UNIVERSIDADE FEDERAL DO PARAN ´A RICARDO PALEARI DA SILVA GLOBAL ANALYTIC HYPOELLIPTICITY FOR A CLASS OF LEFT-INVARIANT OPERATORS ON T

GLOBAL ANALYTIC HYPOELLIPTICITY FOR A CLASS OF LEFT-INVARIANT OPERATORS ONT¹×S³

CURITIBA FEVEREIRO 2020

GLOBAL ANALYTIC HYPOELLIPTICITY FOR A CLASS OF LEFT-INVARIANT OPERATORS ONT¹×S³

Tese apresentada como requisito parcial `a obtenc¸˜ao do grau de Doutor em Matem´atica, no Curso de P´os-Graduac¸˜ao em Matem´atica, Setor de Ciˆencias Exatas, da Universidade Federal do Paran´a.

Orientador: Prof Dr. Alexandre Kirilov

CURITIBA FEVEREIRO 2020

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Apresentamos uma caracterizac¸˜ao completa da hipoeliticidade global anal´ıtica de uma classe de operadores de primeira ordem definidos em alguns produtos de grupos de Lie compactos, principalmenteT¹ ×S³. No caso de coeficientes com valores reais, provamos que o operador ´e conjugado a um operador com coeficientes constantes e que tal conjugac¸˜ao preserva a hipoeliticidade global anal´ıtica. No caso em que a parte imagin´aria n˜ao ´e identicamente nula, n´os mostramos que o operador ´e globalmente anal´ıtico hipoel´ıtico se a condic¸˜ao(P) de Nirenberg-Treves vale em conjunto com uma condic¸˜ao Diofantina. Tamb´em estendemos parte de nossos resultados para uma classe de operadores definidos em produtos da formaT¹×S³× · · · ×S³.

Palavras-chaves: Hipoeliticidade Global Anal´ıtica. S´eries de Fourier em gru- pos de Lie compactos. Condic¸˜oes Diofantinas anal´ıticas. Condic¸˜ao (P) de Nirenberg-Treves.

We present a complete characterization to the global analytic hypoellipticity of a class of first-order operators defined on some products of compact Lie groups, mainlyT¹ ×S³. In the case of real-valued coefficients, we prove that the op- erator is conjugated to a constant coefficient operator and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is not identically zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition (P) holds in addition to a Diophantine condition. We also extend part of our results for a class of operators defined on products of the typeT¹×S³× · · · ×S³.

Keywords: Global analytic hypoellipticity. Fourier Series on compact Lie groups. analytic Diophantine conditions. Nirenberg-Treves condition(P).

Introduction 1

1 Preliminaries: Fourier Analysis on Compact Lie Groups 5

1.1 Representations of Topological Groups. . . 5

1.2 Fourier Series and Trigonometric Polynomials . . . 15

1.3 Function Spaces . . . 17

1.4 Partial Fourier Series . . . 23

2 Preliminaries: Representation Theory and Fourier Analysis onSU(2) 26 2.1 The Lie groupSU(2)and the Euler angles . . . 26

2.2 Invariant differential operators onSU(2) . . . 28

2.3 Irreducible representations ofSU(2) . . . 30

2.4 Fourier Analysis onSU(2) . . . 38

3 Invariant constant-coefficient operators 40 3.1 Global analytic hypoellipticity for constant-coefficient operators . . . 40

3.2 Special classes of operators and equivalent analytic Diophantine conditions . . 43

3.2.1 Constant coefficient operators onT¹ ×S³ . . . 46

4 A class of invariant evolution operators 49 4.1 A necessary condition to the global analytic hypoellipticity . . . 50

4.2 The Nirenberg-Treves condition (P) . . . 53

4.3 Singular solutions . . . 55

4.4 Main Theorem and examples . . . 61

5 Evolution operators with more variables 66

Since the 1970s, the property called Global Hypoellipticity is being studied for diffe- rent classes of (pseudo-)differential operators defined on different manifolds. The pioneering work in this area is the paper [17] of S. Greenfield and N. Wallach, which relates the global hypoellipticity of constant-coefficient vector fields defined on tori with the growth of the opera- tor’s symbol at infinity. In particular, on the 2-torus, this property translates into a Diophantine condition, that is, this study of this property becomes a problem on approximation by rational numbers.

This leads us to one of the major problems in this area, the Greenfield’s and Wallach’s conjecture [18], which claims the following: if a vector field defined on a closed connected orientable manifold is globally hypoelliptic, then the manifold is diffeomorphic to a torus and the vector field is conjugated to a Diophantine constant vector field. There are positive partial answers for this conjecture, for example, it is true in dimensions2and3([21] and [16]).

Then, a lot of different directions are natural to consider. For example, one can con- sider systems of partial differential equations; classes of pseudo-differential operators; or more general classes of regularity, like the classes of Gevrey and Komatsu. In this work, we are inter- ested in investigating the property called Global Analytic Hypoellipticity for a class of operators defined on compact Lie Groups.

