Pascal Auscher
email: pascal.auscher@math.u-psud.fr

Jos É Mar

Í A Martell

José María Martell

WEIGHTED NORM INEQUALITIES FOR FRACTIONAL OPERATORS

Keywords: Mathematics Subject Classification. 42B25, 35J15 Muckenhoupt weights, elliptic operators in divergence form, fractional operators, commutators with bounded mean oscillation functions, good-λ inequalities

We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-λ method that does not use any size or smoothness estimates for the kernels.

Introduction

In [MW] Muckenhoupt and Wheeden resolve the one-weight problem for the classical fractional integrals I α = (-∆) -α/2 and fractional maximal operators M α in R n defined by

M α f (x) = sup r(B) α - B |f (y)| dy,
where the supremum is taken over all balls B of R n that contain x.

Theorem 1.1 ( [MW]). Let w be a weight. Let 0 < α < n, 1 ≤ p < n α and q = n p n-α p , that is, 1/p -1/q = α/n. If p > 1, M α is bounded from L p (w p ) to L q (w q ) if and only if w ∈ A p,q . If p = 1, M α is bounded from L 1 (w) to L q,∞ (w q ) if and only if w ∈ A 1,q . Furthermore, the same estimates for the Riesz potential I α are characterized by the classes A p,q .

The class A p,q , whose definition is recalled below, can be equivalently written as A 1+1/p ′ ∩ RH q where A p and RH q are the standard Muckenhoupt and reverse Hölder classes. These two operators have intimate relations and the estimates for I α follow from the ones for M α . First there is a pointwise domination M α f I α (|f |) and second, although the pointwise converse does not hold, by means of a good-λ inequality, one has for all 0 < p < ∞ and w ∈ A ∞ :

R n |I α f | p w dx R n (M α f ) p w dx (1.1)
and also its corresponding L 1,∞ -L 1,∞ version. Different authors have studied the commutators of the fractional integrals with BMO functions. Unweighted estimates were considered in [Cha] and the weighted estimates were established in [ST] by means of extrapolation. Another proof based on the sharp maximal function was given in [CF].

Here, we consider operators with the same scaling properties as fractional integrals but which may not be representable by kernels with good estimates and that we call fractional operators. We wish to generalize the part of the theorem concerning I α but a direct comparison to M α will not work because we will have a limited range of α. Hence, we are looking for some other technique which could also provide another proof of the sufficiency part of Muckenhoupt-Wheeden theorem for I α .

Our main example is the fractional power of an elliptic operator L on R n , given formally by

L -α/2 = 1 Γ(α/2) ∞ 0 t α/2 e -t L dt t ,
with α > 0 and Lf = -div(A ∇f ), where A is an elliptic n×n matrix of complex and L ∞ -valued coefficients (see Section 3.1 for the precise definition). The operator -L generates a C 0 -semigroup {e -t L } t>0 of contractions on L 2 (dx) = L 2 (R n , dx). There exist p -= p -(L) and p + = p + (L), 1 ≤ p -< 2 < p + ≤ ∞ such that the semigroup {e -t L } t>0 is uniformly bounded on L p (dx) for every p -< p < p + (see Proposition 3.1 below). The unweighted estimate states as follows.

Theorem 1.2 ( [Aus]). Let p -< p < q < p + and α/n = 1/p -1/q. Then L -α/2 is bounded from L p (dx) to L q (dx).

Let us observe that the range of α's in Theorem 1.2 is 0 < α < n/p --n/p + . By (e) in Proposition 3.1 below, for n = 1 or n = 2 or when L has real coefficients, we have pointwise Gaussian domination of the semigroup, hence 0 < α < n. In general, by (f ) in Proposition 3.1 the range of α's always contains the interval (0, 2].

Our first main result in this paper gives sufficient conditions for the weighted norm inequalities of L -α/2 . Theorem 1.3. Let p -< p < q < p + and α/n = 1/p -1/q. Then L -α/2 is bounded from L p (w p ) to L q (w q ) for w ∈ A 1+ 1 p --1 p ∩ RH q ( p + q ) ′ . Notice that if p -= 1 and p + = ∞ (for instance, when L = -∆ or under Gaussian domination), then the condition on w becomes w ∈ A 1+1/p ′ ∩ RH q , that is, w ∈ A p,q (see Proposition 2.1), and our result agrees with that by Muckenhoupt and Wheeden (see Theorem 1.1).

We also obtain estimates for commutators with bounded mean oscillation functions:

Let b ∈ BMO, that is, b BMO = sup B -B |b(x) -b B | dx < ∞,
B. Given f ∈ L ∞ c (dx), set (L -α/2 ) 0 b f = L -α/2 f , and for k ≥ 1, the k-th order commutator (L -α/2 ) k b f (x) = L -α/2 (b(x) -b) k f (x)
. These commutators can be also defined by recurrence:

(L -α/2 ) k b = [b, (L -α/2 ) k-1 b ] where [b, T ]f (x) = b(x) T f (x) -T (b f )(x).
We obtain the following weighted estimates:

Theorem 1.4. Let p -< p < q < p + and α/n = 1/p -1/q. Given k ∈ N, b ∈ BMO and w ∈ A 1+ 1 p - -1 p ∩ RH q ( p + q ) ′ we have (L -α/2 ) k b f L q (w q ) ≤ C b k BMO f L p (w p
) . In the particular case k = 1 and under Gaussian kernel bounds (as in (e) of Proposition 3.1 below) the unweighted estimates were studied in [DY] using the sharp maximal function introduced in [Mar]. A simpler proof, that also yields the weighted estimates, was obtained in [CMP]: a discretization method inspired by [START_REF] Pérez | Sharp L p -weighted Sobolev inequalities[END_REF] allows the authors to extend (1.1) to (L -α/2 ) k b f which is controlled by M L log L,α f (see the definition below) and then use the weighted estimates for the latter which are studied in [CF].

Theorems 1.3 and 1.4 will be proved in Section 3. They depend on a general statement (Theorem 2.2), interesting on its own, based itself upon a good-λ method in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF] developed for operators with the same scaling properties as singular integral operators. This was used in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators[END_REF] for the same class of elliptic operators and also for the Riesz transforms on Riemannian manifolds in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights[END_REF], and we shall see that the very same tools apply as well for fractional operators.

In Section 4 we present a variant of Theorem 2.2 extending earlier results from [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF] and [She] to the context of fractional operators.

While the good-λ method in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF] is valid in all spaces of homogeneous type, the application to fractional operators can be adapted only to those spaces with polynomial growth from below. We comment on this in Section 5.

Weighted estimates for general operators

We introduce some notation and recall known facts on weights. We work in R n . Given a ball B, we write

- B h dx = 1 |B| B h(x) dx.
2.1. Muckenhoupt Weights. Let w be a weight (that is, a non negative locally integrable function) on R n . We say that w ∈ A p , 1 < p < ∞, if there exists a constant C such that for every ball

B ⊂ R n , - B w dx - B w 1-p ′ dx p-1 ≤ C. For p = 1, we say that w ∈ A 1 if there is a constant C such that for every ball B ⊂ R n , - B w dx ≤ C w(y), for a.e. y ∈ B. Finally, A ∞ = ∪ p≥1 A p .
The reverse Hölder classes are defined in the following way: w ∈ RH q , 1 < q < ∞, if there is a constant C such that for any ball B,

- B w q dx 1 q ≤ C - B w dx.
The endpoint q = ∞ is given by the condition w ∈ RH ∞ whenever there is a constant C such that for any ball B,

w(y) ≤ C - B w dx,
for a.e. y ∈ B.

We introduce the classes A p,q that characterize the weighted estimates for the fractional operators (see Theorem 1.1). Given 1 ≤ p ≤ q < ∞ we say that w ∈ A p,q if there exists a constant C such that every ball B ⊂ R n ,

- B w q dx 1 q - B w -p ′ dx 1 p ′ ≤ C, when 1 < p < ∞, and - B w q dx 1 q ≤ C w(x),
for a.e. x ∈ B, when p = 1. We summarize some properties about weights (see [GR], [Gra] and [JN]).

Proposition 2.1.

(i) A 1 ⊂ A p ⊂ A q for 1 ≤ p ≤ q < ∞. (ii) RH ∞ ⊂ RH q ⊂ RH p for 1 < p ≤ q ≤ ∞.
(iii) If w ∈ A p , 1 < p < ∞, then there exists 1 < q < p such that w ∈ A q .

(iv) If w ∈ RH q , 1 < q < ∞, then there exists q < p < ∞ such that w ∈ RH p .

(v) A ∞ = 1≤p<∞ A p = 1<q≤∞ RH q (vi) If 1 < p < ∞, w ∈ A p if and only if w 1-p ′ ∈ A p ′ .
(vii) If 1 ≤ p ≤ ∞ and 1 < q < ∞, then w ∈ A p ∩RH q if and only if w q ∈ A q (p-1)+1 .

(viii) If 1 ≤ p ≤ q < ∞, then w ∈ A p,q if and only if w q ∈ A 1+q/p ′ if and only if w ∈ A 1+1/p ′ ∩ RH q .

(ix) If 1 ≤ p < q < ∞ and α/n = 1/p -1/q then w ∈ A p,q if and only if w q ∈ A q/1 * α where 1 * α = n/(n -α). 2.2. The general statement. Our main statement is based on unweighted estimates relating the fractional operators and their commutators with the corresponding fractional maximal functions.

We introduce some notation in order to state our general result in a way that is valid also for sublinear operators. Given a sublinear operator T and b ∈ BMO, for any k ∈ N we define the k-th order commutator as

T k b f (x) = T (b(x) -b) k f (x), f ∈ L ∞ c (dx), x ∈ R n . Note that T 0 b = T . We claim that if T is bounded from L p 0 (dx) to L s 0 (dx) for some 1 ≤ p 0 ≤ s 0 ≤ ∞ then T k
b f is well defined in L q loc for any 0 < q < s 0 and for any f ∈ L ∞ c (dx): take a cube Q containing the support of f and observe that by sublinearity, for a.e. x ∈ R n ,

|T k b f (x)| ≤ k m=0 C m,k |b(x) -b Q | k-m T (b -b Q ) m f (x) .
John-Nirenberg's inequality implies

Q |b(y) -b Q | m p 0 |f (y)| p 0 dy ≤ C f L ∞ b m p 0 BMO |Q| < +∞.
Hence, T (b-b Q ) m f ∈ L s 0 (dx) and the claim follows by using again John-Nirenberg's inequality.

Theorem 2.2. Let 0 < α < n, 1 ≤ p 0 < s 0 < q 0 ≤ ∞ such that 1/p 0 -1/s 0 = α/n. Suppose that T is a sublinear operator bounded from L p 0 (dx) to L s 0 (dx) and that {A r } r>0 is a family of operators acting from L ∞ c (dx) into L p 0 (dx). Assume that

- B |T (I -A r(B) )f | s 0 dx 1 s 0 ≤ ∞ j=1 α j r(2 j+1 B) α - 2 j+1 B |f | p 0 dx 1 p 0 , (2.1)
and

- B |T A r(B) f | q 0 dx 1 q 0 ≤ ∞ j=1 α j - 2 j+1 B |T f | s 0 dx 1 s 0 , (2.2)
for all f ∈ L ∞ c and all balls B, where r(B) denotes the radius of B. Let p 0 < p < q < q 0 be such that 1/p -1/q = α/n and w

∈ A 1+ 1 p 0 -1 p ∩ RH q ( q 0 q ) ′ . (a) If j≥1 α j < ∞ then T is bounded from L p (w p ) to L q (w q ). (b) Given k ∈ N and b ∈ BMO, if j≥1 j k α j < ∞ then for every f ∈ L ∞ c (dx) we have T k b f L q (w q ) ≤ C b k BMO f L p (w p ) .
(2.3)

The case q 0 = ∞ is understood in the sense that the L q 0 -average in (2.2) is indeed an essential supremum. Thus, the condition for w turns out to be w

∈ A 1+ 1 p 0 -1 p ∩ RH q for p > p 0 . Similarly, if (2.
2) is satisfied for all q 0 < ∞ then the conclusions hold for all p 0 < p < ∞ and w

∈ A 1+ 1 p 0 -1 p ∩ RH q .
Remark 2.3. In case (a) the hypotheses can be slightly relaxed. Namely, instead of (2.1) and (2.2), it suffices that

- B |T (I -A r(B) )f | s 0 dx 1 s 0 ≤ C M α p 0 |f | p 0 dx (x) 1 p 0 , ∀ x ∈ B, (2.4) - B |T A r(B) f | q 0 1 q 0 ≤ C M |T f | s 0 (x) 1 s 0 , ∀ x ∈ B. (2.5)
It is clear that these estimates follow from (2.1) and (2.2) provided j α j < ∞. We prove (a) below (which corresponds to (b) with k = 0) by using these weaker conditions. The proof also shows that the right hand side of (2.5) can be weakened to

C M |T f | s 0 (x) 1 s 0 + C M α p 0 |f | p 0 (x) 1 p 0 where x ∈ B is also arbitrary. Remark 2.4. Equivalent ways to write the condition w ∈ A 1+ 1 p 0 -1 p ∩ RH q ( q 0 q ) ′ are w q ∈ A 1+ q/p 0 (p/p 0 ) ′
∩RH (q 0 /q) ′ or w q ∈ A q/(p 0 ) * α ∩RH (q 0 /q) ′ where (p 0 ) * α = n p 0 /(n-α p 0 ), or w p 0 ∈ A p/p 0 ,q/p 0 and w q ∈ RH (q 0 /q) ′ (see (viii) and (ix) in Proposition 2.1). Note that when p 0 = 1 and q 0 = ∞, then this reduces to w ∈ A 1+1/p ′ ∩ RH q which is equivalent to w ∈ A p,q by (viii) in Proposition 2.1.

