The existence of the F-signature for rings with large -Gorenstein locus

We show that the F-signature of a strongly F-regular local ring of characteristic p exists in the case that the non- -Gorenstein locus is dimension 1, given that a certain bound on zeroth local cohomology modules holds. This bound is shown to hold for rings essentially of finite type over a field. This is the first case in which hypotheses sufficient to prove the existence of the F-signature are not readily sufficient to prove the implication that weak F-regularity implies strong F-regularity.