Gevrey local solvability in locally integrable structures

Abstract

We consider a locally integrable real-analytic structure, and we investigate the local solvability in the category of Gevrey functions and ultradistributions of the complex $\mathrm{d}^{\prime }$ naturally induced by the de Rham complex. We prove that the so-called condition $Y(q)$ on the signature of the Levi form, for local solvability of $\mathrm{d}^{\prime }u=f$, is still necessary even if we take $f$ in the classes of Gevrey functions and look for solutions $u$ in the corresponding spaces of ultradistributions.

1 Introduction and statement of the results

Consider a real-analytic manifold $M$ of dimension $m+n$. A real-analytic locally integrable structure on $M$, of rank $n$, is defined by a real-analytic subbundle $\mathcal V \subset \mathbb C TM$ of rank $n$, satisfying the Frobenius condition. In fact, in the real-analytic category the Frobenius condition implies that the subbundle $T^\prime \subset \mathbb C T^*M$ orthogonal to $\mathcal V$ is locally spanned by exact differentials and hence the structure is locally integrable (see e.g. [3, Theorem I.9.2]). As usual we will denote by $T^0=T^\prime \cap T^*M$ the so-called characteristic set. For any open subset $\Omega \subset M$ and $s>1$ the space $G^s(\Omega , \Lambda ^{p,q})$ of $(p,q)$-forms with Gevrey coefficients of order $s$ is then defined (see Section 3 below and Treves [29]) and the de Rham differential induces a map

$$\begin{aligned} \mathrm{d}^\prime : G^s(\Omega , \Lambda ^{p,q})\rightarrow G^s(\Omega , \Lambda ^{p,q+1}). \end{aligned}$$

Similarly, the de Rham differential induces a complex on the space of ”ultra-currents” $\mathcal D ^{\prime }_s(\Omega , \Lambda ^{p,q})$, i.e. forms with ultradistribution coefficients:

$$\begin{aligned} \mathrm{d}^\prime : \mathcal D ^{\prime }_s(\Omega , \Lambda ^{p,q})\rightarrow \mathcal D ^{\prime }_s(\Omega , \Lambda ^{p,q+1}). \end{aligned}$$

When $\mathcal V \cap \overline{\mathcal{V }}=0$ the structure is called $CR$ and $\mathrm{d}^\prime$ is the so-called tangential Cauchy-Riemann operator.

We are interested in necessary conditions for the Gevrey local solvability problem for the complex $\mathrm{d}^{\prime }$ to hold near a given point $x_0$.

Definition 1.1

We say that the complex $\mathrm{d}^{\prime }$ is locally solvable near $x_0$ and in degree $q,\,1\le q\le n$, in the sense of ultradistribution of order $s$, if for every sufficiently small open neighborhood $\Omega$ of $x_0$ and every cocycle $f\in G^s(\Omega ,\Lambda ^{0,q})$ there exists an open neighborhood $V\subset \Omega$ of $x_0$ and a ultradistribution section $u\in \mathcal D ^\prime _s(V,\Lambda ^{0,q-1})$ solving $\mathrm{d}^\prime u=f$ in $V$.

The analogous problem in the setting of smooth functions and Schwartz distributions has been extensively considered, see e.g. [1–3, 8–12, 20, 22–24, 26, 29], inspired by the results in [16, 17] for scalar operators of principal type; see also [18, 19] as general references for the problem of local solvability of scalar linear partial differential operators.

Several geometric invariants were there introduced, e.g. the signature of the Levi form recalled below, which represent obstructions to the solvability in the sense of distributions, that is, for some smooth $f\in C^\infty (\Omega ,\Lambda ^{0,q})$ there is no distribution solution $u\in \mathcal D ^{\prime }(V,\Lambda ^{0,q-1})$ to $\mathrm{d}^{\prime }u=f$ in $V$, for every neighborhood $V\subset \Omega$ of $x_0$.

It is therefore natural to wonder whether, under the same condition as in the smooth category, $\mathrm{d}^{\prime }$ is still non-solvable even if we choose $f$ in the smaller class of Gevrey functions $G^s(\Omega ,\Lambda ^{0,q})\subset C^\infty (\Omega ,\Lambda ^{0,q})$ and we look for solutions in the larger class of ultradistributions $\mathcal D ^{\prime }_s(V,\Lambda ^{0,q})\supset \mathcal D ^{\prime }(V,\Lambda ^{0,q})$, as in Definition 1.1. In this note we present a result in this direction.

Let us note that general sufficient conditions for local solvability in the Gevrey category have been recently obtained in [5]; see also [4, 21].

We recall that at any point $(x_0,\omega _0)\in T^0$ it is well defined a sesquilinear form $\mathcal B _{(x_0,\omega _0)}:\mathcal V _{x_0} \times \mathcal V _{x_0}\rightarrow \mathbb C$, ($\mathcal V _{x_0}$ is the fiber above $x_0$) by

$$\begin{aligned} \mathcal B _{(x_0,\omega _0)}(\mathbf v _1,\mathbf v _2)=\left\langle \omega _0,(2\iota )^{-1}[V_1,\overline{V_2}]|_{x_0}\right\rangle , \end{aligned}$$

with $\mathbf v _1,\mathbf v _2\in \mathcal V _{x_0}$, where $V_1$ and $V_2$ are smooth sections of $\mathcal V$ such that $V_1|_{x_0}=\mathbf v _1,\,V_2|_{x_0}=\mathbf v _2$. The associated quadratic form $\mathcal V _{x_0} \ni \mathbf v \mapsto \mathcal B _{(x_0,\omega _0)} (\mathbf v ,\mathbf v )$, or $\mathcal B _{(x_0,\omega _0)}$ itself, is known as Levi form.

