# A Transference Result of the [L.sup.p]-Continuity of the Jacobi Littlewood-Paley g-Function to the Gaussian and Laguerre Littlewood-Paley g-Function.

1. PreliminariesIn 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 [L.sup.p]-continuity for the Gaussian-Littlewood-Paley g-function and the L-continuity of the Laguerre-Littlewood-Paley g-function from the [L.sup.p]-continuity of the Jacobi-Littlewood-Paley g-function, in dimension one. We are going to use the normalizations given in G. Szego's book [1], for all classical polynomials.

(i) Jacobi Polynomials. For [alpha], [beta] > -1, the Jacobi polynomials [{[P.sup.([alpha],[beta]).sub.n]}.sub.n[member of]N] are defined (up to a multiplicative constant) as the orthogonal polynomials associated with the Jacobi measure [[mu].sub.[alpha],[beta]] (or beta measure) in (-1, 1), defined as

[[mu].sub.[alpha],[beta]](dx) = [[omega].sub.[alpha],[beta]](x) dx = [[eta].sub.[alpha],[beta]][chi](-1,1) (x)[(1-x).sup.[alpha]] [(1+ x}.sup.[beta]] dx, (1)

where [[eta].sub.[alpha],[beta]] = [1/2.sup[alpha]+[beta]+1] B([alpha]+1, [beta]+1)/ [2.sup.[alpha]+[beta]+1][GAMMA]([alpha] + 1)[GAMMA]([beta] + 1).

The function [[omega].sub.[alpha],[beta]] is called the (normalized) Jacobi weight.

The Jacobi polynomials can be obtained (up to a multiplicative constant) from the polynomial canonical basis {1, x, [x.sup.2], ..., [x.sup.n], ...} using the Gram-Schmidt orthogonalization process with respect to the inner product in [L.sup.2]([[mu].sub.[alpha],[beta]]). Thus we have the orthogonality property of Jacobi polynomials with respect to [[mu].sub.[alpha],[beta]],

[mathematical expression not reproducible] (2)

n, m = 0, 1, 2, ..., where

[mathematical expression not reproducible] (3)

We will denote the normalized Jacobi polynomial of degree n as

[p.sup.([alpha],[beta).sub.n] (x) = [P.sup.([alpha],[beta).sub.n] (x)/[P.sup.([alpha],[beta).sub.n] (1). (4)

On the other hand, the Jacobi polynomial of parameter ([alpha], [beta]) of degree n, [P.sup.([alpha],[beta).sub.n], is a polynomial solution of the Jacobi differential equation, with parameters [alpha], [beta], n,

(l - [x.sup.2]) y" + [[beta] - [alpha] - ([alpha] + [beta] + 2) x] y' + n(n + [alpha] + [beta] + 1) y = 0; (5)

that is, [P.sup.([alpha],[beta).sub.n] is an eigenfunction of the (one-dimensional) second-order diffusion operator

[L.sup.[alpha],[beta]] = -(1 - [x.sup.2]) [d.sup.2]/[dx.sup.2] - ([beta] - [alpha] - ([alpha] + [beta] + 2) x) d/dx, (6)

associated with the eigenvalue [[lambda].sup.[alpha]+[beta].sub.n] = n(n + [alpha] + [beta] + 1). [L.sup.[alpha],[beta]] is called the Jacobi differential operator. Observe that if we choose [[delta].sub.[alpha],[beta]] = [square root of (1 - [x.sup.2])](d/dx), and consider its formal [L.sup.2]([[mu].sub.[alpha],[beta]]-adjoint,

[mathematical expression not reproducible] (7)

then [L.sup.[alpha],[beta] = [[delta].sup.*.sub.[alpha],[beta]][[delta].sub.[alpha],[beta]]. The differential operator [[delta].sub.[alpha],[beta]] 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 (-1, 1), as

[T.sup.[alpha],[beta].sub.t] f (x) = [[integral].sup.1.sub.-1] [p.sup.[alpha],[beta]] (t, x, y) f (y) [[mu].sub.[alpha],[beta]] (dy), (8)

where

[mathematical expression not reproducible] (9)

No simple explicit representation of [p.sup.[alpha],[beta]](t, x, y) is known, since the eigenvalues [[lambda].sub.n] are not linearly distributed; nevertheless there is one obtained by Gasper which is analog of Bailey's [F.sub.4] representation of the kernel of Abel summability for Jacobi series, also called the Jacobi-Poisson integral; see [2]. From that form, taking x = -y = 1, it can be proved that [p.sup.[alpha],[beta]](t, x, y) is a positive kernel.

{[T.sup.[alpha],[beta].sub.t]} is called the Jacobi semigroup and can be proved, that is, a Markov semigroup; for details see [3]. The Jacobi-Poisson semigroup {[P.sup.[alpha],[beta].sub.t]} can be defined, using Bochner's subordination formula,

[mathematical expression not reproducible] (10)

as the subordinated semigroup of the Jacobi semigroup,

[mathematical expression not reproducible] (11)

For a function f [member of] [L.sup.2]([-1, 1], [[mu].sub.([alpha],[beta])] let us consider its Fourier- Jacobi expansion

[mathematical expression not reproducible] (12)

where

<f, [P.sup.([alpha],[beta]).sub.k]> = [[integral].sup.1.sub.-1] f (y) [P.sup.([alpha],[beta]).sub.k] (y) [[mu].sub.[alpha],[beta]] (dy). (13)

Then, the action of [T.sub.t] and [P.sub.t] can be expressed as

[mathematical expression not reproducible] (14)

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

[mathematical expression not reproducible] (15)

where [[nabla].sub.([alpha],[beta])] = ([partial derivative]/[partial derivative]t, [[delta].sub.[alpha],[beta]]) = ([partial derivative]/[partial derivative]t, [square root of (l - [x.sup.2])]([partial derivative]/[partial derivative]x)).

