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My question is about show: Let $N\geq 2$ and $1\leq p<\infty$. Every radial function $u\in W^{1,p}(\mathbb R^ N)$ satisfy $$ |u(x)|\leq C|x|^{-\frac{N-1}{p}}\|u\|_{W^{1,p}(\mathbb R^ N)}, $$ where $C$ is a positive constant depending only on $N$ and $p$.

And why this imply that $u$ is almost everywhere equal to a continuos function for $x\neq 0$?

Raoní Cabral Ponciano
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  • What have you tried? Are you familiar with continuity of Sobolev functions on $\mathbb R$? – Michał Miśkiewicz Apr 12 '18 at 22:38
  • I tried proof that $f(r):=u(|x|)$ (for $r=|x|$) satisfy this inequality and it is continuous in $(0,+\infty)$. But I could not. – Raoní Cabral Ponciano Apr 13 '18 at 08:27
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    Hint. You can rephrase the condition $u \in W^{1,p}$ in terms of $f$ (e.g. $u \in L^p$ corresponds to $r^{(n-1)/p} f(r) \in L^p$). In particular you find that $f \in W^{1,p}_{loc}$, so $f$ is continuous. Applying the fundamental theorem of calculus to $f(r)-f(\infty)$, you should obtain some estimates. – Michał Miśkiewicz Apr 13 '18 at 10:11

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