El Peng Chen

Maati Ouhabaz

Adam Sikora
email: adam.sikora@mq.edu.au

Lixin Yan

Yan Lixin

Spectral Sharp

El Maati Ouhabaz
email: elmaati.ouhabaz@math.u-bordeaux.fr

SHARP SPECTRAL MULTIPLIERS WITHOUT SEMIGROUP FRAMEWORK AND APPLICATION TO RANDOM WALKS

Keywords: Mathematics Subject Classification. 42B15, 42B20, 47F05 Spectral multipliers, polynomial off-diagonal decay kernels, space of homogeneous type, random walk

published or not. The documents may come

1. Introduction 1.1. The main results. 1.2. Notations and assumptions 2. Preliminary results 2.1. A criterion for L p -L q boundedness for linear operators 2.2. Operators with kernels satisfying off-diagonal polynomial decays 3. Spectral multipliers via polynomial off-diagonal decay kernels 3.1. Preparatory results 3.2. Proof of Theorem 3.1 4. Sharp spectral multiplier results via restriction type estimates 5. Applications 5.1. Symmetric Markov chains 5.2. Random walk on Z n 5.3. Fractional Schrödinger operators References 1. Introduction Let (X, d, µ) be a metric measure space, i.e. X is a metric space with distance function d and µ is a nonnegative, Borel, doubling measure on X. Let A be a self-adjoint operator acting on L 2 (X, µ). By the spectral theorem one has

A = ∞ -∞ λdE(λ),
where dE(λ) is the spectral resolution of the operator A. Then for any bounded measurable function F : R → R one can define operator

F(A) = ∞ -∞ F(λ)dE(λ).
It is a standard fact that the operator F(A) is bounded on L 2 with norm bounded by the L ∞ norm of the function F.

The theory of spectral multipliers consists of finding minimal regularity conditions on F (e.g. existence of a finite number of derivatives of F in a certain space) which ensure that the operator F(A) can be extended to a bounded operator on L p (X, µ) for some range of exponents p 2. Spectral multipliers results are modeled on Fourier multiplier results described in fundamental works of Mikhlin [START_REF] Mikhlin | Multidimensional singular integrals and integral equations[END_REF] and Hörmander [25]. The initial motivation for spectral multipliers comes from the problem of convergence of Fourier series or more generally of eigenfunction expansion for differential operators. One of the most famous spectral multipliers is the Bochner-Riesz mean

σ R,α (A) := 1 - A R α + .
When α is large, the function σ R,α is smooth. The problem is then to prove boundedness on L p (X) (uniformly w.r.t. the parameter R) for small values of α. This is the reason why, for general function F with compact support, we study sup t>0 F(tA) p→p ≤ C < ∞. The constant C depends on F and measures the (minimal) smoothness required on the function.

In recent years, spectral multipliers have been studied by many authors in different contexts, including differential or pseudo-differential operators on manifolds, sub-Laplacians on Lie groups, Markov chains as well as operators in abstract settings. We refer the reader to [START_REF] Maati Ouhabaz | Spectral multipliers for Markov chains[END_REF][START_REF]Spectral multipliers on Lie groups of polynomial growth[END_REF]4,[START_REF] Blunck | A Hörmander-type spectral multiplier theorem for operators without heat kernel[END_REF][START_REF] Christ | L p bounds for spectral multipliers on nilpotent groups[END_REF]16,19,[START_REF] Guillarmou | Restriction and spectral multiplier theorems on asymptotically conic manifolds[END_REF][START_REF] Hebisch | A multiplier theorem for Schrödinger operators[END_REF]22,25,[START_REF] Jensen | L p -mapping properties of functions of Schrödinger operators and their applications to scattering theory[END_REF][START_REF] Kunstmann | Spectral multiplier theorems of Hörmander type on Hardy and Lebesgue spaces[END_REF]29,[START_REF] Mikhlin | Multidimensional singular integrals and integral equations[END_REF][START_REF] Sikora | Sharp spectral multipliers for operators satisfying generalized Gaussian estimates[END_REF] and references therein. We mention in particular the recent paper [START_REF] Chen | Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means[END_REF] where sharp spectral multiplier results as well as end-point estimates for Bochner-Riesz means are proved. A restriction type estimate was introduced there in an abstract setting which turns out to be equivalent to the classical Stein-Tomas restriction estimate in the case of the Euclidean Laplacian. Also it is proved there (see also [4]) that in an abstract setting, dispersive or Strichartz estimates for the Schrödinger equation imply sharp spectral multiplier results.

1.1. The main results. There are two main objectives of the present paper. First, in contrast to the previous papers on spectral multipliers where usually decay assumptions are made on the heat kernel or the semigroup, we do not make directly such assumptions and work outside the semigroup framework. The second objective is to replace the usual exponentiel decay of the heat kernel by a polynomial one. All of this is motivated by applications to new settings and examples which were not covered by previous works. In addition most of spectral multipliers proved before can be included in our framework.

In order to state explicitly our contributions we first recall that (X, d, µ) satisfies the doubling property (see Chapter 3, [15]) if there exists a constant C > 0 such that V(x, 2r) ≤ CV(x, r) ∀ r > 0, x ∈ X. (1.1) Note that the doubling property implies the following strong homogeneity property, (1.2)

V(x, λr) ≤ Cλ n V(x, r)

for some C, n > 0 uniformly for all λ ≥ 1 and x ∈ X. In Euclidean space with Lebesgue measure, the parameter n corresponds to the dimension of the space, but in our more abstract setting, the optimal n need not even be an integer. Let τ > 0 be a fixed positive parameter and suppose that A is a bounded self-adjoint operator on L 2 (X, µ) which satisfies the following polynomial off-diagonal decay

(1.3) P B(x,τ) V σ p τ AP B(y,τ) p→2 < C 1 + d(x, y) τ -n-a ∀x, y ∈ X with σ p = 1/p -1/2 and P B(x,τ) is the projection on the open ball B(x, τ). We prove that if a > [n/2] + 1 and F is a bounded Borel function such that F ∈ H s (R) for some s > n(1/p -1/2) + 1/2, then F(A)A p→p ≤ C F H s .
Note that the operator A which we discuss here cannot be, in a natural way, considered as a part of a semigroups framework. See Theorem 3.1 below for more additional information. In the particular case where A = e -τL for some non-negative (unbounded) self-adjoint operator L with constant in (1.3) independent of τ, we obtain for

s > n(1/p -1/2) + 1/2 (1.4) sup τ>0 F(τL) p→p ≤ C F H s .
As mentioned previously, this latter property implies Bochner-Riesz L p -summability with index α > n(1/p -1/2). See Corollary 3.2. Some significant spectral multiplier results for operators satisfying polynomial estimates were considered by Hebisch in [22] and indirectly also in [START_REF] Boutayeb | A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces[END_REF][START_REF] Jensen | L p -mapping properties of functions of Schrödinger operators and their applications to scattering theory[END_REF] by Jensen and Nakamura. Our results are inspired by ideas initiated in [START_REF] Nicola | Sharp L p estimates for Schrödinger groups[END_REF]22,[START_REF] Boutayeb | A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces[END_REF][START_REF] Jensen | L p -mapping properties of functions of Schrödinger operators and their applications to scattering theory[END_REF][START_REF] Mcintosh | Finite propagation speed for first order systems and Huygens' principle for hyperbolic equations[END_REF].

Following [START_REF] Chen | Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means[END_REF] we introduce the following restriction type estimate

(1.5) F(A)AV σ p τ p→2 ≤ C F q , σ p = 1 p - 1 2 .
We then prove a sharper spectral multiplier result under this condition. Namely,

(1.6) AF(A) p→p ≤ C p F W s,q
for F ∈ W s,q (R) for some

s > n 1 p - 1 2 + n 4[a]
.

We refer to Theorem 4.1 for the precise statement. We prove several other results such as bounds for e itA A on L p for p as in (1.3). The proofs are very much based on Littlewood-Paley type theory, commutator estimates and amalgam spaces [10,[START_REF] Nicola | Sharp L p estimates for Schrödinger groups[END_REF][START_REF] Boutayeb | A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces[END_REF].

Our result can be applied to many examples. Obviously, if the A has an exponential decay (e.g. a Gaussian upper bound) then it satisfies the previous polynomial off-diagonal decay. Hence our results apply to a wide class of elliptic operators on Euclidean domains or on Riemannian manifolds. They also apply in cases where the heat kernel has polynomial decay. This is the case for example for fractional powers of elliptic or Schrödinger operators. In the last section we discuss applications to Markov chains. We also study spectral multipliers (and hence Bochner-Riesz means) for random walk on Z n . To be more precise, we consider

A f (d) := 1 2n n i=1 j=±1 f (d + je i )
where e i = (0, . . . , 1, . . . 0). We prove for appropriate function

F sup t>1 F(t(I -A) p→p ≤ C F H s for any s > n|1/p -1/2|. If in addition, if supp F ⊂ [0, 1/n] and sup 1/n>t>0 ηF(t•) H s < ∞
for some s > n|1/p -1/2| and η is a non-trivial C ∞ c (0, ∞) function, then F(I -A) is bounded on L p if 1 < p < (2n + 2)/(n + 3) and weak type (1, 1) if p = 1. This result is similar to the sharp spectral multiplier theorem for the standard Laplacian on R n . Here again the operator A cannot be, in a natural way, included in a semigroups framework.

1.2. Notations and assumptions. In the sequel we always assume that the considered ambient space is a separable metric measure space (X, d, µ) with metric d and Borel measure µ. We denote by B(x, r) := {y ∈ X, d(x, y) < r} the open ball with centre x ∈ X and radius r > 0. We often use B instead of B(x, r). Given λ > 0, we write λB for the λ-dilated ball which is the ball with the same centre as B and radius λr. For x, ∈ X and r > 0 we set V(x, r) := µ(B(x, r)) the volume of B(x, r). We set

V r (x) := V(x, r), r > 0, x ∈ X. (1.7)
We will often write V(x) in place of V 1 (x). For 1 ≤ p ≤ +∞, we denote by f p the norm of f ∈ L p (X), ., . the scalar product of L 2 (X), and if T is a bounded linear operator from L p (X) to L q (X) we write T p→q for its corresponding operator norm. For a given p ∈ [1, 2) we define

σ p = 1 p - 1 2 . (1.8)
Given a subset E ⊆ X, we denote by χ E the characteristic function of E and set

P E f (x) = χ E (x) f (x).
For every x ∈ X and r > 0.

Throughout this paper we always assume that the space X is of homogeneous type in the sense that it satisfies the classical doubling property (1.2) with some constants C and n independent of x ∈ X and λ ≥ 1. In the Euclidean space with Lebesgue measure, n is the dimension of the space. In our results critical index is always expressed in terms of homogeneous dimension n. Note also that there exists c and D, 0 ≤ D ≤ n so that (1.9)

V(y, r)

≤ c 1 + d(x, y) r D V(x, r)
uniformly for all x, y ∈ X and r > 0. Indeed, the property (1.9) with D = n is a direct consequence of triangle inequality of the metric d and the strong homogeneity property. In the cases of Euclidean spaces R n and Lie groups of polynomial growth, D can be chosen to be 0.

Preliminary results

In this this section we give some elementary results which will be used later.

2.1.

A criterion for L p -L q boundedness for linear operators. We start with a countable partitions of X. For every r > 0, we choose a sequence (x i ) ∞ i=1 ∈ X such that d(x i , x j ) > r/4 for i j and sup x∈X inf i d(x, x i ) ≤ r/4. Such sequence exists because X is separable. Set

D = i∈N B(x i , r/4).
Then define Q i (r) by the formula

Q i (r) = B(x i , r/4)         B (x i , r/2) \         j<i B x j , r/2 \ D                 , (2.1) so that (Q i (r)) i is a countable partition of X (Q i (r) ∩ Q j (r) = ∅ if i j). Note that Q i (r) ⊂ B i = B(x i , r
) and there exists a uniform constant C > 0 depending only on the doubling constants in (1.2) such that µ(Q i (r)) ≥ Cµ(B i ).

We have the following Schur-test for the norm T p→q of a given linear operator T .

