# p-Adic Fractional Pseudodifferential Equations and Sobolev Type Spaces over p-Adic Fields

Authors: Bo Wu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

In this paper, we study the solutions of the pseudodifferential equations of typeDαu=v over p-adic field ℚp, where Dα is a p-adic fractional pseudodifferential operator. If v is a Bruhat-Schwartz function, then there exists a distribution Eα, a fundamental solution, such that u=Eα*v is a solution. We also show that the solution u belongs to a certain Sobolev space. Furthermore, we give conditions for the continuity and uniqueness of u.


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## Body

## 1. Introduction

In recent years,p-adic analysis has received a lot of attention due to its applications in mathematical physics; see, for example, [1–11] and references therein. As a consequence, new mathematical problems have emerged, among them, the study of p-adic pseudodifferential equations; see, for example, [10–16] and references therein. In this paper, we study the solutions of p-adic fractional pseudodifferential equations on the Sobolev type spaces.Ap-adic fractional pseudodifferential operator Dα, which is defined by Su in 1992 [17], is an operator of the form
(1)Dαφ(x)=(〈ξ〉αφ∧(ξ))∨(x)
for φ∈S(ℚp), where φ∧ and φ∨ denote the Fourier transform and the inverse Fourier transform, S(ℚp) denotes the ℂ-vector space of the Bruhat-Schwartz functions over p-adic fields ℚp, and α is a positive real number. A p-adic fractional pseudodifferential equation is an equation of type
(2)Dαu=v.
If v∈S(ℚp), then there exists a distribution Eα, a fundamental solution, such that u=Eα*v is a solution.We will also show that (2) has a solution u=Eα*v belonging to a certain Sobolev space. Furthermore, we give conditions for the continuity and uniqueness of u.