Remark 2.5. In the limiting case α = 0, this result corresponds to a special case of [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF]Theorems 3.7 and 3.16]. In such a case, we have p = q and the weight w q turns out to be in A p/p 0 ∩ RH (q 0 /p) ′ . This condition arises naturally when proving weighted norm inequalities for operators with the same scaling properties as singular integral operators -such as those appearing in the functional calculus associated with L, see [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators[END_REF]-whose range of unweighted L p boundedness is (p 0 , q 0 ). Also, these classes of weights admit a variant of the Rubio de Francia extrapolation theorem that is valid for the limited range of exponents (p 0 , q 0 ), see [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF] .

2.3. The technical result. The proof of Theorem 2.2 is a consequence of the following result which appears in a more general form in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF] and is based on a two-parameter good-λ inequality.

Theorem 2.6 ([AM1]). Fix 1 < r ≤ ∞, a ≥ 1 and w ∈ RH s ′ , 1 ≤ s < ∞. Let 1 < p < r s . Assume that F ∈ L 1 (dx), G, H
1 and H 2 are non-negative measurable functions on R n such that for any cube Q there exist non-negative functions G Q and

H Q with F (x) ≤ G Q (x) + H Q (x) for a.e. x ∈ Q and - Q H r Q dx 1 r ≤ a MF (x) + MH 1 (x) + H 2 (x) , ∀ x, x ∈ Q; (2.6)
and

- Q G Q dx ≤ G(x), ∀ x ∈ Q. (2.7)
Then, there exists a constant C = C(p, r, n, a, w, s) such that

MF L p (w) ≤ C G L p (w) + MH 1 L p (w) + H 2 L p (w) .
(2.8)

Note that the assumption F ∈ L 1 (dx) is not used quantitatively. The case r = ∞ is the standard one: the L r -average appearing in the hypothesis is understood as an essential supremum and the L p (w) estimate holds for any 1 < p < ∞, no matter the value of s, that is, for any w ∈ A ∞ .

2.4. Proof of Theorem 2.2, Part (a). As mentioned in Remark 2.3, we can relax the hypotheses by assuming (2.4) and (2.5), which we do. We consider the case q 0 < ∞, the other one is left to the reader.

Let f ∈ L ∞ c (dx), so F = |T f | s 0 ∈ L 1 (dx)
. We fix a cube Q (we switch to cubes for the proof). As T is sublinear, we have

F ≤ G Q + H Q ≡ 2 s 0 -1 |T (I -A r(Q) )f | s 0 + 2 s 0 -1 |T A r(Q) f | s 0 .
Then (2.4) and (2.5) yield respectively (2.7) and (2.6) with r = q 0 /s 0 ,

H 1 = H 2 ≡ 0, a = 2 s 0 -1 C s 0 and G = 2 s 0 -1 C s 0 M α p 0 |f | p 0 s 0 /p 0 .
By Remark 2.4, w q ∈ RH (q 0 /q) ′ and one can pick 1 < s < q 0 /q so that w q ∈ RH s ′ . Thus, Theorem 2.6 with q/s 0 in place of p (notice that 1 < q/s 0 < r/s) yields

T f s 0 L q (w q ) ≤ MF L q s 0 (w q ) ≤ C G L q s 0 (w q ) = C M α p 0 |f | p 0 s 0 p 0 L q p 0 ((w p 0 ) q/p 0 ) ≤ C |f | p 0 s 0 p 0 L p p 0 ((w p 0 ) p/p 0 ) = C f s 0 L p (w p ) .
In the last estimate we have used that M α p 0 maps L p p 0 ((w p 0 ) p/p 0 )) into L q p 0 ((w p 0 ) q/p 0 )) by Theorem 1.1 from w p 0 ∈ A p/p 0 ,q/p 0 (see Remark 2.4) and the easily checked conditions 0 < α p 0 < n, 1 < p/p 0 < n/(α p 0 ) and 1/(p/p 0 ) -1/(q/p 0 ) = α p 0 /n. 2.5. Proof of Theorem 2.2, Part (b). Before starting the proof, let us introduce some notation (see [BS] for more details). Let φ be a Young function:

φ : [0, ∞) -→ [0, ∞) is continuous, convex, increasing and satisfies φ(0+) = 0, φ(∞) = ∞. Given a cube Q we define the localized Luxemburg norm f φ,Q = inf λ > 0 : - Q φ |f | λ ≤ 1 ,
and then the maximal operator

M φ f (x) = sup Q∋x f φ,Q .
In the definition of • φ,Q , if the probability measure dx/|Q| is replaced by dx and Q by R n , then one has the Luxemburg norm • φ which allows one to define the Orlicz space L φ . Some specific examples needed here are φ(t) ≈ e t r for t ≥ 1 which gives the classical space expL r and φ(t) = t (1 + log + t) α with α > 0 that gives the space L (log L) α . In the particular case α

= k -1 with k ≥ 1, it is well known that M L(log L) k-1 f ≈ M k f where M k is the k-iteration of M.
We also need fractional maximal operators associated with an Orlicz space: given 0 < α < n we define

M φ,α f (x) = sup Q∋x ℓ(Q) α f φ,Q .
John-Nirenberg's inequality implies that for any function b ∈ BMO and any cube

Q we have b -b Q expL,Q
b BMO . This yields the following estimates: First, for each cube Q and x ∈ Q

- Q |b -b Q | k s 0 |f | s 0 ≤ b -b Q k s 0 expL,Q |f | s 0 L (log L) k s 0 ,Q b k s 0 BMO M L (log L) k s 0 |f | s 0 )(x) b k s 0 BMO M [k s 0 ]+2 |f | s 0 )(x), (2.9) where [s] is the integer part of s (if k s 0 ∈ N, one can take M [k s 0 ]+1 ). Second, for each j ≥ 1 and each Q, b -b 2 Q expL,2 j Q ≤ b -b 2 j Q expL,2 j Q + |b 2 j Q -b 2 Q | b BMO + j-1 l=1 |b 2 l+1 Q -b 2 l Q | b BMO + j-1 l=1 - 2 l+1 Q |b -b 2 l+1 Q | j b BMO .
(2.10)

The following auxiliary result allows us to assume further that b ∈ L ∞ (dx). The proof is postponed until the end of this section.

Lemma 2.7. Let 1 ≤ p 0 < s 0 < ∞, p 0 < p < q < ∞, k ∈ N and w q ∈ A ∞ . Let T be a sublinear operator bounded from L p 0 (dx) to L s 0 (dx). (i) If b ∈ BMO ∩ L ∞ (dx) and f ∈ L ∞ c (dx), then T k b f ∈ L s 0 (dx). (ii) Assume that for any b ∈ BMO ∩ L ∞ (dx) and for any f ∈ L ∞ c (dx) we have that T k b f L q (w q ) ≤ C 0 b k BMO f L p (w p ) , (2.11)
where C 0 does not depend on b and f . Then for all b ∈ BMO, (2.11) holds with constant 2 k C 0 instead of C 0 .

Part (ii) in this result ensures that it suffices to consider the case b ∈ L ∞ (dx) (provided the constants obtained do not depend on b). So from now on we assume that b ∈ L ∞ (dx) and obtain (2.11) with C 0 independent of b and f . Note that by homogeneity we can also assume that b BMO = 1.

We proceed by induction. Note that the case k = 0 corresponds to (a). We write the case k = 1 in full detail and indicate how to pass from k -1 to k as the argument is essentially the same. Let us fix p 0 < p < q 0 and w q ∈ A 1+ q/p 0 (p/p 0 ) ′ ∩ RH (q 0 /q) ′ (see Remark 2.4). We assume that q 0 < ∞, for q 0 = ∞ the main ideas are the same and details are left to the interested reader.

Case k = 1: We use the ideas in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF] (see also [START_REF] Pérez | Endpoint estimates for commutators of singular integral operators[END_REF]). Let f ∈ L ∞ c (dx) and set

F = |T 1 b f | s 0 . Note that F ∈ L 1 (dx) by (i) in Lemma 2.7 (this is the only place in this step where we use that b ∈ L ∞ (dx)). Given a cube Q, we set f Q,b = (b 4 Q -b) f and decompose T 1 b as follows: |T 1 b f (x)| = |T (b(x) -b) f (x)| ≤ |b(x) -b 4 Q | |T f (x)| + |T (b 4 Q -b) f (x)| ≤ |b(x) -b 4 Q | |T f (x)| + |T (I -A r(Q) )f Q,b (x)| + |T A r(Q) f Q,b (x)|.
With the notation of Theorem 2.6, we observe that

F ≤ G Q + H Q where G Q = 4 s 0 -1 G Q,1 + G Q,2 = 4 s 0 -1 |b -b 4 Q | s 0 |T f | s 0 + |T (I -A r(Q) )f Q,b | s 0 and H Q = 2 s 0 -1 |T A r(Q) f Q,b | s 0 .
We first estimate the average of G Q on Q. Fix any x ∈ Q. By (2.9) with k = 1,

- Q G Q,1 = - Q |b -b 4 Q | s 0 |T f | s 0 b s 0 BMO M [s 0 ]+2 |T f | s 0 )(x).
Using (2.1), (2.9) and (2.10),

- Q G Q,2 1 s 0 = - Q |T (I -A r(Q) )f Q,b | s 0 1 s 0 ∞ j=1 α j ℓ(2 j+1 Q) α - 2 j+1 Q |f Q,b | p 0 1 p 0 ≤ ∞ j=1 α j b -b 4 Q expL,2 j+1 Q M L (log L) p 0 ,α p 0 |f | p 0 ) 1 p 0 (x) b BMO M L (log L) p 0 ,α p 0 |f | p 0 )(x) 1 p 0 ∞ j=1 α j j M L (log L) p 0 ,α p 0 |f | p 0 ) 1 p 0 (x), since j α j j < ∞. Hence, for any x ∈ Q - Q G Q ≤ C M [s 0 ]+2 |T f | s 0 )(x) + M L (log L) p 0 ,α p 0 |f | p 0 )(x) s 0 p 0 ≡ G(x).
We next estimate the average of H r Q on Q with r = q 0 /s 0 . Using (2.2) and proceeding as before

- Q H r Q 1 q 0 = 2 (s 0 -1)/s 0 - Q |T A r(Q) f Q,b | q 0 1 q 0 ∞ j=1 α j - 2 j+1 Q |T f Q,b | s 0 1 s 0 ≤ ∞ j=1 α j - 2 j+1 Q |T 1 b f | s 0 1 s 0 + j≥1 α j - 2 j+1 Q |b -b 4 Q | s 0 |T f | s 0 1 s 0 (MF ) 1 s 0 (x) + ∞ j=1 α j b -b 4 Q expL,2 j+1 Q M [s 0 ]+2 |T f | s 0 ) 1 s 0 (x) (MF ) 1 s 0 (x) + M [s 0 ]+2 |T f | s 0 ) 1 s 0 (x) ∞ j=1 α j j (MF ) 1 s 0 (x) + M [s 0 ]+2 |T f | s 0 ) 1 s 0 (x),
for any x, x ∈ Q, where we have used that j α j j < ∞. Thus we have obtained

- Q H r Q 1 r ≤ C MF (x) + M [s 0 ]+2 |T f | s 0 )(x) ≡ C MF (x) + H 2 (x) .
As mentioned before F ∈ L 1 . Since w q ∈ RH (q 0 /q) ′ , we can choose 1 < s < q 0 /q so that w q ∈ RH s ′ . Thus, Theorem 2.6 with q/s 0 in place of p (notice that 1 < q/s 0 < r/s) yields

T 1 b f s 0 L q (w q ) ≤ MF L q s 0 (w q ) G L q s 0 (w q ) + H 2 L q s 0 (w q ) M L (log L) p 0 ,α p 0 |f | p 0 ) s 0 p 0 L q s 0 (w q ) + M [s 0 ]+2 |T f | s 0 L q s 0 (w q ) I + II.
We estimate each term in turn. For I, we claim that M L (log L) p 0 ,α p 0 maps L p p 0 (w p ) into L q p 0 (w q ). This implies that

I = M L (log L) p 0 ,α p 0 |f | p 0 ) s 0 p 0 L q p 0 (w q ) f s 0 L p (w p ) .
Let us show our claim. We observe that 1 + q/p 0 (p/p 0 ) ′ = q s 0 = q ( 1 p 0 -α n ). Then, by (iii) in Proposition 2.1, there exists 1 < s < p/p 0 so that w q ∈ A q ( 1 s p 0 -α n ) . Let us observe that the choice of s guarantees that q ( 1 s p 0 -α n ) > 1. We set α = s p 0 α, p = p/(p 0 s) and q = q/(p 0 s). Let us observe that 0 < α < n, 1 < p < n/ α and 1/p -1/q = α/n. Besides by (ix) of Proposition 2.1 we have that w = w p 0 s ∈ A p,q . Therefore, by Theorem 1.1 it follows that M α maps L p( wp ) into L q( wq ).

