Last time Littlewood pale15 6 20

decomposition

is If II Is n.e.agr.uaI Supp PT E I 431 ez

k Supp P.fr El 13h23Recall PF tfFourier multiplier PIG 1413 13

for some function th

Farrier multipliers always commute

just multiplication after f T

for example of a furrier multiplier

write Dd filk da

Dj i INrj

This war Dif 3 3 FtsSomewhat standard notation p t f

134 131 1313 Ffs

Recall PF Y FY E

This mean flat

P f Fe fasf

when tf the

inverse Fourier transform

t iE

Th Ye were a partitin funkyH Hn

A

21413 ElK s

Our specific construction

this 412 3 km

to 3 0113 ko

when 2413131 El

radially decrease 0 13 232

ad 4131 0 13 01lb to

an i

Localization in frequency space

Read Kille Hello tsvplkYI.ttkrfElRn

hot II'tCrucialbut yields equivalent harm

Tim Fft C o sci

llfllcs qyt.IS lPefHwhen AnB means CA EB ECAand here the implied combats depdon

n and s

I e Http FM s

t H Pet Ilo E MTt.INT order at

magnitudeI frequversin Ppf

Remark se space of functions

that are locally Hitler

ft Cf Fate lfo E Cs

Ve will see from proof that it

2 1114916 in feteKs is only for global behaviour

Lemma Bernstein 19 Supp f E 13614

then HTM E R Hello

when A E B hea A ECBhen the implied constant Apids on n

Proof of thm Assume first FEC

P f IT f t CANeel d baud P horns of Pef

kftp.t la syp Sdfx HfHd Is P Yolk aldy Hella

110111 Hello E C Hello

k 4 Coho hence Sti Holstola si

f

Hena for any talk

117 full fatter Hr Ady

YIH HE H dy

se f 11 141 Hsdf HelloRecall 4 3 4 2 3

hence Fick 2n FC2kz

Hello 2 4 12 Al H dy

tells filthiest 1 14 1

i 2 hello S HTM HIDY

Cs 2 Hello

for the other side Assum that

M gyp 2 HPkfko ex

Therf Pref f is

adully Cameogala in Lxcontinuous

KPK Hla E 2KsM

S

yell a 2.5 m Cs M

fix x y t R Tty It HelNeed to prove

IHH Hale Cs Mlt Hsf it Nti s t

2N E x yl E 2 Ntl

Nowno

Ifk ful E 21Pa fly Peftestf o

E ElPxfH Pat t Iu2M 5H

NsCs M 2

M

IM IX Hs

for KEN be Bernstein

H VPefll.no 2 Il Peth

Hera by mean value fhe.ua Vitry

Picky Pickle K H HFPrellEhr t.nl PxfHo

To summarize IX fl v 2

I flirt ftAI E GM 1 H

II I Petti Piatt'tE M It 71st FE K H 2 Il Pio fHo

E M H H't Ix H II 2 n f KsE M H yl t Ix yl µ 2N h

D

E m ix us my

2kts

IH D

Remarks about L theory no details

Bernstein Kuna Supp 4 C IF ERKKIµ

Http n R 11Mt

The function

Sf ZPkfXis called the

squarefunction of P W

11914 n Hsfllp lepers

Fav p 2 trivial equality Mahal

Ifl Sf11

I 217ft n IthIth'Iailnfzai win probs

different frequencies are roughly a armed

512114 f 21144i e Cross hems do not contribute

little carrelatin

Remarks

D llfkcs sy.pt 1114411

11h1mn22241114ftg What about de

If YI 2 HP fth no

fairit ie

for P mama

Hu t.ec IRe

ph qplstfl3Pel 1315 3

up tires on 1314 then Px

lines near 131 v2

Basic lemma Vptf 1361 0

f f t C Fka

ftp.clpfl xE2 ksffHcsh d

when the implied constant depots

on h S nude Pthe function

Prof Pe function as tic frm fha

pelf pic f SPI Biokohence fn any ft IR

I pick fm IS PICH Hx A del

Spin Ifk En fan dy112

c ftp.tlylsdy.HfHcslKHMlcs 2nkflpf2k HltlsdykHlcs 2 ksgfp nl.lyid

Rn

E 2 Hello H

Basic lemur is just about scalingor homogeneity in frequency space

Elliptic regularity of the Laplacian

a IIje IKI

f ftp.ATIFE isiltisjfldj I

14131 1312913

The Laplacian is a Fourier multipliers

The Suppose u C It solves deerThen

rt c necki fach he equals a G function of

1moderate growth plus a CH function

