On Fundamental Solutions of Generalized Schrödinger Operators

Abstract

We consider the generalized Schrödinger operator -Δ+μ, where μ is a nonnegative Radon measure in Rn, n≥3. Assuming that μ satisfies certain scale-invariant Kato conditions and doubling conditions we establish the following bounds for the fundamental solution of -Δ+μ in Rn,ce-ε2d(x, y, μ)x-yn-2≤Γμ(x, y)≤Ce-ε1d(x, y, μ)x-yn-2, where d(x, y, μ) is the distance function for the modified Agmon metric m(x, μ)dx2 associated with μ. We also study the boundedness of the corresponding Riesz transforms ∇(-Δ+μ)-1/2 on Lp(Rn, dx). © 1999 Academic Press.