Abstract
We provide criteria for local unsolvability of firstorder 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: topdegree, Levinondegenerate structures and corank 1 structures.
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1 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 NirenbergTreves (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 firstorder linear operators, namely systems induced by locally integrable structures over a \(\mathcal {C}^\infty \)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 \(\Omega \subset \mathbb {R}^N\) with coefficients in \(\mathcal {C}^\infty (\Omega )\) and \(\mathcal {C}^1(\Omega )\), respectively, \(C = [\overline{P},P]\), and if, at a given point \(x_0 \in \Omega \), the following conditions hold:

(i)
There is \(\xi \in \mathbb {R}^N\) such that
$$\begin{aligned} P_1(x_0, \xi ) = 0, \quad C_1(x_0,\xi ) \ne 0, \end{aligned}$$(1) 
(ii)
The equation
$$\begin{aligned} P(x,D)u = Q(x,D)f \end{aligned}$$(2)has a solution \(u \in \mathcal {D}^\prime (\Omega )\) for every \(f \in \mathcal {C}_c^\infty (\Omega )\),
where \(C_1\) is term of order 1 in C, then, there is a constant \(\mu \in \mathbb {C}\) such that \(^tQ(x_0,D) = \mu \, ^tP(x_0,D)\). 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 DenckerWittsten [11] to the setting of pseudodifferential operators of principal type and microlocal solvability (condition (1) is thus replaced by the failure of NirenbergTreves (\(\Psi \))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 topdegree, via the failure of CordaroHounie \(\mathrm {(P}_{n1}\mathrm {)}\)condition introduced in [8];

(ii)
in Levinondegenerate structures: in [2] (see Section 5.17 (c)), it is shown that if the Levi form at a point \(p_0\) of a hypersurface \(\mathcal {M}\subset \mathbb {C}^{n+1}\) has q positive eigenvalues and \(nq\) negative eivenvalues, then the \({\bar{\partial }}_b\)complex in \(\mathcal {M}\) is not locally exact at \(p_0\) in degrees (0, q) and \((0,nq)\), 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 corank 1 structures, via Treves condition on the homology of fibers of first integrals introduced in [31].
The CordaroHounie \(\mathrm {(P}_{n1}\mathrm {)}\)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 “peaksets”, i.e., level sets of a minimum. We prove, for instance, that the replacement of condition (1) by the negation of CordaroHounie \(\mathrm {(P}_{n1}\mathrm {)}\)condition (see Definition 4.1) entails the following theorem (see Sect. 4).
Theorem 1.1
Let \(\{L_j:\, 1 \le j \le n\}\) be a \(\mathcal {C}^\infty \)smooth locally integrable system of vector fields in the open neighborhood \(\Omega \subset \mathbb {R}^{n+m}\) of the origin. Let Q be firstorder linear partial differential operator of \(\Omega \) with \(\mathcal {C}^\infty \)smooth coefficients. If for every open neighborhood \(U \subset \Omega \) of the origin there is another neighborhood \(V \subset U\) such that for every \(f \in \mathcal {C}^\infty (U)\) there exists \(u_j \in \mathcal {D}^\prime (V)\), \(1 \le j \le n\), solving the equation
in V, and \(\mathrm {(P}_{n1}\mathrm {)}\)condition fails at the origin, then there exists \(\mu _j \in \mathbb {C}\), \(1 \le j \le n\), such that
When we are dealing with a single locally integrable complex vector field, the CordaroHounie \(\mathrm {(P}_{0}\mathrm {)}\)condition is equivalent to NirenbergTreves \(\mathrm {(P)}\)condition (see [8, 32]). Therefore, Theorem 1.1 extends Hörmander’s Theorem for vector fields replacing condition (1) by the negation of NirenbergTreves \(\mathrm {(P)}\)condition.
2 Preliminaries
In the present section we are going to describe and fix the notation used in the article.
Throughout the paper, we denote by \(\mathcal {M}\) an abstract \(\mathcal {C}^\infty \)smooth manifold of real dimension N and by \(\mathcal {V}\) an involutive \(\mathcal {C}^\infty \)smooth subbundle of \(\mathbb {C}\textrm{T}\mathcal {M}\) (the complexified tangent bundle of \(\mathcal {M}\)) of rank n. We denote by \(\textrm{T}^\prime \subset \mathbb {C}\textrm{T}^*\mathcal {M}\) the annihilator bundle of \(\mathcal {V}\). Let p and q be nonnegative integers. We denote by \(\textrm{T}^{p,q}\) the subbundle of the exterior power bundle \( \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{p+q} \mathbb {C}\textrm{T}^*\mathcal {M}\) whose fibers are given by
The inclusion \(\textrm{T}^{p+1,q1} \subset \textrm{T}^{p,q}\) allow us to define the (p, q)bundle associated to \(\mathcal {V}\). It is just the quotient bundle
(with the convention \(\textrm{T}^{p,1} = 0\), thus \(\Lambda ^{0,0} = \textrm{T}^{0,0} = \mathbb {C}\) is the trivial bundle). Involutivity means that the exterior derivative of a section of \(\textrm{T}^{p,q}\) is a section of \(\textrm{T}^{p,q+1}\), therefore, it induces mappings acting on \(\mathcal {C}^\infty \)smooth sections of \(\Lambda ^{p,q}\) over any open set \(\Omega \subset \mathcal {M}\)
Since De Rham exterior derivative \(\textrm{d}\) is a differential operator, the induced map \(\textrm{d}^\prime \) is a differential operator acting on sections of the (p, q)bundle. We have \(\textrm{d}^\prime \circ \textrm{d}^\prime = 0\) because \(\textrm{d}\circ \textrm{d}= 0\), thus the \(\textrm{d}^\prime \)operator defines a differential complex
for each open set \(\Omega \subset \mathcal {M}\) and each p. We have thus defined the differential complex associated to \(\mathcal {V}\). We may also consider distributionsections of the (p, q)bundle, i.e., (p, q)currents, and the corresponding differential complex
The bundle \(\mathcal {V}\) is a locally integrable structure if \(\textrm{T}^\prime \) is locally generated by exact 1forms. A solution for \(\mathcal {V}\) is a function u such that \(\textrm{d}u\) is a section of \(\textrm{T}^\prime \). Thus, the subbundle \(\mathcal {V}\) is a locally integrable structure if and only if every point of \(\mathcal {M}\) has a neighbourhood where \(m = N  n \;\) \(\mathcal {C}^\infty \)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 \(\mathcal {M}\). Any such set of solutions is called a full set of basic solutions for \(\mathcal {V}\). From now on, we denote by \(m = \textrm{Rank} \, \textrm{T}^\prime \) the corank of \(\mathcal {V}\). 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 1.2
Let \(p_0 \in \mathcal {M}\) be given. There exists a local chart centered at \(p_0\) with coordinates
over an open neighbourhood of the origin \(\Omega \subset \mathbb {R}^N\), and a \(\mathcal {C}^\infty \)smooth function \(\Phi = (\varphi _1, \dots , \varphi _m): \Omega \rightarrow \mathbb {R}^m\) with \(\Phi (0) = 0\) and \(\partial \Phi /\partial x(0) = 0\) such that the functions
define a full set of basic solutions for \(\mathcal {V}\) in the local chart. Moreover, one has a local frame for \(\mathcal {V}\) 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 \((L_1, \dots , L_n, M_1, \dots , M_m)\) is a local frame for \(\mathbb {C}\textrm{T}\mathcal {M}\) consisting of commuting vector fields.
