Amplitude representation and Change of Variables

2.6 AMPLITUDE REPRESENTATION AND CHANGE OF VARIABLES 53

One of the advantages of working with amplitude symbols is the following: LetKPC_c⁸pRⁿˆRⁿq and define

T fpxq:“

ż

Rⁿ

Kpx, yqfpyqdy, for f PSpRⁿq.

We claim that there exists an amplitude symbolcpx, y, ξqsuch thatcPA^m for anymPRand such thatT “T_rcs for allf PS. Indeed, consider ηPC_c⁸pRⁿqsuch that

ż

ηpzqdz‰0 and set

cpx, y, ξq “ ˆż

ηpzqdz

˙_´1

Kpx, yqηpξqe´2πipx´yq¨ξ.

Note thatcPC_c⁸pR³ⁿqand therefore is inA^mfor anymPR. It then follows from Fubini’s theorem that for any f PSpRⁿq we have

T_rcsfpxq “ ż ż

e^{2πipx´yq¨ξ}cpx, y, ξqfpyqdy dξ“ ż

Kpx, yqfpyqdy“T fpxq, and our claim holds.

Let us now discuss how certain pseudo-differential operators behave under smooth coordinate changes. The result that we present here is weaker than L. Hörmander [11, Theorem 18.1.17, p. 81], as we do not use results concerning the Schwartz Kernel of pseudo-differential operators.

Let U,V be open sets of Rⁿ, and let κ be a diffeomorphism of U onto V. Denote by Jκ the Jacobian matrix of κ.

For any uPC_c⁸pVqwe define its pullback by κ by

pκ^˚uqpxq “upκpxqq, xPU . It easily follows thatκ^˚uPC_c⁸pUq.

Now let a P S^mpRⁿˆRⁿq be such that there exists K Ă V compact with supp_xap¨, ξq Ă K for all ξPRⁿ and denote the pseudo-differential operator with symbol aPS^m by A. Then for any f P SpRⁿq, the smooth function Af has compact support in V and thus can be identified as an element of C_c⁸pVq.

Given any ψPC_c⁸pRⁿq such that suppψ ĂU we set M_ψ :SpRⁿq ÑC_c⁸pUq by M_ψpfq:“ψf. Then we may define a linear map with the help of the commutative diagram

C_c⁸pVq C_c⁸pVq

SpRⁿq C_c⁸pUq A

κ^˚ pκ^´1q^˚˝Mψ

A_κ´1ψ

More precisely,

rpA_κ´1ψqfspxq:“Apψf ˝κ^´1q ˝κpxq, for f PSpRⁿq,@xPU . (2.71) Note that pA_κ^´1ψqf P C_c⁸pUq for any f PSpRⁿq, and thus pA_κ^´1ψqf P SpRⁿq after extension by zero on the complement ofU.

We shall show thatpA_κ^´1ψqis a pseudo-differential operator of ordermPRand that its symbols is related to symbolaPS^m.

Theorem 2.21. Let U, V be open subsets ofRⁿ,κ:U ÑV be a diffeomorphism. LetψPC_c⁸pRⁿq be such that suppψ Ă U and suppose a P S^m is such that there exists a compact set K Ă V for which supp_xap¨, ξq ĂK for all ξ PRⁿ. Denote the pseudo-differential operator with symbol aPS^m by A.

2.6 AMPLITUDE REPRESENTATION AND CHANGE OF VARIABLES 55 If for any f PSpRⁿq we define

rpA_κ´1ψqfspxq:“Apψf ˝κ^´1qpκpxqq, @xPU , (2.72) and rpA_κ^´1ψqfs ”0 on U^c, then pA_κ^´1ψq POp S^m.

Moreover, if a_ψ,κ´1 P S^m is the symbol of pA_κ´1ψq, then a_ψ,κ´1px, ξq “ apκpxq,“_Bκ

Bx

‰1

ξqψpxq modulo S^m´1, where “_Bκ

Bx

‰₁

““

Jκpxq^T‰´1

.

