On a restriction problem of de Leeuw type for Laguerre multipliers

George Gasper^{1} and Walter Trebels^{2}

Dedicated to Károly Tandori on the occasion of his 70th birthday

(Published in Acta Math. Hungar. 68(1–2)(1995), 135–149)

Abstract. In 1965 K. de Leeuw [3] proved among other things in the Fourier transform setting: If a continuous function m(ξ_{1}, …, ξ_{n}) on R^{n} generates a bounded transformation on L^{p}(R^{n}), 1 ≤ p ≤ ∞, then its trace m̃(ξ_{1}, …, ξ_{k}) = m(ξ_{1}, …, ξ_{k}, 0, …, 0), k < n, generates a bounded transformation on L^{p}(R^{k}). In this paper, the analogous problem is discussed in the setting of Laguerre expansions of different orders.

Key words. Laguerre polynomials, embeddings of multiplier spaces, projections, transplantation, weighted Lebesgue spaces

AMS(MOS) subject classifications. 33C45, 42A45, 42C10

1 Introduction

The purpose of this paper is to discuss the question: suppose {m_{k}}_{k∈N_{0}} generates a bounded transformation with respect to a Laguerre function expansion of order α on some L^{p}–space, does it also generate a corresponding bounded map with respect to a Laguerre function expansion of order β? To become more precise let us first introduce some notation. 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 < ∞,
L_{w(γ)}^{∞} = {f : ∥f∥_{L_{w(γ)}^{∞}} = ess sup_{x>0}|f(x)e^{−x/2}| < ∞}, p = ∞,

where γ > −1. Let L_{n}^{α}(x), α > −1, n ∈ N_{0}, denote the classical Laguerre polynomials (see Szegö [15, p. 100]) and set

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)}.

1Department of Mathematics, Northwestern University, Evanston, IL 60208, USA. The work of this author was supported in part by the National Science Foundation under grant DMS-9103177.

2Fachbereich Mathematik, TH Darmstadt, Schloßgartenstr.7, D–64289 Darmstadt, Germany.

Associate to f its formal Laguerre series

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

where the Fourier Laguerre coefficients of f are defined by

f̂_{α}(n) = ∫_{0}^{∞} f(x)R_{n}^{α}(x)x^{α}e^{−x} dx  (1)

(if the integrals exist). A sequence m = {m_{k}}_{k∈N_{0}} is called a (bounded) multiplier from L_{w(γ)}^{p} into L_{w(δ)}^{q}, notation m ∈ M_{α; γ, δ}^{p,q}, if

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

for all polynomials f; the smallest constant C for which this holds is called the multiplier norm ∥m∥_{M_{α; γ, δ}^{p,q}}. For the sake of simplicity we write M_{α; γ}^{p,q} := M_{α; γ, γ}^{p,q} if γ = δ and, if additionally p = q, M_{α; γ}^{p} := M_{α; γ}^{p,p}.

We are mainly interested in the question: when is M_{α; α}^{p,q} continuously embedded in M_{β; β}^{p,q}:

M_{α; α}^{p,q} ⊂→ M_{β; β}^{p,q} , 1 ≤ p ≤ q ≤ ∞ ?

The Plancherel theory immediately yields

l^{∞} = M_{α;α}^{2} = M_{β;β}^{2}, α, β > −1.

A combination of sufficient multiplier conditions with necessary ones indicates which results are to be expected. To this end, define the fractional difference operator Δ^{δ} of order δ by

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

(whenever the series converges), the classes wbv_{q,δ} , 1 ≤ q ≤ ∞, δ > 0, of weak bounded variation (see [5]) of bounded sequences which have finite norm ∥m∥_{q,δ} , where

∥m∥_{q,δ} := sup_{k} |m_{k}| + sup_{N∈N_{0}} ( ∑_{k=N}^{2N} |(k + 1)^{δ}Δ^{δ}m_{k}|^{q} \frac{1}{k + 1})^{1/q} , q < ∞,
∥m∥_{∞,δ} := sup_{k} |m_{k}| + sup_{N∈N_{0}} |(k + 1)^{δ}Δ^{δ}m_{k}|  , q = ∞.

