Martin Costabel
email: martin.costabel@univ-rennes1.fr

AND Monique Dauge
email: monique.dauge@univ-rennes1.fr

Serge Nicaise
email: snicaise@univ-valenciennes.fr

MELLIN ANALYSIS OF WEIGHTED SOBOLEV SPACES WITH NONHOMOGENEOUS NORMS ON CONES

On domains with conical points, weighted Sobolev spaces with powers of the distance to the conical points as weights form a classical framework for describing the regularity of solutions of elliptic boundary value problems, cf. papers by Kondrat'ev and Maz'ya-Plamenevskii. Two classes of weighted norms are usually considered: Homogeneous norms, where the weight exponent varies with the order of derivatives, and nonhomogeneous norms, where the same weight is used for all orders of derivatives. For the analysis of the spaces with homogeneous norms, Mellin transformation is a classical tool. In this paper, we show how Mellin transformation can also be used to give an optimal characterization of the structure of weighted Sobolev spaces with nonhomogeneous norms on finite cones in the case of both non-critical and critical indices. This characterization can serve as a basis for the proof of regularity and Fredholm theorems in such weighted Sobolev spaces on domains with conical points, even in the case of critical indices.

When analyzing elliptic regularity in the neighborhood of a conical point on the boundary of an otherwise smooth domain, one is faced with a dilemma:

Near the singular point, the conical geometry suggests the use of estimates in weighted Sobolev spaces with homogeneous norms, and a well-known tool for analyzing them and for obtaining the estimates is the Mellin transformation. This analysis is carried out in the classical paper [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF] by Kondrat'ev.

On the other hand, since this analysis corresponds to a blow-up of the corner, that is a diffeomorphism between the tangent cone and an infinite cylinder, the conical point moves to infinity, and therefore functions in this class of spaces always have trivial Taylor expansions at the corner. Depending on the weight index, they either have no controlled behavior at the corner at all or they tend to zero. If one wants to study inhomogeneous boundary value problems, then smooth right hand sides and the corresponding solutions will require spaces that allow the description of non-trivial Taylor expansions at corner points.

Appropriate spaces have been analyzed using tools from real analysis by Maz'ya and Plamenevskii [START_REF] Maz'ya | Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points[END_REF]. Such spaces can be defined by nonhomogeneous weighted norms, where the weight exponent is the same for all derivatives. The simplest examples are ordinary, non-weighted Sobolev norms. As presented in detail in the book [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF] by Kozlov, Maz'ya and Rossmann, the analysis of these spaces with nonhomogeneous norms shows several peculiarities:

1. For a given space dimension n and Sobolev order m, there is a finite set of exceptional, "critical" weight exponents β, characterized in our notation by the condition

-β -n 2 = η ∈ N ; 0 η m -1 ,
such that for the non-critical case, the space with nonhomogeneous norm splits into the direct sum of a space with homogeneous norm and a space of polynomials, corresponding to the Taylor expansion at the corner. In the critical case, the splitting involves an infinitedimensional space of generalized polynomials. The study of the critical cases is of practical importance, because for example in two-dimensional domains, the ordinary Sobolev spaces with integer order are all in the critical case η = m -1.

2. The relation of the spaces with nonhomogeneous norms with respect to Taylor expansions at the corner is somewhat complicated, depending on the weight and the order: For η < 0, the space with nonhomogeneous norm coincides with the corresponding space with homogeneous norm and contains all polynomials, but has no controlled Taylor expansion. For 0 η < m, the nonhomogeneous norm still allows all polynomials and controls the Taylor expansion of order [η] at the corner. If η m, then the space with nonhomogeneous norm again coincides with the corresponding space with homogeneous norm and has vanishing Taylor expansion of order m -1. Thus there are two (non-disjoint) classes of spaces involved, and the weighted Sobolev spaces with nonhomogeneous norms fall into one or the other of these classes, namely the class of spaces with homogeneous norms on one hand and a class of spaces with weighted norms and nontrivial Taylor expansion on the other hand.

3. Whereas the definition of the non-homogeneous norms is simple, it turns out that for the analysis of the spaces one also needs descriptions by more complicated equivalent norms, where the weight exponent does depend, in a specific way, on the order of the derivatives. Such "step-weighted" Sobolev spaces have been studied by Nazarov [START_REF] Nazarov | Vishik-Lyusternik method for elliptic boundary value problems in regions with conical points. I. The problem in a cone[END_REF][START_REF] Nazarov | The Neumann problem for selfadjoint elliptic systems in a domain with a piecewise-smooth boundary[END_REF].

In [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF], the analysis of the weighted Sobolev spaces with nonhomogeneous norms is presented using real-variable tools, in particular techniques based on Hardy's inequality.

In this paper, we present an analysis of the spaces with nonhomogeneous norms based on Mellin transformation. We show how the three points described above can be achieved in an optimal way. In particular, 1. we characterize the spaces with nonhomogeneous norms via Mellin transformation in the non-critical and in the critical case;

2. we give a natural definition via Mellin transforms of the second class of spaces mentioned in point 2 above, namely the spaces with weighted norms and nontrivial Taylor expansions;

3. we show how the question of equivalent norms can be solved via Mellin transformation.

The analysis in this paper is a generalization of the Mellin characterization of standard Sobolev spaces that was introduced in [START_REF] Dauge | Elliptic Boundary Value Problems in Corner Domains -Smoothness and Asymptotics of Solutions[END_REF] for the analysis of elliptic regularity on domains with corners. For the case of critical weight exponents, we give a Mellin description of the generalized Taylor expansion that was introduced and analyzed with real-variable techniques in [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]. Based on our Mellin characterization, one can obtain Fredholm theorems and elliptic regularity results, in particular analytic regularity results, on domains with conical points. This is developed in the forthcoming work [START_REF] Costabel | Corner Singularities and Analytic Regularity for Linear Elliptic Systems In preparation[END_REF].

NOTATION: WEIGHTED SOBOLEV SPACES ON

CONES A regular cone K ⊂ R n , n 2 is an unbounded open set of the form (2.1) K = x ∈ R n \ {0} : x |x| ∈ G ,
where G is a smooth domain of the unit sphere S n-1 called the solid angle of K. Note that if n = 2, this implies that K has a Lipschitz boundary (excluding domains with cracks), which is not necessarily the case if n 3. Note further that our analysis below is also valid in the case of domains with cracks.

The finite cone S associated with K is simply (2.2) S = K ∩ B(0, 1).

In the one-dimensional case, we consider K = R + and S = (0, 1), which corresponds to

G = {1}. For k ∈ N, • k;O denotes the standard Sobolev norm of H k (O).
2.1. Weighted spaces with homogeneous norms. The spaces on which relies a large part of our analysis are the "classical" weighted spaces of Kondrat'ev. The "originality" of our definition is a new convention for their notation.

Definition 2.1. ⋆ Let β be a real number and let m 0 be an integer. ⋆ β is called the weight exponent and m the Sobolev exponent. ⋆ The weighted space with homogeneous norm K m β (K) is defined by

(2.3) K m β (K) = u ∈ L 2 loc (K) : r β+|α| ∂ α x u ∈ L 2 (K)
, ∀α, |α| m and endowed with semi-norm and norm respectively defined as (2.4)

|u| 2 K m β (K) = |α|=m r β+|α| ∂ α x u 2 0;K and u 2 K m β (K) = m k=0 |u| 2 K k β (K) .
The weighted spaces introduced by Kondrat'ev in [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF] are denoted by

• W m α (K). The correspondence with our notation is • W m α (K) = K m α 2 -m (K) i.e. K m β (K) = 0 W m 2β+2m (K).
These spaces are also of constant use in related works by Kozlov, Maz'ya, Nazarov, Plamenevskii, Rossmann, see the monographs [START_REF] Nazarov | Elliptic Problems in Domains with Piecewise Smooth Boundaries[END_REF][START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF][START_REF] Kozlov | Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations[END_REF] for example. They are denoted by V m β (K) with the following correspondence with our spaces

V m β (K) = K m β-m (K) i.e. K m β (K) = V m β+m (K).
We choose the convention in (2.3) because it simplifies some statements: An obvious, but fundamental property of the scale K β is its monotonicity with respect to m

K m+1 β (K) ⊂ K m β (K), m ∈ N.
This allows a simple definition of C ∞ and analytic functions with weight, see Definition 4.1. Also, in mapping properties of differential operators with constant coefficients, as well as in elliptic regularity theorems ("shift theorem"), the shift in the weight exponent β is independent of the regularity parameter m, in contrast to what happens with the Kondrat'ev or the V m β spaces. The space K m β (S) with its semi-norm

| • | K m β (S) and norm • K m β (S)
is defined similarly by replacing K by S.

2.2.

Weighted spaces with nonhomogeneous norms. Definition 2.2.

⋆ Let β be a real number and m 0 be an integer. ⋆ The weighted space with non-homogeneous norm J m β (S) is defined by (2.5)

J m β (S) = u ∈ L 2 loc (S) : r β+m ∂ α x u ∈ L 2 (S), ∀α, |α| m with its norm u 2 J m β (S) = |α| m r β+m ∂ α x u 2 0;S .
The semi-norm of J m β (S) coincides with the semi-norm of K m β (S):

(2.6) |u| 2 J m β (S) = |u| 2 K m β (S) = |α|=m r β+|α| ∂ α x u 2 0;S .
⋆ The space J m β (K) with its norm and semi-norm is defined in the same way. Our space J m β (S) is the same as the space denoted by W m 2,β+m (S) in [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]. The following properties are obvious consequences of the definitions:

Lemma 2.3.
a) For all β < β ′ we have the embedding

J m β (S) ⊂ J m β ′ (S). b) We have the embeddings for all β ∈ R and m ∈ N (2.7) K m β (S) ⊂ J m β (S) ⊂ K m β+m (S). c) Let α ∈ N n be a multi-index of length |α| = k m. Then the partial differential operator ∂ α x is continuous from J m β (S) into J m-k β+k (S)
. In contrast to the scale K m β , we do not necessarily have the inclusion of the space J m β (S) in J m-1 β (S). We will see (Corollary 3.19) that such an inclusion does hold when m is large enough, which allows the definition of J ∞ β (S) and of the corresponding analytic class. A remarkable and unusual property of the spaces J m β (S) is that we do not, in general, obtain an equivalent norm for J m β (S) if we retain in (2.5) only the semi-norm (|α| = m) and the L 2 norm (|α| = 0). A counter-example for such an equivalence is obtained with the following choice

(2.8) m 2, m < η = -β -n 2 < m + 1, u = x 1 . Then the function r β+m ∂ α
x u is square integrable for |α| = 0 and for |α| 2, but not for α = (1, 0, . . . , 0). See §3.4 for further details.

We shall need more precise comparisons between the K and J spaces than the embeddings (2.7). As we will show later on, the space K m β (S) may be closed with finite codimension in J m β (S) (non-critical case), or not closed with infinite codimension (critical case). In the following lemma we compare the properties of inclusion of the space C ∞ (S) of smooth functions:

Lemma 2.4. Let β ∈ R and m ∈ N. Let η = -β -n 2 . a) The space C ∞ (S) is embedded in K m β (S) if and only if η < 0. b) The space C ∞ (S) is embedded in J m β (S)
if and only if η < m. Proof. Using polar coordinates and the Cauchy-Schwarz inequality, we see that

L ∞ (S) ⊂ L 2 β (S) ⇐⇒ β > -n 2 .
The sufficiency follows by using this for all derivatives of u ∈ C ∞ (S).

We find the necessity of the conditions on η by considering the constant function u = 1 in both cases.

