Tangential Cauchy-Riemann Complex Research Articles

Spectral Multipliers for the Kohn Laplacian on Forms on the Sphere in $$\mathbb {C}^n$$

The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We prove a H\"ormander spectral multiplier theorem for $\Box_b$ with critical index $n-1/2$, that is, half the topological dimension of $\mathbb{S}$. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on $\mathbb{S}$.

Complex vector fields, unique continuation and the maximum modulus principle

We prove unique continuation and maximum modulus principle for solutions to systems of differential equations and inequalities, involving complex vector fields, under conditions that generalize some weak-pseudoconcavity assumptions for the tangential Cauchy-Riemann complex.

Local solvability in top degree of the semilinear Kohn equation

We study the local solvability problem for a class of semilinear equations whose linear part is the Kohn Laplacian, acting on top degree forms. We also study the validity of the Poincare lemma, in top degree, for semilinear perturbations of the tangential Cauchy–Riemann complex.

Laplacians on quotients of Cauchy-Riemann complexes and Szegö map for L2-harmonic forms

We compute the spectra of the Tanaka type Laplacians □ = ∂ ¯ Q ∗ ∂ ¯ Q + ∂ ¯ Q ∂ ¯ Q ∗ and Δ = ∂ ¯ Q ∗ ∂ ¯ Q + ∂ Q ∂ Q ∗ on the Rumin complex Q, a quotient of the tangential Cauchy-Riemann complex on the unit sphere S 2 n−1 in ℂ n . We prove that Szegö map is a unitary operator from a subspace of ( p, q −1)-forms on the sphere defined by the operators △ and the normal vector field onto the space of L 2-harmonic ( p, q)-forms on the unit ball. Our results generalize earlier result of Folland.

Conormal suspensions of differential complexes

§1 An exact sequence for Whitney functions. §2 A generalized Mayer–Vietoris sequence. §3 Carriers and wedge decomposition. §4 Localization. §5 Wedge decomposition for CR manifolds. §6 The de Rham complex associated to a regular assignment of closed sets. §7 The long exact sequence associated to a regular assignment of closed sets. §8 Differential equivalence. §9 Local isomorphisms for the ∆–complex. §10 Tangential complexes. §11 Conormal suspensions. §12 On the mixed de Rham–Dolbeault complex. §13 Poincare lemma and holomorphic extension for the tangential Cauchy–Riemann complex. §14 Solvability wave front sets. §15 Degree q cohomology for q–pseudoconcave CR manifolds. §16 Solvability wave front sets and wedge decomposition for q–pseudoconcave CR manifolds.

Quasi-coherence of Hardy spaces in several complex variables

In this paper, we prove that the Hardy spaceHp(Ω), 1≤p<∞, over a strictly pseudoconvex domain in ℂn with smooth boundary is quasi-coherent. More precisely, we show that Toeplitz tuplesTφ with suitable symbols φ onHp (Ω) have property (βɛ). This proof is based on a well known exactness result for the tangential Cauchy-Riemann complex.

Regularity of the tangential Cauchy-Riemann complex and applications

In a series of papers Webster ([26], [27], [28]) has shown how C-estimates for the tangential Cauchy-Riemann complex can be applied to several non-linear problems in Complex Analysis. For example, he gave a simplification of the proof of Kuranishi’s embedding theorem and an application to the integrability problem for almost CR vector bundles. Connected with Webster’s approach are some regularity assertions, whose parameters follow from the C-estimates. Therefore, an improvement of these estimates would lead to an improvement in the applications. Now the author of this article, together with Lan Ma, has given in [12] such an improvement. In two subsequent papers this has been applied to Webster’s approach. The results are formulated in two theorems at the end of this introduction. Here we want to briefly describe the estimates for the tangential CR equations. Let G be a strictly pseudoconvex domain in C with C-smooth boundary and 0 ∈ bG. By a local homotopy formula for the tangential Cauchy-Riemann operator ∂b we mean the following. There exists a neighborhood base {M} of relatively open sets M , with 0 ∈ M ⊂ bG, and on each M there are given operators Rq (q = 1, 2, . . . , n− 2), which fullfill the equation

Hypoellipticity in the tangential Cauchy-Riemann complex

Hypoellipticity in the tangential Cauchy-Riemann complex

Local regularity for the tangential Cauchy-Riemann complex.

Local regularity for the tangential Cauchy-Riemann complex.

Hypoellipticity on Cauchy-Riemann Manifolds

Using a recent homotopy formula by Trèves, it is shown that the existence of $(q + 1)$-dimensional holomorphic supporting manifolds is a sufficient condition for the hypoellipticity on level $q$ and $n - q$ of a tangential Cauchy-Riemann complex of ${\text {CR}}$-dimension $n$. In the hypersurface case, this result is given microlocally.

Hypoellipticity on Cauchy-Riemann manifolds

Using a recent homotopy formula by Trèves, it is shown that the existence of ( q + 1 ) (q + 1) -dimensional holomorphic supporting manifolds is a sufficient condition for the hypoellipticity on level q q and n − q n - q of a tangential Cauchy-Riemann complex of CR {\text {CR}} -dimension n n . In the hypersurface case, this result is given microlocally.

Homotopy formulas in the tangential Cauchy-Riemann complex

Homotopy formulas in the tangential Cauchy-Riemann complex

On the local solution of the tangential Cauchy-Riemann equations

We study the solution operators P and homotopy formula introduced by G. M. Henkin for the tangential Cauchy-Riemann complex of a suitable small domain D on a strictly pseudoconvex real hypersurface in complex n-space. The main difficulties stem from the fact that P is an integral operator with a rather complicated kernel. For U ⊂⊂ D, we derive a Ck-norm estimate of the form ∥Pφ∥U, k ≦ K∥φ∥D, k, where the constant K blows up as U increases to D. We obtain careful control of the rate of this blow-up and of the dependence of K on the derivatives of the function defining the real hypersurface. Our estimates are sufficient for application to the local CR embedding problem.

L 2 estimates and existence theorems for the tangential Cauchy-Riemann complex

L 2 estimates and existence theorems for the tangential Cauchy-Riemann complex