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

TL;DR

This paper investigates the properties of weighted Green operators associated with second-order elliptic operators, establishing their boundedness, spectral characteristics, and conditions for compactness across various weighted Lebesgue spaces.

Contribution

It introduces a family of weighted Lebesgue spaces for elliptic operators, analyzes the boundedness, spectral properties, and compactness of Green operators, and links these to perturbations of the underlying operator.

Findings

Green operators are bounded on weighted spaces with uniform bounds.

Existence and simplicity of principal eigenfunctions and eigenvalues are established.

Green operators are compact under small perturbations of the elliptic operator.

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 $L^p(\phi_p)$ with their dual spaces, where $1\leq p\leq \infty$. 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 $L^p(\phi_p)$ for any $1\leq p\leq \infty$ 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 $C_0^\infty$, 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 $1\leq p\leq \infty$, the associated Green operator is compact on $L^p(\phi_p)$, and the corresponding spectrum is $p$-independent.