1. Introduction

The concept of p-adic numbers was originally invented by Kurt Hensel who is German mathematician, around the end of the nineteenth century ^[12]. In spite of their being already one hundred years old, these numbers are still today enveloped in an aura of mystery within the scientific community and also play a vital and important role in mathematics.

The fermionic p-adic q-integral in the p-adic integer ring was originally constructed by Kim ^{[2, 6]} who introduced Lebesgue-Radon-Nikodym Theorem with respect to fermionic p-adic q-integral on . The fermionic p-adic q-integral on is used in mathematical physics for example the functional equation of the q-zeta function, the q-stirling numbers and q-mahler theory of integration with respect to the ring together with Iwasawa's p-adic q-L function.

In ^[11], Jang defined q-extension of Hardy-Littlewood-type maximal operator by means of q-Volkenborn integral on . Afterwards, in ^[1], Araci and Acikgoz added a weight into Jang's q-Hardy-Littlewood-type maximal operator and derived some interesting properties by means of Kim's p-adic q-integral on . Now also, we shall consider weighted q-Hardy-Littlewood-type maximal operator on the fermionic p-adic q-integral on . Moreover, we shall analyse q-Hardy-Littlewood-type maximal operator via the fermionic p-adic q-integral on .

Assume that p be an odd prime number. Let be the field of p-adic rational numbers and let be the completion of algebraic closure of .

Thus,

Then is an integral domain to be

or

In this paper, we assume that with as an indeterminate.

The p-adic absolute value , is normally defined by

where with and .

A p-adic Banach space is a Q_p-vector space with a lattice (Z_p-module) separated and complete for p-adic topology, ie.,

For all , there exists , such that . Define

It satisfies the following properties:

Then, defines a norm on such that is complete for and is the unit ball.

A measure on with values in a p-adic Banach space is a continuous linear map

from , (continuous function on ) to . We know that the set of locally constant functions from to is dense in so.

Explicitly, for all , the locally constant functions

Now if , set . Then is given by the following Riemann sums

T. Kim defined as follows:

and this can be extended to a distribution on . This distribution yields an integral in the case .

So, q-Volkenborn integral was defined by T. Kim as follows:

(1.1)

where is a q-extension of which is defined by

Note that cf. ^{[1, 2, 4, 5, 6, 7, 11]}.

Let be a fixed positive integer with . We now set

where satisfies the condition . For ,

see ^[10]

By means of q-Volkenborn integral, we consider below strongly p-adic q-invariant distribution on in the form

where as and is independent of . Let , for any , we assume that the weight function is defined by where with . We define the weighted measure on as follows:

(1.2)

where the integral is the fermionic p-adic q-integral on . From (1.2), we note that is a strongly weighted measure on . Namely,

Thus, we get the following proposition.

Proposition 1. For , then, we have

where are positive constants. Also, we have

where is positive constant.

Let be an arbitrary q-polynomial. Now also, we indicate that is a strongly weighted fermionic p-adic q-invariant measure on . Without a loss of generality, it is sufficient to evidence the statement for .

(1.3)

where

(1.4)

and

(1.5)

By (1.5), we have

(1.6)

By (1.3), (1.4), (1.5) and (1.6), we have the following

For , let and , where , with and

Then, we procure the following

where is positive constant and .

Let be the space of uniformly differentiable functions on with sup-norm

The difference quotient of is the function of two variables given by

for all

A function is said to be a Lipschitz function if there exists a constant such that

The linear space consisting of all Lipschitz function is denoted by . This space is a Banach space with the respect to the norm (for more information, see ^{[3, 4, 5, 6, 7, 8, 9]}). The objective of this paper is to introduce weighted q-Hardy Littlewood-type maximal operator on the fermionic p-adic q-integral on . Also, we show that the boundedness of the weighted q-Hardy-littlewood-type maximal operator in the p-adic integer ring.

2. The Weighted q-Hardy-Littlewood-Type Maximal Operator

In view of (1.2) and the definition of fermionic p-adic q-integral on , we now consider the following theorem.

Theorem 1. Let be a strongly fermionic p-adic q-invariant on and . Then for any and any , we have

(1)

(2)

Proof. (1) By using (1.1) and (1.2), we see the following applications:

(2) By the same method of (1), then, we easily derive the following

Since for our assertion follows.

We are now ready to introduce the definition of the weighted q-Hardy-littlewood-type maximal operator related to fermionic p-adic q-integral on with a strong fermionic p-adic q-invariant distribution in the p-adic integer ring.

Definition 1. Let be a strongly fermionic p-adic q-invariant distribution on and . Then, q-Hardy-littlewood-type maximal operator with weight related to fermionic p-adic q-integral on is defined as

for all .

We recall that famous Hardy-littlewood maximal operator , which is defined by

(2.1)

where is a locally bounded Lebesgue measurable function, is a Lebesgue measure on and the supremum is taken over all cubes which are parallel to the coordinate axes. Note that the boundedness of the Hardy-Littlewood maximal operator serves as one of the most important tools used in the investigation of the properties of variable exponent spaces (see ^[11]). The essential aim of Theorem 1 is to deal mainly with the weighted q-extension of the classical Hardy-Littlewood maximal operator in the space of p-adic Lipschitz functions on and to find the boundedness of them. By means of Definition 1, then, we state the following theorem.

Theorem 2. Let and , we get

(1)

(2)

where

Proof. (1) Because of Theorem 1 and Definition 1, we see

(2) On account of (1), we can derive the following

Thus, we complete the proof of theorem.

We note that Theorem 2 (2) shows the supnorm-inequality for the q-Hardy-Littlewood-type maximal operator with weight on , on the other hand, Theorem 2 (2) shows the following inequality

(2.2)

where . By the equation (2.2), we get the following Corollary, which is the boundedness for weighted q-Hardy-Littlewood-type maximal operator with weight on .

Corollary 1. is a bounded operator from into , where is the space of all p-adic supnorm-bounded functions with the

for all

References

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