Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may want your initial condition (or solution) to some PDE to be in the space $C^{k+2, \alpha}$. This is more restrictive than the space $C^{k+2}$ and I don't know what the payoff is.

How do I know which function spaces to use?