Laurent Moonens

AND Emmanuel Russ
email: emmanuel.russ@ujf-grenoble.fr

Heli Tuominen
email: heli.m.tuominen@jyu.fi

REMOVABLE SINGULARITIES FOR div v = f IN WEIGHTED LEBESGUE SPACES

Let w ∈ L 1 loc (R n ) be a positive weight. Assuming that a doubling condition and an L 1 Poincaré inequality on balls for the measure w(x)dx, as well as a growth condition on w, we prove that the compact subsets of R n which are removable for the distributional divergence in L ∞ 1/w are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for L p 1/w , 1 < p < +∞, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.

Introduction

In the past years, removable singularities of bounded vector fields satisfying div v = 0 in the distributional sense have been studied, e.g. by the first author [START_REF] Moonens | Removable singularities for the equation div v = 0. Real Anal. Exchange[END_REF], Silhavy [START_REF] Šilhavý | Divergence measure fields and Cauchy's stress theorem[END_REF] and Phuc and Torres [START_REF] Nguyen | Characterizations of the existence and removable singularities of divergence-measure vector fields[END_REF]. It has been shown, in particular, that a compact set S ⊆ R n can contain a non void support of the distributional divergence of a bounded vector field on R n , if and only if its (n -1)-dimensional Hausdorff measure is positive. As a matter of fact, all those results have immediate counterparts for vector fields defined on an open subset Ω of R n , satisfying the equation div v = f , where f is a locally integrable function on Ω, in case the latter equation admits at least one solution in L ∞ (Ω).

Given n/(n -1) < p < ∞, Phuc and Torres in [START_REF] Nguyen | Characterizations of the existence and removable singularities of divergence-measure vector fields[END_REF] showed a corresponding result for L p -vector fields. More precisely, given an open set Ω ⊆ R n and a locally integrable function f in Ω for which the equation div v = f is solvable in L p (Ω), their results imply that a compact set S ⊆ Ω contains a non void support of the distributional divergence of an L p -vector field in R n , if and only if cap p ′ (S) > 0, where cap p ′ is the capacity associated to the Sobolev space W 1,p ′ (R n ) (see Definition 4.4 below) and p ′ is the conjugate exponent to p verifying 1/p + 1/p ′ = 1.

On the other hand, given a (bounded) domain Ω ⊂ R n , it may happen that it is not possible to find a constant C > 0 such that given any f ∈ L ∞ (Ω), the equation

div v = f admits a bounded solution v ∈ L ∞ (Ω, R n ) satisfying v ∞ C f ∞ .
In fact, the existence in this context of an integrable weight w > 0 such that the divergence operator acting from the weighted Lebesgue space L ∞ 1/w (Ω, R n ) to the usual space L ∞ (Ω), admits a continuous right inverse, has been shown by Duran, Muschietti, the second author and Tchamitchian in [START_REF] Duran | Divergence operator and Poincaré inequalities on arbitrary bounded domains[END_REF] to be equivalent to the integrability of the geodesic distance (in Ω) to a fixed point x 0 ∈ Ω. Under the latter integrability 1 property, a similar invertibility result also holds when L ∞ 1/w (Ω, R n ) and L ∞ (Ω) are replaced by L p 1/w (Ω, R n ) and L p (Ω), respectively, with 1 < p < ∞. In order to get some understanding of how the introduction of a (locally) integrable weight w influences, in the associated weighted Lebesgue spaces, the set of singularities of a vector field having a prescribed divergence, we shall assume here that Ω = R n and f = 0 in the sequel (leaving the case where Ω is a bounded domain for a future work), and study first (see section 3) the possible sets of singularities of vector fields in L ∞ 1/w (R n , R n ) solving div v = 0 (or div v = f for some locally integrable function f on R n yielding at least a solution in L ∞ 1/w (R n , R n )), and make a similar study in appropriate weighted L p spaces.

More precisely, calling L p 1/w -removable any compact subset of R n that does not support any nonzero distributional divergence of a vector field v ∈ L p 1/w (R n , R n ), we show the following result (which combines our Theorems 3.24 and 4. [START_REF] Kilpeläinen | Weighted Sobolev spaces and capacity[END_REF]).

Theorem 1.1. Assume S is a compact subset of R n .

(i) If p = ∞ and if the weight w is 1-admissible and satisfies the growth condition (3.1) below, then S is L ∞ 1/w -removable if and only if H h (S) = 0, where H h is the Hausdorff outer measure associated to w as in Section 3.1.

(ii) If 1 < p < ∞ and if one has w p ′ -1 ∈ A p ′ , then S is L p 1/w -removable if and only if one has Cap w p ′ -1 p ′ (S) = 0, where Cap w p ′ -1 p ′ is the Sobolev capacity associated to w p ′ -1 appearing in Definition 4.4.

In the previous statement, we mean by saying that a weight is 1-admissible, that it is doubling and satisfies a (1, 1)-Poincaré inequality (see Definition 2.2 below), while A p ′ stands for the Muckenhoupt class introduced (see Definition 2.3). Note that any A 1 weight is 1-admissible (see Remark 2.4). Note that H h is the classical (spherical) Hausdorff measure of dimension n -1 is case w = 1.

A first remark about the previous theorem is that we recover, when w = 1, the result mentioned above stating that L ∞ -removable (compact) sets are exactly those satisfying H n-1 (S) = 0, and similarly in L p .

An interesting case covered by our results is the one when the weight w equals +∞ on a "large" set (e.g. on a set of positive Hausdorff dimension) -allowing the vector fields in L ∞ 1/w to have singular pointwise behaviour on this "large" set. More precisely, as an interesting complement to [Theorem 1.1, (i)], we provide examples of A 1 weights w of the form w(x) := dist(x, C) -α , with α > 0, whose singular set C has positive Hausdorff dimension yet is L ∞ 1/w -removable for the equation div v = 0 for some values of α related to the Hausdorff dimension of C and that of the ambient space. This is our Example 3.26.

The proof of both parts of Theorem 1.1 follow the same structure. Sufficient removability conditions are obtained by truncation arguments as in De Pauw [START_REF] De | On the exceptional sets of the flux of a bounded vectorfield[END_REF] and [START_REF] Moonens | Removable singularities for the equation div v = 0. Real Anal. Exchange[END_REF] (case p = ∞) and [START_REF] Nguyen | Characterizations of the existence and removable singularities of divergence-measure vector fields[END_REF] (case 1 < p < ∞), but extra care is needed for we cannot, in this weighted context, rely on straightforward estimates relating the (weighted) perimeter of a ball B of radius r to the integral 1 r B w. Showing those conditions are also necessary for a set S to be removable is done by solving the

equation div v = µ in L p 1/w (R n , R n
) for some suitable measures µ supported in S, and showing it might admit non trivial solutions in case S does not satisfy the conditions in question. This is done, as in Bourgain and Brezis [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF] and [START_REF] Nguyen | Characterizations of the existence and removable singularities of divergence-measure vector fields[END_REF], by using a simple version of the closed range theorem. Note that the case where p = +∞ cannot be dealt with using capacity arguments, see Remark 3.25 below.

The paper is organized as follows. In Section 2, we give definitions and basic properties of p-admissible weights and introduce the notations used in the paper. In Section 3, we study the removability question for the divergence equation for weighted L ∞ -vector fields. In the proofs, we need theory of functions of bounded variation in the weighted case, the weighted Hausdorff measure of co-dimension one, the boxing inequality and a version of Frostman's lemma. Those, as well as some technical lemmas are presented before the main results of the section. Section 4 contains characterization of removable sets for the divergence equation for weighted L p -vector fields. In this section, important tools are weighted Sobolev spaces, different capacities and some tools from the general theory of L q -capacities, discussed before the main results.

Weights and notation

A locally integrable function w : R n → R is a weight if w(x) > 0 for almost every x ∈ R n . We say that the weight w is doubling if there exists a constant C D 1 (called the doubling constant of w) such that for any x ∈ R n and any r > 0 one has:

B(x,2r) w C D B(x,r) w,
where B(x, r) denotes the Euclidean (open) ball with center x and radius r in R n and where one integrates with respect to Lebesgue measure. An iteration of the doubling inequality then ensures that one has, for all t > 0:

(2.1)

B(x,tr) w C D t s D B(x,r) w,
where s D := log 2 C D is the doubling dimension of the weighted space (R n , w).

For 1 p ∞, the weighted L p -space, L p w (R n , R k ), consists of measurable functions u : R n → R k for which |u| p w ∈ L 1 (R n , R k ), and we let:

u p,w := R n |u| p w 1/p , for p < ∞, and u ∞,w := uw ∞ . We finally let L p (R n ) := L p (R n , R).
In the sequel we shall denote by Lip c (R n ) the set of all compactly supported (real valued) Lipschitz functions in R n . For a weight w, the weighted Euclidean space endowed with the Euclidean metric and the measure dµ = w dx is denoted by (R n , w). Definition 2.1. Let 1 p < +∞ be a real number. We shall say that the weighted space (R n , w) supports a weighted (1, p)-Poincaré inequality in case that there exist constants C P > 0 and τ > 1 such that for any ϕ ∈ Lip c (R n ), any x ∈ R n and any r > 0 we have:

B(x,r) |ϕ -φx,r |w C P r B(x,τ r) |∇ϕ| p w 1/p
, where we let φx,r := B(x,r) ϕw and where B ϕw denotes the mean value 1 B w B ϕw for any Borel set B satisfying |B| > 0.

We use the class of p-admissible weights as in [16, Section 1.1] and [START_REF] Björn | Nonlinear potential theory on metric spaces[END_REF]Definition A.6]. Such weights are important in the nonlinear potential theory developed in [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF], see also [6, Appendix A]. Definition 2.2. Let 1 p < ∞ be a real number. A weight w is said to be p-admissible in case it is doubling and the weighted space (R n , w) supports a (1, p)-Poincaré inequality.

It follows from Heinonen, Kilpeläinen and Martio [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF]Corollary 20.9] (in the second edition of their book) that p-admissible weights for p > 1 satisfy a bunch of other interesting properties, among which the following Poincaré inequality for compactly supported functions (see [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF]Section 1.4] and [34, Corollary 2.1.5]): there exists κ > 0 such that for any ball B = B(x, r) ⊆ R n and any ϕ ∈ Lip c (R n ) supported in B(x, r), we get: An important class of p-admissible weights are the A p -weights, which were defined by Muckenhoupt in [START_REF] Muckenhoupt | Weighted norm inequalities for the Hardy maximal function[END_REF], where he showed that when 1 < p < ∞, the Hardy-Littlewood maximal operator is L p w -bounded if and only if w

∈ A p . Definition 2.3. A weight w is an A p -weight, 1 < p < ∞, if sup B B w B w 1/(1-p) p-1 < ∞,
where the supremum is taken over all balls B ⊂ R n . Note that the

A p -condition implies that w ∈ L 1/(1-p) (R n ) locally. A weight w is an A 1 -weight, if sup B B w ess sup B 1 w < ∞.
Remark 2.4. The fact that A p -weights are p-admissible has been proved in [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF]Theorem 15.21] in case p > 1 and in [START_REF] Björn | Poincaré inequalities for powers and products of admissible weights[END_REF]Theorem 4] in case p = 1. The doubling property follows easily from the A p -condition but the validity of a weighted (1, p)-Poincaré inequality requires more work.

For further properties of A p -weights, see for example [START_REF] Coifman | Weighted norm inequalities for maximal functions and singular integrals[END_REF], [START_REF] Jones | Factorization of A p weights[END_REF], [32, Chapter V] and [START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF]Remark 1.2.4] and for examples of p-admissible weights that are not A p -weights, see for example [START_REF] Chanillo | Poincaré inequalities for a class of non-A p weights[END_REF] and [START_REF] Franchi | Weighted Sobolev-Poincaré inequalities for Grushin type operators[END_REF].

Example 2.5. It is classical that for -n < η 0, w η (x) := |x| η is an A 1 -weight. Moreover any weight of the form w η (x) = |x| η for η > -n is doubling (while it may not be A 1 ). See e.g. [START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF]Example 1.2.5].

