1 Introduction
In 1983, M. S. Baouendi, C. H. Chang and F. Treves introduced the Fourier–Bros–Iagolnitzer (F.B.I.) transform on a hypo-analytic manifold and, with that, they gave a characterisation of holomorphic extendability for solutions; see [Reference Baouendi, Chang and Treves1]. Since then, the Baouendi–Chang–Treves F.B.I. transform was used in several papers concerning holomorphic extendability of CR functions ([Reference Baouendi and Treves3], [Reference Baouendi, Rothschild and Treves2], [Reference Hanges and Treves9], to name a few). In this article, we characterise the Gevrey-s vectors (
$s\geq 1$
) on a hypo-analytic manifold via the decay of the F.B.I. transform (Theorem 3.4). We point out that since the hypo-analytic manifold is in principle only smooth, we do not have the usual Gevrey classes defined on it. Now, if the manifold has Gevrey-s regularity, then both notions coincide; see, for instance, [Reference Ragognette11]. Then we use our characterisation to obtain a propagation of Gevrey singularities (Theorem 4.3).
Now let us state a simplified version of Theorem 3.4. Consider in an open set
$U=V\times W\subset \mathbb {R}^{n+m}$
the following family of smooth functions and complex vector fields:
$$ \begin{align*} Z_{j}(x,t)=x_{j}+i\phi_{j}(t),\quad j=1,\dots,m, \end{align*} $$
where the map
$\phi :W\rightarrow \mathbb {R}^{m}$
satisfies
$\phi (0)=0$
and the vector fields
$\{\mathrm {L}_{1},\dots ,\mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m}\}$
are defined by the relations
$$ \begin{align*} \begin{array}{ll} \mathrm{L}_{j} Z_{k}=0 & \mathrm{M}_{l} Z_{k} = \delta_{l,k}\\ \mathrm{L}_{j} t_{i} = \delta_{j,i} & \mathrm{M}_{l} t_{i}=0. \end{array} \end{align*} $$
The functions
$Z_{j}$
s define a hypo-analytic structure on U. If
$u\in \mathcal {C}^{\infty }(W;\mathcal {E}^{\prime }(V))$
, the F.B.I. transform of u is given by
$$ \begin{align*} \mathfrak{F}[u](t;z,\xi)=\left\langle u(x,t),e^{i(z-Z(x,t)\cdot\xi-|\xi|\langle z-Z(x,t)\rangle^{2})}\det Z_{x}(x,t)\right\rangle_{\mathcal{D}^{\prime}(V),\mathcal{D}(V)}, \end{align*} $$
where
$t\in W$
,
$z\in \mathbb {C}^{m}$
and
$\xi \in \mathbb {R}^{m}$
. Here we use
$\langle u(x,t),\phi (x,t)\rangle _{\mathcal {D}^{\prime }(V),\mathcal {D}(V)}$
to indicate the duality between distributions on the x variable and functions, where the variable t is to be understood as a parameter. Now let
$u\in \mathcal {C}^{\infty }(W;\mathcal {D}^{\prime }(V))$
be such that
$\mathrm {L}_{j} u=0$
,
$j=1,\dots ,n$
. In [Reference Baouendi, Chang and Treves1] the authors proved that u is an analytic vector for
$\{\mathrm {M}_{1},\dots ,\mathrm {M}_{m}\}$
at the origin (see Definition 2.1) if and only if, for every
$\chi \in \mathcal {C}^{\infty }_{c}(V)$
with small support and
$\chi \equiv 1$
near the origin, there are
$\mathcal {O}\subset \mathbb {C}^{m}$
a neighbourhood of the origin and constants
$C,\varepsilon>0$
such that
$$ \begin{align*} \big|\mathfrak{F}[\chi u](0;z,\xi)\big|\leq Ce^{-\varepsilon|\xi|}, \end{align*} $$
for all
$z\in \mathcal {O}$
and
$\xi \in \mathbb {R}^{m}$
. Now we can state the following simplified version of Theorem 3.4.
Theorem 1.1. Let
$u\in \mathcal {C}^{\infty }(W;\mathcal {D}^{\prime }(V))$
be a distribution such that
$\mathrm {L}_{j}u=0$
,
$j=1,\dots ,n$
. Then u is a Gevrey vector for
$\{\mathrm {M}_{1},\dots ,\mathrm {M}_{m}\}$
; that is, there is a constant
$C>0$
such that for every
$\alpha \in \mathbb {Z}_{+}^{m}$
we have the following estimate:
$$ \begin{align*} \sup|\mathrm{M}^{\alpha} u|\leq C^{|\alpha|+1}\alpha!^{s}, \end{align*} $$
if and only if, for every
$\chi \in \mathcal {C}^{\infty }_{c}$
with small support and
$\chi \equiv 1$
near the origin, there are
$V_{0}\times W_{0}\subset U$
neighbourhoods of the origin and constants
$C,\varepsilon>0$
such that for every
$(x,t)\in V_{0}\times W_{0}$
and
$\xi \in \mathbb {R}^{m}$
,
$$ \begin{align*} \big|\mathfrak{F}[\chi u](t;Z(x,t),\xi)\big|\leq Ce^{-\varepsilon|\xi|^{\frac{1}{s}}}. \end{align*} $$
Since our result is very technical, we shall recall the basic definitions and results in Section 2. The results here presented were obtained by the author in his Ph.D. dissertation written at the Institute of Mathematics and Statistics of the University of São Paulo, Brazil.
2 Preliminaries
2.1 Locally integrable structures
In this section we follow closely the first chapter of [Reference Berhanu, Cordaro and Hounie6]. Let
$\Omega \subset \mathbb {R}^{N}$
be an open set. By a locally integrable structure on
$\Omega $
we mean a complex vector bundle
$\mathcal {V}\subset \mathbb {C}\mathrm {T}\Omega $
, such that
$[\mathcal {V},\mathcal {V}]\subset \mathcal {V}$
and at every point
$p\in \Omega $
there are
$Z_{1},\dots ,Z_{m}$
, smooth, complex-valued functions in some open neighbourhood of p in
$\Omega $
, such that
$$ \begin{align*} \begin{cases} \mathrm{d}Z_{1}\wedge\cdots\wedge\mathrm{d}Z_{m}\neq 0;\\ \mathrm{L}Z_{j}=0,\quad\forall \,\mathrm{L}\in\mathcal{V},\;j=1,\dots,m. \end{cases} \end{align*} $$
We denote by
$\mathrm {T}^{\prime }\subset \mathbb {C}\mathrm {T}^{\ast }\Omega $
the orthogonal bundle, with respect to the duality between forms and vectors, of the bundle
$\mathcal {V}$
. Let p be an arbitrary point at
$\Omega $
. Then there exist a local coordinate system vanishing at p on some open set
$U=V\times W$
,
$(x_{1},\dots , x_{m},t_{1},\dots , t_{n})$
and smooth, real-valued functions
$\phi _{1},\dots , \phi _{m}$
, defined on U and satisfying
$\phi (0)=0$
and
$\mathrm {d}_{x}\phi (0)=0$
, such that the differentials of the functions
$$ \begin{align} Z_{k}(x,t)\doteq x_{k}+i\phi_{k}(x,t),\quad k=1,\dots,m, \end{align} $$
span
$\mathrm {T}^{\prime }$
in U. There are also linear independent, pairwise commuting, complex vector fields
$$ \begin{align*} \mathrm{M}_{j}=\sum_{k=1}^{m} a_{j,k}(x,t)\frac{\partial}{\partial x_{k}},\quad j=1,\dots, m, \end{align*} $$
and
$$ \begin{align*} \mathrm{L}_{j}=\frac{\partial}{\partial t_{j}}-i\sum_{k=1}^{m}\frac{\partial\phi_{k}}{\partial t_{j}}(x,t)\mathrm{M}_{k},\quad j=1,\dots,n, \end{align*} $$
satisfying the relations
$$ \begin{align*} \begin{array}{ll} \mathrm{L}_{j} Z_{k}=0 & \mathrm{M}_{l} Z_{k} = \delta_{l,k}\\ \mathrm{L}_{j} t_{i} = \delta_{j,i} & \mathrm{M}_{l} t_{i}=0. \end{array} \end{align*} $$
Now let
$u\in \mathcal {C}^{\infty }(W;\mathcal {D}^{\prime }(V))$
. By an uniform boundedness principle argument, we have that for all compact sets
$K_{1}\Subset V$
and
$K_{2}\Subset W$
, there exist a constant
$C>0$
and an integer
$q>0$
such that
$$ \begin{align} \left|\langle u(x,t), \phi(x) \rangle_{\mathcal{D}^{\prime}(V),\mathcal{D}(V)}\right|\leq C\sum_{|\alpha|\leq q}\sup_{x\in K_{1}}\left|\partial^{\alpha}\phi\right|,\quad\forall\phi\in\mathcal{C}^{\infty}_{c}(K_{1}), \end{align} $$
for every
$t\in K_{2}$
.
Definition 2.1. Let
$s\geq 1$
. We say that a distribution u on U is a Gevrey-s vector for
$\{\mathrm {L}_{1},\dots ,\mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m}\}$
if u is a smooth function on U and for every compact set
$K\subset U$
there exists a constant
$C>0$
such that
$$ \begin{align*} \sup_{(x,t)\in K}|\mathrm{L}^{\alpha}\mathrm{M}^{\beta} u(x,t)|\leq C^{|\alpha|+|\beta|+1}\alpha!^{s}\beta!^{s},\quad\forall \alpha\in\mathbb{Z}_{+}^{n}, \beta\in\mathbb{Z}_{+}^{m}. \end{align*} $$
We denote by
$\mathrm {G}^{s}(U; \mathrm {L}_{1},\dots ,\mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m})$
the space of all Gevrey-s vectors on
$U_{1}$
. If
$s=1$
, we say that u is an analytic vector and we write
$u\in \mathcal {C}^{\omega }(U; \mathrm {L}_{1},\dots ,\mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m})$
.
Remark 2.2. If the structure is real analytic, meaning that the map
$\phi $
is real analytic, then
$\mathrm {G}^{s}(U; \mathrm {L}_{1},\dots ,\mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m}) = \mathrm {G}^{s}(U)$
, for
$s\geq 1$
, even as topological vector spaces; see [Reference Ragognette11].
2.2 Hypo-analytic structures
In this section we introduce the basic definitions and properties of the so-called hypo-analytic structures. We follow closely F. Treves’ book [Reference Treves14]. Let
$\Omega \subset \mathbb {R}^{N}$
be an open set. A hypo-analytic structure on
$\Omega $
is a pair
$\{(U_{\alpha })_{\alpha \in \Lambda },(Z_{\alpha })_{\alpha \in \Lambda }\}$
such that
-
•
$(U_{\alpha })_{\alpha \in \Lambda }$
is an open covering for
$\Omega $
; -
•
$Z_{\alpha }:U_{\alpha }\longrightarrow \mathbb {C}^{m}$
is a smooth map, for every
$\alpha \in \Lambda $
; -
•
$\mathrm {d}Z_{\alpha ,1},\dots ,\mathrm {d}Z_{\alpha ,m}$
are
$\mathbb {C}-$
linear independent on
$U_{\alpha }$
, for every
$\alpha \in \Lambda $
; -
• if
$\alpha \neq \beta $
, then to each
$p\in U_{\alpha }\cap U_{\beta }$
there is a holomorphic map F such that
$Z_{\alpha }=F\circ Z_{\beta }$
, in a neighbourhood of p in
$U_{\alpha }\cap U_{\beta }$
; -
• if
$Z:U\longrightarrow \mathbb {C}^{m}$
is a smooth function such that for every
$p\in U\cap U_{\alpha }$
there exists a holomorphic function F such that
$Z=F\circ Z_{\alpha }$
, then
$(U,Z)=(U_{\beta },Z_{\beta })$
, for some
$\beta \in \Lambda $
.
We call each pair
$(U_{\alpha },Z_{\alpha })$
a hypo-analytic chart.
Definition 2.3. We say that a distribution
$u\in \mathcal {D}^{\prime }(\Omega )$
is hypo-analytic at p if for some
$\alpha \in \Lambda $
such that
$p\in U_{\alpha }$
, there is a holomorphic function F, defined on a complex neighbourhood of
$Z_{\alpha }(p)$
, such that
$u=F\circ Z_{\alpha }$
, in some open neighbourhood of p.
Given a hypo-analytic structure on
$\Omega $
, we can associate a locally integrable structure
$\mathcal {V}$
setting its orthogonal
$\mathrm {T}^{\prime }$
as the complex bundle locally defined by the differentials
$\mathrm {d}Z_{1},\dots ,\mathrm {d}Z_{m}$
. So let
$p\in \Omega $
and
$(U,Z)$
be a hypo-analytic chart, with
$p\in U$
. We can assume that there are local coordinates
$(x_{1},\dots , x_{m},t_{1}, \dots , t_{n})$
in
$U=V\times W$
, as described in the previous section, so the function Z is given by (1). Note that in this coordinate system, the point p is the origin.
In order to prove that the Gevrey regularity imples the decay of the F.B.I. transform, we shall need the following ‘almost-holomorphic’ extension result, proved in [Reference Caetano8] (see also Appendix).
