Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Abstract

This article studies strong $A_{\infty}$-weights and Besov capacities as well as their relationship to Hausdorff measures. It is shown that in the Euclidean space ${\mathbb{R}}^n$ with $n\ge 2$, whenever $n-1 < s \le n$, a function $u$ yields a strong $A_\infty$-weight of the form $w=e^{nu}$ if the distributional gradient $\nabla u$ has sufficiently small $||\cdot||_{{\mathcal L}^{s,n-s}}({\mathbb{R}}^n; {\mathbb{R}}^n)$-norm. Similarly, it is proved that if $2\le n < p < \infty$, then $w=e^{nu}$ is a strong $A_\infty$-weight whenever the Besov $B_p$-seminorm $[u]_{B_p({\mathbb{R}}^n)}$ of $u$ is sufficiently small. Lower estimates of the Besov $B_p$-capacities are obtained in terms of the Hausdorff content associated with gauge functions $h$ satisfying the condition $\int_0^1 h(t)^{p'-1} \frac{dt}{t} < \infty$.