Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces
A boundedness result is established for multilinear singular integral operators on the homogeneous Morrey–Herz spaces. As applications, two corollaries about interesting cases of the boundedness of the considered operators on the homogeneous Morrey–Herz spaces are obtained.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509146388889600 |
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| author | Wu, Jianglong Ву, Янглонг |
| author_facet | Wu, Jianglong Ву, Янглонг |
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| description | A boundedness result is established for multilinear singular integral operators on the homogeneous Morrey–Herz spaces. As applications, two corollaries about interesting cases of the boundedness of the considered operators on the homogeneous Morrey–Herz spaces are obtained. |
| first_indexed | 2026-03-24T02:36:28Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDC 517.9
Jianglong Wu (Mudanjiang Teachers College, China)
BOUNDEDNESS OF MULTILINEAR SINGULAR
INTEGRAL OPERATORS ON THE HOMOGENEOUS
MORREY – HERZ SPACES
OBMEÛENIST| BAHATOLINIJNYX SYNHULQRNYX
INTEHRAL|NYX OPERATORIV NA ODNORIDNYX
PROSTORAX MORREQ – HERCA
A boundedness result is established for the multilinear singular integral operators on the homogeneous
Morrey – Herz spaces. As applications, two corollaries about interesting cases of the boundedness of
considered operators on the homogeneous Morrey – Herz spaces are obtained.
Vstanovleno obmeΩenist\ bahatolinijnyx synhulqrnyx intehral\nyx operatoriv na odnoridnyx
prostorax Morreq – Herca. Qk zastosuvannq, oderΩano dva naslidky pro cikavi vypadky obme-
Ωenosti rozhlqduvanyx operatoriv na odnoridnyx prostorax Morreq – Herca.
1. Introduction and main results. Let Rn , n ≥ 1, be the n-dimensional Euclidean
space and let Sn−1 be the unit sphere in Rn equipped with normalized Lebesgue
measure d d xσ σ= ′( ) . Let Ω be a homogeneous function of degree zero on Rn
and Ω ∈ −L Sr n( )1 for some r ∈ ∞[ ]1, , and Ω( ) ( )′ ′−∫ x d x
Sn σ1 = 0, where ′x =
= x x −1 for any x ≠ 0. If f Lq n∈ ω ( )R , that is
f f x x dxL
q
q
q n
n
ω
ω( )
/
( ) ( )R
R
=
< ∞∫
1
.
The Calderón – Zygmund singular integral operator T is defined by
T f x
y
y
f x y dy
n
n
( ) . .
( )
( )= ′ −∫p v
Ω
R
,
and the truncated maximal operator T ∗ is defined by
T f x
y
y
f x y dy
n
x y
∗
> − >
= ′ −∫( ) sup
( )
( )
ε ε0
Ω
,
where ′ = −y y y 1 ∈ Sn − 1 and f C n∈ ∞
0 ( )R .
© JIANGLONG WU, 2009
1434 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
BOUNDEDNESS OF MULTILINEAR SINGULAR … 1435
In 1971, Muckenhoupt and Wheeden [1] proved that if Ω( ) ( )′ ∈ −x L Sr n 1 , r > 1,
then the operators T and T ∗ are bounded on L
x
q n
β ( )R , 1 < q < ∞, provided that β
be in the interval max (−( n , − − − ′)1 1( ) /n q r , min ( )n q −( 1 , q – 1 + (n – 1) q r/ )′ ) .
Here and in the following, ′r denotes the dual exponent of r, i. e., ′r = r r/( )− 1 ,
and L
x
q n
β ( )R denotes the weighted Lebesgue space. For general Aq weights,
Duoandikoetxea [2] gave the weighted Lq , 1 < q < ∞, boundedness of T and T ∗ .
