We have the fact that (here $u\in H_0^1(\Omega),u>0,\text{ in } \Omega ,\Delta u>0\in L^2 \text{ in } \Omega ,\Omega $ is a bounded domain (smooth),$n=4$) $$ u\le \frac{c}{|x|^2}*|\Delta u| $$ How can we deduce that $$ |\Delta u|_{L^1}^2 \ge c' [u]_{2,w}^2 . $$
I want use the weak Young inequality , but the theorem must satisfies $1<p,r,q<+\infty $.Now I don't have other ideas.