Volume 8 (2007), Issue 2, Article 38, 6 pp.
INTEGRABILITY CONDITIONS PERTAINING TO ORLICZ SPACE
L. LEINDLER
U NIVERSITY OF S ZEGED , B OLYAI I NSTITUTE
A RADI VÉRTANÚK TERE 1,
6720 S ZEGED , H UNGARY
leindler@math.u-szeged.hu
Received 04 September, 2006; accepted 01 June, 2007
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Trigonometric series, Integrability, Orlicz space.
2000 Mathematics Subject Classification. 42A32, 46E30.
1. I NTRODUCTION
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
and
(1.2)
f (x) :=
∞
X
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:
The author was partially supported by the Hungarian National Foundation for Scientific Research under Grant # T042462.
251-06
2
L. L EINDLER
Theorem 1.1. Let Φ(x) ∈ ∆(p, 0) (0 ≤ p). If {λn } ∈ R0+ BV S, and the sequence {γn } is such
that {γn n−1+ε } is almost decreasing for some ε > 0, then
∞
X
γn
Φ(n λn ) < ∞ ⇒ ψ(x) ∈ L(Φ, γ),
(1.3)
n2
n=1
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
∞
X
γn
(1.4)
Φ(n2 λn ) < ∞ ⇒ g(x) ∈ L(Φ, γ).
2+q
n
n=1
2. N EW R ESULT
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
Φ
(2.1)
λν < ∞ ⇒ ψ(x) ∈ L(Φ, γ),
n2
n=1
ν=n
and
∞
2n−1
X
X
γn
Φ
n
λν
n2+q
n=1
ν=n
(2.2)
!
< ∞ ⇒ g(x) ∈ L(Φ, γ),
respectively.
Remark 2.2. It is easy to see that if {λn } ∈ R0+ BV S also holds, then
2n−1
X
λν n λn ,
ν=n
that is, our assumptions are not worse than (1.3) and (1.4).
3. N OTIONS AND N OTATIONS
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
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.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 38, 6 pp.
http://jipam.vu.edu.au/
I NTEGRABILITY C ONDITIONS P ERTAINING TO O RLICZ SPACE
3
A sequence γ of positive terms will be called almost increasing (decreasing) if
K(γ)γn ≥ γm
(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
π
following
way:
γ
:=
γ
n , n ∈ N and K1 (γ)γn+1 ≤ γ(x) ≤ K2 (γ)γn holds for all x ∈
n
π
π
, .
n+1 n
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 space L(Φ, γ) is defined by
t→∞
Z π
L(Φ, γ) := h :
γ(x)Φ(ε|h(x)|)dx < ∞ for some ε > 0 .
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.
4. L EMMAS
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
(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
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
ν=1
holds for all m ∈ N, then
∞
X
ρk Φ
k=1
k
X
!
aν
ν=1
∞
X
Φ
k=1
2k−1
X
!
aν
ρk
ν=k
∞
1 X
ρν
kρk ν=k
!p∗
,
where p∗ := max(1, p).
Proof. Denote by An := n−1
(4.2)
P2n−1
k
X
ν=1
ν=n
aν . Let ξ be an integer such that 2ξ ≤ k < 2ξ+1 . Then
m+1
aν ≤
ξ 2
X
X−1
m=0 ν=2m
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 38, 6 pp.
aν =
ξ
X
2m A2m .
m=0
http://jipam.vu.edu.au/
4
L. L EINDLER
Utilizing the properties of Φ, furthermore (4.1), (4.2) and Lemma 4.2, we obtain that
k
X
Φ
!
Φ
aν
ν=1
ξ
X
!
2m A2m
m=0
Φ
ξ−1
X
!
2m A2m
m=0
ξ−1
X
!p∗
Φ
1/p∗
(2m A2m )
m=0
k
X
!p∗
ν −1 Φ
1/p∗
(ν Aν )
.
ν=1
Hence, by Lemma 4.1, we have
∞
X
k
X
ρk Φ
!
aν
∞
X
ν=1
k=1
ρk
!p∗
ν −1 Φ
1/p∗
(ν Aν )
ν=1
k=1
∞
X
k
X
∗
∗
ρ1−p
(k −1 Φ1/p (k
k
p∗
Ak ))
k=1
∞
X
−1
ρk Φ(k Ak ) (k ρk )
k=1
∞
X
∞
X
!p∗
ρν
ν=k
!p∗
ρν
.
ν=k
Herewith the proof is complete.
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=`
∞
X
|∆ λν | ≥
∞
X
|∆ λν | ≥ λm ,
ν=m
ν=2`
whence
−1
Λ` = `
2`−1
X
ν=`
obviously follows.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 38, 6 pp.
λν k
−1
2k−1
X
λm = Λk
m=k
http://jipam.vu.edu.au/
I NTEGRABILITY C ONDITIONS P ERTAINING TO O RLICZ SPACE
5
5. P ROOF OF T HEOREM 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
λk +
k=1
n
X
k=n+1
∞
X
λk +
|∆ λk Dk (x)| + λn |Dn (x)|
k=n
n
X
k=1
λk + n λ n .
k≥n/2
Hence, λ ∈ M RBV S, and we obtain that
|f (x)| n
X
λk
k=1
also holds.
A similar argument yields
|g(x)| n
X
λk ,
k=1
thus we have
|ψ(x)| (5.1)
n
X
λk ,
k=1
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)dx
γ(x)Φ(|ψ(x)|)dx Φ
λk
0
n=1
∞
X
π/(n+1)
k=1
γn n−2 Φ
n
X
n=1
(5.2)
∞
X
!
λk
k=1
Φ
k=1
2k−1
X
!
λν
γk k −2
ν=k
k γk−1
∞
X
!p∗
γν ν −2
.
ν=k
Since the sequence {γn n−1+ε } is almost decreasing, then
k
γk−1
∞
X
γν ν −2 1,
ν=k
therefore (5.2) proves (2.1).
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 38, 6 pp.
http://jipam.vu.edu.au/
6
L. L EINDLER
To prove (2.2) we follow a similar procedure as above. Then
n
∞
X
X
kxλk + λk sin kx
|g(x)| ≤
k=1
x
n
X
kλk +
k=1
n−1
n
X
k=n+1
∞
X
|∆λk D̃k (x)| + λn |D̃n (x)|
k=n
n
X
kλk +
k=1
n−1
(5.3)
n
X
λk + n λ n
k≥n/2
kλk .
k=1
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
γn n
−2−q
n
X
Φ
n=1
(5.4)
∞
X
π/(n+1)
k=1
!
kλk
k=1
Φ k
2k−1
X
!
λν
γk k −2−q
k 1+q γk−1
ν=k
k=1
By the assumption on {γn },
k 1+q γk−1
∞
X
!p∗
γν ν −2−q
.
ν=k
∞
X
γν ν −2−q 1,
ν=k
and thus (5.4) yields that
Z π
γ(x)Φ(|g(x)|)dx 0
∞
X
γk k −2−q Φ k
k=1
holds, which proves (2.2).
Herewith the proof of Theorem 2.1 is complete.
2k−1
X
!
λν
ν=k
R EFERENCES
[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].
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 38, 6 pp.
http://jipam.vu.edu.au/