Arnaud Rougirel

TIME FRACTIONAL LINEAR PROBLEMS ON L 2 (R d )

Keywords: March 14, 2017. 2010 Mathematics Subject Classification. 35C15, 35E99 Fractional derivatives, Time fractional PDEs, regularizing effect, dissipative inequalities, Fractional Schrödinger equation, Fractional transport equation

In this paper, we propose a theory for linear time fractional PDEs on L 2 (R d ). The order of the time derivatives under consideration is less than 1. We study well-posedness, regularizing effects and dissipative properties. In particular, we give a necessary and sufficient condition for well-posedness. Regarding regularizing effects, we describe quite precisely the equations that have this effect or not. We highligth that, in purely fractional settings, the regularizing effect acts always only up to finite order; unlike to the standard case. π 2 α then the problem is well-posed, has regularizing effects and dissipative properties.

Introduction

Time fractional differential equations have been the subject of many research in the recent years, both in terms of applications ([GARGBA + 12], [START_REF] Laskin | Fractional market dynamics[END_REF], [START_REF] Baleanu | New trends in nanotechnology and fractional calculus applications[END_REF]) and mathematical studies ( [START_REF] Kostić | Abstract Volterra integro-differential equations[END_REF], [START_REF] Gorenflo | Mittag-Leffler functions, related topics and applications[END_REF]).

The aim of this paper is to built a theory for linear time fractional PDEs on L 2 (R d ). It is about to extend the well-known theory of first order time derivative equations. Surprisingly, to our knowledge, such extension has not been made yet. However, there are many results regarding abstract time fractional equations: see for instance [START_REF] Kostić | Systems of abstract time-fractional equations[END_REF], [START_REF] Bazhlekova | The abstract Cauchy problem for the fractional evolution equation[END_REF], [START_REF] Keyantuo | Existence, regularity and representation of solutions of time fractional diffusion equations[END_REF].

For first order time derivative equations of the form

u t = P (D)u, (1.1)
where u = u(x, t) and P (D) is a differential operator acting on the space variable x, the well posedness is equivalent to the condition

sup ξ∈R d Re P (ξ) < ∞,
where P is the symbol of P (D) (see Subsection 2.3 for notation and definitions).

For time fractional equations of the form

D α 0,t u = P (D)u, (1.2)
with α ∈ (0, 1), the above condition is sufficient (see [START_REF] Emamirad | Solution operators of three time variables for fractional linear problems[END_REF]) but no more necessary (see Example 4.3 below). In this paper, we give a quite simple necessary and sufficient condition, namely (4.1), for the above fractional problem to be well posed in L 2 (R d ).

See Theorem 4.2. Roughly speaking, our results state that the critical angle for equation (1.2) is π 2 α in a similar fashion that π/2 is the critical angle for (1.1). Indeed, if | arg P (ξ)| < π 2 α then the Cauchy problem corresponding to (1.2) is generally ill-posed. If | arg P (ξ)| > If | arg P (ξ)| = π 2 α then the problem is well-posed in L 2 (R d ) but, has no regularizing and dissipative properties.

For instance, the fractional Schrödinger equation

D α 0,t u = -i∆u, (1.3)
is hyperbolic for α = 1 in the sense that it has a conservation law and no regularizing effect. In this case, arg P (ξ) = π/2. When α ∈ (0, 1), the critical angle is π 2 α. Since π/2 > π 2 α, there results that (1.3) has regularizing effect and dissipative properties (see Theorem 6.3). In order to recover a hyperbolic behaviour for α ∈ (0, 1), Equation (1.3) may be modified into D α 0,t u = -i α ∆u, see Example 4.6 for details. This equation appears in the poineer work of Naber dedicated to fractional extensions of Schrödinger's equation ( [START_REF] Naber | Time fractional Schrödinger equation[END_REF]). It seems that the proper way time fractional Schrödinger's equation should be formulated is still discussed ( [START_REF] Narahari Achar | Time fractional Schrodinger equation revisited[END_REF]). Let us notice that the same issue holds for transport equations. We hope that the systematic analysis we have made in this paper will be useful for modeling processes using fractional time derivatives.

Regarding the regularizing effect of (1.2), we would like to emphasize that, contrary to the standard case, it acts only up to a finite order even if the right hand side of (1.2) is the Laplacian operator (see Proposition 6.7 and Example 6.8).

In [START_REF] Emamirad | Solution operators of three time variables for fractional linear problems[END_REF], we introduce some "no-initial" value problems. These problems are interesting since (1.2) can by no means be considerated as an autonomous equation if α ∈ (0, 1) (see [START_REF] Emamirad | Solution operators of three time variables for fractional linear problems[END_REF] for details). In this paper, we continue the study of this kind of problems. To our knowledge, this study has also not be made at that time.

The paper is organised as follow. In the next section we introduce some preliminaries which are essential in our forthcoming discussions. This includes the definition of fractional derivatives, mild and strong solutions and Fourier multiplier results.

The third section is devoted to analyse some asymptotic properties of Mittag-Leffler functions. Sections 4 and 6 are devoted to the study of quantitative and qualitative properties of the solutions to D α τ,t u = P (D)u, u(s) = v, where τ and s are real number satisfying τ ≤ s. It has to be noticed that the analysis of the above problem differs whether τ = s or not. In the case τ = s, we give existence and uniqueness results when the initial condition v lies in L 2 (R d ). That extends the results of [START_REF] Emamirad | Solution operators of three time variables for fractional linear problems[END_REF] where v was supposed to be smoother-i.e. lying in the domain of the operator P (D). Our qualitative results concern regularizing effects and dissipative properties of the solutions.

In Section 5, we study the three time variable solution operator corresponding to the above equation.

Preliminaries

2.1. Fractional derivatives. In this subsection, we recall the required background on Caputo's fractional derivatives. Let us introduce the convolution of functions defined on semi-infinite intervals.

Definition 2.1. For each τ ∈ R, let g be a function of L 1 loc ([0, ∞)) and f be an element of L 1 loc ([τ, ∞), X). Then the convolution of g and f is the function of

L 1 loc ([τ, ∞), X) defined by g * τ f (t) = t τ g(t -y)f (y)dy, a.e. t ∈ [τ, ∞).
Remark 2.2. (i) If τ = 0 then we will write g * f instead of g * τ f . (ii) A function f belongs to L 1 loc ([0, ∞), X) iff for each positive T , f lies in L 1 (0, T, X). The following kernel is of fundamental importance in the theory of fractional derivatives.

Definition 2.3. For β ∈ (0, ∞), let us denote by g β the function of L 1 loc ([0, ∞)) defined for a.e. t > 0 by

g β (t) = 1 Γ(β) t β-1 .
Let us notice that g 1 = 1 and for each α ∈ [0, 1), we have

g 1-α (t) = 1 Γ(1 -α) t -α , a.e. t > 0.
Moreover, for each α, β ∈ (0, ∞), the following identity holds.

g α * g β = g α+β , in L 1 loc [0, ∞) . (2.1)
Then we are able to introduce the well known fractional derivative of a function in the sense of Caputo.

Definition 2.4. Let α ∈ (0, 1), τ ∈ R and f ∈ C([τ, ∞), X). We say that f admits a (fractional) derivative of order α in

L 1 loc ([τ, ∞), X) or that f is α-differentiable in L 1 loc ([τ, ∞), X) if g 1-α * τ f ∈ W 1,1 loc ([τ, ∞), X).
In that case, its (fractional) derivative of order α is the function of L 1 loc ([τ, ∞), X) defined by

D α τ,t f := d dt g 1-α * τ (f -f (τ )) .
If α = 1 then D 1 τ,t will denote the usual first order time derivative and f is said to be

1-differentiable in L 1 loc ([τ, ∞), X) if f belongs to W 1,1 loc ([τ, ∞), X). Above, the vector space W 1,1 loc ([τ, ∞), X) is the set of functions u in L 1 loc ([τ, ∞), X) whose time-derivative (in the sense of distributions) belongs to L 1 loc ([τ, ∞), X). Remark 2.5. (i) If f ∈ C([τ, ∞), X), α ∈ (0, 1) and g 1-α * τ f ∈ W 1,1 loc ([τ, ∞), X) then clearly, g 1-α * τ f -f (τ ) ∈ W 1,1 loc ([τ, ∞), X). Hence Definition 2.4 makes sense. (ii) It is well known that the space W 1,1 loc ([τ, ∞), X) is a subset of C([τ, ∞), X). (iii) Analog definition of fractional derivative in C([τ, ∞), X) can be given; see [ER17, Definition 3.3] for a precise statement.
The following results will be useful in the sequel. The statement of Proposition 2.6 is closed to [ABHN11, Proposition 1.3.6] and can be proved in a same way.

Proposition 2.6. For any T > 0, let f ∈ W 1,1 (0, T, X) and g ∈ L 1 (0, T ). Then g * f lies in W 1,1 (0, T, X) and

d dt g * f = g * d dt f + g(•)f (0), in L 1 (0, T, X). (2.2)
By combining (2.2) and (2.1), we prove easily the well known following result.

Corollary 2.7.

Let α ∈ (0, 1], τ ∈ R and f ∈ C([τ, ∞), X) be a α-differentiable function in L 1 loc ([τ, ∞), X). Then g α * τ D α τ,t f = f -f (τ ) in L 1 loc ([τ, ∞), X).
2.2. Abstract linear fractional Problems. Let A : D(A) ⊆ X → X be a closed linear operator on a complex Banach space X, whose norm will be denoted by • X . The domain D(A) is equipped with the standard graph norm, so that it is a Banach space. For α ∈ (0, 1] and any real number τ , we consider the following homogeneous linear fractional problem

D α τ,t u = Au, u(τ ) = v. (2.3)
Following [START_REF] Prüss | Evolutionary integral equations and applications[END_REF], we introduce the definition of strong and mild solutions .

Definition 2.8. Let α ∈ (0, 1], τ ∈ R and v be in D(A). We say that a function u is a strong solution to

(2.3) on [τ, ∞) if (i) u belongs to C([τ, ∞), D(A)) and u(τ ) = v; (ii) u admits a derivative of order α in C([τ, ∞), X); (iii) D α τ,t u = Au in C([τ, ∞), X).
Similarly to the case where α = 1, the fractional differential equation (2.3) can be transformed into an integral equation. More precisely, we have the following well known result, easily proved via Corollary 2.7. Proposition 2.9. Let α ∈ (0, 1], τ ∈ R and v be in D(A). Then the following propositions are equivalent.

(i) u is a strong solution to (2.3) on [τ, ∞). (ii) u belongs to C([τ, ∞), D(A)), and u = v + g α * τ Au, in C [τ, ∞), X .
When v is no more restricted to live in D(A), we are led to consider solutions in a weaker sense, the so-called mild solutions . Definition 2.10. Let α ∈ (0, 1], τ ∈ R and v be in X. We say that a function u is a mild solution to

(2.3) on [τ, ∞) if (i) u belongs to C([τ, ∞), X); (ii) g α * τ u belongs to C([τ, ∞), D(A)); (iii) u = v + A(g α * τ u) in C([τ, ∞), X).
Clearly, A being closed, any strong solution to (2.3) is a mild solution to (2.3). Definition 2.11. Let α ∈ (0, 1] and τ ∈ R. We say that (2.3) is well posed on X if for each v in X, (2.3) has a unique mild solution u on [τ, ∞) and for all positive time T ,

u(t) X ≤ C T v X , ∀t ∈ [τ, τ + T ],
where C T is a real constant independent of t and v.

When the two time parameters problem

D α τ,t u = Au, u(s) = v (2.4)
is considered, we will use Definition 2.12 for solutions to (2.4). Problem (2.4) is a "no-initial" value problems that is generally ill-posed when α = 1 and τ < s. We refer to [START_REF] Emamirad | Solution operators of three time variables for fractional linear problems[END_REF] for a first approach to (2.4).

Definition 2.12. Let α ∈ (0, 1], s, τ ∈ R with τ ≤ s and v be in X. We say that a function

u ∈ C([τ, ∞), X) is a solution to (2.4) on [τ, ∞) if (i) u belongs to L 1 loc ([τ, ∞), D(A)) and u(s) = v; (ii) u admits a derivative of order α in L 1 loc ([τ, ∞), X); (iii) D α τ,t u = Au in L 1 loc ([τ, ∞), X).
Let us notice that Definition 2.12 encapsulates a regularity effect when s = τ . Indeed, the initial condition v belongs to X whereas u(t) lies in D(A) for almost every t larger than τ .

By considering Problem (2.4), it appears that a time shift simplifies the situation by reducing the study to the case τ = 0. More precisely, for τ ≤ s, let us consider the problem D α 0,t w = Aw, w(s -τ ) = v.

