4
ALEKSANDER PEECZYNSK I
O.I I. Classical spaces of analytic functions and the Hilbert transform. Genera l refer -
ences t o thi s par t ar e th e book s o f Dure n [Du] , Hoffman [H] , and th e treatis e o f Zygmun d
[Z].
In th e seque l
C—denotes th e comple x plane ,
D = {z G C : \z\ 1}—the closed uni t disc ,
3D = {z E C : \z\ = l}-th e uni t circl e = the boundar y o f D,
m—denotes (except §11 ) th e normalize d Lebesgu e measur e o n 3D ,
Lp
= L
p(m)
an d L
P
R
= L
p
R(m) ( 0 p ~) ,
,4-stands fo r th e Disc Algebra which i s defined t o b e th e smalles t close d (i n th e topol -
ogy of unifor m convergence ) subspac e o f C(dD) whic h contain s al l th e polynomial s i n z,
P{z) = Z
rJc
=
0ckzk
(^-arbitrar y comple x numbers , k = 0 , 1, . . . , n\ n = 0 , 1, . . . ).
For 0 p ° ° we define th e Hardy space If9 t o b e th e subspac e o f L p consistin g of
all th e function s / whic h ar e limits o f a sequence o f polynomial s (P s) i n th e nor m | I * | | ,
i.e.,\\msfdDlPs-f\pdm=0.
For p ~ °° we pu t
/ T =fGL°°: ^
D
f(z)zn m(dz) = 0 for n = 1,2, . . . }.
Next w e pu t
A0 = zA = { / G A: / = zg fo r som e g G A} ,
HP=ZHP= {fetf:f=zg fo r somegG//?} .
(Here r denote s th e identit y functio n o n 3D. )
Obviously A an d HP for 1 p ° ° are comple x Banac h spaces ; if 0 p 1, then L p
and IP ar e complet e comple x linea r metri c spaces . Clearl y A ca n b e identifie d wit h a
closed linea r subspac e o f H°° via the ma p which assign s to each/ G , 4 it s m-equivalenc e
class.
Let g b e a n analyti c functio n i n th e ope n uni t dis c D\bD suc h tha t fo r m-almos t al l
z G 3D there exist s th e radia l limit , lim
rt x
g(rz). The n th e measurabl e functio n z
lim^u^rz) (define d rc?-almost everywhere o n 3D ) is called th e boundary value function of g.
Recall (cf . [Du , Chapter s 2 and 3] ) tha t ever y f^lF ( 0 p °° ) is a boundar y
value functio n o f a unique analyti c functio n i n D\3D; which i s called th e analytic extension
of/; w e denot e thi s function , unles s otherwis e stated , als o by /. I t satisfie s th e inequalit y
(0.2)
SU
P M (r,f) *
0 r l
where Mp(rt f) = f
dD\f(rz)\p
m(dz) fo r 0 p « an d M„(r, f) = es s sup z^bD\f(rz)l
Conversely, i f / i s an analyti c functio n i n D\3D whic h satisfie s (0.2 ) fo r som e p wit h
0 p oo
?
then ther e exist s a boundary valu e functio n of/an d i t belong s t o H?.
An/G l iff/extend s t o a continuous functio n o n th e dis c D whic h i s analytic i n
D\3D.
l
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