On necessary multiplier conditions for Laguerre expansions

by

George Gasper^{1} and Walter Trebels

Dedicated to P.G. Rooney on the occasion of his 65th birthday.

(Published in Canad. J. Math. 43(1991), 1228–1242)

Abstract

Necessary multiplier conditions for Laguerre expansions are derived and discussed within the framework of weighted Lebesgue spaces.

1 Introduction

The purpose of this paper is to enlighten the structure of multipliers for Laguerre expansions on L^{p} spaces from the point of view of necessary conditions. From the theory of Hankel and Jacobi multipliers (see Gasper and Trebels [6], [7] ) it is known that necessary conditions may very well reflect the behavior of multipliers in so far as they are (up to a natural smoothness gap) comparable with sufficient conditions. Following Görlich and Markett [9] we consider the Lebesgue spaces

L_{w(γ)}^{p} = {f :∥ f ∥_{L_{w(γ)}^{p}}= (∫_{0}^{∞} |f(x)e^{−x/2}|^{p}x^{γ} dx)^{1/p} < ∞} , 1 ≤ p < ∞;

in particular, for γ = αp/2, these are the L_{u(α)}^{p}-spaces in [9]:

L_{u(α)}^{p} = {f :∥ f ∥_{L_{u(α)}^{p}}= (∫_{0}^{∞} |f(x)u(x, α)|^{p} dx)^{1/p} < ∞}, 1 ≤ p < ∞,

where u(x, α) = x^{α/2}e^{−x/2}. Let L_{n}^{α}(x), α > −1, n ∈ N_{0}, be the classical Laguerre polynomials (see Szegö [19, p. 100]),

R_{n}^{α}(x) = L_{n}^{α}(x)/L_{n}^{α}(0), L_{n}^{α}(0) = A_{n}^{α} = ( \begin{matrix}n + α \\ n \end{matrix} ) = \frac{Γ(n + α + 1)}{Γ(n + 1)Γ(α + 1)}.

1Supported in part by the National Science Foundation under grant DMS-9103177.

1980 Mathematics Subject Classifications: 33A65, 42A45, 42C10.

Define the Fourier Laguerre coefficients of a function f ∈ L_{w(γ)}^{p} with respect to the orthogonal system {R_{n}^{α}} by

f̂_{α}(n) = ∫_{0}^{∞} f(x)R_{n}^{α}(x)x^{α}e^{−x} dx,

if the integrals exist. Then the formal Laguerre expansion of f is given by

f(x) ∼ (Γ(α + 1))^{−1} ∑_{k=0}^{∞} f̂_{α}(k)L_{k}^{α}(x).

A sequence {m_{k}} is called a multiplier on L_{w(γ)}^{p}, notation {m_{k}} ∈ M_{w(γ)}^{p}, if

∥ ∑_{k=0}^{∞} m_{k}f̂_{α}(k)L_{k}^{α} ∥_{L_{w(γ)}^{p}}≤ C ∥ f ∥_{L_{w(γ)}^{p}}

for all polynomials f; the smallest constant C for which this holds is called the multiplier norm ∥ {m_{k}} ∥_{M_{w(γ)}^{p}} . Generic positive constants that are independent of the functions (and sequences) will be denoted by C. In the case of Laguerre multipliers on L_{w(α)}^{p} there seems to occur a surprising phenomenon: whereas for 4/3 ≤ p < 2 the necessary conditions quite well reflect the boundedness behavior of the well understood example of the Cesàro means, there is a broadening (towards p = 1) growth/smoothness gap between our (at p = 1 best possible) necessary conditions and the Cesàro multiplier; it seems that the space L_{w(α)}^{4/3} plays a crucial role for the theory of Fourier Laguerre multipliers. The boundedness of the Cesàro means of the Laguerre expansion of f

(C, δ)_{n}^{α}(f, x) = (A_{n}^{δ})^{−1} ∑_{k=0}^{n} A_{n−k}^{δ}f̂_{α}(k)L_{k}^{α}(x)

is discussed in a number of papers by Askey and Wainger [2], Muckenhoupt [16], Poiani [17], Markett [12], and Görlich and Markett [9] ; e.g. there holds for α ≥ 0 and δ ≥ 0

∥ (C, δ)_{n}^{α}f ∥_{L_{w(γ)}^{p}}≤ C ∥ f ∥_{L_{w(γ)}^{p}}, δ > \{ \begin{matrix}2|\frac{1}{p} − \frac{1}{2}| − \frac{1}{2} & if γ = αp/2 \\ (2α + 2)|\frac{1}{p} − \frac{1}{2}| − \frac{1}{2} & if γ = α \end{matrix} (1)

uniformly in n; by interpolation one easily gets results also for other γ-values. By Trebels [20, p.21] this implies in particular that any sequence {m_{k}}, converging to zero and being sufficiently smooth, is a multiplier on L_{w(γ)}^{p}, more precisely,

∥ ∑_{k=0}^{∞} m_{k}f̂_{α}(k)L_{k}^{α} ∥_{L_{w(γ)}^{p}}≤ C ∑_{k=0}^{∞} A_{k}^{δ}|Δ^{δ+1}m_{k}| ∥ f ∥_{L_{w(γ)}^{p}}  (2)

for all polynomials f when δ and γ satisfy the conditions in (1). Here the fractional difference of order δ is defined by

Δ^{δ}m_{k} = ∑_{j=0}^{∞} A_{j}^{−δ−1}m_{k+j}

whenever the sum converges. Within the setting of the L_{w(α)}^{p}-spaces our main result reads

Theorem 1.1 If f ∈ L_{w(α)}^{p}, 1 ≤ p ≤ 2, and α > −1, then

sup_{k} |(k + 1)^{λ+(α+1)/q}Δ^{λ}f̂_{α}(k)| ≤ C ∥ f ∥_{L_{w(α)}^{p}},

provided

a) 0 < λ ≤ (2α + \frac{4}{3}) (\frac{1}{p} − \frac{1}{2}) − \frac{1}{3} if 1 ≤ p < \frac{4}{3},

b) 0 < λ < (2α + 2) (\frac{1}{p} − \frac{1}{2}) − \frac{1}{2} if \frac{4}{3} ≤ p < 2.

