Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers

Vladimir G Maz'ya Department of Mathematics, Linköping University Igor E Verbitsky Department of Mathematics, Wayne State University

TBD mathscidoc:1701.332829

Arkiv for Matematik, 33, (1), 81-115, 1993.4
Some new characterizations of the class of positive measures γ on$R$^{$n$}such that$H$_{$p$}^{$l$}∉L_{$p$}(γ) are given where$H$_{$p$}^{$l$}(1<$p$<∞ 0<$l$∞) is the space of Bessel potentials This imbed ding as well as the corresponding trace inequality $$||J_l u||_{L_p (\gamma )} \leqslant C||u||_{L_p } $$ for Bessel potentials$J$_{$l$}=(1-Δ)^{-1/2}is shown to be equivalent to one of the following conditions(a)$J$_{$l$}($J$_{$l$γ})^{$p$}≤$CJ$_{$lγ$}a e(b)$M$_{$l$}($M$_{$l$γ})^{$p’$}≤$CM$_{$lγ$}a e(c)For all compact subsets$E$of$R$^{$n$} $$\int_E {(J_{l\gamma } )^p dx} \leqslant C{\text{ }}cap (E H_p^l )$$ where 1/$p$+1/$p'$=1$M$_{$l$}is the fractional maximal operator and cap ($H$_{$p$}^{$l$}) is the Bessel capacity In particular it is shown that the trace inequality for a positive measure \gg holds if and only if it holds for the measure$(J$_{$l\gg$})^{$p'$}$dx$Similar results are proved for the Riesz potentials$I$_{$l$γ}=|$x$|^{$l-n$}* γ
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@inproceedings{vladimir1993capacitary,
  title={Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers},
  author={Vladimir G Maz'ya, and Igor E Verbitsky},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203545101885638},
  booktitle={Arkiv for Matematik},
  volume={33},
  number={1},
  pages={81-115},
  year={1993},
}
Vladimir G Maz'ya, and Igor E Verbitsky. Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers. 1993. Vol. 33. In Arkiv for Matematik. pp.81-115. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203545101885638.
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