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The Banach Space H1
Functional Analysis: Surveys and Recent Results I l l K.-D. Bierstedt and 6. Fuchssteiner (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984
1
THE BANACH SPACE
H.,
P . Wojtaszczyk
I n s t i t u t e of M a t h e m a t i c s P o l i s h Academy o f S c i e n c e s 00 - 9 5 0 Warszawa, P o l a n d
W e present the linear -topological properties
of t h e c l a s s i c a l Hardy s p a c e
H
1 '
INTRODUCTION
The aim o f t h i s p a p e r i s t o g i v e a n e x p o s i t i o n o f l i n e a r - t o p o l o g i c a l and i s o m e t r i c p r o p e r t i e s o f t h e c l a s s i c a l Hardy s p a c e importance o f t h e space
H 1 ( D ) and i t s v e r s i o n s l i k e
H1(D).
The
ReHl ( T ) i n
a n a l y s i s s t e m s from t h e f a c t t h a t many i n t e g r a l o p e r a t o r s which a r e unbounded on
L1
a r e bounded on
HI.
W e w i l l n o t e l a b o r a t e on t h i s
p o i n t h e r e . While w r i t i n g t h i s p a p e r w e had two g o a l s i n mind. F i r s t , w e w a n t e d t o show t o a f u n c t i o n a l a n a l y s t an i n t e r e s t i n g and n a t u r a l example o f Banach s p a c e . I n o u r o p i n i o n t h e Banach s p a c e p r o p e r t i e s H (D) a r e n o t w e l l u n d e r s t o o d . T h e r e a r e many i n t e r e s t i n g and 1 d i f f i c u l t problems. Secondly t o a c l a s s i c a l a n a l y s t t h e f u n c t i o n a l
of
a n a l y t i c p o i n t of v i e w may be t h e s o u r c e o f new p r o b l e m s . Our p r e s e n t a t i o n of t h e s u b j e c t i s l i m i t e d t o
H1(D).
The f a c t t h a t
H ( D ) i s embedded i n t o t h e s c a l e of
H spaces, 0 < p i m is receiving 1 P v e r y l i t t l e a t t e n t i o n . L e t u s remark h e r e t h a t t h e s c a l e of H spaP c e s , 0 < p z m seems t o b e n i c e r t h a n t h e much more i n v e s t i g a t e d s c a l e
of L s p a c e s , 0 < p < m . I n p a r t i c u l a r t h e p a s s a g e from p '1 to P than for L s p a c e s . The P < I i s much more n a t u r a l f o r H P P g e n e r a l i s a t i o n from Banach s p a c e c a s e t o p-Banach s p a c e s , p < 1 i s much e a s i e r t o u n d e r s t a n d
i f one t h i n k s i n t e r m s o f
A l s o t h e d u a l and p r e d u a l o f
H,(D)
H -spaces. P receives relatively l i t t l e
a t t e n t i o n . T h i s i s n o t i n t e n d e d t o mean t h a t t h e s p a c e BMO d e s e r v e s smaller a t t e n t i o n than
H,(D)
does.
2
P. Wojtaszczyk
Now some e x p l a n a t i o n a b o u t t h e s t y l e o f e x p o s i t i o n . W e u s u a l l y do n o t g i v e p r o o f s o f p r e s e n t e d r e s u l t s , however w e d i d o u r b e s t t o g i v e d e t a i l e d r e f e r e n c e s t o t h e l i t e r a t u r e where t h e p r o o f c a n be found. When t h e p r o o f i s g i v e n , i t i s u s u a l l y a s k e t c h . A s a r u l e t h i s happens when t h i s p a r t i c u l a r p r o o f o r r e s u l t i s n o t e x p l i c i t l y s t a t e d i n t h e l i t e r a t u r e , b u t i s a n e a s y c o n s e q u e n c e of known r e s u l t s o r methods. The p a p e r - i s d i v i d e d i n t o s e v e n s e c t i o n s of v e r y u n e q u a l l e n g t h . The f i r s t two g i v e t h e n e c e s s a r y a n a l y t i c b a c k g r o u n d , t h e n e x t f o u r a r e devoted
t o t h e subject proper of our exposition while
t h e l a s t one f o r m u l a t e s some a d d i t i o n a l
directions for possible
future research. Now w e i n d i c a t e t h e c o n t e n t of p a r t i c u l a r s e c t i o n s . S e c t i o n 1
describes the space
H 1 ( D ) as a s p a c e o f a n a l y t i c f u n c t i o n s on
t h e u n i t d i s c while S e c t i o n 2 p r e s e n t s t h e real v a r i a b l e approach
t o t h e s a m e s p a c e . S e c t i o n 3 g i v e s some r e s u l t s on g e n e r a l s u b s p a c e s of
H 1 ( D ) . Short Section 4 i s
devoted
t o i s o m e t r i c problems.
S e c t i o n 5 c o n t a i n s t h e d e s c r i p t i o n o f b a s e s and u n c o n d i t i o n a l b a s e s in
HI
and c l o s e l y r e l a t e d r e s u l t s a b o u t isomorphisms between
H1(D)
and v a r i o u s o t h e r s p a c e s . F i n a l l y S e c t i o n 6 d i s c u s s e s complemented subspaces of
H , (D).
SECTION 1 ; Complex f u n c t i o n a p p r o a c h t o
H,(D).
I t i s w e l l known t h a t t h e f u n c t i o n IzI < I }
f
(2)
analytic i n
can have an e x t r e m a l l y i r r e g u l a r b e h a v i o u r a s
D ={z EE : IzI
approa-
c h e s 1 . So i t i s a n a t u r a l i d e a t o c o n s i d e r f u n c t i o n s w i t h somehow r e s t r i c t e d b e h a v i o u r c l o s e t o t h e b o u n d a r y . Hardy s p a c e s i s one s u c h p o s s i b i l i t y , which t u r n e d o u t t o b e e x t r e m e l y s u c c e s s f u l . For 0
we define
f(z), /zl
H ( D ) as t h e space o f a l l a n a l y t i c f u n c t i o n s P such t h a t
So w e r e s t r i c t t h e mean g r o w t h o f a f u n c t i o n . The f i r s t i m p o r t a n t
H ( D ) i s t h a t it l e a v e s a good t r a c e P T h i s is summarised i n t h e f o l l o w i n g
p r o p e r t y o f a f u n c t i o n from on
T.
3
The Banach space H 1 THEOREM 1 . 1 . m
f(z)
Let
=
C anz n =O
n
and l e t
EHp(D)
fr(eie)
a) l i m f r+l and in
f(re
f ( ei 0 ) E L p ( T )
r < I . Then t h e r e e x i s t s a f u n c t i o n
for all
ie)
be d e f i n e d a s
such t h a t
almost e v e r y w h e r e
=f LP(T)
a,
c) i f
p
~
f(eiO)
1t h e n
has t h e F o u r i e r series
C aneine.
n =O
These a r e by no means t r i v i a l f a c t s . From now on w e w i l l v e r y o f t e n i d e n t i f y
H ( D ) w i t h a subspace o f P L p ( T ) . Hidden i n t h o s e s t a t e m e n t s i s a n i m p o r t a n t F.M.Riesz Theorem.
p =I.
L e t u s d i s c u s s it i n more d e t a i l . C o n s i d e r
a bounded f a m i l y i n
Then
(fr)r
is
L 1 ( T ) . I f w e look a t t h e Fourier series of
t h o s e f u n c t i o n s w e e a s i l y see t h a t (f ) c o n v e r g e s i n t h e fweak r m t o p o l o g y t o a m e a s u r e whose F o u r i e r series i s C anei n 0 That n=O
.
L , ( T ) f u n c t i o n is a c o n t e n t of t h e
t h i s measure i s a c t u a l l y a n THEOREM 1 . 2 . ( F . M . R i e s z ) Let
p
b e a m e a s u r e on
2T
Je-inedu(0)
for
=O
T
such t h a t
...
n =1,2,3,
0
p
Then
i s a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t t o t h e Lebesgue
measure. A f u n c t i o n a l a n a l y t i c c o r o l l a r y of t h i s t h e o r e m i s t h a t
a *weak c l o s e d s u b s p a c e o f
we can say a l i t t l e m o r e . Let W
t h e set
{einOln=,
and l e t
H1(D)
is
M ( T ) , so it i s a d u a l space. A c t u a l l y
.A HZ
denote denote
t h e closure i n the
C(T) of
a(Lw,L1) closure of
t h e same s e t . Those s p a c e s c a n b e i d e n t i f i e d w i t h t h e s p a c e s o f a l l a n a l y t i c f u n c t i o n s which a r e u n i f o r m l y c o n t i n u o u s i n in
D,
respectively.
D
o r bounded
4
P. Wojtaszczyk
COROLLARY 1 . 3 . H ~ ( D )is isometric to the dual of isometric to
L_ ( T ) /H:.
C(T)/Ao
and
H1(D)*
is
Now we will explain the canonical factorization. We first introduce
three classes of functions.
(1) Blaschke products. Let (an)n=, be a sequence of numbers from m
such that
m
C
n=1
D
(l-lanl) < a .
Then B(z) = II n=l
janl an-z a n I-: z n
is an analytic function (called Blaschke product) such that (a) the zeros of B(z) are exactly (b) IB(z)/ < I
for
z ED
and
m
counting multiplicity,
IB(eie)l = I
a.e.
( 2 ) Singular inner functions are the functions of the form
for some positive singular measure
of singular inner functions are
for
T. Basic properties
Z E D
for =I
on
z ED
a.e.
CAUTION : S(z)-’
is not bounded, it does not even belong to any
Hp(D).
For $(t) defined on T such that $(t) 20, we define an outer function of the class H to be
( 3 ) Outer functions.
log $ EL1, )I ELP
P
5
The Banach space H 1
where
i s a r e a l number. I t i s e a s y t o see t h a t
y
Obviously
F ( z ) h a s no z e r o s i n
ie) I = $ ( e ) .
(F(e
D.
Now w e a r e r e a d y t o s t a t e THEOREM 1 . 4 .
(Canonical f a c t o r i z a t i o n t h e o r e m ) .
Every f u n c t i o n
f E H ( D ) a d m i t s a u n i q u e f a c t o r i z a t i o n i n t h e form P where B i s a B l a s c h k e p r o d u c t , S i s a
f(z) =B(z).S(z).F(z)
s i n g u l a r i n n e r f u n c t i o n and
F
i s a n o u t e r f u n c t i o n of c l a s s
Converselyeverysuch product belongs to
Hp(D).
%'
The f o l l o w i n g e a s y c o r o l l a r y from t h e C a n o n i c a l F a c t o r i z a t i o n Theorem i s v e r y u s e f u l . COROLLARY 1 . 5 .
f E H D ( D ) c a n be w r i t t e n a s
a ) Every
no z e r o s i n
D
and
f =h
llhl[lr
1
+h2
where
hl,h2
'
have
- =- +-1 t h e n e v e r y f E H ( D ) c a n b e w r i t t e n a s f = g - h , P P cs r ' in g EHq(D), h EHr(D) and f Br ( D 1 ' p a r t i c u l a r f o r p = I w e c a n have r =q = 2 . b) L e t
-
I]
The p r o o f s a r e so e a s y t h a t w e c a n g i v e them h e r e . L e t For a ) p u t h =(B-l).S.F 1 h =FP/'. The s p e c i a l case o f two of
H2(D)
H1(D).
and
h 2 = S . F . For b ) p u t
b ) t h a t every
H1(D)
and
function i s a product of
f u n c t i o n s o p e n s up a new d e s c r i p t i o n of d u a l and p r e d u a l I do n o t want t o go i n t o d e t a i l s ,
here. L e t me say only
t h a t it i s p o s s i b l e t o r e p r e s e n t t h e p r e d u a l o f C(T)/Ao,
f =B*S.F.
g =B*S*F
H1(D),
the space
a s a s p a c e of so c a l l e d compact Hankel o p e r a t o r s on a
H i l b e r t s p a c e . The p r o o f of t h i s a s s e r t i o n c a n be f o u n d i n
[K - P I o r
[ S a r ] and a d e t a i l e d e x p o s i t i o n o f t h e t h e o r y o f t h o s e o p e r a t o r s i s given i n
[H - P I
and
[Pow].
