On the range of unsolvable systems induced by complex vector fields

We provide criteria for local unsolvability of first-order differential systems induced by complex vector fields employing techniques from the theory of locally integrable structures. Following Hörmander’s approach to study locally unsolvable equations, we obtain analogous results in the differential complex associated to a locally integrable structure provided that it is not locally exact in three different scenarios: top-degree, Levi-nondegenerate structures and co-rank 1 structures.


Introduction
The existence of linear partial differential operators that fail to be locally solvable (see the classic Lewy [17] and Mizohata [19] operators and Treves' example [28,29] for an operator with real coefficients) gave rise to two major problems in mathematical analysis.The first one is to find necessary and sufficient conditions for local solvability (see for instance Hörmander's condition [13,14] and the striking Nirenberg-Treves (P)-condition [22]) and the second one is to understand to which extent unsolvable operators are bad behaved, i.e., to describe or give some information about the range of them (see [12,21] for characterizations of the ranges of the Lewy and Mizohata operators, respectively; [7] for a characterization of the range of the Lewy complex; [26] for a characterization of the range of generalized Mizohata operators).Here we address the second problem for certain general systems of first-order linear operators, namely systems induced by locally integrable structures over a C ∞ -smooth manifold.Solvability results for such systems were extensively studied (see for instance [2,6,8,9,18,22,32]).In the present article we study analogues of a classical theorem of Hörmander (see Theorem 6.2.1 of [15]) which states that if P(x, D) and Q(x, D) are firstorder linear partial differential operators on a given open set ⊂ R N with coefficients in C ∞ ( ) and C 1 ( ), respectively, C = [P, P], and if, at a given point x 0 ∈ , the following conditions hold: (i) There is ξ ∈ R N such that (ii) The equation has a solution u ∈ D ( ) where C 1 is term of order 1 in C, then, there is a constant μ ∈ C such that t Q(x 0 , D) = μ t P(x 0 , D).Notice that under condition (1) the operator P(x, D) cannot be locally solvable (see Theorem 6.1.1 of [15] or Theorem 1 of [13]).Hörmander's Theorem was generalized by Wittsten [34] and Dencker-Wittsten [11] to the setting of pseudodifferential operators of principal type and microlocal solvability (condition ( 1) is thus replaced by the failure of Nirenberg-Treves ( )-condition introduced in [23][24][25]).In the context of locally integrable systems, we consider different unsolvability conditions to provide new analogues of that result, namely, we consider unsolvability of the differential complex induced by a locally integrable structure in three scenarios: (i) in top-degree, via the failure of Cordaro-Hounie (P n−1 )-condition introduced in [8]; (ii) in Levi-nondegenerate structures: in [2] (see Section 5.17 (c)), it is shown that if the Levi form at a point p 0 of a hypersurface M ⊂ C n+1 has q positive eigenvalues and n − q negative eivenvalues, then the ∂b -complex in M is not locally exact at p 0 in degrees (0, q) and (0, n − q), this result was extended to higher codimensional generic submanifolds of complex manifolds in [1] (see Theorem 3) and Treves extended it to locally integrable structures in his book [33] (see Theorem VIII.3.1);(iii) in co-rank 1 structures, via Treves condition on the homology of fibers of first integrals introduced in [31].
The Cordaro-Hounie (P n−1 )-condition asserts that locally around a point the real part of solutions of the homogeneous equations of a locally integrable system cannot have compact sets as "peak-sets", i.e., level sets of a minimum.We prove, for instance, that the replacement of condition (1) by the negation of Cordaro-Hounie (P n−1 )-condition (see Definition 4.1) entails the following theorem (see Sect. 4).
Theorem 1.1 Let {L j : 1 ≤ j ≤ n} be a C ∞ -smooth locally integrable system of vector fields in the open neighborhood ⊂ R n+m of the origin.Let Q be first-order linear partial differential operator of with C ∞ -smooth coefficients.If for every open neighborhood U ⊂ of the origin there is another neighborhood V ⊂ U such that for every f ∈ C ∞ (U ) there exists u j ∈ D (V ), 1 ≤ j ≤ n, solving the equation n j=1 L j u j = Q f , in V , and (P n−1 )-condition fails at the origin, then there exists μ j ∈ C, 1 ≤ j ≤ n, such that When we are dealing with a single locally integrable complex vector field, the Cordaro-Hounie (P 0 )-condition is equivalent to Nirenberg-Treves (P)-condition (see [8,32]).Therefore, Theorem 1.1 extends Hörmander's Theorem for vector fields replacing condition (1) by the negation of Nirenberg-Treves (P)-condition.

