Thomas Cometx
email: thomas.cometx@math.u-bordeaux.fr

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

Littlewood-Paley-Stein functionals: an R-boundedness approach

Keywords: Littlewood-Paley-Stein functionals, Riesz transforms, Kahane-Khintchine inequality, spectral multipliers, Schrödinger operators, elliptic operators

Let L = ∆ + V be a Schrödinger operator with a non-negative potential V on a complete Riemannian manifold M . We prove that the vertical Littlewood-Paley-Stein functional associated with L is bounded on

We also introduce and study more general functionals. For a sequence of functions m k : [0, ∞) → C, we define

.

Under fairly reasonable assumptions on M we prove boundedness of H on L p (M ) in the sense

p for some constant C independent of (f k ) k . A lower estimate is also proved on the dual space L p ′ . We introduce and study boundedness of other Littlewood-Paley-Stein type functionals and discuss their relationships to the Riesz transform. Several examples are given in the paper.

Introduction

Let M be a complete non-compact Riemannian manifold and denote by ∇ and ∆ the corresponding gradient and the (positive) Laplace-Beltrami operator, respectively. One of the classical problems in harmonic analysis on manifolds concerns the boundedness on L p (M) of the Riesz transform R := ∇∆ -1/2 . By integration by parts, it is obvious that ∇u 2 = ∆ 1/2 u 2 for all u ∈ W 1,2 (M). Therefore, the operator R, initially defined on the range of ∆ 1/2 (which is dense in L 2 (M)) has a bounded extension to L 2 (M). Note that R takes values in L 2 (M, T M) where T M is the tangent space. Alternatively, the Riesz transform may also be defined by d∆ -1/2 where d is the exterior derivative. In this case R takes values in the L 2 space of differential forms of order 1. It is a singular integral operator with a kernel which may not be smooth. For this reason it is a difficult problem to know whether R extends to a bounded operator on L p (M) for some or all p ∈ (1, ∞). This problem has been studied by several authors during the last decades. We do not give an account on the subject and we refer the reader to [START_REF] Auscher | Riesz transform on manifolds and heat kernel regularity[END_REF][START_REF] Bakry | Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée[END_REF][START_REF] Carron | Riesz transform and L pcohomology for manifolds with Euclidean ends[END_REF][START_REF] Peng Chen | The Hodge-de Rham Laplacian and L p -boundedness of Riesz transforms on non-compact manifolds[END_REF][START_REF] Coulhon | Riesz transforms for 1 ≤ p ≤ 2[END_REF][START_REF] Coulhon | Riesz transform and related inequalities on noncompact Riemannian manifolds[END_REF][START_REF] Guillarmou | Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds (II)[END_REF] and the references therein. If the Riesz transform is bounded on L p (M), then it follows immediately from the analyticity of the heat semigroup that

√ t∇e -t∆ L(L p (M ),L p (M,T M ))
≤ C ∀t > 0.

(1.1)

A natural question is whether (1.1) is already sufficient to obtain the boundedness of the Riesz transform. This question is still open in general and only few results in this direction are known. It was proved by P. Auscher, Th. Coulhon, X.T. Duong and S. Hofmann [START_REF] Auscher | Riesz transform on manifolds and heat kernel regularity[END_REF] that for a manifold M satisfying the volume doubling condition and L 2 -Poincaré inequalities then (1.1) for some p > 2 implies that the Riesz transform is bounded on L r (M) for r ∈ (1, p). See also F. Bernicot and D. Frey [START_REF] Bernicot | Riesz transforms through reverse Hölder and Poincaré inequalities[END_REF] and Th. Coulhon, R. Jiang, P. Koskela and A. Sikora [START_REF] Thierry Coulhon | Gradient estimates for heat kernels and harmonic functions[END_REF] for related recent results. Note that under the volume doubling property, the L 2 -Poincaré inequalities are equivalent to Gaussian upper and lower bounds for the corresponding heat kernel. The sole Gaussian upper bound together with the volume doubling condition imply the boundedness of the Riesz transform on L p (M) for p ∈ (1, 2] (cf. Th. Coulhon and X.T. Duong [START_REF] Coulhon | Riesz transforms for 1 ≤ p ≤ 2[END_REF]).

The study of the Riesz transform is closely related to the study of the Littlewood-Paley-Stein functional

H ∇ (f ) := ∞ 0 |∇e -t∆ f | 2 dt 1/2 (1.2)
or its variant defined in terms of the Poisson semigroup e -t √ ∆ . It is known (see Th. Coulhon and X.T. Duong [START_REF] Coulhon | Riesz transform and related inequalities on noncompact Riemannian manifolds[END_REF] or Proposition 5.3 below) that if H ∇ is bounded on L p (M) then (1.1) is satisfied. One might then ask whether (1.1) is in turn equivalent to the boundedness of H ∇ . To the best of our knowledge, this question is also open in general. The starting point of the present paper is that if we strengthen the property that { √ t∇e -t∆ , t > 0} is uniformly bounded on L p (M) (i.e., (1.1)) into { √ t ∇ e -t∆ , t > 0} is R-bounded on L p (M) (Rademacher-bounded or randomized bounded) then H ∇ is bounded on L p (M). We prove that the converse is also true. Recall that { √ t ∇ e -t∆ , t > 0} is R-bounded on L p (M) if for every t k > 0, f k ∈ L p (M), k = 1, ..., n,

E n k=1 r k √ t k ∇ e -t k ∆ f k p ≤ C E n k=1 r k f k p
with a constant C > 0 independent of t k , f k and n. Here, (r k ) k is a sequence of independent Rademacher variables.

Actually we deal with more general versions of the Littlewood-Paley-Stein functional and we also consider Schrödinger operators L = ∆ + V instead of the sole Laplacian. We introduce the functional

H Γ ((f k )) := k ∞ 0 |Γm k (tL)F (tL)f k | 2 dt 1/2 (1.3)
where m k and F are bounded holomorphic functions on a sector Σ(ω p ) of the right halfplane with some angle ω p and Γ is either ∇ or multiplication by √ V . We prove that a-the Riesz transform ΓL -1/2 is bounded on

L p , ⇓ b -{ √ t Γ e -tL , t > 0} is R-bounded on L p , c-the Littlewood-Paley-Stein functional H Γ in (1.3) is bounded on L p .
We do not need the general form of the square function (1.3) for the implication c ⇒ b, see Theorem 3.1. We do not know whether b ⇒ a is true in general but we hope that putting into play the R-boundedness idea will shed some new light on the problem of boundedness of the Riesz transform. Before describing in a more explicit way some other contributions in this paper we recall some known results on H ∇ . A classical result of E.M. Stein [START_REF] Elias | Singular Integrals and Differentiability Properties of Functions[END_REF]Chapter IV] states that H ∇ is bounded L p (R N ) for all p ∈ (1, ∞). This was extended to the case of sub-Laplacians on Lie groups in [START_REF] Elias | Topics in Harmonic Analysis Related to the Littlewood-Paley Theory[END_REF]. On Riemannian manifolds, the boundedness on L p (M) was also considered. N. Lohoué [START_REF] Lohoué | Estimation des fonctions de Littlewood-Paley-Stein sur les variétés riemanniennes à courbure non positive[END_REF] proved several results in the setting of Cartan-Hadamard manifolds. See also J.C. Chen [START_REF] Chen | Heat Kernels on Positively Curved Manifolds and Applications[END_REF]. For p ∈ [START_REF] Assaad | Riesz transforms of Schrödinger operators on manifolds[END_REF][START_REF] Auscher | Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials[END_REF], the method of Stein works in the general setting of any complete Riemannian manifold as pointed out by Th. Coulhon, X.T. Duong and X.D. Li in [START_REF] Thierry Coulhon | Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2[END_REF]. More precisely, it is proved there that H ∇ is bounded on L p (M) for all p ∈ [START_REF] Assaad | Riesz transforms of Schrödinger operators on manifolds[END_REF][START_REF] Auscher | Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials[END_REF] and if in addition the manifold satisfies the doubling condition (4.6) and a Gaussian upper bound (4.8) for the corresponding heat kernel then H ∇ is of weak type [START_REF] Assaad | Riesz transforms of Schrödinger operators on manifolds[END_REF][START_REF] Assaad | Riesz transforms of Schrödinger operators on manifolds[END_REF]. We also refer to [START_REF] Thierry Coulhon | Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2[END_REF] for references to other related works. These questions are also studied for elliptic operators in divergence form, we refer to the work of P. Auscher, S. Hofmann and J.M. Martell [START_REF] Auscher | Vertical versus conical square functions[END_REF] for recent advance and references. For a given Schrödinger operator L = ∆ + V with a non-negative potential V ∈ L 1 loc (M), the method of Stein can be used to prove that the functional

H(f ) := ∞ 0 |∇e -tL f | 2 dt 1/2 + ∞ 0 | √ V e -tL f | 2 dt 1/2
is bounded on L p (M) for all p ∈ [START_REF] Assaad | Riesz transforms of Schrödinger operators on manifolds[END_REF][START_REF] Auscher | Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials[END_REF]. See E.M. Ouhabaz [START_REF] Maati | Littlewood-Paley-Stein functions for Schrödinger operators[END_REF]. The situation is different for p > 2 and negative results, even for M = R N , are given in [START_REF] Maati | Littlewood-Paley-Stein functions for Schrödinger operators[END_REF].

We mention that there are the so-called horizontal Littlewood-Paley-Stein functionals.

These functionals are of the form

∞ 0 |ϕ(tL)f | 2 dt t 1/2
for a bounded holomorphic function ϕ in a certain sector of C + . They do not involve the gradient term or multiplication by √ V . Such functionals are easier to handle and their boundedness on L p can be obtained from the bounded holomorphic functional calculus. See M. Cowling, I. Doust, A. McIntosh and A. Yagi [START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF] or Ch. Le Merdy [START_REF] Le | On square functions associated to sectorial operators[END_REF] and the references therein.

In the present paper we prove in a general setting that for all p ∈ (1, 2] and

F such that |F (z)| ≤ C |z| δ as z → ∞ and |F ′ (z)| ≤ C |z| 1-ǫ as z → 0 for some δ > 1 2
and ǫ > 0, then H Γ is bounded on L p (M) in the sense that there exists a constant C > 0, independent of (f k ), such that

H Γ ((f k )) p ≤ C sup k m k H ∞ (Σ(ωp)) k |f k | 2 1/2 p , (1.4)
where

m k H ∞ (Σ(ωp)) = sup z∈Σ(ωp) |m k (z)|.
See Theorem 4.1 below. The particular case k = 1, m 1 = 1 and F (z) = e -z corresponds to the standard Littlewood-Paley-Stein functional which we discussed before. This result holds for p ∈ (2, ∞) under the assumption that { √ t ∇ e -t∆ , t > 0} is R-bounded on L p (M). We also prove a similar result for the functional

G Γ ((f k )) := k ∞ 0 |Γm k (tL)f k | 2 dt 1/2
for compactly supported functions m k which belong to a certain Sobolev space (see Theorem 4.3). There is a standard duality argument which provides a reverse inequality on the dual space for the classical Littlewood-Paley-Stein functional. We adapt the argument to our general setting and prove a reverse inequality in L q (M) (1 q + 1 p = 1) for the previous Littlewood-Paley-Stein functionals. See Theorem 6.1.

