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DIFFERENTIAL PROPERTIES OF LOGARITHMIC POTENTIALS
Choriyev Shoxro`z Faxriddin o`g`li
(magistr)
National University of Uzbekistan, Tashkent, Uzbekistan;
https://doi.org/10.5281/zenodo.15796838
Abstract:
In this paper, Luzin’s
k
C
properties are proved. Logarithmic
potential in the space
n
, i.e. it is proved that outside some open set with small
Lebesgue measure, the logarithmic potential is
n
times continuously
differentiable.
Key words:
potential, measure, capacity, singular integral, theorem
Cartan, theorem Kishi, logarithmic potentials
Let us consider the logarithmic potential in the space
n
( )
ln
( ),
,
n
l
U
x
x
y d
y
x
where
is a finite Borel measure with compact support
n
S
.
In Cartan’s work [1] (see also [2, p. 231]) the following analogue of
Luzin’s
C
property (see [3, p. 309] ) was proved for Riesz potentials:
,
,0
.
n
n p
n
d
y
U
x
x
p
n
x
y
Theorem 1
(Cartan). For any
0
there exists an open set
n
O
with
capacity
(
)
n р
C
O
such that the potential
( )
n р
U
x
is continuous in the
complement
\
n
O
.
In [4] M. Kishi proved Cartan’s theorem in the case of a more general
potential
( )
( , )
( )
K
U
x
K x y d
y
,
n
x
.
Here, also the measure
with compact support and kernel
( , )
K x y
,
( , )
n
n
x y
, has the following properties:
1)
( , )
K x y
a positive-valued polycontinuous from below a
function such that
( , )
,
n
K x x
x
and continuous
in
(
) (
)
\
,
,
n
n
n
x x x
,
2)
( , )
K x y
is symmetric, i.e.
( , )
( , )
K x y
K y x
.
According to condition 2) the reciprocity law is always satisfied: for any вс
and
measure and with compact supports,
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K
K
U d
U d
.
A measure
is called admissible on a compact set
n
E
, if
S
E
and
1
K
U
x
everywhere in
n
. The family of all admissible measures on
E
is
denoted by
.
E
M
To each compact set
E
we associate a number
K
C
E
,
defined as sup
( )
E
for all
( ).
E
M
Using this set function
( )
K
C
E
, we define
the inner and outer capacities of an arbitrary set мы
M
as follows: the inner
capacity
(
)
i
K
C
M
is equal to
sup
( )
K
C E
for all compact sets
,
E
M
and the outer
capacity
(
)
e
K
C
M
is equal to
inf
( )
i
K
C
G
for all open sets
.
G
M
It follows
immediately that
( )
( )
i
K
K
C
E
C
E
for any compact set
E
,
(
)
(
)
i
e
K
K
C
M
C
M
for
any set
M
and
( )
( )
i
e
K
K
C
G
C
G
for any open set G. When
(
)
(
)
i
e
K
K
C
M
C
M
we
say that
M
is capacious and we denote by
M
the common value of these two
capacities by
(
)
K
C M
which call the capacity of
M
.
Theorem 2
(Kishi). For any
0
there exists an open set
n
O
with capacity
(
)
K
C O
such that the potential of the continuous
( )
K
U
x
complement
\
n
O
.
Note that in particular the potential
K
U
x
can be a Riesz potential or a
logarithmic potential with kernel
2
,
ln
,
,
0,
0,
,
R
K x y
x y
B
R
B
R
x
y
where
(0, )
:
n
B
R
x
x
R
(see, for example, [2],[4]).
In this work, we study the Luzin
k
C
properties of logarithmic potentials.
Main result. Theorem 3.
For any
0
there exists an open set
n
O
with Lebesgue measure
(
)
m O
such that the logarithmic potential
( )
l
U
x
belongs to the class
\
(
)
n
n
C
O
continuously differentiable functions.
Literature:
1.
Cartan H. Theori du Potential newtonien: energie, capacite, suites de
potentials. Bull. Soc. Math. France. 1945. V. 73. P. 74-106.b
2.
Landkof N.S. Foundation of modern potential theory (in Russian).
Мoskov,1966.
3.
Колмогоров А.Н. Фомин С.В. Элементы теории функции и
функционального анализа. 7-е издание.-М.: ФИЗМАТЛИТ, 2009.
ACADEMIC RESEARCH IN MODERN SCIENCE
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4.
Kishi M. Capacities of borelian sets and the continuity of potentials.
Nagoya Mathematical Journal. 1957. V. 12. P. 195-219.
