CR Extension from Manifolds of Higher Type

Abstract

Abstract. This paper deals with the extension of CR functions from a manifold M ⊂ C n into directions produced by higher order commutators of holomorphic and antiholomorphic vector fields. It uses the theory of complex "sectors" attached to real submanifolds introduced in recent joint work of the authors with D. Zaitsev. In addition, it develops a new technique of approximation of sectors by smooth discs.

Lie algebras of infinitesimal CR automorphisms of weighted homogeneous and homogeneous CR-generic submanifolds of CN

“…The real analyticity of the functions Ξ j enables us to employ their Taylor series for computing the associated algebras of infinitesimal CR-automorphisms (see the next subsection). For a real analytic CR-generic manifold M represented by the above defining functions as (5) and for each p ∈ M , it is well-known ( [33]) that the space T (0,1) p M is generated by the following holomorphic vector fields, tangent to M :…”

“…The main idea. Now, let us explain briefly the main strategy used in [34] for computing the Lie algebra aut CR (M ) associated to an arbitrary real analytic generic CR manifold M ⊂ C n+k , represented as the graph of the k complex defining equations as (5). For this, we shall proceed to do successively the following steps:…”

CR extension from hypersurfaces of higher type

“…Let h be of class C k+2 , and be written as in (2) for g homogeneous of degree k and let w(τ ) = m 0 m=1 a m (1 − τ ) mα ie 2 , τ ∈ Δ. Then for any there is δ such that if h C 1,α δ, w F α δ, |x 0 1 | δ, then the equation…”

“…At the last step, in order to pass from F (k−1)α to C 1,β , we have to notice that u ∈ F (k−1)α implies |u (2) …”

“…In the present paper, we investigate this problem and for this purpose introduce a new family of analytic discs with singular boundary which can be naturally "attached" to sectors of hypersurfaces. This theory was first introduced in [8,9] and next developed in [2]. We give a simple presentation of it which is appropriate for the hypersurface case.…”