a transference result of lp continuity from the jacobi riesz transform to the gaussian and laguerre...

A transference result of the Lp continuity from Jacobi Riesz transform to the Gaussian and Laguerre Riesz transforms

A transference result of the Lp continuity fromJacobi Riesz transform to the Gaussian and

Laguerre Riesz transforms

Eduard Navas (UNEFM, Coro)Wilfredo Urbina (Roosevelt University, Chicago)

Western Spring Sectional Meeting University of New Mexico, Albuquerque, NM

April 4-6, 2014


The Classical orthogonal polynomials, their semigroup and Riesz Transforms.

The Jacobi case

The Jacobi case.

Let us consider the P(α,β)n n∈N, which are orthogonal polynomials with

respect to the Jacobi measure µα,β in (−1, 1), with α, β > −1,

µα,β(dx) = ωα,β(x)dx = χ(−1,1)(x)(1− x)α(1 + x)β

2α+β+1B(α+ 1, β + 1)dx (1)

= ηα,βχ(−1,1)(x)(1− x)α(1 + x)βdx,

with ηα,β = 12α+β+1B(α+1,β+1)

= Γ(α+β+2)2α+β+1Γ(α+1)Γ(β+1)

.i.e. ∫ ∞

−∞P(α,β)

n (y)P(α,β)m (y)µα,β(dy) = ηα,βh(α,β)

n δn,m = hn(α,β)

δn,m, (2)



The Jacobi case.

where n,m = 0, 1, 2, · · · ,

h(α,β)n =

2α+β+1

(2n + α+ β + 1)

Γ (n + α+ 1) Γ (n + β + 1)

Γ (n + 1) Γ (n + α+ β + 1), (3)

and

hn(α,β)

=1

(2n + α+ β + 1)

Γ(α+ β + 2)Γ (n + α+ 1) Γ (n + β + 1)

Γ(α+ 1)Γ(β + 1)Γ (n + 1) Γ (n + α+ β + 1)

= ‖P(α,β)n ‖2

2,(α,β)



The Jacobi case.

The Jacobi polynomial P(α,β)n is also a polynomial solution of the Jacobi

differential equation, with parameters α, β, n,(1− x2) y′′ + [β − α− (α+ β + 2) x] y′ + n (n + α+ β + 1) y = 0, (4)

thus, P(α,β)n is an eigenfunction of the (one-dimensional) Jacobi differential

operator

Lα,β = −(1− x2)d2

dx2 − (β − α− (α+ β + 2)x)ddx,

associated with the eigenvalue λα+βn = n(n +α+ β + 1). Moreover, Lα,β is

a diffusion since it is a second order differential operator with non-constantterm. Observe that if we choose δα,β =

√1− x2 d

dx , and consider its formalL2(µα,β)-adjoint,

δ∗α,β = −√

1− x2 ddx

+ [(α+12

)

√1 + x1− x

− (β +12

)

√1− x1 + x

]I,

then Lα,β = δ∗α,βδα,β . The differential operator δα,β is considered the“natural” notion of derivative in the Jacobi case.



The Jacobi case.

The operator semigroup associated to the Jacobi polynomials is defined forpositive or bounded measurable Borel functions of (−1, 1), as

Tα,βt f (x) =

∫ 1

−1pα,β(t, x, y)f (y)µα,β(dy). (5)

where

p(α,β)(t, x, y) =∑

k

e−k(k+α+β+1)t

hk(α,β)

P(α,β)k (x)P(α,β)

k (y).

The explicit representation of p(α,β)(t, x, y) is very complicated since theeigenvalues λn are not linearly distributed and was obtained by G. Gasper.Tα,βt is called the Jacobi semigroup and can be proved that is a Markovsemigroup.The Jacobi-Poisson semigroup Pα,βt can be defined, using Bochner’ssubordination formula,

e−λ1/2t =

1√π

∫ ∞0

e−u√

ue−

λt24u du,



The Jacobi case.

as the subordinated semigroup of the Jacobi semigroup,

P(α,β)t f (x) =

1√π

∫ ∞0

e−u√

uT(α,β)

t2/4u f (x)du.

For a function f ∈ L2([−1, 1] , µ(α,β)

)let us consider its Fourier- Jacobi

expansion

f =

∞∑k=0

〈f ,Pα,βk 〉

hk(α,β)

Pα,βk , (6)

where

〈f ,Pα,βk 〉 =

∫ 1

−1f (y)Pα,βk (y)µα,β(dy).

