Arithmetic partial differential operators on ramified extensions of ${\text{ZZ}}_p$

The notion of $p$-derivation as introduced by Buium has a rich history of applications in arithmetic geometry. Working over ${\text{ZZ}}_p$, Buium-Ralph-Simanca showed that arithmetic differential operators built from these determine $p$-adic analytic functions and vice versa. Notably, over ${\text{ZZ}}_p$, there is only one $p$-derivation. In this article, we consider the same question for ramified extensions $A_{\pi }$ with uniformizer $\pi$. This has the effect of passing from ordinary to partial differential operators, since such extensions can enjoy multiple $\pi$-derivations. We introduce a notion of analytic functions which are naturally determined by these operators in a way analogous to that in the unramified case, but with a significantly richer structure. We show that under mild conditions, analytic functions and partial differential operators determine each other in this setting.