Annihilators of distributions associated with algebraic local cohomology of a hypersurface

Let f be a real-valued real analytic function in several variables. We associate with each algebraic local cohomology class u with support in f 0 a distribution (generalized function) rho(u) in terms of the residue of f(+)(lambda) with respect to lambda at a negative integer. Then rho constitutes a homomorphism of modules over the sheaf of analytic functions but not over the sheaf of differential operators in general. We compare the annihilator of rho(u) in the ring of differential operators with that of u: we give sufficient conditions, together with examples, for each inclusion between the two annihilators.