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On weighted transplantation and multipliers for

arXiv:math/9307203v1 [math.CA] 8 Jul 1993On weighted transplantation and multipliers forLaguerre expansionsKrzysztof Stempak 1 and Walter Trebels2(April 22, 1993 version)Abstract. Using the standard square–function method (based on the Poissonsemigroup), multiplier conditions of H¨ormander type are derived for Laguerre expan-sions in Lp–spaces with power weights in the Ap-range; this result can be interpretedas an “upper end point” multiplier criterion which is fairly good for p near 1 ornear ∞.

A weighted generalization of Kanjin’s [10] transplantation theorem allows toobtain a “lower end point” multiplier criterion whence by interpolation nearly “op-timal” multiplier criteria (in dependance of p, the order of the Laguerre polynomial,the weight).Key words. Laguerre polynomials, sufficient multiplier conditions, transplanta-tion, fractional differences, weighted Lebesgue spacesAMS(MOS) subject classifications.

42A45, 42B25, 42C101IntroductionIn the last fifteen years a considerable activity initiated by a series of papers of Marketthas taken place to study the Harmonic Analysis for Laguerre expansions.There are several ways of studying Laguerre expansions and corresponding multipliers.A principal one is based on the system {Lαk}∞k=0 of Laguerre polynomials, see Szeg¨o [25,p. 100], which are orthogonal in L2(R+, xαe−xdx), α > −1.

Weighted Lp-inequalities,proved for this system by Muckenhoupt [14], turn out to be very useful and implyLp-estimates for Laguerre function expansions.However, as far as multipliers areconcerned, this system is hard to deal with.For instance, Askey and Hirschmanproved that uniform boundedness of Ces`aro multiplier sequences of any positive orderis restricted to p = 2 only.1Institute of Mathematics, University of Wroc law, Wroc law, Poland. The work of this authorwas done during a stay at the Fachbereich Mathematik of the TH Darmstadt and was supportedin part by the TH Darmstadt and in part by the Deutsche Forschungsgemeinschaft under grant436POL/115/1/0.

.2Fachbereich Mathematik, TH Darmstadt, D–6100 Darmstadt, Germany.1

Another principal type of expansion was discussed by G¨orlich and Markett [8], [11].To become more precise, let us introduce the Lebesgue spacesLpv(γ) = {f : ∥f ∥Lpv(γ)= (Z ∞0|f(x)|pxγ dx)1/p < ∞} ,1 ≤p < ∞,γ > −1.If we define the Laguerre function system {lαk} bylαk(x) = (k!/Γ(k + α + 1))1/2e−x/2Lαk(x),α > −1,n ∈N0,then it is orthonormal in L2(R+, xαdx). If γ < p(α + 1) −1 we can associate tof ∈Lpv(γ) the Laguerre seriesf(x) ∼∞Xk=0aklαk(x),ak =Z ∞0f(x)lαk (x)xαdx.Let m = {mk} be a sequence of real or complex numbers and associate to m theoperatorTmf(x) =∞Xk=0mkaklαk (x)for all f of the type f(x) = p(x)e−x/2, p polynomial (the set of these f is dense inLpv(γ), see [14]).

The sequence m is called a bounded multiplier on Lpv(γ), notationm ∈Mpα,γ, if∥m ∥Mpα,γ:= inf{C : ∥Tmf ∥Lpv(γ)≤C ∥f ∥Lpv(γ)}is finite for all f as before. We note the duality property (1/p + 1/p′ = 1)Mpα,γ = Mp′α,αp′−γp′/p ,−1 < γ < p(α + 1) −1,1 < p < ∞.

(1)The purpose of this paper is to obtain a better insight into the structure of theLaguerre multiplier space Mpα,γ from the point of view of sufficient conditions. Wemention that Gasper and Trebels [7] discussed this question from the point of viewof necessary conditions.Remarks.

1) In [7] another scaling of the orthogonal system {lαk } is used, but itfollows directly that our Mpα,γ-space coincides with the space Mpw(γ) in [7]. The lαk ’shave the great advantage to possess a nice convolution structure as shown by G¨orlichand Markett [8].2) The standard orthonormal system on L2(R+, dx) is given byLαk(x) = (k!/Γ(k + α + 1))1/2xα/2e−x/2Lαk(x),k ∈N0.

(2)Though the Parseval identity follows at once, its disadvantage is to be seen in the factthat no nice convolution structure is available. If we denote by Mpα,γ the multiplier2

space with respect to the system {Lαk} (introduced analogously to Mpα,γ), then onecan easily show the following connection between the two multiplier spacesMpα,γ = Mpα,γ+αp/2 . (3)3) Markett [11] and Thangavelu [24] considered another orthonormalized system onL2(R+, dx), namely {ϕαk},ϕαk(x) = Lαk(x2)(2x)1/2,k ∈N0.Again one makes sure that the associated multiplier space coincides with the pre-viously introduced Mpα,γ+αp/2+p/4−1/2.

