Partial Fourier series on compact Lie groups

arXiv:1909.12824v1 [math.AP] 27 Sep 2019

ALEXANDRE KIRILOV, WAGNER A. A. DE MORAES, AND MICHAEL RUZHANSKY

Abstract. 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 onT¹×S³ and we will show that some properties of this evolution equation can be obtained from a constant coefficient equation.

Contents

1. Introduction 1

2. Fourier analysis on compact Lie groups 2

3. Partial Fourier series 5

4. Example: Global solvability of a vector field onT¹×S³ 13

4.1. Normal form 18

Acknowledgments 20

References 20

1. 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 [2,3,7,14–18,20–22] 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 [1,4,5,9–11,19] and several others references.

Date: September 30, 2019.

1991Mathematics Subject Classification. Primary 22E30, 43A77 ; Secondary 58D25, 43A25 . Key words and phrases. compact Lie group, partial Fourier series, evolution equation.

1

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 operatorP :C^∞(T¹×S³)→ C^∞(T¹×S³) defined by

(1.1) P u(t, x) :=∂tu(t, x) +a(t)Xu(t, x),

whereX is a left-invariant vector field onS³ anda∈C^∞(T¹) 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 globalC^∞-solvability of the operator (1.1) through its partial Fourier coefficients.

Then, we present a conjugation that transform the operator (1.1) in a constant coefficient operator.

2. Fourier analysis on compact Lie groups

In this section we introduce most of the notations and preliminary results necessary for the develop- ment 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 [12] and [23].

LetGbe a compact Lie group and let Rep(G) be the set of continuous irreducible unitary representa- tions ofG. SinceGis compact, every continuous irreducible unitary representationφis finite dimensional and it can be viewed as a matrix-valued functionφ:G→C^dφ×dφ, wheredφ = dimφ. We say thatφ∼ψ if there exists an unitary matrixA∈C^dφ×dφ such that Aφ(x) =ψ(x)A, for all x∈G. We will denote byGbthe quotient of Rep(G) by this equivalence relation.

Forf ∈L¹(G) we define the group Fourier transform off atφas

(2.1) fb(φ) =

Z

G

f(x)φ(x)^∗dx,

wheredxis the normalized Haar measure on G. By the Peter-Weyl theorem, we have that

(2.2) B:=np

dimφ φij; φ= (φij)^d_i,j=1^φ ,[φ]∈Gbo ,

is an orthonormal basis for L²(G), where we pick only one matrix unitary representation in each class of equivalence, and we may write

f(x) = X

[φ]∈Gb

dφTr(φ(x)fb(φ)).

Moreover, the Plancherel formula holds:

(2.3) kfkL²(G)=



X

[φ]∈Gb

dφ kfb(φ)k²HS





1/2

=:kfbk_ℓ²₍G)b, where

kfb(φ)k²HS= Tr(fb(φ)fb(φ)^∗) =

dφ

X

i,j=1

bf(φ)ij².

LetL^G be the Laplace-Beltrami operator ofG. For each [φ]∈G, its matrix elements are eigenfunctionsb ofL^G correspondent to the same eigenvalue that we will denote by−ν[φ], whereν[φ]≥0. Thus

−LGφij(x) =ν[φ]φij(x), for all 1≤i, j≤dφ, and we will denote by

hφi:= 1 +ν[φ]1/2

the eigenvalues of (I−L^G)^1/2.We have the following estimate for the dimension ofφ(Proposition 10.3.19 of [23]): there existsC >0 such that for all [ξ]∈Gb it holds

(2.4) dφ≤Chφi^dim2G.

Forx∈G, X∈gandf ∈C^∞(G), define LXf(x) := d

dtf(xexp(tX)) _t=0.

Notice that the operatorLX is left-invariant. Indeed, πL(y)LXf(x) =LXf(y⁻¹x) = d

dtf(y⁻¹xexp(tX))

t=0

= d

dtπL(y)f(xexp(tX))

t=0

=LXπL(y)f(x), for allx, y∈G.

When there is no possibility of ambiguous meaning, we will write onlyXf instead ofLXf.

Let Gbe a compact Lie group of dimension n and let {Xi}ⁿi=1 be a basis of its Lie algebra. For a multi-indexα= (α1,· · ·αn)∈Nⁿ₀, we define the left-invariant differential operator of order|α|

∂^α:=Y1· · ·Y|α|, with Yj ∈ {Xi}ⁿi=1, 1 ≤ j ≤ |α| and P

j:Yj=X_k

1 = αk for every 1 ≤ k ≤ n. It means that ∂^α is a composition of left-invariant derivatives with respect to vectorsX1, . . . , Xn such that eachXk enters∂^α exactlyαk times. We do not specify in the notation ∂^α the order of vectors X1, . . . , Xn, but this will not be relevant in the arguments that we will use in this work.

We equippedC^∞(G) with the usual Frchet space topology defined by seminormspα(f) = max

x∈G|∂^αf(x)|. Thus, the convergence onC^∞(G) is just the uniform convergence of functions and all their derivatives:

fk → f in C^∞(G) if ∂^αfk(x) → ∂^αf(x), for all x ∈ G, due to the compactness ofG. We define the space of distributionsD^′(G) as the space of all continuous linear functional on C^∞(G). Foru∈ D^′(G), we define the distribution∂^αuas

h∂^αu, ψi:= (−1)^|α|hu, ∂^αψi, for allψ∈C^∞(G).

