I have problems with my homework. First of all: $\mathcal{H}^2$ means the Sobolev space, right? Because in the script we are using a $W$ for Sobolev spaces and the $\mathcal{H}^2$ can't be found anywhere. My actual problem is this question:
Let $\Omega_{\beta} := B_1(0)_{||\cdot||_\infty}\setminus\left\{z = r\exp(i\phi) \in \mathbb{C}\mid r > 0, 0 \leq \phi \leq \beta \right\}$, $0\leq\beta\leq 2\pi$ an open and bounded domain, $\Gamma_\beta = \{z \mid r > 0, \phi = \beta \} \cap \partial\Omega_{\beta}$ part of its boundary. Furthermore, let $\alpha \in \mathbb{R}^+$, $f_\alpha(r) = r^\alpha \sin(\alpha\phi)$.
For what values of $\alpha$ is $f_\alpha(r) \in\mathcal{H}^2(\Omega_{\beta})$ for all $\beta$?
So, assuming I am right and $H$ describes the Sobolev space. Then it means that $f$ has to to have 2nd weak derivatives in $L^2$. So is that really all I have to do: Try to calculate the 2nd weak derivatives and see whether the are in $L^2$ or is there something more I have overseen so far?