Moreover, observe that [g.sup.([alpha],[beta])] f can be written as a singular integral with values in the Hilbert space [L.sup.2]((0, [infinity]), dt/t)

[mathematical expression not reproducible] (16)

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

[mathematical expression not reproducible] (17)

the time Jacobi-Littlewood-Paley function, and

[mathematical expression not reproducible] (18)

the spatial Jacobi-Littlewood-Paley function.

The [L.sup.p]([[mu].sub.[alpha],[beta])]-boundedness of the Jacobi-Littlewood-Paley [g.sup.([alpha],[beta])]-function was proved by Nowak and Sjogren in [4].

Theorem 1. Assume that 1 < p < [infinity] and [alpha], [beta] [member of] [[-1/2, [infinity]).sup.d]. There exists a constant [c.sub.p] such that

[[parallel[g.sup.([alpha],[beta])] f[parallel].sub.p,([alpha],[beta])] [less than or equal to] [c.sub.p] [[parallel]f[parallel].sub.p,([alpha],[beta])]. (19)

Observe that, from Theorem 1, we get immediately the [L.sup.p]([[mu].sub.[alpha],[beta]])-boundedness of [g.sup.([alpha],[beta]).sub.1] and of [g.sup.([alpha],[beta]).sub.2].

Now, if we define

[T.sup.([alpha],[beta])] f(x, t) = t[[nabla].sub.([alpha],[beta])][P.sup.[alpha],[beta].sub.t] f (x), (x, t) [member of] (-1, 1) x (0, [infinity]), (20)

then, the [L.sup.p]([[mu].sub.[alpha],[beta]])-boundedness of [g.sup.([alpha],[beta])] f is equivalent to the boundedness of [T.sup.([alpha],[beta])] from [L.sup.p]([[mu].sub.[alpha],[beta]]) into [mathematical expression not reproducible] (for the case of [g.sup.([alpha],[beta]).sub.1] that could be also linked to [L.sup.2]((0, [infinity]), dt/t)-Jacobi multipliers).

(ii) Hermite Polynomials. The Hermite polynomials, [{[H.sub.n]}.sub.n], are defined as the orthogonal polynomials associated with the Gaussian measure in R, [mathematical expression not reproducible], that is,

[[integral].sup.[infinity].sub.-[infinity]] [H.sub.n] (y) [H.sub.m] (y) [gamma] (dy) = [2.sup.n]n![[delta].sub.n,m], (21)

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

[H.sub.2n+1] (0) = 0,

[H.sub.2n] (0) = [(-1).sup.n] (2n)!/n!. (22).

We have

[H'.sub.n] (x) = 2n[H.sub.n-1] (x), [H".sub.n] (x) - 2x[H'.sub.n] (x) + 2n[H.sub.n] (x) = 0; (23)

thus [H.sub.n] is an eigenfunction of the one-dimensional Ornstein-Uhlenbeck operator (or harmonic oscillator operator),

L = 1/2 [d.sup.2.sub.[dx.sup.2] + x d/dx, (24)

associated with the eigenvalue [[lambda].sub.n] = n. Observe that if we choose [[delta].sub.[gamma]] = (1/[square root of (2))(d/dx), and consider its formal [L.sup.2]([gamma])-adjoint,

[[delta].sup.*.sub.[gamma]] = 1/[square root of (2)] d/dx + [square root of (2x)]I, (25)

then L = [[delta].sup.*.sub.[gamma]][[delta].sub.[gamma]]. The differential operator [[delta].sub.y] is considered the "natural" notion of derivative in the Hermite case.

The Gaussian-Littlewood-Paley [g.sup.[gamma]] function can be defined as

[g.sup.[gamma]] f (x) = [([[integral].sup.[infinity].sub.0] t [[absolute value of ([[nbla].sub.[gamma]] [P.sup.[gamma].sub.t] f (x))].sup.2] dt).sup.1/2], (26)

where [[nabla].sub.[gamma]] = ([partial derivative]/[partial derivative]t, [[delta].sub.[gamma]]) = ([partial derivative]/ [partial derivative]t, (1/[square root of (2)])([partial derivative]/[partial derivative]x)) and {[P.sup.[gamma].sub.t]} is the Poisson-Hermite semigroup, that is, the subordinated semigroup to the Ornstein-Uhlenbeck semigroup; for more information see [3].

Moreover, observe that [g.sup.[alpha]] f can be written as a singular integral with values in the Hilbert space [L.sup.2]((0, [infinity]), dt/t)

[mathematical expression not reproducible] (27)

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

[mathematical expression not reproducible] (28)

the time Gaussian-Littlewood-Paley, and

[mathematical expression not reproducible] (29)

the spatial Gaussian-Littlewood-Paley.

The [L.sup.p]-continuity of the Gaussian-Littlewood-Paley [g.sup.[gamma]] function was proved by Gutierrez in [5].

Theorem 2. Assume that 1 < p [infinity]. There exists a constant [c.sub.p] such that

[[parallel][g.sup.[gamma]] f[parallel].sub.p,[gamma]] [less than or equal to] [c.sub.p] [[parallel]f[parallel].sub.p,[gamma]]. (30)

Observe that, from Theorem 2, we get immediately the [L.sup.p]([gamma])-boundedness of [g.sup.[gamma].sub.1] and of [g.sup.[gamma].sub.2].

Now, if we define

[T.sup.[gamma]] f (x, t) = t[[nabla].sub.[gamma]][P.sup.[gamma].sub.t] f (x), (x, t) [member of] R x (0, [infinity]), (31)

then, the [L.sup.p]([gamma])-boundedness of [g.sup.[gamma]] f is equivalent to the boundedness of [T.sup.[gamma]] from [L.sup.p]([gamma]) into [mathematical expression not reproducible].

