Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is generated by a distribution). In characteristic $p$, you can still talk about $D$-modules (although things are more complicated), and it would be nice/interesting to know if there were some similar kind of natural object that filled the same role as distributions do in char. zero. One may of course have other reasons for asking this question, as "generating $D$-modules" is just one of many things distributions do.
The problem is of course that the construction (continuous dual of compactly supported smooth functions) doesn't make sense in characteristic $p$, and I have no idea what kind of analogous objects fill the same kind of roles as the ingredients in characteristic 0.
Question: Is there any such thing? Has this been studied?