Theorem 2.4. Let
$s>1$
and u be a distribution on U and
$U_{1}=V_{1}\times W_{1}\Subset U$
, where
$V_{1}$
and
$W_{1}$
are open balls centred at the origin. Suppose that
$u|_{U_{1}}\in \mathrm {G}^{s}(U_{1};\mathrm {L}_{1},\dots \mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m})$
. Then there are an open neighbourhood
$\mathcal {O}$
of
$\{Z(x,t)\;:\;x\in V_{1},t\in W_{1}\}$
on
$\mathbb {C}^{m}$
and a smooth function
$F\in \mathcal {C}^{\infty }(\mathcal {O}\times W_{1})$
such that
$$ \begin{align} \begin{cases} u(x,t)=F(Z(x,t),t),\quad (x,t)\in V_{1}\times W_{1}\\ \big|\partial_{\overline{z}}F(z,t)\big|\leq C^{k+1}k!^{s-1}\mathrm{dist}(z;\mathfrak{W}_{t})^{k},\quad \forall k\in\mathbb{Z}_{+}, z\in\mathcal{O}, t\in W_{1}, \end{cases} \end{align} $$
where C is a positive constant and
$$ \begin{align*} \mathfrak{W}_{t}=\{Z(x,t)\;:\;x\in V_{1}\}. \end{align*} $$
Despite the difference between Gevrey and analytic vectors being a power of s in their definition, they have very different properties. To illustrate this difference, let us recall some well-known properties of hypo-analytic functions (see [Reference Treves14]).
Theorem 2.5. Let u be a distribution on U such that
$\mathrm {L}_{j} u=0$
,
$j=1,\dots ,n$
. Then the following are equivalent:
-
1. u is hypo-analytic at the origin.
-
2. The restriction of u to a maximally real submanifold, passing through the origin, is hypo-analytic at the origin (with respect to the induced hypo-analytic structure).
-
3. u is an analytic vector in some open neighbourhood of the origin.
Let us prove that 2.
$\Rightarrow $
1. So let
$\mathcal {H}$
be a maximally real submanifold such that
$u|_{\mathcal {H}}$
is hypo-analytic at p. Then there exists
$U_{\mathcal {H}}$
an open neighbourhood of p on
$\mathcal {H}$
and a holomorphic function F defined on
$\mathcal {O}$
, an open neighbourhood of
$Z(U_{\mathcal {H}})$
on
$\mathbb {C}^{m}$
, such that
$$ \begin{align*} u(p^{\prime})=F(Z(p^{\prime})),\quad\forall p^{\prime}\in U_{\mathcal{H}}. \end{align*} $$
Now set
$\tilde {u}\doteq F\circ Z$
, defined in some neighbourhood of p on
$\Omega $
. Since F is holomorphic we have that
$L \tilde {u}=0$
, for every
$L\in \mathcal {V}$
, so the same is valid for
$u-\tilde {u}$
and
$u-\tilde {u}$
vanishes on a neighbourhood of p on
$\mathcal {H}$
. By a standard uniqueness result, based on the Baouendi–Treves approximation formula, we have that
$u-\tilde {u}$
vanishes on some neighbourhood of p; that is, u is hypo-analytic at p.□
Note that a key ingredient of this argument is that the composition of a holomorphic function with the first integrals
$Z_{1},\dots , Z_{m}$
is a solution in a full neighbourhood of p. So in the Gevrey scenario, where the function F would be a Gevrey function such that
$\overline {\partial _{z}}F$
is flat on
$Z(U_{\mathcal {H}})$
, we do not have this same phenomenon anymore; that is,
$F\circ Z$
is not a solution on a full neighbourhood of p, just on
$U_{\mathcal {H}}$
, so we cannot apply the uniqueness result in this case.
A crucial notion on a hypo-analytic structure defined by F. Treves on his book [Reference Treves14] is the real structure bundle. If
$(V\times W,x_{1},\dots ,x_{m},t_{1},\dots ,t_{n})$
is a local coordinate system as before, for every
$t\in W$
we define
$\mathfrak {W}_{t}\subset \mathbb {C}^{m}$
as the following maximally real submanifold:
$$ \begin{align*} \mathfrak{W}_{t}\doteq\{Z(x,t)\,:\,x\in V\}\subset \mathbb{C}^{m}. \end{align*} $$
For each
$ \mathfrak {W}_{t}$
we define its real structure bundle
$\mathbb {R}\mathrm {T}^{\prime }|_{\mathfrak {W}_{t}}$
as
$$ \begin{align*} \mathbb{R}\mathrm{T}^{\prime}|_{\mathfrak{W}_{t}}=\{(Z(x,t),{}^{t}Z_{x}(x,t)^{-1}\xi\,:\, x\in V, \xi\in\mathbb{R}^{m}\setminus 0)\}. \end{align*} $$
Shrinking V and W if necessary, we can assume that
$$ \begin{align} |\phi_{j}(x,t)|\leq C(|x|^{3}+|t|),\qquad j=1,\dots,m, \end{align} $$
for some positive constant C and
$$ \begin{align} |\phi(x,t)-\phi(x^{\prime},t)|\leq\mu|x-x^{\prime}|, \end{align} $$
with
$0<\mu $
as small as we want; for instance, less than
$1$
(see p. 433 of [Reference Treves14]). Under this assumption, we can assume that for some
$0<\kappa <1$
and
$c>0$
we have that for every
$t\in W$
,
$z,z^{\prime }\in \mathfrak {W}_{t}$
and
$\zeta \in \big (\mathbb {R}\mathrm {T}^{\prime }|_{\mathfrak {W}_{t}}\big )|_{z}\cup \big (\mathbb {R}\mathrm {T}^{\prime }|_{\mathfrak {W}_{t}}\big )|_{z^{\prime }}$
, the following estimate holds:
$$ \begin{align} \begin{cases} |\mathrm{Im}\zeta|<\kappa|\mathrm{Re}\zeta|;\\ \mathrm{Im}\left\{\zeta\cdot(z-z^{\prime})+i\langle\zeta\rangle\langle z-z^{\prime}\rangle^{2}\right\}\geq c|\zeta||z-z^{\prime}|^{2}, \end{cases} \end{align} $$
where
$\langle \zeta \rangle ^{2} = \zeta \cdot \zeta $
. This inequality plays an indispensable role in the proof of Theorem 3.4. One immediate consequence of the first inequality of (6) is that for every
$(z,\zeta )\in \mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}$
the following is valid:
$$ \begin{align} \mathrm{Re}\langle\zeta\rangle\geq\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}|\zeta|\quad\text{and}\quad \mathrm{Im}\langle\zeta\rangle\leq |\zeta|. \end{align} $$
3 An F.B.I. characterisation of Gevrey vectors
Since the main result of this section, Theorem 3.4, is a local result, we shall fix an arbitrary point at
$\Omega $
and for simplicity we shall call it the origin. As we saw in the previous section, there is a hypo-analytic chart
$(U,Z_{1}(x,t),\dots ,Z_{m}(x,t))$
, with
$0\in U$
, and we can assume that the
$Z_{j}$
s are given by
$$ \begin{align*} Z_{j}(x,t)=x_{j}+i\phi_{j}(x,t),\qquad j=1,\dots,m, \end{align*} $$
with
$(x,t)\in U$
, where the map
$\phi (x,t)=(\phi _{1}(x,t),\dots ,\phi _{m}(x,t))$
is smooth, real-valued, and
$\phi (0)=0$
,
$\mathrm {d}_{x}\phi (0)=0$
and
$\phi $
satisfies (4) and (5), so we also have (6). As in Subsection 2.1, we have a family of complex vector fields
$\{\mathrm {M}_{1},\dots ,\mathrm {M}_{m},\mathrm {L}_{1},\dots ,\mathrm {L}_{n}\}$
satisfying
$$ \begin{align*} \begin{array}{l l} \mathrm{L}_{j}Z_{k}=0 & \mathrm{M}_{j}Z_{k}=\delta_{j,k} \\ \mathrm{L}_{j}t_{k}=\delta_{j,k} & \mathrm{M}_{j} t_{k}=0. \end{array} \end{align*} $$
By a similar argument presented in [Reference Treves14], in order to prove Proposition I.4.3, one can prove that if
$u\in \mathcal {D}(V\times W)$
is such that
$\mathrm {L}_{j} u \in \mathcal {C}^{\infty }(V\times W)$
, for every
$j=1,\dots ,n$
, then for some
$V^{\prime }\subset V$
and
$W^{\prime }\subset W$
, two open neighbourhoods of the origin
$u|_{V^{\prime }\times W^{\prime }} \in \mathcal {C}^{\infty }(W^{\prime };\mathcal {D}^{\prime }(V^{\prime }))$
. Therefore, for our purposes, it is enough to consider distributions on the space
$\mathcal {C}^{\infty }(W;\mathcal {D}^{\prime }(V))$
.
Definition 3.1. Let
$u\in \mathcal {C}^{\infty }(W;\mathcal {E}^{\prime }(V))$
. We define the F.B.I. transform of u as
$$ \begin{align*} \mathfrak{F}[u](t;z,\zeta)= \bigg\langle u(x,t),e^{i\zeta\cdot(z-Z(x,t))-\langle\zeta\rangle\langle z-Z(x,t)\rangle^{2}}\Delta(z-Z(x,t),\zeta))\det Z_{x}(x,t)\bigg\rangle_{\mathcal{D}^{\prime}(V),\mathcal{D}(V)}, \end{align*} $$
for
$z\in \mathbb {C}^{m}$
and
$\zeta \in \mathfrak {C}_{1}\setminus 0$
, where
$\mathfrak {C}_{\kappa } = \{\eta \in \mathbb {C}^{m} \; : \; |\mathrm {Im} \eta | <\kappa |\mathrm {Re} \eta |\}$
, for
$0<\kappa \leq 1$
.
Since u has compact support in x, we have that
$\mathfrak {F}[u](t;z,\zeta )$
is holomorphic with respect to
$(z,\zeta )\in \mathbb {C}^{m}\times \mathfrak {C}_{1}\setminus 0$
and
$\mathcal {C}^{\infty }$
with respect to t. Similarly, with the usual Fourier transform of compactly supported distributions, we have the following bound for the F.B.I. transform.
Lemma 3.2. Let
$u\in \mathcal {C}^{\infty }(W,\mathcal {E}^{\prime }(V))$
. Then there exist
$C>0$
and
$k\in \mathbb {Z}_{+}$
such that
$$ \begin{align} |\mathfrak{F}[u](t;z,\zeta)|\leq C(1+|\zeta|)^{k},\quad\forall(z,\zeta)\in\mathbb{R}\mathrm{T}^{\prime}_{\mathfrak{W}_{t}}. \end{align} $$
Note that the bound (8) is only valid for
$(z,\zeta )\in \mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}$
. Every characterisation via control of the decay/growth of the F.B.I. transform is based on an inversion formula. The one that we will use here is not quite the same as Lemma IX.
$4.1.$
of [Reference Treves14], but its proof is essentially the same. But before enunciating it, we need to extend the function
$Z(x,t)$
with respect to the variable x to the whole
$\mathbb {R}^{m}$
.
Let
$V_{1}\Subset V$
and let
$\psi \in \mathcal {C}^{\infty }_{c}(V)$
satisfying
$$ \begin{align*} &0\leq\psi\leq 1\\ &\psi\equiv 1\quad \text{in}\;V_{1}. \end{align*} $$
Define
$\widetilde {Z}(x,t)\doteq x+i\psi (x)\phi (x,t)$
. Then
$\widetilde {Z}$
defines the hypo-analytic structure in
$V_{1}\times W$
, but
$\widetilde {Z}(x,t)$
is defined for all
$x\in \mathbb {R}^{m}$
. Also note that
${}^{t}\widetilde {Z}_{x}(x,t)^{-1}=(\mathrm {Id}-i{}^{t}(\psi \phi )_{x}(x,t))(\mathrm {Id}+{}^{t}(\psi \phi )_{x}(x,t)^{2})^{-1}$
. We can choose
$V_{1}, W_{1}$
small enough so that
${}^{t}\widetilde {Z}_{x}(x,t)$
is invertible for all
$x\in \mathbb {R}^{m}$
and
$t\in W_{1}$
. From now on we shall write
$Z(x,t)$
instead of
$\widetilde {Z}(x,t)$
and V and W instead of
$V_{1}$
and
$W_{1}$
. So now
$$ \begin{align*} \mathbb{R}\mathrm{T}^{\prime}_{\mathfrak{W}_{t}}=\{(z,\zeta)\in\mathbb{C}^{m}\times\mathfrak{C}_{1}\;:\;z=Z(x,t),\;\zeta={}^{t}Z_{x}(x,t)^{-1}\xi,\;\mbox{for some}\;(x,\xi)\in\mathbb{R}^{m}\times \mathbb{R}^{m}\}, \end{align*} $$
and we can also assume that the inequality (5) is valid for all
$x\in \mathbb {R}^{m}$
. Note that (6) is still valid for
$(x,t)\in V\times W$
. Now we state the inversion formula for the F.B.I. transform that we shall use in the following.