This theorem and an extension of it are proved in Section 2. The proof relies heavily on the particularly simple formula for fractional differences of the R_{n}^{α} polynomials

Δ^{λ}R_{k}^{α}(x) = \frac{Γ(α + 1)}{Γ(α + λ + 1)}x^{λ}R_{k}^{α+λ}(x), x > 0, λ > −(α + 1/2)/2,  (3)

which is just formula 6.15(4) in [4] when setting c = α+1, c′ = α+λ+1 and observing that R_{k}^{α}(x) = _{1}F_{1}(−k; α + 1; x) = Φ(−k, α + 1; x).

Corollary 1.1 Let 1 ≤ p ≤ 2, λ > 0, and {m_{k}} be a Fourier Laguerre multiplier sequence on L_{w(α)}^{p}. Then

sup_{k} |(k + 1)^{λ}Δ^{λ}m_{k}| ≤ C ∥ {m_{k}} ∥_{M_{w(α)}^{p}},

provided λ satisfies the conditions in Theorem 1.1.

Remarks. 1) This result is best possible for p = 1 and α ≥ 0 in the sense that there is a uniformly bounded multiplier family which satisfies the above necessary condition only for λ ≤ α+1/3. For consider the multiplier sequence {m_{k}(t)}, m_{k}(t) = e^{−t/2}R_{k}^{α}(t), which is uniformly bounded in t > 0 (see [9]). By (3) it follows that

|k^{λ}Δ^{λ}m_{k}(t)| = C|k^{λ}t^{λ}e^{−t/2}R_{k}^{α+λ}(t)| ≈ |k^{−α}t^{λ}e^{−t/2}L_{k}^{α+λ}(t)|.

The sup-norm over t of the last expression behaves like k^{λ−α−1/3} by the fourth case of Markett’s Lemma 1 in [14], hence it diverges when λ > α + 1/3.

2) Corollary 1.1 gives unboundedness of the Cesàro means in the p interval 1 ≤ p < 4/3 only for δ < (2α + 4/3)(1/p − 1/2) − 1/3, whereas the correct critical index δ_{c} at which still divergence happens is δ_{c} = (2α + 2)(1/p − 1/2) − 1/2 (see [9]), i.e., there is a considerable gap between the real range of unboundedness and the one given by Corollary 1.1 in the case 1 ≤ p < 4/3 for the Cesàro test multiplier. This is in contrast to the Jacobi and Hankel multiplier case (see [6], [7]) where, except for the endpoint, the correct range for the unboundedness of the Cesàro means is given by the general necessary conditions. We note that for 4/3 ≤ p ≤ 2 Corollary 1.1 gives divergence for δ < δ_{c} with the right divergence order.

3) In summability theory for numerical series the following result is well known (see [22, p. 105]): The factor sequence {m_{k}} maps each C_{δ} summable series ∑ u_{k} into a C_{δ} summable series ∑ m_{k} u_{k} if and only if the sequence is bounded and

∑_{k=0}^{∞} A_{k}^{δ}|Δ^{δ+1}m_{k}| < ∞.

If one wants to discuss this problem in a Banach space setting (see [20]) one may decompose the Banach space X when assuming the existence of a sequence of projections {P_{k}}_{k∈N_{0}} ⊂ [X], where [X] is the space of all bounded linear operators from X to X, with the following properties:

i) the projections are mutually orthogonal: P_{k}P_{j} = δ_{j,k}P_{k},

ii) they are total: P_{k}f = 0 for all k implies f = 0,

iii) the linear span of the ranges P_{k}(X) is dense in X.

iv) the Cesàro means

(C, δ)_{n}f = (A_{n}^{δ})^{−1} ∑_{k=0}^{n} A_{n−k}^{δ}P_{k}f

are uniformly bounded for some δ ≥ 0 :

∥ (C, δ)_{n}f ∥≤ C ∥ f ∥ ∀f ∈ X.  (4)

If we introduce multipliers analogous to the above Laguerre case, then an analog to the sufficient direction holds for such Cesàro bounded expansions (see [20, p. 21]). But one cannot expect that the converse is also true since concrete orthogonal expansions in general satisfy additional properties, e.g. they are (C, δ) bounded for all δ greater than a critical index but not for the critical index itself. Nevertheless, motivated by the case of Jacobi expansions (or Hankel transforms) one may look at the following problem in the above Banach space setting:

Suppose that

a) (4) holds for all δ > δ_{c} > 0,

b) for some f ∈ X one has

lim sup_{n→∞} ∥ (C, δ_{c})_{n}f ∥= ∞.

Is it true that the multiplier norm of a sequence {m_{k}} can, up to a constant, be estimated from below by sup_{k} |k^{λ}Δ^{λ}m_{k}| for all λ, 0 < λ < δ_{c}?

Corollary 1.1 answers this question with no: the (C, α + 1/2)_{n}^{α} means of the above Laguerre expansion are not uniformly bounded (see [9]) so that according to (1) the critical index in L_{w(α)}^{1} is α+1/2, whereas only λ ≤ α+1/3 is admitted by the example in Remark 1.

4) According to a written communication of C. Markett there exists, apart from the obvious sufficient condition of type (2), the following unpublished result due to V. Dietrich, E. Görlich, G. Hinsen, and C. Markett

∥ ∑_{k=0}^{∞} m_{k}f̂_{α}(k)L_{k}^{α} ∥_{L_{w(α)}^{p}}≤ C \{sup_{k} |m_{k}| + sup_{n} (∑_{k=n}^{2n} |A_{k}^{γ}Δ^{γ}m_{k}|^{2} \frac{1}{k} )^{1/2}\} ∥ f ∥_{L_{w(α)}^{p}}

provided 1 < p < ∞ and γ ≥ α + 1 ≥ 1. This condition is comparable with the necessary one in Corollary 1.1 (see [5]); in particular, their combination gives

Corollary 1.2 If the sequence {m_{k}} ∈ M_{w(α)}^{p} for all α ≥ 0 and some fixed p ≠ 2, then {m_{k}} ∈ M_{w(α)}^{p} for all α ≥ 0 and for all p, 1 < p < ∞.

