Last week Sm is 7 6 20the class of symbol alx resellg 6 VfB

Didion1 131

m lisis bold

Main property lalr.si Eltf3ImSm is a frechet space

Def We say that asequence dud

belongs f 8h with uniform estimates

if Harp F Aarp SC

ldfdfdpcr.si EA pCltHYfor all ad k

We associate an operation f at 87

ah D 9 64 fair Dei 9131dg

ArtesAKA Dp Ia Ta differed

hofafin

tDO of order E m

We proved for m o thatalarm I i bolt

H UK H CRI bdlWe pronish also in Hitler nom nextweek I prord fats OpLD is cul ins

todays Relation between 5 fordifferent m

Properties

1 5 E5 sur

US all symbolshet

whatsoever

5 no Mgm infinitelySh thing

MEX HDDsVIcompactly supp.tl inssymbols

2 at Sm best abeshtlas by 6h Leibnitz rule

kid ask.frPrliiadYblFbrivF.derhEC2lttkl

h lrlfi qyl lPHIrl

E Hlnhtt 1131

3 It 131 THE S V mEE

our usual representative frm Sm

Ki Yi Iiimm

Haa Idf Ala Z El 1131local oper.hr only if mis is era

Hine

at Sm 1h14 Tha ES4 If at Sm then

did dare c Sm1131

5 Suppose at 5 ad rtcillkn.IEwith Hop L then

dear 3 a Krs Ther E3is a company supported symbol alit is in Sm with estimatesuniform in E

t.Y.ae Arg acres pinkin

at this tea

Tae Y k Talkprod lost fine

smh lithiffts FIEFThe If at Sm then

Dp Ia H Hk mis Ill

f K2 hair m 03

Impartial for m 20

If at gGET'k fha

OpEa I HEH't Eck

S if at SA flu

opted V E Cinfinitely su fhing

Prost for n o pront last time

1 135 aka C SHea He corresponding nper.hr is bdlin Hk

h2 e FY tf

Hh game Hist finds III mrE aight

E 11911rem SHH 191349

inlithi loyal

Hea a good V ytf.tl gHIy

Htt fakes eiHgNd3Hµk n E HMH

B

A TDD of narder un gainsin derivaties in L sense

Holle nextneek

TDD symbol calculus and oscillatoryintegrals

composing Hbs and conjugation

Then 1 for at Sm besemulti

OpLa Op ID Op Icfor C E Smtd denoted by

c a b

What is ftp.dwb ruleExauplei multipliers alt D all

just multiply the synths

t.in it t aim cp

t.IE'iminiatinnianfar at Sm base

a bar 3 aka bars En

E is one degree sun the Smt I

or decays one order fasho in 3

In fact there is asymptotic expansion

Haitian dial b iegntl N

A formula In the composition is forinstance Crain

a bars fig.Vfeitask.yjbfy.pk

R'ypn

where delta af 3 HER E3ad recede 861 1 Any

Remark We will actually prove thatf E Smt l N with estimates that

depend only on the estimates of a

art b and an N m l n

I

a s dial b t E

Ftl In gmtl N

Thh2 Conjugative at Sm

play 9 t Tf Op43TThe Op Ia Op 1A't where

A't C Sm with

0 4.3 figitifeitt in death

E.it didiacxsT ESn N

Brief renal alert oscillatory integraland stationary phases

Seirolan

Ms

Hyde

ms ibvtin cones from around ofpoint will Flohr D

Because if Follman can integrate

by pahs

I einola eino

Renato oth intr a 5hrs1 x aka bars

TE

EtrOPED OPED iop

3qbtfttsa.si

i.EEiFsdoE.I

W.fXqXs when whrai da

Fron mu n ve prove the compositionrule alal asymptotic expansion

Step suppose at Sm lesthe compactly supported in rant 3Then alx D is an integral operator