Letα∈Nⁿ₀. The symbol of∂^αat x∈Gandξ∈Rep(G) is

(2.5) σ∂^α(x, φ) :=φ(x)^∗(∂^αφ)(x)∈C^dφ×dφ.

Notice that the symbol of∂^α does not depends ofx∈Gbecause ∂^α is a left-invariant operator. There existsC0>0 such that

(2.6) kσ∂^α(φ)k^op≤C₀^|α|hφi^|α|,

for all α∈Nⁿ₀ and [φ] ∈Gb (see [8], Proposition 3.4), where k · kop stands for the usual operator norm and, from Chapter 2 of [13], we have

(2.7) kσ∂^α(φ)kop≤ kσ∂^α(φ)kHS≤p

dφkσ∂^α(φ)kop. For every integerM ≥ ^dimG2 there existsCM >0 such that

(2.8) kφijkL^∞(G)≤CMhφi^M,

for all [φ]∈G, 1b ≤i, j≤dφ (see [8], Lemma 3.3).

Smooth functions on G can be characterized in terms of their Fourier coefficients comparing with powers of the eigenvalues of (I− LG)^1/2. Precisely, the following statements are equivalent:

(i) f ∈C^∞(G);

(ii) for eachN >0, there existsCN >0 such that

kfb(φ)kHS≤CNhφi^−N, for all [φ]∈G;b

(iii) for eachN >0, there existsCN >0 such that

|fb(φ)ij| ≤CNhφi^−N, for all [φ]∈G,b 1≤i, j≤dφ.

By duality, we have a similar result for distributions onG, where we define b

u(φ)ij:=u(φji).

This definition agrees with (2.1) when the distribution comes from a L¹(G) function. The following statements are equivalent:

(i) u∈ D^′(G);

(ii) there existC, N >0 such that

kbu(φ)kHS≤Chφi^N, for all [φ]∈G;b

(iii) there existC, N >0 such that

|bu(φ)ij| ≤Chφi^N, for all [φ]∈G,b 1≤i, j≤dφ.

3. Partial Fourier series

Let G1 and G2 be compact Lie groups, and set G = G1 ×G2. Let ξ ∈ Hom(G1,Aut(V1)) and η∈Hom(G2,Aut(V2)). The external tensor product representationξ⊗η ofGonV1⊗V2 is defined by

ξ⊗η: G1×G2 → Aut(V1⊗V2)

(x1, x2) 7→ (ξ⊗η)(x1, x2) : V1⊗V2 → V1⊗V2

(v1, v2) 7→ ξ(x1)(v1)⊗η(x2)(v2)

We point out that the external tensor product of unitary representation is also unitary. More- over, if ξ ∈Hom(G,U(dξ)) and η ∈Hom(G,U(dη)) are matrix unitary representations, then ξ⊗η ∈ Hom(G,U(dξdη)) is also a matrix unitary representation and

ξ⊗η(x1, x2) =ξ(x1)⊗η(x2)∈C^{dξdη×dξdη}, whereξ(x1)⊗η(x2) is the Kronecker product of these matrices, that is,

ξ(x1)⊗η(x2) =







ξ(x1)11η(x2) · · · ξ(x1)1dξη(x2) ... . .. ... ξ(x1)d_ξ1η(x2) · · · ξ(x1)d_ξd_ξη(x2)





.

It is enough to study continuous irreducible unitary representations of G1 and G2 to obtain the elements of G, since for every [φ]b ∈ G, there exist [ξ]b ∈ Gc1 and [η] ∈ Gc2 such that φ ∼ ξ⊗η, that is, [φ] = [ξ⊗η] ∈ Gb and dφ = dξ·dη. Moreover, [ξ1⊗η1] = [ξ2⊗η2] if and only if [ξ1] = [ξ2] and [η1] = [η2]. The proof of this fact can be found on [6] (Chapter II, Proposition 4.14). Therefore, the map [ξ⊗η]7→([ξ],[η]) is a bijection fromGb toGc1×Gc2.

It is easy to see thatL^G=L^G1+L^G2, soν[ξ⊗η]=ν[ξ]+ν[η]. Therefore we have

(3.1) 1

2(hξi+hηi)≤ hξ⊗ηi ≤ hξi+hηi, for all [ξ]∈Gc1 and [η]∈Gc2.

Letf ∈L¹(G) and [φ]∈G. Let [ξ]b ∈Gc1 and [η]∈Gc2 such that [φ] = [ξ⊗η]. Notice that fb(ξ⊗η) =

Z

G

f(x)(ξ⊗η)(x)^∗dx

= Z

G2

Z

G1

f(x1, x2)(ξ(x1)⊗η(x2))^∗dx1dx2

= Z

G2

Z

G1

f(x1, x2)ξ(x1)^∗⊗η(x2)^∗dx1dx2.

Thusfb(ξ⊗η)∈C^{dξdη×dξdη} with elements fb(ξ⊗η)ij =

Z

G2

Z

G1

f(x1, x2)(ξ(x1)^∗⊗η(x2)^∗)ijdx1dx2

= Z

G2

Z

G1

f(x1, x2)ξ(x1)nmη(x2)srdx1dx2

where 1≤m, n≤dξ, 1≤r, s≤dη are given by

m = j

i−1 dη

k+ 1,

n = j

j−1 dη

k+ 1,

r = i−j

i−1 dη

kdη, s = j−j

j−1 dη

kdη.

Similarly foru∈ D^′(G), we have b

u(ξ⊗η)ij =D

u,(ξ⊗η)ji

E=

u, ξnm×ηsr ,

where (ξnm×ηsr)(x1, x2) :=ξ(x1)nmη(x2)sr.

