In the book *Sobolev Spaces with Application* of Maz'ya, $\mathring {L^k\_p}(\Omega)$ is defined to be the completion of $\mathcal D(\Bbb R^n)$ under the norm $||\nabla^ku||\_{L\_p(\Omega)}$.


For nice domain (and correct values of $k,p,q,n$), Sobolev's embedding theory tell us that $\mathring {L^k\_p}(\Omega)\hookrightarrow L\_q(\Omega)$ continuously. However, I am having a trouble with understanding the space for more general cases. Actually, in the same book we have this theorem:



> 
> Let $\Omega=\Bbb R^n$, $n<kp$, $p>1$. For a multi-index $\alpha$ of order $|\alpha|\le l-n/p$, there exists a sequence $u\_{m}\in\mathcal D(\Bbb R^n)$ such that $u\_{m}\to 0$ in $\mathring {L^k\_p}(\Bbb R^n)$ while $u\_{m}\to x^{\alpha}$ in $\mathcal D'(\Bbb R^n)$.
> 
> 
> 


The way I interpreted this is that for these particular values of $n,p,k$; some "polynomials" (that are obviously not in any $L\_q(\Bbb R^n)$) are arbitrary close to $0$ of $\mathring {L^k\_p}(\Omega)$.


Here are some of my questions:


1. How does a typical element of $\mathring {L^k\_p}(\Omega)$ look like?
2. We may define, for $u\in\mathring {L^k\_p}(\Omega)$, $$|u|\_{\mathring {L^k\_p}}:=\lim\_{m\to\infty} ||\nabla^ku\_m||\_{L\_p}$$ where $u\_m\in\mathcal D(\Omega)$ converges to $u$ in the above sense. Since $\nabla^ku\_m$ actually converges in $L\_p$, we may even write its limit suggestively as $\nabla^ku\in L\_p$ but what does it mean when $u\notin \mathcal D'(\Omega)$? Surely it cannot be the weak derivative of $u$ right?
3. $|u|\_{\mathring {L^k\_p}}$ as defined above is just a seminorm right (generally)? Is this space even Hausdorff?



Typical elements of the space $\mathring {L^k_p}(\Omega)$
The answer presented here is copied from the paper [2]. Many similar results (sometimes with more complicated proofs) can be found in [1].
Let the space $L^{k,p}$ be defined by:
$$
L^{k,p}(\mathbb{R}^n)=\{ f\in \mathcal{D}'(\mathbb{R}^n):\, \nabla^kf\in L^p(\mathbb{R}^n)\},
\quad
\Vert f\Vert\_{L^{k,p}}=\Vert \nabla^kf\Vert\_p.
$$



> 
> **Theorem** (Theorem 4 in [2]). *Let $1\leq p<\infty$ and $k=1,2,\ldots$ Then $\mathcal{D}(\mathbb{R}^n)$ is dense in $L^{k,p}$ if and only if
>  $n>1$ or $p>1$.*
> 
> 
> 


Here density is with respect to the seminorm $\Vert \nabla^kf\Vert\_p$. That is for any $f\in L^{k,p}(\mathbb{R}^n)$ there is a sequence $f\_i\in \mathcal{D}(\mathbb{R}^n)$ such that $\Vert \nabla^k(f-f\_i)\Vert\_p\to 0$.
Therefore the space coincides with the completion that is described in the question i.e., 
$\mathring {L^k\_p}(\mathbb{R}^n)=L^{k,p}(\mathbb{R}^n)$.
Formally, the spaces $L^{k,p}$ are defined as distributions and only their $k$-th order derivatives are functions. However we have
$$
L^{k,p}(\mathbb{R}^n)\subset W^{k,p}\_{\rm loc}(\mathbb{R}^n)=\{f\in L^p\_{\rm loc}:\, \nabla^\ell f\in L^p\_{\rm loc} \text{ for all $0\leq\ell\leq k$\}}.
$$
This is a classical result, but a short and self-contained proof is given in this post <https://mathoverflow.net/a/296464/121665>.


**This gives an answer to questions 2 and a partial answer to question 1:**



> 
> $$ \mathring {L^k\_p}(\mathbb{R}^n)=L^{k,p}(\mathbb{R}^n)\subset
>  W^{k,p}\_{\rm loc}(\mathbb{R}^n). $$
> 
> 
> 


Clearly $u$ is a function and $\nabla^k u$ defined as a limit of approximations (as in question 2) is a weak derivative of $u$.


**So how does a typical element of $L^{k,p}$ look like?**


If $\Omega$ is a bounded domain, then by the Poincare inequality 
$\Vert u\Vert\_{W^{k,p}}\leq \Vert \nabla^k u\Vert\_{p}$ for 
$u\in\mathcal{D}(\mathbb{R}^n)$. This gives



> 
> If $\Omega$ is a bounded domain, then $\mathring
>  {L^k\_p}(\Omega)=W^{k,p}\_0(\Omega)$.
> 
> 
> 


A precise description of the space $\mathring {L^k\_p}(\mathbb{R}^n)$ is available when $kp<n$.


If $kp<n$ or $k=n$, $p=1$ we define the homogeneous Sobolev space by 
$$
\mathring W^{k,p}(\mathbb{R}^n)=\left\{f:\, \nabla^\ell f\in L^{p\_{\ell}^\*}(\mathbb{R}^n),\ \ 
p\_\ell^\*=\frac{np}{n-(k-\ell)p},\ \ \ 0\leq\ell\leq k\right\}.
$$
This space is equipped with the norm
$$
\Vert f\Vert\_{\mathring W^{k,p}} =\sum\_{\ell=0}^k \Vert\nabla^\ell f\Vert\_{p\_\ell^\*}.
$$


It follows from the Sobolev embedding that $W^{k,p}\subset \mathring W^{k,p}$, but the later space is larger. The next result provides a **complete 
answer to question 1**
in the case when $kp<n$ and $1\leq p<\infty$.



> 
> **Theorem.** (Theorem 5 in [2]). Let $kp<n$ and $1\leq p<\infty$. The for every $f\in L^{k,p}(\mathbb{R}^n)$ there exists a unique
>  polynomial $P^{k-1}f\in \mathcal{P}^{k-1}$ of degree $\leq k-1$ such that $f-P^{k-1}f\in
>  \mathring W^{k,p}(\mathbb{R}^n)$. Morover $$ \Vert f- P^{k-1}f\Vert\_{
>  \mathring W^{k,p}}\leq C\Vert\nabla^k f\Vert\_p. $$
> 
> 
> 


In other words $L^{k,p}= \mathring W^{k,p}+\mathcal{P}^{k-1}$.


For other related results, see [1], [2] and [3].


[1] O. V. Besov, V. P. Ilʹin, S. M. Nikolʹskiĭ, *Integral representations of functions and imbedding theorems.* Vol. I & II. Scripta Series in Mathematics. Edited by Mitchell H. Taibleson. V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1979.


[2] P. Hajłasz, A. Kałamajska, [Polynomial asymptotics and approximation of Sobolev functions](http://www.pitt.edu/~hajlasz/OriginalPublications/HajlaszK-Polynomial-StudiaMath-113-1995-55-64.pdf), *Studia Math.* 113 (1995), 55-64. 


[3] P. J. Rabier, 
$L^p$ regularity of homogeneous elliptic differential operators with constant coefficients on $\mathbb{R}^N$. 
*Rev. Mat. Iberoam.* 34 (2018), 423–454. 