Notice that as s > 1 we have that t (1 + log + t) p 0 t s for every t ≥ 1. Thus,

M L (log L) p 0 ,α p 0 g(x) = sup Q∋x ℓ(Q) α p 0 g L (log L) p 0 ,Q sup Q∋x ℓ(Q) α p 0 g L s ,Q = M α p 0 s |g| s )(x) 1 s = M α |g| s )(x) 1 s ,
and therefore we conclude the desired estimate

M L (log L) p 0 ,α p 0 g L q p 0 (w q ) M α |g| s ) 1 s L q p 0 (w q ) = M α |g| s ) 1 s L q ( w q ) |g| s 1 s L p( w p) = g L p p 0 (w p )
.

For II as before we observe that 1 + q/p 0 (p/p 0 ) ′ = q s 0 . Besides, 1/p -1/q = α/n = 1/p 0 -1/s 0 implies that 1/s 0 -1/q = 1/p 0 -1/p > 0 and therefore q/s 0 > 1. Consequently, M (hence, M 2 , M 3 , . . . ) is bounded on L q s 0 (w q ) which gives

II = M [s 0 ]+2 |T f | s 0 L q s 0 (w q ) T f s 0 L q (w q ) f s 0 L p (w p ) ,
where in the last inequality we have used (a) (which is the case k = 0). Collecting the obtained estimates for I and II we conclude as desired

T 1 b f s 0 L q (w q ) f s 0 L p (w p ) .
Case k: We now sketch the induction argument. Assume that we have already proved the cases m = 0, . .

. , k -1. Let f ∈ L ∞ c (dx). Given a cube Q, write f Q,b = (b 4 Q -b) k f and decompose T k b as follows: |T k b f (x)| = |T (b(x) -b) k f (x)| ≤ k-1 m=0 C k,m |b(x) -b 4 Q | k-m |T m b f (x)| + |T (b 4 Q -b) k f (x)| k-1 m=0 |b(x) -b 4 Q | k-m |T m b f (x)| + |T (I -A r(Q) )f Q,b (x)| + |T A r(Q) f Q,b (x)|.
Following the notation of Theorem 2.6, we set

F = |T k b f | s 0 ∈ L 1 (dx) by (i) in Lemma 2.7. Observe that F ≤ G Q + H Q where G Q = 4 s 0 -1 C k-1 m=0 |b -b 4 Q | k-m |T m b f | s 0 + |T (I -A r(Q) )f Q,b | s 0 and H Q = 2 s 0 -1 |T A r(Q) f Q,b | s 0 .
Proceeding as before we obtain for any

x ∈ Q - Q G Q ≤ C k-1 m=0 M [(k-m) s 0 ]+2 |T m b f | s 0 )(x) + M L (log L) k p 0 ,α p 0 |f | p 0 )(x) s 0 p 0 ≡ G(x),
and for r = q 0 /s 0

- Q H r Q 1 r ≤ C MF (x) + k-1 m=0 M [(k-m) s 0 ]+2 |T m b f | s 0 (x) ≡ C MF (x) + H 2 (x) .
Therefore, as F ∈ L 1 , Theorem 2.6 gives us as before

T k b f s 0 L q (w q ) ≤ MF L q s 0 (w q ) G L q s 0 (w q ) + H 2 L q s 0 (w q ) M L (log L) k p 0 ,α p 0 |f | p 0 ) s 0 p 0 L q s 0 (w q ) + k-1 m=0 M [(k-m) s 0 ]+2 |T m b f | s 0 L q s 0 (w q ) f s 0 L p (w p ) + k-1 m=0 T m b f s 0 L q (w q ) f s 0 L p (w p ) ,
where have proceeded as in the estimates of I and II in the case k = 1 and we have used the induction hypothesis on T m b , m = 0, . . . , k -1. Let us point out again that none of the constants involved in the proof depend on b and f . Proof of Lemma 2.7. We use an argument similar to that in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights[END_REF] (see also [START_REF] Pérez | Endpoint estimates for commutators of singular integral operators[END_REF]). Fix f ∈ L ∞ c (dx). Note that (i) follows easily observing that

|T k b f (x)| k m=0 |b(x)| m-k |T (b m f )(x)| ≤ k m=0 b m-k L ∞ |T (b m f )(x)| ∈ L s 0 (dx), since b ∈ L ∞ (dx), f ∈ L ∞ c (dx) imply that b m f ∈ L ∞ c (dx) ⊂ L p 0 (dx) and, by assump- tion, T (b m f ) ∈ L s 0 (dx).
To obtain (ii), we fix b ∈ BMO and f ∈ L ∞ c (dx). Let Q 0 be a cube such that supp f ⊂ Q 0 . We may assume that b

N (x) = b(x) when -N ≤ b(x) ≤ N, b N (x) = N when b(x) > N and b(x) = -N when b(x) < -N. Then, it is immediate to see that |b N (x) -b N (y)| ≤ |b(x) -b(y)| for all x, y. Thus, b N BMO ≤ 2 b BMO . As b N ∈ L ∞ (dx)
we can use (2.11) and

T k b N f L q (w q ) ≤ C 0 b N k BMO f L p (w p ) ≤ C 0 2 k b k BMO f L p (w p ) < ∞.
To conclude, by Fatou's lemma, it suffices to show that

|T b N j f (x)| -→ |T k b f (x)
| for a.e. x ∈ R n and for some subsequence {N j } j such that N j → ∞.

As

|b N | ≤ |b| ∈ L p (Q 0 ) for any 1 ≤ p < ∞, the dominated convergence theorem yields that (b N ) m f -→ b m f in L p 0 (dx) as N → ∞ for all m = 0, . . . , k. Therefore, the fact that T is bounded from L p 0 (dx) to L s 0 (dx) yields T (b N ) m f -b m f -→ 0 in L s 0 (dx). Thus, there exists a subsequence N j → ∞ such that T (b N j ) m f - b m f (x) -→ 0 for a.e.
x ∈ R n and for all m = 1, . . . , k. In this way we obtain

|T k b N j f (x)| -|T k b f (x)| T (b N j (x) -b N j ) k -(b(x) -b) k f (x) k m=0 |b N j (x)| k-m T (b N j ) m f -b m f (x) + b N j (x) k-m -b(x) k-m T (b m f )(x) ,
and as desired we get that

|T b N j f (x)| -→ |T k b f (x)| for a.e. x ∈ R n .
3. Proof of Theorems 1.3 and 1.4

We first introduce our class of elliptic operators and state some needed properties. Then we present an auxiliary lemma which leads us to prove the weighted estimates for L -α/2 and the corresponding commutators.

3.1. The Class of Elliptic Operators. Let A be an n×n matrix of complex and L ∞valued coefficients defined on R n . We assume that this matrix satisfies the following ellipticity (or "accretivity") condition: there exist 0 < λ ≤ Λ < ∞ such that

λ |ξ| 2 ≤ Re A(x) ξ • ξ and |A(x) ξ • ζ| ≤ Λ |ξ| |ζ|,
for all ξ, ζ ∈ C n and almost every x ∈ R n . We have used the notation ξ

• ζ = ξ 1 ζ 1 + • • • + ξ n ζ n and therefore ξ • ζ is the usual inner product in C n . Note that then A(x) ξ • ζ = j,k a j,k (x) ξ k ζj .
Associated with this matrix we define the second order divergence form operator Lf = -div(A ∇f ), which is understood in the standard weak sense as a maximal-accretive operator on L 2 (dx) with domain D(L) by means of a sesquilinear form.

The operator -L generates a C 0 -semigroup {e -t L } t>0 of contractions on L 2 (dx). Define ϑ ∈ [0, π/2) by,

ϑ = sup arg Lf, f : f ∈ D(L) .
Then the semigroup has an analytic extension to a complex semigroup {e -z L } z∈Σ π/2-ϑ of contractions on L 2 (dx). Here we have written for 0 < θ < π,

Σ θ = {z ∈ C * : | arg z| < θ}.
We need to recall some properties of the generated semigroup {e -t L } t>0 (the reader is referred to [Aus] and [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part II: Off-diagonal estimates on spaces of homogeneous type[END_REF] for more details and complete statements). In what follows we set d(E, F ) = inf{|x -y| : x ∈ E, y ∈ F } where E, F are subsets of R n . Proposition 3.1. Given L as above, there exist p -= p -(L) and p + = p + (L), 1 ≤ p -< 2 < p + ≤ ∞ such that:

(a) The semigroup {e -t L } t>0 is uniformly bounded on L p (dx) for every p -< p < p + .

(b) The semigroup {e -t L } t>0 satisfies L p -L q off-diagonal estimates for every p -< p ≤ q < p + : For 1 ≤ p ≤ q ≤ ∞, L p -L q off-diagonal estimates mean that for some c > 0, for all closed sets E and F , all f and all t > 0 we have

F |e -t L (χ E f )| q dx 1 q t -1 2 ( n p -n q ) e -c d 2 (E,F ) t E |f | p dx 1 p . (3.1)
(c) For every m ∈ N and 0 < µ < π/2 -ϑ, the complex family {(zL) m e -z L } z∈Σµ is uniformly bounded on L p (dx) for p -< p < p + and satisfies L p -L q off-diagonal estimates for every p -< p ≤ q < p + (in (3.1) one replaces t by |z|).

(d) The interval (p -, p + ) is maximal for any of the properties above up to end-points, that is, none of them can hold outside [p -, p + ].

(e) If n = 1 or 2, or L has real coefficients, then p -= 1 and p + = ∞. In those cases, one has the stronger Gaussian domination |e -tL f | ≤ Ce ct∆ |f | for all f ∈ L 1 (dx)∪ L ∞ (dx) and t > 0 with constants c, C > 0. This implies uniform boundedness and off-diagonal estimates in the whole interval [1, ∞]. Other instances of a Gaussian domination occur for complex, continuous and periodic coefficients in any dimension, see [ERS].

(f ) If n ≥ 3, p -< 2n n+2 and p + > 2n n-2 .
Let us make some relevant comments. In the Gaussian factors of the off-diagonal estimates the value of c is irrelevant as long as it remains positive. When q = ∞ in (3.1), one should adapt the definitions in the usual straightforward way. One can prove that L 1 -L ∞ off-diagonal estimates are equivalent to pointwise Gaussian upper bounds for the kernels of the family (see [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part II: Off-diagonal estimates on spaces of homogeneous type[END_REF]). In dimensions n ≥ 3, it is not clear what happens at the endpoints for either boundedness or off-diagonal estimates.

3.2. Auxiliary Lemma. The proofs of Theorems 1.3 and 1.4 will use the following auxiliary lemma.

Lemma 3.2. Let p -< p 0 < s 0 < q 0 < p + so that 1/p 0 -1/s 0 = α/n. Fix a ball B with radius r. For f ∈ L ∞ c (dx) and m large enough we have

- B |L -α/2 (I -e -r 2 L ) m f | s 0 dx 1 s 0 ≤ ∞ j=1 g 1 (j) (2 j+1 r) α - 2 j+1 B |f | p 0 dx 1 p 0 , (3.2)
and for

1 ≤ l ≤ m - B |L -α/2 e -l r 2 L f | q 0 dx 1 q 0 ≤ ∞ j=1 g 2 (j) - 2 j+1 B |L -α/2 f | s 0 dx 1 s 0 , (3.3)
where g 1 (j) = C 2 -j (2 m-n/s 0 ) and g 2 (j) = C e -c 4 j .

