Global ${\mathcal {M}}-$Hypoellipticity, Global ${\mathcal {M}}-$Solvability and Perturbations by Lower Order Ultradifferential Pseudodifferential Operators

Abstract

We introduce a new class of ultradifferentiable pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity, in ultradifferentiable classes, with a finite loss of derivatives of a system of pseudodifferential operators, is stable under perturbations by lower order pseudodifferential operators whose order depends on the loss of derivatives. The key point in our study is our definition of loss of derivatives. We also give an easy proof of the fact that if a system of pseudodifferential operators is globally ${\mathcal {M}}$-hypoelliptic then its transpose is globally solvable in $D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$. Finally we present an application of our results.

1 Introduction

If A is an elliptic properly supported pseudodifferential operator of order m in a $C^\infty$ manifold then $Au \in H^{\text {loc}}_{(s)}$ implies $u\in H^{\text {loc}}_{(s+m)}$ (cf. Hörmader, [8], Theorem 18.1.29). There are cases where A is not elliptic but the condition $Au \in H^{\text {loc}}_{(s)}$ implies $u\in H^{\text {loc}}_{(s+m-r)}$ for some $r>0$. When $Au \in H^{\text {loc}}_{(s)}$ implies that $u \in H^{\text {loc}}_{(s+m-r)}$ for some $r\ge 0$ we say that the operator A is hypoelliptic with loss of r derivatives. There are many results in the literature on this problem. For instance, we refer the reader to Parmegiani [12], Ferra and Petronilho [6], Chinni and Cordaro [5] and to the references in these papers.

Our main purpose in this paper is to study the above problem in the periodic ultradifferentiable frame, which includes the periodic Gevrey case, as P. D. Cordaro suggested to us.

We start by recalling that in the paper “On Global Analytic and Gevrey Hypoellipticity on the Torus and the Métivier Inequality”, see [5], the authors G. Chinni and P. D. Cordaro introduced a new theory about analytic pseudodifferential operators on the N-dimensional torus $\mathbb {{\mathbb {T}}}^N$. One question analyzed by them is the following: assuming that P(x, D) is a linear partial differential operator defined on $\mathbb {{\mathbb {T}}}^N$ with real-analytic coefficients, that P(x, D) is $\epsilon$-subelliptic for some $\epsilon >0$ and that P(x, D) is globally analytic hypoelliptic on $\mathbb {{\mathbb {T}}}^N$ they ask when is it true that P(x, D) remains globally analytic hypoelliptic when one adds to it an analytic pseudodifferential operator on $\mathbb {{\mathbb {T}}}^N$ of order $< \epsilon$.

In order to start working on the Cordaro’s question we first, in Sect. 2, describe the theory of the ultradifferentiable functions on the torus that we will need. We would like to point out that throughout this paper we work with the space of the periodic ultradifferentiable functions of Roumieu type and we do not distinguish if our weight sequence is quasianalytic or non-quasianalytic. In Sect. 3 we introduce a new class of $\mathcal {M}$-ultradifferentiable pseudodifferential operators defined on $\mathbb {{\mathbb {T}}}^N$ (cf. Definition 3.3), where $\mathcal {M}$ is a weight sequence (see Sect. 2), that generalizes the class of analytic pseudodifferential operators introduced by [4] and [5], in fact, for ${\mathcal {M}}=\{n!\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ we obtain the class of [5]. Also, our class does not involves finite differences in the dual variable $\xi$, as in Ruzhansky M. and Turunen V., see [14].

We would like to point out that in [4] and [5] the authors work in the distributions frame while we work in the $\mathcal {M}$-ultradistributions classes, and therefore we generalize [4] and [5] even in the analytic case. Due to the fact that we are working with $\mathcal {M}$-ultradistributions, in general, the statements and proofs are more sophisticated.

For example, in order to use the calculus of our class of ultradifferential pseudodifferential operators to study the perturbation problem the first key point is the appropriate caracterization of the $\mathcal {M}$-ultradistributions introduced in Sect. 4, (see the subspace $(\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} } ({\mathbb {T}}^N)$ of the $\mathcal {M}$-ultradistribution given in the Definition 4.1), where we have defined a norm and we characterize when a $\mathcal {M}$-ultradistribution belongs either to the space $(\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} } ({\mathbb {T}}^N)$ or belongs to the space ${\mathcal {E}}_{{\mathcal {M}}}\left( {\mathbb {T}}^N\right)$ (see Definition 2.2). This norm, in some sense, mixes the ultradifferentiable usual norm and the usual Sobolev norms (see Definition 4.1). The ultradifferentiable usual norm appears, for instance, when we take the Gevrey classes, i.e., we replace $m_n$ by $n^{s-1}$ in the expression of the supremum in Definition 4.1. We also would like to stress that in the Definition 4.1 we allow $\delta <0$ in order to introduce norms in the space of ultradistributions.

The other key point is to use the norm introduced in the space $(\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} } ({\mathbb {T}}^N)$ to give our definition of a system of ultradifferentiable pseudodifferental operators to be globally ${\mathcal {M}}$-hypoelliptic with loss of $r\ge 0$ derivatives, (cf. Definition 5.1), given in Sect. 5. In order to compare our Definition 5.1 with the definition of a properly supported pseudodifferential operator to be hypoelliptic with loss of r derivatives (with $0\le r <\infty$), in the local smooth case, we refer, e.g., the reader to Parmegiani [12]. We now point out that in [4] and [5] some of the results are for $\varepsilon -$subelliptic operators with $\varepsilon = m -r>0$ where m is the order of the operator and $r\ge 0$ is the loss of derivatives while in our case we can have $m-r<0$ as one can see in our application in Sect. 6.

The theory introduced in Sects. 4 and 5 and the important Proposition 5.5 allows us to prove the following result: “Assuming that ${\mathcal {A}}=\{a_j(x,D)\}_{j=1}^m$ is a system of $\mathcal {M}$-ultradifferentiable pseudodifferental operators of order $\sigma \in \mathbb {{\mathbb {R}}}$, where $\mathcal {M}$ is a weight sequence (cf. Definition 3.3) that is globally ${\mathcal {M}}$-hypoelliptic on $\mathbb {{\mathbb {T}}}^N$ with loss of $r\ge 0$ derivatives (cf. Definition 5.1) and therefore is ${\mathcal {M}}$-hypoelliptic on $\mathbb {{\mathbb {T}}}^N$ (cf. Theorem 5.3), then the system ${\mathcal {C}}\dot{=}\{c_j(x,D)=a_j(x,D) + b_j(x,D)\}_{j=1}^m$ in $\mathfrak {D}_{\mathfrak {p}_{\sigma }}^{\mathcal {M}}({\mathbb {T}}^N)$ is globally ${\mathcal {M}}$-hypoelliptic, provided that ${\mathcal {B}}=\{b_j(x,D)\}_{j=1}^m$ is a system of pseudodifferential operators in $\mathfrak {D}_{\mathfrak {p}_{\tau }}^{\mathcal {M}}({\mathbb {T}}^N)$, with $\tau <\sigma -r.$

In Sect. 6 we turn attention to our second question: it is well known that if P(x, D) is a linear partial differential operator that is hypoelliptic, then its transpose is locally solvable, see, e.g., Treves [15], Theorem 52.2. For the Gevrey version of this result, see Albanese, Corli and Rodino [1] and for a global version in the Gevrey classes, see Albanese and Zanghirati [2]. In this section, we shall prove an analogue of these results for our class of ultradifferentiable pseudodifferential operators. More precisely we prove that if ${\mathcal {A}}=\{a_j(x,D)\}_{j=1}^m$ is a system of $\mathcal {M}$-ultradifferentiable pseudodifferental operators of order $\sigma \in \mathbb {{\mathbb {R}}}$ is globally ${\mathcal {M}}$-hypoelliptic on ${\mathbb {T}}^N$ then $\ker {\mathcal {A}}$ has finite dimension and $\;\!^t {\mathcal {A}}$ is globally solvable in $D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$ (cf. Theorem 6.7), thus extending to the ultradifferentiable classes the well-known analogous result in the corresponding Gevrey classes (see [2]) and our proof is easier than the proof given by [2]. We would like to point out that the proof of this last result was inspired by [3].

Finally in the last section we present an application of our results which was considered in [5] in the analytic frame. More precisely we consider a linear partial differential operator that is of constant strength (cf. [7] Vol. II) given by $a(x,D)=P_0(D)+\sum _{j=1}^ma_j(x)P_j(D),$ where $a_{j}(x) \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$ and $P_{0}(D), P_{1}(D) \ldots P_{m}(D)$ are linear partial differential operators with constant coefficients in ${\mathbb {T}}^N$ and we prove that if $\beta _0$ is the order of operator $P_0(D)$ then a(x, D) is globally $\mathcal {M}$-hypoelliptic on ${\mathbb {T}}^N$ with loss of $\beta _{0} + M$ derivatives, where M is the constant that appears in the Greenfield-Wallach condition on $P_0(D)$. As a consequence of our results we conclude that a(x, D) is globally $\mathcal {M}$-hypoelliptic on $\mathbb {T}^N$, its kernel has finite dimension and its transpose $\;\!^t {\mathcal {A}}$ is globally solvable in $D'_{\mathcal {M}}(\mathbb {T}^N)$. Furthermore, these properties remain valid if we perturb the operator a(x, D) by any ultradifferentiable pseudodifferential operator of order $\tau < -M$. (cf. Corollary 7.3).

2 Ultradifferentiable Functions on Torus

Throughout this paper $\mathbb {T}^N$ is going to denote the N-dimensional torus. We say that a sequence of positive real numbers ${\mathcal {M}}=\{m_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ is a weight sequence if it satisfies the following properties:

$$\begin{aligned}&m_0=m_1=1, \end{aligned}$$

(2.1)

$$\begin{aligned}&m_n^2 \le m_{n-1}m_{n+1}, \ \forall \ n \in {{\,\mathrm{{\mathbb {N}}}\,}}, \end{aligned}$$

(2.2)

$$\begin{aligned}&\sup _{j,k\in {{\,\mathrm{{\mathbb {N}}}\,}}} \left( \frac{m_{j+k}}{m_j m_k}\right) ^{\frac{1}{j+k}} < H, \ \ \ \text {with} \ H >1. \end{aligned}$$

(2.3)

We recall that a weight sequence ${\mathcal {M}}=\{m_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ is called quasianalytic if

$$\begin{aligned} \sum _{k=1}^\infty \frac{m_{k-1}}{m_k}=\infty . \end{aligned}$$

If the sum if finite, then ${\mathcal {M}}=\{m_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ is called a non-quasianalytic weight sequence.

Remark 2.1

Throughout this paper we do not distinguish if our weight sequence is quasianalytic or non-quasianalytic.

Definition 2.2

Let $\mathcal {M}=\{m_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ be a weight sequence. We say that a function $f \in C^\infty \left( {\mathbb {T}}^N\right)$ is periodic ultradifferentiable of class $\{{\mathcal {M}}\}$ if there exist constants $C,h>0$ such that

$$\begin{aligned} \left| D^\alpha f(x) \right| \le C h^{|\alpha |} m_{|\alpha |}|\alpha |!, \ \ \forall \ x \in {\mathbb {T}}^N, \ \forall \,\, \alpha \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+^N. \end{aligned}$$

The space of the periodic ultradifferentiable functions of class $\{{\mathcal {M}}\}$ will be denoted by ${\mathcal {E}}_{{\mathcal {M}}}\left( {\mathbb {T}}^N\right)$. This class is also known as the space of the periodic ultradifferentiable functions of Roumieu type.

We would like to point out that for $m_n=n!^{s-1}$ we have the spaces of periodic Gevrey functions, in particular for $s=1$ we have the space of periodic analytic functions.

The reader should compare this definition with the one given in [10], where the author used the sequence $M_n=m_n \cdot n!$ to define the space of ultradifferentiable functions (see also the next remark).

Remark 2.3

Note that (2.2) is equivalent to say that the sequence $\left\{ \frac{m_n}{m_{n-1}}\right\} _{n\in {{\,\mathrm{{\mathbb {N}}}\,}}}$ is increasing. Also if the sequence $\{m_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ satisfies (2.2) then the sequence given by $M_n=m_n n!$ also satisfies (2.2). Thus our condition (2.2) is stronger than the condition (M.1) of Komatsu [ [10], p.26], which says that the sequence $\{M_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ satisfies (2.2). It is not difficult to see that our condition (2.3) is equivalent to condition (M.2) of [10] given by

$$\begin{aligned} M_p \le A H^p \min _{0\le q \le p} M_q M_{p-q}, \ \ \ \forall \ p \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+, \ \text{ for } \text{ some } A,H>0. \end{aligned}$$

(2.4)

We also point out that several basic properties that we shall need are also valid if we replace (2.3) by the weaker condition

$$\begin{aligned} \sup _{j\in {{\,\mathrm{{\mathbb {N}}}\,}}} \left( \frac{m_{j+1}}{m_j}\right) ^{\frac{1}{j}} <\infty , \end{aligned}$$

(2.5)

which is equivalent to say that the space ${\mathcal {E}}_{{\mathcal {M}}}\left( {\mathbb {T}}^N\right)$ is closed under differentiation and it is also equivalent to condition $(M2)'$ of [10].

For a weight sequence $\mathcal {M}=\{m_n\}_{n\in {{\,\mathrm{{\mathbb {N}}}\,}}}$ we define the associated weight function $\omega _{\mathcal {M}} : [0,\infty ]\longrightarrow [0,\infty ]$ by

$$\begin{aligned} \omega _{\mathcal {M}} (t) = \sup _{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+} \ln \left( \frac{t^n}{m_n n!}\right) , \ \text{ if } t>0 \text{ and }\ \omega _{\mathcal {M}} (0) = 0. \end{aligned}$$

(2.6)

We now list some basic properties of weight sequences that we will need.

Proposition 2.4

If $\mathcal {M}=\{m_n\}_{n\in {{\,\mathrm{{\mathbb {N}}}\,}}}$ is a weight sequence then we have the following properties:

• (i) $m_n \ge 1$ for all $n \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$;

• (ii) the sequence $\{m_n^{1/n}\}_{n\in {{\,\mathrm{{\mathbb {N}}}\,}}}$ is increasing;

• (iii) $m_j m_k \le m_{j+k}$ for all $j,k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$;

• (iv) for each $k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ we can find a constant $A_k\ge 1$ such that

  $$\begin{aligned} m_{n+k} (n+k)! \le A_k^{n+1} m_n n!, \ \forall \ n \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+; \end{aligned}$$

  (2.7)

• (v) for each $t \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ the supremum in (2.6) is assumed in $n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ that depends on t;

• (vi) for each $k>0$ we have

  $$\begin{aligned} \omega _{\mathcal {M}}(k\rho )-\omega _{\mathcal {M}}(\rho ) \ge \frac{\ln (k) \ln (\rho )}{\ln (H_0)}, \ \ \forall \ \rho>0,\,\,\text {for some}\,\,H_0>1. \end{aligned}$$

  (2.8)

For more results on weight sequences we refer the reader to Komatsu [10], Pilipović [13], Kirilov and Victor [9] and to the references in these papers.

Remark 2.5

It is clear from item (i) of Proposition 2.4 that ${\mathcal {E}}_{{\mathcal {M}}}\left( {\mathbb {T}}^N\right)$ contains the space of all periodic analytic functions $C^\omega \left( {\mathbb {T}}^N\right)$ since $|\alpha |! \le m_{|\alpha |} |\alpha |!$ for all $\alpha \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+^N.$

Remark 2.6

Let ${\mathcal {M}}=\{m_n\}_{n\in {\mathbb {Z}}_+}$ and ${\mathcal {L}} = \{\ell _n\}_{n\in {\mathbb {Z}}_+}$ be two weight sequences. It can be proved that ${\mathcal {E}}_{{\mathcal {M}}}\left( {\mathbb {T}}^N\right) \subset {\mathcal {E}}_{{\mathcal {L}}}\left( {\mathbb {T}}^N\right)$ if and only if

$$\begin{aligned} \sup _{n\in {\mathbb {Z}}_+} \left( \frac{m_n}{\ell _n}\right) ^{\frac{1}{n}}<\infty . \end{aligned}$$

(2.9)

Definition 2.7

For $h>0$ we set

$$\begin{aligned} {\mathcal {E}}_{{\mathcal {M}}, h}({\mathbb {T}}^N) = \left\{ f \in {\mathcal {E}}_{{\mathcal {M}}}({\mathbb {T}}^N): \ \left\| f \right\| _{{\mathcal {M}},h} < \infty \right\} , \end{aligned}$$

where $\left\| f \right\| _{{\mathcal {M}},h} := \sup \left\{ \frac{|D^{\alpha } f(x)|}{h^{|\alpha |} m_{|\alpha |} |\alpha |!} : x \in {\mathbb {T}}^N,\, \alpha \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+^{N} \right\} .$

It can be proved that ${\mathcal {E}}_{{\mathcal {M}}, h}({\mathbb {T}}^N)$ is a Banach space. Furthermore, if $h<h'$ , ${\mathcal {E}}_{{\mathcal {M}}, h}({\mathbb {T}}^N) \subset {\mathcal {E}}_{{\mathcal {M}}, h'}({\mathbb {T}}^N)$ with compact inclusion. We equip ${\mathcal {E}}_{{\mathcal {M}}}({\mathbb {T}}^N) = \displaystyle \bigcup _{j=1}^\infty {\mathcal {E}}_{{\mathcal {M}}, h_j}({\mathbb {T}}^N)$ with the inductive limit of any strictly increasing $\{h_j\}_{j \in {{\,\mathrm{{\mathbb {N}}}\,}}}$ that tends to infinity. In particular, ${\mathcal {E}}_{{\mathcal {M}}}({\mathbb {T}}^N)$ is a DFS-space (i.e. locally convex spaces whose topology can be described as injective limits of sequences of Banach spaces with compact inclusion maps).

