F Pellarin

R B Perkins

ON CERTAIN GENERATING FUNCTIONS IN POSITIVE CHARACTERISTIC

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit connection with certain deformations of the Carlitz logarithm introduced by M. Papanikolas and involve the Anderson-Thakur function and the Carlitz exponential function. They collect certain functional identities in families for a new class of L-functions introduced by the first author. This paper also deals with specializations at roots of unity of these generating functions, producing a link with Gauss-Thakur sums. Positive characteristic and Carlitz module and Anderson generating functions and L-series and Special values and Periodic functions

Introduction

Let A be the ring F q [θ] of polynomials in an indeterminate θ with coefficients in the finite field of q elements F q , and let A + ⊆ A denote the subset of monic elements in θ. We denote by K the fraction field of A and by K ∞ := F q ((1/θ)) the completion of K at the infinite place. We also denote by C ∞ the completion of an algebraic closure K ac ∞ of K ∞ ; we endow it with the norm | • | determined by |θ| = q. Let T s be the standard Tate algebra in the indeterminates t 1 , . . . , t s with coefficients in C ∞ . Explicitly, it is the ring of formal series i1,...,is≥0 c i1,...,is t i1

1 • • • t is s ∈ C ∞ [[t 1 , . . . , t s ]]
such that c i1,...,is → 0 as i 1 + • • • + i s → ∞. It can also be viewed as the completion of C ∞ [t 1 , . . . , t s ] for the Gauss norm associated to the absolute value | • | of C ∞ ; see §3 for the precise definition. The Tate algebra T s is endowed with the open and continuous F q [t 1 , . . . , t s ]-linear automorphism τ : T s → T s defined by τ (c) = c q , for c ∈ C ∞ . For each i ∈ Σ s := {1, 2, . . . , s}, let χ ti : A → F q [t i ] ⊆ T s be the F q -algebra morphism determined by θ → t i , and note that much of what follows depends on these choices of algebra generators. Finally, we denote by E s the subalgebra of T s whose elements are entire in the all variables t 1 , . . . , t s . When s = 1, we will write t 1 = t, T 1 = T and E 1 = E. The purpose of this note is to introduce new methods into the study of the properties of the T s -valued functions here (and elsewhere in this article) the primed summation indicates the exclusion of 0 ∈ A from the sum, and second equality follows from the fact that χ t (0) = 0. For all positive integers s, the series ψ s converges in T s for all z ∈ C ∞ \ (A \ {0}), and, in particular, ψ s converges in an open neighborhood of z = 0. These functions are a positive characteristic analog of the Hurwitz zeta function and were introduced by the second author as generating functions, when s is fixed and n ≡ s (mod q -1), for the following multivariate L-functions

L(χ t1 • • • χ ts , n) := a∈A + a -n χ t1 (a) • • • χ ts (a);
see Lemma 20 below. For example, when s = 1, for all z ∈ C ∞ such that |z| < 1, we have ψ 1 (z) = k≥0 z k(q-1) L(χ t , 1 + k(q -1)).

The L-functions above were introduced by the first author in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF] and converge for all positive integers n and s to non-zero elements of T s (K ∞ ), the standard Tate algebra in t 1 , . . . , t s with coefficients in the local field K ∞ . They may be regarded as complementary objects to the global L-functions in equal positive characteristic introduced by Goss (see [START_REF] Pellarin | Interpolating Carlitz Zeta Values (Analytic Number Theory: Arithmetic Properties of Transcendental Functions and their Applications)[END_REF]).

The authors plan to use the functions ψ s in a crucial way in a forth coming work developing the theory of T s -valued vectorial modular forms introduced by the first author in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF]. Indeed, these functions allow us to introduce a notion of regularity at infinity for weak T s -valued vectorial modular forms on the Drinfeld upper-half plane which guarantees the finite rank of the T-modules of such forms of a given weight and type. Additionally, we think that the functions ψ s carry their own intrinsic interest, and we intend to demonstrate this by cataloging some of their finer properties in the sequel.

As a product of our labor, we find a new proof of a basic relation that exists between the L-series value L(χ t , 1) evaluated in t at roots of unity and Gauss-Thakur sums. The precise statement is given in Theorem 4 below.

1.1. Main Functions. Let d 0 = 1 and, for all i ≥ 1, let

d i := i-1 j=0 (θ q i -θ q j ). Define exp C (z) := i≥0 d -1 i z q i and π := -λ q θ i≥1 (1 -θ 1-q i ) -1 ∈ K ∞ (λ θ ),
respectively, the Carlitz exponential function and a fundamental period of the Carlitz exponential, the latter being unique up to the choice of (q -1)-th root λ θ of -θ in the algebraic closure K ac of K. The Carlitz exponential defines an F q -linear endomorphism of C ∞ , and its kernel is πA. Finally, set e C (z) := exp C ( πz). This function is entire, A-periodic, and has as its only zeroes the elements of A with multiplicity one. We consider the following special case of an Anderson generating function for the Carlitz module studied by the first author in [START_REF] Pellarin | Aspects de l'indépendance algébrique en caractéristique non nulle[END_REF],

f t (z) := j≥0 e C (zθ -j-1 )t j = n≥0 ( πz) q n (θ q n -t)d n ∈ T,
where the equality was first established; see also [START_REF] El-Guindy | Identities for Anderson generating functions for Drinfeld modules[END_REF]. From the equality above we see that f t is an entire function of z and a meromorphic function of t with simple poles at t = θ, θ q , . . . . From the definition, we see that, for each z ∈ C ∞ , f t (z) satisfies the τ -difference equation

(1) τ (f t (z)) = e C (z) + (t -θ)f t (z).
In particular, if z = 1, we obtain f t (1) = ω(t), the Anderson-Thakur function, which satisfies

(2)

τ (ω) = (t -θ)ω,
as e C (1) = 0, and generates the rank one

F q [t]-submodule of T of solutions X ∈ T of the equation τ (X) = (t -θ)X.
With this, one readily shows that

ω(t) = λ θ i≥0 1 - t θ q i -1 ∈ T,
as shown in [2, Lemma 2.5.4]; see also [START_REF] Anderson | t-motives[END_REF]Lemma 3.1.4]. The function ω is in many ways a cousin of Euler's gamma function, see e.g. the discussion in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF].

The difference equation ( 1) also allows for a connection with the following function introduced by Papanikolas in [START_REF] Papanikolas | Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms[END_REF]

L α (t) := α + j≥1 α q j (t -θ q )(t -θ q 2 ) • • • (t -θ q j ) .
This function converges in T for each α ∈ C ∞ such that |α| < q q q-1 . The function L α (t) is a deformation of the Carlitz logarithm function; indeed, one may evaluate the variable t at θ to recover the Carlitz logarithm log C (α) = L α (θ) in this way. The function L α (t) plays a crucial role in Papanikolas' proof of a "folklore conjecture:" a K-linearly independent set of Carlitz logarithms of non-zero elements of K ac is algebraically independent over K. Indeed, L α (t) is the main ingredient in the construction of a rigid analytic trivialization of Papanikolas' logarithm motive. The reader may consult [START_REF] Papanikolas | Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms[END_REF] for the exact details of its use.

The following connection made by El-Guindy and Papanikolas [START_REF] El-Guindy | Identities for Anderson generating functions for Drinfeld modules[END_REF] between L α (t) and f t (z) will be of special interest. Let z ∈ C ∞ be such that |z| < 1. Letting α = e C (z), one has an identity of elements of T

(3) L α (t) = (θ -t)f t (z);
note that the right side above converges for all z ∈ C ∞ .

1.1.1. A zeta function realization of Papanikolas' functions. The next result was first obtained in [START_REF] Perkins | Explicit formulae for L-values in positive characteristic[END_REF] by the second author using interpolation polynomials and τ -difference formalism, and we shall give a new proof by using the ultrametric maximum principle. We wish to stress here the connection with the function of Papanikolas described above. The proof of the following result appears in §4.2.

Theorem 1. Fix z ∈ C ∞ such that |z| < 1, and let α = e C (z). The following identity holds in T,

(4) e C (z)ψ 1 (z) = πL α (t) (θ -t)ω(t)
.

As the title of this section suggests, the identity of the previous theorem gives a zeta function realization of the function L α akin to the identity ( 5)

L(χ t , 1) = π (θ -t)ω(t) ,
discovered by the first author in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF], which gave a zeta function realization for the rigid analytic trivialization

Ω := τ (ω) -1 = 1 (t -θ)ω of the dual Carlitz motive. 1.2. Main Results.
1.2.1. Functional identities for ψ s with s arbitrary. Without the restriction on s necessary in [START_REF] Perkins | Explicit formulae for L-values in positive characteristic[END_REF], we shall prove a qualitative generalization of the explicit identities of the second author [15, Theorem 1.1], and, in particular, Theorem 1; see §4.2.

Clearly, the following result may also be stated in terms of Papanikolas' functions, but we leave the explicit details to the reader.

To follow, any empty product shall be understood to equal 1, and for a finite set I, we denote its cardinality by |I|. We will prove the next result in §4.1.

Theorem 2. Let s be a positive integer. For each subset I ⊂ Σ s = {1, 2, . . . , s}, I = Σ s , there exists a polynomial g I ∈ K(t 1 , . . . , t s )[X] of degree at most |Σs\I| q in X, such that g I (e C ) j∈Σs\I ω(t j ) -1 ∈ E s [e C ] and, for all z ∈ C ∞ ,

1 π ω(t 1 ) • • • ω(t s )e C (z)ψ s (z) = i∈Σs f ti (z) + e C (z) I Σs g I (e C (z)) i∈I f ti (z).
We are unable to explicitly describe the polynomials g I guaranteed by Theorem 2 above; we do however give an explicit description of the functions g I (e C ) in the proof of Theorem 2 below. In contrast with the explicit results of the second author [15, Theorem 1.1], we are led to pose the following question. Question 3. Are the polynomials g I (X) of Theorem 2 actually elements of K(t 1 , . . . , t s )?

The entireness, in all variables, of the function g I (e C )/ j∈Σs\I ω(t j ) ∈ E s [e C ] tells us that the poles of the coefficients of g I (X) can only occur at t j = θ q i , for j ∈ Σ s \ I and i ≥ 0. Conceivably, with the methods of [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF], we can give an upper bound on the powers of θ which can occur, but we do not pursue such questions in this article.

Further, we do not yet know if there is a connection of the identity of Theorem 2 above with the theory of t-motives. Rather, our primary interest in proving the theorem above is that it demonstrates how the functional identities of [3, Theorem 1] assemble in families when s is fixed and α varies in the class of s (mod q -1); see Lemma 20. We shall also discuss the growth of the functions ψ s and f t as |z| ℑ := inf κ∈K∞ |z -κ| → ∞; clearly, ψ s tends toward zero, and by Remark 16, f t is unbounded. Thus the identity of Theorem 2 above demonstrates a rich interplay between the functions e C , f t1 , . . . , f ts when |z| ℑ is large.