Concerning spaces of finite regularity, it follows from the definition that the standard Sobolev space H m without weight coincides with J m -m . For the Sobolev spaces H m we have the embeddings corresponding to (2.7), namely (2.9) K m -m (S) ⊂ H m (S) ⊂ K m 0 (S). In addition we know from the Sobolev embedding theorem that if k is a non-negative integer such that k < m -n 2 , we have the embeddings

H m (S) ⊂ C k (S) ⊂ H k (S).
In particular, for elements of H m (S) all derivatives of length |α| k have a trace at the vertex 0. On the other hand, by density of smooth functions which are zero at the vertex, the elements of K m β (S), as soon as they have traces, have zero traces at the vertex. One can expect that the spaces J have vertex traces similar to the standard Sobolev spaces. The investigation of this question will be the key to the comparison between the J spaces and the K spaces.

Using the same simple argument as in the proof of Lemma 2.4, we find the conditions for the inclusion of polynomials in the weighted Sobolev spaces.

We denote by P M (S) the space of polynomial functions of degree M on S and by P M (S) the space of homogeneous polynomials of degree M.

Lemma 2.5. Let β ∈ R and m, k ∈ N. Let η = -β -n 2 . a) P k (S) ⊂ K m β (S) ⇐⇒ P 0 (S) ⊂ K m β (S) ⇐⇒ η < 0. b) P k (S) ⊂ J m β (S) ⇐⇒ P 0 (S) ⊂ J m β (S) ⇐⇒ η < m.
This complete similarity between the K spaces and the J spaces is no longer present if we refine the probe by considering the space of homogeneous polynomials. Still using the same simple argument based on finiteness of norms, we now get

Lemma 2.6. Let β ∈ R and m, k ∈ N. Let η = -β -n 2 . a) P k (S) ⊂ K m β (S) ⇐⇒ η < k. b) If k m, then P k (S) ⊂ J m β (S) ⇐⇒ η < k. c) If k m -1, then P k (S) ⊂ J m β (S) ⇐⇒ P 0 (S) ⊂ J m β (S) ⇐⇒ η < m.
As we will show in the following, the question of inclusion of polynomials completely characterizes the structure of the spaces J m β (S) and their corner behavior.

CHARACTERIZATIONS BY MELLIN TRANSFORMATION TECHNIQUES

The homogeneous weighted Sobolev norms can be expressed by Mellin transformation, which is the Fourier transformation associated with the group of dilations. We first recall this characterization from Kondrat'ev's classical work [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF]. Then we generalize it to include non-homogeneous weighted Sobolev norms, based on the observation that the non-homogeneous norms are defined by sums of homogeneous semi-norms.

3.1. Mellin characterization of spaces with homogeneous norms. In this section, we recall the basic results from [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF].

For a function u in C ∞ 0 ((0, ∞)) the Mellin transform M [u] is defined for any complex number λ by the integral

(3.1) M [u](λ) = ∞ 0 r -λ u(r) dr r .
The function λ → M [u](λ) is then holomorphic on the entire complex plane C. Note that M [u](λ) coincides with the Fourier-Laplace transform at iλ of the function t → u(e t ). Now any function u defined on our cone K can be naturally written in polar coordinates as R + × G ∋ (r, ϑ) -→ u(x) = u(rϑ). If u has a compact support which does not contain the vertex 0, the Mellin transform of u at λ ∈ C is the function

M [u](λ) : ϑ → M [u](λ, ϑ) defined on G by (3.2) M [u](λ, ϑ) = ∞ 0 r -λ u(rϑ) dr r , ϑ ∈ G.
If we define the function u on the cylinder R × G by u(t, ϑ) = u(e t ϑ), we see that the Mellin transform of u at λ is the partial Fourier-Laplace transform of u at -iλ.

Hence the Mellin transform of a function u ∈ C ∞ 0 (K) is holomorphic with values in C ∞ 0 (G). On the other hand, if u is simply in L 2 (K), the function e More generally, the Mellin transformation extends to functions u given in a weighted space K 0 β (K) with a fixed real number β: Since r β u belongs to L 2 (K), the function u in turn satisfies that e (β+ n 2 )t u belongs to

L 2 (R × G). Therefore λ → M [u](λ) defines an L 2 function on the line Re λ = -β -n 2 . If u belongs to K m β (K)
, then there appear parameterdependent H m norms for its Mellin transform, which motivates the following definition. Definition 3.1. Let G be the solid angle of a regular cone K. Let m ∈ N.

⋆ For λ ∈ C, the parameter-dependent H m norm on G is defined by

(3.3) U 2 m; G; λ = m k=0 |λ| 2m-2k U 2 k; G . ⋆ Let λ → U(λ) be a function with values in H m (G), defined for λ in a strip b 0 < Re λ < b 1 . Then for any b ∈ (b 0 , b 1 ), we set (3.4) N m G (U, b) = Re λ=b U(λ) 2 m; G; λ d Im λ 1 2
,

and N m G (U, [b 0 , b 1 ]) = sup b∈(b 0 ,b 1 ) N m G (U, b).
Later on, we will use the following observation:

Let λ → U(λ) be meromorphic for b 0 < Re λ < b 1 with values in H m (G). If N m G (U, [b 0 , b 1 ]) is finite, then U is actually holomorphic. In fact, if U has a pole in λ 0 , then N m G (U, b) is bounded from below by C |b -Re λ 0 | -1 .
As a consequence of the isomorphism between G (U, η). The inverse Mellin transform can be written as:

K m β (K) and H m β+ n 2 (R × G),
(3.5) u(x) = 1 2iπ Re λ=η r λ M [u](λ)(ϑ) dλ, x = rϑ.
From this Theorem, we see immediately that if u belongs to the intersection of two weighted spaces K m β (K) and K m β ′ (K) with β < β ′ , the Mellin transform of u is defined on two different lines in C. Since u belongs also to all intermediate spaces K m β ′′ (K) for β β ′′ β ′ , the Mellin transform is defined in a complex strip. In fact, the Mellin transform of u is holomorphic in this strip, and this characterizes the intersection of weighted spaces with different weights, as stated in the following theorem.

Theorem 3.3. Let β < β ′ two real numbers and m ∈ N. Let η := -β -n 2 and η ′ := -β ′ -n 2 . a) Let u ∈ K m β (K) ∩ K m β ′ (K). Then the Mellin transform U := M [u] of u is holo- morphic in the open strip η ′ < Re λ < η with values in H m (G)
and satisfies the following boundedness condition:

(3.6) N m G (U, [η ′ , η]) C u K m β (K) + u K m β ′ (K) . b) Let U be a holomorphic function in the open strip η ′ < Re λ < η with values in H m (G), satisfying N m G (U, [η ′ , η]) < ∞.
Then the mapping

(3.7) b -→ (ξ, ϑ) → U(b + iξ, ϑ)
has limits as b → η and b → η ′ , and the inverse Mellin transforms

(3.8) u ′ = 1 2iπ Re λ=η ′ r λ U(λ) dλ and u = 1 2iπ Re λ=η r λ U(λ) dλ,
coincide with each other and define an element of

K m β (K) ∩ K m β ′ (K).
In the following theorem, we recall the close relation between asymptotic expansions and meromorphic Mellin transforms.

Theorem 3.4. Let β < β ′ be two real numbers and set

η = -β -n 2 and η ′ = -β ′ -n
2 . Let λ 0 be a complex number such that η ′ < Re λ 0 < η. Let q be a non-negative integer and ϕ 0 , . . ., ϕ q be fixed elements of L 2 (G).

a) Let u ′ ∈ K 0 β ′ (K) such that the identity (3.9) u(x) = u ′ (x) + r λ 0 q j=0 1 j! log j r ϕ j (ϑ)
defines a function u in K 0 β (K). Then the Mellin transform U of u ′ , defined for Re λ = η ′ , has a meromorphic extension to the strip η ′ < Re λ < η such that the function V defined as

(3.10) V (λ) := U(λ) - q j=0 ϕ j (λ -λ 0 ) j+1 is holomorphic in η ′ < Re λ < η with values in L 2 (G) and satisfies the bounded- ness condition (3.11) N 0 G (V, [η ′ , η]) < ∞.
b) Conversely, let U be a meromorphic function with values in L 2 (G), such that V defined by (3.10) is holomorphic in the strip η ′ < Re λ < η and satisfies the boundedness condition (3.11). Then, like in the holomorphic case, the mapping (3.7) has limits at η and η ′ , and the inverse Mellin formulas

(3.8) define u ∈ K 0 β (K) and u ′ ∈ K 0 β ′ (K).
They satisfy the relation (3.9), which can be also written in the form of a residue formula:

(3.12) u(x) -u ′ (x) = 1 2iπ C r λ U(λ) dλ,
for a contour C surrounding λ 0 and contained in the strip η ′ < Re λ < η.

Mellin characterization of semi-norms.

The principle of our Mellin analysis is to apply to a function u and some of its derivatives ∂ α x u the Mellin characterization of Kweighted spaces from Theorems 3.2, 3.3, and 3.4. Definition 3.5.

⋆ For any α ∈ N n , we denote by D α the differential operator in polar coordinates satisfying

(3.13) r |α| ∂ α x = D α (ϑ; r∂ r , ∂ ϑ ). ⋆ For any m ∈ N and λ ∈ C, let the parameter dependent semi-norm | • | m; G; D(λ) be defined on H m (G) by (3.14) |V | 2 m; G; D(λ) = |α|=m D α (ϑ; λ, ∂ ϑ )V 2 0; G . Lemma 3.6. Let β < β 0 be two real numbers. We set η = -β -n 2 and η 0 = -β 0 -n 2 . Let m ∈ N. Let u ∈ K 0 β 0 (K)
with support in B(0, 1) such that its K m β (K) semi-norm is finite. Then the Mellin transform of u is holomorphic for Re λ < η 0 and has a meromorphic extension U to the half-plane Re λ < η. Its poles are contained in the set of integers {0, . . . , m -1} ∩ (η 0 , η) and U satisfies the estimates, with two constants c, C > 0 independent of u

(3.15) c |u| K m β (K) sup b∈(η 0 ,η) Re λ=b |U(λ)| 2 m; G; D(λ) d Im λ 1 2 C |u| K m β (K) .
Proof. As u ∈ K 0 β 0 (K), by Theorem 3.2 its Mellin transform λ → M [u](λ) is defined for all λ on the line Re λ = η 0 . We set

v m := r m ∂ m r u and w α := r m ∂ α x u, |α| = m.
By assumption, the functions w α for |α| = m all belong to K 0 β (K). Using the identity

r k ∂ k r = |β|=k k! β! x β ∂ β x , we obtain v m = |α|=m m! α! ϑ α w α , with ϑ α = x α r m , hence v m belongs to K 0 β (K) too. Therefore the Mellin transforms λ → M [v m ](λ) and λ → M [w α ](λ)
are defined for all λ on the line Re λ = η, and we have the estimates

(3.16a) c |u| 2 K m β (K) Re λ=η M [v m ](λ) 2 0; G + |α|=m M [w α ](λ) 2 0; G d Im λ and (3.16b) Re λ=η M [v m ](λ) 2 0; G + |α|=m M [w α ](λ) 2 0; G d Im λ C |u| 2 K m β (K) .
Since u, and thus v m and w α , have compact support, their Mellin transforms extend holomorphically to the half-planes Re λ < η 0 for u, and Re λ < η for v m and w α . Moreover, due to the condition of support, estimate (3.16b) holds with the same constant C if we replace the integral over the line Re λ = η with the integral over any line Re λ = b, η 0 < b < η:

(3.17) sup b∈(η 0 ,η) Re λ=b M [v m ](λ) 2 0; G + |α|=m M [w α ](λ) 2 0; G d Im λ C|u| 2 K m β (K) .
Using the identity

(3.18) r m ∂ m r = r∂ r (r∂ r -1) • • • (r∂ r -m + 1)
, we find for all λ, Re λ η 0 , the following relation between Mellin transforms:

M [v m ](λ) = λ(λ -1) • • • (λ -m + 1)M [u](λ).
Hence we define a meromorphic extension U of M [u] by setting

(3.19) U(λ) = M [v m ](λ) λ(λ -1) • • • (λ -m + 1)
for Re λ η.

