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I apologize for repeating the same question from ME, but it seems more subtle than I expected. Let me fix the notations here first: \begin{equation} C^\infty_c(0,1):= \{ f : (0,1) \to \mathbb{C} \mid ...

Sobolev inequalities show us when we can embed a Sobolev space into another. However, I wonder if these inclusions are always proper. More specifically, let $\Omega \subset \mathbb{R}^n$ be a bounded ...

Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that \begin{equation} V \subset H \subset V' \end{equation} where $V'$ is the dual of $V$ and the inclusions are ...

Let $\Omega$ be a bounded domain in $\mathbb R^2$. By the Sobolev embedding theorem, if $k>\frac np$ (in this case $k>\frac 2p$) then $u\in W^{k,p}(U) \implies u\in C^{k-[\frac 2 p]-1,\gamma}(...