Arjen K Lenstra

Thorsten Kleinjung

Emmanuel Thomé

Universal security from bits and mips to pools, lakes -and beyond

The relation between cryptographic key lengths and security depends on the cryptosystem used. This leads to confusion and to insecure parameter choices. In this note a universal security measure is proposed that puts all cryptographic primitives on the same footing, thereby making it easier to get comparable security across the board.

Current security levels

The security of a cryptographic primitive is measured by the effort to break it. If that effort is 2 k , measured in some agreed upon unit, then it offers k-bit security and it is said to have security level k. Although this looks easy, it turns out to be impractical. For symmetric cryptosystems it is indeed easy: with a k-bit key they are supposed to offer k-bit security, measured in the most basic application of the cryptosystem, else they are no good. Cryptographic hash functions are harder: if the hash length is h they should offer and cannot offer more than h/2-bit security, measured in basic applications of the hash function. Given comparable speeds of the systems in similar environments -both in software or both in hardwaretheir security levels can easily be compared in a meaningful manner.

With public key primitives it gets confusing. For RSA the runtimes to factor an RSA modulus can be measured using experiments on small numbers. The results are then extrapolated to derive estimates for larger moduli: if factoring an n-bit modulus takes time T , then the time to factor an m-bit modulus (for m not much bigger than n) is estimated as N (m) N (n) T , where N (k) = exp(1.923(ln 2 k ) 1/3 (ln ln 2 k ) 2/3 ). In 1988 the first factorization of a 100-digit modulus required "100 MIPS years": equivalent to a century of computing time on a computer that performs a million "instructions" per second. In 1999, using a better factoring method, a 512-bit challenge modulus took about 40 years of computing time on a then current core (running at several hundred MHz, in four months in parallel on many cores), which was, somewhat questionably, estimated as 8400 MIPS years. The 2009 factorization of a 768-bit modulus took roughly a year on 2000 cores running at 2GHz. With N (1024) N (768) ≈ 1200, a 1024-bit modulus can be factored within two million 2GHz core-years. Further extrapolation is trickier, but it is reasonable to estimate that 2048-bit RSA moduli are a billion times harder to factor than 1024-bit ones. Using generic figures for the speed of software implementations, it follows that 768-, 1024-, and 2048-bit RSA have approximate security levels 66, 76, and 106, respectively.

For discrete logarithm public key cryptosystems additional parameters further confuse the picture. Let G be a well-chosen prime order q group for which the group operation is 2 c times slower than a basic application of a symmetric cryptosystem (or cryptographic hash function). It may be assumed that 0 ≤ c ≤ 10. Discrete logarithms in G offer c + log 2 √ q-bit security, assuming a large enough finite field F if G ⊂ F * , the size of which is estimated in the same way as the security of RSA, with a small and uncertain twist τ > 0 and with the potential of all kinds of trouble if an extension field is used. Thus, if q is a 160-bit prime and G an elliptic curve group, the security level is c+80. If, for a similar q-value, G ⊂ F * for a cardinality p finite field F , then the security offered by F * needs to be taken into account as well: the security level is min(c + 80, 76 + τ ) = 76 + min(c + 4, τ ) if p is a 1024-bit prime, and min(c + 80, 106 + τ ) = c + 80 for 2048-bit p.

Lattice-based public key cryptosystems still escape proper analysis. Despite enthusiastically riding post-quantum waves and fully homomorphic hot air balloons, they have not found wide-spread application and are not further discussed.

Summarizing the above, for symmetric cryptosystems the key length and the security level are the same, for cryptographic hash functions the security level is half the hash length, for RSA the security level is an obscure but small and quickly decreasing fraction of the modulus length, and for discrete logarithms it is half the size of the largest prime dividing the group order, unless one works in a multiplicative group of a finite field in which case extra care needs to be taken. This is easy enough for those in the know, but incomprehensible to most: it is not uncommon for people to proudly declare that they are using 128-bit RSA moduli for their 128-bit security cryptosystems -or indeed for crypto-practitioners to use 256-bit AES, SHA-512, and 512-bit RSA thinking they get an overall security level of min(256, 512 2 , 512 2 ) = 256.

Intuitive security levels

The problem is that for all these cryptosystems key length and security level are measured in bits, but that the relationship between the two varies wildlyfrom trivial, to simple, to rather contrived. If this is not well understood, it is tempting to use just the trivial and simple relationships and to forget about the complicated ones, with potentially disastrous consequences. This has occurred in practice, on multiple occasions and involving major corporations.

To address this problem the relationships between key length and security level must be put on the same footing for all cryptosystems. Because the complicated relationships cannot be simplified, all must be made equally complicated. Below a new definition of security level is proposed that does away with "k-bit security" for "security level k". It has the additional advantage that it allows a more intuitive interpretation of what security actually means.

The new approach was inspired by a remark made by the third author during his presentation of the factorization of the 768-bit RSA challenge at Crypto 2010: We estimate that the energy required for the factorization would have sufficed to bring two 20 • C Olympic size swimming pools to a boil. This amount of energy was estimated as half a million kWh. Thus, a cryptosystem is said to offer pool security if breaking it requires as much energy as it takes to boil a single Olympic size swimming pool (i.e., 2500 cubic meters of water). It follows that 65-bit symmetric cryptosystems, 130-bit cryptographic hash functions, and 745-bit RSA currently all offer pool security (because 768-bit RSA offers 66-bit security, 66 -1 = 65, 130 2 = 65, and N (768) N (745) ≈ 2). Larger quantities of water need to be considered to fully appreciate the security offered by practically relevant cryptosystems. On average, The Netherlands enjoys a healthy daily average of 82 million cubic meters of rain (i.e., 2 15 pools); with 65 + 15 = 80 and N (1130) N (768) ≈ 2 14 it is found that 80-bit symmetric cryptosystems, 160-bit cryptographic hash functions, and 1130-bit RSA offer rain security or dagelijkse neerslagverdampingsenergiebehoeftezekerheid. Equivalently, with each German citizen boiling a cubic meter of water, one may refer to rain security as German security or JedeRtausendliterbierverdampfungssicherheit1 .

Slightly more than 2 25 pools suffice to fill the lake of Geneva (89 cubic kilometers of water), so 90-bit symmetric cryptosystems, 180-bit cryptographic hashes, and 1440-bit RSA all offer lake security or sécurité lémanique. Boiling all water on the planet (including all starfish) amounts to about 2 24 lakes of Geneva and leads to global security: 114-bit symmetric cryptosystems, 228-bit cryptographic hashes, and 2380-bit RSA. This needs to be done 16 thousand times to break AES-128, SHA-256, or 3064-bit RSA.

With all water evaporated and no bodies of water left to fire the imagination, the above new security levels have run out of steam. Solar security can still be defined as offered by cryptosystems that can be broken in a year requiring that year's total solar energy: 140-bit symmetric cryptosystems, 280-bit cryptographic hash functions, and 3730-bit RSA. Reaching further in an intuitively appealing manner requires a different approach.

To infinity -and beyond

There is a big gap between the energy required to bring a gram of water to a boil (as usual starting at 20 • C), namely 93 • 10 -6 kWh, and the mass-energy of that same gram of water, namely 25 million kWh. With 25•10 6 93•10 -6 ≈ 2 38 , cryptanalysis suddenly looks a lot easier -maybe a bit too easy. Breaking AES-128, SHA-256, or 3064-bit RSA using the mass-energy of the lake of Geneva (where 90 + 38 = 128) still looks more or less reasonable, but that is mostly because it is inconceivable. What about a 0.02 gram scrap of paper having the mass-energy (namely 0.02 • 25 • 10 6 kWh = 500 000 kWh) needed to break 768-bit RSA? It implies paper-thin security for 1024-bit RSA: five A4 sheets of 80g/m2 paper together weigh 25 grams, 25/0.02 ≈ N (1024) N (768) , and therefore have enough massenergy to break 1024-bit RSA. This could have been cooked up by Certicom's PR department, but does not feel intuitively right.

Finally, with the estimated mass-energy of the observable universe 2 190 times what is required to boil two pools, 190 + 66 = 256-bit symmetric cryptosystems, 512-bit cryptographic hash functions, and 14954-bit RSA offer universal security.

Summary

Determining the newly proposed security levels requires a trained professional. This considerably raises the bar for uneducated guesswork and thereby significantly diminishes the risk of insecure parameter choices. Worldwide adoption of these intuitive security levels will have a beneficial effect on Internet security.

The table lists the new 2 security levels3 , with "sea" referring to the Mediterranean Sea. For metrically-challenged prism-enabled readers it is noted that using the same numbers of gallons or cubic miles as the table's liters or cubic kilometers, respectively, requires adding two to the figures in the "symmetric" column, adding four to the figures in the "hash" column, and replacing n in the "RSA" column by the m for which N (m) N (n) ≈ 4. Note, however, that this requires longer showers, a larger lake, etc., which is only appropriate.

Table 1 .

 1 Intuitive security levels.

	bit-lengths

The "2+"-bit difference between bringing to a boil and Verdampfung (which Alexander Kruppa kindly reminded us of) easily falls within the margin of Moore's eternal law which ensures that keys requiring evaporation a few years ago can now be boiled.

Unfortunately, but not surprisingly, cloud security is not well-defined.

Going further down the scale than in Table1would lead to meat security, the amount of computing humans can be expected to do, but we rather see computers as meat's next step on the evolutionary ladder (instruct.westvalley.edu/lafave/meat_in_ space.html).

Acknowledgements We gratefully acknowledge usage of Wikipedia and gladly blame its contributors for any errors in our data.


Baptiste Devyver

A spectral result for Hardy inequalities

Let P be a linear, elliptic second order symmetric operator, with an associated quadratic form q, and let W be a potential such that the Hardy inequality

holds with (non-negative) best constant λ0. We give sufficient conditions so that the spectrum of the operator 1 W P is [λ0, ∞). In particular, we apply this to several well-known Hardy inequalities: (improved) Hardy inequalities on a bounded convex domain of R n with potentials involving the distance to the boundary, and Hardy inequalities for minimal submanifolds of R n .

Introduction

Let P be a linear elliptic, second order, symmetric, non-negative operator on a domain Ω, and let q be the quadratic form associated to P . Following Carron [START_REF] Carron | Inégalités de Hardy sur les variétés riemaniennes non-compactes[END_REF] and Tertikas [START_REF] Tertikas | Critical Phenomena in Linear Elliptic Problems[END_REF], we will call Hardy inequality for P with weight W ≥ 0 and constant λ > 0, the following inequality:

λ Ω W u 2 ≤ q(u), ∀u ∈ C ∞ 0 (Ω). (1.1)
We denote by λ 0 = λ 0 (Ω, P, W ) the best constant λ for which inequality (1.1) is valid. By convention, if (1.1) does not hold for any λ > 0, we will let λ 0 = 0. The inequality (1.1) aims to quantify the positivity of P : for instance, inequality (1.1) with W ≡ 1 is equivalent to the positivity of the bottom of the spectrum of (the Friedrichs extension of) P . Let us give a celebrated example of Hardy inequality for P = -∆, which will be a guideline for us in this paper (see [START_REF] Marcus | On the best constant for Hardy's inequality in R n[END_REF] for the convex case, and [START_REF] Barbatis | A unified approach to improved L p Hardy inequalities with best constants[END_REF], [START_REF] Psaradakis | L 1 Hardy Inequalities with Weights[END_REF] for the mean convex case):

Example 1.1. If Ω is a C 2 , bounded, mean convex domain of R n , and δ is the distance to the boundary of Ω, then

1 4 Ω u 2 δ 2 ≤ Ω |∇u| 2 , ∀u ∈ C ∞ 0 (Ω). (1.2)
We recall that Ω is called mean convex if the mean curvature of its boundary is non-negative.

Let us return to the general case.

1.1. The best constant and the existence of minimizers. A natural question is, for a given weight W ≥ 0 such that the Hardy inequality (1.1) holds, to compute the best constant λ 0 and to discuss whether λ 0 is attained by a minimizer in the appropriate space or not. More precisely, define D 1,2 to be the completion of C ∞ 0 (Ω) with respect to the norm √ q. The variationnal problem associated to the Hardy inequality (1.1) is

λ 0 = inf u∈D 1,2 \{0}
q(u)

Ω W u 2 .

