Some Fourier inequalities for orthogonal systems in Lorentz–Zygmund spaces

R E S E A R C H Open Access

Some Fourier inequalities for orthogonal systems in Lorentz–Zygmund spaces

G. Akishev^1,2, L.E. Persson^3*and A. Seger³

*Correspondence:

larserik6pers@gmail.com

3Department of Computer Science and Computational Engineering Campus Narvik, The Artic University of Norway, Narvik, Norway Full list of author information is available at the end of the article

Abstract

A number of classical inequalities and convergence results related to Fourier coefficients with respect to unbounded orthogonal systems are generalized and complemented. All results are given in the case of Lorentz–Zygmund spaces.

MSC: 42A16; 42B05; 26D15; 26D20; 46E30

Keywords: Inequalities; Fourier series; Fourier coefficients; Unbounded orthogonal systems; Lorentz–Zygmund spaces

1 Introduction

Letq∈(1, +∞),r∈(0, +∞) andα ∈R. Moreover, letLq,r(logL)^α denote the Lorentz–

Zygmund space, which consists of all measurable functionsf on [0, 1] such that

fq,r,α:=

1 0

f^∗(t)r

1 +|lnt|αr

·t^qr–1dt ¹_r

< +∞,

wheref^∗is a nonincreasing rearrangement of the function|f|(see e.g. [1]).

If α = 0, then the Lorentz–Zygmund space coincides with the Lorentz space:

Lq1,q2(logL)^α=Lq1,q2. If α= 0 andq1 =q2=q, then Lq1,q2(logL)^α space coincides with the Lebesgue spaceLq[0, 1] (see e.g. [2]) with the norm

fq:=

₁

0

f(x)^qdx

1 q

, 1≤q< +∞.

Moreover,L_∞[0, 1] denotes the space, which consists of all measurable function on [0, 1]

such that

f∞:=ess sup

x∈[0,1]

f(x)<∞.

We consider an orthogonal system{ϕ_n}inL₂[0, 1] such that

ϕ_ns≤Mn, n∈N, (1)

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and

μ_n=sup

n k=1

ckϕ_k s

: n

k=1

c²_k= 1

, ρ_n= _∞

k=n

|ak|^{2 1}₂

, (2)

for somes∈(2, +∞]. HereMn↑andMn≥1 (see [3], [4, p. 313]).

An orthonormal system{u_n}is called uniformly bounded if there is a constantM> 0 such that un∞≤M,∀n∈N. Note that any uniformly bounded system{un} satisfies condition (1) but the reversed implication is false.

For one variable function Marcinkiewicz and Zygmund [4] proved the following theo- rems.

Theorem A(see [4]) Let the orthogonal system{ϕ_n}satisfy the condition(1)and2≤p<s.

If the real number sequence{an}satisfies the condition ∞

n=1

|a_n|^pM^(p–2)

s

n s–2n^{(p–2)s–1s–2}< +∞,

then the series ∞

n=1

anϕ_n(x)

converges in L_pto some function f ∈L_p[0, 1]and

fp≤C_{p,s ∞}

n=1

|a_n|^pM^(p–2)

s

n s–2n^{(p–2)s–1s–2 1}_p

.

Theorem B(see [4]) Let the orthogonal system{ϕ_n}satisfy the condition(1),and _s–1^s = μ<p≤2.Then the Fourier coefficients an(f)of the function f ∈Lp[0, 1]with respect to the system{ϕ_n}satisfy the inequality

_∞

n=1

a_n(f)^pM^(p–2)

s

n s–2n^{(p–2)s–1s–2 1}_p

≤C_p,sfp.

Nowadays there are several generalizations of TheoremsAandBfor different spaces and systems (see e.g. [5–8] and the corresponding references).

Here we just mention that Flett [8] generalized this to the case of Lorentz spaces and that Maslov [5] proved generalizations of TheoremAand TheoremBin Orlicz spaces.

The problem concerning the summability of the Fourier coefficients by bounded or- thonormal system with functions from some Lorentz spaces were investigated e.g. by Stein [9], Bochkarev [10], Kopezhanova and Persson [11] and Kopezhanova [12] (cf. also Persson [13]).

