Revista Matemática Iberoamericana

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

Șerban Costea

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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 $\int_0^1 h(t)^{p'-1} \frac{dt}{t} < \infty$.

Article information

Source
Rev. Mat. Iberoamericana, Volume 23, Number 3 (2007), 1067-1114.

Dates
First available in Project Euclid: 27 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1204128311

Mathematical Reviews number (MathSciNet)
MR2414503

Zentralblatt MATH identifier
1149.46028

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 31C99: None of the above, but in this section
Secondary: 30C99: None of the above, but in this section

Keywords
strong $A_{\infty}$-weights Besov spaces capacity

Citation

Costea, Șerban. Strong $A_{\infty}$-weights and scaling invariant Besov capacities. Rev. Mat. Iberoamericana 23 (2007), no. 3, 1067--1114. https://projecteuclid.org/euclid.rmi/1204128311


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