  We provide a method for obtaining upper estimates of the resolvent kernel of
the Laplacian on a post-critically finite self-similar fractal that relies on a
self-similar series decomposition of the resolvent. Decay estimates on the
positive real axis are proved by analyzing functions satisfying an interior
eigenfunction condition with positive eigenvalue. These lead to estimates on
the complement of the negative real axis via the Phragmen-Lindelof theorem.
Applications are given to kernels for functions of the Laplacian, including the
heat kernel, and to proving the existence of a self-similar series
decomposition for the Laplacian resolvent on fractal blowups.