Preliminaries
In the present section we are going to describe and fix the notation used in the article.
Throughout the paper, we denote by M an abstract C ∞ -smooth manifold of real dimension N and by V an involutive C ∞ -smooth subbundle of CTM (the complexified tangent bundle of M) of rank n.We denote by T ⊂ CT * M the annihilator bundle of V. Let p and q be non-negative integers.We denote by T p,q the subbundle of the exterior power bundle p+q CT * M whose fibers are given by T p,q The inclusion T p+1,q−1 ⊂ T p,q allow us to define the ( p, q)-bundle associated to V. It is just the quotient bundle p,q (with the convention T p,−1 = 0, thus 0,0 = T 0,0 = C is the trivial bundle).Involutivity means that the exterior derivative of a section of T p,q is a section of T p,q+1 , therefore, it induces mappings acting on C ∞ -smooth sections of p,q over any open set Since De Rham exterior derivative d is a differential operator, the induced map d is a differential operator acting on sections of the ( p, q)-bundle.We have d for each open set ⊂ M and each p.We have thus defined the differential complex associated to V. We may also consider distribution-sections of the ( p, q)-bundle, i.e., ( p, q)currents, and the corresponding differential complex The bundle V is a locally integrable structure if T is locally generated by exact 1-forms.A solution for V is a function u such that du is a section of T .Thus, the subbundle V is a locally integrable structure if and only if every point of M has a neighbourhood where m = N − n C ∞ -smooth solutions with linearly independent differentials are defined, i.e., the maximum number of solutions with linearly independent differentials do exist around every point of M. Any such set of solutions is called a full set of basic solutions for V. From now on, we denote by m = Rank T the co-rank of V.The following proposition gives us special local coordinates for every locally integrable structure.In this text, any set of local coordinates with the properties of Proposition 2.1 is called a set of coarse regular coordinates.The reader may consult Corollary I.10.2 in [5] or section I.7 of [33] for a proof of it.
Proposition 2.1 Let p 0 ∈ M be given.There exists a local chart centered at p 0 with coordinates over an open neighbourhood of the origin ⊂ R N , and a C ∞ -smooth function define a full set of basic solutions for V in the local chart.Moreover, one has a local frame for V in a possibly smaller neighbourhood of the origin in the coordinate chart given by the vector fields where is the complex vector field characterized by the conditions and (L 1 , . . ., L n , M 1 , . . ., M m ) is a local frame for CTM consisting of commuting vector fields.
For any involutive structure V (locally integrable or not) the characteristic set at a point p 0 ∈ M is defined as the real vector space The dimension of T 0 p 0 may vary with p 0 .In the following, we state a sharper version of Proposition 2.1 that encompasses information on the characteristic set.In this text, any set of local coordinates with the properties of Proposition 2.2 is called a set of fine regular coordinates.The reader may consult Theorem I.10.1 in [5] or section I.7 of [33] for a proof of it.Proposition 2.2 Let p 0 ∈ M be given and let d = dim T 0 p 0 .There exists a local chart centered at p 0 with coordinates over an open neighbourhood of the origin ⊂ R N , and there exists a C ∞ -smooth function = (ϕ 1 , . . ., ϕ d ) : → R d with (0) = 0 and D (0) = 0 such that the functions define a full set of basic solutions for V in the local chart.Thus, n = ν + μ.Moreover, one has T 0 0 = ds 1 | 0 , . . ., ds d | 0 , and a local frame for V in a possibly smaller neighbourhood of the origin in the coordinate chart is given by and is the complex vector field characterized by the conditions In the context of Proposition 2.2, set where I m is the m × m identity matrix and and since ∂ /∂x(0) = 0 we can solve it for M locally around the origin.The same computation holds for the coefficients μ kr in Proposition 2.2.
If V is locally integrable, one can apply Proposition 2.1 or Proposition 2.2 to describe local sections of p,q and the action of d on them as follows.Let us assume that we have the local coarse-regular coordinates (x, t) of Proposition 2.1 near a fixed point of M. Consider a ( p + q)-form where the sum is carried over the set of all ordered multi-indexes I , J of length p and q, respectively, and for I = (i 1 , . . ., i p ), J = ( j 1 , . . ., j q ) we write The form f belongs to T p,q , thus it represents a section [ f ] of p,q .Conversely, every section of p,q is represented by a unique such ( p + q)-form.The representative of d [ f ] is given by The operator L so defined verifies L • L = 0.One can define a Fréchet space structure on C ∞ ( , p,q ) by means of the semi-norms where K ⊂ is compact and r ∈ Z + (in the sum, we have (α, β) ∈ Z m + × Z n + ).Analogously, in the fine-regular coordinates (x, y, s, t) of Proposition 2.2, the sections of p,q are uniquely represented by forms of the following kind and the representative of d The same Fréchet space structure on C ∞ ( , p,q ) is defined by the semi-norms where K ⊂ is compact and r ∈ Z + (in the sum, we have (α, β, γ, δ) The uniquely determined representatives of sections of p,q given above are called standard representatives of sections of the ( p, q)-bundle in regular coordinates.From now on we identify sections of the ( p, q)-bundle with their standard representatives.
Let ω ∈ T p,q p 0 and η ∈ T m− p,n−q p 0 , for fixed p 0 ∈ , be given.
, then ω ∧ η = 0 (indeed, we have T m+1,q = T p,n+1 = 0 for all p, q).Therefore, we have a well-defined product ∧ : p,q × m− p,n−q → m,n = T m,n using representatives.We consider over ⊂ R N the Lebesgue measure dx dt (or dx dy ds dt if the chosen regular coordinates are fine).We have, thus, bilinear forms These bilinear forms extend to bilinear forms acting on currents ) (and vice-versa) and identify E ( , m− p,n−q ) with the dual of C ∞ ( , p,q ) (and vice-versa) (see Proposition VIII.1.2 in [33]).The elements of D ( , p,q ) are called ( p, q)-currents and in regular coordinates they can be written as formal expressions such as (3) or ( 4) where the coefficients are distributions (or compactly supported distributions for E ( , therefore, the transpose of d acting on ( p, q)-forms with respect to the duality pairing above is (−1) p+q+1 d .

