On the embedding of Waterman class in the class Hpω
In this paper the necessary and sufficient condition for the inclusion of class ЛBV in the class Hpω is found.
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| author | Goginava, U. Гогінава, У. |
| author_facet | Goginava, U. Гогінава, У. |
| author_sort | Goginava, U. |
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| description | In this paper the necessary and sufficient condition for the inclusion of class ЛBV in the class Hpω is found. |
| first_indexed | 2026-03-24T02:47:29Z |
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K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDC 517.5
U. Goginava (Tbilisi Univ., Georgia)
ON THE EMBEDDING OF WATERMAN CLASS
IN THE CLASS Hp
ωω
PRO VKLGÇENNQ KLASU VATERMANA DO KLASU Hp
ωω
In this paper the necessary and sufficient condition for the inclusion of class ΛBV in the class Hp
ω is
found.
Znajdeno neobxidnu i dostatng umovu dlq vklgçennq klasu ΛBV do klasu Hp
ω .
1. Introduction. The notion of function of bounded variation was introduced by
Jordan [1]. Generalized this notion Wiener [2] has considered the class Vp of
functions. Young [3] introduced the notion of functions of Φ -variation. In [4]
Waterman has introduced the following concept of generalized bounded variation.
Definition 1. Let Λ = { λn : n ≥ 1 } be an increasing sequence of positive num-
bers such that ( )/1
1
λnn=
∞∑ = ∞ . A function f is said to be of Λ -bounded varia-
tion ( f ∈ ΛBV ) , if for every choice of nonoverlapping intervals { In : n ≥ 1 } w e
have
n
n
n
f I
=
∞
∑
1
( )
λ
< ∞ ,
where In = [ an , bn ] ⊂ [ 0, 1 ] and f ( In ) = f ( bn ) – f ( an ) .
If f ∈ ΛBV, then Λ-variation of f is defined to be the supremum of such sums,
denoted by VΛ ( f ) .
Properties of functions of the class ΛBV as well as the convergence and summabi-
lity properties of their Fourier series were investigated in [4 – 10].
For everywhere bounded 1-periodic functions, Chanturia [11] introduced the con-
cept of the modulus of variation.
If ω ( δ ) is a modulus of continuity, then Hp
ω , p ≥ 1, denotes the class of
functions f ∈ L
p ( [ 0, 1 ] ) for wich ω ( δ, f ) p = O ( ω ( δ )) as δ → 0 +, where
ω δ( , )f p = sup ( ) ( )
/
0 0
1 1
< ≤
∫ + −
h
p
p
f x h f x dx
δ
.
The relation between different classes of generalized bounded variation was taken into
account in the works of Avdispahic[12], Kovacik [13], Belov [14], Chanturia [15],
Akhobadze [16], Medvedeva [17], Kita, Yoneda [18], Goginava [19, 20].
2. Main result. The main result of this paper is presented in the following propo-
sition:
Theorem 1. ΛBV ⊂ Hp
ω for some p ∈ [ 1, ∞ ) if and only if
lim max
( / ) /
/
/
n p m n
p
ii
mn n
m
→∞ ≤ ≤
=∑
1
1 1
1 1
1
1
ω λ
< + ∞ . (1)
© U. GOGINAVA, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11 1557
1558 U. GOGINAVA
3. Proof. The sufficiency of Theorem 1 follows immediately from the following
theorem:
Theorem A [21]. Let f ∈ ΛBV and p ∈ [ 1, ∞ ) . Then
ω 1
n
f
p
,
≤ V f
n
m
m n
ii
m p
p
Λ( ) max
/
/
1
1
1
1
1
≤ ≤
=∑( )
λ
.
Necessity. We suppose that condition (1) is not satisfied. As an example, we
construct function from ΛBV which is not in Hp
ω .
Since condition (1) is not satisfied, there exists a sequence of integers { γk : k ≥ 1 }
such that
lim max
( / ) /
/
/
k
k k
p m
p
ii
m
k
m
→∞ ≤ ≤
=∑
1
1 1
1 1
1
1
ω γ γ λγ
= ∞ .
