The function space $H^{\alpha} (\Omega)$ for $0 < \alpha \le 1$, is the set of functions:
$\{ f \in C^0(\Omega) : \sup_{x \neq y} \dfrac{|f(x) - f(y)|}{|x-y|^{\alpha}} < \infty \}$
with the metric $d_{H^{\alpha}} = || f - g ||_{H^{\alpha}}$, where $||f||_{H^{\alpha}} = ||f||_{sup} + [f]_{H^{\alpha}} \text{ , } [f]_{H^{\alpha}} = \sup_{x \neq y} \dfrac{|f(x) - f(y)|}{|x-y|^{\alpha}} $
Now, if $0 < \alpha < \beta \le 1$, then
$[f]_{H^{\alpha}} \le 2 ||f||_{sup}^{1-\frac{\alpha}{\beta}} [f]_{H^{\beta}}^{\frac{\alpha}{\beta}} \space \forall f \in H^{\beta}$
And also, there is some constant $M$ so that:
$||f||_{H^{\alpha}} \le M ||f||_{sup}^{1-\frac{\alpha}{\beta}} ||f||_{H^{\beta}}^{\frac{\alpha}{\beta}} \space \forall f \in H^{\beta}$
These were some questions on a problem set: I have checked that $d_{H^{\alpha}}$ is a metric, and proved the two properties (in the second I found that $M = 2$ is sufficient). However, rather blindly. It's easy to show from the first that if $0 < \alpha < \beta \le 1$, then $H^{\beta} \subset H^{\alpha}$.
What else do these formulas mean? Are they just some useful inequalities, or do they establish some connection between $H^{\beta}$ and $H^{\alpha}$?
Thanks.