Observing the duality (see [14])

M_{α; γ}^{p} = M_{α; αp′−γp′/p}^{p′} , −1 < γ < p(α + 1) − 1, 1 < p < ∞,  (2)

where 1/p + 1/p′ = 1, we may restrict ourselves to the case 1 < p < 2. The Corollary 1.2 b) in [14] gives the embedding

M_{α; α}^{p} ⊂→ wbv_{p′,s}, s = (2α + 2/3)(1/p − 1/2), α > −1/3,  (3)

when (2α + 2)(1/p − 1/2) > 1/2. Theorem 5 in [5] gives the first embedding in

wbv_{p′,s} ⊂→ wbv_{2,s} ⊂→ M_{β; β}^{p},

whereas the last one follows from Corollaries 1.2 and 4.5 in [14] provided s > max{(2β+ 2)(1/p − 1/2), 1}, β > −1. Hence, choosing γ = α in (2), we obtain

Proposition 1.1 Let 1 < p < ∞ and α be such that (2α+2/3)|1/p−1/2| > 1. Then

M_{α; α}^{p} ⊂→ M_{β; β}^{p} , −1 < β < α − 2/3.

In the same way we can derive a result for M^{p,q}–multipliers. The necessary condition in [6, Cor. 1.3] can easily be extended in the sense of [6, Cor. 2.5 b)] to

sup_{k} |(k + 1)^{σ}m_{k}| + sup_{n}(∑_{k=n}^{2n} |(k + 1)^{σ+s}Δ^{s}m_{k}|^{q′}/k)^{1/q′} ≤ C∥m∥_{M_{α; α}^{p,q}} ,

where α > −1/3, 1/q = 1/p − σ/(α + 1), 1 < p < q < 2, (α + 1)(1/q − 1/2) > 1/4, and s = (2α + 2/3)(1/q − 1/2) > 0. Using this and the sufficient condition for M_{β; β}^{p,q}–multipliers given in [4, Cor. 1.2], which is proved only for β ≥ 0, we obtain

M_{α; α}^{p,q} ⊂→ M_{β; β}^{p,q} , 0 ≤ β < α − 2/3, (2α + 2/3)(1/q − 1/2) > 1, 1 < p < q < 2.

In this context let us mention that the same technique yields for 1 < p, q < 2

M_{α; α}^{p} ⊂→ M_{β; β}^{q} , (2α + 2/3)(1/p − 1/2) > max{(2β + 2)(1/q − 1/2), 1}. (4)

This embedding is in so far interesting as it allows to go from p, 1 < p < 2, to q ≠ p, 1 < q < 2, connected with a loss in the size of β if q < p or a gain in β if 1 < p < q < 2 ; e.g.

M_{10; 10}^{4/3} ⊂→ M_{5; 5}^{q} , 1.08 ≤ q ≤ 2, or M_{2; 2}^{8/7} ⊂→ M_{4; 4}^{q} , 3/2 ≤ q ≤ 2.

Improvements of (4) can be expected by better necessary conditions and/or better sufficient conditions; but this technique cannot give something like

M_{α; α}^{p} ⊂→ M_{β; β}^{q}, (α + 1)(1/p − 1/2) > (β + 1)(1/q − 1/2), 1 < p, q < 2,

which is suggested by (4) when choosing “large” α with p near 2 since then the number 4(1/p−1/2)/3, which describes the smoothness gap between the necessary conditions and the sufficient conditions in [14, Cor. 1.2], is “negligible”.

Concerning the general problem “When does M_{α; γ_{1}, δ_{1}}^{p,q} ⊂→ M_{β; γ_{2}, δ_{2}}^{p,q} hold?”, we mention results in Stempak and Trebels [14, Cor. 4.3]: For 1 < p < ∞ there holds

M_{β;βp/2+δ}^{p} = M_{0;δ}^{p} if \{ \begin{matrix}−1 − βp/2 < δ < p − 1 + βp/2 , & −1 < β < 0 , \\ −1 < δ < p − 1 , & 0 ≤ β, \end{matrix}

which for δ = 0 contains half of Kanjin’s [9] result and for δ = p/4−1/2 Thangavelu’s [16]. In particular, there holds for −1 < β < α, 1 < p < ∞,