Now we define the ultradistributions with respect to the sequence $\{m_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$: the space $D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$ is the topological dual of ${\mathcal {E}}_{{\mathcal {M}}}({\mathbb {T}}^N)$, that is $u \in D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$ when $u:{\mathcal {E}}_{{\mathcal {M}}}({\mathbb {T}}^N)\longrightarrow {{\,\mathrm{{\mathbb {C}}}\,}}$ is a continuous linear functional.

Finally we end this section with a characterization of the space of ultradifferentiable functions and ultradistributions in terms of the Fourier transform. For $u \in D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$, we set

$$\begin{aligned} {\hat{u}}(\xi ) = \frac{1}{(2\pi )^N} \left\langle u, e^{-i\left\langle x,\xi \right\rangle }\right\rangle , \xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N, \end{aligned}$$

which is well defined by Remark 2.5.

Theorem 2.8

Let $\mathcal {M}=\{m_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ be a weight sequence.

• (a) A function $\varphi \in C^\infty \left( {\mathbb {T}}^N\right)$ belongs to ${\mathcal {E}}_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$ if and only if there exists $C,h>0$ such that

  $$\begin{aligned} \left| {\hat{\varphi }}(\xi )\right| \le C \displaystyle \inf _{n \in {\mathbb {Z}}_{+}}\left( \frac{h^n m_n n!}{(1+|\xi |)^n} \right) , \ \ \forall \ \xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N. \end{aligned}$$

  (2.10)

  Moreover, if $\{C_\xi \}_{\xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N}$ is a sequence such that (2.10) holds true with $C_\xi$ in place of ${\hat{\varphi }}(\xi )$, then there exits an unique function $\varphi \in {\mathcal {E}}_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$ such that ${\hat{\varphi }}(\xi )=C_\xi$ for all $\xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N$.

• (b) If $u \in D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$, for each $\epsilon >0$ we can find $C_\epsilon >0$ such that

  $$\begin{aligned} \left| {\hat{u}}(\xi )\right| \le C_\epsilon \sup _{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+} \left( \frac{\epsilon ^n (1+|\xi |)^n}{m_n n!}\right) , \ \forall \ \xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N. \end{aligned}$$

  (2.11)

  Moreover, if $\{C_\xi \}_{\xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N}$ is a sequence such that (2.11) holds true with $C_\xi$ in place of $\hat{u}(\xi )$, then there exits an unique ultradistribution $u \in D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$ such that ${\hat{u}}(\xi )=C_\xi$ for all $\xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N$ and we have that

  $$\begin{aligned} u(x) = \sum _{\xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} {\hat{u}}(\xi ) e^{i\left\langle x,\xi \right\rangle } \end{aligned}$$

  where the limit is taken in the weak topology of $D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$.

3 Pseudodifferential Operators on Torus

In this section, we introduce a new class of ultradifferentiable pseudodifferential operators defined on the N-dimensional torus ${\mathbb {T}}^N$ which map ${\mathcal {E}}_{{\mathcal {M}}}({\mathbb {T}}^N)$ into itself and whose calculus allows us to study the corresponding perturbation problem in a far more general context. Our theory was inspired by the papers [4] and [5] where the authors introduced an analytic class of pseudodifferential operators on torus and by the work [6] where the smooth case was considered.

We start by recalling that the discrete symbol of a continuous and linear operator $A:C^\infty ({\mathbb {T}}^N) \longrightarrow C^\infty ({\mathbb {T}}^N)$ is the function $a:{\mathbb {T}}^N\times {\mathbb {Z}}^N\longrightarrow {\mathbb {C}}$ defined by $a(x,\eta ) = e^{-i\left\langle x,\eta \right\rangle } A\left( e^{i\left\langle x,\eta \right\rangle }\right)$ and we shall use the notation $A=a(x,D)$. Notice that if $\varphi \in C^\infty ({\mathbb {T}}^N)$ then by linearity and continuity we have

$$\begin{aligned} a(x,D)\varphi (x) = \sum _{\xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} a(x,D)\left( e^{i\left\langle x,\xi \right\rangle }\right) {\widehat{\varphi }}(\xi ) = \sum _{\xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} e^{i\left\langle x,\xi \right\rangle }a(x,\xi ){\widehat{\varphi }}(\xi )\in C^\infty (\mathbb {T}^N).\nonumber \\ \end{aligned}$$

(3.1)

We first note that if $\{\widehat{a}(\xi ,\eta )\}_{\xi \in \mathbb {Z}^N}$ denotes the sequence of the Fourier coefficients of the function $x \mapsto a(x,\eta )$ then it is easy to check that

$$\begin{aligned} \widehat{(a(x,D)\varphi )}(\xi ) = \sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} {\widehat{a}}(\xi -\eta ,\eta ){\widehat{\varphi }}(\eta ), \ \ \forall \ \varphi \in C^\infty ({\mathbb {T}}^N),\,\, \xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N \end{aligned}$$

(3.2)

and, if $k \in D'\left( {\mathbb {T}}^N\times {\mathbb {T}}^N\right)$ denotes the Schwartz distribution kernel of a(x, D) then we have the following important relation

$$\begin{aligned} (2\pi )^N{\widehat{k}}(\xi ,\eta ) = {\widehat{a}}(\xi +\eta ,-\eta ), \ \forall \ (\xi ,\eta ) \in {{\,\mathrm{{\mathbb {Z}}}\,}}^{2N}. \end{aligned}$$

(3.3)

We now will show a relation between the Schwartz distribution kernel k(x, y) and the discrete symbol of the operator a(x, D). More precisely we have the following:

Proposition 3.1

Let $a(x, D): C^{\infty }({\mathbb {T}}^N) \rightarrow C^{\infty }({\mathbb {T}}^N)$ be a continuous, linear operator and consider $k \in D'({\mathbb {T}}^N \times {\mathbb {T}}^N)$ its distribution kernel. If $\mathcal {M}=\{m_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ is a weight sequence and $\sigma \in {\mathbb {R}}$, then the following statements are equivalent:

• (i) There exist positive constants $C_1$ and $h_1$ such that

  $$\begin{aligned} |D_{x}^{\alpha } a(x, \eta )| \le C_1 h_1^{|\alpha |} m_{|\alpha |} |\alpha |! (1+|\eta |)^{\sigma },\ \forall \,\, x \in {\mathbb {T}}^{N}, \ \forall \,\, \eta \in {\mathbb {Z}}^N, \ \forall \,\, \alpha \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+^{N}.\nonumber \\ \end{aligned}$$

  (3.4)

• (ii) There exist positive constants $C_2, h_2 >0$ such that

  $$\begin{aligned} |{\widehat{a}}(\xi , \eta )| \le \frac{C_{2} h_{2}^{k} m_{k} k! (1+ |\eta |)^{\sigma }}{ (1+|\xi |)^{k} }, \ \forall \,\, k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+, \ \forall \,\, (\xi , \eta ) \in {\mathbb {Z}}^{2N}. \end{aligned}$$

  (3.5)

• (iii) There exist positive constants $C_3, h_3 >0$ such that

  $$\begin{aligned} |{\widehat{k}}(\xi , \eta )| \le \frac{C_{3} h_{3}^{k} m_{k} k! (1+ |\eta |)^{\sigma }}{(1+|\xi + \eta |)^{k}}, \ \ \forall \,\, k \in {\mathbb {Z}}_+, \ \forall \,\, (\xi , \eta ) \in {\mathbb {Z}}^{2N}. \end{aligned}$$

  (3.6)

Proof

We begin the proof by showing that (3.4) implies (3.5). If $\xi =0$ then

$$\begin{aligned} |{\widehat{a}}(\xi , \eta )| = |{\widehat{a}}(0, \eta )|=\frac{1}{(2\pi )^N} \left| \int _{{\mathbb {T}}^N} a(x,\eta )dx \right| \le C_1 (1+|\eta |)^\sigma , \ \forall \, \eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N. \end{aligned}$$

(3.7)

For $\xi \ne 0$ we choose $p \in \{1,\ldots ,N\}$ such that $\displaystyle \max _{1\le q \le N} |\xi _q| = |\xi _p|$. If $k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ and $e_p$ is the p-th vector of the canonical basis of ${{\,\mathrm{{\mathbb {R}}}\,}}^N$, then $\beta = k e_p \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+^N$ and $|\beta |=k$. We also have

$$\begin{aligned} |\xi |^{2k} = \Big (\displaystyle \sum _{j = 1}^{N} |\xi _{j}|^2\Big )^{k} \le (N |\xi _{p}|^{2})^{k} \le N^{k} |\xi _{p}|^{2k} \le N^{k} \big |\xi ^{\beta } \big |^{2}. \end{aligned}$$

(3.8)

The last inequality, (3.4) and the fact that we have $\xi \ne 0$ give us that

$$\begin{aligned} |{\widehat{a}}(\xi , \eta )| (1+|\xi |)^{k}&\le 2^k |\xi |^k |{\widehat{a}}(\xi , \eta )| \le N^{k/2} 2^k |\xi ^\beta | |{\hat{a}}(\xi , \eta )| = (2 \sqrt{N})^{k} |\widehat{D_x^\beta a}(\xi , \eta )| \\&\le \frac{1}{(2 \pi )^{N}} (2 \sqrt{N})^{k} \left| \displaystyle \int _{{\mathbb {T}}^N} e^{-i\left\langle x,\xi \right\rangle } D_x^\beta a(x,\eta ) dx \right| \\&\le (2 \sqrt{N})^{k} C_1 h_1^{|\beta |} m_{|\beta |} |\beta |! (1+|\eta |)^\sigma \\&= C_1 (2 \sqrt{N}h_1)^{k} m_{k} k! (1+|\eta |)^\sigma , \ \forall \ (\xi ,\eta ) \in {{\,\mathrm{{\mathbb {Z}}}\,}}^{2N}, k\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+. \end{aligned}$$

Hence, if we take $C_2=C_1$ and $h_2=\max \{1,2\sqrt{N} h_1\}$, the last inequality and (3.7) give us (3.5).

Reciprocally, let us prove that (3.5) implies (3.4). We have

$$\begin{aligned} \left| D_{x}^{\alpha } a(x, \eta ) \right|&= \big | \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} \xi ^{\alpha } e^{i \left\langle x,\xi \right\rangle } {\widehat{a}}(\xi , \eta ) \big | \le \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} (1+|\xi |)^{|\alpha |} |{\widehat{a}}(\xi , \eta )| \\&\le \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} \left( (1+|\xi |)^{|\alpha | +2N} |{\hat{a}}(\xi ,\eta )| \right) \displaystyle \frac{1}{(1+|\xi |)^{2N}}. \end{aligned}$$

By using (3.5) it follows from the last inequality that

$$\begin{aligned} \left| D_{x}^{\alpha } a(x, \eta ) \right|&\le \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} C_2 h_2^{|\alpha |+2N} \left( m_{|\alpha |+2N} (|\alpha |+2N)!\right) (1+|\eta |)^\sigma \displaystyle \frac{1}{(1+|\xi |)^{2N}} \\&\le \Big ( C_{2} h_{2}^{|\alpha |+2N} \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} \displaystyle \frac{1}{(1+|\xi |)^{2N}}\Big ) \underbrace{A_{2N}^{|\alpha |+1} m_{|\alpha |} |\alpha |!}_{(2.7)} (1+|\eta |)^\sigma \\&\le \Big ( A_{2N} C_{2} h_{2}^{2N}\displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} \displaystyle \frac{1}{(1+|\xi |)^{2N}}\Big ) (h_2 A_{2N})^{|\alpha |} m_{|\alpha |} |\alpha |! \cdot (1+|\eta |)^\sigma . \end{aligned}$$

By taking $C_{1} = A_{2N} C_{2} h_{2}^{2N}\displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} \displaystyle \frac{1}{(1+|\xi |)^{2N}}$ and $h_{1} = h_2 A_{2N}$, we obtain (3.4).

The equivalence between (3.4) and (3.6) follows immediately from (3.3). $\square$

We will now proceed in the opposite direction to that we have done so far in this section and discuss classes of symbols that correspond to an operator as that one defined in (3.1).

We start by the following definition.

Definition 3.2

Let $\mathcal {M}$ be a weight sequence and $\sigma \in {\mathbb {R}}$. We consider $S^\sigma _\mathcal {M}({\mathbb {T}}^N\times {{\,\mathrm{{\mathbb {Z}}}\,}}^N)$ as the space of all functions $a(x,\eta ) \in C^\infty ({\mathbb {T}}^N, {\mathbb {Z}}^N)$, i.e., when function $a(\cdot , \xi )$ is smooth on ${\mathbb {T}}^N$ for all $\xi \in {\mathbb {Z}}^N$, satisfying (3.4).

Definition 3.3

If $a(x,\xi ) \in S^\sigma _\mathcal {M}({\mathbb {T}}^N\times {{\,\mathrm{{\mathbb {Z}}}\,}}^N)$ we define, in $C^\infty ({\mathbb {T}}^N)$, the operator

$$\begin{aligned} a(x,D) \varphi (x) = \sum _{\xi \in {\mathbb {Z}}^N} e^{i x\cdot \xi } a(x,\xi ) \widehat{\varphi }(\xi ). \end{aligned}$$

(3.9)

We say that a(x, D) is a $\mathcal {M}$-ultradifferentiable pseudodifferential operator of order $\sigma$, i.e, $a(x,D)\in {\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$.

By using the fact that $\varphi \in C^\infty ({\mathbb {T}}^N)$ and estimate (3.4) one can easy prove the following

Proposition 3.4

Let $\varphi \in C^\infty ({\mathbb {T}}^N)$. Then $a(x,D)\varphi$ in (3.9) is well defined and $a(x,D)\varphi \in C^\infty ({\mathbb {T}}^N)$. Moreover, operator $a(x,D) : C^\infty ({\mathbb {T}}^N) \rightarrow C^\infty ({\mathbb {T}}^N)$ is continuous.

In the rest of this section we will present the main properties of this class of ultradifferential pseudodifferential operators and we will only present the proof of some of them. The first property is given by

Proposition 3.5

Let $a(x,D) \in \mathfrak {D}^{\mathcal {M}}_{\mathfrak {p}_{\sigma }} ({\mathbb {T}}^N)$ and $b(x, D) \in \mathfrak {D}^{\mathcal {M}}_{\mathfrak {p}_{\sigma '}}({\mathbb {T}}^N)$. The composition $a(x,D) \circ b(x,D)$ is an element of $\mathfrak {D}^{\mathcal {M}}_{\mathfrak {p}_{\sigma + \sigma '}} ({\mathbb {T}}^N)$ whose discrete symbol is given by

$$\begin{aligned} (a \circ b)(x, \eta ) = \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} e^{i\left\langle x,\xi -\eta \right\rangle } a(x,\xi ) {\widehat{b}} (\xi -\eta , \eta ),\,\,\forall \,\, x\in {\mathbb {T}}^{N},\,\,\forall \, \, \eta \in {\mathbb {Z}}^{N}.\nonumber \\ \end{aligned}$$

(3.10)

The next auxiliary result will be used in the proof that any pseudodifferential operator in ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$ can be extended as a continuous and linear operator in $D'_\mathcal {M}\left( {\mathbb {T}}^N\right)$ and its proof will be omitted.

Lemma 3.6

• (a) There exists a positive constant $\lambda$ depending only on the dimension N of ${\mathbb {T}}^N$ such that for all $h>0$ and all $f \in \mathcal {E}_{\mathcal {M}, h}({\mathbb {T}}^N)$ we have

  $$\begin{aligned} |\widehat{f}(\xi )| \le \frac{\left\| f \right\| _{\mathcal {M}, h} (\lambda h)^{k} m_{k} k!}{(1+|\xi |)^{k} }, \ \ \forall \, k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+, \ \forall \, \xi \in {\mathbb {Z}}^{N}. \end{aligned}$$

  (3.11)

• (b) For each $C,h>0$ we can find positive constants $C'$ and $h'$ depending only on h, ${\mathcal {M}}$ and the dimension N of ${\mathbb {T}}^N$ such that if $g \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$ and g satisfies

  $$\begin{aligned} |\widehat{g}(\xi )| \le \frac{C h^{k} m_{k} k!}{(1+|\xi |)^{k} }, \ \ \forall \, k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+, \ \forall \, \xi \in {\mathbb {Z}}^{N}, \end{aligned}$$

  (3.12)

  then $\left\| g \right\| _{\mathcal {M}, h'} \le C'C$.

Theorem 3.7

If $a(x,D) \in {\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$, then a(x, D) defines a continuous, linear operator on $\mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$.

Proof

In order to prove this result it suffices to show that given $h>0$ there exists $h'>0$ and $C>0$ such that $\left\| a(x,D) f \right\| _{\mathcal {M}, h'} \le C\left\| f \right\| _{\mathcal {M}, h}$, for all $f\in \mathcal {E}_{\mathcal {M}, h}({\mathbb {T}}^N)$.

