Hölder space with negative exponent

Question

The definition of the Hölder space with positive exponent is well-known. Namely we say that a function $f\colon\mathbb{R}\to\mathbb{R}$ belong to the Hölder space $\mathcal {C}^{\alpha}$ ($\alpha>0$), if $f$ has $[\alpha]$ continuous derivatives and it's $[\alpha]$-th derivative (that we denote here by $f^{([\alpha])}$) satisfies $$ \|f\|_\alpha:=\|f\|_\infty+\sup_{s,t\in\mathbb{R}}\frac{|f^{([\alpha])}(t)-f^{([\alpha])}(s)|}{|t-s|^{\alpha-[\alpha]}}<\infty. $$

As usual, $[\alpha]$ denotes the integer part of a real $\alpha$.

It is also clear that if $f\in\mathcal{C}^\alpha$, then $f'\in\mathcal{C}^{\alpha-1}$, $\alpha>1$.

My question is how can one extend this definition for negative $\alpha$. Let $f\in\mathcal{C}^{1/2}$. Then its derivative $g:=f'$ is a generalized function (distribution). How one can define $\mathcal{C}^{-1/2}$, so that $f'\in\mathcal{C}^{-1/2}$?

Answer 1

The straightforward approach would be to say that a distribution $f\in D'$ belongs to the class $C^{-\alpha}$ for $\alpha\in(0,1)$ there exists $F\in C^{1-\alpha}$ such that the derivative of $F$ in the sense of distributions is $f$.

However, I do not know to what kind of problems this approach presents - most notably, what you use as a norm in this space.

Answer 2

Negative Holder spaces can be defined and are also used. One definition is through Besov spaces with $p, q = \infty.$ Another is straightforward through scaling properties. It turns out that a positive Holder functions scales as $\lambda^{\alpha}$ and a negative one as $\lambda^{-\alpha},$ for $\alpha \in (0,1)$ in a certain sense that can be made precise either (humbly) by my two questions here and here or for example in this paper here at page 10.