Christophe Raufaste

Benjamin Dollet

Simon Cox

Yi Jiang

François Graner

Yield drag in

come

I. INTRODUCTION

Multiphase materials such as colloids, emulsions, polymer or surfactant solutions, wet granular systems and suspensions of deformable objects like red blood cells are characterized by a complex mechanical behaviour [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF], due to the interaction of their constitutive entities. The concentration is one of the key parameters which control the rheology, determining especially the transition from liquid-like to solid-like properties [START_REF] Saint-Jalmes | [END_REF]. Amongst these complex fluids, liquid foams provide a convenient model experimental system for laboratory studies of the interplay between structure, concentration and rheology. This is because the bubbles which constitute the foam's internal structure can be easily visualised and manipulated. The mechanical behaviour of foams is very diverse: they appear elastic, plastic or viscous depending on the deformation and velocity * Address for correspondence: craufast@ujf-grenoble.fr † present address: Physics of Fluids, University of Twente, The Netherlands ‡ UMR 5588 CNRS and Université Joseph Fourier.

gradient [START_REF] Weaire | The physics of foams[END_REF][START_REF] Höhler | [END_REF].

A liquid foam consists of gas bubbles separated by a connected network of liquid boundaries. This liquid phase occupies a fraction Φ of the volume of the foam.

The "dry foam" limit, in which Φ tends to zero, corresponds to polyhedral bubbles separated by thin walls. It is associated with a divergence of certain contributions to the viscous dissipation [5]. However, the foam's nondissipative properties (such as surface energy [6], shear modulus or yield stress [7,8]) usually tend to a regular, finite limit when the liquid fraction Φ tends to zero.

The total "yield drag" F t Y is the minimal force observed when there exists (or, equivalently, required to create) a movement of the foam relative to an obstacle [9]. It is a global, geometry-dependent quantity directly measurable in experiments and in practical applications of foams, for instance when a foam flows through a porous medium [10], or when one introduces an object into a foam (analogous to sticking one's finger into shaving cream).

In the low-velocity limit (in which viscous dissipation is neglected [11,[START_REF] Denkov | Subramanian and A. Lips[END_REF]) the total yield drag F t Y has two contributions. These are due to the pressure inside the bubbles, denoted F p Y , and the network of bubble walls (i.e., soap films with surface tension), F n Y . Thus

F t Y = F p Y + F n Y . (1)
Here we consider the network contribution F n Y and show how it is affected by the liquid content of the foam.

We consider a single layer of equal-area bubbles to facilitate preparation and analysis of experiments, as well as numerical and analytical modelling [START_REF]Workshop on Foam Rheology In Two dimensions (FRIT)[END_REF]. Section II presents a 2D flow of a quasi-2D foam (a bubble monolayer) around a fixed circular obstacle within a long channel: this is the historical experiment of Stokes, already adapted to foams both in 2D [9,[START_REF] Asipauskas | [END_REF] and 3D [START_REF] Cox | Proc. 3rd Euroconf. on Foams, Emulsions and Applications[END_REF][START_REF] De Bruyn | [END_REF]17,18] flows. We compare them with truly 2D simulations using two physically equivalent but differently optimised software packages (Section III). The simulation methods allow easy variation of the geometrical parameters such as bubble, obstacle and channel size and better control of bubble area. In Section IV, we discuss the issue of a common, unambiguous definition of liquid fraction for all systems, in theory, experiments and simulations. Section V presents our results: we show that the yield drag displays the expected dependence with the bubble, obstacle and channel size, and increases when the liquid fraction Φ decreases. The discussion in Section VI emphasises that taking into account the effect of liquid fraction allows all data to be plotted on a single master-curve and that, although they cover different ranges of Φ, the results of both simulation and experiment are consistent with a simple model.

II. EXPERIMENTAL METHODS

A. Foam channel

Our bulk soap solution is de-ionised water with 1% Teepol, a commercial dish-washing liquid. Its surface tension, measured with the oscillating bubble method (that is, imaging the interface shape), is γ = 26.1 ± 0.2 mN m -1 , and its kinematic viscosity, measured with a capillary viscometer, is 1.06 ± 0.04 mm 2 s -1 . The experimental set-up [9] confines the foam between a liquid reservoir and a glass lid ("liquid-glass" set-up [19]). A 1 m long, w c = 10 cm wide tank is filled with soap solution, leaving below the glass lid a free space of thickness h which we can adjust. We will call this parameter the "foam thickness" for simplicity. At its centre is a circular obstacle of diameter d 0 = 3 (Fig. 1) or 4.8 cm. At the entrance to the channel, nitrogen is blown at a computer-controlled flow rate, which varies between 5 and 500 ml min -1 . A typical value of the average velocity is 3 mm s -1 , for a 3.5 mm thickness and a flow rate of 50 ml min -1 .

The resulting foam consists of a horizontal monolayer of bubbles. It exits freely at atmospheric pressure at the open end of the channel, P = P atm . In the absence of the obstacle, it yields a two-dimensional plug flow. With the obstacle present, the flow remains twodimensional (even though the foam itself is not exactly 2D [19]): there is no vertical component of the velocity.

Due to the presence of the obstacle, there is a velocity gradient. There are many bubble rearrangements (so called "T1s" or neighbour-swapping events): two three-fold vertices contact, merge and re-separate. We observe well-separated T1s; that is, between two T1s, there is enough time for the foam to relax to an equilibrium state. The present flow is slow enough that we can extrapolate the results to the low velocity limit [9],

where a comparison with quasi-static calculations and simulations makes sense [20]. (Although note that this is distinct from the zero-velocity case, that is the absence of flow.) The data presented below are all in this limit, extracted from the experiments as described in Appendix A 1.

The bubble walls meet the solid boundaries of the foam (glass plate, lateral channel walls, obstacle itself) at a 90 • angle [21]. The surface density of bubbles is 1/A, where A is the average area per bubble (including its walls). The foam is monodisperse: the area variation at the channel entrance is less than 5%. The average area is fixed at a value ranging from 0.121 cm 2 to 0.393 cm 2 ; most experiments have A = 0.160 cm 2 . Despite the low velocity, and hence the long transit time, we detect neither bubble coalescence nor coarsening. The effect of foam ageing on rheology [22,23] is thus negligible.

B. Force measurements

Total yield drag

The obstacle floats just below the top glass surface and is free to move, without solid friction. However, it is linked to a fixed base through a calibrated elastic fibre.

We track the obstacle displacement from its position at rest using a CCD camera which images the foam flow from above. We thus measure the force exerted by the flowing foam on the obstacle (precision better than 0.1 mN) [9].

We check that the lift (spanwise component of the resultant force) is consistently zero, within fluctuations, as expected by symmetry (data not shown). After a transient, the total drag F t (streamwise component of the resultant force) fluctuates around a steady value: we record the average and standard deviation of these steady flow data. The extrapolation to the low velocity limit (or zero-velocity intercept) of the force-velocity curve defines the yield drag F t Y . It is independent of the bulk solution viscosity [24], and increases with the obstacle to bubble size ratio [9].

In this paper, we reanalyse the data already published in [9] at various bubble areas, and we present new data for another control parameter: the foam thickness. These data are presented in Appendix A 1. As explained in Sec. IV B, the foam thickness provides a means by which to vary the liquid fraction in experiments.

Network contribution to the yield drag

We measure F n Y as follows. Each bubble wall in contact with the obstacle pulls it with a force equal to its line tension λ (the energy per unit length, which is of order 2γh, see Appendix A 2). The elastic contribution of the wall network to the drag is then the vectorial sum of all these individual forces, which all have the same modulus λ. As mentioned above, in a quasi-static flow each wall touches the obstacle at 90 • angle. Thus it suffices to find the contact points between bubble walls and the obstacle and sum all outward normal vectors to the obstacle vectorially at these contact points (which is easy to determine for a circular obstacle). If the downstream geometry of the foam was the same as that upstream, the drag would be zero. Since bubbles are squashed upstream and stretched downstream, the asymmetry means that there are more bubble walls pulling the obstacle downstream, and we measure a downstream elastic contribution to the drag, F n Y /λ. The Surface Evolver [25,26] offers the possibility to reach a true quasi-static limit, that is a succession of exact equilibrium states, through a deterministic minimisation of the foam's energy. It yields precise details of the foam structure. The lateral sides of the channel are rigid and do not interact with the foam, ensuring free-slip boundary conditions for the flow, resulting in a 90 • angle where a bubble wall meets the side. We adopt a periodic boundary condition in the direction of motion: bubbles that exit at the end of the channel are fed back into the entrance of the channel. We stop the simulation when each bubble has passed the obstacle no more than once.

We begin with a rectangular lattice of 30 × 25 monodisperse bubbles of area slightly larger than the required area A. We randomly perturb this lattice so that all the unstable four-fold vertices dissociate into pairs of three-fold vertices and the whole foam structure relaxes towards equilibrium. We then choose one Bubbles coloured in white are without topological defect:

6-sided bulk bubbles, or 5-sided bubbles touching a lateral wall or the obstacle [6]. Bubbles with fewer neighbours are in dark grey, bubbles with more are in light grey.

bubble to be the circular obstacle, and slowly increase its area to the required value (and correspondingly reduce the bubble areas to A) and constrain its edges to lie on a circle. The centre of the circular obstacle is then moved to the centre of the channel and the structure again relaxed to equilibrium.

Simulation of the flow

With the obstacle in the desired location and the foam close to equilibrium, we start the quasi-static iteration procedure. This requires that we move the foam past the obstacle, in a direction which we denote by x;

the difficulty is in doing this with the periodic boundary conditions without fixing any vertices or bubble shapes.

Our method is to choose a continuous line of consecutive bubble walls from one side of the channel to the other. Joining this to a line at x = 0 with lines along the channel walls defines a plane region with a certain area, which we constrain. At each iteration we choose a convenient line of consecutive walls, and increment the target area of the region formed by a small amount dA (equal to 0.05 cm 2 in all simulations), resulting in a slight movement of a line of films without modifications to the bubble areas. The total perimeter of the structure is then reduced until it converges to a constant value (Fig. 2a), so that measurements can be performed.

We have double-precision values for the network geometry. We measure the network contribution F n Y to the yield drag as in experiments (section II B 2). It is the sum of the unit vectors of the bubble wall with one end attached to the obstacle, expressed in units of the line tension (hence as a dimensionless number). Here too, we check that the lift is consistently zero within fluctuations (data not shown).

With the area increment dA = 0.05 cm 2 , the transient lasts for about 600 iteration steps (Fig. 3a). This is comparable to, but still smaller than, the total simulation time that is reasonably accessible. After this transient, the drag fluctuates around a steady value. Such

B. Stochastic simulations (Potts Model)

To simulate a larger number of bubbles, the Potts model adapted for foam rheology [29] also minimises the same energy, but stochastically (Monte-Carlo), which increases the simulation speed. It thus provides more statistics on F n Y and allows quicker variation of the geometrical parameters.

Principle of the Potts Model

The Potts model is derived from a large-Q Potts model run at zero temperature, a model widely used to model grains in crystals [30]. It has been also applied to different domains of foam physics, including rheology, by enforcing the conservation of bubble size and applying an external force [29].

We consider a 2D square lattice. Each site i has an integer index σ i . The k th bubble is defined as the domain consisting of all sites with the same index value σ i = k. Thus bubbles tile the plane without gaps or overlaps. The evolution is driven by the minimisation of a total energy H (strictly speaking, it is a Hamiltonian), which has the same three physical ingredients as in the Surface Evolver: interfacial energy, area constraints, external forcing of the flow. Since the calculations are performed on a lattice, we have

H = λ i,j neighbours [1 -δ(σ i , σ j )] +χ bubbles k A k -A t k 2 + b sites i x i . (2)
The first term represents the contribution of the energy of the interfaces between the bubbles. Minimising this term leads to perimeter minimisation. Here δ is the Kronecker symbol: 1 -δ is equal to 1 if the neighbouring sites i, j belong to different bubbles (σ i = σ j ); else it equals zero. We choose to evaluate this term with the fourth nearest neighbour interactions to obtain an isotropic line tension insensitive to the details of the lattice [31].

The second term keeps each bubble area A k (the number of sites with the same index) close to its predefined target value A t k . Here χ is the compressibility, which we choose to be high enough to keep bubble areas constant to within a few pixels. The balance between this term and the preceding one simulates a foam relaxing towards mechanical equilibrium.

The third term is a bias term that describes an energy gradient, hence a homogeneous external force field.

Here b is the bias intensity and x the site's coordinate along the flow. Without obstacle, the resulting velocity profile would be a plug flow.

We use a Metropolis algorithm to evolve the foam: we randomly select a site at a bubble boundary, change its index to the value of a neighbour if and only if this decreases the total energy (eq. 2). Several independent changes are tried successively; a Monte Carlo Step (MCS) is defined conventionally as a number of tries equal to the total number of lattice sites. We then switch the bias on again, and use the smallest bias b for which the foam flows, which is constant and independent of parameters such as bubble diameter. We perform measurements at intervals of 1500 MCS (during which a bubble moves a few pixels).

We measure the network contribution to the drag using the same method as in the experiments and Surface Evolver simulations. It fluctuates around a steady value: we record the average and standard deviation of these plateau (steady-flow) data (Fig. 3b). We run each simulation for a total of 600,000 MCS, during which a bubble passes completely through the channel but no bubble passes the obstacle twice. One simulation takes about 12 hours on a Pentium IV 2.8 GHz processor.

IV. LIQUID FRACTION

In ideal 2D foams (Sec. IV A), given A, the liquid fraction scales as the square of the vertex radius. This radius in turn relates to a cut-off length L c , that is the length at which a bubble edge becomes unstable and the surrounding bubbles undergo a T1. This length L c is always defined, and is relevant to the mechanical properties investigated here. In experiments, L c depends on the actual (3D) shape of bubbles; in simulations, L c is an input parameter.

