Regularity of CR mappings of abstract CR structures

We study the C∞ regularity problem for CR maps from an abstract CR manifold M into some complex Euclidean space CN ′ . We show that if M satisfies a certain condition called the microlocal extension property, then any C k-smooth CR map h : M → CN , for some integer k, which is nowhere C∞-smooth on some open subset Ω of M , has the following property: for a generic point q of Ω, there must exist a formal complex subvariety through h(q), tangent to h(M) to infinite order, and depending in a C 1 and CR manner on q. As a consequence, we obtain several C∞ regularity results generalizing earlier ones by Berhanu–Xiao and the authors (in the embedded case).


Introduction and Results for CR Structures of Hypersurface Type
The purpose of this paper is to extend our recent study [16] of the regularity problem for CR mappings between smooth CR submanifolds of C N and C N , respectively, to the case of maps from an abstract CR manifold with values in some complex Euclidean space C N . More precisely, the question we are interested in is the following. Given an abstract CR manifold M and a C ∞ -smooth CR manifold M ⊂ C N , under which conditions can we guarantee that there exists an integer k such that h * (T 0 p M ) = T 0 p M. We should mention that the notion of strictly noncharacteristic map coincides with the well-known condition of CR transversality (see e.g. [10,13]) when M and M have the same CR codimension (see [16]). If h is as above, the singular support of h, denoted SingSupp h, is the locus of points p in M such that h is not C ∞ -smooth in any neighborhood of p.
In order to state our first main result, we shall briefly recall an extension of the notion of finite type which was used in [16], building on the original concept introduced by D'Angelo [6]. If X ⊂ C N is a set, we denote the ideal of germs at q ∈ X of smooth functions vanishing along X by I q (X) = {ϕ ∈ C ∞ (C N , q) : ϕ| X = 0}. For ψ ∈ C ∞ (C, 0), denote by ν 0 ψ the order of vanishing of ψ at 0. For p ∈ X, we define the 1-type of X at p as Δ(X, p) = sup where the supremum is taken over all holomorphic curves γ : Δ = {ζ ∈ C : |ζ| < 1} → C N . We say that p is a D'Angelo finite-type point of X if Δ(X, p) < ∞, and a D'Angelo infinite-type point of X otherwise. We denote the set of infinitetype points in X by E X and recall that, in the case where X is a smooth real hypersurface, then E X is closed in X by [7,8].
Our main result for abstract CR manifolds of hypersurface type is the following. When E M = ∅, we obtain as an immediate consequence the following regularity result. Note that it follows from the proof that any map as in Theorem 1.1 must be automatically CR immersive and hence Theorem 1.2 appears also as a regularity result of CR embeddings. Our approach in this paper will allow us to get more general versions of Theorems 1.1 and 1.2 for CR manifolds of arbitrary CR codimension (see Sec. 2).

Statement of Results for Abstract CR Manifolds of Any CR Codimension
The results in Sec. 1 follow from more general results that we shall now describe.
To this end, we first introduce some notation to be used throughout the paper, and define a number of notions (some of them are not necessarily standard). Let (M, V ) be an abstract CR manifold of CR dimension n and CR codimension d. For every p ∈ M , the Levi map of M is the (vector-valued) Hermitian form In (2.1), the definition of the Levi form L p is independent of the choice of the vectors fields X and Y extending X p and Y p in a neighborhood of p.
We say that M is Levi-nondegenerate if for every p ∈ M , L p (X p , Y p ) = 0 for all Y p ∈ V p implies that X p = 0. If M is of hypersurface type, then the Levi map is a Hermitian form and we say that M is strongly pseudoconvex if the Levi form L p is positive definite at every p (or negative definite at every p).
If (M, V ) and (M , V ) are abstract CR manifolds, and h : M → M is a map of class C 1 , then we say that h is CR provided that dh(V ) ⊂ V , or, equivalently, if h * (T M ) ⊂ T M (we again abuse notation by identifying bundles with sections here). If M = C N , h = (h 1 , . . . , h N ) is CR if and only if each h j a CR function on M (for details, see e.g. [1]). We shall denote by Γ p (M ) the set of all germs at p of CR vector fields of M . Now we come to a notion that will be important throughout the remainder of this paper. We are going to recall the classical notion of wavefront set WF (u) in Definition 4.1. In Sec. 4, we will discuss the microlocal extension property in more detail, exhibit important instances of CR manifolds for which this property is satisfied, and relate it to almost analytic extendability. The reader should compare this notion with the work of Berhanu and Xiao [3,4]. We also note that we require that the (usual) wavefront set of a CR function/distribution in Ω is a priori contained in a convex cone, which however is allowed to change as Ω changes; this is a bit in contrast with the hypoanalytic wavefront set in the embedded setting (for this concept, we refer the reader to [23]).
Recall that for a subset X ⊂ C N , and for every q ∈ X, we denote by I q (X) ⊂ C ∞ (C N , q) the ideal of all germs at q of C ∞ -smooth functions ρ : (C N w , q) → C that vanish on X near q. For r ∈ {1, . . . , N }, we define the regular r-type of X ⊂ C N at q as follows: Here, Δ r = {t = (t 1 , . . . , t r ) ∈ C r : |t j | < 1, j = 1, . . . , r} and the supremum is taken over all holomorphic maps α : Δ r → C N which are of full rank r at 0 and satisfy α(0) = q. Note the inequalities where Δ(X, q) is the 1-type already defined in (1.2). We shall say that q is of r-regular infinite-type if Δ r (X, q) = ∞, and we denote the set of points in X which are of r-regular infinite-type by E r X . We therefore have We say that a formal holomorphic subvariety Γ ⊂ C N through the point p is formally contained in X at p ∈ X if for every C N -valued formal power series ϕ(t), t ∈ C, with ϕ(0) = p, one has It follows from this definition that if there exists a nontrivial formal subvariety through p which is formally contained in X, then p is a D'Angelo infinite-type point. If there exists a formal (holomorphic) submanifold of dimension r through p which is formally contained in X, then p ∈ E r X . The next definition introduces the key geometrical concept to be used in our main theorem.

