A lower estimate for the Lebesgue constants of linear means of Laguerre expansions

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

Dedicated to P. L. Butzer on the occasion of his 70th birthday in gratitude

(Oct. 14, 1997 version)

Abstract. S. G. Kal’neĭ derived in [5], [6] a quite sharp necessary condition for the multiplier norm of a finite sequence in the setting of Fourier-Jacobi series on L^{1} with “natural weight” (which ensures a nice convolution structure). In this paper, Kalneĭ’s problem is considered in the setting of Laguerre series on weighted L^{1}-spaces; the admitted scale of weights contains in particular the appropriate “natural weights” occurring in transplantation and convolution.

Key words. Laguerre polynomials, linear means, Lebesgue constants, weighted Lebesgue spaces

AMS(MOS) subject classifications. Primary: 33C45, 42A45, 42C10; Secondary: 39A70

1 Introduction

The purpose of this paper is to derive a good lower bound for the L^{1}-Laguerre multiplier norm of a finite sequence m = {m_{k}} , m_{k} = 0 when k ≥ n + 1. Such sequences occur when one considers linear summability methods generated by a lower triangular numerical matrix Λ = {λ_{k}^{n}}. A very important example is the Cesàro method for which the general necessary criteria in [9], [2] only give a constant as a lower bound at the critical index whereas discussing the Cesàro means (at the critical index) directly gives a logarithmic divergence (see [3], [4]). To become more precise let us first introduce some notation. Consider the Lebesgue spaces

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

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-9401452.

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

where α ≥ γ > −1. Let L_{n}^{α}(x), n ∈ N_{0}, denote the classical Laguerre polynomials (see Szegö [10, 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)} .

Then one can 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 on L_{w(γ)}^{p}, notation m ∈ M_{α; γ}^{p}, if

∥ ∑_{k=0}^{∞} m_{k}f̂_{α}(k)L_{k}^{α}∥_{p,γ} ≤ C∥ ∑_{k=0}^{∞} f̂_{α}(k)L_{k}^{α}∥_{p,γ}  (2)

for all polynomials f; the smallest constant C for which this holds is called the multiplier norm ∥m∥_{M_{α,γ}^{p}}.

We are interested in good lower estimates of ∥m∥_{M_{α,γ}^{1}} for finite sequences m = {m_{k}} , m_{k} = 0 when k ≥ n + 1. By the definition of the multiplier norm we have

∥m∥_{M_{α,γ}^{1}} = sup_{∥f∥_{1,γ}≤1} ∥ ∑_{k=0}^{n} m_{k}f̂_{α}(k)L_{k}^{α}∥_{1,γ} ≥ C(n + 1)^{γ−α}∥ ∑_{k=0}^{n} m_{k}L_{k}^{α}∥_{1,γ} .  (3)

Here the particular test function Φ_{n} is given via its coefficients (Φ_{n})_{ˆ α}(k) = ϕ(k/n) where ϕ is a smooth cut-off function with ϕ(t) = 1 for 0 ≤ t ≤ 1 and = 0 for t ≥ 2. For this function there holds by [2], formula (9) in I, ∥Φ_{n}∥_{1,γ} ≤ C(n+1)^{α−γ} , α ≥ γ > −1, hence (3). Generic positive constants that are independent of the parameter n and of the sequence m will be denoted by C. Our main result now reads

Theorem 1. Suppose α ≥ 0 , α/2 ≤ γ ≤ α. Then for any finite sequence m = {m_{k}} , m_{k} = 0 for k ≥ n + 1, there holds

∥m∥_{M_{α,γ}^{1}} ≥ C(n + 1)^{γ−α} ∑_{k=0}^{n} |m_{k}|\frac{(k + 1)^{γ+1/2}}{(n + 1 − k)^{2γ−α+3/2}} ,  (4)

where C is a constant independent of n.

In the following remarks we discuss this result for the standard weight γ = α/2 , which is the natural setting for transplantation theorems (see Kanjin [7]), and the weight γ = α , which is the natural setting for a nice convolution structure (see Görlich and Markett [4]). Remarks. 1) Taking only the (k = n)–term in (4) and γ = α one arrives at the Cohen type inequality (1.8) in [9], whereas γ = α/2 leads to (1.9) in [9].

