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 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 \(\mathcal {C}^\infty \)-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(xD) and Q(xD) are first-order 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:

  1. (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)
  2. (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(xD) 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 (\(\Psi \))-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:

  1. (i)

    in top-degree, via the failure of Cordaro-Hounie \(\mathrm {(P}_{n-1}\mathrm {)}\)-condition introduced in [8];

  2. (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 \(\mathcal {M}\subset \mathbb {C}^{n+1}\) has q positive eigenvalues and \(n-q\) negative eivenvalues, then the \({\bar{\partial }}_b\)-complex in \(\mathcal {M}\) 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);

  3. (iii)

    in co-rank 1 structures, via Treves condition on the homology of fibers of first integrals introduced in [31].

The Cordaro-Hounie \(\mathrm {(P}_{n-1}\mathrm {)}\)-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 “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 \(\mathrm {(P}_{n-1}\mathrm {)}\)-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 first-order 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

$$\begin{aligned} \sum _{j=1}^n L_j u_j = Qf, \end{aligned}$$

in V, and \(\mathrm {(P}_{n-1}\mathrm {)}\)-condition fails at the origin, then there exists \(\mu _j \in \mathbb {C}\), \(1 \le j \le n\), such that

$$\begin{aligned} \,^tQ_1|_0 = \sum _{j=1}^n \mu _jL_j|_0 \end{aligned}$$

When we are dealing with a single locally integrable complex vector field, the Cordaro-Hounie \(\mathrm {(P}_{0}\mathrm {)}\)-condition is equivalent to Nirenberg-Treves \(\mathrm {(P)}\)-condition (see [8, 32]). Therefore, Theorem 1.1 extends Hörmander’s Theorem for vector fields replacing condition (1) by the negation of Nirenberg-Treves \(\mathrm {(P)}\)-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 non-negative 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

$$\begin{aligned} \textrm{T}^{p,q}_{p_0} = \textrm{span}\, \big \{ \omega _1 \wedge \cdots \wedge \omega _{p+q} : \omega _1, \dots , \omega _p \in \textrm{T}^\prime _{p_0}, \; \omega _{p+1}, \dots , \omega _{p+q} \in \mathbb {C}\textrm{T}^*_{p_0} \mathcal {M}\big \}, \quad p_0 \in \mathcal {M}. \end{aligned}$$

The inclusion \(\textrm{T}^{p+1,q-1} \subset \textrm{T}^{p,q}\) allow us to define the (pq)-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 \(\Omega \subset \mathcal {M}\)

$$\begin{aligned} \textrm{d}^\prime : \mathcal {C}^\infty (\Omega , \Lambda ^{p,q}) \rightarrow \mathcal {C}^\infty (\Omega , \Lambda ^{p,q+1}),\quad p,q \in \mathbb {Z}_+. \end{aligned}$$

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 (pq)-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

figure a

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 distribution-sections of the (pq)-bundle, i.e., (pq)-currents, and the corresponding differential complex

figure b

The bundle \(\mathcal {V}\) is a locally integrable structure if \(\textrm{T}^\prime \) is locally generated by exact 1-forms. 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 co-rank 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

$$\begin{aligned} (x,t) = (x_1, \dots , x_m, t_1, \dots , t_n),\qquad \qquad N = m + n, \end{aligned}$$

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

$$\begin{aligned} Z_k(x,t) = x_k + i \varphi _k(x,t),\qquad \qquad 1 \le k \le m, \end{aligned}$$

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

$$\begin{aligned} L_j = \dfrac{\partial }{\partial t_j} - i \displaystyle \sum _{k=1}^m \dfrac{\partial \varphi _k}{\partial t_j}(x,t) M_k,\qquad \qquad 1 \le j \le n, \end{aligned}$$

where

$$\begin{aligned} M_k = \sum _{r=1}^m \mu _{kr}(x,t)\dfrac{\partial }{\partial x_r},\qquad \qquad 1 \le k \le m, \end{aligned}$$

is the complex vector field characterized by the conditions

$$\begin{aligned} M_k Z_r = {\left\{ \begin{array}{ll} 1, \text { if } k = r, \\ 0, \text { otherwise,} \end{array}\right. }\qquad \qquad 1 \le k, r \le m, \end{aligned}$$

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

$$\begin{aligned} \ T^0_{p_0} = \textrm{T}^\prime _{p_0} \cap \textrm{T}^*_{p_0} \mathcal {M}. \end{aligned}$$

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

$$\begin{aligned} (x,y,s,t) = (x_1, \dots , x_\nu , y_1, \dots , y_\nu , s_1, \dots , s_d, t_1, \dots , t_\mu ), \qquad \qquad {N = 2\nu + d + \mu ,} \end{aligned}$$

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

$$\begin{aligned} \begin{array}{l} Z_j(x,y,s,t) = x_j + i y_j,\\ W_k(x,y,s,t) = s_k + i \varphi _k(x,y,s,t), \end{array} \qquad \qquad { \begin{array}{r} 1\le j \le \nu ,\\ 1 \le k \le d, \end{array} } \end{aligned}$$

define a full set of basic solutions for \(\mathcal {V}\) in the local chart. Thus, \(n = \nu + \mu \). Moreover, one has

$$\begin{aligned} \textrm{T}^0_0 = \langle \textrm{d}s_1|_0, \dots , \textrm{d}s_d|_0 \rangle , \end{aligned}$$

and a local frame for \(\mathcal {V}\) in a possibly smaller neighbourhood of the origin in the coordinate chart is given by

$$\begin{aligned} \begin{array}{l} L_j = \dfrac{\partial }{\partial {\bar{z}}_j} - i \displaystyle \sum _{k=1}^d \dfrac{\partial \varphi _k}{\partial {\bar{z}}_j}(x,y,s,t) N_k, \qquad \qquad \quad 1 \le j \le \nu , \\ L_{\nu +\ell } = \dfrac{\partial }{\partial t_\ell } - i \displaystyle \sum _{k=1}^d \dfrac{\partial \varphi _k}{\partial t_\ell }(x,y,s,t) N_k, \qquad \qquad 1 \le \ell \le \mu , \end{array} \end{aligned}$$

where

$$\begin{aligned} \dfrac{\partial }{\partial {\bar{z}}_j} = \dfrac{1}{2} \bigg ( \dfrac{\partial }{\partial x_j} + i \dfrac{\partial }{\partial y_j} \bigg ), \qquad \qquad {1 \le j \le \nu ,} \end{aligned}$$

and

$$\begin{aligned} N_k = \sum _{r=1}^d \mu _{kr}(x,y,s,t)\dfrac{\partial }{\partial s_r}, \qquad \qquad {1 \le k \le d,} \end{aligned}$$

is the complex vector field characterized by the conditions

$$\begin{aligned} N_k W_r = {\left\{ \begin{array}{ll} 1, \text { if } k = r, \\ 0, \text { otherwise,} \end{array}\right. } \qquad \qquad {1 \le k, r \le d.} \end{aligned}$$

In the context of Proposition 2.2, set

$$\begin{aligned} \begin{array}{l} M_j = \dfrac{\partial }{\partial z_j} - i \displaystyle \sum _{r=1}^d \dfrac{\partial \varphi _r}{\partial z_j}(x,y,s,t) N_r, \qquad \qquad 1 \le j \le \nu ,\\ M_{\nu + k} = N_k, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad 1 \le k \le d, \end{array} \end{aligned}$$

where

$$\begin{aligned} \dfrac{\partial }{\partial z_j} = \dfrac{1}{2} \bigg ( \dfrac{\partial }{\partial x_j} - i \dfrac{\partial }{\partial y_j} \bigg ), \qquad \qquad {1 \le j \le \nu ,} \end{aligned}$$

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

$$\begin{aligned} I_m = M(x,t)\bigg (I_m+i \bigg [\dfrac{\partial \Phi }{\partial x}(x,t)\bigg ]^t \, \bigg ), \end{aligned}$$

where \(I_m\) is the \(m \times m\) identity matrix and

$$\begin{aligned} \begin{matrix} M(x,t) = \begin{bmatrix} \mu _{11}(x,t) &{} \cdots &{} \mu _{1m}(x.t) \\ \vdots &{} &{} \vdots \\ \mu _{m1}(x,t) &{} \cdots &{} \mu _{mm}(x,t) \end{bmatrix}, &{} \bigg [\dfrac{\partial \Phi }{\partial x}(x,t)\bigg ] = \begin{bmatrix} \dfrac{\partial \varphi _1}{\partial x_1}(x,t) &{} \cdots &{} \dfrac{\partial \varphi _1}{\partial x_m}(x,t) \\ \vdots &{} &{} \vdots \\ \dfrac{\partial \varphi _m}{\partial x_1}(x,t) &{} \cdots &{} \dfrac{\partial \varphi _m}{\partial x_m}(x,t) \end{bmatrix}, \end{matrix} \end{aligned}$$

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 coarse-regular coordinates (xt) of Proposition 2.1 near a fixed point of \(\mathcal {M}\). Consider a \((p+q)\)-form

$$\begin{aligned} f = \sum _{{\begin{matrix} |I| = p \\ |J| = q \end{matrix}}} f_{IJ}(x,t) \, \textrm{d}Z_I \wedge \textrm{d}t_J \end{aligned}$$
(3)

where the sum is carried over the set of all ordered multi-indexes IJ of length p and q, respectively, and for \(I = (i_1, \dots , i_p)\), \(J = (j_1, \dots , j_q)\) we write

$$\begin{aligned} \textrm{d}Z_I&= \textrm{d}Z_{i_1} \wedge \cdots \wedge \textrm{d}Z_{i_p}, \\ \textrm{d}t_J&= \textrm{d}t_{j_1} \wedge \cdots \wedge \textrm{d}t_{j_q}. \end{aligned}$$

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

$$\begin{aligned} L f = \sum _{{\begin{matrix} 1 \le j \le n \\ |I| = p \\ |J| = q \end{matrix}}} L_jf_{IJ}(x,t) \, \textrm{d}t_j \wedge \textrm{d}Z_I \wedge \textrm{d}t_J. \end{aligned}$$

