INTEGRABILITY CONDITIONS PERTAINING TO
ORLICZ SPACE
Integrability Conditions
L. LEINDLER
University of Szeged, Bolyai Institute
Aradi vértanúk tere 1,
6720 Szeged, Hungary
EMail: leindler@math.u-szeged.hu
L. Leindler
vol. 8, iss. 2, art. 38, 2007
Title Page
Contents
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 the R0+ BV S class. These results are extended such
that the R0+ BV S class is replaced by the M RBV S class.
Acknowledgements:
The author was partially supported by the Hungarian National Foundation for
Scientific Research under Grant # T042462.
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Contents
1
Introduction
3
2
New Result
5
3
Notions and Notations
6
Integrability Conditions
4
Lemmas
8
L. Leindler
vol. 8, iss. 2, art. 38, 2007
5
Proof of Theorem 2.1
11
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1.
Introduction
There are many known and classical theorems pertaining to the integrability of formal sine and cosine series
∞
X
(1.1)
g(x) :=
λn sin nx,
n=1
Integrability Conditions
and
L. Leindler
f (x) :=
(1.2)
∞
X
vol. 8, iss. 2, art. 38, 2007
λn cos nx.
n=1
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
sequence λ := {λn } 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:
R0+ BV
Theorem 1.1. Let Φ(x) ∈ ∆(p, 0) (0 ≤ p). If {λn } ∈
S, and the sequence
−1+ε
{γn } is such that {γn n
} is almost decreasing for some ε > 0, then
(1.3)
∞
X
γn
n=1
n2
Φ(n λn ) < ∞ ⇒ ψ(x) ∈ L(Φ, γ),
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where a function ψ(x) is either a sine or cosine series.
Theorem 1.2. Let Φ(x) ∈ ∆(p, q) (0 ≤ q ≤ p). If {λn } ∈ R0+ BV S, and the
sequence {γn } is such that {γn n−(1+q)+ε } is almost decreasing for some ε > 0, then
(1.4)
∞
X
γn
Φ(n2 λn ) < ∞ ⇒ g(x) ∈ L(Φ, γ).
2+q
n
n=1
Integrability Conditions
L. Leindler
vol. 8, iss. 2, art. 38, 2007
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2.
New Result
Now, we formulate our result in a terse form.
Theorem 2.1. Theorems 1.1 and 1.2 can be improved when the condition {λn } ∈
R0+ BV S is replaced by the assumption {λn } ∈ M RBV S. Furthermore the conditions of (1.3) and (1.4) may be modified as follows:
!
2n−1
∞
X
X
γn
(2.1)
Φ
λν < ∞ ⇒ ψ(x) ∈ L(Φ, γ),
2
n
ν=n
n=1
Integrability Conditions
L. Leindler
vol. 8, iss. 2, art. 38, 2007
and
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(2.2)
∞
2n−1
X
X
γn
Φ
n
λν
n2+q
n=1
ν=n
!
< ∞ ⇒ g(x) ∈ L(Φ, γ),
respectively.
Remark 1. It is easy to see that if {λn } ∈
2n−1
X
R0+ BV
S also holds, then
λν n λ n ,
ν=n
that is, our assumptions are not worse than (1.3) and (1.4).
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3.
Notions and Notations
A null-sequence c := {cn } (cn → 0) of positive numbers satisfying the inequalities
∞
X
|∆ cn | ≤ K(c)cm ,
(∆ cn := cn − cn+1 ), m = 1, 2, . . .
n=m
with a constant K(c) > 0 is said to be a sequence of rest bounded variation, in brief,
c ∈ R0+ BV S.
A null-sequence c of nonnegative numbers possessing the property
∞
X
|∆ cn | ≤ K(c)m
n=2m
−1
2m−1
X
cν
L. Leindler
vol. 8, iss. 2, art. 38, 2007
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ν=m
is called a sequence of mean rest bounded variation, in symbols, c ∈ M RBV S.
It is clear that the class M RBV S includes the class R0+ BV S.
The author is grateful to the referee for calling his attention to an inaccurancy in
the previous definition of the class M RBV S and to some typos.
