What does $ C_{-\infty}$mean

Question

This question refers to the link: https://mathoverflow.net/questions/260854/a-roadmap-to-hairers-theory-for-taming-infinities/260941#260941 In the answer of Abdelmalek Abdesselam the following equation is mentioned. $$ C_{-\infty}(f,g)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^d}\frac{\overline{\widehat{f}(\xi)} \widehat{g}(\xi)}{|\xi|^{2}} d^d\xi $$

My question is what does $C_{-\infty}$ mean, specifically the ${-\infty}$ part. Few words and explanation of the equation would be highly appreciated.

Answer 1

$C_{-\infty}$ in that answer is a kind of covariance between $f$ and $g$. The subscript $-\infty$ is used because $C_{-\infty}$ is meant to be understood as a kind of limit of $C_r$, defined for $r \in \mathbb{Z}$ later in the answer, as $r \to -\infty$.

Answer 2

It may vary, but typically $C_0$ is the set of all continuous functions on $\mathbb{R}$, $C_1$ the set of all functions with continuous first derivative, $C_2$ the set of all functions with continuous second derivative, and so forth, with $C_\infty$ being the set of all functions where all derivatives are continuous. I would assume, then, that $C_{-\infty}$ is the set of all functions which are infinitely integrable, i.e. the functions that you can keep integrating over and over again without worrying about pathological behaviour.