Definition 3.1 Let ⊂ M be an open set. A linear partial differential operator
Furthermore, we say that V is locally weakly Q-exact at the point p 0 ∈ if it preserves d -closed forms and for every open neighbourhood p 0 ∈ U ⊂ there is a smaller open neighbourhood p 0 ∈ V ⊂ U with the following property: The following proposition is the analogue of Lemma VIII.1.1 in [33] in the context of solvability in the range.

linear partial differential operator that preserves d -closed forms defined on an open domain of regular coordinates
U ,V -condition holds, then for every compact set K ⊂ V there is a compact set K ⊂ U and there are constants C > 0, r ∈ Z + such that the following estimate holds ) be the subspace of all the d -closed forms endowed with the closed subspace Frechét topology.Let the compact set K ⊂ V be given and set The functions are well defined and turn E into a metrizable space.The bilinear function is well defined and is separately continuous.Indeed, if v ∈ Z, then let f ∈ F be given and let u ∈ D (V , p,q−1 ) be a solution of the equation and v ∧ u is compactly supported, Stokes' formula implies thus the bilinear form is well defined.For a fixed v ∈ E 0 , the function F f → v∧Q f ∈ C is continuous.On the other hand, for a fixed f ∈ F let u ∈ D (V , p,q−1 ) be a solution of the equation d u = Q f .For the given compact set K there are constants A > 0 and r ∈ Z + such that the estimate thus the bilinear form ( 6) is separately continuous.Therefore, it is continuous by the Banach-Steinhaus theorem (see the corollary of Theorem 34.1 in [30]) and the proof is complete.
As a consequence of Proposition 3.2, we have the following (see Theorem VIII.1.1 in [33]).

linear partial differential operator that preserves d -closed forms defined on an open domain of regular coordinates
Proof Apply Proposition 3.2 with the choice K = supp v.For every ρ > 0, let We apply estimate (5).The right hand side tends exponentially to 0 as ρ → ∞.Indeed, since with some constants C K ,r , C r , c > 0. Let us assume that the coordinates in are the coarseregular (x, t) of Proposition 2.1.If we write the operator t Q acts on v ρ by an expression of the form where P I J RS is a linear partial differential operator for each (I , J , R, S).Applying Leibniz' formula we get where R I J RS is a polynomial in the ρ-variable with C ∞ -smooth functions as coefficients.Therefore we set We have removed the dependence in the ρ-parameter and the proof is complete.
Remark 3. 4 Since every section of p,n is already d -closed for all 1 ≤ p ≤ m, the conclusion of Proposition 3.3 still holds if q = n and there is a non-empty open set ω ⊂ V , and functions satisfying d h = 0 and estimates (7), i.e., in this case, the functions h and f can be defined in smaller open sets and f may be assumed to have compact support.

