Partial Fourier series on compact Lie groups
Alexandre Kirilov Wagner Augusto Almeida de Moraes Michael Ruzhansky

	
	Partial Fourier Series on Compact Lie Groups]
	Partial Fourier Series on Compact Lie Groups
	


Alexandre Kirilov]Alexandre Kirilov

	Universidade Federal do Paraná, 
	Departamento de Matemática,
	C.P.19096, CEP 81531-990, Curitiba, Brazil

akirilov@ufpr.br






Wagner de Moraes]Wagner A. A. de Moraes

	Universidade Federal do Paraná,
	Programa de Pós-Graduação de Matemática,
	C.P.19096, CEP 81531-990, Curitiba, Brazil

wagneramat@gmail.com



Michael Ruzhansky]Michael Ruzhansky
Ghent University, 
	Department of Mathematics: Analysis, Logic and Discrete Mathematics, 
	Ghent, Belgium 
	and 
	Queen Mary University of London, 
	School of Mathematical Sciences, 
	London, United Kingdom

Michael.Ruzhansky@ugent.be
	


	
	
	Primary 22E30, 43A77 ; Secondary 58D25, 43A25 

	
	
	
	In this note we investigate the partial Fourier series on a product of two compact Lie groups. We give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. 
	As applications, we will study conditions for the global solvability of an evolution equation defined on  and we will show that some properties of this evolution equation can be obtained from a constant coefficient equation.
	
	
	
	[
    [
    December 30, 2023
=====================



§ INTRODUCTION


The study of global properties for vector fields and systems of vector fields defined on closed manifolds has made a great advance in the last decades, especially in the case where the manifold is a torus or a product of a torus by another closed manifold. See, for example, the impressive list of articles <cit.> and references therein, discussing problems related to the global solvability and global hypoellipticity in the smooth, analytical and ultradifferentiable senses.

One of the main tools in these references is the characterization of the functional environments via the asymptotic behavior of its Fourier coefficients, especially of the partial Fourier coefficients with respect to one or more variables.

For example, in the case of evolution equations, the use of partial Fourier series in the spatial variable can reduce the problem to find solutions to a sequence of ordinary differential equations depending on the time, for which the methods of control of asymptotic behavior are much more developed and understood. This technique is used in 
<cit.> and several others references.

A natural step in the development of the theory is to extend such techniques to the study of global properties of vector fields in compact Lie Groups. For instance, consider the operator P: C^∞ (𝕋^1 ×) → C^∞(𝕋^1×)   defined by

    Pu(t,x) := ∂_tu(t,x) + a(t)Xu(t,x),

where X is a left-invariant vector field on  and a ∈ C^∞(𝕋^1) is a real-valued function. 

Taking the Fourier series with respect to the second variable, we obtain a system of ordinary differential equations on t, which can be easily solved. The analysis of the behavior of these solutions at infinity will allow us to say if the corresponding Fourier partial series converges to a smooth (or distributional) solution of the problem. 

In this work we define precisely the partial Fourier coefficients of functions and distributions, and we obtain the characterization of functions and distributions by its  partial Fourier series on a product of compact Lie groups.

The paper is organized as follows.	In Section 2 we present some classical results about Fourier analysis on compact Lie groups and fixed the notation that will be used henceforth. In Section 3 first we establish the relation of the Fourier analysis of a product of two compact Lie groups with the Fourier analysis of each compact Lie group. Then we define the partial Fourier coefficients and we give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. 
In section 4 we present two applications of the theory developed in the previous section. First we will study the global C^∞-solvability of the operator (<ref>) through its partial Fourier coefficients. Then, we present a conjugation that transform the operator (<ref>) in a constant coefficient operator. 


§ FOURIER ANALYSIS ON COMPACT LIE GROUPS

	
In this section we introduce most of the notations and preliminary results necessary for the development of this work. A very careful presentation of these concepts and the demonstration of all the results presented here can be found in the references <cit.>  and <cit.>.

Let G be a compact Lie group and let (G)  be the set of continuous irreducible unitary representations of G. Since G is compact, every continuous irreducible unitary representation ϕ is finite dimensional and it can be viewed as a matrix-valued function ϕ: G →^d_ϕ× d_ϕ, where d_ϕ = ϕ. We say that ϕ∼ψ if there exists an unitary matrix A∈ C^d_ϕ× d_ϕ such that Aϕ(x) =ψ(x)A, for all x∈ G. We will denote by G the quotient of (G) by this equivalence relation.
	
	For f ∈ L^1(G) we define the group Fourier transform of f at ϕ as
	
    f(ϕ)=∫_G f(x) ϕ(x)^*   dx,

	where dx is the normalized Haar measure on G.
	By the Peter-Weyl theorem, we have that 
	
    ℬ := {√(ϕ) ϕ_ij ;  ϕ=(ϕ_ij)_i,j=1^d_ϕ, [ϕ] ∈G},

	is an orthonormal basis for L^2(G), where we pick only one matrix unitary representation in each class of equivalence, and we may write
	
    f(x)=∑_[ϕ]∈Gd_ϕ(ϕ(x)f(ϕ)).

	Moreover, the Plancherel formula holds:
	
    f_L^2(G)=(∑_[ϕ] ∈G  d_ϕ f(ϕ)_^2)^1/2=:
    	f_ℓ^2(G),

	where 
	
    f(ϕ)_^2=(f(ϕ)f(ϕ)^*)=∑_i,j=1^d_ϕ|f(ϕ)_ij|^2.

	Let ℒ_G be the Laplace-Beltrami operator of G. For each [ϕ] ∈G, its matrix elements are eigenfunctions of ℒ_G correspondent to the same eigenvalue that we will denote by -ν_[ϕ], where ν_[ϕ]≥ 0. Thus
	
    -ℒ_G ϕ_ij(x) = ν_[ϕ]ϕ_ij(x),   for all  1 ≤ i,j ≤ d_ϕ,

	and we will denote by
	
    ϕ := (1+ν_[ϕ])^1/2

	the eigenvalues of (I-ℒ_G)^1/2. We have the following estimate for the dimension of ϕ (Proposition 10.3.19 of <cit.>): there exists C>0 such that for all [ξ] ∈G it holds

    d_ϕ≤ C ϕ^ G/2.

	
		For x∈ G, X∈𝔤 and f∈ C^∞(G), define
	
    L_Xf(x):=d/dt f(xexp(tX))|_t=0.


Notice that the operator L_X is left-invariant. Indeed,

    π_L(y)L_Xf(x) = L_Xf(y^-1x)    = d/dt f(y^-1xexp(tX))|_t=0
       = d/dtπ_L(y)f(xexp(tX))|_t=0
       = L_Xπ_L(y)f(x),

for all x,y∈ G.

