Weak q-concavity conditions for $CR$ manifolds

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(1)Annali di Matematica DOI 10.1007/s10231-017-0638-3. Weak q-concavity conditions for CR manifolds. of. Mauro Nacinovich1 · Egmont Porten2,3. 1 2 3. 4 5. Pro. Received: 2 November 2016 / Accepted: 25 January 2017 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2017. Abstract We introduce various notions of q-pseudo-concavity for abstract CR manifolds, and we apply these notions to the study of hypoellipticity, maximum modulus principle and Cauchy problems for CR functions. Keywords CR-manifolds · q-concavity conditions · CR-hypoelliptic · CR functions · Cauchy problem Mathematics Subject Classification Primary 32V20; Secondary 32V05 · 32V25 · 32V30 · 32V10 · 32W10 · 32D10 · 35H10 · 35H20 · 35A18 · 35A20 · 35B65 · 53C30. 8. Contents. 11 12 13 14 15 16 17. 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 CR-and Z-manifolds: preliminaries and notation . . 2.1 Scalar and vector-valued Levi forms . . . . . . 3 Hypoellipticity and the maximum modulus principle 4 The kernel of the Levi form and the (H) property . . 5 Malgrange’s theorem and some applications . . . . 6 Hopf lemma and some consequences . . . . . . . . 6.1 Hopf lemma . . . . . . . . . . . . . . . . . . . 6.2 The complex Hessian and the operators ddc , Pτ. Rev. 9 10. ised. 7. 6. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 1 The Witt index of a Hermitian form of signature ( p, q) is min{ p, q}.. B. Mauro Nacinovich nacinovi@mat.uniroma2.it Egmont Porten Egmont.Porten@miun.se. 1. Dipartimento di Matematica, II Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy. 2. Department of Mathematics, Mid Sweden University, 85170 Sundsvall, Sweden. 3. Instytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Kielce, Poland. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(2) M. Nacinovich, E. Porten. 29. 1 Introduction. 22 23 24 25 26 27. 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. Pro. 21. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. The definition of q-pseudo-concavity for abstract CR manifolds of arbitrary CR-dimension and CR-codimension, given in [20], required that all scalar Levi forms corresponding to non-characteristic codirections have Witt index1 larger or equal to q. Important classes of homogeneous examples (see, for example, [1,3–5,33,35]) show that these conditions are in fact too restrictive and that weaker notions of q-pseudo-concavity are needed. For example, the results on the non-validity of the Poincaré lemma for the tangential Cauchy–Riemann complex in [10,23] only involve scalar Levi forms of maximal rank. In [21], the classical notion of 1-pseudo-concavity was extended by a trace condition that was further improved in [2,18,22]. These notions are relevant to the behavior of CR functions, being related to hypoellipticity, weak and strong unique continuation, hypoanaliticity (see [38]) and the maximum modulus principle. In this paper, we continue these investigations. A key point of this approach is the simple observation that the Hermitian-symmetric vector-valued Levi form L of a CR manifold M defines a linear form on T 1,1 M = T 1,0 M ⊗ M T 0,1 M. Our notion of pseudo-concavity is the request that its kernel contains elements τ which are positive semidefinite. To such a τ, we can associate an invariantly defined degenerate elliptic real partial differential operators Pτ , which turns out to be related to the ddc operator of [32]. By consistently keeping this perspective, we prove in this paper some results on C ∞ hypoellipticity, the maximum modulus principle, and undertake the study of boundary value problems for CR functions on open domains of abstract CR manifolds, testing the effectiveness of a new notion of weak two-pseudo-concavity by its application to the Cauchy problem for CR functions. The general plan of the paper is the following. In the next section, we define the notion of Z-structure that generalizes CR structures insofar that all formal integrability and rank conditions can be dropped, while our focus is CR functions, only considered as solutions of a homogeneous overdetermined system of first-order p.d.e.’s, and set the basic notation that will be used throughout the paper. In particular, we introduce the kernel [kerL] of the Levi form as a subsheaf of the sheaf of germs of semipositive tensors of type (1, 1). In Sect. 3, we show how the maximum modulus principle relates to C ∞ -regularity and weak and strong unique continuation of CR functions. We also make some comments on generic points of non-embeddable CR manifolds, where, by using our results of [38], we can prove, in Proposition 3.5, a result of strong unique continuation and partial hypoanaliticity (cf. [47]). In Sect. 4, we show that to each semipositive tensor τ in the kernel of the Levi form we can associate a real degenerate elliptic scalar p.d.o. of the second-order Pτ . Real parts of CR functions are Pτ -harmonic, and the modulus of a CR function is Pτ -subharmonic at points. ised. 20. . . . . . . . . . . .. Rev. 19. of. 28. 6.3 Real parts of CR functions . . . . . . . . . . . . . . . . . . . . . . . 6.4 Peak points of CR functions and pseudo-convexity at the boundary . . 6.5 1-convexity/concavity at the boundary and the vector-valued Levi form 7 Convex cones of Hermitian forms . . . . . . . . . . . . . . . . . . . . . . 7.1 Convexity in Euclidean spaces . . . . . . . . . . . . . . . . . . . . . 7.2 Convex cones in the space of Hermitian-symmetric forms . . . . . . . 8 Notions of pseudo-concavity . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Convexity/concavity at the boundary and weak pseudo-concavity . . . 9 Cauchy problem for CR functions—uniqueness . . . . . . . . . . . . . . . 10 Cauchy problem for CR functions existence . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(3) Weak q-concavity conditions for CR manifolds. 95. 2 CR-and Z-manifolds: preliminaries and notation. 96. Let M be a real smooth manifold of dimension m.. 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93. 97 98. 99 100 101 102 103. 104 105. 106. Pro. 69 70. ised. 67 68. ∞ -submodule Z of the sheaf XC of Definition 2.1 A Z-structure on M is the datum of a CM M germs of smooth complex vector fields on M. It is called. • • • • •. Rev. 66. of. 94. where it is different from zero. Then, by using some techniques originally developed for the generalized Kolmogorov equation (cf. [25,26,29]), we are able to enlarge, in comparison with [2], the set of vector fields enthralled by Z. Thus, we can improve, by Theorem 4.2, some hypoellipticity result of [2], and, by Theorem 4.7, a propagation result of [22], for the case in which this hypoellipticity fails. In Sect. 5, we prove the CR analogue of Malgrange’s theorem on the vanishing of the top degree cohomology under some subellipticity condition. Our result slightly generalizes previous results of [9,30,31], also yielding a Hartogs-type theorem on abstract CR manifolds, to recover a CR function on a relatively compact domain from boundary values satisfying some momentum condition (Proposition 5.3). In Sect. 6, we use the ddc -operator of [32] to show that the operators Pτ are invariantly defined in terms of sections of [kerL] (Corollary 6.8). The Hopf Lemma for Pτ is used to deduce pseudo-convexity properties of the boundary of a domain where a CR functions has a peak point (Proposition 6.15). This leads to a notion of convexity/concavity for points of the boundary of a domain (Definition 6.4). Most of these notions can be formulated in terms of the scalar Levi forms associated with the covectors of a half-space of the characteristic bundle. Thus, in Sect. 7, we have found it convenient to consider properties of convex cones of Hermitian-symmetric forms satisfying conditions on their indices of inertia, which are preliminary to the definitions of the next section. In Sect. 8, we propose various notions of weak-q-pseudo-concavity, give some examples, and show in Proposition 8.7 that on an essentially 2-pseudo-concave manifold strong-1convexity/concavity at the boundary becomes an open condition, i.e., stable under small perturbations. This is used in the last two sections to discuss existence and uniqueness for the Cauchy problem for CR functions, with initial data on a hypersurface. In Sect. 9, after discussing uniqueness in the case of a locally embeddable CR manifold, we turn to the case of an abstract CR manifold, proving, via Carleman-type estimates, that the uniqueness results of [13,21,22] can be extended by using some convexity condition (see Proposition 9.9). In Sect. 10, an existence theorem for the Cauchy problem is proved for locally embeddable CR manifolds, under some convexity conditions.. 