N Badr
email: badr@math.univ-lyon1.fr

G Dafni
email: gdafni@mathstat.concordia.ca

An atomic decomposition of the Haj lasz Sobolev space M 1 1 on manifolds *

Keywords: Hardy-Sobolev spaces, atomic decomposition, metric measure spaces, Haj lasz-Sobolev spaces MS Classification (2010): Primary: 42B30; Secondary: 46E35, 58D15

Several possible notions of Hardy-Sobolev spaces on a Riemannian manifold with a doubling measure are considered. Under the assumption of a Poincaré inequality, the space M 1 1 , defined by Haj lasz, is identified with a Hardy-Sobolev space defined in terms of atoms. Decomposition results are proved for both the homogeneous and the nonhomogeneous spaces.

Introduction

The aim of this paper is to compare different definitions of Hardy-Sobolev spaces on manifolds. In particular, we consider characterizations of these spaces in terms of maximal functions, atomic decompositions, and gradients, some of which have been shown in the Euclidean setting, and apply them to the L 1 Sobolev space defined by Haj lasz.

In the Euclidean setting, specifically on a domain Ω ⊂ R n , Miyachi [START_REF] Miyachi | Hardy-Sobolev spaces and maximal functions[END_REF] shows that for a locally integrable function f to have partial derivatives ∂ α f (taken in the sense of distributions) belonging to the real Hardy space H p (Ω), is equivalent to a certain maximal function of f being in L p (Ω). Earlier work by Gatto, Jiménez and Segovia [START_REF] Gatto | On the solution of the equation ∆ m F = f for f ∈ H p[END_REF] on Hardy-Sobolev spaces, defined via powers of the Laplacian, used a maximal function introduced by Calderón [START_REF] Calderón | Estimates for singular integral operators in terms of maximal functions, in: Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI[END_REF] in characterizing Sobolev spaces for p > 1 to extend his results to p ≤ 1. Calderón's maximal function was subsequently studied by Devore and Sharpley [START_REF] Devore | Maximal functions measuring smoothness[END_REF], who showed that it is pointwise equivalent to the following variant of the sharp function. For simplicity we only give the definition in the special case corresponding to one derivative in L 1 , which is what this article is concerned with. We will call this function the Sobolev sharp maximal function (it is also called a "fractional sharp maximal function" in [START_REF] Kinnunen | Pointwise behaviour of M 1,1 Sobolev functions[END_REF]): Another definition of Hardy-Sobolev spaces on R n , using second differences, is given by Strichartz [START_REF] Strichartz | H p Sobolev spaces[END_REF], who also obtains an atomic decomposition. Further characterizations of Hardy-Sobolev spaces on R n by means of atoms are given in [START_REF] Cho | Atomic decomposition on Hardy-Sobolev spaces[END_REF] and [START_REF] Lou | An atomic decomposition for the Hardy-Sobolev space[END_REF]. For related work see [START_REF] Janson | On functions with derivatives in H 1[END_REF].

Several recent results provide a connection between Hardy-Sobolev spaces and the p = 1 case of Haj lasz's definition of L p Sobolev spaces on a metric measure space (X, d, µ): Definition 1.2 (Haj lasz). Let 1 ≤ p ≤ ∞. The (homogeneous) Sobolev space Ṁ1 p is the set of all functions u ∈ L 1,loc such that there exists a measurable function g ≥ 0, g ∈ L p , satisfying |u(x)u(y)| ≤ d(x, y)(g(x) + g(y)) µa.e.

(

) 1
We equip Ṁ1 p with the semi-norm

u Ṁ 1 p = inf g satisfies(1)
g p .

In the Euclidean setting, Haj lasz [START_REF] Haj | Sobolev spaces on an arbitrary metric space[END_REF] showed the equivalence of this definition with the usual one for 1 < p ≤ ∞. For p ∈ (n/n + 1, 1], Koskela and Saksman [START_REF] Koskela | Pointwise characterizations of Hardy-Sobolev functions[END_REF] proved that Ṁ1

p (R n ) coincides with the homogeneous Hardy-Sobolev space Ḣ1 p (R n ) defined by requiring all first-order partial derivatives of f to lie in the real Hardy space H p (the same space defined by Miyachi [START_REF] Miyachi | Hardy-Sobolev spaces and maximal functions[END_REF]). In recent work [START_REF] Koskela | A characterization of Haj lasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions[END_REF], the Haj lasz Sobolev spaces Ṁs p , for 0 < s ≤ 1 and n n+s < p < ∞, are characterized as homogeneous grand Triebel-Lizorkin spaces.

In the more general setting of a metric space with a doubling measure, Kinnunen and Tuominen [START_REF] Kinnunen | Pointwise behaviour of M 1,1 Sobolev functions[END_REF] show that Haj lasz's condition is equivalent to Miyachi's maximal function characterization, extending to p = 1 a previous result of Haj lasz and Kinnunen [START_REF] Haj Lasz | Hölder quasicontinuity of Sobolev functions on metric spaces[END_REF] for p > 1:

Theorem 1. 3 ([17], [START_REF] Kinnunen | Pointwise behaviour of M 1,1 Sobolev functions[END_REF]). For 1 ≤ p < ∞

Ṁ1 p = {f ∈ L 1,loc : Nf ∈ L p } with f Ṁ 1 p ∼ Nf p . Moreover, if f ∈ L 1,loc and Nf ∈ L 1 , then f satisfies |f (x) -f (y)| ≤ Cd(x, y)(Nf (x) + Nf (y)) (2)
for µa.e. x, y.

We now restrict the discussion to a complete Riemannian manifold M satisfying a doubling condition and a Poincaré inequality (see below for definitions). In this setting, Badr and Bernicot [START_REF] Badr | Abstract Hardy-Sobolev spaces and Interpolation[END_REF] defined a family of homogeneous atomic Hardy-Sobolev spaces ḢS 1 t,ato and proved the following comparison between these spaces: Theorem 1.4. ( [START_REF] Badr | Abstract Hardy-Sobolev spaces and Interpolation[END_REF]) Let M be a complete Riemannian manifold satisfying a doubling condition and a Poincaré inequality (P q ) for some q > 1. Then ḢS 1 t,ato ⊂ ḢS 1 ∞,ato for every t ≥ q and therefore ḢS 1 t 1 ,ato = ḢS 1 t 2 ,ato for every q ≤ t 1 , t 2 ≤ ∞. In particular, under the assumption of the Poincaré inequality (P 1 ), for every t > 1 we can take 1 < q ≤ t for which (P q ) holds, so all the atomic Hardy-Sobolev spaces ḢS 1 t,ato coincide and can be denoted by ḢS 1 ato . The main result of this paper is to identify this atomic Hardy-Sobolev space with Haj lasz's Sobolev space for p = 1: Theorem 1.5. Let M be a complete Riemannian manifold satisfying a doubling condition and the Poincaré inequality (P 1 ). Then

Ṁ1 1 = ḢS 1 ato .
The definition of the atomic Hardy-Sobolev spaces, as well as the doubling condition, the Poincaré inequality, and other preliminaries, can be found in Section 2. The proof of Theorem 1.5, based on the characterization given by Theorem 1.3 and a Calderón-Zygmund decomposition, follows in Section 3. In Section 4, a nonhomogeneous version of Theorem 1.5 is obtained. Finally, in Section 5, we characterize our Hardy-Sobolev spaces in terms of derivatives. In particular, we show that the space of differentials df of our Hardy-Sobolev functions coincides with the molecular Hardy space of differential one-forms defined by Auscher, McIntosh and Russ [START_REF] Auscher | Hardy spaces of differential forms on Riemannian manifolds[END_REF] (and by Lou and McIntosh [START_REF] Lou | Hardy spaces of exacts forms on R n[END_REF] in the Euclidean setting).

Preliminaries

In all of this paper M denotes a complete non-compact Riemannian manifold. We write T x M for the tangent space at the point x ∈ M, •, • x for the Riemannian metric at x, and µ for the Riemannian measure (volume) on M. The Riemannian metric induces a distance function ρ which makes (M, ρ) into a metric space, and B(x, r) will denote the ball of radius r centered at x in this space.

Let T * x M be the cotangent space at x, ΛT * x M the complex exterior algebra, and d the exterior derivative acting on C ∞ 0 (ΛT * M). We will work only with functions (0-forms) and hence for a smooth function f , df will be a 1-form. In fact, in most of the paper we will deal instead with the gradient ∇f , defined as the image of df under the isomorphism between T *

x M and T x M (see [START_REF] Warner | Foundations of Differentiable Manifolds and Lie Groups[END_REF], Section 4.10). Since this isomorphism preserves the inner product, we have

df, df x = ∇f, ∇f x (3)
, g ∈ C ∞ 0 (M), using (3), ∆f, g L 2 (M ) = M ∆f, g x dµ = M df, dg x dµ = ∇f, ∇g L 2 (M ) .
We will use Lip(M) to denote the space of Lipschitz functions, i.e. functions f satisfying, for some C < ∞, the global Lipschitz condition

|f (x) -f (y)| ≤ Cρ(x, y) ∀ x, y ∈ M.
The smallest such constant C will be denoted by f Lip . By Lip 0 (M) we will denote the space of compactly supported Lipschitz functions. For such functions the gradient ∇f can be defined µ-almost everywhere and is in L ∞ (M), with ∇f ∞ ≈ f Lip (see [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF] for Rademacher's theorem on metric measure spaces and also the discussion of upper gradients in [START_REF] Haj Lasz | Sobolev met Poincaré[END_REF], Section 10.2).

The doubling property

Definition 2.1. Let M be a Riemannian manifold. One says that M satisfies the (global) doubling property (D) if there exists a constant C > 0, such that for all x ∈ M, r > 0 we have

µ(B(x, 2r)) ≤ Cµ(B(x, r)). (D) Observe that if M satisfies (D) then diam(M) < ∞ ⇔ µ(M) < ∞ (see [1]). Therefore if M is a complete non-compact Riemannian manifold satisfying (D) then µ(M) = ∞.
Lemma 2.2. Let M be a Riemannian manifold satisfying (D) and let s = log 2 C (D) . Then for all x, y ∈ M and θ ≥ 1 where

µ(B(x, θR)) ≤ Cθ s µ(B(x, R)). (4)
f E := - E f dµ := 1 µ(E) E f dµ. Then for every 1 < p ≤ ∞, M is L p bounded
and moreover it is of weak type [START_REF] Ambrosio | Calculus of variations : topics from the mathematical heritage of E. De Giorgi[END_REF][START_REF] Ambrosio | Calculus of variations : topics from the mathematical heritage of E. De Giorgi[END_REF]. Consequently, for r ∈ (0, ∞), the operator M r defined by

M r f (x) := [M(|f | r )(x)] 1/r
is of weak type (r, r) and L p bounded for all r < p ≤ ∞.