(1.3)

If (1.3) is not realized by a function in D 1,2 , we will say that the Hardy inequality (1.1) with best constant λ 0 does not have a minimizer. Another interesting quantity, related to the existence of minimizers, is the best constant at infinity. It is defined as follows (see [START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators in Schrödinger operators[END_REF], [START_REF] Marcus | On the best constant for Hardy's inequality in R n[END_REF] and [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]): Definition 1.2. The best constant at infinity λ ∞ = λ ∞ (Ω, P, W ) is the supremum of the set of α ≥ 0 such that

α Ω\Kα W u 2 ≤ q(u), ∀u ∈ C ∞ 0 (Ω \ K α ),
for some K α ⊂⊂ Ω compact subset of Ω.

Closely related notions have been introduced (under different names) in [START_REF] Tertikas | Critical Phenomena in Linear Elliptic Problems[END_REF] and [START_REF] Filippas | Optimizing improved Hardy inequalities[END_REF].

Example 1.3 (Example 1.1, continued). For the Hardy inequality (1.2) with W = 1 δ 2 , the best constant and the best constant at infinity are both equal to 1 4 . Furthermore, inequality (1.2) does not have a minimizer. 1.2. Improving Hardy inequalities. A very natural question is the following: for the Hardy inequality (1.1) with the best constant λ 0 , can we improve it by adding another non-negative potential V to the left-hand side, i.e. is there a positive constant µ and a non-negative, non-zero potential V such that

λ 0 Ω W u 2 + µ Ω V u 2 ≤ q(u), ∀u ∈ C ∞ 0 (Ω) (1.4)
Of course, such an improvement -if it exists -is not at all unique, and finding a potential V which is "as large as possible" is important.

Results by Filippas-Tertikas [START_REF] Filippas | Optimizing improved Hardy inequalities[END_REF], Agmon [START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators in Schrödinger operators[END_REF], Pinchover [START_REF] Pinchover | On principal eigenvalues for indefinite-weight elliptic problems[END_REF], [START_REF] Pinchover | Large scale properties of multiparameter oscillation problems[END_REF], Marcus-Mizel-Pinchover [START_REF] Marcus | On the best constant for Hardy's inequality in R n[END_REF], Pinchover-Tintarev [START_REF] Pinchover | Existence of minimizers for Schrödinger operators under domain perturbations with application to Hardy's inequality[END_REF] among others show that the best constant at infinity, as well as the existence of minimizers, play an important role in this problem: indeed, a general result obtained by Agmon [START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators in Schrödinger operators[END_REF] (see also Pinchover [START_REF] Pinchover | Large scale properties of multiparameter oscillation problems[END_REF], Lemma 4.6 for an easier and more general proof) shows that if the best constant at infinity is strictly greater than the best constant in the inequality (1.1), then no improvement by a non-negative, non-zero potential V is possible.

Also, concerning the minimizers, Pinchover-Tintarev show (Lemma 1.1 in [START_REF] Pinchover | Existence of minimizers for Schrödinger operators under domain perturbations with application to Hardy's inequality[END_REF]) that if W > 0 and if there is a minimizer for the Hardy inequality (1.1), then no improvement by a non-negative, non-zero potential V is possible. In fact, if a minimizer exists then it is a ground state (in the sense of Agmon) of P -W .

The possibility of adding a potential V 0 in the left-hand side of the Hardy inequality (1.1) with best constant λ 0 has to do with the criticality of the operator P -λ 0 W : there exists V 0 such that the improved inequality (3.5) is valid (with µ = 1) if and only if P -λ 0 W is subcritical in Ω. See for example [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] for details and references on this.

In the case that the best constant at infinity is strictly larger than the best constant, or that there is a minimizer, the potential W "does not grow fast enough at infinity" in a certain sense. Let us illustrate this by the following example, related to Example 1.1:

Example 1.4. Let P = -∆ on a smooth, bounded, mean convex domain Ω, and let us define, for α ∈ R, W α = δ -α , where we recall that δ is the distance to the boundary of Ω. Then,

λ ∞ (Ω, -∆, W α ) =    +∞, α ∈ (-∞, 2) 1 4 , α = 2 0, α ∈ (2, +∞)
More generally, if W is a small perturbation of P (see [START_REF] Pinchover | Criticality and ground states for second-order elliptic equations[END_REF]), then λ ∞ (Ω, P, W ) = ∞.

Definition 1. [START_REF] Anderson | The compactification of a minimal submanifold in Euclidean space by the Gauss map[END_REF]. A non-negative potential W , satisfying the Hardy inequality (1.1), is said to belong to the class of admissible potentials A(Ω, P ) if λ 0 (Ω, P, W ) = λ ∞ (Ω, P, W ), and if there is no minimizer of the associated variational problem (1.3). When the dependence with respect to Ω and P will be clear, we will write A instead of A(Ω, P ).

For instance, it follows from the results of Marcus-Mizel-Pinchover [START_REF] Marcus | On the best constant for Hardy's inequality in R n[END_REF] that if Ω is a C 2 , bounded, mean convex domain, then the potential δ -2 belongs to the class A(Ω, -∆). In this article, we will focus on Hardy inequalities (1.1) with admissible potentials. A Hardy inequality (1.1) with admissible potential can sometimes be improved, but not always. Actually, if W is admissible, deciding whether the Hardy inequality (1.1) with best constant λ 0 can be improved or not, is a delicate question. We define a subclass of A: Definition 1.6. A non-negative potential W , satisfying the Hardy inequality (1.1), is said to belong to the class of optimal potentials O(Ω, P ) if W ∈ A(Ω, P ) and if the Hardy inequality (1.1) with best constant λ 0 cannot be improved, i.e. if there is no V 0, µ > 0 such that inequality (3.5) holds. Equivalently, W ∈ O(Ω, P ) if and only if W ∈ A(Ω, P ) and P -λ 0 W is critical (see [START_REF] Pinchover | A ground state alternative for singular Schrödinger operators[END_REF]). Furthermore, P -λ 0 W is critical if and only if it does not have a ground state in the sense of Agmon (see [START_REF] Pinchover | A ground state alternative for singular Schrödinger operators[END_REF]); this of course includes an obvious case when such a ground state is a true minimizer).

Example 1.7. The potential 1 |x| 2 is an optimal potential for P = -∆ on R n (or equivalently, on R n \ {0}) for n ≥ 3 (see [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] for a short proof). Indeed, the operator -∆ -n-2

n -2 2 2 R n u 2 (x) |x| 2 ≤ R n |∇u| 2 , ∀u ∈ C ∞ 0 (R n ).
However, the Hardy inequality (1.2) of Example 1.1 can be improved. The first improvement of inequality (1.2) was obtained by Brezis and Vazquez [START_REF] Brezis | Blow-up solutions of some nonlinear elliptic problems[END_REF], for

V ≡ 1, however 1 / ∈ A(Ω, -∆ -1 4δ 2 ) -in fact, V ≡ 1 is a small perturbation of -∆ -1 4δ 2 , and thus λ ∞ (Ω, -∆ -1 4δ 2 , 1) = ∞.
Later, an improvement by a potential in the class A(Ω, -∆ -1 4δ 2 ) was obtained by Brezis and Marcus [START_REF] Brezis | Hardy's inequalities revisited[END_REF]. Let us introduce the normalized logarithm function, defined by

X 1 (t) := (1 -log t) -1 .
A consequence of the work of Brezis and Marcus is the following: Example 1.8. Let Ω be a smooth, bounded, mean convex domain, then we have the improved Hardy inequality

1 4 Ω u 2 δ 2 X 2 1 δ D ≤ Ω |∇u| 2 - 1 4 Ω u 2 δ 2 , ∀u ∈ C ∞ 0 (Ω), (1.5)
where D is any constant such that

D ≥ sup x∈Ω δ(x).
Furthermore, 1 4 is the best constant and the best constant at infinity, and there is no minimizer. In particular, δ -2 X 2 1 δ D belongs to the class A(Ω, -∆-

1 4δ 2 ).
More recently, Barbatis, Filippas and Tertikas [START_REF] Barbatis | Series expansion for L p Hardy inequalities Indiana[END_REF] have obtained a series of successive improvements of the Hardy inequality (1.2) with admissible potentials, generalizing the improved Hardy inequality (1.5) obtained by Brezis and Marcus. In order to present their result, let us define for i ≥ 1,

X i (t) := X 1 (X i-1 (t)),
and by convention X 0 ≡ 1. Let us also define

W i := 1 4δ 2 i k=0 X 2 0 δ D • • • X 2 k δ D ,
and

J i := W i -W i-1 = 1 4δ 2 X 2 0 δ D • • • X 2 i δ D .
The result of Barbatis, Filippas and Tertikas obtained in [START_REF] Barbatis | Series expansion for L p Hardy inequalities Indiana[END_REF] is presented in the following example:

Example 1.9. Let Ω be a C 2 , bounded, mean convex domain, then we can choose D > sup x∈Ω δ(x) big enough (see Remark 3.4) so that for every i ≥ 1, the following improved Hardy inequality holds

Ω J i u 2 ≤ Ω |∇u| 2 - Ω W i-1 u 2 , ∀u ∈ C ∞ 0 (Ω). (1.6)
Furthermore, 1 is the best constant and the best constant at infinity, and there is no minimizer. In particular, J i belongs to the class A(Ω, -∆ -W i-1 ).

1.3. The spectrum of 1 W P . In this subsection, we collect some results concerning the spectrum of operators of the type 1 W P . For more details on this, see [START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators in Schrödinger operators[END_REF], Section 3 in [START_REF] Marcus | On the best constant for Hardy's inequality in R n[END_REF], or Proposition 4.2 in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF].

If the potential W in the Hardy inequality (1.1) is positive, then there is a spectral interpretation of the best constant λ 0 (Ω, P, W ) and of the best constant at infinity λ ∞ (Ω, P, W ). Consider (the Friedrichs extension of) the operator 1 W P , which is self-adjoint on L 2 (Ω, W dx). Then λ 0 (Ω, P, W ), the best constant in (1.1) is the infimum of the spectrum of 1 W P , and λ ∞ (Ω, P, W ), the best constant at infinity is the infimum of the essential spectrum of 1 W P (the result concerning λ ∞ comes from the Persson's formula, see [START_REF] Persson | Bounds on the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF], [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators Mathematical Notes[END_REF]). As an immediate consequence, the following result holds: Lemma 1.10. Assume that λ ∞ (Ω, P, W ) = +∞. Then the spectrum of

1 W P is discrete. Example 1.11.
Let Ω be a smooth, bounded domain. Then for every α ∈ (2, ∞), the spectrum of δ α is discrete. Indeed, for every α

∈ (2, ∞), λ ∞ (Ω, -∆, δ -α ) = ∞.
Also, there is a minimizer of the variational problem (1.3) if and only if λ 0 (Ω, P, W ) is an eigenvalue of 1 W P . In [START_REF] Marcus | On the best constant for Hardy's inequality in R n[END_REF], p.3246, the authors attributed the following result to Agmon: Claim 1.12 (Agmon). On a smooth, bounded, mean convex domain, the spectrum of δ 2 (-∆) is [ 1 4 , ∞). Furthermore, without the mean convexity assumption on the domain, the essential spectrum of δ 2 (-∆) is [ 1 4 , ∞). To the author's knowledge, Agmon never published this result, but proved a closely related result in [START_REF] Agmon | A representation theorem for solutions of Schrödinger type equations on noncompact Riemannian manifolds Méthodes semi-classiques[END_REF]. The validity of Agmon's claim 1.12 implies at once that if Ω is a smooth, bounded, mean convex domain, the best constant and the best constant at infinity for the Hardy inequality (1.1) are both equal to 1 4 . Agmon's claim has to be compared to Example 1.11. 1.4. Optimal Hardy inequalities and the supersolution construction. In the article [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF], starting from a general subcritical operator P on a punctured domain Ω ⋆ = Ω \ {0}, we have constructed an optimal potential W satisfying the Hardy inequality (1.1) with best constant 1. This is actually a generalization of Example 1.7, since for the case

P = -∆ with Ω = R n , n ≥ 3, the constructed potential W is equal to n-2 2 2 1 |x| 2 .
The most remarkable property of these constructed potentials W is the criticality of P -λ 0 W : as we have mentionned above, if W ∈ A(Ω, P ), the criticality of P -λ 0 W is a delicate property.