(iii) Laguerre Polynomials. For [alpha] > -1, the Laguerre polynomials {[L.sup.[alpha].sub.k]} are defined as the orthogonal polynomials associated with the Gamma measure on (0, [infinity]), [[mu].sub.[alpha](dx) = [[chi].sub.(0,[infinity])](x)([x.sup.[alpha][e.sup.-x]/[GAMMA]([alpha] + 1))dx, that is,

[mathematical expression not reproducible] (32)

n,m = 0, 1, 2, ... We have

[mathematical expression not reproducible] (33)

thus [L.sup.[alpha].sub.k] is an eigenfunction of the (one-dimensional) Laguerre differential operator

[L.sup.[alpha]] = -x [d.sup.2]/[dx.sup.2] - ([alpha] + 1 - x) d/dx, (34)

associated with the eigenvalue [[lambda].sub.k] = k. Observe that if we choose [[delta].sub.[alpha]] = [square root of (x)](d/dx), and consider its formal [L.sup.2]([alpha])-adjoint,

[[delta].sup.*.sub.[alpha]] = -[square root of (x)] d/dx + [[alpha] + 1/2/[square root of (x)] + [square root of (x)]] I, (35)

then [L.sup.[alpha]] = [[delta].sup.*.sub.[alpha]][[delta].sub.[alpha]]. The differential operator is considered the "natural" notion of derivative in the Laguerre case.

The Laguerre-Littlewood-Paley g function can be defined as

[g.sup.[alpha]] f (x) = [([[integral].sup.[infinity].sub.0] t [[absolute value of ([[nabla].sub.[alpha]][P.sup.[alpha].sub.t] f (x))].sup.2] dt).sup.1/2], (36)

where [[nabla].sub.[alpha]] = ([partial derivative]/[partial derivative]t, [[delta].sub.[alpha]]) = ([partial derivative]/[partial derivative]t, [square root of (x)]([partial derivative]/[partial derivative]x)) and {[P.sup.[alpha].sub.t]} is the Poisson-Laguerre semigroup, that is, the subordinated semigroup to the Laguerre semigroup; for more information see [3].

Moreover, observe that [g.sup.[alpha]] f can be written as a singular integral with values in the Hilbert space [L.sup.2]((0, [infinity]), dt/t)

[mathematical expression not reproducible] (37)

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

[g.sup.[alpha].sub.1] f (x) = [([[integral].sup.[infinity].sub.0] t [[absolute value of ([partial derivative][P.sup.[alpha].sub.t]f/[partial derivative]t (x))].sup.2] dt).sup.1/2], (38)

the time Laguerre Littlewood-Paley, and

[mathematical expression not reproducible] (39)

the spatial Laguerre Littlewood-Paley.

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

Theorem 3. Assume that 1 < p < [infinity] and [alpha] [member of] [[1/2, [infinity]).sup.d]. There exists a constant [c.sub.p] such that

[[parallel][g.sup.[alpha]][parallel].sub.p,[alpha]] [less than or equal to] [c.sub.p] [[parallel]f[parallel].sub.p,[alpha]]. (40)

Observe that, from Theorem 3, we get immediately the [L.sup.p]([[mu].sub.[alpha]])-boundedness of [g.sup.[alpha].sub.1] and of [g.sup.[alpha].sub.2].

Now, if we define

[T.sup.[alpha]] f (x, t) = t[[nabla].sub.[alpha]][P.sup.[alpha].sub.t] f (x), (x, t) [member of] [(0, [infinity]).sup.2], (41)

then, the [L.sup.p]([[mu].sub.[alpha]])-boundedness of [g.sup.[alpha]] f is equivalent to the boundedness of [T.sup.[alpha]] from [L.sup.p]([[mu].sub.[alpha]]) into [mathematical expression not reproducible].

(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,

[mathematical expression not reproducible] (42)

where {[C.sup.[lambda].sub.n](x)} are the Gegenbauer polynomials defined as

[C.sup.[lambda].sub.n] (x) = [GAMMA] ([lambda] + 1/2) [GAMMA] (n + 2[lambda])/ [GAMMA] (2[lambda]) [GAMMA] (n + [lambda] + 1/2) [P.sup.([lambda]-1/2,[lambda]-1/2).sub.n] (x). (43)

(ii) For Laguerre polynomials,

[mathematical expression not reproducible] (44)

Both relations hold uniformly in every closed interval of R.

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 f [member of] [L.sup.2](R, [gamma]) and define [f.sub.[beta]](x) = f([square root of ([lambda])]x)[[chi].sub.-1, 1]](x); then [f.sub.[lambda]] [member of] [L.sup.2]([-1, 1], [[mu].sub.[lambda]] and

[mathematical expression not reproducible] (45)

(ii) Let f [member of] [L.sup.2](R, [[mu].sub.[alpha]]) and define [f.sub.[beta]](x) = f(([beta]/2)(1 - x))[[chi].sub.[-1,1]](x); then [f.sub.[beta]] [member of] [L.sup.2] ([-1, 1], [[mu].sub.([alpha],[beta])] and

[mathematical expression not reproducible] (46)

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

(i) let f [member of] [L.sup.2](R, [gamma]), and then

[mathematical expression not reproducible] (47)

(ii) let f [member of] [L.sup.2](R, [[mu].sub.[alpha]]), and then

[mathematical expression not reproducible] (48)

The results of this paper follow the same scheme of the proof given in [7], where this transference method was used to obtain the [L.sup.p]-boundedness for the Riesz transform in the Hermite and Laguerre case from the [L.sup.p]-boundedness for the Riesz transform in the Jacobi case. Unfortunately due to the nonlinearity of the Littlewood-Paley g-function, the computations for the case p [not equal to] 2 are more involved.

2. Main Results

We want to obtain the [L.sup.p]-continuity for the Gaussian-Littlewood-Paley g and the [L.sup.p]-continuity for the Laguerre-Littlewood-Paley g from the [L.sup.p]-continuity of the Jacobi-Littlewood-Paley g, in the one-dimensional case (d = 1), 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 p = 2; more precisely we want to prove the following.