Theorem 3.3. Let
$u\in \mathcal {C}^{\infty }(W;\mathcal {E}^{\prime }(V))$
. Then
$$ \begin{align} u(x,t)=\lim_{\varepsilon\to 0^{+}}\frac{1}{(2\pi^{3})^{\frac{m}{2}}}\iint_{\mathbb{R}\mathrm{T}^{\prime}_{\mathfrak{W}_{t}}} e^{i\zeta\cdot(Z(x,t)-z^{\prime})-\langle\zeta\rangle\langle Z(x,t)-z^{\prime}\rangle^{2}-\varepsilon\langle\zeta\rangle^{2}}\mathfrak{F}[u](t;z^{\prime},\zeta)\langle\zeta\rangle^{\frac{m}{2}}\mathrm{d}z^{\prime}\wedge\mathrm{d}\zeta, \end{align} $$
where the convergence takes place in
$\mathcal {C}^{\infty }(W;\mathcal {D}^{\prime }(V))$
Now we can state the main theorem of this work.
Theorem 3.4. Let
$\mathcal {V}$
be a locally integrable structure on
$\Omega \subset \mathbb {R}^{N}$
, let
$p\in \Omega $
be an arbitrary point and let
$s>1$
. Consider
$(V\times W, x_{1},\dots , x_{m}, t_{1},\dots t_{n})$
a local coordinate system vanishing at p, as described above. Let
$u\in \mathcal {C}^{\infty }(W;\mathcal {D}^{\prime }(V))$
be a solution of
$$ \begin{align*} \begin{cases} \mathrm{L}_{1} u = f_{1},\\ \qquad\vdots\\ \mathrm{L}_{n} u = f_{n}, \end{cases} \end{align*} $$
where
$f_{j}\in \mathrm {G}^{s}(U;\mathrm {L}_{1},\dots ,\mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m})$
,
$j=1,\dots ,n$
. The following are equivalent:
-
1. There exist
$V_{0}\subset V$
,
$W_{0}\subset W$
open balls containing the origin such that
$u|_{V_{0}\times W_{0}}\in \mathrm {G}^{s}(V_{0}\times W_{0};\mathrm {L}_{1},\dots ,\mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m})$
. -
2. There exists
$V_{1}\Subset V$
an open ball centred at the origin such that for every
$\chi \in \mathcal {C}_{c}^{\infty }(V_{1})$
, with
$0\leq \chi \leq 1$
and
$\chi \equiv 1$
in some open neighbourhood of the origin, there exist
$\widetilde {V}\subset V_{1}$
,
$\widetilde {W}\subset W$
, open balls centred at the origin and constants
$C,\varepsilon>0$
such that where
$$ \begin{align*} |\mathfrak{F}[\chi u](t;z,\zeta)|\leq Ce^{-\varepsilon|\zeta|^{\frac{1}{s}}},\qquad \forall t\in \widetilde{W}, (z,\zeta)\in\left.\mathbb{R}\mathrm{T}^{\prime}_{\mathfrak{W}_{t}}\right|{}_{\widetilde{V}}\setminus 0, \end{align*} $$
$(z,\zeta )\in \left .\mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}\right |{}_{\widetilde {V}}$
means that
$z=Z(x,t)$
,
$\zeta ={}^{\mathrm {t}}Z_{x}(x,t)^{-1}\xi $
,
$\xi \in \mathbb {R}^{m}\setminus 0$
and
$x\in \widetilde {V}$
.
-
3. For every
$\chi \in \mathcal {C}_{c}^{\infty }(V)$
, with
$0\leq \chi \leq 1$
and
$\chi \equiv 1$
in some open neighbourhood of the origin, there exist
$\widetilde {V}\subset V$
,
$\widetilde {W}\subset W$
, open balls centred at the origin, constants
$C,\varepsilon>0$
such that (10)where
$$ \begin{align} |\mathfrak{F}[\chi u](t;z,\zeta)|\leq Ce^{-\varepsilon|\zeta|^{\frac{1}{s}}},\qquad \forall t\in \widetilde{W}, (z,\zeta)\in\left.\mathbb{R}\mathrm{T}^{\prime}_{\mathfrak{W}_{t}}\right|{}_{\widetilde{V}}\setminus 0, \end{align} $$
$(z,\zeta )\in \left .\mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}\right |{}_{\widetilde {V}}$
means that
$z=Z(x,t)$
,
$\zeta ={}^{\mathrm {t}}Z_{x}(x,t)^{-1}\xi $
,
$\xi \in \mathbb {R}^{m}\setminus 0$
and
$x\in \widetilde {V}$
.
Before proving Theorem 3.4, we would like to compare it with the real analytic (hypo-analytic) case.
Theorem 3.5. Let
$\mathcal {V}$
be a locally integrable structure on
$\Omega \subset \mathbb {R}^{N}$
and let
$p\in \Omega $
be an arbitrary point. Consider
$(V\times W, x_{1},\dots , x_{m}, t_{1},\dots t_{n})$
a local coordinate system vanishing at p, as described before. Let
$u\in \mathcal {C}^{\infty }(W;\mathcal {D}^{\prime }(V))$
be such that
$\mathrm {L}_{j} u =0$
, for
$j=1,\dots ,n$
. Then the following are equivalent:
-
1. u is hypo-analytic at the origin.
-
2. There exists
$V_{1}\Subset V$
an open ball centred at the origin such that for every
$\chi \in \mathcal {C}_{c}^{\infty }(V_{1})$
, with
$0\leq \chi \leq 1$
and
$\chi \equiv 1$
in some open neighbourhood, there exist
$\mathcal {O}\subset \mathbb {C}^{m}$
, an open neighbourhood of the origin,
$0<\kappa ^{\prime }<1$
and
$C,\varepsilon>0$
such that
$$ \begin{align*} \left|\mathfrak{F}[\chi u](0;z,\zeta)\right|\leq Ce^{-\varepsilon|\zeta|},\quad\forall (z,\zeta)\in\mathcal{O}\times\mathfrak{C}_{\kappa^{\prime}}. \end{align*} $$
-
3. There exists
$V_{1}\Subset V$
an open ball centred at the origin such that for every
$\chi \in \mathcal {C}_{c}^{\infty }(V_{1})$
, with
$0\leq \chi \leq 1$
and
$\chi \equiv 1$
in some open neighbourhood, there exist
$\mathcal {O}\subset \mathbb {C}^{m}$
, an open neighbourhood of the origin, such that
$\mathfrak {F}[\chi u](0;z,\xi )$
is bounded by an integrable function with respect to
$\xi \in \mathbb {R}^{m}$
, uniformly in
$z\in \mathcal {O}$
.
This theorem can be derived from Theorem IX.3.1. and Proposition IX.5.3. (see Proposition 4.1) of F. Treves’ book [Reference Treves14]. So if
$u\in \mathcal {C}^{\infty }(W;\mathcal {D}^{\prime }(V))$
is such that
$\mathrm {L}_{j} u =0$
, for
$j=1,\dots ,n$
and it satisfies condition 2. of Theorem 3.4 but for
$t=0$
, z in some open neighbourhood of the origin
$\mathcal {O}\subset \mathbb {C}^{m}$
and
$\zeta =\xi \in \mathbb {R}^{m}\setminus 0$
, then actually u is hypo-analytic at the origin and not only a Gevrey-s vector. This shows the importance of the real structure bundle in the study of Gevrey regularity on hypo-analytic structures. In the proof of Theorem 3.4, we shall need the following formula for the derivatives of the Gaussian, which follows from Faà di Bruno’s formula (see, for instance, [Reference Bierstone and Milman7]).
Lemma 3.6. Let
$\lambda>0$
and
$\alpha \in \mathbb {Z}_{+}^{m}$
. Then
$$ \begin{align} \partial_{x}^{\alpha} e^{-\lambda|x|^{2}}=\sum_{l=(l^{\prime},l^{\prime\prime})}\frac{\alpha!}{l!}(-\lambda)^{|l|}(2x_{1},\dots,2x_{m})^{l^{\prime}}e^{-\lambda|x|^{2}}, \end{align} $$
where the sum is taken on the set
$$ \begin{align*} \{l=(l_{1}^{1},l_{2}^{1},\dots,l_{1}^{m},l_{2}^{m})\,:\,l_{1}^{j}+2l_{2}^{j}=j,\, j=1,\dots,m\}, \end{align*} $$
and
$l^{\prime }=(l_{1}^{1},\dots ,l_{1}^{m})$
and
$l^{\prime \prime }=(l_{2}^{1},\dots ,l_{2}^{m})$
.
Proof of the Theorem 3.4
$\textit {1}.\Rightarrow \textit {2}.$
:
By Theorem 2.4 we have that there exist
$\mathcal {O}\subset \mathbb {C}^{m}$
an open neighbourhood of
$\{Z(x,t)\;:\;x\in V_{0},\; t\in W_{0}\}$
on
$\mathbb {C}^{m}$
and
$F(z,t)\in \mathcal {C}^{\infty }(\mathcal {O}\times W_{0})$
such that
$$ \begin{align*} \begin{cases} F(Z(x,t),t)=u(x,t),\quad\forall (x,t)\in V_{0}\times W_{0};\\ \left|\partial_{\overline{z}}F(z,t)\right|\leq C^{k+1}k!^{s-1}\mathrm{dist}\,(z,\mathfrak{M}_{t})^{k},\quad\forall k>0, z\in\mathcal{O}, t\in W_{0}, \end{cases} \end{align*} $$
where C is a positive constant, as in (3). Set
$V_{1}=V_{0}$
and let
$\chi \in \mathcal {C}^{\infty }_{c}(V_{1})$
be such that
$0\leq \chi \leq 1$
and
$\chi \equiv 1$
in
$V_{2}\Subset V_{1}$
, an open ball centred at the origin. We shall estimate
$$ \begin{align*} \mathfrak{F}[\chi u](t;z,\zeta)=\int e^{i\zeta\cdot(z-Z(x,t))-\langle\zeta\rangle\langle z-Z(x,t)\rangle^{2}}\chi(x)u(x,t)\Delta(z-Z(x,t),\zeta)\mathrm{d}Z. \end{align*} $$
To do so, we shall deform the contour of integration. So let
$(z,\zeta )\in \left .\mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {M}_{t}}\right |{}_{\widetilde {V}}$
be fixed, where
$t\in \widetilde {W}$
and
$\widetilde {V}\Subset V$
,
$\widetilde {W}\Subset W$
are open balls centred at the origin to be chosen later. Let
$\lambda>0$
such that the image of the map
$$ \begin{align*} V_{1}\times W_{0}\ni (y,t)\mapsto \Theta_{\lambda}(y,t)\doteq Z(y,t)-i\lambda \mathbb{E}_{V_{2}}(y)\frac{\zeta}{\langle\zeta\rangle} \end{align*} $$
is contained in
$\mathcal {O}$
, where
$\mathbb {E}_{V_{2}}$
is the characteristic function of
$V_{2}$
. By Stokes’ theorem we obtain
$$ \begin{align*} &\mathfrak{F}[\chi u](t;z,\zeta)=\\ & \int_{V_{1}\setminus V_{2}}e^{i\zeta\cdot(z-Z(x,t))-\langle\zeta\rangle\langle z-Z(x,t)\rangle^{2}}\chi(x)u(x,t)\Delta(z-Z(x,t),\zeta)\mathrm{d}Z\\ &+\int_{ V_{2}}e^{i\zeta\cdot(z-\Theta_{\lambda}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\lambda}(x,t)\rangle^{2}}F(\Theta_{\lambda}(x,t),t)\Delta(z-\Theta_{\lambda}(x,t),\zeta)\mathrm{d}Z\\ &+(-1)^{m-1}2i\int_{0}^{\lambda}\int_{V_{2}}e^{i\zeta\cdot(z-\Theta_{\sigma}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\sigma}(x,t)\rangle^{2}}\overline{\partial_{z}}F(\Theta_{\sigma}(x,t),t)\cdot\frac{\overline{\zeta}}{\overline{\langle\zeta\rangle}}\cdot\\ &\quad\cdot\Delta(z-\Theta_{\sigma}(x,t),\zeta)\mathrm{d}Z\mathrm{d}\sigma-\\ &-\int_{0}^{\lambda}\int_{\partial V_{2}}e^{i\zeta\cdot(z-\Theta_{\sigma}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\sigma}(x,t)\rangle^{2}}F(\Theta_{\sigma}(x,t),t)\cdot\\ &\quad\cdot\Delta(z-\Theta_{\sigma}(x,t),\zeta)\mathrm{d}S_{\mathfrak{M}_{t}}\mathrm{d}\sigma, \end{align*} $$
where
$\mathrm {d}S_{\mathfrak {M}_{t}}$
is the surface measure in
$\{Z(x,t)\,:\,x\in \partial V_{2}\}$
. We shall estimate these four integrals separately, and we refer to them as (I), (II), (III) and (IV). Since the estimates for (I) and (IV) are very similar, we will estimate them first. We start by writing
$\widetilde {V}=B_{r}(0)$
, so
$z=Z(x_{0},t)$
for some
$x_{0}\in B_{r}(0)$
. In view of (6) and
$$ \begin{align*} |z-Z(x,t)|\geq |x_{0}-x|, \end{align*} $$
for every x, we have that
$$ \begin{align*} \mathrm{Im}\{\zeta\cdot (z-Z(x,t))+i\langle\zeta\rangle\langle z-Z(x,t)\rangle^{2}\}\geq c(r_{2}-r)^{2}|\zeta|, \end{align*} $$
for every
$x\in V_{1}\setminus V_{2}$
, where
$V_{2}=B_{r_{2}}(0)$
and we choose
$r<r_{2}$
. Therefore, (I) can be estimated by
$Ce^{-c(r_{2}-r)^{2}|\zeta |}$
.