(2.5) Then we have the following result. Proposition 2.13. Let α ∈ (0, 1], τ ≤ s, v ∈ X, u ∈ C([τ, ∞), X) and w ∈ C([0, ∞), X). We suppose that

u(t + τ ) = w(t), ∀t ≥ 0.
Then u is a solution to (2.4) on [τ, ∞) if and only if w is a solution to (2.5) on [0, ∞).

The proof of Proposition 2.13 is easy and analog to the one of [ER17, Proposition 4.2], so we omit it. Notice that the latter is concerned with smoother solutions since the initial condition is supposed to belong to D(A).

Remark 2.14. Statements analog to Proposition 2.13 hold for strong and mild solutions.

When τ = 0 and A = P (D) is a differential operator with constant coefficients on X := L 2 (R d ), the strong solution to (2.3) has a representation of the form

u(t) = S α (t)v,
where the Fourier multiplier of the solution operator S α (t) : D(P ) → D(P ) is explicitly computed (see [START_REF] Emamirad | Solution operators of three time variables for fractional linear problems[END_REF]). From this computation, it is clear that S α (t) can be extended into a bounded operator on L 2 (R d )-see (5.3).

Since, for v in L 2 (R d ), S α (t)v does not belong generically to D(P ), S α (•)v cannot be a strong solution. Thus the issue is to find an equation whose solution is S α (•)v. Proposition 2.13 below brings an answer by means of mild solutions . Proposition 2.15. Let A : D(A) ⊆ X → X be a closed densily defined linear operator on X. Let us assume that (i) for each v in D(A), there exists a unique strong solution to

D α 0,t u = Au, u(0) = v;
(2.6) (ii) for each T > 0, there exists a constant C T such that for all v in D(A) and t in [0, T ], the solution u to (2.6) satisfies

u(t) X ≤ C T v X .
(2.7)

Then for each τ ∈ R and v in X, (2.3) has a unique mild solution u on [τ, ∞). Moreover, if (v n ) n≥0 ⊂ D(A) converges toward v in X then for each positive time T ,

u n (• -τ ) → u in C [τ, τ + T ], L 2 (R d ) , (2.8)
where u n denotes the strong solution to (2.6) with initial condition v n .

Proof. It relies on (2.1), Proposition 2.13 and on arguments developed in [START_REF] Prüss | Evolutionary integral equations and applications[END_REF] after Definition 1.2. Let us notice that we do not need to use Titchmarsh's Theorem (see for instance [START_REF] Doss | An elementary proof of Titchmarsh's convolution theorem[END_REF]) here. We use (2.1) instead.

2.3. Pseudo-differential operators on L 2 (R d ). The Fourier transform on L 2 (R d ), denoted by F, is defined for f ∈ L 1 (R d ) ∩ L 2 (R d ), by Ff (ξ) = f (ξ) := R d f (x)e -iξ•x dx, a.e. ξ ∈ R d .
Then its inverse F -1 satisfies

F -1 (f )(x) := (2π) -d R d f (ξ)e iξ•x dξ, a.e. x ∈ R d .
Moreover, according to Plancherel's identity, we have for all f, g ∈ L 2 (R d ),

R d f (ξ) g(ξ)dξ = (2π) d R d f (x)g(x)dx.
(2.9)

For any function P : R d → C continuous on R d , let

D(P ) := {u ∈ L 2 (R d ) | P (•)û ∈ L 2 (R d )}
and P (D) :

D(P ) ⊆ L 2 (R d ) → L 2 (R d
), the so-called pseudo-differential operator with constant coefficients, defined for all u ∈ D(P ) by

P (D)u := F -1 P (•)û .
The continuous function P is called the symbol of the pseudo-differential operator P (D). In the literature, the symbol of a pseudo-differential operator is supposed to be a smooth function on R d . In this paper, we extend this definition to any continuous symbol P . That allows us to encapsulate the case of fractional Laplacian operators, nowaday extensively studied-see Example 2.16 below.

The domain

D(P ) contains F -1 (D(R d )), where D(R d ) is the space of C ∞ -functions on R d with compact support. Since F -1 (D(R d )) is dense in L 2 (R d ) (see
the proof of Lemma 2.17), there results that P (D) is a closed and densely defined linear operator on L 2 (R d ).

If P is a complex polynomial on R d i.e.

P : R d → C, ξ → |β|≤M p β ξ β , with β = (β 1 , . . . , β d ) ∈ N d , |β| = β 1 + • • • + β d , M ∈ N, p β ∈ C,
then P (D) is a differential operators with constant coefficients. More precisely, let us introduce Hörmander's notation for partial derivatives, that-is-to-say for k = 1, . . . , d, put

D k := 1 i ∂ x k , D := (D 1 , . . . , D d ),
where i 2 = -1 and ∂ x k denotes the usual partial derivative in the direction x k . Then

P (D) := |β|≤M p β D β .
Also

D(P ) = {u ∈ L 2 (R d ) | P (D)u ∈ L 2 (R d )} (2.10) = {u ∈ L 2 (R d ) | R d |P (ξ)û(ξ)| 2 dξ < ∞}.
In (2.10), P (D)u is understood in the sense of distributions.

Example 2.16. let β ∈ (0, ∞) and

P 2β : R d → C, ξ = (ξ 1 , . . . , ξ d ) → -|ξ| 2β := - d k=1 ξ 2 k β . If β = 1 then -P 2 (D) is the Laplacian operator, i.e. -P 2 (D) = d k=0 D 2 k = - d k=0 ∂ 2 x k = -∆.
If β ∈ (0, 1) then the pseudo-differential operator -P 2β (D) is called the fractional Laplacian operator and is denoted by (-∆) β . See for instance [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF]. Its domain is the fractional Sobolev space

H 2β (R d ) = W 2β,2 (R d ), and
D -(-∆) β = D(P 2β ) = H 2β (R d ) = {u ∈ L 2 (R d ) | R d |ξ| 4β |û(ξ)| 2 dξ < ∞}.

2.4.

A Fourier Multiplier Result. We will use the following classical result in Fourier multiplier Theory.

Recall that D(R d ) denotes the space of C ∞ -functions on R d with compact support. Then F -1 (D(R d )) is the space of L 2 -functions whose Fourier transform is in D(R d ).
Also, for X a complex Banach space, we denote by L(X) the space of linear continuous maps from X into X.

Lemma 2.17. Let a : R d → C be a continuous function on R d and

A : F -1 (D(R d )) → L 2 (R d ), v → Av := F -1 (av).
Then the following propositions are equivalent.

(i) A has a unique extension in L 2 (R d ), still labelled A. (ii) There exists C ≥ 0 such that Av L 2 (R d ) ≤ C v L 2 (R d ) for all v in F -1 (D(R d )). (iii) The Fourier multiplier a(•) belongs to L ∞ (R d ).
Moreover, if one of the above properties holds then

A L(L 2 (R d )) = sup ξ∈R d |a(ξ)|. (2.11) Proof. It is clear that (i) implies (ii), with C := A L(L 2 (R d ))
. By [BTW75, Theorem I.2.2 and proof], (ii) implies (iii) and sup ξ∈R d |a(ξ)| ≤ C. Finally, (iii) implies (i) and

A L(L 2 (R d )) ≤ sup ξ∈R d |a(ξ)|. That follows from the fact that F -1 (D(R d )) is dense in L 2 (R d ). Indeed, if a function f in L 2 (R d ) satisfies R d F -1 ϕ(x)f (x) dx = 0, ∀ϕ ∈ D(R d ).
Then by Plancherel's identity, we have

R d ϕ(ξ) f (ξ) dξ = 0. Thus f = 0 and F -1 (D(R d )) is dense in L 2 (R d ).
There results that the equivalences hold. Then (2.11) follows easily.

Mittag-Leffler functions

In fractional differential problems, Mittag-Leffler functions play the role of exponential functions in differential equations. In this section, we will give all the material on Mittag-Leffler functions needed to solve the time fractional problems under consideration in this paper. For further information, we refer the reader to [START_REF] Podlubny | Fractional differential equations[END_REF] or [START_REF] Gorenflo | Mittag-Leffler functions, related topics and applications[END_REF].

E α,β (z) = ∞ k=0 z k Γ(kα + β)
, ∀z ∈ C.

If β = 1 then we put E α := E α,1 and E α is called the Mittag-Leffler function of order α.

For all α > 0 and λ ∈ C, the Mittag-Leffler function E α satisfies

D α 0,t E α (t α λ) = λE α (t α λ), ∀t ≥ 0 (3.1) d dt E α (t α λ) = 1 t E α,0 (t α λ). (3.2)
Moreover, the function E α is increasing and positive on R (see [START_REF] Pollard | The completely monotonic character of the Mittag-Leffler function E a (-x)[END_REF]). Also we have the following deep result that will be extensively used in this paper.

Theorem 3.2. [Pod99, Theorem 1.3 and 1.4] Let α ∈ (0, 1], β ∈ R and α ∈ (α, 2). Then E α,β (z) = - 1 Γ(β -α) 1 z + O 1 z 2 , as |z| → ∞, | arg(z)| ≥ π 2 α . (3.3) Moreover, if α < 2α then E α,β (z) = 1 α z 1-β α exp z 1/α + O 1 z , as |z| → ∞, | arg(z)| < π 2 α . (3.4)
In this paper, it is understood that the function arg(•) ranges in (-π, π].

3.2.

Estimates of the Mittag-Leffler functions. For any continuous function P : R d → C and α ∈ (0, 1], let us consider the following property: there exists M P ≥ 0 such that for each ξ ∈ R d satisfying P (ξ) = 0, the following implication holds: if

| arg P (ξ)| < π 2 α then Re P (ξ) 1/α ≤ M P . (3.5)
Of course, for z ∈ C, we have set

z 1/α := |z| 1/α exp i arg z α if z = 0 0 if z = 0 ,
and Re (P (ξ) 1/α ) denotes the real part of

P (ξ) 1/α . Lemma 3.3. Let P : R d → C be a continuous function on R d and α ∈ (0, 1]. If (3.5) holds then, for each T > 0, sup t∈[0,T ] sup ξ∈R d E α t α P (ξ) < ∞. (3.6)
If for each ξ ∈ R d satisfying P (ξ) = 0, one has

| arg P (ξ)| ≥ π 2 α then sup t≥0 sup ξ∈R d E α t α P (ξ) < ∞. (3.7) Proof. Let t ∈ [0, ∞) and α ∈ (α, 2α). By (3.3), (3.4), there exist R 1 > 0 and C > 0 such that for each ξ ∈ R d , |t α P (ξ)| > R 1 , | arg P (ξ)| < π 2 α =⇒ E α t α P (ξ) ≤ C exp tP (ξ) 1/α + C t α |P (ξ)|
(3.8) and

|t α P (ξ)| > R 1 , | arg P (ξ)| ≥ π 2 α =⇒ E α t α P (ξ) ≤ C t α |P (ξ)| .
(3.9)

We will consider four cases. (i) If |t α P (ξ)| > R 1 and | arg P (ξ)| ≥ π 2 α then by (3.9), one has

E α t α P (ξ) ≤ C R 1 . (ii) If |t α P (ξ)| > R 1 and | arg P (ξ)| ∈ [ π 2 α, π 2 α ) then exp tP (ξ) 1/α = exp t|P (ξ)| 1/α cos arg P (ξ) α .
However,

π 2 ≤ arg P (ξ) α ≤ π 2 α α < π, thus exp tP (ξ) 1/α ≤ 1. Going back to (3.8), we get E α t α P (ξ) ≤ C + C R 1 .
(iii) If |t α P (ξ)| ≤ R 1 then by continuity of the Mittag-Leffler function, we may find a constant C(R 1 ) such that

|E α t α P (ξ) | ≤ C(R 1 ).
Hence (3.7) follows from (i), (ii) and (iii). (iv) Finally, if t ∈ [0, T ], |t α P (ξ)| > R 1 and | arg P (ξ)| < π 2 α then, according to (3.5), exp tP (ξ) 1/α = exp tRe P (ξ) 1/α ≤ exp(T M P ).

Hence, with (3.8),

|E α t α P (ξ) | ≤ C exp(T M P ) + C R 1 .
Hence (3.7) follows from (i)-(iv).

It turns out that (3.5) and (3.6) are equivalent if P is continuous. In that case, the following result is the converse of Lemma 3.3. Lemma 3.4. Let P : R d → C be continuous on R d and α ∈ (0, 1]. Suppose that there exists a sequence (ξ n ) n≥0 ⊂ R d such that for each n ≥ 0, one has P (ξ n ) = 0,

| arg P (ξ n )| < π 2 α and Re P (ξ n ) 1/α ---→ n→∞ ∞.
Then for each t > 0, sup

ξ∈R d E α t α P (ξ) = ∞. (3.10)
Proof. We have

exp tP (ξ n ) 1/α = exp tRe P (ξ n ) 1/α ---→ n→∞ ∞.
Moreover,

Re (P (ξ n ) 1/α ) → ∞ implies |P (ξ n )| → ∞.
Thus, with (3.4), we deduce

E α t α P (ξ n ) → ∞.
That proves (3.10).