When there is no possibility of ambiguous meaning, we will write only Xf instead of L_Xf. 

	Let G be a compact Lie group of dimension n and let {X_i}_i=1^n be a basis of its Lie algebra. For a multi-index α =(α_1,⋯α_n)∈_0^n, we define the left-invariant differential operator of order |α|
	
    ∂^α := Y_1 ⋯ Y_|α|,

	with Y_j ∈{X_i }_i=1^n, 1 ≤ j ≤ |α| and ∑_j:Y_j=X_k1=α_k for every 1≤ k ≤ n. It means that ∂^α is a composition of left-invariant derivatives with respect to vectors X_1, …, X_n such that each X_k enters ∂^α exactly α_k times. We do not specify in the notation ∂^α the order of vectors X_1,…,X_n, but this will not be relevant in the arguments that we will use in this work. 
	
	We equipped C^∞(G) with the usual Fréchet space topology defined by seminorms p_α(f) = max_x ∈ G |∂^α f(x)|. Thus, the convergence on C^∞(G) is just the uniform convergence of functions and all their derivatives: f_k → f in C^∞(G) if ∂^α f_k(x) →∂^α f(x), for all x ∈ G, due to the compactness of G.	We define the space of distributions 𝒟'(G) as the space of all continuous linear functional on C^∞(G).	
	For u ∈, we define the distribution ∂^α u as
	
    ∂^α u, ψ := (-1)^|α|u,∂^αψ,

	for all ψ∈ C^∞(G).
	
	Let α∈_0^n. The symbol of ∂^α at x ∈ G and  ξ∈ (G) is
	
    σ_∂^α(x, ϕ) := ϕ(x)^*(∂^αϕ)(x) ∈^d_ϕ× d_ϕ.

	Notice that the symbol of ∂^α does not depends of x∈ G because ∂^α is a left-invariant operator. 
	There exists C_0>0 such that 
	
    σ_∂^α(ϕ)_op≤ C_0^|α|ϕ^|α|,

	for all α∈_0^n and [ϕ] ∈G (see <cit.>, Proposition 3.4), where ·_op stands for the usual operator norm and, from Chapter 2 of <cit.>, we have

    σ_∂^α(ϕ)_op≤σ_∂^α(ϕ)_≤√(d_ϕ)σ_∂^α(ϕ)_op.


For every integer M ≥ G/2 there exists C_M>0 such that 
	
    ϕ_ij_L^∞(G)≤ C_M ϕ^M,

	for all [ϕ] ∈G, 1≤ i,j ≤ d_ϕ (see <cit.>, Lemma 3.3).
	
	Smooth functions on G can be characterized in terms of
	their Fourier coefficients comparing with powers of the eigenvalues of (I-ℒ_G)^1/2. Precisely, the following statements are equivalent:
	 
		
  * f ∈ C^∞(G);
		
  * for each N>0, there exists C_N > 0 such that
		
    f(ϕ)_≤ C_N ϕ^-N,

		for all [ϕ] ∈G;
		
  * for each N>0, there exists C_N > 0 such that
		
    |f(ϕ)_ij| ≤ C_N ϕ^-N,

		for all [ϕ] ∈G,  1≤ i,j ≤ d_ϕ.
	

By duality, we have a similar result for distributions on G, 	where we define 
    u(ϕ)_ij:=u(ϕ_ji).

This definition agrees with (<ref>) when the distribution comes from a L^1(G) function. The following statements are equivalent:
	
		
  * u ∈;
		
  * there exist C,   N > 0 such that
		
    u(ϕ)_≤ C ϕ^N,

		for all [ϕ] ∈G;
		
  * there exist C,   N > 0 such that
		
    |u(ϕ)_ij| ≤ C ϕ^N,

		for all [ϕ] ∈G,   1≤ i,j ≤ d_ϕ.
	




§ PARTIAL FOURIER SERIES


Let G_1 and G_2 be compact Lie groups, and set G=G_1× G_2.
Let  ξ∈G_1(V_1) and η∈G_2(V_2). The  external tensor product representation ξ⊗η of G on V_1⊗ V_2 is defined by

    [                    ξ⊗η:               G_1 × G_2                       →             (V_1 ⊗ V_2)                        ;                                       (x_1,x_2)                       ↦         (ξ⊗η)(x_1,x_2):               V_1 ⊗ V_2                       →               V_1 ⊗ V_2;                                       (v_1,v_2)                       ↦ ξ(x_1)(v_1)⊗η(x_2)(v_2) ]



We point out that the external tensor product of unitary representation is also unitary. Moreover, if ξ∈G(d_ξ) and η∈G(d_η) are matrix unitary representations, then ξ⊗η∈G(d_ξ d_η) is also a matrix unitary representation and 

    ξ⊗η(x_1,x_2) = ξ(x_1) ⊗η(x_2) ∈^d_ξ d_η× d_ξ d_η,

where  ξ(x_1) ⊗η(x_2) is the Kronecker product of these matrices, that is,

    ξ(x_1) ⊗η(x_2) = ([      ξ(x_1)_11η(x_2)                    ⋯    ξ(x_1)_1d_ξη(x_2);                    ⋮                    ⋱                    ⋮;    ξ(x_1)_d_ξ1η(x_2)                    ⋯ ξ(x_1)_d_ξ d_ξη(x_2) ]).


It is enough to study continuous irreducible unitary representations of G_1 and G_2 to obtain the elements of G, since for every [ϕ] ∈G, there exist [ξ] ∈G_1 and [η] ∈G_2 such that
	ϕ∼ξ⊗η,
	that is, [ϕ] = [ξ⊗η] ∈G  and d_ϕ = d_ξ· d_η. Moreover,  [ξ_1 ⊗η_1]= [ξ_2 ⊗η_2] if and only if [ξ_1] = [ξ_2] and [η_1] = [η_2].
	The proof of this fact can be found on <cit.> (Chapter II, Proposition 4.14). Therefore, the map [ξ⊗η] ↦ ([ξ],[η]) is a bijection from G to G_1×G_2.


It is easy to see that ℒ_G = ℒ_G_1 + ℒ_G_2, so ν_[ξ⊗η] = ν_[ξ] + ν_[η]. Therefore we have

    1/2 (ξ + η) ≤ξ⊗η≤ξ + η,

	for all [ξ] ∈G_1 and [η] ∈G_2.


Let f ∈ L^1(G) and [ϕ] ∈G. Let [ξ] ∈G_1 and [η] ∈G_2 such that [ϕ] = [ξ⊗η]. Notice that

    f(ξ⊗η)    = ∫_G f(x) (ξ⊗η)(x)^*   dx 
       = ∫_G_2∫_G_1f(x_1,x_2)(ξ(x_1)⊗η(x_2))^*   dx_1dx_2 
       = ∫_G_2∫_G_1f(x_1,x_2)ξ(x_1)^*⊗η(x_2)^*   dx_1dx_2 .


