Journal of Function Spaces
Volume 2018 (2018), Article ID 9304964, 29 pages
https://doi.org/10.1155/2018/9304964

Research Article

A Transference Result of the -Continuity of the Jacobi Littlewood-Paley -Function to the Gaussian and Laguerre Littlewood-Paley -Function

Eduard Navas¹ and Wilfredo O. Urbina²

¹Departamento de Matemáticas, Universidad Nacional Experimental Francisco de Miranda, Punto Fijo, Venezuela
²Department of Mathematics and Actuarial Sciences, Roosevelt University, Chicago, IL 60605, USA

Correspondence should be addressed to Wilfredo O. Urbina; wurbinaromero@roosevelt.edu

Received 4 October 2017; Accepted 11 February 2018; Published 1 August 2018

Academic Editor: Yoshihiro Sawano

Copyright © 2018 Eduard Navas and Wilfredo O. Urbina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We develop a transference method to obtain the -continuity of the Gaussian-Littlewood-Paley -function and the -continuity of the Laguerre-Littlewood-Paley -function from the -continuity of the Jacobi-Littlewood-Paley -function, in dimension one, using the well-known asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials.

1. Preliminaries

In the theory of classical orthogonal polynomials, the asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials are well known. Using those asymptotic relations we develop a transference method to obtain -continuity for the Gaussian-Littlewood-Paley -function and the -continuity of the Laguerre-Littlewood-Paley -function from the -continuity of the Jacobi-Littlewood-Paley -function, in dimension one. We are going to use the normalizations given in G. Szegö’s book [1], for all classical polynomials.

(i) Jacobi Polynomials. For , the Jacobi polynomials are defined (up to a multiplicative constant) as the orthogonal polynomials associated with the Jacobi measure (or beta measure) in , defined as

where .

The function is called the (normalized) Jacobi weight.

The Jacobi polynomials can be obtained (up to a multiplicative constant) from the polynomial canonical basis using the Gram-Schmidt orthogonalization process with respect to the inner product in . Thus we have the orthogonality property of Jacobi polynomials with respect to ,

, where

We will denote the normalized Jacobi polynomial of degree as

On the other hand, the Jacobi polynomial of parameter of degree , , is a polynomial solution of the Jacobi differential equation, with parameters ,

that is, is an eigenfunction of the (one-dimensional) second-order diffusion operator

associated with the eigenvalue . is called the Jacobi differential operator. Observe that if we choose , and consider its formal -adjoint,

then The differential operator is considered the “natural" notion of derivative in the Jacobi case.

The operator semigroup associated with the Jacobi polynomials is defined for positive or bounded measurable Borel functions of , as

where

No simple explicit representation of is known, since the eigenvalues are not linearly distributed; nevertheless there is one obtained by Gasper which is analog of Bailey’s representation of the kernel of Abel summability for Jacobi series, also called the Jacobi-Poisson integral; see [2]. From that form, taking , it can be proved that is a positive kernel.

is called the Jacobi semigroup and can be proved, that is, a Markov semigroup; for details see [3]. The Jacobi-Poisson semigroup can be defined, using Bochner’s subordination formula,

as the subordinated semigroup of the Jacobi semigroup,

For a function let us consider its Fourier- Jacobi expansion

where

Then, the action of and can be expressed as

Following the classical case, the Jacobi-Littlewood-Paley function can be defined as

where

Moreover, observe that can be written as a singular integral with values in the Hilbert space

We could also consider the following Jacobi-Littlewood-Paley functions:

the time Jacobi-Littlewood-Paley function, and

the spatial Jacobi-Littlewood-Paley function.

The -boundedness of the Jacobi-Littlewood-Paley -function was proved by Nowak and Sjögren in [4].

Theorem 1. Assume that and . There exists a constant such that

Observe that, from Theorem 1, we get immediately the -boundedness of and of .

Now, if we define

then, the -boundedness of is equivalent to the boundedness of from into (for the case of that could be also linked to -Jacobi multipliers).

(ii) Hermite Polynomials. The Hermite polynomials, , are defined as the orthogonal polynomials associated with the Gaussian measure in , , that is,

, with the normalization

We have

thus is an eigenfunction of the one-dimensional Ornstein-Uhlenbeck operator (or harmonic oscillator operator),

associated with the eigenvalue . Observe that if we choose , and consider its formal -adjoint,

then The differential operator is considered the “natural" notion of derivative in the Hermite case.

The Gaussian-Littlewood-Paley function can be defined as

where and is the Poisson-Hermite semigroup, that is, the subordinated semigroup to the Ornstein-Uhlenbeck semigroup; for more information see [3].

