My question is about the following compact embedding: \begin{equation} C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T). \end{equation} what condition should be put on $Q_T=\Omega \times (0, T)$ where $\Omega\subset \mathbb{R}$ be an open bounded interval and $T<\infty$, so that the above relation is correct? If this relation is correct, please give me a valid reference.

We denote by $C^{m+\alpha, \beta}_{x, t}(Q_T)$ ($m$ integer $\geq 0$, $0<\alpha, \beta <1$) the space of functions $u(x, t)$ with finite norm \begin{equation} \Vert u \Vert_{C^{m+\alpha, \beta}_{x, t}(Q_T)}=\sum_{\vert l \vert=0}^{m} \Big[ \sup _{Q_T}\vert D^{l}_{x}u \vert +\langle D^{l}_{x}u \rangle^{(\alpha)}_{x, Q_T}+\langle D^{l}_{x}u \rangle^{(\beta)}_{t, Q_T}\Big] \end{equation} where \begin{equation} \langle w \rangle^{(\alpha)}_{x, Q_T}=\sup_{(x, t), (y, t)\in {Q_T}} \frac {\vert w(x, t)-w(y, t)\vert}{\vert x-y \vert^\alpha}, \end{equation} \begin{equation} \langle w \rangle^{(\beta)}_{t, Q_T}=\sup_{(x, t), (x, \tau)\in {Q_T}} \frac {\vert w(x, t)-w(x, \tau)\vert}{\vert t-\tau \vert^\beta}. \end{equation} We denote by $C^{\alpha+2, \beta+1}_{x, t}(Q_T)$ the space of functions $u(x, t)$ with norm \begin{equation} \Vert u \Vert_{C^{\alpha+2, \beta}_{x, t}(Q_T)}+\Vert u_t \Vert_{C^{\alpha, \beta}_{x, t}(Q_T)}. \end{equation}

  • $\begingroup$ Do you mean to say that $\Omega$ is a bounded interval here? If so, isn't it the case that all the domains you're considering here are bounded rectangles (where you'll almost certainly have a compact embedding)? Unless I'm missing something, I think you should be able to get what in that setting using something like the interpolation inequality trick described here: books.google.co.uk/… $\endgroup$
    – DCM
    Nov 6 '18 at 19:47
  • $\begingroup$ @DCM Yes, $\Omega$ is an open bounded interval. Do you mean that if $Q_T$ be a bounded domain, then the above embedding is compact? $\endgroup$
    – VirgoMath
    Nov 6 '18 at 19:59
  • $\begingroup$ I think you should be ok if $\Omega \subset \mathbb{R}$ is a finite union of bounded open intervals (I think you may want some sort of boundary smoothness for compactness of this sort of embedding in general). I'm not sure you'll find the statement you want explicitly, but you might be able to derive it from suitable interpolation estimates. Might be worth looking what google turns up on interpolation inequalities in parabolic Holder spaces. $\endgroup$
    – DCM
    Nov 6 '18 at 21:14
  • $\begingroup$ Hi again. I'm not sure I'm quite clear on the definitions of your spaces. Could you perhaps confirm (or give an indication to the contrary) that all the functions in your first space $C^{2+\alpha,\beta}(Q_T)$ have $u, \dfrac{\partial u}{\partial x}, \dfrac{\partial u}{\partial t}\in C^{\alpha,\beta}(\bar Q_T)$? I think this is likely to be all you need (provided $\Omega$ is an interval or similar) to get a compact embedding here, but I'm reluctant to work on an interpolation inequality until I'm clear on what spaces you're considering. Feel free to refer me to a reference if you like :) $\endgroup$
    – DCM
    Nov 8 '18 at 0:01
  • $\begingroup$ The $C^{2+\alpha,\beta}(Q_T)$ in my last comment should have been a $C^{2+\alpha,1+\beta}(Q_T)$, and what I'm really interested in is whether you agree that, for $u$ in this space, $u$ and its first order $x$ and $t$ derivatives are Lipschitz continuous with respect to the metric given by $d((x,t),(x',t')) = |x-x'|^\alpha + |t-t'|^\beta$. I suspect that'll suffice, but will check and post an answer once I'm clearer on your definitions. $\endgroup$
    – DCM
    Nov 8 '18 at 0:31

You may want to check the details, but I think that the following argument (or something like it) is enough to give you compactness. I am assuming throughout that your smaller space is contained in $C^1(\bar Q_T)$; it looks to me like it will be, but I'm not clear enough on your definitions to say for certain.

Provided $\Omega$ is Whitney regular, there will be constants $C_1,C_2>0$ depending on $\alpha,\beta,\Omega$ and $T$ such that

$$ \langle u \rangle_{x,Q_T}^{(\alpha)} \leq C_1\Vert \nabla u \Vert_\infty^\alpha \Vert u \Vert_\infty^{1-\alpha} $$

$$ \langle u \rangle_{t,Q_T}^{(\beta)} \leq C_2\left\Vert \dfrac{\partial u}{\partial t} \right\Vert_\infty^\beta \Vert u \Vert_\infty^{1-\beta} $$

for all $u\in C^1(\bar Q_T)$, where $\Vert.\Vert_\infty$ is the uniform norm over $\bar Q_T$. Thus

$$ \Vert u \Vert_{C^{\alpha,\beta}(Q_T)} \leq \Vert u \Vert_\infty +C_1\Vert \nabla u \Vert_\infty^\alpha \Vert u \Vert_\infty^{1-\alpha} + C_2\left\Vert \dfrac{\partial u}{\partial t}\right \Vert_\infty^\beta \Vert u \Vert_\infty^{1-\beta} $$

for all $u\in C^1(\bar Q_T)$. If you take a bounded sequence $(u_n)$ in your smaller space, Ascoli-Arzela gives you a subsequence which converges in $C(\bar Q_T)$ (i.e. with respect to $\Vert.\Vert_\infty$). Since the terms $\Vert \nabla u_n\Vert_\infty$ and $\Vert \partial u_n/\partial t\Vert_\infty$ terms are bounded with $n$, the inequality above gives convergence of this subsequence in $C^{\alpha,\beta}(Q_T)$. This argument gives you compactness of the embedding $C^1(\bar Q_T)\to C^{\alpha,\beta}(Q_T)$, which I think is enough for what you want.

You can replace 'Whitney regular' with 'convex' if you don't need that level of generality.


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