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 ${\bar{\mathrm{\partial }}}_b$-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 Calderón-Zygmund theory to obtain subelliptic and maximal-$L^p$ estimates, respectively.