超距离域上函数的整体光滑性和沿曲线的光滑性
Smoothness of functions global and along curves over ultra-metric fields
摘要：
研究了无穷域上具有非平凡乘法超模的变元函数$f(x_1,...,x_m)$的光滑性，其中$m\ge 2$。研究了有界区域上连续或有界一致连续的函数的$C^n$或$C^n_b$光滑类的性质，给出了$n$阶以下的部分差分函数的光滑类.从$f\circ u\in C^n({\bf K},{\bf K}^l)$或$f\circ u\in C^n_b({\bf K},{\bf K}^l)$证明，对于每个$C^{\infty}$或$C^{\infty }_b$曲线$u: {\bf K}\to {\bf K}^m$，分别遵循$f\in C^n({\bf K}^m,{\bf K}^l)$或$f\in C^n_b({\bf K}^m,{\bf K}^l)$。此外，光滑类$C^{n,r}$和$C^{n,r}_b$和更一般的意义下的Lipschitz偏差分方程被认为是和他们的定理证明。
英文摘要：
The article is devoted to the investigation of smoothness of functions $f(x_1,...,x_m)$ of variables $x_1,...,x_m$ in infinite fields with non-trivial multiplicative ultra-norms, where $m\ge 2$. Theorems about classes of smoothness $C^n$ or $C^n_b$ of functions with continuous or bounded uniformly continuous on bounded domains partial difference quotients up to the order $n$ are investigated. It is proved, that from $f\circ u\in C^n({\bf K},{\bf K}^l)$ or $f\circ u\in C^n_b({\bf K},{\bf K}^l)$ for each $C^{\infty}$ or $C^{\infty }_b$ curve $u: {\bf K}\to {\bf K}^m$ it follows, that $f\in C^n({\bf K}^m,{\bf K}^l)$ or $f\in C^n_b({\bf K}^m,{\bf K}^l)$ respectively. Moreover, classes of smoothness $C^{n,r}$ and $C^{n,r}_b$ and more general in the sense of Lipschitz for partial difference quotients are considered and theorems for them are proved.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