These optimal potentials are obtained through a construction that we have called in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] the supersolution construction, and that we recall now: Proposition 1.13 (Supersolution construction, see [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF], Lemma 5.1 and Corollary 5.2). Assume that P is subcritical in Ω, and that there exist u 0 and u 1 two linearly independent, positive supersolutions of P . Then the non-negative potential

W (u 0 , u 1 ) := 1 4 ∇ log u 0 u 1 2
satisfies the Hardy inequality (1.1) with λ = 1.

In fact, the above norm | • | is a norm associated with P , as we shall explain later. Moreover, the results of [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] show that if u 0 and u 1 are positive solutions of P u = 0, in Ω \ {0}, and if moreover

lim x→∞ u 0 (x) u 1 (x) = 0, lim x→0 u 0 (x) u 1 (x) = ∞,
then W (u 0 , u 1 ) is an optimal potential (having a singularity at 0). Actually, u 0 as above is necessarily the minimal, positive Green function of P with pole 0. These optimal potentials obtained via the supersolution construction have another interesting property: if W (u 0 , u 1 ) is positive, then the spectrum of the operator

1 W (u 0 ,u 1 ) P is [1, ∞).
The proof of this last fact relies on the existence of generalized eigenfunctions of exponential type for 1 W P -λ, for every λ ≥ 1.

However, the optimal potentials W (u 0 , u 1 ) have two drawbacks: first, the existence of two linearly independent positive solutions u 0 , u 1 of P u i = 0 does not always hold: one generally needs to remove one point in Ω in order to guarantee this condition, and the potential obtained will have a singularity at that point. Secondly, finding the asymptotic of W (u 0 , u 1 ) at infinity, or even a lower bound of W (u 0 , u 1 ) in order to get a more explicit Hardy inequality, is a difficult problem. Actually, it is an open problem to give sufficient conditions guaranteeing that W (u 0 , u 1 ) is positive in a neighborhood of infinity. This is an interesting question, since the interpretation of the best constant and the best constant at infinity in term of the spectrum of the operator 1 W (u 0 ,u 1 ) P holds only if W (u 0 , u 1 ) is positive.

It is sometimes more natural to work with non-optimal (but good enough) potentials given by an explicit formula. In the supersolution construction, instead of two positive solutions u i of P u = 0, one can take two (well chosen) positive supersolutions. For example, the potential 1 4δ 2 of the Hardy inequality of Example 1.1 is obtained by applying the supersolution construction with u 0 = 1 and u 1 = δ (which is a supersolution, but not a solution of P = -∆). Furthermore, the potentials of the Examples 1.1, 1.8 and 1.9 are not optimal, because the corresponding Hardy inequality can be improved, yet they are admissible potentials. We will also see (in Proposition 3.2) that each improvement of the Hardy inequality of Example 1.9 can also be obtained using the supersolution construction, with explicit supersolutions u 0 and u 1 which are functions of δ. It is thus tempting to ask whether the property that the spectrum of 1 W (u 0 ,u 1 ) P is [1, ∞), valid in the case of the optimal potentials obtained in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF], remains true for only admissible potentials (that is, potentials in the class A(Ω, P ). In particular, it is interesting to ask the following question: Question 1.14. For the Hardy inequalities of Examples 1.1 and 1.9 obtained respectively by Brezis-Marcus and Barbatis-Filippas-Tertikas, does it hold that the spectrum of

1 W (u 0 ,u 1 ) P is [1, ∞)?
1.5. Our results and organization of the article. In this article, we generalize the results of [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] concerning the spectrum of 1 W (u 0 ,u 1 ) P for optimal potentials W (u 0 , u 1 ), to non-optimal potentials W (u 0 , u 1 ) obtained by the supersolution construction, provided that the function u 0 and u 1 are optimal approximate solutions of P at infinity. We will define later what optimal approximate solutions of P at infinity precisely means (see Definition 2.16), since it is technical. One of the main results of this article can then be roughly stated as follows (for a precise formulation, see Theorem 2.17): Theorem 1.15. Let (u 0 , u 1 ) be a pair of optimal approximate solutions at infinity for P . Recall that W (u 0 , u 1 ) := 1 4 ∇ log u 0 u 1

2

. Then the essential spectrum of 1 W (u 0 ,u 1 ) P is equal to [1, ∞), and its spectrum below 1 consists at most of a finite number of eigenvalues with finite multiplicity.

Roughly speaking, the proof relies on the fact that there are approximate generalized eigenfunctions of exponential type for 1 W (u 0 ,u 1 ) P . As a main corollary of Theorem 1.15, we will be able to answer Question 1.14 (see Theorem 3.5):

Theorem 1.16. Let Ω be a C 2 , bounded, mean convex domain of R n , and assume that D is chosen as in Example 1.9. Then for every i ≥ 1, the spectrum of

1 J i (-∆ -W i-1 ) is [1, ∞)
. Furthermore, without the mean convexity assumption on Ω, the essential spectrum of

1 J i (-∆ -W i-1 ) is [1, ∞)
, and the intersection of the spectrum of 1 J i (-∆ -W i-1 ) with (-∞, 1) consists of (at most) a finite number of eigenvalues with finite multiplicity.

In particular, this proves Agmon's claim 1.12 about the spectrum of 1 δ 2 (-∆). As an immediate corollary, we recover the value of the best constant and of the best constant at infinity in the improved Hardy inequalities (1.6), a result already proved in [START_REF] Barbatis | Series expansion for L p Hardy inequalities Indiana[END_REF].

Another result that we obtain as a consequence of Theorem 1.15 concerns Hardy inequalities on minimal submanifolds M n ֒→ R N . Let us fix x 0 ∈ R N , and denote by r := d R N (x 0 , •), where d R N is the Euclidean distance. Carron obtained in [START_REF] Carron | Inégalités de Hardy sur les variétés riemaniennes non-compactes[END_REF] the following Hardy inequality

n -2 2 2 M u 2 r 2 ≤ M |∇u| 2 , ∀u ∈ C ∞ 0 (M ). (1.7)
Let us denote by II the second fundamental form of the isometric immersion M n ֒→ R N , and denote by ∆ M the (negative) Laplacian on M . As a consequence of Theorem 1.15, we get the following result (see Theorem 4.1):

Theorem 1.17. Assume that the total curvature of the minimal isometric immersion

M n ֒→ R N is finite, i.e. that M |II| n 2 < ∞.
Then the spectrum of

r 2 (-∆ M ) is n-2 2
2 , ∞ .

To conclude this introduction, we propose the following open problem:

Question 1.18. Are there embedded eigenvalues for the operators 1 J i (-∆ -W i-1 ) and r 2 (-∆ M ), appearing respectively in Theorem 1.16 and Theorem 1.17?

The structure of the article is as follows: in Section 2, we establish the general result (Theorem 1.15) that will be the key to studying the spectrum of our operators. In Section 3, we apply this result to the Hardy inequality (1.6), and we prove Theorem 1.16. In Section 4, we consider the case of the Hardy inequality on minimal submanifolds of R n , and we prove Theorem 1.17. In Section 5, we study an exponential volume growth property, which shows up naturally for Hardy inequalities with admissible potentials.

General theory

2.1. Preliminaries. Throughout the paper, the potentials appearing in the various Hardy inequalities will always be non-negative. As usual, C will denote a generic constant, whose value can change from line to line.

Notation: For two positive functions f and g, we will write f ≍ g if there is a positive constant C such that

C -1 f ≤ g ≤ Cf.
Let Ω, n ≥ 2 be a smooth domain in R n (or more generally, a smooth, connected manifold of dimension n). The infinity of Ω is the ideal point in the one-point compactification of Ω. From this, we derive the notions of neighborhood of infinity in Ω, of convergence at infinity for a real function defined in Ω, etc... Let ν be a positive measure on Ω. Consider a symmetric second-order elliptic operator L on Ω with real coefficients in divergence form of the type

P u = -div A∇u + cu,
(2.1) Here, -div is the formal adjoint of the gradient with respect to the measure ν. We assume that for every x ∈ Ω the matrix A(x) = a ij (x) i,j is symmetric and that the real quadratic form

ξ, A(x)ξ := n i,j=1 ξ i a ij (x)ξ j ξ ∈ R n (2.2)
is positive definite. We will denote the norm associated to this quadratic form by

| • | A , that is |ξ| 2 A := ξ, A(x)ξ .
Moreover, it is assumed that P is locally uniformly elliptic, and that A and c are locally bounded in Ω.

We denote by q the quadratic form associated to P , defined by

q(u) = Ω A∇u, ∇u + cu 2 dν, ∀u ∈ C ∞ 0 (Ω).
We will assume that q is nonnegative, and consider the Friedrichs extension of P , that we will also denote P . It is a self-adjoint operator on L 2 (Ω, dν). If u ∈ W 1,2 loc (Ω) and f ∈ L ∞ loc (Ω), we will say that the equation

P u = f holds in Ω in the weak sense if for every ϕ ∈ C ∞ 0 (Ω), Ω ( A∇u, ∇ϕ + cuϕ) dν = Ω f ϕ dν.
In a similar way, we define the notion of weak supersolutions/subsolutions for P .

We will make use of the following two constructions. The first one is called the h-transform and is defined as follows: if h > 0 is in C 1,α loc (Ω) and P h ∈ L ∞ loc (Ω), we define the operator

P h := h -1 P h,
which is self-adjoint on L 2 (Ω, h 2 dν). Also, the following formula holds in the weak sense:

P h u = -div h (A∇u) + P h h u, (2.3)
where

div h (X) = div(X) + 2 h -1 ∇h, X
is the divergence with respect to the measure h 2 dν.

We call the second construction the change of measure. Let W > 0, then we can consider the operator 1 W P . It is a self-adjoint operator on L 2 (Ω, W dν), and its quadratic form is the same as P . If div W is the divergence with respect to the measure W dν, then

1 W P u = -div W A W ∇u + c W u, (2.4)
which follows from the formula

div W (X) = 1 W div(W X).

2.2.

A spectral result. We now present a general spectral result concerning operators of the form 1 W P . It is probable that this result, maybe in a weaker form, is already known to experts, even if we have been unable to find a reference in the literature that covers such a general case.

Theorem 2.1. Assume that W 1 and W 2 are two positive (in a neighborhood of infinity) potentials in L ∞ loc (Ω), such that

W 1 (x) ∼ W 2 (x) when x → ∞.
Then σ ess ( 1

W 1 P ) = σ ess ( 1 W 2 P ).
Proof. Let λ ∈ σ ess ( 1 W 1 P ). Let (Ω n ) n∈N be an exhaustion of Ω. Then, using a ground state transform, it follows essentially from the decomposition principle for the essential spectrum (see [START_REF] Donnelly | Pure point spectrum and negative curvature for noncompact manifolds[END_REF] or [START_REF] Glazman | Direct methods of qualitative spectral analysis of singular differential operators[END_REF]) that for every compact set K ⋐ Ω, there is a Weyl sequence (u n ) n∈N associated to λ, orthonogal in L 2 (W 1 dν), such that for every n ∈ N, the support of

u n is included in Ω \ Ω n . By definition of a Weyl sequence, if we denote by || • || the norm in L 2 (W 1 dν), there holds lim n→∞ ||( 1 W 1 P -λ)u n || ||u n || = 0. Since W 1 (x) ∼ x→∞ W 2 (x)
and given the hypothesis on the support of u n , one has, when n → ∞,

||u n || ∼ ||u n || L 2 (W 2 dν) .
Also, using the hypothesis on the support of u n , and the fact that W 1 and W 2 are equivalent at infinity, one has for n big enough,

||( 1 W 2 P -λ)u n || L 2 (W 2 dν) = Ω |(P -λW 2 )u n | 2 dν W 2 1/2 ≤ Ω |(P -λW 1 )u n | 2 dν W 2 1/2 +|λ| Ω |(W 1 -W 2 )u n | 2 dν W 2 1/2 ≤ 2||( 1 W 1 P -λ)u n || +|λ| Ω W 1 -W 2 W 2 2 u 2 n W 2 dν 1/2 ≤ o(||u n ||).
Consequently, u n is also a Weyl sequence for 1 W 2 P , associated to λ. The hypothesis on the support of u n now implies that λ is in the essential spectrum of 1 W 2 P .