Thus f(ξ⊗η) ∈^d_ξ d_η× d_ξ d_η with elements

    f(ξ⊗η)_ij   =  ∫_G_2∫_G_1f(x_1,x_2)(ξ(x_1)^*⊗η(x_2)^*)_ij  dx_1dx_2 
       =∫_G_2∫_G_1 f(x_1,x_2)ξ(x_1)_nm η(x_2)_sr  dx_1 dx_2

where 1 ≤ m,n ≤ d_ξ, 1 ≤ r,s ≤ d_η are given by

    [               m               = ⌊i-1/d_η⌋  +1 ,;               n               = ⌊j-1/d_η⌋  +1 , ]    [                r                = i-⌊i-1/d_η⌋ d_η,;                s                = j-⌊j-1/d_η⌋ d_η. ]


Similarly for u ∈, we have

    u(ξ⊗η)_ij = u, (ξ⊗η)_ji = u, ξ_nm×η_sr,

where (ξ_nm×η_sr)(x_1,x_2):= ξ(x_1)_nmη(x_2)_sr.

		Let G_1 and G_2 be compact Lie groups and, set G=G_1× G_2. Let f ∈ L^1(G), ξ∈(G_1), and x_2 ∈ G_2. The  ξ-partial Fourier coefficient of f at x_2 is defined by 
	
    f(ξ, x_2) = ∫_G_1 f(x_1,x_2) ξ(x_1)^*   dx_1 ∈^d_ξ× d_ξ,

	with components
	
    f(ξ, x_2)_mn = ∫_G_1 f(x_1,x_2) ξ(x_1)_nm  dx_1,    1 ≤ m,n≤ d_ξ.

	Similarly, for η∈(G_2) and x_1 ∈ G_1, we define the  η-partial Fourier coefficient of f at x_1 as
	
    f(x_1,η) = ∫_G_2 f(x_1,x_2) η(x_2)^*   dx_2 ∈^d_η× d_η,

	with components
	
    f(x_1, η)_rs = ∫_G_2 f(x_1,x_2) η(x_2)_sr  dx_1,    1 ≤ r,s≤ d_η.



By the definition, the function
	
    [ f(ξ,  ·  )_mn:            G_2              ⟶               ;                           x_2              ⟼   f(ξ, x_2)_mn ]

	belongs to L^1(G_2) for all ξ∈(G_1), 1 ≤ m,n≤ d_ξ. Similarly, the function

    [ f( ·  , η )_rs:             G_1               ⟶                ;                             x_1               ⟼     f(x_1,η)_rs ]

	belongs to L^1(G_1) for all η∈(G_2), 1 ≤ r,s≤ d_η. 

Let ξ∈(G_1) and η∈(G_2). Since f(ξ, ·  )_mn∈ L^1(G_2) for all 1≤ m,n ≤ d_ξ, we can take its η-coefficient of Fourier:

    f (ξ,η)_mn:= ∫_G_2f(ξ, x_2)_mnη(x_2)^*   dx_2 ∈^d_η× d_η

with components

    f (ξ,η)_mn_rs   = ∫_G_2f(ξ, x_2)_mnη(x_2)_sr  dx_2 
       = ∫_G_2∫_G_1 f(x_1,x_2)ξ(x_1)_nm η(x_2)_sr  dx_1 dx_2,

for 1 ≤ r,s ≤ d_η. 
Similarly, since f( · ,η)_rs∈ L^1(G_1) for all 1≤ r,s ≤ d_η, we can take its ξ-coefficient of Fourier:

    f (ξ,η)_rs:= ∫_G_1f(x_1,η)_rsξ(x_1)^*   dx_1 ∈^d_ξ× d_ξ

with components

    f (ξ,η)_rs_mn   = ∫_G_1f(x_1, η)_rsξ(x_1)_nm  dx_1 
       = ∫_G_1∫_G_2 f(x_1,x_2)ξ(x_1)_nm η(x_2)_sr  dx_2 dx_1,

for 1 ≤ m,n ≤ d_ξ. 

Notice that

    f (ξ,η)_mn_rs =  f (ξ,η)_rs_mn =  f(ξ⊗η)_ij,

with

    i = d_η(m-1)+ r ,     
    j =  d_η(n-1) + s ,

for all 1≤ m,n ≤ d_ξ and 1 ≤ r,s ≤ d_η.

	Let G_1 and G_2 be compact Lie groups, and set G=G_1× G_2. Let u ∈, ξ∈(G_1) and 1≤ m,n ≤ d_ξ. The mn-component of  the ξ-partial Fourier coefficient of u is the linear functional defined by
	
    [                u(ξ,  ·  )_mn:                      C^∞(G_2)                             ⟶                              ;                                                           ψ                             ⟼ u(ξ,  · )_mn,ψ := u,ξ_nm×ψ_G. ]

	In a similar way, for η∈(G_2) and 1 ≤ r,s≤ d_η, we define the rs-component of the η-partial Fourier coefficient of u as
	
    [                u(  · ,η )_rs:                      C^∞(G_1)                             ⟶                              ;                                                           φ                             ⟼ u(  · ,η)_rs,φ := u,φ×η_sr_G. ]



		By definition, u(ξ,  ·  )_mn∈𝒟'(G_2) and u(  · ,η )_rs∈𝒟'(G_1) for all ξ∈(G_1), η∈(G_2), 1 ≤ m,n≤ d_ξ and 1 ≤ r,s≤ d_η.


Let ξ∈(G_1) and η∈(G_2). Since u(ξ, ·  )_mn∈𝒟'(G_2) for all 1≤ m,n ≤ d_ξ, we can take its η-coefficient of Fourier:

    u (ξ,η)_mn:= u(ξ, ·  )_mn,η^*∈^d_η× d_η

with components

    u (ξ,η)_mn_rs = u(ξ, ·  )_mn,η_sr = u,ξ_nm×η_sr_G = u,ξ_nm×η_sr_G,

for all 1≤ r,s ≤ d_η. Now, since u(  · ,η )_rs∈𝒟'(G_1) for all 1 ≤ r,s≤ d_η we can take its ξ-coefficient of Fourier:

    u (ξ,η)_rs:= u(  · ,η )_rs,ξ^*∈^d_ξ× d_ξ

with components

    u (ξ,η)_rs_mn = u(  · ,η )_rs,ξ_mn = u,ξ_nm×η_sr_G = u,ξ_nm×η_sr_G,

for all 1≤ m,n ≤ d_ξ. Notice that

    u (ξ,η)_mn_rs =  u (ξ,η)_rs_mn =  u(ξ⊗η)_ij,

with

    i = d_η(m-1)+ r ,    
    j  =  d_η(n-1) + s ,

for all 1≤ m,n ≤ d_ξ and 1 ≤ r,s ≤ d_η.

