1 Introduction

In the archimedean context, the theory of pseudo-differential operators had their origins in the mid-sixties with the works of Grothendieck, H\(\ddot{o}\)rmander, Kohn and Nirenberg, as a tool for studying the problems of partial differential equations, see e.g. [14, 18, 26]. Since then, there have been various applications of these types of operators in different fields, see e.g. [13, 19, 31,32,33].

In the no-archimedean context (specifically the p-adic numbers), it was presented in [6,7,8] by Avetisov, Bikulov, Kozyrev and Osipov, a preamble to a new quantum theory where the p-adic pseudo-differential operators are protagonists due to their multiple applications in mathematical physics, cellular neural networks, spread of infectious diseases, oil formation, probability theory, among others, see e.g., [1, 3,4,5, 7, 9, 10, 15,16,17, 20,21,25, 27, 29, 35,36,41, 43].

In this article, we introduced two new classes of p-adic pseudo-differential operators which we denote, respectively, by \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) and \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\). The pseudo-differential operators \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) are defined as

$$\begin{aligned} \left( {\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\varphi \right) (x)&:=-\int \limits _{{\mathbb {Q}}_{p}^{n}}\chi _{p}(-x\cdot \xi )\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}{\widehat{\varphi }}(\xi )d^{n}\xi \\&=-{\mathcal {F}}_{\xi \rightarrow x}^{-1}\left( \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}{\widehat{\varphi }}(\xi ) \right) , \ \ \varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n}), \ \ x\in {\mathbb {Q}}_{p}^{n}, \end{aligned}$$

where \({\mathbb {Q}}_{p}^{n}\) denotes the n-dimensional p-adic numbers, l is a nonnegative real number, \({\mathcal {D}}({\mathbb {Q}}_{p}^{n})\) is the space of locally constant functions on \({\mathbb {Q}}_{p}^{n}\) with compact support, \({\widehat{\varphi }}\) is the Fourier transform of \(\varphi \), \({\mathcal {F}}^{-1}_{\xi \rightarrow x}\) denotes the inverse Fourier transform and \(\left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}\) is the symbol associated to the operator; on the other hand, the pseudo-differential operators \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\) are given by

$$\begin{aligned} {\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }(f)(x)= & {} -{\mathcal {F}}^{-1}_{\xi \rightarrow x}\left( \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}{\widehat{f}}(\xi ) \right) \\= & {} -\int \limits _{{\mathbb {Q}}_{p}^{n}}\chi _{p}(-x\cdot \xi )\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}{\widehat{f}}(\xi ) d^{n}\xi , \end{aligned}$$

where \(\alpha \) is a nonnegative real number, \(\left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}\) is the symbol of the operator and f is a function belonging to a new class of p-adic Sobolev space. It will be seen in this work that, due to the characteristics of the symbols introduced here, our pseudo-differential operators are not part of the family of operators studied in [38].

With these new pseudodifferential operators we make an important contribution to the theory of parabolic p-adic pseudo-differential equations and its applications to p-adic theoretical physics and probability theory. Recently in [40], the study of p-adic pseudo-differential operators satisfying the positive maximum principle on the space of the continuous functions that are annulled in the infinite (i.e., \(C_{0}({\mathbb {Q}}_{p}^{n})\)) was initiated by the authors Torresblanca-Badillo and Zúñiga-Galindo. Since then, there has been a great interest in finding new non-archimedean pseudo-differential operators that satisfy this principle. Motivated by this, we show that pseudo-differential operators \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) satisfy the positive maximum principle on \(C_{0}({\mathbb {Q}}_{p}^{n})\), see Theorem 2. This result and the theory of dissipative operators, allowed us to show that operators \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) are connected with Feller semigroups \(\{T_{t}\}_{t\ge 0}\), see Theorem 4.

Also, in recent years, there has been a great interest in studying the connections between the p-adic pseudo-differential operators and stochastic processes, see e.g. [1, 3, 4, 9, 10, 15, 17, 21, 41]. In our case, denoting by \({\mathcal {P}}({\mathbb {Q}}_{p}^{n})\) and \(D_{{\mathbb {Q}}_{p}^{n}}[0,\infty )\), respectively, the family of Borel probability measures on \({\mathbb {Q}}_{p}^{n}\) and the space of right continuous functions \(f:[0,\infty )\rightarrow {\mathbb {Q}}_{p}^{n}\) with left limits, we show that for each \(\upsilon \in {\mathcal {P}}({\mathbb {Q}}_{p}^{n})\), there exists a Markov process \({\mathfrak {X}}\) corresponding to \(\{T_{t}\}_{t\ge 0}\) with initial distribution \(\upsilon \) and sample paths in \(D_{{\mathbb {Q}}_{p}^{n}}[0,\infty )\). Moreover, we prove that the procces \({\mathfrak {X}}\) is strong Markov with respect to the filtration \({\mathcal {F}}_{t^{+}}^{{\mathfrak {X}}}=\bigcap _{\epsilon >0}{\mathcal {F}}_{t+\epsilon }^{{\mathfrak {X}}}\). We also prove that for each \(x\in {\mathbb {Q}}_{p}^{n}\) there exists a Markov process \({\mathfrak {X}}_{x}\) corresponding to \(\{T_{t}\}_{t\ge 0}\) with initial distribution \(\delta _{x}\) and sample paths in \(D_{{\mathbb {Q}}_{p}^{n}}[0,\infty )\), see Theorem 5.

On the other hand, in this article we introduce a class of Sobolev spaces denoted by \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) which is the natural domain of the pseudo-differential operators \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\). Since our Sobolev spaces are Hilbert spaces and they are contained in \(L^{2}({\mathbb {Q}}_{p}^{n})\) (the set \(L^{2}({\mathbb {Q}}_{p}^{n})\) is the set of all square integrable functions on \({\mathbb {Q}}_{p}^{n}\)), by using the theory of m-dissipative operators we show that the operators \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\) are the infinitesimal generators of contraction semigroups \(\left\{ T(t):t\ge 0\right\} \) in \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\). Moreover, we prove that for any \(T>0\), \(f\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) and \(g\in C([0,T],B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}))\), the function

$$\begin{aligned} u(t):=T(t)f+\int \limits _{0}^{t}T(t-s)g(s)ds, \ \ \text { for all } t\in [0,T], \end{aligned}$$

is the unique solution of the inhomogeneous equation

$$\begin{aligned} \left\{ \begin{array}{ll} u\in C\left( [0,T],B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\right) \cap C^{1}\left( [0,T],B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\right) ; &{} \\ &{} \\ \frac{\partial u}{\partial t}(x,t)={\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }u(t)+g(t), \ \ t\in [0,T], \\ &{} \\ u(0)=f, \end{array} \right. \end{aligned}$$

see Theorems 7 and 8.

Finally, for \(t_{0}\ge 0\) and \(T>t_{0}\), we consider a function \(f:[t_{0},T]\times B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\rightarrow B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\), which is assumed to be continuous in all \(t\in [t_{0},T]\) and uniformly Lipschitz continuous (with constant L) on \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\), and the semilinear initial value problem:

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}(t)-{\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }u(t)=f(t,u(t)), &{} t>t_{0}, \\ &{} \\ u(t_{0})=u_{0}, \ \ u_{0}\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}) \text {.} &{} \end{array} \right. \end{aligned}$$
(1.1)

We prove that there exists a unique function \(u\in C([t_{0},T]:B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}))\) that satisfies the integral equation

$$\begin{aligned} u(t)=T(t-t_{0})u_{o}+\int \limits _{t_{0}}^{t}T(t-s)f(s,u(s))ds, \end{aligned}$$

which is a mild solution of the semilinear initial value problem in (1.1), see Theorem 9.

