CUBO A Mathematical Journal Vol.19, No¯ 02, (11–31). June 2017 On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck semigroup Iris A. López 1 Departamento de Matemáticas Puras y Aplicadas, Universidad Simón Bolivar, Aptdo 89000. Caracas 1080-A. Venezuela. iathamaica@usb.ve ABSTRACT The aim of this paper is to prove the hypercontractive propertie of the Dunkl-Ornstein- Uhlenbeck semigroup, {e(tLk)}t≥0. To this end, we prove that the Dunkl-Ornstein- Uhlenbeck differential operator Lk with k ≥ 0 and associated to the Zd2 group, satisfies a curvature-dimension inequality, to be precise, a C(ρ,∞)-inequality, with 0 ≤ ρ ≤ 1. As an application of this fact, we get a version of Meyer’s multipliers theorem and by means of this theorem and fractional derivatives, we obtain a characterization of Dunkl-potential spaces. RESUMEN El objetivo de este art́ıculo es demostrar la propiedad hipercontractiva del semigrupo de Dunkl-Ornstein-Uhlenbeck, {e(tLk)}t≥0. Para lograr esto, probamos que el operador diferencial de Dunkl-Ornstein-Uhlenbeck Lk con k ≥ 0 y asociado al grupo Zd2, satisface una desigualdad de curvatura-dimensión, para ser precisos, una C(ρ,∞)-desigualdad, con 0 ≤ ρ ≤ 1. Como una aplicación de este hecho, obtenemos una versión del teo- rema de multiplicadores de Meyer y a través de este teorema y derivadas fraccionales, obtenemos una caracterización de espacios Dunkl-potenciales. Keywords and Phrases: Dunkl-Ornstein-Uhlenbeck operator, generalized Hermite polynomial, squared field operator, Meyer’s multiplier theorem, Dunkl-potential space, fractional integral, frac- tional derivative. 2010 AMS Mathematics Subject Classification: 33C45, 6A33, 33C52. 1The author’s research was partially supported by DID-USB-CB-004-17 12 Iris A. López CUBO 19, 2 (2017) 1 Preliminaries In this section we collect some notations and results in the Dunkl theory (see [5]), but particularly for the Zd2 group. Let ν = (ν1, . . . ,νd) ∈ Zd+ be a multi-index, where Z+ = {0,1,2, . . .}, so ν! = ∏d j=1 vj! and |ν| = ∑d j=1 νj. For x = (x1, . . . ,xd) ∈ Rd, we set xν = x ν1 1 . . .x νd d and |x| 2 2 = ∑d j=1 x 2 j . In what follows, we denote ∂j = ∂/∂xj, for each 1 ≤ j ≤ d, and ∂ν = ∂ν11 . . .∂ νd d . Also, 〈., .〉 denotes the Euclidean inner product in Rd and finally, △ and ∇ denote the usual Laplacean and the usual gradient, respectively. Let us consider the finite reflection group generated by σj(x) = x − 2 〈x,ej〉 |ej| 2 2 ej, where (ej) d j=1 are the standard unit vectors of R d. So, for each j = 1, . . . ,d, σj(x1, . . . ,xj, . . . ,xd) = (x1, . . . ,−xj, . . . ,xd) and isomorphic to Zd2 = {0,1} d. The reflection σj is in the hyperplane orthogonal to ej. Then, we consider the root system R and the positive root system R+, respectively, as R = {± √ 2ej : j = 1, . . . ,d}, R+ = { √ 2ej : j = 1, . . . ,d} and let k be a nonnegative multiplicity function k : R+ → [0,∞), which is Z d 2-invariant. Then, we set k = (k1, . . . ,kd), where kj = αj + (1/2) and αj ≥ −1/2, for each j = 1, . . . ,d. Thus, in this particular case, the Dunkl differential difference operators, Tkj , are given by Tkj f(x) = ∂jf(x) + kj ( f(x) − f(σjx) xj ) , j = 1, . . . ,d with f ∈ C1(Rd) and in the following, the operator △k = d∑ j=1 (Tkj ) 2, given explicitty by △kf(x) = d∑ j=1 { ∂2jf(x) + 2kj xj ∂jf(x) − kj ( f(x) − f(σjx) x2j )} is called ”the generalized Laplacian” or ”Dunkl-Laplacian” associated to Zd2 and k. Then the Dunkl-Ornstein-Uhlenbeck differential operator is defined as, Lk = △k 2 − 〈x,∇x〉 (1.1) CUBO 19, 2 (2017) On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck... 13 and therefore, from (1.1), the Dunkl-Ornstein-Uhlenbeck differential operator can be written as Lkf(x) = d∑ j=1 1 2 { ∂2jf(x) + 2kj xj ∂jf(x) − kj ( f(x) − f(σjx) x2j )} − xj∂jf(x). (1.2) Here, the corresponding weight function is defined by wk(x) = ∏d j=1 |xj| 2kj and we con- sider the Hilbert space L2(mk), where the probability measure, mk, is defined by mk(dx) = ck exp(−|x| 2 2)wk(x)dx, with x ∈ Rd and ck = (∫ Rd exp(−|x|22)wk(x)dx )−1 . Now, we consider