Notice that 

    u(ξ⊗η)_^2 = ∑_i,j=1^d_ξ d_η| u(ξ⊗η)_ij|^2 = ∑_m,n=1^d_ξ∑_r,s=1^d_η|  u (ξ,η)_mn_rs|^2 =:  u (ξ,η)_^2,

for all [ξ] ∈G_1 and [η] ∈G_2 whenever u ∈ L^1(G) or u∈.

It follows from the (<ref>) and (<ref>) that we have the following characterization of smooth functions and distributions using the method of taking the partial Fourier coefficients twice that was discussed above.
 

	Let G_1 and G_2 be compact Lie groups, and set G=G_1× G_2 . The following statements are equivalent:
	
		
  * f ∈ C^∞(G);
		
  * For every N>0, there exists C_N>0 such that
	
    f (ξ,η)_≤ C_N(ξ+η)^-N,

	for all [ξ] ∈G_1 and [η] ∈G_2;
	
  * For every N>0, there exists C_N>0 such that
	
    | f (ξ,η)_mn_rs| ≤ C_N (ξ+η)^-N,

	for all [ξ] ∈G_1, [η] ∈G_2, 1 ≤ m,n ≤ d_ξ,  and 1 ≤ r,s ≤ d_η.




	Let G_1 and G_2 be compact Lie groups, and  set G=G_1× G_2. The following statements are equivalent:
	
		
  * u ∈;
		
  * There exist C,   N >0 such that
	
    u (ξ,η)_≤ C(ξ+η)^N,

	 for all [ξ] ∈G_1 and [η] ∈G_2;
	
  * There exist C,   N >0 such that
	
    | u (ξ,η)_mn_rs| ≤ C (ξ+η)^N,

	for all [ξ] ∈G_1, [η] ∈G_2, 1 ≤ m,n ≤ d_ξ, and 1 ≤ r,s ≤ d_η.
	


In the next results we will investigate when a sequence of partial Fourier coefficients can define a smooth function or a distribution.


	Let G_1 and G_2 be compact Lie groups, G=G_1× G_2, and let {f( · , η)_rs} be a sequence of functions on G_1. Define
	
    f(x_1,x_2) := ∑_[η]∈G_2 d_η∑_r,s=1^d_ηf(x_1, η)_rsη_sr(x_2).
 
	Then f ∈ C^∞(G) if and only if f( · ,η)_rs∈ C^∞(G_1), for all [η]∈G_2, 1 ≤ r,s ≤ d_η and for every β∈_0^n and ℓ >0 there exist C_βℓ > 0 such that
	
    |∂^βf(x_1,η)_rs| ≤ C_βℓ⟨η⟩ ^-ℓ,   ∀ x_1∈ G_1,  [η]∈G,  1≤ r,s≤ d_η.



() It is sufficient to consider N∈ in Theorem <ref> to conclude that f∈ C^∞(G). Notice that

    ℒ_G_1g(ξ)_mn = ℒ_G_1g,ξ_nm = g,ℒ_G_1ξ_nm = -ν_[ξ]g,ξ_nm = -ν_[ξ]g(ξ)_mn,

for all g ∈ C^∞(G_1), [ξ] ∈G_1, and 1 ≤ m,n ≤ d_ξ. In particular, for N∈ we obtain

    ν_[ξ]^N | f (ξ,η)_rs_mn|    = |ℒ_G_1^Nf(ξ,η)_rs_mn|
       = |∫_G_1ℒ_G_1^Nf(x_1,η)_rsξ(x_1)_nm  dx_1 | 
       ≤∫_G_1  |ℒ_G_1^Nf(x_1,η)_rs| |ξ(x_1)_nm|  dx_1
       ≤(∫_G_1  |ℒ_G_1^Nf(x_1,η)_rs|^2  dx_1)^1/2(∫_G |ξ(x_1)_nm|^2  dx_1 )^1/2
       ≤1/√(d_ξ)∑_|β|=2Nmax_x_1∈ G_1 | ∂^βf(x_1,η)_rs|  
       ≤∑_|β|=2Nmax_x_1∈ G_1 | ∂^βf(x_1,η)_rs|.


Notice that there exists C>0 such that ξ≤ C ν_[ξ] for all non-trivial [ξ] ∈G_1. Thus for all ℓ=N we have 

    | f (ξ,η)_rs_mn| ≤ C_Nξ^-Nη^-N≤ C_N2^N (ξ + η)^-N.

Therefore f ∈ C^∞(G).

() Let E_2:=(I-ℒ_G_2)^1/2. Since f∈ C^∞(G), for all β∈ N_0^n and N ∈_0 we have ∂^β E_2^Nf ∈ C^∞(G) and then, by the compactness of G, there exists C_β N≥0 such that

    |∂^β E_2^N f(x_1,x_2)| ≤ C_β N,     ∀ (x_1,x_2) ∈ G_1 × G_2.

Fix η∈(G_2), 1≤ r,s ≤ d_η. We already know that f( · ,η)_rs∈ C^∞(G_1). Moreover


    |η^N∂^βf(x_1,η)_rs|    = |∂^βE_2^Nf(x_1,η)_rs| 
       = |∂^β∫_G_2 E_2^Nf(x_1,x_2) η(x_2)_sr  dx_2| 
       ≤∫_G_2|∂^β E_2^Nf(x_1,x_2)| |η(x_2)_sr|   dx_2 
       ≤(∫_G_2 |∂^β E_2^Nf(x_1,x_2)|^2   dx_2)^1/2(∫_G_2 |η(x_2)_sr|^2   dx_2 )^1/2
       (<ref>)≤1√(d_η)C_β N.

Therefore,

    |∂^βf(x_1,η)_rs| ≤ C_β Nη^-N,

for all x_1 ∈ G_1, [η] ∈G_2, 1≤ r,s≤ d_η.



		Let G_1 and G_2 be compact Lie groups, set G=G_1× G_2, and let {u( · ,η)_rs} be a sequence of distributions on G_1.  Define 
    u=∑_[η]∈G d_η∑_r,s=1^d_ηu( · ,η)_rsη_sr.
 
	Then u ∈ if and only if there exist K ∈ and C>0 such that
	
    | u(·,η)_rs,φ| ≤ C   p_K(φ) ⟨η⟩ ^K,
 
	for all φ∈ C^∞(G_1) and [η]∈G, where p_K(φ) := ∑_|β| ≤ K∂^βφ_L^∞(G_1).