Finally, we must mention a paper of Gekeler [START_REF] Gekeler | On the de Rham isomorphism for Drinfeld modules[END_REF], wherein he uses functions akin to ψ 1 to establish surjectivity of the de Rham morphism for the cohomology of Drinfeld modules. Given a positive integer h, a lattice Λ ⊆ C ∞ with exponential function e Λ , and an A-linear function X : Λ → C ∞ , Gekeler considers the quasiperiodic functions

F h,X (z) := λ∈Λ X (λ)(e Λ (z)/(z -λ)) q h ,
which can be shown to converge uniformly for all z ∈ C ∞ to F q -linear functions. Further, they interpolate the values of X on Λ and satisfy, for each a ∈ A, the functional identities

F h,X (az) -aF h,X (z) = η a (e Λ (z))
for some η a in the ring of twisted polynomials τ C ∞ {τ }, giving the functions A-linearity upon restriction of z to Λ. Thus, perhaps Theorem 2 above suggests an extension of Gekeler's results for the multilinear pointwise products z →

X 1 (z)X 2 (z) • • • X s (z) of the A-linear functions X 1 , X 2 , . . . , X s : Λ → C ∞ .
We hope to return to this question in the future.

1.2.2. Evaluation at roots of unity. Another basic feature of these multivariate Lseries, and hence the functions ψ s , is the possibility of evaluating the variables t 1 , . . . , t s at roots of unity; see e.g. [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF][START_REF] Anglès | Universal Gauss-Thakur sums and L-series[END_REF][START_REF] Anglès | Arithmetic of positive characteristic L-series values in Tate algebras[END_REF][START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF]. As a highlight of such considerations, detailed below, we recover the following connection, due originally to Anglès and the first author [START_REF] Anglès | Universal Gauss-Thakur sums and L-series[END_REF], between the Anderson-Thakur function ω and Thakur's Gauss sums, from [START_REF] Thakur | Gauss sums for Fq[t][END_REF],

g(χ) := a∈(A/pA) × χ(a) -1 e C (a/p),
where χ(a) denotes the image of a under the F q -algebra map χ determined by θ → ζ for some fixed root ζ of p. The Gauss-Thakur sum g(χ) is non-zero for all such characters χ. Letting ℓ 0 := 1 and ℓ i := (θ -θ q i )ℓ i-1 ∈ A, for all positive integers i, we have the following result whose proof appears in §5.1 below.

Theorem 4. For all ζ ∈ F ac q with minimal polynomial in A of degree d, we have

ω| t=ζ = -χ(ℓ d-1 )g(χ).
It is worth noting that the sign of g(χ) is quickly obtained from the previous theorem, and the absolute value of g(χ) is readily seen from the definition above and will be clear from Cor. 30 below. See [START_REF] Thakur | Behaviour of function field Gauss sums at infinity[END_REF] where such questions are handled in full generality.

In the case of more indeterminates, we let p 1 , . . . , p s be primes (that is, irreducible monic polynomials) of A; we also set m

= p 1 • • • p s . Let us choose, for all i = 1, . . . , s, a root ζ i of p i in F ac q , the algebraic closure of F q in C ∞ . Let ev m : T s → C ∞ be the evaluation map sending a formal series i1,...,is c i1,...,is t i1 1 • • • t is s ∈ T s to i1,...,is c i1,...,is ζ i1 1 • • • ζ is s .
We emphasize the dependence on ζ 1 , . . . , ζ s , rather than on p 1 , . . . , p s .

One easily checks that, for each a ∈ A, the functions ψ s satisfy the following simple transformation rule,

ψ s (z + a) = I⊆Σs b∈A (z -b) -1 i∈I χ ti (b) j∈Σs\I χ tj (a).
Now, while the ψ s themselves are not A-periodic, for any such evaluation map ev m , we have ev m (ψ s (z + ma)) = ev m (ψ s (z)), for all a ∈ A.

Furthermore, letting

(6) u m (z) := 1 π a∈A 1 z -ma = 1 me C (z/m) ,
Proposition 25 below gives

π -1 ev m (ψ s ) ∈ K ac (u m ) ∩ K ac [[u m ]],
and convergence of the formal series holds for z ∈ C ∞ such that |z| ℑ is big enough. Explicitly computing the rational functions above appears to be a difficult task. However, we can say something more when s = 1.

To state the final result we would like to highlight, we define, ad hoc, g(χ -1 ) := (-1) d p/g(χ), which is is the Gauss-Thakur sum for the character χ -1 and is an element of A[ζ, e C (1/p)] by [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF]Prop. 15.2]. For more precision, in the case of a single prime p with root ζ we now write ev ζ for the map ev p described above.

Theorem 5. When |z| ℑ is sufficiently large, we have

p ev ζ (ψ 1 / π) = u |p|(q-1)/q p i≥0 a i u i p ∈ A[ζ, e C (p -1 )][[u p ]],
with

a 0 = (-1) d+1 g(χ -1 )χ(ℓ d-1 ) -1 .
The proof of Theorem 5 appears in §5.1 below.

Acknowledgements. Both authors wish to thank D. Goss for his continued interest in our project and helpful comments on previous versions of this work. We especially thank the anonymous referees for their careful reading and numerous helpful comments. It was their keen eyes which led us to discover a more satisfactory and correct proof of Theorem 2.

Entire functions with values in Banach algebras

Let R be a Chapter 13] for basic definitions and results. We identify

C ∞ -Banach algebra with norm | • | R extending the norm | • | of C ∞ , i.e. such that for all z ∈ C ∞ we have |z • 1| R = |z|; see [16,
C ∞ with a subalgebra of R via C ∞ • 1 ⊂ R. Throughout this section, we assume that |R| R = |C ∞ | = q Q ∪ {0} ( 1 ), that is, the set of norms |f | R for f ∈ R is equal to the set of norms |x|, x ∈ C ∞ . Definition 6. A function f : C ∞ → R is called R-entire (or simply entire) if, on every bounded subset B of C ∞ , f can be obtained as a uniform limit of polynomial functions f i ∈ R[z], f i : B → R.
In particular, for such an R-entire function f and for all z ∈ C ∞ , we can write

f (z) = i≥0 c i z i , c i ∈ R, and lim i→∞ |c i | 1/i R = 0. The identification of C ∞ with C ∞ • 1 ⊂ R allows every C ∞ -
entire function to be identified with an R-entire function.

Remark 7. Similarly, a natural notion of a R-rigid analytic function exists but will be marginal for the purposes of the present paper. Proof. We follow Schikhof's [START_REF] Schikhof | Ultrametric Calculus: An Introduction to p-Adic Analysis[END_REF]Theorems 42.2 and 42.6]. We denote by B 0 (1) the set of the z ∈ C ∞ such that |z| < 1. We note that if

f : B 0 (1) → R is defined by f (x) = i≥0 a i x i for a i ∈ R, then (7) sup{|f (x)| R ; |x| ≤ 1} = max{|a n | R ; n ≥ 0}
(cf. loc. cit. Lemma 42.1). Already, taking x = 1, we see that max{|a n | R ; n ≥ 0} is well defined and equals the supremum of the same set. We may assume that this maximum is one by rescaling f by an element c ∈ C × ∞ (because of our assumption that |R| R = |C ∞ |). We then have:

sup{|f (x)| R ; |x| < 1} ≤ sup{|f (x)| R ; |x| ≤ 1} ≤ max{|a n | R ; n ≥ 0} = 1
and we need to show that we have equalities everywhere. If |a 0 | R = 1, we have |f (0)| R = 1 and we are done. Otherwise, let N be the smallest integer j such that

|a j | R = 1. We have that N > 0. Let us choose ǫ > 0 such that ǫ < 1 - max{|a 0 | R , |a 1 | R , . . . , |a N -1 | R }. Since |C × ∞ | = q Q is dense in the positive real numbers, there exists x ∈ C × ∞ such that 1 -ǫ < |x N | < 1. Then, |f (x)| R = |x N | ≥ 1 -ǫ.
Now let f be R-entire and let us consider r ∈ q Q . We deduce from [START_REF] Gekeler | On the de Rham isomorphism for Drinfeld modules[END_REF] with f replaced by f (ax

) with |a| = r that sup{|f (x)| R ; |x| ≤ r} = max{|a n | R r n ; n ≥ 0}. If there exists M ∈ R ≥0 such that |f (x)| R ≤ M for all x ∈ C ∞ , then, for all r as above, max{|a n | R r n ; n ≥ 0} ≤ M which implies a 1 = a 2 = • • • = 0 and f (x) = a 0 for all x ∈ C ∞ . We choose r ∈ q Q . We denote by B 0 (r) the set {z ∈ C ∞ ; |z| < r}. A function f : B 0 (r) → R is said R-analytic (on B 0 (r)) if there exist c 0 , c 1 , . . . ∈ R such that |c i | R r i → ∞ as i → ∞,
and such that, for all z ∈ B 0 (r), we have the equality

f (z) = i≥0 c i z i . Proposition 9. Let f 1 , f 2 , . . . and f be functions B 0 (r) → R. Let us assume that f 1 , f 2 , . . . are R-analytic on B 0 (r) and that f = lim i→∞ f i uniformly on B 0 (r). Then, f is R-analytic on B 0 (r). In particular, if f 1 , f 2 , . . . and f are functions C ∞ → R such that f 1 , f 2 , . . . are R-entire and such that, for all r > 0, f = lim i→∞ f i uniformly on B 0 (r), then f is R-entire.
Proof. This follows from an easy adaptation of the uniform closedness principle [16, Theorem 42.2, (ii)] that we leave to the reader.

2.2.

A-periodic R-entire functions. We need a few preliminary lemmas.

Lemma 10. Let z be in C ∞ . Then, one of the following two properties holds.

(1) There exists a ∈ A such that |z -a| < 1.

(

) |z| ℑ ≥ 1. 2
Proof. Assume that for all a ∈ A, |z -a| ≥ 1, and let α be an element of K ∞ . Then, we can write (uniquely) α = a + β with a ∈ A and β

∈ 1 θ F q [[ 1 θ ]]. We observe that |β| < 1. Hence, |β| = |z -a| and |z -α| = |z -a -β| = max{|z -a|, |β|} ≥ 1.
In particular, if the property 1 of the Lemma does not hold, then, we have that |z| ℑ ≥ 1.

Lemma 11. Let z be an element of C ∞ .

(1) If there exists a ∈ A such that |z -a| < 1, then |e C (z)| < |z -a|q q q-1 .

(2) There exists κ 0 > 1, independent of z, with the following property. If

|z| ℑ ≥ 1, then q -κ0 ≤ |e C (z)| ≤ q -1 .
Proof. If there exists a ∈ A such that |z -a| < 1, the conclusion is easily obtained as it is well known from its Newton polygon that exp

C : C ∞ → C ∞ induces an isometric F q -automorphism of the disk {z ′ ∈ C ∞ ; |z ′ | < | π|}; recall that | π| = q q q-1 .
The second case of the lemma follows easily from [8, Proposition 9].

Let P ! (R) be the R-algebra of R-entire functions which are A-periodic, that is, the R-algebra of R-entire functions f such that f (z + a) = f (z) for all z ∈ C ∞ and a ∈ A.
The following result will be used in the proof of Theorem 2 below. We continue with the assumption |R| R = |C ∞ |.

