# Existence and Continuity of Solutions to a Class of Pseudodifferential Equations overp-Adic Field

Authors: Bo Wu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

We study the pseudodifferential operatorTα and the pseudodifferential equations of type Tαu+u=δxk over p-adic field ℚp, where δxk is the Dirac delta function. We discuss the existence and uniqueness of solutions to the equations. Furthermore, we give conditions for the continuity of the solutions uk when u belongs to the space L2(ℚp).


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## Body

## 1. Introduction

In recent yearsp-adic analysis has received a lot of attention due to its applications in mathematical physics; see, for example, [1–13] and references therein.A good example of the applications is the pseudodifferential equations on the fieldℚp. In 60s of the last century, Gibbs first gives logic derivative on dyadic field. Since then, Vladimirov et al. [13] extended logic derivative to p-adic field and made the certain correction of the original Gibbs definition. This kind of derivative referred to Vladimirov pseudodifferential operator. Kuzhel and Albeverio et al. used Vladimirov operator to study pseudodifferential equations; and see, for example, [14–16]. However, the Vladimirov pseudodifferential operator, as a kind of operation, is not closed in the test function space S(ℚp). This makes the definition of Vladimirov operator difficult to be applied to distribution space S′(ℚp). In 1992, Su [17] defined derivative and integral operator Tα on locally compact Vilenkin groups G. The space of test functions space S(G) and its distribution S′(G) with the operation of Tα is closed. Tα is defined as a pseudodifferential operator with the symbol 〈ξ〉α(1)Tαφ=(〈ξ〉αφ∧(ξ))∨,
where 〈ξ〉=max{1,|ξ|}, α∈ℝ. The new definition plays a role in promoting the development of theory for p-adic analysis.In this paper, we study a class of pseudodifferential equations of type(2)Tαu+u=δxk,u∈L2(ℚp),xk∈ℚp,
where Tα is pseudodifferential operator defined by Su and ℚp is the field of p-adic numbers.Our aim is to show that (2) has a unique solution uk belonging to L2(ℚp) if α>1/2 and has no solutions if α≤1/2.Moreover, we give the condition for the continuity of the solutionsuk with the index α.


## 2. Preliminaries

We use the notations indicated in Taiblesons book [18]. Let p≥2 be a prime number. The field ℚp of p-adic numbers is a topologically complete space of rational numbers ℚ with respect to p-adic norm |·|p (non-Archimedean norm), which is defined as follows:ifx=0, then |0|p=0;ifx≠0 is an arbitrary rational number, we define |x|p=p-r. And x can be represented as x=pr(m/n), where r=orderpx∈ℤ and integers m and n are relatively primes and 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 in the topologically complete space ℚp can be presented as series uniquely:
(3)x=∑i=m+∞‍xipi,0≤xi≤p-1,i=m,m+1,…,m∈ℤ,xm≠0.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, totally disconnected, and complete topological field.Denote byB0={x∈ℚp:|x|p≤1} the ring of integers in ℚp which is the subring of ℚp. B0 is the compact subring and compact subspace of locally compact field ℚp. With the addition operation of p-adic field ℚp, there exists the Haar measure dx on ℚp such that ∫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 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 according to the following definition:  φk→ε0, k→∞ if and only if for any compact subset E⊂ℚp, φk→0,  k→∞ holds uniformly on E. The convergence in S has the following meaning:  φk→S0, k→∞ if and only if there exists 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 test function space.The Fourier transform ofφ∈S(ℚp) is defined by the formula
(6)φ∧(ξ)=∫ℚp‍φ(x)χp¯(ξx)dx,ξ∈ℚp,
and 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 S(ℚp) onto S(ℚp).Denote byS′=S′(ℚp) the distribution space of test function space S. S′ is a complete topological linear space under the dual topology. The convergence in S′ according to the following definition: fk→S′0, k→∞ if and only if 〈fk,φ〉→0, k→∞ holds for any φ∈S.It follows from the definition ofS(ℚp) that any test function φ∈S(ℚp) is continuous on ℚp. This means the Dirac delta function 〈δx,φ〉=φ(x) is well posed for any point x∈ℚp.Denote byL2(ℚp) the set of the measurable functions f on ℚp with the condition ∫ℚp‍|f(x)|2dx<∞. And set L2(ℚp) is a Hilbert space satisfying the scalar product (f,g)L2(ℚp)=∫ℚp‍f(x)g(x)¯dx.LetE be compact set, and 1E is the indicative function of set E. Then there exist standard sequences Δk(x) of 1 satisfying Δk(x)=1Bk, k∈ℤ, x∈ℚp.