Lemma 2.1. Let T be a linear operator and 1 ≤ p ≤ q ≤ ∞. For every r > 0,

T p→q ≤ sup j i P Q i (r) T P Q j (r) p→q + sup i j P Q i (r) T P Q j (r) p→q ,
where

(Q i (r)) i is a countable partition of X.
Proof. The proof is inspired by [START_REF] Guillarmou | Restriction and spectral multiplier theorems on asymptotically conic manifolds[END_REF]. Given a function f ∈ L p (X), we have

T f q q = i, j P Q i (r) T P Q j (r) f q q = i j P Q i (r) T P Q j (r) f q q ≤ i         j P Q i (r) T P Q j (r) p→q P Q j (r) f p         q =: j a i j c j q q ,
where a i j = P Q i (r) T P Q j (r) p→q and c j = P Q j (r) f p . Next note that, for all 1 ≤ p ≤ ∞,

(2.2) k        l |a lk c l |        p ≤        max        sup l k |a lk |, sup k l |a lk |               p n |c n | p ,
with the obvious meaning for p = ∞, where c lk , a l are sequences of real or complex numbers. Indeed, for p = 1 or p = ∞, (2.2) is easy to obtain. Then we obtain (2.2) for all 1 ≤ p ≤ ∞ by interpolation. Observe that

T f q ≤ (a i j ) p → q c j p ≤ (a i j ) p → p f p .
The lemma follows from (2.2) and the above inequality.

2.2.

Operators with kernels satisfying off-diagonal polynomial decays. For a given function W : X → R, we denote by M W the multiplication operator by W, that is

(M W f )(x) = W(x) f (x).
In the sequel, we will identify the operator M W with the function W. This means that, if T is a linear operator, we will denote by

W 1 T W 2 the operators M W 1 T M W 2 .
In other words,

W 1 T W 2 f (x) := W 1 (x)T (W 2 f )(x).
Following [START_REF] Blunck | Weighted norm estimates and maximal regularity[END_REF]8,9], we introduce the following estimates which are interpreted as polynomial offdiagonal estimates.

Definition 2.2. Let A be a self-adjoint operator on L 2 (X) and τ > 0 be a constant. For 1 ≤ p < 2 and a > 0, we say that A satisfies the property (PVE a p,2 (τ)) if there exists a constant C > 0 such that for all x, y ∈ X,

(PVE a p,2 (τ)) P B(x,τ) V σ p τ AP B(y,τ) p→2 < C 1 + d(x, y) τ -n-a with σ p = 1/p -
τ)) holds for some p < 2. Then (PVE a q,2 ) holds for every q ∈ [p, 2]. This can be shown by applying complex interpolation to the family

F(z) = P B(x,τ) V z σ p τ AP B(y,τ) .
For z = 1 + is we use (PVE a p,2 (τ)) and for z = is we use (2.4).

In the sequel, for a given τ > 0 we fix a countable partition {Q (τ)} ∞ =1 of X and and a sequence (x ) ∞ =1 ∈ X as in Section 2.1. First, we have the following result.

Proposition 2.4. Let 1 ≤ p ≤ 2, and A be a self-adjoint operator on L 2 (X). Assume that condition (PVE a p,2 (τ)) holds for some τ > 0 and a > 0. There exists a constant C > 0 independent of τ > 0 such that

V σ p τ A p→2 ≤ C. (2.5)
As a consequence, the operator A is a bounded operator on L p (X), and A p→p + A 2→2 ≤ C. Proof. By Lemma 2.1 and condition (PVE a p,2 (τ)), one is lead to estimate

sup j i 1 + d(x i , x j ) τ -n-a ≤ C < ∞. (2.6) Note that for every k ≥ 1, sup j, τ # i : 2 k τ ≤ d(x i , x j ) < 2 k+1 τ ≤ sup j, τ sup {i:2 k τ≤d(x i ,x j )<2 k+1 τ} V(x i , 2 k+3 τ) V(x i , τ) ≤ C2 kn < ∞. (2.7)
This implies that for every j ≥ 1 and k ≥ 1,

i: 2 k τ≤d(x i ,x j )<2 k+1 τ 1 + d(x i , x j ) τ -n-a ≤ C(1 + 2 k ) -n-a 2 kn ≤ C2 -ka
and we sum over k to get (2.6).

The boundedness of the operator A on L p is proved in the same way by applying (2.3).

Note that when the operator A has integral kernel K A (x, y) satisfying the following pointwise estimate

|K A (x, y)| ≤ CV(x, τ) -1 1 + d(x, y) τ -n-a
for some τ > 0 and all x, y ∈ X, then A satisfies the property (PVE a p,2 (τ)) with p = 1. Conversely, we have the following result. Proposition 2.5. Suppose that a > D where D is the constant in (1.9). If the operator A satisfies the property (PVE a p,2 (τ)) for some τ > 0 and p = 1, then the operator A 2 has integral kernel K A 2 (x, y) satisfying the following pointwise estimate: For any > 0, there exists a constant C > 0 independent of τ such that

|K A 2 (x, y)| ≤ CV(x, τ) -1 1 + d(x, y) τ -(a-D-) (2.8)
for all x, y ∈ X.

Proof. For every x, y ∈ X and τ > 0, we write

P B(x,τ) A 2 P B(y,τ) = P B(x,τ) AP B(x,τ) AP B(y,τ) + ∞ =0 i:2 τ≤d(x i ,x)<2 +1 τ P B(x,τ) AP B(x i ,τ) AP B(y,τ) .
From the property (PVE a p,2 (τ)) with p = 1 we get

P B(x,τ) A 2 P B(y,τ) L 1 →L ∞ ≤ CV(x, τ) -1 1 + d(x, y) τ -n-a +C ∞ =0 i:2 τ≤d(x i ,x)<2 +1 τ V(x i , τ) -1 1 + d(x, x i ) τ -n-a 1 + d(x i , y) τ -n-a .
This, in combination with the fact that V(x i , τ) -1 ≤ C2 D V(x, τ) -1 for every ≥ 0 and 2 τ ≤ d(x i , x) < 2 +1 τ, and the property (2.7), implies that

P B(x,τ) A 2 P B(y,r) L 1 →L ∞ ≤ CV(x, τ) -1 1 + d(x, y) τ -(a-D-)
for any > 0. Hence it follows that (2.8) holds. This completes the proof of Proposition 2.5.

Finally, we mention that if A is the semigroup e -tL generated by (minus) a non-negative selfadjoint operator L, then the condition PVE a p,2 (t 1/m ) holds for some m ≥ 1 if the corresponding heat kernel K t (x, y) has a polynomial decay (2.9)

|K t (x, y)| ≤ CV(x, t 1/m ) -1 1 + d(x, y) t 1/m -n-a
.

it is known that the heat kernel satisfies a Gaussian upper bound, for a wide class of differential operators of order m ∈ 2N on Euclidean domains or Riemannian manifolds (see for example [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF]).

In this case (2.9) holds with any arbitrary a > 0.

Spectral multipliers via polynomial off-diagonal decay kernels

In this section we prove spectral multiplier results corresponding to compactly supported functions in the abstract setting of self-adjoint operators on homogeneous spaces. Recall that we assume that (X, d, µ) is a metric measure space satisfying the doubling property and n is the homogeneous dimension from condition (1.2). We use the standard notation H s (R) for the Sobolev space

F H s = (I -d 2 /dx 2 ) s/2 F 2 . Theorem 3.1. Suppose that A is a bounded self-adjoint operator on L 2 (X) which satisfies condition (PVE a p,2 (τ)) for some τ > 0, 1 ≤ p < 2 and a > [n/2] + 1. If F is a bounded Borel function such that F ∈ H s (R) for some s > n(1/p -1/2) + 1/2, then there exists constant C > 0 such that F(A)A p→p ≤ C F H s .
The constant C above does not depend on the choice of τ > 0. In addition, if we assume that A is a nonnegative self-adjoint operator and supp F ⊂ [-1, 1], then there exists a constant C > 0 which is also independent of τ such that

F(-log A) p→p ≤ C F H s .
The proof of Theorem 3.1 will be given at the end of this section after a series of preparatory results. The following statement is a direct consequence of Theorem 3.1.

Corollary 3.2. Suppose that L is a non-negative self-adjoint operator on L 2 (X) that for a given τ > 0 the semigroup operator e -τL satisfies condition (PVE a p,2 (τ)) for some

1 ≤ p < 2 and a > [n/2] + 1. If F is a bounded Borel function such that supp F ⊂ [-1, 1] and F ∈ H s (R) for some s > n(1/p -1/2) + 1/2, then there exists constant C > 0 such that F(τL) p→p ≤ C F H s .
As a consequence if operators e -τL satisfies condition (PVE a p,2 (τ)) with constant independent of τ then

sup τ>0 F(τL) p→p ≤ C F H s
for the same range of exponents p and s.

Proof. We apply Theorem 3.1 to the operator A = e -τL for all τ > 0. Then the theorem follows from the fact that the constants in statement of Theorem 3.1 are independent of τ.

Remark 3.3. It is possible to obtain a version of Theorem 3.1 under the weaker assumption a > 0. However, this requires a different approach which we do not discuss here. Related results can be found in [22] and [29]. We will consider this more general case in a forthcoming paper [START_REF] Chen | Hörmander type spectral multipliers for operators generating slowly decaying semigroups[END_REF].

3.1. Preparatory results. The following result plays a essential role in the proof of Theorem 3.1.

Theorem 3.4. Suppose that A is a bounded self-adjoint operator on L 2 (X) and satisfies condition (PVE a p,2 (τ)) for 1 ≤ p < 2, a > [n/2] + 1 and for some τ > 0. Then there exists a constant C > 0 such that for all t ∈ R,

e itA A p→p ≤ C(1 + |t|) nσ p (3.1)
where σ p = 1/p -1/2. Remark 3.5. It is interesting to compare the above statement with Theorem 1.1 of [START_REF] Nicola | Sharp L p estimates for Schrödinger groups[END_REF]. Note that in Theorem 3.4 we assume condition (PVE a p,2 (τ)) for one fixed exponent τ but conclusion is valid for all t ∈ R.

In order to prove Theorem 3.4 we need two technical lemmas. First, we state the following known formula for the commutator of a Lipschitz function and an operator T on L 2 (X) on metric measure space X. Recall our notation convention ηT = M η T . Lemma 3.6. Let T be a self-ajoint operator on L 2 (X). Assume that for some η ∈ Lip(X), the commutator

[η, T ], defined by [η, T ] f = M η T f -T M η f , satisfies that for all f ∈ Dom(T ), η f ∈ Dom(T ) and [η, T ] f 2 ≤ C f 2 ,
where Dom(T ) denotes the domain of T . Then the following formula holds:

[η, e itT ] f = it 1 0 e istT [η, T ]e i(1-s)tT f ds ∀t ∈ R, ∀ f ∈ L 2 (X).
Proof. The proof of Lemma 3.6 follows by integration by parts. See for example, [START_REF] Mcintosh | Finite propagation speed for first order systems and Huygens' principle for hyperbolic equations[END_REF]Lemma 3.5].

Next we recall some useful results concerning amalgam spaces [10, [START_REF] Nicola | Sharp L p estimates for Schrödinger groups[END_REF][START_REF] Boutayeb | A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces[END_REF]. For a given τ > 0, we recall that {Q (τ)} ∞ =1 is a countable partition of X as in Section 2.1. For 1 ≤ p, q ≤ ∞ and a non-negative number τ > 0, consider a two-scale Lebesgue space X p,q τ of measurable functions on X equipped with the norm

f X p,q τ :=        ∞ =1 P Q (τ) f p q        1/p . (3.2) (with obvious changes if q = ∞).
Notice that when q = p these spaces are just the Lebesgue spaces X p,p τ = L p for every τ > 0 and p. Note also that for 1 ≤ p 1 < p 2 ≤ ∞, we have that

X p 1 ,q τ ⊆ X p 2 ,q τ with f X p 2 ,q τ ≤ f X p 1 ,q τ .
The following result gives a criterion for a linear operator A to be bounded on X p,2 τ , 1 ≤ p ≤ 2. We define a family of functions {η (x)} by

η (x) := d(x, x ) τ , = 1, 2, • • • (3.3)
i.e., the distance function between x and x (up to a constant).