Our choice is natural in the sense that the standard approach used by Greenfield and Wallach is based in Fourier analysis in tori; and on compact Lie Groups, there is a very well established Fourier theory.

The sense of an operator being globally hypoelliptic or being globally analytic hypoel- liptic in this work are the following.

Definition 0.1. LetGbe a compact Lie group. A linear operatorP :D(G)→ D(G)is called globally hypoelliptic (GH) if the following condition is true:

u∈ D(G)andP(u)∈C^∞(G)⇒u∈C^∞(G).

Similarly, we say thatP is globally analytic hypoelliptic (GAH) if the following condi- tion is true:

u∈ D(G)andP(u)∈C^ω(G)⇒u∈C^ω(G).

In order to give a friendly version of our results at this moment, avoiding concepts that will be carefully introduced in the first two chapters of this work, we will state our results in T¹ ×S³. However, we note that, under certain conditions, it is possible to obtain more general versions.

We start by considering first-order operators of form

P =∂_t+ (a+ib)(t)∂₀+q, (1) whereq ∈ C,a, b :T¹ → Rare real-analytic functions and∂₀ is the left-invariant vector field onS³known as the neutral operator.

Global analytic (and smooth) hypoellipticity of vector fields and system of vector fields has been extensively studied in tori, within which we cite as most important and inspiring for this project [2,3,4,7,12,17,20,21].

In the specific case of global analytic hypoellipticity on the torus T², Bergamasco proved in [2] that∂_t+ (a(t) +ib(t))∂_x,is (GAH) if and only if eitherb(t)does not change sign orb ≡0and the real numbera₀ = ₂¹_{π 2}π

0 a(t)dtis neither rational nor an exponential-Liouville number.

Recall that an irrational numberλis said to be an exponential Liouville number if there exists >0such that the inequality|λ−p/q| ≤e^−qhas infinitely many rational solutionsp/q.

Next, in [8], Bergamasco and Zani proved that, if there exists a non-singular, glob- ally analytic hypoelliptic vector field Lon a compact surface M, then M is real analytically diffeomorphic toT² and, either, the Nirenberg-Treves condition (P) holds inM, or there are coordinates on which we can writeL =g(x, t)(∂_t+λ∂_x), whereg = 0everywhere andλis a real number which is neither rational nor exponential-Liouville.

Our results in T¹ × S³ recover part of the behavior identified by Bergamasco and Zani in dimension 2, involving the Nirenberg-Treves condition (P) and an analytic Diophantine condition, suggesting the existence of a real analytic version of the famous Greenfield’s and Wallach’s conjecture, see [16].

In the case of constant-coefficients operators, our main result is as follows.

Theorem 0.2. Let c, q ∈ C. The operatorL = ∂_t+c∂₀ +q is globally analytic hypoelliptic if and only if the following condition holds: for allB > 0, there isK_B > 0such that for all k, ∈Z,

k+ ¹₂c−iq≥K_Be^−B(|k|+||). (ADC3) For example, writingc=a+ib, witha, b∈ R, ifb = 0andRe(q)/b /∈ ¹₂Z, thenLis (GAH). And whenb = 0 andiq ∈ Z, thenLis (GAH) if and only if ais neither rational nor exponential-Liouville.

This theorem follows directly from Propositions3.1 and3.5, Remark3.8and Lemma 3.9, while the details of the above example are given in Example3.10.

For the general case (1), we introduce the following notation:

P₀ .

=∂_t+ (a₀+ib₀)∂₀+q, where

a₀ = 1 2π

₂π 0

a(s)ds and b₀ = 1 2π

₂π 0

b(s)ds.

Theorem 0.3. The operator P = ∂_t + (a +ib)(t)∂₀ +q is (GAH) if and only if one of the following conditions holds:

1. ifb≡0thenbdoes not change sign; and either Re(q)

b₀ ∈/ ¹₂Z or Im(q) + Re(q)a₀ b₀ ∈/ Z. 2. ifb≡0, then the condition(ADC3)holds; and either

Re(q)= 0 or Im(q)∈/ Z+ a₀ 2 Z.

The previous theorem is a consequence of Proposition4.5and Theorems4.3,4.10and 4.11. Part of our proofs follows the ideas for the smooth case used in [4, 12,13] which relies heavily on the use of cut-off functions. One of the difficulties of adapting such arguments is that in the analytic case there are no such functions. To overcome this problem we draw on ideas used by Bergamasco, Nunes and Zani in [6] to construct a singular solution, and results of Sjostrand, see [26], about the asymptotic behavior at the infinity of sequence of integrals involving analytic functions. Let us add one more remark about the theorem we just stated. It is not exactly the same one as the reader will find later in this thesis, but another formulation of it.

It is easy to see that the algebraic conditionN0 =∅together with the hypothesisb₀ = 0imply the so called (ADC) condition (see Example3.10), so Theorem0.3will hold.