## 3. Pseudodifferential OperatorTα

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 Tα (α∈ℝ) which is defined as
(8)Tαφ(x)=(〈ξ〉αφ∧(ξ))∨(x),
for φ∈S(ℚp). It is easy to see that Tαφ∈S(ℚp). With
(9)〈Tαf,φ〉=〈f,Tαφ〉,f∈S′(ℚp),φ∈S(ℚp),
the defined domain of Tα can be extended to the space S′(ℚp). Thus, we also have Tαf=(〈ξ〉αf∧(ξ))∨ with f∈S′(ℚp) and Tαf∈S′(ℚp).Definition 1.
Ifα>0, then Tα is defined as p-adic derivatives of the order α on S′(ℚp). And if α<0, then Tα is defined as p-adic integral of the order -α on S′(ℚp). If α=0, T0f=f for any f∈S′, then T0 is called the identity operator.In what follows we considerTα as an operator in Hilbert space L2(ℚp)=L2. Obviously, the set of functions D(Tα)={f∈L2:〈ξ〉αf∧∈L2} is the domain of definition of Tα on the space L2(10)Tαf=(〈ξ〉αf∧)∨.In [19], Qiu and Su have recently studied the spectrum of Tα and constructed the set of eigenfunctions of Tα:
(11)ψNjI(x)=p-N/2χ(pN-1jx)Δ0(pNx-zI),N∈ℤ,I=zI+B0∈ℚpB0,j=1,2,…,p-1
which forms an orthonormal basis in L2(ℚp) (p-adic wavelet basis) such that
(12)TαψNjI(x)={p(1-N)αψNjI(x),N<1,ψNjI(x),N≥1.Theorem 2.
The functionf is continuous in D(Tα) if and only if α>1/2.Proof.
Letf∈D(Tα); we expand the function f under the p-adic wavelet basic (11) as follows:
(13)f(x)=∑N=1∞‍∑j=1p-1‍∑I‍(f,ψNjI)ψNjI(x)+∑N=-∞0‍∑j=1p-1‍∑I‍(f,ψNjI)ψNjI(x).
Evidently,ψNjI(x) belong to the domain of definition of Tα, and then, the functions ψNjI(x) are continuous on ℚp. Thus, it is sufficient to prove the continuity of f(x) with verifying that the series (13) converges uniformly.
First, we note that there is at most onezI such that ψNjI(x)≠0 for the fixed N and x. Indeed, if there are zI1 and zI2 satisfying ψNjI(x)≠0, then we have Δ0(pNx-zIi)=1. But |pNx-zI1|≤1 and |pNx-zI2|≤1. With the strong triangle inequality |zI1-zI2|≤1. From the condition I=zI+B0∈ℚp/B0, we can get the equality I1=I2.
Thus, the sum with the parameterI consists of at most one nonzero term for fixed N and x.
Further, following from (11) and (13), we obtain
(14)|ψNjI(x)|≤p-N/2,|(f,ψNjI)|≤∥f∥L2(Q)p.
The relations (14) obtain the following estimate if N≥1(15)|∑j=1p-1‍∑I‍(f,ψNjI)ψNjI(x)|≤p-N/2∥f∥L2(Q)p(p-1).
The estimate gives the uniform convergency of the first series in (13).
The conditionf∈D(Tα) and (12) imply (f,ψNjI)=p(N-1)(Tαf,ψNjI) for N<1. For a fixed N<1, using Cauchy-Schwarz equality and (14), we obtain
(16)|∑j=1p-1‍∑I‍(f,ψNjI)ψNjI(x)|=|∑j=1p-1‍∑I‍p(N-1)(Tαf,ψNjI)ψNjI(x)|≤{∑j=1p-1‍∑I‍|(Tαf,ψNjI)|2}1/2×{∑j=1p-1‍∑I‍p2(N-1)|ψNjI(x)|2}1/2≤∥Tαf∥L2(Q)p{∑j=1p-1‍p-N+2(N-1)}1/2.
The estimate obtained above means the second series in (13) is uniformly convergent if α>1/2. Thus, function f is continuous in D(Tα) on ℚp for α>1/2. Theorem 2 is proved.Next whenα≤1/2, we will give an example in which the function belongs to D(Tα) but is not continuous on ℚp.Example 3.
The function(17)f(x)=∑-∞-1‍1|N|p(N-1)/2ψN10(x)
belongs to D(Tα) but f(x) is not continuous on ℚp  for α≤1/2.It is easy to seef∈L2(ℚp) and its Fourier transform is
(18)f∧(ξ)=∑-∞-1‍1|N|p(N-1)/2ψN10∧(ξ).
From (8) and (12), 〈ξ〉αψN10∧(ξ)=pα(1-N)ψN10∧(ξ). Thus
(19)〈ξ〉αf∧(ξ)=∑-∞-1‍1|N|p(N-1)/2pα(1-N)ψN10∧(ξ).〈ξ〉αf∧(ξ)∈L2(ℚp) for α≤1/2 and 〈ξ〉αf∧(ξ)∉L2(ℚp) for α>1/2 with the orthonormal basis {ψN10}N<0. So, f(x)∈D(Tα) for α≤1/2.Next, We will show thatf(x) is not continuous on ℚp. First, using (11), we rewrite the definition (17) of f as
(20)f(x)=∑-∞-1‍1|N|p-1/2χ(pN-1x)Δ0(pNx).
We consider the sequence xn=pn, n∈ℕ. It is easy to see xn→0, (n→∞) in the p-adic norm |·|p. Moreover, Δ0(pNxn)=Δ0(pN+n)=0 when N+n≤-1. On the other hand, when N+n≤1, pN-1xn become an integer p-adic number, and then χ(pN-1xn)=χ(pN+n-1)=1. From the above relations and (20) we can get that
(21)f(xn)=f(pn)=p-1/2(χ(p-1)n+∑N=-n+1-1‍1|N|)⟶∞asn⟶∞.
Therefore, f(x) cannot be continuous at x=0.


