Existence and Continuity of Solutions to a Class of Pseudodifferential Equations over 𝑝-Adic Field

Abstract

We study the pseudodifferential operator 𝑇^𝛼 and the pseudodifferential equations of type 𝑇^𝛼𝑢+𝑢=𝛿_𝑥𝑘 over 𝑝-adic field ℚ_𝑝, where 𝛿_𝑥𝑘 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 𝑢_𝑘 when 𝑢 belongs to the space 𝐿₂(ℚ_𝑝).

1. Introduction

In recent years 𝑝-adic analysis has received a lot of attention due to its applications in mathematical physics; see, for example, [1–13] and references therein.

A good example of the applications is the pseudodifferential equations on the field ℚ_𝑝. 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(ℚ_𝑝). This makes the definition of Vladimirov operator difficult to be applied to distribution space 𝑆^′(ℚ_𝑝). In 1992, Su [17] defined derivative and integral operator 𝑇^𝛼 on locally compact Vilenkin groups 𝐺. The space of test functions space 𝑆(𝐺) and its distribution 𝑆^′(𝐺) with the operation of 𝑇^𝛼 is closed. 𝑇^𝛼 is defined as a pseudodifferential operator with the symbol ⟨𝜉⟩^𝛼𝑇^𝛼𝜑=𝜉^𝛼𝜑^∧(𝜉)^∨, (1) 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 𝑇^𝛼𝑢+𝑢=𝛿_𝑥𝑘,𝑢∈𝐿₂ℚ_𝑝,𝑥_𝑘∈ℚ_𝑝, (2) where 𝑇^𝛼 is pseudodifferential operator defined by Su and ℚ_𝑝 is the field of 𝑝-adic numbers.

Our aim is to show that (2) has a unique solution 𝑢_𝑘 belonging to 𝐿₂(ℚ_𝑝) if 𝛼>1/2 and has no solutions if 𝛼≤1/2.

Moreover, we give the condition for the continuity of the solutions 𝑢_𝑘 with the index 𝛼.

2. Preliminaries

We use the notations indicated in Taiblesons book [18]. Let 𝑝≥2 be a prime number. The field ℚ_𝑝 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: if 𝑥=0, then |0|_𝑝=0; if 𝑥≠0 is an arbitrary rational number, we define |𝑥|_𝑝=𝑝^−𝑟. And 𝑥 can be represented as 𝑥=𝑝^𝑟(𝑚/𝑛), where 𝑟=order_𝑝𝑥∈ℤ and integers 𝑚 and 𝑛 are relatively primes and not divisible by 𝑝.

The 𝑝-adic norm |⋅|_𝑝 satisfies the strong triangle inequality |𝑥+𝑦|_𝑝≤max(|𝑥|_𝑝,|𝑦|_𝑝).

Any 𝑝-adic number 𝑥≠0 in the topologically complete space ℚ_𝑝 can be presented as series uniquely: 𝑥=^+∞_𝑖=𝑚𝑥_𝑖𝑝^𝑖,0≤𝑥_𝑖≤𝑝−1,𝑖=𝑚,𝑚+1,…,𝑚∈ℤ,𝑥_𝑚≠0. (3)

Define the bitwise operation of addition and multiplication of 𝑥 in ℚ_𝑝 (either from left to right carry, or not carry); then ℚ_𝑝 is a locally compact, totally disconnected, and complete topological field.

Denote by 𝐵₀={𝑥∈ℚ_𝑝∶|𝑥|_𝑝≤1} the ring of integers in ℚ_𝑝 which is the subring of ℚ_𝑝. 𝐵₀ is the compact subring and compact subspace of locally compact field ℚ_𝑝. With the addition operation of 𝑝-adic field ℚ_𝑝, there exists the Haar measure 𝑑𝑥 on ℚ_𝑝 such that ∫_𝐵0𝑑𝑥=1. Denote by 𝐵_𝑟=𝑥∈ℚ_𝑝∶|𝑥|_𝑝≤𝑝^𝑟,𝑆_𝑟=𝑥∈ℚ_𝑝∶|𝑥|_𝑝=𝑝^𝑟, (4) respectively, the ball and the sphere of radius 𝑝^𝑟 with the center at 0∈ℚ_𝑝. Obviously, ∫_𝐵𝑟𝑑𝑥=𝑝^𝑟, ∫_𝑆𝑟𝑑𝑥=𝑝^𝑟(1−𝑝⁻¹).