Let $h>0$ be given. For $f \in \mathcal {E}_{\mathcal {M}, h}({\mathbb {T}}^N)$ we shall analyze the term $\widehat{(a(x,D)f)}(\xi )$ given in (3.2). Take $p\in {\mathbb {Z}}_+$ such that $\sigma \le p$. By Lemma 3.6 item (a) and (3.5) there exist $\lambda , C_2, h_2>0$ such that, for $j,k\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$, we have

$$\begin{aligned}&\sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} |{\hat{a}}(\xi -\eta ,\eta )||{\hat{f}}(\eta )| \le \sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} \frac{C_2 h_2^{j} m_{j} j! (1+|\eta |)^\sigma }{(1+|\xi -\eta |)^{j}} \\&\quad \times \frac{\left\| f \right\| _{\mathcal {M}, h} (\lambda h)^{k+p} m_{k+p} (k+p)!}{(1+|\eta |)^{k+p}}, \,\, \forall \, \xi \in {\mathbb {Z}}^N. \end{aligned}$$

Since $\sigma < p$ and taking advantage from (2.7), Proposition 2.4, we have

$$\begin{aligned}&\sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} |{\hat{a}}(\xi -\eta ,\eta )||{\hat{f}}(\eta )| \le C_3 \left\| f \right\| _{\mathcal {M}, h} \sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} \frac{h_2^{j} m_{j} j!}{(1+|\xi -\eta |)^{j}} \nonumber \\&\quad \times \frac{(A_p \lambda h)^k m_{k} k!}{(1+|\eta |)^{k}},\,\, \forall \, \xi \in {\mathbb {Z}}^N, j, k \in {\mathbb {Z}}_+, \end{aligned}$$

(3.13)

where $C_3 = C_2 A_{p} (\lambda h)^p$.

Now we will analyze the term $(I)\, \dot{=} \displaystyle \sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} \frac{h_2^{j} m_{j} j!}{(1+|\xi -\eta |)^{j}} \frac{(A_p \lambda h)^k m_{k} k!}{(1+|\eta |)^{k}}.$

Note that

$$\begin{aligned} (I)= & {} \sum _{|\eta | \le \frac{|\xi |}{2}} \frac{h_2^{j} m_{j} j!}{(1+|\xi -\eta |)^{j}} \frac{(A_p \lambda h)^k m_{k} k!}{(1+|\eta |)^{k}}\\&+\sum _{|\eta | > \frac{|\xi |}{2}} \frac{h_2^{j} m_{j} j!}{(1+|\xi -\eta |)^{j}} \frac{(A_p \lambda h)^k m_{k} k!}{(1+|\eta |)^{k}} \dot{=} (II)+(III). \end{aligned}$$

If $|\eta | \le \frac{|\xi |}{2}$ then we have $|\xi | \le |\xi -\eta |+\frac{|\xi |}{2}$ which in turns implies that $(1+|\xi |)\le 2(1+|\xi -\eta |)$. Thus, by taking $k=2N$ we can infer that

$$\begin{aligned} (II)= & {} \sum _{|\eta | \le \frac{|\xi |}{2}} \frac{h_2^j m_{j} j!}{(1+|\xi -\eta |)^{j}} \frac{(A_p \lambda h)^k m_{k} k!}{(1+|\eta |)^{k}} \nonumber \\\le & {} \sum _{|\eta | \le \frac{|\xi |}{2}} \frac{(2h_2)^j m_{j} j!}{(1+|\xi |)^j} \frac{(A_p \lambda h)^{2N} m_{2N} (2N)!}{(1+|\eta |)^{2N}}\nonumber \\\le & {} \Big (\sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} \frac{1}{(1+|\eta |)^{2N}}\Big ) (A_p \lambda h)^{2N} m_{2N} (2N)! \displaystyle \frac{(2 h_{2})^j m_{j} j!}{(1+|\xi |)^{j}}, \, \forall \ j \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+,\, \xi \,\in {{\,\mathrm{{\mathbb {Z}}}\,}}^N.\nonumber \\ \end{aligned}$$

(3.14)

If $|\eta | > \frac{|\xi |}{2}$ then we have $(1+|\xi |) \le 2(1+|\eta |)$ and by taking $j=2N$, we obtain

$$\begin{aligned} (III)= & {} \sum _{|\eta |> \frac{|\xi |}{2}} \frac{h_2^j m_{j} j!}{(1+|\xi -\eta |)^{j}} \frac{(A_p \lambda h)^k m_{k} k!}{(1+|\eta |)^{k}} \nonumber \\\le & {} \sum _{|\eta | > \frac{|\xi |}{2}} \frac{h_{2}^{2N} m_{2N} (2N)!}{(1+|\xi -\eta |)^{2N}} \frac{(A_p \lambda h)^k m_{k} k! 2^k}{(1+|\xi |)^{k}}\nonumber \\\le & {} \Big (\sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} \frac{1}{(1+|\eta |)^{2N}} \Big ) h_{2}^{2N} m_{2N} (2N)! (2A_p \lambda h)^k \frac{m_{k} k!}{(1+|\xi |)^{k}}, \ \forall \ k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+,\, \xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N,\nonumber \\ \end{aligned}$$

(3.15)

since

$$\begin{aligned}&\sum _{|\eta | > \frac{|\xi |}{2}} \frac{1}{(1+|\xi -\eta |)^{2N}} \le \sum _{\eta \in \mathbb {Z}^N} \frac{1}{(1+|\xi -\eta |)^{2N}} \\&\quad = \sum _{\eta \in \mathbb {Z}^N} \frac{1}{(1+|\eta |)^{2N}}\,\,\text {for every fixed}\,\, \xi \in \mathbb {Z}^N. \end{aligned}$$

If we denote $C_4 = \Big (\displaystyle \sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N}\frac{1}{(1+|\eta |)^{2N}}\Big ) (A_p \lambda h)^{2N} m_{2N} (2N)!$, $C_5=\Big (\displaystyle \sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N}\frac{1}{(1+|\eta |)^{2N}} \Big )h_{2}^{2N} m_{2N} (2N)!$, and $h_1=\max \{2 h_{2}, 2 A_p\lambda h\}$, it follows from (3.14) and (3.15) that

$$\begin{aligned} (I) \le (C_4+C_5) \frac{h_1^k m_{k} k!}{(1+|\xi |)^{k}}, \ \ \forall \ k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+, \ \xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N. \end{aligned}$$

From the last inequality and (3.13) we conclude that

$$\begin{aligned} \sum _{\eta \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N} |{\hat{a}}(\xi -\eta ,\eta )||\hat{f}(\eta )| \le C_3 \left\| f \right\| _{\mathcal {M}, h} (C_4+C_5) \frac{h_1^k m_{k} k!}{(1+|\xi |)^{k}}, \ \forall \ k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+,\, \xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N. \end{aligned}$$

Hence, if we set $C=C_3(C_4+C_5)$ we conclude that for all $f \in \mathcal {E}_{\mathcal {M},h}({\mathbb {T}}^N)$ we have

$$\begin{aligned} \left| \widehat{(a(x,D)f)}(\xi )\right| \le C \left\| f \right\| _{\mathcal {M}, h} \frac{h_1^{n} m_n n!}{(1+|\xi |)^n}, \ \ \forall \ \xi \in {\mathbb {Z}}^N. \end{aligned}$$

If $f\not \equiv 0$, then we use item (b) of Lemma 3.6 with $h=h_1$ in order to obtain $C',h'>0$ (also take $g=a(x,D){\tilde{f}}$, where ${\tilde{f}} = f/ \left\| f \right\| _{\mathcal {M},h}$) such that $\left\| a(x,D) f \right\| _{\mathcal {M}, h'} \le C' C \left\| f \right\| _{\mathcal {M}, h}.$ $\square$

We now recall that the transpose operator, ${}^ta(x, D)$, of the operator $a(x,D) \in {\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$ is given by

$$\begin{aligned} \left\langle {}^ta(x, D) u, \varphi \right\rangle = \left\langle u, a(x,D) \varphi \right\rangle , \ \forall \, u\in D'({\mathbb {T}}^N),\,\varphi \in C^\infty ({\mathbb {T}}^N), \end{aligned}$$

(3.16)

and we will prove the following result.

Proposition 3.8

If $a(x,D) \in {\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$ then its transpose ${}^ta(x,D)$ given by (3.16) belongs to the space ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$.

Proof

We denote by $\left\langle \cdot ,\cdot \right\rangle _0$ the inner product in $L^2({\mathbb {T}}^N)$. The formal adjoint in $L^2$ of a(x, D) is the operator $a^*(x,D)$ such that

$$\begin{aligned} \left\langle a(x,D) f,g\right\rangle _0 = \left\langle f, a^*(x,D) g\right\rangle _0, \ \forall \ f,g \in C^\infty ({\mathbb {T}}^N). \end{aligned}$$

Note that if $f,g \in C^\infty ({\mathbb {T}}^N)$ then $\left\langle f,g\right\rangle _0 = \int _{{\mathbb {T}}^N} f(x) \overline{g(x)} dx = \left\langle f, {{\overline{g}}}\right\rangle .$

Since $\langle f,\;\!^ta(x,D) g \rangle = \left\langle a(x,D) f, g\right\rangle = \left\langle a(x,D) f, {\overline{g}}\right\rangle _0 = \left\langle f, a^*(x,D) ({\overline{g}})\right\rangle _0 = \left\langle f, \overline{a^*(x,D) ({\overline{g}})}\right\rangle$ for all $f,g \in C^\infty ({\mathbb {T}}^N)$, it follows that $\;\!^ta(x,D) g = \overline{a^*(x,D) {\overline{g}}}$.

If we take $g_\xi (x) = e^{-i\left\langle x,\xi \right\rangle }$ in the last equality we obtain

$$\begin{aligned} \;\!^ta(x,\eta )= & {} e^{-i\left\langle x,\eta \right\rangle } \left( \;\!^t a(x,D) e^{i\left\langle x,\eta \right\rangle }\right) = e^{-i\left\langle x,\eta \right\rangle } \overline{a^*(x,D) e^{-i\left\langle x,\eta \right\rangle }}\\= & {} \overline{e^{i\left\langle x,\eta \right\rangle }a^*(x,D) e^{-i\left\langle x,\eta \right\rangle }} = \overline{a^*(x,-\eta )}. \end{aligned}$$

Thus

$$\begin{aligned} \widehat{\;\!^ta}(\xi ,\eta )= & {} (2\pi )^{-N} \int _{{\mathbb {T}}^N} e^{-i\left\langle x,\xi \right\rangle }\,\, \;\!^ta(x,\eta ) dx = (2\pi )^{-N} \int _{{\mathbb {T}}^N} e^{-i\left\langle x,\xi \right\rangle } \overline{a^*(x,-\eta )} dx\\= & {} \overline{(2\pi )^{-N} \int _{{\mathbb {T}}^N} e^{i\left\langle x,\xi \right\rangle } a^*(x,-\eta ) dx} = \overline{\widehat{a^*}(-\xi ,-\eta )}. \end{aligned}$$

Since $\widehat{a^*}(\xi ,\eta ) = \overline{{\hat{k}}(\eta ,-(\xi +\eta ))}$, (see [5]), it follows from (3.3) that

$$\begin{aligned} \widehat{\;\!^ta}(\xi ,\eta ) = \overline{\widehat{a^*}(-\xi ,-\eta )} = {\hat{k}}(-\eta ,\xi +\eta )=\frac{1}{(2\pi )^N}\widehat{a}(\xi ,-(\xi +\eta )). \end{aligned}$$

Since $a(x,D) \in {\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$ then it follows from Proposition 3.1 that we can find $C,h>0$ such that for all $j\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ we have

$$\begin{aligned} |\widehat{\;\!^t a}(\xi ,\eta )| \le \frac{1}{(2\pi )^N} \displaystyle \frac{C h^{j} m_j j!}{(1+|\xi |)^j} (1+|\xi +\eta |)^\sigma ,\,\, \forall \ (\xi , \eta ) \in \mathbb {Z}^{2N}. \end{aligned}$$

Now we take $p\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ such that $|\sigma | \le p$. By using (2.7) and the following inequality

$$\begin{aligned} \Big (\frac{1+|\xi +\eta |}{1+|\eta |} \Big )^\sigma \le (1+|\xi |)^{|\sigma |} \end{aligned}$$

we conclude that, for all $n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ we have

$$\begin{aligned} |\widehat{\;\!^t a}(\xi ,\eta )|\le & {} \frac{1}{(2\pi )^N}\frac{C h^{n+p} m_{n+p} (n+p)!}{(1+|\xi |)^{n+p}} (1+|\xi +\eta |)^\sigma \\\le & {} \frac{1}{(2\pi )^N} \frac{C h^{n+p} A_p^{n+1} m_n! n!}{(1+|\xi |)^{n+p}} (1+|\xi |)^{|\sigma |} (1+|\eta |)^\sigma \\\le & {} \frac{C h^p A_p}{(2\pi )^N} \frac{(A_ph)^n m_n n!}{(1+|\xi |)^n} (1+|\eta |)^\sigma , \ \forall \ (\xi ,\eta )\in {{\,\mathrm{{\mathbb {Z}}}\,}}^{2N} \end{aligned}$$

which concludes the proof by Proposition 3.1. $\square$

Remark 3.9

It follows from the last proposition and from the fact that ${\mathcal {E}}_{{\mathcal {M}}}({\mathbb {T}}^N)$ is a Montel space that we can extend $a(x,D) \in {\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$ to a continuous and linear operator on $D'_\mathcal {M}\left( {\mathbb {T}}^N\right)$ by setting

$$\begin{aligned} \left\langle a(x,D) u, \varphi \right\rangle = \left\langle u, {}^ta(x,D)\varphi \right\rangle , \ \forall \ u \in D'_\mathcal {M}\left( {\mathbb {T}}^N\right) , \varphi \in \mathcal {E}_\mathcal {M}({\mathbb {T}}^N), \end{aligned}$$

and the same can be done with ${}^ta(x,D)$ instead of a(x, D).

4 Properties of the Space $(\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} } ({\mathbb {T}}^N) \subset D_{\mathcal {M}}'({\mathbb {T}}^N)$

In this section, we shall introduce a key definition that will allow us prove our main result of the next section. We start by setting the following

Definition 4.1

For $\delta , \sigma \in {\mathbb {R}},$ with $\delta \in {\mathbb {R}}$, we denote by $(\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} } ({\mathbb {T}}^N)$ the space of all ultradistributions $u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$ such that:

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma \right\} }^{2}:= \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{2 r(\delta )} (1+|\xi |)^{2 \sigma } |\hat{u}(\xi )|^{2} \ < +\infty , \end{aligned}$$

where $r(\delta ) = sgn(\delta ) ={\left\{ \begin{array}{ll} 1,\quad \text{ if } \delta \ge 0\\ -1,\quad \text{ if }\, \delta <0. \end{array}\right. }$

Remark 4.2

• (i) For $\delta =0$, in the definition 4.1, we recover the Sobolev spaces that are used to study the $C^\infty$ global hypoellipticity, which we are not interested here, since it has been done in [6]. Thus, here we will work with the case $\delta \ne 0$.

• (ii) For $m_n=n!^{s-1}$, with $s\ge 1$, we are in the Gevrey spaces and we can prove that for $\delta >0$ the condition $\left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma \right\} } < +\infty$ is equivalent to existence of $\delta _1 > 0$ such that $\left\| u \right\| _{\left\{ \delta _1, s, \sigma \right\} } = \sum _{\xi \in {\mathbb {Z}}^N} e^{2 \delta _1 |\xi |^{1/s}}(1+|\xi |)^{2 \sigma }|\widehat{u}(\xi )|^2 <+\infty .$

We shall need the following basic property whose proof will be omitted.

Proposition 4.3

Let $\delta _{1}, \delta _{2}, \sigma _{1}, \sigma _{2}$ be real numbers with $\delta _{1} \le \delta _{2}$ and $\sigma _{1} \le \sigma _{2}$, with $\delta _1, \delta _2 \ne 0$. Then

$$\begin{aligned} (\mathcal {DE})_{\left\{ \mathcal {M}, \delta _{2}, \sigma _{2} \right\} } ({\mathbb {T}}^N) \subset (\mathcal {DE})_{\left\{ \mathcal {M}, \delta _{1}, \sigma _{1} \right\} } ({\mathbb {T}}^N). \end{aligned}$$

Moreover, if $u \in (\mathcal {DE})_{\left\{ \mathcal {M}, \delta _{2}, \sigma _{2} \right\} } ({\mathbb {T}}^N)$, then

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta _{1}, \sigma _{1} \right\} } \le \left\| u \right\| _{\left\{ \mathcal {M}, \delta _{2}, \sigma _{2} \right\} }. \end{aligned}$$

We now will prove the following important properties.

Proposition 4.4

Let $u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$. Then the following statements hold true:

• 1. $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^{N})$ if and only if there exist $\delta > 0$ and $\sigma \in {\mathbb {R}}$ such that $u \in (\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} }({\mathbb {T}}^N)$.

• 2. $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^{N})$ if and only if there exists $\delta > 0$ such that $u \in (\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} }({\mathbb {T}}^N)$ for all $\sigma \in {\mathbb {R}}$.

• 3. $u \in (\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} }({\mathbb {T}}^N)$, for all $\delta <0$ and $\sigma \in {\mathbb {R}}$.