a complete system of orthogonal polynomials, with respect to the measure mk, which is known as generalized Hermite polynomials. In dimension one, for the reflection group Z2, the corresponding generalized Hermite polynomials are defined as { Hk2n(x) = (−1) n22nn!L α−(1/2) n (x 2), Hk2n+1(x) = (−1) n22n+1n!xL α+(1/2) n (x 2), where Lαn are the Laguerre polynomials of degree n and order α, (see [11] and [4, pages 156,157]). In the multidimensional case the generalized Hermite polynomials are defined by taking tensor products of the one-dimensional Hkn; that is, H k ν(x) = ∏d j=1 H kj νj(xj), x ∈ Rd, ν ∈ Zd+. This way, we will denote hkν = 2 −|ν|/2Hkν, ν ∈ Zd+, from now on. Hkν is a polynomial of degree |ν| and {h k ν}ν∈Zd + forms an orthonormal basis of L2(mk). The gen- eralized Hermite polynomials satisfy the following important identity which is known as Mehler’s formula. For r ∈ C with |r| < 1, ∑ ν∈Zd + hkν(x)h k ν(y)r |ν| = 1 (1 − r2)|k|+d/2 exp ( − r2(|x|22 + |y| 2 2) 1 − r2 ) Ek ( 2rx 1 − r2 ,y ) , where the sum is absolutely convergent and the Dunkl kernel, Ek(x,y), replaces the usual exponen- tial function, exp〈x,y〉. Then, from [11] the generalized Hermite polynomials are eigenfunctions of Lk; Lk(h k ν) = −|ν|h k ν, ∀ν ∈ Zd+. (1.3) Also, let Ckn be the closed subspace of L 2(mk) generate by linear combination of {h k ν : |ν| = n} and we denote by Jkn the orthogonal projection of L 2(mk) onto C k n. If f is a polynomial, then Jknf = ∑ |ν|=n ckν(f)h k ν, where given a function f ∈ L2(mk), its Dunkl-Fourier coefficient is defined by ckν(f) = ∫ Rd f(x)hkν(x)mk(dx) and therefore, if f ∈ L2(mk), its Dunkl-Hermite expansion is given by f = ∑ ∞ n=0 J k nf. Thus, the operator Lkf = ∞∑ n=0 −nJknf, 14 Iris A. López CUBO 19, 2 (2017) defined on the domain D2(Lk) = { f ∈ L2(mk) : ∑ ∞ n=0 ∑ |ν|=n |c k ν(f)| 2 < ∞ } , is a self-adjoint ex- tension of Lk considered on C ∞ c (R d). More precisely, Lk has a clousure which also will be denoted by Lk. Now, following [11, 13], the generalized heat kernel, Γk(t,x,y), is given by Γk(t,x,y) = ck exp{−(|x| 2 2 + |y| 2 2)/4t} (4t)|k|+d/2 Ek ( x√ 2t , y√ 2t ) , (1.4) where x,y ∈ Rd and t > 0. Therefore, from [12] the Dunkl-Ornstein-Uhlenbeck integral operator is defined as Oktf(x) = ∫ Rd Γk ( (1 − e−2t) 4 ,e−tx,y ) f(y)wk(y)dy. But, (1.4) allows us to express exp(|y|22)Γk ( (1 − e−2t) 4 ,e−tx,y ) = ck exp ( − e−2t(|x|22+|y| 2 2) 1−e−2t ) (1 − e−2t)|k|+d/2 Ek ( √ 2e−tx√ 1 − e−2t , √ 2e−ty√ 1 − e−2t ) , and since, Ek ( √ 2e−tx√ 1−e−2t , √ 2e−ty√ 1−e−2t ) = Ek ( 2e−tx 1−e−2t ,y ) , by using Mehler formula, we get explicity: Oktf(x) = ∫ Rd f(y)Okt(x,y)mk(dy), where, Okt(x,y) = ∑ ν∈Zd + e−|ν|thkν(x)h k ν(y). Besides, {O k t}t≥0 is a positive, strongly continuous contraction semigroup on C0(R d) with generator Lk. Thus, formally, we write O k t = e (tLk) and following [12, 14], if we consider Mkt(x,dy) = Γk ( (1 − e−2t) 4 ,e−tx,y ) wk(y)dy, (1.5) form, (together with the trivial kernel Mk0), a semigroup of Markov kernels. Also, the corresponding Dunkl-Poisson semigroup {Pkt }t≥0 is defined, by means of subordination principle, as Pktf(x) = 1√ π ∫ ∞ 0 exp(−u)√ u Okt2/4uf(x)du = ∫ Rd f(y)Pkt (x,y)mk(dy), where the kernel Pkt (x,y) is defined as Pkt (x,y) = 1√ π ∫ ∞ 0 exp(−u)√ u Okt2/4u(x,y)du. Again, {Pkt }t≥0 is a positive, strongly continuous semigroup with infinitesimal generator (−Lk) 1/2 and it is a Markov process (see [12, 16]). In particular, by (1.3) we obtain that Okt(h k ν) = e −|ν|thkν and P k t (h k ν) = e −t √ |ν|hkν CUBO 19, 2 (2017) On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck... 15 also, if f is a polynomial Oktf = ∑ n≥0 e−ntJknf and P k tf = ∑ n≥0 e−t √ nJknf. Following [2, 3], let us consider the squared field operator, Γ, associated to Lk, as Γ(f,g) = 1 2 [Lk(fg) − fLk(g) − gLk(f)], ∀f,g ∈ A, (1.6) where we choose A as the set of all polynomials on Rd, which is a dense subspace in L2(mk). Besides, let us consider the operator Γ2 defined as Γ2(f,g) = 1 2 [LkΓ(f,g) − Γ(f,Lkg) − Γ(g,Lkf)], ∀f,g ∈ A × A (1.7) and throughout this