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

Abstract

In this paper, we study the solutions of the pseudodifferential equations of type 𝐷^𝛼𝑢=𝑣 over 𝑝-adic field ℚ_𝑝, where 𝐷^𝛼 is a 𝑝-adic fractional pseudodifferential operator. If 𝑣 is a Bruhat-Schwartz function, then there exists a distribution 𝐸_𝛼, a fundamental solution, such that 𝑢=𝐸_𝛼∗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.

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.

A 𝑝-adic fractional pseudodifferential operator 𝐷^𝛼, which is defined by Su in 1992 [17], is an operator of the form 𝐷^𝛼𝜑(𝑥)=𝜉^𝛼𝜑^∧(𝜉)^∨(𝑥) (1) for 𝜑∈𝑆(ℚ_𝑝), where 𝜑^∧ and 𝜑^∨ denote the Fourier transform and the inverse Fourier transform, 𝑆(ℚ_𝑝) denotes the ℂ-vector space of the Bruhat-Schwartz functions over 𝑝-adic fields ℚ_𝑝, and 𝛼 is a positive real number. A 𝑝-adic fractional pseudodifferential equation is an equation of type 𝐷^𝛼𝑢=𝑣. (2) If 𝑣∈𝑆(ℚ_𝑝), then there exists a distribution 𝐸_𝛼, a fundamental solution, such that 𝑢=𝐸_𝛼∗𝑣 is a solution.

We will also show that (2) has a solution 𝑢=𝐸_𝛼∗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 ℚ_𝑝 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=𝑝^𝑟(𝑚/𝑛), where 𝑟=ord_𝑝(𝑥)∈ℤ and integers 𝑚 and 𝑛 are 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 can be uniquely presented as series 𝑥=^+∞_𝑖=𝑚𝑥_𝑖𝑝^𝑖,0≤𝑥_𝑖≤𝑝−1,𝑖=𝑚,𝑚+1,…,𝑚∈ℤ,𝑥_𝑚≠0, (3) convergent in 𝑝-adic norm (the canonical presentation of 𝑥).

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, nondiscrete, complete, and totally disconnected topological field.

Denote by 𝐵₀ the ring of integers in ℚ_𝑝, 𝐵₀={𝑥∈ℚ_𝑝∶|𝑥|≤1}. Let 𝑑𝑥 denote the Haar measure on ℚ_𝑝 normalized by the condition ∫_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 center 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 has the following definition: 𝜑→−_𝑘^𝜀0, 𝑘→∞ if and only if for any compact subset 𝐸⊂ℚ_𝑝,  𝜑_𝑘→0,  𝑘→∞ hold uniformly on 𝐸. The convergence in 𝑆 has the following meaning: 𝜑→−_𝑘^𝑆0, 𝑘→∞ if and only if there exist 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 Bruhat-Schwartz function space.

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

The Fourier transform of 𝜑∈𝑆(ℚ_𝑝) is defined by the following formula: 𝜑^∧(𝜉)=_ℚ𝑝𝜑(𝑥)𝜒_𝑝(𝜉𝑥)𝑑𝑥,𝜉∈ℚ_𝑝, (6) and the 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 from 𝑆 into 𝑆.

The Fourier transform 𝑓^∧ of a distribution 𝑓∈𝑆^′(ℚ_𝑝) is defined by the standard relation ⟨𝑓^∧,𝜑⟩=⟨𝑓,𝜑^∧⟩, 𝜑∈𝑆(ℚ_𝑝).

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 𝐷^𝛼(𝛼∈ℝ) which is defined as 𝐷^𝛼𝜑(𝑥)=𝜉^𝛼𝜑^∧(𝜉)^∨(𝑥), (8) for 𝜑∈𝑆(ℚ_𝑝). It is easy to see that 𝐷^𝛼𝜑∈𝑆(ℚ_𝑝). With 𝐷^𝛼𝑓,𝜑=𝑓,𝐷^𝛼𝜑,𝑓∈𝑆^′ℚ_𝑝,𝜑∈𝑆ℚ_𝑝, (9) the defined domain of 𝐷^𝛼 can be extended to the space 𝑆^′(ℚ_𝑝). Thus, we also have 𝐷^𝛼𝑓=(⟨𝜉⟩^𝛼𝑓^∧(𝜉))^∨ with 𝑓∈𝑆^′(ℚ_𝑝) and 𝐷^𝛼𝑓∈𝑆^′(ℚ_𝑝).

