Let $f\in W_0^{1,p}(U)$, for $U$ a bounded domain and $p < n/(n-1)$. I am trying to prove that there is an inequality of the form
$\|f\|_{W^{1,p}} \leq C \int_{\Omega} |\Delta f| $
where $\Delta$ is the Laplacian. I tried applying the Sobolev embedding theorem, followed by the inequality
$\|D^2 f\|_p \leq C \|\Delta f\|_p$
but this does not hold for $p = 1$. I also tried using elliptic estimates, also to no avail. Does anyone have any suggestions?
EDIT: Perhaps it is possible to deduce this from the $L^p$ inequality
$\|D^2 f\|_{L^p} \leq C \|\Delta f\|_{L^p}$
for $u\in W_0^{2,p}(\Omega)$, $1 < p < \infty$?