Regularity gain for one plus an even power of the Laplacian
= Regularity gain for one plus an even power of the Laplacian
If $(\Delta^m+1)u=f$ distributionally on $\mathbb R^n$, $m$ is even, and $f\in H^r$, then
$$
\widehat u(\xi)=\frac{\widehat f(\xi)}{1+|\xi|^{2m}}
$$
and $u\in H^{r+2m}(\mathbb R^n)$.