Let $\Omega$ be an open set in $R^n$, and $f \in L^1_{loc}(\Omega)$, such that foreach multiindex $\alpha\in N^n$, $|\alpha| = l$ f has weak derivative $D^\alpha f$ in $L^p(\Omega)$, with $1\leq p\leq \infty$.

In general it is not true that $f\in L^p(\Omega)$, but it have to be true that $f\in L^p_{loc}(\Omega)$. How can be shown that?

`$f\in W^{1,p}\subset L^p$`

. Here we don't know that, hence it seems a bit harder. I suspect, though, that an approximation argument using convolution with a mollifier and using Poincaré for each approximand will resolve it. – Harald Hanche-Olsen Apr 4 '10 at 22:30