TRANSFERENCE OF L p -BOUNDEDNESS BETWEEN HARMONIC ANALYSIS OPERATORS FOR LAGUERRE AND HERMITE SETTINGS

Transcription

1 REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 5, Número 2, 29, Páginas TRANSFERENCE OF L -BOUNDEDNESS BETWEEN HARMONIC ANALYSIS OPERATORS FOR LAGUERRE AND HERMITE SETTINGS JORGE J. BETANCOR Abstract. In this aer we discuss a transference method of L -boundedness roerties for harmonic analysis oerators in the Hermite setting to the corresonding oerators in the Laguerre context. As a byroduct of our rocedure we obtain new characterizations of certain classes of Banach saces and Köethe saces. 1. Introduction From May 26 to June 6, 28, was held in La Falda (Córdoba, Argentina) the congress CIMPA School 28. In this aer we collect the main results that aeared in the talk resented by the author there. This roject research about transference of L boundedness roerties for harmonic analysis oerators in the Hermite and Laguerre setting has been develoed jointly with José Luis Torrea, from the Universidad Autónoma de Madrid, Juan Carlos Fariña, Lourdes Rodríguez- Mesa and Alejandro Sanabria, from the Universidad de La Laguna (Tenerife). Most of the results with their roofs can be encountered in [2], [3] and [4]. In the monograh of Stein [28] harmonic analysis oerators (maximal oerators, multiliers, Riesz transforms, Littlewood-Paley g-functions,...) related to semigrou of oerators are defined. Here we consider the oerators associated with semigrous in the Hermite and Laguerre context. We denote by H the second order Hermite oerator defined by H = d2 dx 2 + x2 = 1 [( ) ( ) ( ) ( )] d d d d 2 dx + x dx x + dx x dx + x, x R. (1.1) This oerator is symmetric with resect to the Lebesgue measure on R. For every n N, we have that Hh n = (2n + 1)h n, where h n (x) = ( π2 n n!) 1 2 e x2 2 H n (x), x R, reresents the n- th Hermite function and H n is the n-th Hermite olynomial (see [33]). 2 Mathematics Subject Classification. 42C5 (rimary), 42C15 (secondary). This aer is artially suorted by MTM27/

2 4 J. BETANCOR The heat semigrou {W t } t associated with the sequence {h n } n N of eigenfunctions for the Hermite oerator H is defined by being W t (f)(x) = W t (x, y) = = + W t (x, y)f(y)dy, x R, f L 2 (R) and t >, e (2n+1)t h n (x)h n (y) n= ( 1 e t π 1 e 4t ) 1 2 e 1 2 (x2 +y 2 ) 1+e 4t 2xye 2t 1 e 4t + 1 e 4t, x, y R, t >. As usual, the maximal oerator W of the heat semigrou {W t } t is W (f) = su W t (f), f L 2 (R). t> According to the Stein s ideas ([28]) the factorization (1.1) of H suggests to define the Riesz transform in the Hermite setting by ( d ) R(f)(x) = dx + x H 1/2 f(x) ( 2n ) 1/2hn 1 = (x)a n (f), f C c (R), 2n + 1 n= where, for every n N and f L 2 (R), a n (f) = f(x)h n(x)dx. C c (R reresents the sace of C (R) having comact suort. The fractional integral H 1/2 is defined by using the heat semigrou as follows H 1/2 (f)(x) = 1 W t (f)(x)t 1/2 dt, f L 2 (R). π From the results established by Muckenhout ([21]) we can deduce that, for every f L 2 (R), being R(f)(x) = lim R(x, y)f(y)dy, a.e. x R, ε x y >ε R(x, y) = 1 π ( d dx + x ) W t (x, y)t 1/2 dt, x, y R. The subordinated Poisson semigrou associated with {h n } n N is given by being P t (x, y) = P t (f)(x) = t 2 π P t (x, y)f(y)dy, f L 2 (R), u 3/2 e t2 /(4u) W u (x, y)du, t (, ), x, y R.

