Let $0<\alpha<1$ and $f \in C^{\alpha}$ be a Hölder function (either with compact support on $\mathbb R^n$ or on a closed Riemaniann manifold). From what I understand, the derivative of $f$ in the sense of distributions is in the Besov space $B_{\infty,\infty}^{\alpha-1}$.
Let $\beta>1-\alpha$. Do we have $B_{\infty,\infty}^{\alpha-1} \subset B_{1,1}^{\beta}$?
Or is the derivative of $f$ an element of the dual of $C^{\beta}$? (I don't know whether this question is weaker than the previous one or not)
In particular, if $\alpha>1/2$, can we pair $f$ with its derivative?
I would be happy to just have the answer to the last question if the other ones are harder. From what I saw, this is not directly stated in Triebel's classical books.