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Let $U$ be a bounded subset of $\mathbb{R}^n$. Let $p>n$. Let $W^{2,1}(U_T)$ be the Banach space of functions $u:U\times[0,T]\rightarrow\mathbb{R}$ with the norm $\|u\|_{W^{2,1}(U_T)}=\sum_{2s+|\kappa|\leq 2}\|\partial_tD^\kappa_xu\|_{L^p(U_T)}$.

For a continuous function $u:U_T\rightarrow\mathbb{R}$ and $0<\alpha<1$. Define $$\langle u\rangle_{C^{\alpha,\alpha/2}(U_T)} = \sup_{x\not=y,t}\frac{|u(x,t)-u(y,t)|}{|x-y|^\alpha} + \sup_{x,t\not=s}\frac{|u(x,t)-u(x,s)|}{|t-s|^{\alpha/2}}.$$

Are there constants $C>0$, $T_0$ and $\delta>0$ such that $\langle u\rangle_{C^{\alpha,\alpha/2}(U_{T_0})} \leq CT_0^\delta\|u\|_{W^{2,1}_p(U_{T_0})}$ for all $u\in W^{2,1}_p(U_{T_0})$?

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    $\begingroup$ If $U$ is a nice enough domain there are embedding theorems for anysothropic Sobolev spaces into anysotropic Holder spaces, see Besov, Ilin, Nikolskii "Integral representations of functions and imbedding theorems". $\endgroup$ – Andrew Dec 30 '18 at 9:33
  • $\begingroup$ I found the book you mentioned before reading your comment. As you mentioned, the book is really nice. Anyway, thanks. $\endgroup$ – Ngu Dec 30 '18 at 10:49
  • $\begingroup$ Is constant $T_0$ essential in the estimate? Putting $C_1=T_0^\delta$ removes it. $\endgroup$ – Andrew Dec 30 '18 at 17:34
  • $\begingroup$ The constant in the right-hand side depends on $T$. It is bounded for $T$ small. You are right. We have a uniform constant $C$ for all $T\leq T_0$. $\endgroup$ – Ngu Dec 31 '18 at 1:48

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