I want to do the following tasks
Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following
a) $u_p \in L^1(G)$, if $\rho>-2$
b) $u_p\in W^{(1,1)}(G)$, if $\rho>-1$ , where $W^{k,p}$ is the Sobolev-space.
For a):
I used substitution: $x=rcos(\phi)$, $y=rsin(\phi)$, to obtain:
$\int_G||x||_2^{\rho}dx=\int_0^{2\pi}\int_0^1 r^p\cdot r drd\phi=\int_0^{2\pi}\int_0^1 r^{p+1}\cdot drd\phi$ which is finite if $\rho>-2$.
I guess this should be correct.
But I don't know how to show the other task.
I have to show that the partial derivatives of $u_p$ are in $L^1$?