Recall that an operator T is of weak type (p, p) if there is C > 0 such that for any α > 0, µ({x

: |T f (x)| > α}) ≤ C α p f p p .

Poincaré inequality

Definition 2.4 (Poincaré inequality on M). We say that a complete Riemannian manifold M admits a Poincaré inequality (P q ) for some q ∈ [1, ∞) if there exists a constant C > 0 such that, for every function f ∈ Lip 0 (M) and every ball B of M of radius r > 0, we have

- B |f -f B | q dµ 1/q ≤ Cr - B |∇f | q dµ 1/q . (P q )
We also recall the following result Theorem 2.5. ( [START_REF] Haj | Sobolev spaces on metric-measure spaces[END_REF], Theorem 8.7) Let u ∈ Ṁ1 1 and g ∈ L 1 such that (u, g) satisfies (1). Take s s+1 ≤ q < 1 and λ > 1. Then (u, g) satisfies the following Sobolev-Poincaré inequality: there is a constant C > 0 depending on (D) and λ, independent of (u, g) such that for all balls B of radius r > 0,

- B |u -u B | q * dµ 1/q * ≤ Cr - λB g q dµ 1/q , ( 5
)
where q * = sq s-q .

Applying this together with Theorem 1.3, for u ∈ Ṁ1 1 we have

- B |u -u B | q * dµ 1/q * ≤ Cr - λB (Nu) q dµ 1/q (6)
for all balls B.

Comparison between N f and |∇f |

The following Proposition shows that the maximal function Nf controls the gradient of f in the pointwise almost-everywhere sense. In the Euclidean setting this result was demonstrated by Calderón (see [START_REF] Calderón | Estimates for singular integral operators in terms of maximal functions, in: Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI[END_REF], Theorem 4) for his maximal function N(f, x) (denoted by f ⋆ in Section 4.2 below), which was shown to be pointwise equivalent to our Nf by Devore and Sharpley (see also the stronger inequality (5.5) in [START_REF] Miyachi | Hardy-Sobolev spaces and maximal functions[END_REF], which bounds the maximal function of the partial derivatives).

Recall that if u ∈ C ∞ 0 (M), given any smooth vector field Φ with compact support, we can write, based on (3) and the definition of

d * , M ∇u, Φ x dµ := M du, ω Φ x dµ = M u(d * ω Φ )dµ,
where ω Φ is the 1-form corresponding to Φ under the isomorphism between the tangent space T x M and the co-tangent space T *

x M (see [START_REF] Warner | Foundations of Differentiable Manifolds and Lie Groups[END_REF], Section 4.10). Denoting d * ω Φ by div Φ, we can define, for u ∈ L 1,loc , the gradient ∇u in the sense of distributions by

∇u, Φ := - M u(div Φ)dµ (7)
for all smooth vector fields Φ with compact support (see [START_REF] Miranda | Heat semigroup and functions of bounded variation on Riemannian manifolds[END_REF]). When M is orientable, div Φ is given by * d * ω Φ with * the Hodge star operator (see [START_REF] Warner | Foundations of Differentiable Manifolds and Lie Groups[END_REF]), and in the Euclidean case this corresponds to the usual notion of divergence of a vector field.

Proposition 2.6. Assume that M satisfies (D), and suppose u ∈ L 1,loc with Nu ∈ L 1 .

Then ∇u, initially defined by [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF], is given by an L 1 vector field and satisfies |∇u| ≤ CNu µa.e.

Proof. Fix r > 0. We begin with a covering of M by balls

B i = B(x i , r), i = 1, 2... such that 1. M ⊂ ∪ i B i , 2. i 1 1 6B i ≤ K.
Note that the constant K can be taken independent of r. Then we take {ϕ i } i a partition of unity related to the covering

{B i } i such that 0 ≤ ϕ i ≤ 1, ϕ i = 0 on (6B i ) c , ϕ i ≥ c on 3B i and i ϕ i = 1.
The ϕ i 's are C/r Lipschitz. For details concerning this covering we refer to [START_REF] Franchi | Definitions of Sobolev classes on metric spaces[END_REF], [START_REF] Kinnunen | Pointwise behaviour of M 1,1 Sobolev functions[END_REF], [START_REF] Heinonen | Lectures on analysis on metric spaces[END_REF], [START_REF] Coifman | Extensions of Hardy spaces and their use in analysis[END_REF]. Now let (see [START_REF] Franchi | Definitions of Sobolev classes on metric spaces[END_REF], p. 1908 and [START_REF] Kinnunen | Pointwise behaviour of M 1,1 Sobolev functions[END_REF], Section 3.1)

u r (x) = j ϕ j (x)u 3B j . (8)
The sum is locally finite and defines a Lipschitz function so we can take its gradient and we have, for µ-almost every x,

|∇u r (x)| = | j ∇ϕ j (x)u 3B j | = | {j:x∈6B j } ∇ϕ j (x)(u 3B j -u B(x,9r) )| ≤ CK 1 r - B(x,9r) |u -u B(x,9r) |dµ ≤ CKNu(x). (9)
We used the fact that ∇φ j = 0 and that for x ∈ 6B j , 3B j ⊂ B(x, 9r). To see that u r → u µa.e. and moreover in L 1 when r → 0 (see also [START_REF] Franchi | Definitions of Sobolev classes on metric spaces[END_REF],p. 1908), write, for x a Lebesgue point of µ,

|u r (x) -u(x)| ≤ j |ϕ j (x)||u(x) -u 3B j | ≤ {j:x∈6B j } |u(x) -u 3B j | ≤ CKrM q (Nu)(x)
where s s+1 ≤ q < 1. The last inequality follows from estimates of |u(x)u B(x,9r) | and |u 3B ju B(x,9r) |, x ∈ 6B j , which are the same as estimates ( 12)- [START_REF] Gatto | On the solution of the equation ∆ m F = f for f ∈ H p[END_REF] in the proof of Lemma 1 in [START_REF] Kinnunen | Pointwise behaviour of M 1,1 Sobolev functions[END_REF], using the doubling property and [START_REF] Calderón | Estimates for singular integral operators in terms of maximal functions, in: Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI[END_REF]. Now let Φ be a smooth vector field with compact support. Using the convergence in L 1 , the fact that div Φ ∈ C ∞ 0 (M), and the estimate on |∇u r | above, we have

| M ∇u, Φ x dµ| = | M u(div Φ)dµ| = | lim r → 0 M u r (div Φ)dµ| ≤ lim sup r → 0 M |∇u r ||Φ|dµ ≤ CK |Nu||Φ|dµ.
Taking the supremum of the left-hand-side over all such Φ with |Φ| ≤ 1, we get that the total variation of u is bounded (see [START_REF] Miranda | Heat semigroup and functions of bounded variation on Riemannian manifolds[END_REF], (1.4), p. 104), i.e.

|Du|(M) ≤ C Nu L 1 (M ) < ∞,
hence u is a function of bounded variation on M, and |Du| defines a finite measure on M. We can write the distributional gradient as

∇u, Φ = M X u , Φ x d|Du|
for some vector field X u with |X u | = 1 a.e. (see again [START_REF] Miranda | Heat semigroup and functions of bounded variation on Riemannian manifolds[END_REF], p. 104 where this is expressed in terms of the corresponding 1-form σ u ). Moreover, from the above estimates and the fact that Nu ∈ L 1 , we further deduce that the measure |Du| is absolutely continuous with respect to the Riemannian measure µ, so there is an L 1 function g such that we can write ∇u = gX u , and |∇u| ≤ CNu, µa.e.

Corollary 2.7. Assume that M satisfies (D). Then

Ṁ1 1 ⊂ Ẇ 1 1 .
Proof. The result follows from Proposition 2.6 and Theorem 1.3.

Hardy spaces

We begin by introducing the maximal function characterization of the real Hardy space H 1 .

Definition 2.8. Let f ∈ L 1,loc (M). We define its grand maximal function, denoted by f + , as follows:

f + (x) := sup ϕ∈T 1 (x)
f ϕdµ [START_REF] Coifman | Extensions of Hardy spaces and their use in analysis[END_REF] where T 1 (x) is the set of all test functions ψ ∈ Lip 0 (M) such that for some ball B := B(x, r) containing the support of ψ,

ψ ∞ ≤ 1 µ(B) , ∇ψ ∞ ≤ 1 rµ(B) . ( 11
)
Set H 1,max (M) = {f ∈ L 1,loc (M) : f + ∈ L 1 (M)}.
While this definition assumes f to be only locally integrable, by taking an appropriate sequence ϕ ǫ ∈ T 1 (x), the Lebesgue differentiation theorem implies that

|f (x)| = lim ǫ → 0 f ϕ ǫ ≤ f + (x) for µ-a.e. x, (12)
so

H 1,max (M) ⊂ L 1 (M).
Another characterization is given in terms of atoms (see [START_REF] Coifman | Extensions of Hardy spaces and their use in analysis[END_REF]).

Definition 2.9.

Fix 1 < t ≤ ∞, 1 t + 1 t ′ = 1. We say that a function a is an H 1 -atom if 1. a is supported in a ball B, 2. a t ≤ µ(B) -1
t ′ , and

3. adµ = 0.
We say f lies in the atomic Hardy space H 1,ato if f can be represented, in L 1 (M), by

f = λ j a j ( 13
)
for sequences of H 1 -atoms {a j } and scalars {λ j } ∈ ℓ 1 . Note that this representation is not unique and we define

f H 1,ato := inf |λ j |,
where the infimum is taken over all atomic decompositions [START_REF] Franchi | Definitions of Sobolev classes on metric spaces[END_REF].

A priori this definition depends on the choice of t. However, we claim Proposition 2.10. Let M be a complete Riemannian manifold satisfying (D). Then

H 1,ato (M) = H 1,max (M) with equivalent norms f H 1,ato ≈ f + 1
(where the constants of proportionality depend on the choice of t).

