A question on Holder spaces

Question

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. I've just encountered with another definition of Holder condition in the Tsybakov's book $$|f^{(l)}(x)-f^{(l)}(x')|\leq L|x-x'|^{\beta-l}$$ where $l=\lfloor\beta\rfloor$. Are they related?

Answer 1

The first definition concerns $C^\beta$, also written as $C^{0,\beta}$. These are zeroth-order Hölder spaces, which contain functions that have an appropriate modulus of continuity, but are not necessarily differentiable.

The second definition describes what could be written as $C^{l,\beta-l}$, the space of order $l$, which is a more general concept. When $l=0$, it's the same as above. When $l$ is positive, we require the existence of $l$ derivatives, and also the highest derivative must have an appropriate modulus of continuity.

When one wants to turn these spaces into a normed vector space, the boundedness of lower order derivatives is also required. Thus, you may well find another definition in the third book you pick up.