NOTE. A l l f a c t s p r e s e n t e d i n t h i s S e c t i o n a r e c l a s s i c a l and c a n b e f o u n d i n
6
P. Wojtaszczyk
any of t h e books [ D u r ] , [ H o ] , [ K o o ] , [ K a t ] , [ Z y g ] . SECTION 2 :
.
H1 (D)
Real v a r i a b l e approach t o
The t h e o r y s k e t c h e d i n t h e p r e v i o u s s e c t i o n depended v e r y much on t o o l s ( l i k e Blaschke p r o d u c t s ) which a r e p e c u l i a r t o one-dimensional s i t u a t i o n , and are i m p o s s i b l e t o g e n e r a l i s e t o s e v e r a l v a r i a b l e s i t u a t i o n . Thus a t t e m p t s t o g e n e r a l i s e t h e t h e o r y r e q u i r e d new t o o l s . I t i s a remarkable f a c t t h a t t h o s e new t o o l s i n v e n t e d f o r g e n e r a l i s a -
t i o n had a l s o a tremendous impact on t h e c l a s s i c a l t h e o r y . I n t h i s s e c t i o n w e i n t e n d t o d e s c r i b e t h i s r e a l v a r i a b l e approach t o L e t us t a k e f l I H 1 (D)
mines
f ( z ) EH,(D).
if3
f(e
and
i f and o n l y i f b o t h
Imf(eie) e x i s t
Ref(eie)
and
on
Imf(eie)
Since
T.
belong t o
s a y t h a t w e are i n t e r e s t e d i n harmonic f u n c t i o n s L,(T).
Ref(z)
deter-
up t o a c o n s t a n t . Moreover it f o l l o w s from Theorem 1 . 1
Ref(ei*)
t h a t both
H1(D).
) E L T ( T ) , and
= J T l f ( e i e ) ld0. I t i s a l s o w e l l known t h a t
Imf(z)
t h a t both
W e know t h a t
I / f l I H , (D) L1(T)
h(z), z ED
' w e may such
and i t s harmonic c o n j u g a t e have boundary v a l u e s i n
h
I t i s known ( c f . [ K a t ~ , ~ K o o ] , [ Z y gt ~ ha ) t for
€ EL1(T)
there
e x i s t s a harmonic e x t e n s i o n v i a t h e P o i s s o n formula t o a harmonic r-L f u n c t i o n f ( 2 ) and i t s harmonic c o n j u g a t e f(z) h a s boundary v a l u e s on T g i v e n b t h e p r i n c i p a l v a l u e of t h e f o l l o w i n g improper i n t e g r a l
F(eit)
=/f(t-.c T
This j u s t i f i e s t h e following d e f i n i t i o n : ReHl ( T )
i s t h e s p a c e of a l l f u n c t i o n s
f EL1(T)
such t h a t
-
f EL1(T)
w i t h t h e norm d
IlflIReH1(T)
fllLl
T) + l I f 1 I L 1 ( T )
(T) b a s i c a l l y c o n s i s t s o f r e a l v a l u e d f u n c t i o n s b u t 1 t h e r e i s no d i f f i c u l t y i n c o n s i d e r i n g complex f u n c t i o n s as w e l l .
This space
ReH
S i n c e it i s known t h a t f o r
we infer that
f E L ( t ) ,1 < p < P
N
f
also is i n
Lp(T)
7
The Banach space H 1
U
L (T)C ReHl !T)C L 1 (T).
p>l p
Up till now it may be seen merely as atranslation but the point is that in this way we put our space H,(D) into entirely new perspec-
tive. The fundamental development in this context is the Fefferman duality theorem. We start by introducing the space BMO of functions
of bounded mean oscillation. Let f (T). For every interval if. We say that f EBMO(T) if we put fI
=h
IC T
The quantity above is not a norm (it is zero for the constant function) so we define the norm by the formula
It follows from the John-Nierenberg inequality (cf.[J-N],[Ner]) that the above norm is equivalent (for all p, 1 < p < - ) to
JJf/
T
SUP(^
ICT
1
In particular
J J f- € I
1
BMO(T
The fundamental Fefferman duality theorem asserts THEOREM 2.1. The dual of ReH1 (T) is BMO(T); more precisely, for every bounded linear functional x* [ReH, (T)] * there exists a unique function cp EBMO(T) such that for f tL2(T) we have
x*(f) =lf(t)q(t)dt T
and conversely there exists a constant BMO(T) and f EL2(T) we have
cp
C
such that for every
8
P: Wojtaszczyk
REMARK.
f EL2(T) i n t h i s statement i s t h a t
The r e a s o n f o r i n v o k i n g f ( t ) V. ( t )
for
f t R e H 1 ( T ) and
cp
E B M O ( T ) need n o t b e Lebesgue
integrable. The above Theorem 2 . 1 .
implies t h e following description of
H 1 (D)*.
THEOREM 2 . 2 .
H1 (D)*
BMOA = H 1 (D) n B M O ( T ) .
c a n be i d e n t i f i e d w i t h t h e s p a c e
Now l e t us e x p l a i n a n o t h e r e q u i v a l e n t form o f t h e D u a l i t y Theorem. W e w i l l call the function
either
a(t) E l
a ( t ) d e f i n e d on
a n atom i f
or
we h a v e
for some interval
Ic T
supp a ( t ) CI
l a ( t )1 2 - q
and
T
'
and
i a ( t ) d t =O. T
THEOREM 2 . 3 . A function
f ( t )belongs t o
R e H l ( T ) i f and o n l y i f
f =E?, . a
where
~j
over a l l t h e a ' s are a t o m s and C \ X j \ < m . Moreover i n f Z 1 A . I j 7 a l l a t o m i c r e p r e s e n t a t i o n s o f f i s a norm e q u i v a l e n t t o
I I I I ReHl
(T) '
PROOF. An e a s y c a l c u l a t i o n shows t h a t
a t o m s h a v e u n i f o r m l y bounded norm i n
I L,
(T)
f o r a l l a t o m s , so a l l
ReH.,(T). To show
the other
i n c l u s i o n it i s enough t o show t h a t t h e s e t o f a l l atoms norms BMO(T).This f o l l o w s i m m e d i a t e l y from t h e o b s e r v a t i o n t h a t mean z e r o functions i n
L_
o f norm
<1 -
norm mean z e r o f u n c t i o n s i n
L1.
Two comments a b o u t Theorem 2 . 3 are i n o r d e r . a ) T h i s i s p o t e n t i a l l y a v e r y u s e f u l t h e o r e m . I t g i v e s a k i n d of " e x t r e m e p o i n t " d e s c r i p t i o n of
ReH1(T).
Obviously i n o r d e r t o
e s t a b l i s h t h e c o n t i n u i t y o f a n o p e r a t o r d e f i n e d on enough t o c h e c k t h a t i t i s
ReH1(T) it i s
u n i f o r m l y bounded o n a l l a t o m s . S i n c e
atoms a r e r a t h e r s i m p l e f u n c t i o n s t h i s t a s k s h o u l d b e much e a s i e r t h a n t h e i n i t i a l problem.
9
The Banach space H 7
b ) The n o t i o n o f a n atom i s v e r y e a s y t o g e n e r a l i s e .
In particular
w e c a n c o n s i d e r atoms on t h e i n t e r v a l a n d w e c a n d e f i n e t h e s p a c e Ha(O,II) a s t h e space of a l l f u n c t i o n s f d e f i n e d on [ O , I I ] such 1 t h a t f =CA.a w h e r e C I A j [ < m and a a r e atoms d e f i n e d on i j j [ O , i l ] . The norm i n H;(O,Il) i s by d e f i n i t i o n i n f C I A . I o v e r a l l
a t o m i c r e p r e s e n t a t i o n s of
I
f.
Now w e e s t a b l i s h a n e a s y p r o p o s i t i o n which w i l l b e u s e d i n S e c t i o n 5 .
For a f u n c t i o n f d e f i n e d o n
we put
[O,II]
PROPOSITION 2 . 4 . For
f EH;(O,fl)
operator
T
we define
Tf = € R e f -EImf + i ( E R e f + & I m f ) . The
e s t a b l i s h e s a n isomorphism b e t w e e n
H;(O,II)
and
H1 ( D ) .
PROOF. I t i s e a s y t o check t h a t
T
i s an a n a l y t i c f u n c t i o n . For
i s a complex l i n e a r map and t h a t a
a n atom on
[O,n]
Ea
Tf
i s a sum of
'L" i s c o n t i n u o u s on ReH1 ( T ) , i s i n R e H 1 ( T ) t h u s T i s continuous. To check t h a t -1 T i s c o n t i n u o u s i t i s enough t o c o n s i d e r f u n c t i o n s o f t h e form h = a +i"a where a i s a r e a l atom o n t h e c i r c l e . W e h a v e h =Tf two atoms on t h e c i r c l e . Moreover
so
€Ref - E I m f
where
€Ref -EImf = a . The norm o f
remembers t h a t
&Ref
is even and
f
i s e a s y t o e s t i m a t e i f one
EImf
i s odd.
NOTES. T h e r e are many p r o o f s of Theorem 2 . 1 .
i n t h e l i t e r a t u r e . The
t h e o r e m w a s announced by C.Fefferman i n [ F e ] a n d t h e p r o o f i n t h e context of
Rn
appeared i n [F-S].
The p r o o f s f o r t h e u n i t d i s c c a n
b e found i n [Koo] a n d [ S a r ] . The J o h n - N i e r e n b e r g i n e q u a l i t y m e n t i o n e d b e f o r e Theorem 2 . 1 .
was p r o v e d i n [ J - N ] .
found i n [ N e r ] o r [ K o o ] .
The s i m p l e r p r o o f s c a n be
The f a c t t h a t Theorem 2 . 1 .
Theorem 2 . 3 w a s o b s e r v e d by C.Fefferman
is equivalent t o
(unpublished). D i r e c t (i.e.
w i t h o u t u s e o f t h e d u a l i t y t h e o r e m ) p r o o f s o f Theorem 2 . 3 . c a n b e found i n [ C o ] and [ W i ] .
A v e r y g e n e r a l e x t e n s i o n of t h e
u s i n g atoms i s p r e s e n t e d i n [C-W].
H -theory
P
P. Wojtaszczyk
10
i
SECTION 3; S t e i n ' s theorem and i t s consequences.
W e s t a r t w i t h t h e f o r m u l a t i o n o f t h e E.M.
S t e i n m u l t i p l i e r theorem.
THEOREM 3 . 1 .
Let
(u ( n ) ) n = O W
b e a s e q u e n c e of complex numbers s u c h t h a t
n=O H1 (D)
a zn
n
-
W
c a n u ( n ) zn n=O
i s a c o n t i n u o u s map from
i n t o i t s e l f a n d i t s norm d e p e n d s o n l y o n
Now w e d e f i n e m u l t i p l i e r s
An,"
2 2 n < k < 22n+1
or
<2Tn-i
-
=1,2,3,
...
C.
as follows
,22n+2
otherwise. W e also put
1
k =0,1,2
0
kL4
. W
I t i s e a s y t o check t h a t
every
E
n
=+1
C An(k) = 1 f o r a l l k and t h a t f o r n=O m m t h e sequence ( C E ~ A ~ ( ~ ) s a) t ~i s f= i e~ s t h e n=O
a s s u m p t i o n s o f Theorem 3 . 1 .
with t h e s a m e
C.
So w e h a v e
COROLLARY 3.2. m
For
f € H I (D)
l y convergent.
we have
C
n=O
An(f) = f
a n d t h e series i s u n c o n d i t i o n a l -
By a s t a n d a r d a p p l i c a t i o n o f t h e K h i n t c h i n e i n e q u a l i t y w e o b t a i n
11
The Banach space H 1
COROLLARY 3.3. For
f EH1(D) we have
Let us recall the following DEFINITION 3 . 4 .