Proof. We first obtain (3.3). We fix f ∈ L ∞ c (dx) and a ball B. We decompose any given function h as

h = j≥1 h j , h j = h χ C j (B) , (3.4)
where C j (B) = 2 j+1 B \ 2 j B when j ≥ 2 and C 1 (B) = 4B. Fix 1 ≤ l ≤ m. Since p -< s 0 < q 0 < p + by Proposition 3.1 part (b) we have

- B |e -l r 2 L h j | q 0 dx 1 q 0 r -n q 0 (l r 2 ) -1 2 ( n s 0 -n q 0 ) e - c d 2 (C j (B),B) l r 2 C j (B) |h| s 0 dx 1 s 0 2 j n/s 0 e -c 4 j - 2 j+1 B |h| s 0 dx 1 s 0 e -c 4 j - 2 j+1 B |h| s 0 dx 1 s 0
and by Minkowski's inequality

- B |e -k r 2 L h| q 0 dx 1 q 0 j≥1 g(j) - 2 j+1 B |h| s 0 dx 1 s 0 (3.5)
with g(j) = e -c 4 j for any h ∈ L s 0 (dx). This estimate with h = L -α/2 f ∈ L s 0 (dx) -here we use that f ∈ L ∞ c (dx) and Theorem 1.2-yields (3.3) since, by the commutation rule, L -α/2 e -l r 2 L f = e -l r 2 L h.

Next we obtain (3.2). We decompose f = j≥1 f j as in (3.4). For j = 1, we use that L -α/2 maps L p 0 (dx) into L s 0 (dx) by Theorem 1.2, and that (I -e -r 2 L ) m is bounded on L p 0 uniformly on r by (a) in Proposition 3.1 as p -< p 0 < p + . Hence,

- B |L -α/2 (I -e -r 2 L ) m f 1 | s 0 dx 1 s 0 |B| -1/s 0 R n |(I -e -r 2 L ) m f 1 | p 0 dx 1 p 0 |B| -1/s 0 4 B |f | p 0 dx 1 p 0 (4 r) α - 4 B |f | p 0 dx 1 p 0 .
(3.6)

Next we estimate the terms j ≥ 2. We first write

L -α/2 (1 -e -r 2 L ) m f j = 1 Γ(α/2) ∞ 0 t α/2 e -t L (1 -e -r 2 L ) m f j dt t = 1 Γ(α/2) ∞ 0 t α/2 ϕ(t, L)f j dt t , (3.7)
where ϕ(t, z) = e -t z (1 -e -r 2 z ) m . The argument will show that the integral in t converges strongly in L s 0 (B). Let µ ∈ (ϑ, π) and assume that ϑ < θ < ν < µ < π/2. Then we have

ϕ(t, L) = Γ + e -z L η + (t, z) dz + Γ - e -z L η -(t, z) dz, (3.8)
where Γ ± is the half ray R + e ±i (π/2-θ) ,

η ± (t, z) = 1 2 π i γ ± e ζ z ϕ(t, ζ) dζ, z ∈ Γ ± ,
with γ ± being the half-ray R + e ±i ν (the orientation of the paths is not needed in what follows so we do not pay attention to it). It is easy to see (see for instance [Aus]) that

|η ± (t, z)| r 2 m (|z| + t) m+1 , z ∈ Γ ± .
Then, since p -< p 0 < s 0 < p + by (c) in Proposition 3.1 we have

- B Γ + η + (t, z) e -z L f j dz s 0 dx 1 s 0 ≤ Γ + - B |e -z L f j | s 0 dx 1 s 0 |η + (t, z)| |dz| Γ + r -n s 0 |z| -1 2 ( n p 0 -n s 0 ) e -c 4 j r 2 |z| C j (B) |f | p 0 dx 1 p 0 |η + (t, z)| |dz| 2 j n/s 0 - 2 j+1 B |f | p 0 dx 1 p 0 ∞ 0 2 j r √ s α e -c 4 j r 2 s r 2 m (s + t) m+1 ds.
The same is obtained when one deals with the term corresponding to Γ -. We plug both estimates into the representation of ϕ(t, L) and use Minkowski's inequality for the integral in the t variable in (3.7) to obtain

- B |L -α/2 (I -e -r 2 L ) m f j | s 0 dx 1 s 0 2 j n/s 0 - 2 j+1 B |f | p 0 dx 1 p 0 ∞ 0 t α/2 ∞ 0 2 j r √ s α e -c 4 j r 2 s r 2 m (s + t) m+1 ds dt t 2 j n/s 0 4 -j m (2 j+1 r) α - 2 j+1 B |f | p 0 dx 1 p 0 ,
(3.9) since, after changing variables and taking

m + 1 > α/2, ∞ 0 ∞ 0 t α/2 2 j r √ s α e -c 4 j r 2 s r 2 m (s + t) m+1 dt t ds = 2 • 4 -j m (2 j r) α ∞ 0 e -c s 2 s 2 m ds s ∞ 0 t α/2 (1 + t) m+1 dt t 4 -j m (2 j+1 r) α .
Gathering (3.6) and (3.9) it follows that

- B |L -α/2 (I -e -r 2 L ) m f | s 0 dx 1 s 0 j≥1 g(j) (2 j+1 r) α - 2 j+1 B |f | p 0 dx 1 p 0
with g(j) = 2 -j(2 m-n/s 0 ) .

3.3. The proofs. We are going to apply Theorem 2.2 to the linear operator T = L -α/2 . Part (a) yields Theorem 1.3 and part (b) gives the estimates of the commutators in Theorem 1.4. Thus, it suffices to establish (2.1) and (2.2) with a sequence {α j } j that decays fast enough. We fix p -< p < q < p + , α so that α/n = 1/p -1/q, and w ∈ A 1+ 1 p --1 p ∩ RH q ( p + q ) ′ . By (iii) and (iv) in Proposition 2.1 there exist p 0 , q 0 , s 0 such that 1/p 0 -1/s 0 = α/n, p -< p 0 < s 0 < q 0 < p + , p 0 < p < q < q 0 and w ∈ A 1+ 1 p 0 -1 p ∩ RH q ( q 0 q ) ′ . Notice that as 1 ≤ p -< p + ≤ ∞ we have that 1 < p 0 < s 0 < q 0 < ∞. By Theorem 1.2, T = L -α/2 maps L p 0 (dx) into L s 0 (dx). We take A r = I-(I-e -r 2 L ) m where m ≥ 1 is an integer to be chosen. By the property (a) of the semigroup in Proposition 3.1, it follows that the family {A r } r>0 is uniformly bounded on L p 0 (dx) (as p -< p 0 < p + ) and so acts from L ∞ c (dx) into L p 0 (dx). We apply Lemma 3.2. Note that (3.2) is (2.1). Also, (2.2) follows from (3.3) after expanding A r = I -(I -e -r 2 L ) m . Then, we have that j≥1 j k g i (j) < ∞ for i = 1, 2 by choosing 2 m > n/s 0 . Consequently applying Theorem 2.2, part (a) if k = 0 and part (b) otherwise, we conclude that T k b maps L p (w p ) into L q (w q ) as desired.

A variant of Theorem 2.2

The next result is an extension to the context of fractional operators of [AM1, Theorem 3.14], itself inspired greatly by [START_REF] Shen | Bounds of Riesz transforms on L p spaces for second order elliptic operators[END_REF]Theorem 3.1].

Theorem 4.1. Let 0 < α < n, 1 ≤ p 0 < s 0 < q 0 ≤ ∞ such that 1/p 0 -1/s 0 = α/n. Suppose that T is a sublinear operator bounded from L p 0 (dx) to L s 0 (dx). Assume that there exist constants α

2 > α 1 > 1, C > 0 such that - B |T f | q 0 dx 1 q 0 ≤ C - α 1 B |T f | s 0 dx 1 s 0 + M α p 0 |f | p 0 (x) 1 p 0 , (4.1) for all balls B, x ∈ B and all f ∈ L ∞ (dx) with compact support in R n \ α 2 B. Let p 0 < p < q < q 0 with 1/p -1/q = α/n and w ∈ A 1+ 1 p 0 -1 p ∩ RH q ( q 0 q ) ′ . Then, there is a constant C such that T f L q (w q ) ≤ C f L p (w p ) for all f ∈ L ∞ c (dx). Proof.
A straightforward modification of Theorem 2.2 is to replace the family {A r } r>0 indexed by radii of balls by {A B } B indexed by balls. For any ball B, let A B f = (1 -χ α 2 B ) f . With this choice, we check (2.4) and the weakened version of (2.5). Fix

f ∈ L ∞ c (dx), a ball B and x, x ∈ B. Using that T is bounded from L p 0 (dx) to L s 0 (dx) we have - α 1 B |T (I -A B )f | s 0 dx 1 s 0 r(B) α - α 2 B |f | p 0 dx 1 p 0 M α p 0 |f | p 0 (x) 1 p 0 . (4.2)
In particular (2.4) holds since α 1 > 1. Next, by (4.1) and since

|A B f | ≤ |f | we have - B |T A B f | q 0 dx 1 q 0 ≤ C - α 1 B |T A B f | s 0 dx 1 s 0 + M α p 0 |f | p 0 (x) 1 p 0 .
Using (4.2) and the sublinearity of T , it follows that

- B |T A B f | q 0 dx 1 q 0 ≤ C M |T f | s 0 1 s 0 (x) + C M α p 0 |f | p 0 (x) 1 p 0 ,
which is the weakened version of (2.5). We conclude on applying the above mentioned variant of Theorem 2.2.

Spaces of homogeneous type

As Theorem 2.6 passes entirely to spaces of homogeneous type -a (quasi-)metric space (X , d) equipped with a Borel doubling measure µ-one may wonder whether Theorems 2.2 and 4.1 can be extended to this setting.

In the Euclidean setting, the classical Riesz potential I α or the fractional maximal operator M α are bounded from L p (dx) to L q (dx) necessarily when 1/p -1/q = α/n. This is caused by the homogeneity of these operators plus the dilation structure of the Lebesgue measure (that is, |B| = c r(B) n ). Concerning the weighted estimates, the boundedness of M α and I α from L p (w p ) to L q (w q ) are modeled by suitable modifications of the Muckenhoupt conditions which are vacuous unless 1/p -1/q = α/n.

Let (X , d, µ) be an space of homogeneous type where it is assumed that d is a distance (see [MS]). We also impose that µ(B) ≥ c r(B) n for some n > 0 -with this assumption, the fractional operators defined below are bounded with the same restriction in p and q as above. In this setting one can define the classes A p , RH q and A p,q by simply replacing the Lebesgue measure by µ. All the properties in Proposition 2.1 hold (to avoid some technicalities we assume that the weights are doubling). Here and in the sequel we understand that the averages are taken with respect to the measure µ.

We consider the following fractional operators that appear, for instance, in the study of subelliptic equations (see [Nag], [SW], [PW] and the references therein): As mentioned before there is a version of Theorem 2.6 in spaces of homogeneous type. Thus, in order to extend Theorem 2.2 and, therefore, Theorem 4.1, one only needs to study the boundedness of the fractional maximal operators M α defined above.

T α f (x) =
Proposition 5.1. Let 0 < α < n, 1 ≤ p < n/α and 1/q = 1/p -α/n. For every w ∈ A p,q , M α maps L p (w p ) into L q (w q ) if p > 1 and L 1 (w) into L q,∞ (w q ) if p = 1.

Proof. The proof follows the classical scheme in [MW] and we give just a few details. Given α, p, q and w as above, using Hölder's inequality and that w ∈ A p,q one can easily obtain that for every 0 ≤ g ∈ L p (w p ) r(B) α -B g dµ q w q (B) r(B) α q µ(B) q p -1 B g p w p dµ q p B g p w p dµ q p , (5.1)

where in the last estimate we have used that µ(B) r(B) n . Given f ∈ L p (w p ) and λ > 0, Vitali's covering lemma yields

E λ = {x ∈ X : M α f (x) > λ} ⊂ j 5 B j
where {B j } j is a family of pairwise disjoint balls such that r(B j ) α -B j |f | dµ > c λ.

Using (5.1) with B = 5 B j and g = |f | χ B j , and that p < q we obtain w q (E λ ) ≤ j w q (5 B j )

j B j |f | p w p dµ q p r(B j ) α - B j |f | dµ -q λ -q j B j |f | p w p dµ q p ≤ λ -q X |f | p w p dµ q p .
This shows that M α maps L p (w p ) into L q,∞ (w q ) for every w ∈ A p,q . When p = 1, this is the desired estimate.

