  This article studies strong A-infinity weights in Ahlfors Q-regular and
geodesic metric spaces satisfying a weak (1,s)-Poincare inequality for some
1<s<=Q, where Q is finite. It is shown that whenever max(1,Q-1)<s<=Q, a
function u yields a strong A-infinity weight of the form w=exp(Qu) if u has a
minimal s-weak upper gradient with sufficiently small Morrey norm. Similarly,
it is proved that if 1<Q<p for some finite p, then w=exp(Qu) is a strong
A-infinity weight whenever u has sufficiently small Besov p-seminorm.
