A Sobolev function

Question

I want to prove that the function $$u(x)=\sum_{k=1}^\infty 2^{-k}|x-r_k|^{-\alpha}$$ is in the Sobolev space $W^{1,p}(B_1(0))$ if $0<\alpha<\frac{n-p}{p}$ where the sequence $ (r_k)$ is dense in the unit ball. I managed to prove this for every single summand of this series but I do not see why is series is even well-defined. Does someone have a hint for me? Thanks a lot in advance.

Answer 1

The $W^{1,p}$ norm of $|r-r_k|^\alpha$ is bounded: $$ \int_{|x|\le1}|x-r_k|^{-\alpha p}\,dx=\int_{|x-r_k|\le1}|x|^{-\alpha p}\,dx\le\int_{|x|\le2}|x|^{-\alpha p}\,dx. $$ Similarly for the gradient. Let $C$ be a bound. Then $$ \bigl\|2^{-k}\,|x-r_k|^{-\alpha}\bigr\|_{1,p}=2^{-k}\,\bigl\||x-r_k|^{-\alpha}\bigr\|_{1,p}\le C\,2^{-k}. $$ Since $\sum 2^{-k}<\infty$ and $W^{1,p}$ is a Banach space, the series converges in $W^{1,p}$ (it also converges pointwise on $B_1(0)\setminus\{r_k\}_{k=1}^\infty$.)