Moreover, Kolyada [6] proved the following improvement of TheoremA.

Theorem C(see [6]) Let an orthogonal system{ϕ_n} satisfy the condition(1),let the se- quence{an} ∈l2andρ_n= (_∞

k=n|ak|²)¹², 2 <q<s≤+∞.If

_q(a) = _∞

n=1

μ

(q–2)s ns–2

ρ_n^q–ρ^q_n+1¹_q

< +∞,

then the series_∞

n=1anϕ_n(x)converges in the space Lq to some function f ∈Lq and the following inequality holds:fq≤Cq,s_q(a).

This result was further generalized by Kirillov [7] as follows.

Theorem D(see [7]) If2 <q<s,r> 0,δ=^r(q–2)s_q(s–2) and the sequence{an} ∈l2satisfies the following condition:

_q,r(a) = _∞

n=1

ρ_n^r–ρ_n+1^r μ^δ_n

¹_r

<∞

μ_n≡μ^(s)_n, ρ_n= _∞

k=n

|ak|^{2 1}₂

,

then the series_∞

n=1a_nϕ_n(x)converges in space L2[0, 1]to some function f and the inequal- ityfq,r≤Cq,r,sq,r(a)holds. (Hereμnandρnare defined by(2).)

The following well-known lemma is used in our proofs.

Lemma E Let0 <p<∞,and{a_k}^∞_k=0and{b_k}^∞_k=0are non-negative sequences.

(i) If ∞

n=k

an≤Cak, k= 0, 1, 2, . . . , (3)

then ∞

n=0

an

_n

k=0

bk

p

≤Cp^p ∞

n=0

anb^p_n.

(ii) If k

n=0

an≤Cak, k= 0, 1, 2, . . . , (4)

then ∞

n=0

an

_∞

k=n

bk

p

≤Cp^p ∞

n=0

anb^p_n,

whereCis a positive number independent ofn.

In this paper we both generalize and complement the statements in TheoremsA–Din various ways and always to the case with Lorentz–Zygmund spaces involved. In partic- ular, in Sect.2such a generalization of TheoremD(and, thus, of TheoremsAandC) is

proved (see Theorem2.1). In Sect.3such a complement of TheoremBto the caseq< 2 is given (see Theorem3.1). Finally, in Sect.4we present and prove some further results for uniformly bounded systems and give some concluding remarks. In particular, we compare our results with some other recent research. For the reader’s convenience we also include a proof of LemmaEin theAppendix.

2 Generalization of TheoremD

In this section we state and prove the following generalization of TheoremD.

Theorem 2.1 Let2 <q<s≤+∞,α∈R,r> 0andδ=^rs(q–2)_q(s–2).If{an} ∈l2and

_q,r,α(a) = _∞

n=1

ρ^r_n–ρ_n+1^r μ^δ_n

1 + 2s

s– 2lnμ_n

αr¹_r

< +∞,

whereρ_nandμ_nare defined by(2),then the series ∞

n=1

a_nϕ_n(x)

with respect to an orthogonal system{ϕ_n}^∞_n=1,which satisfies the condition(1),converges to some function f ∈Lq,r(logL)^αandfq,r,α≤Cq,r,α.

Corollary 2.2 For the caseα= 0,Theorem2.1coincides with TheoremD.

Proof Since the sequence{μ_n}is increasing, let us define the sequence{ν_n}in the following way (see [7]):

ν₁= 1, ν_n+1=min{k∈N:μ_k≥2μνn}, n= 1, 2, 3, . . . . Thenμν_n+1≥2μν_n,μν_n+1–1< 2μν_n,n= 1, 2, . . . .

Lettn=μ^–

2s ns–2,

uj(x) =

ν_j+1–1 k=νj

akϕ_k(x),

Sn(x) = n

k=1

uj(x) and Rn(x) =f(x) –Sn(x).