In the case of a space of homogeneous type (X, d, µ), this was shown in [START_REF] Macias | A decomposition into atoms of distributions on spaces of homogeneous type[END_REF] (Theorem 4.13) for a normal space of order α and in [START_REF] Uchiyama | A maximal function characterization of H p on the space of homogeneous type[END_REF] (Theorem C) under the assumption of the existence of a family of Lipschitz kernels (see also the remarks following Theorem (4.5) in [START_REF] Coifman | Extensions of Hardy spaces and their use in analysis[END_REF]). For the manifold M this will follow as a corollary of the atomic decomposition for the Hardy-Sobolev space below. We first prove the inclusion

H 1,ato (M) ⊂ H 1,max (M). ( 14
)
Proof. We show that if f ∈ H 1,ato then f + ∈ L 1 . Let t > 1 and a be an atom supported in a ball B 0 = B(x 0 , r 0 ). We want to prove that a + ∈ L 1 . First take x ∈ 2B 0 . We have a + (x) = sup

ϕ∈T 1 (x) B aϕdµ ≤ CM(a)(x).
Then by the L t -boundedness of the Hardy-Littlewood maximal function for t > 1 (Theorem 2.3) and the size condition on a,

2B 0 |a + (x)|dµ ≤ µ(B 0 ) 1/t ′ 2B 0 |a + | t dµ 1/t ≤ Cµ(B 0 ) 1/t ′ Ma t ≤ C t µ(B 0 ) 1/t ′ a t ≤ C t . (15)
Note that the constant depends on t due to the dependence of the constant in the boundedness of the Hardy-Littlewood maximal function, which blows up as t

→ 1 + . Now if x ∈ M \ 2B 0 , there exists k ∈ N * such that x ∈ C k (B 0 ) := 2 k+1 B 0 \ 2 k B 0 . Let ϕ ∈ T 1 (x)
and take a ball B = B(x, r) such that ϕ is supported in and satisfies [START_REF] Dafni | Hardy Spaces on Strongly Pseudoconvex Domains in C n and Domains of Finite Type in C 2[END_REF] with respect to B. Using the moment condition for a and the Lipschitz bound on ϕ, we get

B aϕdµ = B∩B 0 a(y)(ϕ(y) -ϕ(x 0 ))dµ(y) ≤ C B∩B 0 |a(y)| d(y, x 0 ) rµ(B) dµ(y) ≤ C r 0 rµ(B) a 1 .
Note that for the integral not to vanish we must have

B ∩ B 0 = ∅. We claim that this implies r > 2 k-1 r 0 and 2 k+1 B 0 ⊂ 8B. ( 16
)
To see this, let

y ∈ B ∩B 0 . Then r > d(y, x) ≥ d(x, x 0 )-d(y, x 0 ) ≥ 2 k r 0 -r 0 ≥ 2 k-1 r 0 . Thus if d(z, x 0 ) < 2 k+1 r 0 then d(z, x) ≤ d(z, x 0 ) + d(x, x 0 ) < 2 k+1 r 0 + 2 k+1
r 0 < 8r and we deduce that 2 k+1 B 0 ⊂ 8B. We then have

µ(2 k+1 B 0 ) ≤ C8 s µ(B)
by [START_REF] Auscher | Hardy-Sobolev spaces on strongly Lipschitz domains of R n[END_REF]. Using this estimate and the fact that a 1 ≤ 1, we have

x / ∈2B 0 |a + |(x)dµ = k≥1 C k (B 0 ) |a + |(x)dµ ≤ C a 1 k≥1 8 s 2 1-k µ(2 k+1 B 0 ) µ(C k (B 0 )) ≤ C8 s k≥1 2 1-k ≤ C. Thus a + ∈ L 1 with a + 1 ≤ C t . Now for f ∈ H 1,
ato , take an atomic decomposition of f as in [START_REF] Franchi | Definitions of Sobolev classes on metric spaces[END_REF]. By the convergence of the series in L 1 , we have, for each x and each ϕ ∈ T 1 (x),

f ϕdµ ≤ |λ j | a j ϕdµ ≤ |λ j |a + j (x) so f + is pointwise dominated by |λ j |a + j , giving f + 1 ≤ j |λ j | a + j 1 ≤ C t j |λ j |.
Taking the infimum over all the atomic decompositions of

f yields f + 1 ≤ C t f H 1 .
The proof of the converse, namely that if f + ∈ L 1 then f ∈ H 1,ato , relies on an atomic decomposition and will follow from the proof of Proposition 3.4 below.

Atomic Hardy-Sobolev spaces

In [START_REF] Badr | Abstract Hardy-Sobolev spaces and Interpolation[END_REF], the authors defined atomic Hardy-Sobolev spaces. Let us recall their definition of homogeneous Hardy-Sobolev atoms. These are similar to H 1 atoms but instead of the usual L t size condition they are bounded in the Sobolev space Ẇ 1 t .

Definition 2.11

([5]). For 1 < t ≤ ∞, 1 t + 1 t ′ = 1, we say that a function a is a homogeneous Hardy-Sobolev (1, t)-atom if 1. a is supported in a ball B, 2. a Ẇ 1 t := ∇a t ≤ µ(B) -1 t ′ ,

and

3. adµ = 0.

They then define, for every 1 < t ≤ ∞, the homogeneous Hardy-Sobolev space ḢS 1 t,ato as follows: f ∈ ḢS 1 t,ato if there exists a sequence of homogeneous Hardy-Sobolev (1, t)-atoms {a j } j such that

f = j λ j a j (17)
with j |λ j | < ∞. This space is equipped with the semi-norm

f ḢS 1 t,ato = inf j |λ j |,
where the infimum is taken over all possible decompositions (17).

Remarks 2.12.

1. Since condition 2 implies that the homogeneous Sobolev Ẇ 1 1 semi-norm of the atoms is bounded by a constant, the sum in [START_REF] Haj Lasz | Hölder quasicontinuity of Sobolev functions on metric spaces[END_REF] converges in Ẇ 1 1 and therefore we can consider ḢS 1 t,ato as its subspace.

2. Since we are working with homogeneous spaces, we can modify functions by constants so the cancellation conditions are, in a sense, irrelevant. As we will see below, and when comparing to other definitions in the literature (see, for example, [START_REF] Lou | An atomic decomposition for the Hardy-Sobolev space[END_REF]), condition 3 can be replaced by one of the following:

3 ′ . a 1 ≤ r(B), or 3 ′′ . a t ≤ r(B)µ(B) -1 t ′ ,
where r(B) is the radius of the ball B. Clearly condition 3 ′′ implies 3 ′ , and conditions 2 and 3 imply 3 ′ (respectively 3 ′′ ) if we assume the Poincaré inequality (P 1 ) (respectively (P t )). It is most common to consider the case t = 2 under the assumption (P 2 ).

3. As mentioned in the introduction, from Theorem 1.4 we have that under (P 1 ) all the spaces ḢS 1 t,ato can be identified as one space ḢS 1 ato . As we will see, in this case the atomic decomposition can be taken with condition 3 ′ instead of 3.

Atomic decomposition of Ṁ 1 1 and comparison with ḢS 1 t,ato

We begin by proving that under the Poincaré inequality (P 1 ), ḢS

1 ato ⊂ Ṁ1
1 . While under this assumption the space ḢS 1 ato is equivalent to any one of the spaces ḢS 1 t,ato defined above, if we want to consider the norms we need to fix some t > 1.

Proposition 3.1. Let M be a complete Riemannian manifold satisfying (D) and (P 1 ). Let 1 < t ≤ ∞ and a be a homogeneous Hardy-Sobolev (1, t)-atom. Then a ∈ Ṁ1

1 with a Ṁ 1 1 ≤ C t , the constant C depending only on t, the doubling constant and the constant appearing in (P 1 ), and independent of a.

Consequently ḢS

1 t,ato ⊂ Ṁ1 1 with f Ṁ 1 1 ≤ C t f ḢS 1 t,ato
.

Proof. Let a be an (1, t)-atom supported in a ball B 0 = B(x 0 , r 0 ). We want to prove that Na ∈ L 1 . For x ∈ 2B 0 we have, using (P 1 ),

Na(x) = sup B: x∈B 1 r(B) - B |a -a B |dµ ≤ C sup B: x∈B - B |∇a|dµ = CM(|∇a|)(x).
Then, exactly as in [START_REF] Haj | Sobolev spaces on an arbitrary metric space[END_REF], by the L t boundedness of M for t > 1 (with a constant depending on t), and properties 1 and 2 of (1, t)-Hardy-Sobolev atoms,

2B 0 |Na(x)|dµ ≤ Cµ(B 0 ) 1/t ′ 2B 0 (M(|∇a|)) t dµ 1/t ≤ C t µ(B 0 ) 1/t ′ ∇a t ≤ C t . Now if x / ∈ 2B 0 , then there exists k ∈ N * such that x ∈ C k (B 0 ) := 2 k+1 B 0 \ 2 k B 0 . Let B = B(y, r(B)) be a ball containing x. Then 1 r(B) - B |a -a B |dµ = 1 r(B) 1 µ(B) B∩B 0 |a -a B |dµ + B∩B c 0 |a B |dµ ≤ 3 r(B) 1 µ(B) B∩B 0 |a|dµ. (18)
From ( 16) we have that

B ∩ B 0 = ∅ implies r(B) > 2 k-1 r 0 and µ(2 k+1 B 0 ) ≤ C8 s µ(B).
This, together with the doubling and Poincaré assumptions (D) and (P 1 ), the cancellation condition 3 for a and the size condition 2 for ∇a, yield

Na(x) ≤ 3 2 k-1 r 0 8 s µ(2 k+1 B 0 ) B 0 |a|dµ ≤ 3 2 k-1 8 s µ(2 k+1 B 0 ) B 0 |∇a|dµ ≤ 3 2 -k+1 8 s µ(2 k+1 B 0 )
.

Note that at this point we could have used condition 3 ′ (see Remarks 2.12) instead of conditions 2, 3, (D) and (P 1 ). Therefore

x / ∈2B 0 |Na|(x)dµ = k≥1 C k (B 0 ) |Na|(x)dµ ≤ C8 s k≥1 2 -k+1 C k (B 0 ) 1 µ(2 k+1 B 0 ) dµ(x) ≤ C8 s k≥1 2 -k+1 = C s . Thus Na ∈ L 1 with Na 1 ≤ C s,t . Now if f ∈ ḢS 1 t,ato
, take an atomic decomposition of f : f = j λ j a j with a j (1, t)-atoms and j |λ j | < ∞. Then the sum j λ j Na j converges absolutely in L 1 so by Theorem 1.3 the sequence of functions f k = k j=1 λ j a j has a limit, g, in the Banach space Ṁ1

1 . By Proposition 2.6, this implies convergence in Ẇ 1 1 . Since (as pointed out in Remarks 2.12) the convergence of the decomposition f = j λ j a j also takes place in Ẇ 1 1 , we get that f = g in Ẇ 1 1 . This allows us to consider f as a (locally integrable) element of Ṁ1

1 , take Nf and estimate

Nf 1 ≤ j |λ j | Na j 1 ≤ C t j |λ j |.
Taking the infimum over all the atomic decompositions of f yields Nf 1 ≤ C t f ḢS 1 . To attain this goal, we need a Calderón-Zygmund decomposition for such functions. We refer to [START_REF] Auscher | Riesz transform on manifolds and Poincaré inequalities[END_REF] for the original proof of the Calderón-Zygmund decomposition for Sobolev spaces on Riemannian manifolds.

Proposition 3.3 (Calderón-Zygmund decomposition). Let M be a complete Riemannian manifold satisfying (D). Let f ∈ Ṁ1

1 , s s+1 < q < 1 and α > 0. Then one can find a collection of balls {B i } i , functions b i ∈ W 1 1 and a Lipschitz function g such that the following properties hold:

f = g + i b i , |∇g(x)| ≤ Cα for µ -a.e. x ∈ M, (19)
supp b i ⊂ B i , b i 1 ≤ Cαµ(B i )r i , ∇b i q ≤ Cαµ(B i ) 1 q , i µ(B i ) ≤ C α Nf dµ, ( 20
)
and i χ B i ≤ K. ( 21
)
The constants C and K only depend on the constant in (D).

