As mentioned, this is largely a question on notation. I'm reading Fractional Integrals and Derivatives: Theory and Applications by Samko, Kilbas, and Marichev. I'm just starting and I'm curious about this notation:
Let $\Omega$ be a finite interval. Denote $H^{\lambda}=H^{\lambda}(\Omega)$ by the space of all Hölder continuous functions on $\Omega$ with coefficient $0\leq \lambda\leq 1$ ($H^0$ is defined as bounded functions). Then define a subspace $h^{\lambda}=h^{\lambda}(\Omega)\subset H^{\lambda}$ by functions $f$ satisfying $$\lim\limits_{x_2\to x_1}\dfrac{f(x_2)-f(x_1)}{|x_2-x_1|^{\lambda}}=0$$
(This is actually a typo in the book; they omit the $\lambda$ in the denominator but I assume this is a mistake but I couldn't find an errata). That $h^{\lambda}\subset H^{\lambda}$ is clear. I was just wondering if the space $h^{\lambda}$ had a name? The book doesn't give a specific name to them; I would assume just the space of "strong" Hölder functions would work, but I couldn't find a reference anywhere.
Has anyone seen this notation before or equivalent notation?