Finally, this thesis is organized as follows. Chapters 1 and 2 are dedicated to prelim- inary definitions and results. In Chapter 1 we discuss how the Fourier theory is defined on compact groups. This is based essentially on representation theory. We state the main defini- tions and results that will be used in this thesis. In Chapter 2 we discuss the particular case of the Lie groupSU(2). It has a very nice and well-understood representation theory, which allows us to explicit every calculation and do Fourier theory by hand. There is no contribution by the author on these two chapters, everything there can be found in the suggested literature.

Next, in Chapter 3, we characterize completely the global analytic hypoellipticity for first-order constant-coefficient operators defined on a product of compact Lie groups. We also obtain equivalent analytic Diophantine conditions that will be important in the next chapters and to construct examples. In Chapter 4 we study the class of invariant evolution operators defined by (1). We obtain necessary and sufficient conditions as announced in Theorem 0.3. Finally, in the last chapter, we extend the results to invariant evolution operators with more variables defined onT¹×S³×...×S³.

Chapter 1

Preliminaries: Fourier Analysis on Compact Lie Groups

In this chapter, we introduce the basic definitions, notations and preliminary results necessary for the development of this thesis. In the first section, we introduce the basic notions of Representation Theory and define the most important types of representations we are con- cerned about in this thesis. We end this section with one of the most important results in that area, which is known as the Peter-Weyl’s Theorem. The second section is shorter and defines the Fourier coefficients of functions and distributions and their Fourier series. The third sec- tion is devoted to state results that show how the rate of decay of Fourier coefficients classify functions and distributions. A very careful presentation of these concepts and complete proofs of all the results presented in the first three sections can be found in the references [15] (chap- ters 1 and 2) and [25] (chapters 7, 8 and 10). The last section of this chapter is dedicated to results concerning partial Fourier coefficients on products of compact Lie groups and the main reference for this part is [23].

1.1 Representations of Topological Groups

IfGis a group, a representation of Gis a pair(V, ϕ)consisting of a (complex) vector spaceV together with a group morphismϕ:G→GL(V). SometimesV is called aG-module.

The dimension of the representation is the dimension of the vector spaceV. If dimV <∞, we will use the notationd_ϕ := dimV. If V has an inner product, we say that (V, ϕ) is unitary if Im(ϕ) ⊂ U(V) = {T : V → V;T^∗ = T⁻¹}. Some of the notations that are commonly used

areg·v :=ϕ(g)(v) =: ϕ_g(v). In this way, it is common to say thatV itself is a representation, without mention the specific mapϕ, which is usually implicit. We also say thatGacts on V. IfW ⊂ V is such thatg·w ∈ W for allw ∈ W, we say thatW is aG-invariant subspace (or a subrepresentation, or aG-submodule of V). Finally, we say that V is irreducible if its only G-invariant subspaces are{0}andV itself.

Example 1.1. LetGbe a group andV =F(G)the vector space of complex-valued functions defined onG. Considerπ_L :G→GL(F(G))defined by

((π_L)(g))(f)(x) :=f(g⁻¹·x),

for eachg, x∈Gandf ∈ F(G). Similarly, we can considerπ_R:G→GL(F(G))defined by (π_R(g))(f)(x) =f(x·g)

forg, x∈Gandf ∈ F(G). It is easy to see thatπ_Landπ_Rare group morphisms, soF(G)is a representation ofGin at least two different ways.

Example 1.2. LetV =Cwith the standard inner product. In this case, we haveGL(C) =C^∗ andU(C) = U(1) = S¹ =T¹. For eachξ ∈ G= Rⁿ, define e_ξ : Rⁿ →S¹ bye_ξ(x) = e^ix·ξ. It is clear that the correspondenceξ →e_ξdefines a unitary representation ofRⁿof dimension 1, in particular, irreducible. Similarly, if ξ ∈ Zⁿ, thene_ξ : Tⁿ → S¹ is well defined and of course is also a morphism of groups. Thus, the correspondenceξ → e_ξ defines a unitary and one-dimensional representation ofTⁿ.

Definition 1.3. LetGbe a group. Ifϕ:G→GL(V)andψ :G→GL(W)are representations ofG, we define a morphism between(ϕ, V)and(ψ, W)as a linear mapA:V →W such that for allg ∈ G, v ∈ V, we have A(ϕ_g(v)) = ψ_g(A(v)). In the short notation of action, this relation is justA(g ·v) = g ·A(v), that is,Ais a linear map that commutes with the actions ofGonV and onW. We say that (ϕ, V)and(ψ, W)are equivalent (or isomorphic), if there exists such a mapAwhich is also an isomorphism of vector spaces. In this case, we will use the notationϕ∼ψ.

Proposition 1.4. If the representations(ϕ, V)and(ψ, W)are irreducible andA : V → W is a morphism between these two representations, thenA= 0orAis an isomorphism.

Proof. It is enough to observe that KerAand ImAare subrepresentations ofV.

