Long ago, I proved that every derivation from $C^{k+1}$ functions to $C^k$ functions is given by a $C^k$ vector field. (The same fact with $\infty$ in place of $k$ and $k+1$ is, of course, classical.) The first (and only, as far as I know) publication is in the book "Manifolds, Tensor Analysis, and Applications" by Abraham, Marsden, and Ratiu (p.235).

The story behind this is that, at the time, I was sharing an office with Bill Floyd; Tudor Ratiu, whose office was just down the hall, was working on this book. Of course, he knew about the $C^\infty$ version of the result, but he stopped by to ask Bill about the $C^k$ version, and I happened to be there too. Neither Bill or I knew anything about it, but later (the same evening, I think, but my memory may be playing tricks here) I worked out a proof. When I told Tudor about it, he asked if he could put it into the book, and I said sure. I think the proof in the book is more streamlined than my original argument.

Long ago, I proved that every derivation from $C^{k+1}$ functions to $C^k$ functions is given by a $C^k$ vector field. (The same fact with $\infty$ in place of $k$ and $k+1$ is, of course, classical.) The first (and only, as far as I know) publication is in the book "Manifolds, Tensor Analysis, and Applications" by Abraham, Marsden, and Ratiu (p.235).

The story behind this is that, at the time, I was sharing an office with Bill Floyd; Tudor Ratiu, whose office was just down the hall, was working on this book. Of course, he knew about the $C^\infty$ version of the result, but he stopped by to ask Bill about the $C^k$ version, and I happened to be there too. Neither Bill nor I knew anything about it, but later (the same evening, I think, but my memory may be playing tricks here) I worked out a proof. When I told Tudor about it, he asked if he could put it into the book, and I said sure. I think the proof in the book is more streamlined than my original argument.