Theorem 6. The [L.sup.2]([[mu].sub.[alpha],[beta]])-boundedness for the Jacobi-Littlewood-Paley g

[[parallel][g.sup.([alpha],[beta])] f[parallel].sub.2,([alpha],[beta])] [less than or equal to] [C.sub.2] [[parallel]f[parallel].sub.2,([alpha],[beta])] (49)

implies

(i) the [L.sup.2]([gamma])-boundedness for the Gaussian-Littlewood-Paley g

[[parallel][g.sup.[gamma]] f[parallel].sub.2,[gamma]] [less than or equal to] [C.sub.2] [[parallel]f[paralel].sub.2,[gamma]], (50)

(ii) the [L.sup.2]([[mu].sub.[alpha]])-boundedness for the Laguerre-Littlewood-Paley g

[[parallel][g.sup.[alpha]] f[parallel].sub.2,[alpha]] [less than or equal to] [C.sub.2] [[parallel]f[paralel].sub.2,[alpha]], (51)

Observe that this result implies that if (49) holds for [alpha] = [beta] big enough, then (50) holds and if (49) holds for [alpha], [beta] > -1 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 f [member of] [L.sup.2](R, [gamma]) and define [f.sub.[lambda]](x) = f([square root of ([lambda])]x); then

[mathematical expression not reproducible] (52)

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

[mathematical expression not reproducible] (53)

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

[mathematical expression not reproducible] (54)

In particular, taking [alpha] = [beta] = [lambda] - 1/2, we obtain

[mathematical expression not reproducible] (55)

Thus, we get using Proposition 4

[mathematical expression not reproducible] (56)

Hence,

[mathematical expression not reproducible] (57)

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

[mathematical expression not reproducible] (58)

Then, using Parseval's identity, we get

[mathematical expression not reproducible] (59)

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

[mathematical expression not reproducible] (60)

Then, for the Gegenbauer case, [alpha] = [beta] = [lambda] - 1/2, we get

[mathematical expression not reproducible] (61)

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

[mathematical expression not reproducible] (62)

Hence,

[mathematical expression not reproducible] (63)

Therefore, using Proposition 4,

[mathematical expression not reproducible] (64)

Now, taking [alpha] = [beta] = [lambda] - 1/2,

[mathematical expression not reproducible] (65)

Now, by the [L.sup.2]-continuity of [g.sup.([lambda]-1/2,[lambda]-1/2)], we have

[mathematical expression not reproducible] (66)

Finally,

[mathematical expression not reproducible] (67)

(ii) Let f [member of] [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]), and define [f.sub.[beta]](x) = f(([beta]/2)(1 - x)) and then analogously as in the Hermite case

[mathematical expression not reproducible] (68)

where

[mathematical expression not reproducible] (69)

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

[mathematical expression not reproducible] (70)

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

[mathematical expression not reproducible] (71)

then we get, using Parseval's identity,

[mathematical expression not reproducible] (72)

Hence,

[mathematical expression not reproducible] (73)

Therefore,

[mathematical expression not reproducible] (74)

and now, by the [L.sup.2]-continuity of [g.sup.([alpha,[beta])], we have

[mathematical expression not reproducible] (75)

Thus, finally,

[mathematical expression not reproducible] (76)

Now we are going to consider the general case p [not equal to] 2. For the proof, we will follow the argument given by Betancor et al. in [8].

Theorem 7. Let [alpha], [beta] > -1 and 1 < p < [infinity]; then the [L.sup.p] ([[mu].sub.[alpha],[beta]])-boundedness for the Jacobi-Littlewood-Paley g-function

[[parallel][g.sup.([alpha],[beta])] f[parallel].sub.p,([alpha],[beta])] [less than or equal to] [C.sub.p] [[parallel]f[parallel].sub.p,([alpha],[beta])] (77)

implies

(i) the [L.sup.p] ([gamma])-boundedness for the Gaussian-Littlewood-Paley g-function

[[parallel][g.sup.[gamma]] f[parallel].sub.p,[gamma]] [less than or equal to] [C.sub.p] [[parallel]f[parallel].sub.p,[gamma]]], (78)

(ii) the [L.sup.p]([[mu].sub.[alpha]])-boundedness for the Laguerre-Littlewood-Paley g-function

[[parallel][g.sup.[alpha]] f[parallel].sub.p,[alpha]] [less than or equal to] [C.sub.p] [[parallel]f[parallel].sub.p,[alpha]]], (79)

From the vector representations of the Littlewood-Paley g-functions, this result could be interpreted then as transference of vector-valued operators (moreover, for the case of the time Littlewood-Paley g-functions, this can be interpreted as transference of [L.sup.2]((0, [infinity]), dt/t)-valued multipliers; see [9]).

Proof. (i) Assume that the operator [g.sup.([lambda]-1/2[lambda]-1/2)] is bounded in [L.sup.p]([-1, 1], [[mu].sub.[lambda]]). Let [phi] [member of] [C.sup.[infinity].sub.0](R), and for each [lambda] > 0 define the function

[[phi].sub.[lambda]] (x) = [phi] ([square root of ([lambda])]x), (80)

x [member of] R. Let [lambda] big enough such that supp [[phi].sub.[lambda]] is contained in [-1, 1]. In what follows, [lambda] will be taken satisfying that condition.