Definition 3.1. Let G1 and G2 be compact Lie groups and, set G =G1×G2. Let f ∈ L¹(G), ξ ∈ Rep(G1), andx2∈G2. Theξ-partial Fourier coefficient of f atx2 is defined by

fb(ξ, x2) = Z

G1

f(x1, x2)ξ(x1)^∗dx1∈C^dξ×dξ, with components

f(ξ, xb 2)mn= Z

G1

f(x1, x2)ξ(x1)nmdx1, 1≤m, n≤dξ.

Similarly, forη ∈Rep(G2)andx1∈G1, we define theη-partial Fourier coefficient of f atx1 as fb(x1, η) =

Z

G2

f(x1, x2)η(x2)^∗dx2∈C^dη×dη, with components

fb(x1, η)rs= Z

G2

f(x1, x2)η(x2)srdx1, 1≤r, s≤dη. By the definition, the function

fb(ξ, ·)mn: G2 −→ C

x2 7−→ fb(ξ, x2)mn

belongs toL¹(G2) for allξ∈Rep(G1), 1≤m, n≤dξ. Similarly, the function fb(·, η)rs: G1 −→ C

x1 7−→ fb(x1, η)rs

belongs toL¹(G1) for allη ∈Rep(G2), 1≤r, s≤dη.

Letξ∈Rep(G1) and η∈Rep(G2). Sincefb(ξ,·)mn∈L¹(G2) for all 1≤m, n≤dξ, we can take its η-coefficient of Fourier:

bbf(ξ, η)mn:=

Z

G2

fb(ξ, x2)mnη(x2)^∗dx2∈C^dη×dη with components

bbf(ξ, η)mnrs = Z

G2

fb(ξ, x2)mnη(x2)srdx2

= Z

G2

Z

G1

f(x1, x2)ξ(x1)nmη(x2)srdx1dx2,

for 1≤r, s≤dη. Similarly, sincefb(·, η)rs∈L¹(G1) for all 1≤r, s≤dη, we can take itsξ-coefficient of Fourier:

bbf(ξ, η)rs:=

Z

G1

fb(x1, η)rsξ(x1)^∗dx1∈C^dξ×dξ with components

bbf(ξ, η)rsmn= Z

G1

fb(x1, η)rsξ(x1)nmdx1

= Z

G1

Z

G2

f(x1, x2)ξ(x1)nmη(x2)srdx2dx1, for 1≤m, n≤dξ.

Notice that

bbf(ξ, η)mnrs = bbf(ξ, η)rsmn=fb(ξ⊗η)ij, with

i=dη(m−1) +r, j=dη(n−1) +s, for all 1≤m, n≤dξ and 1≤r, s≤dη.

Definition 3.2. LetG1andG2be compact Lie groups, and setG=G1×G2. Letu∈ D^′(G),ξ∈Rep(G1) and1 ≤m, n≤dξ. The mn-component of theξ-partial Fourier coefficient of uis the linear functional defined by

b

u(ξ, ·)mn: C^∞(G2) −→ C

ψ 7−→ hbu(ξ, ·)mn, ψi:=

u, ξnm×ψ

G.

In a similar way, forη∈Rep(G2)and1≤r, s≤dη, we define thers-component of theη-partial Fourier coefficient of uas

b

u(·, η)rs: C^∞(G1) −→ C

ϕ 7−→ hbu(·, η)rs, ϕi:=hu, ϕ×ηsriG.

By definition, u(ξ,b ·)mn ∈ D^′(G2) and u(b ·, η)rs ∈ D^′(G1) for all ξ ∈ Rep(G1), η ∈ Rep(G2), 1≤m, n≤dξ and 1≤r, s≤dη.

Letξ∈Rep(G1) and η ∈Rep(G2). Sincebu(ξ,·)mn ∈ D^′(G2) for all 1≤m, n≤dξ, we can take its η-coefficient of Fourier:

bb

u(ξ, η)mn:=hbu(ξ, ·)mn, η^∗i ∈C^dη×dη with components

bb

u(ξ, η)mnrs =hbu(ξ,·)mn, ηsri=

u, ξnm×ηsr

G =

u, ξnm×ηsr

G,

for all 1≤r, s≤dη. Now, sincebu(·, η)rs∈ D^′(G1) for all 1≤r, s≤dη we can take its ξ-coefficient of Fourier:

bb

u(ξ, η)rs:=hbu(·, η)rs, ξ^∗i ∈C^dξ×dξ with components

bb

u(ξ, η)rsmn = b

u(·, η)rs, ξmn

=

u, ξnm×ηsr

G=

u, ξnm×ηsr

G,

for all 1≤m, n≤dξ. Notice that bb

u(ξ, η)mnrs = bbu(ξ, η)rsmn=u(ξb ⊗η)ij, with

i=dη(m−1) +r, j=dη(n−1) +s, for all 1≤m, n≤dξ and 1≤r, s≤dη.

Notice that

(3.2) kbu(ξ⊗η)k²HS=

dξdη

X

i,j=1

|bu(ξ⊗η)ij|²=

dξ

X

m,n=1 dη

X

r,s=1

bbu(ξ, η)mnrs

²=: bbu(ξ, η)²

HS, for all [ξ]∈Gc1 and [η]∈Gc2 wheneveru∈L¹(G) oru∈ D^′(G).

It follows from the (3.1) and (3.2) 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.