We begin this paper by recapitulating some basic concepts and results on the p-adic analysis, see Sect. 2. In Sect. 3, the p-adic pseudo-differential operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) is introduced. We will study some properties corresponding to the convolution kernel \(K_{l}\) associated with these operators, see Theorem 1. Also, we show that the pseudo-differential operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) is dissipative on \(C_{0}({\mathbb {Q}}_{p}^{n})\), see Remark 3; furthermore, it is shown that the operator is closable and its closure \(\overline{{\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}}\) is single-valued and generates a strongly continuous, positive, contraction semigroup \(\{ T_{t}\}_{t\ge 0}\) on \(C_{0}({\mathbb {Q}}_{p}^{n})\), see Theorem 3. Finally, in this section we obtain Feller semigroups and stochastic processes associated with these operators. In Sect. 4, we introduce the Sobolev spaces \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) and the p-adic pseudo-differential operator \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\). We show that the operator \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\) is m-dissipative in \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\), see Theorem 6. According to this result, in that section we will also show that the operator \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\) is the infinitesimal generator of a contraction semigroup in \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\). In consequence, we can introduce and solve new classes of inhomogeneous equations and semilinear initial value problems.

2 Fourier analysis on p-adic numbers

We fix a prime number p. The p-adic absolute value (or p-adic norm) of a rational number x is denoted and defined as follows:

$$\begin{aligned} \left| x\right| _{p}=\left\{ \begin{array}{lll} 0\text {,} &{} \text {if} &{} x=0 \\ &{} &{} \\ p^{-\gamma }\text {,} &{} \text {if} &{} x=p^{\gamma }\frac{a}{b}\text {,} \end{array} \right. \end{aligned}$$

where a and b are integers not divisible by p. The integer \(\gamma :=ord(x) \), with \(ord(0):=+\infty \), is called the \(p-\)adic order of x.

The field of p-adic numbers,

$$\begin{aligned} {\mathbb {Q}}_{p}=\left\{ x=\sum _{i=k}^{\infty }a_{i}p^{i}: k\in {\mathbb {Z}}, \text { } k=ord(x), \text { } a_{i}\in \{0,1,2,\dots ,p-1\}, \text { } a_{0}\ne 0 \right\} . \end{aligned}$$

is the completion of the field of rational numbers (\({\mathbb {Q}}\)) with respect to the p-adic norm. We denote and define the fractional part of \(x\in {\mathbb {Q}}_{p}\) as follows:

$$\begin{aligned} \left\{ x\right\} _{p}:=\left\{ \begin{array}{lll} 0\text {,} &{} \text {if} &{} x=0\text { or } ord(x)\ge 0 \\ &{} &{} \\ \sum _{i=k}^{-ord(x)-1}a_{i}p^{i}\text {,} &{}\quad \text {if} &{} ord(x)<0. \end{array} \right. \end{aligned}$$

We extend the \(p-\)adic norm to \({\mathbb {Q}}_{p}^{n}\) by taking \(||x||_{p}:=\max _{1\le i\le n}|x_{i}|_{p},\text { for }x=(x_{1},\dots ,x_{n})\in {\mathbb {Q}}_{p}^{n}\). The p-adic distance over \({\mathbb {Q}}_{p}^{n}\) is defined as \(d(x,y):=||x-y||_{p}\), for \(x, y \in {\mathbb {Q}}_{p}^{n}\). For \(r\in {\mathbb {Z}}\), denote by \(B_{r}^{n}(a)=\left\{ x\in {\mathbb {Q}}_p^n;||x-a||_{p}\le p^{r}\right\} \) the ball of radius \(p^{r}\) with center at \(a=(a_{1},\dots ,a_{n})\in {\mathbb {Q}}_p^n\), and take \(B_{r}^{n}(0)=:B_{r}^{n}\). Note that \( B_{r}^{n}(a)=B_{r}(a_{1})\times \cdots \times B_{r}(a_{n})\), where \( B_{r}(a_{i}):=\left\{ x\in {\mathbb {Q}}_{p}:|x_{i}-a_{i}|_{p}\le p^{r}\right\} \) is the one-dimensional ball of radius \(p^{r}\) with center at \(a_{i}\in {\mathbb {Q}}_{p}\). The ball \(B_{0}^{n}\) equals the product of n copies of \(B_{0}={\mathbb {Z}}_{p}\), the ring of \(p-\)adic integers of \({\mathbb {Q}}_{p}\). We also denote by \(S_{r}^{n}(a)=\left\{ x\in {\mathbb {Q}}_p^n;||x-a||_{p}=p^{r}\right\} \) the sphere of radius \(p^{r}\) with center at \(a=(a_{1},\dots ,a_{n})\in {\mathbb {Q}}_p^n\), and take \(S_{r}^{n}(0)=:S_{r}^{n}\). The balls and spheres are compact subsets in \({\mathbb {Q}}_{p}^{n}\). Moreover, as a topological space \(({\mathbb {Q}}_p^n,||\cdot ||_{p})\) is a totally disconnected and locally compact topological space. Also, \({\mathbb {Q}}_{p}^{n}\) admits a Haar measure \(d^{n}x\) normalized such that \(\int _{{\mathbb {Z}}_{p}^{n}}d^{n}x=1\).

By \(L^{\rho }({\mathbb {Q}}_{p}^{n})\), \(1\le \rho < \infty \), we denote the space of functions \(g:{\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {C}}\) satisfying \(\int \limits _{{\mathbb {Q}}_{p}^{n}}\left| g\left( x\right) \right| ^{\rho }d^{n}x<\infty \); and by \(C_{0}({\mathbb {Q}}_{p}^{n})\) the space of continuous functions \(f:{\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {C}}\) such that \(\lim _{||x||_{p}\rightarrow \infty }f(x)=0\). The space \(C_{0}({\mathbb {Q}}_{p}^{n})\) is a Banach space with the norm \(||f||=sup_{x\in {\mathbb {Q}}_{p}^{n}}|f(x)|\).

A function \(\varphi :{\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {C}}\) is called locally constant if for any \(x\in {\mathbb {Q}}_{p}^{n}\) there exists an integer \(m:=m(x)\) such that

$$\begin{aligned} \varphi (x^{\prime })=\varphi (x)\text { for all }x^{\prime }\in B_{m}^{n}(x). \end{aligned}$$

A function \(\varphi :{\mathbb {Q}}_p^n\rightarrow {\mathbb {C}}\) is called a Bruhat-Schwartz function or a test function if it is locally constant with compact support. The space of Bruhat-Schwartz functions is denoted by \({\mathcal {D}}({\mathbb {Q}}_{p}^{n})\). Now, if \({\mathcal {D}}'({\mathbb {Q}}_p^n)\) denotes the set of continuous functionals (distributions) on \({\mathcal {D}}({\mathbb {Q}}_p^n)\), then each function f in \(L_{loc}^1({\mathbb {Q}}_p^n)\), the space of locally integrable functions, defines a distribution in \(\mathcal D'({\mathbb {Q}}_p^n)\) by the formula:

$$\begin{aligned} \langle f,\varphi \rangle :=\int \limits _{{\mathbb {Q}}_{p}^{n}}f(x)\varphi (x)d^{n}x, \ \ \varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n}). \end{aligned}$$