Thus, if one wants to study multipliers withrespect to these three orthonormal systems, one only needs to discuss Mpα,γ for vari-ous parameters γ.The sufficient multiplier conditions we have in mind are described in terms of thefollowing “weak-bounded-variation” sequence spaces from Gasper and Trebels [4]wbvq,s = {m ∈l∞: ∥m∥q,s < ∞},1 ≤q ≤∞,s > 0,where the norm on this sequence space is given by∥m∥q,s = ∥m∥∞+ supn 2nXk=nk−1|ks∆smk|q!1/q,(with the standard interpretation for q = ∞) and the difference operator ∆s offractional order s by∆smk =∞Xj=0A−s−1jmk+jwhenever the sum converges. We are now ready to formulate a first sufficient multi-plier criterion for Laguerre expansions (to be proved in Section 3).Theorem 1.1 Let α ≥0, 1 < p < ∞, and m ∈wbv2,s for some s > α + 1.

Thenm ∈Mpα,γ and ∥m ∥Mpα,γ≤C∥m∥2,s , provided(α + 1) max{−p/2, −1} < γ −α < (α + 1) min{p/2, p −1};(4)i.e., under the above assumptions there holds in the sense of continuous embeddingwbv2,s ⊂Mpα,γ .3

Remarks. 1) For γ = α Theorem 1.1 coincides with an unpublished result of Diet-rich, G¨orlich, Hinsen, and Markett (due to a written communication of C. Markett).For integer α = n, a weaker version of Theorem 1.1, namelywbv∞,s(n) ⊂Mpn,n,s(n) = n + 2odd nn + 3even ncan be read offfrom Thangavelu [23].2) By the weighted transplantation theorem to be deduced in Section 4 Theorem 1.1implieswbv2,s0 ⊂Mpα,α,s0 > 1,α ≥0,2α + 2α + 2 < p < 2α + 2α.

(5)Following the lines of Connett and Schwartz [2], interpolation between Theorem 1.1for γ = α with p near 1 and (5) leads to Part a) ofCorollary 1.2 a) Let α ≥0 and 1 < p < ∞, thenwbv2,s ⊂Mpα,α,s > max{sc(p), 1},where the quantity sc(p) = (2α + 2)|1/p −1/2| plays the role of a critical index in themultiplier space Mpα,α.b) Let α > −1/3 and 1 ≤p < 2, thenMpα,α ⊂wbvp′,s′,0 < s′ ≤sc(p) −4(1/p −1/2)/3.Part b) is a slight extension of a result in Gasper and Trebels [7]. The corollarynicely shows where Laguerre multipliers live.

Surprising is the smoothness gap of size4|1/p −1/2|/3, since in other settings a gap of size |1/p −1/2| is well known, see e.g. [6] in the Jacobi case (α, −1/2).If sc(p) ≥1 the result in Part a) is best possible in the following sense (continuousembedding)wbv2,s′ ̸⊂Mpα,α,s′ < sc(p),as the example of the Ces`aro means mn,ν of order ν shows: observingmn,ν(k) =(Aνn−k/Aνn, 0 ≤k ≤n0, k > n,∆s′mn,ν(k) =(Aν−s′n−k /Aνn, 0 ≤k ≤n0, k > n,it immediately follows that mn,ν ∈wbv2,s′ uniformly in n if s′ < ν + 1/2 < sc(p).

Butit is well known that ∥mn,ν∥Mpα,α ≥C(n + 1)sc(p)−ν−1/2 (see e.g. [12]).On the other hand concerning the result in Part b) it is shown in [7] that at leastfor p = 1 and α ≥0 the necessary condition is best possible.

In [7, II] there is given4

a modification of the wbv-condition (not nicely comparable in the wbv-framework)which suggests a smoothness gap of size |1/p −1/2| if one could modify in the samesense the sufficient condition.The plan of the paper is as follows. In Section 2 we develop the required square-function calculus based on the Poisson kernel.In Section 3 we give the relevantestimates for the Poisson means of the multiplier which allow one to use the standardsquare-function method (see e.g.

[2]). In Section 4 we turn to a generalization ofKanjin’s [10] transplantation theorem, thus including Thangavelu’s [24] modification,by admitting more general power weights.

This allows some further insight into thestructure of Laguerre multipliers.Acknowledgements. Some of the ideas occurring in this paper were developed inother work done jointly by George Gasper and the second author.

At this point weexpress our indebtedness to these discussions. The first author thanks the FachbereichMathematik of the TH Darmstadt for hospitality and financial support.2On square-functions for Laguerre expansionsThe main tool we will use is the twisted generalized convolution defined in L1(R+, dµα), dµα(x) =x2α+1dx, α ≥0, byf × g(x) =Z ∞0τxf(y)g(y) dµα(y),where the twisted generalized translation operator τx is given byτxf(y) =Γ(α + 1)π1/2Γ(α + 1/2)Z π0 f((x, y)θ)Jα−1/2(xy sin θ)(sin θ)2α dθ,Jβ(x) = Γ(β + 1)Jβ(x)/(x/2)β, Jβ denoting the Bessel function of order β > −1, and(x, y)θ = (x2 + y2 −2xy cos θ)1/2;this convolution is commutative – for all this see G¨orlich and Markett [8] and Stempak[20].

