Pascal Auscher

Jos É Mar

Í A Martell

WEIGHTED NORM INEQUALITIES, OFF-DIAGONAL ESTIMATES AND ELLIPTIC OPERATORS. PART IV: RIESZ TRANSFORMS ON MANIFOLDS AND WEIGHTS

Keywords: Mathematics Subject Classification. 58J35, 35B65, 35K05, 42B20 Riemannian manifolds, Riesz transforms, Muckenhoupt weights, doubling property, Gaussian upper bounds

This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.

Introduction and main results

On R n , it is well-known that the classical Riesz transforms R j , 1 ≤ j ≤ n, are bounded on L p (R n , dx) for 1 < p < ∞ and are of weak-type (1,1) with respect to dx. As a consequence of the weighted theory for classical Calderón-Zygmund operators, the Riesz transforms are also bounded on L p (R n , w(x)dx) for all w ∈ A p (dx), 1 < p < ∞, and are of weak-type (1,1) with respect to w(x)dx for w ∈ A 1 (dx). Furthermore, it can be shown that the A p condition on the weight is necessary for the weighted L p boundedness of the Riesz transforms (see, for example, [Gra]).

On a manifold, there has been a number of works discussing the validity of the unweighted L p theory depending on the geometry of the manifold. Although some progress has been done in this direction, the general picture is far from clear. A difficulty is that one has to leave the class of Calderón-Zygmund operators. In particular, the Riesz transforms on the manifold may not have Calderón-Zygmund kernels, either because one does not have regularity estimates, or worse because one does not even have size estimates. It turns out also that the range of p for which one obtains L p boundedness may not be (1, ∞). See [ACDH] for a detailed account on all this and Section 2 below.

Here, we wish to develop a weighted theory: we want to obtain weighted L p estimates for a range of p and for Muckenhoupt weights with respect to the volume form. Of course, they must encompass the unweighted estimates so we shall restrict ourselves to situations where the unweighted theory has been developed. Nothing new will be done on the unweighted case (except the commutator result in Section 4).

We assume that the volume form is doubling. In that case, we are able to apply a machinery developed in the first article of our series [AM]. In a sense, the results we obtain could have been included in the latter as an application of the theory there, but we preferred a separate article to focus here on the geometric aspect of manifolds and because it also needed technical estimates or ideas from [ACDH]. As a matter of facts, recent developments in [BZ] allowed us to improve and simplify our result. The reader should have both [ACDH] and [AM] handy from now on.

Let M be a complete non-compact Riemannian manifold with d its geodesic distance. Let ∆ be the positive Laplace-Beltrami operator on M given by ∆f, g = M ∇f • ∇g dµ where ∇ is the Riemannian gradient on M and • is an inner product on T M. The Riesz transform is the tangent space valued operator ∇∆ -1/2 and it is bounded from

L 2 (M, µ) into L 2 (M; T M, µ) by construction.
The manifold M verifies the doubling volume property if the volume form is doubling:

µ(B(x, 2 r)) ≤ C µ(B(x, r)) < ∞, for all x ∈ M and r > 0 where B(x, r) = {y ∈ M : d(x, y) < r}. A Riemannian manifold M equipped with the geodesic distance and a doubling volume form is a space of homogeneous type. Non-compactness of M implies infinite diameter, which together with the doubling volume property yields µ(M) = ∞ (see for instance [Mar]).

One says that the heat kernel p t (x, y) of the semigroup e -t∆ has Gaussian upper bounds if for some constants c, C > 0 and all t > 0, x, y ∈ M,

p t (x, y) ≤ C µ(B(x, √ t)) e -c d 2 (x,y) t .
It is known that under doubling it is a consequence of the same inequality only at y = x [Gri, Theorem 1.1]. We recall the theorem proved in [CD].

Theorem 1.1. Let M be a complete non-compact Riemannian manifold satisfying the doubling volume property and the Gaussian upper bounds. Then

|∇∆ -1/2 f | p ≤ C p f p (R p )
holds for 1 < p < 2 and all f bounded with compact support.

Here, | • | is the norm on T M associated with the inner product. We shall set q + = sup p ∈ (1, ∞) : (R p ) holds . which satisfies q + ≥ 2 under the assumptions of Theorem 1.1. It can be equal to 2 ( [CD]). It is bigger than 2 assuming further the stronger L 2 -Poincaré inequalities ( [AC]). It can be equal to +∞ (see below).

Let us turn to weighted estimates. Properties of Muckenhoupt weights A p and reverse Hölder classes RH s are reviewed in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF]Section 2]

. If w ∈ A ∞ (µ), one can define r w = inf{p > 1 : w ∈ A p (µ)} ∈ [1, ∞) and s w = sup{s > 1 : w ∈ RH s (µ)} ∈ (1, ∞]. Given 1 ≤ p 0 < q 0 ≤ ∞, we introduce the (possibly empty) set W w (p 0 , q 0 ) = p 0 r w , q 0 (s w ) ′ = p : p 0 < p < q 0 , w ∈ A p p 0 (µ) ∩ RH ( q 0 p ) ′ (µ) .
Here, q ′ = q q-1 is the conjugate exponent to q. And note that RH 1 means no condition on the weight (besides A ∞ ).

Theorem 1.2. Let M be a complete non-compact Riemannian manifold satisfying the doubling volume property and Gaussian upper bounds. Let w ∈ A ∞ (µ).

(i) For p ∈ W w (1, q + ), the Riesz transform is of strong-type (p, p) with respect to w dµ, that is,

|∇∆ -1/2 f | L p (M,w) ≤ C p,w f L p (M,w) (1.1)
for all f bounded with compact support.

(ii) If w ∈ A 1 (µ) ∩ RH (q + ) ′ (µ), then the Riesz transform is of weak-type (1, 1) with respect to w dµ, that is,

|∇∆ -1/2 f | L 1,∞ (M,w) ≤ C 1,w f L 1 (M,w) (1.2)
for all f bounded with compact support.

If q + = ∞ then the Riesz transform is bounded on L p (M, w) for r w < p < ∞, that is, for w ∈ A p (µ), and we obtain the same weighted theory as for the Riesz transform on R n : Corollary 1.3. Let M be a complete non-compact Riemannian manifold satisfying the doubling volume property and Gaussian upper bounds. Assume that the Riesz transform has strong type (p, p) with respect to dµ for all 1 < p < ∞. Then the Riesz transform has strong type (p, p) with respect to w dµ for all w ∈ A p (µ) and 1 < p < ∞ and it is of weak-type (1, 1) with respect to w dµ for all w ∈ A 1 (µ).

In [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF]Lemma 4.6], examples of weights in A p (µ) ∩ RH q (µ) are given. The computations are done in the Euclidean setting, but most of them can be carried out in spaces of homogeneous type. In particular, given f, g

∈ L 1 (M, µ) (or Dirac masses) 1 ≤ r < ∞ and 1 < s ≤ ∞, we have that w(x) = M µ f (x) -(r-1) + M µ g(x) 1/s ∈ A p (µ) ∩ RH q (µ) (M µ
is the Hardy-Littlewood maximal function) for all p > r and q < s (and p = r if r = 1 and q = s if s = ∞). Thus, r w ≤ r and s w ≥ s.

We next provide some applications, then proof of our main result and eventually we add a short discussion on how to obtain (new) estimates for commutators with bounded mean oscillation functions.

Applications

Unweighted L p bounds for Riesz transforms in different specific situations were reobtained in a unified manner in [ACDH] and the methods used there are precisely those which allowed us to start the weighted theory. Therefore, it is natural to apply this theory in return to those situations. Let us concentrate on five situations (more is done in [ACDH]). Recall the notion of Poincaré inequalities: Let 1 ≤ p < ∞. One says that M satisfies the L p -Poincaré property, we write M satisfies (P p ), if there exists C > 0 such that, for every ball B and every

f with f, ∇f ∈ L p loc (µ), B |f -f B | p dµ ≤ Cr(B) p B |∇f | p dµ. (P p )
Here, r(B) is the radius of B, f B is the mean value of f over B. It has been proved in [START_REF] Saloff-Coste | A note on Poincaré, Sobolev, and Harnack inequalities[END_REF] that the doubling volume property and (P 2 ) is equivalent to lower and upper Gaussian estimates on the heat kernel. Recall that (P p ) implies (P q ) when q > p (see for instance [HK]).