If \(f\in L^{1}({\mathbb {Q}}_p^n)\), its Fourier transform is defined by

$$\begin{aligned} ({\mathcal {F}} f)(\xi )={\mathcal {F}}_{x\rightarrow \xi }(f):={\widehat{f}}(\xi )=\int \limits _{{\mathbb {Q}}_p^n}\chi _{p}(\xi \cdot x)f(x)d^{n}x, \ \ \xi \in {\mathbb {Q}}_{p}^{n}, \end{aligned}$$

where \(x\cdot \xi :=\sum _{j=1}^{n}x_{j}\xi _{j}\) for \(x=(x_{1},\dots ,x_{n}), \xi =(\xi _{1},\dots ,\xi _{n})\in {\mathbb {Q}}_{p}^{n}\), and \(\chi _{p}(\xi \cdot x)=e^{2\pi i\{\xi \cdot x\}_{p}}\) is an additive character on \({\mathbb {Q}}_{p}^{n}\). The inverse Fourier transform of a function \(f\in L^{1}({\mathbb {Q}}_p^n)\) is

$$\begin{aligned} ({\mathcal {F}}^{-1}f)(x)={\mathcal {F}}^{-1}_{\xi \rightarrow x }(f):={\check{f}}(\xi )=\int \limits _{{\mathbb {Q}}_p^n}\chi _{p}(-x \cdot \xi )f(\xi )d^{n}\xi ,\ \ x \in {\mathbb {Q}}_{p}^{n}. \end{aligned}$$

The set \(L^{2}({\mathbb {Q}}_{p}^{n})\) is the Hilbert space endowed with the scalar product

$$\begin{aligned} \left( f,g\right) =\int \limits _{{\mathbb {Q}}_{p}^{n}}f(x){\overline{g}}(x)d^{n}x, \text { } f,g\in L^{2}({\mathbb {Q}}_{p}^{n}), \end{aligned}$$
(2.1)

so that \(||f||_{L^{2}}=\sqrt{(f,f)}\). If \(f\in L^{2}({\mathbb {Q}}_{p}^{n})\), its Fourier transform is defined as

$$\begin{aligned} ({\mathcal {F}}f)(\xi )=\lim \limits _{k\rightarrow \infty }\int _{||x||\le p^{k}}\chi _{p}(\xi \cdot x)f(x)d^{n}x, \ \ \xi \in {\mathbb {Q}}_p^n, \end{aligned}$$

where the limit is taken in \(L^{2}({\mathbb {Q}}_{p}^{n})\). We recall that the Fourier transform is unitary on \(L^{2}({\mathbb {Q}}_{p}^{n}),\) i.e. \(||f||_{L^{2}({\mathbb {Q}}_{p}^{n})}=||{\mathcal {F}}f||_{L^{2}({\mathbb {Q}}_{p}^{n})}\) for \(f\in L^{2}({\mathbb {Q}}_{p}^{n})\). The Fourier transform \(f\rightarrow {\widehat{f}}\) maps \(L^{2}({\mathbb {Q}}_{p}^{n})\) onto \(L^{2}({\mathbb {Q}}_{p}^{n})\) one-to-one and is mutually continuous. Moreover, the Parseval-Steklov equality holds:

$$\begin{aligned} (f,g)=({\mathcal {F}}f,{\mathcal {F}}g), \ \ ||f||_{L^{2}({\mathbb {Q}}_{p}^{n})}=||{\mathcal {F}}f||_{L^{2}({\mathbb {Q}}_{p}^{n})}, \ \ f,g \in L^{2}({\mathbb {Q}}_{p}^{n}). \end{aligned}$$

For other details, the reader may consult [2, 34, 42].

3 The p-adic pseudo-differential operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) and their applications

In this section, we study new classes of non-archimedean pseudo-differential operators, in the p-adic context, and its connections with contraction semigroups, Feller semigroups and stochastic process. Along this article, we write \({\mathbb {N}}=\left\{ 1,2,\ldots \right\} \) and \({\mathbb {R}}_{+}=\left\{ x\in {\mathbb {R}}: x\ge 0 \right\} \).

Definition 1

(Condition A) We say that the continuous functions \({\varvec{f}}_{i}:{\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {C}}\), \(i=1,2\), satisfies the Condition A if the following properties hold:

  1. (i)

    \({\varvec{f}}_{i}\), \(i=1,2\), are radial functions with \({\varvec{f}}_{1}(\xi )\ne 0\) for all \(\xi \in {\mathbb {Q}}_{p}^{n}\). We use the notation \({\varvec{f}}_{i}(\xi )={\varvec{f}}_{i}(||\xi ||_{p})\), \(i=1,2\), \(\xi \in {\mathbb {Q}}_{p}^{n}\), to indicate that \({\varvec{f}}_{i}\), \(i=1,2\), are radial functions.

  2. (ii)

    \(|{\varvec{f}}_{i}|\), \(i=1,2\), are increasing functions with respect to \(||\cdot ||_{p}\).

  3. (iii)

    There is an integer \(r:=r({\varvec{f}}_{1},{\varvec{f}}_{2})\) such that

    $$\begin{aligned} |{\varvec{f}}_{1}(||\xi ||_{p})|\ge |{\varvec{f}}_{2}(||\xi ||_{p})| \text { if and only if } \xi \in B_{r}^{n}. \end{aligned}$$
  4. (iv)

    There exist \(C_{1}({\varvec{f}}_{2})=:C_{1}\in {\mathbb {R}}_{+}{\setminus } \left\{ 0\right\} \) and \(\beta _{1}({\varvec{f}}_{2})=:\beta _{1}\in {\mathbb {R}}_{+}{\setminus } \left\{ 0\right\} \) such that

    $$\begin{aligned} |{\varvec{f}}_{2}(||\xi ||_{p})|\ge C_{1}||\xi ||_{p}^{\beta _{1}}, \text { for all } \xi \in {\mathbb {Q}}_{p}^{n}\setminus B_{r}^{n}. \end{aligned}$$

Example 1

  1. (i)

    Taking \({\varvec{f}}_{1}(||\xi ||_{p}):=C\) (\(C>0\) is a fixed real constant), and \({\varvec{f}}_{2}(||\xi ||_{p}):=||\xi ||_{p}\), \(\xi \in {\mathbb {Q}}_{p}^{n}\), we have that \({\varvec{f}}_{1}\) and \({\varvec{f}}_{2}\) satisfies Condition A.

  2. (ii)

    Taking \({\varvec{f}}_{1}(||\xi ||_{p}):=e^{||\xi ||_{p}^{\beta _{1}}}\) and \({\varvec{f}}_{2}(||\xi ||_{p}):=e^{||\xi ||_{p}^{\beta _{2}}}\), \(\xi \in {\mathbb {Q}}_{p}^{n}\), with \(\beta _{1}, \beta _{2}\in {\mathbb {N}}\) and \(\beta _{1}\le \beta _{2}\), we have that \({\varvec{f}}_{1}\) and \({\varvec{f}}_{2}\) satisfies Condition A.

Throughout this paper we will assume that the functions \({\varvec{f}}_{1}\) and \({\varvec{f}}_{2}\) both satisfy Condition A. Now, we denote and define the function

$$\begin{aligned} \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}:=\max \{|{\varvec{f}}_{1}(||\xi ||_{p})|,|{\varvec{f}}_{2}(||\xi ||_{p})|\}, \ \ \xi \in {\mathbb {Q}}_{p}^{n}. \end{aligned}$$
(3.1)

Then, from Definition 1, the application \(\left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}:{\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}_{+}\backslash \left\{ 0\right\} \) is a radial, increasing with respect to \(||\cdot ||_{p}\) and continuous function.

