Commutator characterization of pseudo-diﬀerential operators on compact manifolds

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Commutator characterization of pseudo-differential operators on compact manifolds

Artur Henrique de Oliveira Andrade

Dissertação apresentada ao

Instituto de Matemática e Estatística da

Universidade de São Paulo para

obtenção do título de

Mestre em Ciências

Programa: Matemática

Orientador: Prof. Dr. Severino Toscano do Rêgo Melo

Durante o desenvolvimento deste trabalho o autor recebeu auxílio financeiro da CNPq São Paulo, agosto de 2020

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on compact manifolds

Esta versão da dissertação contém as correções e alterações sugeridas pela Comissão Julgadora durante a defesa da versão original do trabalho, realizada em 11/08/2020. Uma cópia da versão original está disponível no Instituto de Matemática e Estatística da Universidade de São Paulo.

Comissão Julgadora:

• Prof. Dr. Severino Toscano do Rêgo Melo - IME-USP

• Prof. Dr. Pedro Tavares Paes Lopes - IME-USP

• Prof. Dr. Gustavo Hoepfner - UFSCar

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Acknowledgments

I would like to thank my family for their constant support not only during my master’s degree, but also throughout my whole life.

I would also like to thank my advisor, professor Severino Toscano do Rego Melo, for allowing me to study under his guidance. His wisdom, patience, and recommendations were paramount for the development of this master’s thesis.

I am grateful to professors Pedro Tavares Paes Lopes and Gustavo Hoepfner who served on my thesis committee and who made valuable suggestions for the final version of this thesis.

I am also grateful to Rodrigo A. H. M. Cabral and his many helpful and insightful discussions that helped me improve this presentation.

My thanks to the National Council for Scientific and Technological Development (CNPq) for the financial support.

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Resumo

Andrade, A. H. O.Caracterização via comutadores de operadores pseudo-diferenciais em variedades compactos. 2020. 72 f. Dissertação (Mestrado) - Instituto de Matemática e Estatís- tica, Universidade de São Paulo, São Paulo, 2020.

Nesse trabalho buscamos apresentar de forma original e detalhada a caracterização de operadores pseudo-diferenciais em variedades compactas via comutadores apresentada em M. Ruzhansky e V.

Turunen [16]. A organização e apresentação também tem como objetivo ser autocontida e acessível a qualquer estudante com conhecimentos básicos em teoria da medida e integração como em G.

Folland [6].

Palavras-chave: Operadores pseudo-diferenciais, comutadores, variedades compactas.

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Abstract

Andrade, A. H. O.Commutator characterization of pseudo-differential operators on compact manifolds. 2020. 72 f. Thesis (Masters) - Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2020.

In this work we seek to present in an original and detailed way the commutator characterization of pseudo-differential operators on compact manifolds presented in M. Ruzhansky and V. Turunen [16]. The organization and presentation also aims to be self-contained and accessible to any student with basic knowledge in measure and integration theory such as in G. Folland [6].

Keywords:Pseudo-differential operators, commutators, compact manifolds.

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Introduction

Roughly speaking, a pseudo-differential operator on Rⁿ is an integral operator acting on a suitable space of functions and is of the form

Aupxq “ ż

Rⁿ

e^2πix¨ξapx, ξqupξqp dξ, (1)

whereupis the Fourier transform of u. The functionain (1) is called thesymbol of A.

It is easier to work with pseudo-differential operators when we impose some quantitative control over its symbol. We shall be concerned with thesymbol class S_1,0^m for somemPR. More specifically, a P S_1,0^m when a: RⁿˆRⁿ Ñ C is smooth on RⁿˆRⁿ and when for all α, β multiindices there exists a constantA_αβ ą0 such that

|B^β_xB_ξ^αapx, ξq|ďA_αβp1`|ξ|q^m´|α| (2) holds for all x, ξ P Rⁿ. We then say that a pseudo-differential operator is of order (less or equal than)mPR if its symbol belongs toS_1,0^m. We denote such class of pseudo-differential operators by Op S^m.

In addition to the symbol classes, there is another way of characterizing pseudo-differential operators onRⁿ, namely, via commutators. In 1977 R. Beals [2] showed how one can characterize the pseudo-differential operators of a given class among the maps fromS toS¹ in terms of boundedness of certain commutators.

In a closely related result, J. Dunau [5, Théorème 1] obtained a characterization of the topology in the space of pseudo-differential operators on compact manifolds in terms of the boundedness of their commutators with differential operators of order at most 1. Then in 1978 R. Coifman and Y. Meyer [3] stated and proved a characterization of pseudo-differential operators of 0-order on compact manifolds via commutators with smooth vector fields. A more general statement for the characterization of pseudo-differential operators of any given order on compact manifolds was formulated by V. Turunen [21]. This later characterization is also presented in M. Ruzhansky and V. Turunen [16].

Although the work of M. Ruzhansky and V. Turunen [16] is a self-contained introduction to the theory of pseudo-differential operators, we believe that a more detailed presentation should be welcomed by those that had no previous exposition to the subject. Thus the main goal of this study is to provide an original and comprehensive presentation, starting from basic facts about Fourier transform and distribution theory, of the result presented by V. Turunen. More specifically, our presentation shall prove a slightly stronger version of the commutator characterization stated by V.

Turunen.

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The structure of this masters thesis is the following:

In chapter 1 we provide the necessary background on distribution theory, Fourier transform, Sobolev spaces and smooth manifolds. All the results are either accompanied by proof or a reference for the interested reader.

Chapter2contains the needed results from pseudo-differential operators onRⁿ. We also provide a commutator characterization that is similar to that presented by R. Beals [2], but for a different class of pseudo-differential operators. All the results presented in this chapter are prerequisites for our commutator characterization on compact manifolds.

Chapter 3 starts with an introduction to distributions and Sobolev spaces over manifolds. We then introduce the definition and some basic properties of pseudo-differential operators on compact manifolds. Lastly, we state and prove our commutator characterization for pseudo-differential operators on compact manifolds.

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Chapter 1

Background and prerequisites

The purpose of this first chapter is to present the necessary background that is needed for a complete understanding of the material.

In section1.1we recall some basic rules of calculus and set up notation that will be used through the rest of the thesis.

Section 1.2gives some elements of the theory of topological vector spaces and distributions on open sets of the Euclidean space. The content and presentation are mostly based on G. Grubb [9].