Here is our result.

Theorem 1.2

Let $(x_0,\omega _0)\in T^0,\,\omega _0\not =0$. Suppose that $\mathcal B _{(x_0,\omega _0)}$ has exactly $q$ positive eigenvalues, $1\le q\le n$, and $n-q$ negative eigenvalues, and that its restriction to $\mathcal V _{x_0}\cap \overline{\mathcal{V }}_{x_0}$ is non-degenerate.

Then, for every $s>1,\,\mathrm{d}^{\prime }$ is not locally solvable in the sense of ultradistributions of order $s$, near $x_0$ and in degree $q$.

The condition that $\mathcal B _{(x_0,\omega _0)}$ does not have neither $q$ positive eigenvalues and $n-q$ negative eigenvalues, nor viceversa, is known in the literature as condition $Y(q)$ (at $(x_0,\omega _0)$). Hence the above result states that the condition $Y(q)$ is necessary for local solvability in degree $q$ in the sense of ultradistributions of order $s$, for every $s>1$, at least when the restriction of the Levi form to $\mathcal V _{x_0}\cap \overline{\mathcal{V }}_{x_0}$ is non-degenerate.

This result therefore strengthens the analogous one in the category of Schwartz distributions, which was proved in [1] for $CR$ manifolds and in [29, Theorem XVIII.3.1] for general locally integrable structure (following closely the proof of [1]). For real-analytic structures with non-degenerate Levi form the condition $Y(q)$ was in fact shown in [28] to be necessary and sufficient for the local solvability in degree $q$ in the frame of Schwartz distributions. See also [15, 25] for partial results when the Levi form is degenerate. As general reference for related results about scalar operators on Gevrey spaces see [27].

2 Preliminaries

2.1 Gevrey functions and ultradistributions

Let us briefly recall the definition of the classes of Gevrey functions and corresponding ultradistributions; see e.g. [27, Chapter 1] for details.

Let $s>1$ be a real number and $\Omega$ be an open subset of $\mathbb R ^{n}$; let $C$ be a positive constant. We denote by $G^s(\Omega ,C)$ the space of smooth functions $f$ in $\mathbb R ^{n}$ such that for every compact $K\subset \Omega$,

$$\begin{aligned} ||f||_{K,C}:=\sup _{\alpha } C^{-|\alpha |}(\alpha !)^{-s}\sup _{x\in K}|\partial ^\alpha f(x)|<\infty . \end{aligned}$$

This is a Fréchet space endowed with the above seminorms. We set $G^s(\Omega )$ for the usual Gevrey space of order $s$, i.e. $f\in G^s(\Omega )$ if $f$ is smooth in $\Omega$ and for every compact $K\subset \Omega$ there exists $C>0$ such that $||f||_{K,C}<\infty$. We will also consider the space $G^s_0(K,C)$ of functions in $G^s(\Omega ,C)$ supported in the compact $K$; it is a Banach space with the norm $||u||_{K,C}$. Finally we set

$$\begin{aligned} G^s_0(\Omega )=\bigcup \limits _{K\subset \Omega ,\ C>0} G^s_0(K,C). \end{aligned}$$

The space of $\mathcal D ^{\prime }_s(\Omega )$ of ultradistributions of order $s$ in $\Omega$ is by definition the dual of $G_0^s(\Omega )$, i.e. an element $u\in \mathcal D ^{\prime }_s(\Omega )$ is a linear functional on $G_0^s(\Omega )$ such that for every compact $K\subset \Omega$ and every constant $C>0$ there exists a constant $C^{\prime }>0$ such that

$$\begin{aligned} |\langle u,f\rangle |\le C^{\prime } \Vert f\Vert _{K,C}, \end{aligned}$$

namely $u\in (G^s_0(K,C))^{\prime }$ for every $K,\,C$. Clearly, $\mathcal D ^{\prime }_s(\Omega )$ contains the usual space $\mathcal D ^{\prime }(\Omega )$ of Schwartz distributions.

We will need the following estimate for Gevrey seminorms of exponential functions.

Proposition 2.1

Let $\psi$ be a real-analytic function in a neighborhood $\Omega$ of $0$ in $\mathbb R ^n$; then for every compact subset $K$ of $\Omega$ and every $C>0,\,s>s^{\prime }>1$, there exists a constant $C^{\prime }>0$ such that

$$\begin{aligned} ||\mathrm{exp}(\iota \rho \psi )||_{K,C}\,\le \, C^{\prime }\mathrm{exp}(a\rho +\rho ^{1/s^{\prime }}) \end{aligned}$$

(2.1)

for every $\rho >0$, where $a=\sup \{-\mathrm{Im}\,\psi (x):\,x\in K\}$.