For the following it is convenient to work with the transformed systemψαk (x) = (2k!/Γ(k + α + 1))1/2e−x2/2Lαk(x2),k ∈N0,which, on account of the orthonormality of {lαk}, is obviously orthonormal on L2(R+, dµα).We introduce the norm∥f∥p,δ =Z ∞0|f(x)|px2δdµα(x)1/p;5

this will not lead to any confusion with the wbv-norm. We note that the ψαk ’s areeigenfunctions, with eigenvalues λk, of the positive symmetric in L2(R+, dµα) differ-ential operator L,L = − d2dx2 + 2α + 1xddx −x2!,λk = 4k + 2α + 2.We mention the following transform property of the twisted generalized convolutionwith respect to ψαk -Laguerre expansions; if f ∼P ckψαk and f × g ∼P ckdkψαk , theng(x) ∼Γ(α + 1)P dkLαk(y2)e−y2/2.

To f ∼P ckψαk we associate its Poisson meansP tf =∞Xk=0exp(−tλk)ckψαk = f × pt,t > 0,with Poisson kernel pt(y) = cαP exp(−tλk)Lαk(y2)e−y2/2, and the square-functiong1(f)2(x) =Z ∞0| ∂∂tP tf(x)|2t dt =Z ∞0|∞Xk=0λk exp(−tλk)ckψαk (x)|2t dt(6)for appropriate f. Since for α ≥0 the semigroup {P t}t>0 forms a positive contractionsemigroup (see [8] and [21]), by the Coifman, Rochberg, and Weiss refinement ofStein’s general Littlewood–Paley theory (see Meda [13])C−1∥f∥p,0 ≤∥g1(f)∥p,0 ≤C∥f∥p,0,1 < p < ∞(7)is true. We want to extend (7) to the weighted case δ ̸= 0.Proposition 2.1 Let 1 < p < ∞, α ≥0, and −(α + 1) < δ < (α + 1)(p −1).

ThenC−1∥f∥p,δ ≤∥g1(f)∥p,δ ≤C∥f∥p,δ.Proof. Assume for the moment that the right inequality holds for the δ’s indicated.We show that then the left one follows.

Since by straightforward calculation, basedon Parseval’s identity, ∥f∥2,0 = C(α)∥g1(f)∥2,0 , polarization and H¨older’s inequalitygiveZ ∞0f1(x)f2(x)dµα(x)≤CZ ∞0g1(f1)(x)x2δ/px−2δ/pg1(f2)(x)dµα(x)≤C∥g1(f1)∥p,δ∥g1(f2)∥p′,−δp′/p.Setting f2(x) = x2δ/ph(x) and using the right hand inequality in Proposition 2.1 oneobtainsZ ∞0f1(x)x2δ/ph(x)dµα(x) ≤C∥g1(f1)∥p,δ∥h∥p′,06

for all h ∈Lp′(dµα) and −(α + 1) < −2δp′/p < (α + 1)(p′ −1) so that finally theconverse of H¨older’s inequality gives the desired left hand side inequality∥f1∥p,δ = ∥f1x2δ/p∥p,0 ≤C∥g1(f1)∥p,δ.To extend the right hand side inequality of (7) to the weighted case δ ̸= 0 we adaptan approach of Stein [17]. The following is a variation of a corresponding remark in[18, p. 271].Lemma 2.2 For α ≥0 let K(x, y) be a homogeneous kernel, K(λx, λy) = λ−(2α+2)K(x, y),satisfyingZ ∞0|K(1, y)|y−(2α+2)/pdµα(y) < ∞.Then the operatorTf(x) =Z ∞0K(x, y)f(y) dµα(y)is bounded on Lp(dµα).The particular kernel in the next lemma is the key of the desired extension.Lemma 2.3 For α ≥0 the kernelK(x, y) = |1 −(x/y)2δ/p|Z π0 (x, y)−(2α+2)θ(sin θ)2α dθsatisfies the properties of Lemma 2.2 provided −(α + 1) < δ < (α + 1)(p −1).Proof.

The homogeneity property of the kernel is clear. For the second propertynote that there are three singularities: 0, ∞, and 1 and the required integrability ofK(1, y) follows once we show thatZ π0 (1, y)−(2α+2)θ(sin θ)2α dθ ≤C1as y →0|1 −y|−1as y →1y−(2α+2)as y →∞.

