Let $U \subset \mathbb{R}^d$ be open, $k \in \mathbb{N}$ and $1\leq p<\infty$. Furthermore we take a function $f$ contained in the Sobolev space, $f \in W^{k,p}(U)$. Take a look at the following assertion:
$g \in C_b^k (U)$ $\Rightarrow$ $gf \in W^{k,p}(U)$.
Anyone with a counterexample or is it really true? I am not sure...
Actually one can find a larger space of $g$, taking into account the Sobolev inequalities: if $f\in W^{k,p}$ and $g\in W^{k,q}$, then for any order of derivation $0\le i\le k$, one has $D^if\in L^{p_i}$ with $1/p_i=1/p -(k-i)/d$, and if $D^{j}g\in L^{q}$, with $1/q_j=1/q -(k-j)/d$. Therefore $D^ifDg^{k-i}\in L^p$, hence $fg\in W^{k,p}$, provided $1/p_i+1/q_{k-i}=1/p$, that is $1\ge q=d/k$ (or $q\ge d/k$ if $U$ is bounded).
For more general statements,especially in the limit cases, you can of course check any treatise on Sobolev Spaces, and several on-line material searching for "multipliers of Sobolev Space".
