Luong Dang Ky
email: dangky@math.cnrs.fr

Torchinsky Strömberg

Here

NEW HARDY SPACES OF MUSIELAK-ORLICZ TYPE AND BOUNDEDNESS OF SUBLINEAR OPERATORS

Keywords: 2010 Mathematics Subject Classification. 42B35 (46E30, 42B15, 42B30) Muckenhoupt weights, Musielak-Orlicz functions, BMOmultipliers, Hardy spaces, atomic decompositions, Hardy-Orlicz spaces, quasi-Banach spaces, sublinear operators

We introduce a new class of Hardy spaces H ϕ(•,•) (R n ), called Hardy spaces of Musielak-Orlicz type, which generalize the Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of García-Cuerva

Introduction

Since Lebesgue theory of integration has taken a center stage in concrete problems of analysis, the need for more inclusive classes of function spaces than the L p (R n )-families naturally arose. It is well known that the Hardy spaces H p (R n ) when p ∈ (0, 1] are good substitutes of L p (R n ) when studying the boundedness of operators: for example, the Riesz operators are bounded on H p (R n ), but not on L p (R n ) when p ∈ (0, 1]. The theory of Hardy spaces H p on the Euclidean space R n was initially developed by Stein and Weiss [START_REF] Stein | On the theory of harmonic functions of several variables. I. The theory of H p -spaces[END_REF]. Later, Fefferman and Stein [START_REF] Fefferman | H p spaces of several variables[END_REF] systematically developed a real-variable theory for the Hardy spaces H p (R n ) with p ∈ (0, 1], which now plays an important role in various fields of analysis and partial differential equations; see, for example, [START_REF] Coifman | Extensions of Hardy spaces and their use in analysis[END_REF][START_REF] Coifman | Compensated compactness and Hardy spaces[END_REF][START_REF] Müller | Hardy space methods for nonlinear partial differential equations[END_REF]. A key feature of the classical Hardy spaces is their atomic decomposition characterizations, which were obtained by Coifman [START_REF] Coifman | A real variable characterization of H p[END_REF] when n = 1 and Latter [START_REF] Latter | A characterization of H p (R n ) in terms of atoms[END_REF] when n > 1. Later, the theory of Hardy spaces and their dual spaces associated with Muckenhoupt weights have been extensively studied by García-Cuerva [START_REF] García-Cuerva | Weighted H p spaces[END_REF], Strömberg and Torchinsky [START_REF] Strömberg | Weighted Hardy spaces[END_REF] (see also [START_REF] Muckenhoupt | On the dual of weighted H 1 of the half-space[END_REF][START_REF] Bui | Weighted Hardy spaces[END_REF][START_REF] García-Cueva | Wavelet characterization of weighted spaces[END_REF]); there the weighted Hardy spaces was defined by using the nontangential maximal functions and the atomic decompositions were derived. On the other hand, as another generalization of L p (R n ), the Orlicz spaces were introduced by Birnbaum-Orlicz in [START_REF] Birnbaum | Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen[END_REF] and Orlicz in [START_REF] Orlicz | Über eine gewisse Klasse von Räumen vom typus B[END_REF], since then, the theory of the Orlicz spaces themselves has been well developed and the spaces have been widely used in probability, statistics, potential theory, partial differential equations, as well as harmonic analysis and some other fields of analysis; see, for example, [START_REF] Astala | Mappings of BMO-bounded distortion[END_REF][START_REF] Iwaniec | H 1 -estimates of Jacobians by subdeterminants[END_REF][START_REF] Martínez | A minimum problem with free boundary in Orlicz spaces[END_REF]. Moreover, the Hardy-Orlicz spaces are also good substitutes of the Orlicz spaces in dealing with many problems of analysis, say, the boundedness of operators.

Let Φ be a Orlicz function which is of positive lower type and (quasi-)concave. In [START_REF] Janson | Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation[END_REF], Janson has considered the Hardy-Orlicz space H Φ (R n ) the space of all tempered distributions f such that the nontangential grand maximal function of f is defined by Remark that these Hardy-Orlicz type spaces appear naturally when studying the theory of nonlinear PDEs (cf. [START_REF] Greco | New inequalities for the Jacobian[END_REF][START_REF] Iwaniec | Weak minima of variational integrals[END_REF][START_REF] Iwaniec | A study of Jacobians in Hardy-Orlicz spaces[END_REF]) since many cancellation phenomena for Jacobians cannot be observed in the usual Hardy spaces H p (R n ). For instance, let f = (f 1 , ..., f n ) in the Sobolev class W 1,n (R n , R n ) and the Jacobians J(x, f )dx = df 1 ∧ • • • ∧ df n , then (see [START_REF] Iwaniec | A study of Jacobians in Hardy-Orlicz spaces[END_REF]Theorem 10.2])

f * (x) = sup
T (J(x, f )) ∈ L 1 (R n ) + H Φ (R n )
where Φ(t) = t/ log(e + t) and T (f ) = f log |f |, since J(x, f ) ∈ H 1 (R n ) (cf. [START_REF] Coifman | Compensated compactness and Hardy spaces[END_REF]) and T is well defined on H 1 (R n ). We refer readers to [START_REF] Rochberg | Derivatives of analytic families of Banach spaces[END_REF][START_REF] Iwaniec | On the operator L(f ) = f log |f[END_REF] for this interesting nonlinear operator T .

In this paper we want to allow generalized Hardy-Orlicz spaces related to generalized Orlicz functions that may vary in the spatial variables. More precisely the Orlicz function Φ(t) is replaced by a function ϕ(x, t), called Musielak-Orlicz function (cf. [START_REF] Musielak | Orlicz spaces and modular spaces[END_REF][START_REF] Diening | Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces[END_REF]). We then define Hardy spaces of Musielak-Orlicz type. Apart from interesting theoretical considerations, the motivation to study function spaces of Musielak-Orlicz type comes from applications to elasticity, fluid dynamics, image processing, nonlinear PDEs and the calculus of variation (cf. [START_REF] Diening | Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces[END_REF][START_REF] Diening | Function spaces of variable smoothness and integrability[END_REF]).

A particular case of Hardy spaces of Musielak-Orlicz type appears naturally when considering the products of functions in BMO(R n ) and H 1 (R n ) (see [START_REF] Bonami | Paraproducts and products of functions in BM O(R n ) and H 1 (R n ) through wavelets[END_REF]); and the endpoint estimates for the div-curl lemma (see [START_REF] Bonami | Endpoint for the div-curl lemma in Hardy spaces[END_REF][START_REF] Bonami | Paraproducts and products of functions in BM O(R n ) and H 1 (R n ) through wavelets[END_REF]). More precisely, in [START_REF] Bonami | Paraproducts and products of functions in BM O(R n ) and H 1 (R n ) through wavelets[END_REF] the authors proved that product of a BMO(R n ) function and a H 1 (R n ) function may be written as a sum of an integrable term and of a term in H log (R n ), a Hardy space of Musielak-Orlicz type related to the Musielak-Orlicz function θ(x, t) = t log(e+|x|)+log(e+t) . Moreover, the corresponding bilinear operators are bounded. This result gives in particular a positive answer to the Conjecture 1.7 in [START_REF] Bonami | On the product of functions in BM O and H 1[END_REF]. By duality, one finds pointwise multipliers for BMO(R n ). Recall that a function g on R n is called a pointwise multiplier for BMO(R n ), if the pointwise multiplication f g belongs to BMO(R n ) for all f in BMO(R n ). In [START_REF] Nakai | Pointwise multipliers for functions of bounded mean oscillation[END_REF], Nakai and Yabuta characterize the pointwise multipliers for BMO(R n ): they prove that g is a pointwise multiplier for BMO(R n ) if and only if g belong to L ∞ (R n ) ∩ BMO log (R n ), where

BMO log (R n ) = =      f ∈ L 1 loc (R n ) : f BM O log := sup B(a,r) | log r| + log(e + |a|) |B(a, r)| B(a,r) |f (x) -f B(a,r) |dx < ∞      .
By using the theory of these new Hardy spaces and dual spaces, we establish that the class of pointwise multipliers for BMO(R n ) is just the dual of L 1 (R n )+ H log (R n ). Remark that the class of pointwise multipliers for BMO(R n ) have also recently been used by Lerner [START_REF] Lerner | Some remarks on the Hardy-Littlewood maximal function on variable L p spaces[END_REF] for solving a conjecture of Diening (see [START_REF] Diening | Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces[END_REF]) on the boundedness of the Hardy-Littlewood maximal operator on the generalized Lebesgue spaces L p(x) (R n ) (a special case of Musielak-Orlicz spaces, for the details see [START_REF] Diening | Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces[END_REF][START_REF] Lerner | Some remarks on the Hardy-Littlewood maximal function on variable L p spaces[END_REF]). Motivated by all of the above mentioned facts, in this paper, we introduce a new class of Hardy spaces H ϕ(•,•) (R n ), called Hardy spaces of Musielak-Orlicz type, which generalize the Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of García-Cuerva, Strömberg, and Torchinsky. Here, ϕ :

R n × [0, ∞) → [0, ∞) is a function such that ϕ(x, •) is an Orlicz function and ϕ(•, t) is a Muckenhoupt weight A ∞ .
In the special case ϕ(x, t) = w(x)Φ(t) with w in the Muckenhoupt class and Φ an Orlicz function, our Hardy spaces are weighted Hardy-Orlicz spaces but they are different from the ones considered by Harboure, Salinas, and Viviani [START_REF] Harboure | A look at BM O ϕ (w) through Carleson measures[END_REF], [START_REF] Harboure | Wavelet expansions for BM O ρ (w)functions[END_REF].

As an example of our results, let us give the atomic decomposition with bounded atoms. Let ϕ be a growth function (see Section 2). A bounded function a is a ϕ-atom if it satisfies the following three conditions i) supp a ⊂ B for some ball B,

ii) a L ∞ ≤ χ B -1 L ϕ , iii) R n a(x)x α dx = 0 for any |α| ≤ [n( q(ϕ)
i(ϕ) -1)], where q(ϕ) and i(ϕ) are the indices of ϕ (see Section 2). We next define the atomic Hardy space of Musielak-Orlicz type

H ϕ(•,•) at (R n ) as those distri- butions f ∈ S ′ (R n ) such that f = j b j (in the sense of S ′ (R n ))
, where b j , s are multiples of ϕ-atoms supported in the balls B j , s, with the property j ϕ(B j , b j L q ϕ (B j ) ) < ∞; and define the norm of f by

f H ϕ(•,•) at = inf Λ ∞ ({b j }) : f = j b j in the sense of S ′ (R n ) ,
where Λ ∞ ({b j }) = inf λ > 0 :

j ϕ B j , b j L ∞ λ ≤ 1 with ϕ(B, t) := B ϕ(x, t
)dx for all t ≥ 0 and B is measurable. Then we obtain

Theorem 1.1. H ϕ(•,•) at (R n ) = H ϕ(•,•) (R n ) with equivalent norms.
The fact that Λ ∞ ({b j }), which is the right expression for the (quasi-)norm in the atomic Hardy space of Musielak-Orlicz type, plays a central role in this paper. It should be emphasized that, even if the steps of the proof of such a theorem are standard, the adaptation to this context is not standard.

On the other hand, to establish the boundedness of operators on Hardy spaces, one usually appeals to the atomic decomposition characterization, see [START_REF] Coifman | A real variable characterization of H p[END_REF][START_REF] Latter | A characterization of H p (R n ) in terms of atoms[END_REF][START_REF] Taibleson | The molecular characterization of certain Hardy spaces[END_REF], which means that a function or distribution in Hardy spaces can be represented as a linear combination of functions of an elementary form, namely, atoms. Then, the boundedness of operators on Hardy spaces can be deduced from their behavior on atoms or molecules in principle. However, caution needs to be taken due to an example constructed in [8, Theorem 2]. There exists a linear functional defined on a dense subspace of H 1 (R n ), which maps all (1, ∞, 0)-atoms into bounded scalars, but however does not extend to a bounded linear functional on the whole H 1 (R n ). This implies that the uniform boundedness of a linear operator T on atoms does not automatically guarantee the boundedness of T from H 1 (R n ) to a Banach space B. Nevertheless, by using the grand maximal function characterization of H p (R n ), Meda, Sjögren, and Vallarino [START_REF] Meda | On the H 1 -L 1 boundedness of operators[END_REF][START_REF] Meda | Atomic decompositions and operators on Hardy spaces[END_REF] proved that if a sublinear operator T maps all (p, q, s)atoms when q < ∞ and continuous (p, ∞, s)-atoms into uniformly bounded elements of L p (R n ) (see also [START_REF] Yang | A boundedness criterion via atoms for linear operators in Hardy spaces[END_REF][START_REF] Bownik | Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators[END_REF] for quasi-Banach spaces), then T uniquely extends to a bounded sublinear operator from H p (R n ) to L p (R n ). In this paper, we study boundedness of sublinear operators in the context of new Hardy spaces of Musielak-Orlicz type which generalize the main results in [START_REF] Meda | On the H 1 -L 1 boundedness of operators[END_REF][START_REF] Meda | Atomic decompositions and operators on Hardy spaces[END_REF]. More precisely, under additional assumption on ϕ(•, •), we prove that finite atomic norms on dense subspaces of H ϕ(•,•) (R n ) are equivalent with the standard infinite atomic decomposition norms. As an application, we prove that if T is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space B, then T uniquely extends to a bounded sublinear operator from

H ϕ(•,•) (R n ) to B.
In a forecoming paper, using the theory of these new Hardy spaces and ideas from [START_REF] Bonami | Paraproducts and products of functions in BM O(R n ) and H 1 (R n ) through wavelets[END_REF], we study and establish some new interesting estimates of endpoint type for the commutators of singular integrals and fractional integrals on Hardy-type spaces.

