What is the meaning of $D^kf$ in this context? I'm a bit confused because I know that $Df(a)$ normally means the total derivative at $a$. Moreover, at some point the text treats $Df(x)$ and $\nabla f(x)$ as the same thing, which seems to confirm my belief. In other part, $D^2f$ appears to be the Hessian of $f$.
However, I don't see how these would work here considering how $[f]_{\alpha}$ is defined.
I also looked up on this Wikipedia page, which confused me even more, because they define the norm $||f||_{k, \alpha}$ involving $D^{\beta}f$, where $\beta$ is a multi-index, and I don't know the meaning of this either.
Usually, in the context of PDE and Hölder spaces, $$ |D^k f|_{0;\Omega} = \sup_{|\beta| = k}\sup_\Omega |D^\beta f|$$ where $\beta=(\beta_1, \ldots, \beta_n)$ is a multiindex, $|\beta| = \sum_i \beta_i$ and $$D^\beta f= \frac{\partial^{|\beta|}}{\partial x^{\beta_1}\dots \partial x^{\beta_n}}f$$.
$[D^k f]_\alpha $ is then the corresponding Hölder Norm for $0 < \alpha\le 1$, $$ \sup_{|\beta| = k}\sup\limits_{x,y\in\Omega, x\neq y} \frac{|D^\beta f(x) - D^\beta f(y)|}{|x-y|^\alpha}$$