INTEGRABILITY CONDITIONS PERTAINING TO ORLICZ SPACE

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Integrability Conditions L. Leindler vol. 8, iss. 2, art. 38, 2007

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INTEGRABILITY CONDITIONS PERTAINING TO ORLICZ SPACE

L. LEINDLER

University of Szeged, Bolyai Institute Aradi vértanúk tere 1,

6720 Szeged, Hungary

EMail:leindler@math.u-szeged.hu

Received: 04 September, 2006

Accepted: 01 June, 2007

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 42A32, 46E30.

Key words: Trigonometric series, Integrability, Orlicz space.

Abstract: Recently S. Tikhonov proved two theorems on the integrability of sine and cosine series with coefficients from theR⁺₀BV Sclass. These results are extended such that theR⁺₀BV Sclass is replaced by theM RBV Sclass.

Acknowledgements: The author was partially supported by the Hungarian National Foundation for Scientific Research under Grant # T042462.

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1. Introduction

There are many known and classical theorems pertaining to the integrability of for- mal sine and cosine series

(1.1) g(x) :=

∞

X

n=1

λ_nsinnx, and

(1.2) f(x) :=

∞

X

n=1

λ_ncosnx.

We do not recall such theorems because a nice short survey of these results with references can be found in a recent paper of S. Tikhonov [3], where he proved two theorems providing sufficient conditions of belonging of f(x) and g(x) to Orlicz spaces. In his theorems the sequence of the coefficients λn belongs to the class of sequences of rest bounded variation. For notions and notations, please, consult the third section.

In the present paper we shall verify analogous results assuming only that the sequenceλ := {λ_n}is a sequence of mean rest bounded variation. We emphasize that the latter sequences may have many zero terms, while the previous ones have no zero term.

Tikhonov’s theorems read as follows:

Theorem 1.1. LetΦ(x) ∈ ∆(p,0) (0 ≤ p). If{λ_n} ∈ R⁺₀BV S, and the sequence {γ_n}is such that{γ_nn^−1+ε}is almost decreasing for someε >0,then

(1.3)

∞

X

n=1

γn

n²Φ(n λ_n)<∞ ⇒ψ(x)∈L(Φ, γ),

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where a functionψ(x)is either a sine or cosine series.

Theorem 1.2. Let Φ(x) ∈ ∆(p, q) (0 ≤ q ≤ p). If {λ_n} ∈ R⁺₀BV S, and the sequence{γ_n}is such that{γ_nn^−(1+q)+ε}is almost decreasing for someε >0,then

(1.4)

∞

X

n=1

γ_n

n^2+qΦ(n²λ_n)<∞ ⇒g(x)∈L(Φ, γ).

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2. New Result

Now, we formulate our result in a terse form.

Theorem 2.1. Theorems 1.1 and1.2 can be improved when the condition {λ_n} ∈ R⁺₀BV S is replaced by the assumption{λ_n} ∈M RBV S.Furthermore the condi- tions of (1.3) and (1.4) may be modified as follows:

(2.1)

∞

X

n=1

γ_n n²Φ

2n−1

X

ν=n

λν

!

<∞ ⇒ψ(x)∈L(Φ, γ), and

(2.2)

∞

X

n=1

γ_n n^2+qΦ n

2n−1

X

ν=n

λ_ν

!

<∞ ⇒g(x)∈L(Φ, γ), respectively.

Remark 1. It is easy to see that if{λ_n} ∈R⁺₀BV S also holds, then

2n−1

X

ν=n

λ_ν n λ_n,

that is, our assumptions are not worse than (1.3) and (1.4).

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3. Notions and Notations

A null-sequencec:={c_n}(c_n→0)of positive numbers satisfying the inequalities

∞

X

n=m

|∆c_n| ≤K(c)c_m, (∆c_n :=c_n−c_n+1), m= 1,2, . . .

with a constantK(c)>0is said to be a sequence of rest bounded variation, in brief, c∈R⁺₀BV S.

A null-sequencecof nonnegative numbers possessing the property

∞

X

n=2m

|∆cn| ≤K(c)m⁻¹

2m−1

X

ν=m

cν

is called a sequence of mean rest bounded variation, in symbols,c∈M RBV S.

It is clear that the classM RBV Sincludes the classR⁺₀BV S.

The author is grateful to the referee for calling his attention to an inaccurancy in the previous definition of the classM RBV Sand to some typos.

