Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$, $F$ is bounded in $L^\infty (\Omega \times (0,T))\cap (\cap\_{k=1}^\infty L^{5/3}(0,T;C^k(\bar{\Omega})))$.  

**Question**: Can we say that $F$ is bounded in $L^q(0,T;C^2(\bar{\Omega}))$ for any $q\in [1,\infty)$?



Is this (interpolation) inequality right?
This is **not an answer**, but just a formulation of a simpler analogous problem, which might help better understand the original question. Let $I=[0,1]$ and $Q=I\times I$ . Let $ 1 < p < +\infty$ , and let $\langle M\_k\rangle\in[1,+\infty[\ ^{\mathbb N}$ satisfy $M\_k +1 < M\_{k+1}$ for all $k\in\mathbb N$ . Let $\|u\|=\sup\lbrace |u(s)|:s\in I\rbrace$ for $u\in C(I)$ . Let $S$ be the set of all measurable $z:Q\to[-1,1]$ having $\lbrace z(t,\cdot):t\in I\rbrace\subseteq C^\infty(I)$ with $\int\_0^1\|{\rm D}^k(z(t,\cdot))\|^p\ {\rm d\ }t\le M\_k$ for all $k\in\mathbb N$ . **Now ask**, whether there are $M,q$ with $ p < q < +\infty$ and $M<+\infty$ , and such that $\int\_0^1\|{\rm D}^k(z(t,\cdot))\|^q\ {\rm d\ }t\le M$ holds for all $z\in S$ and $k\in\lbrace 1,2\rbrace$ .


**Edit.** I answer to the original question: We **can say** that $F$ is bounded in $L^q(0,T;C^2(\bar\Omega))$ for all $q\in[1,+\infty[$ *under the additional assumption* that $\Omega$ is the interior of an open qube and the spaces $C^k(\bar\Omega)$ are suitably defined. See [this preprint](http://arxiv.org/PS_cache/math/pdf/0505/0505380v3.pdf) for some aspects related to this matter. The deduction is essentially based on



> **Proposition 1.**  *Let $K=[0,1]^N$ where $N\in\mathbb N$ . For $k\in\mathbb N\_0$ and $y\in C^k(K)$ , let $\|y\|\_k$ denote the supremum of all $|\partial^\alpha y(s)|$ where $s\in K$ and $\alpha\in\mathbb N\_0^N$ with $|\alpha|=\sum\_{i=0}^{N-1}\alpha\_i\le k$ . For all $i,j\in\mathbb N\_0$ with $0\lt i\lt j$ , it then holds that $\|y\|\_i\le(\frac{17}4)^{2^{j-2}}\|y\|\_0^{1-\frac ij}\|y\|\_j^{\frac ij}$ .* 


Namely, given any $q$ with $\frac 53= p < q < +\infty $ , by taking $k\ge 2p^{-1}q$ , we get $\|y\|\_2^q\le C\|y\|\_k^p$ under the assumption that $\|y\|\_0$ remains bounded. For a more general bounded domain $\Omega$ , whether the same holds, depends on whether we can get for it estimates similar to those in Proposition 1. This depends first of all on the specific definition of the spaces $C^k(\bar\Omega)$ used, cf. the preprint referred to above. Once the definition is fixed, if one can construct an extension operator $C^k(\bar\Omega)\to C^k(Q)$ where $Q$ is some compact qube containing $\overline\Omega$ in its interior, then one gets the same result.


**Remark.** The essential content of Proposition 1 is "well-known". For example in [Richard S. Hamilton's article](http://www.ams.org/journals/bull/1982-07-01/S0273-0979-1982-15004-2/S0273-0979-1982-15004-2.pdf) on "the" Nash−Moser theorem, 2.2.1. Theorem (pp. 143−144) gives a rough sketch of an idea for a proof, without specifying the constant.