Definition 2.2.
Let M be an abstract CR manifold and h : M → C N be a C 1smooth CR map. We say that (Γ q ) q∈M is a CR family of r-dimensional formal (holomorphic) submanifolds through h(M ) if for every p ∈ M we can find a neighborhood U of p such that there exists a map ψ : U → (C t 1 , . . . t r ) N such that for ξ ∈ U , we can write a parametrization of Γ ξ in the form where each ψ α : U → C N is a C 1 -smooth CR map and ∂ t (ψ(ξ))(0) is of rank r for all ξ ∈ U . For every p ∈ M , and every CR map h : M → C N of class C k , we define the following numerical invariant: 3) The complex gradients which correspond to j = 0 in (2.3), and their CR derivatives are considered as vectors in C N . For k ≥ 0, the function p → e k (p) is integer valued and lower semicontinuous, and obviously e k (p) ≤ N for every p ∈ M . We define We may now state our most general result.
In fact, it follows from the proof of Corollary 2.6 that the result also holds for CR maps whose differential is injective on T 1,0 M . Hence, Corollary 2.6 recovers an earlier result by Berhanu-Xiao [3, Theorem 2.5] for strongly pseudoconvex real hypersurfaces M as targets.
In the previous results, C ∞ regularity of the maps follows automatically once the target manifold is of D'Angelo finite type. However, Theorem 2.3 can also be used to establish C ∞ regularity results when the target manifold is everywhere of D'Angelo infinite-type. We will illustrate this by showing how the following other result due to Berhanu-Xiao [4] may also be derived from Theorem 2.3. certain a priori smoothness), we associate a disjoint union of open subsets of M (Sec. 5). Each open subset ω in the obtained decomposition satisfies the following alternative: it is either contained in M \(SingSupp h) (Proposition 5.5) or has the property that for every point q ∈ ω, there is a formal holomorphic submanifold (of fixed positive dimension) through h(q) that is formally contained in h(M ) (Proposition 5.6). Since the union of such open subsets happens to be dense in M , this roughly proves Theorem 2.3. The open subsets decomposition is constructed through the introduction of numerical invariants associated to rings of functions attached to the map h. This strategy is analogous to that carried out in [16] in the case where M is embedded. However, we should point out that the open subset decomposition in [16] uses heavily the minimality assumption on the embedded manifold M (and Tumanov's extension theorem) and therefore cannot be applied in the abstract case tackled in this paper. We instead proceed with a different construction of the rings and invariants attached to the map leading to the desired open subset decomposition. Our present construction, though still similar in spirit with that of [16], has the advantage to make no assumption on M , and hence, is more general than the one given in [16], even in the embedded case. Furthermore, in order to prove Proposition 5.5, we also need to establish a smooth version of the reflection principle of [16, Theorem 3.1] adapted to abstract CR structures. This is achieved in Theorem 4.8 and Corollary 4.10 where the microlocal extension property of M comes into play. In Sec. 4, we prove Theorem 4.8 and discuss in detail the microlocal extension property. Using [3, Theorem 2.9], we give instances of abstract as well as embeddable CR manifolds satisfying this condition, showing in particular that the results of this paper recover those of Berhanu-Xiao [3,4] in the abstract case, and those of the authors [16] in the embedded case. We also relate the microlocal extension property to the notion of almost analytic extension, whose basic properties are recalled in Sec. 3. The proofs of Theorem 2.3 and its consequences are finalized in Sec. 6.

Almost Analytic Extensions, Wedges and Boundary Values
In this section, we recall some standard facts about almost analytic extensions on wedges and boundary values, which will be useful when discussing the microlocal analytic extension property in Sec. 4. We also prove in this section (Proposition 3.5) a Hölder regularity result for∂-bounded extensions of Hölder continuous boundary values on some wedges, which will be used in the proof of the smooth reflection principle given in Theorem 4.8.