Then the action of Tt and Pt can be expressed using the Jacobi expansion as

Ttf =

∞∑k=0

〈f ,Pα,βk 〉

hk(α,β)

e−λk tPα,βk , and Ptf =

∞∑k=0

〈f ,Pα,βk 〉

hk(α,β)

e−√λk tPα,βk ,



The Jacobi case.

Following the classical case, the Jacobi-Riesz transform can be defineformally as

Rα,β = δα,β(Lα,β)−1/2 =√

1− x2 ddx

(Lα,β)−1/2, (7)

where (Lα,β)−ν/2 is the Jacobi-Riesz potential of order ν/2. (Lα,β)−ν/2

can be represented as

(Lα,β)−ν/2f =1

Γ(ν)

∫ ∞0

tν−1P(α,β)t fdt,

and then for f ∈ L2([−1, 1] , µ(α,β)

), Rα,β f has Jacobi expansion

Rα,β f (x) =

∞∑k=1

〈f ,P(α,β)k 〉

hk(α,β)

λ−1/2k

(k + α+ β + 1)

2

√1− x2P(α+1,β+1)

k−1 (x),

(8)where λk = k(k + α+ β + 1).



The Jacobi case.

The Lp-continuity of the Riesz-Jacobi transform Rα,β , was proved by Zh. Li(J. Approx. Theory 86 (1996), no. 2, 179196; MR1400789 (98g:42044)) andL. Caffarelli (Sobre conjugaci—n y sumabilidad de series de Jacobi, Univ.Buenos Aires, 1971) in the case d = 1. In the case d ≥ 1Rα,βi , i = 1, · · · , d, are defined analogously, using partial differentiation in(7), and their Lp-continuity was proved by A. Nowak and P. Sjogren.

TheoremAssume that 1 < p <∞ and α, β ∈ [−1/2,∞)d. There exists a constant cp

such that‖Rα,βi f‖p,(α,β) ≤ cp‖f‖p,(α,β). (9)

for all i = 1, · · · , d.



The Hermite case

The Hermite case.

Now consider the Hermite polynomials Hnn, which are defined as theorthogonal polynomials associated with the Gaussian measure in R,

γ(dx) = e−x2

√π

dx, i.e. ∫ ∞−∞

Hn(y)Hm(y) γ(dy) = 2nn!δn,m, (10)

n,m = 0, 1, 2, · · · , with the normalization

H2n+1(0) = 0, H2n(0) = (−1)n (2n)!

n!. (11)

We have

H′

n(x) = 2nHn−1(x), (12)

H′′

n (x) − 2xH′

n(x) + 2nHn(x) = 0. (13)



The Hermite case.

thus Hn is an eigenfunction of the one dimensional Ornstein-Uhlenbeckoperator or harmonic oscillator operator,

L = −12

d2

dx2 + xddx, (14)

associated with the eigenvalue λn = n. Observe that if we chooseδγ = 1√

2ddx , and consider its formal L2(γ)-adjoint,

δ∗γ = − 1√2

ddx

+√

2xI

then L = δ∗γδγ . The differential operator δγ is considered the “natural”notion of derivative in the Hermite case.



The Hermite case.

The Gaussian-Riesz transform can be defined formally, as

Rγ = δγL−1/2 =1√2

ddx

L−1/2. (15)

Therefore, for f ∈ L2 (R, γ) with Hermite expansion

f =

∞∑k=1

〈f ,Hk〉2kk!

Hk,

its Gaussian-Riesz transform has Hermite expansion

Rγ f (x) =

∞∑k=1

〈f ,Hk〉2kk!

√2kHk−1(x). (16)

The Lp continuity of the of the Gaussian-Riesz transform was proved by B.Muckenhoupt in 1969, in the case d = 1 (Trans. Amer. Math. Soc. 139(1969), 243260; MR0249918 (40 3159)). In the case d ≥ 1Rγi , i = 1, · · · , d, are defined analogously, using partial differentiation in(15), and their Lp continuity has been proved by very different ways, usinganalytic and probabilistic tools, by P. A. Meyer, R. Gundy, S. Perez and F.Soria, G. Pissier, C. Gutierrez and W. Urbina.



The Hermite case.

TheoremAssume that 1 < p <∞. There exists a constant cp such that

‖Rγi f‖p,γ ≤ cp‖f‖p,γ . (17)

for all i = 1, · · · , d.



The Laguerre case.

The Laguerre case.

