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

Abstract

Abstract. 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.

CR Extension from Manifolds of Higher Type

“…In particular, if we evaluate at (λ, x, w 1 , u) = (0, 0, 0, 0), then this is invertible since ∂ x h| 0 = 0. The differentiability with respect to the parameters in the space F α is also clear in view of [16,Proposition 11]. We show that the components u Ii , i ≥ j, of the solution to Bishop's equation, as well as their harmonic conjugates v Ii , are in fact in F mi α for i < j (resp.…”

“…In particular if we evaluate at (λ, x, w 1 , u) = (0, 0, 0, 0), then this is invertible since ∂ x h| 0 = 0. The differentiability with respect to the parameters in the space F α is also clear in view of [15,Prop. 11].…”

“…Let 0 < α < 1 and denote by τ = re iθ the variable in the standard disc ∆. Let us recall from [14], [15] and [16] some basics about attaching analytic discs to M in the subclasses F α of the Hölder classes C α . These are the spaces of real continuous functions σ(θ), θ ∈ [−π, π] which are C 1,α out of 0 and for which the following norm is finite ||σ|| F α := ||σ|| C 0 + ||θσ (1) || C α .…”

Propagation of CR extendibility at the vertex of a complex sector

“…Thus the complex angle for a set to be a propagator reduces from 2π to π. Going on in reduction of the angle, Zaitsev and Zampieri prove in [11] that a sector A α of angle απ > π 2 is a propagator; again, this is not explicitely stated. What about α ≤ 1 2 ?…”

“…What about α ≤ 1 2 ? The positive solution to the question comes from a good balance between the size of α and the flatness of M at p. (As for the result of [11], note that any smooth M is at least "2-flat".) For instance, consider the sector in C…”