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

Authors: Eduard Navas; Wilfredo O. Urbina
Journal: Journal of Function Spaces
 (2018)

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## Abstract

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


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## Body

## 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 obtainLp-continuity for the Gaussian-Littlewood-Paley g-function and the L-continuity of the Laguerre-Littlewood-Paley g-function from the Lp-continuity of the Jacobi-Littlewood-Paley g-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 α,β>-1, the Jacobi polynomials {Pn(α,β)}n∈N are defined (up to a multiplicative constant) as the orthogonal polynomials associated with the Jacobi measure μα,β (or beta measure) in (-1,1), defined as(1)μα,βdx=ωα,βxdx=ηα,βχ-1,1x1-xα1+xβdx, where ηα,β=1/2α+β+1B(α+1,β+1)=Γ(α+β+2)/2α+β+1Γ(α+1)Γ(β+1).The functionωα,β is called the (normalized)Jacobi weight.The Jacobi polynomials can be obtained (up to a multiplicative constant) from the polynomial canonical basis{1,x,x2,…,xn,…} using the Gram-Schmidt orthogonalization process with respect to the inner product in L2(μα,β). Thus we have theorthogonality property of Jacobi polynomials with respect to μα,β,(2)∫-∞∞Pnα,βyPmα,βyμα,βdy=ηα,βhnα,βδn,m=hn^α,βδn,m,n,m=0,1,2,…, where(3)hnα,β=2α+β+12n+α+β+1Γn+α+1Γn+β+1Γn+1Γn+α+β+1,hn^α,β=12n+α+β+1Γα+β+2Γn+α+1Γn+β+1Γα+1Γβ+1Γn+1Γn+α+β+1=Pnα,β2,α,β2.We will denote thenormalized Jacobi polynomial of degree n as(4)pnα,βx=Pnα,βxPnα,β1.On the other hand, the Jacobi polynomial of parameter(α,β) of degree n, Pnα,β, is a polynomial solution of theJacobi differential equation, with parameters α,β,n,(5)1-x2y′′+β-α-α+β+2xy′+nn+α+β+1y=0;that is, Pnα,β is an eigenfunction of the (one-dimensional) second-order diffusion operator(6)Lα,β=-1-x2d2dx2-β-α-α+β+2xddx,associated with the eigenvalue λnα+β=n(n+α+β+1). Lα,β is called theJacobi differential operator. Observe that if we choose δα,β=1-x2d/dx, and consider its formal L2(μα,β)-adjoint, (7)δα,β∗=-1-x2ddx+α+121+x1-x-β+121-x1+xI,then Lα,β=δα,β∗δα,β. The differential operator δα,β 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(8)Ttα,βfx=∫-11pα,βt,x,yfyμα,βdy,where (9)pα,βt,x,y=∑ke-kk+α+β+1thk^α,βPkα,βxPkα,βy. No simple explicit representation of pα,β(t,x,y) is known, since the eigenvalues λn are not linearly distributed; nevertheless there is one obtained by Gasper which is analog of Bailey’s F4 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 pα,β(t,x,y) is a positive kernel.{ T t α , β } is called theJacobi semigroup and can be proved, that is, a Markov semigroup; for details see [3]. The Jacobi-Poisson semigroup {Ptα,β} can be defined, using Bochner’s subordination formula, (10)e-λ1/2t=1π∫0∞e-uue-λt2/4udu, as the subordinated semigroup of the Jacobi semigroup, (11)Ptα,βfx=1π∫0∞e-uuTt2/4uα,βfxdu.For a functionf∈L2-1,1,μ(α,β) let us consider its Fourier- Jacobi expansion(12)f=∑k=0∞f,Pkα,βhk^α,βPkα,β,where (13)f,Pkα,β=∫-11fyPkα,βyμα,βdy. Then, the action of Tt and Pt can be expressed as (14)Ttα,βf=∑k=0∞f,Pkα,βhk^α,βe-λktPkα,β,Ptα,βf=∑k=0∞f,Pkα,βhk^α,βe-λktPkα,β. Following the classical case, the Jacobi-Littlewood-Paley g function can be defined as(15)gα,βfx=∫0∞t∇α,βPtα,βfx2dt1/2,where ∇(α,β)=∂/∂t,δα,β=∂/∂t,1-x2∂/∂x.Moreover, observe thatg(α,β)f can be written as a singular integral with values in the Hilbert space L2((0,∞),dt/t)(16)gα,βfx=t∇α,βPtα,βfL20,∞,dt/t. We could also consider the following Jacobi-Littlewood-Paley functions:(17)g1α,βfx=∫0∞t∂Ptα,βf∂tx2dt1/2, the time Jacobi-Littlewood-Paley function, and(18)g2α,βfx=∫0∞tδα,βPtα,βfx2dt1/2=∫0∞t1-x2∂Ptα,βf∂xx2dt1/2, the spatial Jacobi-Littlewood-Paley function.TheLp(μα,β)-boundedness of the Jacobi-Littlewood-Paley g(α,β)-function was proved by Nowak and Sjögren in [4].Theorem 1.
Assume that1<p<∞ and α,β∈[-1/2,∞)d. There exists a constant cp such that(19)gα,βfp,α,β≤cpfp,α,β.Observe that, from Theorem1, we get immediately the Lp(μα,β)-boundedness of g1(α,β) and of g2(α,β).Now, if we define(20)Tα,βfx,t=t∇α,βPtα,βfx,x,t∈-1,1×0,∞, then, the Lp(μα,β)-boundedness of g(α,β)f is equivalent to the boundedness of T(α,β) from Lp(μα,β) into LL2((0,∞),dt/t)p(μα,β)(for the case of g1(α,β) that could be also linked to L2((0,∞),dt/t)-Jacobi multipliers).(ii) Hermite Polynomials. TheHermite polynomials, {Hn}n, are defined as the orthogonal polynomials associated with the Gaussian measure in R, γ(dx)=e-x2/πdx, that is,(21)∫-∞∞HnyHmyγdy=2nn!δn,m,n,m=0,1,2,…, with thenormalization(22)H2n+10=0,H2n0=-1n2n!n!. We have(23)Hn′x=2nHn-1x,Hn′′x-2xHn′x+2nHnx=0;thus Hn is an eigenfunction of the one-dimensionalOrnstein-Uhlenbeck operator (or harmonic oscillator operator), (24)L=-12d2dx2+xddx,associated with the eigenvalue λn=n. Observe that if we choose δγ=1/2d/dx, and consider its formal L2(γ)-adjoint, (25)δγ∗=-12ddx+2xI, then L=δγ∗δγ. The differential operator δγ is considered the “natural" notion of derivative in the Hermite case.The Gaussian-Littlewood-Paleygγ function can be defined as(26)gγfx=∫0∞t∇γPtγfx2dt1/2,where ∇γ=∂/∂t,δγ=∂/∂t,1/2∂/∂x and {Ptγ} is the Poisson-Hermite semigroup, that is, the subordinated semigroup to the Ornstein-Uhlenbeck semigroup; for more information see [3].Moreover, observe thatgαf can be written as a singular integral with values in the Hilbert space L2((0,∞),dt/t)(27)gγfx=t∇γPtγfL20,∞,dt/t.We could also consider the following Gaussian Littlewood-Paley functions:(28)g1γfx=∫0∞t∂Ptγf∂tx2dt1/2, the time Gaussian-Littlewood-Paley, and(29)g2γfx=12∫0∞t∂Ptγf∂xx2dt1/2, the spatial Gaussian-Littlewood-Paley.TheLp-continuity of the Gaussian-Littlewood-Paley gγ function was proved by Gutiérrez in [5].Theorem 2.
Assume that1<p<∞. There exists a constant cp such that(30)gγfp,γ≤cpfp,γ.Observe that, from Theorem2, we get immediately the Lp(γ)-boundedness of g1γ and of g2γ.Now, if we define(31)Tγfx,t=t∇γPtγfx,x,t∈R×0,∞, then, the Lp(γ)-boundedness of gγf is equivalent to the boundedness of Tγ from Lp(γ) into LL2((0,∞),dt/t)p(γ).(iii) Laguerre Polynomials. For α>-1, theLaguerre polynomials{Lkα} are defined as the orthogonal polynomials associated with the Gamma measure on (0,∞), μα(dx)=χ(0,∞)(x)xαe-x/Γ(α+1)dx, that is,(32)∫0∞LnαyLmαyμαdy=n+αnδn,m=Γn+α+1Γα+1n!δn,m,n,m=0,1,2,…. We have(33)Lkαx′=-Lk-1α+1x,xLkαx′′+α+1-xLkαx′+kLkαx=0;thus Lkα is an eigenfunction of the (one-dimensional)Laguerre differential operator(34)Lα=-xd2dx2-α+1-xddx, associated with the eigenvalue λk=k. Observe that if we choose δα=xd/dx, and consider its formal L2(α)-adjoint, (35)δα∗=-xddx+α+1/2x+xI,then Lα=δα∗δα. The differential operator δα is considered the “natural" notion of derivative in the Laguerre case.The Laguerre-Littlewood-Paleyg function can be defined as(36)gαfx=∫0∞t∇αPtαfx2dt1/2,where ∇α=∂/∂t,δα=∂/∂t,x∂/∂x and {Ptα} is the Poisson-Laguerre semigroup, that is, the subordinated semigroup to the Laguerre semigroup; for more information see [3].Moreover, observe thatgαf can be written as a singular integral with values in the Hilbert space L2((0,∞),dt/t)(37)gαfx=t∇αPtαfL20,∞,dt/t.We could also consider the following Laguerre Littlewood-Paley functions:(38)g1αfx=∫0∞t∂Ptαf∂tx2dt1/2, the time Laguerre Littlewood-Paley, and(39)g2αfx=∫0∞tδαPtαfx2dt1/2=∫0∞tx∂Ptαf∂xx2dt1/2, the spatial Laguerre Littlewood-Paley.TheLp-continuity of the Laguerre-Littlewood-Paley g function was proved by Nowak in [6].Theorem 3.
