Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case A_n
aa r X i v : . [ m a t h . R T ] D ec Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case A n P. Graczyk and P. Sawyer Abstract
In this article, we consider the radial Dunkl geometric case k = 1 corresponding to flatRiemannian symmetric spaces in the complex case and we prove exact estimates for thepositive valued Dunkl kernel and for the radial heat kernel.Dans cet article, nous consid´erons le cas g´eom´etrique radial de Dunkl k = 1 correspon-dant aux espaces sym´etriques riemanniens plats dans le cas complexe et nous prouvons desestimations exactes pour le noyau de Dunkl `a valeur positive et pour le noyau de chaleurradial. Key words
Punkl 33C67, 43A90, 53C35
MSC (2010)
Thanks
The authors thank Laurentian University, Sudbury and the D´efimaths programme of the R´egiondes Pays de la Loire for their financial support.
Finding good estimates of Dunkl heat kernels is a challenging and important subject, developedrecently in [1]. Establishing estimates of the heat kernels is equivalent to estimating the Dunklkernel as demonstrated by equation (3) below.In this paper we prove exact estimates in the W -radial Dunkl geometric case of multiplicity k = 1, corresponding to Cartan motion groups and flat Riemannian symmetric spaces with theambient group complex G , the Weyl group W and the root system A n .We study for the first time the non-centered heat kernel, denoted p Wt ( X, Y ), on Riemanniansymmetric spaces and we provide its sharp estimates. Exact estimates were obtained in [2] in thecentered case Y = 0 for all Riemannian symmetric spaces.We provide exact estimates for the spherical functions ψ λ ( X ) in the two variables X, λ when λ is real and, consequently, for the heat kernel p Wt ( X, Y ) in the three variables t, X, Y .We recall here some basic terminology and facts about symmetric spaces associated to Cartanmotion groups.Let G be a semisimple Lie group and let g = k ⊕ p be the Cartan decomposition of G . Werecall the definition of the Cartan motion group and the flat symmetric space associated withthe semisimple Lie group G with maximal compact subgroup K . The Cartan motion group isthe semi-direct product G = K ⋊ p where the multiplication is defined by ( k , X ) · ( k , X ) = LAREMA, UFR Sciences, Universit´e d’Angers, 2 bd Lavoisier, 49045 Angers cedex 01,France,[email protected] Department of Mathematics and Computer Science, Laurentian University, Sudbury, Canada P3E 2C6,[email protected] k k , Ad ( k )( X ) + X ). The associated flat symmetric space is then M = p ≃ G /K (the actionof G on p is given by ( k, X ) · Y = Ad ( k )( Y ) + X ).The spherical functions for the symmetric space M are then given by ψ λ ( X ) = Z K e λ ( Ad ( k )( X )) dk where λ is a complex linear functional on a ⊂ p , a Cartan subalgebra of the Lie algebra of G . Toextend λ to X ∈ Ad ( K ) a = p , one uses λ ( X ) = λ ( π a ( X )) where π a is the orthogonal projectionwith respect to the Killing form (denoted throughout this paper by h· , ·i ). Note that in [6, 7, 8], λ is replaced by i λ .Throughout this paper, we usually assume that G is a semisimple complex Lie group. Thecomplex root systems are respectively A n for n ≥ p consists of the n × n hermitianmatrices with trace 0), B n for n ≥ p = i so (2 n + 1)), C n for n ≥ p = i sp ( n ))and D n for n ≥ p = i so (2 n )) for the classical cases and the exceptional root systems E , E , E , F and G .The radial heat kernel is considered with respect to the invariant measure µ ( dY ) = π ( Y ) dY on M , where π ( Y ) = Q α> α ( Y ).Note also that in the curved case M = G/K , the spherical functions for the symmetric space M are then given by φ λ ( e X ) = Z K e ( λ − ρ ) H ( e X k ) dk where ρ is the half-sum of the roots counted with their multiplicities and H ( g ) is the abeliancomponent in the Iwasawa decomposition of g : g = k e H ( g ) n . We will be developing a sharp estimate for the spherical function ψ λ ( X ). We introduce thefollowing useful convention. We will write f ( t, X, λ ) ≍ g ( t, X, λ )in a given domain of f and g if there exists constants C > C > t , X and λ such that C f ( t, X, λ ) ≤ g ( t, X, λ ) ≤ C g ( t, X, λ ) in the domain of consideration.We conjecture the following global estimate for the spherical function in the complex case. Conjecture 2.1.