(⟸) Take φ=ξ_nm, [ξ]∈G_1, 1≤ m,n ≤ d_ξ. Let β∈_0^n, |β| ≤ K, with K as in (<ref>). Since the symbol of ∂^β at x_1∈ G_1 and ξ∈(G_1) is given by

    σ_∂^β(x_1,ξ) = ξ(x_1)^*(∂^βξ)(x_1),

we have

    |∂^βξ_nm(x_1)|    = | ∑_i=1^d_ξξ_ni(x) σ_∂^β(ξ)_im| 
       ≤∑_i=1^d_ξ |ξ_ni(x)| |σ_∂^β(ξ)_im| 
       ≤(  ∑_i=1^d_ξ |ξ_ni(x)^2| )^1/2(∑_i=1^d_ξ|σ_∂^β(ξ)_im|^2  )^1/2

Let M∈ satisfying M ≥ G_1/2. By (<ref>) we have

    (∑_i=1^d_ξ |ξ_ni(x)^2|)^1/2≤(∑_i=1^d_ξξ_ni_L^∞(G_1)^2)^1/2≤ C_M√(d_ξ)ξ^M

and by (<ref>) there exists C>0 such that

    d_ξ≤ Cξ^M.

Moreover, notice that

    (∑_i=1^d_ξ|σ_∂^β(ξ)_im|^2  )^1/2≤σ_∂^β(ξ)_≤√(d_ξ)σ_∂^β(ξ)_op≤√(d_ξ)C_0^|β|ξ^|β|,

where the last inequalities come from (<ref>) and (<ref>). Hence

    |∂^βξ_nm(x_1)|    ≤ C ξ^M√(d_ξ)σ_∂^β(ξ)_
       ≤ C ξ^Md_ξσ_∂^β(ξ)_op
       ≤ CC_0^|β|ξ^2M+|β|.

Then

    p_K(ξ_nm) = p_K(ξ_nm) ≤ C ξ^2M+K.

Hence

    |  u (ξ,η)_rs_mn| = | u(·,η)_rs,ξ_nm|    ≤ C   p_K(ξ_nm) ⟨η⟩ ^K
       ≤ C  ξ^2M+K⟨η⟩^2M+K
       ≤  C (⟨ξ⟩ + ⟨η⟩)^2(2M+K).

Therefore u ∈𝒟'(G).

(⟹) Since u ∈𝒟'(G), then there exist C>0 and K∈ such that

    | u (ξ,η)_rs_mn| ≤ C (⟨ξ⟩ + ⟨η⟩)^K,

for all [ξ] ∈G_1, [η] ∈G_2,   1 ≤ r,s≤ d_η, and 1≤ m,n≤ d_ξ
and 

    u=∑_[ξ]∈G_1∑_[η] ∈G d_ξ d_η∑_m,n=1^d_ξ∑_r,s=1^d_η u (ξ,η)_rs_mnξ_nmη_sr.








For φ∈ C^∞(G_1) we have

    |u(·,η)_rs,φ|    = |(u,φ×η_sr)| 
       = |  ∑_[ξ]∈G_1∑_[η] ∈G d_ξ d_η∑_m,n=1^d_ξ∑_k,ℓ=1^d_η u (ξ,η)_kℓ_mnξ_nm,φ_G_1η_ℓ k, η_sr_G_2|.


Notice that η_ℓ k, η_sr_G_2 = 1/d_ηδ_ℓ sδ_kr, since the set ℬ is orthonormal (see (<ref>)).
 Moreover, φ(ξ)_mn =ξ_nm,φ_G_1. So


    |u(·,η)_rs,φ|    =|∑_[ξ]∈G_1d_ξ∑_m,n=1^d_ξ u (ξ,η)_rs_mnφ(ξ)_mn| 
       ≤∑_[ξ]∈G_1d_ξ∑_m,n=1^d_ξ| u (ξ,η)_rs_mn||φ(ξ)_mn| 
       ≤ C∑_[ξ]∈G_1d_ξ∑_m,n=1^d_ξ (ξ+η)^K|φ(ξ)_mn|,

where the last inequality comes from (<ref>). Notice that for all K∈ we have (ξ+η)^K ≤ 2^K ξ^K η^K. In addition, we have

    ∑_m,n=1^d_ξ|φ(ξ)_mn| ≤(d_ξ^2∑_m,n=1^d_ξ|φ(ξ)_mn|^2)^1/2 = d_ξφ(ξ)_.

Since ξ = ξ and the summation is over all G_1, we have

    ∑_[ξ]∈G_1 d_ξ⟨ξ⟩^K∑_m,n=1^d_ξ|φ(ξ)_mn|    =  ∑_[ξ]∈G_1 d_ξ⟨ξ⟩^K∑_m,n=1^d_ξ|φ(ξ)_mn| 
       =  ∑_[ξ]∈G_1 d_ξ⟨ξ⟩^K∑_m,n=1^d_ξ|φ(ξ)_mn|.

 Thus

    |u(·,η)_rs,φ| 
       ≤  Cη^K ∑_[ξ]∈G_1 d_ξ⟨ξ⟩^K∑_m,n=1^d_ξ|φ(ξ)_mn|
       ≤  Cη^K ∑_[ξ]∈G_1 d_ξ^2 ⟨ξ⟩^Kφ(ξ)_.

The series ∑_[ξ]∈G_1 d_ξ^2 ⟨ξ⟩^-2t 
converges if and only if t >  G_1/2 (Lemma 3.1 of <cit.>), which implies that there exists C>0 such that d_ξ≤ C ξ^ G_1, for all [ξ] ∈G_1. Hence,


    |u(·,η)_rs,φ|   = C η^K ∑_[ξ]∈G_1(d_ξ⟨ξ⟩^- G_1) (d_ξξ^K+ G_1φ(ξ)_) 
       ≤  C η^K (∑_[ξ]∈G_1 d_ξ^2 ⟨ξ⟩^-2 G_1)^1/2(∑_[ξ]∈G_1 d_ξ^2ξ^2(K+ G_1)φ(ξ)_^2)^1/2
       ≤ C η^K(∑_[ξ]∈G_1 d_ξξ^2(K+2 G_1)φ(ξ)_^2)^1/2.