Proof

First note that if we prove that the condition in item 1 is sufficient to have $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^{N})$, then the condition in item 2 will also be sufficient to have $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^{N})$. On the other hand, if we prove that the condition in item 2 is necessary to have $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^{N})$, then the condition in item 1 will also be necessary to have $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^{N})$.

Sufficiency of condition 1: By hypothesis there exist $C, \delta >0$ and $\sigma \in {\mathbb {R}}$ such that

$$\begin{aligned} \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{\delta ^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{2} (1+|\xi |)^{2 \sigma } |\hat{u}(\xi )|^{2} = C^{2}. \end{aligned}$$

Let $p\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ such that $-\sigma <p$ and, for $k\in {\mathbb {Z}}_+$, we take $n=k+p$. It follows from the last equality that

$$\begin{aligned} |\hat{u}(\xi )| \le C \frac{m_{k+p} (k+p)!}{\delta ^{k+p} (1+|\xi |)^{k+p}} (1+|\xi |)^{-\sigma } \le C \frac{m_{k+p} (k+p)!}{\delta ^{k+p} (1+|\xi |)^{k}}, \, \forall \, \xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N. \end{aligned}$$

Now by using (2.7) we obtain

$$\begin{aligned} |\hat{u}(\xi )|\le C \frac{A_p^{k+1} m_{k} k!}{\delta ^{k+p} (1+|\xi |)^{k}} = \frac{C A_p}{\delta ^p} \frac{A_p^{k} m_{k} k!}{\delta ^k(1+|\xi |)^{k}}, \ \forall \ \xi \in {{\,\mathrm{{\mathbb {Z}}}\,}}^N, \forall \ k\in {\mathbb {Z}}_+. \end{aligned}$$

By setting $C_1=\frac{C A_p}{\delta ^p}$ and $h=\frac{A_p}{\delta }$ it follows from the last inequality that $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$ by Theorem 2.8, item (a).

Necessity of condition 2: Let $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$, then again it follows from Theorem 2.8, item (a),  that there are positive constants C and h such that

$$\begin{aligned} |\hat{u}(\xi )| (1+|\xi |)^{n} \le C h^{n} m_{n} n!,\ \forall \, n \in {\mathbb {Z}}_+,\, \xi \in {\mathbb {Z}}^N. \end{aligned}$$

(4.1)

For $\xi \in {\mathbb {Z}}^{N}$ and $\sigma \in {\mathbb {R}}$, we denote

$$\begin{aligned} (I) = \left[ \displaystyle \frac{\delta ^{n} (1 + |\xi |)^{n}}{m_{n} n!} \right] ^{2} (1 + |\xi |)^{2\sigma } |\hat{u}(\xi )|^2, \end{aligned}$$

(4.2)

where $\delta >0$ will be chosen later.

By choosing $p \in {\mathbb {Z}}_+$ such that $\sigma \le p$ we can estimate (4.2) as

$$\begin{aligned} (I) \le \left[ \displaystyle \frac{\delta ^{2n} (1+|\xi |)^{2n} |\hat{u}(\xi )|}{(m_{n} n!)^{2}} \left( |\hat{u}(\xi )| (1+|\xi |)^{2(p+N)} \right) \right] \displaystyle \frac{1}{(1+|\xi |)^{2N}}. \end{aligned}$$

It now follows from the last inequality and (4.1) that

$$\begin{aligned} (I)&\le \left[ \displaystyle \frac{\delta ^{2n} (1+|\xi |)^{2n} |\hat{u}(\xi )|}{(m_{n} n!)^{2}} \left( C h^{2(p+N)} m_{2(p+N)} (2(p+N))! \right) \right] \displaystyle \frac{1}{(1+|\xi |)^{2N}}\\&\le \left[ \displaystyle \frac{\delta ^{2n} (1+|\xi |)^{2n} |\hat{u}(\xi )|}{m_{2n} (2n)!} \displaystyle \frac{m_{2n} (2n)!}{(m_{n} n!)^{2}} \right] \displaystyle \frac{C h^{2(p+N)} m_{2(p+N)} (2(p+N))!}{(1+|\xi |)^{2N}}. \end{aligned}$$

By using (2.3) and (4.1), it follows from the last inequality that

$$\begin{aligned} (I)&\le \left[ \displaystyle \frac{(1+|\xi |)^{2n} |\hat{u}(\xi )|}{m_{2n} (2n)!} \right] (2 \delta H)^{2n} \displaystyle \frac{C h^{2(p+N)} m_{2(p+N)} (2(p+N))!}{(1+|\xi |)^{2N}} \\&\le (C^{2} h^{2(p+N)} m_{2(p+N)} (2(p+N))!) (2 h H \delta )^{2n} \displaystyle \frac{1}{(1+|\xi |)^{2N}}. \end{aligned}$$

If $B\dot{=}(C^{2} h^{2(p+N)} m_{2(p+N)} (2(p+N))!)$, then B depends only on u, $\mathcal {M}$, $\sigma$ and N. By taking $0<\delta \le \displaystyle \frac{1}{2 h H},$ we obtain

$$\begin{aligned} (I) \le B \displaystyle \frac{1}{(1+|\xi |)^{2N}}. \end{aligned}$$

(4.3)

Now (4.2) and (4.3) give us

$$\begin{aligned} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{\delta ^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{2} (1 + |\xi |)^{2\sigma } |\hat{u}(\xi )|^2 \le B \displaystyle \frac{1}{(1+|\xi |)^{2N}}, \end{aligned}$$

from where we conclude that

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma \right\} }^{2}= & {} \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{\delta ^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{2} (1+|\xi |)^{2 \sigma } |\hat{u}(\xi )|^{2} \\\le & {} B \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \displaystyle \frac{1}{(1+|\xi |)^{2N}} < \infty . \end{aligned}$$

It proves that $u \in (\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} }({\mathbb {T}}^N)$ for every $\sigma \in {\mathbb {R}}$ and $0<\delta \le \displaystyle \frac{1}{2 h H}$.

We proceed then to item 3: let $u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$, $\delta < 0$ and $\sigma \in {\mathbb {R}}$. Our goal is to prove that $u \in (\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} }({\mathbb {T}}^N)$. From Theorem 2.8, item (b), for each $\varepsilon > 0$ there exists a positive constant $C_{\varepsilon }$ such that

$$\begin{aligned} |\hat{u}(\xi )| \le C_{\varepsilon } \displaystyle \sup _{n \in {\mathbb {Z}}_+} \left( \displaystyle \frac{\varepsilon ^{n} (1 +|\xi |)^{n} }{m_{n} n!} \right) ,\ \forall \,\,\xi \in {\mathbb {Z}}^N. \end{aligned}$$

(4.4)

Fixed $\xi \in {\mathbb {Z}}^N$ and recalling that for $\delta <0$ we have $r(\delta )=-1$ we write

$$\begin{aligned} (1) = \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2} (1+|\xi |)^{2 \sigma } |\hat{u}(\xi )|^{2}. \end{aligned}$$

(4.5)

Again we take $p\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ satisfying $\sigma \le p$. By using (4.4) and Proposition 2.4, item (v), we have for some $n_0\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ depending on $\varepsilon$ and $\xi$ that

$$\begin{aligned} (1)&\le \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2} (1+|\xi |)^{2 p} \left[ C_{\varepsilon } \displaystyle \left( \displaystyle \frac{ \varepsilon ^{n_{0}} (1 +|\xi |)^{n_{0}}}{m_{n_{0}} \cdot n_{0}!} \right) \right] ^{2}. \end{aligned}$$

Since $\left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2}\le \left[ \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2}$ for any $n\in {\mathbb {Z}}_+$ then it follows from the last inequality that

$$\begin{aligned} (1)&\le \left[ \displaystyle \frac{|\delta |^{(n_{0} + p + N)} (1+|\xi |)^{(n_{0} + p + N)}}{m_{(n_{0} + p + N)} (n_{0} + p + N)!} \right] ^{-2} (1+|\xi |)^{2 p} \left[ C_{\varepsilon } \displaystyle \left( \displaystyle \frac{ \varepsilon ^{n_{0}} (1 +|\xi |)^{n_{0}} }{m_{n_{0}} n_{0}!} \right) \right] ^{2} \\&\le \left( \displaystyle \frac{C_{\varepsilon }}{|\delta |^{p+N}} \right) ^{2} \left( \displaystyle \frac{m_{(n_{0} + p + N)} (n_{0} + p + N)!}{m_{n_{0}} n_{0}!}\right) ^{2} \left( \displaystyle \frac{\varepsilon }{|\delta |} \right) ^{2n_{0}} \displaystyle \frac{1}{(1+|\xi |)^{2N}}. \end{aligned}$$

Now we use (2.7) and we obtain

$$\begin{aligned} (1) \le \left( \displaystyle \frac{C_{\varepsilon } A_{p+N}}{|\delta |^{p+N}} \right) ^{2} \left( A_{p+N} \displaystyle \frac{\varepsilon }{|\delta |} \right) ^{2n_{0}} \displaystyle \frac{1}{(1+|\xi |)^{2N}}. \end{aligned}$$

Now we choose $\varepsilon = \displaystyle \frac{|\delta |}{A_{p+N}}$, which depends only on $\sigma , \delta , N$ and ${\mathcal {M}}$. Then, for a new positive constant $C_1=C_{1} (\sigma , \delta , N, {\mathcal {M}})$ we have

$$\begin{aligned} (1) \le C_{1} (\sigma , \delta , N, {\mathcal {M}}) \displaystyle \frac{1}{(1+|\xi |)^{2N}}, \ \forall \ \xi \in {\mathbb {Z}}^N. \end{aligned}$$

(4.6)

From (4.5) and (4.6) it follows that

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M},\delta , \sigma \right\} }^{2}= & {} \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n}(1+|\xi |)^{n}}{m_{n} n! } \right] ^{-2} (1+|\xi |)^{2 \sigma } |\hat{u}(\xi )|^{2} \\\le & {} C_{1} (\sigma , \delta , N, {\mathcal {M}}) \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \displaystyle \frac{1}{(1+|\xi |)^{2N}}, \end{aligned}$$

from where we conclude that $u \in (\mathcal {DE})_{\left\{ \mathcal {M}, \delta , \sigma \right\} }({\mathbb {T}}^N)$.

Since $\delta <0$ and $\sigma$ are arbitrary, the proof is complete. $\square$

Another important property that we need is given by the following:

Lemma 4.5

If $f \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^{N})$, then we can find positive constants $\delta _{0}, B, \rho$ such that

$$\begin{aligned} \left\| f \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \le B \rho ^{k} m_{k} k!, \ \ \forall k \in {\mathbb {Z}}_+, \ \delta \le \delta _{0}. \end{aligned}$$

Proof

For $\xi \in {\mathbb {Z}}^{N}$, $\delta _0 >0$ and $k,n \in {\mathbb {Z}}_+$, we write

$$\begin{aligned} (I) = \left[ \displaystyle \frac{\delta _0^{n} (1 + |\xi |)^{n}}{m_{n} n!} \right] ^{2} (1 + |\xi |)^{2k} |\hat{f}(\xi )|^2, \end{aligned}$$

(4.7)

where $\delta _0$ will be chosen later.

Then

$$\begin{aligned} (I)= \left[ \displaystyle \frac{ |\hat{f}(\xi )| \delta _0^{2n} (1+|\xi |)^{2n}}{m_{2n} (2n!)} \displaystyle \frac{m_{2n} (2n)!}{(m_{n} n!)^{2}} \right] \displaystyle \frac{(1 + |\xi |)^{2(k+N)} |\hat{f}(\xi )|}{(1 + |\xi |)^{2N}} \frac{(m_kk!)^2}{(m_kk!)^2}. \end{aligned}$$

Applying (2.3), the fact that $f \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$, twice, and (2.7) we conclude that there are positive constants C, H and h satisfying

$$\begin{aligned} (I)&\le \frac{ C h^{2n}m_{2n}(2n)! \delta _0^{2n}}{m_{2n}(2n)!}\, (2H)^{2n}\, \frac{C h^{2(k+N)} m_{2(k+N)} (2(k+N))!}{(1 + |\xi |)^{2N}} \frac{(m_kk!)^2}{(m_kk!)^2} \\&\le C h^{2n} \delta _0^{2n}\, (2H)^{2n}\,\frac{C h^{2(k+N)} A_{2N}^{2k+1}}{(1 + |\xi |)^{2N}} \frac{m_{2k}(2k)!}{(m_k k!)^2} (m_k k!)^2 \\&\le C h^{2n}\delta _0^{2n}\, (2H)^{2n}\,\frac{Ch^{2(k+N)} A_{2N}^{2k+1}}{(1+|\xi |)^{2N}} (2H)^{2k}(m_k k!)^2 \\&= \left( C^2 h^{2N} A_{2N} (2 \delta _0 h H)^{2n}\right) (h 2H A_{2N})^{2k} \frac{(m_k k!)^2}{(1 + |\xi |)^{2N}}. \end{aligned}$$

Now we consider $\delta _{0} = \displaystyle \frac{1}{2 h H}$ in the definition of (I) and we obtain

$$\begin{aligned} (I) \le \left( C^2 h^{2N} A_{2N}\right) (h 2H A_{2N})^{2k} \frac{(m_k k!)^2}{(1 + |\xi |)^{2N}}. \end{aligned}$$

(4.8)

It follows from (4.7) and (4.8) that

$$\begin{aligned} \left[ \displaystyle \sup _{n \in Z_+}\displaystyle \frac{\delta _{0}^{n} (1 + |\xi |)^{n}}{m_{n} n!} \right] ^{2} (1 + |\xi |)^{2k} |\hat{f}(\xi )|^2 \le \left( C^2 h^{2N} A_{2N}\right) (h 2H A_{2N})^{2k} \frac{(m_k k!)^2}{(1 + |\xi |)^{2N}}. \end{aligned}$$

Hence, if we set $B \dot{=} \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \displaystyle \frac{1}{(1 + |\xi |)^{2N}}, B_1 \dot{=} (C^2 h^{2N} A_{2N} B)^{\frac{1}{2}}$ and $\rho = h 2H A_{2N}$ we conclude that

$$\begin{aligned} \left\| f \right\| _{\left\{ \mathcal {M}, \delta _{0}, k \right\} }^{2} \le B_1^{2} \rho ^{2k} (m_{k} k!)^{2}, \ \forall \ k \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+ \end{aligned}$$

and now we use the Proposition 4.3 in order to obtain

$$\begin{aligned} \left\| f \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \le B_1 \rho ^{k} m_{k} k!, \ \ \forall \delta \le \delta _{0}, \ k \in {\mathbb {Z}}_+, \end{aligned}$$

which concludes the proof. $\square$

5 Systems of Ultradifferentiable Pseudodifferential Operators and Global Hypoellipticity with Loss of Derivatives

From now on, if $a_1(x,D), \ldots , a_m(x,D)$ belong to ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$, then we will denote by ${\mathcal {A}}$ the system associated to these operators and we say that ${\mathcal {A}}=\{a_j(x,D)\}_{j=1}^m$ is a system of pseudodifferental operators in ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$.

We now state our key definition.

Definition 5.1

Let ${\mathcal {A}}=\{a_j(x,D)\}_{j=1}^m$ be a system of pseudodifferental operators in ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$. We say that ${\mathcal {A}}$ is globally ${\mathcal {M}}-$hypoelliptic with loss of $r\ge 0$ derivatives if there exist constants $B>0$, $C > 0$, $\delta _{0} <0$ and $k_{0} \in {\mathbb {R}}$ such that

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} } \le C \left[ \max _{1\le j \le m}\left\| a_j(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + B^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right] , \end{aligned}$$

for any $k \in {\mathbb {N}}_{0}, \ u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$ and $\delta _{0}< \delta < 0.$

We also have a standard notion of global ultradifferentiable hypoellipticity for systems of pseudodifferential operators given by

Definition 5.2

Let ${\mathcal {A}}=\{a_j(x,D)\}_{j=1}^m$ be a system of pseudodifferental operators in ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$. We say that ${\mathcal {A}}$ is globally ${\mathcal {M}}-$hypoelliptic on ${\mathbb {T}}^N$ if the conditions $u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$ and $a_j(x,D)u \in \mathcal {E}_\mathcal {M}\left( {\mathbb {T}}^N\right)$ for all $j=1,\ldots ,m$ imply $u \in \mathcal {E}_\mathcal {M}\left( {\mathbb {T}}^N\right)$.

As in the $C^\infty$ frame (cf. [12], p.229 for the local case and cf. [6] for the global case) we will prove the correspondent result for the ultradifferentiable version.

Theorem 5.3

If ${\mathcal {A}}=\{a_j(x,D)\}_{j=1}^m$ is a system of pseudodifferental operators in ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$ that is globally ${\mathcal {M}}-$hypoelliptic with loss of $r\ge 0$ derivatives, then ${\mathcal {A}}$ is globally ${\mathcal {M}}-$hypoelliptic on ${\mathbb {T}}^N$.

