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In this manuscript we prove the Bernstein inequality and develop the theoryof holonomic D-modules for rings of invariants of finite groups incharacteristic zero, and for strongly F-regular finitely generated gradedalgebras with FFRT in prime characteristic. In each of these cases, the ringitself, its localizations, and its local cohomology modules are holonomic. Wealso show that holonomic D-modules, in this context, have finite length. Weobtain these results using a more general version of Bernstein filtrations.

Bernstein's inequality and holonomicity for certain singular rings