The purpose of this paper is to consider the boundedness on homogeneous Morrey
– Herz spaces for the multilinear singular integral operators TA m, and TA m,
∗ which
are defined as follows,
T f x R A x y
x y
x y
f yA m m n m
n
, ( ) . ( ; , )
( )
( )=
−
−
+ +∫p. v 1
R
Ω
ddy ,
T f x R A x y
x y
x y
A m m
x y
n, ( ) sup ( ; , )
( )∗
>
+
− >
=
−
−∫
ε ε0
1
Ω
++ m
f y dy( ) ,
where m is positive integer, A has derivatives of order m in BMO( )Rn , R Am + 1( ;
x, y) denotes the ( )m + 1 -th Taylor series remainder of A at x about y, that is
R A x y A x D A y x ym
m
+
≤
= − −∑1
1
( ; , ) ( )
!
( ) ( )
µ
µ µ
µ
,
and f C n∈ ∞
0 ( )R .
Let B Bk
k= ( , )0 2 = x n∈{ R : x k≤ }2 and C B Bk k k= −\ 1 for k ∈ Z . Let
χ χk Ck
= for k ∈ Z be the characteristic function of the set Ck .
Definition 1 [3]. Let α ∈ R , 0 < p ≤ ∞, 0 < q < ∞ and λ ≥ 0. The homoge-
neous Morrey – Herz spaces MK p q
n�
,
, ( )α λ R are defined by
MK f L fp q
n q n
MK p q
n
�
�,
,
(( ) \ :
,
,
α λ
α λR R R= ∈ { }( )loc 0 )) < ∞{ } ,
where
f fMK
k
k k p
k L
p
k
p q
n q n�
,
, ( ) ( )
supα λ
λ α χR Z R
=
∈
−
=0
02 2
−−∞
∑
k p
0
1/
with the usual modifications made when p = ∞.
Now, let us state the main results of this paper.
Theorem 1. Let R A x ym + 1( ; , ) and A be defined as above, r ≥ 1, m ∈ N , λ ≥
≥ 0, 0 < p ≤ ∞, 1 < q < ∞, max /− +( n q λ , − − − ′ + )1 1/ ( )/q n r λ < α <
< min ( / )n q1 1−( + λ , 1 – 1 1/ ( )/q n r+ − ′ + )λ . And let �Ω be a homogeneous
function of degree zero on Rn with �Ω ∈ −L L Sr n(ln ) ( )1 , that is
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1436 JIANGLONG WU
� �Ω Ω( ) ln ( ) ( )′ + ′( ) ′ < ∞
−
∫ x x d x
S
r
n 1
2 σ .
If a sublinear operator �Tb m, is bounded on Lq n( )R and there is a constant C inde-
pendent of f such that
�
�
T f x C
x y
x y
R A x y f y dyb m n m m
n
, ( )
( )
( ; , ) ( )≤
−
− + +
Ω
1
R
∫∫
for any f L n∈ 1( )R with compact support and x f∉ supp , then �Tb m, is also
bounded on MK p q
n�
,
, ( )α λ R .
Throughout this paper, C denotes the constants that are independent of the main
parameters involved but whose value may differ from line to line.
2. Proof of Theorem 1. Let Ω be a homogeneous function of degree zero on
Rn and Ω ∈ −L Sr n( )1 for some r ∈ ∞[ ]1, . The truncated operator St b; , Ω is de-
fined by
S f x t
x y
x y
R A x y f y dyt A
n
m m
x
; , ( )
( )
( ; , ) ( )Ω
Ω
=
−
−
−
+
−
1
yy t<
∫ .
In this section we give two lemmas which are the key to the proof of Theorem 1.
Lemma 1 [4]. Let A x( ) be a function on Rn with derivatives of order m in
Lr n( )R for some r n∈ ∞( ], . Then
R A x y C x y
x y
D A z dzm m n
m r
x y
( ; , )
( , )
( ),
( , )
≤ −
∫
1
Ω Ω
µ
=
∑
1/r
mµ
,
where Ω( , )x y is the cube centered at x with sides parallel to the coordinate axes
and whose side length is 5 n x y− .
