Global hypoellipticity for a class of periodic Cauchy operators
Fernando de Ávila Silva
math.AP 35B10 35B65 35H10 35S05










fernando.avila@ufpr.br



Departamento de Matemática, Universidade Federal do Paraná, 
 Caixa Postal 19081, Curitiba, PR 81531-990, Brazil




This note presents an investigation on the  global hypoellipticity problem for Cauchy operators on 𝕋^n+1 belonging to the class  L = ∏_j=1^m(D_t + c_j(t) P_j(D_x)), where P_j(D_x) is a pseudo-differential operator on 𝕋^n and c_j = c_j(t), a smooth, complex valued  function on 𝕋. The main goal of this investigation consists in establishing connections between the global hypoellipticity of the operators L and its normal form L_0 =  ∏_j=1^m ( D_t + c_0,jP_j(D_x)). In order to do so, the problem is approached   by combining Hörmander's and Siegel's  conditions on the symbols of the operators L_j = D_t + c_j(t) P_j(D_x).




Global hypoellipticity, Pseudo-differential operators,  Fourier series, Cauchy operators, Siegel  conditions
[2010] 35B10 35B65 35H10 35S05











§ INTRODUCTION




This note discusses  the  global hypoellipticity problem for operators 
belonging to the class

    L  = ∏_j=1^m(D_t + c_j(t) P_j(D_x)),  (t, x) ∈𝕋×𝕋^n,

where 𝕋≃ℝ/ 2 πℤ is the  Torus and  P_j(D_x) is a pseudo-differential operator of order ν_j ∈ℝ defined by 

    P_j(D_x) u(x) =  ∑_ξ∈ℤ^ne^i x ·ξ p_j(ξ) u(ξ),

where its  symbol p_j=p_j(ξ)  satisfies

    |p_j(ξ)| ≤ C |ξ|^ν_j,  ∀ξ∈ℤ^n

and u(ξ) are the Fourier coefficients

    u(ξ) = (2π)^-n∫_𝕋^ne^- i x ·ξ u(x) dx,  ξ∈ℤ^n.


In particular, p_j belongs to the class S^ν_j(ℤ^n),  as shown in Ruzhansky and Turunen <cit.>. 

Let us recall that a pseudo-differential operator 𝒫 is globally hypoellitpic ((GH) for short) on the N-dimensional torus  𝕋^N   if conditions u ∈𝒟'(𝕋^N)  and 𝒫 u  ∈ C^∞(𝕋^N) imply u ∈ C^∞(𝕋^N).

We emphasize that the study of global hypoellipticity is a highly non-trivial problem
and it seems impossible to attack it by a unified approach, even for the classes of vector fields, as we can see in A. Bergamasco, P. Cordaro and G. Petronilho <cit.>, G. Petronilho <cit.>, A. Himonas and G. Petronilho <cit.> and J. Hounie <cit.>. Also, the study is considered in different functional spaces, for instance,  analytic and Gevrey classes, see A. Bergamasco <cit.>, A. Albanese <cit.>,  G. Chinni, P. Cordaro  <cit.>  and the references therein.

Furthermore, the problem is analyzed in the class of pseudo-differential operators as proposed by D. Dickson, T. Gramchev and M. Yoshino  in <cit.> where 
the authors characterizes the global hypoellipticity by means of Siegel type conditions on the symbols of these operators. 



Now, since the main goal of this investigation is to establish connections between the global hypoellipticity of  L and its normal form

    L_0 =  ∏_j=1^m ( D_t + c_0,jP_j(D_x)),  (t, x) ∈𝕋^n+1,

where c_0,j = (2 π)^-1∫_0^2 πc_j(t)dt, first we take inspiration
from <cit.>, where R. Gonzalez, A.  Kirilov, C. Medeira and F. Ávila, present a characterization of the  global hypoellipticity for operators in the form

    L_j = D_t+c_j(t)P_j(D_x),  (t, x) ∈𝕋^n+1,

by analyzing the functions 

    t∈𝕋↦ℳ_j(t,ξ)≐ c_j(t)p_j(ξ),  ξ∈ℤ^n,

and its averages

    ℳ_0,j(ξ) = (2 π)^-1∫_0^2 πℳ_j(t, ξ) dt,  ξ∈ℤ^n.


