Harmonic functions, conjugate harmonic functions and the Hardy space $H^1$ in the rational Dunkl setting

In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on a Euclidean space $\mathbb{R}^N$. On the half-space $\mathbb{R}_+\times\mathbb{R}^N$, we consider systems of conjugate $(\partial_t^2+\Delta_{\mathbf{x}})$-harmonic functions satisfying an appropriate uniform $L^1$ condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space $H^1$, can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood-Paley square functions.

One says that u has the L p property if As in the case N = 1, if 1− 1 N < p < ∞ and u 0 (x 0 , x 1 , . . . , x N ) is a harmonic function, there is a system u = (u 0 , u 1 , . . . , u N ) of C 2 functions satisfying (1.1) and (1.2) if and only if belongs to L p (R N ). Here x = (x 1 , . . . , x N ) ∈ R N and similarly x ′ = (x ′ 1 , . . . , x ′ N ). Then u 0 has a limit f 0 in the sense of distributions, as x 0 ց 0, and u 0 is the Poisson integral of f 0 . It turns out that the set of all distributions obtained in this way, which form the so-called real Hardy space H p (R N ), can be equivalently characterized in terms of real analysis (see [12]), namely by means of various maximal functions, square functions or Riesz transforms. An other important contribution to this theory lies in the atomic decomposition introduced by Coifman [5] and extended to spaces of homogeneous type by Coifman and Weiss [7].
The goal of this paper is to study harmonic functions, conjugate harmonic functions, and related Hardy space H 1 for the Dunkl Laplacian ∆ (see Section 2). We shall prove that these objects have properties analogous to the classical ones. In particular the related real Hardy space H 1 ∆ , which can be defined as the set of boundary values of (∂ 2 t + ∆ x )-harmonic functions satisfying a relevant L 1 property, can be characterized by appropriate maximal functions, square functions, Riesz transforms or atomic decompositions. Apart from the square function characterization, this was achieved previously in [2] and [9] in the one-dimensional case, as well as in the product case.

Statement of the results
In this section we first collect basic facts concerning Dunkl operators, the Dunkl Laplacian, and the corresponding heat and Poisson semigroups. For details we refer the reader to [8], [21] and [22]. Next we state our main results.
In the Euclidean space R N , equipped with a scalar product x, y , the reflection σ α with respect to the hyperplane α ⊥ orthogonal to a nonzero vector α is given by A finite set R ⊂ R N \ {0} is called a root system if σ α (R) = R for every α ∈ R. We shall consider normalized reduced root systems, that is, α 2 = 2 for every α ∈ R. The finite group G generated by the reflections σ α is called the Weyl group (reflection group) of the root system. We shall denote by O(x), resp. O(B) the G-orbit of a point x ∈ R N , resp. a subset B ⊂ R N . A multiplicity function is a G-invariant function k : R → C, which will be fixed and ≥ 0 throughout this paper. Given a root system R and a multiplicity function k, the Dunkl operators T ξ are the following deformations of directional derivatives ∂ ξ by difference operators : Here R + is any fixed positive subsystem of R. The Dunkl operators T ξ , which were introduced in [8], commute pairwise and are skew-symmetric with respect to the Ginvariant measure dw(x) = w(x) dx, where Set T j = T e j , where {e 1 , . . . , e N } is the canonical basis of R N . The Dunkl Laplacian associated with R and k is the differential-difference operator ∆ = n j=1 T 2 j , which acts on C 2 functions by where The operator ∆ is essentially self-adjoint on L 2 (dw) (see for instance [1,Theorem 3.1]) and generates the heat semigroup Here the heat kernel h t (x, y) is a C ∞ function in all variables t > 0, x ∈ R N , y ∈ R N , which satisfies h t (x, y) = h t (y, x)> 0 and R N h t (x, y) dw(y) = 1.
Our goal is to study real harmonic functions of the operator The operator L is the Dunkl Laplacian associated with the root system R, considered as a subset of R 1+N under the embedding R ⊂ R N ֒→ R × R N . We say that a system T j u j = 0. In this case each component u j is L-harmonic, i.e., Lu j = 0.
We say that a system u of C 2 real L-harmonic functions on R 1+N + belongs to the Hardy space H 1 if it satisfies both (2.4) and the L 1 condition |u(x 0 , ·)| L 1 (dw) = sup where |u(x 0 , x)| = n j=0 |u j (x 0 , x)| 2 1/2 . We are now ready to state our first main result.
