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The abstract condition to have the Sobolev embeddings on domains in their known form as on Euclidean space is the existence of an extension operator for $D$, that is, a continuous linear mapping $E \c …

Take a look into Maz'Yas book "Sobolev spaces", Chapter 1.4.7, Corollary 2. There it is shown that if $\Omega$ is an extension domain (for $W^{1,p}$ and $L^p$ simultaneously) and $\mu$ is a measure …

The answer is yes if $\Omega$ admits a uniform extension operator for $W^{1,p}$ and $L^p$. I suspect that it is not written down in the Hitchhiker's guide because they seem to avoid interpolation. S …

Yes, your identities are correct. Theorem 1.14.5 in Triebel's book [T] says that $$F = (H,D(A))_{1/2,2},$$ and $(1/2,2)$-real interpolation spaces between Hilbert spaces are in fact exactly the $1/2$- …

For the sake of completeness, an expansion on the comment by Mike Miller: In Evans/Gariepy, Thm. 4.3.1, it is proven that if the boundary of $\Omega$ is Lipschitz, then for $1 \leq p < \infty$ there …

Welcome to MathOverflow. A good reference for such a question is Goldberg, H.; Kampowsky, W.; Tröltzsch, F., On Nemytskij operators in $L^p$-spaces of abstract functions, Math. Nachr. 155, 127-140 (1 …

The paper "Compact embeddings of vector-valued Sobolev and Besov spaces" by Amann should probably cover your questions.

The desired embedding is indeed correct for $\theta = 1-1/p$. This is a classical result in interpolation theory and the theory of evolution equations. See for example the book of Amann [2], Theorem I …

The second compact inclusion also follows from an Aubin-Lions type result. There is for example the famous paper Compact sets in $L^p$ by Simon (it usually ranks high on most-cited-per-year lists) whi …

I think the last desired inequality cannot be right, but a direct counterexample escaped me, so here is an argument with a bit of a detour. Let me maybe first frame this with an abstract result. Lemma …

Your domain is a Lipschitz (graph) domain. The 3-dimensional (!) Lebesgue measure of $\mathcal{F}$ is indeed zero, whereas $\mathcal{F}$ has positive measure for the boundary measure $\sigma$ (which c …

The spatial evaluation (or trace) operator $\mathrm{tr}_L$ at $L$ is well defined and continuous on $H^s(0,L)$ for $s>1/2$; a classic reference is [LM, Chapter 1.9]. (Of course the range $s>1/2$ is ex …

Please see the answer of Mateusz for some fundamental problems with your question. With a bit of good faith however, your question HAS a positive answer for $m=2$ in the sense that $$\bigl[H^2(\Omega …

If you assume that $\Omega$ is a bounded uniform extension domain, then your desired inequality holds true. By uniform extension domain, I mean that there exists a linear extension operator $E$ which …

This should work according to the embedding $$W \hookrightarrow L^{\frac2{1-2s}}\bigl((L^2(\Omega),H^2(\Omega))_{\theta,1}\bigr)$$ where $0 < s < \frac12$ and $0 \leq \theta < 1-s$ as in Amann: Linear …