For any involutive structure \(\mathcal {V}\) (locally integrable or not) the characteristic set at a point \(p_0 \in \mathcal {M}\) is defined as the real vector space
The dimension of \(\textrm{T}^0_{p_0}\) 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 1.3
Let \(p_0 \in \mathcal {M}\) be given and let \(d = \dim \textrm{T}^0_{p_0}\). There exists a local chart centered at \(p_0\) with coordinates
over an open neighbourhood of the origin \(\Omega \subset \mathbb {R}^N\), and there exists a \(\mathcal {C}^\infty \)smooth function \(\Phi = (\varphi _1, \dots , \varphi _d): \Omega \rightarrow \mathbb {R}^d\) with \(\Phi (0) = 0\) and \(\textrm{D}\Phi (0) = 0\) such that the functions
define a full set of basic solutions for \(\mathcal {V}\) in the local chart. Thus, \(n = \nu + \mu \). Moreover, one has
and a local frame for \(\mathcal {V}\) in a possibly smaller neighbourhood of the origin in the coordinate chart is given by
where
and
is the complex vector field characterized by the conditions
In the context of Proposition 2.2, set
where
and we have a local frame \((L_1, \dots , L_n, M_1, \dots , M_m)\) for \(\mathbb {C}\textrm{T}\mathcal {M}\) consisting of commuting vector fields.
Remark 2.3
One can compute explicitly the coefficients \(\mu _{kr}\) of Proposition 2.1 in terms of \(\partial \Phi / \partial x\). Indeed, we have the matrix equation
where \(I_m\) is the \(m \times m\) identity matrix and
and since \(\partial \Phi / \partial x (0) = 0\) we can solve it for M locally around the origin. The same computation holds for the coefficients \(\mu _{kr}\) in Proposition 2.2.
If \(\mathcal {V}\) is locally integrable, one can apply Proposition 2.1 or Proposition 2.2 to describe local sections of \(\Lambda ^{p,q}\) and the action of \(\textrm{d}^\prime \) on them as follows. Let us assume that we have the local coarseregular coordinates (x, t) of Proposition 2.1 near a fixed point of \(\mathcal {M}\). Consider a \((p+q)\)form
where the sum is carried over the set of all ordered multiindexes I, J of length p and q, respectively, and for \(I = (i_1, \dots , i_p)\), \(J = (j_1, \dots , j_q)\) we write
The form f belongs to \(\textrm{T}^{p,q}\), thus it represents a section [f] of \(\Lambda ^{p,q}\). Conversely, every section of \(\Lambda ^{p,q}\) is represented by a unique such \((p+q)\)form. The representative of \(\textrm{d}^\prime [f]\) is given by
The operator L so defined verifies \(L \circ L = 0\). One can define a Fréchet space structure on \(\mathcal {C}^\infty (\Omega , \Lambda ^{p,q})\) by means of the seminorms
where \(K \subset \Omega \) is compact and \(r \in \mathbb {Z}_+\) (in the sum, we have \((\alpha ,\beta ) \in \mathbb {Z}_+^m \times \mathbb {Z}_+^n\)).
Analogously, in the fineregular coordinates (x, y, s, t) of Proposition 2.2, the sections of \(\Lambda ^{p,q}\) are uniquely represented by forms of the following kind
and the representative of \(\textrm{d}^\prime [f]\) is
The same Fréchet space structure on \(\mathcal {C}^\infty (\Omega , \Lambda ^{p,q})\) is defined by the seminorms
where \(K \subset \Omega \) is compact and \(r \in \mathbb {Z}_+\) (in the sum, we have \((\alpha ,\beta ,\gamma ,\delta ) \in \mathbb {Z}_+^\nu \times \mathbb {Z}_+^\nu \times \mathbb {Z}_+^d \times \mathbb {Z}_+^\mu \)). The uniquely determined representatives of sections of \(\Lambda ^{p,q}\) 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 \(\omega \in \textrm{T}^{p,q}_{p_0}\) and \(\eta \in \textrm{T}^{mp,nq}_{p_0}\), for fixed \(p_0 \in \Omega \), be given. If \(\omega \in \textrm{T}^{p+1,q1}_{p_0}\) or \(\eta \in \textrm{T}^{mp+1,nq1}_{p_0}\), then \(\omega \wedge \eta = 0\) (indeed, we have \(\textrm{T}^{m+1,q} = \textrm{T}^{p,n+1} = 0\) for all p, q). Therefore, we have a welldefined product
using representatives. We consider over \(\Omega \subset \mathbb {R}^N\) the Lebesgue measure \(\textrm{d}x \, \textrm{d}t\) (or \(\textrm{d}x \, \textrm{d}y \, \textrm{d}s \, \textrm{d}t\) if the chosen regular coordinates are fine). We have, thus, bilinear forms
given by the same expression
These bilinear forms extend to bilinear forms acting on currents
that indentify \(\mathcal {D}^\prime (\Omega ,\Lambda ^{mp,nq})\) with the dual of \(\mathcal {C}^\infty _c(\Omega ,\Lambda ^{p,q})\) (and viceversa) and identify \(\mathcal {E}^\prime (\Omega ,\Lambda ^{mp,nq})\) with the dual of \(\mathcal {C}^\infty (\Omega ,\Lambda ^{p,q})\) (and viceversa) (see Proposition VIII.1.2 in [33]). The elements of \(\mathcal {D}^\prime (\Omega ,\Lambda ^{p,q})\) 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 \(\mathcal {E}^\prime (\Omega ,\Lambda ^{p,q})\)). If f is a (p, q)form, v is an \((mp,nq1)\)form and one of them has compact support, Stokes’ Theorem implies
therefore, the transpose of \(\textrm{d}^\prime \) acting on (p, q)forms with respect to the duality pairing above is \((1)^{p+q+1}\textrm{d}^\prime \).
3 Local solvability on the range of an operator
In order to extend Hörmander’s Theorem, for the \(\textrm{d}^\prime \)operator it is necessary to impose a compatibility condition on the operator acting on the right side. Thus, we need the following definition.
Definition 1.5
Let \(\Omega \subset \mathcal {M}\) be an open set. A linear partial differential operator
preserves \(\textrm{d}^\prime \)closed forms if \(\textrm{d}^\prime Qf = 0\) for all \(f \in \mathcal {C}^\infty (\Omega , \Lambda ^{p,q})\) satisfying \(\textrm{d}^\prime f = 0\). Furthermore, we say that \(\mathcal {V}\) is locally weakly Qexact at the point \(p_0 \in \Omega \) if it preserves \(\textrm{d}^\prime \)closed forms and for every open neighbourhood \(p_0 \in U \subset \Omega \) there is a smaller open neighbourhood \(p_0 \in V \subset U\) with the following property:
 \((*)^{p,q}_{U,V}\):

“For every \(\textrm{d}^\prime \)closed \(f \in \mathcal {C}^\infty (U, \Lambda ^{p,q})\) there is a solution \(u \in \mathcal {D}^\prime (V, \Lambda ^{p,q1})\) of the equation
$$\begin{aligned} \textrm{d}^\prime u = Qf \end{aligned}$$in V”.