Unsolvability in top-degree
Let p 0 ∈ M. In this section, we consider the d -equation in top-degree where A necessary condition for the local solvability of Eq. ( 8) is given by the Cordaro-Hounie (P n−1 ) condition introduced in [8].We recall the definition below.
We say that V satisfies (P n−1 )-condition at p 0 if there is an open neighborhood U 0 p 0 with the following property: • For every open set U 1 ⊂ U 0 and every h ∈ C ∞ (U 1 ), with d h = 0, the function Re h does not assume a local minimum over any non-empty compact subset of U 1 .
In the ring of germs at p 0 ∈ M of solutions of V, we denote by m p 0 the ideal of those that vanish at p 0 .
Theorem 4.2 Let V be a locally integrable structure over an N -dimensional C ∞ -smooth manifold M and assume that V does not satisfy Cordaro-Hounie (P n−1 )-condition at a point p 0 ∈ M. Let ⊂ M be an open neighbourhood of p 0 and Proof We make use of the regular coordinate system (x, t) = (x 1 , . . ., x m , t 1 , . . ., t n ) of Proposition 2.1 centered at p 0 , thus we assume ⊂ R m × R n and we have the complete set of first-integrals {Z j : 1 ≤ j ≤ m} given in terms of those coordinates.Let U ⊂ be an open neighbourhood of the origin with det Z x (x, t) = 0 in U .Let V ⊂ U be a neighbourhood of the origin such that ( * ) m,n U ,V -condition holds.Since V does not satisfy (P n−1 )-condition at the origin, for every open neighborhood U 0 ⊂ V of the origin the following objects exist: be any function where By Remark 3.4, Proof of Theorem 1.1 Making use of the regular coordinate system (coarse version) of Proposition 2.1 in ⊂ R m ×R n around p 0 and centered at the origin, we have the local frame {L j : 1 ≤ j ≤ n} for V and taking representatives in the (m, n − 1)-and (m, n)-bundles, Eq. ( 8) translates into the following equation where f ∈ C ∞ (U ) is now identified with its single coefficient and the desired solution is an n-tuple of germs of distributions u j ∈ D (0), 1 ≤ j ≤ n.We may as well identify Q with its single entry matrix.Thus, we immediately obtain Theorem 1.1.

Remark 4.3 A peak function for a locally integrable structure
for all p ∈ U \ {p 0 }.It is clear that if V admits a peak function at p 0 , then (P n−1 )-condition at p 0 does not hold and it follows from the proof of Theorem 4.2 that t Q[Z k ] = 0 in an open neighbourhood of the origin for all 1 ≤ k ≤ m.If t Q 0 = 0, then t Q 1 must be a local section of V around p 0 .This phenomenon occurs for instance for the k-Mizohata vector field in the plane for any odd positive integer k, or any lineally convex real hypersurface of C N (see [27]).The reader may consult [4,27] for further results on peak functions in locally integrable structures related to the so-called Borel map.

Example 4.4
Let us consider the co-rank 1 structure in R n+1 , with (fine-regular) coordinates (s, t) = (s, t 1 , . . ., t n ), generated by the first-integral A frame for V is given by the following vector-fields Setting M = ∂/∂s we have a frame (L 1 , . . ., L n , M) for CTR n+1 consisting of commuting vector fields.The real part of the solution h = −i W + W 2 is Thus, condition (P n−1 ) does not hold at the origin (hence V is not locally exact at the origin in degree (0, n − 1)).Let