For a set A ⊂ R n , M + (A) is the set of locally finite (nonnegative) Radon measures supported in A.

The case of weighted L ∞ vector fields

In this section, we study the removability question for the divergence equation for weighted L ∞ -vector fields. We start by defining some tools and proving results needed in the proofs -those include weighted Hausdorff content and measure of codimension 1 and functions of bounded variation in the weighted setting. Our main results in this section hold for doubling weights that satisfy a (1, 1)-Poincaré inequality. In Theorem 3.17, we show that any compact set S ⊆ R n with H h (S) = 0, is L ∞ 1/w -removable for div v = 0. Indeed, vanishing Hausdorff measure almost characterizes removable sets -if the weight w satisfies an additional mild growth condition, then a compact set S ⊆ R n is L ∞ 1/w -removable for div v = 0 if and only if H h (S) = 0, see Theorem 3.24.

3.1. Hausdorff contents. Let w be a weight. Associated to w, define a (spherical) measure function h on (closed) balls B(x, r) by: h(B(x, r)) := 1 r B(x,r) w.

According to the usual Carathéodory construction (see [START_REF] Mattila | Geometry of sets and measures in Euclidean spaces[END_REF]Section 4.1]), we also define a weighted co-dimension 1 (spherical) Hausdorff outer measure (as in Turesson [34, Section 2.3] and Nieminen [START_REF] Nieminen | Hausdorff measures, capacities, and Sobolev spaces with weights[END_REF]) by letting first, for 0 < δ ∞ and E ⊆ R n :

H h δ (E) := inf j∈J h(B(x j , r j )),
where the infimum is taken on all countable coverings of E by balls (B(x j , r j )) j∈J satisfying r j δ for all j ∈ J. We define then

H h (E) := lim δ→0 H h δ (E).
It follows from [START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF]Proposition 2.3.3] that H h δ is an outer measure for any 0 < δ ∞, and that H h is a Borel regular outer measure.

The following straightforward lemma will be useful in the sequel.

Lemma 3.1. Let h be associated to the weight w as above and assume that the compact set S ⊆ R n satisfies H h (S) = 0. Then S is Lebesgue-negligible.

Proof. If η > 0 is an arbitrary positive number, let (B(x j , r j )) j∈J be a finite family of balls covering S, verifying r j 1 for each j ∈ J as well as:

j∈J h(B(x j , r j )) η.
We then compute:

S w j∈J B(x j ,r j ) w j∈J B(x j ,r j ) w = j∈J r j h(B(x j , r j )) j∈J h(B(x j , r j )) η,
so that one has S w = 0, since η > 0 is arbitrary. It follows that the set {x ∈ S : w(x) > 0} is Lebesgue-negligible, and hence that S itself is Lebesgue-negligible for we assumed than one has w > 0 a.e. on R n .

When R n is endowed with a doubling weight which grows fast enough, we have the following version of Frostman's lemma, which is a particular case of [START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF]Theorem 3.4.27].

Lemma 3.2. Assume that w is a doubling weight on R n . If moreover, for any x ∈ R n :

(3.1) lim r→∞ h(B(x, r)) = ∞,
and if B ⊆ R n is a Borel set verifying H h ∞ (B) > 0,
then there exists a nontrivial measure µ ∈ M + (B) satisfying the following inequality:

µ(B(x, r)) Ch(B(x, r)),
for any x ∈ R n and r > 0.

Example 3.3. An easy computation shows that the weight w η defined in Example 2.5 also satisfies condition (3.1) in case 1n < η 0.

In fact, assuming the growth condition (3.1) only for x = 0, it is equivalent for a bounded set B to satisfy H h ∞ (B) = 0 or H h (B) = 0. Lemma 3.4. Assume that w is doubling and that one has:

(3.2) lim r→∞ h(B(0, r)) = ∞.
For any bounded set B ⊆ R n , the equalities

H h (B) = 0 and H h ∞ (B) = 0 are equivalent. Proof. Fix a bounded set B ⊆ R n . Since one has H h ∞ (B) H h (B), it is clear that H h ∞ (B) = 0 provided that H h (B) = 0. Conversely, assume that H h ∞ (B) = 0. Choose R 0 > 0 such that one has B ⊂ B(0, R 0 ). Let f (r) := h(B(0, r)) for all r > 0. Since (3.2) yields lim r→+∞ f (r) = +∞,
we are allowed to choose R > R 0 such that f (r) > 3 s D C D for all r > R, where C D > 0 and s D := log 2 C D are the doubling constants of w (see (2.1) above).

Fix now δ > 0 and let:

c δ := max 3, 2 + R 0 δ .
Choose then ε > 0 with:

ε < min 1, 1 c s D δ C D R B(0,δ)
w .

Since one has H h ∞ (B) = 0, there exists an (at most countable) family of balls (B(x j , r j )) j∈J covering B and satisfying:

j∈J h(B(x j , r j )) = j∈J 1 r j B(x j ,r j ) w ε.
One may assume that, for all j ∈ J, one has B(x j , r j ) ∩ B = ∅, so that one computes |x j | r j + R 0 . By the doubling property (2.1), we get for any j ∈ J such that r j > R 0 :

B(0,r j ) w B(x j ,r j +|x j |) w C D 1 + |x j | r j s D B(x j ,r j ) w C D 2 + R 0 r j s D B(x j ,r j ) w 3 s D C D B(x j ,r j )
w, so that:

j∈J f (r j ) 3 s D C D ε < 3 s D C D .
We hence get r j R for all j ∈ J, for the latter inequality is obvious in case j ∈ J is such that one has r j R 0 < R. Assume now that j ∈ J is such that r j > δ. If moreover one has r j > R 0 , the computations above show that:

B(0,r j ) w 3 s D C D r j • 1 r j B(x j ,r j ) w 3 s D C D Rε c s D δ C D Rε.
In case one has δ < r j R 0 , we compute using again (2.1):

B(0,r j ) w C D 2 + R 0 r j s D B(x j ,r j ) w 2 + R 0 δ s D C D r j ε c s D δ C D Rε.
Hence in both cases we have B(0,r j ) w c s D δ C D Rε. We hence get:

B(0,δ) w B(0,r j ) w c s D δ C D Rε < B(0,δ) w,
which is impossible. Therefore r j δ for each j ∈ J, so that we get:

H h δ (B) j∈J h(B(x j , r j )) ε.
Since ε is arbitrary small, this yields H h δ (B) = 0. Finally, we get H h (B) = 0 for the previous estimates yield H h δ (B) = 0 for any δ > 0. Remark 3.5. In case w is 1-admissible, the previous lemma can be obtained by combing results [START_REF] Kinnunen | Lebesgue points and capacities via the boxing inequality in metric spaces[END_REF]Lemma 7.6,Remark 7.4] by Kinnunen, Korte, Shanmugalingam and the third author.

Miranda's BV -functions.

In the more general context of metric measure spaces, M. Miranda introduced in [START_REF] Miranda | Functions of bounded variation on "good" metric spaces[END_REF] the notion of function with bounded variation. We shall in the sequel particularize some results obtained by Miranda to the context where R n is endowed by an appropriate weight; to this purpose, we need to introduce some terminology.

Following Miranda [START_REF] Miranda | Functions of bounded variation on "good" metric spaces[END_REF], read in this weighted context by Camfield [START_REF] Scott | Comparison of BV norms in weighted Euclidean spaces and metric measure spaces[END_REF], define the metric (weighted) variation of u ∈ L 1 w (R n ) by:

Du w := inf lim k→∞ R n |∇ϕ k |w : (ϕ k ) ⊆ Lip(R n ), ϕ k → u in L 1 w .
The following Theorem, stated here for reader's convenience but unnecessary for our purposes, is a direct consequence of two deep results by Camfield [9, Theorems 3.2.6 and 3.4.5]. It shows that, under some regularity conditions on the weight w, the metric variation of a Lipschitz function is identical to the L 1 w norm of its gradient.

Theorem 3.6 (Camfield). Assume that w is locally integrable and lower semicontinuous. If moreover there exists an H h -negligible set E outside which w is continuous and (strictly) positive, then for any ϕ ∈ Lip c (R n ) we have:

(3.3) Dϕ w = ∇ϕ 1,w = R n |∇ϕ|w.
For 1-admissible weights, the equality (3.3) can be replaced by a comparison between the two quantities involved, at least for Lipschitz functions (see [25, p. 992, (19) and below]). Proposition 3.7. Assume that the weight w is 1-admissible. Then for any ϕ ∈ Lip c (R n ) we have: c ∇ϕ 1,w Dϕ w ∇ϕ 1,w , where c > 0 is independent of ϕ.

The following proposition is a particular case of M. Miranda's Coarea formula [START_REF] Miranda | Functions of bounded variation on "good" metric spaces[END_REF]Proposition 4.2]. The (weighted) perimeter of a Borel set B, denoted by P w (B), is defined by: P w (B) := Dχ B w .

Proposition 3.8. Assume that w is a 1-admissible weight. Then, for any u ∈ L 1 w (R n ) verifying Du w < +∞, we have:

Du w = R P w ({u > t}) dt.
We shall also make use of the following boxing inequality, due (in the more general framework of measure metric spaces) to Kinnunen, Korte, Shanmugalingam and the third author [20, Theorem 3.1]. Theorem 3.9 (Boxing inequality). Assume that w is a 1-admissible weight. There exists a constant C B = C B (C D , C P ) > 0 such that for any open set U ⊆ R n verifying U w < ∞, we can find sequences (x i ) ⊆ U and (r i ) ⊆ (0, +∞) satisfying the following conditions:

(i) B(x i , r i ) ∩ B(x j , r j ) = ∅ for i = j; (ii) U ⊆ i∈N B(x i , 5r i ); (iii) ∞ i=0 h(B(x i , 5r i )) = ∞ i=0 1 5r i B(x i ,5r i ) w C B P w (U).
We are now ready to study removable singularities of divergence-free vector fields 3.3. Removable singularities. Assume in this whole section that w is a weight.

Given vector field v ∈ L ∞ 1/w (R n , R n ) and ϕ ∈ Lip c (R n ), it is clear that we have, a.e. on R n : |v • ∇ϕ| v w |∇ϕ|wχ supp ϕ v ∞,1/w ∇ϕ ∞ wχ supp ϕ .
Since w is locally integrable, one can define the (extended) distributional divergence of v by:

div v, ϕ := - R n v • ∇ϕ, for any ϕ ∈ Lip c (R n ). Definition 3.10. Let S ⊆ R n be compact and f ∈ L 1 loc (R n ). The set S is said to be L ∞ 1/w -removable with respect to the equation div v = f in case for any v ∈ L ∞ 1/w (R n , R n ), the equality (3.4) div v, ϕ = R n f ϕ
for any ϕ ∈ Lip c (R n ) with supp ϕ ∩ S = ∅ (which we shall abbreviate "div v = f outside S") implies that (3.4) also holds for any ϕ ∈ Lip c (R n ) (which we shall abbreviate "div v = f in R n ").

Remark 3.11. Assume that w is an A 1 weight. According to the fact that given

ϕ ∈ Lip c (R n ), one has ϕ ∈ L 1 w (R n ) ∩ L 1 loc (R n ) and ∇ϕ ∈ L 1 w (R n ) ∩ L 1 loc (R n ), [34, Corollary 2.1.5] ensures that the above definition remains unchanged in case Lip c (R n ) is replaced by C ∞ c (R n
); yet it is not clear that the two definitions are equivalent without this extra assumption on w.

Remark 3.12. Assume that f ∈ L 1 loc (R n ) yields a solution v 0 ∈ L ∞ 1/w (R n , R n ) to the equation div v = f , in the sense that div v 0 = f in R n . Assume also that S is L ∞ 1/w -removable for the equation div v = 0. Fix now v ∈ L ∞ 1/w (R n , R n ) and assume that div v = f outside S.
Since it is clear that one has div(vv 0 ) = 0 outside S, the removability assumption made on S ensures that one has div(vv 0 ) = 0 in R n ; we hence get div v = f in R n , and S is L ∞ 1/w -removable for the equation div v = f . Conversely, one shows in a similar fashion that any L ∞ 1/w -removable set for div v = f is also removable for div v = 0. Hence we shall assume in the sequel that f = 0.

