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The Hardy space H1 in the rational Dunkl setting

This paper consists in a first study of the Hardy space H1 in the rational Dunkl setting. Following Uchiyama's approach, we characterizee H1 atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H1. These results are proved here in the one-dimensional case and in the product case.


Introduction
Dunkl theory is a far reaching generalization of Euclidean Fourier analysis, which includes most special functions related to root systems, such as spherical functions on Riemannian symmetric spaces. It started in the late eighties with Dunkl's seminal article [7] and developed extensively afterwards. We refer to the lecture notes [18] for the rational Dunkl theory, to the lecture notes [15] for the trigonometric Dunkl theory, and to the books [4,11] for the generalized quantum theories. This paper deals with the real Hardy space H 1 in the rational Dunkl setting, where the underlying space is of homogeneous type in the sense of Coifman-Weiss. In such a setting, the theory of Hardy spaces goes back to the seventies [6,12]. Here we follow Uchyama's approach [25] and we characterize the Hardy space H 1 in two ways, by means of the heat maximal operator and atomically. The first characterization, which requires precise heat kernel estimates, has lead us to a seemingly new observation, namely that the heat kernel has a rather slow decay in certain directions and is in particular not Gaussian in the present setting (see Remark 2.4). The second characterization is used to prove a Fourier multiplier theorem for H 1 .
Throughout the paper we shall restrict to the one-dimensional case and to the product case. This restriction is due to our present lack of knowledge in general about the behavior of the Dunkl kernel on the one hand and about generalized translations on the other hand.
After this informal introduction, let us introduce some notation and state our main results. On R n we consider the Dunkl operators dv v k−1 (1− v) k e −2xy v 1 F 1 (k; 2k +1; −2 xy) (see for instance [18,Example 2.34]). Here 1 F 1 (a; b; z) is the confluent hypergeometric function, which is also known as the Kummer function and denoted by M (a, b, z). Notice that E(x, y) = e x,y if all multiplicities k j vanish.
Let us first define the Hardy space H 1 by means of the heat maximal operator. The Dunkl laplacian is the infinitesimal generator of the heat semigroup e t L (t > 0), which acts by linear self-adjoint operators on L 2 (R n , dµ) and by linear contractions on L p (R n , dµ), for every 1 ≤ p ≤ ∞, where From this point of view, the Hardy space H 1 consists of all functions f ∈ L 1 (R n , dµ) whose maximal heat transform belongs to L 1 (R n , dµ) and the norm is given by Let us turn next to the atomic definition of the Hardy space H 1 . Notice that R n , equipped with the Euclidean distance d(x, y) = |x − y| and with the measure µ, is a space of homogeneous type in the sense of Coifman-Weiss (see Appendix A). Recall that an atom is a measurable function a : R n → C such that By definition, the atomic Hardy space H 1 atom consists of all functions f ∈ L 1 (R n , dµ) which can be written as f = ℓ λ ℓ a ℓ , where the a ℓ 's are atoms and ℓ |λ ℓ | < +∞, and the norm is given by where the infimum is taken over all atomic decompositions of f . Our first main result is the following theorem.
Theorem 1.8. The spaces H 1 and H 1 atom coincide and their norms are equivalent, i.e., there exists a constant C > 0 such that The Fourier transform in the Dunkl setting is given by It is an isometric isomorphism of L 2 (R n , dµ) onto itself and the inversion formula reads Notice that, if all multiplicities k j vanish, then (1.9) boils down to the classical Fourier transform Our second main result is the following Hörmander type multiplier theorem (see [10] for the original multiplier theorem on L p spaces).
Theorem 1.10. Let χ = χ(ξ) be a smooth radial function on R n such that for some ε > 0, then the multiplier operator is bounded on the Hardy space H 1 and Here W σ 2 (R n ) denotes the classical L 2 Sobolev space on R n , whose norm is given by Notice that the multiplier m is continuous and bounded, as N 2 + ε > n 2 . The theory of classical real Hardy spaces in R n originates from the study of holomorphic functions of one variable in the upper half-plane. We refer the reader to the original works of Stein-Weiss [22], Burkholder-Gundy-Silverstein [3] and Fefferman-Stein [9]. An important contribution to this theory lies in the atomic decomposition introduced by Coifman [5] and extended to spaces of homogeneous type by Coifman-Weiss [6] (see also [12]). More information can be found in the book [21] and references therein.
Our paper is organized as follows. Section 2 is devoted to the heat kernel in dimension 1. There we analyze its behavior thoroughly and we remove a small part, in order to get Gaussian estimates similar to the Euclidean setting. These results are extended to the product case in Section 3. Section 4 is devoted to the proof of Theorem 1.8 and Section 5 to the proof of Theorem 1.10. Section 6 consists of 3 appendices. Appendix A contains information about the measure of balls, which is used throughout the paper. Appendices B and C are devoted to so-called folklore results in connection with Uchiyama's Theorem, which have been used for instance in [8].
This paper results from two independent research works, which were carried out by the first and third authors, respectively by the second and fourth authors, and which have been merged into a joint article.