To conclude that M α is of strong type when p > 1 we use an interpolation argument in [MW]. Having fixed p, q and w ∈ A p,q , we have w = w q ∈ A r with r = 1 + q/p ′ (see (viii) in Proposition 2.1). We define a new operator S α g = M α g wα/n . By Proposition 2.1 part (iii), there exists 1 < p 1 < p such that w ∈ A r 1 where r 1 = 1 + q 1 /p ′ 1 < r and 1/q 1 = 1/p 1 -α/n. Thus, w1/q 1 ∈ A p 1 ,q 1 and the argument above shows that S α is bounded from L p 1 ( w) to L q 1 ,∞ ( w). On the other hand we can find p < p 2 < α/n, then we define 1/q 2 = 1/p 2 -α/n and r 2 = 1+q 2 /p ′ 2 > r and so w ∈ A r 2 . Thus, w1/q 2 ∈ A p 2 ,q 2 and as before we conclude that S α maps L p 2 ( w) into L q 2 ,∞ ( w). By Marcinkiewicz's interpolation theorem, it follows that S α is bounded from L p ( w) into L q ( w) which in turn gives the desired weighted norm inequality M α .

Once we have obtained the weighted norm estimates for M α , the proofs of Theorems 2.2 and 4.1 can be carried out in X . The precise proofs and formulations are left to the interested reader. As a consequence, we show weighted estimates for T α .

Corollary 5.2. Let 0 < α < n, 1 < p < n/α and 1/q = 1/p -α/n. Then, T α maps L p (w p ) into L q (w q ) for all w ∈ A p,q .

Proof. We first notice that it suffices work with the sublinear operator f -→ T α (|f |). Abusing on the notation, we write T α for this new operator. Note that in that case T α f ≥ 0.

We claim that T α maps L p (µ) into L q (µ) for every 1 < p < q < ∞ such that 1/p -1/q = α/n, and also that sup (5.2) for every ball B, and f ∈ L ∞ c with supp f ⊂ X \ 4B. Assuming this, we obtain the desired estimate. Fix p, q, and w ∈ A p,q . Note that w ∈ A 1+1/p ′ ∩ RH q and by (iii) in Proposition 2.1 there exists 1 < p 0 < p such that

  where the supremum is taken over all balls and b B stands for the average of b on

  Q 0 = 0 since otherwise we can work with b = bb Q 0 and observe that T k b = T k b and b BMO = b BMO . Note that for all m = 0, . . . , k, we have that |b m f | and T (b m f ) are finite almost everywhere since |b m f | ∈ L p 0 (dx) and T (b m f ) ∈ L s 0 (dx). Let N > 0 and define b N as follows: b

X

  d(x, y) α µ(B(x, d(x, y))) f (y) dµ(y)for 0 < α < n. The associated maximal operator isM α f (x) = sup B∋x r(B) α -B |f(y)| dµ(y).

  x∈B T α f (x) -B T α f (x) dµ(x)

This work was partially supported by the European Union (IHP Network "Harmonic Analysis and Related Problems" 2002-2006, Contract HPRN-CT-2001-00273-HARP). Part of this work was carried out while the first author was visiting the Universidad Autónoma de Madrid as a participant of the Centre de Recerca Matemàtica research thematic term "Fourier analysis, geometric measure theory and applications". The second author was also supported by MEC "Programa Ramón y Cajal, 2005", by MEC Grant MTM2007-60952, and by UAM-CM Grant CCG07-UAM/ESP-1664. We warmly thank the anonymous referee for the suggestions that enhanced the presentation of this article.

A 1+ 1 p 0 -1 p ∩ RH q . We take s 0 so that 1/p 0 -1/s 0 = α/n, and q 0 = ∞. Then, (5.2) clearly implies (4.1) and thus T α is bounded from L p (w p ) to L q (w q ).

To finish we need to show our claims. First, we obtain the boundedness of T α . Fixed p, q, let 0 < s < ∞ to be chosen. Then, as α > 0,

On the other hand, since 1 < p < n/α,

Collecting the obtained estimates and choosing s = f L p (µ) /Mf (x) p/n we conclude

Let us point out that this estimate in the classical setting was shown by Hedberg [Hed]. From here, that T α maps L p (µ) into L q (µ) follows from the boundedness of M on L p (µ).

Next, we show (5.2). Let B be a ball and f supported on X \ 4 B. For every x, z ∈ B and y / ∈ 4 B we have that d(x, y) ≈ d(z, y) and the doubling condition yields µ(B(x, d(x, y)) ≈ µ (B(z, d(z, y)). Therefore T α f (x) ≈ T α f (z) for every x, z ∈ B and this readily leads to (5.2).


Naji Massad

Vitting Jørgen

 Andersen

October

Defining an intrinsic "stickiness" parameter of stock price returns

Keywords:

We introduce a non-linear pricing model of individual stock returns that defines a "stickiness" parameter of the returns. The pricing model resembles the capital asset pricing model (CAPM) used in finance but has a non-linear component inspired from models of earth quake tectonic plate movements. The link to tectonic plate movements happens, since price movements of a given stock index is seen adding "stress" to its components of individual stock returns, in order to follow the index. How closely individual stocks follow the index's price movements, can then be used to define their "stickiness".

Introduction

When investors want to assess the price of a given stock, in order, for example, to do risk management or portfolio allocation, they often use the basic capital asset pricing model (CAPM) [START_REF] Lintner | The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets[END_REF][START_REF] Sharpe | Capital asset prices: a theory of market equilibrium under condition of risk[END_REF][START_REF] Treynor | Toward a Theory of Market Value of Risky Assets[END_REF] to guide them of a proper price/risk. The CAPM introduces a linear relationship between the return of a given asset, and the risk of the market. The risk of the market, is, in the CAPM framework defined in terms of variances and co-variances between the return of the asset and the excess return of the market. This measure of risk is called the market beta. For a Documents de travail du Centre d'Economie de la Sorbonne -2019.28 longer discussion of the model, as well as some of the empirical failings of the CAPM, see for example [START_REF] Fama | The capital asset pricing model: theory and evidence[END_REF] [START_REF] Fama | The capital asset pricing model: theory and evidence[END_REF].

It is hard to underestimate the importance that the CAPM has had for the financial industry as a tool for pricing assets. [START_REF] Graham | The theory and practice of corporate finance: Evidence from the field[END_REF] [START_REF] Graham | The theory and practice of corporate finance: Evidence from the field[END_REF] found in a survey of 392 CFOs in companies throughout the U.S. and Canada, that 74% of respondents always, or almost always, use the CAPM in estimating the cost of equity capital. In another study [START_REF] Bruner | Best practices in estimating the cost of capital: survey and synthesis[END_REF] Bruner, Eades, Harris, and Higgins (1998) found that 85% of their 27 best-practice firms use the CAPM or a modified version of the CAPM. While the CAPM is popular tool for practitioners to use for pricing of assets, it has in parallel been extensively studied in academic research papers, introducing "a zoo of new factors" [START_REF] Cochrane | Presidential Address: Discount Rates[END_REF]. Given such a plethora of factors and the inevitable data mining, many of the discovered factors probably were deemed «significant» by pure chance [START_REF] Harvey | …and the Cross-Section of Expected Returns[END_REF]. [START_REF] Harvey | …and the Cross-Section of Expected Returns[END_REF] [START_REF] Harvey | …and the Cross-Section of Expected Returns[END_REF] made an extensive testing of at least 316 linear factors models to explain the cross-section of expected returns. They argue that it is a serious mistake to use the usual statistical significance cutoffs in asset pricing tests.

A natural extension of the original CAPM would be to take into account betas that are time varying. Time varying properties can be captured in models like the autoregressive conditionally heteroskedastic (ARCH) model of Engle [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom Inflation[END_REF] and the generalized version (GARCH) by Bollerslev [START_REF] Bollerslev | Generalized autoregressive conditional heteroscedasticity[END_REF]. But such a description within a linear framework has been shown to have its own problems, even when allowing for time varying coefficients. Ghysel [START_REF] Ghysels | On stable factor structures in the pricing of risk: do time varying betas help or hurt?[END_REF] for example showed that models with dynamical varying betas have stability problems. Often they are not able to give a proper description of the dynamics of the betas. The study by Ghysel found that constant beta models in many cases still yield on average better predictions compared to models with time varying betas. In essence, the time varying CAPM models have a tendency to overstate the time variation and as a result produce beta risk that is too volatile and changing Documents de travail du Centre d'Economie de la Sorbonne -2019.28 too rapidly. It highlights the general problem with parameter instability in factor models like the CAPM and the APT (Arbitrage Pricing Theory). Still many would argue that the static CAPM should be replaced by some form of conditional or non-linear version of the CAPM.

One argument for conditional or non-linear versions of the CAPM relates to business-cycles, since market volatility is tightly linked to business-cycles. [START_REF] Lettau | Resurrecting the (C)CAPM: A cross-sectional test when risk premia are time-varying[END_REF] [START_REF] Lettau | Resurrecting the (C)CAPM: A cross-sectional test when risk premia are time-varying[END_REF] and [START_REF] Petkova | Is value riskier than growth[END_REF] [START_REF] Petkova | Is value riskier than growth[END_REF] both suggest that stocks have different exposures to market risk during recessions and expansions. In similar vein, Granger and Silvapulle (2001) [START_REF] Granger | Capital asset pricing model, bear, usual and bull market conditions and beta instability: a value-at-risk approach[END_REF] study whether or not beta responds asymmetrically to good and bad news. They define three market scenarios: bad, usual and good, conditional on the quantiles of the market return. Using the Dow Jones index over 9 years, they found that, for 21 stocks the beta of the stock is higher when the market is in a bearish state compared to a bullish state, while for the remaining 9 stocks the opposite is true. Their results confirm the general view that the betas increase (decrease) when the market is in a bearish (bullish) state. Another example of extending models to a non-linear description of the prices was introduced by Chen et al. [START_REF] Chen | Multi-regime non-linear capital asset pricing models[END_REF]. They introduced a multiple-regime threshold GARCH model, in order to capture asymmetric risk, by allowing the market beta to change discretely between regimes that are driven by market information. In the following we introduce a new threshold model. However, in our case the threshold is not at the level of the market, but is instead introduced as an intrinsic property of a given stock.

2 Theoretical framework: linear factor models, the CAPM, the yard-stick model

In the capital asset pricing model the stock returns are estimated through a linear formula:

E(R ) = E(R ) + β [E(R ) -E(R )] (CAPM) (1)
Here E(R ) is the expected return of the market M, and E(R ) is the expected return of asset i.

E(R ) is the expected return of the interest free rate, which, since we are only considering daily Documents de travail du Centre d'Economie de la Sorbonne -2019.28 returns, will be taken equal to zero in the following. β is given by the covariance of the return of asset i with the market return divided by the variance of the market return: β = ( , ) ( ) .

We don't have access to people's expectation, so in practice the beta's are estimated via timeseries regression:

R (t) = β R (t) (2)
Insert Figure 1 around here Figure 1: "Stickiness" of the stock price movements. Aggregate price movements of the stocks (illustrated by the bar) pull each stock differently depending on the "coupling" constant β and the "stickiness" parameter 𝑅 of each stock. In the case illustrated, the "stickiness" thresholds 𝑅 and 𝑅 prevent stocks 1 and 3 from moving, whereas stock 2 overcomes its threshold 𝑅 and moves.

In the following we will assume the stocks being influenced by the aggregate price movements of the other N-1 stocks in an index, see Fig. 1, which illustrates the idea that each stock is linked to the market through a spring, with a spring constant given by a combination of two factors γ , * 𝛼 , . The first factor γ , measures, like the β of the CAPM, the specific sensitivity of the asset i's return to variation in the market return of the remaining N-1 assets.

The second factor, 𝛼 , in addition weights the importance of the capitalization of a stock for the aggregate price movements, see Eq. represented by the "rod". In order to exclude any self-impact, we consider that the price of stock i moves because of the price movement of the N-1 other stocks constituting the index, with the return of the N-1 stocks, R (t) ≡ ln(

∑ ( ) ∑ ( )
). Here S (t) is the price of stock j at time t (assuming for simplicity an index with equal weights on its assets).

R (t) = α , γ , R (t) ; α , = [ 1 -𝑒 * ] (3)
The constant α , describes how a given stock moves in response to the market, depending on the ratio of capitalization of the given stock, 𝐾 , to the capitalization of the remaining stocks in the index 𝐾 ≡ ∑ 𝐾 . 𝛿 describes the scale of impact that the capitalization 𝐾 of the remaining stocks have on the price movement of stock i. A small 𝛿 ≪ 1 means that the price movement of stock i is dominated by the movement of the remaining stocks. The case 𝛿 ≫ 1 would correspond to a case of stocks moving independently, something we know empirically is not true, so a priori one should expect small values of 𝛿, but how small would define the proper impact of the remaining stocks on the price movement of stock i.