3.4.

A sufficient condition for a set to be L ∞ 1/w -removable. Let us observe first that it suffices, in order to show that a set is removable, that one is able to construct, in any of its neighborhood, an appropriate Lipschitz approximation of its characteristic function. Lemma 3.13. Assume that S is compact and that for any ε > 0, there exists a neighborhood U of S satisfying U w ε together with a Lipschitz function χ ∈ Lip c (R n ) satisfying supp χ ⊆ U, 0 χ 1 in U, χ = 1 in a neighborhood of S as well as ∇χ 1,w ε. Then S is L ∞ 1/w -removable for the equation div v = 0. Proof. Assume that one has div v, ϕ = 0 for any ϕ ∈ Lip c (R n ) satisfying supp ϕ ∩ S = ∅. Fix then ϕ ∈ Lip c (R n ), let ε > 0 and let χ be associated to U and ε as in the above assumption. Observe that one has, by hypothesis:

div v, ϕ = div v, ϕχ v ∞,1/w ∇ϕ ∞ U w + ϕ ∞ ∇χ 1,w ε v ∞,1/w ( ∇ϕ ∞ + ϕ ∞ ).
It then follows that div v, ϕ = 0 for ε > 0 is arbitrary. This establishes that S is L ∞ 1/w -removable for the equation div v = 0.

The following proposition will be useful while showing that compact sets with H h (S) = 0 are removable for the divergence equation.

∈ Lip c (R n ) satisfying supp ϕ ⊆ U, ϕ = 1 on V ′ and ∇ϕ 1,w ε + 2P w (V ). Proof. Start by choosing ψ ∈ Lip c (R n ) satisfying χ V ′ ψ χ V . Choose, according to the definition of P w (V ), a sequence (ϕ k ) ⊆ Lip(R n ) converging in L 1
w (R n ) to χ V and such that for any k ∈ N, one has:

∇ϕ k 1,w ε 4 + P w (V ).
Replacing if necessary ϕ k by min[1, max(ϕ k , 0)] (which does not increase the norm of the gradient of ϕ k on the complement of a negligible set), one can assume that one has 0 ϕ k 1 for each k. Replacing if necessary ϕ k by θϕ k for k ∈ N, where θ ∈ Lip c (R n ) satisfies 0 θ 1, as well as θ = 1 on V and θ = 0 outside U, and observing that one has, for any k ∈ N:

|∇(θϕ k )|w |∇θ||ϕ k -χ V |w + |∇ϕ k |w ∇θ ∞ |ϕ k -χ V |w + |∇ϕ k |w,
we may also assume that supp ϕ k ⊆ U for any k ∈ N. Define now, for k ∈ N:

φk := ψ + (1 -ψ)ϕ k .
It is clear that φk is Lipschitz and has compact support for any k ∈ N; moreover one computes, for a.e. x ∈ R n (recall that one has

|∂V ′ | = |∂V | = 0): |∇ φk (x)|      0 if x ∈ V ′ , 2|∇ϕ k (x)| + ∇ψ ∞ |1 -ϕ k (x)| if x ∈ V \ V ′ , |∇ϕ k (x)| if x ∈ ∁V.
We hence get, a.e. on R n , for any k ∈ N:

|∇ φk |w 2|∇ϕ k |w + ∇ψ ∞ |χ V -ϕ k |w.
Since we have

ϕ k → χ V in L 1 w (R n ), there exists k ∈ N such that: ∇ φk 1,w ε + 2P w (V ).
We can hence take ϕ := φk , and the proof is complete.

The next lemma, taken from [21, Lemma 6.2], will be of some help. We include its proof for the sake of clarity.

Lemma 3.15. Assume that w is a 1-admissible weight. Given x ∈ R n and r > 0, there exists ρ ∈ [r, 2r] such that one has:

P w (B(x, ρ)) Ch(B(x, ρ)),
where C > 0 is independent of x and ρ.

Proof. Fix x ∈ R n and r > 0. Define ϕ ∈ Lip c (R n ) by the formula:

ϕ(y) := max 0, min 2 - 1 r |y -x|, 1 , for y ∈ R n . It is clear that one has |∇ϕ| 1 r χ B(x,2r)\B(x,r) on R n .
Hence we get, using M. Miranda's coarea formula (Proposition 3.8) and Proposition 3.7:

1 0 P w (B(x, (2 -t)r)) dt = R P w ({ϕ > t}) = Dϕ w R n |∇ϕ|w 1 r B(x,2r)\B(x,r) w C D h(B(x, r)).
If we now choose ρ ∈ [r, 2r] such that one has P w (B(x, ρ))

1 0 P w (B(x, (2 -t)r)) dt,
we get:

P w (B(x, ρ)) C D h(B(x, r)) 2C D h(B(x, ρ)),
and the lemma is proved.

The next proposition is a first step towards showing H h -negligible sets are removable.

Proposition 3.16. Assume that w is a 1-admissible weight. If S is compact and satisfies H h (S) = 0, then for any ε > 0 and any neighborhood U of S, one can find an open subset V ⊂⊂ U, satisfying S ⊆ V as well as:

P w (V ) ε.
Moreover V can be chosen to be a finite union of balls.

Proof. Fix η > 0. Let 0 < δ < 1 4 dist(S, ∁U) and observe that H h δ (S) = 0. Hence there are balls B(x j , r j ) for some x j ∈ R n , r j > 0, j ∈ J covering S (since S is compact we can assume J to be finite), satisfying r j δ for each j ∈ J and: j∈J h(B(x j , r j )) η.

According to Lemma 3.15, choose for each j ∈ J a radius ρ j ∈ [r j , 2r j ] for which one has:

P w (B(x j , ρ j )) C ρ j B(x j ,ρ j ) w;
observe that the doubling property of w yields, for any j ∈ J:

P w (B(x j , ρ j )) CC D r j B(x j ,r j ) w = CC D h(B(x j , r j )).
Letting V := j∈J B(x j , ρ j ) ⊂⊂ U, we hence get, using the subadditivity of the weighted perimeter ([25, Proposition 4.7]):

P w (V ) j∈J P w (B(x j , ρ j )) CC D j∈J h(B(x j , r j )) CC D η,
from which the desired inequality readily follows.

The previous proposition together with Lemma 3.13 and Proposition 3.14, yield a sufficient removability condition. Theorem 3.17. Assume that w is a 1-admissible weight. Then any compact set S ⊆ R n verifying H h (S) = 0, is L ∞ 1/w -removable for div v = 0. Proof. According to Lemma 3.1, the set S is Lebesgue-negligible. So we may fix ε > 0, and let U be a neighborhood of S for which one has U w ε (we may moreover assume that U consists of a finite union of balls). Start, according to Proposition 3.16, by choosing an open set V ⊂⊂ U satisfying P w (V ) ε 4 as well as S ⊆ V (and consisting of a finite union of balls). Choose, according to Proposition 3.14, a Lipschitz function ϕ ∈ Lip c (R n ) equal to 1 in a neighborhood V ′ ⊂⊂ V of S and verifying ∇χ 1,w ε 2 + 2P w (V ). One now computes:

∇χ 1,w ε 2 + 2P w (V ) ε,
and it follows from Lemma 3.13 that S is L ∞ 1/w -removable for div v = 0. 3.5. A necessary condition for a set to be L ∞ 1/w -removable. We first state the following estimate for measures satisfying a weighted-Frostman condition. It is similar to [34, Theorem 2.6.3], although we do not require here w to be A 1 .

Proposition 3.18. Assume that w is a 1-admissible weight. Let µ be a (nonnegative) Radon measure on R n and assume that for any x ∈ R n and any r > 0, we have :

(3.5)
µ(B(x, r)) Ch(B(x, r)).

Then for any ϕ ∈ Lip c (R n ) we have:

R n |ϕ| dµ M ∇ϕ 1,w ,
where M > 0 is independent of ϕ.

Proof. Fix t > 0 and let U t be the bounded open set defined by U t := {x ∈ R n : |ϕ(x)| > t}. According to the weighted boxing inequality (Theorem 3.9), there exists (x i ) ⊆ U t and (r i ) ⊆ (0, ∞) satisfying conditions (i) to (iii) in Theorem 3.9, with U t instead of U. Write then

µ(U t ) ∞ i=0 µ(B(x i , 5r i )) C ∞ i=0 1 5r i B(x i ,5r i ) w CC B P w (U t ).
Now we have, using Cavalieri's principle and Miranda's coarea formula (Proposition 3.8):

R n |ϕ| dµ = ∞ 0 µ(U t ) dt CC B ∞ 0 P w (U t ) dt = CC B D|ϕ| w .
According to Proposition 3.7, we have

D|ϕ| w ∇|ϕ| 1,w = ∇ϕ 1,w ,
since one has |∇|ϕ|| = |∇ϕ| almost everywhere. This finishes the proof since one can take M := CC B > 0.

The following lemma, of functional analytic nature, avoids unnecessary technicalities in the sequel, and has been suggested by J. Boël. Lemma 3.19. Let X and Y be two normed spaces, and assume that T : X → Y is linear and isometric, i.e. that one has T (x) Y = x X for every x ∈ X. Then its adjoint map T * : Y * → X * is surjective. Remark 3.20. Observe that one does not need, in the above statement, any kind of completeness to be satisfied, neither by X nor by Y .

Proof. Fix f ∈ X * . It is clear by assumption that T is injective; hence the formula:

g 0 , T (x) := f, x , defines a linear map g 0 : T (X) → R satisfying | g 0 , T (x) | f x X = f T (x) Y
for any x ∈ X. The Hahn-Banach Theorem hence ensures the existence of g ∈ Y * verifying g ↾ T (X) = g 0 , meaning that one has:

T * g, x := g, T (x) = g 0 , T (x) = f, x ,
for any x ∈ X. This shows that T * g = f and establishes the surjectivity of T * .

We are now able to show that any Radon measure satisfying a weighted Frostman condition is the divergence of a vector field in L ∞ 1/w . Theorem 3.21. Let w be a 1-admissible weight. If µ is a (nonnegative) Radon measure verifying condition (3.5) above, then there exists v ∈ L ∞ 1/w (R n , R n ) such that, for any ϕ ∈ Lip c (R n ), one has:

div v, ϕ = R n ϕ dµ;
in particular, div v = µ holds on R n , in the distributional sense.

Proof. It is inspired by [START_REF] Nguyen | Characterizations of the existence and removable singularities of divergence-measure vector fields[END_REF]Theorem 3.3]. Denote by X the space Lip c (R n ) endowed with the norm ϕ X := ∇ϕ 1,w , and define an operator

T : X → L 1 w (R n , R n ), u → -∇u. Since T is clearly isometric, its adjoint operator div : [L 1 w (R n , R n )] * = L ∞ 1/w (R n , R n ) → X * is surjective. Yet Proposition 3.
18 ensures that we have µ ∈ X * . The proof is complete.