Let { ′γ k : k ≥ 1 } be a sequence of integers for which 2 1′ −γ k ≤ γk < 2 ′γ k . Then from
the fact that ω ( δ ) is nondecreasing we write
2
2 2 1
1
1 2
1
1
/
/
/
( )
max
/
p
p
m
p
ii
mk k k
m
ω λγ γ γ− ′ ′ ≤ ≤
=
′ ∑
≥
≥
1
1 1
1 1
1
1
ω γ γ λγ( / ) /
/
/
max
k k
p m
p
ii
m
k
m
≤ ≤
=∑
,
consequently,
lim
( )
max/
/
/k p
m
p
ii
mk k k
m
→∞ − ′ ′ ≤ ≤
=
′ ∑
1
2 2 11 2
1
1
ω λγ γ γ
< + ∞ .
Then there exists a sequence of integers { }:′ ≥n kk 1 ⊂ { }:′ ≥γ k k 1 such that
lim
( )
( ) /
( )
/
k n
ii
m n
k
n
p
k k k
m n
→∞ − ′
=
′ ′∑
′
1
2
1
1 2
1
1
ω λ
< + ∞ , (2)
where
max
/
/1 2
1
1
1≤ ≤
=
′ ∑m
p
ii
mnk
m
λ
=
( ( )) /
( )
/
m nk
p
ii
m nk
′
=
′∑
1
1
1 λ
.
The following three cases are possible:
a) there exists a sequence of integers { }:′ ≥s kk 1 ⊂ { }:′ ≥n kk 1 such that
m sk( )′ < 22 1′ −sk ,
b) there exists a sequence of integers { }:′ ≥q kk 1 ⊂ { }:′ ≥n kk 1 such that
22 1′ −qk ≤ m qk( )′ < 2 1′ − ′ −q qk k ,
c) 2 1′ − ′ −n nk k ≤ m nk( )′ < 2 ′nk for all k ≥ k0 .
First, we consider the case a). We choose a sequence of integers { }:s kk ≥ 1 ⊂
⊂ { }:′ ≥s kk 1 such that
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
ON THE EMBEDDING OF WATERMAN CLASS IN THE CLASS Hp
ω 1559
i
m s
i
k
=
∑
1
1
( )
λ
≥ 22 1s pk − / .
Then from (2) we get
lim /
k s
s p
k
k
→∞
ω 1
2
2 = 0.
Let { }:r kk ≥ 1 ⊂ { }:s kk ≥ 1 such that
ω 1
2
2r
r p
k
k
/ ≤ 4−k . (3)
Consider the function f defined by
f ( x ) =
2 2 1 2 3 2
2 2 2 3 2 2 2 1 2
0
1
1
c x x
c x x j
j
r r r
j
r r r
j j j
j j j
( ), [ , ),
( ), [ , ) , , ,
,
− ∈ ⋅
− − ∈ ⋅ ⋅ = …
− − −
− − −
if
if for
otherwise,
f ( x + l ) = f ( x ) , l = ± 1, ± 2, … ,
where
cj = ω 1
2
2r
r p
j
j
/
.
From the construction of the function f and by (3), we get f ∈ ΛBV. Next, we shall
prove that f ∉ Hp
ω . Since f x f x
rj( ) ( )+ −− −
2
2
= cj /2, for x ∈ [ ],2 5 2
2− − −⋅r rj j
we get
0
1
2
2∫ + −− −
f x f x dx
r p
j( ) ( ) ≥
≥
2
5 2
2
2
2
−
− −⋅
− −∫ + −
rj
rj
jf x f x dx
r p
( ) ( ) =
1
2
2
2
p j
p r
c j− −
.