(8)For small y note that (1, y)θ ≈1, 0 < θ < π whereas (1, y)θ ≈y for large y. Finally,for y ≈1 it follows thatZ π0(sin θ)2α((1 −y)2 + 4y sin2(θ/2))α+1 dθ ≤C|1 −y|−(2α+2)Z |1−y|0(sin θ)2α dθ+CZ π|1−y| sin−2(θ/2) dθ ≤C|1 −y|−1so that (8) is obvious.The modification to twisted generalized convolution operators of Stein’s [17] resultnow reads7

Lemma 2.4 Suppose |A(x)| ≤Cx−(2α+2) and Tf = A×f is bounded on Lp(dµα), 1

We only note thatx2δ/pA × f(x) = A × (y2δ/pf)(x) −Z ∞0 (1 −(x/y)2δ/p)τxA(y)f(y)y2δ/p dµα(y)and, since |Jα−1/2(t)| ≤1, α ≥0, the last integral can be estimated byZ ∞0K(x, y)|f(y)|y2δ/p dµα(y),where K is the kernel from Lemma 2.3, hence the assertion by Lemma 2.2.The above scalar-valued result can be extended to the case of functions taking theirvalues in a Hilbert space, see [18, pp. 45], by a repetition of the arguments given forthe scalar-valued case.Proposition 2.5 Let A(x) be a function on R+ taking values in B(H1, H2), Hi beingseparable Hilbert spaces, and satisfying ∥A∥≤Cx−(2α+2).

Further, if one definesTf = A × f for f ∈Lp(R+, H1) and ifZ ∞0∥Tf(x)∥pH2x2δdµα(x) ≤CZ ∞0∥f(x)∥pH1x2δdµα(x),1 < p < ∞,holds for δ = 0, then the same inequality is valid for all δ’s satisfying −(α + 1) < δ <(α + 1)(p −1).To complete the proof of Proposition 2.1 we choose in the preceding propositionH1 = C, H2 = L2(R+, t dt) and Tf = (∂/∂t)P tf = f × (∂/∂t)pt. It is shown inThangavelu [23, Lemma 3.1] that∥(∂/∂t)pt(x)∥L2(R+,t dt) ≤Cx−(2α+2),where (∂/∂t)pt = cαP λk exp(−tλk)Lαk(y2)e−y2/2.

Since (7) holds, all hypotheses ofProposition 2.5 are satisfied and hence Proposition 2.1 is established.For the proof of the multiplier Theorem 1.1 we need the standard variations on theg1-function, namely gσ-functions and a g∗λ-function.We note that a substitution in (6) givesg1(f)2(x) =Z 10 |∞Xk=0λkrλkckψαk (x)|2| log r| drr .8

Denoting u(x, r) = P tf(x), e−t = r, and following Strichartz [22] we introduce asσ-th derivative of u(x, r)dσu(x, r) =∞Xk=0λσkrλkckψαk (x)and setgσ(f)2(x) =Z 10 |dσu(x, r)|2| log r|2σ−1 drr . (9)For the definiton of the g∗λ-function we need a generalized Euclidean translation (oc-curring in the framework of modified Hankel transforms, cf.

[20])τ Ex f(y) =Γ(α + 1)π1/2Γ(α + 1/2)Z π0 f((x, y)θ)(sin θ)2α dθand its associated convolutionf ∗g(x) =Z ∞0τ Ex f(y)g(y) dµα(y);then the g∗λ-function is defined byg∗λ(f)2(x) =Z 10 K| log r| ∗|d1u(·, r)|2(x)| log r| drr ,where Kt(y) = δ√tK(y), K(y) = (1 + y2)−λ and δuf(y) = u−2(α+1)f(y/u) is anL1(dµα)-invariant dilation.Proposition 2.6 a)gρ(f)(x) ≤Cgσ(f)(x)a.e., 1 ≤ρ ≤σfor all f ∈Lp(x2δdµα) for which the right hand side makes sense.b)∥g∗λ(f)∥p,δ ≤C∥f∥p,δ,λ > α + 1,p ≥2provided −(α + 1) < δ < p(α + 1)(1/2 −1/p).Proof. Assertion a) is proved just as in Strichartz [22].

For the proof of b) we notethat the method in Stein [18, p. 91] works. Start with the basic inequality (M denotesthe Hardy-Littlewood maximal operator in the homogeneous space (R+, dµα, ρ) whereρ is the usual distance on R+).Z ∞0g∗λ(f)2(x)h(x)dµα ≤CZ ∞0g1(f)2(x)Mh(x)dµα,(10)9

which also holds in the present setting on account of the formulae (3.6), (3.8) and(3.13) in Stempak [20]. Here the assumption λ > α + 1 implies K(y) ∈L1(dµα).In the case p = 2 choose h(x) = x2δ in (10) and note that the maximal functionMh(x) = Cx2δMh(1) for −(α + 1) < δ ≤0, hence the assertion by Proposition 2.1.In the case q = p/2 > 1 choose h(x) = x4δ/ph1(x) in (10) and apply H¨older’s inequality(1/q + 1/q′ = 1) to obtainZ ∞0g∗λ(f)2(x)x4δ/ph1(x)dµα ≤CZ ∞0g1(f)2(x)x4δ/px−4δ/pM(h1x4δ/p)dµα≤CZ ∞0g1(f)p(x)x2δdµα2/p Z ∞0M(h1x4δ/p)q′(x)x−4q′δ/pdµα1/q′≤C∥g1(f)∥2p,δ∥h1∥q′,0,for x−4q′δ/p ∈Aq′(dµα) by the assumption on δ (see [20, II]).