Our paper is organized as follows. In Section 2 we give the notation and definitions that we shall use in the sequel. For simplicity we write ϕ for ϕ(•, •). One then introduces Hardy spaces of Musielak-Orlicz type H ϕ (R n ) via grand maximal functions, atomic Hardy spaces H ϕ,q,s at (R n ), finite atomic Hardy spaces H ϕ,q,s fin (R n ) for any admissible triplet (ϕ, q, s), BMO-Musielak-Orlicz-type spaces BMO ϕ (R n ), and generalized quasi-Banach spaces B γ for γ ∈ (0, 1]. In Section 3 we state the main results: the atomic decompositions (Theorem 3.1), the duality (Theorem 3.2), the class of pointwise multipliers for BMO(R n ) (Theorem 3.3), the finite atomic decomposition (Theorem 3.4), and the criterion for boundedness of sublinear operators in H ϕ (R n ) (Theorem 3.5). In Section 4 we present and prove the basic properties of the growth functions ϕ since they provide the tools for further work with this type of functions. In Section 5 we generalize the Calderón-Zygmund decomposition associated to the grand maximal function on R n in the setting of the spaces of Musielak-Orlicz type. Applying this, we further prove that for any admissible triplet (ϕ, q, s), H ϕ (R n ) = H ϕ,q,s at (R n ) with equivalent norms (Theorem 3.1). In Section 6 we prove the dual theorem. By Theorem 2 in [START_REF] Bownik | Boundedness of operators on Hardy spaces via atomic decompositions[END_REF], one has to be careful with the argument "the operator T is uniformly bounded in H p w (R n ) (H ϕ (R n ) here ϕ(x, t) = w(x).t p in our context) on w-(p, ∞)-atoms, and hence it extends to a bounded operator on H p w (R n )" which has been used in [START_REF] García-Cuerva | Weighted H p spaces[END_REF] and [START_REF] Bui | Weighted Hardy spaces[END_REF]. In Section 7 we introduce log-atoms and consider the particular case of H log (R n ). Finally, in Section 8 we prove that • H ϕ,q,s fin and • H ϕ are equivalent quasi-norms on H ϕ,q,s fin (R n ) when q < ∞ and on H ϕ,q,s fin (R n ) ∩ C(R n ) when q = ∞, here and in what follows C(R n ) denotes the set of all continuous functions. Then, we consider generalized quasi-Banach spaces which generalize the notion of quasi-Banach spaces in [START_REF] Yang | A boundedness criterion via atoms for linear operators in Hardy spaces[END_REF] (see also [START_REF] Bownik | Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators[END_REF]), and obtain criterious for boundedness of sublinear operators in H ϕ (R n ).

Throughout the whole paper, C denotes a positive geometric constant which is independent of the main parameters, but may change from line to line. The symbol f ≈ g means that f is equivalent to g (i.e. C -1 f ≤ g ≤ Cf ), and [•] denotes the integer function. By X * we denote the dual of the (quasi-)Banach space X. In R n , we denote by B = B(x, r) an open ball with center x and radius r > 0. For any measurable set E, we denote by χ E its characteristic function, by |E| its Lebesgue measure, and by E c the set R n \ E.

Acknowledgements. The author would like to thank Prof. Aline Bonami and Prof. Sandrine Grellier for many helpful suggestions and discussions. He would also like to thank Prof. Sandrine Grellier for her carefully reading and revision of the manuscript. The author is deeply indebted to them.

Notation and definitions

2.1. Musielak-Orlicz-type functions. First let us recall notations for Orlicz functions.

A function φ : [0, ∞) → [0, ∞) is called Orlicz if it is nondecreasing and φ(0) = 0; φ(t) > 0, t > 0; lim t→∞ φ(t) = ∞. An Orlicz function φ is said to be of lower type (resp., upper type) p, p ∈ (-∞, ∞), if there exists a positive constant C so that φ(st) ≤ Cs p φ(t), for all t ≥ 0 and s ∈ (0, 1) (resp., s ∈ [1, ∞)). One say that φ is of positive lower type (resp., finite upper type) if it is of lower type (resp., upper type) p for some p > 0 (resp., p finite). Obviously, if φ is both of lower type p 1 and of upper type p 2 , then p 1 ≤ p 2 . Moreover, if φ is of lower type (resp., upper type) p then it is also of lower type (resp., upper) p for -∞ < p < p (resp., p < p < ∞). We thus write

i(φ) := sup{p ∈ (-∞, ∞) : φ is of lower type p} I(φ) := inf{p ∈ (-∞, ∞) : φ is of upper type p}
to denote the critical lower type and the critical upper type of the function φ.

Let us generalize these notions to functions ϕ :

R n × [0, ∞) → [0, ∞).
Given a function ϕ : R n × [0, ∞) → [0, ∞) so that for any x ∈ R n , ϕ(x, •) is Orlicz. We say that ϕ is of uniformly lower type (resp., upper type) p if there exists a positive constant C so that (2.1) ϕ(x, st) ≤ Cs p ϕ(x, t), for all x ∈ R n and t ≥ 0, s ∈ (0, 1) (resp., s ∈ [1, ∞)). We say that ϕ is of positive uniformly lower type (resp., finite uniform upper type) if it is of uniformly lower type (resp., uniform upper type) p for some p > 0 (resp., p finite), and denote

i(ϕ) := sup{p ∈ (-∞, ∞) : ϕ is of uniformly lower type p} I(ϕ) := inf{p ∈ (-∞, ∞) : ϕ is of uniformly upper type p}.
We next need to recall notations for Muckenhoupt weights. Let 1 ≤ q < ∞. A nonnegative locally integrable function w belongs to the Muckenhoupt class A q , say w ∈ A q , if there exists a positive constant C so that 1

|B| B w(x)dx 1 |B| B (w(x)) -1/(q-1) dx q-1 ≤ C, if 1 < q < ∞, and 1 |B| B w(x)dx ≤ C ess-inf x∈B w(x), if q = 1,
for all balls B in R n . We say that w ∈ A ∞ if w ∈ A q for some q ∈ [1, ∞).

It is well known that w ∈ A q , 1 ≤ q < ∞, implies w ∈ A r for all r > q. Also, if w ∈ A q , 1 < q < ∞, then w ∈ A r for some r ∈ [1, q). One thus write q w := inf{q ≥ 1 : w ∈ A q } to denote the critical index of w.

Now, let us generalize these notions to functions ϕ :

R n × [0, ∞) → [0, ∞). Let ϕ : R n × [0, ∞) → C be so that x → ϕ(x, t
) is measurable for all t ∈ [0, ∞). We say that ϕ(•, t) is uniformly locally integrable if for all compact set K in R n , the following holds

K sup t>0 |ϕ(x, t)| K |ϕ(y, t)|dy dx < ∞
whenever the integral exists. A simple example for such uniformly locally integrable functions is ϕ(x, t) = w(x)Φ(t) with w a locally integrable function on R n and Φ an arbitrary function on [0, ∞). Our interesting examples are uniformly locally integrable functions ϕ(x, t) = t p (log(e+|x|)+log(e+t p )) p , 0 < p ≤ 1, since they arise naturally in the study of pointwise product of functions in H p (R n ) with functions in BMO(R n ) (cf. [START_REF] Bonami | Paraproducts and products of functions in BM O(R n ) and H 1 (R n ) through wavelets[END_REF]).

Given ϕ : R n × [0, ∞) → [0, ∞) is a uniformly locally integrable function. We say that the function ϕ(•, t) satisfies the uniformly Muckenhoupt condition A q , say ϕ ∈ A q , for some 1 ≤ q < ∞ if there exists a positive constant C so that 1

|B| B ϕ(x, t)dx. 1 |B| B ϕ(x, t) -1/(q-1) dx q-1 ≤ C, if 1 < q < ∞, and 1 |B| B ϕ(x, t)dx ≤ C ess-inf x∈B ϕ(x, t), if q = 1,
for all t > 0 and balls B in R n . We also say that ϕ ∈ A ∞ if ϕ ∈ A q for some q ∈ [1, ∞), and denote

q(ϕ) := inf{q ≥ 1 : ϕ ∈ A q }.
Now, we are able to introduce the growth functions which are the basis for our new Hardy spaces. For ϕ a growth function, we denote m(ϕ) := n q(ϕ) i(ϕ) -1 .

Clearly, ϕ(x, t) = w(x)Φ(t) is a growth function if w ∈ A ∞ and Φ is of positive lower type and of upper type 1. Of course, there exists growth functions which are not of that form for instance ϕ(x, t)

= t α [log(e+|x|)] β +[log(e+t)] γ for α ∈ (0, 1]; β, γ ∈ (0, ∞).
More precisely, ϕ ∈ A 1 and ϕ is of uniformly upper type α with i(ϕ) = α. In this paper, we are especially interested in the growth functions ϕ(x, t) = t p (log(e+|x|)+log(e+t p )) p , 0 < p ≤ 1, since the Hardy spaces of Musielak-Orlicz type H ϕ (R n ) arise naturally in the study of pointwise product of functions in H p (R n ) with functions in BMO(R n ) (see also [START_REF] Bonami | Hankel operators and weak factorization for Hardy-Orlicz spaces[END_REF] in the setting of holomorphic functions in convex domains of finite type or strictly pseudoconvex domains in C n ).

2.2.

Hardy spaces of Musielak-Orlicz type. Throughout the whole paper, we always assume that ϕ is a growth function.

Let us now introduce the Musielak-Orlicz-type spaces.

The Musielak-Orlicz-type space L ϕ (R n ) is the set of all measurable functions f such that R n ϕ(x, |f (x)|/λ)dx < ∞ for some λ > 0, with Luxembourg (quasi-)norm f L ϕ := inf λ > 0 : R n ϕ(x, |f (x)|/λ)dx ≤ 1 .
As usual, S(R n ) denote the Schwartz class of test functions on R n and S ′ (R n ) the space of tempered distributions (or distributions for brevity). For m ∈ N, we define

S m (R n ) = φ ∈ S(R n ) : φ m = sup x∈R n ,|α|≤m+1 (1 + |x|) (m+2)(n+1) |∂ α x φ(x)| ≤ 1 .
For each distribution f , we define the nontangential grand maximal function

f * m of f by f * m (x) = sup φ∈Sm(R n ) sup |y-x|<t |f * φ t (y)|, x ∈ R n . When m = m(ϕ) we write f * instead of f * m(ϕ) . Definition 2.2. The Hardy space of Musielak-Orlicz type H ϕ (R n ) is the space of all distributions f such that f * ∈ L ϕ (R n ) with the (quasi-)norm f H ϕ := f * L ϕ .
Observe that, when ϕ(x, t) = w(x)Φ(t) with w a Muckenhoupt weight and Φ an Orlicz function, our Hardy spaces are weighted Hardy-Orlicz spaces which include the classical Hardy-Orlicz spaces of Janson [START_REF] Janson | Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation[END_REF] (w ≡ 1 in this context) and the classical weighted Hardy spaces of García-Cuerva [START_REF] García-Cuerva | Weighted H p spaces[END_REF], Strömberg and Torchinsky [START_REF] Strömberg | Weighted Hardy spaces[END_REF] (Φ(t) ≡ t p in this context), see also [START_REF] Muckenhoupt | On the dual of weighted H 1 of the half-space[END_REF][START_REF] Bui | Weighted Hardy spaces[END_REF][START_REF] García-Cueva | Wavelet characterization of weighted spaces[END_REF] . Recently, the weighted anisotropic Hardy spaces (see [START_REF] Bownik | Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators[END_REF]) and the Hardy-Orlicz spaces associated with operators (see [START_REF] Jiang | New Orlicz-Hardy spaces associated with divergence form elliptic operators[END_REF]) have also been studied.

Next, to introduce the atomic Hardy spaces of Musielak-Orlicz type below, we need the following new spaces.

Definition 2.3. For each ball B in R n , we denote L q ϕ (B), 1 ≤ q ≤ ∞, the set of all measurable functions f on R n supported in B such that f L q ϕ (B) :=    sup t>0 R n |f (x)| q ϕ(x,t)dx ϕ(B,t) 1/q < ∞ , 1 ≤ q < ∞, f L ∞ < ∞ , q = ∞,
here and in the future ϕ(B, t) := B ϕ(x, t)dx.

Then, it is straightforward to verify that (L q ϕ (B),

• L q ϕ (B) ) is a Banach space.
Now, we are able to introduce the atomic Hardy spaces of Musielak-Orlicz type.

Definition 2.4. A triplet (ϕ, q, s) is called admissible, if q ∈ (q(ϕ), ∞] and s ∈ N satisfies s ≥ m(ϕ). A measurable function a is a (ϕ, q, s)-atom if it satisfies the following three conditions i) a ∈ L q ϕ (B) for some ball B, ii) a L q ϕ (B) ≤ χ B -1 L ϕ , iii) R n a(x)x α dx = 0 for any |α| ≤ s.
In this setting we define the atomic Hardy space of Musielak-Orlicz type H ϕ,q,s at (R n ) as those distributions f ∈ S ′ (R n ) that can be represented as a sum of multiples of (ϕ, q, s)-atoms, that is,

f = j b j in the sense of S ′ (R n ),
where b j , s are multiples of (ϕ, q, s)-atoms supported in the balls B j , s, with the property

j ϕ(B j , b j L q ϕ (B j ) ) < ∞.
We introduce a (quasi-)norm in H ϕ,q,s at (R n ). Given a sequence of multiples of (ϕ, q, s)-atoms, {b j } j , we denote

Λ q ({b j }) = inf λ > 0 : j ϕ B j , b j L q ϕ (B j ) λ ≤ 1
and define

f H ϕ,q,s at = inf Λ q ({b j }) : f = j b j in the sense of S ′ (R n ) .
Let (ϕ, q, s) be an admissible triplet. We denote H ϕ,q,s fin (R n ) the vector space of all finite linear combinations of (ϕ, q, s)-atoms, that is,

f = k j=1 b j ,
where b j 's are multiples of (ϕ, q, s)-atoms supported in balls B j 's. Then, the norm of f in H ϕ,q,s fin (R n ) is defined by

f H ϕ,q,s fin = inf Λ q ({b j } k j=1 ) : f = k j=1 b j .
Obviously, for any admissible triplet (ϕ, q, s), the set

H ϕ,q,s fin (R n ) is dense in H ϕ,q,s at (R n ) with respect to the quasi-norm • H ϕ,q,s at .
We should point out that the theory of atomic Hardy-Orlicz spaces have been first introduced by Viviani [START_REF] Viviani | An atomic decomposition of the Predual of BM O(ρ)[END_REF] in the setting of spaces of homogeneous type. Later, Serra [START_REF] Serra | Molecular characterization of Hardy-Orlicz spaces[END_REF] generalized it to the context of the Euclidean space R n and obtained the molecular characterization. In the particular case, when ϕ(x, t) ≡ Φ(t) the space H ϕ,q,s at (R n ) is the space considered in [START_REF] Serra | Molecular characterization of Hardy-Orlicz spaces[END_REF]. We also remark that when ϕ(x, t) ≡ w(x).t p , 0 < p ≤ 1, w a Muckenhoupt weight, the space H ϕ,q,s at (R n ) is just the classical weighted atomic Hardy space H p,q,s w (R n ) which has been considered by García-Cuerva [START_REF] García-Cuerva | Weighted H p spaces[END_REF], Strömberg and Torchinsky [START_REF] Strömberg | Weighted Hardy spaces[END_REF].

