We give a homotopy formula for the $\partial {̅}_b$ operator on a domain with (q+k)-concave boundary. As a consequence we show that the dimension of the $\partial {̅}_b$-cohomology groups for some CR manifolds q-concave at infinity is finite.

The Hartogs Theorem for holomorphic functions is generalized in two settings: a CR version (Theorem 1.2) and a corresponding theorem based on it for $C^k$$\overline{\partial }$-closed forms at the critical degree, $0\le k\le \infty$ (Theorem 1.1). Part of Frenkel’s lemma in $C^k$ category is also...

Let M be a real-analytic submanifold of ${ℂ}^n$ whose “microlocal” Levi form has constant rank $s_M^{+}+s_M^{-}$ in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees $s_M^{-}$, $s_M^{+}$ (and 0).  This phenomenon is known in the literature as “absence of the Poincaré Lemma” and was already proved in case the Levi form is non-degenerate (i.e. $s_M^{-}+s_M^{+}=n-codimM$). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively....

Given an embeddable $CR$ manifold $M$ and a non-characteristic hypersurface $S\subset M$ we present a necessary condition for the tangential Cauchy-Riemann operator ${\overline{\partial }}_M$ on $M$ to be locally solvable near a point $x_0\in S$in one of the sidesdetermined by $S$.

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.

Let M be a smooth q-concave CR submanifold of codimension k in ${ℂ}^n$. We solve locally the $\partial {̅}_b$-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 $\partial {̅}_b$-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...