$ W_0^{1,p}$ norm bounded by norm of Laplacian 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 $u\in W_0^{2,p}(\Omega)$, $1 < p < \infty$? 

I don't think this estimate is correct. Consider the case where $U$ is the unit disk $U = B_1(0)$ in $\mathbb{R}^2$ and set $f_\epsilon(x) = \log \frac{|x| + \epsilon}{ 1 + \epsilon}$ for $\epsilon > 0$. These functions all vanish on the boundary of $U$.
Setting $r = |x|$, one computes $\nabla f_\epsilon(x) = \frac{x}{r(r+\epsilon)}$ and $\Delta f_\epsilon(x) = \frac{\epsilon}{r(r+\epsilon)^2}$. Using polar coordinates, it is straight forward to show that $\|\Delta f_\epsilon\|_{L^1}$ remains bounded but $\|f_\epsilon\|_{W^{1,2}} \sim \sqrt{\log |\epsilon|} $ as $\epsilon \to 0$.
$L^p$ theory for elliptic equations does not extend to the case $p = 1$.