Since M [w α ](λ) = D α (ϑ; λ, ∂ ϑ )M [u](λ)
for Re λ η 0 , by meromorphic extension we find that

(3.20) M [w α ](λ) = D α (ϑ; λ, ∂ ϑ )U(λ), for Re λ η.
Putting (3.16a), (3.17 In Theorem 3.12 below we will see that under the conditions of the Lemma, the poles of the Mellin transform of u are associated with polynomials, corresponding to the Taylor expansion of u at the origin.

If λ is not an integer in the interval [0, m -1], the semi-norm |V | m; G; D(λ) defines a norm on H m (G) equivalent to the parameter dependent norm V m; G; λ introduced in Definition 3.1. In order to describe this equivalence in a neighborhood of integers, we need to introduce a projection operator on polynomial traces on G:

Definition 3.7. Let k ∈ N.
⋆ By P k (G) we denote the space of restrictions to G of homogeneous polynomial functions of degree k on K. ⋆ Let ϕ k γ |γ|=k be the basis in

P k (G) dual in L 2 (G) of the homogeneous monomials ϑ α /α! |α|=k (ϑ = x |x| ), i.e.: (3.21) G ϑ α α! ϕ k γ (ϑ) dϑ = δ αγ , |α| = |γ| = k.
By P k we denote the projection operator L 2 (G) → P k (G) defined as

(3.22) P k U = |α|=k U, ϕ k α G ϑ α α! .
Lemma 3.8. Let m ∈ N and η 0 , η real numbers such that η 0 < 0 m < η. Let δ ∈ (0, 1 2 ). Then there exist two constants C, c > 0 such that for all V ∈ H m (G) the following estimates hold: a) For λ satisfying Re λ ∈ [η 0 , η] and |λ -k| δ for all k ∈ {0, . . . , m -1}:

(3.23) c|V | m; G; D(λ) V m; G; λ C|V | m; G; D(λ) .
b) For λ satisfying |λ -k| δ for a k ∈ {0, . . . , m -1}:

(3.24) c|V | m; G; D(λ) V -P k V m; G + |λ -k| P k V m; G C|V | m; G; D(λ) .
Proof. Let A be an annulus of the form {x ∈ K, 1 R < |x| < R} for a R > 1. It is not hard to see that one has the following equivalence of the norm • m; G; λ and seminorm | • | m; G; D(λ) with the norm and semi-norm of H m (A) on its closed subspace S λ m (A) of homogeneous functions of the form r λ V (ϑ):

c r λ V m; A V m; G; λ C r λ V m; A c |r λ V | m; A |V | m; G; D(λ) C |r λ V | m; A .
Here the equivalence constants can be chosen uniformly for λ in the whole strip η 0 Re λ η.

• The well-known Bramble-Hilbert lemma implies that the semi-norm

| • | m; A is equiv- alent to the norm • m; A on S λ m (A) if and only if S λ m (A)
does not contain any non-zero polynomial of degree m -1: Thus for all λ ∈ {0, . . . , m -1} there exists C λ such that

r λ V m; A C λ |r λ V | m; A ,
and C λ can be chosen uniformly on the set Re λ ∈ [η 0 , η] with |λ -k| δ for all k ∈ {0, . . . , m -1}; whence estimates (3.23) in case a) of the lemma.

• Let λ be such that |λ -k| δ for a k ∈ {0, . . . , m -1}. The left inequality in (3.24) is easy to prove with the help of the estimate

(1) |P k V | m; G; D(λ) C|λ -k| P k V m; G , which is follows from |P k V | m; G; D(k) = 0.
Concerning the right-hand-side estimate of (3.24), the Bramble-Hilbert lemma argument implies the equivalence of the semi-norm with the norm for functions V such that

P k V = 0: For all V ∈ H m (G) there holds (2) V -P k V m; G C|V -P k V | m; G; D(λ) .
On the other hand, the operator r m ∂ m r is a linear combination of the operators r m ∂ α x , |α| = m, with coefficients bounded on G. Therefore

r m ∂ m r (r λ V ) 0; A C|r λ V | m; A . From (3.18) we get |λ(λ -1) • • • (λ -m + 1)| V 0; G C r m ∂ m r (r λ V ) 0; A , hence |λ -k| V 0; G C r m ∂ m r (r λ V ) 0; A C ′ |V | m; G; D(λ) .
From the continuity of P k in L 2 (G), we deduce

|λ -k| P k V 0; G C|V | m; G; D(λ) .
The equivalence of norms in L 2 (G) and H m (G) on the finite dimensional range of P k yields finally

(3) |λ -k| P k V m; G C|V | m; G; D(λ) .
It remains to bound V -P k V m; G . Using (1), ( 2) and ( 3) we find

V -P k V m; G C |V -P k V | m; G; D(λ) C |V | m; G; D(λ) + |P k V | m; G; D(λ) C |V | m; G; D(λ) + |λ -k| P k V m; G C |V | m; G; D(λ) ,
which completes the proof of the lemma. .

⋆ For N = {k 1 , . . . , k j } ⊂ N ∩ [b 0 , b 1 ]
, and using the norm (3.4), we introduce

(3.25b) N m G (U, [b 0 , b 1 ], N) = max sup b∈B 0 N m G (U, b), sup b∈B 1 N m G (U, b, k 1 ), . . . , sup b∈B j N m G (U, b, k j ) ,
with the sets

B ℓ = (k ℓ -1 2 , k ℓ + 1 2 ) ∩ (b 0 , b 1 ) for ℓ = 1, . . . , j and B 0 = (b 0 , b 1 ) \ ∪ j ℓ=1 B ℓ . ⋆ If N = ∅, the definition (3.25b) becomes, compare Definition 3.1, (3.25c) N m G (U, [b 0 , b 1 ], ∅) = sup b∈(b 0 ,b 1 ) N m G (U, b) = N m G (U, [b 0 , b 1 ]).
Using the continuity of P k G on H m (G), we obtain the estimate

(3.26a) N m G (U, b, k) C b,m,k N m G (U, b)
where the constant C b,m,k does not depend on U. On the other hand, the definition immediately implies the estimate

(3.26b) N m G (U, b) (b -k) -1 N m G (U, b, k), if b = k.
A consequence of the latter two inequalities is that for any fixed real number ρ ∈ (0, 1 2 ], we would obtain an equivalent norm to (3.25b) by defining B ℓ as

(k ℓ -ρ, k ℓ + ρ) ∩ (b 0 , b 1 ) instead of (k ℓ -1 2 , k ℓ + 1 2 ) ∩ (b 0 , b 1 ).
= -β -n 2 , η 0 = -β 0 -n 2 . Let N be a subset of N ∩ [η 0 , η]. ⋆ The functions u ∈ N m β,β 0 ;N (K)
with support in B(0, 1) are the functions whose Mellin transform M [u] is holomorphic in the half-plane Re λ < η 0 and has a meromorphic extension U to the half-plane Re λ < η satisfying the estimate

(3.27) N m G (U, [η 0 , η], N) < ∞ . Let χ ∈ C ∞ (R ν
) be a cut-off function with support in B(0, 1), equal to 1 in a neighborhood of the origin. The elements u of N m β,β 0 ;N (S) are defined by the two conditions that χu ∈ N m β,β 0 ;N (K) and (1χ)u ∈ H m (S). ⋆ In the case when η 0 = min{0, η} and N = N ∩ [η 0 , η], the space N m β,β 0 ;N (K) will alternatively be denoted by J m max, β (K). Note that in this definition, the set of poles N is contained in the interval [η 0 , η] determined by the weight exponents, but N has no relation with the regularity order m. Thus the residues at the poles which, according to Theorem 3.4, give an asymptotic expansion at the origin, can only be identified with the terms of a Taylor expansion in a generalized sense, in general, because the corresponding derivatives need not exist outside of the origin. With m = 0, for example, one gets weighted L 2 spaces with detached asymptotics.

For the maximal J-weighted Sobolev spaces J m max, β , the definition immediately yields the following properties. From Definition 3.10 follows that the poles of the Mellin transform of elements of N m β,β 0 ;N are associated with polynomials and that N m β,β 0 ;N can be split into a sum of a space with homogeneous norm and a space of polynomials. Theorem 3.12. Let m ∈ N and β, β 0 ∈ R such that β β 0 and as usual,

Proposition 3.11. a) For all m 0, β < β ′ implies J m max, β (S) ⊂ J m max, β ′ (S). b) For all β ∈ R, 0 m ′ < m implies J m max, β (S) ⊂ J m ′ max, β (S). c) ∂ α x is continuous from J m max, β (S) into J m-|α| max, β+|α| (
η = -β -n 2 , η 0 = -β 0 -n 2 . Let N be a subset of N ∩[η 0 , η]. Let u ∈ N m β,β 0 ;N (K)
with support in B(0, 1) and let U be its Mellin transform. Then for b ∈ (η 0 , η] \ N, the inverse Mellin transform u ′ of U on the line Re λ = b belongs to K m -b-n 2 (K) and there holds

(3.28) u ′ -u = k ∈ N∩ (η 0 ,b) Res λ = k r λ U(λ) is a polynomial.
The coefficients of the polynomial in (3.28) depend continuously on u in the norm of N m β,β 0 ;N (K). Proof. Let b ∈ (η 0 , η] \ N. By definition of N m β,β 0 ;N (K), we have in particular

N m G (U, b) = Re λ=b U(λ) 2 m; G; λ d Im λ 1 2 < ∞.
Theorem 3.2 then provides the existence of a function

u ′ ∈ K m -b-n 2 (K) such that M [u ′ ](λ) = U(λ), ∀λ, Re λ = b,
and according to Theorem 3.4, there holds

u ′ -u = 1 2iπ C r λ U(λ) dλ = k ∈ N∩ (η 0 ,b) Res λ = k r λ U(λ) .
Here C is a contour surrounding the poles of

U in N ∩ [η 0 , b].
It remains to show that the residual at k ∈ N∩(η 0 , b) is a polynomial. From the finiteness of N m G (U, [η 0 , η], N) follows in particular that sup

|b-k|<1/2 N m G (U, b, k) < ∞
and therefore that both (I -P k G )U(λ) and (λk)P k G U(λ) are holomorphic at k. Hence Res

λ = k r λ U(λ) = Res λ = k r λ P k G U(λ) = r k P k G Res λ = k U(λ) ,
which is a polynomial in x of degree k.

We can now complete the characterization of the K m β semi-norm that was begun in Lemma 3.6. Theorem 3.13. Let β < β 0 be two real numbers. We set η

= -β -n 2 , η 0 = -β 0 -n 2 and N m = {0, . . . , m -1} ∩ (η 0 , η]. Let u ∈ K 0 β 0 (K)
with support in B(0, 1). Let U be its Mellin transform.