Lemma 3.7. Let τ > 0 and {η (x)} ∞ =1 be as above. For a bounded operator T on L 2 (X), the multi-commutator ad k (T ) :

L 2 (X) → L 2 (X) is defined inductively by ad 0 (T ) = T, ad k (T ) = ad k-1 (η T -T η ), ≥ 1, k ≥ 1. (3.4)
Suppose that there exists a constant M > 1 such that for all 1 ≤ < ∞,

ad k (T ) 2→2 ≤ CM k , 0 ≤ k ≤ [n/2] + 1. Then for 1 ≤ p ≤ 2, V σ p τ T V -σ p τ f X p,2 τ ≤ CM nσ p f X p,2 τ with σ p = (1/p -1/2)
for some constant C > 0 depending on n, p and T 2→2 .

Proof. We prove this lemma by using the complex interpolation method. Let S be the closed strip 0 ≤ Rez ≤ 1 in the complex plane. For every z ∈ S, we consider the analytic family of operators:

T z := V z/2 r T V -z/2 τ .
Consider z = 1 + iy, y ∈ R and p = 1. Let N ≥ 1 be a constant large enough to be chosen later. Let {Q (τ)} be a countable partition of X in Section 2.1. By definition of X 1,2 τ ,

T 1+iy f X 1,2 τ = ∞ j=1 P Q j (τ) V 1+iy 2 τ T V -1+iy 2 τ f 2 ≤ ∞ =1 ∞ j=1 P Q j (τ) V 1+iy 2 τ T V -1+iy 2 τ P Q (τ) f 2 =          j: d(x ,x j )>Nτ + j: d(x ,x j )≤Nτ          P Q j (τ) V 1+iy 2 τ T V -1+iy 2 τ P Q (τ) f 2 =: I + II. (3.5)
By the Cauchy-Schwarz inequality, we obtain

II ≤ C ∞ =1          j: d(x ,x j )≤Nr V τ (x j )          1/2          j: d(x ,x j )≤Nτ P Q j (τ) T V -1+iy 2 τ P Q (τ) f 2 2          1/2 ≤ C V(x , Nτ) 1/2 T V -1+iy 2 τ P Q (τ) f 2 ≤ CN n/2 T 2→2 µ(Q (τ)) 1/2 V -1+iy 2 τ P Q (τ) f 2 ≤ CN n/2 T 2→2 f X 1,2 τ . (3.6)
Now we estimate the term I. Let κ = [n/2] + 1. We apply the Cauchy-Schwarz inequality again to obtain

I ≤ C ∞ =1          j: d(x ,x j )>Nτ µ(Q j (τ))η (x j ) -2κ          1/2          j:d(x ,x j )>Nτ η (x j ) 2κ P Q j (τ) T V -1+iy 2 τ P Q (τ) f 2 2          1/2 ≤ C          ∞ k=0 j: 2 k Nτ<d(x ,x j )≤2 k+1 Nτ µ(Q j (τ))η (x j ) -2κ          1/2 η κ T V -1+iy 2 τ P Q (τ) f 2 ≤ C        ∞ k=0 (2 k N) n µ(Q (τ))(2 k N) -2κ        1/2 η κ T V -1+iy 2 τ P Q (τ) f 2 ≤ C N -κ+n/2 µ(Q (τ)) 1/2 η κ T V -1+iy 2 τ P Q (τ) f 2 . (3.7)
To continue we define we let Γ(κ, 0) = 1 for κ ≥ 1, and Γ(κ, m) defined inductively by

Γ(κ, m + 1) = κ-1 =m Γ( , m) for 1 ≤ m ≤ κ -1.
Applying the following known formula for commutators (see Lemma 3.1, [START_REF] Jensen | L p -mapping properties of functions of Schrödinger operators and their applications to scattering theory[END_REF]):

η κ T = κ m=0 Γ(κ, m)ad m (T )η κ-m ,
we obtain

η κ T (1 + η ) -κ 2→2 ≤ C κ m=0 ad m (T )η κ-m (1 + η ) -κ 2→2 ≤ C κ m=0 ad m (T ) 2→2 ≤ CM κ . (3.8)
This implies

I ≤ CN -κ+n/2 M κ µ(Q (τ)) 1/2 V -1+iy 2 τ (1 + η ) κ P Q (τ) f 2 ≤ CN -κ+n/2 M κ P Q (τ) f 2 ≤ CN -κ+n/2 M κ f X 1,2 τ . Next, set N = M T -1/κ 2→2 .
Then above estimates of I and II give

T 1+iy f X 1,2 τ ≤ C N -κ+n/2 M κ + N n/2 T 2→2 f X 1,2 τ ≤ CM n/2 T 1-1/κ 2→2 f X 1,2 τ . (3.9)
On the other hand, we have that for z = iy, y ∈ R,

T iy X 2,2 τ →X 2,2 τ = T iy 2→2 ≤ T 2→2 (3.10)
From estimates (3.9) and (3.10), we apply the complex interpolation method to obtain

V σ p τ T V -σ p τ f X p,2 τ = T 2 p -1 f X p,2 τ ≤ CM nσ p f X p,2 τ , σ p = (1/p -1/
2) for some constant C > 0 depending on n, p and T 2→2 . This finishes the proof of Lemma 3.7. Now we apply Lemmas 3.6 and 3.7 to prove Theorem 3.4.

Proof f Theorem 3.4. The proof is inspired by Theorem 1.3 of [START_REF] Jensen | L p -mapping properties of functions of Schrödinger operators and their applications to scattering theory[END_REF] and Theorem 1.1 of [START_REF] Nicola | Sharp L p estimates for Schrödinger groups[END_REF]. Note that

e itA A p→p ≤ V -σ p τ X p,2 τ →L p V σ p τ e itA V -σ p τ X p,2 τ →X p,2 τ V σ p τ A L p →X p,2 τ . First, we have that V -σ p τ X p,2
τ →L p ≤ C by definition of X p,2 τ and Hölder's inequality. To estimate the term V

σ p τ A L p →X p,2 τ , we recall that ({Q i (τ)} ∞ i=1
) is a countable partition of X as in Section 2.1 and note that

V σ p τ A f X p,2 τ =         ∞ j=1 P Q j (τ) V σ p τ A f p 2         1/p ≤         j        P Q j (τ) V σ p τ AP Q (τ) p→2 P Q (τ) f p        p         1/p .
For every j, , we set

a j = P Q j (τ) V σ p τ AP Q (τ) p→2 and b = P Q (τ) f p . It follows by interpolation that V σ p τ A f X p,2 τ ≤ a j b p ≤ (a j ) p → p b p ≤ (a j ) θ 1 → 1 (a j ) 1-θ ∞ → ∞ f p . Therefore, by condition (PVE a p,2 (τ)) we have (a j ) 1 → 1 = sup j P Q j (τ) V σ p τ AP Q (τ) p→2 ≤ C and (a j ) ∞ → ∞ = sup j P Q j (τ) V σ p τ AP Q (τ) p→2 ≤ C. Thus (3.11) V σ p τ A L p →X p,2 τ ≤ C.
Next we show that

V σ p τ e itA V -σ p τ X p,2 τ →X p,2 τ ≤ C(1 + |t|) nσ p . (3.12)
By Lemma 3.7, it suffices to show for every m ∈ N,

ad k m (e itA ) 2→2 ≤ C(1 + |t|) k , 0 ≤ k ≤ [n/2] + 1. (3.13)
Note that A is a bounded operator on L 2 (X). Recall that (cf. Lemma 3.6) for all m ∈ N:

ad 1 m (e itA ) f = it 1 0 e istA ad 1 m (A)e i(1-s)tA f ds, ∀t ∈ R, f ∈ L 2 (X).
Repeatedly, we are reduced to prove that for every m ∈ N, there exists a constant C > 0 independent of m such that,

ad k m (A) 2→2 ≤ C, 0 ≤ k ≤ [n/2] + 1. (3.14) Fix m ∈ N. By Lemma 2.1, it suffices to show sup j P Q (τ) ad k m (A)P Q j (τ) 2→2 ≤ C, 0 ≤ k ≤ [n/2] + 1 (3.15)
for some constant C > 0 independent of m.

To show (3.15) we note that

P Q (τ) ad k m (A)P Q j (τ) = α+β+γ=k k! α! β! γ! [η m (x ) -η m (x j )] β [η m (•) -η m (x )] α × ×P Q (τ) AP Q j (τ) [η m (x j ) -η m (•)] γ .
Observe that:

• |η m (x ) -η m (x j )| ≤ d(x , x j )/τ; • |η m (x) -η m (x )|χ Q (τ) ≤ d(x, x )χ Q (τ) /τ ≤ 1; • |η m (x j ) -η m (y)|χ Q j (τ) ≤ d(y, x j )χ Q j (τ) /τ ≤ 1.
These, together with estimate (PVE a p,2 (τ)) and a > [n/2] + 1 ≥ k, yield

P Q (τ) ad k m (A)P Q j (τ) 2→2 ≤ C 1 + d(x , x j ) τ k P Q (τ) AP Q j (τ) 2→2 ≤ C
for some constant C > 0 independent of j. Hence, (3.15) is proved. This, in combination with estimates (3.13) and (3.14), implies (3.12). All together, we obtain that e itA A p→p ≤ C(1 + |t|) nσ p where σ p = (1/p -1/2). The proof of Theorem 3.4 is complete.

3.2. Proof of Theorem 3.1. We apply Lemma 3.4 to see that

F(A)A p→p ≤ R | F(ξ)| e iξA A p→p dξ ≤ C R | F(ξ)|(1 + |ξ|) nσ p dξ ≤ C F H s R (1 + |ξ|) 2(nσ p -s) dξ 1/2 ≤ C F H s .
If we also assume that A ≥ 0 and supp F ⊂ [-1, 1], then we may consider G(λ) := F(-log λ)λ -1 so that F(-log A) = G(A)A. Therefore,

F(-log A) p→p ≤ G(A)A p→p ≤ C G H s .
Since supp F ⊂ [-1, 1] we have G H s ≤ C F H s and we obtain F(-log A) p→p ≤ C F H s . The proof of Theorem 3.1 is complete.

Sharp spectral multiplier results via restriction type estimates

The aim of this section is to obtain sharp L p boundedness of spectral multipliers from restriction type estimates. We consider the metric measure space (X, d, µ) with satisfies the doubling condition (1.2) with the homogeneous dimension n. Let q ∈ [2, ∞]. Recall that V r = V(x, r) = µ(B(x, r)). We say that A satisfies the restriction type condition if there exist interval [b, e] for some -∞ < b < e < ∞ and τ > 0 such that for all Borel functions F with supp F ⊂ [b, e], (ST q p,2 (τ))

F(A)AV σ p τ p→2 ≤ C F q , σ p = 1 p - 1 2 .
The above conditions originates and in fact is a version of the classical Stein-Tomas restriction estimates. For more detailed discussion and the rationale of formulation of the above condition we referee readers to [START_REF] Chen | Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means[END_REF][START_REF] Sikora | Sharp spectral multipliers for operators satisfying generalized Gaussian estimates[END_REF]) for a related definition).

The following statement is our main result in this section.

Theorem 4.1. Let 1 ≤ p < 2 ≤ q ≤ ∞. Let A be a bounded self-adjoint operator on L 2 (X) satisfying the property (PVE a p,2 (τ)) for some a > [n/2] + 1 and τ > 0. Suppose also that A satisfies the property (ST q p,2 (τ)) on some interval

[R 1 , R 2 ] for -∞ < R 1 < R 2 < ∞. Let F be a bounded Borel function such that supp F ⊂ [R 1 + γ, R 2 -γ] for some γ > 0 and F ∈ W s,q (R) for some s > n 1 p - 1 2 + n 4[a]
.

Then AF(A) is bounded on L p (X) and

(4.1) AF(A) p→p ≤ C p F W s,q .
Remark 4.2. One can formulate a version of Corollary 3.2 corresponding to Theorem 4.1. See also the statement of Theorem 5.3.

