JournalsrmiVol. 23, No. 3pp. 1067–1114

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

  • Șerban Costea

    Helsinki University of Technology, Finland
Strong $A_{\infty}$-weights and scaling invariant Besov capacities cover
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Abstract

This article studies strong AA_{\infty}-weights and Besov capacities as well as their relationship to Hausdorff measures. It is shown that in the Euclidean space Rn{\mathbb{R}}^n with n2n\ge 2, whenever n1<snn-1 < s \le n, a function uu yields a strong AA_\infty-weight of the form w=enuw=e^{nu} if the distributional gradient u\nabla u has sufficiently small Ls,ns(Rn;Rn)||\cdot||_{{\mathcal L}^{s,n-s}}({\mathbb{R}}^n; {\mathbb{R}}^n)-norm. Similarly, it is proved that if 2n<p<2\le n < p < \infty, then w=enuw=e^{nu} is a strong AA_\infty-weight whenever the Besov BpB_p-seminorm [u]Bp(Rn)[u]_{B_p({\mathbb{R}}^n)} of uu is sufficiently small. Lower estimates of the Besov BpB_p-capacities are obtained in terms of the Hausdorff content associated with gauge functions hh satisfying the condition 01h(t)p1dtt<\int_0^1 h(t)^{p'-1} \frac{dt}{t} < \infty.

Cite this article

Șerban Costea, Strong AA_{\infty}-weights and scaling invariant Besov capacities. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 1067–1114

DOI 10.4171/RMI/524