(1) Then the following two conditions are equivalent a. The semi-norm

|u| K m β (K) is finite. b. u ∈ N m β,β 0 ;Nm (K).
(2) Moreover we have the equivalence of norms

(3.29) c u K 0 β 0 (K) + |u| K m β (K) N m G (U, [η 0 , η], N m ) C u K 0 β 0 (K) + |u| K m β (K) .
Proof. a. ⇒ b. and equivalence (3.29). Let u ∈ K 0 β 0 (K) with finite K m β (K) semi-norm and support in B(0, 1). According to Lemma 3.6 its Mellin transform is defined for Re λ η 0 and has a meromorphic extension U to the half-plane Re λ < η satisfying estimates (3.15), i.e., the semi-norm |u| K m β (K) is equivalent to the norm But Lemma 3.8 reveals that the norm ( 1) is equivalent to

(2) N m G (U, [η 0 , η], N m ) with N m = {0, . . . , m -1} ∩ [η 0 , η]
. Hence Lemma 3.6 yields the equivalence of the norm (2) with the semi-norm |u| K m β (K) . On the other hand, the norm u K 0 Since all norms are equivalent on the range of P k G , we find the estimate

β 0 (K) is equivalent to N 0 G (U, η 0 ). Therefore the norm (3) u K 0 β 0 (K) + |u| K m β (K) present in (3.29) is equivalent to (4) N 0 G (U, η 0 ) + N m G (U, [η 0 , η], N m ). It remains to prove that (4) is equivalent to N m G (U, [η 0 , η], N m ). • If N m = N m (this occurs if η 0 ∈ {0, . . . , m -1}), we have the equality of norms N m G (U, [η 0 , η], N m ) = N m G (U, [η 0 , η], N m ). Moreover N 0 G (U, η 0 ) is
(6) N m G (U, η 0 ) C N 0 G (U, η 0 ) + N m G (U, η 0 , η 0 ) C • norm (4).
Let us choose b ∈ (η 0 , η 0 + 1 2 ). We have ( 7)

N m G (U, b) C(b)N m G (U, [η 0 , η], N m ),
0 , cf. (3.26b)). The finiteness of N m G (U, η 0 ) and N m G (U, b) implies that u ∈ K m β 0 (K) ∩ K m -b-n 2 (K) and by Theorem 3.3: (8) N m G (U, [η 0 , b], ∅) C N m G (U, η 0 ) + N m G (U, b) .
(6)- [START_REF] Nazarov | The Neumann problem for selfadjoint elliptic systems in a domain with a piecewise-smooth boundary[END_REF] gives that N m G (U, [η 0 , b], ∅) is bounded by norm (4), which, in association with (5) implies that N m G (U, [η 0 , η], N m ) is bounded by [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]. The converse estimate is obvious. b. ⇒ a. Let u ∈ K 0 β 0 (K) ∩ N m β,β 0 ;Nm (K) with support in B(0, 1). Since η 0 ∈ N m , we have in particular, cf. (3.26b)

N m G (U, η 0 ) C N m G (U, [η 0 , η], N m ).
Hence we obtain that u belongs to K m β 0 (K). Therefore, for all α, |α| = m, the function w α = r m ∂ α

x u belongs to K 0 β 0 (K). Let W α be its Mellin transform. By construction, and thanks to Lemma 3.8, we find

N 0 G (W α , [η 0 , η], ∅) C N m G (U, [η 0 , η], N m ). Hence w α ∈ K 0 β (K)
, and therefore the K m β (K) semi-norm of u is finite.

3.4.

Spaces defined by weighted semi-norms. We have seen in Theorem 3.13 how a space defined by two weighted seminorms

| • | K 0 β 0 and | • | K m
β has a Mellin characterization described by the space N m β,β 0 ;Nm . We are now generalizing this to the case of spaces given by several weighted semi-norms, and this will eventually lead to the Mellin characterization of the space J m β , which is defined by the m

+ 1 semi-norms | • | K ℓ β+m-ℓ , 0 ℓ m, see Definition 2.2.
Definition 3.14. Let L be a subset of N that includes 0: For each ℓ ∈ L let β ℓ be a weight exponent such that β ℓ decreases as ℓ increases , and denote B = {β ℓ : ℓ ∈ L}. We define the associated norm

(3.30) u J L B (S) = ℓ∈L |α|=ℓ r β ℓ +ℓ ∂ α x u 2 0;S 1 2 ≡ ℓ∈L |u| 2 K ℓ β ℓ (S) 1 2
.

The Hilbert space defined by this norm is denoted by J L B (S). This definition includes the weighted Sobolev spaces with homogeneous norms and those with non-homogeneous norms as obvious special cases:

• We obtain the norm in K m β by choosing β ℓ = β for all ℓ and L any arbitrary subset contained in {0, . . . , m} and containing 0 and m. • According to Definition 2.2, we obtain the norm in J m β by choosing L = {0, . . . , m} , β ℓ = β + mℓ .

• Finally, the space defined by the norm K 0 β 0 and the seminorm K m β simply corresponds to L = {0, m} and B = {β 0 , β}. We can use Theorem 3.13 to obtain a first Mellin characterization of the space J L B (S). We set, as usual, η ℓ = -β ℓ -n 2 . Then we have

(3.31) J L B (S) = ℓ∈L N ℓ β ℓ ,β 0 ;N ℓ (S) with N ℓ = {0, . . . , ℓ -1} ∩ (η 0 , η ℓ ] .
This can be simplified with the following result.

Lemma 3.15. Let L, B and N ℓ be as above. Let m = max L. Then there exists a unique subset N ⊂ N m = {0, . . . , m -1} ∩ (η 0 , η] such that there is a norm equivalence

(3.32) c N m G (U, [η 0 , η], N) max ℓ∈L N ℓ G (U, [η 0 , η ℓ ], N ℓ ) C N m G (U, [η 0 , η], N).
This set N is given by

(3.33) N = N m \ ℓ∈L [ℓ, η ℓ ] .
Proof. Assume that N is such that the norm equivalence (3.32) holds. Then any pole k ∈ N that lies in an interval (η 0 , η ℓ ] must appear as a pole in the corresponding set N ℓ and vice versa. This means

(3.34) N ∩ (η 0 , η ℓ ] ⊂ {0, . . . , ℓ -1} ∀ℓ ∈ L .
In other words, for k ∈ N m there holds k ∈ N if and only if there exists ℓ ∈ L such that

ℓ k η ℓ .
This implies the formula (3.33) for N. Conversely, it is not hard to see that if we define the set N by (3.33), then the norm equivalence (3.32) holds.

Combining (3.31) with Lemma 3.15, we obtain the Mellin characterization of the space J L B (S). J L B (S) = N m β,β 0 ;N (S) . We have seen that with each set of semi-norms given by L and B there is a unique associated set of poles N that characterizes the space N m β,β 0 ;N (S) and therefore the space J L B (S). The converse is not always true, that is, the spaces N m β,β 0 ;N cannot always be defined by a set of weighted Sobolev semi-norms. A necessary condition is that N ⊂ N m . But this is also sufficient:

For fixed m and β, let N be a given subset of N m . We can construct indices L and weight exponents B such that the formula (3.33), and therefore the equality of spaces (3.35) in Proposition 3.16 holds. This can be done by setting

(3.36) L = {0} ∪ N m \ N ∪ {m},
and for all ℓ ∈ L, ℓ = 0, m, (3.37)

β ℓ = -η ℓ -n 2 with η ℓ = ℓ,
and

η 0 = 0 if 0 ∈ N, η 0 < 0 arbitrary if 0 ∈ N.
In this context, the counter-example (2.8), for instance, corresponds to the choice of m 2, L = {0} ∪ {2, . . . , m}, and η ℓ = ηm + ℓ with m < η < m + 1, so that η ℓ ∈ (ℓ, ℓ + 1). From these informations one obtains N = {1}.

From the equality (3.35), we conclude that the space J L B (S) depends only on m, β = β m and on the set of integers N. Several different choices of L and B can therefore lead to the same space. We have already seen this for the space K m β , where the choice of L is arbitrary, as soon as it includes 0 and m. This observation expresses the fact that for the spaces with homogeneous norms, the intermediate semi-norms are bounded by the two extreme semi-norms. The set of poles N is empty in this case.

Also for the space with non-homogeneous norm J m β , several different choices of sets of semi-norms are possible, as we will discuss now.

3.5. Mellin characterization of spaces with non-homogeneous norms. For the weighted Sobolev space with non-homogeneous norm J m β , several different choices of sets {L, B} of semi-norms are possible that lead to the same set of poles N and define therefore, according to Proposition 3.16, the same space. The original definition of J m β corresponds to the choice L = {0, . . . , m} and β ℓ = β + mℓ, ℓ ∈ L, which implies η ℓ+1 = η ℓ + 1. From this information and formula (3.33) one easily deduces that N is either empty or a set of consecutive integers starting with 0. It is non-empty if and only if 0 < η < m, and in this case

(3.38) N = {0, . . . , m -1} ∩ (η -m, η] = {0, . . . M} with M = [η] .
Since N = ∅ corresponds to the space K m β and N = {0, . . . , [η]} to the space J m max, β , we have found the following classification of the space J m β . Proposition 3.17.

Let m ∈ N, β ∈ R and η = -β -n 2 . a) If η < 0, then J m β (S) = J m max, β (S) = K m β (S). b) If 0 η < m, then J m β (S) = J m max, β (S). c) If η m, then J m β (S) = K m β (S)
. The set N of integers (3.38) that characterizes J m β can also be obtained by other choices for the weight indices: We start again with 0 < η < m and L = {0, . . . , m}, but now we fix some integer ℓ 0 in the interval (η, m]. Then we define the weight indices β ℓ in such a way that η ℓ = ηℓ 0 + ℓ for 0 ℓ ℓ 0 and η ℓ = η for ℓ ℓ 0 Since η 0 < 0 and ℓ 0 -1 M, we easily see that this set of weight indices defines the same set of degrees N = {0, . . . M} as in (3.38). In this way, we have proved the "stepweighted" characterization of J m β : Proposition 3.18. Let β ∈ R and m ∈ N such that m > η = -β -n 2 . Let ρ be any real number in the interval (-n 2 , β + m]. Then the norm in the space J m β (S) is equivalent to

(3.39) |α| m r max{β+|α|, ρ} ∂ α x u 2 0;S 1 2 . Corollary 3.19. Let β ∈ R. Set η = -β -n 2 .
Let m be a natural number, m > η. Then J m+1 β (S) ⊂ J m β (S). Proof. Using Proposition 3.18, we note that we can choose the same ρ for J m β (S) and J m+1 β (S). The embedding J m+1 β (S) ⊂ J m β (S) follows. Still another choice giving the same result is possible. When 0 < η < m, it suffices to take L = {0, m}, η m = η, and any η 0 < 0. In this case, N = N m = {0, . . . M}, which corresponds to the identity J m β (S) = J m max, β (S). We obtain the corollary that when 0 < η < m, the intermediate semi-norms in the definition of J m β (S) are indeed bounded by the sum of the two extreme semi-norms. This is not the case, as we have seen, if η > m.

Let us mention another identity that can be obtained from these purely combinatorial arguments, namely K m β (S) = J m β (S) ∩ K 0 β (S) . This can be seen as follows: The intersection J m β (S) ∩ K 0 β (S) is included in the space J L B (S) with L = {0, m} and β 0 = β m = β. Then N = ∅ and we find that this latter space coincides with K m β (S). Remark 3.20. Corollary 3.19 gives a partial response to the question of how to define spaces J s β with non-integer Sobolev index s. If [s] > η, the natural idea is to define the space of index s by Hilbert space interpolation between spaces with integer indices [s] and [s] + 1. The same possibility exists if [s] + 1 η, since for m + 1 η the inclusion J m+1 β (S) ⊂ J m β (S) holds, too, because according to Proposition 3.17 c) the J-weighted spaces coincide with the K-weighted spaces in this range.

For fixed weight β, both scales of spaces K m β (S) m∈N and J m max, β (S) m∈N can be extended in a natural way by interpolation to scales with arbitrary real positive index. This definition, when extended by analogy to the n-1-dimensional conical manifold ∂K, is then also compatible with the trace operator, that is, the trace space of

K m β (K) is K m-1 2 β+ 1 2 (∂K)
and similarly for the J max scale. There is, however, no natural definition of J s β for the remaining non-integer indices s for which

[s] η < [s] + 1. The problem is that if m > η, so that J m β (S) = J m max, β (
S), then the trace space is also of the J max class, because it contains non-zero constant functions. But if m -1 2 < η, then the candidate for the trace space would be J m-

1 2 β+ 1 2
and should be of the K class, which does not contain non-constant functions.