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 semi-norms

$$\begin{aligned} |f|_{K,r} = \sum _{ {\begin{matrix} |\alpha |+|\beta | \le r \\ |I| = p \\ |J| = q \end{matrix}} } \sup _K \big | \partial ^\alpha _x \partial ^\beta _t f_{IJ} \big |, \end{aligned}$$

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 fine-regular coordinates (xyst) of Proposition 2.2, the sections of \(\Lambda ^{p,q}\) are uniquely represented by forms of the following kind

$$\begin{aligned} f = \sum _{{\begin{matrix} |I|+|J| = p \\ |R|+|S| = q \end{matrix}}} f_{IJRS}(x,y,s,t) \, \textrm{d}Z_I \wedge \textrm{d}W_J \wedge \textrm{d}\overline{Z}_R \wedge \textrm{d}t_S, \end{aligned}$$
(4)

and the representative of \(\textrm{d}^\prime [f]\) is

$$\begin{aligned} L f&= \sum _{{\begin{matrix} 1 \le j \le \nu \\ |I|+|J| = p \\ |R|+|S| = q \end{matrix}}} L_jf_{IJRS}(x,y,s,t) \, \textrm{d}\overline{Z}_j \wedge \textrm{d}Z_I \wedge \textrm{d}W_J \wedge \textrm{d}\overline{Z}_R \wedge \textrm{d}t_S \\&\quad \quad + \sum _{{\begin{matrix} 1 \le \ell \le \mu \\ |I|+|J| = p \\ |R|+|S| = q \end{matrix}}} L_{\nu +\ell } f_{IJRS}(x,y,s,t) \, \textrm{d}t_\ell \wedge \textrm{d}Z_I \wedge \textrm{d}W_J \wedge \textrm{d}\overline{Z}_R \wedge \textrm{d}t_S. \end{aligned}$$

The same Fréchet space structure on \(\mathcal {C}^\infty (\Omega , \Lambda ^{p,q})\) is defined by the semi-norms

$$\begin{aligned} |f|_{K,r} = \sum _{ {\begin{matrix} |\alpha |+|\beta |+|\gamma |+|\delta | \le r \\ |I|+|J| = p \\ |R|+|S| = q \end{matrix}} } \sup _K \big | \partial ^\alpha _x \partial ^\beta _y \partial ^\gamma _s \partial ^\delta _t f_{IJRS} \big |, \end{aligned}$$

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 (pq)-bundle in regular coordinates. From now on we identify sections of the (pq)-bundle with their standard representatives.

Let \(\omega \in \textrm{T}^{p,q}_{p_0}\) and \(\eta \in \textrm{T}^{m-p,n-q}_{p_0}\), for fixed \(p_0 \in \Omega \), be given. If \(\omega \in \textrm{T}^{p+1,q-1}_{p_0}\) or \(\eta \in \textrm{T}^{m-p+1,n-q-1}_{p_0}\), then \(\omega \wedge \eta = 0\) (indeed, we have \(\textrm{T}^{m+1,q} = \textrm{T}^{p,n+1} = 0\) for all pq). Therefore, we have a well-defined product

$$\begin{aligned} \wedge : \Lambda ^{p,q} \times \Lambda ^{m-p,n-q} \rightarrow \Lambda ^{m,n} = \textrm{T}^{m,n} \end{aligned}$$

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

$$\begin{aligned} \mathcal {C}^\infty _c(\Omega ,\Lambda ^{p,q}) \times \mathcal {C}^\infty (\Omega ,\Lambda ^{m-p,n-q}) \rightarrow \mathbb {C}, \\ \mathcal {C}^\infty (\Omega ,\Lambda ^{p,q}) \times \mathcal {C}^\infty _c(\Omega ,\Lambda ^{m-p,n-q}) \rightarrow \mathbb {C}, \end{aligned}$$

given by the same expression

$$\begin{aligned} (f,v) \mapsto \int _\Omega f \wedge v. \end{aligned}$$

These bilinear forms extend to bilinear forms acting on currents

$$\begin{aligned} \mathcal {C}^\infty _c(\Omega ,\Lambda ^{p,q})&\times \mathcal {D}^\prime (\Omega ,\Lambda ^{m-p,n-q}) \,\, \rightarrow \mathbb {C}, \\ \mathcal {D}^\prime (\Omega ,\Lambda ^{p,q})&\times \mathcal {C}^\infty _c(\Omega ,\Lambda ^{m-p,n-q}) \rightarrow \mathbb {C}, \\ \mathcal {C}^\infty (\Omega ,\Lambda ^{p,q})&\times \mathcal {E}^\prime (\Omega ,\Lambda ^{m-p,n-q}) \,\,\,\rightarrow \mathbb {C}, \\ \mathcal {E}^\prime (\Omega ,\Lambda ^{p,q})&\times \mathcal {C}^\infty (\Omega ,\Lambda ^{m-p,n-q}) \rightarrow \mathbb {C}, \end{aligned}$$

that indentify \(\mathcal {D}^\prime (\Omega ,\Lambda ^{m-p,n-q})\) with the dual of \(\mathcal {C}^\infty _c(\Omega ,\Lambda ^{p,q})\) (and vice-versa) and identify \(\mathcal {E}^\prime (\Omega ,\Lambda ^{m-p,n-q})\) with the dual of \(\mathcal {C}^\infty (\Omega ,\Lambda ^{p,q})\) (and vice-versa) (see Proposition VIII.1.2 in [33]). The elements of \(\mathcal {D}^\prime (\Omega ,\Lambda ^{p,q})\) are called (pq)-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 (pq)-form, v is an \((m-p,n-q-1)\)-form and one of them has compact support, Stokes’ Theorem implies

$$\begin{aligned} 0 = \int _\Omega \textrm{d}^\prime (f \wedge v) = \int _\Omega (\textrm{d}^\prime f) \wedge v + (-1)^{p+q} \int _\Omega f \wedge \textrm{d}^\prime v, \end{aligned}$$

therefore, the transpose of \(\textrm{d}^\prime \) acting on (pq)-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

$$\begin{aligned} Q: \mathcal {C}^\infty (\Omega , \Lambda ^{p,q}) \rightarrow \mathcal {C}^\infty (\Omega , \Lambda ^{p,q}) \end{aligned}$$

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 Q-exact 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 \(u \in \mathcal {D}^\prime (V, \Lambda ^{p,q-1})\) 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 non-empty 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

$$\begin{aligned} \bigg |\int _\Omega v \wedge Qf \bigg | \le C |f|_{K, r} |\textrm{d}^\prime v|_{K^\prime , r} \end{aligned}$$
(5)

for every \(\textrm{d}^\prime \)-closed form \(f \in \mathcal {C}^\infty (U,\Lambda ^{p,q})\) and every \(v \in \mathcal {C}^\infty _c(V,\Lambda ^{m-p,n-q})\) 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

$$\begin{aligned} E_0&= \{v \in \mathcal {C}^\infty _c(V, \Lambda ^{m-p,n-q}) : \textrm{supp}\, v \subset K^\prime \}, \\ \mathcal {Z}&= \{v \in E_0 : \textrm{d}^\prime v = 0\}, \\ E&= E_0 / \mathcal {Z}. \end{aligned}$$

The functions

$$\begin{aligned} E \ni v \!\!\!\! \mod \mathcal {Z} \mapsto |\textrm{d}^\prime v|_{K^\prime , \ell } \in [0, +\infty ) \end{aligned}$$

are well defined and turn E into a metrizable space. The bilinear function

$$\begin{aligned} E \times F \ni \big ( v \!\!\!\! \mod \mathcal {Z}, f \big ) \mapsto \int _\Omega v \wedge Qf \in \mathbb {C}\end{aligned}$$
(6)

is well defined and is separately continuous. Indeed, if \(v \in \mathcal {Z}\), then let \(f \in F\) be given and let \(u \in \mathcal {D}^\prime (V,\Lambda ^{p,q-1})\) be a solution of the equation \(\textrm{d}^\prime u = Qf\). Since

$$\begin{aligned} \textrm{d}^\prime (v \wedge u) = (\textrm{d}^\prime v) \wedge u + (-1)^{N-p-q} v \wedge \textrm{d}^\prime u, \end{aligned}$$

and \(v \wedge u\) is compactly supported, Stokes’ formula implies

$$\begin{aligned} \int _\Omega v \wedge Qf = \int _\Omega v \wedge \textrm{d}^\prime u = (-1)^{N-p-q+1} \int _\Omega (\textrm{d}^\prime v) \wedge u = 0, \end{aligned}$$

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 \(u \in \mathcal {D}^\prime (V,\Lambda ^{p,q-1})\) 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

$$\begin{aligned} \bigg |\int _\Omega w \wedge u \bigg | \le A |w|_{K^\prime , r}, \end{aligned}$$

holds for every \(w \in \mathcal {C}^\infty _c(V, \Lambda ^{m-p,n-q+1})\). Stokes’ formula implies

$$\begin{aligned} \bigg | \int _\Omega v \wedge Qf \bigg | = \bigg | \int _\Omega (\textrm{d}^\prime v) \wedge u \bigg | \le A |\textrm{d}^\prime v|_{K^\prime , r}, \end{aligned}$$

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. \(\square \)

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 non-empty 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 \(v \in \mathcal {C}^\infty _c(V,\Lambda ^{m-p,n-q})\) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{Re} \, h \le 0, \text { on } \textrm{supp}\, f,\\ \textrm{Re} \, h > 0, \text { on } \textrm{supp}\, \textrm{d}^\prime v, \end{array}\right. } \end{aligned}$$
(7)

then

$$\begin{aligned} \int \,^tQv \wedge f = 0. \end{aligned}$$

Proof

Apply Proposition 3.2 with the choice \(K^\prime = \textrm{supp}\, v\). For every \(\rho > 0\), let

$$\begin{aligned} v_\rho&= e^{-\rho h}v, \\ f_\rho&= e^{\rho h}f. \end{aligned}$$

