1. Introduction

Let be -dimensional Euclidean space of the points , and be a closed ball in of radius with the center at point . Denote by , , a class of all local -power summable functions defined on and by the class of all local bounded functions defined on . By we mean the usual Lebesgue space on , and we denote by the corresponding norm, that is

if , and .

Denote by the totality of all polynomials on whose degrees are equal to or less than .

Let, , , ( is the set of all positive integers). Define the following

functions

Let , , be non-negative integers, . Apply the orthogonalization process by the scalar product

to the system of the power functions , , arranged by partially lexicographic order¹ [1], where is the Lebesque measure of the set . Denote by the obtained orthogonal normed system.

Let . Suppose that ([2],[3]):

.

It is obvious that is a polynomial degree of which is equal or less than .

Denote

for . Let us call local oscillation of -th order of the function on the ball in the metric of .

Note that if then

,and therefore

.

It is known that (cf.[4]) for each polynomial and each ball the inequality

is true, where the positive constant does not depend on and . Hence it follows that

.

It should be mentioned that the theory of spaces defined by local oscillation has been developed by several authors, for instance F.John and L.Nirenberg[5], S.Campanato[6], N.G.Meyers[7], S.Spanne[8], J. Peetre[9], D.Sarason[10] etc. (see also[11],[12]).

Let , . We introduce the following denotations

, , , .

Let be a class of all positive monotonically increasing on functions. Let , . By we denote the set of all the functions , for which

, .

We introduce the norm in the space by the equality

.

If and , , , then , where is the Morrey space, i.e.

.

Let is a positive number. We denote by a set of all such that almost decreases on .

If , , then we denote by the set of all functions such that , where

.

If we consider the class as a subset in the quotient space , then is the norm on . In the introduced norm the space is a Banach space.

If , , then we will denote by the class of all the functions for which the following relation

is valid.

We define the norm on by the equality

.

In particular, if then , where is the space of all local summable functions of bounded mean oscillation. The class for the first time was introduced in[5].

It is easy to see that if , , , then and their norms are equivalent.

Consider also a class which was introduced in[10]: is the class of all for which the relation

is valid. For we define .

Let . By we mean the weak Lebesgue space on , and we denote

.

Potential type integrals play an important role in the mathematical analysis. For the properties of Riesz potentials in terms of mean oscillations we refer the readers to[9],[4],[16] and the related papers for further information.

In this paper we study the properties of Riesz potential of a function in terms of local oscillation of functions when belongs to , or general Morrey type spaces.

The structure of the paper is as follows. In section 2 some inequalities and embedding theorems is proved. The mean results of the paper are given in Theorems 5, 6 and 7, which was proved in section 3.

2. Some Inequalities and Embedding Theorems

Proposition 1. Let , . Then the inequality

is true, where the constant depends only on , and .

Proof. Let . Applying the Hölder’s Inequality we obtain

.

Therefore

, .

The case is obvious.

Corollary 1. If , , , then the inequality

, , is true.

Proposition 2. Let . Then there exists a function such that

(1)

where the constant is independent of and .

Proof. Let and , . Then we have

,

, and similarly

, , where is the unit sphere, is a surface area of . From here we obtain the inequality (1).

Proposition 3. Let , . Then and there exists a constant independent of , such that

(2)

for all and .

Proof. It follows from well known equality (see[17], Chapter 1, Lemma 4.1) that (for any positive constant )

.

Now choosing we obtain

From here we obtain the inequality (1) with .

Remark 1. In the case the inequality (2), in general, is not true. For instance, the function belong to , but .

Proposition 4. Let , , . Then and there exists a constant , depends only on , , , such that

(3)

for all .

Proof. We have

.

Therefore

.

This means that and .

Further we obtain

, ,

where is the unit sphere, is a surface area of . From here

, .

The previous proposition shows that it is impossible to improve the estimation (2).

Proposition 5. Let , . Then the inequality

(4)

is true, where a constant is independent of , .

Proof. At first we consider the case . Then we have

,

where . Further, applying the Minkowski’s Inequality we have

,

where is the volume of the unit ball .

In the case the proof of inequality (4) is obvious.

The following proposition shows that it is impossible to improve the estimation (4) in the case .

Proposition 6. Let . Then there exists the function such that for the inequality

(5)

is true, where a constant is independent of and .

Proof. Let

.

If , , , then we obtain

.

If , , then . Besides, for all ,

.

Thus the function is integrable on the with respect to argument . Further we obtain that if , then

.

It is obvious that

Therefore we obtain that

, .

In the case the arguments are similar.

With help of Proposition 3, Proposition 5 and Corollary 1 we obtain correspondingly, the following theorems.

Theorem 1. Let . Then and

, where , .

Theorem 2. Let , . Then and

Theorem 3. Let , and . Then and

: , where , .

3. Properties of Riesz Potentials

Consider the following potential type integral operator

where

are non-negative integers,

,

,

is the characteristic function of the set .

Operator is a certain modification of the Riesz potential

.

It should be noted that if and , then the integral differs from integral by a polynomial power of which is equal or less than . If , then the potential is defined not for all functions .

Moreover, if and , for example, then for integral absolutely converges almost everywhere.

Note that modified Riesz potential similar to the was considered, for example, in T.Kurokawa[13], T.Shimomura and Y. Mizuta[14] etc. (see also[15]).

The following assertion holds true.

Theorem 4[16]. Let , , , and

.

Then the inequality

(6)

is valid, where , and constant does not depend on , and .

Corollary 2. Let , , , and

.

Then inequality

(7)

is valid, where , and constant does not depend on and .

Theorem 5. Let , . If , then and

where , .

Proof. With help of Proposition 3 and Theorem 4 we have

where . From here the theorem statement easily turns out.

Corollary 3. Let , , , , . Then

Corollary 4. Let , , . Then and

Corollary 5. If , ,

, then , where , and

.

Theorem 6. Let , , , , . Then and

where , . In the case , in addition , , where .

Proof. Taking into account the inequalities (4) and (7), it is easy to obtain the inequality

(8)

where a constant is independent of , . From here we obtain that

If , , then . Therefore if , and , then from estimation (7), in addition, we have .

Corollary 6. Let , , , . Then and

,

where , . In the case , in addition, we have .

Corollary 7. Let . Then and

.

Corollary 8. Let , , If , then and

,

where . If , then in addition, .

Theorem 7. Let , , , and

.

If , then and

, where , .

Let , . Then and . From Theorem 7 we obtain that if , , , , then , i.e. the operator boundedly acts from into .

Corollary 9. If , then .

Corollary 10. If , then .

Corollary 11. If , , , then , where .

Notes

1. It means that precedes if either , or but the first nonzero difference is negative.