Bessel capacity of symmetric generalized Cantor sets (Q1110687)

We obtain upper and lower estimates for the Bessel capacity of symmetric generalized Cantor sets. Namely, we shall prove the following theorem. Let E be the n-dimensional symmetric generalized Cantor set constructed by the system \([\{k_ j\}^{\infty}_{j=1}\), \(\{\ell_ j\}^{\infty}_{j=0}]\) with \(\ell_ 0\leq 1\). If \(\alpha p<n\), then \[ C^{-1}\{\ell_ 0^{(\alpha p-n)/(p-1)}+\sum^{\infty}_{j=1}(k_ 1...k_ j)^{-n/(p-1)}\ell_ j^{(\alpha p-n)/(p-1)}\}^{1-p} \] \[ \leq B_{\alpha,p}(E)\leq C\{\sum^{\infty}_{j=1}(k_ 1...k_ j)^{- n/(p-1)}\ell_ j^{(\alpha p-n)/(p-1)}\}^{1-p} \] and if \(\alpha p=n\), then \[ C^{-1}\{1+(-\log \ell_ 0)+\sum^{\infty}_{j=1}(k_ 1...k_ j)^{-n/(p-1)}(-\log \ell_ j)\}^{1-p} \] \[ \leq B_{\alpha,p}(E)\leq C\{\sum^{\infty}_{j=1}(k_ 1...k_ j)^{- n/(p-1)}(-\log \ell_ j)\}^{1-p}, \] where the number C (\(\geq 1)\) depends only on n, p and \(\alpha\). In case \(p=2\), this theorem is a refinement of \textit{M. Ohtsuka}'s result [Nagoya Math. J. 11, 151-160 (1957; Zbl 0091.275)]. As an application of our estimates we construct a set which belongs to the (\(\beta\),q)-fine topology \(\tau_{\beta,q}\) but not to the (\(\alpha\),p)-fine topology \(\tau_{\alpha,p}\), provided either \(0<\beta q<\alpha p<n\), \(0<\beta q=\alpha p<n\) and \(q>p\) or \(0<\beta q<\alpha p=n\) or \(\beta q=\alpha p=n\) and \(q>p\).