2) The Cesàro means of order δ are generated by the matrix Λ = (λ_{k}^{n}) where λ_{k}^{n} = A_{n−k}^{δ}/A_{n}^{δ}. If one evaluates the ∑_{[n/2]}^{n} … –portion in (4) in the case of the critical index δ_{c} = 1/2 when γ = α/2 or δ_{c} = α + 1/2 when γ = α one obtains

∥{A_{n−k}^{δ_{c}}/A_{n}^{δ_{c}}}∥_{M_{α,γ}^{1}} ≥ C log(n + 1) , α ≥ 0 .

For γ = α/2 this is directly computed in [3], for γ = α in [4]. In the case 0 ≤ δ < δ_{c} = 2γ − α + 1/2 already the (k = n)–term of the sum (cf. the above mentioned Cohen type inequalities) leads to the estimate (for γ = α/2 see [3], for γ = α [4])

∥{A_{n−k}^{δ}/A_{n}^{δ}}∥_{M_{α,γ}^{1}} ≥ C(n + 1)^{δ_{c}−δ} , 0 ≤ δ < δ_{c} .

3) Obviously, by omitting the terms on the right side of (4) with k < [n/2], there holds

∥m∥_{M_{α,γ}^{1}} ≥ C(n + 1)^{δ_{c}} ∑_{k=[n/2]}^{n} |m_{k}|(n + 1 − k)^{−δ_{c}−1} .  (5)

which is equivalent to (4) since M_{α,γ}^{1} ⊂ l^{∞} (just choose f = L_{k}^{α} in (2)). Condition (5) may be compared with the necessary conditions given in [2].

For the proof of the Theorem we follow the lines of Kal’neĭ and first observe that by the converse of Hölder’s inequality we may continue the estimate (3) as follows

∥m∥_{M_{α,γ}^{1}} ≥ C(n + 1)^{γ−α} sup_{∥g∥_{∞,γ}≤1} ∫_{0}^{∞} ∑_{k=0}^{n} m_{k}L_{k}^{α}(x)g(x)e^{−x}x^{α}dx .  (6)

If we choose a particular g we make the right hand side of (6) smaller. Consider the test function g = g_{n},

g_{n}(x) = ∑_{j=0}^{n}(sgn m_{j}) Δ_{2(n+1−j)}^{N}R_{j}^{α}(x)\frac{(j + 1)^{γ+1/2}}{(n + 1 − j)^{2γ−α+3/2}} ,  (7)

where Δ_{k}R_{j}^{α} = R_{j}^{α} − R_{j+k}^{α} , Δ^{N} = Δ(Δ^{N−1}), and N ∈ N is so large that N − 1 ≤ 2γ − α + 1/2 < N. Suppose that the lemma below holds, then g_{n} ∈ L_{w(γ)}^{∞} is obviously true and the assertion of the Theorem immediately follows by the orthogonality of the Laguerre polynomials. Thus there only remains to prove the following result.

Lemma. Suppose that 0 ≤ α/2 ≤ γ ≤ α and set

f_{n}^{α,γ}(x) = ∑_{j=0}^{n} |Δ_{2(n+1−j)}^{N}R_{j}^{α}(x)|\frac{(j + 1)^{γ+1/2}}{(n + 1 − j)^{2γ−α+3/2}} ,

where N − 1 ≤ 2γ − α + 1/2 < N ∈ N. Then there holds ∥f_{n}^{α,γ}∥_{∞,γ} ≤ C with C independent of n.

2 Proof of the Lemma

We use the standard notation ν = 4n + 2α + 2 and note that on account of formulae (2.5) and (2.7) in [8] there holds for α > −1 and some positive ξ

|e^{−x/2}R_{n}^{α}(x)| ≤ C \{ \begin{matrix}1  , & 0 ≤ x ≤ 2/ν , \\ (xν)^{−α/2−1/4}  , & 1/2ν ≤ x ≤ 3ν/4 , \\ (xν)^{−α/2}(ν(ν^{1/3} + |x − ν|))^{−1/4} , & ν/4 ≤ x ≤ 2ν , \\ (xν)^{−α/2}e^{−ξx}  , & 5ν/4 ≤ x . \end{matrix} (8)

Thus, if we choose n_{∗} := [(ν − 40)/20], these estimates are also true for R_{n−k}^{α}(x) and R_{n+k+2}^{α}(x) on the x-intervals [0, 1/ν] , [1/ν, ν/2] , [ν/2, 3ν/2] , [3ν/2, ∞), resp., when k ≤ n_{∗}.