It suggests a coherent definition of the effective liquid fraction, presented below, consistent within and between experiments, simulations and theory. Note that we consider here a foam dry enough to have a non-zero shear modulus [32], that is, below the critical liquid fraction [7] (see eq. 8 for the honeycomb value).

A. Ideal 2D foams

In an ideal 2D foam, a Plateau border is a triangle with concave edges of radius R which match tangentially three straight lines meeting at 120 • (Fig. 4). The area A P B of a Plateau border is [START_REF] Weaire | The physics of foams[END_REF]:

A P B = √ 3 - π 2 R 2 . ( 3
)
The liquid fraction Φ is defined as:

A l = AΦ, (4)
where A l is the area occupied by the liquid. Since each bubble has n Plateau borders, each being shared be- tween 3 bubbles, we have A l = nA BP /3. For a honeycomb array of bubbles, n = 6 and:

Φ = 2 √ 3 -π R 2 A . (5)
Here we consider foams where bubbles have the same area (monodisperse foams), but not necessarily the same number of sides n (topological disorder). Since on average over the whole foam n ≈ 6 [START_REF] Weaire | The physics of foams[END_REF], and since Plateau borders have almost the same size and radius of curvature, eq. ( 5) still holds here approximately.

A T1 is triggered when the distance between these vertices becomes smaller than a cut-off wall length L c , which increases with R, and thus with Φ. To make this observation more quantitative, one possible convention to define L c is the condition that two vertices touch (Fig. 4):

R √ 3 = L c 2 , (6)
so that, together with eq. ( 5):

Φ = 3 2 √ 3 - π 2 L 2 c A ≈ 0.242 L 2 c A . (7)
Given A, the physical information conveyed by R, Φ or L c is the same. For comparison between different experiments or simulations, we use Φ because it is dimensionless.

For an ideal honeycomb without shear, all vertices merge at the same liquid fraction: the hexagons become circular when L c equals the side-length of the hexagons;

the bubbles are circular, with a radius equal to R. This critical liquid fraction is [7]:

Φ c = 1 - π 2 √ 3 = 0.0931. (8)

B. Experiments

In the experiments, the actual (3D) shape of bubbles is determined by the foam thickness. Fig. [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF] shows a foam thickness of 4.5 mm; beyond this thickness, the bubbles undergo a three-dimensional instability and the foam is no longer a monolayer [33]. At the other extreme, at 2 mm and below, the bubbles are circular and separated: both the foam's 2D shear modulus and the yield drag vanish. When h increases, L c decreases, thus Φ decreases too (Fig. 9 in Appendix A).

We show in Appendix B that there is a correspondence between L c and the length L max of the edge of a bubble just attached to the obstacle (Fig. 11c in Appendix B). Since L max is much bigger than L c , it can be estimated with much more precision in experiment (the uncertainty in both L c and L max is one pixel).

We measure on the skeletonized image the length L max several times preceding its disappearance during a T1 process, and keep the average as L max and the standard deviation as δL disp max . The skeletonization itself induces a systematic error in determining the actual position of the vertex centre; we estimate as δL skel max = 1 pixel the systematic error on L max . The total uncertainty on L max is therefore δL max = (δL disp max ) 2 + (δL skel max ) 2 . To deduce Φ from the measurements of L max , we combine Eqs. ( 5) and (B3) to get the following expression as a function of L 2 max /A only:

Φ = 3 2 2 √ 3 -π 2 + √ 3 √ A L max - L max 4 √ 3A 2 . ( 9
)
Its uncertainty is:

δΦ Φ = 2 δL max L max √ A/L max + L max /4 √ 3A √ A/L max -L max /4 √ 3A .

C. Simulations

The Surface Evolver requires that we specify explicitly the cut-off wall length L c at which two three-fold vertices are allowed to contact, merge and re-separate.

Since L c is an input parameter, it is determined without uncertainty. This defines explicitly an effective liquid fraction (eq. 7), at least for small values of Φ. We choose L c to be of the order of 0.1 cm or slightly smaller, reaching Φ = 0.0015, 0.0037, 0.0061 and 0.015. At very small values of Φ < 6 10 -6 , films behind the obstacle would get very stretched and lead to numerical problems. Attempting larger values of Φ > 0.015 would lead to poor convergence in the Surface Evolver and would require that we simulate the actual geometry of the liquid in the vertices (including 4-fold vertices).

In the Potts model, the cut-off distance L c at which two vertices merge is either 1 or 2 pixels. We use this range to define the uncertainty on the value of Φ. Since we will plot the results in log scale, we choose L c ≈ √ 2 and Φ ≈ 0.242 × 2/A ≈ 0.5A -1 to lie in the middle of this interval. Thus the area A of bubbles (that is, the number of pixels per bubble) defines an effective liquid fraction, at least for small values of Φ. The simulated range 64 pix 2 ≤ A ≤ 400 pix 2 corresponds to 0.00125 < Φ < 0.0075, large enough to describe realistically the shape of bubbles, and small enough to keep the computation time reasonable.

V. RESULTS

The experiments and both simulations present qualitatively similar images (Figs. 1,2) and consistent results for the yield drag force, always directed downstream.

There are a priori four lengths in this problem: the channel width w c , the obstacle diameter d 0 , the bubble size √ A; and the cut-off length L c . As far as we can tell, it is safe to assume that the channel length (if long enough) is irrelevant here. These four lengths can be reduced to three dimensionless parameters. We present the results using: d 0 /w c which characterises the flow geometry; d 0 / √ A which describes the foam-obstacle interaction; and L 2 c /A which characterises the threshold for T1 rearrangements, and corresponds to the liquid fraction Φ.

A. Effect of obstacle to channel size ratio

Potts model simulations indicate that the network yield drag is independent of the ratio of obstacle size to channel width, d 0 /w c (Fig. 5a). This ceases to be valid at small d 0 , when the obstacle is comparable in size to a bubble, and at large d 0 , when the distance between the obstacle and the channel side is small [26].

The lack of dependence on d 0 /w c that we find characterises the yielding behaviour of the foam: it means that only a small region near the obstacle is affected by the flow [START_REF] De Bruyn | [END_REF]. Nonetheless, the zone where the obstacle influences the flow is larger in 2D [34] than in 3D [START_REF] De Bruyn | [END_REF], as elastic or hydrodynamic interactions would suggest. of the obstacle's spanwise dimension ("leading edge") [9]. Note that most elastic properties of a foam scale like 1/ √ A [START_REF] Weaire | The physics of foams[END_REF]. In fact, when A increases, the density of bubbles and of bubble walls decreases, and so does a foam's elastic modulus (it would eventually vanish if there were only one large bubble left).

C. Effect of liquid fraction

We need to separate the effects of foam geometry, well rescaled in the range 10 -3 < Φ < 10 -1 (Fig. 6). This is the main result of the present paper.

d 0 / √ A,

VI. DISCUSSION

A. Model

The effect of the liquid fraction on the network drag can be understood as follows. A bubble of area A detaches from the obstacle when its width is of order L c , and thus its length is of order A/L c . When Φ decreases, L c decreases too. Bubbles stretch more downstream, and more bubbles pack behind the obstacle. The num-ber of bubble walls pulling the obstacle downstream increases; simultaneously, the number of walls upstream decreases. This larger up/downstream asymmetry results in an increase in the resulting drag F n Y . The contribution from the network (or bubble walls) increases as their number per unit length along the obstacle boundary, namely L -1 c , and thus scales like 1/ √ Φ.

However, the length of the region on which stretched bubbles act decreases, and the divergence in 1/ √ Φ is in fact softened by a geometrical factor. As shown in Appendix B, we can estimate this factor by integrating the bubble wall contribution around the obstacle. When Φ increases, F n Y decreases; it vanishes for Φ = 0.086. This is close to the rigidity loss value (eq. 8). Eq. (B10) is plotted in Fig. (6), without adjustable parameters.

It shows qualitative agreement with the data over two decades of liquid fraction, suggesting that it captures the essence of the physics.

B. Influence of the control parameters

In the limit of low Φ, the development of the above argument indicates that, provided that the obstacle diameter and the obstacle-wall distances are larger than the bubble diameter, F n Y increases according to:

F n Y = 0.516 Φ 1/4 λ d 0 √ A . (10)
In simulations, if we multiply the bubble and obstacle diameters, expressed in units of the cut-off length, by the same prefactor, the network drag changes (data not shown), due to the change in Φ.

Conversely, increasing only the bubble area A at fixed L c simultaneously decreases both d 0 / √ A and Φ. This has two opposing effects, the former decreasing F n Y , the latter increasing it, resulting in an almost constant F n Y (Fig. 7). (In fact there is a weak dependence on area, varying as A -1/4 .) This shows that the relevant way to vary the liquid fraction in simulations is to modify Direct observation of simulation images of very dry foams (Fig. 8) confirms that the up/downstream asymmetry in the number of bubbles touching the obstacle is around 10, roughly independent of liquid fraction.

L 2 c /A at given d 0 / √ A.
The model seems to correctly describe the squashing and stretching of bubble shapes. However, the interpolation between both extreme values assumes a phenomenological expression (eq. B4). It seems approximately valid only for 10 -3 < Φ < 10 -1 (see Appendix B). It applies to other obstacle shapes, such as an ellipse [35].

D. Yield drag versus yield stress

Princen and Kiss [36] have shown that in three dimensions a foam's yield stress scales as:

σ Y = γ(1 - Φ 3D ) 1/3 Y (Φ 3D )/R 32
, where R 32 is the surface-volume mean radius (Sauter radius), and Y (Φ 3D ) a decreasing function of Φ 3D which is approximately Y (Φ 3D ) ≃ -0.080 -0.114 ln Φ 3D [36].

At this stage, it is worth discussing the fundamental differences between yield stress and yield drag.

The yield stress or yield strain is an intrinsic property of the foam. On the other hand, the yield drag depends on the geometry of the flow: the foam does not yield everywhere around the obstacle, and especially not at angles |θ| ≈ π/4 from the downstream direction, as appears both in experiments (Fig. 1) and in simulations (Figs. 2 and8). This spatial dependence implies that the relation between yield stress and yield drag is nontrivial and Φ-dependent. In particular, in a dryer foam (Fig. 8), the region where bubbles reach their maximal deformation is narrower [34].

Moreover, the flow around an obstacle involves not only shear, but also elongation, especially near the front and back of the obstacle. When Φ decreases, the bubble elongation can become arbitrary large and dominates the contribution to the yield drag. The yield drag is defined as the low-velocity limit of the interaction force between an obstacle and a flowing foam. The network contribution scales as the ratio of obstacle to bubble diameter, as long as this ratio is larger than unity, and is almost independent of the channel width. It increases (because more and more stretched bubbles accumulate behind the obstacle) as a power law when the liquid fraction contained in the foam decreases to 10 -3 , then saturates.

Having found a relevant definition of the liquid fraction, which is appropriate for experiments, simulations and theory, the dependence of yield drag with liquid fraction is well characterized. It is very different from that of local intrinsic properties such as the yield stress or shear modulus. This observation suggests that it will be difficult to deduce one quantity from the other. This should be kept in mind in future simulations, and has to be taken into account when modelling the foam behaviour.

Note that this definition of liquid fraction can be extended to other 2D flows in experiments (quasi-2D foam set-ups [19]) or simulations. Extension to 3D [START_REF] Cox | Proc. 3rd Euroconf. on Foams, Emulsions and Applications[END_REF][START_REF] De Bruyn | [END_REF] should also be simple, especially since the main effect of quasi-2D set-ups -external friction on the glass plate [37,38] -does not seem dominant here. Our present effective liquid fraction based on rheological properties (T1s) facilitates the comparison between 2D and 3D flows (which is difficult when using the actual volume fraction of water [START_REF] Denkov | Subramanian and A. Lips[END_REF]). In simulations too, our definition immediately extends to 3D for both Surface Evolver, which uses as input parameter the cut-off area for a face which undergoes a T1; and Potts model, where the voxel (3D pixel) size plays the same role.

thicknesses.

Notably, for the smallest foam thickness (h = 2.0 mm) the foam is almost decompacted and the drag tends to vanish at low velocity. More precisely, the fit gives an unphysical negative value. This suggests that the rigidity loss transition [32] occurs for a foam thickness between 2.0 and 2.5 mm.

Comparison of network and total yield drags

We measure the line tension directly as the force ex- suggests that λ/h ≈ 110 mN/m. The actual gas-liquid interfaces have a 3D curvature to match tangentially the water surface and the glass plate (Fig. 10). This explains why the the measured value is between a lower bound, λ/h = 2γ ≈ 52 mN/m expected for a vertical soap film (that is, two flat gas/liquid interfaces), and an upper bound, λ/h = (2 + π)γ ≈ 134 mN/m, expected for two films with circular cross section (see [START_REF] Dollet | Écoulement bidimensionnels de mousse autour d'obstacles[END_REF] for details).

Using the measured value of λ, we determine the absolute value of the network yield drag F n Y in experiments. We check (data not shown) that F n Y is consistently of the same order of magnitude, but lower than, the value of F t Y measured directly. The remaining part is attributed to the pressure contribution F p Y , to be described in a further paper [34]. The spatial variation of bubble height h due to pressure differences is always less than 10%, giving an upper limit to the spatial variations of λ.

APPENDIX B: VARIATION OF F n Y WITH THE CUT-OFF LENGTH

We consider here only the bubbles touching the obstacle, and the contribution of their walls to the yield drag.

We assume that (i) the foam is truly 2D and all bubbles have the same area (ideal 2D monodisperse foam);

(ii) pressure P is the same for each bubble, i.e. bubble walls are straight and all Plateau borders have the same radius of curvature R; (iii) the obstacle is much larger than the bubbles ( √ A ≪ d 0 ) so that we can neglect its curvature at the bubble scale. This latter approximation could in principle affect the bubbles upstream, which share a long edge with the obstacle. However, it should not greatly affect the bubbles downstream, which are the main contributors to the drag.