Lemma 1

For any real number \(l>\frac{2n}{\beta _{1}}\) (\(\beta _{1}\) is the constant given in Definition 1), we have

$$\begin{aligned} \left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}\in L^{1}({\mathbb {Q}}_{p}^{n}). \end{aligned}$$

Proof

$$\begin{aligned} \int \limits _{{\mathbb {Q}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}d^{n}\xi&=\int \limits _{B_{r}^{n}}|{\varvec{f}}_{1}(||\xi ||_{p})|^{-\frac{l}{2}}d^{n}\xi +\int \limits _{{\mathbb {Q}}_{p}^{n}\backslash B_{r}^{n}}|{\varvec{f}}_{2}(||\xi ||_{p})|^{-\frac{l}{2}}d^{n}\xi =I_{1}+I_{2}. \end{aligned}$$

Since \(|{\varvec{f}}_{1}(\cdot )|^{-\frac{l}{2}}\) is a continuous function on the compact subset \(B_{r}^{n}\), then \(I_{1}<\infty \). Moreover, by Definition 1 we have

$$\begin{aligned} I_{2}&=\sum _{j=r+1}^{\infty }|{\varvec{f}}_{2}(p^{j})|^{-\frac{l}{2}}\int \limits _{||\xi ||_{p}=p^{j}}d^{n}\xi \\&\le \left( \frac{1}{C_{1}}\right) ^{\frac{l}{2}}\sum _{j=r+1}^{\infty }p^{-jl\beta _{1}/2}\int \limits _{||\xi ||_{p}=p^{j}}d^{n}\xi \\&=\frac{1-p^{-n}}{C_{1}^{l/2}}\sum _{j=r+1}^{\infty }p^{-j\left( \frac{l\beta _{1}}{2}-n\right) }, \end{aligned}$$

which assumes a finite value if \(l>\frac{2n}{\beta _{1}}\) and thus the result is obtained. \(\square \)

In what follows l corresponds to a fixed real number satisfying the condition \(l>\frac{2n}{\beta _{1}}\). Now we introduce the p-adic operator

$$\begin{aligned} \begin{aligned} \left( {\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\varphi \right) (x)&:=-\int \limits _{{\mathbb {Q}}_{p}^{n}}\chi _{p}(-x\cdot \xi )\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}{\widehat{\varphi }}(\xi )d^{n}\xi \\&=-{\mathcal {F}}_{\xi \rightarrow x}^{-1}\left( \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}{\widehat{\varphi }}(\xi ) \right) , \ \ \varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n}), \ \ x\in {\mathbb {Q}}_{p}^{n}. \end{aligned} \end{aligned}$$
(3.2)

Remark 1

Since \(\left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}\) is a continuous function and \({\widehat{\varphi }}\in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})\), with \(\varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})\), we have that \(\left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}{\widehat{\varphi }}\in L^{1}({\mathbb {Q}}_{p}^{n})\). Therefore, by Riemann-Lebesgue Theorem, see [34, Theorem 1.6 - p. 24], we have that \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\varphi \in C_{0}({\mathbb {Q}}_{p}^{n})\). This fact implies that the operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}:{\mathcal {D}}({\mathbb {Q}}_{p}^{n})\rightarrow C_{0}({\mathbb {Q}}_{p}^{n})\) is a well-defined non-archimedean pseudo-differential operator with symbol \(\left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}\).

Definition 2

We denote and define the convolution kernel associated with the p-adic pseudo-differential operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) by

$$\begin{aligned} K_{l}(x){} & {} :=\int \limits _{\mathbf {{\mathbb {Q}}}_{p}^{n}}\chi _{p}(-x\cdot \xi )\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}d^{n}\xi \nonumber \\{} & {} ={\mathcal {F}}_{\xi \rightarrow x}^{-1}\left( \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}\right) , \ \ x\in {\mathbb {Q}}_{p}^{n}. \end{aligned}$$
(3.3)

At this point, we must emphasize that unlike the convolution kernel \(K_{\alpha }\) studied in [35], our functions \({\varvec{f}}_{1}\) and \({\varvec{f}}_{2}\) are not restricted to be negative definite functions, among other conditions. We also want to highlight also that the p-adic pseudo-differential operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) and the convolution kernel \(K_{l}\) introduced in this section are not part of the family of pseudo-differential operators and convolution kernels studied in [35]. Additionally, we do not impose growth conditions to the function \(\left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}\) to guarantee the non-negativity of our convolution kernel \(K_{l}\), among other properties, as we will prove in the next theorem.

Theorem 1

With the hypotheses of Definition 1, the convolution kernel \(K_{l}\) satisfies the following conditions:

  1. (i)

    \(K_{l}(x)\ge 0\) for all \(x\in {\mathbb {Q}}_{p}^{n}\);

  2. (ii)

    there exists a positive constant \(C_{2}\) such that \(K_{l}(x) \le C_{2}||x||_{p}^{-n}\) for all \(x\in {\mathbb {Q}}_{p}^{n}\backslash \left\{ 0\right\} \).

Proof

  1. (i)

    If \(x=0\) the statement is immediate. Let \(x\in {\mathbb {Q}}_{p}^{n}\backslash \left\{ 0\right\} \). Then, with a procedure similar to the proof of [35, Theorem 1-(ii)], we obtain that

    $$\begin{aligned} K_{l}(x)=||x||_{p}^{-n}\left\{ (1-p^{-n})\sum \limits _{j=0}^{\infty }p^{-nj}\left[ ||x||_{p}^{-1}p^{-j}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}-\left[ ||x||_{p}^{-1}p\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}} \right\} .\nonumber \\ \end{aligned}$$
    (3.4)

    Consider the following cases for \(||x||_{p}\) taking into account the constant r given in Definition 1. Case 1 \(||x||_{p}=p^{-r}\). In this case, \(||x||_{p}^{-1}=p^{r}\) and \(||x||_{p}^{-1}p^{-j}\le p^{r}\), for all \(j=0,1,\ldots \) Then by Definition 1 and (3.4) we have that

    $$\begin{aligned} K_{l}(x)&=||x||_{p}^{-n}\left\{ \left( 1-p^{-n}\right) \sum \limits _{j=0}^{\infty }p^{-nj}\left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p^{-j}\right) \right| ^{-\frac{l}{2}}-\left| {\varvec{f}}_{2}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right\} \\&\ge ||x||_{p}^{-n}\left\{ \left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}\right) \right| ^{-\frac{l}{2}}\sum \limits _{j=0}^{\infty }\left( p^{-nj}-p^{-n(j+1)}\right) -\left| {\varvec{f}}_{2}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right\} \\&= ||x||_{p}^{-n}\left\{ \left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}\right) \right| ^{-\frac{l}{2}}-\left| {\varvec{f}}_{2}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right\} \\&\ge ||x||_{p}^{-n}\left\{ \left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}-\left| {\varvec{f}}_{2}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right\} \\&\ge 0. \end{aligned}$$

    Case 2 \(||x||_{p}>p^{-r}\). In this case, \(||x||_{p}^{-1}<p^{r}\), \(||x||_{p}^{-1}p^{-j}\le p^{r}\), for all \(j=0,1,\ldots \) and \(||x||_{p}^{-1}p\le p^{r}\). Then by Definition 1 and (3.4) we have that

    $$\begin{aligned} K_{l}(x)&=||x||_{p}^{-n}\left\{ \left( 1-p^{-n}\right) \sum \limits _{j=0}^{\infty }p^{-nj}\left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p^{-j}\right) \right| ^{-\frac{l}{2}}-\left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right\} \\&\ge ||x||_{p}^{-n}\left\{ \left| {\varvec{f}}_{1}(\left( ||x||_{p}^{-1}\right) \right| ^{-\frac{l}{2}}\sum \limits _{j=0}^{\infty }\left( p^{-nj}-p^{-n(j+1)}\right) -\left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right\} \\&= ||x||_{p}^{-n}\left\{ \left| {\varvec{f}}_{1}(\left( ||x||_{p}^{-1}\right) \right| ^{-\frac{l}{2}}-\left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right\} \\&\ge 0. \end{aligned}$$