Proposition 12. Let f ∈ P ! (R) be such that there exist an integer N and a real number

C > 0 with |f (z)| R ≤ C max{1, |e C (z)| N } for all z ∈ C ∞ . Then, f is a polynomial of R[e C ] of degree at most N .
Proof. We recall that the function e C : C ∞ → R, is R-entire, A-periodic, and vanishes at z = a with order one for all a ∈ A and has no other zeroes. This implies that there exist elements c 0 , . . . , c N of R such that the function g :

C ∞ → R g = f - N i=0 c i e i C e N C is R-entire and A-periodic. We set C 1 = sup |z|<1 |g(z)|. Let z ∈ C ∞ . If there exists a ∈ A such that |z -a| < 1, then |g(z)| R < 1. Otherwise, Lemma 10 implies that |z| ℑ ≥ 1. From Lemma 11, we then deduce |g(z)| R ≤ q c0N max{1, q -N , |c 0 | R , |c 0 | R q -1 , . . . , |c N | R q -N } =: C 2 . For all z ∈ C ∞ , |g(z)| R ≤ max{C 1 , C 2 } and |g| R is bounded over C ∞ ,
∈ C ∞ , |f (z)| R ≤ C max{1, |e C (z)|} N .
We denote by T (R) the R-algebra of tempered R-entire functions. We also set

P(R) = P ! (R) ∩ T (R). Remark 14. Note that by Proposition 12, if |R| R = |C ∞ |, then P(R) = R[e C ].
Lemma 18. For z ∈ C ∞ , we have χ t (z) = 0 if and only if z = 0.

Proof. We denote by e the unique F q [t]-linear endomorphism of T extending the Carlitz exponential function exp C : C ∞ → C ∞ . Following [START_REF] Anglès | Universal Gauss-Thakur sums and L-series[END_REF], this yields a short exact sequence of A[t]-modules

0 → πA[t] → T → C(T) → 0, where C(T) is the unique A[t]-module structure over T extending F q [t]-linearly the Carlitz module θ → θ + τ over C ∞ . If we set f (h) = e( πh), then the kernel of f is A[t] and χ t (z) = f z θ-t f 1 θ-t , z ∈ C ∞ , from which we easily deduce that χ t (z) = 0 if and only if z = 0, if z ∈ C ∞ . Remark 19. The function z → χt(z) z is a non-constant E-entire function C ∞ → E which never vanishes.

The functions ψ s and the proof of Theorem 2

Before giving the proof of Theorem 2, we require three auxiliary results.

Lemma 20. Let z ∈ C ∞ be such that |z| < 1. Let s be a positive integer, and let m be the unique integer 0 < m ≤ q -1 such that m ≡ s (mod q -1). As power series in

( πz) m ω(t1)•••ω(ts) K(t 1 , . . . , t s )[[ πz]], we have zψ s (z) = k≥0 ( πz) m+k(q-1) L(χ t1 • • • χ ts , m + k(q -1))
π m+k(q-1) .

Proof. We have

ψ s (z) = a∈A\{0} χt 1 (a)•••χt s (a) z-a
. Since for a ∈ A \ {0} and |z| < 1, we have 1 z-a = -1 a i≥0 z a i , we deduce

ψ s (z) = - a∈A\{0} a -1 χ t1 (a) • • • χ ts (a) i≥0 z a i = - i≥0 z i λ∈F × q λ s-i-1 a∈A + χ t1 (a) • • • χ ts (a) a i+1 = k≥0 z m-1+k(q-1) L(χ t1 • • • χ ts , m + k(q -1)).
Multiplying by z and applying [3, Theorem 1] finishes the proof.

Remark 21. We observe that for s = 0, the previous lemma reduces to the wellknown identity of Carlitz z a∈A (z -a) -1 = k≥0 z k(q-1) ζ C (k(q -1)), where, for positive integers n, ζ C (n) := a∈A + a -n are the Carlitz zeta values.

Lemma 22. The functions e C , χ t1 , . . . , χ ts are algebraically independent over the field C ∞ ((t 1 , . . . , t s )).

Proof. We set K = C ∞ ((t 1 , . . . , t s )) and L = K((e C )). We choose the canonical embedding of T s → K.

We claim that χ t1 ,..., χ ts are algebraically independent over L. Let us suppose, by contradiction, that there exists P ∈ L[X 1 , . . . , X s ], irreducible, such that P (χ t1 , . . . , χ ts ) = 0, let P be the prime ideal of L[X 1 , . . . , X s ] generated by P . Then, for all a ∈ A, P a = P (X 1 + χ t1 (a), . . . , X s + χ ts (a)) ∈ P, because χ t (z + a) = χ t (z) + χ t (a), for all z ∈ C ∞ and a ∈ A.

Hence, P a = HP for some H ∈ L[X 1 , . . . , X s ] \ {0} (P a is clearly non-zero). The total degrees of P and P a for a ∈ A are equal; then H ∈ L × . The constant term of P a is the polynomial P (χ t1 (a), . . . , χ ts (a)). If R is the constant term of P , we thus have the identity: P (χ t1 (a), . . . , χ ts (a)) = HR, a ∈ A.

But the image of the map A → A s (L) defined by a → (χ t1 (a), . . . , χ ts (a)) is Zariskidense. Hence, P is non-zero and constant, a contradiction. In particular, the functions χ t1 (z), . . . , χ ts (z) are algebraically independent over K(e C ) which is a transcendental extension of K.

Remark 23. In fact, the functions of the previous lemma are even algebraically independent over C ∞ ((t 1 , . . . , t s ))(z), but we do not need this property.

Proposition 24. Let K be a field, let us consider formal series f 0 , f 1 , . . . , f s ∈ K[[Z]], let L/K be a field extension and let us suppose that the series f i are algebraically independent over L. Let P be a polynomial of L[X 0 , . . . , X s ] such that

P (f 0 , . . . , f s ) ∈ K[[Z]]. Then, P ∈ K[X 0 , . . . , X s ].
Proof. Let P ∈ L[X 0 , . . . , X s ] be such that P (f 0 , . . . , f s ) ∈ K[[Z]], and let d be its degree. Since the functions f 0 , . . . , f s are algebraically independent over L, the monic monomials M 0 , . . . , M N in f 0 , . . . , f s of degree ≤ d are linearly independent over L and there is an L-rational linear combination

F = i a i M i ∈ K[[Z]].
The matrix U whose columns are the coefficients of the formal series

M i ∈ K[[Z]]
(it has N + 1 columns and infinitely many rows) has maximal rank and the right multiplication by the column matrix with coefficients a i yields an infinite column of elements of K which cannot be identically zero. Extracting from U a non-singular square matrix of order N + 1 and inverting it, we deduce that the coefficients a 0 , . . . , a N are in K.

4.1.

Proof of Theorem 2. Recall that we have set Σ s := {1, 2, . . . , s}. For subsets I ⊆ Σ s we define χ I (z) := i∈I χ ti (z), and, as before, χ ∅ (z) = 1. For I ⊆ Σ s , we recall that |I| denotes the cardinality of I.

We have

ψ s (z) = a∈A χ 1 (a -z + z) • • • χ s (a -z + z) z -a = πu(z)χ Σs (z) + I⊂Σs χ I (z) a∈A χ I c (a -z) z -a ,
where I ⊂ Σ s runs over all strict subsets of Σ s , and I c := Σ s \ I.

We argue that the sums h I (z) :=

a∈A χ I c (a-z) z-a
, for strict subsets I ⊂ Σ s , define E s -entire, A-periodic, tempered functions. Let r ∈ q Q , and let z ∈ B 0 (r) := {z ∈ C ∞ : |z| < r}. Consider a ∈ A. If |z -a| ≥ q, by Lemma 15, we have [START_REF] El-Guindy | Identities for Anderson generating functions for Drinfeld modules[END_REF] χ

I c (a -z) z -a ≤ max{1, sup z∈B0(r) |e C (z)| |I c | q } |z -a| ,
and the right side tends to zero as |a| → ∞. If |z -a| < q, by Lemma 17, ( 10)

χ I c (a -z) z -a = |z -a| |I c |-1 .
The singularity of χ I c (z-a)

z-a at z = a is clearly removable, and thus this function is E s -entire for all a ∈ A. It follows from Proposition 9 that h I is E s -entire, and clearly h I is A-periodic. Temperedness follows from ( 9) and ( 10) giving, for some positive real number C,

a∈A χ I c (a -z) z -a ≤ C max{1, |e C (z)| |I c | q }.
Thus, for all

I ⊂ Σ s , h I ∈ E s [e C ] of degree in e C at most |I c |/q. For I Σ s , let g I ∈ T s [Z] be such that g I (e C ) π -1 h I j∈I c ω(t j ). It remains to show that g I (e C ) ∈ K(t 1 , . . . , t s )[e C ].
Multiplying out the numerators of the functions h I , this follows easily from Lemma 20 and Proposition 24 applied to the functions e C , f t1 , . . ., f ts which are algebraically independent by Lemma 22 and have their expansions in powers of πz defined over K(t 1 , . . . , t s ).

4.2.

Proof of Theorem 1. Now we may deduce Theorem 1 from Theorem 2 and Lemma 15. By Theorem 2 we have

ψ 1 = πuχ t + h ∅ ,
for some h ∅ ∈ T. By Lemma 15, we see that ψ 1 -πuχ t → 0 as |z| ℑ → ∞. Hence, h ∅ = 0. We make the connection with Papanikolas' function via [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF], and this puts the constraint on z.

Evaluations at roots of unity

Throughout this section, s will be a fixed positive integer, and we retain the notation Σ s = {1, 2, . . . , s}. We recall our notation from the introduction: we let p 1 , . . . , p s be primes (that is, irreducible monic polynomials) of A; we also set

m = p 1 • • • p s . Let us choose, for all i = 1, . . . , s, a root ζ i of p i in F ac q , the algebraic closure of F q in C ∞ . Let ev m : T s → C ∞ be the evaluation map sending a formal series i1,...,is c i1,...,is t i1 1 • • • t is s ∈ T s to i1,...,is c i1,...,is ζ i1 1 • • • ζ is s .
For j ≥ 0, let A(j) be the F q -vector subspace of A consisting of all polynomials of degree strictly less than j; of course, A(0) = {0}. Let Z be an indeterminate over C ∞ , and let K{Z} denote the non-commutative ring of F q -linear polynomials with coefficients in K, with composition as product. For j ≥ 0, let E j (Z) := 1 dj a∈A(j) (Z -a) ∈ K{Z}; see [START_REF] Goss | Basic Structures of Function Field Arithmetic[END_REF]Theorem 3.1.5]. For a ∈ A, let C a (Z) := j≥0 E j (a)Z q j . These are the polynomials arising from the F q -linear map A → A{Z} determined by θ → θZ + Z q , by [10, Corollaries 3.5.3 and 3.5.4].

