Sobolev embeddings *lowering* the order of integrability

Question

All Sobolev embeddings I am aware of increase the order of integrability. What about embeddings into spaces whose order of integrability is lower? Say, is it possible to find $k$ large enough such that $H^k(\Omega)$ embeds into $L^1(\Omega)$? If $\Omega$ is bounded, this is not a problem, since $H^1(\Omega)\hookrightarrow L^2(\Omega)\hookrightarrow L^1(\Omega)$. But what about general, possibly unbounded $\Omega$? Say, $\Omega\subset \mathbb R^3$ with $C^2$-boundary.

Answer 1

This is not true in general, by considering functions of the form $$ u_{a}(x) = (1+|x|^2)^{-a/2} $$ for $a>0.$ Note each $u_{a}$ is smooth in $\Bbb R^n$ and a computation shows that $|\partial^{\beta}u_a(x)| \leq C\, u_a(x)$ for any multi-index $\beta,$ where $C=C(n,a,\beta)$ is a constant. Also we have $$ \int_{\Bbb R^n} |u_{\alpha}|^p \,\mathrm{d}x = \omega_n \int_0^{\infty} \frac{r^{n-1}}{(1+r^2)^{ap/2}} \,\mathrm{d}x, $$ which is finite if and only if $a > \frac np.$

Now let $\Omega \subset \Bbb R^n$ be any domain such that for some $c>0$ we have $|\Omega \cap B_R| \geq c R^n$ for all $R>0;$ then we also have $u_a \in L^p(\Omega)$ if and only if $a>\frac np.$ Then given $q>p$ (say $q=2, p=1$) we can pick $a \in (n/q,n/p)$ and consider $u_a.$ By above we can show $u_a \in W^{k,q}(\Omega)$ for any $k \geq 0,$ but $u_a \not\in L^p(\Omega).$