Assume that1<p<∞ and α∈[1/2,∞)d. There exists a constant cp such that(40)gαfp,α≤cpfp,α.Observe that, from Theorem3, we get immediately the Lp(μα)-boundedness of g1α and of g2α.Now, if we define(41)Tαfx,t=t∇αPtαfx,x,t∈0,∞2, then, the Lp(μα)-boundedness of gαf is equivalent to the boundedness of Tα from Lp(μα) into LL2((0,∞),dt/t)p(μα).(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,(42)limλ→∞λ-n/2Cnλxλ=Hnxn!,where {Cnλ(x)} are the Gegenbauer polynomials defined as (43)Cnλx=Γλ+1/2Γn+2λΓ2λΓn+λ+1/2Pnλ-1/2,λ-1/2x.(ii)
For Laguerre polynomials,(44)limβ→∞Pnα,β1-2xβ=Lnαx.Both relations hold uniformly in every closed interval ofR.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)
Letf∈L2(R,γ) and define fλ(x)=f(λx)χ[-1,1](x); then fλ∈L2([-1,1],μλ) and(45)limλ→∞fλ2,λ=f2,γ. (ii)
Letf∈L2(R,μα) and define fβ(x)=fβ/2(1-x)χ[-1,1](x); then fβ∈L2([-1,1],μ(α,β)) and(46)limβ→∞fβ2,α,β=f2,α.Proposition 5 (inner product relations).
With the same notation as in Proposition4, (i)
letf∈L2(R,γ), and then(47)limλ→∞fλ,λ-k/2Ckλ=f,Hkk!,(ii)
letf∈L2(R,μα), and then(48)limβ→∞fβ,Pkα,β=f,Lkα.The results of this paper follow the same scheme of the proof given in [7], where this transference method was used to obtain the Lp-boundedness for the Riesz transform in the Hermite and Laguerre case from the Lp-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≠2 are more involved.


## 2. Main Results

We want to obtain theLp-continuity for the Gaussian-Littlewood-Paley g and the Lp-continuity for the Laguerre-Littlewood-Paley g from the Lp-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 casep=2; more precisely we want to prove the following.Theorem 6.
TheL2(μα,β)-boundedness for the Jacobi-Littlewood-Paley g(49)gα,βf2,α,β≤C2f2,α,βimplies(i)
theL2(γ)-boundedness for the Gaussian-Littlewood-Paley g(50)gγf2,γ≤C2f2,γ,(ii)
theL2(μα)-boundedness for the Laguerre-Littlewood-Paley g(51)gαf2,α≤C2f2,α.Observe that this result implies that if (49) holds for α=β big enough, then (50) holds and if (49) holds for α,β>-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) Letf∈L2(R,γ) and define fλ(x)=f(λx); then (52)gα,βfλ2,α,β2=∫-11∫0∞t∇α,βPtα,βfλx2dtμα,βdx=∫0∞t∫-11∂∂tPtα,βfλx2+1-x2∂∂xPtα,βfλx2μα,βdxdt.
Let us study the first term, by Parseval’s identity,(53)∫-11∂∂tPtα,βfλx2μα,βdx=∑k=1∞fλ,Pkα,βPkα,β2,α,β2λke-2tλk1/2.
Therefore, interchanging the integral with the series, using Lebesgue’s dominated convergence theorem, we get(54)∫0∞t∫-11∂∂tPtα,βfλx2μα,βdxdt=14∑k=1∞fλ,Pkα,βPkα,β2,α,β2=14fλ2,α,β2.
In particular, takingα=β=λ-1/2, we obtain (55)∫0∞t∫-11∂∂tPtλ-1/2,λ-1/2fλx2μλdxdt=14∫-11fλx2μλdx=14fλ2,λ2.
Thus, we get using Proposition4(56)limλ→∞∫0∞t∫-11∂∂tPtλ-1/2,λ-1/2fλx2μλdxdt=limλ→∞14fλ2,λ2=14f2,γ2=∑k=1∞f,HkHk2,γ2k∫0∞te-2tk1/2dt=∫0∞t∫-∞∞∂∂tPtγfxdγxdt.
Hence,(57)limλ→∞∫0∞t∫-11∂∂tPtλ-1/2,λ-1/2fλx2μλdxdt=∫0∞t∫-∞∞∂∂tPtγfx2dγxdt.
Now, let us consider the second term. First of all, observe that(58)1-x2Pk-1α+1,β+12,α,β2=4α+1β+12α+β+3α+β+3α+β+2Bα+2,β+2×∫-111-xα+11+xβ+1Pk-1α+1,β+1x2dx=4α+1β+1α+β+3α+β+2Pk-1α+1,β+12,α+1,β+12=4kk+α+β+1Pkα,β2,α,β2.
Then, using Parseval’s identity, we get(59)∫-111-x2∂∂xPtα,βfλx2μα,βdx=1-x2∂∂xPtα,βfλ2,α,β2=∑k=1∞fλ,Pkα,βPkα,β2,α,β22e-2tλk1/2k+α+β+1241-x2Pk-1α+1,β+12,α,β2=∑k=1∞fλ,Pkα,βPkα,β222e-2tλk1/2λkPkα,β2,α,β2=∑k=1∞fλ,Pkα,βPkα,β22λke-2tλk1/2.
Now as before, using Lebesgue’s dominated convergence theorem, interchanging the integral with the series, we get(60)∫0∞t∫-111-x2∂∂xPtα,βfλx2μα,βdxdt=∫0∞t∑k=1∞fλ,Pkα,βPkα,β2,α,β2λke-2tλk1/2dt=14∫-11fλx2μα,βdx.
Then, for the Gegenbauer case,α=β=λ-1/2, we get (61)∫0∞t∫-111-x2∂∂xPtλ-1/2,λ-1/2fλx2μλdxdt=14fλ2,λ2.
On the other hand, again using Parserval’s identity, we have(62)∫-∞∞12∂∂xPtγfx2dγx=∑k=1∞f,HkHk2,γ222k2e-2tk1/2Hk-12,γ2=∑k=1∞f,HkHk2,γ222k2e-2tk1/212kHk2,γ2=∑k=1∞f,HkHk2,γ2ke-2tk1/2.
Hence,(63)∫0∞t∫-∞∞12∂∂xPtγfx2dγxdt=∫0∞t∑k=1∞f,HkHk2,γ2ke-2tk1/2dt=14f2,γ2.
Therefore, using Proposition4, (64)limλ→∞∫0∞t∫-111-x2∂∂xPtλ-1/2,λ-1/2fλx2μλdxdt=limλ→∞14fλ2,λ2=14f2,γ2=∫0∞t∫-∞∞12∂∂xPtγfx2dγxdt.
Now, takingα=β=λ-1/2, (65)limλ→∞gλ-1/2,λ-1/2fλ2,λ2=limλ→∞∫0∞t∫-11∂∂tPtλ-1/2,λ-1/2fλx2μλdxdt+limλ→∞∫0∞t∫-111-x2∂∂xPtλ-1/2,λ-1/2fλx2μλdxdt=∫0∞t∫-∞∞∂∂tPtγfx2dγxdt+∫0∞t∫-∞∞12∂∂xPtγfx2dγxdt=gγf2,γ2.Now, by the L2-continuity of g(λ-1/2,λ-1/2), we have (66)gλ-1/2,λ-1/2fλ2,λ≤C2fλ2,λ.
Finally,(67)gγf2,γ=limλ→∞gλ-1/2,λ-1/2fλ2,λ≤C2limλ→∞fλ2,λ=C2f2,γ.
