D-module and F-module length of local cohomology modules

Let R be a polynomial or power series ring over a field k. We study the length of local cohomology modules HjI (R) in the category of D-modules and Fmodules. We show that the D-module length of HjI (R) is bounded by a polynomial in the degree of the generators of I. In characteristic p > 0 we obtain upper and lower bounds on the F-module length in terms of the dimensions of Frobenius stable parts and the number of special primes of local cohomology modules of R/I. The obtained upper bound is sharp if R/I is an isolated singularity, and the lower bound is sharp when R/I is Gorenstein and F-pure. We also give an example of a local cohomology module that has different D-module and F-module lengths.