Consequently, by (3) we get
ω
ω
( )
( )
,f
r
p
r
j
j
2
2
−
− ≥
1
2
1
4 2 2
1/ /( )p
j
r r p
c
j jω − ≥
2
4
1
1
j
p
−
/ → ∞ as j → ∞ .
Now we consider the case b). Let { }:q kk ≥ 1 ⊂ { }:′ ≥q kk 1 such that
1
2
1
1 2
1
1
ω λ( )
( )
/
( )
/
−
=∑
q
ii
m q
k
q
p
k k k
m q
≥ 4
k. (4)
Consider the function gk defined by
gk ( x ) =
h x j x j j
h x j x j j
j m q m q
k
q q q
k
q q q
k k
k k k
k k k
( ), [( ) , ),
( ), [ , ( ) )
( ), ( ) ,
,
/ /
/ /
2 2 1 2 1 2 2 2
2 2 1 2 2 2 1 2
1
0
1
− + ∈ −
− − − ∈ +
= … −
−for
otherwise,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1560 U. GOGINAVA
where
hk =
1
2 1
1
k
jj
m qk /
( ) λ=∑
.
Let
g ( x ) = g xk
k
( )
=
∞
∑
2
, g ( x + l ) = g ( x ) , l = ± 1, ± 2, … .
First, we prove that g ∈ ΛBV. For every choice of nonoverlapping intervals { In :
n ≥ 1 }, we get
j
j
j
g I
=
∞
∑
1
( )
λ
≤ 2
1
1 1i
i
j
m q
j
h
i
=
∞
=
∑ ∑
( )
λ
= 2
1
21i
i
=
∞
∑ = 2.
Hence, we have g ∈ ΛBV.
Next, we shall prove that g ∉ Hp
ω . Since g xk
qk( )+ − −2 1 = g x hk k( ) /+ 2 , for x ∈
∈ [ )( ) , ( )2 1 2 4 1 2 1j jq qk k− −− − − and m ( qk ) ≥ 2 1m qk( )− we obtain
0
1
1
1
2∫ +
−+g x g x dxq
p
k
( ) ≥
≥
j m q
m q
j
j
k q k
p
k
k
q
q
k
k
k
g x g x dx
=
−
−
−
+
− −
− −
∑ ∫ +
−
( )
( )
( )
( )
( )
1
1
1
2 1 2
4 1 2
1
1
2
=
=
h
m q m qk
p
p q k kk2
1
2 1 1+ −−( ( ) ( )) ≥
h m qk
p
p
k
qk2 22+
( )
,
consequently, by (4)
ω
ω
( )
( )
,2
2
−
−
q
p
q
k
k
f
≥
1
2 2 21 2
1
+ −
/
/
( )
( )
p
k
q
k
q
ph m q
k kω
=
=
1
2 2
1
2
1
1 21 2
1
1
+ −
=∑
/ ( )
/
( ) /
( )
p k q
ii
m q
k
q
p
k k k
m q
ω λ
≥
2
21 2
k
p+ / → ∞ as k → ∞ .
Finally, we consider the case c). Let { }:n kk ≥ 1 ⊂ { }:′ ≥n k kk 0 such that
nk ≥ 2 11nk− + , (5)
1
2
1
1 2
1
1
ω λ( ) /
( )
/( )
−
=∑
n
ii
m n
k
n
p
k k k
m n
≥ 22 1n p kk − +/ . (6)
Consider the function ϕk defined by
ϕk ( x ) =
d x j x j j
d x j x j j
j
k
n n n
k
n n n
n n n n
k k k
k k k
k k k k
( ), [( ) , ),
( ), [ , ( ) )
, , ,
,
/ /
/ /
2 2 1 2 1 2 2 2
2 2 1 2 2 2 1 2
2 2 1
0
1 2 1 1
− + ∈ −
− − − ∈ +
= … −
− − −− − −for
otherwise,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
ON THE EMBEDDING OF WATERMAN CLASS IN THE CLASS Hp
ω 1561
where
dk =
1
2 1
1
k
jj
m nk /
( ) λ=∑
.