As a further corollary of the Mellin description of the space J m β , we give an equivalent definition by derivatives in polar coordinates that is valid when η < 1 and will be useful later on:

Lemma 3.21. Let β ∈ R, η = -β -n
2 and m ∈ N, m 1. We assume that η < 1. Then (3.40)

1 ℓ+|γ| m r β (r∂ r ) ℓ ∂ γ ϑ u 2 0;S + r β+1 u 2 0,S 1 2
defines a norm on J m β (S), equivalent to its natural norm. Proof. If η < 0, the statement is clear because, in that case, J m β (S) coincides with K m β (S). Let us suppose that 0 η < 1 and let u be such that its norm (3.40) is finite. Using a cut-off, we can assume that u has the same regularity on K, with support in S. Let M [u] =: U be the Mellin transform of u. By the Parseval identity we have the equivalence (1)

1 ℓ+k m Re λ=η |λ| 2ℓ |U(λ)| 2 k; G d Im λ ≃ 1 ℓ+|γ| m r β (r∂ r ) ℓ ∂ γ ϑ u 2 0;S .
It is easy to see that we have the uniform estimate

m k=1 |U(λ)| k; G; D(λ) C 1 ℓ+k m |λ| ℓ |U(λ)| k; G .
We deduce that u ∈ J m β (S). Conversely, let u ∈ J m β (S). We apply Lemma 3.8. Outside a neighborhood of 0, we have the uniform estimate: Since P 0 U(λ) is a constant, there holds (4)

m k=1 |U(λ)| k; G C U(λ) -P 0 U(λ) m; G .
We deduce from ( 2)-( 4) that

1 ℓ+k m |λ| ℓ |U(λ)| k; G C|U(λ)| m; G; D(λ) .
The boundedness of norm (3.40) follows from (1) and Lemma 3.6.

We can now collect the informations about the Mellin description of the space J m β . For this, we introduce some notation concerning the Taylor expansion at the origin:

For u ∈ C ∞ (S) and M ∈ N, we write T M u ∈ P M (S) for the Taylor part of u of degree M at 0:

(3.41) T M u = |α| M ∂ α x u(0)
x α α! .

By continuity, the coefficients of the Taylor expansion and therefore the corner Taylor operator T M can be defined on the space N m β,β 0 ;N (S), as soon as {0, . . . , M} ⊂ N ⊂ (η 0 , η), see Theorem 3.12.

The proofs of the following two theorems are contained in the results of the preceding section. When M < 0 or m η, the sum of residues collapses to 0, and u belongs to K m β (K). We will call the case η ∈ N critical. In the non-critical case, we can take b = η in the previous result, and we obtain therefore the following relations between the space J m β (S) with non-homogeneous norm and the space K m β (S) with homogeneous norm: J m β (S) = K m β (S) ⊕ P M (S) . • The critical case η ∈ N: The operator T M -1 is continuous on J m β (S), but T M is not. The space J m β (S) contains K m β (S) ⊕ P M (S) as a strict subspace of infinite codimension.

The structure of J m β in the critical case, and the generalization of the Taylor expansion in that case, is the subject of the following section.

= -β -n 2 . ⋆ K ∞ β (K) = m∈N K m β (K).
⋆ We denote by A β (K) the subspace of the functions u ∈ K ∞ β (K) satisfying the following analytic estimates for some C > 0

(4.1) ∃C > 0, ∀k ∈ N, |u| K k β (K) C k+1 k! . ⋆ J ∞ β (S) = k∈N , k>η J k β (S).
⋆ The analytic weighted class B β (S) with non-homogeneous norm is the space of functions u ∈ J ∞ β (S) such that there exists a constant C > 0 with

(4.2) ∀k ∈ N with k > η, |u| K k β (K) C k+1 k!.
Note that in (4.2) the estimates are the same as in (4.1) but only for k > η. This suggests that for η < 0, we have B β (S) = A β (S), which will be proved below.

For generalization of the Taylor expansion in the critical case, we develop the Mellindomain analogue of an idea from [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF], based on the splitting of u M provided by the decomposition

U(λ) = (I -P M )U(λ) + P M U(λ).
The first part is the Mellin transform of a function in K m β (K) and the second one has essentially a one dimensional structure -that is, the most important features of its structure are described by the behavior of functions of one variable -and it can be regularized in such a way that it splits again into two parts, one in the analytic class B β (K), and the remaining part in K m β (K).

4.2.

Mellin regularizing operator in one dimension. The main tool of the following analysis is a one-dimensional Mellin convolution operator: Definition 4.2. We denote by K : v → Kv be the Mellin convolution operator defined by

(4.3) M [Kv](λ) = e λ 2 M [v](λ).
Owing to the strong decay properties of the kernel e λ 2 in the imaginary direction, the operator K has analytic regularizing properties in the scales K m β and J m β . Proposition 4.3. Let β ∈ R and m 1. There holds:

a) If v ∈ K m β (R + ), then Kv ∈ A β (R + ). b) If v ∈ J m -1 2 (R + ) with support in I := [0, 1], then Kv| I belongs to the analytic class B -1 2 (I), and v -Kv ∈ K m -1 2 (R + ). c) If v ∈ J m β (R + ) with support in I := [0, 1]
, and if β < -1 2 so that v is continuous in 0, then Kv is continuous in 0 as well, and Kv(0) = v(0).

The proof of this proposition is based on the following characterization of analytic classes by Mellin transformation:

Lemma 4.4. Let β ∈ R and η = -β -1 2 . We set I = (0, 1). There holds

a) Let v ∈ K 1 β (R + ). Then v belongs to A β (R + ) if and only if V := M [v] satisfies (4.4) ∃C > 0, ∀k 1, Re λ=η |λ| 2k |V (λ)| 2 d Im λ 1 2 C k+1 k! b) Let v ∈ J 1 -1 2 (R + ). Then v| I belongs to B -1 2 (I) if (4.4) is satisfied with η = 0 and V (λ) := λ -1 M [r∂ r v](λ). Proof. a) According to Definition 4.1, v ∈ A β (R + ) if and only if ∃C > 1, ∀k 0, r β+k ∂ k r v 0;R + C k+1 k! Using (3.18), one can see that this is equivalent to ∃C > 1, ∀k 0, r β (r∂ r ) k v 0;R + C k+1 k! Then a) is a consequence of the Parseval equality. b) Let v ∈ J 1 -1 2 (R + ). With V (λ) = λ -1 M [r∂ r v](λ), for any k 1 the function λ k V (λ) is
the Mellin transform of (r∂ r ) k v on the line Re λ = 0. Thus, (4.4) with η = 0 implies the analytic estimates

∃C > 0, ∀k 1, r -1 2 (r∂ r ) k v 0;R + C k+1 k!
Restricting this to I, and using Definition 4.1, we find that v| I ∈ B -1 2 (I).

Proof of Proposition 4.3. a) Let v ∈ K m β (R + ) and let V be the Mellin transform of v. It is defined for Re λ = η and, in particular, the norm

N 0 := Re λ=η |V (λ)| 2 d Im λ 1 2 is finite. The Mellin transform of Kv is λ → e λ 2 V (λ). We have for any k 1 Re λ=η |λ| 2k |e λ 2 V (λ)| 2 d Im λ 1 2 N 0 sup Re λ=η |λ| k |e λ 2 | C(η)N 0 sup ξ 0 ξ k e -ξ 2 = C(η)N 0 k 2e k 2 .
Therefore condition (4.4) is satisfied for the Mellin transform of Kv. By Lemma 4.4 a), Kv belongs to

A β (R + ). b) Let v ∈ J m -1 2 (R + ) with support in I. By Corollary 3.19, v ∈ J 1 -1 2 (R + ). Now, V is
defined as the Mellin transform of r∂ r v divided by λ. Thus V coincides with M [v] where M [v] is well defined, and the Mellin transform of Kv is given by e λ 2 V (λ). With the same arguments as above, we prove that Kv satisfies the assumptions of Lemma 4.4 b), hence

Kv| I ∈ B -1 2 (I). The Mellin transform of v -Kv is (1 -e λ 2 )V (λ). Since r -1 2 (r∂ r ) k v ∈ L 2 (R + ) for k = 1, . . . , m we have (1) m k=1 Re λ=0 |λ| 2k |V (λ)| 2 d Im λ 1 2 < ∞.
The function λ → (1e λ 2 ) is bounded on the line Re λ = 0 and has a double zero at λ = 0. Hence we deduce from (1) that

m k=0 Re λ=0 |λ| 2k |(1 -e λ 2 )V (λ)| 2 d Im λ 1 2 < ∞. Therefore v -Kv ∈ K m -1 2 (R + ). c) Let v ∈ J m β (R + ) with support in I, β < -1 2 .
It is sufficient to consider the case m = 1 and -3 2 < β < -1 2 . With η = -β -1 2 we then have 0 < η < 1. Let V be the Mellin transform of v, and set w = v -Kv. As above we have

M [w](λ) = (1 -e λ 2 )V (λ) = 1 -e λ 2 λ U(λ), where U = M [r∂ r v] .
Since the function u = r∂ r v belongs to K 0 β (R + ) and has support in I, U is holomorphic for Re λ < η, and M [w](λ) has the same property. It follows that w ∈ K 1 β (R + ), which implies that w is continuous at 0 and w(0) = 0. Let us fix a cut-off function χ ∈ C ∞ 0 ((-1, 1)), χ ≡ 1 on [-1 2 , 1 2 ]. Then, using (4.3), the regularizing operator K M u is defined by does not make sense, because the Taylor coefficients ∂ α x u(0) are not bounded with respect to the J m β norm in this case. But one can replace the constants ∂ α x u(0) by "generalized constants", namely the analytic functions K χd α , which means that the homogeneous part of degree M of the Taylor expansion is replaced by K M u, which is not polynomial but belongs to the analytic class B β (S). The "Taylor remainder" then belongs to K m β (K). Theorem 4.8. Let β be such that -β -n 2 = M ∈ N and let u ∈ J m β (K) with support in B(0, 1). Then

n 2 t

 2 u belongs to L 2 on the cylinder R × G and λ → M [u](λ) therefore defines an L 2 function on the line Re λ = -n 2 , with values in L 2 (G).

one gets: Theorem 3 . 2 .

 32 Let β be a real number and m ∈ N. Let η := -β -n 2 and R[η] := λ ∈ C : Re λ = η . The Mellin transformation (3.2) u → M [u] induces an isomorphism from K m β (K) onto the space of functions U : R[η] × G ∋ (λ, ϑ) → U(λ, ϑ) with finite norm N m

  ), (3.20) together and using the semi-norm |•| m; G; D(λ) we have proved the equivalence (3.15).

|

  Putting the norm equivalences (3.23) and (3.24) together, one is led to the following definition of norms of meromorphic H m (G)-valued functions. Definition 3.9. Let λ → U(λ) be a meromorphic function with values in H m (G) for λ in a strip b 0 < Re λ < b 1 . ⋆ For b ∈ (b 0 , b 1 ) and k ∈ N, and with P k G the projection operator (3.22) we set (3.25a) N m G (U, b, k) = Im

3. 3 .

 3 Spaces defined by Mellin norms. The norms defined in (3.25b) suggest the introduction of a class of Sobolev spaces N m β,β 0 ;N with Mellin transforms meromorphic in a strip η 0 < Re λ < η and a fixed set of poles N. Definition 3.10. Let m ∈ N and β, β 0 ∈ R such that β β 0 and as usual, η

  S) for any α ∈ N n , any m |α|, and any β. d) The multiplication by x α is continuous from J m max, β (S) into J m max, β-|α| (S) for any α ∈ N n .

  bounded by the quantity N m G (U, [η 0 , η], N m ). Hence the desired equivalence. • If N m = N m , then η 0 ∈ {0, . . . , m -1} and (5) N m = N m ∪ {η 0 }.

  where C(b) means that this constant depends on b (and would blow up if b approaches η

  Proposition 3.16. Let L and B satisfy the conditions in Definition 3.14. Let m = max L, β = β m and define N by (3.33). Then (3.35)

( 2 )

 2 U(λ) m; G; λ C|U(λ)| m; G; D(λ) , and in a bounded neighborhood of 0 (3) U(λ) -P 0 U(λ) m; G + |λ| U(λ) m; G C|U(λ)| m; G; D(λ) .