A complex-valued function 𝜑∶ℚ_𝑝→ℂ defined on ℚ_𝑝 is called locally constant if for any 𝑥∈ℚ_𝑝 there exists an integer 𝑙(𝑥)∈ℤ satisfying 𝜑𝑥+𝑥^′=𝜑(𝑥),∀𝑥^′∈𝐵_𝑙(𝑥). (5) Denote by 𝜀=𝜀(ℚ_𝑝) the linear space of all the locally constant functions. 𝑆=𝑆(ℚ_𝑝) is defined as the linear space of all locally constant functions with compact support in ℚ_𝑝.

The convergence of the point in 𝜀 according to the following definition:  𝜑→−_𝑘^𝜀0, 𝑘→∞ if and only if for any compact subset 𝐸⊂ℚ_𝑝, 𝜑_𝑘→0,  𝑘→∞ holds uniformly on 𝐸. The convergence in 𝑆 has the following meaning:  𝜑→−_𝑘^𝑆0, 𝑘→∞ if and only if there exists the indices 𝑙 and 𝑁 which do not depend on 𝑘, such that the functions {𝜑_𝑘} with supports in the ball 𝐵_𝑁 and with constant on the coset of 𝐵_𝑙, 𝜑_𝑘→0, 𝑘→∞ hold uniformly in 𝐵_𝑁. Then, 𝜀 and 𝑆 are complete topological linear spaces. Also denote by 𝑆 the test function space.

The Fourier transform of 𝜑∈𝑆(ℚ_𝑝) is defined by the formula 𝜑^∧(𝜉)=_ℚ𝑝𝜑(𝑥)𝜒_𝑝(𝜉𝑥)𝑑𝑥,𝜉∈ℚ_𝑝, (6) and inverse Fourier transform 𝜑^∨ by 𝜑^∨(𝑥)=_ℚ𝑝𝜑(𝜉)𝜒_𝑝(𝜉𝑥)𝑑𝜉,𝑥∈ℚ_𝑝, (7) where 𝜒_𝑝(𝑥)=𝑒^{2𝜋𝑖{𝑥}} is an additive character of the field ℚ_𝑝, with value 1 in 𝐵₀, and {𝑥}=𝑥₋₁𝑝⁻¹+𝑥₋₂𝑝⁻²+⋯+𝑥_𝑚𝑝^𝑚. The Fourier transform and inverse Fourier transform map 𝑆(ℚ_𝑝) onto 𝑆(ℚ_𝑝).

Denote by 𝑆^′=𝑆^′(ℚ_𝑝) 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: 𝑓−→_𝑘^𝑆′0, 𝑘→∞ if and only if ⟨𝑓_𝑘,𝜑⟩→0, 𝑘→∞ holds for any 𝜑∈𝑆.

It follows from the definition of 𝑆(ℚ_𝑝) that any test function 𝜑∈𝑆(ℚ_𝑝) is continuous on ℚ_𝑝. This means the Dirac delta function ⟨𝛿_𝑥,𝜑⟩=𝜑(𝑥) is well posed for any point 𝑥∈ℚ_𝑝.

Denote by 𝐿₂(ℚ_𝑝) the set of the measurable functions 𝑓 on ℚ_𝑝 with the condition ∫_ℚ𝑝|𝑓(𝑥)|²𝑑𝑥<∞. And set 𝐿₂(ℚ_𝑝) is a Hilbert space satisfying the scalar product (𝑓,𝑔)_{𝐿2(ℚ𝑝)}=∫_ℚ𝑝𝑓(𝑥)𝑔(𝑥)𝑑𝑥.

Let 𝐸 be compact set, and 1_𝐸 is the indicative function of set 𝐸. Then there exist standard sequences Δ_𝑘(𝑥) of 1 satisfying Δ_𝑘(𝑥)=1_𝐵𝑘, 𝑘∈ℤ, 𝑥∈ℚ_𝑝.

3. Pseudodifferential Operator 𝑇^𝛼

In 1992, Su [17] has given definitions of the derivative for the 𝑝-adic local fields ℚ_𝑝, including derivatives of the fractional orders and real orders.

