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 function $u(x, t)$ 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}

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}