A sequence γ of positive terms will be called almost increasing (decreasing) if
K(γ)γn ≥ γm
Integrability Conditions
(γn ≤ K(γ)γm )
holds for any n ≥ 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)/xp is nonincreasing and Φ(x)/xq is
nondecreasing.
In this paper a sequence γ := {γn } is associated to a function γ(x) being defined
in the following way: γ πn := γn , n ∈ N and K1 (γ)γn+1 ≤ γ(x) ≤ K2 (γ)γn
π
holds for all x ∈ n+1
, πn .
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A locally integrable almost everywhere positive function γ(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 Φ(t) = +∞. For a weight function γ(x) the weighted Orlicz
t→∞
space L(Φ, γ) is defined by
Z
L(Φ, γ) := h :
π
γ(x)Φ(ε|h(x)|)dx < ∞ for some ε > 0 .
Integrability Conditions
0
Later on Dk (x) and D̃k (x) shall denote the Dirichlet and the conjugate Dirichlet
kernels. It is known that, if x > 0, |Dk (x)| = O(x−1 ) and |D̃k (x)| = O(x−1 ) hold.
We shall also use the notation L R if there exists a positive constant K such
that L ≤ KR.
L. Leindler
vol. 8, iss. 2, art. 38, 2007
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4.
Lemmas
Lemma 4.1 ([1]). If an ≥ 0, ρn > 0, and if p ≥ 1, then
!p
!p
n
∞
∞
∞
X
X
X
X
p
ρn
aν
ρ1−p
ρν .
n an
n=1
ν=1
n=1
ν=n
Lemma 4.2 ([2]). Let Φ ∈ ∆(p, q) (0 ≤ q ≤ p) and tj ≥ 0, j = 1, 2, . . . , n, n ∈ N.
Then
Integrability Conditions
L. Leindler
vol. 8, iss. 2, art. 38, 2007
1. Qp Φ(t) ≤ Φ(Qt) ≤ Qq Φ(t), 0 ≤ Q ≤ 1, t ≥ 0,
!
!p∗
n
n
P
P
∗
2. Φ
tj ≤
Φ1/p (tj )
, p∗ := max(1, p).
j=1
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j=1
Lemma 4.3. Let Φ ∈ ∆(p, q) (0 ≤ q ≤ p). If ρn > 0, an ≥ 0, and if
2m+1
X−1
(4.1)
aν ν=2m
m −1
2X
aν
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ν=1
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holds for all m ∈ N, then
∞
X
k=1
ρk Φ
k
X
ν=1
where p∗ := max(1, p).
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!
aν
∞
X
k=1
Φ
2k−1
X
ν=k
!
aν
ρk
1
kρk
∞
X
ν=k
!p∗
ρν
,
Close
Proof. Denote by An := n−1
Then
k
X
(4.2)
P2n−1
ν=n
aν . Let ξ be an integer such that 2ξ ≤ k < 2ξ+1 .
m+1
aν ≤
ξ 2
X
X−1
aν =
m=0 ν=2m
ν=1
ξ
X
2m A2m .
m=0
Utilizing the properties of Φ, furthermore (4.1), (4.2) and Lemma 4.2, we obtain that
!
!
ξ
k
X
X
Φ
aν Φ
2m A2m
ν=1
m=0
ξ−1
Φ
X
Integrability Conditions
L. Leindler
vol. 8, iss. 2, art. 38, 2007
!
2m A2m
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m=0
ξ−1
X
!p∗
Φ
1/p∗
m
(2 A2m )
m=0
ν=1
k=1
∞
X
k
X
ν
−1
Φ
1/p∗
(ν Aν )
ν −1 Φ
(ν Aν )
∞
X
k=1
Herewith the proof is complete.
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ν=1
∗
∗
ρ1−p
(k −1 Φ1/p (k
k
p∗
Ak ))
k=1
.
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!p∗
1/p∗
Contents
!p∗
ν=1
Hence, by Lemma 4.1, we have
!
∞
k
∞
X
X
X
ρk Φ
aν ρk
k=1
k
X
ρk Φ(k Ak ) (k ρk )−1
∞
X
ν=k
∞
X
!p∗
ρν
ν=k
!p∗
ρν
.