Moreover, observe that can be written as a singular integral with values in the Hilbert space

We could also consider the following Gaussian Littlewood-Paley functions:

the time Gaussian-Littlewood-Paley, and

the spatial Gaussian-Littlewood-Paley.

The -continuity of the Gaussian-Littlewood-Paley function was proved by Gutiérrez in [5].

Theorem 2. Assume that . There exists a constant such that

Observe that, from Theorem 2, we get immediately the -boundedness of and of .

Now, if we define

then, the -boundedness of is equivalent to the boundedness of from into

(iii) Laguerre Polynomials. For , the Laguerre polynomials are defined as the orthogonal polynomials associated with the Gamma measure on , , that is,

. We have

thus is an eigenfunction of the (one-dimensional) Laguerre differential operator

associated with the eigenvalue . Observe that if we choose , and consider its formal -adjoint,

then The differential operator is considered the “natural" notion of derivative in the Laguerre case.

The Laguerre-Littlewood-Paley function can be defined as

where and is the Poisson-Laguerre semigroup, that is, the subordinated semigroup to the Laguerre semigroup; for more information see [3].

Moreover, observe that can be written as a singular integral with values in the Hilbert space

We could also consider the following Laguerre Littlewood-Paley functions:

the time Laguerre Littlewood-Paley, and

the spatial Laguerre Littlewood-Paley.

The -continuity of the Laguerre-Littlewood-Paley function was proved by Nowak in [6].

Theorem 3. Assume that and . There exists a constant such that

Observe that, from Theorem 3, we get immediately the -boundedness of and of .

Now, if we define

then, the -boundedness of is equivalent to the boundedness of from into

(iv) Finally, let us consider the asymptotic relations between Jacobi polynomials and other classical orthogonal polynomials (see [1], (5.3.4) and (5.6.3)):

(i)For Hermite polynomials,

where are the Gegenbauer polynomials defined as

(ii)For Laguerre polynomials,

Both relations hold uniformly in every closed interval of .

Actually these relations are expression of deeper relations between the measures and operators involved. As a consequence of those relations, we have the following technical results that were proved in [7] and are needed to prove Theorem 6.

Proposition 4 (norm relations).

(i)Let and define ; then and

(ii)Let and define ; then and

Proposition 5 (inner product relations). With the same notation as in Proposition 4,

(i)let , and then

(ii)let , and then

The results of this paper follow the same scheme of the proof given in [7], where this transference method was used to obtain the -boundedness for the Riesz transform in the Hermite and Laguerre case from the -boundedness for the Riesz transform in the Jacobi case. Unfortunately due to the nonlinearity of the Littlewood-Paley -function, the computations for the case are more involved.

2. Main Results

We want to obtain the -continuity for the Gaussian-Littlewood-Paley and the -continuity for the Laguerre-Littlewood-Paley from the -continuity of the Jacobi-Littlewood-Paley , in the one-dimensional case (), using a transference method based on the asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials. The case of the transference method in higher dimension is still open.

We will start considering the case ; more precisely we want to prove the following.

Theorem 6. The -boundedness for the Jacobi-Littlewood-Paley

implies

(i)the -boundedness for the Gaussian-Littlewood-Paley

(ii)the -boundedness for the Laguerre-Littlewood-Paley

Observe that this result implies that if (49) holds for big enough, then (50) holds and if (49) holds for big enough, then (51) holds.

Proof. Even though this is the Hilbertian case, we will give some details of the proof since the exact constants involved in the computations are crucial here.
(i) Let and define ; then

Let us study the first term, by Parseval’s identity,

Therefore, interchanging the integral with the series, using Lebesgue’s dominated convergence theorem, we get

In particular, taking , we obtain

Thus, we get using Proposition 4

Hence,

Now, let us consider the second term. First of all, observe that

Then, using Parseval’s identity, we get

Now as before, using Lebesgue’s dominated convergence theorem, interchanging the integral with the series, we get

Then, for the Gegenbauer case, , we get

On the other hand, again using Parserval’s identity, we have

Hence,

Therefore, using Proposition 4,

Now, taking ,

Now, by the -continuity of , we have

Finally,

(ii) Let and define and then analogously as in the Hermite case

where

Let us study the first term. Using Proposition 4 and Lebesgue’s dominated convergence theorem, interchanging the integral with the series, we get

Let us look now at the second term. First of all, observe that

then we get, using Parseval’s identity,

Hence,

Therefore,

and now, by the -continuity of , we have

Thus, finally,

Now we are going to consider the general case For the proof, we will follow the argument given by Betancor et al. in [8].