Proof

Let us fix $u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$ such that $a_j(x,D)u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$ for all $j=1,\ldots ,m$. We will prove that $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$. It follows from our hypothesis and Lemma 4.5 that one can obtain $\delta _0 <0$, positive constants $C, B, B_1, \rho$ and $k_0\in {\mathbb {R}}$ such that

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} }&\le C \max _{1\le j\le m}\left\| a_j(x,D) u \right\| _{\left\{ \mathcal {M}, \delta , k\right\} } + C B^k m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \\&\le C B_1 \rho ^{k} m_{k} k! + C B^k m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }, \end{aligned}$$

for all $k\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$ and $\delta _0<\delta <0$. Since $u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$ it follows from Proposition 4.4, item 3, that $\left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }< \infty .$

Thus, if we set $C_{1} = \max \left\{ C, C B_1 \right\}$ and $h_{1} = \max \left\{ \rho , B \right\}$ then we have

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} } \le C_{1} h_{1}^{k} m_{k} k! \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }\right) ,\ \forall k \in {\mathbb {Z}}_+, \ \delta _0< \delta < 0. \end{aligned}$$

(5.1)

For $j \in {\mathbb {Z}}_+, \ \xi \in {\mathbb {Z}}^N$ and $\delta _{0}< \delta < 0$, we denote

$$\begin{aligned} (I) (\xi ) := \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2} |\hat{u}(\xi )|^{2} (1+|\xi |)^{2j}. \end{aligned}$$

If $q \in {\mathbb {Z}}_+$ satisfies $r - \sigma \le q$ then we can use (5.1) to obtain

$$\begin{aligned} (I)(\xi )&\le \displaystyle \sum _{\eta \in {\mathbb {Z}}^N} \left[ \displaystyle \sup _{n \in {\mathbb {N}}_{0}} \displaystyle \frac{|\delta |^{n} (1+|\eta |)^{n}}{m_{n} n!} \right] ^{-2} |\hat{u}(\eta )|^{2} (1+|\eta |)^{2 (j+q + \sigma -r)} \\&= \left\| u \right\| _{\left\{ \mathcal {M}, \delta , j + q + \sigma -r \right\} }^{2} \\&\le C_{1}^{2} h_{1}^{2(j+q)} [m_{j+q} (j+q)!]^{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }\right) ^{2}. \end{aligned}$$

From (2.7),

$$\begin{aligned} (I)(\xi ) \le (C_{1} h_{1}^{q} A_q)^{2} \left( h_{1} A_q\right) ^{2j} (m_{j} j!)^{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }\right) ^{2}. \end{aligned}$$

Since the right hand side of the last inequality does not depends on $\xi$, if we set $C_2=C_{1} h_{1}^{q} A_q$ and $h_2 = h_1 A_q$ and by taking the square root in the last inequality, we conclude that

$$\begin{aligned} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-1} |\hat{u}(\xi )| (1+|\xi |)^{j} \le C_{2} h_{2}^{j} m_{j} j! \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }\right) ,\nonumber \\ \end{aligned}$$

(5.2)

and therefore

$$\begin{aligned} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-1} \left[ \displaystyle \frac{(1+|\xi |)^{j}}{(2h_{2})^{j} m_{j} j!} \right] |\hat{u}(\xi )| \le C_{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }\right) \displaystyle \frac{1}{2^{j}}, \end{aligned}$$

for every $\xi \in {\mathbb {Z}}^{N}$ and $\delta _{0}< \delta < 0$. By taking the sum in j in the last inequality, we obtain

$$\begin{aligned} \displaystyle \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-1} \left( \sum _{j=0}^{\infty }\left[ \displaystyle \frac{(1+|\xi |)^{j}}{(2h_{2})^{j} m_{j} j!} \right] \right) |\hat{u}(\xi )| \le 2C_{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }\right) , \end{aligned}$$

which ensures that, for every $\xi \in {\mathbb {Z}}^{N}, \ j \in Z_+$ and $\delta _{0}< \delta < 0$, we have

$$\begin{aligned}&\left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-1} \left[ \sup _{j \in {\mathbb {Z}}_+} \displaystyle \frac{(1+|\xi |)^{j} \left( \frac{1}{2h_{2}} \right) ^{j}}{m_{j} j!} \right] |\hat{u}(\xi )| \nonumber \\&\quad \le 2C_{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }\right) . \end{aligned}$$

(5.3)

It follows from Proposition 2.4 (item (v)) that we can find $n_0 \in {\mathbb {Z}}_+$ depending on $\xi$ and $\delta$ such that

$$\begin{aligned} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-1} = \left[ \displaystyle \frac{|\delta |^{n_{0}} (1+|\xi |)^{n_{0}}}{m_{n_{0}} n_{0}!} \right] ^{-1}. \end{aligned}$$

Now we use (5.3) and conclude that, for all $k \in {\mathbb {Z}}_+$ and $\delta _{0}< \delta < 0$ we have

$$\begin{aligned} \left[ \displaystyle \frac{|\delta |^{n_{0}} (1+|\xi |)^{n_{0}}}{m_{n_{0}} n_{0}!} \right] ^{-1} \left[ \displaystyle \frac{(1+|\xi |)^{k + n_{0}} \left( \frac{1}{2h_{2}} \right) ^{k+n_{0}}}{ m_{k+n_{0}} (k+n_{0})!} \right] |\hat{u}(\xi )| \le 2C_{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }\right) , \end{aligned}$$

in turns implies that

$$\begin{aligned} |\hat{u}(\xi )| (1+|\xi |)^{k} \le \left[ 2C_{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right) \right] (2h_{2})^{k} (2h_{2} |\delta |)^{n_{0}} \left[ \displaystyle \frac{ m_{k+n_{0}} (k+n_{0})!}{m_k m_{n_{0}}\,\, k!n_{0}!} \right] m_k k!. \end{aligned}$$

It follows from (2.3) and the last inequality that

$$\begin{aligned} |\hat{u}(\xi )| (1+|\xi |)^{k}&\le \left[ 2C_{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right) \right] (4H h_{2})^{k} (4h_{2} H |\delta |)^{n_{0}} m_{k} k! \end{aligned}$$

Let us fix $\delta$ such that $\delta _0< \delta < 0$ and $-\delta =|\delta | \le \frac{1}{4 H h_{2}}$. Then $(4h_{2} H |\delta |)^{n_{0}} \le 1$ for any $n_{0}$. Thus we can rewrite last inequality as

$$\begin{aligned} \frac{\left( \frac{1}{4H h_{2}}\right) ^{k}(1+|\xi |)^{k}}{m_{k} k!}(1+|\xi |)^{-N}|\hat{u}(\xi )|\le 2C_{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right) (1+|\xi |)^{-N} \end{aligned}$$

in turns implies that

$$\begin{aligned}&\left[ \displaystyle \sup _{k \in {\mathbb {Z}}_+} \displaystyle \frac{\left( \frac{1}{4H h_{2}}\right) ^{k}(1+|\xi |)^{k}}{m_{k} k!}\right] ^2 (1+|\xi |)^{-2N}|\hat{u}(\xi )|^2\\&\quad \le \left[ 2C_{2} \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }\right) \right] ^2(1+|\xi |)^{-2N} \end{aligned}$$

Now by summing in $\xi \in {\mathbb {Z}}^N$ we conclude that

$$\begin{aligned} \left\| u \right\| _{\{\mathcal {M},\frac{1}{4H h_2}, -N\}}^2 <+\infty . \end{aligned}$$

(5.4)

It follows from Proposition 4.4 item 1. that $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^{N})$ and proof is now complete. $\square$

For the proof of our main result of this section we need the following auxiliary result whose proof is easily seen and it will be omitted.

Lemma 5.4

For any $\lambda >0, \ \delta <0$, $\rho< \sigma < \tau$ and $u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$ we have

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma \right\} } \le \lambda \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \tau \right\} } + \lambda ^{-\frac{\sigma -\rho }{\tau - \sigma }} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \rho \right\} }. \end{aligned}$$

The next result is a version of Proposition 1.1 of [CC] with our norms in the ultradifferentiable case and its proof presents several thecnical difficults. So we are going to present a detailed proof of this result here.

Proposition 5.5

Let $a(x, D) \in \mathfrak {D}_{\mathfrak {p}_{\sigma }}^{\mathcal {M}}({\mathbb {T}}^N)$. Then there exist positive constants C and h such that for every $\varepsilon > 0$, we can find $\delta _{\varepsilon } < 0$ in such way that, for every $u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$, $k \in {\mathbb {Z}}_+$ and $\delta _\varepsilon<\delta <0$ we have

$$\begin{aligned} \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \le C \left( \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+\sigma + \varepsilon \right\} } + h^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} } \right) . \end{aligned}$$

Proof

Let $u \in D_{\mathcal {M}}'({\mathbb {T}}^N), k\in {\mathbb {Z}}_+$ and $\delta <0$ be given. By using (3.2) and Remark 3.9 we have

$$\begin{aligned}&\left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }\nonumber \\&\quad = \Bigg (\displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \Bigg [\displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} \cdot (1+|\xi |)^{n}}{m_{n} n!} \Bigg ]^{-2} (1+|\xi |)^{2k} \Big |\displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} ({\hat{a}}(\xi - \eta , \eta ) \hat{u}(\eta ) \Big |^{2} \Bigg )^{1/2} \nonumber \\&\quad = \Bigg (\displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \Bigg | \displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} \Bigg [\displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \Bigg ]^{-1} (1+|\xi |)^{k} {\hat{a}}(\xi - \eta , \eta ) \hat{u}(\eta ) \Bigg |^{2} \Bigg )^{1/2} \nonumber \\&\quad \le \left[ \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \left( \displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-1} (1+|\xi |)^{k} \left| {\hat{a}}(\xi - \eta , \eta ) \hat{u}(\eta ) \right| \right) ^{2}\right] ^{1/2}. \end{aligned}$$

(5.5)

Noticing that

$$\begin{aligned} (1+|\xi |)^k \le \sum _{j=0}^k \left( {\begin{array}{c}k\\ j\end{array}}\right) (1+|\xi - \eta |)^j (1+|\eta |)^{k-j}; \end{aligned}$$

it follows from the (5.5) that

$$\begin{aligned}&\left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \\&\quad \le \Bigg [\displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \Bigg (\displaystyle \sum _{j=0}^{k} \left( {\begin{array}{c}k\\ j\end{array}}\right) \displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-1} \left( 1 + |\xi - \eta |\right) ^{j} \times \\&\qquad \times \left( 1 + |\eta |\right) ^{k-j} \left| {\hat{a}}(\xi - \eta , \eta ) \hat{u}(\eta ) \right| \Bigg )^{2} \Bigg ]^{1/2}. \end{aligned}$$

It follows from the Minkowiski’s inequality and Schwarz’s inequality that we can estimate the last term by

$$\begin{aligned} \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }&\le \sum _{j=0}^{k} \left( {\begin{array}{c}k\\ j\end{array}}\right) \Big [\displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \Big [\displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \Big ]^{-2} \nonumber \\&\quad \times \Big (\displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} \left( 1 + |\xi - \eta |\right) ^{j} \left( 1 + |\eta |\right) ^{k-j} \left| {\hat{a}}(\xi - \eta , \eta ) \hat{u}(\eta ) \right| \Big )^{2} \Big ]^{1/2} \nonumber \\&\le \sum _{j=0}^{k} \left( {\begin{array}{c}k\\ j\end{array}}\right) \Big [\displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \Big [\displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \Big ]^{-2} \\&\quad \times \Big (\underbrace{\displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}}\left( 1 + |\xi - \eta |\right) ^{2j} |{\hat{a}}(\xi - \eta , \eta )| \Big )}_{(A)}\nonumber \\&\qquad \times \Big (\displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} \left( 1 + |\eta |\right) ^{2(k-j)} |{\hat{a}}(\xi - \eta , \eta )| |\hat{u}(\eta )|^{2} \Big )\Big ]^{1/2}.\nonumber \end{aligned}$$

(5.6)

Let $q \in {\mathbb {Z}}_+$ such that $q \ge |\sigma |$. Now we use the Proposition 3.1 and inequality

$$\begin{aligned} (1+|\eta |)^{\sigma }\le (1+|\xi - \eta |)^{|\sigma |} (1+|\xi |)^{\sigma } \end{aligned}$$

(5.7)

in order to conclude that, for some positive constants $C_2$ and $h_2$,

$$\begin{aligned} (A)&\le \displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} \left( 1 + |\xi - \eta |\right) ^{2j} \displaystyle \frac{ C_{2} h_{2}^{(2j + q + 2N)} m_{(2j + q + 2N)} (2j + q + 2N)!}{(1+|\xi - \eta |)^{(2j + q + 2N)}} (1+|\eta |)^{\sigma } \\&\le (1+|\xi |)^{\sigma } \displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} \left( 1 + |\xi - \eta |\right) ^{2j} \displaystyle \frac{ C_{2} h_{2}^{(2j + q + 2N)} m_{(2j + q + 2N)} (2j + q + 2N)!}{(1+|\xi - \eta |)^{(2j + q + 2N)}} \\&\quad \times (1+|\xi - \eta |)^{q}. \end{aligned}$$

Now we set $C_{3} = C_{2} h_{2}^{q+2N} \Big (\displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} \displaystyle \frac{1}{(1+|\eta |)^{2N}}\Big )$ and use (2.7) to obtain

$$\begin{aligned} (A)&\le C_{2} h_{2}^{(2j + q + 2N)} m_{(2j + q + 2N)} (2j + q + 2N)! (1+|\xi |)^{\sigma } \Big (\displaystyle \sum _{\eta \in {\mathbb {Z}}^{N}} \displaystyle \frac{1}{(1+|\xi - \eta |)^{2N}}\Big ) \\&\le C_{3} h_{2}^{2j} m_{(2j + q + 2N)} (2j + q + 2N)! (1+|\xi |)^{\sigma }\\&\le \left( C_{3} A_{q+2N}\right) \left( h_{2} A_{q+2N}\right) ^{2j} m_{2j} (2j)! (1+|\xi |)^{\sigma }. \end{aligned}$$

From (2.3), it follows that

$$\begin{aligned} (A)&\le \left( C_{3} A_{q+2N}\right) \left( h_{2} A_{q+2N}\right) ^{2j} (2H)^{2j} (m_j j!)^2 (1+|\xi |)^{\sigma }\\&\le C_4^2 h_3^{2j} (m_j \cdot j!)^2 (1+|\xi |)^{\sigma }, \end{aligned}$$

where $C_4^2=C_{3} A_{q+2N}$ and $h_3= 2 H h_2 A_{2q+N}$. From this inequality and (5.6) we conclude that

$$\begin{aligned} \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }&\le C_{4} \displaystyle \sum _{j=0}^{k} \left( {\begin{array}{c}k\\ j\end{array}}\right) h_{3}^{j} m_{j} j! (B)^{1/2}, \end{aligned}$$

(5.8)

where

$$\begin{aligned}&(B)^{1/2}=\Bigg [ \displaystyle \sum _{\xi , \eta \in {\mathbb {Z}}^N} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2} \\&\quad \times (1+|\xi |)^{\sigma } \left( 1 + |\eta |\right) ^{2(k-j)} |{\hat{a}}(\xi - \eta , \eta )| |\hat{u}(\eta )|^{2} \Bigg ]^{1/2}. \end{aligned}$$

Let us analyze the term (B). First, note that by applying inequality (5.7), with the roles of $\xi$ and $\eta$ changed, we obtain

$$\begin{aligned} (B) \le \displaystyle \sum _{\eta \in {\mathbb {Z}}^N} \left( 1 + |\eta |\right) ^{2(k-j)+\sigma } |\hat{u}(\eta )|^{2} (C), \end{aligned}$$

(5.9)

where

$$\begin{aligned} (C) = \sum _{\xi \in {\mathbb {Z}}^N} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2} (1+|\xi - \eta |)^{|\sigma |} |{\hat{a}}(\xi - \eta , \eta )|. \end{aligned}$$

For the term (C), observe that by fixing $s \in {\mathbb {N}}$, we can write

$$\begin{aligned} (C)&= \underbrace{\displaystyle \sum _{|\xi | \le \frac{s}{s+1} |\eta |} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2} (1+|\xi - \eta |)^{|\sigma |} |{\hat{a}}(\xi - \eta , \eta )|}_{(C1)} + \nonumber \\&\quad + \underbrace{\displaystyle \sum _{ \frac{s}{s+1} |\eta | < |\xi |} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2} (1+|\xi - \eta |)^{|\sigma |} |{\hat{a}}(\xi - \eta , \eta )|}_{(C2)}. \end{aligned}$$

(5.10)

By using the fact that the supremum that appears on the expression of (C1) is greater or equal to 1 and Proposition 3.1, item (ii), there exist positive constants $C_2$ and $h_2$ such that

$$\begin{aligned} (C1)&\le \displaystyle \sum _{|\xi | \le \frac{s}{s+1} |\eta |} (1+|\xi - \eta |)^{q} |{\hat{a}}(\xi - \eta , \eta )| \\&\le \displaystyle \sum _{|\xi | \le \frac{s}{s+1} |\eta |} (1+|\xi - \eta |)^{q} \displaystyle \frac{C_2 \cdot h_2^{(2n + 2N + q)} m_{(2n + 2N + q)} (2n + 2N + q)!}{(1+|\xi - \eta |)^{(2n + 2N + q)}}\\&\quad \times (1+|\eta |)^{\sigma }. \end{aligned}$$

We notice that $|\xi |\le \frac{s}{s+1}|\eta |$ implies that $|\xi - \eta | \ge \frac{|\eta |}{s+1}$ in turns implies that $\frac{1}{1+|\xi -\eta |} \le \frac{1}{1+\frac{|\eta |}{s+1}}$ and it follows from the above inequality that

$$\begin{aligned}&(C1) \le \Bigg (C_{2} h_2^{2N + q} \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \displaystyle \frac{1}{(1+|\xi - \eta |)^{2N}} \Bigg ) \\&\qquad \displaystyle \times \frac{h_2^{2n} m_{(2n + 2N + q)} (2n + 2N + q)!}{(1+\frac{|\eta |}{s+1})^{2n}} (1+|\eta |)^{\sigma } \\&\quad \le C_{5} \displaystyle \frac{((s+1) h_2)^{2n} m_{(2n + 2N + q)} (2n + 2N + q)!}{(1+|\eta |)^{2n}} \\&\qquad \times (1+|\eta |)^{\sigma }, \end{aligned}$$

where $C_5=C_{2} h_{2}^{2N + q} \displaystyle \sum _{\xi \in {\mathbb {Z}}^N}\displaystyle \frac{1}{(1+|\xi - \eta |)^{2N}}$ and therefore does not depend on $\eta$ and $n \in {\mathbb {Z}}_{+}$. We use again (2.3) and (2.7):

$$\begin{aligned} (C1)&\le C_{5} A_{2N+q} \displaystyle \frac{((s+1) h_{2} A_{2N+q} )^{2n} m_{(2n)} (2n)!}{(1+|\eta |)^{2n}} (1+|\eta |)^{\sigma } \\&\le C_{5} A_{2N+q} \displaystyle \frac{((s+1) h_{2} A_{2N+q} 2H )^{2n} (m_{n} n!)^2}{(1+|\eta |)^{2n}} (1+|\eta |)^{\sigma }. \end{aligned}$$

Now we choose $\delta <0$ such that $|\delta | \le \displaystyle \frac{1}{(s+1) h_{2} A_{2N+q} 2H}$. Then

$$\begin{aligned} (C1) \le C_{5} A_{2N+q} \displaystyle \frac{(m_{n} n!)^{2}}{|\delta |^{2n} (1+|\eta |)^{2n}} (1+|\eta |)^{\sigma }, \ \ \forall n \in {\mathbb {Z}}_{+}, \end{aligned}$$

which implies that

$$\begin{aligned} (C1) \le C_{6} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\eta |)^{n}}{m_{n} n!} \right] ^{-2} (1+|\eta |)^{\sigma }, \end{aligned}$$

(5.11)

for all $\eta \in {\mathbb {Z}}^N$, since $C_6\dot{=}C_{5} A_{2N+q}$ and $\delta$ does not depend on $\eta$.