The proof of Theorem 4.1 uses heavily the fact that L has a bounded holomorphic functional calculus on L p (M) and as a consequence L satisfies square function estimates. In addition, m k (L), k ≥ 1 is R-bounded on L p (M) by a result of N.J. Kalton and L. Weis [START_REF] Kalton | The H ∞ -calculus and sums of closed operators[END_REF]. This does not apply mutatis mutandis to the functional G Γ . Instead we rely on a recent result by L. Deleaval and Ch. Kriegler [START_REF] Deleaval | Maximal Hörmander functional calculus on L p spaces and UMD lattices[END_REF].

We introduce the local Littlewood-Paley-Stein functional and the Littlewood-Paley-Stein functional at infinity defined respectively by

H loc Γ (f ) := 1 0 |Γe -tL f | 2 dt 1/2 and H (∞) Γ (f ) := ∞ 1 |Γe -tL f | 2 dt 1/2
.

We study the boundedness on L p (M) of H loc Γ (respectively, H

Γ ) and their relationship to the local Riesz transform R loc := Γ(L + I) -1/2 (respectively the Riesz transform at infinity R ∞ := ∇L -1/2 e -L ). 1 For example, if L = ∆ and M has Ricci curvature bounded from below, then it is well known that R loc is bounded on L p (M) for all p ∈ (1, ∞) (see D. Bakry [START_REF] Bakry | Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée[END_REF]). As a consequence we obtain that H loc ∇ is bounded on L p (M) for all p ∈ (1, ∞) and the lower bound C f q ≤ e -∆ f q + H loc (f ) q holds for all q ∈ (1, ∞).

We give several examples in Section 7 including Schrödinger operators on R N with a potential in a reverse Hölder class or Schrödinger operators on manifolds. We shall see that for the connected sum M n := R n #R n (n ≥ 2) the Littlewood-Paley-Stein at infinity is not bounded on L p (M n ) for p > n. The fact that the Riesz transform is not bounded on L p (M n ) for p > n was proved by Th. Coulhon and X.T. Duong [START_REF] Coulhon | Riesz transforms for 1 ≤ p ≤ 2[END_REF].

Although we focus on Schrödinger operators on manifolds, our results are also valid for elliptic operators on rough domains. Let Ω be an open subset of R N and consider on L 2 (Ω) an elliptic operator L = -div(A(x)∇•) with real symmetric and bounded measurable coefficients. The operator L is subject to the Dirichlet boundary conditions. Then (1.4) holds on L p (Ω) for all p ∈ (1, 2]. As a particular case of the reverse inequality, we obtain for q ∈ [2, ∞)

C f L q (Ω) ≤ ∞ 0 |∇e -tL f | 2 dt 1/2 L q (Ω) (1.5) and C f L q (Ω) ≤ e -L f L q (Ω) + 1 0 |∇e -tL f | 2 dt 1/2 L q (Ω)
.

(

We point out that no regularity assumption is required on the domain nor on the coefficients of the operator. For another proof of (1.6) and related inequalities on a smooth domain, we refer to a recent paper by O. Ivanovici and F. Planchon [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF]. If Ω = R N , we prove that the lower bounds (1.5) and (1.6) are valid for all q ∈ (1, ∞).

Notation. We denote by d the exterior derivative. We use either ∇L -1/2 or dL -1/2 for the Riesz transform. We often write |∇f (x)| (or |df (x)|) for the norm in T x M (or in T * x M) and we sometimes write |∇f (x)| x to emphasize the dependence of this norm in the point x. We use the notation L p (Λ 1 T * M) := L p (M, T * M) for the L p -space of differential forms of order 1 on M. For a Banach space E, L p (M, E) denotes the L p space of functions with values in E. As usual, the boundedness of the Riesz transform ∇L -1/2 on L p (M) means that ∇L -1/2 , initially defined on the range of L 1/2 , extends to a bounded operator from L p (M) into L p (M, T M). For a given Banach space E, we use . E to denote its norm and the L p -norm will be denoted by . p as usual. We shall use dx for the Riemannian measure on M. Finally, all inessential constants are denoted by C, C ′ , c... Acknowledgements. The authors thank Christoph Kriegler for stimulating discussions and for pointing out the recent paper [START_REF] Deleaval | Maximal Hörmander functional calculus on L p spaces and UMD lattices[END_REF]. They also thank Bernhard Haak for a precious help in the proof of Theorem 4.1 and the anonymous referee for his/her comments. This research is partly supported by the ANR project RAGE, ANR-18-CE-0012-01.

Preliminary results

This section is essentially a preparation for the next ones. We start off by recalling some well known tools on the holomorphic functional calculus, square functions and R-boundedness of a family of operators.

Let ω ∈ (0, π) and set

Σ(ω) := {z ∈ C, z = 0, | arg(z)| < ω}
the open sector of C + with angle ω. We denote by H ∞ (Σ(ω)) the set of bounded holomorphic functions on Σ(ω). By H ∞ 0 (Σ(ω)) we denote the subset

H ∞ 0 (Σ(ω)) = F ∈ H ∞ (Σ(ω)), ∃ C, s > 0 : |F (z)| ≤ C|z| s 1 + |z| 2s ∀z ∈ Σ(ω) .
Consider a closed, densely defined operator A with dense range on a Banach space E and suppose that it satisfies the basic resolvent estimate

(λI -A) -1 ≤ C |λ| ∀ λ / ∈ Σ(ω).
One defines the bounded operator

F (A) for F ∈ H ∞ 0 (Σ(ω)) by the standard Cauchy formula F (A) = 1 2πi γ F (z)(zI -A) -1 dz
on an appropriate contour γ. One says that A has a bounded holomorphic functional calculus with angle ω if for some constant C ω > 0

F (A) L(E) ≤ C Ω F H ∞ (Σ(ω)) := C ω sup z∈Σ(ω) |F (z)|
for all F ∈ H ∞ 0 (Σ(ω)). In this case, for every F ∈ H ∞ (Σ(ω)), F (A) is well defined and satisfies the same estimate as above. We refer to [START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF] for all the details. One of the most important consequences of the holomorphic functional calculus in harmonic analysis concerns square function estimates. Set E = L p (X, µ). For F ∈ H ∞ 0 (Σ(ω)), we define for g ∈ E,

∞ 0 |F (tA)g| 2 dt t 1/2 .
It turns out that this functional is bounded on E, i.e.,

∞ 0 |F (tA)g| 2 dt t 1/2 p ≤ C F g p . (2.1)
We refer to [START_REF] Cowling | Banach space operators with a bounded H ∞ functional calculus[END_REF] and [START_REF] Le | On square functions associated to sectorial operators[END_REF]. Now let L = ∆ + V be a Schrödinger operator with a non-negative V ∈ L 1 loc (M). Since the semigroup (e -tL ) is sub-Markovian, L has a bounded holomorphic functional calculus on L p (M) for all p ∈ (1, ∞). This was proved by many authors and the result had successive improvements during several decades. The most recent and general result in this direction states that L has a bounded holomorphic functional calculus with angle ω p = arcsin | 2 p -1| + ǫ (for any ǫ > 0). We refer to [START_REF] Carbonaro | Functional calculus for generators of symmetric contraction semigroups[END_REF] for the precise statement. In particular, one has the square function estimate (2.1) for F ∈ H ∞ 0 (Σ(ω p )). A well known duality argument which can be found in [37, p. 85] shows that the reverse inequality holds on L q (M), that is for every q ∈ (1, ∞) and F as above

C ′ F g q ≤ ∞ 0 |F (tA)g| 2 dt t 1/2 q . (2.2)
Recall that a subset T of L(L p (M)) is said R-bounded if there exists a constant C > 0 such that for every collection T 1 , .., T n ∈ T and every f 1 , ..., f n ∈ L p (M)

E n k=1 r k T k f k p ≤ C E n k=1 r k f k p .
(2.3)

Here, (r k ) k is a sequence of independent Rademacher variables and E is the usual expectation. By the Kahane-Khintchine inequality, this definition can be reformulated as follows

n k=1 |T k f k | 2 1/2 p ≤ C n k=1 |f k | 2 1/2 p . (2.4)
The notion of R-bounded operators plays a very important role in many questions in functional analysis (cf. [START_REF] Hytönen | Probabilistic Methods and Operator Theory[END_REF]) as well as in the theory of maximal regularity for evolution equations (see [START_REF] Weis | A new approach to maximal regularity[END_REF] or [START_REF] Kalton | The H ∞ -calculus and sums of closed operators[END_REF]).

For L = ∆ + V and Γ = ∇ or multiplication by √ V , we shall use the property that the set Proof. Let T k := √ t k Γe -t k L for t k > 0 and f k ∈ L p (M) for k = 1, ..., n. We have

{ √ t Γe -tL , t > 0} is R-bounded on L p (M). If Γ = ∇, then ∇e -tL f (x) ∈ T x M and hence |∇e -tL f (x)| = |∇e -tL f (x))| x .
E n k=1 r k T k f k p = E ΓL -1/2 n k=1 r k (t k L) 1/2 e -t k L f k p ≤ C E n k=1 r k (t k L) 1/2 e -t k L f k p .
Let φ k (z) := √ t k ze -t k z and observe that the sequence (φ k ) k is uniformly bounded in H ∞ (Σ(ω p )). As we mentioned above, the operator L has bounded holomorphic functional calculus on L p (M) with angle ω p . Therefore, by [START_REF] Kalton | The H ∞ -calculus and sums of closed operators[END_REF] or [START_REF] Hytönen | Probabilistic Methods and Operator Theory[END_REF]Theorem 10.3.4], the set

{φ k (L), k ≥ 1} is R-bounded on L p (M).
Using this in the previous inequality yields (2.3).

It is useful to notice that the R-boundedness of √ t Γ e -tL can be reformulated in terms of the resolvent. More precisely,

Proposition 2.2. Let δ ′ > 1 2 . Then the following assertions are equivalent i) the set { √ t Γ e -tL , t > 0} is R-bounded on L p (M), ii) the set { √ t Γ (I + tL) -δ ′ , t > 0} is R-bounded on L p (M). Proof. Suppose that { √ t Γ e -tL , t > 0} is R-bounded on L p (M) and let δ ′ > 1 2
. By the Laplace transform

√ t Γ (I + tL) -δ ′ = c δ ′ √ t ∞ 0 s δ ′ -1 e -s Γ e -stL ds = c δ ′ ∞ 0 a t (s) √ s Γ e -sL ds with a t (s) := t 1 2 -δ ′ s δ ′ -3 2 e -s/t . Since δ ′ > 1 2 we have ∞ 0 a t (s)ds = c ′ δ ′ . We can then apply [14, Lemma 3.2] to conclude that the set in ii) is R-bounded. Suppose now that ii) is satisfied with some δ ′ > 1 2 . Define for each t > 0, φ t (z) := (1 + tz) δ ′ e -tz . Then (φ t ) t is uniformly bounded in H ∞ (Σ(ω p )). Hence, {φ t (L), t > 0} is R-bounded. Taking the product of the R-bounded operators √ t Γ (1 + tL) -δ ′ and φ t (L) gives assertion i).
We finish this section by the following lemma.