Proof. Letf PSpRⁿq. The right-hand side of (2.72) gives us that rpA_κ^´1ψqfspxq “ lim

εÑ0^`

ż ż

e2πipκpxq´yq¨ξapκpxq, ξqϕpεξq pψf˝κ^´1qpyqdy dξ

“ lim

εÑ0^`

ż ż

e2πipκpxq´κpyqq¨ξapκpxq, ξqψpyq|detJκpyq|ϕpεξqfpyqdy dξ,

(2.73)

where we made the change of variables y ÞÑ κpyq from the first to the second line. Note that this formula holds for all xPRⁿsince aPS^m has compact support inV. We now reason that the main contribution of the above expression comes from when xis close to y.

In order to do that properly, we begin by considering the following: By applying Taylor’s formula to each κj we then may obtain the following equality

κpxq ´κpyq “Mpx, yq ¨ px´yq, (2.74) withMpx, yq given by

Mpx, yq “ ż1

0

Jκpx`tpy´xqqdt. (2.75)

Note that Mpx, yq is well defined on the open set tpx, yq P UˆU : rx, ys P Uu. In particular, this set is a neighbourhood of the diagonal of UˆU and Mpx, yq is a smooth map onx andy on that domain. We now claim thatMpx, yq is invertible on a small neighbourhood of the diagonal.

Indeed, note that for each x PU, Mpx, xq “J_κpxq, hence is invertible. Therefore there exists an open set Ax ĂU ˆU such thatpx, xq PAx and Mpz, yq is invertible for all pz, yq PAx. Taking Ω“Ť

xPUAx, we get thatpx, xq PΩ for allxPU and that M is invertible on this set.

Now consider K¹ “ κ^´1pKq, where K Ă V is the compact set containing the x-support of symbola. We then have that ∆K¹ˆK¹ :“ tpz, zq PUˆU : zPK¹u is compact and is a subset ofΩ.

Hence there exists χ PC_c⁸pU ˆUq such that χ”1 on a neighbourhood of K¹ˆK¹, suppχ ĂΩ, and 0 ďχ ď1. Writing 1 “ p1´χpx, yqq `χpx, yq and inserting this relation into (2.73) we now have to evaluate two different expressions.

Main contribution: Let us first estimate the term which contains the points wherex is close to y. More precisely, let us analyse (2.73) after inserting χpx, yq by writing

lim

εÑ0^`

ż ż

e2πipκpxq´κpyqq¨ξχpx, yqapκpxq, ξqψpyq|detJκpyq|ϕpεξqfpyqdy dξ.

By (2.74) we have lim

εÑ0^`

ż ż

e^2πiMpx,yqpx´yq¨ξχpx, yqapκpxq, ξqψpyq|detJ_κpyq|ϕpεξqfpyqdy dξ.

Making the change of variables Mpx, yq^Tξ ÞÑ ξ and letting Mpx, yq¹ ““

Mpx, yq^T‰´1

we conclude that

lim

εÑ0^`

ż ż

e^{2πipx´yq¨ξ}χpx, yqapκpxq, Mpx, yq¹ξqψpyq|detJκpyq| |detMpx, yq^´1|ϕpεMpx, yq¹ξqfpyqdy dξ.

(2.76)

If we set

c1px, y, ξq “χpx, yqapκpxq, Mpx, yq¹ξqψpyq|detJκpyq| |detMpx, yq^´1|, thenc1 PA^m is an amplitude symbol such that T_rc1sf coincides with (2.76).

By Proposition 2.20 we get that there exists a symbol a₁ P S^m such that T_a1 “ T_rc1s and for which the first term of the asymptotic expansion is given by

c₁px, y, ξq ˇ ˇ

ˇy“x“χpx, xqapκpxq, Mpx, xq¹ξqψpxq|detJ_κpxq| |detMpx, xq^´1|

“apκpxq,

„Bκ Bx

₁

ξqψpxq,

and all other terms from the asymptotic expansion have order less than or equal tom´1.

Off-diagonal: We now analyse (2.73) after inserting 1´χpx, yq, i.e., lim

εÑ0^`

ż ż

e2πipκpxq´κpyqq¨ξp1´χpx, yqqapκpxq, ξqψpyq|detJ_κpyq|ϕpεξqfpyqdy dξ. (2.77) Let us set Lr_ξ“ p4π²|κpxq ´κpyq|²q^´1p∆_ξq. Since for all x, yPU with x‰y we have

Lr_ξpe2πipκpxq´κpyqq¨ξ

q “e2πipκpxq´κpyqq¨ξ,

we can insert this operator N times in (2.77) and, after integration by parts 2N times on the ξ-variable we get

lim

εÑ0^`

ż ż

e2πipκpxq´κpyqq¨ξp1´χpx, yqqψpyq|detJκpyq|pLr_ξq^Nrapκpxq, ξqϕpεξqsfpyqdy dξ.