Now the exponent of (IV) can be written as
$$ \begin{align*} i\zeta\cdot(z-\Theta_{\sigma}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\sigma}(x,t)\rangle^{2}&=i\zeta\cdot(z-Z)-\langle\zeta\rangle\langle z-Z\rangle^{2}+\\ &\quad + \sigma^{2}\langle\zeta\rangle-2i\sigma(z-Z)\cdot\zeta-\sigma\langle\zeta\rangle, \end{align*} $$
where we write
$Z=Z(x,t)$
. Now recall that in (IV) we integrate in
$\sigma $
from
$0$
to
$\lambda $
, so
$\sigma <\lambda $
, and using (6) and (7) we have that
$$ \begin{align*} &\hspace{-12pt}\mathrm{Im}\{\zeta\cdot(z-\Theta_{\sigma}(x,t))+i\langle\zeta\rangle\langle z-\Theta_{\sigma}(x,t)\rangle^{2}\}=\\ &\mathrm{Im}\{\zeta\cdot(z-Z(x,t))+i\langle\zeta\rangle\langle z-Z(x,t)\rangle^{2}\}+\sigma\mathrm{Im}\{i\langle\zeta\rangle(1-\sigma)-2(z-Z(x,t))\cdot\zeta\}\\ &\quad\geq c|z-Z(x,t)|^{2}|\zeta|+\sigma\mathrm{Re}\langle\zeta\rangle(1-\sigma)-2\sigma\mathrm{Im}\{\zeta\cdot(z-Z(x,t))\}\\ &\quad\geq c|z-Z(x,t)|^{2}|\zeta|+\sigma\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\sigma)|\zeta|-2\sigma|\zeta||z-Z(x,t)|\\ &\quad\geq |\zeta||z-Z(x,t)|\left\{c|z-Z(x,t)|-2\lambda\right\}\\ &\quad\geq |\zeta||z-Z(x,t)|\left[c(r_{2}-r)-2\lambda\right]\\ &\qquad\qquad\geq |\zeta|(r_{2}-r)\left[c(r_{2}-r)-2\lambda\right], \end{align*} $$
where we choose
$\lambda $
satisfying
$2\lambda <c(r_{2}-r)$
. Therefore, we can estimate (IV) by
$Ce^{-\varepsilon _{1}|\zeta |}$
, where
$\varepsilon _{1}=(r_{2}-r)[c(r_{2}-r)-2\lambda ]$
. Before estimating (II) and (III), note that the exponent that appears in each of them is similar to the one that we have just estimated. In (II) we have that
$x\in V_{2}$
– that is,
$|x|<r_{2}$
– so the exponentials have the following estimate:
$$ \begin{align*} \left|e^{i\zeta\cdot(z-\Theta_{\lambda}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\lambda}(x,t)\rangle^{2}}\right| \leq e^{-|\zeta|\left\{\lambda\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\lambda)+|z-Z|\left[c|z-Z|-2\lambda\right]\right\}}, \end{align*} $$
where again we write
$Z=Z(x,t)$
. When
$c|z-Z(x,t)|\geq 2\lambda $
, we have that
$$ \begin{align*} \left|e^{i\zeta\cdot(z-\Theta_{\lambda}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\lambda}(x,t)\rangle^{2}}\right|\leq e^{-|\zeta|\lambda\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\lambda)}, \end{align*} $$
and when
$c|z-Z(x,t)|\leq 2\lambda $
,
$$ \begin{align*} \left|e^{i\zeta\cdot(z-\Theta_{\lambda}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\lambda}(x,t)\rangle^{2}}\right|&\leq e^{-|\zeta|\left\{\lambda\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\lambda)-2\lambda|z-Z(x,t)|\right\}}\\ &\leq e^{-|\zeta|\left\{\lambda\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\lambda)-\frac{4\lambda^{2}}{c}\right\}}\\ &\leq e^{-|\zeta|\lambda\left\{\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\lambda)-\frac{4\lambda}{c}\right\}}. \end{align*} $$
Combining these two estimates, we conclude that (II) is bounded by
$Ce^{-\lambda \varepsilon _{2}|\zeta |}$
, where
$\varepsilon _{2}=\sqrt {\frac {1-\kappa ^{2}}{1+\kappa ^{2}}}(1-\lambda )-\frac {4\lambda }{c}>0$
, decreasing
$\lambda $
if necessary. To estimate (III), we reason as before, so for each
$0<\sigma \leq \lambda $
we have that if
$|z-Z(x,t)|\geq 2\sigma /c$
, then
$$ \begin{align*} \left|e^{i\zeta\cdot(z-\Theta_{\sigma}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\sigma}(x,t)\rangle^{2}}\right|\leq e^{-|\zeta|\sigma\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\sigma)}, \end{align*} $$
and if
$|z-Z(x,t)|\leq 2\sigma /c$
,
$$ \begin{align*} \left|e^{i\zeta\cdot(z-\Theta_{\sigma}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\sigma}(x,t)\rangle^{2}}\right|&\leq e^{-|\zeta|\left\{\sigma\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\sigma)-2\sigma|z-Z(x,t)|\right\}}\\ &\leq e^{-|\zeta|\left\{\sigma\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\sigma)-\frac{4\sigma^{2}}{c}\right\}}\\ &\leq e^{-|\zeta|\sigma\left\{\sqrt{\frac{1-\kappa^{2}}{1+\kappa^{2}}}(1-\sigma)-\frac{4\sigma}{c}\right\}}, \end{align*} $$
and since
$\sqrt {\frac {1-\kappa ^{2}}{1+\kappa ^{2}}}(1-\sigma )-\frac {4\sigma }{c}\geq \varepsilon _{2}$
, for
$\sigma <\lambda $
, we have that
$$ \begin{align*} \left|e^{i\zeta\cdot(z-\Theta_{\sigma}(x,t))-\langle\zeta\rangle\langle z-\Theta_{\sigma}(x,t)\rangle^{2}}\right|\leq e^{-\sigma\varepsilon_{2}|\zeta|}, \end{align*} $$
for every
$x\in V_{1}$
. So for every
$k>0$
we can estimate the integral (III) by
$$ \begin{align*} \bigg(\int_{0}^{\lambda} e^{-\sigma\varepsilon_{2}|\zeta|} &\sup_{(x,t)\in V_{0}\times W_{0}}\left|\overline{\partial_{z}}F(\Theta_{\sigma}(x,t),t)\Delta(z-\Theta_{\sigma}(x,t),\zeta)\right|\mathrm{d}\sigma\bigg)\cdot\\ &\quad\cdot2\left|\frac{|\zeta|}{\langle\zeta\rangle}\right|\left|\int_{V_{1}}\left|\mathrm{d}Z(x,t)\right|\right|\leq\\ &\leq C \int_{0}^{\lambda} e^{-\sigma\varepsilon|\zeta|}C^{k+1}k!^{s-1}\mathrm{dist}\,(\Theta_{\sigma}(x,t),\mathfrak{M}_{t})^{k}\mathrm{d}\sigma\\ &\leq C^{k+1}k!^{s-1}\int_{0}^{\infty} e^{-\sigma\varepsilon_{2}|\zeta|}\left|\frac{\sigma|\zeta|}{\langle\zeta\rangle}\right|{}^{k}\mathrm{d}\sigma\\ &\leq C^{k+1}k!^{s-1}\int_{0}^{\infty} e^{-y}\left(\frac{y}{\varepsilon_{2}|\zeta|}\right)^{k}\frac{1}{\varepsilon_{2}|\zeta|}\mathrm{d}y\\ &\leq C^{k+1}\frac{k!^{s}}{(\varepsilon_{2}|\zeta|)^{k+1}}. \end{align*} $$
Since the constant
$C>0$
does not depend on k and the above estimate holds for every
$k>0$
, we have that (III) is bounded by
$Ce^{-\varepsilon _{3}|\zeta |^{\frac {1}{s}}}$
, for some constants
$C, \varepsilon _{3}>0$
. Summing up, we have obtained the required estimate (10), with
$\widetilde {W}=W_{0}$
and
$\widetilde {V}=B_{r}(0)$
, where
$r>0$
is any positive number less than
$r_{2}$
, the radius of
$V_{2}$
.
$\textit {2}.\Rightarrow \textit {3}.$
:
Let
$\chi \in \mathcal {C}^{\infty }_{c}(V)$
and
$\chi _{1}\in \mathcal {C}^{\infty }_{c}(V_{1})$
as in
$\textit {2}.$
and
$\textit {3}.$
Since
$\chi \chi _{1}\in \mathcal {C}^{\infty }_{c}(V_{1})$
and
$\chi \chi _{1}\equiv 1$
in some open neighbourhood of the origin, we have that
$$ \begin{align*} |\mathfrak{F}[\chi\chi_{1} u](t;z,\zeta)|\leq Ce^{-\varepsilon|\zeta|^{\frac{1}{s}}}, \quad t\in \widetilde{W},\, (z,\zeta)\in\left.\mathbb{R}\mathrm{T}^{\prime}_{\mathfrak{W}_{t}}\right|{}_{\widetilde{V}}. \end{align*} $$
Now note that
$\chi -\chi \chi _{1}\equiv 0$
in some open neighbourhood of the origin
$V_{2}\Subset V_{1}$
. Write
$V_{2}=B_{\rho }(0)$
. So if
$x^{\prime }\in B_{\frac {\rho }{2}}(0)$
,
$x\in V\setminus V_{2}$
,
$t\in W$
and
$\zeta \in \left .\mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}\right |{}_{x}$
, we have that
$$ \begin{align*} \mathrm{Im}\{\zeta\cdot(Z-Z^{\prime})+i\langle\zeta\rangle\langle Z-Z^{\prime}\rangle^{2}\}&\geq c|Z-Z^{\prime}|^{2}|\zeta|\\ &\geq c\frac{\rho^{2}}{4}|\zeta|, \end{align*} $$
where write
$Z=Z(x,t)$
and
$Z^{\prime }=Z(x^{\prime },t)$
. Therefore, if we set
$V_{3}=B_{\frac {\rho }{2}}(0)\cap \widetilde {V}$
, we have that
$$ \begin{align*} |\mathfrak{F}[(\chi-\chi\chi_{1})u](t;z,\zeta)|\leq Ce^{-\varepsilon^{\prime}|\zeta|},\quad t\in W,\,(z,\zeta)\in\left.\mathbb{R}\mathrm{T}^{\prime}_{\mathfrak{W}_{t}}\right|{}_{V_{3}}. \end{align*} $$
Combining these two decays, we obtain
$$ \begin{align*} |\mathfrak{F}[\chi u](t;z,\zeta)|\leq Ce^{-\widetilde{\varepsilon}|\zeta|^{\frac{1}{s}}},\quad t\in \widetilde{W},\,(z,\zeta)\in\left.\mathbb{R}\mathrm{T}^{\prime}_{\mathfrak{W}_{t}}\right|{}_{V_{3}}. \end{align*} $$
$\textit {3}.\Rightarrow \textit {1}.$
:
Let
$\widetilde {V}\subset V$
and
$\widetilde {W}$
be open balls centred at the origin,
$\chi \in \mathcal {C}_{c}^{\infty }(V)$
with
$0\leq \chi \leq 1$
and
$\chi \equiv 1$
in an open ball centred at the origin and
$C,\tilde {\varepsilon }>0$
for which the following estimate holds:
$$ \begin{align*} |\mathfrak{F}[\chi u](t;z,\zeta)|\leq Ce^{-\tilde{\varepsilon}|\zeta|^{\frac{1}{s}}}, \end{align*} $$
for every
$z=Z(x,t)$
and
$\zeta ={}^{t}Z_{x}(x,t)^{-1}\xi $
, where
$x\in \widetilde {V}$
,
$t\in \widetilde {W}$
and
$\xi \in \mathbb {R}^{m}\setminus 0$
. Note that we can choose
$\mathrm {supp}\;\chi $
as small as we want, keeping in mind that
$\widetilde {V}$
depends on
$\chi $
. Since we already have that
$\mathrm {L}_{j} u\in \mathrm {G}^{s}(U;\mathrm {L}_{1},\dots ,\mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m})$
, we only have to prove that there exist
$V_{0}\subset V$
and
$W_{0}\subset W$
, open balls centred at the origin, such that, writing
$U_{0}=V_{0}\times W_{0}$
,
$u|_{U_{0}}\in \mathrm {G}^{s}(U_{0};\mathrm {M}_{1},\dots ,\mathrm {M}_{m})$
, since the complex vector fields
$\{\mathrm {L}_{1},\dots ,\mathrm {L}_{n},\mathrm {M}_{1},\dots ,\mathrm {M}_{m}\}$
are pair-wise commuting. We write
$V_{0}=B_{r}(0)$
and
$W_{0}=B_{\delta }(0)$
. By (9), we have that
$$ \begin{align*} \chi(x)u(x,t)=\lim_{\varepsilon\to 0^{+}}\frac{1}{(2\pi^{3})^{\frac{m}{2}}}\iint_{\mathbb{R}\mathrm{T}^{\prime}_{\mathfrak{W}_{t}}}e^{i\zeta\cdot(Z(x,t)-z^{\prime})-\langle\zeta\rangle\langle Z(x,t)-z^{\prime}\rangle^{2}-\varepsilon\langle\zeta\rangle^{2}}\mathfrak{F}[\chi u](t;z^{\prime},\zeta)\langle\zeta\rangle^{\frac{m}{2}}\mathrm{d}z^{\prime}\wedge\mathrm{d}\zeta. \end{align*} $$
We shall split this integral in three regions:
$$ \begin{align*} &Q^{1}_{t}\doteq\{(z^{\prime},\zeta)\,:\, z=Z(x^{\prime},t),\;\zeta={}^{t} Z_{x}(x^{\prime},t)^{-1}\xi,\;|x^{\prime}|< \tilde{r}\;\xi\in\mathbb{R}^{m}\}\\ &Q^{2}_{t}\doteq\{(z^{\prime},\zeta)\,:\,z= Z(x^{\prime},t),\;\zeta={}^{t} Z_{x}(x^{\prime},t)^{-1}\xi, \;\tilde{r}\leq|x^{\prime}|<r_{0}\;\xi\in\mathbb{R}^{m}\}\\ &Q^{3}_{t}\doteq\{(z^{\prime},\zeta)\,:\,z= Z(x^{\prime},t),\;\zeta={}^{t} Z_{x}(x^{\prime},t)^{-1}\xi, \;r_{0}\leq|x^{\prime}|\;\xi\in\mathbb{R}^{m}\}, \end{align*} $$
where
$\tilde {r}$
and
$r_{0}$
are the radii of
$\widetilde {V}$
and V. For
$\varepsilon>0$
and
$j=1, 2, 3$
, we set
$$ \begin{align*} \mathrm{I}_{j}^{\varepsilon}(x,t)\doteq\iint_{Q^{j}_{t}}e^{i\zeta\cdot(Z(x,t)-z^{\prime})-\langle\zeta\rangle\langle Z(x,t)-z^{\prime}\rangle^{2}-\varepsilon\langle\zeta\rangle^{2}}\mathfrak{F}[\chi u](t;z^{\prime},\zeta)\langle\zeta\rangle^{\frac{m}{2}}\mathrm{d}z^{\prime}\wedge\mathrm{d}\zeta, \end{align*} $$
so we can write
$$ \begin{align*} \chi(x)u(x,t)=\lim_{\varepsilon\to 0^{+}}\frac{1}{(2\pi^{3})^{\frac{m}{2}}}\big(\mathrm{I}_{1}^{\varepsilon}(x,t)+\mathrm{I}_{2}^{\varepsilon}(x,t)+\mathrm{I}_{3}^{\varepsilon}(x,t)\big). \end{align*} $$
To prove
$\textit {1}.$
it is enough to prove the following.