paper we denote Γ(f) = Γ(f,f) and Γ2(f) = Γ2(f,f). Again, motivated by [2, 3], we say that the differential operator Lk satisfies a CD(ρ,n)- inequality, (a curvature-dimension inequality with curvature ρ and dimension n), if and only if Γ2(f) ≥ ρΓ(f) + 1 n (Lkf) 2, ∀f ∈ A, where ρ ∈ R and n ∈ [1,∞]. Particularly, Lk satisfies a CD(ρ,∞)-inequality, if and only if Γ2(f) ≥ ρΓ(f) ∀f ∈ A. Finally, we denote a Dirichley form associated to the measure mk by E(f) = ∫ Γ(f)(x)mk(dx) and the Entropy of a positive function f as Ent(f) = ∫ f(x) log(f)(x)mk(dx) − ∫ f(x)mk(dx) log (∫ f(x)mk(dx) ) . In this case, a logarithmic Sobolev inequality, LS(A,C), has the form Ent(f2) ≤ A ∫ f2(x)mk(dx) + CE(f), ∀f ∈ A. Particularly, if A = 0 we say that the logarithmic Sobolev inequality is tight. The logarithmic Sobolev inequalities relate Entropy to the Dirichlet norm (the Energy) and these type of inequalities were introduced by L. Gross to study the hypercontractive propertie of the diffusion semigroups and the Markov semigroups, (see [7, 8]). 2 The results Now, we are ready to present the results of this paper. 16 Iris A. López CUBO 19, 2 (2017) 2.1 Hypercontractivity of Dunkl-Ornstein-Uhlenbeck semigroup Following D. Bakry [2, 3], we turn now to the study of the local structure of the Dunkl-Ornstein- Uhlenbeck differential operator Lk. Then, we started recalling the operators, Γ and Γ2, defined in (1.6) and (1.7) respectively. That is, Γ(f) = 1 2 [ Lk(f 2) − 2fLk(f) ] and Γ2(f) = 1 2 [LkΓ(f) − 2Γ(f,Lkf)] , (2.1) where, Γ(f,Lkf) = 1 2 [ Lk(fLkf) − fLk(Lkf) − (Lkf) 2 ] , (2.2) ∀f ∈ A. Again, we consider A as the space of all polynomials on Rd. Now, from (1.2), let us denote Lkf = d∑ j=1 L j kf, ∀f ∈ A, (2.3) where, for each j = 1, . . . ,d, L j kf(x) = 1 2 { ∂2jf(x) + 2kj xj ∂jf(x) − kj ( f(x) − f(σjx) x2j )} − xj∂jf(x). (2.4) Thus, in order to obtain our results, we prove the following technical Lemmas. Lemma 2.1. Let Lk be the Dunkl-Ornstein-Uhlenbeck differential operator defined as in (1.2). Then Γ(f)(x) = |∇f(x)|2 2 + d∑ j=1 kj ( f(x) − f(σjx) 2xj )2 , ∀f ∈ A. Proof. From (2.1) and (2.3), it is obvious that we can write Γ(f)(x) = 1 2 d∑ j=1 {L j k(f 2)(x) − 2f(x)L j k(f)(x)}, where, considering (2.4), we denote L j kf(x) = L jf(x) + Ω j kf(x), j = 1, . . . ,d with Ljf(x) = 1 2 ∂2jf(x) − xj∂f(x) and Ω j kf(x) = 1 2 [ 2kj xj ∂jf(x) − kj x2j (f(x) − f(σjx)) ] , CUBO 19, 2 (2017) On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck... 17 for each j = 1, . . . ,d. Thus, first we have to compute Lj(f2) and 2fLj(f). We can see that { Lj(f2)(x) = (∂jf) 2(x) + f(x)∂2jf(x) − 2xjf(x)∂jf(x), 2f(x)Lj(f)(x) = f(x)∂2jf(x) − 2xjf(x)∂jf(x), since { ∂j(f 2)(x) = 2f(x)∂jf(x), ∂2j (f 2)(x) = 2(∂jf) 2(x) + 2f(x)∂2jf(x) and therefore Lj(f2)(x) − 2f(x)Ljf(x) = (∂jf) 2 (x). (2.5) On the other hand, Ω j k(f 2)(x) = 1 2 [ 4kj xj f(x)∂jf(x) − kj x2j (f2(x) − f2(σjx)) ] and 2f(x)Ω j kf(x) = 1 2 [ 4kj xj f(x)∂jf(x) − kj x2j (2f2(x) − 2f(x)f(σjx)) ] . Then, we obtain that Ω j k(f 2 )(x) − 2f(x)Ω j k(f 2 )(x) = kj 2x2j (f(x) − f(σjx)) 2 (2.6) and consequently, the result of the Lemma follows from (2.5) and (2.6), since Γ(f)(x) = 1 2 d∑ j=1 {Lj(f2)(x) − 2f(x)Ljf(x)} + {Ω j k(f 2)(x) − 2f(x)Ω j kf(x)} = 1 2 { d∑ j=1 (∂jf) 2(x) + kj 2x2j (f(x) − f(σjx)) 2 } . Lemma 2.2. Let Lk be the Dunkl-Ornstein-Uhlenbeck differential operator defined as in (1.2). Then, ∀f ∈ A, we have Lk(fLkf)(x) = (Lkf) 2(x) + f(x)Lk(Lkf)(x) + 〈∇f(x),∇Lkf(x)〉 + 1 2 d∑ i=1 d∑ j=1 ki x2i (f(x) − f(σix)) ( L j kf(x) − L j kf(σix) ) . Proof. From (2.3) we obtain that Lk(fLkf)(x) = d∑ i=1 d∑ j=1 Lik(fL j kf)(x) 18 Iris A. López CUBO 19, 2 (2017) and by using (2.4) with fL j kf instead of f, we can express Lik(fL j kf)(x) = 1 2 [ ∂2i(fL j kf)(x) + 2ki xi ∂i(fL j kf)(x) − ki x2i ( f(x)L j kf(x) − f(σix)L j kf(σix) ) ] − xi∂i(fL j kf)(x). (2.7) Now, for each j = 1, . . . ,d and i = 1, . . . ,d, we have { ∂i(fL j kf)(x) = ∂if(x)L j kf(x) + f(x)∂i(L j kf)(x), ∂2i(fL j kf)(x) = ∂ 2 if(x)L j kf(x) + 2∂if(x)∂i(L j kf)(x) + f(x)∂ 2 i(L j kf)(x) (2.8) and since, − ki