Proof

By the Faà di Bruno formula (see e.g. [14, page 16]) we have, for $|\alpha |\ge 1$,

$$\begin{aligned} \partial ^\alpha e^{\iota \rho \psi (x)}=\sum _{j=1}^{|\alpha |} \frac{\mathrm{exp}(\iota \rho \psi )}{j!}\sum _{\mathop {\gamma _1+\ldots +\gamma _j=\alpha }\limits _{ |\gamma _k|\ge 1}}\frac{\alpha !}{\gamma _1!\ldots \gamma _j!}\partial ^{\gamma _1}(\iota \rho \psi (x))\ldots \partial ^{\gamma _j} (\iota \rho \psi (x)). \end{aligned}$$

By assumption there exists a constant $C_1>0$ such that $|\partial ^\gamma \psi (x)|\le C_1^{|\gamma |}\gamma !$ for $x\in K,\,|\gamma |\ge 1$. Hence for every $\alpha$,

$$\begin{aligned} \sup _{x\in K}|\partial ^\alpha e^{\iota \rho \psi (x)}|\le e^{a\rho }\alpha !C_2^{|\alpha |}\sum _{j=0}^{|\alpha |}\frac{\rho ^j}{j!} \end{aligned}$$

with $C_2=2^{n+1}C_1$, where we used

$$\begin{aligned} \sum _{\mathop {\gamma _1+\ldots +\gamma _j=\alpha }\limits _{|\gamma _k|\ge 1}} 1\le \prod _{k=1}^n{\alpha _k+j-1 \atopwithdelims ()j-1} \le 2^{|\alpha |+n(j-1)}\le 2^{(n+1)|\alpha |}. \end{aligned}$$

Hence we have

$$\begin{aligned} ||\mathrm{exp}(\iota \rho \psi )||_{K,C}&\le e^{a\rho }(\alpha !)^{1-s}(C_2/C)^{|\alpha |}\sum _{j=0}^{|\alpha |}\frac{\rho ^j}{j!}\\&\le e^{a\rho }(|\alpha |!)^{1-s}(C_2C_3/C)^{|\alpha |}\sum _{j=0}^{|\alpha |}\frac{\rho ^j}{j!}\\&= e^{a\rho }(|\alpha |!)^{s^{\prime }-s}(C_2C_3/C)^{|\alpha |}\sum _{j=0}^{|\alpha |}(|\alpha |!)^{1-s^{\prime }}(j!)^{s^{\prime }-1}\frac{\rho ^j}{(j!)^{s^{\prime }}} \end{aligned}$$

because $|\alpha |!\le n^{|\alpha |}\alpha !$ and $s>1$. Now, we have $(|\alpha |!)^{1-s^{\prime }}(j!)^{s^{\prime }-1}\le 1$ (because $j\le |\alpha |$ and $s^{\prime }>1$), and by Stirling formula the sequence $(|\alpha |!)^{s^{\prime }-s}(C_2C_3/C)^{|\alpha |}$ tends to $0$ as $|\alpha |\rightarrow +\infty$, so that it is bounded: $(|\alpha |!)^{s^{\prime }-s}(C_2C_3/C)^{|\alpha |}\le C^{\prime }$ for a suitable constant $C^{\prime }>0$. Hence

$$\begin{aligned} ||\mathrm{exp}(\iota \rho \psi )||_{K,C}&\le C^{\prime }e^{a\rho }\sum _{j=0}^{|\alpha |}\frac{\rho ^j}{(j!)^{s^{\prime }}}\le C^{\prime }e^{a\rho }\sum _{j=0}^{|\alpha |}\left( \frac{\rho ^{j/s^{\prime }}}{j!}\right) ^{s^{\prime }}\\&\le C^{\prime }e^{a\rho }\left( \sum _{j=0}^{|\alpha |}\frac{\rho ^{j/s^{\prime }}}{j!}\right) ^{s^{\prime }}\le C^{\prime }e^{a\rho }\left( \sum _{j=0}^{\infty }\frac{\rho ^{j/s^{\prime }}}{j!}\right) ^{s^{\prime }}= C^{\prime } e^{a\rho +s^{\prime }\rho ^{1/s^{\prime }}}. \end{aligned}$$

This is essentially (2.1), except for the coefficients $s^{\prime }$ in the last exponent. To eliminate it, fix $1<s^{\prime }<s$, choose $s^{\prime }<s^{\prime \prime }<s$ and apply the last estimate to the pair of exponents $s^{\prime \prime },s$: we obtain

$$\begin{aligned} ||\mathrm{exp}(\iota \rho \psi )||_{K,C}\le C^{\prime } e^{a\rho +s^{\prime \prime }\rho ^{1/s^{\prime \prime }}} \end{aligned}$$

for a constant $C^{\prime }>0$ (depending on $s$ and $s^{\prime \prime }$). On the other hand,

$$\begin{aligned} C^{\prime } e^{s^{\prime \prime }\rho ^{1/s^{\prime \prime }}}\le C^{\prime \prime }e^{\rho ^{1/s^{\prime }}} \end{aligned}$$

for a new constant $C^{\prime \prime }>0$, which proves (2.1).$\square$

2.2 Locally integrable structures

Consider a real-analytic manifold $M$ of dimension $N=m+n$. A real-analytic locally integrable structure on $M$, of rank $n$, is defined by a real-analytic subbundle $\mathcal V \subset \mathbb C TM$ of rank $n$, satisfying the Frobenius condition, namely the commutator of sections of $\mathcal V$ is still a section of $\mathcal V$. As observed in the introduction the subbundle $T^\prime \subset \mathbb C T^*M$ orthogonal to $\mathcal V$ is then locally spanned by exact differentials (see e.g. [3, Theorem I.9.2]). As usual we will denote by $T^0=T^\prime \cap T^*M$ the so-called characteristic set. Let $k$ be a positive integer, we denote by $\Lambda ^k\mathbb C T^*M$ the $k$-th exterior power of $\mathbb C T^*M$. Let us consider complex exterior algebra

$$\begin{aligned} \Lambda \mathbb C T^*M=\oplus _{k=0}^N \Lambda ^k\mathbb C T^*M, \end{aligned}$$

for any pair of positive integers $p,q$ we denote by

$$\begin{aligned} T^{\prime p,q} \end{aligned}$$

the homogeneous of degree $p+q$ in the ideal generated by the $p$-th exterior power of $T^\prime ,\,\Lambda ^p T^\prime$. We have the inclusion

$$\begin{aligned} T^{\prime p+1,q-1}\subset T^{\prime p,q} \end{aligned}$$

which allows us to define

$$\begin{aligned} \Lambda ^{p,q}=T^{\prime p,q}/T^{\prime p+1,q-1}. \end{aligned}$$

