Ole.rs cii Vlan.rskizo1211 5 20

for ft L

go Y L Agent

Howeur if gtf its nottrue in general that f g c

Exam fetidfly 1

1

gecillN Nt i

f g has a heavy tail robin sif og x n for a

Exercise 4 get f gtfg

solution fgtf by def ftp.h fgPlancherel If fe

HEIL GTI KfkThe proof unvlal under assumption tht

f f t l 5

Corollary Planchenel in Intl

for ft L nt HI

H F H rt Kfk

Prod know the result for f cGiven f E L'n E consider

2 fan f in l and in E

Yf full a

Hf fall 0

Then

I I f in L

since Kfi fth a Hfa ft so

2 In EI FmH full rt Hfmll

by planchent for Schwartt function

Also the sequin.me fin m

is a Cauchy sequence in Lh as

Vkinwffm.FI a C2T hh fm fallHence Fg C I with

Fm g in LaSince fu F in Lo necessarily

f g

I I

Claim 9 in L

k g in v

Hence Gim ft Inti7 fm f in L2

Fm I in L

tha f EI119th fig HillEH ftp.lllin k E I4lflk

Dn

Radon transform

for f t L CIR define

l I

p

dm

whenPat fxarjfr.at

By Fubini f art tGis well defined forany DES foralmost any C EIR

If ft 44 1 then

Rf 5 HR 6 is cutand company suppnd

Examplei f HE EEN

f art H E Pae

Thin slice theorem e.g Falconer Hs

If n23 then for almost aes

the function

th it E Pseis confinvors

with explicit bonds for NEY mm

for sa tu

More generally if ft In Lh then

for aln.sk DES the function

th Rf Cartis continuous in t

Exercise If he 6 then its ere

C function

Etat

Rhett GTE

ftp.i.iyn

EsWhats a counter example in H

Kakeya set CE 112

A compact set H c D

fatsFt st 12 Htc left at

Thulin it gets RI c art

I I It 14 0

BUT gao wit sup yalHence y is not continuous

EEE4,5

The F Kakya set in 3Dwill and proud

Convention Dembe

Ef ai _I ftp.tte ittdt

Farrier transform ONLY in thef variable

Lemmy f n D FT is 1 D FT04 RT

for ft L HI THR Ats

If Ca x t.rsProofi fix AES

whoa D e lo o

Use coordinates in IR

IR at It y c Kr IR

Kiif Cta f fly e iii no oh

I

fifteen

fS f.tk es de e dtIR

TRf Art

Io Rf Ca E eitt de

Rf Et R

Corollaryi for ft L 11127if Rf fA Rko

fan vi a

I 4 0

Examplei Integrability is crucial

F sina.bhlen.f.in f in 112 EHi l Flin l He is

i ad

f l Gf D

I

How do Canfruch

f hin in Gn

i

Vg f

Zak man 1182 Uniqueness at ha uniquefor fk Ratu transform

Proposition Parseral for the Palm transform

EN'S Ift I thiamin t

Vfc L.nldm 4

Recall polar coordinates

If f first du cnr dr

co area formula change of variables

xFubini

Sf S f f Fel112 x f t

41 1 1 1

proof of Proposition

Ga ftp.ft flflII

I ro der du t

L Iftroll mn droitW

If fair p

Lemmai Let us Mao f E L 12 1

D H FH en

s they Iti em

III is HFA a GMneeded ii HfHµ E Cam

if a mint 1 HIIn fact f t.se R

If f fall1

E Gm

ad G C depend mly a M

Proofi Why is f ELHEH em IIFHa EnIff HIM 2Mt F t'm

tem

If h I I the integral is convergent

we getHEH E 2Mt M E 3M

Theuton Ft ElkEx

f HI It f f lyeitt dei h.lk true

HAI E HEH E 3M

In order f prone ii

Htt Fts fi far i d

feint eittsl feitlt sl.it y.tt sl

leitt eitsleminls.tlslfE2fxttsDd

I.ec minltx e Xd xd

IFH flsllEIFIIFAY.fi t skhHiller continuity Fts