Let 𝜉∈ℚ_𝑝 and ⟨𝜉⟩=max{1,|𝜉|}. Its role is played by the operator of pseudodifferential operator 𝑇^𝛼 (𝛼∈ℝ) which is defined as 𝑇^𝛼𝜑(𝑥)=𝜉^𝛼𝜑^∧(𝜉)^∨(𝑥), (8) for 𝜑∈𝑆(ℚ_𝑝). It is easy to see that 𝑇^𝛼𝜑∈𝑆(ℚ_𝑝). With 𝑇^𝛼𝑓,𝜑=𝑓,𝑇^𝛼𝜑,𝑓∈𝑆^′ℚ_𝑝,𝜑∈𝑆ℚ_𝑝, (9) the defined domain of 𝑇^𝛼 can be extended to the space 𝑆^′(ℚ_𝑝). Thus, we also have 𝑇^𝛼𝑓=(⟨𝜉⟩^𝛼𝑓^∧(𝜉))^∨ with 𝑓∈𝑆^′(ℚ_𝑝) and 𝑇^𝛼𝑓∈𝑆^′(ℚ_𝑝).

Definition 1. If 𝛼>0, then 𝑇^𝛼 is defined as 𝑝-adic derivatives of the order 𝛼 on 𝑆^′(ℚ_𝑝). And if 𝛼<0, then 𝑇^𝛼 is defined as 𝑝-adic integral of the order −𝛼 on 𝑆^′(ℚ_𝑝). If 𝛼=0, 𝑇⁰𝑓=𝑓 for any 𝑓∈𝑆^′, then 𝑇⁰ is called the identity operator.

In what follows we consider 𝑇^𝛼 as an operator in Hilbert space 𝐿₂(ℚ_𝑝)=𝐿₂. Obviously, the set of functions 𝐷(𝑇^𝛼)={𝑓∈𝐿₂∶⟨𝜉⟩^𝛼𝑓^∧∈𝐿₂} is the domain of definition of 𝑇^𝛼 on the space 𝐿₂𝑇^𝛼𝑓=𝜉^𝛼𝑓^∧∨. (10)

In [19], Qiu and Su have recently studied the spectrum of 𝑇^𝛼 and constructed the set of eigenfunctions of 𝑇^𝛼: 𝜓_𝑁𝑗𝐼(𝑥)=𝑝^−𝑁/2𝜒𝑝^𝑁−1𝑗𝑥Δ₀𝑝^𝑁𝑥−𝑧_𝐼,𝑁∈ℤ,𝐼=𝑧_𝐼+𝐵₀∈ℚ_𝑝𝐵₀,𝑗=1,2,…,𝑝−1 (11) which forms an orthonormal basis in 𝐿₂(ℚ_𝑝) (𝑝-adic wavelet basis) such that 𝑇^𝛼𝜓_𝑁𝑗𝐼(𝑥)=𝑝^{(1−𝑁)𝛼}𝜓_𝑁𝑗𝐼(𝑥)𝜓,𝑁<1,_𝑁𝑗𝐼(𝑥),𝑁≥1. (12)

Theorem 2. The function 𝑓 is continuous in 𝐷(𝑇^𝛼) if and only if 𝛼>1/2.