A Banach space
X
has
an unconditional finite dimensional expansion
of identity if there exists a sequence 0: finite dimensional operators m x EX, x = c Tn(x) and the series is (Tn)n=O such that for every n=O unconditionally convergent. Obviously an unconditional basis is a special example of a finite dimensional expansion of identity (for the definition see Section 5). Now we have the following THEOREM 3.5. L1 (T) with an uncond tional fin te dimensional expansion of identity. Then X is isomorphic to a subspace of H1(D).
Let
X
be a subspace of
PROOF.
-
x
m
C Tn (x) and the n=O series is unconditionally convergent. This implies that ( ( \xi(1 =
We have
m
Tn : X
12)’
X I finite dimensional and
=
1
J ( C ITn(x) is an equivalent norm on X. By a standard T n=O kn perturbation argument we can assume that ImTnCspan{eiJt}. 1=pn We choose integers way that
pn
and
wnI vAs
[pn +pn,pn + k , ] c [22vn,22"n+1I.
Now it is obvious that the map
x
-
.
being all different in such
m
C
e
n=O
ipnt
Tn(x)
is an
a
P. Wojtaszczyk
12
i s o m o r p h i c embedding from
X
into
H1(D).
H (D) c o n t a i n s s u b s p a c e s i s o m o r p h i c t o L p ( T ) , 1 < p 52, 1 d o e s . S i n c e by a t h e o r e m o f R o s e n t h a l [ R o s ] e v e r y r e f l e x i -
I n particular
L1
since
L1 i s isomorphic t o a w e g e t t h a t e v e r y r e f l e x i v e subspace of
v e s u b s p a c e of
s u b s p a c e o f some L1
is
a
Lpl
p 71
s u b s p a c e of
T h i s l a s t f a c t was p r o v e d i n a much s i m p l e r way i n [K-PI.
H1(D).
A s a n e x t a p p l i c a t i o n of S t e i n ' s theorem w e w i l l p r o v e t h e f o l l o w i n g
r e s u l t due t o S.Kwapien a n d A . P e l c z y n s k i
[K-PI.
THEOREM 3 . 6 .
L e t (f,)
be a s e q u e n c e i n
H, (D) s u c h t h a t
11
Canfnll H,
(Dr
1
2 2 (Clan/ )
-
Then t h e r e e x i s t s a s u b s e q u e n c e (f ) s u c h t h a t s p a n ( f ) i s nk "k complemented i n H 1 ( D ) . PROOF.
B y a s t a n d a r d g l i d i n g hump a r g u m e n t w e c a n assume t h a t t h e r e e x i s t a
subsequence ( f a)
(f
"k
=f
nk
) and a sequence o f m u l t i p l i e r s
m
(rk(j))j=l
such t h a t
"k
c ) f o r a l l c h o i c e s of
Let
gk € H I ( D ) *
=*I
t h e sum
CEkrk(j) satisfies the k a s s u m p t i o n s o f Theorem 3 . 7 w i t h t h e c o n s t a n t i n d e p e n d e n t of ( E ~ ) . E~
be s u c h t h a t
ne
a
p r o j e c t i o n o n t o s p a n ( f n ) by t h e f o r m u l a
k
13
The Banach space H 1
1
er that
P
1
is a cont nuous projection onto span
f ). nk
REMARK. f = z kn where kn+l/kn LX > I for all n, n then the above argument works without passing to a subsequence, so we obtain the following classical inequality of Paley [Pa]. Let us observe that if
COROLLARY 3.7.
n f = Z anz m
Let
kn+,/kn A,
>1
and let
n =O
be in
H I ( D ) . We have
1
n
I akn
REMARK Some of the results of this section follow also from the existence of an unconditional basis in H, ( D ) (cf.Section 5 ) . NOTES. Theorem 3 . 1 and Corollaries 3 . 2 and 3 . 3 . were proved in [Stel. A nice proof using atoms can be found in [C-W], Theorem 1 . 2 0 . Theorem 3.5. is a routine observation which, to the best of our knowledge, does not appear in the literature. Theorem 3.6. was proved in [K-P1,Theorem 3 . 1 . The proof given there is different and rather more complicated. We may remark that Theorem 3 . 5 shows that Theorem 3 . 6 holds also for H I (Tn); the H I space on the polydisc. SECTION 4; Isometric questions. We already know several isomorphic representations of H 1 ( D ) and we will see many other later. In this situation the importance of isometric theory is rather diminished. On the other hand there are several interesting results and open problems pertaining to the (in this particular norm),so we decided isometric theory of H 1 ( D ) to present them in our exposition. The existing isometric results
P. Wojraszczyk
14
c o n c e n t r a t e on t w o q u e s t i o n s :
a ) d e s c r i p t i o n of i s o m e t r i e s b) extreme p o i n t s t r u c t u r e . H - s p a c e s on more g e n e r a l P t h a n t h e u n i t d i s c ( c f . [ Ru 1 ] and t h e b i b l i o g r a p h y
The i s o m e t r i e s h a v e b e e n d e s c r i b e d f o r
En
domains i n
g i v e n t h e r e ) . The t h e o r e m f o r
H,(D)
reads a s follows:
THEOREM 4 . 1 .
Every i s o m e t r y
where
I
of
H1(D)
is a non-constant
@
I b ( t ) d t = J ( b o @ )( t )1 g ( t ) I d t T T
i s g i v e n by
H,(D)
inner function i n
D,
q EH,(D)
and
f o r a l l bounded, Bore1 f u n c t i o n s
b(t)
C o n v e r s e l y e v e r y map d e s c r i b e d above i s a n i s o m e t r y
d e f i n e d on
T.
from
into itself.
H.,(D)
into
T h i s theorem was p r o v e d by F . F o r e l l i
[Fo]. For e a r l i e r r e s u l t s t h e
r e a d e r may c o n s u l t [ F o ] , [ R u 3 ] , [ H o ] . I t i s known t h a t a s u b s p a c e o f
i s norm one complemented. A s f a r as w e P know t h e c o m p l e m e n t a t i o n o f s u b s p a c e s of H I (D) i s o m e t r i c t o H , ( D ) L p , p 21
isometric t o
L
wa.s n o t i n v e s t i g a t e d i n g e n e r a l . Some p a r t i a l r e s u l t s are c o n t a i n e d i n [Ba]
. The
o t h e r open problem c o n n e c t e d w i t h i s o m e t r i e s i s t h e
H (D) contain a s u b s p a c e isometric t o H p ( D ) , 1 1 < p < 2 ? I t i s known ( c f . Theorem 3 . 5 . a n d f o l l o w i n g r e m a r k s ) t h a t
following: does H
P
(D),1 < p '2
i s i s o m o r p h i c t o a s u b s p a c e of
HI (D).
The e x i s t i n g i n f o r m a t i o n a b o u t e x t r e m e p o i n t s t r u c t u r e of i s presented i n
[Ho],
Chapter 9 .
H1(D)
L e t u s quote t h e d e s c r i p t i o n of
extreme p o i n t s . THEOREM 4 . 2 .
Let
f
be a f u n c t i o n i n
the unit
b a l l of
H,(D)
H1(D).
Then
i s a n e x t r e m e p o i n t of
f
i f and o n l y i f
o f norm 1 . W e do n o t know a b o u t any f u r t h e r r e s u l t s .
f
is an o u t e r f u n c t i o n
In particular strongly
exposed p o i n t s o r d e n t i n g p o i n t s of t h e u n i t
ball in
H,(D)
are
15
The Banach space H I
n o t d e s c r i b e d . F o r d e f i n i t i o n s of t h o s e n o t i o n s and t h e i r i m p o r t a n c e The problem o f d e s c r i p t i o n of s t r o n g l y
t h e r e a d e r may c o n s u l t [ D - U ] . exposed p o i n t s i s d i s c u s s e d i n
[ G a J ] , IV.5.
general theory (cf.[D-U]) t h a t
H , ( D ) h a s many p o i n t s o f b o t h k i n d s .
I t f o l l o w s from t h e
The l a s t i s o m e t r i c r e s u l t w e want t o m e n t i o n i s t h e f o l l o w i n g t h e o -
r e m o f Newman's [ N e w ] . THEOREM 4 . 3 . Let
m
(fn)n=,
and
11
weakly and
f
fn/l
-
be i n
Assume t h a t
H1(D).
11 f II .
Then
fn
tends t o
fn
tends t o
f
f
i n norm.
Newman named t h i s p r o p e r t y " p s e u d o - u n i f o r m c o n v e x i t y " . I n t h e Banach s p a c e t h e o r y t h i s i s e x p r e s s e d a s " t h e Kadec-Klee norm". Kadec a n d K l e e h a v e shown t h a t e v e r y Banach s p a c e c a n b e renormed t o have t h e For a d i s c u s s i o n o f t h i s n o t i o n
p r o p e r t y d e s c r i b e d i n Theorem 4 . 3 .
i n t h e framework o f g e n e r a l Eanach s p a c e s see
[Di].
SECTION 5 ; Eases a n d v a r i o u s i s o m o r p h i c r e p r e s e n t a t i o n s .
F o r a n a r b i t r a r y Banach s p a c e
t h e s y s t e m of v e c t o r s
X
i s c a l l e d a Schauder b a s i s i f f o r e v e r y element m
a u n i q u e s e q u e n c e o f s c a l a r s ( an ) n=O converges t o
x
i n t h e norm o f
X
x
m
C x:(x) n=O I' system
x-. I1
x;(xm) = O
,
x* € X * n
, so
a
i .e. a system such t h a t
m
o n l y i f t h e c l o s e d l i n e a r span o f {xnln=O e q u a l s
I f t h e b a s i s (x,)
is actually
an
t h e series h a s t h e form
f o r n # m , t h e n such a system is a basis f o r
of p a r t i a l sum o p e r a t o r s
C anxn n=O
X.
It i s a l s o known t h a t i f w e a r e q i v e n
( X ~ , X : ) ; =X~xC X*
there Zxists
s u c h t h a t t h e series
I t i s w e l l known t h a t i n t h i s case e a c h c o e f f i c i e n t
g i v e n by a l i n e a r f u n c t i o n a l
m
( ~ ~X ) ~ = ~
N * P (x) = C x (x)xn n=O n N
X
biorthoqonal x* ( x ) = 1 n n X
and
i f and
and t h e f a m i l y
i s u n i f o r m l y bounded.
has t h e a d d i t i o n a l property t h a t f o r every
x EX
m
t h e series
C x:(x) xn c o n v e r g e s u n c o n d i t i o n a l l y , w e c a l l s u c h a n=O b a s i s u n c o n d i t i o n a l . The b a s i s i s c a l l e d monotone i f IIPNII = I for
P. Wojtaszczyk
16
N =011,2, m
...
and u n c o n d i t i o n a l l y monotone i f f o r e v e r y
x = C x*(x)x n n=O n m
11
w e have
and f o r e v e r y s e q u e n c e of numbers
C cnx:(x)
n=O
x n l l 511x11
E~
long t i m e .
a
for quite
a
1
~
~
=I
1
-
The problem o f t h e e x i s t e n c e o f a S c h a u d e r b a s i s i n constructed
with
H 1 ( D ) was a r o u n d
I t w a s s o l v e d by P . B i l l a r d i n [ B i ] who
basis for
H1(D).
H i s p r o o f was r e a l v a r i a b l e i n
s p i r i t a n d u s e d t h e Haar f u n c t i o n s . The g e n e r a l scheme i s as f o l l o w s . Let
m
(fnInzo
be a n o r t h o n o r m a l s y s t e m o n
L e t us d e f i n e
[o,III w i t h
-.
fo(t) = 1
fi
( a s i n Proposition 2.4.)
n1
fn(t)
tE[O,
fn(-t)
tr[-nlol.
& f n ( t )=
n,
C l e a r l y ( & f n ) n = O i s a n o r t h o g o n a l s y s t e m on
T.