Theorem 2.1 actually allows us to get a first quick proof of Agmon's claim 1.12, relying on the results of [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]. An alternative proof, which works more generally for improved Hardy inequalities (1.6), will be given in Section 3.

Corollary 2.2. If Ω is a bounded, C 2 domain in R n , then the essential spectrum of δ -2 ∆ is [ 1 4 , ∞). Proof. Let W = 1 4 ∇G G
2 be an optimal weight in the sense of [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF], where G is the Green function of ∆ with pole at some fixed point x 0 ∈ Ω. It has been proved in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]Example 13.2] that as x → ∂Ω, W ∼ 1 4δ 2 . Furthermore, according to [15, Theorem 2.2], the (essential) spectrum of

1 W ∆ is [1, ∞). By Theorem 2.1, the essential spectrum of δ -2 ∆ is equal to the essential spectrum of 4 1 W ∆, and thus is equal to [ 1 4 , ∞).
Another direct application of Theorem 2.1 and of the results of [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] is to multipolar Hardy inequalities. Let x 1 , • • • , x N , N ≥ 2 be distinct points in R n , and consider the positive weight

W =   1≤i<j≤N |x i -x j ] 2 |x -x i | 2 |x -x j | 2   .
In [START_REF] Cazacu | Improved multipolar Hardy inequalities[END_REF], the following multipolar Hardy inequality was shown:

R n |∇u| 2 ≥ n -2 N 2 R n W u 2 , ∀u ∈ C ∞ 0 (R n ). (2.5)
It was proved in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]Remark B2 and B.3

] that ∆ -n-2 N 2 W is critical, but the weight W is not optimal, i.e. does not belong to O(R n \ {x 1 , • • • , x N }) (one reason is that the constant n-2 N 2 in (2.5
) is not optimal for test functions supported outisde a ball B(0, R), for R large enough). From Theorem 2.1, one can deduce the following result:

Corollary 2.3 (Spectrum for multipolar Hardy inequalities). Denote C(N ) = N 2 4(N -1) . If N = 2, then the (essential) spectrum of 1 W ∆ is [ n-2 2 2 , ∞). If N > 2, then the essential spectrum of 1 W ∆ is [C(N ) n-2 N 2 , ∞), the bottom of the spectrum of 1 W ∆ is n-2 N 2 ,

and the positive function

v = N i=1 |x - x i | 2-n N is eigenfunction for 1 W ∆, associated to the eigenvalue n-2 N 2 .
Remark 2.4. One can actually show, using arguments similar to the one appearing in the proofs of Lemma 2.13 and Proposition 2.14, that if N > 2, 1 W ∆ has a finite number of eigenvalues, each with finite multiplicity, belonging to

[ n-2 N 2 , C(N ) n-2 N 2
). This in turn implies (see [START_REF] Devyver | On the finiteness of the Morse index for Schrödinger operators[END_REF]) that the eigenspace associated to C(N ) n-2 N 2 is finite dimensional.

Proof. Denote L := 1 W ∆. Let ε > 0 be small enough such that the balls B(x j , ε) are disjoint. Let K 1 be a regular compact set containing the balls B(x j , ε), j = 1, • • • , N , and denote K 2 the complement in K 1 of ∪ N i=1 B(x j , ε). The decomposition principle for the essential spectrum (see [START_REF] Donnelly | Pure point spectrum and negative curvature for noncompact manifolds[END_REF] or [START_REF] Glazman | Direct methods of qualitative spectral analysis of singular differential operators[END_REF]) implies that the essential spectrum of L is equal to the essential spectrum of L on L 2 (R n \ K 1 , W dx), with Neumann boundary conditions on ∂K 1 . Thus, it consists of the union of the essential spectrum of L on B(x j , ε), j = 1, • • • , N with Neumann boundary conditions, and of the essential spectrum of L on R n \ K 1 , with Neumann boundary conditions on

∂K 1 . When x → x j , one has (cf [15, Remark B.3]) n -2 N 2 W ∼ C(N ) -1 C H |x -x j | 2 = C(N ) -1 W opt,j ,
where

C H := n-2 2
2 . Since W opt,j is an optimal weight, by [15, Theorem

2.2], the essential spectrum of W -1 opt,j ∆ is [1, ∞). More precisely, for every λ ∈ [1, ∞), one can find a Weyl sequence for W -1 opt,i ∆ supported in B(x j , ε 2
). This implies that the essential spectrum of

W -1 opt,j ∆ on B(x j , ε), with Neumann boundary conditions, is equal to [1, ∞). Applying Theorem 2.1, one obtains that the essential spectrum of L on B(x j , ε), j = 1, • • • , N with Neumann boundary conditions is equal to [C(N ) n-2 N 2 , ∞). Moreover, when |x| → ∞, it holds that W ≍ 1 |x| 4 , and therefore λ ∞ (R n \ K 1 , ∆, W ) = ∞,
which implies, according to Persson's formula, that L on R n \ K 1 , with Neumann boundary conditions on ∂K 1 , has no essential spectrum. Therefore, the essential spectrum of

L on R n is [C(N ) n-2 N 2 , ∞). The statement that if N > 2, the bottom of the spectrum of L is n-2 N 2 and that v is eigenfunction follows from [15, Remark B.2].
However, it is more delicate to use Theorem 2.1 and [15, Theorem 2.2] in order to prove Theorems 1.16 and 1.17 in a similar way. More precisely, in order to prove Theorem 1.17, one would have to show that as x → ∞ in M ,

1 4 ∇G(x) G(x) 2 ∼ n -2 2 2 1 r 2 (x) , (2.6)
where G is the Green function of the Laplacian on M , with pole at some fixed point of M . In the case of Theorem 1.16, one would have to show that, as x → ∂Ω,

1 4 ∇G i-1 (x) G i-1 (x) 2 ∼ J i (x), (2.7)
where G i-1 is the Green function of ∆ -W i-1 , with pole at some fixed point of Ω. We do not know if the two estimates (2.6) and (2.7) hold. In Section 3 and 4, we will prove respectively Theorem 1.16 and 1.17, without having to prove gradient estimates for Green functions. This is an important point, since in general finding the asymptotic at infinity of |∇ log G| for a Green function G is a difficult task, that can be achieved only in some particular cases. Instead, in our general approach that will lead to the proof of Theorems 1.16 and 1.17, we will not have to estimate the gradient of a Green function. This general approach is based on the results of the next subsection. We have the following elementary technical Lemma:

Lemma 2.5. If σ(χ) = 0, then for every a > 0, ε > 0 and d > 0, there exists b > a such that |b -a| ≥ d and

S(b) V (b) -V (a) < ε.
Conversely, if for every ε > 0, there exists a > 0 and d > 0 such that for every b > a + d, the inequality

S(b) V (b) -V (a) < ε holds, then σ = 0.
Proof. The proof is elementary, but we provide it for the sake of completeness. Assume that σ = 0. We proceed by contradiction: define f (r) := V (r) -V (a), and assume that there is ε > 0 and R big enough such that for every r ≥ R,

f (r + 1) -f (r) f (r) > ε.
Define g(r) := e νr , with ν > 0 chosen so that e ν -1 < ε, then

g(r + 1) -g(r) g(r) = e ν -1 < ε < f (r + 1) -f (r) f (r) .
From this, we deduce at once that there is a constant C such that for every r > 0,

f (r) ≥ Cg(r). But this implies that σ ≥ ν, which is impossible. Conversely, if ε > 0 is fixed and for every r > R, f (r + 1) -f (r) f (r) < ε,
then proceeding as above, by comparison with g(r) := e νr with ν chosen so that e ν -1 > ε, we find that f (r) ≤ Ce νr , which yields that σ ≤ ν. Letting ε → 0, we can let ν → 0 and conclude that σ = 0. Definition 2.6. A positive measure χ ∈ C is said to have subexponential volume growth if it satisfies σ(χ) = 0. In other words, χ has subexponential volume growth iff V (r) = e o(r) .

We will consider push-forward measures, of which we recall the definition: Definition 2.7. If X and Y are measured spaces, f : X → Y is measurable and µ is a measure on X, then the push-forward of µ by f is the measure f ⋆ µ, defined so that for every measurable set B of Y ,

(f ⋆ µ)(B) = µ(f -1 (B)),
where by definition f -1 (B) is the subset of X defined by

f -1 (B) = {x ∈ X : f (x) ∈ B}.
Then, for every g : Y → R measurable, we have the change of variable formula

Y g d(f ⋆ µ) = X g • f dµ.
(2.8)

After these preliminaries, let us now turn to our first general spectral result. Let L be a symmetric, second-order elliptic operator of the form (2.1). We assume that L1 = 0 (that is, c = 0), then by Allegretto-Piepenbrink theory (see [START_REF] Agmon | On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds[END_REF], or Lemma 3.10 in [START_REF] Pigola | Vanishing and finiteness results in geometric analysis: a generalization of the Bochner technique[END_REF]), L extends to a self-adjoint operator on L 2 (Ω, dν), whose spectrum is included in [0, ∞). In the following Proposition, we give conditions guaranteeing that the spectrum of L is the whole [0, ∞). (2.10)

Assume also that the push-forward measure χ := v ⋆ ν is in C and has subexponential volume growth. Then the spectrum of L is [0, ∞): more precisely, for every η ≥ 0, we can construct a Weyl sequence whose support goes to infinity, i.e. a sequence (ϕ n ) n∈N of smooth, compactly supported functions such that

lim n→∞ ||(L -η)ϕ n || 2 ||ϕ n || 2 = 0
and for every compact K ⊂ Ω, there is an N such that the support of ϕ n is in Ω \ K for every n ≥ N .

Remark 2.9.

(1) The infinity is the ideal point in the one-point compactification of Ω.

(2) The condition that v ⋆ ν be in C is quite weak: for example, if ν is the Lebesgue measure on R n , then by the co-area formula, v ⋆ ν ∈ C if and only if the function f (t

) := Vol H n-1 ({v = t}) is locally L 1 and ∞ f (t)dt = ∞.
Here, Vol H n-1 is the volume with respect to H n-1 , the (n -1)-dimensional Hausdorff measure. (3) As follows from the proof, Proposition 2.8 holds under the weaker condition on χ that for every a > 0, ε > 0 and d > 0, there exists b > a such that |b -a| ≥ d and

S(b) V (b) -V (a) < ε.
This is indeed (slightly) weaker than subexponential volume growth, by Lemma 2.5.

