A Henrot

I Lucardesi

G Philippin

ON TWO FUNCTIONALS INVOLVING THE MAXIMUM OF THE TORSION FUNCTION

Keywords: torsional rigidity, first Dirichlet eigenvalue, shape optimization. 2010 MSC: 35P15, 49R05, 35J25, 35B27, 49Q10

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider T (Ω)/(M (Ω)|Ω|) and M (Ω)λ 1 (Ω), where Ω is a bounded open set of R d with finite Lebesgue measure |Ω|, M (Ω) denotes the maximum of the torsion function, T (Ω) the torsion, and λ 1 (Ω) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

Introduction

The two most classical (and most studied) elliptic PDEs are probably the torsion problem, also known as St-Venant problem, and the Dirichlet eigenvalue problem, see (1.3) and (1.4) below. Many estimates and qualitative properties have been obtained for these classical problems, see for example works by G. Pólya, G. Szegö, M. Schiffer, L. Payne, J. Hersch, C. Bandle and many others. In this paper, following these former works, we are interested in finding bounds (if possible optimal) for quantities involving the maximum of the torsion function. We have been particularly inspired by two recent works in [START_REF] Van Den Berg | Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity, New trends in shape optimization[END_REF] and [START_REF] Van Den Berg | On Pólya's Inequality for Torsional Rigidity and First Dirichlet Eigenvalue[END_REF], where the ratio T (Ω)λ 1 (Ω)/|Ω| has been investigated in a similar way. Here T (Ω) denotes the torsion, λ 1 (Ω) the first Dirichlet eigenvalue, and |Ω| the volume of Ω, see Section 1.2 for the precise definitions.

Le M (Ω) be the maximum of the torsion function. In this paper we investigate upper and lower bounds for the shape functionals

F (Ω) := T (Ω) M (Ω)|Ω| , G(Ω) := M (Ω)λ 1 (Ω) ,
defined over the bounded open sets Ω of R d with finite Lebesgue measure. In Section 2, we prove that the obvious upper bound F (Ω) ≤ 1 is actually sharp. Then, we show that for convex domains we have indeed F (Ω) ≤ 2/3 and we give more precise lower and upper bounds for regular plane convex domains in terms of the curvature of their boundaries. In Section 3, we consider the functional G. We prove that the easy lower bound G(Ω) ≥ 1 is actually sharp. For convex domains, we recall the lower bound G(Ω) ≥ π 2 /8 obtained by L. Payne. Finding the optimal upper bound for G seems much more difficult. Using topological derivatives, we prove that no maximizer exists in a wide class of domains. When we restrict to the class of convex domains, we can prove existence of an optimal domain but we cannot identify it. In the plane, we suspect that it is the equilateral triangle (which is definitely better than the disk). At last, we write the shape derivative of G and prove that the equilateral triangle does not cancel this shape derivative, in other words it is not a critical point among all regular open sets. Given a point x ∈ R d and a positive parameter r > 0, we denote by B r (x) the ball of radius r centered in x, and with B r (x) its closure.

We define the minimal width of a set as the minimal distance between two parallel supporting hyperplanes.

We denote by f + the positive part of a scalar function f , namely f + (x) := max{f (x), 0}.

The partial derivative of a scalar function f defined in R d with respect to the i-th variable is denoted either by ∂f /∂x i or by f ,i ; the same notation is used for higher order partial derivatives.

We adopt the convention of summation over repeated indices.

1.2. First properties. Given a bounded open set Ω of R d with finite Lebesgue measure, we denote by u Ω the torsion function of Ω, that is, the solution of

-∆u = 1 in Ω u ∈ H 1 0 (Ω) , (1.3)
and we set

T (Ω) := u Ω L 1 (Ω) , M (Ω) := u Ω L ∞ (Ω) .
It is easy to check that u Ω is C ∞ inside Ω and non negative in Ω, thus

T (Ω) = Ω u Ω dx , M (Ω) = max Ω u Ω .
We denote by λ 1 (Ω) the first eigenvalue of the Dirichlet Laplacian and by ϕ Ω the corresponding (normalized) eigenfunction, that is, the solution of

-∆ϕ = λ 1 (Ω)ϕ in Ω ϕ ∈ H 1 0 (Ω) , (1.4)
with ϕ Ω L 2 (Ω) = 1.
We recall that the functionals T and λ 1 admit the following variational formulations:

T (Ω) = sup v∈H 1 0 (Ω)\{0} Ω v dx 2 Ω |∇v| 2 dx , λ 1 (Ω) = inf v∈H 1 0 (Ω)\{0} Ω |∇v| 2 dx Ω v 2 dx . (1.5)
It follows from the homogeneity relations

T (tΩ) = t d+2 T (Ω) , M (tΩ) = t 2 M (Ω) , λ 1 (tΩ) = t -2 λ 1 (Ω) , t > 0 ,
that both F and G are scale invariant.

In the sequel, when no ambiguity may arise, we will denote the torsion function and the first eigenfunction of the Dirichlet Laplacian of a given set Ω simply by u and ϕ, respectively. Remark 1.1. Note that the functional G can be studied on a larger class of sets, satisfying λ 1 (Ω) > 0 and u Ω ∞ < +∞. In particular, since these two conditions are equivalent (see, e.g., [START_REF] Van Den Berg | Hardy inequality and L p estimates for the torsion function[END_REF]), G is naturally defined on sets on which a Poincaré inequality holds (e.g., sets bounded in one direction).

Bounds for the functional F

2.1. The upper bound. The upper bound F (Ω) ≤ 1 is obvious. Actually, we are going to prove that this bound is sharp. This is not so intuitive since the equality F (Ω) = 1 is only true for constant functions and clearly a torsion function of any domain Ω is a priori far to be constant. The idea is to use the theory of homogenization. Indeed, by performing suitable spherical holes (with the appropriate radius) in a domain Ω, we are able to get a sequence of torsion functions which γ-converges to something which is no longer a torsion function : the "strange term coming from nowhere" in the celebrated paper by D. Cioranescu and F. Murat, [START_REF] Cioranescu | Topics in the mathematical modelling of composite materials[END_REF]. Our theorem is the following. Theorem 2.1. In any dimension, we can find a sequence of domains Ω such that F (Ω ) → 1.

Here we recall the construction of a sequence of perforated domains introduced by Cioranescu-Murat in [START_REF] Cioranescu | Topics in the mathematical modelling of composite materials[END_REF], see also [START_REF] Kacimi | Estimation de l'erreur dans des problèmes de Dirichlet où apparait un terme étrange, Partial differential equations and the calculus of variations[END_REF] for a more precise estimate and convergence result.

Let Ω ⊂ R d , d ≥ 2 a regular (or a convex) domain, and C 0 > 0 be fixed. For every > 0, consider the ball T := B r (0) with a radius r which satisfies

r = C 0 d/(d-2) if d ≥ 3 exp(-C 0 / 2 ) if d = 2 (2.1)
and the perforated domain

Ω := Ω \ ∪ z∈Z d (2 z + T ) . (2.2)
Note that the removed holes form a periodic set in the plane, with period 2 . Now let u denote the torsion function of the perforated domain Ω , extended to zero in the holes. It is proved in [START_REF] Cioranescu | Topics in the mathematical modelling of composite materials[END_REF] that the sequence u converges weakly in H 1 0 (Ω) (and strongly in L 2 (Ω)) to the solution u * of -∆u

* + au * = 1 in Ω u * ∈ H 1 0 (Ω) , where the constant a satisfies a = C d-2 0 2 d d(d -2)ω d if d ≥ 3 π 2C0 if d = 2 , (2.3)
and ω d is the volume of the unit ball in R d . As a consequence we have

Ω u dx → Ω u * dx , |Ω | → |Ω| , as → 0 . (2.4)
Now we want to analyze the asymptotic behavior of the L ∞ norm of the functions u . We cannot hope for uniform convergence of u to u * , nevertheless we can prove the convergence of the L ∞ norms: Theorem 2.2. Let u be the torsion functions of the perforated domains Ω extended to zero in the holes and let u * be their weak limit in H 1 0 (Ω). Then, up to a subsequence,

M (Ω ) = u L ∞ (Ω) → u * L ∞ (Ω) as → 0.
Proof. We are indebted to G. Buttazzo and B. Velichkov of this proof (see the final comment at the end of the paper).

First of all, up to a subsequence, we can assume that u converges pointwise almost everywhere to u * : for a.e. x ∈ Ω , u (x) → u * (x) .

Applying this to a ball centered at a point where u * is maximum, we infer that

u * L ∞ (Ω) ≤ lim inf u L ∞ (Ω) . (2.5)
Now let us assume that the inequality in (2.5) is strict. Then we could find two positive numbers

b 1 < b 2 such that u * L ∞ (Ω) ≤ b 1 < b 2 ≤ lim inf u L ∞ (Ω)
. It is proved in [START_REF] Velichkov | Existence and regularity results for some shape optimization problems[END_REF]Proposition 3.2.34] that for any non negative function v satisfying ∆v + 1 ≥ 0 in Ω, the following inequality holds:

v L ∞ (Ω) ≤ C Ω v(x) dx d/(d+2) , (2.6)
where C is a positive constant which only depends on Ω. By taking in (2.6) the subsolution v := (u -b) + to the torsion equation in Ω, with b := (b 1 + b 2 )/2, we obtain

(u -b) + L ∞ (Ω) ≤ C Ω (u -b) + dx d/(d+2)
.

(2.7)

It comes on the one hand

lim inf (u -b) + L ∞ (Ω) ≥ b 2 -b = (b 2 -b 1 )/2 > 0 ,
while on the other hand, by L 2 convergence,

Ω (u -b) + dx → Ω (u * -b) + dx = 0 , contradicting inequality (2.7).
Now we are in position to prove Theorem 2.1. Let us introduce v * := au * . In the following Lemma, we list some properties of this function.

Lemma 2.3. Let a ∈ R + and v * be the solution of

-a -1 ∆v * + v * = 1 (2.8)
in H 1 0 (Ω). Then 0 < v * ≤ 1 in Ω and, in the limit as a → +∞, v * 1 weakly in L 2 (Ω).

Proof. The positivity of v * in Ω is a simple consequence of the maximum principle. For x 0 ∈ Ω maximum point for v * it holds ∆v * (x 0 ) ≤ 0. In particular, for every x ∈ Ω we have

v * (x) ≤ v * (x 0 ) ≤ -∆v * (x 0 ) + v * (x 0 ) = 1 ,
which proves the upper bound. Exploiting the optimality of v * for the functional

H 1 0 (Ω) v → 1 2 Ω a -1 |∇v| 2 dx + Ω (v 2 -v) dx ,
it is easy to see that v * and a -1/2 ∇v * are uniformly (with respect to a) bounded in L 2 (Ω) and L 2 (Ω; R 2 ), respectively. The former bound implies that, in the limit as a → +∞, up to subsequences, v * weakly converges in L 2 (Ω) to some v * . The latter bound, combined with (2.8), implies that the weak limit v * is 1.

An immediate consequence of the previous Lemma is

lim inf u L ∞ (Ω) = u * L ∞ (Ω) ≤ 1 a .
(2.9)

Therefore, using (2.4) and (2.9), we get, for a subsequence:

lim sup F (Ω ) ≥ Ω u * dx (max u * )|Ω| = Ω v * dx (max v * )|Ω| ≥ Ω v * dx |Ω| → 1 ,
as a → +∞. This finishes the proof of Theorem 2.1.

2.2.

The upper bound for convex sets. The maximizing sequence used in the previous section is very specific, thus we can expect that in the convex case we can significantly improve the upper bound. Indeed, let us prove the following Theorem 2.4. Let Ω be any bounded convex domain in R d , then

F (Ω) ≤ 2 3 . (2.10)
Moreover, inequality (2.10) is sharp.