Definition 1. If 𝛼>0, then 𝐷^𝛼 is defined as 𝑝-adic derivative of the order 𝛼 on 𝑆^′(ℚ_𝑝). And if 𝛼<0, then 𝐷^𝛼 is defined as 𝑝-adic integral of the order −𝛼 on 𝑆^′(ℚ_𝑝). 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 𝐷^𝛼. Consider 𝐷^𝛼𝑢=𝑘_𝛼∗𝑢,for𝑢∈𝑆^′, (10) where 𝑘_𝛼=1−𝑝^𝛼1−𝑝^−𝛼−1𝜋_−𝛼+𝑝^𝛼−1𝑝^𝛼+1Δ−1⁰1,𝛼≠0,−1;𝛿,𝛼=0;1−𝑝1−log_𝑝|𝑥|_𝑝Δ₀,𝛼=−1. (11)

Here, Δ₀ is the indicative function of set 𝐵₀ satisfying Δ₀=1_𝐵0, and 𝜋_𝛼 is a distribution defined as 𝜋_𝛼,𝜑=_ℚ𝑝|𝑥|_𝑝^𝛼−1𝜑(𝑥)−𝜑(0)𝑑𝑥,𝜑∈𝑆. (12)

Lemma 2 (see [19, 20]). 𝑘^∧_𝛼=⟨𝜉⟩^𝛼, and 𝑘_𝛼 satisfies the property of semigroup: 𝑘_𝛼∗𝑘_𝛽=𝑘_𝛼+𝛽.

3. Solutions of the Pseudodifferential Equations on Sobolev Type Spaces

We now consider the following pseudodifferential equation: 𝐷^𝛼𝑢=𝑣,with𝑣∈𝑆ℚ_𝑝,𝛼∈ℝ. (13) We say that 𝐸_𝛼∈𝑆^′(ℚ_𝑝) is a fundamental solution of (13) if 𝐸_𝛼∗𝑣 is a solution.

Lemma 3. If 𝐸_𝛼 is a fundamental solution of (13), then for any constant 𝑐, 𝐸_𝛼+𝑐 is also a fundamental solution.

Proof. Let 𝐸_𝛼 be a fundamental solution for (13), then 𝐷^𝛼𝐸_𝛼+𝑐∗𝑣=𝐷^𝛼𝐸_𝛼∗𝑣+𝑐∗𝑣=𝑢+𝐷^𝛼(𝑐∗𝑣)=𝑢, (14) because 𝑢 and the constant function, 𝑐∗𝑣, are in the domain of 𝐷^𝛼.

Theorem 4. A fundamental solution of (13) is𝐸_𝛼=1−𝑝^−𝛼1−𝑝^𝛼𝜋−1^𝛼+𝑝^−𝛼−1𝑝^−𝛼+1Δ−1⁰1,𝛼≠1;𝛿,𝛼=0;1−𝑝1−log_𝑝|𝑥|_𝑝Δ₀,𝛼=1. (15)

Proof. We use the definition of 𝐷^𝛼 (8), then 𝐷^𝛼𝐸_𝛼∗𝑣=𝜉^𝛼𝐸_𝛼∗𝑣^∧∨=𝜉^𝛼𝐸^∧_𝛼𝑣^∧∨. (16) The existence of a fundamental solution 𝐸_𝛼 is equivalent to the existence of a distribution 𝐸^∧_𝛼 satisfying 𝜉^𝛼𝐸^∧_𝛼=1, (17) as distributions. By Lemma 2, we have 𝐸^∧_𝛼=lim_{𝑠→−𝛼}𝜉^𝑠=lim_{𝑠→−𝛼}𝑘^∧_𝑠=𝑘^∧_−𝛼. (18) Then, 𝐸_𝛼=𝑘_−𝛼. By (11), Theorem 4 is proved.
In what follows we will introduce some relevant spaces over 𝑝-adic fields (see [20]).
The Holder type space 𝐶^𝜎(ℚ_𝑝), 𝜎∈[0,+∞):(1)for 𝜎=0, we define 𝐶⁰(ℚ_𝑝)=𝐶(ℚ_𝑝) as the continuous function space on ℚ_𝑝;(2)for 𝜎>0, we define 𝐶^𝜎(ℚ_𝑝)as the set of all distributions 𝑢∈𝑆^′ with the Littlewood-Paley decomposition 𝑢=∑^∞_𝑗=0𝑢_𝑗 with respect to the following norm: ‖𝑢‖_𝐶^𝜎_(ℚ𝑝)=sup_𝑗𝑝^−𝑗𝜎𝑢_{𝑗𝐿∞(ℚ𝑝)}. (19)Thus, 𝐶^𝜎(ℚ_𝑝) becomes a Banach space with the above norm.
The Sobolev type space 𝑊^𝑠(ℚ_𝑝), 𝜎∈[0,+∞): 𝑊^𝑠ℚ_𝑝=𝑢∈𝑆^′∶‖𝑢‖_𝑊^𝑠_(ℚ𝑝)=⟨⋅⟩^𝑠𝑢^∧(⋅)^∨_𝐿2ℚ_𝑝<∞, (20) where 𝐿₂(ℚ_𝑝) denotes the set of measurable functions 𝑓 on ℚ_𝑝 satisfying the condition ∫_ℚ𝑝|𝑓(𝑥)|²𝑑𝑥<∞.

