Context: http://www.hairer.org/notes/Regularity.pdf, page 8 (between equations 2.9 and 2.10).
given a distribution $\eta \in \mathcal{C}^{\alpha}$ for some $\alpha > -r = |\inf A|, $ [...]
This implies that $\alpha$ is allowed to be negative, and so that $\eta$ may be in, for example, $\mathcal{C}^{-1}$. What does this mean? I've never come across $\mathcal{C}^k$ spaces with negative exponents.
Second, why does the author call $\eta$ a distribution? My understanding is that a distribution is an element of the continuous dual space of a function space. Here, however, it looks like $\eta$ is an element of the function space $\mathcal{C}^{\alpha}$ itself.