Proof. Let f ∈ Ṁ1 1 , s s+1 < q < 1 and α > 0. Consider the open set Ω = {x : M q (Nf )(x) > α}.

If Ω = ∅, then set g = f , b i = 0 for all i so that ( 19) is satisfied according to the Lebesgue differentiation theorem. Otherwise

µ(Ω) ≤ C α M M q (Nf )dµ ≤ C α M (M(Nf ) q ) 1/q dµ ≤ C α M Nf dµ < ∞. ( 22
)
We used the fact the M is L 1/q bounded since 1/q > 1 and Theorem 1.3. In particular Ω = M as µ(M) = +∞.

Let F be the complement of Ω. Since Ω is an open set distinct from M, let {B i } i be a Whitney decomposition of Ω (see [START_REF] Coifman | Extensions of Hardy spaces and their use in analysis[END_REF]). That is, the B i are pairwise disjoint, and there exist two constants C 2 > C 1 > 1, depending only on the metric, such that

1. Ω = ∪ i B i with B i = C 1 B i ,
and the balls B i have the bounded overlap property; 2. r i = r(B i ) = 1 2 d(x i , F ) and x i is the center of B i ;

each ball

B i = C 2 B i intersects F (C 2 = 4C 1 works).
For x ∈ Ω, denote I x = {i : x ∈ B i }. By the bounded overlap property of the balls B i , we have that ♯I x ≤ K, and moreover, fixing

k ∈ I x , 1 r i ≤ r k ≤ 3r i and B i ⊂ 7B k for all i ∈ I x .
Condition ( 21) is nothing but the bounded overlap property of the B i 's and (20) follows from [START_REF] Kinnunen | Pointwise behaviour of M 1,1 Sobolev functions[END_REF] and [START_REF] Koskela | Pointwise characterizations of Hardy-Sobolev functions[END_REF]. Note also that using the doubling property, we have

B i |Nf | q dµ ≤ Cµ(B i ) - B i |Nf | q dµ ≤ µ(B i )M q q (Nf )(y) ≤ Cα q µ(B i ) (23)
for some y ∈ B i ∩ F , whose existence is guaranteed by property 3 of the Whitney decomposition.

Let us now define the functions b i . For this, we construct a partition of unity {χ i } i of Ω subordinate to the covering {B i } i . Each χ i is a Lipschitz function supported in B i with 0 ≤ χ i ≤ 1 and ∇χ i ∞ ≤ C r i (see for example [START_REF] Franchi | Definitions of Sobolev classes on metric spaces[END_REF], p. 1908).

We set b i = (fc i )χ i where c i := 1 χ i (B i ) B i f χ i dµ and χ i (B i ) means B i χ i dµ, which is comparable to µ(B i ). Note that by the properties of the χ i we have the trivial estimate

b i 1 ≤ B i |f -c i |dµ ≤ B i |f |dµ + µ(B i ) χ i (B i ) B i |f |dµ ≤ C 1 1 B i |f |dµ, (24)
but we need a better estimate, as follows:

b i 1 ≤ 1 χ i (B i ) B i B i (f (x) -f (y))χ i (y)dµ(y) dµ(x) ≤ 1 χ i (B i ) B i B i |f (x) -f (y)|dµ(y)dµ(x) ≤ 2 µ(B i ) χ i (B i ) B i |f (x) -f B i |dµ(x) ≤ Cr i B i |Nf | q dµ 1/q µ(B i ) ≤ Cr i M q (Nf )(y)µ(B i ) ≤ Cr i αµ(B i ), (25)
as in [START_REF] Koskela | A characterization of Haj lasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions[END_REF]. Here we have used the Sobolev-Poincaré inequality ( 6) with λ = 4 and the fact that q * > 1.

Together with the estimate on b i 1 , we use the fact that |∇f | is in L 1 (see Proposition 2.6) to bound ∇b i 1 and conclude that b i ∈ W 1 1 :

∇b i 1 ≤ B i |f -c i ||∇χ i |dµ + B i |∇f |dµ ≤ C 1 r i r i µ(B i ) - 4B i |Nf | q dµ 1/q + B i |∇f |dµ ≤ Cαµ(B i ) + B i |∇f |dµ < ∞. (26)
Similarly, we can estimate b i in the Sobolev space Ẇ 1 q ; note again that by Proposition 2.6, |∇f | is in L 1 and can be bounded pointwise µ-a.e. by Nf :

∇b i q ≤ |(f -c i )∇χ i | q + |∇f |χ i q ≤ µ(B i ) 1 q -1 χ i (B i ) B i B i |f (x) -f (y)|χ i (y)|∇χ i (x)|dµ(y)dµ(x) + B i |∇f | q dµ 1/q ≤ C - B i |Nf | q dµ 1/q µ(B i ) 1/q + B i |Nf | q dµ 1/q ≤ Cαµ(B i ) 1/q (27)
by [START_REF] Koskela | A characterization of Haj lasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions[END_REF].

Set now g = fi b i . Since the sum is locally finite on Ω, g is defined almost everywhere on M and g = f on F . Observe that g is a locally integrable function on M. Indeed, let ϕ ∈ L ∞ with compact support. Since d(x, F ) ≥ r i for x ∈ supp b i , we obtain

i |b i | |ϕ| dµ ≤ i |b i | r i dµ sup x∈M d(x, F )|ϕ(x)|
Hence by [START_REF] Lou | An atomic decomposition for the Hardy-Sobolev space[END_REF] and the bounded overlap property,

i |b i ||ϕ|dµ ≤ Cα i µ(B i ) sup x∈M d(x, F )|ϕ(x)| ≤ CKαµ(Ω) sup x∈M d(x, F )|ϕ(x)| .
Since f ∈ L 1,loc , we conclude that g ∈ L 1,loc . It remains to prove [START_REF] Heinonen | Lectures on analysis on metric spaces[END_REF]. Indeed, using the fact that on Ω we have χ i = 1 and ∇χ = 0, we get

∇g = ∇f - i ∇b i = ∇f -( i χ i )∇f - i (f -c i )∇χ i = 1 1 F ∇f - i (f -c i )∇χ i . (28)
From Proposition 2.6, the definition of F and the Lebesgue differentiation theorem, we have that 1

1

F |∇f | ≤ 1 1 F Nf ≤ α, µ-a.e.
We claim that a similar estimate holds for

h = i (f -c i )∇χ i ,
i.e. |h(x)| ≤ Cα for all x ∈ M. For this, note first that by the properties of the balls B i and the partition of unity, h vanishes on F and the sum defining h is locally finite on Ω. Then fix x ∈ Ω and let B k be some Whitney ball containing x. Again using the fact that i ∇χ i (x) = 0, we can replace f (x) by any constant in the sum above, so we can write

h(x) = i∈Ix - 7B k f dµ -c i ∇χ i (x).
For all i, k ∈ I x , by the construction of the Whitney collection, the balls B i and B k have equivalent radii and B i ⊂ 7B k . Thus

c i -- 7B k f dµ ≤ 1 χ i (B i ) B i f -- 7B k f dµ χ i dµ - 7B k |f -f 7B k |dµ r k - 7λB k |Nf | q dµ 1/q αr k . (29)
We used (D), ( 6) , χ i (B i ) ≃ µ(B i ) and ( 23) for 7B k . Hence

|h(x)| i∈Ix αr k (r i ) -1 ≤ CKα. ( 30
)
Proposition 3.4. Let M be a complete Riemannian manifold satisfying (D). Let f ∈ Ṁ1 1 . Then for all s s+1 < q < 1, q * = sq s-q , there is a sequence of homogeneous (1, q * ) Hardy-Sobolev atoms {a j } j , and a sequence of scalars {λ j } j , such that

f = j λ j a j in Ẇ 1 1 , and |λ j | ≤ C q f Ṁ 1 1 . Consequently, Ṁ1 1 ⊂ ḢS 1 q * ,ato with f ḢS 1 q * ,ato ≤ C q f Ṁ 1 1 .
Remark 3.5. Note that for the inclusion Ṁ1 1 ⊂ ḢS 1 q * ,ato , we do not need to assume any additional hypothesis, such as a Poincaré inequality, on the doubling manifold.

Proof of Proposition 3.4. Let f ∈ Ṁ1

1 . We follow the general scheme of the atomic decomposition for Hardy spaces, found in [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF], Section III.2.3. For every j ∈ Z * , we take the Calderón-Zygmund decomposition, Proposition 3.3, for f with α = 2 j . Then

f = g j + i b j i with b j
i , g j satisfying the properties of Proposition 3.3. We want to write

f = ∞ -∞ (g j+1 -g j ) ( 31
)
in Ẇ 1 1 . First let us see that g j → f in as j → ∞. Indeed, since the sum is locally finite we can write

∇(g j -f ) 1 = ∇( i b j i ) 1 ≤ i ∇b j i 1 .
By [START_REF] Macias | A decomposition into atoms of distributions on spaces of homogeneous type[END_REF],

i ∇b j i 1 ≤ CK2 j µ(Ω j ) + K Ω j |∇f |dµ = I j + II j . (32)
When j → ∞, I j → 0 since j 2 j µ(Ω j ) ≈ M q (Nf )dµ < ∞. This also implies M q (Nf ) is finite µ-a.e., hence Ω j = ∅ so II j → 0, since |∇f | ∈ L 1 .