## 4. Pseudodifferential Equation

In this section we will consider the pseudodifferential equation(22)Tαu+u=δxk,u∈L2(ℚp),xk∈ℚp,
where Tα:L2(ℚp)→S′(ℚp) is pseudodifferential operator in the distribution sense.Theorem 4.
Equation (22) has a unique solution u=uk∈L2(ℚp) for α>1/2 and has no solutions belonging to L2(ℚp) for α≤1/2.Proof.
Letφ∈D(Tα). Similar to the proof of Theorem 2, we give the expansion of φ using the uniformly convergent series with respect to the complex-conjugated p-adic wavelet basis {ψNjI¯}. For {ψNjI¯} are continuous functions on ℚp we can write
(23)φ(xk)=∑N=-∞+∞‍∑j=1p-1‍∑I‍(φ,ψNjI¯)ψNjI¯(xk)
for x=xk.
Consider(24)ψNjI¯(xk)=p-N/2χ(pN-1jxk)¯Δ0(pNxk-zI)=p-N/2χ(-pN-1jxk)Δ0(pNxk-zI).
It is easy to see ψNjI¯(xk)≠0 equals |pNxk-zI|≤1. Here I=zI+B0∈ℚp/B0 and by the strong triangle inequality we obtain |pNxk-zI|≤1 equals zI={pNxk}. Then, we obtain the conclusion that
(25)ψNjI¯(xk)={0,zI≠{pNxk},p-N/2χ(-pN-1jxk),zI={pNxk}.
And then
(26)〈δxk,φ〉=φ(xk)=∑N=-∞+∞‍∑j=1p-1‍p-N/2χ(-pN-1jxk)(φ,ψNj{pNxk}¯)=∑N=-∞+∞‍∑j=1p-1‍p-N/2χ(-pN-1jxk)〈ψNj{pNxk},φ〉.
For S(ℚp)⊂D(Tα), the equality (26) implies that
(27)δxk=∑N=-∞+∞‍∑j=1p-1‍p-N/2χ(-pN-1jxk)ψNj{pNxk},
in which series converges uniformly in S′(ℚp).
We suppose that there is a functionuk∈L2(ℚp) which can be represented as a convergent series in L2:
(28)uk(x)=∑N=-∞+∞‍∑j=1p-1‍∑I‍cNjIψNjI(x).
Applying pseudodifferential operator  Tα+I  on the both sides, we obtain a series with (12)
(29)Tαuk+uk=∑N=-∞0‍∑j=1p-1‍∑I‍cNjI(1+pα(1-N))ψNjI+∑N=1+∞‍∑j=1p-1‍∑I‍2cNjIψNjI,
which converges in S′(ℚp) for TαS(ℚp)⊂L2(ℚp). Comparing the two series (27) and (29), we obtain
(30)cNj{pNxk}={12p-N/2χ(-pN-1jxk),N≥1,(pα(1-N)+1)-1p-N/2χ(-pN-1jxk),N<1.
Thus,
(31)uk(x)=∑N=1+∞‍∑j=1p-1‍12p-N/2×χ(-pN-1jxk)ψNj{pNxk}(x)+∑N=-∞0‍∑j=1p-1‍(pα(1-N)+1)-1p-N/2×χ(-pN-1jxk)ψNj{pNxk}(x).
Next, We will show that the series (31) belong to L2(ℚp) for α>1/2. For the general term of the first series we obatin
(32)|12p-N/2χ(-pN-1jxk)|2≤14p-N,N≥1
that means the first series converges in L2(ℚp) for any α∈ℝ.
We estimate the general term of the second series as follows:(33)|(pα(1-N)+1)-1p-N/2χ(-pN-1jxk)|2≤p-N(pα(1-N)+1)-2≤p-2αp-N(1-2α),N<1.
It is easy to see the second series converges in L2(ℚp) for α>1/2.
Thusuk is a unique solution of (31) for α>1/2.
Forpα(1-N)+1≤2pα(1-N) with N<1, we estimate the general term of the second series as follows:
(34)|(pα(1-N)+1)-1p-N/2χ(-pN-1jxk)|2≥14p-2αp-N(1-2α),N<1
that means the series diverges in L2(ℚp)   for α≤1/2. Theorem 4 is proved.Theorem 5.
The solution
u
k
of (22) is continuous on
ℚ
p
for
α
>
1
.Proof.