Lemma 2 [5]. Let R A x ym + 1( ; , ) and A x( ) be defined as above. Let Ω ∈
∈ L Sn∞ −( )1 be a homogeneous function of degree zero on Rn , m ∈ N , r ≥ 1,
µ = m , 1 < q < ∞ and D A nµ ∈ BMO( )R . Set
ξ ξ
ξ ξΩ
Ω Ω
= > +
≤
∞inf : ln0 2 1r .
If max ,− −( n 1 – (n – 1) q r/ ′) < β < min ( )n q −( 1 , q – 1 + (n – 1) q r/ ′) , then the trun-
cated operator St A; , Ω is bounded on L
x
q n
β ( )R with bound
C D A nm
µ
µ
ξ
BMO( )R Ω=∑ .
Here and in what follows, let f p denote f L Sp n( )− 1 . And without loss of ge-
nerality, we may assume that D A nm
µ
µ BMO( )R=∑ = 1.
Proof of Theorem 1. We choose α1 , α2 ∈ R , such that max /−( n q , – 1 / q – (n –
– 1) / ′)r < α1/q < α – λ < α2 /q < min ( / )n q1 1−( , 1 – 1 / q + (n – 1) / ′)r . Write
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
BOUNDEDNESS OF MULTILINEAR SINGULAR … 1437
f x f x x f xj j
jj
( ) ( ) ( ) ( )= ≡
= −∞
∞
= −∞
∞
∑∑ χ .
Then, we have
� �
�T f TA m MK k
k k p
k A
p q
n, ( )
( ) sup
,
,α λ
λ α χ
R Z
=
∈
−
0
02 2 ,, ( )
/
( )m L
p
k
k p
f q nR
= −∞
∑
0
1
≤
≤ C T f
k
k k p
k A m j L
j
k
q nsup ( ), ( )
0
02 2
2
∈
−
= −∞
−
∑
Z R
λ α χ �
= −∞
∑
p
k
k
p
0
1/
+
+ C T f
k
k k p
k A m j L
j k
k
q nsup ( ), ( )
0
02 2
1
1
∈
−
= −
+
∑
Z R
λ α χ �
= −∞
∑
p
k
k
p
0
1/
+
+ C T f
k
k k p
k A m j L
j k
q nsup ( ), ( )
0
02 2
2∈
−
= +
∞
∑
Z R
λ α χ �
= −∞
∑
p
k
k
p
0
1/
≡
≡ E E E1 2 3+ + .
For E2 , by the Lq n( )R boundedness of �TA m, , we have
E C f
k
k k p
k j L
j k
k
q n2
1
1
0
02 2≤
∈
−
= −
+
∑sup
( )Z R
λ α χ
= −∞
∑
p
k
k
p
0
1/
≤
≤ C f
k
k k p
k L
p
k
k p
q nsup
( )
/
0
0
0
2 2
1
∈
−
= −∞
∑
Z R
λ α χ =
= C f MK p q
n�
,
, ( )α λ R .
For E1 , note that when x Ck∈ , j ≤ k – 2, and y C j∈ , then 2 y x≤ . Therefo-
re, for x Ck∈ ,
�
�
T f x C
x y
x y
R A x y f y dyA m j n m m j, ( )
( )
( ; , ) ( )≤
−
− + +
Ω
1
RRn
∫ ≤
≤
C
x
x y
x y
R A x y f y dy CS
n m m j
x y x
�Ω( )
( ; , ) ( )
−
−
≤+
− ≤
∫ 1
3
2
22 1k A jf x+ ; , ( )�Ω .
Let F x Sn
0
1= ∈{ −
: �Ω( )x ≤ 2} and F x Sd
n= ∈{ − 1
: 2d < �Ω( )x ≤ 2 1d + } for
positive integer d. Denote by �Ωd the restriction of �Ω on Fd . Then
S f x S f xk k
dA j A j
d
2 2
0
1 1+ +=
=
∞
∑; , ; ,( ) ( )� �Ω Ω .