Geometry

To model a wet foam, we apply the decoration theorem [32]: the liquid is present only at the vertices which decorate an ideally dry foam. For a bubble touching the obstacle, we denote by L the distance between two neighbouring vertices in contact with the obstacle (Fig. 11a).

When the foam flows, bubbles attach to the obstacle upstream, and detach from it downstream. Visual observation of both experiments (Fig. 1) and simulations (Fig. 2) indicate that bubbles are flattened along the obstacle at the leading side of the obstacle, and that they progressively stretch streamwise at the trailing side.

L reaches its minimum value downstream, where bubbles detach. There, two neighbouring (decorated) vertices come in contact, and L equals the cut-off length 2R (Fig. 11b). and now equals (1 + 1/ √ 3)R. This geometrically determines that the maximum bubble width L max obeys:

A = 1 + 1 √ 3 RL max + L 2 max 4 √ 3 . ( B1
)
Inverting eq. (B1) yields L max :

L max (A, R) = 2 ( √ 3 + 1) 2 R 2 + A √ 3 -2( √ 3 + 1)R. ( B2
)
At low liquid fraction, L max tends to a finite value, namely 4A √ 3; there is no singularity at vanishing R.

Conversely, at high liquid fraction, L max varies greatly with R, so it is preferable to rewrite eq. (B2) and determine R from the measurement of L max :

R(A, L max ) = 1 + 1 √ 3 -1 A L max - L max 4 √ 3 . (B3)

Continuous assumption

We assume that the shape of the bubbles varies smoothly from the configuration of Fig. To compute this integral, we introduce two dimensionless variables, both functions of A and R:

ε = R √ A , (B6)
β = L max 2R . (B7)
The physical meaning of ε is equivalent to the liquid fraction, since

Φ = (2 √ 3 -π)ε 2 . (B8)
On the other hand, β quantifies the amount of up/downstream asymmetry, that is, the squashing and stretching of bubbles. It increases when Φ (or equivalently ε) decreases (eq. B2):

β(Φ) = ( √ 3 + 1) 2 + (6 - √ 3π) Φ -( √ 3 + 1). (B9)
When Φ goes to zero, ε goes to zero too, and β diverges.

Using these variables, eq. (B5) yields

F = λ d 0 L max β √ β -1 arctan β -1 -1 . (B10)
At high liquid fraction, the force F vanishes when β = 1, that is (eq. B9) when:

Φ = 2 √ 3 -π 2 + √ 3 = 0.086. (B11)
At low liquid fraction, we develop eq. (B10) to leading order in β and insert the leading order term of eq. (B9) to obtain eq. ( 10).

FIG. 1 :

 1 FIG. 1: Image of the experiment, with the foam confined between liquid and glass and flowing from top to bottom. Foam thickness h = 4.5 mm; bubble area A = 16 mm 2 ; obstacle diameter d0 = 3 cm; mean velocity v = 5.6 mm s -1 ; effective liquid fraction Φ = 0.06.

1 .

 1 Preparation of the foam We use a mode in which all bubble walls are represented as circular arcs. The Surface Evolver lets these circular arcs evolve in order to minimise the total perimeter (equivalent to the energy, up to the prefactor λ). It enforces the constraint that bubble areas A remain fixed and determines the corresponding Lagrange multipliers, namely each bubble's pressure P . Since we can freely choose the units, we call them "cm" and use bubble size A = 0.16 or 0.353 cm 2 , channel width w c = 10 cm, and obstacle diameters d 0 = 1.5, 3 and 4.8 cm, to reproduce actual experiments.

FIG. 2 : 16 cm 2 ,

 2162 FIG. 2: Images of simulated foam flow. The x-axis is parallel to the flow along the channel, with periodic boundary conditions (exiting bubbles re-enter); axis y is spanwise, with free-slip rigid boundary conditions on either side of the channel. (a) Surface Evolver. The image shows the whole simulation domain of 750 bubbles; the shaded bubbles started in a horizontal line. Here d0 = 4.8 cm, A = 0.16 cm 2 , wc = 10 cm, Lc = 0.05 cm and therefore Φ = 0.0037. (b) Potts model. The image shows the simulated channel's full width (except for a few pixels) of 256 pix, and half its length. Here d0 = 74 pix, A = 100 pix 2 and Φ = 0.005.

FIG. 3 :

 3 FIG. 3: The network force F n Y (expressed in units of the line tension λ) measured in simulations versus time. (a) Surface Evolver data plotted every iteration step; d0 = 4.8 cm, A = 0.16 cm 2 and Lc = 0.05 cm, Φ = 0.004. The plateau value is F n Y = 9.6±1.4. (b) Potts model data plotted every 1500 Monte Carlo Steps; d0 = 74 pix, A = 100 pix 2 , Φ = 0.005. The plateau value is F n Y = 5.4 ± 1.1.

2 .

 2 Simulation of the flow As for the Surface Evolver (section III A 1), we choose a periodic boundary condition in the direction of flow and free-slip rigid boundary conditions on the channel sides. To ensure that it does not affect the steady-state measurements presented below, the total channel length is 4w c , out of which only 2w c are used for measurements and are shown on Fig. (2b). To match the experiments, we choose 16 ≤ d 0 ≤ 148 pix, 64 ≤ A ≤ 400 pix 2 and 64 < w c < 512 pix. Initially, we insert a rigid round obstacle in the centre of the channel, and let a perfectly ordered foam (honeycomb pattern) flow in. We then switch off the bias term by setting b = 0, and relax the foam to ensure that the bubbles recover their (near) equilibrium state. The foam has reached the stationary state at the end of this preparation.

FIG. 4 :

 4 FIG. 4: Picture of two adjacent three-fold vertices with Plateau borders. (a) We apply the decoration theorem [32] to model a wet foam. The liquid is present only at the vertices, and (assuming here straight or nearly-straight walls) the uniformity of pressure P inside bubbles forces each gas/liquid interface to have the same radius of curvature, R. (b) Critical position of the vertices just before the "T1" neighbour-swapping event. It defines the cut-off wall length Lc.

B.FIG. 5 :

 5 FIG. 5: Network contribution to the yield drag F n Y (expressed in unit of λ), measured in Potts model simulations with A = 100 pix 2 (Φ = 0.005). (a) F n Y versus wc, for d0 = 74 pix; the solid line is the average (value 5.43). (b) F n Y versus d0, for wc = 256 pix; the solid line is a linear fit with zero intercept, F n Y = 0.77 d0/ √ A.

FIG. 6 :

 6 FIG. 6: The network contribution to the yield drag F n Y , rescaled by λd0/ √ A, is plotted versus the dimensionless quantity L 2 c /A (bottom scale), that is versus the effective liquid fraction Φ (top scale), in the range 10 -3 < Φ < 10 -1 . All control parameters (d0, wc, A and Φ) are varied. Vertical bars indicate the standard deviation of the force fluctuations in time around the plateau value. Horizontal bars indicate the uncertainty discussed in Sec. IV. Data are from experiments ( ), Surface Evolver (•) and Potts model (△); the solid line denotes the analytical model, from eq. (B10), without adjustable parameters. Note that the horizontal scales are logarithmic and shifted with respect to each other.

FIG. 7 :

 7 FIG. 7: Opposite effects in simulations of F n Y (here with Potts model). When A/L 2 c increases, both d0/ √ A and Φ ≈ A -1 decrease; so that F n Y barely varies (eq. 10). Obstacle diameter d0 equal to 16 (△), 32 ( ), 74 ( ) and 128 (•).

FIG. 8 :

 8 FIG. 8: Surface Evolver simulation of a very dry foam: zoom around the obstacle. (a) Φ = 3.7 10 -3 ; (b) Φ = 6 10 -6 . Here d0 = 4.8 cm, A = 0.16 cm 2 , wc = 10 cm. Flow from top to bottom.

  VII. CONCLUSIONTo summarise, we investigate the two-dimensional flow of a foam around a circular obstacle, within a long channel. Our deterministic (Surface Evolver) and stochastic (Potts model) simulations, as well as our model and experiments, complement and validate each other.

FIG. 9 :

 9 FIG. 9: (a) Drag versus velocity for various distances h between the top plate and the bottom solution: 2.0 (•), 2.5 ( ), 3.0 (•), 3.5 (△), 4.0 ( ) and 4.5 mm (×). Here A = 16.0 mm 2 and d0 = 3 cm. The curves are the best fits by the formula F = F t Y + A × V a . Snapshots of the foam of the smallest (bottom) and highest (top) h are also displayed. (b) Photograph of a single soap film pulling on the left side of the circular obstacle.

  erted on the obstacle by a single soap film, as shown in Fig. (9b), for two thicknesses. Its value is 0.44 mN for h = 4.0 mm, and 0.49 mN for h = 4.5 mm. This

FIG. 10 :

 10 FIG. 10: Shape of the interfaces between bubbles, for bubbles of area 16.0 mm 2 and volume V = 16.0 × 3.5 mm 3 , calculated with the Surface Evolver. Vertical films are shown in light grey and the liquid surface in dark grey. To reproduce the experiment, we enforce the hexagonal symmetry and include the buoyancy. For simplicity, the junction between lateral faces and the top plate is assumed to be orthogonal.

FIG. 11 :

 11 FIG. 11: Model configuration of the bubbles in contact with the obstacle. (a) Equilibrium configuration: the dashed line represents a polygonal bubble at Φ = 0, the dotted line is a circular bubble at Φ = Φc, and the plain line represents the intermediate case (0 < Φ < Φc), with straight edges and curved triangular vertices. (b) Configuration at the limit of detachment: two neighbouring vertices on the boundary of the obstacle come into contact. (c) Configuration at the point of attachment of a new bubble. There is one vertex between bubble 2, bubble 3 and the obstacle, and a second vertex between bubbles 1, 2 and 3. When these two vertices come in contact, bubble 1 attaches to the obstacle.

2 L

 2 (11c) upstream to that of Fig. (11b) downstream: 2R < L < L max .Since the obstacle is much larger than the bubbles, we switch from a discrete to a continuous description of the bubbles. We thus consider L as a continuous function of the ortho-radial angle θ along the obstacle boundary: θ = 0 downstream, π (and -π) upstream. Equivalently, L -1 is the linear density of vertices along the obstacle boundary.Then L(0) = 2R, L(±π) = L max . To interpolate between these values, we assume the following phenomenological dependence, reflecting that all bubbles in the range |θ| ≥ π/2 appear squashed against the obstacle:|θ| ≤ π/2 : L(θ) = R + 1 2 L max + R -1 max cos 2θ |θ| ≥ π/2 : L(θ) = L max . (B4)Since each bubble edge exerts a pulling force of magnitude λ along the outward normal vector of the obstacle boundary, the network contribution to the drag is

Acknowledgements

We gratefully acknowledge the help of Steven Thomas with Potts model simulations. We thank Isabelle Cantat for stimulating discussions, and emphasising the differences between yield drag and yield stress.

CR thanks LANL and SC thanks LSP for hospitality. YJ is supported by US DOE under contract

No. DE-AC52-06NA25396.SC is supported by EPSRC (EP/D071127/1) and his visit to Grenoble was supported by the Ulysses exchange programme. Part of this work was performed during the FRIT workshop [13].

We present here new data concerning the drag exerted by a flowing foam of bubble area A = 16.0 mm 2 on a circular obstacle of diameter d 0 = 3 cm. We measured the drag, as explained in full detail in [9], versus the foam velocity V for six different foam thicknesses (Fig. 9a). As usual, the drag increases with increasing foam velocity.

More importantly for this paper, at given velocity, and especially at the limit of vanishing velocity, the drag increases with increasing foam thickness. This is due to (i) the decrease of liquid fraction with increasing foam thickness, as shown by the snapshots of the two extreme foam thicknesses in Fig. (9a); (ii) the increase in the height of the films with increasing foam thickness.

We fit the data by the formula F = F t Y +AV a to get the values of the total yield drag F t Y for the various foam


Karine Bertin
email: karine.bertin@uv.cl

Sharp adaptive estimation in sup-norm for d-dimensional Hölder classes

Keywords: anisotropic Hölder class, minimax exact constant, uniform norm, white noise Model. β 

come

Sharp adaptive estimation for d-dimensional classes

Introduction

Minimax adaptive estimation of a non-parametric function f from noisy data is the subject of many papers. Assuming that f belongs to a smoothness class Σ β , where β is an unknown smoothness parameter, the aim is to find an estimator of f independent of β and which attains asymptotically optimal behaviour on all the classes Σ β , for β given in a known set B. In this article, we study the problem of adaptive estimation in the Gaussian white noise model in supnorm, assuming that the function f satisfies a Hölder condition. We observe Y t , t ∈ R d , where Y t is a random process defined by the stochastic differential equation

dY t = f (t)dt + σ √ n dW t , t ∈ R d , (1)
where f is an unknown function, n ∈ N, σ > 0 is known and W is a standard Brownian sheet on [0, 1] d . We want to estimate the function f on R d given a realization y = Y t , t ∈ R d . We suppose that f belongs to a d-dimensional anisotropic Hölder class Σ(β, L) where L = (L 1 , . . . , L d ) ∈ (0, +∞) d is known, β = (β 1 , . . . , β d ) ∈ (0, 1] d is unknown and belongs to a finite set B ⊂ (0, 1] d known. The class Σ(β, L) is defined by

Σ(β, L) = f : R d → R : |f (x) -f (y)| ≤ L 1 |x 1 -y 1 | β 1 + • • • + L d |x d -y d | β d , x, y ∈ R d ,
where x = (x 1 , . . . , x d ) and y = (y 1 , . . . , y d ). In the following, P f is the distribution of y under the model ( 1) and E f is the corresponding expectation. An estimator θ n of f is a random function on [0, 1] d taking its values in R, measurable with respect to y. We will evaluate the quality of an estimator θ n by the maximal risk in sup-norm on B R n (θ n ) = sup , C β being a constant depending on β. In the nonadaptive case, i.e. when B contains only one vector, it has been proved that ψ n (β) is the minimax rate of convergence for sup-norm estimation: for d = 1, it was done by [START_REF] Ibragimov | of Applications of Mathematics[END_REF]; for multidimensional case, this fact was shown by [START_REF] Stone | Additive regression and other nonparametric models[END_REF] and [START_REF] Nussbaum | On the nonparametric estimation of regression functions that are smooth in a domain in R k[END_REF] for isotropic setting (β 1 = • • • = β d ) and in [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF] for anisotropic setting considered here. In other set-ups, some results on rate of convergence can be found in [START_REF] Barron | Risk bounds for model selection via penalization[END_REF], [START_REF] Neumann | Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra[END_REF] and [START_REF] Kerkyacharian | Nonlinear estimation in anisotropic multiindex denoising[END_REF]. In an adaptive set-up, [START_REF] Lepski | On problems of adaptive estimation in white Gaussian noise[END_REF] proved that ψ n (β) is the adaptive rate of convergence (cf. [START_REF] Tsybakov | Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes[END_REF] for precise definition of adaptive rate of convergence) for the problem considered here when d = 1.