    Case 3 \(||x||_{p}<p^{-r}\). In this case, \(||x||_{p}^{-1}>p^{r}\) and \(||x||_{p}^{-1}p>p^{r}\). Since \(||x||_{p}^{-1}>p^{r}\), then there exists \(s\in {\mathbb {N}}\) such that \(||x||_{p}^{-1}p^{-j}>p^{r}\) for all \(j\in \left\{ 0,1,\ldots , s-1\right\} \) and \(||x||_{p}^{-1}p^{-j}\le p^{r}\) for all \(j\in \left\{ s,s+1,s+2,\ldots \right\} \). Then by Definition 1 and (3.4) we have

    $$\begin{aligned} K_{l}(x)&=||x||_{p}^{-n}\left\{ \left( 1-p^{-n}\right) \sum \limits _{j=0}^{s-1}p^{-nj}\left| {\varvec{f}}_{2}\left( ||x||_{p}^{-1}p^{-j}\right) \right| ^{-\frac{l}{2}}-\left| {\varvec{f}}_{2}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right. \\&\quad \left. +\left( 1-p^{-n}\right) \sum \limits _{j=s}^{\infty }p^{-nj}\left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p^{-j}\right) \right| ^{-\frac{l}{2}}\right\} \\&\ge ||x||_{p}^{-n}\left\{ \left( 1-p^{-n}\right) \sum \limits _{j=0}^{s-1}p^{-nj}\left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p^{-j}\right) \right| ^{-\frac{l}{2}}-\left| {\varvec{f}}_{2}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right. \\&\quad \left. +\left( 1-p^{-n}\right) \sum \limits _{j=s}^{\infty }p^{-nj}\left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p^{-j}\right) \right| ^{-\frac{l}{2}}\right\} \\&\ge ||x||_{p}^{-n}\left\{ \left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}\right) \right| ^{-\frac{l}{2}}\sum \limits _{j=0}^{\infty }\left( p^{-nj}-p^{-n(j+1)}\right) -|{\varvec{f}}_{2}\left( ||x||_{p}^{-1}p\right) |^{-\frac{l}{2}}\right\} \\&= ||x||_{p}^{-n}\left\{ |{\varvec{f}}_{1}(\left( ||x||_{p}^{-1}\right) |^{-\frac{l}{2}}-|{\varvec{f}}_{2}\left( ||x||_{p}^{-1}p\right) |^{-\frac{l}{2}}\right\} \\&\ge ||x||_{p}^{-n}\left\{ \left| {\varvec{f}}_{1}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}-\left| {\varvec{f}}_{2}\left( ||x||_{p}^{-1}p\right) \right| ^{-\frac{l}{2}}\right\} \\&\ge 0. \end{aligned}$$
  2. (ii)

    By Lemma 1 and the definition of additive character we have

    $$\begin{aligned} |K_{l}(x)|\le \int \limits _{{\mathbb {Q}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}d^{n}\xi <\infty . \end{aligned}$$

    Therefore, by (i) and (3.4)

    $$\begin{aligned} K_{l}(x)&\le ||x||_{p}^{-n}(1-p^{-n})\sum \limits _{j=0}^{\infty }\frac{p^{-nj}}{\left[ ||x||_{p}^{-1}p^{-j}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\frac{l}{2}}}\\&\le \frac{1}{\left[ 0\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\frac{l}{2}}}||x||_{p}^{-n}\sum \limits _{j=0}^{\infty }(1-p^{-n})p^{-nj}\\&=C_{2}||x||_{p}^{-n}, \text { where } C_{2}:=\frac{1}{\left[ 0\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\frac{l}{2}}}. \end{aligned}$$

\(\square \)

Example 2

Consider the functions \({\varvec{f}}_1\) and \({\varvec{f}}_2\) as in (ii) of Example 1, with \(p=2\), \(n=2\), \(\beta _1=2\), \(\beta _2=4\) and \(l=8\). We evaluate the convolution kernel \(K_l\), associated with \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) in (3.2), for illustration of Theorem 1 above. In Fig. 1 it is shown the behavior of \(K_l(x)\), \(x \in {\mathbb {Q}}_p^2\).

Fig. 1
figure 1

Graph of \(K_l\) as \(||x||_p\) increases from zero at the top figure. At the bottom one, the \(K_l(x)\) values with \(x \in B_j^n\), \(j\in {\mathbb {Z}}\)

Full size image

Remark 2

Let \(\varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})\). Then, by [2, Subsections 4.8 and 4.9], Definition 2 and Theorem 1, produce

$$\begin{aligned} ({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\varphi )(x)=-\int \limits _{supp(\varphi )}K_{l}(x-y)\varphi (y)d^{n}y=-\int \limits _{supp(\varphi )}K_{l}(y)\varphi (x-y)d^{n}y, \ \ x\in {\mathbb {Q}}_{p}^{n}. \end{aligned}$$

Definition 3

[12] An operator A on \(C_{0}(\mathbf {{\mathbb {Q}}}_{p}^{n})\) is said to satisfy the positive maximum principle if whenever \(f\in Dom(A)\), \(x_{0}\in \mathbf {{\mathbb {Q}}}_{p}^{n}\), and \(\sup _{x\in \mathbf {{\mathbb {Q}}}_{p}^{n}}f(x)=f(x_{0})\ge 0\), we have \(Af(x_{0})\le 0\).

Theorem 2

The pseudo-differential operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) satisfies the positive maximum principle on \(C_{0}(\mathbf {{\mathbb {Q}}}_p^n)\).

Proof

Let \(\varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})\) be fixed. Without loss of generality we may assume that \(supp(\varphi )=B_{M}^{n}\), \(M\in {\mathbb {Z}}\), and that \(sup_{x\in {\mathbb {Q}}_{p}^{n}}\varphi (x)=\varphi (x_{0})\ge 0\), for some \(x_{0}\in {\mathbb {Q}}_{p}^{n}\). Let us consider the following cases for \(x_{0}\):

If \(\varphi (x_{0})>0\), then \(x_{0}\in supp(\varphi )\) and \(\varphi (x)>0\) for all \(x\in supp(\varphi )\). In this case, by Theorem 1 and Remark 2, we have that

$$\begin{aligned} ({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\varphi )(x_{0})=-\int \limits _{supp(\varphi )}K_{l}(x_{0}-y)\varphi (y)d^{n}y\le 0. \end{aligned}$$

Now, if \(\varphi (x_{0})=0\), then \(x_{0}\notin supp(\varphi )\), i.e., \(||x_{0}||_{p}>p^{M}\). In this case, by Theorem 1, Remark 2 and the ultrametricity of the norm \(||\cdot ||_{p}\), we have that

$$\begin{aligned} \left( {\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\varphi \right) (x_{0})=-\int \limits _{supp(\varphi )}K_{l}(y)\varphi (x_{0}-y)d^{n}y=0. \end{aligned}$$

\(\square \)

Remark 3

A linear operator A on a Banach space \((X,||\cdot ||)\) is said to be dissipative if \(||\lambda f-Af||\ge \lambda ||f||\) for every \(f\in Dom(A)\) and \(\lambda >0\). In our case, by Theorem 2 and [12, Lemma 2.1 - p. 165], the pseudo-differential operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) is dissipative on \(C_{0}({\mathbb {Q}}_{p}^{n})\).