65. formally integrable if [Z, Z] ⊂ Z; of CR type if Z ∩ Z = 0 (the 0-sheaf); almost-CR if Z is of CR type and locally free of constant rank; quasi-CR if it is of CR type and formally integrable; CR if Z is of CR type, formally integrable and locally free of constant rank.. ∞ is a fine A Z-manifold is a real smooth manifold M endowed with a Z-structure. Since CM sheaf, Z can be equivalently described by the datum of the space Z(M) of its global sections.. When M is a smooth real submanifold of a complex manifold X, then Z(M) = {Z ∈ XC (M) | Z p ∈ T p0,1 X, ∀ p ∈ M}. 107. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(4) M. Nacinovich, E. Porten. 109 110 111 112 113. is formally integrable. Hence, Z(M) defines a quasi-CR structure on M, which is CR if the dimension of T p0,1 X ∩ CT p M is constant for p ∈ M. This is always the case when M is a real hypersurface in X. A complex embedding (immersion) φ : M → X of a quasi-CR manifold M into a complex manifold X is a smooth embedding (immersion) for which the Z-structure on M is the pullback of the complex structure of X: Z(M) = {Z ∈ XC (M) | dφ(Z p ) ∈ Tφ(0,1p) X, ∀ p ∈ M}.. 116 117. 118. 119 120. 121 122 123 124 125 126. 127. 128. Example 2.1 Let M = {w = z 1 z¯ 1 + i z 2 z¯ 2 } ⊂ C3w,z 1 ,z 2 = X. We can take the real and imaginary parts of z 1 , z 2 as coordinates on M, which therefore, as a smooth manifold, is diffeomorphic to C2z 1 ,z 2 . The embedding M → C3 yields the quasi-CR structure ∂ ∂ Z(M) = C ∞ (M) z 2 + i z1 ∂ z¯ 1 ∂ z¯ 2. Pro. 115. of. 114. on M. Then, M \ {0} is a CR manifold of CR-dimension 1 and CR-codimension 2, while all elements of Z(M) vanish at 0 ∈ M. A Z-manifold M of CR type contains an open dense subset M˚ whose connected components are almost-CR for the restriction of Z. Likewise, any quasi-CR manifold M contains an open dense subset M˚ whose connected components are CR manifolds. We shall use Ω and A for the sheaves of germs of complex-valued and real-valued alternate forms on M (subscripts indicate degree of homogeneity). Starting with the case of an almost-CR manifold M, we introduce the notation: 0,1 T p M = {Z p | Z ∈ Z(M)} ⊂ CT M, T 1,0 M = T 0,1 M, T 0,1 M = p∈M HM = H p M = {Re Z p | Z p ∈ T p0,1 M} ⊂ T M,. ised. 108. p∈M. JM : H p M → H p M,. 129. (partial complex structure),. 130. H = {Re Z | Z ∈ Z},. 131. 133. 134. 135. 136 137 138 139 140 141. π M : T M → T M/H M (projection onto the quotient), ν I(M) = {α ∈ Ω h (M, C) | α|T 0,1 M = 0}, (I is the ideal shea f ), h=1 0 H0M = H p M = {ξ ∈ T p∗ M | ξ(H p M) = {0}} ⊂ T ∗ M, p∈M 1,1 H 1,1 M = H p M = convex hull of {(Z p ⊗ Z¯ p ) | Z ∈ Z(M)} , p∈M 1,1,(r ) 1,1,(r ) Hp H M= M = {τ ∈ H p1,1 M | rank τ = r } .. Rev. 132. p∈M. Note that H M, T M/H M, H 0 M, H 1,1 M, H 1,1,(r ) M define smooth vector bundles because we assumed that the rank n of Z is constant. This n is called the CR-dimension and the difference k = m − 2n the CR-codimension of M. For a general Z-manifold, we use the same symbols T 0,1 M,. T 1,0 M,. T 0,1 M, T 1,0 M,. 142. 143. X p + i JM X p ∈ T p0,1 M, ∀X p ∈ H p M,. H M, T M/H M,. H 1,1 M,. H 1,1,(r ) M. ˚ ˚ ˚ H M, T M/H M,. ˚ H 1,1 M,. H 1,1,(r ) M˚. for the closures of. ˚ ˚ T 0,1 M, T 1,0 M,. 144. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(5) Weak q-concavity conditions for CR manifolds. 147 148. 149 150 151 152 153 154 155 156 157 158 159 160. 161 162 163. 164. Example 2.2 For the M in Example 2.1, the fiber T p0,1 M has dimension 1 at all points p of M˚ = M \ {0}, while T00,1 M = C[∂/¯z 1 , ∂/∂ z¯ 2 ] has dimension 2. By contrast, as we already observed, all elements of Z(M) vanish at 0. If F is a subsheaf of the sheaf of germs of (complex-valued) distributions on M, an element f of F is said to be CR if it satisfies the equations Z f = 0 for all Z ∈ Z(M). The CR germs of F are the elements of a sheaf that we denote by FOM . We will simply write OM for C ∞OM . We will assume in the rest of this section that M is an almost-CR manifold. The fibers of H 1,1 M are closed convex cones, consisting of the positive semidefinite Hermitian-symmetric tensors in T 0,1 M ⊗ M T 1,0 M. The characteristic bundle H 0 M is the dual of the quotient T M/H M. Let us describe more carefully the bundle structure of H 1,1,(r ) M. Set V = T p0,1 M and consider the non-compact Stiefel space Str (V ) of r -tuples of linearly independent vectors of V . Two different r -tuples v1 , . . . , vr and w1 , . . . , wr in Str (V ) define the same τ p , i.e., satisfy τ p = v1 ⊗ v¯1 + · · · + vr ⊗ v¯r = w1 ⊗ w¯ 1 + · · · + wr ⊗ w¯ r ,. of. 146. in T C M, T C M, T M, T M/H M, T C M ⊗ M T C M, T C M ⊗ M T C M, respectively.. Pro. 145. if and only if there is a matrix a = (a ij ) ∈ U(r ) (the unitary group of order r ) such that w j = i i j a j vi . In fact, the span of v1 , . . . , vr is determined by the tensor τ p , so that w j = j a j vi i for some a = (a j ) ∈ GLr (C) and ⎛ ⎞ r r r r r. i h i h ⎝ wi ⊗ w¯ i = a j a¯ j vi ⊗ v¯ h = a j a¯ j ⎠ vi ⊗ v¯ h i=1. 166 167 168. j=1 i,h=1. i,h=1. j=1. ised. 165. shows that a ∈ U(r ). Hence, H 1,1,(r ) M is the quotient bundle of the non-compact complex Stiefel bundle of r -frames in T 0,1 M by the action of the unitary group U(r ). By using the Cartan decomposition U(r ) × p(r ) (x, X ) − → x · exp(X ) ∈ GLr (C),. 169. 173. 2.1 Scalar and vector-valued Levi forms. 174. The map. 175. 176 177. 178 179. 180. 181 182. 183. Rev. 172. where p(r ) is the vector space of Hermitian-symmetric r × r matrices, we see that H 1,1,(r ) M can be viewed as a rank r 2 real vector bundle on the Grassmannian Grr (M) of r -planes of T 0,1 M. Thus, it is a smooth vector bundle when M is almost-CR.. 170 171. Z p ⊗ Z¯ p −→ −π M (i[Z , Z¯ ] p ), ∀ p ∈ M, ∀Z ∈ Z(M),. (2.1). extends to a linear map. L : H 1,1 M → T M/H M,. (2.2). that we call the vector-valued Levi form. To each characteristic codirection ξ ∈ H p0 M, we associate the Hermitian quadratic form Lξ (Z p , Z¯ p ) = L(Z p ⊗ Z¯ p ) = − ξ|i[Z , Z¯ ] p

(6) , ∀Z ∈ Z(M).. It extends to a convex function on H p1,1 M, which is the evaluation by the covector ξ of the vector-valued Levi form. Thus, the scalar Levi forms are Lξ (τ) = ξ(L(τ)), for p ∈ M, ξ ∈ H p0 M, τ ∈ H p1,1 M.. (2.3). 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(7) M. Nacinovich, E. Porten. 186. 187. 188 189. The range p M of the vector-valued Levi form is a convex cone of T p M/H p M, whose dual cone is 0p M = {ξ ∈ H p0 M | Lξ ≥ 0}. Thus, we obtain Lemma 2.3 An element v ∈ T p M/H p M belongs to the closure or the range of the vectorvalued Levi form if and only if v|ξ

(8) ≥ 0, ∀ξ ∈ H p0 M such that Lξ ≥ 0.. of. 184 185. 190. 191. 193 194. 195. It is convenient to introduce the notation:. 196. [kerL](q) = H 1,1,(q) M ∩ ker L, [kerL] =. 199 200 201 202 203 204 205 206 207 208 209 210 211 212. 215 216 217 218 219. 222. q>0. [kerL](q) .. We note that this definition is at variance with a notion that appears in the literature (see, for example, [12]), where the kernel of the Levi form consists of the (1, 0)-vectors which are isotropic for all scalar Levi forms. These vectors are related to [kerL](1) , which is trivial in several examples of CR manifolds which are not of hypersurface type and have a non-trivial [kerL]. Let Y be a generalized distribution of real vector fields on M and p ∈ U open ⊂ M. The Sussmann leaf of Y through p in U is the set ( p; Y, U ) of points p which are ends of piecewise C ∞ integral curves of Y starting from p and lying in U . We know that ( p; Y, U ) is always a smooth submanifold of U (see [17]). Let H = {Re Z | Z ∈ Z}. A Z-manifold M is called minimal at p if ( p; H, U ) is an open neighborhood of p for all U open ⊂ M and p ∈ U . (This notion was introduced in [46] for embedded CR manifolds.) In the following, by a Sussmann leaf of Z we will mean a Sussmann leaf of H. A smooth real submanifold N of M (of arbitrary codimension ) is said to be noncharacteristic, or generic, at p0 ∈ N , when T p0 N + H p0 M = T p0 M.. (2.5). If this holds for all p ∈ N , then N is a generic CR submanifold of M, of type (n − , k + ), as T p0,1 N = T pC N ∩ T p0,1 N and H p0 N = H p0 M ⊕ JM∗ (T p N )0 for all p ∈ N . To distinguish from the Levi form L of M, we write L N for the Levi form of N . A Sussmann leaf for Z which is not open is characteristic at all points. More generally, when (M) is any distribution of complex-valued smooth vector fields on M, we say that N is -non-characteristic at p0 ∈ N if T p0 N + {Re Z p0 | Z ∈ (M)} = T p0 M.. 220. 221. . Definition 2.2 We call [kerL] the kernel of the Levi form.. 213. 214. q≥0. [kerL](q), [kerL] =. ised. 198. . Rev. 197. . Remark 2.4 Note that p M need not be closed. An example is provided by the quadric M = {Re z 3 = z 1 z¯ 1 , Re z 4 = Re(z 1 z¯ 2 )} ⊂ C4 : the cone 0 M is the union of the origin and of an open half-plane.. Pro. 192. (2.4). In this terminology, non-characteristic is equivalent to Z-non-characteristic. We note that the -non-characteristic points make an open subset of N .. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small. (2.6).