Theorem 3 .-b-n 2 (

 32 22. Let K be a regular cone in R n . Let β ∈ R. We set, as usual η = -β -n 2 and M = [η].Let N = {0, . . . , M} if M 0 and m > η, and N = ∅ in the other cases (either M < 0 or m η). Let u ∈ K 0 β+m (K) with support in B(0, 1). Let U be its Mellin transform. Setη 0 = ηm and β 0 = η 0 -n 2 = β + m. a) Then u ∈ J m β (K) if and only if u ∈ N m β,β 0 ;N (K).Moreover we have the equivalence of norms(3.42) c u J m β (K) N m G (U, [ηm, η], N) C u J m β (K). Furthermore U is meromorphic in the half-plane Re λ < η with only possible poles on natural numbers and the residues of r λ U(λ) are polynomials. b) Let M * = M if η = M and M * = M -1 if η = M. Let b ∈ (M * , η] with b = η if η = M. Then the inverse Mellin transform u ′ of U on the line Re λ = b belongs to K m K) and there holds, with notation (3.41) (3.43) u ′u = M * k=0 Res λ = k r λ U(λ) = -T M * u.

Theorem 3 .

 3 23. Let K ⊂ R n be a cone and S = K ∩ B(0, 1). Let β ∈ R. We set η = -β -n 2 and M = [η]. Let m ∈ N. Then there holds: a) If η < 0, the spaces J m β (S) and K m β (S) coincide. b) If η 0 and m η, the spaces J m β (S) and K m β (S) coincide. c) If η 0 and m > η, then J m β (S) and J m max, β (S) coincide and there are two cases: • The non-critical case η ∈ N: The corner Taylor operator T M defined in (3.41) is continuous from J m β (S) to P M (S) and I -T M is continuous from J m β (S) to K m β (S). The decomposition u = (u -T M u) + T M u gives the direct sum (3.44)

4 .

 4 STRUCTURE OF SPACES WITH NONHOMOGENEOUS NORMS IN THE CRITICAL CASE 4.1. Weighted Sobolev spaces with analytic regularity. Using the monotonicity of embeddings K m+1 β (S) ⊂ K m β (S) for all m and β and J m+1 β (S) ⊂ J m β (S) if M > η = -β -n 2 , we introduce corresponding weighted spaces with infinite and with analytic regularity: Definition 4.1. Let β ∈ R and η

4. 3 .Definition 4 . 5 .

 345 Generalized Taylor expansions. We are now ready for the definition of the splitting which replaces the Taylor expansion in the critical case: Let us assume that the natural number M is critical. We are going to replace the homogeneous partT M u = |α|=M ∂ α x u(0) x α α! ,of the corner Taylor expansion with a new operator u → K M u for which the point traces ∂ α x u(0) are replaced by moments defined thanks to the the dual basis (3.21) ϕ M γ |γ|=M . Let us recall that:dϑ = δ αγ , |α| = |γ| = M, ϑ α = r -M x α ,and this dual basis served to define the projection operator P M : L 2 (G) → P M (Let M ∈ N. For u ∈ C ∞ (K), let T M -1 u be its Taylor expansion at 0 of order M -1, and u M = u -T M -1 u its Taylor remainder of order M, considered in polar coordinates (r, ϑ). With the dual basis (4.5), we define the moments of u M : (4.7) ∀α, |α| = M, d α (r) = r -M u M (r, •), ϕ M α G , r > 0.

Remark 4 . 6 . 1 2( 1 2( 2 m; G; λ d Im λ 1 2 1 2( 1 2(R + ), then Re λ=0 1 +

 461122111 For M = 0, d α ≡ d 0 is the mean value of u(r, •) over G and K 0 u = K χd 0 . In particular, if u is continuous in 0, then both d 0 and K χd 0 are continuous in 0, and K 0 u(0) = u(0), see Proposition 4.3 c). More generally, for sufficiently smooth u, one hasd α (0) = K χd α (0) = ∂ α x u(0).The moments d α are well-defined in the critical case and have the following properties:Proposition 4.7. Let β be real such that -β -n 2 coincides with a non-negative integer M. For m > M, let u ∈ J m β (K) with support in B(0, 1). Then the moments d α as defined in (4.7) satisfy:a) For all |α| = M, χd α ∈ J m -R + ). b) If ∀|α| = M, χd α ∈ K m -R + ), then u -T M -1 u belongs to K m β (K).Proof. We can assume without restriction that χu = u.Let us set v = u -χT M -1 u. Then χd α [u] = d α [v] and v -T M -1 v = v and u -T M -1 u = v -(1χ)T M -1 u.Since (1χ)T M -1 u belongs to K m β (K), we can replace u with v and omit the cut-off χ. We still denote v by u. The Mellin transform U(λ) of u is holomorphic in the half-plane Re λ < M and, by Theorem 3.22, the norm (4.9)sup b ∈ (M -1 2 ,M ) | Im λ| 1 Re λ=b (I -P M )U(λ) 2 m; G + |λ -M| 2 P M U(λ)is bounded by C u J m β (K) . As a mere consequence of the definition of P M , see (4.6), we have the uniform inequality for Re λ < M P M U(λ) m; G; λ C U(λ) m; G; λ .Hence we deduce from estimates (4.9) thatsup b∈(M -1 2 ,M ) Re λ=b (I -P M )U(λ) 2 m; G; λ d Im λ C u 2 J m β (K) .Thus Theorem 3.2 yields that M -1 [(I -P M )U] belongs to K m β (K).Let us set Dα = M [d α ]. We have D α (λ) = M r -M u M , ϕ M α G (λ) = M u M , ϕ M α G (λ + M) = U(λ + M), ϕ M α G. Therefore, by formulas (4.5) and (4.6), we find(1) D α (λ) = P M U(λ + M), ϕ M α G . a)We deduce from (1) and (4.9) that D α is holomorphic in the half-plane Re λ < 0 and thatsup b ∈ (-1 2 ,0) Re λ=b |λ| 2 + |λ| 2m |D α (λ)| 2 d Im λ 1 2is bounded. This allows to prove thatd α ∈ J m -R + ). b) If d α ∈ K m -|λ| 2m |D α (λ)| 2 d Im λis bounded. Since by (1) and (4.5):P M U(λ + M) ; λ d Im λ < ∞, hence M -1 [P M U] ∈ K m β (K). Since M -1 [(I-P M )U] ∈ K m β(K), this ends the proof. We conclude this section with a result about the generalized Taylor expansion at the corner in the critical case. The homogeneous part of critical degree |α|=M ∂ αx u(0) x α α!

( 4 . 1 2( 2 (

 412 10) u -T M -1 u -K M u ∈ K m β (S) and K M u ∈ B β (S). Proof. Let u ∈ J m β (K)with support in B(0, 1). By Proposition 4.7 a), for all |α| = M, χd α belongs to J m -R + ). By Proposition 4.3 b), we deduce that K(χd α ) belongs to B -1 I). Let us consider the function v α : S ∋ x → K(χd α )(r)

and the class B -n 2 (S). This class is associated with η = 0. Therefore, using Lemma 3.21 we deduce that v α ∈ B -n 2 (S) as a direct consequence of the fact that K(χd α ) ∈ B -1 2 (I). Multiplying by x α , we find that

the moments of v defined like in (4.7). We notice that

and the quotient J m β (S)/K m β (S) is infinite dimensional.

CONCLUSION

Theorems 3.22 and 4.8 can advantageously be used for the analysis of second order elliptic boundary value problems in domains Ω with corners: Let L be the interior operator and B the operator on the boundary. L is supposed to be elliptic on Ω and B to cover L on ∂Ω. The order d of B is 0 or 1.

Theorem 3.22 fully characterizes the spaces J m β by Mellin transformation. This is an essential tool for stating necessary and sufficient conditions for (L, B) to define a Fredholm operator:

where Γ ∂Ω denotes the trace operator on ∂Ω. When K m β spaces are involved instead, this condition is the absence of poles for the corner Mellin resolvents on certain lines {Re λ = const}, see [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF]. Theorem 3.22 allows to prove by Mellin transformation that the necessary and sufficient condition associated with spaces J m β is the injectivity modulo polynomials (cf. [START_REF] Dauge | Elliptic Boundary Value Problems in Corner Domains -Smoothness and Asymptotics of Solutions[END_REF][START_REF] Costabel | Corner Singularities and Analytic Regularity for Linear Elliptic Systems In preparation[END_REF]) on similar lines in the complex plane.

Theorem 4.8 allows to prove an analytic shift theorem in J m β spaces for elliptic (L, B) with analytic coefficients: Roughly, this means that if a solution u belongs to J 2 β (Ω) and is associated with a right hand side in RB β (Ω) := B β+2 (Ω) × Γ ∂Ω B β+d (Ω), then u belongs to B β (Ω). This result relies on [START_REF] Costabel | Corner Singularities and Analytic Regularity for Linear Elliptic Systems In preparation[END_REF] The analytic shift theorem in the scale K m β : If u ∈ K 2 β (Ω) and the right hand sides belongs to RA β (Ω), then u ∈ A β (Ω).

(2) The splitting (4.10).

The analytic shift theorem in the scale K m β , that is with homogeneous norms, can be proved by a "standard" technique of dyadic refined partitions towards the corners combined with local analytic estimates in smooth regions. This technique cannot be directly applied to spaces with non-homogeneous norms, hence the utility of the splitting (4.10).

See [1, Part II] for details.


Bahram Djafari-Rouhani
email: bahram.djafari-rouhani@univ-lille.fr

Laurent Carpentier
email: laurent.carpentier2@univ-lille.fr

Yan Pennec
email: yan.pennec@univ-lille.fr

Elastic Metasurface made of elliptic shape pillars for acoustic wave focusing

Keywords: Metasurface, elastic wave, focusing

We numerically propose the design of a metasurface for focusing the flexural Lamb waves propagating on a plate. The metasurface consists of a line of pillars of elliptic shape with a gradient in their ellipticity. We first determine an elliptic pillar that allows the superposition of two resonances (one compresional and one bending) in order to obtain a phase shift of 2π around the resonance. Then, the metasurface is built by chosing the ellipticity of each pillar at a given position based on the generalized Snell's law for the selected focal point. We demonstrate the capacity of this design for subwavelength focusing of the incident plane wave at any targeted point.