Now, from the boundedness of [g.sup.([lambda]-1/2,[lambda]-1/2)] we have

[[parallel][g.sup.([lambda]-1/2,[lambda]-1/2)] [[phi].sub.[lambda]][parallel].sub.p,[lambda]] [less than or equal to] C [[parallel][[phi].sub.[lambda]][parallel].sub.p,[lambda]]; (81)

that is to say,

[mathematical expression not reproducible] (82)

where [[nabla].sub.[lambda]] = [[nabla].sub.([lambda]-1/2,[lambda]-1/2)]. Now making the change of variables x = y/[square root of ([lambda])] and taking Z([lambda]) = ([[lambda].sup.1/2][[[GAMMA]([lambda])].sup.2][2.sup.2[lambda]])/ 2[pi][GAMMA](2[lambda]) we have

[mathematical expression not reproducible] (83)

which implies

[mathematical expression not reproducible] (84)

On the other hand, analogously we have, from the case p = 2,

[mathematical expression not reproducible] (85)

Now, define, for any K [member of] N and [lambda] > 0 such that [square root of ([lambda]) > K, the functions

[mathematical expression not reproducible] (86)

Observe that [F.sub.[lambda],K] = [f.sub.[lambda],K][[OMEGA].sub.[lambda]], where

[mathematical expression not reproducible] (87)

for all K [member of] N and [square root of ([lambda])] > K. Moreover,

[mathematical expression not reproducible] (88)

Therefore, if p [less than or equal to] 2 [absolute value of ([[OMEGA].sub.[lambda](y))] [less than or equal to] 1 and for p > 2

[mathematical expression not reproducible] (89)

hence [[OMEGA].sub.[lambda]] is bounded in [-K, K], Now

[mathematical expression not reproducible] (90)

Thus, making the change of variables x = y/[square root of ([lambda])], we get

[mathematical expression not reproducible] (91)

Therefore,

[mathematical expression not reproducible] (92)

and moreover,

[mathematical expression not reproducible] (99)

On the other hand,

[mathematical expression not reproducible] (94)

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

[[parallel][F.sub.[lambda],K][parallel].sub.2,[gamma]] [less than or equal to] C [[parallel][phi][parallel].sub.2,[gamma]]. (95)

Analogously,

[mathematical expression not reproducible] (96)

Since [[OMEGA].sub.[lambda]] is bounded in [-k, k], we get

[mathematical expression not reproducible] (97)

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

[[parallel][F.sub.[lambda],K][parallel].sub.p,[gamma]] [less than or equal to] C [[parallel][phi][parallel].sub.p,[gamma]]. (98)

for all [square root of ([lambda])] > A. Thus, [F.sub.[lambda],K] is a bounded sequence in [L.sup.2](R, [gamma]) and [L.sup.p](R, [gamma]). By Bourbaki-Alaoglu's theorem, there exists a subsequence [([[lambda].sub.j]).sub.j[member of]N such that [lim.sub.j[right arrow][infinity]][[lambda].sub.j] = [infinity] and functions [F.sub.K] [member of] [L.sup.2] (L, [gamma]) and [f.sub.K] [member of] [L.sup.p] (R, [gamma]) satisfying

(a) [mathematical expression not reproducible], as j [right arrow] [infinity], in the weak topology on [L.sup.2](R, [gamma]),

(b) [mathematical expression not reproducible], as j [right arrow] [infinity], in the weak topology on [L.sup.p](R, [gamma]).

Moreover, supp [F.sub.K] [union] supp [f.sub.K] [subset or equal to] [-K, K], and

[mathematical expression not reproducible] (99)

Analogously, one gets

[[parallel][F.sub.K][parallel].sub.p,[gamma]] [less than or equal to] C [[parallel][phi][parallel].sub.p,[gamma]]. (100)

Observe that, for every k [member of] N,

[[tau].sub.K] (g) = [[integral].sup.[infinity].sub.-[infinity]] g(x) [[chi].sub.[-K,K]] (x) dx; (101)

then, by Cauchy-Schwartz inequality [[tau].sub.K] [member of] ([L.sup.2][(R, [gamma])).sup.*], and therefore,

[mathematical expression not reproducible] (102)

thus, [F.sub.k] = [f.sub.K] a.e. on [-K, K], for all K [member of] N, so [F.sub.K] = [f.sub.K] a.e. on R. Then from (100), we get

[[parallel][F.sub.K][parallel].sub.p,[gamma]] [less than or equal to] C [[parallel][phi][parallel].sub.p,[gamma]]. (103)

and therefore, from (99) and (103), there exists an increasing sequence [{[[lambda].sub.j]}.sub.j[member of]N] [subset] (0, [infinity]), with [lim.sub.j[right arrow][infinity]][[lambda].sub.j] = [infinity], and a function F [member of] [L.sup.p](R, [gamma]) [interseciton] [L.sup.2](R, [gamma]), such that

(a) for each K [member of] N, [mathematical expression not reproducible], as j [right arrow] [infinity], in the weak topology of [L.sup.2](R, [gamma]) and in the weak topology of [L.sup.p](R, [gamma]),

(b) [[parallel][F.sub.K][parallel].sub.p,[gamma]] [less than or equal to] C [[parallel][phi][parallel].sub.p,[gamma]]. (100)

On the other hand,

[mathematical expression not reproducible] (104)

Now, define

[mathematical expression not reproducible] (105)

Let us study first the function [g.sub.1,[lambda]]. For a function [phi] [member of] [L.sup.2] (R, [gamma]),

[mathematical expression not reproducible] (106)

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

[mathematical expression not reproducible] (107)

Then, for the Gegenbauer case, [alpha] = [beta] = [lambda] - 1/2, taking [[??].sub.k] = k(k + 2[lambda])

[mathematical expression not reproducible] (108)

Thus,

[mathematical expression not reproducible] (109)

Therefore,

[mathematical expression not reproducible] (110)

Let us take

[mathematical expression not reproducible] (111)

Thus,

[mathematical expression not reproducible] (112)

Now, we want to prove that, for K [member of] N and [lambda] > 0 such that [square root of ([lambda])] > K,

[mathematical expression not reproducible] (113)

Indeed, by Minkowski integral inequality we have

[mathematical expression not reproducible] (114)

then,

[mathematical expression not reproducible] (115)

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

[mathematical expression not reproducible] (116)

where [p.sup.([alpha],[beta]).sub.i] (x) is defined in (4) and

[mathematical expression not reproducible] (117)

Hence,

[mathematical expression not reproducible] (118)

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

[mathematical expression not reproducible] (119)

Now,

[mathematical expression not reproducible] (120)

where q = max([alpha], [beta]). Thus

[mathematical expression not reproducible] (121)

Therefore, for [alpha] = [beta] = [lambda] - 1/2, we have

[mathematical expression not reproducible] (122)