In order t o obtain
a (complex) o r t h o g o n a l s y s t e m o f a n a l y t i c f u n c t i o n s on
T
we define
T h e theorem o f B i l l a r d [ B i l s a y s t h a t i f w e s t a r t w i t h t h e Haar
s y s t e m o n [ O , I I I a s (fn);=o H1(D).
follows
Let
if
F n ( z ) i s a Schauder b a s i s i n
u s r e c a l l t h a t t h e Haar s y s t e m o n
[O,n] i s d e f i n e d a s
I
h ( t )=- 1 0
then
di?
I
n =2k + j , 0 < j <2k, k =0,1,2
,...
then
I
0
otherwise.
I t c a n be s e e n from P r o p o s i t i o n 2.4.
t h a t t h e theorem o f B i l l a r d can
17
The Banach space H 7
be f o r m u l a t e d as f o l l o w s . THEOREM 5 . 1 .
.
(Billard)
The Haar s y s t e m (hn);=o
is a b a s i s i n
H:(O,ll).
PROOF. Obviously t h e l i n e a r span o f t h e Haar s y s t e m e q u a l s p a r t i a l sum o p e r a t o r N P f = C N j=o
where
1
lIJi
I!s 3
PN
PNa
i s a c o n s t a n t m u l t i p l e of a n atom. I n t h e c a s e
.I
7
4
diam ( s u p p P N a )
so
+4/N
w e see t h a t f o r
f o r a t most two
#O
1 1 PNal I co ( 2 N .
1 1 P N a /I mz1 a /1
a ( t ) , supp a c I . S i n c e /PNa = 0 ,
d i a m ( s u p p P N a ) (111 flI
1
a r e d i s j o i n t i n t e r v a l s c o v e r i n g [ O , l l I and
and
1I
The
XIj
L e t us t a k e a n atom
have
HY(O,II).
i s given as an averaging o p e r a t o r
Obviously
111
Lm1
the function
<%1
111
we
j ' s . T h i s means t h a t
fPNa =O.
Moreover
I
II7J IJ,al 5TT-J-
T h i s shows t h a t a l s o i n t h i s c a s e
c o n s t a n t m u l t i p l e o f an atom.
1
1
7
PNa
is a
I t i s r e l a t i v e l y e a s y t o check ( c f . t h e l a s t P r o p o s i t i o n of [ K - P I )
t h a t t h e Haar system i s n o t an u n c o n d i t i o n a l b a s i s i n q u e s t i o n o f t h e e x i s t e n c e o f an u n c o n d i t i o n a l b a s i s i n
The
H;(O,II). H1(D)
was
r a i s e d by s e v e r a l m a t h e m a t i c i a n s ( c f . [ E I , [ K - P ] , [ P e ] ) . B e f o r e w e p r e s e n t t h e s o l u t i o n t o t h i s problem w e need t o d e f i n e H -space, 1
namely
H.,(d), t h e
new
H1-space c o n n e c t e d w i t h t h e c a n o n i c a l
The g e n e r a l t h e o r y of t h i s s p a c e c a n be found
dyadic martingale. in
a
[ G a r ] . From o u r p o i n t of view t h e most c o n v e n i e n t d e f i n i t i o n i s
t h e following.
For a f u n c t i o n
f
on
[O,II]
w e d e f i n e i t s norm i n
Hl(d) as
The s p a c e of a l l
f
such t h a t
H l ( d ) . I t i s known t h a t
I I f l I H1
(dl
i s f i n i t e i s denoted by
L [ O , I I ] c H 1 (d)CL1[O,II]
for a l l p > I . P Moreover i t i s o b v i o u s t h a t t h e Haar system is an u n c o n d i t i o n a l
P. Wojtaszczyk
18
basis i n
H , ( d ) . The q u e s t i o n a b o u t t h e e x i s t e n c e o f a n u n c o n d i t i o n a l
basis i n
H l ( D ) was a n s w e r e d b y t h e f o l l o w i n g t h e o r e m o f Maurey's
[ Mau
1.
THEOREM 5 . 2 .
The s p a c e s H 1 ( D ) a n d
H ( d ) are isomorphic.
1
Maurey's p r o o f o f Theorem 5 . 2 . h a s
one drawback, i t i s n o t c o n s t r u c -
t i v e . T h i s w a s remedied by C a r l e s o n [ C a ] and t h e a u t h o r [WO~].I n o r d e r t o d e s c r i b e t h e r e s u l t from [ W O ~ w]e need o n e more d e f i n i t i o n .
W e define points
to = O f i f
tn a s f o l l o w s
n =2k + j , k =0,1,2,...,
The F r a n k l i n s y s t e m
( t )=-
f O
I
/
1
i
fn(t)
O s j <2k;
t
n
=(j+1)2-kII.
i s d e f i n e d by t h e f o l l o w i n g c o n d i t i o n s :
,
i
f ( t )is a piecewise l i n e a r f u n c t i o n on [ O , l I ] w i t h nodes a t p o i n t s n t o f tl , . . . , t which i s o r t h o g o n a l t o a l l f j ' s , j < n and I I f n / 1 2 = 1 . n W e have t h e f o l l o w i n g t h e o r e m p r o v e d i n [WO~]. THEOREM 5 . 3 .
-
d e f i n e d by
The o p e r a t o r
T
from
onto
H:(O,II)
map F n ( z )
hn
Hl(d). If
T ( f ) = h n e x t e n d s t o a n isomorphism n F n ( z ) i s d e f i n e d as i n ( 1 ) t h e n t h e
i n d u c e s t h e isomorphism between
H1(D)
and
Hl(d).
I n p a r t i c u l a r t h e F r a n k l i n system i s an u n co n d itio n al b a s i s i n
Ha ( 0 , JI) and t h e c o r r e s p o n d i n g s y s t e m
1 basis i n
Fn ( z ) i s a n u n c o n d i t i o n a l
H1(D).
I t i s c l e a r t h a t o n l y t h e f i r s t a s s e r t i o n n e e d s a p r o o f . T h i s c a n be
found i n [ W O ~ ] and [ C i e ] o r i n a g r e a t e r g e n e r a l i t y i n [ W O ~ ] o r
[s-SI. The r e a d e r may be i n t e r e s t e d t o n o t e t h a t t h e e x i s t e n c e o f a n u n c o n d i t i o n a l b a s i s f o r H (D) i s a phenomenon from t h e i s o m o r p h i c t h e o r y . 1 The c o r r e s p o n d i n g i s o m e t r i c f a c t i s f a l s e as i s g i v e n i n t h e following
19
The Banach space H I
PROPOSITION 5 . 4 . H1(D) is not isometric to a subspace of a Banach space with an unconditionally monotone basis. The outline of the proof can be found in [ W O ~ ] . The following isometric problem concerning
H q (D) is open.
PROBLEM. a) Does
H (D) have a monotone Schauder basis? 1 b) Describe the norm one, finite dimensional projections in
H1(D).
It is quite likely that the answer to a) is negative. I do not
know about any norm one, finite dimensional projection in H1(D) whose rank has dimension greater than 1. It was shown in [ W O ~ ] that any norm one finite dimensional projection in L1/H1 is actually one dimensional.
As the reader may have noticed we have dealt with three different H I spaces, and all of them turned out to be isomorphic. It is only a tip of an iceberg. There is a vast proliferation of HI-spaces important in analysis, cf. [C-W] or [ F o - S ] . T o decide exactly which are isomorphic to H ( D ) and which are not may be 1 of interest. Maurey in [Maul considers a wide class of martingale H -spaces associated with different sequences of a-fields and 1 shows them to be isomorphic to HI (d), He also shows that the Fefferman-Stein spaces H I (Rn) (for definitions see [ F - S ] ) are isomorphic to H,(d). If we adhere to the spirit of complex function theory the picture becomes more complicated. It is possible to consider HI-spaces of functions analytic on relatively wild sets in (I: (cf. [Dur],Chapter 10) but instead we prefer to discuss the situation in several complex variables. There are at least two extremely nice subsets of En, the ball Bn and the polydisc Dn. Both admit natural HI spaces, which can be defined as a closure of analytic polynomials in a certain norm. To get HI (D") we take the norm
20 where
P. Wojtaszczyk
i s a n o r m a l i s e d i n v a r i a n t measure o n a t o r u s
v
s p a c e H1 (B,)
where sphere
o
Tn.
The
i s o b t a i n e d i f w e t a k e a s a norm
i s a n o r m a l i s e d , r o t a t i o n - i n v a r i a n t measure on t h e u n i t S.
About t h o s e s p a c e s w e have t h e f o l l o w i n g THEOREM 5 . 5 .
( a ) H1 ( B n ) (b) I f
is isomorphic t o
H1(Dn)
H1 ( D ) , n =1,2,3,.
i s isomorphic t o
H1 (Dm)
then
..
[WO~].
n =m [Boul],[BouZ].
Both t h o s e f a c t s a r e q u i t e c o m p l i c a t e d t o p r o v e . L e t m e remark o n l y t h a t i n b o t h c a s e s t h e isomorphism between
H,(D)
and
H l ( d ) is a
v i t a l p a r t of t h e p r o o f . W e feel t h a t a l o t r e m a i n s t o b e done i n c o n n e c t i o n w i t h t h i s Theorem.
I n p a r t i c u l a r w e do n o t know
what i s t h e Banach-Mazur d i s t a n c e
H1(B,) and H , ( D ) . A l s o w e do n o t know w h a t i s t h e s i t u a t i o n f o r o t h e r n i c e domains i n En. W e feel t h a t it s h o u l d b e p o s s i b l e
between
t o extend ( a )
t o a l l s t r i c t l y pseudoconvex domains. The methods o f
[ W O ~ ]g i v e many domains which are n o t s t r i c t l y pseudoconvex f o r which ( a ) h o l d s . The o t h e r class o f domains which a d m i t n i c e bounded homogeneous
domains
.
H -spaces a r e 1 Those are f u l l y c l a s s i f i e d a n d r a t h e r
w e l l u n d e r s t o o d , c f . [Hua] ,[H-MI i n v e s t i g a t e and c l a s s i f y
.
I t i s a v e r y i n t e r e s t i n g problem t o
H 1 - s p a c e s o n t h o s e d o m a i n s . Some i n t e r e s -
t i n g p a r t i a l r e s u l t s i n t h i s d i r e c t i o n h a v e b e e n o b t a i n e d by T . Wolniewicz [wall. L e t m e d i s c u s s b r i e f l y t h e method o f p r o o f o f Theorem 5 . 5 .