Proof. Let 𝑓∈𝐷(𝑇^𝛼); we expand the function 𝑓 under the 𝑝-adic wavelet basic (11) as follows: 𝑓(𝑥)=^∞_𝑁=1^𝑝−1_𝑗=1𝐼𝑓,𝜓_𝑁𝑗𝐼𝜓_𝑁𝑗𝐼(𝑥)+⁰_𝑁=−∞^𝑝−1_𝑗=1𝐼𝑓,𝜓_𝑁𝑗𝐼𝜓_𝑁𝑗𝐼(𝑥). (13)
Evidently, 𝜓_𝑁𝑗𝐼(𝑥) belong to the domain of definition of 𝑇^𝛼, and then, the functions 𝜓_𝑁𝑗𝐼(𝑥) are continuous on ℚ_𝑝. 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 one 𝑧_𝐼 such that 𝜓_𝑁𝑗𝐼(𝑥)≠0 for the fixed 𝑁 and 𝑥. Indeed, if there are 𝑧_𝐼1 and 𝑧_𝐼2 satisfying 𝜓_𝑁𝑗𝐼(𝑥)≠0, then we have Δ₀(𝑝^𝑁𝑥−𝑧_𝐼𝑖)=1. But |𝑝^𝑁𝑥−𝑧_𝐼1|≤1 and |𝑝^𝑁𝑥−𝑧_𝐼2|≤1. With the strong triangle inequality |𝑧_𝐼1−𝑧_𝐼2|≤1. From the condition 𝐼=𝑧_𝐼+𝐵₀∈ℚ_𝑝/𝐵₀, we can get the equality 𝐼₁=𝐼₂.
Thus, the sum with the parameter 𝐼 consists of at most one nonzero term for fixed 𝑁 and 𝑥.
Further, following from (11) and (13), we obtain 𝜓_𝑁𝑗𝐼(𝑥)≤𝑝^−𝑁/2,𝑓,𝜓_𝑁𝑗𝐼≤𝑓_{𝐿2(𝑄)𝑝}. (14) The relations (14) obtain the following estimate if 𝑁≥1^𝑝−1_𝑗=1𝐼𝑓,𝜓_𝑁𝑗𝐼𝜓_𝑁𝑗𝐼(𝑥)≤𝑝^−𝑁/2𝑓_{𝐿2(𝑄)𝑝}𝑝−1. (15) The estimate gives the uniform convergency of the first series in (13).
The condition 𝑓∈𝐷(𝑇^𝛼) and (12) imply (𝑓,𝜓_𝑁𝑗𝐼)=𝑝^(𝑁−1)(𝑇^𝛼𝑓,𝜓_𝑁𝑗𝐼) for 𝑁<1. For a fixed 𝑁<1, using Cauchy-Schwarz equality and (14), we obtain ^𝑝−1_𝑗=1𝐼𝑓,𝜓_𝑁𝑗𝐼𝜓_𝑁𝑗𝐼(𝑥)=^𝑝−1_𝑗=1𝐼𝑝^(𝑁−1)𝑇^𝛼𝑓,𝜓_𝑁𝑗𝐼𝜓_𝑁𝑗𝐼(𝑥)≤^𝑝−1_𝑗=1𝐼𝑇^𝛼𝑓,𝜓_𝑁𝑗𝐼^21/2×^𝑝−1_𝑗=1𝐼𝑝^2(𝑁−1)𝜓_𝑁𝑗𝐼(𝑥)^21/2≤𝑇^𝛼𝑓_{𝐿2(𝑄)𝑝}^𝑝−1_𝑗=1𝑝^{−𝑁+2(𝑁−1)1/2}. (16) The estimate obtained above means the second series in (13) is uniformly convergent if 𝛼>1/2. Thus, function 𝑓 is continuous in 𝐷(𝑇^𝛼) on ℚ_𝑝 for 𝛼>1/2. Theorem 2 is proved.

Next when 𝛼≤1/2, we will give an example in which the function belongs to 𝐷(𝑇^𝛼) but is not continuous on ℚ_𝑝.

Example 3. The function 𝑓(𝑥)=⁻¹_−∞1|𝑁|𝑝^(𝑁−1)/2𝜓_𝑁10(𝑥) (17) belongs to 𝐷(𝑇^𝛼) but 𝑓(𝑥) is not continuous on ℚ_𝑝  for 𝛼≤1/2.

It is easy to see 𝑓∈𝐿₂(ℚ_𝑝) and its Fourier transform is 𝑓^∧(𝜉)=⁻¹_−∞1|𝑁|𝑝^(𝑁−1)/2𝜓^∧_𝑁10(𝜉). (18) From (8) and (12), ⟨𝜉⟩^𝛼𝜓^∧_𝑁10(𝜉)=𝑝^{𝛼(1−𝑁)}𝜓^∧_𝑁10(𝜉). Thus 𝜉^𝛼𝑓^∧(𝜉)=⁻¹_−∞1|𝑁|𝑝^(𝑁−1)/2𝑝^{𝛼(1−𝑁)}𝜓^∧_𝑁10(𝜉). (19)⟨𝜉⟩^𝛼𝑓^∧(𝜉)∈𝐿₂(ℚ_𝑝) for 𝛼≤1/2 and ⟨𝜉⟩^𝛼𝑓^∧(𝜉)∉𝐿₂(ℚ_𝑝) for 𝛼>1/2 with the orthonormal basis {𝜓_𝑁10}_𝑁<0. So, 𝑓(𝑥)∈𝐷(𝑇^𝛼) for 𝛼≤1/2.

