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A note on the Analogue of Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on Z p

Abstract

We give the analogue of the Lebesgue-Radon-Nikodym theorem with respect to a weighted p-adic q-measure on Z p . In a special case, when the weight q x is 1, we can derive the same result as Kim et al. (Abstr. Appl. Anal. 2011:637634, 2011). And if q=1, we have the same result as Kim (Russ. J. Math. Phys. 19:193-196, 2012).

MSC:11B68, 11S80.

1 Introduction

Let p be a fixed odd prime number. Throughout this paper, the symbols Z p , Q p , and C p denote the ring of p-adic integers, the field of p-adic rational numbers, and the p-adic completion of the algebraic closure of Q p , respectively. Let ν p be the normalized exponential valuation of C p with |p|= p ν p ( p ) = 1 p and ν p (0)=.

When one speaks of q-extension, q can be regarded as an indeterminate, a complex qC, or a p-adic number q C p . In this paper, we assume that q C p with |1q|<1, and we use the notations of q-numbers as follows:

[ x ] q =[x:q]= 1 q x 1 q and [ x ] q = 1 ( q ) x 1 + q .
(1.1)

For any positive integer N, let

a+ p N Z p = { x Z p | x a ( mod p N ) } ,
(1.2)

where aZ satisfies the condition 0a< p N (see [18]).

It is known that the fermionic p-adic q-measure on Z p is given by Kim as follows:

μ q ( a + p N Z p ) = ( q ) a [ p N ] q = 1 + q 1 + q p N ( q ) a ,(see [1, 6, 9–12]).
(1.3)

Let C( Z p ) be the space of continuous functions on Z p . From (1.3), the fermionic p-adic q-integral on Z p is defined by Kim as follows:

I q (f)= Z p f(x)d μ q (x)= lim N 1 [ p N ] q x = 0 p N 1 f(x) ( q ) x ,
(1.4)

where fC( Z p ) (see [1, 6, 912]).

Let us assume q C p with |q1|<1. By (1.4), we get

Z p q x e [ x ] q t d μ q (x)= n = 0 E n , q t n n ! ,
(1.5)

(see [7, 8, 13]) where E n , q are q-Euler numbers. The q-Euler polynomials, E n , q (x), are also defined by

Z p q y e [ x + y ] q t d μ y (t)= n = 0 E n , q (x) t n n ! .
(1.6)

By (1.5) and (1.6), we get

E n , q (x)= l = 0 n ( n l ) x n l E l , q = ( x + E q ) n ,

with the usual convention of replacing ( E q ) n by E n , q (see [1, 2, 7, 8, 13]),

E n , q = Z p q x [ x ] q n d μ q (x)= [ ι ] q ( 1 q ) n l = 0 n ( 1 ) l ( n l ) 1 [ l ] q .

We will give the analogue of the Lebesgue-Radon-Nikodym theorem with respect to a weighted p-adic q-measure on Z p . In a special case, when the weight q x is 1, we can derive the same result as Kim et al. [9]. And if q=1, we have the same result as Kim [4].

2 Lebesgue-Radon-Nikody-type theorem with respect to a weighted p-adic q-measure on Z p

For any positive integer a and n, with a< p n and fC( Z p ), let us define

μ ˜ f , q ( a + p n Z p ) = a + p n Z p q x f(x)d μ q (x),
(2.1)

where the integral is the fermionic p-adic q-integral on Z p .

From (1.3), (1.4), and (2.1), we note that

(2.2)

By (2.2), we get

μ ˜ f , q ( a + p n Z p ) = [ 2 ] q [ 2 ] q p n ( 1 ) a Z p q p n x f ( a + p n x ) d μ q p n (x).
(2.3)

Therefore, by (2.3), we obtain the following theorem.

Theorem 1 For f,gC( Z p ), we have

μ ˜ α f + β g , q =α μ ˜ f , q +β μ ˜ g , q ,
(2.4)

where α, β are constants.