2.3. BMO-Musielak-Orlicz-type spaces. We also need BMO type spaces, which will be in duality of the Hardy spaces of Musielak-Orlicz type defined above Muckenhoupt and Wheeden [37,[START_REF] Muckenhoupt | On the dual of weighted H 1 of the half-space[END_REF]. There, they proved that BMO w (R n ) is the dual of H 1 w (R n ) (see also [START_REF] Bui | Weighted Hardy spaces[END_REF]).

. A function f ∈ L 1 loc (R n ) is said to belong to BMO ϕ (R n ) if f BM O ϕ := sup B 1 χ B L ϕ B |f (x) -f B |dx < ∞, where f B = 1 |B| B f (x)dx and the supremum is taken over all balls B in R n . Our typical example is BMO ϕ (R n ), called BMO log (R n ), related to ϕ(x, t) = t log(e+|x|)+log(e+t) . Clearly, when ϕ(x, t) ≡ t, then BMO ϕ (R n ) is just the well- known BMO(R n ) of John and Nirenberg. We remark that when ϕ(x, t) = w(x).t with w ∈ A (n+1)/n , then BMO ϕ (R n ) is just BMO w (R n ) was first in- troduced by
2.4. Quasi-Banach valued sublinear operators. Recall that a quasi-Banach space B is a vector space endowed with a quasi-norm • B which is nonnegative, non-degenerate (i.e., f B = 0 if and only if f = 0), homogeneous, and obeys the quasi-triangle inequality, i.e., there exists a positive constant κ no less than 1 such that for all f, g ∈ B, we have f

+ g B ≤ κ( f B + g B ).
Definition 2.5. Let γ ∈ (0, 1]. A quasi-Banach space B γ with the quasi-norm • Bγ is said to be a γ-quasi-Banach space if there exists a positive constant κ no less than 1 such that for all f j ∈ B γ , j = 1, 2, ..., m, we have

m j=1 f j γ Bγ ≤ κ m j=1 f j γ Bγ .
Notice that any Banach space is a 1-quasi-Banach space, and the quasi-Banach spaces ℓ p , L p w (R n ) and H p w (R n ) with p ∈ (0, 1] are typical p-quasi-Banach spaces. Also, when ϕ is of uniformly lower type p ∈ (0, 1], the space

H ϕ (R n ) is a p-quasi-Banach space.
For any given γ-quasi-Banach space B γ with γ ∈ (0, 1] and a linear space Y, an operator T from Y to B γ is called B γ -sublinear if there exists a positive constant κ no less than 1 such that for all f j ∈ Y, λ j ∈ C, j = 1, ..., m, we have

T m j=1 λ j f j γ Bγ ≤ κ m j=1 |λ j | γ T (f j ) γ Bγ .
We remark that if T is linear, then T is B γ -sublinear. We should point out that if the constant κ, in Definition 2.5, equal 1, then we obtain the notion of γ-quasi-Banach spaces introduced in [START_REF] Yang | A boundedness criterion via atoms for linear operators in Hardy spaces[END_REF] (see also [START_REF] Bownik | Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators[END_REF]).

Statement of the results

Our main theorems are the following.

Theorem 3.1. Let (ϕ, q, s) be admissible. Then H ϕ (R n ) = H ϕ,q,s at (R n ) with equivalent norms.
Denote by L ∞ 0 (R n ) the set of all bounded functions with compact support and zero average. As a consequence of Theorem 3.1, we have the following Lemma 3.1. Let ϕ be a growth function satisfying nq(ϕ) < (n+1)i(ϕ). Then,

L ∞ 0 (R n ) is dense in H ϕ (R n ).
We now can present our dual theorem as follows Theorem 3.2. Let ϕ be a growth function satisfying nq(ϕ) < (n + 1)i(ϕ).

Then, the dual space of

H ϕ (R n ) is BMO ϕ (R n ) in the following sense i) Suppose b ∈ BMO ϕ (R n ). Then the linear functional L b : f → L b (f ) := R n f (x)b(x)dx, initially defined for L ∞ 0 (R n ), has a bounded extension to H ϕ (R n ). ii) Conversely, every continuous linear functional on H ϕ (R n ) arises as the above with a unique element b of BMO ϕ (R n ). Moreover b BM O ϕ ≈ L b (H ϕ ) * .
Next result concerns the class of pointwise multipliers for BMO(R n ).

Theorem 3.3. The class of pointwise multipliers for BMO(R n ) is the dual of L 1 (R n ) + H log (R n ) where H log (R n ) is a Hardy space of Musielak-Orlicz type related to the Musielak-Orlicz function θ(x, t) = t log(e+|x|)+log(e+t) .
In order to obtain the finite atomic decomposition, we need the notion of uniformly locally dominated convergence condition. A growth function ϕ is said to be satisfy uniformly locally dominated convergence condition if the following holds:

Given K compact set in R n . Let {f m } m≥1 be a sequence of measurable functions s.t f m (x) tends to f (x) a.e x ∈ R n . If there exists a nonnegative measurable function g s.t |f m (x)| ≤ g(x) and sup t>0 K g(x) ϕ(x,t) K ϕ(y,t)dy dx < ∞, then sup t>0 K |f m (x) -f (x)| ϕ(x,t)
K ϕ(y,t)dy dx tends 0. We remark that the growth functions ϕ(x, t) = w(x)Φ(t) and ϕ(x, t) = t p (log(e+|x|)+log(e+t p )) p , 0 < p ≤ 1, satisfy the uniformly locally dominated convergence condition.

Theorem 3.4. Let ϕ be a growth function satisfying uniformly locally dominated convergence condition, and (ϕ, q, s) be an admissible triplet.

i

) If q ∈ (q(ϕ), ∞) then • H ϕ,q,s fin and • H ϕ are equivalent quasi-norms on H ϕ,q,s fin (R n ). ii) • H ϕ,∞,s fin and • H ϕ are equivalent quasi-norms on H ϕ,∞,s fin (R n ) ∩ C(R n ).
As an application, we obtain criterions for boundedness of quasi-Banach valued sublinear operators in H ϕ (R n ). Theorem 3.5. Let ϕ be a growth function satisfying uniformly locally dominated convergence condition, (ϕ, q, s) be an admissible triplet, ϕ be of uniformly upper type γ ∈ (0, 1], and B γ be a quasi-Banach space. Suppose one of the following holds: i) q ∈ (q(ϕ), ∞), and T :

H ϕ,q,s fin (R n ) → B γ is a B γ -sublinear operator such that A = sup{ T a Bγ : a is a (ϕ, q, s)-atom} < ∞; ii) T is a B γ -sublinear operator defined on continuous (ϕ, ∞, s)-atoms such that A = sup{ T a Bγ : a is a continuous (ϕ, ∞, s)-atom} < ∞.
Then there exists a unique bounded B γ -sublinear operator T from H ϕ (R n ) to B γ which extends T .

Some basic lemmas on growth functions

We start by the following lemma.

∞ j=1 t j ) ≤ C ∞ j=1 ϕ(x, t j ), for all (x, t j ) ∈ R n × [0, ∞), j = 1, 2, ... ii) Let ϕ be a growth function and ϕ(x, t) := t 0 ϕ(x,s) s ds for (x, t) ∈ R n × [0, ∞).
Then ϕ is a growth function equivalent to ϕ, moreover, ϕ(x, •) is continuous and strictly increasing.

iii) A Musielak-Orlicz function ϕ is a growth function if and only if ϕ is of positive uniformly lower type and uniformly quasi-concave, i.e. there exists a constant C > 0 such that

λϕ(x, t) + (1 -λ)ϕ(x, s) ≤ Cϕ(x, λt + (1 -λ)s), for all x ∈ R n , t, s ∈ [0, ∞) and λ ∈ [0, 1].
Proof. i) We just need to consider the case when ∞ j=1 t j > 0. Then it follows from the fact that

t k ∞ j=1 t j ϕ(x, ∞ j=1 t j ) ≤ Cϕ(x, t k )
by ϕ is of uniformly upper type 1.

ii) Since ϕ is a growth function, it is easy to see that ϕ(x, •) is continuous and strictly increasing. Moreover, there exists p > 0 such that ϕ is of uniformly lower type p. Hence,

(4.1) ϕ(x, t) = t 0 ϕ(x, s) s ds ≤ C ϕ(x, t) t p t 0 1 s 1-p ds ≤ Cϕ(x, t).
On the other hand, since ϕ is of uniformly upper type 1, we get

(4.2) ϕ(x, t) = t 0 ϕ(x, s) s ds ≥ C -1 t 0 ϕ(x, t) t ds ≥ C -1 ϕ(x, t).
Combining (4.1) and (4.2), we obtain ϕ ≈ ϕ, and thus ϕ is a growth function.

iii) Suppose ϕ is a growth function. By (ii), ϕ is equivalent to ϕ. On the other hand, ∂ ϕ ∂t (x, t) = ϕ(x,t) t is uniformly quasi-decreasing in t. Hence, ϕ is uniformly quasi-concave, and thus is ϕ.

The converse is easy by taking s = 0. We omit the details.

R n ϕ x, |f (x)| f L ϕ dx = 1 for all f ∈ L ϕ (R n ) \ {0}. ii) lim k→∞ f k L ϕ = 0 if and only if lim k→∞ R n ϕ(x, |f k (x)|)dx = 0.
Proof. Statement i) follows from the fact that the function

ϑ(t) := R n ϕ(x, t|f (x)|)dx, t ∈ [0, ∞), is continuous by the dominated convergence theorem since ϕ(x, •) is continuous. Statement ii) follows from the fact that f L ϕ ≤ C max R n ϕ(x, |f (x)|)dx, R n ϕ(x, |f (x)|)dx 1/p
, and

R n ϕ(x, |f (x)|)dx ≤ C max f L ϕ , ( f L ϕ ) p
for some p ∈ (0, i(ϕ)).

Lemma 4.3. Given c is a positive constant. Then, there exists a constant

C > 0 such that i) The inequality R n ϕ x, |f (x)| λ dx ≤ c, for λ > 0, implies f L ϕ ≤ Cλ.
ii) The inequality j ϕ B j ,

t j λ ≤ c, for λ > 0, implies inf α > 0 : j ϕ B j , t j α ≤ 1 ≤ Cλ.
Proof. The proofs are simple since we may take C = (1 + c.C p ) 1/p , for some p ∈ (0, i(ϕ)), where C p is such that (2.1) holds.

Lemma 4.4. Let (ϕ, q, s) be an admissible triplet. Then there exists a positive constant C such that

∞ j=1 b j L q ϕ (B j ) χ B j L ϕ ≤ CΛ q ({b j }), for all f = ∞ j=1 b j ∈ H ϕ,q,s at (R n
) where b j 's are multiples of (ϕ, q, s)-atoms supported in balls B j 's.

Proof. Since ϕ is of uniformly upper type 1, there exists a positive constant c > 0 such that

ϕ x, b i L q ϕ (B i ) ∞ j=1 b j L q ϕ (B j ) χ B j L ϕ ≥ c b i L q ϕ (B i ) χ B i L ϕ ∞ j=1 b j L q ϕ (B j ) χ B j L ϕ ϕ x, 1 χ B i L ϕ for all x ∈ R n , i ≥ 1. Hence, for all i ≥ 1, ϕ B i , b i L q ϕ (B i ) ∞ j=1 b j L q ϕ (B j ) χ B j L ϕ ≥ c b i L q ϕ (B i ) χ B i L ϕ ∞ j=1 b j L q ϕ (B j ) χ B j L ϕ since B i ϕ x, 1 χ B i L ϕ dx = 1 by Lemma 4.2. It follows that ∞ i=1 ϕ B i , b i L q ϕ (B i ) ∞ j=1 b j L q ϕ (B j ) χ B j L ϕ ≥ c.
We deduce from the above that

∞ j=1 b j L q ϕ (B j ) χ B j L ϕ ≤ CΛ q ({b j }),
with C = (C p /c) 1/p for some p ∈ (0, i(ϕ)), where C p is such that (2.1) holds.

Lemma 4.5. Let ϕ ∈ A q , 1 < q < ∞. Then, there exists a positive constant C such that i) For all ball B(x 0 , r), λ > 0, and t ∈ [0, ∞), we have ϕ(B(x 0 , λr), t) ≤ Cλ nq ϕ(B(x 0 , r), t).

ii) For all ball B(x 0 , r) and t ∈ [0, ∞), we have

B c ϕ(x, t) |x -x 0 | nq dx ≤ C ϕ(B, t) r nq .
iii) For all ball B, f measurable and t ∈ (0, ∞), we have

1 |B| B |f (x)| q ≤ C 1 ϕ(B, t) B |f (x)| q ϕ(x, t)dx.
iii) For all f measurable and t ∈ [0, ∞), we have

R n Mf (x) q ϕ(x, t)dx ≤ C R n |f (x)| q ϕ(x, t)dx,
where M is the classical Hardy-Littlewood maximal operator defined by

Mf (x) = sup x∈B-ball 1 |B| B |f (y)|dy , x ∈ R n .
In the setting ϕ(x, t) = w(x)Φ(t), w ∈ A ∞ and Φ a Orlicz function, the above lemma is well-known as a classical result in the theory of Muckenhoupt weight (cf. [START_REF] García-Cuerva | Weighted norm inequalities and related topics[END_REF]). Since ϕ satisfies uniformly Muckenhoupt condition, the proof of Lemma 4.5 is a slight modification of the classical result. We omit the details.

Atomic decompositions

The purpose of this section is prove the atomic decomposition theorem (Theorem 3.1). The construction is by now standard, but the estimates require the preliminary lemmas. For the reader convenience, we give all steps of the proof, even if only the generalization to our framework is new.

We first introduce a class of Hardy spaces that the Hardy space of Musielak-Orlicz type H ϕ (R n ) containing as a particular case. Definition 5.1. For m ∈ N, we denote by

H ϕ m (R n ) the space of all distributions f such that f * m ∈ L ϕ (R n ) with the (quasi-)norm f H ϕ m := f * m L ϕ . Clearly, H ϕ (R n ) is a special case associated with m = m(ϕ).
5.1. Some basic properties concerning H ϕ m (R n ) and H ϕ,q,s at (R n ). We start by the following proposition.