(ii) Letf∈L2((0,∞),μα), and define fβ(x)=fβ/2(1-x) and then analogously as in the Hermite case (68)gα,βfβ2,α,β2=∫0∞t∫-11∂∂tPtα,βfβx2+1-x2∂∂xPtα,βfβx2μα,βdxdt,where (69)∫0∞t∫-11∂∂tPtα,βfβx2μα,βdxdt=14∫-11fβx2μα,βdx=14fβ2,α,β2,∫0∞t∫-111-x2∂∂xPtα,βfλx2μα,βdxdt=14∫-11fβx2μα,βdx=14fβ2,α,β2.Let us study the first term. Using Proposition 4 and Lebesgue’s dominated convergence theorem, interchanging the integral with the series, we get (70)limβ→∞∫0∞t∫-11∂∂tPtα,βfβx2μα,βdxdt=limβ→∞14fβ2,α,β2=14f2,α2=∫0∞t∑k=1∞f,LkαLkα2,α2ke-2tk1/2dt=∫0∞t∫0∞∂∂tPtαfx2μαdxdt.Let us look now at the second term. First of all, observe that (71)xLk-1α+12,α2=∫0∞xLk-1α+1x2xαe-xΓα+1dx=α+1∫0∞Lk-1α+1x2xα+1e-xΓα+2dx=α+1Lk-1α+12,α+12=α+1Γk-1+α+1+1Γα+2k-1!=kLkα2,α2;then we get, using Parseval’s identity, (72)∫0∞x∂∂xPtαfx2μαdx=∑k=1∞f,LkαLkα2,α22e-2tk1/2xLk-1α+12,α2=∑k=1∞f,LkαLkα2,α2ke-2tk1/2.Hence, (73)limβ→∞∫0∞t∫-111-x2∂∂xPtα,βfβx2μα,βdxdt=limβ→∞14fβ2,α,β2=14f2,α2=∫0∞t∑k=1∞f,LkαLkα2,α2ke-2tk1/2dt=∫0∞t∫0∞x∂∂xPtαfx2μαdxdt.Therefore, (74)limβ→∞gα,βfβ2,α,β2=limβ→∞∫0∞t∫-11∂∂tPtα,βfβx2μα,βdxdt+limβ→∞∫0∞t∫-111-x2∂∂xPtα,βfβx2μα,βdxdt=∫0∞t∫0∞∂∂tPtαfx2μαdxdt+∫0∞t∫0∞x∂∂xPtαfx2μαdxdt=gαf2,α2and now, by the L2-continuity of g(α,β), we have (75)gα,βfβ2,α,β≤C2fβ2,α,β.Thus, finally, (76)gαf2,α=limβ→∞gα,βfβ2,α,β≤C2limβ→∞fβ2,α,β=C2f2,α.Now we are going to consider the general casep≠2. For the proof, we will follow the argument given by Betancor et al. in [8].Theorem 7.
Letα,β>-1 and 1<p<∞; then the Lp(μα,β)-boundedness for the Jacobi-Littlewood-Paley g-function(77)gα,βfp,α,β≤Cpfp,α,β implies (i)
theLp(γ)-boundedness for the Gaussian-Littlewood-Paley g-function(78)gγfp,γ≤Cpfp,γ,(ii)
theLp(μα)-boundedness for the Laguerre-Littlewood-Paley g-function(79)gαfp,α≤Cpfp,α.From the vector representations of the Littlewood-Paleyg-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 L2((0,∞),dt/t)-valued multipliers; see [9]).Proof.
(i) Assume that the operatorg(λ-1/2,λ-1/2) is bounded in Lp[-1,1],μλ. Let ϕ∈C0∞(R), and for each λ>0 define the function (80)ϕλx=ϕλx,x∈R. Let λ big enough such that suppϕλ is contained in [-1,1]. In what follows, λ will be taken satisfying that condition.
Now, from the boundedness ofg(λ-1/2,λ-1/2) we have (81)gλ-1/2,λ-1/2ϕλp,λ≤Cϕλp,λ; that is to say, (82)∫0∞t∇λPtλ-1/2,λ-1/2ϕλ2dt1/2p,λ≤Cϕλp,λ, where ∇λ=∇(λ-1/2,λ-1/2). Now making the change of variables x=y/λ and taking Z(λ)=λ1/2Γ(λ)222λ/2πΓ2λ we have (83)∫-λλ∫0∞t∇λPtλ-1/2,λ-1/2ϕλyλ2dt1/2pZλ1-y2λλ-1/2dy1/p≤Cϕλp,λ,which implies (84)∫-λλ∫0∞t∇λPtλ-1/2,λ-1/2ϕλyλ2dt1/21-y2λλ/p-1/2pey2/ppe-y2πdy1/p≤CZλ-1/pϕλp,λ.
On the other hand, analogously we have, from the casep=2, (85)∫-λλ∫0∞t∇λPtλ-1/2,λ-1/2ϕλyλ2dt1/21-y2λλ/2-1/4ey2/22e-y2πdy1/2≤CZλ-1/2ϕλ2,λ.
Now, define, for anyK∈N and λ>0 such that λ>K, the functions (86)Fλ,Ky=∫0∞t∇λPtλ-1/2,λ-1/2ϕλyλ2dt1/21-y2λλ/2-1/4ey2/2ify≤K0ify>K,fλ,Ky=∫0∞t∇λPtλ-1/2,λ-1/2ϕλyλ2dt1/21-y2λλ/p-1/2pey2/pify≤K0ify>K. Observe that Fλ,K=fλ,KΩλ, where (87)Ωλy=ey2/2-y2/p1-y2λλ/2-λ/p-1/4+1/2p, for all K∈N and λ>K. Moreover, (88)Ωλy≤ey2/2-y2/pe-y2/λλ/2-λ/p-1/4+1/2p=ey21/4λ-1/2pλ. Therefore, if p≤2  |Ωλ(y)|≤1 and for p>2(89)Ωλy≤eK21/4λ-1/2pλ, hence Ωλ is bounded in [-K,K]. Now (90)Zλ-1/pϕλp,λ=Zλ-1/p∫-11ϕλxpλΓλ222λ2πΓ2λ1-x2λ-1/21/pdx. Thus, making the change of variables x=y/λ, we get (91)Zλ-1/pϕλp,λ=Zλ-1/p∫-λλϕλyλpZλ1-y2λλ-1/2dy1/p=∫-λλϕyp1-y2λλ-1/2dy1/p≤C∫-λλϕype-y2πdy1/p. Therefore, (92)limλ→∞Zλ-1/pϕλp,λ≤limλ→∞C∫-λλϕype-y2πdy1/p=C∫-∞∞ϕype-y2πdy1/p=Cϕp,γ, and moreover,(93)Zλ-1/pϕλp,λ≤Cϕp,γ.On the other hand,(94)Fλ,K2,γ=∫RFλ,Ky2e-y2πdy1/2=∫-KKFλ,Ky2e-y2πdy1/2≤∫-λλFλ,Ky2e-y2πdy1/2≤CZλ-1/2ϕλ2,λ.Then, from (93) and (94), we have (95)Fλ,K2,γ≤Cϕ2,γ. Analogously, (96)Fλ,Kp,γ=∫RFλ,Kype-y2πdy1/p=∫-KKfλ,KΩλype-y2πdy1/p. Since Ωλ is bounded in [-k,k], we get(97)Fλ,Kp,γ≤C∫-∞∞fλ,Kype-y2πdy1/p≤CZλ-1/pϕλp,λ.Then, from (93) and (99) we have (98)Fλ,Kp,γ≤Cϕp,γfor all λ>K. Thus, Fλ,K is a bounded sequence in L2R,γ and LpR,γ. By Bourbaki-Alaoglu’s theorem, there exists a subsequence (λj)j∈N such that limj→∞λj=∞ and functions FK∈L2R,γ and fK∈LpR,γ satisfying (a)
Fλj,K→FK, as j→∞, in the weak topology on L2R,γ,(b)
Fλj,K→fK, as j→∞, in the weak topology on LpR,γ.
Moreover,suppFK∪suppfK⊆[-K,K], and(99)FK2,γ≤limj→∞Fλj,K2,γ≤Cϕ2,γ.Analogously, one gets(100)fKp,γ≤Cϕp,γ.Observe that, for every k∈N, (101)τKg=∫-∞∞gxχ-K,Kxdx; then, by Cauchy-Schwartz inequality τK∈L2R,γ∗, and therefore, (102)∫-KKFKxdx=∫-∞∞FKxχ-K,Kxdx=τKFK=limj→∞τKFλj,K=limj→∞∫-∞∞Fλj,Kxχ-K,Kxdx=∫-KKfKxdx; thus, FK=fKa.e.on[-K,K], for all K∈N, so FK=fKa.e. on R. Then from (100), we get(103)FKp,γ≤Cϕp,γ,and therefore, from (99) and (103), there exists an increasing sequence {λj}j∈N⊂(0,∞), with limj→∞λj=∞, and a function F∈LpR,γ∩L2R,γ, such that (a)
for eachK∈N,Fλj,K→F, as j→∞, in the weak topology of L2R,γ and in the weak topology of LpR,γ,(b)
Fp,γ≤Cϕp,γ.