For the proof observe that by duality one can assume without loss of generality that 1 < p < 2. For fixed p < 2 and fixed α′ ≥ 0 the necessary condition guarantees for the multiplier sequence in question a λ smoothness of order greater than α′ + 1 since by hypothesis α may be chosen sufficiently large, and so application of the sufficient condition with respect to the parameter α′ gives the assertion.

Better sufficient conditions would allow better transplantation theorems for multipliers with respect to Laguerre expansions of different parameters.

5) Corollary 1.1 may be extended by considering multipliers acting on L_{w(α)}^{p} into L_{w(α)}^{r}, p ≤ r, i.e., more precisely, we say m ∈ M_{w(α)}^{p,r} if

∥ ∑_{k=0}^{∞} m_{k}f̂_{α}(k)L_{k}^{α} ∥_{L_{w(α)}^{r}}≤ C ∥ f ∥_{L_{w(α)}^{p}}

for all polynomials f, and define ∥ m ∥_{M_{w(α)}^{p,r}} to be the smallest constant C for which the preceding inequality holds.

Corollary 1.3 Let 1 ≤ p ≤ r ≤ 2, λ > 0, and {m_{k}} ∈ M_{w(α)}^{p,r}. Then

sup_{k} |(k + 1)^{λ+σ}Δ^{λ}m_{k}| ≤ C ∥ {m_{k}} ∥_{M_{w(α)}^{p,r}},

where 1/r = 1/p − σ/(α + 1) and

a) 0 < λ ≤ (2α + \frac{4}{3}) (\frac{1}{r} − \frac{1}{2}) − \frac{1}{3} if 1 ≤ r < \frac{4}{3},

b) 0 < λ < (2α + 2) (\frac{1}{r} − \frac{1}{2}) − \frac{1}{2} if \frac{4}{3} ≤ r < 2.

Corollary 1.3 nicely indicates how fractional integration (with multiplier sequence {(k + 1)^{−σ}}) should work.

Theorem 1.1, Corollary 1.1, and Corollary 1.3 are proved in Section 2 along with some extensions. In Section 3 expansions with respect to the orthonormalized Laguerre functions

L_{n}^{α}(x) = (n!/Γ(n + α + 1))^{1/2}x^{α/2}e^{−x/2}L_{n}^{α}(x)

will be considered. We define modified Fourier Laguerre coefficients

f̂_{n} = ∫_{0}^{∞} f(x)L_{n}^{α}(x)x^{α/2}e^{−x/2} dx

(whenever the integrals exist, e.g., when f ∈ L_{u(α)}^{p}, 1 ≤ p < ∞, α ≥ 0) and have the expansion

f(x)x^{α/2}e^{−x/2} ∼ ∑_{n=0}^{∞} f̂_{n}L_{n}^{α}(x).

Since f̂_{n} = (A_{n}^{α}/Γ(α + 1))^{1/2}f̂_{α}(n) we may state the standard Parseval formula in the following form

\frac{1}{Γ(α + 1)} ∑_{k=0}^{∞} A_{k}^{α}|f̂_{α}(k)|^{2} = ∑_{k=0}^{∞} |f̂_{k}|^{2} = ∫_{0}^{∞} |f(x)u(x, α)|^{2} dx = ∫_{0}^{∞} |f(x)e^{−x/2}|^{2}x^{α} dx

(5)

whenever f ∈ L_{u(α)}^{2} = L_{w(α)}^{2}. In Section 3 it is shown that even though the associated multiplier spaces M_{u(α)}^{p} and M_{u(α)}^{p} for expansions of functions in L_{u(α)}^{p} with respect to L_{n}^{α} and, respectively, to L_{n}^{α} coincide, there is an interesting different L_{u(α)}^{1} behavior of the Cesàro kernel χ_{n}^{α,δ}(x) in (7) and the modified Cesàro kernel k_{n}^{α,δ}(x) defined in Section 3.

2 Proofs and extensions

Theorem 1.1 is an immediate consequence of the γ = α case of the following basic

Lemma 2.1 Let α > −1, f ∈ L_{w(γ)}^{p}, 1 ≤ p ≤ 2, and γ > (α + 1)p/2 − 2/3. Then

a)

sup_{k} |(k + 1)^{γ/p+1/(3p)}Δ^{(2γ+4/3)/p−(α+1)}f̂_{α}(k)| ≤ C ∥ f ∥_{L_{w(γ)}^{p}}, 1 ≤ p < 4/3,

provided γ ≥ −1/3 if p = 1 or γ > −1/3 if 1 < p < 4/3.

b)

sup_{k} |(k + 1)^{α+λ+1−(γ+1)/p}Δ^{λ}f̂_{α}(k)| ≤ C ∥ f ∥_{L_{w(γ)}^{p}}, 1 ≤ p ≤ 2,

provided

0 < λ < \{ \begin{matrix}(2γ + 4/3)/p − (α + 1) & if 1 ≤ p < 4/3 \\ (2γ + 2)/p − (α + 3/2) & if 4/3 ≤ p ≤ 2. \end{matrix}

Proof. By the definition of the Fourier Laguerre coefficients, formula (3), and Hölder’s inequality it follows that (1/p + 1/q = 1)

Δ^{λ}f̂_{α}(k) = ∫_{0}^{∞} f(t)Δ^{λ}R_{k}^{α}(t)e^{−t}t^{α} dt
= C ∫_{0}^{∞} f(t)R_{k}^{α+λ}(t)e^{−t}t^{α+λ} dt (= Cf̂_{α+λ}(k))
≤ C ∥ f ∥_{L_{w(γ)}^{p}} \{ \begin{matrix}(∫_{0}^{∞} |R_{k}^{α+λ}(t)e^{−t/2}t^{α+λ−γ}|^{q}t^{γ} dt)^{1/q} & if p > 1 \\ sup_{t>0} |R_{k}^{α+λ}(t)e^{−t/2}t^{α+λ−γ}| & if p = 1 . \end{matrix}

The observation that R_{k}^{α+λ} = L_{k}^{α+λ}/A_{k}^{α+λ} and a direct application of Lemma 1 in [14] now give for λ ≥ 0

|Δ^{λ}f̂_{α}(k)| ≤ C ∥ f ∥_{L_{w(γ)}^{p}} ×  (6)