Next, We will show that 𝑓(𝑥) is not continuous on ℚ_𝑝. First, using (11), we rewrite the definition (17) of 𝑓 as 𝑓(𝑥)=⁻¹_−∞1|𝑁|𝑝^−1/2𝜒𝑝^𝑁−1𝑥Δ₀𝑝^𝑁𝑥. (20) We consider the sequence 𝑥_𝑛=𝑝^𝑛, 𝑛∈ℕ. It is easy to see 𝑥_𝑛→0, (𝑛→∞) in the 𝑝-adic norm |⋅|_𝑝. Moreover, Δ₀(𝑝^𝑁𝑥_𝑛)=Δ₀(𝑝^𝑁+𝑛)=0 when 𝑁+𝑛≤−1. On the other hand, when 𝑁+𝑛≤1, 𝑝^𝑁−1𝑥_𝑛 become an integer 𝑝-adic number, and then 𝜒(𝑝^𝑁−1𝑥_𝑛)=𝜒(𝑝^{𝑁+𝑛−1})=1. From the above relations and (20) we can get that 𝑓𝑥_𝑛=𝑓𝑝^𝑛=𝑝^−1/2𝜒𝑝⁻¹𝑛+⁻¹_{𝑁=−𝑛+1}1|𝑁|⟶∞as𝑛⟶∞. (21) Therefore, 𝑓(𝑥) cannot be continuous at 𝑥=0.

4. Pseudodifferential Equation

In this section we will consider the pseudodifferential equation 𝑇^𝛼𝑢+𝑢=𝛿_𝑥𝑘,𝑢∈𝐿₂ℚ_𝑝,𝑥_𝑘∈ℚ_𝑝, (22) where 𝑇^𝛼∶𝐿₂(ℚ_𝑝)→𝑆^′(ℚ_𝑝) is pseudodifferential operator in the distribution sense.

Theorem 4. Equation (22) has a unique solution 𝑢=𝑢_𝑘∈𝐿₂(ℚ_𝑝) for 𝛼>1/2 and has no solutions belonging to 𝐿₂(ℚ_𝑝) for 𝛼≤1/2.

