Notion of Non-archimedean Pseudo-differential Operators Associated with Fractional Fourier Transform

Abstract

In this manuscript, we define the first type of non-archimedean pseudo-differential operator associated with the fractional Fourier transform and Bessel potentials, denoted by $\mathcal{J}^{\omega},~\omega>1$ and second type of non-archimedean pseudo-differential operator $\mathcal{A}^{\omega}$ on $\mathcal{D}(\mathbb{Q}_{p}).$ We show that these operators holds the positive maximum principle and a strongly continuous, positive, contraction semigroup on $ \mathbb{C}_{0}(\mathbb{Q}_{p}).$ Also, we solve Cauchy problem ( the inhomogeneous initial value problem ) related to fractional Fourier transform and these operators.