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

[GAMMA] (az + b) ~ [square root of (2[pi])][e.sup.-az] [(az).sup.az+b-1/2 ([absolute value of (arg z)] < [pi], a > 0), (123)

we obtain, for [lambda] big enough, that

[mathematical expression not reproducible] (124)

Then, for [lambda] big enough

[mathematical expression not reproducible] (125)

Also,

[mathematical expression not reproducible] (126)

Taking [lambda] = (k - n)[(1 + 2k).sup.2], for 1 [greater than or equal to] 1, we get

[mathematical expression not reproducible] (127)

Now, if 0 < t < 1 we have that, for t near 0, there exists k > 0 big enough such that 1/k [less than or equal to] t; that is, 1/[t.sup.2] [less than or equal to] [k.sup.2]. Hence,

[mathematical expression not reproducible] (128)

Therefore, for [lambda] big enough, there exists C > 0 such that

[mathematical expression not reproducible] (129)

Then, for [lambda] big enough, we have

[mathematical expression not reproducible] (130)

Hence,

[mathematical expression not reproducible] (131)

Thus, {[[integral].sup.[infinity].sub.0] [H.sup.N,1.sub.[lambda],K](t, x) dt} is a bounded sequence on [L.sup.2](R, [gamma]), so, by Bourbaki-Alaoglu's theorem, there exists a sequence [([[lambda].sub.j]).sub.j[member of]N], [lim.sub.j[right arrow][infinity]][[lambda].sub.j] = [infinity], such that, for all [mathematical expression not reproducible] converges weakly in [L.sup.2](R, [gamma]) to a function [H.sup.N,1.sub.K] [member of] [L.sup.2](R, [gamma]). Moreover,

[mathematical expression not reproducible] (132)

Then, there exists a nondecreasing sequence [([N.sub.j]).sub.j[member of]N] such that

[mathematical expression not reproducible] (133)

The study of [g.sub.2,[lambda]] is essentially analogous, so fewer details will be given. For [phi] [member of] [L.sup.2] (R, [gamma]) we have

[mathematical expression not reproducible] (134)

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

[mathematical expression not reproducible] (135)

then

[mathematical expression not reproducible] (136)

Hence,

[mathematical expression not reproducible] (137)

Taking

[mathematical expression not reproducible] (138)

we can write

[mathematical expression not reproducible] (139)

Similarly to the previous case, we want to prove that, for any K [member of] N and [lambda] > 0 such that [square root of ([lambda])] > K,

[mathematical expression not reproducible] (140)

Indeed, by Minkowski integral inequality we have

[mathematical expression not reproducible] (141)

Now,

[mathematical expression not reproducible] (142)

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

[mathematical expression not reproducible] (143)

Thus,

[mathematical expression not reproducible] (144)

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

[mathematical expression not reproducible] (145)

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

[mathematical expression not reproducible] (146)

On the other hand,

[mathematical expression not reproducible] (147)

Hence,

[mathematical expression not reproducible] (148)

Then, for [lambda] big enough, we have

[mathematical expression not reproducible] (149)

Therefore, for [lambda] big enough, we have

[mathematical expression not reproducible] (150)

Similarly to the previous case, taking [lambda] = (k - n)[(1 + 2k).sup.2] we have that for k big enough there exists C > 0 such that

[mathematical expression not reproducible] (151)

Thus, for [lambda] big enough, we have

[mathematical expression not reproducible] (152)

Hence

[mathematical expression not reproducible] (153)

Then, {[[integral].sup.[infinity].sub.0] [H.sup.N,2.sub.[lambda],K](t, x) dt} is a bounded sequence on [L.sup.2](R, [gamma]), so, by Bourbaki-Alaoglu's theorem, there exists a sequence [([[lambda].sub.j]).sub.j[member of]N] [lim.sub.j[right arrow][infinity]][[lambda].sub.j] = [infinity] such that, for all N [member of] N, [mathematical expression not reproducible] converges weakly in [L.sup.2] (R, [gamma]) to a function [H.sup.N,2.sub.K] [member of] [L.sup.2] (R, [gamma]). Moreover,

[mathematical expression not reproducible] (154)

Therefore, there exists a nondecreasing sequence [([N.sub.j]).sub.j[member of]N] such that

[mathematical expression not reproducible] (155)

On the other hand,

[mathematical expression not reproducible] (156)

where

[mathematical expression not reproducible] (157)

Defining, for each [mathematical expression not reproducible], then since [F.sub.[lambda],K] [right arrow] [F.sub.K], as j [right arrow] [infinity] in the weak topology of [L.sup.2](R, [gamma]) and

[mathematical expression not reproducible] (158)

we have, for all m [member of] N,

[mathematical expression not reproducible] (159)

also,

[mathematical expression not reproducible] (160)

Now,

[mathematical expression not reproducible] (161)

Then,

[mathematical expression not reproducible] (162)

so we have

[mathematical expression not reproducible] (163)

Thus,

[mathematical expression not reproducible] (164)

On the other hand, given that

[mathematical expression not reproducible] (165)

we get

[mathematical expression not reproducible] (166)

then,

[mathematical expression not reproducible] (167)

Now, for t [greater than or equal to] 1, we have

[mathematical expression not reproducible] (168)

which converges. Therefore

[mathematical expression not reproducible] (169)

Let 0 < t < 1. For y [member of] [-K, K], let us define

[mathematical expression not reproducible] (170)

We know that, for y [member of] [-K, K],

[mathematical expression not reproducible] (171)

and also

[mathematical expression not reproducible] (172)

Now, for [epsilon] = 1, there exists M([epsilon], y) > 0 such that

[mathematical expression not reproducible] (173)

then, for m > M, we get

Therefore, for y [member of] [-K, K], we have

[[integral].sup.1.sub.0] t (1 + C [(1 + [absolute value of (y)] [e.sup.-t]).sup.2] dt < [infinity]. (175)