If $\phi$ is a smooth section of $T^\prime$ over an open subset $\Omega \subset M$, its exterior derivative $d\phi$ is section of $T^{\prime 1,1}$. In other words

$$\begin{aligned} d T^\prime \subset T^{\prime 1,1}. \end{aligned}$$

It follows at once from this that, if $\sigma$ is a smooth section of $T^{\prime p,q}$ over $\Omega$, then $d\sigma$ is a section of $T^{\prime p, q+1}$ i.e.

$$\begin{aligned} d T^{\prime p,q}\subset T^{\prime p,q+1}. \end{aligned}$$

Let $s>1$, the space $G^s(\Omega , \Lambda ^{p,q})$ of $(p,q)$-forms with Gevrey coefficients of order $s$ is defined, as well as $G^s(\Omega ,C; \Lambda ^{p,q}),\,G_0^s(K,C; \Lambda ^{p,q})$, etc., with notation analogous to the scalar case.

The de Rham differential induces then a map

$$\begin{aligned} \mathrm{d}^\prime : G^s(\Omega , \Lambda ^{p,q})\rightarrow G^s(\Omega , \Lambda ^{p,q+1}). \end{aligned}$$

(see Treves [29, Section I.6] for more details). Similarly, the de Rham differential induces a complex on the space of ”ultra-currents” $\mathcal D ^{\prime }_s(\Omega , \Lambda ^{p,q})$, i.e. forms with ultradistribution coefficients:

$$\begin{aligned} \mathrm{d}^\prime : \mathcal D ^{\prime }_s(\Omega , \Lambda ^{p,q})\rightarrow \mathcal D ^{\prime }_s(\Omega , \Lambda ^{p,q+1}). \end{aligned}$$

Namely, consider for simplicity the case when $\Omega$ is orientable (in fact, in the sequel we will work in a local chart). Stokes’ theorem implies that

$$\begin{aligned} \int \limits _\Omega \mathrm{d}^{\prime }u\wedge v=(-1)^{p+q-1}\int \limits _\Omega u\wedge \mathrm{d}^{\prime }v \end{aligned}$$

if $u\in G^s(\Omega ,\Lambda ^{p,q}),\,v\in G_0^s(\Omega ,\Lambda ^{m-p,n-q-1})$, and accordingly we can define

$$\begin{aligned} \langle \mathrm{d}^{\prime }u, v\rangle =(-1)^{p+q-1}\langle u, \mathrm{d}^{\prime }v\rangle \end{aligned}$$

if $u\in D^\prime _s(\Omega ,\Lambda ^{p,q}),\,v\in G_0^s(\Omega ,\Lambda ^{m-p,n-q-1})$.

3 Local solvability estimates

We now show that local solvability implies an a priori-estimate. This is analogous to the estimates of Hörmander [16], Andreotti, Hill and Nacinovich [1], Treves [29, Lemma VIII.1.1], in the framework of Schwartz distributions.

Proposition 3.1

Suppose that, for some $s>1$, the complex $\mathrm{d}^{\prime }$ is locally solvable near $x_0$ and in degree $q$, in the sense of ultradistributions of order $s$ (see Definition 1.1). Then for every sufficiently small open neighborhood $\Omega$ of $x_0$, every $C_1>0,\,0<\epsilon <C_2$ there exist a compact $K\subset \Omega$, an open neighborhood $\Omega ^{\prime }\subset \subset \Omega$ of $x_0$ and a constant $C^{\prime }>0$, such that

$$\begin{aligned} \left| \int \limits _\Omega f\wedge v\right| \le C^{\prime }\Vert f\Vert _{K,C_1}\Vert \mathrm{d}^{\prime }v\Vert _{\overline{\Omega ^{\prime }},C_2}, \end{aligned}$$

(3.1)

for every cocycle $f\in G^s(\Omega ,C_1;\Lambda ^{0,q})$ and every $v\in G^s_0(\overline{\Omega ^{\prime }},C_2-\epsilon ;\Lambda ^{m,n-q}).$

It will follow from the proof that $\Vert \mathrm{d}^{\prime }v\Vert _{\overline{\Omega ^{\prime }},C_2}<\infty$ if $v\in G^s_0(\overline{\Omega ^{\prime }},C_2-\epsilon ;\Lambda ^{m,n-q}).$

Proof

Let $V_{j+1}\subset V_j\subset \subset \Omega ,\,j=1,2,\ldots$, be a fundamental system of neighborhoods of $x_0$. Fix $C_1>0,\,0<\epsilon <C_2$ and consider the space

$$\begin{aligned} F_j&= \{(f,u)\in G^s(\Omega ,C_1;\Lambda ^{0,q})\times G^s_0(\overline{V_j},C_2;\Lambda ^{0,q-1})^{\prime }:\\ \mathrm{d}^{\prime }f&= 0\ \mathrm{in}\ \Omega ,\ \mathrm{d}^{\prime }u=f \ \mathrm{in}\ G^s_0(\overline{V_j},C_2-\epsilon ;\Lambda ^{0,q})^{\prime }\}. \end{aligned}$$

The last condition means $\langle \mathrm{d}^{\prime }u,v\rangle =\langle f,v \rangle$ for every $v\in G^s_0(\overline{V_j},C_2-\epsilon ;\Lambda ^{m,n-q})$, which makes sense by transposition, because differentiation maps $G^s_0(\overline{V_j},C_2-\epsilon )\rightarrow G^s_0(\overline{V_j},C_2)$ (see e.g. [27, Proposition 2.4.8]) and multiplication by analytic functions preserve the latter space.