Remark 3.22. To obtain the isometric isomorphism between

[L 1 w (R n , R n )] * and L ∞ 1/w (R n , R n ), it suffices to notice that given f ∈ [L 1 w (R n , R n )] * , the formula v → f, v
w defines a bounded linear map on L 1 (R n , R n ) with the same norm as f . Hence there

exists g ∈ L ∞ (R n , R n ) with g ∞ = f such that for any u ∈ L 1 w (R n , R n ) one has f (u) = R n u • gw.
Yet the function h := gw belongs to

L ∞ 1/w (R n , R n ) and h L ∞ 1/w = f .
We now have a necessary condition on a compact set S for it to be L ∞ 1/w -removable. Theorem 3.23. Assume that w is a 1-admissible weight that satisfies condition

(3.1). If the compact set S ⊆ R n is L ∞ 1/w -removable for div v = 0, then H h ∞ (S) = 0. Proof.
To show this, assume that one has H h ∞ (S) > 0. According to the above weighted version of Frostman's lemma (Lemma 3.2), there exists a nontrivial Radon measure µ on R n supported in S and satisfying:

µ(B(x, r)) Ch(B(x, r)),
for any x ∈ R n and r > 0. Since Theorem 3.21 ensures the existence of a vector field

v ∈ L ∞ 1/w (R n , R n ) such that div v = µ, we get div v = 0 while div v, ϕ = 0 holds in case ϕ ∈ Lip c (R n ) satisfies supp ϕ ∩ S = ∅. Hence S cannot be L ∞ 1/w -removable for div v = 0.
According to Lemma 3.4, Theorems 3.17 and 3.23 give a complete characterization of L ∞ 1/w -removable (compact) subsets of R n for the equation div v = 0, in case w satisfies (3.1). Theorem 3.24. Assume that w is a 1-admissible weight and that w satisfies condition (3.1). Let S ⊆ R n be compact. Under those assumptions, S is L ∞ 1/w -removable for div v = 0 if and only if one has H h (S) = 0.

Remark 3.25. The reader may wonder why we did not use the capacity theory to establish Theorem 3.24. Indeed, recall that in the classical unweighted case, H n-1 , Hausdoff measure of dimension n -1 and 1-capacity cap 1 (defined by (4.2) below) have same zero sets, see [START_REF] Evans | Measure theory and fine properties of functions[END_REF]Theorem 3,p. 193]. A corresponding result holds in the setting of metric spaces: by [START_REF] Kinnunen | Lebesgue points and capacities via the boxing inequality in metric spaces[END_REF]Theorem 3.5] Hausdorff content of codimension 1 and 1-capacity are comparable for compact sets. The reason why we cannot use this result here is that, in [START_REF] Kinnunen | Lebesgue points and capacities via the boxing inequality in metric spaces[END_REF], admissible functions for 1-capacity belong to the Sobolev space defined using weak upper gradients, and it is not known if |∇u| is a 1-minimal weak upper gradient of each locally Lipschitz function u, see [START_REF] Björn | Nonlinear potential theory on metric spaces[END_REF]Appendix A2].

Before discussion on L p -analogues of the previous results, let us mention an interesting example.

Example 3.26. Given a cube Q, we denote by σ(Q) the length of its edges. For 0 < λ < 1/2, we let E (Q, λ) stand for the collection of 2 n cubes contained in Q whose edges have length λσ(Q), arranged in such a way that each cube of E (Q, λ) has a common vertex with Q.

Fix a sequence λ = (λ k ) k 1 ⊆ (0, 1/2). Write E 0 := [0, 1] n and E 0 = {E 0 }. For each k 1, define inductively E k (λ 1 , . . . , λ k ) := ∪{E (Q, λ k ) : Q ∈ E k-1 (λ 1 , . . . , λ k-1 )} and E k (λ 1 , . . . , λ k ) := ∪E k (λ 1 , . . . , λ k ) (
or briefly E k and E k when the underlying sequence is clear). In particular, each cube in E k has edges of length σ k := λ 1 . . . λ k . We finally let

E(λ) = ∞ k=1 E k .
Associated to λ, we choose a nondecreasing function h : [0, ∞) → [0, ∞) satisfying h(0) = 0 and h(σ k ) = 2 -kn for each k 1. Recall that the Hausdorff h-measure of E(λ) is positive and finite (see [START_REF] Mattila | Geometry of sets and measures in Euclidean spaces[END_REF]Section 4.9]).

If for example we fix a real number 0 < s < n and take λ k = 2 -n s for each k, this yields σ k = 2 -kn s for each k, and we may take h(t) = t s . Hence in this case the set E := E(λ) has Hausdorff dimension s and is s-Ahlfors regular according to [START_REF] Mackay | Conformal dimension. Theory and application[END_REF]Theorem 8.3.2] (see also [START_REF] Moran | Additive functions of intervals and Hausdorff measure[END_REF] and [START_REF] Hutchinson | Fractals and self-similarity[END_REF]). It hence follows from [START_REF] Aimar | Powers of distances to lower dimensional sets as Muckenhoupt weights[END_REF]Theorem 7] that for any 0 γ < 1, the map w γ defined for a.e. x ∈ R n by

w γ (x) := dist(x, C) γ(s-n) , is an A 1 -weight.
Observing now that E k consists of 2 kn cubes of side length σ k , compute, for any one of those cubes Q and B ⊇ Q, the smallest ball containing Q having the same center:

1

|B| B w γ A γ ess inf B w γ ,
where A γ > 0 is a constant associated with the A 1 -property of w γ . Since the set:

x ∈ B : dist(x, E) 1 4 σ k 2 -σ k+1
has positive Lebesgue measure, this yields:

1 |B| B w γ A γ 4 γ(n-s) σ k 2 -σ k+1 γ(s-n)
.

Since 2 kn of those balls suffice to cover E k , we compute:

H h (E k ) 2 kn h(B) = 2 kn • √ 2 √ nσ k B w γ C(n, γ, s)2 kn σ n-1 k σ k 2 -σ k+1 γ(s-n) = C(n, γ, s)2 kn σ n-1+γ(s-n) k 1 2 -2 -n s γ(n-s) = C ′ (n, γ, s)2 -k (1-γ) n 2 s -n s +(γ-1)n .
The last expression tends to zero for k → ∞, provided that one has:

(3.6) s < n - 1 1 -γ .
This yields H h (E) = 0 if inequality (3.6) is satisfied. In particular, the set E is then L ∞ 1/wγ -removable for the equation div v = 0 while we have w γ = +∞ on E.

The case of weighted L p vector fields

In this section, we study the removability question for the equation div v = 0 for L p 1/w -vector fields, that is measurable functions v : R n → R n such that

v L p 1/w := R n |v(x)| p 1 w(x) dx 1/p < +∞,
1 < p < +∞. In Theorems 4.17 and 4.18, we give a characterization for removability by showing that, under suitable assumptions on the weight w, a compact set S is L p 1/w -removable for div v = 0 if and only if Cap w p ′ -1 p ′ (S) = 0. Before that, we recall the definition of weighted Sobolev spaces, different capacities and prove some lemmas. In the whole section, p and p ′ are conjugate exponents satisfying 1/p + 1/p ′ = 1.

∈ L p ′ -1 loc (R n ), then the distributional divergence of v ∈ L p 1/w (R n , R n ) is well defined by the formula: div v, ϕ := - R n v • ∇ϕ, for ϕ ∈ Lip c (R n ).
Removability in the L p -case is defined similarly as for L ∞ -vector fields.

Definition 4.2. Let S ⊆ R n be compact and 1 < p < ∞. Let w be a weight. The set S is L p w -removable with for the equation div v = 0 (in the distributional sense) if for any v ∈ L p w (R n , R n ), the validity of the equality (4.1) div v, ϕ = 0 for any ϕ ∈ Lip c (R n ) with supp ϕ ∩ S = ∅ implies that (4.1) also holds for any

ϕ ∈ Lip c (R n ). Remark 4.3. Given 1 < p < ∞
c (R n ) is replaced by C ∞ c (R n ). 4.1.
Weighted Sobolev spaces. There are two ways to define weighted Sobolev spaces: using distributional derivatives or the closure of smooth compactly supported functions in the weighted Sobolev norm. The space W 1,p w (R n ) consists of functions u ∈ L p w (R n ) whose distributional derivatives of order one belong to L p w (R n ). The space is equipped with norm 4.2. Potentials and capacities. We need three different capacities: Sobolev capacity and the weighted versions of Riesz and Bessel capacity. The capacity theory in weighted Sobolev spaces has been developed to study nonlinear potential theory, see [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF], [START_REF] David | Weighted nonlinear potential theory[END_REF] and [START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF]. For the properties of Riesz and Bessel potentials, we refer, in the classical case to [START_REF] Adams | Function spaces and potential theory[END_REF], [31, Chapter V] and [START_REF] William | Weakly differentiable functions. Sobolev spaces and functions of bounded variation[END_REF] and in the weighted case to [START_REF] David | Weighted nonlinear potential theory[END_REF], [START_REF] Aikawa | On weighted Beppo Levi functions-integral representations and behavior at infinity[END_REF], [START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF]. As in the unweighted Euclidean space, Riesz and Bessel potentials and the corresponding capacities are closely related to Sobolev spaces.

u W 1,p w (R n ) := u p,w + ∇u p,w , where ∇u is the distributional gradient of u. The space H 1,p w (R n ) is the closure of C ∞ c (R n ) under the norm • W 1,p w (R n ) . For A p -weights, 1 < p < ∞,
R n |ϕ| p w + R n |∇ϕ| p w ,
where the infimum is taken over all functions ϕ ∈ A(E), where we let:

A(E) := ϕ ∈ C ∞ c (R n ) : ϕ ≥ 1 in a neighbourhood of E , where as usual C ∞ c (R n
) denotes the space of all compactly supported smooth functions in R n . Remark 4.5. Assume that E is compact and that w ∈ A p . We claim that

Cap w p (E) = inf R n |ϕ| p w + R n |∇ϕ| p w ,
where the infimum is taken over all functions ϕ ∈ B(E), defined by

B(E) := ϕ ∈ C ∞ c (R n ) : 0 ϕ 1 in
R n and ϕ = 1 in a neighbourhood of E . Indeed, let ϕ ∈ A(E). Define ψ := max(0, inf(ϕ, 1)). The function ψ belongs to Lip c (R n ) with |∇ψ| |∇ϕ| almost everywhere, satisfies 0 ψ 1 in R n and ψ = 1 in a neighbourhood of E and

R n |ψ| p w + R n |∇ψ| p w R n |ϕ| p w + R n |∇ϕ| p w.
According to [START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF]Corollary 2.1.5] (see also [START_REF] Nicholas | Weighted Sobolev spaces and pseudodifferential operators with smooth symbols[END_REF]Lemma 2.4] for the proof in our case where p > 1) and since ψ is compactly supported, for all ε > 0, there exists a function ψ ε belonging to C ∞ c (R n ) (obtained by mollification of ψ with a suitable convolution kernel), satisfying 0 ψ ε 1 in R n and ψ ε = 1 in a neighbourhood of E, such that

ψ ε p p,w ε + ψ p p,w
and ∇ψ ε p p,w ε + ∇ψ p p,w , which proves the claim, for we hence get:

R n |ψ ε | p w + R n |∇ψ ε | p w 2ε + R n |ϕ| p w + R n |∇ϕ| p w.
M + (R n ) is I 1,R µ : R n → [0, ∞], given by I 1,R µ(x) := B(x,R) 1 |x -y| n-1 dµ(y).
Taking R = ∞ gives the usual Riesz potential as defined in [START_REF] Adams | Function spaces and potential theory[END_REF] and [START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF] (note that usually the integrals above are multiplied by a constant γ(n); since we are not interested in the exact values of potentials and capacities, we omit the constant). If I 1,R µ ∈ L p w (R n ), then we say that measure µ has a finite weighted (p, w; R)energy (or finite weighted (p, w)-energy in case R = ∞).

Bessel potential and capacity. The Bessel kernel G 1 is the tempered distribution whose Fourier transform is Ĝ1 (x) = (1 + |x| 2 ) -1/2 . It is actually a function with the same singularity at 0 as the Riesz kernel I 1 (x) = |x| 1-n , but has more rapid decay at infinity. Similarly as for Riesz potentials, the Bessel potential of a measurable function u is convolution

G 1 u = G 1 * u and of measure µ ∈ M + (R n ), G 1 µ = G 1 * µ.
Riesz potentials are easier to handle but Bessel potentials have better mapping properties. The most important properties for us are the inequality

(4.3) 0 < G 1 (x) < CI 1 (x)
for all x ∈ R, the fact that

G 1 (x) = I 1 (x) + o(I 1 (x))
as x → 0 (for these two facts, see [34, Section 3.1.2]), and the following Theorem from [24, Theorem 3.3], [START_REF] Nieminen | Hausdorff measures, capacities, and Sobolev spaces with weights[END_REF] which says that the Bessel potential space equals the weighted Sobolev space for A p -weights.