We start with the case N = 1, thus 2γ − α < 1/2 , and decompose f_{n}^{α,γ} as follows

f_{n}^{α,γ}(x) = (∑_{k=0}^{n_{∗}} + ∑_{k=n_{∗}+1}^{n}) |Δ_{2(k+1)}R_{n−k}^{α}(x)|\frac{(n + 1 − k)^{γ+1/2}}{(k + 1)^{2γ−α+3/2}} =: Σ_{n,1}(x) + Σ_{n,2}(x) .

(9)

Let us first handle the contribution coming from Σ_{n,2}. By [8], formula (2.9), one has ess sup_{x}|R_{n}^{α}(x)x^{α−γ}e^{−x/2}| ≤ C(n + 1)^{γ−α} and, therefore,

ess sup_{x}|Σ_{n,2}(x)x^{α−γ}e^{−x/2}| ≤ C(n + 1)^{α−2γ−3/2} ∑_{k=n_{∗}+1}^{n}(n + 1 − k)^{γ+1/2+(γ−α)} ≤ C (10)

uniformly in n. Concerning the estimate of Σ_{n,1}(x) we first consider the case 0 < x < 1/ν. Use of the identity

Δ_{2k+2}R_{n−k}^{α}(x) = \frac{x}{α + 1}(R_{n−k}^{α+1}(x) + R_{n−k+1}^{α+1}(x) + ⋯ + R_{n+k+1}^{α+1}(x))  (11)

in combination with the first case of (8) gives

|e^{−x/2}Δ_{2(k+1)}R_{n−k}^{α}(x)| ≤ \frac{xe^{−x/2}}{α + 1} ∑_{j=0}^{2k+1} |R_{n−k+j}^{α+1}(x)| ≤ C(k + 1)x ,  (12)

and thus the desired estimate

|x^{α−γ}e^{−x/2}Σ_{n,1}(x)| ≤ Cx^{α−γ+1}(n + 1)^{γ+1/2} ∑_{k=0}^{n_{∗}}(k + 1)^{α−2γ−1/2}

≤ Cx^{α−γ+1}(n + 1)^{α−γ+1} ≤ C .

It is similarly simple to deal with the case x ≥ 3ν/2. In this case we have by (8) (note ξ > 0) that

|x^{α−γ}e^{−x/2}Σ_{n,1}(x)| ≤ C(n + 1)^{γ−α/2+1/2}x^{α/2−γ}e^{−ξx} ∑_{k=0}^{n_{∗}}(k + 1)^{α−2γ−3/2} ≤ C . (13)

To deal with Σ_{n,1}(x) for 1/ν ≤ x ≤ ν/2 we make use of formula (2.5) in [8] and (11) to obtain

|e^{−x/2}Δ_{2k+2}R_{n−k}^{α}(x)| ≤ Cx(k + 1)(xν)^{−α/2−3/4}.

If one considers a fixed x , 1/ν ≤ x ≤ ν/2, one can find a real number λ , −1 < λ < 1, such that ν^{λ}/2 ≤ x ≤ ν^{λ}. Choosing μ = 1/2 − λ/2 > 0 we obtain with the previous asymptotic and the second case in (8)

|x^{α−γ}e^{−x/2}Σ_{n,1}(x)| ≤ C(n+1)^{γ+1/2}x^{α−γ}e^{−x/2} ∑_{k=0}^{[n_{∗}^{μ}]} |R_{n−k}^{α}(x)−R_{n+k+2}^{α}(x)|(k +1)^{α−2γ−3/2}

+(n + 1)^{γ+1/2}x^{α−γ}e^{−x/2} ∑_{k=[n_{∗}^{μ}]+1}^{n_{∗}}(|R_{n−k}^{α}(x)| + |R_{n+k+2}^{α}(x)|)(k + 1)^{α−2γ−3/2}

=: Σ_{n,3}(x) + Σ_{n,4}(x) ≤ C(n + 1)^{γ+1/2−(α/2+3/4)}x^{α−γ+1−(α/2+3/4)} ∑_{k=0}^{[n_{∗}^{μ}]}(k + 1)^{α−2γ−1/2}

+C(n + 1)^{γ+1/2−(α/2+1/4)}x^{α−γ−(α/2+1/4)}(n + 1)^{(α−2γ−1/2)μ} ≤ C

uniformly in n.

