# Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding: $$$$C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T).$$$$ what condition should be put on $$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 $$$$\Vert u \Vert_{C^{m+\alpha, \beta}_{x, t}(Q_T)}=\sum_{\vert l \vert=0}^{m} \Big[ \sup _{Q_T}\vert D^{l}_{x}u \vert +\langle D^{l}_{x}u \rangle^{(\alpha)}_{x, Q_T}+\langle D^{l}_{x}u \rangle^{(\beta)}_{t, Q_T}\Big]$$$$ where $$$$\langle w \rangle^{(\alpha)}_{x, Q_T}=\sup_{(x, t), (y, t)\in {Q_T}} \frac {\vert w(x, t)-w(y, t)\vert}{\vert x-y \vert^\alpha},$$$$ $$$$\langle w \rangle^{(\beta)}_{t, Q_T}=\sup_{(x, t), (x, \tau)\in {Q_T}} \frac {\vert w(x, t)-w(x, \tau)\vert}{\vert t-\tau \vert^\beta}.$$$$ We denote by $$C^{\alpha+2, \beta+1}_{x, t}(Q_T)$$ the space of functions $$u(x, t)$$ with norm $$$$\Vert u \Vert_{C^{\alpha+2, \beta}_{x, t}(Q_T)}+\Vert u_t \Vert_{C^{\alpha, \beta}_{x, t}(Q_T)}.$$$$