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 operatorassociated 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 , aswhere 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 expansionwhere Then, the action of and can be expressed as Following the classical case, the Jacobi-Littlewood-Paley function can be defined aswhere
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 havethus 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 aswhere 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 havethus 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 aswhere 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 4Hence, 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 getThen, 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, andAnalogously, one getsObserve that, for every , then, by Cauchy-Schwartz inequality , and therefore, thus, , for all , so on Then from (100), we getand 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 haveOn 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 haveHence, 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 thatThe 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 haveTherefore, 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 thatOn 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 thatthen, 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 thatFor 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 thatTherefore, similarly to (156), where Defining, for each , , then since , as in the weak topology of and we have, for all , and alsoNow, given that we get Thus, We want to prove thatIndeed, 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 haveNow, 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.