BooksecrCollected Volumepp. 459–489

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

  • Yehuda Pinchover

    Technion - Israel Institute of Technology, Haifa, Israel
On the boundedness and compactness of weighted Green operators of second-order elliptic operators cover

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Abstract

For a given second-order linear elliptic operator LL which admits a positive minimal Green function, and a given positive weight function WW, we introduce a family of weighted Lebesgue spaces Lp(ϕp)L^p(\phi_p) with their dual spaces, where 1p1\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 Lp(ϕp)L^p(\phi_p) for any 1p1\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 (W1)L(-W^{-1})L on C0C_0^\infty, and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if WW is a (semi)small perturbation of LL, then for any 1p1\leq p\leq \infty, the associated Green operator is compact on Lp(ϕp)L^p(\phi_p), and the corresponding spectrum is pp-independent.