what does $\|f\|_{C^{k,\alpha}}$ mean?

Question

I am reading a reference paper by Elgini and Jeong titled "Symmetries and Critical Phenomena in Fluids" (arXiv:1610.09701).

There appeared two expressions: $$u_0\in C^0\left([0,T];C^{k,\alpha}\right)\tag{1}$$ $$\|\nabla u\|_{L^{\infty}}\leq C_{k,\alpha}\|u(t)\|_{C^{k,\alpha}}\tag{2}$$

Question: what does $\|u(t)\|_{C^{k,\alpha}}$ mean?

Thanks a lot!

Answer 1

As noted in the comments: $\newcommand{\n}[1]{\left\| #1 \right\|} \n{f}_{C^{k,\alpha}}$ refers to a norm on $k$-times-continuously differentiable functions with Hölder parameter $\alpha$.

To dive a little deeper (and to give something a little more permanent than a linked-to article on Wikipedia), and give the question a fuller answer, however...

First, some preliminaries:

• Define the Hölder coefficient of $f$ (domain $\Omega$) w.r.t. $\alpha$ by

$$|f|_{C^{0,\alpha}(\Omega)} := \sup_{\substack{x,y \in \Omega \\ x \ne y}} \frac{|f(x) - f(y)|}{ \|x-y\|^\alpha }$$

• A multi-index is a tuple of nonnegative integers, $\beta := (\beta_1,\cdots,\beta_n) \in \mathbb{Z}_{\ge 0}^n$. We denote $|\beta| := \sum \beta_i$. Their primary use is in compactifying derivative notations by writing $\partial^\beta = \partial_1^{\beta_1} \partial_2^{\beta_2} \cdots \partial_n^{\beta_n}$ where $\partial_i$ denotes the partial derivative in the $i$th coordinate.

• We may define a $C^k$ norm for multivariable functions by $$ \n{f}_{C^{k}(\Omega)} := \max_{|\beta| \le k} \left( \sup_{x \in \Omega} \left| \partial^\beta f(x) \right| \right) $$ referencing ordinary absolute values for clarity (analogous to the usual sup-norms applied in the one-dimensional case).

Then we define,

$$\n{f}_{C^{k,\alpha}(\Omega)} := \n{f}_{C^k(\Omega)} + \max_{|\beta|=k} \left| \partial^\beta f \right|_{C^{0,\alpha}(\Omega)}$$