In section 1.3 we discuss the Fourier transform and some of its properties. Detailed proofs and more properties may be found on L. Grafakos [7].

Section 1.4 covers Sobolev spaces on the Euclidean space and their local versions for open subsets. It contains results that may be found on G. Grubb [9] or on F. Trèves [20].

Finally, in section 1.5we present some basic ingredients on the theory of smooth manifolds. Its presentation is based on J. M. Lee [12] and F. Warner [22].

1.1 Notation

We denote by Z the integers, by N the positive integers and by N0 the nonnegative integers.

We denote by R the set of real numbers and by C the set of complex numbers. Rⁿ is the n- dimensional Euclidean space, with pointsx“ px₁, . . . , x_nq and distance distpx, yq “|x´y|, where

|x|“ px²₁` ¨ ¨ ¨ `x²_nq^1{2. Whenx, yPRⁿ, we writex¨y“ř

1ďiďnxiyi. For a,bPRwithaăbwe denote

ttPR : aătăbu “ pa, bq, ttPR : aďtăbu “ ra, bq, ttPR : aătďbu “ pa, bs, ttPR : aďtďbu “ ra, bs.

Ifx, yPRⁿ, thenrx, ysis used to denote the line segment connecting x and y.

Differentiation of functions on R is denoted by B or by Bx in order to emphasize that the differentiation is with respect to thex-variable. Partial differentiation of functions onRⁿis indicated by

B

Bx_j “ Bxj.

We will make use of the multiindex notation. A multiindex is a vector α “ pα₁, . . . , α_nq P Nⁿ₀. GivenαPNⁿ0 we define its length by|α|“ř_n

i“1αi. ForxPRⁿ we define B_x^α “ B^|α|

Bx^α₁¹ . . . Bx^αnⁿ

, x^α“x^α₁¹ . . . x^α_nⁿ.

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We will also use the conventions

β ďα means β_j ďα_j for all 1ďjďn, α!“α₁!¨ ¨ ¨α_n!,

α˘β “ pα1˘β1, . . . , αn˘βnq, ˆα

β

˙

“ α!

β!pα´βq!.

Constants appearing in inequalities will be usually denoted by a capital letter with an index as e.g.

Cαą0 denotes a positive constant that depends onα.

When M ĂV for some setV, we define the indicator function of M over V by 1Mpxq “

#1 for whenxPM , 0 for whenxPVzM .

If f is any real-valued or complex-valued function on a topological space X, we denote the support off by

suppf :“ txPX : fpxq ‰0u.

A functionf :RⁿÑCis said to be of classC^N if it has continuous derivatives up orderN and is said to besmooth or C⁸ if f PC^N for allN PN. Recall Taylor’s formula and Leibniz’s rule:

Theorem 1.1 (Taylor’s formula). Let f PC^NpRⁿq. Then for any x, yPRⁿ we have fpx`yq “ ÿ

|α|ăN

y^α

α!B^αfpxq ` ÿ

|α|“N

N α!y^α

ż₁

0

p1´θq^N´1B^αfpx`θyqdθ.

Proof. X. Saint Raymond [17, Theorem 1.1, p. 2]

Theorem 1.2 (Leibniz’s rule). Let α PNⁿ₀ and ΩĂRⁿ an open set. Then for any f, g PC^|α|pΩq we have

B^αpf gq “ ÿ

βďα

ˆα β

˙

B^α´βfB^βg.

Proof. X. Saint Raymond [17, Theorem 1.2, p. 3].

Recall the following result about smooth functions with compact support.

Theorem 1.3. Let Ω Ă Rⁿ be open and let K Ă Ω be compact. There exists a smooth function φ: ΩÑRsuch that 0ďφď1, φ”1on a neighbourhood of K, andsuppφĂΩ.

Proof. G. Grubb [9, Theorem 2.13, p. 22] and its corollary.

Let pX, dq be a metric space. For xPX and r ą0, we denote the closed ball and the open ball of radiusr by

tzPRⁿ : dpx, zq ďru “Bdpx, rq, tzPRⁿ : dpx, zq ăru “Udpx, rq, respectively. WhenX “Rⁿ we shall omit the subscriptd.

LetV be a vector space over a scalar fieldK“Ror C, and letΣĂK. WhenX, Y are subsets of V, we define

X˘Y :“ tx˘y : xPX, yPYu, ΣX:“ tαx : αPΣ, xPXu.

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1.2 TOPOLOGICAL VECTOR SPACES, FUNCTION SPACES AND DISTRIBUTIONS 3 In particular, for xPV and αPK we write

txu ˘Y “x˘Y, tαuX“αX.

When X is a normed vector space we denote the norm on X by k¨k_X. WhenX, Y are normed spaces we denote the space of bounded linear maps fromX to Y byBpX, Yq.

1.2 Topological vector spaces, Function spaces and Distributions

When X, Y are topological spaces we endow X ˆY with the usual product topology unless otherwise stated.

Definition 1.4. Atopological vector space (T.V.S.) over a scalar fieldK“Ror Cis a vector space X provided with a topology τ having the following properties:

1. A set consisting of one pointtxuis closed.

2. The bilinear maps

XˆXQ px, yq ÞÑx`yPX, KˆXQ pλ, xq ÞÑλxPX, are continuous.

We here follow the terminology of G. Grubb [9] and W. Rudin [15] where condition1is included in the definition of a topological vector space. In this way, a T.V.S. is in particular a topological Hausdorff space (check W. Rudin [15, Theorem 1.12, p. 11]).

Let X be a topological vector space. We associate to each aPX and to each scalar λPKzt0u thetranslation operatorTa and the multiplication operatorM_λ, by the formulas

T_apxq:“x`a, for allxPX, M_λpxq:“λx for allxPX.

It is not hard to see that their inverses are T_´a and M_1{λ, respectively, and that these four maps are continuous. Hence each of them is a homeomorphism of X onto X.

This implies that the topology of a T.V.S.Xistranslation invariant, i.e.,U Pτ ðñ x0`U Pτ for all x₀ PX. The topology is therefore determined from the system of neighbourhoods of 0.

Definition 1.5. A setY ĂX is said to be

1. convex, if for anyy₁, y₂PY and tP p0,1q we have that ty₁` p1´tqy₂PY, 2. balanced, when yPY andλPKwith|λ|ď1 implyλyPY,

3. bounded (with respect to τ), when for every neighbourhood U Q 0 there exists t ą 0 such thatY ĂtU.