Now fix an N PN such that2N ąm`nto obtain that the above expression becomes ż ż

e2πipκpxq´κpyqq¨ξp1´χpx, yqqψpyq|detJ_κpyq|pLr_ξq^Nrapκpxq, ξqsfpyqdy dξ.

Define

Kpx, yq:“

ż

e2πipκpxq´κpyqq¨ξp1´χpx, yqqψpyq|detJκpyq|pLr_ξq^Nrapκpxq, ξqsdξ.

Note that KPC_c⁸pRⁿˆRⁿq and that (2.77) is the same as ż

Kpx, yqfpyqdy.

Therefore there exists an amplitude symbolc2px, y, ξq such thatc2 PA^m for everymPRand such that the above expression coincides with T_rc2sf. By Proposition2.20 there exists a symbola₂ such thata₂ PS^m for every mPRand such that T_a2 “T_rc2s.

Finally, define a_ψ,κ^´1 “ a₁ `a₂. It follows at once that a_ψ,κ^´1px, ξq “ apκpxq,“_Bκ

Bx

‰1

ξqψpxq moduloS^m´1 and that pA_κ´1ψq “T_a

ψ,κ´1, hencepA_κ´1ψq POp S^m.

If AĂRⁿand f :RⁿÑC, thenfæA denotes the restriction off to the subset A.

Corollary 2.21.1. Let U, V be open subsets ofRⁿ and letκ:U ÑV be a diffeomorphism. Suppose we are given φ, ψ P C_c⁸pRⁿq both supported on V and let A :C_c⁸pRⁿq ÑD¹pRⁿq be a linear map such that φAψPOp S^m. If for any f PSpRⁿq we set

pφAψq_κ´1fpxq:“ pφAψqpfæU˝κ^´1qpκpxqq, @xPU ,

2.6 AMPLITUDE REPRESENTATION AND CHANGE OF VARIABLES 57 with extension by zero on U^c, then it follows that pφAψq_κ´1 POp S^m.

Proof. It is clear that the symbol ofφAψ has compactx-support contained in suppφĂV.

Now letψrPC_c⁸pRⁿqbe such thatsuppψrĂU andψr”1on the neighbourhood ofκ^´1psuppψq.

Then for any function f :RⁿÑCwe have

ψrfæU˝κ^´1s “ψrpψfr q ˝κ^´1s.

Thus for any f PSpRⁿq and anyxPU we have

pφAψq_κ^´1fpxq “ pφAψqpfæU˝κ^´1qpκpxqq

“ pφAψqpψfr ˝κ^´1qpκpxqq, (2.78) and the result follows from Theorem 2.21.

Chapter 3

Pseudo-differential Operators on Manifolds

On a compact smooth manifold the pseudo-differential operators can be characterized by taking commutators with smooth functions and vector fields. Such characterization was first stated by R.

Coifman and Y. Meyer [3] in the case of 0-order operators. In 2000 V. Turunen [21] provided a result for pseudo-differential operators of any given order. 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.

The goal of this chapter is to provide the necessary concepts of pseudo-differential operators on manifolds before we prove the main result about commutator characterization of pseudo-differential operators on compact smooth manifolds. Sections are organized as follows:

In section 3.1 we introduce distributions and Sobolev spaces over smooth manifolds. We also show that Sobolev spaces over compact smooth manifolds can be endowed with a Hilbert space structure.

Section3.2shows how to define pseudo-differential operators on compact smooth manifolds and some of their properties.

Finally, section3.3is entirely devoted to our main result about commutator characterization of pseudo-differential operators on compact smooth manifolds.

The references used for the exposition of this chapter were G. Grubb [9], L. Hörmander [10] and [11], M. Ruzhansky and V. Turunen [16], M. Shubin [18], and F. Trèves [20].

