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



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> Is the completion of $C\_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
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> $$\|u\|= \Bigg(\int\_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}, $$
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> well-known space?
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> 


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 $D^{2,2}(\mathbb{R}^N) \subset H$.



Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
"The" completion is not always a space of functions, for $N=1$ or $2$ for example it is a quotient $D^{-2}L^2/P\_1$ (equivalence classes of functions $u\in H^2\_{loc}$ with $\partial\_i \partial\_j u\in L^2$ with $u\sim v$ iff $u(x)-v(x)=a\cdot x+b$.


For $N=3$ and $4$ it is $D^{-1}\dot{H}^1/P\_0$ where the "homogeneous Sobolev space" $\dot{H}^1$ is "the" completion of $C\_0^\infty$ with respect to $\int|\nabla v|^2$ (and $P\_0$ is the space of constants).


Only for $N\ge 5$ is "the" completion with respect to $\int (\Delta v)^2$ a function space: $\dot{H}^2$, the space of tempered distributions $v$ (in fact, functions in $L^{2N/(N-4)}$) whose Fourier transform $\hat{v}\in L^2\_{loc}$ with $\int [|\xi|^2\hat{v}(\xi)]^2<\infty$. Same space as your $D^{2,2}$, because $|\xi|^4=\sum(\xi\_i \xi\_j)^2$.


(If self-advertizing is allowed: maybe have a look at my paper "Splines minimizing rotation-invariant semi-norms in Sobolev spaces", 1977).


