# Holder-Sobolev type inequality

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 $$u\in W^{2,1}_p(U_{T_0})$$?

• 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". – Andrew Dec 30 '18 at 9:33
• I found the book you mentioned before reading your comment. As you mentioned, the book is really nice. Anyway, thanks. – Ngu Dec 30 '18 at 10:49
• Is constant $T_0$ essential in the estimate? Putting $C_1=T_0^\delta$ removes it. – Andrew Dec 30 '18 at 17:34
• 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 $T\leq T_0$. – Ngu Dec 31 '18 at 1:48