Remark 4

In what follows in this subsection we will additionally assume that there exists an integer m, with \(m\le r\) (r is the constant given in Definition 1), such that the function \(|{\varvec{f}}_{1}|\) is constant over the ball \(B_{m}^{n}\), i.e. \(|{\varvec{f}}_{1}(||y||_{p})|=|{\varvec{f}}_{1}(0)|\), for all \(y\in B_{m}^{n}\). In this case, by [35, Lemma 1] we have that the function \(\left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}\) is locally constant on \({\mathbb {Q}}_{p}^{n}\).

Lemma 2

Let \(\lambda \in {\mathbb {R}}_{+}\backslash \left\{ 0\right\} \). Then, the range of the operator \(\lambda -{\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) (which we denote by \(Ran(\lambda -{\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l})\)) is dense in \(C_{0}({\mathbb {Q}}_{p}^{n})\).

Proof

Let \(\lambda \in {\mathbb {R}}_{+}\backslash \left\{ 0\right\} \) to be fixed. Consider the following p-adic pseudo-differential equation

$$\begin{aligned} \left( \lambda -{\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\right) f =\varphi , \ \ \varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n}). \end{aligned}$$

Using [2, Subsections 4.8 and 4.9], by a direct calculation we obtain

$$\begin{aligned} f(x)={\mathcal {F}}_{\xi \rightarrow x}^{-1}\left( \frac{{\widehat{\varphi }}(\xi )}{\lambda +\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{l}{2}}}\right) \end{aligned}$$

which is a solution of the above equation. Moreover, by Remark 4 and [2, Chapter 4], we have that \(f\in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})\). Therefore, since \({\mathcal {D}}({\mathbb {Q}}_{p}^{n})\) is dense in \(C_{0}({\mathbb {Q}}_{p}^{n})\), see e.g. [34, Proposition 1.3], we have that \(Ran(\lambda -{\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l})\) is dense in \(C_{0}({\mathbb {Q}}_{p}^{n})\). \(\square \)

Definition 4

[11] A one-parameter family \(\left\{ T(t):t\ge 0\right\} \) of bounded linear operators on a Banach space X is called a contraction semigroup in X provided that

  1. (a)

    \(||T(t)||\le 1\) for all \(t>0\);

  2. (b)

    \(T(0)=I\);

  3. (c)

    \(T(t+s)=T(t)T(s)\) for all \(s,t\ge 0\);

  4. (d)

    for all \(x\in X\), the function \(t\mapsto T(t)x\) belongs to \(C([0,\infty ),X)\).

Theorem 3

The pseudo-differential operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) is closable and its closure \(\overline{{\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}}\) is single-valued and generates a strongly continuous, positive, contraction semigroup \(\left\{ T_{t}\right\} _{t\ge 0}\) on \(C_{0}({\mathbb {Q}}_{p}^{n})\).

Proof

The result follows from Remark 3, Lemma 2 y [12, Theorem 2.12 - p. 16 and Theorem 2.2 - p. 165]. \(\square \)

Remark 5

Let X be an infinite dimensional Banach space, B[X] the bounded linear operators on X and F a closed subspace of X. An element of the set

$$\begin{aligned} {\mathcal {T}}_{F}:=\left\{ T: T\in B[X], T(F)\subseteq F\right\} \end{aligned}$$

is called a conservative operator. We have that \({\mathcal {D}}({\mathbb {Q}}_{p}^{n})\) is a complete space, see e.g. [2, Subsection 4.3]. Now, by [2, Proposition 4.7.3 and Theorem 4.8.2] we have that

$$\begin{aligned} ({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\varphi )(x)=-K_{l}(x)*\varphi (x)\in {\mathcal {D}}({\mathbb {Q}}_{p}^{n}), \text { for all } \varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n}), \ \ x\in {\mathbb {Q}}_{p}^{n}, \end{aligned}$$

i.e. the pseudo-differential operator \({\mathcal {T}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{l}\) is conservative.

As a consequence of [12, Chapter 4-Subsection 2] and Remark 5 we obtain the following result.

Theorem 4

The operators \(\left\{ T_{t}\right\} _{t\ge 0}\) obtained in Theorem 3 determine a Feller semigroup on \({\mathbb {Q}}_{p}^{n}\). That is, the following conditions are satisfied:

  1. (i)

    \(\{T_{t}\}_{t\ge 0}\) determine a family of bounded linear operators on \(C_{0}({\mathbb {Q}}_{p}^{n})\);

  2. (ii)

    \(T_{t+s}=T_{t}\cdot T_{s}\), \(t,s\ge 0\), and \(T_{0}=I\);

  3. (iii)

    \(\{T_{t}\}_{t\ge 0}\) is strongly continuous in t:

    $$\begin{aligned} \lim _{s\rightarrow 0^{+}}||T_{t+s}f-T_{t}f||=0, \ \ f\in C_{0}({\mathbb {Q}}_{p}^{n}); \end{aligned}$$
  4. (iv)

    \(\{T_{t}\}_{t\ge 0}\) is non-negative and contractive on \(C_{0}({\mathbb {Q}}_{p}^{n})\):

    $$\begin{aligned} f\in C_{0}({\mathbb {Q}}_{p}^{n}), \ \ 0\le f(x)\le 1 \text { on } {\mathbb {Q}}_{p}^{n}\Rightarrow 0\le T_{t}f(x)\le 1 \text { on } {\mathbb {Q}}_{p}^{n}. \end{aligned}$$

We will denote by \({\mathcal {P}}({\mathbb {Q}}_{p}^{n})\) and \(D_{{\mathbb {Q}}_{p}^{n}}[0,\infty )\), respectively, the family of Borel probability measures on \({\mathbb {Q}}_{p}^{n}\) and the space of right continuous functions \(f:[0,\infty )\rightarrow {\mathbb {Q}}_{p}^{n}\) with left limits. A stochastic process \({\mathfrak {X}}\) (or simply a process) with index set \([0,\infty )\) and state space \(({\mathbb {Q}}_{p}^{n},{\mathcal {B}})\) (a measure space) defined on a probability space \((\Omega ,{\mathcal {F}}, P)\) is a function defined on \([0,\infty ) \times \Omega \) with values in \({\mathbb {Q}}_{p}^{n}\) such that for each \(t\in [0,\infty )\), \({\mathfrak {X}}(t,\cdot ):\Omega \rightarrow {\mathbb {Q}}_{p}^{n}\) is an \({\mathbb {Q}}_{p}^{n}\)-valued random variable, that is, \(\left\{ \omega :{\mathfrak {X}}(t,w)\in \Gamma \right\} \in {\mathcal {F}}\) for every \(\Gamma \in {\mathcal {B}}\). On the other hand, a collection \(\left\{ {\mathcal {F}}_{t}\right\} \equiv \left\{ {\mathcal {F}}_{t}, t\in [0,\infty )\right\} \) of \(\sigma \)-algebras of sets in \({\mathcal {F}}\) is a filtration if \({\mathcal {F}}_{t}\subset {\mathcal {F}}_{t+s}\), for \(t,s \in [0,\infty )\). Intuitively, \({\mathcal {F}}_{t}\) corresponds to the information known by an observer at time t. In particular, for a process \({\mathfrak {X}}\) we define \(\left\{ {\mathcal {F}}_{t}^{{\mathfrak {X}}}\right\} \) by \({\mathcal {F}}_{t}^{{\mathfrak {X}}}=\sigma \left( {\mathfrak {X}}(s):s\le t \right) \); that is, \({\mathcal {F}}_{t}^{{\mathfrak {X}}}\) is the information obtained by observing \({\mathfrak {X}}\) up to time t. We say \(\left\{ {\mathcal {F}}_{t}\right\} \) is right continuous if for each \(t\ge 0\), \({\mathcal {F}}_{t}={\mathcal {F}}_{t^{+}}\equiv \bigcap _{\epsilon >0}{\mathcal {F}}_{t+\epsilon }\). A process \({\mathfrak {X}}\) is adapted to a filtration \({\mathcal {F}}_{t}\) (or simply \(\left\{ {\mathcal {F}}_{t}\right\} \)-adapted) if \({\mathfrak {X}}(t)\) is \({\mathcal {F}}_{t}\)-measurable for each \(t\ge 0\). For more details the reader may consult [12].