When j → -∞, we want to show ∇g j 1 → 0. Breaking ∇g up as in [START_REF] Miyachi | Hardy-Sobolev spaces and maximal functions[END_REF], we know that

F j |∇g j | = 1 1 F j |∇f | ≤ {N f ≤2 j } Nf → 0, (33)
since Nf ∈ L 1 . For the other part we have, by [START_REF] Strichartz | H p Sobolev spaces[END_REF],

Ω j |∇g j | = |h(x)| ≤ CK2 j µ(Ω j ) → 0 (34)
from the convergence of 2 j µ(Ω j ), as above. Denoting g j+1g j by ℓ j , we have supp ℓ j ⊂ Ω j so using the partition of unity {χ j k } corresponding to the Whitney decomposition for Ω j , we can write f = j, k ℓ j χ j k in Ẇ 1

1 . Let us compute ℓ j χ j k Ẇ 1 q * . We have

∇(ℓ j χ j k ) = (∇ℓ j )χ j k + ℓ j ∇χ j k .
From the estimate ∇g j ∞ ≤ C2 j it follows that -B j k |∇ℓ j | q * dµ 1/q * ≤ C2 j , while

ℓ j ∇χ j k =   i:B j k ∩B j i =∅ (f -c j i )χ j i - l:B j k ∩B j+1 l =∅ (f -c j+1 l )χ j+1 l   ∇χ j k . (35)
Observe that since Ω j+1 ⊂ Ω j , for a fixed k, the balls B j+1 l with B j k ∩ B j+1 l = ∅ must have radii r j+1 l ≤ cr j k for some constant c. Therefore B j+1 l ⊂ (B j k ) ′ := (1 + 2c)B j k . Moreover, by the properties of the Whitney balls, given λ > 1 we can take c sufficiently large so that (B j k ) ′ contains λB j i for all B j i intersecting B j k . Using this fact as well as ( 6) and ( 23), and proceeding in the same way as in the derivations of ( 25) and ( 29), we get

(r j k ) q * B j k |ℓ j ∇χ j k | q * dµ ≤ K q * -1 B j k i 1 1 B j i |f -c j i | q * + l 1 1 B j+1 l |f -c j+1 l | q * dµ ≤ K q * -1 i:B j k ∩B j i =∅ B j i |f -c j i | q * dµ + K q * -1 (B j k ) ′ l 1 1 B j+1 l |f -f (B j k ) ′ + f (B j k ) ′ -c j+1 l | q * dµ (36) K q * -1 i:B j k ∩B j i =∅ (r j i 2 j ) q * µ(B j i ) + K q * (r j k 2 j ) q * µ((B j k ) ′ ) K q * (r j k 2 j ) q * µ((B j k ) ′ ). Therefore - (B j k ) ′ |ℓ j ∇χ j k | q * dµ 1 q * ≤ CK2 j . ( 37
)
The ℓ j χ j k 's seem to be a good choice for our atoms but unfortunately they do not satisfy the cancellation condition. If we wanted to get atoms with property 3 ′ (see Remarks 2.12) instead of the vanishing moment condition 3, we could use [START_REF] Lou | An atomic decomposition for the Hardy-Sobolev space[END_REF] to bound the L 1 norm of ℓ j χ j k , then normalize as below. However, if we want to obtain the vanishing moment condition, we need to consider instead the following decomposition of the ℓ j 's:

ℓ j = k ℓ j k with ℓ j k = (f -c j k )χ j k - l (f -c j+1 l )χ j+1 l χ j k + l c k,l χ j+1 l , (38)
where

c k,l := 1 χ j+1 l (B j+1 l ) B j+1 l (f -c l+1 j )χ j+1 l χ j k dµ.
First, this decomposition holds since k χ j k = 1 on the support of χ j+1 l and k c k,l = 0. Furthermore, the cancellation condition M ℓ j k dµ = 0 follows from the fact that M (fc j k )χ j k dµ = 0 and the definition of c k,l , which immediately gives

(f -c j+1 l )χ j+1 l χ j k -c k,l χ j+1 l dµ = 0. Noting that ℓ j k is supported in the ball (B j k ) ′ (see above), let us estimate ∇ℓ j k L q * ((B j k ) ′ ) . Write ∇ℓ j k = (∇f )χ j k + (f -c j k )∇χ j k - l (f -c j+1 l )∇χ j+1 l χ j k - l (f -c j+1 l )χ j+1 l ∇χ j k -(∇f )1 1 Ω j+1 χ j k + l c k,l ∇χ j+1 l = ∇f (1 -1 1 Ω j+1 )χ j k + ((f -c j k ) - l (f -c j+1 l )χ j+1 l )∇χ j k - l (f -c j+1 l )∇χ j+1 l χ j k + l c k,l ∇χ j+1 l .
Since the first term, concerning the gradient of f , is supported in B j k ∩ F j+1 , we can use Proposition 2.6, the definition of F j+1 and the Lebesgue differentiation theorem to bound it, namely

B j k |∇f | q * dµ ≤ 2 (j+1)q * µ(B j k ).
Recalling (35), we see that the estimate of the L q * norm of the second term is given by (37). The third term can be handled by the pointwise estimate [START_REF] Strichartz | H p Sobolev spaces[END_REF]:

l (f -c j+1 l )∇χ j+1 l χ j k q * ≤ CK2 j+1 µ(B j k ) 1/q * . For l c k,l ∇χ j+1 l , note first that c k,l = 0 when B j k ∩ B j+1 l = ∅ and |c k,l | ≤ C2 j r j+1 l
thanks to [START_REF] Lou | An atomic decomposition for the Hardy-Sobolev space[END_REF]. By the properties of the partition of unity, this gives |c k,l ∇χ j+1 l | ≤ C2 j for every l, and as the sum has at most K terms at each point we get the pointwise bound

| l c k,l ∇χ j+1 l | ≤ CK2 j , from which it follows that l c k,l ∇χ j+1 l q * ≤ CK2 j µ((B j k ) ′ ) 1/q * . Thus ∇ℓ j k q * ≤ γ2 j µ((B j k ) ′ ) 1/q * . ( 39
)
We now set a j k = γ -1 2 -j µ((B j k ) ′ ) -1 ℓ j k and λ j,k = γ2 j µ((B j k ) ′ ). Then f = j,k λ j,k a j k , with a j k being (1, q * ) homogeneous Hardy-Sobolev atoms and

j,k |λ j,k | = γ j,k 2 j µ((B j k ) ′ ) ≤ γ ′ j,k 2 j µ(B j k ) ≤ γ ′ j 2 j µ({x : M q (Nf )(x) > 2 j }) ≤ C M q (Nf )dµ ≤ C q Nf 1 ∼ f Ṁ 1 1 .
We used that µ((B j k ) ′ ) ∼ µ(B j k ) thanks to (D), and the fact that the B j k are disjoint.

Again by Theorem 1.3, 12) and Corollary 2.7, we have

M 1 1 = {f ∈ H 1,max : Nf ∈ L 1 } , with equivalent norm f M 1 1 = f + 1 + Nf 1 . By (
M 1 1 ⊂ M 1 1 ⊂ W 1 1 .
In [START_REF] Badr | Abstract Hardy-Sobolev spaces and Interpolation[END_REF], the authors also defined the nonhomogeneous atomic Hardy-Sobolev spaces. Let us recall their definition.

Definition 4.3 ([5]). For 1 < t ≤ ∞, we say that a function a is a nonhomogeneous Hardy-Sobolev (1, t)-atom if 1. a is supported in a ball B, 2. a W 1 t := a t + ∇a t ≤ µ(B) -1 t ′ , 3. adµ = 0.
They then define, for every 1 < t ≤ ∞, the nonhomogeneous Hardy-Sobolev space HS 1 t,ato as follows: f ∈ HS 1 t,ato if there exists a sequence of nonhomogeneous Hardy-Sobolev (1, t)-atoms {a j } j such that f = j λ j a j with j |λ j | < ∞. This space is equipped with the norm f HS 1 t,ato := inf

j |λ j |,
where the infimum is taken over all such decompositions. We also recall the following comparison between these atomic Hardy-Sobolev spaces.

Theorem 4.4. ( [START_REF] Badr | Abstract Hardy-Sobolev spaces and Interpolation[END_REF]) Let M be a complete Riemannian manifold satisfying (D) and a Poincaré inequality (P q ) for some q > 1. Then HS 1 t,ato ⊂ HS 1 ∞,ato for every t ≥ q and therefore HS 1 t 1 ,ato = HS 1 t 2 ,ato for every q ≤ t 1 , t 2 ≤ ∞.

Atomic decomposition of M 1 1 and comparison with HS 1 t,ato

As in the homogeneous case, under the Poincaré inequality (P 1 ), HS 1 t,ato ⊂ M 1 1 : Proposition 4.5. Let M be a complete Riemannian manifold satisfying (D) and (P 1 ). Let 1 < t ≤ ∞ and a be a nonhomogeneous Hardy-Sobolev (1, t)-atom. Then a ∈ M 1 1 with a M 1 1 ≤ C t , the constant depending only on t, the doubling constant and the constant appearing in (P 1 ) , but not on a.

Consequently HS 1 t,ato ⊂ M 1 1 with f M 1 1 ≤ C t f HS 1 t,ato .
Proof. The proof follows analogously to that of Proposition 3.1, noting that in the nonhomogeneous case every Hardy-Sobolev (1, t)-atom a is an H 1 atom and so by ( 14) is in H 1,max with norm bounded by a constant. Now for the converse, that is, to prove that M 1 1 ⊂ HS 1 t,ato , we establish, as in the homogeneous case, an atomic decomposition for functions f ∈ M 1 1 using a Calderón-Zygmund decomposition for such functions. Proposition 4.6 (Calderón-Zygmund decomposition). Let M be a complete Riemannian manifold satisfying (D). Let f ∈ M 1 1 , s s+1 < q < 1 and α > 0. Then one can find a collection of balls {B i } i , functions b i ∈ W 1 1 and a Lipschitz function g such that the following properties hold:

f = g + i b i , |g(x)| + |∇g(x)| ≤ Cα for µ -a.e x ∈ M, supp b i ⊂ B i , b i 1 ≤ Cαµ(B i )r i , b i + |∇b i | q ≤ Cαµ(B i ) 1 q , i µ(B i ) ≤ C α (f + + Nf )dµ,
and i χ B i ≤ K.
The constants C and K only depend on the constant in (D).

Proof. The proof follows the same steps as that of Proposition 3.3. We will only mention the changes that occur due to the nonhomogeneous norm. Let f ∈ M 1 1 , s s+1 < q < 1 and α > 0. The first change is that we consider the open set

Ω = {x : M q (f + + Nf )(x) > α}.
We define, as in the homogeneous case, the partition of unity χ i corresponding to the Whitney decomposition