A sequenceγof positive terms will be called almost increasing (decreasing) if K(γ)γn≥γm (γn ≤K(γ)γm)

holds for anyn≥m.

Denote by ∆(p, q) (0 ≤ q ≤ p) the set of all nonnegative functions Φ(x) defined on[0,∞) such that Φ(0) = 0 and Φ(x)/x^p is nonincreasing andΦ(x)/x^q is nondecreasing.

In this paper a sequenceγ :={γ_n}is associated to a functionγ(x)being defined in the following way: γ ^π_n

:= γn, n ∈ N and K1(γ)γn+1 ≤ γ(x) ≤ K2(γ)γn

holds for allx∈ _n+1^π ,^π_n .

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A locally integrable almost everywhere positive functionγ(x) : [0, π] →[0,∞) is said to be a weight function.

Let Φ(t) be a nondecreasing continuous function defined on [0,∞) such that Φ(0) = 0 and lim

t→∞Φ(t) = +∞. For a weight function γ(x) the weighted Orlicz spaceL(Φ, γ)is defined by

L(Φ, γ) :=

h:

Z π

0

γ(x)Φ(ε|h(x)|)dx <∞for someε >0

.

Later onD_k(x)andD˜_k(x)shall denote the Dirichlet and the conjugate Dirichlet kernels. It is known that, ifx >0, |D_k(x)|=O(x⁻¹)and|D˜_k(x)|=O(x⁻¹)hold.

We shall also use the notation L R if there exists a positive constantK such thatL≤KR.

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4. Lemmas

Lemma 4.1 ([1]). Ifa_n ≥0, ρ_n>0,and ifp≥1,then

∞

X

n=1

ρ_n

n

X

ν=1

a_ν

!p

∞

X

n=1

ρ^1−p_n a^p_n

∞

X

ν=n

ρ_ν

!p

.

Lemma 4.2 ([2]). LetΦ∈∆(p, q) (0≤q ≤p)andt_j ≥0, j = 1,2, . . . , n, n∈N. Then

1. Q^pΦ(t)≤Φ(Qt)≤Q^qΦ(t), 0≤Q≤1, t≥0, 2. Φ

n

P

j=1

t_j

!

≤

n

P

j=1

Φ^1/p^∗(t_j)

!p^∗

, p^∗ := max(1, p).

Lemma 4.3. LetΦ∈∆(p, q) (0≤q≤p).Ifρ_n>0, a_n ≥0,and if (4.1)

2^m+1−1

X

ν=2^m

a_ν

2^m−1

X

ν=1

a_ν holds for allm∈N,then

∞

X

k=1

ρkΦ

k

X

ν=1

aν

!

∞

X

k=1

Φ

2k−1

X

ν=k

aν

! ρk

1 kρ_k

∞

X

ν=k

ρν

!p^∗

, wherep^∗ := max(1, p).

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Proof. Denote byA_n :=n⁻¹P2n−1

ν=n a_ν.Letξbe an integer such that2^ξ ≤k <2^ξ+1. Then

(4.2)

k

X

ν=1

a_ν ≤

ξ

X

m=0 2^m+1−1

X

ν=2^m

a_ν =

ξ

X

m=0

2^mA₂^m.

Utilizing the properties ofΦ, furthermore (4.1), (4.2) and Lemma4.2, we obtain that Φ

k

X

ν=1

aν

! Φ

ξ

X

m=0

2^mA2^m

!

Φ

ξ−1

X

m=0

2^mA₂^m

!

ξ−1

X

m=0

Φ^1/p^∗(2^mA₂^m)

!^p

∗

k

X

ν=1

ν⁻¹Φ^1/p^∗(ν A_ν)

!^p^∗ . Hence, by Lemma4.1, we have

∞

X

k=1

ρ_kΦ

k

X

ν=1

a_ν

!

∞

X

k=1

ρ_k

k

X

ν=1

ν⁻¹Φ^1/p^∗(ν A_ν)

!^p^∗

∞

X

k=1

ρ^1−p_k ^∗(k⁻¹Φ^1/p^∗(k A_k))^p^∗

∞

X

ν=k

ρ_ν

!p^∗

∞

X

k=1

ρ_kΦ(k A_k) (k ρ_k)⁻¹

∞

X

ν=k

ρ_ν

!p^∗

. Herewith the proof is complete.

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Lemma 4.4. Ifλ:={λ_n} ∈M RBV SandΛ_n:=n⁻¹P2n−1

ν=n λ_ν,then Λ_k Λ_`

holds for allk ≥2`.

