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−y|n−2⩽Γμ(x, y)⩽Ce−ε1d(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 Lp(Rn, dx).