On flat Riemannian symmetric spaces with complex group G , we have ψ λ ( X ) ≍ e h λ,X i Q α> (1 + α ( λ ) α ( X )) , λ ∈ a + , X ∈ a + . Remark 2.2.
Recall that, denoting δ ( X ) = Q α> sinh α ( X ) , we have φ λ ( e X ) = π ( X ) δ / ( X ) ψ λ ( X ) . (1)2 ince δ / ( X ) ≍ e ρ ( X ) π ( X ) / Q α> (1 + α ( X )) in the complex case, Conjecture 2.1 thereforebecomes φ λ ( e X ) ≍ e ( λ − ρ )( X ) Y α> α ( X )1 + α ( λ ) α ( X ) (2) in the curved complex case.Let us compare the estimate (2) we conjecture for φ λ with the one obtained in [9], cf. also[13]. The estimates in [9] apply in all the generality of hypergeometric functions of Heckman andOpdam. The authors show that there exists constants C ( λ ) > , C ( λ ) > such that C ( λ ) e ( λ − ρ )( X ) Y α> ,α ( λ )=0 (1 + α ( X )) ≤ φ λ ( e X ) ≤ C ( λ ) e ( λ − ρ )( X ) Y α> ,α ( λ )=0 (1 + α ( X )) . Given (1) , corresponding estimates clearly also hold in the flat case for ψ λ ( X ) . The interest ofour result, in the case A n , lies in the fact that our estimate is universal in both λ and X . The results of [9, 13] and our estimates in the A n case strongly suggest that the Conjecture2.1 is true for any complex root system.Note that asymptotics of ψ λ ( t X ) when λ and X are singular and t → ∞ were proven in [4]for all classical complex root systems and the systems F and G .Consider the relationship between the Dunkl kernel E k ( X, Y ) and the Dunkl heat kernel p t ( X, Y ), as given in [10, Lemma 4.5] p t ( X, Y ) = 12 γ + d/ c k t − d − γ e −| X | −| Y | t E k (cid:18) X, Y t (cid:19) , (3)where γ is the number of positive roots and the constant c k is the Macdonald–Mehta–Selbergintegral. The formula (3) remains true for the W -invariant kernels p Wt and E W . In the geometriccases k = , W -invariant formula (3) translates in a similar relationshipbetween the spherical function ψ λ and the heat kernel p Wt ( X, Y ): p Wt ( X, Y ) = 12 γ + d/ c k t − d − γ e −| X | −| Y | t ψ X (cid:18) Y t (cid:19) . (4)A simple direct proof of (4) for k = 1 is given in [4, Remark 2.9].Equation (4) and Conjecture 2.1 bring us to an equivalent conjecture for the heat kernel p Wt ( X, Y ). Conjecture 2.3.
We have p Wt ( X, Y ) ≍ t − d e −| X − Y | t Q α> ( t + α ( X ) α ( Y )) . Consider also the relationship between the heat kernel p Wt ( X, Y ) and the heat kernel ˜ p Wt ( X, Y )in the curved case. We have˜ p Wt ( X, Y ) = e −| ρ | t π ( X ) π ( Y ) δ / ( X ) δ / ( Y ) p Wt ( X, Y ) . (5)3his relation follows directly from the fact that the respective radial Laplacians and radial mea-sures are π − L a ◦ π and π ( X ) dX in the flat case and δ − / ( L a − | ρ | ) ◦ δ / and δ ( X ) dX in thecurved case ( L a stands for the Euclidean Laplacian on a ).In the curved complex case, Conjecture 2.3 becomes˜ p Wt ( X, Y ) ≍ e −| ρ | t t − d e − ρ ( X + Y ) Y α> (1 + α ( X )) (1 + α ( Y ))( t + α ( X ) α ( Y ) e −| X − Y | t . Remark 2.4.
In [5], sharp estimates of W -invariant Poisson and Newton kernels in the complexDunkl case were obtained, by exploiting the method of construction of these W -invariant kernels byalternating sums. When a root system Σ acts in R d , the sharp estimates of [5] have the commonform K W ( X, Y ) ≍ K R d ( X, Y ) Q α> ( | X − Y | + α ( X ) α ( Y )) , X, Y ∈ a + , (6) where K W ( X, Y ) is the W -invariant kernel in Dunkl setting and K R d ( X, Y ) is the classical kernelon R d . Let us observe a common pattern in the appearance of the classical kernels K R d and ofproducts of roots α ( X ) α ( Y ) in formulas (6) and of the Fourier kernel e h λ,X i and the classicalGaussian heat kernel and of products α ( λ ) α ( X ) in the estimates given in Conjecture 2.1 andConjecture 2.3. We start with a practical result.
