Strong A-infinity weights, Besov and Sobolev capacities in metric measure spaces

TL;DR

This paper investigates strong A-infinity weights in metric measure spaces with specific regularity and inequality conditions, establishing new criteria involving upper gradients and Besov seminorms for these weights.

Contribution

It introduces novel conditions linking minimal s-weak upper gradients and Besov p-seminorms to the formation of strong A-infinity weights in metric spaces.

Findings

Strong A-infinity weights are characterized by exponential functions of functions with small Morrey norm.

Exponential weights of functions with small Besov p-seminorm are also strong A-infinity weights.

Results apply to Ahlfors Q-regular, geodesic metric spaces satisfying a weak Poincaré inequality.

Abstract

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.