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

Erdoğan Şen, Mehmet Acikgoz, Serkan Araci*

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

Keywords: fermionic p-adic q-integral on Zp, hardy-littlewood theorem, p-adic analysis, q-analysis

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

Citation: Erdoğan Şen, Mehmet Acikgoz, 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). 4-8. January 2013. doi:10.12691/tjant-1-1-2

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:

[picture] | (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:

[picture] | (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 .

[picture] | (1.3)

where

[picture] | (1.4)

and

[picture] | (1.5)

By (1.5), we have

[picture] | (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

[picture] | (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

[picture] | (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

[1]   | S. Araci and M. Acikgoz, A note on the weighted q-Hardy-littlewood-type maximal operator with respect to q-Volkenborn integral in the p-adic integer ring, J. Appl. Math. & Informatics, Vol. 31(2013), No. 3-4, pp. 365-372.
[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.
[3]   | T. Kim, Lebesgue-Radon-Nikodym theorem with respect to fermionic p-adic invariant measure on Zp, Russ. J. Math. Phys. 19 (2012).
[4]   | T. Kim, Lebesgue-Radon-Nikodym theorem with respect to fermionic q-Volkenborn distribution on μp, Appl. Math. Comp. 187 (2007), 266-271.
[5]   | T. Kim, S. D. Kim, D.W. Park, On Uniformly differntiabitity and q-Mahler expansion, Adv. Stud. Contemp. Math. 4 (2001), 35-41.
[6]   | T. Kim, q-Volkenborn integration, Russian J. Math. Phys. 9 (2002) 288-299.
[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.
[8]   | T. Kim, Note on Dedekind-type DC sums, Advanced Studies in Contemporary Mathematics 18(2) (2009), 249-260.
[9]   | T. Kim, A note on the weighted Lebesgue-Radon-Nikodym Theorem with respect to p-adic invariant integral on Zp, J. Appl. Math. & Informatics, Vol. 30(2012), No. 1, 211-217.
[10]   | T. Kim, Non-archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials, Russ. J. Math Phys. 10 (2003) 91-98.
[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.
[12]   | K. Hensel, Theorie der Algebraischen Zahlen I. Teubner, Leipzig, 1908.
[13]   | N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta Functions, Springer-Verlag, New York Inc, 1977.