Proposition 5.1. For m ∈ N, we have H ϕ m (R n ) ⊂ S ′ (R n ) and the inclusion is continuous. Proof. Let f ∈ H ϕ m (R n ).
For any φ ∈ S(R n ), and x ∈ B(0, 1), we write

f, φ = f * φ(0) = f * ψ(x),
where ψ(y) = φ(yx) = φ(xy) for all y ∈ R n . It is easy to verify that sup

x∈B(0,1),y∈R n 1+|y| 1+|y-x| ≤ 2. Consequently, | f, φ | = |f * ψ(x)| ≤ 2 (m+2)(n+1) φ Sm inf x∈B(0,1) f * m (x) ≤ 2 (m+2)(n+1) φ Sm χ B(0,1) -1 L ϕ f H ϕ m . This implies that f ∈ S ′ (R n
) and the inclusion is continuous.

The following proposition gives the completeness of H ϕ m (R n ). Proposition 5.2. The space H ϕ m (R n ) is complete. Proof. In order to prove the completeness of H ϕ m (R n ), it suffices to prove that for every sequence {f j } j≥1 with f j H ϕ m ≤ 2 -j for any j ≥ 1, the series j f j converges in H ϕ m (R n ). Let us now take p > 0 such that ϕ is of uniformly lower type p. Then, for any j ≥ 1, (5.1)

R n ϕ(x, (f j ) * m (x))dx ≤ C(2 -j ) p R n ϕ x, (f j ) * m (x) 2 -j dx ≤ C2 -jp . Since { j i=1 f i } j≥1 is a Cauchy sequence in H ϕ m (R n )
, by Proposition 5.1 and the completeness of S ′ (R n ), { j i=1 f i } j≥1 is also a Cauchy sequence in S ′ (R n ) and thus converges to some f ∈ S ′ (R n ). This implies that, for every φ ∈ S(R n ), the series j f j * φ converges to f * φ pointwisely. Therefore

f * m (x) ≤ j (f j ) * m (x) and (f -k j=1 f j ) * m (x) ≤ j≥k+1 (f j ) * m (x) for all x ∈ R n , k ≥ 1.
Combining this and (5.1), we obtain

R n ϕ(x, (f - k j=1 f j ) * m (x))dx ≤ C j≥k+1 R n ϕ(x, (f j ) * m (x))dx ≤ C j≥k+1 2 -jp → 0,
as k → ∞, here we used Lemma 4.1. Thus, the series j f j converges to f in H ϕ m (R n ) by Lemma 4.2. This completes the proof.

Corollary 5.1. The Hardy space of Musielak-Orlicz type H ϕ (R n ) is complete.

The following lemma and its corollary show that (ϕ, q, s)-atoms are in H ϕ (R n ). Furthermore, it is the necessary estimate for proving that H ϕ,q,s at (R n ) ⊂ H ϕ (R n ) and the inclusion is continuous, see Theorem 5.1 below. Lemma 5.1. Let (ϕ, q, s) be an admissible triplet and m ≥ s. Then, there exists a constant C = C(ϕ, q, s, m) such that

R n ϕ(x, f * m (x))dx ≤ Cϕ(B, f L q ϕ (B) ),
for all f multiples of (ϕ, q, s)-atom associated with ball B = B(x 0 , r).

Proof. The case q = ∞ is easy and will be omitted. We just consider q ∈ (q(ϕ), ∞). Now let us set B = B(x 0 , 9r), and write

R n ϕ(x, f * m (x))dx = B ϕ(x, f * m (x))dx + ( B) c ϕ(x, f * m (x))dx = I + II.
Since ϕ is of uniformly upper type 1, by Hölder inequality, we get

I = B ϕ(x, f * m (x))dx ≤ C B f * m (x) f L q ϕ (B) + 1 ϕ(x, f L q ϕ (B) )dx ≤ Cϕ( B, f L q ϕ (B) ) + C 1 f L q ϕ (B) B |f * m (x)| q ϕ(x, f L q ϕ (B) )dx 1/q ϕ( B, f L q ϕ (B) ) (q-1)/q ≤ Cϕ(B, f L q ϕ (B) ) + C 1 f L q ϕ (B) f L q ϕ ( B) ϕ( B, f L q ϕ (B) ) ≤ Cϕ(B, f L q ϕ (B) ).
We used the fact f * m (x) ≤ C(m)Mf (x) and Lemma 4.5. To estimate II, we note that since m ≥ s, there exists a constant

C = C(m) such that φ x -y t - |α|≤s ∂ α φ( x-x 0 t ) α! x 0 -y t α ≤ Ct n |y -x 0 | s+1 |x -x 0 | n+s+1 for all φ ∈ S m (R n ), t > 0, x ∈ ( B) c , y ∈ B. Therefore |f * φ t (x)| = 1 t n B f (y) φ x -y t - |α|≤s ∂ α φ( x-x 0 t ) α! x 0 -y t α dy ≤ C B |f (y)| |y -x 0 | s+1 |x -x 0 | n+s+1 dy ≤ C r s+1 |x -x 0 | n+s+1 B |f (y)| q ϕ(y, λ)dy 1/q B [ϕ(y, λ)] -1/(q-1) dy (q-1)/q ≤ C f L q ϕ (B) r |x -x 0 | n+s+1 .
For any λ > 0, we used that B ϕ(y, λ)dy( B [ϕ(y, λ)] -1/(q-1) dy) q-1 ≤ C|B| q since ϕ ∈ A q . As a consequence, we get

f * m (x) ≤ C(m) sup φ∈Sm(R n ) sup t>0 |f * φ t (x)| ≤ C f L q ϕ (B) r |x -x 0 | n+s+1 .
By s ≥ m(ϕ), there exists p ∈ (0, i(ϕ)) such that (n + s + 1)p > nq(ϕ). Hence, by Lemma 4.5,

II = ( B) c ϕ(x, f * m (x))dx ≤ C ( B) c r |x -x 0 | (n+s+1)p ϕ(x, f L q ϕ (B) )dx ≤ Cr (n+s+1)p ϕ( B, f L q ϕ (B) ) (9r) (n+s+1)p ≤ Cϕ(B, f L q ϕ (B)
). This completes the proof.

Corollary 5.2. There exists a constant C = C(ϕ, q, s) > 0 such that a H ϕ ≤ C, for all (ϕ, q, s)-atom a.

Theorem 5.1. Let (ϕ, q, s) be an admissible triplet and m ≥ s. Then

H ϕ,q,s at (R n ) ⊂ H ϕ m (R n ), moreover, the inclusion is continuous.
Proof. For any 0 = f ∈ H ϕ,q,s at (R n ). Let f = j b j be an atomic decomposition of f , with supp b j ⊂ B j , j = 1, 2, ... For all φ ∈ S(R n ), the series j b j * φ converges to f * φ pointwise since f = j b j in S ′ . Hence f * m (x) ≤ j (b j ) * m (x). By applying Lemma 5.1, we obtain

R n ϕ x, f * m (x) Λ q ({b j }) dx ≤ C j R n ϕ x, (b j ) * m (x) Λ q ({b j }) dx ≤ C j ϕ B j , b j L q ϕ (B j ) Λ q ({b j }) ≤ C.
This implies that f H ϕ m ≤ CΛ q ({b j }) (see Lemma 4.3) for any atomic decomposition f = j b j , and thus, f H ϕ m ≤ C f H ϕ,q,s at .

5.2.

Calderón-Zygmund decompositions. Throughout this subsection, we fix m and s so that m, s ≥ m(ϕ). For a given λ > 0, we set Ω = {x ∈ R n : f * m (x) > λ}. Observe that Ω is open. Hence by Whitney's Lemma , there exists x 1 , x 2 , ... in Ω and r 1 , r 2 , ... > 0 such that (i) Ω = ∪ j B(x j , r j ), (ii) the balls B(x j , r j /4), j = 1, 2, ..., are disjoint, (iii) B(x j , 18r j ) ∩ Ω c = ∅, but B(x j , 54r j ) ∩ Ω c = ∅, for any j = 1, 2, ..., (iv) there exists L ∈ N (depending only on n) such that no point of Ω lies in more than L of the balls B(x j , 18r j ), j = 1, 2, ... We fix once for all, a function θ ∈ C ∞ 0 (R n ) such that supp θ ⊂ B(0, 2), 0 ≤ θ ≤ 1, θ = 1 on B(0, 1), and set θ j (x) = θ((xx j )/r j ), for j=1,2,... Obviously, supp θ j ⊂ B(x j , 2r j ), j = 1, 2, ..., and 1 ≤ j θ j ≤ L for all x ∈ Ω. Hence if we set

ζ j (x) = θ j (x)/ ∞ i=1 θ i (x) if x ∈ Ω and ζ j (x) = 0 if x ∈ Ω c , j = 1, 2, ..., then supp ζ j ⊂ B(x j , 2r j ), 0 ≤ ζ j ≤ 1, j ζ j = χ Ω , and L -1 ≤ ζ j ≤ 1 on B(x j , r j ).
The family {ζ j } j forms a smooth partition of unity of Ω. let s ∈ N be some fixed natural number and P s (R n ) (or simply P s ) denote the linear space of polynomials in n variables of degree less than s. For each j, we consider the inner product

P, Q j = 1 R n ζ j (x)dx R n P (x)Q(x)ζ j (x)dx for P, Q ∈ P s . Then (P s , •, • j ) is a finite dimensional Hilbert space. Let f ∈ S ′ . Since f induces a linear functional on P s via Q → 1 R n ζ j (x)dx R n f (x)Q(x)ζ j (x)
dx, by the Riesz theorem, there exists a unique polynomial P j ∈ P s such that for all Q ∈ P s ,

P j , Q j = 1 R n ζ j (x)dx R n f (x)Q(x)ζ j (x)dx.
For each j, j = 1, 2, ..., we define b j = (f -P j )ζ j , and note B j = B(x j , r j ), B j = B(x j , 9r j ). Then, it is easy to see that R n b j (x)Q(x)dx = 0 for all Q ∈ P s . It turns out, in the case of interest, that the series j b j converges in S ′ . In this case, we set g = fj b j , and we call the representation f = g + j b j a Calderón-Zygmund decomposition of f of degree s and height λ associated to f * m . For any j = 1, 2, ..., we denote B j = B(x j , r j ) and B j = B(x j , 9r j ). Then we have the following lemma Lemma A (see [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF]Chapter 3]) There are four constant c 1 , c 2 , c 3 , c 4 , independent of f, j, and λ, such that i) sup

|α|≤N,x∈R n r |α| j |∂ α ζ j (x)| ≤ c 1 .
ii) sup

x∈R n |P j (x)ζ j (x)| ≤ c 2 λ. iii) (b j ) * m (x) ≤ c 3 f * m (x), for all x ∈ B j . iv) (b j ) * m (x) ≤ c 4 λ(r j /|x -x j |) n+ms , for all x / ∈ B j ,
where m s = min{s + 1, m + 1}.

Lemma 5.2. For all f ∈ H ϕ m (R n ), there exists a geometric constant C, independent of f, j, and λ, such that,

R n ϕ x, (b j ) * m (x) dx ≤ C B j ϕ(x, f * m (x))dx.
Moreover, the series j b j converges in H ϕ m (R n ), and

R n ϕ x, ( j b j ) * m (x) dx ≤ C Ω ϕ(x, f * m (x))dx.
Proof. As m, s ≥ m(ϕ), m s = min{s + 1, m + 1} > n(q(ϕ)/i(ϕ) -1). Hence, there exist q > q(ϕ) and 0 < p < i(ϕ) such that m s > n(q/p -1), deduce that (n + m s )p > nq. Therefore, ϕ ∈ A (n+ms)p/n and ϕ is of uniformly lower type p. Thus, there exists a positive constant C, independent of f, j, and λ, such that

( B j ) c ϕ(x, λ(r j /|x -x j |) n+ms )dx ≤ C ( B j ) c r j |x -x j | (n+ms)p ϕ(x, λ)dx ≤ C(r j ) (n+ms)p ϕ( B j , λ) (9r j ) (n+ms)p ≤ C B j ϕ(x, f * m (x))dx, since r j /|x -x j | < 1 and f * m > λ on B j .
Combining this and Lemma A, we get

R n ϕ x, (b j ) * m (x) dx ≤ C B j ϕ(x, f * m (x))dx + ( B j ) c ϕ(x, λ(r j /|x -x j |) n+ms )dx ≤ C B j ϕ(x, f * m (x))dx.
As a consequence of the above estimate, since j χ B j ≤ L and Ω = ∪ j B j , we obtain

j R n ϕ x, (b j ) * m (x) dx ≤ C j B j ϕ(x, f * m (x))dx ≤ C Ω ϕ(x, f * m (x))dx.
This implies that the series

j b j converges in H ϕ m (R n ) by completeness of H ϕ m (R n ). Moreover, R n ϕ x, ( j b j ) * m (x) dx ≤ C Ω ϕ(x, f * m (x))dx. Let q ∈ [1, ∞].
We denote by L q ϕ(•,1) (R n ) the usually weighted Lebesgue space with the Muckenhoupt weight ϕ(x, 1). Then we have the following lemma (see [START_REF] Bownik | Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators[END_REF]Lemma 4.8]). Lemma B. Let q ∈ (q(ϕ), ∞]. Assume that f ∈ L q ϕ(•,1) (R n ), then the series j b j converges in L q ϕ(•,1) (R n ) and there exists a constant C, independent of f, j, and λ such that j |b j | L q ϕ(•,1) ≤ C f L q ϕ(•,1) . Remark 5.1. By Lemma B, the series j |b j |, and thus the series j b j , converges a.e on R n .