Almost analytic extensions
We start the section by recalling that any smooth function possesses an almost analytic extension. This fact is commonly attributed to Nirenberg [21], and we also refer the reader to the paper of Dyn'kin [9].
In the above result and in what follows, we use the standard notation ∂ ∂σ = 1 2 ( ∂ ∂s + i ∂ ∂t ) for the CR operators on R 2d associated to the complex coordinates σ = s + it. Let (with vertex the origin) and r > 0, we denote by Γ r = {t ∈ Γ : t < r}. Consider now an open cone Γ ⊂ R d and a set of the form and define the set B(W r ) to consist of all functions U ∈ C 1 (W r ) which have the following properties: (a) For any compact set K ⊂⊂ D and any compactly contained As in [14], we introduce the families of functions that we are taking as "almost analytic extensions" as follows. We first define the set A(W r ) to consist of all functions U ∈ C ∞ (W r ) with the property that for every α, β ∈ N n and every γ ∈ N d , and such that If, furthermore, U ∈ A(W r ) has the property that, for every compact set K ⊂ D, every subcone Γ ⊂ Γ with Γ ⊂⊂ Γ, every α, β ∈ N n , every γ ∈ N d , every a ∈ N, then we say that U ∈ A ∞ (W r ).

Definition 3.2.
We say that a distribution u defined on an open subset D ⊂ C n × R d possesses a∂-bounded extension (respectively, a regular∂-bounded extension, respectively, an almost analytic extension) to We also refer to such an U as a∂-bounded (respectively, a regular∂-bounded, respectively, an almost analytic) extension of u (to W r ).
If a distribution u on D has one of the types of extensions introduced in We shall consequently drop the index r from consideration when appropriate.
Observe also that u possesses one of the extensions introduced in Definition 3.2 to D+iΓ if and only ifū possesses the same type of extension to D−iΓ = D+i(−Γ).

Remark 3.3.
We observe that A ∞ (W r ) is a subalgebra of C ∞ (W r ) and therefore, distributions u 1 , u 2 which have almost analytic extensions U 1 and U 2 , respectively, can be multiplied by setting u 1 u 2 = bv(U 1 U 2 ).

Remark 3.4.
The preceding remark also shows that for a vector field X on D, whose coefficients are boundary value distributions of functions in A ∞ (W ), thus extending to a vector field X + on W , for u = bv W U with U ∈ A ∞ (W ) the distribution Xu is defined, and Xu = bv W (X + U ). In particular, derivatives with respect to smooth vector fields of functions/distributions on D which extend almost analytically to W also extend almost analytically to W .

A priori regularity for∂-bounded extensions
Our goal in this section is to prove a Hölder regularity result for extensions of Höder continuous functions which are∂-bounded and whose derivative is of slow growth. We are following in our approach the paper of Coupet [5]. However, we need a slightly more general result than what is stated in [5], which we could not locate in the literature. We therefore include the details of the proof.

Regularity of CR mappings of abstract CR structures
We first recall that a continuous function f : Ω → C is Höder continuous on a set Ω ⊂ R d with Höder exponent α ∈ (0, 1] if there exists a constant C > 0 such that The space of all Höder continuous functions with Höder exponent α is denoted by C 0,α (Ω). If Ω is compact, it becomes a Banach space if endowed with the norm In what follows, we write for some constant C > 0 and some integer k ≥ 1. If h| t=0 ∈ C 0,α (R d ) for some 0 < α < 1 and β := α α+k , then there exists a universal constant A > 0, depending only on α and the support of h such that Proof. As in the aforementioned paper of Coupet (which treats the case k = 1), we divide the proof in several steps. First note that without loss of generality, we may assume that Γ = R d + . In each of the following steps, for σ = s + it and σ = s + it with t, t ∈ Γ, we estimate |h(σ) − h(σ )| satisfying different restrictions in each step.
Step 1. s = s , t = λt , λ > 0. We consider the map ϕ : Note that ϕ(i t ) = σ, ϕ(iλ t ) = σ . The function u = h • ϕ is defined on H + , continuous up to R, and u| R ∈ C 0,α . Furthermore, ∂u ∂ζ is bounded (by C) on H + , and C 1 (H + ). We can therefore apply the generalized Cauchy formula to write the function u as The first integral can be estimated via We estimate the second integral as follows. Let us assume that supp u ⊂ D R := {ζ ∈ C : |ζ| < R}. First note that for any r > 2, we have that where C 2 depends on the support of u and α, but not on u.
We combine the preceding estimates to obtain Step 2. t = t . First assume that s − s 1 α+k ≤ t . Then, we use the mean value theorem to estimate If on the other hand, s − s By Step 1, (3.6), we can estimate Therefore, using these estimates and (3.7), we have Without loss of generality, we assume that C 4 > 1, to obtain now for all s, s that (3.8) Step 3. s = s . As in [5], we define the points , and the functions h j (s + it) := h(s + iP j +s + it). Now note that for every j, by Step 2, the function h j |t =0 is Höder regular of order α α+k , with We can apply Step 1 (note that h j is also∂-bounded, with the same constant C), Step 4. We can now estimate which finishes the proof.