Theorem 7. Let and ; then the -boundedness for the Jacobi-Littlewood-Paley -function

implies

(i)the -boundedness for the Gaussian-Littlewood-Paley -function

(ii)the -boundedness for the Laguerre-Littlewood-Paley -function

From the vector representations of the Littlewood-Paley -functions, this result could be interpreted then as transference of vector-valued operators (moreover, for the case of the time Littlewood-Paley -functions, this can be interpreted as transference of -valued multipliers; see [9]).

Proof. (i) Assume that the operator is bounded in . Let , and for each define the function

Let big enough such that is contained in In what follows, will be taken satisfying that condition.
Now, from the boundedness of we have

that is to say,

where Now making the change of variables and taking we have

which implies

On the other hand, analogously we have, from the case ,

Now, define, for any and such that , the functions

Observe that where

for all and . Moreover,

Therefore, if    and for

hence is bounded in Now

Thus, making the change of variables , we get

Therefore,

and moreover,

On the other hand,

Then, from (93) and (94), we have

Analogously,

Since is bounded in , we get

Then, from (93) and (99) we have

for all Thus, is a bounded sequence in and By Bourbaki-Alaoglu’s theorem, there exists a subsequence such that and functions and satisfying

(a), as , in the weak topology on ,(b), as , in the weak topology on .

Moreover, and

Analogously, one gets

Observe that, for every ,

then, by Cauchy-Schwartz inequality , and therefore,

thus, , for all , so on Then from (100), we get

and therefore, from (99) and (103), there exists an increasing sequence , with , and a function , such that

(a)for each , as , in the weak topology of and in the weak topology of ,(b)

On the other hand,

Now, define

Let us study first the function . For a function ,

which converges absolutely. Then, taking the Cauchy product, we obtain

Then, for the Gegenbauer case, , taking

Thus,

Therefore,

Let us take

Thus,

Now, we want to prove that, for and such that ,

Indeed, by Minkowski integral inequality we have

then,

Using Gasper’s linearization of the product of Jacobi polynomials (see [10]),

where is defined in (4) and

Hence,

Then, by Parseval’s identity and Cauchy-Schwartz inequality, we have

Now,

where Thus

Therefore, for , we have

On the other hand, using Stirling’s approximation formula for the Gamma function,

we obtain, for big enough, that

Then, for big enough

Also,

Taking , for , we get

Now, if we have that, for near , there exists big enough such that ; that is, . Hence,

Therefore, for big enough, there exists such that

Then, for big enough, we have

Hence,

Thus, is a bounded sequence on so, by Bourbaki-Alaoglu’s theorem, there exists a sequence , such that, for all , converges weakly in to a function . Moreover,

Then, there exists a nondecreasing sequence such that

The study of is essentially analogous, so fewer details will be given. For we have

which converges absolutely. Then, again taking the Cauchy product, we obtain

then

Hence,

Taking

we can write

Similarly to the previous case, we want to prove that, for any and such that ,

Indeed, by Minkowski integral inequality we have

Now,

Again, using Gasper’s linearization of the product of Jacobi polynomials [10], we may write

Thus,

Then, using Parseval’s identity and Cauchy-Schwartz inequality, we have

Then, by a similar argument as in the previous case, we get

On the other hand,

Hence,

Then, for big enough, we have

Therefore, for big enough, we have

Similarly to the previous case, taking we have that for big enough there exists such that

Thus, for big enough, we have

Hence

Then, is a bounded sequence on so, by Bourbaki-Alaoglu’s theorem, there exists a sequence such that, for all , converges weakly in to a function . Moreover,

Therefore, there exists a nondecreasing sequence such that

On the other hand,

where

Defining, for each , , then since , as in the weak topology of and

we have, for all ,

also,

Now,

Then,

so we have

Thus,

On the other hand, given that

we get

then,

Now, for , we have

which converges. Therefore

Let For , let us define

We know that, for ,

and also

Now, for , there exists such that

then, for , we get

Therefore, for , we have

Let us then define

which is integrable in ; now for all we have

then, by Lebesgue’s dominated convergence theorem, we have

Similarly, we have

that is,

therefore, from (160) and (180), we get

(ii) The proof is essentially analogous to (i), so fewer details will be provided. Assume that the operator is bounded in . Let and , and then we have

where for ; that is,

Now making the change of variable we have

thus,

where Analogously, we get

Now, for each and , such that , define the functions

From the previous inequalities both series converge for all , and where

for all and . Now, as

we conclude that is bounded in . On the other hand,

and making the change of variable we get

Then,

Moreover,

Now,

and therefore, from (193) for and (196), we get

Analogously, using the fact that is bounded in ,

Then, from (193) and (196),

for all . Therefore is a bounded subsequence in with Thus by Bourbaki-Alaoglu’s theorem, there exists an increasing sequence with and functions and satisfying that

(a), as , in the weak topology of ,(b), as , in the weak topology of

Then, as in (i), we can conclude that there exists an increasing sequence such that , and a function , such that

(a)for each , as , in the weak topology on and in the weak topology on ,(b).