Sincetn↓0 forn→+∞, by the property of nonincreasing rearrangement of the function (see [14, p. 83]), we get

f^r_q,r,α= ∞

n=1

_tn

tn+1

f^∗∗(t)r

1 +|lnt|αr

t^rq–1dt

≤C _∞

n=1

tn tn+1

S_n^∗∗(t)r

1 +|lnt|αr

t^rq–1dt

+ ∞

n=1

tn tn+1

R^∗∗_n(t)r

1 +|lnt|αr

t^qr–1dt

:=C[I₁+I₂] (5)

and, moreover,

S^∗∗_n(t)≤1 t

n j=1

t 0

u^∗_j(y)dy.

By applying Hölder’s inequality we obtain t

0

u^∗_j(y)dy≤t^1–1sρν_jμνj+1–1.

Therefore,

S^∗∗_n(t)≤t^–1s n

j=1

ρν_jμν_j+1–1.

By using this estimate we find that

I₁≤ ∞

n=1

tn tn+1

_n

j=1

ρν_jμν_j+1–1

r

1 +|lnt|αr

t^{r(1q–1s)–1}dt

≤C ∞

n=1

_n

j=1

ρν_jμν_j+1–1

r

1 +|lntn|αr t^r(

1 q–¹_s) n –t^r(

1 q–¹_s) n+1

.

Thus, by taking into account the definition oft_n, we can conclude that

I1≤ ∞

n=1

_n

j=1

ρν_jμν_j+1–1

r 1 +

2s s– 2lnμνn

^αrμ^–r

2(s–q) q(s–2)

ν_n . (6)

Since for anyε> 0 the functiont^–εlnt↓0 fort→+∞, according to the definition of the numbersν_n, we see that

∞ k=n

1 +

2s s– 2lnμν_k

^αrμ^–r

2(s–q) q(s–2) ν_k

≤

1 +|_s–2^2s lnμ_νn| μ^ε_νn

αr

μ^–r(

2(s–q) q(s–2)–εα) ν_n

∞ k=n

2^–(k–n)r(

2(s–q) q(s–2)–εα)

.

Now choose the numberεsuch that^2(s–q)_q(s–2)–εα> 0. Then ∞

k=n

2^–(k–n)r(

2(s–q) q(s–2)–εα)≤

∞ l=0

2^–lr(

2(s–q)

q(s–2)–εα)< +∞.

Hence, ∞

k=n

1 +

2s s– 2lnμν_k

^αrμ^–r

2(s–q) q(s–2)

ν_k ≤C

1 +

2s s– 2lnμνn

^αrμ^–r

2(s–q) q(s–2) νn .

Therefore, by LemmaE, we have ∞

n=1

_n

j=1

ρν_jμν_j+1–1

r 1 +

2s s– 2lnμν_n

^αrμ^–r

2(s–q) q(s–2) νn

≤C ∞

n=1

(ρν_nμν_n+1–1)^r

1 + 2s

s– 2lnμν_n

^αrμ^–r

2(s–q) q(s–2) νn .

Thus, from (6) it follows that

I1≤C ∞

n=1

(ρν_nμν_n)^r

1 + 2s

s– 2lnμν_n

^αrμ^δ_νn, (7)

whereδ=r^2(s–q)_q(s–2). Sinceρ_n→0 forn→+∞, it yieldsρ_ν^r

n=_∞

k=n(ρ_ν^r_k–ρ_ν^r

k+1). Therefore, by changing the order of summation, we get

∞ n=1

ρ_ν^r_n

1 + 2s

s– 2lnμνn

^αrμ^δ_νn= ∞

k=1

ρ_ν^r

k–ρ^r_ν

k+1

^k

n=1

1 +

2s s– 2lnμνn

^αrμ^δ_νn. (8)

Sinceδ> 0 andμνn+1≥2μνn, we havek

n=1μ^δ_νn≤Cμ^δ_ν

k. Hence, by again using LemmaE, from (8) it follows that

∞ n=1

ρ_ν^r_n

1 + 2s

s– 2lnμνn

^αrμ^δ_νn≤C ∞

k=1

ρ_ν^r

k

1 +

2s s– 2lnμν_k

^αrμ^δ_ν

k. (9)