3 TRANSFERENCE OF L -BOUNDEDNESS 41 For every 1 < q <, we denote by gq H the Littlewood-Paley function for the Poisson semigrou for H, that is, ( gq H (f)(x) = t t P t(f)(x) q dt ) 1/q. t Similar definitions for the heat semigrou for H can be made. Harmonic analysis associated with the Hermite olynomials was began by Muckenhout ([2] and [21]). Maximal oerators, Riesz transforms and Littlewood-Paley g-functions in the Hermite setting have been investigated by Torrea and Stemak ([3], [31] and [32]) and Thangavelu ([34]). For every α > 1 we consider the Laguerre differential oerator L α = 1 2 ( d2 dx 2 + x2 + 1 x 2 ( α This oerator can be factorized in the following way )), x (, ). L α = 1 2 D α D α + α + 1, (1.2) ( where D α f = α + 1/2 + x + d ) f = x α+ d 1 2 x dx dx (x α 1 2 f)+xf, and D α denotes the formal adjoint of D α in L 2 ((, ), dx). For every n N, we have that L α ϕ α n = (2n + α + 1)ϕ α n, where the Laguerre function ϕ α n is defined by ( ) 1 2Γ(n + 1) ϕ α 2 n(x) = e x2 2 x α+ 1 2 L α Γ(n + α + 1) n (x 2 ), x (, ), and L α n denotes the n-th Laguerre olynomial of tye α ([33,. 1] and [34,. 7]). The heat semigrou {W α t } t generated by the oerator L α takes the form Wt α (f)(x) = Wt α (x, y)f(y)dy, x (, ), f L2 ((, ), dx), where, for every t, x, y (, ), W α t (x, y) = n= ( 2e t = 1 e 2t e t(2n+α+1) ϕ α n (x)ϕα n (y) ) 1 2 ( 2xye t 1 e 2t ) 1 2 Iα ( 2xye t 1 e 2t ) e 1 2 (x2 +y 2 ) 1+e 2t 1 e 2t. Here I α reresents the modified Bessel function of the first kind and order α. The maximal oerator associated with the heat semigrou {Wt α } t is defined by W α (f) = su t> Wt α (f). The Riesz transform in the Laguerre setting is defined by (see (1.2)) R α (f) = D α Lα 1/2 (f), f C c (, ).

4 42 J. BETANCOR This oerator is actually a rincial value integral oerator given by being R α (f)(x) = lim R α (x, y)f(y)dy, a.e. x R, ε, x y >ε R α (x, y) = 1 D α,x W α π t (x, y)t 1/2 dt, x, y (, ). The Poisson semigrou {Pt α} t generated by L α is given in terms of {Wt α} t by the subordination formula. The Littlewood-Paley gq Lα -function, when 1 < q <, for the Poisson semigrou {Pt α } t is defined by ( gq Lα (f)(x) = t t P t α (f)(x) q dt ) 1/q. t L -boundedness roerties for maximal oerators, Riesz transforms and Littlewood-Paley functions in the Laguerre setting have been established by [8], [13], [14], [16], [2], [22], [24], [29], and [34], amongst others. In [2], [3] and [4] a new method is emloyed to study L boundedness roerties for the harmonic analysis oerators in the Laguerre setting by using corresonding roerties for oerators in the Hermite context. Our rocedure allows to get new and shorter roofs of known results and also to obtain new results. The idea of transferring boundedness roerties from Hermite to Laguerre settings was used also by Gutiérrez, Incognito, and Torrea [13] (see also [12] and [16]). They emloyed formulae relating Hermite olynomials in dimension n with Laguerre olynomials in dimension 1 and α = n 2 1. The rocedure develoed in [13] only allows us to obtain boundedness roerties for oerators in the Laguerre setting for these secial values of α. Then, it is necessary to use some kind of translantation result to extend the result to other values of α. Our method is esentially different to the one considered in [13]. We connect the harmonic analysis oerators in the Laguerre setting for every α > 1 with the corresonding oerators in the Hermite context. More recisely, the kernel of the oerator under consideration is broken in the art close to the diagonal (local art) and in the art far away from the diagonal (global art). In the local art the transference between Hermite and Laguerre contexts works. In the global art the oerators can be controlled by ositive oerators whose L -boundedness roerties are wellknown. In the following sections of this aer we show how our method can be alied to analyze L -boundedness roerties for the maximal oerator associated with the heat semigrou, Riesz transform and Littlewood-Paley g functions for the Poisson semigrou in the Laguerre setting. Moreover, we can obtain new characterizations of certain geometric roerties (UMD, Hardy-Littlewood and Lusin tye and cotye) by using the harmonic analysis oerators in the Laguerre context. Throughout this aer by C we always denote a ositive constant that can change from one line to the other line.

