Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds

Consider the bi-harmonic differential expression of the form $ A=\triangle _{M}^{2}+q\ $ on a manifold of bounded geometry (M,g) with metric g, where △M is the scalar Laplacian on M and q≥0 is a locally integrable function on M. In the terminology of Everitt and Giertz, the differential expression A is said to be separated in Lp(M), if for all u∈Lp(M) such that Au∈Lp(M), we have qu∈Lp(M). In this paper, we give sufficient conditions for A to be separated in Lp(M),where 1