By now combining inequalities (7) and (9) we obtain

I1≤C ∞

k=1

ρ_ν^r

k

1 +

2s s– 2lnμν_k

^αrμ^δ_ν

k. (10)

Next we estimateI2. By using Hölder’s inequality we find thatR^∗∗_n (t)≤Ct^–12Rn2. There- fore,

I2≤C ∞

n=1

Rn

2

_tn

tn+1

1 +|lnt|αr

t^{r(1q–12)–1}dt

≤C ∞

n=1

R_n

2

1 +|lnt_n|αr tn tn+1

t^{r(1q–12)–1}dt

≤C ∞

n=1

ρ_ν^r

n+1

1 +

2s

s– 2lnμν_n+1

^αrμ^δ_ν

n+1. (11)

Next, by repeating the proof of Eq. (9) we obtain ∞

n=1

ρ_ν^r

n+1

1 +

2s

s– 2lnμν_n+1

^αrμ^δ_ν

n+1≤C ∞

k=1

ρ_ν^r

k

1 +

2s s– 2lnμν_k

^αrμ^δ_ν

k. (12)

By combining the inequalities (11) and (12) we have

I2≤C ∞

k=1

ρ_ν^r

k

1 +

2s s– 2lnμν_k

^αrμ^δ_ν

k. (13)

Moreover, in view of inequalities (10) and (13), from (5) it follows that

f^rq,r,α≤C ∞

k=1

ρ_ν^r

k

1 +

2s s– 2lnμν_k

^αrμ^δ_ν

k (14)

in the caseα> 0. Sinceα> 0 andμ_n↑, we see that ∞

n=1

ρ_n^r–ρ_n+1^r μ^δ_n

1 + 2s

s– 2lnμ_n

αr

= ∞

k=1 ν_k+1–1

n=νk

ρ_n^r–ρ_n+1^r μ^δ_n

1 + 2s

s– 2lnμ_n

αr

≥ ∞

k=1

μ^δ_ν

k

1 + 2s

s– 2lnμν_k

αrν_k+1–1 n=ν_k

ρ_n^r–ρ_n+1^r

= ∞

k=1

μ^δ_ν

k

1 + 2s

s– 2lnμν_k αr

ρ_ν^r

k–ρ_ν^r

k+1

.

Hence, from the inequality (14) it follows that f^r_q,r,α≤C

∞ n=1

ρ_n^r–ρ_n+1^r μ^δ_n

1 + 2s

s– 2lnμ_n

αr

(15)

in the caseα> 0.

Letα< 0. Then, for any numberε> 0, the functiony^ε(1 +lny)^rα increases on (1,∞).

Therefore, by taking into account thatμ_n↑, we obtain ∞

n=1

ρ_n^r–ρ_n+1^r μ^δ_n

1 + 2s

s– 2lnμ_n

αr

= ∞

k=1 ν_k+1–1

n=νk

ρ_n^r–ρ_n+1^r μ^δ_n

1 + 2s

s– 2lnμn αr

≥ ∞

k=1

μ^ε_ν

k

1 + 2s

s– 2lnμν_k

αrν_k+1–1 n=ν_k

ρ_n^r–ρ_n+1^r

μ^δ–ε_n . (16)

Chooseε> 0 such thatδ–ε> 0. Sinceμ^δ–ε_n ↑, according to the inequality (16), we have ∞

n=1

ρ_n^r–ρ_n+1^r μ^δ_n

1 + 2s

s– 2lnμn αr

≥ ∞

k=1

μ^δ_ν

k

1 + 2s

s– 2lnμν_k

αrν_k+1–1 n=νk

ρ_n^r–ρ_n+1^r

in the caseα< 0. Therefore (15) holds also for caseα< 0 and the proof is complete.

Corollary 2.3 Let{ϕ_n}^∞_n=1be an uniformly bounded orthogonal system and let2 <q< +∞, α∈Rand r> 0.

