PUB - Publikationen an der Universität Bielefeld

Given a (local) Kato measure mu on R-d\{0}, d >= 2, letH(0)(Delta+mu)(U) be the convex cone of all continuous real solutions u >= 0 to the equation Delta u + u mu = 0 on the punctured unit ball U satisfying lim(vertical bar x vertical bar>1) u(x) = 0. It is shown that H-0(Delta+mu) (U) not equal {0} if and only if the operator f bar right arrow integral(U) G(., y) f (y) d mu (y), where G denotes the Green function onU, is bounded on L-2(U, mu) and has a norm which is at most one. Moreover, extremal rays in H-0(Delta+mu) (U) are characterized and it is proven that Delta + mu satisfies the Picard principle on U, that is, that H-0(Delta+mu) (U) consists of one ray, provided there exists a suitable sequence of shells in U such that, on these shells, mu 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 L-2-(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.