{B i } i of Ω, the functions b i = (f -c i )χ i with c i := 1 χ i (B i ) B i f χ i dµ, and g = f -b i .
In addition to the previous estimates ( 25) -( 27) for b i and ∇b i , we need here to estimate b i q . We begin by showing that for x ∈ Ω,

|c i | ≤ Cα (42) for every i ∈ I x . Set ϕ i = γ χ i χ i (B i )
. From the properties of χ i , in particular since χ i (B i ) ≈ µ(B i ), we see that we can choose γ (independent of i) so that ϕ i ∈ T 1 (y) and thus

|c i | ≤ γ -1 f + (y) for all y ∈ B i .
Recall that the ball B i = C 2 B i has nonempty intersection with F . Taking y 0 ∈ F ∩B i , we get, by integrating the inequality above,

|c i | ≤ γ -1 - B i (f + ) q dµ 1 q ≤ C - B i (f + ) q dµ 1 q ≤ CM q (f + )(y 0 ) ≤ Cα.
Combining this with (12), we have

b i q ≤ B i |f -c i | q 1 q ≤ - B i |f + | q dµ 1 q µ(B i ) 1 q + |c i |µ(B i ) 1 q ≤ Cαµ(B i ) 1 q .
For g, we need to prove that g ∞ ≤ Cα. We have

g = f 1 1 F + i c i χ i . (43)
For the first term we have |f | ≤ f + ≤ M q (f + ) at all Lebesgue points and thus |f 1 1 F | ≤ α µ-a.e. For the second term, thanks to the bounded overlap property and (42), we get the desired estimate. Proposition 4.7. Let M be a complete Riemannian manifold satisfying (D). Let f ∈ M 1 1 . Then for all s s+1 < q < 1, there is a sequence of (1, q * ) (q * = sq s-q ) nonhomogeneous atoms {a j } j , and a sequence of scalars {λ j } j , such that f = j λ j a j in W 1 1 , and

|λ j | ≤ C q f M 1 1 . Consequently, M 1 1 ⊂ HS 1 q * ,ato with f HS 1 q * ,ato ≤ C q f M 1 1 .
Proof. Again, we will only mention the additional properties that one should verify in comparison with the proof of Proposition 3.4. First let us see that [START_REF] Uchiyama | A maximal function characterization of H p on the space of homogeneous type[END_REF] holds in the nonhomogeneous Sobolev space W 1 1 . We already showed convergence in the homogeneous Ẇ 1 1 norm so we only need to verify convergence in L 1 . By (24)

g j -f 1 ≤ i b j i 1 ≤ C i 1 1 B j i |f |dµ ≤ CK Ω j |f |dµ → 0, (44)
as j → ∞. Here we've used the properties of the χ j i , the bounded overlap property of the B j i , the fact that f ∈ L 1 and that Ω j = ∅ since M q (f + + Nf ) is finite µ-a.e. Taking now j → -∞, we write, by (43), (42), and the bounded overlap property

|g j | ≤ F j |f | + i |c j i |χ j i ≤ {Mq(f + )≤2 j } M q (f + ) + CK2 j |Ω j | → 0. ( 45
)
Remark 4.10. As discussed previously, condition 3 in Definition 4.9 is a substitute for the cancellation condition 3 in Definition 2.11. Assuming a Poincaré inequality (P t ), LS 1 t,ato -atoms corresponding to small balls (with r(B) bounded above) can be shown (see [START_REF] Dafni | Hardy Spaces on Strongly Pseudoconvex Domains in C n and Domains of Finite Type in C 2[END_REF], Appendix B) to be elements of Goldberg's local Hardy space (defined by restricting the supports of the test functions in Definition 2.8 to balls of radii r < R for some fixed R -see [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF], Section III.5.17), so that LS 1 t,ato is a subset of the "localized" space H 1,loc .

As in the homogeneous case, under the Poincaré inequality (P 1 ), LS 1 t,ato ⊂ M 1 1 :

Proposition 4.11. Let M be a complete Riemannian manifold satisfying (D) and (P 1 ). Let 1 < t ≤ ∞ and a be an LS 1 t,ato -atom. Then a ∈ M 1 1 with a M 1 1 ≤ C t , the constant C depending only on t, the doubling constant and the constant appearing in (P 1 ), and independent of a.

Consequently

LS 1 t,ato ⊂ M 1 1 with f M 1 1 ≤ C t f LS 1 t,ato . Proof.
The proof follows analogously to that of Proposition 3.1, noting that we can use Remark 3.2 thanks to property 3 in Definition 4.9, and that this property also implies every atom a is in L 1 . Now for the converse, that is, to prove that M 1 1 ⊂ LS 1 t,ato , we again establish an atomic decomposition for functions f ∈ M 1 1 . In order to do that we must introduce an equivalent maximal function f ⋆ , which is a variant of the one originally defined by Calderón [START_REF] Calderón | Estimates for singular integral operators in terms of maximal functions, in: Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI[END_REF] and denoted by N(f, x) (here we are only defining it in the special case q = 1 and m = 1, where for x a Lebesgue point of f , the constant P (x, y) in Calderón's definition is equal to f (x), and we are allowing for the balls not to be centered at x):

Definition 4.12. Let f ∈ L 1,loc (M). Suppose x is a Lebesgue point of f , i.e. lim r → 0 - B(x,r) |f (y) -f (x)|dµ(y) = 0.
We define

f ⋆ (x) := sup B: x∈B 1 r(B) - B |f (y) -f (x)|dµ(y).
Then f ⋆ is defined µ-almost everywhere.

We now show the equivalence of f ⋆ and Nf . As discussed in the Introduction, the following Proposition was proved in [START_REF] Devore | Maximal functions measuring smoothness[END_REF] (see also [START_REF] Miyachi | Hardy-Sobolev spaces and maximal functions[END_REF]) in the Euclidean case: Proposition 4.13. Let M be a complete Riemannian manifold satisfying (D). Then, there exist constants C 1 , C 2 such that for all f ∈ L 1,loc (M)

C 1 Nf ≤ f ⋆ ≤ C 2 Nf
pointwise µ-almost everywhere.

Proof. Let f ∈ L 1,loc and x be a Lebesgue point of f , so that there exists a sequence of balls B n = B(x, r n ) with r n → 0 and f Bn → f (x). Given a ball B containing x, take n sufficiently large so that B n ⊂ B. Since x ∈ B, there is a smallest k ≥ 1 such that 2 k B n = B(x, 2 k r n ) ⊃ B, and for this k we have 2 k r n ≤ 4r(B), so

|f B -f Bn (x)| ≤ - B |f -f 2 k Bn |dµ + k j=1 |f 2 j Bn -f 2 j-1 Bn | ≤ µ(2 k B n ) µ(B) - 2 k Bn |f -f 2 k Bn |dµ + k j=1 µ(2 j B n ) µ(2 j-1 B n ) - 2 j Bn |f -f 2 j Bn |dµ ≤ 2C 2 (D) k j=1 2 j r n Nf (x) ≤ 16C 2 (D) r(B)Nf (x).
Taking the limit as n → ∞, we see that

|f B -f (x)| ≤ Cr(B)Nf (x) so that - B |f (y) -f (x)|dµ(y) ≤ - B |f (y) -f B |dµ(y) + |f B -f (x)| ≤ Cr(B)Nf (x).
Dividing by r(B) and taking the supremum over all balls B containing x, we conclude that f ⋆ (x) ≤ CNf (x).

For the converse, again take any Lebesgue point x and let B be a ball containing

x. Writing |f (y) -f B | ≤ |f (y) -f (x)| + | -B f -f (x)|, we have - B |f (y) -f B |dµ(y) ≤ 2 - B |f (y) -f (x)|dµ(y) ≤ 2r(B)f ⋆ (x).
Taking the supremum over all balls B containing x, we deduce that Nf (x) ≤ 2f ⋆ (x). Proposition 4.14 (Calderón-Zygmund decomposition). Let M be a complete Riemannian manifold satisfying (D). Let f ∈ M 1 1 , s s+1 < q < 1 and α > 0. Then one can find a collection of balls {B i } i , functions b i ∈ W 1 1 and a Lipschitz function g such that the following properties hold:

f = g + i b i ,
Now we claim that for an appropriate constant c q (to be chosen independent of i and α), there exists a point x i ∈ E i with

f ⋆ (x i ) ≤ c q α. (53)
Again, suppose not. Then we have, by (52),

(c q α) q µ(E i ) ≤ E i (f ⋆ ) q dµ ≤ α q µ(B i ),
implying that µ(E i ) ≤ c -q q µ(B i ). Taking c q > (1 -2 -q ) -1/q , we get a contradiction. Thanks to our choice of x i , we now have

|c i | = |f (x i )| ≤ 2α and b i 1 ≤ C B i |f (y) -f (x i )|dµ(y) ≤ Cµ(B i )r i f ⋆ (x i ) ≤ Cc q r i αµ(B i ).
Moreover for b i q , one has, by (51),

b i q ≤ C B i |f -c i | q dµ 1 q ≤ C B i |f | q dµ 1 q + C2αµ(B i ) 1 q ≤ Cαµ(B i ) 1 q .
Finally, for ∇b i , we can estimate the L 1 norm by

∇b i 1 ≤ (f -c i )∇χ i | 1 + (∇f )χ i 1 ≤ B i |f (x) -f (x i )||∇χ i (x)|dµ(x) + B i |∇f |dµ ≤ Cµ(B i )f ⋆ (x i ) + B i |∇f |dµ ≤ Cc q αµ(B i ) + B i |∇f |dµ, (54)
showing (since |∇f | in L 1 by Proposition 2.6) that b i ∈ W 1 1 , and the L q norm by

∇b i q q ≤ (f -c i )∇χ i | q q + (∇f )χ i q q ≤ µ(B i ) 1-q B i |f (x) -f (x i )||∇χ i (x)|dµ(x) q + B i |∇f | q dµ ≤ Cµ(B i )f ⋆ (x i ) q + B i |Nf | q dµ ≤ C(c q α) q µ(B i ) + B i |f ⋆ | q dµ ≤ Cα q µ(B i ),
where we used Propositions 2.6 and 4.13, and (52). Taking the 1/q-th power on both sides, we get (48). It remains to prove (47). First note that g ∞ ≤ Cα since

g = f 1 1 F + i c i χ i
and for the first term, by the Lebesgue differentiation theorem, we have |f 1 1 F | ≤ M q (f )1 1 F ≤ α µ-a.e., while for the second term, thanks to the bounded overlap property and |c i | ≤ 2α, we get the desired estimate. Now for the gradient, we write, as in [START_REF] Miyachi | Hardy-Sobolev spaces and maximal functions[END_REF],

∇g = 1 1 F (∇f ) - i (f -f (x i ))∇χ i .
Again we have, by Propositions 2.6 and 4.13, that 1 1

F (|∇f |) ≤ C1 1 F (Nf ) ≤ C1 1 F (f ⋆ ) ≤ Cα µ-a.e. Let h = i (f -f (x i ))∇χ i .
We will show |h(x)| ≤ Cα for all x ∈ M. Note first that the sum defining h is locally finite on Ω and vanishes on F . Then take x ∈ Ω and a Whitney ball B k containing x.

As before, since i ∇χ i (x) = 0, we can replace f (x) in the sum by any constant so

h(x) = i∈Ix (f (x k ) -f (x i )) ∇χ i (x).
Recall that for all i, k ∈ I x , by the construction of the Whitney collection, the balls B i and B k have equivalent radii and

B i ⊂ 7B k . Thus |f (x k ) -f (x i )| ≤ |f 7B k -f (x k )| + |f 7B k -f (x i )| (55) ≤ - 7B k |f -f (x k )|dµ + - 7B k |f -f (x i )|dµ ≤ 7r k (f ⋆ (x k ) + f ⋆ (x i )) ≤ 14r k c q α, by (53)
. Therefore we again get the estimate [START_REF] Strichartz | H p Sobolev spaces[END_REF].

