A note on Halász's Theorem in ${\mathbb{F}}_q[t]$

In the setting of the integers, Granville, Harper and Soundararajan showed that the upper bound in Halász’s Theorem can be improved for smoothly supported functions. We derive the analogous result for Halász’s Theorem in ${\mathbb{F}}_q[t]$, and then consider the converse question of when the general upper bound in this version of Halász’s Theorem is actually attained.