On the boundedness and compactness of weighted Green operators of second-order elliptic operators
Yehuda PinchoverTechnion - Israel Institute of Technology, Haifa, Israel
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For a given second-order linear elliptic operator which admits a positive minimal Green function, and a given positive weight function , we introduce a family of weighted Lebesgue spaces with their dual spaces, where . 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 for any 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 on , and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if is a (semi)small perturbation of , then for any , the associated Green operator is compact on , and the corresponding spectrum is -independent.