(9) Weak q-concavity conditions for CR manifolds. 226 227 228. 229 230. 231. 232 233 234 235 236 237 238 239. 240. In [38], we proved that, for locally embedded CR manifolds, the hypoellipticity of its tangential Cauchy–Riemann system is equivalent to the holomorphic extendability of its CR functions. Thus, hypoellipticity may be regarded as a weak form of pseudo-concavity. The regularity of CR distributions implies a strong maximum modulus principle for CR functions (see [21, Theorem 6.2]). Proposition 3.1 Let M be a Z-manifold. Assume that all germs of CR distributions on M that are locally L 2 are smooth. Then, for every open connected subset of M, we have. of. 224 225. 3 Hypoellipticity and the maximum modulus principle. | f ( p)| < sup | f |, ∀ p ∈ , for all non-constant f ∈ OM ( ).. Proof We prove that an f ∈ OM ( ) for which | f | attains a maximum value at some inner point p0 of is constant. Assume that p0 ∈ and | f ( p0 )| = sup | f |. If f ( p0 ) = 0, then f is constant and equal to zero on . Assume that f ( p0 ) = 0. After rescaling, we can make f ( p0 ) = | f ( p0 )| = 1. 2 topology. By the hypoellipticity assumpLet E be the space OM ( ) endowed with the L loc tion, E is Fréchet. Then, by Banach open mapping theorem, the identity map E → OM ( ) is an isomorphism of topological vector spaces. In particular, for all compact neighborhoods K of p0 in , there is a constant C K > 0 such that |u( p0 )|2 ≤ C K |u|2 dλ, ∀u ∈ OM ( ). K. Applying this inequality to. f ν,. we obtain that 2ν 1≤ | f | dλ ≤ dλ.. ised. 241. 242. K. 243 244 245 246 247 248 249 250. (3.1). Pro. 223. K. The sequence { f ν } is compact in OM ( ), because, by the hypoellipticity assumption and the Ascoli-Arzelà theorem, restriction to a relatively compact subset of CR functions is a compact map. Hence, we can extract from { f ν } a sequence that converges to a CR function φ, which is nonzero because it has a positive square-integral on every compact neighborhood of p0 . We note now that |φ| is continuous and takes only the values 1, at points where | f | = 1, and 0 at points where | f | < 1. Since φ = 0, we have |φ| ≡ 1 on and hence | f | ≡ 1 on . By applying the preceding argument to p → 21 (1 + f ( p)), we obtain that |1 + f ( p)| = 2 on . Hence, Re f ≡ 1, which yields f ≡ 1, on . . 254. Proposition 3.2 Assume that. 255 256. 257 258. 259 260 261. Rev. 253. Under the assumptions of Proposition 3.1, a CR function f ∈ OM ( ) is constant on a neighborhood of any point where | f | attains a local maximum. Then, we have. 251 252. (i) all germs of CR distribution on M are smooth; (ii) the weak unique continuation principle for CR functions is valid on M. Then, any CR function f, defined on a connected open subset of M, for which | f | attains a local maximum at some point of , is constant. We recall that the weak unique continuation principle (ii) means that a CR function f ∈ OM ( ) which is zero on an open subset U of is zero on the connected component of U in .. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(10) M. Nacinovich, E. Porten. 265 266 267 268. 269 270 271. 272 273 274 275 276 277 278 279 280. 281 282 283. For a locally CR-embeddable CR manifold M, the implication (H ) ⇒ (WUC) is a consequence of [38]. In fact, (H ) implies minimality, which implies (WUC) when M is locally CR-embeddable (see [46,48]). In fact, in this case (H ) implies the strong unique continuation principle for CR functions. Proposition 3.3 Assume that M is a CR submanifold of a complex manifold X and that M has property (H ). Then, a CR function, defined on a connected open subset of M and vanishing to infinite order at a point p0 of , is identically zero in .. of. 264. Definition 3.1 We say that M has property (H ) if (i) holds, and property (WUC) if (ii) holds. We say that (H ) (or (WUC)) holds at p if it holds when M is substituted by a sufficiently small open neighborhood of p in M.. Proof Let f ∈ OM ( ). It is sufficient to prove that the set of points where f vanishes to infinite order is open in . This reduces the proof to a local statement, allowing us to assume that the embedding M → X is generic. By Nacinovich and Porten [38], any CR function f extends to a holomorphic function f˜, defined on a connected open neighborhood U of p in X. By the assumption that M → X is generic, f˜ is uniquely determined by the Taylor series of f at p in any coordinate chart and thus vanishes to infinite order at a point p ∈ U ∩. if and only if f does. Hence, f vanishes to infinite order at p if and only if f˜ vanishes on U , and this is equivalent to the fact that f vanishes identically on U ∩ . The proof is complete. . Pro. 263. When M is not locally embeddable, there should be smaller local rings of CR functions, so that in fact properties of regularity and unique continuation should even be more likely true. Let us shortly discuss this issue. Set T p∗ 1,0 M = {ζ ∈ CT p∗ M | ζ(Z ) = 0, ∀Z ∈ T p0,1 M}.. 284. 285. Note that, with. T p∗ 0,1 M = T p∗ 1,0 M = {ζ ∈ CT p∗ M | ζ( Z¯ ) = 0, ∀Z ∈ T p0,1 M},. 286. 287. the intersection. 288. 290 291. is the complexification of the fiber of the characteristic bundle and therefore different from zero, unless Z is an almost complex structure. Differentials of smooth CR functions are sections of the bundle T ∗ 1,0 M. Thus, for a fixed p, we can consider the map OM, p f −→ d f ( p) ∈ T p∗. 292. 293. 294 295. 296. 297. 298 299. 300 301. T p∗ 1,0 M ∩ T p∗ 0,1 M = CH p0 M. Rev. 289. ised. 262. 1,0. M.. (3.2). Clearly, we have. Lemma 3.4 A necessary and sufficient condition for M to be locally CR-embeddable at p is that (3.2) is surjective. We can associate with the map (3.2) a pair (ν p , k p ) of nonnegative integers, with k p = dimC {d f ( p) | f ∈ OM, p } ∩ CH p0 M, and ν p + k p = dimC {d f ( p) | f ∈ OM, p }. The numbers ν p and ν p + k p are upper semicontinuous functions of p and hence locally constant on a dense open subset M˚ of M. Thus, we can introduce Definition 3.2 We call generic the points of the open dense subset M˚ of M, where ν p and ν p + k p are locally constant.. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(11) Weak q-concavity conditions for CR manifolds. 303 304 305 306. 307 308 309 310. Proposition 3.5 Assume that M has property (H ). Then, the strong unique continuation principle is valid at generic points p0 of M. This means that f ∈ OM, p0 is the zero germ if and only if it vanishes to infinite order at p0 . Moreover, there are finitely many germs f 1 , . . . , f μ ∈ OM, p0 , vanishing at p0 , such that, for every f ∈ OM, p0 , we can find F ∈ O Cμ ,0 such that f = F( f 1 , . . . , f μ ). Proof By the assumption that p0 is generic, we can fix a connected open neighborhood U of p0 in M and functions f 1 , . . . , f μ ∈ O M (U ), vanishing at p0 , such that d f 1 ( p) ∧ · · · ∧ d f μ ( p) = 0 for all p ∈ U and d f 1 ( p), . . . , d f μ ( p) generate the image of (3.2) for all p ∈ U . Then, by shrinking U , if needed, we can assume that. of. 302. φ : U p −→ ( f 1 ( p), . . . , f μ ( p)) ∈ N ⊂ Cμ. 311. 329. 4 The kernel of the Levi form and the (H) property. 330. To a finite set Z 1 , . . . , Z r of vector fields in Z(M), we associate the real-valued vector field. 317 318 319 320 321 322 323 324 325 326 327. 331. 332. 333. 334. 335. ised. 315 316. Y0 =. r 1 [Z j , Z¯ j ]. 