The idea behind Proposition 2.8 is that L, restricted to the set of "radial" functions, that is functions of the form f (v), has spectrum [0, ∞). Thus it can be thought of in a way as a result about a "one-dimensional operator". In fact, for every η ≥ 0, the function e i √ η is an approximate solution at infinity of (L -η)u = 0, and the Weil sequence will be constructed as a sequence of functions that approwimate e i √ η . Let us turn to the detailed proof of Proposition 2.8:

Proof. Without loss of generality, we can assume that |∇v(x)| A → 1 when x → ∞. We will use the fact that since L1 = 0, L satisfies the two formulae (in the weak sense):

L(f (u)) = f ′ (u)Lu -f ′′ (u)|∇u| 2 A (2.11)
and if g and h are C 1,α loc (Ω),

L(gh) = gL(h) + hL(g) -2A∇g • ∇h.
(2.12) Fix η ≥ 0, and let µ := √ η. Define ϕ = e iµv . Then, using formula (2.11), we get that in the weak sense,

Lϕ = iµ(Lv)ϕ + µ 2 |∇v| 2 A ϕ, that is (L -η)ϕ = iµ(Lv)ϕ + η(|∇v| 2
A -1)ϕ. We want to define ϕ n := ψ n (v)ϕ, where ψ n (v) is going to play the role of a "radial" cut-off function. We first compute (using fomulae (2.11) and (2.12)) that in the weak sense,

(L -η)ϕ n = ψ n (v) [(L -η)ϕ] + ϕ(Lψ n (v)) -2 A∇ϕ, ∇ψ n (v) = iµϕ n (Lv) + η(|∇v| 2 A -1)ϕ n +ϕ ψ ′ n (v)(Lv) -ψ ′′ n (v)|∇v| 2 A -2iµψ ′ n (v)ϕ|∇v| 2 A
(2.13) We now define the real function ψ n : we take

ψ n (t) equal to 1 if t ∈ [a n + 1, b n -1], 0 if t / ∈ [a n , b n ]
-a n and b n to be chosen later -, and such that there is a constant C independent of n satisfying

|ψ ′ n | + |ψ ′′ n | ≤ C.
We are now ready to estimate each of the terms appearing in the computation of (L -η)ϕ n : we first have, using the property of change of variable formula (2.8) of the push-forward measure,

||ϕ n || 2 2 = ||ψ n (v)ϕ|| 2 2 = ψ 2 n (t)dχ(t) ≥ bn-1 an+1 dχ(t).
Moreover, using that Lv and |∇v| A are bounded, and the fact that ψ ′ n and ψ ′′ n are supported in the union of the intervals [a n , a n + 1] ∪ [b n -1, b n ], we get (using again the change of variable formula (2.8))

||ϕ ψ ′ n (v)(Lv) -ψ ′′ n (v)|∇v| 2 A -2iµψ ′ n (v)|∇v| 2 A ϕ|| 2 ≤ C [an,an+1]∪[bn-1,bn] dχ(t).
Furthermore, since |∇v| 2 A → 1 at infinity, we have

||η(|∇v| 2 A -1)ψ n (v)ϕ|| 2 2 = o ψ 2 n (t)dχ(t) when n → ∞.
We now choose the sequences (a n , b n ) n∈N inductively: suppose that (a n-1 , b n-1 ) is defined, then take a n > min(b n-1 , n) such that |Lv| ≤ 1 n on the set {v ≥ a n } (here we use hypotheses (2.9) and (2.10)), and take b n big enough such that

[an,an+1]∪[bn-1,bn] dχ(t) bn-1 an+1 dχ(t) ≤ 1 n .
This is possible by Lemma 2.5, since χ has subexponential volume growth. Collecting all the estimates, we get that

lim n→∞ ||(L -η)ϕ n || 2 ||ϕ n || 2 = 0,
which concludes the proof.

2.4. Spectral result for Hardy inequalities. We now apply Proposition 2.8 to study Hardy inequalities. We consider a symmetric operator P on L 2 (Ω, dν) of the form (2.1). We take u 0 , u 1 positive functions on Ω which, for some α ∈ (0, 1) and K ⊂⊂ Ω compact subset of Ω, are C 1,α loc (Ω \ K). Denote

V i = P u i u i .
Recall that X 1 is defined by

X 1 (t) := (1 -log(t)) -1 ,
and consider the non-negative weight

W (u 0 , u 1 ) := 1 4 ∇ log u 0 u 1 2 A = 1 4 ∇X -1 1 u 0 u 1 2 A
We emphasize that here and everywhere in the paper, X -1 1 is a notation for 1 X 1 and not for the inverse of X 1 . We know from the supersolution construction of [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] that in the case where u 0 , u 1 are solutions of P , then u 1/2 := √ u 0 u 1 and u 1/2 X -1

1 u 0 u 1
are solutions of P -W (u 0 , u 1 ). Furthermore, if u 0 and u 1 are only supersolutions of P , then u 1/2 is supersolution of P -W (u 0 , u 1 ). In this subsection, we will be interested in the case where u 0 and u 1 are approximate solutions of P . In the rest of this subsection, we will denote W (u 0 , u 1 ) by W . We will need the following general computational lemma (see [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] for the proof of the first equality):

Lemma 2.10. The following equalities hold, in the weak sense:

P - 1 2 (V 0 + V 1 ) -W u 1/2 = 0, and
P - 1 2 (V 0 + V 1 ) + (V 0 -V 1 )X 1 u 0 u 1 -W u 1/2 X -1 1 u 0 u 1 = 0.
For the rest of this section, we assume that W is positive in a neighborhood of infinity in Ω: more precisely, we will assume that W > 0 on Ω \ K. From the assumption that u 0 and u 1 belong to C 1,α (Ω \ K), it follows that W and 1 W are continuous, and in particular locally bounded, on Ω \ K. In order to study the spectral properties of 1 W P , we perform simultaneously a h-transform and a change of measure: we consider the operator

L := u -1 1/2 W -1 P -1 u 1/2 , which is symmetric on L 2 (Ω, u 2
1/2 W dν) and unitarily equivalent to W -1 P -1 . We compute from the formulae (2.3) and (2.4) that

L = -div A W ∇• + W -1 P -1 u 1/2 u 1/2 ,
where the divergence is for the measure u 2 1/2 W dν. Let us denote by V the potential

V := W -1 P -1 u 1/2 u 1/2 = 1 2W (V 0 + V 1 ),
(we have used here the first equality in Lemma 2.10), and by L the symmetric operator

L := -div A W ∇•
acting on L 2 (Ω, u 2 1/2 W dν), so that L = L + V. We have then the following consequence of Proposition 2.8: Proposition 2.11. Assume that the following conditions are satisfied:

(1)

lim x→∞ u 0 (x) u 1 (x) = 0, (2)
lim x→∞ 1 2W (V 0 + V 1 )X -1 1 u 0 (x) u 1 (x) = 0, (3)
lim x→∞ V 0 -V 1 W = 0, ( 4
)
The push-forward measure X -1

1 u 0 u 1 ⋆ (u 0 u 1 W dν) is in C and has subexponential volume growth. Then the essential spectrum of W -1 P on L 2 (Ω, W dν) is [1, +∞).
Remark 2.12.

(1) It will be clear from the proof that if W = W in a neighborhood of infinity of Ω, then the essential spectrum of W -1 P is also [1, ∞).

(2) Concerning the hypotheses made in Proposition 2.11: condition (1) expresses the fact that u 0 has "minimal growth" at infinity; conditions (2) and (3) express in a quantitative way that u 0 and u 1 are "approximate solutions" of P at infinity; condition (4) is satisfied for optimal potentials obtained by the supersolution construction, i.e. if u 0 and u 1 are solutions of P : indeed, in this case, X -1

1 u 0 u 1 ⋆
(u 0 u 1 W dν) has linear volume growth (see [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]).

Therefore, the hypotheses (1)-( 4) can be considered to express in a quantitative way that W is "optimal at infinity". We will discuss with greater details the relevance of condition (4) in Section 5.

Proof. Since W -1 P -1 and L are unitarily equivalent, it is enough to show that the essential spectrum of L is [0, ∞). By Lemma 2.10, we have Let us remark that by definition of W ,

LX -1 1 u 0 u 1 = 1 2W (V 0 + V 1 )X -1 1 u 0 u 1 + V 0 -V 1 W . Define v := X -1 1 u 0 u 1 , which is C 1,α loc (Ω \ K) for some K compact
|∇v| 2 A/W = W -1 |∇v| 2 A = 4.
We can apply Proposition 2.8 to L (notice that the matrix A W appearing in the definition of L is locally bounded in Ω \ K): for every η ≥ 0, there is a Weyl sequence (ϕ n ) n∈N for L -η, with the support of ϕ n going to infinity. Since lim x→∞ V (x) = 0, we conclude that (ϕ n ) n∈N is also a Weyl sequence for L -η. Hence the essential spectrum of L contains [0, ∞). For the inverse inclusion, it is enough to prove that 0 is the infimum of the essential spectrum of L. By Persson's formula (see [START_REF] Persson | Bounds on the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF] or [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators Mathematical Notes[END_REF]), λ ∞ (L), the infimum of the essential spectrum of L, is given by

λ ∞ (L) = sup{λ : ∃K ⊂⊂ Ω, ∃u λ > 0, s.t. (L -λ)u λ = 0 on Ω \ K}. (2.14)
Let ε > 0. Since V tends to 0, we can find a compact set K 0 containing K such that

|V | ≤ ε on Ω \ K 0 .
Since L1 = 0, by Allegretto-Piepenbrink L is nonnegative. Therefore, again by Allegretto-Piepenbrink one can find a positive function u, solution of

(L + ε)u = ( L + V + ε)u = 0 on Ω \ K 0 . Persson's formula (2.14) now implies that λ ∞ (L) ≥ -ε. Letting ε → 0, we conclude that λ ∞ (L) ≥ 0.
We will be also interested in the part of the essential spectrum below the essential spectrum. For this, we will use the following Lemma: Lemma 2.13. Let L be an operator of the form (2.1). Assume that in a neighborhood of infinity there is a positive C 1,α loc function v such that when x → ∞, there holds

|∇v(x)| A ≥ C > 0 and Lv v + L1 = o(v -2 ).
Then the negative spectrum of L consists of (at most) a finite number of eigenvalues, each with finite multiplicity.

Proof. We apply Lemma 2.10 to get, in a neighborhood of infinity,

L √ v = 1 2 Lv v + L1 + 1 4 |∇v| 2 A v 2 √ v.
Using the hypotheses, we see that in a neighborhood of infinity,

L √ v ≥ 0.
According to [START_REF] Devyver | On the finiteness of the Morse index for Schrödinger operators[END_REF], the existence of a positive supersolution of L outside a compact set of Ω is equivalent to the fact that the negative spectrum of L consists of (at most) a finite number of eigenvalues, with finite multiplicity, hence the conclusion.

We apply this in the framework of Proposition 2.11:

Proposition 2.14. Let P , W , u 0 , u 1 , V 0 , V 1 be as defined above Proposition 2.11. Assume that when x → ∞, (1)
lim x→∞ 1 2W (V 0 + V 1 ) = o X 2 1 u 0 (x) u 1 (x) , (2)
lim x→∞ V 0 -V 1 W = o X 1 u 0 (x) u 1 (x) ,
Then the spectrum of W -1 P strictly below 1 consists (at most) of a finite number of eigenvalues of finite multiplicity.

Remark 2.15. It is clear from the proof that the same conclusion also holds for W -1 P , if W = W in a neighborhood of infinity of Ω.

Proof. It is a direct consequence of Lemma 2.13, applied with

L = L = u -1 1/2 ( 1 W P -1)u 1/2 and v = X -1 1 u 0 (x) u 1 (x) .
The results of this section lead us naturally to the concept of "approximate solutions", as announced in the introduction. We introduce the following definition:

Definition 2.16. Recall that W (u 0 , u 1 ) := 1 4 ∇ log u 0 u 1 2 A
, and assume that W (u 0 , u 1 ) is positive in a neighborhood of infinity. We say that (u 0 , u 1 ) is a pair of optimal approximate solutions at infinity for P if the following conditions are satisfied:

X 1 (t) := (1 -log(t)) -1
, and for all i ≥ 1,

X i+1 (t) = X 1 (X i (t)).
Then for every i ≥ 1, X i ((0, 1]) = (0, 1], and

X i (1) = 1, lim t→0 X i (t) = 0.
We will use the following formula which gives, for i ≥ 1, the derivative of X i , and which is extracted from [START_REF] Barbatis | Series expansion for L p Hardy inequalities Indiana[END_REF]:

X ′ i (t) = 1 t X 1 (t) • • • X i-1 (t)X 2 i (t)
. Now, we have the following computation, which is a direct consequence of Lemma 2.10, and which implies the Hardy inequality (1.6) obtained in [START_REF] Barbatis | Series expansion for L p Hardy inequalities Indiana[END_REF] (related computations have been performed in [START_REF] Filippas | Optimizing improved Hardy inequalities[END_REF]). Proposition 3.2. Let i ≥ 0, and define

U 0,i := δ D X -1 0 δ D • • • X -1 i δ D 1/2 , U 1,i := U 0,i X -1 i+1 δ D , R i := i k=1 X 1 δ D • • • X k δ D (R 0 = 0 by convention),
and

H i := 1 4 i+1 k=1 ∇X -1 k δ D 2 .
Then in Ω,

-∆ - -∆δ 2δ (1 -R i ) -H i U 0,i = 0, (3.1)
and

-∆ - -∆δ 2δ (1 -R i ) + -∆δ δ X 1 δ D • • • X i+1 δ D -H i U 1,i = 0.
(3.2) Moreover, if we denote

W i := 1 4δ 2 i k=0 X 2 0 δ D • • • X 2 k δ D , then
H i = W i a.e.
on Ω, and if U is the neighborhood of ∂Ω given by Lemma 3.1, then

H i = W i on U. Corollary 3.3. If Ω is mean-convex, and if D is chosen such that in Ω, one has R i = i k=1 X 1 δ D • • • X k δ D ≤ 1,
then the Hardy inequality (1.6) takes place on Ω.