Remark 3.5. (i) If α = 1 then (3.5) is equivalent to

sup ξ∈R d Re P (ξ) < ∞. (3.11)
That is the usual assumption on symbols of differential operators with constant coefficients.

(ii) If α ∈ (0, 1) then (3.11) implies (3.5). Indeed, for each ξ ∈ R d with P (ξ) = 0, assuming

Re P (ξ) ≤ M and | arg P (ξ)| < π 2 α,
we deduce

|P (ξ)| ≤ M 1 + tan 2 ( π 2 α).
Thus, for some finite constant M P , Re P (ξ) 1/α ≤ M P .

(iii) In general, (3.5) does not imply (3.11): see Example 4.3 below.

In order to obtain a polynomial decay of Fourier multipliers at infinity, we will assume instead of (3.5), that P satisfies the following stronger assumption: for each α in (0, 1], there exist R 0 ≥ 0 and α ∈ (α, 2) such that for each ξ ∈ R d , one has the following implication:

if |P (ξ)| > R 0 then | arg P (ξ)| ≥ π 2 α .
(3.12) Lemma 3.6. Let P : R d → C be continuous on R d , α ∈ (0, 1] and β ∈ R. If (3.12) holds then for all T > 0, there exists a constant C T such that

|E α,β t α P (ξ) | ≤ C T 1 + t α |P (ξ)| , ∀t ∈ [0, T ], ∀ξ ∈ R d . (3.13)
Moreover, if R 0 = 0 in (3.12) then there exists a constant C such that

|E α,β t α P (ξ) | ≤ C 1 + t α |P (ξ)| , ∀t ≥ 0, ∀ξ ∈ R d . (3.14)
Proof. We start to prove (3.13). Let T > 0 be fixed.

If t ∈ [0, T ] and ξ ∈ R d are such that t α |P (ξ)| > R 0 T α then (3.12) implies | arg P (ξ)| ≥ π 2 α .
Then, by (3.3), there exists a positive number R 1,T larger than R 0 T α such that for

t α |P (ξ)| > R 1,T , one has |E α,β t α P (ξ) | ≤ C t α |P (ξ)| . Besides, if t α |P (ξ)| ≤ R 1,T then, for some constant C(R 1,T ), |E α,β t α P (ξ) | ≤ C(R 1,T ).
Combining the two above estimates, we get (3.13) by choosing

C T := max C( 1 R 1,T + 1), (1 + R 1,T )C(R 1,T ) .
If R 0 = 0 in (3.12) then R 1,T can be choosen independently of T . Hence, C T becomes independent of T and (3.14) follows.

4. Mild Solutions and well-posedness on L 2 (R d ) For P : R d → C, a continuous function on R d , we consider the pseudo-differential operator P (D) whose symbol is the function P as defined in Subsection 2.3. The fundamental hypothesis on P is (3.5) i.e. there exists

M P ≥ 0 such that for all ξ ∈ R d with P (ξ) = 0, one has if | arg P (ξ)| < π 2 α then Re P (ξ) 1/α ≤ M P . (4.1) Theorem 4.1. Let α ∈ (0, 1], s ∈ R and P : R d → C be a continuous function on R d . Then for each v ∈ L 2 (R d ), the problem D α s,t u = P (D)u, u(s) = v, (4.2)
has a unique mild solution u. Moreover, u admits the following representation for all t in [s, ∞),

u(t) = F -1 E α (t -s) α P (ξ) v(ξ) , in L 2 (R d ). (4.3)
Proof. Let us start to show that, for each v in D(P ), the function u defined by

u(t) = F -1 E α t α P (ξ) v(ξ) , (4.4) is the unique solution on [0, ∞) to u ∈ C([0, ∞), D(P )) u = v + g α * P (D)u, in C [0, ∞), L 2 (R d ) . (4.5)
The uniqueness can be established as in the proof of [ER17, Theorem 6.1]. Regarding the existence of solution to (4.5), we may prove with Lemma 3.3 that the function u defined by (4.4), belongs to C([0, ∞), D(P )). In order to show that

u = v + g α * P (D)u, in C [0, ∞), L 2 (R d ) ,
we introduce for each ξ ∈ R d , the function w : [0, ∞) → C defined for all t ≥ 0 by

w(t) := F(u(t))(ξ) = E α t α P (ξ) v(ξ).
With (3.1) and Proposition 2.9, we have

w(t) = v(ξ) + g α * P (ξ)w (t), ∀t ≥ 0.
Then the issue is to pass from the above equality which holds in [0, ∞) to an equality in C [0, ∞), L 2 (R d ) . This is done by using the facts that

• w(t) -v(ξ) = F(u(t)) -v (ξ) • g α * P (ξ)w (t) = g α * P F(u(•)) (t)(ξ) • F(u(•)) -v ∈ C [0, ∞), L 2 (R d ) • g α * P F(u(•)) ∈ C [0, ∞), L 2 (R d ) .
Hence we deduce

F(u(•)) -v = g α * P F(u(•)), in C [0, ∞), L 2 (R d ) .
Consequently, u satisfies (4.5). Then Proposition 2.9 implies that the function u defined by (4.4) is the unique strong solution to D α 0,t u = P (D)u, u(0) = v. Moreover, the estimate (3.6) in Lemma 3.3 implies that u satisfies the assumption (2.7) in Proposition 2.15. The later proposition gives existence and uniqueness of the mild solution u to (4.2) on [s, ∞). Finally, the estimate (3.6), (4.4) and the limit process (2.8) yield (4.3).

In the case where P is continuous, the assumption (4.1) is necessary to have a well posed problem in the sense of Definition 2.11. Proof. W.l.o.g. we may assume that s = 0. Let us assume that (4.1) holds. Then Theorem 4.1 yields that (4.2) has a unique mild solution. The estimate for wellposedness follows from (4.3) and (3.6).

Conversely, for any v ∈ F -1 (D(R d )), we obtain by Fourier transform that the mild solution u to (4.2) satisfies for all t > 0,

u(t) := F -1 E α t α P (ξ) v(ξ) , in L 2 (R d ).
By the estimate for well-posedness and Lemma 2.17, we deduce that E α (t α P (•)) is bounded on R d . Then, by Lemma 3.4, we must have (4.1). Recall that for convinence of reading, (4.1) is just a rewriting of (3.5).

Example 4.3. For d = 1, a > 0 and θ ∈ R, let us put

P (ξ) := iae iθ ξ, ∀ξ ∈ R.
Then, in view of Subsection 2.3, the polynomial P is the symbol of the operator P (D) with domain D(P ) = H 1 (R) and defined by

P (D) = iae iθ D = ae iθ ∂ x . We claim that if |θ| ≤ π 2 (1 -α) and α ∈ (0, 1]
then the problem

D α 0,t u = iae iθ Du = ae iθ ∂ x u, u(0) = v ∈ L 2 (R d ), (4.6)
has a unique mild solution u whose Fourier transform satisfies

F u(t) (ξ) = E α t α iae iθ ξ v(ξ), ∀t ≥ 0, ∀ξ ∈ R d .
Indeed, by Theorem 4.1, it is enough to check (4.1). For, we observe that

arg P (ξ) = θ + π 2 if ξ > 0 θ -π 2 if ξ < 0 .
Thus we deduce easily that

| arg P (ξ)| ≥ π 2 α, ∀ξ = 0.
Hence (4.1) holds.

Let us notice that if α = 1 and θ = 0 then we recover the usual transport equation on L 2 (R d ), namely u t = iaDu = a∂ x u.

More generally, if α ∈ (0, 1] and θ = -π 2 (1 -α) then the equation in (4.6) reads

D α 0,t u = ai α Du = ai α-1 ∂ x u. Besides, if α ∈ (0, 1) then Re P (ξ) = a cos π 2 α ξ ---→ ξ→∞ ∞.
Thus (3.11) does not hold. That shows that (4.1) (or (3.5)) and (3.11) are not equivalent-see Remark 3.5 (iii). Also let us notice that, in view of Lemma 2.17, the problem

u t = ai α-1 ∂ x u, u(0) = v.
is not well posed on L 2 (R d ) in the sense of Definition 2.11.

In the standard case α = 1, linear hyperbolic equations like transport equations or Schrödinger equations induce conservation laws. The symbol P of these equations satisfies | arg P (ξ)| = π/2. This is the critical angle since, roughly speaking, if | arg P (ξ)| < π/2 then the problem is generally ill-posed. If | arg P (ξ)| > π/2 then the problem is well-posed, has regularizing effects and dissipative properties i.e. u(t) → 0 as t → ∞. If | arg P (ξ)| = π/2 then the problem is well-posed in L 2 (R d ) but, has no regularizing and dissipative properties; in particular t → u(t) oscillates.

When α ∈ (0, 1], we will highlight that the critical angle is π 2 α. Indeed, we have seen in Theorem 4.2 that this angle characterizes the well posedness. Also, if | arg P (ξ)| = π 2 α, Theorem 4.4 below states that the solution oscillates at infinity, like in the case α = 1.

For simplicity, we will consider w.l.o.g. the problem

D α 0,t u = P (D)u, u(0) = v ∈ L 2 (R d ). (4.7) Theorem 4.4. Let α ∈ (0, 1], v ∈ L 2 (R d )
and assume that (i) P is continuous on R d ;

(ii) for almost every ξ ∈ R d , one has P (ξ) = 0 and

| arg P (ξ)| = π 2 α. (4.8)
Then the mild solution u to (4.7) satisfies

u(t) -w(t) ---→ t→∞ 0 in L 2 (R d ), (4.9)
where w : [0, ∞) → L 2 (R d ) is the function with Fourier transform

F w(t) (ξ) = 1 α exp tP (ξ) 1/α v(ξ), ∀ξ ∈ R d . (4.10)
In particular,

u(t) L 2 (R d ) ---→ t→∞ 1 α v L 2 (R d ) . (4.11)
Let us notice that the function w above is the mild solution to

d dt w = P 1/α (D)w, w(0) = 1 α v.
Also, combining (4.8) and (4.10), we may write

F w(t) (ξ) = 1 α exp ± it|P (ξ)| 1/α v(ξ), ∀ξ ∈ R d .
Proof. According to Theorem 4.1, we know that (4.7) has a unique mild solution given by (4.3) with s = 0. Let us first show that

û(t) -ŵ(t) ---→ t→∞ 0 in L 2 (R d ).
(4.12)

For, let ξ ∈ R d , with P (ξ) = 0. Then

û(t)(ξ) -ŵ(t)(ξ) = E α t α P (ξ) - 1 α exp tP (ξ) 1/α v(ξ) = O 1 t α P (ξ) v(ξ),
according to the asymptotic expansion (3.4) and (4.8). Thus, since P (ξ) = 0 for a.e. ξ ∈ R d , we get û(t) -ŵ(t) ---→ t→∞ 0 a.e. in R d .

Moreover, by estimate (3.7) in Lemma 3.3, there exists a constant C such that

|û(t)(ξ) -ŵ(t)(ξ)| ≤ C|v(ξ)|, ∀t ≥ 0, a.e. ξ ∈ R d .
Then Lebesgue's convergence Theorem implies (4.12); hence (4.9) follows.

In order to prove (4.11), denoting by •, • 2 the inner product of L 2 (R d ), we observe that

u(t) 2 L 2 (R d ) -w(t) 2 L 2 (R d ) = u(t) -w(t), (u(t) -w(t)) + 2w(t) 2 ≤ u(t) -w(t) 2 L 2 (R d ) + 2 u(t) -w(t) L 2 (R d ) w(t) L 2 (R d )
. By (4.9) and w(t

) L 2 (R d ) = 1 α v L 2 (R d )
, we obtain (4.11). In the critical case where | arg P (ξ)| = π 2 α, we cannot expect that the equation in (4.7) possesses a regularizing effect. More precisely, the following result holds. Proposition 4.5. Let α ∈ (0, 1] and assume that (i) P : R d → C is continuous on R d and satisfies (4.1);

(ii) there exists a sequence

(ξ n ) n≥1 ⊂ R d and β ∈ N d such that |ξ β n | → ∞, |P (ξ n )| → ∞ as n → ∞ and |P (ξ n )| = 0, | arg P (ξ n )| = π 2 α, ∀n ≥ 1.
Then for each t ≥ 0, there is no finite constant C = C(t, β) satisfying

D β u(t) L 2 (R d ) ≤ C v L 2 (R d ) , ∀v ∈ F -1 (D(R d )), (4.13)
where u is the mild solution to (4.7) with initial condition v.