β∈B sup f ∈Σ(β,L) E f θ n -f ∞ ψ n (β) p , (2)
Our goal is to study the asymptotics of the minimax risk in sup-norm on B (i.e. the adaptive minimax risk), in others words to study the asymptotics of

inf θ n R n (θ n ).
We want to prove that there exist optimal rate adaptive estimators on the scale of classes {Σ(β, L)} β∈B for the L ∞ norm and to find an estimator f n and the constant C β with ψ n (β) =

C β log n n β 2β+1
, such that we have

lim n→∞ inf θn R n (θ n ) = lim n→∞ R n ( f n ) = 1, (3)
where inf θn stands for the infimum over all the estimators. To obtain that there exist optimal rate adaptive estimators, it is enough to have that there exist an estimator θn and a positive constant C such that lim sup n→∞ R n ( θn ) ≤ C.

An estimator f n that satisfies (3) is called asymptotically exact adaptive estimator on the scale of classes {Σ(β, L)} β∈B for the L ∞ norm, and C β is called exact adaptive constant.

In the non-adaptive case (cf. [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF]), the relation ( 3) is satisfied by a constant C 0 (β) which depends on β, L and σ 2 , and for fβ a kernel estimator with kernel close to the kernel K β . The kernel K β is defined for u = (u 1 , . . . , u d ) ∈ R d by

K β (u 1 , . . . , u d ) = β + 1 α(β)β 2 (1 - d i=1 |u i | β i ) + , with α(β) = 2 d d i=1 Γ( 1 β i ) Γ( 1 β ) d i=1 β i ,
Γ denotes the gamma function and (x) + = max(0, x). The constant C 0 (β) satisfies

C 0 (β) =   σ 2β L * (β) β + 1 α(β)β and L * (β) = d i=1 L 1/β j j β .
In this article, we prove that there exist optimal rate adaptive estimators on {Σ(β, L)} β∈B for the L ∞ norm. We give precise upper and lower bounds for any finite set B. When B contains only two vectors, we improve the upper bound.

For particular forms of the set B (including sets B of isotropic classes), we prove it exist an estimator fn and a constant C β satisfying (3). The constant C β we have found is equal to the constant C 0 (β)(which was the solution of (3) in [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF]) multiplied by a constant larger than 1, depending on the set B and p. As a consequence, for this case, we lose efficiency in the constant under adaptation. We will see that the estimator satisfying (3) is obtained using the Lepski method. This method is introduced in [START_REF] Lepski | On problems of adaptive estimation in white Gaussian noise[END_REF] for the problem considered here with d = 1 and it has been studied by many authors. In particular, for d > 1, [START_REF] Klemelä | Sharp adaptive estimation of linear functionals[END_REF] have used it for the estimation of linear functionals in isotropic settings. [START_REF] Tsybakov | Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes[END_REF] used this method to study pointwise and sup-norm estimation on Sobolev classes.

For more general forms of the set B, we construct optimal rate adaptive estimators using o generalization of the Lepski method proposed in [START_REF] Lepski | Adaptive minimax estimation of infinitely differentiable functions[END_REF].

Main results

In this section, we introduce the notation and the assumptions about the set B, we define three families of estimators and we give our results.

The set B

We suppose that the set B = β (1) , . . . , β (l) contains l vectors belonging to (0, 1] d . The coordinates of β (i) are denoted by β (i) j , j = 1, . . . , d:

β (i) = (β (i) 1 , . . . , β (i) d ) ∈ (0, 1] d . (4)
We define the real β (i) by

β (i) =   d j=1 1 β (i) j   -1
(5)

and we denote by B the set: l) .

B = β (1) , . . . , β (
We suppose that β (i) = β (j) for all i, j ∈ {1, . . . , l} such that i = j. As a consequence, a β ∈ B is matched to a unique β ∈ B via the relations (4) and (5). We define the following relation of order in B: the vectors β and γ satisfy

β ≤ γ if and only if β ≤ γ.
This is a relation of total order in B, thus the notion of maximum and minimum in B are welldefined. We denote

β max = max{β (i) , i = 1, • • • , l} (respectively β min = min{β (i) , i = 1, • • • , l})
and β max (respectively β min ) the associated real number in B via (5).

Remark. In the isotropic setting (i.e. for all β = (β 1 , . . . ,

β d ) ∈ B, β 1 = • • • = β d ), this order is the order β = (β 1 , . . . , β d ) ≤ γ = (γ 1 , . . . , γ d ) if and only if β 1 ≤ γ 1 .

Three families of estimators

Here we define three families of kernel estimators ( fβ,1 ) β∈B , ( fβ,2 ) β∈B and ( fβ,3 ) β∈B . They are close to the asymptotically exact estimator of [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF], but the kernel is somewhat different at the boundary. This is a consequence of the fact that the observations here are for t ∈ R d whereas they were for t ∈ [0, 1] d in [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF]. The estimators are defined in the following way. For β ∈ B and j ∈ {1, 2, 3},

fβ,j (t) = R d K β,j (t -u) dY u , defined for t = (t 1 , . . . , t d ) ∈ [0, 1] d , where for u = (u 1 , . . . , u d ) ∈ R d K β,j (u) = 1 h 1,j (β) • • • h d,j (β) K β u 1 h 1,j (β) , . . . , u d h d,j (β)
.

For j ∈ {1, 2, 3}, the bandwidth

h j = (h 1,j (β), . . . , h d,j (β)) satisfies for i ∈ {1, . . . , d} h i,j (β) = C β λ j (β) L i log n n β/(2β+1) 1/β i , where λ j (β) =          1 for j = 1, 2 -2β 2β+1 c 2 (β) c 1 (β) β 2β+1 for j = 2, 2 -2β 2β+1 for j = 3, C β =   σ 2β L * (β) c 1 (β)(β + 1) α(β)β 3 β   1 2β+1 , (6)
c 1 (β) = 1 + p(β max -β) 2β max + 1 , c 2 (β) = 1 + pµ(β)(2β + 1)
and

µ(β) = max λ>β λ 2λ + 1 - min γ<λ,i=1,...,d γ 2γ + 1 , λ i γ γ i (2γ + 1)
,

where the maximum is taken for λ = (λ 1 , . . . , λ d ) ∈ B, and the minimum for γ = (γ 1 , . . . , γ d ) ∈ B.

Lemma 1. The kernel

K β satisfies [-1,1] d K β (u)du = 1 and [-1,1] d K 2 β (u)du = 2(β+1) βα(β)(2β+1) .
This a consequence of formula of Gradshteyn and Ryzhik (1965) (cf Appendix of Bertin ( 2004)) For t ∈ [0, 1] d , j ∈ {1, 2, 3} and for a function f belonging to one of the classes Σ(β, L), define the bias of fβ,j b β,j (t, f

) = E f ( fβ,j (t)) -f (t) = K β,j * f (t) -f (t)
and the stochastic term of the error

Z β,j (t) = fβ,j (t) -E f ( fβ,j (t)) = σ √ n K β,j (t -u)dW u .
The symbol * stands here the convolution of functions defined on R d . The quantity K β,j * f is well-defined since K β,j is continuous with compact support and f is continuous.

A lower bound for anisotropic classes

In the following, we will use the maximal risk in sup-norm on B defined in (2) with

ψ n (β) = C β log n n β 2β+1
, C β defined in (6) and p > 0. We have the following lower bound which is valid for any finite set B satisfying the conditions of Subsection 2.1.

Theorem 1. The minimax risk in sup-norm on B satisfies

lim inf n→∞ inf θn R n (θ n ) ≥ 1.

Exact asymptotics for particular forms of the set B

For β ∈ B and j ∈ {1, 2, 3}, we consider

η j (β) = 2 K β 2 2 σ 2 c j (β) n d i=1 h i,j (β)(2β + 1) log n 1/2
, where c 3 (β) = c 1 (β). We select the vector β(p) ∈ B defined by

β(p) = max β ∈ B : ∀γ < β, fβ,1 -fγ,1 ∞ ≤ η 1 (γ)
We have the following theorem for particular forms of the set B, which include the sets B of isotropic classes.

Theorem 2. We suppose that the set B satisfies the property (P ):

(P ) For all β = (β 1 , . . . , β d ), γ = (γ 1 , . . . , γ d ) ∈ B, if β ≤ γ then for all i = 1, . . . , d β i < γ i .
Then, the estimator f (p) = f β(p) ,1 is asymptotically exact adaptive, i.e. it satisfies the condition (3) with the constant C β defined in (6):

lim n→∞ inf θn R n (θ n ) = lim n→∞ R n ( f (p) ) = 1
Remark. The estimator f (p) is obtained using the method of Lepski. The constant C β has the form of the constant obtained by [START_REF] Lepski | On problems of adaptive estimation in white Gaussian noise[END_REF] in the case d = 1 where we replace β by β.

Upper bounds for anisotropic classes

If the set B does not satisfy Condition (P), Theorem 2 is not true anymore. In this subsection, Theorem 3 gives an upper bound for any finite set B ⊂ (0, 1] d and Theorem 4 improves this upper bound when B contains only two vectors.

We consider new estimators defined for β and γ ∈ B and t

∈ [0, 1] d by fβ * γ,2 (t) = K β * γ,2 (t -u)dY u where K β * γ,2 = K β,2 * K γ,2 .
We consider the vector βani ∈ B defined by

βani = max β ∈ B : ∀γ < β, fβ * γ,2 -fγ,2 ∞ ≤ η 2 (γ) .
Theorem 3. The estimator f ani = f βani ,2 satisfies for p > 0 lim sup

n→∞ sup β∈B sup f ∈Σ(β,L) E f f ani -f p ∞ (ψ n (β)M 2 (β)) -p ≤ 1, (7)
where, for j ∈ {2, 3} and β ∈ B,

M j (β) = c j (β) c 1 (β) β 2β+1 2 1/(2β+1) + β2 2β/(2β+1)
2β + 1 .

The relation (7) imply that f ani is an optimal rate adaptive estimator on {Σ(β, L)} β∈B for the L ∞ norm, for any finite set B ⊂ (0, 1] d .

For l = 2, we have a better result. We suppose that B = {γ, β} with γ < β. We consider a new estimator defined for t ∈ [0, 1] d by

fβ * γ,1 (t) = K β * γ,1 (t -u)dY u where K β * γ,1 = K β,1 * K γ,3 .
We select the estimator f ani2 defined by

f ani2 = fβ,1 if fβ * γ,1 -fγ,3 ∞ ≤ η 3 (γ) or fβ * γ,1 -fβ,1 ∞ ≥ λ 3 (γ)ψn(γ) 2γ+1 (1 + ρ n ) fγ,3 otherwise, where ρ n = ψn((β+γ)/2) ψn(γ) and ψ n (λ), for λ = (λ 1 , . . . , λ d ) ∈ (0, 1] d ∩ B c , is defined by ψ n (λ) = log n n λ 2λ+1 with λ = d i=1 1/λ i -1
. Here B c denotes the complement of the set B. Then we have the following theorem Theorem 4. The estimator f ani2 satisfies for p > 0 lim sup

n→∞ sup f ∈Σ(β,L) E f f ani2 -f p ∞ ψ -p n (β) ≤ 1 (8) and lim sup n→∞ sup f ∈Σ(γ,L) E f f ani2 -f p ∞ ψ -p n (γ) ≤ (M 3 (γ)) p . ( 9
)
2.6 Some remarks

1. In the model here, we choose to have observations in R d to simplify the calculations. To obtain the results, it is enough to have observations in a neighborhood in [0, 1] d .

2. These results can be generalized to others models such regression with regular design. This can be done by following the same proofs with minor modifications.

3. In this article, we have considered a finite set B such that its cardinality, card(B), does not depend on n. We could have considered a set B such that card(B) grows as n → ∞ and there exists M > 0 such that, for all β ∈ B, β > M . In Theorems 2 and 4, following the same proofs, it is possible to have the same results taking B such card(B) ≤

(log n) 1 2(2βmax+1) (
The last condition is necessary to have lim n→∞ R 1 (β) = 0 in Section 5 and the relation ( 49)). A weaker condition is possible on B. If, for j ∈ {1, 2, 3}, we replace c j (β) by c j (β) + 1 log log n in η j (β), the results of Theorem 2, Theorem 3 and Theorem 4 can be obtained for B such that card(B) ≤ (log n) a with a > 0.