2.1. Example without (P 2 ). Consider two copies of R n minus the unit ball glued smoothly along their unit circles with n ≥ 2. It is shown in [CD] that this manifold has doubling volume form and Gaussian upper bounds. (P 2 ) does not hold: in fact, it satisfies (P p ) if and only if p > n (see [HK] in the case of a double-sided cone in R n , which is the same). If n = 2, (R p ) holds if and only if p ≤ 2 ( [CD]). If n > 2, (R p ) holds if and only if p < n ( [CCH]). In any case, we have q + = n. Hence, W w (1, q + ) = (r w , n/(s w ) ′ ) is (contained in) the range of L p boundedness for a given weight provided this is not empty. In other words, if 1 < p < n and w ∈ A p (µ) ∩ RH (n/p) ′ (µ) then one has strong type (p, p) with respect to w dµ. For p = 1, one has weak-type (1, 1) with respect to

w dµ if w ∈ A 1 (µ) ∩ RH n ′ (µ).
2.2. Manifolds with non-negative Ricci curvature. In this case, the Riesz transform is bounded on (unweighted) L p for 1 < p < ∞ ([Ba1], [START_REF] Bakry | The Riesz transforms associated with second order differential operators[END_REF]). Thus q + = ∞. Such manifolds are known to have doubling volume form (see [START_REF] Chavel | Riemannian geometry: a modern introduction[END_REF]Theorem 3.10]), (P 2 ) and even (P 1 ) [Bus] (see, for instance, [HK] or [START_REF] Saloff-Coste | Parabolic Harnack inequality for divergence-form second-order differential operators, Potential theory and degenerate partial differential operators (Parma)[END_REF] for other references). By Corollary 1.3, we obtain strong-type (p, p) for 1 < p < ∞ and A p (µ) weights and weak-type (1,1) for A 1 (µ) weights.

2.3. Co-compact covering manifolds with polynomial growth deck transformation group. In this case, one has the doubling volume property and (P 2 ) (see [START_REF] Saloff-Coste | Parabolic Harnack inequality for divergence-form second-order differential operators, Potential theory and degenerate partial differential operators (Parma)[END_REF]) * . That the Riesz transform is of unweighted strong type (p, p) for 1 < p ≤ 2 is due to [CD]. For 2 < p < ∞ this is first done in [Dun] and hence q + = ∞. By Corollary 1.3, we obtain strong-type (p, p) for 1 < p < ∞ and A p (µ) weights and weak-type (1,1) for A 1 (µ) weights.

2.4. Conical manifolds with compact basis without boundary. As mentioned in [ACDH], this is not strictly speaking a smooth manifold but it is stochastically complete and this is what is needed to develop the unweighted theory for the Riesz transform: it is shown in [Li] that q + is a finite value related to the bottom of the spectrum on the Laplace operator on the compact basis. Also, one has doubling and (P 2 ) by [Li] and [CL] (and even (P 1 ) by using the methods in [GS]). Hence, W w (1, q + ) = (r w , q + /(s w ) ′ ) is (contained in) the range of L p boundedness for a given weight provided this is not empty. In other words, if 1 < p < q + and w ∈ A p (µ) ∩ RH (q + /p) ′ (µ) then one has strong type (p, p) with respect to w dµ. For p = 1, one has weak-type (1, 1) with respect to w dµ if w ∈ A 1 (µ) ∩ RH (q + ) ′ (µ).

2.5. Lie groups with polynomial volume growth endowed with a sublaplacian. One starts with left-invariant vector fields X j satisfying the Hörmander condition and µ is the left (and right) invariant Haar measure. The sublaplacian is ∆ = -n j=1 X 2 j . One has the doubling volume property and (P 2 ) (and even (P 1 )) (see [Var] or [START_REF] Haj | Sobolev met Poincaré[END_REF]p. 70] for a statement and references). The statement of Theorem 1.2 applies with no change to the Riesz transforms X j ∆ -1/2 . In this case, q + = ∞ * (P 1 ) also holds by a discretization method [START_REF] Coulhon | Variétés riemanniennes isométriques à l'infini[END_REF]Théorème 7.2] and Poincaré inequalities for discrete groups (see [START_REF] Haj | Sobolev met Poincaré[END_REF]p.76]).

from [Ale]. By Corollary 1.3, the weighted theory for these Riesz transforms is the same as the ones in R n for 1 ≤ p < ∞: strong type (p, p) with respect to w dµ holds for w ∈ A p (µ) and 1 < p < ∞, and weak-type (1, 1) with respect to w dµ for w ∈ A 1 (µ).

Proof of the main result

We assume that M satisfies the doubling volume property and Gaussian upper bounds.

We first introduce some notation. Given a ball B we set C j (B) = 4B for j = 1 and C j (B) = 2 j+1 B \ 2 j B for j ≥ 2, where λ B is the ball co-centered with B and radius λr(B). We use the notation

- B hdµ = 1 µ(B) B h dµ, - C j (B) h dµ = 1 µ(2 j+1 B) C j (B)
h dµ.

We write D µ for the doubling order of µ: µ(λB) ≤ C λ Dµ µ(B) for every λ > 1. In the sequel M µ is the Hardy-Littlewood maximal function with respect to the measure µ on M. We state a particular case of [AM, Theorem 3.1] † that is used in the proof of Theorem 1.2, (i).

Theorem 3.1. Fix 1 < q < ∞, a ≥ 1 and v ∈ RH s ′ (µ), 1 < s < q. Then, there exist C and K 0 ≥ 1 with the following property: Assume that F , G and H 1 are non-negative measurable functions on M such that for any ball B there exist non-negative functions G B and H B with F (x) ≤ G B (x) + H B (x) for a.e. x ∈ B and, for all x, x ∈ B,

- B H q B dµ 1 q ≤ a M µ F (x) + H 1 (x), - B G B dµ ≤ G(x). (3.1)
If 1 < r ≤ q/s and F ∈ L 1 (M, µ) (this assumption being only qualitative) we have

M µ F L r (M,v) ≤ C G L r (M,v) + C H 1 L r (M,v) . (3.2)
Proof of Theorem 1.2, (i). The argument borrows some ideas from [BZ] which are adapted to the present situation. Fix w ∈ A ∞ (µ) and p ∈ W w (1, q + ). Then (see, for instance, [AM, Proposition 2.1]) there exist p 0 , q 0 such that 1 < p 0 < p < q 0 < q + and w ∈ A p p 0

(µ) ∩ RH ( q 0 p ) ′ (µ).
By [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF]Lemma 4.4] we have that

v = w 1-p ′ ∈ A p ′ /q ′ 0 (µ) ∩ RH (p ′ 0 /p ′ ) ′ (µ)
. We write T = ∇∆ -1/2 and observe that the boundedness of T from L p (M, w) to L p (M; T M, w) is equivalent to that of T * from L p ′ (M; T M, v) to L p ′ (M, v) -we notice that T takes scalar valued functions on M to functions valued in the tangent space (sections) and T * the opposite, T M being equipped with the inner product arising in the definition of the Laplace-Beltrami operator-.

Fix f ∈ L ∞ c (M; T M, µ) ‡ , and write h = T * f and F = |h| q ′ 0 . Notice that F ∈ L 1 (M, µ) since T * is bounded from L q ′ 0 (M; T M, µ) to L q ′ 0 (M, µ) as 1 < q 0 < q + and † It is stated in the Euclidean setting, see [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF]Section 5] for the extension to spaces of homogeneous type ‡ Here and subsequently, the subscript c means with compact support.

T is bounded from L q 0 (M, µ) to L q 0 (M; T M, µ). We pick A r = I -(I -e -r 2 ∆ ) m with m large enough. Given a ball B we write r B for its radius. Then,

F ≤ G B + H B ≡ 2 q ′ 0 -1 |(I -A r B ) * h| q ′ 0 + 2 q ′ 0 -1 |A * r B h| q ′ 0 .
We first estimate H B . Set q = p ′ 0 /q ′ 0 and observe that by duality there exists g ∈ L p 0 (B, dµ/µ(B)) with norm 1 such that for all x ∈ B -

B H q B dµ 1 q q ′ 0 µ(B) -1 M |h| |A r B g| dµ ∞ j=1 2 j Dµ - C j (B) |h| q ′ 0 dµ 1 q ′ 0 - C j (B) |A r B g| q 0 dµ 1 q 0 ≤ M µ F (x) 1 q ′ 0 ∞ j=1 2 j Dµ - C j (B) |A r B g| q 0 dµ 1 q 0 .
To estimate the summands we use the Gaussian upper bound on p t (x, y), so that for any fixed integer m there exist c, C > 0 such that for all j ≥ 1, all ball B, all

g ∈ L 1 (M, µ) supported in B and all 1 ≤ k ≤ m, sup C j (B) |e -k r 2 B ∆ g| ≤ C e -c 4 j - B |g| dµ. (3.3) Then expanding A r = I -(I -e -r 2 ∆ ) m we conclude that - B H q B dµ 1 q q ′ 0 M µ F (x) 1 q ′ 0 ∞ j=1 2 j Dµ e -c 4 j - B |g| q 0 dµ 1 q 0 M µ F (x) 1 q ′ 0 . (3.4)
We next estimate G B . Using duality there exists g ∈ L q 0 (B, dµ/µ(B)) with norm 1 such that for all

x ∈ B - B G B dµ 1 q ′ 0 µ(B) -1 M |f | |T (I -A r B )g| dµ ∞ j=1 2 j Dµ - C j (B) |f | q ′ 0 dµ 1 q ′ 0 - C j (B) |T (I -A r B )g| q 0 dµ 1 q 0 ≤ M µ (|f | q ′ 0 )(x) 1 q ′ 0 ∞ j=1 2 j Dµ - C j (B) |T (I -A r B )g| q 0 dµ 1 q 0 .
To estimate the terms in the sum we use the following auxiliary result whose proof is given below.

