Some classes of non-archimedean pseudo-differential operators related to Bessel potentials

Abstract

In this article, we study two classes of non-archimedean pseudo-differential operators related to Bessel potentials. The first class of these operators, which we will denote as $$J^{\alpha }$$ , $$\alpha >n$$ , are associated via Fourier transform to the Bessel potentials. These operators are of the form $$\begin{aligned} (J^{\alpha }\varphi )(x)={\mathcal {F}}_{\xi \rightarrow x}^{-1} \left[ (\max \{1,||\xi ||_{p}\})^{-\alpha }\widehat{\varphi }(\xi )\right] , \quad \varphi \in {\mathcal {D}}({\mathbb {Q}}_p^n),\ x\in {\mathbb {Q}}_p^n. \end{aligned}$$ The second class of these operators, which we will denote as $${\mathcal {A}}^{\alpha }$$ , $$\alpha >0$$ , are a linear combination of the operators $$-J^{\alpha }$$ , $$\alpha >0$$ , and the operator identity I. These operators are of the form $$\begin{aligned} \left( {\mathcal {A}}^{\alpha }\varphi \right) (x) ={\mathcal {F}}_{\xi \rightarrow x}^{-1} \left[ \left\{ 1-\left( \max \{1,||\xi ||_{p}\}\right) ^{-\alpha }\right\} \widehat{\varphi }(\xi )\right] ,\quad \varphi \in {\mathcal {D}} ({\mathbb {Q}}_p^n),\ x\in {\mathbb {Q}}_p^n. \end{aligned}$$ We study the Cauchy problem naturally associated with these classes of operators. We show that the heat equation associated to the operator $$J^{\alpha }$$ , $$\alpha >n$$ , describes the cooling (or loss of heat) in a given region over time, since the fundamental solution Z(x, t) (explicitly represented), of real positive time variable and p-adic spatial variables satisfies $$Z(x,t)\le 0$$ . Next, we study the positive maximum principle, the generators of certain families of semigroups and the existence of a mild solution for a certain inhomogeneous initial value problem. With respect to the operator $${\mathcal {A}}^{\alpha }$$ , $$\alpha >0$$ , we find explicit a representations of the fundamental solutions $$\widetilde{Z}(x,t)$$ , also of real positive time variable and p-adic spatial variables. Via the theory of distributions we will show that these fundamental solutions are related to a transition functions of strong Markov processes whose paths are right continuous and have no discontinuities other than jumps.