5 TRANSFERENCE OF L -BOUNDEDNESS Maximal oerator for the Laguerre heat semigrou. In this section we analyze the L -boundedness roerties for the maximal oerator of the heat semigrou in the Laguerre setting. Also we establish new characterizations of certain Banach lattices with the Hardy-Littlewood roerty. The key roerty is the following ointwise estimate involving the kernels of the heat semigrou for the Hermite and Laguerre oerators. Proosition 2.1. let α > 1. There exists C > such that (i) W α t (x, y) Cy α+1/2 x α 3/2, t > and < y < x/2. (ii) W α t (x, y) Cxα+1/2 y α 3/2, t > and < 2x < y. (iii) W α t (x, y) W t/2(x, y) Cx 1, t > and x/2 < y < 2x. Note that the comarison between the kernels Wt α and W t/2 for the Laguerre and Hermite heat semigrous, resectively, is got in the local region (close to the diagonal), as it is shown in (iii). The estimates obtained in (i) and (ii) say that the maximal oerator W α restricted to the global region is controlled by Hardy tye oerators. L -boundedness roerties of the Hardy tye oerators are wellknown ([22] and [8]). Then, L -boundedness of W α is imlied by the L -boundedness of W. Suose that B is a Banach sace consisting of equivalence classes, modulo equality almost everywhere, of locally integrable real functions on a comlete σ- finite measure sace (Ω, Σ, µ). This class of Banach saces is named Köethe function saces ([18] and [26]) when the following two conditions are satisfied. (a) If f(w) g(w), a.e. w Ω, with f measurable and g B, then f B and f B g B. (b) For every A Σ with µ(a) < the characteristic function χ A of A belongs to B. Each Köethe function sace is a Banach lattice with the natural order (f f(w), a.e. w Ω). This lattice is σ-order comlete. Banach lattices and Köethe function saces with the Hardy-Littlewood roerty were introduced in [1]. If f is a real locally integrable B-valued function, where B is a Köethe sace, and J is a finite subset of Q +, the set of ositive rational numbers, we define 1 M J (f)(x) = su f(y) dy, x R n. r J B(x, r) B(x,r) We say that B has the Hardy-Littlewood roerty ([1]) when for a certain 1 < < there exists C > such that M J f L B (R) C f L B (R), f L B (R), for every finite subset J of Q +. Maximal oerators associated with heat semigrou for the Ornstein-Uhlenbeck oerator (in the Hermite olynomial setting) was used in [15] to characterize Köethe saces with the Hardy-Littlewood roerty. The corresonding roerties in the Hermite and Laguerre context is included in the following roosition. In its roof the ointwise estimates established in Proosition 2.1 lay a fundamental role.

6 44 J. BETANCOR Proosition 2.2. Let B be a Köethe function sace and α > 1. The following roerties are equivalent. (i) B has the Hardy-Littlewood roerty. (ii) The maximal oerator W is bounded from L B (R, w(x)dx) into itself, for every 1 < < and w A (R). (iii) The maximal oerator W is bounded from L 1 B (R, w(x)dx) into L1, B (R, w(x)dx), for every w A 1 (R). (iv) The maximal oerator W α is bounded from L B ((, ), dx) into itself for some 1 < <, when α > 1/2, and for 2/(2α + 3) < < 2/(2α + 1), when 1 < α 1/2. (v) The maximal oerator W α is bounded from L B ((, ), xσ dx) into itself for every 1 < < and 1 (α + 1/2) < σ < (α + 3/2) Laguerre Riesz transforms The UMD roerty for Banach saces was introduced by Burkholder ([7]) in a robability setting. For a Banach sace B the UMD roerty is equivalent to the fact that the Hilbert transform admits a B valued extension to L B (R) as a bounded oerator of L B (R) for some (any) 1 < < ([6] and [7]). Recently, Abu-Falahah and Torrea ([1]) have obtained a characterization of UMD Banach saces in terms of the Riesz transform R associated with the Hermite oerator. In [2, Theorem 4.1] we obtain the corresonding result for the Riesz transform in the Laguerre setting. To rove this roerty are crucial the next ointwise estimates involving the kernels for the Laguerre and Hermite Riesz transforms. Proosition 3.1. Let α > 1. Then (i) R α (x, y) Cy α+1/2 x α 3/2, < y < x/2. (ii) R α (x, y) Cx α+3/2 y α 5/2, 2x < y. (iii) R α (x, y) ( d dx + x)w t/2(x, y) dt C ) (1 + (xy)1/4, < x/2 < t y x y 1/2 y < 2x. Note that the Laguerre and Hermite kernels differ in the local art, at most, by an absolutely integrable function on (, ). By using Proosition 3.1 the following result can be roved. Proosition 3.2. Let α > 1 and let B be a Banach sace. Then the following statements are equivalent. (i) B has the UMD roerty. (ii) Riesz transform R α admits a bounded extension from L B ((, ), dx) into itself, for some such that max{1, 2/(2α + 3)} < <. (iii) Riesz transform R α admits a bounded extension from L B ((, ), xσ dx) into itself, for every 1 < < and (α + 3/2) 1 < σ < (α + 3/2) 1.