Analogously to the Hermite case, we have

Now. let us define

Let us first consider . For a function ,

Therefore, as in the previous case,

where

We want to prove that, for and ,

Now,

Then, again by Gasper’s linearization of the product of Jacobi polynomials [10] and using Parseval’s identity, we have

Using analogous boundedness argument as in the Hermite case, we get

Now, for big enough,

Thus, for big enough,

Taking , then, is big enough if is big enough and we have two cases.
(i) Case . In this case, we have

Therefore, for big enough, there exists such that

thus,

(ii) Case . For near , there exists big enough such that ; that is, . Then, analogously to the Hermite case,

Hence,

Therefore,

Thus, is a bounded sequence on , so, by Bourbaki-Alaoglu’s theorem, there exists a sequence such that, for all , converges weakly in to a function . Moreover,

Then, there exists a nondecreasing sequence , such that

For the function , we get similar estimates. Given ,

Thus,

where

Again, we want to prove that, for and ,

Let us see the following:

Thus, using Parseval’s identity,

Analogously to the previous case,

Also, given that

we get

Then we have for big enough

Hence, for big enough,

then, analogous to (211), we have

therefore,

Thus, is a bounded sequence on so, by Bourbaki-Alaoglu’s theorem, there exists a sequence such that, for all , converges weakly in to a function . Moreover,

Then, there exists a nondecreasing sequence , such that

Therefore, similarly to (156),

where

Defining, for each , , then since , as in the weak topology of and

we have, for all ,

and also

Now, given that

we get

Thus,

We want to prove that

Indeed, initially we have

From (8.22.1) of [1], for y , we have

Thus, there exist such that

hence,

On the other hand, since for big enough , then, there exist such that

Thus,

Then, for , we have

Now, analogously to the Hermite case, we have

For given that , we get for

where

then, by Lebesgue’s dominated convergence theorem, we have

Similarly, as before,

Thus,

hence, from (236) and (240), we have

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

G. Szegö, Orthogonal Polynomials, vol. 23 of Colloq. Publ., American Mathematical Society, Providence, RI, USA, 1959.

W. N. Bayley, “Generalized Hypergeometric Series,” in Cambridge Tracs in Mathematics and Mathematical Physics # 32, New York, NY, USA, 1964.

W. Urbina, Operators Semigroups associated to Classical Orthogonal Polynomials and Functional Inequalities, Lecture Notes of the French Mathematical Society (SMF), 2008.

A. Nowak and P. Sjögren, “Transform for Jacobi expansions,” Journal d'Analyse Mathématique, vol. 104, pp. 341–369, 2008.

C. E. Gutiérrez, “On the Riesz transforms for Gaussian measures,” Journal of Functional Analysis, vol. 120, no. 1, pp. 107–134, 1994.

A. Nowak, “On Riesz transforms for Laguerre expansions,” Journal of Functional Analysis, vol. 215, no. 1, pp. 217–240, 2004.

E. Navas and W. O. Urbina, “A Transference Result of the Lp Continuity from Jacobi Riesz Transform to the Gaussian and Laguerre Riesz Transforms,” Journal of Fourier Analysis and Applications, vol. 19, no. 5, pp. 910–942, 2013.

J. J. Betancor, J. C. Fariña, L. Rodriguez, and A. Sanabria, “Transferring boundedness from conjugate operators associated with Jacobi, Laguerre, and Fourier-Bessel expansions to conjugate operators in the Hankel setting,” Journal of Fourier Analysis and Applications, vol. 14, no. 4, pp. 493–513, 2008.

O. Blasco and P. Villarroya, “Transference of vector-valued multipliers on weighted Lp-spaces,” Canadian Journal of Mathematics, vol. 65, no. 3, pp. 510–543, 2013.

G. Gasper, “Linearization of the product of Jacobi polynomials,” Canadian Journal of Mathematics, vol. 22, pp. 171–175, 1970.