# Compact embedding for Sobolev space involving time

Let $d \in \mathbb{N}$ and $\Omega$ be a bounded domain of $\mathbb{R}^d$. Consider $m,n,p,q \in \mathbb{N}$ and $T>0$.

Is the space $W^{m,p}([0,T],W^{n,q}(\Omega))$ compactly embedded in any space of continuous functions (such as $C^k([0,T],C^l(\Omega))$ with $k,l\in\mathbb{N}$ ) ?

I guess it is true if $n=m$ and $p=q$, in which case we have

$W^{m,p}([0,T],W^{m,p}(\Omega)) = W^{m,p}(]0,T[\times\Omega)$

and we can use classical compact Sobolev embeddings of Rellich-Kondrachov (right?). However, I did not find any references concerning the general case.

Less general results involving compact embeddings for spaces like $L^p([0,T],W^{n,q}(\Omega))$ or $L^{\infty}([0,T],W^{n,q}(\Omega))$ could be a good start too.

Thanks !

## 1 Answer

The paper "Compact embeddings of vector-valued Sobolev and Besov spaces" by Amann should probably cover your questions.