Proof. We use the maximum principle for P -functions. Following [START_REF] Payne | Bounds for the maximum Stress in the St-Venant problem[END_REF] (for regular convex domains) or [START_REF] Philippin | On extending some maximum principles to convex domains with nonsmooth boundaries[END_REF] (for general convex domains), it is known that the function ψ := |∇u| 2 + 2u takes its maximum at a critical point of u, namely at the point where u is maximum. Note that √ u has only one critical point since it is strictly concave (see, e.g., [START_REF] Kennington | Power concavity and boundary value problems[END_REF] for the concavity property and [START_REF] Korevaar | Convex solutions of certain elliptic equations have constant rank Hessians[END_REF]Introduction] for the strict one). Therefore, for every x ∈ Ω, we have

|∇u(x)| 2 + 2u(x) ≤ 2M (Ω) ; (2.11) in particular, integrating (2.11) over Ω yields 3T (Ω) ≤ 2M (Ω)|Ω|.
In order to prove the sharpness of the inequality, let us consider the sequence of rectangles in the plane Ω n := (-n, n) × (0, 1). The same construction holds in any dimension d, using the sequence of parallelepipeds Ω n := (-n, n) d-1 × (0, 1). Let us denote by u n the torsion function of Ω n . By the maximum principle, we have

u n (x, y) ≤ 1 2 y(1 -y) .
(2.12)

The function 1 2 y(1 -y) can be seen as the torsion function of the unbounded strip {0 < y < 1}. Therefore (2.12) implies that M (Ω n ) ≤ 1 8 . Now, in view of (1.3), it is easy to check that the torsion admits the variational formulation

- 1 2 T (Ω n ) = min v∈H 1 0 (Ωn) 1 2 Ωn |∇v| 2 dx - Ωn v dx . (2.13)
Let us introduce the function ψ n (x) defined as

ψ n (x) :=        1 if x ∈ [-n + 1, n -1] n -x if x ∈ [n -1, n] x -n if x ∈ [-n, -n + 1] 0 if |x| > n (2.14)
and let us choose as a test function in (2.13) the function v(x, y)

:= ψ n (x) 1 2 y(1 -y), which is an element of H 1 0 (Ω n ). We immediately get Ωn v dx = n -n ψ n (x) dx 1 0 1 2 y(1 -y) dy = n 6 - 1 12 ≥ n 6 -1 . Since |ψ n (x)| = 1 if x ∈ (-n, -n + 1) ∪ (n -1, n)
and it is 0 otherwise, and y(1 -y)/2 < 1 for every y ∈ (0, 1), we have

Ωn |∇v| 2 dx ≤ 2 + n -n ψ 2 n (x) dx 1 0 ( 1 2 -y) 2 dy = 2 + n 6 - 1 9 ≤ n 6 + 2 . Thus - 1 2 T (Ω n ) ≤ - n 12 + 2 ,
which implies that

F (Ω n ) ≥ 2 3 - 16 n → 2/3 when n → +∞ .
For strictly convex and regular domains in the plane, one can improve this upper bound by the following: Theorem 2.5. Let Ω be a strictly convex bounded domain of class C 2 in R 2 . Let us introduce the quantity β which depends only on the geometry of Ω (actually its curvature k):

β = 2 - 1 4 min ∂Ω k max ∂Ω k 3 ≤ 2 .
Then we have

F (Ω) ≤ β β + 1 ≤ 2 3 .
We postpone the proof of this Theorem to Section 2.4, where the proof for a similar lower bound will also be given at the same time.

2.3. The lower bound. Clearly, by the positivity of T , M , and Lebesgue measure, the infimum of F is greater than or equal to zero. It is easy to show that the lower bound 0 is optimal: consider the sequence of sets

Ω n := B 1 (0) n i=1 B rn (x i ) , n ∈ N ,
with x 1 , . . . , x n = 0 distinct points in a compact set and r n > 0 a small parameter (whose precise value will be chosen later). In this case, u n := u Ωn is the sum of the torsion functions associated to every single connected component of Ω n , namely

u n = n i=0 u i with u 0 := u B1(0) and u i := u B n -1/4 (xi) , i = 1 . . . , n. Since u 0 (x) = 1 -|x| 2 2d and u i = r 2 n -|x -x i | 2 2d ,
it is easy to see that

T (Ω n ) = ω d d 2 (d + 2) (1 + nr d+2 n ) , M (Ω n ) = 1 2d , |Ω n | = ω d d (1 + nr d n ) ,
where ω d is the volume of the unit ball in R d . By taking r n = n -1/(2d) , we infer that

F (Ω n ) = 2 d + 2 n 1 2 - 1 d + 1 n 1 2 + 1 ∼ n -1 d → 0 , implying that inf F = 0.
2.4. The lower bound for convex sets. By strict concavity of √ u when Ω is convex, see e.g. [START_REF] Korevaar | Convex solutions of certain elliptic equations have constant rank Hessians[END_REF], it is easy to get a lower bound for convex sets: Theorem 2.6. Let Ω be any bounded convex set in R d , then

F (Ω) ≥ 1 (d + 1) 2 .
(2.15)

Proof. Since √ u is concave, its graph is above the cone of basis Ω and vertex M 0 the maximum point of √ u. Therefore, by comparison of the volumes:

Ω u(x) dx ≥ M (Ω)|Ω| d + 1 .
By taking the square of the previous inequality and using Cauchy-Schwarz inequality for the left-hand side

Ω u(x) dx 2 ≤ |Ω| Ω u(x) dx ,
we get the desired inequality.

We believe that inequality (2.15) is not optimal. For example, in the plane, we conjecture: Conjecture: For any plane convex domain, the following lower bound holds:

F (Ω) ≥ 1 3 .
Moreover, this inequality should be optimal, and a minimizing sequence could be a sequence of isosceles triangles degenerating to a segment. Let us remark that when u is concave, for instance when Ω is an ellipse, we obtain exactly in the same way

T (Ω) |Ω|M (Ω) ≥ 1 3 .
Sufficient conditions on the geometry of Ω to insure the concavity of u have been established by Kosmodem'yanskii in [START_REF] Kosmodem | yanskii: Sufficient conditions for the concavity of the solution of the Dirichlet problem for the equation ∆u = -1[END_REF].

Let us conclude this section with a theorem in the spirit of Theorem 2.5.

Theorem 2.7. Let Ω be a strictly convex bounded domain of class C 2 in R 2 . Then we have

F (Ω) ≥ 1 4 min ∂Ω k max ∂Ω k 3 , (2.16)
where k is the curvature of Ω.

Set for brevity α := min ∂Ω k/ max ∂Ω k 3 /4. Note that inequality (2.16) is better than the general inequality (2.15) when α > 1/8, which occurs when min ∂Ω k > max ∂Ω k/ 3 √ 2. In [START_REF] Payne | Isoperimetric inequalities in the torsion and clamped membrane problems for convex plane domains[END_REF], Payne and Philippin have derived sharp upper bounds for |∇u|. The goal of this section is to derive new lower bounds for these quantities by using the same approach as in [START_REF] Payne | Isoperimetric inequalities in the torsion and clamped membrane problems for convex plane domains[END_REF] . For the torsion problem one can associate an auxiliary function involving the curvature k of the level lines {u = const.}. Properly chosen, the auxiliary function turns out to satisfy some minimum principles, implying the convexity of the level sets of u, under suitable convexity assumptions on Ω. These results have been derived by Makar-Limanov in [START_REF] Makar-Limanov | Solution of the Dirichlet's problem for the equation ∆u = -1 in a convex region[END_REF] for the torsion problem with the associate function

P (x) := k|∆u| 3 + u[(∆u) 2 -u ,ij u ,ij ] . (2.17)
Proof of Theorems 2.5 and 2.7. Making use of normal coordinates with respect to the level lines {u = const.} we have

|∇u| 2 = u ,i u ,i = u 2 n , ∆u = u nn + ku n , u ,ij u ,ij = u 2 nn + k 2 u 2 n + 2u 2 ns
, where an index n stands for the outward normal derivative and an index s stands for the derivative along the level lines {u = const.}, and k is the curvature of the level lines defined as

k := - u ,i |∇u| ,i = u ,ij u ,i u ,j -|∇u| 2 ∆u |∇u| 3 .
The Makar-Limanov function P introduced in (2.17) may be rewritten in terms of normal coordinates as

P = |∇u| 3 k -2[ku n + k 2 u 2 n + u 2 ns ]
. Makar-Limanov's result is based on the fact that P is super-harmonic. It then follows that P takes its minimum value P min on ∂Ω, so that the following quadratic inequality for k holds:

|∇u| 3 k -2u[ku n + k 2 u 2 n + u 2 ns ] ≥ P min , x ∈ Ω . (2.18)
Omitting the term containing u 2 ns and solving (2.18) for k, we obtain

Φ 4u {1 - √ 1 -z} ≤ k|∇u| ≤ Φ 4u {1 + √ 1 -z} , (2.19)
with

Φ := |∇u| 2 + 2u , (2.20)
z := 8P min u Φ 2 . We note that z ≤ 1 in view of the inequality

P ≤ 1 8u Φ 2 in Ω ,
derived in [START_REF] Philippin | Isoperimetric inequalities and overdetermined problems for the Saint-Venant equation[END_REF]. Multiplying (2.19) by -2|∇u| √ u, we obtain

- Φ|∇u| 2 √ u {1 + √ 1 -z} ≤ √ u ∂Φ ∂n ≤ - Φ|∇u| 2 √ u {1 - √ 1 -z} , (2.21)
in view of ∂Φ ∂n = -2k|∇u| 2 .
For convenience we set

θ := Φ √ u = |∇u| 2 √ u + 2 √ u . ( 2
- 1 2 du u ≤ - dθ √ θ 2 -8P min ≤ 1 2 du u . (2.23)
These inequalities link the functions u and θ to their differentials along the orthogonal trajectories of the level lines (also called fall lines of u). Rewriting (2.23) in the form

- 1 2 d(log u) ≤ -d(log[θ + θ 2 -8P min ]) ≤ 1 2 d(log u)
and integrating from a point x ∈ Ω to the maximum point x 0 of u along the fall line joining these points, we obtain

u(x) u(x 0 ) ≤ θ(x) + θ 2 (x) -8P min θ 0 + θ 2 0 -8P min ≤ u(x 0 ) u(x) , (2.24)
with θ 0 := θ(x 0 ) = 2 u(x 0 ) .

Multiplying (2.24) by (θ 0 + θ 2 0 -8P min ) √ u, replacing back θ by Φ, and recalling that u(x 0 ) = M (Ω), we obtain

2u 1 + 1 -2Pmin M (Ω) -Φ ≤ √ Φ 2 -8P min u ≤ 2M (Ω) 1 + 1 -2Pmin M (Ω) -Φ .
(2.25) Squaring (2.25) and solving for Φ, we obtain

2P min 1 + 1 -2Pmin M (Ω) + u 1 + 1 - 2P min M (Ω) ≤ Φ ≤ 2P min u M (Ω) 1 + 1 -2Pmin M (Ω) + M (Ω) 1 + 1 - 2P min M (Ω) .
(2.26)

Replacing (2.20) in (2.26), after some reduction we obtain the basic inequalities

α(M (Ω) -u) ≤ |∇u| 2 ≤ β(M (Ω) -u) in Ω , (2.27)
with

α := 1 -1 - 2P min M (Ω) , (2.28)
β := 1 + 1 - 2P min M (Ω) . ( 2

.29)

The upper bound for |∇u| 2 in (2.27) was already derived in [START_REF] Payne | Isoperimetric inequalities in the torsion and clamped membrane problems for convex plane domains[END_REF]. The lower bound is nontrivial only for strictly convex Ω, whereas the upper bound makes sense even for nonconvex Ω. However in this case P min is negative, and β is greater than two. We note that inequalities (2.27) are exact when α = β, i.e. when 2P min (M (Ω)) -1 = 1. This is the case if and only if Ω is a disk. For practical use of (2.27) a computable positive lower bound for the quantity 2P min (M (Ω)) -1 is needed. To this end, we write

P min = min ∂Ω (|∇u| 3 k) ≥ min ∂Ω |∇u| 3 min ∂Ω k (2.30)
and make use of the inequalities min

∂Ω |∇u| ≥ 1 2 max ∂Ω k , (2.31)
M (Ω) = max Ω u ≤ 1 2 ρ 2 ≤ 1 2 min ∂Ω k -2 , (2.32)
derived in [START_REF] Payne | Some special maximum principles with applications to isoperimetric inequalities, Maximum principles and eigenvalue problems in partial differential equations[END_REF], [START_REF] Payne | Some remarks on the problems of elastic torsion and of torsional creep, Some Aspects of Mechanics of Continua, part 1[END_REF], where ρ is the inradius of Ω. Using (2.30), (2.31), and (2.32), for strictly convex Ω we have

2P min M (Ω) ≥ 2 min ∂Ω |∇u| 3 min ∂Ω k M (Ω) ≥ 1 2 min ∂Ω k max ∂Ω k 3 = 2α . ( 2
T (Ω) |Ω|M (Ω) : α α + 1 ≤ T (Ω) |Ω|M (Ω) ≤ β β + 1 ≤ 2 3 .
The upper bound proves Theorem 2.5.