Secondly, by denoting 
Z_ℳ_j = {ξ∈ℤ^n;  ℳ_0,j(ξ) ∈ℤ} and L_0,j = D_t + c_0,jP_j(D_x), then the  results from <cit.>, that  relate to this note,  can be summarized by the following:

 Set c_j(t) = a_j(t) + i b_j(t) and 
	p_j(ξ) = α_j(ξ) + i β_j(ξ).
	
	
		
  (a) if L_j is globally hypoelliptic, then  Z_ℳ_j is finite and L_0,j is globally hypoelliptic;
		
		
		
		
  (b) if L_0,j is  globally hypoelliptic  and the functions 
		
    𝕋∋ t ↦ℳ_j(t,ξ) = a_j(t)β_j(ξ) + b_j(t)α_j(ξ)

		do not change sign, for a sufficiently large |ξ|, then L_j is (GH).
		
		
  (c)  the following statements are equivalent:
		
			
  i) L_0,j  is globally hypoelliptic;
			
  ii)  there exist positive constants C_j, M_j and R_j such that
			
    |τ+ℳ_0,j(ξ)|⩾ C_j|(τ,ξ)|^-M_j,  for all |τ| + |ξ| ⩾ R_j;

			
  iii) there exist positive constants C_j, M_j and R_j such that
			
    |1-e^±2π iℳ_0,j(ξ)|⩾C_j|ξ|^-M_j,  for all  |ξ|⩾R_j.

		
		
	 
	






In view of the main goal, the first step in our investigations is the study of 
the constant coefficients operators belonging to the class:

    L =  ∏_j=1^m ( D_t + α_j P_j(D_x)),  (t, x) ∈𝕋^n+1,  α_j ∈ℂ.


In Section <ref>, we present an analysis for this class that combines Theorem <ref> and a generalization of Siegel's condition. This condition was introduced by D. Dickson, T. Gramchev and M. Yoshino in <cit.>. As it shall be stated in Theorem  <ref>, necessary and sufficient conditions for global hypoellipticity are presented  by analyzing the behavior of the symbol L(τ, ξ).

In section <ref>, the case of variable coefficients is studied.
A first investigation is performed through  Theorem <ref> and also by assuming the following commutative condition:

    [L_j, L_k] = L_j  L_k - L_k  L_j = 0,  j, k ∈{ 1, … m},

that will be shown in  Theorem <ref>.

In section <ref>, a second investigation is presented and the approach is motivated by a condition proposed in Hörmander <cit.>: there exists positive constants η_j such that

    ℳ_j(t, ξ) ≥ -η_j,  t ∈𝕋,  ∀ξ∈ℤ^n.


Under this hypothesis, in  Theorem <ref>  necessary and sufficient conditions for the global hypoellipticity of the operator L_j will be exhibited.
Moreover, in Theorem <ref>, connections between the hypoellipticity of operators L and L_0 will be established. Also, in Theorem <ref>
conditions for the global  solvability of the operator L_j will be presented and, in particular, obstructions for the global hypoellipticity of operator L will be shown in Theorem <ref>.

Furthermore, also in section <ref>, an interesting fact shall present itself: in view of  condition (<ref>), it becomes possible to obtain a new class of globally hypoelliptic  operators, so that the functions ℳ_j(·,ξ) change sign for infinitely many indexes ξ∈ℤ^n. This fact will be discussed at the beginning of the section, where an example will also be proposed.