2i. Rev. 314. Pro. 328. is a smooth real vector bundle on a generic CR submanifold N of Cμ , of CR-dimension νp0 and CR-codimension k p0 . In fact, we can assume that Re d f 1 , . . . , Re d f μ , Im(d f 1 ), . . . , Im(d f ν ) are linearly independent on U . We can fix local coordinates x1 , . . . , xm centered at p0 with x1 , . . . , xμ+ν equal to Re f 1 , . . . , Re f μ , Im f 1 , . . . , Im f ν . By the assumption, in these local coordinates Im f ν+1 , . . . , Im f μ are smooth functions of x1 , . . . , xμ+ν and this yields a parametric representation of N as a graph of Cν × Rμ−ν in Cμ , which is therefore locally a generic CR-submanifold of type (ν, μ − ν) of Cμ . The map φ : U → N is CR, and therefore, the pullback of germs of continuous CR function on N defines germs of continuous CR function on M. If M has property (H ), then the C ∞ regularity of their pullbacks implies the C ∞ regularity of the germs on N . Thus, N also has property (H ), and, since it is embedded in Cμ , by [38], all CR functions on an open neighborhood ω0 of 0 in N are the restriction of homomorphic functions on a full open neighborhood ω˜ 0 of 0 in Cμ , with ω0 = ω˜ 0 ∩ N . Since f i = φ∗ (z i ) for the holomorphic coordinates z 1 , . . . , z μ of Cμ , we obtain that all germs of CR functions at p0 ∈ M are germs of holomorphic functions of f 1 , . . . , f μ . This clearly implies the validity at p0 of the strong unique continuation principle. The proof is complete. . 312 313. Any CR function u on M satisfies the degenerate-Schrödinger-type equation Su = 0,. S = −iY0 +. 1 2. r. with. (Z j Z¯ j + Z¯ j Z j ) = −iY0 +. j=1. 336. (4.1). j=1. (4.2) 2r. X 2j ,. (4.3). j=1. where X j = Re Z j , X j+r = Im Z j for 1 ≤ j ≤ r. In fact, by (4.1), we have S=. 337. 1 2. r. Z¯ j Z j ,. i=1. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(12) M. Nacinovich, E. Porten. 339 340 341 342 343. and thus the operator S belongs to the left ideal, in the ring of scalar linear partial differential operators with complex smooth coefficients, generated by Z(M). We note that S is of the second order, with a real principal part which is uniquely determined by τ = Z 1 ⊗ Z¯ 1 + · · · + Z r ⊗ Z¯ r ∈ (H 1,1 M), while a different choice of the Z j ’s would yield a new Y0 , differing form Y0 by the addition of a linear combination of X 1 , . . . , X 2r . If we assume that τ ∈ ker(L), then r. 344. [Z j , Z¯ j ] = L¯ 0 − L 0. i=1. 348 349 350 351 352. 353. 354 355 356 357. 358. Pro. 347. for some L ∈ Z(M), which is uniquely determined by τ modulo a linear combination with C ∞ coefficients of Z 1 , . . . , Z r . Thus, the distributions of real vector fields ⎧ ⎪ ⎨ Q1 (τ) = Re Z 1 , . . . , Re Z r , Im Z 1 , . . . , Im Z r

(13) , V1 (τ) = L(Q1 (τ)), (4.5) ⎪ ⎩ V2 (τ) = L(Q1 (τ) + Re L 0 ), are uniquely determined by τ and Z. By L(...), we indicate the formally integrable distribution of real vector fields, which is generated by the elements of the set inside the parentheses and their iterated commutators. Note that V1 (τ) ⊆ V2 (τ), and while Y0 = Im L 0 ∈ V1 (τ), the vector field X 0 = Re L 0 may not belong to V1 (τ). We also introduce, for further reference, the distributions of complex vector fields (τ ) = Z 1 , . . . , Z r

(14) and = τ∈[kerL] (τ), (4.6) ˜ ) = (τ ) + L 0

(15) and ˜ ˜ = τ∈[kerL] (τ) (τ. ised. 345 346. (4.4). of. 338. When there is a τ ∈ [kerL]( open ), we utilize (4.4) to show that the real and imaginary parts of CR functions or distributions on ⊂ M are solutions of a real degenerate elliptic scalar second-order differential equation. Indeed, if f is a CR function, or distribution, in. , then r. L 0 f = 0, Z j f = 0 ⇒ ( L¯ 0 + L 0 ) f = (Z j Z¯ j + Z¯ j Z j ) f. i=1. 360. This is a consequence of the algebraic identities r r r. 1 ¯ j + Z¯ j Z j ) − ( L¯ 0 + L 0 ) = ¯ j Z j − L0 = (Z Z j Z¯ j − L¯ 0 . Z Z j 2 i=1. 361 362. 364. 366 367 368 369. i=1. (4.7). i=1. It terms of the real vector fields X 0 = Re L 0 and X j = Re Z j , X r + j = Im Z j , for 1 ≤ j ≤ r , the linear partial differential operator of (4.7) is Pτ = −X 0 +. 363. 365. Rev. 359. 2r. X 2j ,. (4.8). i=1. which has real-valued coefficients and is degenerate elliptic according to [8]. Thus, the real and imaginary parts of a CR function, or distribution, both satisfy the homogeneous equation Pτ φ = 0. Actually, Pτ is independent of the choice of Z 1 , . . . , Z r in the representation of τ, as we will later show in Proposition 6.6, by representing Pτ in terms of the ddc operator on M. We also have (see [22]):. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(16) Weak q-concavity conditions for CR manifolds. Lemma 4.1 If u ∈ OM ( ), then Pτ |u| ≥ 0, on ∩ {u = 0}.. 371. 372 373 374. 375. (4.9). Proof On a neighborhood of a point where u = 0, we can consistently define a branch of log u. This still is a CR function, and from the previous observation, it follows that Pτ (log |u|) = Pτ (Re log u) = 0 on ∩ {u = 0}. Hence, r. 2 Pτ |u| = Pτ exp(log |u|) = |u| Pτ (log |u|) + |Z j (log |u|)|. of. 370. i=1. = |u|. 376. 381. 382 383 384. 385. there.. Pro. 379 380. |Z j (log |u|)|2 ≥ 0. i=1. 377 378. r. We can use the treatment of the generalized Kolmogorov equation in [25, §22.2] to slightly improve the regularity result of [2, Corollary 1.15]. Let us set V2 = L V2 (τ) , Y = L(V2 ; H), (4.10) τ∈[kerL]. where we use L(V2 ; H) for the V2 -Lie module generated by H, which consists of the linear combinations, with smooth real coefficients, of the elements of H and their iterated commutators with elements of V2 : L(V2 ; H) = H + [V2 , H] + [V2 , [V2 , H]] + [V2 , [V2 , [V2 , H]]] + · · ·. (4.11). Note that V2 ⊂ L(V2 ; H) and that both V2 and Y are fine sheaves.. 387. Theorem 4.2 M has property (H ) at all points p where {Y p | Y ∈ Y(M)} = T p M.. 388. 390 391. 392. Before proving the theorem, let us introduce some notation. For > 0, we denote by S (M) the set of real vector fields Y ∈ X(M) such that for every p ∈ M, there is a neighborhood U open M of p, a constant C ≥ 0, τ1 , . . . , τh ∈ [kerL](M) and complex vector fields Z 1 , . . . , Z ∈ Z(M) such that ⎛ ⎞ h . Y f −1 ≤ C ⎝ Pτ j f 0 + Z j f 0 + f 0 ⎠ , ∀ f ∈ C0∞ (U ). (4.12) j=1. 395 396. 397. 398 399 400 401. 402. i=1. The Sobolev norms of real order (and integrability two) in (4.12) are of course computed after fixing a Riemannian metric on M. Different choices of the metric yield equivalent norms (see, for example, [2,16] for technical details). Beware that the Z j in the right-hand side of (4.12) are not required to be related to those entering the definition of the Pτ j ’s. Set S(M) = S (M). (4.13). Rev. 393 394. ised. 386. 389. . >0. Theorem 4.2 will follow from the inclusion Y(M) ⊂ S(M). The following Lemmas 4.3 and 4.4 were proved in [2,21]. 2r 2 Lemma 4.3 If τ ∈ [kerL](M) and Pτ = −X 0 + i=1 X i , then X 1 , . . . , X 2r ∈ S1 (M), and for every U open M, there is a constant C > 0 and Z 1 , . . . , Z ∈ Z(M) such that ⎞ ⎛ 2r . X i f 0 ≤ C ⎝ f 0 + Z j f 0 ⎠ , ∀ f ∈ C0∞ (U ). (4.14) i=1. j=1. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(17) M. Nacinovich, E. Porten. Set V1 = L. 406 407. . τ∈[kerL] V1 (τ). and. L(V1 ; H) = H + [V1 , H] + [V1 , [V1 , H]] + [V1 , [V1 , [V1 , H]]] + · · · .. 404. 405. . Lemma 4.4 We have the inclusion L(V1 ; H) ⊂ S . To prove Theorem 4.2, we add the following lemma. 2r 2 Lemma 4.5 Let τ ∈ [kerL](M), with Pτ = −X 0 + i=1 X i . Then, [X 0 , S (M)] ⊂ S/4 (M).. 408. of. 403. (4.15). 416 417. (Q τ f | f )0 = (Q τ f | f )0 ≥ 0, ∀ f ∈ C0∞ (U ).. 411 412 413 414. 418 419 420 421 422. 423. i([Q τ , Y ] f |A f ) = ((Q τ − Q ∗τ )Y f |A f )0 + ((Q τ − Q τ ) f |Y ∗ A f )0. = (Q τ Y f |A f )0 − (Y f |Q τ A f )0 + (Q τ f |Y ∗ A f )0 − (Q τ f |Y ∗ A f )0 .. 424 425. 428. 429. 