Non-archimedean generalized Bessel potentials and their applications

Abstract

This article describes a class of pseudo-differential operators \begin{equation*} (\mathcal{A}^{\alpha}\varphi)(x)=\mathcal{F}^{-1}_{\xi \rightarrow x}\left(\left[\max\{|\boldsymbol{\psi}_{1}(||\xi||_{p})|,|\boldsymbol{\psi}_{2}(||\xi||_{p})|\}\right]^{-\alpha}\widehat{\varphi}(\xi)\right), \end{equation*} $\varphi\in \mathcal{D}(\mathbb{Q}_{p}^{n})$ and $\alpha\in\mathbb{C}$; here $\left[\max\{|\boldsymbol{\psi}_{1}(||\xi||_{p})|,|\boldsymbol{\psi}_{2}(||\xi||_{p})|\}\right]^{-\alpha}$ is the symbol of the operator $\mathcal{A}^{\alpha}$. These operators can be seen as a generalization of the Bessel potentials in the $p$-adic context. We show that the family $\left(K_{\alpha}\right)_{\alpha>0}$ of convolution kernels attached to generalized Bessel potentials $\mathcal{A}^{\alpha}$, $\alpha>0$, determine a convolution semigroup on $\mathbb{Q}_{p}^{n}$. Imposing certain conditions we have that $K_{\alpha}$, $\alpha>0$, is a probability measure on $\mathbb{Q}_{p}^{n}$. Moreover, we will study certain properties corresponding to the Green function of the operator $\mathcal{A}^{\alpha}$ and we show that heat equations, naturally associated to these operators, describes the cooling (or loss of heat) in a given region over time.