Theorem on uniform continuity of Logarithmic potential Текст научной статьи по специальности «Философия, этика, религиоведение»

ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ

THEOREM ON UNIFORM CONTINUITY OF LOGARITHMIC

POTENTIAL

Nguyen Van Quynh Email: [email protected]

Nguyen Van Quynh - Candidate of Physical and Mathematical Sciences, Senior Lecturer, FACULTY OF FUNDAMENTAL SCIENCE, INDUSTRY UNIVERSITY, HANOI, SOCIALIST REPUBLIC OF VIETNAM

Abstact: logarithmic potential J ln|£ — z | dy(£) is important in the theory of

subharmonic and S -subharmonic functions. Classical properties were presented in many monographs, for example in [1]. Not that AF Grishin and A. Chouigui studied the case where the measure у is the restriction of the Lebesgue measure in the half-plane. The

paper considers the case: measure у in the plane. For any zeC we consider the Logarithmic potential as an element of the spaces Lp {у, C). In this article we give a

sufficient condition on a measure у the function J*In | £ — z \ dy(£) G Lp(y, C) to be

uniformly continious in the parameter z in С. We give example of measures у, which satisfy this condition.

Keywords: logarithmic potential, Borel measure, uniform continuity, Minkowski inequality, Lebesgue measure.

ТЕОРЕМА О РАВНОМЕРНОЙ НЕПРЕРЫВНОСТИ ЛОГАРИФМИЧЕСКОГО ПОТЕНЦИАЛА Нгуен Ван Куинь

Нгуен Ван Куинь - кандидат физико-математических наук, старший преподаватель, факультет фундаментальной науки, Университет промышленности, г. Ханой, Социалистическая Республика Вьетнам

Аннотация: логарифмический потенциал J ln | £ — z | dy(£) играет важную роль в

теории субгармонических и S -субгармонических функций. Классические свойства были представлены во многих монографиях, например в [1]. Отметим, что А.Ф. Гришин и А. Шуиги изучили случай, когда мера у есть ограничение меры Лебега на полуплоскости. В работе рассматривается случай: мера у в плоскости. Для любого ze С мы рассматриваем логарифмический потенциал как элемент пространств Lp {/', С). В этой статье приводится достаточное условие на меру у для того,

чтобы функция j*ln | £ — Z \ dy(£) G Lp(y,C) была равномерно непрерывной по

параметру Z в С. Приводится пример конкретных мер у, которые удовлетворяют приведенному условию.

Ключевые слова: логарифмический потенциал, борелевская мера, равномерная непрерывность, неравенство Минковского, мера Лебега.

УДК 517.518.14

The study of the theory of potential and related questions of mathematical physics since the nineteenth century has been the focus of mathematicians. In particular, in the study of subharmonic and 8 -subharmonic functions, reasoning using the methods of potential theory plays an important role.

Assumptions in the present paper can be considered as variants of some theorems from monographs of N.S.Landkof [1] and V.S. Azarin [2]. Note also the article of A.F. Grishin, Nguyen van Quynh, and I. Podiedtseva [3], where the theorem on the representation of 8 -subharmonic functions of finite order in the form of Logarithmic potentials is proved. And also the article of A.F. Grishin and A. Chouigui [6], in which various types of convergence of sequences of 8 -subharmonic functions. The results of our article allow us to simplify the constructions from these articles somewhat.

We will use the following notation:

B(z0,R) = {z e C: \z-z0\< R};CB(z0,R) = C/B(z0,R).

In the theory of functions that are subharmonic and 5-subharmonic in the plane C, an important role is played by the Logarithmic potential

K(z) = \\n\C-z\dy(C),

c

Where ;/ is a positive finite Borel measure with compact support that satisfies the condition supp;/ cz C .

In this paper, we consider this Logarithmic potential K( z ) as a map from the space C into the space Lp (C; dy(^j). In this case, we can write

We state several well-known results obtained in this area. We have already mentioned H'ormander's proposition [7]. H'ormander considered the multidimensional case; we consider the case where the dimension of the space is m = 2 , although the requirement m = 2 is not essential in many of our arguments. It is important for what follows that

the dependence of K(z) on z be uniformly continuous.

Theorem 1. Let p > 1 be an arbitrary fixed number. Suppose that ;/ is a positive finite Borel measure with compact support that satisfies the condition supp ;/ cz C . In addition, suppose that

sup| J \\n\z-C\\p dy(C)\zGC ^0; (1)

[b(z-s) J

(8^ 0).

Then the function K(z) : C —> Lp (y) is uniformly continuous with respect to the

variable z in the space C.

Proof. We divide the proof into several stages.