Lemma 5 (see [21, 22]). For 𝑠>1/2, it holds that 𝑊^𝑠ℚ_𝑝⊂𝐶^𝑠−1/2ℚ_𝑝. (21)

Lemma 6. For any 𝛼>0 and 𝑠≥0, the mapping 𝐷^𝛼∶𝑊^𝑠+𝛼→𝑊^𝑠 is a well-defined continuous mapping between Banach spaces.

Proof. Let 𝑢∈𝑆. We have that 𝐷^𝛼𝑢²_𝑊𝑠=_ℚ𝑝𝜉^𝑠𝐷^𝛼𝑢^∧∨2=𝑑𝜉_ℚ𝑝𝜉^𝑠𝜉^𝛼𝑢^∧∨2=𝑑𝜉_ℚ𝑝𝜉^𝑠+𝛼𝑢^∧∨2𝑑𝜉=‖𝑢‖²_𝑊^𝑠+𝛼. (22) The result follows from the fact that 𝑆 is dense in 𝑊^𝑠+𝛼.

Theorem 7. Let 𝐷^𝛼(𝛼>0) be a 𝑝-adic fractional pseudodifferential operator. Let 𝑠 be a positive real number satisfying 𝑠≥1/2−𝛼. Then, the equation 𝐷^𝛼𝑢=𝑣,𝑣∈𝑆, (23) has a unique uniformly continuous solution 𝑢∈𝑊^𝑠+𝛼.

Proof. Let 𝑣∈𝑆, as 𝑆 is dense in 𝑊^𝑠 and we have ‖𝑢‖²_𝑊^𝑠+𝛼=_ℚ𝑝𝜉^𝑠+𝛼(𝑢)^∧∨2=𝑑𝜉_ℚ𝑝𝜉^𝑠+𝛼𝐸_𝛼∗𝑣^∧∨2=𝑑𝜉_ℚ𝑝𝜉^𝑠+𝛼𝐸^∧_𝛼𝑢^∧∨2𝑑𝜉=_ℚ𝑝𝜉^𝑠𝑢^∧∨2𝑑𝜉=‖𝑣‖²_𝑊^𝑠. (24) In this way, we established the existence of 𝑢∈𝑊^𝑠+𝛼. By Lemma 5, 𝑢∈𝐶^{𝑠+𝛼−1/2} is uniformly continuous for 𝑠+𝛼−1/2≥0.
Finally, we show that 𝑢 is unique. Indeed, if 𝐷^𝛼𝑢^′=𝑣, then 𝐷^𝛼𝑢−𝑢^′=0,i.e.,𝜉^𝛼𝑢−𝑢^′∧=0, (25) And, thus, (𝑢−𝑢^′)^∧=0. Then, (𝑢−𝑢^′)𝜒_𝑥(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∈ℚ_𝑝. Theorem 7 is proved.

4. Conclusion

In this work, the pseudodifferential equations of type 𝐷^𝛼𝑢=𝑣 over 𝑝-adic field ℚ_𝑝 were investigated, where 𝐷^𝛼 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 𝑊^𝑠+𝛼 were obtained when 𝑠≥1/2−𝛼, using the explicit calculation of fundamental solutions approach.

Acknowledgments

The author would like to thank the Academic Editor Thabet Abdeljawad 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 NSFC nos. 11071109 and 11001119 and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).