Unsolvability in Levi-nondegenerate structures
Our first result in dealing with an intermediate degree in the differential complex induced by V is based on a necessary condition for local exactness related with the Levi form.For a given point p 0 ∈ M and a characteristic direction ( p 0 , σ ) ∈ T 0 p 0 , the Levi form at ( p 0 , σ ) is the Hermitian form in V p 0 given by where v, v ∈ V p 0 and L, L are any sections of V in a neighborhood of p 0 such that L| p 0 = v, L | p 0 = v .When V is a CR-structure, the map σ → L ( p 0 ,σ ) can be identified with the usual Levi map in V p 0 (as it is given for instance in section 2.2 of [3]).In [2] (see Section 5.17 (c)), it is shown that if the Levi form at a point p 0 of a hypersurface M ⊂ C n+1 has q positive eigenvalues and dim V p 0 − q = dim CR M − q = n − q negative eivenvalues, then the differential complex associated to M is not locally exact at p 0 in degrees (0, q) and (0, n − q).This result was extended to higher codimensional generic submanifolds of complex manifolds in [1] (see Theorem 3, p. 383).Later on (see Theorem VIII.3.1 of [33]) an analogous result in the setting of general locally integrable structures was proved by Treves, which we state below.
Theorem 5.1 ([33], p. 364) Let p 0 ∈ M and 1 ≤ q ≤ n be given and let ( p 0 , σ ) be a direction in the characteristic set of V. Suppose the following condition holds ( * ) q σ The Levi form of V at ( p 0 , σ ) has q positive eigenvalues and n − q negative eigenvalues, and its restriction to Then the differential complex associated to V is not locally exact at p 0 in degree (0, q) (and thus also in degree (m, n − q)).
In order to state the result of the present section, we need to introduce some notation.Let ⊂ M be an open neighbourhood of p 0 and let be a first-order linear partial differential operator.We denote by t Q 1 ( p 0 , σ ) the principal symbol of t Q at ( p 0 , σ ).It is the linear map where λ is any local section of m,n−q near p 0 with λ| p 0 = v and g is any C ∞ -smooth function vanishing at p 0 verifying dg| p 0 = σ (the reader may consult Section 3.3 of [20] for the background on differential operators between vector bundles).We now introduce a direct sum decomposition of m,n−q p 0 to state in an invariant way the theorem of this section.We denote by V + p 0 and V − p 0 the positive and the negative space of L ( p 0 ,σ ) , respectively.If the Levi form at ( p 0 , σ ) is non-degenerate, it induces the following identification We single out the factor m T p 0 ⊗ n−q V − p 0 and denote by ι − and π − the natural inclusion and projection maps Analogously, we may consider a first-order linear partial differential operator and if L ( p 0 ,σ ) is non-degenerate, we have the identification 0,q p 0 j+ =q and we may single out the factor q V + p 0 and denote by i + and π + the natural inclusion and projection maps Under the notation above we may state our result for this section.
Theorem 5.2 Let p 0 ∈ M be a fixed point and let ⊂ M be an open neighbourhood of p 0 .Let 1 ≤ q ≤ n be an integer and let be a first-order linear partial differential operator that preserves d -closed forms.Let ( p 0 , σ ) be a direction in T 0 p 0 .If the ( * ) q σ -condition hold and V is locally weakly Q-exact at p 0 , then Analogously, for a given first-order linear partial differential operator that preserves d -closed forms, if ( * ) q σ -condition hold and V is locally Q-exact at p 0 , then To prove Theorem 5.2, we start by following closely the proof of Theorem 5.1 given in [33].Then we proceed applying Hörmander's scheme in his proof by choosing suitable phases and forms.
Proof We make use of the regular coordinate system (fine version) (see Proposition 2.2) Recall that in our coordinate system we have W k (x, y, s, t) = s k + iϕ k (x, y, s, t), 1 ≤ k ≤ d, and we may write σ = d k=1 σ k ds k | 0 , where σ k ∈ R for 1 ≤ k ≤ d.The Levi form of V at (0, σ ) in this coordinate system is given by where L, L are any sections of V in a neighborhood of 0 such that

We choose the following basis for
Thus, we may express L (0,σ ) as a block matrix After a real linear change of coordinates in the s-variable we may assume without loss of generality σ = ds d | 0 .Since the restriction of L (0,σ ) to V 0 ∩ V 0 is non-degenerate, after another linear change of coordinates we may assume (see the proof of Proposition I.9.1 of [33])