Lemma C. (see [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF]Lemma 3.19]) Suppose that the series j b j converges in S ′ (R n ). Then, there exists a positive constant C, independent of f, j, and λ, such that for all x ∈ R n ,

g * m (x) ≤ Cλ j r j |x -x j | + r j n+ms + f * m (x)χ Ω c (x). Lemma 5.3. For any q ∈ (q(ϕ), ∞) and f ∈ H ϕ m (R n ). Then g * m ∈ L q ϕ(•,1) (R n ),
and there exists a positive constant C, independent of f, j, and λ, such that

R n [g * m (x)] q ϕ(x, 1)dx ≤ Cλ q max{1/λ, 1/λ p } R n ϕ(x, f * m (x))dx.
Proof. For any j = 1, 2, ... and x ∈ R n , we have

r j |x -x j | + r j n = 1 |B(x j , |x -x j | + r j )| B(x j ,|x-x j |+r j ) χ B j (y)dy ≤ M(χ B j )(x) since B j ⊂ B(x j , |x -x j | + r j ).
Therefore, by L rq ϕ(•,1) -boundedness of vector-valued maximal functions [1, Theorem 3.1], where r := (n + m s )/n > 1, we obtain that

R n j r j |x -x j | + r j n+ms q ϕ(x, 1)dx ≤ R n j (M(χ B j )(x)) r 1/r rq ϕ(x, 1)dx ≤ C s,q R n j (χ B j (x)) r 1/r rq ϕ(x, 1)dx ≤ C s,q L Ω ϕ(x, 1)dx ≤ C max{1/λ, 1/λ p } R n ϕ(x, f * m (x))dx
for some p ∈ (0, i(ϕ)) since ϕ ∈ A q ⊂ A rq and f * m > λ on Ω. Combine this, Lemma C and Hölder inequality, we obtain

R n [g * m (x)] q ϕ(x, 1)dx ≤ Cλ q max{1/λ, 1/λ p } R n ϕ(x, f * m (x))dx + C Ω c [f * m (x)] q ϕ(x, 1)dx ≤ Cλ q max{1/λ, 1/λ p } R n ϕ(x, f * m (x))dx, since f * m ≤ λ on Ω c , here one used ϕ(x, λ)/λ q ≤ Cϕ(x, f * m (x))/[f * m (x)] q on Ω c .
Proposition 5.3. For any q ∈ (q(ϕ), ∞) and m ≥ m(ϕ). The subspace

L q ϕ(•,1) (R n ) ∩ H ϕ m (R n ) is dense in H ϕ m (R n ).
Proof. Let f be an arbitrary element in H ϕ m (R n ). For each λ > 0, let f = g λ + j b λ j be the Calderon-Zygmund decomposition of f of degree m(ϕ), and height λ associated with f * m . Then by Lemma 5.2 and Lemma 5.3,

g λ ∈ L q ϕ(•,1) (R n ) ∩ H ϕ m (R n ), moreover, R n ϕ(x, (g λ -f ) * m (x))dx ≤ C f * m (x)>λ ϕ(x, f * m (x))dx → 0, as λ → ∞. Consequently, g λ -f H ϕ m → 0 as λ → ∞ by Lemma 4.2. Thus L q ϕ(•,1) (R n ) ∩ H ϕ m (R n ) is dense in H ϕ m (R n ).

The atomic decompositions

H ϕ m (R n ).
Recall that m, s ≥ m(ϕ), and f is a distribution such that f * m ∈ L ϕ (R n ). For each k ∈ Z, let f = g k + j b k j be the Calderón-Zygmund decomposition of f of degree s and height 2 k associated with f * m . We shall label all the ingredients in this construction as in subsection 5.2, but with superscript k ′ s: for example,

Ω k = {x ∈ R n : f * m (x) > 2 k }, b k j = (f -P k j )ζ k j , B k j = B(x k j , r k j )
. Moreover, for each k ∈ Z, and i, j, let P k+1 i,j be the orthogonal projection of (f -P k+1 j )ζ k i onto P s with respect to the norm associated to ζ k+1 j , namely, the unique element of P s such that for all

Q ∈ P s , R n (f (x) -P k+1 j (x))ζ k i (x)Q(x)ζ k+1 j (x)dx = R n P k+1 i,j (x)Q(x)ζ k+1 j (x)dx.
For convenience, we set Bk j = B(x k j , 2r k j ). Then we have the following lemma. Lemma D. (see [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF]Chapter 3]

) i) If Bk+1 j ∩ Bk i = ∅ the r k+1 j < 4r k i and Bk+1 j ⊂ B(x k i , 18r k i ).
ii) For each j there are at most L (depending only on n as in last section) values of i such that Bk+1 j ∩ Bk i = ∅. iii) There is a constant C > 0, independent of f, i, j, and k, such that sup

x∈R n |P k+1 i,j (x)ζ k+1 j (x)| ≤ C2 k+1 .
iv) For every k ∈ Z, i ( j P k+1 i,j ζ k+1 j ) = 0, where the series converges pointwise and in S ′ (R n ).

We now give the necessary estimates for proving that

H ϕ m (R n ) ⊂ H ϕ,∞,s at (R n ), m ≥ s ≥ m(ϕ)
, and the inclusion is continuous.

Lemma 5.4. Let f ∈ H ϕ m (R n ), and for each k ∈ Z, set Ω k = {x ∈ R n : f * m (x) > 2 k }.
Then for any λ > 0, there exists a constant C, independent of f and λ, such that

∞ k=-∞ ϕ Ω k , 2 k λ ≤ C R n ϕ x, f * m (x) λ dx.
Proof. Let p ∈ (0, i(ϕ)) and C p is such that (2.1) holds. We now set

N 0 = [(log 2 C p )/p] + 1 so that 2 N 0 p > C p . For each ℓ ∈ N, 0 ≤ ℓ ≤ N 0 -1, we consider the sequence U ℓ m = m k=-m ϕ Ω N 0 k+ℓ , 2 N 0 k+ℓ λ . Obviously, {U ℓ m } m∈N is an increasing sequence. Moreover, for any m ∈ N, U ℓ m = m k=-m ϕ Ω N 0 (k+1)+ℓ , 2 N 0 k+ℓ λ + m k=-m ϕ Ω N 0 k+ℓ , 2 N 0 k+ℓ λ -ϕ Ω N 0 (k+1)+ℓ , 2 N 0 k+ℓ λ ≤ C p 1 2 N 0 p U ℓ m + ϕ Ω N 0 (m+1)+ℓ , 2 N 0 (m+1)+ℓ λ + ϕ Ω N 0 (-m)+ℓ , 2 N 0 (-m)+ℓ λ + + m k=-m Ω N 0 k+ℓ \Ω N 0 (k+1)+ℓ ϕ x, f * m (x) λ dx ≤ C p 2 N 0 p U ℓ m + 2 C p 2 N 0 p + 1 R n ϕ x, f * m (x) λ dx. This implies that U ℓ m ≤ 3 1-Cp/(2 N 0 p ) R n ϕ x, f * m (x) λ dx. Consequently, ∞ k=-∞ ϕ Ω k , 2 k λ = N 0 -1 ℓ=0 lim m→∞ U ℓ m ≤ C R n ϕ x, f * m (x) λ dx, where C = 3N 0 1-Cp/(2 N 0 p ) independent of f and λ. Theorem 5.2. Let m ≥ s ≥ m(ϕ). Then, H ϕ m (R n ) ⊂ H ϕ,∞,s at (R n ) and the inclusion is continuous. Proof. Suppose first that f ∈ L q ϕ(•,1) (R n ) ∩ H ϕ m (R n ) for some q ∈ (q(ϕ), ∞). Let f = g k + j b k j be the Calderón-Zygmund decompositions of f of degree s with height 2 k , for k ∈ Z associated with f * m . By Proposition 5.3, g k → f in H ϕ m (R n ) as k → ∞, while by [9, Lemma 4.10], g k → 0 uniformly as k → -∞ since f ∈ L q ϕ(•,1) (R n ). Therefore, f = ∞ -∞ (g k+1 -g k ) in S ′ (R n ). Using [17, Lemma 3.27] together with the equation i ζ k i b k+1 j = χ Ω k b k+1 j = b k+1 j by suppb k+1 j ⊂ Ω k+1 ⊂ Ω k , we get g k+1 -g k = (f - j b k+1 j ) -(f - i b k i ) = i b k i - j b k+1 j + i j P k+1 i,j ζ k+1 j = i b k i - j ζ k i b k+1 j -P k+1 i,j ζ k+1 j = i h k i
where all the series converge in S ′ (R n ) and almost everywhere. Furthermore, (5.2)

h k i = (f -P k i )ζ k i - j (f -P k+1 j )ζ k i -P k+1 i,j ζ k+1 j .
From this formula it is obvious that R n h k i (x)P (x)dx = 0 for all P ∈ P s . Moreover,

h k i = ζ k i f χ (Ω k+1 ) c -P k i ζ k i + ζ k i j P k+1 j ζ k+1 j + j P k+1 i,j ζ k+1 j , by j ζ k+1 j = χ Ω k+1 . But |f (x)| ≤ C(m)f * m (x) ≤ C2 k+1
for almost every x ∈ (Ω k+1 ) c , so by [17, Lemma 3.8 and 3.26], and j ζ k+1 j ≤ L,

(5.3) h k i L ∞ ≤ C2 k+1 + C2 k + CL2 k+1 + CL2 k+1 ≤ C2 k , Lastly, since P k+1 i,j = 0 unless Bk i ∩ Bk+1 j = ∅, it follows from (5.2) and [17, Lemma 3.24] that h k i is supported in B(x k i , 18r k i ).
Thus h k i is a multiple of (ϕ, ∞, s)-atom. Moreover, by (5.3) and Lemma 5.4, for any λ > 0,

k∈Z i ϕ B(x k i , 18r k i ), h k i L ∞ λ ≤ k∈Z Lϕ(Ω k , C2 k /λ) ≤ C R n ϕ x, f * m (x) λ dx < ∞.
Thus the series k∈Z i h k i converges in H ϕ,∞,s at (R n ) and defines an atomic decomposition of f . Moreover,

k∈Z i ϕ B(x k i , 18r k i ), h k i L ∞ f H ϕ m ≤ C R n ϕ x, f * m (x) f H ϕ m dx ≤ C. Consequently, f H ϕ,∞,s at ≤ Λ ∞ ({h k i }) ≤ C f H ϕ m by Lemma 4.3. Now, let f be an arbitrary element of H ϕ m (R n ). There exists a sequence {f ℓ } ℓ≥1 ⊂ L q ϕ(•,1) (R n ) ∩ H ϕ m (R n ) such that f = ∞ ℓ=1 f ℓ in H ϕ m (R n ) (thus in S ′ (R n )) and f ℓ H ϕ m ≤ 2 2-ℓ f H ϕ m for any ℓ ≥ 1.
For any ℓ ≥ 1, let f ℓ = j b j,ℓ be the atomic decomposition of f ℓ , with supp b j,ℓ ⊂ B j,ℓ constructed above. Then f = ∞ ℓ=1 j b j,ℓ is an atomic decomposition of f , and

∞ ℓ=1 j ϕ B j,ℓ , b j,ℓ L ∞ f H ϕ m ≤ ∞ ℓ=1 i ϕ B j,ℓ , b j,ℓ L ∞ 2 ℓ-2 f ℓ H ϕ m ≤ ∞ ℓ=1 C p 1 (2 ℓ-2 ) p =: C, where C p is such that (2.1) holds. Thus f ∈ H ϕ,∞,s at (R n ), moreover, f H ϕ,∞,s at ≤ Λ ∞ ({b j,ℓ }) ≤ C f H ϕ
H ϕ,∞,s at (R n ) ⊂ H ϕ,q,s at (R n ) ⊂ H ϕ,q,m(ϕ) at (R n ) ⊂ H ϕ (R n ) ⊂ H ϕ s (R n ) ⊂ H ϕ,∞,s at (R n )
and the inclusions are continuous. Thus H ϕ (R n ) = H ϕ,q,s at (R n ) with equivalent norms.

Dual spaces

In this Section, we give the proof of Theorem 3.2. In order to do this, we need the lemma, which can be seen as a consequence of the fact that ϕ(•, t) is uniformly locally integrable, below. We omit the details here. We next note that if b ∈ BMO ϕ (R n ) is real-valued and

b N (x) =      N if b(x) > N, b(x) if |b(x)| ≤ N, -N if b(x) < -N,
then by using the fact

f BM O ϕ ≤ sup B-ball 1 χ B L ϕ 1 |B| B×B |f (x) -f (y)|dxdy ≤ 2 f BM O ϕ , we obtain that b N BM O ϕ ≤ 2 b BM O ϕ for all N > 0. Proof of Theorem 3.2. i) It is sufficient to prove it for b ∈ BMO ϕ (R n ) real- valued since b ∈ BMO ϕ (R n ) iff b = b 1 +ib 2 with b j ∈ BMO ϕ (R n ) real-valued, j = 1, 2, moreover b BM O ϕ ≈ b 1 BM O ϕ + b 2 BM O ϕ . Suppose first that b ∈ BMO ϕ (R n ) ∩ L ∞ (R n ). Then, the functional L b (f ) = R n f (x)b(x)dx is well defined for any f ∈ L ∞ 0 (R n ) since b ∈ L 1 loc (R n ). Furthermore, since f ∈ L ∞ 0 (R n ) ⊂ L 2 (R n ) ∩ H 1 (R n ),
we remark that the atomic decomposition f = k∈Z i h k i in the proof of Theorem 5.2 is also the classical atomic decomposition of f in H 1 (R n ), so that the series converge in H 1 (R n ) and thus in L 1 (R n ). Combining this with the fact b ∈ L ∞ (R n ), we obtain

L b (f ) = R n f (x)b(x)dx = k∈Z i R n h k i (x)b(x)dx.
Therefore, by Lemma 4.4 and the proof of Theorem 5.2,

|L b (f )| = R n f (x)b(x)dx ≤ k∈Z i R n h k i (x)b(x)dx = k∈Z i B(x k i ,18r k i ) h k i (x)(b(x) -b B(x k i ,18r k i ) (x))dx ≤ b BM O ϕ k∈Z i h k i L ∞ χ B(x k i ,18r k i ) L ϕ ≤ C b BM O ϕ Λ ∞ ({h k i }) ≤ C b BM O ϕ f H ϕ . Now, let b be an arbitrary element in BMO ϕ (R n ). For any f ∈ L ∞ 0 (R n ), it is clear that |f |.|b ℓ | ≤ |f |.|b| ∈ L 1 (R n ), ℓ ≥ 1, and f (x)b ℓ (x) → f (x)b(x),
as ℓ → ∞, a.e x ∈ R n . Therefore, by the Lebesgue dominated convergence theorem, we obtain

|L b (f )| = R n f (x)b(x)dx = lim ℓ→∞ R n f (x)b ℓ (x)dx ≤ C b BM O ϕ f H ϕ , since b ℓ BM O ϕ ≤ 2 b BM O ϕ for all ℓ ≥ 1. Because of the density of L ∞ 0 (R n ) in H ϕ (R n ), the functional L b can be ex- tended to a bounded functional on H ϕ (R n ), moreover, L b (H ϕ ) * ≤ C b BM O ϕ .
ii) Conversely, suppose L is a continuous linear functional on H ϕ (R n ) ≡ H ϕ,q,0 at (R n ) for some q ∈ (q(ϕ), ∞). For any ball B, denote by L q ϕ,0 (B) the subspace of L q ϕ (B) defined by