Lemma 3.2. For all β ∈ [1, q+ ) ∪ [1, 2], one has the following estimate: for all m ≥ 1, there exists C > 0 such that for all j ≥ 2, all ball B, all g ∈ L 1 (M, µ) with support in B, -

C j (B) |∇∆ -1/2 (I -e -r(B) 2 ∆ ) m g| β dµ 1 β ≤ C4 -jm - B |g| dµ. (3.5)
Here, q+ is defined as the supremum of those p ∈ (1, ∞) such that for all t > 0,

|∇e -t ∆ f | p ≤ C t -1/2 f p . (3.6)
By analyticity of the heat semigroup, one always have q+ ≥ q + . Under the doubling volume property and (P 2 ), it is shown in [START_REF] Auscher | Riesz transforms on manifolds and heat kernel regularity[END_REF]Theorem 1.3] that q + = q+ . We do not know if the equality holds or not under doubling and Gaussian upper bounds. This lemma allows us to conclude right away with the terms where j ≥ 2. When j = 1, we use that T is bounded from L q 0 (M, µ) to L q 0 (M; T M, µ) as 1 < q 0 < q + . Also, applying (3.3) it follows easily that

- 4 B |T (I -A r B )g| q 0 dµ 1 µ(4 B) B |g| q 0 dµ + ∞ j=1 C j (B) |A r B g| q 0 dµ - B |g| q 0 dµ ∞ j=1 2 j Dµ e -c 4 j - B |g| q 0 dµ. (3.7)
Using this and (3.5) (with β = q 0 < q + ≤ q+ ) we conclude the estimate for G B :

- B G B dµ 1 q ′ 0 M µ (|f | q ′ 0 )(x) 1 q ′ 0 ∞ j=1 2 j Dµ 4 -j m - B |g| q 0 dµ 1 q 0 ≤ C M µ (|f | q ′ 0 )(x) 1 q ′ 0 = G(x) 1 q ′ 0 , (3.8)
provided m > D µ /2. With these estimates in hand we can use Theorem 3.1 with r = p ′ /q ′ 0 , q = p ′ 0 /q ′ 0 and H 1 ≡ 0. Notice that v ∈ RH s ′ (µ) with s = p ′ 0 /p ′ , 1 < s < q < ∞ and r = q/s. Hence, using v ∈ A r (µ) we obtain the desired estimate

T * f q ′ 0 L p ′ (M,v) ≤ M µ F L r (M,v) M µ (|f | q ′ 0 ) L r (M,v) |f | q ′ 0 L p ′ (M,v) . (3.9)
Proof of Lemma 3.2. First, this estimate is known for β = 2 (see [ACDH]). Also, the inequality for a fixed β 0 implies the same one for all β with 1 ≤ β ≤ β 0 . It suffices to treat the case β > 2, which happens only if q+ > 2.

We use a trick from [START_REF] Auscher | Riesz transforms on manifolds and heat kernel regularity[END_REF]Proof of Lemma 3.1]. Fix a ball B, with radius r, and f ∈ L ∞ (M, µ) supported in B. We have

∇∆ -1/2 (I -e -r 2 ∆ ) m f = ∞ 0 g r (t) ∇e -t ∆ f dt where g r : R + → R is a function such that ∞ 0 |g r (t)| e -c 4 j r 2 t dt √ t ≤ C m 4 -j m . (3.10)
By definition of q+ and the argument of [START_REF] Auscher | Riesz transforms on manifolds and heat kernel regularity[END_REF]p. 944] we have

M |∇ x p t (x, y)| β e γ d 2 (x,y) t dµ(x) 1/β ≤ C √ t µ(B(y, √ t)) 1-1/β ,
for all t > 0 and y ∈ M, with γ > 0 depending on β. This implies that for all j ≥ 2, y ∈ B and all t > 0,

- C j (B) |∇ x p t (x, y)| β dµ(x) 1/β 1 √ t e -c 4 j r 2 t 1 µ(B(y, √ t)) 1-1/β µ(2 j+1 B) 1/β .
Using the doubling property, µ(2 j+1 B) ∼ µ(B(y, 2 j+1 r)) uniformly in y ∈ B and

µ(B(y, 2 j+1 r)) µ(B(y, √ t)) max 1, 2 j r √ t Dµ .
Hence, with another c > 0,

- C j (B) |∇ x p t (x, y)| β dµ(x) 1/β 1 √ t e -c 4 j r 2 t 1 µ(2 j+1 B) ≤ 1 √ t e -c 4 j r 2 t 1 µ(B)
.

We conclude using Minkowski's integral inequality and (3.10) that the left hand side of (3.5) is bounded by

∞ 0 |g r (t)| B |f (y)| - C j (B) |∇ x p t (x, y)| β dµ(x) 1/β dµ(y) dt ∞ 0 |g r (t)| 1 √ t e -c 4 j r 2 t dt - B |f | dµ 4 -j m - B |f | dµ.
The following result, used to prove Theorem 1.2, (ii), is taken from [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF]Theorem 8.8 & Remark 8.10] (see also [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF]Section 8.4] for the extension to spaces of homogeneous type) Theorem 3.3. Let 1 ≤ p 0 < q 0 ≤ ∞ and w ∈ A ∞ (µ). Let T be a sublinear operator defined on L 2 (M, µ) and {A r } r>0 be a family of operators acting from L ∞ c (M, µ) into L 2 (M, µ). Assume the following conditions:

(a) There exists q ∈ W w (p 0 , q 0 ) such that T is bounded from L q (M, w) to L q,∞ (M, w). (b) For all j ≥ 1, there exist a constant α j such that for any ball B with r(B) its radius and for any f ∈ L ∞ c (M, µ) supported in B, -

C j (B) |A r(B) f | q 0 dµ 1 q 0 ≤ α j - B |f | p 0 dµ 1 p 0 .
(3.11) (c) There exists β > (s w ) ′ , i.e. w ∈ RH β ′ (µ), with the following property: for all j ≥ 2, there exist a constant α j such that for any ball B with r(B) its radius and for any f ∈ L ∞ c (M, µ) supported in B and for j ≥ 2,

- C j (B) |T (I -A r(B) )f | β dµ 1 β ≤ α j - B |f | p 0 dµ 1 p 0 . (3.12) (d) j α j 2 Dw j < ∞ for α j in (b) and (c)
, where D w is the doubling constant of w dµ.

If w ∈ A 1 (µ) ∩ RH (q 0 /p 0 ) ′ (µ) then T is of weak-type (p 0 , p 0 ) with respect to w dµ, that is, for all f ∈ L ∞ c (M, µ), T f L p 0 ,∞ (M,w) ≤ C f L p 0 (M,w) .
Proof of Theorem 1.2, (ii). We are going to apply Theorem 3.3 with p 0 = 1 and q 0 = q + . Thus we need to check that the four items hold. Fix w ∈ A 1 (µ) ∩ RH (q + ) ′ (µ). By [AM, Proposition 2.1], there exists 1 < q < q + such that w ∈ A q (µ) ∩ RH (q + /q) ′ (µ). This means that q ∈ W w (1, q + ), therefore by (i), T = |∇∆ -1/2 | is bounded on L q (M, w) and so (a) holds.

We pick A r = I -(I -e -r 2 ∆ ) m with m large enough to be chosen. Notice that expanding A r , (3.3) yields (b) with α j = C e -c 4 j . To see (c) we apply Lemma 3.2 with (s w ) ′ < β -notice that such β exist: we have q + ≤ q+ and w ∈ RH (q + ) ′ (µ) implies (s w ) ′ < q + -. Then, we obtain (c) with α j = C 4 -j m . Finally, we pick m > D w /2 so that (d) holds and therefore Theorem 3.3 gives the weak-type (1, 1) with respect to w dµ.

Commutators

Let us write again T = ∇∆ -1/2 and take b ∈ BMO(M, µ) (the space of bounded mean oscillation functions on M). We define the first order commutator T

1 b g = [b, T ]g = b T g -T (b g), and for k ≥ 2 the k-th order commutator is T k b = [b, T k-1 b
]. Here g, b are scalar valued and T k b g is valued in the tangent space. Theorem 4.1. Under the assumptions of Theorem 1.2, T k b satisfies (1.1) for each k ≥ 1, that is, it is bounded from L p (M, w) into L p (M; T M, w) under the same conditions on w, p.