A better lower bound for T (Ω) |Ω|M (Ω) may be derived by integrating the inequality

P (x) = u ,ij u .i u .j -|∇u| 2 ∆u + u[(∆u) 2 -u .ij u ,ij ] ≥ P min over Ω. Making use of Ω u ,ij u ,i u ,j dx = - Ω u u ,ij u ,ij dx , we obtain Ω P (x) dx = Ω u[(∆u) 2 -2u ,ij u ,ij ] dx + T (Ω) ≥ P min |Ω| . Since (∆u) 2 -2u ,ij u ,ij ≤ 0, it follows that T (Ω) ≥ P min |Ω| .
This inequality, together with (2.33), gives the lower bound F (Ω) ≥ α, concluding the proof of Theorem 2.7.

3.

Bounds for the functional G 3.1. The upper bound. The best available upper bound for G, still not optimal, is very recent and is due to Vogt: in [29, Theorem 1.5], exploiting semigroups techniques, the author proved that

G(Ω) ≤ d 8 + 1 4 5(1 + ln 2/4) √ d + 1 .
Among the upper bounds obtained before Vogt, let us mention the one obtained in [5, Theorem 1], stating that G(Ω) ≤ 3d ln 2 + 4 , and the one in [START_REF] Van Den Berg | On the torsion function with Robin or Dirichlet boundary conditions[END_REF], based on a variational technique, simple and interesting at te same time.

Finding the optimal upper bound suggests to look at the shape optimization problem:

P G sup{G(Ω) , Ω ⊂ R d }.
Even if it looks like as a standard shape optimization problem, the existence of a solution is not clear for us. We believe that a maximizer does not exist and a partial result in this direction is given by the following

Proposition 3.1. Let Ω ⊂ R d be a bounded open set of class C 2 .
Assume that G is differentiable at Ω (i.e., the shape derivative of G at Ω exists and is given by (3.12)). Then Ω is not a maximizer for G.

The proof is postponed to §3.5 and is based on a topological derivative argument: under suitable regularity assumptions on Ω, removing a small hole near the boundary makes G increase.

In order to further investigate P G , other useful tools are represented by numerical tests and the theory of shape derivatives. The former technique suggests that, in the case of polygons in the plane, the optimum should be non convex. The latter, that we detail in §3.5, provides a necessary condition for critical shapes; in particular, it turns out that the equilateral triangle, even if strictly better than the disk, is not optimal for sup G in dimension d = 2 (see Corollary 3.7 below).

Let us now consider the restricted class of convex domains, for which the equilateral triangle could be a maximizer.

The maximization problem in the convex framework.

Unlike what happens in the general case, if we add the convex constraint, the existence of a maximizer for G is guaranteed.

Theorem 3.2. The shape functional G admits a maximizer in the class of bounded convex sets of R d .

Proof. Let Ω n be a maximizing sequence of convex subsets of R d . By the scale invariance of G, without loss of generality, we may assume that the elements of the sequence are all contained in a fixed bounded set K. By the Blaschke selection theorem, there exists a subsequence (not relabeled) converging to some convex set Ω ⊂ K in the Hausdorff metric.

We claim that the minimal width w n of Ω n does not vanish as n → +∞, so that the limit set Ω has non-empty interior: choosing a suitable reference frame in R d , we may assume that Ω n is contained in the strip {x ∈ R d : 0 ≤ x d ≤ w n }; by the maximum principle, it is easy to see that the torsion function

u n of Ω n satisfies u n (x) ≤ x d (w n -x d )/2 in Ω n , in particular M (Ω n ) ≤ w 2 n 8 ; (3.1)
on the other hand, in [6, formula (4.8)] the authors provide the following upper bound for λ 1 in terms of the minimal width:

λ 1 (Ω n ) ≤ π 2 w 2 n 1 + c n (3/2 + 3/2 1/3 + 2 1/3 ) , (3.2)
with c n = c n (Ω n ) a positive constant vanishing as w n → 0 (see [START_REF] Makar-Limanov | Solution of the Dirichlet's problem for the equation ∆u = -1 in a convex region[END_REF] in [START_REF] Van Den Berg | On Pólya's Inequality for Torsional Rigidity and First Dirichlet Eigenvalue[END_REF]). By combining (3.1), (3.2), and the lower bound (3.4), we conclude that G(Ω n ) → π 2 /8. This gives a contradiction, since π 2 /8 is clearly not the maximum of G.

In order to conclude the proof, we need to show that the limit set Ω is a maximizer for G. To this aim, we prove that M , λ 1 , and hence G, are continuous in the class of bounded open convex sets, with respect to the Hausdorff metric.

Let {Ω h } h be a sequence of bounded convex sets converging to some convex set Ω, with respect to the Hausdorff metric. Under these assumptions, there exists a sequence of positive numbers t h → 1 such that t h Ω h ⊂ Ω for every h. By comparison, such inclusion readily implies M (t h Ω h ) ≤ M (Ω). On the other hand, by applying [14, Proposition 3.6.1] (see also the comments below its proof), we get

M (Ω) ≤ M (t h Ω h )+d H (t h Ω h , Ω), being d H the Hausdorff distance. Thus M (t h Ω h ) → M (Ω) as h → +∞. Finally, since M (t h Ω h ) = t 2
h M (Ω h ) and t h → 1, we conclude that M (Ω h ) → M (Ω). The continuity of λ 1 is more classical and can be found, e.g., in [START_REF] Bucur | Variational methods in shape optimization problems[END_REF] or [START_REF] Henrot | Variation et Optimisation de Formes[END_REF].

The problem of finding an optimal set is still open. We conjecture that in dimension d = 2 the maximizer of G among the convex sets is the equilateral triangle T , namely, for every Ω convex, G(Ω) ≤ 4 27 π 2 = G(T ) (see the computations in the proof of Corollary 3.7 below). 3.3. The lower bound. The lower bound G(Ω) ≥ 1 is obvious: indeed, making use of (1.3) and (1.4), we get

Ω ϕ dx = Ω ∇u • ∇ϕ dx = λ 1 (Ω) Ω uϕ dx ≤ G(Ω) Ω ϕ dx .
An alternative proof can also be found in [START_REF] Bueno | Solutions of the Cheeger problem via torsion functions[END_REF]Proposition 6], [START_REF] Van Den | Estimates for the Torsion Function and Sobolev Constants[END_REF]Theorem 5].

Exploiting the same strategy used for the upper bound of F , we show that the constant 1 is sharp.

Theorem 3.3. In any dimension, we can find a sequence of domains Ω such that G(Ω ) → 1.

Proof. Let > 0 be fixed and consider the perforated domain Ω defined in (2.2), obtained by removing to a given regular set Ω periodic spherical holes of period 2 and radius r . The value of the constant C 0 appearing in (2.1) will be chosen later. Let A : L 2 (Ω) → L 2 (Ω) be the resolvent operator of the Dirichlet Laplacian on Ω , which associates to f ∈ L 2 (Ω) the unique solution u ∈ H 1 0 (Ω ) to -∆u = f , extended by zero outside Ω . By applying Theorem 2.5 in [START_REF] Kacimi | Estimation de l'erreur dans des problèmes de Dirichlet où apparait un terme étrange, Partial differential equations and the calculus of variations[END_REF], we infer that, for every f ∈ L 2 (Ω), A (f ) strongly converges to A(f ) in L 2 (Ω), where A is the resolvent operator of -∆ + a in H 1 (Ω) with Dirichlet boundary conditions, being a (function of C 0 ) defined in (2.3). In particular, in view of [13, Theorem 2.3.2], the eigenvalues of A converge to the corresponding eigenvalue of A; in other words, we have

λ 1 (Ω ) → λ 1 (Ω) + a ,
as → 0. On the other hand, as already noticed in (2.9), we have lim inf

M (Ω ) ≤ 1 a . Thus lim inf G(Ω ) = lim inf λ 1 (Ω )M (Ω ) ≤ 1 a (λ 1 (Ω) + a) = 1 + λ 1 (Ω) a . (3.3)
By choosing a suitable C 0 (small enough in the case of d = 2 and large enough in the case of d ≥ 3), the parameter a can be taken arbitrarily large, so that the right-hand side of (3.3) is arbitrarily close to 1. This fact, together with the trivial lower bound G ≥ 1, concludes the proof.

3.4. The lower bound for convex sets. In the convex setting, the optimal lower bound for G was provided by Payne in 1981: for every bounded convex domain Ω of R d , we have

G(Ω) < π 2 8 , (3.4)
and the bound is sharp (see Theorem I and formula (3.12) in [START_REF] Payne | Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function[END_REF]). The optimality of the constant can be checked, e.g., by considering the sequence of parallelepipeds Ω n := (-n, n) d-1 × (0, 1). Indeed, as already seen in the proof of Theorem 2.4, by comparing the torsion function of Ω n with the function

x d (1 -x d )/2 we get M (Ω n ) ≤ 1/8 ; (3.5)
on the other hand, recalling the definition (2.14) of ψ n and taking

v(x 1 , . . . , x d ) := sin(πx d )Π d-1 j=1 ψ n (x j ) ∈ H 1 0 (Ω n )
as test function in the variational formulation (1.5) of λ 1 (Ω n ), we get

λ 1 (Ω n ) ≤ Ωn |∇v| 2 dx Ωn v 2 dx = d -1 n -2/3 + π 2 . (3.6)
From (3.5) and (3.6) we obtain the inequality

G(Ω n ) ≤ π 2 8 + d -1 8(n -2/3) ,
whose right-hand side is arbitrarily close to π 2 /8 as n → +∞.

3.5. Optimality conditions via shape derivatives. In this section we derive optimality conditions by computing the first order shape derivative of G. Namely, given Ω ⊂ R d bounded, open, regular or convex, connected set, we study the limit (when the latter exists)

G (Ω, V ) := lim t→0 G(Ω t ) -G(Ω) t ,
with Ω t := (I + tV )(Ω), I being the identity map and V : R d → R d an arbitrary C 1 vector field.

Recalling that G(Ω) = M (Ω)λ 1 (Ω), if the shape derivative exists, it reads

G (Ω, V ) = M (Ω, V )λ 1 (Ω) + M (Ω)λ 1 (Ω, V ) . (3.7)
It is well known (see, e.g., [14, Théorème 5.7.1]) that

λ 1 (Ω, V ) = - ∂Ω ∂ϕ ∂n 2 V • n dH d-1 , (3.8)
where ϕ is the (normalized) first eigenfunction and n denotes the unit outer normal to ∂Ω.

