On Shiba - Waterman space
We give a necessary and sufficient condition for the inclusion of $\Lambda BV^{(p)}$ in the classes $H^q_{\omega}$.
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| author | Hormozi, M. Гормозі, М. |
| author_facet | Hormozi, M. Гормозі, М. |
| author_sort | Hormozi, M. |
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| description | We give a necessary and sufficient condition for the inclusion of $\Lambda BV^{(p)}$ in the classes $H^q_{\omega}$. |
| first_indexed | 2026-03-24T02:25:59Z |
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UDC 517.5
M. Hormozi (Chalmers Univ. Technology and Univ. Gothenburg, Sweden)
ON SHIBA – WATERMAN SPACE
ПРО ПРОСТIР ШИБИ – УОТЕРМЕНА
We give a necessary and sufficient condition for the inclusion of ΛBV (p) in the classes Hq
ω.
Наведено необхiдну та достатню умову належностi ΛBV (p) класам Hq
ω.
In 1980 M. Shiba [9] introduced the class ΛBV (p), 1 ≤ p < ∞, expanding a fundamental concept
of bounded Λ-variation formulated and usefully applied by D. Waterman in 1972 [13].
The main objective of this note is to find a necessary and sufficient condition for the embedding
ΛBV (p) ⊂ Hq
ω.
1. Introduction and preliminaries. Let Λ = (λi) be a nondecreasing sequence of positive
numbers such that
∑ 1
λi
= +∞ and let p be a number greater than or equal to 1. A function
f : [a, b] → R is said to be of bounded p-Λ-variation on a not necessarily closed subinterval P ⊂
⊂ [a, b] if
V (f ;P ) := sup
(
n∑
i=1
|f(Ii)|p
λi
)1/p
< +∞,
where the supremum is taken over all finite families {Ii}ni=1 of nonoverlapping subintervals of P
and where f(Ii) := f(sup Ii) − f(inf Ii) is the change of the function f over the interval Ii.
The symbol ΛBV (p) denotes the linear space of all functions of bounded p-Λ-variation with domain
[0, 1]. We will write V (f) instead of V (f, P ) if P = [0, 1]. The Shiba – Waterman class ΛBV (p) was
introduced in 1980 by M. Shiba in [9] and it clearly is a generalization of the well-known Waterman
class ΛBV. Some of the basic properties of functions of class ΛBV (p) were discussed by R. G. Vyas
in [11] recently. More results concerned with the Shiba – Waterman classes and their applications can
be found in [1, 2, 4, 6 – 8, 10, 12]. ΛBV (p) equipped with the norm ‖f‖Λ, p := |f(0)| + V (f) is a
Banach space.
Functions in a Shiba – Waterman class ΛBV (p) are regulated [11] (Theorem 2), hence integrable,
and thus it makes sense to consider their integral modulus of continuity
ωq(δ, f) = sup
0≤h≤δ
1−h∫
0
|f(t+ h)− f(t)|q
1/q
dt,
for 0 ≤ δ ≤ 1. However, if f is defined on R instead of on [0, 1] and if f is 1-periodic, it is
convenient to modify the definition and put
ωq(δ, f) = sup
0≤h≤δ
1∫
0
|f(t+ h)− f(t)|q
1/q
dt,
c© M. HORMOZI, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2 245
246 M. HORMOZI
since the difference between the two definitions is then nonessential in all applications of the concept.
We will use the second definition in our note, and thus the main Theorem 2.1 will actually deal with
1-periodic functions.
A function ω : [0, 1]→ R is said to be a modulus of continuity if it is nondecreasing, continuous,
subadditive and ω(0) = 0. If ω is a modulus of continuity, then Hq
ω denotes the class of functions
f ∈ Lq[0, 1] for which ωq(δ, f) = O(ω(δ)) as δ → 0 + .
2. On the imbedding of ΛBV (p) class in the class Hq
ω. In [3] Goginava gave a necessary
and sufficient condition for the inclusion ΛBV in Hq
ω. Also Wang [15] by using an interesting
method found a necessary and sufficient condition for the embedding Hq
ω ⊂ ΛBV. Here, we give
a necessary condition for the inclusion ΛBV (p) in Hq
ω. This work uses [3] and [5] as the bases. If
ω(δ) is a modulus of continuity, then the following theorem is true.
