# A Note on the Lebesgue-Radon-Nikodym Theorem with respect to Weightedp-adic Invariant Integral on ℤp

Authors: Joohee Jeong; Seog-Hoon Rim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We will give the Lebesgue-Radon-Nikodym theorem with respect to weightedp-adic q-measure on ℤp. In special case, q=1, we can derive the same result as Kim, 2012; Kim et al., 2011.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper, the symbols ℤp,ℚp, and ℂp denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of ℚp, respectively. The p-adic norm |·|p is defined by |x|p=p-vp(x)=p-r for x=pr(s/t) where s and t are integers with (p,s)=(p,t)=1 and r∈ℚ (see [1–12]).When one speaks ofq-extension, q can be regarded as an indeterminate, a complex q∈ℂ, or a p-adic number q∈ℂp. In this paper, we assume that q∈ℂp with |q-1|p<p-1/(p-1) and we use the notations of q-numbers as follows: (1.1)[x]q=[x:q]=1-qx1-q,[x]-q=1-(-q)x1+q.LetC(ℤp) be the space of continuous functions on ℤp. The fermionic invariant measure on ℤp is defined by Kim as follows:(1.2)μ-1(a+pnZp)=(-1)a,
where (1.3)a+pnZp={x∈Zp∣x≡a(mod⁡pn)},anda∈ℤ with 0≤a<pn (see [2–4, 7]).From (1.2), the fermionic p-adic invariant integral on ℤp is defined by Kim as follows:(1.4)I(f)=∫Zpf(x)dμ-1(x)=lim⁡N→∞∑x=0pN-1f(x)(-1)x,
where f∈C(ℤp) (see [2–4, 6–9]).The idea for generalizing the fermionic integral is replacing the fermionic Haar measure with weakly (strongly) fermionic measureℤp satisfying (1.5)|μ-1(a+pnZp)-μ-1(a+pn+1Zp)|p≤δn,
(see [4, 5, 10]), where δn→0, a is a element of ℤp, and δn is independent of a (for strongly fermionic measure, δn is replaced by Cp-n, where C is a positive constant).Letf(x) be a function defined on ℤp. The fermionic integral of f with respect to a weakly fermionic measure μ-1 is (1.6)∫Zpf(x)dμ-1(x)=lim⁡n→∞∑x=0pn-1f(x)μ-1(x+pnZp),if the limit exists.Ifμ-1 is a weakly fermionic measure on ℤp, then we can define the Radon-Nikodym derivative of μ-1 with respect to the Haar measure on ℤp as follows: (1.7)fμ-1(x)=lim⁡n→∞μ-1(x+pnZp)
(see [4, 11]). Note that fμ-1 is only a continuous function on ℤp. Let UD(ℤp) be the space of uniformly differentiable functions on ℤp. For f∈UD(ℤp), let us define μ-1,f as follows: (1.8)μ-1,f(x+pnZp)=∫x+pnZpf(x)dμ-1(x)
(see [4, 11]), where the integral is the fermionic p-adic invariant integral. From (1.8), we can easily note that μ-1,f is a strongly fermionic measure on ℤp. Since then (1.9)|μ-1,f(x+pnZp)-μ-1,f(x+pn+1Zp)|p=|∑x=0pn-1f(x)(-1)x-∑x=0pnf(x)(-1)x|p=|f(pn)pn||pn|≤Cp-n,whereC is positive constant.The purpose of this paper is to derive a Lebesgue-Radon-Nikodym type theorem with respect to the fermionic weightedp-adic q-measure on ℤp.