Heat kernel estimates in dimension 1
Consider first the one-dimensional Dunkl kernel E(x, y) = E k (x, y). As the case k = 0 is trivial, we may assume that k > 0.
(c) E(x, y) has the following symmetry and rescaling properties : (d) For every y ∈ C, x → E(x, y) is an eigenfunction of the Dunkl operator . Proof. The first four properties are known to hold in general. In dimension 1, they can be also deduced from the explicit expression (1.3), as does (e). As already observed in [20,Section 2] (see also [18,Example 5.1]), the asymptotics of E(x, y) at infinity follow from the asymptotics of the confluent hypergeometric function, which read, let say for 0 < a < b, as z → +∞ and [1, (13.5.1)] or [14, (13.7.2)]).
Consider next the one-dimensional heat kernel (c) h t (x, y) has the following symmetry and rescaling properties : (e) The heat kernel has the following global behavior : and the following asymptotics : The following gradient estimates hold for the heat kernel : Proof. The first five properties follow from the expression (2.2) and from Lemma 2.1. Let us turn to the proof of (f). By differentiating (2.2) with respect to y and by using the well-known formula d dz 1 F 1 (a; b; z) = a b 1 F 1 (a +1; b +1; z) (see for instance [1, (13.4.8)] or [14, (13.3.15)]), we get for every t > 0 and x ∈ R. Observe in particular that the heat kernel has no global Gaussian behavior and decays rather slowly in certain directions. This phenomenon is even more striking in the product case Let us eventually introduce a variant of the heat kernel with a Gaussian behavior. Given two smooth bump functions χ 1 and consider the smooth cutoff function and the truncated heat kernel Remark 2.5. The truncated heat kernel H t (x, y) inherits the following properties of the heat kernel h t (x, y) : Theorem 2.6. The following estimates hold for the truncated heat kernel H t (x, y).
(a) On-diagonal estimate : (c) Gradient estimate : (d) Lipschitz estimates : with the following improvement, if |y−y ′ | ≤ 1 2 |x−y | : Here c denotes some positive constant and the ball measure has the following behavior, according to Appendix A : Proof. As far as (a), (b), (c) are concerned, the case x = 0 follows immediately from the previous heat kernel estimates. Thus we may assume that x = 0 and reduce furthermore to x = 1 by rescaling.
(a) is immediate : Let us next prove (b).
• Subcase 1.1. Assume that t is bounded above, say t ≤ 1 2 . Then • Case 4. Assume that y ≤ −t (< 0) and that y stays away from −1, say y / The proof of (c) follows the same pattern. To begin with, observe that the derivative of the cut-off is bounded and vanishes unless y ∈ −3, −2 ∪ − 1 2 , 0 and t ≤ 1. According to the subcases 1.1, 4.2 and 4.3 above, the contribution of Thus it remains for us to estimate the contribution of Then Eventually, (d) is an immediate consequence of (c). For every y ′′ ∈ [ y, y ′ ], we have indeed Moreover, if |y −y ′ | ≤ 1 2 |x−y |, then |x−y ′′ | ≥ |x−y | − |y −y ′′ | ≥ |x−y | − |y −y ′ | ≥ 1 2 |x−y |, Remark 2.7. Contrarily to h t (x, y), H t (x, y) is not symmetric in the space variables x, y . Nevertheless, according to the following result, we may replace µ(B(x, √ t )) by µ(B(y, √ t )) in the estimates (b), (c) and in the second estimate (d).
Lemma 2.8. For every ε > 0, there exists C > 0 such that Proof. By rescaling (see Appendix A), we can reduce to the case t = 1. The estimate is obvious if x and y are bounded or if |x|/|y| is bounded from above. In the remaining case, let say when |x| ≥ 1+ 2|y|, we have |x| ≤ |x− y| + |y| ≤ |x− y| + 1 2 |x|, hence |x| ≤ 2 |x− y|. Furthermore, as |x−y| ≥ |x| − |y| ≥ 1, we have |x| ≤ 2 (x−y) 2 . Thus Next proposition, which will be used in the proof of Theorem 1.8, shows that the truncated heat kernel H t (x, y) captures the main features of the heat kernel h t (x, y). Proposition 2.9. The maximal operator Proof. It suffices to check that The case y = 0 is trivial, as χ t (x, 0) and hence Q t (x, 0) vanish, for every t > 0 and x ∈ R. Consider next the case y ∈ R * , which reduces to y = 1 by rescaling. Then χ t (x, 1) and Q t (x, 1) vanish, unless t < 9 and −3 < x < − 1 3 , and in this range (see Proposition 2.3)

