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Semi-global solutions of ∂b with Lp (1 ≤ p ≤ ∞) bounds on strongly pseudoconvex real hypersurfaces in Cn (n ≥ 3).

C. H. Chang, H. P. Lee (1999)

Publicacions Matemàtiques

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.

Some applications of a new integral formula for ̅ b

Moulay-Youssef Barkatou (1998)

Annales Polonici Mathematici

Let M be a smooth q-concave CR submanifold of codimension k in n . We solve locally the ̅ b -equation on M for (0,r)-forms, 0 ≤ r ≤ q-1 or n-k-q+1 ≤ r ≤ n-k, with sharp interior estimates in Hölder spaces. We prove the optimal regularity of the ̅ b -operator on (0,q)-forms in the same spaces. We also obtain L p estimates at top degree. We get a jump theorem for (0,r)-forms (r ≤ q-2 or r ≥ n-k-q+1) which are CR on a smooth hypersurface of M. We prove some generalizations of the Hartogs-Bochner-Henkin extension...

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