This theorem applies in particular to the five situations described in Section 2. Note that even the unweighted L p estimates for the commutators are new.

Proof. The proof is similar to that of Theorem 1.2 using again the ideas in [BZ] and we point out the main changes. We only consider the case k = 1: the general case follows by induction and the details are left to the reader (see [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF]Section 6.2] for similar arguments). As in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF]Lemma 6.1] it suffices to assume qualitatively b ∈ L ∞ (M, µ) and quantitatively b BMO(M,µ) = 1 and get uniform bounds.

We proceed as before working with T 1 b in place of T . Write F = |(T 1 b ) * f | q ′ 0 with f ∈ L ∞ c (M; T M, µ) and observe that F ∈ L 1 (M, µ) as b ∈ L ∞ (M, µ) and T * is bounded from L q ′ 0 (M; T M, µ) into L q ′ 0 (M, µ) (as 1 < q 0 < q + ) -we observe that this is the only place where we use that b ∈ L ∞ (M, µ)-.

F = |(T 1 b ) * f | q ′ 0 = |(T * ) 1 b f | q ′ 0 ≤ 2 q ′ 0 -1 | b T * f | q ′ 0 + 2 q ′ 0 -1 |T * ( b f )| q ′ 0 ≤ 2 q ′ 0 -1 | b T * f | q ′ 0 + 4 q ′ 0 -1 |(I -A r B ) * T * ( b f )| q ′ 0 + 4 q ′ 0 -1 |A * r B T * ( b f )| q ′ 0 = G B + H B .
We estimate H B . By duality we pick g as before and obtain F dµ

- B H q B dµ 1 q q ′ 0 = C µ(B) -1 M T * ( b f ) A r B g dµ = C µ(B) -1 M -(T * ) 1 b f + b T * f A r B g
1 q ′ 0 - C j (B)
|A r B g| q 0 dµ 1 q 0 M µ F (x)

1 q ′ 0 .

  Fixing B we write b = b-b B and decompose T 1 b as T 1 b g = -T ( bg) + b T g. Using this equality one sees (T 1 b ) * = -(T * ) 1 b . Then we have

  * ) 1 b f | |A r B g| dµ + µ(B) -1 M | b| |T * f | |A r B g| dµ = I + II.The estimate for I follows as in (3.4) by using (3.3): for all x ∈ B,I ∞ j=1 2 j Dµ -C j (B)

This work was partially supported by the European Union (IHP Network "Harmonic Analysis and Related Problems" 2002-2006, Contract HPRN-CT-2001-00273-HARP). The second author was also supported by MEC "Programa Ramón y Cajal, 2005" and by MEC Grant MTM2007-60952.

We warmly thank T. Coulhon for interest and help in finding appropriate references. We also want to express our deep gratitude to F. Bernicot and J. Zhao for letting us use their unpublished work.

We pick 1 < q < ∞ and use Hölder's inequality to obtain that for all x ∈ B II ∞ j=1 2 j Dµ -

|A r B g| q 0 s dµ

where we have used (3.3) and John-Nirenberg's inequality. Collecting I and II we conclude the first estimate in (3.1) with

Let us write G B,1 and G B,2 for each of the terms that define G B and we estimate them in turn. Take δ > 1 to be chosen and use John-Nirenberg's inequality: for any x ∈ B we have

To estimate G B,2 we proceed as with G B in the proof of Theorem 1.2. Let g be the corresponding dual function and use again John-Nirenberg's inequality

where in the last estimate we have proceeded as in (3.8) using (3.5) for j ≥ 2 and (3.7) for j = 1. Gathering what has been obtained for G B,1 and G B,2 we conclude the second estimate in (3.1) with

1 δ . We apply Theorem 3.1 as in the proof of Theorem 1.2. In this case, we observe that as v ∈ A r (µ), there exists 1 < δ < r so that v ∈ A r/δ (µ). Then, the desired estimate follows

where we have used that T * is bounded from L p ′ (M; T M, v) into L p ′ (M, v) (see (3.9)).


DYNAMICS IN THE PRESENCE OF UNILATERAL CONTACTS AND DRY FRICTION

This paper is essentially devoted to mechanical systems with finite degree of freedorn. However, as a test for adapting the proposed numerical methods to continuous systems, the representation of a one-dimensional medium by a finite chain of mass points is presented in fine.

The theoretical background has been defined by one of the authors in the Volume of the last meeting [START_REF] Moreau | Standard inelastic shocks and the dynamics of unilate ral constraints[END_REF] and will only be recalled under some simplifying assumptions in the sequel. This background involves in particular that the possible shocks, occurring if two parts of the system suddenly enter into contact, are inelastic. The paper referred to was restricted to frictionless contacts; in contrast dry friction is taken here into account, under the traditional law of Coulomb or some extension of it to anisotropie contacting surfaces. Mathematically, this law results in some non-smooth relation between the involved variables, i.e. not everywhere differentiable ; the same is true for the expression of unilateral constraints. Treating both non-smooth aspects jointly is made possible here by using simple concepts of Convex Analysis, widely known today. Of course, some smooth dissipative actions -viscosityare easy to be introduced additionally.

Through the frictional effects as well as the inelasticity of possible shocks, irreversibility emerges as a dominant feature in the present approach.

No attention is paid in this paper to the existence and uniqueness study of the investigated motions. It has long been observed [START_REF] Delassus | Considérations sur le frottement de glissement[END_REF] [3] tqat when bodies exhibit dry friction in their possible points of contact, time intervals of smooth motion may end with instants at which some velo city jump occurs necessarily; this is called a tangential shock. Suffi cient conditions imposed to the system constants and to initial data have recently been given [START_REF] Jean | A system of rigid bodies with dry friction[END_REF] , in order that the motion begins with a time interval exempt from such an accident (and from contact breaking off as well) ; thereby a local existence and uniqueness theorem has been obtained.

Since the algorithms presented in the sequel are devised to treat shocks on an equal footing with regular motions, they prove able to include such events in the computation.

An example presented in Sect. 8 is that of the chattering motion of a solid body elastically driven along a plane wall (a piece of chalk pushed at an angle against a blackboard). Oscillations with intermittent breaking of contact are exhibited. The slip velocity may drop recurrently to zero ; contrary to common belief, this "stick-slip" phenomenon is not essentially connected with the possible dependence of the friction coef ficient on slip velocity. Anyway, the proposed algorithm allows one to take s�ch a dependence into account, even under its extreme form : distin guishing between the "static" (i.e. for zero slip velocity) and "dynamic" friction coefficients.

Another example presented in section 6 is that of the motion of a chain of material points in contact with a wall moving at constant velo city along one of the axes. The points are connected together by springs and dashpots and each of them is pushed against the wall by a single pair of springs and dashpots. The points are submitted to Coulomb friction from the wall. The example presented shows a "stick-slip" phenomenon.

ANALYTICAL SETTING

As usual in Analytical Dynamics, the totality of the possible posi tions of the system S, regardless of the unilateral constraints which will be imposed further, constitutes a differential manifold Q, with dimension i equal ta the freedom of S • Let (q i ) , i = 1,2, ... ,i denote (at least local) coordinates in Q. For every differentiable motion t � q(t) the derivatives q i (t) are the components of the veloc i ty q(t) , an element of the tangent space ta Q at the point q(t) . The kinetic energy of S is supposed expressed as a time i ndependent positive definite quadratic form of q, namely • I • i. j T(q,q) = -2 a .. (q)q q iJ where a .. = a .. iJ Ji are known functions of I i q=(q , ... ,q )

( 2. I)

For simplicity we shall make in the whole paper the following assumptians.

ASSUMPTION I • S i s partitioned into a finite collection of subsystems (S) , a= I, ... ,n, which may interact through configuration -dependent a and/or velocity -dependent forces, but without any mutual hinge or contact.

Therefore Q equals the produc t of the corresponding man ifolds � and T(q,q) splits i nto the sum of the respective kinetic energies, narnely for subsystem ASSUMPTION 2. For every et , it is supposed that, at least loca:ly in ma � ifold � the coordinate system 1 2

(q ,q ' ... ) This means that the Riemann metric defined on manifold � by the follo wing expression of arc-length ds actually turns out to be Euclidean.

Such is trivially the case if subsystem S a consists of a single mass-point moving through some region of the physical space.

