Lecture 2

Pseudo - Differential operators

Calderari - Vaillancourt theorem

Consider a symbol a = alu , 3) Et

✗ € Rd,

Es C- Rd,d } I

We want to sturdy Op ( a) Op

( Op /a) u ) ( x) =,d) e

" " -

Saks)Û /Mds.

pgd

• 2f a = I then Op (1) = Id

Fourier rhwuntn theoreme

• 2f alx,5) = I a ✗ (n) ( is)

"

KIEN

a,E C J ( Rd )

the space of co functions which are founded

as roll as all their dérivative .

Lfv E I (Rd) S = S

ai v E J

then Op (a) = En a ✗ OÎKIEN

In this lecture : a = a ( ×,5)

a E CI ( IR"

; e)

meam that : t ✗ End, tp e Nd

sup / À Osta (x, 5) | < + oo.

Rdx Rd

1. Action on the Schwartz space

2f ne SCRD),then Û c- SIR') so their?

I to alx,5) Ûls) belongs to SIR ! ) .

Aime SUR ' ) C L' ( Rd )

,(On /a) a) ( x) is

roll defined . In addition Lebesgue's différentiationtheorem impurs Qplalu E C° (Rd)

.

Roy .

ta E CI ( Red ),tu C- SUR ' )

,

Oplalu E SIRH .

• op (a) : SUR') → SIR ' ) is continuous.

Prof Op (a) ulx) = 1-(2M)' de

" "

{ ↳ 5) Û / 5) dg .

top /a) ulx) / E | la / × , 5) I tût) / dsR'

c- " ah; / Y,Ï÷Ë+Ïû " > tas£ Hall

pq,>

Il IHISÂ"

tût» / llpç / µÇ÷+ ,

µ E SUR') ⇒ Û E SCIR") ⇒ (1+151)"

'û c- LÇ .

top / a) u /{ • E C Hatty Ndt, ( û ) c- f1✗ Hs))

dt,

< to.

TÉN'p( v) = Imp s' v15 ) / .

KK-pgc-p.at/lplEpNp ( Û) E C Np + ( dti)

( a )

Opta) : f (Rd) → [• ( Rd)

.

•We daim that

It x

" 0¥ Op /a) u Hyo , ,p;)E Cap Npk.pt/n )

Cqp depends on A DÎ § all↳ ( Red) ,

181+11

Etait / PI .

We woe :

Qj ( Op (a) u ) = Op / dxja ) u t Op /a) dxju{✗ j Op (a) u = Opta ) (xju) + iopldsja ) n

Exercise : verify

✗je

" " }

= ! dsj e" " }

+ integrale ly

parts .

It fellas that

✗"d! @plat u ) is a htnear combination of terms of

the fam : Opel did! a) ( x"

o!- tu )

A ×

"

d! / opta) u ) H(°

/R' )

E C Z Il OÎ ç'a 11¥ , Nan (à-%!

-

tu ) .r,r

Gims the wonted estimait :

Np ( Orla ) u ) estimated in terms of semi - noms

of a in CI ( R") and semi - nains of u in SIRH

.

D

-

Z.

Boundedness on [ ( Rd).

Theoreme ( Calderari - Vaillancourt ).

fa E Cf ( IR"),

op (a) E L ( L' IRM) .

(Opca) entends in a unique way as au quator founded from E to L? )

( E-④ - E )

Prof : • since CI ( Rd) C S ( Rd) is deux dn L' IR ' ).

( C : ( Rd) is dense ( measure theory ) + convolution )

to it is enouqhto proue an estimait : 3- c > o / tu ES/Rdl,

Il Op (a) µ Ily

E C Nutty .

then linear extension.

•For simplicity of votations , D= 1

, 2,3 .

• Prop by Hwang ( T.A.MS.

1987 )

Vey nue prof based on Wigner Transform /

Wave packet transform .

Heuustics Opca) is founded on L' UR? ).

"

W*( a Wu )Qpla) u ~"

where W : EUR ? ) → [ ( IR? ✗ RÇ )isometry.ae

Gabor transform / Wick quantizatim / Wigner . . .

Fourier Transform lœalèzed .

K"

Wu ( ×,5) = | e-

" Y" Y / × - g) u (y ) dy .

Rd

4=1 → Wu ( x, 5) = Û (s)

i) Q E C : ii) Q / g) = e-' S ''

④Iii) Hwang Q ( S) = z k=d

.