Concerning an estimate of Σ_{n,1}(x) for x ∈ [ν/2, 3ν/2] we may restrict ourselves to x ∈ [ν/2, ν] since the interval [ν, 3ν/2] is handled in the same way. We start with fixed x , ν − 2ν^{1/3} ≤ x ≤ ν, hence x ≈ n + 1, and use the preceding decomposition |x^{α−γ}e^{−x/2}Σ_{n,1}(x)| ≤ Σ_{n,3}(x) + Σ_{n,4}(x) with μ = 1/3. By (8), third case, there follows

Σ_{n,4}(x) ≤ C(n + 1)^{1/2−1/3} ∑_{k=[n_{∗}^{1/3}]+1}^{n_{∗}}(k + 1)^{α−2γ−3/2} ≤ C

uniformly in n. In order to dominate Σ_{n,3}(x) we have to use precise asymptotics for the orthonormal Laguerre functions

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

as given in Askey and Wainger [1, p. 699]. We first observe that

Σ_{n,3}(x) ≤ C(n + 1)^{α/2+1/2} ∑_{k=0}^{[n_{∗}^{1/3}]} |\frac{L_{n−k}^{α}(x)}{\sqrt{L_{n−k}^{α}(0)}} − \frac{L_{n+k+2}^{α}(x)}{\sqrt{L_{n+k+2}^{α}(0)}}| (k + 1)^{α−2γ−3/2}

≤ C(n + 1)^{1/2} ∑_{k=0}^{[n_{∗}^{1/3}]} |L_{n−k}^{α}(x) − L_{n+k+2}^{α}(x)|(k + 1)^{α−2γ−3/2}

+C(n + 1)^{α/2+1/2} ∑_{k=0}^{[n_{∗}^{1/3}]} |Δ_{2}(L_{n−k}^{α}(0))^{−1/2}||L_{n+k+2}^{α}(x)|(k + 1)^{α−2γ−3/2}

=: Σ′_{n,3}(x) + Σ′′_{n,3}(x)

(note that x ≈ (n + 1) ). That |Σ′′_{n,3}(x)| ≤ C holds is obvious when one uses the third case of (8) and observes that for 0 ≤ k ≤ n^{∗} one has

|\frac{1}{\sqrt{L_{n−k}^{α}(0)}} − \frac{1}{\sqrt{L_{n+k+2}^{α}(0)}}| ≤ \frac{C(k + 1)}{(n + 1)^{α/2+1}} .

The crucial term Σ′_{n,3}(x) also turns out to be uniformly bounded when we use the fourth asymptotic in [1, p. 699].

Σ′_{n,3}(x) ≤ C(n + 1)^{1/2} ∑_{k=0}^{[n_{∗}^{1/3}]} ∑_{j=0}^{k} |Δ_{2}L_{n+2j−k}^{α}(x)|(k + 1)^{α−2γ−3/2}

≤ C(n + 1)^{1/2−2/3} ∑_{k=0}^{[n_{∗}^{1/3}]}(k + 1)^{α−2γ−1/2} ≤ C.

Let us now consider the remaining x , ν/2 ≤ x ≤ ν − 2ν^{1/3}. Then there exists a λ , 1/3 < λ < 1 , such that ν − 2ν^{λ} ≤ x ≤ ν − ν^{λ} . Associate to λ the number μ , 4μ = (1 − λ)/(2γ − α + 1/2); then obviously 0 < μ < 1. Analogously to the above we decompose Σ_{n,1}(x) in the following way

|x^{α−γ}e^{−x/2}Σ_{n,1}(x)| ≤ C(n + 1)^{1/2} ∑_{k=0}^{[n_{∗}^{μ}]} ∑_{j=0}^{k} |Δ_{2}L_{n+2j−k}^{α}(x)|(k + 1)^{α−2γ−3/2}

+C(n + 1)^{α/2+1/2} ∑_{k=0}^{[n_{∗}^{μ}]} |Δ_{2}(L_{n−k}^{α}(0))^{−1/2}||L_{n+k+2}^{α}(x)|(k + 1)^{α−2γ−3/2}

+C(n + 1)^{α+1/2}e^{−x/2} ∑_{k=[n_{∗}^{μ}]+1}^{n_{∗}}(|R_{n−k}^{α}(x)| + |R_{n+k+2}^{α}(x)|)(k + 1)^{α−2γ−3/2}

=: Σ′_{n,3}(x) + Σ′′_{n,3}(x) + Σ_{n,4}(x) .