Lemma 2.3. Let I be an interval of R and suppose that for each t ∈ I, S t is a bounded operator on

L p (M) (with values in L p (M) or in L p (M, T M)). Then the set {S t , t ∈ I} is R-bounded on L p (M) if and only if there exists a constant C > 0 such that I |S t u(t)| 2 dt 1/2 p ≤ C I |u(t)| 2 dt 1/2 p for all u ∈ L p (M, L 2 (I)).
This lemma is proved in [START_REF] Weis | A new approach to maximal regularity[END_REF] (see 4.a) in the case where S t : L p (M) → L p (M) for each t > 0. Here M is any σ-finite measured space. In our case, these operators may take values in L p (M, T M) as in the case of S t = √ t∇e -tL . Here, |S t u(t, x)| is actually |S t u(t, x)| x where |.| x is again the norm in the tangent space T x M at the point x. For the proof one can either repeat the argument in [START_REF] Weis | A new approach to maximal regularity[END_REF] or argue by taking projection on each e j where {e 1 , ..., e m } is an orthonormal basis of T x M.

Littlewood-Paley-Stein inequalities and R-boundedness

Let L = ∆ + V with 0 ≤ V ∈ L 1 loc (M).
We use again the notation Γ for the gradient ∇ or the multiplication by √ V .

Theorem 3.1. Let H Γ (f ) = ∞ 0 |Γe -tL f | 2 dt 1/2 and p ∈ (1, ∞). Then H Γ is bounded on L p (M) if and only if the set { √ t Γe -tL , t > 0} is R-bounded on L p (M).
Proof. Suppose that H Γ is bounded on L p (M). We prove that { √ t Γe -tL , t > 0} is Rbounded on L p (M). For the converse we shall prove a more general result in the next section and hence we do not give the details here in order to avoid repetition. Let t k ∈ (0, ∞) and f k ∈ L p (M) for k = 1, ..., N. We start by estimating the quantity

I := E k r k √ t k Γ e -t k L f k 2 .
Using (twice) the independence of the Rademacher variables we have

I = - ∞ 0 d dt E|Γe -tL k r k √ t k e -t k L f k | 2 dt = 2 ∞ 0 E (Γe -tL k r k √ t k e -t k L f k ) • (Γe -tL k r k √ t k Le -t k L f k ) dt = 2 ∞ 0 E k Γe -tL r k √ t k e -t k L f k • Γe -tL r k √ t k Le -t k L f k dt = 2 ∞ 0 E k Γe -tL r k e -t k L f k • Γe -tL r k (t k L)e -t k L f k dt = 2 ∞ 0 E (Γe -tL k r k e -t k L f k ) • (Γe -tL k r k (t k L)e -t k L f k ) dt.
Next, by the Cauchy-Schwarz inequality,

I ≤ 2 ∞ 0 E|Γe -tL k r k e -t k L f k | 2 1/2 E|Γe -tL k r k (t k L)e -t k L f k | 2 1/2 dt ≤ ∞ 0 E|Γe -tL k r k e -t k L f k | 2 dt + ∞ 0 E|Γe -tL k r k (t k L)e -t k L f k | 2 dt.
Therefore,

I ≤ E H Γ ( k r k e -t k L f k ) 2 + E H Γ ( k r k (t k L)e -t k L f k ) 2 . (3.1)
In order to continue, we look at H Γ as the norm in L 2 ((0, ∞), dt) so that

E H Γ ( k r k e -t k L f k ) 2 = E k r k Γe -tL e -t k L f k 2 L 2 ((0,∞),dt)
.

Hence by the Kahane inequality,

c p √ I ≤ E H Γ ( k r k e -t k L f k ) p 1/p + E H Γ ( k r k (t k L)e -t k L f k ) p 1/p (3.2)
for some constant c p > 0. Now we use the assumption that H Γ is bounded on L p (M) and obtain

√ I p ≤ C E k r k e -t k L f k p p 1/p + E k r k (t k L)e -t k L f k p p 1/p ≤ C ′ E k r k e -t k L f k p + E k r k (t k L)e -t k L f k p
where we used again the Kahane inequality. On the other hand, it is easy to see by the Kahane inequality that

√ I p is equivalent to E k r k √ t k Γ e -t k L f k p
. Since the operator L has a bounded holomorphic functional calculus on L p (M), it follows from [START_REF] Kalton | The H ∞ -calculus and sums of closed operators[END_REF] or [START_REF] Hytönen | Probabilistic Methods and Operator Theory[END_REF]Theorem 10.3.4] that (e -tL ) t>0 and (tLe -tL ) t>0 are R-bounded on L p (M). This and the previous estimates give

E k r k √ t k Γ e -t k L f k p ≤ C E k r k f k p with a constant C independent of t k and f k . This proves that { √ t Γe -tL , t > 0} is R- bounded on L p (M).
We have the following corollary which is valid on any complete Riemannian manifold M.

Corollary 3.2. Let p ∈ (1, 2]. Then the set { √ t Γe -tL , t > 0} is R-bounded on L p (M).
Proof. As already mentioned in the introduction, H Γ is always bounded on L p (M) for all p ∈ (1, 2] (cf. [START_REF] Maati | Littlewood-Paley-Stein functions for Schrödinger operators[END_REF] for Schrödinger operators and Γ = ∇ or √ V and [START_REF] Thierry Coulhon | Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2[END_REF] for H ∇ and L = ∆). The corollary is then a consequence of the previous theorem.

Remark 3.3. For Γ = √ V we have the following alternative proof for the R-boundedness of { √ t √ V e -tL , t > 0} on L p (M) for p ∈ (1, 2]. We have t 0 √ s √ V e -sL |f |ds ≤ t √ 2 t 0 | √ V e -sL |f || 2 ds 1/2 ≤ t √ 2 ∞ 0 | √ V e -sL |f || 2 ds 1/2 . It follows from the fact that f → ∞ 0 | √ V e -sL |f || 2 ds 1/2 is bounded on L p (M) for p ∈ (1, 2] that sup t>0 1 t t 0 √ s √ V e -sL |f |ds p ≤ C f p .
From this, the positivity of √ s √ V e -sL and [START_REF] Weis | A new approach to maximal regularity[END_REF]

(4.c) it follows that { √ t √ V e -tL , t > 0} is R-bounded.

Generalized Littlewood-Paley-Stein functionals

In this section we prove new Littlewood-Paley-Stein inequalities for L = ∆ + V . The first inequality involves the holomorphic functional calculus of L on L p (M) and the second one spectral multipliers with compactly supported functions.

We have already mentioned and used that L has a bounded holomorphic functional calculus with angle ω p ∈ (arcsin

| 2 p -1|, π 2 ) on L p (M) for p ∈ (1, ∞).
In particular, F (L) is a bounded operator on L p (M) for F ∈ H ∞ (Σ(ω p )). Let again Γ be either ∇ or multiplication by √ V . Our first result is the following. and ǫ 0 ∈ (0, 1),

|F (z)| ≤ C |z| δ as z → ∞, |F ′ (z)| ≤ C |z| 1-ǫ 0 as z → 0 and |m ′ k (z)| ≤ C |z| 1-ǫ 0 as z → 0. Set M 0 = sup k m k H ∞ (Σ(wp)) and M 1 = sup k z → z 1-ǫ 0 m ′ k (z) H ∞ (Σ(wp)) . 1) Given p ∈ (1, 2]. Then there exists a constant C F > 0, independent of m k , such that for all f k ∈ L p (M), k ∞ 0 |Γ m k (tL)F (tL)f k | 2 dt 1/2 p ≤ C F M 0 + M1 k |f k | 2 1/2 p . (4.1)
In particular, the functional

H F Γ (f ) := ∞ 0 |Γ F (tL)f | 2 dt 1/2 is bounded on L p (M). 2) If p ∈ (2, ∞) we assume in addition that { √ t Γ e -tL , t > 0} is R-bounded on L p (M).
Then the same conclusions as before hold on L p (M).

The sums over k used here can be taken up to some K ∈ N, the estimate is independent of K.

Proof. By a simple density argument we can assume that

f k ∈ L 2 (M) ∩ L p (M). Let f ∈ L 2 (M)∩L p (M) and set I(x) := ∞ 0 |Γ F (tL)f (x)| 2 dt 1/2 (if Γ = ∇ then actually, I(x) := ∞ 0 |∇ F (tL)f (x)| 2 x dt 1/2
but we ignore the subscript x for the norm | • |). By integration by parts,

I 2 = lim t→∞ t|ΓF (tL)f | 2 -2 ∞ 0 tΓ LF ′ (tL)f.Γ F (tL)f dt = -2 ∞ 0 tΓ LF ′ (tL)f.Γ F (tL)f dt ≤ 2 ∞ 0 |Γ tLF ′ (tL)f | 2 dt 1/2 I. (4.2)
In order to justify the second equality we note that Γg 2 ≤ L 1/2 g 2 and hence by the spectral resolution of

L M t|Γ F (tL)f | 2 dx = √ tΓ F (tL)f 2 2 ≤ √ tL 1/2 F (tL)f 2 2 = ∞ 0 |H(tλ)| 2 dE λ (f, f ) where |H(z)| 2 = |z||F (z)| 2 .
Since F decays as 1 |z| δ at infinity with some δ > 1 2 , it follows that |H(z)|2 is bounded and |H(tλ)| 2 → 0 as t → ∞ for all λ ∈ (0, ∞). Note that if λ = 0, then H(0) = 0. We conclude by the dominated convergence theorem that

∞ 0 |H(tλ)| 2 dE λ (f, f ) → 0 as t → ∞. After extraction of a subsequence if necessary we obtain (4.2). Set G(z) := zF ′ (z). It follows from (4.2) that ∞ 0 |Γ F (tL)f | 2 dt 1/2 ≤ 2 ∞ 0 |Γ G(tL)f | 2 dt 1/2 . (4.3)
The gain here is that the function G on the RHS has decay at 0 (and also at infinity) whereas F was not assumed to have such decay at 0. This will allow us to use square function estimates.