Let L ∈_0 such that K+2 G_1 ≤ 2L. So

    |u(·,η)_rs,φ|   ≤ C η^K(∑_[ξ]∈G_1 d_ξξ^4Lφ(ξ)_^2)^1/2
       = C η^K(∑_[ξ]∈G_1 d_ξE^2L_1φ(ξ)_^2)^1/2
        =   Cη^K E^2L_1φ_L^2(G_1)
       ≤  C E^2L_1φ_L^2(G_1)η^2L,

where E_1 = (I-ℒ_G_1)^1/2, and the last equality comes from the Plancherel formula (<ref>). Notice that

    E^2L_1φ_L^2(G_1)≤E^2L_1φ_L^∞(G_1) = (I-ℒ_G_1)^Lφ_L^∞(G_1)≤ Cp_2L(ϕ).

Therefore,

    |u(·,η)_rs,φ| ≤ Cp_2L(ϕ)η^2L.





§ EXAMPLE: GLOBAL SOLVABILITY OF A VECTOR FIELD ON 𝕋^1 ×

The 3-sphere  is a Lie group with respect to the quaternionic 
product of ℝ^4, and it is globally diffeomorphic and isomorphic to the
group (2) of unitary 2× 2 matrices of determinant one,
with the usual matrix product.

Let a(t)∈ℝ for t∈𝕋^1. Let X be a normalized left-invariant vector field on .
We consider the operator

    L=∂_t+a(t)X.

We are interested in solvability properties for the vector field L on 𝕋^1×.
Let  be the unitary dual of . It consists of the
equivalence classes [^ℓ] of the continuous irreducible unitary representations
^ℓ:→^(2ℓ+1)× (2ℓ+1), ℓ∈1/2_0,
of matrix-valued functions satisfying
^ℓ(xy)=^ℓ(x)^ℓ(y) and ^ℓ(x)^*=^ℓ(x)^-1 for all
x,y∈.
We will use the standard convention of enumerating the matrix elements
^ℓ_mn of ^ℓ using indices m,n ranging between
-ℓ to ℓ with step one, i.e. we have -ℓ≤ m,n≤ℓ
with ℓ-m, ℓ-n∈_0.

For a function
f∈ C^∞() we can define its Fourier coefficient at ℓ∈1/2_0 by

    f(ℓ):=∫_ f(x) ^ℓ(x)^*dx∈,

where the integral is (always) taken with respect to the Haar measure on ,
and with a natural extension to distributions.
The Fourier series becomes

    f(x)=∑_ℓ∈1/2_0 (2ℓ+1) ^ℓ(x)f(ℓ),

with the Plancherel's identity taking the form

    f_L^2()=∑_ℓ∈1/2_0 (2ℓ+1) 
    	f(ℓ)_^2^1/2=:
    f_ℓ^2(),

which we take as the definition of the norm on the Hilbert space
ℓ^2(), and where 
f(ℓ)_^2=(f(ℓ)f(ℓ^*)) is the
Hilbert–Schmidt norm of the matrix f(ℓ).

Smooth functions and distributions on  can be characterized in terms of
their Fourier coefficients. Thus, we have 

    f∈ C^∞()⟺∀ N  ∃ C_N such that f(ℓ)_≤ C_N (1+ℓ)^-N, ∀ℓ∈1/2_0.
 
Also, for distributions, we have

    u∈𝒟'()
    ⟺∃ M  ∃ C  such that u(ℓ)_≤ C(1+ℓ)^M, ∀ℓ∈1/2_0.


Given an operator T:C^∞()→ C^∞()
(or even T:C^∞()→𝒟'()), we define its matrix symbol by

    σ_T(x,ℓ):=^ℓ(x)^* (T ^ℓ)(x)∈,
 where
T ^ℓ means that we apply T to the matrix components of ^ℓ(x).
In this case we can prove that 

    Tf(x)=∑_ℓ∈1/2_0 (2ℓ+1) (^ℓ(x)σ_T(x,ℓ)f(ℓ)).

The correspondence between operators and symbols is one-to-one, and
we will write T_σ for the operator given by 
(<ref>) corresponding to the symbol σ(x,ℓ).
The quantization (<ref>) has been extensively studied in
<cit.>, to which we
refer for its properties and for the corresponding symbolic calculus.

Using rotation on , without loss of generality we may assume that the vector
field X has the symbol 

    σ_X(ℓ)_mn=imδ_mn,
     -ℓ≤ m,n≤ℓ,
 
with δ_mn standing for the Kronecker's delta.
Consequently, taking the Fourier transform of
Lu=∂_tu+a(t)Xu with respect to x, we get 

    Lu(t,ℓ)= ∂_tu(t,ℓ)+ia(t)Xu(t,ℓ).

Writing this in terms of matrix coefficients, we obtain

    Lu(t,ℓ)_mn= ∂_tu(t,ℓ)_mn+ia(t) m
    u(t,ℓ)_mn,    
    -ℓ≤ m,n≤ℓ.

Consequently, the equation Lu=f, where f ∈ C^∞(𝕋^1×) is reduced to

    ∂_tu(t,ℓ)_mn+ia(t) mu(t,ℓ)_mn=f(t,ℓ)_mn,  ℓ∈1/2_0, 
    -ℓ≤ m,n≤ℓ.


We say that an operator P is global hypoelliptic if the conditions Pu ∈ C^∞(𝕋^1×) and u ∈𝒟'(𝕋^1×) imply u ∈ C^∞(𝕋^1×). 

We point out that the operator L defined in (<ref>) is not globally hypoelliptic for any a∈ C^∞(𝕋^1). Indeed, define u by its partial Fourier coefficients:

    u(t,ℓ)_mn = {[           1, ℓ∈_0  m=n=0;;           0,            . ].

Notice that by Theorem <ref> we have u ∉ C^∞() and by Theorem <ref> we have u ∈𝒟'(). Moreover, the function defined by (<ref>) is a homogeneous solution of (<ref>) for all ℓ∈1/2_0, -ℓ≤ m,n ≤ℓ. Thus Lu=0 ∈ C^∞() which implies that L is not globally hypoelliptic.

Let us turn our attention now to study the solvability of the operator L. Denote by 
    a_0= 1/2π∫_0^2π a(t)  dt
 and define

    A(t)=∫_0^t a(s)   ds - a_0t.


Put

    v(t,ℓ)_mn:= e^imA(t)u(t,ℓ)_mn.

In this way v(t,ℓ)_mn satisfies the equation

    ∂_tv(t,ℓ)_mn+ima_0v(t,ℓ)_mn=g(t,ℓ)_mn:= e^imA(t)f(t,ℓ)_mn.


	Let λ∈ and consider the equation
	
    d/dtu(t)+λ u(t)=f(t),

	where f ∈ C^∞(𝕋^1).
	
	If λ∉ i then the equation (<ref>) has a unique solution that can be expressed by
	
    u(t)=1/1-e^-2πλ∫_0^2π e^-λ s f(t-s)  ds,

	or equivalently,
	
    u(t)=1/e^2πλ-1∫_0^2π e^λ r f(t+r)  dr.