Proposition 2.5.
Let α i be the simple roots and let A α i be such that h X, A α i i = α i ( X ) for X ∈ a .Suppose X ∈ a + and w ∈ W \ { id } . Then we have Y − w Y = r X i =1 a wi ( Y ) | α i | A α i (7) where a wi is a linear combination of positive simple roots with non-negative integer coefficients foreach i .Proof. Refer to [5].
Remark 2.6.
Note that a wi ( Y ) / | α i | is bounded by C max k | α k ( Y ) | where C is a constant de-pending only on w ∈ W and, ultimately, on W . Corollary 2.7.
Let Y ∈ a + and w ∈ W . Consider the decomposition (7) of Y − wY . If a wk ( Y ) = 0 then α k appears in a wk , i.e. a wk = P ri =1 n i α i with n k > .Proof. Refer to [5].
Proposition 2.8.
Let δ > . Suppose α i ( λ ) α j ( X ) ≤ δ for all i , j . Then ψ λ ( X ) ≍ e λ ( X ) (theconstants involved only depend on δ ). roof. Let K ( X, Y ) be the kernel of the Abel transform. Recall that K ( X, Y ) dY is a probabilitymeasure supported on C ( X ), the convex envelope of the orbit W · X . Notice that e w min λ ( X ) ≤ ψ λ ( X ) = Z C ( X ) e λ ( Y ) K ( X, Y ) dY ≤ e λ ( X ) (8)where w min is the element of the Weyl group giving the minimum value of w λ ( X ). Now, usingProposition 2.5 and Remark 2.6 with Y = λ , we see that for any w ∈ We λ ( X ) ≥ e w λ ( X ) = e h wλ − λ,X i e h λ,X i = r Y i =1 e − awi ( λ ) | αi | α i ( X ) e h λ,X i ≥ r Y i =1 e − C (max k α k ( λ )) α i ( X ) e h λ,X i ≥ r Y i =1 e − C δ e h λ,X i . Remark 2.9.
This case and this method apply for any radial Dunkl case; it suffices to replace K ( X, Y ) dY by the so-called R¨osler measure µ X ( dY ) in the integral in (8) , see [11]. Proposition 2.10.
A spherical function ψ λ ( X ) on M is given by the formula ψ λ ( X ) = π ( ρ )2 γ π ( λ ) π ( X ) X w ∈ W ǫ ( w ) e h wλ,X i , (9) where ρ = P α ∈ Σ + m α α = P α ∈ Σ + α and γ = | Σ + | is the number of positive roots (refer to [8,Chap. IV, Proposition 4.8 and Chap. II, Theorem 5.35]). Proposition 2.11.
Suppose α ( λ ) α ( X ) ≥ (log | W | ) / for all α > . Then ψ λ ( X ) ≍ e λ ( X ) π ( λ ) π ( X ) . We are assuming here that | α i | ≥ for each i .Proof. Suppose w ∈ W is not the identity. In that case, a wi ( λ ) is not equal to 0 for some i . By Proposition 2.5 with y = λ and Corollary 2.7, λ ( X ) − w λ ( X ) ≥ a wi ( λ ) α i ( X ) / | α i | ≥ α i ( λ ) α i ( X ) ≥ log | W | . Each term e h wλ,X i in the alternating sum (9) corresponding to w = id isbounded by e − log | W | e λ ( X ) = e λ ( X ) / | W | . Hence, since only half the terms in the sum are negative, | W | e λ ( X ) ≥ X w ∈ W ǫ ( w ) e h wλ,X i ≥ e λ ( X ) − | W | e λ ( X ) / | W | = 12 e λ ( X ) . The conjecture in the case of the root system A n We will prove the conjecture in the case of the root system of type A . Theorem 3.1.