On the other hand,(104)Fλ,Ky=χ-K,Ky∫0∞t∇λPtλ-1/2,λ-1/2ϕλyλ21-y2λλ-1/2ey2dt1/2=χ-K,Ky∫0∞t∂∂tPtλ-1/2,λ-1/2ϕλyλ2dt1-y2λλ-1/2ey2+∫0∞t1-y2λ∂∂xPtλ-1/2,λ-1/2ϕλyλ2dt1-y2λλ-1/2ey21/2. Now, define (105)g1,λϕy=χ-K,Ky∫0∞t∂∂tPtλ-1/2,λ-1/2ϕλyλ2dt1-y2λλ-1/2ey2,g2,λϕy=χ-K,Ky∫0∞t1-y2λ∂∂yPtλ-1/2,λ-1/2ϕλyλ2dt1-y2λλ-1/2ey2. Let us study first the function g1,λ. For a function ϕ∈L2(R,γ), (106)t∂∂tPtα,βϕλyλ=-∑k=1∞ϕλ,Pkα,βPkα,β2,α,β2tλk1/2e-tλk1/2Pkα,βyλ which converges absolutely. Then, taking the Cauchy product, we obtain (107)t∂∂tPtα,βϕλyλ2=∑k=1∞∑n=1kϕλ,Pnα,βPnα,β22ϕλ,Pk-nα,βPk-nα,β22tλn1/2λk-n1/2e-tλk1/2+λk-n1/2Pnα,βyλPk-nα,βyλ. Then, for the Gegenbauer case, α=β=λ-1/2, taking λk^=k(k+2λ)(108)t∂∂tPtλ-1/2,λ-1/2ϕλyλ2=∑k=1∞∑n=1kϕλ,Cnλϕλ,Ck-nλtλ^n1/2λ^k-n1/2e-tλ^k1/2+λ^k-n1/2CnλyλCk-nλyλ×Γ2λΓn+λ+1/2Γλ+1/2Γn+2λΓ2λΓk-n+λ+1/2Γλ+1/2Γk-n+2λ22n+λΓλ+1/22n!Γn+2λΓ2λ+1Γn+λ+1/2×2k-n+λΓλ+1/22k-n!Γk-n+2λΓ2λ+1Γk-n+λ+1/2=∑k=1∞∑n=1kϕλ,Cnλϕλ,Ck-nλtλ^n1/2λ^k-n1/2e-tλ^k1/2+λ^k-n1/2CnλyλCk-nλyλ×n+λn!k-n!k-n+λλ22λn2λk-n. Thus, (109)t∂∂tPtλ-1/2,λ-1/2ϕλyλ2=∑k=1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2tλ^n1/2λ^k-n1/2e-tλ^n1/2+λ^k-n1/2CnλyλCk-nλyλ. Therefore, (110)g1,λϕy=χ-K,Ky∫0∞∑k=1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2tλ^n1/2λ^k-n1/2e-tλ^n1/2+λ^k-n1/2×CnλyλCk-nλyλ1-y2λλ-1/2ey2dt. Let us take (111)Hλ,KN,1t,y=χ-K,Ky∑k=N+1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2tλ^n1/2λ^k-n1/2e-tλ^n1/2+λ^k-n1/2×CnλyλCk-nλyλ1-y2λλ/2-1/4ey2/2. Thus, (112)g1,λϕy=χ-K,Ky∫0∞∑k=1N∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2tλ^n1/2λ^k-n1/2e-tλ^n1/2+λ^k-n1/2×CnλyλCk-nλyλ1-y2λλ-1/2ey2dt+χ-K,Ky∫0∞∑k=N+1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2tλ^n1/2λ^k-n1/2e-tλ^n1/2+λ^k-n1/2×CnλyλCk-nλyλ1-y2λλ-1/2ey2dt=∑k=1N∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2λ^n1/2λ^k-n1/2λ^n1/2+λ^k-n1/22CnλyλCk-nλyλ×1-y2λλ-1/2ey2+∫0∞Hλ,KN,1t,ydt1-y2λλ/2-1/4ey2/2.Now, we want to prove that, for K∈N and λ>0 such that λ>K, (113)∫-∞∞∫0∞Hλ,KN,1t,ydt2e-y2dyπ≤θeN+1. Indeed, by Minkowski integral inequality we have (114)∫-∞∞∫0∞Hλ,KN,1t,ydt2e-y2dyπ1/2≤∫0∞∫-∞∞Hλ,KN,1t,y2e-y2dyπ1/2dt; then, (115)∫-∞∞Hλ,KN,1t,y2e-y2dyπ≤1π∫-11∑k=N+1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2tλ^n1/2λ^k-n1/2e-tλ^k1/2+λ^k-n1/2×CnλxCk-nλx21-x2λ-1/2dx. Using Gasper’s linearization of the product of Jacobi polynomials (see [10]),(116)
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, where pi(α,β)(x) is defined in (4) and (117)νi,k-n,n=1piα,β2,α,β2∫-11piα,βxpk-nα,βxpnα,βxμα,βdx. Hence, (118)∫-∞∞Hλ,KN,1t,y2e-y2dyπ≤1π∫-11∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2tλ^n1/2λ^k-n1/2e-tλ^k1/2+λ^k-n1/2×Γλ+1/22Γn+2λΓk-n+2λΓ2λ2Γn+λ+1/2Γk-n+λ+1/2×Pnλ-1/2,λ-1/21Pk-nλ-1/2,λ-1/21νi,n,k-npiλ-1/2,λ-1/2x21-x2λ-1/2dx. Then, by Parseval’s identity and Cauchy-Schwartz inequality, we have (119)
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Now,(120)νi,k-n,npiα,β2,α,β≤1piα,β2,α,β∫-11piα,βxpk-nα,βxpnα,βxμα,βdx≤n+qnpiα,β2,α,βPnα,β1Pk-nα,β1piα,β2,α,βPk-nα,β2,α,β=n+qnPk-nα,β2,α,βPnα,β1Pk-nα,β1, where q=max(α,β). Thus (121)Pnα,β1Pk-nα,β12νi,k-n,n2piα,β2,α,β2≤n+qn2Pk-nα,β2,α,β2.Therefore, for α=β=λ-1/2, we have(122)∫-∞∞Hλ,KN,1t,y2e-y2dyπ≤1π2πΓ2λλΓλ222λ∑k=N+1∞∑n=1k∑i=k-2nkϕλ2Ck-nλ2Cnλ2ϕλ,CnλCnλ2t2λ^nλ^k-ne-2tλ^k1/2+λ^k-n1/2×Γλ+1/22Γn+2λΓk-n+2λΓ2λ2Γn+λ+1/2Γk-n+λ+1/22n+λ-1/2n2×Γ2λ2Γn+λ+1/22Γλ+1/22Γn+2λ2Ck-nλ2×Γλ+1/22Γk-n+2λ2Γ2λ2Γk-n+λ+1/22Γn+λ+1/22Γλ+1/22n!2Γ2λn+λn!λΓn+2λ=1π2πΓ2λλΓλ222λ∑k=N+1∞∑n=1k∑i=k-2nkϕλ2ϕλ,CnλCnλ2t2λ^nλ^k-ne-2tλ^k1/2+λ^k-n1/2×Γk-n+2λ2Γn+λ+1/22n+λΓ2λΓk-n+λ+1/22n!λΓn+2λ.
On the other hand, using Stirling’s approximation formula for the Gamma function,(123)Γaz+b~2πe-azazaz+b-1/2arg⁡z<π,a>0, we obtain, for λ big enough, that (124)Γk-n+2λ2Γn+λ+1/22n+λΓ2λΓk-n+λ+1/22n!λΓn+2λ~2πe-4λ2λ4λ+2k-2n-12πe-2λλ2λ+2nn+λ2πe-2λ2λ2λ-1/22πe-2λλ2λ+2k-2n2πe-2λ2λ2λ+n-1/2λn!=22k23nλn-1n+λn!.
Then, forλ big enough (125)∫-∞∞Hλ,KN,1t,y2e-y2dyπ≤1π2πΓ2λϕλ2λΓλ222λ∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2t2λ^nλ^k-n×e-2tλ^k1/2+λ^k-n1/222k23nλn-1n+λn!.
Also,(126)t2λ^nλ^k-ne-2tλ^k1/2+λ^k-n1/222k23nλn-1n+λn!≤e-te-tλ^k-n1/222kλnn/λ+1t223nn!.
Takingλ=(k-n)(1+2k)2, for t≥1, we get (127)1etλ^k-n1/2≤1ek-n1/2k-n+2k-n1+2k21/2≤1ek-n1+21+2k21/2≤1ek-n1ek1ek.
Now, if0<t<1 we have that, for t near 0, there exists k>0 big enough such that 1/k≤t; that is, 1/t2≤k2. Hence, (128)1etλ^k-n1/2≤1e1/kk-n1/2k-n+2k-n1+2k21/2≤1ek-n1ek1ek.
Therefore, forλ big enough, there exists C>0 such that (129)t2λ^nλ^k-ne-2tλ^k1/2+λ^k-n1/222k23nλn-1n+λn!≤e-tCek.
Then, forλ big enough, we have(130)∫-∞∞Hλ,KN,1t,y2e-y2dyπ≤1π2πΓ2λλΓλ222λ∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2t2λ^nλ^k-n×e-2tλ^k1/2+λ^k-n1/222k23nλn-1n+λn!≤e-tπ2πΓ2λλΓλ222λMCϕλ2eN+1∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2≤2πΓ2λλΓλ222λe-tMCϕλ2,λ4eN+1≤e-tMCϕ2,γ4eN+1.