The following proposition is the analogue of Lemma VIII.1.1 in [33] in the context of solvability in the range.
Proposition 1.6
Let \( Q: \mathcal {C}^\infty (\Omega ,\Lambda ^{p,q}) \rightarrow \mathcal {C}^\infty (\Omega ,\Lambda ^{p,q}) \) be a linear partial differential operator that preserves \(\textrm{d}^\prime \)closed forms defined on an open domain of regular coordinates \(\Omega \subset \mathcal {M}\). If \(U,V \subset \Omega \) are nonempty open sets, \(V \subset U\), such that \((*)^{p,q}_{U,V}\)condition holds, then for every compact set \(K^\prime \subset V\) there is a compact set \(K \subset U\) and there are constants \(C > 0\), \(r \in \mathbb {Z}_+\) such that the following estimate holds
for every \(\textrm{d}^\prime \)closed form \(f \in \mathcal {C}^\infty (U,\Lambda ^{p,q})\) and every \(v \in \mathcal {C}^\infty _c(V,\Lambda ^{mp,nq})\) with \(\textrm{supp}\, v \subset K^\prime \).
Proof
Let \(F \subset \mathcal {C}^\infty (U, \Lambda ^{p,q})\) be the subspace of all the \(\textrm{d}^\prime \)closed forms endowed with the closed subspace Frechét topology. Let the compact set \(K^\prime \subset V\) 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 \(v \in \mathcal {Z}\), then let \(f \in F\) be given and let \(u \in \mathcal {D}^\prime (V,\Lambda ^{p,q1})\) be a solution of the equation \(\textrm{d}^\prime u = Qf\). Since
and \(v \wedge u\) is compactly supported, Stokes’ formula implies
thus the bilinear form is well defined. For a fixed \(v \in E_0\), the function \(F \ni f \mapsto \int v \wedge Qf \in \mathbb {C}\) is continuous. On the other hand, for a fixed \(f \in F\) let \(u \in \mathcal {D}^\prime (V,\Lambda ^{p,q1})\) be a solution of the equation \(\textrm{d}^\prime u = Qf\). For the given compact set \(K^\prime \) there are constants \(A > 0\) and \(r \in \mathbb {Z}_+\) such that the estimate
holds for every \(w \in \mathcal {C}^\infty _c(V, \Lambda ^{mp,nq+1})\). Stokes’ formula implies
thus the bilinear form (6) is separately continuous. Therefore, it is continuous by the BanachSteinhaus theorem (see the corollary of Theorem 34.1 in [30]) and the proof is complete. \(\square \)
As a consequence of Proposition 3.2, we have the following (see Theorem VIII.1.1 in [33]).
Proposition 1.7
Let \( Q: \mathcal {C}^\infty (\Omega ,\Lambda ^{p,q}) \rightarrow \mathcal {C}^\infty (\Omega ,\Lambda ^{p,q}) \) be a linear partial differential operator that preserves \(\textrm{d}^\prime \)closed forms defined on an open domain of regular coordinates \(\Omega \subset \mathcal {M}\). Let \(U,V \subset \Omega \) be nonempty open sets, \(V \subset U\), satisfying the \((*)^{p,q}_{U,V}\)condition. If there exist a solution \(h \in \mathcal {C}^\infty (U)\), a \(\textrm{d}^\prime \)closed form \(f \in \mathcal {C}^\infty (U,\Lambda ^{p,q})\) and a compactly supported form \(v \in \mathcal {C}^\infty _c(V,\Lambda ^{mp,nq})\) such that
then
Proof
Apply Proposition 3.2 with the choice \(K^\prime = \textrm{supp}\, v\). For every \(\rho > 0\), let
We apply estimate (5). The right hand side tends exponentially to 0 as \(\rho \rightarrow \infty \). Indeed, since \(\textrm{d}^\prime h = 0\) we have \(\textrm{d}^\prime v_\rho = e^{\rho h} \, \textrm{d}^\prime v\) thus
with some constants \(C_{K,r},C_r,c > 0\). Let us assume that the coordinates in \(\Omega \) are the coarseregular (x, t) of Proposition 2.1. If we write
the operator \(\,^tQ\) acts on \(v_\rho \) by an expression of the form
where \(P^{IJ}_{RS}\) is a linear partial differential operator for each (I, J, R, S). Applying Leibniz’ formula we get
where \(\mathcal {R}^{IJ}_{RS}\) is a polynomial in the \(\rho \)variable with \(\mathcal {C}^\infty \)smooth functions as coefficients. Therefore we set
where
Since \(I(\rho )/\rho ^k \rightarrow 0\), \(k \in \mathbb {Z}_+\), as \(\rho \rightarrow \infty \), we have
We have removed the dependence in the \(\rho \)parameter and the proof is complete. \(\square \)
Remark 3.4
Since every section of \(\Lambda ^{p,n}\) is already \(\textrm{d}^\prime \)closed for all \(1 \le p \le m\), the conclusion of Proposition 3.3 still holds if \(q = n\) and there is a nonempty open set \(\omega \subset V\), and functions \(h \in \mathcal {C}^\infty (\omega )\) and \(f,v \in \mathcal {C}^\infty _c(\omega )\) satisfying \(\textrm{d}^\prime h = 0\) 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.
4 Unsolvability in topdegree
Let \(p_0 \in \mathcal {M}\). In this section, we consider the \(\textrm{d}^\prime \)equation in topdegree
where \(f \in \mathcal {C}^\infty (U, \Lambda ^{m,n})\), the desired solutions \(u \in \mathcal {D}^\prime (V, \Lambda ^{m,n1})\) are \((m,n1)\)currents and \(V \subset U \subset \mathcal {M}\) are neighbourhoods of \(p_0\).
A necessary condition for the local solvability of Eq. (8) is given by the CordaroHounie \(\mathrm {(P}_{n1}\mathrm {)}\) condition introduced in [8]. We recall the definition below.
Definition 1.9
A real valued continuous function f defined in an open subset \(U \subset \mathcal {M}\) is said to assume a local minimum over a compact set \(K \subset U\) if there exists a value \(a \in \mathbb {R}\) and an open set \(U^\prime \), \(K \subset U^\prime \subset U\), allowing the following split
We say that \(\mathcal {V}\) satisfies \(\mathrm {(P}_{n1}\mathrm {)}\)condition at \(p_0\) if there is an open neighborhood \(U_0 \ni p_0\) with the following property:

For every open set \(U_1 \subset U_0\) and every \(h \in \mathcal {C}^\infty (U_1)\), with \(\textrm{d}^\prime h = 0\), the function Re h does not assume a local minimum over any nonempty compact subset of \(U_1\).