Thus, by Lemma 2 and the conditions in Theorem 1, we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1438 JIANGLONG WU
S f S fk
x
q n
k
d
x
q nA j
L A j
L2 21
2
1
2
+ +≤; , ( ) ; , (
� �Ω Ω
α αR R ))d =
∞
∑
0
≤
≤ C f C f
d
x
q n
x
q n
d
j L j L
ξ
α α
Ω
=
∞
∑ ≤
0 2 2
( ) ( )R R
. (1)
In obtaining the last inequality, we use the fact that ξΩd
C
d
≤
=
∞∑ 0
. In fact, our hy-
pothesis on �Ω now says that d d rd
�Ω
>∑ 0
< ∞. Set ξd = d d r
�Ω + 2−d . It is
obvious that
� � �
�
Ω Ω Ω
Ω
d r
d
d
d
d r
d r
d
dξ ξ
ln ln2 22 1+
≤ ( )∞ + ≤≤ C .
This in turn implies that ξΩd
≤ C d d r
d�Ω +( )−2 . Therefore,
ξ ξ ξΩ Ω Ω Ω
d d
d d
d r
d
d
d
C C d C C= + ≤ + + ≤
≥ > >
−
>
∑ ∑ ∑� �
0
0 0 0 0
2∑∑ .
Thus, by (1), it follows that
E C T f
k
k k q p
k A m j L
x
q1
0
0 2
2
2 2≤
∈
− −sup ( )( / )
,
Z
λ α α χ
α
�
(( )
/
Rn
j
k p
k
k
p
= −∞
−
= −∞
∑∑
2
1
0
≤
≤ C S f
k
k k q p
A j
L
k
x
q
sup ( )( / )
; ,
0
0 2
1
2
2 2 2∈
− −
+
Z
λ α α
α
�Ω (( )
/
Rn
j
k
p
k
k
p
= −∞
−
= −∞
∑∑
2
1
0
≤
≤ C f
k
k k q p
j L
j
k
x
q nsup ( / )
( )
0
0 2
2
2 2
2
∈
− −
= −∞
−
∑
Z R
λ α α
α
= −∞
∑
p
k
k
p
0
1/
≤
≤ C
k
k k p
k
k
j ksup ( ) ( – /
0
0
0
22 2 2
∈
−
= −∞
− +∑
Z
λ λ λ α α qq j
j
k
) −
= −∞
−
∑
λ
2
×
× 2
1
1
l p
l L
p
l
j p p p
f q n
α
( )
/
/
R
= −∞
∑
≤
≤ C f
k
k k p j k q
j
k
Msup ( ) ( – / )
0
0 22 2 2
2
∈
− − +
= −∞
−
∑
Z
λ λ λ α α
��K
p
k
k
p
p q
n
,
, ( )
/
α λ R
= −∞
∑
0
1
≤
≤ C f
k
k k p
k
k p
MK p q
sup
/
,
,
0
0
0
2 2
1
∈
−
= −∞
∑
Z
λ λ
α λ� (( )Rn ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
BOUNDEDNESS OF MULTILINEAR SINGULAR … 1439
≤ C f
k
k k
MK p q
nsup
,
, ( )
0
0 02 2
∈
−
Z R
λ λ
α λ� = C f MK p q
n�
,
, ( )α λ R .
Now, let us turn to estimate for E3. Note that when x Ck∈ , j ≥ k + 2, and
y C j∈ , then 2 x y≤ and
�
�
T f x C
x y
x y
R A x y f y dyA m j n m m j, ( )
( )
( ; , ) ( )≤
−
− + +
Ω
1
RRn
∫ ≤
≤ C
x y
x y
R A x y f y dyjn
m
x y
m j
j
2
2
1
1
−
− ≤
+
−
−+
∫
�Ω( )
( ; , ) ( ) ≤≤ +CS f xj A j2 1; , ( )�Ω .
