Compact embeddings of \(p\)-Sobolev-like cones of nuclear operators (Q2073150)

The \(p\)-Sobolev-like cone, denoted by \(\mathcal{W}^{1,p}\), is the set of nuclear operators that possess eigenvalues \(v_i\), \(i\in \mathbb{N}\), and their corresponding eigenvectors \(\psi_i\) satisfying \(\sum_{i\in \mathbb{N}}|v_i|\int_{\Omega}[|\nabla \psi_i|^p+V(x)|\psi_i|^p]dx <\infty\), where \(\Omega\) is a smooth bounded subset of \(\mathbb{R}^N\) In this paper, it is proved that \(\mathcal{W}^{1,p}\) is compact for \(p\ge2\). This result extends the compactness of \(\mathcal{W}^{1,2}\) obtained by \textit{J. Dolbeault} et al. [Monatsh. Math. 155, No. 1, 43--66 (2008; Zbl 1151.81014)]. Such extension is done following the conventional argument based on the Hölder inequality. Consequently, the compactness of \(\mathcal{W}^{1,p}\) is used to show that a certain type of free energy functionals have operator ground states.