For each subset J ⊂ Σ s define

M J m (Z) := b∈A/mA C m (Z -e C (b/m)) (Z -e C (b/m)) j∈J b(ζ j ) ∈ A[ζ 1 , . . . , ζ s , e C (m -1 )][Z].
Observe that the degree in Z of M J m (Z) is strictly less than |m| and that M J m is the Lagrange interpolation polynomial for the data e C (a/m) → m j∈J a(ζ j ), defined for a ∈ A/mA. For J = Σ s we simply write M Σs m = M m . Proposition 25. For all z ∈ C ∞ \ A, square-free monic polynomials m and subsets J ⊆ Σ s , the following identity holds,

π -1 ev m (ψ J (z)) = 1 m M J m (e C (z/m)) C m (e C (z/m)) ∈ K(ζ 1 , . . . , ζ s , e C (m -1 ))(u m (z)),
and when |z| ℑ is sufficiently large, we have

π -1 ev m (ψ J )(z) ∈ u m (z)A[ζ 1 , . . . , ζ s , e C (m -1 )][[u m (z)]].
Proof. We have

ev m (ψ J )(z) = b∈A (z -b) -1 j∈J b(ζ j ) = b∈A/mA j∈J b(ζ j ) a∈mA (z -b -a) -1 = π b∈A/mA u m (z -b) j∈J b(ζ j ) = π m b∈A/mA j∈J b(ζ j ) e C (z/m) -e C (b/m) . ( 11
)
From the third equality to the fourth, we have used the identity [START_REF] Carlitz | On certain functions connected with polynomials in a Galois field[END_REF], and the final identity comes from direct comparison with the definition of M J m combined with the observation that C m (Z -e C (b/m)) = C m (Z), for all b ∈ A/mA.

Returning to [START_REF] Papanikolas | Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms[END_REF], and assuming that |z| ℑ is sufficiently large so that, for all b ∈ A/mA we have |me C (b/m)u m | < 1, the second claim follows from the computation

ev m (ψ J / π) = b∈A/mA u m j∈J b(ζ j ) 1 -me C ( b m )u m = k≥0 u k+1 m b∈A/mA (me C ( b m )) k j∈J b(ζ j ). ( 12
)
Corollary 26. We suppose that s = 1. For every prime m = p and all z ∈ C ∞ , we have ev p (χ t (z)) = p -1 M p (e C (z/p)).

Proof. This is immediate from Theorem 1 and the previous proposition.

One would like to describe the precise order of vanishing in u m of the evaluations ev m (ψ s ) for all square-free conductors m and positive integers s, or equivalently to determine the degree in Z of M m (Z). This appears to be difficult in general, but it is possible in the case of s = 1. We tackle this problem in the next section.

We quickly mention that we may give an upper bound on the order of vanishing of ev p (ψ 1 ) analytically using Remark 16 and Theorem 1. Starting from the identity ψ 1 = πuχ t , for all z such that |z| ℑ is big enough, we have ψ 1 = C|u| 1-1/q , for some positive real constant C. Hence, using the characterization of the Gauss norm f = sup |t|≤1 |f (t)|, for p a prime, we obtain | ev p (ψ 1 )| ≤ C|u p | |p| q (q-1) , for a new constant C provided that |z| ℑ is big enough. We now know that ev p (ψ

1 ) ∈ C ∞ [[u p ]],
and it follows that ev p (ψ 1 ) ∈ u

|p| q (q-1) p C ∞ [[u p ]].
5.1. Gauss-Thakur sums and the polynomial M p . We now focus our attention completely on the interpolation polynomials M p in order to prove Theorems 4 and 5 of the introduction.

Let p be a monic irreducible of degree d, and let λ = e C (p -1 ) be a fixed element of Carlitz p-torsion. Choose a root ζ of p, and let χ be the Teichmüller character determined by θ → ζ.

The following Gauss sum was first introduced by Thakur in [START_REF] Thakur | Gauss sums for Fq[t][END_REF] and already appeared in the introduction,

g(χ) := ′ a∈A(d) χ(a) -1 C a (λ).
The sum g(χ) is a non-zero element of A[ζ, λ]. We recall setting g(χ -1 ) := (-1) d p/g(χ), which is the Gauss-Thakur sum for the character χ -1 and is an element of A[ζ, λ] by [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF]Prop. 15.2]. We also recall that for each a ∈ (A/pA) × , there is a unique element σ a of the Galois group of K(ζ, λ) over K(ζ) satisfying σ a (λ) = C a (λ) = e C (a/p) and that g(χ) satisfies [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF] σ a (g(χ)) = a(ζ)g(χ)

for all a ∈ (A/pA) × .

Lemma 27. The following identity holds in g(χ -1 )A

[ζ][Z], ( 14
) M p (Z) = (-1) d g(χ -1 ) d-1 j=0 Z q j a∈A(d)\A(j) a(ζ) -1 E j (a).

Proof. Define the coefficients a

i ∈ A[ζ] via g(χ) := |p|-2
i=0 a i λ i ; one easily sees here that only those indices i ≡ 1 (mod q -1) can appear. We observe immediately from (13) that the polynomial g(χ) -1 i≥0 a i Z i takes the value a(ζ) for Z = C a (λ) for all a ∈ A(d) and takes the value zero at Z = 0. Further its degree is strictly bounded above by

|A(d)| = |p|. Since p -1 M p (Z) is the Lagrange interpolation polynomial of the data C a (λ) → a(ζ) for a ∈ A(d), we must have p -1 M p (Z) = g(χ) -1 |p|-2
i=0 a i Z i , by consideration on the degrees in Z and uniqueness of the Lagrange interpolating polynomial. Replacing g(χ) -1 with (-1) d p -1 g(χ -1 ), we obtain M p (Z) = (-1) d g(χ -1 ) |p|-2 i=0 a i Z i . Now we derive an expression for the coefficients a i appearing above. From the definition of g(χ), we immediately obtain

g(χ) = ′ a∈A(d) a(ζ) -1 d-1 j=0 E j (a)λ q j = d-1 j=0 λ q j ′ a∈A(d) a(ζ) -1 E j (a).
Remark 28. It is an interesting question to determine the F q -vector subspace of C ∞ of roots to the polynomial M p (Z).

We can give a more compact expression for the coefficients of Z and Z |p|/q in M p . We require a lemma, and some additional notation.

Let x and y be indeterminates over F q and write a(x) ∈ F q [x], resp. a(y) ∈ F q [y], for the image of a ∈ A under θ → x, resp. θ → y. We recall that for j ≥ 1, we have set ℓ j := (θ -θ q j )(θ -θ q j-1 ) • • • (θ -θ q ) ∈ A and ℓ 0 := 1.

Lemma 29. For all integers d ≥ 1, we have

d-1 j=0 ℓ j (x) -1 j-1 k=0 (y -x q k ) = ℓ d-1 (x) -1 d-1 j=1 (y -x q j ).
Proof. We prove this claim by induction on d. When d = 1, we have two empty products, both equal to 1 by convention. For d = 2, we have 1 + (y -x)

x -x q = (y -x q )

x -x q , as required. Now assume the result holds for some d ≥ 2. For such d we have

d j=0 ℓ j (x) -1 j-1 k=0 (y -x q k ) = 1 ℓ d (x) d-1 k=0 (y -x q k ) + d-1 j=0 1 ℓ j (x) j-1 k=0 (y -x q k ) = 1 ℓ d (x) d-1 k=0 (y -x q k ) + 1 ℓ d-1 (x) d-1 j=1 (y -x q j ) = 1 ℓ d-1 (x) ( y -x x -x q d + 1) d-1 j=1 (y -x q j ) = 1 ℓ d (x) d j=1 (y -x q j ).
Hence, the result holds for d + 1 as well and proves the claim.

Corollary 30. The coefficient of Z |p|/q in M p equals (-1) d+1 g(χ -1 )χ(ℓ d-1 ) -1 , and the coefficient of Z equals (-1)

d+1 pχ(ℓ d-1 ) -1 (θ -ζ) -1 g(χ -1 ); both are elements of A[ζ, λ].
Proof. For non-negative integers j, let A + (j) denote the set of all elements of A + of degree j in θ.

From ( 14), the coefficient of Z |p|/q equals (-1) d+1 g(χ -1 )

a∈A + (d-1) a(ζ) -1 = (-1) d+1 g(χ -1 )χ(ℓ d-1 ) -1 ;
the latter equality follows from the work of Carlitz [6, (9.09)].

Similarly, the coefficient of Z equals (-1) d+1 g(χ -1 )

d-1 j=0 a∈A + (j) a(ζ) -1 a.
One may easily calculate that

a∈A + (j) a(y) a(x) = 1 ℓ j (x) j-1 k=0 (y -x q k ),
by comparing zeros of the left and right hand sides above using [6, (9.15)] and the next displayed equation therein and comparing the coefficients of y j , which is again given by [6, (9.09)].

Summing over 0 ≤ j ≤ d -1, applying Lemma 29, and replacing x by ζ and y by θ, we obtain the expression above for the coefficient of Z.

5.1.1. Proof of Theorem 4. It suffices to compare the coefficients of z on both the left and right sides of the identity of Cor. 26. On the left side we have π ev ζ ((θt)ω(t)) -1 , while on the right we have (-1) d+1 πp -1 (θ -ζ) -1 χ(ℓ d-1 ) -1 g(χ -1 ), by Cor. 30. Hence, using the definition of g(χ -1 ) = (-1) d pg(χ) -1 , we obtain ev ζ (ω) = -χ(ℓ d-1 )g(χ), as desired. 5.1.2. Proof of Theorem 5. This is an immediate consequence of Prop. 25 and Cor. 30 just above.

Remark 31. Let p, d and χ be as in Theorem 5. From [START_REF] Pellarin | Aspects de l'indépendance algébrique en caractéristique non nulle[END_REF] and Theorem 5, one may deduce the vanishing of the sums b∈(A/pA) × e C (b/p) k χ(b), for k = 0, 1, , . . . , |p|(q-1) q -2, and the identity g(χ -1 ) = (-1) d+1 p |p|(q-1) q χ(ℓ d-1 )

b∈(A/pA) × e C (b/p) |p|(q-1) q -1 χ(b).

We leave the easy details to the reader.

5.2. Questions in closing. We conclude with a speculative subsection. Let us fix an integer s ≥ 1. As Theorem 2 shows, the functions ψ s are elements of the T s -algebra R s := T s [u, u -1 , {χ J } J⊆Σs ]. It follows from Corollary 26 that for all monic m = p 1 • • • p s , as above, and all h ∈ R s , we have ev m (h) ∈ C ∞ (u m ) for every choice of roots ζ i of p i , as we saw for ψ s . We define the sub-T s -algebra R 0 s ⊆ R s whose elements are the h ∈ R s such that ev m (h) ∈ C ∞ [[u m ]] for all p 1 , . . . , p s and all roots ζ i of p i . As proven in Proposition 25, we have ψ J ∈ R 0 s , for all J ⊆ Σ s . We denote by T s,i the completion of C ∞ [t 1 , . . . , t i , . . . , t s ] ⊂ T s for the Gauss norm (the hat denotes an omitted term and the completion is canonically identified to a subring of T s ). We also denote by ϕ i the T s,i -linear endomorphism of T s uniquely defined by ϕ i (t i ) = t q i .

Lemma 32. The ring R 0 s is ϕ i -stable for all i. That is, for all f ∈ R 0 s and all i = 1, . . . , s, we have ϕ i (f ) ∈ R 0 s .