Theorem 3.3. LetG1 andG2 be compact Lie groups, and setG=G1×G2 . The following statements are equivalent:

(i) f ∈C^∞(G);

(ii) For everyN >0, there existsCN >0 such that

kbbf(ξ, η)kHS≤CN(hξi+hηi)^−N, for all[ξ]∈Gc1 and[η]∈Gc2;

(iii) For every N >0, there existsCN >0 such that

bbf(ξ, η)mnrs

≤CN(hξi+hηi)^−N, for all[ξ]∈Gc1,[η]∈Gc2,1≤m, n≤dξ, and1≤r, s≤dη.

Theorem 3.4. Let G1 andG2 be compact Lie groups, and setG=G1×G2. The following statements are equivalent:

(i) u∈ D^′(G);

(ii) There exist C, N >0 such that

kbbu(ξ, η)kHS ≤C(hξi+hηi)^N, for all[ξ]∈Gc1 and[η]∈Gc2;

(iii) There exist C, N >0 such that

bbu(ξ, η)mnrs

≤C(hξi+hηi)^N, for all[ξ]∈Gc1,[η]∈Gc2,1≤m, n≤dξ, and1≤r, s≤dη.

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

Theorem 3.5. Let G1 andG2 be compact Lie groups,G=G1×G2, and let{fb(·, η)rs} be a sequence of functions on G1. Define

f(x1, x2) := X

[η]∈Gc2

dη dη

X

r,s=1

fb(x1, η)rsηsr(x2).

Then f∈C^∞(G)if and only iffb(·, η)rs∈C^∞(G1), for all[η]∈Gc2,1≤r, s≤dη and for every β∈Nⁿ₀ andℓ >0 there existCβℓ>0 such that

∂^βfb(x1, η)rs

≤Cβℓhηi^−ℓ, ∀x1∈G1, [η]∈G,b 1≤r, s≤dη.

Proof. (⇐= ) It is sufficient to consider N ∈N in Theorem3.3 to conclude that f ∈C^∞(G). Notice that

L[G1g(ξ)mn=

LG1g, ξnm

=

g,LG1ξnm

=−ν[ξ]

g, ξnm

=−ν[ξ]bg(ξ)mn, for allg∈C^∞(G1), [ξ]∈Gc1, and 1≤m, n≤dξ. In particular, forN∈Nwe obtain

ν_[ξ]^N|fbb(ξ, η)rsmn|= \

L^NG1fb(ξ, η)rsmn

= Z

G1

L^NG1fb(x1, η)rsξ(x1)nmdx1

≤ Z

G1

|L^NG1fb(x1, η)rs||ξ(x1)nm|dx1

≤ Z

G1

|L^NG1fb(x1, η)rs|²dx1

1/2Z

G|ξ(x1)nm|²dx1

1/2

≤ 1 pdξ

X

|β|=2N

xmax1∈G1|∂^βfb(x1, η)rs|

≤ X

|β|=2N

xmax1∈G1|∂^βf(xb 1, η)rs|.

Notice that there existsC >0 such that hξi ≤Cν[ξ] for all non-trivial [ξ]∈Gc1. Thus for all ℓ=N we have

|fbb(ξ, η)rsmn| ≤CNhξi^−Nhηi^−N ≤CN2^N(hξi+hηi)^−N. Thereforef ∈C^∞(G).

( =⇒) Let E2 := (I− LG2)^1/2. Since f ∈C^∞(G), for all β ∈N₀ⁿ andN ∈ N₀ we have ∂^βE₂^Nf ∈ C^∞(G) and then, by the compactness ofG, there existsCβN ≥0 such that

(3.3) |∂^βE₂^Nf(x1, x2)| ≤CβN, ∀(x1, x2)∈G1×G2.

Fixη∈Rep(G2), 1≤r, s≤dη. We already know thatfb(·, η)rs∈C^∞(G1). Moreover

|hηi^N∂^βfb(x1, η)rs|=|∂^βE[₂^Nf(x1, η)rs|

= ∂^β

Z

G2

E₂^Nf(x1, x2)η(x2)srdx2

≤ Z

G2

|∂^βE₂^Nf(x1, x2)||η(x2)sr|dx2

≤ Z

G2

|∂^βE₂^Nf(x1, x2)|²dx2

1/2Z

G2

|η(x2)sr|²dx2

1/2

(3.3)

≤ √¹

dη

CβN.

Therefore,

|∂^βfb(x1, η)rs| ≤CβNhηi^−N,

for allx1∈G1, [η]∈Gc2, 1≤r, s≤dη.

Theorem 3.6. LetG1andG2be compact Lie groups, setG=G1×G2, and let b

u(·, η)rs be a sequence of distributions onG1. Define

u= X

[η]∈Gb

dη dη

X

r,s=1

b

u(·, η)rsηsr.

Then u∈ D^′(G) if and only if there existK∈NandC >0 such that (3.4) hbu(·, η)rs, ϕi≤C pK(ϕ)hηi^K, for allϕ∈C^∞(G1)and[η]∈G, whereb pK(ϕ) := P

|β|≤Kk∂^βϕkL^∞(G1).

