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

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.