In the ring of germs at \(p_0 \in \mathcal {M}\) of solutions of \(\mathcal {V}\), we denote by \({\mathfrak {m}}_{p_0}\) the ideal of those that vanish at \(p_0\).
Theorem 1.10
Let \(\mathcal {V}\) be a locally integrable structure over an Ndimensional \(\mathcal {C}^\infty \)smooth manifold \(\mathcal {M}\) and assume that \(\mathcal {V}\) does not satisfy CordaroHounie \(\mathrm {(P}_{n1}\mathrm {)}\)condition at a point \(p_0 \in \mathcal {M}\). Let \(\Omega \subset \mathcal {M}\) be an open neighbourhood of \(p_0\) and
be a linear partial differential operator of order \(r \ge 1\) with \(\mathcal {C}^\infty \)smooth coefficients. If \(\mathcal {V}\) is locally weakly Qexact at \(p_0\) then
for every \(u \in {\mathfrak {m}}_{p_0}^r\). In particular, if Q is a firstorder linear partial differential operator, then
Proof
We make use of the regular coordinate system \((x,t) = (x_1, \dots , x_m,t_1, \dots , t_n)\) of Proposition 2.1 centered at \(p_0\), thus we assume \(\Omega \subset \mathbb {R}^m \times \mathbb {R}^n\) and we have the complete set of firstintegrals \(\{Z_j: 1 \le j \le m\}\) given in terms of those coordinates. Let \(U \subset \Omega \) be an open neighbourhood of the origin with \(\det Z_x(x,t) \ne 0\) in U. Let \(V \subset U\) be a neighbourhood of the origin such that \((*)^{m,n}_{U,V}\)condition holds.
Since \(\mathcal {V}\) does not satisfy \(\mathrm {(P}_{n1}\mathrm {)}\)condition at the origin, for every open neighborhood \(U_0 \subset V\) of the origin the following objects exist:

An open set \(U_1 \subset U_0\);

A nonempty compact set \(K \subset U_1\);

A function \(H \in \mathcal {C}^\infty (U_1)\), with \(\textrm{d}^\prime H = 0\), such that

(i)
\(\textrm{Re} \, H = 0\), in K;

(ii)
\(\textrm{Re}\, H > 0\), in \(U_1 \setminus K\).

(i)
Let \(\zeta \in \mathcal {C}^\infty _c(U_1)\) be a positive valued function with \(\zeta = 1\) in a neighborhood of K. Let \(\varepsilon > 0\) be any strict lower bound for \(\textrm{Re}\, H\) in \(U_1 {\setminus } \zeta ^{1}(\{0,1\})\). Let \(\psi \in \mathcal {C}^\infty _c(V_\varepsilon )\) be any function where
Notice that \(K \subset V_\varepsilon \subset \zeta ^{1}(\{0,1\})\). Fix any germ \(u \in {\mathfrak {m}}_{p_0}^r\) and set
Set \(h = H  \varepsilon /2\). Since \(\textrm{d}^\prime v = u \, \textrm{d}^\prime \zeta \), we have
By Remark 3.4,
therefore
since \(V_\varepsilon \subset \zeta ^{1}(\{0,1\})\) and \(\textrm{d}Z \wedge \textrm{d}t = \det Z_x(x,t) \, \textrm{d}x \wedge \textrm{d}t\).
The function \(\psi \in \mathcal {C}^\infty _c(V_\varepsilon )\) is arbitrary, therefore \( \,^tQ[u] = 0\), in \(V_\varepsilon \). In particular, \(\,^tQ[u]\) vanishes in every point of K. Letting \(U_0 \rightarrow 0\), we have \(\,^tQ_r[u](0) = 0\). \(\square \)
Proof of Theorem 1.1
Making use of the regular coordinate system (coarse version) of Proposition 2.1
in \(\Omega \subset \mathbb {R}^m \times \mathbb {R}^n\) around \(p_0\) and centered at the origin, we have the local frame \(\{L_j: 1 \le j \le n\}\) for \(\mathcal {V}\) and taking representatives in the \((m,n1)\) and (m, n)bundles, Eq. (8) translates into the following equation
where \(f \in \mathcal {C}^\infty (U)\) is now identified with its single coefficient and the desired solution is an ntuple of germs of distributions \(u_j \in \mathcal {D}^\prime (0)\), \(1 \le j \le n\). We may as well identify Q with its single entry matrix. Thus, we immediately obtain Theorem 1.1. \(\square \)
Remark 4.3
A peak function for a locally integrable structure \(\mathcal {V}\) at \(p_0 \in \mathcal {M}\) is a local solution h for \(\mathcal {V}\) defined in an open neighbourhood U of \(p_0\) verifying
for all \(p \in U \setminus \{p_0\}\). It is clear that if \(\mathcal {V}\) admits a peak function at \(p_0\), then \(\mathrm {(P}_{n1}\mathrm {)}\)condition at \(p_0\) does not hold and it follows from the proof of Theorem 4.2 that \(\,^tQ[Z_k] = 0\) in an open neighbourhood of the origin for all \(1 \le k \le m\). If \(\,^tQ_0 = 0\), then \(\,^tQ_1\) must be a local section of \(\mathcal {V}\) around \(p_0\). This phenomenon occurs for instance for the kMizohata vector field
in the plane for any odd positive integer k, or any lineally convex real hypersurface of \(\mathbb {C}^N\) (see [27]). The reader may consult [4, 27] for further results on peak functions in locally integrable structures related to the socalled Borel map.
Example 4.4
Let us consider the corank 1 structure in \(\mathbb {R}^{n+1}\), with (fineregular) coordinates \((s,t) = (s,t_1,\dots ,t_n)\), generated by the firstintegral
A frame for \(\mathcal {V}\) is given by the following vectorfields
Setting \(M = \partial /\partial s\) we have a frame \((L_1,\dots ,L_n,M)\) for \(\mathbb {C}\textrm{T}\mathbb {R}^{n+1}\) consisting of commuting vector fields. The real part of the solution \(h = iW + W^2\) is
Thus, condition \(\mathrm {(P}_{n1}\mathrm {)}\) does not hold at the origin (hence \(\mathcal {V}\) is not locally exact at the origin in degree \((0,n1)\)). Let \(a_j, b \in \mathcal {C}^\infty (\mathbb {R}^{n+1})\) with \(b(0) \ne 0\), \(1 \le j \le n\). Theorem 4.2 ensures that there exist \(f \in \mathcal {C}^\infty (\mathbb {R}^{n+1})\) such that the equation
does not admit any solution \((u_1, \dots , u_n) \in \big ( \mathcal {D}^\prime (\mathbb {R}^{n+1}) \big )^n\).
