Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$.
It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the Sobolev-to-Lebesgue space embedding $I \colon W^1_p \to L_q$ is continuous and compact. It is also well known, that in the limit case $q = p^*$, the embedding is continuous but not (ever) compact.
Now let
- $\alpha$ denote Kuratowski's measure of noncompactness; defined on a metric space $(X,d)$ by $$\alpha(M) = \sup \{ \delta > 0 \colon \exists_{M_1, \dotsc, M_n \subset M} \colon M \subseteq \bigcup_{i=1}^n M_i\text, \operatorname{diam}(M_i) \le \delta\}$$ for any $M \subset X$ where $\operatorname{diam}(M) = \sup \{ d(x,y) \colon x, y \in y \}$.
- $\beta$ denote Istrățescu's spreading measure of noncompactness; defined on a metric space $(X,d)$ by $$\beta(M) = \sup \{ \varepsilon > 0 \colon \exists_{(x_n)^{\mathbb N} \in X^{\mathbb N}} \forall_{m \ne n} \colon d(x_n,x_m) > \varepsilon \}$$ for any $M \subset X$,
- $\gamma$ denote Hausdorff's measure of noncompactness; defined on a metric space $(X,d)$ by $$\gamma(M) = \sup \{ \varepsilon > 0 \colon \exists_{x_1, \dotsc, x_n \in X} \colon M \subseteq \bigcup_{i=1}^n B(x_i,\varepsilon) \}$$ for any $M \subset X$ where $B(x,\varepsilon) = \{ y \in X \colon d(x,y) < \varepsilon \}$, and
I'm interested in the quantity $$ [I \colon W^1_p \to L_{p^*}]_\phi = \inf \{ k > 0 \colon \phi(I(M)) \le k \phi(M)\ \text{for every bounded $M \subset W^1_p$} \} $$ with $\phi \in \{ \alpha, \beta, \gamma \}$.
Q: Since $I \colon W^1_p \to L_{p^*}$ is not compact, $[I]_\phi$ is non-zero (i.e. positive). What else is known about these (three) quantities? Can we compute or estimate $[I]_\phi$?
I've already found a few puzzle pieces that allow for answers in that direction but I'm not very confident about them (afraid I might be missing something), and the fact that this particular question appears never to have been explicitly addressed (not even in a corollary -- is the answer so obvious?) gives me the impression I could be missing something. Here's what I've found:
There is a connection between what V. Maz'ya calls localisation moduli, the essential operator norm of such embeddings, and the aforementioned noncompactness reduction quantity.
- The essential operator norm $\operatorname{ess}\|T\|$ of an operator $T$ is the quotient norm in the space of bounded operators with the compact operators factored out. It satisfies $$ [T]_\phi \le \operatorname{ess}\|T\| \le \|T\| $$ (examples show that each inequality can be strict).
- We have $[I \colon W^k_p \to L_p]_\gamma = \operatorname{ess}\|I \colon W^k_p \to L_p \|$ whenever the embedding is continuous (i.e., the smoothness of the boundary suffices for $k$). If this holds also for $q \ne p$, I do not know. [1; Theorem 3]
- We have $$[I \colon W^k_p \to L_q/P_0]_\gamma = \limsup_{\lambda(D) \to 0} \sup_{\|u\|_{W^k_p} = 1} \|u\|_{L_q(D)}$$ whenever the embedding is continuous (i.e., the smoothness of the boundary suffices for $k$ and $q$ is not too large); here, $P_0$ denotes the space of constant functions and $\lambda$ stands for the $n$-dimensional Lebesgue measure. [1; Theorem 1]
- We have $$[I \colon W^1_2 \to L_2]_\gamma = \lim_{\varepsilon \to 0} \sup_{\|u\|_{W^1_2} = 1} \|u\|_{L_2(\Omega_\varepsilon)}$$ with $\Omega_\varepsilon = \{ x \in \Omega \colon d(x, \partial \Omega) < \varepsilon \}$. If this holds for more general $p$ and $q$ I do not know. [2; Theorem 4] [1; Remark 3].
Whenever we have the Poincaré-type inequality $$ \|u - u_\Omega\|_{L_p} \le k \|\nabla u\|_{L_p} $$ with $u_\Omega = \int_\Omega u$ for every $u \in W^1_p$, then we have the bounds $$[I \colon W^1_p \to L_p]_\beta \le \left( 1 - \frac 1{1+k^p} \right)^{1/p}$$ and $$[I \colon W^1_p \to L_p]_\gamma = \operatorname{ess}\|I \colon W^1_p \to L_p \| \le \left( 1 - \frac 1{1+2^{p-1}k^p} \right)^{1/p}$$ with the same $k$. [1; Theorem 4, Remark 4]
For the related space $L^1_p$ of distributions with derivatives in $L_p$, equipped with the norm $\|u\|_{L^1_p} = \|\nabla u\|_{L_p} + \|u\|_{L_p(\omega)}$, where $\omega$ is a non-empty open subset of $\Omega$ with $\bar \omega \subset \Omega$ and $\partial \Omega$ is $C^1$-smooth, we have $$\operatorname{ess}\|I \colon L^1_p \to L_{p^*}\| = \lim_{\varepsilon \to 0} \sup_{\substack{\|u\|_{L^1_p} = 1\\\text{$u = 0$ on $\Omega \setminus \Omega_\varepsilon$}}} \|u\|_{L_{p^*}} = \lim_{\rho \to 0} \sup_{x \in \partial \Omega} \sup_{\substack{\|u\|_{L^1_p} = 1\\\operatorname{supp} u \subset B(x, \rho)}} \|u\|_{L_{p^*}} $$ and $$\operatorname{ess}\|I \colon L^1_p \to L_{p^*}\| = 2^{1/n} c(p,n)$$ where $c(p,n)$ is the best constant in the Sobolev inequality $$ \|u\|_{L_{p^*}(\mathbb R^n)} \le c \|\nabla u\|_{L_p(\mathbb R^n)} $$ [3; Theorem 8.3].
Edit: If $\Omega$ has a $C^1$ boundary (which seems rather restrictive; one would expect this to hold for any extension domain), then [3; Theorem 8.3] thus tells us at least $$ [I \colon W^1_p \to L_{p^*}]_\phi \le \underbrace{\|I \colon W^1_p \to L^1_p \|}_1 [I \colon L^1_p \to L_{p^*}]_\phi \le \operatorname{ess}\|I \colon L^1_p \to L_{p^*}\| = 2^{1/n} c(p,n)\text. $$
[1] Yerzakova, Nina A. The measure of noncompactness of Sobolev embeddings. Integral Equations Operator Theory 19 (1994), no. 3, 349--359. MR1280128
[2] Amick, Charles J. Some remarks on Rellich's theorem and the Poincaré inequality. J. London Math. Soc. (2) 18 (1978), no. 1, 81--93. MR0502660
[3] Maz'ya, Vladimir. Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. xxviii+866 pp. ISBN: 978-3-642-15563-5 MR2777530