Proof. For each t ≥ 0, let us set

a t (ξ) := ξ β E α t α P (ξ) , ∀ξ ∈ R d . Then a t (•) is continuous on R d and, for each v in F -1 (D(R d )), the function ξ → a t (ξ)v(ξ) belongs to L 2 (R d ). Moreover, by (4.3), D β u(t) = F -1 a t (•)v , so that D β u(t) L 2 (R d ) is well defined in (4.13). Let t > 0. Since | arg P (ξ n )| = π 2 α, we have thanks (3.4), a t (ξ n ) = ξ β n α exp tP (ξ n ) 1/α + O 1 t α P (ξ n ) = ξ β n α exp ± it|P (ξ n )| 1/α + O 1 t α P (ξ n ) . Moreover, |P (ξ n )| → ∞, hence |a t (ξ n )| ∼ |ξ β n | α , when n → ∞.
Since |ξ β n | → ∞ by assumption, we obtain sup

ξ∈R d |a t (ξ)| = ∞, ∀t ≥ 0,
and the impossibility of (4.13) follows from Lemma 2.17.

u(t) L 2 (R d ) ---→ t→∞ 1 α v L 2 (R d ) .
This result was originally stated in [START_REF] Naber | Time fractional Schrödinger equation[END_REF]. More precisely, Theorem 4.4 yields

u(t) -w(t) ---→ t→∞ 0 in L 2 (R d ),
where

F w(t) (ξ) = 1 α exp it|ξ| 2/α v(ξ), a.e. ξ ∈ R d .
Moreover, we claim that Equation (4.14) has no regularizing effect, that is, for each β ∈ N d , β = 0, there is no finite constant C such that

D β u(t) L 2 (R d ) ≤ C v L 2 (R d ) , ∀v ∈ F -1 (D(R d )).
Indeed, by setting ξ n := n(1, . . . , 1) ∈ R d , we see that (ξ n ) n≥1 ⊂ R d satisfies the assumption (ii) in Proposition 4.5. Then the claim follows by Proposition 4.5.

The results featured in this example are well known and easily proved in the case α = 1. Thus (4.14) is a suitable generalization of Schrödinger's equation.

Solution operators and regularizing effects

For α ∈ (0, 1], τ ≤ s, a symbol P and v ∈ L 2 (R d ), we will study the solution operator of the following fractional problem

D α τ,t u = P (D)u, u(s) = v.
(5.1)

Our main tool is Fourier multipliers. Regularizing effects are obtained by assuming (3.12). The study of solution operators to (4.5) differs whether s is equal to τ or not. We will start with the simpler case τ = s.

5.1.

The case where τ = s. In Proposition 4.5 below, D 2 denotes the subset of R 2 defined by

D 2 := (t, s) ∈ R 2 | t > s .
(5.2)

Proposition 5.1. Let α ∈ (0, 1] and P be a continuous function on R d satisfying (3.12). Then for t, s in R with t ≥ s, the operator T α (t, s) defined, for each v in L 2 (R d ) by

T α (t, s)v = F -1 E α (t -s) α P (•) v(•) (5.3) belongs to L(L 2 (R d ))
. Moreover, T α is continuous on D 2 with values in L(L 2 (R d ), D(P )), and for each T > 0, there exists a constant C T such that

T α (t, s) L(L 2 (R d )) ≤ C T , ∀ 0 ≤ t -s ≤ T (5.4) P (D)T α (t, s) L(L 2 (R d )) ≤ C T (t -s) α , ∀ 0 < t -s ≤ T. (5.5) Remark 5.2. According to Theorem 4.1, we know that if v ∈ L 2 (R d ) then T α (•, s)v
is the unique mild solution to

D α s,t u = P (D)u, u(s) = v.
(5.6)

For all purposes, we recall that L(L 2 (R d )) denotes the complex Banach space of linear and continuous maps from L 2 (R d ) into itself.

Estimate (5.5) shows that the solution operator T α (t, s) has a regularizing effect. In the case where α = 1 and s = 0, (5.5) holds for (abstract) analytic semi-groups. Let us notice that when P = 0 and α ∈ (0, 1), S α (•) := T α (•, 0) is never a semi-group: see [START_REF] Emamirad | Solution operators of three time variables for fractional linear problems[END_REF]. However, by (5.3), T α is translation invariant for each α ∈ (0, 1] i.e.

T α (t, s) = T α (t -s, 0), ∀t ≥ s.
Proof of Proposition 5.1. Estimate (5.4) is a consequence of Lemma 2.17 and (3.6). Regarding (5.5), we have for v ∈ L 2 (R d ) and 0 < t -s ≤ T ,

P (D)T α (t, s)v L 2 (R d ) = (2π) -d/2 P (ξ)E α (t -s) α P (ξ) v L 2 (R d ) ≤ C T (t -s) α (2π) -d/2 v L 2 (R d ) (by (3.13)) = C T (t -s) α v L 2 (R d ) .

Hence (5.5) follows.

There remains to prove that T α belongs to C(D 2 , L(L 2 (R d ), D(P ))). For, we will first show that P (D)T α lies in C(D 2 , L(L 2 (R d ))). By translation invariance of T α , it is enough to show that

P (D)T α (•, 0) ∈ C ( 1 T , T ), L L 2 (R d ) ,
for each T > 1. For, let t ∈ ( 1 T , T ) and (t n ) n≥0 ⊂ ( 1 T , T ) be a sequence converging toward t. Fourier multiplier Theory yields that

P (D) T α (t, 0) -T α (t n , 0) L(L 2 (R d )) = sup ξ∈R d f ξ (t) -f ξ (t n ) P (ξ) ,
where for each ξ ∈ R d , f ξ denotes the function

f ξ : [0, ∞) → C, t → E α t α P (ξ) .
(5.7)

Thus there remains to prove that

sup ξ∈R d f ξ (t) -f ξ (t n ) P (ξ) ---→ n→∞ 0.
(5.8)

For, by the mean value theorem, we have for each ξ ∈ R d ,

f ξ (t) -f ξ (t n ) P (ξ) ≤ sup τ ∈[ 1 T ,T ] f ξ (τ )P (ξ) |t -t n | ≤ sup τ ∈[ 1 T ,T ] E α,0 τ α P (ξ) P (ξ) τ |t -t n | (by (3.2)) ≤ sup τ ∈[ 1 T ,T ] C T P (ξ) τ 1 + τ α P (ξ) |t -t n | (by (3.13)) ≤ C T T 1+α |t -t n |.
Hence (5.8) follows. Thus

P (D)T α (•, •) lies in C(D 2 , L(L 2 (R d ))
). In the same way, we may prove that

T α (•, •) belongs to C(D 2 , L(L 2 (R d ))
. That completes the proof of the proposition.

5.2.

The case where τ ≤ s. In that case, the Fourier multiplier is

ξ → E α (t -τ ) α P (ξ) E α (s -τ ) α P (ξ) .
(5.9)

In order to control this multiplier, we will need the following assumptions.

P : R d → R is continuous on R d (5.10) P (ξ) ≤ M P for each ξ ∈ R d .
(5.11)

The fact that P is real valued warrants that the denominator in (5.9) does not vanish. Under assumptions (5.10) and (5.11), we deduce from (3.3) the following fundamental estimates about the Mittag-Leffler function: for each α in (0, 1), there exists R 1 > 0 such that for t α P (ξ) ≤ -R 1 , one has

1 √ 2Γ(1 -α) 1 t α |P (ξ)| ≤ E α t α P (ξ) ≤ √ 2 Γ(1 -α) 1 t α |P (ξ)| .
(5.12)

It is essential that α = 1 since the latter inequality is obviously false when α = 1.

The following result gives estimates on the Fourier multiplier (5.9).

Lemma 5.3. Let α ∈ (0, 1). Under assumptions (5.10) and (5.11), one has

sup ξ∈R d E α t α P (ξ) E α s α P (ξ) ≤ C max s t α , 1, E α (t α M P ) , ∀t, s > 0 (5.13) sup ξ∈R d E α t α P (ξ) = E α (t α M P ), ∀t ≥ 0, (5.14)
where the constant C in (5.13) is independent of s and t.

Proof. Equality (5.14) is a straightforward consequence of (5.10), (5.11) and the monotonicity of Mittag-Leffler's function on R. Let s and t be positive and ξ ∈ R d . In order to establish (5.13), we will consider four cases. (i) s α P (ξ) ≤ -R 1 , t α P (ξ) ≤ -R 1 . Then, by (5.12),

E α t α P (ξ) E α s α P (ξ) ≤ 2 s t α . (ii) s α P (ξ) ≤ -R 1 , t α P (ξ) > -R 1 .
Then, by (5.12),

1 E α s α P (ξ) ≤ √ 2Γ(1 -α)s α |P (ξ)|.
(5.15)

Moreover, in the present case, we have -R 1 t -α < P (ξ) and

-R 1 s t α < s α P (ξ) ≤ -R 1 .
Thus, with (5.15),

1 E α s α P (ξ) < √ 2Γ(1 -α)R 1 s t α .
Regarding E α t α P (ξ) , we notice that

s α P (ξ) ≤ -R 1 implies that P (ξ) ≤ 0. Thus E α t α P (ξ) ≤ 1. Then E α t α P (ξ) E α s α P (ξ) ≤ √ 2Γ(1 -α)R 1 s t α . (iii) s α P (ξ) > -R 1 , t α P (ξ) ≤ -R 1 . Then, since E α is increasing on R, E α t α P (ξ) ≤ E α (-R 1 ), E α s α P (ξ) > E α (-R 1 )
.

Thus E α t α P (ξ) E α s α P (ξ) ≤ 1. (iv) s α P (ξ) > -R 1 , t α P (ξ) > -R 1 .
Then, in view of (5.11),

E α t α P (ξ) E α s α P (ξ) ≤ E α t α M P E α (-R 1 )
.

By gathering these four cases, we get (5.13).

By Fourier multipliers Theory and Lemma 5.3, we easily obtain the following result. In Corollary 5.4 below, D 3 is the subset of R 3 defined by

D 3 := (t, s, τ ) ∈ R 3 | t > τ, s > τ ∪ (t, s, s) ∈ R 3 | t ≥ s .
(5.16)

Corollary 5.4. Let α ∈ (0, 1) and P satisfy (5.10), (5.11). Then for (t, s, τ ) in D 3 , the function

ξ → E α (t -τ ) α P (ξ) E α (s -τ ) α P (ξ) is a Fourier multiplier on L 2 (R d ).
Hence the operator T α (t, s, τ ) defined by

T α (t, s, τ )v = F -1 E α (t -τ ) α P (•) E α (s -τ ) α P (•) v in L 2 (R d ), (5.17)
belongs to L(L 2 (R d )). Moreover, for some constant C, one has

T α (t, s, τ ) L(L 2 (R d )) ≤ C max s -τ t -τ α , 1, E α ((t -τ ) α M P ) , ∀ t > τ, s ≥ τ (5.18) T α (t, s, s) L(L 2 (R d )) ≤ E α (t -s) α M P ) , ∀ t ≥ s. (5.19)
For α ∈ (0, 1), let us assume that P satisfies (5.10) and (5.11). Then, with (5.12), we may prove that there exists a constant C > 0 such that for each v in D(P ) and each (t, s, τ ) in D 3 , we have

E α (t -τ ) α P (ξ) E α (s -τ ) α P (ξ) v(ξ) ≤ CE α (t -τ ) α M P (s -τ ) α P (ξ)v(ξ) , ∀ξ ∈ R d .
Thus setting

T (t, s, τ )v = F -1 E α (t -τ ) α P (•) E α (s -τ ) α P (•) v , ∀v ∈ D(P ), (5.20)
we define a linear and continuous operator T (t, s, τ

) from D(P ) into L 2 (R d ). More- over, T (t, s, τ ) L(D(P ),L 2 (R d )) ≤ CE α (t -τ ) α M P (s -τ ) α ,
(5.21) where C is independent of t, s and τ .

Of course, T (t, s, τ )v = T α (t, s, τ )v, for each (t, s, τ ) in D 3 and v in D(P ); we refer to Corollary 5.4 for the definition of T α . Hence

T : D 3 × D(P ) → L 2 (R d ), T α : D 3 × L 2 (R d ) → L 2 (R d ) and T = T α on D 3 × D(P ). Let us also precise that D 3 is the closure of D 3 , that is D 3 = (t, s, τ ) ∈ R 3 | t ≥ τ, s ≥ τ .
The next result states that the operator T α has no continuous extension on D 3 when P (D) is unbounded.

Proposition 5.5. Let us assume that P satisfies (5.10), (5.11) and is unbounded from below. Then, for α ∈ (0, 1) and s > τ ,

T α (t, s, τ ) L(L 2 (R d )) → ∞ as t → τ + . (5.22)
Proof. By translation invariance, it is enough to prove (5.22) for s > 0 and τ = 0.