Heat kernel estimates in the product case
According to (1.5) and (1.2), the heat kernel in the product case splits up into onedimensional heat kernels : By expanding t (x j , y j ) and P t (x, y) is the sum of all possible products t (x j , y j ), and at least one factor p (j) and similarly for the other product kernels. The following estimates follow from the onedimensional case (see Theorem 2.6 and Remark 2.7).

Theorem 3.2.
(a) On-diagonal estimate : for every t > 0 and for every x, y ∈ R n . (c) Gradient estimate : for every t > 0 and x, y ∈ R n . (d) Lipschitz estimates : for every t > 0 and x, y, y ′ ∈ R n , with the following improvement, if |y−y ′ | ≤ 1 2 |x−y| : Let us turn to the analog of Proposition 2.9 in the product case.

Proposition 3.3. The maximal operator
Proof. We will show again that but the proof will be more involved in the product case than in the one-dimensional case. Let us begin with some observations. First of all, by using the symmetries and by interchanging variables, we may reduce to products of the form where 1 ≤ n ′ ≤ n and 0 ≤ y 1 ≤ . . . ≤ y n ′ . Next we may assume that, for every 1 ≤ j ≤ n ′ , because otherwise χ t (x j , y j ) and hence Q (j) t (x j , y j ) vanish. Eventually, by rescaling, we may reduce to the case y 1 = 1. Consequently, t is bounded by x 2 1 < 9 y 2 1 = 9 and each factor Q . Thus, on the one hand, the integral is bounded, uniformly in y ′ . On the other hand, let us prove the uniform boundedness of when n ′′ = n − n ′ > 0. For this purpose, let us deduce from the Gaussian estimate Assume first that |x ′′ − y ′′ | ≥ √ t with 0 < t < 9. Then, by using (6.4), Assume next that 0 < |x ′′ − y ′′ | ≤ √ t (≤ 3). Then, by using again (6.4), By summing up over j ∈ Z, we obtain the uniform boundedness of I ′′ (y ′′ ).

