# A Question about compactness of an embedding into $L^p$ spaces

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

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

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.

• in the `improve Hardy inequality' you need to renormalize the $L^p$ term; ie. it should be $\| u \|_{L^p}^2$. Take a look at this paper regarding $H(\Omega)$ , I think it says that some of the earlier papers regarding $H$ may have had some flaw??? arxiv.org/pdf/1102.5661v1.pdf Further here is a paper regarding improved $L^2$ Hardy inequaliteis : arxiv.org/abs/0805.0610 – Math604 Oct 8 '15 at 18:54
• @Math604: thanks. Did you know any other article about improved case of rellich inequality and similar space H as completion with respect to norm $$\bigg( \int_{\Omega} \Big((\Delta u)^2 - \dfrac{N^2(N-4)^2}{16} \dfrac{u^2}{|x|^4}\Big) \,\mathrm{d}x \bigg)^{\frac{1}{2}}$$ – Finish Oct 9 '15 at 18:31
• there should be a bunch of "improved Hardy-Rellich papers". Try googling that phrase (also Amir Moradifam wrote some papers on this in the radial case). Regarding the space $H$ I don't really know of many references.... Do you need to understand $H$ for a particular reason or ?? – Math604 Oct 9 '15 at 18:47
• Thanks. I must give a seminar in the classroom in this regard. I want to minimize a function in this space and I need an compact embedding. – Finish Oct 9 '15 at 18:52
• Maybe you can argue directly. Let $u_m$ be bounded by say $1$ in $H$ and use the improved imbedding to see that $u_m$ is bounded in $L^2(\Omega)$. Then directly from the inequality again we see that $u_m$ should be bounded in $H^1_0_{loc}( overline{\Omega} \backslash \{0\} )$ (trying to say its bounded in $H^1$ except near origin). By diagonal argument there should be some $u in H^1_{loc} (remove near origin) such that$ u_m \rightharpoonup u$in$H^1_{loc}(away from origin). Now try and show that $u_m$ converges in $L^1(\Omega)$ (so in $L^1$ it can't concentrate at origin). Then... – Math604 Oct 9 '15 at 20:01