123
where the numbers 0 ≤ λ ≤ μ and 0 ≤ κ ≤ ν verify κ + λ = q.Thus, in our coordinate system, the positive and negative spaces of L (0,σ ) are given by We denote to simplify the expressions below.The isomorphism between V 0 and V * 0 induced by L (0,σ ) identifies ∂/∂ z j | 0 with ±dz j | 0 and ∂/∂t | 0 with ±dt | 0 for each 1 ≤ j ≤ ν and 1 ≤ ≤ μ.Thus, the spaces V + 0 and V − 0 are identified with the subspaces and of V * 0 , respectively.Therefore, the vector dZ ∧ dW ∧ dZ ∧ dt | 0 spans m T 0 ⊗ n−q V − 0 under the identification induced by L (0,σ ) .Let τ > 0 be small number to be chosen later and consider the change of scale One may take as first-integrals of the push-forward * V the functions

Since ( * )
q σ -condition is invariant under positive multiples of σ , we may delete all the tildes (i.e.rename the variables and the functions) to obtain Let ε > 0 be another small number and define and in U .We set Thus and for a small constant a > 0 we have and thus it suffices to apply again the fact 123 We need to reduce once more the neighbourhood U in order to later apply Lebesgue's dominated convergence theorem.In the following, we assume ρ > 1 and that the domain is an open ball centered at the origin.We have and since in the reduced U .We reduce it again to ensure in U .Now we are ready to apply the hypothesis of local Q-exactness at the origin: there exists an open neighbourhood of the origin V U such that for every d -closed f ∈ C ∞ (U , 0,q ) there is a solution u ∈ D (V , 0,q−1 ) to the equation We apply Proposition 3.2 to the pair (U , V ).Thus, for every compact K ⊂ V there is a compact set K ⊂ U and constants C > 0 and r ∈ Z + such that be a cut-off function that equals 1 in a neighborhood of the origin and set ψ = χ W 2 d .We define for each ρ > 1.
We apply the estimate (12) choosing K = supp χ.For the right-hand side we have d f ρ = 0 thanks to (10) and Identities (10) also imply for some constant b > 0. Therefore, the right-hand side of (12) goes to 0 as ρ → ∞.
For any given section v ∈ C ∞ (U , m,n−q ), we may write so the operator t Q acts on v by an expression of the form where P I J RS is a first-order linear partial differential operator for each (I , J , R, S).
for I 0 = (κ + 1, . . ., ν) and J 0 = (λ + 1, . . ., μ), since dZ = dz I 0 and dt = dt J 0 .For each (R, S) we split in its principal part and its zeroth-order part, thus Setting P I 0 J 0 I 0 J 0 = P, we have where the ±-sign of the integral comes from reordering of 1-forms and is immaterial in the following.If we change scale dx dy ds dt = ρ m+n 2 dx dy ds dt, we get (after renaming of variables by removing primes) where Notice that since W d (0) = 0, we have α(0) = γ (0) = 0 and since we also have β(0) = 0.For any X , Y in the set otherwise.(11), we can apply Lesbegue's Dominated Convergence Theorem, to conclude

If we write
and the proof is complete.
Example 5. 3 Let us consider the co-rank 1 structure in R 3 , with (fine-regular) coordinates (s, t) = (s, t 1 , t 2 ), generated by the first-integral A frame for V is given by the following vector-fields and setting M = ∂/∂s we have a frame (L 1 , L 2 , M) for CTR 3 consisting of commuting vector fields.The matrix of the Levi form at (0, ds) with respect to the basis Thus, condition ( * ) 1 ds of Theorem 5.1 holds (hence V is not locally exact at the origin in degree (0, 1)).Let a, b and c be a solutions for V with c(0) = 0.The operator preserves d -closed forms.Theorem 5.2 ensures that d is not locally Q-exact at the origin in degree (0, 0), i.e., there exists a d -closed form f = f 1 dt 1 + f 2 dt 2 such that the system of equations does not admit any distribution solution u.