Proof of Theorem 1.8
Theorem 1.8 relies on the following result due to Uchiyama [25].
Theorem 4.1. Assume that a set X is equipped with • a quasi-distance d i.e. a distance except that the triangular inequality is replaced by the weaker condition • a measure µ whose values on quasi-balls satisfy r A ≤ µ( B (x, r)) ≤ r ∀ x ∈ X , ∀ r > 0 , • a continuous kernel K r (x, y) ≥ 0 such that, for every r > 0 and x, y, y ′ ∈ X,
Here A ≥ 1 and δ > 0. Then the following definitions of the Hardy space H 1 (X) are equivalent : • Maximal definition : H 1 (X) consists of all functions f ∈ L 1 (X) such that belongs to L 1 (X) and the norm f H 1 is comparable to K * f L 1 .
• Atomic definition : H 1 (X) consists of all functions f ∈ L 1 (X) which can be written as f = ℓ λ ℓ a ℓ , where the a ℓ 's are atoms ( 1 ) and ℓ |λ ℓ | < +∞, and the norm f H 1 is comparable to the infimum of ℓ |λ ℓ | over all such representations. Going back to X = R n , equipped with the Euclidean distance d(x, y) = |x − y| and the measure (1.4), set d(x, y) = inf µ(B) ∀ x, y ∈ R n , where the infimum is taken over all closed balls B containing x and y, and where t = t(x, r) is defined by µ(B (x, √ t )) = r. In Appendixes B and C, we check that these data satisfy the assumptions of Uchiyama's Theorem with δ = 1 N . Actually the conditions in Theorem 4.1 are obtained up to constants and they can be achieved by considering suitable multiples of µ and K r (x, y). Thus the conclusion of Uchiyama's Theorem hold for the quasidistance d and for the maximal operator K * .
On the one hand, d and d define the same Hardy space H 1 , as balls and quasi-balls are comparable. Let us elaborate. For every x, y ∈ R n and t > 0, we have where r = µ(B (x, √ t )). The first implication is an immediate consequence of the definition of d and the second one is obtained by combining Lemma 6.6.(b) in Appendix B with (6.4) in Appendix A. Hence there exists a constant c > 0 such that and these sets have comparable measures, according to Appendix A.
On the other hand, the maximal operators K * and H * coincide and they define the same Hardy space H 1 as the heat maximal operator h * , according to Propositions 2.9 and 3.3. Indeed, for every f ∈ L 1 (R n , dµ), the integrals differ at most by a multiple of f L 1 , which is itself controlled by either integral above, as h t (x, y) dµ(y) and H t (x, y) dµ(y) are approximations of the identity.
In conclusion, the atomic Hardy space H 1 associated with Euclidean balls coincide with the Hardy space H 1 defined by the heat maximal operator h * .

Proof of Theorem 1.10
The proof of Theorem 1.10 requires some weighted estimates in Dunkl analysis, which are well-known in the Euclidean setting corresponding to k = 0. Let us first prove a weak analog of the Euclidean estimate Lemma 5.1. For every ℓ ∈ N and r > 0, there is a constant C = C ℓ,r > 0 such that for every f ∈ C ℓ (R n ) with supp f ⊂ B(0, r). 1 Recall that an atom is a measurable function a : X → C such that a is supported in a quasi-ball B , Proof. By using the Riemann-Lebesgue lemma for the Fourier transform (1.9), we get The last expression is bounded by f C ℓ as, by induction on ℓ, supp(D ℓ j f ) ⊂ B(0, r) and Corollary 5.2. For every ℓ ∈ N, r > 0 and ε > 0, there is a constant C = C ℓ,r,ε > 0 such that Proof. This result is deduced from Lemma 5.1, by using on the left hand side the finiteness of the integral and on the right hand side the Euclidean Sobolev embedding theorem.
Proof. Let χ ∈ C ∞ c (R n ). Following an argument due to Mauceri-Meda [13], this result is obtained by interpolation between the L 2 estimate which is deduced from Plancherel's formula, and the following estimate for ℓ ∈ N large, which is deduced from Corollary 5.2 : By using the Cauchy-Schwartz inequality, we deduce eventually the following result.
We conclude by estimating Lemma 5.6. For every δ > 0, there is a constant C > 0 such that, for every y ∈ R n and r > 0, is the orbit of the ball B(y, r) under the group generated by the reflections (1.1).
Proof. As R n O(y, r) is contained in the union of the sets we have As |z| ≥ |z j | ≥ ||x j | − |y j || > n −1/2 r when x ∈ A j and z ∈ I x,y , the latter expression is bounded above by We conclude by using the uniform estimate Thus it remains for us to show that For this purpose, let us introduce a dyadic partition of unity on the Dunkl transform side. More precisely, given a smooth radial function ψ on R n such that Then e −t|ξ| 2 m(ξ) = ℓ∈Z m t,ℓ (2 −ℓ ξ). Consider the convolution kernel Lemma 5.9.
(a) On the one hand, for every 0 ≤ δ < ε, we have Proof. On the one hand, as Lemma 5.1 yields the estimate which holds for any d ∈ N and which is uniform in t > 0 and ℓ ∈ Z. On the other hand, Corollary 5.4 yields the estimate which holds uniformly in ℓ ∈ Z. By resuming the proof of Lemma 5.5, we deduce that We reach our first conclusion by rescaling and by using Lemma 5.6 : Let us turn to the proof of (b). This time we factorize and accordingly On the one hand, by resuming the proof of (5.10), we get On the other hand, h(x, y) = (τ −y h)(x) is the heat kernel at time t = 1, which satisfies R n dµ(x) |h(x, y) − h(x, y ′ )| |y − y ′ | ∀ y, y ′ ∈ R n , according to next lemma. After rescaling, we reach our second conclusion : Lemma 5.11. The following gradient estimate holds for the heat kernel : Proof. We can reduce to the one-dimensional case and moreover to t = 1 by rescaling. It follows from our gradient estimates for the heat kernel in dimension 1 (see Proposition 2.3) that  1) dµ(y) F t,ℓ (x, y) a(y) Then (5.8) follows from Lemma 5.9.  Recall that k 1 , . . . , k n ≥ 0 and N = n + n j=1 2k j . On R n , equipped with the Euclidean distance, the product measure (1.4) dµ(x) = dµ 1 (x 1 ) . . . dµ n (x n ) = |x 1 | 2k 1 . . . |x n | 2kn dx 1 . . . dx n has the following rescaling properties : In particular, µ is doubling, i.e., Let us prove (6.3) and (6.4). In dimension n = 1, we have µ(B(x, r)) = |x|+r |x|−r dy |y| 2k .
The product case follows from the one-dimensional case, since the ball B(x, r) and the cube Q(x, r) = n j=1 B(x j , r) have comparable measures. More precisely, we have with µ(Q(x, r √ n )) ≍ µ(Q(x, r)) ≍ r n n j=1 (|x j |+ r) 2k j .