Remark 4.3. Note that if A satisfies (PVE a p,2 (τ)) for some a > [n/2] + 1 and τ > 0 then A satisfies (ST q p,2 (τ)) with q = ∞. Indeed, by Proposition 2.4, we have F(A)AV

σ p τ p→2 ≤ F(A) 2→2 AV σ p τ p→2 ≤ C F ∞ .
As a consequence of Theorem 4.1 we obtain under the sole assumption

(PVE a p,2 (τ)) (4.2) AF(A) p→p ≤ C p F W s,∞
for every F ∈ W s,∞ and some s >

n 1 p -1 2 + n 4[a] .
Before we start the proof of Theorem 4.1, we need some preliminary result. For a given r > 0, we recall that (Q i (r)) i is a countable partition of X. For a bounded operator T and a given r > 0, we decompose the operator T into the on-diagonal part [T ] <r and the off-diagonal part [T ] >r as follows:

[T ] <r := i j:d(x i ,x j )≤5r

P Q j (r) T P Q i (r) (4.3) and [T ] >r := i j:d(x i ,x j )>5r P Q j (r) T P Q i (r) . (4.4)
For the on-diagonal part [T ] <r , we have the following result.

Lemma 4.4. Assume that T is a bounded operator from L p (X) to L q (X) (p ≤ q). Then the ondiagonal part [T ] <r is bounded on from L p (X) to L q (X) and there exists a constant C = C(n) > 0 independent of r such that

[T ] <r p→q ≤ C T p→q .

Proof. Note that i j:d(x i ,x j )≤5r

P Q j (r) T P Q i (r) f q q = j i:d(x i ,x j )≤5r P Q j (r) T P Q i (r) f q q ≤ C j i:d(x i ,x j )≤5r P Q j (r) T P Q i (r) f q q ≤ C i j:d(x i ,x j )≤5r T q p→q P Q i (r) f q p ≤ C T q p→q i P Q i (r) f q p ≤ C T q p→q f q p . This proves Lemma 4.4. Proof of Theorem 4.1. Let φ ∈ C ∞ c (R) be a function such that supp φ ⊆ {ξ : 1/4 ≤ |ξ| ≤ 1} and ∈Z φ(2 -λ) = 1 for all λ > 0. Set φ 0 (λ) = 1 -∞ =1 φ(2 -λ).
By the Fourier inversion formula, we can write

AF(A) = ∞ =0 AF ( ) (A), (4.5)
where

F (0) (λ) = 1 2π +∞ -∞ φ 0 (s) F(s)e isλ ds (4.6) and F ( ) (λ) = 1 2π +∞ -∞
φ(2 -s) F(s)e isλ ds. (4.7)

We now estimate L p -L p norm of the operator AF ( ) (A), ≥ 0. Let N ≥ 1 be a constant to be chosen later. For every ≥ 0, we follow (4.3) and (4.4) to write

AF ( ) (A) =: [AF ( ) (A)] <Nτ + [AF ( ) (A)] >Nτ . (4.8) Estimate of the term [AF ( ) (A)] <Nτ . Let ψ ∈ C ∞ c (R) be a function such that ψ(λ) = 1 for λ ∈ [R 1 + γ/2, R 2 -γ/2] and supp ψ ⊂ [R 1 , R 2 ]. We write [AF ( ) (A)] <Nτ = [ψ(A)AF ( ) (A)] <Nτ + [(1 -ψ(A))AF ( ) (A)] <Nτ . Observe that [ψ(A)AF ( ) (A)] <Nτ p p ≤ C j λ:d(x j ,x λ )≤5Nτ P Q λ (Nτ) Aψ(A)F ( ) (A)P Q j (Nτ) f p p .
We use the Hölder inequality to obtain

P Q λ (Nτ) Aψ(A)F ( ) (A)P Q j (Nτ) f p ≤ µ(Q λ (Nτ)) σ p P Q λ (Nτ) Aψ(A)F ( ) (A)P Q j (Nτ) f 2 ≤ Cµ(Q j (Nτ)) σ p F ( ) q V 1/2-1/p τ P Q j (Nτ) f p .
Note that in the last inequality we used the condition d(x j , x λ ) ≤ 5Nτ and the fact that supp ψF

( ) ⊂ [R 1 , R 2 ]
, and it follows from the property (ST q p,2 (τ)) that Aψ(A)F ( ) (A)V

σ p τ p→2 ≤ F ( ) q . Hence, [ψ(A)AF ( ) (A)] <Nτ p p ≤ CN npσ p F ( ) p q j µ(Q j (τ)) pσ p V -σ p τ P Q j (Nτ) f p p ≤ CN npσ p F ( ) p q j P Q j (Nτ) f p p ≤ CN npσ p F ( ) p q f p p . (4.9)
On the other hand, we apply Lemma 4.4 and Theorem 3.1 to obtain that for some s > n/2 + 1 and every large number M > 0,

[(1 -ψ(A))AF ( ) (A)] <Nτ p→p ≤ C (1 -ψ(A))AF ( ) (A) p→p ≤ C (1 -ψ(•))F ( ) (•) H s ≤ C σ,M 2 -M F q .
This, together with (4.9), yields

[AF ( ) (A)] <Nτ f p p ≤ CN npσ p F ( ) p q + 2 -M p F p q f p p . (4.10)
Estimate of the term [AF ( ) (A)] ≥Nτ . From (4.6) and (4.7), we have that

[AF ( ) (A)] >Nτ = 1 2π +∞ -∞ φ(2 -t) F(t)[Ae itA ] >Nτ dt. (4.11)
To estimate the term [Ae itA ] >Nτ , we write

Ae itA = e itA [A] >Nτ/2 + [e itA ] >Nτ/2 [A] <Nτ/2 + [e itA ] <Nτ/2 [A] <Nτ/2 =: T 1 + T 2 + T 3 .
It is easy to see that i j:d(x i ,x j )>5Nτ P Q j (Nτ) T 3 P Q i (Nτ) = 0, and so [Ae itA ] >Nτ =:

[T 1 ] >Nτ + [T 2 ] >Nτ .
By Lemma 4.4 with r = Nτ, we obtain

[Ae itA ] >Nτ p→p ≤ T 1 -[T 1 ] <Nτ p→p + T 2 -[T 2 ] <Nτ p→p ≤ (1 + C) T 1 p→p + (1 + C) T 2 p→p , (4.12)
and hence we have to estimate T i p→p , i = 1, 2.

Let us estimate the term T 1 p→p . We write

T 1 p→p = e itA [A] >Nτ/2 p→p ≤ V -σ p τ X p,2 τ →L p V σ p τ e itA V -σ p τ X p,2 τ →X p,2 τ V σ p τ [A] >Nτ/2 L p →X p,2 τ . (4.13) We have that V -σ p τ X p,2
τ →L p ≤ C by definition of X p,2 and Hölder's inequality. Also it follows from (3.12) that

V σ p τ e itA V -σ p τ X p,2 τ →X p,2 τ ≤ C(1 + |t|) nσ p . (4.14)
To handle the term V

σ p τ [A] >Nτ/2 L p →X p,2 τ , we note that V σ p τ [A] >Nτ/2 f X p,2 τ =         ∞ j=1 P Q j (τ) V σ p τ [A] >Nτ/2 f p 2         1/p ≤          j          :d(x ,x j )>Nτ/2 P Q j (τ) V σ p τ AP Q (τ) p→2 P Q (τ) f p          p          1/p .
For every j, , we set

a j = P Q j (τ) V σ p τ AP Q (τ) p→2 and b = P Q (τ) f p . This gives V σ p τ [A] >Nτ/2 f X p,2 τ ≤ a j b p ≤ (a j ) p → p b p ≤ (a j ) θ 1 → 1 (a j ) 1-θ ∞ → ∞ f p by interpolation. It then follows from condition (PVE a p,2 (τ)) that (a j ) 1 → 1 = sup j:d(x ,x j )>Nτ/2 P Q j (τ) V σ p τ AP Q (τ) p→2 ≤ CN -a and (a j ) ∞ → ∞ = sup j :d(x ,x j )>Nτ/2 P Q j (τ) V σ p τ AP Q (τ) p→2 ≤ CN -a ,
and so

V σ p τ [A] >Nτ/2 L p →X p,2 τ ≤ CN -a .
This, in combination with (4.14) and (4.13), shows that

T 1 p→p = e itA [A] >Nτ/2 p→p ≤ CN -a (1 + t) nσ p . (4.15)
Next we estimate the term T 2 p→p . We write

T 2 p→p = [e itA ] >Nτ/2 [A] <Nτ/2 p→p ≤ V -σ p τ X p,2 τ →L p V σ p τ [e itA ] >Nτ/2 V -σ p τ X p,2 τ →X p,2 τ V σ p τ [A] <Nτ/2 L p →X p,2 τ . (4.16) We have that V -σ p τ X p,2
τ →L p ≤ C by definition of X p,2 and Hölder's inequality. Also, we follow a similar argument as in (3.11) to obtain

V σ p τ [A] <Nτ/2 L p →X p,2 τ ≤ C. (4.17)
In order to deal with the term V

σ p τ [e itA ] >Nτ/2 V -σ p τ X p,2 τ →X p,2
τ , we apply the complex interpolation method. To do it, we let S denote the closed strip 0 ≤ Rez ≤ 1 in the complex plane. For z ∈ S, we consider an analytic family of operators:

T z = V z 2 τ [e itA ] >Nτ/2 V -z 2 τ .
Case 1: z = 1 + iy, y ∈ R and p = 1. In this case, we observe that

T 1+iy f X 1,2 τ = ∞ j=1 P Q j (τ) V 1+iy 2 τ [e itA ] >Nτ/2 V -1+iy 2 τ f 2 ≤ ∞ =1 ∞ j=1 P Q j (τ) V 1+iy 2 τ [e itA ] >Nτ/2 V -1+iy 2 τ P Q (τ) f 2 ≤ ∞ =1 j: d(x ,x j )>Nτ/2 P Q j (τ) V 1+iy 2 τ e itA V -1+iy 2 τ P Q (τ) f 2 . (4.18)
By the Cauchy-Schwarz inequality, j: d(x ,x j )>Nτ/2

P Q j (τ) V 1+iy 2 τ e itA V -1+iy 2 τ P Q (τ) f 2 ≤          j: d(x ,x j )>Nτ/2 µ(Q j (τ))η (x j ) -2[a]          1/2          d(x ,x j )>Nτ/2 η (x j ) 2[a] P Q j (τ) e itA V -1+iy 2 τ P Q (τ) f 2 2          1/2
, where η (x) := d(x, x )/τ. Since

j: d(x ,x j )>Nτ/2 µ(Q j (τ))η (x j ) -2[a] ≤ C ∞ i=0 j: 2 i Nτ<d(x ,x j )≤2 i+1 Nτ µ(Q j (τ))η (x j ) -2[a] ≤ C ∞ i=0 (2 i N) n µ(Q (τ))(2 i N) -2[a] ≤ CN -2[a]+n µ(Q (τ))
and

j: d(x ,x j )>Nτ/2 η (x j ) 2[a] P Q j (τ) e itA V -1+iy 2 τ P Q (τ) f 2 2 ≤ C η [a] e itA V -1+iy 2 τ P Q (τ) f 2 2 ,
we have

T 1+iy f X 1,2 τ ≤ C ∞ =1 N -[a]+n/2 µ(Q (τ)) 1/2 η [a] e itA V -1+iy 2 τ P Q (τ) f 2
Following an argument as in (3.8) and (3.13) we obtain [a] . From this, it follows that

η [a] e itA (1 + η ) -[a] 2→2 ≤ C(1 + t)
T 1+iy f X 1,2 τ ≤ CN -[a]+n/2 (1 + t) [a] i µ(Q (τ)) 1/2 V -1+iy 2 τ (1 + η ) [a] P Q (τ) f 2 ≤ CN -[a]+n/2 (1 + t) [a] i P Q (τ) f 2 ≤ CN -[a]+n/2 (1 + t) [a] f X 1,2
τ . Case 2: z = iy, y ∈ R and p = 2. In this case, we note that

T iy X 2,2 τ →X 2,2 τ = T iy 2→2 ≤ [e itA ] >Nτ/2 2→2 ≤ (1 + C) e itA 2→2 ≤ 1 + C.
From Cases 1 and 2, we apply the complex interpolation method to obtain

V σ p τ [e itA ] >Nτ/2 V -σ p τ f X p,2 τ = T 2 p -1 f X p,2 τ ≤ C p N -[a]+n/2 (1 + t) [a] θ f X p,2 τ
where θ = 2/p -1. This implies

T 2 p→p ≤ CN (-[a]+n/2)(2/p-1) (1 + t) [a](2/p-1)
Substituting this and estimate (4.15) back into (4.12), yields

[Ae itA ] >Nτ p→p ≤ C T 1 p→p + C T 2 p→p ≤ CN (-[a]+n/2)(2/p-1) (1 + t) [a](2/p-1) .
This, in combination with (4.11) and (4.10), shows

AF (A) p→p ≤ CN n(1/p-1/2) F ( ) (λ) q + 2 -M F q + CN (-[a]+n/2)(2/p-1) 2 +1 2 |φ(2 -t)|| F(t)|(1 + t) [a](2/p-1) dt.
We take N = 2 (1+p/(2[a](2-p))) to obtain

AF (A) p→p ≤ C2 n(1/p-1/2)(1+p/(2[a](2-p))) F ( ) (λ) q + 2 -M F q + φ F 2 .
After summation in , we obtain

AF(A) p→p ≤ C F W s,q
as in (4.1), whenever

s > n 1 p - 1 2 1 + p 2[a](2 -p) = n 1 p - 1 2 + n 4[a]
.

