On the Picard principle for ∆ + μ

Given a (local) Kato1 measure μ on Rd \ {0}, d ≥ 2, let H 0 (U) be the convex cone of all continuous real solutions u ≥ 0 to the equation ∆u+ uμ = 0 on the punctured unit ball U satisfying lim|x|→1 u(x) = 0. It is shown that H 0 (U) 6= {0} if and only if the operator f 7→ ∫ U G(·, y)f(y) dμ(y), where G denotes the Green function on U , is bounded on L2(U, μ) and has a norm which is at most one. Moreover, extremal rays in H 0 (U) are characterized and it is proven that ∆ + μ satisfies the Picard principle on U , that is, that H 0 (U) consists of one ray, provided there exists a suitable sequence of shells in U such that, on these shells, μ is either small or not too far from being radial. Further, it is shown that the verification of the Picard principle can be localized. Several results on L2-(sub)eigenfunctions and 3G-inequalities which are used in the paper, but may be of independent interest, are proved at the end of the paper. Mathematics subject classification (2000): 31D05, 35J10, 35J15