Let us then define

[mathematical expression not reproducible] (176)

which is integrable in (0, [infinity]); now for all m > 0 we have

[mathematical expression not reproducible] (177)

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

[mathematical expression not reproducible] (178)

Similarly, we have

[mathematical expression not reproducible] (179)

that is,

[mathematical expression not reproducible] (180)

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

F (y) = [g.sup.[gamma]] (phi] (y)). (181)

(ii) The proof is essentially analogous to (i), so fewer details will be provided. Assume that the operator [g.sup.([alpha],[beta])] is bounded in [L.sup.p]([-1, 1], [[mu].sub.[alpha],[beta]]). Let [phi] [member of] [C.sup.[infinity].sub.0] (0, [infinity]) and [beta] > 0, and then we have

[mathematical expression not reproducible] (182)

where [[phi].sub.[beta]](x) = [phi](([beta]/2)(1 - x)), for x [member of] [-1, 1]; that is,

[mathematical expression not reproducible] (183)

Now making the change of variable x = 1 - (2/[beta])y, we have

[mathematical expression not reproducible] (184)

thus,

[mathematical expression not reproducible] (185)

where [Z.sub.[alpha],[beta]] = [GAMMA]([alpha] + [beta] + 2)/[GAMMA] ([alpha] + 1)[[beta].sup.[alpha]+2][GAMMA]([beta]). Analogously, we get

[mathematical expression not reproducible] (186)

Now, for each K [member of] N and [beta] >0, such that [beta] > K, define the functions

[mathematical expression not reproducible] (187)

From the previous inequalities both series converge for all y [member of] (0, [beta]), and [F.sub.[beta],K] = [f.sub.[beta],K] [[OMEGA].sub.[beta]], where

[[OMEGA].sub.[beta]] (y) = [e.sup.y/2-y/p] [(1 - y/[beta]).sup.[beta]/2-[beta]/p] (188)

for all K [member of] N and [beta] > K. Now, as

[mathematical expression not reproducible] (189)

we conclude that [[OMEGA].sub.[beta]] is bounded in (0, K). On the other hand,

[mathematical expression not reproducible] (190)

and making the change of variable x = 1 - (2/[beta])y we get

[mathematical expression not reproducible] (191)

Then,

[mathematical expression not reproducible] (192)

Moreover,

[mathematical expression not reproducible] (193)

Now,

[mathematical expression not reproducible] (194)

and therefore, from (193) for p = 2 and (196), we get

[mathematical expression not reproducible] (195)

Analogously, using the fact that [[OMEGA].sub.[beta]] is bounded in (0, K),

[mathematical expression not reproducible] (196)

Then, from (193) and (196),

[mathematical expression not reproducible] (197)

for all [beta] > K. Therefore {[F.sub.[beta],K]} is a bounded subsequence in [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]) with [L.sup.p]((0, [infinity]), [[mu].sub.[alpha]]). Thus by Bourbaki-Alaoglu's theorem, there exists an increasing sequence [{[[beta].sub.j]}.sub.j[member of]N] with [lim.sub.j[right arrow][infinity]][[beta].sub.j] = [infinity] and functions [F.sub.K] [member of] [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]) and [f.sub.K] [member of] [L.sup.p]((0, [infinity]), [[mu].sub.[alpha]]) satisfying that

(a) [mathematical expression not reproducible], as j [right arrow] [infinity], in the weak topology of [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]),

(b) [mathematical expression not reproducible], as j [right arrow] [infinity], in the weak topology of [L.sup.p]((0, [infinity]), [[mu].sub.[alpha]]).

Then, as in (i), we can conclude that there exists an increasing sequence [([[beta].sub.j]).sub.j[member of]N] [subset] (0, [infinity]) such that [lim.sub.j[right arrow][[beta].sub.j] = [infinity], and a function F [member of] [L.sup.p]((0, [infinity]), [[mu].sub.[alpha]]) [intersection] [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]), such that

(a) for each [mathematical expression not reproducible], as j [right arrow] [infinity], in the weak topology on [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]) and in the weak topology on [L.sup.p] ((0, [infinity]), [[mu].sub.[alpha]]),

(b) [mathematical expression not reproducible].

Analogously to the Hermite case, we have

[mathematical expression not reproducible] (198)

Now. let us define

[mathematical expression not reproducible] (199)

Let us first consider [g.sub.1,[beta]]. For a function [phi] [member of] [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]),

[mathematical expression not reproducible] (200)

Therefore, as in the previous case,

[mathematical expression not reproducible] (201)

where

[mathematical expression not reproducible] (202)

We want to prove that, for K [member of] N and [beta] > K,

[mathematical expression not reproducible] (203)

Now,

[mathematical expression not reproducible] (204)

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

[mathematical expression not reproducible] (205)

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

[mathematical expression not reproducible] (206)

Now, for [beta] big enough,

[mathematical expression not reproducible] (207)

Thus, for [beta] big enough,

[mathematical expression not reproducible] (208)

Taking [beta] = (k - n)[(1 + [2k.sup.2]).sup.2], then, k is big enough if [beta] is big enough and we have two cases.

(i) Case t [greater than or equal to] 1. In this case, we have

[mathematical expression not reproducible] (209)

Therefore, for k big enough, there exists C > 0 such that

[mathematical expression not reproducible] (210)

thus,

[mathematical expression not reproducible] (211)

(ii) Case 0 < t < 1. For t near 0, there exists k > 0 big enough such that 1/k [less than or equal to] t; that is, 1/[t.sup.2] [less than or equal to] [k.sup.2]. Then, analogously to the Hermite case,

[mathematical expression not reproducible] (212)

Hence,

[mathematical expression not reproducible] (213)

Therefore,

[mathematical expression not reproducible] (214)

Thus, {[[integral].sup.[infinity].sub.0] [H.sup.N,1.sub.[beta],K] (t, x)dt} is a bounded sequence on [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]), so, by Bourbaki-Alaoglu's theorem, there exists a sequence [([[lambda].sub.j]).sub.j[member of]N] [lim.sub.j[right arrow][infinity]] = [infinity] such that, for all N [member of] N, [mathematical expression not reproducible] converges weakly in [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]) to a function [H.sup.N,1.sub.K] [member of] [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]). Moreover,