Lemina Gurren u C- SUR") set

Wu (xp) = | e-" Y "

gÎ¥ykdy (× , 5) ERIK"-

Md

D Wu E CI ( R"

) Fx tp tre Nd

sup ( 1+1×15 151'

I did! (Wu) ( ×, 5) | ( to .

M' ✗ Rd

a) 3- A > 0,tu E SUR ' )

,

Il Wu Il[ (pad)

= A Du //[ (pa)

3) tre Nd,3- As> 0 / H À /Wu ) /lpçppça) E At Null Ecpa) -

We Resume at 5:10.

D

s'

d.io?wulx.s)=sroio!fe-iY-Sl#-yFMHdggre-itS=1.idyIe-ib. }

s' ÊçPWu Ixis) = / 1- idylle- it ' ) t - iy )

P À / Équipes .

( intégration by parts )

=[ * je

"" ojfytuk.NO" "

ftp.g.ldrJ'+811=8

À ¥12 £ p)' + r E ¥2

A ÇB meurs A E C B C universal constant.

1+1×12 E ( It I x - s1' ) ( Itty )

' ) ( triangle inequality )

⇒ i÷.

⇐

It / y /2

Ô""

(ÉE ) E TE

2) Fix x E Rd.

Notice that

Wu IX. D= / e-i"

,Î¥÷ dir

Rd

=F•⇒ l

Plancherel 's theorem :

flwulx.si/'d5=lzR)dfpay+Y?y-pdy .

Rd

Fubini :

f) lwulx.SI/'d5dx=(znjdfftu /b) l'

Rdxpzd

dy dy

RIR "

=À] tu / g) l' dy

pd

wheredis

À= ( 2M)

" | c to since DE 3

Rd

iii) " À Wu Ity E Ar Hully .

Lemmy .

tu E SIIRM,t ✗ C- R

'f5 E IR

':

ûls) =e-

" " ' s

( z -D,) ( e"?

/Wu ) / ×,» )

zu) = e-i " '

( z - ☐×) ( e

" "

(WÈ ) / s,x ) )

.

R.

Here we use the Special Choice fer 4 .

ix. 5

Prof . Write ( I - ☐g) e" "? ( z - Ê O

'

j'= , ¥) e

= (1 + 1×12) ein- S

.

Now write :

e

" ""

û (s ) = JÉ""-

July ) dy

to e

""

ûts) = / ( I - Ds ) (e" '"" " ) 1- ulyldy .

It Ix - y /2

=(I - Ds ) Je

""""

,+Ê%- dir= (I - Ds) ( e

""Wu ( ×

,s) )

.

ùlx) same ! starting fromFourier invasion formula ( exercise )

( Q=e-

's"WÛIS

,

- × ) Wrlxis ) )

Prof cf the theorem.

3- c> o / Il opta ) u Ily E C Hully a c- SIRI.

By duahty 11f14 = surffv-dx.tv/lyl-lto it is sufficient to

pierrethat te > o / tu c- SIR?

fr E S / IRM,

III E C Hully Hully

Z : = ( Opca ) u , r) = / (Oplululx ) ) J /x) dx

I = #-), f) e

""

ah,5) n' (5) Jlx) dsdx

Stup 1 : ux

e

""

n' (g) = (z - Ds ) / e""

Wu /×,s ) )

-

-

so

I = ça f) alx , 5) (I - Bg ) (e""Wulx

,s) ) J /x) dsdx

integrale by parts in I

I =

(§, f) ((Z - Ds ) a) Wulx , 5) ei

"

J /×) dsdx

f) §, b) a ds = | a Dsb ds }Step na the fansub for e

""

it)

I =

, f) KI - Ds ) a) (Wu ) ( I - Dx ) ( e""

WÈ /Ex ) ) dsdx

integrale by parts in x.

I = [ * µ Être - Des)a À"

Wu ei "" WÎ dsdxlritlr "/12

Ditu v1 = A✗u u + u Dirt LG u -R

, v

r"

/ Il E E I Ét - Ds) allLaç ,

" ! Wally , pia )"WÈ /{4x4)

rir"

IZI E Cca ] / -2 HÉ Nutty ) HWÈ ApIHH

But : Hdi Nutty E C " " "e

and Il WÀ Ily = A MÊME = A" Itv "

Lypd )

to :

IZI E C Hully Hully . as wankd.

☐.

n = W ! I A Want .

lol, a) Wu .

GI b) u = -. .

-

loplahluil_Jadpn@Mesun.fOplalun, nul → Jadµ .

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