Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says

For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} \frac{u^2}{|x|^2} \, \mathrm{d}x \leq \int_{\Omega} |\nabla u|^2 \, \mathrm{d}x $$ Where $ \Lambda=\frac{(N-2)^2}{4} $ is optimal constant.

Brezis and Vazquez has extended this inequality as follow:

For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ and $ 1 \leq p < \frac{2N}{N-2} $we have

$$ \int_{\Omega} |\nabla u|^2 \, \mathrm{d}x \geq \Lambda \int_{\Omega} \frac{u^2}{|x|^2} \, \mathrm{d}x + C \int_{\Omega} u^p \, \mathrm{d}x $$

See This Article.

From the above inequality we deduce that $$ \|\phi\|= \bigg( \int_{\Omega} |\nabla \phi|^2 \, \mathrm{d}x - \Lambda \int_{\Omega} \frac{\phi^2}{|x|^2} \, \mathrm{d}x \bigg)^{\frac{1}{2}} $$

defines a norm on the space $W_0^{1,2}(\Omega)$. Now Let $H(\Omega)$ be the compeletion of $C_0^{\infty}(\Omega)$ with respect to above norm.

Now since

$$ \bigg( \int_{\Omega} |\nabla \phi|^2 \, \mathrm{d}x - \Lambda \int_{\Omega} \frac{\phi^2}{|x|^2}\bigg)^{\frac{1}{2}} \leq \bigg( \int_{\Omega} |\nabla \phi|^2 \, \mathrm{d}x \bigg)^{\frac{1}{2}} $$ we can see that $ W_0^{1,2}(\Omega) \subset H(\Omega) $, and also converse inclusion fails.

Also the above inequality results that $ H(\Omega) $ is embedded in $ L^p(\Omega)$, for any $ 1 \leq p < \frac{2N}{N-2} $.

Now I want to investigate that, is this embedding compact for $ 1 \leq p < \frac{2N}{N-2} $ or not? I know some criterion for precompactness in $L^p$ spaces, but I can't use it to show compactness of embedding.

I am thanksed for any hint, or comments.