× \{ \begin{matrix}k^{−γ/p−1/(3p)} & if λ ≥ (2γ + 4/3)/p − (α + 1), 1 ≤ p < 4/3 \\ k^{−(α+λ+1)+(γ+1)/p} & if λ < (2γ + 4/3)/p − (α + 1), 1 ≤ p < 4/3 \\ k^{1/2−(γ+1)/p} & if λ > (2γ + 2)/p − (α + 3/2), 4/3 < p ≤ ∞ \\ k^{1/2−(γ+1)/p}(log k)^{1−1/p} & if λ = (2γ + 2)/p − (α + 3/2), 4/3 ≤ p ≤ ∞ \\ k^{−(α+λ+1)+(γ+1)/p} & if λ < (2γ + 2)/p − (α + 3/2), 4/3 ≤ p ≤ ∞, \end{matrix}

where, as usual, k and log k on the right hand side are replaced by positive constants when k = 0 or 1. The assertion of the Lemma is now evident.

In order to deduce necessary multiplier conditions from Lemma 2.1 we need on the one hand boundedness of the multipliers involved and on the other control over suitable test functions; the latter will be guaranteed by Lemma 2.2 Let 1 ≤ p ≤ 2, γ > −1, and let N be a fixed integer greater than 2(γ + 1)/p − (α + 3/2). If {g_{k}} ∈ l^{∞} has compact support and g(x) = ∑_{k=0}^{∞} g_{k}L_{k}^{α}(x), then

∥ g ∥_{L_{w(γ)}^{p}}≤ C ∑_{k=0}^{∞}(k + 1)^{N+(α+1)−(γ+1)/p}|Δ^{N+1}g_{k}|.

Proof. Start with the Cesàro kernel given by

χ_{n}^{α,δ}(x) = (A_{n}^{δ}Γ(α + 1))^{−1} ∑_{k=0}^{n} A_{n−k}^{δ}L_{k}^{α}(x) = (A_{n}^{δ}Γ(α + 1))^{−1}L_{n}^{α+δ+1}(x).  (7)

Then g may be represented as

g(x) = Γ(α + 1) ∑_{k=0}^{∞} A_{k}^{N}(Δ^{N+1}g_{k})χ_{k}^{α,N}(x).

Since the third case of Lemma 1 in [14] gives

∥ χ_{k}^{α,δ} ∥_{L_{w(γ)}^{p}}≤ C (k + 1)^{(α+1)−(γ+1)/p}, δ > 2(γ + 1)/p − (α + 3/2),  (8)

when 1 ≤ p ≤ 2, α + δ > −2, and γ > −1, Lemma 2.2 follows after taking the L_{w(γ)}^{p}-norm of g(x).

Consider a monotone decreasing C^{∞}-function ϕ(x) with

ϕ(x) = \{ \begin{matrix}1 & if 0 ≤ x ≤ 2 \\ 0 & if x ≥ 4 \end{matrix} , ϕ_{i}(x) = ϕ(x/2^{i}).

Then

∑_{j=0}^{∞}(j + 1)^{N+(α+1)−(γ+1)/p}|Δ^{N+1}ϕ_{i}(j)| ≤ C (2^{i})^{(α+1)−(γ+1)/p},

which can be easily verified by using a slight modification of Lemma 3.6 in [20], and it follows by applying Lemma 2.2 to the function

Φ^{(i)}(x) = ∑_{j=0}^{∞} ϕ_{i}(j)L_{j}^{α}(x)

that

∥ Φ^{(i)} ∥_{L_{w(γ)}^{p}}≤ C (2^{i})^{(α+1)−(γ+1)/p}  (9)

when 1 ≤ p ≤ 2, α > −1 and γ > −1.

Let us turn to the problem of dominating the l^{∞}-norm of the multiplier sequence in question by its multiplier norm. First observe that by the second and fifth case of formula (6) there holds

|f̂_{α}(k)| ≤ C(k + 1)^{(γ+1)/p−(α+1)} ∥ f ∥_{L_{w(γ)}^{p}}

if γ > (α + 1)p/2 − 2/3 when 1 ≤ p < 4/3, and if γ > (α + 3/2)p/2 − 1 when 4/3 ≤ p ≤ 2. So if one considers m = {m_{k}} ∈ M_{w(γ)}^{p} and replaces f̂_{α}(k) by m_{k}ϕ_{i}(k) one obtains by the previous estimate and (9) that

sup_{k≤2^{i+1}} |m_{k}| ≤ sup_{k} |m_{k}ϕ_{i}(k)| ≤ C ∥ m ∥_{M_{w(γ)}^{p}}

with constant independent of i, thus

M_{w(γ)}^{p} ⊂ l^{∞} when γ > \{ \begin{matrix}(α + 1)p/2 − 2/3 & if 1 ≤ p < 4/3 \\ (α + 3/2)p/2 − 1 & if 4/3 ≤ p ≤ 2 \end{matrix}  (10)

in the sense of continuous embedding.

Lemma 2.3 If α > −1 and m ∈ M_{w(γ)}^{p}, 1 ≤ p ≤ 2, then

a) sup_{k} |(k + 1)^{(2γ+4/3)/p−(α+1)}Δ^{(2γ+4/3)/p−(α+1)}m_{k}| ≤ C ∥ m ∥_{M_{w(γ)}^{p}} when 1 ≤ p < 4/3 and γ > (α + 1)p/2 − 2/3, and

b) sup_{k} |(k + 1)^{λ}Δ^{λ}m_{k}| ≤ C ∥ m ∥_{M_{w(γ)}^{p}} when λ satisfies the conditions of Lemma 2.1 and γ those of (10) .