x2i [f(x)L j kf(x) − f(σix)L j kf(σix)] = − ki x2i (f(x) − f(σix))L j kf(x) − ki x2i f(x) ( L j kf(x) − L j kf(σix) ) + ki x2i (f(x) − f(σix)) ( L j kf(x) − L j kf(σix) ) , (2.9) then substituing (2.8) and (2.9) in (2.7) we obtain explicitly Lik(fL j kf)(x) = ( 1 2 [ ∂2if(x) + 2ki xi ∂if(x) − ki x2i (f(x) − f(σix)) ] − xi∂if(x) ) L j kf(x)+ ( 1 2 [ ∂2i(L j kf)(x) + 2ki xi ∂i(L j kf)(x) − ki x2i (L j kf(x) − L j kf(σix)) ] − xi∂i(L j kf)(x) ) f(x)+ ∂if(x)∂i(L j kf)(x) + ki 2x2i (f(x) − f(σix)) ( L j kf(x) − L j kf(σix) ) and we can express Lik(fL j kf)(x) = Likf(x)L j kf(x) + f(x)L i k(L j kf)(x) + ∂if(x)∂i(L j kf)(x)+ ki 2x2i (f(x) − f(σix))(L j kf(x) − L j kf(σix)), for each j = 1, . . . ,d and i = 1, . . . ,d. Therefore, taking the sum with respect to i and j, we obtain the result of the Lemma. CUBO 19, 2 (2017) On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck... 19 Consequently, the identity (2.2) and the Lemma 2.2 allows us to write Γ2(f)(x) = 1 2 [ Lk(Γ(f))(x) − 〈∇f(x),∇Lkf(x)〉 − d∑ i=1 d∑ j=1 ki 2x2i (f(x) − f(σix)) ( L j kf(x) − L j kf(σix) ) ] (2.10) and in the following result we obtain an explicit expression for the operator Γ2(f). More precisely, Proposition 2.3. Let Lk be the Dunkl-Ornstein-Uhlenbeck differential operator defined as in (1.2). Then, ∀f ∈ A, the operator Γ2(f) can be rewritten as Γ2(f)(x) = d∑ i=1 { (∂2if) 2(x) 4 + (∂if) 2(x) 2 + ki 4 [ (∂if(x) + ∂if(σix)) xi − (f(x) − f(σix)) x2i ]2 + ki 2 [ ∂if(x) xi − (f(x) − f(σix)) 2x2i ]2 + ki 4x2i (f(x) − f(σix)) 2 } + d∑ i=1 d∑ j=1,j6=i { (∂2ijf) 2(x) 4 + ki 8x2i (∂jf(x) − ∂jf(σix)) 2 + kj 8x2j (∂if(x) − ∂if(σjx)) 2 + kikj 16x2ix 2 j [(f(x) − f(σix)) − (f(σjx) − f(σiσjx))] 2 } . Proof. We have to compute each term in equation (2.10). As first step in this argument, observe that by using (2.3) and the Lemma 2.1 we get Lk(Γf) = d∑ i=1 d∑ j=1 L j k(Γif) = d∑ i=1 Lik(Γif) + d∑ i=1 d∑ j=1,j6=i L j k(Γif), where we denote Γif(x) = (∂if) 2(x) 2 + ki ( f(x) − f(σix) 2xi )2 . (2.11) Using the identity (2.4) with Γif instead of f, we can express L j k(Γif)(x) = 1 2 { ∂2j (Γif)(x) + 2kj xj ∂j(Γif)(x) − kj ( Γif(x) − Γif(σjx) x2j )} − xj∂jΓif(x). (2.12) Then, let us consider two cases: i = j and i 6= j, with 1 ≤ i ≤ d and 1 ≤ j ≤ d. If i = j, differentiating (2.11) with respect to xi we obtain ∂i(Γif)(x) = ∂if(x)∂ 2 if(x) + ki 2x2i (f(x) − f(σix))(∂if(x) + ∂if(σix)) − ki 2x3i (f(x) − f(σix)) 2 (2.13) 20 Iris A. López CUBO 19, 2 (2017) and ∂2i(Γif)(x) = ∂if(x)∂ 3 if(x) + (∂ 2 if) 2 (x) + ki 2x2i (∂if(x) + ∂if(σix)) 2 + ki 2x2i (f(x) − f(σix))(∂ 2 if(x) − ∂ 2 if(σix)) − 2ki x3i (f(x) − f(σix))(∂if(x) + ∂if(σix)) + 3ki 2x4i (f(x) − f(σix)) 2, (2.14) (note that ∂i(σix) = −1). Otherwise, considering (2.11) with σix instead of x, we can write Γif(σix) = (∂if) 2(σix) 2 + ki ( f(σix) − f(x) 2xi )2 , 1 ≤ i ≤ d, since σi(σix) = x and we obtain that ki 2x2i (Γif(x) − Γif(σix)) = ki 4x2i ((∂if) 2(x) − (∂if) 2(σix)). (2.15) Therefore, if i = j, replacing (2.13), (2.14) and (2.15) in (2.12), we get that Lik(Γif)(x) = (∂2if) 2(x) 2 + ∂if(x)∂ 3 if(x) 2 − xi∂ 2 if(x)∂if(x) + ki 4x2i (∂if(x) + ∂if(σix)) 2 + ki 4x2i (f(x) − f(σix))(∂ 2 if(x) − ∂ 2 if(σix)) − ki 4x2i ((∂if) 2(x) − (∂if) 2(σix)) + [ k2i 2x3i − ki x3i − ki 2xi ] (f(x) − f(σix))(∂if(x) + ∂if(σix)) + ki xi ∂2if(x)∂if(x) + [ 3ki 4x4i − k2i 2x4i + ki 2x2i ] (f(x) − f(σix)) 2. (2.16) Now, if i 6= j, again differentiating (2.11) with respect to xj we obtain ∂j(Γif)(x) = ∂if(x)∂ 2 ijf(x) + ki 2x2i (f(x) − f(σix))(∂jf(x) − ∂jf(σix)) (2.17) and ∂2j (Γif)(x) = (∂ 2 ijf) 2 (x) + ∂if(x)∂ 3 ijjf(x) + ki 2x2i (∂jf(x) − ∂jf(σix)) 2 + ki 2x2i (f(x) − f(σix))(∂ 2 jf(x) − ∂ 2 jf(σix)). (2.18) CUBO 19, 2 (2017) On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck... 21 Again, considering (2.11) with σjx instead of x we have that kj 2x2j (Γif(x) − Γif(σjx)) = kj 4x2j ((∂if) 2(x) − (∂if) 2(σjx)) + kjki 8x2jx 2 i [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] . (2.19) Therefore, if i 6= j, replacing the equations (2.17), (2.18) and (2.19) in (2.12) we can see that L j k(Γif) can