Let
ϕ ( x ) = ϕk
k
x( )
=
∞
∑
3
, ϕ ( x + l ) = ϕ ( x ) , l = ± 1, ± 2, … .
For every choice of nonoverlapping intervals { In : n ≥ 1 }, we get
j
j
j
I
=
∞
∑
1
ϕ
λ
( )
≤ 2
1
2 1
2 1 1
di
i jj
n ni i
=
∞
=
∑ ∑
− −−
λ
≤ 2
1
2 1
di
i jj
m ni
=
∞
=
∑ ∑ λ
( )
≤ 2
1
22
i
i=
∞
∑ = 1.
Hence, we have ϕ ∈ ΛBV .
Next, we shall prove that ϕ ∉ Hp
ω . From (5) we write
0
1
1
1
2∫ +
−+ϕ ϕx x dxn
p
k
( ) ≥
≥
j j
j
k n k
p
nk nk
nk nk
n
n
k
k
k
x x dx
=
−
−
−
+
− − −
− −
−
− −
∑ ∫ +
−
2
2 1
2 1 2
4 1 2
1
1 2
1
1
1
2
( )
( )
( )ϕ ϕ ≥
≥
2 1
2 2
1
2
n n
p
k
p
n
k k
k
d−
+
− −
≥ c
d m nk
p
n
k
nk k2 21−
( )
.
Consequently, by (6)
ω
ω
( , )
( )
2
2
−
−
n
p
n
k
k
f
≥ c
d m nk
n p
k
n
p
nk k k2 2
1
21
1
−
−/
/( )
( )ω
=
=
c m n
n p k j
j
m n
k
n
p
nk
k
k k2
1
2
1
21
1
1 1
− +
=
−
−∑
/
( ) /
/
( )
( )
λ
ω
≥ c n pk2 1− → ∞/ as k → ∞ .
Therefore we get ϕ ω∉ Hp and the proof of Theorem 1 is complete.
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ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
1562 U. GOGINAVA
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P. 230 – 235.
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Vekua Inst. Appl. Math. – 1988. – 3. – P. 11 – 13 (in Russian).
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S. 534 – 554 (in Russian).
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(in Russian).
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Received 24.06.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 11
|
| id | umjimathkievua-article-3707 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:47:29Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4a/1964d2a57d49a18ddebc2a1282382e4a.pdf |
| spelling | umjimathkievua-article-37072020-03-18T20:02:37Z On the embedding of Waterman class in the class Hpω Про включення класу Ватермана до класу Hpω Goginava, U. Гогінава, У. In this paper the necessary and sufficient condition for the inclusion of class ЛBV in the class Hp&omega; is found. Знайдено необхідну i достатню умову для включення класу KBV до класу Hp&omega;. Institute of Mathematics, NAS of Ukraine 2005-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3707 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 11 (2005); 1557–1562 Український математичний журнал; Том 57 № 11 (2005); 1557–1562 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3707/4139 https://umj.imath.kiev.ua/index.php/umj/article/view/3707/4140 Copyright (c) 2005 Goginava U. |
| spellingShingle | Goginava, U. Гогінава, У. On the embedding of Waterman class in the class Hpω |
| title | On the embedding of Waterman class in the class Hpω |
| title_alt | Про включення класу Ватермана до класу Hpω |
| title_full | On the embedding of Waterman class in the class Hpω |
| title_fullStr | On the embedding of Waterman class in the class Hpω |
| title_full_unstemmed | On the embedding of Waterman class in the class Hpω |
| title_short | On the embedding of Waterman class in the class Hpω |
| title_sort | on the embedding of waterman class in the class hpω |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3707 |
| work_keys_str_mv | AT goginavau ontheembeddingofwatermanclassintheclasshpō AT gogínavau ontheembeddingofwatermanclassintheclasshpō AT goginavau provklûčennâklasuvatermanadoklasuhpō AT gogínavau provklûčennâklasuvatermanadoklasuhpō |