Remark that ∂ϕ ∂n is well defined as soon as Ω is regular or convex, since ϕ ∈ H 2 (Ω) in that case. The computation of M is more delicate and requires additional assumptions. Proposition 3.4. Let Ω ⊂ R d be a bounded open convex set. Then, for every V ∈ C 1 (R d ), the shape derivative of M at Ω in direction V exists and is given by

M (Ω, V ) = u (x 0 ) ,
where x 0 is a maximum point for u and u is the solution of ∆u = 0 in Ω u + ∇u • V ∈ H 1 0 (Ω) . Proof. Let u t denote the torsion function of Ω t , for t > 0, and consider the function ψ : R×R d → R d defined as (t, x) → ψ(t, x) := ∇u t (x) . By optimality of x 0 for u, we have ψ(0, x 0 ) = 0. Moreover, the matrix D x ψ(0, x 0 ) is invertible: indeed, setting v := √ u, we have

D x ψ(0, x 0 ) = Hess u (x 0 ) = 2 M (Ω)Hess v (x 0 ) ,
and the matrix Hess v is negative definite everywhere in Ω (see [START_REF] Korevaar | Convex solutions of certain elliptic equations have constant rank Hessians[END_REF]). Thus, by the implicit function theorem, we infer that, in a neighborhood of x 0 , for t small enough, there exists a unique x t such that ∇u t (x t ) = 0; furthermore, t → x t is differentiable. We claim that, for every t, u t (x t ) = M (Ω t ) , (3.9) and that, as t → 0,

u t (x 0 ) -u(x 0 ) t → u (x 0 ) , (3.10)
u t (x t ) -u t (x 0 ) t → 0 . (3.11)
Once proved the claims we are done, indeed we have

M (Ω t ) -M (Ω) t = u t (x t ) -u(x 0 ) t = u t (x t ) -u t (x 0 ) t + u t (x 0 ) -u(x 0 ) t → u (x 0 ) ,
which concludes the proof. Let us show (3.9). As already pointed out in the proof of Theorem 3.2, exploiting [14, Proposition 3.6.1] we infer that the sequence u t uniformly converges to u, extended to zero in a larger common domain independent of t. In particular, every sequence of maximizers y t for u t converges to the unique minimizer x 0 of u (see, e.g., [START_REF] Maso | An introduction to Γ-convergence[END_REF]Proposition 5.2,Proposition 7.24] applied to the sequence -u t ). Finally, since x t are isolated critical points in a neighborhood of x 0 , they must coincide with y t , so that u t (x t ) = u t (y t ) = M (Ω t ).

Assertion (3.10) follows by applying the mean value property to the harmonic functions u t -u and u : choose R > 0 such that B R (x 0 ) ⊂ Ω t for every t << 1, then we have

u t (x 0 ) -u(x 0 ) t -u (x 0 ) = - B R (x0) u t (x) -u(x) t -u (x 0 ) dx ≤ C (u t -u)/t -u L 2 (B R (x0)) .
The right-hand side vanishes as t → 0, since the map t → u t ∈ L 2 (R d ) is differentiable at 0 with derivative d dt u t=0 = u (see, for instance, [14, Théorème 5.3.1]). Similarly, property (3.11) follows by combining the mean value property of ∇u t , the differentiability of t → x t , and the strong convergence of u t to u in H 1 (R d ):

u t (x t ) -u t (x 0 ) t = ∇u t (ξ t ) • x t -x 0 t = - B R (ξt) ∇u t (x) dx • x t -x 0 t -→ ∇u(x 0 ) • v 0 = 0 , as t → 0 ,
with ξ t a suitable intermediate point between x 0 and x t , R a positive radius such that B R (ξ t ) ⊂ Ω t for every t << 1, and v 0 the derivative d dt x t t=0 . Remark 3.5. We point out that Proposition 3.4 is valid in a more general setting, when the torsion function u Ω belongs to W 1,∞ (Ω) and has a unique maximum point which is non degenerate (i.e. invertible Hessian at the point). These properties are ensured by the convexity of Ω.

Theorem 3.6. Let Ω ⊂ R d be a bounded open convex set. Then, for every V ∈ C 1 (R d ), the shape derivative of G at Ω in direction V exists and is given by

G (Ω, V ) = ∂Ω λ 1 (Ω) ∂u ∂n ∂φ x0 ∂n -M (Ω) ∂ϕ ∂n 2 V • n dH d-1 , (3.12)
where x 0 ∈ Ω is a maximum point of u and φ x0 is the (Green function) solution of

-∆φ x0 = δ x0 in D (Ω) φ x0 = 0 on ∂Ω .
Proof. First, we rewrite in terms of φ x0 the derivative M (Ω, V ), whose existence is ensured by Proposition 3.4:

M (Ω, V ) = -∆φ x0 , u = - ∂Ω u ∂φ x0 ∂n dH d-1 = ∂Ω ∂u ∂n ∂φ x0 ∂n V • n dH d-1 . (3.13)
Formula (3.12) follows by combining (3.7), (3.8), and (3.13).

As a consequence of Theorem 3.6 we obtain the following optimality condition: if Ω ⊂ R d bounded open convex set is a critical shape for G, then Now we partially answer a question raised in [START_REF] Van Den | Estimates for the Torsion Function and Sobolev Constants[END_REF] where the author asked whether the disk could be the maximizer for G.

Corollary 3.7. The equilateral triangle gives a better value than the disk; however it is not a critical shape for G.

Proof. Let T ⊂ R 2 be the equilateral triangle with side of length 1 and vertices in

(-1/2, -1/(2 √ 3)), (1/2, -1/(2 √ 3
)), and (0, 1/ √ 3), so that the center is at the origin. In this case, u, ϕ, M , and λ 1 can be explicitly computed and read

u(x, y) = 1 2 √ 3 y + 1 2 √ 3 y - √ 3x - 1 √ 3 y + √ 3x - 1 √ 3 = 1 2 √ 3 y 3 -3x 2 y - √ 3 2 y 2 - √ 3 2 x 2 + 1 6 √ 3 ϕ(x, y) = 2 √ 3 3/2 sin 4π 3 1 - √ 3y -2 cos(2πx) sin 2π 3 1 - √ 3y
M (T ) = u(0, 0) = 1 36 , λ 1 (T ) = 16 3 π 2 .
Therefore G(T ) = 4π 2 /27 1.4622 while, for the unit disk D, we have G(D) = j 2 0,1 /4 1.4458 which proves the first part of the claim. Now assume by contradiction that T is a critical shape for G. Since the normal derivative of u on ∂T is never zero except at the vertices (where both ∇u and ∇ϕ vanish), we may recast the optimality condition (3.14) as ∂φ 0 ∂n = h a.e. on ∂T ,

with h := M (Ω) λ 1 (Ω) -1 ∂u ∂n -1 ∂ϕ ∂n 2 .
In particular, if we multiply both sides by an arbitrary harmonic function w and integrate over ∂T , we obtain -w(0) = ∂T wh dH 1 .

(3.15)

By taking as test functions w ≡ 1 and w = Re(z 6 ), we get

       ∂T h dH 1 = -1 ∂T Re(z 6 )h dH 1 = 0 .
Exploiting the symmetry of the domain and of the functions involved, these conditions can be rephrased as follows:

       Σ h dH 1 = - 1 3 Σ Re(z 6 )h dH 1 = 0 ,
where Σ denotes the basis of the triangle, i.e. the segment Σ = [-1/2, 1/2] × {-1/(2 √ 3)}. This system may be rewritten as

         σ := 1/2 0 (1 + cos(2πx)) 2 x 2 -1/4 dx = - 27 8 τ := 1/2 0 P (x) (1 + cos(2πx)) 2 x 2 -1/4 dx = 0 , (3.16)
with

P (x) : = Re(z 6 ) Σ = (x 6 -15x 4 y 2 + 15x 2 y 4 -y 6 ) y=-1/(2 √ 3) = x 6 - 5 4 x 4 + 5 48 x 2 - 1 1728
.

Making use of the factorization

P (x) = x 2 - 1 4 x 4 -x 2 - 7 48 - 1 27
,

we obtain τ = 1/2 0 x 4 -x 2 - 7 48 (1 + cos(2πx)) 2 dx - σ 27 .
It is easy to check that (3.16), which in turn implies τ + σ/27 = -1/8. Therefore we conclude that the equilateral triangle is not a critical shape. Remark 3.8. We point out that the choice of any test function of the form w = Rez n , for n = 1, . . . , 5, in (3.15) does not provide any contradiction. Moreover the numerical values of σ and τ defined in the above proof are not so far of what appears in (3.16). In some sense, the equilateral triangle is not far from being a critical point.

We conclude the Section with the proof of Proposition 3.1.

Proof of Proposition 3.1. Assume by contradiction that Ω is a maximizer for G. Given x ∈ Ω and > 0 a small parameter, we set Ω (x) the perforated domain Ω \ B (x) and we denote by u ,x the associated torsion function.

In the limit as → 0, we have the following asymptotic expansions for λ 1 (Ω (x)) and u ,x in terms of λ 1 (Ω) and u (cf. [START_REF] Flucher | Approximation of Dirichlet eigenvalues on domains with small holes[END_REF] and [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF]Chapter 8]): (here we have used the symmetry of the Green function: φ x (x 0 ) = φ x0 (x)). To get a contradiction, it suffices to find a point x in which R(x) > 0. Taking x close to the boundary, say x = x 1 -δn(x 1 ) for some x 1 ∈ ∂Ω and 0 < δ << ∂n is strictly negative on the boundary ∂Ω, therefore R(x) is strictly positive. In particular, in view of (3.17) we conclude that, if x is close enough to the boundary, G(Ω (x)) > G(Ω) , which is in contradiction with the maximality of Ω.

λ 1 (Ω (x)) =    λ 1 (Ω) + 2π | log | ϕ 2 (x) + o(1/| log |) if d = 2 λ 1 (Ω) + d-2 (d -2)ω d ϕ 2 (x) + o( d-2 ) if d > 2 u ,x (y)
Final comment. In the conference "Shape optimization and isoperimetric and functional inequalities", held in CIRM -Luminy (France) on November 21-25, 2016, we had the occasion to discuss with G. Buttazzo and B. Velichkov, who kindly gave us the proof of Theorem 2.2.

In the same conference, we announced our results to M. van den Berg, since they disproved some conjectures he gave in [START_REF] Van Den | Estimates for the Torsion Function and Sobolev Constants[END_REF]. Few weeks later, M. van den Berg posted [START_REF] Van Den | Spectral bounds for the torsion function[END_REF] on ArXiv, where he gave a different proof of our Theorem 3.3. Finally, we point out that the recent contribution [START_REF] Della Pietra | On functionals involving the torsional rigidity related to some classes of nonlinear operators[END_REF] generalizes some of our results to more general operators.

1. 1 .

 1 Notations. We adopt standard notations for Lebesgue and Sobolev spaces on a bounded open set of R d , for example L 2 (Ω) and H 1 (Ω) (space of functions in L 2 whose derivative, in the sense of distributions, are still in L 2 ). The boundary values of a Sobolev function are always intended in the sense of traces. The (d -1)-dimensional Hausdorff measure is denoted by H d-1 . Given a bounded open set Ω ⊂ R d , we denote by |Ω| its Lebesgue measure, by -Ω the average integral over it, and by D(Ω) the space of C ∞ functions having compact support contained into Ω. The closure of D(Ω) in H 1 (Ω) is denoted by H 1 0 (Ω). If the open set Ω has Lipschitz boundary, we denote by n the outer unit normal vector to ∂Ω, defined a.e. on the boundary.