Thus, by (1), similar to the proof of E1, we have
E C T f
k
k k q p
k A m j L
x
q3
0
0 1
1
2 2≤
∈
− −sup ( )( / )
,
Z
λ α α χ
α
�
(( )
/
Rn
j k
p
k
k
p
= +
∞
= −∞
∑∑
2
1
0
≤
≤ C S f
k
k k q p
A j
L
j
x
q
sup ( )( / )
; ,
0
0 1
1
1
2 2 2∈
− −
+
Z
λ α α
α
�Ω (( )
/
Rn
j k
p
k
k
p
= +
∞
= −∞
∑∑
2
1
0
≤
≤ C f
k
k k q p
j L
j k x
q nsup ( )( / )
( )
0
0 1
1
2 2
2∈
− −
= +
∞
Z R
λ α α
α
∑∑∑
= −∞
p
k
k
p
0
1/
≤
≤ C
k
k k p
k
k
k jsup ( ) ( /
0
0
0
12 2 2
∈
−
= −∞
− − −∑
Z
λ λ α λ α qq j
j k
) −
= +
∞
∑
λ
2
×
× 2
1
1
l p
l L
p
l
j p p p
f q n
α
( )
/
/
R
= −∞
∑
≤
≤ C f
k
k k p k j q
MK p q
sup ( ) ( / )
,
,
0
0 12 2 2
∈
− − − −
Z
λ λ α λ α
α λ� (( )
/
Rn
j k
p
k
k
p
= +
∞
= −∞
∑∑
2
1
0
≤
≤ C f
k
k k p
k
k p
MK p q
sup
/
,
,
0
0
0
2 2
1
∈
−
= −∞
∑
Z
λ λ
α λ� (( )Rn ≤
≤ C f
k
k k
MK p q
nsup
,
, ( )
0
0 02 2
∈
−
Z R
λ λ
α λ� = C f MK p q
n�
,
, ( )α λ R .
Therefore, Theorem 1 is proved.
3. Corollary. The above proof of Theorem 1 also indicates that the boundedness
of operator �TA m, on Lebesgue spaces with power weights implies its boundedness on
homogeneous Morrey – Herz spaces. Similar result is proved by Lu and Xu in [3].
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1440 JIANGLONG WU
The following lemma for the boundedness of operators TA m, and TA m,
∗ on the
Lebesgue spaces with power weights can be found in [5].
Lemma 3 [5]. Let R A x ym + 1( ; , ) and A be defined as above, m ∈ N , r > 1,
and Ω ∈ L L Sr n(ln ) ( )− 1 . If max −( n , – 1 – (n – 1) q / ′)r < β < min ( )n q −( 1 , q –
– 1 + (n – 1) q / ′)r and 1 < q < ∞, then
T f C fA m L L
x
q n
x
q n, ( ) ( )
β βR R≤
and
T f C fA m
L L
x
q n
x
q n, ( ) ( )
∗ ≤
β βR R .
As a simple corollary of Theorem 1 and Lemma 3, when r > 1, we have the follo-
wing result.
Corollary 1. Let R A x ym + 1( ; , ) and A be defined as above, m ∈ N , r > 1, 1 <
< q < ∞, 0 < p ≤ ∞, and Ω ∈ L L Sr n(ln ) ( )− 1 . I f max /−( +n q λ , – 1 / q – (n –
– 1) ′ + )r λ < α < min ( / )n q1 1−( + λ , 1 – 1 / q + (n – 1) ′ + )r λ , then TA m, and
TA m,
∗ are bounded on MK p q
n�
,
, ( )α λ R .
