  This note investigates the regularity of zero-extensions of $L^p$ functions
from bounded domains. Simple examples show the possibility of a loss in
smoothness and our goal is to quantify this loss more generally.
  For the unit cube $\mathcal{Q}=[0,1]^d$, one of our main results is a bound
for the $L^p$ modulus of continuity of zero-extensions. Using this, we prove
that nonconstant functions in the Besov space $B^\alpha_{p,q}(\mathcal{Q})$
have zero-extensions in $B^\beta_{p,r}(\mathbb{R}^d)$ with
$\beta=\frac\alpha{\alpha p+1}$ and $r=q(1+\alpha p)$. This seems to be new
when $\frac1p\leq\alpha<1$. The key idea behind the main estimate is to use
piecewise constant approximation on dyadic subcubes. This technique can likely
be sharpened, even for the unit cube, and extended to less regular domains.