Proof. (⇐=) Take ϕ=ξnm, [ξ]∈Gc1, 1≤m, n≤dξ. Letβ ∈Nⁿ₀,|β| ≤K, withK as in (3.4). Since the symbol of∂^β atx1∈G1 andξ∈Rep(G1) is given by

σ∂^β(x1, ξ) =ξ(x1)^∗(∂^βξ)(x1), we have

|∂^βξnm(x1)|=

dξ

X

i=1

ξni(x)σ∂^β(ξ)im

≤

d_ξ

X

i=1

|ξni(x)||σ_∂β(ξ)im|

≤





dξ

X

i=1

|ξni(x)²|





1/2



dξ

X

i=1

|σ∂^β(ξ)im|²





1/2

LetM ∈ZsatisfyingM ≥ ^dimG₂ ¹. By (2.8) we have





dξ

X

i=1

|ξni(x)²|





1/2

≤





dξ

X

i=1

kξnik²L^∞(G1)





1/2

≤CM

pdξhξi^M and by (2.4) there existsC >0 such that

dξ ≤Chξi^M.

Moreover, notice that





dξ

X

i=1

|σ_∂^β(ξ)im|²





1/2

≤ kσ_∂^β(ξ)kHS≤p

dξkσ_∂^β(ξ)kop≤p

dξC₀^|β|hξi^|β|,

where the last inequalities come from (2.6) and (2.7). Hence

|∂^βξnm(x1)| ≤Chξi^Mp

dξkσ∂^β(ξ)kHS

≤Chξi^Mdξkσ∂^β(ξ)k^op

≤CC₀^|β|hξi^2M+|β|.

Then

pK(ξnm) =pK(ξnm)≤Chξi^2M+K. Hence

bbu(ξ, η)rsmn

=

bu(·, η)rs, ξnm≤C pK(ξnm)hηi^K

≤Chξi^2M+Khηi^2M+K

≤C(hξi+hηi)^2(2M+K). Thereforeu∈ D^′(G).

(=⇒) Sinceu∈ D^′(G), then there existC >0 andK∈Nsuch that

(3.5)

bbu(ξ, η)rsmn

≤C(hξi+hηi)^K,

for all [ξ]∈Gc1,[η]∈Gc2, 1≤r, s≤dη, and 1≤m, n≤dξ and u= X

[ξ]∈Gc1

X

[η]∈Gb

dξdη dξ

X

m,n=1 dη

X

r,s=1

bb

u(ξ, η)rsmnξnmηsr.

Forϕ∈C^∞(G1) we have

|hu(·, η)b rs, ϕi|=|(u, ϕ×ηsr)|

=

X

[ξ]∈Gc1

X

[η]∈Gb

dξdη d_ξ

X

m,n=1 d_η

X

k,ℓ=1

bb

u(ξ, η)kℓ_mnhξnm, ϕi_G1hηℓk, ηsri_G2 .

Notice thathηℓk, ηsriG2 = _d¹

ηδℓsδkr, since the setB is orthonormal (see (2.2)). Moreover, ϕ(ξ)b mn = hξnm, ϕiG1. So

|hbu(·, η)rs, ϕi|=

X

[ξ]∈cG1

dξ dξ

X

m,n=1

bb

u(ξ, η)rsmnϕ(ξ)b mn

≤ X

[ξ]∈cG1

dξ d_ξ

X

m,n=1

bbu(ξ, η)rsmn

bϕ(ξ)mn

≤C X

[ξ]∈Gc1

dξ dξ

X

m,n=1

(hξi+hηi)^K bϕ(ξ)mn

,

where the last inequality comes from (3.5). Notice that for allK∈Nwe have (hξi+hηi)^K ≤2^Khξi^Khηi^K. In addition, we have

d_ξ

X

m,n=1

|ϕ(ξ)b mn| ≤



d²_ξ

d_ξ

X

m,n=1

|ϕ(ξ)b mn|²





1/2

=dξkϕ(ξ)b kHS.

Sincehξi= ξ

and the summation is over allGc1, we have X

[ξ]∈cG1

dξhξi^K

dξ

X

m,n=1

|ϕ(ξ)b mn|= X

[ξ]∈Gc1

d_ξhξi^K

dξ

X

m,n=1

|ϕ(ξ)b mn|

= X

[ξ]∈Gc1

dξhξi^K

d_ξ

X

m,n=1

|ϕ(ξ)b mn|.

Thus

|hbu(·, η)rs, ϕi| ≤Chηi^K X

[ξ]∈Gc1

dξhξi^K

dξ

X

m,n=1

|ϕ(ξ)b mn|

≤Chηi^K X

[ξ]∈Gc1

d²_ξhξi^Kkϕ(ξ)b kHS. The series P

[ξ]∈Gc1

d²_ξhξi^−2tconverges if and only ift > ^dim₂^G1 (Lemma 3.1 of [8]), which implies that there existsC >0 such thatdξ ≤Chξi^dimG1, for all [ξ]∈Gc1. Hence,

|hbu(·, η)rs, ϕi|=Chηi^K X

[ξ]∈Gc1

dξhξi^−dimG1 dξhξi^K+dimG1kϕ(ξ)b kHS

≤Chηi^K



 X

[ξ]∈Gc1

d²_ξhξi^{−2 dimG1}





1/2

 X

[ξ]∈Gc1

d²_ξhξi^2(K+dimG1)kϕ(ξ)b k²HS





1/2

≤Chηi^K



 X

[ξ]∈Gc1

dξhξi^{2(K+2 dimG1)}kϕ(ξ)b k²HS





1/2

.

