Spectral theory of some degenerate elliptic operators with local singularities

This paper is based on our previous results (Haroske and Skrzypczak (2008) 23], Haroske and Skrzypczak (in press) 25]) on 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. Our approach also extends (Haroske and Triebel (1994) 21]) in various ways. We obtain eigenvalue estimates of degenerate pseudodifferential operators of type b₂○p(x,D)○b₁ where bi∈Lri(Rn,wi), wi∈A_∞, i=1,2, and , ?>0. Finally we deal with the ‘negative spectrum’ of some operator Hγ=A−γV for γ→∞, where the potential V may have singularities (in terms of Muckenhoupt weights), and A is a positive elliptic pseudodifferential operator of order ?>0, self-adjoint in L₂(Rn). This part essentially relies on the Birman-Schwinger principle. We conclude this paper with a number of examples, also comparing our results with preceding ones.