3.1 Distributions on Manifolds

Before we talk about pseudo-differential operators on compact smooth manifolds we need to discuss some vector spaces which are suitable for the development of the theory.

LetM be a smoothn-manifold. Recall that a functionf :M ÑCis inC⁸pMq if for any given compatible chartpU, κq on M we have

fκ:“f˝κ^´1 is inC⁸pκpUqq.

Note that ifpUα, καq and pUβ, κβq are two compatible charts onM such thatUαXUβ ‰ H, then for all f PC⁸pMq we have

fκα “fκβ ˝ pκβ˝κ^´1_α q inκαpUαXUβq. (3.1) We now introduce the definition of distributions on a smooth n-manifold. To motivate our definition, we use the compatibility condition (3.1) to describe smooth functions on manifolds.

Indeed, let M be a smoothn-manifold with smooth structureF “ tpU_α, κ_αquαPA. Lettf_αuαPA

59

be a family of smooth functions such that f_α P C⁸pκ_αpU_αqqand such that for every α, β PA we have

f_α “f_β˝ pκ_β˝κ^´1_α q inκ_αpU_αXU_βq.

Then there exists one and only onef PC⁸pMqsuch thatf_α “f ˝κ_α for every αPA.

Conversely, if f P C⁸pMq, then by definition the family tf_καuαPA satisfies the compatibility condition (3.1).

In analogy to this description we define distributions on a smoothn-manifold as follows:

Definition 3.1. Let M be a smooth n-manifold with smooth structure F “ tpUα, καquαPA. Let T_pκ

β˝κ^´1α q:D¹pκ_αpU_αXU_βqq ÑD¹pκ_βpU_αXU_βqqbe as in Definition 1.28.

If for every chart pUα, καq we are given a distributionuαPD¹pκαpUαqqand the familytuαuαPA

is such that

T_pκ

β˝κ^´1α qu_α “u_β inκ_βpU_αXU_βq, for every α, βPA, (3.2) then we call the familytu_αuαPAa distributionuinM. The set of all distributions inM is denoted by D¹pMq.

With this definition we have that D¹pMq appears as a natural extension of C⁸pMq when we identify a function f PC⁸pMq with the family of smooth functionstfαuαPA.

Let us show that this definition of distribution in M is consistent with the definition for open subsets ofRⁿ. First, we show that distributions on a smoothn-manifold can be identified by using only an atlas. For that we need the following lemma.

Lemma 3.2. Let f : Ω₁ Ñ Ω₂ and g : Ω₂ Ñ Ω₃ be diffeomorphisms of open subsets of Rⁿ. If uPD¹pΩ1q, then

T_pg˝fqu“T_gpT_fuq.

Proof. It follows from the chain rule that the Jacobian matrix of g˝f is given by J_pg˝fq “ pJg˝fqJf.

Now, for anyϕPC_c⁸pΩ₃qwe have

xT_pg˝fqu, ϕyΩ3 “ xu,

detJ_pg˝fq

pϕ˝ pg˝fqqyΩ1

“ xu, |detJ_g˝f||detJ_f|ppϕ˝gq ˝fqyΩ1

“ xT_fu,|detJg|pϕ˝gqyΩ2

“ xTgpTfuq, ϕyΩ3, which proves the lemma.

Theorem 3.3. Let M be a smooth n-manifold and let A:“ tpUi, κiquiPI be a compatible atlas on M. If for every chart pU_i, κ_iq P A we have a distribution u_i PD¹pκ_ipU_iqq and the family tu_iuiPI is such that (3.2) holds for all charts inA, then tu_iuiPI defines a unique distribution uPD¹pMq.

Proof. LetF “ tpU_α, κ_αquαPA denote the smooth structure on M. For any chartpU_α, κ_αq we have that

καpUαq “ ď

iPI

καpUαXUiq.

Since Ais a compatible atlas onM we have that for eachiPI the map κ_α˝κ^´1_i :κ_ipU_αXU_iq Ñ καpUαXUiq is a diffeomorphism and thus

T_pκ

α˝κ^´1_i qu_i PD¹pκ_αpU_αXU_iqq.