The proof of Theorem 4.1 is complete.

Applications

As an illustration of our results we shall discuss some examples. Our main results, Theorems 3.1 and 4.1. can be applied to all examples which are discussed in [19], [START_REF] Chen | Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means[END_REF] and [START_REF] Sikora | Sharp spectral multipliers for operators satisfying generalized Gaussian estimates[END_REF].

5.1. Symmetric Markov chains. In [START_REF] Maati Ouhabaz | Spectral multipliers for Markov chains[END_REF] Alexopoulos considers bounded symmetric Markov operator A on a homogeneous space X whose powers A k have kernels A k (x, y) satisfying the following Gaussian estimates

(5.1) 0 ≤ A k (x, y) ≤ C V(x, k 1/2 ) exp - d(x, y) 2 k
for all k ∈ N. The operator I -A is symmetric and satisfies for every f ∈ L 2 (X),

(I -A) f, f = 1 2 ( f (x) -f (y)) 2 A(x, y)dµ(x)dµ(y) ≥ 0.
Thus, I -A is positive. In addition, (

I -A) f 2 ≤ f 2 + A f 2 ≤ 2 f 2 .
Hence, I -A admits the spectral decomposition (see for example, [START_REF] Mcintosh | Operators which have an H ∞ functional calculus[END_REF]) which allows to write

I -A = 2 0 (1 -λ)dE A (λ).
Let F be a bounded Borel measurable function. Then by the spectral theorem we can define the operator

F(I -A) = 2 0 F(1 -λ)dE A (λ).
Note that I -A is bounded on L 2 (X) and F(I -A) 2→2 ≤ F ∞ . In [START_REF] Maati Ouhabaz | Spectral multipliers for Markov chains[END_REF] Alexopoulos obtained the following spectral multiplier type result. In the sequel, let us consider a function 0 ≤ η ∈ C ∞ (R) and let us assume that η(t) = 1 for t ∈ [START_REF] Maati Ouhabaz | Spectral multipliers for Markov chains[END_REF][START_REF]Spectral multipliers on Lie groups of polynomial growth[END_REF] and that η(t) = 0 for t [1/2, 4].

Theorem 5.1. Assume that F is a bounded Borel function with supp F ⊂ [0, 1/2] and that

sup 0<t≤1 η(•)F(t•) W n/2+ ,∞ < ∞
for some > 0. Then under above assumption on the operator A, the spectral multiplier F(I -A) extends to a bounded operator on L p for 1 ≤ p ≤ ∞.

Our approach allows us to prove a version of Alexopoulos' result under the weaker assumption of polynomial decay rather than the exponential one. We start with the following statement.

Theorem 5.2. Let 1 ≤ p < 2 ≤ q ≤ ∞. Suppose that (X, d, µ) satisfies the doubling condition with the doubling exponent n from (1.2). Assume next that A is a bounded self-adjoint operator and there exists k ∈ N such that the kernel of the operator A k exists and satisfies the following estimate

(5.2) |A k (x, y)| ≤ C 1 max(V(x, k 1/2 ), V(y, k 1/2 )) 1 + d(x, y) 2 k -N for some N > n + [n/2] + 1. Then for every 1 ≤ p ≤ ∞, (5.3)
F(A k )A k p→p ≤ C min F W s,∞ , F H s for any s > nσ p + n 4[a]
and any s > nσ p + 1 2 . If in addition the restriction type bounds (ST q p,2 (τ)) with τ = k 1/2 are valid on the interval (R 1 , R 2 ) for some -∞ < R 1 < R 2 < ∞ and suppF ⊂ (R 1 + γ, R 2 -γ) for some γ > 0 then

(5.4) F(A k )A k p→p ≤ C F W s,q s > nσ p + n 4[a]
. Proof. It follows from (5.2) that the operator A k satisfies the property (PVE a p,2 (τ)) with a = Nn and τ = k 1/2 . The theorem follows from Theorems 3.1, 4.1 and Remark 4.3.

The following result is a direct consequence of the above theorem.

Theorem 5.3. Let 1 ≤ p < 2 ≤ q ≤ ∞. Suppose that (X, d, µ) satisfies the doubling condition with the doubling exponent n from (1.2) and that N > n + [n/2] + 1. Assume next that A is a bounded self-adjoint operator and that for all k ∈ N the kernel of the operator A k exists and satisfies the following estimate

(5.5) |A k (x, y)| ≤ C 1 max(V(x, k 1/2 ), V(y, k 1/2 )) 1 + d(x, y) 2 k -N
with the constant C independent of k. In addition we assume that the restriction type bounds (ST q p,2 (τ)) with τ = k 1/2 are valid on the interval (R 1 , R 2 ) ⊂ [-1, 1] for all k ∈ N. Then for function F with supp F ⊂ [1/4, 1/2] (5.6)

F(k(I -A)) p→p ≤ C F W s,q
for any s > nσ p + n 4[a] .

Proof. Note that by (5.5) we have that A k 2→2 ≤ C < ∞ for some constant C independent of k. It follows that the spectrum of A is contained in the interval [-1, 1]. For a given k ∈ N, we define a function G as

G(λ) = F(k(1 -λ 1/k )) λ .
and so G(λ k )λ k = F(k(1 -λ)). It follows from Theorem 5.2 that for s

> nσ p + n 4[a] , G(A k )A k p→p ≤ C G W s,q , which yields F(k(I -A)) p→p = G(A k )A k p→p ≤ C G W s,q = C F(k(1 -λ 1/k )) λ W s,q
.

Note that supp F ⊂ [1/4, 1/2], we have

F(k(I -A)) p→p ≤ C F(k(1 -λ 1/k )) λ W s,q ≤ C F W s,q .
This completes the proof of Theorem 5.3.

Remark 5.4. 1) Note that we do not assume that the operator A is Markovian.

2) Using similar technique as in [START_REF] Sikora | Sharp spectral multipliers for operators satisfying generalized Gaussian estimates[END_REF] one can obtain the singular integral version of Theorem 5.2 stated in Theorem 5.1. We do not discuss the details here.

5.2.

Random walk on Z n . In this section, we consider random walk on the n-dimensional integer lattice Z n . Define the operator A acting on l 2 (Z n ) by the formula

A f (d) = 1 2n n i=1 j=±1 f (d + je i )
where e i = (0, . . . , 1, . . . 0) and 1 is positioned on the i-coordinate. The aim of this section is to prove the following result. Recall that η is the auxiliary nonzero compactly supported function η ∈ C ∞ c [1/2, 4] as in Theorem 5.1.

Theorem 5.5. Let 1 < p < ∞. Let A be the random walk on the integer lattice defined above. Suppose that supp F ⊂ (0, 1/n). Then

sup t>1 F(t(I -A) p→p ≤ C F H s (5.7)
for any s > n|1/p -1/2|.

Next we assume that a bounded Borel function F : R + → C satisfies supp F ⊂ [0, 1/n] and

sup 1/n>t>0 ηF(t•) H s < ∞ (5.8)
for some s > n|1/p -1/2|. Then the operator F(I -A) is bounded on L p if 1 < p < (2n + 2)/(n + 3) and weak type (1, 1) if p = 1.

The proof of Theorem 5.5 is given at the end of this section and it is based on the following restriction type estimate. Proposition 5.6. Let A be defined as above and dE(λ) be the spectral measure of A. Then for

λ ∈ [1 -1/n, 1] dE(λ) p→p ≤ C(1 -λ) n(1/p-1/p )/2-1 (5.9) for 1 < p < (2n + 2)/(n + 3).
The proof of Proposition 5.6 is based on the following result due to Bak and Seeger [3, Theorem 1.1].

Lemma 5.7. Consider a probability measure µ on R n . Assume that for positive constants 0 < a < n,

0 < b ≤ a/2, M i ≥ 1, i = 1, 2, µ satisfies (5.10) sup r B ≤1 µ(B(x B , r B )) r a B ≤ M 1
where the supremum is taken over all balls with radius ≤ 1 and

(5.11) sup |ξ|≥1 |ξ| b | dµ| ≤ M 2 .
Let p 0 = 2(n-a+b) 2(n-a)+b . Then

(5.12)

f 2 dµ ≤ CM b n-a+b 1 M n-a n-a+b 2 f 2 L p 0 ,2 (R n ) ,
where L p 0 ,2 is the Lorentz space.

Proof of Proposition 5.6. Let T n be the n-dimensional torus (note that n is equal to the homogeneous dimension of T n ). For any function f ∈ l 2 (Z n ), one can define the Fourier series F f :

T n → C of f by F f (θ) = d∈Z n f (d)e i d,θ .
Then the inverse Fourier series F -1 f : Z n → C is defined by

F -1 f (d) = 1 (2π) n T n f (θ)e -i d,θ . Define the convolution of f, g ∈ L 2 (Z n ) by f * g(d) = d 1 ∈Z n f (d -d 1 )g(d 1 ). Note that F (A f )(θ) =         1 n n j=1 cos θ j         F f (θ) = (G(θ)) F f (θ), where
G(θ) = 1 n n j=1 cos θ j .
Hence for any continuous function F

1 -1 F(λ)dE(λ) f = F(A) f (d) = F -1 F(G(θ)) f (θ) = 1 (2π) n R σ λ F(λ)e i d,θ 1 |∇G| dσ λ (θ)dλ * f = R F(λ) (Γ λ * f ) dλ,
where σ λ is the level set defined by the formula

σ λ = {θ : 1 n n j=1 cos θ j = λ} ⊂ T n and Γ λ (d) = 1 (2π) n σ λ e i d,θ 1 |∇G| dσ λ (θ) for all λ ∈ [-1, 1]. Thus dE(λ) f = f * Γ λ .
For the range λ ∈ (1 -1/n, 1) considered in the proposition, changing variable yields

Γ λ (d) = (1 -λ) (n-1)/2 (2πi) n σ λ e i d, (1-λ) 1/2 θ 1 |∇G|((1 -λ) 1/2 θ) d σ λ (θ),
where

σ λ = {θ : 1 n n j=1 cos((1 -λ) 1/2 θ j ) = λ} ⊂ T n .
Next we define a probability measure µ λ on the surface σ λ by the formula

dµ λ = 1 |∇G|((1 -λ) 1/2 θ)N(λ) d σ λ (θ),
where

N(λ) = σ λ 1 |∇G|((1 -λ) 1/2 θ) d σ λ (θ).
For λ ∈ (1 -1/n, 1) we define the restriction type operator R λ :