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

$$\begin{aligned} |f_\rho |_{K,r}&\le C_{K,r} \rho ^r,\\ |\textrm{d}^\prime v_\rho |_{K^\prime ,r}&\le C_r \rho ^r e^{-c \rho }, \end{aligned}$$

with some constants \(C_{K,r},C_r,c > 0\). Let us assume that the coordinates in \(\Omega \) are the coarse-regular (xt) of Proposition 2.1. If we write

$$\begin{aligned} v = \sum _{ {\begin{matrix} |I| = m-p,\\ |J| = n-q \end{matrix}} } v_{IJ} \, \textrm{d}Z_I \wedge \textrm{d}t_J, \end{aligned}$$

the operator \(\,^tQ\) acts on \(v_\rho \) by an expression of the form

$$\begin{aligned} \,^tQ v_\rho = \sum _{ {\begin{matrix} |I| = |R| = m-p,\\ |J| = |S| = n-q \end{matrix}} } P^{IJ}_{RS} \big [e^{-\rho h}v_{IJ} \big ] \, \textrm{d}Z_R \wedge \textrm{d}t_S, \end{aligned}$$

where \(P^{IJ}_{RS}\) is a linear partial differential operator for each (IJRS). Applying Leibniz’ formula we get

$$\begin{aligned} P^{IJ}_{RS} \big [e^{-\rho h}v_{IJ} \big ] = e^{-\rho h} \big ( P^{IJ}_{RS} [v_{IJ}] + \rho \, \mathcal {R}^{IJ}_{RS}(\rho ) \big ) \end{aligned}$$

where \(\mathcal {R}^{IJ}_{RS}\) is a polynomial in the \(\rho \)-variable with \(\mathcal {C}^\infty \)-smooth functions as coefficients. Therefore we set

$$\begin{aligned} I(\rho ) = \int \big (\,^tQv_\rho \big ) \wedge f_\rho = \int \widetilde{v} \wedge f + \int \big (\,^tQv \big ) \wedge f, \end{aligned}$$

where

$$\begin{aligned} \widetilde{v} = \sum _{ {\begin{matrix} |I| = |R| = m-p,\\ |J| = |S| = n-q \end{matrix}} } \rho \, \mathcal {R}^{IJ}_{RS}(\rho ) \, \textrm{d}Z_R \wedge \textrm{d}t_S. \end{aligned}$$

Since \(I(\rho )/\rho ^k \rightarrow 0\), \(k \in \mathbb {Z}_+\), as \(\rho \rightarrow \infty \), we have

$$\begin{aligned} \int \widetilde{v} \wedge f = 0. \end{aligned}$$

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 non-empty 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 top-degree

Let \(p_0 \in \mathcal {M}\). In this section, we consider the \(\textrm{d}^\prime \)-equation in top-degree

$$\begin{aligned} \textrm{d}^\prime u = f, \end{aligned}$$
(8)

where \(f \in \mathcal {C}^\infty (U, \Lambda ^{m,n})\), the desired solutions \(u \in \mathcal {D}^\prime (V, \Lambda ^{m,n-1})\) are \((m,n-1)\)-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 Cordaro-Hounie \(\mathrm {(P}_{n-1}\mathrm {)}\) 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

$$\begin{aligned} K&= \{q \in U^\prime : f(q) = a\}, \\ U^\prime \setminus K&= \{q \in U^\prime : f(q) > a\}. \end{aligned}$$

We say that \(\mathcal {V}\) satisfies \(\mathrm {(P}_{n-1}\mathrm {)}\)-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 non-empty 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 N-dimensional \(\mathcal {C}^\infty \)-smooth manifold \(\mathcal {M}\) and assume that \(\mathcal {V}\) does not satisfy Cordaro-Hounie \(\mathrm {(P}_{n-1}\mathrm {)}\)-condition at a point \(p_0 \in \mathcal {M}\). Let \(\Omega \subset \mathcal {M}\) be an open neighbourhood of \(p_0\) and

$$\begin{aligned} Q: \mathcal {C}^\infty (\Omega , \Lambda ^{m,n}) \rightarrow \mathcal {C}^\infty (\Omega , \Lambda ^{m,n}) \end{aligned}$$

be a linear partial differential operator of order \(r \ge 1\) with \(\mathcal {C}^\infty \)-smooth coefficients. If \(\mathcal {V}\) is locally weakly Q-exact at \(p_0\) then

$$\begin{aligned} \,^tQ_r[u](p_0) = 0, \end{aligned}$$

for every \(u \in {\mathfrak {m}}_{p_0}^r\). In particular, if Q is a first-order linear partial differential operator, then

$$\begin{aligned} \,^tQ_1|_{p_0} \in \mathcal {V}_{p_0}. \end{aligned}$$

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 first-integrals \(\{Z_j: 1 \le j \le m\}\) 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 \(\mathrm {(P}_{n-1}\mathrm {)}\)-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 non-empty compact set \(K \subset U_1\);

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

    1. (i)

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

    2. (ii)

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

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

$$\begin{aligned} V_\varepsilon = \big \{ (x,t) \in U_1: \textrm{Re}\, H(x,t) < \varepsilon /2 \big \}. \end{aligned}$$

Notice that \(K \subset V_\varepsilon \subset \zeta ^{-1}(\{0,1\})\). Fix any germ \(u \in {\mathfrak {m}}_{p_0}^r\) and set

$$\begin{aligned} v(x,t)&= \zeta (x,t) u(x,t),\\ f(x,t)&= \dfrac{\psi (x,t)}{\det Z_x(x,t)} \, \textrm{d}Z \wedge \textrm{d}t. \end{aligned}$$

Set \(h = H - \varepsilon /2\). Since \(\textrm{d}^\prime v = u \, \textrm{d}^\prime \zeta \), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{Re}\, h < 0, \text { on } \textrm{supp}\, f,\\ \textrm{Re}\, h > 0, \text { on } \textrm{supp}\, \textrm{d}^\prime v. \end{array}\right. } \end{aligned}$$

By Remark 3.4,

$$\begin{aligned} \int \,^tQv \wedge f = 0, \end{aligned}$$

therefore

$$\begin{aligned} 0&= \int _{V_\varepsilon } \,^tQ[\zeta u](x,t) \dfrac{\psi (x,t)}{\det Z_x(x,t)} \, \textrm{d}Z \wedge \textrm{d}t \\&= \int _{V_\varepsilon } \psi (x,t) \,^tQ[u](x,t) \, \textrm{d}x \, \textrm{d}t, \end{aligned}$$

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

$$\begin{aligned} (x,t) = (x_1, \dots , x_m,t_1, \dots , t_n) \end{aligned}$$

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 \((m,n-1)\)- and (mn)-bundles, Eq. (8) translates into the following equation

$$\begin{aligned} \sum _{j=1}^n L_j u_j = f, \end{aligned}$$

where \(f \in \mathcal {C}^\infty (U)\) is now identified with its single coefficient and the desired solution is an n-tuple 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

$$\begin{aligned} \textrm{Re} \, h(p_0) < \textrm{Re}\, h(p), \end{aligned}$$

for all \(p \in U \setminus \{p_0\}\). It is clear that if \(\mathcal {V}\) admits a peak function at \(p_0\), then \(\mathrm {(P}_{n-1}\mathrm {)}\)-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 k-Mizohata vector field

$$\begin{aligned} M(x,t,D_x,D_t) = D_t - it^kD_x \end{aligned}$$

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 so-called Borel map.

Example 4.4

Let us consider the co-rank 1 structure in \(\mathbb {R}^{n+1}\), with (fine-regular) coordinates \((s,t) = (s,t_1,\dots ,t_n)\), generated by the first-integral

$$\begin{aligned} W(s,t) = s + i \sum _{j=1}^n t_j^2. \end{aligned}$$

A frame for \(\mathcal {V}\) is given by the following vector-fields

$$\begin{aligned} L_j = \dfrac{\partial }{\partial t_j} - 2it_j \dfrac{\partial }{\partial s}, \qquad \qquad {1 \le j \le n.} \end{aligned}$$

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

$$\begin{aligned} s^2 + \bigg (1 - \sum _{j=1}^n t_j^2 \bigg ) \sum _{j=1}^n t_j^2. \end{aligned}$$

Thus, condition \(\mathrm {(P}_{n-1}\mathrm {)}\) does not hold at the origin (hence \(\mathcal {V}\) is not locally exact at the origin in degree \((0,n-1)\)). 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

$$\begin{aligned} \sum _{j=1}^n \dfrac{\partial u_j}{\partial t_j} - 2it_j \dfrac{\partial u_j}{\partial s} = \sum _{j=1}^n a_j(s,t) \bigg ( \dfrac{\partial f}{\partial t_j} - 2it_j \dfrac{\partial f}{\partial s} \bigg ) + b(s,t) \dfrac{\partial f}{\partial s} \end{aligned}$$

does not admit any solution \((u_1, \dots , u_n) \in \big ( \mathcal {D}^\prime (\mathbb {R}^{n+1}) \big )^n\).

5 Unsolvability in Levi-nondegenerate 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

$$\begin{aligned} \mathcal {L}_{(p_0,\sigma )}(v,v^\prime ) = \dfrac{1}{2i} \sigma ([L,\overline{L^\prime }])(p_0), \end{aligned}$$

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 CR-structure, 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 \(\dim \mathcal {V}_{p_0} - q = \dim _{\textrm{CR}}\mathcal {M}- q = n-q\) negative eivenvalues, then the differential complex associated to \(\mathcal {M}\) 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 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 \(n-q\) 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 \((m,n-q)\)).