Abstract CR Manifolds and the Microlocal Extension Property
Our goal in this section is to discuss in detail the microlocal extension property introduced in Sec. 2 for abstract CR manifolds. We will furthermore show that CR mappings on such CR manifolds satisfying certain "regular" systems of smooth equations are actually smooth. Even though in spirit this follows our recent paper [16], the abstract case, as already indicated, poses specific problems, which need special treatment that we address in this section.

The microlocal extension property
Let M be an abstract CR manifold of CR dimension n and of real dimension 2n+ d.
In what follows, we set N = n + d. We first recall the definition of the wavefront set of a function u : M → R (since we shall only deal with smooth wavefront sets here, we drop it from the notation).
We note that the preceding definition does not depend on the (bundle) coordinates used for defining the Fourier transform, and that WF(u) is a closed subset of T * M \{0} (meaning that the image of the zero section is removed). We also note that (x, ξ) ∈ WF(u) if and only if (x, −ξ) ∈ WF(ū).
The (closed) set of points where u is not smooth (i.e. the set of points p for which there does not exist a neighborhood U on which u is smooth) is called the singular support of u and coincides with the projection of the wavefront set of u to M :  For all of the (pseudodifferential) operators P which we will consider in this paper, it will hold that for any distribution u on M . We also have the elliptic regularity theorem (see e.g. [12]), which states that A direct consequence of the elliptic regularity theorem is that for any CR function (or distribution) u on M , we have WF(u) ⊂ T 0 M , where T 0 M is the characteristic bundle of M . Indeed, the CR vector fields, considered as pseudodifferential operators, have the property that the intersection of their characteristic sets coincides with T 0 M (see [2]). We will therefore, when speaking about wavefront sets of CR distributions, only consider them as subsets of T 0 M .
We now introduce the notion of standard coordinate patch which will be useful for us in order to characterize the microlocal extension property of an abstract CR manifold.
We say that an open subset D ⊂ M is a standard coordinate patch if there exists an open subset D ⊂ C n × R d , a C ∞ -smooth diffeomorphism Φ : D → D with the property that, for every z 0 ∈ π 1 (D), where π 1 : D → C n is the canonical projection, the submanifold N z0 := Φ({z = z 0 }) of D is totally real and transverse to the complex tangent directions in M . On a standard coordinate patch D, for p ∈ D, we have T 0 p M ∼ = T * p N π1(p) (induced by the restriction of evaluation of the forms) and we require that this yields a well-defined identification T 0 D ∼ = D × R d . We shall refer to any such a choice of local coordinates as standard coordinates. It is quite simple to see that for every p ∈ M , there exists a standard coordinate patch D ⊂ M containing p: Starting with any smooth chart Ψ :D → R 2n+d centered at p. Then dΨ(T c p M ) ⊂ T 0 R 2n+d = R 2n+d is a 2n-dimensional subspace, and we can choose smooth coordinates (x, y, s) in R 2n+d such that dΨ(T c p M ) = {s = 0}. We claim that for (x, y) = z ∈ C n close by the origin and for a small ε > 0, the map γ x+iy : (−ε, ε) s → Ψ −1 (x, y, s) parametrizes a smooth submanifold N z of M which is actually totally real. Indeed, dΨ −1 ({s = 0} ⊕ {z = 0}) = T c p M ⊕ T p N 0 , and this direct sum decomposition necessarily stays stable for small perturbations of (x, y, s), i.e.
where q = Ψ −1 (x 0 , y 0 , s 0 ). Now set D to be a neighborhood of the origin for which the above equation holds, Φ = Ψ −1 | D , and D = Φ(D).
Therefore, any abstract CR manifold can be covered by standard coordinate patches. Furthermore, any distribution u defined in some standard coordinate patch D will be identified as a distribution over D ⊂ R 2n × R d (through the associated diffeomorphism Φ : D → D) .
The dual cone of a cone Γ ⊂ R d is defined by We also recall that ( The basic result about almost analytic extensions and the wavefront set we are going to use is the following well-known result (see e.g. [18] or the proof of [2, Theorem 5.3.7]).