5 Unsolvability in Levinondegenerate structures
Our first result in dealing with an intermediate degree in the differential complex induced by \(\mathcal {V}\) is based on a necessary condition for local exactness related with the Levi form. For a given point \(p_0 \in \mathcal {M}\) and a characteristic direction \((p_0,\sigma ) \in \textrm{T}_{p_0}^0\), the Levi form at \((p_0,\sigma )\) is the Hermitian form in \(\mathcal {V}_{p_0}\) given by
where \(v,v^\prime \in \mathcal {V}_{p_0}\) and \(L, L^\prime \) are any sections of \(\mathcal {V}\) in a neighborhood of \(p_0\) such that \(L_{p_0} = v\), \(L^\prime _{p_0} = v^\prime \). When \(\mathcal {V}\) is a CRstructure, the map \(\sigma \mapsto \mathcal {L}_{(p_0,\sigma )}\) can be identified with the usual Levi map in \(\mathcal {V}_{p_0}\) (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 \(\mathcal {M}\subset \mathbb {C}^{n+1}\) has q positive eigenvalues and \(\dim \mathcal {V}_{p_0}  q = \dim _{\textrm{CR}}\mathcal {M} q = nq\) negative eivenvalues, then the differential complex associated to \(\mathcal {M}\) is not locally exact at \(p_0\) in degrees (0, q) and \((0,nq)\). 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 1.13
([33], p. 364) Let \(p_0 \in \mathcal {M}\) and \(1 \le q \le n\) be given and let \((p_0,\sigma )\) be a direction in the characteristic set of \(\mathcal {V}\). Suppose the following condition holds
 \((*)_\sigma ^q\):

The Levi form of \(\mathcal {V}\) at \((p_0,\sigma )\) has q positive eigenvalues and \(nq\) negative eigenvalues, and its restriction to \(\mathcal {V}_{p_0} \cap {\overline{\mathcal {V}}}_{p_0}\) is nondegenerate.
Then the differential complex associated to \(\mathcal {V}\) is not locally exact at \(p_0\) in degree (0, q) (and thus also in degree \((m,nq)\)).
In order to state the result of the present section, we need to introduce some notation. Let \(\Omega \subset \mathcal {M}\) be an open neighbourhood of \(p_0\) and let
be a firstorder linear partial differential operator. We denote by \(\,^tQ_1(p_0,\sigma )\) the principal symbol of \(\,^tQ\) at \((p_0,\sigma )\). It is the linear map
given by
where \(\lambda \) is any local section of \(\Lambda ^{m,nq}\) near \(p_0\) with \(\lambda _{p_0} = v\) and g is any \(\mathcal {C}^\infty \)smooth function vanishing at \(p_0\) verifying \(\textrm{d}g_{p_0} = \sigma \) (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 \(\Lambda _{p_0}^{m,nq}\) to state in an invariant way the theorem of this section. We denote by \(\mathcal {V}_{p_0}^+\) and \(\mathcal {V}_{p_0}^\) the positive and the negative space of \(\mathcal {L}_{(p_0,\sigma )}\), respectively. If the Levi form at \((p_0,\sigma )\) is nondegenerate, it induces the following identification
thus
We single out the factor \(\Lambda ^m \textrm{T}_{p_0}^\prime \otimes \Lambda ^{nq} \mathcal {V}_{p_0}^\) and denote by \(\iota ^\) and \(\pi ^\) the natural inclusion and projection maps
Analogously, we may consider a firstorder linear partial differential operator
thus
and if \(\mathcal {L}_{(p_0,\sigma )}\) is nondegenerate, we have the identification
and we may single out the factor \(\Lambda ^q \mathcal {V}_{p_0}^+\) and denote by \(i^+\) and \(\pi ^+\) the natural inclusion and projection maps
Under the notation above we may state our result for this section.
Theorem 1.14
Let \(p_0 \in \mathcal {M}\) be a fixed point and let \(\Omega \subset \mathcal {M}\) be an open neighbourhood of \(p_0\). Let \(1 \le q \le n\) be an integer and let
be a firstorder linear partial differential operator that preserves \(\textrm{d}^\prime \)closed forms. Let \((p_0, \sigma )\) be a direction in \(\textrm{T}_{p_0}^0\). If the \((*)_\sigma ^q\)condition hold and \(\mathcal {V}\) is locally weakly Qexact at \(p_0\), then
Analogously, for a given firstorder linear partial differential operator
that preserves \(\textrm{d}^\prime \)closed forms, if \((*)_\sigma ^q\)condition hold and \(\mathcal {V}\) is locally Qexact 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)
in \(\Omega \subset \mathbb {R}^\nu \times \mathbb {R}^\nu \times \mathbb {R}^d \times \mathbb {R}^\mu = \mathbb {R}^N\) centered at \(p_0\). Let \((0,\sigma ) \in \textrm{T}_0^0\). Recall that in our coordinate system we have \(W_k(x,y,s,t) = s_k + i \varphi _k(x,y,s,t)\), \(1 \le k \le d\), and we may write \(\sigma = \sum _{k=1}^d \sigma _k \textrm{d}s_k_0\), where \(\sigma _k \in \mathbb {R}\) for \(1 \le k \le d\). The Levi form of \(\mathcal {V}\) at \((0,\sigma )\) in this coordinate system is given by
where \(L, L^\prime \) are any sections of \(\mathcal {V}\) in a neighborhood of 0 such that \(L_0 = v\), \(L^\prime _0 = v^\prime \) and
We choose the following basis for \(\mathcal {V}_0\)
Thus, we may express \(\mathcal {L}_{(0,\sigma )}\) as a block matrix
After a real linear change of coordinates in the svariable we may assume without loss of generality \(\sigma = \textrm{d}s_d_0\). Since the restriction of \(\mathcal {L}_{(0,\sigma )}\) to \(\mathcal {V}_0 \cap {\overline{\mathcal {V}}}_0\) is nondegenerate, after another linear change of coordinates we may assume (see the proof of Proposition I.9.1 of [33])
where the numbers \(0 \le \lambda \le \mu \) and \(0 \le \kappa \le \nu \) verify \(\kappa + \lambda = q\). Thus, in our coordinate system, the positive and negative spaces of \(\mathcal {L}_{(0,\sigma )}\) are given by
We denote
to simplify the expressions below. The isomorphism between \(\mathcal {V}_0\) and \(\mathcal {V}_0^*\) induced by \(\mathcal {L}_{(0,\sigma )}\) identifies \(\partial /\partial {\bar{z}}_j_0\) with \(\pm \textrm{d}{\bar{z}}_j_0\) and \(\partial /\partial t_\ell _0\) with \(\pm \textrm{d}t_\ell _0\) for each \(1 \le j \le \nu \) and \(1 \le \ell \le \mu \). Thus, the spaces \(\mathcal {V}_0^+\) and \(\mathcal {V}_0^\) are identified with the subspaces
and
of \(\mathcal {V}_0^*\), respectively. Therefore, the vector
spans \(\Lambda ^m \textrm{T}^\prime _0 \otimes \Lambda ^{nq} \mathcal {V}^{}_0\) under the identification induced by \(\mathcal {L}_{(0,\sigma )}\).