4. For d = 1, in the Lepski method, for a given family of estimators ( fβ ) β∈B , we choose β, the largest

β ∈ B ⊂ R + such that fβ -fγ ∞ ≤ cψ n (γ) for all γ ≤ β, with a constant c > 0.
This choice is based on the fact that, if f ∈ Σ(β v , L) and γ ≤ β ≤ β v , the bias of fβfγ is upper-bounded by a term of order ψ n (γ). This property is still valid for anisotropic settings when B satisfies Condition (P ), but it does not work for general anisotropic settings. Indeed, for anisotropic settings, we do not have such a property on the bias of fβfγ and this bias can be very large. [START_REF] Kerkyacharian | Nonlinear estimation in anisotropic multiindex denoising[END_REF] give a new criteria which permits to obtain results in anisotropic Besov classes. Rather than comparing fβ to fγ for γ ≤ β, they compare, for γ ≤ β, fγ to a kernel estimator fβγ with bandwidth (h 1 (β, γ), . . . , h d (β, γ))

where

h i (β, γ) = max(h i (β), h i (γ)) and (h 1 (β), . . . , h d (β)), respectively (h 1 (γ), . . . , h d (γ)),
is the bandwidth of fβ , respectively fγ . This comparison permits to have a new criteria for selecting β. In Theorem 3 and Theorem 4, we use another estimator fβ * γ,j to do this kind of comparison. The different choice of fβ * γ,j is motivated by our goal to have better constants.

5. Our results are adaptation with respect to β and we suppose that L is fixed and known.

To use our method of estimation, the statistician need to know L. Here we suppose the statistician does not know L. We suppose that β belongs to B and

L ∈ {L (1) , . . . , L (l) } ⊂ ]0, +∞[ d . We note for j = 1, . . . , l, L (j) = (L (j) 1 , . . . , L (j) d ). Our method can be applied to select β ∈ B with L = (max j=1,...,d L (j) 1 , . . . , max j=1,...,d L (j) d ).
6. Theorem 2 gives a result of exact estimation. In Theorem 3 and in Theorem 4 in the case f ∈ Σ(γ, L), the difference between the lower bound and the upper bound is the factor (M 2 (β)) p and (M 3 (γ)) p respectively. The following plot represents the quantity M 3 (β)(on the vertical axis) as a function of β ∈ (0, 1/2]. We can see that, for β ∈ (0, 1/2], 1.06 ≤ M 3 (β) ≤ 2 and then it implies that

1.06 ≤ 1.06 c 2 (β) c 1 (β) β 2β+1 ≤ M 2 (β) ≤ 2 c 2 (β) c 1 (β) β 2β+1
.

The following sections are devoted to preliminary results and to the proofs of the theorems.

Some preliminary results

In the following, D i , with i = 1, 2, . . ., denote positive constants, except otherwise mentioned. These constants can depend on β ∈ B but we do not indicate explicitly the dependence on β. This does not have consequences on the proofs since B is a finite set. The quantity η j (β) satisfies the following lemma which will be proved in Section 8.

Lemma 2. For β ∈ B and j ∈ {1, 2}, we have

η j (β) + jψ n (β)λ j (β) 2β + 1 = M j (β)ψ n (β),
where M 1 (β) = 1. Moreover, for β ∈ B, we have

η 3 (β) + 2ψ n (β)λ 3 (β) 2β + 1 = M 3 (β)ψ n (β).
We have the following results for the families of estimators ( fβ,1 ) β∈B , ( fβ,2 ) β∈B and ( fβ,3 ) β∈B .

Proposition 1. For β ∈ B and j ∈ {1, 2, 3}, we have

sup f ∈Σ(β,L) b β,j (•, f ) ∞ ≤ ψ n (β)λ j (β) 2β + 1 . Proposition 2. For β ∈ B, j ∈ {1, 2, 3} and t ∈ [0, 1] d , we have E f Z 2 β,j (t) ≤ σ 2 K β 2 2 n hj (β) , and for b ≥ D 0 n hj (β)| log( hj (β))|
, we have

P f [ Z β,j ∞ ≥ b] ≤ D 1 hj (β) exp - b 2 n hj (β) 2 K β 2 2 σ 2 exp    D 2 (β)b n hj (β) (log( hj (β))) 1/2    ,
where hj (β) = d i=1 h i,j (β).

Proposition 3. For p > 0, β ∈ B and j ∈ {1, 2, 3}, we have

lim sup n→∞ sup f ∈Σ(β,L) E f fβ,j -f ∞ (M j (β)ψ n (β)) -1 p ≤ 1 (10) lim n→∞ sup f ∈Σ(β,L) E f fβ,j -f ∞ ψ -1 n (β) p I { fβ,j -f ∞≥(1+ε(n))Mj (β)ψn(β)} = 0, (11)
where ε(n) satisfies ε(n) ≥ (log n) -1/4 , and I A denotes the indicator function of a set A.

Proposition 1, respectively Proposition 2, can be obtained following the proof of Proposition 1, respectively Lemma 4, of [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF]. The proof of Proposition 3 can be deduced from the proofs of [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF] and we add some elements of proof in Section 8. Define for t ∈ [0, 1] d , j ∈ {1, 2} and a function f the bias term of fβ * γ,j

b β * γ,j (t, f ) = E f ( fβ * γ,j (t)) -f (t),
and the stochastic term of fβ * γ,j

Z β * γ,j (t) = fβ * γ,j (t) -E f ( fβ * γ,j (t)) = σ √ n K β * γ (t -u)dW u .
The estimator fβ * γ,j satisfies the two lemmas.

Lemma 3. For β, γ ∈ B, we have

sup f ∈Σ(β,L) b β * γ,2 (•, f ) -b γ,2 (•, f ) ∞ ≤ sup f ∈Σ(β,L) b β,2 (•, f ) ∞ ≤ ψ n (β)λ 2 (β) 2β + 1 . When B = {γ, β} with γ < β we have sup f ∈Σ(β,L) b β * γ,1 (•, f ) -b γ,3 (•, f ) ∞ ≤ sup f ∈Σ(β,L) b β,1 (•, f ) ∞ ≤ ψ n (β) 2β + 1 and sup f ∈Σ(γ,L) b β * γ,1 (•, f ) -b β,1 (•, f ) ∞ ≤ sup f ∈Σ(γ,L) b γ,3 (•, f ) ∞ ≤ ψ n (γ)λ 3 (γ) 2γ + 1 . Lemma 4. For β, γ ∈ B such that γ < β and j ∈ {1, 2}, we have for all t ∈ [0, 1] d E f (Z β * γ,j ) 2 (t) ≤ σ 2 K β 2 2
n hj (β) ,

and for b ≥ D 3 n hj (β)| log( hj (β))| P f [ Z β * γ,j ∞ ≥ b] ≤ D 4 hj (β) exp - b 2 n hj (β) 2 K β 2 2 σ 2 exp    D 5 b n hj (β) (log( hj (β))) 1/2    . ( 12
)
Lemma 5. Let j ∈ {1, 2, 3}. We have, for γ, β ∈ B such γ < β,

lim n→∞ sup f ∈Σ(β,L) E f Z γ,j p ∞ ψ -p n (β)I { Z γ,j ∞ >τ n,j (γ)} = 0,
where

τ n,j (γ) = 2 K γ 2 2 σ 2 (1 + pβ max ) log n n d i=1 h i,j (γ)(2γ + 1) 1/2
.

Remark: τ n,j (γ) is of the same order of ψ n (γ).

These three lemmas will be proved in Section 8.

Proof of Theorem 1

The proof of the lower bound is similar to the proof of lower bounds in [START_REF] Tsybakov | Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes[END_REF]The difficulties consist in finding a good subclass of Σ(β, L) and good parameters to apply Theorem 6 of [START_REF] Tsybakov | Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes[END_REF]. We have

∆ n = inf Tn sup β∈B sup f ∈Σ(β,L) E f T n -f p ∞ ψ -p n (β) ≥ max β∈B {∆ n (β)}, where ∆ n (β) = inf Tn sup f ∈Σ(β,L) E f T n -f p ∞ ψ -p n (β) .
By Theorem 1 of [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF], we know that lim inf n→∞ ∆ n (β max ) ≥ 1. This comes from the fact that the loss function w(x) = x p satisfies the conditions required in this theorem (i.e. w is a non-decreasing function which admits a polynomial majorant and such that w(0) = 0), and the exact constant C βmax in the rate of convergence is the same as the exact constant in the non-adaptive case. Now, in order to prove the theorem, it is enough to prove that

lim inf n→∞ ∆ n (β) ≥ 1 for all β ∈ B \βmax . Let 0 < ε < 1/2. Let β ∈ B \βmax .
In the following several quantities depend on β, but we do not indicate this dependence to simplify the notations. We consider the set of functions

f j,β (•) defined, for j ∈ {0, • • • , M }, by f j,β = ψ n (β)(1 -ε) 1 -d i=1 t i -a j,i h i,1 (β) β i + , j = 1, . . . , M f 0,β = 0,
where the a j = (a j,1 , . . . , a j,d ) form a grid of points in [0, 1] d . This grid is defined in the following manner. For

m i = 1 2h i,1 (2 1/β + 1) -1
with [x] the integer part of x and M = d i=1 m i , we consider the points a(l 1 , . . . , l d ) ∈ [0, 1] d for l i ∈ {1, . . . , m i } and i ∈ {1, . . . , d}, such that:

a(l 1 , . . . , l d ) = 2(2 1 β + 1) (h 1,1 l 1 , . . . , h d,1 l d ) .
To simplify the notation, we denote these points a 1 , . . . , a M and each a j takes the form:

a j = (a j,1 , . . . , a j,d ).
The functions f j,β satisfy the following lemma which can be proved as in [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF].

Lemma 6. 1-f j,β ∈ Σ(β, L), 2-f j,β 2 2 = σ 2 c 1 (β) log n(1-ε) 2 n(2β+1)
3-the functions f j,β have disjoint support.

Here we come back to the study of ∆ n . Let j ∈ {1, . . . , M } and T n an estimator. We have, since

f j,β (a k ) = (1 -ε)ψ n (β)δ j,k , ψ -1 n (β) T n -f j,β ∞ ≥ψ -1 n (β) max 1≤k≤M |T n (a k ) -f j,β (a k )| ≥(1 -ε) max 1≤k≤M θk -δ j,k ,
where δ j,k is the Kronecker delta and θk = T n (a k )ψ -1

n (β) 1-ε . As a consequence ψ -1 n (β) T n -f j,β ∞ ≥ d( θ, θ j ),
where θ = ( θ1 , . . . , θM ), θ j = (δ 1,j , . . . , δ M,j ), and

d(u, v) = (1 -ε) max 1≤k≤M |u k -v k | for two vectors u = (u 1 , . . . , u M ) and v = (v 1 , . . . , v M ) of R M .
In the same way

ψ -1 n (β) T n -f 0,β ∞ ≥ψ -1 n (β) max 1≤k≤M |T n (a k )| ≥ψ -1 n (β max ) max 1≤k≤M |T n (a k )| ≥ ψ n (β) ψ n (β max ) d( θ, θ 0 ),
where θ 0 is the null-vector of R M . Then we have

∆ n (β) ≥ inf T n max 0≤k≤M E k T n -f k,β p ∞ ψ -p n (β) ≥ inf θ∈R d max E 0 ψ n (β) ψ n (β max ) p d p ( θ, θ 0 ) , max 1≤k≤M E k d p ( θ, θ k ) , (13)
where E k is the expectation E k = E f k,β . We denote also P k = P f k,β .

Here we apply Theorem 6 of [START_REF] Tsybakov | Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes[END_REF] to the relation (13). We consider the M + 1 parameters {θ 0 , . . . , θ M } ⊂ [0, 1] M , the family of probability measures {P θ j = P j }, the loss function w(x) = x p and the distance d previously defined. The parameters θ j satisfy d(θ i , θ k ) ≥ 1ε for i, k ∈ {0, . . . , M } and i = k. We are now going to prove that there exists α n , with 0 ≤ α n ≤ 1 and lim n→∞ α n = 0, such that

Q dP 0 dQ ≥ τ n ≥ 1 -α n , (14)
where

Q = 1 M M k=1 P k , τ n = n -ν , ν = pβ max 2β max +1 -pβ 2β+1 -εc 1 (β) 2β+1 (1 -ε/2
) and ε is chosen small enough to have ν > 0. Indeed, if we prove ( 14), Theorem 6 of Tsybakov implies that

∆ n (β) ≥ (1 -α n )τ n (1 -2ε) p ψn(β)ε ψn(βmax) p (1 -2ε) p + τ n ψn(β)ε ψn(βmax) p . We have lim n→∞ τ n ψ n (β)ε ψn(βmax) p = +∞. Then lim inf n→∞ ∆ n (β) ≥ (1 -2ε) p .
Since ε can be arbitrarily small, we obtain that lim inf n→∞ ∆ n (β) ≥ 1 and it proves the theorem.

