Compact embedding between parabolic Hölder spaces

Question

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}

Answer 1

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.