Remark 3.4. Actually, D in Corollary 3.3 can be chosen independently of i: it is a consequence of the fact that the series

∞ k=0 X 1 (t) • • • X k (t)
converges for every t ∈ [0, 1). For a proof of this fact (kindly provided to us by A. Tertikas), see the Appendix.

Proof of Corollary 3.3. The hypothesis on D gives that R i ≤ 1, so that by Proposition 3.2 and the fact that -∆δ ≥ 0, we have (in the weak sense)

(-∆ -H i ) U 0,i ≥ 0.
Given that U i,0 > 0, this implies by Allegretto-Piepenbrink theory that -∆ -H i ≥ 0, which is equivalent to saying that the following Hardy inequality holds:

Ω H i u 2 ≤ Ω |∇u| 2 , ∀u ∈ C ∞ 0 (Ω).
Since δ is Lipschitz on Ω, it is not hard to see that W i and H i are in L ∞ (Ω). By Proposition 3.2, there is a set A ⊂ Ω of zero measure such that

H i = W i on Ω \ A. Let us fix u ∈ C ∞ 0 (Ω); then Ω\A W i u 2 + A H i u 2 ≤ Ω |∇u| 2 .
Since δ is Lipschitz on Ω, it is not hard to see that H i and W i are in L ∞ (Ω), and thus A H i u 2 = A W i = 0. Therefore,

Ω W i u 2 ≤ Ω |∇u| 2 .
Consequently, the Hardy inequality (1.6) holds on Ω.

Proof of Proposition 3.2. The proof of the formulae (3.1) and (3.2) is by induction on i, using the construction described in Lemma 2.10. Indeed, to pass from (3.1) and (3.2) of index i to (3.1) and (3.2) of index i + 1, just apply Lemma 2.10 with P = -∆ + H i , u 0 = U 0,i and u 1 = U i,1 . In order to initialize the induction, we apply Lemma 2.10 with P = -∆, u 0 = δ D , V 0 = -∆δ δ and u 1 = 1, V 1 = 0, and we get exactly the formulae (3.1) and (3.2) for i = 0. Now assume that the formulae (3.1) and (3.2) are true for the index i, and apply Lemma 2.10 with P = -∆ + H i , u 0 = U 0,i and u 1 = U i,1 . By the induction hypothesis,

V 0 = -∆δ 2δ (1 -R i ) ,
and

V 1 = -∆δ 2δ (1 -R i ) - -∆δ δ X 1 δ D • • • X i+1 δ D .
The formulae of Lemma 2.10 gives (3.1) and (3.2) for the index i + 1, upon noticing that

X 1 u 0 u 1 = X 1 X i δ D = X i+1 δ D .
The fact that W i = H i on U follows from the following computation:

∇X -1 k+1 δ D = 1 D X ′ k+1 δ D X -2 k+1 δ D = 1 D D δ X 1 δ D • • • X k δ D X 2 k+1 δ D X -2 k+1 δ D = 1 δ X 1 δ D • • • X k δ D , and therefore W i = H i on U .
We now prove the announced spectral result for the Hardy inequality (1.6), as a consequence of Proposition 2.11. Recall that for i ≥ 1,

J i = W i -W i-1 ,
and by convention

J 0 = 1 4δ 2 .
Theorem 3.5. For every i ≥ 0, the essential spectrum of the operator

L i := J -1 i (-∆-W i-1 ) in L 2 (Ω, J i dx) is [1, ∞).
Furthermore, the spectrum of L i strictly below 1 consists (at most) of a finite number of eigenvalues with finite multiplicity.

In the case where Ω is mean-convex and if D is chosen as in Corollary 3.3, then the spectrum of L i in L 2 (Ω, J i dx) is [1, ∞), and in particular 1 is the best constant at infinity in the Hardy inequality (1.6).

Proof. By simplicity, we will assume that

H i = W i-1
on all Ω (and not only in a neighborhood of infinity). If it is not the case, one has to use the Remarks 2.12 and 2.15 . The modifications are left to the reader.

For k ≥ 1, define Y k := X -1 k δ D . Fix i ≥ 0.
We have seen in the proof of Proposition 3.2 that the improved Hardy inequality (1.6) at step i is obtained by applying the construction of Lemma 2.10 with P = -∆ -W i-1 , u 0 = U 0,i-1 , u 1 = U 0,i-1 Y i (for i = 0, we have to take P = -∆, u 0 = δ D and u 1 = 1). For this choice of u 0 and u 1 , we have

V 0 := P u 0 u 0 = O(δ -1 )
and

V 1 := P u 1 u 1 = O(δ -1 ).
We want to apply Proposition 2.11 to P , u 0 , u 1 , W := J i . We have to check the three corresponding conditions. Notice that

u 0 u 1 = X i δ D → 0 when δ → 0.
Next, remark that

X -1 1 u 0 u 1 = X -1 1 X i δ D = Y i+1 .
Therefore,

1 2J i (V 0 + V 1 )X -1 1 u 0 u 1 = 1 2J i Y i+1 O(δ -1 ) = O(δ 1-ε ).
Also,

1 2J i (V 0 -V 1 ) = O(δ 1-ε ),
so the second and the third conditions of Proposition 2.11 are satisfied. It remains to verify the condition on the measure. Noticing that u 0 u 1 = U 2 0,i , we see that the measure is dν = J i U 2 0,i dx. We compute

J i U 2 0,i = 1 4δ 2 X 2 1 • • • X 2 i × δ D X -1 1 • • • X -1 i = 1 4D 1 δ X 1 • • • X i = 1 4D |∇Y i+1 | Therefore, by the coarea formula, {a≤Y i+1 ≤b} J i U 2 0,i dx = 1 4D {a≤Y i+1 ≤b} |∇Y i+1 |dx = 1 4D b a {Y i+1 =t} dσ dt But Y i+1 = t if and only if δ = ϕ(t), for a decreasing function ϕ such that lim t→∞ ϕ(t) = 0.
When ε → 0, the surface measure of the level set {δ = ε} is equivalent to |∂Ω|, and therefore we obtain that when a and b go to +∞,

{a≤Y i+1 ≤b} J i U 2 0,i dx ≍ (b -a).
Applying now Proposition 2.8 gives that the essential spectrum of L i is [1, ∞). Concerning the finiteness of the spectrum below 1, it is immediate to see that the previous computations imply that the hypotheses of Proposition 2.14 are satisfied, which gives the result.

The analogue of Proposition 3.2 and Theorem 3.5 holds for the improved Hardy inequalities considered in [START_REF] Filippas | Optimizing improved Hardy inequalities[END_REF], where δ is replaced by ϕ(x) := |x| 2-n , which is harmonic (and not only superharmonic). Let us explain this: for n ≥ 3, we consider Ω a bounded domain of R n containing 0, and we define

Z i := ϕ(x)X -1 0 |x| D • • • X -1 i |x| D 1/2 , we have for D ≥ sup Ω |x|, -∆ - 1 4 i+1 k=1 ∇X -1 k |x| D 2 Z i = 0, (3.3)
and

-∆ - 1 4 i+1 k=1 ∇X -1 k |x| D 2 Z i X -1 i+1 |x| D = 0. (3.4)
Equation (3.3) has already been obtained in [START_REF] Filippas | Optimizing improved Hardy inequalities[END_REF]. We then have the following result, which is proved exactly like Theorem 3.5, considering what is happening around zero rather than at the boundary of Ω:

Theorem 3.6. For n ≥ 3, let Ω is a bounded domain of R n containing 0. Define

H i := 1 4 i+1 k=1 ∇X -1 k |x| D 2 and R i := H i -H i-1 .
Then the (essential) spectrum of the operator

R -1 i (-∆-W i-1 ) in L 2 (Ω, R i dx is [1, ∞).
In particular, 1 is the best constant around zero in the improved Hardy inequality

Ω R i u 2 ≤ Ω |∇u| 2 - Ω H i-1 u 2 , ∀u ∈ C ∞ 0 (Ω).
Remark 3.7. The fact that 1 is the best constant around zero was already shown in [START_REF] Filippas | Optimizing improved Hardy inequalities[END_REF].

Minimal immersions of the Euclidean space

We consider a minimal isometric immersion M n ֒→ R N , for n ≥ 3. We will denote by II the second fundamental form of this immersion. Let x 0 be any point of R N , and let r = d R N (x 0 , •) be the Euclidean distance. G. Carron [START_REF] Carron | Inégalités de Hardy sur les variétés riemaniennes non-compactes[END_REF] has shown the following Hardy inequality:

n -2 2 2 M u 2 r 2 ≤ M |∇u| 2 , ∀u ∈ C ∞ 0 (M ). (4.1)
Of course, if x 0 is a point of M , then an easy argument using test functions localized close to x 0 shows that n-2 This result is not so surprising, since by results of Anderson [START_REF] Anderson | The compactification of a minimal submanifold in Euclidean space by the Gauss map[END_REF], we know that the condition on the second fundamental form implies that M is asymptotically Euclidean. In fact, we will use some estimates obtained in [START_REF] Anderson | The compactification of a minimal submanifold in Euclidean space by the Gauss map[END_REF] in order to establish Theorem 4.1.

Proof. The proof is once again an application of Proposition 2.11. First, let us recall Carron's computation that leads to (4.1). We have

∆r 2 = n, which implies that -∆ - 1 r 2 n -2 2 2 + (2 -n)(n + 2) 4 1 -|∇r| 2 r 2-n 2 = 0. (4.2)
Here, ∇ is the gradient on M (and not on R n ). Notice that |∇r| ≤ 1, which implies by the above equation that

-∆ - n -2 2 2 1 r 2 r 2-n 2 ≥ 0.
By Allegretto-Piepenbrink theory, this yields the Hardy inequality (4.1). We will use the following Lemma, consequence of the work of Anderson [START_REF] Anderson | The compactification of a minimal submanifold in Euclidean space by the Gauss map[END_REF] and Shen-Zhu [START_REF] Shen | On stable complete minimal hypersurfaces in R n+1[END_REF]:

Lemma 4.2.
The volume of M is Euclidean at infinity, i.e. if B r (x 0 ) is the Euclidean ball centered in x 0 and V (r) the volume of M ∩ B R (x 0 ), then as r → ∞, V (r) is comparable to r n . Also, M has a finite number of ends, and at infinity in each end, M tends to a linear subspace of dimension of n; moreover, the second fundamental form satisfies the estimate

|II| = O(r -n/2
), and as r → ∞,

|∇r| 2 -1 = O(r -1 ).
For the estimate of |∇r| 2 -1, see in particular the proof of Lemma 2.4 in [START_REF] Anderson | The compactification of a minimal submanifold in Euclidean space by the Gauss map[END_REF].

Define V 0 := (2 -n)(n + 2) 4 1 -|∇r| 2 r 2 , then (-∆ -V 0 -W )r 2-n 2
= 0, and by Lemma 2.10,

-∆ - n(n -2)(1 -|∇r| 2 ) r 2 X 1 (r 2-n ) -V 0 -W r 2-n 2 X -1 1 (r 2-n ) = 0.
Furthermore,

1 W ∇X -1 1 (r 2-n ) 2 → 1 when r → ∞.
The proof of Proposition 2.11, which relies on Proposition 2.8, shows that it is enough to prove the following three properties:

(1)

lim r→∞ V 0 W X -1 1 (r 2-n ) = 0, (2) lim r→∞ 1 W n(n -2)(1 -|∇r| 2 ) r 2 = 0, (3)
The measure r 2-n W dx on M satisfies the hypothesis (4) of Proposition 2.8. The first two claims are consequences of the estimate of 1 -|∇r| 2 given by Lemma 4.2. For the last one, since

X -1 1 (r 2-n ) ∼ (n -2
) log r when r → ∞, we see that it is enough to prove the estimate on the measure with v = log r. By the co-area formula, a≤log r≤b r 2-n W = a≤log r≤b r 2-n r -2 ≍ a≤log r≤b r -n |∇r| ≍ e b e a t -n dV (t), so that the corresponding measure on R + is dχ(t) = t -n dV (t). We have to check that for any a big enough,

lim b→∞ b b-1 t -n dV (t) b a t -n dV (t) = 0.
Integrating by part gives the formula

d c t -n dV (t) = d -n V (d) -c -n V (c) + n d c t -n-1 V (t)dt.
Using the hypothesis that the volume growth in M is Euclidean at infinity (Lemma 4.2), we see that

b b-1 t -n dV (t) b a t -n dV (t) ≍ 1 log(b)
, hence goes to 0 when b → ∞. This gives that χ is in the class C of measure with subexponential volume growth.