Here we prove (14). In the same way as Tsybakov (1998), we have that under P i

dP k dP 0 = exp ξ k v n + v 2 n k = i exp ξ k v n -v 2 n k = i,
where the ξ k are i.i.d. N (0, 1) variables and

v 2 n = n σ 2 f k,β 2 2 = 2 log n 2β+1 c 1 (β)(1 -ε) 2
. Now, by the independence of the ξ i , we have

P i 1 M M k=1 dP k dP 0 < 1/τ n ≥ P i   1 M M k=1,k =i dP k dP 0 < 1 2τ n   P i 1 M dP i dP 0 < 1 2τ n . ( 15
)
The probabilities above satisfy, since the ξ k are N (0, 1) variables,

P i   1 M M k=1,k =i dP k dP 0 < 1 2τ n   =P i   1 M M k=1,k =i exp ξ k v n -v 2 n < 1 2τ n   =1 -P i   M k=1,k =i exp ξ k v n -v 2 n ≥ M 2τ n   (16) ≥1 - 2τ n (M -1) M E i exp ξ k v n -v 2 n =1 - 2τ n (M -1) M , (17)
and

P i 1 M dP i dP 0 < 1 2τ n =P i exp ξ i v n + v 2 n < M 2τ n =1 -Φ 1 v n -log M + log(2τ n ) + v 2 n /2 , ( 18
)
where Φ is the Gaussian c.d.f. By replacing M , τ n and v n by their values in terms of n and β,

since c 1 (β) 2β+1 = 1 2β+1 + pβ max 2β max +1 -pβ 2β+1 , we obtain that 1 v n -log M + log(2τ n ) + v 2 n /2 = log n v n - 1 2β + 1 - pβ max 2β max + 1 + pβ 2β + 1 + εc 1 (β) 2β + 1 (1 -ε/2) + c 1 (β) 2β + 1 (1 -2ε + ε 2 ) + o(1) = - log n v n c 1 (β)ε 2β + 1 (1 -ε/2) + o(1) . ( 19
)
Using the relations ( 15), ( 17), ( 18), ( 19), noting that 2τn(M -1) M tends to 0 and that

-log n vn c 1 (β)ε 2β+1 (1 -ε/2) + o(1) tends to -∞, as n tends to ∞, we deduce that lim n→∞ Q dP 0 dQ ≥ τ n = 1.
This implies the existence of a sequence α n that satisfies (14) and lim n→∞ α n = 0, which concludes the proof of the lower bound.

Proof of Theorem 2

Since Theorem 1 is true, in particular for set B satisfying Condition (P ), to prove Theorem 2, it is enough to show that

lim sup n→∞ sup β∈B sup f ∈Σ(β,L) E f f (p) -f p ∞ ψ -p n (β) ≤ 1. (20)
Here we prove (20). Since the cardinal of B is finite, we are going to prove that for all β ∈ B,

lim sup n→∞ ∆(β, n) ≤ 1, where ∆(β, n) = sup f ∈Σ(β,L) E f f (p) -f p ∞ ψ -p n (β) ≤ R 1,n (β) + R 2,n (β), with R 1,n (β) = sup f ∈Σ(β,L) E f f (p) -f p ∞ ψ -p n (β)I { β(p) <β} and R 2,n (β) = sup f ∈Σ(β,L) E f f (p) -f p ∞ ψ -p n (β)I { β(p) ≥β} .
This will be obtained by proving that

lim n→∞ R 1,n (β) = 0 (21) and that lim sup n→∞ R 2,n (β) ≤ 1. (22)
•Proof of ( 21)

Let β ∈ B. We have R 1,n (β) ≤ γ∈B,γ<β sup f ∈Σ(β,L) E f fγ,1 -f p ∞ ψ -p n (β)I { β(p) =γ} .
We fix now γ ∈ B with γ < β. The event β(p) = γ satisfies

β(p) = γ ⊂ β ′ ∈B,β ′ <γ ′ fγ ′ ,1 -fβ ′ ,1 ∞ > η 1 (β ′ ) ,
where γ ′ = min{λ ∈ B such that λ > γ}. Then we have

sup f ∈Σ(β,L) E f fγ,1 -f p ∞ ψ -p n (β)I { β(p) =γ} ≤ β ′ ∈B,β ′ <γ ′ sup f ∈Σ(β,L) E f fγ,1 -f p ∞ ψ -p n (β)I A 1,n (β ′ ) , with A 1,n (β ′ ) = fγ ′ ,1 -fβ ′ ,1 ∞ > η 1 (β ′ ) .
We do not indicate the dependence of the set A 1,n (β ′ ) on γ ′ because we argue for fixed γ in B and then for fixed γ ′ . To prove the result (21), since the cardinal of B is finite, it is enough to prove that the following quantity, for

β ′ ∈ B with β ′ < γ ′ , sup f ∈Σ(β,L) E f fγ,1 -f p ∞ ψ -p n (β)I A 1,n (β ′ ) , (23)
tends to 0 as n → ∞.

Here we study the quantity (23). Let f ∈ Σ(β, L). We fix β ′ ∈ B such that β ′ < γ ′ . We have

E f fγ,1 -f p ∞ ψ -p n (β)I A 1,n (β ′ ) ≤ 2 p ψ -p n (β) b γ,1 (•, f ) p ∞ P f (A 1,n (β ′ )) + E f Z γ,1 p ∞ I A 1,n (β ′ ) .
To study the quantity above, we need the following lemma which will be proved in Section 8.

Lemma 7. We have for n large enough sup f ∈Σ(β,L)

P f A 1,n (β ′ ) ≤ D 6 (log n) -1 2β ′ +1 n - p(β max -β ′ ) (2β max +1)(2β ′ +1) .
The bias b γ,1 satisfies, for all

f ∈ Σ(β, L), b γ,1 (•, f ) ∞ ≤ K γ (u) d i=1 L i |u i h i,1 (γ)| β i du.
Since for all i ∈ {1, • • • , d}, γ i < β i , we have that

sup f ∈Σ(β,L) b γ,1 (•, f ) = o(ψ n (γ)),
as n → ∞. As a consequence, we deduce from Lemma 7 that

sup f ∈Σ(β,L) ψ -p n (β) b γ,1 (•, f ) p ∞ P f (A 1,n (β ′ )) ≤ D 7 (log n) -1 2β ′ +1 + pγ 2γ+1 -pβ 2β+1 n - pβ max 2β max +1 + pβ ′ 2β ′ +1 -pγ 2γ+1 + pβ 2β+1 .
(24) Moreover,

E f ψ -p n (β) Z γ,1 p ∞ I A 1,n (β ′ ) ≤ (τ n,1 (γ)) p ψ -p n (β)P f (A 1,n (β ′ )) + E f Z γ,1 p ∞ ψ -p n (β)I { Z γ,1 ∞>τn,1(γ)} . (25)
From Lemma 7, we know that

P f (A 1,n (β ′ )) is at most of order (log n) -1 2β ′ +1 n - pβ max 2β max +1 + pβ ′ 2β ′ +1 thus (τ n,1 (γ)) p ψ -p n (β)P f (A 1,n (β ′ )) ≤ D 8 (log n) -1 2β ′ +1 + pγ 2γ+1 -pβ 2β+1 n - pβ max 2β max +1 + pβ ′ 2β ′ +1 -pγ 2γ+1 + pβ 2β+1 .
Since β ′ ≤ γ < β ≤ β max , we obtain that

lim n→∞ sup f ∈Σ(β,L) ψ -p n (β) b γ,1 (•, f ) p ∞ P f (A 1,n (β ′ )) = 0 (26)
and

lim n→∞ sup f ∈Σ(β,L) (τ n,1 (γ)) p ψ -p n (β)P f (A 1,n (β ′ )) = 0.
Using Lemma 5, we deduce that

lim n→∞ sup f ∈Σ(β,L) E f ψ -p n (β) Z γ,1 p ∞ I A 1,n (β ′ ) = 0. (27)
From ( 26) and ( 27), we conclude that the quantity (23) tends to 0 as n → ∞.

•Proof of ( 22)

Let δ n = 1 log n 1/4 and β ∈ B. We have R 2,n (β) ≤ (1 + δ n ) p + sup f ∈Σ(β,L) E f f (p) -f p ∞ ψ -p n (β)I { β(p) ≥β}∩{ f (p) -f ∞ψ -1 n (β)>1+δn} ≤ (1 + δ n ) p + γ∈B,γ≥β sup f ∈Σ(β,L) Q 2,n (β, γ, f ), where Q 2,n (β, γ, f ) = E f fγ,1 -f p ∞ ψ -p n (β)I { β(p) =γ}∩{ fγ,1 -f ∞ψ -1 n (β)>1+δn}
. By Proposition 3, we have lim n→∞ sup f ∈Σ(β,L) Q 2,n (β, β, f ) = 0. To prove ( 22), since the cardinal of B is finite and lim n→∞ δ n = 0, it is enough to prove, for γ ∈ B with γ > β, that

lim n→∞ sup f ∈Σ(β,L) Q 2,n (β, γ, f ) = 0. ( 28
)
Here we prove the result (28

). Let γ ∈ B such that γ > β and f ∈ Σ(β, L). If β(p) = γ, since γ > β, we have fγ,1 -fβ,1 ∞ ≤ η 1 (β). Then, fγ,1 -f ∞ I { β(p) =γ} ≤ fγ,1 -fβ,1 ∞ + fβ,1 -f ∞ I { β(p) =γ} ≤ η 1 (β) + fβ,1 -f ∞ I { β(p) =γ} . ( 29
)
The quantity Q 2,n (β, γ, f ) is upper bounded by

E f fγ,1 -f 2p ∞ ψ -2p n (β)I { β(p) =γ} 1/2 P f { β(p) = γ} ∩ { fγ,1 -f ∞ ψ -1 n (β) > 1 + δ n } 1/2 .
Then, (29) and Proposition 3 imply that

sup f ∈Σ(β,L) E f fγ,1 -f 2p ∞ ψ -2p n (β)I { β(p) =γ} 1/2
is bounded above by a positive constant for n large enough and we deduce that

sup f ∈Σ(β,L) Q 2,n (β, γ, f ) ≤ D 9 P f { β(p) = γ} ∩ { fγ,1 -f ∞ ψ -1 n (β) > 1 + δ n } 1/2 , ( 30
)
for n large enough. Now we are going to prove that the right hand side of inequality (30) tends to 0 as n → ∞. We have

fγ,1 -f ∞ I { β(p) =γ} ≤ ( b γ,1 (•, f ) -b β,1 (•, f ) ∞ + b β,1 (•, f ) ∞ + Z γ,1 ∞ ) I { β(p) =γ} ≤ b γ,1 (•, f ) -b β,1 (•, f ) ∞ + ψ n (β) 2β + 1 + Z γ,1 ∞ I { β(p) =γ} , (31)
where the last line is a consequence of Proposition 1.

We

have b γ,1 (•, f )-b β,1 (•, f ) ∞ = E f fγ,1 -E f fβ,1 ∞ . The function φ : t -→ E f fγ,1 (t) - E f fβ,1 (t) is a continuous function on [0, 1] d which admits a non-random maximum x 0 satis- fying b γ,1 (•, f ) -b β,1 (•, f ) ∞ = φ(x 0 ) ≤ fγ,1 (x 0 ) -fβ,1 (x 0 ) + |ϕ 1 | ≤ fγ,1 -fβ,1 ∞ + |ϕ 1 | ≤ η 1 (β) + |ϕ 1 |, (32)
where ϕ 1 = fγ,1 (x 0 ) -fβ,1 (x 0 ) -E f fγ,1 (x 0 ) -fβ,1 (x 0 ) . Since ϕ 1 is a N (0, π 2 n ) variable, by Proposition 2, we deduce that its variance π 2 n satisfies

π 2 n ≤ 2E f (Z β,1 ( x 0 )) 2 + 2E f (Z γ,1 (x 0 )) 2 ≤ 2 K β 2 2 σ 2 (1 + o(1)) n h1 (β) . ( 33
)
Then using ( 31) and ( 32) we deduce that

fγ,1 -f ∞ I { β(p) =γ} ≤ η 1 (β) + ψ n (β) 2β + 1 + Z γ,1 ∞ + |ϕ 1 | I { β(p) =γ} ≤ (ψ n (β) + Z γ,1 ∞ + |ϕ 1 |) I { β(p) =γ} ,
the last line being a consequence of Lemma 2. Thus,

P f { β(p) = γ} ∩ { fγ,1 -f ∞ > (1 + δ n )ψ n (β)} ≤P f ( Z γ,1 ∞ + |ϕ 1 | > δ n ψ n (β)) ≤P f Z γ,1 ∞ > δ n 2 ψ n (β) + P f |ϕ 1 | > δ n 2 ψ n (β) . (34)
As ϕ 1 is a N (0, π 2 n ) variable and using (33), we have

P f |ϕ 1 | > δ n 2 ψ n (β) ≤ exp - ψ 2 n (β)δ 2 n n h1 (β) 16 K β 2 2 σ 2 .
The quantity ψ 2 n (β)δ 2 n n h1 (β) is of order √ log n, and then

lim n→∞ P f |ϕ 1 | > δ n 2 ψ n (β) = 0. ( 35
)
Using Proposition 2, we have

P f Z γ,1 ∞ > δ n 2 ψ n (β) ≤ D 1 h1 (γ) exp - ψ 2 n (β)δ 2 n n h1 (γ) 8 K β 2 2 σ 2 exp    D 2 δ n ψ n (β) n h1 (γ) 2(log h1 (γ)) 1/2    .
The quantity

ψ 2 n (β)δ 2 n n h1 (γ) is of order ψ 2 n (β)ψ -2 n (γ)(log n) D 10 with some D 10 ∈ R and h1 (γ) is of order log n n 1/(2γ+1)
. Hence, since β < γ, this implies that

lim n→∞ P f Z γ,1 ∞ > δ n 2 ψ n (β) = 0. ( 36
)
Using ( 35), ( 36) and that the fact that the right hand side of inequality (34) does not depend on f , we deduce that

lim n→∞ sup f ∈Σ(β,L) P f { β(p) = γ} ∩ { fγ,1 -f ∞ > (1 + δ n )ψ n (β)} = 0.
Then we obtain the result (28) which implies ( 22).

