Spectral theory of some degenerate elliptic operators with local singularities
The present paper extends the results from [\textit{D.\,D.\thinspace Haroske} and \textit{L.\,Skrzypczak}, Rev.\ Mat.\ Complut.\ 21, No.\,1, 135--177 (2008; Zbl 1202.46039)] and [\textit{D.\,Haroske} and \textit{H.\,Triebel}, Math.\ Nachr.\ 168, 109--137 (1994; Zbl 0829.46020)] regarding compact embeddings of Muckenhoupt weighted function spaces of Besov and Triebel-Lizorkin type with example weights of polynomial growth near infinity and near some local singularity. The authors obtain eigenvalue estimates for degenerate pseudodifferential operators of type \(b_{2}\circ p(x,D)\circ b_{1}\), where \(b_i \in L_{r_i}(\mathbb R^n, w_i),\) \(w_i \in \mathcal A_{\infty },\) \(i=1,2,\) and \(p(x, D) \in \psi ^{-x}_{1,0},\) \(x > 0.\) Moreover, by means of the Birman-Schwinger principle, the ``negative spectrum'' of operators like \(H_{\gamma }=A - \gamma V\) for \(\gamma \rightarrow \infty \) is studied, where the potential \(V\) may have singularities (in terms of Muckenhoupt weights) and \(A\) is a positive elliptic pseudodifferential operator of order \(x>0,\) which is selfadjoint in \(L_2(\mathbb R^n)\). Various examples and comparisons to already existing results are given as well.
scientific article; zbMATH DE number 5768948
scientific article