If u ∈ H 1 , we shall prove that the limit f (x) = lim x 0 →0 u 0 (x 0 , x) exists in L 1 (dw) and u 0 (x 0 , x) = P x 0 f (x). This leads to consider the so-called real Hardy space equipped with the norm the nontangential maximal function associated with the Poisson semigroup P t = e −t √ −∆ . According to Theorem 2.5, H 1 ∆ coincides with the following subspace of L 1 (dw) : Moreover, the norms f H 1 ∆ and M P f L 1 (dw) are equivalent. Our task is to prove other characterizations of H 1 ∆ by means of real analysis. A. Characterization by the heat maximal function. Let be the nontangential maximal function associated with the heat semigroup H t = e t∆ and set  Section 8), are bounded operators on L p (dw), for every 1 < p < ∞ (cf. [3]). In the limit case p = 1, they turn out to be bounded operators from H 1 ∆ into H 1 ∆ ⊂ L 1 (dw). This leads to consider the space are equivalent.
D. Characterization by atomic decompositions. Let us define atoms in the spirit of [15]. Given a Euclidean ball B in R N , we shall denote its radius by r B and its Gorbit by O(B). For any positive integer M, let D(∆ M ) be the domain of ∆ M as an (unbounded) operator on L 2 (dw).
Definition 2.12. Let 1 < q ≤ ∞ and let M be a positive integer. A function a ∈ L 2 (dw) is said to be a (1, q, M)-atom if there exist b ∈ D(∆ M ) and a ball B such that In this case, set where the infimum is taken over all representations (2.14). In the one-dimensional case and in the product case considered in [2] and [9], the Dunkl kernel can be expressed explicitly in terms of classical special functions (Bessel functions or the confluent hypergeometric function). Thus its behavior is fully understood, and consequently all kernels involved in the definitions above. In the general case considered in the present paper, no such information is available. Therefore an essential part of our work consists in deriving estimates of the Dunkl kernel, of the heat kernel, of the Poisson kernel, and of their derivatives (see the end of Section 3, Section 4 and Section 5). These estimates, which are in a spirit of analysis on spaces of homogeneous type, allow us to build up the theory of the Hardy space H 1 in the Dunkl setting.
Let us briefly describe the organization of the proofs of the results. Clearly, , which is actually the inclusion H 1 ∆ ⊂ H 1 max,P , is presented in Section 7, see Proposition 7.12. The proof is based on L-subharmonicity of certain function constructed from u (see Section 6). The converse to Proposition 7.12 is proved at the very end of Section 11.

Dunkl kernel, Dunkl transform and Dunkl translations
The purpose of this section is to collect some facts about the Dunkl kernel, the Dunkl transform and Dunkl translations. General references are [8], [21], [22]. At the end of this section we shall derive estimates for the Dunkl translations of radial functions. These estimates will be used later to obtain bounds for the heat kernel and for the Poisson kernel, as well as for their derivatives, and furthermore upper and lower bounds for the Dunkl kernel. We begin with some notation. Given a root system R in R N and a multiplicity function k ≥ 0, let (3.1) γ = α∈R + k(α) and N = N + 2γ. The number N is called the homogeneous dimension, because of the scaling property w(B(tx, tr)) = t N w(B(x, r)).
Thus the measure w is doubling, that is, there is a constant C > 0 such that Moreover, there exists a constant C ≥ 1 such that, for every x ∈ R N and for every R ≥ r > 0,
Finally, let d(x, y) = min σ∈G x−σ(y) denote the distance between two G-orbits O(x) and O(y). Obviously, {y ∈ R N | d(y, x) < r} = O(B(x, r)) and Dunkl kernel. For fixed x ∈ R N , the Dunkl kernel y −→E(x, y) is the unique solution to the system The following integral formula was obtained by Rösler [19] : 1 The symbol ∼ between two positive expressions means that their ratio is bounded between two positive constants.
where µ x is a probability measure supported in the convex hull conv O(x) of the G-orbit of x. The function E(x, y), which generalizes the exponential function e x,y , extends holomorphically to C N × C N and satisfies the following properties : The following properties hold for the Dunkl transform (see [16], [22]): • The Dunkl transform is a topological automorphisms of the Schwartz space S(R N ). • (Inversion formula) For every f ∈ S(R N ) and actually for every f ∈ L 1 (dw) such that F f ∈ L 1 (dw), we have • (Plancherel Theorem) The Dunkl transform extends to an isometric automorphism of L 2 (dw). • The Dunkl transform of a radial function is again a radial function.
Dunkl translations and Dunkl convolution. The Dunkl translation τ x f of a function f ∈ S(R N ) by x ∈ R N is defined by Notice the following properties of Dunkl translations : • each translation τ x is a continuous linear map of S(R N ) into itself, which extends to a contraction on L 2 (dw), • (Commutativity) the Dunkl translations τ x and the Dunkl operators T ξ all commute, The latter formula allows us to define the Dunkl translations τ x f in the distributional sense for f ∈ L p (dw) with 1 ≤ p ≤ ∞. In particular, Finally, notice that τ x f is given by (3.4), if f ∈ L 1 (dw) and F f ∈ L 1 (dw).