[mathematical expression not reproducible] (215)

Then, there exists a nondecreasing sequence [([N.sub.j]).sub.j[member of]N], such that

[mathematical expression not reproducible] (216)

For the function [g.sub.2,[beta]], we get similar estimates. Given [phi] [member of] [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]),

[mathematical expression not reproducible] (217)

Thus,

[mathematical expression not reproducible] (218)

where

[mathematical expression not reproducible] (219)

Again, we want to prove that, for K [member of] N and [beta] > K,

[mathematical expression not reproducible] (220)

Let us see the following:

[mathematical expression not reproducible] (221)

Thus, using Parseval's identity,

[mathematical expression not reproducible] (222)

Analogously to the previous case,

[mathematical expression not reproducible] (223)

Also, given that

[mathematical expression not reproducible] (224)

we get

[mathematical expression not reproducible] (225)

Then we have for [beta] > 0 big enough

[mathematical expression not reproducible] (226)

Hence, for [beta] > 0 big enough,

[mathematical expression not reproducible] (227)

then, analogous to (211), we have

[mathematical expression not reproducible] (228)

therefore,

[mathematical expression not reproducible] (229)

Thus, {[[integral].sup.[infinity].sub.0] [H.sup.N,2.sub.[beta],K]] (t, x)dt} is a bounded sequence on [L.sup.2]((0, [infinity])), [[mu].sub.[alpha]]), so, by Bourbaki-Alaoglu's theorem, there exists a sequence [([[lambda].sub.j]).sub.j[member of]N] [lim.sub.j[right arrow][infinity]][[lambda].sub.j] = [infinity] such that, for all [mathematical expression not reproducible] converges weakly in [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]) to a function [H.sup.N,2.sub.K] [member of] [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]). Moreover,

[mathematical expression not reproducible] (230)

Then, there exists a nondecreasing sequence [(N.sub.j]).sub.j[member of]N], such that

[mathematical expression not reproducible] (231)

Therefore, similarly to (156),

[mathematical expression not reproducible] (232)

where

[mathematical expression not reproducible] (233)

Defining, for each [mathematical expression not reproducible], then since [F.sub.[beta],K] [right arrow] [F.sub.K], as j [right arrow] [infinity] in the weak topology of [L.sup.2]((0, [infinity]), [[mu].sub.[alpha]]) and

[mathematical expression not reproducible] (234)

we have, for all m [member of] N,

[mathematical expression not reproducible] (235)

and also

[mathematical expression not reproducible] (236)

Now, given that

[mathematical expression not reproducible] (237)

we get

[mathematical expression not reproducible] (238)

Thus,

[mathematical expression not reproducible] (239)

We want to prove that

[mathematical expression not reproducible] (240)

Indeed, initially we have

[mathematical expression not reproducible] (241)

From (8.22.1) of [1], for [alpha] y x > 0, we have

[mathematical expression not reproducible] (242)

Thus, there exist M > 0 such that

[mathematical expression not reproducible] (243)

hence,

[mathematical expression not reproducible] (244)

On the other hand, since for k big enough [[parallel][L.sup.[alpha].sub.k][parallel].sup.2.sub.[alpha]] [approximately equal to] [k.sup.[alpha]]/([GAMMA]([alpha] + 1)), then, there exist [N.sub.1] > 0 such that

[mathematical expression not reproducible] (245)

Thus,

[mathematical expression not reproducible] (246)

Then, for t [greater than or equal to] 1, we have

[mathematical expression not reproducible] (247)

Now, analogously to the Hermite case, we have

[mathematical expression not reproducible] (248)

For 0 < t < 1, given that [absolute value of (([partial derivative]/[partial derivative]t) [P.sup.[alpha].sub.t]([phi](y)))] [less than or equal to] C(1 + [absolute value of (y)])[e.sup.-t], we get for y [member of] (0, K)

[mathematical expression not reproducible] (249)

where

[mathematical expression not reproducible] (250)

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

[mathematical expression not reproducible] (251)

Similarly, as before,

[mathematical expression not reproducible] (252)

Thus,

[mathematical expression not reproducible] (253)

hence, from (236) and (240), we have F(y) = [g.sup.[alpha]]{([phi]{y)).

https://doi.org/10.1155/2018/9304964

Conflicts of Interest

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

References

[1] G. Szego, in Orthogonal Polynomials, vol. 23 of Colloq. Publ., American Mathematical Society, 1959.

[2] W. N. Bayley, "Generalized Hypergeometric Series," in Cambridge Tracs in Mathematics and Mathematical Physics # 32, New York, NY, USA, 1964.

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

[4] A. Nowak and P. Sjogren, "Transform for Jacobi expansions," Journal d'Analyse Mathematique, vol. 104, pp. 341-369, 2008.

[5] C. E. Gutierrez, "On the Riesz transforms for Gaussian measures," Journal of Functional Analysis, vol. 120, no. 1, pp. 107-134, 1994.

[6] A. Nowak, "On Riesz transforms for Laguerre expansions," Journal of Functional Analysis, vol. 215, no. 1, pp. 217-240, 2004.

[7] 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.

[8] J. J. Betancor, J. C. Farina, 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.

[9] 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.

[10] G. Gasper, "Linearization of the product of Jacobi polynomials," Canadian Journal of Mathematics, vol. 22, pp. 171-175, 1970.

Eduard Navas (1) and Wilfredo O. Urbina (iD) (2)

(1) Departamento de Matematicas, Universidad Nacional Experimental Francisco de Miranda, Punto Fijo, Venezuela

(2) Department 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

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Title Annotation: | Research Article |
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Author: | Navas, Eduard; Urbina, Wilfredo O. |

Publication: | Journal of Function Spaces |

Date: | Jan 1, 2018 |

Words: | 7178 |

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