Theorem 4.8. Let 1 < p < ∞. Let w be an A p -weight. A function u belongs to

H 1,p w (R n ) if and only if there is a function f ∈ L p w (R n ) such that u = G 1 f . Moreover, there is a constant C = C(n, p, w) > 0 such that 1 C f p,w ≤ u H 1,p w (R n ) ≤ C f p,w . Definition 4.9. Let 1 < p < ∞. Let w be a weight. The weighted Bessel capacity of a set E ⊂ R n is B w 1,p (E) = inf f p p,w : f ≥ 0, G 1 f ≥ 1 on E . 4
> 0 such that 1 C Cap w p (K) ≤ B w 1,p (K) ≤ C Cap w p (K) for all compact sets K ⊂ R n .
Concerning Riesz capacities, we have, by (4.3), that R w 1,p (E) ≤ CB w 1,p (E) for all sets E ⊂ R n . In the unweighted case, Bessel and Riesz capacities have same sets of finite capacity by [START_REF] Adams | Quasi-additivity and sets of finite L p -capacity[END_REF]Theorem 1]. This Theorem says that if p < n, then

B 1,p (E) ≤ C R 1,p (E) + R 1,p (E) n/(n-p) .
In the weighted case, Bessel and Riesz capacities are locally comparable if the A pweight satisfies an additional integrability condition, see [START_REF] Aikawa | On weighted Beppo Levi functions-integral representations and behavior at infinity[END_REF]Lemma 20]. The following lemma (resulting from [34, Theorem 3.3.7 and Lemma 3.3.8]) together with Theorem 4.10 implies that for A p -weights, weighted Sobolev, Bessel and (localized) Riesz capacities are comparable. Lemma 4.11. Let 1 < p < ∞ and R > 0 be real numbers. Let w be an A p -weight. There is a constant

C R > 0 such that 1 C R R w 1,p;R (K) ≤ B w 1,p (K) ≤ C R R w 1,p;R (K),
for all compact sets K ⊂ R n .

Remark 4.12. We could use the general theory for L q -capacities and dual definitions of capacity from [3, Chapter 2] for weighted Riesz capacities, see also [START_REF] David | Weighted nonlinear potential theory[END_REF] and [START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF]Chapter 3]. We do not repeat the details of the theory because the only property we need is that for each (compact) set S ⊆ B(0, R) ⊆ R n of positive weighted (3R)-localized Riesz capacity, there is a nonnegative non-zero measure supported in S with finite energy. Indeed, by [3, Theorem 2.5.3], for each compact set K ⊂ R n , there is a measure

µ K ∈ M + (K) such that (4.4) Cap g,q (K) = µ K (K) = R n R n g(x, y) dµ K (x) q ′ dν(y)
where 1 < q < ∞, 1/q + 1/q ′ = 1, capacity Cap g,q is defined using a kernel function g(x, y) and ν is a positive measure in R n . Let S ⊆ B(0, R) be a compact set with Cap w p ′ -1 p ′ (S) > 0. A non-zero measure supported in S with finite (p, 1/w; 3R) energy is obtained using (4.4) and selecting q ′ = p, because for dν = w p ′ -1 dy and g(x, y) = χ B(0,3R) (xy) for each ϕ ∈ Lip c (R n ).

|x -y| n-1 w(y) 1-p ′ , we have R n R n g(x, y) dµ(x) p dν(y) = R n R n χ B(0,3R) (x -y) |x -y| n-1 w(y) 1-p ′ dµ(x) p w(y) p ′ -1 dy = R n R n χ B(0,3R) (x -y) |x -y| n-1 dµ(x) p w(y) (1-p ′ )p-1+p ′ dy = R n I 1,3R µ p w(y)
Proof. Assume first that ϕ ∈ Lip c (R n ) satisfies supp ϕ ⊆ B(0, 2R). Fix x ∈ B(0, R).

Since for any y ∈ B(0, R) one has xy ∈ B(0, 3R), and since, according to [31, (18) Hence we have, using the Fubini Theorem and the Hölder inequality,

R n |ϕ| dµ ≤ C(n) R n I 1,3R |∇ϕ| dµ = C(n) R n |∇ϕ|I 1,3R µ ≤ C(n) R n |∇ϕ| p ′ w p ′ -1 1/p ′ R n
(I 1,3R µ) p w -1 1/p , and the claim follows in case one has supp ϕ ⊆ B(0, 2R). Given a general ϕ ∈ Lip c (R n ), choose χ ∈ Lip c (R n ) satisfying χ B(0,R) χ χ B(0,2R) as well as ∇ϕ ∞ 2/R. According to the computations before and to the fact that supp µ ⊆ B(0, R), we know that one has: Since we have ∇(ϕχ) = χ∇ϕ + ϕ∇χ a.e. on R n and since χ = 0 outside B(0, 2R), we get:

∇(ϕχ) p ′ ,w p ′ -1 ∇ϕ p ′ ,w p ′ -1 + 2 R B(0,2R) |ϕ| p ′ w p ′ -1 1/p ′ .
Yet w p ′ -1 being p ′ -admissible, we get:

B(0,2R) |ϕ| p ′ w p ′ -1 κ(2R) p ′ R n |∇ϕ| p ′ w p ′ -1 ,
where κ is a constant associated to the p ′ -admissibility of the weight w p ′ -1 according to Poincaré inequality (2.2). This finally gives rise to the following inequality:

∇(ϕχ) p ′ ,w p ′ -1 (1 + 4κ 1/p ′ ) ∇ϕ p ′ ,w p ′ -1 and the proof is complete.

Remark 4.14. We have assumed that w p ′ -1 ∈ A p ′ . By the definition of conjugate exponents 1/p + 1/p ′ = 1 and A q -weights, this means that sup B B w p ′ -1 B w -1 1/(p ′ -1) < ∞ and hence that 1/w is locally integrable on R n .

A counterpart of Theorem 3.21 in the L p -case says that any nonnegative Radon measure with finite (p, 1/w)-energy is the divergence of a vector field in L p 1/w .

Theorem 4.15. Let w ∈ L p ′ -1 loc (R n ) be a weight and let R > 0 and 1 < p < ∞ be real numbers. Assume that w p ′ -1 is p ′ -admissible. If µ ∈ M + (B(0, R)) is a measure with finite (p, 1/w; 3R)-energy, then there exists v ∈ L p 1/w (R n , R n ) such that div v = µ in R n in the distributional sense.

Proof. Let X := Lip c (R n ) be endowed with the norm ϕ X = ∇ϕ p ′ ,w p ′ -1 (notice that it is clear that this norm is well defined), and let T : X → L p ′ w p ′ -1 (R n , R n ), T u = -∇u. Since T is a linear isometry, Lemma 3.19 implies that the adjoint operator

T * : L p ′ w p ′ -1 (R n , R n ) * → X *
is surjective. A standard argument of functional analysis recalled below shows that L p ′ w p ′ -1 (R n , R n ) * = L p 1/w (R n , R n ). Since

T * v, ϕ = v, T ϕ = - R n v • ∇ϕ
for all ϕ ∈ X, we have that T * = div in the distributional sense. The claim follows because µ ∈ X * by Proposition 4.13.

Remark 4.16. To obtain the isometric isomorphism [L p ′ w p ′ -1 (R n , R n )] * = L p 1/w (R n , R n ), it suffices to notice that given f ∈ [L p ′ w p ′ -1 (R n , R n )] * , the formula v → f, v w 1/p defines a bounded linear map on L p ′ (R n , R n ) with the same norm as f . Hence there exists g ∈ L p (R n , R n ) with g L p (R n ) = f such that for any u ∈ L p ′ w p ′ -1 (R n , R n ) one has

f, u = R n u • gw 1/p .
Yet the function h := gw 1/p belongs to L p 1/w (R n , R n ) and h L p 1/w = f . Theorem 4.17. Let 1 < p < ∞ be a real number. Let w ∈ L p ′ -1 loc (R n ) be a weight such that w p ′ -1 ∈ A p ′ . Let S ⊂ R n be a compact set. If Cap w p ′ -1 p ′ (S) = 0, then S is L p 1/w -removable for (4.5) div v = 0. wish to thank the institutes for the kind hospitality. The first author was partially supported by the French ANR project "GEOMETRYA" no. ANR-12-BS01-0014. The second author was partially supported by the French ANR project "HAB" no. ANR-12-BS01-0013). The third author was supported by the Academy of Finland, grants no. 135561 and 272886.

Proposition 3 . 14 .

 314 Let w be a 1-admissible weight and assume that V ′ ⊂⊂ V ⊂⊂ U ⊂⊂ R n are open sets, and assume that |∂V | = |∂V ′ | = 0. For any ε > 0, there exists a Lipschitz function ϕ

Remark 4 . 1 .

 41 It follows from the Hölder inequality that if w

  H and W definitions of weighted Sobolev spaces give the same space by [19, Theorem 2.5].

Definition 4 . 4 .

 44 Let w be a weight and let 1 ≤ p < ∞. The weighted Sobolev p-capacity of a set E is Cap w p (E) := inf

Remark 4 . 6 .

 46 Assume that 1 < p < n is a real number and that w is a p-admissible weight. According to[START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF] Corollary 2.39], the weighted Sobolev capacity Cap w p has the same zero sets as the capacity cap w p defined by(4.2) cap w p (E) := inf R n |∇ϕ| p w ,where the infimum is taken over all functions ϕ ∈ A(E).Riesz potential and capacity. Let 0 < R ∞. The (R-truncated) Riesz potential (of order 1) of a nonnegative, locally integrable function u is the function I 1,R u : R n → [0, ∞] given byI 1,R u(x) := B(x,R)u(y) |x -y| n-1 dy, and the (R-truncated) Riesz potential of a measure µ ∈

Definition 4 . 7 .

 47 Let w be a weight and let 1 < p < ∞. For 0 < R ∞, we define the (R-truncated) weighted Riesz capacity of a set E ⊂ R n by R w 1,p;R (E) := inf f p p,w : f ≥ 0, I 1,R f ≥ 1 on E , and we let R w 1,p (E) := R w 1,p;∞ (E).

  p. 125], there is a constant C = C(n) > 0 such that, for all x ∈ R n , ϕ(x) = C(n) R n ∇ϕ(y) • (xy) |x -y| n dy,we get, using the fact that ϕ vanishes outside B(0, 2R) that, for all x ∈ R n ,ϕ(x) = C(n) R n ∇ϕ(y) • (xy)|x -y| n χ B(0,3R) (xy) dy.

  dµ C(n) I 1,3R µ p,w -1 ∇(ϕχ) p ′ ,w p ′ -1 .

  and an A p weight w, we can again (as in Remark 3.11) rely on [34, Corollary 2.1.5] (see[START_REF] Nicholas | Weighted Sobolev spaces and pseudodifferential operators with smooth symbols[END_REF] Lemma 2.4] for a proof in this case) to claim that the above definition remains unchanged in case Lip

  .3. Connection between different capacities. In the proof of Theorem 4.18, we need the property that weighted Sobolev, Bessel and Riesz capacities have same zero sets. Connection between Sobolev and Bessel capacities is much stronger; Theorem 4.8 implies that the weighted Bessel and Sobolev capacities for an A p -weight are comparable. For a proof, see[START_REF] Ove | Nonlinear potential theory and weighted Sobolev spaces[END_REF] Theorem 3.5.2].

Theorem 4.10. Let 1 < p < ∞. Let w be an A p -weight. There is a constant C

  Proposition 4.13. Let w be a weight and let 1 < p < ∞ and R > 0 be real numbers. Assume that w p ′ -1 is p ′ -admissible. If µ ∈ M + (B(0, R)) has a finite (p, 1/w; 3R)energy, thenR n |ϕ| dµ ≤ C(n) I 1,3R µ p,w -1 ∇ϕ p ′ ,w p ′ -1

-1 dy. 4.4. Removable singularities. The following result is a counterpart of Proposition 3.18 in the L p -case. In the unweighted case, compare it with [30, Theorem 3.2].