I. INTRODUCTION

Metasurfaces are sub-wavelength planar metamaterials that allow the manipulation and control of incident acoustic/elastic waves, in particular to achieve anomalous refraction and reflection scatterings. They have known a great deal of attention during the last decade, in particular for acoustic and elastic waves [START_REF] Cummer | Controlling sound with acoustic metamaterials[END_REF][START_REF] Chen | Design of acoustic/elastic phase gradient metasurfaces: principles, functional elements, tunability and coding[END_REF][START_REF] Assouar | Acoustic metasurfaces[END_REF]. The metasurface is in general constituted by a repetition of a set of unit cells where each unit produces a specific phase shift of the incident wave such that the phase shifts of all the units span a range of 2π. Therefore, the application of a generalized Snell-Descartes law allows the realization of various functionalities in the behavior of the scattered waves. In this work, we are interested by the capacity of a pillared metasurface for the focusing of flexural Lamb waves propagating along a thin plate. The metasurface consists of a set of pillars, each producing a sufficiently high transmission and a specific phase shift of the incident wave, with a span of all the phase shifts over 2π. We recently proposed such a design exhibiting sub-wavelength focusing by using cylindrical pillars of varying heights [START_REF] Jin | Elastic metasurfaces for deep and robust subwavelength focusing and imaging[END_REF]. The principle of the design consists of starting from a reference pillar where the geometrical parameters are adapted such as to superimpose two resonances of the pillar, the second bending and the first compressional modes in our case [START_REF] Jin | Pillar-type acoustic metasurface[END_REF]. This is an essential feature of the structure because the transmission phase shift at resonance reaches 2π which cannot be achieved with separated resonances. The functionality of the design was also demonstrated experimentally [START_REF] Wang | Experimental realization of a pillared metasurface for flexural wave focusing[END_REF]. However, from a technological point of view, the fabrication of pillars with varying heights is difficult to achieve at the micro or nanoscale. Therefore, in this work, we propose a different approach based on the utilization of pillars of elliptical shape and of constant height. In this numerical study, we show that the ellipticity can be used as the adjustable parameter to bring the bending and compressional modes in superposition. The design of the proposed metasurface consists of an array of resonant pillars of different ellipticity parameters, allowing to realize a gradient phase shift fully covering a 2π range

II. NORMAL MODES FOR A LINE OF IDENTICAL PILLARS

The numerical analysis was performed with the help of the finite element method (COMSOL MULTIPHYSICS). In these calculations, the plate and pillars are made of silicon and the plate thickness is e = 145µm. We determine the frequencies of the lowest resonant modes, hereafter called C1 for the compressional mode and B1 and B2 for the two bending modes, as a function of the pillars parameters by calculating the transmission through a line of identical pillars. The plate is submitted to an incident antisymmetric Lamb wave A0 whose main component is out-of-plane uz at the frequency of 7.48MHz, corresponding to a wavelength of λ = 445µm (as discussed below, this frequency corresponds to the resonance frequency associated with the superposition of the 2 modes B2 and C1). One of the parameters of the ellipse is fixed to the value e1 = 20µm while the other e2 can vary along the direction of propagation of the incident wave. All the geometrical parameters are reported in figure 1.

The method we used to compute the resonances is described as follows: an oscillating force is applied normally to the surface of the plate, along a line [AB] parallel to the crystallographic direction (100) of the silicon (referred as to x axis in the following) and the resulting displacements are recorded as a function of frequency, on a line [CD] located on the other side of the pillars row. An infinite line of identical pillars is obtained by using the Periodic Boundary Conditions along the direction perpendicular to the incident wave and Perfect Matched Layers (PML) located at each end of the plate are used to avoid any reflection from the boundaries of the plate. In the explored frequency range (0-10MHz), the three first resonance modes manifest themselves by more or less sharp dips in the transmission curve and can be unambiguously ascribed to eigenmodes corresponding to the two first bending modes B1 and B2, and to the first compressional mode C1 (figure . 2). While B1 and C1 are rather insensitive to the parameter e2 defining the elliptic shape of the pillar, B2 mode shifts rapidly toward high frequencies when e2 increases (figure 3.a). Finally, the B2 and C1 modes end up overlapping, at the frequency of 7.48MHz, for the value e2 = 50µm. (figure 3.b) As a consequence, for this specific value of e2 for which two resonance modes are superimposed, a span of the phase shift over 2π can be produced by varying the ellipticity of the pillars around the preceding value. Therefore, we can now design a gradient-phase meta-line of pillars to realize anomalous transmitted wave fronts based on the generalized Snell law:

- = ( ) (1)
III. SUBWAVELENGTH FOCUSING As an illustration of anomalous transmission, we present a structure allowing to focus a transmitted Lamb wave A0 at any point of the surface M(x0,y0). To do that, a line of 21 elliptic pillars spaced by 150µm is arranged along the x direction perpendicular to wave propagation. The phase profile Φ(x) along this direction is determined according to the relation:

( ) -( = 0) = ( -) + -( + ) (2)
Where λ = 445µm is the wavelength of the A0 Lamb mode at 7.48MHz. The phase of the central pillar (x = 0) is set to -π, corresponding to the strongest resonant status. The discrete theoretical phases Φ(x) of the 20 other pillars in the line are calculated from the previous equation as a function of their positions x and compared to the numerical calculation of the phase as a function of e2. A span of the phase shift over 2π is obtained for e2 values ranging from 10µm to 210µm. Figure 4 shows the focusing effect at a point M with coordinates (0,F) where F is the focal length fixed to F = λ = 445µm. Figure 5 shows the transmitted intensity around the focal point which highlights a significant focusing effect with a sub-wavelength FWHM (∼ 0.37λ).

VI. CONCLUSION

We proposed a type of pillared elastic metasurface consisting of a line of gradient resonant pillars with elliptic shape base of subwavelength dimension, on a homogeneous plate, which is able to provide a 2π phase range manipulation and high transmissions. A particularly striking advantage of this structure is that, due to their constant height, it is possible to fabricate a microscale sample using nanofabrication facilities. The experimental validation is ongoing.

x(λ)

FIG. 1 .

 1 FIG. 1. Model used for finite element calculations of an infinite line of identical pillars. A time-periodic vertical force is applied along [AB]. PML are placed at the ends of the plate and periodic boundary conditions are applied on the lateral sides to simulate an infinite line of pillars. The amplitude of displacement uz as a function of frequency is monitored along [CD].

Frequency 2 FIG. 2 .

 22 FIG. 2. Resonance frequencies of the pillar appearing as dips in the transmission coefficient through a line of pillars and displacement field of the individual elliptic pillar for e2 = 40µm.

FIG. 3 .

 3 FIG. 3. (a) Resonance frequencies of the elliptic pillar as a function of the parameter e2 ranging from 20µm to 60µm. (b) For e2 = 50µm, both the second bending mode and the first compressional mode fall at ν = 7.48MHz and the corresponding dips in the transmission coefficient overlap.

FIG. 5 .

 5 FIG.5. Intensity of the transmitted wave along the x axis (red line -figure4) crossing the focusing spot.

ACKNOWLEDGMENT

This work was supported by the French National Research Agency ANR as part of the project "RANdom", ANR-19-CE24-0014.


Nathan Lombard
email: nathan.lombard@umontpellier.fr

Questioning the quantum world? A priori analysis of an SRP at the interface between mathematics and quantum mechanics

Keywords: Teachers' and students' practices at university level, Novel approaches to teaching, Teaching and learning

This work is at the confluence of two groups of studies: on the one hand, research on didactic phenomena (and their consequences for teaching and learning) when mathematics is taught in an engineering or a physics class; on the other hand, research on conditions and constraints allowing to carry out the evolution from the prevalent didactic paradigm of visiting works to the novel paradigm of questioning the world, which are both brought out by the Anthropological Theory of the Didactic (ATD). In this dual context, we present in this paper the preliminary and a priori analyses of a Study and Research Path (SRP) set at the interface between mathematics and introductory quantum mechanics, in third year of bachelor's degree, at the University of Montpellier.

INTRODUCTION

In recent years, much work has been carried out on the teaching and learning of mathematics for engineers or on the role of physical modelling in mathematics curricula (see for instance Hausberger, Bosch & Chellougi, 2020, TWG2). However, the relative roles of mathematics and advanced physics are quite different: rather than seing mathematics merely as being used, both fields are then more on an equal footing (Lombard & Hausberger, to appear). In this work, we focus on the relationship between algebraic structures and theoretical physics, based on the example of Hilbert spaces in quantum mechanics. The study is carried out in the framework of the Anthropological Theory of the Didactic (ATD), which "aim (…) is the elucidation of human societies' relation to "the didactic," that is to say, to all the possible factors of learning. By adopting an anthropological point of view, it purports to embrace the didactic wherever it may show itself around us, paying special attention to the institutional constructions of knowledge and the conditions established to disseminate it." (Chevallard & Bosch, 2020a, p. 53).

Within the research program of the ATD falls the identification of "study paradigms", and the "contribution (…) to the paradigm of questioning the world" (Chevallard & Bosch, 2020a, p. 59; see also the next section). The study and development of Study and Research Paths (SRP) plays a key role in this endeavour [START_REF] Bosch | How to integrate study and research paths into university courses? Teaching formats and ecologies[END_REF]. In this context, we have undertaken to set up an SRP at the interface between mathematics and introductory quantum mechanics at the University of Montpellier. More specifically, this SRP is set in the context of quantum computing, and starts with the following generating question Q 0 : In what respect are quantum computers indeed quantum? It is designed for students that are in third year of bachelor's degree (sixth semester). It consists of nine two-to threehour sessions, distributed into three lab sessions and six classroom sessions. In this paper we present the results of the first two phases of the implementation of an SRP: the preliminary and a priori analyses.

Our work is thus in line with the past INDRUM conference "instructional proposals for university mathematics to move towards a change of paradigm, such as problemposing activities, interdisciplinary projects or study and research paths" (Hausberger et al., 2020, p. 167). In this context, it aims at addressing the following issues (Hausberger et al.,:

How to find a "good" generating question for an SRP? Can the design and implementation of SRPs help us to rethink the contents of the course? […] How to look at university mathematics curricula from an interdisciplinary approach? How can the perspective of mathematical modelling contribute to it?

THEORETICAL FRAMEWORK AND METHODOLOGY

SRPs as part of ATD

In the paradigm of visiting works, also called paradigm of visiting monuments, works under study at school or university (theorems, formulae, methods) are "approached as (…) monument(s) stand(ing) on (their) own" (Chevallard 2015, p. 3). Notably, the raison d'être of these pieces of knowledge is never specified. The paradigm of questioning the world, as its name indicates, focuses rather on questions. In this paradigm, one sees pieces of knowledge as answers to be considered only when judged relevant to solve given problems, hence highlighting their raison d'être.

Study and Research Paths [START_REF] Winsløw | Study and research courses as an epistemological model for didactics[END_REF]) constitute a contribution to the advent of this "oncoming counter paradigm" [START_REF] Chevallard | Teaching Mathematics in Tomorrow's Society: A Case for an Oncoming Counter Paradigm[END_REF] in teaching institutions. "An SRP is an inquiry process which starts from an open-ended question and leads to a combination of investigation activities -to explore the question, and study processes -to obtain new information that will help in the research" [START_REF] Bartolomé | A 'study and research path' enriching the learning of mechanical engineering[END_REF].

Throughout an SRP, students are looking for answers to intermediary questions in lectures, in (text)books, on the internet, etc. All these "social system pretending to inform (…) some group of people about the natural or social world" are media in the sense of ATD, and they can become components of the inquiry "milieu", under the "dialectic of media and milieus" (Chevallard, 2006, p. 9). During the study of questions, the notion of systems and models (Barquero, Bosch & Gascón, 2019, p. 9) is also to play an important role, especially in the context of interface between disciplines we consider in this paper. The dialectics of media and milieu and systems and models should be important theoretical tools to monitor our experimentation.

Research questions

By means of this theoretical framework, we will now consider the following research issues: what are the conditions and constraints imposed on the implementation of an SRP at this level of studies, at the interface of mathematics and physics? What lies behind the choice of its generating question? How to avoid visiting works while maintaining research and teaching objectives?

Methodology

In order to prepare this module, we proceeded according to the Didactic Engineering methodology as it was applied to SRPs by [START_REF] Bartolomé | A 'study and research path' enriching the learning of mechanical engineering[END_REF]: see fig. 1. First, we performed a preliminary analysis consisting in three parts:

-the study of epistemological aspects underlying the project, based on the study of primary and secondary sources in history of quantum physics and mathematics -the institutional conditions of the SRP, based on interviews with professors at the University of Montpellier -the ecological context of the SRP, based on the same interviews as well as analyses of relevant course material Then, we carried out the a priori analysis which "includes the specific design of the SRP, including the selection of a generating question starting the study process, taking into consideration the conditions and constraints identified in the first phase." (Bartolomé et al., 2018, p. 6).