Now, by direct inspection one sees that $F_j$ is a closed subspace of $G^s(\Omega ,C_1;\Lambda ^{0,q-1})\times G^s_0(\overline{V_j},C_2;\Lambda ^{0,q})^{\prime }$, therefore Fréchet.

Let

$$\begin{aligned} \pi _{j}:F_j\rightarrow \{f\in G^s(\Omega ,C_1;\Lambda ^{0,q-1}): \mathrm{d}^{\prime }f=0\} \end{aligned}$$

be the canonical projection $(f,u)\mapsto f$. The assumption of local solvability implies that

$$\begin{aligned} \{f\in G^s(\Omega ,C_1;\Lambda ^{0,q-1}): \mathrm{d}^{\prime }f=0\}=\cup _{j} \pi _j(F_j). \end{aligned}$$

By the Baire theorem, there exists $j_0$ such that $\pi _{j_0}(F_{j_0})$ is of second category. By the open mapping theorem, we see that $\pi _{j_0}$ is onto and open: there exists a compact $K\subset \Omega$ and a constant $C^{\prime }>0$ such that for every cocycle $f\in G^s(\Omega ,C_1;\Lambda ^{0,q})$, there exists $u\in G_0^s(\overline{V_{j_0}},C_2;\Lambda ^{0,q-1})^{\prime }$ satisfying $\mathrm{d}^{\prime }u=f$ in $G_0^s(\overline{V_{j_0}},C_2-\epsilon ;\Lambda ^{0,q})^{\prime }$ and

$$\begin{aligned} |u|_{{\overline{V_{j_0}},C_2}}:=\sup \limits _{||v||_{\overline{V_{j_0}},C_2}=1}|\langle u,v\rangle |\le C^{\prime } ||f||_{K,C_1}. \end{aligned}$$

Consider now the bilinear functional $(f,v)\mapsto \int _\Omega f\wedge v=\langle f,v\rangle$, for $f\in G^s(\Omega ,C_1;\Lambda ^{0,q})$ cocycle, and $v\in G_0^s(\overline{V_{j_0}},C_2-\epsilon ;\Lambda ^{m,n-q})$. Given such a $f$, we take $u$ as before, and we get

$$\begin{aligned} |\langle f,v\rangle |=|\langle \mathrm{d}^{\prime }u,v\rangle |=|\langle u,\mathrm{d}^{\prime }v\rangle |\le |u|_{{\overline{V_{j_0}},C_2}}\Vert \mathrm{d}^{\prime }v\Vert _{{\overline{V_{j_0}},C_2}}\le C^{\prime } \Vert f\Vert _{K,C_1}\Vert \mathrm{d}^{\prime }v||_{{\overline{V_{j_0}},C_2}}. \end{aligned}$$

$\square$

4 Proof of Theorem 1.2

We work in a sufficiently small neighborhood $\Omega$ of the point $x_0$ (to be chosen later), where local solvability holds. We also take $x_0$ as the origin of the coordinates, i.e. $x_0=0$. Moreover we make use of the special coordinates, whose existence is proved in see section I.9 of [29]. Namely, let $n=\mathrm{dim}_\mathbb C \mathcal V _0,\,d=\mathrm{dim}_\mathbb R T^0_0,\,\nu =n-\mathrm{dim}_\mathbb C (\mathcal V _0\cap \overline{\mathcal{V }}_0)$. We have the following result.

Proposition 4.1

Let $(0,\omega _0)\in T^0,\,\omega _0\not =0$, and suppose that the restriction of the Levi form $\mathcal B _{(0,\omega _0)}$ to $\mathcal V _0\cap \overline{\mathcal{V }}_0$ is non-degenerate. There exist real-analytic coordinates $x_j,y_j$,$s_k$ and $t_l,\,j=1,\ldots ,\nu ,\,k=1,\ldots , d,\,l=1,\ldots ,n-\nu$, and smooth real-valued and real-analytic functions $\phi _k(x,y,s,t),\,k=1,\ldots d$, in a neighborhood $\mathcal O$ of $0$, satisfying

$$\begin{aligned} \phi _k|_0=0\ \ \mathrm{and}\ \ d\phi _k|_0=0, \end{aligned}$$

(4.1)

such that

$$\begin{aligned} \left\{ \begin{array}{l} z_j:=x_j+\iota y_j,\ j=1,\ldots ,\nu ,\\ w_k:=s_k+\iota \phi _k(x,y,s,t),\ k=1,\ldots ,d, \end{array}\right. \end{aligned}$$

define a system of first integrals for $\mathcal V$, i.e. their differential span $T^\prime |_\mathcal O$.