7 TRANSFERENCE OF L -BOUNDEDNESS Littlewood-Paley g-functions for the Laguerre Poisson semigrou. Xu ([35]) introduced the Lusin cotye and tye roerties for a Banach sace B as follows. Let f be in L 1 B (T), where T denotes the one dimensional torus. We consider, for every 1 < q <, the generalized Littlewood-Paley g function defined by ( 1 ) g q (f)(z) = (1 r) q P r f(z) q dr 1 q B, 1 r where P r (θ) reresents the Poisson kernel on T. We say that B has a Lusin cotye q 2 when for some (1, ) it has g q (f) L (T) C f L B (T), f L B (T). It is said that B has a Lusin tye 1 q 2 when for some (1, ) the following inequality holds ( f L B (T) C ˆf() ) B + g q (f) L (T). The Lusin tye and Lusin cotye of B do not deend on (1, ). Moreover Xu roved in [35, Theorem 3.1] that a Banach sace B has Lusin cotye q (Lusin tye q) if and only if B has a martingale cotye q (martingale tye q). We recall that the double inequality 1 C f L B (T) ˆf() + g 2 (f) L B (T) C f L P (T), (4.1) holds when B = C, that is, when f is a scalar valued function. For a Banach sace B the above double inequality holds if and only if B is isomorhic to a Hilbert sace ([17]). Recently, Martínez, Torrea and Xu [19] have extended the results obtained by Xu in [35] to subordinated Poisson semigrous of general symmetric difusion Markovian semigrous. Also, Harboure, Torrea and Viviani [15] characterized the Lusin cotye of a Banach sace by using Littlewood-Paley g-function in the Ornstein-Uhlenbeck setting. In [3] it is shown that our method of comarison between Laguerre and Hermite contexts allows us to describe the martingale (Lusin) cotye and tye in terms of the Littlewood-Paley g functions associated with the Hermite and Laguerre Poisson semigrous. Proosition 4.2. Let B a Banach sace, q 2 and α > 1. We denote Ω α = ( (1, ), when α > 1 2, and Ω 2 α = 2α + 3, 2 ), when 1 < α 1 2α The following assertions are equivalent. (i) B has q-martingale cotye. (ii) For every (or, equivalently, for some) Ω α there exists C > such that g α q (f) L (, ) C f L B (, ), f L B (, ). (iii) For every (or, equivalently, for some) Ω α there exists C > such that g q (f) L (R) C f L B (R), f L B (R).

8 46 J. BETANCOR To establish the above result is fundamental to rove a ointwise estimate involving the kernels of the Laguerre and Hermite Litlewood-Paley functions. Proosition 4.3. Let B a Banach sace, 1 < q 2 and α > 1. Ω α is defined as in Theorem 4.2. Then the following assertions are equivalent. (i) B has q-martingale tye. (ii) For every (or, equivalently, for some) Ω α there exists C > such that f L B (, ) C g α q (f) L (, ). (iii) For every (or, equivalently, for some) Ω α there exists C > such that f L B (R) C g q (f) L (R). 5. Other oerators in the Laguerre setting. Our rocedure also works for other harmonic analysis oerators in the Laguerre context. The factorization (1.2) suggests to define (formally), for every k N, the k-th Riesz transform R α (k) associated with the Laguerre oerator by α = Dk α L k/2 α. Here L β α, β >, denotes the β-th ower of the oerator L α defined by L β α (f)(x) = 1 Γ(β) t β 1 W α t (f)(x)dt, x (, ). In the Laguerre olynomial context, Graczyk, Loeb, Lóez, Nowak and Urbina [12] investigated the corresonding higher Riesz transform α when k N and α = n 2 1, n N. They use the connection between n-dimensional Hermite olynomial and Laguerre olynomials of order α = n 2 1, that had been exloited by Incognito, Gutiérrez and Torrea [13]. Also, Nowak and Stemak [25] studied weighted L -boundedness roerties for R α (k) by using Calderón-Zygmund theory. An alication of our method, by comaring R α (k) with the corresonding k-th order Riesz transform in the Hermite setting, allows us to imrove the results in [25] in the one dimensional case. In [4] the following result was established. The class of weights admitted in Proosition 5.1 bellow is wider than the one considered in [25] when k is odd. Also it is obtained a reresentation of the higher order Riesz transform R α (k) as rincial value integral oerators. Proosition 5.1. Let α > 1 and k N. For every φ C c where α φ(x) = w k φ(x) + lim ε + α 1 (x, y) = ( ) Γ k 2, x y >ε α (x, y)φ(y)dy, t k 2 1 D k α W t α (x, y)dt, and w k =, when k is odd and w k = 2 k 2, when k is even. (, ) it has that x (, ), x, y (, ),