Finally, analogously for the Laguerre polynomials Lαk , α > −1 which aredefined as the orthogonal polynomials associated with the Gamma measureon (0,∞), µα(dx) = χ(0,∞)(x) xαe−x

Γ(α+1) dx, i.e.∫ ∞0

Lαn (y)Lαm(y)µα(dy) =

(n + α

n

)δn,m =

Γ(n + α+ 1)

Γ(α+ 1)n!δn,m, (18)

n,m = 0, 1, 2, · · · . We have

(Lαk (x))′ = −Lα+1k−1 (x). (19)

x(Lαk (x))′′ + (α+ 1− x)(Lαk (x))′ + kLαk (x) = 0. (20)

thus Lαk is an eigenfunction of the (one-dimensional) Laguerre differentialoperator

Lα = −xd2

dx2 − (α+ 1− x)ddx,

associated with the eigenvalue λk = k.Observe that if we choose δα =√

x ddx ,

and consider its formal L2(γ)-adjoint,



The Laguerre case.

δ∗α = −√

xddx

+ [α+ 1/2√

x+√

x]I

then L = δ∗αδα. The differential operator δα is considered the “natural”notion of derivative in the Laguerre case.The Laguerre-Riesz transform can be defined formally, as

Rα = δα(Lα)−1/2 =√

xddx

(Lα)−1/2. (21)

Therefore for f ∈ L2 ((0,∞), µα) with Laguerre expansion

f =

∞∑k=0

Γ(α+ 1)k!

Γ(k + α+ 1)〈f ,Lαk 〉Lαk

its Laguerre-Riesz transform has Laguerre expansion

Rαf (x) = −∞∑

k=1

Γ(α+ 1)k!

Γ(k + α+ 1)(√

k)−1√x〈f ,Lαk 〉Lα+1k−1 (x). (22)



The Laguerre case.

The Lp continuity of the Laguerre-Riesz transform was proved by B.Muckenhoupt, for the case d = 1 (Trans. Amer. Math. Soc. 147 (1970),403418; MR0252945 (40 6160)). In the case d ≥ 1 Rαi , i = 1, · · · , d, aredefined analogously, using partial differentiation in (21), and their Lp

continuity was proved by A. Nowak, using Littlewood-Paley theory.

TheoremAssume that 1 < p <∞ and α ∈ [−1/2,∞)d. There exists a constant cp

such that‖Rαi f‖p,α ≤ cp‖f‖p,α. (23)

for all i = 1, · · · , d,.



Asymptotic relations

Asymptotic relations.Now, it is well know the asymptotic relations of the Jacobi polynomials withother classical orthogonal polynomials:

i) The asymptotic relation with the Hermite polynomials is

limλ→∞

λ−n/2Cλn (x/√λ) =

Hn(x)

n!. (24)

ii) The asymptotic relation with the Laguerre polynomials is

limβ→∞

P(α,β)n (1− 2x/β) = Lαn (x). (25)


Main Results

In the case of the Riesz transforms, we have proved (see ArXiv:1202.5728)that the Lp-continuity of the Gaussian-Riesz transform and the Lp-continuityof the Laguerre-Riesz transform can be obtained from the Lp-continuity ofthe Jacobi-Riesz transform using the asymptotic relations.

TheoremLet α, β > −1 and 1 < p <∞, then the Lp(µα,β) boundedness for theJacobi-Riesz transform

‖Rα,β f‖p,(α,β) ≤ Cp‖f‖p,(α,β) (26)

implies

i) the Lp(γ)-boundedness for the Gaussian-Riesz transform

‖Rγ f‖p,γ ≤ Cp‖f‖p,γ . (27)

ii) the Lp(µα)-boundedness for the Laguerre-Riesz transform

‖Rαf‖p,α ≤ Cp‖f‖p,α. (28)


Main Results

For the proof of the theorem we need the following technical results,

Proposition

i) Let fλ(x) = f (√λx)1[−1,1](x) where f ∈ L2(R, γ). Then

fλ ∈ L2([−1, 1], µλ) and

limλ→∞

‖fλ‖2,λ = ‖f‖2,γ

ii) Similarly, Let fβ(x) = f(β2 (1− x)

)1[−1,1](x) where

f ∈ L2(R, µα). Then fβ ∈ L2([−1, 1], µ(α,β)) and

limβ→∞

‖fβ‖2,(α,β) = ‖f‖2,α


Main Results.

and

Proposition

i) Let f ∈ L2(R, γ). Then

limλ→∞

〈fλ, λ−k/2Cλk 〉 = 〈f , Hk

k!〉

ii) Similarly, Let f ∈ L2(R, µα). Then

limβ→∞

〈fβ ,P(α,β)k 〉 = 〈f ,Lαk 〉


Main Results.