Spectrum and Agmon metric

In this section, we study the relationship between good Hardy inequalities and some weak "hyperbolicity" properties. The motivation comes from the following example: in the case of the Euclidean unit ball B, consider the Hardy inequality

1 4 B u 2 δ 2 ≤ B |∇u| 2 , ∀u ∈ C ∞ 0 (B).
Define the following metric (the associated Agmon metric, see the paragraph below):

ds 2 = 1 δ 2 dx 2
, then ds 2 is the hyperbolic metric on the ball B. We ask the following question: Question 5.1. For a general Hardy inequality with an admissible potential, does the corresponding Agmon metric retain some (weak) hyperbolicity properties, i.e. properties similar to the one of the hyperbolic metric 1 δ 2 dx 2 on B?

We will show in this section that in the case of a general good Hardy inequalities, the Agmon metric satisfies a property of exponential growth of volume, similar to the exponential volume growth of the hyperbolic space. We will then apply this to the example of improved Hardy inequalities (1.9) on a mean convex domain of the Euclidean space. In passing, we will show a connection between this property of exponential growth of volume, and the condition (4) (the condition of subexponential volume growth of the measure) in Proposition 2.11. 5.1. General case. In this section, we consider a general Hardy inequality. Let W be a positive potential, and P of the form (2.1), such that the following Hardy inequality takes place for some λ > 0:

λ Ω W u 2 dν ≤ q(u), ∀u ∈ C ∞ 0 (Ω), (5.1)
where q is the quadratic form of P . Our first result is a direct generalization Brooks' results [START_REF] Brooks | A relation between growth and the spectrum of the Laplacian[END_REF], [START_REF] Brooks | On the spectrum of noncompact manifolds with finite volume[END_REF], who proved an estimate for the bottom of the essential spectrum of the Laplace-Beltrami operator on a complete Riemannian manifold, in term of the exponential growth of the volume of the geodesic balls. Brooks' results are consequence of Agmon's work on exponential decay of solutions of second-order elliptic equations [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators Mathematical Notes[END_REF]. Here we will show that Brooks' results can be formulated in the more general context of Hardy inequalities: our result is an estimate of the best constant at infinity in the Hardy inequality (5.1), in term of the exponential volume growth of some measure. Before presenting our result, we need to introduce some definitions and notations. Let ϕ be a positive solution of P ϕ = 0, and let us perform a h-transform with respect to ϕ: define P := ϕ -1 P ϕ, which is self-adjoint on L 2 (Ω, ϕ 2 dν). By formula (2.3), P is given by P u = -div ϕ 2 (A∇u). Now perform a change of measure: introduce the measure dµ := ϕ 2 W dν, and define

L := 1 W P ,
which is self-adjoint on L 2 (Ω, ϕ 2 W dν), so that the Hardy inequality is equivalent to

λ Ω u 2 dµ ≤ Ω |∇u| 2 A W dµ, ∀u ∈ C ∞ 0 (Ω), (5.2)
where |ξ| 2

A W := A W ξ, ξ . The term on the right-hand side is the quadratic form associated to L. Furthermore, the best constant λ 0 (resp. best constant at infinity λ ∞ ) in (5.1) is equal to the best constant (resp. best constant at infinity) in (5.2). The Hardy inequality (5.2) expresses that L has a spectral gap, indeed as we have already indicated in the introduction, the bottom of the spectrum (resp. essential spectrum) of L is λ 0 (resp. λ ∞ ). Let us define the Agmon metric

|ξ| 2 Ag := W A -1 ξ, ξ . (5.3)
Denote by ρ the distance function in this Agmon metric. If µ(Ω) = ∞, then for a fixed x 0 ∈ Ω, denote by V (r) the volume for the measure µ of the ball B Ag (x 0 , r) of center x 0 and of radius r (with respect to the distance ρ). If µ(Ω) < ∞, then define V (r) -1 to be the volume of Ω \ B Ag (x 0 , r) for the measure µ instead. Finally, define σ by σ := lim r→∞ sup 1 r log V (r).

(5.4)

If µ(Ω) = ∞, then σ is the exponential rate of volume growth for the measure µ, in the Agmon metric (5.3). If µ(Ω) < ∞, then σ is the exponential rate of convergence to µ(Ω) of the volume of balls in the Agmon metric. This definition does not depend on the choice of x 0 . Our result is Theorem 5.2. Assume that the Agmon metric is complete. Then the following inequality takes place:

λ ∞ ≤ σ 2 4 .
Proof. The proof follows closely Brook's proofs in [START_REF] Brooks | A relation between growth and the spectrum of the Laplacian[END_REF], [START_REF] Brooks | On the spectrum of noncompact manifolds with finite volume[END_REF], once the reduction to inequality (5.2) has been made. The need to use the Agmon metric (5.3) instead of the Riemannian metric on Ω comes from the following fact, that we extract from [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators Mathematical Notes[END_REF], Theorem 1.4:

Lemma 5.3. The distance ρ for the Agmon metric (5.3) satisfies

|∇ρ(x 0 , •)| 2 A W ≤ 1.
With this at hand, the adaptation of Brook's proof is quite straightforward, replacing the Riemannian metric by the Agmon metric and the Riemannian volume form by the measure µ. We leave the details to the reader. A for some real-valued Lipschitz function h, then we have the following formula for the distance in the Agmon metric (5.3) (for a proof, see Lemma 10.5 in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]): ρ(x, y) = |h(x) -h(y)|.

(5.5)

The completeness of Ω for the Agmon metric is then equivalent to

lim x→∞ |h(x)| = ∞ (see Lemma A1.2 in [3]
). In the case of the optimal potentials obtained by the supersolution construction in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF], the Agmon metric is complete (Lemma 10.5 in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]) and the inequality of Theorem 5.2 is an equality (see Lemma 7.2 in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]).

In fact, following the argument of Li and Wang [19, Theorem 1.3], more can be said: Theorem 5.5. For every ε > 0, the following holds:

(1) if µ(Ω) < ∞, then there is a constant C such that V (∞) -V (r) ≤ C exp -(2 -ε) λ ∞ , where V (∞) := µ(Ω). (2) if µ(Ω) = ∞, then there is a constant C such that V (r) ≥ C exp (2 -ε) λ ∞ .
Furthermore, if the spectrum of L below λ ∞ consists of a finite number of eigenvalues, each with finite multiplicity, then we can take ε = 0 in the above inequalities.

To conclude this general subsection, we present a result about the discrete spectrum case, related to Lemma 1.10. Let P be of the form (2.1) qith quadratic form q, and let W be a positive potential. Define D 1,2 (Ω, P, W ) to be the completion of C ∞ 0 (Ω) for the norm

q(u) 2 + Ω |u| 2 W dν 1/2 .
Theorem 5.6. Assume that for some λ > 0, the Hardy inequality (1.1) holds. Let V be a positive potential such that

lim x→∞ V (x) W (x) = 0.
Then λ ∞ (Ω, P, V ) = +∞. In particular, the spectrum of V -1 P consists of an increasing sequence of eigenvalues, tending to +∞, and if λ ∈ R does not belong to the spectrum of 1 W P , then the resolvent ( 1 W P -λ) -1 is compact. If moreover the Agmon metric

|ξ| 2 Ag := W A -1 ξ, ξ is complete and V ∈ L ∞ loc (Ω), then D 1,2 (Ω, P, V ) injects compactly into L 2 (Ω, V dν).
Proof. Let us prove the first part of the theorem. Let ε > 0. Then there is a compact set K such that for every

x ∈ Ω \ K, V (x) W (x) ≤ ε.
Therefore, using the fact that the Hardy inequality (1.1) is satisfied by assumption, we obtain that for every u

∈ C ∞ 0 (Ω \ K), λε -1 Ω\K V u 2 dν ≤ λ Ω\K W u 2 dν ≤ q(u).
Therefore,

λ ∞ (Ω, P, W ) ≥ λε -1 .
Letting ε → 0, we get that λ ∞ (Ω, P, W ) = +∞.

This concludes the proof of the first part of the theorem.

Let us now prove the second part. First, using a h-transform, we can assume that P 1 = 0. We denote in all the proof D 1,2 (Ω, P, W ) = D 1,2 . Fix a point x 0 ∈ Ω, and let ε > 0. By hypothesis, there exists a compact set K 0 such that for every

x ∈ Ω \ K 0 , V (x) W (x) ≤ ε.
Let K compact set of Ω, containing K 0 in its interior. As a consequence of the completeness of the Agmon metric, there is a constant C (indepedant of ε) such that for every R > 0 big enough, there exists a C ∞ 0 cut-off function ϕ R equal to 1 on B Ag (x 0 , R) \ K, zero outside B Ag (x 0 , R + 1), and zero in

K 0 , such that ||ϕ R || ∞ + |∇ϕ R | A W ≤ C.
Let (w n ) n∈N be a bounded sequence in D 1,2 (Ω, P, W ). Up to a subsequence, one can assume that (w n ) n∈N converges weakly in D 1,2 (Ω, P, W ), and by substracting the limit, one can assume that (w n ) n∈N converges weakly to zero in D 1,2 (Ω, P, W ). Let us compute

Ω (A∇(ϕ R w n ) • ∇(ϕ R w n )) = Ω (A∇w n • ∇w n )ϕ 2 R +2 Ω (A∇w n • ∇ϕ R )w n ϕ R + Ω (A∇ϕ R • ∇ϕ R )w 2 n ≤ C Ω (A∇w n • ∇w n ) + Ω (A∇ϕ R • ∇ϕ R )w 2 n +2C Ω (A∇w n • ∇w n ) 1/2 (A∇ϕ R • ∇ϕ R ) 1/2 w n ≤ C Ω (A∇w n • ∇w n ) + C Ω W w 2 n +2C(A∇w n • ∇w n ) 1/2 W 1/2 w n ≤ C Ω (A∇w n • ∇w n ) +2C Ω (A∇w n • ∇w n ) 1/2 Ω W w 2 n 1/2 ≤ C Ω (A∇w n • ∇w n ) ≤ C,
where C is independent of R. Here, we have successively used that (A∇ϕ R , ∇ϕ R ) ≤ CW , and the Hardy inequality (1.1), satisfied by hypothesis. Also, using the hypothesis on V , the definition of ϕ R and the Hardy inequality (1.1),

Ω\K 0 V (ϕ R w n ) 2 ≤ ε Ω\K 0 W (ϕ R w n ) 2 ≤ εq(ϕ R w n ) = ε Ω (A∇(ϕ R w n ) • ∇(ϕ R w n ))
(recall that we have assumed that P 1 = 0). Therefore,

Ω\K 0 V (ϕ R w n ) 2 ≤ εC. Since C is independent of R, we get, letting R → ∞: Ω\K V w 2 n ≤ εC.
By local uniform ellipticity of P and the Rellich theorem, (w n

) n∈N converges to zero in L 2 loc (Ω). Since V ∈ L ∞ loc (Ω), lim n→∞ K V w 2 n = 0. Finally, we obtain that lim sup n→∞ Ω V w 2 n ≤ εC.
(3) µ has linear volume growth:

µ(B Ag (x 0 , R)) ≍ R, ∀R ≥ 1
, where we recall that B Ag (x 0 , R) is the geodesic ball of center x 0 and radius R in the Agmon metric dx 2 4δ 2 . In the rest of this subsection, we will show that properties (1), ( 2 In the rest of the proof, we will denote W (u 0 , u 1 ) by W , and we let V i := P u i u i . As in section 5.1, we successively perform an h-transform and a change of measure. But this time, the h-transform is performed with respect to an approximate solution of P -W , and not to a solution: we define two operators L i := 1 W u -1 i P u i , where i ∈ {0; 1}, where L i is self-adjoint on L 2 (Ω, u 2 i W dν). By formulae (2.3) and (2.4), we have

L i = -div i A W • + V i W ,
where the divergence div i is for the measure µ i . Denote by Li the operator Li := -div i A W

• .