Proposition 4.15. Let M be a complete Riemannian manifold satisfying (D). Let f ∈ M 1 1 . Then for all s s+1 < q < 1, there is a sequence of LS 1 q * ,ato -atoms {a j } j (q * = sq s-q ), as in Definition 4.9, and a sequence of scalars {λ j } j , such that

f = j λ j a j in W 1 1 , and |λ j | ≤ C q f M 1 1 . Consequently, M 1 1 ⊂ HS 1 q * ,ato with f LS 1 q * ,ato ≤ C q f M 1 1 .
Proof. Here as well we will only mention the additional properties that one should verify in comparison with Proposition 3.4 and 4.7. We use the Calderón-Zygmund decomposition (Proposition 4.14) above with Ω j corresponding to α = 2 j , and denote the resulting functions by g j and b j i , recalling that for the definition of the constant c j i we have c j i = f (x j i ) for a specially chosen point x j i ∈ B j i . First let us see that g j → f in W 1 1 . For the convergence in L 1 we just repeat (44) and (45) from the nonhomogeneous case, replacing f + by |f |. For the convergence in Ẇ 1 1 , we can estimate i ∇b j i 1 exactly as in [START_REF] Warner | Foundations of Differentiable Manifolds and Lie Groups[END_REF], using (54) instead of ( 26), and replacing Nf by f ⋆ and M q (Nf ) by M q (|f | + f ⋆ ). This gives ∇g j → ∇f in L 1 as j → ∞. For the convergence of ∇g j to 0 as j → -∞, we imitate (33) and (34), using ( 28) and ( 30) with f ⋆ and our new choice of c j i . We define the functions ℓ j = g j+1g j as in Proposition 3.4 but this time we just use ℓ j k := ℓ j χ j k for the "pre-atoms", since we no longer need to have the moment condition ℓ j k = 0 (see Remark 3.6). From the L ∞ bounds (47) on g j and ∇g j in Proposition 4.14, we immediately get ℓ j k 1 ≤ C2 j µ(B j k ) and |∇ℓ j |χ j k q * ≤ C2 j µ(B j k ) 1/q * . We need a similar estimate on ℓ j |∇χ j k | q * in order to bound ∇ℓ j k q * . As in (36), write

r j k - B j k |ℓ j ∇χ j k | q * dµ 1/q * ≤ C - B j k i 1 1 B j i |f -c j i | + l 1 1 B j+1 l |f -c j+1 l | q * dµ 1/q * Expanding |f -c j i | = |f -f B j k +f B j k -c j k +c j k -c j i
| and using the bounded overlap property of the balls, the Sobolev-Poincaré inequality (6), Proposition 4.13, and properties (53) and (55) of the constants c j i = f (x j i ), we have for the integral of the first sum on the right-hand-side:

- B j k i 1 1 B j i |f -c j i | q * dµ 1/q * ≤ K - B j k |f -f B j k | q * dµ 1/q * + K|f B j k -c j k | +   - B j k B j i ∩B j k =∅ 1 1 B j i |c j k -c j i | q * dµ   1/q * ≤ CKr j k - B j k (Nf ) q 1/q + Kr j k f ⋆ (x j k ) + CKr j k 2 j ≤ CKr j k 2 j .
where the supremum is taken over all pairs ϕ ∈ T

1 (x), Φ ∈ C 1 0 (M, T M) such that Φ ∞ ≤ 1 and div Φ ∞ ≤ 1 r
for the radius r of the same ball B containing x for which ϕ satisfies [START_REF] Dafni | Hardy Spaces on Strongly Pseudoconvex Domains in C n and Domains of Finite Type in C 2[END_REF]. We equip this space with the semi-norm

f ḢS 1 max = (∇f ) + 1 .
Note that in case both ϕ and Φ are smooth, the quantity ∇ϕ, Φ + ϕdiv Φ represents the divergence of the product ϕΦ, so the definition coincides with that of the maximal function M (1) f given in [START_REF] Auscher | Hardy-Sobolev spaces on strongly Lipschitz domains of R n[END_REF] for the case of domains in R n , but here we want to allow for the case of Lipschitz ϕ. Proposition 5.2. Let f ∈ ḢS 1 max . Then ∇f , initially defined by [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF], is given by an L 1 function and satisfies

|∇f | ≤ C(∇f ) + µ -a.e.
Consequently, ḢS

1 max ⊂ Ẇ 1 1 with f Ẇ 1 1 ≤ C f ḢS 1 max .
Proof. We follow the ideas in the proof of Proposition 2.6. Let Ω be any open subset of M and consider the total variation of u on Ω, defined by

|Df |(Ω) := sup | ∇f, Φ | ,
where the supremum is taken over all vector fields Φ ∈ C 1 0 (Ω, T M) with Φ ∞ ≤ 1. For such a vector field Φ, take r > 0 sufficiently small so that div Φ ∞ ≤ r -1 and dist(supp(Φ), M \ Ω) > 12r. As in the proof of Proposition 2.6, take a collection of balls B i = B(x i , r) with 6B i having bounded overlap (with a constant K independent of r), covering M, and a Lipschitz partition of unity

{ϕ i } i subordinate to {6B i } i , with 0 ≤ ϕ i ≤ 1 and |∇ϕ i | ≤ r -1 . Then for all x ∈ B i , ϕ i /µ(B i ) ∈ T 1 (x), so f [ ∇ϕ i , Φ + ϕ i div Φ]dµ ≤ (∇f ) + (x)µ(B i ). Hence f [ ∇ϕ i , Φ + ϕ i div Φ]dµ ≤ B i (∇f ) + (x)dµ.
Summing up over i such that 6B i ⊂ Ω, by the choice of r we still get ϕ i = 1 on the support of Φ, hence ∇ϕ i = 0, so using the bounded overlap of the balls we have

f div Φ dµ ≤ {i:6B i ⊂Ω} B i (∇f ) + dµ ≤ K Ω (∇f ) + dµ ≤ K (∇f ) + 1 < ∞.
The rest of the proof proceeds as in the proof of Proposition 2. We would like to prove the reverse inclusion. However, this would require some tools such as Lemma 6 in [START_REF] Koskela | Pointwise characterizations of Hardy-Sobolev functions[END_REF] or Lemma 10 in [START_REF] Auscher | Hardy-Sobolev spaces on strongly Lipschitz domains of R n[END_REF] (solving div Ψ = φ with Ψ having compact support) which are particular to R n .

Another possible maximal function we can use, following the ideas in [START_REF] Kinnunen | Pointwise behaviour of M 1,1 Sobolev functions[END_REF] (see Section 4.1), is given by Definition 5.4.

M * (∇f )(x) := sup j |∇f r j | with the "discrete convolution" f r j defined as in [START_REF] Cho | Atomic decomposition on Hardy-Sobolev spaces[END_REF], corresponding to an enumeration of the positive rationals {r j } j , where for each j we have a covering of M by balls {B j i } i of radius r j , and a partition of unity ϕ j i subordinate to this covering.

We have already shown in the proof of Proposition 2.6 (see [START_REF] Coifman | Analyse harmonique sur certains espaces homogènes[END_REF]) that Lemma 5.5. Let f ∈ L 1,loc . Then at µ-almost every point of M, M * (∇f ) ≤ Nf.

Derivatives of molecular Hardy spaces

As noted in the previous section, on a manifold, obtaining a decomposition with atoms of compact support from a maximal function definition is not obvious. In [START_REF] Auscher | Hardy spaces of differential forms on Riemannian manifolds[END_REF], the authors considered instead Hardy spaces generated by molecules. We begin by recalling their definition of H mol,1 (∧ 1 T * M) (a special case with N = 1 of H 1 mol,N (∧T * M) in Definition 6.1 of [START_REF] Auscher | Hardy spaces of differential forms on Riemannian manifolds[END_REF], where we have dropped the superscript 1 for convenience). If in addition the heat kernel on M satisfies Gaussian upper bounds, this space coincides with the space H 1 (∧T * M), which also has a maximal function characterization (see [START_REF] Auscher | Hardy spaces of differential forms on Riemannian manifolds[END_REF], Theorem 8.4).

A sequence of non-negative Lipschitz functions {χ k } k is said to be (a partition of unity) adapted to a ball B of radius r if supp χ 0 ⊂ 4B, supp Summing in k, this implies that a L 2 (∧ 1 T * M ) ≤ 2(µ(B)) -1/2 and b L 2 ≤ 2r(µ(B)) -1/2 . Moreover, there exists a constant C ′ , depending only on the doubling constant in (D), such that

χ k ⊂ 2 k+2 B \ 2 k-1 B for all k ≥ 1, ∇χ k ∞ ≤ C2 -k r -1 (56)
1 1 2 k+2 B\2 k-1 B b 2 ≤ k+3 l=k-3 χ l b 2 ≤ C ′ r2 -k (µ(2 k+2 B)) -1/2 . ( 58
)
Definition 5.6 ([3]). We say that f ∈ H mol,1 (∧ 1 T * M) if there is a sequence {λ j } j ∈ ℓ 1 and a sequence of 1-molecules {a j } j such that f = j λ j a j Now for the converse, let g ∈ H mol,1 (∧ 1 T * M). Write g = j λ j a j := j λ j db j where j |λ j | < ∞, for every j, a j is a 1-molecule associated to a ball B j , and the convergence is in L 1 . Let {χ k j } k be the partition of unity adapted to B j . Then g = since the sum is locally finite and k χ k j = 1. We claim that for every j, k, β k j := 2 k-1 γb j χ k j , with γ a constant to be determined, satisfies properties 1, 2 and 3 ′′ (see Definition 2.11 and Remarks 2.12) of a (1, 2)homogeneous Hardy-Sobolev atom. Indeed, β k j is supported in the ball 2 k+2 B j with

β k j 2 ≤ 2 k-1 γ 2 -k r j µ(2 k B j ) 1 2 ≤ 2 k+2 r j µ(2 k+2 B j ) 1 2
for an appropriate choice of γ depending only on the doubling constant in (D). Furthermore, by (57), (56), and (58),

∇β k j 2 = 2 k-1 γ d(b j χ k j ) 2 ≤ 2 k-1 γ( a j χ k j 2 + b j dχ k j 2 ) ≤ 2 k-1 γ(2 -k (µ(2 k B j )) -1/2 + C2 -k r -1 i 1 1 2 k+2 B j \2 k-1 B j b j 2 ) ≤ µ(2 k+2 B j ) -1/2 .
Here we again chose γ conveniently, depending only on the doubling constant, and used the fact that k ≥ 0.

Since j,k |λ j |γ -1 2 1-k ≤ 4γ -1 j |λ j | < ∞, the sum f := j λ j k γ -1 2 1-k β k j defines an element of ḢS Corollary 5.9. In the Euclidean case, we then obtain

H mol,1 (R n , ∧ 1 ) = H 1 d (R n , ∧ 1 ) = d( Ṁ1 1 ) = d( ḢS 1
t,ato ) for all t > 1. (For details on H 1 d (R n , ∧ 1 ), see [START_REF] Lou | Hardy spaces of exacts forms on R n[END_REF]).

Definition 1 . 1 .