= 2 M

 2 ) -2π | log | u(x)φ x (y) + o(1/| log |) if d = 2 u(y) -d-2 (d -2)ω d u(x)φ x (y) + o( d-2 ) if d > 2, where ω d is the measure of the (d -1)-sphere and φ x (y) is the Green function of the Laplace operator vanishing on the boundary of Ω. In particular, choosing x different from the maximum point x 0 of u and evaluating u ,x at x 0 , we obtainM (Ω (x)) ≥    M (Ω) -2π | log | u(x)φ x (x 0 ) + o(r( )) if d (Ω) -d-2 (d -2)ω d u(x)φ x (x 0 ) + o(r( )) if d > 2 ,where r( ) = 1/| log | if d = 2 and d-2 otherwise. In particular, we obtain the lower boundG(Ω (x)) ≥ G(Ω) + R(x)r( ) + o(r( )) ,(3.17)with R(x) := M (Ω)ϕ 2 (x) -λ 1 (Ω)u(x)φ x0 (x)

  Since π is not algebraic, the last relation is in contradiction with

				= -	13 80	+	15 64π 2 +	189 128π 4 .
	0	1/2	(1 + cos(2πx)) 2 dx =	3 4	;
	0	1/2		x 2 (1 + cos(2πx)) 2 dx =	1 16	-	15 32π 2 ;
	0	1/2		x 4 (1 + cos(2πx)) 2 dx =	3 320	-	15 64π 2 +	189 128π 4 .
	Thus we get
	τ +	σ 27	=	3 320	-	15 64π 2 +	189 128π 4 -	1 16	+	15 32π 2 -	7 48	•	3 4
				=	3 320	-	1 16	-	7 64	+	15 32π 2 -	15 64π 2 +	189 128π 4

  [START_REF] Van Den | Estimates for the Torsion Function and Sobolev Constants[END_REF], and recalling that ϕ vanishes on ∂Ω, we may write∂ 2 ϕ ∂n 2 (x 1 ) + o(δ 2 ) . Furthermore, by combining (1.4) with the relation ∆ϕ = ∆ ∂Ω ϕ + H ∂Ω ∂ϕ ∂n + ∂ 2 ϕ ∂n 2 on ∂Ω, we get ∂ 2 ϕ ∂n 2 (x 1 ) = -H ∂Ω (x 1 )Arguing in the same way for u and φ x0 , we obtain the developments ∂ 2 φ x0 ∂n 2 (x 1 ) + o(δ 2 ) , and the equalities∂ 2 u ∂n 2 (x 1 ) = -1 -H ∂Ω (x 1 )In view of these computations, we infer that R(x) = δ 2 + δ 3 H ∂Ω (x 1 ) M (Ω) ∂ϕ ∂n ) + o(δ 3 ) .

	2 (x 1 By optimality of Ω, the equality (3.14) holds true at x 1 , so that -λ 1 (Ω) ∂u ∂n ∂φ x0 ∂n (x 1 ) -δ 3 2 λ 1 (Ω) ∂φ x0 ∂n
		R(x) = -	δ 3 2	λ 1 (Ω)	∂φ x0 ∂n	(x 1 ) + o(δ 3 ) .
	By the Hopf's principle	∂φx 0
		ϕ(x) = -δ	∂ϕ ∂n	(x 1 ) +	δ 2 2
						∂ϕ ∂n	(x 1 ) .
		u(x) = -δ	∂u ∂n	(x 1 ) +	δ 2 2	∂ 2 u ∂n 2 (x 1 ) + o(δ 2 ) ,
		φ x0 (x) = -δ	∂φ x0 ∂n	(x 1 ) +	δ 2 2
						∂u ∂n	(x 1 ) ,

∂ 2 φ x0 ∂n 2 (x 1 ) = -H ∂Ω (x 1 ) ∂φ x0 ∂n (x 1

) .

Acknowledgments. We thank the referee for his/her careful reading of the paper and for the useful suggestions.


Pascal Lef Èvre
email: pascal.lefevre@univ-artois.fr

M ÜNTZ SPACES AND SPECIAL BLOCH TYPE INEQUALITIES

Keywords: 2010 Mathematics Subject Classification. 30B10 Müntz spaces, Power series, lacunary sequences, Bloch type inequalities

In this paper, we are interested in classical Müntz spaces and their properties. We first reprove Müntz theorem and obtain the Clarkson-Erdös-Schwartz expansion in the non dense case for some general abstract Müntz spaces without invoking any blow up property, but using classical complex variables tools. We introduce new Bloch type Müntz spaces and investigate the validity of Bloch type inequalities in this framework.

Introduction

The Müntz theorem (see [START_REF] Müntz | Über den Approximationssatz von Weierstrass[END_REF] for the original result and [START_REF] Almira | Müntz type theorems I[END_REF] for a good recent survey) gives an answer to a very natural question on the extension of the Weierstrass theorem in approximation theory in the framework of L p (0, 1) and C([0, 1]). It is a natural question to understand the geometry (and the behavior of their operators) of the Banach spaces spanned by monomials in the non dense case: that is the purpose of the monograph [START_REF] Gurariy | Geometry of Müntz spaces and related questions[END_REF], and a series of recent results [START_REF] Alalam | Essential norms of Volterra and Cesàro operators on Müntz spaces[END_REF], [START_REF] Chalendar | Embeddings theorems for Müntz spaces[END_REF], [START_REF] Fricain | Müntz spaces as model spaces[END_REF], [START_REF] Gaillard | Lacunary Müntz spaces: isomorphisms and Carleson embeddings[END_REF], [START_REF] Ludkowski | On the geometry of Müntz spaces[END_REF] or [START_REF] Noor | Embeddings of Mntz spaces: the Hilbertian case[END_REF].

More precisely, in the sequel, we shall use functions e λ : t ∈ (0, 1) → t λ for λ ∈ R + . Given Λ ⊂ R + , we denote by M Λ , the vector space spanned by the functions e λ where λ ∈ Λ. For p ≥ 1, we denote by M p Λ (resp.M ∞ Λ ) the closure of M Λ in L p (0, 1) (resp. in C([0, 1])), equipped with their usual norm. When Λ = {λ n } (concretely we shall always work with an increasing sequence in the sequel) satisfies the

Müntz condition

λn =0 1 λ n < ∞
then we get a strict subspace of L p (0, 1) and C([0, 1]): this is the conclusion of the classical Müntz theorem.

Let us emphasize that, as soon as we restrict to the case Λ ⊂ N, the functions in the space spanned by the e λ are "true" (algebraic) polynomials. In this case, there are already many open questions on the structure of the spaces M p Λ , which turns out to be some spaces of analytic functions on the complex unit disk. Indeed the Clarkson-Erdös-Schwartz theorem (see [START_REF] Gurariy | Geometry of Müntz spaces and related questions[END_REF] for instance, and section 2 below) states that, up to the gap condition (see (GC) below), functions of Müntz spaces M p Λ (resp. M ∞ Λ ) are precisely functions f ∈ L p (0, 1) (resp. in C([0, 1])) which are analytic on (0, 1) with an expansion f (t) = a n t λn for almost every t ∈ [0, 1).

Up to a change of our original function on a set with measure 0, we shall assume in sections 3 and 4 that this expansion is valid on [0, 1). In particular our functions have derivatives on [0, 1).

Let us give more details on the framework. We are going to work mainly (at least in the three first sections) with some "classical" functions spaces over (0, 1), that is Banach spaces X of functions on (0, 1) with the following properties (i) C 0 ([0, 1]) = {f ∈ C([0, 1]) | f (0) = 0} ⊂ X continuously: there exists k > 0 such that for every f ∈ C 0 ([0, 1]),

f X ≤ k f ∞ .
(ii) C 0 ([0, 1]) is dense in X.

Keep mainly in mind the examples of spaces C 0 ([0, 1]) and L p (0, 1) = L p ([0, 1], m), where m is the Lebesgue measure. In another direction, we shall introduce some Bloch type spaces in section 4 and we shall see that the Clarkson-Erdös-Schwartz expansion holds too in this framework under the Müntz condition, although (i) and (ii) do not hold.

In section 2, we are going to recover Müntz theorem and Clarkson-Erdös-Schwartz for rather general Banach spaces of functions on [0, 1), containing polynomials. The gap condition

(GC) inf λ n+1 -λ n > 0.
will be sometimes involved.

With our approach based on Blaschke products, we do not need to invoke any blow up property to get Müntz theorem and Clarkson-Erdös-Schwartz expansion for classical spaces. We shall say that X has the blow up property if the blow up operator T

ρ : f ∈ X -→ f (ρ •) ∈ X is bounded for every ρ ∈ (0, 1) and (buc) sup ρ 0 <ρ<1
T ρ X < ∞ for any ρ 0 ∈ (0, 1).

Exploiting this property turns out to be very efficient to deduce from the case X = C 0 ([0, 1]) Müntz and Clarkson-Erdös-Schwartz theorems for more general spaces, using a duality argument, but also some complementary technical tools (see [START_REF] Gurariy | Geometry of Müntz spaces and related questions[END_REF]). In the present paper, we use a more direct argument to get the main results for classical spaces. We only use partially the blow up property in the context of weighted Bloch type spaces with large parameter (see the proof of Th.4.2 below), but avoiding the requirement of more machinery.

In section 3, we focus on the Gurariy-Macaev theorem. We point out that this nice result implies that the Bergman norm and the L p (0, 1)-norm are equivalent on the space of lacunary Taylor series. We also revisit an almost isometric version of this theorem in the hilbertian framework and give a very elementary proof of this version.

In section 4, we focus on Bloch type inequalities for Müntz spaces. We first introduce a Bloch type space of function on (0, 1) and prove a version of the Clarkson-Erdös-Schwartz theorem in this framework. We compare this space to classical Müntz spaces M p Λ , equivalently we consider special inequalities between the L p norm of a Müntz polynomial and a weighted norm of its derivative. Such inequalities hold in the lacunary case but we shall construct some counterexamples showing that they do not hold in general for a given Λ. We also show that for some Λ ⊂ N, there exists some function in the associated Müntz space M ∞ Λ such that the derivative does not belong to the weak L 1 space over (0, 1).

We did no try to adopt the most general point of view. For instance in the L p framework, we could consider e λ when λ > -1 p (or even complex exponent) and some phenomenom when λ is close to -1 p are similar when λ → ∞. Actually, as already mentioned, many natural and interesting questions are still completely open when Λ ⊂ N.

In the sequel, we shall sometimes use the notation A B, which means that there exists k > 0 such that A ≤ kB where k depends maybe on p (or on Λ). In the same way, A ≈ B means that A B and B A.

Müntz theorem and Clarkson-Erdös-Schwartz Theorem

In this section, we are interested in proving in our framework two classical results of the theory of Müntz spaces. The spirit and the tools of the following proofs are not really new, nevertheless it is slightly different from the usual presentation, as chosen in [START_REF] Gurariy | Geometry of Müntz spaces and related questions[END_REF] for instance, where the proof uses Cauchy determinant whereas we use a Blaschke product. The spirit of the proof is partially inspired from part of the proof of the Müntz theorem given in [START_REF] Rudin | Real and complex analysis[END_REF] in the special case X = C([0, 1]) (see [4, E.13, p.192] too). We push this idea farther and adapt it to get both Müntz Theorem and Clarkson-Erdös-Schwartz expansion for Banach spaces of functions on (0, 1) whose norm dominates the L 1 norm. As far as we know, this strategy is new and our approach relies on the following proposition.

Proposition 2.1. Assume that 1 λ n < +∞.

• For every m > -1, we have d L 1 e m , span e λ ; λ ∈ Λ \ {m} > 0.

• When (GC) is fulfilled, there exists a function γ : Λ → R + satisfying γ(m) → 0 (when m → +∞) and ∀m ∈ Λ , d L 1 e m , span e λ ; λ ∈ Λ \ {m} ≥ e -(m+1)γ(m) .

Here d L 1 (f, A) = inf{ f -a 1 ; a ∈ A} where f ∈ L 1 (0, 1) and A ⊂ L 1 (0, 1).

Proof. In the sequel for any a ∈ R, we denote C a = {z ∈ C; Re(z) > a}.

