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AREA OF SEPARATE-HARMONICITY
Satlikov G.R.
1
, Choriev Sh.F.
2
1
Urgench State University,
2
National University of Uzbekistan, University St.,4, Tashkent, 100174
Abstract:
The paper considers the concept of separate harmonicity of functions of several
variables with respect to disjoint groups of variables. The separate harmonicity hulls for
domains are studied. In particular, it is proven that a domain possessing barrier functions on a
dense subset of the boundary is a domain of separate harmonicity.
Keywords:
Harmonic function, separately harmonic function, separate harmonicity hull, barrier,
domain of separate harmonicity
Аннотация:
В работе рассматривается понятие сепаратно-гармоничности функций
многих переменных по раздельным группам переменных. Исследуются оболочки
сепаратно-гармоничности для областей. В частности, доказано, что область обладающего
барьерными функциями на всюду плотном подмножестве границы, является областью
сепаратно-гармоничности.
Ключевые слова:
Гармоническая функция, сепаратно-гармоническая функция, оболочка
сепаратно-гармоничности, барьер, область сепаратно-гармоничности.
1.Introduction
It is known that for any flat domain in
D
there exists a function holomorphic in
D
and
not analytically continued beyond this domain, i.e. every flat domain is a domain of
holomorphy. In contrast, in the space
n
,
1
n
>
, there exist domains from which any
holomorphic function can be analytically continued into a wider domain. For example, a non-
simply connected domain
{
}
:1
2
n
z
z
< <
is an example of such domains (see [1, p.148,
Osgood-Brown theorem]). This means that in
n
,
1
n
>
, every domain is not a domain of
holomorphy, i.e. the class holomorphic functions of several complex variables have the effect
of forced analytic continuation (see [1, p.153]).
It turns out that the class of separately harmonic functions also has the property of forced
analytic continuation. In this paper, we study the domains of separately harmonicity.
2. Separate harmonic functions
Let
R
n
x
,
R
m
y
,
R
R
n
m
D G
— be a domain,
D E
and
G
F
—
some subsets. Suppose that, following the function
( , )
u x y
, initially defined on the set
E F
,
has the properties:
a) for any fixed
0
x
E
, the function
0
( , )
u x y
continues harmonically in
G
;
b) for any fixed
0
y
F
the function
0
( , )
u x y
continues harmonically in
D
.
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In this case, the specified extensions and
( , )
u x y
define a certain function on the set
(
) (
)
X
E G
D F
=
U
, which is called a separate harmonic function on
X
.
In the case when in
E D
=
and
F G
=
, the function
( , )
u x y
is called separately harmonic in
the region
X D G
=
, i.e. harmonic with respect to the groups of variables separately.
The set
X
is not domain in general. For an arbitrary domain that cannot be represented as a
product of two domains, the separately harmonic function is defined as follows: if the function
( , )
u x y
is defined in the domain
( )
( )
W
n
m
x
y
R
R
,
,
1
n m
>
, and has the following
properties.
1) for any
{
}
0
0
:
=
W
I
x
x x
, function
0
( , )
u x y
harmonic with respect to variable
y
on
the section
{
}
0
=
W
I
x x
;
2) for any
{
}
0
0
:
=
W
I
y
y y
, the function
0
( , )
u x y
is harmonic in the variable
x
on the
section
{
}
0
=
W
I
y y
, then it is called a separately harmonic function in region
W
.
The above defined class of separately harmonic functions is denoted by
( ) ( ( ))
nm
nm
h
h X
W
.
The well-known Hartogs theorem (see
[1]
) states that if a function
( , )
f z w
is holomorphic in
the domain
n
D
for fixed
w
and it is holomorphic in
m
G
holomorphic in the
domain
( )
( )
n
m
D G
z
w
with respect to the set of variables.
In 1961, P. Lelong [2] proved the following analogue of this theorem for separately harmonic
functions: if the function
( , )
u x y
is separately harmonic in the domain
( )
( )
n
m
D G
x
y
R
R
, then the function
( , )
u x y
is harmonic in
D G
with respect to
the set of variables.
