Subelliptic and Maximal $L^p$ Estimates for the Complex Green Operator on non-pseudoconvex domains

We prove subelliptic estimates for ethe complex Green operator $ K_q $ at a specific level $ q $ of the $ \bar\partial_b $-complex, defined on a not necessarily pseudoconvex CR manifold satisfying the commutator finite type condition. Additionally, we obtain maximal $ L^p $ estimates for $ K_q $ by considering closed-range estimates. Our results apply to a family of manifolds that includes a class of weak $ Y(q) $ manifolds satisfying the condition $ D(q) $. We employ a microlocal decomposition and Calder\'on-Zygmund theory to obtain subelliptic and maximal-$ L^p $ estimates, respectively.