The Dunkl convolution of two reasonable functions (for instance Schwartz functions) is defined by Dunkl translations of radial functions. The following specific formula was obtained by Rösler [20] for the Dunkl translations of (reasonable) radial functions f (x) =f ( x ) : Here A(x, y, η) = x 2 + y 2 − 2 y, η = x 2 − η 2 + y − η 2 and µ x is the probability measure occurring in (3.3), which is supported in conv O(x).
In the remaining part of this section, we shall derive estimates for the Dunkl translations of certain radial functions. Let us begin with the following elementary estimates (see, e.g., [3]), which hold for x, y ∈ R N and η ∈ conv O(x) : Hence and, more generally, Similarly, ifθ ∈ C ∞ (R) is an even inhomogeneous symbol of order m ∈ R, i.e., d dx Consider the radial function Notice thatq(x) = c M (1+x 2 ) −M/2 is an even inhomogeneous symbol of order −M. The following estimate holds for the translates q t (x, Proposition 3.9. There exists a constant C > 0 (depending on M) such that Proof. By scaling we can reduce to t = 1. Fix x, y ∈ R N . By continuity, the function y ′ −→ q 1 (x, y ′ ) reaches a maximum K = q 1 (x, y 0 ) ≥ 0 on the ball For every y ′ ∈B, we have Here we have used (3.8) and the elementary estimate Hence if y ′ ∈B ∩ B(y 0 , r) with r = 1 2M . Moreover, as w(B ∩ B(y 0 , r)) ∼ w(B), we have We conclude by using the symmetry q 1 (x, y) = q 1 (y, x) .
Consider next a radial function f satisfying ( 3 ) with M > N and κ ≥ 0. Then the following estimate holds for the translates Corollary 3.10. There exists a constant C > 0 such that Proof. By scaling we can reduce to t = 1. By using (3.5), (3.6) and Proposition 3.9, we get Notice that the space of radial Schwartz functions f on R N identifies with the space of even Schwartz functionsf on R, which is equipped with the norms Proposition 3.12. For every κ ≥ 0, there exist C ≥ 0 and m ∈ N such that, for all even Schwartz functionsψ {1} ,ψ {2} and for all even nonnegative integers ℓ 1 , ℓ 2 , the convolution kernel for every s, t > 0 and for every x, y ∈ R N .
Proof. By continuity of the inverse Dunkl transform in the Schwartz setting, there exists a positive even integer m and a constant C > 0 such that for every even functionf ∈ C m (R) with f Sm < ∞. Consider first the case 0 < s ≤ t = 1. Then S m+ℓ 1 +ℓ 2 s ℓ 1 . According to Corollary 3.10, we deduce that where N = ψ {1} S m+ℓ 1 +ℓ 2 ψ {2} S m+ℓ 1 +ℓ 2 . In the case s = 1 ≥ t > 0, we have similarly The general case is obtained by scaling.

Heat kernel and Dunkl kernel
Via the Dunkl transform, the heat semigroup H t = e t∆ is given by Alternately (see, e.g., [22]) where the heat kernel h t (x, y) is a smooth positive radial convolution kernel. Specifically, for every t > 0 and for every x, y ∈ R N , Upper heat kernel estimates. We prove now Gaussian bounds for the heat kernel and its derivatives, in the spirit of spaces of homogeneous type, except that the metric x − y is replaced by the orbit distance d(x, y). In comparison with (4.2), the main difference lies in the factor t N/2 , which is replaced by the volume of appropriate balls.
for every t > 0 and for every x, y ∈ R N . (b) Hölder bounds : for any nonnegative integer m, there are constants C, c > 0 such that for every t > 0 and for every x, y, y ′ ∈ R N such that y−y ′ < √ t . (c) Dunkl derivative : for any ξ ∈ R N and for any nonnegative integer m, there are constants C, c > 0 such that for all t > 0 and x, y ∈ R N .
(d) Mixed derivatives : for any nonnegative integer m and for any multi-indices α, β, there are constants C, c > 0 such that, for every t > 0 and for every x, y ∈ R N , for every t > 0 and for every x, y ∈ R N .
Proof. The proof relies on the expression and on the properties of A(x, y, η).
(a) Consider first the case m = 0. By scaling we can reduce to t = 1. On the one hand, we use (3.6) to estimate On the other hand, it follows from Proposition 3.9 and Corollary 3.10 that for any fixed c > 0 . Hence By differentiating (4.8) and by using (4.9), we deduce that We conclude by using the case m = 0.