Proof. We know that for any choice of ζ 1 , . . . , ζ s ∈ F ac q , for f ∈ R 0 s , we have

f | ti=ζi ∈ C ∞ [[u m ]]
, where p i is the prime of A vanishing at ζ i (for i = 1, . . . , s) and m = p 1 • • • p s . Let us write g = ϕ j (f ) for a fixed j ∈ {1, . . . , s}. Since ζ q j is conjugate to ζ j over F q , we again get, thanks to the hypothesis on f , that g| ti=ζi ∈ C ∞ [[u m ]]. This means that g ∈ R 0 s . Example 33. We set s = 1, case in which we write t = t 1 , T = T 1 and ϕ = ϕ 1 . Observe that ϕ(f ) = (τ -1 (f )) q . From this and (1), ( 2), (3), and (4) we deduce τ (ψ 1 ) = ( πu) q-1 (ψ 1 -L(χ t , 1)), and ϕ(ψ 1 ) = ( πu) 1-q ψ q 1 + L(χ q t , 1). Since ϕ(ψ 1 ), L(χ q t , 1) both belong to R 0 1 , we deduce that ψ q 1 u 1-q belongs to R 0 1 . More generally, the reader can check that for all d ≥ 0, and with L = L(χ t , 1) and µ = πu, µ 1-q d ψ q d + µ 1-q d-1 ϕ(L) q d-1 + µ 1-q d-2 ϕ 2 (L) q d-2 + • • • + µ 1-q ϕ d-1 (L) q ∈ R 0 1 . One interesting consequence of the τ and ϕ difference equations satisfied by ψ 1 is that by comparing the expansions in T[[z]] of the left and right sides one may obtain recursion relations satisfied by the L-functions L(χ t , n) with n ≡ 1 (mod q -1).

The structure of the algebras R 0 s is likely to be quite intricate. Question 34. Determine explicitly a set of generators of R 0 s . Another natural question is the following. Question 35. Is the algebra R 0 s a τ -stable algebra, that is, for all f ∈ R 0 s , τ (f ) ∈ R 0 s ?

In this direction we observe, if s = 1, that the sub-T-algebra

R 1 1 := T[ϕ k (ψ 1 ); k ≥ 0] of R 0
1 is also τ -stable. One may hope that R 0 1 = R 1 1 , but we are unable to show it presently.

  ψ s (z) := a∈A χ t1 (a) • • • χ ts (a) z -a = ′ a∈A χ t1 (a) • • • χ ts (a) z -a ;Date: January 2016. Part of this research occurred while the first author was supported by the ANR HAMOT. The second author is supported by the Alexander von Humboldt Foundation.

2. 1 .

 1 R-entire functions. Proposition 8. A bounded R-entire function is constant.

hence constant by Proposition 8 .

 8 But g(0) = 0 and g = 0.Definition 13. An R-entire function f : C ∞ → R is called tempered if there exists a real number C > 0 and an integer N ≥ 0 depending on f such that for all z

Where q Q = {x ∈ R >0 ; x n = q m , for some m, n ∈ Z \ {0}}.

Anderson generating functions

For the remainder of this paper, R will either be T s or E s equipped with the Gauss norm | • | R = • , given by i c i t i = max i {|c i |}. Both of these rings

We observe that the function z → ω(t) -1 f t (z) extends the map χ t : [START_REF] Perkins | Explicit formulae for L-values in positive characteristic[END_REF]Th. 2.19]. Thus, to follow we shall write χ t (z) := ω(t) -1 f t (z). From the definition of f t (z), it is easy to see that χ t (z) is F q -linear.

For all z ∈ C ∞ \ A we also set

.

Lemma 15. The map χ t : C ∞ → E is the unique F q -linear, E-entire function which restricts on A to the F q -algebra isomorphism A → F q [t] determined by the assignment θ → t, and satisfies

Proof. That χ t (z) is F q -linear, has image in E and is E-entire, follows easily from the definitions. From ( 1) and ( 2) we deduce that χ t (z) satisfies the τ -difference equation ( 8)

Thus, if χ t (z) > 1, then τ (χ t (z)) = χ t (z) q > χ t (z) , and from (8) we obtain

q . In all cases, we have

q }, yielding our inequality. Now, let g : C ∞ → E be another E-entire, F q -linear function, interpolating the map χ t over A. Then, the function f (z) = χ t (z) -g(z) is E-entire, A-periodic, vanishing over A, and such that for all z ∈ C ∞ , f (z) ≤ max{1, |e C (z)| q -1 }. Then, f is constant, therefore zero because vanishing at the elements of A.

Remark 16. When |z| ℑ is sufficiently large, it is possible to show that χ t (z) > 1, and we obtain the equality

We will also employ the following property.

Lemma 17. If z ∈ C ∞ , |z| < q, then χ t (z) = |z|. In particular, χ t induces an injective F q -linear map {z : |z| < q} → {f ∈ E : f < q}.

Proof. By definition, the Newton polygon of the function f t (z) (as a series of T[[z]] and with the Gauss norm over T) is equal to the Newton polygon of e C ( z θ ). The Lemma follows from the identity π θω = 1. We mention a final interesting result concerning the function χ t (z) though we do not need it in the sequel.


Olivier Poisson
email: poisson@latp.univ-mrs.fr

Carleman estimates for the heat equation with discontinuous diffusion coefficients

Keywords: AMS classification scheme numbers: 35K05, 35K55, 35R05, 35R30

We consider the heat equation with a diffusion coefficient that is discontinuous at an interface. We give global Carleman estimates for solutions of this equation, even if the jump of the coefficient across the interface has not a constant sign.

Introduction

The heat Equation with a discontinuous diffusion coefficient

Let Ω ⊂ IR n be a bounded connected open set with boundary Γ of class C 2 . Let a(•) be a measurable function of the variable x ∈ Ω and satisfying Assumption 1.1 There exist two constants a min , a max so that 0 < a min ≤ a(x) ≤ a max < ∞.

We denote by A := -div(a∇•) the formal stationary differential operator on Ω. We consider for a time T > 0 the formal heat operator L := ∂ t + A in the cylinder Q = (0, T ) × Ω and we define L as the heat operator L = L with domain D(L) := {q : q ∈ L 2 (0, T ; H 1 0 (Ω)), ∂ t q ∈ L 2 (0, T ; H -1 (Ω)), Lq ∈ L 2 (Q)} . For q 0 ∈ L 2 (Ω) we consider the following equations:

     Lq = f in Q, q(0, •) = q 0 in Ω .
(1.1)

The function q then satisfies the Dirichlet homogeneous condition q(t, x) = 0 for (t, x) ∈ Σ := (0, T ) × Γ. We know that if f ∈ L 2 (t 0 , T ; H -1 (Ω)) then (1.1) admits an unique solution we denote by Y (f, q 0 ) ∈ L 2 (0, T ; H 1 0 (Ω)), and if f ∈ L 2 (Q) then Y (f, q 0 ) ∈ D(L). (We refer to [1, Theorems 1.1, 4.1 and Remark 4.3; Chap. 3] and to [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] for a complete proof). We make the following assumptions on the diffusion coefficients a(•): a(x) is piecewise smooth in Ω; more precisely, Ω admits a partition in two non empty open sets, Ω 0 and Ω 1 = Ω \ Ω 0 , such that a j ≡ a |Ω j ∈ C 1 (Ω j ), j = 0, 1. Moreover the interface S = Ω 0 ∩ Ω 1 is of class C 2 . We denote by n the outward unit normal to Ω 1 at the points of S and also the outward unit normal to Ω at the points of Γ. Formally, the jump of a quantity q on S is defined by [q] := q 0 -q

1 ≡ (q |Ω 0 ) |S -(q |Ω 1 ) |S . So S is partitioned in S = S + ∪ S -∪ S 0 with S ± := {x ∈ S; ±[a](x) > 0}, S 0 = {x ∈ S; [a](x) = 0} .
We shall analyse (1.1) in the large framework S = S 0 , that is, when a is discontinuous across S + ∪ S -= ∅. We look for global Carleman estimates on Y (f, q 0 ) with source terms in ω T = (0, T ) × ω, where ω ⊂ Ω is open, non empty, and satisfies

Assumption 1.2 Each connected component of Ω 0 contains part of ω.
Then there are two different situations that will be analysed in this work.

• Case 1: Ω 0 ⊂⊂ Ω (see Fig. 1);

• Case 2: Ω 1 ⊂⊂ Ω (see Fig. 2).

The main goal of this paper is to obtain a Carleman estimate for q solution of (1.1) in cases 1 or 2.

Main Results

In order to state the main result of this work, we need some geometrical conditions (see Figure 2).

Geometrical Conditions and Technical Assumptions

Assumption 2.1 on (Ω 1 , S); corresponding to case 1. There exists a vector field ζ :

Ω 1 → IR n , ζ ∈ C 1 (Ω 1 ), so that ζ(x) • n(x) < 0 ∀x ∈ Γ , ζ(x) • n(x) > 0 ∀x ∈ S , ζ(x) = 0 ∀x ∈ Ω 1 ,
and if we consider the characteristics associated to ζ

     dx(t) dt = ζ(x(t)) , t > 0 , x(0) = x 0 , (2.1)
with x 0 ∈ Γ, we also assume that for some time T 1 > 0 and for every x 0 ∈ Γ, there exists t 1 (x 0 ) < T 1 such that the solution x(t) of (2.1) verifies x(t) ∈ Ω 1 for 0 < t < t 1 (x 0 ), and x(t 1 (x 0 )) ∈ S.

(2.2)

See figure 1

Ω 1 → IR n , ζ ∈ C 1 (Ω 1 ), so that ζ(x) • n(x) > 0 ∀x ∈ S , ζ(x) • n(x) > 0 ∀x ∈ O , ζ(x) = 0 ∀x ∈ Ω 1 \ O ,
and for the characteristics associated to ζ

     dx(t) dt = -ζ(x(t)) , t > 0 , x(0) = x 0 , (2.3)
with x 0 ∈ S, we also assume that for some time T 2 > 0 and for every x 0 ∈ S, there exists t 2 (x 0 ) < T 2 such that the solution x(t) of (2.3) verifies x(t) ∈ Ω 1 \ O for 0 < t < t 2 (x 0 ), and x(t 2 (x 0 )) ∈ ∂O.

(2.4)

See figure 2 and [3, page 625].

The main restriction of this framework lies in We shall need the technical assumption 2.4 below. We denote by ∇ the tangential derivative on S and we set S 00 := S + ∩ S -⊂ S 0 .

Assumption 2.4 There exist a neighbourhood V ⊂ S of S 00 and a function

h 0 ∈ C 2 (V ) such that (∇ h 0 (x) = 0, ∀x ∈ V ), and
(±h 0 (x) > 0, ∀x ∈ V ∩ S ± ). Equivalently, S 00 is a (n -2)-dimensional submanifold of S separating S -∩ V = h -1 0 (IR - * ) and S + ∩ V = h -1 0 (IR + * ). Assumption 2.4 is checked if [a(•)] is of class C 2 in a neighbourhood of S 00 and if ∇ [a(x)] = 0 for any x ∈ S 00 : in such a case we can choose h 0 (x) = [a(x)].
Remark 2.2 If S + = ∅ then S 00 = ∅ and so assumption 2.4 is validated. However it possibly happens that sets S ± are non empty but S 00 = ∅. In such a case ∇ [a] = 0 on S 0 = ∂S + ∪ ∂S -.

