On the Smoothness of Zero-Extensions

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 $B_{p,q}^{\alpha }(\mathcal{Q})$ have zero-extensions in $B_{p,r}^{\beta }({\mathbb{R}}^d)$ with $\beta =\frac{\alpha }{\alpha p+1}$ and $r=q(1+\alpha p)$. This seems to be new when $\frac{1}{p}\le \alpha <1$. 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.