By differentiating (4.8) and by using (3.8) and (4.4), we estimate In order to conclude, notice that under the assumption y− y ′ < √ t and let us show that, for every c > 0, there exists C ≥ 1 such that As long as d(x, y) = O( √ t ), all expressions in (4.12) are indeed comparable to 1. On the other hand, if d(x, y) ≥ √ 32 t , then (c) By symmetry, we can replace T ξ,x by T ξ,y . Consider first the contribution of the directional derivative in T ξ,y . By differentiating (4.8) and by using (4.10) and (4.4), we estimate as above

Consider next the contributions
of the difference operators in T ξ,y . If | α, y | > t/2 , we use (4.4) and estimate separately each term in (4.13). If | α, y | ≤ t/2 , we estimate again In the last step we have used (4.11) and (4.12), which hold as 2t A(x, y, η) . Firstly, by differentiating (4.8) and by using (4.14), we obtain by using (4.15) and by symmetry, we get . We conclude by using (4.4).
Lower heat kernel estimates. We begin with an auxiliary result.

Lemma 4.16. Letf be a smooth bump function on
.
Proof. All claims follow from (3.5) and (3.6). Let us prove the last one. On the one hand, by translation invariance, On the other hand, This proves (4.17) with c 1 = w(B(0,1/2)) |G| . Proposition 4.18. There exist positive constants c 2 and ε such that Proof. By scaling it suffices to prove the proposition for t = 2. According to Lemma 4.16, applied toh 1 f ( 4 ), there exists c 3 > 0 and, for every This estimate holds true around y(x), according to (4.5), Specifically, there exists 0 < ε < 1 (independent of x) such that By using the semigroup property and the symmetry of the heat kernel, we deduce that By using the fact that the balls B(y(x), ε), B(x, 1), B(x, √ 2) have comparable volumes and by using again (4.5), we conclude that for all x, y ∈ R N sufficiently close.
A standard argument, which we include for the reader's convenience, allows us to deduce from such a near on diagonal estimate the following global lower Gaussian bound.
for every t > 0 and for every x, y ∈ R N .
Proof. We resume the notation of Proposition 4.18. Assume that x − y 2 /t ≥ 1 and . By using the semigroup property, Proposition 4.18 and the behavior of the ball volume, we estimate We conclude by symmetry.
By combining (4.4) and (4.20), we obtain in particular the following near on diagonal estimates. Notice that the ball volumes w(B(x, √ t )) and w(B(y, √ t )) are comparable under the assumptions below.
for every t > 0 and x, y ∈ R N such that x−y ≤ c √ t .
Estimates of the Dunkl kernel. According to (4.1), the heat kernel estimates (4.4) and (4.20) imply the following results, which partially improve upon known estimates for the Dunkl kernel. Notice that x can be replaced by y in the ball volumes below.
for all x, y ∈ R N . In particular, • for every ε > 0, there exist C ≥ 1 such that for all x, y ∈ R N satisfying x − y < ε ; • there exist c > 0 and C > 0 such that for all λ≥ 1 and for all x, y ∈ R N with x = y = 1.

Poisson kernel in the Dunkl setting
The Poisson semigroup P t = e −t √ −∆ is subordinated to the heat semigroup H t = e t∆ by (2.2) and correspondingly for their integral kernels This subordination formula enables us to transfer properties of the heat kernel h t (x, y) to the Poisson kernel p t (x, y). For instance, The following global bounds hold for the Poisson kernel and its derivatives.
for every t > 0 and for every x, y ∈ R N . (b) Dunkl gradient : for every ξ ∈ R N , there is a constant C > 0 such that for all t > 0 and x, y ∈ R N . (c) Mixed derivatives : for any nonnegative integer m and for any multi-index β, there is a constant C ≥ 0 such that, for every t > 0 and for every x, y ∈ R N , Moreover, for any nonnegative integer m and for any multi-indices β, β ′ , there is a constant C ≥ 0 such that, for every t > 0 and for every x, y ∈ R N , . Notice that, by symmetry, (5.6) holds also with T ξ,x instead of T ξ,y .
Proof. (a) The Poisson kernel bounds (5.5) are obtained by inserting the heat kernel bounds (4.4) and (4.20) in the subordination formula (5.1). For a detailed proof we refer the reader to [11,Proposition 6].
for every positive integer m and for every nonnegative integer β. By using (3.5), (3.6), (5.2), (5.3) and (5.9), we estimate and similarly for every positive integer m. Finally (5.8) is deduced from (5.7) by using the semigroup property. More precisely, by differentiating by using (5.7) and by symmetry, we obtain Notice the following straightforward consequence of the upper bound in (5.5) : where M HL denotes the Hardy-Littlewood maximal function on the space of homogeneous type (R N , x−y , dw). Likewise, (4.4) yields Observe that the Poisson kernel is an approximation of the identity in the following sense.