In order to continue let H := L 2 ((0, ∞), dt t ) 2 and set

J := k ∞ 0 |Γ m k (tL)F (tL)f k | 2 dt 1/2 p .
We apply (4.3) to the function z → F (z)m k (z). Then J is bounded (up to a constant) by

J 1 + J 2 with J 1 := k ∞ 0 |Γ G(tL)m k (tL)f k | 2 dt 1/2 p and J 2 := k ∞ 0 |Γ (tL)F (tL)m ′ k (tL)f k | 2 dt 1/2 p .
We first estimate J 1 . By the Kahane inequality

J p 1 ≤ k ∞ 0 |Γ G(tL)m k (tL)f k | 2 dt 1/2 p p = k √ t Γ G(tL)m k (tL)f k 2 H 1/2 p p ≈ E k r k √ t Γ G(tL)m k (tL)f k p H 1/p p p = E √ t Γ (I + tL) -δ ′ (I + tL) δ ′ G(tL) k r k m k (tL)f k H p p where δ ′ ∈ ( 1 2 , δ). If p ∈ (1, 2], then { √ t Γ e -tL , t > 0} is R-bounded on L p (M) by Corollary 3.2. If p ∈ (2,
∞) this R-boundedness was assumed in the theorem. Hence by Proposition 2.2 and Lemma 2.3 the very last term is bounded (up to a constant) by

E (I + tL) δ ′ G(tL) k r k m k (tL)f k H p p .
Set H(tL) := (I + tL) δ ′ G(tL) and use again the Kahane inequality to obtain

J 1 ≤ C E k r k m k (tL)H(tL)f k H p . (4.4)
In order to continue we approximate the norm k r k m k (tL)H(tL)f k H by the discrete sum

j∈Z k r k m k (2 j/N L)H(2 j/N L)f k 2 1/2
, when N → +∞. We deal with the term

J1,N := E j∈Z k r k m k (2 j/N L)H(2 j/N L)f k 2 1/2 p .
We use a double randomisation argument. Take another independent Rademacher variables rj with expectation Ẽ and apply the Kahane-Khintchine inequality to see that

J1,N ≈ E Ẽ j rj k r k m k (2 j/N L)H(2 j/N L)f k p .
Since L p has Pisier's contraction property, the double expectation is equivalent to a single one with a doubly indexed Rademacher variables rj,k (with expectation Ẽ) as follows

Ẽ j,k rj,k m k (2 j/N L)H(2 j/N L)f k p .
See [27, Propositions 7.5.3 and 7.5 [START_REF] Kalton | The H ∞ -calculus and sums of closed operators[END_REF] or [START_REF] Hytönen | Probabilistic Methods and Operator Theory[END_REF]Theorem 10.3.4] and the fact that L has bounded holomorphic functional calculus on L p (M). Using this in the forgoing estimates, we arrive at

.4]. If sup k m k H ∞ (Σ(wp)) < ∞, then {m k (L), k ≥ 1} is R-bounded on L p (M) by
J1,N ≤ C sup k m k H ∞ (Σ(wp)) Ẽ j,k rj,k H(2 j/N L)f k p .
We argue as before by using again Pisier's contraction property to see that the term on the RHS is equivalent to

E Ẽ j rj k r k H(2 j/N L)f k p ≈ E j k r k H(2 j/N L)f k 2 1/2 p .
We let N → ∞ and we obtain the bound

J 1 ≤ C E ∞ 0 H(tL) k r k f k 2 dt t 1/2 p . Let ω ′ ∈ (arcsin | 2 p -1|, ω p ).
Using the fact that F has decay 1 |z| δ at infinity, it follows easily from the Cauchy formula that F ′ (z) decays at least as 1 |z| 1+δ for z ∈ Σ(ω ′ p ). This implies that the function H(z) := (1+z) δ ′ G(z) = (1+z) δ ′ zF ′ (z) decays at least as 1 |z| δ-δ ′ at infinity. On the other hand, since

|F ′ (z)| ≤ C |z| 1-ǫ 0 as z → 0 it follows that H ∈ H ∞ 0 (Σ(ω ′ p )
). Therefore, we can use the square function estimate (2.1) for H(tL) on L p (M) and we obtain

J 1 ≤ C sup k m k H ∞ (Σ(wp)) k |f k | 2 1/2 p , (4.5)
which gives the desired estimate for J 1 .

Next we estimate J 2 . The proof is similar to the previous one and hence we will not repeat all the details. Let again δ ′ ∈ ( 1 2 , δ) and take ǫ ∈ (0, ǫ 0 ) such that δ ′ + ǫ < δ. Set H(tL) := (I + tL) δ ′ (tL) ǫ F (tL). The function H has decay both at 0 and infinity and hence H(tL) satisfies a square function estimate. Similarly to (4.4) we have

J 2 ≤ C E k r k (tL) 1-ǫ m ′ k (tL)H(tL)f k H p .
We use the double randomisation argument as before as well as the square function estimate for H(tL) to obtain (4.5) for

J 2 with z → z 1-ǫ m ′ k (z) at the place of m k . Since ǫ < ǫ 0 it is obvious that |z 1-ǫ m ′ k (z)| ≤ M 1 for |z| ≤ 1. For |z| ≥ 1, the inequality, |z 1-ǫ m ′ k (z)| ≤ m k H ∞ (Σ(wp))
follows from the Cauchy formula. Thus, we obtain J 2 ≤ C(M 0 + M 1 ). This finishes the proof of the theorem. Remark 4.2. In the estimate of J 2 , the role of (tL) ǫ is to ensure that the function H(z) = (1 + z) δ ′ z ǫ F (z) has decay at zero which then allows to use a square function estimate for H(tL). If the function F itself has a decay at zero, i.e., F ∈ H ∞ 0 (Σ(w p )), then the term (tL) ǫ is not needed. This means that we can take ǫ = 0 in the previous proof. Next, the term

zm ′ k (z) is bounded by m k H ∞ (Σ(wp)
) by the Cauchy formula. Thus, (4.1) can be replaced in this case by

k ∞ 0 |Γ m k (tL)F (tL)f k | 2 dt 1/2 p ≤ C F m k H ∞ (Σ(wp)) k |f k | 2 1/2
p for all bounded holomorphic functions m k on Σ(w p ).

In the next result we aim to consider functions m k which are not holomorphic. We will take such functions in a Sobolev space W δ,2 on the half-line (0, ∞). In order to so, we make some assumptions on the manifold M. We assume that M satisfies the volume doubling property

v(x, 2r) ≤ Cv(x, r), (4.6)
where v(x, r) denotes the volume of the ball of centre x ∈ M and radius r > 0. The constant C is independent of x and r. Note that (4.6) implies the existence of C, N > 0 such that for all x in M, r > 0 and λ ≥ 1

v(x, λr) ≤ Cλ N v(x, r). (4.7)
Next, we assume that the heat kernel p t (x, y) of ∆ satisfies the Gaussian upper bound

p t (x, y) ≤ C v(x, t 1/2 ) e -c d 2 (x,y) t (4.8)
for some positive constants c and C and all x, y ∈ M and t > 0. Since V is non-negative, it is a standard fact that the semigroup e -tL is pointwise dominated by e -t∆ (see, e.g. [33, Section 4.5]) and in particular, the heat kernel k t (x, y) associated with L = ∆ + V satisfies the same Gaussian upper bound. We have

k ∞ 0 |∇m k (tL)f k | 2 dt 1/2 p + k ∞ 0 | √ V m k (tL)f k | 2 dt 1/2 p ≤ C sup k m k W δ,2 k |f k | 2 1/2 p (4.9) for all f k ∈ L p (M). For a given p ∈ (2, ∞), suppose in addition that { √ t∇ e -tL , t > 0} is R-bounded on L p (M). Then k ∞ 0 |∇m k (tL)f k | 2 dt 1/2 p ≤ C sup k m k W δ,2 k |f k | 2 1/2 p (4.10) for all f k ∈ L p (M). If { √ t √ V e -tL , t > 0} is R-bounded on L p (M), then the same estimate holds with √ V in place of ∇.
Proof. Recall that by Corollary 3.2, the set

{ √ t Γ e -tL , t > 0} is R-bounded on L p (M) for all p ∈ (1, 2]. Define J := k ∞ 0 |Γ m k (tL)f k | 2 dt 1/2 p .
As in the proof of Theorem 4.1 we use the Kahane inequality to obtain

J p = k √ t Γ m k (tL)f k 2 H 1/2 p p ≈ E √ t Γ k r k m k (tL)f k p H 1/p p p = E √ t Γ e -tL k r k ϕ k (tL)f k H p p
where ϕ k (λ) = e λ m k (λ). Using the R-boundedness of √ t Γ e -tL and Lemma 2.3 we obtain

J p ≤ C E k r k ϕ k (tL)f k H p p .
Hence (use Kahane again)

J ≤ C M E k r k ϕ k (tL)f k p H 1/p ≤ C ′ M k ϕ k (tL)f k 2 H p/2 1/p = C ′ k ϕ k (tL)f k 2 H 1/2 p . (4.11)
Now, since L satisfies the Gaussian upper bound (4.8) and M satisfies the doubling condition (4.7) (in which we take N to be the smallest possible), then it is known that L satisfies spectral multiplier theorems. In particular, since ϕ k has compact support, one has

ϕ k (L) is bounded on L p (M) provided ϕ k ∈ W α,2 for some α > N| 1 2 -1 p | + 1 2 .
See [START_REF] Duong | Plancherel-type estimates and sharp spectral multipliers[END_REF] or [START_REF] Chen | Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means[END_REF], Theorem A, and the references therein. Finally, [20, Theorem 3.1] asserts that the RHS term in (4.11) is bounded by (up to a constant)

sup k ϕ k W δ,2 k |f k | 2 1/2 p with δ = α + 1. Since the support of m k is contained in [ 1 2 , 2]
, the quantities ϕ k W δ,2 and m k W δ,2 are equivalent. This proves (4.9). For p > 2 the proof is the same since we assume here that { √ t Γ e -tL , t > 0} is R-bounded on L p (M). Remark 4.4. 1-In the proof we have taken δ = α + 1 with α > N| 1 2 -1 p | + 1 2 . The latter value is the order required in the regularity of spectral multipliers under the sole conditions (4.7) and (4.8). There are however many situations where one has sharp spectral multiplier results and hence a smaller order α. This is the case if for example L satisfies the so-called restriction estimate or if the corresponding Schrödinger group e itL satisfies global Strichartz estimates. We refer to [START_REF] Duong | Plancherel-type estimates and sharp spectral multipliers[END_REF] and [START_REF] Chen | Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means[END_REF]. 2-We assumed in the previous theorem that the functions m k are compactly supported. For more general functions, we may use Corollary 3.3 from [START_REF] Deleaval | Maximal Hörmander functional calculus on L p spaces and UMD lattices[END_REF] and obtain the same results under the condition

n sup k λ → √ 2 n λm k (2 n λ)φ 0 (λ) W δ,2 < ∞
for some auxiliary non trivial function φ 0 having compact support in (0, ∞).