In another usual case, S will consist in a rigid body, allowed a to perform only motions parallel to some fixed plane. Locating each of its possible positions by the Cartesian (orthonormal) coordinates X,Y of the center of mass G and by the rotation angle E) ' one obtains

1 • 2 2 1 • 2 T = -m(X +Y)+ -I 0 a 2 2 (2.3)
Here m denotes the mass of the rigid body and I its moment of inertia about the axis drawn through G, normal to the plane of motion. This makes [aa J a diagonal matrix, mn constant with regard to the triplet

I 2 3
(q ,q ,q ) = ( X,Y,0) a a a

As already said, the investigated unilateral constraints are not taken into account at the stage of defining the parametrization of S Such superimposed constraints will consist in impenetrability effects about which we shall make now some simplifying hypotheses.

ASSUMPTION 3 . Unilateral contacts never occur between two different members of the previously defined partition ; they take place either 1n side some S or between some of these subsystems and some external body a with prescribed motion. In addition, it is supposed that the geometric effect of impenetrability upon the concerned subsystem by a single scalar inequality

I 2 f (t,q ,q , ••. ) � 0 a a a is expressed (2.4)
where the real function f is c 1 a with nonzero gradient (at least in a neighborhood of the hypersurface f a = 0) This inequality defines for every t a closed subset L a of manifold �-Under these assumptions, let us first write the Lagrange equations governing the motion of the system S during a time interval of smooth motion, i.e. an interval, if any, where the functions i t + q (t) happen to be twice differentiable. Since the matrix a a mn tant with regard to t relative to subsystem and to that a.

the Lagrange equations corresponding to the respective members Sa. of the partition of S are coupled together.

Under the writing (2.5) is summarized a set of scalar equations whose number equals the freedom of subsystem a.

There is to join with them the geometric inequality (2.4) and some mechanical information we shall state in the section to corne, about the force of unilateral constraint a. r

LAWS OF FRICTIONAL UNILATERAL CONTACT

The primary stateme nt concern ing the force of constra int introdu ced in (2.5) , is that it vanishe s at every instan t t where the corresp onding contact is not in effect, i.e. when "' I 2 the position inequality. q (t)-( q (t),q (t), ... ) a a a verifies (2.4) as a strict

On the contra ry, every positi on q a of S a such that (2.4) holds as an equalit y corresp onds to effecti ve contact . The governi ng laws have to be made precise in the framewo rk of the 3-dimen sional physica l space after wards in sect. 4 we shall transcr ibe tqem in terms of analyti cal variabl es.

Let us first conside r the case where the investi gated contact takes place between some part B I of s a and an externa l body B with pres- cribed motion. Suc� a contact will always be suppose d located at a single point M and the definit ion of a plane is assumed , which will be declare d DECOMPOSED FORM (see more details in [START_REF] Moreau | Sur les lois de frottement, de plasticité et de visco sité[END_REF] [6]).

The reaction vector R lets itself be uniquely decomposed into

(3.3)
It is first stated that the real number R N satisfies (3.4)

i.e. no adhesive effect takes place between the bodies in contact.

Secondly there is given a closed convex subset v 1 of T • con- taining the zero of E 3 . For every R )) ; see e.g. [START_REF] Moreau | Application of convex analysis to some problems of dry friction[END_REF] [io] .

N � o put V(R N ) = R N V l ( 3 .
The traditional law of Coulomb for isotropie contact is simply obtained by taking as V 1 the closed disk in T, with center at the origin and radius equal ta the friction coefficient.

CONICAL FORM.

Standard arguments of Convex Analysis [START_REF] Moreau | Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel ; essais numériques[END_REF] Then the previous formulation 1s found equivalent ta

(3. 7)
Here proj T denotes the orthogonal projection mapping from E The contact actions exerted by B 2 upon B 1 are supposed represented by a single force R applied to M l ; hence, in view of Newton's third law, the contact actions exerted by B I upon B z are represented by the single force -R applied to M z The law of dry friction stipula tes a relation between R and u ' exactly in the same terms as in the preceding case.

Let us finally take notice of the simplifying circumstance of two dimensional contact. In fact, it may happen that, due to the primitive constraints of the system (i.e. the perfect constraints which have per mitted its parametrization) the set of the values of V(M 1 ) -V(M 2 ) (resp. V(M 1 ) in case of external contact) which,regardless to contact permanence,are possible, is not the whole of E 3 but only a twodimensional linear subspace E 2 , a priori depending on q a Then the role of T is played by some one-dimensional subspace of E

2

; the cone C simply reduces to an angular region whose boundary consists of two half-lines v l and v 2 (possibly non-symmetric with respect to the normal direction N: this reflects difference in the friction coefficients associated with either directions of sliding).

The active part of the reaction R consists now in the orthogonal projection of this vector on E 2 • Similar formulations as above then hold, with E 3 replaced by E 2 , but two-dimensionality brings the following simplification.

Let A 1 and A 2 denote generators of V 1 and V 2 B = A 2 -A 1 cannot be parallel to N; then, for every is readily found equivalent to

RE C if B.U > o if B.U < D
The vector

UET (3. 7) (3.9)

EQUATIONS OF SMOOTH MOTIONS

The law of frictional unilateral contact formulated above has now ta be transcribed into the analytical setting of Sect. 2

Let us first consider the case of the contact occurring between a part B I of s CL B 0

The contacting particle and some external body Ml of B I a priori depends on the position q CL E � attained by the system at the considered instant and also, since B has been assu-CL 0 med to possess a certain prescribed motion, on the time t as independent variable. Supposing t fixed and given the position q CL (t) E � , let us consider a chain of positions of s CL depending in a differentiable way on some real variable T . Using local coordinates

1 2 (q , q , ••• ) in CL CL Q define it by CL ' i i q CL = K (T) ,
with real functions i

T 1-+ K (T) ( 4. 1)
such that i = q (t)

CL

,i =l,2, ... this is conventionally called a virtual displacement at fixed time t.

If some particle P of S has been identified, every value of CL yields a position of P, relatively to the reference frame F adopted in physical space. When q CL is expressed by (q ,q , .•• )

Cl. Cl.

aP\o)

have been defined, one may obtain for the vector i ÔT an expression in which the real derivatives �(0) ÔT figures linearly (using here ô instead of d to denote differentials is traditional in order to prevent confusion between virtual displacements and proper motions ). This is true in particular if P equals the particle This calculation is first used to express the covariant components (or "generalized components") of the force R exerted on the particle M 1 of B • By definition these components are the real numbers r 1 ,r 2 , .•. such that, for �very virtual displacement as above,

The real numbers r. may be viewed as of the dual of E the linear space Cl.

at point q (t) Denoting by <. '. > Cl.

and E* and a "'-a dinates as differentiable functions of time. Then, for every identified particle P of the system, the position of P , relative to the reference frame F becomes a t-dependent point of this frame, whose vector derivativeconstitutes the velocity vector V(P) • For simplicity, we restrict ourselves in the sequel to scleronomic systems, i.e. the position of p in F is determined by "' 1 2 q -(q 'q ' ••. ) a a a independently of time; this is in fact the only practical case where the expression of kinetic energy reduces to the form assumed in (2.1) Then expressing

• 1 • 2 q a' q a' • • • V(P) from the t-derivatives
requires exactly the same calculation as was made above for expressing o/;o, in some virtual displa- OK 1 OK 2 cernent, from the ,-derivatives OT OT ' ... (in the absence of the scleronomy assumption, there would appear in V(P) some additional term).

Hence, by making P equal to the particle M 1 one obtains

The contacting particle of external body a priori depends on the position q a E � of S a and also on t as independent variable, since, by assumption, B exhibits some given motion. Hence the velocity 0 V(M ) 0 appears as a vector function u e of 1 2 (t,q ,q ' ... ) a a which has to be calculated from geometric inspection. This finally yields an expression of the slip velocity (3.1 ) under the form

u G(q) + U (t,q ). a e a
Of course the linear mapping G: E a + E 3 depends on q En a --u

(4.5)
and also on time t as independent variable, since these elements are needed in order to identify the contact particle M 1 • What precedes makes sense only for t and such that the contact is effective, i.e. inequality (2.4) holds as an equality. In the sequel we shall need the definition of G and U to be extended in a smooth e arbitrary way to every (t,q a ) , at least in some neighborhood of hypersurface f (t,q) = 0. a a

The case of contact occurring between two parts B 1 of S a lets itself be treated in the same way. Then both opposite reactions R and -R experienced by the contact particles M 1 and M 2 are forces acting on the system. Thus the analytical treatment will involve the covariant components of this pair of forces, i.e. the real numbers r 1 ,r 2 , .. such that, instead of (4. 3) one has

F F o M 1 o M 2 O K i R.
---;fr -R. Furthermore\ the no-contact case the same writing by putting : f(t,q ) > 0 may be embodied in

Cl CONVENTION 2
In the case f (t,q) > 0, the value of the multia a function cou is declared to be the singleton {O} whichever is U

Cl

Incidentally, the geometric impenetrability condition could also be included by agreeing that the multifunction assumes the value 0 whenever it is not satisfied. We shall avoid this trick for it does not prove con venient when discussing numerical approximation.

