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?

  • 1
    $\begingroup$ I'd call it the "little-oh" Holder space. (As in $|f(x)-f(y)|=o(|x-y|^\lambda)$.) $\endgroup$ – David C. Ullrich Aug 18 '15 at 23:57
  • $\begingroup$ @DavidC.Ullrich That sounds good. Fits in with $H^{\lambda}$ as a "big oh" space. Thanks! $\endgroup$ – 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 $C^\alpha$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.