Giuseppe Zampieri

Extremal discs and holomorphic extension from convex hypersurfaces

Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f holomorphically extends to D. The result partially answers a conjecture by Globevnik and Stout of 1991.

Extension of CR-functions into weighted wedges through families of nonsmooth analytic discs

The goal of this paper is to develop a theory of nonsmooth analytic discs attached to domains with Lipschitz boundary in real submanifolds of C n . We then apply this technique to establish a propagation principle for wedge extendibility of CR-functions on these domains along CR-curves and along boundaries of attached analytic discs. The technique from this paper has been also extensively used by the authors recently to obtain sharp results on wedge extension of CR-functions on wedges in prescribed directions extending results of BOGGESS-POLKING and EASTWOOD-GRAHAM.

L 2-estimates with Levi-singular weight, and existence for\(\overline \partial \)

We give a symplectic proof of the link between pseudoconvexity of domains ofCn and of their boundaries (cf. [7, Th. 2.6.12]). Our approach also allows us to treat boundaries of codimension >1. We then extend the estimates by Hörmander in [7, Ch. 4, 5] and [6] toL2-norms which haveC1 but notC2 weights and under a less restrictive assumption of weakq-pseudoconvexity. (A special trick is needed as a substitute for the method of thelowest positive eigenvalue of [6].)

Hypersurfaces through higher-codimensional submanifolds of ℂn with preserved Levi-kernelwith preserved Levi-kernel

For a “generic” submanifoldS of a complex manifoldX, we show that there exists a hypersurfaceM⊃S which has the same number of negative (or positive) Levi-eigenvalues asS at one prescribed conormal (cf. also [9]). When ranksLS is constant, thenM may be found such thatLM andLS have the same number of negative eigenvalues at any common conormal. Assuming the existence of a hypersurfaceM with the above property, we then discuss the link between complex submanifolds ofS whose tangent plane belongs to the null-space of the Levi-formLS ofS (of all complex submanifolds whenLS is semi-definite), and complex submanifolds ofTS*X. As an application we give a simple result on propagation of microanalyticity for CR-hyperfunctions along complex,LS-null, curves (cf. [3]).

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

Let X be a complex analytic manifold. Consider S⊂M⊂Xreal analytic submonifolds with codium R MS=1,and let ω be a connected component of M\S. Let p∈S XMTM *X where T* Xdenotes the conormal bundle to M in X, and denote by ν(p) the complex radial Euler field at p. Denote by μ*(Ox) (for * = M, ω) the microlocalization of the sheaf of holomorphic functions along *. Under the assumption dimR(TpTM *X⋂ ν(p)) = 1, a theorem of vanishing for the cohomology groups HjμM(Ox)p is proved in [K-S 1, Prop. 11.3.1], j being related to the number of positive and negative eigenvalue for the Levi form of M. Under the hypothesis dimR(TpTS *X∩ν(p))=1, a similar result is proved here for the cohomology groups of the complex of microfunctions at the boundary μω(Ox).Stating this result in terms of regularity at the boundary for CR–hyperfunctions a local Bochner–type theorem is then obtained.

Separate real analyticity and CR extendibility

Abstract It is proved here that a function in ℝ2 which is separately real analytic in one variable and CR extendible in the other (that is separately holomorphically extendible to a uniform strip), is real analytic. It is also considered the case when the CR extendibility occurs only on one side. The proof is obtained by bringing the problem into the frame of CR geometry.

Regularity at the boundary for v onQ-pseudoconvex domainsonQ-pseudoconvex domains

AbstractSolvability for

Loss of derivatives for systems of complex vector fields and sums of squares

\bar \partial