Corollary 1.5(Schur Lemma). If ϕ : G → GL(V) is an irreducible representation of finite dimension andA :V →V is a morphism of representations, then there existsλ ∈Csuch that A =λ·IdV.

Proof. Since dimV < ∞, A has some eigenvalue λ ∈ C, so A−λIdV is not invertible. But A−λIdV is also a morphism of representations, so by the above proposition we must have A−λIdV = 0.

Corollary 1.6. LetGbe an abelian group. Ifϕ:G→GL(V)is an irreducible representation of finite dimension, thendimV = 1.

Proof. For each fixedg ∈G, the mapϕ_G :V →V if a morphism of representations because ϕ_g(h·x) =ϕ_h(ϕ_g(x)) =ϕ_gh(x) =ϕ_hg(x) = h·ϕ_g(x).

By Schur Lemma, there existsc ∈ Csuch that ϕ_g = cIdV, so ifv ∈ V, v = 0, thenϕ_g(v) = cv. Hence, span{v} is a subrepresentation ofV. Since span{v} = {0}, we must have V = span{v}.

By the above corollary, we have in particular that every irreducible unitary representa- tion ofTⁿmust be a morphism of groupsf :Tⁿ→S¹.

Theorem 1.7. Iff : Tⁿ → S¹ ⊂ C is a group morphism such that f ∈ L¹(Tⁿ), then there existsξ ∈Zⁿwithf =e_ξ.

Proof. Suppose for now that n = 1. We can think the function f as a 2π-periodic function f :R → S¹. Sincef ≡ 0, there existsλ > 0such thatΛ := λ

0 f(τ)dτ = 0. So, ifx ∈ T¹ is fixed, we have

f(x) = Λ⁻¹ λ

0

f(x)f(τ)dτ = Λ⁻¹ λ

0

f(x+τ)dτ = Λ⁻¹ x+λ

x

f(τ)dτ.

From the above formula, it follows by induction thatf ∈ C^∞(T¹). Taking the derivative of f we get

f(x) = Λ⁻¹f(x+λ)−Λ⁻¹f(x) = Λ⁻¹f(x).f(λ)−Λ⁻¹f(x) = (Λ⁻¹f(λ)−Λ⁻¹)f(x). So the functionf satisfies the differential equationf =C₀f, whereC₀ = Λ⁻¹(f(λ)−1)∈C. Hence, we must have f(x) = f(0)e^C0x and since |f(0)| = 1, we have |f(x)| = |e^C0x| = e^Re(C0x). But|f(x)| = 1for all x ∈ R, so Re(C₀) = 0, which impliesC₀ = 2πiξ for some

ξ ∈ R. Finally, sincef(x) = e^i2πx·ξ is periodic, we conclude thatξ ∈ Z. For the general case we write

f(x) =f(x₁e₁+...+x_ne_n) =f(x₁e₁)·...·f(x_ne_n).

Each functionx → f(x·e_j)is a group morphismf_j : T¹ → S¹. Iff is integrable, then each functionf_j ∈ L¹(T¹). We apply the already proved result for eachj and conclude the proof of the Theorem.

Definition 1.8. If G is a topological group, a representation ϕ : G → GL(V) is strongly continuous if for eachv ∈ V the mapϕ_v : G → V, g → ϕ_v(g) .

= ϕ(g, v), is continuous. We say thatϕis topologically irreducible if the only closed subrepresentations ofV are{0}andV itself. We say thatϕis cyclic if there existsv ∈V such thatV =span(ϕ(G)(v)). Such a vector will be called a cyclic vector.

The proof of the next Proposition can be done using standard arguments of Zorn’s Lemma.

Proposition 1.9. Ifϕ : G→ GL(V)is strongly continuous, then there exist a family{V_λ}λ∈Λ

of cyclic subrepresentations ofV such thatV =

λ∈Λ

V_λ.

It is a well known fact that every compact topological groupGhas a unique normal- ized Haar measure, that is, a measure μthat is bi-invariant (with respect to all left and right translations), invariant by inversion andμ(G) = 1. In the case whereGis a compact Lie group, this existence is much more easier to prove. The idea is as follows. First, take any left-invariant metric onG(take any inner product on its Lie algebra and spread it on the whole group using left-translations) and consider the associated (positive) left-invariant volume form. Then we do the standard average process, that is, we define a new metric now using right-translations and integrate the resulting function overG. It is easy to see that the resulting metric is bi-invariant.

Finally, one just normalized this metric to get the Haar measure ofG. All groups we are going to deal with in this thesis will be compact Lie groups. We will assume that any integral taken over any compact groupGwill be with respect to its Haar measure. Besides that, from now on, every representationϕ :G→ U(H)will be unitary withHbeing a Hilbert space.

Lemma 1.10. LetGbe a compact group andϕ:G→ U(H)be an unitary strongly continuous representation. Suppose thatw∈ U is a cyclic vector withw= 1. Then

u, v_ϕ :=

G

ϕ(x)u, w_Hw, ϕ(x)v_Hdμ_G

defines another inner product inH. Moreover,ϕ is also unitary with respect to this new inner product anduϕ ≤ u_Hfor allu∈ H.