X is said to be locally convex, when it has a local basis of neighbourhoods at 0 consisting of convex sets.

X is said to bemetrizable, when there exists a metricd:XˆXÑ r0,8qsuch that the topology onXis identical with the topology defined by this metric. A metric is said to betranslation invariant when

dpx, yq “dpx`z, y`zq for all x, y, zPX.

A Cauchy sequence in a T.V.S. X is a sequence txnunPN such that for any neighbourhood U Q 0 there exists NU PNso that xn´xmPU for allm, něNU.

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In a metric space pM, dq, a Cauchy sequence is a sequence such that dpx_n, x_mq Ñ 0 in R as n, m Ñ 8. In particular, if the topology of a T.V.S. is given by atranslation invariant metric d, then those concepts of Cauchy sequences coincide.

A metric space is called complete when every Cauchy sequence is convergent. More generally we call a T.V.S.sequentially complete, when every Cauchy sequence is convergent.

Banach spaces and Hilbert spaces are examples of complete metrizable topological vector spaces.

It is possible to define some topological vector spaces through the use of seminorms; we shall now take a closer look at this method.

Definition 1.6. LetX be a vector space. Aseminorm is a mapp:XÑ r0,8qsuch that 1. ppx`yq ďppxq `ppyq @x, yPX,

2. ppλxq “|λ|ppxq @λPK, @xPX.

A familytpiuiPI is calledseparating, when pipxq “0 for alliPI imply x“0.

Theorem 1.7. Let X be a vector space and lettpiuiPI be a separating family of seminorms on X.

Define the topology on X by taking, as a local basis B for the system of neighbourhoods at 0, the sets

Vppi;εq:“ txPX:pipxq ăεu, εą0, together with their finite intersections

Wpp_i₁, . . . , p_i_N;ε_i₁, . . . , ε_i_Nq “

N

č

k“1

Vpp_i_k;ε_i_kq; (1.1) and letting a local basis for the system of neighbourhoods at eachxPX consist of the translated sets x`Wppi1, . . . , piN;εi1, . . . , εiNq.

With this topology, X is a topological vector space. The seminorms tpiuiPI are continuous maps from X into R. Moreover, a set E Ă X is bounded if and only if for every i P I there exists a constant Ci ą0 such that pipEq ďCi for every xPE.

Proof. G. Grubb [9, Theorem B.5, p. 434].

Note that the sets defined in (1.1) are open, convex and balanced sets, and thus any T.V.S.

endowed with a topology such as in the above theorem is locally convex.

Additionally, this theorem implies that a sequence txkukPN converges to x in X if and only if for every i PI we have p_ipx_k´xq Ñ 0 as k Ñ 8. Furthermore, a sequence tx_kukPN is a Cauchy sequence inX if and only if for everyiPI we havep_ipx_n´x_mq Ñ0 asn, mÑ 8.

We also have the following characterization for the continuity of linear maps

Lemma 1.8. Let X, Y be topological vector spaces with topologies given as in Theorem 1.7 by separating families of seminorms tpiuiPI andtqjujPJ, respectively. Then we have the following:

1^o A linear functional Λ :XÑK is continuous if and only if there exist a constantC ą0 and i₁, . . . , i_N PI such that

|Λpxq|ďC

N

ÿ

k“1

pikpxq @xPX.

2^oA linear map T :X ÑY is continuous if and only if for everyjPJ, there arei1, . . . , iNj PI and a constantC_j ą0 such that

q_jpT xq ďC_j

Nj

ÿ

k“1

p_i_kpxq @xPX.

Proof. G. Grubb [9, Lemma B.7 and Remark B.8, p. 436].

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1.2 TOPOLOGICAL VECTOR SPACES, FUNCTION SPACES AND DISTRIBUTIONS 5 In particular, the above lemma implies that a continuous linear map T :X ÑY takes bounded sets of X into bounded sets of Y. Moreover, if Z is another T.V.S. with topology given as in Theorem 1.7 by a separating family of seminorms tr_`u`PL, then a bilinear map B : XˆY Ñ Z is continuous if and only if for every ` P L, there are i₁, . . . , i_N_` P I, and j₁, . . . , j_M_` P J, and a constant C` ą0 such that

r`pBpx, yqq ďC`

ÿ

1ďkďN`

1ďnďM`

pikpxqqjnpyq @xPX, @yPY.

Definition 1.9. A T.V.S. is called aFréchet space, whenXis metrizable with a translation invariant metric, is complete and is locally convex.

We have the following characterization of Fréchet spaces

Theorem 1.10. WhenX is a T.V.S. where the topology is defined by a countable separating family of seminormstpnunPN, thenX is locally convex and metrizable, and the topology onX is determined by the invariant metric

dpx, yq “

8

ÿ

n“1

1 2ⁿ

pnpx´yq 1`pnpx´yq. Moreover, if X is complete in this metric, thenX is a Fréchet space.

Proof. G. Grubb [9, Theorem B.9, p. 437].

We now present some important examples of Fréchet spaces. Let ΩĂ Rⁿ be an open set and for each jPNlet K_j be the compact subset ofΩ given by

K_j :“ txPΩ : xPBp0, jq,distpx,Ω^cq ě1{ju. (1.2) (If Ω^c “ ∅, then the condition distpx,Ω^cq ě 1{j is left out). Note that tKjujPN is an increasing sequence of compact sets such that Ť

jPN intpK_jq “Ω.

Let 1 ď p ď 8. Denote by L^p_locpΩq the space of all complex-valued functions on Ω whose restrictions to compact subsets ofΩare inL^p

L^p_locpΩq:“ tf is measurable : 1Kf PL^ppKq, for all compactK ĂΩu.

This is a Fréchet space when provided with the family of seminorms q_jpfq:“k1_K_jfk_LppΩq, for jPN.

Moreover, one has L^p_locpΩq ĂL^q_locpΩq for all pąq, with continuous injection.

We also have spaces of differentiable functions over open subsets of Rⁿ. Define the family of seminorms

ρ_k,jpfq:“supt|B^αfpxq|: |α|ďk, xPKju, for kPN0,jPN.

Lemma 1.11. 1^o For each k P N0, C^kpΩq is a Fréchet space when provided with the family of seminormstρ_k,jujPN.