Proof. Set λ = (2γ + 4/3)/p − (α + 1). From Part a) of Lemma 2.1, it follows that

C(2^{i})^{(α+1)−(γ+1)/p} ∥ m ∥_{M_{w(γ)}^{p}}≥ C ∥ ∑ m_{k}ϕ_{i}(k)L_{k}^{α} ∥_{L_{w(γ)}^{p}}

≥ sup_{2^{i−1}≤k≤2^{i}} |k^{γ/p+1/3p}Δ^{λ}(m_{k}ϕ_{i}(k))| ≥ sup_{2^{i−1}≤k≤2^{i}} |k^{γ/p+1/3p}Δ^{λ}m_{k}|

− sup_{2^{i−1}≤k≤2^{i}} |k^{γ/p+1/3p} ∑_{j=2^{i}}^{∞} A_{j}^{−λ−1}{ϕ_{i}(k + j) − 1}m_{k+j}|.

Hence

∥ m ∥_{M_{w(γ)}^{p}}≥ C2^{iλ} sup_{2^{i−1}≤k≤2^{i}} |Δ^{λ}m_{k}| − C sup_{k} |m_{k}|

and therefore, by (10) ,

sup_{2^{i−1}≤k≤2^{i}} |k^{λ}Δ^{λ}m_{k}| ≤ C ∥ m ∥_{M_{w(γ)}^{p}}

uniformly in i, whence Part a); Part b) follows analogously.

Remarks. 1) Corollary 1.1 is the γ = α case of Lemma 2.3. Corollary 1.3 can be derived analogously from Lemma 2.1 (with γ = α) and (9) when observing that

sup_{k} |(k + 1)^{λ+α+1−(α+1)/r}Δ^{λ}(m_{k}ϕ_{i}(k))|

≤ C ∥ m ∥_{M_{w(α)}^{p,r}}∥ Φ^{(i)} ∥_{L_{w(α)}^{p}}≤ C(2^{i})^{α+1−(α+1)/p} ∥ m ∥_{M_{w(α)}^{p,r}} .

For historical reasons (e.g., see the convolution structure in [8]) and for later use we state a special case (γ = αp/2) of Lemmas 2.1 and 2.3 (using the notation M_{u(α)}^{p} := M_{w(αp/2)}^{p} and 1/p + 1/q = 1)

Corollary 2.4 If α > −1 and 1 ≤ p < 4/3, then

a) sup_{k} |(k + 1)^{λ+α/2+1/q}Δ^{λ}f̂_{α}(k)| ≤ C ∥ f ∥_{L_{u(α)}^{p}}, 0 < λ ≤ \frac{1}{3} − \frac{4}{3q},

b) sup_{k} |(k + 1)^{λ}Δ^{λ}m_{k}| ≤ C ∥ m ∥_{M_{u(α)}^{p}}, 0 < λ ≤ \frac{1}{3} − \frac{4}{3q}.

For sufficient multiplier conditions on L_{u(α)}^{p} comparable with (of the same type as) the necessary ones, see the Corollary for n = 1 in Długosz [3]. Using the transplantation result of Kanjin [10] one can improve Długosz’s result to

∥ ∑_{k=0}^{∞} m_{k}f̂(k)L_{k}^{α} ∥_{L_{u(α)}^{p}}≤ C \{sup_{k} |m_{k}| + sup_{n} (∑_{k=n}^{2n} |(k + 1)Δ^{1}m_{k}|^{2} \frac{1}{k} )^{1/2}\} ∥ f ∥_{L_{u(α)}^{p}} .

for all α ≥ 0 and 1 < p < ∞; namely, Kanjin’s result implies

M_{u(α)}^{p} = M_{u(0)}^{p} = M_{w(0)}^{p}, α ≥ 0, 1 < p < ∞,

and the assertion follows by the above mentioned result of Dietrich, Görlich, Hinsen, and Markett.

2) There arises the question whether Lemma 2.3 can be improved by interpolation. Observe that from Lemma 2.1 with p = 1, γ = (α + λ)/2 − 1/6 and λ > 0, we have

sup_{k} |\sqrt{A_{k}^{α+λ}}(k + 1)^{1/6}Δ^{λ}f̂_{α}(k)| ≤ C ∫_{0}^{∞} |f(x)e^{−x/2}x^{(α+λ)/2−1/6}| dx,

and from (5) with α replaced by α + λ and the formula in the first lines of the proof to Lemma 2.1 we have

(∑_{k=0}^{∞} |\sqrt{A_{k}^{α+λ}}Δ^{λ}f̂_{α}(k)|^{2})^{1/2} ≤ C (∫_{0}^{∞} |f(x)e^{−x/2}x^{(α+λ)/2}|^{2} dx)^{1/2} .

Then application of the Stein and Weiss interpolation theorem (see [18]), where we set T f = \{T f(k)\}, T f(k) = \sqrt{A_{k}^{α+λ}}Δ^{λ}f̂_{α}(k), gives

(∑_{k=0}^{∞} |(k + 1)^{(2/p−1)/6}T f(k)|^{q})^{1/q} ≤ C (∫_{0}^{∞} |f(x)e^{−x/2}x^{(α+λ)/2−(2/p−1)/6}|^{p} dx)^{1/p} .

In particular this implies Part a) of the following Corollary 2.5 Let α > 0 , 1 ≤ p ≤ 2, and (α + 1)(1/p − 1/2) > 1/4. Then, with λ = (2α + 2/3)(1/p − 1/2),

a) (∑_{k=0}^{∞} |(k + 1)^{λ+α/q}Δ^{λ}f̂_{α}(k)|^{q})^{1/q} ≤ C ∥ f ∥_{L_{w(α)}^{p}},

b) sup_{n} (∑_{k=n}^{2n} |(k + 1)^{λ}Δ^{λ}m_{k}|^{q} \frac{1}{k} )^{1/q} ≤ C ∥ m ∥_{M_{w(α)}^{p}} .

Part b) follows along the lines of the proof of Lemma 2.3; observe that the restriction on p comes from (10). Part b) does not contain Corollary 1.1 and vice versa, which may be seen by the examples of the Cesàro multiplier family {A_{n−k}^{δ}/A_{n}^{δ}} (where, e.g., at p = 4/3 Corollary 1.1 gives a greater δ-domain where divergence happens) and sequences of type {e^{ik}/(k +1)^{ν}} (where, e.g., at p = 4/3 Corollary 2.5 leads to a greater ν-domain in which this sequence cannot be a multiplier). The embedding results in [5] lead to the conjecture that λ = (2α + 1/3)(1/p − 1/2) + 1/6 for 1 < p < 4/3 and λ = (2α + 1)(1/p − 1/2) for 4/3 ≤ p < 2 should be the best possible λ-parameters. One possibility to get these is to try to improve the inequality in Corollary 2.5 a) at the point p = 4/3.