Hence,(131)∫-∞∞∫0∞Hλ,KN,1t,ydt2e-y2dyπ1/2≤∫0∞e-tMCϕ2,γ4eN+11/2dt=MC1/2ϕ2,γ2eN+1/2∫0∞e-t/2dt=MC1/2ϕ2,γ2eN+1/2.
Thus,∫0∞Hλ,KN,1(t,·)dt is a bounded sequence on L2R,γ, so, by Bourbaki-Alaoglu’s theorem, there exists a sequence (λj)j∈N,limj→∞λj=∞, such that, for all N∈N, ∫0∞Hλj,KN,1(t,·)dtj∈N converges weakly in L2R,γ to a function HKN,1∈L2R,γ. Moreover,(132)∫-∞∞HKN,1y2e-y2πdy≤MC1/2ϕ2,γ2eN+1/2.
Then, there exists a nondecreasing sequence(Nj)j∈N such that(133)HKNj,1y⟶0,a.e.y∈R.
The study ofg2,λ is essentially analogous, so fewer details will be given. For ϕ∈L2(R,γ) we have (134)∂∂yPtα,βϕλyλ=-∑k=1∞ϕλ,Pkα,βPkα,β2,α,β2e-tλk1/2k+α+β+12Pk-1α+1,β+1yλ,which converges absolutely. Then, again taking the Cauchy product, we obtain (135)∂∂yPtλ-1/2,λ-1/2ϕλyλ2=∑k=1∞∑n=1kϕλ,Cnλϕλ,Ck-nλe-tλ^n1/2+λ^k-n1/2Cn-1λ+1yλCk-n-1λ+1yλn+2λk-n+2λ4×Γ2λ+2Γn+λ+1/2Γλ+3/2Γn+2λ+1Γ2λ+2Γk-n+λ+1/2Γλ+3/2Γk-n+2λ+1Γ2λΓn+λ+1/2Γλ+1/2Γn+2λ×Γ2λΓk-n+λ+1/2Γλ+1/2Γk-n+2λ2n+λΓλ+1/22n!Γn+2λΓ2λ+1Γn+λ+1/22×2k-n+λΓλ+1/22k-n!Γk-n+2λΓ2λ+1Γk-n+λ+1/22=∑k=1∞∑n=1kϕλ,Cnλϕλ,Ck-nλe-tλ^n1/2+λ^k-n1/2Cn-1λ+1yλCk-n-1λ+1yλ×4n+λn!k-n+λk-n!2λn2λk-n;then (136)t1-y2λ∂∂tPtλ-1/2,λ-1/2ϕλyλ2=1-y2λ∑k=1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2te-tλ^n1/2+λ^k-n1/2Cn-1λ+1yλCk-n-1λ+1yλ.
Hence,(137)g2,λϕy=χ-K,Ky∫0∞∑k=1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2te-tλ^n1/2+λ^k-n1/2×Cn-1λ+1yλCk-n-1λ+1yλ1-y2λλ+1/2ey2dt.
Taking(138)Hλ,KN,2t,y=χ-K,Ky∑k=N+1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2te-tλ^n1/2+λ^k-n1/2×Cn-1λ+1yλCk-n-1λ+1yλ1-y2λλ/2+1/4ey2/2,we can write (139)g2,λϕy=χ-K,Ky∫0∞∑k=1N∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2te-tλ^n1/2+λ^k-n1/2×Cn-1λ+1yλCk-n-1λ+1yλ1-y2λλ+1/2ey2dt+χ-K,Ky∫0∞∑k=N+1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2te-tλ^n1/2+λ^k-n1/2×Cn-1λ+1yλCk-n-1λ+1yλ1-y2λλ+1/2ey2dt1-y2λλ+1/2ey2dt=∑k=1N∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2λ^n1/2+λ^k-n1/22×Cn-1λ+1yλCk-n-1λ+1yλ1-y2λλ+1/2ey2+∫0∞Hλ,KN,2t,ydt1-y2λλ/2+1/4ey2/2.
Similarly to the previous case, we want to prove that, for anyK∈N and λ>0 such that λ>K, (140)∫-∞∞∫0∞Hλ,KN,2t,ydt2e-y2dyπ≤θeN+1.
Indeed, by Minkowski integral inequality we have(141)∫-∞∞∫0∞Hλ,KN,2t,ydt2e-y2dyπ1/2≤∫0∞∫-∞∞Hλ,KN,2t,y2e-y2dyπ1/2dt.
Now,(142)∫-∞∞Hλ,KN,2t,y2e-y2dyπ≤1π∫-11∑k=N+1∞∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2te-tλ^k1/2+λ^k-n1/2×Cn-1λ+1xCk-n-1λ+1x21-x2λ-1/2dx.
Again, using Gasper’s linearization of the product of Jacobi polynomials [10], we may write (143)Cn-1λ+1xCk-n-1λ+1x=Γλ+3/22Γn+2λ+1Γk-n+2λ+1Γ2λ+22Γn+2λ+3/2Γk-n+2λ+3/2×Pn-1λ+1/2,λ+1/21Pk-n-1λ+1/2,λ+1/21pn-1λ+1/2,λ+1/2xpk-n-1λ+1/2,λ+1/2x=Γλ+3/22Γn+2λ+1Γk-n+2λ+1Γ2λ+22Γn+2λ+3/2Γk-n+2λ+3/2×Pn-1λ+1/2,λ+1/21Pk-n-1λ+1/2,λ+1/21∑i=k-2nk-2νi,n-1,k-n-1piλ+1/2,λ+1/2x.
Thus,(144)∫-∞∞Hλ,KN,2t,y2e-y2dyπ≤1π∫-11∑k=N+1∞∑n=1k∑i=k-2nk-2ϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2te-tλ^n1/2+λ^k-n1/2×Γλ+3/22Γn+2λ+1Γk-n+2λ+1Γ2λ+22Γn+2λ+3/2Γk-n+2λ+3/2Pn-1λ+1/2,λ+1/21Pk-n-1λ+1/2,λ+1/21×νi,n-1,k-n-1piλ+1/2,λ+1/2x21-x2λ-1/2dx.
Then, using Parseval’s identity and Cauchy-Schwartz inequality, we have(145)∫-∞∞Hλ,KN,2t,y2e-y2dyπ≤1π2πΓ2λ+1λ+1Γλ+1222λ+1∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2ϕλ2Cnλ2Ck-nλ2×16λ4t2e-2tλ^k1/2+λ^k-n1/2Γλ+3/22Γn+2λ+1Γk-n+2λ+1Γ2λ+22Γn+2λ+3/2Γk-n+2λ+3/22×Pn-1λ+1/2,λ+1/21Pk-n-1λ+1/2,λ+1/212νi,n-1,k-n-12piλ+1/2,λ+1/22.
Then, by a similar argument as in the previous case, we get(146)∫-∞∞Hλ,KN,2t,y2e-y2dyπ≤1π2πΓ2λ+1λ+1Γλ+1222λ+1∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2ϕλ2Cnλ2Ck-nλ2×16λ4t2e-2tλ^k1/2+λ^k-n1/2Γλ+3/22Γn+2λ+1Γk-n+2λ+1Γ2λ+22Γn+2λ+3/2Γk-n+2λ+3/22×n+λ-1/2n-12Γ2λΓk-n+λ+1/2Γλ+1/2Γk-n+2λ22λ+22λ+1k-nλ+1/22k-n+2λCk-nλλ2.
On the other hand,(147)Γλ+3/22Γn+2λ+1Γk-n+2λ+1Γ2λ+22Γn+2λ+3/2Γk-n+2λ+3/22×n+λ-1/2n-12Γ2λΓk-n+λ+1/2Γλ+1/2Γk-n+2λ22λ+22λ+1k-nλ+1/22k-n+2λCk-nλλ2=2Γn+2λ2Γn+λ+1/22Γk-n+λ+1/22Ck-nλλ22λ4Γ2λ2Γn+2λ+1/22Γk-n+2λ+1/22Γn2.
Hence,(148)∫-∞∞Hλ,KN,2t,y2e-y2dyπ≤1π2πΓ2λ+1λ+1Γλ+1222λ+1∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2ϕλ2t2e-2tλ^k1/2+λ^k-n1/2Cnλ2×2Γn+2λ2Γn+λ+1/22Γk-n+λ+1/22Ck-nλλ2Γ2λ2Γn+2λ+1/22Γk-n+2λ+1/22Γn2=1π2πΓ2λ+1ϕλ2λ+1Γλ+1222λ+1∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2t2e-2tλ^k1/2+λ^k-n1/2×2Γn+2λΓn+λ+1/22Γk-n+λ+1/22Γ2λΓn+2λ+1/22Γk-n+2λ+1/22Γn2n+λn!λ.Then, for λ big enough, we have(149)2Γn+2λΓn+λ+1/22Γk-n+λ+1/22Γ2λΓn+2λ+1/22Γk-n+2λ+1/22Γn2n+λn!λ~22πe-2λ2λ2λ+n-1/22πe-2λλ2λ+2n2πe-2λλ2λ+2k-2nn+λn!2πe-2λ2λ2λ-1/22πe-4λ2λ4λ+2n2πe-4λ2λ4λ+2k-2nλn-1!2=2n+1λn-1n+λn!22k+8λλ4λe-4λn-1!2≤nλ+1λnnλ4λn-1!.