Unsolvability in co-rank 1 structures
Our last section deals with another scenario where a necessary condition for local exactness in the differential complex associated with a locally integrable structure is known: the co-rank 1 case.This condition was introduced in [10] for locally integrable structures of hypersurface type.In order to state it properly we recall some definitions in the particular case of co-rank 1 structures (we refer the reader to [10] or sections VIII.4-6 of [33] for more details).Let V be a locally integrable structure of rank n = N − 1 on a C ∞ -smooth N -manifold M and let p 0 ∈ M be a distinguished point in M. We are going to assume that the structure V is not elliptic at p 0 (otherwise the associated differential complex is locally exact at p 0 , see section VI.7 of [33]).Let W be a local C ∞ -smooth solution of V defined in an open neighbourhood ⊂ M of p 0 with T 0 p 0 = dW | p 0 .Let w 0 ∈ C be a regular value of W and let S = W −1 (w 0 ) be the corresponding level set, thus S is a C ∞ -smooth (N −2)-submanifold of M. As a consequence of Baouendi-Treves approximation formula the germs of S at its points are invariants of the locally integrable structure V (see Corollary II.3.1 of [33]).We are going to associate to every pair V ⊂ U ⊂ of open neighbourhoods of p 0 , every level set S and every degree (0, q) a relative intersection number I q U ,V ,S : H q (S ∩ U ) × H q (S ∩ V ) → C where H * and H * denote the reduced singular cohomology and homology (with complex coefficients), respectively.By Poincaré duality, we identify singular homology with the cohomology with compact support By reduced we mean that the space H 0 (S ∩ V ) is computed as follows (see Definition 2.1 of [10]) , while the remaining homology spaces are computed in the usual fashion.Under this identification, the intersection number is defined by where the brackets denote the usual projections on equivalence classes.Under the notation above we state the main theorem of [10] in the case of co-rank 1 structures (see Theorem 2.1 of [10]).
Theorem 6.1 Let V be a locally integrable structure of rank n = N − 1 on a C ∞ -smooth Nmanifold M and let p 0 ∈ M be a distinguished point in M. Let us assume that the structure V is not elliptic at p 0 and let W be a local C ∞ -smooth solution of V defined in an open neighbourhood ⊂ M of p 0 with T 0 p 0 = dW | p 0 .If the differential complex associated to V is locally exact at p 0 in degree (0, q), then for every open neighbourhood U ⊂ of p 0 there is another open neighbourhood V ⊂ U of p 0 such that I q−1 U ,V ,S ≡ 0 for every level set S that is noncritical in V .Remark 6. 2 The property I q−1 U ,V ,S ≡ 0 in Theorem 6.1 is equivalent to the following assertion (see Section 5 in [10]): "The natural map induced by the inclusion S ∩ V → S ∩ U vanishes".In [9], it is proved that this condition is also sufficient for local exactness in degree (0, q).
Our result for this section is based on a technical device employed in [10] to prove Theorem 6.1.We now briefly recall it.The proof of Theorem 6.1 is by contradiction: one considers a particular fundamental system U of open neighbourhoods of p 0 and assume that there is U ∈ U such that for every V ∈ U with V ⊂ U there is a level set S and closed forms [γ ]) = 0.One then constructs sections of the associated bundles (β) ∈ C ∞ (U , 0,q ) and ϒ(γ ) ∈ C ∞ c (V , 1,n−q ) (by a procedure to be described later) such that d (β) = 0 and The procedure behind the maps and ϒ is carefully built to produce also a solution h ∈ C ∞ ( ) of V such that Re h ≤ 0, on supp (β), Re h > 0, on supp d ϒ(γ ).
Finally, Lemma 3.2 in [10] (also Theorem VIII.1.1 in [33]) entails a contradiction.Since Proposition 3.3 generalizes this lemma we combine it with the proof of Theorem 6.1 (i.e.we apply the maps and ϒ) to get our criterion for local weak Q-exactness at p 0 .Before we state our result we describe how the maps and ϒ are defined (see section 4 of [10]).
By Proposition 2.2, we can choose regular coordinates denoted by (s, t) = (s, t 1 , . . ., t n ) around p 0 centered at the origin.Thus we have an open neighbourhood of the origin ⊂ R × R n and a C ∞ -smooth function ϕ : → R with ϕ(0) = 0 and Dϕ(0) = 0 such that our local solution of V is given in coordinates by From now on we identify S with S 0 via the map (s As in the proof of Proposition 3.2 of [10], there are forms g [γ ]) = 0 ensures that at least one of the integrals above does not vanish.Let us assume S 0 g − ∧ u + = 0 and set g = g − and u = u + (on the other case, one would set g = g + and u = u − ).
Let ρ > 0 be a positive real number such that we may estimate where π is the projection (s, t) → t and π 0,q is the projection q CT * U = 0,q ⊕T 1,q−1 → 0,q .We point out that 1,n−q = T 1,n−q since Rank T = 1.
In [10], it is shown that d (β) = 0 and that if the parameters ε and δ are small enough then there exists a solution h ∈ C ∞ ( ) of V such that Re h ≤ 0, on supp (β), Re h > 0, on supp d ϒ(γ ).Theorem 6.3 Let V be a locally integrable structure of rank n = N − 1 on a C ∞ -smooth N -manifold M and let p 0 ∈ M be a distinguished point in M. Let us assume that the structure V is not elliptic at p 0 and let W be a local C ∞ -smooth solution of V defined near p 0 with T 0 p 0 = dW | p 0 .Let 1 ≤ q ≤ n be an integer and let be a first-order linear partial differential operator that preserves d -closed forms.If V is locally weakly Q-exact in p 0 , then there is a fundamental system U of neighbourhoods of p 0 such that for every U , V ∈ U with V ⊂ U and every level set S of W , noncritical in V , and for every pair of closed forms β ∈ Proof Apply Proposition 3.3.