Remark 2.3

In assumption 2.4 the function h 0 can be extended to a function h0 ∈ C 2 (W ), where W is some neighbourhood of S 00 in Ω, and such that the vector field

∇ h 0 ∈ (C 1 (V )) n-1 extends to the vector field ∇ h0 ∈ (C 1 (W )) n satisfying ∇ h0 (x) = 0 for all x ∈ W . The set h-1 0 ({0}) ⊂ W separates W into W + = h-1 0 (IR + * ) and W -= h-1 0 (IR + * ). Thus if S is a C 2 extension of S + , then S ∩ W + = S + ∩ W = S + ∩ V , S ∩ W -= S -∩ W = S -∩ V , h0 (x) < 0 for all x ∈ (S \ S + ) ∩ W and h0 (x) > 0 for all x ∈ S + ∩ W . Furthermore we can choose V , W sufficiently small so that | h0 | ≤ 1 2 in W . Ω 0 Ω 1 b) a) Ω c S Γ Figure 1. Case 1: Assumption 2.1 is not fulfilled in situation a). Situation in b)
shows a third case.

Ω 1 Ω 0 Γ S' S + ω S' Ω 0 ' Ω 1 ' S - Γ ω Figure 2. Case 1: Assumptions 2.1, 1.2

and 2.3 are fulfilled

The arrow means that we extend S + to S .

Ω 1 Ω 0 Γ ω S S + S' Ω 1 ' Ω 0 ' ω Γ S' S -

Carleman weight functions

Carleman estimates on q solution of (1.1) requires Carleman weight functions that we present now. Lemma 2.1 Assume that we have the geometrical situation of case 1. Under assumptions 1.2, 2.1, 2.3, 2.4, there exist functions β i ∈ C 0 (Ω), i = 1, 2 that satisfy the following properties.

(i) β i ∈ C 0 (Ω), β k i ≡ β i |Ω k ∈ C 2 (Ω k ), k = 0, 1; div(a∇β i ) ∈ L ∞ (Ω).
(Thus the following transmission conditions are checked on S:

[β i ] = 0 and [a∂ n β i ] = 0). (ii) β i (x) > 0, ∀x ∈ Ω.
(iii) There exists r > 0 so that

|∇β i | ≥ r in Ω \ ω, i = 1, 2. (iv) ∂ n β i (x) < 0 for any x ∈ Γ, i = 1, 2. (v) β 1 = β 2 on S 00 , β 1 (x) > β 2 (x) for any x ∈ S -, β 2 (x) > β 1 (x) for any x ∈ S + .
(vi) For x ∈ S, let the symmetrical matrix M(x) := i=1,2 M i (x) of size n + 1 where we set:

M i :=       -ρa∂ n β i ∇ β i |∇ β i | 2 -ρ(a∂ n β i ) 2 t ∇ β i -(a∂ n β i )I n-1 t 0 IR n-1 |∇ β i | 2 -ρ(a∂ n β i ) 2 0 IR n-1 2a∂ n β i {|∇ β i | 2 -ρ(a∂ n β i ) 2 }       . (2.

5)

Here I n-1 denotes the unit matrix of size n -1 and ρ(x) := (a 0 (x)a 1 (x)) -1 .

Then M 1 ≤ 0 on S -and M 2 ≥ 0 on S + . Furthermore there exists a neighbourhood V of S 00 in S so that M = 0 on V .

Lemma 2.2 Assume that we have the geometrical situation of case 2. Under assumptions 1.2, 2.2, 2.3, 2.4, there exist functions β i ∈ C 0 (Ω), i = 1, 2 that satisfy properties (i), (ii), (iv), (v), (vi) of lemma 2.1. The function β 2 satisfies (iii) of lemma 2.1 and β 1 satisfies:

(iii) there exists r > 0 so that

|∇β 1 | ≥ r in Ω \ (ω ∪ B),
where B is an opened ball such that B ⊂⊂ O. Moreover β 1 and β 2 satisfy

(vii) β 2 ≥ 2β 1 in B.
Remark 2.4 In all cases 1, 2, we can construct β i so that β i |Γ are constant, i = 1, 2.

Remark 2.5 In all cases 1, 2, we shall construct β i , i = 1, 2, so that in V we have ρ(a∂ n β i ) 2 -|∇ β i | 2 = 0, M = 0, and

M i =       -ρa∂ n β i ∇ β i 0 t ∇ β i -(a∂ n β i )I n-1 t 0 IR n-1 0 0 IR n-1 0       . (2.6)
Thus the eigenvalues of M i (x) are 0 (with multiplicity n) and -(ρ(x)+n-1)(a∂ n β i )(x).

Lemmas 2.1 and 2.2 are proved in appendix.

Global Carleman Estimates

We set β := 5 4 max i=1,2 max Ω β i . We can assume that max i=1,2

max

Ω β i (x) < β < 2 min i=1,2 min Ω β i (x) , (2.7)
since if not, we replace β i by β i + 4β and then β by 6β. We set

ϕ i (t, x) := e λβ i (x) t(T -t)
, η i (t, x) := e λβ -e λβ i (x) t(T -t) ,

Π j := i=1,2 s j t -j (T -t) -j e -2sη i , j ∈ IN .
In order to state the main result of this work in common statements for cases 1 and 2, we set

B = O = ∅ in case 1.
Theorem 2.1 Assume that conditions of lemma 2.1 in case 1 (respectively: of lemma 2.2 in case 2) are fulfilled. There exists λ 1 (Ω, ω, c) > 0 such that for each λ > λ 1 there exists a positive constant C that only depend on Ω, ω, O and a, and s 1 (λ) > 0 so that the following estimate holds

Q Π 3 |q| 2 + Q Π 1 |∇q| 2 ≤ C ω T Π 3 |q| 2 + C Q Π 0 |Lq| 2 (2.8)
for all q ∈ D(L) and s ≥ s 1 . Moreover, s 1 is of the form s 1 = σ 1 (T 2 + T ), where σ 1 is a positive constant that only depends on Ω, ω, O, a and λ.

Theorem 2.2 Assume that conditions of lemma 2.1 in case 1 (respectively: of lemma 2.2 in case 2) are fulfilled. Let f ∈ (L 2 (Q)) n and q ∈ L 2 (Q) satisfying Lq = divf , that is for any g ∈ L 2 (Q) we have

Q qg dxdt = - Q f • ∇z dxdt + Ω q 0 z(t 0 ) dx , (2.9)
where z ∈ D(L * ) is solution of L * z = g, z(T ) = 0. Then there exists λ 1 (Ω, ω, a) > 0 such that for each λ > λ 1 there exists a positive constant C that only depend on Ω, ω, and a, and s 1 (λ) > 0 so that the following estimate holds

Q Π 3 |q| 2 ≤ C ω T Π 3 |q| 2 + C Q Π 2 |f | 2 (2.10) for s ≥ s 1 . Moreover, s 1 is of the form s 1 = σ 1 (T 2 + T )
, where σ 1 is a positive constant that only depends on Ω, ω, ), a and λ.

Applications

These Carleman estimates have many applications.

• Firstly, as described in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF], semilinear parabolic equations as

     Lq + f (q) = vχ ω + g in Q, q(0, x) = q 0 (x) in Ω,
with unknown q ∈ D(L) and under additional assumptions on the functional f are exactly controllable to the trajectories. This is [3, theorem 2.1] if S + = ∅.

• Secondly the inverse problem related to equation 1.1 and stated in terms of uniqueness and Hölder stability for a(•), as described in [START_REF] Poisson | Uniqueness and Hölder stability of discontinuous diffusion coefficients in three related inverse problems for the heat equation[END_REF], is solved in our extended framework.

Comments

Global Carleman inequalities were first used in [START_REF] Fursikov | Controllability of evolution equations[END_REF] to obtain inequality of observability for parabolic equations with C 1 coefficients. By this approach the problem of null controllability for semilinear parabolic equations can be proved: see [START_REF] Barbu | Exact Controllability of the superlinear heat equation[END_REF], [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF], [START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. In the case of a discontinuous coefficient, results on global Carleman equalities and on null controllability recently appear thanks to the developments in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF]. In the n dimensional case [START_REF] Benabdallah | On the controllability of linear parabolic equations with an arbitrary control location for a stratified media[END_REF] gives a positive answer to the controllability question for a class of discontinuous coefficients, with separated variables that are smooth w.r.t. to all but one variables, which includes the case of stratified media. The proof relies on the global Carleman inequalities pointed in [START_REF] Benabdallah | On the controllability of linear parabolic equations with an arbitrary control location for a stratified media[END_REF], [START_REF] Rousseau | Carleman estimate and controllability results for the one -dimensional heat equation with bv coefficients[END_REF] for the one -dimensional case. The proof for null controllability for the linear parabolic heat equation is then achieved in [START_REF] Rousseau | Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF] for the case of a coefficient that exhibits jump of arbitrary sign at an interface. Our work is about Global Carleman inequalities for the heat equation as in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] where the null controllability problem for semilinear parabolic equations is solved. In fact authors of [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] prove inequalities (2.8) and (2.10), but with some restrictions that we remove. Let us precise several points about this.

1. They assume that a j ∈ C 2 (Ω j ), j = 1, 2, which is too strong. Indeed, the Carleman functions they build do not depends on all the a k (x), x ∈ Ω, k = 0, 1, but only on the fraction (a 1 /a 0 )(x), x ∈ S. Their condition - , then a j ∈ C 1 (Ω j ), j = 1, 2 suffices. Moreover, outside S, the assumption a j ∈ C 1 (Ω j ) is only used to ensure that (div(a j ∇β))

a j ∈ C 2 (Ω j ) -is only used in [3,
|Ω j ∈ C 0 (Ω j ).
But in fact that we only need is div(a∇β) ∈ L ∞ (Ω). Hence the results (about Carleman estimates) in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] still holds if the assumption a j ∈ C 2 (Ω j ) is replaced by

a j ∈ W 1,∞ (Ω j ), j = 0, 1.
2. They assume that q belongs to the following restricted set Z 0 ⊂ D(L):

Z 0 := {q : q |Ω j ∈ C 2 ([t 0 , T ] × Ω j ), j = 0, 1, [q] = 0, [c∂ n q] = 0, q = 0 on Σ} . (2.11)
We can extend the results to q ∈ D(L). See [START_REF] Poisson | Uniqueness and Hölder stability of discontinuous diffusion coefficients in three related inverse problems for the heat equation[END_REF] for a proof.

3. They assume that [a] ≤ 0, or equivalently S + = ∅, in both cases 1, 2. Besides, regarding the case 2 with S + = ∅ and considering remark 2.1, assumption [3, Condition 2.2] appears more sophisticated than our assumption 2.2. (Precisely: two sets -O i , i = 1, 2 -in their work, only one -O -in our's) .

The paper is organized as follows. In Section 3 we first recall the main computations of [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] to prove (2.8) in the case 1, then we analyse the sign of the quadratic form coming from integrals on S. In Sections 4 and 5 we study two particular situations, where S 00 = ∅. We show that we need only one function β that we describe there.

In Section 6 we derive from section 3 and from [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] the proof for the case 2.

Proof for the Case 1

We prove the part of theorems 2.1, 2.2 corresponding to Case 1.