Therefore, forλ big enough, we have (150)
∫
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∞
∞
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t
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Similarly to the previous case, takingλ=(k-n)(1+2k)2 we have that for k big enough there exists C>0 such that (151)t2e-2tλ^k1/2+λ^k-n1/2nλ+1λnnλ4λn-1!≤e-tCek.
Thus, forλ big enough, we have (152)∫-∞∞Hλ,KN,2t,y2e-y2dyπ≤ϕλ2π2πΓ2λ+1λ+1Γλ+1222λ+1×∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2t2e-2tλ^k1/2+λ^k-n1/2nλ+1λnnλ4λn-1!≤e-tπ2π2λ+12λΓ2λλ+1λ2Γλ222λ22MCϕλ2eN+1∑k=N+1∞∑n=1k∑i=k-2nkϕλ,CnλCnλ2≤2πΓ2λλΓλ222λe-tMCϕλ2,λ4eN+1≤e-tMCϕ2,γ4eN+1.
Hence(153)∫-∞∞∫0∞Hλ,KN,2t,ydt2e-y2dyπ1/2≤∫0∞e-tMCϕ2,γ4eN+11/2dt=MC1/2ϕ2,γ2eN+1/2. Then, ∫0∞Hλ,KN,2(t,·)dt is a bounded sequence on L2R,γ, so, by Bourbaki-Alaoglu’s theorem, there exists a sequence (λj)j∈Nlimj→∞λj=∞ such that, for all N∈N, ∫0∞Hλj,KN,2(t,·)dtj∈N converges weakly in L2R,γ to a function HKN,2∈L2R,γ. Moreover,(154)∫-∞∞HKN,2y2e-y2πdy≤MC1/2ϕ2,γ2eN+1/2.Therefore, there exists a nondecreasing sequence (Nj)j∈N such that(155)HKNj,2y⟶0,a.e.y∈R.On the other hand,(156)Fλ,Ky2=g1,λϕy+g2,λϕy=∑k=1N∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2λ^n1/2λ^k-n1/2λ^n1/2+λ^k-n1/22CnλyλCk-nλyλ×1-y2λλ-1/2ey2+∫0∞Hλ,KN,1t,ydt1-y2λλ/2-1/4ey2/2+∑k=1N∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2λ^n1/2+λ^k-n1/22×Cn-1λ+1yλCk-n-1λ+1yλ1-y2λλ+1/2ey2+∫0∞Hλ,KN,2t,ydt1-y2λλ/2+1/4ey2/2=Fλ,KNy+∫0∞Hλ,KN,1t,ydt1-y2λλ/2-1/4ey2/2+∫0∞Hλ,KN,2t,ydt1-y2λλ/2+1/4ey2/2,where (157)Fλ,KNy=∑k=1N∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ2λ^n1/2λ^k-n1/2λ^n1/2+λ^k-n1/22CnλyλCk-nλyλ×1-y2λλ-1/2ey2+∑k=1N∑n=1kϕλ,CnλCnλ2ϕλ,Ck-nλCk-nλ24λ2λ^n1/2+λ^k-n1/22×Cn-1λ+1yλCk-n-1λ+1yλ1-y2λλ+1/2ey2. Defining, for each m∈N, FKNm=F2-HKNm,1-HKNm,2, then since Fλ,K→FK, as j→∞ in the weak topology of L2R,γ and (158)Fλ,KN=Fλ,K2-∫0∞Hλ,KN,1t,ydt1-y2λλ/2-1/4ey2/2-∫0∞Hλ,KN,2t,ydt1-y2λλ/2+1/4ey2/2,we have, for all m∈N, (159)Fλj,KNm⟶FKNm,weaklyinL2R,γ,asj⟶∞; also,(160)FKNmy⟶F2y,asm⟶∞a.e.y∈R.
Now,(161)
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, so we have (163)limλ→∞Fλ,KNy=∫0∞t∑k=1Nk1/2e-tk1/2ϕ,HkHk2Hky2dt+∫0∞t12∑k=1N2ke-tk1/2ϕ,HkHk2Hk-1y2dt.
Thus,(164)FKNmy=∫0∞t∑k=1Nmk1/2e-tk1/2ϕ,HkHk2Hky2dt+∫0∞t12∑k=1Nm2ke-tk1/2ϕ,HkHk2Hk-1y2dt. On the other hand, given that (165)Hkx≤ηex2/22k/2k!,η=1.086435,we get (166)k1/2e-tk1/2ϕ,HkHk2Hky≤k1/2e-tk1/2ϕηey2/22k/2k!π1/42k/2k!=ηϕey2/2π1/4k1/2e-tk1/2;then,(167)t∑k=1Nmk1/2e-tk1/2ϕ,HkHk2Hky2≤η2ϕ2ey2π1/2t∑k=1Nmk1/2e-tk1/22≤η2ϕ2ey2π1/2t∑k=1∞k1/2e-tk1/22.Now, for t≥1, we have (168)∑k=1∞k1/2e-tk1/2-1/2≤∑k=1∞k1/2e-k1/2+1/2,which converges. Therefore (169)∫1∞t∑k=1∞k1/2e-tk1/22dt≤∫1∞te-t∑k=1∞k1/2e-tk1/2-1/22dt≤∫1∞te-t∑k=1∞k1/2e-k1/2+1/22dt≤C∫1∞te-tdt<∞.
Let0<t<1. For y∈[-K,K], let us define (170)τmy=∑k=1Nmk1/2e-tk1/2ϕ,HkHk2Hky.
We know that, fory∈[-K,K], (171)τmy⟶∑k=1∞k1/2e-tk1/2ϕ,HkHk2Hky,ifm⟶∞,
and also(172)∑k=1∞k1/2e-tk1/2ϕ,HkHk2Hky=∂∂tPtγϕy.
Now, forε=1, there exists M(ε,y)>0 such that(173)τmy-∂∂tPtγϕy<1,ifm≥M;then, for m≥M, we get (174)τmy<1+∂∂tPtγϕy<t1+C1+ye-t.
Therefore, fory∈[-K,K], we have (175)∫01t1+C1+ye-t2dt<∞.
Let us then define(176)ψt=η2ϕ2ey2π1/2t∑k=1∞k1/2e-tk1/22ift≥1t1+C1+ye-t2if0<t<1 which is integrable in (0,∞); now for all m>0 we have (177)t∑k=1Nmk1/2e-tk1/2ϕ,HkHk2Hky2≤ψt,forallt>0; then, by Lebesgue’s dominated convergence theorem, we have (178)limm→∞∫0∞t∑k=1Nmk1/2e-tk1/2ϕ,HkHk2Hky2dt=∫0∞t∑k=1∞k1/2e-tk1/2ϕ,HkHk2Hky2dt=∫0∞t∂∂tPtγϕy2dt.
