Nuclearity of imbedding operators of Sobolev classes into weighted spaces

O. G. Parfenov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 25, Tome 247 (1997), pp. 156-165

Résumé

Let $\Omega$ be an open set in $\mathbf R^m$. Denote by $d_x$ the distance from a point $x$ to the boundary of $\Omega$: $$ d_x=\inf_{y\in\partial\Omega}|x-y|; $$ if $\Omega=\mathbf R^m$, then $d_x=1+|x|$. Define the class $\overset{\circ}{\mathbf W}{}_{p,\lambda}^l(\Omega)$ as the closure of $\mathbf C^\infty_0(\Omega)$ with respect to the norm $$ \|f\|_{\overset{\circ}{\mathbf W}{}_{p,\lambda}^l(\Omega)}=\left(\int\limits_\Omega\left(\sum_{|\beta|=l}|D^\beta f|^p d^{-\lambda}_x+|f|^p d^{-pl-\lambda}_x\right)dx\right)^{1/p}; $$ here $l=1,2$; $1\le p<\infty$; $\lambda\in(-\infty,\infty)$. Let $\mu$ be a measure in $\Omega$ and $\mathbf L_q(\mu)$ the Lebesgue space. A criterion for the nuclearity of the imbedding of $\overset{\circ}{\mathbf W}{}_{p,\lambda}^l(\Omega)$ into $\mathbf L_q(\Omega)$ is given for $l>m$.