2^o The space C⁸pΩq “Ş

kPN0C^kpΩq is a Fréchet space when provided with the family of seminorms tρk,jukPN0,jPN

Proof. G. Grubb [9, Lemma 2.4, p. 13].

Lemma 1.12. 1^o Let α PNⁿ₀ be a multiindex. The mapping B^α :φÞÑ B^αφ is a continuous linear operator on C⁸pΩq.

2^o Let f PC⁸pΩq. The mappingMf :φÞÑf φis a continuous linear operator on C⁸pΩq.

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Proof. Clearly both mapsB^α and M_f are linear from C⁸pΩq to itself.

For the continuity of B^α we see that for eachkPN0,j PNwe have ρ_k,jpB^αφq “supt|B^βB^αφpxq|: |β|ďk, xPKju

ďsupt|B^γφpxq|: |γ|ďk`|α|, xPKju

“ρk`|α|,jpφq.

For continuity of M_f we have by Leibniz’s rule and for each kPN0,j PNthat ρ_k,jpf φq “supt|B^αpf φqpxq|: |α|ďk, xPK_ju

ďsup

! ÿ

βďα

ˆα β

˙

B^α´βfpxqB^βφpxq

: |α|ďk, xPK_j )

ďCkρk,jpfqρk,jpφq,

for a suitably large constant C_k ą 0. Since f P C⁸pΩq, we have that ρ_k,jpfq ă 8 for all k PN0, jPN and this finishes the proof.

Similarly, we may define the family of seminorms

ρ_kpfq:“supt|B^αfpxq| : |α|ďk, xPRⁿu, for kPN0, and the space of bounded differentiable functions of orderk

C_B^kpRⁿq:“ tf PC^kpRⁿq : ρkpfq ă 8u.

It is not hard to check thatC_B^kpRⁿq endowed with the seminormρ_k becomes a Banach space and thatC_B⁸pRⁿq “Ş

kPN0C_B^kpRⁿqendowed with the family tρkukPN0 becomes a Fréchet space.

Definition 1.13. The Schwartz space or Rapidly decreasing function space SpRⁿq is the set of infinitely differentiable complex-valued functions f on Rⁿ such that for all multiindices α, β we have

p_α,βpfq:“ sup

xPRⁿ

|x^βB^αfpxq|ă 8. (1.3)

Thus, the functions inSpRⁿqare those functions which together with their derivatives fall off more quickly than the inverse of any polynomial.

One can directly check thatpα,βas in (1.3) defines a seminorm onSpRⁿqand we shall call them theSchwartz seminorms.

Theorem 1.14. The vector spaceSpRⁿqwith the natural topology induced by the family of Schwartz seminormstp_α,β : α, β PNⁿ₀u is a Fréchet space.

Proof. M. Reed and B. Simon [14, Theorem V.9, p. 133].

An important observation is that for all 1 ď p ď 8 we have that SpRⁿq Ă L^ppRⁿq with continuous injection. Indeed, for a each 1ďpă 8, takeM PN such thatn{2ăM p. Then for all φPS we have

ż

Rⁿ

|φpxq|^pdx“ ż

Rⁿ

1

p1`|x|²q^{M p}rp1`|x|²q^M|φpxq|s^pdx ď

ˆ sup

xPRⁿ

p1`|x|²q^Mφpxq

˙_p ż

Rⁿ

1 p1`|x|²q^{M p}

dx.

Thus there exists a constant C_M ą 0 such that kφk_Lp ď C_M sup_xP_Rn|p1`|x|²q^Mφpxq| and the right-hand side of this expression is bounded by a linear combination of Schwartz seminorms. The casep“ 8is immediate.

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1.2 TOPOLOGICAL VECTOR SPACES, FUNCTION SPACES AND DISTRIBUTIONS 7 Definition 1.15. The vector spaceO_MpRⁿqconsists of complex-valued functionspPC⁸pRⁿqwhich together with all of its derivatives has at most polynomial growth. More precisely, pPO_MpRⁿq if for each αPNⁿ₀ there exists C_αą0 and N_αPN such that

|B^αppxq|ďCαp1`|x|²q^N^α @xPRⁿ.

Lemma 1.16. 1^o Let α PNⁿ₀ be a multiindex. The mapping B^α :φÞÑ B^αφ is a continuous linear operator on SpRⁿq.

2^o Let pPO_MpRⁿq. The mapping M_p :φÞÑpφ is a continuous linear operator on SpRⁿq. 3^o Let χ P C_c⁸pRⁿq. The mapping Mχ : φÞÑ χφ is a continuous linear map from C⁸pRⁿq to SpRⁿq.

Proof. It is clear that those maps are linear, so let us check continuity.

The proof of 1^o follows from the fact that for anyf PS and any γ PNⁿ0 we have p_γ,βpB^αfq “ sup

xPRⁿ

x^βB^γB^αfpxq

“ sup

xPRⁿ

x^βB^γ`αfpxq

“p_γ`α,βpfq

The proof of 2^o follows by Leibniz’s rule, as for eachpPO_M there exists a constantCą0 such that

x^βB^αpppxqfpxqq

“|x^β|ÿ

δďα

ˆα δ

˙

B^δppxq B^α´δfpxq

ď|x^β|ÿ

δďα

ˆα δ

˙

|C_δp1`|x|²q^N^δB^α´δfpxq| ď|x^β| ÿ

|γ|ď2Nδďα _δ

C_δγα|x^2γB^α´δfpxq|

ďC ÿ

δďα γď2Nδ

|x^β`2γB^α´δfpxq|

ďC ÿ

δďα γď2Nδ

p_{α´δ,β`2γ}pfq.

Taking the supremum overxPRⁿ we conclude 2^o.

For 3^o we note that there exists j0 PN such that suppB^αχ Ăsuppχ ĂBp0, j0q. By Leibniz’s rule we get

x^βB^αpχpxqfpxqq

“|x^β|ÿ

δďα

ˆα δ

˙

B^δχpxq B^α´δfpxq

ď1_Bp0,j₀_qpxq|x^β|ÿ

δďα

ˆα δ

˙

kB^δχk_L⁸|B^α´δfpxq|

ďj₀^|β| ÿ

δďα

ˆα δ

˙

kB^δχk_L8ρ_|α|,j₀pfq.

Taking the supremum overxPRⁿ we conclude our result.