If

_q,r(a) = _∞

n=1

ρ_n^r–ρ_n+1^r n

r(q–2)

2q (1 +lnn)^{αr 1}_r

<∞,

whereρ_nare defined by(2),then the series _∞

n=1a_nϕ_n(x)converges to some function f ∈ Lq,r(logL)^αandfq,r,α≤C·q,r,α.

Proof Since{ϕ_n}^∞_n=1is an uniformly bounded orthogonal system, we haves= +∞. There- fore

s→+∞lim

rs(q– 2)

q(s– 2) =r(q– 2) q . Now, given thatMn≤M,μ_n≤√

nM,n∈N, we have ∞

n=1

ρ_n^r–ρ_n+1^r μ^δ_n

1 + 2s

s– 2lnμ_n

αr

≤C ∞

n=1

ρ_n^r–ρ_n+1^r n

r(q–2)

2q (1 +lnn)^αr (17)

ifα≥0.

Ifα< 0, then we choose a numberεsuch that 0 <ε<^(q–2)_q . Then, by considering the function (1 +lnt)^αt^ε↑on [1, +∞), we can verify that the inequality (17) holds also for α< 0. Consequently, by Theorem2.1, the statement is true.

3 A complement of TheoremB. The caseq< 2

In this section we prove a result which was formulated but not proven in [15]. It may be regarded as a complement of TheoremBrelevant for a more general situation.

Theorem 3.1 Let s∈(2, +∞], _s–1^s <q< 2,r> 1,α∈Randδ=^r(q–2)s_q(s–2).If f ∈Lq,r(logL)^α, then the inequality

_∞

n=1

_ν

n+1–1 k=νn

a²_k(f) ^r₂

(1 +logμν_n)^rαμ^δ_ν

n

¹_r

≤Cfq,r,α

holds,whereμ_νnare defined by(2)and an(f)denote the Fourier coefficients of f with respect to an orthogonal system{ϕ_n}^∞_n=1satisfying condition(1).

Remark3.2 Theorem3.1was formulated, but not proved, in [15]. Here we present the details of the proof.

Proof Choose an increasing sequence{ν_n}of natural numbers such that ν₁= 1,ν_n+1= min{k:μ_k≥2μν_n}, n= 1, 2, 3, . . . . Thenμν_n+1≥2μν_n, μν_n+1–1< 2μν_n. Since the system

{ϕ_n}is orthogonal we have ₁

0

f(x)g(x)dx =

∞ k=1

a_k(f)b_k(g)

for any functiong∈Lq,r(logL)^–α,¹_r+_r¹ = 1 and¹_q+_q¹ = 1. Let

b_k:=

_∞

n=1

_νn+1–1

k=νn

a²_k(f) ₂^r

(1 +logμ_νn)^rαμ^δ_νn –_r¹

× _νn+1–1

k=νn

a_k(f)^2r–2₂

(1 +logμνn)^rαμ^δ_νna_k(f) (18)

fork=ν_n, . . . ,ν_n+1– 1,n= 1, 2, . . . , and consider a functiong∈Lq,r(logL)^–2with Fourier coefficientsbk(g) =bk. Then

₁

0

f(x)g(x)dx =

∞ n=1

ν_n+1–1 k=νn

ak(f)bk(g)

= _∞

n=1

_ν

n+1–1 k=νn

a²_k(f) ^r₂

(1 +logμν_n)^rαμ^δ_ν

n

¹_r

. (19)

Taking into account thatrr=r+r, by Theorem2.1and (18), we have

gq,r,–α≤C _∞

n=1

_ν

n+1–1

k=νn

b²_k(g) ^r₂

(1 +logμν_n)^–rαμ

s(q–2) (s–2)q·r νn

¹

r

=C _∞

n=1

_ν

n+1–1

k=νn

a²_k(f) ₂^r

(1 +logμν_n)^rαμ^δ_νn –¹

r

× _∞

n=1

_ν

n+1–1

k=νn

a²_k(f) ^r_2ν

n+1–1 k=νn

a²_k(f) ^r₂(r–2)

×(1 +logμ_νn)^–rαμ

s(q–2) (s–2)q·r

νn (1 +logμ_νn)^rrαμ

s(q–2) (s–2)qrr νn

¹

r

=C _∞

n=1

_νn+1–1

k=νn

a²_k(f) ₂^r

(1 +logμνn)^rαμ^δ_νn –_r¹

× _∞

n=1

_νn+1–1

k=νn

a²_k(f) ^r₂

(1 +logμνn)^rαμ^δ_νn ¹

r

=C.