Proof. For each fixedu∈ H, consider the functionf_u(x) :=ϕ(x)u, w_H. Then,

|f_u(x)−f_u(y)| = |ϕ(x)u, w_H− ϕ(y)u, w_H|

= |(ϕ(x)−ϕ(y))u, w_H|

≤ (ϕ(x)−ϕ(y))u_Hw_H

= (ϕ(x)−ϕ(y))uH.

But(ϕ(x)−ϕ(y))u_H → 0asx→ysinceϕis strongly continuous, sof_u ∈C⁰(G). In this way,u, v_ϕ =

Gf_uf_vis well defined. It is clear that this pairing is bilinear, anti-hermitian and non-negative. Also, ifu, u_ϕ = 0, then sincef₀ is continuous we must haveϕ(x)u, w_H = 0 for allx∈ G. Sinceϕis unitary, this is equivalent toϕ(x⁻¹)w, u_H = 0for allx∈ G. Since wis cyclic, this implies thatu= 0. So·,·_ϕis an inner product. Now, ifu∈ U, we have

u²_ϕ =

G

|ϕ(x)u, w|² ≤

G

ϕ(x)u²_H w²_H

=1

=

G

u²_H=u²_H,

souϕ ≤ uH. Finally, let us prove thatϕis unitary with respect to·,·_ϕ: u, ϕ^∗(y), v_ϕ = ϕ(y)u, v_ϕ

=

Gϕ(xy)u, ww, ϕ(x)vdμ(x)

=

Gϕ(xy)u, ww, ϕ(xy)ϕ(y⁻¹)vdμ(x)

=

Gϕ(x)u, ww, ϕ(x)(ϕ(y⁻¹v))dμ_x

= u, ϕ(y⁻¹)v_ϕ.

Lemma 1.11. Let·,·_ϕ be the inner product defined in the above lemma. There exists a com- pact, self-adjoint, positive-definitive operator A ∈ B(H), which is also a morphism of repre- sentations that satisfiesu, Av_H =u, v_ϕ.

Proof. For each fixedv ∈ H, the mapF_v(u) :=u, vϕ is a continuous linear functional:

|F_v(u)| ≤ uϕvϕ ≤ u_Hv_H,

that is,F_vϕ ≤ vϕ. By Riesz’s Lemma, there exists an uniqueA(v)∈ Hsuch that F_v(u) = u, A(v)_Hfor allu∈ H.

• Ais bounded;

Av²_H=Av, Av_H =Av, v_ϕ ≤ Avϕvϕ ≤ Av_Hv_H, So, ifAv= 0, thenAv_H≤ v_H.

• Ais self-adjoint;

u, A^∗v_H =Au, v_H=v, Au_H=v, u_ϕ =u, v_ϕ =u, Av_H, soA^∗ =A.

• Ais positive-definite; Just note thatu, Au_H=u, u_ϕ.

• Ais a morphism of representations;

u, A(ϕ(y)v)_H = u, ϕ(y)v_ϕ

= ϕ⁻¹(y)u, vϕ

= ϕ⁻¹(y)u, Av_H

= u, ϕ(y)Av_H, for allu, v ∈ H, soA◦ϕ(y) = ϕ(y)◦A. for ally∈G.

• Ais compact;

LetB = {u∈ H;u_H ≤ 1}, our goal is to prove thatA(B)⊂ His compact. Let(v_j) be a sequence on A(B) and(u_j) a sequence on B such that A(u_j) = v_j for all j. By Banach-Alaoglu’s Theorem, B is weakly compact, so there exists a sub-sequence(u_jk)k

which is weakly-convergent to some u ∈ B, that is,u_jk, v_H → u, v_Hfor allv ∈ H. Let us prove thatv_jk →Au.

v_jk −Au²_H = A(u_jk)−Au²_H=A(u_jk−u), A(u_jk −u)_H

= A(u_jk −u), u_jk−u_ϕ =

Gg_kdμ_G, whereg_k(x) =ϕ(x)A(u_jk −u), w_Hw, ϕ(x)(u_jk−u)_H. Note that

|g_k(x)| = |ϕ(x)A(u_jk −u), w_H| · |w, ϕ(x)(u_jk−u)_H|

= |u_jk −u, Aϕ(x⁻¹)w_H| · |ϕ(x⁻¹)w, u_jk −u_H|

≤ u_jk −uH AH

≤1

wH

=1

u_jk −uH

≤ u_jk −u²_H≤2( u_jk²_H

=1

+ u²_H

=1

) = 4.

So the sequence of functions {g_k}k is dominated by a constant function, which lives in L¹(G) because μ(G) = 1 < ∞. Besides that, u_jk−u, v_H → 0 for all v ∈ H, so g_k→0pointwise. Hence, we can apply Dominated Convergence Theorem and conclude that

Gg_k →0, which impliesv_jk →Au.

