On removable singularities for integrable cr functions

Abstract

The subject of removable singularities for the boundary values of holomorphic functions of several variables has been intensively studied in recent years, see works: [1], [14], [17], [18], [19], [21], [23], [24], [25], [26], [28], [32], [34], and the excellent surveys [11], [40].The present paper is devoted to a further step into this direction. Its main intention is to illustrate the way how, on a CR-manifold M of arbitrary codimension, removable singularity theorems may be understood in terms of the analysis of CR-orbits of M.Let M be a locally embeddable CR manifold,We give various conditions in order that Φ is L 1 -removable, i.e. REMOVABLE SINGULARITIES FOR INTEGRABLE CR FUNCTIONSbecome classical and easy in case n = 0, i.e. M is an open set Ω in C m , m ≥ 2. Finally, there is obstruction to removability in case N is not generic at any point, similar to N being a complex hypersurface in Ω if n = 0.Remark. After the reduction to M being globally minimal, the assumption that M\Φ is globally minimal too is essential and cannot be dropped (see below).Remark. Theorem 3 is treated in [29] and the following will appear in [33]: (iv) If m ≥ 2, codim M N = 1 and N is generic, then every closed set K ⊂ N which does not contain any CR orbit of N is L 1 -removable.Finally, using the theory of CR orbits, we extend also [24].Theorem 5. Let M be C 2,α globally minimal. Then every C λ peak set S satisfies H d (S) = 0 and is L 1 -removable.Now, we explain the terminology, compare our results to the codimension one case and give some motivations.The general feature of our work is that L p -removablity is linked with W-removability, i.e. with envelope of holomorphy results.a. About the methods. Of course, different approaches for proving (L p , ∂ b )-removability are conceivable, and the distinguished role of holomorphic hulls in our context is not clear yet.For instance, one could consider the problem from the viewpoint of the general theory of removable singularities for solutions of linear partial differential operators. Let Ω ⊂ R n be a domain, let K ⊂⊂ Ω be a compact and let P = |β|≤e a β (x)∂ β x be such an operator. K is called (L p , P )-removable if any u ∈ L p (Ω) with P u ≡ 0 on Ω\K satisfies P u ≡ 0 on Ω (in the distributional sense). Of course, the notion makes sense by replacing L p (Ω) with other differentiability classes, e.g. C 0 (Ω), C k (Ω), or even D ′ (Ω).Harvey and Polking [14] have proved removable singularity theorems for general P , but they give results only in case p > 1. Indeed, their main theorem 4.1 [14], states that K is (L p , ∂ b )-removable if the Hausdorff measure H n−p ′ (K) < ∞, p ′ = p/(p − 1), e = deg P . The authors further point out that this result cannot be improved in terms of Hausdorff measures in the class of all first order differential operators. Especially information about L 1 -removability is never available on this level (more precisely, in our special setting results for p < e/(e − 1) cannot be derived from the above theorem).In fact, one of the main argument here (cf. [14],...

On wedge extendability of CR-meromorphic functions

“…In continuation with our previous works [MP1,2,3], we study the wedge removability of metrically thin singularities of CR functions and its application to the local extendability of CR-meromorphic functions defined on CR manifolds of arbitrary codimension.…”

“…This notion generalizes the concept of local minimality in the sense of Tumanov, cf. [Trp], [Tu1,2], [J1,2], [M], [MP1]. A wedge W with edge M ⊂ M is a set of the form W = {p + c : p ∈ M , c ∈ C}, where C ⊂ C m+n is a truncated open cone with vertex in the origin.…”

Totally real discs in non-pseudoconvex boundaries

“…First, we consider the case when K' is contained in a finite union of boundaries 0A~,...,0As~. As K' cannot contain the whole boundary of a disc A~, K' is contained in a finite union A of proper subarcs of c3A~l,..., 0A~ k. According to Theorem 4 of [25], A is removable in the sense of one-sided analytic extension, and we get a contradiction to the definition of K'. (If M=OD, a combination of the theorem of G. Stolzenberg [26] on polynomial convexity of arcs and a removability theorem of C. Laurent-Thi4baut [20] can also be applied.)…”

On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

“…Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Trépreau-Tumanov) if it consists of a single CR orbit (see ~Tr 1~ , [Tr2], [Tul], [Tu2], [MP1] [Stl] , [St2] [DP1], [DP2] (see also [V], [Sha], [PV]) where the extension as a mapping is derived from the extension as a correspondence.…”

“…This family Aql will be our starting point to study the envelope of holomorphy of (a certain subdomain of) the union of D together with a neighborhood Q of M1 and with an arbitrarily thin neighborhood of (see Figure 3 To deform these discs by applying the classical works on analytic discs and because Banach spaces are necessary, we shall work in the regularity class ck,a, where k &#x3E; 1 is arbitrary and where 0 a 1, which is sufficient for our purposes. Let T1 denote the Hilbert transform vanishing at 1 (see [Tul], [Tu2], [Tu3], [MP1], [MP2] depending only on the Jacobian matrix of the mapping in 3) at 0 x 1 such that c-1 b ~ c6. Let 0 ( 1, b) denote the disc of radius 6 centered at 1 E C. Furthermore, since the boundary of the disc ApI,o is transversal to then after shrinking a bit "7 if necessary, we can assume that the set contains and foliates by half analytic discs the whole lower side An (0, 2TI) n E" (see Figure 6).…”