There exists a sequence
$\{\varepsilon _{j}\}_{j\in \mathbb {Z}_{+}}$
with
$\varepsilon _{j}\to 0$
such that
$\mathrm {I}_{2}^{\varepsilon _{j}}$
and
$\mathrm {I}_{3}^{\varepsilon _{j}}$
converge to analytic vectors for
$\mathrm {M}_{1},\dots ,\mathrm {M}_{m}$
and
$\mathrm {I}_{1}^{\varepsilon }$
converges to a Gevrey vector for
$\mathrm {M}_{1},\dots ,\mathrm {M}_{m}$
. To do so, we shall prove that there exist
$\mathrm {G}_{2}^{\varepsilon }(z,t)$
,
$\mathrm {G}_{3}^{\varepsilon }(z,t)$
,
$\mathrm {G}_{2}(z,t)$
and
$\mathrm {G}_{3}(z,t)$
, holomorphic functions in some open neighbourhood of the origin such that
$\mathrm {I}_{2}^{\varepsilon }(x,t)=\mathrm {G}_{2}^{\varepsilon }(Z(x,t),t)$
,
$\mathrm {I}_{3}^{\varepsilon }(x,t)=\mathrm {G}_{3}^{\varepsilon }(Z(x,t),t)$
and
$\mathrm {G}_{2}^{\varepsilon _{j}}(z,t)\longrightarrow \mathrm {G}_{2}(z,t)$
and
$\mathrm {G}_{3}^{\varepsilon _{j}}(z,t)\longrightarrow \mathrm {G}_{3}(z,t)$
uniformly in z, for some sequence
$\{\varepsilon _{j}\}_{j\in \mathbb {Z}_{+}}$
satisfying
$\varepsilon _{j}\to 0$
, and we shall also prove that there exists a positive constant C such that
$$ \begin{align*} |\mathrm{M}^{\alpha} \mathrm{I}_{1}^{\varepsilon}(x,t)|\leq C^{|\alpha|+1}\alpha!^{s},\quad\forall\alpha\in\mathbb{Z}_{+}^{m}, \end{align*} $$
for all
$(x,t)\in U_{0}$
and
$\varepsilon>0$
.
Let us then begin with the term
$\mathrm {I}^{\varepsilon }_{2}(x,t)$
. Let
$(z^{\prime },\zeta )\in Q_{t}^{2}$
. Since
$z^{\prime }=Z(x^{\prime },t)$
, with
$x^{\prime }\in V$
, we can use (6) and (5) to obtain
$$ \begin{align*} \mathrm{Im}\{\zeta\cdot(Z(0,t)-Z(x^{\prime},t))+&i\langle\zeta\rangle\langle Z(0,t)-Z(x^{\prime},t)\rangle^{2}\}\geq\\ &c|\zeta||Z(0,t)-Z(x^{\prime},t)|^{2}\\ &\geq c|\zeta|(1-\mu^{2})|x^{\prime}|^{2}\\ &\geq c(1-\mu^{2})\tilde{r}|\zeta|; \end{align*} $$
in other words,
$$ \begin{align*} \inf_{(z^{\prime},\zeta)\in Q_{t}^{2}}\frac{ \mathrm{Im}\{\zeta\cdot(Z(0,t)-z^{\prime})+i\langle\zeta\rangle\langle Z(0,t)-z^{\prime}\rangle^{2}\}}{|\zeta|}\geq c(1-\mu^{2})\tilde{r}, \end{align*} $$
and this is valid for every
$t\in W$
. So there are
$\mathcal {O}_{1}\subset \mathbb {C}^{m}$
an open neighbourhood of the origin and
$W_{1}\Subset W$
an open neighbourhood of the origin, such that
$$ \begin{align*} \inf_{(z^{\prime},\zeta)\in Q_{t}^{2}}\frac{ \mathrm{Im}\{\zeta\cdot(z-z^{\prime})+i\langle\zeta\rangle\langle z-z^{\prime}\rangle^{2}\}}{|\zeta|}\geq \frac{ c(1-\mu^{2})\tilde{r}}{2},\quad\forall z\in\mathcal{O}_{1}, t\in W_{1}. \end{align*} $$
Now using (8) we obtain
$$ \begin{align} \left|e^{i\zeta\cdot(z-z^{\prime})-\langle\zeta\rangle\langle z-z^{\prime}\rangle^{2}}\mathfrak{F}[\chi u](t;z^{\prime},\zeta)\langle\zeta\rangle^{\frac{m}{2}}\right|\leq C(1+|\zeta|)^{k+\frac{m}{2}}e^{-\frac{c(1-\mu^{2})\tilde{r}}{2}|\zeta|}, \end{align} $$
for some
$k\geq 0$
and for all
$z\in \mathcal {O}_{1}$
,
$(z^{\prime },\zeta )\in Q_{t}^{2}$
and
$t\in W_{1}$
. Now set
$$ \begin{align*} \mathrm{G}_{2}^{\varepsilon}(z,t)\doteq\iint_{Q^{2}_{t}}e^{i\zeta\cdot(z-z^{\prime})-\langle\zeta\rangle\langle z-z^{\prime}\rangle^{2}-\varepsilon\langle\zeta\rangle^{2}}\mathfrak{F}[\chi u](t;z^{\prime},\zeta)\langle\zeta\rangle^{\frac{m}{2}}\mathrm{d}z^{\prime}\wedge\mathrm{d}\zeta \end{align*} $$
and
$$ \begin{align*} \mathrm{G}_{2}(z,t)\doteq\iint_{Q^{2}_{t}}e^{i\zeta\cdot(z-z^{\prime})-\langle\zeta\rangle\langle z-z^{\prime}\rangle^{2}}\mathfrak{F}[\chi u](t;z^{\prime},\zeta)\langle\zeta\rangle^{\frac{m}{2}}\mathrm{d}z^{\prime}\wedge\mathrm{d}\zeta, \end{align*} $$
for
$\varepsilon>0$
,
$z\in \mathcal {O}_{1}$
and
$t\in W_{1}$
. Let
$V_{1}\Subset V$
and
$W_{2}\Subset W_{1}$
such that
$\{Z(x,t)\;:\;(x,t)\in V_{1}\times W_{2}\}\subset \mathcal {O}_{1}$
, so
$\mathrm {G}_{2}^{\varepsilon }(Z(x,t),t)=\mathrm {I}_{2}^{\varepsilon }(x,t)$
for every
$(x,t)\in V_{1}\times W_{2}$
. Define
$\mathrm {I}_{2}(x,t)\doteq \mathrm {G}_{2}(Z(x,t),t)$
, for
$(x,t)\in V_{1}\times W_{2}$
. In view of (12), we have that
$\mathrm {G}_{2}^{\varepsilon }(z,t)$
and
$\mathrm {G}_{2}(z,t)$
are holomorphic with respect to z and
$\mathrm {G}_{2}^{\varepsilon }(z,t)\longrightarrow \mathrm {G}_{2}(z,t)$
uniformly on
$\mathcal {O}_{1}\times W_{1}$
.