Acknowledgements: This research was conducted during the visits of the authors to Laboratoire de Mathématiques d'Orsay, the Institut Fourier of the Université de Grenoble and Department of Mathematics and Statistics of University of Jyväskylä and to Forschungsinstitut für Mathematik of ETH Zürich. The authors

Proof. Since Cap w p ′ -1 p ′ (S) = 0, by Remark 4.5, there are functions u k ∈ C ∞ 0 (R n ), k ∈ N, such that 0 ≤ u k ≤ 1 and u k = 1 in a neighborhood of S for all k,

Hence, using the Hölder inequality, we obtain

Proof. Since S is compact, there exists R > 0 such that S ⊆ B(0, R).

If Cap w p ′ -1 p ′ (S) > 0, then, by Theorem 4.10 and Lemma 4.11, R w p ′ -1 1,p ′ ;3R (S) > 0. Hence, by Remark 4.12, there is a nonnegative non-zero measure µ S supported in S such that:

Theorem 4.15 implies that there exists v ∈ L p 1/w (R n , R n ) such that div v = µ S in R n in the distributional sense. This shows that S is not removable for div v = 0. Hence we have that Cap

In summary, we proved the following result. Theorem 4.19. Let 1 < p < ∞ be a real number. Let w ∈ L p ′ -1 loc (R n ) be a weight such that w p ′ -1 ∈ A p ′ . Let S ⊂ R n be a compact set. Under those assumptions, the set S is L p 1/w -removable for div v = 0 if and only if one has Cap w p ′ -1 p ′ (S) = 0.


Flavio Pressacco

Laura Ziani

A Fibonacci Approach to Weighted Majority Games

Keywords: Weighted majority games, homogeneous representation, minimal winning coalition, type weight vector, satellite games, Fibonacci numbers

come   L'archive ouverte pluridisciplinaire

Introduction

In this paper we wish to explore the connections between game theory, precisely constant sum homogeneous weighted majority games 1 , and Fibonacci numbers.

A bridge between these two topics has been thrown by Isbell [START_REF] Isbell | A class of majority games[END_REF] in a vintage paper going back to the early stage of development of cooperative game theory. The bridge was embedded in a couple of propositions. In the first ( [START_REF] Isbell | A class of majority games[END_REF], Cor. [START_REF] Neumann | Theory of games and economic behaviour[END_REF], p. 185), the author showed that, weakly ordering players of a n person constant sum homogeneous weighted majority game from bottom to top, the individual weight (in the minimal homogeneous representation) of the player number i could not exceed the corresponding Fibonacci number (more formally w i ≤ f i for any i = 1, . . . , n). The second proposition ( [START_REF] Isbell | A class of majority games[END_REF], third indent, p. 185) claimed that a game whose individual weights satisfied w i = f i for any i = 1 . . . , n -2, w n-1 = f n-2 , w n = f n-1 , described, for any n > 2, a constant sum homogeneous weighted majority game with minimal winning quota q = f n .

Hence, in such games all the first n Fibonacci numbers are associated to all individual weights and to the winning quota. On the ground of this connection we suggest to call Fibonacci-Isbell the set of games defined by the second proposition.

Thinking in terms of type weights rather than of individual ones and keeping account that in any n person Fibonacci-Isbell game there are t = n -2 types of players, we argue that such games satisfy another slightly different "bridge" property: the bottom-top (strictly) ordered n-1 (or t+1) dimension vector of type weights and winning quota is a string of n -1 consecutive Fibonacci numbers starting from f 2 = 1.

In order to generalize this property to other feasible (n, t) combinations, we suggest to define Fibonacci games the subset of constant sum homogeneous weighted majority games whose bottom-top increasing sequence of type weights and winning quota (in their minimal homogeneous representation) is a string of consecutive Fibonacci numbers 2 . Henceforth, it is convenient to use the (first order) "delayed" Fibonacci sequence g defined, for any natural n, by the relation g n = f n+1 , and delayed Fibonacci subsequences g odd = g o = (g 1 , g 3 , . . . , g 2m-1 , . . .) and g even = g e = (g 2 , g 4 , . . . , g 2m , . . .); moreover, following Rosenmüller ([5], p. 311), we keep into consideration the "profile" of a constant sum homogeneous weighted majority game with t types denoted by the ordered vector k = (k 1 , . . . , k j , . . . , k t ) whose component k j is the number of players of type j in the game.

After that, it is clear that Fibonacci games are unequivocally described by feasible triplets (k; g; g t+1 ) in which the profile k and the delayed Fibonacci sequence g share the dimension t, g j is the weight of players of type j, g t+1 is the winning quota and a triplet is feasible if it meets the homogeneity conditions (for details see [START_REF] Rosenmüller | Homogeneous games: recursive structure and Mathematics of Operations Research[END_REF] and [START_REF] Neumann | Theory of games and economic behaviour[END_REF]).

In our paper we give a simple but insightful characterization of the feasible profiles k of Fibonacci games with t types. The proof of such a characterization largely exploits fundamental properties of the delayed Fibonacci sequence (as for the necessary conditions) and the satellite game approach introduced in [4], Theor. 1.4 (in the sufficiency part).

It follows that, for any positive integer t, there are altogether 3 ⌊(t + 1)/2⌋ Fibonacci games, i.e. the Fibonacci-Isbell one and other ⌊(t -1)/2⌋ games.

Moreover, it turns out that any two Fibonacci games with the same t have a different number n of non dummy players. Hence, it is expressive to associate, for any t, the Fibonacci games to the set of integers z = 1, . . . , ⌊(t + 1)/2⌋ in such a way that the number n(t, z) of players in the Fibonacci game with t types and index z is an increasing function of z.

A closed form description of the behavior of the function n(t, z) is also provided. It is amazing to check that the derived function ∆(t, z) = n(t, z +1)-n(t, z) is resumed by a matrix whose columns are the entire delayed Fibonacci sequence (whose starting point is properly shifted down with z), while the rows are, in backward order, the delayed Fibonacci subsequences coherent with the parity of t (properly truncated, so as the row t has ⌊(t -1)/2⌋ components).

We do not discuss here applications of the Fibonacci games, but anticipate that we reached some (not yet published) preliminary evidences that interesting applications to the weighted voting systems in parliamentary elections may be obtained.

The plan of the paper is as follows: section 2 gives a short description of the basic notations used in the paper and recalls well known concepts of homogeneous weighted majority games; section 3 defines Fibonacci games and resumes the main results of the paper; section 4 provides an explicit description of the profiles of Fibonacci games for some small values of t; the behaviour of the n(t, z) and of ∆(t, z) functions and their connections with the delayed Fibonacci (sub)sequences are presented and discussed 2 Of course other definitions, based on more or less restrictive conditions on Fibonacci numbers, could have been chosen. For details on this alternative approach see Gambarelli, Gnocchi and Pressacco (unpublished) communication in the section "Power indices" of the conference: Models of collusion, games and decisions for applications to judging, selling and voting. Oldofredi Castle, Monte Isola 18-19 June 2012. 3 As usual ⌊x⌋ denotes the floor(x). In particular, for t integer even (odd), ⌊(t + 1)/2⌋ = t/2 (or (t + 1)/2).

in section 5; all the proofs are grouped in sections 6 and 7; conclusions follow in the final section 8.

Notations

Let Ω = {1, . . . , n} denote the set of non dummy players of a simple constant sum game in characteristic function form. A simple game is a mapping v : P(Ω) → {0, 1} such that v(∅) = 0 and v(Ω) = 1 and a coalition S ∈ P(Ω) is winning if its payoff v(S) = 1 and losing otherwise. A simple game is constant sum if v(S) + v(Ω/S) = 1 for any S. Moreover, S is minimal winning if, for any player i ∈ S, v(S/i) = 0. A simple weighted majority game is described by the pair (w n ; q) where4 w n = {w 1 , . . . , w n } is the (weakly) ordered vector (w i ≤ w j for i < j) of individual weights and q the winning quota of the game. Thus w(S) = i∈S w i is the weight of S and w(S) ≥ q ⇔ v(S) = 1.

A simple constant sum homogeneous weighted majority game is described by its minimal homogeneous representation, that is the ordered vector h = (w n ; q) which meets the homogeneity conditions, that is h ∈ N n+1 , w 1 = 1, q = (1+w(Ω)) 2 and w(S) = q for any minimal winning coalition. The vector w n of the minimal homogeneous representation induces a decomposition of Ω in equivalence classes K 1 , . . . , K j , . . . , K t . Each class groups all players of the same type, sharing the same individual weight and the corresponding (strongly) ordered type weight vector is w t = {w 1 , . . . , w j , . . . , w t } with w 1 = 1.

Coherently, h = (w t ; q) ∈ N t+1 , but to unequivocally describe a game it is necessary to introduce also the profile k t = (k 1 , . . . , k j , . . . , k t ) with k j the number of players of type j in the game. Thus, a constant sum homogeneous weighted majority game is described by (k t ; w t ; q) ∈ N 2t+1 . Finally, we denote by s t = (s 1 , . . . , s j , . . . , s t ) with s j = |S ∧ K j | ≤ k j the "profile" of the coalition S, so that w(S) = t j=1 s j w j is an alternative formalization of the weight of S and, by homogeneity, w(Ω) = t j=1 k j w j = 2q -1.

Fibonacci games: main results

The Fibonacci sequence f is defined by the well known finite difference equation:

f n = f n-1 + f n-2
holding for any natural n > 2 with initial conditions f 1 = f 2 = 1. Henceforth, we exploit the "delayed" Fibonacci sequence g n = f n+1 for any n and denote, for any integer m, by g m ∈ N m the vector (g 1 , g 2 , . . . , g m ). Coherently, the denumerable sequence g ∞ = (g 1 , . . . , g n , . . .), and g ∞ = g o ∨g e with g o = (g 1 , g 3 , . . . , g 2m-1 , . . .) g e = (g 2 , g 4 , . . . , g 2m , . . .)

the two denumerable (sub)sequences obtained extracting from g ∞ all elements of odd or respectively even index.

Definition 3.1. The set of Fibonacci games is the subset of constant sum homogeneous weighted majority games with h = g t+1 , that is (k t ; w t ; q) = (k t ; g t ; g t+1 ).

Remark 3.1. Feasibility conditions require that in any Fibonacci game with t types, t j=1 k j g j = 2g t+1 -1 constant for any k t : the winning quota is the same for all Fibonacci games with the same t.

Proof. Homogeneity requires that w(Ω) = 2q -1, i.e. t j=1 k j g j = 2g t+1 -1.

Theorem 3.1. Both for t = 1 and t = 2 there is a unique Fibonacci game; their k t are respectively k 1 = (3), and k 2 = (3, 1); for any t > 2, a game is a Fibonacci game if and only if its k t = k(t, z) is given, for any value of the counter z = 1, . . . , ⌊(t -1)/2⌋ and with j 0 = t + 1 -2z, by:

k(t, z) = (k 1 , k ′ , k j 0 , k ′′ ) = (2 + g t-1 -g j 0 , 1 j 0 -2 , 2, 1 2z-1 ) (3.1)
or, putting z = ⌊(t + 1)/2⌋ in order to have a complete indexation of the games, by:

k(t, ⌊(t + 1)/2⌋) = (2 + g t-1 , 1 t-1 ) (3.2)
Corollary 3.1. For any t, there are exactly

Φ(t) = ⌊(t + 1)/2⌋ (3.3)
Fibonacci games with t types.

Let us denote by n(t, z) the total number of non dummy players in a Fibonacci game with t types and index z.

Theorem 3.2. For any t, z 1 < z 2 ⇔ n(t, z 1 ) < n(t, z 2 ).
Proofs of Theorems 3.1 and 3.2 follow in sections 6 and 7.