Recall that ifA∈ B(H)is compact and self-adjoint, then

• σ(A)is countable;

• dim Ker(A−λId)<∞if0=λ∈σ(A);

• σ(A)\ {0}is discrete;

• H=

λ∈σ(A)Ker(A−λId).

Corollary 1.12. Let G be a compact group and ϕ : G → U(H) be a strongly continuous unitary representation on a Hilbert spaceH. Then there exists a decomposition ofHin a direct sum (as representations) of a family of subrepresentations ofHwhich are irreducible and finite dimensional.

Proof. We already saw that every strongly continuous representation is a direct sum of cyclic representations, so we can suppose without loss of generality that ϕ is cyclic. Consider the operator A from the above lemma. Since A is positive-definite, we have KerA = 0, so dim(KerA− λId) < ∞ for all λ ∈ σ(A) and H =

λ∈σ(A)(KerA−λId). Since A is a morphism of representations, each subspace KerA−λId is a subrepresentation. Besides that, each subspace Ker−λId can also be decomposed as a sum of irreducible subrepresentations since they are finite dimensional, so isH.

Corollary 1.13. Irreducible strongly continuous unitary representations of compact groups are finite dimensional.

Definition 1.14. IfGis compact, we denote byRep(G)the set of all irreducible strongly conti- nuous unitary representations of G. IfG is just locally compact, in the definition of the set Rep(G)we just change the condition strongly continuous by continuous. The unitary dual ofG isG =Rep(G)/∼, where we identify isomorphic representations.

It is important to know that the setGis countable ifGis compact. This is a very well known fact and it has some generalizations, see for example [14].

Example 1.15. We have thatRⁿ ∼= Rⁿ. In fact, the correspondence Rⁿ → Rⁿ,ξ → [e_ξ], is a bijection. Similarly, we haveTⁿ ∼=Zⁿsince the correspondenceZⁿ → Tⁿ, ξ →[e_ξ], is also a bijection.

Proposition 1.16. LetGbe a compact group. Ifξ ∈ G, with dimξ = m, thenξ has a matrix representative, that is, there exists a continuous morphism of groupsϕ : G → U(m)such that ξ= [ϕ].

Proof. Letψ ∈ ξ, ψ : G → U(H), dimH < ∞, and fix an orthonormal basisβ ={e_j}^mj=1 of H. Define, for eachx∈G, the matrix

ϕ_ij(x) := ψ(x)e_j, e_i,1≤i, j ≤m,

which is the matrix that representsψ(x) ∈ U(H)in the basis β, so ϕ = (ϕ_ij) : G → U(m) is a unitary representation. Let us see thatξ ∼ ϕ. If {˜e_j}^mj=1 denoted the canonical basis of C^m, let A : H → C^m the unique isomorphism of vector spaces such that A(e_j) = ˜e_j for all j = 1,2, ..., m. Ifv =

jλ_je_j ∈ H, then ϕ(x)Av=ϕ(x)

m j=1

λ_je˜_j = m

j=1

λ_j(ϕ(x)˜e_j) = m

j=1

λ_j m

i=1

ψ(x)e_j, e_i˜e_i.

On the other hand,

A(ψ(x)v) = A

ψ(x) m

j=1

λ_je_j

=A _m

j=1

λ_jψ(x)e_j

= A

_m

j=1

λ_j m

i=1

ψ(x)e_j, e_ie_i

= m

i=1

⎛

⎝^m

j=1

λ_jψ(x)e_j, e_iAe_i

=˜e_i

⎞

⎠=ϕ(x)Av.

Lemma 1.17. LetGbe a compact group,ξ, η ∈G, ξϕ = (ϕ_ij)^mij,=1 andηψ = (ψ_kl)ⁿk,l=1. Then,

ϕ_ij, ψ_klL2(G) =

⎧⎨

⎩

0 if ξ =η

1

mδ_ikδ_jl if ξ =η .

Proof. Fix1≤j ≤m,1≤l≤n, and defineE_pq = 1ifp=j, q =landE_pq = 0in any other case, withE ∈C^m×n. We also define

A:=

G

ϕ(y)Eψ(y)⁻¹dμ_G(y)∈C^m×n. Note thatAis a morphism of representations because:

ϕ(x)A =

G

ϕ(x)ϕ(y)Eψ(y⁻¹)dμ(y)

=

G

ϕ(xy)Eψ((xy)⁻¹x)dμ(y)

=

G

ϕ(y)Eψ(y⁻¹x)dμ(y)

=

G

ϕ(y)Eψ(y⁻¹)ψ(x)dμ(y) =Aψ(x).

By Schur’s Lemma, sinceϕandψare irreducible of finite dimension we must have A=

⎧⎨

⎩

0 if ϕ∼ψ λ·Id if ϕ=ψ for someλ∈C. On the other hand,

A_ik =

G

p

q

ϕ_ip(y)E_pqψ_qk(y⁻¹)dμ=

G

ϕ_ij(y)ψ_lk(y⁻¹)dμ

=

G

ϕ_ij(y)ψ_lk(y)^∗dμ=ϕ_ij, ψ_klL2(G).