 11 For f ∈ L 1,loc , define Nf by Nf (x) = sup B: x∈B 1 r(B) -B |ff B |dµ,where B denotes a ball, r(B) its radius and f B the average of f over B.

Letting

  L p := L p (M, µ), 1 ≤ p ≤ ∞, and denoting by | • | the length induced by the Riemannian metric on the tangent space (forgetting the subscript x for simplicity), we can define ∇f p := |∇f | Lp(M,µ) and, in view of (3), df p = ∇f p . If d * denotes the adjoint of d on L 2 (ΛT * x M), then the Laplace-Beltrami operator ∆ is defined by dd * + d * d. However since d * is null on 0-forms, this simplifies to ∆f = d * df on functions and we have, for f

Theorem 2 . 3 (

 23 Maximal theorem,[START_REF] Coifman | Analyse harmonique sur certains espaces homogènes[END_REF]). Let M be a Riemannian manifold satisfying (D). Denote by M the non-centered Hardy-Littlewood maximal function over open balls of M, defined by Mf (x) := sup B ball x∈B |f | B ,

Proposition 5 . 3 . 1 max≤ C f Ṁ 1 1 .

 5311 [START_REF] Calderón | Estimates for singular integral operators in terms of maximal functions, in: Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI[END_REF], replacing Nf by (∇f ) + . Let f ∈ L 1,loc . Then at every point of M,(∇f ) + ≤ Nf. Proof. Let f ∈ L 1,loc and x ∈ M. Take ϕ ∈ T 1 (x), Φ ∈ C 1 0 (M, T M) as in Definition 5.1. Then ( ∇ϕ, Φ + ϕdiv Φ)dµ = 0 so we can write f ( ∇ϕ, Φ + ϕdiv Φ)dµ = (ff B )( ∇ϕ, Φ + ϕdiv Φ)dµ ≤ 1 rµ(B) |ff B |dµ ≤ Nf (x).

and k χ k = 1 on M. A 1 -

 1 form a ∈ L 2 (∧ 1 T * M) is called a 1-molecule if a = db for some b ∈ L 2 (M) and there exists a ball B with radius r, and a partition of unity {χ k } k adapted to B, such that for all k ≥ 0χ k a L 2 (∧ 1 T * M ) ≤ 2 -k (µ(2 k B)) -1/2 (57) and χ k b 2 ≤ 2 -k r(µ(2 k B)) -1/2 .

1 2 ,

 2 ato , with the convergence being in Ẇ 1 1 . This means that in L 1 we havedf = d j,k λ j (b j χ k j ) = j λ j k d(b j χ k j ) = g. Therefore g = df = d j,k λ j (b j χ k j ) , with f ḢS 1 2,ato ≤ 4γ -1 j |λ j |.Taking the infimum over all such decompositions of g, we see thatinf df =g f ḢS 1 2,ato ≤ 4γ -1 g H mol,1 (∧ 1 T * M ) .

Council, Canada, the

Remark 3.6. As pointed out in the proof following (37), we can get an atomic decomposition as in Proposition 3.4, but replacing the vanishing moment condition 3 of the atoms from Definition 2.11 by condition 3 ′ in Remarks 2.12. This does not assume a Poincaré inequality.

Conclusion:

Let M be a complete Riemannian manifold satisfying (D). Then 1. for all s s+1 < q < 1, Ṁ1 1 ⊂ ḢS 1 q * ,ato . 2. (Theorem 1.5) If moreover we assume (P 1 ), then

for all t > 1.

The nonhomogeneous case

We begin by recalling the definitions of the nonhomogeneous versions of the spaces considered above.

Definition 4.1. ( [START_REF] Haj | Sobolev spaces on metric-measure spaces[END_REF]) Let 1 ≤ p ≤ ∞. The Sobolev space M 1 p is the set of all functions u ∈ L p such that there exists a measurable function g ≥ 0, g ∈ L p , satisfying

That is,

From Theorem 1.3, we deduce that for 1 ≤ p ≤ ∞,

We define the Hardy-Sobolev space M 1 1 as the set of all functions u ∈ H 1,max such that there exists a measurable function g ≥ 0, g ∈ L 1 , satisfying

We equip M 1 1 with the norm

We have

For the functions ℓ j = g j+1g j , we have

since by Proposition 4.6, g j ∞ ≤ C2 j . This estimate also applies when we replace ℓ j χ j k by the moment-free "pre-atoms"

For the second and third terms, we use (42) and the bounded overlap property of the B j+1 l . Finally, that

follows by arguing as in the proof of (42), since

) can be considered as a multiple of some ϕ ∈ T 1 (x) for every x ∈ B j+1 l , due to the fact that

from which we conclude, as

1 q * . The rest of the proof is exactly the same as that of Proposition 3.4. Now we can state the converse inclusion from Theorem 2.10:

we follow the steps outlined in the proofs of Propositions 4.6 and 4.7, which use only the maximal function f + , while ignoring the estimates on the gradients from the proofs of Proposition 3.3 and 3.4, which are the only ones involving Nf . From the L ∞ bound (46) we are able to obtain atoms satisfying the conditions of Definition 2.9 with t = ∞, hence for every other t with uniform bounds.

Conclusion:

Let M be a complete Riemannian manifold satisfying (D). Then 1. for all s s+1 < q < 1, M 1 1 ⊂ HS 1 q * ,ato .

2. If we moreover assume (P 1 ), then

for all t > 1.

Atomic decomposition for the Sobolev space M 1 1

For this we need to define new nonhomogeneous atomic spaces LS 1 t,ato , where the L is used to indicate that the atoms will now be in L 1 but not necessarily in H 1 . Let us define our atoms. Definition 4.9. For 1 < t ≤ ∞, we say that a function a is an LS 1 t,ato -atom if

We then say that f belongs to LS 1 t,ato if there exists a sequence of LS 1 t,ato -atoms {a j } j such that f = j λ j a j in W 1 1 , with j |λ j | < ∞. This space is equipped with the norm

where the infimum is taken over all such decompositions.

The constants C and K only depend on the constant in (D).

Proof. The proof follows the same steps as that of Propositions 3.3 and 4.6. Again we will only mention the changes that occur. Let f ∈ M 1 1 , s s+1 < q < 1 and α > 0. By Proposition 4.13, we have f ⋆ ∈ L 1 with norm equivalent to Nf 1 . Thus if we consider the open set Ω = {x :

its Whitney decomposition {B i } i , and the corresponding partition of unity {χ i } i , we get immediately (50) and ( 49) by the bounded overlap property and the boundedness of the maximal function in L 1/q . We again define b i = (fc i )χ i but this time we set c i = f (x i ) for some x i ∈ B i chosen as follows. Recall that B i = 4B i contains some point y of F = M \ Ω so that

as well as -

Let

x is a Lebesgue point of f and |f | q , and |f (x)| ≤ 2α}.

We claim that

Otherwise we would have µ(B i \ E i ) > 2 -q µ(B i ) and so, since f and |f | q are locally integrable and the set of points which are not their Lebesgue points has measure zero,

contradicting (51).

The analogous estimate holds for the integral of the second sum, in l, since as pointed out previously, when B j+1 l ∩ B j k = ∅ we have that r j+1 l ≤ cr j k . This gives

as desired. The rest of the proof follows in the same way as that of Propositions 3.4 and 4.7.

Conclusion:

Let M be a complete Riemannian manifold satisfying (D). Then

1. for all s s+1 < q < 1,

2. If moreover we assume (P 1 ), then

for all t > 1.

5 Comparison between Ṁ 1 1 and Hardy-Sobolev spaces defined in terms of derivatives

Using a maximal function definition

In the Euclidean case, the homogeneous Hardy-Sobolev space ḢS

1 consists of all locally integrable functions f such that ∇f ∈ H 1 (R n ) (i.e. the weak partial derivatives D j f = ∂f ∂x j belong to the real Hardy space H 1 (R n )). In [START_REF] Miyachi | Hardy-Sobolev spaces and maximal functions[END_REF], it was proved that this space is nothing else than {f ∈ L 1,loc (R n ) : Nf ∈ L 1 }, which also coincides with the Sobolev space Ṁ1

1 ([22]). Does this theory extends to the case of Riemannian manifolds? If this is the case, which hypotheses should one assume on the geometry of the manifold? We proved an atomic characterization of Ṁ1

1 but we would like to clarify the relation with Hardy-Sobolev spaces defined using maximal functions. Definition 5.1. We define the (maximal) homogeneous Hardy-Sobolev space ḢS 1 max as follows: ḢS

where ∇f is the distributional gradient, as defined in [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF], and the corresponding maximal function is defined, analogously to [START_REF] Coifman | Extensions of Hardy spaces and their use in analysis[END_REF], by

, with the norm defined by

Here the infimum is taken over all such decompositions. The space H mol,1 (∧ 1 T * M) is a Banach space.

Proposition 5.7. Let M be a complete Riemannian manifold satisfying (D) and (P 1 ).

We then have

. Consequently, in this case we have an atomic decomposition for H mol,1 (∧ 1 T * M) (this was already proved in [START_REF] Auscher | Hardy spaces of differential forms on Riemannian manifolds[END_REF], after Theorem 8.4).

Remark 5.8. As pointed out in Remarks 3.2 and 3.6, we can define the atomic Hardy-Sobolev space ḢS 1 2,ato (M) by using (1, 2)-atoms satisfying condition 3 ′′ of Remarks 2.12 instead of condition 3 of Definition 2.11. As will be seen from the proof below, if we restrict ourselves to this kind of atoms we do not require the hypothesis (P 1 ) for (59). Under the assumption (P 1 ), we actually get the stronger conclusion

1 ) for all t > 1.

Proof. Take f ∈ ḢS 1 2,ato . There exists a sequence {λ j } j ∈ ℓ 1 and (1, 2)-atoms b j such that f = j λ j b j in Ẇ 1 1 . This means j λ j ∇b j converges in L 1 to ∇f , and by the isometry between the vector fields and the 1-forms, we have df = j λ j db j in L 1 (∧ 1 T * M).

We claim that a j = db j are 1-molecules. Indeed, fix j, take B j to be the ball containing the support of b j and let {χ k j } k be a partition of unity adapted to B j . Then χ 0 j a j 2 ≤ db j 2 = ∇b j 2 ≤ 1 µ(B j )

and by condition 3 ′′ of Remarks 2.12 (alternatively condition 3 of Definition 2.11 and (P 1 )) we get

.

For k ≥ 1, there is nothing to do since supp b j ⊂ B j and supp χ k j ⊂ 2 k+2 B j \ 2 k-1 B j ⊂ (B j ) c . Consequently, df ∈ H mol,1 (∧ 1 T * M) with df H mol,1 (∧ 1 T * M ) ≤ j |λ j |. Taking the infimum over all such decompositions, we get df H mol,1 (∧ 1 T * M ) ≤ f ḢS 1 2,ato .