Idea of the proof for the theorem of the Riesz transform.Let us consider first the case p = 2 then by Parseval’s identity, forf ∈ L2 ([−1, 1] , µλ)∥∥∥R(α,β)f

∥∥∥2

2,(α,β)=

∞∑k=1

|〈f ,Pα,βk 〉

hn(α,β)

|2 (k + α+ β + 1)2

4λk

∥∥∥√1− x2P(α+1,β+1)k−1

∥∥∥2

2,(α,β).

and ∥∥∥√1− x2P(α+1,β+1)k−1

∥∥∥2

2,(α,β)

=4(α+ 1)(β + 1)

(α+ β + 3)(α+ β + 2)

∥∥∥P(α+1,β+1)k−1

∥∥∥2

2,(α+1,β+1)

=4k

(k + α+ β + 1)

∥∥∥P(α,β)k

∥∥∥2

2,(α,β)


The Main Result.

and therefore,∥∥∥R(α,β)f∥∥∥2

2,(α,β)=

∞∑k=1

|〈f ,Pα,βk 〉

hn(α,β)

|2 (k + α+ β + 1)2

4k(k + α+ β + 1)

∥∥∥√1− x2P(α+1,β+1)k−1

∥∥∥2

2,(α,β)

=

∞∑k=1

|〈f ,Pα,βk 〉

hn(α,β)

|2 (k + α+ β + 1)2

4k(k + α+ β + 1)

× 4k(k + α+ β + 1)

∥∥∥P(α,β)k

∥∥∥2

2,(α,β)

=

∞∑k=1

|〈f ,Pα,βk 〉|2

× (2k + α+ β + 1)Γ(α+ 1)Γ(β + 1)Γ(k + 1)Γ(k + α+ β + 1)

Γ(α+ β + 2)Γ(k + α+ 1)Γ(k + β + 1)


Main Results

∥∥Rλf∥∥2

2,λ =

∞∑k=1

|〈f ,P(λ−1/2,λ−1/2)k 〉|2 (2k + 2λ) [Γ(λ+ 1/2)]

2Γ(k + 1)Γ(k + 2λ)

Γ(2λ+ 1) [Γ(k + λ+ 1/2)]2

=

∞∑k=1

|〈f ,Cλk 〉|2(2k + 2λ) [Γ(λ+ 1/2)]

2Γ(k + 1)Γ(k + 2λ)

Γ(2λ+ 1) [Γ(k + λ+ 1/2)]2

× [Γ(2λ)]2

[Γ(k + λ+ 1/2)]2

[Γ(λ+ 1/2)]2

[Γ(k + 2λ)]2

≥∞∑

k=1

|〈f , λ−k/2Cλk 〉|2k!

2√π(2 + 1/λ)(2 + 2/λ) . . . (2 + (k − 1)/λ)


Main Results

On the other hand, again using Parseval’s identity, we have that the L2-normof the Gaussian Riesz transform for f ∈ L2 (R, γ) is given by

‖Rγ f‖22,γ =

∞∑k=1

|〈f ,Hk〉|2

π22k(k!)2 2k ‖Hk−1‖22,γ =

∞∑k=1

|〈f ,Hk〉|2

k!2k√π.

Then, taking fλ(x) = f (√λx)1[−1,1](x) and using the asymptotic relation

and the previous proposition, we get

‖Rγ f‖22,γ =

∞∑k=1

1k!2k√π|〈f ,Hk〉|2 =

∞∑k=1

1(k!)2

k!

2√π2(k−1)

|〈f ,Hk〉|2

= limλ→∞

∞∑k=1

k!

2√π(2 + 1

λ )(2 + 2λ ) . . . (2 + (k−1)

λ )|〈fλ, λ−k/2Cλk 〉|2

≤ limλ→∞

∥∥Rλfλ∥∥2

2,λ ≤ C2 limλ→∞

‖fλ‖22,λ = C2 ‖f‖2

2,γ .


Main Results.