CR manifolds satisfying the microlocal extension property
We want to discuss here some classes of CR manifolds for which the microlocal extension property does hold. Though this is not the focus of this present paper, let us first look at the case of embedded CR manifolds. Recall that an embedded CR submanifold M ⊂ C N is said to be minimal at a point p ∈ M if there is no proper CR submanifold Σ ⊂ M through p with the same CR dimension as that of M (see e.g. [1,24]). We have the following: Proof. In order to prove the proposition, we use the characterization given by Proposition 4.4 to show that the almost analytic extension property holds for any continuous CR function defined in any neighborhood of p. To this end, we just note that by Tumanov's theorem [24], for any neighborhood Ω of p in M , there exists a wedge of edge M at p in C N to which all continuous CR functions on Ω extend holomorphically. The desired almost analytic extension property for all CR functions on Ω then follows by using standard coordinates attached to generic submanifolds in complex space such as in [16,Proposition 3.2]. The details are left to the reader.
To the authors' knowledge, there is no known characterization of the microlocal extension property for abstract CR manifolds. However, the following general sufficient condition for abstract CR manifolds with nondegenerate Levi form (analogous to Lewy's extension theorem [17]) is due to Berhanu and Xiao [3, Theorem 2.9], and is also, to the best of our knowledge, currently the only such result.

Smooth edge-of-the-wedge theory
Let D be a standard coordinate patch in an abstract CR manifold M . If u : D → C is a CR function that admits an almost analytic extension U ∈ A ∞ (W ), with W = D + iΓ for some cone Γ ⊂ R d , and if furthermore u ∈ C k (D), then, by a result of Rosay theorem). The version we will need for the purpose of this paper can be stated as follows.

A smooth reflection principle on abstract CR manifolds
We now turn to an important a priori regularity result for CR maps, which, for embedded CR manifolds, is contained in [16]. Let us stress that even though h is assumed to be CR, g need not be CR in Theorem 4.8. We are going to exploit this in a more specific result that follows from Theorem 4.8. In order to state it, we introduce the following notion (that will appear later in this paper).

Definition 4.9.
Let M be an abstract CR manifold, r ∈ N, and let f : Ω → C be a C r -smooth function on some open subset Ω ⊂ M . We say that f is C r -admissible if there exists an integer ≥ 0, a C r+ -smooth CR function F : Ω → C, and C ∞smooth (1, 0) vector fields X 1 , . . . , X defined on Ω such that f = X 1 . . . X F . This notion of admissible functions extends obviously to C n -valued maps by requiring that each component be admissible. Regularity of CR mappings of abstract CR structures C N -valued map, defined in a neighbhourhood of (p, h(p), g(p)) ∈ M × C N × C k , holomorphic in its last variable, satisfying for q ∈ M near p the following properties: , h(p), h(p), g(p)) = 0. Proof of Theorem 4.8. Let (z,z, s) be standard coordinates defined on D ⊂ R 2n × R d , in which p may be assumed to be the origin. We may also assume that h(0) = 0, g(0) = 0. Since the conclusion of the theorem is local, we shall do a number of steps each of which requires us to possibly shrink D; we shall do so without explicitly mentioning it, and will not rename D. We will apply the same policy when denoting constants, that may change from one line to the other. The proof given here is an adaptation of the proof of [16,Theorem 3.1], that includes the appropriate changes needed to deal with abstract CR manifolds. First, we consider R d s and C N w as totally real subspaces of C d σ and C 2N Z,ζ by σ = s + it, Z =ζ = w, respectively, i.e. Re w = Re ( Z+ζ 2 ), Im w = Re ( Z−ζ 2i ). Proposition 3.1 yields, for appropriate open subsets Ω ⊂ C N and O ⊂ C k containing 0, an extension R(z,z, s, t, Z,Z, ζ,ζ, R(z,z, s, 0, w,w,w, w, Λ) = r(z,z, s, w,w, Λ,Λ), and such that for every m, ∈ N, there exists a C = C(m, ) > 0 such that, denoting for every partial derivative of the form, it holds that shrinking D, Ω and O if necessary. The Jacobian of the map R with respect to Z (considered as a map R 2N to R 2N ) at the origin is computed to be which has a nonzero determinant by assumption. We can therefore apply the implicit function theorem to obtain a map Φ(z,z, s, t, ζ,ζ, Λ), holomorphic in its  z, s, t, ζ,ζ, Λ). In particular, one has for (z, s) ∈ D h(z,z, s) = Φ (z,z, s, 0, h(z,z, s), h(z,z, s), g(z,z, s)). (4.4) We now claim that, for j = 1, . . . , d and k = 1, . . . , N , the Φσ j and Φζ k are controlled in the following way.
This can be seen in the following way: First, we have that Then any partial derivative (in (z,z, s, ζ,ζ)) acting on this equation gives (by the chain rule) rise to expressions which can be controlled in the claimed way using (4.3). Now, by assumption, h extends almost analytically to D + iΓ. Let us denote an almost analytic extension of h by h + (z,z, s, t), defined for (z, s) ∈ D and t ∈ Γ, and similarly for g. In the remainder of the proof, we will show that h − (z,z, s, t) := Φ(z,z, s, t, h + (z,z, s, −t), h + (z,z, s, −t), g + (z,z, s, −t)) (4.5) is an almost analytic extension of h to D − iΓ (here t is small). An application of Theorem 4.7 then yields that h is smooth near the origin. Let us first check that h − is of slow growth. Let α, β ∈ N n and γ ∈ N d , K ⊂ D compact, and Γ ⊂⊂ −Γ be given. By assumption, h + and g + are of slow growth in D + iΓ. In particular, there exist constants C = C(α, β, γ) > 0 and k = k(α, β, γ) ∈ N, such that for (z, s) ∈ K and t ∈ −Γ α ,β ∈N n ,γ ∈N d |α |+|β |+|γ |≤|α|+|β|+|γ| Regularity of CR mappings of abstract CR structures chain rule, (4.6) and (4.5) then imply that there exists a combinatorial factorΔ such that for (z, s) ∈ K and t ∈ −Γ with t sufficiently small; i.e. h − is of slow growth as claimed.
we need some preparation. Fix K and Γ as before. We will first establish the following.