3-

The assumption of the theorem for p > 2 is valid if the Riesz transform ∇L -1/2 is bounded on L p (M). This latter property may not be satisfied in some case even for L = ∆, especially when p > m where m is the dimension of M (see [START_REF] Coulhon | Riesz transforms for 1 ≤ p ≤ 2[END_REF]). For L = ∆ + V we may have boundedness of the corresponding Riesz transform (together with √ V L -1/2 ) on L p under some integrability conditions on V (cf. [START_REF] Assaad | Riesz transforms of Schrödinger operators on manifolds[END_REF] or [START_REF] Devyver | Heat kernel and Riesz transform of Schrödinger operators[END_REF]). In the Euclidean case M = R m , ∇L -1/2 is bounded on L p for a range of p > 2 if V is in an appropriate reverse Hölder class (cf. [START_REF] Auscher | Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials[END_REF] or [START_REF] Shen | L p estimates for Schrödinger operators with certain potentials[END_REF]). We shall come back to these examples again in Section 7 in which we will see that the Littlewood-Paley-Stein functional might be unbounded outside the range of p for which we have boundedness of the Riesz transform. 4-In [START_REF] Maati | Littlewood-Paley-Stein functions for Schrödinger operators[END_REF], it is shown for a class of potentials V that the boundedness on L p (R m ) for some p > m of the Littlewood-Paley-Stein functional

H ∇ (f ) = ∞ 0 |∇e -tL f | 2 dt 1/2 implies V = 0.

Other Littlewood-Paley-Stein functionals

Following [START_REF] Chen | Quasi transformées de Riesz, espaces de Hardy et estimations sousgaussiennes du noyau de la chaleur[END_REF], the local Riesz tranform for L is defined by R loc := ∇(L + I) -1/2 and the Riesz transform at infinity is R ∞ := ∇L -1/2 e -L . Then (cf. [START_REF] Chen | Quasi transformées de Riesz, espaces de Hardy et estimations sousgaussiennes du noyau de la chaleur[END_REF] Theorem 1.5), the Riesz transform is bounded on L p (M) if and only if R loc and R ∞ are both bounded on L p (M). The direct implication is obvious. For the converse, we write

∇L -1/2 f p ≤ ∇L -1/2 e -L f p + ∇(L + I) -1/2 (L + I) 1/2 L -1/2 (I -e -L )f p ≤ C f p + (L + I) 1/2 L -1/2 (I -e -L )f p .
(5.1)

Since (L + I) 1/2 L -1/2 (I -e -L ) = ϕ(L) with ϕ(z) = √ z + 1 1-e -z √ z
we use the boundedness of the holomorphic functional calculus on L p and obtain ∇L -1/2 f p ≤ C f p . The same observation is also valid for √ V in place of ∇.

We define the local vertical Littlewood-Paley-Stein functional and the vertical Littlewood-Paley-Stein functional at infinity associated with L by

H loc Γ (f ) := 1 0 |Γe -tL f | 2 dt 1/2
and

H (∞) Γ (f ) := ∞ 1 |Γe -tL f | 2 dt 1/2 .
We restrict our selves in this section to these Littlewood-Paley-Stein functionals but we can also deal with general ones as in Theorem 4.1 at least for functions F which have some exponential decay at infinity. As we have already remarked in the introduction, these functionals are always bounded on L p (M) for p ∈ (1, 2]. Thus, we consider in the sequel the case p > 2, only.

Proposition 5.1. Let Γ be either ∇ or multiplication by

√ V and let p ∈ (2, ∞). 1) If the set { √ t Γ e -tL , t ∈ (0, 1]} is R-bounded on L p (M), then the local vertical Littlewood- Paley-Stein functional H loc Γ is bounded on L p (M). 2) Similarly, if the set { √ t -1 Γ e -tL , t > 1} is R-bounded on L p (M), then H (∞) Γ is bounded on L p (M).
Proof. The arguments are similar to the proof of Theorem 4.1. For assertion 1), the same proof as (4.3) gives

1 0 |Γ e -tL f | 2 dt 1/2 ≤ 2 |Γ e -L f | + 1 0 |Γ tLe -tL f | 2 dt 1/2 .
(5.2)

Note that the R-boundedness assumption implies that Γ e -L is a bounded operator on L p (M). The second term on the RHS of (5.2) coincides (up to a constant) with

1 0 | t 2 Γ e -t 2 L t 2 Le -t 2 L f | 2 dt 1/2 . Since { t 2 Γ e -t 2 L , t ∈ (0, 1]} is R-bounded we apply Lemma 2.3. Note that the term t 2 Le -t 2 L f is in L 2 ((0, ∞), dt
) by a square function estimate. In order to prove assertion 2), we start by writing

∞ 1 |Γ e -tL f | 2 dt = (t -2)|Γ e -tL f | 2 ∞ 1 + 2 ∞ 1 (t -2)ΓLe -tL f.Γe -tL f dt
and proceed exactly as in the proof of (4.3) to obtain

∞ 1 |Γ e -tL f | 2 dt 1/2 ≤ 2 |Γ e -L f | + ∞ 1 |(t -2)Γ Le -tL f | 2 dt 1/2 . (5.3) Next, since Γe -tL has L p -norm bounded by C √ t , the part 2 1 |(t -2)Γ Le -tL f | 2 dt 1/2
is obviously bounded on L p (M). It remain to deal with the part involving t ≥ 2. This part coincides with (up to constant)

∞ 2 | t 2 -1 Γ e -t 2 L t 2 -1 Le -t 2 L f | 2 dt 1/2
. Now, we use the R-boundedness of { t 2 -1 Γ e -t 2 L , t > 2}, Lemma 2.3 and a square function estimate for the term t 2 -1 Le -t 2 L f to obtain 2).

We have the following version of Proposition 2.1.

Proposition 5.2. Let p ∈ (1, ∞). If the local Riesz transform Γ(L + I) -1/2 is bounded on L p (M), then { √ t Γ e -tL , t ∈ (0, 1]} is R-bounded on L p (M). Similarly, if the Riesz transform at infinity ΓL -1/2 e -L is bounded on L p (M), then { √ t -1 Γ e -tL , t > 1} is R-bounded on L p (M).
Proof. The proof of the first assertion is exactly the same as for Proposition 2.1. We prove the second one. Let f k ∈ L p (M) and t k > 1 for k = 1, ..., n. We have

E n k=1 r k √ t k -1 Γ e -t k L f k p = E Γ L -1/2 e -L n k=1 r k ((t k -1)L) 1/2 e -(t k -1)L f k p ≤ C E n k=1 r k ((t k -1)L) 1/2 e -(t k -1)L f k p .
We finish the proof by appealing again to the R-boundedness of the holomorphic functional calculus.

It is an interesting question whether the boundedness of the Littlewood-Paley-Stein functional implies the boundedness of the Riesz transform. For L = ∆ on R m this is true and very easy to prove (see [START_REF] Elias | Topics in Harmonic Analysis Related to the Littlewood-Paley Theory[END_REF], p. 52-54). Note however that this uses heavily the fact that ∇ and ∆ commute, a fact which is rarely satisfied outside the Euclidean context. If L = ∆ and M satisfies (4.7) and L 2 -Poincaré inequalities, then the L p -boundedness of H implies boundedness of the Riesz transform on L r for r ∈ (1, p). Indeed, the boundedness of H implies that ∇e -t∆ p ≤ C √ t by Proposition 5.3 below. The latter inequality implies the boundedness of the Riesz transform on L r (M) for r < p, see [START_REF] Auscher | Riesz transform on manifolds and heat kernel regularity[END_REF] or [START_REF] Bernicot | Riesz transforms through reverse Hölder and Poincaré inequalities[END_REF]. In general, we do not have an answer to the previous question but we make some observations below. Let d be the exterior derivative on differential forms and let d * its formal adjoint. One defines the Hodge-de Rham Laplacian ∆ on 1-differential forms by the formal expression ∆ = dd * + d * d. It is the self-adjoint operator associated with the symmetric bilinear form

a(w, η) = M dw.dη + M d * w.d * η,
for w and η in the Sobolev space of 1-forms such that |w| 2 , |dw| 2 + |d * w| 2 are integrable on M. As a self-adjoint, -∆ generates a C 0 -semigroup (e -t ∆ ) on the L 2 -space of 1-forms. We also recall the commutation property d ∆ = ∆d. The reader can consult [START_REF] Driver | Heat equation derivative formulas for vector bundles[END_REF] and the references there.

Let p ∈ (1, ∞) and suppose that ∆ satisfies the (weak) lower square function estimate

e -∆ w p ≤ C ∞ 1 | ∆ 1/2 e -t ∆ w| 2 dt 1/2 p . (5.4)
Then the boundedness on L p (M) of the Littlewood-Paley-Stein functional at infinity implies the boundedness on L p (M) of Riesz transform at infinity (compare with [16, Theorem 5.1]). Indeed, we chose w = d∆ -1/2 f for f in the range of ∆ 1/2 , and notice that e -∆ d∆ -1/2 f = d∆ -1/2 e -∆ f and ∆ 1/2 e -t ∆ d∆ -1/2 f = de -t∆ f . Then (5.4) gives

R ∞ f p ≤ C ∞ 1 |de -t∆ f | 2 dt 1/2 p ≤ C ′ f p .
If for example the Ricci curvature is bounded from below, then the local Riesz transform is bounded on L p (M) for all p ∈ (1, ∞) (cf. [START_REF] Bakry | Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée[END_REF]). This together with the observation (5.1) imply the boundedness of the Riesz transform on L p (M).

The next observation is that if we have the following Littlewood-Paley-Stein estimate for ∆ on 1-forms

∞ 0 |d * e -t ∆ w| 2 dt 1/2 p ≤ C w p , (5.5)
then the Riesz transform d∆ -1/2 is bounded on L q (M), where 1 p + 1 q = 1. Indeed, using the lower square function estimate for ∆ and the commutation property we obtain

d * w p ≤ C ∞ 0 |∆ 1/2 e -t∆ d * w| 2 dt 1/2 p = C ∞ 0 |d * e -t ∆ ∆ 1/2 w| 2 dt 1/2 p ≤ C ′ ∆ 1/2 w p .
This means that the Riesz transform d * ∆ -1/2 is bounded on L p (Λ 1 T * M) into L p (M). The adjoint is then bounded on L q (M). But the adjoint is exactly the Riesz transform d∆ -1/2 (use the commutation property again).

We also mention the following related result. It is taken from [START_REF] Coulhon | Riesz transform and related inequalities on noncompact Riemannian manifolds[END_REF] for L = ∆ and [START_REF] Maati | Littlewood-Paley-Stein functions for Schrödinger operators[END_REF] for L = ∆ + V . We reproduce the proof for the sake of completeness.

Proposition 5.3. Let p ∈ (1, ∞) and set Γ = ∇ or √ V . Suppose that ∞ 0 |Γe -tL f | 2 dt 1/2 p ≤ C f p (5.6) for all f ∈ L p (M). Then Γf p ≤ C ′ Lf 1/2 p f 1/2 p (5.7)
for f in the domain of L, seen as an operator on L p (M).