M_{β; β}^{p} = M_{β; βp/2+βp(1/p−1/2)}^{p} = M_{α; αp/2+βp(1/p−1/2)}^{p} , (2β + 2)|1/p − 1/2| < 1. (5)

These results are based on Kanjin’s [9] transplantation theorem and its weighted version in [14]. The latter gives further insight into our problem in so far as it implies that the restriction β < α − 2/3 in Proposition 1.1 is not sharp. To this end we first note that the following extension of Corollary 4.4 in [14] holds

wbv_{2,s} ⊂→ M_{α; αp/2+η(p/2−1)}^{p}, 0 ≤ η ≤ 1, 1 < p ≤ 2, s > 1/p.

(For the proof observe that for α = 0 the parameter γ = η(p/2 − 1), 0 ≤ η ≤ 1, is admissible in [14, Theorem 1.1] and then follow the argumentation of [14, Cor. 4.4].) This combined with (3) yields for s = (2α + 2/3)(1/p − 1/2) > 1/p

M_{α; α}^{p} ⊂→ wbv_{2,s} ⊂→ M_{α; αp/2+p/2−1}^{p} , 1 < p ≤ 2, α > (p + 1)/(6 − 3p).

Thus, by interpolation with change of measure,

M_{α; α}^{p} ⊂→ M_{α; αp/2+δ}^{p} , p/2 − 1 ≤ δ ≤ α − αp/2, α > (p + 1)/(6 − 3p).

Since (5) gives

M_{α; αp/2+βp(1/p−1/2)}^{p} = M_{β; β}^{p}

we arrive at

Proposition 1.2 Let 1 < p ≤ 2 and α > (p + 1)/(6 − 3p). Then

M_{α; α}^{p} ⊂→ M_{β; β}^{p} , (2β + 2)(1/p − 1/2) < 1, −1 < β < α.

The first restriction on β is equivalent to β < (2p − 2)/(2 − p). This combined with the restriction on α gives α − β > (7 − 5p)/(6 − 3p), the latter being decreasing in p and taking the value 2/3 at p = 1. Hence Proposition 1.2 is an improvement of the previous one for all 1 < p < 2 provided (p + 1)/(6 − 3p) < α ≤ (2p − 2)/(2 − p).

For big α’s, Proposition 1.1 is certainly better. If in the transplantation theorem in [14] higher exponents could be allowed in the power weight – which is possible in the Jacobi expansion case as shown by Muckenhoupt [12] – the technique just used would give the embedding when −1 < β < α, 1 < p < 2, and α > (p + 1)/(6 − 3p). Summarizing, it is reasonable to

conjecture M_{α; α}^{p,q} ⊂→ M_{β; β}^{p,q}, −1 < β < α, 1 ≤ p ≤ q ≤ ∞.

Apart from the above fragmentary results, so far we can only prove the conjecture in the extreme case when q = ∞ and β ≥ 0; the latter restriction arises from the fact that we have to make use of the twisted Laguerre convolution (see [7]) which is proved till now only for Laguerre polynomials L_{n}^{α}(x) with α ≥ 0. Our main result is

Theorem 1.3 If 1 ≤ p ≤ ∞, then

M_{α; α}^{p,∞} ⊂→ M_{β; β}^{p,∞}, 0 ≤ β < α.

Remarks. 1) One could speculate that an interpolation argument applied to

M_{α; α}^{2} = M_{β; β}^{2} , M_{α; α}^{∞} = M_{α; α}^{1} ⊂→ M_{β; β}^{1} = M_{β; β}^{∞} , β < α,

could give the open case M_{α; α}^{p} ⊂→ M_{β; β}^{p} , 1 < p < 2. In this respect we mention a result of Zafran [17, p. 1412] for the Fourier transform pointed out to us by A. Seeger:

Denote by M^{p}(R) the set of bounded Fourier multipliers on L^{p}(R) and by M^{∧}(R) the set of Fourier transforms of bounded measures on R. Then M^{p}(R), 1 < p < 2, is not an interpolation space with respect to the pair (M^{∧}(R), L^{∞}(R)).