1. We set

= J" Iln I zi-С I -ln I -С I (dy(£)

vc

F{z) = [\\\n\z-CWP dy{C)

Since the Logarithmic potential as a function from C into M , is not bounded, we need to prove the convergence of the integrals introduced above. We do this at the first stage. It follows from the Minkowski inequality (see ) that

f y p / y p

F(z) <

J |ln|z-\| | pdy(C)

V B (z;5)

J |ln|z-\| | pdy(C)

V CB(z;S)

= J + J 2

where ( 7i( z; 8) = C \ /i(z; c>) . we prove that the inequality 1

|ln|z-\ || < ln

£

(2)

holds for sufficiently small 8 and for C, e (supp y) f] C6(z; c)). From the inequality

1

(2) we obtain that J2 < ln —(/(C)) P . By the hypothesis of the theorem, ./, < 1 for

8

sufficiently small 8 . Hence, l' (z) is finite. What has been proved can also be stated as follows: for any zeC the function K{z) is an element of the space L (y). The inequality y/(z{, z2 ) < F(^ ) + F(z2 )

implies that y/(zx; z2 ) is finite. 2. we prove that the inequality

V, ln | z-C I <-.

I C 1 i s

holds for |z -f \ >S.

Let z = * + iy; C = V +

(3)

we

have

(ln| -

X-ц

; (ln| - W

У

From these estimates it is easy to obtain inequality

VC ln| z-CI=f( ln| z-C\)]2 +f( In I z-C\)'

- - 1 <1 " " | z-C | ~8 '

Thus the inequality (3) is proved.

3. This is the most essential part of the proof. We claim that ¥(z'; zo) — 0 (z —

Let 8 be an arbitrary number. We assume that the inequality |z — z0| <8 holds. Consecutive application of the Minkowski inequality [4] and the inclusion B(z0 ;28) c B(z;38) yields

i уp f Y/p

j jlnjz-^j | pdy(C) + j jlnjz -a- lnizo-^" d^)

у CB( z;3S)

Г V/p

v

CS( z0;2S)

+

J^ I J 2 I J*

j |ln|Zo — C| | Pdy(C)

yCB(zo;28) ,

If B(z0 ;28) and |z — z0| <8, then the inequalities |w—\|>8 and

ln | z — C | | < 1 hold for any w e B(z0;|z — z01) . The estimate of the gradient implies that n\z-C\-ln\z0-C\\<^\z-z0\;J2<^(/(C))Vp\z-z0\. From the hypothesis of the theorem we obtain that

J ^ 0; J3 ^ 0 (8^ 0),

which proves the required assertion. Thus the theorem is proved. In this theorem we consider the case where the dimension of the space is m = 2, although the requirement m = 2 is not essential in many of our arguments. It is important for what follows theorem:

Theorem 2. Let p > 1 be an arbitrary fixed number. Suppose that y is a positive finite

Borel measure with compact support that satisfies the condition supp y c: K'" (in > 3). In addition, suppose that

sup J (Hx-jp^/WiyGr Uo; (4)

{B(r,S) J

(8^ 0).

Then the function — j j| : M™ —> Lp (y) is uniformly continuous with respect to

the variable y in the space Mm

Note that the proof of this theorem can be carried out along the lines of proof of the preceding theorem.

Example. Let y be a Lebesgue measure in the space C. We will use the standard notation dy = dxdy. Consider the integral in the condition (1) and, making a parallel translation, we obtain the equality for \ = x + iy:

J = J |ln |z — \\P dxdy = J |ln |C||P dxdy.

B ( z;8) B(0;8)

f x = r cos^

We introduce polar coordinates: \ .

[ y = r sin^

2n 8 8

We obtain the equality: J = J Jr|ln r|P drdq = 2njr|ln r^ dr.

0 0

0

It is obvious that condition holds for arbitrary p :

J ^ 0; (S^ 0) .

References / Список литературы

1. Landkof N.S., 1966. Foundations of modern potential theory. GRFML. Science (Russian).

2. Azarin V.S., 2009. Growth theory of subharmonic functions Birkhanser. Basel. Boston. Berlin.

3. Grishin A.F., Nguyen Van Quynh, Poedintseva I.V., 2014. Representation theorems of S -subharmonic Functions, Visnyk of V.N.Karazin Kharkiv National University. Ser. "Mathematics, Applied Mathematics and Mechanics. № 1133. P. 56-75. (Russian).

4. Kadets V.M., 2006. A course of Functional Analysis, Kharkov National University (Russian).

5. Van Quynh Nguyen, 2015. Various Types of Convergence of Sequences of Subharmonic Functions, Zh. Mat. Fiz. Anal. Geom,Volume 11. № 1. 63-74.

6. Grishin A.F., Chouigui А., 2008. Various types of convergence of sequences of S -subharmonic functions, Math. sbornyk. № 199. 27-48.

7. Hormander L., 1983. The Analysis of Linear Partial Di®erential Operators. II. Differential Operators with Constant Coefficients. Springer-Verlag, Berlin, Heidelberg. New York. Tokyo, 1983.