非阿基米德伪微分方程的径向解
Radial Solutions of Non-Archimedean Pseudo-Differential Equations
摘要：
考虑一类具有分数阶微分算子$D^α$，$α>0$的方程，其复值函数$x\mapsto f(|x|_K)$在非阿基米德局部域$K$上只依赖于绝对值$|\cdot |_K$ .我们引入一个右逆$I^α$ $D^α$，这样的变化的未知函数$u=I^αv$减少了柯西问题的方程与$D^α$（径向函数）的积分方程的性质类似于那些经典的Volterra方程。这与$D^α$在其他类函数上的复杂行为形成了对比。
英文摘要：
We consider a class of equations with the fractional differentiation operator $D^α$, $α>0$, for complex-valued functions $x\mapsto f(|x|_K)$ on a non-Archimedean local field $K$ depending only on the absolute value $|\cdot |_K$. We introduce a right inverse $I^α$ to $D^α$, such that the change of an unknown function $u=I^αv$ reduces the Cauchy problem for an equation with $D^α$ (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. This contrasts much more complicated behavior of $D^α$ on other classes of functions.
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