Proof. Set P t := e -t √ L the Poisson semigroup associated with L and fix f ∈ L 2 (M)∩D(L). By integration by parts,

∇P t f 2 2 + √ V P t f 2 2 = L 1/2 P t f 2 2 . In particular, ΓP t f 2 ≤ C t f 2 → 0 as t → +∞.
The same arguments show that t ΓL 1/2 P t f 2 → 0 as t → +∞. Therefore,

|Γf | 2 = - ∞ 0 d dt |ΓP t f | 2 dt = -t d dt |ΓP t f | 2 ∞ 0 + ∞ 0 d 2 dt 2 |ΓP t f | 2 t dt ≤ ∞ 0 d 2 dt 2 |ΓP t f | 2 t dt = 2 ∞ 0 (|ΓL 1/2 P t f | 2 + ΓLP t f.ΓP t f )t dt =: 2(I 1 + I 2 ).
On the other hand, (5.6) implies by the subordination formula for the Poisson semigroup

e -t √ L that G(f ) := ∞ 0 |Γe -t √ L f (x)| 2 t dt 1/2
is also bounded on L p (M). Observe that √ I 1 = G(L 1/2 f ) and by the Cauchy-Schwartz inequality

|I 2 | ≤ ∞ 0 (|ΓLP t f | 2 t dt 1/2 ∞ 0 (|ΓP t f | 2 t dt 1/2 ≤ G(Lf )G(f ).
Hence for any ǫ > 0

|Γf | ≤ √ 2(G(L 1/2 f ) + ǫG(f ) + 1 ǫ G(Lf )).
Taking the L p -norm yields

Γf p ≤ C ( L 1/2 f p + ǫ f p + 1 ǫ Lf p ). We chose ǫ = √ Lf p √ f p
and we obtain

Γf p ≤ C L 1/2 f p + f 1/2 p Lf 1/2 p .
It is well known that L 1/2 f p is bounded (up to a constant) by f

1/2 p Lf 1/2 p
(see, e.g., [START_REF] Komatsu | Fractional powers of operators[END_REF], Proposition 5.5). Hence (5.7) is proved for f ∈ D(L) ∩ L 2 (M). In order to extend this for all f ∈ D(L) we take a sequence f n ∈ L 2 (M) ∩ L p (M) which converges in the L p -norm to f . We apply (5.7) to e -tL f n (for t > 0) and then let n → +∞ and t → 0.

The standard argument of Stein which allows to prove that the functional

H(f ) = ∞ 0 |∇e -tL f | 2 dt 1/2 + ∞ 0 | √ V e -tL f | 2 dt 1/2
is always bounded on L p (M) for p ∈ (1, 2] can be used to prove the following proposition3

Proposition 5.4.

Let p ∈ (1, 2]. Then ∞ 0 ∇e -tL f 2 p dt + ∞ 0 √ V e -tL f 2 p dt ≤ C f 2 p (5.8) for all f ∈ L p (M). For q ∈ [2, ∞) we have C f 2 q ≤ ∞ 0 ∇e -tL f 2 q dt + ∞ 0 √ V e -tL f 2 q dt (5.9) for all f ∈ L q (M) ∩ L 2 (M).
Proof. It is enough to consider non-negative (and non-trivial

) f ∈ L 1 (M) ∩ L 2 (M).
Hence by irreducibility, e -t∆ f > 0 (a.e. on M). We have

∇e -tL f p p = M |∇e -tL f | p (e -tL f ) p(p-2) 2 (e -tL f ) p(2-p) 2 dx ≤ M |∇e -tL f | 2 (e -tL f ) p-2 dx p 2 M (e -tL f ) p dx 2-p 2 ≤ M |∇e -tL f | 2 (e -tL f ) p-2 dx p 2 f p(2-p) 2 p .
The same inequality holds when ∇ is replaced by √ V . Hence

∇e -tL f 2 p + √ V e -tL f 2 p ≤ M |∇e -tL f | 2 + | √ V e -tL f | 2 (e -tL f ) p-2 dx f 2-p p ≤ C M - ∂ ∂t (e -tL f ) p dx f 2-p p .
We integrate over t ∈ [0, τ ] to obtain

τ 0 ∇e -tL f 2 p + √ V e -tL f 2 p dt ≤ C M τ 0 - ∂ ∂t (e -tL f ) p dx f 2-p p = C f p p -e -τ L f p p f 2-p p ≤ C f 2 p .
Letting τ → ∞ gives the desired result. The proof of the lower estimate (5.9) is postponed to the next section (see (6.6)).

We have formulated the previous proposition for Schrödinger operators on manifolds but it is also true for elliptic operators with non-smooth coefficients on domains.

Lower bounds

In this section we prove reverse inequalities for the Littlewood-Paley-Stein functionals. The strategy is classical and it is based on a duality argument which goes back at least to [37, p. 85].

Let L = ∆ + V be again a Schrödinger operator with a non-negative potential V . We shall need the assumption that 0 in not an eigenvalue of L as an operator on L 2 (M). Otherwise, if M is compact without boundary, then ∇e -t∆ 1 = ∇1 = 0, and hence a lower estimate cannot hold for H ∇ .

Observe that if Lf = 0, then taking the scalar product with f yields ∇f = 0 and √ V f = 0. Hence, f is constant. Therefore, 0 cannot be an eigenvalue of L if M has infinite volume or if V is not identically zero.

The main result of this section reads as follows.

Theorem 6.1. Suppose that 0 in not an eigenvalue of L as an operator on L 2 (M). Let

m k : [0, ∞) → C in L 2 (0, ∞) ∩ L ∞ (0, ∞) and such that inf k m k 2 2 > 0. (6.1)
Let p ∈ (1, ∞) and q its conjugate number. Suppose that there exists a constant

C > 0 such that k ∞ 0 |∇m k (tL)f k | 2 dt 1/2 p + k ∞ 0 | √ V m k (tL)f k | 2 dt 1/2 p ≤ C k |f k | 2 1/2 p (6.2)
for all f k ∈ L p (M). Then there exists C ′ > 0 such that

C ′ k |g k | 2 1/2 q (6.3) ≤ k ∞ 0 |∇m k (tL)g k | 2 dt 1/2 q + k ∞ 0 | √ V m k (tL)g k | 2 dt 1/2 q for all g k ∈ L q (M) ∩ L 2 (M).
Proof. We may assume without loss of generality that k runs over {1, ..., n} for some finite n (the constants C and C ′ are then independent of n). Let f k ∈ L p (M) ∩ L 2 (M) and g k ∈ L q (M) ∩ L 2 (M). Set F = (f 1 , ..., f n ) and g = (g 1 , ..., g n ). We denote by ., . the usual scalar product in C n . Then we have

∞ 0 M ∇(m 1 (tL)f 1 , ..., m n (tL)f n ), ∇(m 1 (tL)g 1 , ..., m n (tL)g n ) dx dt + ∞ 0 M √ V (m 1 (tL)f 1 , ..., m n (tL)f n ), √ V (m 1 (tL)g 1 , ..., m n (tL)g n ) dx dt = ∞ 0 M (Lm 1 (tL)f 1 , ..., Lm n (tL)f n ), (m 1 (tL)g 1 , ..., m n (tL)g n ) dx dt = ∞ 0 M (L|m 1 | 2 (tL)f 1 , ..., L|m n | 2 (tL)f n ), (g 1 , ..., g n ) dx dt.
The first equality is obtained by integration by parts (with respect to x ∈ M) in each coordinate and the second one uses the duality and the basic fact that the adjoint of

m k (tL) is m k (tL). For each k, set M k (λ) := ∞ λ |m k (s)| 2 ds. Then M k (λ) → 0 as λ → ∞ and hence M k (tL)f → 0 in L 2 (M) as t → ∞ for all f ∈ L 2 (M).
In order to see this, we write by the spectral resolution

M k (tL)f 2 2 = (|M k | 2 (tL)f, f ) = ∞ 0 |M k (tλ)| 2 dE λ (f, f ).
Since 0 is not an eigenvalue of L, the latter integral is taken over (0, ∞). Now, the positive measure

dE λ (f, f ) is finite, |M k (.)| 2 is bounded on (0, ∞) (remember m k ∈ L 2 (0, ∞)),
and M k (tλ) → 0 as t → ∞ for all λ ∈ (0, ∞), then the result follows by the dominated convergence theorem. Using again the spectral resolution we see that

d dt M k (tL) = -L|m k | 2 (tL). From this we obtain ∞ 0 M (L|m 1 | 2 (tL)f 1 , ..., L|m n | 2 (tL)f n ), (g 1 , ..., g n ) dx dt = M ∞ 0 - d dt (M 1 (tL)f 1 , ..., M n (tL)f n ), (g 1 , ..., g n ) dt dx = M (f 1 , ..., f n ), (M 1 (0)g 1 , ..., M n (0)g n ) dx.
Using the forgoing equalities, the Cauchy-Schwarz inequality (for t) and Hölder's inequality

(in L r (C n )) yields M (f 1 , ..., f n ), (M 1 (0)g 1 , ..., M n (0)g n ) dx ≤ M ∞ 0 |∇(m 1 (tL)f 1 , ...m n (tL)f n )| 2 dt 1/2 ∞ 0 |∇(m 1 (tL)g 1 , ...m n (tL)g n )| 2 dt 1/2 + M ∞ 0 | √ V (m 1 (tL)f 1 , ...m n (tL)f n )| 2 dt 1/2 ∞ 0 | √ V (m 1 (tL)g 1 , ...m n (tL)g n )| 2 dt 1/2 ≤ k ∞ 0 |∇m k (tL)f k | 2 dt 1/2 p + k ∞ 0 | √ V m k (tL)f k | 2 dt 1/2 p × k ∞ 0 |∇m k (tL)g k | 2 dt 1/2 q + k ∞ 0 | √ V m k (tL)g k | 2 dt 1/2 q ≤ C k |f k | 2 1/2 p × k ∞ 0 |∇m k (tL)g k | 2 dt 1/2 q + k ∞ 0 | √ V m k (tL)g k | 2 dt 1/2 q .
Hence, for

J := k ∞ 0 |∇m k (tL)g k | 2 dt 1/2 q + k ∞ 0 | √ V m k (tL)g k | 2 dt 1/2 q ,
we have proved

| M F, (M 1 (0)g 1 , ..., M n (0)g n ) dx| ≤ C F L p (M,C n ) J.
This implies

(M 1 (0)g 1 , ..., M n (0)g n ) L q (M,C n ) ≤ C J.
Finally, we use (6.1) to finish the proof.

A particular case of the above theorem shows that if

H(f ) := ∞ 0 |∇e -tL f | 2 dt 1/2 + ∞ 0 | √ V e -tL f | 2 dt 1/2 (6.4)
is bounded on L p (M), then there exists a constant C > 0 such that C f q ≤ H(f ) q (6.5)

for f ∈ L q (M) ∩ L 2 (M). As we already mentioned in the introduction, H is bounded on L p (M) for p ∈ [START_REF] Assaad | Riesz transforms of Schrödinger operators on manifolds[END_REF][START_REF] Auscher | Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials[END_REF]. Therefore, the lower bound (6.5) holds for all q ∈ [2, ∞). This together with the triangle inequality for the L q 2 -norm implies

C f 2 q ≤ ∞ 0 ∇e -tL f 2 q dt + ∞ 0 √ V e -tL f 2 q dt. (6.6)
This is the lower bound stated in Proposition 5.4.