Theorem 2.1. For some p, q ∈ [1,∞), the inclusion ΛBV (p) ⊂ Hω
q holds if and only if
lim sup
n→∞
1
ω(1/n)n1/pq
max
1≤m≤n
m1/pq(∑m
i=1
1
λi
)1/p
< +∞. (1)
Proof. Sufficiency. We prove an inequality which gives us the sufficiency:
ω
(
1
n
, f
)
q
≤ V (f)
1
np
max
1≤m≤n
m1/p(∑m
i=1
1/λi
)q/p
1/q
.
First we recall the following lemma and corollary from [5]:
Lemma 2.1. Consider the following problem:
F (x) =
n∑
i=1
xqi → max under the condition
(
n∑
i=1
xi
λi
)
≤ 1 and
x1 ≥ x2 ≥ x3 ≥ . . . ≥ xn ≥ 0. (L)
Then the solution x = (x1, x2, . . . , xn) of problem (L) is among vectors that satisfy conditions
n∑
i=1
xi
λi
= 1
x1 = x2 = . . . = xk > xk+1 = xk+2 = . . . = xn = 0 with some k, 1 ≤ k ≤ n.
Corollary 2.1. The extermal value of problem (L) is max1≤k≤n
k(∑k
i=1
1/λi
)q .
Now, we return to the proof of inequality
(
ωq
(
1
n
, f
))q
≤ sup
0<h≤1/n
1∫
0
|f(x+ h)− f(x)|q dx =
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON SHIBA – WATERMAN SPACE 247
= sup
0<h≤1/n
n∑
k=1
k/n∫
(k−1)/n
|f(x+ h)− f(x)|q dx =
= sup
0<h≤1/n
n∑
k=1
1/n∫
0
∣∣∣∣f (x+
k − 1
n
+ h
)
− f
(
x+
k − 1
n
)∣∣∣∣q dx =
= sup
0<h≤1/n
1/n∫
0
n∑
k=1
∣∣∣∣f (x+
k − 1
n
+ h
)
− f
(
x+
k − 1
n
)∣∣∣∣q dx ≤
≤ sup
0<h≤1/n
1/n∫
0
n1−1/p
(
n∑
k=1
∣∣∣∣f (x+
k − 1
n
+ h
)
− f
(
x+
k − 1
n
)∣∣∣∣pq
)1/p
dx.
Where the last inequality has been obtained by Hölder inequality. Under the condition |h| ≤ 1
n
and fixed x, the segment Ik(x) do not overlap each other and their union does not exceed P. Let
enumerate the intervals Ik in decreasing of values |f(Ik)| we get
|f(I1)| ≥ |f(I2)| ≥ . . . ≥ |f(In)|,
(
n∑
k=1
|f(Ik)|p
λk
)1/p
≤ V (f).
Therefore taking into account the Lemma 2.1 we get
(
ωq
(
1
n
, f
))q
≤ sup
0<h≤1/n
1/n∫
0
n1−1/p
(
n∑
k=1
∣∣∣∣f (x+
k − 1
n
+ h
)
− f
(
x+
k − 1
n
)∣∣∣∣pq
)1/p
dx ≤
≤ n1−1/p
1/n∫
0
V q(f) max
1≤k≤n
k1/p(∑k
i=1
1/λi
)q/p dx =
=
1
np
V q(f) max
1≤k≤n
k1/p(∑k
i=1
1/λi
)q/p .
Necessity. Our proof uses Goginava’s paper as a basis. Assume the condition (1) is not satisfied.
As an example, we construct a function from ΛBV (p) that is not in Hω
q . Since condition (1) is not
satisfied, there exists a sequence of integers {γk, k ≥ 1} such that
lim
k→∞
1
ω(1/γk)γ
1/(pq)
k
max
1≤m≤γk
m1/(pq)(∑m
i=1
1/λi
)1/p
=∞.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
248 M. HORMOZI
Let {γ′k, k ≥ 1} be a sequence of integers for which 2γ
′
k−1 ≤ γk < 2γ
′
k . Since ω(δ) is nondecreasing,
we have
21/(pq)
ω(2−γ
′
k)2γ
′
k/(pq)
max
1≤m≤2
γ′
k
m1/(pq)(∑m
i=1
1/λi
)1/p
≥ 1
ω(1/γk)γ
1/(pq)
k
max
1≤m≤γk
m1/(pq)(∑m
i=1
1/λi
)1/p
,
whence
lim
k→∞
1
ω(2−γ
′
k)2−γ
′
k/(pq)
max
1≤m≤2
γ′
k
m1/(pq)(∑m
i=1
1/λi
)1/p
= +∞.