For anyjPN, let K_j ĂΩbe as in (1.2). We define the space of smooth functions with compact support inKj ĂΩas

C_K⁸_jpΩq:“ tφPC⁸pΩq : suppφĂKju.

This space provided with the inherited topology of C⁸pΩq is also a Fréchet space.

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We now turn our attention to another space of differentiable functions which is a cornerstone in the theory of distributions and is not a Fréchet space. The space of infinitely differentiable functions with compact support inΩ

C_c⁸pΩq:“ tφPC⁸pΩq: suppφis a compact subset of Ωu.

Clearly, C_K⁸_jpΩq ĂC_c⁸pΩq Ă C⁸pΩq for any j PN. Now, if we provide C_c⁸pΩq with the topology inherited fromC⁸pΩq, we get an incomplete metric space.

Therefore we prefer to provide C_c⁸pΩq with a stronger and somewhat more complicated topology that makes it locally convex and sequentially complete as a topological vector space. For the definition and several important properties of such topology on C_c⁸pΩq we recommend G. Grubb [9] or W. Rudin [15].

The topological properties of this space that we shall need are summed up in the following theorem.

Theorem 1.17. The topology on C_c⁸pΩq has the following properties:

1. A sequence tφ_kukPN ĂC_c⁸pΩq converges to an element φ in C_c⁸pΩq if and only if there is a jPN such that suppφ_kĂK_j for all kPN, suppφĂK_j, and for all αPNⁿ₀ we have

sup

xPKj

|B^αφ_kpxq ´ B^αφpxq|Ñ0 as kÑ 8.

2. A linear functional Λ : C_c⁸pΩq Ñ C is continuous if and only if for each j P N and all φPC_c⁸pΩq with suppφPK_j, there is an N_j PN0 and a constant C_j ą0 such that

3. Let Y be a locally convex topological vector space. A linear map T is continuous from C_c⁸pΩq toY if and only if for each jPNthe linear map T :C_K⁸

jpΩq ÑY is continuous.

Differentiation and multiplication by smooth functions are examples of continuous linear operators onC_c⁸pΩq.

Theorem 1.18. 1^o Let αPNⁿ₀ be a multiindex. The mapping B^α :φÞÑ B^αφis a continuous linear operator on C_c⁸pΩq.

2^o Let f PC⁸pΩq. The mappingM_f :φÞÑf φis a continuous linear operator on C_c⁸pΩq. Proof. G. Grubb [9, Theorem 2.6, p. 15].

Theorem 1.19. 1^o C_c⁸pΩq is dense in C⁸pΩq.

2^o C_c⁸pΩq is dense in L^p_locpΩq for all 1ďpă 8. 3^o C_c⁸pΩq is dense in L^ppΩq for all 1ďpă 8. Proof. G. Grubb [9, Theorem 2.15, p. 22]

SinceC_c⁸pRⁿq ĂSpRⁿq, it follows from the above theorem that SpRⁿq is dense inC⁸pRⁿq, and inL^p_locpRⁿq and L^ppRⁿq for all 1ďpă 8.

Lemma 1.20. C_c⁸pRⁿq ĂSpRⁿq with continuous dense embedding.

Proof. G. Grubb [9, Lemma 5.9, p. 103]

Definition 1.21. Adistribution onΩis a continuous linear functionalΛ :C_c⁸pΩq ÑC. The vector space of distributions onΩis denoted byD¹pΩq. WhenΛPD¹pΩqwe denote its value onφPC_c⁸pΩq by either xΛ, φyor Λpφq.

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1.2 TOPOLOGICAL VECTOR SPACES, FUNCTION SPACES AND DISTRIBUTIONS 9 The space D¹pΩq itself is provided with a weak topology defined by the family of seminorms

p_φpuq:“|xu, φy|, for φPC_c⁸pΩq.

We here use Theorem 1.7noticing that the family tp_φuφPC⁸_c pΩq is separating since u “0 inD¹pΩq if and only ifxu, φy “0for all φPC_c⁸pΩq.

An important example of distribution is the following: When f PL¹_locpΩq the linear map Λ_f : φÞÑ

ż

Ω

fpxqφpxqdx

from C_c⁸pΩqto C is a distribution. Indeed, for each jPN and allφPC_c⁸pΩq withsuppφPKj we have

|Λ_fpφq|“ ż

Ω

fpxqφpxqdx

ď

˜ż

Kj

|fpxq|dx

¸

supt|φpxq| : xPK_ju. (1.4) Here one can in fact identifyf withΛ_f in view of the following lemma:

Lemma 1.22. When f PL¹_locpΩq with ż

Ω

fpxqφpxqdx“0 @φPC_c⁸pΩq, then f ”0 in L¹_locpΩq.

Thus by (1.4) and the above Lemma we have that the linear map L¹_locpΩq Qf ÞÝÑΛ_f PD¹pΩq is continuous and injective.

WhenT is a continuous linear operator onC_c⁸pΩq, andΛPD¹pΩq, we have that the composition mapΛ˝T is a distribution. Define the adjoint map of operatorT to be the mapT¹: ΛÞÑΛ˝T. Theorem 1.23. Let T be a continuous linear operator on C_c⁸pΩq. The adjoint map T¹ defined by

xT¹Λ, φy:“ xΛ, T φy, for ΛPD¹pΩq, φPC_c⁸pΩq is a continuous linear operator on D¹pΩq.

Proof. G. Grubb [9, Theorem 3.8, p. 35].

Definition 1.24. 1^o For anyαPNⁿ₀, we define the differentiation operatorB^α:D¹pΩq ÑD¹pΩq by xB^αΛ, φy:“ xΛ,p´1q^|α|B^αφy, for φPC_c⁸pΩq.

2^o When f PC⁸pΩq, we define the multiplication operator M_f :D¹pΩq ÑD¹pΩqby xM_fΛ, φy:“ xΛ, f φy, for φPC_c⁸pΩq.

Both of these maps are continuous by Theorem 1.23. Moreover, it can be shown that both of these operators are extensions of the operators in Theorem1.18 originally defined onC_c⁸pΩq. Theorem 1.25(Convergence inD¹pΩq). A sequence of distributionstu_kukPNis convergent inD¹pΩq if and only if for everyφPC_c⁸pΩqthe sequence xu_k, φyis convergent in C. The limit ofu_k inD¹pΩq is the distributionu defined by

xu, φy “ lim

kÑ8xu_k, φy, for φPC_c⁸pΩq.