	
	If λ∈ i and ∫_0^2π e^λ sf(s)  ds=0 then we have that
	
    u(t)=e^-λ t∫_0^t e^λ sf(s)   ds

	is a solution of the equation (<ref>).

To prove this lemma observe that E=(1-e^-2πλ)^-1e^λ t is the fundamental solution of the operator d/dt+λ, when λ∉ i. The equivalence between (<ref>) and (<ref>) follows from the change of variable s ↦ -r+2π. We point out that the solution (<ref>) is not unique.

By Lemma <ref>, if ma_0 ∉  the equation (<ref>) has a unique solution that can be written as

    v(t,ℓ)_mn = 1/1-e^-2π ima_0∫_0^2π e^-ima_0 s g(t-s,ℓ)_mn  ds

and then

    u(t,ℓ)_mn = 1/1-e^-2π ima_0∫_0^2π e^-imH(t,s)f(t-s,ℓ)_mn  ds,

where H(t,s) := ∫_t-s^t a(θ)   dθ.

If ma_0 ∈, a solution for the equation (<ref>) can be expressed as

    v(t,ℓ)_mn = e^-ima_0t∫_0^t e^ima_0 s g(s,ℓ)_mn  ds

and then

    u(t,ℓ)_mn = e^-imH(t,t)∫_0^t e^imH(s,s)f(s,ℓ)_mn  ds.

Here, one can see a condition on f for the existence of a solution u(t,ℓ)_mn on 𝕋^1. If m a_0 ∈, then

    ∫_0^2π e^imH(t,t)f(t,ℓ)_mn  dt=0.

This condition appears in Lemma <ref> in order to guarantee that the solution is well-defined in 𝕋^1.

So, if f ∈ C^∞() does not satisfy the condition above, the equation (<ref>) has no solution on 𝕋^1. We will denote by 𝒦 the set of smooth functions on  that satisfy the condition (<ref>).


	We say that the operator L is globally C^∞-solvable if for any f ∈𝒦, there exists u ∈ C^∞() such that Lu=f.



	In the literature it is common to define the global C^∞-solvability for f in the set
	
    {h∈ C^∞(); w,h=0;  ∀ w ∈  ^tL },

	(e.g. <cit.>, <cit.>). For the operator L that we are studying in this work, this set coincides with the set 𝒦 defined previously.

Let us investigate if the operator L defined in (<ref>) is globally C^∞-solvable. First, let us prove a technical result about derivatives of the exponential function.

	For any α∈_0, there exists C_α such that
	
    |∂^α_t e^imH(t,t)| ≤ C_α |m|^α,

	for all t ∈𝕋^1.


	By Faà di Bruno's Formula, we have
	
    ∂_t^α e^imH(t,t) = ∑_γ∈Δ(α)α! /γ! (im)^|γ| e^imH(t,t)∏_j=1^α(∂_t^j H(t,t)/j!)^γ_j,

	where Δ(α) = {γ∈_0^α;  ∑_j=1^α jγ_j= α}.
	
	Notice that H(t,t)=∫_0^t a(θ)   dθ, so ∂^j_tH(t,t) = ∂_t^j-1 a(t), for all j≥ 1. Hence
	
    |∂^α_t e^imH(t,t)| ≤∑_γ∈Δ(α)α! /γ! |m|^|γ|∏_j=1^γ(|∂_t^j H(t,t)|/j!)^γ_j≤ C_α |m|^α.




Assume that f ∈ C^∞(𝕋^1×). By Theorem <ref>, for all β∈_0 and >0 there exists C_β N>0 such that

    |∂_t^βf(t,ℓ)_mn| ≤ C_β N(1+ℓ)^-N,

for all t ∈𝕋^1, ℓ∈1/2_0, and -ℓ≤ m,n ≤ℓ.

Let us determine when u defined by the partial Fourier coefficients (<ref>) and (<ref>) belongs to C^∞(𝕋^1×).

Let α∈_0 and N>0. If ma_0 ∈, we have

    ∂_t^αu(t,ℓ)_mn = ∑_β≤ααβ∂_t^α - β{e^-imH(t,t)}∂_t^β{∫_0^t e^imH(s,s)f(s,ℓ)_mn  ds}.


Notice that for β≥ 1 we have 


    ∂_t^β{∫_0^t e^imH(s,s)f(s,ℓ)_mn  ds}   = ∂_t^β-1{ e^imH(t,t)f(t,ℓ)_mn}
       = ∑_γ≤β-1β-1γ∂^(β-1)-γ_t e^imH(t,t)∂_t^γf(t,ℓ)_mn.

For each γ, by (<ref>) there exists C_γα N>0 such that 
    |∂_t^γf(t,ℓ)_mn| ≤ C_γα N(1+ℓ)^-(N+α).

By Lemma <ref>, since |m| ≤ℓ,  we have

    | ∂^(β-1)-γ_t e^imH(t,t)| ≤ C_β |m|^(β-1)-γ≤ C_β (1+ℓ)^β.

So, there exists C_αβ N>0 such that 

    | ∂_t^β{∫_0^t e^imH(s,s)f(s,ℓ)_mn  ds}| ≤ C_αβ N (1+ℓ)^-(N+α)+β.

Again by Lemma <ref> we have

    |∂_t^α - β{e^-imH(t,t)}| ≤ C_αβ(1+ℓ)^α-β

Therefore, from estimates (<ref>) and (<ref>), we obtain C_α N >0 such that

    |∂_t^αu(t,ℓ)_mn| ≤ C_α N (1+ℓ)^-N.


Let us assume now that ma_0 ∉. With a slight change in the proof of Lemma <ref> we obtain the same estimate

    |∂_t^α e^imH(t,s)| ≤ C_α |m|^α,

for all t,s ∈𝕋^1.

By the same arguments of the previous case we obtain for all α∈_0 and N>0

    |∂_t^αu(t,ℓ)_mn| ≤ |1-e^-2π im a_0|^-1 C_α N(1+ℓ)^-N.


Now, we need to estimate |1-e^-2π im a_0|^-1 in order to apply Theorem <ref> to conclude that u ∈ C^∞(𝕋^1×). 
In the case where m a_0 ∉, there exist C,M>0 such that 

    |1-e^-2π im a_0| ≥ C|m|^-M,

if and only if a_0 ∈ or a_0 is an irrational non-Liouville number (see <cit.>, Lemma 3.4)

This way, for either a_0 ∈ or a_0 an irrational non-Liouville number, we obtain constants C,M>0 such that 


    |1-e^-2π im a_0|^-1≤ C|m|^M≤ Cℓ^M,

when ma_0 ∉. So, adjusting the constants if necessary, for all α∈_0 and N>0, there exists C_α N>0 such that

    |∂^α_t u(t,ℓ)_mn| ≤ C_α N (1+ℓ)^-N,

for all t ∈𝕋^1, ℓ∈1/2_0, and -ℓ≤ m,n ≤ℓ. Therefore, by Theorem <ref> we have u ∈ C^∞(𝕋^1×) and by the uniqueness of Fourier coefficients, we conclude that Lu=f.


