Remark 6.4
In the definition of the maps and ϒ we have some freedom in the choice of the cut-off function G. Now we exploit this fact to provide finer necessary conditions for local weak Q-exactness.Let us consider the same coordinate system (s, t) as above and the same fundamental system of neighbourhoods of the origin U.As above, let β ∈ C ∞ S 0 ∩ U 0 , q−1 CT * S 0 and γ ∈ C ∞ c S 0 ∩ V 0 , n−q CT * S 0 be a pair of closed forms verifying I q−1 U 0 ,V 0 ,S 0 ([β], [γ ]) = 0 and u ∈ C ∞ c V 0 , n−q CT * R n , g ∈ C ∞ U 0 , q−1 CT * R n be the corresponding forms in the t-space.We may write in our coordinates where P I R is a first-order linear partial differential operator for each (I , R).Since where ε j R J ∈ {−1, 0, 1} is characterized by Therefore, letting λ → 0, we have  Therefore, if V is Q-exact at p 0 , then for every level set W −1 (s 0 + ir 0 ) with non-vanishing intersection number the integral above given in coordinates over the perturbed level set W −1 (s 0 + ir 1 ) must vanish.

Definition 4 . 1 A
real valued continuous function f defined in an open subset U ⊂ M is said to assume a local minimum over a compact set K ⊂ U if there exists a value a ∈ R and an open set U , K ⊂ U ⊂ U , allowing the following split Let U be the set of all open neighbourhoods of the origin of the form I × O ⊂ where I is a open interval around 0 ∈ R and O is an open ball centered at the origin of R n .Let U , V ∈ U be fixed neighbourhoods with V ⊂ U and w 0 = s 0 + ir 0 ∈ C be a regular value for W noncritical in V .We write V = B × V 0 ⊂ R × R n where B is an open interval around 0 ∈ R and V 0 is an open ball centered at the origin of R n and set u I (t)dt I , g = |J |=q−1 g J (t)dt J ,where the sum runs over ordered multi-indexes and the coefficients are C ∞ -smooth functions.Let 0 < δ < η /2 and 0 < ε < ρ/4 as before and choose C ∞ -smooth functions ψ, ζ : R → [0, +∞) withsupp ψ = [−δ, δ], supp ζ = [r 0 + ρ/2 − ε, r 0 + ρ/2 + ε],and > 0 in the interior of the supports and ψ = 1.Notice that for each 0 < λ < 1 the functionG λ : C z → ψ (Re z − s 0 )/λ λ ζ(Im z) ∈ [0, +∞),has support in the rectangleR = z ∈ C : |Re z − s 0 | ≤ δ, |Im z − r 0 − ρ/2| ≤ ε ,and G λ > 0 in the interior of R. For this family of choices of G we haveλ (β) = π 0,q (G λ • W )| U dW ∧ π * g = π 0,q G λ (s + iϕ(s, t)) |J |=q−1 g J (t) dW ∧ dt J , ϒ(γ ) = χ(s) dW ∧ π * u = χ(s) |I |=n−q u I (t) dW ∧ dt I ,The operator t Q acts on ϒ(γ ) by an expression of the formt Q ϒ(γ ) = |I |=|R|=n−q P I R χu I dW ∧ dt R ,