Moreover, with respect to the basis

$$\begin{aligned} \left\{ \left. \frac{\partial }{\partial \overline{z}_j}\right| _0,\left. \frac{\partial }{\partial t_l}\right| _0; j=1,\ldots ,\nu ,\ l=1,\ldots ,n-\nu \right\} \end{aligned}$$

of $\mathcal V _0$ the Levi form $\mathcal B _{(0,\omega _0)}$ reads

$$\begin{aligned} \sum _{j=1}^{p^{\prime \prime }}|\zeta _j|^2-\sum _{j=p^{\prime \prime }+1}^{\nu }|\zeta _j|^2+ \sum _{l=1}^{p^\prime }|\tau _l|^2-\sum _{l=p^\prime +1}^{n-\nu }|\tau _l|^2. \end{aligned}$$

(4.2)

Remark 4.2

In particular

$$\begin{aligned} \mathrm{d}^\prime z_j=0,\ \mathrm{d}^\prime w_k=0,\quad j=1,\ldots ,\nu ;\ k=1,\ldots ,d. \end{aligned}$$

(4.3)

In these coordinates we have $T_0^0=\mathrm{span}_\mathbb R \{ds_k|_0;\ k=1,\ldots ,d\}$, so that $\omega _0=\sum _{k=1}^d \sigma _k ds_k|_0$, with $\sigma _k\in \mathbb R$. By (I.9.2) of [29] we have $\mathcal B _{(0,\omega _0)}(\mathbf v _1,\mathbf v _2)=\sum _{k=1}^d \sigma _k(V_1\overline{V}_2\phi _k)|_0,$ with $V_1$ and $V_2$ smooth sections of $\mathcal V$ extending $\mathbf v _1$ and $\mathbf v _2$, respectively. Upon setting $\Phi =\sum _{k=1}^d\sigma _k\phi _k$ we can suppose, in addition, that

$$\begin{aligned} \Phi&= \sum _{j=1}^{p^{\prime \prime }}|z_j|^2-\sum _{j=p^{\prime \prime }+1}^{\nu }|z_j|^2+ \frac{1}{2}\sum _{l=1}^{p^\prime }t_l^2- \frac{1}{2}\sum _{l=p^\prime +1}^{n-\nu }t_l^2+O(|s|(|z|+|s|\nonumber \\&+\,|t|)+|z|^3+|t|^3); \end{aligned}$$

(4.4)

see [29, Section I.9] and [29, (XVIII.3.2)] for details.

We can now prove Theorem 1.2. We may assume, without loss of generality, that $\sigma =(1,0,\ldots ,0)$. Consequently, from (4.4) (after the change of variables $t\mapsto t/\sqrt{2}$) we have

$$\begin{aligned} \phi _1(x,y,s,t)=|z^\prime |^2-|z^{\prime \prime }|^2+|t^\prime |^2-|t^{\prime \prime }|^2+O(|s|(|z|+|s|+|t|)+|z|^3+|t|^3),\qquad \end{aligned}$$

(4.5)

where we set

$$\begin{aligned} \left\{ \begin{array}{l} z^\prime =(z_{1},\ldots ,z_{p^{\prime \prime }}),\\ z^{\prime \prime }=(z_{p^{\prime \prime }+1},\ldots ,z_{\nu })\\ t^\prime =(t_1,\ldots ,t_{p^\prime }),\\ t^{\prime \prime }=(t_{p^\prime +1},\ldots , t_{n-\nu }). \end{array}\right. \end{aligned}$$

Moreover, we choose a function $\chi (x,y,s,t)$ in $G^s_0(\mathbb R ^{2\nu +d+(n-\nu )}),\,\chi =0$ away from a neighborhood $V\subset \subset \Omega$ of $0$ and $\chi =1$ in a neighborhood $U\subset \subset V$ of $0$, where $V$ and $U$ will be chosen later on. We set, for $\rho >0, \lambda >0$,

$$\begin{aligned} f_{\rho ,\lambda }&= e^{\rho h_{1,\lambda }} d\overline{z}^\prime \wedge dt^\prime \\ v_{\rho ,\lambda }&= \rho ^{(m+n)/2}\chi e^{\rho h_{2,\lambda }} d\overline{z}^{\prime \prime }\wedge dt^{\prime \prime }\wedge dz\wedge dw, \end{aligned}$$

where, with $\lambda >1$,

$$\begin{aligned} h_{1,\lambda }:=-\iota s_1+\phi _1-2|z^\prime |^2-2|t^\prime |^2-\lambda \sum _{k=1}^d(s_k+\iota \phi _k)^2, \end{aligned}$$

(4.6)

and

$$\begin{aligned} h_{2,\lambda }:=\iota s_1-\phi _1-2|z^{\prime \prime }|^2-2|t^{\prime \prime }|^2-\lambda \sum _{k=1}^d(s_k+\iota \phi _k)^2. \end{aligned}$$

(4.7)

Now, we have $\chi \in G_0^s(\overline{V},C/2)$, for some $C>0$. We then apply Proposition 3.1 with $f_{\rho ,\lambda }$ and $v_{\rho ,\lambda }$ in place of $f$ and $v$, respectively, and $C_1=C_2=C,\,\epsilon =C/2$, taking $V$ small enough to be contained in the neighborhood $\Omega ^{\prime }$ which arises in the conclusion of Proposition 3.1. Observe that, in fact, $f_{\rho ,\lambda }\in G^s (\Omega ,C;\Lambda ^{0,q})$ since this form is real-analytic and $p^{\prime \prime }+p^\prime =q$ by hypothesis, whereas $v_{\rho ,\lambda }\in G^s_0(\overline{V},C/2;\Lambda ^{m,n-q})\subset G^s_0(\overline{\Omega ^{\prime }},C/2;\Lambda ^{m,n-q})$ (recall, $m=\mathrm{dim} M-n$). We prove now that $f_\rho$ is a cocycle (i.e. $\mathrm{d}^\prime f_{\rho ,\lambda }=0$), so that Proposition 3.1 can in fact be applied. However, we will show that (3.1) fails for every choice of $C^{\prime }$ when $\rho \rightarrow +\infty$, if $\lambda$ is large enough, obtaining a contradiction.