Similarly, we have(179)limm→∞∫0∞t12∑k=1Nm2ke-tk1/2ϕ,HkHk2Hk-1y2dt=∫0∞t12∂∂yPtγϕy2dt; that is,(180)FKNmy⟶gγϕy2,asm⟶∞a.e.y∈R;therefore, from (160) and (180), we get (181)Fy=gγϕy.(ii) The proof is essentially analogous to (i), so fewer details will be provided. Assume that the operator g(α,β) is bounded in Lp[-1,1],μα,β. Let ϕ∈C0∞(0,∞) and β>0, and then we have (182)gα,βϕβLp-1,1,μα,β≤CϕβLp-1,1,μα,β, where ϕβ(x)=ϕβ/2(1-x), for x∈[-1,1]; that is, (183)∫0∞t∇α,βPtα,βϕβ2dt1/2Lp-1,1,μα,β≤CϕβLp-1,1,μα,β. Now making the change of variable x=1-2/βy, we have (184)ηα,β2α+β+1βα+1∫0β∫0∞t∇α,βPtα,βϕβ1-2βy2dt1/21-yββ/pey/ppyαe-ydy1/p≤CϕβLp-1,1,μλ; thus, (185)∫0β∫0∞t∇α,βPtα,βϕβ1-2βy2dt1/21-yββ/pey/ppyαe-ydy1/p≤CZα,β-1ϕβLp-1,1,μλ,where Zα,β=Γ(α+β+2)/Γ(α+1)βα+2Γ(β). Analogously, we get (186)∫0β∫0∞t∇α,βPtα,βϕβ1-2βy2dt1/21-yββ/2ey/22yαe-ydy1/2≤CZα,β-1ϕβL2-1,1,μλ.Now, for each K∈N and β>0, such that β>K, define the functions (187)Fβ,Ky=χ0,Ky∫0∞t∇α,βPtα,βϕβ1-2βy2dt1/21-yββ/2ey/2,fβ,Ky=χ0,Ky∫0∞t∇α,βPtα,βϕβ1-2βy2dt1/21-yββ/pey/p. From the previous inequalities both series converge for all y∈(0,β), and Fβ,K=fβ,KΩβ, where (188)Ωβy=ey/2-y/p1-yββ/2-β/p for all K∈N and β>K. Now, as (189)Ωβy=ey/2-y/p1-yββ/2-β/p≤ey/2-y/pe-y/ββ/2-β/p=1 we conclude that Ωβ is bounded in (0,K). On the other hand, (190)Zα,β-1/pϕβLp-1,1μα,β=Zα,β-1/pηα,β∫-11ϕβxp1-xα1+xβdx1/p, and making the change of variable x=1-2/βy we get (191)Zα,β-1/pηα,β∫-11ϕβxp1-xα1+xβdx1/p=Zα,β-1/pηα,β2β∫0βϕβ1-2βyp2yβα2-2yββdy1/p≤C∫0βϕypyα1-yββdy1/p≤C∫0∞ϕypyαe-ydy1/p=CϕLp0,∞,μα. Then, (192)limβ→∞Zα,β-1/pϕβLp-1,1,μα,β≤limβ→∞C∫0∞ϕypyαe-ydy1/p=CϕLp0,∞,μα. Moreover,(193)Zα,β-1/pϕβLp-1,1,μα,β≤CϕLp0,∞,μα.Now,(194)Fβ,KL20,∞,μα≤CZα,β-1/2ϕβL2-1,1,μα,βand therefore, from (193) for p=2 and (196), we get (195)Fβ,KL20,∞,μα≤CϕL20,∞,μα. Analogously, using the fact that Ωβ is bounded in (0,K),(196)Fβ,KLp0,∞,μα≤C1Γα+1∫0kfβ,Kypyαe-ydy1/p≤CZα,β-1/pϕβLp-1,1,μα,β.Then, from (193) and (196), (197)Fβ,KLp0,∞,μα≤CϕLp0,∞,μα, for all β>K. Therefore {Fβ,K} is a bounded subsequence in L2(0,∞),μα with Lp(0,∞),μα. Thus by Bourbaki-Alaoglu’s theorem, there exists an increasing sequence {βj}j∈N with limj→∞βj=∞ and functions FK∈L2(0,∞),μα and fK∈Lp(0,∞),μα satisfying that(a)
Fβj,K→FK, as j→∞, in the weak topology of L2(0,∞),μα,(b)
Fβj,K→fK, as j→∞, in the weak topology of Lp(0,∞),μα.
Then, as in (i), we can conclude that there exists an increasing sequence(βj)j∈N⊂(0,∞) such that limj→∞βj=∞, and a function F∈Lp(0,∞),μα∩L2(0,∞),μα, such that(a)
for eachK∈N,Fβj,K→F, as j→∞, in the weak topology on L2(0,∞),μα and in the weak topology on Lp(0,∞),μα,(b)
FLp(0,∞),μα≤CϕLp(0,∞),μα.
Analogously to the Hermite case, we have(198)Fβ,Ky=χ0,Ky∫0∞t∂∂tPtα,βϕβ1-2βy2dt1-yββey+∫0∞tβ∂∂yPtα,βϕβ1-2βy2dt4y1-yββ+1ey1/2. Now. let us define (199)g1,βϕy=χ0,Ky∫0∞t∂∂tPtα,βϕβ1-2βy2dt1-yββey,g2,βϕy=χ0,Ky∫0∞tβ∂∂yPtα,βϕβ1-2βy2dt4y1-yββ+1ey. Let us first consider g1,β. For a function ϕ∈L2((0,∞),μα), (200)t∂∂tPtα,βϕβ1-2βy2=∑k=1∞∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22tλn1/2λk-n1/2×e-tλk1/2+λk-n1/2Pnα,β1-2βyPk-nα,β1-2βy. Therefore, as in the previous case, (201)g1,βϕy=∑k=1N∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22λn1/2λk-n1/2λk1/2+λk-n1/22×Pnα,β1-2βyPk-nα,β1-2βy1-yββey+∫0∞Hβ,KN,1t,ydt1-yββ/2ey/2, where (202)Hβ,KN,1t,y=χ0,Ky∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22tλn1/2λk-n1/2×e-tλk1/2+λk-n1/2Pnα,β1-2βyPk-nα,β1-2βy1-yββ/2ey/2. We want to prove that, for K∈N and β>K, (203)∫0∞∫0∞Hβ,KN,1t,ydt2yαe-ydyΓα+1≤Cϕ2,α4eN+1. Now, (204)∫0∞Hβ,KN,1t,y2yαe-ydyΓα+1≤βα2α+βΓα+1∫0∞∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22tλn1/2λk-n1/2e-tλk1/2+λk-n1/2×Pnα,β1-2βyPk-nα,β1-2βy22yβα2β1-yββdy=βα+12α+β+1Γα+1∫-11∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22tλn1/2λk-n1/2e-tλk1/2+λk-n1/2×Pnα,βxPk-nα,βx21-xα1+xβdx.Then, again by Gasper’s linearization of the product of Jacobi polynomials [10] and using Parseval’s identity, we have (205)∫0∞Hβ,KN,1t,y2yαe-ydyΓα+1≤βα+1ηα,β2α+β+1Γα+1∑k=N+1∞∑n=1k∑i=k-2nkϕβ,Pnα,βPnα,β222ϕβ,Pk-nα,βPk-nα,β222t2λnλk-n×Pnα,β1Pk-nα,β12e-2tλk1/2+λk-n1/2νi,k-n,n2piα,β22.Using analogous boundedness argument as in the Hermite case, we get (206)
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. Now, for β big enough, (207)1Pnα,β22=2n+α+β+1Γα+1Γβ+1Γn+1Γn+α+β+1Γα+β+2Γn+α+1Γn+β+1≤Cn!α+1n,n+qn=n+βn=Γn+β+1Γβ+1n!~βnn!.Thus, for β big enough, (208)
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. Taking β=(k-n)1+2k22, then, k is big enough if β is big enough and we have two cases.
(i) Case t ≥ 1. In this case, we have (209)1etλk-n1/2≤1ek-n1/2k-n+k-n1+2k22+α+11/2≤1ek-n1ek.
Therefore, fork big enough, there exists C>0 such that (210)∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β222nβ2ne-tλk-n1/2t2α+1n≤CeN∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β22; thus,(211)∫0∞Hβ,KN,1t,y2yαe-ydyΓα+1≤Ce-tϕ2,α4eN+1.(ii) Case0<t<1. For t near 0, there exists k>0 big enough such that 1/k≤t; that is, 1/t2≤k2. Then, analogously to the Hermite case, (212)1etλk-n1/2≤1ek-n1ek1ek. Hence, (213)∫0∞Hβ,KN,1t,y2yαe-ydyΓα+1≤Ce-tϕ2,α4eN+1. Therefore, (214)∫0∞∫0∞Hβ,KN,1t,ydt2yαe-ydyΓα+11/2≤∫0∞e-tCϕ2,α4eN+11/2dt=C1/2ϕ2,α2eN+1/2. Thus, ∫0∞Hβ,KN,1(t,·)dt is a bounded sequence on L2(0,∞),μα, so, by Bourbaki-Alaoglu’s theorem, there exists a sequence (λj)j∈Nlimj→∞λj=∞ such that, for all N∈N, ∫0∞Hβj,KN,1(t,·)dtj∈N converges weakly in L2(0,∞),μα to a function HKN,1∈L2(0,∞),μα. Moreover,(215)∫0∞HKN,1y2yαe-ydyΓα+1≤CeN+1/2.Then, there exists a nondecreasing sequence (Nj)j∈N, such that(216)HKNj,1y⟶0,a.e.y∈R.For the function g2,β, we get similar estimates. Given ϕ∈L2((0,∞),μα), (217)tβ∂∂yPtα,βϕβ1-2βy2=∑k=1∞∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22λnλk-n4nk-n×Pn-1α+1,β+11-2βyPk-n-1α+1,β+11-2βytβe-tλk1/2+λk-n1/2. Thus, (218)g2,βϕy=∑k=1N∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22Pn-1α+1,β+11-2βyPk-n-1α+1,β+11-2βy×yβnk-nλnλk-nλk1/2+λk-n1/221-yββ+1ey+∫0∞Hβ,KN,2t,ydty1/21-yββ+1/2ey/2, where (219)Hβ,KN,2t,y=χ0,Ky∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22tβe-tλk1/2+λk-n1/2λnλk-nnk-n×Pn-1α+1,β+11-2βyPk-n-1α+1,β+11-2βyy1/21-yββ+1/2ey/2. Again, we want to prove that, for K∈N and β>K, (220)∫0∞∫0∞Hβ,KN,2t,ydt2yαe-ydyΓα+1≤C1/2ϕ2,α2eN+1/2. Let us see the following: (221)∫0∞Hβ,KN,2t,y2yαe-ydyΓα+1≤βα+22α+β+3Γα+1∫-11∑k=N+1∞∑n=1k∑i=k-2nk-2ϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22tβ×e-tλk1/2+λk-n1/2λnλk-nnk-nPn-1α+1,β+11Pk-n-1α+1,β+11νi,k-n-1,n-1piα+1,β+1x2×1-xα+11+xβ+1dx.