Thus de Leeuw’s result mentioned at the beginning cannot be proved by interpolation.

2) It is perhaps amazing to note that the wbv–classes do not play only an auxiliary role in dealing with the above formulated general problem. In the framework of onedimensional Fourier transforms/series this was shown by Muckenhoupt, Wheeden, and Wo-Sang Young [13]. That this phenomenon also occurs in the framework of Laguerre expansions can be seen from the following two theorems.

Theorem 1.4 If α > −1, α ≠ 0, then

wbv_{2,1} ⊂→ M_{α; α+1}^{2} .

In the case −1 < α < 0 the multiplier operator is defined only on the subspace {f ∈ L_{w(α+1)}^{2} : f̂_{α}(0) = 0}.

Theorem 1.5 If α > −1, then

M_{α; α+1}^{2} ⊂→ wbv_{2,1} .

A combination of these two results leads to

M_{α; α+1}^{2} = M_{β; β+1}^{2} = wbv_{2,1} , α, β > −1, α, β ≠ 0,  (6)

and a combination with [14, (19)] gives

M_{α; α+1}^{2} ⊂→ M_{α; α}^{p} , α ≥ 0, (2α + 2)/(α + 1) < p ≤ 2.

2 Proof of Theorem 1.3

Theorem 1.3 is an immediate consequence of the combination of the following two theorems.

Theorem 2.1 Let f ∈ L_{w(α)}^{p} with α > −1 when 1 ≤ p < ∞ and α ≥ 0 when p = ∞. Then there exists a function g ∈ L_{w(β)}^{p}, −1 < β < α, with

g(x) ∼ (Γ(β + 1))^{−1} ∑_{k=0}^{∞} f̂_{α}(k)L_{k}^{β}(x), ∥g∥_{L_{w(β)}^{p}} ≤ C∥f∥_{L_{w(α)}^{p}} .

Proof

First let 1 ≤ p < ∞ and, without loss of generality, let f be a polynomial (these are dense in L_{w(α)}^{p}). We recall the projection formula (3.31) in Askey and Fitch [2]

e^{−x}L_{n}^{β}(x) = \frac{1}{Γ(α − β)} ∫_{x}^{∞}(y − x)^{α−β−1}e^{−y}L_{n}^{α}(y) dy, −1 < β < α.

Then the following computations are justified.

∥g∥_{L_{w(β)}^{p}} = C (∫_{0}^{∞} | ∑_{k=0}^{∞} f̂_{α}(k)L_{k}^{β}(x)e^{−x/2}|^{p}x^{β}dx)^{1/p}

= C (∫_{0}^{∞} |∫_{x}^{∞}(y − x)^{α−β−1}e^{−y} ∑_{k=0}^{∞} f̂_{α}(k)L_{k}^{α}(y) dy|^{p} x^{β}e^{xp/2}dx)^{1/p}

≤ C ∫_{1}^{∞}(t − 1)^{α−β−1} (∫_{0}^{∞} | ∑_{k} f̂_{α}(k)L_{k}^{α}(xt)x^{α−β+β/p}e^{−x(t−1/2)}|^{p}dx)^{1/p} dt

after a substitution and application of the integral Minkowski inequality. Additional substitutions lead to

∥g∥_{L_{w(β)}^{p}} ≤ C ∫_{0}^{∞} s^{α−β−1}(s + 1)^{β/p′−α−1/p} ×

(∫_{0}^{∞} | ∑_{k} f̂_{α}(k)L_{k}^{α}(y)e^{−y/2}y^{(α−β)/p′}e^{−ys/2(s+1)}|^{p}y^{α}dy)^{1/p} ds