Taking the supremumover all h1, ∥h1∥q′,0 ≤1 gives∥g∗λ(f)∥2p,δ = ∥g∗λ(f)x4δ/p∥q,0 ≤C∥g1(f)∥2p,δ ≤C∥f∥2p,δ.3Proof of Theorem 1.1.Since we follow the standard method (see e.g. [2, p.73]), we only indicate the mainsteps.

We use the notationf ∼Xckψαk ,Smf ∼Xmkckψαk ,and work on a dense subset of Lp(x2δdµα) (see [14]) such that we can write = insteadof ∼in the preceding formulae. We note that the multiplier space associated to ∥·∥p,δcoincides with Mpα,α+δ.

Thus all we need to show, under the assumptions of Theorem1.1, is∥Smf∥p,δ ≤C∥m∥2,s∥f∥p,δ,δ = γ −α. (11)If we assume for the moment thatgs+1(Smf)(x) ≤C∥m∥2,sg∗s(f)(x)a.e.

(12)holds, then (11) is proved in the case p ≥2, −(α + 1) < δ < p(α + 1)(1/2 −1/p) bythe following chain of norm inequalities if we choose λ = s > α + 1 in Proposition 2.6b)∥Smf∥p,δ ≤C∥g1(Smf)∥p,δ ≤C∥gs+1(Smf)∥p,δ≤C∥m∥2,s∥g∗s(f)∥p,δ ≤C∥m∥2,s∥f∥p,δ.10

In the case p = 2 the result (11) extends at once to −(α + 1) < δ < α + 1 by duality(1). Repeating the interpolation and duality arguments in Hirschman [9, p.50] yieldsTheorem 1.1 provided (12) holds.Let us turn to the proof of (12).

First we note that (up to a constant) we don’t changethe gσ-function (9) if we substitute r2 for r. By the properties of twisted convolutionwe haveds+1Smu(x, r2) = dsM(·, r) × d1u(·, r)(x) =Z ∞0τxd1u(y, r)dsM(y, r)dµα(y),wheredsM(y, r) = cα∞Xk=0λskmkrλkLαk(y2)e−y2/2.Basic properties of dsM(y, r), we need for the proof of (12), are contained inProposition 3.1 Let α ≥0 and m ∈wbv2,s for s > α + 1. Thena)supy |dsM(y, r)| ≤Cr2α+2(1 −r)−s−α−1∥m∥∞,b)Z ∞0|ysdsM(y, r)|2dµα(y) ≤Cr4α+4(1 −r)−s−α−1∥m∥22,s.Suppose that Proposition 3.1 is proved, then one obtains by the Cauchy–Schwarzinequalitygs+1(Smf)2(x) =Z 10 | Z √1−r0+Z ∞√1−r!τxd1u(y, r)dsM(y, r)dµα(y)|2| log r|2s+1drr≤C∥m∥2∞Z 10r4α+4(1 −r)α+2s+1Z √1−r0|τxd1u(y, r)|2dµα(y)| log r|2s+1drr+C∥m∥22,sZ 10r4α+4(1 −r)α+s+1Z ∞√1−r |y−sτxd1u(y, r)|2dµα(y)| log r|2s+1drr≤C∥m∥2∞Z 10r4α+4| log r|2s(1 −r)α+2s+1Z √1−r0τ Ex (|d1u(y, r)|2)dµα(y)| log r|drr+C∥m∥22,sZ 10r4α+4| log r|2s(1 −r)α+s+1Z ∞√1−r y−2sτ Ex (|d1u(y, r)|2)dµα(y)| log r|drr11

because |τxf(y)| ≤τ Ex (|f|)(y), see [20]. Now we use in the first integral y2/| log r| ≤1if 0 < y < √1 −r and r4(| log r|/(1 −r))2s+α+1 ≤C for 0 < r < 1, and in the secondr| log r|sy−2s ≤C(1 + y2/| log r|)−s if 1 −r < y2 and arrive atgs+1(Smf)2(x)≤C{∥m∥2∞+ ∥m∥22,s}Z 10 K| log r| ∗|d1u(·, r)|2(x)| log r|drr≤C∥m∥22,sg∗s(f)2(x)by the definition of the wbv-norm.Thus there remains only to prove Proposition 3.1.

On account of [11, Lemma 1], 5thcase, there holdssupy |Lαk(y2)e−y2/2| ≤C(k + 1)αand hence Part a)supy |dsM(y, r)| ≤C∥m∥∞r2α+2∞Xk=0(k + 1)s+αr4k ≤C∥m∥∞r2α+2(1 −r)−s−α−1,since P(k + 1)γrk ≤C(1 −r)−γ−1 is true for γ > −1.To prove Part b) we use a weighted Parseval formula. First we note that the coeffi-cients ck in the ψαk -expansion are related to the Fourier Laguerre coefficients ˆgα(k) of[7] in the following wayck = Γ(α + 1)(Γ(k + α + 1)/2k!