9 TRANSFERENCE OF L -BOUNDEDNESS 47 The oerator α α f(x) = w k f(x) + lim ε + can be extended, defining it by, x y >ε α (x, y)f(y)dy, a.e. x (, ), as a bounded ( oerator ) from L( ((, ), ) x δ dx) into itself, for 1 < < and (a) α < δ < α , when k is odd; ( ) ( ) (b) α < δ < α , when k is even; and as a bounded oerator from L 1 ((, ), x δ dx) into L 1, ((, ), x δ dx) when (c) α 5 2 δ α + 1 2, when k is odd; (d) α 3 2 δ α + 1 2, for α 1 2, and 1 < δ 1, for α = 1 2, when k is even. We can use our method also to study L -boundedness roerties of other Littlewood-Paley functions in the Laguerre setting. In [5] the behaviour of the area Littlewood-Paley function for the Poisson and heat semigrous for the Laguerre oerator is studied on weighted Lebesgue saces. To use our rocedure it is needed in a first ste to establish the corresonding results for the area Littlewood-Paley functions for the Poisson and heat semigrous for the Hermite oerator. Sasso [27] investigated L -boundedness roerties for the sectral multiliers of Laguerre exansions when the multilier is the Lalace transform of a bounded function. After writing the multilier oerator in terms of the Poisson of heat kernel, our method can be used to establish weighted L -boundedness for that oerators by using the corresonding roerties for the sectral multilier associated with Hermite exansions. Recently, Garrigós, Harboure, Signes, Torrea and Viviani [11] have studied ower weighted inequalities for Mihlin multiliers associated with Laguerre exansions. They used the ideas develoed in [13] and translantation theorems. It is an interesting question to analyze the alicability of our method in the roblem considered in [11] by using the Mihlin-Hormander multilier theorem for Hermite exansions (see [34]). References [1] I. Abu-Falahah, and J.L. Torrea, Hermite function exansions versus Hermite olynomial exansions, Glasgow Math. J., 48 (26), [2] J.J. Betancor, J.C. Fariña, L. Rodríguez-Mesa, A. Sanabria-García and J.L. Torrea, Transference between Laguerre and Hermite settings, J. Funct. Anal. 254 (28), [3] J.J. Betancor, J.C. Fariña, L. Rodríguez-Mesa, A. Sanabria-García and J.L. Torrea, Lusin tye and cotye and Laguerre g-functions, rerint, 28. [4] J.J. Betancor, J.C. Fariña, L. Rodríguez-Mesa, and A. Sanabria-García, Higher order Riesz transforms for Laguerre exansions, rerint, 28. [5] J.J. Betancor, S.M. Molina and L. Rodríguez-Mesa, Area Littlewood-Paley functions associated with Hermite and Laguerre oerators, in rearation. [6] J. Bourgain, Some remarks on Banach saces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983),

