SEMI-GLOBAL SOLUTIONS OF ∂̄b WITH Lp (1 ≤ p ≤ ∞) BOUNDS ON STRONGLY PSEUDOCONVEX REAL

Let M be an open subset of a compact strongly pseudoconvex hypersurface {ρ = 0} defined by M = D × Cn−m ∩ {ρ = 0}, where 1 ≤ m ≤ n − 2, D = {σ(z1,..., zm) < 0} ⊂ Cm is strongly pseudoconvex in Cm. For ∂̄b closed (0, q) forms f on M, we prove the semi-global existence theorem for ∂̄b if 1 ≤ q ≤ n−m−2, or if q = n−m − 1 and f satisfies an additional “moment condition”. Most importantly, the solution operator satisfies Lp estimates for 1 ≤ p ≤ ∞ with p = 1 and ∞ included