We now are going to estimate the term (C2). By hypothesis,

$$\begin{aligned} \displaystyle \frac{s}{s+1} |\eta |< |\xi | \ \Rightarrow \ 1 + \displaystyle \frac{s |\eta |}{s+1} < (1+|\xi |). \end{aligned}$$

Thus

$$\begin{aligned} |\delta | (1+|\xi |)> & {} |\delta | \left( 1 + \displaystyle \frac{s |\eta |}{s+1} \right) = \left( \displaystyle \frac{s}{s+1} |\delta | \right) \left( \displaystyle \frac{s+1}{s} + |\eta | \right) \\> & {} \left( \displaystyle \frac{s}{s+1} |\delta | \right) \left( 1 + |\eta | \right) \end{aligned}$$

and then

$$\begin{aligned} (C2)&=\displaystyle \sum _{ \frac{s}{s+1} |\eta | < |\xi |} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right] ^{-2} (1+|\xi - \eta |)^{|\sigma |} |{\hat{a}}(\xi - \eta , \eta )|\\&\le \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} \Bigg [\displaystyle \sup _{n \in {\mathbb {Z}}_+} \frac{\Big (\displaystyle \frac{s}{s+1} |\delta | \Big )^{n} \left( 1 + |\eta | \right) ^{n}}{m_{n} n!} \Bigg ]^{-2} (1+|\xi - \eta |)^{|\sigma |} |{\hat{a}}(\xi - \eta , \eta )| \\&\le \Bigg [\displaystyle \sup _{n \in {\mathbb {Z}}_+} \frac{\Big (\displaystyle \frac{s}{s+1} |\delta | \Big )^{n} \left( 1 + |\eta | \right) ^{n}}{m_{n} n!} \Bigg ]^{-2} \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} (1+|\xi - \eta |)^{q} |{\hat{a}}(\xi - \eta , \eta )|. \end{aligned}$$

By Proposition 3.1, item (ii), we have

$$\begin{aligned} \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} (1+|\xi - \eta |)^{q} |{\hat{a}}(\xi - \eta , \eta )| \le \displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} \displaystyle \frac{C_{2} h_{2}^{q + 2N} m_{q+2N} (q +2N)!}{(1+|\xi - \eta |)^{2N}} (1+|\eta |)^{\sigma } \end{aligned}$$

and then by setting $B_{2} \dot{=} C_{2} h_{2}^{q + 2N} m_{q+2N} (q +2N)! \Big (\displaystyle \sum _{\xi \in {\mathbb {Z}}^{N}} \displaystyle \frac{1}{(1+|\xi |)^{2N}} \Big )$ we obtain

$$\begin{aligned} (C2) \le B_2 \Bigg [\displaystyle \sup _{n \in {\mathbb {Z}}_+} \frac{\Big (\displaystyle \frac{s}{s+1} |\delta | \Big )^{n} \left( 1 + |\eta | \right) ^{n}}{m_{n} n!} \Bigg ]^{-2} \left( 1 + |\eta | \right) ^{\sigma }. \end{aligned}$$

From (2.6), we may rewrite the last inequality as

$$\begin{aligned} (C2) \le B_{2} \exp \left[ -2 \omega _{\mathcal {M}}\left( \displaystyle \frac{s}{s+1} (|\delta | (1+|\eta |) \right) \right] (1+|\eta |)^{\sigma }. \end{aligned}$$

(5.12)

Now we use (2.8) with $k = \displaystyle \frac{s}{s+1}$ and $\rho = |\delta | (1+|\eta |)$, in order to get

$$\begin{aligned} \omega _{\mathcal {M}}\left( \displaystyle \frac{s}{s+1} (|\delta | (1+|\eta |) \right) \ge \omega _{\mathcal {M}} \left( |\delta | (1+|\eta |)\right) + \displaystyle \frac{\log (|\delta | (1+|\eta |)) \log \left( \displaystyle \frac{s}{s+1}\right) }{\log H_0}. \end{aligned}$$

Multiplying both sides of the last equation by $-2$ and taking the exponential, we obtain

$$\begin{aligned} \exp \left[ -2\omega _{\mathcal {M}}\left( \displaystyle \frac{s}{s+1} (|\delta | (1+|\eta |) \right) \right] \le \exp \left[ -2 \omega _{\mathcal {M}} (|\delta | (1+|\eta |) \right] \big [ |\delta | (1+|\eta |) \big ]^{ \frac{2\log \left( \frac{s+1}{s}\right) }{\log H_0}}. \end{aligned}$$

If $|\delta |\le 1$ then

$$\begin{aligned} \exp \left[ -2\omega _{\mathcal {M}}\left( \displaystyle \frac{s}{s+1} (|\delta | (1+|\eta |) \right) \right] \le \exp \left[ -2 \omega _{\mathcal {M}} (|\delta | (1+|\eta |) \right] \left( 1+|\eta | \right) ^{\frac{2\log \left( 1 +\frac{1}{s}\right) }{\log H_0}}. \end{aligned}$$

Now, for a given $\varepsilon > 0$, we take $s \in {\mathbb {Z}}_+$ such that $\displaystyle \frac{\log \left( 1 + \frac{1}{s}\right) }{\log H_0} \le \varepsilon$ and the last inequality tells us that

$$\begin{aligned} \exp \left[ -2\omega _{\mathcal {M}}\left( \displaystyle \frac{s}{s+1} (|\delta | (1+|\eta |) \right) \right] \le \exp \left[ -2 \omega _{\mathcal {M}} (|\delta | (1+|\eta |) \right] \left( 1+|\eta | \right) ^{2 \varepsilon }.\nonumber \\ \end{aligned}$$

(5.13)

Puting (5.12) into (5.13) and using (2.6), we obtain

$$\begin{aligned} (C2) \le B_{2} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\eta |)^{n}}{m_{n} n!} \right] ^{-2} (1+|\eta |)^{\sigma + 2 \varepsilon }. \end{aligned}$$

(5.14)

In short, (5.10) , (5.11) and (5.14) give us

$$\begin{aligned} C = (C1) + (C2) \le B_3^{2} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\eta |)^{n}}{m_{n} n!} \right] ^{-2} (1+|\eta |)^{\sigma + 2 \varepsilon }, \end{aligned}$$

(5.15)

where $B_3^2= \max \left\{ C_{6}, B_{2} \right\}$ for all $-1<\delta <0$, with $|\delta | \le \displaystyle \frac{1}{(s+1) h_{2} A_{2N+q} 2H}$. Now, by associating (5.9) and (5.15), we conclude that

$$\begin{aligned} (B)\le & {} B_3^{2} \displaystyle \sum _{\eta \in {\mathbb {Z}}^N} \left[ \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\eta |)^{n}}{m_{n} n!} \right] ^{-2} \left( 1 + |\eta |\right) ^{2 (k-j+\sigma + \varepsilon )} |\hat{u}(\eta )|^{2}\\= & {} B_3^2 \left\| u \right\| ^2_{\left\{ \mathcal {M}, \delta , k - j + \sigma + \varepsilon \right\} } \end{aligned}$$

which, together with (5.8), ensures us that

$$\begin{aligned} \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \le B_3 C_{4} \displaystyle \sum _{j=0}^{k} \left( {\begin{array}{c}k\\ j\end{array}}\right) h_{3}^{j} m_{j} j! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k - j + \sigma + \varepsilon \right\} }. \end{aligned}$$

(5.16)

Let us fix $1 \le j \le k-1$. Since $\sigma + \varepsilon< k - j + \sigma + \varepsilon < k + \sigma + \varepsilon$, we use Lemma 5.4 with $\lambda _{j} = \displaystyle \frac{1}{\left( {\begin{array}{c}k\\ j\end{array}}\right) (2h_{3})^{j} m_{j} j!}$ to conclude that

$$\begin{aligned}&\left\| u \right\| _{\left\{ \mathcal {M}, \delta , k - j + \sigma + \varepsilon \right\} } \le \displaystyle \lambda _{j} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k + \sigma + \varepsilon \right\} } + \lambda _{j}^{\frac{(\sigma + \varepsilon ) - (k- j + \sigma + \varepsilon ) }{(k + \sigma + \varepsilon ) - (k - j + \sigma + \varepsilon ) }} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} } \\&\quad \le \displaystyle \lambda _{j} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k + \sigma + \varepsilon \right\} } + \lambda _{j}^{\frac{j-k}{j}} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} } \\&\quad = \lambda _{j} \left( \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k + \sigma + \varepsilon \right\} } + \lambda _{j}^{\frac{-k}{j}} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} } \right) . \end{aligned}$$

By the choice of $\lambda _j$,

$$\begin{aligned} \left( {\begin{array}{c}k\\ j\end{array}}\right) h_{3}^{j} m_{j} j! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k - j + \sigma + \varepsilon ) \right\} } \le \displaystyle \frac{1}{2^{j}} \cdot \left( \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k + \sigma + \varepsilon \right\} } + \lambda _{j}^{\frac{-k}{j}} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} } \right) .\nonumber \\ \end{aligned}$$

(5.17)

Because $\frac{k!}{(k-j)!} \le k^j$ and $k!\le k^k \le e^k \cdot k!$, it follows from Proposition 2.4 (item (ii)) that

$$\begin{aligned} \lambda _{j}^{\frac{-k}{j}}&= \left( \left( {\begin{array}{c}k\\ j\end{array}}\right) (2h_{3})^{j} m_{j} j!\right) ^{\frac{k}{j}} \le \left( k^{j} (2h_{3})^{j} m_{j} \right) ^{\frac{k}{j}} \le k^{k} (2h_{3})^{k} \left( m_{j}^{{\frac{1}{j}}} \right) ^{k} \nonumber \\&\le k^{k} (2h_{3})^{k} \left( m_{k}^{{\frac{1}{k}}} \right) ^{k} \end{aligned}$$

(5.18)

$$\begin{aligned}&\le k^{k} (2h_{3})^{k} m_{k} \le e^{k} k! (2h_{3})^{k} m_{k}= (2 h_{3} e)^{k} m_{k} k!=h_4^k m_k k!, \end{aligned}$$

(5.19)

where $h_{4} = 2 h_{3} e$.

Now we join the informations of (5.16), (5.17), (5.18) and the fact that $h_3\le h_4$, to conclude that

$$\begin{aligned} \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }&\le B_{3} C_{4} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k + \sigma + \varepsilon \right\} } \\&\quad + B_3 C_4\sum _{j=1}^{k-1} \displaystyle \frac{1}{2^{j}} \Big (\left\| u \right\| _{\left\{ \mathcal {M}, \delta , k + \sigma + \varepsilon \right\} } + h_{4}^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} } \Big ) \\&\quad + B_3 C_4 h_4^k m_k k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} }, i.e., \\ \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }&\le B_{3} C_{4} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k + \sigma + \varepsilon \right\} }\\&\quad + (B_3 C_{4} 2) \left( \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k + \sigma + \varepsilon \right\} } + h_{4}^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} } \right) \\&\quad + B_3 C_4 h_4^k m_k k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} }. \end{aligned}$$

By taking $C \dot{=} 2(B_3 C_{4} 2)$ we have from last inequality:

$$\begin{aligned} \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \le C \left( \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k + \sigma + \varepsilon \right\} } + h^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma + \varepsilon \right\} } \right) ,\ \forall k \in {\mathbb {Z}}_+, \end{aligned}$$

for all $-\delta _\epsilon< \delta <0$, where $\delta _{\varepsilon } = \min \left\{ 1, \displaystyle \frac{1}{(s+1) h_{2} A_{2N+q} 2H}\right\}$ (recall that s is a natural number with $\displaystyle \frac{\log \left( 1 + \frac{1}{s}\right) }{\log H_0} \le \varepsilon$). The proof is now complete. $\square$

One of the main reasons for introducing this class is the following result:

Theorem 5.6

Let ${\mathcal {A}}=\{a_j(x,D)\}_{j=1}^m$ be a system of pseudodifferential operators in $\mathfrak {D}_{\mathfrak {p}_{\sigma }}^{\mathcal {M}}({\mathbb {T}}^N)$ that is globally ${\mathcal {M}}$-hypoelliptic with loss of $r\ge 0$ derivatives. If ${\mathcal {B}}=\{b_j(x,D)\}_{j=1}^m$ is a system of pseudodifferential operators in $\mathfrak {D}_{\mathfrak {p}_{\tau }}^{\mathcal {M}}({\mathbb {T}}^N)$, with $\tau <\sigma -r$, then the system ${\mathcal {C}}\dot{=}\{c_j(x,D)=a_j(x,D) + b_j(x,D)\}_{j=1}^m$ in $\mathfrak {D}_{\mathfrak {p}_{\sigma }}^{\mathcal {M}}({\mathbb {T}}^N)$ is globally ${\mathcal {M}}$-hypoelliptic.

Proof

Let $u \in D_{\mathcal {M}}'({\mathbb {T}}^N)$ such that $c_j(x,D)u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$ for all $j=1,\ldots ,m$. We will prove that $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$. By hypothesis about ${\mathcal {A}}$ and the triangular inequality, one can find constants $C_{1}, \ B_1 > 0, \ \delta _{0} <0$ and $k_{0} \in {\mathbb {R}}$ such that:

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} }\le & {} C_{1} \left[ \max _{1\le j \le m}\left\| c_j(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \right. \\&\left. + \max _{1\le j \le m}\left\| b_j(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + B_1^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right] , \end{aligned}$$

for all $k \in {\mathbb {Z}}_+$ and $\delta _{0}< \delta < 0$.

It follows from Lemma 4.5 the existence of positive constants $\beta _0, C_2, h_1$ such that for $\delta \le \beta _0$ we have

$$\begin{aligned} \max _{1\le j \le m} \left\| c_j(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \le C_2 h_1^{k} m_{k} k!,\ \forall k \in {\mathbb {Z}}_+, \end{aligned}$$

in particular it also holds true for $\delta _0<\delta <0$.

Hence, if we set $C_{3} = \max \{C_1,C_2\}$ and $h_2 = 2\max \{h_1, B_1\}$, we can rewrite the last inequality as

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} }&\le C_{1} \left[ C_2 h_{1}^{k} m_{k} k! + \max _{1\le j \le m}\left\| b_j(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }\right. \\&\quad \left. + B_{1}^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right] \\&\le C_3\left[ h_2^{k} m_k k! + \max _{1\le j \le m}\left\| b_j(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } +h_2^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right] \end{aligned}$$

for all $\delta _0< \delta <0$ and $k \in {\mathbb {Z}}_+$.