Let \(\tau > 0\) be small number to be chosen later and consider the change of scale
One may take as firstintegrals of the pushforward \(\Psi _*\mathcal {V}\) the functions
Since \((*)^q_\sigma \)condition is invariant under positive multiples of \(\sigma \), we may delete all the tildes (i.e. rename the variables and the functions) to obtain
Let \(\varepsilon > 0\) be another small number and define
Since
and \(\varphi _k = O(z^2+s^2+t^2)\) for all \(1 \le k \le d\), there is an open neighbourhood \(U \subset \Omega \) of the origin and a constant \(C>0\) such that the estimate
holds in U. We reduce U, if necessary, to get the estimates \(z^2+t^2 < 1\) and \(s^2 < 1/(4C)\) in U. Now we choose \(\tau \) and \(\varepsilon \) such that \(\tau < \varepsilon /(4C)\) and \(\tau + \varepsilon < 1/(2C)\) to get
or
in U. We set
Thus
and for a small constant \(a > 0\) we have
in U (reduced, if necessary), for \(i=1,2\). Indeed, from (9) we estimate
and
thus it suffices to apply again the fact \(\varphi _k = O(z^2+s^2+t^2)\) for all \(1 \le k \le d\).
We need to reduce once more the neighbourhood U in order to later apply Lebesgue’s dominated convergence theorem. In the following, we assume \(\rho > 1\) and that the domain \(\Omega \) is an open ball centered at the origin. We have
and since \(\varphi _k = O(z^2+s^2+t^2)\) for all \(1 \le k \le d\), there is a constant \(C^\prime > 0\) such that the following estimate holds
in the reduced U. We reduce it again to ensure \(z^2+s^2+t^2 < 1/(2C^\prime )\) in U. Therefore
in U.
Now we are ready to apply the hypothesis of local Qexactness at the origin: there exists an open neighbourhood of the origin \(V \Subset U\) such that for every \(\textrm{d}^\prime \)closed \(f \in \mathcal {C}^\infty (U, \Lambda ^{0,q})\) there is a solution \(u \in \mathcal {D}^\prime (V, \Lambda ^{0,q1})\) to the equation
in V. We apply Proposition 3.2 to the pair (U, V). Thus, for every compact \(K^\prime \subset V\) there is a compact set \(K \subset U\) and constants \(C > 0\) and \(r \in \mathbb {Z}_+\) such that
for every \(v \in \mathcal {C}^\infty (V,\Lambda ^{m,nq})\) with \(\textrm{supp}\, v \subset K^\prime \) and every \(\textrm{d}^\prime \)closed \(f \in \mathcal {C}^\infty (U,\Lambda ^{0,q})\).
Let \(\chi \in \mathcal {C}^\infty _c(V)\) be a cutoff function that equals 1 in a neighborhood of the origin and set \(\psi = \chi W_d^2\). We define
and
for each \(\rho > 1\).
We apply the estimate (12) choosing \(K^\prime = \textrm{supp}\, \chi \). For the righthand side we have \(\textrm{d}^\prime f_\rho = 0\) thanks to (10) and
Identities (10) also imply
we have
for some constant \(b>0\). Therefore, the righthand side of (12) goes to 0 as \(\rho \rightarrow \infty \).
For any given section \(v \in \mathcal {C}^\infty (U,\Lambda ^{m,nq})\), we may write
so the operator \(\,^tQ\) acts on v by an expression of the form
where \(P^{IJ}_{RS}\) is a firstorder linear partial differential operator for each (I, J, R, S).
Thus
for \(I_0 = (\kappa +1, \dots , \nu )\) and \(J_0 = (\lambda +1, \dots , \mu )\), since \(\textrm{d}\overline{Z^{\prime \prime }} = \textrm{d}{\bar{z}}_{I_0}\) and \(\textrm{d}t^{\prime \prime } = \textrm{d}t_{J_0}\). For each (R, S) we split
in its principal part and its zerothorder part, thus
Setting \(P_{I_0J_0}^{I_0J_0} = P\), we have
where the ±sign of the integral comes from reordering of 1forms and is immaterial in the following. If we change scale
we get (after renaming of variables by removing primes)
where
Notice that since \(W_d(0) = 0\), we have \(\alpha (0) = \gamma (0) = 0\) and since
we also have \(\beta (0) = 0\). For any X, Y in the set
the product rule entails
If we write
then
Thus Taylor’s formula implies
therefore the integrand in (13) converges pointwise to
as \(\rho \rightarrow \infty \) since \(\beta (0)=\gamma (0)=0\). By the estimate (11), we can apply Lesbegue’s Dominated Convergence Theorem, to conclude
as \(\rho \rightarrow \infty \). Therefore \(c_d(0) = 0\). Since \(\,^tQ_1(0,\sigma )(\omega _0) = \,^tQ_1(W_d\omega )_0\) for any \(\omega \in \mathcal {C}^\infty (0,\Lambda ^{m,nq})\), if we choose \(\omega = \textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z^{\prime \prime }} \wedge \textrm{d}t^{\prime \prime }\) we get
and the proof is complete. \(\square \)
Example 5.3
Let us consider the corank 1 structure in \(\mathbb {R}^3\), with (fineregular) coordinates \((s,t) = (s,t_1,t_2)\), generated by the firstintegral
A frame for \(\mathcal {V}\) is given by the following vectorfields
and setting \(M = \partial /\partial s\) we have a frame \((L_1,L_2,M)\) for \(\mathbb {C}\textrm{T}\mathbb {R}^3\) consisting of commuting vector fields. The matrix of the Levi form at \((0, \textrm{d}s)\) with respect to the basis \((L_1_0, L_2_0)\) of \(\mathcal {V}_0\) is given by
Thus, condition \((*)^1_{\textrm{d}s}\) of Theorem 5.1 holds (hence \(\mathcal {V}\) is not locally exact at the origin in degree (0, 1)). Let a, b and c be a solutions for \(\mathcal {V}\) with \(c(0) \ne 0\). The operator \(Q: \mathcal {C}^\infty (\mathbb {R}^3, \Lambda ^{0,1}) \rightarrow \mathcal {C}^\infty (\mathbb {R}^3, \Lambda ^{0,1})\) given in coordinates by
preserves \(\textrm{d}^\prime \)closed forms. Theorem 5.2 ensures that \(\textrm{d}^\prime \) is not locally Qexact at the origin in degree (0, 0), i.e., there exists a \(\textrm{d}^\prime \)closed form \(f = f_1 \textrm{d}t_1 + f_2 \textrm{d}t_2\) such that the system of equations
does not admit any distribution solution u.
6 Unsolvability in corank 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 corank 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 corank 1 structures (we refer the reader to [10] or sections VIII.46 of [33] for more details).
Let \(\mathcal {V}\) be a locally integrable structure of rank \(n = N1\) on a \(\mathcal {C}^\infty \)smooth Nmanifold \(\mathcal {M}\) and let \(p_0 \in \mathcal {M}\) be a distinguished point in \(\mathcal {M}\). We are going to assume that the structure \(\mathcal {V}\) 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 \(\mathcal {C}^\infty \)smooth solution of \(\mathcal {V}\) defined in an open neighbourhood \(\Omega \subset \mathcal {M}\) of \(p_0\) with \(\textrm{T}^0_{p_0} = \langle \textrm{d}W_{p_0} \rangle \). Let \(w_0 \in \mathbb {C}\) be a regular value of W and let \(\mathcal {S} = W^{1}(w_0)\) be the corresponding level set, thus \(\mathcal {S}\) is a \(\mathcal {C}^\infty \)smooth \((N2)\)submanifold of \(\mathcal {M}\). As a consequence of BaouendiTreves approximation formula the germs of \(\mathcal {S}\) at its points are invariants of the locally integrable structure \(\mathcal {V}\) (see Corollary II.3.1 of [33]). We are going to associate to every pair \(V \subset U \subset \Omega \) of open neighbourhoods of \(p_0\), every level set \(\mathcal {S}\) and every degree (0, q) a relative intersection number
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 \(\textrm{H}_0(\mathcal {S} \cap V)\) 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 corank 1 structures (see Theorem 2.1 of [10]).