Hallen EHnfp.aar.pe3 paid

f Gath facto eisk Hd Mady

Ivykak.tl

Therefore Tao Ts is agot an integraloperator with kernel

K Kil f ka Cret KsltindtI t

f4Ti Jak g eight H biz Deis CtAddy

att f cars eis k ndt

we wafThis would imply that Ta Tg Towhen

4th 6Th falxipblt.DE A dyd

A iy.hr t t iz It A i3 Ct yiy x t ig Z x

i ly 3 t t

so

cfx.pe nJeil HH Malxq1blysldydyIR

is sunk at compactly Suppwht Today

T.ly D f e in't bi 3 dyH

1Forum in the fash variable So

clap Eth ftp.eit 4 hacxrylbly s.ge

Nerd what to do it at ore nah

compactly sypnrtd

Fix rt CTHNIA with riot i

decks acres Kerrey C Sm

bear 3 b Hrs Herrey Eslwith uniform estimates in E

Define

Ce de be rel

LeiHH

afripbelyndydy

We will prove that Cg c gate

with the asymphic expansion with estimates

uniform in E

Justification of the limit e so buildexercise Coe converges pointwise

upb subsequence 6 sore symbol C CSmt

with the som asymptotic expansion at

Opic OpLa Op Ibas operators on f

Now we will p.nu the asymptotic expansion

uniformly in E

Steps Assure a Esm besewith Sfx y if It Kol 2

far son find t.HR µ

The Harp

IX d b Ks ECHHYwith implicit combats deperly m Lrp ArtIn other ants in terms of the hornI In n f in the t variable

KN l be l 3 E 1 1314Recall 19 In E 19ham Hua

TN

II 3 In E Clt 131ii e f Nci114.311 E ftp.IHHYwhen In deped al na and b in additin

Gs T

I stop writing subscript E for lateness

C Crs 645 Jafar.TT limb dyR tTaylor series in y

edrstHTJIE aI n Rn

eiMbGMdnWhn Rn Putts y

remainderin Taylor

Each sunward in a gives us a fern in

the asymptotic expansion

staff 05141 yheimba.si dn

d

diafn.txifeimoibln.mgtil dig di blah

D

We need 6 bound the renailer in Taylors

formula Since ldfafx.si ElltlBlHau v n N

IRnlrs.nl enfl9MlHHllyldsf141N ht IR

Nem

13th Enough to promthe asymptotic expansinfor N m

I Asianet

In niter f c phh the pond of afarexpansion in Shep 2 a net to lmkah

EKA canst Tzu dial bat p.nu that Et Seth

Nyiff.uaestimates

finale rHN

al

IEHDfEHYSRnks.yleimbh.si dy112

II

Maksym ne

f

Finidy

TN_Nt

Hitt t fFunidn rare

HayN

nTo

CHHY.fri lntrn

E 1 131 htt N

This proves the 2 p n inequality from

6h desird Et Smet n

Next pick multiidaices QB alhedfindftp.m s1ltly

mtl N M

Reduce nutters to the case a p o

We know I b f Sl 1131still

suppnihl in Ix x tea hence kiN l Ipl

ldiblynlef.IN HHditties

Rn thy was the remainder in Taylorformula for alt 3 4alx 3 iyf E.IEMyERrlrr3ry

Thutan fgm Ipl

did Rn t 3 y is the reunitein Taylor formula for IED a 4 seyHau similar eshinh

105dg'RE.sn ep4MlHHmn

4kI 19T 94M

I 191

So if we differ.fmN m

did's Earp did

flank.am eiHblqDdyhe get a conshah amber if srmnals each

tiddly fly yltn N IRI

The main contribution will kr l Ip rt

him I at p.etfrm Rr

E ft 131th n Ipl

T sun arin we proved that a bC sail with 6h asymptotic expansion

Im 6h case blares o for It talk

steps far general at 8h lots fixdecoupnsr b ab to Elk

b Noth y b

when yallin It x let

l 0 in ft Kol 72

y N It'D

Now b t Sl satisfies the

assumption of step 2 Therefnn

Tao Tb Te c c sadasymphticexpansion

What remains is A b K ah

c a ba

Ned 6 show that f NI

Id D car 3 centwith estimate uniform in X art e

b ly 3 D for ly x lol

Crs HT ki tak.rpbdy.gl

dyr

yeilt A G y Ix ypeiko H GyJakoph by pals thyHmu TN

yj eiko AH H v

ctx.it Hnf alry1klyHdHylpnko ryv

HstiYfAyalxo.ylpi

bales eiG y laAndy

n r

I A eik A h D

eiko Ah N Ith N

carpet.MEIaHoIHtEAITbrgo.fm

lK tHlHl3 ypk

diet eHo Tol G yl Iggy

Karnes i eKillin It k yDTµ

M rn 11ftIf x I

Integrate w.nl y

g1 131 f dy

1pm IHHm m11th yDK

4 191711 13 4 zit in i

a the faint

1 131 NIL

1

ft NYThis finishes the case F p's in

the estimik for C Cto D

This the sane as did c'CrayL

is again a sun of Pimm nayderirohn at

geilt H.ly 3 proem se Jdydhominids of

alxropbft.SIwhich is still suppnhl

an Iko H 21

San analysis applies at shows that

I

rtoidiilx.nl leiIiiiT

Adam

A Ae 1 4 bhinylh