The Blaschke product of the right half plane

B(z) = ∞ n=0 λ n + 1 -z λ n + 1 + z is defined for every z ∈ C -1 and B is holomorphic on C -1 . Moreover B is inner: for every z ∈ C 0 , we have |B(z)| ≤ 1.
Therefore, the function

F (z) = 2 (1 + z) 2 B(z) defines an holomorphic function on C -1 such that s ∈ R -→ F (s) = F (is) belongs to L 1 (R)
, with norm less than 2π. Thanks to the Cauchy formula, we have

(1) ∀z ∈ C 0 , F (z) = 1 2π +∞ -∞ F (s) z -is ds and the function (2) µ Λ (x) = 1 2π +∞ -∞ F (s)x -is ds for x ∈ [0, 1]
defines a continuous function with µ Λ ∞ ≤ 1. The continuity at point 0 follows from the Riemann-Lebesgue lemma, since actually µ Λ is the Fourier transform of F at point ln(x).

The crucial point is the property

(3) ∀z ∈ C 0 , 1 0 x z-1 µ Λ (x) dx = F (z) = 2 (1 + z) 2 ∞ n=0 λ n + 1 -z λ n + 1 + z ,
which follows from Fubini's theorem. Indeed, for every z ∈ C 0 , we have

1 0 x z-1 µ Λ (x) dx = 1 2π +∞ -∞ 1 0 x z-1 x -is dx F (s) ds = 1 2π +∞ -∞ 1 z -is F (
x m µ Λ (x) dx = 2 (m + 2) 2 ∞ n=0 λ n -m λ n + m + 2
which vanishes if and only m ∈ Λ. Now, consider Λ m = Λ \ {m}. For every polynomial P ∈ span e λ ; λ ∈ Λ m , we have, thanks to (4),

1 0 (x m -P (x))µ Λm (x) dx = 1 0 x m µ Λm (x) dx = 2 (m + 2) 2 n≥0 λn =m λ n -m λ n + m + 2 • A fortiori, for every real number m > -1, (5)
e m -P 1 ≥ e m -P 1 • µ Λm ∞ ≥ 2 (m + 2) 2 n≥0 λn =m |λ n -m| λ n + m + 2 > 0
and the first part of the proposition is proved. Now, we assume that the gap condition is fulfilled: inf λ n+1 -λ n ≥ δ for some δ ∈ (0, 1). We focus on (5) when m ∈ Λ and aim to get a lower bound estimate of the right hand side. We split the product into three part and estimate each of them.

We first easily check that

n≥0 λn>2m λ n -m λ n + m + 2 = exp - λn>2m ln 1+ 2(m + 1) λ n -m ≥ exp -4(m+1)ρ 2m (Λ)
where ρ s (Λ) stands for

λn>s 1 λ n . Denote N s = {n ∈ N; λ n ≤ s} . We have N s = o(s) (when s → +∞) because 1/λ n < +∞.
Let n be an integer such that m < λ n ≤ 2m. We point out that λ n -m ≥ δ(n -d) when m = λ d . We get this way distincts integers n -d running over {1, . . . , N 2m -N m }.

Using the inequality k! ≥ k k e -k (which is valid for any integer k ≥ 1), we obtain

m<λn≤2m λ n -m λ n + m + 2 ≥ (N 2m -N m )! δ N 2m -Nm (3m + 2) N 2m -Nm ≥ δ N 2m -Nm exp -(m + 1)θ((N 2m -N m )/(m + 1))
where the function θ is defined by θ(t) = t ln(3e/t) when t > 0 and θ(0) = 0.

In the same way, we have

n≥0 m>λn m -λ n λ n + m + 2 ≥ δ Nm exp -(m + 1)θ((N m -1)/(m + 1))
We define

γ(m) = -ln(2) + 2 ln(m + 2) m + 1 +4ρ 2m (Λ)- N 2m ln(δ) m + 1 +θ N 2m -N m (m + 1) +θ N m -1 (m + 1)
and clearly γ(m) -→ 0 when m → ∞ and the conclusion follows.

As an immediate corollary, we recover the non dense part of the Müntz theorem for some general classes of Banach spaces.

Theorem 2.2. [abstract Müntz theorem] Let X be a Banach space of functions on (0, 1) such that the following inequalities hold continuously:

C 0 ([0, 1]) ⊂ X ⊂ L 1 (0, 1) and such that C 0 ([0, 1]) is dense in X. Let Λ = {λ n } be an increasing sequence of positive numbers.
Then span{e λn } is dense in X if and only if

1 λ n = +∞.
Moreover, when the series converges, for every m / ∈ Λ, we have e m / ∈ span{e λn }

Let us mention that the space C([0, 1]) does not fulfill all the hypothesis of this theorem. Nevertheless, it suffices to include the constant functions (i.e. to add e 0 in the span

) since C([0, 1]) = Ce 0 + C 0 ([0, 1]).
Proof. Assume that 1/λ n < +∞, then Prop.2.1 implies that for every m not belonging to Λ, the function e m does not belong to M 1 Λ . It implies that e m does not belong to M X Λ neither, which is a strong negation of the density. Now assume that 1/λ n = +∞, then we know that span{e λn } is dense in C 0 ([0, 1]) by the classical version of the Müntz theorem, but for sake of completeness let us reproduce here the (now classical) argument from [START_REF] Rudin | Real and complex analysis[END_REF]: assume that a measure µ on (0, 1) satisfies

Φ(z) = (0,1) t z dµ(t) = 0 for every z ∈ Λ.
It means that the function Φ which is actually defined, bounded and holomorphic on the right half plane C 0 , vanishes on Λ. Therefore, Φ belongs to the space H ∞ (C 0 ) (the Hardy space H ∞ of the right half plane) and the condition 1/λ n = +∞ means that the Blaschke condition on zeroes of non trivial such functions is not satisfied, hence Φ = 0 and a fortiori Φ actually vanishes on positive integers. The Weierstrass theorem implies that µ = 0 and

span{e λn } is dense in C 0 ([0, 1]). Since C 0 ([0, 1]) is dense in X, the "if" part is proved.
We are going to deduce too the Clarkson-Erdös-Schwartz Theorem from Prop.2.1. We first need Lemma 2.3. Let Λ satisfying the gap condition (GC) and 1/λ n < +∞. For every ε > 0, there exists C ε > 0 (depending on Λ) satisfying:

For every P ∈ span{e λn }, written P = a n e λn , we have for every n,

|a n | ≤ C ε (1 + ε) λn P 1 .
In particular, the linear map e * λ : P ∈ span{e λn }, • 1 -→ a λ ∈ C (where P = a λ e λ ) is continuous, hence extends to a (bounded) linear functional on M 1 Λ . We shall still denote e * λ this functional in the sequel.

Proof. Fix m = λ n (where a n = 0) and apply Prop.2.1 to P = -a -1 n j =n a j e λ j . We get a -1 n P 1 = e λn -P 1 ≥ e -(λn+1)γ(λn) .

We know that e γ(λn) is bounded (by C say) on one hand and is less than 1 + ε for n large enough (say n ≥ n ε ) on the other hand. We denote

C ε = C max 1, sup n≤nε e γ(λn)
1 + ε λn and we get the conclusion.

We are going to recover in the following statement the existence of a Taylor type expansion for usual spaces, without requiring the blow up property. This makes our proof of (i) below different from the one in [START_REF] Gurariy | Geometry of Müntz spaces and related questions[END_REF] and more direct.

Theorem 2.4 (Clarkson-Erdös-Schwartz). Let Λ = {λ n } be an increasing sequence of positive numbers satisfying the gap condition (GC) and 1/λ n < +∞. Let X be a Banach space of functions on (0, 1) such that the following inequality holds (continuously):

X ⊂ L 1 (0, 1). Let f ∈ X. (i) f ∈ M X Λ =⇒ ∃(a n ) n ∈ C N , f (t) = n a n t λn
for a.e. t ∈ [0, 1).

(ii) If moreover, X satisfies (buc), X ∩ C([0, 1]) is dense in X and M ∞ Λ ⊂ X (continuously) then ∃(a n ) n ∈ C N , f (t) = n a n t λn for a.e. t ∈ [0, 1) =⇒ f ∈ M X Λ .
Proof. Thanks to Lemma 2.3, we know that the functionals e * λ are bounded on M X Λ for every λ ∈ Λ and that e * λ ≤ C ε (1 + ε) λ for every ε ∈ (0, 1). Let us prove (i). Assume that f ∈ M X Λ . By hypothesis, there exists k > 0 (depending on X only) such that f

1 ≤ k f X . Fix x ∈ [0, 1) and choose ε such that r = (1 + ε)x ∈ [0, 1), we have |e * λn (f )x λn | ≤ C ε (1 + ε) λn x λn f 1 ≤ C ε (1 + ε) λn x λn f X = C ε r λn f X where C ε = kC ε ,
and we are then allowed to consider

f (x) = n e * λn (f )x λn , where x ∈ [0, 1)
There exists a sequence (P j ) in span{e λn } converging to f in X by definition. We see that, for every x ∈ [0, 1), we have

P j (x) -f (x) = n e * λn (P j ) -e * λn (f ) x λn ≤ C ε n r λn P j -f X .
But the gap condition (GC) implies that n r λn < ∞, since λ n ≥ λ 0 + δn, where δ = inf{λ n+1 -λ n }. We obtain that lim P j (x) exists and is equal to f (x). On the other hand, (P j ) converges to f in L 1 so Fatou's lemma implies that lim P j = f a.e. We conclude that f = f a.e.

Let us prove (ii). By hypothesis, there exists k 0 such that k 0 h ∞ ≥ h X for every h ∈ M ∞ Λ . Fix ε > 0. We first claim that there exists some ρ such that T ρ (f ) -f X ≤ ε. Indeed we know that C = sup ρ>1/2 T ρ < ∞ and that there exists some g ∈ C([0, 1]) with f -g X ≤ ε/2(C + 1). Therefore, for every ρ > 1/2, ]) and there exists some J such that

T ρ (f ) -f X ≤ T ρ (f -g) X + T ρ (g) -g X + f -g X ≤ ε 2 + T ρ (g) -g X and T ρ (g) -g X ≤ k 0 T ρ (g) -g ∞ ≤ ε/2 if ρ
(f ) = f (ρ •) belongs to C([0, 1
T ρ (f ) - n≤J a n ρ λn e λn ∞ ≤ ε k 0 •
We conclude that f -n≤J a n ρ λn e λn X ≤ 2ε.

Around the Gurariy-Macaev theorem

An important consequence of the Clarkson-Erdös-Schwartz theorem is that a Müntz space associated to an increasing sequence Λ is a space of analytic functions on (0, 1). When we focus on the particular case of a set of integers Λ, it can also be seen as a space of analytic functions on the open unit disk D. Having in mind classical Hardy spaces H p and Bergman spaces A p , it is natural to wonder whether they are comparable to Müntz spaces, and (more precisely) compare the L p norm, the norm on H p and the norm on A p of lacunary polynomials. Consider the L p norm and take a lacunary sequence (of integers, to guarantee the analycity), the Gurariy-Macaev theorem (see below) shows that the norm of a linear combination of the e λ behaves like a weighted p norm of the coefficient, whereas in H p the norm of a linear combination of the z λ behaves like the 2 norm of the coefficients for the Hardy space H p . Concerning the Bergman norm of such a linear combination, we shall see below that it behaves like its L p norm.

