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

Erdoğan Şen, Mehmet Acikgoz, Serkan Araci

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

Erdoğan Şen1, Mehmet Acikgoz2, Serkan Araci3,

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

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

3Atatürk 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 Zp. Finally, we derive some interesting properties involving this-type maximal operator.

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 Zp." 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 Zp. 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 Zp." 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 Qp-vector space with a lattice (Zp-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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[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:
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[3]  T. Kim, Lebesgue-Radon-Nikodym theorem with respect to fermionic p-adic invariant measure on Zp, Russ. J. Math. Phys. 19 (2012).
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[4]  T. Kim, Lebesgue-Radon-Nikodym theorem with respect to fermionic q-Volkenborn distribution on μp, 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.
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[6]  T. Kim, q-Volkenborn integration, Russian J. Math. Phys. 9 (2002) 288-299.
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[7]  T. Kim, On a q-analogue of the p-adic log Gamma functions and related integrals, Journal of Number Theory 76 (1999), 320-329. doi:
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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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[10]  T. Kim, Non-archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials, Russ. J. Math Phys. 10 (2003) 91-98.
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[11]  L-C. Jang, On the q-extension of the Hardy-littlewood-type maximal operator related to q -Volkenborn integral in the p-adic integer ring, Journal of Chungcheon Mathematical Society, Vol. 23, No. 2, June 2010.
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[12]  K. Hensel, Theorie der Algebraischen Zahlen I. Teubner, Leipzig, 1908.
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[13]  N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta Functions, Springer-Verlag, New York Inc, 1977.
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