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When u P D¹pΩq and Ω¹ is an open subset of Ω, we define the restriction of u to Ω¹ as the elementuæ_Ω¹ PD¹pΩ¹q defined by

xuæ_Ω¹, ϕy_Ω¹ “ xu, ϕy_Ω for ϕPC_c⁸pΩ¹q.

For the sake of precision, we here indicate the duality betweenD¹pωqand C_c⁸pωqby x,yω, whenω is an open set.

When u1 PD¹pΩ1qand u2 PD¹pΩ2q, andω ĂΩ1XΩ2 is open, we say thatu1 “u2 onω, when u₁æω´u₂æω “0 as an element ofD¹pωq.

Theorem 1.26. Let tω_iuiPI be an arbitrary family of open sets in Rⁿ and let Ω “Ť

ω_i. Assume that there is given a family of distributionsu_iPD¹pω_iq with the property that u_i “u_j on ω_iXω_j for each pair of indicesi, j PI. Then there exists one and only one distribution uPD¹pΩq such thatui

is the restriction of u to ω_i for every iPI.

Theorem 1.27. For any u PD¹pΩq the exists a sequence tujujPN Ă C_c⁸pΩq such that uj Ñ u in D¹pΩq.

In other words, the space C_c⁸pΩq issequentially dense inD¹pΩq.

Let Ω,Ω¹ be open sets of Rⁿ, and let κ be a diffeomorphism ofΩ onto Ω¹. More precisely,κ is a bijective map

ΩQx“ px1, . . . , xnq ÞÝÑ pκ1pxq, . . . , κnpxqq PΩ¹,

where eachκj is aC⁸-function fromΩto Rand the inverse mapκ^´1 : Ω¹ ÑΩ is likewise smooth.

TheJacobian matrix ofκ is given by

J_κpxq “

»

—

— –

Bκ1

Bx1

Bκ1

Bx2 . . . _Bx^Bκ¹

n

Bκ2

Bx1

Bκ2

Bx2 . . . _Bx^Bκ² .. n

. ... . .. ...

Bκn

Bx1

Bκn

Bx2 . . . ^Bκ_Bxⁿ

n

fi ffi ffi ffi ffi ffi fl .

Define Jpxq :“ |detJκpxq| and note that Jpxq ą 0 for all x P Ω and that both J and 1{J are C⁸-functions. For φPC_c⁸pΩ¹q we have that φ˝κPC_c⁸pΩq. The following definition then extends C⁸-coordinate change fromC_c⁸ to the space of distributions.

Definition 1.28. We define the coordinate change mapT_κ by

xT_κΛ, φy_Ω¹ :“ xΛ, Jpφ˝κqy_Ω, for ΛPD¹pΩq,φPC_c⁸pΩ¹q.

It defines a continuous linear map from D¹pΩq to D¹pΩ¹q.

Definition 1.29. A tempered distribution is a continuous linear functional Λ : SpRⁿq Ñ C. The vector space of tempered distributions is denoted by S¹pRⁿq or simply by S¹. When Λ P S¹ we denote its value on φPS by either xΛ, φy or Λpφq.

In a similar fashion to what we have done for the space D¹pΩq, we provide S¹pRⁿq with a weak topology defined by the family of seminorms

p_φpuq:“|xu, φy|, for φPSpRⁿq.

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1.3 FOURIER TRANSFORMATION ON DISTRIBUTIONS 11 Note that Lemma 1.20implies that the restriction ofΛPS¹pRⁿqonC_c⁸pRⁿqdefines an element inD¹pRⁿq. More precisely

Theorem 1.30. The linear map J : ΛÑΛ¹ from S¹pRⁿq toD¹pRⁿq defined by restriction ofΛ on C_c⁸pRⁿq,

xΛ¹, φy:“Λpφq, @φPC_c⁸pRⁿq,

is injective, and hence allows an identification of JrS¹pRⁿqs with a subspace of D¹pRⁿq, also called S¹pRⁿq.

Lemma 1.31. For 1ďpď 8 andf PL^ppRⁿq, the linear map S QφÞÝÑ

ż

Rⁿ

fpxqφpxqdxPC

defines a tempered distribution. In this way one has a continuous injection of L^ppRⁿq into S¹pRⁿq for each1ďpď 8.

The above results combined with Theorem 1.27 give us that C_c⁸pRⁿq, and thus SpRⁿq, are sequentially dense in S¹pRⁿq.

The operators B^α and M_p for pPC⁸pΩq in Definition 1.24 are continuous linear operators on D¹pΩq. Concerning their act onS¹ we have

Lemma 1.32. 1^o For anyαPNⁿ₀,B^α mapsS¹ continuously intoS¹. 2^o When pPO_M, Mp mapsS¹ continuously intoS¹.

Proof. G. Grubb [9, Lemma 5.13, p. 106]

Proposition 1.33. Let X, Y be Banach spaces such that X, Y ĂS¹ with continuous injection and such that SĂX with continuous dense embedding. Let A:S¹ ÑS¹ be a continuous linear operator such that ApSq ĂY and such that there exists a constantC ą0 for which

kAfk_Y ďCkfk_X @f PS. (1.5)

Then A defines a continuous linear map from X to Y with operator norm at most equal to the constant Cą0.

Proof. Let f P X and let tfnunPN Ă S be such that fn Ñ f in X. Inequality (1.5) shows that tAfnunPNis a Cauchy sequence inY, and therefore converges to some elementgPY. On the other hand, if f_n Ñ f inX, then f_n Ñ f inS¹ by hypothesis. Since A:S¹ Ñ S¹ is a continuous linear map we have that Afn Ñ Af in S¹. This implies that Af “g in S¹ and thus also in Y, finishing the proof.

1.3 Fourier Transformation on Distributions

Definition 1.34. Givenf inL¹pRⁿq we define theFourier transform of f to be Fpfqpξq ”fppξq:“

ż

Rⁿ

fpxqe^´2πix¨ξdx.

Theorem 1.35. 1^o The mapping F :f ÞÑ Fpfq is a bounded linear transformation from L¹pRⁿq into L⁸pRⁿq. In fact kfpk_L8 ďkfk_L1.

