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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|
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|
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