Since \({\mathbb {Q}}_{p}^{n}\) is separable, see e.g. [42, Chapter 1-I-3], then, as a consequence of [2, Subsection 1.10], [12, Theorem 2.7-Corollary 2.8-Chapter 4] and Theorem 3, we obtain the following result.

Theorem 5

Let \(\{T_{t}\}_{t\ge 0}\) be the Feller semigroup obtained in the Theorem 4. Then, the following statements are true:

  1. (i)

    For each \(\upsilon \in {\mathcal {P}}({\mathbb {Q}}_{p}^{n})\), there exists a Markov process \({\mathfrak {X}}\) corresponding to \(\{T_{t}\}_{t\ge 0}\) with initial distribution \(\upsilon \) and sample paths in \(D_{{\mathbb {Q}}_{p}^{n}}[0,\infty )\). Moreover, \({\mathfrak {X}}\) is a strong Markov processes with respect to the filtration \({\mathcal {F}}_{t^{+}}^{{\mathfrak {X}}}=\bigcap _{\epsilon >0}{\mathcal {F}}_{t+\epsilon }^{{\mathfrak {X}}}\).

  2. (ii)

    There exists, for each \(x\in {\mathbb {Q}}_{p}^{n}\), a Markov process \({\mathfrak {X}}_{x}\) corresponding to \(\{T_{t}\}_{t\ge 0}\) with initial distribution \(\delta _{x}\) and with sample paths in \(D_{{\mathbb {Q}}_{p}^{n}}[0,\infty )\).

4 New classes of p-adic Sobolev spaces, the pseudo-differential operator \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\) and their applications

In this section, we introduce new classes of p-adic Sobolev spaces and we define certain types of pseudo-differential operators on these spaces. The connections of these operators with contraction semigroups, the theory of m-dissipative operators and initial value problems are also studied.

We assume throughout this section that \(|{\varvec{f}}_{1}(||\xi ||_{p})|\ge 1\), for all \(\xi \in {\mathbb {Q}}_{p}^{n}\). From Definition 1 and (3.1) one gets

$$\begin{aligned} \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}\ge 1 \text { for all } \xi \in {\mathbb {Q}}_{p}^{n}. \end{aligned}$$
(4.1)

For \(\varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})\) and \(\alpha \in {\mathbb {R}}_{+}\), we define the norm

$$\begin{aligned} ||\varphi ||_{{\varvec{f}}_{1},{\varvec{f}}_{2},\alpha }^{2}:=\int \limits _{\mathbf {{\mathbb {Q}}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{2\alpha }|\widehat{\varphi }(\xi )|^{2}d^{n}\xi . \end{aligned}$$

We will denote by \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\), \(\alpha \ge 0\), the Sobolev space corresponding to the completion of \({\mathcal {D}}({\mathbb {Q}}_{p}^{n})\) with respect to \(||\cdot ||_{{\varvec{f}}_{1},{\varvec{f}}_{2},\alpha }\).

Remark 6

If \(\alpha ^{\prime }\ge \alpha \ge 0\), then \(||\cdot ||_{{\varvec{f}}_{1},{\varvec{f}}_{2},\alpha }\le ||\cdot ||_{{\varvec{f}}_{1},{\varvec{f}}_{2},\alpha '}\) and consequently \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha '}({\mathbb {Q}}_{p}^{n})\hookrightarrow B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\). In particular, \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\hookrightarrow B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{0}({\mathbb {Q}}_{p}^{n})=L^{2}({\mathbb {Q}}_{p}^{n})\), for all \(\alpha \ge 0\), where as it was mentioned before \(L^{2}({\mathbb {Q}}_{p}^{n})\) is the Hilbert space with the scalar product given in (2.1). Moreover, by construction, \({\mathcal {D}}({\mathbb {Q}}_{p}^{n})\) is dense in \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) for all \(\alpha \ge 0\), and by [34, (3.8) - p. 38] we have that \({\mathcal {D}}({\mathbb {Q}}_{p}^{n})\) is dense in \(L^{2}({\mathbb {Q}}_{p}^{n})\). Therefore, \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\), \(\alpha \ge 0\), is dense in \(L^{2}({\mathbb {Q}}_{p}^{n})\).

For \(f \in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\), \(\alpha \ge 0\), we introduce the p-adic pseuodo-differential operator

$$\begin{aligned} \begin{aligned} {\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }(f)(x)&=-{\mathcal {F}}^{-1}_{\xi \rightarrow x}\left( \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}{\widehat{f}}(\xi ) \right) \\&=-\int \limits _{{\mathbb {Q}}_{p}^{n}}\chi _{p}(-x\cdot \xi )\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}{\widehat{f}}(\xi ) d^{n}\xi . \end{aligned} \end{aligned}$$
(4.2)

Remark 7

For \(f \in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\), \(\alpha \ge 0\), note that

$$\begin{aligned} ||{\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }(f)||_{{\varvec{f}}_{1},{\varvec{f}}_{2},\alpha }^{2}&=\int \limits _{{\mathbb {Q}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{2\alpha }\left| \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}{\widehat{f}}(\xi ) \right| ^{2}d^{n}\xi \\&=\int \limits _{{\mathbb {Q}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }|{\widehat{f}}(\xi ) |^{2}d^{n}\xi \\&\le \int \limits _{{\mathbb {Q}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{2\alpha }|{\widehat{f}}(\xi ) |^{2}d^{n}\xi \\&=||f||_{{\varvec{f}}_{1},{\varvec{f}}_{2},\alpha }^{2}. \end{aligned}$$

Therefore, \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }:B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\rightarrow B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\), with \(\alpha \ge 0\) fixed, is a well-defined bounded pseudo-differential operator with symbol \(\left[ ||\cdot ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}\).

Remark 8

An operator A with domain D(A) in a Hilbert space X and with scalar product \(\left( \cdot ,\cdot \right) \) is called dissipative if

$$\begin{aligned} re\left( Af,f\right) \le 0,\text { for all } f\in D(A), \end{aligned}$$

see e.g. [11, 28]. In our case, the pseudo-differential operator \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\) is dissipative in \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\). Indeed, for \(f\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\), \(\alpha \ge 0\), and by the Parseval-Steklov equality, see e.g. [2, Theorem 5.3.1], we have that

$$\begin{aligned} \left( {\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }(f),f\right) = -\left( \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}{\widehat{f}},{\widehat{f}} \right) =-\int \limits _{{\mathbb {Q}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}|{\widehat{f}}(\xi )|^{2}d^{n}\xi \le 0. \end{aligned}$$

Definition 5

[11, Definition 2.2.2] An operator A in a Banach space X (endowed with the norm \(||\cdot ||\)) is \(m-\)dissipative if

  1. (i)

    A is dissipative;

  2. (ii)

    for all \(\lambda >0\) and all \(f\in X\) there exists \(u\in D(A)\) such that \(u-\lambda Au=f\).