Recall from Section 5 the local Littlewood-Paley-Stein functional

H loc (f ) := 1 0 |∇e -tL f | 2 dt 1/2 + 1 0 | √ V e -tL f | 2 dt 1/2 .
We have seen that the boundedness on L p (M) of the local Riesz transforms ∇(L + I) -1/2 and √ V (L + I) -1/2 imply the boundedness on L p (M) of H loc . This together with the standard fact that the semigroup e -tL acts (as a contraction) on L p (M) imply that the functional

Q(f ) := |e -L f | + H loc (f ) (6.7)
is also bounded on L p (M). The next proposition shows that a lower bound is also true for Q. More precisely, Proposition 6.2. Let p ∈ (1, ∞) and suppose that H loc is bounded on L p (M). Then there exists a constant C > 0 such that

C g q ≤ Q(g) q for all g ∈ L q (M) ∩ L 2 (M), 1 p + 1 q = 1. Proof. Let f ∈ L p (M) ∩ L 2 (M) and g ∈ L q (M) ∩ L 2 (M).
We have

1 0 M ∇e -tL f.∇e -tL g + √ V e -tL f. √ V e -tL g dx dt = 1 0 M (Le -2tL f )g dx dt = - 1 2 M 1 0 d dt (e -2tL f )g dx dt = 1 2 M f g dx - 1 2 M (e -L f )(e -L g) dx.
Therefore,

M f g dx ≤ M |e -L f ||e -L g| dx + 2 M H loc (f )H loc (g) dx ≤ 2 M (|e -L f | + H loc (f ))(|e -L g| + H loc (g)) ≤ 2 Q(f ) p Q(g) q ≤ C f p Q(g) q .
The latter inequality extends by density to all f ∈ L p (M) and the proposition follows.

The final observation in this section is that if the Littlewood-Paley-Stein functional at infinity

H (∞) (f ) := ∞ 1 |∇e -tL f | 2 dt 1/2 + ∞ 1 | √ V e -tL f | 2 dt 1/2
is bounded on L p (M) for some p ∈ (1, ∞), then C e -2L g q ≤ H (∞) g q for g ∈ L q (M) ∩ L 2 (M), 1 p + 1 q = 1. The proof is very similar to the previous one. When we integrate over t on [1, ∞) we obtain

∞ 1 M ∇e -tL f.∇e -tL g + √ V e -tL f. √ V e -tL g dx dt = - 1 2 M ∞ 1 d dt (e -2tL g)f dx dt = 1 2 M f e -2L g dx
and we proceed as before.

Examples and counter-examples

In this section we discuss several examples. We also give a short review of some known results on the Riesz transform. The boundedness of the Riesz transform implies the boundedness of the Littlewood-Paley-Stein functionals. We shall see that the examples for which the Riesz transform is not bounded are also examples for which the Littlewood-Paley-Stein functionals are unbounded.

The Laplacian.

We start with the case L = ∆ the (positive) Laplace-Beltrami operator on a manifold M.

We give examples of manifolds for which the Riesz transform ∇∆ -1/2 is bounded on L p (M) (with values in the L p -space of vector fields). This subject has been studied for many years and it is impossible to provide comprehensive bibliography here. Recall that if the Riesz transform is bounded then the Littlewood-Paley-Stein estimates of Section 4 are satisfied on L p (M). The lower bounds of Section 6 are then satisfied.

-Manifold with non-negative Ricci curvature. If M has non-negative Ricci curvature then it is well known that ∇∆ -1/2 is bounded on L p (M) for all p ∈ (1, ∞) (cf. [START_REF] Bakry | Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée[END_REF]).

-Ricci curvature bounded from below. In this case, the local Riesz transform ∇(I + ∆) -1/2 is bounded on L p (M) for all p ∈ (1, ∞) (cf. [START_REF] Bakry | Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée[END_REF]). It then follows from Section 5 that the local Littlewood-Paley-Stein functional

H loc (f ) := 1 0 |∇e -tL f | 2 dt 1/2
is bounded on L p (M) for all p ∈ (1, ∞). By Proposition 6.2, the lower bound (for some C q > 0)

C q f q ≤ e -∆ f q + 1 0 |∇e -tL f | 2 dt 1/2 q (7.1)
holds for all q ∈ (1, ∞). Note that (7.1) holds for q ∈ [2, ∞) on any Riemannian manifold since H ∇ (and hence H loc ) is always bounded on L p (M) for p ∈ (1, 2].

-Manifolds with doubling and Gaussian bound. Recall that if M satisfies (4.7) and (4.8) then the Riesz transform is bounded on L p (M) for all p ∈ (1, 2] and it is weak type (1, 1) (cf. [START_REF] Coulhon | Riesz transforms for 1 ≤ p ≤ 2[END_REF]). The case p > 2 is more complicate and there are counter-examples (see [START_REF] Coulhon | Riesz transforms for 1 ≤ p ≤ 2[END_REF]). One needs extra assumptions on M in order to have the Riesz transform bounded on L p (M) for p > 2. See for example [START_REF] Auscher | Riesz transform on manifolds and heat kernel regularity[END_REF][START_REF] Carron | Riesz transform and L pcohomology for manifolds with Euclidean ends[END_REF][START_REF] Peng Chen | The Hodge-de Rham Laplacian and L p -boundedness of Riesz transforms on non-compact manifolds[END_REF][START_REF] Coulhon | Riesz transform and related inequalities on noncompact Riemannian manifolds[END_REF][START_REF] Guillarmou | Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds (II)[END_REF] and the references therein. We recall the following result from [START_REF] Peng Chen | The Hodge-de Rham Laplacian and L p -boundedness of Riesz transforms on non-compact manifolds[END_REF]. Theorem 7.1 ([12], Theorem 4.1). Let M be a complete Riemannian manifold with the doubling property (4.7) and the Gaussian upper estimate (4.8). Suppose that the negative part R -of the Ricci curvature satisfies the following estimate

1 0 |R -| 1/2 v(., t 1/2 ) 1/r 1 r 1 dt t 1/2 + ∞ 1 |R -| 1/2 v(., t 1/2 ) 1/r 2 r 2 dt t 1/2 < ∞ (7.2)
for some r 1 > 2, r 2 > 3. Then the Riesz transform is bounded on L p (M) for p ∈ (1, r 2 ).

As a consequence, Theorems 4.1 and 4.3 apply to L = ∆ on L p (M) for p ∈ (2, r 2 ).

-

M n = R n #R n with n ≥ 2.
We consider the manifold M n which consists of two copies of R n \B(0, 1) endowed with the euclidean metrics and smoothly glued along the unit balls. It is proved in [START_REF] Coulhon | Riesz transforms for 1 ≤ p ≤ 2[END_REF] that for such manifold, the Riesz transform is unbounded on L p (M n ) for p > n. We also refer to [START_REF] Carron | Riesz transform and L pcohomology for manifolds with Euclidean ends[END_REF] and [START_REF] Guillarmou | Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds (II)[END_REF] for more general and precise results. In particular, it is proved in [START_REF] Carron | Riesz transform and L pcohomology for manifolds with Euclidean ends[END_REF] that the Riesz transform is bounded on L p (M n ) for p ∈ (1, n) and this is sharp. Therefore, the Littlewood-Paley-Stein estimates of Section 4 are satisfied on L p (M n ) for p ∈ (1, n). Similarly to the Riesz transform, this interval is optimal in the sense that H ∇ in (1.2) is unbounded on L p (M n ) for p > n. In order to see this, recall that M n satisfies the global Sobolev inequality

|f (x) -f (y)| ≤ C d(x, y) 1-n/p ∇f p . (7.3)
It is also known that there exist positive constants c and C such that cr n ≤ v(x, r) ≤ Cr n uniformly in r > 0 and x ∈ M n . Now, if the Littlewood-Paley-Stein functional is bounded on L p (M) for some p > n, then it follows from Proposition 5.3 and the analyticity of the semigroup that ∇e -t∆ f p ≤ C t 1/2 f p . We apply this inequality with f = p t (., y) (the heat kernel associated with ∆) and notice that p t (., y) = e -t 2 ∆ p t 2 (., y) and then combine this with (7.3) to obtain

|p t (x, y) -p t (y, y)| ≤ Cd(x, y) 1-n/p ∇p t (., y) p ≤ Cd(x, y) 1-n/p t -1/2+n/2p v(y, t 1/2 ) .
A well known chain argument allows to obtain from this inequality a Gaussian lower bound

p t (x, y) ≥ C e -c d(x,y) 2 t v(y, t 1/2 ) .
This lower bound is not true for M n . We refer to [START_REF] Coulhon | Riesz transforms for 1 ≤ p ≤ 2[END_REF] for the details and additional information.

Set g := ∆u + V u = -2∇φ.∇v + v∆φ. Suppose that H is bounded on L p 0 (R n ). Then ∇f 2 p 0 ≤ C f p 0 Lf p 0 by Proposition 5.3. Therefore,

∇u 2 p 0 = ∇L -1 g 2 p 0 ≤ C g p 0 u p 0 < ∞
since u and g are in L p 0 (they are bounded and compactly supported). But ∇u is not in

L p 0 (R n ) because |∇u| ∼ 1 |x| n/p 0 as x ∼ 0.
-Schrödinger operators on manifolds. Riesz transforms associated with Schrödinger operators have been also studied on Riemannian manifolds M. As we already mentioned before, if M satisfies (4.6) and (4.8) then ∇L -1/2 and √ V L -1/2 are bounded on L p (M) for p ∈ [START_REF] Assaad | Riesz transforms of Schrödinger operators on manifolds[END_REF][START_REF] Auscher | Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials[END_REF]. Here the potential V is non-negative and locally integrable. See [START_REF] Duong | Endpoint estimates for Riesz transforms of magnetic Schrödinger operators[END_REF] where this is stated on R N but the proof works on manifolds having (4.6) and (4.8). The case p > 2 is again complicate (even if M = R N ). We recall the following result which deals also with potentials having a non-trivial negative part. Theorem 7.4 ([1], Theorem 3.9). Suppose that M satisfies (4.7) and (4.8). Suppose in addition that V -is subcritical with rate α ∈ (0, 1), i.e., for all suitable f in L 2 (M) we have

M V -f 2 dx ≤ α M (|∇f | 2 + V + f 2 )dx. (7.5)
Assume there exist r 1 , r 2 > 2 such that

1 0 |V | 1/2 v(., s 1/2 ) 1/r 1 r 1 ds s 1/2 + ∞ 1 |V | 1/2 v(., s 1/2 ) 1/r 2 r 2 ds s 1/2 < ∞. (7.6)
Let r = inf(r 1 , r 2 ). If N ≤ 2, then the operators ∆ 1/2 L -1/2 and V 1/2 L -1/2 are bounded on L p for p ∈ (1, r). If N > 2, the same operators are bounded on L p for p ∈ (p ′ 0 , p 0 r p 0 +r ) where

p 0 = N N -2 2 1- √ 1-α .
In particular, if the Riesz transform ∇∆ -1/2 is bounded on L p with p in this range, then ∇L -1/2 is also bounded.