PRELIMINARY ANALYSIS Epistemological aspects

From a study of historical epistemology (in the sense of « Histories of epistemic things » (Feest and Sturm, 2011, p. 288)) dedicated to the interplay between quantum theory and functional analysis during their respective early developments , we could extract several aspects that seem particularly relevant for the design of this SRP.

First, mathematics and physics exerted a mutual influence throughout the development of quantum mechanics. In particular, from 1925-1927, mathematicians could draw results from questionings arising from physics [START_REF] Lacki | From Matrices to Hilbert Spaces: The Interplay of Physics and Mathematics in the Rise of Quantum Mechanics[END_REF]. This culminated in von Neumann's introduction of Hilbert spaces as an abstract structure encompassing the variety of theories of quantum mechanics known at that time. So, a physical context may be fruitful to introduce higher level mathematics via the dialectics of questions and answers.

Then, models and formulations were abundant at the interface with successive attempts at unifications and simplifications, which the structuralist stance in mathematics finally helps to achieve (von Neumann, 1955, p. 28). We should thus expect the dialectics of objects and structures [START_REF] Hausberger | La dialectique objets-structures comme cadre de référence pour une étude didactique du structuralisme algébrique[END_REF] will play a role.

Finally, interviews we performed with professors of physics and mathematical physics at the University of Montpellier lead us to consider such phenomena still occur in their day-to-day activity. More precisely, though there seems to exist a vivid practice at the interface between mathematics and physics when considering "scholarly knowledge", this does not seem to be the case anymore regarding "taught knowledge", showing a lack in the "didactic transposition" at the interface (Chevallard & Bosch, 2020b). This is one of the issues this SRP attempts to address.

Quantum computing appeared to be a subject that could put at play the aforementioned epistemological aspects. In addition, as the setting of an SRP should "be regarded -by the students, by their teachers (…) -as crucial to a better understanding and mastery of their lived world" (Chevallard, 2006, p. 7-8), quantum computing seemed to fit all the more. Lastly, specific institutional conditions in Montpellier would facilitate the implementation of an SRP on this topic.

Institutional conditions

Indeed, in 2019, the technology company IBM initiated a partnership with the University of Montpellier, in which several members of both the mathematics and physics departments are involved. This is why, in the first place, quantum computing emerged as a potentially workable setting for the upcoming SRP. However, it was yet to be found how to include such a project into the sequence of teaching units (TU) taught in this university. In Montpellier, physics and mathematics curricula are quite detached (see fig. 2). The bachelor (Licence) lasts three years (L1-L3), and it is divided into six semesters (S1-S6). Some TU taken by physics students are nevertheless taught by mathematicians: we filled them in blue. Lastly, our experimentation takes place during the sixth semester (S6), as a mixed TU (see below). It is shown in purple in the figure.

Several constraints weighted on the institutional implementation of the SRP, as it was to develop at the interface between mathematics and quantum mechanics. Firstly, physics students take their first quantum mechanics class at the fifth semester (S5, Mécanique analytique et quantique). Then, during the sixth semester, on the one hand, mathematics students have to take a "common knowledge" class (Culture générale), whereas, on the other hand, physicists take a TU devoted to doing an experimental research project supervised by a professor (Projet tuteuré). So, we could set up the SRP as a mixed teaching unit registered with both math and physics departments. It is the only such TU at this level of studies at the University of Montpellier. This way we could project nine two-to three-hour time-slots. With this organisation came further conditions and constraints, mainly from the physics department. For instance, evaluation should include a peer-reviewing process among students taking this TU. This was actually a favorable condition. Indeed, defining the recipients of the answer to be given to the generating question is a crucial step. So, we extended the physics instructions to all students: to write a report and make a presentation their third-year colleagues could read and understand.

Ecological context

Physics students has already had a quantum mechanics course during the first semester, whereas students from mathematics never did (at least at the University of Montpellier). The content of this quantum mechanics course corresponded to parts of Mechanics, vol. 1 by Cohen-Tannoudji, Diu & Laloe (1991) which is often used in introductory quantum mechanics courses in France. We could analyse its content in a study which provides additional inputs into the preliminary analysis (Lombard et Hausberger, to appear).

For instance, a tension exists between the abstractness of the Hilbert space formalism and the necessity for students to develop operative skills in order to study actual physical systems or models. In particular, the raison d'être of some elements of the formalism is never specified (why an infinite number of dimensions? why Hilbert spaces and not Banach spaces or even pre-Hilbertian spaces, which are studied in second year by math students?). Of course, this is consistent with the fact this course provides an example of the paradigm of visiting works. By shifting towards the paradigm of questioning the world, this is another aspect this SRP wishes to address. This tension particularly transpires when it comes to the passage from finite-to infinite-dimensional Hilbert spaces, as was actually acknowledged by a professor we interviewed. As a matter of fact, the course we studied began with the infinitedimensional case, even though the students never encountered them, especially to solve eigenvalue problems as is customary in quantum mechanics. In this SRP we chose to go the other way, starting with what students may already be familiar with and going from finite-to infinite-dimensional models (for such a treatment in the common paradigm of study, see for instance Le Bellac ( 2013)).

A PRIORI ANALYSIS

Resources and assessment

In light of the previous considerations we undertook the a priori analysis of an SRP at the interface between mathematics and quantum mechanics, with quantum computing as its background. It would include both classroom sessions (six) and computing sessions (three), the latter being taught by an IBM representative already working with the University of Montpellier (denoted by P qc ). Besides, to enforce the conveyance of mathematics and physics content, we chose to let one professor from each field (P𝜑 and P µ for physics and mathematics, respectively) teach once during the SRP. Finally, in order to comply with the constraints imposed on this TU by each department, we decided students would work in groups of three, by curriculum. So, the set of students may be denoted by X = {g 3 µ1 , g 3 µ2 , g 3 𝜑 1 , g 3 𝜑 2 } (where g 3 µ and g 3 𝜑 are math and physics trios, respectively) and the team of teachers by Y = {P qc , P𝜑, P µ }. So, we end up considering the didactic system S(X,Y,Q 0 ) (Chevallard 2019, p. 72), where Q 0 is our generating question (see next section).

Before addressing the choice of the generating question, we can complete the answers to the following questions in our meta-SRP: "Q2) Human and physical resources" and "Q3) Student assessment" (Bartolomé et al., 2018, p. 6-7, and fig. 1 here), which otherwise almost fully ensue from our preliminary analyses. First, we planned on collecting question-answer maps per group in order to monitor the evolution of students' questioning, actually letting students draft them themselves.

Then, complying with the institutional constraints coming from physics, we opted for a final answer in the form of a written report and an oral presentation. This implied, in particular, to let a significant part of the investigation on the "other stage" (Chevallard, 1998, p. 17). This is a questionable choice, as our research would gain in monitoring as closely as possible students' questioning. On the other hand, it enforces the customisation of the final answer A ♥ produced by students, making it indeed close to their hearts (Chevallard, 2019, p. 100).

Generating question

We may now answer the "Q1) The SRP structure" block of the meta-SRP. The most critical point here is our objectives are always two-fold, as this SRP is at the same time a course and a research experiment (see also Markulin, Bosch, Florensa & Montañola, 2022, p. 3). For instance, regarding Hilbert spaces, we wish to put two phenomena at play: on the one hand, in relation to the paradigm shift we investigate, we wish to reinstall the raison d'être of their use in quantum mechanics; on the other hand, in the context of a PhD devoted to the didactics of algebraic structures, we wish to see them play a unifying and simplifying role. Actually, our epistemological analyses lead to the conclusion we could meet both these targets at once, provided the use of the structures comes as an answer to the generating question or to a question in the process.

As was already mentioned, the passage from the finite-to the infinite-dimensional settings is a crucial step in quantum mechanics, and on in which the Hilbert space structure plays an important role. The latter structure is indeed the good framework where the practice acquired in low dimension can the most easily be transposed to infinite dimension. Though it is common in introductory quantum mechanics to cover several situations where both finite and infinite-dimensional frameworks play a role, even though the peculiarities of the infinite-dimensional case are often hidden -and with it the necessity of a more general, hence abstract, framework (see Lombard & Hausberger, to appear). This is particularly so when it comes to quantisation (eigenvalue problems), as show for instance most treatments of the infinite quantum well, the simplest realistic model of a quantum computer. That is, quantisation in this case can only be accounted for on the base of truly quantum mathematics.

Actually, the context of quantum computing puts at play numerous models to describe the machine, from two-level systems to anharmonic oscillators. Each time, the question of quantisation is crucial, as quantum bits are quantised states spanning two levels (usually marked |0〉 and |1〉). So, working on models of quantum computers surely brings about many epistemological aspects our analyses have highlighted so far. Consequently, we put forward the following generating question Q 0 : In what respect are quantum computers indeed quantum? As it stands, the questioning is however quite open, so we chose to add three questions, as is for instance done in Barquero (2009, p. 198 All that being said, a pitfall consists in wishing students go through certain questions for the sake of our research (for instance), thus leading us back to the "monumentalist" paradigm we wished to quit (on this point, see Chevallard, 2006, p. 8). Consequently, given this setting, we can only hope students will opt for a mathematically-leaning answer to our generating question.

Media and models

The choice of media gives us further latitude though, especially regarding how open the SRP will be. Indeed, in the case students actually enrich their milieu with them, media could channel the questioning towards given works, be they visited or not. So, media both increase the numbers of models of quantum computers under study and decrease the openness of the generating question, in order to balance its scope. During lab sessions, students will manipulate the software Qiskit. During class sessions, book excerpts or videos will enunciate facts about physical or mathematical models of quantum computers (such as the Block sphere, abstract two-level systems or quantum wells). In addition, the professors taking part in the experiment should play an important role in the media environment (see fig. 3). Lastly, we plan on producing tailor-made pieces of media, for instance to encourage the process of questioning about the links between the various models so introduced.

Fig. 1 :

 1 Fig. 1: The Didactic Engineering methodology applied to the implementation of an SRP (Bartolomé et al., 2018, p. 5)

Fig. 2 :

 2 Fig.2: Overview of some TU from the "General Mathematics" and "Fundamental Physics" programs that are taken by students participating in the experimentation

  ): Q 0a : What are quantum bits and how can you calculate with them? Q 0b : What phenomena allow quantum computers to operate? Q 0c : What characteristics of quantum systems are shared by quantum computers?

Fig. 3: Practicable sequence of sessions and prefiguration of media and models at play throughout the SRP

CONCLUSION

In this article, we have given a concrete instance of the implementation of both the preliminary and a priori analyses of an SRP set up at the interface between mathematics and quantum mechanics. We have shown by way of example that careful studies of the institutional and epistemological contexts are in order. Lastly, we have described the rationale underlying our choice of the generating question, which is a crucial step in the design of an SRP.

The module we set up is "a subject totally organised as an SRP" [START_REF] Bosch | How to integrate study and research paths into university courses? Teaching formats and ecologies[END_REF], which means in particular that no course adopting the paradigm of visiting works supports it. As a consequence, it seems necessary to find a balance between leaving the students' questioning totally open (which amounts to leave aside tailored planning of learning as well as research goals) and closing it so much that the generating question Q 0 "becomes a mere decorative and opportunistic artefact" (Barquero, 2009, p. 93). So, we have arranged the conditions for a cohabitation of the two main paradigms. Indeed, as put Chevallard (2011, p. 40), the development of the paradigm of questioning the world "does not cancel the paradigm of visiting pieces of knowledge (le paradigme de la visite des savoirs), but rather places it otherwise, at both epistemological and didactical levels". Organising this space needs careful preparation.