LetL∈N₀ such thatK+ 2 dimG1≤2L. So

|hbu(·, η)rs, ϕi| ≤Chηi^K



 X

[ξ]∈Gc1

dξhξi^4Lkϕ(ξ)b k²HS





1/2

=Chηi^K



 X

[ξ]∈Gc1

dξkE\₁^2Lϕ(ξ)k²HS





1/2

=Chηi^KkE^2L₁ ϕkL²(G1)

≤CkE₁^2LϕkL²(G1)hηi^2L,

whereE1= (I− L^G1)^1/2, and the last equality comes from the Plancherel formula (2.3). Notice that kE₁^2LϕkL²(G1)≤ kE₁^2LϕkL^∞(G1)=k(I− LG1)^LϕkL^∞(G1)≤Cp2L(φ).

Therefore,

|hbu(·, η)rs, ϕi| ≤Cp2L(φ)hηi^2L.

4. Example: Global solvability of a vector field on T¹×S³

The 3-sphere S³ is a Lie group with respect to the quaternionic product of R⁴, and it is globally diffeomorphic and isomorphic to the group SU(2) of unitary 2×2 matrices of determinant one, with the usual matrix product.

Let a(t) ∈R for t ∈ T¹. Let X be a normalized left-invariant vector field on S³. We consider the operator

(4.1) L=∂t+a(t)X.

We are interested in solvability properties for the vector fieldLonT¹×S³. LetcS³ be the unitary dual of S³. It consists of the equivalence classes [t^ℓ] of the continuous irreducible unitary representations t^ℓ:S³→C(2ℓ+1)×(2ℓ+1),ℓ∈ ¹2N₀, of matrix-valued functions satisfyingt^ℓ(xy) =t^ℓ(x)t^ℓ(y) andt^ℓ(x)^∗= t^ℓ(x)⁻¹ for all x, y ∈ S³. We will use the standard convention of enumerating the matrix elements t^ℓ

mn of t^ℓ using indices m, nranging between −ℓ to ℓ with step one, i.e. we have −ℓ ≤m, n ≤ℓ with ℓ−m, ℓ−n∈N₀.

For a functionf ∈C^∞(S³) we can define its Fourier coefficient atℓ∈ ¹₂N₀ by fb(ℓ) :=

Z

S³

f(x)t^ℓ(x)^∗dx∈C(2ℓ+1)×(2ℓ+1),

where the integral is (always) taken with respect to the Haar measure onS³, and with a natural extension to distributions. The Fourier series becomes

f(x) = X

ℓ∈¹2N0

(2ℓ+ 1)Tr

t^ℓ(x)fb(ℓ) ,

with the Plancherel’s identity taking the form (4.2) kfkL²(S³)=



 X

ℓ∈¹2N0

(2ℓ+ 1)kfb(ℓ)k²HS





1/2

=:kfbkℓ²(S³),

which we take as the definition of the norm on the Hilbert spaceℓ²(cS³), and wherekfb(ℓ)k²HS= Tr(fb(ℓ)fb(ℓ^∗)) is the Hilbert–Schmidt norm of the matrixfb(ℓ).

Smooth functions and distributions onS³ can be characterized in terms of their Fourier coefficients.

Thus, we have

f ∈C^∞(S³)⇐⇒ ∀N ∃CN such thatkfb(ℓ)k^HS≤CN(1 +ℓ)^−N,∀ℓ∈ 1 2N₀. Also, for distributions, we have

u∈ D^′(S³)⇐⇒ ∃M ∃C such thatkbu(ℓ)kHS≤C(1 +ℓ)^M,∀ℓ∈ 1 2N₀.

Given an operator T : C^∞(S³) → C^∞(S³) (or even T : C^∞(S³) → D^′(S³)), we define its matrix symbol by

σT(x, ℓ) :=t^ℓ(x)^∗(Tt^ℓ)(x)∈C(2ℓ+1)×(2ℓ+1),

whereTt^ℓ means that we applyT to the matrix components oft^ℓ(x). In this case we can prove that

(4.3) T f(x) = X

ℓ∈¹2N0

(2ℓ+ 1)Tr

t^ℓ(x)σT(x, ℓ)fb(ℓ) .

The correspondence between operators and symbols is one-to-one, and we will writeTσfor the operator given by (4.3) corresponding to the symbolσ(x, ℓ). The quantization (4.3) has been extensively studied in [23,24], to which we refer for its properties and for the corresponding symbolic calculus.

Using rotation onS³, without loss of generality we may assume that the vector fieldX has the symbol σX(ℓ)mn=imδmn, −ℓ≤m, n≤ℓ,

with δmn standing for the Kronecker’s delta. Consequently, taking the Fourier transform of Lu =

∂tu+a(t)Xuwith respect tox, we get

Lu(t, ℓ) =c ∂tbu(t, ℓ) +ia(t)dXu(t, ℓ).

Writing this in terms of matrix coefficients, we obtain

(4.4) Lu(t, ℓ)c mn=∂tu(t, ℓ)b mn+ia(t)mbu(t, ℓ)mn, −ℓ≤m, n≤ℓ.

Consequently, the equationLu=f, wheref ∈C^∞(T¹×S³) is reduced to (4.5) ∂tbu(t, ℓ)mn+ia(t)mu(t, ℓ)b mn=fb(t, ℓ)mn, ℓ∈ 1

2N₀, −ℓ≤m, n≤ℓ.

We say that an operator P is global hypoelliptic if the conditions P u ∈ C^∞(T¹ ×S³) and u ∈ D^′(T¹×S³) implyu∈C^∞(T¹×S³).

We point out that the operator L defined in (4.1) is not globally hypoelliptic for any a ∈C^∞(T¹).

Indeed, defineuby its partial Fourier coefficients:

(4.6) bu(t, ℓ)mn=





1, ifℓ∈N₀ andm=n= 0;

0, otherwise.