6 Proof of Theorem 3

The scheme of proof is similar to the proof of relation ( 20) in the proof of Theorem 2. To prove Theorem 3, we will prove that, for all β ∈ B,

lim n→∞ R 3,n (β) = 0 (37) and lim sup n→∞ R 4,n (β) ≤ (M 2 (β)) p , ( 38
)
where

R 3,n (β) = sup f ∈Σ(β,L) E f f ani -f p ∞ ψ -p n (β)I { βani <β} , R 4,n (β) = sup f ∈Σ(β,L) E f f ani -f p ∞ ψ -p n (β)I { βani ≥β} .
•Proof of ( 37)

Let β, γ ∈ B such that γ < β. The event βani = γ satisfies βani = γ ⊂ β ′ ∈B,β ′ <β A 2,n (β ′ ), where A 2,n (β ′ ) = fβ * β ′ ,2 -fβ ′ ,2 ∞ > η 2 (β ′ ) .
Following the same reasoning as in the proof of relation ( 20), to prove (37), it is enough to prove that, for all β ′ ∈ B with β ′ < β,

lim n→∞ sup f ∈Σ(β,L) E f fγ,2 -f p ∞ ψ -p n (β)I A 2,n (β ′ ) = 0. ( 39
)
The event A 2,n (β ′ ) satisfies the following lemma:

Lemma 8. We have for n large enough

sup f ∈Σ(β,L) P f A 2,n (β ′ ) ≤ D 11 (log n) -1 2β ′ +1 n -pµ(β ′ ) .
The bias b γ,2 satisfies for all

f ∈ Σ(β, L) b γ,2 (•, f ) ∞ ≤ K γ (u) d i=1 L i |u i h i,2 (γ)| β i du, then sup f ∈Σ(β,L) b γ,2 (•, f ) ∞ is at most of order log n n min β i γ γ i (2γ+1)
, where the minimum is taken on i = 1, . . . , d. Therefore

sup f ∈Σ(β,L) ψ -p n (β) b γ,2 (•, f ) p ∞ P f (A 2,n (β ′ ))
is upper bounded by

D 12 (log n) -pω(β,γ)-1 2β ′ +1 n -p(µ(β ′ )-ω(β,γ))
where

ω(β, γ) = β 2β + 1 -min i=1,...,d β i γ γ i (2γ + 1)
.

If B satisfies Condition (P ), then ω(β, γ) < β 2β + 1 - γ 2γ + 1 .
Since β > γ and µ(β ′ ) > 0, this implies that

lim n→∞ sup f ∈Σ(β,L) ψ -p n (β) b γ,2 (•, f ) p ∞ P f (A 2,n (β ′ )) = 0. (40)
If B do not satisfy Condition (P ), there exists i ∈ {1, . . . , d} such that

β i /γ i ≤ 1, then µ(β ′ ) ≥ ω(β, γ) ≥ β 2β + 1 - γ 2γ + 1 > 0.
This implies that lim

n→∞ sup f ∈Σ(β,L) ψ -p n (β) b γ,2 (•, f ) p ∞ P f (A 2,n (β ′ )) = 0. (41)
Reasoning as in the proof of (21), using the decomposition (25), Lemma 5, Lemma 8 and that

β 2β + 1 - γ 2γ + 1 ≤ µ(β ′ ),
we deduce that lim

n→∞ sup f ∈Σ(β,L) E f ψ -p n (β) Z γ,2 p ∞ I A 2,n (β ′ ) = 0. (42)
From ( 41), ( 40) and ( 42), we obtain (37).

•Proof of (38)

The proof will be similar to the proof in the proof of ( 22). We fix

δ n = 1 log n 1/4 . We have R 4,n (β) ≤ (1 + δ n ) p (M 2 (β)) p + γ∈B,γ≥β sup f ∈Σ(β,L) Q 4,n (β, γ, f ), where Q 4,n (β, γ, f ) = E f fγ,2 -f p ∞ ψ -p n (β)I { βani =γ}∩C 1,n ,
and

C 1,n = { fγ,2 -f ∞ ψ -1 n (β) > M 2 (β)(1 + δ n )}.
By Proposition 3, we have lim n→∞ sup f ∈Σ(β,L) Q 4,n (β, β, f ) = 0. To prove (38), it is enough to prove that, for all γ ∈ B such that γ > β,

lim n→∞ sup f ∈Σ(β,L) Q 4,n (β, γ, f ) = 0. (43)
Let γ ∈ B, such that γ > β, and f ∈ Σ(β, L). Following the same reasoning as in the proof of ( 22) from ( 31) to (33), using Proposition 1, Lemma 2 and Lemma 3, we deduce that

fγ,2 -f ∞ I { βani =γ} ≤( b β * γ,2 (•, f ) -b γ,2 (•, f ) ∞ + b β * γ,2 (•, f ) -b β,2 (•, f ) ∞ + b β,2 (•, f ) ∞ + Z γ,2 ∞ )I { βani =γ} ≤ 2ψ n (β)λ 2 (β) 2β + 1 + η 2 (β) + Z γ,2 ∞ + |ϕ 2 | I { βani =γ} , ≤ (M 2 (β) + Z γ,2 ∞ + |ϕ 2 |) I { βani =γ} (44)
where

ϕ 2 = fβ * γ,2 (x 0 ) -fβ,2 (x 0 ) -E f fβ * γ,2 (x 0 ) -fβ,2 (x 0 ) , with some x 0 ∈ [0, 1] d . Using
Lemma 4 and Proposition 2, we have that ϕ 2 is a N (0, π 2 n ) variable, with variance π 2 n satisfying

π 2 n ≤ 2 K β 2 2 σ 2 (1 + o(1)) n h1 (β) .
We have

E f Z γ,2 2p ∞ ψ -2p n (β) ≤ (τ n,2 (γ)) 2p ψ -2p n (β) + ψ -2p n (β)E f Z γ,2 2p
∞ I { Z γ,2 ∞>τn,2(γ)} .
Reasoning as in the proof of Lemma 5, we have that

lim n→∞ (ψ n (β)) -2p sup f ∈Σ(β,L) E f Z γ,2 2p ∞ I { Z γ,2 ∞>τn,2(γ)} = 0.
Since γ > β, we deduce that

lim n→∞ sup f ∈Σ(β,L) E f Z γ,2 2p ∞ ψ -2p n (β) = 0. (45)
Moreover, the variable ϕ 3 satisfies the properties

P f [|ϕ 2 | > δ n ψ n (β)t] ≤ exp -D 13 t 2 log n , t ≥ 0, (46)
lim n→∞ sup f ∈Σ(β,L) E f |ϕ 2 | 2p ψ -2p n (β) = 0. (47)
The property (46) comes from the fact that ϕ 2 is a N (0, π 2 n ) variable, and the property (47) can be proved as ( 45) using ( 46) and the proof of Lemma 5.

The relations ( 45) and ( 47) and Proposition 3 imply that

sup f ∈Σ(β,L) E f fγ,2 -f 2p ∞ ψ -2p n (β)I { βani =γ} 1/2
is bounded above by a positive constant for n large enough. Now to have (43), it is enough to prove that lim n→∞ sup f ∈Σ(β,L)

P f C 1,n ∩ { βani = γ} = 0. (48)
Using the relations ( 44) and ( 46), and following the proof of ( 22) from ( 34) to (36), we deduce (48), which finishes the proof.

7 Proof of Theorem 4

•Proof of ( 8)

By Proposition 3, we have

lim sup n→∞ sup f ∈Σ(β,L) E f fβ,1 -f p ∞ ψ -p n (β) ≤ 1
Then to have (8), it is enough to prove that

lim n→∞ sup f ∈Σ(β,L) E f fγ,3 -f p ∞ ψ -p n (β)I { f ani2 = fγ,3 } = 0 (49)
The event f ani2 = fγ,3 satisfies f ani2 = fγ,3 ⊂ A 3,n ∩ A 4,n , where

A 3,n = fβ * γ,1 -fγ,3 ∞ > η 3 (γ) , A 4,n = fβ * γ,1 -fβ,1 ∞ < ψ n (γ)λ 3 (γ) 2γ + 1 (1 + ρ n ) ,
The event A 3,n satisfies the lemma:

Lemma 9. We have for n large enough

sup f ∈Σ(β,L) P f [A 3,n ] ≤ D 14 (log n) -1 2γ+1 n - p(β-γ) (2β+1)(2γ+1) .
The proof of Lemma 9 is similar to the proof of Lemma 7 and 8. Let f ∈ Σ(β, L). We have

fγ,3 -f ∞ I A 3,n ∩A 4,n ≤ fβ * γ,1 -fγ,3 ∞ + fβ * γ,1 -fβ,1 ∞ + fβ,1 -f ∞ I A 3,n ∩A 4,n
Using Proposition 3 and Lemma 9 we deduce that

lim n→∞ sup f ∈Σ(β,L) E f fβ,1 -f p ∞ ψ -p n (β)I A 3,n ∩A 4,n = 0,
Moreover, using the definition of A 4,n , we have

E f fβ * γ,1 -fβ,1 p ∞ ψ -p n (β)I A 3,n ∩A 4,n ≤ λ 3 (γ)ψ n (γ)ψ -1 n (β) 2γ + 1 p (1 + ρ n ) p P f (A 3,n ).
Using Lemma 9, we deduce that, for n large enough,

ψ -p n (β)ψ p n (γ)(1 + ρ n ) p P f (A 3,n ) is at most of order (log n) -1 2γ+1 + pγ 2γ+1 -pβ 2β+1 , therefore, since γ < β, lim n→∞ sup f ∈Σ(β,L) E f fβ * γ,1 -fβ,1 p ∞ ψ -p n (β)I A 3,n ∩A 4,n = 0
To prove (49), it remains to prove that

lim n→∞ sup f ∈Σ(β,L) E f fβ * γ,1 -fγ,3 p ∞ ψ -p n (β)I A 3,n ∩A 4,n = 0.
We have

fβ * γ,1 -fγ,3 ∞ I A 3,n ∩A 4,n ≤ ( b β * γ,1 (•, f ) -b γ,3 (•, f ) ∞ + Z γ,3 ∞ + Z β * γ,1 ∞ ) I A 3,n ∩A 4,n .
Reasoning as in the proof of ( 21), using the decomposition (25) and applying Lemma 5 to Z γ,3 , we deduce that lim

n→∞ sup f ∈Σ(β,L) E f ( Z γ,3 ∞ ) p ψ -p n (β)I A 3,n ∩A 4,n = 0.
Since Z β * γ,1 satisfies Lemma 4, a similar reasoning permits to have

lim n→∞ sup f ∈Σ(β,L) E f ( Z β * γ,3 ∞ ) p ψ -p n (β)I A 3,n ∩A 4,n = 0.
Appying Lemma 3, we obtain that the quantity

sup f ∈Σ(β,L) b β * γ,1 (•, f ) -b γ,3 (•, f ) ∞ is of order ψ n (β)
, and finally we have by Lemma 9 that

lim n→∞ sup f ∈Σ(β,L) b β * γ,1 (•, f ) -b γ,3 (•, f ) ∞ ψ -p n (β)P f (A 3,n ∩ A 4,n ) = 0,
which prove (49) and then (8).

•Proof of ( 9)

The proof will be similar to the proof in the proof of ( 22). Let f ∈ Σ(γ, L). We fix δ n = 1 log n 1/4 . We have

E f f ani2 -f p ∞ ψ -p n (γ) ≤(1 + δ n ) p (M 3 (γ)) p + E f f ani2 -f p ∞ ψ -p n (γ)I C 2,n , ≤(1 + δ n ) p (M 3 (γ)) p + E f fβ,1 -f p ∞ ψ -p n (γ)I C 2,n ∩An +E f fγ,3 -f p ∞ ψ -p n (γ)I C 2,n ∩A c n , where C 2,n = f ani2 -f ∞ ψ -1 n (γ) > (1 + δ n )M 3 (γ) ,
and

A n = (A 3,n ∩ A 4,n ) c
. By Proposition 3, we have

lim n→∞ sup f ∈Σ(γ,L) E f fγ,3 -f p ∞ ψ p n (γ)I C 2,n ∩A c = 0.
The event A 4,n satisfies the lemma Lemma 10. For n large enough, we have sup f ∈Σ(γ,L)

P f A c 4,n ≤ D 15 h1 (β) exp -D 16 (log n) D 17 n D 18 exp -D 19 (log n) D 17 n D 18 , with D 17 ∈ R.
Lemma 10 implies that

lim n→∞ sup f ∈Σ(γ,L) E f fβ,1 -f p ∞ ψ -p n (γ)I C 2,n ∩A c 4,n = 0.
Then to have (9), it is enough to prove that

lim n→∞ sup f ∈Σ(γ,L) E f fβ,1 -f p ∞ ψ -p n (γ)I A 5,n = 0, (50)
where

A 5,n = C 2,n ∩ A c 3,n ∩ A 4,n On A c 3,n ∩ A 4,n , we have fβ,1 -f ∞ ≤ fβ,1 -fβ * γ,1 ∞ + fβ * γ,1 -fγ,3 ∞ + fγ,3 -f ∞ ≤η 3 (γ) + ψ n (γ)λ 3 (γ)(1 + ρ n ) 2γ + 1 + fγ,3 -f ∞ .
Using Proposition 3, we deduce that, for n large enough,

sup f ∈Σ(γ,L) E f fβ,1 -f 2p ∞ ψ -2p n (γ)I A 5,n
is upper bounded by a constant. Now to have (50), it is enough to prove that

lim n→∞ sup f ∈Σ(γ,L) P f [A 5,n ] = 0. (51)
We have by Lemma 2 and Proposition 1,

fβ,1 -f ∞ I A 5,n ≤ fβ,1 -fβ * γ,1 ∞ + b β * γ,1 (•, f ) -b γ,3 (•, f ) ∞ + b γ,3 (•, f ) ∞ + Z β * γ,1 ∞ I A 5,n ≤ b β * γ,1 (•, f ) -b γ,3 (•, f ) ∞ + 2λ 3 (γ)ψ n (γ) 2γ + 1 + Z β * γ,1 ∞ .
Following the same argument as in the proof of ( 22) from ( 31) to (33), we deduce that

b β * γ,1 (•, f ) -b γ,3 (•, f ) ∞ ≤ η 3 (γ) + |ϕ 3 | (52)
where

ϕ 3 = fβ * γ,1 (x 0 ) -fγ,3 (x 0 ) -E f fβ * γ,1 (x 0 ) -fγ,3 (x 0 ) , with some x 0 ∈ [0, 1] d . Using
Lemma 4, we have that ϕ 3 is a random variable N (0, π 2 n ), where π 2 n satisfies

π 2 n ≤ 2 K γ 2 2 σ 2 (1 + o(1)) n h1 (γ) .
Using Lemma 3 and Lemma 4, we obtain finally that

fβ,1 -f ∞ I A 5,n ≤ η 3 (γ) + 2λ 3 (γ)ψ n (γ) 2γ + 1 + Z β * γ,1 ∞ + |ϕ 3 | I A 5,n ≤ (M 3 (γ)ψ n (γ) + Z β * γ,1 ∞ + |ϕ 3 |) I A 5,n
Now following the proof of ( 22) from ( 34) to (36), we deduce (51), which finishes the proof of (50).