Theorem 1.16
Let \(\mathcal {V}\) be a locally integrable structure of rank \(n = N1\) on a \(\mathcal {C}^\infty \)smooth Nmanifold \(\mathcal {M}\) and let \(p_0 \in \mathcal {M}\) be a distinguished point in \(\mathcal {M}\). Let us assume that the structure \(\mathcal {V}\) is not elliptic at \(p_0\) and let W be a local \(\mathcal {C}^\infty \)smooth solution of \(\mathcal {V}\) defined in an open neighbourhood \(\Omega \subset \mathcal {M}\) of \(p_0\) with \(\textrm{T}^0_{p_0} = \langle \textrm{d}W_{p_0} \rangle \). If the differential complex associated to \(\mathcal {V}\) is locally exact at \(p_0\) in degree (0, q), then for every open neighbourhood \(U \subset \Omega \) of \(p_0\) there is another open neighbourhood \(V \subset U\) of \(p_0\) such that \(I^{q1}_{U,V,\mathcal {S}} \equiv 0\) for every level set \(\mathcal {S}\) that is noncritical in V.
Remark 6.2
The property \(I^{q1}_{U,V,\mathcal {S}} \equiv 0\) in Theorem 6.1 is equivalent to the following assertion (see Section 5 in [10]):
“The natural map
induced by the inclusion \(\mathcal {S} \cap V \hookrightarrow \mathcal {S} \cap U\) 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 \(\mathcal {U}\) of open neighbourhoods of \(p_0\) and assume that there is \(U \in \mathcal {U}\) such that for every \(V \in \mathcal {U}\) with \(V \subset U\) there is a level set \(\mathcal {S}\) and closed forms \(\beta \in \mathcal {C}^\infty \big (\mathcal {S}\cap U, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q1}\mathbb {C}\textrm{T}^*\mathcal {S} \big )\) and \(\gamma \in \mathcal {C}^\infty _c \big (\mathcal {S} \cap V, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{nq} \mathbb {C}\textrm{T}^*\mathcal {S} \big )\) such that \(I^{q1}_{U,V,\mathcal {S}} ([\beta ],[\gamma ]) \ne 0\). One then constructs sections of the associated bundles \(\Phi (\beta ) \in \mathcal {C}^\infty (U, \Lambda ^{0,q})\) and \(\Upsilon (\gamma ) \in \mathcal {C}^\infty _c(V,\Lambda ^{1,nq})\) (by a procedure to be described later) such that \(\textrm{d}^\prime \Phi (\beta ) = 0\) and
The procedure behind the maps \(\Phi \) and \(\Upsilon \) is carefully built to produce also a solution \(h \in \mathcal {C}^\infty (\Omega )\) of \(\mathcal {V}\) such that
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 \(\Phi \) and \(\Upsilon \)) to get our criterion for local weak Qexactness at \(p_0\). Before we state our result we describe how the maps \(\Phi \) and \(\Upsilon \) are defined (see section 4 of [10]).
By Proposition 2.2, we can choose regular coordinates denoted by \((s,t) = (s,t_1,\dots ,t_n)\) around \(p_0\) centered at the origin. Thus we have an open neighbourhood of the origin \(\Omega \subset \mathbb {R}\times \mathbb {R}^n\) and a \(\mathcal {C}^\infty \)smooth function \(\varphi : \Omega \rightarrow \mathbb {R}\) with \(\varphi (0) = 0\) and \(\textrm{D}\varphi (0) = 0\) such that our local solution of \(\mathcal {V}\) is given in coordinates by
Let \(\mathcal {U}\) be the set of all open neighbourhoods of the origin of the form \(I \times O \subset \Omega \) where I is a open interval around \(0 \in \mathbb {R}\) and O is an open ball centered at the origin of \(\mathbb {R}^n\). Let \(U,V \in \mathcal {U}\) be fixed neighbourhoods with \(V \subset U\) and \(w_0 = s_0 + i r_0 \in \mathbb {C}\) be a regular value for W noncritical in V. We write \(V = B \times V_0 \subset \mathbb {R}\times \mathbb {R}^n\) where B is an open interval around \(0 \in \mathbb {R}\) and \(V_0\) is an open ball centered at the origin of \(\mathbb {R}^n\) and set
From now on we identify \(\mathcal {S}\) with \(\mathcal {S}_0\) via the map \((s_0,t) \mapsto t\). Let \(\beta \in \mathcal {C}^\infty \big (\mathcal {S}_0 \cap U_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q1}\mathbb {C}\textrm{T}^*\mathcal {S}_0\big )\) and \(\gamma \in \mathcal {C}^\infty _c\big (\mathcal {S}_0 \cap V_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{nq}\mathbb {C}\textrm{T}^*\mathcal {S}_0\big )\) be a pair of closed forms verifying \(I^{q1}_{U_0,V_0,\mathcal {S}_0} ([\beta ],[\gamma ]) \ne 0\). As in the proof of Proposition 3.2 of [10], there are forms \(g_+, g_ \in \mathcal {C}^\infty \big ( U_0, \bigwedge ^{q1} \mathbb {C}\textrm{T}^*\mathbb {R}^n \big )\) and \(u_+,u_ \in \mathcal {C}^\infty _c \big ( V_0,\bigwedge ^{nq} \mathbb {C}\textrm{T}^*\mathbb {R}^n \big )\) such that:
the decomposition
holds and
The condition \(I^{q1}_{U_0,V_0,\mathcal {S}_0} ([\beta ],[\gamma ]) \ne 0\) ensures that at least one of the integrals above does not vanish. Let us assume \(\int _{\mathcal {S}_0}g_ \wedge u_+ \ne 0\) and set \(g = g_\) and \(u = u_+\) (on the other case, one would set \(g = g_+\) and \(u = u_\)).
Let \(\rho > 0\) be a positive real number such that we may estimate
for every \(t \in \textrm{supp}\, \textrm{d}u\). Taylor’s formula in the svariable ensures the existence of a constant \(A > 0\) such that the following estimate holds
therefore we may choose a small positive real number \(\eta > 0\), with \((s_0\eta ,s_0+\eta ) \subset B\), such that the following estimate holds
for every \((s,t) \in (s_0\eta ,s_0+\eta ) \times \textrm{supp}\, \textrm{d}u\).