430 431 432. While estimating the summands in the last expression, we shall indicate by C1 , C2 , . . . positive constants independent of the choice of f in C0∞ (U ). Let us first consider the second and third summands. We have |(Y f |Q τ A f )0 | ≤ Y f −1 Q τ A f 1− ≤ Y f −1 (AQ τ f 1− + [A, Q τ ] f 1− ) ⎡ ⎛ ⎤ ⎞ 2r. ⎣ ≤ C1 Y f −1 ⎝Q τ f − 2 + X 2j ⎦ f + f − 2 ⎠ . A, j=1 We have. 434. 436. 437. 438. 439. 440 441. . A,. 433. 435. Rev. 426 427. (). This is the single requirement for our choice of c. In [2], it was shown that [X i , S ] ⊂ S 2 for i = 1, . . . , 2r and all > 0. Then, (4.15) is equivalent to the inclusion [Q τ , S ] ⊂ S 4 . Let Y ∈ S (M) and U open M. We need to estimate [Q τ , Y ] f 4 −1 for f ∈ C0∞ (U ). Let A be any properly supported pseudo-differential operator of order 2 − 1. We have. ised. 410. Pro. 415. Proof Let Q τ = Pτ +c, for a suitable nonnegative real constant c, to be precised later. We decompose Q τ into the sum Q τ = Q τ + i Q τ , where Q τ = 21 (Q τ + Q ∗τ ) and Q τ = 1 Q ∗τ ) are self-adjoint. In particular, Q ∗τ = Q τ − i Q τ . We can rewrite Q τ as a sum 2i (Q τ − ∗ Q τ = − 2r j=1 X j X j + i T + c, for a p.d.o. T of order ≤ 1, whose principal part of order 1 is a linear combination with C ∞ coefficients of X 1 , . . . , X 2r . Moreover, we note that Pτ − Pτ ∗ = Q τ − Q ∗τ . The advantage in dealing with Q τ instead of Pτ is that, for c positive and sufficiently large,. 409. 2r. j=1. 1−. 2r 2[X j , A]X j + [X j , [X j , A]] . X 2j = − j=1. Since [X j , A] and [X j , [X j , A]] have order 2 − 1, and Pτ and Q τ differ by a constant, we obtain 2r |(Y f |Q τ A f )0 | ≤ C2 Y f 2 −1 Pτ f − 2 + f − 2 + X j f − 2 . j=1. + Y ∗). Analogously, for the third summand we have, since (Y has order zero, ∗ ∗ ∗ |(Q τ f |Y A f )0 | ≤ Q τ f 0 AY f 0 + [Y , A] f 0 ≤ C2 ( Pτ f 0 + f 0 ) Y f 2 −1 + f 2 −1 .. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(18) Weak q-concavity conditions for CR manifolds. 442. Next we consider |(Q τ Y f |A f )0 | = |(Y f |Q τ A f )| ≤ |(Y f |AQ τ f )0 | + |(Y f |[Q τ , A] f )0 |.. 443 444. 446 447. 448 449 450 451. 452. Let us first estimate the second summand in the last expression. 2r We have Q τ = i=1 X i2 + R0 for a first-order p.d.o. R0 whose principal part is a linear combination of X 1 , . . . , X 2r . Hence,. [Q τ , A] = [R0 , A] + (2[X i , A]X i + [X i , [X i , A]]) , with pseudo-differential operators [R0 , A], [X i , A], [X i , [X i , A]] of order ≤ ( 4 − 1). Thus, we obtain ⎛ ⎞ 2r. |(Y f |[Q τ , A] f )0 | ≤ C3 Y f −1 ⎝ f − 4 + X j f − ⎠ .. of. 445. j=1. 454. 455. 456. Because of (∗), we have the Cauchy inequality ! |(Q τ f 1 | f 2 )| ≤ (Q τ f 1 | f 1 ) (Q τ f 2 | f 2 ), for f 1 , f 2 ∈ C0∞ (U ). Hence,. |(Y f |AQ τ f )0 |2 = |(Q τ f |A∗ Y f )0 |2 ≤ (Q τ A∗ Y f |A∗ Y f )0 (Q τ f | f )0 ,. 457. |(Q τ f |Y ∗ A f )0 | ≤ (Q τ Y ∗ A f |Y ∗ A f )0 (Q τ f | f )0 .. 458. We have, for the second factor on the right-hand sides,. (Q τ f | f )0 = (Q τ f | f )0 ≤ Q τ f 0 f 0 ≤ ( Pτ f 0 + |c| f 0 ) f 0 .. 461. ised. 460. Let us estimate the first factors. We get. (Q τ A∗ Y f |A∗ Y f )0 = (Q τ A∗ Y f |A∗ Y f ) ≤ Q τ A∗ Y f − 2 A∗ Y f 2 ≤ A∗ Y f 2 A∗ Y Q τ f − 2 + [A∗ Y, Q τ ] f − 2 ≤ C3 Y f −1 Q τ f 0 + [A∗ Y, Q τ ] f − 2 .. 462. 463. 464 465 466 467. 468. We need to estimate the second summand inside the parentheses in the last expression. We note that [A∗ Y, Q τ ] = [A∗ Y, Pτ ] = −[A∗ Y, X 0 ] +. 469 470. Since the operators. 471. 472. 473. 474. 475 476. Rev. 459. 4. Pro. 453. Finally,. 2r. . 2[A∗ Y, X j ]X j + [X j , [A∗ Y, X j ]] .. j=1. X j ], [X j , [A∗ Y, X j ]] have order 2 , we obtain ⎛ ⎞ 2r. ∗ [A Y, Q τ ] f − 2 ≤ C4 ⎝ f 0 + X j f 0 ⎠ .. [A∗ Y,. X 0 ],. [A∗ Y,. j=1. (Q τ Y ∗ A f |Y ∗ A f )0 = (Q τ Y ∗ A f |Y ∗ A f ) ≤ Y ∗ A f 2 Y ∗ AQ τ f − 2 + [Q τ , Y ∗ A] f − 2 ≤ C5 Y ∗ A f 2 Q τ f 0 + [Q τ , Y ∗ A] f − 2 .. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(19) M. Nacinovich, E. Porten. 477. 478. Since [Y ∗ A, Q τ ] = [Y ∗ A, Pτ ] = −[Y ∗ A, X 0 ] +. and the operators [Y ∗ A, X 0 ], [Y ∗ A, X j ], [X j , [Y ∗ A, X j ]] have order 2 , we obtain that ⎛ ⎞ 2r. [Q τ , Y ∗ A] f − 2 ≤ C6 ⎝ f 0 + X j f 0 ⎠ .. of. 481. 2[Y ∗ A, X j ]X j + [X j , [Y ∗ A, X j ]]. j=1. 479 480. 2r. . j=1. Moreover,. Y ∗ A = −AY + (Y + Y ∗ )A + [A, Y ],. 483 484 485. with {(Y + Y ∗ )A+[A, Y ]} of order ≤ ( 2 − 1), because Y + Y ∗ has order 0. Hence, Y ∗ A f 2 ≤ C7 (Y f −1 + f 0 ) .. 486. 487. 488. Pro. 482. Putting all these inequalities together, we conclude that ⎛ |([X 0 , Y ] f |A f )0 | ≤. C8 ⎝ f 20. + Y. f 2−1. + Pτ f 20. +. 2r. ⎞. X j f 20 ⎠ ,. ∀ f ∈ C0∞ (U ).. j=1. 489 490. By taking A = 2 −1 [X 0 , Y ] for an elliptic properly supported pseudo-differential operator 2 −1 of order 2 − 1, we deduce that. 491. ised. . [X 0 , Y ] f 4 −1 ≤ C9. f 0 + Y f −1 + Pτ f 0 +. 2r. X i f 0. i=1. 492. and therefore, since X 1 , . . . , X 2r ∈ S1 (M) and Y ∈ S (M), that [X 0 , Y ] ∈ S 4 .. 493. Corollary 4.6 We have. L(V2 ; S) ⊂ S .. 494. 495. 497 498. 499. j=1. 500 501 502. (4.16) . Proof of Theorem 4.2 By the assumption, {Yq | Y ∈ S(M)} = Tq M for all q in an open neighborhood of p in M. Thus, there are p ∈ U open M, τ1 , . . . , τh ∈ [kerL](M), Z 1 , . . . , Z ∈ Z(M) and C > 0 such that ⎛ ⎞ h . f ≤ C ⎝ f 0 + Pτ j f 0 + Z i f 0 ⎠ , ∀ f ∈ C0∞ (U ). (4.17). Rev. 496. . i=1. 2r j. 2 , with Z Let Pτj = −X 0, j + s=1 X s, s, j = X s, j + i X s+r j , j ∈ Z(M) for 1 ≤ s ≤ r j , j and let Z 0, j be the vector field in Z(M) with Re Z 0, j = X 0, j . If A is a properly supported pseudo-differential operator, then. [Pτj , A] = −[X 0, j , A] +. 503. 2r j. . 2X s, j , [X s, j , A] + [[X s, j , A], X s, j ] .. s=1. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(20) Weak q-concavity conditions for CR manifolds. 506. 507. If A has order δ and is zero outside a compact subset K of U , and χ is a smooth function with compact support which equals one neighborhood of K , then we obtain Pτj A(χ f )0 ≤ A(χ Pτj f )0 + [Pτj , A](χ f )0 ⎛ ⎞ 2r j. ⎝ χ ≤ C Pτj f δ + χ f δ + X s [X s , A](χ f )0 ⎠ s=1. 508. ≤C. . χ Pτj f δ + χ f δ +. ≤C. χ Pτj f δ + χ f δ +. 512. 513. C , C , C ,. 515. 516. This shows that for any pair of functions χ1 , χ2 ∈ C0∞ (U ) with supp(χ1 ) ⊂ {χ2 > 0}, we obtain the estimate ⎛ ⎞ h . χ1 f +δ ≤ const ⎝χ2 f δ + χ2 Pτ j f δ + χ2 Z i f 0 ⎠ , ∀ f ∈ C ∞ (U ),. 520 521. 522 523 524 525 526 527 528. 529 530 531. 532 533 534 535 536. i=1. ised. 519. , ∀ f ∈ C ∞ (U ),. for some constant const = const(χ1 , χ2 ) ≥ 0. By [15], this inequality is valid for all δ,2 δ,2 δ,2 f ∈ Wloc (U ) with Pτj f, Z i f ∈ Wloc (U ), where Wloc (U ) is the space of distributions φ in ∞ U such that, for all χ ∈ C0 (U ), the product χ · φ belongs to the Sobolev space of order δ. δ,2 and integrability two. This implies in particular that any CR distribution which is in Wloc (U ) δ+,2 belongs in fact to Wloc (U ), and this implies property (H ). . Let us consider the case where L(V2 ; H) does not contain all smooth real vector fields. In this case, we have a propagation phenomenon along the leaves of V2 . Let τ ∈ [kerL](M), and X 0 , Y0 , X 1 , . . . , X 2r the vector fields introduced above for a given representation of τ = Z 1 ⊗ Z¯ 1 +· · ·+ Z r ⊗ Z¯ r . As we already noticed, while Y0 = Im [Z i , Z¯ i ] belongs to the Lie subalgebra of X(M) generated by X 1 , . . . , X 2r , the real part X 0 of L 0 = X 0 +iY0 ∈ Z(M) may not belong to V1 (τ). Thus, the following result improves [22, Theorem 5.2], where only the smaller distribution V1 (τ) was involved.. Rev. 518. χ Z s, j f δ. i=1. j=1. 517. . for suitable positive constants uniform with respect to f . By using similar argument to estimate Z i A f 0 , we obtain that ⎛ ⎞ h . A(χ f ) ≤ const ⎝χ f δ + χ Pτ j f δ + χ Z i f 0 ⎠ , ∀ f ∈ C ∞ (U ). j=1. 