Thus, the functiong0:=C^–1g∈Lq,r(logL)^–αandg0q,r,–α≤1. Next, by the property of the norm in the Lorentz–Zygmund space and using equality (19), we get

fq,r,α sup

g∈Lq,r(logL)^–α g_q,r,–α≤1

₁

0

f(x)g(x)dx ≥

₁

0

f(x)g₀(x)dx

=C^–1 _∞

n=1

_νn+1–1

k=νn

a²_k(f) ^r₂

μ^δ_νn ¹_r

.

The proof is complete.

4 Further results and concluding remarks

In this section we first prove some results which are closely related to but not covered by the results in the previous sections (Propositions4.1and4.2). After that, we present some results of a similar kind (see [11,12] and TheoremF) and in remarks we point out how these results can be compared with our results in some special cases when such a comparison is possible.

Proposition 4.1 Let{ϕ_n}^∞_n=1be an uniformly bounded orthogonal system and2 <q< +∞, α∈Rand r> 1.If

q,r,α(a) = _∞

n=1

|an|^rn^{r(1–1q–1r)}(1 +lnn)^{αr 1}_r

<∞,

then the series _∞

n=1anϕ_n(x)converges to some function f ∈Lq,r(logL)^α and fq,r,α ≤ Cq,r,α(a).

Proof Sinceρ_n↓0 whenn→+∞, we can choose numbersn₁= 1,

nk+1=min

n∈N:ρ_nk+1≤1 2ρ_nk

, k= 1, 2 . . . .

Therefore, ifα≥0, it yields ∞

n=1

ρ_n^r–ρ_n+1^r

n^r(q–2)2q (1 +lnn)^αr= ∞

k=2

(n_k– 1)^r(q–2)2q (1 +lnn_k)^αr ρ^r_n

k–1–ρ_n^r

k

. (20)

For any numbersk= 2, 3, . . . , the following inequality holds:

ρ_n^r

k–1–ρ_n^r

k≤ρ_n^r

k–1≤2^r ρ_n²

k–1

^r₂

. (21)

Sinceρ_nk+1≤¹₂ρ_nk≤¹₂ρ_nk–1, we have

ρ_n²

k–1–ρ_n²

k+1≥ρ_n²

k–1– 1

2ρ_nk–1

2

=3 4ρ_n²

k–1. (22)

By using (21) and (22), we can obtain the following inequality:

ρ_n^r

k–1–ρ_n^r

k≤2^r 4

3

r 2_nk+1–1

n=nk–1

|an|^{2 r}₂

. (23)

Therefore, from (20) it follows that ∞

n=1

ρ_n^r–ρ_n+1^r

n^r(q–2)2q (1 +lnn)^αr≤C_r ∞

k=2

(n_k– 1)^r(q–2)2q (1 +lnn_k)^αr _nk+1–1

n=nk–1

|a_n|² ₂^r

(24)

whenα≥0.

Ifα< 0, then we can choose a numberεwhich satisfies 0 <ε<^q–2_2q . We note that (1 + lnn)^αn^ε↑and we obtain the following inequality:

nk–1 n=n_k–1

ρ_n^r–ρ_n+1^r

n^r(q–2)2q (1 +lnn)^αr

=

n_k–1

n=nk–1

ρ_n^r–ρ^r_n+1 n

r(q–2) 2q –rε

(1 +lnn)^αn^εr

≤(nk– 1)

r(q–2) 2q –rε

1 +ln(nk– 1)α

(nk– 1)^εr n_k–1 n=nk–1

ρ_n^r–ρ_n+1^r

= (nk– 1)^r(q–2)2q

1 +ln(nk– 1)αr ρ_n^r

k–1–ρ_n^r

k

. (25)

By now combining the inequalities (20), (23) and (25), we conclude that (24) holds also for the caseα< 0.