≤ C ∫_{0}^{∞} s^{(α−β)/p−1}(s + 1)^{−(α+1)/p} (∫_{0}^{∞} | ∑_{k} f̂_{α}(k)L_{k}^{α}(y)e^{−y/2}|^{p}y^{α}dy)^{1/p} ds,

where we used the inequality y^{(α−β)/p′}e^{−ys/2(s+1)} ≤ C((s + 1)/s)^{(α−β)/p′}. Since −1 < β < α it is easily seen that the outer integration only gives a bounded contribution. If f ∈ L_{w(α)}^{∞} then |(k + 1)^{−1/2}f̂_{α}(k)| ≤ C∥f∥_{L_{w(α)}^{∞}} by [10, Lemma 1] and, therefore, the Abel-Poisson means of an arbitrary f ∈ L_{w(α)}^{∞} can be represented by

P_{r}f(x) = (Γ(α + 1))^{−1} ∑_{k} r^{k}f̂_{α}(k)L_{k}^{α}(x), 0 ≤ r < 1, x ≥ 0,

and, by the convolution theorem in Görlich and Markett [7, p. 169],

∥P_{r}f∥_{L_{w(α)}^{∞}} ≤ C∥f∥_{L_{w(α)}^{∞}}, 0 ≤ r < 1, α ≥ 0.

A slight modification of the argument in the case 1 ≤ p < ∞ shows that

∥g_{r}∥_{L_{w(β)}^{∞}} := ∥(Γ(β + 1))^{−1} ∑_{k} r^{k}f̂_{α}(k)L_{k}^{β}∥_{L_{w(β)}^{∞}} ≤ C∥P_{r}f∥_{L_{w(α)}^{∞}} ≤ C∥f∥_{L_{w(α)}^{∞}}.

By the weak^{∗} compactness there exists a function g ∈ L_{w(β)}^{∞} with ĝ_{β}(k) = f̂_{α}(k) and ∥g∥_{L_{w(β)}^{∞}} ≤ lim inf_{k→∞} ∥g_{r_{k}}∥_{L_{w(β)}^{∞}} for a suitable sequence r_{k} → 1^{−}; hence also the assertion in the case p = ∞.

Theorem 2.2 For α ≥ 0 there holds

i) M_{α; α}^{1,p} = M_{α; α}^{p′,∞} = (L_{w(α)}^{p}) , 1 < p ≤ ∞,

ii) M_{α; α}^{1,1} = M_{α; α}^{∞,∞} = {m = {m_{k}}_{k∈N_{0}} : ∥P_{r}(m)∥_{L_{w(α)}^{1}} = O(1), r → 1^{−}}, where P_{r}(m)(x) = (Γ(α + 1))^{−1} ∑_{k} r^{k}m_{k}L_{k}^{α}(x).

Proof

The first equalities in i) and ii) are the standard duality statements. Let us briefly indicate the second equalities (which are also more or less standard). If m = {m_{k}}_{k∈N_{0}} are the Fourier Laguerre coefficients of an L_{w(α)}^{p}–function, 1 < p ≤ ∞, or in the case p = 1 of a bounded measure with respect to the weight e^{−x/2}x^{α}, then Young’s inequality in Görlich and Markett [7] (or a slight extension of it to measures in the case p = 1) shows that m ∈ M_{α;α}^{p′,∞}. Conversely, associate formally to a sequence m = {m_{k}} an operator T_{m} by

T_{m}f(x) ∼ (Γ(α + 1))^{−1} ∑_{k=0}^{∞} m_{k}f̂_{α}(k)L_{k}^{α}(x).  (7)

Then, in essentially the notation of Görlich and Markett [7],

T_{m}(P_{r}f)(x) = P_{r}(m) ∗ f(x) = ∫_{0}^{∞} T_{x}^{α}(P_{r}(m)(y))f(y)e^{−y}y^{α}dy,

where T_{x}^{α} is the Laguerre translation operator. If ∥f∥_{L_{w(α)}^{p′}} = 1 then

∥T_{m}(P_{r}f)∥_{L_{w(α)}^{∞}} ≤ ∥m∥_{M_{α; α}^{p′,∞}}∥P_{r}f∥_{L_{w(α)}^{p′}} ≤ C∥m∥_{M_{α; α}^{p′,∞}} ,

and hence, by the converse of Hölder’s inequality,

sup_{∥f∥_{L{p′ \atop w(α)}}=1} |∫_{0}^{∞} T_{x}^{α}(P_{r}(m)(y))e^{−y/2}y^{α/p}f(y)e^{−y/2}y^{α/p′}dy|

= ∥T_{x}^{α}(P_{r}(m))∥_{L_{w(α)}^{p}} ≤ C∥m∥_{M_{α; α}^{p′,∞}}

for x ≥ 0, 0 ≤ r < 1. In particular, for x = 0 we obtain

∥P_{r}(m)∥_{L_{w(α)}^{p}} ≤ C∥m∥_{M_{α; α}^{p′,∞}} , 0 ≤ r < 1.