Proposition 5.11. Given any compact subset K ⊂ R N , any r > 0 and any ε > 0, there exists t 0 = t 0 (K, r, ε) > 0 such that, for every 0 < t < t 0 and for every x ∈ K, Proof. Let K be a compact subset of R N and let r, ε > 0. Fix In particular, for every 0 < t < t 0 and for every x ∈ B(x 0 , r/4), we have A straightforward compactness argument allows us to conclude.
Corollary 5.13. Let f be a bounded continuous function on R N . Then its Poisson integral u(t, x) = P t f (x) is also bounded and continuous on [0, ∞) × R N .
Remark 5.15. The assertion of Proposition 5.11 remains valid with the same proof if
Consequently, for every α ∈ R, (6.10) We shall need the following auxiliary lemma.
with real entries a ij one has where A HS denotes the Hilbert-Schmidt norm of A.
Proof. The lemma was proved in [9]. For the convenience of the reader we present a short proof. The inequality is known for trace zero symmetric A (see Stein and Weiss [26, Lemma 2.2]). By homogeneity we may assume that A HS = 1. Assume that the inequality does not hold. Then there is ε > 0 such that for every n > 0 there is Thus there is a subsequence n s such that A ns → A, A HS = 1 and But then A = A * and trA = 0, and so, A 2 ≥ A 2 HS . This contradicts the already known inequality.
We now state and prove the main theorem of Section 6, which is the analog in the Dunkl setting of a Euclidean subharmonicity property (see [24, Chapter VII, Section 3.1]) and which was proved in the product case in [9, Proposition 4.1]. Recall (2.3) that L = T 2 0 + ∆. Theorem 6.12. There is an exponent 0 < q < 1 which depends on k such that if u = (u 0 , u 1 , . . ., u N ) ∈ C 2 satisfies the Cauchy-Riemann equations (2.4), then the function |F | q is L-subharmonic, that is, L(|F | q )(t, x) ≥ 0 on the set where |F | > 0.

Harmonic functions in the Dunkl setting.
In this section we characterize certain L-harmonic functions in the half-space R 1+N + by adapting the classical proofs (see, e.g., [12], [24] and [26]). Let us first construct an auxiliary barrier function.
Barrier function. For fixed δ > 0 let v 1 , . . ., v s ∈ R N be a set of vectors of the unit sphere in S N −1 = {x ∈ R N : x = 1} which forms a δ-net on S N −1 . Let M, ε > 0. Define By Corollary 4.21, Maximum principle and the mean value property. As we have already remarked in Section 2, the operator L is the Dunkl-Laplace operator associated with the root system R as a subset of R 1+N = R × R N . We shall denote the element of R 1+N by x = (x 0 , x). The associated measure will be denoted by w. Clearly, dw(x) = w(x) dx dx 0 . Moreover, E(x, y) = e x 0 y 0 E(x, y). We shall slightly abuse notation and use the same letter σ for the action of the group G in R 1+N , so σ(x) = σ(x 0 , x) = (x 0 , σ(x)).
The following weak maximum principle for L-subharmonic functions was actually proved in Theorem 4.2 of Rösler [18]. Proof. The proof is identical to that of Stein [24]. Clearly, the Poisson integral of a bounded function is bounded and L-harmonic. To prove the converse assume that u is L-harmonic and bounded, so |u| ≤ M. Set f n (x) = u( 1 n , x) and u n (x 0 , x) = P x 0 f n (x). Then U n (x 0 , x) = u(x 0 + 1 n , x) − u n (x 0 , x) is L-harmonic, |U n | ≤ 2M, continuous on [0, ∞) × R N , and U n (0, x) = 0. We shall prove that U n ≡ 0. Fix (y 0 , y) ∈ R 1+N + . Set and consider the function U on the closure of the set Ω = (0, ε −1 ) × B(0, R), with ε > 0 small and R large enough. Then U is L-harmonic in Ω, continuous onΩ, and positive on the boundary of the ∂Ω. Thus, by the maximum principle, U is positive inΩ, so Letting ε → 0 we obtain U n (y 0 , y) ≥ 0. The same argument applied to −u gives −U n (y 0 , y) ≥ 0, so U n ≡ 0, which can be written as Clearly |f n | ≤ M, so by the *-weak compactness, there is a subsequence n j and f ∈ L ∞ (R N ) such that for ϕ ∈ L 1 (dw), we have lim j→∞ ϕ(y)f n j (y) dw(y) = ϕ(y)f (y) dw(y).