1 (Z n ) → L 2 ( σ λ , µ λ ) by (R λ f )(θ) = F f (θ(1 -λ) 1/2 ) = d∈Z n f (d)e i d, (1-λ) 1/2 θ , θ ∈ σ λ .
Then the dual operator R * λ is given by

(R * λ f )(d) = σ λ e -i(1-λ) 1/2 d,θ f (θ) |∇G|((1 -λ) 1/2 θ)N(λ) d σ λ (θ). Hence dE(λ) f = N(λ)((1 -λ) 1/2 ) n-1 (2π) n R * λ R λ f.
Following the standard approach on the Euclidean space we study boundedness of the operator R * λ acting from L 2 ( σ λ , µ λ ) to p (Z n ). Next we define the operator R *

λ : L 2 ( σ λ , µ λ ) → L ∞ (R n ) by ( R * λ f )(ξ) = σ λ e -i ξ,(1-λ) 1/2 θ dµ λ (θ).
By the Plancherel-Pólya inequality (cf. [36, Section 1. There exists also a constant C > 0 such that for all λ ∈ (1 -1/n, 1) where C just depends on n and does not depend on λ and θ. Now, it is not difficult to check that surfaces σ λ and measures µ λ satisfy assumptions of Lemma 5.7. The required exponent for (5.10) is equal to a = n -1. In addition dµ λ satisfies (5.11) with b = (n -1)/2 uniformly in λ ∈ (1 -1/n, 1). Hence by Lemma 5.7

3.3]) (5.13) (R * λ f )(d) L p (Z n ) ∼ R * λ f (ξ) L p (R n ) for all 1 ≤ p ≤ ∞.
(1 -λ) 1/2 ≥ |∇G|((1 -λ) 1/2 θ) = 1 n         n j=1 sin 2 ((1 -λ) 1/2 θ j )         1/2 ≥ C(1 -λ)
σ λ f 2 dµ λ ≤ C f 2 L p (R n ) Hence R * λ f (ξ) L p (R n ) ≤ C(1 -λ) -n 2p f L 2 ( σ λ , µ λ ) .
Thus by the Plancherel-Pólya inequality (5.13) Remark 5.8. For n = 2 it is enough to assume that supp F ⊂ (0, 1). Remark 5.9. There is another approach to the proof of Theorem 5.5 via transference type statements on equivalence of L p boundedness of the Fourier integral and the Fourier series multipliers under suitable condition on the multiplier support (cf. [START_REF] Chen | [END_REF]). 5.3. Fractional Schrödinger operators. Let n ≥ 1 and V, W be locally integrable non-negative functions on R n . Consider the fractional Schrödinger operator with a potentials V and W: L = (-∆ + W) α + V(x), α ∈ (0, 1].

R * λ f (d) L p (Z n ) ≤ C(1 -λ) -n
The particular case α = 1 2 is often referred to as the relativistic Schrödinger operator. The operator L is self-adjoint as an operator associated with a well defined closed quadratic form. By the classical subordination formula together with the Feynman-Kac formula it follows that the semigroup kernel p t (x, y) associated to e -tL satisfies the estimate 0 ≤ p t (x, y) ≤ Ct -n 2α 1 + t -1 2α |x -y|

-(n+2α)
for all t > 0 and x, y ∈ R n . See page 195 of [START_REF] Ouhabaz | Analysis of heat equations on domains[END_REF]. Hence, estimates (PVE a p,2 (τ)) hold for p = 1 and a = 2α. If n = 1 and α > 1 2 then we can apply Corollary 3.2 and obtain a spectral multiplier result for L.

2 .

 2 Hence, it suffices to studyR * λ f (ξ), ξ ∈ R n . Set G(θ) = 1 n n j=1 cos((1 -λ) 1/2 θ j ).Denote H( G) the Hessian corresponding to G. Then the Gaussian curvature for an implicitly defined surface corresponding to the equation G(θ) = λ is given by the following formulaK = -H( G) ∇ GT ∇ G 0 |∇ G| -(n+1) = (-1) n+1 ((1 -λ) 1/2 ) n-1 n j=1 cos((1 -λ) 1/2 θ j ) n j=1 sin(θ j (1 -λ) 1/2 ) tan((1 -λ) 1/2 θ j ) Note that if λ ∈ (1 -1/n, 1), then cos((1 -λ) 1/2 θ j ) > 0 for all j and cos((1 -λ) 1/2 θ j ) ≥ 1 -(1 -λ)n. 1 -λ) 1/2 θ j ) < n -1 + 1 -(1 -λ)n n = λwhich contradicts θ ∈ σ λ . It follows that for every n ≥ 2 there exists a positive constant C n > 0 which does not depend on λ and θ such that |K| ≥ C n .

1 / 2 so

 12 N(λ) ∼ (1 -λ) -1/2. Then from Stein [35, Page 360, Section 5.7 of Chapter VIII], we know that| dµ λ | ≤ C(1 + |ξ|) (1-n)/2 ,(5.14)

2p f L 2 ( 1 [ 1 ( 2 F 2 ,

 21122 σ λ , µ λ ) for 1 ≤ p ≤ 2n+2 n+3 . By dualitydE(λ) p→p ≤ N(λ)(1 -λ) R λ p→p ≤ C(1 -λ) n(1/p-1/p )/2-1 , for 1 ≤ p ≤ 2n+2 n+3 and λ ∈ (1 -1/n, 1). This completes the proof of Proposition 5.6.Now we are able to conclude the proof of Theorem 5.5.Proof of Theorem 5.5. For every k ∈ N, we denoteA k (d 1 , d 2 ) the kernel of A k for k ∈ Z. Note that V(x, k) ∼ k n .It is well-known (see e.g.[23]) that A k (d 1 , d 2 ) satisfies the following Gaussian type upper estimate:A k (d 1 , d 2 ) ≤ Ck -n/2 exp -|d 1d 2 | 2 ck . (5.15)Next we verify that the operators A k satisfy condition (ST 2 p,2 (τ)) with τ = k 1/2 uniformly for all k ∈ N for all bounded Borel functions F such that supp F ⊂ (1 -1/n, 1). By T * T argument and Proposition 5.6,F(A k )A k 2 p→2 = |F| 2 (A k )A 2k p→p ≤ (n-1)/n] 1/k |F| 2 (λ k )λ 2k dE A (λ) p→p dλ ≤ C n-1)/n |F| 2 (λ)λ 2 (1 -λ 1/k ) n(1/p-1/p )/2-1 dλ 1/kas desired. Now (5.7) follows from Theorem 5.3. The L p boundedness of F(I -A) for functions F satisfying condition (5.8) follows from[START_REF] Sikora | Sharp spectral multipliers for operators satisfying generalized Gaussian estimates[END_REF] Theorem 3.3]. This completes the proof of Theorem 5.5.

Acknowledgments. P. Chen was supported by NNSF of China 11501583, Guangdong Natural Science Foundation 2016A030313351. The research of E.M. Ouhabaz is partly supported by the ANR project RAGE ANR-18-CE40-0012-01. A. Sikora was supported by Australian Research Council Discovery Grant DP160100941. L. Yan was supported by the NNSF of China, Grant No. 11871480, and Guangdong Special Support Program. The authors would like to thank Xianghong Chen for useful discussions on the proof of Theorem 5.5.


M Othman

D Lollman

K Aguir

W Belkacem

N Mliki

Response Enhancement of WO 3 Gas Sensors by Metallic Nanograins

In the present work, we have studied the effect of metal nanoparticles on the enhancement of the sensor response. These nanoparticles were deposited on the surface of WO3 sensors. In a second step, we have focused on a theoretical study to simulate the response of the sensor to determine influenced parameters related to response time.

INTRODUCTION (HEADING 1)

Metal oxide WO3 exhibit anionic vacancies that give it ntype semiconductor character. The sensing principle is based on the variation of the resistance (or conductance) in the presence of oxidizing or reducing gases. The change of electrical properties of the metal-oxide semiconductor is directly related to the adsorption of gas molecules which can generate surfaces states and then gives rise to an electronic charges transfer between the conduction band and these states.

It is therefore of great interest to study and implement gas sensors using tungsten trioxide as sensitive layers for the detection of gaseous species. This is particularly true at the level of sensitivity and selectivity which are two of the main challenges to the for developers of metal-oxide gas sensors and until now limit the performance of these devices. A possible approach for enhancing the sensitive and selective properties of gas sensors are quoted the use is speculated by additive catalysts based on noble metals (Pd, Au, Ag ...) [START_REF] Eason | On certain integrals of Lipschitz-Hankel type involving products of Bessel functions[END_REF]. This surface modification which leads to a new chemical reactivity by establishing chemical bonds of metal-metal oxides has been implemented to improve the response and the sensitivity of tungsten trioxide gas sensors.

The aim of this work is to analyze the influence of the metal additive on the gas sensor performance. In order to achieve this objective, three different metallic nanoparticles (gold, silver and cobalt) are used to change the surface properties of the tungsten oxide sensors. The catalytic effect on the sensor characteristic is usually associated to a distribution of the additive nanoparticles through the whole sensing material surface.

II. SENSORS DESIGN

The sensors are formed by a micro-hotplate platform and a WO3 sensing layer. The heater consists of a silicon substrate with a SiO2-SiNx bi-layer membrane, on which a Pt heater is placed. Above the heater, an insulating layer is deposited followed by interdigitated electrodes and then a thin layer of WO3 . (Fig. 1).

The WO3 sensing thin film was prepared by reactive radio frequency (13.56 MHz) magnetron sputtering. The film is sputtered on the top of the interdigitated electrodes using a 99.9% pure tungsten target in a reactive atmosphere under controlled oxygen-argon mixtures. After the deposition and a lift-off of the sensitive layer, the film was annealed at 400°C in air for 90 minutes in order to stabilize the chemical structure.

Fig. 1. WO3 gas sensor with the micro-hotplate

The sensors are introduced in a test chamber allowing investigating of resistance variation under different gas concentrations after exposure to dry air as reference gas. The typical operating temperature giving the highest sensitivity for these sensors is about 270°C.

Ozone is generated by oxidizing oxygen molecules of a dry air exposed to a UV lamp. In the other hand, ethanol vapors were generated by blowing dry air through an ethanol /water mixture in thermostatic bath kept at 34 ºC. The resultant air will contain humidity and evaporated ethanol with a concentration in air that follows the Dubowski's formula.

III. RESULTS AND DISCUSSION

Among the important sensing characteristics in gas sensing are quoted gas response and response time. Gas response is defined as the ratio of the resistance of a sample on exposure to a test gas to the resistance in air for oxidizing gases, but as the reverse ratio when the resistance decreases in reducing gases. It can be written as.

S= Rgas/Ro

for oxidizing gases

S= R0/Rgas

for reducing gases

where Rgas and R0 are the resistances in the presence of test gas and in air, respectively. Response time is defined as the time needed for a sensor to attain 90% of maximum change in resistance upon exposure to a test gas.

A. Gold nanoparticles

The electrical resistance of pure and gold-modified WO3 thin films was measured as a function of time at 300°C????270°C ???. Figure 2 shows the variation of logarithm of the film resistance under various ozone concentrations before and after gold deposition as a function of time. The gold nanoparticles were deposited onto the WO3 surface by vacuum evaporation with an equivalent thikness of 15 Å. Figure 2 shows that there is a systematic decrease in baseline resistance from 8 MΩ to 2 MΩ after surface modification with gold nanoparticles. Stabilization time, which is about 20 minutes before gold deposition, was reduced also to 4 minutes.

Figure 3, representing the evolution of gas response mentioned above (for oxidizing gases) as a function of ozone concentration, demonstrates that the response of the WO3 material can be improved by incorporating gold nanoparticles onto the oxide film.

It is clear from figure 3 that for each tested concentration, the response has been enhanced. It should be noted here that the addition of the 15Å gold film resulted in an increase in sensor response by more than five orders of magnitude??? for 300 and 400 ppb ozone concentration. Implementing gold nanoparticles can also reduce slightly the response time, from 25 s to 21 s for 60 ppb. The increase in resistance under ozone is considered to arise from electron transfer from the WO3 film to the Au due the difference between electron affinity of tungsten trioxide (3.33-3.94 eV) [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] and the Au work function (5.1 eV) and can also be assigned to the good dispersion of gold nanograins on the WO3 surface (Fig. 4).