In exploiting the above set of conditions, one may wish to eliminate the variables R and U. This finally yields a a q N -Q a lt,q,q ) E G�( co u (t,q ,G(q) + U (t,q ))) � On the contrary one may keep the reactions as primary unknowns. One has the following set of equations :

eif a is the index of a subsystem with contact at time t Q�(t,q,q ) + G*(R) where aw is the subdifferential of the indicator function of the convex

"' Cl Cl U = G (q ) + U a a a ea R = R T + R N N Cl Cl Cl Cl R N � Cl -u E aw<Rr ) Cl (4.9 )
set V(R N ) = R V 1 a N a
eif a is the index of a subsystem with no contact

The initial conditions q(t ),q( t) being fixed, let us write as

0 0 U = A (R )
Cl CL the result of the elimination of between (4.9 ) and where the dot refers to scalar product in E 3

Hence conditions (4.9 ) to (4.13 ) are finally equivalent to a system of quasi-variational inequalities.

In usual cases, due to the positiveness of inertia matrices and the convexity of elastic potential, the mapping

'\, A -(A 1 , ..• ,A 0 , ••• )
turns out to be monotone in the sense of Minty and the mapping Id -p A is a contraction for �very p > 0 in some interval.

Finally the problem may be formulated as the following fixed point condition -p A (R))

a together with (4.11),(4.12)

ALGORITHM OF CONICAL TYPE

By introducing the additional unknown vector function of t one gives to the differential inclusion (4.8) the form Let a a û -Q a (t,q,q) E c*(cou (t,q ,G(u ) + U (t,q ))) a which refers to the cons idered member of the parti tian of S denote by q i , u i some approximants of vectors (5.1) is discretized into q. I = q. + h u. l. + l.

l.

q (t.) 'u (t.) a 1. a 1.

we shall

Then

(5.

3)

The discretization of (5.2) has to be performed in such a way that each step, starting with known q . l. and yields a computable value of In simple cases, this value is expected to be uniquely defined (anyway, it has long been recognized [START_REF] Delassus | Considérations sur le frottement de glissement[END_REF] that, in the dynamics of sys tems with dry friction, uniqueness of the motion consequent to initial datais not granted in general).

Recall that q , q in Q a , on the left-hand side of (5.2 ) , refer to the whole system S a and not only to the rnember of the partition. When cornputing an approximant of Q a one may assign to t and q in this function the starting values of the considered step or, alternatively, some anticipated values such as q i + l or ( q . + q . 1 ) / 2 l.

l. + the influence of this choice upon numerical precision is discussed in [START_REF] Moreau | Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel ; essais numériques[END_REF] As for q in Q a , we decide to adopt the s tarting values of the step; in that respect, our computation looks like an "implicit" scheme : this is justified by the assumed smoothness of the function Q a .

For the sequel, let us denote by constructed in that way.

Q!

l.

the approximant of vector Incidentally, in the absence of assumption 2 made in sect. 2

for sirnplicity, there would appear in the left-hand members of Lagrange equations some quadratic terms in q : the same would apply to them.

Similarly, the linear mapping G of E a and the velocity U e E E 3 of a possible external obstacle, which are smooth known functions of t and q, will be calculated from the starting values q . or through some anticipation as above.

l.

t. '

l.

In contrast, the Coulomb multifunction behaves critically since, with regard to variables t and q , it is discontinuous on the boundary a of the permitted region. Strictly speaking, cou is not defined if a f (t, q ) > 0 a a however, 1.n the course• of computation, a certain amount of violation of the unilateral constraint may be faced, so wé agree to extend in a smooth arbitrary way the definition of cou a at least when condition f (t,q) � 0 is moderately violated. For instance, in a a the case of the traditional (isotropie) Coulomb law, a cone of revolution in E 3 will be considered, with axis and angle depending continuously on t and so as to reduce to the proper Coulomb cone when the contact condition f(t,q ) = 0 holds. The definition of cou derives a a from this cone in the way explained in sect. 3

Recall that cou (t,q ,U) = {O} if f (t,q ) < 0 . Hence every a a a a discretization step rests on the consideration of some te1.it pain;t (t! ,q!) l l

an expedient choice proves to be q! = q, + h U, /2 .

l l l t! = t. + h/2 , l l
If f (t!,q!) < 0 it will be admitted that cou = {0} for the If, on the contrary, <u!,g> > 0, l.

the nontrivial multifunction U + cou (t,q ,U) has to be considered, with (t,qa) conveniently fixed Cl Cl at the same test point (t ! 'q ! ) l. l.

as above, and shortly written cou(U) .

In order to yield a precise characterization of u i + l , the discretiza tion has to involve this multifunction through sorne "implicit" procedure, i.e. the value substituted for u in the right-hand side of (5.2)

Cl will be the unknown u i+l itself. Since the set cou(U) is essentially conic, one rnay equivalently multiply both rnembers by the positive number h, thus obtaining and the positive definite matrix a may be made unit; this amounts to identify both spaces with a single copy of Euclidean JR. 3 (at a more elaborate level it may be said that, due to the Riemannian structure of manifold � , the tangent and cotangent spaces at each point q a merge into a single Euclidean linear space;

this standpoint is systematically used in [IJ , [START_REF] Moreau | Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel ; essais numériques[END_REF])

Recall that we are now in the case where <u! ,g> , i.e. the scalar l.

product u! .g l.

in the sense of Euclidean JR. this is precisely a di-One readily concludes in the existence of a single point u, l E T 1+ satisfying (5.6) , namely :the. n.e.aJt.e..&t po-<.n-t -<.n. S ta :the. aJug-<.n. 06 :m. Also observe that, at each step of our algorithm,there is no difficulty in making the Coulomb mapping depend on u. this accounts in particular for the usual empirical distinction between the "static" and "dynamic" friction coefficients.

ALGORITHM BASED ON REACTIONS

In presenting the main features of this algorithm we shall restrict ourselves to the case where the subsystems S are points, connected CL wi th springs and dashpots moving in a plane ox 1 x 2 (see fig. 2 )

All of them are confined in the region x 2 � 0

The obstacle B has 0 rectilinear boundary x 2 = 0 , moving along the x 1 axis at velocity E.

We consider small displacements so that the equation (2.5) governing the system may be written as

P (t) + R CL CL V ( " )
CL q represents the viscosity forces from the dashpots

K (q)

CL presents the elastic forces from the springs P (t) is some force

into JR

2n , with the identity matrix on the a 6 ,s block and zero elsewhere.

There is a wide range of methods to compute an approximation of the solution to an ordinary differential e q uation such as (6.1).In numerous cases, explici t or implicit methods provide two matrices � ' ]Bh , ( h denotes the step length t . 1 t.) deriving an approximate value 1+ 1 q i+l of q( t.) and qi+l of q ( t i+l ) from an approximate value q. 1 1 of q ( t.) and q . of q (t.) namely 1 1 1

q i+l q . 1 =A h + q i+l q . 1 If) p + c* il p is an approximant of p R on [ t i ,t i+l]
One has approximants of q and q mant of the slip velocity on q = ( 1 -s) q i + s q i+l ( l-s) q

0 • + s q 1 i+l ]Bh h � on also on [t.,t. l ] 1 1+ to choose [t. ,t. ,1 1 1+ [t., t. ,J 1 1+ . and R is the values in order A possible s E [0,1] (6. 2)
an approximant of and ...... of q q

to get an approxichoice 1S

The choice s = 1 proves the best one regarding stability or smoothing effects. The velocity resulting from ( 6. 2) may be written as = r + B'h G 'J:. R Here r contains terms with and P The term 'J:. - G R can be decomposed into G 'J:. R = G 'J:. R + G 'J:. R T T N N where R T . is the X column of tangential components of R and X column of normal components of R transpose matrices of a* and G 'J:. N T X T X ... X T and from lR T T h Rr))

(6. 6)
Observe that the impulse hR emerges as the natural variable. We shall develop in section 7 below how this allows for treating shocks as well as smooth motions. Of course the method applies to more general situations than those related to systems of points. The projectors have to be compu ted at each step. Nevertheless in the case of a plane obstacle the compu tation may be avoided in checking whether the set of contact points at some step is the same set as the one met at the previous step. If a motion with persistent contact is expected then the projectors may be computed once for all. The algorithms may be constructed in taking advantage of simplifications allowed by special forms of the matrix M and of the projectors. For instance M or G N B'G�, G ,J 3'G�, G yB'G� nal matrices (see for instance [12]).

may be diago-

The system of figure 3 is composed of ten material points. The points are connected together by viscoelastic dipoles (involving spring and dash

•pot). Each of them is submitted to a constant vertical force (gravity)

andis pushed against the obstacle by a viscoelastic dipole. The angle of rake is such that when a point moves to the right, the pressure against the obstacle increases. For this system one may observe a motion with persistent contact. The velocities and tangential components of the reaction as functions of time are plotted for points l, 3, S on figure 3 . A stick-slip phenomenon is in evidence for points 3 and S (see arrows f)

The velocity settles at the value E of the obstacle velocity while a drop of the tangential cornponent of the reaction occurs. .i.J.. 11 .i.J. 1L

� �Pc=JPc=JPc=J� 2 10

-Figure 3 -

2 10 E . 1 1 1 ql 2 q2 ••••. q4 2 q l q2 q4 \ \ \ \ 1 ', ', '• ', \ •", '• ', '• \ ', -)
.,

--Figure 4 -

SHOCK CALCULATION

All what precedes concerns motions smooth enough for the second derivative q to exist. Such motions f (t,q ) � 0

CL CL either as a strict inequality or as an equality. In particular, some episode with persistent equality may be followed by contact break without any jump in velocity.