For any number M greater than |M P | and R 1 s α , where R 1 is the positive constant involved in (5.12), let ξ 0 ∈ R d be such that P (ξ 0 ) = -M (recall that P is continuous and unbounded from below). Let also t 0 > 0 be such that t α 0 M = 1. Then for each t ∈ [0, t 0 ], one has

E α t α P (ξ 0 ) E α s α P (ξ 0 ) ≥ E α (-1) E α -s α M .
Since s α M > R 1 , (5.12) implies that

E α t α P (ξ 0 ) E α s α P (ξ 0 ) ≥ 1 √ 2 Γ(1 -α)E α (-1)s α M. Thus, since s > 0, sup ξ∈R d E α t α P (ξ) E α s α P (ξ) → ∞ as t → 0 + .
Then (5.22) follows by Fourier multiplier Theory (see Lemma 2.17).

The following lemma will be useful in the definition of weak solution to Problem (5.1).

Lemma 5.6. Under assumptions (5.10) and (5.11), let s, τ ∈ R be such that τ ≤ s. Then the operator T (τ, s, τ ) :

D(P ) ⊆ L 2 (R d ) → L 2 (R d ) is self-adjoint.
Proof. In view of the representation (5.20), it is enough to consider the operator T (0, s, 0) for s > 0. Let us first show that the domain of the adjoint operator T (0, s, 0) * is equal to D(P ). For, let ϕ be in D(T (0, s, 0) * ). Then by Definition of D(T (0, s, 0) * ), there exists a constant C such that for each v ∈ D(P ), one has

| T (0, s, 0)v, ϕ 2 | ≤ C v L 2 (R d ) .
(5.23)

In view of (5.20) and Plancherel's identity (2.9), we get

T (0, s, 0)v, ϕ 2 = (2π) -d R d v(ξ) E α s α P (ξ) φ(ξ)dξ.
(5.24)

Since P is real valued, we deduce from these two later relations that φ

E α s α P (•) ∈ L 2 (R d ).
(5.25)

Thus with (5.11), (5.12) and the constant R 1 involved in (5.12), we have

R d |s α P (ξ) φ(ξ)| 2 dξ ≤ [-R 1 ≤s α P (ξ)] max(R 1 , s α M P ) 2 | φ(ξ)| 2 dξ + C [s α P (ξ)<-R 1 ] φ(ξ) E α s α P (ξ) 2 dξ ≤ C(s) φ 2 L 2 (R d ) + C φ E α s α P 2 L 2 (R d ) .
There results that ϕ ∈ D(P ) since s > 0.

Conversely, let ϕ ∈ D(P ). Then, by the monotonicity of E α ,

R d φ(ξ) E α s α P (ξ) 2 dξ ≤ [-R 1 ≤s α P (ξ)] | φ(ξ)| 2 E α (-R 1 ) 2 dξ + C [s α P (ξ)<-R 1 ] |s α P (ξ) φ(ξ)| 2 dξ ≤ 1 E α (-R 1 ) 2 φ 2 L 2 (R d ) + Cs 2α P φ 2 L 2 (R d ) .
Thus (5.25) holds, and then (5.23) too, which yields that ϕ belongs to D(T (0, s, 0) * ).

We have proved that D(T (0, s, 0) * ) = D(P ).

Finally, since P is real valued, (5.24) yields

T (0, s, 0)v, ϕ 2 = v, T (0, s, 0)ϕ 2 , ∀v, ϕ ∈ D(P ).
Then the proof is complete.

6. Regularizing effect for solutions 6.1. The case where τ = s. The problem under consideration becomes

D α s,t u = P (D)u, u(s) = v ∈ L 2 (R d ), (6.1)
where P : R d → C is the symbol of some pseudo-differential operator. In order to get a regularizing effect, we have to strengthen our assumptions on P (see Proposition 4.5). So we will replace (4.1) by (3.12). That will allow us to get that u(t) belongs to D(P ) for t > 0 even if u(0) does not belong to D(P ).

Theorem 6.1. Let α ∈ (0, 1], s ∈ R, v ∈ L 2 (R d ) and assume that P is continuous on R d and satisfies (3.12). Then (6.1) has a unique solution u on [s, ∞), in the sense of Definition 2.12. Moreover, u lies in C((0, ∞), D(P )) and admits the following representation for t ≥ s:

u(t) = F -1 E α (t -s) α P (•) v(•) . (6.2)
Proof. By Proposition 2.13, we may assume w.l.o.g. that s = 0. Existence of solution is proved via a continuity argument. More precisely, let (v n ) n≥0 ⊂ D(P ) be a sequence converging toward v in L 2 (R d ). Recalling the definition of T α given in Proposition 5.1, we set

u := T α (•, 0)v, u n := T α (•, 0)v n , on [0, ∞).
By (5.5),

P (D)u n ---→ n→∞ P (D)u, in L 1 (0, T, L 2 (R d )). (6.3)
By [ER17, Theorem 6.2], we know that u n is the strong solution to

D α 0,t u n = P (D)u n , u n (0) = v n ∈ D(P ). (6.4) Hence u n lies in C([0, T ], L 2 (R d ))
. Besides, Lebesgue's Theorem and (5.4) yield that u belongs to C([0, T ], L 2 (R d )) and

u n → u, in C([0, T ], L 2 (R d )).
Then

g 1-α * (u n -u n (0)) → g 1-α * (u -v), in L 1 (0, T, L 2 (R d ))
. Moreover, by (6.4) and (6.3),

d dt g 1-α * u n -u n (0) = P (D)u n → P (D)u, in L 1 (0, T, L 2 (R d )).
Since the operator d dt with domain W 1,1 (0, T, L 2 (R d )) is closed in L 1 (0, T, L 2 (R d )), the two latter limits yield that g 1-α * (u -v) belongs to W 1,1 (0, T, L 2 (R d )) and

D α s,t u = P (D)u, in L 1 (0, T, L 2 (R d )).
There results that u is solution to (6.1) in the sense of Definition 2.12. Since T α (•, 0) belongs to C (0, ∞), L(L 2 (R d ), D(P )) by Proposition 5.1, we derive easily that u lies in C((0, ∞), D(P )).

Finally, in view of Corollary 2.7, each solution u to (6.1) satisfies

u = v + g α * P (D)u, in L 1 loc ([0, ∞), L 2 (R d ))
. Hence uniqueness is proved as in the case where the initial condition belongs to D(P ), so we refer to the uniqueness part of the proof of Theorem 6.1 in [START_REF] Emamirad | Solution operators of three time variables for fractional linear problems[END_REF] for details. Then P satisfies the assumption (3.12). More precisely, it is geometrically clear that for each α ∈ (1, 2), there exists R 0 = R 0 (α ) such that

|P (ξ)| > R 0 =⇒ | arg P (ξ)| ≥ π 2 α .
Also, if p 0 is a non positive real number then R 0 = 0. Thus Theorem 6.1 yields that Equation (6.5) supplemented with the initial condition u(0) = v ∈ L 2 (R d ) has a unique solution u. Moreover, u belongs to C((0, ∞), H 2β (R d )) and

F u(t) (ξ) = E α t α (-|ξ| 2β + p 0 ) v(ξ), ∀t ≥ 0, a.e. ξ ∈ R d .
If the initial condition is more regular then the corresponding solution gains regularity. For simplicity we will state this property in the case s = 0, that is we will consider the problem D α 0,t u = P (D)u, u(0) = v. (6.6) Theorem 6.3. Let α ∈ (0, 1], and P , Q : R d → C be continuous on R d . Besides, assume that P fufills (3.12). Then for each v in D(Q), the solution u to (6.6) belongs to C (0, ∞), D(P Q) ∩ C [0, ∞), D(Q) and, for each T > 0, there exists a constant C T independent of v such that for all t ∈ (0, T ], one has

P Q(D)u(t) L 2 (R d ) ≤ C T t α Qv L 2 (R d ) . (6.7)
Moreover, if R 0 = 0 in (3.12), that is

| arg P (ξ)| ≥ π 2 α
for each ξ ∈ R d with P (ξ) = 0, then there exists a constant C independent of v such that

P Q(D)u(t) L 2 (R d ) ≤ C t α Qv L 2 (R d ) , ∀t > 0. (6.8)
Proof. By Theorem 6.1, we know that (6.6) has a unique solution u. In particular, u lies in C([0, ∞), L 2 (R d )). Let us show that for each T > 1,

P Q(D)u ∈ C ( 1 T , T ), L L 2 (R d ) , ∀v ∈ D(Q). (6.9)
For, by (3.13), we have

P Q(ξ)E α t α P (ξ) v(ξ) ≤ C T t α |Q(ξ)v(ξ)| (6.10) ≤ C T T α |Q(ξ)v(ξ)|, ∀t ∈ ( 1 T , T ), a.e ξ ∈ R d .
Thus we deduce with Lebesgue's Theorem, that (6.9) holds. Hence u belongs to C((0, ∞), D(P Q)). In the same way, we prove that u lies in C([0, ∞), D(Q)).

Estimate (6.7) is a consequence of (6.10). (6.8) is obtained as (6.7) by using (3.14) instead of (3.13). That completes the proof of the theorem. Example 6.4. Under the notation of Example 6.2, any solution u to (6.5) on [0, ∞) satisfies for each T > 0,

P (D)u(t) L 2 (R d ) ≤ C T t α u(0) L 2 (R d ) , ∀t ∈ (0, T ].
Hence Equation (6.5) has a regularizing effect. Moreover, if p 0 is a non positive real number then R 0 = 0 according to Example 6.2; hence the following dissipative estimate holds.

P (D)u(t) L 2 (R d ) ≤ C t α u(0) L 2 (R d ) , ∀t > 0.
That results from Theorem 6.3.

Example 6.5. Let us consider the fractional transport equation

D α 0,t u = iae iθ Du = ae iθ ∂ x u, u(0) = v ∈ L 2 (R d ).
Under the notation of Example 4.3, let α ∈ (0, 1) and α ∈ (α, 1]. Regarding the angle θ, we assume that

|θ| ≤ π 2 (1 -α ).
Then the symbol P (ξ) := iae iθ ξ satisfies (3.12) with R 0 = 0. Thus Theorem 6.3 yields that

∂ x u(t) L 2 (R d ) ≤ C t α v L 2 (R d )
, ∀t > 0. Thus the above fractional transport equation has a regularizing effect and fullfils a dissipative estimate for α ∈ (0, 1). In particular, by choosing α = 1, we deduce that the equation D α 0,t u = a∂ x u is dissipative and has regularizing effect. That contrasts with the case α = 1, since it is well known that the standard transport equation has no regularizing effect.

The following result states that if the order of derivative is small enough, then the partial derivatives of the solution satisfy some dissipative estimates. Proposition 6.6. Let us assume that (i) α ∈ (0, 1]; P is continuous on R d and satisfies (3.12) with R 0 = 0; Then, if |β| ≤ m, one has

D β u(t) L 2 (R d ) ≤ C t α |β| m v L 2 (R d ) . (6.12) Moreover, if v ∈ D(P ) and m ≤ |β| ≤ 2m then D β u(t) L 2 (R d ) ≤ C t α( |β| m -1) P (D)v L 2 (R d ) , ∀t > 0. ( 6
≤ C t α |P (ξ)| |β| m 1 + t α |P (ξ)| t -α |β| m |v(ξ)| ≤ Ct -α |β| m |v(ξ)| (since |β| ≤ m).
Thus (6.12) follows. In order to prove (6.13), we put m 1 := |β| -m. Then in view of (6.14), we have

|ξ β û(t)| ≤ C |P (ξ)| m 1 m 1 + t α |P (ξ)| |P (ξ)v(ξ)| ≤ Ct -α m 1 m |P (ξ)v(ξ)| (since m 1 ≤ m).
Hence (6.13) follows.

Unlike to the standard case α = 1, purely fractional equations lack of regularizing effect for high order derivatives even if (3.12) holds. Proposition 6.7. Let us assume that (i) α ∈ (0, 1);

(ii) P is continuous on R d and satisfies (3.12); (iii) there exists a sequence

(ξ n ) n≥1 ⊂ R d and β ∈ N d such that |P (ξ n )| → ∞, | |ξ β n | |P (ξ n )| → ∞ as n → ∞.
Then for each time t ≥ 0, there is no finite constant C = C(t) satisfying

D β u(t) L 2 (R d ) ≤ C v L 2 (R d ) , ∀v ∈ F -1 (D(R d )), (6.15)
where u is the solution to (6.6) with initial condition v.

Notice that the left hand side of (6.15) is well defined since, by applying Theorem 6.3 with Q

(ξ) = ξ β , we get that D β u(•) belongs to C([0, ∞), L 2 (R d )), since v ∈ F -1 (D(R d )) ⊂ D(Q).
Proof. It relies on the proof of Proposition 4.5. For each t ≥ 0, let us set

a t (ξ) := ξ β E α t α P (ξ) , ∀ξ ∈ R d .
Then a t (•) is continuous on R d for each t ≥ 0. Thus, by Lemma 2.17, it is enough to prove that a t (•) is unbounded on R d .

