On any complete Riemannian manifold M and for all p∈ [2 , ∞) , we prove a family of second-order L p -interpolation inequalities that arise from the following simple L p -estimate valid for every u∈ C ∞ (M) : ‖∇u‖_p≤‖uΔ_pu‖_1∈[0,∞],where Δ_p denotes the p-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for L p -solutions of the Poisson equation for all p∈ (1 , ∞) , and new global Sobolev regularity results for the singular magnetic Schrödinger semigroups.
Güneysu, B., Pigola, S. (2019). L-p-interpolation inequalities and global Sobolev regularity results (with an appendix by Ognjen Milatovic). ANNALI DI MATEMATICA PURA ED APPLICATA, 198(1), 83-96 [10.1007/s10231-018-0763-7].
L-p-interpolation inequalities and global Sobolev regularity results (with an appendix by Ognjen Milatovic)
Pigola , Stefano
2019
Abstract
On any complete Riemannian manifold M and for all p∈ [2 , ∞) , we prove a family of second-order L p -interpolation inequalities that arise from the following simple L p -estimate valid for every u∈ C ∞ (M) : ‖∇u‖_p≤‖uΔ_pu‖_1∈[0,∞],where Δ_p denotes the p-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for L p -solutions of the Poisson equation for all p∈ (1 , ∞) , and new global Sobolev regularity results for the singular magnetic Schrödinger semigroups.
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