Proofs of the lemmas and propositions

Proof of Lemma 2

Since K β 2 2 = 2(β+1) βα(β)(2β+1) , we have for j ∈ {1, 2, 3},

η j (β) = log n n β 2β+1 C -1 2β β 2c j (β) K β 2 2 σ 2 L 1/β * 2β + 1 1/2 (λ j (β)) -1 2β , = log n n β 2β+1 C -1 2β β 4β 2 c j (β)(β + 1)σ 2 L 1/β * (2β + 1) 2 α(β)β 3 1/2 (λ j (β)) -1 2β .
For j=1, we deduce that

η 1 (β) = log n n β 2β+1 C -1 2β β C 2β+1 2β β 2β 2β + 1 = 2β 2β + 1 ψ n (β).
This implies the result of the lemma for j = 1. For j ∈ {2, 3}, we have

η j (β) = 2β 2β + 1 ψ n (β)2 -1/(2β+1) c j (β) c 1 (β) β 2β+1
, which permits to have the result for j ∈ {2, 3}.

Proof of Proposition 3

Following the proof of Theorem 1 of [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF] with the loss function w(x) = x p , studying the bias term and the stochastic term of fβ,j , we can deduce, for j ∈ {1, 2, 3}, that

lim sup n→∞ sup f ∈Σ(β,L) P f fβ,j -f ∞ > (1 + ε n )ψ n (β) λ j (β) 2β + 1 + η j (β) ψ n (β)(c j (β)) 1/2 = 0 and lim sup n→∞ sup f ∈Σ(β,L) E f fβ,j -f p ∞ ψ -p n (β) ≤ λ j (β) 2β + 1 + η j (β) ψ n (β)c j (β) p . For j = 1, using that η 1 (β) = 2β 2β+1 ψ n (β) (cf. proof of Lemma 2), and that c 1 (β) ≥ 1, we have that λ 1 (β) 2β + 1 + η 1 (β) ψ n (β)c 1 (β) = 1 2β + 1 + 2β (2β + 1)(c 1 (β)) 1/2 ≤ 1. For j ∈ {2, 3}, using that η j (β) = 2β 2β+1 ψ n (β)2 -1/(2β+1) c j (β) c 1 (β) β 2β+1 (
cf. proof of Lemma 2), and that c j (β) ≥ 1, we have that

λ j (β) 2β + 1 + η j (β) ψ n (β)c j (β) = c j (β) c 1 (β) β 2β+1   2 -2β 2β+1 2β + 1 + 2β2 -1 2β+1 (2β + 1)(c 2 (β)) 1/2   ≤ M j (β).
The above lines imply the results of Proposition 3.

Proof of Lemma 3

Here we prove the first result of the lemma. Let β = (β 1 , . . . , β d ) ∈ B, γ ∈ B, f ∈ Σ(β, L) and x = (x 1 , . . . , x d ), y = (y 1 , . . . , y d ) ∈ R d . We have

K γ,2 * f (x) -K γ,2 * f (y) = K γ (u) {f (x 1 -u 1 h 1,2 (γ), . . . , x d -u d h d,2 (γ)) -f (y 1 -u 1 h 1,2 (γ), . . . , y d -u d h d,2 (γ))} du ≤ K γ (u)du d i=1 L i |x i -y i | β i = d i=1 L i |x i -y i | β i Then, K γ,2 * f belongs to Σ(β, L). As a consequence, we have b β * γ,2 (•, f ) -b γ,2 (•, f ) ∞ = b β,2 (•, K γ,2 * f ) ∞ ≤ sup f ∈Σ(β,L) b β,2 (•, f ) ∞ .
Thus, from Proposition 1, we deduce that

sup f ∈Σ(β,L) b β * γ,2 (•, f ) -b γ,2 (•, f ) ∞ ≤ ψ n (β)λ 2 (β) 2β + 1 .
The two other results can be proved exactly in the same way.

Proof of Lemma 4

We prove here the result for j = 2. The result for j = 1 can be proved exactly in the same way. Let β = (β 1 , . . . , β d ), γ ∈ B such that γ < β and t ∈ [0, 1] d . We have

E f Z 2 β * γ,2 (t) = σ 2 n h2 2 (β) h2 2 (γ) K β t -u -v h 2 (β) K γ v h 2 (γ) dv 2 du,
where the notation u t , for two vectors u = (u 1 , . . . , u d ) and t = (t 1 , . . . , t d ), represents the vector (u 1 /t 1 , . . . , u d /t d ). By the generalized Minkowskii inequality (cf. Appendix), we deduce that

E f Z 2 β * γ,2 (t) ≤ σ 2 n h2 2 (β) h2 2 (γ) K 2 β t -u -v h 2 (β) K 2 γ v h 2 (γ) du 1/2 dv 2 ≤ σ 2 K β 2 2 n h2 2 (β) h2 2 (γ) K γ v h 2 (γ) dv 2 = σ 2 K β 2 2 n h2 (β) .
Now the proof of ( 12) is similar to the proof of Proposition 3, and then to that of Lemma 4 of [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF]. The process Z β * γ satisfies as β) . Furthermore, to apply that argument, we need to bound the quantity

Z β , E f Z 2 β * γ (t) ≤ σ 2 K β 2 2 n h2 (
E f Z 2 β * γ (t) -Z 2 β * γ (s) 2 by a multiple of 1 n h2 (β) d i=1 s i -t i h i,2 (β)
β i for s = (s 1 , . . . , s d ), t = (t 1 , . . . , t d ) in [0, 1] d .
Here we look at this quantity.

Let s, t ∈ [0, 1] d , we have

E f Z 2 β * γ (t) -Z 2 β * γ (s) 2 = σ 2 n h2 2 (β) h2 2 (γ) K γ v h 2 (γ) K β t -u -v h 2 (β) -K β s -u -v h 2 (β) dv 2 du ≤ σ 2 n h2 2 (β) h2 2 (γ)   K 2 γ v h 2 (γ) K β t -u -v h 2 (β) -K β s -u -v h 2 (β) 2 du 1/2 dv   2 ≤ σ 2 D 20 n h2 (β) 1 h2 (γ) K γ v h 2 (γ) dv 2 d i=1 s i -t i h i,2 (β)
β i = σ 2 D 20 n h2 (β) d i=1 s i -t i h i,2 (β) β i ,
the second line being obtained by the generalized Minkowskii inequality (cf. Appendix) and the third line coming from the fact K β satisfies an Hölder condition of order β.

As a consequence, following the proof of Lemma 4 of [START_REF] Bertin | Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes[END_REF], we will obtain the result for j = 2 of Lemma 4.

Proof of Lemma 5

Let f ∈ Σ(β, L) and j ∈ {1, 2, 3}. We have, by a change of variables, (1 + pβ max ) log n exp D 21 t log n t p-1 dt.

E f Z γ,j p ∞ ψ -p n (β)I { Z γ,j ∞>τn,j (γ)} = ψ -p n (β) +∞ 0 P f Z γ,j p ∞ I { Z γ,j ∞>τn,j (γ)} > t dt =ψ -p n (β) (τ n,j (γ)) p +∞ 0 P f Z γ,j p ∞ I { Z γ,j ∞>τn,j (γ)} > t (τ n,j (
By using several integrations by parts, one can find that this integral is at most of order n log n

P f fγ ′ ,1 -fβ ′ ,1 ∞ > η 1 β ′ ≤ P f Z γ ′ ,1 ∞ + Z β ′ ,1 ∞ > η 1 β ′ (1 + κ n ) ≤ P 1 (n) + P 2 (n),
where κ n is of order n -δ with a δ > 0 and

P 1 (n) = P f Z γ ′ ,1 ∞ > ψ n ((β ′ + γ ′ )/2)(1 + κ n ) , P 2 (n) = P f Z β ′ ,1 ∞ > η 1 β ′ (1 + κ n ) 1 - ψ n ( β ′ +γ ′ 2 ) η 1 (β ′ )
.

Using Proposition 2, since

η 2 1 (β ′ )n h1 (β ′ ) 2 K β ′ 2 2 σ 2 = 1 2β ′ +1
1 + p β max -β ′ 2β max +1 log n, we obtain that for n large enough

P 2 (n) ≤ D 24 (log n) -1 2β ′ +1 exp - p 2β ′ + 1 β max -β ′ 2β max + 1 log n . ( 55
)
Using Proposition 2, it can be proved that P 1 (n) is negligible with respect to P 2 (n) as n → ∞.

The relation (55) implies the lemma.

Proof of Lemma 8

Let f ∈ Σ(β, L). By Lemma 3, we have that b β ′ * β,2 (•, f )b β ′ ,2 (•, f ) ∞ is at most of order ψ n (β). Since β ′ < β, ψ n (β) is negligible with respect to η 2 (β ′ ) and therefore

P f fβ * β ′ ,2 -fβ ′ ,2 ∞ > η 2 β ′ ≤ P f Z β * β ′ ,2 ∞ + Z β ′ ,2 ∞ > η 2 β ′ (1 + κ n ) ≤ P 3 (n) + P 4 (n),
where κ n is of order ψ n (β)/η 2 (β ′ ),

P 3 (n) = P f Z β * β ′ ,2 ∞ > ψ n ((β ′ + β)/2)(1 + κ n ) , and
P 4 (n) = P f Z β ′ ,2 ∞ > η 2 β ′ (1 + κ n ) 1 - ψ n ( β ′ +β 2 ) η 2 (β ′ )
.

Using Proposition 2, since

η 2 2 (β ′ ) d i=1 h i,2 (β ′ ) 2 K β ′ 2 2 σ 2 = 1 2β ′ + 1 + pµ(β ′ ) log n,
we obtain that for n large enough

P 4 (n) ≤ D 11 (log n) -1 2β ′ +1 exp -pµ(β ′ ) log n . ( 56
)
Using Lemma 4, it can be proved that P 3 (n) is negligible with respect to P 4 (n) as n → ∞. The relation (56) implies the lemma.

Proof of Lemma 10

Let f ∈ Σ(γ, L). By Lemma 3, we have that

b β * γ,1 (•, f ) -b β,1 (•, f ) ∞ ≤ ψ n (γ)λ 3 (γ) 2γ + 1 . Then P f fβ * γ,1 -fβ,1 ∞ ≥ ψ n (γ)λ 3 (γ) 2γ + 1 (1 + ρ n ) ≤ P 5 (n) + P 6 (n),
where P 5 (n) = P f Z β * γ,1 ∞ > ψ n (γ)λ 3 (γ)ρ n 2(2γ + 1) , and

P 6 (n) = P f Z β,1 ∞ > ψ n (γ)λ 3 (γ)ρ n 2(2γ + 1) .
Using Proposition 2, since ρ n = ψn((β+γ)/2) ψn(γ)

, we obtain that (57) Using Lemma 4, we have that P 5 (n) satisfies the same inequality as (57) but with different constants D 1 and D 2 . Now, since β > γ, we have (cf. [START_REF] Besov | Integral representations of functions and imbedding theorems[END_REF] for a proof)

  where g ∞ = sup t∈[0,1] d |g(t)|, p > 0 and ψ n (β) has the form ψ n (β) = C β

  γ)) p dt = ψ -p n (β) (τ n,j (γ)) p P f ( Z γ,j ∞ > τ n,j (γ))+ψ -p n (β) (τ n,j (γ)) p +∞ 1 P f ( Z γ,j p ∞ > t (τ n,j (γ)) p ) dt.(53) We are going to prove that the two elements of the sum (53) tend to 0 as n tends to ∞. We obtain by Proposition 2 thatP f ( Z γ,j p ∞ > t (τ n,j (γ)) p ) ≤ D 1 p (1 + pβ max ) 2γ + 1 log n exp t 1/p D 21 log n .(54) Using the formula (54) with t = 1, we deduce that ψ -p n (β) (τ n,j (γ)) p P f ( Z γ,j ∞ > τ n,j (γ)) log n) D 22 with some D 22 ∈ R, and then it tends to 0 as n tends to ∞ since γ ≤ β ≤ β max . Moreover by (54) we have, after a change of variables,

∞

  > t (τ n,j (γ)) p ) dt tends to 0 as n tends to ∞.Proof of Lemma 7Let f ∈ Σ(β, L). Using the same argument as in the proof of (21), we can deduce thatb γ ′ ,1 (•, f ) ∞ = o(ψ n (γ ′ )) and b β ′ ,1 (•, f ) ∞ = o(ψ n (β ′ )) for all f ∈ Σ(β, L). As a consequence for all f ∈ Σ(β, L) b γ ′ ,1 (•, f )b β ′ ,1 (•, f ) ∞ = o(ψ n (β ′ )), since β ′ < γ ′ ,and therefore

=

  D 16 (log n) D 17 n D 18 ,with D 17 ∈ R, which includes the lemma.Appendix: Minkowskii inequalityFor all function g measurable on R d × R d , we have g