Interpolation of weighted Sobolev spaces (Q5938483)

Let \(\Omega\) be a domain of the space \(\mathbb R^n\), let \(\omega(x)\) and \(\{\omega_\alpha(x)\}\) be positive continuous functions on \(\Omega\), and let \(H^m_{p\psi}(\Omega)\) and \(L_{p,\omega}(\Omega)\) be weighted spaces with the respective norms \[ \begin{gathered} \|u\|_{H^m_{p,\psi}(\Omega)}= \left(\sum_{|\alpha|\leq m}\omega_\alpha(x)|D^\alpha u(x)|^p dx\right)^{1/p}, \\ \|u\|_{L_{p,\omega}(\Omega)}= \left(\sum_\Omega\omega(x) D^\alpha u(x) ^p dx\right)^{1/p}. \end{gathered} \] Next, let \((H^m_{p\psi}(\Omega),L_{p,\omega}(\Omega))_{\theta,p}\) be the interpolation space constructed by real interpolation. The main results of the article consist in describing the properties of these spaces. These properties can be used for studying eigenvalue problems with a sign-changing weighted function.