On Fundamental Solutions of Generalized Schrodinger Operators

摘要

We consider the generalized Schrodinger operator -triangle open+#mu#, where #mu# is a nonnegative Radon measure in R~n, n>=3. Assuming that #mu# satisfies certain scale-invariant Kato conditions and doubling conditions we establish the following bounds for the fundamental solution of -triangle open+#mu# in R~n, ce~(-#epsilon#_2d(x, y, #mu#))/|x-y|~(n-2) <=#GAMMA#_(#mu#)(x, y)<=Ce~(-#epsilon#_2d(x, y, #mu#))/|x-y|~(n-2), where d(x, y, #mu#) is the distance function for the modified Agmon metric m(x, #mu#) dx~2 associated with #mu#. We also study the boundedness of the corresponding Riesz transforms nabla(-triangle open+#mu#)~(-1/2) on L~p(R~n, dx).