514. s=0. 510 511. Z s, j [X s , A](χ f )0. Pro. 509. . s=0 rj. . rj. of. 504 505. Theorem 4.7 Let open ⊂ M and assume that V2 has constant rank in . If f ∈ OM ( ) and | f | attains a maximum at a point p0 of , then f is constant on the leaf through p0 of V2 in . Proof On the integral manifold N of V2 through p0 in , we can consider the Z -structure ˆ Indeed, the CR functions defined by the span of the restrictions to N of the elements of . on restrict to CR functions for Z on the leaf N . By Corollary 4.6 and Theorem 4.2, the Z -manifold N has property (H ), and therefore, the statement is a consequence of Proposition 3.1. . 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(21) M. Nacinovich, E. Porten. 538 539 540 541 542 543 544 545. 5 Malgrange’s theorem and some applications In this section, we state the obvious generalization of Malgrange’s vanishing theorem and its corollary on the extension of CR functions under momentum conditions, slightly generalizing results of [9,30,31] to the case where M has property (SH ). In this section, we require that M is a CR manifold. We recall that the tangential Cauchy–Riemann complex can be defined as the quotient of the de Rham complex on the powers of the ideal sheaf (for this presentation, we refer to [19]): since d I ⊂ I, we have d I a ⊂ I a for all nonnegative integers a and the tangential CR-complex (Qa,∗ , ∂¯ M ) on a-forms is defined by the commutative diagram 0 −−−−→ I a+1 −−−−→ ⏐ ⏐ d#. I a −−−−→ Q a −−−−→ 0. ⏐ ⏐. d#. ⏐ ⏐. ∂¯ M #. (5.1). Pro. 546. of. 537. 0 −−−−→ I a+1 −−−−→ I a −−−−→ Q a −−−−→ 0,. 551 552 553 554 555 556 557. 558 559 560. 561 562 563 564 565 566. ised. 549 550. a is the quotient I a /I a+1 . In turn, ∂¯ where Q$ M is a degree 1 derivation for a Z-grading n a,q , where the elements of Q a,q are equivalence classes of forms having Qa = q=0 Q C . representatives in I a ∩ A a+q We denote by E the sheaf of germs of smooth complex-valued functions on M. The Q a,q are all locally free sheaves of E -modules, and therefore, we can form the corresponding sheaves and cosheaves of functions and distributions. We will consider the tangential Cauchy–Riemann complexes (D a,∗ , ∂¯ M ) on smooth forms with compact support, (E a,∗ , ∂¯ M ) = (Qa,∗ , ∂¯ M ) on smooth forms with closed support, (D a,∗ , ∂¯ M ) on form distributions, (E a,∗ , ∂¯ M ) on form distributions with compact support. We use the notation H q (F a,∗ ( ), ∂¯ M ) for the cohomology group in degree q on open ⊂ M, for F equal to either one of E , D , D , E .. Proposition 5.1 If M has property (SH ), and either M is compact or has property (WUC), then ∂¯ M : E a,0 (M) −→ E a,1 (M) and ∂¯ M : D a,0 (M) −→ D a,1 (M) have closed range for all integers a = 0, . . . , m. Proof We can assume that M is connected. It is convenient to fix a Riemannian metric on M, and smooth Hermitian products on the complex linear bundles Q a,q M corresponding to the sheaves Qa,q , to define L 2 and Sobolev norms, by using the associated smooth regular Borel measure. By property (SH ), we have a subelliptic estimate: for every K M, we can find constants C K ≥ 0, c K > 0, K > 0 such that ∂¯ M u20 + C K u20 ≥ c K u2 K , ∀u ∈ D a,0 (K ).. 567. 568. 569. 570. Rev. 547 548. (5.2). In a standard way, we deduce from (5.2) that u ∈ D. a,0. r + K a,1 ˚ r a,1 (M), ∂¯ M u ∈ [Wloc ] (M) ⇒ u| K˚ ∈ [Wloc ] ( K ), ∀K M,. (5.3). and that for all K M and real r , there are constants Cr,K ≥ 0, cr.K > 0 such that ∂¯ M ur2 + Cr,K ur2 ≥ cr,K ur2+ K ,. 571. (M) | ∂¯ M u ∈ [Wr ]a,1 (M), supp(u) ⊂ K }.. 572 573. ∀u ∈ {u ∈ E . 574. This suffices to obtain the thesis when M is compact.. a,0. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small. (5.4).

(22) Weak q-concavity conditions for CR manifolds. 579 580 581 582 583 584 585. 586 587 588. a,0. of. 577 578. Let us consider the case where M is connected and non-compact. Let {u ν } be a sequence in (M) such that all ∂¯ M u ν have support in a fixed compact subset K of M and there is r ∈ R such that {∂¯ M u ν } ⊂ [Wr ](M), supp(∂¯ M u ν ) ⊂ K for all ν and ∂¯ M u ν → f in [Wr ]a,1 (M). We can assume that M\K has no compact connected component. Then, since M has property (WUC), it follows that supp(u ν ) ⊂ K for all ν, because the u ν | M\K define elements of O M (M\K ) which vanish on a non-empty open subset of each connected component of M\K , and thus on M\K . Moreover, this also implies that (5.4) holds with Cr,K = 0. Then, {u ν } is uniformly bounded in [Wr + ]a,0 (M) and hence contains a subsequence which weakly converges to a solution u ∈ [Wr + ]a,0 (M) of ∂¯ M u = f . The closedness of the image of ∂¯ M in D a,1 (M) follows from the already proved result for a,1 E (M) and the hypoellipticity of ∂¯ M on (a, 0)-forms. E. We remind that if M is embedded and has property (H ), or is (abstract and) essentially pseudo-concave, then it has property (WUC). As in [9], one obtains. Pro. 575 576. 591. Proposition 5.2 Assume that M is a connected non-compact CR manifold of CR-dimension n which has properties (SH ) and (WUC). Then, H n (E a,∗ (M), ∂¯ M ) and H n (D a,∗ (M), ∂¯ M ) are 0 for all a = 0, . . . , m.. 592. Proof By Proposition 5.1, the sequences. 589 590. ∂¯ M. 0 −−−−→ D a,0 (M) −−−−→ D a,1 (M),. 593. ∂¯ M. 0 −−−−→ E a,0 (M) −−−−→ E a,1 (M) 595 596. 597. are exact and all maps have closed range. Assume that M is oriented. Then, we can define duality pairings between D a,q (M) and n+k−a,n−q D (M) and between E a,q (M) and E n+k−a,n−q (M), extending [α], [β]

(23) = α ∧ β,. ised. 594. M. 598 599 600. (M)∩ I a (M) has compact support and is a representative of [α]. where α ∈ A a+q ∈ D a,q (M) and β ∈ A m−a−q (M) ∩ I n+k−a (M) a representative of [β] ∈ E n+k−a,n−q (M). Then, by duality (see, for example, [43]) we obtain exact sequences ∂¯ M. Rev. 601. 0 ←−−−− D n+k−a,n (M) ←−−−− D n+k−a,n−1 (M), ∂¯ M. 0 ←−−−− E n+k−a,n (M) ←−−−− E n+k−a,n−1 (M),. 602 603 604 605. 606. proving the statement in the case where M is orientable. If M is not orientable, then we can take its oriented double covering π : M˜ → M, which is a CR-bundle with the total space M˜ being a CR manifold of the same CR dimension and codimension. From the exact sequences ∂¯ M˜. ˜ ←−−−− D n+k−a,n−1 ( M), ˜ 0 ←−−−− D n+k−a,n ( M) ∂¯. M˜ ˜ ←−− ˜ −− E n+k−a,n−1 ( M), 0 ←−−−− E n+k−a,n ( M). 607. we deduce that statement for the non-orientable M by averaging on the fibers.. 608. We also obtain the analogue of the Hartogs-type theorem in [30].. . 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(24) M. Nacinovich, E. Porten. 609 610 611. 612. Proposition 5.3 Let open M be relatively compact, orientable, and with a piecewise smooth boundary ∂ . If u 0 is the restriction to ∂ of an (a, 0)-form u˜ 0 of class C 2 on M, with ∂¯ u˜ 0 vanishing to the second order on ∂ , and u 0 ∧ φ = 0, ∀φ ∈ ker(∂¯ M : E n+k−a,n−1 (M ) → E n+k−a,n (M )), ∂. 615 616 617 618 619 620 621 622 623 624 625. 626. Proof We restrain for simplicity to the case a = 0. The general case can be discussed in an analogous way. If M is not orientable, then the inverse image of in the double covering π : M˜ → M consists of two disjoint open subsets, both CR-diffeomorphic to . Thus, we can and will assume that M is orientable. Let E be a discrete set that intersects each relatively compact connected component of ¯ in a single point and M = M \ E. Note that M has been chosen in such a way that M \. no connected component of M \ is compact. Extending ∂¯ M u˜ 0 by 0 outside of , we define a ∂¯ M -closed element f of E 0,1 (M ), with ¯ The map ∂¯ M : E 0,0 (M ) → E 0,1 (M ) has a closed image by support contained in . Proposition 5.1. Hence, to get existence of a solution v ∈ E 0,0 (M ) to ∂¯ M v = f , it suffices to prove that f is orthogonal to the kernel of ∂¯ M : E n+k,n−1 (M ) → E n+k,n (M ). This is the case because ¯ f ∧ φ = (∂ M u˜ 0 ) ∧ φ = (du 0 ) ∧ φ = u0φ − u 0 dφ. of. 614. ¯ with ∂¯ M u = 0 on and u = u 0 on ∂ . then there is u ∈ Qa,0 ( ) ∩ C 1 ( ). M. 629 630 631 632. 