§ OPERATORS WITH CONSTANT COEFFICIENTS 





In this section we consider the constant coefficient  operator 

    L =  ∏_j=1^m ( D_t + α_j P_j(D_x)),  (t, x) ∈𝕋^n+1,  α_j ∈ℂ,

and study the global hypoellipticity by analyzing the growth of its symbol

    L(τ, ξ) =   ∏_j=1^m(τ - α_j p_j(ξ)),  (τ, ξ) ∈ℤ×ℤ^n.



By following the approach introduced by Greenfield and Wallach in <cit.>, we shall characterize the global hypoellipticity of L
by means of a control in L(τ, ξ):



	The operator L is globally hypoelliptic if and only if there are positive constants C, M and R so that
	
    |L(τ, ξ)|  ≥ C |(τ,ξ)|^-M,  |τ| + |ξ| ≥ R.
	




Now, in order to present a complete characterization for the global hypoellipticity, we introduce a Siegel condition,  inspired by  D. Dickson, T. Gramchev and M. Yoshino in <cit.>.



	We say that a sequence {c(ξ)}_ξ∈ℤ^n satisfies the generalized Siegel condition if there exists positive constants C, M and R  such that
	
    |τ - c(ξ)| ≥ C |(τ, ξ)|^-M,  |τ|+| ξ| ≥ R,  (τ, ξ) ∈ℤ×ℤ^n.

	
	Furthermore, a set Γ={{c_j(ξ)}_ξ∈ℤ^n,  j = 1, …, m }
	satisfies  the simultaneous generalized Siegel condition  if each {c_j(ξ)}_ξ∈ℤ^n satisfies (<ref>). In this case we can write  Γ∈  (𝒢𝒮)_∞.




	Let L be the operator defined in (<ref>) and set 
	
    σ_L = {{α_j p_j(ξ)}_ξ∈ℤ^n,  j = 1, …, m }.

	
	Then, L is globally hypoelliptic if and only if σ_L∈ (𝒢𝒮)_∞.  



	Since the coefficients of L_j are constant,  we obtain [L_j, L_k] = 0, for 
	j,k ∈{1, …, m}. Hence, L is globally hypoelliptic if and only if each L_j is also globally hypoelliptic.  Then, if L is (GH),  it follows from Theorem <ref> that all sequences {α_j p_j(ξ)} satisfy (<ref>).
	
	
	Conversely, if σ_L ∈  (𝒢𝒮)_∞ it is possible to  obtain constants C_j, R_j and M_j so that |τ - α_j p_j(ξ)| ≥ C_j |(τ,ξ)|^-M_j, for all |τ| + |ξ|≥ R_j, hence
	
    |L(τ, ξ)| = ∏_j=1^m|τ - α_j p_j(ξ)| ≥   C |(τ,ξ)|^-M,  |τ| + |ξ|≥ R,

	where C = ∏ C_j, M = ∑ M_j  and R = max R_j. The proof is then complete, in view of Theorem <ref>.	
	







	Let L be 	the differential operator 
	
    L =  ∏_j=1^m (D_t - α_j D_x),  (t,x) ∈𝕋^2.

	
	In this case, it is possible to write:
	
    L(τ, ξ) =  (τ - α_1 ξ) … (τ - α_mξ)

	and thus L is not globally hypoelliptic if and only if one of the numbers α_j is either a rational  or  Liouville. It can be noted here that this example recaptures results in <cit.>.





	It is possible to consider operators belonging to the class
	
    𝒫 =D_t^m + ∑_j=1^mP_j(D_x) D_t^m-j,  (t, x) ∈𝕋^n+1,  c_j ∈ℂ,

	rewriting  their symbols as
	
    𝒫 (τ, ξ) =   ∏_j=1^m(τ - σ_j(ξ)),  (τ, ξ) ∈ℤ×ℤ^n,

	where σ_j(ξ) satisfies 𝒫(σ_j(ξ), ξ) = 0. It follows from Theorem <ref> that 𝒫 is globally hypoelliptic provided that
	
    σ_𝒫 = {{σ_j(ξ)}_ξ∈ℤ^n,  j = 1, …, m }∈ (𝒢𝒮)_∞.