(b)
( c f . [Bou 1 1 ) s i n c e i t g i v e s a v e r y i n t e r e s t i n g i n f o r m a t i o n a b o u t
H1(D). W e w i l l f o r m u l a t e i t i n t e r m s o f t h e Haar s y s t e m i n H . , ( d ) . N L e t H ( d ) d e n o t e t h e s p a n of t h e f i r s t N H a a r f u n c t i o n s . W e s a y 1 ( f o l l o w i n g J . B o u r g a i n [ B o u l ] ) t h a t a map 5 :H 1N ( d ) H1 ( d ) i s order inversing i f c ( h , ) = C akhk and t h e A n ' s a r e f i n i t e , p a i r w i s e
-
kEAn
d i s j o i n t sets w i t h min
Ak >max
As
whenever
k < s . I t is a n easy
e x e r c i s e ( c f - T h e o r e m 3 . 5 . ) t h a t t h e r e e x i s t o r d e r i n v e r s i n g isomorHNl ( d ) i n t o H l ( d ) w i t h u n i f o r m c o n s t a n t s . The
p h i c embeddings o f
21
The Banach space H 1
-
main r e s u l t o f [ B o u l ] a s s e r t s however t h a t e v e r y p r o j e c t i o n o n t o t h e r a n g e o f s u c h o r d e r i n v e r s i n g i s o m o r p h i s m s h a s b i g norm. More p r e c i s e l y : f o r every
C >1
there e x i s t s a function
f o r e v e r y o r d e r i n v e r s i n g isomorphism
I I f l l 'IIcfl/ HI (d) onto
N
cp(N)
6 :Hy(d)
m’N
such t h a t
m
Hl(d)
with
l C l l f l [ t f t H l ( d ) , t h e n o r m o f e v e r y p r o j e c t i o n from N < ( H I ( a ) ) is g r e a t e r than cp(N).
w e wish t o conclude t h i s s e c t i o n w i t h t h e d e s c r i p t i o n of a r e c e n t b e a u t i f u l and d e e p r e s u l t o f P . J o n e s [ J o n ] . L e t u s s t a r t w i t h t h e d e f i n i t i o n of t h e uniform approximation p r o p e r t y . A Banach s p a c e
X
i s s a i d t o have t h e uniform approximation p r o p e r t y
(cf.[P-R]) i f there e x i s t
a constant
C
t h a t f o r e v e r y f i n i t e system of v e c t o r s e x i s t s a f i n i t e dimensional operator
T
and a f u n c t i o n
-
XI,
:X
... , x N X
in
cp(N)
X
such
there
such t h a t
T h i s i s a p r o p e r t y s t r o n g e r t h a n t h e bounded a p p r o x i m a t i o n p r o p e r t y s i n c e t h e bound
on t h e d i m e n s i o n of t h e r a n g e of a n o p e r a t o r
T
is
imposed. P . J o n e s proved i n [ J o n ] t h a t
BMO
h a s t h e uniform approximation
p r o p e r t y . I t was p r e v i o u s l y unknown ( c f . [ P e l ] ) w h e t h e r
BMO
has the
approximation p r o p e r t y . S i n c e uniform approximation p r o p e r t y i s a self-dual property (i.e. X [ H e i ] , w e see t h a t
has i t i f and o n l y i f
X*
h a s it),c f .
H (D) h a s t h e u n i f o r m a p p r o x i m a t i o n p r o p e r t y . 1
SECTION 6 ; Complemented s u b s p a c e s . I n t h i s s e c t i o n w e a r e i n t e r e s t e d i n complemented s u b s p a c e s o f
H1(D).
i s complemented i f t h e r e e x i s t s a p r o j e c t i o n ( a n i d e m p o t e n t map) w i t h a r a n g e e q u a l t o X. L e t us
remind t h a t a s u b s p a c e
XCH1(D)
G e n e r a l l y , b e i n g a complemented s u b s p a c e o f a g i v e n s p a c e i s a much s t r o n g e r r e s t r i c t i o n t h e n merely b e i n g a s u b s p a c e . To i l l u s t r a t e
P. Wojtaszczyk
22
t h i s l e t m e remark t h a t f o r e v e r y
a subspace of and
H1(D)
p, 1 ( p 5 2 1 i s isomorphic t o P ( c f . Remarks a f t e r Theorem 3 . 5 . ) b u t o n l y 1,
l 2 a r e isomorphic t o
complemented s u b s p a c e s o f
i s immediate from C o r o l l a r y 2 . 1 .
of
r e f l e x i v e complemented s u b s p a c e o f
[K-PI which s a y s t h a t t h e o n l y H1(D)
is
12.
W e s t a r t however w i t h a more h a r m o n i c - a n a l y t i c
projections i n
H
1
problem: what a r e t h e
-
( D ) which commute w i t h r o t a t i o n s ? I t i s e a s y t o
see t h a t such a p r o j e c t i o n i s d e t e r m i n e d by a s u b s e t numbers, and i s g i v e n by t h e formula
terise
t h e idempotent sets, i . e .
o u s map on
H
This
H1(D).
c" anzn n=O
sets
A
of n a t u r a l n C anz To c h a r a c nEA A
.
f o r which it i s a c o n t i n u -
1 ( D ) i s a n i n t e r e s t i n g u n s o l v e d problem. The f o l l o w i n g
remarks a r e obvious: (1)
each f i n i t e set i s idempotent
(2)
if
A
and
are idempotent t h e n
B
A U B,
A nB
and
N\B
are
idempotent (3)
if
A
i s idempotent t h e n f o r e v e r y
(A-k) n N
k EN
the translate
i s idempotent.
(4) a p e r i o d i c sequence is i d e m p o t e n t . L e t us remark t h a t t h e above f a c t s ( e x c e p t ( 3 ) ) a r e v a l i d a l s o
for
L1(T).
I n t h i s case t h e i d e m p o t e n t s e t s have been d e s c r i b e d by
Helson [ H e l l as p e r i o d i c s e q u e n c e s mod a f i n i t e s e t . For
H1(D)
lary 3.7.)
t h e s i t u a t i o n i s d i f f e r e n t . By t h e P a l e y theorem ( C o r o l t h e r e is an i n v a r i a n t p r o j e c t i o n onto a H i l b e r t space.
The theorem o f Rudin [Ru4] d e s c r i b e s t h e i n v a r i a n t p r o j e c t i o n s o n t o H i l b e r t spaces i n
H ( D ) a s g i v e n by a f i n i t e sum of l a c u n a r y sets. 1
The above mentioned theorem of Kwapien and p e l c z y n s k i ([K-P],Coroll a r y 2.1.) s a y t h a t e v e r y complemented r e f l e x i v e subspace of
H1(D)
i s isomorphic t o a H i l b e r t s p a c e . The above mentioned f a c t s l e a d If
A
t o t h e following conjecture.
i s an i d e m p o t e n t s e t t h e n t h e r e e x i s t a p e r i o d i c s e t
B
and
l a c u n a r y s e t s ( i . e . s e t s s a t i s f y i n g t h e a s s u m p t i o n s of C o r o l l a r y 3 . 7 . ) ( C j ) j=1
and ( D . ) S 3 1=1
such t h a t
23
The Banach space H 1
A =B U(
k
S
( U D.).
U C.)
j=1 1
j=1
7
L e t us remark t h a t i n v a r i a n t p r o j e c t i o n s f o r
Hp(D),
p <1
have been
d e s c r i b e d i n [K-TI a s c o r r e s p o n d i n g t o a p e r i o d i c s e t mod a f i n i t e
s e t . F o r t h e i n v a r i a n t , norm one p r o j e c t i o n s on
H1(D)
the situation
i s c l e a r . W e have PROPOSITION 6.1. There is a c o n s t a n t
c >I
t i o n on
[lPli
form
HI (D) with
P is an i n v a r i a n t p r o j e c i s g i v e n by a s e t A of t h e
such t h a t i f
A ={a +bk, k = 0 , 1 , 2 ,
then
- .. I
P
f o r some
a
PROOF. I f the set p z q
p,q,
i s n o t of t h i s form t h e n t h e r e e x i s t i n t e g e r s
A
such t h a t
p €A
and e x a c t l y one o f
p+q
and
p -q
is i n
So w e i n f e r t h a t t h e r e e x i s t s a n i n v a r i a n t p r o j e c t i o n o f norm
from
span {zP-q,zP,zP+q~ onto
span {zP-q,zP~ o r onto
A.
(c
span
Both t h o s e cases r e d u c e t o t h e n a t u r a l p r o j e c t i o n from span{z,l,z} onto
s p a n {!,I}.
I t i s a n e l e m e n t a r y c a l c u l a t i o n t o show
t h a t t h e norm o f t h i s p r o j e c t i o n i s s t r i c t l y g r e a t e r t h a n 1 . Now w e t u r n t o o t h e r c l a s s o f d i s t i n g u i s h e d s u b s p a c e s o f
namely t o s u b s p a c e s i n v a r i a n t f o r m u l t i p l i c a t i o n by specified to
t e d B e u r l i n g theorem, [Koo], 1 V . E . )
reads a s follows: L e t
H (D)
Then t h e r e e x i s t s a n i n n e r f u n c t i o n p r o d u c t and
a
1
XCH I
1
HI (D),
z. The celebra-
(cf.[Dur],Sect.7.3.
( D ) be s u c h t h a t
or
z-XCX.
( i . e . a p r o d u c t of a Blaschke
s i n g u l a r i n n e r f u n c t i o n ) such t h a t
X =I*H, (D).
The f o l l o w i n g t h e o r e m h o l d s THEOREM 6 . 2 .
The m u l t i p l i c a t i o n i n v a r i a n t s u b s p a c e H1(D)
i f and o n l y i f
I
X =I-H,(D)
i s complemented i n
i s a B l a s c h k e p r o d u c t whose z e r o s form a
Carleson sequence. L e t m e r e c a l l t h a t a s e q u e n c e ( z )c D measure g i v i n g e a c h o f t h e measure.
2:s
n
is a C a r l e s o n sequence i f t h e
t h e m a s s ( 1 -lzn
I 2)
i s a Carleson
P. Wojtaszczyk
24
The p r o o f o f t h i s Theorem i s a l m o s t t h e same b u t s i m p l e r t h a n t h e one g i v e n f o r t h e a n a l o g o u s r e s u l t i n t h e d i s c a l g e b r a c a s e i n [ c-P-s
I.
Now w e t u r n t o t h e ( p o s s i b l y open e n d e d ) p r o b l e m o f c l a s s i f y i n g i n f i n i t e d i m e n s i o n a l complemented s u b s p a c e s o f
H1(D).
To do i t
f o r a g i v e n Banach s p a c e i s one o f t h e f a v o r i t e o c c u p a t i o n s i n t h e geometry o f Banach s p a c e s . The case l i t e r a t u r e , t h e r e s u l t s o f [B -R -S ]
p '1
Lp'
has
generated a v a s t
i n d i c a t e t h a t such a c l a s s i f i c a -
t i o n i n t h i s c a s e h a s t o be very complicated or i s i m p o s s i b l e a t
a l l . I n contrast, t h e case
L1
seems t o be much s i m p l e r ; o n l y two
i s o m o r p h i c t y p e s o f complemented s u b s p a c e s o f namely
1,
and
L1.
L1
a r e known,
I t i s a w e l l known open p r o b l e m i f
that is all.
Our aim now i s t o w r i t e down a l l i s o m o r p h i c t y p e s of complemented H 1 ( D ) w e know o f a n d t o i n d i c a t e t h a t t h e y a r e i n d e e d
subspaces of different
.
W e have t h r e e b u i l d i n g b l o c k s ;
( a ) by P a l e y ' s t h e o r e m ( C o r o l 1 a r y 3 . 7 . ) complemented s u b s p a c e o f
l2
i s isomorphic t o a
H1(D)
( b ) l 1 i s i s o m o r p h i c t o a complemented s u b s p a c e of
H1(D)
(this
i s very easy) (c) H , ( D ) HY
h a s an u n c o n d i t i o n a l b a s i s ;
denote
t h e s p a n of t h e f i r s t
f i x a F r a n k l i n b a s i s and l e t n
elements of t h i s b a s i s .
W e w i l l put those blocks together with t h e a i d of t h e following
THEOREM 6.3.
H 1 ( D ) i s isomorphic t o i t s
The p r o o f c a n be found i n
THEOREM 6 . 4 .
(CH,(D))l.
[ W O ~ ] . A l s o t h e isomorphism
provides the a l t e r n a t i v e proof, since f o r easy.
I.,-sum
H
1
H1 (D)-'H,
( d ) t h e theorem i s
The f o l l o w i n g s p a c e s a r e complemented i n
H1(D):
(d)
25
The Banach space H I
The arrow
X
-
Y
means t h a t
X
embeds i n t o
Y
a s a complemen-
t e d s u b s p a c e . All t h o s e t e n s p a c e s a r e p a i r w i s e n o n - i s o m o r p h i c . PROOF.