In the case of the root system of type A n in the complex case, we have ψ λ ( e X ) ≍ e h λ,X i Q i Formula (11) represents the action of the root system A n − on R n . If we assume P nk =1 x k = 0 = P nk =1 λ k , we have then the action of the root system A n − on R n − . We can alsoconsider the action of A n − on any R m with m ≥ n − by considering formula (9) and deciding onwhich entries x k , the Weyl group W = S n acts. These considerations do not affect the conclusionof Theorem 3.1. A n Before proving the conjecture in the case A n , we will prove an interesting “factorization”. Proposition 3.4. For n ≥ , consider the root system A n on R n +1 . Let λ, X ∈ a + ⊂ R n +1 and X ′ = [ X , . . . , X n ] . Define I ( n ) = I ( n ) ( λ ; X ) = Z x n x n +1 Z x n − x n · · · Z x x Z x x e − λ ( X ′ − Y ) Y i Since u/ (1 + u ) is an increasing function, we clearly have I ( n ) ≤ Z x n x n +1 Z x n − x n · · · Z x x Z x x e − λ ( X ′ − Y ) Y i 12 ( x i − y j ) ( λ i − λ j )1 + ( x i − y j ) ( λ i − λ j ) . B ( n ) k = Z ( x k + x k +1 ) / x k +1 e − ( λ k − λ n +1 ) ( x k − y k ) k − Y j =1 ( x j − y k ) ( λ j − λ k )1 + ( x j − y k ) ( λ j − λ k ) dy k and note that I ( n ) k = A ( n ) k + B ( n ) k .Now, using the change of variable 2 w = x k − y k , we have B ( n ) k = 2 Z ( x k − x k +1 ) / x k − x k +1 ) / e − λ k − λ n +1 ) w k − Y j =1 ( x j − x k + 2 w ) ( λ j − λ k )1 + ( x j − x k + 2 w ) ( λ j − λ k ) dw ≤ Z ( x k − x k +1 ) / x k − x k +1 ) / e − λ k − λ n +1 ) w k − Y j =1 ( x j − x k + w ) ( λ j − λ k )1 + ( x j − x k + w ) ( λ j − λ k ) dw ≤ Z ( x k − x k +1 ) / e − ( λ k − λ n +1 ) w k − Y j =1 ( x j − x k + w ) ( λ j − λ k )1 + ( x j − x k + w ) ( λ j − λ k ) dw = 4 A ( n ) k , where the last equality comes from the change of variable w = x k − y k in the expression for A ( n ) k .Therefore I ( n ) k = A ( n ) k + B ( n ) k ≤ A ( n ) k . The result follows.The next proposition gives an inductive way of estimating I ( n +1) , knowing I ( n ) and I ( n − . Proposition 3.5. Consider the root system A n +1 on R n +2 . Let λ, X ∈ a + ⊂ R n +2 . Assume α ( X ) ≥ α n +1 ( X ) . Then I ( n +1) ( λ ; X ) ≍ I ( n ) ( λ , . . . , λ n , λ n +2 ; x , . . . , x n +1 ) ( x − x n +1 )( λ − λ n +1 )1 + ( x − x n +1 )( λ − λ n +1 ) I ( n ) ( λ , . . . , λ n +1 , λ n +2 ; x , . . . , x n +2 ) I ( n − ( λ , . . . , λ n , λ n +2 ; x , . . . , x n +1 ) . Proof. We start with an outline of the proof.(i) I ( n +1) is estimated by a product of n + 1 factors I ( n +1) k ( λ ; X ).(ii) The product of the first n factors I ( n +1)1 ( λ ; X ), . . . , I ( n +1) n ( λ ; X ) give an estimate of the term I ( n ) ( λ , . . . , λ n , λ n +2 ; X ′ ) by Proposition 3.4.(iii) In the last factor I ( n +1) n +1 ( λ ; X ), we “draw off” one term from under the integral, using the addi-tional hypothesis α ( X ) ≥ α n +1 ( X ). The remaining integral corresponds to I ( n ) n ( λ , . . . , λ n +2 ; x , . . . , x n +2 ).(iv) The last factor I ( n ) n of I ( n ) is estimated by I ( n ) /I ( n − , up to a change of variables (we re-usethe idea of (ii)). 