be expressed as L j k(Γif)(x) = (∂2ijf) 2(x) 2 + ∂if(x)∂ 3 ijjf(x) 2 − xj∂jf(x)∂ 2 ijf(x) + ki 4x2i (∂jf(x) − ∂jf(σix)) 2 + ki 4x2i (f(x) − f(σix))(∂ 2 jf(x) − ∂ 2 jf(σix)) + kj xj ∂if(x)∂ 2 ijf(x) + [ kikj 2x2ixj − kixj 2x2i ] (f(x) − f(σix))(∂jf(x) − ∂jf(σix)) − kj 4x2j ((∂if) 2 (x) − (∂if) 2 (σjx)) − kikj 8x2ix 2 j [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] . (2.20) Thus, taking the sum with respect to i and j in (2.16) and (2.20) we obtain explicitly that Lk(Γf)(x) = d∑ i=1 { (∂2if) 2(x) 2 + ∂if(x)∂ 3 if(x) 2 − xi∂ 2 if(x)∂if(x) + ki 4x2i (∂if(x) + ∂if(σix)) 2 + ki 4x2i (f(x) − f(σix))(∂ 2 if(x) − ∂ 2 if(σix)) − ki 4x2i ((∂if) 2(x) − (∂if) 2(σix)) + [ k2i 2x3i − ki x3i − ki 2xi ] (f(x) − f(σix))(∂if(x) + ∂if(σix)) + ki xi ∂2if(x)∂if(x) + [ 3ki 4x4i − k2i 2x4i + ki 2x2i ] (f(x) − f(σix)) 2 } + d∑ i=1 d∑ j=1,j6=i { (∂2ijf) 2(x) 2 + ∂if(x)∂ 3 ijjf(x) 2 − xj∂jf(x)∂ 2 ijf(x) + ki 4x2i (∂jf(x) − ∂jf(σix)) 2 + ki 4x2i (f(x) − f(σix))(∂ 2 jf(x) − ∂ 2 jf(σix)) + kj xj ∂if(x)∂ 2 ijf(x) + [ kikj 2x2ixj − kixj 2x2i ] (f(x) − f(σix))(∂jf(x) − ∂jf(σix)) − kj 4x2j ((∂if) 2(x) − (∂if) 2(σjx)) − kikj 8x2ix 2 j [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] } . (2.21) Now, we will develop the second part of the proof of this proposition calculating the operator 22 Iris A. López CUBO 19, 2 (2017) 〈∇f,∇Lkf〉, such can be written as 〈∇f(x),∇Lkf(x)〉 = d∑ i=1 ∂if(x)∂i(Lkf)(x) = d∑ i=1 d∑ j=1 ∂if(x)∂i(L j kf)(x) = d∑ i=1 ∂if(x)∂i(L i kf)(x) + d∑ i=1 d∑ j=1,j6=i ∂if(x)∂i(L j kf)(x). Again, we consider i = j and i 6= j. If i = j, from (2.4) we obtain that ∂i(L i kf)(x) = ∂3if(x) 2 − ∂if(x) − xi∂ 2 if(x) + ki xi ∂2if(x) − ki x2i ∂if(x) − ki 2x2i (∂if(x) + ∂if(σix)) + ki x3i (f(x) − f(σix)). (2.22) But, if i 6= j, we get ∂i(L j kf)(x) = ∂3jjif(x) 2 + kj xj ∂2jif(x) − kj 2x2j (∂if(x) − ∂if(σjx)) − xj∂ 2 jif(x). (2.23) Thus, taking the sum with respect to i and j in (2.22) and (2.23) we express 〈∇f(x),∇Lkf(x)〉 = d∑ i=1 { ∂3if(x)∂if(x) 2 − (∂if) 2(x) − xi∂ 2 if(x)∂if(x) + ki xi ∂2if(x)∂if(x) − ki x2i (∂if) 2(x) − ki 2x2i (∂if(x) + ∂if(σix))∂if(x) + ki x3i (f(x) − f(σix))∂if(x) } + d∑ i=1 d∑ j=1,j6=i { ∂3jjif(x)∂if(x) 2 + kj xj ∂2jif(x)∂if(x) − kj 2x2j (∂if(x) − ∂if(σjx))∂if(x) − xj∂ 2 jif(x)∂if(x) } . (2.24) Finally as third step, we turn now to compute explicitly the terms of the expression d∑ i=1 d∑ j=1 ki 2x2i (f(x) − f(σix)) ( L j kf(x) − L j kf(σix) ) . Then, if i = j, by using (2.4) with σix instead of x, we can write Likf(σix) = ∂2if(σix) 2 − ki xi ∂if(σix) − ki 2x2i (f(σix) − f(x)) + xi∂if(σix), CUBO 19, 2 (2017) On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck... 23 (remind that σix = (x1, . . . ,−xi, . . . ,xd) and σi(σix) = x), therefore, Likf(x) − L i kf(σix) = (∂2if(x) − ∂ 2 if(σix)) 2 + ki xi (∂if(x) + ∂if(σix)) − ki x2i (f(x) − f(σix)) − xi(∂if(x) + ∂if(σix)). (2.25) But, if i 6= j, we get L j kf(x) − L j kf(σix) = (∂2jf(x) − ∂ 2 jf(σix)) 2 + kj xj (∂jf(x) − ∂jf(σix)) + kj 2x2j (f(σjx) − f(x) + f(σix) − f(σjσix)) − xj(∂jf(x) − ∂jf(σix)). (2.26) So, from (2.25) and (2.26) we can conclude that d∑ i=1 d∑ j=1 ki 2x2i (f(x) − f(σix)) ( L j kf(x) − L j kf(σix) ) = d∑ i=1 { ki 4x2i (∂2if(x) − ∂ 2 if(σix))(f(x) − f(σix)) + k2i 2x3i (∂if(x) + ∂if(σix))(f(x) − f(σix)) − k2i 2x4i (f(x) − f(σix)) 2 − ki 2xi (∂if(x) + ∂if(σix))(f(x) − f(σix)) } + d∑ i=1 d∑ j=1,j6=i { ki 4x2i (∂2jf(x) − ∂ 2 jf(σix))(f(x) − f(σix)) + kikj 2x2ixj (∂jf(x) − ∂jf(σix))(f(x) − f(σix)) + kikj 4x2ix 2 j (f(σjx) − f(x) + f(σix) − f(σjσix))(f(x) − f(σix)) − kixj 2x2i (∂jf(x) − ∂jf(σix))(f(x) − f(σix)) } . (2.27) Then, at this point in our argument, replacing the identities (2.21), (2.24) and (2.27) in (2.10) and simplifying the terms that are equal, we can write Γ2(f)(x) = E1(x) + E2(x), 24 Iris A. López CUBO 19, 2 (2017) where we denote by E1(x) = 1 2 { d∑ i=1 (∂2if) 2(x) 2 + (∂if) 2(x) + ki 4x2i (∂if(x) + ∂if(σix)) 2 − ki x3i (∂if(x) + ∂if(σix))(f(x) − f(σix)) + [ 3ki 4x4i + ki 2x2i ] (f(x) − f(σix)) 2 − ki 4x2i ((∂if) 2(x) − (∂if) 2(σix)) + ki x2i (∂if) 2(x) + ki 2x2i (∂if(x) + ∂if(σix))∂if(x) − ki x3i (f(x) − f(σix))∂if(x) } and E2(x) = 1 2 { d∑ i=1 d∑ j=1,j6=i (∂2ijf) 2(x) 2 + ki 4x2i (∂jf(x) − ∂jf(σix)) 2 − kj 4x2j [(∂if) 2(x) − (∂if) 2(σjx)] − kjki 8x2jx 2 i [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] + kj 2x2j (∂if(x) − ∂if(σjx))∂if(x) − kikj 4x2ix 2 j [(f(σjx) − f(x) + f(σix) − f(σjσix))(f(x) − f(σix))] } . Therefore, we only need to express E1(x) and E2(x) more easily. First, we consider E1(x) and associating the terms, we see that ki 4x2i (∂if(x) + ∂if(σix)) 2 + ki 2x2i (∂if(x) + ∂if(σix))∂if(x) − ki 4x2i ((∂if) 2 (x) − (∂if) 2 (σix)) = ki 2x2i (∂if(x) + ∂if(σix)) 2. (2.28) Now, taking the identity (2.28) and completing squares in E1(x) we obtain that ki 2x2i (∂if(x) + ∂if(σix)) 2 − ki x3i (∂if(x) + ∂if(σix))(f(x) − f(σix)) = ki 2 [ (∂if(x) + ∂if(σix)) 2 x2i − 2 (∂if(x) + ∂if(σix)) xi (f(x) − f(σix)) x2i ± (f(x) − f(σix)) 2 x4i ] = ki 2 [ ( ∂if(x) + ∂if(σix) xi ) − ( f(x) − f(σix) x2i ) ]2 − ki 2x4i (f(x) − f(σix)) 2. (2.29) This way, from (2.29) we can write E1(x) = 1 2 { d∑ i=1 (∂2if) 2(x) 2 + (∂if) 2(x) + ki 2 [ ( ∂if(x) + ∂if(σix) xi ) − ( f(x) − f(σix) x2i ) ]2 + [ ki 4x4i + ki 2x2i ] (f(x) − f(σix)) 2 + ki x2i (∂if) 2(x) − ki x3i (f(x) − f(σix))∂if(x) } , CUBO 19, 2 (2017) On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck... 25 since, (3ki/4x 4 i) − (ki/2x 4 i) = ki/4x 4 i . Then, associating the terms in the above expression, we have ki x2i (∂if) 2(x) − ki x3i (f(x) − f(σix))∂if(x) + ki 4x4i (f(x) − f(σix)) 2 = ki [ ∂if(x) xi − ( f(x) − f(σix) 2x2i )]2 and therefore, we can conclude that E1(x) = 1 2 { d∑ i=1 (∂2if) 2(x) 2 + (∂if) 2(x) + ki 2 [ ( ∂if(x) + ∂if(σix) xi ) − ( f(x) − f(σix) x2i ) ]2 + ki [ ∂if(x) xi − ( f(x) − f(σix) 2x2i )]2 + ki 2x2i (f(x) − f(σix)) 2 } . (2.30) Now, we consider E2(x). Once more, we observe that − kj 4x2j [(∂if) 2(x) − (∂if) 2(σjx)] + kj 2x2j (∂if(x) − ∂if(σjx))∂if(x) = kj 4x2j [∂if(x) − ∂if(σjx)] 2. (2.31) Moreover, associating the terms − kjki 4x2jx 2 i (f(σjx) − f(x) + f(σix) − f(σjσix))(f(x) − f(σix)) = kjki 8x2jx 2 i [ 2(f(x) − f(σix)) 2 − 2(f(σjx) − f(σjσix))(f(x) − f(σix)) ] , then − kjki 8x2jx 2 i [ (f(x) − f(σix)) 2 − (f(σjx) − f(σiσjx)) 2 ] + kjki 8x2jx 2 i [ 2(f(x) − f(σix)) 2 − 2(f(σjx) − f(σjσix))(f(x) − f(σix)) ] = kjki 8x2jx 2 i [(f(x) − f(σix)) − (f(σjx) − f(σiσjx))] 2 . (2.32) Therefore, replacing (2.31) and (2.32) in E2(x), we express E2(x) = 1 2 { d∑ i=1 d∑ j=1,j6=i (∂2ijf) 2(x) 2 + ki 4x2i [∂jf(x) − ∂jf(σix)] 2+ kj 4x2j [∂if(x) − ∂if(σjx)] 2 + kjki 8x2jx 2 i [(f(x) − f(σix)) − (f(σjx) − f(σiσjx))] 2 } (2.33) and finally, the sum of (2.30) and (2.33) allows us to obtain the result of the Proposition. In consecuense, we are able to prove that the Dunkl-Ornstein-Uhlenbeck differential opera- tor, Lk, defined as in (1.2) and associated with the Z d 2 group, satisfies a CD(ρ,∞)-inequality, if 0 ≤ ρ ≤ 1. 26 Iris A. López CUBO 19, 2 (2017) Theorem 2.4. Let Lk be the Dunkl-Ornstein-Uhlenbeck differential operator defined as in (1.2). Then, if 0 ≤ ρ ≤ 1, the CD(ρ,∞)-inequality is satisfied. Proof. From Lemma 2.1 and the Proposition 2.3, we have that Γ2(f)(x) ≥ ρΓ(f)(x) is true, if and only if, d∑ i=1 { (∂2if) 2(x) 4 + (1 − ρ) (∂if) 2(x) 2 + ki 4 [ (∂if(x) + ∂if(σix)) xi − (f(x) − f(σix)) x2i ]2 + ki 2 [ ∂if(x) xi − (f(x) − f(σix)) 2x2i ]2 + (1 − ρ) ki 4x2i (f(x) − f(σix)) 2 } + d∑ i=1 d∑ j=1,j6=i { (∂2ijf) 2(x) 4 + ki 8x2i (∂jf(x) − ∂jf(σix)) 2 + kj 8x2j (∂if(x) − ∂if(σjx)) 2 + kikj 16x2ix 2 j [(f(x) − f(σix)) − (f(σjx) − f(σiσjx))] 2 } ≥ 0. Then, we only need to choose 0 ≤ ρ ≤ 1 to obtain the result. Now, again we consider the family of measures Mkt(x,dy) defined in (1.5). If the measures mk are replaced by Mkt(x,dy), then the logarithmic Sobolev inequalities LS(A,C) can be rewritten as Okt(f 2 logf2) − Okt(f 2) logOkt(f 2) ≤ A(t)Okt(f2) + c(t)Okt(Γf) and if A = 0, Okt(f 2 logf2) − Okt(f 2) logOkt(f 2) ≤ c(t)Okt(Γf), (2.34) which are known as local Log-Sobolev inequalities and local tight-Log-Sobolev inequalities respec- tively, (see [2]). Therefore, from the general criterion of D. Bakry and M. Emery