10 48 J. BETANCOR [7] D.L. Burkholder, A geometrical characterization of Banach saces in which martingale difference sequences are unconditional, Ann. Prob., 9 (1981), [8] A. Chicco Ruiz and E. Harboure, Weighted norm inequalities for the heat-diffusion Laguerre s semigrous, Math. Z. 257 (27), [9] R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains esaces homogenes, Etude de certaines integrales singulieres, Lecture Notes in Mathematics, Vol Sringer-Verlag, Berlin-New York, [1] J. García-Cuerva, R. Macías, and J. L. Torrea, The Hardy-Littlewood roerty of Banach lattices, Israel J. Math., 83 (1993), [11] G. Garrigós, E. Harboure, T. Signes, J.L. Torrea, and B. Viviani, A shar weighted translantation theorem for Laguerre function exansions, J. Funct. Anal. 244 (27), no. 1, [12] P. Graczyk, J.-L. Loeb, I.A. Lóez, A. Nowak and W. Urbina, Higher order Riesz transforms, fractional derivatives and Sobolev saces for Laguerre exansions, J. Math. Pures Al. (9) 84 (25), [13] C. Gutiérrez, A. Incognito and J.L. Torrea, Riesz tranforms, g-functions and multiliers for the Laguerre semigrou, Houston J. Math. 27 (21), [14] E. Harboure, C. Segovia, J.L. Torrea and B. Viviani, Power weighted L -inequalities for Laguerre-Riesz transforms, Arkiv Mat., 46 (28), [15] E. Harboure, J. L. Torrea, and B. Viviani, Vector valued extensions of oerators related to the Ornstein-Uhlenbeck semigrou, J. Analyse Math., 91 (23), [16] E. Harboure, J.L. Torrea and Viviani, Riesz transforms for Laguerre exansions, Indiana Univ. Math. J. 55 (26), [17] S. Kwaien, Isomorhic characterizations of inner roduct saces by orthogonal series with vector valued coefficients, Collection of articles honoring the comletion by Antoni Zygmund of 5 years of scientific activity, VI. Studia Math., 44 (1972), [18] J. Lindenstrauss, and L. Tzafriri, Classical Banach saces, II. Function saces, Sringer Verlag, Berlin, [19] T. Martínez, J. L. Torrea, and Q. Xu, Vector-valued Littlewood-Paley-Stein theory for semigrous, Adv. Math., 23 (26), no. 2, [2] B. Muckenhout, Poisson integrals for Hermite and Laguerre exansions, Trans. Amer. Math. Soc. 139 (1969), [21] B. Muckenhout, Hermite conjugate exansions, Trans. Amer. Math. Soc. 139 (1969), [22] B. Muckenhout, Conjugate functions for Laguerre exansions, Trans. Amer. Math. Soc. 147 (197), [23] B. Muckenhout, Hardy s inequality with weights, Studia Math. 44 (1972), [24] A. Nowak and K. Stemak, Riesz transforms and conjugacy for Laguerre function exansions of Hermite tye, J. Funct. Anal. 244 (27), [25] A. Nowak and K. Stemak, Riesz transforms for multi-dimensional Laguerre function exansions, Adv. Math. 215 (27), no. 2, [26] J. L. Rubio de Francia, Martingale and integral transforms of Banach sace valued functions, Lecture Notes in Math., Sringer, 1221 (1986), [27] E. Sasso, Sectral multiliers of Lalace transform tye for the Laguerre oerator, Bull. Austral. Math. Soc., 69 (24), no. 2, [28] E. M. Stein, Toics in harmonic analysis related to the Littlewood-Paley theory, Ann. of Math. Studies 63, Princenton Univ. Press, Princenton, 197.

11 TRANSFERENCE OF L -BOUNDEDNESS 49 [29] K. Stemak, Heat-diffusion and Poisson integrals for Laguerre exansions, Tohoku Math. J.(2) 46 (1994), [3] K. Stemak and J.L. Torrea, Poisson integrals and Riesz transforms for Hermite function exansions with weights, J. Functional Anal. 22 (23), [31] K. Stemak and J.L. Torrea, Higher Riesz transforms and imaginary owers associated to the harmonic oscillator, Acta Math. Hungar. 111 (26), [32] K. Stemak and J.L. Torrea, On g-functions for Hermite function exansions, Acta Math. Hungar., 19 (25), no. 1-2, [33] G. Szegö, Orthogonal olynomials, Colloquium Publications, Vol. XXIII, American Math. Soc., Providence, R.I., [34] S. Thangavelu, Lectures on Hermite and Laguerre exansions, Mathematical Notes, 42, Princeton University Press, Princeton, [35] Q. Xu, Littlewood-Paley theory for functions with values in uniformly convex saces, J. Reine Angew. Math., 54 (1998), J. Betancor Deartamento de Análisis Matemático Universidad de la Laguna Camus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n La Laguna (Sta. Cruz de Tenerife), Sain jbetanco@ull.es Recibido: 1 de agosto de 28 Acetado: 18 de diciembre de 28