To estimate Σ′_{n,3}(x) we note that by [1, p. 699]

|Δ_{2}L_{n+2j−k}^{α}(x)| ≤ C(n + 2j − k)^{−3/4}|4(n + 2j − k) + 2α + 2 − x|^{1/4} ≤ C(n + 1)^{−3/4+λ/4}

since n+2j −k ≥ n−k ≥ n/2 and |4(n+2j −k)+2α+2−x| ≤ C|8j −4k+ν^{λ}| ≤ Cν^{λ} (observe that k ≤ [n_{∗}^{μ}] ≤ Cn^{λ} , μ ≤ λ , and λ > 1/3). Thus

Σ′_{n,3}(x) ≤ C(n + 1)^{1/2}(n + 1)^{−3/4+λ/4} ∑_{k=0}^{[n_{∗}^{μ}]}(k + 1)^{α−2γ−1/2} ≤ C.

Analogously we have that Σ′′_{n,3}(x) is uniformly bounded in n on the interval [ν − 2ν^{λ}, ν − ν^{λ}]. To dominate Σ_{n,4}(x) we note that by (8), third case, the worst contribution estimate comes from |R_{n−k}^{α}(x)| so that the k-range has to be examined in order to know which asymptotic to use. Since |4(n − k) + 2α + 2 − x| ≈ |ν^{λ} − 4k| we further split up

Σ_{n,4}(x) ≤ C(n + 1)^{α+1/2}e^{−x/2} ∑_{k=[n_{∗}^{μ}]+1}^{n_{∗}} … =: Σ′_{n,4}(x) + Σ′′_{n,4}(x)

where in Σ′ only over those k is summed for which |ν^{λ} − 4k| ≥ ν^{λ}/2, thus the summation variable k in the sum associated to Σ′′ runs from [ν^{λ}/8] to [3ν^{λ}/8]. Then, dealing with Σ′ and observing that for these k there holds |x^{α/2}e^{−x/2}R_{n−k}^{α}(x)| ≤ C(n + 1)^{−1/4−α/2}ν^{−λ/4} we obtain

Σ′_{n,4}(x) ≤ C(n + 1)^{1/4}ν^{−λ/4} ∑_{k=[n_{∗}^{μ}]+1}^{n_{∗}}(k + 1)^{α−2γ−3/2} ≤ C

by the choice of λ and μ since α ≤ 2γ by hypothesis. Turning to Σ′′ we note that for these k in any case |x^{α/2}e^{−x/2}R_{n−k}^{α}(x)| ≤ C(n + 1)^{−1/3−α/2} holds so that

Σ′′_{n,4}(x) ≤ C(n + 1)^{1/2}(n + 1)^{−1/3} ∑_{k=[ν^{λ}/8]}^{[3ν^{λ}/8]}(k + 1)^{α−2γ−3/2} ≤ C

since λ > 1/3.

Now let α , γ be such that N − 1 ≤ 2γ − α + 1/2 < N , N ∈ N , N ≥ 2. We make again a decomposition analogous to (9), this time choosing n_{∗} := [(ν − 40)/100N] , replacing Δ by Δ^{N} , and denoting the two resulting sums by Σ_{n,1,N}(x) and Σ_{n,2,N}(x). The estimate analogous to (10) remains valid on account of the triangle inequality. If instead of (12) one uses the estimate |Δ_{2(k+1)}^{N}R_{n−k}^{α}(x)| ≤ C(k + 1)^{N}x^{N} it is clear that |x^{α−γ}e^{−x/2}Σ_{n,1,N}(x)| ≤ C holds for 0 < x < 1/ν. Also the analog of (13) is obvious for x ≥ 3ν/2 so that there is only to discuss the case 1/ν ≤ x ≤ 3ν/2.

