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Now I am interested in the calculus on Banach space valued function, especially the function with value in a certain Sobolev space. I want to prove that $$\bigcap_{k=0}^m C^k([0,T];H^{m-k}(\Omega))\subset C^{m-[\frac{n}{2}]-1}(\overline{Q_T}),\tag{1}$$by Sobolev imbedding theorem. Here $\Omega\subset\mathbb{R}^n$ and $Q_T:=(0,T)\times\Omega$. Since I'm not familiar with the theory of Banach space valued function (only know some basic concepts), I wish to see the detail proof of $(1)$. Any reference which contain the detail proof of $(1)$ is exceedingly welcome!

Any answer and reference will be appreciated!

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  • $\begingroup$ The space in the left hand side can be shown to be in $H^{m}(Q_T)$, but the exponent $k=m-[\frac{n}2]-1$ is equal to $m-\frac{n+1}2$ when $n$ is odd, which means that we cannot prove the assertion by the (false) embedding $H^{m}(Q_T)\subset C^k(Q_T)$. So if the assertion is true, one must use a direct approach. What is the reason you believe that the assertion is true? $\endgroup$ – timur Aug 16 '13 at 21:27
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Sweeping out the trash, reformulate what indeed is needed to be established: $$ \bigcap_{k=0}^{m-[\frac{n}{2}]-1}C^k\bigl([0,T];H^{m-k}(\Omega)\bigr)\subset C^{m-[\frac{n}{2}]-1}\Bigl(\overline{Q}_T\Bigr), $$ which in fact immediately follows by an a priori inclusion $$ \bigcap_{k=0}^{l}C^k\bigl([0,T];C^{l-k}(\overline{\Omega})\bigr)\subset C^{l}\Bigl(\overline{Q}_T\Bigr)$$ due to the embedding $\,H^{j}(\Omega)\hookrightarrow C^{j-[\frac{n}{2}]-1} \bigl(\overline{\Omega}\bigr)\,$ valid for all integer $j\geqslant [\frac{n}{2}]+1$.

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