L q ϕ,0 (B) := f ∈ L q ϕ (B) : R n f (x)dx = 0 . Obviously, if B 1 ⊂ B 2 then (6.1) L q ϕ (B 1 ) ⊂ L q ϕ (B 2 ) and L q ϕ,0 (B 1 ) ⊂ L q ϕ,0 (B 2 ). Moreover, when f ∈ L q ϕ,0 (B) \ {0}, a(x) = χ B -1 L ϕ f -1 L q ϕ (B) f (x) is a (ϕ, q, 0)- atom, thus f ∈ H ϕ,q,0 at (R n ) and f H ϕ,q,0 at ≤ χ B L ϕ f L q ϕ (B) .
Since L ∈ (H ϕ,q,0 at (R n )) * , by the above,

|L(f )| ≤ L (H ϕ,q,0 at ) * f H ϕ,q,0 at ≤ L (H ϕ,q,0 at ) * χ B L ϕ f L q ϕ (B) ,
for all f ∈ L q ϕ,0 (B). Therefore, L provides a bounded linear functional on L q ϕ,0 (B) which can be extended by the Hahn-Banach theorem to the whole space L q ϕ (B) without increasing its norm. On the other hand, by Lemma 6.1 and Lebesgue-Nikodym theorem, there exists h ∈ L 1 (B) such that

L(f ) = R n f (x)h(x)dx,
for all f ∈ L q ϕ,0 (B), and thus f ∈ L ∞ ϕ,0 (B) since L ∞ ϕ,0 (B) ⊂ L q ϕ,0 (B). We now take a sequence of balls {B j } j≥1 such that

B 1 ⊂ B 2 ⊂ • • • ⊂ B j ⊂ • • • and ∪ j B j = R n .
Then, there exists a sequence {h j } j≥1 such that

h j ∈ L 1 (B j ) and L(f ) = R n f (x)h j (x)dx, for all f ∈ L ∞ ϕ,0 (B j ), j = 1, 2, ... Hence, for all f ∈ L ∞ ϕ,0 (B 1 ) ⊂ L ∞ ϕ,0 (B 2 ) (by 6.1), R n f (x)(h 1 (x)-h 2 (x))dx = R n f (x)h 1 (x)dx- R n f (x)h 2 (x)dx = L(f )-L(f ) = 0. As f B 1 = 0 if f ∈ L ∞ ϕ,0 (B 1 ), we have R n f (x) (h 1 (x) -h 2 (x)) -(h 1 -h 2 ) B 1 dx = 0 for all f ∈ L ∞ ϕ,0 (B 1 )
, and thus for f ∈ L ∞ ϕ (B 1 ). Hence,

h 1 (x) -h 2 (x) = (h 1 -h 2 ) B 1 , a.e x ∈ B 1 .
By the similar arguments, we also obtain (6.2)

h j (x) -h j+1 (x) = (h j -h j+1 ) B j
a.e x ∈ B j , j = 2, 3, ... Consequently, if we define the sequence { h j } j≥1 by

h 1 = h 1 h j+1 = h j+1 + ( h j -h j+1 ) B j , j = 1, 2, ...
then it follows from (6.2) that h j ∈ L 1 (B j ) and h j+1 (x) = h j (x)

a.e x ∈ B j , j = 1, 2, ... Thus, we can be define the function b on R n by b

(x) = h j (x) if x ∈ B j for some j ≥ 1 since B 1 ⊂ B 2 ⊂ • • • ⊂ B j ⊂ • • • and ∪ j B j = R n .
Let us now show that b ∈ BMO ϕ (R n ) and

L(f ) = R n f (x)b(x)dx, for all f ∈ L ∞ 0 (R n ). Indeed, for any f ∈ L ∞ 0 (R n ), there exists j ≥ 1 such that f ∈ L ∞ ϕ,0 (B j ). Hence, L(f ) = R n f (x) h j (x)dx = B j f (x) h j (x)dx = R n f (x)b(x)dx.
On the other hand, for all ball B, one consider f = sign(bb B ) where signξ = ξ/|ξ| if ξ = 0 and sign0 = 0. Then, In this subsection, we give as an interesting application that the class of pointwise multipliers for BMO We first introduce log-atoms. A measurable function a is said to be log-atom if it satisfies the following three conditions

a = 1 2 χ B -1 L ϕ (f -f B )χ B is a (ϕ, ∞, 0)-atom. Consequently, |L(a)| = 1 2 χ B -1 L ϕ R n b(x)(f (x) -f B )χ B (x)dx = 1 2 1 χ B L ϕ B (b(x) -b B )f (x)dx = 1 2 1 χ B L ϕ B |b(x) -b B |dx ≤ L (H ϕ ) * a H ϕ ≤ C L (H ϕ ) * since L ∈ (H ϕ (R n )) *
(R n ) is just the dual of L 1 (R n )+H log (R n ) where H log (R n ) is
• a supported in B for some ball B in R n ,

• a L ∞ ≤ log(e + • R n a(x)dx = 0.

To prove Theorem 3.3, we need the following two propositions.

Proposition 7.1. There exists a positive constant C such that if f is a θ-atom (resp., log-atom) then C -1 f is a log-atom (resp., θ-atom).

Proposition 7.2. On BMO log (R n ), we have

f BM O log ≈ sup B-ball log(e + 1 |B| ) + sup x∈B log(e + |x|) |B| B |f (x) -f B |dx < ∞.
We first note that θ is a growth function that satisfies nq(θ) < (n + 1)i(θ) in Theorem 3.2. More precisely, θ ∈ A 1 and θ(x, •) is concave with i(θ) = 1.

Proof of Proposition 7.1. Let f be a log-atom. By the above remark, to prove that there exists a constant C > 0 (independent of f and which may change from line to line) such that C -1 f is a θ-atom, it is sufficient to show that there exists a constant C > 0 such that B θ x, log(e + However it is true. For instance we may take C = 3.

Proof of Proposition 7.2. It is sufficient to show that there exists a constant C > 0 such that

C -1 (| log r|+log(e+|x|)) ≤ log e+ 1 |B(x, r)| + sup y∈B(x,r) log(e+|y|) ≤ C(| log r|+log(e+|x|)).
The first inequality is easy and shall be omited. For the second, one first consider the 1 dimensional case. Then by symmetry, we just need to prove that

(R n )) * ≡ BMO log (R n ). We deduce that, the class of pointwise multipliers for BMO(R n ) is the dual of L 1 (R n ) + H log (R n ).

Finite atomic decompositions and their applications

We first prove the finite atomic decomposition theorem.

Proof of Theorem 3.4. Obviously, H ϕ,q,s fin

(R n ) ⊂ H ϕ (R n ) and for all f ∈ H ϕ,q,s fin (R n ), f H ϕ ≤ C f H ϕ,q,s fin .
Thus, we have to show that for every q ∈ (q(ϕ), ∞) there exists a constant

C > 0 such that f H ϕ,q,s fin ≤ C f H ϕ
for all f ∈ H ϕ,q,s fin (R n ) and that a similar estimate holds for q = ∞ and all

f ∈ H ϕ,∞,s fin (R n ) ∩ C(R n ).
Assume that q ∈ (q(ϕ), ∞], and by homogeneity, f ∈ H ϕ,q,s fin (R n ) with f H ϕ = 1. Notice that f has compact support. Suppose that supp f ⊂ B = B(x 0 , r) for some ball B. Recall that, for each k ∈ Z,

Ω k = {x ∈ R n : f * (x) > 2 k }. Clearly, f ∈ L q ϕ(•,1) (R n ) ∩ H ϕ (R n
) where q = q if q < ∞, q = q(ϕ) + 1 if q = ∞. Hence, by Theorem 5.2, there exists a atomic decomposition f

= k∈Z i h k i ∈ H ϕ,∞,s at (R n ) ⊂ H ϕ,q,s at (R n )
where the series converges in S ′ (R n ) and almost everywhere. Moreover,

(8.1) Λ q ({h k i }) ≤ Λ ∞ ({h k i }) ≤ C f H ϕ = C.
On the other hand, it follows from the second step in the proof of Theorem 6.2 in [START_REF] Bownik | Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators[END_REF] that there exists a constant C > 0, depending only on m(ϕ), such that f * (x) ≤ C inf y∈B f * (y) for all x ∈ B(x 0 , 2r) c . Hence, we have

f * (x) ≤ C inf y∈B f * (y) ≤ C χ B -1 L ϕ f * L ϕ ≤ C χ B -1 L ϕ
for all x ∈ B(x 0 , 2r) c . We now denote by k ′ the largest integer k such that

2 k < C χ B - 1
L ϕ . Then, (8.2) Ω k ⊂ B(x 0 , 2r) for all k > k ′ .
Next we define the functions g and ℓ by

g = k≤k ′ i h k i and ℓ = k>k ′ i h k i ,
where the series converge in S ′ (R n ) and almost everywhere. Clearly, f = g + ℓ and supp ℓ ⊂ ∪ k>k ′ Ω k ⊂ B(x 0 , 2r) by (8.2). Therefore, g = f = 0 in B(x 0 , 2r) c , and thus supp g ⊂ B(x 0 , 2r). Let 1 < q < q q(ϕ) , then ϕ ∈ A q/ q . Consequently,

1 |B| B |f (x)| q dx 1/ q ≤ C   1 ϕ(B, 1) B |f (x)| q ϕ(x, 1)dx   1/q < ∞
by Lemma 4.5 if q < ∞ and it is trivial if q = ∞. Observe that supp f ⊂ B and that f has vanishing moments up to order s. By the above, we obtain that f is a multiple of a classical (1, q, 0)-atom and thus f * ∈ L 1 (R n ). Hence, it follows from (8.2) that

R n k>k ′ i |h k i (x)x α |dx ≤ C(|x 0 | + 2r) s k>k ′ 2 k |Ω k | ≤ C(|x 0 | + 2r) s f * L 1 < ∞,
for all |α| ≤ s. This together with the vanishing moments of h k i implies that ℓ has vanishing moments up to order s and thus so does g by g = fℓ.

In order to estimate the size of g in B(x 0 , 2r), we recall that (8.3)

h k i L ∞ ≤ C2 k , supp h k i ⊂ B(x k i , 18r k i ) and i χ B(x k i ,18r k i ) ≤ C.
Combining the above and the fact χ B L ϕ ≈ χ B(x 0 ,2r) L ϕ , we obtain

g L ∞ ≤ C k≤k ′ 2 k ≤ C2 k ′ ≤ C C χ B - 1
L ϕ ≤ C χ B(x 0 ,2r) -1 L ϕ .
This proves that (see Definition 2.4) (8.4) C -1 g is a (ϕ, ∞, s)-atom. Now, we assume that q ∈ (q(ϕ), ∞) and conclude the proof of (i). We first verify k>k ′ i h k i ∈ L q ϕ (B(x 0 , 2r)). For any x ∈ R n , since R n = ∪ k∈Z (Ω k \ Ω k+1 ), there exists j ∈ Z such that x ∈ Ω j \ Ω j+1 . Since supp h k i ⊂ Ω k ⊂ Ω j+1 for k ≥ j + 1, it follows from (8.3) that

k>k ′ i |h k i (x)| ≤ C k≤j 2 k ≤ C2 j ≤ Cf * (x).
Since f ∈ L q ϕ (B) ⊂ L q ϕ (B(x 0 , 2r)), we have f * ∈ L q ϕ (B(x 0 , 2r)). As ϕ satisfies uniformly locally dominated convergence condition, we further obtain k>k ′ i h k i converges to ℓ in L q ϕ (B(x 0 , 2r)). Now, for any positive integer K, set F K = {(i, k) : k > k ′ , |i| + |k| ≤ K} and ℓ K = (i,k)∈F K h k i . Observe that since k>k ′ i h k i converges to ℓ in L q ϕ (B(x 0 , 2r)), for any ε > 0, if K is large enough, we have ε -1 (ℓℓ K ) is a (ϕ, q, s)-atom. Thus, f = g + ℓ K + (ℓℓ K ) is a finite linear atom combination of f . Then, it follows from (8.1) and (8.4) that f H ϕ,q,s fin ≤ C(C + Λ q ({h k i } (i,k)∈F K ) + ε) ≤ C, which ends the proof of (i).

To prove (ii), assume that f is a continuous function in H ϕ,∞,s fin (R n ), and thus f is uniformly continuous. Then, h k i is continuous by examining its definition. Since f is bounded, there exists a positive integer k ′′ > k ′ such that Ω k = ∅ for all k > k ′′ . Consequently, ℓ = k ′ <k≤k ′′ i h k i . Let ε > 0. Since f is uniformly continuous, there exists δ > 0 such that if |x -y| < δ, then |f (x)f (y)| < ε. Write ℓ = ℓ ε 1 + ℓ ε 2 with

ℓ ε 1 ≡ (i,k)∈F 1 h k i and ℓ ε 2 ≡ (i,k)∈F 2 h k i
where F 1 = {(i, k) : Cr k i ≥ δ, k ′ < k ≤ k ′′ } and F 2 = {(i, k) : Cr k i < δ, k ′ < k ≤ k ′′ } with C > 36 the geometric constant (see [START_REF] Meda | Atomic decompositions and operators on Hardy spaces[END_REF]). Notice that the remaining part ℓ ε 1 will then be a finite sum. Since the atoms are continuous, ℓ ε 1 will be a continuous function. Furthermore, ℓ ε 2 L ∞ ≤ C(k ′′k ′ )ε (see also [START_REF] Meda | Atomic decompositions and operators on Hardy spaces[END_REF]). This means that one can write ℓ as the sum of one continuous term and of one which is uniformly arbitrarily small. Hence, ℓ is continuous, and so is g = fℓ.

To find a finite atomic decomposition of f , we use again the splitting ℓ = ℓ ε 1 + ℓ ε 2 . By (8.1), the part ℓ ε 1 is a finite sum of multiples of (ϕ, ∞, s)-atoms, and (8.5)

ℓ ε 1 H ϕ,∞,s fin ≤ Λ ∞ ({h k i }) ≤ C f H ϕ = C. By ℓ, ℓ ε
1 are continuous and have vanishing moments up to order s, and thus so does ℓ ε 2 = ℓℓ ε 1 . Moreover, supp ℓ ε 2 ⊂ B(x 0 , 2r) and ℓ ε 2 L ∞ ≤ C(k ′′k ′ )ε.

So we can choose ε small enough such that ℓ ε 2 into an arbitrarily small multiple of a continuous (ϕ, ∞, s)-atom. Therefore, f = g + ℓ ε 1 + ℓ ε 2 is a finite linear continuous atom combination of f . Then, it follows from (8.4) and (8.5) that

f H ϕ,∞,s fin ≤ C( g H ϕ,∞,s fin + ℓ ε 1 H ϕ,∞,s fin + ℓ ε 2 H ϕ,∞,s fin ) ≤ C.
This finishes the proof of (ii) and hence, the proof of Theorem 3.4.