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

$$\begin{aligned} Q: \mathcal {C}^\infty (\Omega , \Lambda ^{0,q}) \rightarrow \mathcal {C}^\infty (\Omega , \Lambda ^{0,q}) \end{aligned}$$

be a first-order 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

$$\begin{aligned} \,^tQ_1(p_0,\sigma ): \Lambda _{p_0}^{m,n-q} \rightarrow \Lambda _{p_0}^{m,n-q}, \end{aligned}$$

given by

$$\begin{aligned} \,^tQ_1(p_0,\sigma )[v] = \,^tQ[g\lambda ](p_0), \qquad v \in \Lambda _{p_0}^{m,n-q}, \end{aligned}$$

where \(\lambda \) is any local section of \(\Lambda ^{m,n-q}\) 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 \(\Lambda _{p_0}^{m,n-q}\) 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 non-degenerate, it induces the following identification

$$\begin{aligned} \Lambda _{p_0}^{0,n-q} \simeq \Lambda ^{n-q} \mathcal {V}_{p_0}^*\underset{ {\begin{matrix} \uparrow \\ \mathcal {L}_{(p_0,\sigma )} \end{matrix}} }{\simeq } \Lambda ^{n-q}\mathcal {V}_{p_0} = \Lambda ^{n-q}\big (\mathcal {V}_{p_0}^+ \oplus \mathcal {V}_{p_0}^-\big ), \end{aligned}$$

thus

$$\begin{aligned} \Lambda ^{m,n-q}_{p_0} \simeq \bigoplus _{j+\ell = n-q} \Lambda ^m\textrm{T}_{p_0}^\prime \otimes \Lambda ^j \mathcal {V}_{p_0}^+ \otimes \Lambda ^\ell \mathcal {V}_{p_0}^-. \end{aligned}$$

We single out the factor \(\Lambda ^m \textrm{T}_{p_0}^\prime \otimes \Lambda ^{n-q} \mathcal {V}_{p_0}^-\) and denote by \(\iota ^-\) and \(\pi ^-\) the natural inclusion and projection maps

figure c

Analogously, we may consider a first-order linear partial differential operator

$$\begin{aligned} Q: \mathcal {C}^\infty (\Omega , \Lambda ^{m,n-q}) \rightarrow \mathcal {C}^\infty (\Omega , \Lambda ^{m,n-q}), \end{aligned}$$

thus

$$\begin{aligned} \,^tQ_1(p_0,\sigma ): \Lambda _{p_0}^{0,q} \rightarrow \Lambda _{p_0}^{0,q}, \end{aligned}$$

and if \(\mathcal {L}_{(p_0,\sigma )}\) is non-degenerate, we have the identification

$$\begin{aligned} \Lambda ^{0,q}_{p_0} \simeq \bigoplus _{j+\ell = q} \Lambda ^j \mathcal {V}_{p_0}^+ \otimes \Lambda ^\ell \mathcal {V}_{p_0}^- \end{aligned}$$

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

figure d

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

$$\begin{aligned} Q: \mathcal {C}^\infty (\Omega ,\Lambda ^{0,q}) \rightarrow \mathcal {C}^\infty (\Omega ,\Lambda ^{0,q}) \end{aligned}$$

be a first-order 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 Q-exact at \(p_0\), then

$$\begin{aligned} \pi ^- \circ \,^tQ_1(p_0,\sigma ) \circ \iota ^- = 0. \end{aligned}$$

Analogously, for a given first-order linear partial differential operator

$$\begin{aligned} Q: \mathcal {C}^\infty (\Omega , \Lambda ^{m,n-q}) \rightarrow \mathcal {C}^\infty (\Omega , \Lambda ^{m,n-q}) \end{aligned}$$

that preserves \(\textrm{d}^\prime \)-closed forms, if \((*)_\sigma ^q\)-condition hold and \(\mathcal {V}\) is locally Q-exact at \(p_0\), then

$$\begin{aligned} \pi ^+ \circ \,^tQ_1(p_0,\sigma ) \circ \iota ^+ = 0. \end{aligned}$$

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)

$$\begin{aligned} (x,y,s,t) = (x_1, \dots , x_\nu , y_1, \dots , y_\nu , s_1, \dots , s_d, t_1, \dots , t_\mu ) \end{aligned}$$

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

$$\begin{aligned} \mathcal {L}_{(0,\sigma )}(v,v^\prime ) = \sum _{k=1}^d \sigma _k L \, \overline{L^\prime }[\varphi _k](0) = L\overline{L^\prime }[\varphi ](0), \quad v, v^\prime \in \mathcal {V}_0, \end{aligned}$$

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

$$\begin{aligned} \varphi = \sum _{k=1}^d \sigma _k\varphi _k. \end{aligned}$$

We choose the following basis for \(\mathcal {V}_0\)

$$\begin{aligned} e = \bigg ( \dfrac{\partial }{\partial t_1}\bigg |_0, \dots , \dfrac{\partial }{\partial t_\mu }\bigg |_0, \dfrac{\partial }{\partial {\bar{z}}_1} \bigg |_0, \dots , \dfrac{\partial }{\partial {\bar{z}}_\nu } \bigg |_0 \bigg ). \end{aligned}$$

Thus, we may express \(\mathcal {L}_{(0,\sigma )}\) as a block matrix

$$\begin{aligned} \begin{bmatrix} \dfrac{\partial ^2\varphi }{\partial ^2 t}(0) &{} \dfrac{\partial ^2\varphi }{\partial t \partial z}(0) \\ {} &{} \\ \dfrac{\partial ^2\varphi }{\partial t \partial {\bar{z}}}(0) &{} \dfrac{\partial ^2\varphi }{\partial {\bar{z}} \partial z}(0) \end{bmatrix}. \end{aligned}$$

After a real linear change of coordinates in the s-variable 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 non-degenerate, after another linear change of coordinates we may assume (see the proof of Proposition I.9.1 of [33])

$$\begin{aligned} \varphi _d(z, {\bar{z}}, s, t)= & {} \sum _{\ell =1}^{\lambda } t_\ell ^2 - \sum _{\ell =\lambda +1}^\mu t_\ell ^2 + \sum _{j=1}^{\kappa } |z_j|^2 - \sum _{j=\kappa +1}^{\nu } |z_j|^2 \\ {}{} & {} + O\big (|s|(|z|+|s|+|t|)+|z|^3+|t|^3 \big ), \end{aligned}$$

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

$$\begin{aligned} \mathcal {V}_0^+&= \bigg \langle \dfrac{\partial }{\partial {\bar{z}}_j}\bigg |_0, \; \dfrac{\partial }{\partial t_\ell }\bigg |_0: \begin{matrix} 1 \le j \le \kappa \\ 1 \le \ell \le \lambda \end{matrix} \bigg \rangle , \\ \mathcal {V}_0^-&= \bigg \langle \dfrac{\partial }{\partial {\bar{z}}_j}\bigg |_0, \; \dfrac{\partial }{\partial t_\ell }\bigg |_0: \begin{matrix} \kappa +1 \le j \le \nu \\ \lambda +1 \le \ell \le \mu \end{matrix} \bigg \rangle . \end{aligned}$$

We denote

$$\begin{aligned} z^\prime&= (z_1, \dots , z_\kappa ),\\ z^{\prime \prime }&= (z_{\kappa +1}, \dots , z_\nu ),\\ t^\prime&= (t_1, \dots , t_\lambda ),\\ t^{\prime \prime }&= (t_{\lambda +1}, \dots , t_\mu ), \end{aligned}$$

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

$$\begin{aligned} \big \langle \textrm{d}{\bar{z}}_j|_0, \textrm{d}t_\ell |_0:\, 1 \le j \le \kappa , \;\; 1 \le \ell \le \lambda \big \rangle , \end{aligned}$$

and

$$\begin{aligned} \big \langle \textrm{d}{\bar{z}}_j|_0, \textrm{d}t_\ell |_0:\, \kappa + 1 \le j \le \nu , \;\; \lambda + 1 \le \ell \le \mu \big \rangle , \end{aligned}$$

of \(\mathcal {V}_0^*\), respectively. Therefore, the vector

$$\begin{aligned} \big (\textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z^{\prime \prime }} \wedge \textrm{d}t^{\prime \prime } \big )|_0 \end{aligned}$$

spans \(\Lambda ^m \textrm{T}^\prime _0 \otimes \Lambda ^{n-q} \mathcal {V}^{-}_0\) 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

$$\begin{aligned} \Psi : {\left\{ \begin{array}{ll} \widetilde{z} = z/\tau , \\ \widetilde{s} = s/\tau ^2, \\ \widetilde{t} = t/\tau . \end{array}\right. } \end{aligned}$$

One may take as first-integrals of the push-forward \(\Psi _*\mathcal {V}\) the functions

$$\begin{aligned} \widetilde{Z}_j \big (\widetilde{x},\widetilde{y},\widetilde{s},\widetilde{t}\,\big )&= \widetilde{x}_j + i\widetilde{y}_j, \\ \widetilde{W}_k \big (\widetilde{x},\widetilde{y},\widetilde{s},\widetilde{t}\,\big )&= \widetilde{s}_k + i \widetilde{\varphi }_k \big (\widetilde{x},\widetilde{y},\widetilde{s},\widetilde{t}\,\big ) = \widetilde{s}_k + i \dfrac{ \varphi _k\big (\tau \widetilde{x}, \tau \widetilde{y}, \tau ^2 \widetilde{s}, \tau \widetilde{t}\,\big ) }{\tau ^2}. \end{aligned}$$

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

$$\begin{aligned} \varphi _d(z, {\bar{z}}, s, t) = |t^\prime |^2 - |t^{\prime \prime }|^2 + |z^\prime |^2 - |z^{\prime \prime }|^2 + \tau \, O\big (|s|(|z|+|s|+|t|)+|z|^3+|t|^3 \big ). \end{aligned}$$

Let \(\varepsilon > 0\) be another small number and define

$$\begin{aligned} h = -iW_d + \varepsilon \sum _{k=1}^dW_k^2. \end{aligned}$$

Since

$$\begin{aligned} \sum _{k=1}^d W_k^2 = \bigg (|s|^2 - \sum _{k=1}^d \varphi _k^2 \bigg ) + 2i\sum _{k=1}^d s_k \varphi _k, \end{aligned}$$

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

$$\begin{aligned} \Big |\textrm{Re}\, h - |t^\prime |^2 + |t^{\prime \prime }|^2 - |z^\prime |^2 + |z^{\prime \prime }|^2 - \varepsilon |s|^2 \Big | \le C \big ( \tau (|z|^2+|s|^2+|t|^2)+\varepsilon (|z|^4+|s|^4+|t|^4) \big ) \end{aligned}$$