3.1 DISTRIBUTIONS ON MANIFOLDS 61 Moreover, it follows from the hypothesis and Lemma3.2that for any αPA, and anyi, jPI

T_pκ

α˝κ^´1_i qui “T_pκ

α˝κ^´1_i qT_pκ

i˝κ^´1_j quj

“T_pκ

α˝κ^´1_j qu_j

onκ_αpU_αXU_iXU_jq “κ_αpU_αXU_iq Xκ_αpU_αXU_jq. By Theorem1.26we conclude that there exists a unique distributionuαPD¹pκαpUαqqsuch thatuα “T_pκ

α˝κ^´1_i qui on καpUαXUiq for every iPI.

To conclude that the family tuαuαPA defines a distribution in M we use Lemma 3.2 and note that for everyiPI and for any α, βPA we have

T_pκ

β˝κ^´1α qu_α “T_pκ

β˝κ^´1α qT_pκ

α˝κ^´1_i qu_i

“T_pκ

β˝κ^´1_i qui

“u_β,

onκβpUβXUαXUiq. Now given anyϕPC_c⁸pκβpUαXUβqqwe can choose a finite family of compact setsK1, . . . , KN withKj Ăκ_βpUαXU_βXUjqand so thatsuppϕĂŤ_N

j“1Kj. By Lemma1.61there exist smooth functionstρju^N_j“1 such that suppρj ĂκβpUαXUβXUjqand ř

jďNρj ”1onsuppϕ.

This implies that

xT_pκ

β˝κ^´1α qu_α, ϕy “

N

ÿ

j“1

xT_pκ

β˝κ^´1α qu_α, ρ_jϕy

“

N

ÿ

j“1

xuβ, ρjϕy

“ xuβ, ϕy which is the desired equality.

Corollary 3.3.1. IfΩis an open subset ofRⁿ, then Definition3.1coincides with the usual definition of a distribution onΩ.

Proof. It follows at once by considering the trivial atlas pΩ, Iq, where I : Ω Ñ Ω is the identity map, and then applying Theorem 3.3.

Note that if ψ P C⁸pMq and u P D¹pMq, then ψu P D¹pMq. Indeed, just note that in local coordinates we get

pψuqi “ pψ˝κ^´1_i qui,

which is a well-defined distribution inD¹pκipUiqq. It is not hard to check that the familytpψuqiuiPI

satisfies the compatibility condition (3.2), and thus defines an element in D¹pMq.

Now recall that ifΩĂRⁿis open we haveH_loc^s pΩq ĂD¹pΩqfor anysPR. Moreover, Proposition 1.50 shows that ifκ: ΩÑΩ¹ is a diffeomorphism between open sets of Rⁿ, then the induced map Tκ :H_loc^s pΩq ÑH_loc^s pΩ¹q is an isomorphism. This allow us to introduce the following definition for Sobolev spaces H_loc^s pMq when M is a smoothn-manifold.

Definition 3.4. Let s P R. Let M be a smooth n-manifold and let F “ tpUα, καquαPA be its smooth structure. We denote by H_loc^s pMq the space of distributions u “ tuαuαPA in M such that u_α PH_loc^s pκ_αpU_αqqfor every αPA.

By the above observations we have that H_loc^s pMqis a well-defined vector space.

Suppose that M is a compact smooth n-manifold. In that case we adopt H^spMq :“ H_loc^s pMq and L²pMq :“ H_loc⁰ pMq. The compactness of M will enable us to equip H^spMq with a Hilbert space structure. Namely, let A:“ tpUi, κiqu^N_i“1 be a finite compatible atlas onM; we may assume that the κipUiq are mutually disjoint. It follows from Theorem 3.3 that a family of distributions

u_i P D¹pκ_ipU_iqq that satisfies the compatibility condition (3.2) for every 1 ď i ď N suffices to describe uPD¹pMq, and hence suffices to describe uPH^spMq for anysPR. By Lemma1.62there exists a partition of unitytρ_iu^N_i“1 strictly subordinate to tU_iu^N_i“1. Now define

xu, vy_H^s_pMq:“

ÿN

i“1

x pρ_i˝κ^´1_i qu_i,pρ_i˝κ^´1_i qv_iyH^spRⁿq. (3.3) Proposition 3.5. LetsPR. IfM is a compact smooth n-manifold and we endowH^spMq with the above structure, then (3.3) defines an inner product on H^spMq that makes it a Hilbert space.

