  We study how smashing Bousfield localizations behave under various
equivariant functors. We show that the analogs of the smash product and
chromatic convergence theorems for the Real Johnson-Wilson theories
$E_{\mathbb{R}}(n)$ hold only after Borel completion. We establish analogous
results for the $C_{2^n}$-equivariant Johnson-Wilson theories constructed by
Beaudry, Hill, Shi, and Zeng. We show that induced localizations upgrade the
available norms for an $N_\infty$-algebra, and we determine which new norms
appear. Finally, we explore generalizations of our results on smashing
localizations in the context of a quasi-Galois extension of $E_\infty$-rings.
