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 ${\int }_0^{1}h(t{)}^{p^{\prime }-1}\frac{dt}{t}<\infty$.