Proof. It is clear from the definition that (3.3) is a sesquilinear form onH^spMq ˆH^spMq and that xu, uy_H^s_pMqě0for alluPH^spMq. To conclude thatx,y_H^s_pMq defines an inner product onH^spMq we need to show that

xu, uy_H^s_pMq“0 ðñ u“0 inH^spMq.

Clearly u“0inH^spMq impliesxu, uy_H^s_pMq“0. For the converse we note that

xu, uy_H^s_pMq“0 ùñ pρj˝κ^´1_j quj “0 inD¹pκjpUjqqfor every 1ďjďN. (3.4) Now for each 1ďiďN and every ϕPC_c⁸pκipUiqqwe use that tρju^N_j“1 is a partition of unity and the compatibility condition u_i “T_pκ

i˝κ^´1_j qu_j inκ_ipU_iXU_jq to write xui, ϕy “

N

ÿ

j“1

xui,pρj˝κ^´1_i qϕy

“

N

ÿ

j“1

xT_pκ

i˝κ^´1_j quj,pρj˝κ^´1_i qϕy.

(3.5)

Letρ˜j PC_c⁸pUjq be such thatρ˜j ”1on a neighbourhood of suppρj to obtain xui, ϕy “

N

ÿ

j“1

xT_pκ

i˝κ^´1_j quj,pρj˝κ^´1_i qϕy

“ ÿN

j“1

xpρ_j˝κ^´1_i qT_pκ

i˝κ^´1_j qu_j,p˜ρ_j ˝κ^´1_i qϕy

“

N

ÿ

j“1

xT_pκ

i˝κ^´1_j qrpρ_j ˝κ^´1_j qu_js,pρ˜_j˝κ^´1_i qϕy

(3.6)

It follows from (3.4) that each term on the sum in the right-hand side of (3.6) is zero. This implies ui “ 0 in D¹pκipUiqq for each i and therefore u “0 in H^spMq. This shows that (3.3) is an inner product on H^spMq.

It now remains to verify the completeness of H^spMqwith respect to the norm kuk_HspMq“

ˆ N

ÿ

i“1

pρi˝κ^´1_i qui

2 H^spRⁿq

˙_1{2 .

Lettu_nunPN be a Cauchy sequence with respect tok¨k_HspMq. Then for each1ďiďN the sequence tpρ_i ˝ κ^´1_i qu_i,nunPN is Cauchy in H^spRⁿq. By completeness there exists v_i P H^spRⁿq such that pρi˝κ^´1_i qui,nÑviinH^spRⁿq. Note that if we letViĂκipUiqbe an open neighbourhood ofsuppρi, thenvi ”0on RⁿzVi because each pρi˝κ^´1_i qui,n is zero onRⁿzVi. In particular, we may identify v_i as an element of D¹pκ_ipU_iqq.

Now for each 1 ď i ď N, every n P N and for any ϕ P C_c⁸pκipUiqq we use (3.5) and (3.6) to

3.1 DISTRIBUTIONS ON MANIFOLDS 63 write

xu_i,n, ϕy “

N

ÿ

j“1

xT_pκ

i˝κ^´1_j qrpρ_j˝κ^´1_j qu_j,ns,pρ˜_j˝κ^´1_i qϕy.

Taking the limit asnÑ 8and using the continuity ofT_pκ

i˝κ^´1_j q:D¹pκ_jpU_iXU_jqq ÑD¹pκ_ipU_iXU_jqq we get

nÑ8limxui,n, ϕy “

N

ÿ

j“1

xT_pκ

i˝κ^´1_j qvj,pρ˜j˝κ^´1_i qϕy.

Hence by Theorem 1.25there exists a distribution u˜_i PD¹pκ_ipU_iqqsuch thatu˜_i “lim_nÑ8u_i,n and x˜u_i, ϕy “

N

ÿ

j“1

xT_pκ

i˝κ^´1_j qv_j,pρ˜_j˝κ^´1_i qϕy forϕPC_c⁸pκ_ipU_iqq.

We shall denote u˜i :“ř_N

j“1pρ˜j˝κ^´1_i qT_pκ

i˝κ^´1_j qvj.