From (2.2) and (2.4), we note that

| μ ˜ f , q ( a + p n Z p ) | M f q ,
(2.5)

where f q = sup x Z p | q x f(x)| and M is some positive constant.

Now, we recall the definition of the strongly fermionic p-adic q-measure on Z p . If μ q satisfies the following equation:

| μ q ( a + p n Z p ) μ q ( a + p n + 1 Z p ) | δ n , q ,
(2.6)

where δ n , q 0 and n and δ n , q is independent of a, then μ q is called a weakly fermionic p-adic q-measure on Z p .

If δ n , q is replaced by C p ν p ( 1 q n ) (C is some constant), then μ q is called a strongly fermionic p-adic q-measure on Z p .

Let P(x) C p [ [ x ] q ] be an arbitrary q-polynomial with a i [ x ] q i . Then we see that μ P , q is a strongly fermionic p-adic q-measure on Z p . Without loss of generality, it is enough to prove the statement for P(x)= [ x ] q k .

Let a be an integer with 0a< p n . Then we get

μ ˜ P , q ( a + p n Z p ) = [ 2 ] q [ 2 ] q p n ( 1 ) a lim m 1 [ p m n ] q p n i = 0 p m n 1 [ a + i p n ] q k ( 1 ) i q p n i ,
(2.7)

and

q p n i = l = 0 i ( i l ) [ p n ] q l ( q 1 ) l .

By (2.7), we easily get

μ ˜ P , q ( a + p n Z p ) [ 2 ] q [ 2 ] q p n ( 1 ) a [ a ] q k ( mod [ p n ] q ) [ 2 ] q [ 2 ] q p n ( 1 ) a P ( a ) ( mod [ p n ] q ) .
(2.8)

Let x be arbitrary in Z p with x x n (mod p n ) and x x n + 1 (mod p n + 1 ), where x n and x n + 1 are positive integers such that 0 x n < p n and 0 x n + 1 < p n + 1 . Thus, by (2.8), we have

| μ ˜ P , q ( a + p n Z p ) μ ˜ P , q ( a + p n + 1 Z p ) | C p ν p ( 1 q p n ) ,
(2.9)

where C is some positive constant and n0.

Let

f μ ˜ P , q (a)= lim n μ ˜ P , q ( a + p n Z p ) .
(2.10)

Then by (2.5), (2.7), and (2.8), we get

f μ ˜ P , q (a)= [ 2 ] q 2 ( 1 ) a [ a ] q k = [ 2 ] q 2 ( 1 ) a P(a).
(2.11)

Since f μ ˜ P , q (x) is continuous on Z p , it follows, for all x Z p ,

f μ ˜ P , q (x)= [ 2 ] q 2 ( 1 ) x P(x).
(2.12)

Let gC( Z p ). By (2.10), (2.11), and (2.12), we get

Z p g ( x ) d μ ˜ P , q ( x ) = lim n i = 0 p n 1 g ( i ) μ ˜ P , q ( i + p n Z p ) = [ 2 ] q 2 lim n i = 0 p n 1 g ( i ) ( q ) i [ i ] q k = Z p q x g ( x ) [ x ] q k d μ q ( x ) .
(2.13)

Therefore, by (2.13), we obtain the following theorem.

Theorem 2 Let P(x) C p [ [ x ] q ] be an arbitrary q-polynomial with a i [ x ] q i . Then μ ˜ P , q is a strongly fermionic weighted p-adic q-measure on Z p , and for all x Z p ,

f μ ˜ P , q = ( 1 ) x [ 2 ] q 2 P(x).
(2.14)

Furthermore, for any gC( Z p ), we have

Z p g(x)d μ ˜ P , q (x)= Z p q x g(x)P(x)d μ q (x),
(2.15)

where the second integral is a fermionic p-adic q-integral on Z p .

