Typical elements of the space $\mathring {L^k_p}(\Omega)$

Question

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?

Answer 1

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, 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.