Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$, $F$ is bounded in $L^\infty (\Omega \times (0,T))\cap L^{5/3}(0,T;C^k(\bar{\Omega}))$ for an arbitrary integer $k\geq 0$(\cap_{k=1}^\infty L^{5/3}(0,T;C^k(\bar{\Omega})))$.
Question: Can we say that $F$ is bounded in $L^q(0,T;C^2(\bar{\Omega}))$ for any $q\in [1,\infty)$?
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Is this (interpolation) inequality right?Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$, $F$ is bounded in $L^\infty (\Omega \times (0,T))\cap L^{5/3}(0,T;C^k(\bar{\Omega}))$ for an arbitrary integer $k\geq 0$.
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