E v e r y t h i n g e x c e p t t h e l a s t s t a t e m e n t f o l l o w s from p r e v i o u s comments. The p r o o f o f t h e l a s t c l a i m u s e s a l o t o f Banach s p a c e t h e o r y b u t o t h e r w i s e i s q u i t e b o r i n g . N e v e r t h e l e s s l e t u s g i v e some h i n t s . W e s t a r t from t h e t o p .
i s t h e o n l y r e f l e x i v e s p a c e on t h e l i s t
l2
l1
i s t h e o n l y one which
1; s
does n o t c o n t a i n
u n i f o r m l y complemented
l 1+ 1 2
h a s a unique up t o p e r m u t a t i o n u n c o n d i t i o n a l b a s i s
[E-W],which i s c l e a r l y
d i f f e r e n t from a l l o t h e r s p a c e s
cm
(
c 1:)1
n=l
a l s o h a s a u n i q u e up t o p e r m u t a t i o n u n c o n d i t i o n a l eJ
b a s i s ( c f . [B-C-L-T
)
so i s not isomorphic t o (
m
( C H:)l
n=l
does n o t c o n t a i n
n
C HI),
n=l
P. Wojtaszczyk m
n
( C 1 2 )+ 11 2
m
complemented copy o f m
( C
n=l
l;)l
(C12)
does n o t c o n t a i n
n=l
n
(Assume it d o e s , t h e n by [ W o l ]
C H,)l
(
n=l
would c o n t a i n
(
and does n o t c o n t a i n a
C HY)l n= 1
complemented, so b o t h would
be i s o m o r p h i c a n d i t i s n o t s o . ) m
l2
i n t o ( C HY) i s compact, by n= 1 m [E-W] w e o b t a i n t h a t e v e r y u n c o n d i t i o n a l b a s i s i n ( C HY) + l2 n=l
s i n c e e v e r y o p e r a t o r from
m
s p l i t s i n t o a p a r t spanning
l2 m
but the natural basis i n
( C H:)l
n= 1
a n d a p a r t s p a n n i n g ( Z H n1) 1 ' n= 1
+
does n o t have t h i s
(C12)1
property (C12)1
h a s a u n i q u e up t o p e r m u t a t i o n u n c o n d i t i o n a l b a s i s ,
see [B-C-L-TI m
( C HY)
+
1 < p (2,
but
n= 1
RGMARK
(El2)
does n o t c o n t a i n a subspace isomorphic t o
H, ( 0 )
1 P'
h a s such subspaces.
.
The i s o m o r p h i c t y p e of t h e s p a c e ( C H?) n= 1
d o e s n o t depend o n a
choice of b a s i s . This i s a s t a n d a r d decomposition argument. The v e r y n a t u r a l q u e s t i o n r a i s e d by t h e above d i s c u s s i o n i s t o c o n s t r u c t more complemented s u b s p a c e s of
HI (D)
.
I n view o f t h e mul-
t i t u d e of a p p a r e n t l y d i f f e r e n t i s o m o r p h i c r e p r e s e n t a t i o n s f o r
H1(D)
(see S e c t i o n 5 ) w e b e l i e v e t h a t such a c o n s t r u c t i o n i s p o s s i b l e . I t i s s t i l l an open problem r a i s e d i n [ C a s l i f
u s r e c a l l t h a t a Banach s p a c e tion X.
X =X 1 + X 2
X
H1(D)
is primary. L e t
i s p r i m a r y i f f o r e v e r y decomposi-
w e h a v e a t l e a s t o n e of
X1
or
X2
isomorphic t o
I t i s o u r b e l i e f t h a t t h e answer t o t h i s p r o b l e m i s p o s i t i v e .
The following r e s u l t s e e m s to j u s t i f y t h i s b e l i e f .
27
The Banach space H I
THEOREM 6 . 5 .
Let
b e a s u b s p a c e of
X
Than
H1(D). H
1
HI(D).
Assume t h a t
c o n t a i n s a s m a l l e r subspace
X
i s isomorphic t o
X
complemented i n
Y
.
H I (D)
( D ) and isomorphic t o
The p r o o f o f t h i s t h e o r e m i s a v e r b a t i m r e p e t i t i o n of t h e p r o o f of t h e analogous statement f o r
L
1
P'
g i v e n i n [J-M-S-T],pp.265-70.
Now w e w i l l d i s c u s s b r i e f l y t h e s p a c e of p o l y n o m i a l s . To be p r e c i s e ,
let
Pn
denote
spanil , z , .
the
.., z n }
in
H I (D). I t i s w e l l known
t h a t i n i t s n a t u r a l p o s i t i o n a s a s u b s p a c e of
H1(D)
the space
Pn
i s b a d l y complemented, more p r e c i s e l y t h e norm of t h e b e s t p r o j e c t i o n is of o r d e r
l o g ( n + l ) . T h i s r e s u l t however d e p e n d s o n t h e
p a r t i c u l a r p o s i t i o n of
Pn
in
It is a natural question
H1(D).
( c f . [Wo41 ) i f t h e P ' s a r e i s o m o r p h i c t o u n i f o r m l y complemented
n
subspaces of
H 1 ( D ) . A p o s i t i v e answer t o t h i s was g i v e n by
J.Bourgain and A.Pelczynski.
w e l l known
and
T h e i r argument r u n s a s f o l l o w s : A s i s
H 1 ( D ) is i s o m o r p h i c t o
qn:H1 ( D )
+
H I (D)
-
Pn
m
I I i n I\ 5 2
and
A c t u a l l y S.V.Bo&ariov
1 I qnl I
51
[Bo] has
i s uniformly isomorphic t o
Now w e d e f i n e
by
k m k n n-k q n ( C a k z , C b z ) = C - akzk k=O k=O k=O Since
H1(D)+ H 1 ( D ) .
Hn"
n
+ c
k=O
and
n-k
bkzn-k'
q n o in= i d
t h e c l a i m follows.
shown more. H e h a s p r o v e d t h a t
.
Pn
T h i s i s i n c o m p a r a b l y more c o m p l i c a -
ted. SECTION 7 ; C o n c l u d i n g Remarks. I n t h e previous sections t h e p a t i e n t reader has
f o u n d many o p e n
problems connected t o t h e r e s u l t s w e have been d i s c u s s i n g . I n t h i s s e c t i o n w e want t o p r e s e n t some d i r e c t i o n s f o r p o s s i b l e f u r t h e r
P. Wojtaszczyk
28
research. 1 ) M u l t i p l i e r s . I t i s a w e l l known open problem t o g i v e good m
c r i t e r i a f o r a s e q u e n c e ( p ( n ) )n=O
into
H I (D)
i s a f i n e example. Our knowledge
The Theorem 3.1.
H1(D).
t o b e a m u l t i p l i e r from
however i s s t i l l q u i t e s m a l l a s i s w i t n e s s e d by t h e p r o b l e m o f d e s c r i p t i o n of 0 , l v a l u e d m u l t i p l i e r s ( i . e . i n v a r i a n t p r o j e c t i o n s ) . There i s a v a s t l i t e r a t u r e o n m u l t i p l i e r s b o t h from i t s e l f and from
H1(D)
H (D) into 1 i n t o o t h e r n a t u r a l s p a c e s . W e do n o t want
t o d i s c u s s it h e r e . L e t u s m e n t i o n o n l y t h a t m u l t i p l i e r s from H1(D)
into
and from
H2(D)
H1(D)
into
l1
a r e f u l l y described
(cf.[Dur],Th.6.4 a n d [ S l - S t ] , r e s p ) . However, as f a r a s w e k n o w , a l l t h e e x i s t i n g t h e o r e m s d e a l w i t h b o u n d e d n e s s o r c o m p a c t n e s s of m u l t i p l i e r s . I t i s o u r b e l i e f t h a t i t i s w o r t h w h i l e t o i n v e s t i g a t e what m u l t i p -
l i e r s b e l o n g t o o t h e r n a t u r a l and i m p o r t a n t o p e r a t o r i d e a l s . The theory of operator i d e a l s ( c f . [ P i e ] ) i s a very important t o o l i n
its applications t o the d i s c algebra a r e d i s c u s s e d i n [ P e l ] . I n t h i s s i t u a t i o n w e b e l i e v e t h a t some
v a r i o u s p a r t s of a n a l y s i s ,
t h e o r e m s o f t h i s t y p e may t u r n o u t t o b e v e r y u s e f u l . 2 ) F i n i t e d i m e n s i o n a l s t r u c t u r e . The problem b a s i c a l l y i s t o d e v e l o p
t h e l o c a l theory of
H1(D).
I n p a r t i c u l a r to investigate various
p a r a m e t e r s a s s o c i a t e d w i t h i t s main f i n i t e d i m e n s i o n a l b u i l d i n g b l o c k s ,
H?. The p r e s e n t a t i o n o f some p a r t s of l o c a l t h e o r y
i.e. the spaces
o f Banach s p a c e ( i n t h e c o n t e x t o f c l a s s i c a l s p a c e s ) c a n b e found i n [ P e l l ] . The o n l y p a p e r which c o n t a i n s some r e s u l t s of t h i s H1(D) i s [G-R].
n a t u r e connected w i t h
3 ) Convergent T a y l o r s e r i e s . I t i s w e l l known t h a t t h e T a y l o r s e r i e s
n C anz n=O
of an
H1(D)
function
f
need n o t c o n v e r g e t o
f
in
norm. W e f e e l t h a t t h e s p a c e of c o n v e r g e n t T a y l o r s e r i e s , i . e . s p a c e c o n s i s t i n g of a l l
f =
m
C an?€ n=O
H1(D)
the
such t h a t
n=o
N
may be a n i n t e r e s t i n g o b j e c t t o s t u d y . I do n o t know o f a s i n g l e work d e a l i n g w i t h t h i s s p a c e . The a n a l o g o u s s p a c e f o r
p
=m,
i.e.
t h e s p a c e o f u n i f o r m l y c o n v e r g e n t T a y l o r series r e c e i v e d much a t t e n t i o n i n r e c e n t y e a r s . The b a s i c r e f e r e n c e f o r t h i s i s [ V i n ] . W e f e e l however t h a t t h e r e i s a v e r y s m a l l c o n n e c t i o n between p =1
and
p
=m
i n t h i s case.
29
The Banach space H I
REFERENCES. J.A.Bal1, Hardy space expectation operators and reducing subspaces, Proc. A.M.S. vol. 47 NO2 (1975) pp.351-7 P.Billard, Bases dans H 1 et bases de sous-espaces de dimension finie dans A, Proc. Conf. Oberwolfach (August 14-22, 7971), ISNM vol. 20, Birkhzuser,
Basel
and Stuttgart 1972
S.V.BoEkariov, to appear
J.Bourgain, The non-isomorphism of
1
H -spaces in one and
several variables, J. of Funct. Anal. vol. 46 NO1 (1982)
[ Bou2
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[ Cas I [ c-s-P 1
[Cal [ Cie I
pp. 45-57 J.Bourgain, The non-isomorphism of H 1 -spaces in different number of variables (to appear)
J.Bourgain, P.Casazza, J.Lindenstrauss, L.Tzafriri, Banach spaces with unique up to permutation unconditional basis (to appear) (The reader may also consult exposes I V and X of Sgminaire d’analyse fonctionnelle 1980-81.) J.Bourgain, H.P.Rosentha1, G.Schechtman, An ordinal
L -index for Banach spaces, with application to P Annals of Math. vol. 114 complemented subspaces of L P’ (1981) pp.193-228 P.Casazza, Complemented subspaces of A,H1 and H a , in:99 unsolved problems of linear and complex analysis, Zapiski LOMI 81. P.Casazza, R.W.Pengra, C.Sundberg, Complemented ideals in the disc algebra,Israel J. of Math. vol. 37 (1980) pp. 76-83 L.Carleson, An explicit unconditional basis in H I , Bull. des Sciences Mathgmatiques 104 (1980) pp.405-16 2-Ciesielski, The Franklin orthogonal system as unconditional basis in ReHl and VMO, in:Functional Analysis and Approximation, ISNM vol.60 Birkhzuser, Basel and Stuttgart, 1981 pp.117-128. R.R.Coifman, A real characterisation of H P’ Studia Math. 5 1 (1974) pp.269-274 R.R.Coifman, G.Weiss, Extensions of Hardy spaces and their use in analysis, Bull. A.M.S. 83 (1977) pp.569-645 J.Dieste1, Geometry of Banach spaces-Selected topics, Lecture Notes in Math. vol. 485, Springer-Verlag 1975
P. Wojtaszczyk
.