On the contrary, if an episode with f < 0 ends at some instant CL t s such that f(t ,q (t )) = 0 , the continuity of at t s cannot

S CL S
be expected in general. We shall assume the existence of the right-limit + u CL and of the left-limit u = q (t-)

CL CL S
This is a shock, the physical circumstances of which are inevitably intricate. Sorne insight can only be gained from relaxing, at least quali tatively, the idealization previously adopted. A certain amount of defor mation will be accepted for the bodies in presence, so that the sudden velocity change develops through a little longer duration than the single instant t s By equalling the integrals of both members of the equations of Dynamics over this duration, one classically obtains an expression for the momentum change in terms of the percussion of constraint.

In the no-friction case, it is usually admitted that this percussion is normal to the colliding surfaces (this might be discussed see e.g.

[11] , sect. 9.7.c , Remarque). Even so the above balance of momentum does not provide enough information + u a.

The complementary condition is commonly derived from assuming that the shock is elastic or, alternatively, inelastic.

Let us restrict ourselves here to the same simple case as in previous sections the system is supposed scleronomic and the function f does a.

not depend on time. Then the definition of an elastic shock simply reduces to asserting that the kinetic energy of the system is preserved, while inelastic shock is characterized by condition + <u , grad f (q ) > a.

a. a.

•

One proves that the latter essentially involves energy dissipation (cf.

[1] , where the concept of an inelastic shock is also extended to the case of constraints expressed by several inequalities).

In practice, physical arguments concerning the energy balance of a shock are generally uncertain. Even if the slighlty deformable material of which the system is built may safely be treated as elastic, one cannot a priori assert energy preservation . In fact the slight deformatiot re sulting from the impact is not usually restricted to the small zone of contact; it propagates to the whole of colliding parts and possibly also to other connected bodies, whether internal or external to the system. Therefore, after bounce, various parts may remain in a state of vibration which, at the macroscopic scale, amounts to energy loss.

In contrast, it seems easier to characterize some physical situations where the inelasticity of shocks is a pretty safe guess. A familiar example is that of a wood or metal ball landing on a floor of beaten earth.

Frictional effects between the colliding bodies still complicate the situation. A theory of frictional shock was proposed by G. Darboux and further improved by J. Pérès (see e.g. p. 316-328) . It consists in studying the evolution of the velocity as a function of some "micro scopie time" ordering the very short duration of the shock, while position is treated as constant. This is interesting as an early example of a multiple scaling method but the physical relevance seems difficult to assess ; in particular the estimation of shock end proves critical.

We shall adopt a cruder approach of frictional shock by assuming that the percussion of contact relates to the final velocity through the same generalized Coulomb law as does the force of contact in smooth motions.

The mathematical background of "nonsmooth Dynamics" has been formu lated in [!] under the assumption that velocity u is a function of t with locally bounded variation, non differentiable in general. Then the role of the acceleration is played by the differential measure du , a vector measure on the considered time interval. Accordingly, forces are replaced by vector measures, comprising possible percussions ; in smooth cases these measures admit, relative to the Lebesgue measure dt , some density functions whose values are forces in the proper sense.

In that context, the differential inclusion (5. Thus the decision of relating frictional percussion to the right limit of velocity appears to parallel exactly the choice previously made of an implicit discretization scheme in treating the differential inclu sion, as well as the assumption of shock inelasticity.

In our opinion, this approach has the following methodological v1r tues. It standardizes a type of irreversible process, resulting in evolu tion problems with strong theoretical and numerical consistency. Sorne practical instances certainly exist where the assumptions made are in satisfactory agreement with reality. But, above all, this ideal situation provides a safe basis to which further empirical corrections could be added in rational order. This looks similar, for instance, to the concept of standard dissipative process in thermoplasticity [START_REF] Halphen | Sur les matériaux standards généra lisés[END_REF] or to the use of Lax's entropy conditions when studying shockwaves in hyperbolic conservation laws ( see e.g. U4]) -Figure 8

  least local) coordinates of the point q et Unlike Latin indices, the repetition of Greek indices will not be understood as implying summation.

5 )

 5 The writing on the left-hand sicle is that of product matrix, i.e. the covector with components a. of constraint arising from the possible unilateral contact experienced by the function expresses all other forces, a priori depending on the position and velocity of the whole sys tem S and possibly also on t as independent variable, that subsystem S undergoes from any part. It is only through such terms as Q a.

a0

  common tangent plane at the point M to both boundar ies of B 0 and B I This does not preclud e bodies present ing edges or vertice s ; one of them may even reduce to a single materia l point, provide d the boundar y of the other one is a regular surface . Anyway , let us denote by M0 and the respective particles of B and which happen to lie at point at the instant in view. The contact actions exerted by B 0 are supposed represented by a single force R applied to particle M I ; hence any torque expressing "resistance to rolling" is neglec ted. Kinematically, for every imagined differentiable motion of the system, passing at time by the considered position, the difference of the velocity vectors of and is referred to as the slip velocity of B I relative to B 0 Let us denote by E 3 the linear space of the vectors of physical space. Let NE E 3 be the unit vector normal to the common tangent plane at point M directed toward B I • This Section is devoted to motions smooth enough for Lagrange's equa tions to rnake sense. The existence of accelerations is then postulated; therefore velocities are continuous functions of time. Elementary kinema tics entails in that case that the slip velocity satisfies N.U = û (3. 2) i.e. U belongs to the two-dimensional subspace T of E 3 , tangent to both contacting bodies. The admitted law of dry friction stipulates a relation between the elements U ET and REE 3 Here are two equivalent formulations, generalizing to possibly anisotropie contacting surfaces the traditional law of Coulomb.

5 ) 3 . 6 )

 536 and denote by iµ the indicator function of the subset V(R N ) of T • i.e. This is a lower semi-continuous convex function defined in T Recall that for every .XE T the subdifferential aiµ(X) , relative to the Euclidean scalar product in T, equals the set of the elements Y of T which are outward normal to the closed convex set V(R N ) at point X in the conventional sense of Convex Analysis. This subdifferential is empty if X� V(R N ) ; otherwise it contains at least the zero of T and reduces to this single element in case X is interior to V(R N ) Then the expected relation writes down as (For every fixed R N this is a resistance law of the rate-independent sort, also familiar in Plasticity theory; some equivalent forms are classically given ta such laws, in terms of a "principle of maximal dissi pation" or also in terms of a "dissipation function" (actually the support fonction of the set -V(R N

3 ) 3 (

 33 permit ta translate the above formulation into a statement avoidinB the decomposition (3.Let us denote by C the closed convex cane in E with vertex at the origin) generated by the set V 1 + N. In the special case of isotropie Coulomb friction, this set is the closed region delimited in E 3 by what is traditionally called the cane of friction : a conical surface of revo lution about vector N.

  of Convex Analysis in E 3 , the set aw c (R) constitutes the outward normal cane ta C at the point R. Since this set is ernpty when R f/. C one observes in particular that the no-adhesion assurnption ( 3 . 4) 1s involved in such a writing. What precedes has been presented for a part B 1 of S in contact a with some external body. The formulation is quite similar in the case where equality f = 0 a expresses the contact, occuring at some point M of physical space, between two bodies B 1 and B2 , both parts of subsystem s a Again the definition of a common tangent plane at M to the respective boundaries is assumed ; M l and M 2 denote the respec- tive particles of B I and B 2 which happen to lie at M ' at the con- sidered instant, and u the slip velocity of B I relative to B 2 ' i.e. u (3. 8)

( 4 . 1 )

 41 as a function of T the said position of P becomes a T-dependent point in F that we shall denote by F T ---+-p (,) • Analysing the geometry of the system and the way parameters derivative 1 2

M 1

 1 identified above ; one obtains an expression of the form Here OK ÔT' with components i �(O) ÔT tangent space E Cl. to the manifold n ù a certain linear mapping of E Cl. into (4. 2) constitutes an element of the and G denotes at point q (t) Cl.