The Laguerre case is essentially analogous.For the general case p 6= 2 we will follow the argument given by Betancouret al.Take φ ∈ C∞0 (R), and φλ(x) = φ(

√λx), for λ > 0 and x ∈ R . If λ big

enough sopφλ ⊂ [−1, 1], then

‖Rλφλ‖Lp([−1,1]µλ) ≤ C‖φλ‖Lp([−1,1]µλ)

i.e. ∥∥∥∥∥∞∑

n=1

φλ(n)r(λ)n

√1− x2Cλ+1

n−1

∥∥∥∥∥Lp([−1,1]µλ)

≤ C‖φλ‖Lp([−1,1]µλ)


Main Results.

Taking the change of variable x = y√λ

, we get

∫ √λ−√λ

∣∣∣∣∣∞∑

n=1

φλ(n)r(λ)n

√1− y2

λCλ+1

n−1 (y√λ

)

∣∣∣∣∣p

Z(λ)(1− y2

λ)λ−1/2dy

1/p

≤ C‖φλ‖Lp([−1,1]µλ)

where Z(λ) = λ1/2[Γ(λ)]222λ

2πΓ(2λ)∫ √λ−√λ

∣∣∣∣∣∞∑

n=1

φλ(n)r(λ)n

(1− y2

λ

)λ/p−1/2p+1/2

ey2

p Cλ+1n−1 (

y√λ

)

∣∣∣∣∣p

e−y2

√π

dy

1/p

≤ C(Z(λ))−1/p‖φλ‖Lp([−1,1]µλ),


Main Results.

and also we have∫ √λ−√λ

∣∣∣∣∣∞∑

n=1

φλ(n)r(λ)n

(1− y2

λ

)λ/2−1/4+1/2

ey2

2 Cλ+1n−1 (

y√λ

)

∣∣∣∣∣2

e−y2

√π

dy

1/2

≤ C(Z(λ))−1/2‖φλ‖L2([−1,1]µλ)

Let k ∈ N and λ > 0 such that√λ > k define

Fλ,k(y) =

∞∑

n=1φλ(n)r(λ)

n Cλ+1n−1 ( y√

λ)(

1− y2

λ

)λ/2−1/4+1/2e

y2

2 if |y| ≤ k

0 si |y| > k

and

fλ,k(y) =

∞∑

n=1φλ(n)r(λ)

n Cλ+1n−1 ( y√

λ)(

1− y2

λ

)λ/p−1/2p+1/2e

y2

p if |y| ≤ k

0 si |y| > k


Main Results.

for y ∈ [−√λ,√λ], both series converges and Fλ,k = fλ,kΩλ, where

Ωλ(y) = ey2

2 −y2

p

(1− y2

λ

)λ/2−λ/p−1/4+1/2p

.

Ωλ is bounded in [−k, k].


Main Results.

On the other hand, it can be proved that

limλ→∞

(Z(λ))−1/p‖φλ‖Lp([−1,1]µλ) ≤ limλ→∞

C

∫ √λ−√λ

|φ(y)|p e−y2

√π

dy

1/p

= C

∫ ∞−∞|φ(y)|p e−y2

√π

dy

1/p

= C‖φ‖Lp(R,γ),

and moreover,

(Z(λ))−1/p‖φλ‖Lp([−1,1]µλ) ≤ C‖φ‖Lp(R,γ). (29)

Then,

‖Fλ,k‖L2(R,γ) ≤ C‖φ‖L2(R,γ).

and

‖Fλ,k‖Lp(R,γ) ≤ C‖φ‖Lp(R,γ)

for all√λ > k.


Main Results.

Thus, Fλ,k is a bounded sequence in L2 (R, γ) and in Lp (R, γ) . ByBourbaki-Alaoglu’s theorem,there exists a subsequence (λj)j∈N such thatlimj→∞ λj =∞ and functions Fk ∈ L2 (R, γ) and fk ∈ Lp (R, γ) satisfying

I Fλj,k → Fk, as j→∞, in the weak topology of L2 (R, γ)

I Fλj,k → fk, as j→∞, in the weak topology of Lp (R, γ).

Moreover, sopFk ∪ sopfk ⊆ [−k, k], and

‖Fk‖L2(R,γ) ≤ limj→∞‖Fλj,k‖L2(R,γ) ≤ C‖φ‖L2(R,γ). (30)

Analogously one gets,

‖fk‖Lp(R,γ) ≤ C‖φ‖Lp(R,γ). (31)

and moreover Fk = fk a.e (−k, k), and as k is arbitrary we have Fk = fka.e. so we get

‖Fk‖Lp(R,γ) ≤ C‖φ‖Lp(R,γ). (32)


Main Results.

