Abstract
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<p<∞.
Highlights
In the terminology of Everitt and Giertz, the concept of separation of differential operators was first introduced in [1]
Several results of the separation problem are given in a series of pioneering papers [2,3,4,5]
For more backgrounds concerning to our problem, see
Summary
In the terminology of Everitt and Giertz, the concept of separation of differential operators was first introduced in [1]. Atia et al [9] have studied the separation property of the bi-harmonic differential expression A =. Atia [10] has studied the sufficient conditions for the magnetic bi-harmonic differential operator B of the form B =. Riemannian manifold M, g with metric g , where E is the magnetic Laplacian on M and q ≥ 0 is a locally square integrable function on M. L2 (M) is the space of complex-valued square integrable functions on M with the inner product: Atia Journal of the Egyptian Mathematical Society (2019) 27:24. We use the notation L2 1T∗M for the space of complex-valued square integrable 1-forms on M with the inner product:. For 1 ≤ p < ∞, Lp (M) is the space of complex-valued p-integrable functions on M with the norm:. Assume that the manifold (M, g) has bounded geometry, that is
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