Proof. Let 𝜑∈𝐷(𝑇^𝛼). 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 {𝜓_𝑁𝑗𝐼}. For {𝜓_𝑁𝑗𝐼} are continuous functions on ℚ_𝑝 we can write 𝜑𝑥_𝑘=^+∞_𝑁=−∞^𝑝−1_𝑗=1𝐼𝜑,𝜓_𝑁𝑗𝐼𝜓_𝑁𝑗𝐼𝑥_𝑘 (23) for 𝑥=𝑥_𝑘.
Consider 𝜓_𝑁𝑗𝐼𝑥_𝑘=𝑝^−𝑁/2𝜒𝑝^𝑁−1𝑗𝑥_𝑘Δ₀𝑝^𝑁𝑥_𝑘−𝑧_𝐼=𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘Δ₀𝑝^𝑁𝑥_𝑘−𝑧_𝐼. (24) It is easy to see 𝜓_𝑁𝑗𝐼(𝑥_𝑘)≠0 equals |𝑝^𝑁𝑥_𝑘−𝑧_𝐼|≤1. Here 𝐼=𝑧_𝐼+𝐵₀∈ℚ_𝑝/𝐵₀ and by the strong triangle inequality we obtain |𝑝^𝑁𝑥_𝑘−𝑧_𝐼|≤1 equals 𝑧_𝐼={𝑝^𝑁𝑥_𝑘}. Then, we obtain the conclusion that 𝜓_𝑁𝑗𝐼𝑥_𝑘=0,𝑧_𝐼≠𝑝^𝑁𝑥_𝑘,𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘,𝑧_𝐼=𝑝^𝑁𝑥_𝑘. (25) And then 𝛿_𝑥𝑘,𝜑=𝜑𝑥_𝑘=^+∞_𝑁=−∞^𝑝−1_𝑗=1𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝜑,𝜓_{𝑁𝑗{𝑝𝑁𝑥𝑘}}=^+∞_𝑁=−∞^𝑝−1_𝑗=1𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝜓_{𝑁𝑗{𝑝𝑁𝑥𝑘}},𝜑. (26) For 𝑆(ℚ_𝑝)⊂𝐷(𝑇^𝛼), the equality (26) implies that 𝛿_𝑥𝑘=^+∞_𝑁=−∞^𝑝−1_𝑗=1𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝜓_{𝑁𝑗{𝑝𝑁𝑥𝑘}}, (27) in which series converges uniformly in 𝑆^′(ℚ_𝑝).
We suppose that there is a function 𝑢_𝑘∈𝐿₂(ℚ_𝑝) which can be represented as a convergent series in 𝐿₂: 𝑢_𝑘(𝑥)=^+∞_𝑁=−∞^𝑝−1_𝑗=1𝐼𝑐_𝑁𝑗𝐼𝜓_𝑁𝑗𝐼(𝑥). (28) Applying pseudodifferential operator  𝑇^𝛼+𝐼  on the both sides, we obtain a series with (12) 𝑇^𝛼𝑢_𝑘+𝑢_𝑘=⁰_𝑁=−∞^𝑝−1_𝑗=1𝐼𝑐_𝑁𝑗𝐼1+𝑝^{𝛼(1−𝑁)}𝜓_𝑁𝑗𝐼+^+∞_𝑁=1^𝑝−1_𝑗=1𝐼2𝑐_𝑁𝑗𝐼𝜓_𝑁𝑗𝐼, (29) which converges in 𝑆^′(ℚ_𝑝) for 𝑇^𝛼𝑆(ℚ_𝑝)⊂𝐿₂(ℚ_𝑝). Comparing the two series (27) and (29), we obtain 𝑐_{𝑁𝑗{𝑝𝑁𝑥𝑘}}=12𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝑝,𝑁≥1,_{𝛼(1−𝑁)}+1⁻¹𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘,𝑁<1. (30) Thus, 𝑢_𝑘(𝑥)=^+∞_𝑁=1^𝑝−1_𝑗=112𝑝^−𝑁/2×𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝜓_{𝑁𝑗{𝑝𝑁𝑥𝑘}}(𝑥)+⁰_𝑁=−∞^𝑝−1_𝑗=1𝑝^{𝛼(1−𝑁)}+1⁻¹𝑝^−𝑁/2×𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝜓_{𝑁𝑗{𝑝𝑁𝑥𝑘}}(𝑥). (31) Next, We will show that the series (31) belong to 𝐿₂(ℚ_𝑝) for 𝛼>1/2. For the general term of the first series we obatin 12𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘²≤14𝑝^−𝑁,𝑁≥1 (32) that means the first series converges in 𝐿₂(ℚ_𝑝) for any 𝛼∈ℝ.
We estimate the general term of the second series as follows: 𝑝^{𝛼(1−𝑁)}+1⁻¹𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘²≤𝑝^−𝑁𝑝^{𝛼(1−𝑁)}+1⁻²≤𝑝^−2𝛼𝑝^{−𝑁(1−2𝛼)},𝑁<1. (33) It is easy to see the second series converges in 𝐿₂(ℚ_𝑝) for 𝛼>1/2.
Thus 𝑢_𝑘 is a unique solution of (31) for 𝛼>1/2.
For 𝑝^{𝛼(1−𝑁)}+1≤2𝑝^{𝛼(1−𝑁)} with 𝑁<1, we estimate the general term of the second series as follows: 𝑝^{𝛼(1−𝑁)}+1⁻¹𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘²≥14𝑝^−2𝛼𝑝^{−𝑁(1−2𝛼)},𝑁<1 (34) that means the series diverges in 𝐿₂(ℚ_𝑝)   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 ℚ_𝑝 if 𝛼>1 and converges uniformly on the ball not containing 𝑥_𝑘 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.
When 𝑁<1 the general term (31) of the second series satisfies 𝑝^{𝛼(1−𝑁)−1}𝑝^−𝑁≤𝑝^−𝛼𝑝^{−𝑁(1−𝛼)}. (35) 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.