2^o If f PL¹pRⁿq then fpis uniformly continuous

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Theorem 1.36 (Riemann–Lebesgue). If f PL¹pRⁿq then fppξq Ñ 0 as |ξ|Ñ 8; thus, in view of the above result, we can conclude that fpPC0pRⁿq, where

C0pRⁿq:“ tf :RⁿÑC : f is continuous and fpxq Ñ0 as |x|Ñ 8u.

A more suitable space to work with the Fourier transform is the Schwartz spaceSpRⁿq. In this space we have the following properties

Proposition 1.37. Givenf PSpRⁿq, αPNⁿ₀ and yPRⁿ we have 1. FpB^α_xfqpξq “ p2πiξq^αfppξq,

2. pB_ξ^αfpqpξq “Fpp´2πixq^αfpxqqpξq, 3. Fpe^2πxÿfpxqqpξq “fppξýq, 4. Fpfpxýqqpξq “e^´2πξÿfppξq, 5. fpPSpRⁿq.

Proof. L. Grafakos [7, Proposition 2.2.11, p. 100] for the proof of these results and many others.

Definition 1.38. Givenf PSpRⁿq, we define theinverse Fourier transform as Fpfqpxq ”fpxqq :“

ż

Rⁿ

fpyqe^2πiy¨xdy.

Note that, apart from some minor sign changes, the inverse Fourier transform shares the same properties stated in Proposition 1.37 as the Fourier transform. The next theorem shows that the inverse Fourier transform is indeed the inverse operator ofF inSpRⁿq. This property is referred to asFourier inversion formula.

Theorem 1.39. Given f, gPSpRⁿq we have 1.

ż

Rⁿ

fppxqgpxqdx“ ż

Rⁿ

fpxqpgpxqdx 2. Fpfq “p f “Fpfqq (Fourier Inversion) 3.

ż

Rⁿ

fpxqgpxqdx“ ż

Rⁿ

fppxqpgpxqdx (Parseval’s identity) 4. kfk_L2 “kfkp _L2 “kfqk_L2 (Plancherel’s identity)

Proof. L. Grafakos [7, Theorem 2.2.14, p. 102]

Theorem 1.40. The Fourier transform is a homeomorphism from SpRⁿq onto itself.

Proof. L. Grafakos [7, Corollary 2.2.15, p. 103]

Definition 1.41. LetuPS¹pRⁿq be a tempered distribution. We define the Fourier transform of a tempered distribution uto be

xpu, fy:“ xu,fpy for allf PSpRⁿq.

Theorem 1.40 implies that the Fourier transform of a tempered distribution is a well-defined tempered distribution.

Additionally, Theorems 1.23,1.30, and 1.40assure that the linear map F :S¹ ÑS¹ given by xFu, φy:“ xu, Fφy, for φPSpRⁿq

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1.4 SOBOLEV SPACES 13 is a continuous linear homeomorphism fromS¹pRⁿq onto itself.

Moreover, if we also invoke 1of Theorem1.39, we can see thatF :S¹ ÑS¹ extends the Fourier transform defined initially onSpRⁿq.

Furthermore, properties 1-4 presented in Proposition 1.37 also hold in the space of tempered distributions.

1.4 Sobolev Spaces

Definition 1.42 (Sobolev spaces). For each sPR, the Sobolev spaceH^spRⁿq is defined by H^spRⁿq:“ tf PS¹pRⁿq | p1`|ξ|²q^s{2fppξq PL²pRⁿqu.

It is a Hilbert space with inner product given by xf, gys “

ż

Rⁿ

p1`|ξ|²q^sfppξqpgpξqdξ, and norm given by

kfk²_Hs “ ż

Rⁿ

p1`|ξ|²q^s|fppξq|²dξ.

It is not hard to check that H^spRⁿq ĂH^s¹pRⁿqfor s¹ăsand that the inclusion is a continuous injection. Moreover, if mPN0, then the Sobolev spaceH^mpRⁿqmay be defined by

H^mpRⁿq:“ tuPL²pRⁿq : B^αuPL²pRⁿq for|α|ďmu,

where B^α is applied in the distribution sense. In this case H^mpRⁿq is equipped with the scalar product and norm

xu, vy_H^m :“ ÿ

|α|ďm

xB^αu, B^αvy_L², kuk_Hm :“

ˆ

xu, uy_H^m

˙1{2

.

Using the Fourier inversion formula it is not hard to obtain that both definitions induce equivalent norms on H^mpRⁿq whenevermPN0.

Lemma 1.43. C_c⁸pRⁿq is dense in H^spRⁿq for every sPR. Proof. G. Grubb [9, Lemma 6.10, p. 129]

In particular, SpRⁿq is dense in H^spRⁿq for every s P R. Moreover, it can be shown that for each sPRwe have that H^spRⁿq ĂS¹pRⁿq with continuous dense embedding.

Theorem 1.44 (Sobolev embedding theorem). Let kě0 be an integer, and let sąk`n{2. Then H^spRⁿq ĂC_B^kpRⁿq,

with continuous injection. More precisely, there exists a constant C ą 0 such that for any u P H^spRⁿq,

supt|B^αupxq| : |α|ďk, xPRⁿu ďCkuk_Hs. Proof. G. Grubb [9, Lemma 6.11, p. 130]

Lemma 1.45. Let sPR.

1^o For each αPNⁿ₀, B^α is a continuous linear map from H^spRⁿq toH^s´|α|pRⁿq.

2^o For eachf PSpRⁿq, the multiplication byf defines a continuous linear operator onH^spRⁿq.

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Definition 1.46. LetsPRand letK ĂRⁿ be a compact set. We denote H^spKq:“ tuPH^spRⁿq : xu, φy “0for all φPC_c⁸pK^cq u.

In other words, the set H^spKq is composed by those elements ofH^spRⁿq which vanish outside of K ĂRⁿ.

Note that H^spKqis a closed linear subspace ofH^spRⁿq, hence a Hilbert space for the inherited structure.

Now let κ be a diffeomorphism from Ωonto Ω¹ and suppose thatK ĂΩ. LetK¹ :“κpKq be a compact subset ofΩ¹ and consider the mapping Tκ :D¹pΩq ÑD¹pΩ¹qfrom Definition 1.28. It easily follows that for anyuPH^spKq we have T_κuPH^spK¹q. Moreover, we have

Proposition 1.47. Let sPR and let K be an arbitrary compact subset of Ω. There is a constant C_spKq ą0 such that

kT_κuk_Hs ďCspKqkuk_Hs @uPH^spKq.