)1/2 ey/2f(y1/2)αˆ(k),k ∈N0;then it follows from Gasper and Trebels [7, I], formulae (3) and (5) there, that∞Xk=0Aα+sk|∆s(ck√k!/qΓ(k + α + 1))|2 ≈Z ∞0|f(x)|2x2α+2s+1dx,hence for the particular case f = dsM(·, r)Z ∞0|ysdsM(y, r)|2dµα(y) ≤C∞Xk=0Aα+sk|∆s(λskmkrλk)|2 =: I.We have to dominate I. Since similar computations for integer s are contained in [2,p.

69] we only sketch the proof in that case. First note that ∆κrk = (1−r)κrk, κ > 0,and that|∆j(λskr4k)|2 ≤CjXi=0(1 −r)2ir8k(k + 1)2(s−i−j);then use these formulae in Leibniz’ formula for differencesI ≤Cr4α+4sXj=0∞Xk=0Aα+sk|∆jmk|2|∆s−j(λsk+jr4k+4j)|212

≤Cr4α+4sXj=0∥m∥22,j(1 −r)−α−s−1 ≤Cr4α+4∥m∥22,s(1 −r)−α−s−1by the embedding properties of the wbv-spaces, see [4].If s is strictly fractional, similar computations have been carried through in the proofof [5, Lemma 1] and again we only sketch the proof. We use Peyerimhoff’s [15] versionof Leibniz’ formula for fractional differences which in our instance reads:Let s = [s] + κ, 0 < κ < 1.

Then∆s(λskrλkmk) = r2α+2[s]Xi=0 si!∆i(λskr4k)∆s−imk+i + r2α+2mk∆s(λskr4k)+(−1)[s]r2α+2Rk,where the remainder term Rk is given by∞Xi=k+1+[s]A−s−1i−k (mi −mk)i−[s]Xj=k+1A−[s]−1i−[s]−j{∆[s](λsjr4j) −∆[s](λskr4k)}.Up to the terms which contain ∆s−imk+i no smoothness of the sequence m is requiredand in that case |mk| can be crudely estimated by ∥m∥∞. To give an idea of the typeof analysis required let us look at|∆s(λskr4k)| ≤C∆s−[s+1][s+1]Xl=0|∆lr4k| |∆[s+1]−lλsk|≤C[s+1]Xl=0(1 −r)lkXi=0+∞Xi=k+1A[s]−sir4(k+i)(k + i + 1)s−[s+1]+l.For 0 ≤i ≤k one has (k + i + 1) ≈(k + 1) and thusPk0 .

. .

≤Cr4k(k + 1)l−1; fori > k one can replace (k + i + 1) by (i + 1) and one obtains∞Xi=k+1A[s]−sir4(k+i)(k + i + 1)s−[s+1]+l ≤Cr4k(r4(k+1)((k + 1)(1 −r4k))−1/2 , l = 0r4(k+1)(1 −r4k)−l, l ̸= 0,hence∞Xk=0Aα+sk|r2α+2mk∆s(λskr4k)|2 ≤Cr4α+4(1 −r)−α−s−1∥m∥22,s.One should note that in these estimates essentially s > α + 1 ≥1/2 is used. Thecontribution of the error term is estimated analogously.13

Let us conclude with estimating the terms which require smoothness of the multipliersequence in question. Observe that (k + 1 + i) ≈(k + 1) if 0 ≤i < s. Then∞Xk=1|r2α+2∆i(λskr4k)∆s−imk+i|2≤C∞Xn=12n−1Xk=2n−1(k + 1)α+siXl=0(1 −r)2lr8k(k + 1)2l+1|(k + i)s−i∆s−imk+i|2(k + i)−1≤CiXl=0(1 −r)2l∥m∥22,s−i∞Xn=12n(α+s+2l+1)r4 2n≤C∥m∥22,s−i(1 −r)−α−s−1 ≤C∥m∥22,s(1 −r)−α−s−1again by the embedding properties of the wbv-spaces.Thus Proposition 3.1 holds and Theorem 1.1 is established.4A weighted transplantation theorem.In this context it is convenient to work with the Laguerre functions {Lαk}, introducedin (2) at the beginning.

The following transplantation theorem for Laguerre functionexpansions has recently been proved by Kanjin [10].Theorem A. Let α, β > −1 and ε = min{α, β}.

If ε ≥0, then∥XbkLαk∥Lp(R+,dx) ≤C∥XbkLβk∥Lp(R+,dx)for 1 < p < ∞, where C is a constant independent of f. If −1 < ε < 0, then theassertion remains true provided p satisfies (1 + ε/2)−1 < p < −2/ε.Thangavelu [24] gave a modification of Kanjin’s result by replacing the Lebesguemeasure dx by xp/4−1/2dx (under the assumption ε ≥−1/2). Here we admit moregeneral power weights.Theorem 4.1 Let 1 < p < ∞, α, β > −1 and ε = min{α, β}.

