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q-Hardy-Littlewood-Type Maximal Operator with Weight Related to Fermionic p-Adic q-Integral on Z_{p}

**Erdoğan Şen**^{1}, **Mehmet Acikgoz**^{2}, **Serkan Araci**^{3,}

^{1}Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, Tekirdağ, Turkey

^{2}Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, Gaziantep, Turkey

^{3}Atatürk Street, 31290 Hatay, TURKEY

### Abstract

The q-extension of Hardy-littlewood-type maximal operator in accordance with q Volkenborn integral in the p-adic integer ring was recently studied . A generalization of Jang's results was given by Araci and Acikgoz . By the same motivation of their papers, we aim to give the definition of the weighted *q*-Hardy-littlewood-type maximal operator by means of fermionic p-adic q-invariant distribution on Z_{p}. Finally, we derive some interesting properties involving this-type maximal operator.

**Keywords:** fermionic p-adic q-integral on Z_{p}, hardy-littlewood theorem, p-adic analysis, q-analysis

*Turkish Journal of Analysis and Number Theory*, 2013 1 (1),
pp 4-8.

DOI: 10.12691/tjant-1-1-2

Received September 14, 2013; Revised September 21, 2013; Accepted September 23, 2013

**Copyright:**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Şen, Erdoğan, Mehmet Acikgoz, and Serkan Araci. "q-Hardy-Littlewood-Type Maximal Operator with Weight Related to Fermionic p-Adic q-Integral on Z
_{p}."*Turkish Journal of Analysis and Number Theory*1.1 (2013): 4-8.

- Şen, E. , Acikgoz, M. , & Araci, S. (2013). q-Hardy-Littlewood-Type Maximal Operator with Weight Related to Fermionic p-Adic q-Integral on Z
_{p}.*Turkish Journal of Analysis and Number Theory*,*1*(1), 4-8.

- Şen, Erdoğan, Mehmet Acikgoz, and Serkan Araci. "q-Hardy-Littlewood-Type Maximal Operator with Weight Related to Fermionic p-Adic q-Integral on Z
_{p}."*Turkish Journal of Analysis and Number Theory*1, no. 1 (2013): 4-8.

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### 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)

**Proo****f**. (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.

**T****heorem 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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In article | |||

[2] | S. Araci, M. Acikgoz and E. Şen, On the extended Kim's p-adic q-deformed fermionic integrals in the p-adic integer ring, Journal of Number Theory 133 (2013) 3348-3361. doi: | ||

In article | CrossRef | ||

[3] | T. Kim, Lebesgue-Radon-Nikodym theorem with respect to fermionic p-adic invariant measure on Z_{p}, Russ. J. Math. Phys. 19 (2012). | ||

In article | |||

[4] | T. Kim, Lebesgue-Radon-Nikodym theorem with respect to fermionic q-Volkenborn distribution on μ_{p}, Appl. Math. Comp. 187 (2007), 266-271. doi: | ||

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[5] | T. Kim, S. D. Kim, D.W. Park, On Uniformly differntiabitity and q-Mahler expansion, Adv. Stud. Contemp. Math. 4 (2001), 35-41. | ||

In article | |||

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[8] | T. Kim, Note on Dedekind-type DC sums, Advanced Studies in Contemporary Mathematics 18(2) (2009), 249-260. | ||

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In article | |||

[12] | K. Hensel, Theorie der Algebraischen Zahlen I. Teubner, Leipzig, 1908. | ||

In article | |||

[13] | N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta Functions, Springer-Verlag, New York Inc, 1977. | ||

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