Under the assumptions on V i made, an argument involving Persson's formula (see the proof of Proposition 2.11) shows that for i = 1, 2, the bottom of the essential spectrum of L i is equal to the bottom of the essential spectrum of Li . But since L i is unitarily equivalent to 1 W P , its bottom of the spectrum is λ ∞ , and therefore the bottom of the spectrum of Li is λ ∞ . We can now apply Theorem 5.2 to Li to get the result.

We now turn to the reverse inequalities for σ 0 , σ 1 . This, as we shall see, requires conditions on the growth of the measure µ = u 0 u 1 W (u 0 , u 1 ) ν. These conditions generalize Property (3) (linear growth of µ) of Example 5.7, and are linked with the condition (4) appearing in Proposition 2.11. Denote by χ the push-forward of the metric µ by X -1

1 u 0 u 1 : χ := X -1 1 u 0 u 1 ⋆ µ.
The measure χ is the measure which appears in the condition (4) of Proposition 2.11.

Definition 5.9. We will say that µ has at most ε-exponential growth and decay if there exists a constant C > 0 such that for every t > 0,

C -1 e -εt ≤ χ(t) ≤ C e εt Remark 5.10. The inequality χ(t) ≤ C e εt is related -but not equivalent -to subexponential growth for χ (that is, is related to the condition (4) in Proposition 2.11). For example, µ = dx 4δ of Example 5.7, has ε-exponential growth and decay, for every ε > 0.

With this definition, we show the following result: Proposition 5.11. Assume that u 0 u 1 is bounded from above, that lim x→∞ u 0 (x) u 1 (x) = 0, and that for some 0 < ε < 1, µ has at most ε-exponential growth and decay. Then µ 0 (Ω) < ∞, µ 1 (Ω) = ∞, and for i = 0, 1, the reverse inequality

σ 2 i 4 ≤ 1 holds.
As a direct consequence of Proposition 5.8 and Proposition 5.11, we get the following corollary: Corollary 5.12. Assume that u 0 u 1 is bounded from above, that assumptions of Proposition 2.11 are satisfied, that there is 0 < ε < 1 and some constant µ 0 (Ω) = µ 0 1 ≤ X -1

1 u 0 u 1 < ∞ = ∞ 1 ψ(t)dχ(t),
and

µ 1 (Ω) = µ 1 1 ≤ X -1 1 u 0 u 1 < ∞ = ∞ 1 dχ(t) ψ(t) .
Since ψ(t) ∼ e -t as t → ∞, µ 0 (Ω) < ∞ (resp. µ 1 (Ω) = ∞) is equivalent to ∞ 1 e -t dχ(t) < ∞ (resp.

∞ 1 e t dχ(t) = ∞). But by the integration by part formula, valid for Stieljes measures, This proves that µ 0 (Ω) < ∞. For µ 1 (Ω), we have again by integration by parts which yields µ 1 (Ω) = ∞. Now we turn to the estimates on σ 0 and σ 1 . Since the Agmon metric is given by formula (5.5) with h = X -1 1 u 0 u 1 , we see that in the definition of σ i we can replace the ball B(x 0 , r) in the Agmon metric by the set 2 ≤ X -1 1 u 0 u 1 ≤ 2r . Thus, using the change of variable formula (2.8), and the fact that ψ(t) ∼ e -t when t → ∞, we see that But using as above the integration by parts formula for Stieljes measures and the the hypothesis on χ, we see that there is a constant c > 0 such that for r > 0 big enough, This implies at once that σ i ≤ 2, which concludes the proof.

σ 0 = lim
5.3. Volume growth for the improved Hardy inequalities on a convex set. In this subsection, we show how the general theory developped in subsection 5.2 applies to the particular example of the improved Hardy inequalities on a bounded domain Ω of R n . Fix i ≥ 0, and define

P := -∆ -W i-1
(W -1 = 0 by convention), where W i is the weight

W i := 1 4δ 2 i k=0 X 2 0 δ D • • • X 2 k δ D ,
(W -1 = 0 by convention). Recall also the definition of

J i := W i -W i-1 = 1 4δ 2 X 2 0 δ D • • • X 2 i δ D .
From Section 3, recall the definition of U 0,j and U 1,j , and define u 0 := U 0,i-1 ,

u 1 := U 1,i-1 .
Define as in subsection 5.2, for k = 0, 1 µ i,k := u 2 k J i dx. Define also the associated volume growth rate σ i,k , for k = 0, 1. Then as a consequence of Corollary 5.12, we have Theorem 5.13. For every i ≥ 0, the measure µ i,0 (resp. µ i,1 ) has finite (resp. infinite) mass, and the convergence of volumes is exponential: for k = 0, 1,

σ 2 i,k 4 = 1.

2. 3 .

 3 Another spectral result. We first introduce some definitions and notations. Denote by C the set of positive measures on [0, ∞) which are absolutely continuous with respect to the Lebesgue measure, and have infinite mass. For χ ∈ C and r ≥ 0, denote V (r) := σ, the exponential rate of volume growth of χ by σ(χ) := lim r→∞ sup 1 r log V (r).

Proposition 2 . 8 .

 28 Assume that (outside a compact set) there exists a function v ∈ C 1,α loc , α ∈ (0, 1) such that, for some positive constant C, lim x→∞ |∇v(x)| A → C, and such that lim x→∞ v(x) = +∞ (2.9) and lim x→∞ Lv(x) = 0 (pointwise).

2 2r 2 4 . 1 . 2 2 2 , 2 2

 2241222 is the best constant in the Hardy inequality (4.1). An interesting question is to what extent the weight W := has the "best behavior" at infinity in M . We show that Theorem Let us assume that the total curvature M |II| n/2 is finite. Then the operator r 2 (-∆) has spectrum [ n-∞) (here, the Laplacian is on M ). In particular, n-2 is the best constant at infinity in the Hardy inequality (4.1).

Remark 5 . 4 .

 54 If P = -div(A∇•) and W = |∇h| 2

2 : 5 . 8 . 1 W

 2581 ) of Example 5.7 hold in more general situations. Let us begin to show an analogue of Theorem 5.Proposition Assume that W (u 0 , u 1 ) is positive, that lim x→∞ u 0 (x) u 1 (x) = 0,and that for i = 0, 1, lim x→∞ (u 0 , u 1 )P u i u i = 0.Then for i = 1, 2, the inequality The completeness of the Agmon metric follow from Remark 5

∞ 1 e 1 e 1 e

 111 -t dχ(t) = e -t χ(t) -t χ(t)dt, and given the hypothesis on χ, lim t→∞ e -t χ(t) = 0, and ∞ -t χ(t)dt < ∞.

∞ 1 e 1 e 1 e

 111 t dχ(t) = e t χ(t) t χ(t)dt, and given the hypothesis on χ,∞ t χ(t)dt = ∞,

2r 1 e

 1 t dχ(t) ≤ c e (2-ε)r and ∞ 2r e -t dχ(t) ≥ c -1 e -(2-ε)r .

The author wishes to thank Y. Pinchover for many interesting discussions and inspiring questions, and A. Tertikas, G. Carron for useful comments. He also thank E. Zuazua for a question about multipolar Hardy inequalities, and the referee for his careful reading of the manuscript. The author acknowledges the support of the Israel Science Foundation (grant No. 963/11) founded by the Israel Academy of Sciences and Humanities, and the Technion.

(1) lim x→∞ u 0 (x) u 1 (x) = 0.

(2) When x → ∞, 1 W (u 0 , u 1 )

u 0 (x) u 1 (x) .

(3) When x → ∞, 1 W (u 0 , u 1 )

(4) The push-forward measure X -1

(u 0 u 1 W dν) is in C and has subexponential volume growth.

Summarizing the results of this section, we obtain one of the main results of this paper (as a consequence of Propositions 2.11 and 2.14): Theorem 2.17. Let P be a symmetric, second-order elliptic operator of the form (2.1). Let (u 0 , u 1 ) be a pair of optimal approximate solutions at

. Then the essential spectrum of the operator

, and furthermore the spectrum below 1 consists at most of a finite number of eigenvalues with finite multiplicity.

Bounded domains of R n

In this section, Ω will be a general (not necessarily mean convex, otherwise stated) C 2 bounded domain of R n , and δ is the distance to the boundary of Ω.

In this section, we study the spectrum of the operator J -1 i (∆ -W i-1 ), associated with the Hardy inequality (1.6) obtained in [START_REF] Barbatis | Series expansion for L p Hardy inequalities Indiana[END_REF]. We will make use of the following properties of the function δ (see [START_REF] Marcus | On the best constant for Hardy's inequality in R n[END_REF] and [START_REF] Psaradakis | L 1 Hardy Inequalities with Weights[END_REF], and references therein):

The equivalence between mean-convexity and non-negativity of -∆δ ≥ 0 goes back to Gromov, and was established for the first time in [START_REF] Psaradakis | L 1 Hardy Inequalities with Weights[END_REF].

Let us recall the definition of the functions X i : we set by convention

and the proof of the second part of the theorem is complete.

5.2. The case of approximate solutions and the role of subexponential growth. In this subsection, we investigate what happens for a Hardy inequality with a weight obtained by the supersolution construction of [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]. We let u 0 and u 1 be positive supersolutions of P , and we recall the notation

. By Lemma 5.1 in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF], the following Hardy inequality takes place for λ = 1:

where q is the quadratic form of P , and define λ 0 and λ ∞ to be respectively the best constant and the best constant at infinity in (5.6). Define also the measures µ i for i = 0, 1 by

ν, and the measure

where we recall that ν is the underlying measure. Define the Agmon metric

and let ρ be the distance for the Agmon metric. By Remark 5.4, ρ is given by formula (5.5) with h = 1 2 X -1 1 u 0 u 1 . Define also σ i for i = 1, 2, being the exponential rate of volume growth of σ i , as in Definition 5.4 with µ replaced by µ i . Let us first consider an example, which introduces the results that we want to present in this subsection: Example 5.7. Let Ω be a bounded, smooth domain, with

Moreover, elementary computations show that:

(1) µ 0 has finite volume, µ 1 has infinite volume.

(

C such that µ has at most ε-exponential growth and decay, and moreover that for i = 0, 1,

Then µ 0 (Ω) < ∞, µ 1 (Ω) = ∞, the Agmon metric (5.7) is complete and for i = 0, 1,

Proof. By normalization of u 0 and u 1 , we will assume without loss of generality that

We will denote W := W (u 0 , u 1 ) and V i := P u i u i . Let us also denote by ψ the inverse function of X -1

1 . Since

Thus, using the change of variable formula (2.8), we see that

and

By the hypothesis that lim x→∞ u 0 u 1 = 0, we have

Proof. Denote dχ the push-forward measure of u 0 u 1 J i dx by X -1 1 u 0 u 1 , then the computations done in the proof of Theorem 3.5 show that χ has linear growth. Also, for k = 0, 1, denote

Then by Proposition 3.2, in a neighborhood of ∂Ω we have

and

It is immediate to check that for k = 0, 1,

Thus we can apply Corollary 5.12 to µ i,k , which gives the result.

Appendix

Here, we give a proof of the fact that the series

converges for every t ∈ [0, 1). We thank A. Tertikas for having provided us with the proof. For every ϕ defined on the unit ball B 1 , we have the following Hardy inequality (as a simple consequence of Allegretto-Piepenbrink theory, or by direct integration by parts)

We make the choice

(|x|), and compute (see Lemma 6.3 in [START_REF] Filippas | Optimizing improved Hardy inequalities[END_REF])

Applying it for n ≥ 3, we conclude the the convergence of the required series for t ∈ (0, 1).

Acknowledgments