Proof. It is clear that ifm≥2`,then

`⁻¹

2`−1

X

ν=`

λ_ν

∞

X

ν=2`

|∆λ_ν| ≥

∞

X

ν=m

|∆λ_ν| ≥λ_m, whence

Λ_` =`⁻¹

2`−1

X

ν=`

λ_ν k⁻¹

2k−1

X

m=k

λ_m = Λ_k obviously follows.

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5. Proof of Theorem 2.1

Proof of Theorem2.1. Let x ∈ _n+1^π ,^π_n

. Using Abel’s rearrangement, the known estimate ofD_k(x)and the fact thatλ∈M RBV S,we obtain that

|f(x)| ≤

n

X

k=1

λ_k+

∞

X

k=n+1

λ_kcoskx

≤

n

X

k=1

λ_k+

∞

X

k=n

|∆λ_kD_k(x)|+λ_n|D_n(x)|

n

X

k=1

λ_k+

n

X

k≥n/2

λ_k+n λ_n. Hence,λ∈M RBV S, and we obtain that

|f(x)|

n

X

k=1

λ_k also holds.

A similar argument yields

|g(x)|

n

X

k=1

λ_k, thus we have

(5.1) |ψ(x)|

n

X

k=1

λ_k,

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whereψ(x)is eitherf(x)org(x).

By Lemma4.4, the condition (4.1) withλ_ν in place ofa_ν is satisfied, thus we can apply Lemma4.3, consequently (5.1) and some elementary calculations give that

Z π

0

γ(x)Φ(|ψ(x)|)dx

∞

X

n=1

Φ

n

X

k=1

λk

!Z π/n

π/(n+1)

γ(x)dx

∞

X

n=1

γ_nn⁻²Φ

n

X

k=1

λ_k

!

∞

X

k=1

Φ

2k−1

X

ν=k

λ_ν

!

γ_kk⁻² k γ_k⁻¹

∞

X

ν=k

γ_νν⁻²

!p^∗

. (5.2)

Since the sequence{γ_nn^−1+ε}is almost decreasing, then k γ_k⁻¹

∞

X

ν=k

γ_νν⁻² 1, therefore (5.2) proves (2.1).

To prove (2.2) we follow a similar procedure as above. Then

|g(x)| ≤

n

X

k=1

kxλ_k+

∞

X

k=n+1

λ_ksinkx

x

n

X

k=1

kλ_k+

∞

X

k=n

|∆λ_kD˜_k(x)|+λ_n|D˜_n(x)|

n⁻¹

n

X

k=1

kλ_k+

n

X

k≥n/2

λ_k+n λ_n n⁻¹

n

X

k=1

kλ_k. (5.3)

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Using Lemmas4.2,4.3,4.4and the estimate (5.3), we obtain that Z π

0

γ(x)Φ(|g(x)|)dx

∞

X

n=1

Φ n⁻¹

n

X

k=1

kλ_k

!Z π/n

π/(n+1)

γ(x)dx

∞

X

n=1

γ_nn^−2−qΦ

n

X

k=1

kλ_k

!

∞

X

k=1

Φ k

2k−1

X

ν=k

λ_ν

!

γ_kk^−2−q k^1+qγ_k⁻¹

∞

X

ν=k

γ_νν^−2−q

!p^∗

(5.4) .

By the assumption on{γ_n},

k^1+qγ_k⁻¹

∞

X

ν=k

γ_νν^−2−q 1, and thus (5.4) yields that

Z π

0

γ(x)Φ(|g(x)|)dx

∞

X

k=1

γ_kk^−2−qΦ k

2k−1

X

ν=k

λ_ν

!

holds, which proves (2.2).

Herewith the proof of Theorem2.1is complete.

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References

[1] L. LEINDLER, Generalization of inequalities of Hardy and Littlewood, Acta Sci. Math. (Szeged), 31 (1970), 279–285.

[2] M. MATELJEVICAND PAVLOVIC,L^p-behavior of power series with positive coefficients and Hardy spaces, Proc. Amer. Math. Soc., 87 (1983), 309–316.

[3] S. TIKHONOV, On belonging of trigonometric series of Orlicz space, J. Inequal.

Pure and Appl. Math., 5(2) (2004), Art. 22. [ONLINE:http://jipam.vu.

edu.au/article.php?sid=395].