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

### Yehuda Pinchover

Technion - Israel Institute of Technology, Haifa, Israel

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## 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 $L_{p}(ϕ_{p})$ 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 $L_{p}(ϕ_{p})$ 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 $L_{p}(ϕ_{p})$, and the corresponding spectrum is $p$-independent.