cf. [1] we have that the curvature inequality C(ρ,∞) is equivalent to the local tight-Log-Sobolev inequality with c(t) = 1−e −2ρt ρ , (for details, we refer the reader to [2, Proposition 2.6]). Thus, from Corolario 2.4 we obtain that inequality (2.34) is true and therefore, ∫ Rd Okt(f 2 logf2)(x)mk(dx) − ∫ Rd Okt(f 2)(x) logOkt(f 2)(x)mk(dx) ≤ c(t) ∫ Rd Okt(Γf)(x)mk(dx), where Γ is defined as in the Lemma 2.1. Then, by using the propertie ∫ Rd Oktf(x)mk(dx) = ∫ Rd f(x)mk(dx), we can conclude that Ent(f2) ≤ CE(f), ∀f ∈ A. CUBO 19, 2 (2017) On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck... 27 In a general context, L. Gross [8, 7] proved that Logarithmic Sobolev inequality is equivalent to the fact that for any t > 0 and p ∈ (1,∞), ‖Htf‖q(t) ≤ em(t)‖f‖p, where (Ht)t is a diffusion semigroup or a symmetric Markov semigroup and the functions q(t) and m(t) are defined by q(t) − 1 p − 1 = exp(4t/C) and m(t) = A 16 ( 1 p − 1 q(t) ) , (see [8, Theorems 1 and 2]). Particularly, tight-Logarithmic-Sobolev inequality is equivalent to the hypercontractivity prop- erty. In consecuense, we can conclude that the Dunkl-Ornstein-Uhlenbeck semigroup, {Okt}t≥0, is hypercontractive, ∀t > 0 and 1 < p < ∞. This means, ‖Oktf‖q(t),mk ≤ ‖f‖p,mk, ∀f ∈ L p(mk), where, exp(4t/C) = (q(t) − 1)/(p − 1) for some positive constant C. By using subordination formula we obtain the same result for the Dunkl-Poisson semigroup {Pkt }t≥0. 2.2 Applications As a consequence of the hypercontractivity propertie of {Okt}t≥0 semigroup, we obtain the L p(mk)- continuity of Jkn operators for every 1 < p < ∞ and n = 0,1,2, ... The reasoning is similar as in the case of classical Ornstein-Uhlenbeck semigroup and we refer the reader to [17, Lemma 1.1], where the identity Okt(J k nf) = e −ntJknf, if f ∈ A, is a key condition in the argument. Moreover, for 1 < p < ∞ and n ∈ N, there exist a constant Cp,n > 0, such that, ‖Okt(I − · · · − Jkn−1)f‖p,mk ≤ Cp,ne−nt‖f‖p,mk, (2.35) and since the development is similar to the classical Ornstein-Uhlenbeck semigroup, we omit the details and refer to [17, Lemma 1.2]. Then we extend the celebrated P.A Meyer’s multiplier the- orem to Dunkl-Ornstein-Uhlenbeck semigroup and the Zd2 group. A first version of this theorem, associated with Hermite expansions, has be obtained in [17, Theorem 1.1] (see also, [18]). After- wards, similar versions to Laguerre and Jacobi setting have been obtained in [6, Theorem 3.4] and [10, Theorem 4.1], respectively. Theorem 2.5 (Meyer’s multiplier theorem). Let {Okt}t≥0 be the Dunkl-Onstein-Uhlenbeck semi- group. Assume that h is a function, which is analytic in a neighborhood of the origin. Let {ψ(n)}n∈N 28 Iris A. López CUBO 19, 2 (2017) be a sequence of real numbers, such that ψ(n) = h(n−β), ∀n ≥ n0 and some β ∈ (0,1]. Then, the operator Tψf = ∑ n≥0 ψ(n)Jknf, f = ∑ n≥0 Jknf, defined initially in L2(mk), has a unique continuous linear extension to each of the spaces L p(mk), for 1 < p < ∞. Next, we consider the fractional integrals, the fractional derivatives and the Bessel potentials associated to the differential operator Lk and the Z d 2 group. Since, Lk is symmetric and has a self- adjoint extension, these can be defined by standarts ways, by example by spectral representation of Bochner subordination. However, the use of these fractional operators together with Meyer’s multipliers theorem allows us to obtain a characterization of Dunkl-potential spaces, similar to the classic case (see [15]). Then, Dunkl-fractional integral of order s > 0, associated to Dunkl-Ornstein-Uhlenbeck dif- ferential operator and the Zd2 group, is defined by Isk = (−Lk) −s/2Π0, where Π0 denotes the orthogonal projection onto the orthogonal complement of the subspace spanned by the constant functions. Immediately, from (1.3) we have Isk(h k ν) = |ν| −s/2hkν, |ν| > 0, f ∈ A, and an integral representation of Iskf can be obtained Iskf = 1 Γ(s) ∫ ∞ 0 ts−1Pkt (I − J k 0)fdt, (2.36) which makes sense, for all f ∈ Lp(mk), by means of (2.35) and subordination formula, because ‖Iskf‖p,mk ≤ Ap‖f‖p,mk for s > 0, and 1 < p < ∞, (we refer the reader to [6, 9] and [10]). Also, we introduce the fractional derivative in the Zd2-Dunkl setting which is given formally by Dsk = (−Lk) s/2. For the generalized Hermite polynomials we have Dsk(h k ν) = |ν| s/2hkν, ∀s > 0 and therefore, by using the density of polynomials in Lp(mk), the derivative D s k can be extended to Lp(mk). Particularly, if 0 < s < 1, we can write Dskf = 1 Cs ∫ ∞ 0 ts−1(Pktf − f)dt, where, Cs = ∫ ∞ 0 u−s−1(e−u − 1)du. (2.37) CUBO 19, 2 (2017) On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck... 29 The identity (2.37) may be regarded as the definiton of Dsk, with 0 < s < 1, for all f ∈ C2B(Rd), or for all f for which the corresponding integral is absolutely convergent. Moreover, if f is a polynomial, we get Dsk(I s kf) = I s k(D s kf) = Π0f. Now, the Dunkl-Bessel potential operator, associated to the Dunkl-Ornstein-Uhlenbeck dif- ferential operator and the Zd2 group, is defined as (I − Lk) −s/2f = ∞∑ n=0 (1 + n)−s/2Jknf, f ∈ A and we defined the Dunkl-potential spaces L p,s k (mk), associated with generalized Hermite expan- sions, as the completion of the space of all polynomials with respect to the norm ‖f‖p,s = ∥ ∥ ∥ (I − Lk) s/2f ∥ ∥ ∥ p,mk . By means of Meyer’s multiplier theorem, we can observe that the Dunkl-Bessel potential op- erator extends to a continuous linear operator on Lp(mk), (for a similar argument see e.g Lemma 6.1 in [6]). Also, the potential spaces have the following properties: i) If 1 ≤ p ≤ q, then Lq,sk (mk) ⊂ L p,s k (mk), for each s ≥ 0. ii) If 0 ≤ s ≤ r, then Lp,rk (mk) ⊂ L p,s k (mk), for each 1 < p < ∞. Moreover, the embeddings in i) and ii) are continuous. Again, we omit the proofs of these two facts, but we refer the reader to the Proposition 2.2 in [9] and the Proposition 6.3 in [6]. Finally, the following theorem allows us to extend the Dunkl-fractional derivative, Dsk, to the potential spaces L p,s k (mk), for 1 < p < ∞, s > 0 and associated to generalized Hermite expansions, where we consider the Zd2 group. Thus, the union of these spaces; Lsk(mk) = ⋃ p>1 L p,s k (mk) make up a natural domain of Dsk. Similar versions of this theorem has been obtained in [9, Theorem 2.2], where we consider classical Hermite expansions, in [6, Theorem 6.4] related to Laguerrre expansions and afterwards, in [10, Theorem 5.1] in the Jacobi context. Theorem 2.6. Let s ≥ 0 and 1 < p < ∞. i) If {Pn}n is a sequence of polynomials such that limn→∞ Pn = f in L p s(mk), then limn D s kPn exists in L p,s k (mk) and does not depend on the choice of a sequence {Pn}n. If f ∈ Lp,sk (mk) ∩ L p,r k (mk), then the limit does not depend on the choice of p or r. Thus, Dskf = limn→∞ D s kPn in L s,p k (mk), limn→∞ Pn = f in L p,s k (mk), f ∈ Lsk(mk), is well defined. 30 Iris A. López CUBO 19, 2 (2017) ii) f ∈ Lp,sk (mk) if and only if Dskf ∈ Lp(mk). Moreover, Bp,s‖f‖p,s ≤ ‖Dskf‖p,mk ≤ Ap,s‖f‖p,s. References [1] D. Bakry, M. Emery, Hypercontractivité de semi-groupes de diffusion, C.R Acad. Sci. Paris Sér, 1299, (15), 775-778, (1984). [2] D. 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Voit, Dunkl theory, convolution algebras and related Markov processes, in: Harmonic and stochastic analysis of Dunkl processes, eds. P. Graczyk, M. Rösler, M. Yor, Travaux en cours 71, 1-112, Hermann, Paris, (2008). [15] E. Stein, The Characterization of Functions Arising as Potentials I, Bull. Amer. Math. Soc. 97, 102-104, (1961). II (ibid) 68, 577-582, (1962). [16] E. Stein, Topics in Harmonic Analysis related to the Littlewood-Paley Theory, Princenton Univ. Press. Princenton (1971). [17] H. Sugita, Sobolev spaces of Wiener functionals and Malliavin’s calculus, J. Math. Kyoto Univ, 25-1, 31-48, (1985). [18] S. Watanabe, M. Gopalan Nair, B. Rajeev Lectures on Stochastic differential equations and Malliavin Calculus, Tata Institute of Fundamental Research, Vol. 73, Springer Verlag, Berlin/Heidelberg/New York/Tokyo, (1984). Preliminaries The results Hypercontractivity of Dunkl-Ornstein-Uhlenbeck semigroup Applications