Claim 2.
There exist β > 0 and C > 0 such that In order to establish Claim 2, note that by Rosay's result already mentioned in the beginning of Sec. 4.3, we have that h + is actually C 1 (D × (Γ ∪ {0})) for any cone Γ ⊂⊂ Γ, i.e. it is C 1 up to the edge. Since g + is an almost holomorphic extension of g, we thus see that (4.8), (4.5) and (4.4) imply that h − (z,z, s, t) is ā ∂-bounded extension to D − iΓ of the function h + | t=0 ∈ C 1 (D). Choose a cutoff function χ such that for an > 0 we have supp χ ⊂ D × (−2 , 2 ) d , χ| K×(− , ) d = 1, and which is almost holomorphic (in the σ j 's).
Since Γ ⊂⊂ Γ is a compact subcone, there exists a finite number of closed convex cones Γ 1 , . . . , Γ e , each of which is linearly equivalent to R d + , and an open convex cone Γ ⊂⊂ Γ such that For z 0 ∈ R 2n sufficiently close to 0, considering h z0,j = (χh − )| {z0}×R d ×−Γj , we see that h z0,j satisfies the conditions of Proposition 3.5, since h is C 1 over D and (4.7) and (4.6) hold. So that for some β > 0 (independent of z 0 and j) we have that A priori C depends on z 0 and j, but the concrete form of the estimates in Proposition 3.5, and the fact mentioned above that h + ∈ C 1 (D × (Γ ∪ {0}), ensures that C is actually independent of z 0 and j. We thus conclude that for z ∈ K and t ∈ −Γ , t small enough, we have that Now, similarly to [16], Claims 1 and 2 can be used to show in particular that we can control the partial derivatives of Φσ and Φζ in the following way: For every K, Γ as above, and m, ∈ N, there exists a C > 0 such that if |α|+|γ|+|γ|+|δ|+| | ≤ m, then for (z, s) ∈ K, t ∈ Γ with t small enough, we have ∂ |α|+|β|+|γ|+|δ|+| | ∂z αzβ s γ ζ δζ Φσ(z,z, s, t, h + (z,z, s, −t), h + (z,z, s, −t), g + (z,z, s, −t)) (4.9) We can now complete the proof of the theorem largely similar to [16]: If we apply a partial derivative of the form P = ∂ |α|+|β|+|γ| ∂z αzβ s γ to (4.8) yields an expression which we can control because it is a sum of products each of which contains only factors which are of slow growth towards t = 0, and at least one factor which vanishes to infinite order in t at t = 0; this is obvious for terms containing derivatives with respect toσ of h + and g + , and for the other terms we can use (4.9). The details are analogous to those of [16] and are therefore left to the reader.

Invariants of CR Maps and Associated Open Subsets Decomposition
In this section, we carry out the construction of the open subsets decomposition associated to any given CR map mentioned in Sec. 2. To this end, we at first introduce a number of rings of functions attached to the map as well as study numerical invariants related to these rings. We follow the lines of thought of the construction done for embedded CR manifolds in [16], but the present construction for abstract CR manifolds needs a number to substantial changes that we explain. In fact, even in the embedded case, our construction in this paper improves the one carried out in [16] (as the required minimality assumption in a number of properties in [16,Secs. 4 and 5] is no longer necessary with the present new construction). We indicate here the main differences and will drop proofs and refer to [16] where they are very similar to the embedded case.