If t = 0 then a t (•) is clearly unbounded since β = 0. Thus (6.15) can not hold. If t > 0 then the sequence (ξ n ) n≥0 satisfies by hypothesis

|P (ξ n )| → ∞, |ξ β n | |P (ξ n )| → ∞ as n → ∞.
Thus recalling that P fullfils (3.12), we may use (3.3), to derive

a t (ξ n ) ∼ Cξ β n t α P (ξ n ) → ∞, as n → ∞.
Then a t (•) is unbounded. Hence (6.15) can not hold, which completes the proof of the proposition.

Example 6.8. Let us consider the following time fractional heat equation on R d , D α 0,t u = ∆u, u(0) = v ∈ L 2 (R d ). This equation has a regularizing effect up to order two when α ∈ (0, 1). Indeed, Proposition 6.6 yields that, for all indices i, j in [1, d], one has

∂ x i u(t) L 2 (R d ) ≤ C t α/2 v L 2 (R d ) ∂ x i x j u(t) L 2 (R d ) ≤ C t α v L 2 (R d )
, ∀t > 0. However, there is no regularizing effect at order three. Indeed, for all indices i, j, k in [1, d], Proposition 6.7 with ξ n := n(1, . . . , 1), yields that

sup v∈F -1 (D(R d )), v =0 ∂ x i x j x k u(t) L 2 (R d ) v L 2 (R d ) = ∞.
Remark 6.9. The lack of regularizing effect featured in Proposition 6.7 is specific to time fractional (non integer) derivatives. Indeed, it is well known that the standard heat equation has a regularizing effect at any order and that the following dissipative estimate holds.

D β u(t) L 2 (R d ) ≤ C β t |β|/2 u(0) L 2 (R d ) ,
for all multi-integer β. If v belongs to D(P ) then we may prove that T (•, s, τ )v is the solution to (6.16) on [τ, ∞) in the sense of Definition 2.12 (see the proof of [ER17, Theorem 6.1] and use Lemma 5.3). The issue is to solve (6.16) when v belongs more generally to L 2 (R d ). Of course the expected solution is T α (•, s, τ )v (defined in Corollary 5.4).

Because of the behaviour of the Fourier multiplier (5.9) at high frequencies, solutions to (6.16) have to be understood in a weak sense. In this respect, we adapt to our framework Prüss' Definition of weak solutions (see [Prü93, Definition 1.1 Chap. 1]). Definition 6.10. Let α ∈ (0, 1), s, τ ∈ R with τ ≤ s and P be a symbol satisfying (5.10), (5.11). Let also v ∈ L 2 (R d ) and u be in L 1 loc ([τ, ∞), L 2 (R d )). We say that the function u is a weak solution to (6.16) on [τ, ∞) if u(•), ϕ 2 = v, T (τ, s, τ )ϕ 2 + g α * τ u, P (D)ϕ 2 , (6.17) in L 1 loc ([τ, ∞)), for all ϕ ∈ D(P ). Remark 6.11. In (6.17), T (τ, s, τ ) is defined on D(P ) by (5.20), and in view of (5.21), T (τ, s, τ )ϕ belongs to L 2 (R d ) if ϕ ∈ D(P ). It should be clear that, in general, T (τ, s, τ )ϕ = ϕ provided s > τ .

A priori, the adjoints of P (D) and T (τ, s, τ ) should appear in (6.17). However, these operators are self-adjoint since P is real valued (see Lemma 5.6).

With Corollary 2.7 and Lemma 5.6, we may prove easily that, for each v in D(P ), the solution to (6.16) in the sense of Definition 2.12, namely T (•, s, τ )v is a weak solution to (6.16). Theorem 6.12. Let α ∈ (0, 1), P : R d → R satisfy (5.10) and (5.11). Let also τ ≤ s and v ∈ L 2 (R d ). Then (6.16) has a unique weak solution u. Moreover, u(t) = T α (t, s, τ )v = F -1 E α (t -τ ) α P (•) E α (s -τ ) α P (•) v , a.e t ∈ (τ, ∞), (6.18)

and T α (•, s, τ )v belongs to C((τ, ∞), L 2 (R d )).

Proof. Let us start to prove the existence part. Let T > τ and (v n ) n≥0 ⊂ D(P ) be a sequence converging toward v in L 2 (R d ). By [ER17, Theorem 6.1], the function u n defined for each t > τ by u n (t) := T (t, s, τ )v n = F -1 E α (t -τ ) α P (•) E α (s -τ ) α P (•) vn is the solution to (6.16). Thus, by Remark 6.11, u n is a weak solution to (6.16); that is to say u n , ϕ 2 = v, T (τ, s, τ )ϕ 2 + g α * τ u n , P (D)ϕ 2 , ∀ϕ ∈ D(P ), (

in L 1 (τ, T ). By (5.18), the function u defined by (6.18) belongs to L 1 (τ, T, L 2 (R d )) and Thus we may pass to the limit in (6.19) to get that u is a weak solution.

u n → u in L 1 (τ, T, L 2 (R d )). ( 6
Regarding uniqueness, by linearity it is enough to show that any function u ∈ L 1 (τ, T, L 2 (R d )) satisfying u, ϕ 2 = g α * τ u, P (D)ϕ 2 , ∀ϕ ∈ D(P ), (6.21) vanishes. Since P (D) is self-adjoint, we deduce from (6.21), that g α * τ u lies in L 1 (τ, T, D(P )) and

u = P (D) g α * τ u , in L 1 τ, T, L 2 (R d ) .
Taking the Fourier transform, we deduce û(t, ξ) = P (ξ)g α * τ û(•, ξ)(t) a.e t ∈ [τ, T ], a.e ξ ∈ R d . (6.22)

Since u ∈ L 1 (τ, T, L 2 (R d )), we know that û(•, ξ) lies in L 1 (τ, T ) for each ξ ∈ R d . That fact together with (6.22) allows us to show that u = 0: see [ER17, Proof of Theorem 6.1] for details. Finally, by Lebesgue's Theorem and (5.18), we prove that T α (•, s, τ )v belongs to C((τ, T ], L 2 (R d )). That completes the proof of the Theorem.

3. 1 .

 1 Definition and classical results. Definition 3.1. For α > 0 and β ∈ R, we define the generalised Mittag-Leffler function, E α,β by

Theorem 4. 2 .

 2 Let α ∈ (0, 1], s ∈ R and P : R d → C be a continuous function on R d . Then (4.2) is well posed if and only if (4.1) holds.

Example 4. 6 .

 6 Let us consider the following fractional Schrödinger equation introduced in [Nab04] D α 0,t u = -i α ∆u = -i α P 2 (D)u. (4.14) By setting P (ξ) = -i α P 2 (ξ) = i α |ξ| 2 , we see by Theorem 4.1 that for each v ∈ L 2 (R d ), Equation (4.14) supplemented with the initial condition u(0) = v has a unique mild solution u. Since arg P (ξ) = π 2 α, ∀ξ = 0, Theorem 4.4 gives

Example 6. 2 .

 2 Let α ∈ (0, 1] and p 0 ∈ C. Under the notation of Example 2.16, we set P (ξ) := P 2β (ξ) + p 0 = -|ξ| 2β + p 0 , ∀ξ ∈ R d . We consider the following fractional heat equation D α 0,t u = P (D)u = P 2β (D)u + p 0 u = -(-∆) β u + p 0 u. (6.5)

  (ii) there exist m ∈ N * and c > 0 such that c|ξ| m ≤ |P (ξ)|, ∀ξ ∈ R d ; (6.11) (iii) the initial data v belongs to L 2 (R d ) and u denotes the solution to (6.6); (iv) the multi-integer β = (β 1 , . . . , β d ) ∈ N d has his length denoted by |β|.

6. 2 .

 2 The case where τ ≤ s. The problem under consideration becomesD α τ,t u = P (D)u, u(s) = v ∈ L 2 (R d ).(6.16)

  α * τ u L 1 (τ,T,L 2 (R d )) ≤ g α L 1 (0,T -τ ) u L 1 (τ,T,L 2 (R d )) .

  .13) Proof. Let us prove (6.12). For a.e. ξ ∈ R d ,|ξ β û(t)| ≤ |ξ| |β| |E α t α P (ξ) ||v(ξ)|

	≤ C	|β| m 1 + t α |P (ξ)| |P (ξ)|	|v(ξ)|	(by (6.11) and (3.14))	(6.14)


Kamil K Kolincio

Ramzy Daou

Olivier Pérez

Laurent Guérin

Pierre Fertey

Alain Pautrat

Giant Nernst effect in the incommensurate charge density wave state of P 4 W 12 O 44

Keywords: numbers: 71.45.Lr, 72.15.Jf, 73.20.-r, Keywords: Nernst effect, charge density waves, thermoelectric properties

We report the study of Nernst effect in quasi-low dimensional tungsten bronze P4W12O44 showing a sequence of Peierls instabilities. We demonstrate that both condensation of the electronic carriers in the CDW state and the existence of high-mobility electrons and holes originating from the small pockets remaining in the incompletely nested Fermi surface give rise to a Nernst effect of a magnitude similar to that observed in heavy fermion compounds.

When a longitudinal thermal gradient ∇T x is applied to a solid in a presence of perpendicular magnetic field, it leads to the generation of a transverse electric field E y . This is recognized as the Nernst effect. Until the discovery of non-zero Nernst coefficient N = Ey ∇Tx in the pseudogap phase of high-T c cuprate superconductors (SC) [1], this phenomenon mostly attracted the attention of researchers exploring the motion of the vortices in the mixed state [2]. More recently, much effort has been devoted to elucidate the key parameters which control the magnitude of the Nernst effect. From general arguments based on a semiclassical model of electronic transport, its magnitude is expected to grow as the ratio of electronic mobility over Fermi energy, µ E F [3]. One can then distinguish the two factors favorable for the increase of N : small Fermi energy, or large electronic mobility. Both of these factors: large µ and reduced E F have been found to give a rise to giant features in the Nernst effect in peculiar heavy fermion compounds. The key examples are URu 2 Si 2 [4], CeCoIn 5 [5] and PrFe 4 P 12 [6]. Recently, strong N has also been observed in a number of low dimensional materials as organic Bechgaard salts [7,8], Li 0.9 Mo 6 O 17 [9] or graphene [10,11]. In these cases, the N values were above those expected within a simple diffusive picture of electronic transport.

The charge density wave (CDW) state of a quasi-low dimensional solid shows, to some extent, similarities with dilute anisotropic metals. The Peierls instability is associated to the nesting of the Fermi surface (FS) accompanied by condensation of electronic carriers. While in an ideal 1D material the Fermi surface is expected to be completely destroyed, the nesting in 2D systems is not perfect and small FS pockets containing high mobility carriers can remain [12,13]. Despite these strongly favorable conditions, the Nernst effect was explored only in a few of the CDW materials as NbSe 2 [14], where the Peierls instability involves the nesting of the minor part of the Fermi Surface. In addition to that, in NbSe 2 , the positive signal due to superconducting fluctuations dominates over the component driven by quasiparticles as soon as the temperature approaches T c , similarly to the features found in cuprate superconductors [15,16], where the relevance of CDW formation was recently emphasized [17].

The promising features standing behind the Peierls instability motivated us to explore the Nernst effect in a purely CDW material with strong FS nesting and absence of a superconducting transition. For this study, we have chosen P 4 W 12 O 44 , which belongs to the monophosphate bronze family of quasi-2D metals known to exhibit Peierls instabilities. In P 4 W 12 O 44 , a sequence of three CDW transitions lead to the nesting of the major part of the high temperature Fermi surface [18][19][20] with several pockets left at low temperatures. Those were found to contain light free electronic carriers with the mobility high enough to allow observation of Shubnikov de Haas (SdH) [21] and de Haas-van Alpen [22] oscillations in this material. This predisposes P 4 W 12 O 44 to exhibit large Nernst coefficient.

To characterize the CDW transition, we have performed a high resolution thermal X-ray diffuse scattering experiment on a single crystal of P 4 W 12 O 44 showing an adequate crystalline quality (plate-like single crystals with typical size of 5 mm x 1 mm x 0.5 mm were grown by chemical vapor transport [23]). The experiment was performed at beamline Crystal of synchrotron Soleil, using monochromated radiation with wavelength λ = 0.50718 Å and beam size of 200 µm x 200 µm. Sample was cooled in a stream of cold He gas. Frames were collected above T P 1 = 120 K (temperature expected for the first transition), in between T P 1 and T P 2 = 62 K (temperature expected for the second transition), and below T P 2 ; however it was not possible to decrease the temperature of the sample below T = 36 K and perform a diffraction experiment at temperatures lower than T P 3 30 K. Data were then gathered at T = 293 K, 200 K, 140 K, 100 K, 72 K and 45 K respectively. The different sets of images were treated using Crysalis [24]. For the different temperatures, orientation matrix and cell parameters were calculated and oriented planes were assembled from the experimental frames. At T = 293 K, above T P 1 , P 4 W 12 O 44 exhibits an orthorhombic sym- The positions of diffraction peaks on the experimental diffraction frames collected at T = 72 K and 45 K were extracted using Crysalis and introduced into Jana2006 [25]. A subroutine of this program, allowing a manual cell determination, is useful to extract the average periodicity and superperiodicities. Once the average cell is defined from the intense reflections, all the diffraction peaks are projected in the reciprocal origin cell. For a regular crystal clouds of peaks are expected only at the node of the cell. At T = 72 K, additional clouds of reflections are observed along the a edges of the cell (see figure 3 b); the indexation of these reflections is requiring the introduction of a wave vector determined and refined with the Jana2006 subroutine as follow: q 1 = 0.3825(2)a . The irrational value of the component of q 1 reveals an one dimensional incommensurate modulated structure in this state.