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

$$\begin{aligned} \Big |\textrm{Re}\, h - |t^\prime |^2 + |t^{\prime \prime }|^2 - |z^\prime |^2 + |z^{\prime \prime }|^2 - \varepsilon |s|^2 \Big | \le \dfrac{1}{2} \big ( |z|^2 + \varepsilon |s|^2 + |t|^2 \big ),\\ \end{aligned}$$

or

$$\begin{aligned} \dfrac{1}{2}(|z^\prime |+\varepsilon |s|^2+|t^\prime |^2) - \dfrac{3}{2}(|z^{\prime \prime }|^2 + |t^{\prime \prime }|^2) \le \Re \, h \le \dfrac{3}{2}(|z^\prime |+\varepsilon |s|^2+|t^\prime |^2) - \dfrac{1}{2}(|z^{\prime \prime }|^2 + |t^{\prime \prime }|^2),\nonumber \\ \end{aligned}$$
(9)

in U. We set

$$\begin{aligned} h_1&= h - 4(|z^\prime |^2 + |t^\prime |^2) - 4 \sum _{k=1}^d W_k^2, \\ h_2&= -h - 4(|z^{\prime \prime }|^2 + |t^{\prime \prime }|^2). \end{aligned}$$

Thus

$$\begin{aligned} \begin{matrix} L_j h_1 = 0, \text { if } \kappa +1 \le j \le \nu &{} \text {or } \nu +\lambda +1 \le j \le n,\\ L_j h_2 = 0, \text { if } 1 \le j \le \kappa \quad \;\;\; &{} \text {or } \nu +1 \le j \le \nu + \lambda , \end{matrix} \end{aligned}$$
(10)

and for a small constant \(a > 0\) we have

$$\begin{aligned} \textrm{Re}\, h_i(z,{\bar{z}},s,t) \le -a(|z|^2+|s|^2+|t|^2), \end{aligned}$$

in U (reduced, if necessary), for \(i=1,2\). Indeed, from (9) we estimate

$$\begin{aligned} \textrm{Re}\, h_1&= \textrm{Re}\, h - 4(|z^\prime |^2 + |t^\prime |^2) - 4 \bigg (|s|^2 - \sum _{k=1}^d \varphi _k^2 \bigg ) \\&\le -\dfrac{1}{2}\big (|z|^2+|t|^2\big ) + \dfrac{3\varepsilon - 8}{2}|s|^2 + 4 \sum _{k=1}^d \varphi _k^2, \end{aligned}$$

and

$$\begin{aligned} \textrm{Re}\, h_2&= - \textrm{Re}\, h - 4(|z^{\prime \prime }|^2 + |t^{\prime \prime }|^2) \\&\le -\dfrac{1}{2}\big (|z|^2+|t|^2\big ) - \dfrac{\varepsilon }{2}|s|^2, \end{aligned}$$

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

$$\begin{aligned} \textrm{Re}\, \rho (h_1 + h_2) \bigg ( \dfrac{x}{\sqrt{\rho }},\dfrac{y}{\sqrt{\rho }},\dfrac{s}{\sqrt{\rho }},\dfrac{t}{\sqrt{\rho }} \bigg )= & {} -4\big (|z|^2+|s|^2+|t|^2\big ) \\ {}{} & {} + \,4 \rho \sum _{k=1}^d \varphi _k^2 \bigg ( \dfrac{x}{\sqrt{\rho }},\dfrac{y}{\sqrt{\rho }},\dfrac{s}{\sqrt{\rho }},\dfrac{t}{\sqrt{\rho }} \bigg ), \end{aligned}$$

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

$$\begin{aligned} \rho \sum _{k=1}^d \varphi _k^2 \bigg ( \dfrac{x}{\sqrt{\rho }},\dfrac{y}{\sqrt{\rho }},\dfrac{s}{\sqrt{\rho }},\dfrac{t}{\sqrt{\rho }} \bigg ) \le \dfrac{C^\prime }{\rho } \big (|z|^2 + |s|^2 + |t|^2\big )^2, \end{aligned}$$

in the reduced U. We reduce it again to ensure \(|z|^2+|s|^2+|t|^2 < 1/(2C^\prime )\) in U. Therefore

$$\begin{aligned} \textrm{Re}\, \rho (h_1 + h_2) \bigg ( \dfrac{x}{\sqrt{\rho }},\dfrac{y}{\sqrt{\rho }},\dfrac{s}{\sqrt{\rho }},\dfrac{t}{\sqrt{\rho }} \bigg ) \le -2(|z|^2+|s|^2+|t|^2), \end{aligned}$$
(11)

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 \(V \Subset U\) such that for every \(\textrm{d}^\prime \)-closed \(f \in \mathcal {C}^\infty (U, \Lambda ^{0,q})\) there is a solution \(u \in \mathcal {D}^\prime (V, \Lambda ^{0,q-1})\) to the equation

$$\begin{aligned} \textrm{d}^\prime u = Qf \end{aligned}$$

in V. We apply Proposition 3.2 to the pair (UV). 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

$$\begin{aligned} \Bigg | \int _\Omega v \wedge Qf \Bigg | \le C \sum _{|\alpha | \le r}\sup _K |\partial ^\alpha f| \sum _{|\alpha | \le r}\sup _{K^\prime } \big |\partial ^\alpha [\textrm{d}^\prime v] \big |, \end{aligned}$$
(12)

for every \(v \in \mathcal {C}^\infty (V,\Lambda ^{m,n-q})\) 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 cut-off function that equals 1 in a neighborhood of the origin and set \(\psi = \chi W_d^2\). We define

$$\begin{aligned} f_\rho&= e^{\rho h_1(z,{\bar{z}},s,t)} \textrm{d}\overline{Z^\prime } \wedge \textrm{d}t^\prime \\ v_\rho&= \rho ^{\frac{m+n}{2}}e^{\rho h_2(z,{\bar{z}},s,t)} \psi (z,{\bar{z}},s,t) \, \textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z^{\prime \prime }} \wedge \textrm{d}t^{\prime \prime } \end{aligned}$$

and

$$\begin{aligned} I(\rho ) = \int v_\rho \wedge Qf_\rho = \int (\,^tQ v_\rho ) \wedge f_\rho , \end{aligned}$$

for each \(\rho > 1\).

We apply the estimate (12) choosing \(K^\prime = \textrm{supp}\, \chi \). For the right-hand side we have \(\textrm{d}^\prime f_\rho = 0\) thanks to (10) and

$$\begin{aligned} \sum _{|\alpha | \le r } \sup _{\overline{V}} |\partial ^\alpha [f_\rho ]| \le \mathrm {Const.}\rho ^{\frac{m+n}{2}+r}. \end{aligned}$$

Identities (10) also imply

$$\begin{aligned} \textrm{d}^\prime v_\rho = \rho ^{\frac{m+n}{2}}e^{\rho h_2} \textrm{d}^\prime [\psi ] \wedge \, \textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z^{\prime \prime }} \wedge \textrm{d}t^{\prime \prime }, \end{aligned}$$

we have

$$\begin{aligned} \sum _{|\alpha | \le r } \sup |\partial ^\alpha [\textrm{d}^\prime v_\rho ]| \le \mathrm {Const.}\rho ^{\frac{m+n}{2}+r}e^{-b\rho } \end{aligned}$$

for some constant \(b>0\). Therefore, the right-hand side of (12) goes to 0 as \(\rho \rightarrow \infty \).

For any given section \(v \in \mathcal {C}^\infty (U,\Lambda ^{m,n-q})\), we may write

$$\begin{aligned} v = \sum _{|I|+|J|=n-q} v_{IJ}(z,{\bar{z}},s,t) \, \textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z}_I \wedge \textrm{d}t_J, \end{aligned}$$

so the operator \(\,^tQ\) acts on v by an expression of the form

$$\begin{aligned} \,^tQ v = \sum _{|R|+|S|=|I|+|J|=n-q} P^{IJ}_{RS}[v_{IJ}] \, \textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z}_R \wedge \textrm{d}t_S, \end{aligned}$$

where \(P^{IJ}_{RS}\) is a first-order linear partial differential operator for each (IJRS).

Thus

$$\begin{aligned} \,^tQ v_\rho = \sum _{|R|+|S|=n-q} P^{I_0 J_0}_{RS} \big [ \rho ^{\frac{m+n}{2}}e^{\rho h_2}\psi \big ] \, \textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z}_R \wedge \textrm{d}t_S, \end{aligned}$$

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 (RS) we split

$$\begin{aligned} P^{I_0J_0}_{RS} = \big (P^{I_0J_0}_{RS}\big )_1 + \big (P^{I_0J_0}_{RS}\big )_0 \end{aligned}$$

in its principal part and its zeroth-order part, thus

$$\begin{aligned} P^{I_0J_0}_{RS}\big [\rho ^{\frac{m+n}{2}}e^{\rho h_2}\psi \big ] = \rho ^{\frac{m+n}{2}}e^{\rho h_2} \Big ( \rho \big (P^{I_0J_0}_{RS}\big )_1[h_2]\psi + \big (P^{I_0J_0}_{RS}\big )_1[\psi ] + \big (P^{I_0J_0}_{RS}\big )_0\psi \Big ). \end{aligned}$$

