1. Introduction

Let be dimensional Euclidean space,

Note that the quantity

is called harmonic oscillation (see, for instance, [1], [2]). In work [2] it has proven that

where

denotes the volume of ball and constants in the relation depend only on dimension (For positive functions and we will use the notation if there exist positive constants and such that

Let

where is a characteristic function of the set It is easy to see that

In the papers of some other authors (see e.g. [3], [4]) the quantity

is chosen as a characteristic to determine homogeneous classes of Besov. We can write the quantity in the following form

where

It is obvious that

Thus,

Hence we have

In the present paper, for the principal characteristic we take the quantity

where is said to be -oscillation of the function in the ball [2].

It is known that maximal functions measuring smoothness play an important role in the study of properties of integral operators and other objects of Harmonic Analysis. The main topic of this paper is the study of certain generalized maximal function (-maximal function) measuring smoothness.

The paper is organized as follows. Section 2 has auxiliary character and presents the basic definitions, some notation and well-known facts. In section 3 the relations between maximal function and metric characteristic are investigated and some useful inequalities were obtained. In section 4 estimations between -maximal function and maximal function was obtained. The main results are given in Propositions 3.1, 3.3, 4.1, 4.3 and 4.4.

2. Some Definition and Auxiliary Facts

Let the function be defined on the set takes only positive values, and monotone increases with respect to the argument on the interval We denote the class of all functions with the above mentioned properties by

Let Let’s introduce the following Φ-maximal function

We also introduce the following metric Φ-characteristic

Consider the known special cases of the introduced maximal function

1) If then where is the maximal function which is introduced in the paper [5].

2) If then The maximal function was mentioned in the papers [6], [7]. In paper [8] the function was investigated.

3) If then the maximal function may be found in the papers [9], [10], [11], [12], [13] and others.

Now let’s consider special cases of metric Φ-characteristic

1) If then (see section 3), where

Note that the function was first introduced in the paper [14] (see also, [15], [16]).

2) Let where is the Poisson kernel, i.e.

where Global variant of the characteristic (more precisely, the equivalent characteristic to it which is called a modulus of harmonic oscillation) for periodic functions of one variable may be found in the paper [1].

It is known that Hardy-Littlewood’s maximal function is determined by the equality

For case of the following maximal function is also considered [17]

It is easy to see that if then

From the definition of a maximal function it follows that

Thus,

(2.1)

It is known that (see e.g. [18]) if then

Hence, from (2.1) we get

The last relation means that the operator is the operator of the type for

It is also known [18] that if then there exists a number such that for any

where denotes the Lebesgue measure of the set Hence, from (2.1) we get

Thus, if then there exist the number such that for any

The last relation means that the operator is the operator of weak type (1,1).

In the case we denote the function by Then for the function we have

where Thus,

(2.2)

From the inequality (2.2), the Hardy-Littlewood maximal theorem and theorem 2 of chapter 3 [18] we get the following facts.

If then for

and for we have

where the positive constant is independent on and

Thus, at the indicated conditions on the function the operator is the operator of type for and is also weak type (1,1) operator.

3. Relations between Maximal Function and Metric Characteristic. Some Inequalities

In this section we’ll assume that

Proposition 3.1. If then the following equality is satisfied

(3.1)

Proof: From the definition of the function we get

(3.2)

On the other hand, for any and we have

Hence it follows that

therefore

So,

From the last inequality we get

(3.3)

Equality (3.1) is obtained from inequalities (3.2) and (3.3).

Lemma 3.1. Let and

(3.4)

Then for any constant the following inequality is true

(3.5)

where the positive constant depends only on the dimension and on the quantity

Proof. Let be any constant. Then we have

Thus, for all and

(3.6)

On the other hand, by means of condition (3.4) we get

Hence

(3.7)

Using this inequality, from (3.6) we get

Proposition 3.2. Let and condition (3.4) be satisfied. Then the following inequality is true

(3.8)

Proof. If we take, then the validity of inequality (3.8) is obtained from relation (3.7).

Remark 3.1. Note that for the function satisfying condition (3.4) we can take, for instance, the following functions:

1)

2)

3) where

Verify, that if then

(3.9)

Indeed, if then

Therefore for this function we have

Hence, equality (3.9) is obtained. We note that the quantity

is said to be mean oscillation of the function in the ball

Remark 3.2. In the case of the quantity

is called a harmonic oscillation of the function (see [1]). In the paper [2] it has been proven that

where the constants in the relation depend only on the dimension Hence it is obtained that

where

(see [2]).

Let's show that the relation

(3.10)

takes place for wider class of functions

Proposition 3.3. Let and condition (3.4) is satisfied. Then the relation (3.10) is true, where the constants in the relation depend only on the constant and dimension

Proof. For convenience we will introduce the following notations:

Then we get

Thus

(3.11)

On the other hand,

Thus

(3.12)

Inequalities (3.11) and (3.12) prove the required relation (3.10).

4. Estimations of -Maximal Functions by Maximal Functions

Proposition 4.1. If and the function satisfies condition (3.4), then the following inequality is true

(4.1)

where and is а constant from inequality (3.4).

Proof. By means of Proposition 3.1 and inequality (3.8) we get

Proposition 4.2. [2]. Let Then the following inequality is true

(4.2)

where the constant is independent on and

Lemma 4.1. If is a non-negative, monotone increasing function on the interval and

then the function

also monotone increases on interval

Proof. Let and Then we have

i.e.

Proposition 4.3. Let

and

(4.3)

Then the following inequality holds

(4.4)

where and the positive constant does not depend on and

Proof. By means of relations (3.1), (4.2) and (4.3), we have

Corollary 4.1. Let and

(4.5)

Then the following inequality holds

(4.6)

where the positive constant is independent on and

Proof. If condition (4.5) is satisfied, then by virtue of proposition 4.3 the inequality (4.4) holds. Furthermore, from a condition (4.5) follows that

Taking this into account, we have

where is a constant from inequality (4.4).

Proposition 4.4. Let and

(4.7)

Then there exists a function such that for any function

Proof. Consider the function

In the paper [14] it is shown that

From the last inequality it follows that

Further, for we have

where denotes the area of the surface of a unit sphere Thus, for the equality is true. Therefore, for any function

Corollary 4.2. Let

and condition (4.5) satisfied. Then there exist the numbers such that

where the constants and are independent on and

Now consider the case of the function where is a Poisson kernel. It is easy to see that there exist the numbers such that for all the relation.

holds. That is where Hence it follows that if and then the following relations are true

By means of these considerations, from corollary 4.2 we get

Corollary 4.3. Let be a Poisson kernel, and

Then the following relation is true

5. Conclusions

Maximal functions play an important role in the study of differentiation of functions, almost everywhere convergence of singular integrals, mapping properties of singular integral operators and potential type integral operators.

Maximal functions measuring smoothness are useful in the study of smoothness of functions and the mapping properties of various operators of Harmonic Analysis on smoothness spaces.

The main theme of this paper is to study certain maximal functions and Φ-maximal functions measuring smoothness. Relations between maximal and Φ-maximal functions measuring smoothness are studied. These relations allow to unite and compare the results received in terms of various characteristics.