At T = 45 K, the diffraction phenomenon becomes mores complex; additionally to the peaks associated to q 1 , the (hk0) plane (figure 2) exhibits new diffraction features. The application of the Jana2006 procedure, described above, reveals their actual character; this is summarized in the figure 3 c. Two types of diffuse scattering running along c are observed: one is located at a 2 and the other one at 0.26 a + 0.07 b . Supplementary peaks requiring the introduction of a second wave vector q 2 = 0.3099(4)a + 0.2908(3)b are also observed; they are characteristic of the second transition occurring below T P 2 . The full diffraction pattern at T = 45 K is compatible with a two dimensional incommensurate modulated structure where all the satellite reflections can be indexed using q 1 , q 2 and q 1 ± q 2 . q 1 and q 2 are in agreement with wave vectors reported by Foury et al. from X-ray diffuse scattering investigations [26]; the diffuse line observed at T = 45 K and located at 0.26 a + 0.07 b (figure 3 c) could be correlated with the wave vector associated to the third transition expected below 30 K [18,26].

To investigate further the CDW transitions in P 4 W 12 O 44 , we have studied the transport properties of this material. Figure 4 displays the in-plane resistivity measured both in absence, and presence of magnetic field using a standard four-probe method. The anomalies at T P 1 and T P 2 are pronounced in ρ ab as metal -metal transitions, which is typical of the imperfect CDW nesting in a 2D metal. No anomaly is observed at T P 3 = 30 K. This temperature was previously identified from X-ray data [26], thermopower [27] and Hall effect [28]. Above

T P 1 the magnetoresistance (M R = ρ(B)-ρ(B=0) ρ(B=0)
) is negligibly small. Upon cooling below T P 1 , the presence of magnetic field increases the resistance by a few percent. Below T P 2 , M R increases significantly reaching even ≈ 40% at low temperatures (see inset of fig. 4). No significant anomaly is observed at T P 3 . The gradual increase of magnetoresistance upon the transitions at T P 1 and T P 2 illustrates the process of subsequent reduction of the high temperature Fermi surface (fig. 3 a) to the small pockets containing high mobility carriers. The relevance of high mobility (several 10 3 cm 2 V -1 s -1 ) of both electrons and holes remaining in the unnested regions of the Fermi surface has been confirmed by the Hall effect studies of this compound [27,28].

The Nernst effect measurements were conducted in the PPMS cryostat with a custom sample holder and electronics, using the standard one-heater two-thermometer technique. While the sample is glued to the substrate A c c e p t e d m a n u s c r i p t FIG. 3. a) Combined Fermi Surface (FS) of P4W12O44 from [18]; the dashed blue lines show its decomposition into 3 FS and the q1 and q2 nesting vectors are reported. b) Projection into the origin reciprocal cell of all the peaks extracted from the experimental X-ray diffraction frames collected at T = 72 K. The cloud of peaks located on the a edge of the cell requires the q1 vector to be indexed. c) Projection into the origin reciprocal cell of all the peaks extracted from the experimental X-ray diffraction frames collected at T = 45 K. The clouds of peaks associated to q1 and q2 are clearly identified. The segments, purple circled, are diffuse scattering and correspond to the diffraction features drawn with burgundy ellipsoids in the right part of the figure 2. (SrTiO 3 ) with GE varnish to homogenize the thermal gradient, the temperature probes are attached directly to the sample. We are confident that the measured gradients are therefore indicative of those present in the sample, not the substrate. The use of an insulating, nonmagnetic substrate to establish the thermal profile also ensures that the thermal gradient is independent of magnetic field. The magnetic field was applied parallel to c and swept continuously from 9 T to -9 T. The Seebeck and Nernst voltage components, respectively symmetric and antisymmetric with the magnetic field have been separated. Typical results are shown in figure 5.

The Nernst coefficient measured in the (ab) plane, plotted against temperature is shown in fig. 6. For T > T P 1 , N is roughly constant with T and close to zero. Upon cooling below T P 2 , N becomes visibly negative (we use the vortex convention) and its magnitude increases A c c e p t e d m a n u s c r i p t as T decreases. For T < T P 2 one can observe a significant downturn of N (T ), with a small minimum at T ≈ 40 K, close to third CDW transition (T P 3 = 30 K) which is apparently easier to detect with Nernst effect than with resistivity. A contrast between the slight change of N (T ) for T P 2 < T < T P 1 and a steep decrease of that value below T P 2 reflects well the anomalies observed in magnetoresistance and is in agreement with reported Hall data measured in the (ab) plane [28], where the significant increase of negative Hall voltage below T P 2 was attributed to opening of the small electronic pockets in Fermi surface, containing high mobility carriers. Interestingly, a similar behavior was observed in thermopower with ∇T applied in a crystallographic direction [START_REF] Hess | the International Conference on Science and Technology of Synthetic Metals[END_REF].

The relatively sharp change of N (T ) in P 4 W 12 O 44 at T P 2 (and to a lesser extent T P 1 ) contrasts with the behavior seen in another CDW material, quasi-2D NbSe 2 [14], where no sharp feature is detected in N (T ) at T CDW . This is also true of the crossovers seen in the Nernst effect at the pseudogap line of the cuprate superconductors [15]. CDW order seems to be a universal feature of the cuprate phase diagram but it is currently unclear whether it is the relevant order parameter underlying the pseudogap phase [17]. In contrast to recalled examples, P 4 W 12 O 44 does not undergo a superconducting transition. Therefore we observe the Nernst signal of the pristine CDW state, not interrupted by superconducting fluctuations or involved in the SC-CDW competition. Comparing our case to NbSe 2 , one should also emphasize the different impact of nesting properties in both materials. P 4 W 12 O 44 undergoes three Peierls transitions leading to almost complete destruction of the Fermi surface (3 a) accompanied with the condensation of the majority of electronic carriers, while the CDW instability in NbSe 2 leads only to a marginal reconstruction of FS [30]. We also observe, that N in P 4 W 12 O 44 is sublinear with B over the whole temperature range below T P 1 , which was also seen in the NbSe 2 in the CDW state.

The value of ν ≈ 2.4 µVK -1 T -1 obtained by us at T = 15 K and B = 1 T parallels the scale of Nernst effect in heavy fermion compounds as URu 2 Si 2 (maximum of normal quasiparticle signal of 4 µVK -1 T -1 at 4 K) [4], and even exceeds the magnitude of ≈ 1 µVK -1 T -1 reported in CeCoIn 5 at 3 K Our result is also close to the Nernst coefficient found in PrFe 4 P 12 (≈ 3 µVK -1 T -1 ) at a similar temperature. At this point, we would like to mention that we have not observed any sign of saturation of N (T ). We suggest that it is reasonable to assume, that the magnitude of N increases even further at lower temperatures. It was recently shown in URu 2 Si 2 that the low-temperature maximum of N can have a strong sample dependence [31]. While in this case it has been attributed to exotic superconducting fluctuations, this highlights the difficulty in obtaining the correct value of the normal state Nernst signal in the low temperature limit. In our case there is no sign of superconductivity, but other effects such as phonon drag are not well understood and may cause a sample dependent signal at low temperatures [START_REF] Behnia | Fundamentals of thermoelectricity[END_REF].

One approximate approach [START_REF] Behnia | [END_REF] predicts that the values of N follow the scaling law:

N BT = π 2 k B µ 3eT F (1)
where µ is the electronic mobility and T F the Fermi temperature. The value of N BT ≈ 0.13 µVK -2 T -1 at T = 15 K was evaluated using µ from Ref. 27 for relevant temperature and T F estimated from the Fermi wavevector corresponding to a pocket revealed by SdH oscillations in this material [21,22]. Interestingly, the predicted value stands in a very good agreement with the experimental result N BT = 0.15 µVK -2 T -1 . This shows that the magnitude of the Nernst effect in the low temperature state is well accounted for by considering the reduction of the carrier number. The Nernst effect can also be enhanced in well-compensated materials, as was seen in NbSe 2 [14], where the Nernst signal is maximised just as the Hall coefficient changes sign. P 4 W 12 O 44 has also been reported to be a compensated metal in the CDW state with FS containing both electron and hole pockets in the CDW state [20,27,28]. In a perfectly compensated material, the Nernst coefficient can be described with the following expression [34]:

N = 2π 2 3 k 2 B T τ eh 1 (k F l B ) 2 (2)
Inserting the magnetic length l B = h eB and the scattering time τ = lem * k F , one obtains:

N = 2π 2 3 k 2 B T Bl e m * h3 k 3 F (3)
To estimate the mean free path l e , we have fit the T < 50 K part of N (T ) plot at B = 1 T using values of m * = 0.45 times the free electron mass (m 0 ) and k F = 0.64 nm -1 evaluated from Shubnikov -de Haas oscillations reported in this material [22]. We have found the value of l e = 265 Å which compares very well to the quantity of 260 Å estimated from Dingle temperature. Since this model is valid only in the limit of low temperatures, and does not account for phonon effects, this surprisingly good agreement should not be overinterpreted. Nevertheless it is worth mentioning that both values of l e are an order of magnitude greater than the lattice constants a and b.

One shall also mention that, his large mean free path is sufficiently high to observe quantum oscillations in this compound.

In conclusion, we report on the study of Nernst effect in a quasi-2D metal showing charge density wave instabilities. This effect induces the partial destruction of the Fermi surface, and the reduction of the carrier concentration. This results in the electrical compensation due to both electron and hole pockets left in the FS and the high mobility of the carriers originating from them, leading to the enhanced Nernst effect with a magnitude as large as those reported in heavy fermion compounds. We also suggest, that since several other members of the monophosphate bronzes family undergo the Peierls transition above 300 K [35], the search of large Nernst effect at room temperature in these materials appears promising.

Financial support by the French National Research Agency ODACE ANR-Project number ANR-11-BS04-004 is gratefully acknowledged.

A c c e

  p t e d m a n u s c r i p t metry (space group P 2 1 2 1 2 1 ) with lattice parameters: a = 5.3032(8) Å, b = 6.5783(11) Å and c = 23.603(11) Å. Above T P 1 diffuse segments running along a are observed for the position a 2 (figure 1, T = 140 K). At T = 100 K satellite reflections appear from both sides of the diffuse segments; the intensity of these reflections is enhanced when the temperature decreases (figure 1, T = 100 K and T = 72 K). At T = 45 K, below T P 2 , (h0l) plane remains unchanged (see left part of figure 2) but additional diffraction spots are observed on the (hk0) plane (see middle part of figure 2); these spots are drawn using green circles and burgundy ellipsoids in the scheme on the right part of the figure 2.

FIG. 1 . 6 FIG. 2 .

 162 FIG. 1. Region of the (h0l) plane assembled from frames collected at T = 140 K, T = 100 K, T = 72 K. The diffuse scattering observed for the position a 2 weakened noticeably above TP 1 while satellite reflections appear in irrational position.

FIG. 4 .

 4 FIG.4. Resistivity as a function of temperature measured with (red color) and without (blue color) magnetic field applied perpendicularly to the (ab) plane. Inset: Magnetoresistance vs temperature measured at B = 14 T.

FIG. 5 .FIG. 6 .

 56 FIG. 5. Field dependence of Nernst (UN -(a))and Seebeck (US -(b)) signal components extracted from the experimental data.


Laurent Ettouati

Alain Ahond

Christiane Poupat

Pierre Potier

Prof Pierre Potier

Taxane-Type Diterpenes from a New Caledonia endemic member of Taxaceae, Austrotaxus spicata Compton

Austrotaxus spicata Compton is an uncommon austral Taxaceae native to New Caledonia. The study of its leaves and trunk bark has allowed the isolation of 31 taxanes. Most of them are alkaloids owing to the presence of a dimethylaminophenylpropionic acid side-chain.

Contrary to the Taxus species, Austrotaxus has shown the presence of alkaloids in the trunk bark, but no oxetane compounds were found.