Now, we have

$$\begin{aligned} \mathrm{d}^\prime f_{\rho ,\lambda }= \rho e^{\rho h_{1,\lambda }}\mathrm{d}^{\prime }h_{1,\lambda }\wedge d\overline{z}^\prime \wedge dt^\prime =0. \end{aligned}$$

(4.8)

In fact, we have $h_{1,\lambda }=-\iota w_1-2|z^{\prime }|^2-2|t^{\prime }|^2-\lambda \sum _{k=1}^d w_k^2$, so that, modulo sections of $T^{\prime 1,0}$, that is $\mathrm{span}\,\{dz_1,\ldots ,dz_\nu ,dw_1,\ldots dw_d\}$, we have

$$\begin{aligned} \mathrm{d}^\prime h_{1,\lambda }= dh_{1,\lambda }= -\sum _{j=1}^{p^{\prime \prime }} z_j d\overline{z}_j-2\sum _{l=1}^{p^{\prime }}t_j\,\mathrm{d}t_j, \end{aligned}$$

and the wedge product in (4.8) therefore vanishes.

Similarly

$$\begin{aligned} \mathrm{d}^\prime v_{\rho ,\lambda }=\rho ^{(m+n)/2}e^{\rho h_{2,\lambda }}\mathrm{d}^\prime \chi \wedge d\overline{z}^{\prime \prime }\wedge dt^{\prime \prime }\wedge dz\wedge dw. \end{aligned}$$

(4.9)

In order to estimate the right hand side of (3.1) we observe that, by (4.5) and (4.6)

$$\begin{aligned} \mathrm{Re}\, h_{1,\lambda }=-|z^\prime |^2-|z^{\prime \prime }|^2-|t|^2-\lambda |s|^2+\mathcal R (z,s,t)+ O(|z|^3+|t|^3)+\lambda O(|z|^4+|t|^4), \end{aligned}$$

where

$$\begin{aligned} |\mathcal R (z,s,t)|=O(|s|(|z|+|s|+|t|))\le \tilde{C}\left( \frac{\epsilon }{2}(|z|+|s|+|t|)^2+\frac{1}{2\epsilon }|s|^2\right) , \end{aligned}$$

(4.10)

for every $\epsilon >0$. Hence, if $\epsilon$ and then $1/\lambda$ are small enough we see that, possibly after replacing $\Omega$ with a smaller neighborhood,

$$\begin{aligned} \sup _{\Omega }\,\mathrm{Re}\,h_{1,\lambda }\le 0. \end{aligned}$$

(4.11)

Similarly,

$$\begin{aligned} \mathrm{Re}\,h_{2,\lambda }&= -|z^\prime |^2-|z^{\prime \prime }|^2-|t|^2-\lambda |s|^2\\&\quad +\mathcal R ^\prime (z,s,t)+O(|z|^3+|t|^3)+\lambda O(|z|^4+|t|^4), \end{aligned}$$

with $\mathcal R ^\prime$ satisfying the same estimate (4.10). Therefore if $\lambda$ is sufficiently large, in $\Omega$ we have

$$\begin{aligned} \mathrm{Re}\, h_{2,\lambda }\le -\frac{1}{2} (|z|^2+|t|^2+\lambda |s|^2)+\tilde{C}_1 (|z|^3+|t|^3)+\tilde{C}_2\lambda (|z|^4+|t|^4). \end{aligned}$$

Hence, possibly for a smaller $V$, since $U\subset \subset V$ is a neighborhood of $0$, there exists a constant $c>0$ such that

$$\begin{aligned} \sup _{\overline{V}\setminus U}h_{2,\lambda }(z,s,t)\le -c. \end{aligned}$$

(4.12)

As a consequence of Proposition 2.1 and (4.11), (4.12), for every compact subset $K\subset \Omega$ it turns out

$$\begin{aligned}&\Vert f_{\rho ,\lambda }\Vert _{K,C}\le C^\prime e^{\rho ^{1/s^{\prime }}},\end{aligned}$$

(4.13)

$$\begin{aligned}&\Vert \mathrm{d}^\prime v_{\rho ,\lambda }\Vert _{K,C}\le C^{\prime \prime }\rho ^{(m+n)/2} e^{-c\rho +\rho ^{1/s^{\prime }}}, \end{aligned}$$

(4.14)

for any $1<s^{\prime }<s$, where the constants $C^\prime , C^{\prime \prime }$ are independent of $\rho$. It follows that

$$\begin{aligned} \Vert f_{\rho ,\lambda }\Vert _{K,C}\Vert \mathrm{d}^\prime v_{\rho ,\lambda }\Vert _{K,C}\le C^{\prime }C^{\prime \prime }\rho ^{(m+n)/2} e^{-c\rho +2\rho ^{1/s^{\prime }}}\longrightarrow 0\ \mathrm{as}\ \rho \rightarrow +\infty , \end{aligned}$$

(4.15)

because $s^{\prime }>1$. On the other hand, we now verify that the left-hand side of (3.1) satisfies

$$\begin{aligned} \int f_{\rho ,\lambda }\wedge v_{\rho ,\lambda }\longrightarrow c^{\prime }\not =0, \end{aligned}$$

(4.16)

as $\rho \rightarrow +\infty$, which together with (4.15) contradicts (3.1).