See also [START_REF] Devyver | Heat kernel and Riesz transform of Schrödinger operators[END_REF] for related results and additional information.

The above integrability condition in (7.6) gives then a range of p's for which the Littlewood-Paley-Stein functionals are bounded on L p (M). Finally, we mention the following negative result (see [START_REF] Maati | Littlewood-Paley-Stein functions for Schrödinger operators[END_REF] in the case M = R N ).

(x) -f (x ′ )| ≤ C x,x ′ d(x, x ′ ) 1-N/p ∇f p .
Suppose also that there exists a positive bounded function ψ such that e -tL ψ = ψ for all t ≥ 0. If the Littlewood-Paley-Stein functional

H ∇ (f ) = ∞ 0 |∇e -tL f | 2 dt 1/2
is bounded on L p (M) for some p > max(N, 2), then V = 0. Here N is the constant from the doubling condition (4.7).

Proof. Assume that H is bounded on L p , then for suitable f ,

∇f p ≤ C f 1/2 p Lf 1/2 p .
Taking f = e -tL g and using the analyticity of the semigroup we obtain for all g ∈ L p ∇e -tL g p ≤ C t 1/2 g p .

We conclude using Theorem 6.1 in [START_REF] Peng Chen | The Hodge-de Rham Laplacian and L p -boundedness of Riesz transforms on non-compact manifolds[END_REF]. Note that in this reference, it is assumed that M satisfies Poincaré inequalities, which in turn imply the above local Sobolev inequality. It is this later inequality which is used in the proof there.

Elliptic operators on domains

We have chosen to write the previous sections in the framework of Schrödinger operators on manifolds. The results remain valid for elliptic operators with real bounded measurable coefficients and subject to Dirichlet boundary conditions on a domain. The proofs, after a little adaptation, are the same.

Let Ω be an open subset of R N (N ≥ 1). We consider for k, l ∈ {1, ..., N} bounded measurable functions a kl = a lk : Ω → R. We suppose the usual ellipticity condition From this and little modifications in the proofs in Section 6 we obtain the lower bounds on L q (Ω) ∩ L 2 (Ω) for all q ∈ [2, ∞). In particular,

C f L q (Ω) ≤ ∞ 0 |∇e -tL f | 2 dt 1/2 L q (Ω) , (8.3)
and

C f L q (Ω) ≤ e -L f L q (Ω) + 1 0 |∇e -tL f | 2 dt 1/2 L q (Ω)
.

(8.4)

It is remarkable that no regularity assumption is required on the domain nor on the coefficients of the operator. For another proof of (8.4) and related inequalities on a smooth domain, we refer to [START_REF] Ivanovici | Square function and heat flow estimates on domains[END_REF].

In the next result we show that if Ω = R N , the previous lower bounds hold for all q ∈ (1, ∞). Proposition 8.1. Let L = -div(A(x)∇•) be a self-adjoint elliptic operator with real bounded measurable coefficients a kl . Then for all q ∈ (1, ∞)

C f L q (R N ) ≤ ∞ 0 |∇e -tL f | 2 dt 1/2 L q (R N ) (8.5) and C f L q (R N ) ≤ e -L f L q (R N ) + 1 0 |∇e -tL f | 2 dt 1/2 L q (R N )
.

(8.6)

Proof. Because of (8.3) and (8.4) we consider the case q ∈ (1, 2) only.

Since the semigroup e -tL is sub-Markovian, L has a bounded holomorphic functional calculus on L p (R N ). Therefore, it has bounded square functions on L p (R N ) for all p ∈ (1, ∞).

A standard duality argument gives then (for q ∈ (1, ∞))

C f L q (R N ) ≤ ∞ 0 |L 1/2 e -tL f | 2 dt 1/2 L q (R N ) . (8.7)
On the other hand, it follows from [3, Theorem 2, p.115] that there exists a Calderón-Zygmund operator U such that L 1/2 f = U∇f . Therefore,

C f L q (R N ) ≤ ∞ 0 |U∇e -tL f | 2 dt 1/2 L q (R N )
The operator U is bounded on L q (R N ). We then apply the same strategy of proof as in Theorem 4.1 and use the Kahane inequality to bound the RHS term by

C ′ ∞ 0 |∇e -tL f | 2 dt 1/2 L q (R N )
.

This proves (8.5).

In order to prove (8.6) for q ∈ (1, 2), we write

∞ 0 |∇e -tL f | 2 dt 1/2 L q (R N ) ≤ 1 0 |∇e -tL f | 2 dt 1/2 L q (R N ) + ∞ 1 |∇e -tL f | 2 dt 1/2 L q (R N ) = 1 0 |∇e -tL f | 2 dt 1/2 L q (R N ) + ∞ 0 |∇e -tL e -L f | 2 dt 1/2 L q (R N ) ≤ 1 0 |∇e -tL f | 2 dt 1/2 L q (R N ) + C ′′ e -L f p .
Note that in the last inequality we use the boundedness of the Littlewood-Paley-Stein functional on L q (R N ) for q ∈ (1, 2). Now, (8.6) follows from (8.5) and the previous inequality.

We finish this section by mentioning another sort of Littlewood-Paley-Stein functionals, called conical vertical square functions, and defined by .

It is proved in [START_REF] Auscher | Vertical versus conical square functions[END_REF], among other things, that S is bounded on L p (R N ) for all p ∈ (1, ∞). Thus, the functionals S and H have different behavior on L p (R N ) for p > 2. It is of interest to study the corresponding conical functionals S for Schrödinger operators on manifolds or for elliptic operators on arbitrary domains of R N . This will be done in a forthcoming project.

Theorem 4 . 1 .

 41 Let m k , F ∈ H ∞ (Σ(ω p )) for k = 1, 2, ... and assume that for some δ > 1 2

Theorem 4 . 3 . 2 , 2 ]

 4322 Suppose that M satisfies (4.6) and (4.8). Let m k : [0, ∞) → C with support contained in [ 1 for every k. Let p ∈ (1, 2]. Then there exist C > 0, independent of m k , and δ > 0 such that

Proposition 7 . 5 .

 75 Assume that M satisfies (4.7), (4.8) and the local Sobolev inequality |f

a4 2 , 2 ]- 1 p

 221 kl (x)ξ k ξ l ≥ ν|ξ| 2 for all ξ = (ξ 1 , ..., ξ n ) ∈ R N , where ν > 0 is a constant independent of x. Set A(x) := (a kl (x)) 1≤k,l≤N . We define the elliptic operator L = -div(A(x)∇•) with Dirichlet boundary conditions. It is the operator associated with the symmetric forma(u, v) = N k,l=1 Ω a kl ∂ k u∂ l v, u, v ∈ W 1,2 0 (Ω).It is known that the heat kernel of L satisfies a Gaussian upper bound and the Riesz transform ∇L -1/2 is bounded on L p (Ω) for all p ∈ (1, 2]. In addition, L satisfies spectral multiplier theorems. See [33, Chapters VI and VII]. The fact that Ω, endowed with the Euclidean distance and Lebesgue measure, may not satisfy the doubling property (4.6) 4 can be bypassed in the proofs of the boundedness of the Riesz transform and spectral multipliers. Thus, the Littlewood-Paley-Stein estimates (4.1) and (4.10) hold on L p (Ω) for all p ∈ (1, 2]. except if Ω is bounded and has smooth boundary, Lipschitz boundary is enough.for bounded holomorphic functions m k and F on a sector of angle θ > 0 5 . If the functions m k are supported in [ 1 and belong to the Sobolev space W δ,2 (R) for some δ > N| 1 2

  |∇ y e -tL f (y)| 2 dy dt t

The quasi-Riesz transforms R loc and R ∞ were studied by L. Chen[START_REF] Chen | Quasi transformées de Riesz, espaces de Hardy et estimations sousgaussiennes du noyau de la chaleur[END_REF] for the Laplace-Beltrami operator.

in the sequel, for a given g ∈ H, we use the notation g(t) H instead of g H or g(.) H . This makes reading easier since the variable t appears at several places.

we owe this observation to Sylvie Monniaux.

here we may take any θ > 0 and not necessarily ω p as in Theorem 4.1. The reason is that the Gaussian upper bound implies the existence of a bounded holomorphic functional with angle θ. This follows readily from the fact that L satisfies spectral multiplier theorems.

Note that M n has Ricci curvature bounded from below. Therefore, the local Littlewood-Paley-Stein functional is bounded on L p (M n ) for all p ∈ (1, ∞). It is then the Littlewood-Paley-Stein at infinity which is not bounded on L p (M n ) for p > n.

Schrödinger operators.

-Potentials in the Reverse Hölder class. We consider L = ∆ + V on L p (R n ) for some n ≥ 3. We assume that the non-negative potential V belongs to the Reverse Hölder class B q , that is, there exists a constant C > 0 such that for every ball B in R n ,

It is known that this property self-improves in the sense that there exists ǫ > 0 such that

This result was improved in [START_REF] Auscher | Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials[END_REF] by considering the cases n < 3 or q ≥ n and also the boundedness of

2-If V ∈ B q for some q ≥ n/2 and q > 1, then ∇L -1/2 is bounded on L p (R n ) for all p ∈ (1, q * + ǫ) if q < n and for all p ∈ (1, ∞) if q ≥ n. Here q * = nq n-q . We apply Theorems 4.1 and 4.3 to obtain general Littlewood-Paley-Stein estimates on L p (R n ) for p in one of intervals given in Theorem 7.2. Their reverse inequalities proved in Section 6 hold on the dual space.

It is also proved in [START_REF] Shen | L p estimates for Schrödinger operators with certain potentials[END_REF] that the above range is optimal for the boundedness of the Riesz transform. One may then ask whether this range is optimal for the boundedness of the Littlewood-Paley-Stein functional as well. This is indeed the case. Proposition 7.3. There exists V ∈ B q with n/2 ≤ q < n such that the Littlewood-Paley-Stein functional H is not bounded on L p (R n ) for any p > q * + ǫ.

Proof. We follow exactly the same arguments as in [START_REF] Shen | L p estimates for Schrödinger operators with certain potentials[END_REF]Section 7]. Let q 0 > n/2 and set V (x) := 1 |x| n/q 0 . Then V ∈ B q for all q < q 0 . Therefore, the Littlewood-Paley-Stein function H is bounded on L p (R n ) for all p such that 1 p > 1 q 0 -1 n . We show that it is false for p = p 0 with 1 p 0 = 1 q 0 -1 n . Let v be the function defined by

with µ = 2 -n q 0 . One has by a direct computation ∆v + V v = 0.

Set u := φv where φ is a smooth non-negative compactly supported function with φ(x) = 1 if |x| ≤ 1. We have ∆u + V u = -2∇φ.∇v + v∆φ.