	





	Consider Δ_x = -∑_j=1^n∂^2_x_j defined on 𝕋^n and set 
	
    𝒫 = D^2_t - 2α D_t (-Δ_x)^1/2 + β^2 Δ_x,

	with α, β∈ℝ. In this case, 
	𝒫(τ, ξ)   =  ( τ  - |ξ| ρ_1 ) (  τ  - |ξ| ρ_2 ), 	where
	
    ρ_1,2 =  -α±√(α^2 - β^2).

	
	
	If 	α^2 - β^2 <0, then ρ_1,2  is not real  and 𝒫 is globally hypoelliptic. However, the case of real roots could be much more complicated since, in general, the usual approximations by rational numbers
	can't be applied, once
	τ / |ξ| may be irrational. For instance, when α=β we have
	𝒫 = (D_t - α (-Δ_x)^1/2)^2 and 
	
    𝒫(τ, ξ) = |ξ|^2(τ|ξ|  - α)^2.

	
	The reader can find a complete discussion and examples for this type of approximations in <cit.>.
	








§ OPERATORS WITH VARIABLE COEFFICIENTS 




In this section we study the global hypoellipticity of the operator (<ref>), which is recalled by

    L  = ∏_j=1^m(D_t + c_j(t) P_j(D_x)),  (t, x) ∈𝕋×𝕋^n.


Given that the main goal is to  exhibit connections  between the hypoellipticity of operators L and 

    L_0 =  ∏_j=1^m ( D_t + c_0,jP_j(D_x)),
 
we start by observing that:


	
  (i) when each L_j is globally hypoelliptic,  then L and L_0 are globally hypoelliptic and the sets Z_ℳ_j are finite;  
	
	
  (ii)  L_0 is  globally hypoelliptic  if and only if each L_0,j is globally hypoelliptic;
	
	
  (iii)  let ρ be a permutation of the set {1, …, m} and define 
	
    L_ρ = ∏_j=1^m L_ρ(j) = L_ρ(1)∘…∘ L_ρ(m).

	If some L_k is not globally hypoelliptic, then for any permutation β of {1, …, m}∖{k} we obtain L_β∘ L_k not (GH). This is the case for L when L_m is not globally hypoelliptic.
	





	Consider L_ρ an operator as in (<ref>).
	
	 
		
  (a) If L_ρ is globally hypoelliptic for any permutation ρ, then L_0 is also globally hypoelliptic. In particular, all sets Z_ℳ_k are finite.
		
		
  (b) If some L_ρ is  globally hypoelliptic, then at least one of the operators L_0,k is globally hypoelliptic.  
	
	



	Assume that every  L_ρ is (GH) and consider the equation L_k u = f ∈ C^∞(𝕋^n+1), for some k.  If  β is a permutation of {1, …, m}∖{k}, then the operator L_β∘ L_k is (GH), which implies u ∈ C^∞(𝕋^n+1), since L_β(C^∞(𝕋^n+1)) ⊂ C^∞(𝕋^n+1). It follows from Theorem <ref> that  L_0,k is globally hypoelliptic and a) is proved.
	
	To verify b) it is enough to observe that if  L_ρ is (GH), then L_ρ(m) is necessarily globally hypoelliptic and consequently L_0,ρ(m).
	




We point out that, by Proposition <ref>,  when L is globally hypoelliptic, then at least L_0,m is  (GH). The next result improves this conclusion if  commutative conditions are admitted.




	Assume  that L satisfies the commutative hypothesis
	
    [L_j, L_k] = L_j  L_k - L_k  L_j = 0,

	for every j, k ∈{1, …, m}.	 Then:
	
	
		
  (a) L is (GH) if and only if every L_k is (GH);
		
		
  (b) if L is (GH), then L_0 is (GH).
	