633 634 635 636. 637. 638 639 640 641 642 643 644 645 646 647 648 649. ∂. n+k (M ), and the last summand in the last term for all φ ∈ = AC m−1 (M ) ∩ I 0,0 ¯ vanishes when dφ = ∂ M φ = 0. A v ∈ E (M ) satisfying ∂¯ M v = f defines a CR function ¯ ¯ Thus, on M \ that vanishes on some open subset of each connected component of M \ .. ised. 628. E n+k,n−1 (M ). for (WUC) and the regularity (5.3), which are consequences of (SH ), the solution v is C 1 ¯ In particular, it vanishes on ∂ and therefore u = u˜ 0 − v satisfies the and has support in . thesis. . Remark 5.4 An analogue of this momentum theorem for functions on one complex variable states that a function u 0 , defined and continuous on the boundary of a rectifiable Jordan curve % c, is the boundary value of a holomorphic function on its enclosed domain if and only if c u 0 (z) p(z)dz = 0 for all holomorphic polynomials p(z) ∈ C[z].. 6 Hopf lemma and some consequences. Rev. 627. Pro. 613. In complex analysis, properties of domains are often expressed in terms of the indices of inertia of the complex Hessian of its exhausting function. Trying to mimic this approach in the case of an (abstract) CR manifold M, we are confronted with the fact that pluri-harmonicity and pluri-subharmonicity are well defined only for sections of a suitable vector bundle T (see [6,32,42]), which can be characterized in terms of 1-jets when M is embedded. We will avoid here this complication, by defining the complex Hessian ddc ρ as an affine subspace of Hermitian-symmetric forms on T 1,0 M. As we did for the Levi form, we shall consider its extension to H 1,1 M, and note that it is an invariantly defined function on [kerL]. Since a CR function canonically determines a section of T, we will succeed in making a very implicit use of the sheaf T of transversal 1-jets of [32]. In this section, we shall consider the Pτ of Sect. 4, exhibit their relationship to the complex Hessian, and, by using the fact that they are degenerate elliptic operators, draw, from. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(25) Weak q-concavity conditions for CR manifolds. 651. their boundary behavior at non-characteristic points, consequences on the properties of CR functions on M.. 652. 6.1 Hopf lemma. 653 654. 655 656 657. The classical Hopf Lemma also holds for degenerate elliptic operators. We have, from [14, Lemma 4.3]: ¯ R) satisfy Pτ u ≥ 0 on , for the Proposition 6.1 Let be a domain in M and u ∈ C 1 ( , 2r 2 operator Pτ = −X 0 + i=1 X j of (4.8). Assume that p0 ∈ ∂ is a C 2 non-characteristic point of ∂ for Pτ and that there is an open neighborhood U of p0 in M such that. 658 659 660. u( p) < u( p0 ), ∀ p ∈ ∩ U.. (6.1). du( p0 ) = 0.. (6.2). Then,. 663. Pro. 661. 662. of. 650. The condition that ∂ is non-characteristic at p0 for Pτ means that, if is represented by 2r ρ < 0 near p0 , with ρ ∈ C 2 and d ρ( p0 ) = 0, then i=1 |X j ρ( p0 )|2 > 0.. 666. Remark 6.2 If M has property (H ), then (6.1) is automatically satisfied if u = | f |, for ¯ when u( p0 ) is a local maximum and f is not constant on a halff ∈ OM ( ) ∩ C 0 ( ), neighborhood of p0 in .. 667. ¯ p0 ∈ ∂ with Corollary 6.3 Let be an open subset of M and f ∈ OM ( ) ∩ C 2 ( ),. 664 665. | f ( p)| < | f ( p0 )|, ∀ p ∈ .. 668. 670 671 672 673. 674. If ∂ is smooth and -non-characteristic at p0 , then d| f |( p0 ) = 0.. ised. 669. (6.3). Proof By the assumption that ∂ is -non-characteristic at p0 , the function u = | f | is, for some open neighborhood U of p0 in M, a solution of Pτ u ≥ 0 on ∩U, for an operator Pτ of the form (4.8), obtained from a section τ of [kerL](U ), and for which ∂ is non-characteristic at p0 .. 6.2 The complex Hessian and the operators dd c , Pτ. 680. Lemma 6.4 If α ∈ A 1 ( ), then we can find ξ ∈ A 1 ( ) such that α + iξ ∈ I1 ( ).. 681. Proof The sequence. 677 678. 682. 683. 684 685 686 687. Rev. 679. Denote by A1 the sheaf of germs of smooth real-valued 1-forms on M, by J1 its subsheaf of germs of sections of H 0 M and by I1 the degree 1-homogeneous elements of the ideal sheaf of M. The elements of I1 are the germs of smooth complex-valued 1-forms vanishing on T 0,1 M. Let be an open subset of M.. 675 676. i·. Re. 0 −−−−→ J1 −−−−→ I1 −−−−→ A 1 −−−−→ 0. of fine sheaves is exact and thus splits on every open subset of M.. . If ρ si a smooth, real-valued function on open ⊂ M, by Lemma 6.4, we can find ξ ∈ A1 ( ) such that d ρ + iξ ∈ I1 ( ). If Z ∈ Z(M), then d ρ(Z ) = −iξ(Z ), d ρ( Z¯ ) = iξ( Z¯ ), and we obtain Z Z¯ ρ = Z (d ρ( Z¯ )) = i Z [ξ( Z¯ )], Z¯ Z ρ = Z¯ (d ρ(Z )) = −i Z¯ [ξ(Z )].. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(26) M. Nacinovich, E. Porten. 688. Hence,. [Z Z¯ + Z¯ Z ]ρ = i(Z [ξ( Z¯ )] − Z¯ [ξ(Z )]) = idξ(Z , Z¯ ) + iξ([Z , Z¯ ]).. 689. 691. We note that ξ is only defined modulo the addition of a smooth section η ∈ J1 ( ) of the characteristic bundle H 0 M, for which. 692. idη(Z , Z¯ ) = −i η([Z , Z¯ ]) = Lη (Z , Z¯ ), ∀Z ∈ Z(M).. 693. Definition 6.1 The complex Hessian of ρ at p0 is the affine subspace ρ ρ Hess1,1 p0 ( ) = {idξ p0 | ξ ∈ A 1 ( ), d + iξ ∈ I1 ( )}.. 694. of. 690. (6.4). 698. Lemma 6.5 For every p ∈ N , we have. { ξ| N | ξ ∈ T p∗ M | dρ( p) + iξ ∈ T p∗ 1,0 M} ⊂ H p0 N .. 699. 700 701. 702 703 704. 705 706. The left-hand side of (6.5) is an affine hypersurface in H p0 M.. 708. ddc ρ p0 (τ) := 2i dξ(τ), ∀τ ∈ [kerL] p0 .. dρ(Z j ) + iξ(Z j ) = 0 ⇒ dρ( Z¯ j ) − iξ( Z¯ j ) = 0 ⇒ idξ(Z j , Z¯ j ) = i Z j ξ( Z¯ j ) − Z¯ j ξ(Z j ) − ξ([Z j , Z¯ j ]) = Z j d ρ( Z¯ j ) + Z¯ j dρ(Z j ) − iξ([Z j , Z¯ j ]). 712. = (Z j Z¯ j + Z¯ j Z j )ρ − iξ([Z j , Z¯ j ]).. 713. 717. We recall that. r. i=1 [Z j ,. Z¯ j ] = L¯ 0 − L 0 = 2i Im L 0 , with L 0 ∈ Z( ). We have. Rev. 716. (6.6) r. ¯ Let τ = Z 1 ⊗ Z¯ 1 + · · · + Z r ⊗ Z r ∈ [kerL]( ), with L¯ 0 − L 0 = i=1 [Z j , Z j ] and L 0 , Z 1 , . . . , Z r ∈ Z( ). Let ξ ∈ A 1 ( ) be such that d ρ + iξ ∈ I1 ( ). Then,. 711. 715. with associated vector space. Definition 6.2 If ρ is a smooth real-valued function defined on a neighborhood of a point p0 ∈ N and ξ ∈ A 1 ( ) is such that dr + iξ ∈ I1 ( ), then we set. 710. 714. (d ρ + iξ)(L 0 ) = 0 ⇒ d ρ(Re L 0 ) = ξ(Im L 0 ), d ρ(Im L 0 ) = −ξ(Re L 0 ) and therefore. 2ddc ρ(τ) =. =. 718. r. i=1 r. idξ(Z j , Z¯ j ) =. r. (Z j Z¯ j + Z¯ j Z j )ρ − iξ. i=1. r. [Z j , Z¯ j ]. i=1. (Z j Z¯ j + Z¯ j Z j )ρ + 2ξ(Im L 0 ). i=1. =. 719. r. (Z j Z¯ j + Z¯ j Z j )ρ − 2dρ(Re L 0 ) = 2Pτ ρ.. i=1. 720 721. (6.5). Proof When Z ∈ Z(U ) is tangent to N , we obtain 0 = dρ(Z p ) = −iξ(Z p ) and hence ξ(Re Z p ) = ξ(Im Z p ) = 0 because ξ is real. This gives ξ| N ∈ H p0 N . The last statement is a consequence of the previous discussion of the complex Hessian. . 707. 709. H p0 N ,. ised. 696. Pro. 697. Fix a point p0 where dρ( p0 ) ∈ / H p00 M, i.e., ∂¯ M ρ( p0 ) = 0, and consider the level set N = { p ∈ U | ρ( p) = ρ( p0 )}, in a neighborhood U of p0 in where ∂¯ M ρ( p) is never 0. Then N is a smooth real hypersurface and a CR-submanifold, of type (n−1, k +1).. 695. As a consequence, we obtain:. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(27) Weak q-concavity conditions for CR manifolds. 725 726. 727 728. Corollary 6.7 The operator Pτ only depends on the section τ of [kerL] and is independent of the choice of the vector fields Z 1 , . . . , Z r ∈ Z in (4.7). Corollary 6.8 Let open ⊂ M. If ρ1 , ρ2 ∈ C ∞ ( ) are real-valued functions which agree to the second order at p0 ∈ , then. of. 724. Proposition 6.6 If ρ is a real-valued smooth function on the open set of M and τ ∈ [kerL]( ), then ddc ρ(τ) = Pτ ρ on . (6.7). ddc ρ1 (τ0 ) = ddc ρ2 (τ0 ), ∀τ0 ∈ [kerL] p0 .. 729. 731. 732 733 734 735. 736. 737 738. 739 740 741. 742. In particular, ddc ρ is well defined and continuous on the fibers of [kerL] for functions ρ which are of class C 2 . Remark 6.9 There is a subtle distinction between ddc ρ, which is the (1, 1)-part of an alternate form of degree two, and Hess1,1 (ρ), which is the (1, 1)-part of a symmetric bilinear form. In fact, we multiplied by (i/2) the differential in (6.6) and identified the two concepts, as multiplication by i interchanges skew-Hermitian and Hermitian-symmetric matrices. We have:. Lemma 6.10 Let ρ be a smooth real-valued function defined on a neighborhood of p0 ∈ M, with d ρ( p0 ) = 0 and N = { p | ρ( p) = ρ( p0 )}. The following statements: ρ (i) every h ∈ Hess1,1 p0 ( ) has a nonzero positive index of inertia; (ii) there exists τ ∈ [kerL] p0 ∩ H p1,1 N such that ddc ρ p0 (τ) > 0; 0 1,1 (iii) the restriction of every h ∈ Hess p0 (ρ) to T p0,1 N has a nonzero positive index of inertia; 0 are related by. (ii) ⇐⇒ (iii) ⇒ (i).. 743. 744. Set U − = { p ∈ U | ρ( p) < ρ( p0 )}.. 745. Definition 6.3 We set. 746. (6.8). Pro. 730. ised. 722 723. 0 − HM, p0 (U ) =. . λ>0. {ξ| N | ξ ∈ T p∗0 ∂U − | λd ρ( p0 ) + iξ ∈ T ∗ 1,0 p M}.. (6.9). 749. 6.3 Real parts of CR functions. 750 751 752 753 754 755 756. 757 758. Rev. 748. 0 − This is an open half-space in H p0 N . Note that HM, p0 (U ) does not depend on the choice of the defining function ρ.. 747. In this subsection, we try to better explain the meaning of ddc by defining a differential operator dλc which associates with a real smooth function a real one form. Its definition depends on the choice of a CR-gauge λ on M, but [dλc]’s corresponding to different choices of λ differ by a differential operator with values in J, so that all the ddλc agree with our ddc on [kerL]. A CR function (or distribution) f is a solution to the equation du ∈ I1 . In this subsection, we study the characterization of the real parts of CR functions. Lemma 6.11 Let be open in M. If M is minimal, then a real-valued f ∈ OM ( ) is locally constant.. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(28) M. Nacinovich, E. Porten. 760. 761 762. Proof A real-valued f ∈ OM ( ) satisfies X f = 0 for all X ∈ (M, H M) and therefore is constant on the Sussmann leaves of (M, H M). We have an exact sequence of fine sheaves (the superscript C means forms with complexvalued coefficients). 764 765. 1,0 λ : CT M −→ T ∗ M. ¯ : CT M α −→ λ(α) ¯ ∈ T ∗ 0,1 M, λ. Pro. we have. 770. 772. Note that. (6.13). ¯ (α) = 21 α, ∀α ∈ J1C . λ(α) = λ. (6.14). ¯ = λ ¯ ◦ λ. ¯ (I1 ) ⊂ J1 , λ(I¯1 ) ⊂ J1 , λ ◦ λ λ. 774. 776. 777. Explicitly, the splitting of (6.10) is provided by. 780. Furthermore, we get. ¯ ⊕ J1 C ⊕ ker λ, I1 = ker λ ¯ ⊕ J1 C , I¯1 = J1 C ⊕ ker λ, A1C= ker λ ¯, λ(α) = α, ∀α ∈ ker λ. ¯ (d f )), ∀ f ∈ C ∞ (M). dλc f = 1i (λ(d f ) − λ. We note that dλc is real: this means that dλc u is a real-valued form when u is a real-valued function. Indeed, for a real-valued u ∈ C ∞ (M), we have ¯ (du)). dλc u = 2 Im λ(du) = −2 Im( λ. 784. 785. 786 787. Lemma 6.12 We have ddλc u ∈J2 for every u ∈ A 0 . Proof For any germ of real-valued smooth function u, the differential ddλc u is real and we have ¯ (du)) = d(2λ(du) − du) = 2d λ(du) ∈ I2 , i ddλc u = d(λ(du) − λ ¯ (du) = −2d λ ¯ (du) ∈I¯2 , = d(du − 2 λ. 788 789 790 791. (6.15). Rev. 783. ¯ (α) = α, ∀α ∈ ker λ, λ(α) = λ ¯ (α) = 21 α, ∀α ∈ J1 C . λ. Let us introduce the first-order linear partial differential operator. 781. 782. ¯. ¯. (α,β)→ λ(α)−λ(β) α→(λ(α), λ(α)) 0 −−−−→ A 1C −−−−−−−−−→ I1 ⊕ I¯1 −−−−−−−−−−−→ J1C −−−−→ 0.. 778 779. ised. 775. (6.12). ¯ (α), ∀α ∈ A 1C , α = λ(α) + λ. 771. 773. (6.11). of C-linear bundles which defines a special splitting of (6.10): with. 768. 769. (6.10). In [32, §2A], the notion of a balanced real CR-gauge was introduced. It was shown that it is possible to define a smooth morphism. 766. 767. (α,β)→α+β. α→(α,−α). 0 −−−−→ J1 C −−−−−−→ I1 ⊕ I¯1 −−−−−−−→ A1C −−−−→ 0.. 763. of. 759. so that ddλc u ∈ I2 ∩I¯2 ∩ A 2 = J2 .. . 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.

(29) Weak q-concavity conditions for CR manifolds. 792 793 794. Proposition 6.13 Let be a simply connected open set in M. A necessary and sufficient condition for a real-valued u ∈ C ∞ ( ) to be the real part of an f ∈ OM ( ) is that there exists a section ξ ∈ J1 ( ) such that d[dλc u + ξ] = 0 on .. 795. 799 800 801 802 803. Proof Assume that (6.16) is satisfied by some ξ ∈ J1 ( ). Then, dλc u + ξ = dv for some real-valued v ∈ C ∞ ( ), and with f = u + iv, we obtain. of. 797 798. If M is minimal, then ξ is uniquely determined.. ¯ (du) = i[λ(dv) + λ ¯ (dv) − ξ] ⇒ λ ¯ (d f ) = λ(du − idv) − iξ ∈ J1 C ( ) λ(du) − λ ⇒ d f ∈ I1 ( ) ⇐⇒ f ∈ OM ( ). Assume vice versa that f = u + iv ∈ OM ( ), with u and v real-valued smooth functions. ¯ (α) = 0. From Write d f = du + idv = α + ζ, with α ∈ I1 ( ), ζ ∈ J1 C ( ), and λ ¯ (du) + i λ ¯ (dv) = 21 ζ ⇒ λ(du) − i λ(dv) = 21 ζ¯, λ. 804. 805. 806. Pro. 796. (6.16). we obtain. ¯ (du) = i λ(dv) + 21 ζ¯ + i λ¯ (dv) = i dv − 21 (ζ − ζ¯) idλc u = λ(du) − λ. 812. The Aeppli complex for pluri-harmonic functions on the CR manifold M is. 808 809 810. ised. 811. This is (6.16) with ξ = (i/2)(ζ − ζ¯). To complete the proof, we note that if ξ ∈ J1 ( ) and dξ = 0, then ξ = dφ for some real-valued function φ ∈ C ∞ ( ). If ξ p0 = 0 for some p0 ∈ , then {φ( p) = φ( p0 )} defines a germ of smooth hypersurface through p0 which is tangent at each point to the distribution H M, contradicting the minimality assumption. . 807. (u,ξ)→ddλc u+dξ. 813. d. −−−−→. d. d. 0 −−−−→ A 0 ⊕ J1 −−−−−−−−−→ J2 −−−−→ J3. d. · · · −−−−→ Jm−1 −−−−→ Jm −−−−→ 0.. We note that J1 = 0 if M is a complex manifold (we reduce to the classical case) and Jq = Aq for q > 0 if M is totally real. In general, the terms of degree ≥ k +2 make a. 816. subcomplex of the de Rham complex.. 817. 6.4 Peak points of CR functions and pseudo-convexity at the boundary. 814. 818 819. 820 821 822. 823 824 825 826 827. 828. Rev. 815. A non-characteristic point of the boundary of a domain, where the modulus a CR function attains a local maximum, is pseudo-convex, in a sense that will be explained below. ¯ such that | f | Lemma 6.14 Let open ⊂ M and assume there is f ∈ OM ( ) ∩ C 2 ( ) attains a local isolated maximum value at p0 ∈ ∂ . If ∂ is smooth, non-characteristic at ∂. 0 p0 , and moreover, d| f ( p0 )| = 0, then there is a nonzero ξ ∈ HM, p0 ( ) with Lξ ≥ 0. Proof Let U be an open neighborhood of p0 in M, and ρ ∈ C ∞ (U, R) a defining function for near p0 , with U − = ∩ U = { p ∈ U | ρ( p) < 0}, and d ρ( p) = 0 for all p ∈ U . We can assume that f ( p0 ) = | f ( p0 )| > 0 and exploit the fact that the restriction of u = Re f to ∂ takes a maximum value at p0 . Since d∂ u( p0 ) = 0, the real Hessian of u on ∂ is well defined at p0 , with hess(u)(X p0 , Y p0 ) = (X Y u)( p0 ), ∀X, Y ∈ X(∂ ),. 123 Journal: 10231 Article No.: 0638. TYPESET. DISK. LE. CP Disp.:2017/2/14 Pages: 39 Layout: Small.