## 2. The Lebesgue-Radon-Nikodym Theorem with Respect to the Weightedp-adic q-Measure

For any positive integera and n with a<pn and f∈UD(ℤp), we define μ̃f,-q, weighted fermionic measure on ℤp as follows:(2.1)μ̃f,-q(a+pnZp)=∫a+pnZpqxf(x)dμ-1(x),
where the integral is the fermionic p-adic invariant integral on ℤp.From (2.1), we note that(2.2)μ̃f,-q(a+pnZp)=lim⁡m→∞∑x=0pm-1f(a+pnx)(-1)a+pnxqa+pnx=(-1)aqalim⁡m→∞∑x=0pm-n-1f(a+pnx)(-1)xqpnx=(-1)a∫Zpf(a+pnx)qa+pnxdμ-1(x).By (2.2), we get (2.3)μ̃f,-q(a+pnZp)=(-1)a∫Zpf(a+pnx)qa+pnxdμ-1(x).Thus, by (2.3), we have (2.4)μ̃αf+βg,-q=αμ̃f,-q+βμ̃g,-q,
where f,g∈UD(ℤp) and α,β are positive constants.By (2.1), (2.2), (2.3), and (2.4), we get (2.5)|μ̃f,-q(a+pnZp)|p≤‖fq‖∞,
where ∥fq∥∞=supx∈ℤp|f(x)qx|p.LetP(x)∈ℂp[[x]q] be an arbitrary q-polynomial. Now we show that  μ̃P,-q is a strongly weighted fermionic p-adic invariant measure on ℤp. Without a loss of generality, it is enough to prove the statement for P(x)=[x]qk.Fora∈ℤ with 0≤a<pn, we have (2.6)μ̃P,-q(a+pnZp)=lim⁡m→∞(-q)a∑i=0pm-n-1qpni[a+ipn]qk(-1)i.
Note that (2.7)qpni=∑l=0i(il)[pn]ql(q-1)l,
and
(2.8)[a+ipn]qk=([a]q+qa[pn]q[i]qpn)k.By (2.6) and (2.8), we easily get (2.9)μ̃P,-q(a+pnZp)≡(-1)aqa[a]qk(mod⁡[pn]q)≡(-1)aP(a)qa(mod⁡[pn]q).Forx∈ℤp, let x≡xn(modpn) and x≡xn+1(modpn+1), where xn,xn+1∈ℤ with 0≤xn<pn and 0≤xn+1<pn+1. Then we have (2.10)|μ̃P,-q(a+pnZp)-μ̃P,-q(a+pn+1Zp)|p≤Cp-vp(1-qpn),
where C is positive constant and n≫0.Let(2.11)fμ̃P,-q(a)=lim⁡n→∞μ̃P,-q(a+pnZp).Then, by (2.9) and (2.10), we see that (2.12)fμ̃P,-q(a)=(-1)aqa[a]qk=(-1)aqaP(a).Sincefμ̃P,-q(x) is continuous function on ℤp. For x∈ℤp, we have (2.13)fμ̃P,-q(x)=(-1)xqx[x]qk,(k∈Z+).Letg∈UD(ℤp). Then, by (2.10), (2.12), and (2.13), we get (2.14)∫Zpg(x)dμ̃P,-q(x)=lim⁡n→∞∑x=0pn-1g(x)μ̃P,-q(x+pnZp)=lim⁡n→∞∑x=0pn-1g(x)qx[x]qk(-1)x=∫Zpg(x)qx[x]qkdμ-1(x).Therefore, by (2.14), we obtain the following theorem.Theorem 2.1.
LetP(x)∈ℂp[[x]q] be an arbitrary q-polynomial. Then μ̃P,-q is a strongly weighted fermionic p-adic invariant measure on ℤp; that is,
(2.15)fμ̃P,-q(x)=(-1)xqxP(x),∀x∈Zp.
Furthermore, for any g∈UD(ℤp),
(2.16)∫Zpg(x)dμ̃P,-q(x)=∫Zpg(x)P(x)qxdμ-1(x),
where the second integral is weighted fermionic p-adic invariant integral on ℤp.Letf(x)=∑n=0∞an,q(xn)q be the Mahler q-expansion of continuous function on ℤp, where (2.17)(xn)q=[x]q[x-1]q⋯[x-n+1]q[n]q!.Then we note thatlimn→∞|an,q|=0.Let(2.18)fm(x)=∑i=0mai,q(xi)q∈Cp[[x]q].Then(2.19)‖f-fm‖∞≤sup⁡n≤m|an,q|.