Then, there exists a monotone increasing sequence (λj)j∈N ⊂ (0,∞) suchthat limj→∞ λj =∞, and a function F ∈ Lp (R, γ) ∩ L2 (R, γ) , satisfying

I For each k ∈ N, Fλj,k → F, as j→∞, in the weak topology ofL2 (R, γ) and in the weak topology of Lp (R, γ)

I ‖F‖Lp(R,γ) ≤ C‖φ‖Lp(R,γ).

Then, to finish the proof we have to prove F is the Gaussian-Riesz transformof φ,

F(y) = Rγφ(y), a.e.,

and then‖Rγ‖Lp(R,γ) = ‖F‖Lp(R,γ) ≤ C‖φ‖Lp(R,γ).

The Laguerre case is essentially analogous.


Main Results

We can also obtain the Lp-continuity of the Gaussian-Littlewood-Paley gfunction and the Lp-continuity of the Laguerre-Littlewood-Paley g functionfrom the Lp-continuity of the Jacobi–Littlewood-Paley g function using theasymptotic relations (but that is another talk!).

TheoremLet α, β > −1 and 1 < p <∞, then the Lp(µα,β) boundedness for theJacobi-Littlewood-Paley g function

‖g(α,β)f‖p,(α,β) ≤ Cp‖f‖p,(α,β) (33)

implies

i) the Lp(γ) boundedness for the Gaussian-Littlewood-Paley g function

‖gγ f‖p,γ ≤ Cp‖f‖p,γ . (34)

ii) the Lp(µα) boundedness for the Laguerre-Littlewood-Paley g function

‖gαf‖p,α ≤ Cp‖f‖2,α. (35)


References

ReferencesI Betancor, J., Farina, J., Rodriguez, L., Sanabria, A. Transferring

boundedness from conjugate operators associated whit Jacobi,Laguerre and Fourier-Bessel expansions to conjugate operators in theHankel setting. J. Fourier Anal. Appl. 14 (2008), no. 4, 493–513.

I Cafarelli, L. Sobre conjugacion y sumabilidad de series de Jacobi,Univ. Buenos Aires, 1971

I Gutierrez, C. On the Riesz transforms for the Gaussian measure. J.Func. Anal. 120 (1) (1994) 107-134.

I Li, Z. Conjugate Jacobi series and conjugate functions. J. Approx.Theory 86 (1996), no. 2, 179196; MR1400789 (98g:42044)

I Meyer, P. A. Transformations de Riesz pour les lois Gaussiennes.Lectures Notes in Math 1059 (1984) Springer-Verlag 179-193.

I Muckenhoupt, B. Hermite conjugate expansions. Trans. Amer. Math.Soc. 139 (1969), 243260; MR0249918 (40 3159)

I Muckenhoupt, B. Conjugate functions for Laguerre expansions. Trans.Amer. Math. Soc. 147 (1970), 403418; MR0252945 (40 6160)

I Muckenhoupt, B. & Stein, E.M. Classical Expansions. Trans. Amer.Math. Soc. 147 (1965) 17-92.


References

I Navas, E & Urbina, W. A transference result of the Lp continuity of theJacobi Riesz transform to the Gaussian and Laguerre Riesz transforms.J. Fourier Anal. Appl. 19 (2013), no. 5, 910942. arXiv: 0441166.

I Navas, E & Urbina, W. A transference result of the Lp continuity of theJacobi Littlewood-Paley g function to the Gaussian and LaguerreLittlewood-Paley g function. (2014) In preparation.

I Nowak, A. On Riesz transforms for Laguerre expansions. J. Funct.Anal. 215 (2004), no. 1, 217–240.

I Nowak, A. & Stempak, K. L2-theory of Riesz transforms for orthogonalexpansions. J. Fourier Anal. Appl. 12 (2006), no. 6, 675–711.

I Novak, A.; Sogren, P. Riesz Transforms for Jacobi expansions J. Anal.Math. 104 (2008), 341369.

I Stein, E.M. Topics in Harmonic Analysis related to theLittlewood-Paley Theory. Princeton Univ. Press. Princeton (1970).

I Szego, G. Orthogonal polynomials. Colloq. Publ. 23. Amer. Math.Soc. Providence (1959).

I Urbina, W. On singular integrals with respect to the Gaussian measure.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 4, 531567.

a transference result of lp continuity from the jacobi riesz transform to the gaussian and laguerre...

Science