Ifϕ ∼ ψ, thenA= 0, which impliesϕ_ij, ψ_kl= 0for alli, j, k, l. Ifϕ =ψ, thenm =n and ϕ_ij, ϕ_kl=A_ik = 0ifi=k becauseAis diagonal. Now

tr(A) =

G

tr(ϕEϕ⁻¹)dμ=

G

tr(E)dμ=δ_jl=λ·m.

hence

A_ii =ϕ_ij, ϕ_il=λ = δ_jl m.

Recall the basic representations π_L : G → GL(F(G)), π_R : G → GL(F(G)), given by translations of functions. Iff ∈L²(G)andg ∈G, then

π_L(g)(f)²₂ =

G

|f(g⁻¹x)|²dμ=

G

|f(x)|²dμ=f²₂,

soL²(G)is aG-invariant sub-space ofF(G). Similarly, π_Ralso restricts toL²(G). The same calculation above also showed that these representations are unitary. In this way, we can con- sider the representationsπ_L : G → U(L²(G))andπ_R : G → U(L²(G)). Also, from now on,

we will follow a certain convention. When one gives an elementξ ∈ G, we will choose once and for all a matrix representativeϕ : G → U(d_ξ)in the class ofξ. The notationξ = [ϕ]will be used suggesting the notation of the already chosen matrix representativeϕin that class.

Theorem 1.18(Peter-Weyl). LetGbe a compact group. Then B={

d_ϕ·ϕ_ij, ϕ:G→ U(d_ϕ); [ϕ]∈G}, is an orthonormal basis ofL²(G). Moreover, for each[ϕ]∈G, we have

• H^ϕi :=span{ϕ_ij; 1≤j ≤d_ϕ} ⊂L²(G)isπ_R-invariant,

• ϕ ∼π_R H^ϕ_i

;

• L²(G) =

[ϕ]∈G d_ϕ

i=1

Hi^ϕ;

• π_R ∼

[ϕ]∈G d_ϕ

i=1

ϕ.

Proof. Fix1≤i≤d_ϕ. The fact thatH^ϕi isπ_R-invariant is direct.

• ϕ ∼π_R|_H^ϕ_i;

Let {e_i} be the canonical basis ofC^dϕ, so ϕ(y)e_j =

kϕ_kj(y)e_k. Consider the linear isomorphism A : C^dϕ → H^ϕi such that A(e_j) = ϕ_ij for allj = 1, ..., d_ϕ. ThenA is a morphism between representations because

π_R(y)A(e_j) = π_R(y)(ϕ_ij) =

k

ϕ_kj(y)ϕ_ik, and

Aϕ(y)(e_j) =A

k

ϕ_kj(y)e_k

=

k

ϕ_kj(y)ϕ_ik, soϕ∼π_R|_H^ϕ_i.

• Bis an orthonormal basis ofL²(G);

We already know thatB is an orthonormal set. LetH :=

[ϕ]∈G

d_ϕ

i=1H^ϕi and suppose that H = L²(G), so H^⊥ = {0}is π_R-invariant. We know that π_R|_H^⊥ is a sum of irre- ducible finite dimensional unitary representations, so there exists a no trivialE ⊂ H^⊥and

a matrix unitary representationϕ = (ϕ_ij) withϕ ∼ π_R|E. Let{f_j}^di=1^E be the orthonor- mal basis ofEsuch thatπ_R(y)f_j =

iϕ_ij(y)f_i. This last equality happens inL²(G), in particular, for almost allx∈Gwe havef_j(xy) =

iϕ_ij(y)f_i(x). Consider now the sets N(y) = {x∈G;f_j(xy)=

iϕ_ij(y)f_i(x)}

M(x) = {y ∈G;f_j(xy)=

iϕ_ij(y)f_i(x)}

K = {(x, y)∈G×G;f_j(xy)=

iϕ_ij(y)f_i(x)},

so we know thatμ_G(N(y)) = 0. Note that N(y)is they-section ofK andM(x)is the x-section ofK. By Fubini’s Theorem,

μ_G(K) =

G

μ_G(M(x)) ≥0

dμ(x) =

G

μ(N(y)) =0

dμ_G= 0,

soμ_G(M(x)) = 0for almost allx∈G. Letx₀be a point such thatμ_G(M(x₀)) = 0, then f_j(x₀y) =

i

ϕ_ij(y)f_i(x₀) holds for almost ally∈G. Ifz =x₀y, we will have

f_j(z) =

i

ϕ_ij(x⁻¹₀ z)f_i(x₀) =

i

j

ϕ_ik(x⁻¹₀ )ϕ_kj(z)f_i(x₀)

=

k

ϕ_kj(z)

⎛

⎜⎜

⎜⎜

⎝

i

ϕ_ik(x⁻¹₀ )f_i(x₀)

:=λ_k

⎞

⎟⎟

⎟⎟

⎠, sof_j(z) =

kλ_kϕ_kj(z)holds for almost allz ∈G. Hence, f_j ∈span{ϕ_kj; 1≤k ≤d_E} ⊂

d_E

k=1

H^ϕk ⊂ H,

soE ⊂ H ∩ H^⊥ ={0}, which is contradiction.