## 2. Preliminaries

We use the notations indicated in Taibleson’s book [18]. Let us fix a prime number p≥2. The field ℚp of p-adic numbers is defined as the completion of the field of rational numbers ℚ with respect to p-adic norm |·|p, which is defined as follows:|0|p=0;|x|p=p-r if an arbitrary rational number x≠0 is represented as x=pr(m/n), where r=ordp(x)∈ℤ and integers m and n are not divisible by p.Thep-adic norm |·|p satisfies the strong triangle inequality |x+y|p≤max(|x|p,|y|p).Anyp-adic number x≠0 can be uniquely presented as series
(3)x=∑i=m+∞‍xipi,0≤xi≤p-1,i=m,m+1,…,m∈ℤ,xm≠0,
convergent in p-adic norm (the canonical presentation of x).Define the bitwise operation of addition and multiplication ofx in ℚp (either from left to right carry, or not carry), then ℚp is a locally compact, nondiscrete, complete, and totally disconnected topological field.Denote byB0 the ring of integers in ℚp, B0={x∈ℚp:|x|≤1}. Let dx denote the Haar measure on ℚp normalized by the condition ∫B0‍dx=1. Denote by
(4)Br={x∈ℚp:|x|p≤pr},Sr={x∈ℚp:|x|p=pr},
respectively, the ball and the sphere of radius pr with the center at center 0∈ℚp. Obviously, ∫Br‍dx=pr, ∫Sr‍dx=pr(1-p-1).A complex-valued functionφ:ℚp→ℂ defined on ℚp is called locally constant if for any x∈ℚp there exists an integer l(x)∈ℤ satisfying
(5)φ(x+x′)=φ(x),∀x′∈Bl(x).
Denote by ε=ε(ℚp) the linear space of all the locally constant functions. S=S(ℚp) is defined as the linear space of all locally constant functions with compact support in ℚp.The convergence of the point inε has the following definition: φk→ε0, k→∞ if and only if for any compact subset E⊂ℚp,  φk→0,  k→∞ hold uniformly on E. The convergence in S has the following meaning: φk→S0, k→∞ if and only if there exist the indices l and N which do not depend on k, such that the functions {φk} with supports in the ball BN and with constant on the coset of Bl, φk→0, k→∞ hold uniformly in BN. Then, ε and S are complete topological linear spaces. Also, denote by S the Bruhat-Schwartz function space.Denote byS′=S′(ℚp) the distribution space of Bruhat-Schwartz function space S. S′ is a complete topological linear space under the dual topology. The convergence of the point in S′ has the following definition: fk→S′0,k→∞ if and only if 〈fk,φ〉→0, k→∞ hold for any φ∈S.The Fourier transform ofφ∈S(ℚp) is defined by the following formula:
(6)φ∧(ξ)=∫ℚp‍φ(x)χp¯(ξx)dx,ξ∈ℚp,
and the inverse Fourier transform φ∨ by
(7)φ∨(x)=∫ℚp‍φ(ξ)χp(ξx)dξ,x∈ℚp,
where χp(x)=e2πi{x} is an additive character of the field ℚp, with value 1 in B0 and {x}=x-1p-1+x-2p-2+⋯+xmpm. The Fourier transform and inverse Fourier transform map from S into S.The Fourier transformf∧ of a distribution f∈S′(ℚp) is defined by the standard relation 〈f∧,φ〉=〈f,φ∧〉, φ∈S(ℚp).In 1992, Su [17] has given definitions of the derivative for the p-adic local fields ℚp, including derivatives of the fractional orders and real orders.Letξ∈ℚp and 〈ξ〉=max{1,|ξ|}. Its role is played by the operator of pseudodifferential operator Dα(α∈ℝ) which is defined as
(8)Dαφ(x)=(〈ξ〉αφ∧(ξ))∨(x),
for φ∈S(ℚp). It is easy to see that Dαφ∈S(ℚp). With
(9)〈Dαf,φ〉=〈f,Dαφ〉,f∈S′(ℚp),φ∈S(ℚp),
the defined domain of Dα can be extended to the space S′(ℚp). Thus, we also have Dαf=(〈ξ〉αf∧(ξ))∨ with f∈S′(ℚp) and Dαf∈S′(ℚp).Definition 1.
Ifα>0, then Dα is defined as p-adic derivative of the order α on S′(ℚp). And if α<0, then Dα is defined as p-adic integral of the order -α on S′(ℚp). If α=0, D0f=f for any f∈S′, then D0 is called the identity operator.
In [19, 20], Qiu and Su constructed the convolution kernel of the operator Dα. Consider
(10)Dαu=kα*u,foru∈S′,
where
(11)kα={(1-pα1-p-α-1π-α+pα-1pα+1-1)Δ0,α≠0,-1;δ,α=0;(1-1p)(1-logp|x|p)Δ0,α=-1.Here,Δ0 is the indicative function of set B0 satisfying Δ0=1B0, and πα is a distribution defined as
(12)〈πα,φ〉=∫ℚp‍|x|pα-1(φ(x)-φ(0))dx,φ∈S.Lemma 2 (see [19, 20]).
k
α
∧
=
〈
ξ
〉
α, and kα satisfies the property of semigroup: kα*kβ=kα+β.


