategies for finding texts on the web. The results distinguish no clear winner, complementary approaches are called for. In light of these results, it seems possible to replace or at least to complement the existing BootCaT approach. It is understandable why search engine queries have been considered a useful data source. However, I revealed that they lack diver-sity at some point, which apart from their impracticality may provide sufficient impetus to look for alternatives.

I discussed how I address several issues in order to design robust processing tools which (combined to the diversity of sources and usable metadata) enable researchers to get a better glimpse of the course a crawl may take. The problem of link diversity has not been well-studied in a corpus linguistics context; I presented metrics to help quantify it and I showed a possible way to go in order to gather a corpus using several sources leading to a satisfying proportion of different domain names and hosts.

As a plea for a technicalities-aware corpus creation, I wish to bring to linguists' attention that the first step of web corpus construction in itself can change a lot of parameters. I argue that a minimum of web science knowledge among the corpus linguistics community could be very useful to fully comprehend all the issues at stake when dealing with corpora from the web.

The toolchain used to perform these experiments is open-source and can be found online. 7 The resulting URL directory, which includes the metadata used in this article, is available upon request. The light scouting approach allows for regular updates of the URL directory. It could also take advantage of the strengths of other tools in order to suit the needs of different communities.

Table 2 :

 2 As expected, the number of different domain names on the Open Directory project is high, giv-URLs extracted from a blend of social networks crawls (FriendFeed, identi.ca, and Reddit) with no language target. 738,476 URLs analyzed, 73,271 URLs retained in the global process.

		% in target URLs	Length	Tokens	Different
		retained	mean median	(total)	IPs (%)
	Dutch	0.6 465	7,560	4,162	470,841	68.8
	French	5.9 4,320	11,170	5,126	7,512,962	49.7
	Indonesian	0.5 336	6,682	4,818	292,967	50.9
	Swedish	1.1 817	13,807	7,059	1,881,970	58.5

Table 3 :

 3 URLs extracted from DMOZ and Wikipedia

	reason why

Note that the proponents of the BootCaT method seem to acknowledge this evolution, see for example Marco Baroni's talk at this year's BootCaTters of the world unite (BOTWU) workshop: "My love affair with the Web... and why it's over!"

http://commoncrawl.org/

This is the reason why Marco Baroni states in the talk mentioned above that his "love affair with the web" is over.

http://www.etools.ch/

http://www.dmoz.org/

The slow URL collection is explained by the cautious handling of this free and reliable source, implying a query rate limiting on my side. The scouting approach by itself is a matter of hours.

Acknowledgments

This work has been partially supported by an internal grant of the FU Berlin as well as machine power provided by the COW (COrpora from the Web) project at the German Grammar Department. Thanks to Roland Schäfer for letting me use the URLs extracted from E-Tools and DMOZ.