Next we give the proof for Theorem 3.5.

Proof of Theorem 3.5. Suppose that the assumption (i) holds. For any f ∈ H ϕ,q,s fin (R n ), by Theorem 3.4, there exists a finite atomic decomposition f = k j=1 λ j a j , where a j 's are multiples of (ϕ, q, s)-atoms with supported in balls B j 's, such that Λ q ({λ j a j } k j=1 ) = inf λ > 0 :

k j=1 ϕ B j , |λ j | χ B j -1 L ϕ λ ≤ 1 ≤ C f H ϕ .
Recall that, since ϕ is of uniformly upper type γ, there exists a constant C γ > 0 such that (8.6) ϕ(x, st) ≤ C γ s γ ϕ(x, t) for all x ∈ R n , s ≥ 1, t ∈ [0, ∞).

If there exist j 0 ∈ {1, ..., k} such that C γ |λ j 0 | γ ≥ k j=1 |λ j | γ , then The above means that k j=1

|λ j | γ 1/γ ≤ C 1/γ γ Λ q ({λ j a j } k j=1 ) ≤ C f H ϕ .
Therefore, by assumption (i), we obtain that

T f Bγ = T k j=1 λ j a j Bγ ≤ C k j=1 |λ j | γ 1/γ ≤ C f H ϕ .
Since H ϕ,q,s fin (R n ) is dense in H ϕ (R n ), a density argument gives the desired result.

The case (ii) is similar by using the fact that H ϕ,∞,s fin (R n ) ∩ C(R n ) is dense in H ϕ,∞,s fin (R n ) in the quasi-norm • H ϕ , see the lemma below.

We end the paper by the following lemma. Lemma 8.1. Let ϕ be a growth function satisfying uniformly locally dominated convergence condition, and (ϕ, ∞, s) be an admissible triplet. Then, H ϕ,∞,s

fin (R n ) ∩ C ∞ (R n ) is dense in H ϕ,∞,s fin (R n ) in the quasi-norm • H ϕ .
Proof. We take q ∈ (q(ϕ), ∞) and φ ∈ S(R n ) satisfying supp φ ⊂ B(0, 1), R n φ(x)dx = 1. Then, the proof of the lemma is simple since it follows from the fact that for every (ϕ, ∞, s)-atom a supported in ball B(x 0 , r), lim t→0 aa * φ t L q ϕ (B(x 0 ,2r)) = 0 as ϕ satisfies uniformly locally dominated convergence condition.

( 1 +

 1 |f * φ t (y)|, for all x ∈ R n , here and in what follows φ t (x) := t -n φ(t -1 x), withA N = φ ∈ S(R n ) : sup x∈R n |x|) N |∂ α x φ(x)| ≤ 1 for α ∈ N n , |α| ≤ Nwith N = N(n, Φ) taken large enough, belongs to the Orlicz space L Φ (R n ).

Definition 2 . 1 .

 21 We say that ϕ : R n × [0, ∞) → [0, ∞) is a growth function if the following conditions are satisfied. (1) The function ϕ is a Musielak-Orlicz function that is (a) the function ϕ(x, •) : [0, ∞) → [0, ∞) is an Orlicz function for all x ∈ R n , (b) the function ϕ(•, t) is a Lebesgue measurable function for all t ∈ [0, ∞). (2) The function ϕ belongs to A ∞ . (3) The function ϕ is of positive uniformly lower type and of uniformly upper type 1.

Lemma 4 . 1 .

 41 i) Let ϕ be a growth function. Then ϕ is uniformly σ-quasisubadditive on R n × [0, ∞), i.e. there exists a constant C > 0 such that ϕ(x,

Remark 4 . 1 .Lemma 4 . 2 .

 4142 Let us observe that the results stated in Section 3 are invariant under change of equivalent growth functions. By Lemm 4.1, in the future, we always consider a growth function ϕ of positive uniformly lower type, of uniformly upper type 1 (or, equivalently, uniformly quasi-concave), and so that ϕ(x, •) is continuous and strictly increasing for all x ∈ R n . Let ϕ be a growth function. Then i)

m by Lemma 4 . 3 .

 43 This completes the proof. Proof of Theorem 3.1. By Theorem 5.1 and Theorem 5.2, we obtain

Lemma 6 . 1 .

 61 Given a ball B, and {B j } j be a sequence of measurable subsets of B such that lim j→∞ |B j | = 0. Then the following holds lim j→∞ sup t>0 ϕ(B j , t) ϕ(B, t) = 0.

7 .

 7 and Corollary 5.2. As B is arbitrary, the above implies b ∈ BMO ϕ (R n ) and b BM O ϕ ≤ C L (H ϕ ) * . The uniqueness (in the sense b = b if bb = const) of the function b is clear. And thus the proof is finished. The class of pointwise multipliers for BMO(R n )

  a Hardy space of Musielak-Orlicz type related to the Musielak-Orlicz function θ(x, t) = t log(e+|x|)+log(e+t) .

  log(e + 1/(ba)) + sup x∈[a,b] log(e + |x|) ≤ C(| log(ba)/2| + log(e + |a + b|/2)) for all b > 0, a ∈ [-b, b) ⊂ R. However, this follows from the basic two inequalities: log(e + 1/(ba)) ≤ 2(| log(ba)/2| + log(e + |a + b|/2)) and log(e + b) ≤ 5 log(e + b)/2 ≤ 5(| log(ba)/2| + log(e + |a + b|/2)).

For

  the general case R n , by the 1-dimensional result, we obtain log e ir, x i + r]| ≤ C n i=1 (| log r| + log(e + |x i |)) ≤ C(| log r| + log(e + |x|)) where c n = |B(0, 1)|, and sup y∈B(x,r) log(e + |y|) ≤ n i=1 sup y i ∈[x i -r,x i +r] log(e + |y i |) ≤ C n i=1 (| log r| + log(e + |x i |)) ≤ C(| log r| + log(e + |x|)) where x = (x 1 , ..., x n ), y = (y 1 , ..., y n ) ∈ R n . This finishes the proof. Proof of Theorem 3.3. By Theorem 3.1, Theorem 3.2, Proposition 7.1, and Proposition 7.2, we obtain (H log

1 L 1 L

 11 |λ j | γ ) 1/γ ≥ ϕ(B j 0 , χ B j 0 -|λ j | γ ) 1/γ ≥ k j=1 |λ j | γ k j=1 |λ j | γ ϕ(B j , χ B j ϕ ) = 1.

  since θ ∈ A 1 . However, the last inequality is obvious.Conversely, suppose that f is a θ-atom. Similarly, we need to show that there exists a constant C > 0 such that

				1 |B| ) + sup x∈B log(e + |x|) |B|	dx ≤ C
	or, equivalently,
				log(e+ 1 |B| )+sup x∈B log(e+|x|)
	log(e +	|B| |B| )+sup x∈B log(e+|x|) log(e+ 1 |B|	) + sup x∈B log(e + |x|)	|B| ≤ C,
		θ x, C	log(e + 1 |B| ) + sup x∈B log(e + |x|) |B|	dx ≥ 1
	B
	or, equivalently,
	log(e + C	log(e+ 1 |B| )+sup x∈B log(e+|x|) |B| |B| ) + sup x∈B log(e + |x|) |B| )+sup x∈B log(e+|x|) C log(e+ 1	|B| ≥ 1.


Fernand Meyer

Digital morphological curvature ‡ow

Keywords: Mathematical morphology, digital geodesics, digital curvature, level sets, contour, thinning, opening

A digital de…nition of curvatures is given, valid for 2D images in any raster and connectivity. 3 equivalent algorithms are presented for implementing the curvature ‡ow. The …rst two operate on successive thresholds of the image. The third operates on a grey tone image.

Introduction

At ICIP 2009, A. Ciomaga et al presented an elegant paper entitled "Level lines shortening yields an image curvature microscope" [START_REF] Ciomaga | Level lines shortening yields an image curvature microscope[END_REF]. An extended version may be found in the open archive (http://hal.archives-ouvertes.fr/hal-00525297/). Their algorithm …rst extracts all level lines of a digital image. Then it simulates an image evolution by applying to all its level line a curve shortening (alternatively an a¢ ne curve sortening). The evolved image is eventually reconstructed from its evolved level lines. They prove in a paper in preparation that the image reconstructed for the evolved level lines is a viscosity solution of the mean curvature motion (resp. a¢ ne curvature motion). Their paper also gives a thorough historical review of the numerous approaches to the computation and …ltering of curvatures in digital images.

One may wonder if there are morphological tools able to produce similar results. The basic morphological tool, dealing with curvatures, is the opening by a disk B or radius : The set (X) is the union of all disks of radius included in X: If X possesses a piece of contour of a curvature higher than ; then the disk B cannot reach this zone. The dual operator, the closing ' with the same disk copes with the concavities of high curvature. In order to obtain a scale space analysis one applies sequences of an opening followed by a closing, for increasing sizes of disks. These …lters are called alternate sequential …lters (see the corresponding chapter in [START_REF] Serra | chapter Alternating Sequential Filters[END_REF]). Unfortunately, if the object is narrow or possesse narrow isthmuses, being for instance a thin stripe, then the disk B is also unable to enter X: For this reason, as they cannot di¤erenciate between zones of high curvature and narrow objects, alternate sequential …lters do not constitute a "morphological curvature ‡ow".

This paper presents a digital de…nition of curvature based on digital geodesics. It proposes 3 algorithms for evolving the contours according a curvature ‡ow. The last one treats all grey tones at the same time and may easily be extended to grey-tone images. Based on the same framework, many variants are possible, for isotropic or anisotropic …ltering (only in some prede…ned directions) for …ltering only dark or white details, or for inhibiting the …ltering on the zones of high gradient.

2 Digital geodesics and digital curvature 2.1 Digital geodesics.

2D images are de…ned on a raster, and the neighborhood relations are illustrated in the …gure 1. The digital geodesics are the digital shortest paths between two pixels. They derive from the neighborhood relations. Consider two nodes O and x ; if the direction of the vector ! Ox is in between two consecutive directions ! i ;

! j of the raster, then any geodesics between O and x is a chain of vectors

( ! k 1 ; ! k 2 ; ! k 3 ..., ! k n ), where each k i is equal to ! i or ! j :
The geodesics are not unique and belong to a parallelogram as illustrated in 2. The sum of the vectors equal to ! i and the sum of vectors equal to ! j is the same for all geodesics. The contour of a binary images may be decomposed in overlapping maximal geodesics, as we will see below. If a portion of contours between two pixels x and y is not a geodesics it contains curvature zones, which will be detected and submitted to a curvature ‡ow.

Digital curvature

If a portion of curvature is not a geodesic path between 2 pixels x and y; it contains several geodesics with distinct directions following each other, with some overlapping zones. Two successive geodesics may have several edges in common or only a node. Figure 3 shows two geodesics in the hexagonal grid which follow each other. The …rst, of direction (0 and =3) between x and b ; the second of direction (0 and =3) between a and y. The common zone contains the segment ab; containing 3 pixels and 2 edges. This overlapping zone of two geodesics with adjacent directions constitutes a zone of digital curvature. The number of edges of this zone is the measure of the curvature (its radius).

The …gure 4 shows a piece of contour (x; y) made of a geodesics of direction ( =3 and 2 =3) between x and a, followed by a geodesics of direction (0 and =3) between a and y. In this case both geodesics have only the pixel a in common, forming an angular con…gurations, that is a also zone of high curvature. (0,3)

(0,2) (0,1) (0,0) (.) (1)
(0,1) : identity (0,0) : identity

Figure 5: Inventory of the angles between two successive edges of the hexagonal raster and rewriting rules in order to simplify the contour

The principle of the curvature ‡ow

The curvature ‡ow has to suppress in priority the angular zones. Two successive edges can only form a limited number of angles on a digital grid. Figure 5 presents the possible angles in the hexagonal grid ; the …rst two present a curvature which is resorbed by simplifying the contour as shown on the right. The encoding (0; 3) means that two edges in directions 0 and 3 =3 are simply suppressed, leaving just a node. In the second example two edges in directions 0 and 2 =3 are replaced by a single edge in direction =3. The last two con…gurations of …gure 5 are not angular con…gurations as they belong to a geodesics, hence they are not modi…ed. Consider now the …g.6, where the direction of the successive edges is encoded according to …g.1 giving for the portion between a and f the encoding (21221166): It contains a …rst geodesic (212211), in red, between a and d followed by a second (1166), in green, between c and f . Their overlapping zone is (cd) ancoded (11). This portion is suppressed from the contour. The last edge before the suppressed portion is (bc) and the …rst edge after is (de). These two edges of direction 2 and 6 constitute an angular con…guration of type b as illustrated in …gure 5. Hence they are suppressed and replaced by an edge in the median direction 1: Like that one obtains the encoding (2121116), for the contour illustrated on the right part of …gure 6. This new contour possesse a larger overlapping zone of geodesics. However, the …rst geodesic has lost the edge bc and the secon the edge de: For this reason, the curvature of the central zone becomes lower and the curvatures of the adjacent zones become higher. The shortening of the adjacent zones makes them earlier candidates for a curvature ‡ow. This mechanism ensures a balanced treatment of the curvatures all along the contour. Figure 7 shows 3 steps of the simpli…cation of a contour on the square raster and 8 connexity.

Curvature ‡ow of a binary set by rewriting its contour encoding

A binary shape may be represented by its Freeman code, starting from some pixel and encoding the directions of the successive edges one has to follow clockwise in order to reach again the initial pixel. Such an encoding has been used in …g.6 and 9. Figure 8 shows a complete binary shape in hexagonal raster submitted to successive stages of …ltering. The initial image and its encoding is on the left. The coloured segments below the encoding show the maximal geodesics of the contour, and theirs ovelapping zones. The central …gure shows the result of stages of simpli…cation and the right image shows the result of 5 supplementary simpli…cation stages. The …ltering occurs by rewriting rules for the angular points according to the rules presented in the …rst two con…gurations of …gure 5. The zones of high curvature, are then …ltered as explained in the previous section. The curvature is measured by the length of the overlapping zone between two successive and adjacent geodesics. Zones of high curvature are …ltered before zones of low curvature. Figure 9 shows successive simpli…cation of a zone of high curvature. The …rst stage produces an angular point. 3 steps of angular ‡ow are necessary before a zone of higher curvature appears.