B. Silver nanoparticles

A thin silver film (15 Å) was deposited onto the surface of another WO3 film to investigate the effect of the Ag nanograins on the performances of the sensor (response, time response, etc.).

Resistance measurements were carried out in the test chamber under ozone at 300°C. Fig 5 shows the variation of resistance in the pure/Ag-modified WO3. The Ag behavior is the same compared to the gold case (i.e. in terms of response) but it is clear that both stabilization time and baseline resistance were increased. To investigate response enhancement to ozone, the ratio Rgas/Ro was plotted as a function of ozone concentration. As shown in Fig 6 . The presence of Ag nanograins in the surface of the WO3 improves, like the gold film case, the response to ozone. Fig. 6. Sensor response before (black square) and after gold deposition (red circles) to different ozone concentrations.

From Fig 6 shows that there is an increase in sensor response by more than two orders of magnitude for 300 ppb. The enhancement is small compared to the case of gold particles for the same ozone concentration; also the increase in both baseline resistance and time stabilization can be noticed to the oxidation of Ag. In fact, the transfer of electrons from WO3 to Ag during adsorption promotes the ionization of oxygen on the Ag. Another possible reason that can explain the modification mentioned above is the degradation of the WO3 sensitive layer as shown in Fig 7 . Thus, the degradation of the film when the sensor is in operating mode can limit the density of Ag nanograins on the WO3 surface and then minimize the sensing mechanism.

C. Cobalt nanoparticles

In addition to gold and silver additives, a nanostructured cobalt film (30 Å) was deposited onto the surface of another sensor. This shows Co-WO3 behavior under reducing gas and also the particular effect of cobalt for oxidizing gas [START_REF] Jacobs | Fine particles, thin films and exchange anisotropy[END_REF]. The testing gas in this case is ethanol with different concentrations. The resistance has been plotted as a function of time as shown in Fig. 8. Fig 8 demonstrates that the interaction of cobalt nanograins with WO3 surface modified in a radical way the response of the sensor. Thus, the Co-WO3 sensor presents an increase in resistance. This behavior is normally the signature of oxidizing gases with n-type semiconductor or that for p-type semiconductor. It should be noted that similar effect was observed in the case of ozone which is an oxidizing gas [START_REF] Jacobs | Fine particles, thin films and exchange anisotropy[END_REF].

A possible explanation of this behavior is related to the formation of cobalt oxide which is a p-type semiconductor that can dominate at the surface. This tends to prove that the conduction is essentially a surface conduction. Fig. 4. SEM micrograph showing surface morphology of cobalt nanoparticles dispersed on WO3 surface.

Fig. 2 .

 2 Fig. 2. Response under ozone of gold (15Å) modified sensor surface.

Fig. 3 .

 3 Fig. 3. Comparison between the sensor response before and after gold deposition versus ozone concentrations.

Fig. 4 .

 4 Fig. 4. SEM micrograph showing surface morphology of gold nanoparticles dispersed on WO3 surface.

Fig. 5 .

 5 Fig. 5. Response under ozone of silver (15Å) modified sensor surface. According to Fig 2, the increase in the baseline value is about 2 orders of magnitudes. Time stabilization changes from 10 minutes for the undoped layer to 17 minutes for Ag modified sensor.

Fig. 7 .

 7 Fig. 7. SEM image showing surface morphology of silver nanoparticles (left) and degradation of the film (right).

Fig. 8 .

 8 Fig. 8. Response under ethanol of: a) unmodified WO3 and b) Co/WO3 sensor surface.


A novel SnO2 sensor and its selectivity improvement with catalytic filters J. Jońca 1 , K. Fajerwerg 1, 2 , M. L. Kahn 1 , A. Ryzhikov 1 , L. Joanny 1 , J. Harmel 1 , P. J. Debouttière 1

Summary

We have developed a new organometallic approach for the preparation of SnO2 gas sensitive layer. The investigated sensor has shown to be very sensitive to CO at 500°C and to NO2 at 350°C. Additionally, an improvement of the sensor selectivity was achieved by the use of catalytic filter materials directly integrated into the device. The filers are made of zinc oxide and titanium dioxide supported gold nanopatricles (Au@ZnO, Au@TiO2), which were prepared by deposition-precipitation by urea (DPU) method. This catalytic filters, operated at room temperatures and under humid atmosphere (HR 50%), allowed to fully eliminate 100 ppm CO response, whereas the sensor always responds to 100 ppm C3H8.

Motivation and results

Despite the intense research work in the field of metal oxide semi-conductor sensors, the need for selective gas sensing devices remains a challenge. One strategy to improve the selectivity is to integrate catalytic filters into the device. Few years ago, gold nanoparticles (< 5 nm) based on reducible metal oxides supports (e.g. TiO2, CeO2 and, MgO) have been investigated in several oxidation reactions [START_REF] Haruta | Novel gold catalysts for the oxidation of carbon monoxide at a temperature far below 0°C[END_REF][START_REF] Delannoy | Supported gold catalysts for the decomposition of VOC : Total oxidation of propene in low concentration as model reaction[END_REF]. Such gold catalysts, prepared by a method initially developed by C. Louis and co-workers [START_REF] Zanella | Mechanism of deposition of gold precursors onto TiO2 during the precipitation by cation absorption and deposition-precipitation with NaOH and urea[END_REF], exhibit a very high catalytic activity for CO even at room temperature [START_REF] Milone | Deposition-precipitation with urea to prepare Au/Mg(OH)2 catalysts: Influence of the preparation conditions on metal size and load[END_REF]. Drawing on that pioneer work, we synthetized the Au@ZnO and Au@TiO2 nano-composites. The catalytic activity towards CO for both materials was then evaluated as catalytic filters (i.e. at room temperature under 50% of humidity).

Concerning the SnO2 sensor, a new sensitive layer was developed in our team by an organometallic approach. The pristine nanoparticles are made of Sn3O2(OH)2. They were deposited on miniaturized silicon gas sensor substrates by an ink-jet method. Upon in situ heating, the layer was transformed into SnO2. The sensor exhibited extremely high response to 1 ppm NO2, especially at low temperatures (S=66% at 350°C) (Fig. 1a), while the other tested gases (i.e. CO, C3H8, and NH3) were almost non-detected in these conditions. Such device allows therefore to selectively detect NO2 gas at low temperature. At high temperatures, the sensor exhibited good CO sensitivity (Fig. 1b) with a detection limit below 2 ppm (S=9.8% at 500°C). However, other reducing gases were also well detectable at this temperature. The catalytic filters were therefore integrated into the device in order to increase the selectivity. The device is presented on Figure 2. The sensor was tightly connected with a holder containing 100 mg of Au@ZnO or Au@TiO2 powders, forcing the gas mixture to pass through the filter before it reaches the SnO2 sensitive layer. Figure 3a presents the sensor response to 100 ppm CO and C3H8 in the presence or in the absence of the catalytic filter. A complete removal of 100 ppm CO from the gas mixture was observed, whereas the C3H8 always passed through the filter. Superior reactivity towards CO at the applied conditions was obtained for Au@ZnO (Fig. 3b). These results open the way towards improved arrays of selective gas sensors. Teflon membrane (0,2 µm)

Figure 1 .

 1 Figure 1. Sensor response to (a) NO2 (1 ppm) and (b) CO (2, 10, 20 ppm) at different temperatures.

Figure 3 .

 3 Figure 3. a) SnO2 sensor response towards 100 ppm CO and C3H8 in the absence (1) and presence (2) of Au@ZnO filter, b) comparison of sensor response towards CO at different concentrations in the presence of Au@ZnO or Au@TiO2 filters.

Figure 2 .

 2 Figure 2. Developed device (a) with the SnO2 sensor (b) and holder containing the catalytic filter (c). Image TEM of the Au@ZnO filer (d) with diameter of Au NPs of ca. 2.5 nm.

  , B. Chaudret 3 , A. Chapelle 4 , P. Menini 4 , P. Fau 1 1: Laboratoire de Chimie de Coordination (LCC), CNRS, Toulouse, France 2: Université Paul Sabatier, UT III, Toulouse, France 3: Laboratoire de Physique et de Chimie des Nano-objets (LPCNO), INSA, UPS, CNRS, Toulouse, France

4: Laboratoire d'Analyse et d'Architecture des Systèmes (LAAS), CNRS, Toulouse, France


Carlos Vaca-Garcia
email: carlos.vacagarcia@ensiacet.fr

I n the wake of eco-design methodology development, the time has come to propose a sustainable process design that integrates not only environmental evaluation but also economic and social issues. All strategic decisions should respect the principles of sustainability. Th e environmental, economic, and social assessment of engineering choices is thus necessary and the agroindustry is not exempt. Design or re-design of any product relies on the choice of processes, which naturally leads to the development of an integrated product-process approach to agro-industrial processes.

Among the means of assessment processes and their consequences for the environment, the economy, and society, Life Cycle Assessment (LCA) derived from Life Cycle Th inking, has become the best adapted method at such a decision level. LCA was fi rst developed to reduce the environmental impacts of a product, a process, or a service. [START_REF]ISO 14040. Environmental Management -Life Cycle Assessment Principles and Framework[END_REF] Since then, Life Cycle Costing and Social Life Cycle Assessment were developed to account for economic and social aspects in the LCA of a system. Historically oriented to the environmental impacts of a product, LCA has naturally extended its fi eld of investigation to social, economic, and engineering aspects to become a Life Cycle Sustainability Assessment (LCSA) of product. [START_REF] Klöpffer | Life cycle sustainability assessment of products[END_REF] Th ere are no mutually agreed standards, however, only guidelines from the United Nations Environment Programme (UNEP). In parallel, enterprise strategies constantly adapt to our rapidly changing world. In order to face these fl uctuations, enterprises need more and more models that permit them to formalize, describe, and simulate their behavioral face to the external environment. General LCSA is a semi-formalized framework that includes system modeling but it needs other disciplines and other business competencies to analyze and explain the link between the system and its impacts.

A process-product-enterprise sustainability assessment approach guided by Life Cycle Th inking has recently been formalized by Busset et al. [START_REF] Busset | Approach for evaluation of systems sustainability guided by life cycle thinking: application to olive oil agro-industry[END_REF][START_REF] Busset | Computer aided process engineering for sustainability analysis of food production[END_REF] Th is approach is part of a decision-making process in the context of system design or re-design. Environmental, social, and economic impacts of the enterprise occur at all levels of design: (bio)molecule, process, product, and enterprise. Th e proposed approach off ers the opportunity to make LCSA possible through the federated integration of process, product, and enterprise models and their interaction with the external environment as well as through the proposal of an ontology from which these models are built. Considering the ISO 15704 standards that defi ne an enterprise-architecture framework, 5 they place LCSA, process and system engineering, and businessprocess modeling within three dimensions: diff erent levels of specifi city (generic, partial, particular), diff erent points of view (organization, information, resources, function), and diff erent life cycle stages.

Th ey propose an approach to design of the agro-process guided by sustainable evaluation. Th e approach allows systematically taking into account sustainable issues within agro-process design or re-design. Agro-process design is supported by a thermodynamic model based on mass and energy balances. Finally, the coupling between process simulation and sustainability assessment is realized through input and output data exchange. It aims at characterizing the environmental, economic, and social consequences of agro-process. Th e proposed approach is addressed to decision-makers. It may allow (i) a technologic choice of agro-industrial process, (ii) a choice of supply chain, (iii) a choice of input.

Even though some diffi culties still remain with energy balances evaluation and social impacts estimation, this approach seems to be the most advanced, the most comprehensive, and the best adapted to the bioeconomy. Further investigations are in process to improve these two main issues. We will wait impatiently…


Paméla Yoboue

Philippe Ménini

André Maisonnat

Myrtil Kahn

Katia Fajerwerg

Bruno Chaudret

Pierre Fau

When organometallic chemistry and metal oxide nanoparticles meet optimized silicon based gas sensor

A robust silicon gas sensor chip (platinum heater, low stress membrane) has been designed and successfully operated with various metal oxide nanoparticles synthesized by organometallic route (SnO2, ZnO) deposited by a generic ink jet method. High quality and micron thick layers can be obtained and gas sensitivity (under CO) is presented.