Profiles of Fibonacci games

According to Definition 3.1, given t there are no degrees of freedom in the choice of the sequence of type weights and of the winning quota. Hence, a Fibonacci game is unequivocally described by its k(t, z) vector which follows the rules described in Formulas (3.1) and (3.2). Table 1 We distinguish between two patterns of profiles: for z = ⌊(t + 1)/2⌋, (the last vector of each row) all but the bottom5 type components are 1, while the bottom component plays a balancing role (according to Formula (3.2)); for all the other z there is just one non bottom component equal to 2 in place j 0 = j(t, z) = t + 1 -2z, all the other non bottom components are still 1 and, again, the bottom component plays the balancing role (according to Formula (3.1)). [START_REF] Rosenmüller | Homogeneous games: recursive structure and Mathematics of Operations Research[END_REF] The behavior of n(t, z) The following properties hold for any feasible combination of (t, z):

Property 5.1. n(t, 1) = t + 2 (5.1) Property 5.2. ∆(t, z) = n(t, z + 1) -n(t, z) = g t-2z
(5.2)

Property 5.3. n(t, 2) -n(t -1, ⌊t/2⌋) = 2 (5.3)
Proofs of the previous Properties are given in section 7.

Here, we distinguish between two kinds of Fibonacci games: the first kind corresponds to z = 1 and Formula (5.1) implies that there is just one Fibonacci game (of this first kind) for any n > 2. Such games have been described by Isbell ([1], p. 185) while studying the class of coalitionally Parsimonious games 6 .

For z > 1 (second kind), there is one Fibonacci game (of this second kind) for sparse values of n. Summing up, there are values of n corresponding to two Fibonacci games (one of the first and one of the second kind), while for other values of n there is just one Fibonacci game of the first kind.

At first glance the values of n associated to the second kind of games seem to follow a chaotic rule. A regularity emerges once we consider the following table of the differences generated by Property 5.2 (see also Remark 7.1). Note that the rows corresponding to odd (even) values of t are backward initial strings with ⌊(t -1)/2⌋) elements of the sequence g o (g e ). In turn, the columns are nothing but the whole sequence g ∞ whose starting point is shifted at the row t = 1 + 2z.

6 Proofs of Theorem 3.1

Proof. For t = 1: q = g 2 = 2 w(Ω) = k 1 • g 1 = k 1 = 2q -1 = 3
Hence there are three players with weight g 1 = 1, k 1 = (3) and Φ(1) = 1.

For t = 2:

q = g 3 = 3 w(Ω) = 2 j=1 k j g j = 2q -1 = 5 Hence, k 1 • g 1 + k 2 • g 2 = k 1 • 1 + k 2 • 2 = 5.
There are two solutions k 2 = (3, 1) and k 2 = (1, 2) but the second is not feasible because the profile s 2 = (0, 2) identifies a minimal winning coalition S with w(S) = 4 > q, incompatible with the homogeneous character. On the other side, the first solution is feasible with profiles (3, 0) and (1, 1) of the minimal winning coalitions, so that Φ(2) = 1.

For t > 2, we show at first that the conditions (3.1) or (3.2) are necessary for Fibonacci games and then that they are also sufficient.

Proof of necessity

Property 6.1. In any Fibonacci game with t > 1, the coalition S whose profile s t = (0 t-2 , 1 2 ), i.e. made by a top and a last but top player, is minimal winning.

Proof.

w(S) = s t-1 g t-1 + s t g t = g t-1 + g t = g t+1 = q Property 6.2. In any Fibonacci game with t > 1, k t = 1.

Proof. Otherwise the coalition S : s t = (0 t-1 , 2) would be minimal winning with w(S) = 2g t > g t + g t-1 = q, a contradiction with the homogeneous character of the game. Property 6.3. In any Fibonacci game with t > 2, k t-1 ≤ 2.

Proof. Otherwise the coalition S : s t = (0 t-2 , 3, 0) would be minimal winning as 2g t-1 < g t-1 + g t = q with w(S) = 3g t-1 > 2g t-1 + g t-2 = g t-1 + g t = q. Property 6.4. In any Fibonacci game with t > 3, k t-2 = 1.

Proof. Suppose on the contrary, k t-2 > 1 and consider the coalition S : s t = (0 t-3 , 2, 0, 1). S would be minimal winning too as g t +g t-2 < g t +g t-1 = q with w(S) = g t +2g t-2 > g t +g t-2 +g t-3 = g t +g t-1 = q. A contradiction. Properties 6.3 and 6.4 may be generalized to types (th) with h odd and respectively h even. Property 6.5. In any Fibonacci game it is k t-h ≤ 2 for h odd < t -1.

Proof. Proof of Property 6.5 is based on the following lemma: Lemma 6.1. In a Fibonacci game consider any player of type j 0 = (th) with h odd < (t -1); there exists a minimal winning coalition S in which this player is the weakest. The profile of S is given by: s t = (s 1 , . . . , s j , . . . , s t ) with s j = 0 for j < j 0 ; s j 0 = 1; s j = 1 for j > j 0 and j = (j 0 + 1) mod 2 and s j = 0 for j > j 0 and j = j 0 mod 2.

Proof.

w(S) = g j 0 + j>j 0 and j=(j 0 +1) mod 2

g j = g t-1 + g t = g t+1 = q
Now to prove Property 6.5 suppose on the contrary k j 0 > 2 and consider the coalition S ′ obtained from S by replacement of the player type j 0 + 1 with two additional players of type j 0 . S ′ would be minimal winning with w(S ′ ) > q, a contradiction. Property 6.6. In any Fibonacci game it is k t-h = 1 for h even < t -1.

Proof. Proof of Property 6.6 is based on the previous Lemma 6.1 and on the following one: Lemma 6.2. Suppose in a Fibonacci game there is a minimal winning coalition S and a j 0 = (t-h) > 2 such that s j = 0 for j = 1, . . . , j 0 -1, s j 0 > 0 (so that a player of type j 0 > 2 is the weakest in S), then k j 0 -1 = 1.

Proof. Suppose on the contrary k j 0 -1 > 1 and consider the coalition S ′ obtained from S by replacement of a player of type j 0 with two players of type j 0 -1. S ′ would be minimal winning with w(S ′ ) > q, a contradiction. This Lemma holds independently from the parity of h. Lemma 6.3. If a player of type j = th > 2 is the weakest in a minimal winning coalition S, then there exists a minimal winning coalition S ′ such that a player of type j -2 is the weakest in S ′ .

Proof. In S replace the player of type j with the unique (by Lemma 6.2) player of type j -1 and one player of type j -2. Also this Lemma holds independently from the parity of h. Lemma 6.4. Suppose in a Fibonacci game k j 0 = 2 for j 0 = th with h odd < (t -1), then: a) k j 0 -1 = 1 (by Property 6.6) and b) there is a minimal winning coalition S ′ such that the player of type (j 0 -1) is its weakest player.

Proof. (of part b)) S

′ is obtained from the coalition S in Lemma 6.1 by replacement of the player of type t -(h -1) with one additional player of type th and the player of type t -(h + 1). Clearly S ′ is minimal winning with w(S ′ ) = q. Lemma 6.5. If in a Fibonacci game k j 0 = 2 for j 0 = th with h odd < (t -1), then k j = 1 for any 1 < j < j 0 .

Proof. Lemma 6.4 and Lemma 6.3 imply that there exists a set of minimal winning coalitions such that each player of type j, 1 < j ≤ j 0 , is the weakest in a minimal winning coalition of the set. After that Lemma 6.5 comes as an immediate corollary of Lemma 6.2. Now, it is possible to give a more precise statement of Property 6.5 concerning the behaviour of k t-h for h odd < t -1. Indeed, Property 6.5 and Lemma 6.5 give immediately: Property 6.7. In a Fibonacci game there is at most one odd h < t -1 such that k j 0 = k t-h = 2 and k j = 1 for all j > 1 and j = j 0 . Now by Remark 3.1 it is:

k 1 = 2g t+1 -1 - t j=2 k j g j (6.1)
and moreover 7 :

g t+2 - t j=1 g j = 2 (6.2)
Furthermore, Properties 6.6 and 6.7 imply that, for any j = 2, . . . , t, k t satisfies conditions (3.1) or (3.2). In the first case and putting j 0 = t + 1 -2z, we obtain:

k 1 = k 1 (t, z) = 2g t+1 -1 -( t j=2 g j + g j 0 ) = = g t+1 + (g t + g t-1 ) - t j=1 g j -g j 0 = = g t+2 + g t-1 - t j=1 g j -g j 0 = = 2 + g t-1 -g j 0 (6.3)
7 By induction: indeed it is immediate to check that (6.2) is true for t = 1 and, if true for t, is true also for t + 1.

In particular, for z = 1, k 1 = 2. In the second case, we get immediately:

k 1 = k 1 (t, ⌊(t + 1)/2⌋) = 2 + g t-1 (6.4)
To resume, by Properties 6.6 and 6.7, and by Formulas (6.3) and (6.4), necessary conditions to be satisfied by the k t vector of a Fibonacci game with t > 2 are those described in Formulas (3.1) and (3.2).

Proof of sufficiency

The proof is based on an adaptation to our problem of the test for homogeneity of a weighted majority games developed as Basic Lemma in [START_REF] Rosenmüller | Weighted majority games and a the matrix of Homogeneity[END_REF] (Theorem 1.4) and [START_REF] Rosenmüller | Homogeneous games: recursive structure and Mathematics of Operations Research[END_REF] (pp. 312).

Henceforth we use the following definitions and notations:

• G 0 = (k 0 t ; g 0 t ; λ 0 = g t+1 ), "seed" game with k 0 t given by Formulas (3.1) or (3.2). In particular, G 0,z , seed game as a function of z = 1, . . . , ⌊(t + 1)/2⌋.

• G r = (k r t ; g r t ; λ r ), any game of a r generation 8 derived by the seed G 0 . Note that the dimension t of G r depends on r. Whenever useful, we utilize t(r). In particular, G r,z , game of a r generation derived by the seed G 0,z .

• Feasible (not feasible) game: a G r for which t j=2 k r j • g r j ≥ λ r (< λ r ).

• For any given type index j 0 > 1, an "intermediate" player in the game G r is a player of type j 0 , a "large" player is a player of any type j > j 0 , a "small" player is any player of type 1 < j < j 0 .

• No bottom coalition of a feasible game G r : a coalition S r whose profile s r has s r 1 = 0.

• Dominant coalition 9 of a feasible game G r a coalition S r such that:

1. S r is no bottom 2. there is an index j 0 (G r ) > 1 (or shortly j r 0 > 1) and a positive integer c r , such that s r = (s r j = 0 for j < j r 0 ; s r j = c r (1 ≤ c r ≤ k r j 0 ); s r j = k r j for j > j r 0 )

i.e. S r is made by all large players and by some (may be all) intermediate players, so that j r 0 is the type index corresponding to the weakest player in the dominant coalition S r . 3. w(S r ) = t(r) j=1 s r j g r j = λ r .

• Generation rules. Let G r a feasible game, S r its dominant coalition and j r 0 the corresponding type index. To any j r 0 ≤ j ≤ t(r), associate the game G r+1 (j) = (k r+1 , g r+1 , λ r+1 ) in which k r+1 = k r+1 (j):

for j > j r 0 and k r j 0c r > 0:

k r+1 j 0 = (k r 1 , k r 2 , . . . , k r j 0 -1 , k r j 0 -c r )
, the dimension of g r+1 is j 0 and λ r+1 = g j 8 Note that the winning quota λ r of G r may well be greater than t j=1 k r j g j r+1 2

, i.e. for r > 0, G r may not be constant sum. 9 It is the counterpart of the "largest min win" coalition (in backward lexicographic order) of a weighted majority game defined in [START_REF] Rosenmüller | Homogeneous games: recursive structure and Mathematics of Operations Research[END_REF], pp. 312.

for j = j r 0 or j > j r 0 and k r j 0c r = 0:

k r+1 j 0 -1 = (k r 1 , k r 2 , . . . , k r j 0 -1
), the dimension of g r+1 is j 0 -1 and λ r+1 = g j Verbally the dominant coalition S r generates as many "satellite" games G r+1 (j) as the number of types j included in the coalition. The players of the satellite game associated to a type j are all players of type i < j not belonging to S r ; the winning quota is g j . The idea is that any satellite game looks for the dominant coalition of the satellite able to replace one player of type j in the generating game while preserving the homogeneous character of the "seed" game. In turn, the dominant coalition gives rise to other satellite games of next generation and so on. Now, our adapted version of Basic Lemma is: Lemma 6.6. A seed weighted majority game G 0 is homogeneous if, in the set of all G r games obtained by the generation rules, do not exist feasible games lacking the dominant coalition.