Let f(x)= n = 0 a n , q ( x n ) q be the q-Mahler expansion of a continuous function on Z p , where

( x n ) q = [ x ] q [ x 1 ] q [ x n + 1 ] q [ n ] q ! ( see [4] ) .
(2.16)

Then we note that lim n | a n , q |=0.

Let

f m (x)= i = 0 m a i , q ( x i ) q C p [ [ x ] q ] .
(2.17)

Then

f f m sup m n | a n , q |.
(2.18)

Writing f= f m +f f m , we easily get

(2.19)

From Theorem 2, we note that

| μ ˜ f f m , q ( a + p n Z p ) | f f m C 1 p ν p ( 1 q p n ) ,
(2.20)

where C 1 is some positive constant.

For m0, we have f = f m .

So,

| μ ˜ f m , q ( a + p n Z p ) μ ˜ f m , q ( a + p n + 1 Z p ) | C 2 p ν p ( 1 q p n ) ,
(2.21)

where C 2 is also some positive constant.

By (2.20) and (2.21), we see that

(2.22)

If we fix ϵ>0 and fix m such that f f m ϵ, then for n0, we have

| f ( a ) μ ˜ f , q ( a + p n Z p ) | ϵ.
(2.23)

Hence, we have

f μ f , q (a)= lim n μ ˜ f , q ( a + p n Z p ) = [ 2 ] q 2 ( 1 ) a f(a).
(2.24)

Let m be a sufficiently large number such that f f m p n .

Then we get

μ ˜ f , q ( a + p n Z p ) = μ ˜ f m , q ( a + p n Z p ) + μ ˜ f f m , q ( a + p n Z p ) = μ ˜ f m , q ( a + p n Z p ) = ( 1 ) a [ 2 ] q [ 2 ] q p n f ( a ) ( mod [ p n ] q ) .
(2.25)

For any gC( Z p ), we have

Z p g(x)d μ ˜ f , q (x)= Z p q x f(x)g(x)d μ q (x).
(2.26)

Assume that f is the function from C( Z p , C p ) to Lip( Z p , C p ). By the definition of μ ˜ q , we easily see that μ ˜ q is a strongly p-adic q-measure on Z p , and for n0,

| f μ ˜ q ( a ) μ ˜ q ( a + p n Z p ) | C 3 p ν p ( 1 q p n ) ,
(2.27)

where C 3 is some positive constant.

If μ ˜ 1 , q is associated with strongly fermionic weighted p-adic q-measure on Z p , then we have

| μ ˜ 1 , q ( a + p n Z p ) f μ ˜ q ( a ) | C 4 p ν p ( 1 q p n ) ,
(2.28)

where n0 and C 4 is some positive constant.

From (2.28), we get

(2.29)

where K is some positive constant.

Therefore, μ ˜ q μ ˜ 1 , q is a q-measure on Z p . Hence, we obtain the following theorem.

Theorem 3 Let μ ˜ q be a strongly fermionic weighted p-adic q-measure on Z p , and assume that the fermionic weighted Radon-Nikodym derivative f μ ˜ q on Z p is a continuous function on Z p . Suppose that μ ˜ 1 , q is the strongly fermionic weighted p-adic q-measure associated to f μ ˜ q . Then there exists a q-measure μ ˜ 2 , q on Z p such that

μ ˜ q = μ ˜ 1 , q + μ ˜ 2 , q .
(2.30)

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Acknowledgements

This research was supported by the Kyungpook National University Research Fund 2012. The authors would like to thank professor T. Kim who suggested this problem to us.

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Correspondence to Seog Hoon Rim.

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Authors’ contributions

HP carried out the q-analogue version of similar material studies. SR conceived of the study and participated in its design and coordination. JJ participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.

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Pak, H.K., Rim, S.H. & Jeong, J. A note on the Analogue of Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on Z p . J Inequal Appl 2013, 15 (2013). https://doi.org/10.1186/1029-242X-2013-15

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Keywords

  • p-adic q-measure
  • Lebesgue-Radon-Nikodym