  components of some element r E* cotangent to the manifold Cl. ù the bilinear pairing between E a equivalently Since the choice of functions i T ---+-K (,) allows one ta identify with an arbitrary element of E a this is equivalent to (4.4) where G* : E 3 + ( the space E 3 E* a is the transpose of the linear mapping G : E a of the vectors of the physical space is paired with itself through the Euclidean scalar product). The same linear mapping G is involved in the calculation of the velocity of particle M 1 in the course of some motion of S Such a a motion is defined by giving the position q E O , i.e. the local coor-

1 OT( 4 . 6 )( 4 . 7 )

 14647 ---;fr = r. 6T i for every virtual displacement. Here a linear mapping G be found, such that o• M F = G(�) OT OT This finally yields the same writing as in (4.4) . Besides, supposing again the system scleronomic, one finds as expression of the slip velociu = which is the same as (4 .5) , with U = 0 . e It is now easy to list a complete set of equations governing any possible smooth motion of the system. This consists in writing, for every subsystem S , the Lagrange equations (2.7) , the expression (4.5)' et or (4.6) of the slip velocity U the expression (4.4) of r from the reaction R and, finally, the law of frictional contact as stated in Sect. 3 . Let us symbolize the latter as RE cou (t,q ,U) et et Here the right-hand member represents the subset of E 3 (resp. E 2 in the case of two-dimensional contact) consisting of the values of the contact reaction R which agree with the formulated dry friction law if, the position of s et being at time t ' the slip velocity happens to assume the value U. It is an essential feature of this law that it cannot be "solved" in order to express R as a single-valued function of U (nor U as a single-valued function of R) . Instead, the symbol cou et represents a multivalued mapping or multifunction that we propose to call the Coulomb multifunction associated with the possible contact affecting subsystem s et The description of friction, in sect. 3 , was developed under such smoothness assumptions that U essentially belongs to the linear subspace T of E 3 , tangent to contacting bodies. Let us put explicitely CONVENTION 1 . In the case of contact, i.e. f (.t, q ) = 0 , the value Cl Cl of the multifunction cou is defined equal to the empty set whenever Cl U E;l T In other words, the existence of some R verifying (4.7) implies UET .

  (4.10) • This defines the mapping A ' Cl under the usual assumptions ensuring existence and uniqueness for the solution to the differential equations (4.9 ) • Furthermore, the inclusion (4.13 ) may equivalently be written as a variational inequality, expressi�g the "principle of maximal dissipa tion

1 t

 1 dynamics of the system in the absence of contact.Since a is an invertible matrix, this yields as end of the computation step In any case, •we shall denote by u! q!) � 0, computation goes on as follows.a a.nd U=O.e If <u!,g > � 0 we estimate that motion takes place without contact l action and adopt again (5.5) as step end.

6 )aC

 6 We place ourselves in the regular case where the linear rnapping G Ea + E 3 (resp. E 2 in bidimensional cases) is surjective, i.e. every value U = G(qa) in E 3 (resp. E 2 ) of the relative velocity of contacting bodies at point M without regard to contact permanence nor impenetrability, may be obtained by choosing suitably the time-derivative Then the transpose G :t: is injective ; in addition, the follo wing geometrical remark will be of use. If contact holds in the considered position of the system, a value u of q(l is compatible with further preservation of condition f (q (t)) � 0 if <u,g> � 0. Now f � 0 expresses that, in physical Cl Cl Cl space, the subsequent imagined motion does not make the bodies inter penetrate. This proves the implication <u,g> � 0 � G(u).N � 0 (i.e. <u,G :t: (N)> � 0) which, in view of the unilateral version of Lagrange multiplier theorem, due to injectivity of c* . This shows in particular that the element U = G(u) lies in the common tangent space T to contacting bodies if and only u belongs to E* a orthogonal to the element g of T ' the subspace of E How inclusion (5.6) may determine u i + l will only be demonstrated here by the simple example of two-cli.meYll.>ional 6.tu.c.:tion, as formulated at the end of sect. 3 (another very clear instance, developed in [7] , is that of a single mass-point, unilaterally confined by some moving bounda-= c*(C) is generated by a couple of elements A 1 , A 2 equals the flat cone in E a whose edges are the half-lines n 1 and n 2 respectively generated by a = 2 Due to (5. 7) C contains -g put In view of the form (3.9) , (3.10) , (3.11) that relation and R E cou(U) takes in two-dimensional cases, the relation x s E G (cou(G(u))), holds between some elements s E E* and u ET if and only if clarity let us imagine it to be 3 through the choice of adapted bases in E a

3

 3 is strictly positive. Therefore the (non homogeneous) cone with the linear subspace T u! + C has nonempty intersection l.

Figure this intersection is

  Figure

3 'REMARK 1 .REMARK 2 .

 312 schemes are derived from such formulations, they can only render equili-

  given as a function of time. R CL is written only if CL is the index of a point in contact with the obstacle. In that case nothing but the identity mapping from E 2 onto itself. The above equa tions may also be written as Mq + Vq + Kq = P(t) + G*R_ (6. 1) whC!rC! the inertia matrix M is diagonal ; V and K are symmetric non n•:;�:1 Li VC! matrices. Let us denote by n the number of points and X the number of contact points, assuming that for some interval of time, these points remain in contact. Contact points are indexed by SE {l, ... , x} and they have a corresponding index in the a list, that we denote a 6 The 2 X vector R is the column of the X two-dimensional vectors R a c* 1s thus a mapping from lR 2 X

3 )

 3 us express that the velocity (q a ) i+ 1 lies in the tangent space Since in most cases B' is an invertible matrix (in fact B is close to M) , and function of R T . the above equation yields an expres-The slip velocity expresses as (6.4) By [E] is denoted the x column wi th components E the velocity of the moving wall. The matrix Inequali ties G B'G* T T is symmetric positive definite. (6.5) are tested. The inclusion may be equivalently transformed into the following fixed point condition

Figure 4 Figure 2

 42 Figure4shows a similar system except for the angle of rake of the di poles pushing against the obstacle. When a point moves to the right, the pressure against the obstacle decreases. The x 1 and x 2 coordinates of the points l, 2, 3, 4, S as functions of time are plotted on figu re 4 . One sees (arrow -) that the point 3 leaves smoothly the obstacle. Then the point cornes back into contact (arrow �) with a shock after which it slides again. By plotting velocity one could see that stick -slip is also present in this case.

2 )

 2 receives a precise generalized meaning. It turns out that the discretization algorithrns of sections 5 and 6 yield approximate solutions of the corresponding problems through exactly the same computation procedures as in smooth cases. If sorne discretization interval shock, it only cornes that the difference (t.,t. 1 ) i i+ happens to include has no more the same order of magnitude as integral of the measure h = t. l -t. 1+ l this difference du over the considered time interval.

Figures 6 , 7 , 8 , 5 AFigure 5 y -b = 0 Here are the driving forces : 1 ° 2 °

 67855012 Figures 6, 7, 8, have been produced by a computer the algorithm of sect. 5 A rigid body B performs a motion parallel to the (x,y)-plane. The plotter draws the profile of B in its initial position the significant part of it consists of an arc of circle with center A and radius b . Using as position parameters the coordinates X, Y of the center of mass G and some rotation angle 8, one gives to the kinetic energy of B the form (2.3) . In this computation b = m = I = 1 ; AG= 1,732.The body is confined in the region y� 0 its possible contact with the fixed wall y= 0 involves dry friction with coefficient 0.5 y

Figure 6

 6 Figure 6 is obtained for C r 700 discretization step length 0.002 ; position plotted every 4th step. Then B does not get loose from the wall. One observes angular oscillations ; also oscillations with longer period affect the space-lag from the pull-back center, as measured for instance by the variable i = x(C) -x(G) The program might be used to draw additionally the two-dimensional trajectories con cerning any couple among the 6 variables i , Y , 8 , i , Y , 8

Figure 7

 7 Figure 7 corresponds to C = -200 ; step length r 0.001 position plotted every 8th step. The motion, in variables i, Y, 8 1snow periodic. Each period consists of a contact-free episode, ending with a shock of B against the wall. The program is also able to display the slip velocity : its value following the shock is found zero ; the body gets loose immediately from the wall in a smooth way.

Figure 8

 8 Figure 8 corresponds to C r -100 step-length 0.0005 position plotted every 15th step. The motior: is again periodic but with some more complicated pattern. Every period exhibits two different episo des of contact-free motion. An episode of the first sort ends with a shock, followed by � nonzero time-interval with persistent contact and positive slip velocity. Afterwards the body B gets loose from the wall with zero normal velocity and performs the second sort of contact-freeepisode. This ends with a second shock, immediately followed by an episode of the first sort. The slip velocity resulting from the second shock is positive, but the normal velocity is zero, in agreement with our inelas ticity assumption.