	



	From the commutative hypothesis we have L = L_ρ, for any  permutation ρ of {1, …, m }. If L is  globally hypoelliptic, then  each L_k is also globally hypoelliptic by the preceding result. In particular, every L_0,k is  (GH), and  consequently so is L_0.
	








 §.§ Hörmander conditions 





In this section we study the global hypoelliticity problem by considering operators
L_j satisfying  the following condition:  there exists positive  constants η_j such that

    ℳ_j(t, ξ) ≥ -η_j,  t ∈𝕋,  ∀ξ∈ℤ^n.



Notice that there are no novelties if η_j = 0 is admitted, since in this case 
ℳ_j(·, ξ) does not change sign for every ξ, and we may apply Theorem <ref>. Otherwise, by assuming η_j>0, the existence of a globally hypoelliptic operator L_j,  with  ℳ_j(t,·) changing sign, becomes possible, as follows from the next result.


	An operator L_j satisfying  (<ref>) is globally hypoellitpic if and only if L_0,j is globally hypoelliptic. 


Before the proof of this theorem, we present a direct consequence. 




	Admit that each operator L_j satisfies  (<ref>). If  L_0 is globally hypoelliptic, then  L is globally hypoelliptic.



	If L_0 is globally hypoelliptic, then each L_0,j is also (GH) and, by the previous theorem, each L_j is globally hypoelliptic.
	



	We emphasize that the  phenomena of hypoellipticity with changes of sign 
	in the imaginary part ℳ_j(t, ξ),  for operators with symbols  satisfying the logarithm growth 
	
    |p_j(ξ)| = O(log(|ξ|)),   when  |ξ|→∞,

	is discussed in <cit.>, as the reader can see in Sections 4 and 5 of that paper.
	
	In the next  example, we exhibit an operator satisfying (<ref>) which  cannot be captured by the approach presented in <cit.>.	




	Let P(D_x) be a pseudo-differential operator  on 𝕋 defined by the symbol p(ξ) =α(ξ) + iβ(ξ), where
	
    α(ξ) = 
    	{[ ξ^-1,   if  ξ<0    is odd,; |ξ|,   if  ξ<0    is even,;            0,   if   ξ≥ 0, ]. 
    	  and β(ξ) = 
    	{[ 1,   if   ξ<0,; ξ,   if  ξ≥ 0. ].

	Clearly, both real and imaginary parts of p(ξ) do not satisfy the logarithm condition.
	
	Now, let a(t) and b(t) be two real, smooth  and  positive functions on 𝕋 with disjoint supports
	and define  Q = D_t + (a(t)+ ib(t))P(D_x). In this case, we have ℳ(t, ξ) =0 when t ∉ supp(a) ∪ supp(b), for any ξ∈ℤ. On the other hand, 
	
    t ∈ supp(a) ⇒ℳ(t, ξ) = a(t) β(ξ),

	and
	
    t ∈ supp(b) ⇒ℳ(t, ξ) = b(t) α(ξ).

	
	Note that ℳ(·, ξ) changes sing if ξ<0 is odd. Indeed, 
	in this case t ∈ supp(a) implies ℳ(t, ξ) = a(t)>0,
	while ℳ(t, ξ) = b(t) ξ^-1 <0 when t ∈ supp(b).
	
	
	Also, condition (<ref>) is fulfilled by choosing 
	η = max_t ∈𝕋b(t). It follows that Q is globally hypoelliptic if and only if Q_0 is globally hypoelliptic.
	




To prove Theorem <ref> we make use of a standard result in literature:


	Let u ∈𝒟'(𝕋^n+1) be a distribution and  its x-Fourier coefficients be
	
    u(t, ξ) = (2 π)^-n <u(t, ·),  e^-i xξ>,  ξ∈ℤ^n.

