On Fundamental Solutions of Generalized Schrödinger Operators

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^-ε₂^{d(x, y, μ)}x-y^n-2≤Γ_μ(x, y)≤Ce^-ε₁^{d(x, y, μ)}x-y^n-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 L^p(Rⁿ, dx).