8ince x n +2 ≤ y n +1 ≤ x n +1 and x n +1 − x n +2 ≤ x − x , we get x − x n +1 ≤ x − y n +1 ≤ x − x n +2 ≤ x − x n +1 ) and we have I ( n +1) n +1 ≍ Z x n +1 x n +2 e − ( λ n +1 − λ n +2 ) ( x n +1 − y n +1 ) ( x − y n +1 ) ( λ − λ n +1 )1 + ( x − y n +1 ) ( λ − λ n +1 ) n Y j =2 ( x j − y n +1 ) ( λ j − λ n +1 )1 + ( x j − y n +1 ) ( λ j − λ n +1 ) dy n +1 ≍ ( x − x n +1 )( λ − λ n +1 )1 + ( x − x n +1 )( λ − λ n +1 ) Z x n +1 x n +2 e − ( λ n +1 − λ n +2 ) ( x n +1 − y n +1 ) n Y j =2 ( x j − y n +1 ) ( λ j − λ n +1 )1 + ( x j − y n +1 ) ( λ j − λ n +1 ) dy n +1 . Hence, noting that I ( n +1)1 ( λ ; X ) · · · I ( n +1) n ( λ ; X ) ≍ I ( n ) ( λ , . . . , λ n , λ n +2 ; X ′ ), we have I ( n +1) ( λ ; X ) ≍ I ( n ) ( λ , . . . , λ n , λ n +2 ; X ′ ) ( x − x n +1 ) ( λ − λ n +1 )1 + ( x − x n +1 ) ( λ − λ n +1 ) Z x n +1 x n +2 e − ( λ n +1 − λ n +2 ) ( x n +1 − y n +1 ) n Y j =2 ( x j − y n +1 ) ( λ j − λ n +1 )1 + ( x j − y n +1 ) ( λ j − λ n +1 ) dy n +1 . Finally, Z x n +1 x n +2 e − ( λ n +1 − λ n +2 ) ( x n +1 − y n +1 ) n Y j =2 ( x j − y n +1 ) ( λ j − λ n +1 )1 + ( x j − y n +1 ) ( λ j − λ n +1 ) dy n +1 = Q nk =1 R x k +1 x k +2 e − ( λ k +1 − λ n +2 ) ( x k +1 − y k +1 ) Q k − j =1 ( x j +1 − y k +1 ) ( λ j +1 − λ k +1 )1+( x j +1 − y k +1 ) ( λ j +1 − λ k +1 ) dy k +1 Q n − k =1 R x k +1 x k +2 e − ( λ k +1 − λ n +2 ) ( x k +1 − y k +1 ) Q k − j =1 ( x j +1 − y k +1 ) ( λ j +1 − λ k +1 )1+( x j +1 − y k +1 ) ( λ j +1 − λ k +1 ) dy k +1 = I ( n ) ( λ , . . . , λ n +1 , λ n +2 ; x , . . . , x n +2 ) I ( n − ( λ , . . . , λ n , λ n +2 ; x , . . . , x n +1 ) . Remark 3.6. When n = 1 , the result of Proposition 3.5 remains valid if we set I (0) = 1 . We now prove our main result. Proof of Theorem 3.1. We use induction on the rank. In the case of A , we have ψ λ ( e X ) = e λ ( x + x ) ( x − x ) − Z x x e ( λ − λ ) y dy = e λ ( x + x ) ( x − x ) − e ( λ − λ ) x − e ( λ − λ ) x λ − λ = e λ x + λ x − e − ( λ − λ ) ( x − x ) ( λ − λ ) ( x − x ) ≍ e λ x + λ x 11 + ( λ − λ ) ( x − x )9ince 1 − e − u ≍ u/ (1 + u ) for u ≥ A r , 1 ≤ r ≤ n , n ≥ 1. Using (11) and the inductionhypothesis, we have for r = 1, . . . , n + 1 and λ, X in positive Weyl chamber in R r +1 π ( X ) π ( λ ′ ) e − λ ( X ) ψ λ ([ x , . . . , x r , x r +1 ])= r ! π ( λ ′ ) e − λ ( X ) e λ r +1 P r +1 k =1 x k Z x r x r +1 · · · Z x x ψ λ ( e Y ) Y i In [1, Theorems 4.1 p. 2372 and 4.4, p. 2377] the following estimates were proven for the heatkernel p t ( X, Y ) in the Dunkl setting on R n . There exists positive constants c , c , C and C suchthat for all X, Y ∈ a + C e − c | X − Y | /t min { w ( B ( X, √ t )) , w ( B ( Y, √ t )) } ≤ p t ( X, Y ) ≤ C e − c | X − Y | /t max { w ( B ( X, √ t )) , w ( B ( Y, √ t )) } (14)where w is the W -invariant reference measure (in our paper w = π ( X ) dX ) and the w -volume ofa ball satisfies the estimate ([1, p. 2365]) w ( B ( X, r )) ≍ r n Y α> ( r + α ( X )) k ( α ) . The same estimates follow for p Wt ( X, Y ). Our sharp estimates in Corollary 3.2 for k ( α ) = 1 in the W -radial case A n suggest that c = c = 1 / t + α ( X ) α ( Y )) k ( α ) are natural in place of separate terms w ( B ( X, √ t )) and w ( B ( Y, √ t )). On the other hand, estimates(14) and in Corollary 3.2 