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We establish a theory for the existence and regularity of solutions to thecohomological equation over an accessible, partially hyperbolic diffeomorphism.As a by-product of our techniques, we show that for $r>1$, any $C^r$homogeneous, locally compact submanifold of a $C^r$ manifold is in fact a $C^r$submanifold.

The cohomological equation for partially hyperbolic diffeomorphisms