Now we shall deal with the term
$\mathrm {I}_{3}^{\varepsilon }(x,t)$
. We can deform the domain of integration with respect to the variable
$\zeta $
(using Stokes’ theorem), moving the contour of the integration from
$\left .\mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}\right |{}_{Z(x^{\prime },t)}$
to
$\mathbb {R}^{m}$
. Now for every
$\varepsilon>0$
we set
$$ \begin{align} \mathrm{G}_{3}^{\varepsilon}(z,t)\doteq\int_{\mathbb{R}^{m}}\int_{r_{0}\leq|x^{\prime}|}\Big\langle \widetilde{\chi u}(z^{\prime\prime},t), |\xi|^{\frac{m}{2}}\Delta(Z^{\prime}-z^{\prime\prime},\xi) e^{i\xi\cdot(z-z^{\prime\prime},t)))-|\xi|\big[\langle z-Z^{\prime}\rangle^{2}+\langle Z^{\prime}-z^{\prime\prime}\rangle^{2}\big]-\varepsilon|\xi|^{2}}\Big\rangle\mathrm{d}Z^{\prime}\mathrm{d}\xi, \end{align} $$
for
$z\in \mathbb {C}^{m}$
and
$t\in W_{2}$
, where we write
$Z^{\prime }=Z(x^{\prime },t)$
and
$\widetilde {\chi u}(z^{\prime \prime },t)=\chi (x^{\prime \prime })u(x^{\prime \prime },t)$
, for
$z^{\prime \prime }=Z(x^{\prime \prime },t)$
. As usual, we begin estimating the exponential, but first, for
$z=Z(0,t)$
:
$$ \begin{align*} \Big|e^{i\xi\cdot(z- z^{\prime\prime})-|\xi|\big[\langle z- Z^{\prime}\rangle^{2}+\langle Z^{\prime}-z^{\prime\prime}\rangle^{2}\big]}\Big|&\leq e^{|\xi||\phi(0,t)-\phi(x^{\prime\prime},t)|-|\xi|\big[|x^{\prime}|^{2}-|\phi(0,t)-\phi(x^{\prime},t)|^{2}\big]}\\ &\quad\cdot e^{-|\xi|\big[|x^{\prime}-x^{\prime\prime}|^{2}-|\phi(x^{\prime},t)-\phi(x^{\prime\prime},t)|^{2}\big]}\\ &\leq e^{-|\xi|\big[(1-\mu^{2})|x^{\prime}-x^{\prime\prime}|^{2}+(1-\mu^{2})|x^{\prime}|^{2}-\mu|x^{\prime\prime}|\big]}, \end{align*} $$
where
$z^{\prime \prime }=Z(x^{\prime \prime },t)$
with
$x^{\prime \prime }\in \textrm {supp}\,\chi $
and
$r_{0}\leq |x^{\prime }|$
. Note that the previous argument (for
$\mathrm {I_{2}^{\varepsilon }}$
) does not depend on the ‘size’ of
$\mathrm {supp}\,\chi $
; therefore, we can shrink it as we want to. So we can assume that
$|x^{\prime \prime }|$
is small enough so
$$ \begin{align*} \left|e^{i\xi\cdot(Z(0,t)- z^{\prime\prime})-|\xi|\big[\langle Z(0,t)- Z(x^{\prime},t)\rangle^{2}+\langle Z(x^{\prime},t)-z^{\prime\prime},t)\rangle^{2}\big]}\right|\leq e^{-|\xi|(1-\mu^{2})|x^{\prime}|^{2}}. \end{align*} $$
Now, for
$z\in \mathbb {C}^{m}$
, we have that
$$ \begin{align*} \bigg|&e^{i\xi\cdot(z- z^{\prime\prime})-|\xi|\big[\langle z- Z(x^{\prime},t)\rangle^{2}+\langle Z(x^{\prime},t)-z^{\prime\prime}\rangle^{2}\big]}\bigg|= \\ &\hspace{2cm}=\left|e^{i\xi\cdot(Z(0,t)- z^{\prime\prime})-|\xi|\big[\langle Z(0,t)- Z(x^{\prime},t)\rangle^{2}+\langle Z(x^{\prime},t)-z^{\prime\prime}\rangle^{2}\big]}\right|\cdot\\ &\hspace{2cm}\cdot \left|e^{i\xi\cdot(z-Z(0,t))-|\xi|\big[\langle z-Z(0,t)\rangle^{2}+2i(z-Z(0,t))\cdot(Z(0,t)-Z(x^{\prime},t))\big]}\right|\\ &\hspace{2cm}\leq e^{-|\xi|(1-\mu^{2})|x^{\prime}|^{2}}e^{|\xi||z-Z(0,t)|\big[1+|z-Z(0,t)|+2|Z(0,t)-Z(x^{\prime},t)|\big]}\\ &\hspace{2cm}\leq e^{-|\xi|(1-\mu^{2})|x^{\prime}|^{2}}e^{|\xi||z-Z(0,t)|\big[1+|z-Z(0,t)|+2(1+\mu)|x^{\prime}|\big]}. \end{align*} $$
By continuity, we can choose
$\rho>0$
such that if
$|z-Z(0,t)|<\rho $
, then
$$ \begin{align*}\frac{(1-\mu^{2})}{2}|x^{\prime}|^{2}-|z-Z(0,t)|\big[1+|z-Z(0,t)|+2(1+\mu)|x^{\prime}|\big]\geq 0,\quad\forall|x^{\prime}|\geq r_{0}.\end{align*} $$
We can shrink, if necessary,
$W_{2}$
, such that
$\sup _{t\in W_{2}}|Z(0,t)|<\rho $
. So if we define
$\mathcal {O}_{2}\subset \mathbb {C}^{m}$
as
$$ \begin{align*} \mathcal{O}_{2}\doteq\left\{z\in\mathbb{C}^{m}\;:\sup_{t\in W_{2}}|z-Z(0,t)|<\rho\right\}, \end{align*} $$
then for every
$z\in \mathcal {O}_{2}$
,
$t\in W_{2}$
and
$r_{0}\leq |x^{\prime }|$
, we have that
$$ \begin{align*} \left|e^{i\xi\cdot(z- z^{\prime\prime})-|\xi|\big[\langle z- Z(x^{\prime},t)\rangle^{2}+\langle Z(x^{\prime},t)-z^{\prime\prime}\rangle^{2}\big]}\right|\leq e^{-|\xi|\frac{(1-\mu^{2})}{2}|x^{\prime}|^{2}}, \end{align*} $$
where, again,
$z^{\prime \prime }=Z(x^{\prime \prime },t)$
and
$x^{\prime \prime }\in \mathrm {supp}\,\chi $
. Since
$\mathrm {supp}\,\chi $
and
$\overline {W_{2}}$
are compact sets, there exist
$k\in \mathbb {Z_{+}}$
and
$C>0$
such that
$$ \begin{align*} \Big|\Big\langle \widetilde{\chi u}(z^{\prime\prime},t),|\xi|^{\frac{m}{2}}&\Delta(Z(x^{\prime},t)-z^{\prime\prime},\xi)e^{i\xi\cdot(z-Z(x^{\prime\prime},t)))-\varepsilon|\xi|^{2}}\cdot\\ &\cdot e^{-|\xi|\big[\langle z-Z(x^{\prime},t)\rangle^{2}+\langle Z(x^{\prime},t)-Z(x^{\prime\prime},t)\rangle^{2}\big]}\Big\rangle\Big|\leq\\ &\leq C_{1}\sum_{|\alpha|\leq k}\sup_{x^{\prime\prime}\in\mathrm{supp}\,\chi}\Big|\partial_{x^{\prime\prime}}^{\alpha}\Big\{\chi(x^{\prime\prime})|\xi|^{\frac{m}{2}}\Delta(Z(x^{\prime},t)-Z(x^{\prime\prime},t),\xi)\cdot\\ &\hspace{2cm}\cdot \det Z_{x}(x^{\prime\prime},t)e^{i\xi\cdot(z-Z(x^{\prime\prime},t)))-\varepsilon|\xi|^{2}}\cdot\\ &\hspace{2cm}\cdot e^{-|\xi|\big[\langle z-Z(x^{\prime},t)\rangle^{2}+\langle Z(x^{\prime},t)-Z(x^{\prime\prime},t)\rangle^{2}\big]}\Big\}\Big|\\ &\leq C_{2}|\xi|^{k+\frac{m}{2}}e^{-|\xi|\frac{(1-\mu^{2})}{2}|x^{\prime}|^{2}}, \end{align*} $$
for every
$z\in \mathcal {O}_{2}$
and
$t\in W_{2}$
, where the constant
$C_{1}>0$
is given from (2), so the constant
$C_{2}$
depends on
$\mathrm {supp}\,\chi $
and k. Therefore, the integrand in (13) is dominated by
$$ \begin{align} C_{1}|\xi|^{k+\frac{m}{2}}e^{-|\xi|\frac{(1-\mu^{2})}{4}|x^{\prime}|^{2}}e^{-|\xi|\frac{(1-\mu^{2})}{4}r_{0}^{2}}. \end{align} $$
Now since the integral of
$e^{-|\xi |\frac {(1-\mu ^{2})}{4}|x^{\prime }|^{2}}$
, with respect to
$x^{\prime }$
, is bounded by a constant times
$|\xi |^{-\frac {m}{2}}$
, we have that (14) is an integrable function with respect to
$(x^{\prime },\xi )$
in
$\mathbb {R}^{m}\times \mathbb {R}^{m}$
. Therefore, by Montel’s theorem, we have that there exists a sequence
$\{\varepsilon _{j}\}_{j\in \mathbb {Z}_{+}}$
, with
$\varepsilon _{j}\to 0$
, such that
$\mathrm {G}_{3}^{\varepsilon _{j}}(z,t)$
converges to
$\mathrm {G}_{3}(z,t)$
uniformly in
$\mathcal {O}_{2}\times W_{2}$
and
$\mathrm {G}_{3}(z,t)$
is holomorphic with respect to z and is given by
$$ \begin{align*} \mathrm{G}_{3}(z,t)\doteq&\iint\Big\langle u(x^{\prime\prime},t), \chi(x^{\prime\prime})|\xi|^{\frac{m}{2}}\Delta(Z(x^{\prime},t)-Z(x^{\prime\prime},t),\xi)e^{i\xi\cdot(z-Z(x^{\prime\prime},t))}\cdot\\ &\cdot e^{-|\xi|\big[\langle z-Z(x^{\prime},t)\rangle^{2}+\langle Z(x^{\prime},t)-Z(x^{\prime\prime},t)\rangle^{2}\big]}\det Z_{x}(x^{\prime\prime},t)\Big\rangle\mathrm{d}x^{\prime}\mathrm{d}\xi, \end{align*} $$
where the integral is taken on
$\mathbb {R}^{m}\times \{|x^{\prime }|\geq r_{0}\}$
. So if we take
$V_{2}\subset V_{1}$
and
$W_{3}\subset W_{2}$
neighbourhoods of the origin, such that
$$ \begin{align*} \{Z(x,t)\;:\;x\in V_{2},t\in W_{3}\}\subset\mathcal{O}_{2}, \end{align*} $$
we have that
$\mathrm {I}_{3}^{\varepsilon _{j}}(x,t)\longrightarrow \mathrm {G}_{3}(Z(x,t),t)$
, on
$(x,t)\in V_{2}\times W_{3}$
.
Finally, we shall analyse the term
$\mathrm {I}_{1}^{\varepsilon }(x,t)$
. Let
$(x,t)\in B_{r}(0)\times B_{\delta }(0)$
and
$\alpha \in \mathbb {Z}_{+}^{m}$
. Then
$$ \begin{align*} \mathrm{M}^{\alpha} \mathrm{I}_{1}^{\varepsilon}(x,t)= \iint_{Q^{1}_{t}}&\mathrm{M}^{\alpha} \left\{e^{i\zeta\cdot(Z(x,t)-z^{\prime})-\langle\zeta\rangle\langle Z(x,t)-z^{\prime}\rangle^{2}}\right\}e^{-\varepsilon\langle\zeta\rangle^{2}}\cdot\\ &\cdot\mathfrak{F}[\chi u](t;z^{\prime},\zeta)\langle\zeta\rangle^{\frac{m}{2}}\mathrm{d}\zeta\mathrm{d}z^{\prime}. \end{align*} $$
Since
$\{\mathrm {M}_{1},\dots ,\mathrm {M}_{m}\}$
are pairwise commuting and
$\mathrm {M}_{j}Z_{k}(x,t)=\delta _{j,k}$
, we can use formula (11) to calculate
$\mathrm {M}^{\alpha } \left \{e^{i\zeta \cdot (Z(x,t)-z^{\prime })-\langle \zeta \rangle \langle Z(x,t)-z^{\prime }\rangle ^{2}}\right \}$
, obtaining
$$ \begin{align*} \mathrm{M}^{\alpha} \mathrm{I}_{1}^{\varepsilon}(x,t)&= \sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\iint_{Q^{1}_{t}}\mathrm{M}^{\alpha-\beta} e^{i\zeta\cdot(Z(x,t)-z^{\prime})}\mathrm{M}^{\beta} e^{-\langle\zeta\rangle\langle Z(x,t)-z^{\prime}\rangle^{2}}\cdot\\ &\cdot e^{-\varepsilon\langle\zeta\rangle^{2}}\mathfrak{F}[\chi u](t;z^{\prime},\zeta)\langle\zeta\rangle^{\frac{m}{2}}\mathrm{d}\zeta\mathrm{d}z^{\prime}\\ &=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\sum_{l^{1}_{1}+2l^{1}_{2}=\beta_{1}}\cdots\sum_{l^{m}_{1}+2l^{m}_{2}=\beta_{m}}\frac{\beta!}{l^{1}_{1}!l^{1}_{2}!\cdots l^{m}_{1}!l^{m}_{2}!}\cdot\\ &\cdot \iint_{Q^{1}_{t}} e^{i\zeta\cdot(Z(x,t)-z^{\prime})-\langle\zeta\rangle\langle Z(x,t)-z^{\prime}\rangle^{2}-\varepsilon\langle\zeta\rangle^{2}}\mathfrak{F}[\chi u](t;z^{\prime},\zeta)\langle\zeta\rangle^{\frac{m}{2}}\cdot\\ &\cdot(-\langle\zeta\rangle)^{l^{1}_{1}+l^{1}_{2}+\cdots+ l^{m}_{1}+l^{m}_{2}}(i\zeta)^{\alpha-\beta}(2(Z_{1}(x,t)-z^{\prime}_{1}))^{l^{1}_{1}}\cdots\\ &\cdots(2(Z_{m}(x,t)-z^{\prime}_{m}))^{l^{m}_{1}}\mathrm{d}\zeta\mathrm{d}z^{\prime}.\\ \end{align*} $$
Therefore, by (10) there exists
$\tilde {\varepsilon }>0$
such that
$$ \begin{align*} \left| \mathrm{M}^{\alpha} \mathrm{I}_{1}^{\varepsilon}(x,t)\right|&\leq \sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\sum_{l=(l^{\prime},l^{\prime\prime})}\frac{\beta!}{l!}\iint_{Q^{1}_{t}}e^{-(1-\kappa)|\zeta||Z(x,t)-z^{\prime}|^{2}}\cdot\\ &\cdot |\zeta|^{|\alpha-\beta|+l^{1}_{1}+l^{1}_{2}+\cdots+ l^{m}_{1}+l^{m}_{2}+\frac{m}{2}}\left|\mathfrak{F}[\chi u](t;z^{\prime},\zeta)\right||\mathrm{d}\zeta\mathrm{d}z^{\prime}|\\ &\leq C_{1}^{|\alpha|+1}\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\sum_{l=(l^{\prime},l^{\prime\prime})}\frac{\beta!}{l!}\iint_{Q^{1}_{t}}e^{-\tilde{\varepsilon}|\zeta|^{\frac{1}{s}}}\cdot\\ &\cdot|\zeta|^{|\alpha-\beta|+l^{1}_{1}+l^{1}_{2}+\cdots+ l^{m}_{1}+l^{m}_{2}+\frac{m}{2}}|\mathrm{d}\zeta\mathrm{d}z^{\prime}|\\ &\leq C_{2}^{|\alpha|+1}\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\sum_{l=(l^{\prime},l^{\prime\prime})}\frac{\beta!}{l!}\frac{\alpha!^{s}}{\beta!^{s}}(l^{1}_{1}+l^{1}_{2})!^{s}\cdots(l^{m}_{1}+l^{m}_{2})!^{s}\\ &\leq C_{3}^{|\alpha|+1}\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\sum_{l=(l^{\prime},l^{\prime\prime})}\frac{\beta!}{(l^{\prime},2l^{\prime\prime})!}\frac{\alpha!^{s}}{\beta!^{s}}(l^{1}_{1}+2l^{1}_{2})!^{s}(l^{m}_{1}+2l^{m}_{2})!^{s}\\ &\leq C_{4}^{|\alpha|+1}\alpha!^{s}, \end{align*} $$
where the constant
$C_{4}$
does not depend on
$\varepsilon $
(see Lemma
$4.2$
of [Reference Bierstone and Milman7] for estimating this binomial).