Thus, using Parseval’s identity,(222)∫0∞Hβ,KN,2t,y2yαe-ydyΓα+1≤βα+2ηα+1,β+12α+β+3Γα+1∑k=N+1∞∑n=1k∑i=k-2nk-2ϕβ,Pnα,βPnα,β222ϕβ,Pk-nα,βPk-nα,β222tβ2×e-2tλk1/2+λk-n1/2λnλk-nnk-nPn-1α+1,β+11Pk-n-1α+1,β+112×νi,k-n-1,n-12piα+1,β+1α+1,β+12.
Analogously to the previous case,(223)∫0∞Hβ,KN,2t,y2yαe-ydyΓα+1≤βα+2ϕβ2ηα+1,β+12α+β+3Γα+1∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β22t22n-1β2Pnα,β22Pk-nα,β22×e-2tλk1/2+λk-n1/2λnλk-nnk-n2.n-1+qn-12Pk-n-1α+1,β+1α+1,β+12. Also, given that (224)Pk-n-1α+1,β+1α+1,β+12=α+β+3α+β+2k-nα+1β+1k-n+α+β+1Pk-nα,βα,β2,we get (225)∫0∞Hβ,KN,2t,y2yαe-ydyΓα+1≤βα+2ϕβ2ηα+1,β+12α+β+3Γα+1∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β22t22e-2tλk1/2+λk-n1/2λn2λk-nβ2n×n-1+qn-12α+β+3α+β+2α+1β+1Pnα,β22≤e-tβα+1ϕβ2ηα,β2α+β+1Γα+1∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β222e-tλk-n1/2n+α+β+1t2βn-1+qn-12Pnα,β22. Then we have for β>0 big enough (226)n-1+qn-12n+α+β+1βPnα,β22≤Cnβ2n-1α+1n≤Cnβ2nα+1n. Hence, for β>0 big enough, (227)∫0∞Hβ,KN,2t,y2yαe-ydyΓα+1≤Ce-tβα+1ϕβ2ηα,β2α+β+1Γα+1∑k=N+1∞∑n=1kϕβ,Pnα,βPnα,β222nβ2ne-tλk-n1/2t2α+1n; then, analogous to (211), we have (228)∫0∞Hβ,KN,2t,y2yαe-ydyΓα+1≤Ce-tϕ2,α4eN+1; therefore, (229)∫0∞∫0∞Hβ,KN,2t,ydt2yαe-ydyΓα+11/2≤C1/2ϕ2,α2eN+1/2. Thus, ∫0∞Hβ,KN,2(t,·)dt is a bounded sequence on L2(0,∞),μα, so, by Bourbaki-Alaoglu’s theorem, there exists a sequence (λj)j∈Nlimj→∞λj=∞ such that, for all N∈N, ∫0∞Hβj,KN,2(t,·)dtj∈N converges weakly in L2(0,∞),μα to a function HKN,2∈L2(0,∞),μα. Moreover,(230)∫0∞HKN,2y2yαe-ydyΓα+1≤CeN+1/2.Then, there exists a nondecreasing sequence (Nj)j∈N, such that(231)HKNj,2y⟶0,a.e.y∈0,∞.Therefore, similarly to (156), (232)Fβ,Ky2=g1,βϕy+g2,βϕy=Fβ,KNy+∫0∞Hβ,KN,1t,ydt1-yββ/2ey/2+∫0∞Hβ,KN,2t,ydty1/21-yββ+1/2ey/2, where (233)Fβ,KNy=∑k=1N∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22λn1/2λk-n1/2λk1/2+λk-n1/22×Pnα,β1-2βyPk-nα,β1-2βy1-yββey+∑k=1N∑n=1kϕβ,Pnα,βPnα,β22ϕβ,Pk-nα,βPk-nα,β22λnλk-nλk1/2+λk-n1/22×yβnk-nPn-1α+1,β+11-2βyPk-n-1α+1,β+11-2βy1-yββ+1ey.Defining, for each m∈N, FKNm=F2-HKNm,1-HKNm,2, then since Fβ,K→FK, as j→∞ in the weak topology of L2(0,∞),μα and (234)Fβ,KN=Fβ,K2-∫0∞Hβ,KN,1t,ydt1-yββ/2ey/2-∫0∞Hβ,KN,2t,ydty1/21-yββ+1/2ey/2, we have, for all m∈N, (235)Fβj,KNm⟶FKNm,weaklyinL20,∞,μα,asj⟶∞, and also(236)FKNmy⟶F2y,asm⟶∞a.e.y∈0,∞.Now, given that (237)limβ→∞Pn-1α+1,β+11-2βy=limβ→∞Pn-1α+1,β+11-2yβ+1β+1β=Ln-1α+1y,limβ→∞1Pnα,βα,β2=limβ→∞2nβ+αβ+1+1βn!Γα+1Γn+α+1βα+1Γβ+1Γα+β+2Γβ+n+α+1βαΓn+β+1=n!Γα+1Γn+α+1=1Lnαα2,we get (238)
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Thus,(239)FKNmy=∫0∞t∑k=1Nmϕ,LnαLnαα2k1/2e-tk1/2Lnαy2dt+∫0∞t∑k=1Nmϕ,LnαLnαα2ye-tk1/2Ln-1α+1y2dt. We want to prove that(240)FKNmy⟶gαϕy2,asm⟶∞a.e.y∈R.Indeed, initially we have (241)∑k=1Nmϕ,LkαLkαα2k1/2e-tk1/2Lkαx≤ϕ∑k=1∞k1/2e-tk1/2LkαxLkαα. From (8.22.1) of [1], for α y x>0, we have (242)Lkαx=π-1/2ex/2x-α/2-1/4cos2kx1/2-απ/2-π/4+Okα/2-3/4. Thus, there exist M>0 such that (243)Lkαx-π-1/2ex/2x-α/2-1/4kα/2-1/4cos2kx1/2-απ/2-π/4≤Mkα/2-3/4; hence, (244)Lkαx≤kα/2-1/4M+π-1/2ex/2x-α/2-1/4. On the other hand, since for k big enough Lkαα2≈kα/Γ(α+1), then, there exist N1>0 such that (245)LkαxLkαα≤M+π-1/2ex/2x-α/2-1/4Γk+11/2k1/4,foranyk≥N1. Thus, (246)∑k=1∞k1/2e-tk1/2LkαxLkαα≤M+π-1/2ex/2x-α/2-1/4Γα+11/2∑k=1N1k1/4e-tk1/2kα/2Lkαα+M+π-1/2ex/2x-α/2-1/4Γα+11/2∑k=1∞k1/2e-tk1/2. Then, for t≥1, we have(247)t∑k=1Nmϕ,LkαLkαα2k1/2e-tk1/2Lkαx2≤M+π-1/2ex/2x-α/2-1/42Γα+1∑k=1N1k1/4e-tk1/2kα/2Lkαα2t+M+π-1/2ex/2x-α/2-1/42Γα+1∑k=1N1k1/4e-k1/2kα/2Lkααt∑k=1∞k1/2e-tk1/2+M+π-1/2ex/2x-α/2-1/42Γα+1t∑k=1∞k1/2e-tk1/22.
Now, analogously to the Hermite case, we have(248)∫1∞tM+π-1/2ex/2x-α/2-1/42Γα+1∑k=1N1k1/4e-tk1/2kα/2Lkαα2+M+π-1/2ex/2x-α/2-1/42Γα+1∑k=1N1k1/4e-k1/2kα/2Lkαα∑k=1∞k1/2e-tk1/2+M+π-1/2ex/2x-α/2-1/42Γα+1∑k=1∞k1/2e-tk1/22dt≤M+π-1/2ex/2x-α/2-1/42Γα+1∑k=1N1e-k1/2-1/2kα/2+1/4Lkαα2+∑k=1N1e-k1/2kα/2+1/4Lkαα∑k=1∞k1/2e-k1/2-1+∑k=1∞k1/2e-k1/2-1/22∫1∞te-tdt<∞. For 0<t<1, given that ∂/∂tPtαϕy≤C1+ye-t, we get for y∈(0,K)(249)t∑k=1Nmϕ,LkαLkαα2k1/2e-tk1/2Lkαx2<t1+∂∂tPtαϕy2<t1+C1+ye-t2, where (250)∫01t1+C1+ye-t2dt<∞; then, by Lebesgue’s dominated convergence theorem, we have (251)limm→∞∫0∞t∑k=1Nmϕ,LkαLkαα2k1/2e-tk1/2Lkαx2dt=∫0∞t∑k=1∞ϕ,LkαLkαα2k1/2e-tk1/2Lkαx2dt=∫0∞t∂∂tPtαϕy2dt. Similarly, as before, (252)limm→∞∫0∞t∑k=1Nmϕ,LnαLnαα2ye-tk1/2Ln-1α+1y2dt=∫0∞ty∂∂yPtαϕy2dt. Thus, (253)FKNmy⟶gαϕy2,asm⟶∞a.e.y∈R; hence, from (236) and (240), we have Fy=gαϕy.



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*Source: 9304964-2018-08-01.xml*