Let us now consider the following general problem: let in
n
E D
R
,
m
F G
R
and
( , )
u x y
separately harmonic in the set it is required to describe the
(
) (
)
X
E G
D F
=
U
the harmonic domain functions
( , )
u x y
be .
This problem was studied in the works
[3]
,
[4]
),
[6]
, [8], [9].
In 1982, A. Zeryakhi [4] obtained the following result: let the domain of satisfying
D G
—
space
2
2
( )
( )
x
y
R
R
and
E D
,
F G
— be compact sets the conditions of
H
regularity in classes of harmonic polynomials. Then any separately harmonic function on
the set
(
) (
)
X
E G
D F
=
U
is harmonically continued into the domain
{
}
*
*
( , )
:
( , , )
( , , ) 1
sh
sh
X
x y D G
x E D
y F G
w
w
=
+
<
.
Usually, to continue harmonic functions, one first goes to holomorphic functions and then uses
the principles of holomorphic continuations.
The following result plays a fundamental role in the study of continuation of harmonic
functions.
Preposition
1
(see
[6]).
Consider
the
space
( )
n
x
R
,
embedded
in
( )
( )
( )
n
n
n
z
x i
y
=
+
C
R
R
, where
(
)
1
,...,
n
z
z
z
=
,
j
j
j
z
x
i y
=
+
,
1,...,
j
n
=
, and let
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102
D
be some bounded domain from
( )
n
x
R
. Then there exists a domain
( )
n
D
z
C
such that
in
D D
and for any function
( )
( )
u x
h D
there exists a holomorphic function
( )
u
f z
in
D
such that
u D
f
u
=
is a subdomain. In addition, for any number
1
M
>
there exists
M
D
D
,
M
D D
, such that
M
u D
D
f
M u
,
( )
( )
u h D
L D
"
I
(here
{
}
sup ( ) :
D
u
u x x D
=
).
Теорема 1
([6]). Let
n
E D
R
and
m
F G
R
be sets that are not pluripolar
compact in the sense of subsets of the spaces
( )
( )
( )
n
n
n
z
x i
y
=
+
C
R
R
and
m
m
m
i
=
+
C
R
R
. Any separately harmonic function
( , )
u x y
on the set
(
) (
)
X
E G
D F
=
U
can be harmonically extended to the domain
{
}
*
*
( , )
: ( , , )
( , , ) 1
X
x y D G
x E D
y F G
w
w
=
+
<
.
Thus, if the compact sets
{
}
1
:| |
n
E
x
x R
=
R
,
{
}
2
:| |
m
F
x
x R
=
R
), then the
function
( , )
u x y
is continued into some neighborhood of
E F
.
Here
(
)
( )
{
}
, ,
sup ( ) :
, |
0, | 1
E
D
z E D
u z u psh D u
u
w
=
.
( , , ) lim ( , , )
z z
z E D
z E D
w
w
*
®
=
,
z D
,
is called the
P
-measure (plurisubharmonic measure) of the set
E D
with respect to the
domain in
n
D
(see [5], [7]).
3
.
Main theorems
From the above theorems it is easy to verify that for some domains with
n m
+
W
there exists
a corresponding, wider domain of harmonic €
W W
such that the class of separately functions
domain is called the envelope of separately-harmonicty to
W
.
Definition 1.
The
n m
+
W
domain is called a domain of separate harmonicity for class
( )
nm
h
W
, if there exists some function
( , )
( )
nm
u x y
h
W
that does not continue at any point
outside
W
which is harmonically.
Definition 2.
We will say that at the boundary point
( , )
x h
¶W
to the domain
n m
+
W
there is a barriers if there exists a function
( , )
( )
nm
x y
h
J
W
,
unbounded at the point
( , )
x h
, i.е.
