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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.

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    $\begingroup$ Could you point to some of the particular materials since context might matter? Generally though, if $\gamma$ is not an integer and one has $\gamma = n + a$ for $n \in \mathbb{N}$ and $0 <a < 1$ then $C^\gamma$ is the space of $n$-times differentiable functions whose $n$-th derivative is $a$-Holder continuous. In the case where $\gamma$ is an integer, it is common in the SPDE literature to set $C^\gamma$ to be the slightly larger space of $(\gamma-1)$-times differentiable functions whose $(\gamma - 1)$-th derivative is Lipschitz. $\endgroup$ – Rhys Steele Jan 13 at 17:10
  • $\begingroup$ @RhysSteele Yes, the material for example the notes in the page mi.fu-berlin.de/math/groups/stoch/teaching/_Stochastics/… on page 32 and 33, the definition is not clear when $\gamma$ is an ingeter. $\endgroup$ – Inuyasha Jan 13 at 17:24
  • $\begingroup$ @RhysSteele Or for example in the following paper arxiv.org/abs/1801.04596 in section 2.3 on page 13, he used notation $C^{\gamma}$ but no further definition. $\endgroup$ – Inuyasha Jan 13 at 18:02
  • $\begingroup$ I'd guess that both of these authors are likely to mean the same thing as I do by $C^\gamma$ (they both have works in the singular SPDE literature where this notation is certainly standard). In fact, Perkowski even clarifies that in the case $\gamma = 1$ his notation means what I have written in the first paragraph on page 33. You should expect this to be the default definition for anyone using either paracontrolled calculus or regularity structures (e.g. the paper of Hao Shen uses discretisations of regularity structures). $\endgroup$ – Rhys Steele Jan 13 at 18:53
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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.

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