  We investigate under what conditions holomorphic forms defined on the regular
locus of a reduced complex space extend to holomorphic (or logarithmic) forms
on a resolution of singularities. We give a simple necessary and sufficient
condition for this, whose proof relies on the Decomposition Theorem and Saito's
theory of mixed Hodge modules. We use it to generalize the theorem of
Greb-Kebekus-Kov\'acs-Peternell to complex spaces with rational singularities,
and to prove the existence of a functorial pull-back for reflexive
differentials on such spaces. We also use our methods to settle the "local
vanishing conjecture" proposed by Musta\c{t}\u{a}, Olano, and Popa.