Appendix B : Distances.
The following result, which is used in Section 4, is certainly known among specialists. We include nevertheless a proof, for lack of reference and for the reader's convenience. Lemma 6.6. Let (X, d, µ) be a metric measure space such that balls have finite positive measure and satisfy the doubling property, i.e., ∃ C > 0, ∀ x ∈ X, ∀ r > 0, µ(B(x, 2 r)) ≤ C µ(B(x, r)) .

Set
d(x, y) = inf µ(B), where the infimum is taken over all closed balls B containing x and y . Then (a) d is a quasi-distance, (b) d(x, y) ≍ µ(B(x, d(x, y))) ∀ x, y ∈ X, Moreover, if the measure µ has no atoms and µ(X) = +∞, then (c) µ( B(x, r)) ≍ r, for every x ∈ X and r > 0, where B(x, r) denotes the closed quasi-ball with center x and radius r.
Proof. Let us first prove (b). Set R = d(x, y). On the one hand, we have d(x, y) ≤ µ(B(x, R)), as x and y belong to B(x, R). On the other hand, if x and y belong to a ball B = B(z, r), then R ≤ 2 r, hence B(x, R) ⊂ B(z, 3r) and µ(B(x, R)) ≤ µ(B(z, 3r)) ≍ µ(B(z, r)). By taking the infimum over all balls B containing both x and y , we conclude that µ(B(x, R)) d(x, y). Let us next prove (a). For every x, y, z ∈ X, we have d(x, y) ≤ d(x, z) + d(z, y). Assume that r = d(x, z) ≥ d(z, y). Then x, y ∈ B(z, r). By using (b), we conclude that d(x, y) ≤ µ(B(z, r) ≍ d(z, x) ≤ max { d(x, z), d(z, y)} ≤ d(x, z) + d(z, y) .
Here we have used our additional assumptions. Let x ∈ X and r > 0. On the one hand, for every y ∈ B(x, r), we have µ(B(x, d(x, y)) ≍ d(x, y) ≤ r. Hence R = sup {d(x, y)| y ∈ B(x, r)} < +∞ .

Appendix C : Kernel bounds.
Recall from Section 4 that the kernels K r (x, y) and H t (x, y) are related by where r = µ(B (x, √ t )). In this appendix, we check that the Gaussian estimates of H t (x, y) in Theorem 3.2 imply the estimates of K r (x, y) required in Uchiyama's Theorem (Theorem 4.1). This result is certainly well-known among specialists. We include nevertheless a proof, for lack of reference and for the reader's convenience.
According to Appendices A and B, we may consider the quasi-distance d on R n associated with the Euclidean distance d(x, y) = |x−y| and the product measure (1.4). The on-diagonal lower estimate (6.7) K r (x, x) ≥ C 1 r is an immediate consequence of Theorem 3.2.(a). For every δ > 0, the upper estimate