If p = 1 then u is a Poisson integral of a bounded measure ω if and only if u is Lharmonic and
(7.10) sup Proof. Assume that either (7.9) or (7.10) holds. Then, by Theorem 7.4, for every ε > 0 (7.11) sup Set f n (x) = u( 1 n , x). From Theorem 7.5 we conclude that u( 1 n + x 0 , x) = P x 0 f n (x). Moreover, there is a subsequence n j such that f n j converges weakly-* to f ∈ L p (dw) (if 1 < p < ∞) or to a measure ω (if p = 1). In both cases u is the Poisson integral either of f or ω. If additionally u * ∈ L 1 (dw), then the measure ω is absolutely continuous with respect to dw.
Proof of a part of Theorem 2.5. We are now in a position to prove a part of Theorem 2.5, which is stated in the following proposition. The converse is proven at the very end of Section 11 (see Proposition 11.27).
Proposition 7.12. Assume that u ∈ H 1 k . Then x). Then, by Theorem 7.4, the L-harmonic function u j,ε (x 0 , x) is bounded and continuous on the closed set [0, ∞) × R N . In particular f j,ε ∈ L ∞ ∩ L 1 (dw) ∩ C 2 . By Theorem 7.5, It is not difficult to conclude using (5.8) Let 0 < q < 1 be as in Theorem 6.12 and p = q −1 > 1. Observe that the function So, by Theorem 6.12 and the maximum principle (see Theorem 7.3), Then, by (7.14) and (5.10), as ε → 0 and the convergence is monotone, we use the Lebesgue monotone convergence theorem and get (7.13). From Theorem 7.8 and Proposition 7.12 we obtain the following corollary.
Corollary 7.15. If u ∈ H 1 k , then there are f j ∈ L 1 (dw), j = 0, 1, . . ., N, such that They are bounded operators on L 2 (dw). Clearly, and the convergence is in L 2 (dw) for f ∈ L 2 (dw). It follows from [3] that R j are bounded operators on L p (dw) for 1 < p < ∞.
Our task is to define R j f for f ∈ L 1 (dw). To this end we set It is not difficult to check that if ϕ ∈ T k , then ϕ ∈ C 0 (R N ) and R j ϕ ∈ C 0 (R N ) ∩ L 2 (dw). Moreover, for fixed y ∈ R N the function p t (x, y) belongs to T k . Now R j f for f ∈ L 1 (dw) is defined in a weak sense as a functional on T k , by Proof of Theorem 2.11. Assume that f ∈ L 1 (dw) is such that R j f belong to L 1 (dw) for j = 1, 2, . . ., N.
Thus u ∈ H 1 k and We turn to prove the converse. Assume that f 0 ∈ H 1 ∆ . By the definition of H 1 ∆ there is a system u = (u 0 , u 1 , . . ., u N ) ∈ H 1 k such that f 0 (x) = lim x 0 →0 u 0 (x 0 , x) (convergence in L 1 (dw)). Set f j (x) = lim x 0 →0 u j (x 0 , x), where limits exist in L 1 (dw) (see Corollary 7.15). We have u j (x 0 , x) = P x 0 f j (x). It suffices to prove that R j f 0 = f j . To this end, for ε > 0, let f j,ε (x) = u j (ε, x), u j,ε (x 0 , x) = u j (x 0 + ε, x). Then f j,ε ∈ L 1 (dw) ∩ C 0 (R N ). In particular f j,ε ∈ L 2 (dw). Set g j = R j f 0,ε , v j (x 0 , x) = P x 0 g j (x). Then v = (u 0,ε , v 1 , . . ., v N ) satisfies the Cauchy-Riemann equations (2.4). Therefore, In this section we show that the atomic space H 1 (1,q,M ) with M > N is contained in the Hardy space H 1 ∆ and there exists C = C k,q,M such that (1,q,M ) . According to Theorem 2.11, it is enough to show that R j f ∈ L 1 (dw) and R j f L 1 (dw) ≤ C f H 1 (1,q,M ) . By the definition of the atomic space there is a sequence a j of (1, q, M) atoms and λ i ∈ C such that f = i λ i a i and i |λ i | ≤ 2 f H 1 (1,q,M ) . Observe that the series converges in L 1 (dw), hence R j f = i λ j R j a j in the sense of distributions. Therefore it suffices to prove that there is a constant C > 0 such R j a L 1 (dw) ≤ C for every a being a (1, q, M)-atom. Our proof follows ideas of [15]. Let b ∈ D(∆ M ) and B(y 0 , r) be as in the definition of (1, q, M) atom. Since R j is bounded on L q (dw), by the Hölder inequality, we have R j a L 1 (O(B(y 0 ,4r))) ≤ C.