Lemma 4 [5]. Let R A x ym + 1( ; , ) and A be defined as above, m ∈ N , r = 1,
and Ω ∈ L L Sn(ln ) ( )− 1 . If – 1 < β < q – 1 and 1 < q < ∞, then
T f C fA m L L
x
q n
x
q n, ( ) ( )
β βR R≤
and
T f C fA m
L L
x
q n
x
q n, ( ) ( )
∗ ≤
β βR R .
As a simple corollary of Theorem 1 and Lemma 4, when r = 1, we have the follo-
wing result.
Corollary 2. Let R A x ym + 1( ; , ) and A be defined as above, m ∈ N , 1 < q <
< ∞, 0 < p ≤ ∞, and Ω ∈ L L Sn(ln ) ( )− 1 . If – 1 < β < q – 1, then TA m, and TA m,
∗
are bounded on MK p q
n�
,
, ( )α λ R .
According to [3] (Theorem 2.1), Corollaries 1 and 2 are proved easily.
1. Muckenhoupt B., Wheeden R. L. Weighted norm inequalities for singular and fractional integrals //
Trans. Amer. Math. Soc. – 1971. – 161. – P. 249 – 258.
2. Duoandikoetxea J. Weighted norm inequalities for homogeneous singular integrals // Ibid. – 1993.–
336. – P. 869 – 880.
3. Lu S. Z., Xu L. F. Boundedness of rough singular integral operators on the homogeneous Morrey –
Herz spaces // Hokkaido Math. J. – 2005. – 34, # 2. – P. 299 – 314.
4. Cohen J., Gosselin J. A BMO estimate for multilinear singular integral // Ill. J. Math. – 1986. – 30.
– P. 445 – 464.
5. Xu L. F. Boundedness of multilinear singular integral operators on the homogeneous Herz // J.
Beijing Normal Univ. (Natur. Sci.). – 2004. – 40, # 3. – P. 297 – 303.
Received 24.12.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
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| id | umjimathkievua-article-3112 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:36:28Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c1/93db0372c784ea038db7605f974cc5c1.pdf |
| spelling | umjimathkievua-article-31122020-03-18T19:45:28Z Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces Обмеженість багатолінійних сингулярних інтегральних операторів на однорідних просторах Моррея - Герца Wu, Jianglong Ву, Янглонг A boundedness result is established for multilinear singular integral operators on the homogeneous Morrey–Herz spaces. As applications, two corollaries about interesting cases of the boundedness of the considered operators on the homogeneous Morrey–Herz spaces are obtained. A boundedness result is established for multilinear singular integral operators on the homogeneous Morrey–Herz spaces. As applications, two corollaries about interesting cases of the boundedness of the considered operators on the homogeneous Morrey–Herz spaces are obtained. Institute of Mathematics, NAS of Ukraine 2009-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3112 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 10 (2009); 1434-1440 Український математичний журнал; Том 61 № 10 (2009); 1434-1440 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3112/2972 https://umj.imath.kiev.ua/index.php/umj/article/view/3112/2973 Copyright (c) 2009 Wu Jianglong |
| spellingShingle | Wu, Jianglong Ву, Янглонг Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces |
| title | Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces |
| title_alt | Обмеженість багатолінійних сингулярних інтегральних операторів на однорідних просторах Моррея - Герца |
| title_full | Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces |
| title_fullStr | Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces |
| title_full_unstemmed | Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces |
| title_short | Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces |
| title_sort | boundedness of multilinear singular integral operators on the homogeneous morrey–herz spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3112 |
| work_keys_str_mv | AT wujianglong boundednessofmultilinearsingularintegraloperatorsonthehomogeneousmorreyherzspaces AT vuânglong boundednessofmultilinearsingularintegraloperatorsonthehomogeneousmorreyherzspaces AT wujianglong obmeženístʹbagatolíníjnihsingulârnihíntegralʹnihoperatorívnaodnorídnihprostorahmorreâgerca AT vuânglong obmeženístʹbagatolíníjnihsingulârnihíntegralʹnihoperatorívnaodnorídnihprostorahmorreâgerca |