Usually the Holder space is denoted by the sdandard notation $C^{k,\alpha}$ or $C^{k+\alpha}$ where $\alpha \in (0,1]$ and $k$ is an integer, but when I read some materials on SPDE, they use the notation $C^{\gamma}$ where $\gamma > 0$. What is the precise meaning of this notation? I am very confused when $\gamma$ takes integer value.
2026-03-30 13:37:21.1774877841
Notation of Holder space used in stochastic analysis
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For the purpose of this question having an answer, I recreate my comments here. In the SPDE literature (especially in singular SPDE) it is usual to define $C^\gamma$ to be the space of functions that are $\lceil \gamma \rceil - 1$ times differentiable whose $(\lceil \gamma \rceil - 1)$-th derivative is $(\gamma - \lceil \gamma \rceil + 1)$-Holder continuous.
In particular, if $\gamma$ is an integer then this space is larger than the space of $\gamma$ times continuously differentiable functions.