Interpolation of Weighted Sobolev Spaces

S. G. Pyatkov

Matematičeskie trudy, Tome 4 (2001) no. 1, pp. 122-173

Résumé

In the present article, we describe the spaces $\bigl(H_{p,\Psi}^m(\Omega),L_{p,\omega}(\Omega)\bigr)_{\theta,p}$, where the norms on $H_{p,\Psi}^m(\Omega)$ and on $L_{p,\omega}(\Omega)$ are defined as follows: \begin{align*} \|u\|_{H_{p,\Psi}^m(\Omega)}^p=\int_{\Omega}\sum_{|\alpha|\le m}\omega_{\alpha}\bigl|D^{\alpha}u(x)\bigr|^p\,dx, \\ \|u\|_{L_{p,\omega}(\Omega)}^p=\int_{\Omega}\omega(x)\bigl|u(x)\bigr|^p\,dx, \end{align*} with $\omega_{\alpha}$, $\omega$ continuous positive functions on $\Omega$. The results obtained are applicable to studying elliptic eigenvalue problems with an indefinite weight function.