Insert Figure 2 around here (3) using same data set. Each point corresponds to a daily return of a given stock i (x-axis), and its prediction (y-axis). The data points show the results of out-of-sample simulations using first an in-sample period (from 23/05/2016 to 13/10/2016) to determine the α 's of Eq.( 2) and the γ , , 𝛿 of Eq.( 3). For a given fixed value of 𝛿, the set of γ , 's that minimizes the error of forecasting of R can be obtained directly from Eq.( 3). The optimal value of delta was found in-sample to be very small, as expected, 𝛿 = 0.00102. In order to smooth out the fluctuations of γ , , we used aggregate returns over a large window of size T=250. As seen from Fig. 2b the mean error in predicting R is larger compared to the CAPM case, but the skewness and kurtosis of the error is smaller compared to the CAPM case.

In [START_REF] Martinez | Pricing Stocks with Yardsticks and Sentiments[END_REF] we have instead pointed out the importance of a relative sentiment measure of a given stock to its peers. The idea is that people use heuristics, or "rules of thumb", in terms of "yard sticks" from the performance of the other stocks in a stock index. The under-/over-performance with respect to a yard stick happens because of a general negative/positive sentiment of the market participants towards a given stock. The bias created in such cases does not necessarily have a psychological origin but could be due to insider information. Insiders having superior information about the state of a company reduce/increase their stock holding gradually causing a persisting bias over time. The introduction of a measure for the relative sentiment of a stock has allowed us to come up with another one factor model, very similar in structure to the CAPM model [START_REF] Martinez | Pricing Stocks with Yardsticks and Sentiments[END_REF]:

Documents de travail du Centre d'Economie de la Sorbonne -2019.28 R (t) = γ R (t) γ = ( ) ( ) ; R (t) ≡ ln( ∑ ( ) ∑ ( ) ), (4)
In Eq.( 4) γ now describes the relative volatility of the stock to the volatility of the market of the remaining N-1 stocks. Therefore the basic pricing mechanism via Eq.( 4) is given by traders using the Sharpe ratio of a stock and comparing it to the Sharpe ratio of the market as a yard-stick for whether a stock is under/overvalued. Figure 2b shows daily performances of the stocks to have a clear tendency to cluster around the measures introduced by the yard sticks in accordance with the pricing formula Eq.( 4). However, the error of the prediction of R has a larger mean and higher order moments larger compared to the CAPM case. In other out-ofsample intervals we found larger error and standard deviations, but smaller skewness and kurtosis compared to the CAPM case.

The CAPM, Eq.( 3), and Eq.( 4) presented above are examples of one-factor models which despite their simplicity often are able to capture quite specific behavior. Cizeau et al. [START_REF] Cizeau | Correlation structure of extreme stock returns[END_REF] showed how correlation between stock returns in extreme conditions could be satisfactorily described by time independent factors without having to invoke correlations that changes over time. It is important to note a qualitative property of the factor models presented above, namely that although the market is built from the fluctuations of the stocks, it is the market that is the fundamental quantity, not the stocks themselves. Therefore, you can't explain statistical properties of the market returns from those of the stocks returns within such models. In that respect we would like to point out the difference here between stocks fluctuating because they are coupled to an index, and fluctuations across markets worldwide. In the case of spillover effects across markets, it does make sense to believe that the rise of correlations is caused by increasing fear by investors, something also pointed out by Balogh et al. [START_REF] Balogh | Persistent collective trend in stock markets[END_REF]. Their argument Documents de travail du Centre d'Economie de la Sorbonne -2019.28

for correlations across markets happens via prospect theory [START_REF] Kahneman | Prospect theory: an analysis of decision-making under risk[END_REF], a very different mechanism compared to stocks following almost automatically a given index.

3 Theoretical framework: a non-linear stick-slip factor model,

We now suggest to go beyond a linear description by introducing a "stickiness" parameter of individual stock returns. As illustrated in Fig. 1, a small price movement of the market (illustrated by the moving rod) does not necessarily induce a price movement of all individual stocks prices, e.g. in Fig. 1 the stock 1 and 3 don't move (they "stick") whereas stock 2 follows the market price movement (it "slips"). As mentioned in [START_REF] Andersen | Pricequakes" shaking the world's stock exchanges[END_REF], a psychological reason why the small price movements get ignored could be because of the so-called "change blindness" [START_REF] Jones | Change detection model for serially correlated data[END_REF],

a behavioral trait which reflects the tendency of humans to reply in a nonlinear fashion to changes. This is consistent with experiments made in psychology which have shown that humans react disproportionally to big changes, whereas small changes go unnoticed [START_REF] Jones | Change detection model for serially correlated data[END_REF][START_REF] Rensink | Change detection[END_REF][START_REF] Lewin | Change blindness, the metacognitive error of estimating change detection ability[END_REF][START_REF] Bellenzier | Contagion in the world's stock exchanges seen as a set of coupled oscillators[END_REF].

Another reason for a non-linear reaction of market participants could be insider knowledge:

people with knowledge of negative news for a stock, could persistently use positive market movements as a selling opportunity, creating inertia in individual stock price movements. To take into account such effects we suggest the following equations:

𝑅 (𝑡) =( ( ) ( ) ×ɵ([ ( ) ( ) |𝑅 (𝑡)| > 𝑅 ]) × 𝑅 (𝑡) (5)
𝑅 (𝑡) =(1-ɵ([ ( ) ( ) |𝑅 (𝑡 -1)| > 𝑅 ]) × 𝑅 (𝑡 -1) + 𝑅 (𝑡) (6)
Documents de travail du Centre d'Economie de la Sorbonne -2019.28 Top plots show the cumulative market return (red line), stock return (blue line) for the stock American Express, and prediction of the stock return Eq.( 5)-( 6 The pricing model resembles the capital asset price model (CAPM) used in finance but has a non-linear component with origin from models of earth quake tectonic plate movements [25].

The link to tectonic plate movements happens, since price movements of a given stock index is seen adding "stress" to its components of individual stock returns, in order to follow the index.

How closely individual stocks follow the index's price movements, can then be used to define its "stickiness".

using Eqs.( 5)-( 6) to obtain 𝑅 (𝑡).

Insert Figure 4 around here The blue line in Fig. 4 shows the error of the model, Eq.( 7), as a function of the "stickiness"

parameter, 𝑅 for one given stock, the Boing Co. stock. As can be seen, for this particular stock, for small values of 𝑅 the prediction of 𝑅 (𝑡) using Eq.5-6 does not seem very different from the yard-stick model Eq.( 2), which corresponds to the case where 𝑅 ≡ 0. The question however is whether our finding of a 𝑅 > 0 is due to the inherent noise of the time series. In order to determine this, we applied boots trapping. By reshuffling the original return price time series (excluding the first 20 days that serve to calibrate the volatility ratio), we break any temporal correlations (but still keep any overall trend of the market), that could contain information used to predict the price returns via our model. The red line in Fig. 4 represents the level of the 10 th percentile, whereas the green line represents the level of 90 th percentile using in total 100 samples. We checked that using instead 1000 samples we obtained the same qualitative behavior. As can be seen for the BA stock we were able to find a 𝑅 below the 10 th noise level. We then applied this method to the remaining 29 stocks of the Dow index, see table 1. First column gives the noise level for the yard-stick model Eq.( 4) which by definition corresponds to the 𝑅 ≡ 0 case of Eqs.( 5)-( 6). Second column gives the level of the 10 th percentile found from bootstrapping, third column gives 𝑅 , and forth column then indicates the value of the "stickiness" 𝑅 for those stocks where we were able to define a such. In order to define a "stickiness" 𝑅 of a stock we required 𝐸 𝑅 < 𝐸 (𝑅 ≡ 0) AND 𝐸 𝑅 < 𝐸 (𝑅 ) with 𝐸 () the error of stock i, and 𝐸 denoting the noise level of the 10 th percentile. That is, in order to claim to have found a "stickiness" 𝑅 of a stock i, it needs to give a better prediction compared to the case of the model without threshold, (𝑅 ≡ 0), as well as have a noise level that is below the 10 th percentile found via the boots trap method. 3). It should come as no surprise that we find the stickiness (as defined above) changes over time, just as example given correlations changes over time. 3) we find only 5 stocks having a stickiness parameter 𝑅 >0. However out of the 5 stocks, 4 were found also to have a stickiness parameter in the first time period. We find one stock with a stickiness parameter over all three time periods, Verizon Communications Inc. A priori it would seem reasonable to expect internet related companies, to be more "detached" from the general price movements of the index, just as we found for the Verizon stock. However, the fact that we are not able (over the given time periods) to define a stickiness parameter for neither IBM, nor Cisco, seems to contradict the results found for Verizon. Finally, we tried the method in the case where one replace the ratio which we could define a stickiness parameter, so whether using the covariance, or just simple ratios of the standard deviations, seem less relevant when it comes to define a stickiness parameter of a given stock.

Discussion

We have introduced and studied two linear pricing models of individual stock returns and compared with the well-known capital asset price model (CAPM) in finance. After an initial discussion on linear pricing models, we then extended our study considered a non-linear pricing model of individual stock returns that enables us to define a "stickiness" parameter of the returns. The pricing model resembles the CAPM, but has a non-linear component inspired from models of earth quake tectonic plate movements. The link to tectonic plate movements happens, since price movements of a given stock index is seen adding "stresses" to its components of individual stock returns, in order to follow the index. How closely individual stocks follow the index's price movements, can then be used to define their "stickiness". Using a bootstrap method, we attributed a statistical significance measure in order to define a stickiness parameter for each individual stock. In this way we were able to attribute a stickiness parameter for a number of stocks in the Dow Jones Industrial average. Our hope is that the introduction of stickiness allow for an enlarged understanding in the pricing process of stocks.

In practical terms, one could for example imagine it be used as a tool in connection with the detection of insider trading, where insider knowledge introduces "friction" in price formation which could be captured in terms of our stickiness measure.

  [START_REF] Treynor | Toward a Theory of Market Value of Risky Assets[END_REF] below. The stock price (represented by a small square) in this picture moves because of the aggregate price movement of the market Documents de travail du Centre d'Economie de la Sorbonne -2019.28

Figure 2 :

 2 Figure 2: Performance of linear pricing models. The plot a) shows the CAPM hypothesis

Figure 3 :

 3 Figure 3: Oscillations of the "stress" field 𝑹 𝒎 𝒊 𝒄𝒖𝒎 (𝒕) for three different stickiness values 𝑹 𝑪 𝒊 .

  ) (green line) over the time period 10/04/2015-07/07/2015. The plots a)-c) correspond to three different values of the stickiness parameter 𝑅 . The bottom plots show the oscillations of the "stress" field 𝑅 (𝑡) .

4)

  Results: defining intrinsic "stickiness" of stocksIn order to test the model, we first applied equations Eqs.(5)-[START_REF] Bruner | Best practices in estimating the cost of capital: survey and synthesis[END_REF] to the Dow Jones Industrial Index over the period 23/03/2015 to 19/04/2016. The initial choice of this special period was because the composition of the Dow did not change during that period, other periods were subsequently chosen to test for stationarity of the results. Since we are studying daily data, and since the trading volume between the opening and the close by far exceeds the trading volume between the close and the open, we only consider the daily return between the open and the close[START_REF] Martinez | Pricing Stocks with Yardsticks and Sentiments[END_REF]. For each stock of the Dow Jones, we applied the discrete equations Eqs.(5)-(6was calculated using a running window of 20 days. In order to determine the "stickiness" parameter, 𝑅 , of each stock i, we minimized the error function, 𝐸 : Documents de travail du Centre d'Economie de la Sorbonne -2019.28 𝐸 (𝑅 ) = ∑ (𝑅 (𝑡) -𝑅 (𝑡))

Figure 4 :

 4 Figure 4: Bootstrap to define confidence levels of the stickiness parameter. Blue line shows

  For larger values of 𝑅 a global minimum, 𝑅 ≈ 0,0052, of the error function 𝐸 appear before the error finally increases for large values of 𝑅 . It is easy to understand the large prediction errors that happens for large values of 𝑅 , by looking at figure 3. A large value of 𝑅 means that the stock only reacts for large cumulative price movements of the market, thereby "overshooting" the stock's observed price movement, when the condition in the Theta function in Eqs.(5)-(6) is finally met. We therefore use the observed minimum of 𝑅 as a measure of what we call intrinsic "stickiness" of a stock. Documents de travail du Centre d'Economie de la Sorbonne -2019.28

  Eq.5-6. In this case 𝑅 ≡ 0, then coincides with the CAPM model . In general, several overlaps were found in the stocks for Documents de travail du Centre d'Economie de la Sorbonne -2019.28

Table 1

 1

	. 14 stocks have a Rc that is above 0 (23/03/2015-19/04/2016)
	Stock	Value for Rc=0	Noise level	Min error	Rc
	AAPL	0.0462	0.0459	0.0460	-
	AXP		0.0330	0.0330	-

Documents de travail du Centre d'Economie de la Sorbonne -2019.28

Table 2 .