Setting \(P_{I_0J_0}^{I_0J_0} = P\), we have

$$\begin{aligned} I(\rho )&= \pm \int \rho ^{\frac{m+n}{2}}e^{\rho (h_1 + h_2)} \Big ( \rho P_1[h_2]\psi + P_1[\psi ] + P_0\psi \Big ) \, \textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z} \wedge \textrm{d}t \\&= \pm \int \rho ^{\frac{m+n}{2}}e^{\rho (h_1 + h_2)} \Big ( \rho P_1[h_2]\psi + P_1[\psi ] + P_0\psi \Big ) \det \bigg (I_d + \frac{\partial \Phi }{\partial s}\bigg ) \textrm{d}x \, \textrm{d}y \, \textrm{d}s \, \textrm{d}t, \end{aligned}$$

where the ±-sign of the integral comes from reordering of 1-forms and is immaterial in the following. If we change scale

$$\begin{aligned} (x^\prime ,y^\prime ,s^\prime ,t^\prime )&= \sqrt{\rho } \, (x,y,s,t),\\ \textrm{d}x^\prime \, \textrm{d}y^\prime \, \textrm{d}s^\prime \, \textrm{d}t^\prime&= \rho ^{\frac{m+n}{2}} \textrm{d}x \, \textrm{d}y \, \textrm{d}s \, \textrm{d}t, \end{aligned}$$

we get (after renaming of variables by removing primes)

$$\begin{aligned} I(\rho )&= \pm \int e^{\rho (h_1 + h_2)\big (\frac{x}{\sqrt{\rho }},\frac{y}{\sqrt{\rho }},\frac{s}{\sqrt{\rho }},\frac{t}{\sqrt{\rho }}\big )} \Bigg ( \rho \, \alpha \bigg (\frac{x}{\sqrt{\rho }},\frac{y}{\sqrt{\rho }},\frac{s}{\sqrt{\rho }},\frac{t}{\sqrt{\rho }}\bigg ) \nonumber \\&\quad \, \,+ \beta \bigg (\frac{x}{\sqrt{\rho }},\frac{y}{\sqrt{\rho }},\frac{s}{\sqrt{\rho }},\frac{t}{\sqrt{\rho }}\bigg ) + \gamma \bigg (\frac{x}{\sqrt{\rho }},\frac{y}{\sqrt{\rho }},\frac{s}{\sqrt{\rho }},\frac{t}{\sqrt{\rho }}\bigg ) \Bigg ) \textrm{d}x \, \textrm{d}y \, \textrm{d}s \, \textrm{d}t \end{aligned}$$
(13)

where

$$\begin{aligned} \alpha&= P_1[h_2]\psi \det \bigg (I_d + \frac{\partial \varphi }{\partial s}\bigg ),\\ \beta&= P_1[\psi ] \det \bigg (I_d + \frac{\partial \varphi }{\partial s}\bigg ),\\ \gamma&= P_0\psi \det \bigg (I_d + \frac{\partial \varphi }{\partial s}\bigg ). \end{aligned}$$

Notice that since \(W_d(0) = 0\), we have \(\alpha (0) = \gamma (0) = 0\) and since

$$\begin{aligned} P_1[\psi ]&= P_1[\chi W_d^2] \\&= \Big (P_1[\chi W_d] + \chi P_1[W_d]\Big )W_d, \end{aligned}$$

we also have \(\beta (0) = 0\). For any XY in the set

$$\begin{aligned} \bigg \{\frac{\partial }{\partial x_j}, \frac{\partial }{\partial y_j},\frac{\partial }{\partial t_\ell }, \frac{\partial }{\partial s_k}: 1 \le j \le \nu , \;\; 1 \le \ell \le \mu , \;\; 1 \le k \le d \bigg \} \end{aligned}$$

the product rule entails

$$\begin{aligned} X\alpha (0)&= 0, \\ YX\alpha (0)&= {\left\{ \begin{array}{ll} P_1[h_2](0), &{}\text { if } X = Y = \dfrac{\partial }{\partial s_d} \\ 0, &{}\text { otherwise.} \end{array}\right. } \end{aligned}$$

If we write

$$\begin{aligned} P_1 = \sum _{j=1}^\nu \bigg \{ a_jL_j + a_j^\prime M_j \bigg \} + \sum _{\ell =1}^\mu b_\ell L_{\nu + \ell } + \sum _{k=1}^d c_kM_{\nu +k}, \end{aligned}$$

then

$$\begin{aligned} P_1[h_2](0)&= P_1\big [-h - 4(|z^{\prime \prime }|^2 + |t^{\prime \prime }|^2)\big ](0) \\&= P_1[-h](0) \\&= P_1\bigg [iW_d - \varepsilon \sum _{k=1}^d W_k^2\bigg ](0) \\&= P_1[iW_d](0) \\&= ic_d(0). \end{aligned}$$

Thus Taylor’s formula implies

$$\begin{aligned} \alpha (x,y,s,t) = ic_d(0) s_d^2 + O\big (|x|^3+|y|^3+|s|^3+|t|^3\big ), \end{aligned}$$

therefore the integrand in (13) converges pointwise to

$$\begin{aligned} e^{-4(|x|^2+|y|^2+|s|^2+|t|^2)} ic_d(0) s_d^2, \end{aligned}$$

as \(\rho \rightarrow \infty \) since \(\beta (0)=\gamma (0)=0\). By the estimate (11), we can apply Lesbegue’s Dominated Convergence Theorem, to conclude

$$\begin{aligned} I(\rho ) \; \longrightarrow \; \pm \int e^{-4(|x|^2+|y|^2+|s|^2+|t|^2)} ic_d(0) s_d^2 \, \textrm{d}x \, \textrm{d}y \, \textrm{d}s \, \textrm{d}t, \end{aligned}$$

as \(\rho \rightarrow \infty \). Therefore \(c_d(0) = 0\). Since \(\,^tQ_1(0,\sigma )(\omega |_0) = \,^tQ_1(W_d\omega )|_0\) for any \(\omega \in \mathcal {C}^\infty (0,\Lambda ^{m,n-q})\), 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

$$\begin{aligned} \pi _- \Big ( \,^tQ_1 \big ( W_d \, \textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z^{\prime \prime }} \wedge \textrm{d}t^{\prime \prime } \big )|_0 \Big ) = P_1(W_d)(0) \big (\textrm{d}Z \wedge \textrm{d}W \wedge \textrm{d}\overline{Z^{\prime \prime }} \wedge \textrm{d}t^{\prime \prime } \big )|_0 = 0, \end{aligned}$$

and the proof is complete. \(\square \)

Example 5.3

Let us consider the co-rank 1 structure in \(\mathbb {R}^3\), with (fine-regular) coordinates \((s,t) = (s,t_1,t_2)\), generated by the first-integral

$$\begin{aligned} W(s,t) = s + i \big (t_1^2 - t_2^2 \big ). \end{aligned}$$

A frame for \(\mathcal {V}\) is given by the following vector-fields

$$\begin{aligned} L_1&= \dfrac{\partial }{\partial t_1} - 2it_1 \dfrac{\partial }{\partial s}, \\ L_2&= \dfrac{\partial }{\partial t_2} + 2it_2 \dfrac{\partial }{\partial s}, \end{aligned}$$

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

$$\begin{aligned} \big [ \mathcal {L}_{(0,\textrm{d}s)} \big ] = \begin{bmatrix} 1 &{} 0 \\ 0 &{} -1 \end{bmatrix}. \end{aligned}$$

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

$$\begin{aligned}{}[Q] = \begin{bmatrix} aL_1 + bL_2 + cM &{} 0 \\ 0 &{} aL_1 + bL_2 + cM \end{bmatrix}, \end{aligned}$$

preserves \(\textrm{d}^\prime \)-closed forms. Theorem 5.2 ensures that \(\textrm{d}^\prime \) is not locally Q-exact 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

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u}{\partial t_1} - 2it_1 \dfrac{\partial u}{\partial s} = a(s,t)\dfrac{\partial f_1}{\partial t_1} + b(s,t)\dfrac{\partial f_1}{\partial t_2} + \big (-2i(t_1a(s,t) - t_2b(s,t)) + c(s,t) \big ) \dfrac{\partial f_1}{\partial s}, \\ \dfrac{\partial u}{\partial t_2} + 2it_2 \dfrac{\partial u}{\partial s} = a(s,t)\dfrac{\partial f_2}{\partial t_1} + b(s,t)\dfrac{\partial f_2}{\partial t_2} + \big (-2i(t_1a(s,t) - t_2b(s,t)) + c(s,t) \big ) \dfrac{\partial f_2}{\partial s}, \end{array}\right. } \end{aligned}$$

does not admit any distribution solution u.

6 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 \(\mathcal {V}\) be a locally integrable structure of rank \(n = N-1\) on a \(\mathcal {C}^\infty \)-smooth N-manifold \(\mathcal {M}\) 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 \((N-2)\)-submanifold of \(\mathcal {M}\). As a consequence of Baouendi-Treves 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

$$\begin{aligned} I_{U,V,\mathcal {S}}^q: \textrm{H}^q(\mathcal {S} \cap U) \times \textrm{H}_q(\mathcal {S} \cap V) \rightarrow \mathbb {C}\end{aligned}$$

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

$$\begin{aligned} \textrm{H}_q(\mathcal {S} \cap V) \simeq \textrm{H}^{N-2-q}_c(\mathcal {S} \cap V) = \textrm{H}^{n-1-q}_c(\mathcal {S} \cap V). \end{aligned}$$

By reduced we mean that the space \(\textrm{H}_0(\mathcal {S} \cap V)\) is computed as follows (see Definition 2.1 of [10])

$$\begin{aligned} \textrm{H}^{n-1}_c(\mathcal {S} \cap V) = \dfrac{\big \{ \sigma \in \mathcal {C}^\infty _c \big ( \mathcal {S} \cap V, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{n-1}\mathbb {C}\textrm{T}^*\mathcal {S} \big ): \int _\mathcal {S} \sigma = 0 \big \}}{\big \{ \textrm{d}\tau : \tau \in \mathcal {C}^\infty _c \big ( \mathcal {S} \cap V, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{n-2} \mathbb {C}\textrm{T}^*\mathcal {S} \big ) \big \}}, \end{aligned}$$

while the remaining homology spaces are computed in the usual fashion. Under this identification, the intersection number is defined by

$$\begin{aligned} I_{U,V,\mathcal {S}}^q([\beta ],[\gamma ]) = \int _\mathcal {S} \beta \wedge \gamma , \qquad \qquad {([\beta ],[\gamma ]) \in \textrm{H}^q(\mathcal {S} \cap U) \times \textrm{H}^{n-1-q}_c(\mathcal {S} \cap V),} \end{aligned}$$

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 1.16

Let \(\mathcal {V}\) be a locally integrable structure of rank \(n = N-1\) on a \(\mathcal {C}^\infty \)-smooth N-manifold \(\mathcal {M}\) 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 \(I^{q-1}_{U,V,\mathcal {S}} \equiv 0\) for every level set \(\mathcal {S}\) that is noncritical in V.