Then there exists a sequence of integers {n′k : k ≥ 1} ⊂ {γ′k : k ≥ 1} such that
lim
k→∞
1
ω(2−n
′
k)
1(∑m(n′k)
i=1
1/λi
)1/p
(
m(n′k)
2n
′
k
)1/(pq)
= +∞, (2)
where
max
1≤m≤2
n′
k
m1/(pq)(∑m
i=1
1/λi
)1/p
=
(m(n′k))
1/(pq)(∑m(n′k)
i=1
1/λi
)1/p
.
The following three cases are possible:
(a) there exists a sequence of integers {s′k : k ≥ 1} ⊂ {n′k : k ≥ 1} such that
m(s′k) < 22s′k−1 ;
(b) there exists a sequence of integers {z′k : k ≥ 1} ⊂ {n′k : k ≥ 1} such that
22z′k−1 ≤ m(z′k) < 2z
′
k−z
′
k−1 ;
(c) 2n
′
k−n
′
k−1 ≤ m(n′k) < 2n
′
k for all k ≥ k0.
First, consider case (a). We choose a sequence of integers {sk : k ≥ 1} ⊂ {s′k : k ≥ 1} such thatm(sk)∑
i=1
1
λi
1/p
≥ 22sk−1/(pq).
Then relation (2) yields
lim
k→∞
ω
(
1
2sk
)
2sk/(pq) = 0.
Let {rk : k ≥ 1} ⊂ {sk : k ≥ 1} be such that
ω
(
1
2rk
)
2rk/(pq) ≤ 4
−k
p . (3)
Consider the function f defined as follows:
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON SHIBA – WATERMAN SPACE 249
f(x) =
2cj(2
rjx− 1) if x ∈ [2−rj , 3.2−rj−1),
−2cj(2
rjx− 2) if x ∈ [3.2−rj−1), 2.2−rj ) for j = 1, 2, . . . ,
0 otherwise,
f(x+ l) = f(x), l = ±1,±2, . . . ,
where
cj =
√
ω
(
1
2rj
)
2rj/(pq).
Relation (3) leads that f ∈ ΛBV (p).
Now consider case (b). Let {zk : k ≥ 1} ⊂ {z′k : k ≥ 1} be such that
1
ω(2−zk)
1(∑m(zk)
i=1
1/λi
)1/p
(
m(zk)
2zk
)1/(pq)
≥ 4k. (4)
Consider the function gk defined as follows:
gk(x) =
hk(2
zkx− 2j + 1), x ∈ [(2j − 1)/2zk , 2j/2zk),
−hk(2zkx− 2j − 1), x ∈ [2j/2zk , (2j + 1)/2zk)
for j = m(zk−1), . . . ,m(zk)− 1,
0 otherwise,
where
hk =
1
2k
∑m(zk)
j=1
1/λj
.
Let
g(x) =
∞∑
k=2
gk(x), g(x+ l) = g(x), l = ±1,±2, . . . .
First, we prove that g ∈ ΛBV (p). For every choice of nonoverlapping intervals {In : n ≥ 1}, we
get
∞∑
j=1
|g(Ij)|p
λj
≤ 2p
∞∑
i=1
hpi
m(zi)∑
j=1
1
λj
≤
≤ 2p
∞∑
i=1
hi
m(zi)∑
j=1
1
λj
= 2p
∞∑
i=1
1
2i
<∞.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
250 M. HORMOZI
Hence g ∈ ΛBV (p). Finally, consider case (c). Let {nk : k ≥ 1} ⊂ {n′k : k ≥ k0} be such that
nk ≥ 2nk−1 + 1,
1
ω (2−nk)
1(∑m(nk)
i=1
1/λi
)1/p
(
m(nk)
2nk
)1/(pq)
≥ 22nk−1/(pq)+k.
Consider the function φk defined as follows:
φk(x) =
dk(2
nkx− 2j + 1), x ∈ [(2j − 1)/2nk , 2j/2nk),
−dk(2nkx− 2j − 1), x ∈ [2j/2nk , (2j + 1)/2nk)
for j = 2nk−1−nk−2 , . . . , 2nk−nk−1−1 − 1,
0 otherwise,
where
dk =
1
2k
∑m(nk)
j=1
1/λj .
Let
φ(x) =
∞∑
k=3
φk(x), φ(x+ l) = φ(x), l = ±1,±2, . . . .
