The Cohomological Equation for Jointly Integrable Partially Hyperbolic Diffeomorphisms on 3-Manifolds

For a jointly integrable partially hyperbolic diffeomorphism $f$ on a 3-manifold $M$ with virtually solvable fundamental group which satisfies Diophantine condition along the center foliation, we show that the cohomological equation $\phi =u\circ f-u+c$ has a continuous solution $u$ if and only if $\phi$ has trivial periodic cycle functional.