J. Diestel I J.J. Uhl jr , Vector Measures, Mathematical Surveys 15, Providence 1977 P.L.Duren, Theory of H -spaces, Academic Press, P New York-London 1970
P.Enflo, in:Problems of present day mathematics, Mathematical developments arising from Hilbert problems, Proc. Symp. in Pure Math. XXVIII, Providence 1976 I.S.Edelstein, P.Wojtaszczyk, On projections and uncon-
ditional bases in direct sums of Banach spaces, Studia Math. 56 (1976) pp.263-76 C.Fefferman, Characterisations of bounded mean
oscillation, Bul1.A.M.S. 77(1971) pp.587-8 C.Fefferman, E.M.Stein, H -spaces of several variables, P Acta Mathematica 129 ( 1972) pp. 1 37-9 3 G.B.Folland, E.M.Stein, Hardy spaces on homogeneous groups, Princeton University Mathematical Notes 28,
Princeton 1982 F.Forelli, The isometries of H Can. J.Math. 16(1964) P’ pp.721-28 J.B.Garnett, Bounded analytic functions, Academic Press, New York-London 1981 A.M.Garsia, Martingale Inequalities, Seminar notes on recent progress, W.A.BenJamin Inc. 1973 K.T.Hahn, J.Mitchel1, Representation of linear functionals in H spaces over bounded symmetric domains in P J.Math.Ana1. Appl. 56(1976) pp.374-396
En,
S.Heinrich, Finite representability and super-ideals of operators, Dissertationes Mathematicae 172 (1980) H.Helson, Note on harmonic functions, Proc. A.M.S. 4(1953) pp . 6 86-91 K.Hoffman, Banach spaces of analytic functions, PrenticeHall 1962
[ Hua I
S.V.HruHEov, V.V.Peller, Hankel operators, best approximation and stationary Gaussian processes, Uspehi Mat. Nauk 37 N0 l(1982) pp.53-124 (in Russian) L.K.Hua, Harmonic analysis of functions of several complex variables in the classical domains, A.M.S. Transl. Math. Monographs 6,1963 F.John, L.Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961) pp.415-26
31
The Banach space H 1
1 J-M-S-T 1
W.B.Johnson,
B.Maurey,
G.Schechtman,
L.Tzafriri,
S y m m e t r i c s t r u c t u r e s i n Banach s p a c e s , A.M.S.
M e m o i r 2 1 7 , 1979 P.W.Jones,
BMO a n d t h e Banach s p a c e a p p r o x i m a t i o n p r o b l e m ,
Mittag-Leffler N.J.Kalton,
(1983)
Report NO2
D.A. T r a u t m a n , Remarks o n s u b s p a c e s o f
when 0 < p < 1 , M i c h i g a n M a t h . J . -P p p 163-170
H
Y.Katznelson, J.Wiley
&
(1982)
An i n t r o d u c t i o n t o h a r m o n i c a n a l y s i s ,
Sons I n c .
1968
I n t r o d u c t i o n to
P.Koosis,
vol. 29 N O 2
London M a t h . S o c . P L e c t u r e N o t e s S e r i e s 4 0 , Cambridge 1 9 8 0 S.Kwapien, A . P e l c z y n s k i ,
H -spaces,
Some l i n e a r t o p o l o g i c a l
p r o p e r t i e s o f t h e Hardy s p a c e s
HpI
Compositio
Math. vol. 33 ( 1 9 7 6 ) pp.261-88 B.Maurey,
Isomorphismes e n t r e e s p a c e s
H1,
A c t a Math.
145 ( 1 9 8 0 ) pp.79-120. U.Neri,
Some p r o p e r t i e s of f u n c t i o n s w i t h bounded mean
o s c i l l a t i o n , S t u d i a Math. 6 1 ( 1 9 7 7 ) p p . D.J.Newman,
Pseudo-uniform
63-75
convexity i n
H1,
P r o c A.M.S.
1 4 ( 1 9 6 3 ) pp.676-79 R.E.A.C.
P a l e y , On t h e l a c u n a r y c o e f f i c i e n t s of power
s e r i e s , A n n a l s o f Math. 34 ( 1 9 3 3 ) pp.615-16 A . P e l c z y n s k i , Banach s p a c e s o f a n a l y t i c f u n c t i o n s a n d a b s o l u t e l y summing o p e r a t o r s , CBMS r e g i o n a l c o n f e r e n c e
s e r i e s No [pel11
30, P r o v i d e n c e 1977
A.Pelczynski,
Geometry o f F i n i t e D i m e n s i o n a l Banach
spaces and o p e r a t o r i d e a l s , e d i t e d by H.E.Lacey,
i n : N o t e s i n Banach s p a c e s ,
U n i v e r s i t y of Texas P r e s s , A u s t i n
1980. A - P e l c z y n s k i , H.P.Rosentha1, L -spaces,
[ POW 1
P S.C.Power,
Hankel o p e r a t o r s o n H i l b e r t s p a c e ,
o f t h e London Math.Soc.
[pie]
A.Pietsch,
1 Ros I
H.P.Rosentha1,
[Rul I
Bull.
v o l . 1 2 ( 1 9 8 0 ) pp.422-442.
O p e r a t o r I d e a l s , B e r l i n 1978 On s u b s p a c e s of
97 ( 1 9 7 3 ) pp.344-73 W.Rudin,
Localisation techniques i n
S t u d i a Math. 5 2 ( 1 9 7 5 ) pp.263-289
L
P'
A n n a l s of Math.
F u n c t i o n t h e o r y i n t h e u n i t b a l l of
Springer-Verlag,
N e w York, H e i d e l b e r g ,
a?,
B e r l i n 1980
P. Wojtaszczyk
W.Rudin, Function theory in polydiscs, W.A.Benjamin Inc. New York 1969.
W.Rudin, L -isometrics and equimeasurability, Indiana P Univ. Math. J. 25 (1976) pp.215-28 W.Rudin, Remarks on a theorem of Paley, J.London Math. SOC. 32 (1957) pp.307-311
D.Sarason, Function theory on the unit circle, Notes
Is1
Stl
[wi [ wo
1
for lectures of a conference at Virginia Polytechnic Institute andstate University, Blacksburg Virginia, 1978 E.M.Stein, Classes H multiplicateurs et fonctions P’ de Littlewood-Paley, C.R.A.S. Paris 263 (1966) pp.716-19 P.Sjolin, J-O.Stromberg, Basis properties of Hardy spaces, Stockholms University preprint 1981 W.T.Sledd. D.A.Stegenga, An H I multiplier theorem, Arkiv for Math. v o l . 19 (1981) pp.265-70 J.M.Wilson, A simple proof of the atomic decomposition n for Hp(R ),0< p < 1, Studia Math. 74(1982) pp.25-33
P.Wojtaszczyk, On projections and unconditional bases in direct sums of Banach spaces 11, Studia Math.62 (1978)
[WO21
pp.193-201 P.Wojtaszczyk, On projections in spaces of bounded analytic functions, with applications, Studia Math.
[wo31
P.Wojtaszczyk, Decompositions of
[w041
65 (1979) pp.147-173
H -spaces, Duke Math. P
J. 46 (1979) pp.635-44 P.Wojtaszczyk, Finite dimensional operators in spaces of analytic functions, in:99 unsolved problems of linear and complex analysis, Zapiski LOMI 8 P.Wojtaszczyk, The Franklin system is an unconditional basis in H 1 , Arkiv for Math. vol. 20 (1982) pp.293-300 < 1 , and spline systems, Studia P.Wojtaszczyk, H -spaces,p -
.
W05 1 w06 1
w07 ]
P
Math. (to appear) P.Wojtaszczyk, Hardy spaces on the complex ball are isomorphic to Hardy spaces on the disc 1 (p < m ,
Annals of Math.(to appear) P.Wojtaszczyk, Projections and isomorphisms of the ball algebra, J. London Math. SOC. (to appear) T.Wolniewicz, in preparation
The Banach space H I
[ Vin I
[ ZYgI
S.V.Vinogradov, Strenghening of Kolmogorov theorem on conjugate function and interpolating properties of uniformly convergent Taylor series, Trudy Steklov Inst 1 6 0 , Leningrad 1 9 8 1 , pp. 7-41 (in Russian) A.Zygmund, Trigonometric series, Cambridge University Press, 1 9 7 9 .
33
1
THE BANACH SPACE
H.,
P . Wojtaszczyk
I n s t i t u t e of M a t h e m a t i c s P o l i s h Academy o f S c i e n c e s 00 - 9 5 0 Warszawa, P o l a n d
W e present the linear -topological properties
of t h e c l a s s i c a l Hardy s p a c e
H
1 '
INTRODUCTION
The aim o f t h i s p a p e r i s t o g i v e a n e x p o s i t i o n o f l i n e a r - t o p o l o g i c a l and i s o m e t r i c p r o p e r t i e s o f t h e c l a s s i c a l Hardy s p a c e importance o f t h e space
H 1 ( D ) and i t s v e r s i o n s l i k e
H1(D).
The
ReHl ( T ) i n
a n a l y s i s s t e m s from t h e f a c t t h a t many i n t e g r a l o p e r a t o r s which a r e unbounded on
L1
a r e bounded on
HI.
W e w i l l n o t e l a b o r a t e on t h i s
p o i n t h e r e . While w r i t i n g t h i s p a p e r w e had two g o a l s i n mind. F i r s t , w e w a n t e d t o show t o a f u n c t i o n a l a n a l y s t an i n t e r e s t i n g and n a t u r a l example o f Banach s p a c e . I n o u r o p i n i o n t h e Banach s p a c e p r o p e r t i e s H (D) a r e n o t w e l l u n d e r s t o o d . T h e r e a r e many i n t e r e s t i n g and 1 d i f f i c u l t problems. Secondly t o a c l a s s i c a l a n a l y s t t h e f u n c t i o n a l
of
a n a l y t i c p o i n t of v i e w may be t h e s o u r c e o f new p r o b l e m s . Our p r e s e n t a t i o n of t h e s u b j e c t i s l i m i t e d t o
H1(D).
The f a c t t h a t
H ( D ) i s embedded i n t o t h e s c a l e of
H spaces, 0 < p i m is receiving 1 P v e r y l i t t l e a t t e n t i o n . L e t u s remark h e r e t h a t t h e s c a l e of H spaP c e s , 0 < p z m seems t o b e n i c e r t h a n t h e much more i n v e s t i g a t e d s c a l e
of L s p a c e s , 0 < p < m . I n p a r t i c u l a r t h e p a s s a g e from p '1 to P than for L s p a c e s . The P < I i s much more n a t u r a l f o r H P P g e n e r a l i s a t i o n from Banach s p a c e c a s e t o p-Banach s p a c e s , p < 1 i s much e a s i e r t o u n d e r s t a n d
i f one t h i n k s i n t e r m s o f
A l s o t h e d u a l and p r e d u a l o f
H,(D)
H -spaces. P receives relatively l i t t l e
a t t e n t i o n . T h i s i s n o t i n t e n d e d t o mean t h a t t h e s p a c e BMO d e s e r v e s smaller a t t e n t i o n than
H,(D)
does.