THE TAXACEAE FAMILY

Review-In the Old World, the European yew tree (Taxus baccata L.) has been known for a long time, due mainly to the toxicity of its foliage especially for cattle and sometimes for human beings. Therefore, it has been the subject of early phytochemical studies (Lythgoe 1968, Miller 1980). From the European yew tree, a member of the Taxaceae family, nitrogenous taxane diterpenes have been isolated, e.g. taxines A 1 (Graf et al. 1986) and B 2 (Graf et al. 1982, Ettouati et al. 1991a) and, in particular from the Pacific yew tree, Taxus brevifolia Nutt., a well-known antitumoral compound, taxol 3 (Wani et al. 1971, Kingston 1991). All these compounds share the same diterpene skeleton called taxane 4. However, in addition to Taxus, several less common genera belong to the Taxaceae family i.e. Amentotaxus Pilger, Pseudotaxus Cheng, Nothotaxus Florin, Torreya Arnold and finally, Austrotaxus Compton (Cope 1998). Biology of Austrotaxus spicata-In contrast to the first four genera mentioned above which are mainly septentrional, the last one Austrotaxus thrives only in the South hemisphere on the island of New Caledonia. Austrotaxus spicata Compton (De Laubenfelds 1972), the only species of the genus, is an uncommon tree of 3 to 25 meters high with a fibrous red-brown bark and linear leaves.This species grows in the shady forest of the septentrional part of New Caledonia to a height of 500 to 1350 meters.

 Conclusion-So far, only Taxus, Torreya (Li et al. 2003)and Amentotaxus (Chen et al. 2006) had been the subject of phytochemical works, but taxanes had only been isolated from yew trees. Therefore, the close relationship of Austrotaxus spicata with yew tree species prompted us to study this plant.

PHYTOCHEMICAL STUDY OF AUSTROTAXUS (Ettouati et al., 1988(Ettouati et al., , 1989(Ettouati et al., , 1991b) ) Methods-Preliminary tests had shown the presence of alkaloids in Austrotaxus leaves and trunk bark. Accordingly, dried powdered samples of trunk bark and leaves were extracted by dichloromethane after basification (aqueous ammonia). After an acid-base extraction, the crude alkaloidal extract was then separated by a combination of chromatography on silica gel column, preparative TLC, and the structure of the isolated compounds determined by classical analytical methods, mainly from 1 H and 13 C NMR data. Anemonin-Besides taxanes, anemonin 5, a non-alkaloid compound derived from ranunculin, a glycoside mainly known in Renonculaceae (Ruijgrok 1966, Bai et al. 1996), has been found in very large amounts (nine percent of the crude basic extract) in the leaves. This presence is an outstanding example of biochemical parallelism. Classification and biogenetic background-Overall, 31 taxanes have been isolated from Austrotaxus leaves and trunk bark. The majority are alkaloids, owing to the presence of a 3dimethylamino 3-phenylpropionic acid ester 6, also called Winstertein acid, as in the major alkaloid of the European yew tree, taxine B 2. The instability of the yew leaves alkaloidal mixture has been well known for a long time, due to the desamination of this side-chain 6 in basic medium to give cinnamoylated compound (Lythgoe 1968). However, if the presence of alkaloids in yew tree has been known for a century, it is only in 1982 that Graf et al.

established the structure of taxine A 2 and shortly thereafter that of taxine B 2 (Graf et al. 1982, Graf et al. 1986, see also Ettouati et al. 1991a). The same instability has been noticed for several Austrotaxus alkaloids bearing the same Winstertein acid side-chain 6 (e.g. 2'desacetoxyaustrospicatine 12). In 1987, Guéritte-Voegelein, Guénard and Potier proposed a biochemical classification for the known taxanes based on the taxane skeleton substitution at the 4 and/or 5 position : group A with an exomethylene in 4 position, group B with an epoxide in 4 position, group C with an oxetane in 4,5 position (Guéritte-Voegelein et al. 1987). Thus, only taxanes bearing an epoxide or an exomethylene have been isolated from Austrotaxus. A preliminary study looking for the presence of taxol in leaves and trunk bark was not conclusive (Colin 1986). Taxanes from Austrotaxus leaves-19 taxanes have been isolated from the leaves of Austrotaxus spicata; 16 are alkaloids and new (Tables 1, 2 and3). Their alkaloidal nature is due to the presence of a Winstertein acid ester 6, hydroxylated or acetoxylated in 2' position.

The majority of those taxanes, such as austrospicatine 8, the major isolated compound (3.5 percent of the crude alkaloids), belong to the group A of the classification of Guéritte-Voegelein et al.. Another example of this group, austrotaxine 20, as four related compounds, has in addition a ketone in 10 position and one acetoxyle in 14 position. Finally, two taxanes with an epoxide β in 4, spicataxine 28 and nicaustrine 33 bearing a additional nicotinoyl substituent in the 9 position, have also been extracted from the leaves. It can be stressed that it is the first time that nicotinic ester taxanes have been isolated. (Kingston et al. 1982, Shi andKiyota 2005) and Austrotaxus. But one could notice that all taxanes from Austrotaxus have the same skeleton substitutions in the 4 position that taxanes isolated from yew tree species. More important is the difference in taxane distribution throughout the plant. Indeed, contrary to Austrotaxus no alkaloids have been isolated so far from the trunk bark of yew tree species even if alkaloids have been found in stem barks of several Taxus species (Baloglu and Kingston 1999). Conversely, Austrotaxus has not been observed to contain oxetane compounds. For the yew tree species, a transformation of the alkaloids into taxol-type compounds in the leaves, following the biogenetic pathway of Guéritte-Voegelein et al., may take place with subsequent carrying of these secondary metabolites to the trunk bark. Indeed, the content of taxol 3 and related compounds in the leaves is strikingly low compared to the non-esterified 10-desacetylbaccatine III (Chauvière et al. 1981). This might be a detoxification process, oxetane-type taxanes with a phenylisoserine side-chain such as taxol 3 being very powerful antimitotic components.

Taxanes from

CONCLUSION

The isolation of taxanes from Austrotaxus confirms the close botanic relationship between this one and yew tree species in spite of a remote austral habitat. Recent phylogenetic analyses have corroborated these observations (Cheng et al. 2000). However, the structures of Austrotaxus taxanes are limited to non-oxetanic compounds, mainly alkaloidal ones.

Austrotaxus does not seem an interesting source of taxanes for possible use in hemisynthesis of taxol, due to the limited habitat of this species, contrary to the European yew tree ( Ettouati et al. 1991c, Poujol et al. 1997)

  Austrotaxus trunk bark-Overall, 16 taxanes have been found in the trunk bark of Austrotaxus; 12 of them are alkaloidal due to the same Winstertein acid side-chain, as those extracted from the leaves, and belong to groups A and B of the classification of Guéritte-Voegelein et al.. Spicaledonine 25 is unique among those taxanes, in that it contains an intramolecular bond between carbons 3 and 11, as austrocaledonine 26 isolated from the leaves. Comparative distribution of taxanes between yew trees and Austrotaxus-5decinnamoyl-taxinine J 17 and comptonine 19 are the only taxanes shared by yew tree species

.

  • Bai, Y.; Benn, M.H.; Majak, W.; McDiarmid, R. 1996. Extraction and HPLC Determination of Ranunculin in Species of the Buttercup Family. Journal of Taxus Alkaloids. In: Brossi A., ed. The alkaloids. Vol. 39. Chap. 6. San Diego. Academic Press: 195-238. • Chauvière, G.; Guénard D.; Picot, F.; Sénilh, V.; Potier, P. 1981. Analyse structurale et étude biochimique de produits isolés de l'if: Taxus baccata L. (Taxacées). Comptes rendus des Séances de l'Académie des Sciences. 293, série II, (7): 501-503. Ettouati, L.; Ahond, A.; Poupat, C.; Potier, P. 1991c. Première Hémisynthèse d'un composé de Type Taxane porteur d'un Groupement Oxétane en 4(20), 5. Tetrahedron.

	47(47): 9823-9838.
	• Graf, E.; Kirfell, A.; Wolff, G.J.; Breitmaier, E. 1982. Die Aufklärung von Taxin A
	aus Taxus baccata L. Liebigs Annalen der Chemie. (2): 376-381. Agricultural and Food Chemistry. 44(8): 2235-2238. • Baloglu, E.; Kingston, D.G.I. 1999. The Taxane Diterpenoids. Journal of Natural • Graf, E.; Weinandy, S.; Koch, B.; Breitmaier, E. 1986. 13 C-NMR-Untersuchung von
	Products. 62(10): 1448-1472. Taxin B aus Taxus baccata L. Liebigs Annalen der Chemie. (7): 1147-1151.
	• Guéritte-Voegelein, F.; Guénard, D.; Potier, P. 1987. Taxol and derivatives: a biogenetic hypothesis. Journal of Natural Products. 50(1): 9-18. • Kingston, D.G.I.; Hawkins, D.R.; Ovington L. 1982. New Taxanes From Taxus brevifolia. Journal of Natural Products. 45(4): 466-470. • Kingston, D.G.I. 1991. The Chemistry of Taxol. Pharmacology and Therapeutics. 52(1): 1-34. • Wani, M.C.; Taylor, H.L.; Wall, M.E.; Coggon, P.; Mc Phail, A.T. 1971. Plant Antitumor Agents. VI. The Isolation and Structure of Taxol, a Novel Antileukemic • Blechert, S.; Guénard, D. 1990. • Ettouati, L.; Ahond, A.; Poupat, C.; Potier, P. 1991a. Révision structurale de la taxine and Antitumor Agent from Taxus brevifolia. Journal of the American Chemical
	B, alcaloïde majoritaire des feuilles de l'if d'Europe, Taxus baccata L. Journal of Society. 93(9): 2325-2327.
	Natural Products. 54(5): 1455-1458.
	• Ettouati, L.; Ahond, A.; Poupat, C.; Potier, P. 1991b. L'austrocalédonine, nouvel
	alcaloïde extrait d'un conifère endémique de Nouvelle-Calédonie, Austrotaxus

• Chen, H.-L.; Wang, L.-W.; Su, H.J.; Wei, B.-L.; Yang, S.Z.; Lin, C.N. 2006. New Terpenoids of Amentotaxus formosana. Organic Letters. 8(4): 753-756. • Cheng, Y; Nicholson, R.G.; Tripp, K.; Chaw, S.-M. 2000. Phylogeny of Taxaceae and Cephalotaxaceae Genera Inferred from Chloroplast matK Gene and Nuclear rDNA ITS Region. Molecular Phylogenetics and Evolution. 14(3): 353-365 • Colin, M. 1986. Personal communication. Institut de Chimie des Substances Naturelles, C.N.R.S., 91198, Gif-sur-Yvette cedex, France. • Cope, E.A. 1998. Taxaceae : The Genera and Cultivated Species. The Botanical Reviews, 64(4): 291-322. • De Laubenfelds D.J. 1972. Flore de la Nouvelle-Calédonie et dépendances. 4-Gymnospermes. Paris: Muséum National d'Histoire Naturelle: 11-14. • Ettouati, L.; Ahond, A.; Convert, O.; Laurent, D.; Poupat, C.; Potier, P. 1988. Plantes de Nouvelle-Calédonie. 114. Taxanes isolés des feuilles d'Austrotaxus spicata Compton (Taxacées). Bulletin de la Société Chimique de France. série II. (4): 749-755. • Ettouati, L.; Ahond, A.; Convert, O.; Poupat, C.; Potier, P. 1989. Plantes de Nouvelle Calédonie. 124. Taxanes isolés des écorces de tronc d'Austrotaxus spicata Compton (Taxacées). Bulletin de la Société Chimique de France. série II. (5): 687-694. spicata. [Abstract] in: Third Pacific-Asia Symposium on biologically active natural products, Nouméa, New Caledonia; 1991 august 26-30: 45. • • Li, S.-H; Zhang, H.-J.; Nui, X.M.; Yao, P.; Sun, H.-D.; Fong, H.H.S. 2003. Chemical Constituents From Amentotaxus yunnanensis and Torreya yunnanensis. Journal of Natural Products. 66(1): 1002-1005. • Lythgoe, B. 1968. The Taxus alkaloids. In: Manske R.H.F., ed. The alkaloids. Vol. 10. Chap. 14. New York: Academic Press: 597-626. • Miller, R.W. 1980. A Brief Survey of Taxus Alkaloids and other Taxane Derivatives. Journal of Natural Products. 43(4): 425-437. • Poujol, H; Ahond, A.; Al Mourabit, A.; Chiaroni, A.; Poupat, C.; Riche, C.; Potier, P. 1997. Taxoïdes: Nouveaux analogues du 7-déshydroxydocétaxel préparés à partir des alcaloïdes de l'If. Tetrahedron. 53(14): 5169-5184. • Ruijgrok, H.W.L. 1966. Distribution of ranunculine and cyanogenetic compounds in the Renunculaceae. In: Swain T. ed. Comparative Phytochemistry. London: Academic Press : 175-186. • Shi, Q.-W.; Kiyota, H. 2005. New Natural Taxane Diterpenoids from Taxus Species since 1999. Chemistry and Biodiversity. 2: 1597-1623.