Remark 6.2

The property \(I^{q-1}_{U,V,\mathcal {S}} \equiv 0\) in Theorem 6.1 is equivalent to the following assertion (see Section 5 in [10]):

“The natural map

$$\begin{aligned} \textrm{H}_{q-1}(\mathcal {S} \cap V) \rightarrow \textrm{H}_{q-1}(\mathcal {S} \cap U) \end{aligned}$$

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 \(\beta \in \mathcal {C}^\infty \big (\mathcal {S}\cap U, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q-1}\mathbb {C}\textrm{T}^*\mathcal {S} \big )\) and \(\gamma \in \mathcal {C}^\infty _c \big (\mathcal {S} \cap V, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{n-q} \mathbb {C}\textrm{T}^*\mathcal {S} \big )\) such that \(I^{q-1}_{U,V,\mathcal {S}} ([\beta ],[\gamma ]) \ne 0\). One then constructs sections of the associated bundles \(\Phi (\beta ) \in \mathcal {C}^\infty (U, \Lambda ^{0,q})\) and \(\Upsilon (\gamma ) \in \mathcal {C}^\infty _c(V,\Lambda ^{1,n-q})\) (by a procedure to be described later) such that \(\textrm{d}^\prime \Phi (\beta ) = 0\) and

$$\begin{aligned} \int _U \Phi (\beta ) \wedge \Upsilon (\gamma ) \ne 0. \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{Re}\, h \le 0, \text { on } \textrm{supp}\, \Phi (\beta ),\\ \textrm{Re}\, h > 0, \text { on } \textrm{supp}\, \textrm{d}^\prime \Upsilon (\gamma ). \end{array}\right. } \end{aligned}$$

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 Q-exactness 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

$$\begin{aligned} W(s,t) = s + i \varphi (s,t). \end{aligned}$$

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

$$\begin{aligned} \mathcal {S}&= W^{-1}(w_0) = \{(s_0,t) \in \Omega : \varphi (s_0,t) = r_0 \}, \\ \mathcal {S}_0&= \{t \in \mathbb {R}^n : (s_0,t) \in \Omega \text { and } \varphi (s_0,t) = r_0 \}, \\ U_0&= \{t \in \mathbb {R}^n : (s_0,t) \in U\}, \\ U_0^+&= \{t \in U_0 : \varphi (s_0,t) > r_0\}, \\ U_0^-&= \{t \in U_0 : \varphi (s_0,t) < r_0\}. \end{aligned}$$

From now on we identify \(\mathcal {S}\) with \(\mathcal {S}_0\) via the map \((s_0,t) \mapsto t\). Let \(\beta \in \mathcal {C}^\infty \big (\mathcal {S}_0 \cap U_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q-1}\mathbb {C}\textrm{T}^*\mathcal {S}_0\big )\) and \(\gamma \in \mathcal {C}^\infty _c\big (\mathcal {S}_0 \cap V_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{n-q}\mathbb {C}\textrm{T}^*\mathcal {S}_0\big )\) be a pair of closed forms verifying \(I^{q-1}_{U_0,V_0,\mathcal {S}_0} ([\beta ],[\gamma ]) \ne 0\). As in the proof of Proposition 3.2 of [10], there are forms \(g_+, g_- \in \mathcal {C}^\infty \big ( U_0, \bigwedge ^{q-1} \mathbb {C}\textrm{T}^*\mathbb {R}^n \big )\) and \(u_+,u_- \in \mathcal {C}^\infty _c \big ( V_0,\bigwedge ^{n-q} \mathbb {C}\textrm{T}^*\mathbb {R}^n \big )\) such that:

$$\begin{aligned} \textrm{supp}\, \textrm{d}g_- \cup \textrm{supp}\, \textrm{d}u_-&\subset U_0^-, \\ \textrm{supp}\, \textrm{d}g_+ \cup \textrm{supp}\, \textrm{d}u_+&\subset U_0^+, \end{aligned}$$

the decomposition

$$\begin{aligned} \beta = g_+|_{\mathcal {S}_0} + g_-|_{\mathcal {S}_0} \end{aligned}$$

holds and

$$\begin{aligned} \int _{\mathcal {S}_0}g_- \wedge \gamma&= \int _{U_0^+} g_- \wedge \textrm{d}u_+ = \pm \int _{\mathcal {S}_0}g_- \wedge u_+, \\ \int _{\mathcal {S}_0}g_+ \wedge \gamma&= \int _{U_0^-} g_+ \wedge \textrm{d}u_- = \pm \int _{\mathcal {S}_0}g_+ \wedge u_-. \end{aligned}$$

The condition \(I^{q-1}_{U_0,V_0,\mathcal {S}_0} ([\beta ],[\gamma ]) \ne 0\) 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

$$\begin{aligned} \varphi (s_0,t) > r_0 + \rho \end{aligned}$$

for every \(t \in \textrm{supp}\, \textrm{d}u\). Taylor’s formula in the s-variable ensures the existence of a constant \(A > 0\) such that the following estimate holds

$$\begin{aligned} |\varphi (s,t) - \varphi (s_0,t)| \le A|s - s_0|, \end{aligned}$$

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

$$\begin{aligned} \varphi (s,t) > r_0 + \dfrac{3 \rho }{4}, \end{aligned}$$

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

$$\begin{aligned} \chi : (s_0 - \eta ^\prime , s_0 + \eta ^\prime ) \rightarrow [0, +\infty ), \end{aligned}$$

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

$$\begin{aligned} {\mathfrak {R}} = \big \{z \in \mathbb {C}: |\textrm{Re}\, z - s_0| \le \delta , \; |\text {Im} \, z - r_0 - \rho /2| \le \varepsilon \big \} \end{aligned}$$

with \(G > 0\) in the interior of \({\mathfrak {R}}\). We define \(\Phi (\beta ) \in \mathcal {C}^\infty (U, \Lambda ^{0,q})\) and \(\Upsilon (\gamma ) \in \mathcal {C}^\infty _c(V,\Lambda ^{1,n-q})\) by the expressions

$$\begin{aligned} \Phi (\beta )&= \pi ^{0,q}\big [ (G \circ W )|_U \, \textrm{d}\overline{W} \wedge \pi ^*g \big ], \\ \Upsilon (\gamma )&= \chi (s) \, \textrm{d}W \wedge \pi ^*u, \end{aligned}$$

where \(\pi \) is the projection \((s,t) \mapsto t\) and \(\pi ^{0,q}\) is the projection \( \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^q\mathbb {C}\textrm{T}^*U = \Lambda ^{0,q} \oplus \textrm{T}^{1,q-1} \rightarrow \Lambda ^{0,q}\). We point out that \(\Lambda ^{1,n-q} = \textrm{T}^{1,n-q}\) 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

$$\begin{aligned} {\left\{ \begin{array}{ll} \textrm{Re}\, h \le 0, \text { on } \textrm{supp}\, \Phi (\beta ),\\ \textrm{Re}\, h > 0, \text { on } \textrm{supp}\, \textrm{d}^\prime \Upsilon (\gamma ). \end{array}\right. } \end{aligned}$$

Theorem 1.18

Let \(\mathcal {V}\) be a locally integrable structure of rank \(n = N-1\) on a \(\mathcal {C}^\infty \)-smooth N-manifold \(\mathcal {M}\) 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

$$\begin{aligned} Q: \mathcal {C}^\infty (\Omega ,\Lambda ^{0,q}) \rightarrow \mathcal {C}^\infty (\Omega ,\Lambda ^{0,q}) \end{aligned}$$

be a first-order linear partial differential operator that preserves \(\textrm{d}^\prime \)-closed forms. If \(\mathcal {V}\) is locally weakly Q-exact 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 \(\beta \in \mathcal {C}^\infty \big (\mathcal {S}_0 \cap U_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q-1}\big )\) and \(\gamma \in \mathcal {C}^\infty _c\big (\mathcal {S}_0 \cap V_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{n-q}\big )\) verifying \(I^{q-1}_{U_0,V_0,\mathcal {S}_0} ([\beta ],[\gamma ]) \ne 0\) we have

$$\begin{aligned} \int _U \,^tQ \big [\Upsilon (\gamma )\big ] \wedge \Phi (\beta ) = 0. \end{aligned}$$