If ε ≥0,then∥XbkLαk∥Lpv(δ) ≤C∥XbkLβk∥Lpv(δ)(13)for −1 < δ < p −1, where C is a constant independent of f. If −1 < ε < 0, then(13) holds for −1 −εp/2 < δ < p −1 + εp/2.14

Transplantation results were proved for various orthogonal expansions (cf. [10] for abrief exposition).Proof.

Looking at Kanjin’s proof one discerns two lines in the argumentation.The first one consists in pointwise reformulations, estimates, and tools like the pro-jection formulaLµ+νk(x) = Γ(k + µ + ν + 1)Γ(ν)Γ(k + µ + 1)Z 10 yµ(1 −y)ν−1Lµk(yx) dy,Re µ > −1, Re ν > 0, discussion of the smoothness properties of the involved “adjust-ing” multiplier sequences, verification of the Calderon-Zygmund property of a kernel,etc.The second one concerns norm estimates. While Stein’s interpolation theorem foranalytic families of operators and a multiplier criterion of Butzer, Nessel, and Trebels[1] do not depend on a particular norm (as long as the hypotheses of these theoremsare satisfied), there is a dependance of the norm in the case of the Calderon-Zygmundtheory, Hardy’s inequality, and D lugosz’ multiplier theorem.If we now follow Kanjin’s proof we only have to pay attention to the norm estimatesand provide the necessary substitutes.

We refer continuously to the notation usedin [10]. For instance, by M we will denote an admissible function, i.e.

a positivefunction M(θ), −∞< θ < ∞, that satisfiessupθ∈R e−a|θ| log M(2θ) < ∞with some 0 < a < π. Also ϕ(θ) = {ϕn(θ)}, θ ∈R, will denote the sequence definedbyϕn(θ) = Γ(n + α + 1)Γ(n + α + 1 + iθ)!1/2;the “adjusting” operator T βα,ϕ(θ) and the transplantation operator T βα are given byT βα,ϕ(θ)f ∼Xϕn(θ)⟨f, Lβn⟩Lαn,T βα = T βα,ϕ(0)where ⟨, ⟩stands for the usual scalar product in L2(R+, dx).Let us begin with the case α, β ≥0, temporarily assuming max{−p/2, −1} < δ

[10, Proposition 2]) is sufficient to prove Theorem 4.1.Proposition 4.2 Let α ≥0, 1 < p < ∞, and k = 0, 2. Then∥T α+k+iθα,ϕ(θ) f∥Lpv(δ) ≤M(θ)∥f∥Lpv(δ)(14)for all δ satisfying max{−p/2, −1} < δ < min{p/2, p −1} with an admissible Mindependently of f ∈C∞c .15

To see, for instance, how (14) implies a weighted analogon of [10, Proposition 1,I]note that {(ϕn(θ))−1} ∈wbv2,s, s > α + 1, whatever α ≥0 is and, moreover,∥{(ϕn(θ))−1}∥2,[α+2] ≤C(1 + |θ|[α+2])with C independent of θ. This follows from the calculus for the wbv-spaces [4] sinceby [10, Lemma 2]supx>0xj djdxj Γ(x + α + 1/2 + iθ)Γ(x + α + 1/2)!1/2 ≤C(α, j)(1 + |θ|j),j ∈N0, α > −1/2,with C independent of θ.

Therefore, by combining (3) and Theorem 1.1,∥T α+k+iθαf∥Lpv(δ) = ∥X⟨f, Lα+k+iθn⟩Lαn∥Lpv(δ)≤C(1 + |θ|[α+2]) ∥T α+k+iθα,ϕ(θ) f∥Lpv(δ) ≤M(θ)∥f∥Lpv(δ)provided (14) holds.Proof of (14). To estimate ∥T α+iθα,ϕ(θ)f∥Lpv(δ) we just follow line by line Section 3 of [10]making use in appropriate places of weighted Hardy’s inequalityZ ∞0Z ∞xf(y) dypxδdx1/p≤pδ + 1Z ∞0 (yf(y))pyδdy1/pvalid for f ≥0 and δ > −1 (cf.

[18, p. 272]) and of weighted inequality for singularintegral operators.Recall that the interval (−1, p −1) characterizes those δ’s forwhich the function |x|δ belongs to Ap(R). Therefore we can apply weighted singularintegral inequalities.Exactly the same means are used to estimate ∥T α+2+iθα,ϕ(θ) f∥Lpv(δ), cf.

Section 4 of Kanjin’spaper, except for the fact that we also need a weighted version of [10, (4.1)], i.e., theinequality∥MαΛ(f)∥Lpv(δ) ≤C∥Λ∥bqc∥f∥Lpv(δ). (15)Here one can apply a result of Poiani [16, Corollary, p. 11], which gives the uniformboundedness of the Ces`aro means of order 1 on weighted Lp(xδdx)-space, −1 < δ

To settle the open cases let us look back at the previous outline and assume withoutloss of generality that α < β. The restrictions on α, β as well as on δ were caused bythe application of Theorem 1.1.