The proof of (4.16) is achieved by the same argument which appears at the end of the proof of [29, Theorem XVIII.3.1]. We sketch it here for the sake of completeness.

We have

$$\begin{aligned} \int f_{\rho ,\lambda }\wedge v_{\rho ,\lambda }=c\rho ^{(m+n)/2}\int e^{\rho (h_{1,\lambda }+h_{2,\lambda })} \chi \mathrm{det}\left( \mathrm{Id}_{d\times d}+\iota \frac{\partial \phi }{\partial s}\right) dx\,dy\,ds\,dt,\qquad \end{aligned}$$

(4.17)

where $0\not =c\in \mathbb R$. Observe that

$$\begin{aligned} (h_{1,\lambda }+h_{2,\lambda })(x,y,s,t)=-2(|z|^2+|t|^2)-2\lambda |s|^2+O(|z|^3+|s|^3+|t|^3). \end{aligned}$$

Hence, since $\lambda >1$, if $V$ is small enough (recall $\mathrm{supp}\chi \subset \overline{V}$) we have

$$\begin{aligned} \mathrm{Re}(h_{1,\lambda }+h_{2,\lambda })(x,y,s,t)\le -(|z|^2+|t|^2+|s|^2) \quad \mathrm{in}\ V. \end{aligned}$$

(4.18)

We now perform the change of variables $(x,y,s,t)\rightarrow \rho ^{-1/2}(x,y,s,t)$ in (4.17). Then $\rho \cdot (h_{1,\lambda }+h_{2,\lambda })(\rho ^{-1/2}x,\rho ^{-1/2}y,\rho ^{-1/2}s, \rho ^{-1/2}t)$ converges pointwise to

$$\begin{aligned} -2(|z|^2+|t|^2)-2\lambda |s|^2 \end{aligned}$$

as $\rho \rightarrow +\infty$. Hence, by virtue of (4.17), (4.1), (4.18) and the Lebesgue convergence theorem we deduce

$$\begin{aligned} \int f_{\rho ,\lambda }\wedge v_{\rho ,\lambda }\longrightarrow c\int e^{-2(|z|^2+|t|^2)-2\lambda |s|^2}dx\,dy\,ds\,dt\not =0, \end{aligned}$$

which proves (4.16).

Remark 4.3

The above machinery can be applied to prove other necessary conditions for Gevrey local solvability in the spirit of analogous results valid in the framework of smooth functions and Schwartz distributions.

As an example, consider the special case of local solvability when $q=n$, namely in top degree. In this case, the Cordaro–Hounie condition $(\mathcal P )_{n-1}$ (see [7] and [9]) is known to be necessary for the local solvability in the smooth category, and it is conjectured to be sufficient as well. Consider the following analytic variant. Let $L_j,\,j=1,\ldots ,n$, be real-analytic independent vector fields which generates $\mathcal V$ at any point of $\Omega$.

We say that the real-analytic condition $(\mathcal P )_{n-1}$ is satisfied at $x_0$ if there exists an open neighborhood $U\subset \Omega$ of $x_0$ such that, given any open set $V\subset U$ and given any real-analytic $h\in C^\infty (V)$ satisfying $L_j h=0,\,j=1,\ldots ,n$, then $\mathrm{Re}\, h$ does not assume a local minimum ¹ over any nonempty compact subset of $V$.

One could then prove that if the real-analytic condition $(\mathcal P )_{n-1}$ is not satisfied at $x_0$, then for every $s>1,\,\mathrm{d}^{\prime }$ is not locally solvable in the sense of ultradistributions of order $s$, near $x_0$ and in degree $n$. For the sake of brevity we omit the proof, which goes on along the same lines as that in [9, Theorem 1.2], using the local solvability estimates in Proposition 3.1, combined with Proposition 2.1.

Remark 4.4

The above results would seem to suggest the following conjecture:

When $\mathrm{d^{\prime }}$ is not locally solvable at $x_0$ in the sense of Schwartz distributions (i.e. for every neighborhood $\Omega$ belonging to a fundamental system of neighborhoods of $x_0$, there exists a $\mathrm{d}^{\prime }$ -closed $(0,q)$ -form $f$ in $\Omega$ for which the system ${\mathrm{d}^{\prime }} u=f$ has no distribution solutions in any neighborhood $V\subset \Omega$ of $x_0$ ), then it is not solvable even in the sense of ultradistributions of order $s$ , for every $s>0$ (Definition 1.1).

A similar result, but on the positive side, is proved in [5, Theorem 5.1].

This conjecture is also supported by the special case of non-vanishing vector fields (or, more generally, operators of principal type), where the Nirenberg-Treves’ condition $\mathcal P$ is in fact necessary and sufficient for local solvability in the frame of Schwartz distributions ([3, Chapter 4]) and the same condition is still necessary (and of course sufficient) for the solvability in the sense of ultradistributions for order $s$, for every $s>1$ ([27, Theorem 3.5.8]). The case of scalar operators is however very special and we do not have strong elements in favor of the above conjecture yet. Further interesting speculations seem to be needed.