We will show that the series (31) converges uniformly on ℚp if α>1 and converges uniformly on the ball not containing xk if 1/2<α≤1.
Indeed, the general term (31) of the first series does not exceed (1/2)p-N by (14) when N≥1. Thus, the first series converges uniformly with N≥1.
WhenN<1 the general term (31) of the second series satisfies
(35)(pα(1-N))-1p-N≤p-αp-N(1-α).
The estimate we obtained above implies that for α>1 the subseries of (31) converges uniformly with N<1. Hence, the series (31) converges uniformly for α>1. Theorem 5 is proved.Letα>1/2 and fixed points x1,x2,…,xn∈ℚp. Then, define Sp{uk}1n as the linear span of solutions uk (k=1,2,…,n) of (22). Hence, we have the following.Theorem 6.
S
p
{
u
k
}
1
n
∩
D
(
T
α
/
2
)
=
{
0
} for 1/2<α≤1 and Sp{uk}1n⊂D(Tα/2) for α>1.Proof.
The solutionuk of (22) is obtained by (31). Considering the expansion (31) and semigroup property (c.f. [19])
(36)Tα1Tα2=Tα1+α2,α1,α2>0
of Tα. Obviously, uk∈D(Tα/2) if and only if the following series converge in L2(ℚp):
(37)∑N=1+∞‍∑j=1p-1‍12p-N/2χ(-pN-1jxk)ψNj{pNxk}(x)+∑N=-∞0‍∑j=1p-1‍(pα(1-N)+1)-1p-N/2×χ(-pN-1jxk)p(α/2)(1-N)ψNj{pNxk}(x).
For the general term of the first series,we have the estimate as follows:
(38)|12p-N/2χ(-pN-1jxk)|2≤14p-N,N≥1
that means it converges in L2(ℚp) for any α∈ℝ.
Similarly, for the general term of the second series, we have(39)|(pα(1-N)+1)-1p-N/2χ(-pN-1jxk)p(α/2)(1-N)|2≤Cp(α-1)N,N<1.
Hence the series converges when α>1. Then Sp{uk}1n⊂D(Tα/2) if α>1.
Forpα(1-N)+1≤2pα(1-N) with N<1, the general term of the second series can be estimated below
(40)|(pα(1-N)+1)-1p-N/2χ(-pN-1jxk)p(α/2)(1-N)|2≥14p-αp-N(1-α),N<1
that means it diverges in L2(ℚp) for α≤1.
Henceuk∉D(Tα/2). Since the estimate (40) does not depend on the choice of uk. And considering the functions {ψNj{pNxk}(x)} (N<1) of the basis {ψNjI(x)} in the uk (k=1,2,…,n) of (31) are different for any small negative N, we obtain that Sp{uk}1n∩D(Tα/2)={0} if 1/2<α≤1. Theorem 6 is proved.


## 5. Conclusions

In this paper, solutions of pseudodifferential equations with the typeTαu+u=δxk over p-adic field ℚp are considered. First, we give the condition for the continuity of arbitrary function in the domain of definition D(Tα) with the index α>1/2 and give the example to show noncontinuity when α≤1/2. Then, we obtain the existence of solutions to the pseudodifferential equations. The results show that the equation has a unique solution belonging to L2(ℚp) for α>1/2 and has no solutions for α≤1/2. Furthermore, we show the continuity of solutions when α>1. Finally, the embedding from the linear span of solutions to the domain of definition D(Tα) is obtained.



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