Ifr> 2, then, by using Hölder’s inequality withθ=₂^r, ¹_θ+_θ¹ = 1, we obtain

nk+1–1 n=nk–1

|an|²≤ _nk+1–1

n=nk–1

|an|^rn^{r(1–1q)–1}

²_rnk+1–1

n=nk–1

n^{θ(1θ–2(1–1q)) 1}

θ

. (26)

Since 2 <q, we have 1 +θ(¹_θ– 2(1 –¹_q)) =θ(²_q– 1) < 0. Therefore,

nk+1–1 n=n_k–1

n^{θ(1θ–2(1–1q))}≤Cr,q

n_k+1 n_k–1

t^{θ(1θ–2(1–1q))}dt

≤ Cr,q

θ(2(1 –¹_q) –¹_θ) – 1(nk– 1)^{1+θ(θ1–2(1–1q))} (27) fork= 2, 3, . . . . From inequalities (26) and (27), we can derive the following inequality:

n_k+1–1 n=nk–1

|an|²≤C(nk– 1)^{θ1+1θ–2(1–1q)} _n

k+1–1

n=nk–1

|an|^rn^{r(1–1q)–1 2}_r

(28)

fork= 2, 3, . . . , in the case of 2 <r<∞.

Now, by combining (26) and (28), we obtain the following inequality:

∞ n=1

ρ_n^r–ρ_n+1^r

n^r(q–2)2q (1 +lnn)^αr

≤ ∞

k=2

(nk– 1)

r(q–2)

2q (1 +lnnk)^αr(nk– 1)^{2r(1–2(1–1q))}

n_k+1–1 n=nk–1

|an|^rn^{r(1–q1)–1} (29)

in the case of 2 <r<∞, 0 <α<∞.

Since r(q– 2)

2q +r 2

1 – 2

1 –1 q

= 0, it follows from (29) that

∞ n=1

ρ_n^r–ρ_n+1^r

n^r(q–2)2q (1 +lnn)^αr≤C ∞

k=2 nk+1–1 n=nk–1

|an|^rn^{r(1–1q)–1}(1 +lnn)^αr (30)

in the case 2 <r<∞, 0 <α<∞. Furthermore,

∞ k=2

n_k+1–1 n=nk–1

|an|^rn^{r(1–1q)–1}(1 +lnn)^αr

≤ ∞

k=2 nk–1

n=n_k–1

|a_n|^rn^{r(1–1q)–1}(1 +lnn)^αr+ ∞

k=2 nk+1–1

n=n_k

|a_n|^rn^{r(1–1q)–1}(1 +lnn)^αr

≤2 ∞

n=1

|an|^rn^{r(1–1q)–1}(1 +lnn)^αr (31)

in the case 2 <r<∞, 0 <α<∞.

If α< 0, then we choose a numberεwhich satisfies 0 <ε< ^q–2_2q . By using the Hölder inequality, we obtain (θ=₂^r,¹_θ+_θ¹ = 1)

n_k+1–1 n=nk–1

|an|²≤ _n

k+1–1

n=nk–1

|an|^rn^{r(1–1q)–εθ–1 2}_rn

k+1–1

n=nk–1

n^{θ(1θ–2(1–1q)+ε) 1}

θ

. (32)

According to the choice of the numberεit shows that 1 +θ

1 θ – 2

1 –1

q +ε =θ 2

q– 1 +ε < 0.

Therefore (as in the case ofα> 0) we obtain the following inequality:

nk+1–1 n=nk–1

n^{θ(1θ–2(1–1q)+ε)}≤Cr,q

_nk+1

nk–1

t^{θ(1θ–2(1–1q)+ε)}dt

≤ Cr,q

θ(2(1 –¹_q) – 1 –ε)(nk– 1)^{1+θ(1θ–2(1–1q)+ε)} (33)