3) Formula (3) is equivalent to the Laguerre expansion (9) in Askey [1] after the latter is corrected by replacing the ratio Γ(n − k + γ − α + 1)/Γ(γ − α + 1) in it by Γ(n − k + γ − α)/Γ(γ − α). By arguing as on pages 251 – 252 of Tricomi [21] it can be shown that the fractional difference formula in (3) also holds for x > 0 when λ > −1−min(α, α/2−1/4). When the more restrictive condition λ > −(α+1/2)/2 is satisfied, the infinite series for the function Δ^{λ}R_{k}^{α}(x) on the left side of (3) converges absolutely for x > 0 .

3 Expansions with respect to the orthonormalized Laguerre functions

The orthonormalized Laguerre functions were introduced at the end of the Introduction. A multiplier sequence in this new setting, notation {m_{k}} ∈ M_{u(α)}^{p}, satisfies

(∫_{0}^{∞} | ∑_{k=0}^{∞} m_{k}f̂_{k}L_{k}^{α}(x)|^{p} dx)^{1/p} ≤ C ∥ f ∥_{L_{u(α)}^{p}}

for all polynomials f. Since Γ(α + 1)f̂_{k}L_{k}^{α}(x) = f̂_{α}(k)L_{k}^{α}(x)u(x, α) it is clear that M_{u(α)}^{p} = M_{u(α)}^{p} and thus, that Corollary 2.4 b) holds. But it is not obvious that an analogue of Corollary 2.4 a) holds with f̂_{α}(k) replaced by f̂_{k} . For consider the modified Cesàro kernel

k_{n}^{α,δ}(x) = (A_{n}^{δ})^{−1} ∑_{j=0}^{n} A_{n−j}^{δ}L_{j}^{α}(x)

which differs from (7), apart from the weight u(x, α), by the additional factor (Γ(α + 1)A_{j}^{α})^{−1/2} inside the sum. Since (8) implies

sup_{n} (A_{n}^{α})^{−1/2} ∥ χ_{n}^{α,δ} ∥_{L_{u(α)}^{1}}≤ C, δ > 1/2,  (11)

the following lemma comes as a surprise.

Lemma 3.1 For α > 0 and δ ≥ 0 there holds

sup_{n} ∫_{0}^{∞} |k_{n}^{α,δ}(x)| dx = ∞.

Observe however that on account of (1) there still holds {A_{n−j}^{δ}/A_{n}^{δ}} ∈ M_{u(α)}^{1} = M_{u(α)}^{1} with its multiplier norm uniformly bounded in n. Let us first give an upper bound for ∫_{0}^{∞} |k_{n}^{α,δ}(x)| dx in the case δ ≥ 1. Since

k_{n}^{α,δ}(x) = ∑_{k=0}^{n}(A_{n−k}^{δ}/A_{n}^{δ})L_{k}^{α}(x)
= ∑_{k=0}^{n} m_{k}(A_{n−k}^{δ}/A_{n}^{δ})L_{k}^{α}(x)u(x, α),

with m_{k} = (Γ(α + 1)A_{k}^{α})^{−1/2}, we have that

k_{n}^{α,δ}(x) = Γ(α + 1)(A_{n}^{δ})^{−1} ∑_{k=0}^{n} A_{k}^{1}Δ^{2}(m_{k}A_{n−k}^{δ})χ_{k}^{α,1}(x)u(x, α),

where χ_{n}^{α,δ} is defined by (7). Hence taking the L^{1}(0, ∞)-norm and observing (11) leads to

∫_{0}^{∞} |k_{n}^{α,δ}(x)| dx ≤ C(A_{n}^{δ})^{−1} ∑_{k=0}^{n} A_{k}^{1}(k + 1)^{α/2}|Δ^{2}(m_{k}A_{n−k}^{δ})|.

Leibniz’ formula for differences gives

|Δ^{2}(m_{k}A_{n−k}^{δ})| ≈ (k + 1)^{−α/2−2}A_{n−k}^{δ} + (k + 1)^{−α/2−1}A_{n−k}^{δ−1} + (k + 1)^{−α/2}A_{n−k}^{δ−2}

and the hypothesis δ ≥ 1 guarantees that we have only positive terms. Split up the resulting three sums into 0 ≤ k ≤ n/2 and n/2 ≤ k ≤ n summations. Then the first term with summation over 0 ≤ k ≤ n/2 gives a log (n + 1) contribution, and all other terms only give (uniformly in n) bounded contributions. Hence, for δ ≥ 1,

∫_{0}^{∞} |k_{n}^{α,δ}(x)| dx ≤ C log (n + 1).

Of course, this is no proof of Lemma 3.1; but by a similar argument its proof can be reduced to the problem of showing that when α > 0 the modified Poisson kernel

p_{r}^{α}(x) = ∑_{j=0}^{∞} r^{j}L_{j}^{α}(x)  (12)

has no uniformly in r, 0 < r < 1, bounded L^{1}(0, ∞)-norm, i.e.

sup_{0<r<1} ∫_{0}^{∞} |p_{r}^{α}(x)| dx = ∞, α > 0.  (13)

Take (13) for the moment for granted and assume that Lemma 3.1 is not true for some α and δ. Since

p_{r}^{α}(x) = ∑_{j=0}^{∞} A_{j}^{δ}(Δ^{δ+1}r^{j})k_{j}^{α,δ}(x)

and ∑ A_{j}^{δ}|Δ^{δ+1}r^{j}| ≤ C for all r, 0 < r < 1 (see Chapter 3 in [20]), we immediately get a contradiction, for if we take L^{1}(0, ∞)-norms on both sides of the last equation, the right hand side is uniformly bounded by assumption, whereas the left hand side is not bounded.

