L’équation dans certains domaines faiblement convexes
Levi-equation in higher dimensions
We announce some results concerning the Dirichlet problem for the Levi-equation in . We consider for the sake of simplicity the case .
Line bundle convexity of pseudoconvex domains in complex manifolds.
Lösungsoperatoren für den Cauchy-Riemann-Komplex mit Ck- Abschätzungen.
Lp estimates for local solutions of ...b on strongly pseudo-convex CR manifolds.
Matrix inequalities and the complex Monge-Ampère operator
We study two known theorems regarding Hermitian matrices: Bellman's principle and Hadamard's theorem. Then we apply them to problems for the complex Monge-Ampère operator. We use Bellman's principle and the theory for plurisubharmonic functions of finite energy to prove a version of subadditivity for the complex Monge-Ampère operator. Then we show how Hadamard's theorem can be extended to polyradial plurisubharmonic functions.
Maximal subextensions of plurisubharmonic functions
In our earlier paper [CKZ], we proved that any plurisubharmonic function on a bounded hyperconvex domain in with zero boundary values in a quite general sense, admits a plurisubharmonic subextension to a larger hyperconvex domain. Here we study important properties of its maximal subextension and give informations on its Monge-Ampère measure. More generally, given a quasi-plurisubharmonic function on a given quasi-hyperconvex domain of a compact Kähler manifold , with well defined Monge-Ampère...
Mesures de Carleson d’ordre et solutions au bord de l’équation
Microlocalisation et estimations pour ... Dans quelques hypersurfaces pseudoconvexes.
Microlocalisation et estimations pour
Mittag-Leffler methods in analysis.
In this survey we present two Mittag-Leffler lemmas and several applications to topics as varied as the delta-equation, Fréchet algebras, inductive limits of Banach spaces and quasi-normable Fréchet spaces.
Moduli for pointed convex domains.
Monge-Ampère boundary measures
We study swept-out Monge-Ampère measures of plurisubharmonic functions and boundary values related to those measures.
Monge-Ampère Equations, Geodesics and Geometric Invariant Theory
Existence and uniqueness theorems for weak solutions of a complex Monge-Ampère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed....
Multipliers and weighted ∂ operator estimates.
We study estimates for the solution of the equation du=f in one variable. The new ingredient is the use of holomorphic functions with precise growth restrictions in the construction of explicit solution to the equation.
Newton numbers and residual measures of plurisubharmonic functions
We study the masses charged by at isolated singularity points of plurisubharmonic functions u. This is done by means of the local indicators of plurisubharmonic functions introduced in [15]. As a consequence, bounds for the masses are obtained in terms of the directional Lelong numbers of u, and the notion of the Newton number for a holomorphic mapping is extended to arbitrary plurisubharmonic functions. We also describe the local indicator of u as the logarithmic tangent to u.
Nondeformability of the complex projective space.
Non-solvability of the tangential ∂̅-system in manifolds with constant Levi rank
Let M be a real-analytic submanifold of whose “microlocal” Levi form has constant rank in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees , (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. ). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively....
Non-solvability of the tangential -systems
We prove that for a real analytic generic submanifold of whose Levi-form has constant rank, the tangential -system is non-solvable in degrees equal to the numbers of positive and negative Levi-eigenvalues. This was already proved in [1] in case the Levi-form is non-degenerate (with non-necessarily real analytic). We refer to our forthcoming paper [7] for more extensive proofs.
On a class of -equations without solutions
In this note we construct -equations (inhomogeneous Cauchy-Riemann equations) without solutions. The construction involves Bochner-Martinelli type kernels and differentiation with respect to certain parameters in appropriate directions.