Let 𝛼>1/2 and fixed points 𝑥₁,𝑥₂,…,𝑥_𝑛∈ℚ_𝑝. Then, define 𝑆𝑝{𝑢_𝑘}^𝑛₁ as the linear span of solutions 𝑢_𝑘 (𝑘=1,2,…,𝑛) of (22). Hence, we have the following.

Theorem 6. 𝑆𝑝{𝑢_𝑘}^𝑛₁∩𝐷(𝑇^𝛼/2)={0} for 1/2<𝛼≤1 and 𝑆𝑝{𝑢_𝑘}^𝑛₁⊂𝐷(𝑇^𝛼/2) for 𝛼>1.

Proof. The solution 𝑢_𝑘 of (22) is obtained by (31). Considering the expansion (31) and semigroup property (c.f. [19]) 𝑇^𝛼1𝑇^𝛼2=𝑇^𝛼1+𝛼2,𝛼₁,𝛼₂>0 (36) of 𝑇^𝛼. Obviously, 𝑢_𝑘∈𝐷(𝑇^𝛼/2) if and only if the following series converge in 𝐿₂(ℚ_𝑝): ^+∞_𝑁=1^𝑝−1_𝑗=112𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝜓_{𝑁𝑗{𝑝𝑁𝑥𝑘}}(𝑥)+⁰_𝑁=−∞^𝑝−1_𝑗=1𝑝^{𝛼(1−𝑁)}+1⁻¹𝑝^−𝑁/2×𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝑝^{(𝛼/2)(1−𝑁)}𝜓_{𝑁𝑗{𝑝𝑁𝑥𝑘}}(𝑥). (37) For the general term of the first series,we have the estimate as follows: 12𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘²≤14𝑝^−𝑁,𝑁≥1 (38) that means it converges in 𝐿₂(ℚ_𝑝) for any 𝛼∈ℝ.
Similarly, for the general term of the second series, we have 𝑝^{𝛼(1−𝑁)}+1⁻¹𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝑝^{(𝛼/2)(1−𝑁)2}≤𝐶𝑝^{(𝛼−1)𝑁},𝑁<1. (39) Hence the series converges when 𝛼>1. Then 𝑆𝑝{𝑢_𝑘}^𝑛₁⊂𝐷(𝑇^𝛼/2) if 𝛼>1.
For 𝑝^{𝛼(1−𝑁)}+1≤2𝑝^{𝛼(1−𝑁)} with 𝑁<1, the general term of the second series can be estimated below 𝑝^{𝛼(1−𝑁)}+1⁻¹𝑝^−𝑁/2𝜒−𝑝^𝑁−1𝑗𝑥_𝑘𝑝^{(𝛼/2)(1−𝑁)2}≥14𝑝^−𝛼𝑝^{−𝑁(1−𝛼)},𝑁<1 (40) that means it diverges in 𝐿₂(ℚ_𝑝) for 𝛼≤1.
Hence 𝑢_𝑘∉𝐷(𝑇^𝛼/2). Since the estimate (40) does not depend on the choice of 𝑢_𝑘. And considering the functions {𝜓_{𝑁𝑗{𝑝𝑁𝑥𝑘}}(𝑥)} (𝑁<1) of the basis {𝜓_𝑁𝑗𝐼(𝑥)} in the 𝑢_𝑘 (𝑘=1,2,…,𝑛) of (31) are different for any small negative 𝑁, we obtain that 𝑆𝑝{𝑢_𝑘}^𝑛₁∩𝐷(𝑇^𝛼/2)={0} if 1/2<𝛼≤1. Theorem 6 is proved.

5. Conclusions

In this paper, solutions of pseudodifferential equations with the type 𝑇^𝛼𝑢+𝑢=𝛿_𝑥𝑘 over 𝑝-adic field ℚ_𝑝 are considered. First, we give the condition for the continuity of arbitrary function in the domain of definition 𝐷(𝑇^𝛼) 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(ℚ_𝑝) 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(𝑇^𝛼) is obtained.

Acknowledgments

The author would like to thank the Academic Editor Guo-cheng Wu and all anonymous reviewers for their kind support which helped the author to improve the paper considerably. This work was supported by the National Natural Science Foundation of China under NSFS nos. 11071109, 11001119 and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).