Remark 3.7. Note that for the implications
$\textit {1}.\Rightarrow \textit {2}.\Rightarrow \textit {3}.$
we can take
$\widetilde {W}=W_{0}$
. Also, by a closer inspection of the proof of
$\textit {1}.\Rightarrow \textit {2}.$
, we can take
$V_{1}$
as V, so that if
$\chi \in \mathcal {C}^{\infty }_{c}(V)$
such that
$\chi \equiv 1$
on
$V_{2}$
an open ball centred at the origin, then the inequality (10) is valid for every open ball
$\widetilde {V}\Subset V_{2}$
centred at the origin.
4 Propagation of singularities
In 1983, N. Hanges and F. Treves [Reference Hanges and Treves9] proved that hypo-analytic regularity propagates along elliptic submanifolds, and in their proof they actually showed that the decay of the F.B.I. transform propagates. But since then, all of the propagation of singularities results concerning systems of complex vector fields were obtained in the setting of CR geometry; for instance, holomorphic extendability of CR functions, propagation along CR orbits, sector extendability (see [Reference Tumanov15], [Reference Trépreau12] and [Reference Baracco and Zampieri4]) and so on. We did not find any other result concerning propagation of Gevrey singularities in this setup in the literature.
We shall consider only analytic tube structures; that is, locally the hypo-analytic structure is given by
$Z(x,t)=x+i\phi (t)$
, defined on
$U=V\times W$
, and
$\phi (t)$
is analytic. One of the reasons we are only dealing with tube structures is that the real structure bundle,
$\mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}$
, is trivial for every t; that is, it is equal to
$Z(U)\times \mathbb {R}^{m}$
. Now we will recall a simple comparison result for the F.B.I. transform for solutions (see proposition IX.
$5.3$
., p.
$436$
of [Reference Treves14]):
Proposition 4.1. There are open balls
$V_{0}\Subset V_{1}\Subset V$
in
$\mathbb {R}^{m}$
and
$W_{0}\Subset W$
in
$\mathbb {R}^{n}$
, all centred at the origin, and constants
$r,\kappa ,R>0$
such that, if
$\chi \in \mathcal {C}_{c}^{\infty }(V_{1})$
is equal to
$1$
in
$V_{0}$
, then, to every solution u in
$U=V\times W$
, there is a constant
$C>0$
such that
$$ \begin{align*} |\mathfrak{F}[\chi u](t;z,\zeta)-\mathfrak{F}[\chi u](t^{\prime};z,\zeta)|\leq Ce^{-|\zeta|/R} \end{align*} $$
in the region
$$ \begin{align*} t, t^{\prime}\in W_{0}, z\in\mathbb{C}^{m}, |z|<r, \zeta\in\mathfrak{C}_{\kappa}. \end{align*} $$
This proposition can be used to show that hypo-analyticity propagates along connected fibres (recall that a fibre is locally a level set of the map
$Z(x,t)$
). Let us just indicate how it is done. Suppose that u is hypo-analytic at the origin and let
$t_{0}\in W_{0}$
be such that
$Z(0,t_{0})=0$
. To show that u is hypo-analytic at
$(0,t_{0})$
, it is enough to show that
$u|_{\mathcal {H}_{t_{0}}}$
is hypo-analytic at
$(0,t_{0})$
, where
$\mathcal {H}_{t_{0}}=\{(x,t_{0})\,:\,x\in V_{0}\}$
, but this is equivalent to
$$ \begin{align*} |\mathfrak{F}[\chi u](t_{0};z,\zeta)|\leq Ce^{-\varepsilon|\zeta|}, \end{align*} $$
for some
$C,\varepsilon>0$
and z in some open neighbourhood of the origin and
$\zeta \in \mathfrak {C}_{\kappa }$
, for some
$0<\kappa <1$
. But since u is hypo-analytic at the origin, we have that
$$ \begin{align*} |\mathfrak{F}[\chi u](0;z,\zeta)|\leq Ce^{-\varepsilon|\zeta|}, \end{align*} $$
for some
$C,\varepsilon>0$
and z in some open neighbourhood of the origin and
$\zeta \in \mathfrak {C}_{\kappa }$
, for some
$0<\kappa <1$
; therefore, we have the desired decay at
$t_{0}$
in view of Proposition 4.1. One can follow the end of the proof of the Theorem 4.3 to globalise this argument to connected fibres. So why can we not use this same argument for Gevrey vectors? First, we do not have the property that ensures the desired regularity by only looking to restrictions on maximally real submanifolds (for instance, the maximally real submanifold
$\{t=0\}$
). Second, in order to measure Gevrey regularity using the F.B.I. transform, the point
$(z,\zeta )$
must belong to the real structure bundle
$\mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}$
, which depends on t. So to avoid this dependence, we restrict ourselves to tube structures. To deal with the problem that ‘restricting to maximally real submanifolds is not enough’, we need some sort of foliation near the ‘propagators’, and for that it is important for the structure to be analytic.
4.1 Propagation of Gevrey regularity for solutions of the nonhomogeneous system
In this section we will define the sets that will propagate the Gevrey regularity – the ‘propagators’ – and then exhibit the proof of the second main theorem of this work. Let
$\Sigma \subset \Omega $
be a connected subset of
$\Omega $
satisfying the following properties:
-
1. For every
$p\in \Sigma $
there is
$(U,Z)$
, a hypo-analytic chart, with
$p\in U$
, such that
$\Sigma \cap U\subset Z^{-1}(0)$
. -
2. In the same situation as above, for every
$q\in \Sigma \cap U$
and
$\widetilde {U}_{1}\Subset U$
, an open neighbourhood of p, there is
$\widetilde {U}_{2}\Subset U$
, an open neighbourhood of q, such that the connected component of the fibre
$Z^{-1}(Z(q^{\prime }))$
that contains
$q^{\prime }$
intersects
$\widetilde {U}_{1}$
, for every
$q^{\prime }\in \widetilde {U}_{2}$
. -
3. The map
$\Sigma \ni p\mapsto \sup \{r>0\;:\;B_{r}(p)\subset U\}$
is continuous.
Condition 2. implies that for every
$q^{\prime }\in \widetilde {U}_{2}$
there is a curve
$\gamma _{q^{\prime }}:[0,1]\longrightarrow U$
satisfying
-
•
$\gamma _{q^{\prime }}(0)=q^{\prime }$
; -
•
$Z(\gamma _{q^{\prime }}(\sigma ))=Z(q^{\prime })$
, for every
$0\leq \sigma \leq 1$
; -
•
$\gamma _{q^{\prime }}(1)\in \widetilde {U}_{1}$
.
Since the structure is analytic, the level sets of
$Z(x,t)$
are subanalytic sets; therefore, the curves
$\{\gamma _{q^{\prime }}\}_{q^{\prime }\in \widetilde {U}_{2}}$
have bounded length; see, for instance, Section
$8$
of [Reference Hardt10] or p.
$39$
of [Reference Treves13] (in the Appendix written by B. Teissier). Let
$p\in \Sigma $
and let
$(U,Z)$
be the hypo-analytic chart described above. Take local coordinates in
$(U$
,
$x_{1}, \dots , x_{m}, t_{1}, \dots , t_{n})$
, such that in this coordinate
$p=0$
,
$U=V\times W$
and the real structure bundle on V,
$\left .\mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}\right |{}_{V}$
, is well positioned for every
$t\in W$
; that is, there exists
$c_{0}>0$
such that
$$ \begin{align*} \mathrm{Im}\{\xi\cdot(Z(x,t)-Z(y,t))+i|\xi|\langle Z(x,t)-Z(y,t)\rangle^{2}\}\geq c_{0}|\xi||Z(x,t)-Z(y,t)|^{2}, \end{align*} $$
for every
$x,y\in V$
,
$t\in W$
and
$\xi \in \mathbb {R}^{m}$
.
Lemma 4.2. Let
$\rho>0$
be such that
$B_{\rho }(0)\Subset V$
and let
$f\in \mathcal {C}^{\infty }(W;\mathcal {E}^{\prime }(K\setminus B_{\rho }(0)))$
, where
$B_{\rho }(0)\subset K\Subset V$
is a compact set. Then
$$ \begin{align*} |\mathfrak{F}[f](t;Z(x,t),\xi)|\leq C e^{-\varepsilon|\xi|},\quad \forall x\in B_{\rho/2}(0),\, t\in W,\,\xi\in\mathbb{R}^{m}. \end{align*} $$
The proof of this lemma can be found in the proof of Proposition IX.
$5.3$
. of [Reference Treves14] or in the proof of 2.
$\Rightarrow $
3. of Theorem 3.4. Now we can state the second main theorem of this work.
Theorem 4.3. Let
$\Omega \subset \mathbb {R}^{n+m}$
be an open set endowed with an analytic hypo-analytic structure of tube type. Let
$\Sigma \subset \Omega $
be a connected submanifold as described above. If
$u\in \mathcal {D}^{\prime }(\Omega )$
is such that
$\mathbb {L}u\in \mathrm {G}^{s}(\Omega )$
, then
$\textrm {singsupp}_{s}\,u\cap \Sigma =\emptyset $
or
$\Sigma \subset \textrm {singsupp}_{s}\,u$
.
Proof. In view of Remark 2.2, we can use the F.B.I. transform to characterise Gevrey-s regularity as in Theorem 3.4. So let
$p\in \Sigma $
and suppose that
$p\notin \textrm {singsupp}_{s}\,u$
. Let
$(U,Z)$
be the hypo-analytic chart described before. Consider in U the local coordinates
$(x_{1}, \dots , x_{m}, t_{1}, \dots , t_{n})$
as in the previous section. In this coordinate system,
$p=0$
, and we write
$U=V\times W$
, where
$V\subset \mathbb {R}^{n}$
and
$W\subset \mathbb {R}^{m}$
are both open neighbourhoods of the origin. We also have that
$$ \begin{align*} Z_{k}(x,t)=x_{k}+i\phi_{k}(t),\quad k=1,\dots, m, \end{align*} $$
and
$$ \begin{align*} \begin{array}{ll} \mathrm{L}_{j} Z_{k}=0 & \mathrm{M}_{l} Z_{k} = \delta_{l,k}\\ \mathrm{L}_{j} t_{i} = \delta_{j,i} & \mathrm{M}_{l} t_{i}=0. \end{array} \end{align*} $$
We can also assume that the real structure bundle
$\mathbb {R}\mathrm {T}^{\prime }_{\mathfrak {W}_{t}}$
is well positioned, for every
$t\in W$
; that is, (6) is valid on U. Now let
$\rho>0$
be such that
$B_{\rho }(0)\subset V$
and
$\chi \in \mathcal {C}^{\infty }_{c}(V)$
be such that
$\chi \equiv 1$
on
$B_{\rho }(0)$
. Since
$\mathbb {L}u\in \mathrm {G}^{s}(\Omega )$
, we have that
$\mathrm {L}_{j} u\in \mathrm {G}^{s}(U; \mathrm {L}_{1}, \dots , \mathrm {L}_{n}, \mathrm {M}_{1}, \dots , \mathrm {M}_{m})$
; then
$u\in \mathcal {C}^{\infty }(W; \mathcal {D}^{\prime }(V))$
. By Theorem 3.4 and Remark 3.7, there exist constants
$C,\varepsilon _{1}>0$
such that
$$ \begin{align} \big|\mathfrak{F}[\chi\mathrm{L}_{j}u](t;Z(x,t),\xi)\big|\leq Ce^{-\varepsilon_{1}|\xi|^{\frac{1}{s}}},\quad j=1,\dots,n, \end{align} $$
for every
$x\in B_{\rho ^{\prime }}(0)$
,
$t\in W$
and
$\xi \in \mathbb {R}^{m}$
, where
$\rho /2\leq \rho ^{\prime }<\rho $
. We assume that
$u|_{U_{0}}\in \mathrm {G}^{s}(U_{0}; \mathrm {L}_{1}, \dots , \mathrm {L}_{n}, \mathrm {M}_{1}, \dots , \mathrm {M}_{m})$
, for some open neighbourhood of the origin,
$U_{0}=V_{0}\times W_{0}$
. Then by Theorem 3.4 and Remark 3.7 there exist
$V_{1}\Subset V$
, an open neighbourhood of the origin and positive constants
$C,\varepsilon _{2}$
, such that
$$ \begin{align} |\mathfrak{F}[\chi u](t;Z(x,t),\xi)|\leq Ce^{-\varepsilon_{2}|\xi|^{\frac{1}{s}}},\quad\forall (x,t)\in V_{1}\times W_{0},\,\forall \xi\in\mathbb{R}^{m}. \end{align} $$
By condition 2., for every
$(x_{0},t_{0})\in \Sigma \cap \left ( B_{\rho /2}(0)\times W\right )$
there exists
$\widetilde {V}\times \widetilde {W}\subset B_{\rho /2}\times W$
, an open neighbourhood of
$(x_{0},t_{0})$
, such that, for every
$(x^{\prime },t^{\prime })\in \widetilde {V}\times \widetilde {W}$
, there is a curve
$\gamma _{(x^{\prime },t^{\prime })}:[0,1]\longrightarrow U$
satisfying
-
•
$\gamma _{(x^{\prime },t^{\prime })}(0)=(x^{\prime },t^{\prime })$
; -
•
$Z(\gamma _{(x^{\prime },t^{\prime })}(\sigma ))=Z(x^{\prime },t^{\prime })$
, for every
$0\leq \sigma \leq 1$
; -
•
$\gamma _{(x^{\prime },t^{\prime })}(1)\in V_{1}\times W_{0}$
; -
• There exists
$C_{1}>0$
such that for every
$$ \begin{align*} \int_{0}^{1}\|\gamma^{\prime}_{(x^{\prime},t^{\prime})}(\sigma)\|\mathrm{d}\sigma\leq C_{1}, \end{align*} $$
$(x^{\prime },t^{\prime })\in \widetilde {V}\times \widetilde {W}$
.