In order to estimate R j a on the set O (B(y 0 , 4r)) c we write Further, using (4.6) with m = 0 together with (3.2), we get y 0 , r)) .

Maximal functions
Let Φ(x) be a radial continuous function such that . Then, by Corollary 3.10, If a = 1, then we simply write M Φ . We say that Definition 10.1. For a > 0, λ > N, and a function u(t, x) denote The tent space N is defined by {u(t, x) : u Na = u * a L 1 (dw) < ∞}. If a = 1, then we write N , u N , and u * (cf. (2.6)).
Lemma 10.2. There are constants C, C λ , c λ > 0 such that The proofs are the same as those in [25,Chapter II] and [13, page 114] .
If Ω ⊂ R N is an open set, then the tent over Ω is given by The space N admits the following atomic decomposition (see [25]).

Definition 10.5. A function A(t, x) is an atom for
Clearly, A N ≤ 1 for every atom A for N . Moreover, every u ∈ N can be written as u = j λ j A j , where A j are atoms for N , λ j ∈ C, and j |λ j | ≤ C u N .
From Proposition 5.4 and the definition of N atoms we conclude that Q t/2 A j (t, x) N ≤ C.
Calderón reproducing formula. Fix a positive integer m sufficiently large. Let Θ ∈ C m (R) be an even function such that Θ S m < ∞ (see (3.11)). Set Θ(x) =Θ( x ). Assume that R N Θ(x) dw(x) = 0. The Plancherel theorem for the Dunkl transform implies . By duality, Then Ψ is radial and real-valued, Moreover, for n = 0, 1, 2, . . ., and f ∈ L 2 (dw) we have the Calderón reproducing formulae: where the integrals converge in the L 2 -norm. Moreover, by Lemma 10.2, there is a We are in a position to state the main results of this section.
Theorem 11.4. For every positive integer M there is a constant C M > 0 such that every element f ∈ H 1 max,H ∩ L 2 (dw) = H 1 max,P ∩ L 2 (dw) can be written as Proof. The proof is a straightforward adaptation of [27] with the difference that tents are now constructed with respect to the orbit distance d(x, y). We include details for the convenience of readers unfamiliar with [27]. More experienced readers may skip the proof and jump to Lemma 11.21. Without loss of generality, we may assume that M is an even integer > 2 N.
The next lemma will help us to remove the extra assumption that f ∈ L 2 (dw). Lemma 11.21. Assume that f ∈ H 1 max,P . Then P t f ∈ L 2 (dw) for every t > 0 and (11.22) lim Proof. Proposition 5.4 implies that P t f ∈ L 2 (dw). To prove (11.22) we follow, e.g., [10, proof of (6.5)]. First observe that there is a constant C > 0 such that for every A > 0 and t > 0 we have sup x−y <s,s>At To see (11.23) fix z ∈ R N . For s > At, thanks to (5.5), we have w(B(z, s + d(x, z))) −1 , (11.24) which implies (11.23).
In order to finish the proof of (11.22) assume that f ∈ H 1 max,P . Then .
Having Lemma 11.21 we are in a position to complete the proof of the atomic decomposition of H 1 max,P functions. This is stated in the theorem below. Theorem 11.25. There is a constant C > 0 such that every function f ∈ H 1 max,P can be written as where a j are (1, ∞, M)-atoms, Proof. Take a sequence t n → 0, n = 0, 1, . . ., such that n=1 g n , with convergence in L 1 (dw). The functions g n ∈ L 2 (dw) ∩ H 1 max,P , so, by Theorem 11.4 they admit atomic decompositions into (1, ∞, M)-atoms with the required control of their atomic norms.
The following theorem is a direct consequence of Lemma 11.2 and Theorem 11.25.
Theorem 11.26. There is a constant C > 0 such that every element f ∈ H 1 max,H can be written as We are in a position to complete the proof of Theorem 2.5, by proving the following proposition, which is the converse to Proposition 7.12.

Inclusion
Assume that a is a (1, q, 1)-atom associated with a set B = σ∈G B(σ(y 0 ), r). Then there is a function b ∈ D(∆) such that a = ∆b, supp ∆ j b ⊂ B, ∆ j b L q (dw) ≤ r 2−2j w(B) 1 q −1 , j = 0, 1. Set u(t, x) = e t 2 ∆ a(x). Observe that (see (2.6) for the definition of u * ). Thus, by the doubling property of the measure dw(x) dx and the Hölder inequality, We turn to estimate u * (x) on d(x, y 0 ) > 8r. Clearly, (12.1) Recall that b L 1 (dw) ≤ r 2 and note that Hence, In order to estimate J 2 , we observe from (4.4) that for t > d(x, y) and d(y, y 0 ) < r < t we have d ds h s (x ′ , y) s=t 2 ≤ C t 2 w(B(y 0 , d(y 0 , x))) Consequently,

Square function characterization
In this section we prove Theorem 2.9. More precisely we show that the atomic Hardy space H 1 (1,2,M ) coincides with the Hardy space defined by the square function (2.8) with −∆ . This is achieved by mimicking arguments in [15]. The proof for Q t = t 2 (−∆) e t 2 ∆ is similar.
