I have a question that I could not find it any where.

Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm

$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}, $$

well-known space?

I know that completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm below $$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} | \nabla u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}, $$ is well-known space $D^{1,2}(\mathbb{R}^N)=\Big \{ u \in L^{\frac{2N}{N-2}}(\mathbb{R}^N); \,D^{\alpha}u \in L^2(\mathbb{R}^N) \,\,\,\,for \, all \, |\alpha|=1\} $

and also I know that completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm below $$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} | D^2 u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}, $$ is $D^{2,2}(\mathbb{R}^N)=\Big \{ u \in L^{\frac{2N}{N-4}}(\mathbb{R}^N); \,D^{\alpha}u \in L^2(\mathbb{R}^N) \,\,\,\,for \, all \, |\alpha|=2\}. $

It is obvious if I name the completion space with $H$ then $D^{2,2}(\mathbb{R}^N) \subset H$.