To prove the Lemma 6.6 we show that any game of the set G r , (r = 0, 1, . . .) generated by a seed G 0 coherent with formulae (3.1) and (3.2) either is not feasible or admits a dominant coalition.

In the proof we will exploit the following relation10 concerning Fibonacci numbers:

g t-2 + 2g t-1 + h j=0 g t+j = g t+h+2 (6.5)
Proof. Let us distinguish three cases: Case α): z = ⌊(t + 1)/2⌋

Let us denote G 0,⌊(t+1)/2⌋ = G 0 = (k 0 t ; g 0 t ; g t+1 ) any seed game with k 0 t = (k 0 1 , 1 t-1 ). The profile of the dominant coalition S 0 is s 0 t = (0 t-2 , 1 2 ) with k 0 t-1c = 0. Hence the satellite games associated to type t -1 and t are G 1,a = G 1 (t -1) = (k 1 , 1 t-3 ; g t-2 ; g t-1 ) and G 1,b = G 1 (t) = (k 1 , 1 t-3 ; g t-2 ; g t ). By Formula (6.2), G 1,b is not feasible, while G 1,a is feasible for any t = t(0) > 4. Let shortly write G 1,a = G 1 ; it is immediate to check that G 1 mimics the structure of G 0 : only the dimension is different: it is t = t(0) for G 0 and t = t(1) = t(0) -2 for G 1 . The profile of the dominant coalition S 1 of G 1 is s 1 = (0 t(1)-2 , 1 2 ) = (0 t-4 , 1 2 ) and the procedure may be recursively repeated until the integer r for which also G 1,r is no more feasible. This is resumed by the following Property 6.8. Let G 0 be a seed game of type α with t = t(0) > 2. For any generation r = 0, 1, . . . , ⌊(t -3)/2⌋ there is a unique feasible G r = (k r t ; g r t ; λ r ). The k r t of such game is k 0 t truncated at dimension t(r) = t(0) -2r, which is coherently the dimension of g r t , while λ r = g t(r)+1 . Feasible G r games have dominant coalition S r whose s r t(0)-2r = (0 t(0)-2(r+1) , 1 2 ). Case β): z = 1 Let G 0,z = G 0,1 = (k 0 t ; g 0 t+1 ; g t+1 ) any seed game with k 0 t = (k 0 1 , 1 t-3 , 2, 1). The profile of the dominant coalition S 0 is still s 0 t = (0 t-2 , 1 2 ) with k 0 t-1 -c = 1. Hence, the satellite games associated to type t -1 and t are G 1,a = G 1 (t -1) = (k 1 , 1 t-3 ; g t-2 ; g t-1 ) and G 1,b = G 1 (t) = (k 1 , 1 t-2 ; g t-1 ; g t ). Both games share the properties of G 0 of case α), that is behave as seed games and generate for r = 2, . . . sequences G r,a and G r,b according to Property 6.8. Of course the dimension of G 1,a = t(1, a) = t(0)-2,

and of G 1,b = t(1, b) = t(0) -1. Case γ): 1 < z < ⌊(t -1)/2⌋ Let us write G 0,z = (k 0 1 , 1 t-1-2z , 2, 1 2z-1
). The profile of the dominant coalition S 0,z is s 0,z t = (0 t-2 , 1 2 ) with k 0 t-1c = 0. Let us denote the two satellite games by 1) ; hence recursively we go back to G 0,1 , i.e. the seed game of case β).

G 1z,a = G 1z (t -1) = (k 1z,a t ; g 1z,a t ; λ 1z,a ) and G 1z,b = G 1z (t) = (k 1z,b t ; g 1z,b t ; λ 1z,b ). It is k 1z,a t-2 = k 1z,b t-2 = (k 0 1 , 1 t-1-2z , 2, 1 2z-3 ), λ 1z,a = g t-1 , λ 1z,b = g t . Note that G 1z,a mimics G 0,(z-
In turn, as by Formula (6.5),

j s 1z,b j g j = g t , G 1z,b has dominant coalition with profile s 1z,b t-2 = (0 t-2z-1 , 1, 2, 1 2z-3 ). Hence G 1z,b gives rise to 2z -1 satellite games G 2z,b (j) for j = t(0) -2z, . . . , t(0) -2. It is G 2z,b (j) = (k 0
1 , 1 t-2z-2 ; g t-2z-1 ; g j ). By Formula (6.2), for all j, except j * = t(0) -2z, the G 2z,b (j) are not feasible; the survival feasible G 2z,b (j * ) mimics the behavior of the G 0 seed of the case α).

Summing up, in all three cases all G r games either are not feasible, or if feasible admit a dominant coalition and the conditions of the Basic Lemma are verified.

Proofs of Properties of Section 5

By definition:

n(t, z) = k 1 (t, z) + t j=2 k j (t, z) (7.1)
Proof. Proof of Property 5.1

For z = ⌊(t + 1)/2⌋ it is t j=2 k j (t, z) = t -1, by Formula (3.2) and k 1 = 2 + g t-1 by Formula (6.4) so that:

n(t, z) = t + 1 + g t-1 (7.2)
For z < ⌊(t + 1)/2⌋ it is t j=2 k j (t, z) = t, by Formula (3.1) and k 1 = 2 + g t-1g t+1-2z by Formula (6.3) so that:

n(t, z) = t + 2 + g t-1 -g t+1-2z (7.3)
and in particular n(t, 1) = t + 2

Proof. Proof of Property 5.2.

For (z + 1) < ⌊(t + 1)/2⌋ by Formula (7.3):

n(t, z + 1) -n(t, z) = (t + 2 + g t-1 -g t+1-2(z+1) ) -(t + 2 + g t-1 -g t+1-2z ) = g t-2z+1 -g t-2z-1 = g t-2z
(7.4)

For (z + 1) = ⌊(t + 1)/2⌋ by Formulas (7.2) and (7.3):

n(t, z + 1) -n(t, z) = (t + 1 + g t-1 ) -(t + 2 + g t-1 -g t+1-2z ) = g t+1-2z -1 (7.5)
but z = ⌊(t -1)/2⌋ = (t/2) -1 for t even and (t -1)/2 for t odd so that:

t + 1 -2z = 3 for t even and t + 1 -2z = 2 for t odd which also means, on one side:

t -2z = 2 for t even and t -2z = 1 for t odd and on the other one: g t+1-2z = 3 for t even and g t+1-2z = 2 for t odd and finally g t+1-2z -1 = 2 for t even and g t+1-2z -1 = 1 for t odd so that in both cases still n(t, z + 1)n(t, z) = g t-2z (7.6) which completes the proof of Property 5.2. Note that this property implies the strict monotony of n(t, z) for any t and then, as a by product, gives a proof of Theorem 3.2.

Proof. Proof of Property 5.3.

Preliminarily note that putting t ′ = t -1 it is ⌊(t ′ + 1)/2⌋ = ⌊(t/2)⌋ so that by Formula (7.2): n(t ′ , ⌊(t ′ + 1)/2⌋) = t ′ + 1 + g t ′ -1 (7.7) equivalent to n(t -1, ⌊(t/2)⌋) = t + g t-2

Moreover, it is by Formula (7.3) for t > 4, or by Formula (7.2) for t = 3 and 411 n(t, 2) = t + 2 + g t-1g t-3 = t + 2 + g t-2

and immediately: n(t, 2)n(t -1, ⌊t/2⌋) = 2 (7.8)

Remark 7.1. Jointly Prop. 5.2 and 5.3 imply that, putting in lexicographic order the set Ξ of values of n(t, z), z > 1, a strictly monotone sequence is obtained. By Formulas (7.4), (7.6) and (7.8), all members of the sequence of first differences ∆ of the Ξ sequence are Fibonacci numbers (see Table 3).

Conclusions

In this paper we introduce the class of Fibonacci games. They are the subset of constant sum homogeneous weighted majority games whose sequence of all type weights and the minimal winning quota is a string of consecutive Fibonacci numbers. Exploiting properties of Fibonacci sequence, we give closed form results able to provide a simple and insightful characterization of such games. In detail, we compute the total number of Fibonacci games for any given value of t (type of players in the game); we describe, for any (t, z) with z a proper counter, the profile of any game, i.e. the vector whose components are the number of non dummy players of each type in the game; we provide elegant connections between Fibonacci numbers and the overall number of non dummy players in the game.

Compared to other weighted majority games, the Fibonacci games combine two specific characteristics: the presence of some, or perhaps many, "peones" (players with minimum weight), along with an almost total ranking (with one tie at most) of all the other players, whose individual power grows at the speed of the Fibonacci sequence. It seems, by some preliminary analysis, that this feature can be particularly useful in applications to weighted voting systems in parliamentary elections. This would be the object of further research.

Table 1 :

 1 helps to understand the profile vectors of Fibonacci games for a set of small values of t. The profile k(t, z) for a set of small values of t.

	t	Φ(t)	k(t, 1)	n(t, 1)	k(t, 2)	n(t, 2)	k(t, 3)	n(t, 3)	k(t, 4)	n(t, 4)
	1	1	(3)	3
	2	1	(3, 1)	4
	3	2	(2, 2, 1)	5	(4, 1, 1)	6
	4	2	(2, 1, 2, 1)	6	(5, 1, 1, 1)	8
	5	3	(2, 1, 1, 2, 1)	7	(5, 2, 1, 1, 1)	10	(7, 1, 1, 1, 1)	11
	6	3	(2, 1, 1, 1, 2, 1)	8	(7, 1, 2, 1, 1, 1)	13	(10, 1, 1, 1, 1, 1)	15
	7	4	(2, 1, 1, 1, 1, 2, 1)	9	(10, 1, 1, 2, 1, 1, 1)	17	(13, 2, 1, 1, 1, 1, 1)	20	(15, 1, 1, 1, 1, 1, 1)	21
	8	4	(2, 1, 1, 1, 1, 1, 2, 1)	10	(15, 1, 1, 1, 2, 1, 1, 1)	23	(20, 1, 2, 1, 1, 1, 1, 1)	28	(23, 1, 1, 1, 1, 1, 1, 1)	30
	...	...	...	...	...	...	...	...	...	...

Table 2

 2 shows the behavior of the function n(t, z).

				z
	t	1	2	3	4	5
	1	3
	2	4
	3	5	6
	4	6	8
	5	7	10	11
	6	8	13	15
	7	9	17	20	21
	8	10	23	28	30
	9	11	32	40	43	44
	10	12	46	59	64	66
	...	...	...	...	...	...

Table 2 :

 2 List of the values n(t, z).

Table 3 :

 3 Table of the function ∆(t, z).

			z
	t	1	2	3	4
	3	1
	4	2
	5	3	1
	6	5	2
	7	8	3	1
	8	13	5	2
	9	21	8	3	1
	10	34	13	5	2
	...	...	... ... ...

Henceforth the subscript of the vector is its dimension, not to be confused with the subscript of a scalar which denotes a component of the vector.

The first component of the sequence corresponds to the bottom type, i.e. to the weakest non dummy players.

Parsimoniuos games are the subset of constant sum homogeneous weighted majority games characterized by the parsimony property to have, for any given number n of non dummy players in the game, the smallest number, i.e. exactly n, of minimal winning coalitions. For details see also[START_REF] Pressacco | Bilateral symmetry and modified Pascal triangles in Parsimonious games[END_REF].

By induction: it is true for j = 0, and if true for j it is true for j + 1.

Note that for t = 3 or 4 it is 1 + gt-2 = gt-1. Hence for such t: t + 2 + gt-2 = t + 1 + gt-1 so that both (7.2) and (7.3) describe n(t, ⌊(t + 1)/2⌋).