On the boundedness and compactness of weighted Green operators of second-order elliptic operators

Abstract

For a given second-order linear elliptic operator L which admits a positive minimal Green function, and a given positive weight function W , we introduce a family of weighted Lebesgue spaces L p (φp) with their dual spaces, where 1 ≤ p ≤ ∞. We study some fundamental properties of the corresponding (weighted) Green operators on these spaces. In particular, we prove that these Green operators are bounded on L p (φp) for any 1 ≤ p ≤ ∞ with a uniform bound. We study the existence of a principal eigenfunction for these operators in these spaces, and the simplicity of the corresponding principal eigenvalue. We also show that such a Green operator is a resolvent of a densely defined closed operator which is equal to (−W −1 )L on C ∞ 0 , and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if W is a (semi)small perturbation of L, then for any 1 ≤ p ≤ ∞, the associated Green operator is compact on L p (φp), and the corresponding spectrum is p-independent.