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What do the symbols $C^{1,2}(\Sigma)$ and $C^{2,\alpha}(\Sigma)$ mean where $\Sigma$ is the domain? I know that $C^m(\Sigma)$ means $m$ times continuously differentiable functions. Also, what does $C^{\infty}_0(\Sigma)$ mean?

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Are you studying parabolic differential equations? In this context one usually writes $C^{1, 2}$ to mean "$C^1$ with respect to the time variable, $C^2$ with respect to the space variables" (see e.g. Evans' book on PDEs). –  Giuseppe Negro Sep 2 '13 at 16:30

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(Working under the assumption that $\Sigma \subseteq \mathbb{R}^n$ is open.)

As you correctly state, $C^m(\Sigma)$ is the collection of all $m$ times continuously differentiable functions on $\Sigma$ - for this to make sense, it is clear that we need $m$ to be a non-negative integer. The spaces $C^{m, \alpha}(\Sigma)$, with parameter $\alpha \in (0, 1]$ are similar. They consist of $m$ times continuously differentiable functions on $\Sigma$ such that the $m^{\textrm{th}}$ partial derivatives are Hölder continuous with exponent $\alpha$. In particular $C^{m,\alpha}(\Sigma) \subset C^m(\Sigma)$. The spaces $C^{m, \alpha}(\Sigma)$ are often called Hölder spaces. You can also extend the definition to spaces $C^{m, \alpha}(\Sigma)$ with $\alpha > 1$.

A function $f : \Sigma \to \mathbb{R}$ is Hölder continuous with exponent $\alpha$ if there is a non-negative constant $C$ such that $$|f(x) - f(y)| \leq C|x - y|^{\alpha}$$ for all $x, y \in \Sigma$.

As with $C^m(\Sigma)$, $C^{\infty}(\Sigma)$ denotes the collection of all infinitely continuously differentiable functions - actually, as we can always can continue to differentiate, the word 'continuously' is redundant so we just say 'infinitely differentiable'. Now $C^{\infty}_0(\Sigma) \subset C^{\infty}(\Sigma)$ denotes the collection of all infinitely differentiable functions with compact support.

The support of a function $f : \Sigma \to \mathbb{R}$ is $\mathrm{supp}(f) = \overline{\{x \in \Sigma\ |\ f(x) \neq 0\}}$.

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