IFYI.fi IeIIElHlH d

th h

2Mt fmFH Hhdynalhi.eYu

ant EciµF

If h 2 14 is convenphECM

i 22 h 2 Dn

Lemmai Let us Mao f E L IR si

D Il Fk en

s I they Iti en

Thr t is L Miller forLi min Il I

Proof of the slice theorem Given

ft L'n EUR need to showflab for almost any DES

th Rf Art is a HitlerCantin.rs

for L cz E min 11,133

By the lemma it suffices toshow that for almost c S

1 Iglehart lth It ex

2 IIRflattldt a

However G holds foes byFubini

Idries.tl dtEIfRIflKa.tldt

a

fubinifl

Navy i follows from Parseral

S Ilkka.nl lil h du'tS Glo

2 stiff 1ft 2x

1KHence Glo is integrable al in

particular fish a e Dr

Whale's correct in 3Dand then use it to prove flat

dime C f Kalayaveraged

A variant a swathed version of RfFm du te IHR

Etr

Rrf Art Is SRffes Ist r

Rr L It is bdl of norm

Why Etf

IRrflo.tl eIsSRlfllEs1dst f

E t.IR Hla.slds Ir.HfH

Denotepff g Sfp Riflort

The 21 substitute for Iia 6hm

Fm ft L E IN h 2 A

Rff ECTl.GE lltll illflk is

The proof uses Parseral like before

and the ID lemma

Lemmai If G K lR Mao

1 Kalla E Mx

4 Stahl kids emno

then ft

f hae

D eiaft.bgii r

e cm NytProof

gaa ei't di

x

ei.it y sinInWun

M m2 I

Divide the integral inks 3 regions

4 HIEI Eff 16hrs1 di E 4M

14h

f

it Haifa.int

multiplyand dine

9M log't3 1h12 t

d Staffa1h12

f 312

ftp.T.it Timiiriec f

E M cf e CM or

The 21 substitute for Iia 6hm

F u ft I E IN h't B

Rff ecfl.GE lltll iKflk

Pn Enough to prove for fefbecause

Hrf RrGH e fr tf GN

H Rft Rf yl at Hf gilTake It f Hf gk e fo HelldS Rft e Is Skiing

From now on assum ft

Rtu il h

Rif aseep Roflort

f I c EWe will pray that fats112ft lay echytftfhiffk.at HID

infrom here by integrating a 5 adusing Parseral for Radon

I

inh.egn.es

54 e c ly't Kelli 114117

from Cauchy Schwartz

Ift early't Hell 11111kThe prod ot ft follows from lemmawith

n Kelli 1 141 1121 1 oh

GA Hor satisfies hc f

a Hallo E KI Ha e HN en

s Iolaall Hd Em

Hence G satisfies the assumptions offhe lemma

Relatin 6 Riff att Rf a aFourier inversion is legimitak

Rf At ffffcqyeittd.amC tf ACH

Rsf Et to 5 Reloadsto

fr I Ian ei's dads

T Iaa ends da

eixi e I

Iaa e

By henna F te IR

Rrf Cattle in ITTake sup our f

112ft a e m Neng's

whirl prnus fr Dr

ThmCDa H Kakeya sets in 217

am of Hausdorff dimension 2

In 3D its conj to Lau din 3

but ash known

Proof Fe 112 compact Kalay setx

Suppose FEU BicF I f E

Ned f frou that Facs

II diam Bio s car 20

Case i All balls have the same

radius of o

May assume by ca patness that

f E Bk

Lwe will show NZcr.TT t.glthis implies

II dial B Y N Nk G j zcrNotation By Ball with sam

135 center as Bk

bff fait

fr Ix HII Fief Ix HdF contains a unit segment in

each direction

F confrain a Irlrerchangle in each direction

Hence for C S

Ritter CHI

sup R Her art

124kt will be at least 1

on an interval of radius of

FE Bic fret BIBy the H substitute for slice 6hm

f Ifr

K 2T RI da ecifyI AHHHHI

lectI Hotta

teary Notre

If NTH then rally to prom

Tf nd et fh

ftp.fyi.rs c NzcItha Nz

by't

ly gI