Notice that by Theorem3.5 we have u /∈C^∞(T¹×S³) and by Theorem3.6 we have u∈ D^′(T¹×S³).

Moreover, the function defined by (4.6) is a homogeneous solution of (4.5) for allℓ∈ ¹2N₀,−ℓ≤m, n≤ℓ.

ThusLu= 0∈C^∞(T¹×S³) which implies thatLis not globally hypoelliptic.

Let us turn our attention now to study the solvability of the operatorL. Denote by a0= 1

2π Z 2π

0

a(t)dt

and define

(4.7) A(t) =

Z t 0

a(s)ds−a0t.

Put

v(t, ℓ)mn:=e^imA(t)bu(t, ℓ)mn.

In this wayv(t, ℓ)mnsatisfies the equation

(4.8) ∂tv(t, ℓ)mn+ima0v(t, ℓ)mn=g(t, ℓ)mn:=e^imA(t)fb(t, ℓ)mn. Lemma 4.1. Let λ∈Cand consider the equation

(4.9) d

dtu(t) +λu(t) =f(t), wheref ∈C^∞(T¹).

If λ /∈iZthen the equation (4.9)has a unique solution that can be expressed by

(4.10) u(t) = 1

1−e^−2πλ Z 2π

0

e^−λsf(t−s)ds, or equivalently,

(4.11) u(t) = 1

e^2πλ−1 Z 2π

0

e^λrf(t+r)dr.

If λ∈iZandR2π

0 e^λsf(s)ds= 0 then we have that

(4.12) u(t) =e^−λt

Z t 0

e^λsf(s)ds is a solution of the equation (4.9).

To prove this lemma observe thatE= (1−e^−2πλ)⁻¹e^λtis the fundamental solution of the operator

d

dt+λ, when λ /∈ iZ. The equivalence between (4.10) and (4.11) follows from the change of variable s7→ −r+ 2π. We point out that the solution (4.12) is not unique.

By Lemma4.1, ifma0∈/ Zthe equation (4.8) has a unique solution that can be written as v(t, ℓ)mn= 1

1−e^−2πima0 Z 2π

0

e^−ima0sg(t−s, ℓ)mnds

and then

(4.13) bu(t, ℓ)mn= 1

1−e^−2πima0 Z 2π

0

e^−imH(t,s)fb(t−s, ℓ)mnds, whereH(t, s) :=Rt

t−sa(θ)dθ.

Ifma0∈Z, a solution for the equation (4.8) can be expressed as v(t, ℓ)mn=e^−ima0t

Z t 0

e^ima0sg(s, ℓ)mnds

and then

(4.14) u(t, ℓ)b mn=e^−imH(t,t) Z t

0

e^imH(s,s)fb(s, ℓ)mnds.

Here, one can see a condition onf for the existence of a solutionu(t, ℓ)b mn onT¹. Ifma0∈Z, then (4.15)

Z 2π 0

e^imH(t,t)f(t, ℓ)b mndt= 0.

This condition appears in Lemma4.1in order to guarantee that the solution is well-defined inT¹. So, iff ∈C^∞(T¹×S³) does not satisfy the condition above, the equation (4.5) has no solution onT¹. We will denote byKthe set of smooth functions onT¹×S³ that satisfy the condition (4.15).

Definition 4.2. We say that the operator L is globally C^∞-solvable if for any f ∈ K, there exists u∈C^∞(T¹×S³)such that Lu=f.

Remark 4.3. In the literature it is common to define the globalC^∞-solvability forf in the set {h∈C^∞(T¹×S³);hw, hi= 0; ∀w∈Ker^tL},

(e.g. [21], [22]). For the operator L that we are studying in this work, this set coincides with the setK defined previously.

Let us investigate if the operator L defined in (4.1) is globally C^∞-solvable. First, let us prove a technical result about derivatives of the exponential function.

Lemma 4.4. For any α∈N₀, there existsCα such that

|∂_t^αe^imH(t,t)| ≤Cα|m|^α,

for allt∈T¹.

Proof. By Fa`a di Bruno’s Formula, we have

∂_t^αe^imH(t,t)= X

γ∈∆(α)

α!

γ!(im)^|γ|e^imH(t,t) Yα j=1

∂_t^jH(t, t) j!

!γj

,

where ∆(α) =

γ∈N^α₀; P^α

j=1

jγj=α

. Notice thatH(t, t) =Rt

0a(θ)dθ, so∂^j_tH(t, t) =∂_t^j−1a(t), for allj ≥1. Hence

|∂_t^αe^imH(t,t)| ≤ X

γ∈∆(α)

α!

γ!|m|^|γ|

Yγ j=1

|∂_t^jH(t, t)| j!

!γj

≤Cα|m|^α.

Assume thatf ∈C^∞(T¹×S³). By Theorem3.5, for allβ∈N₀ andN>0 there existsCβN >0 such that

(4.16) |∂_t^βfb(t, ℓ)mn| ≤CβN(1 +ℓ)^−N, for allt∈T¹,ℓ∈¹₂N₀, and−ℓ≤m, n≤ℓ.

Let us determine when u defined by the partial Fourier coefficients (4.13) and (4.14) belongs to C^∞(T¹×S³).

Letα∈N₀ andN >0. Ifma0∈Z, we have

∂_t^αu(t, ℓ)b mn=X

β≤α

α β

∂_t^α−β{e^−imH(t,t)}∂_t^β Z t

0

e^imH(s,s)fb(s, ℓ)mnds

.