Proof We need 6 show thatd u c Ces for lat E 2

Enough for 121 2 first derivationare C su GL

Neel 6 prove that for i.j i.vn

Di Dja E Ces

dijuBy thin

enough to show that

f Kel1 1 H P ID D a Ily EI Hmcs

Reid DID u T.IR DiDju tPolDiDY

when P Dil a DID u OIis A Sh tl of moderate growth

How to prove 1 1 Ya 131113

PxlDiDj this 3i3 lil

tick3iVID

pits p 2 3

th s 412 3when 1313 4174where p E with plot p

Hence from basic humor Fka

HPrldimulln llk.HN y

2k

11 Nlcswith implied constant deputy 0 phence only on h als Dw

Halli Halfa max Kddflks1412

Next weeks 4Dos local issues atelliptic regularay

Remark If E C at An V

then heKH s

be

because Ih V lol k

trJar a der

the TS

Ks

If f u use Sobolev norms ellipticregularity of Laplacian is easier

Pry Let att satisfies Ia v

Cin f't then Ifs o

VE Hs af Hst2Proof Recall 11M fat 13 4 1964llsince Ia v 131 13 Tls and

Hullfish SCHIST Iii dK

Eff It 131kt 14311 di

Ell alls't 131 leitsyth

Halli 131 KianiHall 5 131 linthE Hall t Hull ns te

po

Ia V u is 2 derivations

swanker tha V h Sobolev sense

ad in Hotter sense

The Sobolev embedding Muret's lean

o sci f f f µts

Hell E HNHartsar Harts E

s

Pr Neel V Kzo

1114dB ft HAH EJ Ilftp.EF.vkzi

11141N Hb E HIDEKI diIR

I

Eif s.IT

DlHslldszEiefyezktiHlDliifFii.T

2 143 4141

j EH ffp2 43 12141

E j 1 4 1 yst A

Remarks

1 off HEt

EctsPr Flat Ek

Idf C HE Pf Els2 How can we recover the exponents

in Sobolev inequality

Hello.is IKIIIYslEHllHsE6rf

Ex1 f X r

Haft r 1144

Il Grf Hasta r Y fHµsthzone over meters to the s

3 an c C at Ceonly for 0L SLI

An example 6kt due 2 but11 daftly x for 14 2

Exercise Art GCH but a K Cu

An example

Example ulx.tt x f log GIF1kV Hal rt the h f Trh Fb

Hhle1 h 2x ay

EX Hy 2

no c C 8 Nis in La

butu 2loylxHt X III

1 th y H t4

Http2hyfxht 7 ae t8EEl

Http

and thank bad

Another example of a multiplier operatorin addition b Laplacian

Hilbert transform

f pr Kr E

this f G fig 59 dxIH2E

f 3 it syncs

Def for ft define the Hilbert

transfer

HY F MEI 4

HT i syncs 9131

cnn.luhim at si win 5 goes b patch

Claim HML 11411 Heesiad H Id

Pa.f

HYHI fz N nkftisgulsldlssl.dz

fluids 11411

Corollary H is exhaled by continuity6 a isometry H Lt

why H 2hbean feint i syncs fisgdillf

9Claim tf E f o ski

H H Has Hfllest Hfll

Hence H Csnl Cs is

well defid at continuous

Proof By the Hiller iron 6hm

ned 4 prove flat floss

ftp.Hf loE2sk

ll9kstlKlHV

kH.HPxHfHoltrlDlHfHinPic f

where pick 1413 fisyalsand the hatin fits with the basic

Khay 3 Hj 3

So if u set

a PG i syn 13 413 pH's

a iz o You you

synch Iiat inlet p G pG s

tick f isg il

Thus

I Pelt tha 11 Pic HI E 2 Hells

fromkbasic

team

For k o

ftp.HHI.ES P.Hf1Id.lssg lflslld3

IR 191349

E 4 Il Illa HellOn

Corollary for any ft Csn L at

any Elk

CH Hf G figg ffk dyKl E

is well held at ftp 2 s 6 is

in Cs

Prost Why is correct for ftsHf F f tf f tf

Therefore f IT put't

Hf x f txt 44K

fi de

Hence H h.lk true for ftFor the general case Hf is

continuous in f in Snl horan

we need to slow that also the RHSat ft P Cnhinors in CSL horn

Htm I feign Sfh dyIfl E

Fix x c IR say to Ned

f I sling FYI is conf Chi

Ottley

ti I t t.itin

Land in

6 ibecause

di i

fig ff.ttdyE4tlccginheynkHello fin di

E Cs 1144 B

why is Hilbert fransfann useful

Props Suppose f is a holomorphic function

in It c Gi In H2o ad

1Mt Ditta as t so

Then on 112

H Ref Inf

HI Int Ref