Controlling the curvature ‡ow.

The contour simpli…cation occurs by successive traversal and simpli…cation of the contour encoding. First the angular points are …ltered then the zones of higher curvature. The simpli…cation of a zone of higher curvature may produce new angular points, which are to be suppressed as they apppear.

The zones of higher curvature may then be processed according several rules:

One proceeds by increasing measures of curvature : zones of high curvature are suppressed before zones of lower curvature

As the curvature ‡ow, all curvature zones are treated at the same time, but the speed is proportional to the curvature. To this e¤ect, one iterates the simpli…cation of high curvature more frequently as the zones of low curvature.

The curvature ‡ow is an increasing operator

We now prove that the curvature ‡ow is an increasing operator: for two binary sets verifying X Y; applying the same simpli…cation steps to both sets preserves the relation X Y: This property permits to use curvature ‡ow on the successive thresholds of grey-tone images, producing the successive thresholds of the simpli…ed image.

Study of the angular points

Angular points may be convex or concave.

Convex angular points

We suppose X Y and consider a pixel x 2 X: We have to show that if x is suppressed as angular point from Y it is also suppressed as angular point from X: As a matter of fact if the two contour edges adjacent to x form an angle Y in Y; then the contour edges of X incident to X form an angle X Y : As the angular con…gurations are suppressed in the order of increasing angles, when the pixel x is suppressed from Y it has already been suppressed from X: Let us take an example (see …g.10) of a angular pixel b, obtained by a rotation of 2 =3 between the vectors ! ab and ! bc: Concave angular points We have to show that if a pixel is added to X in ordert to suppress a concave angular zone, it will also be added to Y . Due to the contour following inside the particles, there are less concave con…gurations as convex con…gurations. Let us take an example in hexagonal raster and 12 connectivity, illustrated by …gure 11, where the arrows ( ! ab; ! bc) belong to the X and the pixels x and y belong to X: The di¤erent con…gurations are the following:

x 2 Y and y 2 Y : then after adding x and y to X; we still have X Y:

Overlapping of geodesics

Convex zone Figure 6 presents an overlapping zone cd of two geodesics for the set Y: The simpli…cation suppresses this segment, the new contour going directly from b to e: If cd \ X = ?; we still have X Y after the suppression of segment cd: On the contrary if cd \ X 6 = ?; the overlapping zone of geodesics for X is smaller, being of length 1 or even reduced to an angular point. In both cases will the contour of X be simpli…ed before Y; and X Y always holds.

Concave zone

We suppose that the segment cd belongs to X in …g.12. By resorbing the concavity, the contour becomes bxye by suppression of the segment cd and adjunction of the pixels x and y: The possible con…gurations after the …ltering of X are: Figure 12: Case where y belongs to Y and x belongs not.

Ultimate vanishing of the closed curves.

As in the continuous space, the curvature ‡ow transforms each closed curve in a digital circle whose radius becomes smaller before vanishing. Fig. 13 gives an example.

Figure 13: Last steps before the vanishing of a connected particle

Implementation of the curvature ‡ow

We have presented the principle of the curvature ‡ow applied to binary images, whose contour has been encoded. The ‡ow is implemented by local rewriting rules of the contour code. The resulting code can be transformed back into the …ltered binary set. As the operator is increasing, it is also possible to use it on the successive thresholds of a grey tone image.

Discussion: the …ltering part is extremely fast, as it consists in recursive local transformations of the contour code. However, the contour code has …rst to be produced by a contour following algorithm, and after the tranformation, translated back into a binary set. We present now two implementations which directly apply on the images. The …rst on binary images, the second on grey tone images.

Curvature ‡ow by thinning of binary images

We illustrate the algorithm in the horizontal direction, for the hexagonal grid in …g.14. We want to detect an upper boundary segment of n nodes and suppress it. The top …gure presents the initial set. The …gure below shows the result of a thinning with the structuring element on the right. It detects the inside pixels of the upper boundary ‡at zones. The next thinnings detect respectively the left and the right borders of such zones. Taking the union of these three thinnings produces a complete upper border, including the segment we have to detect but also other upper ‡at zones. The upper segment is the only possessing both a left boundary and a right boundary pixels. Reconstructing all upper borders …rst from the left boundary and the result from the right boundary produces the desired result. This algorithm has detected all upper segments, whatever their length. The last steps consists in selecting by their length the segments we are interested in and eliminate all others.

This …rst stage has treated the convex parts in the horizontal direction. We have to apply it on all directions, …rst for the convex parts, and then, by duality on the concave parts.

Illustration

We show the result of …ltering on an extremely noisy image in …g. 4.2). Fig. 4.2 presents the e¤ect of a conventional alternate sequential …lter (a series of increasing pairs of a hexagonal opening followed by a hexagonal closing. Figure 4.2 is the result of a curvature folow suppressing the convexities and concavities successively of sizes 1, 2 and 3. For each size we apply 3 iterations. Figure 4.2. shows the residual image, containing mainly noise, but also zones with high convexities and concavities, as could be expected. However, the …ltering destroys much less image structures than the alternate sequential …lter.

Curvature ‡ow on grey tone images

We now present a method which is able to run the curvature ‡ow for all grey tone levels at the same time ; it works direction after direction, concavities after convexities. It is based on simple grey tone morphological operators operating on the lines of the image. The tools used for the upper convexitites in the horizontal direction are the following:

1. Erosion with a two successive horizontal pixels, centred right (the result being written on the right pixel) 3 last steps for …ltering the upper boundary segments of size 4

The complete algorithms

We have presented the tool for the upper convexities. By rotation of the structuring elements one obtains a tool for the convexities in all 6 directions of the grid. By duality one obtains a tool for the concavities.

It may be used in a symmetrical way for convexities and concavities ; or for convexities alone (resp. concavities alone). It is easy to introduce an unbalance between convexities and concavities. One may iteratie the tool up to idempotence for each size of repeat it only a …xed number of times. By privileging some directions, it may be used for anisotropic …ltering.

Opening Only convexities are …ltered, the concavities being untouched. The operator is applied iteratively for increasing sizes. It may be repeated up to idempotence or for a …xed number of iterations.

Closing Dual operator of the opening for the concavities Alternate sequential …lter For each size one iterates a …ltering for convexities and a …ltering for concavities. The size is progressively increased, after idempotence has been reached for the lower size.

Curvature ‡ow One uses a cumulative version of the …lter where all sizes of segments up to a limit size are treated simultaneously. Like that, as the size of the cumulative …lter increases, the lower sizes are revisited. In fact, any succession of sizes may be used.

Illustration

We illustrate now the …ltering on a series of images, showing the result of intermediate sizes.

Autodual until idempotence and residue of the largest size Autodual …ltering. Each operator is applied on the result of the preceding one, for sizes between 1 and 5.

Each operator is iterated until idempotence.

Residue image between the initial and the last operator.

Edges of the previous image, showing the level line shortening due to the …ltering.

Autodual with 2 iterations for each sizel Autodual …ltering. Each operator is applied on the result of the preceding one, for sizes between 1 and 5.

Each operator is iterated 2 times.

Residue image between the initial and the last operator.

iterations with …ltering of the black details

Filtering of the black details. Each operator is applied on the result of the preceding one, for sizes between 1 and 5. Each operator is iterated 2 times.

Residue image between the initial and the last operator.

iterations with …ltering of the white details

Filtering of the white details. Each operator is applied on the result of the preceding one, for sizes between 1 and 5. Each operator is iterated 2 times.

Residue image between the initial and the last operator.

Gradient inhibited curvature ‡ow

The heat equation is an isotropic di¤usion, smoothing an image in a uniform way. Perona and Malik have introduced the so-called "anisotropic di¤usion" inhibiting the di¤usion accross zones of high gradient. The name "anisotropic" is badly chosen, as the …ltering is the same in all directions. It would be more appropriate to call it "gradient inhibited di¤usion". In our case, up to now,we have presented directional …lters which may be used in an isotropic way when all directions are …ltered in an equal way or in an anisotropic way if one restricts the …ltering to a few directions of the grid.

We now show how to modify the scheme so as to inhibit the …ltering of zones of high gradient. To this e¤ect we apply the grey tone algorithm described just above and at the same time estimate the gradient ; we then modify only the contours in zones of low gradient.

Consider again the algorithm of the preceding section and its result in …gure 4.3.1 the image N ew_segments; representing all segments where the initial image is to be modi…ed. For inhibiting the …ltering on zones of high gradient, one estimates the gradient on each of these N ew_segments and retains only those where the gradient is below some threshold. The following algorithm is the continuation of the algorithm given for grey-tone curvature ‡ow and keeps the same notations. Each operator is iterated until idempotence Residue image between the initial and the last operator.

itérations

22

Autodual …ltering. Each operator is applied on the result of the preceding one, for sizes between 1 and 5 with a constant threshold for the gradient inhibition.

Each operator is iterated 2 times.

Residue image between the initial and the last operator.

Filtering only the dark structures

Filtering of the dark structures. Each operator is applied on the result of the preceding one, for sizes between 1 and 5 with a constant threshold for the gradient inhibition. Each operator is iterated 2 times.

Residue image

between the initial and the last operator.

Filtering only the white structures

Filtering of the white structures. Each operator is applied on the result of the preceding one, for sizes between 1 and 5 with a constant threshold for the gradient inhibition. Each operator is iterated 2 times.

Residue image

between the initial and the last operator.

Conclusion and future work

We have presented a method for implementing the digital curvature ‡ow on 2d images on any grid and connexity. We have illustrated it mainly on the hexagonal raster in 6 connexity. The …ltering by rewriting the contour encoding is fast and elegant as it treats all directions, convexities and concavities at the same time. However it requires encoding and decoding the contours, and works only level by level.

The two other methods treat each direction separately and convexities and concavities also separately. This slows the implementation but has the advantage that the operators may be combined in any order. When only a subset of operators is used, one may implement non isotropic …ltering, with a reduced number of directions or …lter only white or dark details.

By inhibiting the …ltering of zones of high gradient, the …ltering may be made more selective, concentrating on noise and preserving meaningful details.

Finally, as the operators used for …ltering grey tone images work line by line independantly, it would be easy to construct their 3D counterpart for …ltering 3D grey-tone images. The required operators would then operate plane by plane, in all directions of a 3D cristallographic grid.

Figure 2 :

 2 Figure 2: Geodesics between x et y; in the angular sector ( ! 1 ; ! 2 ):

Figure 3 :

 3 Figure 3: Overlapping between a …rst geodesics between x and a and a second between a and y

Figure 4 :

 4 Figure 4: Two successive geodesics having only the pixel a in common

Figure 6 :

 6 Figure 6: Left: overlapping zone of two geodesics. Right : scimpli…cation of the ontour by suppression of a zone of curvature 2.

Figure 7 :

 7 Figure 7: 3 successive simpli…cations of the contour in square raster and 8 connectivity

Figure 8 :

 8 Figure 8: Left: initial image with its contour encoding. The color segments identify the geodesics. Center and right: 2 steps of curvature ‡ow with intermediate steps of simpli…cation of the contour encoding.

Figure 10 :Figure 11 :

 1011 Figure 10: Suppression of a convex angular point.

x 2 Y

 2 but y = 2 Y : the contour for Y forms a sharper concave angular zone ( ! yb; ! bc), which is suppressed for Y before being suppressed for X. The con…guration y 2 Y and x = 2 Y is symmetrical x = 2 Y and y = 2 Y : X and Y having the same neighborhood con…guration are treated likewise.

  If x and y already belong to Y; the relation X Y is still veri…ed. If y 2 Y and x = 2 Y; then the contour of Y follows bzty and constitutes an overlapping of geodesics shorter in Y than in X ; hence it will be resorbed earlier in Y: If both pixels x and y do not belong to Y; and X Y; then X and Y have locally the same boundary and are submitted simultaneously to the same simpli…cation.

Figure 14 :Remark 1

 141 Figure 14: Algorithm for detecting an upper convex boundary based on thinnings.

2 .

 2 Dilation with two successive horizontal pixels centred left 3. Opening by iteration of n erosions followed by n dilations 4. Horizontal grey tone reconstruction 5. Operator using an equilateral triangle with an horizontal upper edge. The maximal value of the upper corners is a¤ected to the lower corner 4.3.1 Detection of the upper convexities. We suppose that the convexities of length n 1 have been suppressed and desire suppressing those of length n: The algorithm is illustrated in …gures 4.3.1 and 4.3.1 for the upper horizontal convexities. We are looking for horizontal segments of length equal to n (or smaller, if one desires to combine the treatment of several sizes). Such segments are lowered by an opening of size n + 1: The algorithm steps are the following: Image = Initial image Opened = horizontal opening of size n + 1 applied to Image Segments = Image for all pixels verifying Opened < Image U pper_border = maximal value of the two upper neighbors of each pixels (tool 5 in the above list) M arkers = (Opened _ U pper_border) ^Segments N ew_segments = reconstruction opening of Segments from the M arkers N ew_Image = result of the replacement of Segments by N ew_segments

  High_value = mean value of Image in each ‡at zone of N ew_segments Low_value = mean value of U pper_border in each ‡at zone of N ew_segments Low_value = Low_value^high_value (only upwards descending gradients interest us) Gradient_estimation = High_value Low_value N ew_segment = N ew_segment when Gradient_estimation is below a given threshold Image = result of the replacement of Segments by N ew_segments 4.4.1 Illustration of the gradient inhibited curvature ‡ow Autodual up to idempotence and residue image 20 Autodual …ltering. Each operator is applied on the result of the preceding one, for sizes between 1 and 5 with a constant threshold for the gradient inhibition. Each operator is iterated until idempotence Residue image between the initial and the last operator. Contours of the preceding …ltered image. 21 Autodual …ltering. Each operator is applied on the result of the preceding one, for sizes between 1 and 5 with a constant threshold for the gradient inhibition.

  All geodesics have the same length ; the two extremal geodesics, at the boundary of the parallelogram regroup all moves in one direction, followed by all moves in the other direction.

	3	2	6	5	4	3	2	2		4	3	2
	4	1	7					1	3	1	5	1
	5	6	8	9 10 11	12	4		6	7	8
	Figure 1: Encoding of the directions for the hexagonal and square grids, in 6,
	12, 4 and 8 connectivity
	In …gure 2 the vector ! xy is between the directions ! i : ! j denotes the scalar product of ! i and ! j , ( ! ! 1 and 1 : ! xy) moves in direction ! 2 of the raster. 1 and ( If ! ! 2 : ! xy) moves in direction ! 2 are necessary for going from x to y: