CHAPTER 1
Introduction
The wav e packe t analysi s o f this boo k originate d i n th e celebrate d wor k o f L .
Carleson [5 ] o n almos t everywher e convergenc e o f Fourie r serie s o f L 2 functions .
Wave packe t analysi s i s not name d i n Carleson' s paper , bu t appear s a s a n a d ho c
means o f solving th e difficul t proble m o f a.e . convergenc e o f Fourie r series . I n th e
decade followin g Carleson' s paper , severa l author s picke d u p th e idea s an d refine d
the analysis . P . Billar d [1] proved th e analogu e fo r Wals h Fourie r series , R . Hun t
[15] extende d th e resul t t o L
p
wit h 1 p 2, P. Sjoli n [37 ] gav e a n extensio n t o
higher dimensions , an d C . Fefferman [9 ] gave an influential ne w proof o f Carleson' s
theorem based on a decomposition o f the relevant Carleso n operator. Severa l book s
have been devote d t o Carleson' s theorem .
Despite th e wid e recognitio n o f Carleson' s theore m an d manifol d researc h ac -
tivity, th e profoun d idea s o f wav e packe t analysi s hav e fo r quit e som e tim e onl y
been applie d t o problem s ver y closel y relate d t o th e origina l proble m motivatin g
the wor k o f Carleson . Consequently , interes t i n Carleson' s idea s ha d becom e les s
pronounced b y the ninetee n nineties , when M . Lacey [16] and Lace y an d C . Thiel e
in a serie s o f paper s [20],[21],[22],[23] revive d interes t i n wav e packe t analysi s a s
a too l t o proo f bound s o n th e bilinea r Hilber t transform . Th e bilinea r Hilber t
transform ha d bee n know n t o A . Caldero n i n the sixties , who encountered i t i n hi s
investigations o f th e socalle d firs t Caldero n commutator . Caldero n succeede d i n
estimating al l commutator s appearin g i n th e multilinea r expansio n o f th e Cauch y
integral on Lipshitz curves [3] , [4] without th e use of the bilinear Hilber t transform .
The problem o f proving L p bound s for the bilinear Hilber t transfor m remaine d a s a
separate challeng e to the analysi s community an d wa s not seriousl y advance d unti l
the abov e mentione d work .
For a very brie f momen t i n histor y i t ma y hav e appeare d a s i f Carleson' s the -
orem an d bound s o n the bilinea r Hilber t transfor m woul d for m ne w smal l equiva -
lence clas s o f problem s withou t application s t o a wide r fiel d o f Mathematics . Bu t
subsequent development s hav e show n tha t a t leas t thi s equivalenc e clas s i s rathe r
large with man y interestin g ramifications . Moreover , question s arisin g fro m differ -
ent field s hav e bee n approache d an d solution s bee n produced , th e mos t strikin g
example bein g th e estimate s b y Lace y an d L i [19] on th e Hilber t transfor m alon g
vectorfields, a proble m wit h a n independen t history . Also , connection s wit h er -
godic theory [17] and scatterin g theor y [29] , [30], [34 ] have been studie d an d ma y
hopefully bea r furthe r frui t i n th e future .
The main part of these lecture notes provides a snapshot o f wave packet analysi s
in the mi d ninetee n nineties , a t whic h tim e Carleson' s theore m an d bound s o n th e
bilinear Hilber t transfor m wer e known , albei t i n a languag e tha t ha s develope d
since. I t certainl y woul d b e desireabl e t o hav e a comprehensiv e discussio n o f th e
work on wave packet analysi s since, but suc h a project ha s turned ou t t o be beyon d
l
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