INTRODUCTION

Air quality control in confined places (automotive cabin, transportation and offices or working places) is getting more and more pregnant in the actual pollution levels due to new industrial environment. Despite the intense research work in the field of new silicon sensors and sensitive elements, the need for a high accuracy and low cost metal oxide gas sensors remains a challenge. Generally, the gas sensor substrate is based on commercial alumina support which comprises sensitive layer contact electrodes and an integrated platinum heater for heating purpose. These devices present high power consumption and are not adapted for devices miniaturization. Since few years, an alternative to alumina substrate is brought by microelectronics processes with the development of micromachined silicon devices together with polysilicon heaters. The generally admitted drawbacks of silicon chip gas sensors is due to resistance drift of the polysilicon heating element, harsh mechanical stress of the thermally insulating membrane and deactivation of the thin oxide sensitive layer. All the defects can lead to the progressive lost of sensor performance upon time. We have developed a new generation of metal oxide gas sensor, based on the combination of optimized micromachined silicon substrates and highly sensitive nanosized metal oxides derived from organometallic synthesis route [1,2]. An integrated approach of nanoparticles combination with silicon based structures has been employed [START_REF] Ph | Eurosensors XXII proceedings[END_REF]. A platinum heater structure will insure very stable thermal performances of the device. This new sensor design has been built in accordance with the upstream constraints coming from the deposition method required by the liquid nature of the nano-sized sensitive material.

EXPERIMENTAL

Silicon platform

Our SGS (semiconducting gas sensor) device is a global system that operates a high performance silicon micromachined substrate and a high sensitivity and stability semiconducting oxide. We present here a specific design which has been set up for the development of an optimized gas sensor adapted to sensitive layers prepared in a liquid form (colloidal solutions). The improvements of such silicon platform are (figure 1a): -Small size: reduced sensitive layer area (200x200 microns).

-Interdigitated electrodes: compatible with high resistance sensitive layers (nano-sized oxides).

-Thermo-mechanical performance: low thermal conductivity and low stress SiO2-SiNx insulating membrane. -Specific round shape insulating membrane: this is achieved by a deep reactive ion etching (DRIE) of the silicon backside. This shape allows a very low membrane bending upon heating.

-Reliability: robust miniaturized platinum heater meander, with distinct tracks for heater and measurement area. A micron thick silicon oxide layer in between the two platinum layers insures the electrical insulation.

Sensitive layers synthesis

The sensitive metal oxide materials for sensing layers are prepared by organometallic routes according to previously described procedures [START_REF][END_REF]. This non aqueous chemistry allows preparing highly controlled nanoparticles (low size dispersion) in very clean conditions and halide free metal precursors. The thermolysis of tin amidure [Sn((NMe)2)2]2 precursor performed in presence of ppm controlled amounts of water in anisole solvent leads to the formation of a collection of partially oxidized tin nanoparticles. The stabilization of the material is achieved by the growth of a protective oxide shell around the metallic tin core. The mean size of these tin/tin oxide nanoparticles is close to 20 nm, i.e. which allow a high sensitivity taking into account the accepted 3 nm depletion layer depth for tin oxide [5]. The full oxidation of tin/tin oxide composite into pure and well crystallized tin dioxide nanoparticles is directly achieved on the sensor surface through in-situ heating at 500°C by the platinum heater means. Zinc oxide nanoparticles formation relies on a different synthesis procedure although based on similar organometallic chemistry principle [2]. A solution of the dicyclohexylzinc (ZnCy2) in tetrahydrofurane (THF) is reacted with controlled amounts of water (2 molar equivalents) in presence of alkyamine organic ligand (octylamine, 1 molar equivalent) at room temperature. Dicyclohexylzinc is spontaneously transformed in the reaction mixture into zinc oxide nanoparticles with zincite structure, stabilized by long alkyl chain ligands. Mean particles size is 3nm in diameter (figure 6b) and they display a round shaped form. After evaporation of the THF solvent, the particles are re-suspended in anisole solvent for liquid dispense purpose.

Sensitive layer deposition

In both cases, tin and zinc oxides colloidal solutions are implemented as organic inks within an ink jet deposition system (Microdrop AG. MD series). This system is adapted to the deposition of controlled number of drops down to few tens of microns in diameter with ADK401 pipette (figure 1b). A sequential deposition process (solvent evaporation/dispense) allow the formation of the desired oxide layer just in the center of the contact electrode of the silicon device.

RESULTS AND DISCUSSION

Silicon platform characterization

Commercial integrated heaters on silicon platforms are generally built with n-doped (phosphorus) or p-doped (boron) polysilicon layers. The desired resistance level of the polysilicon layer is achieved by the high temperature diffusion (1000°C) of the doping elements inside the silicon grains. One of the major drawbacks of such device is linked to the progressive drift of their electrical properties upon sensor operation temperature. At a constant temperature above 350°C, the doping elements start to diffuse out from the polysilicon grains to the grain boundary where they become useless. The net result at constant voltage rate is a decrease of the heating power of the device upon time giving rise to a decrease of target gas sensitivity. Platinum based heater structures offer a much better thermo-electrical behavior since their operational temperature can be adjusted up to 650°C [START_REF] Ph | Eurosensors XXII proceedings[END_REF] without noticeable change in heater resistance value with time. The thermal response of the heater is presented in figure 2a and2b. The operational temperature is measured in the center of the sensitive layer area by a calibrated infra-red camera. A mean temperature of 500°C is obtained with a 6.5V voltage at a power consumption of only 60 mW. Constant temperature drift measurements of this device have been evaluated over a period of more than 400 days (figure 3). It appears that this heater element is very stable over such a period and even at very high temperature close to 600°C. The thermally insulating membrane with a rounded shape brings a better mechanical behavior to the sensor (figure 4). Previous geometries based on square or rectangular membrane generates a high level of stress resulting in large deformations of the structure upon temperature operation (from -2 to +3 micron height). Membrane deformation of round membranes remains limited to less than 1 micron height at a temperature of 400°C (voltage 6V). This will give rise to low induced stress in the deposited sensitive layers and therefore participates to the reliability of the device.

Drop deposition of sensitive layers

Metal oxide nanoparticles are obtained in organic solvent medium and therefore used as organic inks by a classical ink jet deposition system. The tip diameter and piezo actuated device allow the precise control of very small droplets containing metal oxide nanoparticles. The deposition process is based on a repetitive sequence of drop deposition, solvent evaporation and further drop deposition. The final sensitive layer thickness is micron large (figure 5a andb) and a controlled thermal treatment up to 500°C of the sensitive layer is operated through the integrated heater. This final sensor preparation allows cleaning the sensitive layer by removing all traces of residual organics in the layer (solvent, and by-products from the nanoparticles synthesis). 6a is a scanning electron microscopy (SEM) image of the final thick tin dioxide sensitive layers. It reveals the highly porous nature of the layer which is a consequence of the used deposition technique. Indeed, liquid deposition process induces large amounts of solvents as a carrier medium with the nanoparticles, leading to an important release of porosity in the deposited layers due to solvent evaporation. Final foam-like structure with high degree of open porosity is obtained and will give rise to large gases diffusion pathways inside the sensing layer. Figure 6b is a SEM image of the drop deposition of a very thin layer of colloidal zinc oxide on silicon substrates and shows up the regular pavement of the surface by nano ZnO particles.

Figure 6: SEM images of a) highly porous 1 micron thick tin dioxide layer made of 20 nm large nanoparticles (G=x20'000), and b) zinc oxide nanoparticles thin layer deposited on silicon substrate (G=x200'000).

CO sensitivity of SnO2 and ZnO nanostructured layers

Tests under gases have been performed in an automated test bench with nine available position tests in parallel. For this first attempt, standard test conditions have been set up and no determination of optimized parameters were conducted. The temperature of the sensitive layers (tin dioxide and zinc oxide) is maintained at 500°C (i.e. 6.5V operating voltage), relative humidity is hold at a constant level of 50% and CO contents of 50, 200 and 500 ppm in air atmosphere are tested. An initial stabilization period of 8 hours under operating conditions (air, 50% HR) is necessary before the sensor signal reaches a stable baseline. Once achieved, the sensors stabilization time are much shorter (less than an hour of air stabilization). Figure 7a presents the response of both zinc and tin dioxide gas sensors upon increasing CO levels. The first observation is the large difference in sensitive layer resistances under air. Zinc oxide sensor resistance is close to 1Mohms whereas tin dioxide is only 30KOhms. These two materials have a similar band gap width (3,37eV for ZnO versus 3,5eV for SnO2), and are contacted by the same interdigitated platinum contact geometry. The smaller size of zinc oxide grains (3 nm) compare to tin oxide (20 nm) can be a part of the explanation for the very high resistance level observed for zinc oxide. In fact, depletion layers width are supposed to extend up to few nanometers (1-3 nm for zinc oxide according to Takahashi [6]). Therefore, this nano-sized zinc oxide layer presents the case of an almost fully electron-depleted oxide grains. A key parameter for an improved response of the sensor is the highly porous nature of the sensitive layer and the number of electrical contact between oxide grains. The sensitivity level is measured as the ratio of air resistance on gas resistance (R air/ R gas). Despite a much lower grain size, zinc oxide presents a lower sensitivity to CO gas exposure than tin dioxide (figure 7b). Sensitivity is a very complex parameter and cannot be simply rely on physical parameters like the size of grains and width of depletion layer. Gas sensors are governed by complex catalytic reactions and the chemical nature of the grain surface, crystallographic orientation, grain boundaries or intra-grains neck formation, metal electrode/oxide grains junctions, are as well key parameters for gas reactions with oxide surfaces. This catalytic point of view of SGS devices is a part of our actual research areas on new gas sensors. Finally, undoped nanosized tin oxide sensors present a very high sensitivity (R air / R CO = 6) to 500 ppm CO. This is a promising result in view of a future improvement of the sensitive layers through doping by metal catalysts or mixing with other nanosized metal oxides. New tests on nano ZnO and SnO2 are currently under investigation.

CON CLUSION

We have presented the development of an optimized silicon based SGS gas sensor working with micron thick layers of tin and zinc oxides nanoparticles. Drop deposition technique is a powerful way in order to prepare sensitive layers on narrow contact electrodes on silicon surfaces. All our future developments will be done in combination with the reliable, electrically stable and low power consumption silicon device described in this paper.

Figure 1 :

 1 Figure 1: a) Silicon platform with integrated platinum heater and round shaped membrane for liquid implementation of the sensitive layer and b) drop deposition on by an ink-jet system.

Figure 2 :

 2 Figure 2: Temperature of a silicon device versus a) voltage and b) power (heater resistance 440 Ohms).

Figure 3 :

 3 Figure 3: Power consumption stability of sensors operating at constant voltage

Figure 4 :

 4 Figure 4: Membrane deformation with heater voltage in the case of first generation sensor with a rectangular shape and second generation with rounded geometry.

Figure 5 :

 5 Figure 5: a) Optical images of a) tin dioxide b) zinc oxide layers on a silicon platform

Figure

  Figure6ais a scanning electron microscopy (SEM) image of the final thick tin dioxide sensitive layers. It reveals the highly porous nature of the layer which is a consequence of the used deposition technique. Indeed, liquid deposition process induces large amounts of solvents as a carrier medium with the nanoparticles, leading to an important release of porosity in the deposited layers due to solvent evaporation. Final foam-like structure with high degree of open porosity is obtained and will give rise to large gases diffusion pathways inside the sensing layer. Figure6bis a SEM image of the drop deposition of a very thin layer of colloidal zinc oxide on silicon substrates and shows up the regular pavement of the surface by nano ZnO particles.

Figure 7

 7 Figure 7: a) CO response of nano tin and zinc oxides and b) Sensitivity values for increasing CO exposure

ACKNOWLEDGMENTS

The authors thank CNRS and University Paul Sabatier for its support. We also gratefully thank the TEMSCAN service and especially M. Vincent Collière for SEM images.