1. Introduction
The concept of padic 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 padic qintegral in the padic integer ring was originally constructed by Kim ^{[2, 6]} who introduced LebesgueRadonNikodym Theorem with respect to fermionic padic qintegral on . The fermionic padic qintegral on is used in mathematical physics for example the functional equation of the qzeta function, the qstirling numbers and qmahler theory of integration with respect to the ring together with Iwasawa's padic qL function.
In ^{[11]}, Jang defined qextension of HardyLittlewoodtype maximal operator by means of qVolkenborn integral on . Afterwards, in ^{[1]}, Araci and Acikgoz added a weight into Jang's qHardyLittlewoodtype maximal operator and derived some interesting properties by means of Kim's padic qintegral on . Now also, we shall consider weighted qHardyLittlewoodtype maximal operator on the fermionic padic qintegral on . Moreover, we shall analyse qHardyLittlewoodtype maximal operator via the fermionic padic qintegral on .
Assume that p be an odd prime number. Let be the field of padic 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 padic absolute value , is normally defined by
where with and .
A padic Banach space is a Q_{p}vector space with a lattice (Z_{p}module) separated and complete for padic 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 padic 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, qVolkenborn integral was defined by T. Kim as follows:
 (1.1) 
where is a qextension 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 qVolkenborn integral, we consider below strongly padic qinvariant 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 padic qintegral 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 qpolynomial. Now also, we indicate that is a strongly weighted fermionic padic qinvariant 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 supnorm
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 qHardy Littlewoodtype maximal operator on the fermionic padic qintegral on . Also, we show that the boundedness of the weighted qHardylittlewoodtype maximal operator in the padic integer ring.
2. The Weighted qHardyLittlewoodType Maximal Operator
In view of (1.2) and the definition of fermionic padic qintegral on , we now consider the following theorem.
Theorem 1. Let be a strongly fermionic padic qinvariant 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 qHardylittlewoodtype maximal operator related to fermionic padic qintegral on with a strong fermionic padic qinvariant distribution in the padic integer ring.
Definition 1. Let be a strongly fermionic padic qinvariant distribution on and . Then, qHardylittlewoodtype maximal operator with weight related to fermionic padic qintegral on is defined as
for all .
We recall that famous Hardylittlewood 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 HardyLittlewood 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 qextension of the classical HardyLittlewood maximal operator in the space of padic 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 supnorminequality for the qHardyLittlewoodtype 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 qHardyLittlewoodtype maximal operator with weight on .
Corollary 1. is a bounded operator from into , where is the space of all padic supnormbounded functions with the
for all
References
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