By definition we have Δ_{2k+2}^{N} = Δ_{2k+2}Δ_{2k+2}^{N−1}. Following Kal’neĭ [6] we use the equality

Δ_{2k+2}^{N−1}R_{n−k}^{α}(x) = ∑_{m=0}^{k} ⋯ ∑_{l=0}^{k} Δ_{2}^{N−1}R_{n−k+2l+···+2m}^{α}(x)  (14)

with (N − 1) summations, and also observe that, by formula (3) in Part I of [2],

Δ_{2}^{N−1}R_{k}^{α}(x) = ∑_{j=0}^{N−1} C_{j,N}Δ_{1}^{N−1}R_{k+j}^{α}(x) = C_{α,N} ∑_{j=0}^{N−1} C_{j,N}x^{N−1}R_{k+j}^{α+N−1}(x) .

Hence, when we need to work with Δ_{2k+2}^{N}R_{n−k}^{α}(x) , 0 ≤ k ≤ n_{∗} = [(ν − 40)/100N] , it suffices to replace this by a linear combination (in j , 0 ≤ j ≤ N − 1) , of (k + 1)^{N−1} terms (in i coming from (14)) of the type

x^{N−1}Δ_{2k+2}R_{n+j+2i−k}^{α+N−1}(x) , 0 ≤ j ≤ N − 1 , 0 ≤ i ≤ k(N − 1) , 0 ≤ k ≤ n_{∗} .

We note, since n_{∗} is chosen so small, that on 1/ν ≤ x ≤ 3ν/2 the R_{n−k+2i}^{α} have the same asymptotics for the relevant k, i as well as R_{n+j+2i−k}^{α+N−1} for the relevant k, j, i.

Let us now consider Σ_{n,1,N}(x) on 1/ν ≤ x ≤ ν/2. As in the case N = 1 we fix x, can find a real λ , −1 < λ < 1, such that ν^{λ}/2 ≤ x ≤ ν^{λ} and choose μ = 1/2 − λ/2 > 0. Then, by the preceding discussion and with the abbreviation N_{k}^{∗} := (2k + 1)(N − 1),

|x^{α−γ}e^{−x/2}Σ_{n,1,N}(x)| ≤ C(n + 1)^{γ+1/2}x^{α−γ}e^{−x/2} ∑_{k=0}^{[n_{∗}^{μ}]} |Δ_{2k+2}^{N}R_{n−k}^{α}(x)|(k + 1)^{α−2γ−3/2}

+C(n + 1)^{γ+1/2}x^{α−γ}e^{−x/2} ∑_{k=[n_{∗}^{μ}]+1}^{n_{∗}} sup_{0≤j≤N} |R_{n+2j(k+1)−k}^{α}(x)|(k + 1)^{α−2γ−3/2}

≤ C(n + 1)^{γ+1/2}x^{α−γ+(N−1)}e^{−x/2} ∑_{k=0}^{[n_{∗}^{μ}]} sup_{0≤j≤N_{k}^{∗}} |Δ_{2k+2}R_{n+j−k}^{α+N−1}(x)|(k + 1)^{α−2γ−3/2+(N−1)}

+C(n + 1)^{γ+1/2−(α/2+1/4)}x^{α−γ−(α/2+1/4)} ∑_{k=[n_{∗}^{μ}]+1}^{n_{∗}} sup_{0≤j≤N}(k + 1)^{α−2γ−3/2}

≤ C(n + 1)^{γ−α/2−N/2+1/4}x^{α/2−γ+N/2−1/4} ∑_{k=0}^{[n_{∗}^{μ}]}(k + 1)^{α−2γ−3/2+N} + C ≤ C

uniformly in n.

To complete the proof of Lemma we may restrict ourselves, as in the case N = 1, to discussing the x-interval [ν/2, ν]. We start with ν − 2ν^{1/3} ≤ x ≤ ν and set 6μ := 1/(2γ − α + 1/2). Then, as in the case 1/ν ≤ x ≤ ν/2,

|x^{α−γ}e^{−x/2}Σ_{n,1,N}(x)|

≤ C(n + 1)^{α+N−1/2}e^{−x/2} ∑_{k=0}^{[n_{∗}^{μ}]} sup_{0≤j≤N_{k}^{∗}} |Δ_{2k+2}R_{n+j−k}^{α+N−1}(x)|(k + 1)^{α−2γ−3/2+(N−1)}