Using the notation of the above Theorem, we will denote byH^ϕ the spaced_ϕ i=1H^ϕi.

1.2 Fourier Series and Trigonometric Polynomials

Here we introduce the trigonometric polynomials and finally define the Fourier coeffi- cients ofL² functions on compact groups.

Definition 1.19. Let G be a compact group and B = {

d_ϕ ·ϕ_ij;ϕ = (ϕ_ij),[ϕ] ∈ G} the basis given by Peter-Weyl’s Theorem. The space of trigonometric polynomials is defined by Trig Pol(G) =span(B).

Example 1.20. Iff ∈Trig Pol(Tⁿ), then f(x) =

ξ∈Zⁿ

f(ξ)e^2πix·ξ,

wheref(ξ)= 0at most on a finite number ofξ ∈Zⁿ.

Theorem 1.21. The spaceTrig Pol(G)is a dense sub-algebra inC⁰(G).

The proof of this Theorem consists in to show that Trig Pol(G)is an involutive subal- gebra ofC⁰(G)that separate points. Hence the result follows by Stone-Weierstrass Theorem.

Corollary 1.22. In a compact groupG, for allf ∈L²(G)we can write

f =

[ϕ]∈G

d_ϕ

d_ϕ

i,j=1

f, ϕ_ijL2(G)·ϕ_ij,

which will be called the Fourier Series off. Moreover, a Plancherel equality type holds f²₂ =

[ϕ]∈G

d_ϕ

d_ϕ

i,j=1

|f, ϕ_ijL2(G)|².

Definition 1.23. LetGbe a compact group,f ∈ L¹(G)andϕ = (ϕ_ij)^di,j^ϕ=1 with[ϕ] ∈ G. We define theϕ-coefficient of the Fourier series off as the matrix

(f(ϕ))ij :=

G

f(x)ϕ^∗_ijdμ_G = f, ϕ^∗_ij

=f, ϕ_jiL2(G), for1≤i, j ≤d_ϕ.

Proposition 1.24. LetGbe a compact group andf ∈L²(G), then f(x) =

[ϕ]∈G

d_ϕtr

f(ϕ)·ϕ(x)

,

and this series converges for almost allx ∈ G, and also in L²(G). The Plancherel’s formula becomes

f²₂ =

[ϕ]∈G

d_ϕtr

f(ϕ)f(ϕ)^∗

.

Proof. Just note that

tr(f(ϕ)ϕ(x)) =

i

f(ϕ)ϕ(x)

ii=

i,j

f(ϕ)ijϕ_ji(x) =

i,j

f, ϕ_jiϕ_ji(x),

which conclude the first part. For the second part, just recall that tr(AA^∗) =

i,j|A_ij|². Example 1.25. Forf ∈L²(Tⁿ), its Fourier Series is given by

f(x) =

k∈Zⁿ

f(k)e^i2πx·k

wheref(k) =

Tⁿf(x)e^−i2πx·kdx.

1.3 Function Spaces

Now we present results about characterizations of smooth functions and distributions in terms of their Fourier coefficients. So, from now on,G will denote a compact Lie group.

Recall that we already fixed a bi-invariant volume form onG, so that it defines the Haar measure onG. One particular operator that helps the characterization of smooth functions onGis the Laplacian operator. One way to define it is by taking the general Laplacian-Beltrami operator associated with the fixed bi-invariant Riemannian metric on G. A more constructive way is the following. IfGis semi-simple, then the Laplace operator on Gcan be identified with its Casimir elementΩ ∈ U(G), where U(G)denotes the universal enveloping algebra ofG. IfG is not semisimple, one decomposes its Lie algebra as a direct sum of its semisimple part with its center, take a basis of the center{U_i}^ki=1of its Lie algebra, and considerL= Ω +

iU_i². It can be proved thatLcoincides with the Laplacian-Beltrami operator ofG(see pg 331 of [19]).

The following Proposition is very important when one considers decay properties of Fourier coefficients.

Proposition 1.26. For every[ξ] ∈ G, the space H^ξ is an eigenspace ofL and−L|_H^ξ = λ_ξId, for someλ_ξ ≥0.

Proof. Recall that by definition the operatorLis bi-invariant, so it commutes with bothπ_R(x) andπ_L(x)for allx∈G. By Peter-Weyl Theorem, it then commutes with allξ ∈G(every repre- sentation in Rep(G)is the restriction ofπ_Rto some finite-dimensional space). So Lpreserves both the spaces generated by lines and by columns ofH^ξ, and since the set of functions{ξ_ij}