Now weak^{∗} compactness gives the desired converse embedding.

3 Proof of Theorems 1.4 and 1.5

The proof relies heavily on the Parseval formula

\frac{1}{Γ(α + 1)} ∑_{k=0}^{∞} A_{k}^{α}|f̂_{α}(k)|^{2} = ∫_{0}^{∞} |f(x)e^{−x/2}|^{2}x^{α}dx  (8)

and its extension

∑_{k=0}^{∞} A_{k}^{α+λ}|Δ^{λ}f̂_{α}(k)|^{2} ≈ ∫_{0}^{∞} |f(x)e^{−x/2}|^{2}x^{α+λ}dx, λ ≥ 0,  (9)

which is a consequence of the formula

Δ^{λ}f̂_{α}(k) = C_{α,λ}f̂_{α+λ}(k)  (10)

(see e.g. the proof of Lemma 2.1 in [6]). For the proof of Theorem 1.4 we further need the following discrete analog of the p = 2 case of a weighted Hardy inequality in Muckenhoupt [11] whose proof can at once be read off from [11] by replacing the integrals there by sums and using the fact that

a ≤ 2(a + b)^{1/2}[(a + b)^{1/2} − b^{1/2}]

when a, b ≥ 0; also see the extensions in [1, Sec. 4].

Lemma 3.1 Let {u_{k}}_{k∈N_{0}} , {v_{k}}_{k∈N_{0}} be non-negative sequences (if v_{k} = 0 we set v_{k}^{−1} = 0). Then

a)  ∑_{k=0}^{∞} | ∑_{j=0}^{k} a_{j}|^{2}u_{k} ≤ C sup_{N} ( ∑_{k=N}^{∞} u_{k} ∑_{k=0}^{N} v_{k}^{−1}) ∑_{j=0}^{∞} |a_{j}|^{2}v_{j}.

b)  ∑_{k=0}^{∞} | ∑_{j=k}^{∞} a_{j}|^{2}u_{k} ≤ C sup_{N} ( ∑_{k=0}^{N} u_{k} ∑_{k=N}^{∞} v_{k}^{−1}) ∑_{j=0}^{∞} |a_{j}|^{2}v_{j}.

Proof of Theorem 1.4. Using (9) and the operator T_{m} defined in (7), we obtain

∫_{0}^{∞} |T_{m}f(x)e^{−x/2}|^{2}x^{α+1}dx ≈ ∑_{k=0}^{∞} A_{k}^{α+1}|Δ(m_{k}f̂_{α}(k))|^{2}.

Since

Δ(m_{k}f̂_{α}(k)) = m_{k}Δf̂_{α}(k) + f̂_{α}(k + 1)Δm_{k}  (11)

we first observe that

∑_{k=0}^{∞} A_{k}^{α+1}|m_{k}|^{2}|Δf̂_{α}(k)|^{2} ≤ ∥m∥_{∞}^{2} ∑_{k=0}^{∞} A_{k}^{α+1}|Δf̂_{α}(k)|^{2} ≤ C∥m∥_{∞}^{2}∥f∥_{L_{w(α+1)}^{2}}^{2} .

To dominate the term containing Δm_{k} we deduce from (8) that for α ≥ 0 the Fourier Laguerre coefficients tend to zero as k → ∞. Hence

∑_{k=0}^{∞} A_{k}^{α+1}|f̂_{α}(k + 1)Δm_{k}|^{2} = ∑_{k=0}^{∞} A_{k}^{α+1}|Δm_{k}|^{2} | ∑_{j=k+1}^{∞} Δf̂_{α}(j)|^{2} =: I.