We have proved the following proposition:

	The operator L=∂_t + a(t)X is globally C^∞-solvable if a_0 ∈ or a_0 is an irrational non-Liouville number, where a_0 = 1/2π∫_0^2π a(s)   ds.





















 §.§ Normal form



Notice that the global solvability of the operator L is strictly related with the number a_0. This is not a coincidence because we can conjugate the operator L to the operator L_a_0 = ∂_t + a_0 X. We say that the operator L_a_0 is the normal form of L. In this section we will construct this conjugation and show that the sufficient conditions on Proposition <ref> for the global C^∞–solvability of L are actually necessary conditions.

Define

    Ψ_a u(t,x) := ∑_ℓ∈1/2_0 (2ℓ +1) ∑_m,n=1^2ℓ+1 e^imA(t)u(t,ℓ)_mn^ℓ(x)_nm.

 Ψ_a is an automorphism in 𝒟'(𝕋^1×) and in C^∞(𝕋^1×).


It is easy to see that Ψ_a is linear and has inverse Ψ_-a, therefore we only need to prove that Ψ_a (C^∞()) = C^∞() and Ψ_a (𝒟'()) = 𝒟'().

Let β∈_0 and u∈ C^∞(). We will use  Theorem <ref> to show that Ψ_a u ∈ C^∞(). Notice that Ψ_a u(t,ℓ)_mn = e^i m A(t)u(t,ℓ)_mn for all ℓ∈1/2_0, ℓ≤ m,n ≤ℓ and t ∈𝕋^1. In this way

    |∂^βΨ_a u(t,ℓ)_mn|    = | ∂^β (e^imA(t)u(t,ℓ)_mn)| 
       = |∑_γ≤ββγ∂^β-γe^imA(t)∂^γu(t,η)_mn|
       ≤∑_γ≤ββγ |∂^β-γe^imA(t)||∂^γu(t,η)_mn| 
       ≤∑_γ≤βC_β |m|^β|∂^βu(t,ℓ)_mn| ,

where the last inequality comes from an adaptation of Lemma <ref>. Since u ∈ C^∞() and |m| ≤ℓ, again by Theorem <ref>, it is easy to see that given N>0, there exists C_β N such that

    |∂^βΨ_a u(t,ℓ)_mn| ≤ C_β N (1+ℓ)^-N.

Therefore Ψ_a u ∈ C^∞(). The distribution case is analogous.




	Let
	L_a_0=∂_t+a_0 X.
	Then

    Ψ_a∘ L = L_a_0∘Ψ_a.




 We will show that for every u ∈ C^∞(𝕋^1×) we have 
    Ψ_a (Lu)(t,ℓ)_mn=L_a_0(Ψ_a u)(t,ℓ)_mn,
 for all t ∈𝕋^1, ℓ∈1/2_0 and -ℓ≤ m,n ≤ℓ.
Indeed, we have

    L_a_0(Ψ_a u)(t,ℓ)_mn    =  ∂_t Ψ_a u(t,ℓ)_mn+ia_0mΨ_a u(t,ℓ)_mn
       = ∂_t {e^imA(t)u(t,ℓ)_mn} + ia_0 me^imA(t)u(t,ℓ)_mn
       = im(a(t)-a_0)e^imA(t)u(t,ℓ)_mn + e^imA(t)∂_tu(t,ℓ)_mn+ia_0me^imA(t)u(t,ℓ)_mn
       =e^imA(t) (∂_t u(t,ℓ)_mn+ia(t)mu(t,ℓ)_mn)
       = e^imA(t)Lu(t,ℓ)_mn
       =Ψ_a(Lu)(t,ℓ)_mn.

Therefore Ψ_a∘ L = L_a_0∘Ψ_a. The same holds when  it is considered u ∈𝒟'(𝕋^1 ×).


Let a_0 be an irrational Liouville number. The Fourier coefficients of L_a_0u, for u∈𝒟'(𝕋^1×) are expressed by

    L_a_0u(τ,ℓ)_mn = i(τ + a_0m)u(τ,ℓ)_mn,

with τ∈, ℓ∈1/2_0, -ℓ≤ m,n ≤ℓ.

Since a_0 is an irrational Liouville number, for every N∈, there exists τ_M ∈ and ℓ_M ∈ such that

    0 < |τ_M+a_0ℓ_M| ≤ (|τ_M|+|ℓ_M|)^-M.

Define

    f (τ,ℓ)_mn =  {[ τ_M+a_0ℓ_M,           ,;          0,             ].


It is easy to see that (<ref>) and Theorem <ref> imply that f ∈𝒦_a_0 (see Definition <ref>). If L_a_0u=f for some u∈ C^∞(𝕋^1×), the expression (<ref>) gives us 

    |  u (τ_M,ℓ_M)_ℓ_Mℓ_M|=1,

which contradicts the fact that u is a smooth function. Therefore L_a_0 is not globally C^∞–solvable when a_0 is an irrational Liouville number. Notice that u ∈𝒟'(𝕋^1×) (see Theorem <ref>). Since f ∈𝒦_a_0, when ma_0∈ we obtain

    0 = ∫_0^2π e^ima_0tf(t,ℓ)_mn  dt    = ∫_0^2πe^imH(t,t)e^-imA(t)f(t,ℓ)_mn  dt 
       = ∫_0^2πe^imH(t,t)Ψ_-af(t,ℓ)_mn  dt.

So Ψ_-af ∈𝒦. Assume that there exists some u ∈ C^∞(𝕋^1×) such that Lu=Ψ_-af. By Proposition <ref> we obtain

    f = Ψ_a Lu = L_a_0Ψ_a u.

By what was discussed previously and the fact that Ψ_a is an automorphism of C^∞(𝕋^1×) and 𝒟'(𝕋^1×), we conclude that u∈𝒟'(𝕋^1×)∖ C^∞(𝕋^1×). We have proved the following proposition:

	The operator L=∂_t + a(t)X is globally C^∞-solvable if and only if a_0 ∈ or a_0 is an irrational non-Liouville number, where a_0 = 1/2π∫_0^2π a(s)   ds.



§ ACKNOWLEDGMENTS


This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The last
author was also supported by the FWO Odysseus grant, by the Leverhulme Grant RPG-2017-151, and by EPSRC Grant EP/R003025/1.
abbrv