Let us analyze the term $\max _{1\le j \le m}\left\| b_j(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }$. By taking $\varepsilon >0$ such that $\tau + \varepsilon < \sigma - r$, it follows Proposition 5.5 that there exist constants $C_4, h_{3} > 1$ and $\delta _{\varepsilon } < 0$ such that

$$\begin{aligned}&\max _{1\le j \le m}\left\| b_j(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \\&\quad \le C_4 \left( \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \tau + \varepsilon \right\} } + h_{3}^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \tau + \varepsilon \right\} } \right) , \ \delta _{\varepsilon }< \delta < 0, k \in {\mathbb {Z}}_+. \end{aligned}$$

Thus, we conclude that

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} }\le & {} C_3\Big [ h_2^{k} m_k k! + C_4 \left( \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \tau + \varepsilon \right\} } + h_{3}^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \tau + \varepsilon \right\} } \right) \\&+ h_2^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \Big ], \end{aligned}$$

for all $\delta _1\dot{=}\max \{\delta _0, \delta _\varepsilon \}< \delta <0$ and $k \in {\mathbb {Z}}_+$.

We may suppose, without loss of generality, that $k_{0} \ge \tau + \varepsilon$. By Proposition 4.3,

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} }&\le C_3\left[ h_2^{k}m_k k! + C_4 \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \tau + \varepsilon \right\} } \right. \nonumber \\&\quad \left. + \left( C_4 h_{3}^{k} m_{k} k! + h_2^{k} m_{k} k! \right) \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right] \nonumber \\&\le C_5 h_4^k m_k k!\left( 1+\left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_0 \right\} }\right) + C_5 \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \tau + \varepsilon \right\} } \end{aligned}$$

(5.20)

for all $k \in {\mathbb {Z}}_+$ and $\delta _1<\delta <0$, $C_5=\max \{2C_3C_4, 2C_3, 1\}$ and $h_4=\max \{h_2,h_3\}$.

By choice of $\varepsilon$, we have $\tau + \varepsilon - 1< k + \tau + \varepsilon < k + \sigma - r$ for any $k\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+$. By taking $\lambda = \displaystyle \frac{1}{2C_{5}}$, Lemma 5.4 ensures us that

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \tau + \varepsilon \right\} }&\le \displaystyle \frac{1}{2C_{5}} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \sigma -r \right\} } + \left( \displaystyle \frac{1}{2C_{5}} \right) ^{\frac{-1 - k}{\sigma - r - (\tau + \varepsilon )}} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \tau + \varepsilon -1 \right\} } \\&= \displaystyle \frac{1}{2C_5} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \sigma -r \right\} } + (2C_5)^{\frac{k+1}{\sigma - r - (\tau + \varepsilon )}} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \tau + \varepsilon -1 \right\} } \\&\le \displaystyle \frac{1}{2C_5} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \sigma -r \right\} } + [(2C_5)^{n_{0}}]^{k+1} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \tau + \varepsilon -1 \right\} }, \end{aligned}$$

where $n_{0}$ is a fixed natural number satisfying $n_{0} \ge \displaystyle \frac{1}{\sigma - r - (\tau + \varepsilon )}$ (also note that $C_5\ge 1$). By setting $h_5=(2C_5)^{n_0}$, one can infer that

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \tau + \varepsilon \right\} } \le \displaystyle \frac{1}{2C_5} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k+ \sigma -r \right\} } +(h_5)^{k+1} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }, \ \forall \ k \in {\mathbb {Z}}_+.\nonumber \\ \end{aligned}$$

(5.21)

Putting (5.21) into (5.20) we obtain

$$\begin{aligned} \displaystyle \frac{1}{2} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} } \le C_5 h_{4}^{k} m_{k} k! \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right) + C_5 (h_5)^{k+1} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} }. \end{aligned}$$

Since $m_k\ge 1$ for all $k\in {\mathbb {Z}}_+$, if we take $h_{6} = \max \left\{ h_{4}, h_5, 1 \right\}$ then it follows that

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} } \le 2 h_6^k m_k k! \left( C_5+2h_6C_5\left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right) . \end{aligned}$$

(5.22)

By taking $C_{6} = 4C_5 h_6$, we obtain from (5.22):

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \sigma +k -r \right\} } \le C_{6} h_{6}^{k} m_{k} k! \left( 1+ \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \right) ,\ \forall k \in {\mathbb {Z}}_+, \ \delta _1< \delta < 0.\nonumber \\ \end{aligned}$$

(5.23)

Since (5.1) is similar to (5.23) then following the lines of the proof of Theorem 5.3 we are able to show that $u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$. The proof is complete. $\square$

6 The Solvability Theorem

It is well known that if P(x, D) is a globally hypoelliptic differential operator on ${\mathbb {T}}^N$, its transpose is globally solvable, see, e.g., Treves [15]. For the Gevrey version of this result, see Albanese, Corli and Rodino [1] and for a global version in the Gevrey classes, see Albanese and Zanghirati [2]. In this section, following the lines of [3] where the author requires a weak version of global hypoellipticity in Gevrey classes, we prove an analogue of these results in ultradifferentiable classes, on the torus ${\mathbb {T}}^N$, for our class of ultradifferential pseudodifferential operators.

Let ${\mathcal {A}}=\{a_1(x,D),\ldots ,a_m(x,D)\}$ be a system of pseudodifferential operators in ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$. We also use ${\mathcal {A}}$ to denote the map ${\mathcal {A}}:\mathcal {E}_{\mathcal {M}}\left( {{\mathbb {T}}^N}\right) (D_{\mathcal {M}}'({\mathbb {T}}^N))\longrightarrow \left( \mathcal {E}_{\mathcal {M}}\left( {{\mathbb {T}}^N}\right) \right) ^{m} \left( \left( D_{\mathcal {M}}'({\mathbb {T}}^N)\right) ^m\right)$ given by

$$\begin{aligned} {\mathcal {A}}(\varphi ) = (a_1(x,D)\varphi ,\ldots ,a_m(x,D)\varphi ), \ \forall \ \varphi \in \mathcal {E}_{\mathcal {M}}\left( {{\mathbb {T}}^N}\right) \,\,\left( u \in D_{\mathcal {M}}'({\mathbb {T}}^N)\right) . \end{aligned}$$

By using the identification $\left( \left( \mathcal {E}_{\mathcal {M}}\left( {{\mathbb {T}}^N}\right) \right) ^{m}\right) '= \left( D'_\mathcal {M}\left( {\mathbb {T}}^N\right) \right) ^m$, the transpose operator of ${\mathcal {A}}$, denoted by $\;\!^t {\mathcal {A}}:\left( D'_\mathcal {M}\left( {\mathbb {T}}^N\right) \right) ^m \longrightarrow D'_\mathcal {M}\left( {\mathbb {T}}^N\right)$, is given by

$$\begin{aligned} \;\!^t {\mathcal {A}}(u_1,\ldots ,u_m) = \sum _{j=1}^m \;\!^ta_j(x,D)u_j, \ \forall \ (u_1,\ldots ,u_m) \in \left( D'_\mathcal {M}\left( {\mathbb {T}}^N\right) \right) ^m. \end{aligned}$$

Suppose that $w \in D'_\mathcal {M}\left( {\mathbb {T}}^N\right)$ and $\;\!^t {\mathcal {A}}(u_1,\ldots ,u_m) = w$ for some $(u_1,\ldots ,u_m) \in \left( D'_\mathcal {M}\left( {\mathbb {T}}^N\right) \right) ^m$. If $\varphi \in \ker {\mathcal {A}}$, with $\varphi \in \mathcal {E}_{\mathcal {M}}\left( {{\mathbb {T}}^N}\right)$, then $a_j(x,D)\varphi =0$ for all $j=1,\ldots ,m$. Hence

$$\begin{aligned} \left\langle w,\varphi \right\rangle = \sum _{j=1}^m \left\langle \;\!^t a_j(x,D)u_j,\varphi \right\rangle = \sum _{j=1}^m\left\langle u_j,a_j(x,D)\varphi \right\rangle =0. \end{aligned}$$

That is, $w \in (\ker {\mathcal {A}})^\perp$. In other words,

$$\begin{aligned} {{\,\mathrm{ran}\,}}(\;\!^t {\mathcal {A}}) \subset ({{\,\mathrm{ker}\,}}{\mathcal {A}})^\perp . \end{aligned}$$

Also recall that by arguments of functional analysis, ${{\,\mathrm{ran}\,}}\;\!^t {\mathcal {A}}$ is closed if and only if ${{\,\mathrm{ran}\,}}(\;\!^t {\mathcal {A}}) = ({{\,\mathrm{ker}\,}}{\mathcal {A}})^\perp$ (see Lemma 2.2 of [3] and Theorem 9 of [11]). This motivates the following

Definition 6.1

We say that the transpose $\;\!^t {\mathcal {A}}$ is globally solvable in $D'_\mathcal {M}\left( {\mathbb {T}}^N\right)$ if $\;\!^t {\mathcal {A}}$ has closed image, which is equivalent to say that ${{\,\mathrm{ran}\,}}(\;\!^t {\mathcal {A}}) = ({{\,\mathrm{ker}\,}}{\mathcal {A}})^\perp$.

Remark 6.2

The definition above means that for all $w\in D'_\mathcal {M}\left( {\mathbb {T}}^N\right)$ such that $\langle w, \varphi \rangle =0,\,\,\text {with}\,\,\varphi \in \mathcal {E}_{\mathcal {M}}\left( {{\mathbb {T}}^N}\right)$   and   ${\mathcal {A}} \varphi =0$, there exists $u=(u_1,\dots ,u_m)\in \left( D'_\mathcal {M}\left( {\mathbb {T}}^N\right) \right) ^m$ such that $\;\!^t {\mathcal {A}} (u)=w.$

In order to use the technique introduced in [3], which is the key point in the proof of our main result of this section, we need some notations and results.

If ${\mathcal {M}}=\{m_n\}_{n\in {{\,\mathrm{{\mathbb {Z}}}\,}}_+}$ is a weight sequence, we set

$$\begin{aligned} \ell _n = m_n n!, \ \forall \ n \in {{\,\mathrm{{\mathbb {Z}}}\,}}_+ \ \ \text{ and } \ \mathscr {L}_{\mathcal {M}}=\{\ell _n\}_{n\in {\mathbb {Z}}_+}. \end{aligned}$$

Since $\mathscr {L}_{\mathcal {M}}$ is given by the product of two weight sequences it is easy to see that $\mathscr {L}_{\mathcal {M}}$ is a weight sequence. Furthermore,

$$\begin{aligned} \sup _{n\in {\mathbb {Z}}_+} \left( \frac{m_n}{\ell _n}\right) ^{\frac{1}{n}} = \sup _{n\in {\mathbb {Z}}_+} \left( \frac{1}{n!}\right) ^{\frac{1}{n}} = 0 \ \text{ and }\ \sup _{n\in {\mathbb {Z}}_+} \left( \frac{\ell _n}{m_n}\right) ^{\frac{1}{n}} = \sup _{n\in {\mathbb {Z}}_+} \left( n!\right) ^{\frac{1}{n}}=\infty . \end{aligned}$$

It follows from Remark 2.6 that $\mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N) \subset \mathcal {E}_{\mathscr {L}_{\mathcal {M}}}({\mathbb {T}}^N)$. In fact we have a more precise information, given by the

Lemma 6.3

$\mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N) \subset \mathcal {E}_{\mathscr {L}_{\mathcal {M}}, 1}({\mathbb {T}}^N)$ and the inclusion $i:\mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N) \hookrightarrow \mathcal {E}_{\mathscr {L}_{\mathcal {M}}, 1}({\mathbb {T}}^N)$ is continuous.

Proof

It suffices to prove that for a given $h >0$, $\mathcal {E}_{\mathcal {M}, h}({\mathbb {T}}^N) \hookrightarrow \mathcal {E}_{\mathscr {L}_{\mathcal {M}}, 1}({\mathbb {T}}^N)$ continuously.

If $f \in \mathcal {E}_{\mathcal {M}, h}({\mathbb {T}}^N)$ we have

$$\begin{aligned} |D^{\alpha }f(x)| \le \left\| f \right\| _{\mathcal {M}, h} h^{|\alpha |} m_{|\alpha |} |\alpha |!, \ \forall \,\, x \in {\mathbb {T}}^N, \alpha \in {\mathbb {Z}}_+. \end{aligned}$$

(6.1)

Since $\displaystyle \lim _{k \rightarrow +\infty } \displaystyle \frac{h^k}{k!} = 0$, we can find $k_{0} \in {\mathbb {Z}}_+$ such that $k! \ge h^{k}$ for every $k \ge k_{0}$ and therefore (6.1) reads as

$$\begin{aligned} |D^{\alpha }f(x)| \le \left\| f \right\| _{\mathcal {M}, h} \left( |\alpha |! m_{|\alpha |} \right) |\alpha |! = \left\| f \right\| _{\mathcal {M}, h} \ell _{|\alpha |} |\alpha |!,\ \forall \,\, x \in {\mathbb {T}}^N, \ |\alpha | \ge k_{0}.\nonumber \\ \end{aligned}$$

(6.2)

By taking $C = \max \{1, \max \{h^p: 0\le p \le k_0\}\}$, it follows from (6.1) and (6.2) that

$$\begin{aligned} |D^{\alpha }f(x)| \le C \left\| f \right\| _{\mathcal {M}, h} \ell _{|\alpha |} |\alpha |!, \ \ \forall x \in {\mathbb {T}}^N, \ \alpha \in {\mathbb {Z}}_+, \end{aligned}$$

(6.3)

which implies that

$$\begin{aligned} \left\| f \right\| _{\mathscr {L}_{\mathcal {M}}, 1} \le C(h) \left\| f \right\| _{\mathcal {M}, h}. \end{aligned}$$

The proof is complete. $\square$

We now introduce a weak version of global hypoellipticity in ultradifferentiable classes that was inspired by [3].

Definition 6.4

Let ${\mathcal {A}}=\{a_1(x,D),\ldots ,a_m(x,D)\}$ be a system of pseudodifferential operators in ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$. We say that ${\mathcal {A}}$ is globally $(\mathscr {L}_{\mathcal {M}}-\mathcal {M})$-hypoelliptic on $\mathbb {\mathbb {T}}^N$ if the following condition holds true:

$$\begin{aligned} u \in \mathcal {E}_{\mathscr {L}_{\mathcal {M}}}({\mathbb {T}}^N) \ \text{ and } \ a_j(x,D)u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N) \ \text{ for } \text{ all }\ j=1,\ldots ,m \ \text{ imply } \ u \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N). \end{aligned}$$

Remark 6.5

It follows immediately from Lemma 6.3 that $D'_{\mathscr {L}_{\mathcal {M}}}({\mathbb {T}}^N) \hookrightarrow D'_\mathcal {M}\left( {\mathbb {T}}^N\right)$ and therefore every globally $\mathcal {M}$-hypoelliptic system of pseudodifferential operators in ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$ is also globally $(\mathscr {L}_{\mathcal {M}}-\mathcal {M})$-hypoelliptic on $\mathbb {\mathbb {T}}^N$.

Lemma 6.6

Let ${\mathcal {A}}$ be a globally $(\mathscr {L}_{\mathcal {M}}-\mathcal {M})$-hypoelliptic on $\mathbb {\mathbb {T}}^N$ system of pseudodifferential operators in ${\mathfrak {D}}^{\mathcal {M}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$ and let $\Gamma$ be the graph of ${\mathcal {A}}: \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N) \rightarrow \left( \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)\right) ^m$. Then $\Gamma$ is a closed subspace of $\mathcal {E}_{\mathscr {L}_{\mathcal {M}}}({\mathbb {T}}^N) \times \left( \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)\right) ^m$.

Proof

By noticing that $\mathcal {E}_{{\mathscr {L}}_\mathcal {M}}({\mathbb {T}}^N) \hookrightarrow D'_{\mathscr {L}_{\mathcal {M}}}({\mathbb {T}}^N),\, D_{\mathscr {L}_{\mathcal {M}}}'({\mathbb {T}}^N) \hookrightarrow D_{\mathcal {M}}'({\mathbb {T}}^N)$ and that the operators $a_j(x,D)\in {\mathfrak {D}}^{\mathscr {L}_{\mathcal {M}}}_{{\mathfrak {p}}_\sigma } ({\mathbb {T}}^N)$, one can complete the proof easily. $\square$

Now we can state and prove our main result of global solvability of this section.

Theorem 6.7

If ${\mathcal {A}}$ is globally ${\mathcal {M}}$-hypoelliptic on ${\mathbb {T}}^N$ then $\ker {\mathcal {A}}$ has finite dimension and $\;\!^t {\mathcal {A}}$ is globally solvable in $D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$.

Proof

If ${\mathcal {A}}$ is globally ${\mathcal {M}}$-hypoelliptic on ${\mathbb {T}}^N$ then ${\mathcal {A}}$ is globally $({\mathscr {L}}_\mathcal {M}- \mathcal {M})$-hypoelliptic on $\mathbb {\mathbb {T}}^N$ (see Remark 6.5). In particular, we may apply Lemma 6.6 and conclude that the graph $\Gamma$ of ${\mathcal {A}}: \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N) \rightarrow \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)^m$ is closed when we view $\Gamma$ as a subspace of $\mathcal {E}_{\mathscr {L}_{\mathcal {M}}}({\mathbb {T}}^N) \times \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)^m$. Hence, it follows from our Lemma 6.3 and Theorem 2.5 in [3] $\big (\text {putting}\,\, E_0=\mathcal {E}_{\mathscr {L}_{\mathcal {M}},1}({\mathbb {T}}^N)\big )$, that $\ker {\mathcal {A}}$ has finite dimension and ran$({\mathcal {A}})$ is closed in $\left( \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)\right) ^m$. Now applying Lemma 2.2 in [3] we conclude that ran$(\;\!^t {\mathcal {A}})$ is closed in $D'_{\mathcal {M}}({\mathbb {T}}^N)$ and therefore $\;\!^t {\mathcal {A}}$ is globally solvable in $D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$. The proof is complete. $\square$

We finish this section with a result that is consequence of Theorem 5.6 and Theorem 6.7.