 2 8 stocks have a Rc that is above 0 (22/03/2016-18/04/2017) Documents de travail du Centre d'Economie de la Sorbonne -2019.28

	Stock	Value for Rc=0	Noise level	Min error	Rc
	AAPL	0.0216	0.0208	0.0205	0.0058
	AXP	0.0193	0.0191	0.0191	0.00213
	BA	0.0290	0.0288	0.0289	-
	CAT	0.0373	0.0366	0.0370	-
	CSCO	0.0126	0.0121	0.0122	-
	CVX	0.0215	0.0211	0.0213	-
	DIS	0.0118	0.0111	0.0118	-
	DWDP	0.0300	0.0297	0.0296	0.00215
	GE	0.0129	0.0126	0.0129	-
	GS	0.0322	0.0321	0.0322	-
	HD	0.0190	0.0185	0.0184	0.00588
	IBM	0.0152	0.0150	0.0148	0.00382
	INTC	0.0204	0.0197	0.0197	-
	JNJ	0.0145	0.0141	0.0142	-
	JPM	0.0146	0.0144	0.0145	-
	KO	0.0153	0.0149	0.0152	-
	MCD	0.0176	0.0168	0.0170	-
	MMM	0.0074	0.0072	0.0072	-
	MRK	0.0464	0.0460	0.0460	0.00426
	MSFT	0.0155	0.0153	0.0155	-
	NKE	0.0415	0.0405	0.0399	0.00633
	PFE	0.0249	0.0242	0.0246	-
	PG	0.0163	0.0154	0.0160	-
	TRV	0.0165	0.0161	0.0165	-
	UNH	0.0265	0.0256	0.0261	-
	UTX	0.0097	0.0096	0.0097	-
	V	0.0222	0.0216	0.0220	-
	VZ	0.0234	0.0230	0.0226	0.0082
	WMT	0.0279	0.0264	0.0269	-
	XOM	0.0205	0.0201	0.0205	-

Table 3 .

 3

	5 stocks have a Rc that is above 0 (21/03/2017 to 12/03/2018)
	Stock	Value for Rc=0	Noise level	Min error	Rc
	AAPL	0.0360	0.0339	0.0355	-
	AXP	0.0150	0.0148 0.0149	-
	BA	0.0461	0.0444	0.0455	-
	CAT	0.0294	0.0282	0.0291	-
	CSCO	0.0181	0.0177	0.0180	-
	CVX	0.0259	0.0251	0.0254	-
	DIS	0.0412	0.0393	0.0408	-
	DWDP	0.0302	0.0286	0.0297	-
	GE	0.1070	0.1060	0.1048	0.0097
	GS	0.0383	0.0375	0.0381	-

Documents de travail du Centre d'Economie de la Sorbonne -2019.28

In the second time period, 22/03/2016-18/04/2017, (table

2

) we are only able to identify 8 stocks with a stickiness parameter 𝑅 >0. Out of the 8 stocks, 3 were also identified having a stickiness in the first time period. In the third time period, 21/03/2017-12/03/2018 (table

Documents de travail du Centre d'Economie de la Sorbonne -2019.28

Documents de travail du Centre d'Economie de la Sorbonne -2019.28 Documents de travail du Centre d'Economie de la Sorbonne -2019.28 Documents de travail du Centre d'Economie de la Sorbonne -2019.28 Documents de travail du Centre d'Economie de la Sorbonne -2019.28

Acknowledgments

Documents de travail du Centre d'Economie de la Sorbonne -2019.28 This work was carried out in the context of the Laboratory of Excellence on Financial Regulation (Labex ReFi) supported by PRES heSam under the reference ANR-10-LABX-0095. It benefitted from French government support managed by the National Research Agency (ANR) as part of the project Investissements d'Avenir Paris Nouveaux Mondes (invesiments for the future, Paris-New Worlds) under the reference ANR-11-IDEX-0006-02.

Eq.5 describes the "change blindness" part since the stock i only gets a contribution from the remaining N-1 stocks when the normalized cumulative return (or "stress"),

exceeds a certain threshold 𝑅 which is specific to stock i. Eq.( 6) describes how the "stress" on stock i, from the remaining N-1 stocks, keeps on cumulating until it exceeds 𝑅 , after which it is reset to zero: the effect of the price movements of the N-1 stocks becomes "prized in". For a very small 𝑅 the condition in the Theta function is always fulfilled, so the model in this case coincide with the yard-stick model Eq.4. As stressed before, this case corresponds to traders comparing the Sharpe ratio of a stock to the Sharpe ratio of the market, in order to decide whether a stock is over/under-valued. A large value 𝑅 instead means that the price of stock i only changes when the aggregate price movements of the market exceeds a certain threshold, determined by the Sharpe ratio of the aggregate market movement and a threshold Sharpe ratio of stock i given by 𝑅 /σ(𝑅 ). As seen from Eq.6 after a price change has happened at say time t-1, 𝑅 is reset to zero (the market price movement has become "priced-in") before another market price movement eventually again adds to 𝑅 at time t. Therefore one can consider the variable 𝑅 (𝑡) as an oscillator: consecutive market movements keeps on adding to 𝑅 up to the value (positive or negative) determined by the threshold condition in the Theta function of Eq.5,6, after which 𝑅 is set to zero and another oscillation can begin. The period of the oscillation is determined by the fraction of the Sharpe ratio of the aggregate market movement, and the threshold Sharpe ratio of stock i, given by the 𝑅 /σ(𝑅 ).


domain: spi phys

text:
Pierre Doublet
email: pierre.doublet@onera.fr

Christine Lempereur

Mikael Orain

Virginel Bodoc

Pierre Gajan

Saturated Laser-Induced Fluorescence in liquid kerosene seeded with a dye: influence of temperature and excitation intensity

Keywords: fluorescence, temperature, saturation, liquid phase

Rapid communication

STATEMENT

The content of this manuscript submitted to "Applied Physics B" is unpublished material that is not being submitted for publication elsewhere.

I. Introduction

Droplet spray characterization often relies on laser techniques and image processing that provide information such as droplet size, velocity and concentration. In the literature, experimental investigations of droplets have been performed at ambient and / or high temperatures and pressures [START_REF] Lefebvre | Atomization and Sprays[END_REF][START_REF] Wang | Influence of ambient air pressure on pressure swirl atomization[END_REF] although there is a constant need for additional data under such harsh conditions to help validate numerical simulations. However, no such investigations have been conducted for altitude relight conditions of aircraft engines (low fuel temperature and sub-atmospheric pressure). Moreover, the combination of reduced temperatures and pressures considerably increases certain physical properties, such as surface tension and viscosity, thus altering spray atomization [START_REF] Lefebvre | Atomization and Sprays[END_REF][START_REF] Yao | Effect of viscosities on structure and instability of sprays from a swirl atomizer[END_REF]. In these conditions, the fuel injector generates larger droplets, liquid fuel evaporation is limited and consequently spray ignition is more difficult. Fine characterization of spray properties in these challenging conditions is needed to help engine manufacturers improve injector design and ultimately ignition performances.

Among the different spray characterization techniques, Planar Droplet Sizing (PDS) became popular in the last decade because of its ability to yield an instantaneous 2D-map of droplet Sauter mean diameter: this technique greatly reduces data acquisition time compared to Phase Doppler Anemometry (PDA) [START_REF] Lempereur | Etude de la technique de granulométrie Planar Droplet Sizing sur un brouillard issu d'un injecteur aéronautique[END_REF][START_REF] Charalampous | Method to reduce errors of droplet sizing based on the ratio of fluorescent and scattered light intensities (laser induced fluorescence/Mie technique)[END_REF]. This method is based on the ratio between laser-induced fluorescence and Mie scattering images [START_REF] Kamimoto | Diagnostic of transient sprays by means of laser sheet techniques[END_REF]. Until now, most of the work on PDS was carried out in water sprays using different types of dye. Domman et al [START_REF] Domann | A study of the influence of absorption on the spatial distribution of fluorescence intensity within large droplets using Mie theory, geometrical optics and imaging experiments[END_REF] and Chaze et al. [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF] used Rhodamine as a dye in their experiments while Pyranine was added to water in the experiments by Mishra et al [START_REF] Mishra | Reliable LIF/Mie droplet sizing in sprays using structured laser illumination planar imaging[END_REF]. Other dyes such as fluorescein with water as a solvent were used in the Chaze experiments. Jermy et al. [START_REF] Jermy | Planar dropsizing by elastic and fluorescence scattering in sprays too dense for phase Doppler measurement[END_REF] implemented the PDS technique in a cooling spray that was too dense for PDA measurements by using sodium salicylate as the dye mixed with water. Some work was also performed with biofuels such as ethanol. For example, Lempereur et al. [START_REF] Lempereur | Etude de la technique de granulométrie Planar Droplet Sizing sur un brouillard issu d'un injecteur aéronautique[END_REF] implemented the PDS technique to an ethanol spray seeded with rhodamine 6G. Koegl [START_REF] Koegl | Analysis of LIF and Mie signals from single micrometric droplets for instantaneous droplet sizing in sprays[END_REF] studied LIF and Mie signals from an ethanol micrometric droplet as a function of its diameter, with eosin as a dye. Eosin was also used to study ethanol and butanol sprays with the two-phase structured laser illumination planar imaging droplet sizing [START_REF] Koegl | Analysis of ethanol and butanol direct-injection sprak-ignition sprays using two-phase structured laser illumination planar imaging droplet sizing[END_REF]. A relatively small number of studies were performed on realistic fuels and these were mainly dedicated to automotive fuels. For example, tetramethyl-p-phylene diamine (TMPD) was used by Kamimoto et al. [START_REF] Kamimoto | Diagnostic of transient sprays by means of laser sheet techniques[END_REF] and Yeh et al. [START_REF] Yeh | Fluorescence/scattering image technique for particle sizing in unsteady diesel spray[END_REF] and 3 pentanone was added to iso-octane fuel by Stojkovic et al. [START_REF] Stojkovic | Evolution and impingement of an automotive fuel spray investigated with simultaneous Mie/LIF techniques[END_REF]. Regarding applications of PDS to aviation fuels, very few publications could be found in the literature. Le Gal et al. [START_REF] Gal | Laser sheet dropsizing of dense sprays[END_REF] used p-Terphenyl (PTP) as a dye in mineral spirit (kerosene without its fluorescing components), at ambient temperature and pressure. This suggests a need to develop the PDS technique for kerosene fuel under challenging operating conditions such as altitude relight. The current paper aims at providing a contribution in the field of Planar Droplet Sizing measurements applied to realistic aviation fuels.

II. Theoretical background

Fluorescence intensity depends on several parameters and can be expressed as follows:

𝐼𝐼 𝑓𝑓 (𝑇𝑇)~𝐼𝐼 0 𝜖𝜖(𝑇𝑇)𝛷𝛷 𝑓𝑓 (𝑇𝑇) 𝐶𝐶 (1)
where 𝐼𝐼 0 is the incident laser intensity, 𝐶𝐶 the concentration of the fluorescent species, 𝛷𝛷 𝑓𝑓 its quantum yield (𝑄𝑄𝑄𝑄) and 𝜖𝜖 its absorption cross section. The last two parameters depend on the fluorescent specie considered. For example, Chaze et al. [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF] studied the influence of temperature on the fluorescence of several dyes and reported different temperature behaviours. On one hand, fluorescence increases with temperature for fluorescein disodium because of an increase in the absorption cross section. By contrast, fluorescence decreases for rhodamine 6G due to the decrease of the 𝑄𝑄𝑄𝑄 as temperature increases (quenching effect). The solvent containing the dye can be an additional parameter that also affects the temperature dependence of the dye fluorescence.