Function rings attached to a CR map and numerical invariants
Let (M, V ) be an abstract CR manifold of CR dimension n and CR codimension d. We introduce here a sequence of local numerical invariants attached to a (germ of a) CR map at p ∈ M . In what follows, if X is a real manifold, x 0 ∈ X and Regularity of CR mappings of abstract CR structures ∈ Z + ∪ {∞}, we denote by C (X, x 0 ) the ring of germs of C -smooth functions at x 0 and by C (X) the ring of C -functions over X.
For p ∈ M and k ∈ Z + ∪ {∞}, we denote by C k CR (M, p) the ring of germs of C k -smooth CR functions at p. Analogously to Definition 4.9, we say that a germ at p of a C m -smooth function g, m ∈ N, is admissible if there exists an integer ∈ N, a germ at p of C +m -smooth CR function G and C ∞ -smooth (1, 0) vector fields defined near p, X 1 , . . . , X , such that g = X 1 , . . . , X G (as germs at p). Note that germs at p of CR functions are obviously admissible. This notion extends in the obvious way to C n -valued mappings by requiring that every component be admissible.
Let h : M → C N w be a C 1 CR map. Even though the notation for the rings and invariants associated to h is the same as we used in [16], let us stress that the rings and invariants introduced here are different from those introduced in the above-mentioned paper.
Definition 5.1. Let M and h be as above, p ∈ M , and μ, j ∈ N with 0 ≤ j ≤ μ.
Observe that if h is C μ−j -smooth, then for any ψ ∈ F j,μ p there is a neighborhood of p in M such that for any q in that neighborhood, (the germ at q of) ψ ∈ F j,μ q . For every p ∈ M , we define the following vector subspace of C N : Next, suppose that we are given a neighborhood U p of p in M as before, and since V is CR. This completes the proof of part (i) of the lemma; part (ii) follows by very similar arguments.

Open subset decomposition and its properties
Let M be as above and h : M → C N be a C 1 -smooth CR map. For μ, j integers satisfying 0 ≤ j ≤ μ ≤ N , since the functions M p → S μ j (p) are all lower semicontinuous and integer valued, the set (5.6) and for ν = N − + k, we set We further decompose each of the sets Ω k,ν defined for k, , ν ∈ N with k ≤ ν ≤ N − + k by either (5.6) or (5.7) into Proof. If there are j, μ such that S μ j (p) = N , then we can find (g, r 1 ), . . . , (g, r N ) ∈ A j,μ p such that for q ∈ M near p, r j (q, h(q), h(q), g(q)) = 0, j = 1, . . . , N , and such that the Jacobian r w of the map r = (r 1 , . . . , r N ) is invertible at p. Hence we may apply Corollary 4.10 to conclude that h is C ∞ -smooth in a neighborhood of p. The remainder of the proposition follows in a straightforward way. The proof is complete.
Once we have dealt with what is happening for all points belonging to each of the open subsets Ω ,N k,ν , we now turn to describing how the existence of points in one of the sets Ω ,m k,ν for m < N impacts the CR geometry of the set h(M ). This is explained in the following result. The proof of Proposition 5.6 follows along the lines of the arguments from [16]; we highlight in what follows the main steps of the proof adapted to the abstract case studied in this paper. Throughout the rest of this section, we fix k, , m, ν ∈ N as in Proposition 5.6.
For p ∈ Ω ,m k,ν , we have by definition dim V ν,N − +k p = N − m. One of the main properties which our construction needs is that locally one can find a basis of CR vector fields which span V ν,N − +k q for q close to p. Before we formulate the proposition, let us recall the following result [16,Lemma 5.4], which also holds in the abstract setting. Then there exist N − δ germs at p of C N -valued mappings, with components in Then for every p ∈ Ω ,m k,ν , there exist a neighborhood U p ⊂ Ω ,m k,ν of p and CR maps V j : U p → C N of class C N − +k−ν , j = 1, . . . , N − m, whose components belong to F ν,N − +k p , such that {V 1 (q), . . . , V N − (q)} forms a basis of V ν,N − +k q for every q ∈ U p . Furthermore, for every q ∈ U p , we have V ν,N − +k q = V ν+1,N − +k q and for every (g, r) ∈ A ν+1,N − +k q defined on a neighborhoodŨ q ⊂ U p of q, it holds that V j (q) · r w (q, h(q), h(q), g(q)) = 0,q ∈Ũ q , j = 1, . . . , N − m.
Once we have established all the useful properties of the spaces V ν,N − +k p , we can now construct in the next proposition the formal holomorphic submanifolds that appear in the statement of Proposition 5.6. Note that the proposition does not make any assumption on M ; in particular no microlocal extension property on M is required.
Then the following hold : (a) For each α ∈ N N −m , d α is a well-defined CR map on U p and of class The proof of Proposition 5.9 is carried out exactly the same way as the proof of [16,Proposition 5.5] by using the following properties: (iii) for each i = 1, . . . , N − m, the conjugate V i also belongs to F ν,N − +k q for q ∈ U p (since V i is CR and therefore admissible), (iv) an appropriate use of the chain rule (e.g. [16,Lemma 5.7]).
Proof of Proposition 5.6. A direct application of Proposition 5.9 yields the (local) existence of Γ q , for every q ∈ Ω ,m k,ν , with the required property. We just have to check that the construction from Proposition 5.9 actually yields the same formal submanifolds on overlaps of neighborhoods U p ∩ Up ⊂ Ω ,m k,ν with p,p ∈ Ω ,m k,ν (the following argument also yields that the Γ q are independent of the chosen basis V in Proposition 5.9).
On such an intersection, by construction of U p and Up, if we denote the local bases used in Proposition 5.9 by V = (V 1 , . . . , V N −m ) andṼ = (Ṽ 1 , . . . ,Ṽ N −m ), respectively, there exist an invertible matrix A = (A r s ) N −m r,s=1 whose components belong to C N − +k−ν CR (U p ∩ Up) (and F ν,N − +k q for every q ∈ U p ∩ Up) such that Denote by D(q; t) the parametrization obtained from Proposition 5.9 for the formal submanifolds Γ q 's, associated to V, andD(q;t) the one corresponding to theΓ q 's associated toṼ, for q ∈ U p ∩ Up. The reader may check that, using the chain rule, one may easily construct, for every q ∈ U p ∩ Up, a formal (holomorphic) invertible map Φ q : (C N −m , 0) → (C N −m , 0), whose coefficients depend C N − +k−ν on q, such that Φ q (t) = A(q)t + · · · ,D(q; Φ q (t)) = D(q; t).
Hence for q ∈ U p ∩ Up, the formal submanifolds Γ q andΓ q coincide. The proof of the proposition is complete. Ω k,ν is dense too in M , where each Ω k,ν is given by (5.6) and where the union is a disjoint union. We therefore get that the disjoint union Proof of Corollary 2.6. As in [16,Lemma 6.2], one can show that the Levinondegeneracy assumption on M and the fact that dh is injective on T 1,0 M imply that e 1 ≥ N − n + n. Now applying Theorem 2.3 with k = 1 and = N − n + n, we reach the desired result.

