Optimal regularity for [`(?)] b\bar \partial _b on CR manifoldson CR manifolds

In this paper a new explicit integral formula is derived for solutions of the tangential Cauchy-Riemann equations on CR q-concave manifolds and optimal estimates in the Lipschitz norms are obtained.     

Conditions for the [`(?)] \bar \partial -closedness of differential forms

The $ \bar \partial $ \bar \partial -closed differential forms with smooth coefficients are studied in the closure of a bounded domain D ⊂ ℂ ⁿ . It is demonstrated that the condition of $ \bar \partial $ \bar \partial -closedness can be replaced with a weaker differential condition in the domain and differential conditions on the boundary. In particular, for the forms with harmonic coefficients the $ \bar \partial $ \bar \partial -closedness is equivalent to some boundary relations. This allows us to treat the results as conditions for the $ \bar \partial $ \bar \partial -closedness of an extension of a form from the boundary.     

Regularity of[`(?)] b\bar \partial _b , complex on nondegenerate Cauchy-Riemann manifolds, complex on nondegenerate Cauchy-Riemann manifolds

For each point ξ in a CR manifold M of codimension greater than 1, the CR structure of M can be approximated by the CR structure of a nilpotent Lie group G_ξ of step two near ξ. G_ξ varies with ξ. \square _b \square _b and [`(?)] <sub>b</sub>\bar \partial _b on M can be approximated by \square <sub>b</sub> \square _b and [`(?)] _b\bar \partial _b on the nilpotent Lie group G_ξ We can construct the parametrix of \square _b \square _b on M by using the parametrix of \square _b \square _b on nilpotent group of step two, and define a quasidistance on M by the approximation. The regularity of \square _b \square _b and [`(?)] _b\bar \partial _b follows from the Harmonic analysis on M.     

Local analyticity for the [`(?)]\overline \partial -Neumann problem in completely decoupled pseudoconvex domains-Neumann problem in completely decoupled pseudoconvex domains

In this paper we prove local analyticity of solutions to the -Neumann problem up to the boundary of rigid, completely decoupled pseudoconvex domains with real-analytic boundary. These are domains that are locally of the form Imw > Σ |h _k (z _k )|² with eachh _k holomorphic and vanishing only at 0. As in those earlier papers, we use purelyL ² methods and must construct a special holomorphic vector fieldM and then use carefully balanced polynomials inM to localize high powers ofT = ∂/∂t effectively, wheret = Rew.     

Polynomial hulls and H￥H^\infty control for a hypoconvex constraint

We say that a subset of is hypoconvex if its complement is the union of complex hyperplanes. Let be the closed unit disk in , . We prove two conjectures of Helton and Marshall. Let be a smooth function on whose sublevel sets have compact hypoconvex fibers over . Then, with some restrictions on , if Y is the set where is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector valued functions with boundary in Y. Furthermore, we show that the infimum is attained by a unique bounded analytic f which in fact is also smooth on . We also prove that if varies smoothly with respect to a parameter, so does the unique f just found. Received: 18 December 1998 / Published online: 28 June 2000     

Vojta’s conjecture on blowups of \mathbbPn{\mathbb{P}^n}, greatest common divisors, and the abc conjecture

We will prove some cases of Vojta’s conjecture on blowups of \mathbbPⁿ{\mathbb{P}^n}, using Schmidt’s subspace theorem. The results can be stated as inequalities of greatest common divisors. Moreover, from Vojta’s conjecture on one further blowup at an infinitely near point, we derive a still-open special case of the abc-conjecture.     

Global existence of solutions for a nonlinearly perturbed Keller–Segel system in \mathbbR2{\mathbb{R}}^2

We show the global existence of small solution to the perturbed Keller–Segel system of simplified version. Our system has a perturbed nonlinear term of worse sign, therefore the existence and uniqueness of solution is not really obvious. The local existence theorem is obtained by a variational observation for the elliptic part.     

Localisation des résidus de Baum-Bott, courbes généralisées et K-théorie (I: feuilletages dans \Bbb C2 {\Bbb C}^2 )

Let v be a holomorphic vector field in a neighborhood of a point m ₀ in , which is a non dicritical isolated singularity. Let f = 0 be a reduced equation of the maximal separatrix V through m ₀, v _f the vector field , and the union of separatrices and pseudo-separatrices (i.e. the set of points where v and v _f are colinear). Assuming the foliations defined by v and v _f to be distinct, we prove that the Baum-Bott residue BB(c ₁ ² , v) of v at m ₀, as well as the difference PH(v) - μ between the Poincaré-Hopf index and the Milnor number of V at m ₀, are "localised" near the separatrices and pseudo-separatrices. (The particular case of generalized curves has already been studied in details in [CLS] and [Br]). We also interpret in K-theory the difference PH - μ as well as the GSV index of Gomez Mont-Seade-Verjovski, and we give a caracterisation of generalized curves in this framework, which will enable us to extend this concept in higher dimension. Received: August 25, 2000