Computation of integrals

As in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] we restrict the proof to q ∈ Z 0 defined above. The general case q ∈ D(L) is completed by following the proof of [4, §6.1].

We particularly watch all the integrals on surfaces Γ or S. Let q ∈ Z 0 and f := Lq ∈ L 2 (Q). We set ψ i := e -2sη i q, i = 1, 2. Then we have

M 1 ψ i + M 2 ψ i = f i,s (3.1)
where

M 1 ψ i := div(a(x)∇ψ i ) + s 2 λ 2 ϕ 2 i |∇β i | 2 a(x)ψ i + s∂ t η i ψ i , M 2 ψ i := ∂ t ψ i -2sλϕ i a(x)∇β i • ∇ψ i -2sλ 2 ϕ i a(x)|∇β i | 2 ψ i , f i,s := e -sη i f + sλϕ i div(a(x)∇β i )ψ i -2sλ 2 ϕ i a(x)|∇β i | 2 ψ i .
From (3.1), we obtain

M 1 ψ i 2 2 + M 2 ψ i 2 2 + 2(M 1 ψ i , M 2 ψ i ) = f i,s 2 2 , (3.2)
where (•, •) denotes the scalar product in L 2 (Q). We compute the scalar product in the left hand side of (3.2), with index i omitted. We set S T = (0, T ) × S and Q = Q \ S T .

We have

(M 1 ψ, M 2 ψ) = 3 k=1,l=1 I kl , (3.3)
where we set

• I 11 := Q div(a(x)∇ψ)∂ t ψ = 0 , • I 12 := -2sλ Q ϕdiv(a(x)∇ψ)a(x)∇β • ∇ψ = J 12 + K 12 + H 12 ,
where we set

J 12 := 2sλ Q a∇(ϕa∇β • ∇ψ) • ∇ψ, K 12 := 2sλ S T ϕ[∇β • ∇ψa 2 ∂ n ψ], H 12 := -2sλ Σ ϕ∇β • ∇ψ a 2 ∂ n ψ. We compute J 12 = 2sλ 2 Q ∇(ϕa∇β • ∇ψ)(a∇β • ∇ψ) + 2sλ Q ϕa∂ j ψ ∂ j (a∂ l β) ∂ l ψ + 2sλ Q ϕa 2 ∂ j ψ ∂ 2 jl ψ ∂ l β = 2sλ 2 Q ϕ|a∇β • ∇ψ| 2 + 2sλ Q ϕa∂ j ψ ∂ j (a∂ l β) ∂ l ψ + sλ Q ϕa 2 ∇(|∇ψ| 2 ) • ∇β = 2sλ 2 Q ϕ|a∇β • ∇ψ| 2 + 2sλ Q ϕa∂ j ψ ∂ j (a∂ l β) ∂ l ψ -sλ Q |∇ψ| 2 div(ϕa 2 ∇β) + K 12 + H 12 ,
where we set

K 12 := -sλ S T ϕ[|∇ψ| 2 a 2 ∂ n β], H 12 := sλ Σ |∇ψ| 2 ϕa 2 ∂ n β. Finally since H 12 = -1 2 H 12 we get I 12 := -sλ 2 Q ϕ|a∇β| 2 |∇ψ| 2 + 2sλ 2 Q ϕ(a∇ψ • ∇β) 2 + K 12 + K 12 + 1 2 H 12 + X 1 , X 1 := 2sλ Q ϕa∂ j ψ∂ j (a∂ l β)∂ l ψ -sλ Q ϕ|∇ψ| 2 div(a 2 ∇β) . • I 13 := -2sλ 2 Q div(a∇ψ)ϕa|∇β| 2 ψ = 2sλ 2 Q a∇ψ∇(ϕa|∇β| 2 ψ) + K 13 , with K 13 := 2sλ 2 S T ϕ[a 2 |∇β| 2 ψ∂ n ψ].
Thus we obtain

I 13 = 2sλ 2 Q ϕ|a∇β| 2 |∇β| 2 + X 2 + K 13 , with X 2 := 2sλ 2 Q ϕa∇(a|∇β| 2 )∇ψ ψ + 2sλ 3 Q ϕa 2 |∇β| 2 ∇β • ∇ψ ψ. • I 21 := s 2 λ 2 Q ϕ 2 |∇β| 2 a∂ t ψ ψ = -s 2 λ 2 Q ϕ∂ t ϕ |∇β| 2 a|ψ| 2 , • I 22 := -s 3 λ 3 Q ϕ 3 a 2 |∇β| 2 ∇β • ∇ψψ = 3s 3 λ 4 Q ϕ 3 a 2 |∇β| 4 |ψ| 2 + K 22 + X 3 , with K 22 := s 3 λ 3 S T ϕ 3 [a 2 |∇β| 2 ∂ n β]|ψ| 2 , X 3 := s 3 λ 3 Q ϕ 3 div(a 2 |∇β| 2 ∇β)|ψ| 2 . • I 23 := -2s 3 λ 4 Q ϕ 3 a 2 |∇β| 4 |ψ| 2 , • I 31 := s Q ∂ t η ψ∂ t ψ = - 1 2 Q ∂ 2 t η |ψ| 2 ,
• I 32 := -2s

The quadratic form on the interface

Thus we write (3.3) in the form (M 1 ψ i , M 2 ψ i ) = J i + K i + H i , with J i : integrals on Q , K i : integrals on S T , H i : integrals on Σ. We have

J i := S T {Q i nn |a∂ n ψ i | 2 + 2Q i nτ • ∇ ψ i (a∂ n ψ i ) + Q i τ τ |∇ ψ i | 2 (3.4)
+ (Q i 00 + X i 00 )|ψ i | 2 + 2X i n0 (a∂ n ψ i )ψ i } , with

Q i nn := sλϕ i [∂ n β i ] , Q i nτ := sλϕ i [a∇ β i ] , Q i τ τ := -sλϕ i [a 2 ∂ n β i ] , X i n0 := sλ 2 ϕ i [a|∇β i | 2 ] , Q i 00 := s 3 λ 3 ϕ 3 i [a 2 ∂ n β i |∇β i | 2 ]
, X i 00 := s 2 λϕ i ∂ t η i [a∂ n β i ] . We write (3.4) by using ψ i := e -sη i q and ∇ψ i = e -sη i ∇q + sλϕ i ∇β i q. Thus we have

J i := S T e -2sη i {Q i nn |a∂ n q| 2 + 2Q i nτ • ∇ q(a∂ n q) + Q i τ τ |∇ q| 2
+ (2Q i 00 + Y i 00 )|q| 2 + 2(Q i n0 + X i n0 )(a∂ n q)q} , with Y i 00 := 2s -1 ϕ -1 i Q i 00 + X i 00 , Q i n0 := s 2 λ 2 ϕ 2 i [a|∇β i | 2 ]. We set

R i :=       Q i nn Q i nτ Q i n0 Q i nτ Q i τ τ I n-1 t 0 IR n-1 Q i n0 0 IR n-1 2Q i 00       , X i :=       0 0 IR n-1 X i n0 t 0 IR n-1 0 n-1 t 0 IR n-1 X i n0 0 IR n-1 Y i 00       ,

Assumption 2 . 3

 23 The set S + can be extended to a variety S of class C 2 which is the interface between two open sets Ω 0 , Ω 1 ⊂ Ω, with Ω 1 ⊂ Ω 0 , Ω 1 ⊂ Ω 0 and each connected component of Ω 0 = Ω \ Ω 1 contains part of ω. See figures 1 and 2.

Remark 2 . 1

 21 If S + = ∅ then assumption 2.3 holds for the case 1 with S = ∅ and Ω 1 = ∅, and for the case 2 with S = Γ, Ω 1 = ∅.

Figure 3 .

 3 Figure 3. Case 2: Assumptions 2.2, 1.2 and 2.3 are fulfilled. The arrow means that we extend S + to S .

  and [3, page 624].Assumption 2.2 on (Ω 1 , S); corresponding to case 2. There exists an open set O ⊂⊂ Ω 1 (with always a unit outward normal n) and a vector field ζ :

  Step 2, page 656] and similarly in [3, Step 4 page 659]. But if we use [12, Volume I, Corollary 1.3.4]

  η a|∇β| 2 |ψ| 2 -s 2 λ 2 Q ϕ∂ t ϕ a|∇β| 2 |ψ| 2 + K 32 with K 32 := s 2 λ S T ϕ ∂ t η [a∂ n β]|ψ| 2 .

2 λ Q ϕ∂ t η a∇β • ∇ψ ψ = s 2 λ Q ϕdiv(a∇β)|ψ| 2 + s 2 λ 2 Q ϕ∂ t • I 33 := -2s 2 λ 2 Q ϕa ∂ t η|∇β| 2 |ψ| 2 .

and R i := e -2sη i R i , X i := e -2sη i X i , R := i=1,2 R i , X := i=1,2 X i , P := R + X . We prove that P ≥ 0 on S for s sufficiently large. Firstly, note that if x ∈ S 0 then P = 0 n+1 , since [a(x)] = 0. Let L i (x), i = 1, 2, be the (n + 1) sized strictly positive diagonal matrix with L i jj = 1, j ≤ n, and L i n+1,n+1 (x) = sλϕ i (x) for x ∈ S. Thus R i = [a] sλϕ i L i M i L i . Let x ∈ V ∩ S + . Note that X i 00 = 0 since [a∂ n β i (x)] = 0, and that X i n0 = Q i 00 = Y i 00 = 0. Thus X (x) = 0 n+1 . Since [a(x)] ≥ 0, M 2 (x) ≥ 0 and M 1 (x) = -M 2 (x) ≤ 0, we thus have R 2 (x) ≥ 0 and R 1 (x) ≤ 0.. From β 2 (x) ≥ β 1 (x) we obtain ϕ 2 (x) ≥ ϕ 1 (x), η 2 (x) ≤ η 1 (x), e -2sη 2 (x) ≥ e -2sη 1 (x) and L 2 (x) ≥ L 1 (x). We thus obtain R 2 (x) ≥ -R 1 (x), R (x) ≥ 0, and P (x) ≥ 0. Similarly we obtain P (x) ≥ 0 for any x ∈ V ∩ S -. Let x ∈ S + \ V . We have M 2 (x) ≥ r > 0 for some r ≥ 0 that do not depend of x and X 2 = [a] λL 2 Y 2 L 2 where the matrix Y 2 does not depend on s and λ. So s -1 |Y 2 (x)| ≤ 1 4 M 2 (x) for s sufficiently large, X 2 is a perturbation term for R 2 of order s -1 ϕ -1 2 , and R 2 + X 2 ≥ 3 4 R 2 . Then we have M 2 (x) ≥ r > 0 and β 2 (x) ≥ β 1 (x) + r for some r ≥ 0 that does not depend of x. Thus L 2 (x) ≥ L 1 (x) and e -2sη 1 (x) is a perturbation term for e -2sη 2 (x) of order s -1 (at least). Consequently we have

Similarly we obtain P (x) ≥ 0 for any x ∈ S -\ V . Consequently, if s is sufficiently large, then i=1,2 J i ≥ 0.

Then we can complete the computations in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] and the proof of (2.8).

A Particular situation for the case 1

The following result corresponds to the case S -= ∅.