In order to apply Lemma 3.1 b), we choose u_{k} = A_{k}^{α+1}|Δm_{k}|^{2} and v_{k} = A_{k}^{α+1}, and observe that when M ∈ N, 2^{M−1} ≤ N < 2^{M}, we have that

( ∑_{k=0}^{N} u_{k} ∑_{k=N}^{∞} v_{k}^{−1}) ≤ C(N + 1)^{−α} ∑_{j=0}^{M} ∑_{k=2^{j}−1}^{2^{j+1}−2}(k + 1)|Δm_{k}|^{2} \frac{A_{k}^{α+1}}{k + 1}

≤ C(N + 1)^{−α} ∑_{j=0}^{M}(2^{j})^{α}∥m∥_{2,1}^{2} ≤ C∥m∥_{2,1}^{2}

uniformly in N if α > 0. Then Lemma 3.1 b) gives

I ≤ C∥m∥_{2,1}^{2} ∑_{j=0}^{∞} A_{j}^{α+1}|Δf̂_{α}(j)|^{2} ≤ C∥m∥_{2,1}^{2}∥f∥_{L_{w(α+1)}^{2}}^{2}

by (9). Thus there remains to consider the case −1 < α < 0. For the same choice of u_{k} and v_{k} one easily obtains

( ∑_{k=N}^{∞} u_{k} ∑_{k=0}^{N} v_{k}^{−1}) ≤ C∥m∥_{2,1}^{2} .

Now assume that f̂(0) = 0. Then we have

∑_{k=0}^{∞} A_{k}^{α+1}|f̂_{α}(k + 1)Δm_{k}|^{2} = ∑_{k=0}^{∞} A_{k}^{α+1}|Δm_{k}|^{2} | ∑_{j=0}^{k} Δf̂_{α}(j)|^{2} ≤ C∥m∥_{2,1}^{2}∥f∥_{L_{w(α+1)}^{2}}^{2} ,

where the last estimate follows by Lemma 3.1 a); thus Theorem 1.4 is established.

The proof of Theorem 1.5 is essentially contained in [6]. As in [6], 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 the ϕ_{i}(k) are the Fourier Laguerre coefficients of an L_{w(α+1)}^{2}-function Φ^{(i)} with norm ∥Φ^{(i)}∥_{L_{w(α+1)}^{2}} ≤ C (2^{i})^{α/2} and

∑_{k=2^{i}}^{2^{i+1}} A_{k}^{α+1}|Δm_{k}|^{2} = ∑_{k=2^{i}}^{2^{i+1}} A_{k}^{α+1}|Δ(m_{k}ϕ_{i}(k))|^{2} ≤ ∑_{k=0}^{2^{i+2}} A_{k}^{α+1}|Δ(m_{k}ϕ_{i}(k))|^{2}

≤ C∥T_{m}Φ^{(i)}∥_{L_{w(α+1)}^{2}}^{2} ≤ C∥m∥_{M_{α; α+1}^{2}}^{2}∥Φ^{(i)}∥_{L_{w(α+1)}^{2}}^{2} ≤ C2^{iα}∥m∥_{M_{α; α+1}^{2}}^{2}.

This immediately leads to

∥m∥_{∞} + (∑_{2^{i}}^{2^{i+1}} |(k + 1)Δm_{k}|^{2} \frac{1}{k + 1})^{1/2} ≤ C∥m∥_{M_{α; α+1}^{2}},

uniformly in i, since by [6, (10)] there holds ∥m∥_{∞} ≤ C∥m∥_{M_{α; α+1}^{2}}; thus Theorem 1.5 is established.

Remark. 3) (Added on Aug. 10, 1994) The characterization (6) can easily be extended to

M_{α,α+l}^{2} = wbv_{2,l}, α > −1, α ≠ 0, …, l − 1, l ∈ N.  (12)

In the case α < l − 1 the multiplier operator is defined only on the subspace {f ∈ L_{w(α+l)}^{2} : f̂_{α}(k) = 0, 0 ≤ k < (l − 1 − α)/2}.