Notice that forβ ≥1 we have

∂t^β

Z t 0

e^imH(s,s)f(s, ℓ)b mnds

=∂t^β−1

n

e^imH(t,t)f(t, ℓ)b mn

o

= X

γ≤β−1

β−1 γ

!

∂t^{(β−1)−γ}e^imH(t,t)∂t^γf(t, ℓ)b mn.

For eachγ, by (4.16) there exists CγαN >0 such that

|∂_t^γfb(t, ℓ)mn| ≤CγαN(1 +ℓ)^−(N+α). By Lemma4.4, since|m| ≤ℓ, we have

|∂_t^{(β−1)−γ}e^imH(t,t)| ≤Cβ|m|^{(β−1)−γ}≤Cβ(1 +ℓ)^β. So, there existsCαβN >0 such that

(4.17)

∂^β_t

Z t 0

e^imH(s,s)fb(s, ℓ)mnds≤CαβN(1 +ℓ)^−(N+α)+β. Again by Lemma4.4we have

(4.18) |∂_t^α−β{e^−imH(t,t)}| ≤Cαβ(1 +ℓ)^α−β Therefore, from estimates (4.17) and (4.18), we obtainCαN >0 such that

|∂_t^αu(t, ℓ)b mn| ≤CαN(1 +ℓ)^−N.

Let us assume now that ma0 6∈ Z. With a slight change in the proof of Lemma 4.4 we obtain the same estimate

|∂_t^αe^imH(t,s)| ≤Cα|m|^α, for allt, s∈T¹.

By the same arguments of the previous case we obtain for allα∈N₀ andN >0

|∂_t^αbu(t, ℓ)mn| ≤ |1−e^−2πima0|⁻¹CαN(1 +ℓ)^−N.

Now, we need to estimate |1−e^−2πima0|⁻¹ in order to apply Theorem 3.5 to conclude that u ∈ C^∞(T¹×S³). In the case wherema0∈/ Z, there existC, M >0 such that

|1−e^−2πima0| ≥C|m|^−M,

if and only ifa0∈Qora0 is an irrational non-Liouville number (see [4], Lemma 3.4)

This way, for eithera0 ∈Qor a0 an irrational non-Liouville number, we obtain constantsC, M >0 such that

|1−e^−2πima0|⁻¹≤C|m|^M ≤Cℓ^M,

whenma0∈/ Z. So, adjusting the constants if necessary, for allα∈N₀andN >0, there existsCαN >0 such that

|∂_t^αu(t, ℓ)b mn| ≤CαN(1 +ℓ)^−N,

for allt ∈T¹, ℓ∈ ¹₂N0, and−ℓ≤m, n≤ℓ. Therefore, by Theorem3.5 we haveu∈C^∞(T¹×S³) and by the uniqueness of Fourier coefficients, we conclude thatLu=f.

We have proved the following proposition:

Proposition 4.5. The operatorL=∂t+a(t)X is globally C^∞-solvable if a0∈Qor a0 is an irrational non-Liouville number, where a0= _2π¹ R2π

0 a(s)ds.

4.1. Normal form.

Notice that the global solvability of the operatorLis strictly related with the number a0. This is not a coincidence because we can conjugate the operatorL to the operator La0 =∂t+a0X. We say that the operatorLa0 is the normal form ofL. In this section we will construct this conjugation and show that the sufficient conditions on Proposition4.5for the globalC^∞–solvability ofLare actually necessary conditions.

Define

Ψau(t, x) := X

ℓ∈¹2N0

(2ℓ+ 1)

2ℓ+1X

m,n=1

e^imA(t)u(t, ℓ)b mnt^ℓ(x)nm.

Proposition 4.6. Ψa is an automorphism in D^′(T¹×S³)and inC^∞(T¹×S³).

Proof. It is easy to see that Ψa is linear and has inverse Ψ−a, therefore we only need to prove that Ψa(C^∞(T¹×S³)) =C^∞(T¹×S³) and Ψa(D^′(T¹×S³)) =D^′(T¹×S³).

Letβ ∈N₀andu∈C^∞(T¹×S³). We will use Theorem3.5to show that Ψau∈C^∞(T¹×S³). Notice thatΨdau(t, ℓ)mn=e^imA(t)bu(t, ℓ)mn for allℓ∈ ¹₂N₀,ℓ≤m, n≤ℓandt∈T¹. In this way

|∂^βΨdau(t, ℓ)mn|=|∂^β(e^imA(t)u(t, ℓ)b mn)|

=

X

γ≤β

β γ

∂^β−γe^imA(t)∂^γu(t, η)b mn

≤X

γ≤β

β γ

|∂^β−γe^imA(t)||∂^γbu(t, η)mn|

≤X

γ≤β

Cβ|m|^β∂^βbu(t, ℓ)mn

,

where the last inequality comes from an adaptation of Lemma4.4. Sinceu∈C^∞(T¹×S³) and|m| ≤ℓ, again by Theorem3.5, it is easy to see that givenN >0, there existsCβN such that

|∂^βΨdau(t, ℓ)mn| ≤CβN(1 +ℓ)^−N. Therefore Ψau∈C^∞(T¹×S³). The distribution case is analogous.

Proposition 4.7. Let La0=∂t+a0X.Then

Ψa◦L=La0◦Ψa. Proof. We will show that for everyu∈C^∞(T¹×S³) we have

Ψ\a(Lu)(t, ℓ)mn=La\0(Ψau)(t, ℓ)mn,