suggest that the following conjecture is true in the Dunkl setting. Conjecture 4.1. The Weyl-invariant heat kernel for a root system Σ in R d satisfies the followingestimates p Wt ( X, Y ) ≍ t − d e −| X − Y | t Q α> ( t + α ( X ) α ( Y )) k ( α ) . (15) Formula (3) then implies that the W -invariant Dunkl kernel satisfies the estimate E Wk ( X, Y ) ≍ e λ ( X ) Q α> (1 + α ( X ) α ( λ )) k ( α ) . p Wt ( X, Y ) Let us finish by giving formulas relating the heat kernel p Wt ( X, Y ) with the spherical functions ψ iλ and φ iλ . These formulas can be useful in further study of the kernel p Wt ( X, Y ). Proposition 5.1. (a) In the flat Riemannian symmetric case, the following formula holds: p Wt ( X, Y ) = C Z a e −| λ | t ψ i λ ( X ) ψ − i λ ( Y ) π ( λ ) dλ, C > . (16) (b) In the curved non-compact Riemannian symmetric case the following formula holds p Wt ( X, Y ) = C Z a ∗ e − ( | λ | + | ρ | ) t φ i λ ( X ) φ − i λ ( Y ) dλ | c ( λ ) | (17) where c ( λ ) is the Harish-Chandra c -function (refer to [8] for details). The constant C canbe given explicitly. roof. We will prove (b). We show that the right hand side of equation (16) satisfies the definitionof the heat kernel. For a test function f , consider u ( X, t ) = C Z a Z a e − ( | λ | + | ρ | ) t φ i λ ( X ) φ − i λ ( Y ) K | c ( λ ) | − dλ f ( Y ) | c ( λ ) | − dY where K | c ( λ ) | − dλ is Plancherel measure.The fact that ∆ u ( X, t ) = ∂∂t u ( X, t ) where ∆ is the radial Laplacian follows easily from thefact that ∆ φ i λ ( X ) = − ( | λ | + | ρ | ) φ i λ ( X ) and ∂∂t e − ( | λ | + | ρ | )] t = − ( | λ | + | ρ | ) e −| λ | t . Now, usingFubini’s theorem, u ( X, t ) = C K Z a e − ( | λ | + | ρ | ) t (cid:20)Z a φ − i λ ( Y ) f ( Y ) | c ( λ ) | − dY (cid:21) φ i λ ( X ) π ( λ ) dλ = C K Z a e − ( | λ | + | ρ | ) t ˜ f ( λ ) φ i λ ( X ) π ( λ ) dλ which tends to f ( X ) as t → Remark 5.2. The heat kernel estimates of h Wt ( X ) = p Wt ( X, on symmetric spaces ([2] andreferences therein) are based on the inverse spherical Fourier transform formula which is a specialcase of (17) when Y = 0 . Thus one may hope that estimates of p Wt ( X, Y ) can be deduced from (17) . Remark 5.3. The passage from h Wt ( X ) to p Wt ( X, Y ) is well understood at the group level: p Wt ( g, h ) = h Wt ( h − g ) , which is equivalent to p Wt ( X, Y ) = Z K h Wt ( e − Y k − e X ) dk and to p Wt ( X, Y ) = Z a h Wt ( H ) k ( H, − Y, X ) π ( H ) dH, (18) where the last formula contains the product formula kernel k which is defined by Z a ψ λ ( e H ) k ( H, X, Y ) π ( H ) dH = ψ λ ( e X ) ψ λ ( e Y ) = Z K ψ λ ( e X k e Y ) dk. Similarly, ˜ p Wt ( X, Y ) = Z a ˜ h Wt ( H ) ˜ k ( H, − Y, X ) δ ( H ) dH, (19) where the last formula contains the product formula kernel ˜ k which is defined by Z a φ λ ( e H ) ˜ k ( H, X, Y ) δ ( H ) dH = φ λ ( e X ) φ λ ( e Y ) = Z K φ λ ( e X k e Y ) dk. eferences [1] J.-P. Anker, J. Dziuba´nski, A. Hejna. Harmonic Functions,Conjugate Harmonic Functionsand the Hardy Space H in the Rational Dunkl Setting , Journal of Fourier Analysis andApplications(2019) 25:2356-2418.[2] J.-P. Anker and L. Ji. 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