Subelliptic and Maximal $L^p$ Estimates for the Complex Green Operator on non-pseudoconvex domains
Joel Coacalle
TL;DR
The paper addresses the problem of gaining regularity for the complex Green operator $K_q$ on CR manifolds that are not assumed to be pseudoconvex. It introduces a novel comparable weak $Y(q)$ condition, together with finite-type hypotheses, and uses a microlocal decomposition together with Calderón–Zygmund theory to obtain $\varepsilon$-subelliptic estimates in $L^2$-Sobolev spaces and maximal $L^p$ estimates at a fixed form level under a closed-range assumption. The main contributions include a local-to-global subelliptic framework, explicit non-isotropic scaling to handle the geometry, and detailed regularity results for the Szegő projection and the Complex Green operator, including kernel bounds and cancellation properties. These results extend regularity theory beyond the pseudoconvex setting to a broader class of weak $Y(q)$ manifolds and provide sharp kernel and $L^p$-theoretic control with potential applications to CR analysis and several complex variables. The work thereby broadens the scope of subelliptic and maximal $L^p$ regularity results for boundary value problems in CR geometry, leveraging microlocal tools to handle nonelliptic systems.
Abstract
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) $. We employ a microlocal decomposition and Calderón-Zygmund theory to obtain subelliptic and maximal-$ L^p $ estimates, respectively.