	
	Given a sequence {c_ξ(t)}_ξ∈ℤ^n of smooth functions on 𝕋, the formal series u = ∑_ξ∈ℤ^Nc_ξ(t) e^i  x ·ξ 	converges in 𝒟'(𝕋^n+1) if and only if for any α∈ℤ_+ there exist positive constants N, C and R such that
	
    |∂^α_t c_ξ(t)| ≤ C |ξ|^-N,  |ξ|≥ R.

	Moreover, u ∈ C^∞(𝕋^n+1) if and only if estimate (<ref>) holds true for every N>0. In both cases, 
	c_ξ(·) = u(·, ξ).







 §.§ Proof of Theorem <ref>


The necessary part  is a consequence of Theorem <ref>. Conversely, let u ∈𝒟'(𝕋^n+1) be a  solution of equation L_ju = f ∈ C^∞(𝕋n+1). By taking the x-Fourier coefficients of u and f we get the O.D.E's	

    ∂_t u(t, ξ) +    iℳ_j(t, ξ) u(t,ξ) = f(t, ξ),   t ∈𝕋,  ξ∈ℤ^n.


Since L_0,j is globally hypoelliptic, then  ℳ_0, j(ξ) satisfies the  Siegel condition (<ref>) and  the set  Z_ℳ_j is finite. When 	ξ∉ Z_ℳ_j, the solutions (<ref>) can be written in the form

    u(t, ξ) = 1/e^ 2 π iℳ_0, j(ξ) - 1∫_0^2πexp(i∫_t^t+sℳ_j(r,  ξ)   dr) f(t+s, ξ)ds.



Recall that by  Theorem <ref>, there are positive constants C_1, N_1 and R_1 such that

    |1-e^2π iℳ_0,j(ξ)|^-1≤ C_1|ξ|^N_1,  for all  |ξ|≥ R_1.



Now, given α∈ℤ_+ and N>0 there are positive constants C_2, C_3  and R_2 such that: |p_j(ξ)|≤ C_2 |ξ|^ν_j (see (<ref>)) and

    ∑_β=0^ααβ |∂_t^α -βf(t, ξ)| ≤ C_3 |ξ|^-N,  ∀ |ξ|≥ R_2.


Hence, we get  

    |∂_t^αu(t, ξ)|    ≤ C_1C_2C_3C_4|ξ|^N_1 - N_2 + ν_j∫_0^2πexp(-∫_t^t+sℳ_j(r,  ξ)   dr) ds 
       ≤ 2 π C_1C_2C_3C_4|ξ|^N_1 - N + ν_j + 2πη_j,

with C_4 depending on α and c_j(t).


Finally, for a fixed α∈ℤ_+ it is possible to obtain for any N>0, positive constants C and R such that

    |∂_t^αu(t, ξ)| ≤ C |ξ|^-N,  t ∈𝕋,  |ξ|≥ R.


It follows from Lemma <ref> that u ∈ C^∞(𝕋^n+1) and, finally, the proof is finished.





	
	We emphasize that condition (<ref>) may be replaced by 
	
    ℳ_j(t, ξ) ≤η_j,  t ∈𝕋,  ∀ξ∈ℤ^n,

	for j ∈{1, … m }, since 
	
    u(t, ξ) = 1/1 - e^-  2 π iℳ_0,j(ξ)∫_0^2πexp(-i∫_t-s^tℳ_j(r,  ξ)   dr) f(t-s, ξ)ds

	is an equivalent expression for (<ref>).  In this case, 
	
    sup_s ∈[0, 2π]|exp(-i∫_t-s^tℳ_j(r,  ξ)   dr) |     =  
    	sup_s ∈[0, 2π]|exp(∫_t-s^tℳ_j(r,  ξ)   dr) |  
       ≤  e^2πη_k.
	







 §.§ A link with the solvability problem


To introduce this section, let us consider the operator L = L_1 ∘ L_2. 
It is already possible to state that L is not globally hypoelliptic when L_2 is not globally hypoelliptic.  Hence, all that is needed is a search for conditions so that  the same conclusion holds true when  L_1 is not globally hypoelliptic.

