On the Regularity of Cr Mappings in Higher Codimension

We give a proof of the regularity of Hölder CR homeomorphisms of strictly pseudo convex CR manifolds of higher codimension. The purpose of this paper is to complete and simplify the proof of one of the main results of [T] on the regularity of CR mappings of strictly pseudo convex CR manifolds of higher codimension. In [T] we introduce a local theory of extremal discs for such manifolds and apply it to the regularity problem. Theorem 7.1 of [T] and Theorem 1 of this paper are the same result. The proof of Theorem 7.1 relies on Theorem 6.10, whose proof is not given in full detail in [T]. Theorem 6.10 says that the union of certain exceptional discs can cover at most a set of measure zero. Thus, the regularity result is proved in [T] only for manifolds for which the conclusion of Theorem 6.10 is true. Instead of giving the details of the proof of Theorem 6.10, which appears to be quite involved, we give a direct proof of Theorem 7.1. The proof in this paper does not pass through Theorem 6.10, and therefore Theorem 6.10 is no longer needed for proving the regularity of CR maps. This paper is written as a continuation of [T] and assumes that the reader is familiar with that paper. We use the same notation and refer to the definitions, statements, and formulas of [T] the way they are numbered there, that is by indicating two numbers separated by a period. We refer to objects from this paper by indicating single numbers. Thus, we prove the following Theorem 1. Let M1 and M2 be C ∞ smooth generic strictly pseudoconvex normal (see Definition 6.6) CR manifolds in C with generating Levi forms, and let F :M1 →M2 be a homeomorphism such that both F and F are CR and satisfy a Lipschitz condition with some exponent 0 < α < 1. Then F is C smooth. We refer to [T] for a discussion on this and related results. The hypothesis that M1 and M2 are normal can be slightly relaxed. That would require reworking Definition 1 6.6, Lemma 6.7, and Proposition 6.8 by replacing normal discs by non-defective ones. In Definition 6.6 the matrices A1, ..., Ak would be linearly independent as linear operators rather than quadratic forms. However, the improvement would be slight, because the author does not even know any examples of manifolds that are not normal. We leave the details to the reader. The key idea of the proof of Theorem 1 is contained in Proposition 7.5, which says that F preserves the lifts of extremal discs. One of the assumptions of that statement is that the disc f2 in the target space is normal. We will prove Proposition 7.5 without that assumption. Then the proof of Theorem 7.1 given in [T] will go through without the use of Theorem 6.10. Thus, we need the following refinement of Proposition 7.5. Proposition 2. Let M1 and M2 be smooth generic manifolds in C N , and let F : M1 →M2 be a homeomorphism such that both F and F −1 are CR and satisfy a Lipschitz condition with some exponent 0 < α < 1. Let F1 be the extension of F to a domain D as in Proposition 7.3. Let f1 be a small stationary disc attached to M1 such that f(∆) ⊂ D, and let f 1 be a supporting lift of f1. Then the disc f2 = F1 ◦ f1 is also stationary and f 2 = f ∗ 1 (F ′ 1 ◦ f1) −1 (where f 1 and f ∗ 2 are considered row vectors) is a lift of f2. The proof is based on a new extremal property of extremal analytic discs which is more suitable for application to CR mappings than Definition 3.1. We call p a real trigonometric polynomial if it has the form p(ζ) = ∑m j=−m ajζ j , where a−j = āj. We call a real trigonometric polynomial p positive if p(ζ) > 0 for |ζ| = 1. We put ∆r = {ζ ∈ C : |ζ| < r}; ∆ = ∆1. We put Resφ = Res(φ, 0), the residue of φ at 0. Definition 3. Let f be an analytic disc attached to a generic manifold M ⊂ C . Let f : ∆ \ {0} → T (C ) be a holomorphic map with a pole of order at most 1 at 0. We say that the pair (f, f) has a special extremal property (SEP) if there exists δ > 0 such that for every positive trigonometric polynomial p there exists C ≥ 0 such that for every analytic disc g : ∆ → C attached to M such that ||g − f ||C(∆̄) < δ we have ReRes(ζ〈f, g − f〉p) + C||g − f ||2C(∆̄1/2) ≥ 0. (1) The above extremal property is close to the one introduced by Definition 3.1 in [T]. In particular, we note (Lemma 4) that stationary discs with supporting lifts have SEP. Conversely, we prove (Proposition 8) that SEP implies that f is a lift of f . 2 In formulating SEP we no longer restrict to the discs g with fixed center g(0) = f(0). This helps prove Proposition 8 in case f is defective; see remark after Lemma 6. The radius 1/2 plays no special role here. In Definition 3, we could even consider f defined only in a neighborhood of 0 and replace 1/2 by a smaller number. Then SEP would still imply that f is a lift of f . Proof of Proposition 2. Since f1 has a supporting lift f ∗ 1 , then the pair (f1, f ∗ 1 ) has SEP. Then we prove (Lemma 5) that the pair (f2, f ∗ 2 ) also has SEP. Then by Proposition 8, SEP implies that f 2 is a lift of f2 and the proposition follows. Lemma 4. Let f be a stationary disc attached to a generic manifold M ⊂ C . Let f be a supporting lift of f . Then the pair (f, f) has SEP with C = 0. Proof. For every analytic disc g attached to M (not necessarily close to f), we have Re〈f, g−f〉 ≥ 0 on the unit circle b∆. Multiplying by a positive trigonometric polynomial p and integrating along the circle we immediately get (1) with C = 0. The lemma is proved. Lemma 5. Under assumptions of Proposition 2, the pair (f2, f ∗ 2 ) has SEP. Proof. For every small disc g2 attached to M2, we put g1 = F2 ◦ g2, where F2 is the extension of F as in Proposition 7.3. If g2 is close to f2 in the sup-norm, then for ζ ∈ ∆̄1/2 we have |g1(ζ)− f1(ζ)| ≤ C1|g2(ζ)− f2(ζ)|, where C1 is the maximum of ||F ′ 2||, the norm of the derivative of F2 in a neighborhood of the compact set f2(∆̄1/2). Then for ζ ∈ ∆̄1/2 we have g2(ζ)− f2(ζ) = F1(g1(ζ))− F1(f1(ζ)) = F ′ 1(f1(ζ))(g1(ζ)− f1(ζ)) +R(ζ)|g1(ζ)− f1(ζ)| , where |R(ζ)| ≤ C2, and C2 is the maximum of ||F ′′ 1 || in a neighborhood of the compact set f1(∆̄1/2). For every positive trigonometric polynomial p, recalling that f ∗ 2 = f ∗ 1 (F ′ 1◦f1) , we obtain |Res(ζ〈f 2 , g2 − f2〉p)−Res(ζ 〈f 1 , g1 − f1〉p)| = ∣