For the proof of (14) in the case k = 2 and θ = 0,∥T α+2αf∥Lpv(δ) ≤C∥f∥Lpv(δ),(16)Theorem 1.1 is not needed and a restriction can only come into play by the quasi-convexity criterion, i.e. by Poiani’s [16, Corollary, p. 11] result which implies in ourcase, if α < 0, that −1 −αp/2 < δ < p −1 + αp/2, hence (16) holds for these δ’s.On the other hand, Theorem 1.1 is true for −1 < δ = γ −α < p −1 if α is sufficientlylarge, say α > A > 0 so that no restriction on δ happens in the transplantation theo-rem if α, β ≥A.

If we choose N ∈N0 so large that α + 2N ≥A, the transplantationtheorem proved so far, gives∥T β+2Nα+2N f∥Lpv(δ) ≤C∥f∥Lpv(δ). (17)The rest of the assertion now follows by (16), (17), duality, the semigroup propertyof T βα (see [10]) fromT βα = T α+2α◦· · · ◦T α+2Nα+2N−2 ◦T β+2Nα+2N ◦T β+2N−2β+2N◦· · · ◦T ββ+2.Hence, Theorem 4.1 is completely established.An immediate consequence of Theorem 4.1 in combination with (3) yields for LaguerremultipliersCorollary 4.3 Let α, p, δ be as in Theorem 4.1.

ThenMpα,αp/2+δ = Mpα,δ = Mp0,δ = Mp0,δ.(18)Remarks. 1) For δ = 0 the formula (18) is just Kanjin’s [10] multiplier result, whilefor δ = p/4 −1/2 we cover Thangavelu’s [24] statement on multipliers.2) If we choose α = 0 and p = 2 then Mpα,αp/2 and Mpα,αp/2+p/4−1/2 coincide withM20,0 = l∞.

Since wbv2,s0 ⊂l∞, s0 > 1/2, interpolation between Theorem 1.1 forα = 0, p > 1, p ≈1 and wbv2,s0 ⊂M20,0 along the lines of [2] yields the followingimprovement of the the sufficient multiplier criteria given in [10] and [24]:Corollary 4.4 Let α > −1, thenwbv2,s ⊂Mpα,αp/2 ,wbv2,s ⊂Mpα,αp/2+p/4−1/2 ,s > max{1/p, 1 −1/p},17

provided p and δ satisfy the conditions of Theorem 4.1 where δ = 0 and δ = p/4−1/2,resp.In particular, if m(x) is a bounded one time differentiable function on R+ satisfyingsupx |m(x)|2 + supNZ 2NN|m′(x)|2x dx ≤B2,then, if we set mk = m(k), there holds∥Tmf∥Lpv(αp/2) ≤CB∥f∥Lpv(αp/2),∥Tmf∥Lpv(αp/2+p/4−1/2) ≤CB∥f∥Lpv(αp/2+p/4−1/2).3) To prove (5) we choose in (3) γ = αp(1/p −1/2). By the transplantation Theorem4.1 we obtainMpα,α = Mpα,αp(1/p−1/2) = Mp0,αp(1/p−1/2) ⊃wbv2,s0,s0 > 1,(19)where the last inclusion follows from Theorem 1.1 provided max{−p/2, −1} < αp(1/p−1/2) < min{p/2, p −1} .

But the latter right inequality leads to a restriction onp, p ≤2, namelyαp(1/p −1/2) < p −1⇐⇒2α + 2α + 2 < p ≤2,thus, by duality, (5) is established.4) Interpolation between wbv2,s0 ⊂Mpα,α, s0 > 1, α ≥0, with p as in the precedingremark and l∞⊂M2α,α leads towbvq,s ⊂Mpα,α,s > (2α + 2)|1/p −1/2|,q < 2/s,2α + 2α + 2 < p < 2α + 2α. (20)If one applies Corollary 1.2 and the latter criterion upon the multiplier sequencemζ,η = {mζ,η(k)}, mζ,η(k) = k−ζη exp(ikη), k ̸= 0, η > 0, one obtains mζ,η ∈Mpα,α, α ≥0, provided ζ > (2α + 2)|1/p −1/2|.5) Let us conclude with a multiplier criterion for Mpα,α in the case −1 < α < 0.Recalling (19), observing −1 −αp/2 < α −αp/2 < p −1 + αp/2 for all p, 1 < p < ∞,and using (20) in the case α = 0, Theorem 4.1 (β = 0) yieldsCorollary 4.5 Let −1 < α < 0, 1 < p < ∞.

Thenwbvq,s ⊂Mpα,α ,s > 2|1/p −1/2|,1/q > |1/p −1/2|.This may be looked at as a weak supplement to Corollary 1.2 a) in so far as, if oneconsiders the Ces`aro means of order ν in the case α = −1/2 (cf. the discussion toCorollary 1.2), one only obtains uniform boundedness if ν > 1/4 for p > 4/3, pnear 4/3, whereas Muckenhoupt [14] has shown in this instance even the uniformboundedness of the partial sums (ν = 0).18

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