What is the definition of Hölder space $C^{\alpha}(\mathbb{R})$.

Are we saying that $\sup _{x, y \in \mathbb{R} \atop x \neq y}\left\{\frac{|u(x)-u(y)|}{|x-y|^{\alpha}}\right\}$ $<\infty$ $(0<\alpha<1)$?

So a smooth function on $\mathbb{R}$ is not necessarily in $C^{\alpha}(\mathbb{R})$?

Solution 1:

The definition is fine and the quantity $\sup _{x, y \in \mathbb{R} \atop x \neq y}\left\{\frac{|u(x)-u(y)|}{|x-y|^{\alpha}}\right\}$ is a seminorm.

Note that a smooth function defined on $\mathbb{R}$ is not necessarily Hölder continuous and this is because the domain in unbounded. Clearly $\mathcal{C}^{\infty}([a,b]) \subset \mathcal{C}^{0,\alpha}([a,b])$, but this condition fails in unbounded domains.

More generally note that the following holds (where $\text{Lip}(\Omega)$ is $\mathcal{C}^{0,\alpha}(\Omega)$ with $\alpha =1$):

Let $\Omega \subset \mathbb{R}^n$ bounded convex and open set, then $\mathcal{C}^{1}(\Omega) \subset \text{Lip}(\Omega)$.

Nevertheless, note that even if $\Omega$ is not convex, then this innocent inclusion fails: $$\text { Let } \Omega=\left\{(x, y) \in \mathbb{R}^{2}: y<\sqrt{|x|}, x^{2}+y^{2}<1\right\}$$ and $$ f(x,y)= \operatorname{sign}(\mathrm{x})\left\{\begin{array}{l} y^{\beta} \quad \text { if } y>0 \\ 0 \quad \text { if } y \leqslant 0 \end{array} \text {, with } 1<\beta<2 \text {. Then } f \in \mathrm{C}^{1}(\bar{\Omega}) \backslash \operatorname{Lip}(\Omega)\right. \text {. } $$