Let $\mathbb{B}_1(0)$ be the unit ball centered at origin. Note that $n$ is the dimension of the space. For what relation of $k,p,n,a $ is $$f(x) = |x|^a$$ in $W^{k,p} (\mathbb{B}_1(0))$ and $W^{k,p} (\mathbb{R}^n \setminus \mathbb{B}_1(0))$?
My work: So far, I have considered the case where $a$ is negative and $k=1$. Assuming these two conditions, I have found that $f\in W^{1,p}(B_1(0))$ if and only if $a > \frac{p-n}{p}$. Additionally, I have found that $f\notin W^{1,p}(B_0(1))$ if $p\geq n$.
The questions that I currently do not know how to handle are
- How does $k>1$ change the problem?
- How does the space $\mathbb{R}^n\setminus B_1(0)$ affect the weak derivative?
For question 1, my problem mostly comes from not fully understanding the notion of $D^{\alpha}f(x)$ and how to find such a weak derivative. For question 2, since $f$ has a continuous derivative away from $0$, can I take the weak derivative to be the normal derivative in $\mathbb{R}^n\setminus B_1(0)$? If so, determining for which $p,n,a$ (if any) is $f \in W^{1,p}(\mathbb{R}^n\setminus B_1(0))$ may not be too difficult, but, once I consider $k>1$, I'm back to question 1.