Now let
$(x^{\prime },t^{\prime })\in \widetilde {V}\times \widetilde {W}$
be fixed. We write
$\gamma _{(x^{\prime },t^{\prime })}(\sigma )=(\gamma ^{(1)}_{(x^{\prime },t^{\prime })}(\sigma ),\gamma ^{(2)}_{(x^{\prime },t^{\prime })}(\sigma ))$
. By the fundamental theorem of calculus, we have that
$$ \begin{align*} \mathfrak{F}[\chi u](t^{\prime};&Z(x^{\prime},t^{\prime}),\xi)-\mathfrak{F}[\chi u](\gamma^{(2)}_{(x^{\prime},t^{\prime})}(1), Z(\gamma_{(x^{\prime},t^{\prime})}(1)),\xi)=\\ &=\int_{0}^{1}\frac{\partial}{\partial \sigma}\mathfrak{F}[\chi u](\gamma^{(2)}_{(x^{\prime},t^{\prime})}(\sigma);Z(x^{\prime},t^{\prime}),\xi)\mathrm{d}\sigma\\ &=\int_{0}^{1}\sum_{j=1}^{n}\mathfrak{F}[\mathrm{L}_{j}(\chi u)](\gamma^{(2)}_{(x^{\prime},t^{\prime})}(\sigma);Z(x^{\prime},t^{\prime}),\xi)\frac{\mathrm{d}}{\mathrm{d}\sigma}{\gamma^{(2)}_{(x^{\prime},t^{\prime})}}_{j}(\sigma)\mathrm{d}\sigma\\ &=\int_{0}^{1}\sum_{j=1}^{n}\mathfrak{F}[u\mathrm{L}_{j}\chi ](\gamma^{(2)}_{(x^{\prime},t^{\prime})}(\sigma);Z(x^{\prime},t^{\prime}),\xi)\frac{\mathrm{d}}{\mathrm{d}\sigma}{\gamma^{(2)}_{(x^{\prime},t^{\prime})}}_{j}(\sigma)\mathrm{d}\sigma+\\ &+\int_{0}^{1}\sum_{j=1}^{n}\mathfrak{F}[\chi\mathrm{L}_{j}u](\gamma^{(2)}_{(x^{\prime},t^{\prime})}(\sigma);Z(x^{\prime},t^{\prime}),\xi)\frac{\mathrm{d}}{\mathrm{d}\sigma}{\gamma^{(2)}_{(x^{\prime},t^{\prime})}}_{j}(\sigma)\mathrm{d}\sigma. \end{align*} $$
Now we analyse these two terms separately. First, we note that
$\mathrm {L}_{j}\chi $
vanishes on
$B_{\rho }(0)$
, for
$j=1,\dots , n$
; therefore, by Lemma 4.2, we have that there exist
$C,\varepsilon _{3}>0$
such that
$$ \begin{align*} |\mathfrak{F}[u\mathrm{L}_{j}\chi ](t;Z(x,t),\xi)|\leq Ce^{-\varepsilon_{3}|\xi|},\quad \forall x\in B_{\rho/2}(0),\,t\in W,\, \xi\in\mathbb{R}^{m}; \end{align*} $$
therefore,
$$ \begin{align*} |\mathfrak{F}[u\mathrm{L}_{j}\chi ](\gamma^{(2)}_{(x^{\prime},t^{\prime})}(\sigma);Z(x^{\prime},t^{\prime}),\xi)|\leq Ce^{-\varepsilon_{3}|\xi|},\quad 0\leq\sigma\leq 1,\, \forall\xi\in\mathbb{R}^{m}. \end{align*} $$
Now in view of (15) and (16), we have that
$$ \begin{align*} |\mathfrak{F}[\chi\mathrm{L}_{j}u](\gamma^{(2)}_{(x^{\prime},t^{\prime})}(\sigma);Z(x^{\prime},t^{\prime}),\xi)|\leq Ce^{-\varepsilon_{1}|\xi|^{\frac{1}{s}}},\quad 0\leq\sigma\leq 1,\, \forall\xi\in\mathbb{R}^{m}, \end{align*} $$
and
$$ \begin{align*} |\mathfrak{F}[\chi u](\gamma^{(2)}_{(x^{\prime},t^{\prime})}(1), Z(\gamma_{(x^{\prime},t^{\prime})}(1)),\xi)| \leq Ce^{-\varepsilon_{2}|\xi|^{\frac{1}{s}}},\, \forall \xi\in\mathbb{R}^{m}. \end{align*} $$
Summing up, we have obtained
$$ \begin{align*} |\mathfrak{F}[\chi u](t^{\prime};Z(x^{\prime},t^{\prime}),\xi)|&\leq |\mathfrak{F}[\chi u](\gamma^{(2)}_{(x^{\prime},t^{\prime})}(1), Z(\gamma_{(x^{\prime},t^{\prime})}(1)),\xi)|+ Ce^{-\varepsilon_{4}|\xi|^{\frac{1}{s}}}\\ &\leq Ce^{-\varepsilon|\xi|^{\frac{1}{s}}} \end{align*} $$
for every
$(x^{\prime },t^{\prime })\in \widetilde {V}\times \widetilde {W}$
, since
$\gamma _{(x^{\prime },t^{\prime })}(1)\in V_{1}\times W_{0}$
, where
$\varepsilon _{4}=\min \{\varepsilon _{1},\varepsilon _{3}\}$
and
$\varepsilon =\min \{\varepsilon _{2},\varepsilon _{4}\}$
. In conclusion, we have proved the following.
For every
$p\in \Sigma $
, if we take
$(V\times W, x_{1},\dots ,x_{m},t_{1},\dots , t_{n})$
the local coordinate system described earlier and, in particular, with respect to these coordinates, the point p is the origin, then for every
$\rho>0$
such that
$B_{\rho }(0)\subset V$
, the following holds true:
$$ \begin{align*} 0\notin \textrm{singsupp}_{s}\,u \Longrightarrow (x,t)\notin\textrm{singsupp}_{s}\,u,\quad\forall (x,t)\in\big(B_{\rho/2}(0)\times W\big)\cap\Sigma. \end{align*} $$
Now since the radii of V and W can be chosen to vary continuously with p, we can also impose the function
$\Sigma \ni p \mapsto \rho (p)$
to be continuous. We denote by
$\mathcal {U}(p)$
the open set
$B_{\rho (p)/2}\times W(p)$
.
We claim that
$\Sigma \cap \,\textrm {singsupp}_{s}\,u$
is an open set. So take
$\{p_{k}\}_{k\in \mathbb {Z}_{+}}$
a sequence on
$\Sigma \cap \complement \,\textrm {singsupp}_{s}\,u$
, such that
$p_{k}\rightarrow p\in \Sigma $
. Since the diameter of
$\mathcal {U}(p)$
is a continuous function of p and the set
$\{p_{k}\}_{k\in \mathbb {Z}_{+}}\cup \{p\}$
is compact, there exists
$\delta>0$
such that the open set
$\mathcal {U}(p_{k})$
, as described above, contains a ball, centred at
$p_{k}$
, of radius at least
$\delta $
, for every k (for the minimum of the diameters of
$\mathcal {U}(p_{k})$
is strictly positive). So there exists
$k_{0}>0$
such that
$p\in {\mathcal {U}}(p_{k_{0}})$
. Since
$p_{k_{0}}\notin \textrm {singsupp}_{s}\,u$
, we have that
$\Sigma \cap \mathcal {U}(p_{k_{0}})\subset \complement \,\textrm {singsupp}_{s}\,u$
; that is,
$p\notin \textrm {singsupp}_{s} u$
. Clearly,
$\Sigma \cap \,\textrm {singsupp}_{s} u$
is closed. Therefore,
$\Sigma \cap \,\textrm {singsupp}_{s} u=\emptyset $
or
$\Sigma \subset \textrm {singsupp}_{s} u$
.
Appendix
Since Theorem 2.4 is only available in Paulo Caetano’s Ph.D. thesis ([Reference Caetano8]), here we shall expose a sketch of its proof.
Let
$\mathcal {H}\subset \mathbb {C}^{m}$
be a maximally real submanifold and assume that
$\mathcal {H}$
contains the origin. Then after a biholomorphism, we can assume that locally
$\mathcal {H}$
is given by the image of the map
$Z(x)=x+i\phi (x)$
, where the map
$\phi $
is real-valued,
$\phi (0)=0$
and
$\mathrm {d}\phi (0)=0$
. So there is U a neighbourhood of the origin such that
$$ \begin{align} \begin{cases} u=x;\\ v=y-\phi(x) \end{cases} \end{align} $$
defines a coordinate system at U (here we identify
$\mathbb {C}^{m}$
with
$\mathbb {R}^{2m}$
), and in this new coordinate system,
$\mathcal {H}\cap U=\{v=0\}$
. We also have that in the new coordinates we can write the vector fields
$\partial /\partial \overline {z}_{j}$
as
$$ \begin{align} \frac{\partial}{\partial \overline{w}_{j}}-\frac{1}{2}\sum_{k=1}^{m}\frac{\partial \phi_{k}}{\partial x_{j}}\frac{\partial}{\partial v_{k}}, \end{align} $$
for
$j=1,\dots , m$
. We also have that the complex vector fields
$\{\mathrm {M}_{1},\dots ,\mathrm {M}_{m}\}$
that are tangent to
$\mathcal {M}$
and dual to
$\{\mathrm {d}Z_{1},\dots ,\mathrm {d}Z_{m}\}$
are given by
$$ \begin{align*} \mathrm{M}_{j}=\sum_{k=1}^{m}\mu_{jk}\frac{\partial}{\partial u_{k}}, \end{align*} $$
where
$(\mu _{ji})$
is the inverse of the matrix
$\left (\delta _{ij}+i\frac {\partial \phi _{i}}{\partial x_{j}}\right )$
. After multiplying the vector fields (18) by the invertible matrix
$2/i(\mu _{jk})$
, we get the following family of first-order linear differential operators:
$$ \begin{align} L_{j}=\frac{\partial}{\partial v_{j}}-i\mathrm{M}_{j}. \end{align} $$
For this system of first-order equations it is quite simple to construct a formal solution to the problem
$$ \begin{align} \begin{cases} L_{j}F^{\sharp}(u,v)=0;\\ F^{\sharp}(u,0)=f(u), \end{cases} \end{align} $$
where the function
$f(u)$
is a Gevrey-s vector for
$\{\mathrm {M}_{1},\dots ,\mathrm {M}_{m}\}$
and
$s>1$
. In fact, the formal solution
$F^{\sharp }$
is given by
$$ \begin{align} F^{\sharp}(u,v)=\sum_{\alpha}\frac{\mathrm{M}^{\alpha} f(u)}{\alpha!}v^{\alpha}. \end{align} $$
Here we point out that the vector fields
$\mathrm {M}_{j}$
s only depend on the variable u, so it is easy to extend the action of the operators
$L_{j}$
s to formal power series on v with coefficients depending on u. Now using the same techniques used to produce a Gevrey approximate solution from a formal solution used by R. F. Baristichi and G. Petronilho in [Reference Barostichi and Petronilho5] (see Proposition 4.3), we can obtain a function
$F(u,v)$
satisfying
$$ \begin{align} \begin{cases} \big|L_{j}F(u,v)\big|\leq C^{k}k!^{s-1}|v|^{k},\quad j=1,\dots,m,\forall k;\\ F(u,0)=f(u). \end{cases} \end{align} $$
After composing the function F with the inverse of the diffeomorphism (17), we obtain the desired almost analytic extension.
Acknowledgements
I express my gratitude to Prof. Paulo D. Cordaro for his careful guidance during my Ph.D., and I also thank the reserach group at the University of São Paulo (São Paulo and São Carlos) and at the Federal University of São Carlos for the helpful seminars and conversations. I thank Luis F. Ragognette for his careful reading of the preprint. Finally, I thank CNPq for financial support.
Competing Interests
None.
