In an earlier paper (A. N. Kochubei, {\it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirov's fractional differentiation operator $D^\alpha$, $\alpha >0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^\alpha$ that the appropriate change of variables reduces equations with $D^\alpha$ (for radial functions) to integral equations whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we begin an operator-theoretic investigation of the operator $I^\alpha$, and study a related analog of the Laplace transform.

中文翻译：

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非阿基米德径向微积分：Volterra 算子和拉普拉斯变换

在较早的一篇论文（AN Kochubei, {\it Pacif. J. Math.} 269 (2014), 355--369）中，作者考虑了 Vladimirov 的分数微分算子 $D^\alpha$, $\alpha > 0$，到非阿基米德域上的径向函数。特别是，发现具有这样的右逆 $I^\alpha$，变量的适当变化将具有 $D^\alpha$（对于径向函数）的方程简化为积分方程，其性质类似于经典 Volterra 方程的性质。换句话说，我们在非阿基米德伪微分算子的框架中发现了常微分方程的对应物。在本文中，我们开始对算子 $I^\alpha$ 进行算子理论研究，并研究拉普拉斯变换的相关模拟。