Consider the following problem:
Let $U \subset \mathbb{R}^{n}$ be an bounded open set. Find conditions on $p$ for which $W^{2,2}(U) \subset W^{1,p}(U)$.
What I have done: I have proved that $W^{2,2}(U) \subset W^{1,p}(U)$ for all $p \in [1,2]$. This comes from the basic inclusion $L^{2}(U) \subset L^{p}(U)$ for $0 < p \leq 2$ (here I use that $U$ has finite Lebesgue measure). However I don't have any idea in what to do in case $p > 2$. Can one always construct a function which is in $W^{2,2}(U)$ but not in $W^{1,p}(U)$ for $p > 2$?