I am reading "Holder and Locally Holder Continuous Functions and Open Sets of Class $C^k,C^{k,\lambda}$" by Renato Fiorenza.
I have seen that a function is Locally Holder Continuous on a set if it is Holder Continuous on every open subset of that set. Also I have read that Locally Holder Continuous does not imply Holder Continuous (but that the inverse is obviously true).
My question relates to the definition of Holder Spaces and specifically these sentences:
Does this imply that $C^{k,\lambda}(\Omega)$ and $C_{loc}^{k,\lambda}(\Omega)$ are equivalent? Why is it that after every mention of $C^{k,\lambda}(\Omega)$, $C_{loc}^{k,\lambda}(\Omega)$ appears in brackets afterwards, as if to signify that everything stated also applies to it. There is also this line which seems to state that the two spaces are equivalent:
Also I would like to clarify that a function is in $C^{k,\lambda}(\Omega)$ if it is $k-$times continuously differentiable and that the function AND its k derivatives are Holder Continuous. Does the function being Holder Continuous when $k=0$ mean that the function itself is Holder Continuous. If so, why does the author treat the cases for $k=0$ and $k >0 $ separately. Why not just say that the function must be Holder continuous for all $k \geq 0$.