## 3. Solutions of the Pseudodifferential Equations on Sobolev Type Spaces

We now consider the following pseudodifferential equation:(13)Dαu=v,withv∈S(ℚp),α∈ℝ.
We say that Eα∈S′(ℚp) is a fundamental solution of (13) if Eα*v is a solution.Lemma 3.
IfEα is a fundamental solution of (13), then for any constant c, Eα+c is also a fundamental solution.Proof.
LetEα be a fundamental solution for (13), then
(14)Dα((Eα+c)*v)=Dα(Eα*v+c*v)=u+Dα(c*v)=u,
because u and the constant function, c*v, are in the domain of Dα.Theorem 4.
A fundamental solution of (13) is(15)Eα={(1-p-α1-pα-1πα+p-α-1p-α+1-1)Δ0,α≠1;δ,α=0;(1-1p)(1-logp|x|p)Δ0,α=1.Proof.
We use the definition ofDα (8), then
(16)Dα(Eα*v)=(〈ξ〉α(Eα*v)∧)∨=((〈ξ〉αEα∧)v∧)∨.
The existence of a fundamental solution Eα is equivalent to the existence of a distribution Eα∧ satisfying
(17)〈ξ〉αEα∧=1,
as distributions. By Lemma 2, we have
(18)Eα∧=lims→-α〈ξ〉s=lims→-αks∧=k-α∧.
Then, Eα=k-α. By (11), Theorem 4 is proved.
In what follows we will introduce some relevant spaces overp-adic fields (see [20]).
The Holder type spaceCσ(ℚp), σ∈[0,+∞):(1)
forσ=0, we define C0(ℚp)=C(ℚp) as the continuous function space on ℚp;(2)
forσ>0, we define Cσ(ℚp)as the set of all distributions u∈S′ with the Littlewood-Paley decomposition u=∑j=0∞‍uj with respect to the following norm:
(19)∥u∥Cσ(ℚp)=supj{p-jσ∥uj∥L∞(ℚp)}.Thus, Cσ(ℚp) becomes a Banach space with the above norm.
The Sobolev type spaceWs(ℚp), σ∈[0,+∞):
(20)Ws(ℚp)={∥(<·>su∧(·))∨∥L2(ℚp)u∈S′:∥u∥Ws(ℚp)1m=∥(〈·〉su∧(·))∨∥L2(ℚp)<∞},
where L2(ℚp) denotes the set of measurable functions f on ℚp satisfying the condition ∫ℚp‍|f(x)|2dx<∞.Lemma 5 (see [21, 22]).
Fors>1/2, it holds that
(21)Ws(ℚp)⊂Cs-1/2(ℚp).Lemma 6.
For anyα>0 and s≥0, the mapping Dα:Ws+α→Ws is a well-defined continuous mapping between Banach spaces.Proof.
Letu∈S. We have that
(22)∥Dαu∥Ws2=∫ℚp‍|(〈ξ〉s(Dαu)∧)∨|2dξ=∫ℚp‍|(〈ξ〉s〈ξ〉αu∧)∨|2dξ=∫ℚp‍|(〈ξ〉s+αu∧)∨|2dξ=∥u∥Ws+α2.
The result follows from the fact that S is dense in Ws+α.Theorem 7.
LetDα(α>0) be a p-adic fractional pseudodifferential operator. Let s be a positive real number satisfying s≥1/2-α. Then, the equation
(23)Dαu=v,v∈S,
has a unique uniformly continuous solution u∈Ws+α.Proof.
Letv∈S, as S is dense in Ws and we have
(24)∥u∥Ws+α2=∫ℚp‍|(〈ξ〉s+α(u)∧)∨|2dξ=∫ℚp‍|(〈ξ〉s+α(Eα*v)∧)∨|2dξ=∫ℚp‍|(〈ξ〉s+αEα∧u∧)∨|2dξ=∫ℚp‍|(〈ξ〉su∧)∨|2dξ=∥v∥Ws2.
In this way, we established the existence of u∈Ws+α. By Lemma 5, u∈Cs+α-1/2 is uniformly continuous for s+α-1/2≥0.
Finally, we show thatu is unique. Indeed, if Dαu′=v, then
(25)Dα(u-u′)=0,i.e.,〈ξ〉α(u-u′)∧=0,
And, thus, (u-u′)∧=0. Then, (u-u′)χx(ξ)=0 almost everywhere, and a fortiori (u-u′)=0 almost everywhere, and by the continuity of u-u′, u(ξ)=u′(ξ) for any ξ∈ℚp. Theorem 7 is proved.


## 4. Conclusion

In this work, the pseudodifferential equations of typeDαu=v over p-adic field ℚp were investigated, where Dα is a p-adic fractional pseudodifferential operator. The fundamental solution of the equation was obtained. And the continuity and uniqueness of solution u which belongs to the Sobolev type space Ws+α were obtained when s≥1/2-α, using the explicit calculation of fundamental solutions approach.



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