In order to make more precise these remarks in the sequel, we need and recall first the following theorem due to Gurariy-Macaev [START_REF] Gurariy | Lacunary power sequences in the spaces C and L p[END_REF]. The full statement involves the space c of convergent sequences (equipped with the usual sup norm • ∞ ). This space is isomorphic to the space c 0 of null sequences. • Λ is lacunary. In other words, when Λ is lacunary, there exist C 1 , C 2 > 0 such that

C 1 k≥0 |a k | p 1 p ≤ k≥0 a k (pλ k + 1) 1 p e λ k L p ≤ C 2 k≥0 |a k | p 1 p and C 1 sup m≥0 m k=0 a k ≤ k≥0 a k e λ k ∞ ≤ sup m≥0 m k=0 a k 3.1.
Müntz spaces and Bergman spaces. The Gurariy-Macaev theorem states that, when Λ is lacunary, M p Λ is isomorphic to p and looks like the (whole) Bergman space A p from the geometry of Banach space viewpoint, but not only: the norms on L p (0, 1) and on A p are equivalent for lacunary Taylor series, but not for every Taylor series. Proposition 3.2. Let p ≥ 1.

(i) There exists f ∈ A p such that f |(0,1) / ∈ L p (0, 1). (ii) Let Λ ⊂ N be a lacunary set. There exist C, C > 0 such that for every polynomial f spanned by the functions z → z λ with λ ∈ Λ, we have

C f A p ≤ f L p (0,1) ≤ C f A p .
Proof. (i) follows quickly from the fact that the contrary would mean that the Carleson embedding A p → L p (ν) would be bounded, where the measure ν is defined by ν(B) = m(B ∩ (0, 1)) (where B is a Borel subset of D). Hence this measure ν would be 2-Carleson, but clearly, considering the Carleson window

S(1, h) = {z ∈ D| |1 -z| ≤ h}, we have ν(S(1, h)) = h, which is not O(h 2 ).
(ii) follows from Gurariy-Macaev theorem. Indeed for every θ ∈ (0, 1), we have for any polynomial

f = k a k z λ k C 1 k≥0 |a k | p pλ k + 2 1 p ≤ k≥0 a k e 2iπλ k θ e λ k + 1 p L p (0,1) ≤ C 2 k≥0 |a k | p pλ k + 2 1 p
where C 1 and C 2 are the constants given by Th.3.1.

Now, recall that

f p A p = 2 1 0 1 0 k≥0
a k r λ k e 2iπλ k θ p r dr dθ and we get

k a k z λ k A p ≈ k |a k | p pλ k + 2 1 p ≈ k |a k | p pλ k + 1 1 p ≈ f L p (0,1)
applying again Th.3.1.

It is not known whether M p

Λ is also isomorphic to p for any Λ. It is the case if Λ is a finite union of lacunary sequence (i.e. quasi-lacunary, see [START_REF] Gurariy | Geometry of Müntz spaces and related questions[END_REF]), but in that case, the isomorphism is not the "natural" one underlying in Gurariy-Macaev theorem as soon as Λ is not lacunary.

f L p (0,1) f H p .
It is immediate that the norms are not equivalent: the L p norm of z n satisfies z n p ≈ 1/n 1 p whereas z n H p = 1. 3.3. Back to Gurariy-Macaev theorem. Let us mention that an almost isometric version of Gurariy-Macaev theorem is proved in [START_REF] Gaillard | Lacunary Müntz spaces: isomorphisms and Carleson embeddings[END_REF] (see [START_REF] Gaillard | Asymptotic isometry for lacunary Müntz spaces and applications[END_REF] for cases p ∈ {1, +∞} too, and [START_REF] Fricain | Müntz spaces as model spaces[END_REF] for an alternative approach in the hilbertian framework for complex exponents), which is relative to the arithmetical condition of super-lacunarity: an increasing sequence of positive numbers (p n ) n is super-lacunary if lim n→+∞ p n+1 p n = +∞.

Before giving the statement of the theorem, we recall Definition 3.3. Let x n n≥1 be a normalized sequence in a Banach space X.

• The sequence x n n≥1 is almost isometric to the canonical basis of p in X if

1 -ε n k≥n |a k | p 1 p ≤ k≥n a k x k X ≤ 1 + ε n k≥n |a k | p 1 p where ε n → 0 • The sequence x n n≥1 is almost isometric to the summing basis of c in X if 1 -ε n sup m≥n m k=n a k ≤ k≥n a k x k X ≤ 1 + ε n sup m≥n m k=n a k where ε n → 0
When X is a Hilbert space (and p = 2), we usually rather say that the sequence x n n≥1 is asymptotically orthonormal. • Λ is super-lacunary.

We mention

See [START_REF] Gaillard | Lacunary Müntz spaces: isomorphisms and Carleson embeddings[END_REF] and [START_REF] Gaillard | Asymptotic isometry for lacunary Müntz spaces and applications[END_REF] for a proof, but we incorporate below a very short and elementary proof in the case p = 2 (less technical than the original one) and complete it with the equivalence to a (strong) variant of almost isometry. This is the counterpart in Müntz spaces of the same argument given in [START_REF] Fricain | Müntz spaces as model spaces[END_REF] for a particular case of a result of Volberg on model spaces.

Proposition 3.5. TFAE

(1) There exists n 0 ≥ 1 and a sequence (ε n ) n≥n 0 in (0, 1) with ε n → 0 such that

n≥n 0 (1 -ε n |a n | 2 1 2 ≤ n≥n 0 a n (2λ n + 1) 1 2 e λn L 2 ≤ n≥n 0 (1 + ε n |a n | 2 1 2
(2) There exists a sequence ε n → 0 such that

1 -ε n k≥n |a k | 2 1 2 ≤ k≥n a k (2λ k + 1) 1 2 e λ k L 2 ≤ 1 + ε n k≥n |a k | 2 1 2
(3) Λ is super-lacunary.

Proof. (1) ⇒ (2) is clear.

(2) ⇒ (3) it is not new but we give the following easy argument for completeness: for every t > 0, we have

(1 + ε n )(1 + t 2 ) 1 2 ≥ 2λ n + 1)
1 2 e λn + t 2λ n+1 + 1)

1 2 e λ n+1 ≥ 2λ n + 1)
1 2 e λn + t 2λ n+1 + 1)

1 2 e λ n+1 , 2λ n + 1)

1 2 e λn ≥ 1 + 2t λ n+1 λn 1 + λ n+1 λn which implies that λ n+1 λ n → +∞.
(3) ⇒ (1) Here is the novelty. Let us fix a super-lacunary sequence (λ n ), which means that λ n+1 λ n → +∞. Equivalently q n = 2λ n + 1 1 2 is super-lacunary. Now we introduce ρ n = inf j≥n q j+1 q j and δ n = ρ

-1 3 n-1 > 0.
Choose n 0 such that δ n 0 < 1/2. Now take any finitely supported sequence of scalars (a k ) and develop k≥n 0

a k 2λ k + 1) 1 2 e λ k 2 = k,l≥n 0 a k a l q k q l e λ k , e λ l = k,l≥n 0 2a k a l q k q l q 2 k + q 2 l = a 2 2 + k,l≥n 0 k =l 2a k a l q k q l q 2 k + q 2 l ( * )
We focus on the sum in the right hand side of ( * ). Write a n = a n δ n , so that

k,l≥n 0 k =l 2a k a l q k q l q 2 k + q 2 l = k,l≥n 0 k =l 2a k a l q k q l q 2 k + q 2 l δ -1 k δ -1 l • Using the inequality 2a k a l ≤ |a k | 2 + |a l | 2 , we get k,l≥n 0 k =l 2a k a l q k q l q 2 k + q 2 l ≤ 2 a 2 2 sup l≥n 0 k≥n 0 k =l q k q l q 2 k + q 2 l δ -1 k δ -1 l .
For every l ≥ n 0 , we have

k≥n 0 k =l q k q l q 2 k + q 2 l δ -1 k δ -1 l ≤ 2 k>l q l q k δ -1 k δ -1 l , but ∀k > l , q k ≥ ρ k-1 . . . ρ l q l and q l q k δ -1 k δ -1 l ≤ ρ 1 3 k-1 ρ 1 3 l-1 ρ k-1 . . . ρ l ≤ 1 ρ 1 3 (k-l) l • Hence k,l≥n 0 k =l 2a k a l q k q l q 2 k + q 2 l ≤ 4 ρ 1 3 n 0 -1 a 2 2 • ( * * ) Writing ε n = 4 ρ 1 3 n 0 -1 δ 2
n , we easily check that ε n ∈ (0, 1) and we get the result.

Let us point out that if one want to prove directly (3) ⇒ (2), the preceding argument is easily simplified (avoiding the trick with a k and δ k ). Actually in that case, we can get directly an upper bound estimate in ( * * ) with 4 a 2 2 /(ρ n 0 -1).

Bloch type spaces and inequalities

In this section, we introduce the Müntz-Bloch space B α Λ , and study the link with classical Müntz spaces. Definition 4.1. Let α > 0. We introduce the space B α as the space

B α = f ∈ C 1 ([0, 1)) | sup [0,1) (1 -t) α |f (t)| < ∞ equipped with the norm f B α = |f (0)| + sup [0,1) (1 -t) α |f (t)|
It is easy to check that B α is a Banach space. When α = 1, we shall simply write B instead of B 1 .

Let us focus on the behavior of monomials in this space. For every λ ≥ 1, the function e λ belongs to B α with

e λ B α = λα α (λ -1) λ-1 (λ + α -1) λ+α-1
and in particular e λ B = 1 -

1 λ λ-1
. We also mention that [START_REF] Gaillard | Lacunary Müntz spaces: isomorphisms and Carleson embeddings[END_REF] lim

λ→+∞ e λ 1 λ B α = 1 .
From our version of the Clarkson-Erdös-Schwartz theorem (Th.2.4), we get a nice Taylor type expansion for functions belonging to

B α Λ = M Λ B α .
Theorem 4.2. Let α > 0 and Λ = {λ n } satisfying the gap condition (GC) and 1/λ n < +∞.

For every f ∈ B α Λ , we have

∃(a n ) n ∈ C N , f (t) = n a n t λn for every t ∈ [0, 1).
Proof. When α ∈ (0, 2), it suffices to apply Th.2.4. We only have to justify that there exists some C > 0 such that

f 1 ≤ C f B α . Indeed, fix f ∈ B α .
For every x ∈ [0, 1), we can write:

f (x) = f (0) + x 0 f (s) ds so that f 1 ≤ |f (0)| + 1 0 x 0 |f (s)| ds dx = |f (0)| + 1 0 (1 -s)|f (s)| ds via Fubini. Now we have 1 0 (1 -s)|f (s)| ds ≤ sup (0,1) (1 -t) α |f (t)| 1 0 (1 -s) 1-α ds = 1 2 -α sup (0,1) (1 -t) α |f (t)|
which gives the conclusion when α < 2.

Of course, t ∈ [0, 1) → 1 1 -t belongs to B 2 , but not to L 1 (0, 1). To prove the theorem when α ≥ 2, we use another argument (which would work for any value of α). Fix ϕ in the unit ball of the dual of B α . Fix a polynomial P = n a n e λn . The map

ρ ∈ [0, 1] -→ n a n ρ λn ϕ e λn ,
is continuous. But we already proved the Clarkson-Erdös-Schwartz theorem for M ∞ Λ and we get for every ε > 0 the existence of C ε (depending on Λ and ε only) such that

a n ϕ e λn ≤ C ε (1 + ε) λn sup ρ∈[0,1] n a n ρ λn ϕ e λn
P (ρ.) B α . But P (ρ.) B α = |P (0)| + sup (0,1) (1 -t) α ρ|P (ρt)| ≤ |P (0)| + sup (0,1) (1 -ρt) α |P (ρt)| ≤ P B α .
We get a n e λn B α ≤ C ε (1 + ε) λn P B α and thanks to [START_REF] Gaillard | Lacunary Müntz spaces: isomorphisms and Carleson embeddings[END_REF], we even have some C ε (depending on Λ and ε only) such that

a n ≤ C ε (1 + ε) λn P B α
Now, we only have to end the proof like for Th.2.4.