+C(n + 1)^{α+1/2}e^{−x/2} ∑_{k=[n_{∗}^{μ}]+1}^{n_{∗}} sup_{0≤j≤N} |R_{n+2j(k+1)−k}^{α}(x)|(k + 1)^{α−2γ−3/2}

=: Σ_{n,3,N}(x) + Σ_{n,4,N}(x)

That Σ_{n,4,N} is uniformly bounded follows by our choice of μ when we use the third line of (8). Now

Σ_{n,3,N}(x) ≤ C(n + 1)^{(α+N)/2} ∑_{k=0}^{[n_{∗}^{μ}]} sup_{0≤j≤N_{k}^{∗}} |Δ_{2k+2} \frac{L_{n+j−k}^{α+N−1}(x)}{\sqrt{L_{n+j−k}^{α+N−1}(0)}}|(k + 1)^{α−2γ−5/2+N}

≤ C(n + 1)^{1/2} ∑_{k=0}^{[n_{∗}^{μ}]} sup_{0≤j≤N_{k}^{∗}} |Δ_{2k+2}L_{n+j−k}^{α+N−1}(x)|(k + 1)^{α−2γ−5/2+N}

+C(n + 1)^{(α+N)/2} ∑_{k=0}^{[n_{∗}^{μ}]} sup_{0≤j≤N_{k}^{∗}} |Δ_{2k+2}(L_{n+j−k}^{α+N−1}(0))^{−1/2}||L_{n+j+k+2}^{α+N−1}(x)|(k + 1)^{α−2γ−5/2+N}

≤ C(n + 1)^{−1/6} ∑_{k=0}^{[n_{∗}^{μ}]}(k + 1)^{α−2γ−3/2+N} + C(n + 1)^{−1/2−1/3} ∑_{k=0}^{[n_{∗}^{μ}]}(k + 1)^{α−2γ−3/2+N} ≤ C

uniformly in n since N ≥ 2. Hence there remains to consider fixed x ∈ [ν/2, ν −2ν^{1/3}]. As in the (N = 1)-case there is a λ , 1/3 < λ < 1, such that ν − 2ν^{λ} ≤ x ≤ ν − ν^{λ}; choose μ , 4μ = (1 − λ)/(2γ − α + 1/2). Since 1 ≤ N − 1 ≤ 2γ − α + 1/2 we obviously have μ < 1/6. Make the same decomposition as in the preceding [ν − 2ν^{1/3}, ν]-case. Concerning the ∑_{k=0}^{[n_{∗}^{μ}]}–contribution we use the third line of the asymptotics in [1, p. 699] and observe that |x−4(n+j −k)−2α−2| ≤ Cν^{λ} since 0 ≤ j ≤ (2k+1)(N −1) ≤ Cn^{μ}, thus

Σ_{n,3,N}(x) ≤ C(n + 1)^{1/2} ∑_{k=0}^{[n_{∗}^{μ}]} sup_{0≤j≤N_{k}^{∗}} ∑_{i=0}^{k} |Δ_{2}L_{n+j+2i−k}^{α+N−1}(x)|(k + 1)^{α−2γ−5/2+N}

+C(n + 1)^{−1/2} ∑_{k=0}^{[n_{∗}^{μ}]} sup_{0≤j≤N_{k}^{∗}} |L_{n+j+k+2}^{α+N−1}(x)|(k + 1)^{α−2γ−3/2+N}

≤ C(n + 1)^{1/2−3/4+λ/4} ∑_{k=0}^{[n_{∗}^{μ}]}(k + 1)^{α−2γ−3/2+N}

+C(n + 1)^{−1/2−1/4−λ/4} ∑_{k=0}^{[n_{∗}^{μ}]}(k + 1)^{α−2γ−3/2+N} ≤ C .

In order to dominate the ∑_{k=[n_{∗}^{μ}]}^{n_{∗}}-contribution we use the method, analogous to the corresponding (N = 1)-case. Hence, we split Σ_{n,4,N} into a sum Σ′ where the summation variable k also satisfies the inequality |ν^{λ} − 4k| ≥ ν^{λ}/2 and a sum Σ′′ over the remaining k’s. Then, as in the corresponding (N = 1)-case, both contributions turn out to be uniformly bounded.

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