Corollary 6.8

Let ${\mathcal {A}}=\{a_j(x,D)\}_{j=1}^m$ be a system of pseudodifferential operators in $\mathfrak {D}_{\mathfrak {p}_{\sigma }}^{\mathcal {M}}({\mathbb {T}}^N)$ that is globally ${\mathcal {M}}$-hypoelliptic with loss of $r\ge 0$ derivatives. If ${\mathcal {B}}=\{b_j(x,D)\}_{j=1}^m$ is a system of pseudodifferential operators in $\mathfrak {D}_{\mathfrak {p}_{\tau }}^{\mathcal {M}}({\mathbb {T}}^N)$, with $\tau <\sigma -r$, then the kernel of the system ${\mathcal {C}}\dot{=}\{c_j(x,D)=a_j(x,D) + b_j(x,D)\}_{j=1}^m$ has finite dimension and $\;\!^t {\mathcal {C}}$ is globally solvable in $D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right)$.

7 Application: Operators with Constant Strenght

In this section, we consider a class of linear partial differential operators that was considered by [5] in the analytic setup, given by

$$\begin{aligned} a(x,D) = P_{0}(D) + \displaystyle \sum _{j = 1}^{m} a_{j}(x) P_{j}(D), \end{aligned}$$

(7.1)

where $a_{j}(x) \in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$ and $P_{0}(D), P_{1}(D) \ldots P_{m}(D)$ are differential operators with constant coefficients in ${\mathbb {T}}^N$, satisfying:

• 1. There exist positive constants C, M and R such that

  $$\begin{aligned} \left| P_{0} (\xi ) \right| \ge \displaystyle \frac{C}{(1+|\xi |)^{M}}, \ \ \forall \,\, \xi \in {\mathbb {Z}}^N,\, |\xi | \ge R. \end{aligned}$$

  (7.2)

• 2. For each $j \in \left\{ 1, 2, \ldots , m \right\}$ there exist positive constants $\beta _{j}, c_{j}$ such that

  $$\begin{aligned} |P_{j}(\xi )| \le c_{j} \cdot \displaystyle \frac{|P_{0}(\xi )|}{(1+|\xi |)^{\beta _{j}}}, \ \ \forall \,\, \xi \, \in {\mathbb {Z}}^N, |\xi | \ge R. \end{aligned}$$

  (7.3)

Notice that under such conditions the operator a(x, D) defined in (7.1) is of constant strengh (cf. [7]).

Theorem 7.1

Let a(x, D) be the operator in (7.1) and $\beta _0$ be the order of operator $P_0(D)$. Then a(x, D) is globally $\mathcal {M}$-hypoelliptic on ${\mathbb {T}}^N$ with loss of $\beta _{0} + M$ derivatives.

Proof

We start by noticing that operator a(x, D) given in (7.1) belongs to the space $\mathfrak {D}^{\mathcal {M}}_{\mathfrak {p}_{\beta _{0}}} ({\mathbb {T}}^N)$.

For $u \in D'_{\mathcal {M}}({\mathbb {T}}^N), \delta <0$ and $k \in {\mathbb {Z}}_+$ we have

$$\begin{aligned} \left\| u \right\| ^2_{\left\{ \mathcal {M}, \delta , k - M \right\} }&= \displaystyle \sum _{|\xi | < R} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{2k - 2M} |\hat{u}(\xi )|^{2} \\&\quad + \displaystyle \sum _{|\xi | \ge R} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{2k - 2M} \displaystyle \frac{|\widehat{P_{0}(D)u}(\xi )|^{2}}{|P_{0}(\xi )|^{2}}, \end{aligned}$$

where R comes from (7.3).

We now use (7.2) and the last inequality in order to obtain

$$\begin{aligned} \left\| u \right\| ^2_{\left\{ \mathcal {M}, \delta , k - M \right\} }&\le (1+R)^{2k} \displaystyle \sum _{|\xi | < R} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{- 2M} |\hat{u}(\xi )|^{2} \\&\quad +C^{-2} \displaystyle \sum _{|\xi | \ge R} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{2k} |\widehat{P_{0}(D)u}(\xi )|^{2}. \end{aligned}$$

By setting $B_{1} = \max \left\{ (1+R), C^{-1} \right\}$ it follows from the last inequality that

$$\begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k - M \right\} } \le B_{1}^{k} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} } + B_{1} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k\right\} }. \end{aligned}$$

(7.4)

For any $\ell >0$, we have

$$\begin{aligned} \left\| P_{j}(D)u \right\| _{\left\{ \mathcal {M}, \delta , \ell \right\} }^{2}&= \underbrace{\displaystyle \sum _{|\xi | < R} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{2\ell } |P_{j}(\xi )|^{2} |\hat{u}(\xi )|^{2}}_{(A)} + \\&\quad + \underbrace{\displaystyle \sum _{|\xi | \ge R} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{2\ell } |P_{j}(\xi )|^{2} |\hat{u}(\xi )|^{2}}_{(B)}. \end{aligned}$$

Let us consider $\tau >0$ and $C'>0$ satisfying

$$\begin{aligned} |P_{j}(\xi )| \le C' (1+|\xi |)^{\tau }, \ \ \forall \xi \in {\mathbb {Z}}^N, \ j=1,\ldots ,m. \end{aligned}$$

Thus

$$\begin{aligned} (A)&\le \displaystyle \sum _{|\xi |< R} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \displaystyle \frac{\left( 1+|\xi | \right) ^{2\ell +2M}}{\left( 1+|\xi | \right) ^{2M}} C'^{2} (1+|\xi |)^{2\tau } |\hat{u}(\xi )|^{2} \\&\le C'^{2} (1+R)^{2 (\tau + \ell + M)} \displaystyle \sum _{|\xi | < R} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} \cdot n!} \right) ^{-2} \displaystyle \frac{1}{\left( 1+|\xi | \right) ^{2M}} |\hat{u}(\xi )|^{2} \\&\le \left[ C' (1+R)^{\tau + M} \right] ^{2} B_{1}^{2\ell } \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} }^{2}. \end{aligned}$$

We also obtain from (7.3) that

$$\begin{aligned} (B)&\le c_{j}^{2} \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{2\ell } \frac{|P_{0}(\xi )|^{2} }{(1+|\xi |)^{2 \beta _{j}}} |\hat{u}(\xi )|^{2}\\&\le c_{j}^{2} \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{2\ell - 2 \beta _{j}} |\widehat{P_{0}(D)u}(\xi )|^{2}. \end{aligned}$$

If we choose $B_{2} = \max \left\{ c_{1}, c_{2}, \ldots c_{m}, C' (1+R)^{\tau + M} \right\}$ and $\gamma = \min \left\{ \beta _{1}, \beta _{2}, \ldots , \beta _{m} \right\}$ then we obtain from the last two inequalities that

$$\begin{aligned} \left\| P_{j}(D)u \right\| _{\left\{ \mathcal {M}, \delta , \ell \right\} }\le & {} B_{2} B_{1}^{\ell } \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} } \nonumber \\&+ B_{2} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , \ell - \gamma \right\} }, j = 1, 2 \ldots , m. \end{aligned}$$

(7.5)

Since we have $a(x,D) = P_{0}(D) + \displaystyle \sum _{j = 1}^{m} a_{j}(x) P_{j}(D)$, by using the triangular inequality we can write

$$\begin{aligned} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \le \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + \displaystyle \sum _{j=1}^{m} \left\| a_{j}(x) P_{j}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }. \end{aligned}$$

(7.6)

Now since $a_{j}\in \mathcal {E}_{\mathcal {M}}({\mathbb {T}}^N)$, it defines an element of $\mathfrak {D}_{\mathfrak {p}_{0}}^{\mathcal {M}}({\mathbb {T}}^N)$. In particular, we can use Proposition 5.5 (with $\sigma =0$ and $\varepsilon =\frac{\gamma }{2}$) in order to conclude the existence of constants $B_{3}, h > 0$ and $\delta _{\gamma } < 0$, such that, for every $k\in {\mathbb {Z}}_+$, $\delta _{\gamma }< \delta < 0$, we have

$$\begin{aligned} \left\| a_{j}(x) P_{j}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }\le & {} B_{3} \left\| P_{j}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k + \frac{\gamma }{2}\right\} } \nonumber \\&+ B_{3} h^{k} m_{k} k! \left\| P_{j}(D)u \right\| _{\left\{ \mathcal {M}, \delta , \frac{\gamma }{2}\right\} }. \end{aligned}$$

(7.7)

From (7.6) and (7.7), we obtain

$$\begin{aligned} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }&\le \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + \displaystyle \sum _{j=1}^{m} B_{3} \left( \left\| P_{j}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k + \frac{\gamma }{2}\right\} } \right) + \\&\quad + B_{3} h^{k} m_{k} k! \displaystyle \sum _{j=1}^{m} \left\| P_{j}(D)u \right\| _{\left\{ \mathcal {M}, \delta , \frac{\gamma }{2}\right\} }. \end{aligned}$$

By using (7.5) twice, one with $\ell =\frac{\gamma }{2}$ and the other with $\ell =k+\frac{\gamma }{2}$, it follows from above inequality that

$$\begin{aligned}&\left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }\le \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }\\&\quad + \displaystyle \sum _{j=1}^{m} B_{3} B_{2} \left( B_{1}^{k +\frac{\gamma }{2}} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} }+\left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k- \frac{\gamma }{2} \right\} } \right) \\&\quad + B_{3} B_{2} h^{k} m_{k} k! \displaystyle \sum _{j=1}^{m}\left( B_{1}^{\frac{\gamma }{2}} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} } +\left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , - \frac{\gamma }{2} \right\} }\right) . \end{aligned}$$

If we set $B_{4} = \max \{2m B_{2} B_{3}, 2m B_{2} B_{3} B_{1}^{\frac{\gamma }{2}}, 1\}$ and $h_{2} = \max \left\{ h, B_{1} \right\}$, then we deduce that

$$\begin{aligned} \begin{aligned} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }&\le \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + B_{4} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k - \frac{\gamma }{2}\right\} } + \\&\quad + B_{4} h_{2}^{k} m_{k} k! \left( \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , -\frac{\gamma }{2}\right\} } + \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} } \right) . \end{aligned} \end{aligned}$$

(7.8)

By taking $\omega \in {\mathbb {R}}$ such that $\omega < - \displaystyle \frac{\gamma }{2}$, we obtain $\omega< k - \displaystyle \frac{\gamma }{2} < k$ for each k em ${\mathbb {Z}}_+$ and use Lemma 5.4 (with $\lambda = \displaystyle \frac{1}{2B_{4}}$) to conclude that

$$\begin{aligned} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k - \frac{\gamma }{2}\right\} } \le \displaystyle \frac{ \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }}{2B_{4}} + \left( 2B_{4} \right) ^{\frac{2 \left( k - \omega - \frac{\gamma }{2}\right) }{\gamma }} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , \omega \right\} }\nonumber \\ \end{aligned}$$

(7.9)

Putting (7.9) into (7.8),

$$\begin{aligned} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }&\le \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + \displaystyle \frac{ \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }}{2} \ \nonumber \\&\quad + \left( 2B_{4} \right) ^{\frac{2 \left( k - \omega \right) }{\gamma }} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , \omega \right\} } \nonumber \\&\quad + B_{4} h_{2}^{k} m_{k} k! \left( \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , -\frac{\gamma }{2}\right\} } + \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M\right\} } \right) \nonumber \\&\le \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + \displaystyle \frac{ \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }}{2}\nonumber \\&\quad + C_{1} r^{k} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , \omega \right\} } \nonumber \\&\quad + B_{4} h_{2}^{k} m_{k} k! \left( \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , -\frac{\gamma }{2}\right\} } + \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} } \right) , \end{aligned}$$

(7.10)

where $C_1=(2B_4)^{-\frac{2\omega }{\gamma }}$ and $r=(2B_4)^{\frac{2}{\gamma }}$.

Absorving the term $\frac{ \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }}{2}$ in the left hand side of (7.10) we obtain

$$\begin{aligned} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} }&\le 2\Big [\left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + C_{1} r^{k} \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , \omega \right\} } \nonumber \\&\quad + B_{4} h_{2}^{k} m_{k} k! \left( \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , -\frac{\gamma }{2}\right\} } + \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} } \right) \Big ].\nonumber \\ \end{aligned}$$

(7.11)

If we set $B_{5} = 2 \max \left\{ C_{1}, B_{4} \right\}$ and $s_{1} = \max \left\{ h_{2}, r \right\}$, then by using that $\omega < - \displaystyle \frac{\gamma }{2}$ and Proposition 4.3 it follows from (7.11) that

$$\begin{aligned}&\left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \le 2\Big [\left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } \nonumber \\&\quad + B_{5} s_{1}^{k} m_{k} k! \left( \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , -\frac{\gamma }{2}\right\} } + \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} } \right) \Big ]. \end{aligned}$$

(7.12)

It follows from (7.4) and (7.12) that

$$\begin{aligned} \begin{aligned} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k - M \right\} }&\le 2\Big [\left( B_{1}^{k} + B_{1} B_{5} s_{1}^{k} m_{k} k! \right) \left\| u \right\| _{\left\{ \mathcal {M}, \delta , - M \right\} } + B_{1} \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + \\&\quad + B_{1} B_{5} s_{1}^{k} m_{k} k! \left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , -\frac{\gamma }{2}\right\} }\Big ] . \end{aligned} \end{aligned}$$

(7.13)

Since $P_{0}(D) \in \mathfrak {D}^{\mathcal {M}}_{\mathfrak {p}_{\beta _{0}}} ({\mathbb {T}}^N)$, we have

$$\begin{aligned}&\left\| P_{0}(D)u \right\| _{\left\{ \mathcal {M}, \delta , -\frac{\gamma }{2}\right\} }^{2} = \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{-\gamma } |P_{0}(\xi )|^{2} |\hat{u}(\xi )|^{2} \nonumber \\&\quad \le C_{2}^{2} \displaystyle \sum _{\xi \in {\mathbb {Z}}^N} \left( \displaystyle \sup _{n \in {\mathbb {Z}}_+} \displaystyle \frac{|\delta |^{n} (1+|\xi |)^{n}}{m_{n} n!} \right) ^{-2} \left( 1+|\xi | \right) ^{-\gamma } (1+|\xi |)^{2 \beta _{0}} |\hat{u}(\xi )|^{2} \nonumber \\&\quad \le C_{2}^{2} \left\| u \right\| _{\left\{ \mathcal {M}, \delta , \beta _{0} -\frac{\gamma }{2}\right\} }^{2}. \end{aligned}$$

(7.14)

Hence, if we take

$$\begin{aligned} s_{2}= & {} \max \left\{ B_{1}, s_{1} \right\} , B_{6} = 3 \max \left\{ 1,B_{1} B_5 , B_1 B_5 C_2, 2B_1\right\} \,\, \text {and}\,\,\\ k_{0}= & {} \max \left\{ \beta _{0} - \displaystyle \frac{\gamma }{2}, - M \right\} \end{aligned}$$

we may utilize Proposition 4.3, (7.13) and (7.14) in order to infer that

$$\begin{aligned}&\left\| u \right\| _{\left\{ \mathcal {M}, \delta , \beta _0 +k - (\beta _0+M) \right\} }=\left\| u \right\| _{\left\{ \mathcal {M}, \delta , k - M \right\} }\\&\quad \le 2\Big [\frac{B_{6}}{6} \left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + B_6s_{2}^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \Big ]\\&\quad \le 2B_{6} \Big [\left\| a(x,D)u \right\| _{\left\{ \mathcal {M}, \delta , k \right\} } + s_{2}^{k} m_{k} k! \left\| u \right\| _{\left\{ \mathcal {M}, \delta , k_{0} \right\} } \Big ] \end{aligned}$$

for any $k\in {\mathbb {Z}}_+$, $u\in D'_{\mathcal {M}}({\mathbb {T}}^N)$, $\delta _\gamma<\delta <0$ and the proof is finally complete. $\square$

Remark 7.2

In [5] the hyphotesis (7.2) is the following: There exist positive constants c, R and $\varepsilon >0$ such that

$$\begin{aligned} \left| P_{0} (\xi ) \right| \ge \displaystyle c(1+|\xi |)^{\varepsilon }, \ \ \forall \,\, \xi \in {\mathbb {Z}}^N,\, |\xi | \ge R, \end{aligned}$$

thus our result generalizes that one in [5].

Corollary 7.3

Let a(x, D) be the operator in (7.1). Then a(x, D) is globally $\mathcal {M}$-hypoelliptic on $\mathbb {T}^N$, its kernel has finite dimension and its transpose $\;\!^t {\mathcal {A}}$ is globally solvable in $D'_{\mathcal {M}}(\mathbb {T}^N)$. Furthermore, these properties remain valid if we perturb the operator a(x,D) by any ultradifferentiable pseudodifferential operator of order $\tau < -M$.

Proof

It is a direct consequence of Theorem 7.1, Theorem 5.3, Theorem 5.6, Theorem 6.7 and Corollary 6.8. $\square$