( , )
j
j
x y
J
®
of the sequence ( , )
j
j
x y
W
,
( , )
( , )
j
j
x y
x h
®
at
j
® +
.
Theorem 2.
If on an everywhere dense set of points of the boundary of a domain
W
there
exists a barrier from class
( )
nm
h
W
, then
W
is a domain of separate harmonicity for class
( )
nm
h
W
.
Proof.
Let the domain
n m
+
W
have an everywhere dense countable set of barrier points
{
}
1
1
2
2
( , ),( , ),...,( , ),...
j
j
x h
x h
x h
¶W
. Then, by definition, there exists a sequence of functions
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( , ) ( , )
( , )
( ),
lim
( , )
.
j
j
j
nm
j
x y
x y
h
x y
x h
J
J
®
W
= +
We take a sequence of domains
j
W
:
1
j
j
+
W
W
W
Р
Р
,
1
j
j
+
=
W = W
U
and consider the sum of the following series
2
1
1
( , )
( , )
j
j
j
x y
x y
j c
J
J
+
=
=
,
{
}
max
( , ) : ( , )
j
j
j
с
x y
x y
J
=
W
.
It is clear that the function
( , )
x y
J
belongs to the class
( )
nm
h
W
is not harmonically continued at
any point outside
W
.
The theorem 2 is proved.
Let
n
D
and
m
D
be domains from
n
and
m
,
n
n
D
O
,
m
m
D
O
respectively, an
opening subset,and
(
) (
)
nm
n
m
n
m
X
O D
D O
=
U
an open set of cross type. Denote by
€
nm
X
the
maximal open set such that
€
nm
nm
X
X
,
€
(
)
(
)
nm
nm
nm
nm
h X
h X
(see Theorem 1).
Theorem 3
. The domain
n m
+
W
is a domain of separate harmonicity if and only if for any
open set of
nm
X
type crosses belonging to
W
, its separate harmonic hulls also belong to
W
:
€
nm
nm
X
X
W
W
.
Corollary 1.
Any domain of type
n m
n
m
D D
+
is domain of separate harmonicity for the
class
(
)
nm
n
m
h D D
.
Corrolary 2.
Any convex domain
n m
+
W
is a domain of separate harmonicity for class
( )
nm
h
W
.
LITERATURE:
1.
Шабат Б.В. Введение в комплексный анализ. Часть II. Функции нескольких
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2.
Lelong P., Fonctions plurisousharmoniques et fonctions analytiques de variables reelles,
Ann. Inst. Fourier. Paris, 1961, vol. 11, pp. 515-562.
3.
Hecart Jean-Marc, Ouverts d`harmonicite pour les fonctions separement harmoniques,
Potential Anal., 2000, vol. 13, no. 2, pp. 115-126.
4.
Zeriahi A., Bases communes dans certains espaces de fonctions harmoniques
et fonctions separement harmoniques sur certains ensembles de
n
, Ann. Fac.
Sci. Toulouse. Math., 1982, ser. 5, vol. 4. pp. 75-102.
5.
Захарюта В.П. Сеператно аналитические функции, обобщённые теоремы
Хартогса и оболочки голоморфности // Мат. сб. — 1976. —Т. 101, №1. —
С. 57-76.
6.
Sadullaev A., Imomkulov S. A., Extension of holomorphic and pluriharmonic
functions with subtle singularities on parallel sections, Proc. Steklov Inst.
Math.,253(2006), pp. 144-159.
7.
Садуллаев А.С. Плюрисубгармонические функции // Итоги науки и
техники. Современные проблемы математики. Фундаментальные
направления. — М.: ВИНИТИ. — 1985. — Т. 8. — С. 65-111.
8.
Sachiko Hamano, Hartogs-Osgud theorem for separately harmonic functions,
Proc. Japan Acad., 83, Ser. A (2007), pp. 16-18.
9.
Имомкулов С.А., Абдикадиров С.М. Продолжение сепаратно-
гармонических функции вдоль фиксированного направления // Научный
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