Proof. F. Trèves [20, Proposition 25.7, p. 230].

Definition 1.48. Let sPR and let ΩĂRⁿ be an open set. The spaceH_loc^s pΩq is defined as the set of distributionsuPD¹pΩq for whichφuPH^spRⁿqfor allφPC_c⁸pΩq (whereφuis understood to be extended by zero outside ofΩ).

ConsiderK_j ĂΩsuch as in (1.2) and consider a sequence of smooth functionstφ_jujPNsatisfying 0ďφď1,φ”1 inKj andsuppφj ĂintpKj`1q. Then it can be shown thatH_loc^s pΩq is a Fréchet space with family of seminorms given by

pjpuq:“kφ_juk_Hs, for jPN.

Lemma 1.49. Let sPR. Let f PC⁸pΩq and αPNⁿ₀, then the operatoruÞÑfB^αu is a continuous linear mapping from H_loc^s pΩq into H_loc^s´|α|pΩq.

Denoting the restriction of T_κ:D¹pΩq ÑD¹pΩ¹q on H_loc^s pΩq byT_κ we have

Proposition 1.50. Letκ be a diffeomorphism fromΩontoΩ¹. Then the mappingT_κ is an isomor- phism from H_loc^s pΩq onto H_loc^s pΩ¹q. Its inverse is the mappingT_κ^´1.

Proof. F. Trèves [20, Proposition 25.8, p. 231].

1.5 Smooth Manifolds

Definition 1.51. Ann-dimensional manifold is a Hausdorff spaceM with countable basis and for which each point has a neighbourhood homeomorphic to an open subset of Rⁿ. If κ :U Ñ κpUq is a homeomorphism from an open set U Ă M onto an open set κpUq Ă Rⁿ, then κ is called a coordinate map, the pairpU, κq is calledcoordinate chart or simplychart, and the open setU ĂM is calledchart neighbourhood.

In particular, since any open subset ofRⁿcan be written as a union of countably many compact subsets, the same holds for any n-dimensional manifold.

Definition 1.52. A smooth structure F on ann-dimensional manifoldM is a collection of charts tpU_α, κ_αquαPAsatisfying the following three properties:

1. M ĂŤ

αPAUα

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1.5 SMOOTH MANIFOLDS 15 2. κ_β ˝κ^´1_α :κ_αpU_αXU_βq Ñκ_βpU_αXU_βqis a diffeomorphism for all α, β PA.

3. The collection F is maximal with respect to 2; that is, if pU, κq is a chart such that κ˝κ^´1_α and κ_α˝κ^´1 are diffeomorphisms for all αPA, thenpU, κq PF.

Property 2of the above definition is usually referred ascompatibility condition and we say that two chartspUα, καq and pU_β, κ_βq arecompatible if

κβ ˝κ^´1_α :καpUαXUβq ÑκβpUαXUβq and

κα˝κ^´1_β :κβpUαXUαq ÑκαpUαXUβq are diffeomorphisms.

Proposition 1.53. Let M be ann-dimensional manifold. IfF₀ is a family of charts satisfying con- ditions1and2 of Definition1.52, then there exists one and only one smooth structureF containing F₀.

Proof. LetF₀ :“ tU_i, κ_iuiPI. Define the collection of charts

F :“ t pU, κq : U ĂM is open,κ˝κ^´1_i and κ_i˝κ^´1 are diffeomorphisms for allκ_i PF₀u.

Note that F₀ Ă F, thus F forms an open cover of M. Because every pU, κq P F satisfies the compatibility condition with the elements of F₀, it follows that κ defines a homeomorphism from U to κpUq ĂRⁿ. Moreover, ifpUα, καq and pUβ, κβq are inF, then for all pUi, κiq inF₀ we have

κ_α˝κ^´1_β “ pκ_α˝κ^´1_i q ˝ pκ_i˝κ^´1_β q on κ_βpU_αXU_βXU_iq,

and therefore onκ_βpU_αXU_βq. Since the same holds if we exchange the roles ofαandβ, we conclude that the familyF satisfies the compatibility condition. NowF is maximal by construction, and so F is a smooth structure containing F₀. Uniqueness follows from the observation that any other family of charts on M that contains F₀ and satisfies the compatibility condition is contained in F.

Definition 1.54. Let M be a n-dimensional manifold and let A denote a family of charts. Then A is called anatlas on M, when it satisfies conditions 1 and 2 of Definition 1.52. Two atlases are calledcompatible or equivalent if they define the same smooth structure onM.

Definition 1.55. A smooth n-manifold is an n-dimensional manifold endowed with a smooth structure F. We shall usually denote a smoothn-manifold pM,Fqsimply by M.

If M is a smooth n-manifold, then a compatible atlas on M is an atlas A that is compatible with the smooth structure on M.

Let M be a smooth n-manifold and letF :“ tpU_α, κ_αquαPA be its smooth structure. If U is an open subset of M, thenU can be given a smooth structure

F_U :“ tpUXU_α, κ_αæUquαPA.

With this smooth structureU is called anopen submanifold ofM. Open subsets of smooth manifolds will always be given this natural smooth structure.

Let us recall some concepts from general topology. Let X be a topological space and suppose that one-point sets are closed in X. Then X is said regular if for each pair consisting of a point x and a closed setB disjoint from x, there exist disjoint open sets containingx and B, respectively.

The space X is said to be normal if for each pair A, B of disjoint closed sets of X, there exist disjoint open sets containing A and B, respectively. X is said locally compact, when every point x PX has a compact neighbourhood, i.e., there exists an open set U and a compact set K, such thatxPU ĂK.

Commutator characterization of pseudo-diﬀerential operators on compact manifolds

Commutator characterization of pseudo-differential operators on compact manifolds

Artur Henrique de Oliveira Andrade

Dissertação apresentada ao

Instituto de Matemática e Estatística da

Universidade de São Paulo para

obtenção do título de

Mestre em Ciências

Programa: Matemática

Orientador: Prof. Dr. Severino Toscano do Rêgo Melo

on compact manifolds

Acknowledgments

Resumo

Abstract

Contents

Introduction

Chapter 1

Background and prerequisites

1.1 Notation

1.2 Topological vector spaces, Function spaces and Distributions

1.3 Fourier Transformation on Distributions

1.4 Sobolev Spaces

1.5 Smooth Manifolds