For every choice of nonoverlapping intervals {In, n ≥ 1}, we get
∞∑
j=1
|φ(Ij)|p
λj
≤ 2p
∞∑
i=2
dpi
2ni−ni−1−1∑
j=1
1
λj
≤
≤ 2p
∞∑
i=2
di
2ni−ni−1−1∑
j=1
1
λj
≤
≤ 2p
∞∑
i=2
di
m(ni)∑
j=1
1
λj
≤ 2p
∞∑
i=2
1
2i
<∞.
Hence φ ∈ ΛBV (p), Now similar to [3] (Theorem 1) we have f, g and φ do not belong to Hq
ω.
Therefore, the theorem is proved. For p ≥ q Theorem 2.1 can be simplified.
To achieve this, we need to prove the following lemma.
Lemma 2.2. Whenever p ≥ q
n(∑n
k=1
1/λk
)q/p ≤ max
1≤m≤n
m(∑m
k=1
1/λk
)q/p ≤ n(∑n+1
k=2
1/λk
)q/p .
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
ON SHIBA – WATERMAN SPACE 251
Proof. The left inequality is obvious, and the right inequality is proved below.
Let λ : [1,∞)→ R be an increasing, continuous, piecewise-linear function defined by the values
λ(k) = λk, k ≥ 1, and let
Φ(x) :=
x∫
1
dt
λ(t)
, H(x) :=
Φ(x+ 1)
xδ
, δ :=
p
q
≥ 1.
Since Φ′ decreases, we conclude that
H(x) =
1
xδ−1
1∫
0
Φ′(1 + tx)dt
also decreases. If, in addition, we take into account that, for m ≥ 2,
n∑
k=2
1
λk
≤
m−1∑
k=2
k+1∫
k
dt
λ(t)
= Φ(m) ≤
m−1∑
k=1
1
λk
,
then, for m ≤ n, we get
mδ∑m
k=1
1/λk
≤ mδ
Φ(m+ 1)
=
1
H(m)
≤ 1
H(n)
=
nδ
Φ(n+ 1)
≤ nδ∑n+1
k=2
1/λk
.
Now using Theorem 2.1 and Lemma 2.2 we have the following corollary.
Corollary 2.2. For some p, q ∈ [1,∞) such that p ≥ q, the inclusion ΛBV (p) ⊂ Hω
q holds if
and only if
lim sup
n→∞
1
ω(1/n)
1(∑n+1
i=2
1/λi
)1/p
< +∞.
Applying Corollary 2.2, we see the following corollary.
Corollary 2.3. For some p, q ∈ [1,∞) such that p ≥ q, the inclusion {kβ}BV (p) ⊂ Hω
q holds
if and only if
lim sup
n→∞
1
ω(1/n)
1(∑n+1
k=2
1/kβ
)1/p
< +∞.
Acknowledgment. The author is grateful to the referee for his valuable comments that helped
to improve the presentation of the paper. The author also would like to thank Professor Grigori
Rozenblum for comments on previous version.
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Received 16.07.11,
after revision — 13.02.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
|
| id | umjimathkievua-article-2571 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:59Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c4/840b002d714ecb5c2545440e907254c4.pdf |
| spelling | umjimathkievua-article-25712020-03-18T19:29:46Z On Shiba - Waterman space Про простiр Шиби – Уотермена Hormozi, M. Гормозі, М. We give a necessary and sufficient condition for the inclusion of $\Lambda BV^{(p)}$ in the classes $H^q_{\omega}$. Наведено необхiдну та достатню умову належностi $\Lambda BV^{(p)}$ класам $H^q_{\omega}$. Institute of Mathematics, NAS of Ukraine 2012-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2571 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 2 (2012); 245-252 Український математичний журнал; Том 64 № 2 (2012); 245-252 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2571/1901 https://umj.imath.kiev.ua/index.php/umj/article/view/2571/1902 Copyright (c) 2012 Hormozi M. |
| spellingShingle | Hormozi, M. Гормозі, М. On Shiba - Waterman space |
| title | On Shiba - Waterman space |
| title_alt | Про простiр Шиби – Уотермена |
| title_full | On Shiba - Waterman space |
| title_fullStr | On Shiba - Waterman space |
| title_full_unstemmed | On Shiba - Waterman space |
| title_short | On Shiba - Waterman space |
| title_sort | on shiba - waterman space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2571 |
| work_keys_str_mv | AT hormozim onshibawatermanspace AT gormozím onshibawatermanspace AT hormozim proprostiršibiuotermena AT gormozím proprostiršibiuotermena |