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 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 (st) as above and the same fundamental system of neighbourhoods of the origin \(\mathcal {U}\). As above, let \(\beta \in \mathcal {C}^\infty \big (\mathcal {S}_0 \cap U_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q-1}\mathbb {C}\textrm{T}^*\mathcal {S}_0\big )\) and \(\gamma \in \mathcal {C}^\infty _c\big (\mathcal {S}_0 \cap V_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{n-q}\mathbb {C}\textrm{T}^*\mathcal {S}_0\big )\) be a pair of closed forms verifying \(I^{q-1}_{U_0,V_0,\mathcal {S}_0} ([\beta ],[\gamma ]) \ne 0\) and \(u \in \mathcal {C}^\infty _c\big (V_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{n-q} \mathbb {C}\textrm{T}^*\mathbb {R}^n\big )\), \(g \in \mathcal {C}^\infty \big (U_0, \mathchoice{{\textstyle \bigwedge }}{{\bigwedge }}{{\textstyle \wedge }}{{\scriptstyle \wedge }} ^{q-1}\mathbb {C}\textrm{T}^*\mathbb {R}^n\big )\) be the corresponding forms in the t-space. We may write in our coordinates

$$\begin{aligned} u&= \sum _{|I|=n-q} u_I(t) \textrm{d}t_I, \\ g&= \sum _{|J|=q-1} g_J(t) \textrm{d}t_J, \end{aligned}$$

where the sum runs over ordered multi-indexes 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

$$\begin{aligned} \textrm{supp}\, \psi&= [- \delta , \delta ], \\ \textrm{supp}\, \zeta&= [r_0 + \rho /2 - \varepsilon , r_0 + \rho /2 + \varepsilon ], \end{aligned}$$

and \(>0\) in the interior of the supports and \(\int \psi = 1\). Notice that for each \(0< \lambda < 1\) the function

$$\begin{aligned} G_\lambda : \mathbb {C}\ni z \mapsto \dfrac{\psi \big ((\textrm{Re}\, z - s_0)/\lambda \big )}{\lambda }\zeta (\text {Im} \, z) \in [0,+\infty ), \end{aligned}$$

has support in the rectangle

$$\begin{aligned} {\mathfrak {R}} = \big \{z \in \mathbb {C}: |\textrm{Re}\, z - s_0| \le \delta , \; |\text {Im} \, z - r_0 - \rho /2| \le \varepsilon \big \}, \end{aligned}$$

and \(G_\lambda > 0\) in the interior of \({\mathfrak {R}}\). For this family of choices of G we have

$$\begin{aligned} \Phi _\lambda (\beta )&= \pi ^{0,q} \big [ (G_\lambda \circ W )|_U \, \textrm{d}\overline{W} \wedge \pi ^*g \big ] = \pi ^{0,q} \bigg [ G_\lambda (s+i\varphi (s,t)) \sum _{|J|=q-1} g_J(t) \, \textrm{d}\overline{W} \wedge \textrm{d}t_J \bigg ], \\ \Upsilon (\gamma )&= \chi (s) \, \textrm{d}W \wedge \pi ^*u = \chi (s) \sum _{|I|=n-q} u_I(t) \, \textrm{d}W \wedge \textrm{d}t_I, \end{aligned}$$

The operator \(\,^tQ\) acts on \(\Upsilon (\gamma )\) by an expression of the form

$$\begin{aligned} \,^tQ \big [ \Upsilon (\gamma ) \big ] = \sum _{|I|=|R|=n-q} P^I_R\big [\chi u_I \big ] \, \textrm{d}W \wedge \textrm{d}t_R, \end{aligned}$$

where \(P^I_R\) is a first-order linear partial differential operator for each (IR). Since

$$\begin{aligned} \,^tQ \big [ \Upsilon (\gamma ) \big ] \wedge \Phi _\lambda (\beta ) = \,^tQ \big [ \Upsilon (\gamma ) \big ] \wedge \big ( (G_\lambda \circ W )|_U \, \textrm{d}\overline{W} \wedge \pi ^*g \big ), \end{aligned}$$

we have by Theorem 6.3

$$\begin{aligned} 0&= \int \,^tQ \big [ \Upsilon (\gamma ) \big ] \wedge \Phi _\lambda (\beta ) \\&= \int \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1 \end{matrix}} } (-1)^{n-q} P^I_R\big [\chi u_I\big ](s,t) \dfrac{\psi ((s-s_0)/\lambda )}{\lambda } \zeta (\varphi (s,t)) g_J(t) \textrm{d}W \wedge \textrm{d}\overline{W} \wedge \textrm{d}t_R \wedge \textrm{d}t_J \\&= \int \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1\\ 1 \le j \le n \end{matrix}} } (-1)^{n-q} \dfrac{\psi \big ((s-s_0)/\lambda \big )}{\lambda } g_J(t) \bigg ( P^I_R\big [\chi u_I\big ] (\zeta \circ \varphi ) (-2i)\dfrac{\partial \varphi }{\partial t_j} \bigg )\\&\quad (s,t) \, \textrm{d}s \wedge \textrm{d}t_j \wedge \textrm{d}t_R \wedge \textrm{d}t_J \\&= \int _{V_0} \int _{s_0-\frac{\delta \eta ^\prime }{2}}^{s_0+\frac{\delta \eta ^\prime }{2}} \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1\\ 1 \le j \le n \end{matrix}} } \varepsilon _{jRJ} \dfrac{\psi \big ((s-s_0)/\lambda \big )}{\lambda } g_J(t) \bigg ( P^I_R\big [\chi u_I\big ] (\zeta \circ \varphi ) (-2i)\dfrac{\partial \varphi }{\partial t_j} \bigg ) (s,t) \, \textrm{d}s \, \textrm{d}t \\&= \int _{V_0} \int _{-\frac{\eta ^\prime }{2}}^{\frac{\eta ^\prime }{2}} \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1\\ 1 \le j \le n \end{matrix}} } \varepsilon _{jRJ} \psi (s^\prime ) g_J(t) \bigg ( P^I_R\big [\chi u_I\big ] (\zeta \circ \varphi ) (-2i)\dfrac{\partial \varphi }{\partial t_j} \bigg ) (s_0 + \lambda s^\prime , t) \, \textrm{d}s^\prime \, \textrm{d}t, \end{aligned}$$

where \(\varepsilon _{jRJ} \in \{-1,0,1\}\) is characterized by

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon _{jRJ} = 0, &{}\text { if } j \in R \cup J \text { or } R \cap J \ne \varnothing ; \\ (-1)^{n-q}\textrm{d}t_j \wedge \textrm{d}t_R \wedge \textrm{d}t_J = \varepsilon _{jRJ} \, \textrm{d}t, &{}\text { otherwise.} \end{array}\right. } \end{aligned}$$

Therefore, letting \(\lambda \rightarrow 0\), we have

$$\begin{aligned} \int&\,^tQ \big [ \Upsilon (\gamma ) \big ] \wedge \Phi _\lambda (\beta ) \\&\longrightarrow \;\; \int _{V_0} \int _{-\frac{\eta ^\prime }{2}}^{\frac{\eta ^\prime }{2}} \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1\\ 1 \le j \le n \end{matrix}} } \varepsilon _{jRJ} \psi (s^\prime ) g_J(t) P^I_R\big [\chi u_I\big ](s_0,t) \zeta (\varphi (s_0,t)) (-2i)\dfrac{\partial \varphi }{\partial t_j}(s_0,t) \, \textrm{d}s^\prime \, \textrm{d}t \\&= \int _{V_0} \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1\\ 1 \le j \le n \end{matrix}} } \varepsilon _{jRJ} g_J(t) P^I_R\big [\chi u_I\big ](s_0,t) \zeta (\varphi (s_0,t)) (-2i)\dfrac{\partial \varphi }{\partial t_j}(s_0,t) \, \textrm{d}t \\&= \int _{V_0} \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1\\ 1 \le j \le n \end{matrix}} } (-1)^{n-q} g_J(t) P^I_R\big [\chi u_I\big ](s_0,t) \zeta (\varphi (s_0,t)) (-2i)\dfrac{\partial \varphi }{\partial t_j}(s_0,t) \, \textrm{d}t_j \wedge \textrm{d}t_R \wedge d t_J \\&= -2i \int _{V_0} \omega \wedge \textrm{d}\varphi _{s_0} \wedge g, \end{aligned}$$

where

$$\begin{aligned} \omega&= i_{s_0}^*\big ( (\zeta \circ \varphi ) \,^tQ \big [ \Upsilon (\gamma ) \big ] \big ), \\ \varphi _{s_0}&= i_{s_0}^*(\varphi ), \\ i_{s_0}(t)&= (s_0,t), \end{aligned}$$

for all \(t \in U_0\). Thus

$$\begin{aligned} \int _{V_0} \omega \wedge g \wedge \textrm{d}\varphi _{s_0} = 0. \end{aligned}$$
(14)

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 \(\mathcal {D}^\prime \big ((r_0 + \rho /2 - \varepsilon , r_0 + \rho /2 + \varepsilon )\big )\)

$$\begin{aligned} \lim _{\tau \rightarrow 0} \zeta _\tau = \delta _{r_1}. \end{aligned}$$

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])

$$\begin{aligned} \lim _{\tau \rightarrow 0} \varphi _{s_0}^*\zeta _\tau = \varphi _{s_0}^*\delta _{r_1} = \dfrac{1}{|\textrm{D} (\varphi _{s_0})|}\,\textrm{d}\sigma , \end{aligned}$$

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

$$\begin{aligned} |\textrm{D} (\varphi _{s_0})| = \bigg ( \sum _{j=1}^n \bigg |\dfrac{\partial \varphi }{\partial t_j}(s_0,\,\cdot \,)\bigg |^2 \bigg )^{\frac{1}{2}}. \end{aligned}$$

Equation (14) becomes

$$\begin{aligned} 0&= \int (\varphi _{s_0}^*\zeta _\tau ) \, i_{s_0}^*\big (\,^tQ \big [ \Upsilon (\gamma ) \big ]\big ) \wedge \textrm{d}\varphi _{s_0} \wedge g \\&= \int _{V_0} (\varphi _{s_0}^*\zeta _\tau )(t) \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1\\ 1 \le j \le n \end{matrix}} } \varepsilon _{jRJ} P^I_R\big [\chi u_I\big ](s_0,t) \dfrac{\partial \varphi }{\partial t_j}(s_0,t) g_J(t) \, \textrm{d}t \\&= \Bigg \langle \varphi _{s_0}^*\zeta _\tau , \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1\\ 1 \le j \le n \end{matrix}} } \varepsilon _{jRJ} P^I_R\big [\chi u_I\big ](s_0,\,\cdot \,) \dfrac{\partial \varphi }{\partial t_j}(s_0,\,\cdot \,) g_J(\,\cdot \,) \Bigg \rangle \\&\longrightarrow \int _{\varphi _{s_0}^{-1}(r_1)} \dfrac{1}{|\textrm{D} (\varphi _{s_0})|} \sum _{ {\begin{matrix} |I|=|R|=n-q\\ |J| = q-1\\ 1 \le j \le n \end{matrix}} } \varepsilon _{jRJ} P^I_R\big [\chi u_I\big ](s_0,\,\cdot \,) g_J(\,\cdot \,) \dfrac{\partial \varphi }{\partial t_j}(s_0,\,\cdot \,) \,\textrm{d}\sigma . \end{aligned}$$

Therefore, if \(\mathcal {V}\) 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.