The function f(x) can be rewritten as f=fm+f-fm. Thus, by (2.4) and (2.19), we get (2.20)|μ̃f,-q(a+pnZp)-μ̃f,-q(a+pn+1Zp)|≤max⁡⁡{|μ̃fm,-q(a+pnZp)-μ̃fm,-q(a+pn+1Zp)|,|μ̃f-fm,-q(a+pnZp)-μ̃f-fm,-q(a+pn+1Zp)|}.From Theorem2.1, we note that (2.21)|μ̃f-fm,-q(a+pnZp)|p≤‖f-fm‖∞≤C1p-2vp(1-qpn),
where C1 are positive constants. For m≫0, we have ∥f∥∞=∥fm∥∞. So, we see that(2.22)|μ̃fm,-q(a+pnZp)-μ̃fm,-q(a+pn+1Zp)|p=|fm([pn]q)qpn[pn]q2|p|[pn]q2|p≤‖fmqx‖∞|[pn]q2|p≤C2p-2vp(1-qpn),
where C2 is a positive constant.By (2.21), we get (2.23)|(-1)af(a)qa-μ̃f,-q(a+pnZp)|p≤max⁡{|qaf(a)-fm(a)qa|p,|qafm(a)-μ̃fm,-q(a+pnZp)|p,|μ̃f-fm,-q(a+pnZp)|p}≤max⁡{|f(a)-fm(a)|p,|fm(a)-μ̃fm,-q(a+pnZp)|p,‖f-fm‖∞}.Let us assume that we fixϵ>0 and fix m such that ∥f-fm∥<ϵ. Then we have (2.24)|(-q)af(a)-μ̃f,-q(a+pnZp)|p≤ϵ,forn≫0.Thus, by (2.24), we have (2.25)fμ̃f,-q(a)=lim⁡n→∞μ̃f,-q(a+pnZp)=(-1)aqaf(a).Letm be the sufficiently large number such that ∥f-fm∥∞≤p-n. Then we get (2.26)μ̃f,-q(a+pnZp)=μ̃fm,-q(a+pnZp)+μ̃f-fm,-q(a+pnZp)=(-1)aqaf(a)(mod⁡[pn]q2).Forg∈UD(ℤp), we have (2.27)∫Zpg(x)dμ̃f,-q(x)=∫Zpf(x)g(x)qxdμ-1(x).Letf be the function from UD(ℤp) to Lip(ℤp). We easily see that qxμ-1(x+pnℤp) is a strongly weighted p-adic invariant measure on ℤp and (2.28)|(fq)μ-1(a)-qaμ-1(a+pnZp)|p≤C3p-2vp(1-qpn),wherefq(x)=f(x)qx and C3 is positive constant and n∈ℤ+.Ifμ̃1,-q is associated with strongly weighted fermionic invariant measure on ℤp, then we have (2.29)|μ̃1,-q(a+pnZp)-(fq)μ-1(a)|p≤C4p-2vp(1-qpn),wheren>0 and C4 is positive constant.Forn≫0, we have (2.30)|qaμ-1(a+pnZp)-μ̃1,-q(a+pnZp)|p≤|qaμ-1(a+pnZp)-(fq)μ̃-1(a)|p+|(fq)μ̃-1(a)-μ̃1,-q(a+pnZp)|p≤K,
where K is positive constant.Hence,qμ-1-μ̃1,-q is a weighted measure on ℤp. Therefore, we obtain the following theorem.Theorem 2.2.
Letqμ-1 be a strongly weighted p-adic invariant measure on ℤp, and assume that the fermionic weighted Radon-Nikodym derivative (fq)μ-1 on ℤp is uniformly differentiable function. Suppose that μ̃1,-q is the strongly weighted fermionic p-adic invariant measure associated with (fq)μ-1. Then there exists a weighted measure μ̃2,-q on ℤp such that
(2.31)qxμ-1(x+pnZp)=μ̃1,-q(x+pnZp)+μ̃2,-q(x+pnZp).



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*Source: 696720-2012-03-18.xml*