By construction we have pρi˝κ^´1_i qu˜i “vi inD¹pκipUiqq and since both have compact support in κ_ipU_iq we have the same equality in H^spRⁿq for every 1 ďi ďN. Moreover, u˜_i P H_loc^s pκ_ipU_iqq and for any1ďi, lďN we have by Lemma3.2that

T_pκ

l˝κ^´1_i qu˜_i “T_pκ

l˝κ^´1_i q

˜ _N ÿ

j“1

pρ˜_j˝κ^´1_i qT_pκ

i˝κ^´1_j qv_j

¸

“

N

ÿ

j“1

pρ˜_j˝κ^´1_l qT_pκ

l˝κ^´1_i qT_pκ

i˝κ^´1_j qv_j

“

N

ÿ

j“1

pρ˜_j˝κ^´1_l qT_pκ

l˝κ^´1_j qv_j

“u˜_l on κ_lpU_iXU_lq.

Therefore the familytu˜iu^N_i“1 defines an elementuPH^spMq. We claim that unÑu inH^spMq.

Indeed,

ku´u_nk²_HspMq “

N

ÿ

i“1

pρ_i˝κ^´1_i qu˜_i´ pρ_i˝κ^´1_i qu_i,n

2 H^spRⁿq

“

N

ÿ

i“1

v_i´ pρ_i˝κ^´1_i qu_i,n

2 H^spRⁿq, which goes to zero as nÑ 8.

Let us show that C⁸pMq is dense in H^spMq for every s P R. Indeed, let s P R and consider uPH^spMq. It follows from Lemma1.43that for each1ďiďN there exists a sequencetv_i,nunPNĂ C_c⁸pRⁿq such that v_i,n Ñ pρ_i˝κ^´1_i qu_i in H^spRⁿq as nÑ 8. Observe that since pρ_i˝κ^´1_i qu_i ”0 on the complement of any open set containing suppρi, we may take the vi,n’s to be compactly supported onκ_ipU_iq.

Now set

˜ u_i,n:“

N

ÿ

j“1

r˜ρ_jpv_j,n˝κ_jqs ˝κ^´1_i “

N

ÿ

j“1

T_pκ

i˝κ^´1_j qrpρ˜_j ˝κ^´1_j qv_j,ns.

Note that for each 1 ď i ď N and for every n P N we have u˜i,n P C⁸pκipUiqq. Furthermore, for each nPNwe have that the family t˜ui,nu^N_i“1 satisfies the compatibility condition and thus defines an element u˜_nPC⁸pMq. Finally, we claim that u˜_nÑu inH^spMq.

To see this, note that for every nPNwe have pρ_i˝κ^´1_i qu˜_i,n“

N

ÿ

j“1

pρ_i˝κ^´1_i qT_pκ

i˝κ^´1_j qrpρ˜_j˝κ^´1_j qv_j,ns

“

N

ÿ

j“1

T_pκ

i˝κ^´1_j qrpρ˜_jρ_i˝κ^´1_j qv_j,ns.

Now for eachi, j we letK_i,j ĂU_iXU_j be any compact set such thatsuppρ_iXsupp ˜ρ_j ĂintpK_i,jq. By proposition1.47we have that T_pκ

i˝κ^´1_j q maps H^spκjpKi,jqqcontinuously intoH^spκipKi,jqq, and thus

nÑ8limpρi˝κ^´1_i qu˜i,n“ lim

nÑ8 N

ÿ

j“1

T_pκ

i˝κ^´1_j qrp˜ρjρi˝κ^´1_j qvj,ns

“

N

ÿ

j“1

T_pκ

i˝κ^´1_j qrp˜ρjρi˝κ^´1_j qujs

“ pρ_i˝κ^´1_i qu_i inH^spRⁿq. Sinceku´u˜nk²_HspMq“ř_N

i“1

pρi˝κ^´1_i qui´ pρi˝κ^´1_i qu˜i,n

2

H^spRⁿq we conclude our result.

Remark: Note that formula (3.3) depends on many choices and is not "canonical" in general.

However, it follows from the compatibility conditions and Proposition 1.47 that if we change our choice of finite compatible atlas onM or that of the partition of unity, we replace the inner product in (3.3) by another one that induces an equivalent norm onH^spMq.

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