This inquiry is  equivalent to the following problem: if u is a singular solution of L_1, namely,  u ∈𝒟'(𝕋^n+1) ∖ C^∞(𝕋^n+1) and L_1 u  ∈ C^∞(𝕋^n+1), under which conditions is there u ∈  L_2(𝒟'(𝕋^n+1))? 


Clearly, a solution for this question is connected with solvability  properties and we discuss this subject in the next few paragraphs. The first step consists in introducing the following:

	Let 𝔼_j be the space of distributions v ∈𝒟'(𝕋^n+1) such that v(·, ξ) ∈ C^∞(𝕋), for each ξ∈ℤ^n, and 
	
    ∫_0^2 πexp(i ∫_0^tℳ_j(r,ξ)dr ) v(t, ξ)dt = 0,  if  ξ∈ Z_ℳ_j.







	Let L_j be a globally hypoelliptic operator satisfying  (<ref>). Thus, for each v ∈𝔼_j, there exist u ∈𝒟'(𝕋^n+1) such that L_j u = v.




	By the hypothesis,  L_0,j is globally hypoelliptic, ℳ_0, j(ξ) satisfies  condition (<ref>) and the set  Z_ℳ_j is finite. For ξ∉ Z_ℳ_j it is possible to define
	u(t, ξ) by expression (<ref>),  while  in the case of ξ∈ Z_ℳ_j we set 
	
    u(t, ξ) = exp( -i ∫_0^tℳ_j(r, ξ) dr ) ∫_0^texp(i ∫_0^sℳ_j(r,ξ)dr ) v(s, ξ)ds.

	
	Since v ∈𝔼_j, we obtain
	u(·, ξ) ∈ C^∞(𝕋) and
	
    ∂_t u(t, ξ) +    iℳ_j(t, ξ) u(t,ξ) = v(t, ξ),   t ∈𝕋,  ξ∈ℤ^n.

	
	Now, we shall prove that {u(t, ξ)}_ξ∈ℤ^n defines an element u ∈𝒟'(𝕋^n+1). Firstly, since the set Z_ℳ_j is finite, no estimates for (<ref>) are needed. For the general case, an argument similar to the one in the proof of Theorem <ref>  shows that 
	
    |∂_t^αu(t, ξ)|  ≤ 2 π C_1C_2C_3C_4|ξ|^N_1 - N_2 + ν_j + 2πη_j,

	where C_3 and R_2 now satisfy
	
    ∑_β=0^ααβ |∂_t^α -βv(t, ξ)| ≤ C_3 |ξ|^-N_2,  ∀ |ξ|≥ R_2,

	for some N_2. Then, it is possible to choose positive constants C, N and R such that
	
    |∂_t^αu(t, ξ)| ≤ C |ξ|^-N,  t ∈𝕋,  |ξ|≥ R,

	which implies  u = ∑_ξ∈ℤ^nu(t, ξ) e^i  x ·ξ∈𝒟'(𝕋^n+1). Clearly, L_ju = v and the proof is done.
	



	Admit that each L_j satisfies  (<ref>). If  some L_k has a singular solution u_k satisfying
	
    u_k+j = L_k+j+1 u_k+j+1,  j ∈{0, …, m-k-1 },

	then L is not globally hypoelliptic. 
	



	It is enough to consider the case m=2 and k=1. By hypothesis, we have L_1 u_1 = f ∈ C^∞(𝕋^n+1) with u_1 ∉ C^∞(𝕋^n+1). 
	Since u_1 ∈𝔼_2 we obtain u_2 ∈𝒟'(𝕋^n+1) such that L_2 u_2 = u_1. Clearly u_2 ∉ C^∞(𝕋^n+1) and f = L_1 ∘ L_2 (u_2), hence the proposition is proved.
	