Let \(0< \eta ^\prime < \eta \) and choose a \(\mathcal {C}^\infty \)smooth function with compact support
with \(\chi (s) = 1\) if \(s  s_0 < \eta ^\prime /2\). Let \(0< \delta < \eta ^\prime /2\) and \(0< \varepsilon < \rho /4\) and choose any \(\mathcal {C}^\infty \)smooth function \(G: \mathbb {C}\rightarrow [0,+\infty )\) with support in the rectangle
with \(G > 0\) in the interior of \({\mathfrak {R}}\). We define \(\Phi (\beta ) \in \mathcal {C}^\infty (U, \Lambda ^{0,q})\) and \(\Upsilon (\gamma ) \in \mathcal {C}^\infty _c(V,\Lambda ^{1,nq})\) by the expressions
where \(\pi \) is the projection \((s,t) \mapsto t\) and \(\pi ^{0,q}\) is the projection \( \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^q\mathbb {C}\textrm{T}^*U = \Lambda ^{0,q} \oplus \textrm{T}^{1,q1} \rightarrow \Lambda ^{0,q}\). We point out that \(\Lambda ^{1,nq} = \textrm{T}^{1,nq}\) since \(\textrm{Rank} \, \textrm{T}^\prime = 1\).
In [10], it is shown that \(\textrm{d}^\prime \Phi (\beta ) = 0\) and that if the parameters \(\varepsilon \) and \(\delta \) are small enough then there exists a solution \(h \in \mathcal {C}^\infty (\Omega )\) of \(\mathcal {V}\) such that
Theorem 1.18
Let \(\mathcal {V}\) be a locally integrable structure of rank \(n = N1\) on a \(\mathcal {C}^\infty \)smooth Nmanifold \(\mathcal {M}\) and let \(p_0 \in \mathcal {M}\) be a distinguished point in \(\mathcal {M}\). Let us assume that the structure \(\mathcal {V}\) is not elliptic at \(p_0\) and let W be a local \(\mathcal {C}^\infty \)smooth solution of \(\mathcal {V}\) defined near \(p_0\) with \(\textrm{T}^0_{p_0} = \langle \textrm{d}W_{p_0} \rangle \). Let \(1 \le q \le n\) be an integer and let
be a firstorder linear partial differential operator that preserves \(\textrm{d}^\prime \)closed forms. If \(\mathcal {V}\) is locally weakly Qexact in \(p_0\), then there is a fundamental system \(\mathcal {U}\) of neighbourhoods of \(p_0\) such that for every \(U, V \in \mathcal {U}\) with \(V \subset U\) and every level set \(\mathcal {S}\) of W, noncritical in V, and for every pair of closed forms \(\beta \in \mathcal {C}^\infty \big (\mathcal {S}_0 \cap U_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q1}\big )\) and \(\gamma \in \mathcal {C}^\infty _c\big (\mathcal {S}_0 \cap V_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{nq}\big )\) verifying \(I^{q1}_{U_0,V_0,\mathcal {S}_0} ([\beta ],[\gamma ]) \ne 0\) we have
Proof
Apply Proposition 3.3. \(\square \)
Remark 6.4
In the definition of the maps \(\Phi \) and \(\Upsilon \) we have some freedom in the choice of the cutoff function G. Now we exploit this fact to provide finer necessary conditions for local weak Qexactness. Let us consider the same coordinate system (s, t) as above and the same fundamental system of neighbourhoods of the origin \(\mathcal {U}\). As above, let \(\beta \in \mathcal {C}^\infty \big (\mathcal {S}_0 \cap U_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q1}\mathbb {C}\textrm{T}^*\mathcal {S}_0\big )\) and \(\gamma \in \mathcal {C}^\infty _c\big (\mathcal {S}_0 \cap V_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{nq}\mathbb {C}\textrm{T}^*\mathcal {S}_0\big )\) be a pair of closed forms verifying \(I^{q1}_{U_0,V_0,\mathcal {S}_0} ([\beta ],[\gamma ]) \ne 0\) and \(u \in \mathcal {C}^\infty _c\big (V_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{nq} \mathbb {C}\textrm{T}^*\mathbb {R}^n\big )\), \(g \in \mathcal {C}^\infty \big (U_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q1}\mathbb {C}\textrm{T}^*\mathbb {R}^n\big )\) be the corresponding forms in the tspace. We may write in our coordinates
where the sum runs over ordered multiindexes and the coefficients are \(\mathcal {C}^\infty \)smooth functions. Let \(0< \delta < \eta ^\prime /2\) and \(0< \varepsilon < \rho /4\) as before and choose \(\mathcal {C}^\infty \)smooth functions \(\psi , \zeta : \mathbb {R}\rightarrow [0,+\infty )\) with
and \(>0\) in the interior of the supports and \(\int \psi = 1\). Notice that for each \(0< \lambda < 1\) the function
has support in the rectangle
and \(G_\lambda > 0\) in the interior of \({\mathfrak {R}}\). For this family of choices of G we have
The operator \(\,^tQ\) acts on \(\Upsilon (\gamma )\) by an expression of the form
where \(P^I_R\) is a firstorder linear partial differential operator for each (I, R). Since
we have by Theorem 6.3
where \(\varepsilon _{jRJ} \in \{1,0,1\}\) is characterized by
Therefore, letting \(\lambda \rightarrow 0\), we have
where
for all \(t \in U_0\). Thus
In the reasoning above, we started with a cuttoff function \(\psi \) and modified it to a get a Dirac family centered at \(s_0\). Now we are going to choose a Dirac family \(\zeta _\tau \) centered at an arbitrary point \(r_1 \in (r_0 + \rho /2  \varepsilon , r_0 + \rho /2 + \varepsilon )\) that is a regular value for the function \(\varphi _{s_0}\) (those values are dense in that interval). Thus, we have the following limit in \(\mathcal {D}^\prime \big ((r_0 + \rho /2  \varepsilon , r_0 + \rho /2 + \varepsilon )\big )\)
Since \(i_{s_0}^*(\zeta _\tau \circ \varphi ) = \varphi _{s_0}^*\zeta _\tau \), continuity of pullback implies (see Theorems 6.1.2 and 6.1.5 in [16])
where \(\textrm{d}\sigma \) is the Euclidean surface measure in \(\varphi _{s_0}^{1}(r_1) = \{t \in U_0: \varphi (s_0,t) = r_1\}\), and
Equation (14) becomes
Therefore, if \(\mathcal {V}\) is Qexact at \(p_0\), then for every level set \(W^{1}(s_0+ir_0)\) with nonvanishing intersection number the integral above given in coordinates over the perturbed level set \(W^{1}(s_0+ir_1)\) must vanish.
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Acknowledgements
I would like to thank Prof. Dr. Paulo Domingos Cordaro for suggestion and guidance through my doctoral studies. I am also grateful to Prof. Dr. Bernhard Lamel and all of the members of the Vienna research group for the hospitality during the year I visited University of Vienna. I would like to thank the anonymous reviewers for the careful reading of the manuscript and for the valuable suggestions that improved the presentation of the final version. Finally, I would like to thank CNPq (Process 141883/20170) and CAPES (Process 88887.364957/201900) for the financial support.
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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.
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da Silva, A.V. On the range of unsolvable systems induced by complex vector fields. Partial Differ. Equ. Appl. 4, 41 (2023). https://doi.org/10.1007/s42985023002600
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DOI: https://doi.org/10.1007/s42985023002600