# Notation for subspace of Hölder Space

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?

• I'd call it the "little-oh" Holder space. (As in $|f(x)-f(y)|=o(|x-y|^\lambda)$.) – David C. Ullrich Aug 18 '15 at 23:57
• @DavidC.Ullrich That sounds good. Fits in with $H^{\lambda}$ as a "big oh" space. Thanks! – Moya Aug 19 '15 at 3:05

As David C. Ullrich said, these are called "little Hölder spaces" (a search will bring up a bunch of papers where this term is used). A more conventional notation is $C^\alpha$ for Hölder spacce and $c^\alpha$ for little Hölder space.
One reason to consider these is that $c^\alpha$ is separable (unlike $C^\alpha$); indeed, it can be equivalently defined as the closure of smooth functions in $C^\alpha$.