The necessity part carries over immediately (see also [6]). The sufficiency part will be proved by induction. Thus suppose that (12) is true for l = 1, …, n and α’s as indicated. Then, as in the case n = 1, by (9)

∫_{0}^{∞} |T_{m}f(x)e^{−x/2}|^{2}x^{α+n+1}dx ≈ ∑_{k=0}^{∞} A_{k}^{α+n+1}|Δ^{n}Δ(m_{k}f̂_{α}(k))|^{2}

≤ C ∑_{k=0}^{∞} A_{k}^{α+n+1}|Δ^{n}(m_{k}Δf̂_{α}(k))|^{2} + C ∑_{k=0}^{∞} A_{k}^{α+n+1}|Δ^{n}(f̂_{α}(k + 1)Δm_{k})|^{2} =: I + II

By the assumption and (10)

I ≤ C∥m∥_{wbv_{2,n}}^{2} ∑_{k=0}^{∞} A_{k}^{α+n+1}|Δ^{n}f̂_{α+1}(k))|^{2} ≤ C∥m∥_{wbv_{2,n+1}}^{2} ∫_{0}^{∞} |f(x)e^{−x/2}|^{2}x^{α+n+1}dx

on account of the embedding properties of the wbv–spaces [5]. Analogously II can be estimated by

II ≤ C∥\{(k + 1)Δm_{k}\}∥_{wbv_{2,n}}^{2} ∑_{k=0}^{∞} A_{k}^{α+n+1}|Δ^{n}(\frac{f̂_{α}(k + 1)}{k + 1})|^{2}.

By the Leibniz formula for differences there holds

Δ^{n}(\frac{f̂_{α}(k + 1)}{k + 1}) ≤ C ∑_{j=0}^{n} |Δ^{j}f̂_{α}(k + 1)| |Δ^{n−j} \frac{1}{j + k + 1}|
≤ C ∑_{j=0}^{n}(j + k + 1)^{j−n−1}|Δ^{j}f̂_{α}(k + 1)|.

Hence we have to dominate for j = 0, …, n

II_{j} := ∑_{k=0}^{∞} A_{k}^{α−n−1+2j}|Δ^{j}f̂_{α}(k + 1)|^{2}.

If α > n then c_{j} := −α − 2j + n + 1 < 1 for all j = 0, …, n, Δ^{j}f̂_{α}(k + 1) = ∑_{i=k+1}^{∞} Δ^{j+1}f̂_{α}(i), and we can apply [8, Theorem 346] repeatedly to obtain

II_{j} ≤ C ∑_{k=0}^{∞} A_{k}^{α−n−1+2j}|(k + 1)Δ^{j+1}f̂_{α}(k + 1)|^{2} ≈ ∑_{k=0}^{∞} A_{k}^{α−n+2j+1}|Δ^{j+1}f̂_{α}(k + 1)|^{2}

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

Since ∥{(k + 1)Δm_{k}}∥_{wbv_{2,n}} ≤ C∥m∥_{wbv_{2,n+1}}, this gives the assertion for the weight x^{n+1} in the case α > n.

If α < n, α ≠ 0, …, n, then some c_{j} > 1. For the application of [8, Theorem 346] one needs c_{j} ≠ 1; this is guaranteed by the hypothesis α ≠ 0, …, n (in the case of an additional weight x^{n+1}). For the j for which c_{j} > 1 we have to use the representation

Δ^{j}f̂_{α}(k + 1) = − ∑_{i=0}^{k} Δ^{j+1}f̂_{α}(i), if Δ^{j}f̂_{α}(0) = 0,

i.e., the first (j + 1) Fourier-Laguerre coefficients have to vanish to ensure this representation. But 0 ≤ j ≤ j_{0}, where j_{0} is choosen in such a way that c_{j_{0}} > 1 and c_{j_{0}+1} < 1, hence j_{0} = [(n − α)/2] (with respect to the additional weight x^{n+1}); here we used the standard notation for [a], a ∈ R, to be the greatest integer ≤ a. Hence the condition that the first [(n−α)/2]+1 Fourier-Laguerre coefficients have to vanish is needed if the additional weight is x^{n+1}. A repeated application of [8, Theorem 346] with appropriate c > 1 or c < 1 now gives the assertion.

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