Let us point out that λ α-1 e λ B α ∼ α α e -α when λ → +∞. In particular, we have the asymptotic behavior (pλ + 1) (1 -t)

1+ 1 p f p ?
Another natural question is closely related in the infinite case:

Problem 2. Let Λ satisfying the Müntz condition and (GC). Does the derivatives of the functions of the Müntz space M ∞ Λ belong to the weak-L 1 space L 1 w (0, 1) ?

We give an answer to Problem 1: Theorem 4.3 (see also the remark after its statement) shows that there exists Λ such that, for any value of p, the answer to the question in problem 1 is negative. We give also an answer to Problem 2 in Theorem 4.5. Actually we show that for both questions, there exists an arithmetical constraint on Λ. A negative answer to Problem 2 implies a negative to Problem 1 for p = +∞. Nevertheless, we shall give separate proofs, through it relies on the same kind of ideas, the negative answer to Problem 1 being a bit faster to prove (and stated for every p ∈ [1, +∞]).

We wish to mention that actually a negative answer to problem 1 in the case p = +∞ can also be found in the monograph [START_REF] Borwein | Polynomials and Polynomial Inequalities[END_REF]: see E.9 p.318. There, another arithmetical restriction is given in this case: λ n ≥ cρ n for some c > 0 and some ρ > 1. Our argument will be independent, more elementary (according to us), fits to any value of p, but leads to a different kind of arithmetical condition.

Let us add that we have a positive answer to problem 1 when Λ is lacunary: it is easy to see (and more or less part of the folklore). It follows from the fact that in this case: f (t) = n a n t λn where t ∈ (0, 1) and |a n | ≤ Cλ 1 p n f p for some constant C depending on the lacunarity of Λ only (it is a direct consequence of Th.3.1 for instance). Then it suffices to see that for every t ∈ [0, 1) and α > 0, we have

λ α n t λn-1 1 (1 -t) α •
This can be viewed as the counterpart in our framework of the following well known fact: every function of H ∞ belongs to the (classical) Bloch space of functions f , analytic on D, such that sup

z∈D (1 -|z|)|f (z)|.
On the other hand, we must mention too that it was stated recently in [START_REF] Ludkowski | Approximation and existence of Schauder bases in Müntz spaces of L 1 functions[END_REF], [START_REF] Ludkowski | Approximation in Müntz spaces MΛ,p of L p functions for 1 < p < ∞ and bases[END_REF], and [START_REF] Ludkowski | Approximation and Schauder bases in Müntz spaces MΛ,C of continuous functions[END_REF] that problem 1 and 2 always have a positive answer. Actually the mistake in these papers is the following: a Taylor series with radius 1 whose restriction to [0, 1) has a limit at point 1 needs not to be bounded on the disk centered at 1/2, with radius 1/2. These inequalities were the conerstone of the proof of the existence of a Schauder basis for every Müntz spaces. Therefore the counterexamples below show that the problem of existence of basis for Müntz spaces is still completely open.

We give now our counterexamples. (1 -t)

1+ 1 p |f (t)| > C f p . • M p Λ ⊂ B 1+ 1 p
This theorem shows that there exists a set Λ of integers such that, for every p ∈ [1, +∞], the answer to problem 1 is negative: choose a sequence of integers such that a n+1 > na n then the Müntz and the gap conditions are fulfilled and the set Λ = n≥1 a n , . . . , na n makes the job.

Proof. We begin with the following simple fact Fact: given A > 0, there exists a polynomial Q such that |Q(t)| ≤ t and |Q (1/e)| ≥ A. One can argue for example by standard abstract functional analysis: the derivation at point 1/e cannot be bounded on (R[X], . ∞ ). This provides some polynomial P such that P ∞ ≤ 1 and |P (1/e)| ≥ A. Actually such a polynomial would suffice to work with when p = +∞ but we shall need the condition |Q(t)| ≤ t on [0, 1] in the case p ∈ [1, +∞). It suffices to take a polynomial P associated to e(A + 1) (instead of A), then take Q(X) = X.P (X). The fact is proved. Now, assume that p ∈ [1, +∞] is fixed and there exists C > 0 such that ∀t ∈ (0, 1) ,

|f (t)| ≤ C (1 -t) 1+ 1 p f p .
Take A = 2Ce and apply the fact. Denote by d the degree of our polynomial Q, consider the function f (t) = Q t an for t ∈ [0, 1]. This function f belongs to M Λ when n ≥ d. We test the inequality at point t = 1 -1 a n for n large enough and we get

a n 1 - 1 a n an-1 Q 1 - 1 a n an ≤ Ca 1+ 1 p n f p
But, for p = +∞, we already know that f ∞ ≤ 1 and, when p is finite, we have

f p p ≤ 1 0 t pan dt = 1 pa n + 1 ≤ 1 a n so in every case, 1 - 1 a n an-1 Q 1 - 1 a n an ≤ C
Letting n going to infinity, we get e -1 |Q (1/e)| ≤ C and a contradiction. Now we focus on Problem 2. When Λ is lacunary, we already mentioned that every f ∈ M ∞ Λ belongs to B 1 , so that its derivative f belongs to L 1 w (0, 1). For our counterexample, we first need the following lemma. Lemma 4.4. Let ε ∈ (0, 1) and A > 0. There exists a polynomial P such that P (0) = 0, P ∞ = 1 and the set E = x ∈ (0, 1)| |P (x)| < A has Lebesgue measure less than ε.

Proof. First consider a function h ∈ C 1 ([0, 1]) such that 0 ≤ h ≤ 1 with h(0) = 0, h ∞ = 1 and |h | = 2A + 1 everywhere except on a finite number (say N ≈ A) of intervals (centered at points k/(2A + 1)) whose length is ε/N . Denote N ε the union of these N intervals. N ε has Lebesgue measure less than ε. Now there exists some polynomial Q 1 such that Q 1 -h ∞ ≤ 1. Take the polynomial Q such that Q = Q 1 and Q(0) = 0. We have Q -h ∞ ≤ 1 so that Q ∞ ≤ 2. Moreover, for every x / ∈ N ε , |Q (x)| ≥ |h (x)| -1 ≥ 2A. At last P = Q/ Q ∞ is the desired polynomial. Theorem 4.5. Assume that there exists an increasing sequence (a n ) n with a n → +∞, such that n≥1 a n , . . . , na n ⊂ Λ.

Then there exists f ∈ M ∞ Λ such that f / ∈ L 1 w (0, 1).

Using the same set than in the remark after Theorem 4.3, we have a negative answer to Problem 2.

Proof. The proof follows the spirit of the one of Th.4.3.

Claim: for any C > 0, there exists some f ∈ M ∞ Λ with uniform norm equal to 1 such that f L 1 w (0,1) ≥ C. We fix C > 0 and apply Lemma 4.4 with A = 4eC and ε = 1/2e. We are given a norm 1 polynomial P with some degree d 0 ≥ 1 such that E = x ∈ (0, 1)| |P (x)| < A has Lebesgue measure less than ε.

We consider now d ≥ d 0 and f (t) = P t a d for t ∈ [0, 1]. It belongs to M ∞ Λ , satisfies f (0) = 0 and has norm 1. We may assume that a d ≥ 1.

We We can conclude the proof by a standard argument via the closed graph theorem. Indeed, if the conclusion of our theorem were not true, the map

M ∞ Λ J -→ L 1
w (0, 1) f -→ f would be defined. Let us show that J has a closed graph in that situation. Let (f n ) be a sequence in M ∞ Λ such that f n → f in M ∞ Λ and f n → g in L 1 w (0, 1). Then we know that f n → f uniformly on compact subsets of [0, 1) hence for every ε > 0 and every y ∈ [0, 1)

m t ∈ [0, y] |f (t) -g(t)| ≥ ε ≤ m t ∈ [0, y] |f (t) -f n (t)| ≥ ε/2 + m t ∈ [0, y] |f n (t) -g(t)| ≥ ε/2 ≤ 2 ε f n -f L 1 (0,y) + 2 ε f n -g L 1 w
Letting n going to infinity, we get that the measure of t ∈ [0, y] |f (t) -g(t)| ≥ ε is equal to 0 for every ε > 0. Therefore f = g a.e. on [0, y] hence on [0, 1) since y was arbitrary. The graph of J is closed as announced.

The closed graph theorem (L 1 w (0, 1) is a quasi-Banach space) implies that the map J is continuous, which contradicts our claim.

  s) ds and (3) follows now from (1) and the definition of F . Applying (3) with z = m + 1 where m ∈ C -1 , we get (4) for every m ∈ C -1 , 1 0

Theorem 3 . 1 ( 1 p

 311 Gurariy-Macaev). Let p ≥ 1. TFAE • The normalized sequence (pλ n +1) e λn n is equivalent to the canonical basis of p in L p (0, 1). • The sequence e λn n is equivalent to the summing basis of c in C([0, 1]).

3. 2 .

 2 Müntz spaces and Hardy spaces. Since ν(B) = m(B ∩ (0, 1)) (where B is a Borel subset of D) defines a Carleson measure on D, we have the following inequality for every Taylor polynomial f : (6)

Theorem 3 . 4 .

 34 [START_REF] Gaillard | Lacunary Müntz spaces: isomorphisms and Carleson embeddings[END_REF][START_REF] Gaillard | Asymptotic isometry for lacunary Müntz spaces and applications[END_REF] Let p ≥ 1. TFAE• The normalized sequence (pλ n + 1)1 p e λn n is almost isometric to the canonical basis of p in L p (0, 1)• The sequence e λn n is almost isometric to the summing basis of c in C([0, 1]).

and since n a n ρ λn ϕ e λn ≤ n a n ρ

  λn e λn B α , we obtain by Hahn-Banacha n e λn B α ≤ C ε (1 + ε) λn sup ρ∈(0,1)

Theorem 4 . 3 .

 43 Let p ∈ [1, +∞] and Λ satisfying the Müntz condition such that there exists an increasing sequence (a n ) n with a n → +∞, such that n≥1 a n , . . . , na n ⊂ Λ.Then• for every C > 0, there exists f ∈ M p Λ satisfying sup t∈(0,1)

  is close enough to 1 by uniform continuity of g. The claim is proved.Since the Taylor series n a n X λn has convergence radius at least 1, T ρ

  ≥ 1, we cannot have the (continuous) embedding M p Λ ⊂ B α when α < 1 + 1/p. The preceding phenomenom occurs too in the case p = +∞ and M ∞ Λ ⊂ B 1 is the best we could expect. It is then natural to ask the following questions: Problem 1. Let p ∈ [1, +∞] and Λ satisfying both Müntz condition and (GC).

		1/p e λ B 1+ 1 p ∼	1+ 1 p p 1+ 1 (p + 1) p e	• Therefore, for any
	p Do we have M p Λ ⊂ B In other words, does there exist C > 0, such that, for every f ∈ M p 1+ 1 p (continuously) ? Λ ,
	∀t ∈ (0, 1) ,	|f (t)| ≤	C

  ∈ (0, 1) ; |f (t)| ≥ 4a d C ≥ m t ∈ exp -1/a d , 1 ; P t a d ≥ 4eC = 1 -exp -1 a d -m t ∈ exp -1/a d , 1 ; t a d ∈ EBut, using the change of variable formula,m t ∈ exp -1/a d , 1 ; t a d ∈ E We get (when d → +∞) 4a d C m t ∈ (0, 1) ; |f (t)| ≥ 4a d C ≥ 4a d C 1 -exp -1/a d -

	=	1 a d	1 1/e	1 I E (u)u	1 a d	-1 du ≤	e a d	m(E) ≤	1 2a d	•
								1 2a d	∼ 2C
	so that, for d large enough,
	f L 1

have t ∈ (0, 1) ; |f (t)| ≥ 4a d C = t ∈ (0, 1) ; t a d -1 P t a d ≥ 4C hence m t w (0,1) ≥ 4a d C m t ∈ (0, 1) ; |f (t)| ≥ 4a d C ≥ C

which was our claim.

Acknowledgment. We wish to thank the referee for her/his suggestions in order to improve the quality of this paper.