As a first assumption for Planar Droplet Sizing, the fluorescence intensity emitted by a droplet 𝐼𝐼 𝑓𝑓 is proportional to its volume [START_REF] Kamimoto | Diagnostic of transient sprays by means of laser sheet techniques[END_REF]. Nonetheless, if the dye concentration is high or if the absorption cross section is large, this assumption needs to be re-evaluated. In such a case, droplet fluorescence intensity can then be written as [START_REF] Domann | A study of the influence of absorption on the spatial distribution of fluorescence intensity within large droplets using Mie theory, geometrical optics and imaging experiments[END_REF][START_REF] Charalampous | Numerical evaluation of droplet sizing based on the ratio of Fluorescent and scattered light intensities (lif/mie technique)[END_REF]:

𝐼𝐼 𝑓𝑓 = a f 𝑑𝑑 𝐹𝐹(𝐶𝐶) (2)
where 𝑎𝑎 𝑓𝑓 is a constant, 𝑑𝑑 the droplet diameter and 𝐹𝐹 a parameter which depends on the dye concentration 𝐶𝐶. For low absorption, 𝐹𝐹(𝐶𝐶) is close to 3 (i.e. the theoretical value), but strongly decreases as absorption increases [START_REF] Charalampous | Numerical evaluation of droplet sizing based on the ratio of Fluorescent and scattered light intensities (lif/mie technique)[END_REF]. Moreover, as can be seen in equation ( 1), the quantum yield 𝛷𝛷 𝑓𝑓 and the absorption cross section 𝜖𝜖 depend on temperature. In some cases, this temperature dependence can be high which alters the reliability of the PDS technique [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF].

A way of limiting the quenching effect consists in working with saturated laser-induced fluorescence.

This method is used in particular to measure OH or NO species concentrations in flames at high pressure [START_REF] Naik | Laser-saturated and linear laser-induced fluorescence measurements of nitric oxide in counterflow diffusion flames under non-sooting oxygen-enriched conditions[END_REF][START_REF] Desgroux | Measurements of OH concentrations in flames at high pressure by two-optical path laser induced fluorescence[END_REF]. However, to the best of the authors' knowledge, the saturated laser-induced fluorescence is mainly applied to gases and the only study conducted in liquids is that of [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF].

In the present paper, the saturated laser-induced fluorescence method is carried out in liquid kerosene in order to overcome the temperature dependence of the dye added to the fuel. This allows to perform accurate Planar Droplet Sizing measurements under challenging conditions, which is the ultimate goal of this study.

III. Experimental setup and measurement protocol

The tests are carried out in a calibration cell in order to analyze the influence of temperature and excitation intensity on the fluorescence measurements. Pyrromethene 597-8C9 (PM from now on) is used as a dye in liquid kerosene because of its high miscibility with kerosene, ethanol, methanol and oils [START_REF] Sanned | Study of fluorescence dyes for tracing lubrification oil inside marine engines[END_REF].

By contrast, despite having useful fluorescence properties as a dye for PDS applications with water, rhodamine 6G cannot be used with kerosene due to non-miscibility between these two compounds.

PM concentration is fixed to 1.2 mg/L in order to minimize absorption along the laser path. This value, which was carefully selected from a parametric investigation of PM fluorescence signal versus concentration, is sufficient to study realistic kerosene sprays. The seeded liquid is placed into a 10 mm-size quartz test cell from HELLMA, and then introduced into a Luma 40 cuvette holder from Quantum Northwest in order to regulate its temperature between -30 and 50 °C. Both measurement devices record the fluorescence above the excitation wavelength (𝜆𝜆 > 540 nm) spatially integrated over a small area (10 × 10 mm²). The transmission of the high-pass filter has a flat profile above 542 nm with values in excess of 99% and therefore does not require any correction.

However, the data cannot be quantitatively compared. The first reason is that the spectrometer provides calibrated spectral data whereas the camera spectrally integrates the fluorescence without correction for its own spectral response. Additionally, both devices have different measurement sensitivities. Nevertheless, the data can be compared after normalization as shown later in the text.

The interesting feature lies in the acquisition modes of the intensified camera. The gate width (i.e. the integration time) can generally be chosen to include all the duration of the fluorescence emission with a constant delay with respect to the laser pulse (Fig. 2. (a)). However, the gate width can also be set to a smaller value (here 3 ns) and moved gradually at increasing delays after the laser pulse in order to measure the temporal evolution of fluorescence (Fig. 2. (b)). Typically, a few tens of pulses are necessary to describe the time evolution of PM fluorescence. In this acquisition mode, the laser intensity has to remain stable over a few shots. The measurement protocol was fixed as follows: the laser is switched on for a few tens of seconds without hitting the cell, until its intensity is stabilized (rms fluctuation < 5 %). The test cell is then illuminated during the required number of pulses only, in order to minimize the photodamage of PM. Actually, after this short laser exposure, the fluorescence level does not recover its initial value because of bleaching. As a result, the fluid is changed in the test cell after each measurement sequence.

IV. Influence of temperature on the fluorescence of PM-seeded kerosene

The temperature conditions for altitude relight of aeroengines engines range from -40 to 0 °C.

However, a wider exploration has been conducted, up to 50 °C, which can be useful for other purposes.

The objective is to quantify the temperature sensitivity of PM fluorescence to define the conditions in which the variations of this parameter can be considered as negligible on the fluorescence measurement.

Measurements are first conducted with a low laser energy chosen to yield a linear fluorescent emission during the whole pulse duration. The emission spectrum, measured above 540 nm, is broadband with a maximum at 600 nm. For each temperature, the corresponding spectrum is normalized by the maximum value of the spectrum at -28 °C. Fluorescence quenching is observed when temperature increases (Fig. 3). The profiles normalized by their maximum value (not shown here) do not exhibit any noticeable shape difference (e.g, redshift) with the wavelength.

The integration of the fluorescence signal over the emission spectral range is plotted as a function of temperature in Fig. 4. The measurement was also performed with the camera, using a gate width of 50 ns to include all the fluorescence emission duration. Both curves, normalized by their value at T = 0 °C, are in good agreement. In order to quantify the temperature dependence of the dye emission, a sensitivity coefficient s [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF] can be introduced:

𝐼𝐼 𝑓𝑓 (𝑇𝑇) = 𝐼𝐼 𝑓𝑓 (𝑇𝑇 0 )𝑒𝑒 -𝑠𝑠(𝑇𝑇-𝑇𝑇 0 ) (3)
where 𝑇𝑇 0 is an arbitrary temperature, chosen here as 𝑇𝑇 0 = 0 °C, and 𝑠𝑠 is estimated at -1.3 % / °C, which can be compared to that of rhodamines. For example, Rh6G is often chosen for its low temperature sensitivity (𝑠𝑠(𝑅𝑅ℎ6𝐺𝐺) = -0.07 % /°C) enabling temperature-independent fluorescence measurements, while RhB is much more sensitive (𝑠𝑠(𝑅𝑅ℎ𝐵𝐵) = -2.3 %/ °C) and can be used for temperature measurements in liquids [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF]. The sensitivity of PM lies between that of these two rhodamines and cannot be neglected since, for instance, the effect of the temperature on Pyrromethene fluorescence measurement can cause a reduction of 40% between -30 °C and 0 °C.

V. Influence of the laser intensity: saturation of fluorescence

When pulsed lasers are used to excite fluorescence, their irradiance usually exceeds the saturation intensity of common fluorescent dyes. The fluorescence signal therefore loses its linear dependence on the laser irradiance [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF]. This phenomenon will later be illustrated by the temporal profiles of fluorescence for increasing laser intensity. The laser irradiance is regulated via the QSwitch / Flash

Lamp internal delay ∆. The maximum irradiance value, approximately 6×10 6 W/cm², is obtained for ∆ 0 =176 µs and decreases for increasing delays. The data used to estimate irradiance is affected by some uncertainties due to the estimation of the temporal profile of the laser pulse and spatial intensity profile of the laser sheet. The temporal evolution of the laser pulse is measured by using a band pass filter (BP 532/25, Lavision) placed in front of the camera. Given the FWHM (i.e. 25 nm) of the filter, a fluorescence signal is also acquired by the camera but this signal is negligible compared to the one emitted by the laser. The sequential acquisition mode of the camera is used to obtain the temporal evolution of the laser pulse. For the delay used in the previous paragraph (∆ 2 = 320 µs, which corresponds to a laser irradiance of 2×10 6 W/cm²), fluorescence intensity increases with the laser energy and then decreases with a time shift induced by the fluorescence lifetime. On the other hand, when the energy is higher (∆ 1 = 300 µs, approximately 3×10 6 W/cm²), a saturation clearly appears, as shown in Fig. 5 (see the dashed black marks). In this case, fluorescence intensity remains constant until the laser energy is lower than a given excitation threshold.

The fluorescence saturation phenomenon can also be visualized on the graph of fluorescence intensity versus excitation intensity of the laser (Fig. 6). The lower part of each curve follows the rising edge of the laser pulse whereas the upper part represents the falling edge. The fluorescence lifetime is illustrated by the fact that, for a given excitation intensity, the fluorescence intensity is greater on the falling edge. In the non-saturated case, fluorescence intensity increases with the excitation up to its maximum value and then decreases. In the saturated case, the fluorescence intensity reaches its maximum value when the excitation intensity is equal to 60 % of the maximum excitation intensity in this case. For higher excitation intensities, a saturation of the fluorescence intensity is also observed.

This saturation phenomenon is interesting and can be exploited to suppress the temperature dependence of PM fluorescence for measurements in sprays. Once the laser intensity threshold is reached, the fluorescence intensity remains constant for the length of time during which the laser irradiance is larger than this threshold value, regardless of the temperature.

The temporal evolution of fluorescence signal with temperature is presented in Fig. 7. Three measurement timings can be analyzed to illustrate the saturation behaviour. The first one, labelled (a),

shows that if the fluorescence signal is integrated over the non-linear part of the curve, between 70 and 80 ns (blue diamonds in Fig. 8) then it is temperature independent. Next, for measurement timing (b), if fluorescence is integrated over the falling edge, when the phenomenon becomes linear again between 85 and 105 ns (green triangles in Fig. 8), then the fluorescence emission exhibits the same decrease with temperature as in Fig. 4. In particular, the fluorescence lifetime also depends on temperature for molecules having a temperature-dependent quantum yield. Finally, for measurement timing (c), if fluorescence is integrated over its full duration (red dots in Fig. 8) then the measured emission is globally less dependent upon temperature. This decrease in dependence is due to the fact that the first temporal part of the curves, which is temperature independent, is integrated in the measurement. A summary of these results is presented in Fig. 8.

The time range in which fluorescence intensity is integrated is a key parameter. Measurement accuracy strongly depends on the acquisition timing of the camera. For example, Fig. 8 shows that for the minimum and maximum temperatures (i.e. -28 °C or 50 °C here), the difference between the measurements carried out in the temperature dependent and independent regions, can reach up to ±30 %. This shows that it is extremely important to know exactly in which regime (e.g. temperature dependent or independent) measurements are performed, to obtain reliable results in future PDS applications. Finally, it is important to note that this methodology using gated acquisitions is only possible with an intensified camera. If the PDS technique is carried out with conventional CCD or CMOS cameras, the gate time cannot be adjusted to fulfill this requirement.

VI. Perspectives: Extension to a laser sheet for planar measurements

In the present case, a parallel light sheet with homogeneous fluence within the field of view (10 × 10 mm²) is obtained with a collimator placed at the exit of the laser. For planar measurements such as Planar Droplet Sizing, the field of view is about 100 × 100 mm² and the laser sheet spatial distribution can introduce a measurement bias. The spatial distribution of the laser sheet intensity exhibits a Gaussian profile. As a result, fluorescence saturation can occur for the droplets located in the center of the laser sheet, ie. with the highest energy, as opposed to the droplets located in the laser sheet edges.

This aspect is currently under investigation.

VII. Conclusion

This

  A frequency-doubled Nd:YAG pulsed laser (Big Sky, Quantel) emitting light at 532 nm is used to excite fluorescence. An energy meter is employed to quantify the pulse-to-pulse variations of the laser energy. The beam is then converted into a collimated sheet passing through the cell. A spectrometer (HR2000+, Ocean Optics) and an intensified camera (PIMAX 4, Gen III, Roper Scientific) are placed at 90° from the laser propagation, on opposite sides of the test cell, in order to perform simultaneous fluorescence measurements. A long-pass edge filter (BLP01-532R, Semrock) enables the acquisition of the fluorescence signal close to the laser line with a deep laser-line blocking (Fig.1.).

FIGURESFig. 1 Fig. 2 Fig. 3

 123 FIGURES

Fig. 4 Fig. 5

 45 Fig. 4 Evolution of fluorescence intensity with temperature measured by the spectrometer and by the

Fig. 6 Fig. 7

 67 Fig. 6 Normalized fluorescence intensity versus normalized excitation intensity for non-saturated (∆ 2 = 320 µs, 2×10 6 W/cm² ) and saturated case (∆ 1 = 300 µs, 3×10 6 W/cm²). Each fluorescence signal is normalized by the maximum value.

Fig. 8

 8 Fig.8 Evolution of fluorescence intensity with temperature measured by the camera for three

Acknowledgements

This work is conducted between ONERA and SAFRAN AIRCRAFT ENGINES. The authors gratefully acknowledge the ANRT (Association Nationale de la Recherche et de la Technologie) which supports part of the PhD.