Proofs of Theorem 2.3 and its Consequences
For the proof of Corollary 2.7, we need the following version of [16, Proposition 7.1] in the abstract case for which we briefly sketch its proof. Proposition 6.1. Let M be an abstract strongly pseudoconvex CR manifold of hypersurface type, of CR dimension n, and M ⊂ C n +1 be (connected) C ∞ -smooth Levi-nondegenerate of signature , n > n ≥ 1. Assume that there exists a point p ∈ M and a germ at p of strictly noncharacteristic CR map h : (M, p) → M of class C 2 satisfying the following: there exists a neighborhood V ⊂ M of p, and for every ξ ∈ V, a smooth complex curve Υ ξ in C n +1 containing h(ξ), depending in a C 1 manner on ξ ∈ V, such that the order of contact of Υ ξ with M at h(ξ) is greater or equal to 3. Then necessarily n < n − < n .
Sketch of proof of Proposition 6.1. The proof of [16,Proposition 7.1] is obtained by adapting the arguments of [19,Proposition 3.1]. We claim that the same arguments may be used in the present situation (of an abstract strongly pseudoconvex CR manifold M of hypersurface type). Indeed, using the fact that there exists an integrable strongly pseudoconvex CR structureV on M near p whose CR bundle agrees with V to infinite order at p (see [23,Theorem IV.1.3]), one obtains, analogously to what is done in [4], first order normalization conditions for the map h at p. Once these normalizations are obtained, the proof of [19,Proposition 3.1] can be carried out in the same way (with obvious adjustments) to prove Proposition 6.1 (see also [15,Lemma 6.7] for similar arguments). We leave the details to the reader.
Proof of Corollary 2.7. By [3, Theorem 2.9], M satisfies the microlocal extension property (at every point). Furthermore, without loss of generality, we may assume that M is connected. If there is point on M whose Levi form has one positive and one negative eigenvalue, then the same holds at every point on M , and by [3, Theorem 2.9] and Theorem 4.7, h is smooth all over M . Hence, we may assume that M is strongly pseudoconvex. Now using again the fact that there exists an integrable strongly pseudoconvex CR structureV on M near p whose CR bundle agrees with V to infinite order at p, one can show that, since h is strictly noncharacteristic, h must be CR immersive too (see [4]). Therefore, as in the proof of Corollary 2.6, we have that e 1 ≥ n + 1. Applying Theorem 2.3 with k = 1 and = n + 1, we get that if (SingSupp h) • were not empty, there would exist a nonempty open subset V of M and a family of formal holomorphic curves Γ q , for q ∈ V , depending on a C 1 -smooth and CR manner on q, such that h(q) ∈ Γ q and Γ q is formally contained in M (at h(q)). From such a family, we easily get another family Υ q , for q ∈ V , of smooth complex curves passing through h(q), depending in a C 1 manner on q, that are tangent to order ≥ 3 to M at h(q). It follows from Proposition 6.1 that n < n − , a contradiction. The proof is complete.

Proof of Corollary 2.5
We shall prove Corollary 2.5 by establishing the following more general result, which contains Corollary 2.5 as a special case for k = 1. First let us recall that an abstract CR manifold M is called k-finitely nondegenerate at a point p ∈ M , σ ∈ Z + , if the Lie derivatives LK 1 · · · LK j ϑ(p), j ≤ k, ϑ ∈ Γ(M, T 0 M ),K ν ∈ Γ(M, V ) span T p M , and it is called k-finitely nondegenerate if it is k-finitely nondegenerate at each point. Furthermore, M is 1-finitely nondegenerate if and only if it is Levinondegenerate (see [1] for more on this).