Tent spaces T p 2 on spaces of homogeneous type. The square function characterization of the Hardy space H 1 (1,2,M ) can be related with the so called tent space T 1 2 .
The tent spaces on Euclidean spaces were introduced in [6] and then extended on spaces of homogeneous type (see, e.g. [23]). For more details we refer the reader to [25]. For a measurable function F (t, x) on (0, ∞) × R N let Clearly, by the doubling property, Remark 13.3. By (10.8) and (13.2) the operator π Ψ maps continuously the space T 2 The tent space T 1 2 on the space of homogenous type admits the following atomic decomposition (see, e.g., [23]).
2 if and only if there are sequences A j of T 1 2 -atoms and λ j ∈ C such that where the convergence is in T 1 2 norm and a.e. The Hölder inequality immediately gives that there is a constant C > 0 such that for every function A(t, x) being a T 1 2 -atom one has A T 1 2 ≤ C. Observe that for f ∈ L 1 (dw) the function F (t, x) = Q t f (x) is well defined. Moreover, AF (x) = Sf (x) and Sf L 1 (dw) = F T 1 2 .
Remark 13.5. According to the proof of atomic decomposition of T 1 2 presented in [23], the function λ j A j can be taken of the form λ j A j (x, t) = χ S j (x, t), where S j are disjoint, R N +1 + = S j , and S j is contained in a tent B j . So, if F ∈ T 1 2 ∩ T 2 2 , then F can be decomposed into atoms such that F (t, x) = j λ j A j (x, t) and the convergence is in T 1 2 , T 2 2 , and pointwise.
Lemma 13.6. The map (P s F )(t, x) = p s (x, y)F (t, y) dw(y) is bounded on T 1 2 . Moreover, there is a constant C > 0 independent of s > 0 such that P s F T 1 2 ≤ C F T 1 2 . Proof. Let F (t, x) = j λ j A j (t, x) be an atomic decomposition of F ∈ T 1 2 as described above. Since p s (x, y) ≥ 0, it suffices to prove that there is a constant C > 0 such that for every atom A of T 1 2 . To this end let B = B(x 0 , r) be a ball associated with A. Obviously, P s |A|(t, x ′ ) = 0 for t > r.
Lemma 13.8. The family P s forms approximate of identity in T 1 2 , that is, lim s→0 P s F − F T 1 2 = 0.
Proof. According to Lemma 13.6, it suffices to establish that for every A being a T 1 2atom we have (13.9) lim s→0 P s A − A T 1 2 = lim s→0 A(P s A − A) L 1 (dw) = 0.
Let A be such an atom and let B = B(x 0 , r) be its associated ball. To prove (13.9) it suffices to consider 0 < s < r.
Proposition 13.11. Let M be a positive integer. Assume that for f ∈ L 1 (dw) the function F (t, x) = Q t f (x) belongs to T 1 2 . Then there are λ j ∈ C and a j being (1, 2, M)atoms such that f = j λ j a j , and j |λ j | ≤ C F T 1 2 .
The constant C depends on M but it is independent of f . Proof. We start our proof under the additional assumption f ∈ L 2 (dw). Then F (t, x) = Q t f (x) ∈ T 1 2 ∩ T 2 2 . The proof in this case is the same as that of [15,Theorem 4.1]. The only difference is to control support of functions ∆ s b j . For the convenience of the reader we provide its sketch.
Let F = j λ j A j be a T 1 2 atomic decomposition of the function Q t f (x) as it is described in Remark 13.5. In particular, j |λ j | ≤ C Sf L 1 (dw) . Let Ψ be chosen such that ∞ 0 Ψ t Q t dt t forms a Calderón reproducing formula, with Ψ = ∆ M +1 Ψ {1} , where Ψ {1} is a radial C ∞ function supported by B(0, 1/4). By Remark 13.3 we have (13.12) f = π Ψ F = j λ j π Ψ A j and the series converges in L 2 (dw). Let B j = B(y j , r j ) be a ball associated with A j .
To remove the additional assumption f ∈ L 2 (dw) we use Lemma 13.8 together with the fact that P s f ∈ L 2 (dw) for f ∈ L 1 (dw), and apply the same arguments as those in the proof of Theorem 11.25.