lecture 2 - talazard.perso.math.cnrs.frtalazard.perso.math.cnrs.fr/lecture-2.pdf · lecture 2...
Post on 24-Aug-2021
10 Views
Preview:
TRANSCRIPT
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 .
>
top related