If \(\alpha =0\) note that \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{0}\) is the identity operator. Therefore, in what follows we will consider a fixed \(\alpha >0\).

Lemma 3

The pseudo-differential operator \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\) is self-adjoint, i.e.

$$\begin{aligned} \left( {\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }f,g\right) =\left( f,{\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }g\right) ,\text { for all } f,g\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}). \end{aligned}$$

Proof

For \(f,g\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) and the Parseval-Steklov equality, we have that

$$\begin{aligned} \left( {\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }f,g\right)&=\left( -{\mathcal {F}}_{\xi \rightarrow x}^{-1}\left[ \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}{\widehat{f}} (\xi )\right] ,g\right) \\&=-\int \nolimits _{{\mathbb {Q}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}{\widehat{f}}(\xi )\overline{ {\widehat{g}}}(\xi )d^{n}\xi \\&=-\int \nolimits _{{\mathbb {Q}}_p^n}\overline{\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}}{\widehat{f}}(\xi )\left[ \int \nolimits _{{\mathbb {Q}}_p^n}\chi _{p}(x\cdot \xi ){\overline{g}}(x)d^{n}x\right] d^{n}\xi \\&=-\int \nolimits _{{\mathbb {Q}}_p^n}{\widehat{f}}(\xi )\overline{\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}} \overline{{\widehat{g}}}(\xi )d^{n}\xi \\&=\left( f,-{\mathcal {F}}_{\xi \rightarrow x}^{-1}\left[ \left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}{\widehat{g}}(\xi )\right] \right) =\left( f,{\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }g\right) . \end{aligned}$$

\(\square \)

Theorem 6

The pseudo-differential operator \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\), is m-dissipative in \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\).

Proof

The result follow from Remark 8, Lemma 3 and [11, Corollary 2.4.8]. \(\square \)

Definition 6

[11] The generator of \(\left\{ T(t):t\ge 0\right\} \) is the linear operator L defined by

$$\begin{aligned} D(L)=\left\{ x\in X:\frac{T(t)x-x}{h} \text { has a limit in X as} h \rightarrow 0^{+}\right\} , \end{aligned}$$

and

$$\begin{aligned} Lx=\lim \limits _{h\rightarrow 0^{+}}\frac{T(t)x-x}{h}, \end{aligned}$$

for all \(x\in D(L)\).

Theorem 7

The pseudo-differential operator \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\) is the generator of a contraction semigroup \(\left\{ T(t):t\ge 0\right\} \) in \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\). Moreover, for any \(T>0\), \(f\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) and \(g\in C([0,T],B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}))\), the function

$$\begin{aligned} u(t):=T(t)f+\int \limits _{0}^{t}T(t-s)g(s)ds, \ \ \text { for all } t\in [0,T], \end{aligned}$$
(4.3)

is the unique solution of the inhomogeneous equation

$$\begin{aligned} \left\{ \begin{array}{ll} u\in C([0,T],B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}))\cap C^{1}([0,T],B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})); &{} \\ &{} \\ \frac{\partial u}{\partial t}(x,t)={\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }u(t)+g(t), \ \ t\in [0,T], \\ &{} \\ u(0)=f. \end{array} \right. \end{aligned}$$

Proof

The result follows from Lemma 6 and [11, Theorem 3.4.4 and Subsection 4.1]. \(\square \)

Definition 7

[30, Definition 2.1] A semigroup \({{\textbf {T}}}(t)\), \(0\le t<\infty \), of bounded linear operators on \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) is a strongly continuous semigroup of bounded linear operators if

$$\begin{aligned} \lim _{t\rightarrow 0^{+}}{{\textbf {T}}}(t)f=f, \text { for every } f\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}). \end{aligned}$$

A strongly continuous semigroup of bounded linear operators on \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) will be called a semigroup of class \(C_{0}\) or simply a \(C_{0}\) semigroup.

Lemma 4

For all fixed \(\lambda _{0}>0\), \(Ran(\lambda _{0}I-{\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha })=B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\).

Proof

Let \(f\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\). For \(\lambda _{0}>0\) arbitrary but fixed consider the equation \(\lambda _{0}u-{\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }u=f\). By Remark 6 and [2, Section 5.3] we have that the function \(u(x)={\mathcal {F}}^{-1}_{\xi \rightarrow x}\left( \frac{{\widehat{f}}(\xi )}{\lambda _{0}+\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}}\right) \) satisfies the above equation. Now, by using (4.1) we have

$$\begin{aligned} ||u ||_{{\varvec{f}}_{1},{\varvec{f}}_{2},\alpha }^{2}= & {} \int \limits _{{\mathbb {Q}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{2\alpha }\left| \frac{{\widehat{f}}(\xi )}{\lambda _{0}+\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{-\frac{\alpha }{2}}}\right| ^{2}d^{n}\xi \\\le & {} \int \limits _{{\mathbb {Q}}_{p}^{n}}\left[ ||\xi ||_{p}\right] _{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{2\alpha }|{\widehat{f}}(\xi )|^{2}d^{n}\xi \\= & {} ||f||_{{\varvec{f}}_{1},{\varvec{f}}_{2},\alpha }^{2}<\infty , \end{aligned}$$

i.e. \(u\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\). Therefore \(Ran(\lambda _{0}I-{\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha })=B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\).

\(\square \)

Theorem 8

The contraction semigroup in Theorem 7 is a \(C_{0}\) semigroup and \({\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }\) is the infinitesimal generator.

Proof

The result follows from Remark 8, Lemma 4, [30, Theorem 4.3], by using well-known results of semigroup theory, see e.g. [30, Sections 1.2\(-\)1.3]. \(\square \)

Consider the following semilinear initial value problem

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}(t)-{\mathcal {J}}_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }u(t)=f(t,u(t)), &{} t>t_{0}, \ \ t_{0}\ge 0 \text {,\ } \\ &{} \\ u(t_{0})=u_{0}, \ \ u_{0}\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}) \text {,} &{} \end{array} \right. \end{aligned}$$
(4.4)

where \(f:[t_{0},T]\times B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\rightarrow B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) is continuous in t and satisfies a Lipschitz condition in u; and the integral equation

$$\begin{aligned} u(t)=T(t-t_{0})u_{o}+\int \limits _{t_{0}}^{t}T(t-s)f(s,u(s))ds. \end{aligned}$$
(4.5)

A continuous solution u of the integral Eq. (4.5) will be called a mild solution of the initial value problem (4.4).

Theorem 9

Let \(f:[t_{0},T]\times B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\rightarrow B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) be continuous in t on \([t_{0},T]\) and uniformly Lipschitz continuous (with constant L) on \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\). Then for every \(u_{0}\in B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) the initial value problem (4.4) has a unique mild solution \(u\in C([t_{0},T]:B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}))\). Moreover, the mapping \(u_{0}\rightarrow u\) is Lipschitz continuous from \(B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n})\) into \(C([t_{0},T]:B_{{\varvec{f}}_{1},{\varvec{f}}_{2}}^{\alpha }({\mathbb {Q}}_{p}^{n}))\).

Proof

The result follows from Theorem 8 and [30, Theorem 1.2-Chapter 6]. \(\square \)