Lemma 4.1 We assume that Γ and S are of class C 2 and S + = S \ S 0 satisfies assumption 2.3. Then there exists a function β ∈ C 0 (Ω) that satisfies the following properties.

(1)

(2) [β] = 0 and [a∂ n β i ] = 0;

(3) β(x) > 0, ∀x ∈ Ω.

(4) There exists r > 0 so that |∇β| ≥ r in Ω \ ω.

(5) β is constant on Γ and ∂ n β(x) < 0 for any x ∈ Γ.

(6) There exists r > 0 so that if we set

then M(x) ≥ r > 0 on S + .

Carleman estimates for the heat equation with discontinuous diffusion coefficients 11

We prove this lemma in appendix.

Remark 4.1 We built β independently of the fact that [a] = 0 in S \ S + . This will be used above (see the Proof of lemma 2.1).

We straightforwardly obtain: , η(t, x) := e λβ -e λβ(x) t(T -t) ,

then theorems 2.1 and 2.2 still hold in case 1 with S -= ∅.

A second Particular situation for the case 1

In this part we assume S -and S + non empty, and S 00 empty, that is, d(S -, S + ) = 0. Thus for any ε ∈ (0, 1) there exists two open sets

where β is built as in lemma 4.1 (with the modification S + = S\(S -∪S 0 )). See also remark 4.1. Let β 1 = ψ 1 β where β is defined as in [3, (43) page 629] (with the modification that S + is non empty). See also remark 4.1.

We choose ε > 0 such that

Then we can easily check that

for some r > 0 and for s sufficiently large. Hence with this modified function β, the conclusion of corollary 4.1 holds again in this situation.

Proof for the case 2

We prove the part of theorems 2.1, 2.2 corresponding to Case 2. Thanks to the computations in [3, Proof of theorem 3.4] and to the properties of the Carleman weights that originate from lemma 2.2, we can complete the proof of these theorems in the case 2. Remark 6.1 If S -is empty or, more generally, if S 00 is empty, then we are as in the first or the second particular situations above -see sections 4, 5 but corresponding to case 2. Then, by following lemma 4.1 and section 5, we can prove theorems 2.1 and 2.2 (in case 2) by introducing only one function β. This is as corollary 4.1, but in the situation of case 2.

Carleman estimates for the heat equation with discontinuous diffusion coefficients 12 7. Appendix

Proof of lemma 4.1

Step 1 -Let S be an extension of S + inside Ω 0 of class C 2 , that separates Ω in two open sets Ω 0 and Ω 1 , with ω ⊂⊂ Ω 0 ⊂⊂ Ω. We denote by n the unit vector field on S with n (x) pointing towards Ω 0 . We are thus in the situation of [3, case 2, page 1] with the prime suppressed, but a j ∈ W 1,∞ (Ω j ) only (that is not a problem as we pointed in comment 1. of §2.5). So at this point we could conclude with two β j functions. In fact, we can use only one. This has a the great advantadge for the inverse problem treatment: one measurement instead of two (see [START_REF] Poisson | Uniqueness and Hölder stability of discontinuous diffusion coefficients in three related inverse problems for the heat equation[END_REF]). Indeed, we construct a function β ∈ C 0 (Ω 0 ) satisfying

All these points can be checked by following [3, Proof of lemma 3.2, Steps 4, 5, 6]. (Concerning the last point, we use there the fact that each connected component of Ω 0 contains part of ω, thanks to assumption 2.3). Note that we have ∂ n β (x) < 0 for any x ∈ S ∩ S , since n = -n on S ∩ S . We set β = β in Ω 1 .

Step 2 -Following [3, Proof of Lemma 3.1, Steps 2, 3, 4], we then extend β in Ω 0 such that

(Thus β satisfies the transmission conditions on S: [β] = 0 and [a∂ n β] = 0).

(ii) β(x) > 0, ∀x ∈ Ω.

(iii) There exists r > 0 so that |∇β| ≥ r in Ω \ ω.

(Concerning point (iii), we use there the fact that each connected component of Ω 0 contains part of ω, thanks to assumption 1.2). We thus have ∇ β(x) = 0 for any x ∈ Γ ∪ S + , and, for any x ∈ S + ⊂ S ∩ S , M(x) ≥ r > 0. (Here, setting α ≡ a∂ n β, the value

is suitable).

Proof of lemma 2.1

We will use the following lemma:

Carleman estimates for the heat equation with discontinuous diffusion coefficients 13 Lemma 7.1 Let r > 0 and, for any ε sufficiently small, let φ ε be a C 2 diffeomorphism defined in a neighbourhood U of S such that φ ε -Id ∞ < ε, |J(φ ε ) -1| < ε, where J(φ ε ) denotes the jacobian of φ ε . We assume that S ε := φ ε (S) ⊂ U 1 , U 1 := U ∩ Ω 1 , and we denote by n ε the unit normal on S ε pointing towards S.

Let h 1 , h 2 be two functions in C 2 (Ω 1 ) with h 1 -h 2 • φ ε ≥ r, g 1 ≥ r on S and g 2 ≥ r on S ε , where we set

Then there exists ε 0 > 0 such that for any ε ∈ (0,

Denote by p the orthogonal projection on S, defined in a neighbourhood U of S in Ω. Then p ∈ C 2 (U ). We set q := Id -p ∈ C 2 (U ). Let U ε := {tp(x) + (1 -t)x; 0 < t < 1; x ∈ S ε }. Note that for any z ∈ U ε there exists an unique (t, x)

Let us prove lemma 2.1 in 6 steps. We successively construct

Step 1 -Let S be the interface between Ω 0 and Ω 1 that extends S + inside Ω 0 . Thus S 00 ⊂ S ∩ S . Let h 0 be the function of assumption 2.4 defined in V ⊂ S and let h0 ∈ C 2 (W ) be an extension to W ⊂ Ω, as explained in remark 2.3.

We set V = W ∩ S of S 00 and h = 2 + h0 in V . Then h can be extended to S so that

Note that h = 2 on S 00 , h (x) < 2 for all x ∈ V ∩ S -, and ∇ h (x) = 0 for all x ∈ V . We then consider g ∈ C 1 (S ) defined on V by

and satisfying g (x) > 0 for all x ∈ S .

Carleman estimates for the heat equation with discontinuous diffusion coefficients 14

Step 2 -Corollary 1.3.4 of [START_REF] Hormander | The Analysis of Linear Partial Differential Equations[END_REF]Volume I] shows that there exists a positive function β defined in a neighbourhood

Thanks to assumption 2.3 and as in Proof of Lemma 4.1, Step 1, we then extend β into Ω 0 , so that

Step 3 -Thanks to assumption 1.2 and as in Proof of Lemma 4.1, Step 2, we then extend β 2 into Ω 0 , so that

• There exists r > 0 so that

Note that (a∂ n β 2 )(x) = -g (x) < 0 for any x ∈ S + .

Step 4 -Thanks to the properties of h (cf. step 1) we have 3 ≥ β 2 (x) > 2 for x ∈ S + and β 2 (x) < 2 for x ∈ V ∩S -. Thus there exists h ∈ C 2 (S) with the following properties:

Note that h = 2 on S 00 , h(x) < β 2 (x) for any x ∈ S + , h(x) > β 2 (x) for any x ∈ S -, and ∇ β 2 (x) = 0 for all x ∈ V . We then consider g ∈ C 1 (S) defined in V by

Step 5 -Corollary 1.3.4 of [12, Volume I] implies that there exists a positive function

Thanks to assumption 1.2 and as in Proof of Lemma 4.1, Step 2, we then extend β into Ω 0 so that

• There exists r > 0 so that |∇β| ≥ r in Ω 0 \ ω.

Carleman estimates for the heat equation with discontinuous diffusion coefficients 15

Step 6 -Corollary 1.3.4 of [START_REF] Hormander | The Analysis of Linear Partial Differential Equations[END_REF]Volume I] implies that there exists a positive function α defined in a neighbourhood U 1 ⊂ Ω 1 of S with α ∈ C 2 (U 1 ), α |S = h and ∂ n α = g on S. Moreover we can choose U 1 so that α(x) > 0 and ∇α(x) = 0 for x ∈ U 1 . For ε > 0 we set

ε and S ε := {x ∈ Ω 1 ; d(x, S) = ε}. We notice that if ε is sufficiently small then S ε satisfies the condition in lemma 7.1 (with some φ ε ) and (Ω ε 1 , S ε ) satisfies assumption 2.1. The proof of [3, Lemma 3.1, Step 1] with (Ω 1 , S) replaced by (Ω ε 1 , S ε ) shows that there exists a function

ε in such a way that β 3 ∈ C 2 (Ω 1 ) and β 3 (φ ε (x)) < α(x) for x ∈ S. Then we apply lemma 7.1 with h 1 = α, h 2 = β 3 , and we denote by

In fact, in such a way, the property β 1 |Ω 1 ∈ C 2 (Ω 1 ) fails, but we can use the argument at the end of [3, proof of Lemma 3.1, Step 1]. This shows that we can extend β 1 |Ω 0 into Ω 1 as we wish.

Proof of lemma 2.2

We follow the proof of lemma 2.1 by pointing the modifications. We consider an interface S between Ω 0 and Ω 1 that extends S + inside Ω 0 .

Step 1 -We construct a neighbourhood V ⊂ S of S 00 , h ∈ C 2 (S ) and g ∈ C 1 (S ) so that

Step 2 -Thanks to assumption 2.3, there exists a positive function β defined in Ω 0 so that • β ∈ C 0 (Ω 0 ) and β (x) > 0 for all x ∈ Ω 0 ;

• There exists r > 0 so that |∇β | ≥ r in Ω 0 \ ω.

We set β 2 = β in Ω 1 .

Carleman estimates for the heat equation with discontinuous diffusion coefficients 16

Step 3 -Thanks to assumption 1.2 and as in Proof of Lemma 4.1, Step 2, we then extend β 2 into Ω 0 , so that

• β 2 |Γ is constant and ∂ n β 2 (x) < 0 for any x ∈ Γ.

• There exists r > 0 such that |∇β 2 | ≥ r in Ω \ ω.

Note that (a∂ n β 2 )(x) = -g (x) < 0 for any x ∈ S + .

Step 4 -There exist functions h ∈ C 2 (S), g ∈ C 1 (S) so that:

• h = 2 on S 00 , h = 4 -β 2 in S + ∪ V ;

• h(x) < β 2 (x) for all x ∈ S + , h(x) > β 2 (x) for all x ∈ S -, h(x) > 0 for all x ∈ S;

in V and g(x) > 0 for any x ∈ S.

Step 5 -Following Step 5 of the proof of lemma 2.1, there exist r > 0 and a function β 1 ∈ C 2 (Ω 0 ) so that

• β 1 |S = h and ∂ n β 1 = 1 a 0 g on S; • β 1 |Γ is constant and ∂ n β 1 (x) < 0 for all x ∈ Γ;

• β 1 (x) > 0 for any x ∈ Ω 0 ;

• |∇β 1 |(x) ≥ r > 0 for any x ∈ Ω 0 \ ω.

Step 6 -We then extend β 1 into Ω 1 so that Step 7 -As in [3, Step 3, Proof of Lemma 3.2] we can construct β 1 so that β 1 ≤ 1 2 β 2 in O.