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Lemma 4.4. If λ := {λn } ∈ M RBV S and Λn := n−1
P2n−1
ν=n
λν , then
Λk Λ`
holds for all k ≥ 2`.
Proof. It is clear that if m ≥ 2`, then
−1
`
2`−1
X
λν ν=`
whence
Λ` = `−1
∞
X
ν=2`
2`−1
X
ν=`
obviously follows.
|∆ λν | ≥
λν k −1
∞
X
Integrability Conditions
|∆ λν | ≥ λm ,
ν=m
2k−1
X
L. Leindler
vol. 8, iss. 2, art. 38, 2007
λm = Λk
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m=k
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5.
Proof of Theorem 2.1
π
Proof of Theorem 2.1. Let x ∈ n+1
, πn . Using Abel’s rearrangement, the known
estimate of Dk (x) and the fact that λ ∈ M RBV S, we obtain that
n
∞
X
X
|f (x)| ≤
λk + λk cos kx
k=1
≤
n
X
k=n+1
λk +
k=1
n
X
k=1
∞
X
λk +
L. Leindler
|∆ λk Dk (x)| + λn |Dn (x)|
k=n
n
X
λk + n λn .
vol. 8, iss. 2, art. 38, 2007
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k≥n/2
Contents
Hence, λ ∈ M RBV S, and we obtain that
|f (x)| n
X
λk
k=1
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also holds.
A similar argument yields
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|g(x)| n
X
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λk ,
k=1
thus we have
(5.1)
Integrability Conditions
|ψ(x)| n
X
k=1
λk ,
Close
where ψ(x) is either f (x) or g(x).
By Lemma 4.4, the condition (4.1) with λν in place of aν is satisfied, thus we can
apply Lemma 4.3, consequently (5.1) and some elementary calculations give that
!Z
Z π
∞
n
π/n
X
X
γ(x)Φ(|ψ(x)|)dx Φ
λk
γ(x)dx
0
n=1
∞
X
π/(n+1)
k=1
n
X
γn n−2 Φ
n=1
(5.2)
∞
X
!
Integrability Conditions
λk
L. Leindler
k=1
2k−1
X
Φ
k=1
!
λν
γk k −2
k γk−1
ν=k
∞
X
vol. 8, iss. 2, art. 38, 2007
!p∗
γν ν −2
ν=k
.
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Since the sequence {γn n−1+ε } is almost decreasing, then
k γk−1
∞
X
Contents
γν ν −2 1,
ν=k
therefore (5.2) proves (2.1).
To prove (2.2) we follow a similar procedure as above. Then
∞
n
X
X
|g(x)| ≤
kxλk + λk sin kx
k=1
x
n
X
kλk +
k=1
(5.3)
n
−1
n
X
k=1
I
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k=n
k≥n/2
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|∆λk D̃k (x)| + λn |D̃n (x)|
kλk +
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k=n+1
∞
X
n
X
JJ
λk + n λ n n
−1
n
X
k=1
kλk .
Using Lemmas 4.2, 4.3, 4.4 and the estimate (5.3), we obtain that
!Z
Z π
∞
n
π/n
X
X
γ(x)Φ(|g(x)|)dx Φ n−1
kλk
γ(x)dx
0
n=1
∞
X
n
X
γn n−2−q Φ
n=1
(5.4)
∞
X
k=1
π/(n+1)
k=1
!
kλk
k=1
Φ k
2k−1
X
Integrability Conditions
!
λν
γk k −2−q
k 1+q γk−1
∞
X
L. Leindler
.
vol. 8, iss. 2, art. 38, 2007
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k 1+q γk−1
∞
X
Contents
γν ν −2−q 1,
ν=k
and thus (5.4) yields that
π
γ(x)Φ(|g(x)|)dx 0
γν ν −2−q
ν=k
ν=k
By the assumption on {γn },
Z
!p∗
∞
X
γk k
−2−q
k=1
holds, which proves (2.2).
Herewith the proof of Theorem 2.1 is complete.
Φ k
2k−1
X
!
λν
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ν=k
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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. MATELJEVIC AND PAVLOVIC, Lp -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].
Integrability Conditions
L. Leindler
vol. 8, iss. 2, art. 38, 2007
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