In order to prove (13), we first observe that from the generating function [4, 10.12(17)]

∑_{n=0}^{∞} r^{n}L_{n}^{α}(x) = (1 − r)^{−α−1}e^{−xr/(1−r)}, |r| < 1,

and the special case of the beta integral [4, 1.5(1)]

∫_{0}^{1} r^{n}(1 − r)^{a−1}dr = \frac{Γ(n + 1)Γ(a)}{Γ(n + a + 1)}, a > 0, n ≥ 0,

it follows, formally, by termwise integration that

∑_{n=0}^{∞}(A_{n}^{a})^{−1}L_{n}^{α}(x) = a ∫_{0}^{1}(1 − r)^{a−α−2}e^{−xr/(1−r)}dr = g^{a,α}(x)

with

g^{a,α}(x) = ae^{x} ∫_{1}^{∞} t^{α−a}e^{−xt}dt = ae^{x}x^{a−α−1} ∫_{0}^{∞} t^{α−a}e^{−t}dt,

where x > 0 and 0 < a < α + 1. Notice that since, by use of the Laplace transform [4, 10.12(32)],

∫_{0}^{∞} g^{a,α}(x)L_{n}^{α}(x)e^{−x}x^{α}dx = a ∫_{1}^{∞} t^{α−a}(∫_{0}^{∞} L_{n}^{α}(x)e^{−xt}x^{α}dx)dt

= \frac{aΓ(n + α + 1)}{Γ(n + 1)} ∫_{1}^{∞}(t − 1)^{n}t^{−a−n−1}dt = Γ(α + 1)A_{n}^{α}/A_{n}^{a}

for x > 0 and 0 < a < α + 1, we have that

g^{a,α}(x) ∼ ∑_{n=0}^{∞}(A_{n}^{a})^{−1}L_{n}^{α}(x).

Also notice that, from the above integral representations for g^{a,α},

g^{a,α}(x) = O(x^{a−α−1}) as x → 0^{+},

and, by [4, 6.9(21)] and [11, 4.7(2)],

g^{a,α}(x) = ax^{a−α−1}ψ(a − α, a − α; x) = O(1/x) as x → ∞.

From these estimates it follows that we have

Lemma 3.2 Let 0 < a < α + 1, c > 0, and 0 < p < ∞. Then

∫_{0}^{∞} |g^{a,α}(x)|^{p}e^{−cx}x^{γ}dx < ∞ if and only if γ > (1 + α − a)p − 1.

In particular, g^{a,α} ∈ L_{u(α)}^{1}, i.e.

∫_{0}^{∞} |g^{a,α}(x)|e^{−x/2}x^{α/2} dx < ∞,

if and only if a > α/2.

Let α > 0. We will now use Lemma 3.2 to show that

∑_{n=0}^{∞}(Γ(α + 1)A_{n}^{α})^{−1/2}L_{n}^{α}(x)

is the Laguerre series of a function g^{α} that is not in L_{u(α)}^{1}. Observe that, by [4, 1.18(4)], Γ(α + 1)A_{n}^{α} ≃ n^{α} and

(Γ(α + 1)A_{n}^{α})^{−1/2} = (Γ(α/2 + 1)A_{n}^{α/2})^{−1} + c_{α}(A_{n}^{α/2+1})^{−1} + E_{n}^{α}

with E_{n}^{α} = O((n + 1)^{−α/2−2}). From the above lemma, the function g_{1}^{α} = (Γ(α/2 + 1))^{−1}g^{α/2,α} is in L_{w(α)}^{1} , but it is not in L_{u(α)}^{1}. The function g_{2}^{α} = c_{α}g^{α/2+1,α} is in both L_{w(α)}^{1} and L_{u(α)}^{1}, and a termwise use of [14, Lemma 1] shows that the function

g_{3}^{α} = ∑_{n=0}^{∞} E_{n}^{α}L_{n}^{α}(x)

is also in both L_{w(α)}^{1} and L_{u(α)}^{1}. Hence, the function g^{α} = g_{1}^{α} + g_{2}^{α} + g_{3}^{α} is in L_{w(α)}^{1}, it has the Laguerre expansion

∑_{n=0}^{∞}(Γ(α + 1)A_{n}^{α})^{−1/2}L_{n}^{α}(x),

but it is not in L_{u(α)}^{1}. By Lemma 4 and Theorem 3 in Muckenhoupt [15], the Poisson integral g^{α}(r, x) of g^{α}(x) has the Laguerre expansion

∑_{n=0}^{∞} r^{n}(Γ(α + 1)A_{n}^{α})^{−1/2}L_{n}^{α}(x)

and tends to g^{α}(x) almost everywhere as r → 1^{−}. In view of Parseval’s formula (5),

∑_{n=0}^{∞} r^{n}(Γ(α + 1)A_{n}^{α})^{−1/2}L_{n}^{α}(x)

is the Laguerre series of an L_{u(α)}^{2} function when 0 ≤ r < 1. Application of the asymptotic formula [19, 8.22.1)] shows that the above series converges for x > 0 when 0 ≤ r < 1. Since, by L^{2} theory, it converges to g^{α}(r, x) in the L_{u(α)}^{2}-norm, it must also converge to g^{α}(r, x) almost everywhere when 0 ≤ r < 1. Then, using Fatou’s Lemma,

∞ = ∫_{0}^{∞} |g^{α}(x)|e^{−x/2}x^{α/2} dx ≤ lim inf_{r→1^{−}} ∫_{0}^{∞} |p_{r}^{α}(x)| dx ≤ sup_{0<r<1} ∫_{0}^{∞} |p_{r}^{α}(x)| dx

when α > 0, which proves (13) and hence completes the proof of Lemma 3.1. So it is not obvious that the following analogue of Corollary 2.4 a) holds.

Theorem 3.1 If f ∈ L_{u(α)}^{p}, 1 ≤ p < 4/3, α ≥ 0 when p = 1, and α > 2/p − 2 when p > 1, then (1/p + 1/q = 1)

sup_{n} |(n + 1)^{λ+1/q}Δ^{λ}f̂_{n}| ≤ C ∥ f ∥_{L_{u(α)}^{p}}, λ = \frac{1}{3} − \frac{4}{3q} .

Proof. Use (3) to write

Δ^{λ}f̂_{n} = ∫_{0}^{∞} f(x)Δ^{λ}L_{n}^{α}(x)u(x, α) dx
= (Γ(α + 1))^{−1/2} ∫_{0}^{∞} f(x)Δ^{λ} (\sqrt{A_{n}^{α}}R_{n}^{α}(x)) u^{2}(x, α) dx
= C ∫_{0}^{∞} f(x)u(x, α) ∑_{j=0}^{∞} A_{j}^{−λ−1}R_{n+j}^{α}(x)\sqrt{A_{n}^{α}} u(x, α) dx
+ C ∫_{0}^{∞} f(x)u(x, α) ∑_{j=0}^{∞} A_{j}^{−λ−1}R_{n+j}^{α}(x) (\sqrt{A_{n+j}^{α}} − \sqrt{A_{n}^{α}}) u(x, α) dx
= I + II.

From (3) and Hölder’s inequality it follows that if 1 ≤ p < 4/3 and λ ≥ 1/3 − 4/3q then

|I| ≤ C ∥ f ∥_{L_{u(α)}^{p}} \sqrt{A_{n}^{α}} ∥ x^{λ}R_{n}^{α+λ}(x) ∥_{L_{u(α)}^{q}}
≤ C(n + 1)^{−α/2−λ} ∥ f ∥_{L_{u(α)}^{p}}∥ L_{n}^{α+2λ−λ} ∥_{L_{u(α+2λ)}^{q}}
≤ C(n + 1)^{−α/2−λ} ∥ f ∥_{L_{u(α)}^{p}} (n + 1)^{(α+2λ)/2−1/3+1/3q}

where the latter inequality follows from the fourth case of Lemma 1 in Markett [14]. Hence, if f ∈ L_{u(α)}^{p}, 1 ≤ p < 4/3 and λ ≥ 1/3 − 4/3q, then

|I| ≤ C(n + 1)^{−1/3+1/3q} ∥ f ∥_{L_{u(α)}^{p}} .

In order to estimate II first note that |\sqrt{A_{n+j}^{α}} − \sqrt{A_{n}^{α}}| ≈ j(n + j + 1)^{α/2−1}. Then, by Hölder’s inequality and the fifth case of Lemma 1 in [14],

|II| ≤ C ∫_{0}^{∞} |f(x)|u(x, α) ∑_{j=1}^{∞} j^{−1−λ}|L_{n+j}^{α}(x)|(n + j)^{−α}|\sqrt{A_{n+j}^{α}} − \sqrt{A_{n}^{α}}| u(x, α) dx
≤ C ∥ f ∥_{L_{u(α)}^{p}} ∑_{j=1}^{∞} j^{−λ}(n + j)^{−α/2−1} ∥ L_{n+j}^{α} ∥_{L_{u(α)}^{q}}
≤ C(n + 1)^{−λ−1/q} ∥ f ∥_{L_{u(α)}^{p}} .

If we now set λ = 1/3 − 4/3q, then the combination of the above inequalities completes the proof of Theorem 3.1.

There is the question in how far supplementary necessary conditions exist which reflect a behavior as shown by the modified Cesàro kernel; this is closely connected with the problem to gain control over additional test multipliers as one has, e.g., in the case of radial Fourier multipliers.

References

[1] Askey, R., Dual equations and classical orthogonal polynomials, J. Math. Anal. Appl. 24 (1968), 677 – 685.

[2] Askey, R. and S. Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695 – 708.

[3] Długosz, J., L^{p}-multipliers for the Laguerre expansions, Colloq. Math. 54 (1987), 285 – 293.

[4] Erdelyi, A. et al., Higher Transcendental Functions, vols. I & II, McGraw Hill, New York 1953.

[5] Gasper, G. and W. Trebels, A characterization of localized Bessel potential spaces and applications to Jacobi and Hankel multipliers, Studia Math. 65 (1979), 243 – 278.

[6] Gasper, G. and W. Trebels, A Hausdorff-Young inequality and necessary multiplier conditions for Jacobi expansions, Acta Sci. Math. 42 (1980), 247 – 255.

[7] Gasper, G. and W. Trebels, Necessary conditions for Hankel multipliers, Indiana Univ. Math. J. 31 (1982), 403 – 414.

[8] Görlich, E. and C. Markett, Estimates for the norm of the Laguerre translation operator, Numer. Funct. Anal. Optim. 1 (1979), 203 – 222.

[9] Görlich, E. and C. Markett, A convolution structure for Laguerre series, Indag. Math. 44 (1982), 161 – 171.

[10] Kanjin, Y., A transplantation theorem for Laguerre series, to appear.

[11] Luke, Y.L., The Special Functions and Their Approximations, vols. I & II, Academic Press, New York, 1969.

[12] Markett, C., Norm estimates for Cesàro means of Laguerre expansions, in Approximation and Function Spaces (Gdansk 1979), pp. 419 – 435, North Holland, Amsterdam 1981.

[13] Markett, C., Nikolskii-Type inequalities for Laguerre and Hermite expansions, Coll. Math. Soc. János Bolyai 35, pp. 811 – 834, Budapest (1980).

[14] Markett, C., Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8 (1982), 19 – 37.

[15] Muckenhoupt, B., Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 139 (1969), 231 – 242.

[16] Muckenhoupt, B., Mean convergence of Hermite and Laguerre series II, Trans. Amer. Math. Soc. 147 (1970), 433 – 460.

[17] Poiani, E.L., Mean Cesàro summability of Laguerre and Hermite series, Trans. Amer. Math. Soc. 173 (1972), 1 – 31.

[18] Stein, E.M. and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159 – 172.

[19] Szegö, G., Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Providence, R.I. 1975.

[20] Trebels, W., Multipliers for (C, α)-Bounded Fourier Expansions in Banach Spaces and Approximation Theory, Springer Lecture Notes in Math. 329, Springer, Berlin 1973.

[21] Tricomi, F. G., Vorlesungen über Orthogonalreihen, Springer, 1970.

[22] Zeller, K. und W. Beekmann, Theorie der Limitierungsverfahren, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 15, Springer, Berlin 1970.

Gasper: Northwestern University, Evanston, IL 60208 Trebels: Technische Hochschule, Darmstadt, Germany