2
P. Wojtaszczyk
Now some e x p l a n a t i o n a b o u t t h e s t y l e o f e x p o s i t i o n . W e u s u a l l y do n o t g i v e p r o o f s o f p r e s e n t e d r e s u l t s , however w e d i d o u r b e s t t o g i v e d e t a i l e d r e f e r e n c e s t o t h e l i t e r a t u r e where t h e p r o o f c a n be found. When t h e p r o o f i s g i v e n , i t i s u s u a l l y a s k e t c h . A s a r u l e t h i s happens when t h i s p a r t i c u l a r p r o o f o r r e s u l t i s n o t e x p l i c i t l y s t a t e d i n t h e l i t e r a t u r e , b u t i s a n e a s y c o n s e q u e n c e of known r e s u l t s o r methods. The p a p e r - i s d i v i d e d i n t o s e v e n s e c t i o n s of v e r y u n e q u a l l e n g t h . The f i r s t two g i v e t h e n e c e s s a r y a n a l y t i c b a c k g r o u n d , t h e n e x t f o u r a r e devoted
t o t h e subject proper of our exposition while
t h e l a s t one f o r m u l a t e s some a d d i t i o n a l
directions for possible
future research. Now w e i n d i c a t e t h e c o n t e n t of p a r t i c u l a r s e c t i o n s . S e c t i o n 1
describes the space
H 1 ( D ) as a s p a c e o f a n a l y t i c f u n c t i o n s on
t h e u n i t d i s c while S e c t i o n 2 p r e s e n t s t h e real v a r i a b l e approach
t o t h e s a m e s p a c e . S e c t i o n 3 g i v e s some r e s u l t s on g e n e r a l s u b s p a c e s of
H 1 ( D ) . Short Section 4 i s
devoted
t o i s o m e t r i c problems.
S e c t i o n 5 c o n t a i n s t h e d e s c r i p t i o n o f b a s e s and u n c o n d i t i o n a l b a s e s in
HI
and c l o s e l y r e l a t e d r e s u l t s a b o u t isomorphisms between
H1(D)
and v a r i o u s o t h e r s p a c e s . F i n a l l y S e c t i o n 6 d i s c u s s e s complemented subspaces of
H , (D).
SECTION 1 ; Complex f u n c t i o n a p p r o a c h t o
H,(D).
I t i s w e l l known t h a t t h e f u n c t i o n IzI < I }
f
(2)
analytic i n
can have an e x t r e m a l l y i r r e g u l a r b e h a v i o u r a s
D ={z EE : IzI
approa-
c h e s 1 . So i t i s a n a t u r a l i d e a t o c o n s i d e r f u n c t i o n s w i t h somehow r e s t r i c t e d b e h a v i o u r c l o s e t o t h e b o u n d a r y . Hardy s p a c e s i s one s u c h p o s s i b i l i t y , which t u r n e d o u t t o b e e x t r e m e l y s u c c e s s f u l . For 0
we define
f(z), /zl
H ( D ) as t h e space o f a l l a n a l y t i c f u n c t i o n s P such t h a t
So w e r e s t r i c t t h e mean g r o w t h o f a f u n c t i o n . The f i r s t i m p o r t a n t
H ( D ) i s t h a t it l e a v e s a good t r a c e P T h i s is summarised i n t h e f o l l o w i n g
p r o p e r t y o f a f u n c t i o n from on
T.
3
The Banach space H 1 THEOREM 1 . 1 . m
f(z)
Let
=
C anz n =O
n
and l e t
EHp(D)
fr(eie)
a) l i m f r+l and in
f(re
f ( ei 0 ) E L p ( T )
r < I . Then t h e r e e x i s t s a f u n c t i o n
for all
ie)
be d e f i n e d a s
such t h a t
almost e v e r y w h e r e
=f LP(T)
a,
c) i f
p
~
f(eiO)
1t h e n
has t h e F o u r i e r series
C aneine.
n =O
These a r e by no means t r i v i a l f a c t s . From now on w e w i l l v e r y o f t e n i d e n t i f y
H ( D ) w i t h a subspace o f P L p ( T ) . Hidden i n t h o s e s t a t e m e n t s i s a n i m p o r t a n t F.M.Riesz Theorem.
p =I.
L e t u s d i s c u s s it i n more d e t a i l . C o n s i d e r
a bounded f a m i l y i n
Then
(fr)r
is
L 1 ( T ) . I f w e look a t t h e Fourier series of
t h o s e f u n c t i o n s w e e a s i l y see t h a t (f ) c o n v e r g e s i n t h e fweak r m t o p o l o g y t o a m e a s u r e whose F o u r i e r series i s C anei n 0 That n=O
.
L , ( T ) f u n c t i o n is a c o n t e n t of t h e
t h i s measure i s a c t u a l l y a n THEOREM 1 . 2 . ( F . M . R i e s z ) Let
p
b e a m e a s u r e on
2T
Je-inedu(0)
for
=O
T
such t h a t
...
n =1,2,3,
0
p
Then
i s a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t t o t h e Lebesgue
measure. A f u n c t i o n a l a n a l y t i c c o r o l l a r y of t h i s t h e o r e m i s t h a t
a *weak c l o s e d s u b s p a c e o f
we can say a l i t t l e m o r e . Let W
t h e set
{einOln=,
and l e t
H1(D)
is
M ( T ) , so it i s a d u a l space. A c t u a l l y
.A HZ
denote denote
t h e closure i n the
C(T) of
a(Lw,L1) closure of
t h e same s e t . Those s p a c e s c a n b e i d e n t i f i e d w i t h t h e s p a c e s o f a l l a n a l y t i c f u n c t i o n s which a r e u n i f o r m l y c o n t i n u o u s i n in
D,
respectively.
D
o r bounded
4
P. Wojtaszczyk
COROLLARY 1 . 3 . H ~ ( D )is isometric to the dual of isometric to
L_ ( T ) /H:.
C(T)/Ao
and
H1(D)*
is
Now we will explain the canonical factorization. We first introduce
three classes of functions.
(1) Blaschke products. Let (an)n=, be a sequence of numbers from m
such that
m
C
n=1
D
(l-lanl) < a .
Then B(z) = II n=l
janl an-z a n I-: z n
is an analytic function (called Blaschke product) such that (a) the zeros of B(z) are exactly (b) IB(z)/ < I
for
z ED
and
m
counting multiplicity,
IB(eie)l = I
a.e.
( 2 ) Singular inner functions are the functions of the form
for some positive singular measure
of singular inner functions are
for
T. Basic properties
Z E D
for =I
on
z ED
a.e.
CAUTION : S(z)-’
is not bounded, it does not even belong to any
Hp(D).
For $(t) defined on T such that $(t) 20, we define an outer function of the class H to be
( 3 ) Outer functions.
log $ EL1, )I ELP
P
5
The Banach space H 1
where
i s a r e a l number. I t i s e a s y t o see t h a t
y
Obviously
F ( z ) h a s no z e r o s i n
ie) I = $ ( e ) .
(F(e
D.
Now w e a r e r e a d y t o s t a t e THEOREM 1 . 4 .
(Canonical f a c t o r i z a t i o n t h e o r e m ) .
Every f u n c t i o n
f E H ( D ) a d m i t s a u n i q u e f a c t o r i z a t i o n i n t h e form P where B i s a B l a s c h k e p r o d u c t , S i s a
f(z) =B(z).S(z).F(z)
s i n g u l a r i n n e r f u n c t i o n and
F
i s a n o u t e r f u n c t i o n of c l a s s
Converselyeverysuch product belongs to
Hp(D).
%'
The f o l l o w i n g e a s y c o r o l l a r y from t h e C a n o n i c a l F a c t o r i z a t i o n Theorem i s v e r y u s e f u l . COROLLARY 1 . 5 .
f E H D ( D ) c a n be w r i t t e n a s
a ) Every
no z e r o s i n
D
and
f =h
llhl[lr
1
+h2
where
hl,h2
'
have
- =- +-1 t h e n e v e r y f E H ( D ) c a n b e w r i t t e n a s f = g - h , P P cs r ' in g EHq(D), h EHr(D) and f Br ( D 1 ' p a r t i c u l a r f o r p = I w e c a n have r =q = 2 . b) L e t
-
I]
The p r o o f s a r e so e a s y t h a t w e c a n g i v e them h e r e . L e t For a ) p u t h =(B-l).S.F 1 h =FP/'. The s p e c i a l case o f two of
H2(D)
H1(D).
and
h 2 = S . F . For b ) p u t
b ) t h a t every
H1(D)
and
function i s a product of
f u n c t i o n s o p e n s up a new d e s c r i p t i o n of d u a l and p r e d u a l I do n o t want t o go i n t o d e t a i l s ,
here. L e t me say only
t h a t it i s p o s s i b l e t o r e p r e s e n t t h e p r e d u a l o f C(T)/Ao,
f =B*S.F.
g =B*S*F
H1(D),
the space
a s a s p a c e of so c a l l e d compact Hankel o p e r a t o r s on a
H i l b e r t s p a c e . The p r o o f of t h i s a s s e r t i o n c a n be f o u n d i n
[K - P I o r
[ S a r ] and a d e t a i l e d e x p o s i t i o n o f t h e t h e o r y o f t h o s e o p e r a t o r s i s given i n
[H - P I
and
[Pow].
NOTE. A l l f a c t s p r e s e n t e d i n t h i s S e c t i o n a r e c l a s s i c a l and c a n b e f o u n d i n
6
P. Wojtaszczyk
any of t h e books [ D u r ] , [ H o ] , [ K o o ] , [ K a t ] , [ Z y g ] . SECTION 2 :
.
H1 (D)
Real v a r i a b l e approach t o
The t h e o r y s k e t c h e d i n t h e p r e v i o u s s e c t i o n depended v e r y much on t o o l s ( l i k e Blaschke p r o d u c t s ) which a r e p e c u l i a r t o one-dimensional s i t u a t i o n , and are i m p o s s i b l e t o g e n e r a l i s e t o s e v e r a l v a r i a b l e s i t u a t i o n . Thus a t t e m p t s t o g e n e r a l i s e t h e t h e o r y r e q u i r e d new t o o l s . I t i s a remarkable f a c t t h a t t h o s e new t o o l s i n v e n t e d f o r g e n e r a l i s a -
t i o n had a l s o a tremendous impact on t h e c l a s s i c a l t h e o r y . I n t h i s s e c t i o n w e i n t e n d t o d e s c r i b e t h i s r e a l v a r i a b l e approach t o L e t us t a k e f l I H 1 (D)
mines
f ( z ) EH,(D).
if3
f(e
and
i f and o n l y i f b o t h
Imf(eie) e x i s t
Ref(eie)
and
on
Imf(eie)
Since
T.
belong t o
s a y t h a t w e are i n t e r e s t e d i n harmonic f u n c t i o n s L,(T).
Ref(z)
deter-
up t o a c o n s t a n t . Moreover it f o l l o w s from Theorem 1 . 1
Ref(ei*)
t h a t both
H1(D).
) E L T ( T ) , and
= J T l f ( e i e ) ld0. I t i s a l s o w e l l known t h a t
Imf(z)
t h a t both
W e know t h a t
I / f l I H , (D) L1(T)
h(z), z ED
' w e may such
and i t s harmonic c o n j u g a t e have boundary v a l u e s i n
h
I t i s known ( c f . [ K a t ~ , ~ K o o ] , [ Z y gt ~ ha ) t for
€ EL1(T)
there
e x i s t s a harmonic e x t e n s i o n v i a t h e P o i s s o n formula t o a harmonic r-L f u n c t i o n f ( 2 ) and i t s harmonic c o n j u g a t e f(z) h a s boundary v a l u e s on T g i v e n b t h e p r i n c i p a l v a l u e of t h e f o l l o w i n g improper i n t e g r a l
F(eit)
=/f(t-.c T
This j u s t i f i e s t h e following d e f i n i t i o n : ReHl ( T )
i s t h e s p a c e of a l l f u n c t i o n s
f EL1(T)
such t h a t
-
f EL1(T)
w i t h t h e norm d
IlflIReH1(T)
fllLl
T) + l I f 1 I L 1 ( T )
(T) b a s i c a l l y c o n s i s t s o f r e a l v a l u e d f u n c t i o n s b u t 1 t h e r e i s no d i f f i c u l t y i n c o n s i d e r i n g complex f u n c t i o n s as w e l l .
This space
ReH
S i n c e it i s known t h a t f o r
we infer that
f E L ( t ) ,1 < p < P
N
f
also is i n
Lp(T)
7
The Banach space H 1
U
L (T)C ReHl !T)C L 1 (T).
p>l p
Up till now it may be seen merely as atranslation but the point is that in this way we put our space H,(D) into entirely new perspec-
tive. The fundamental development in this context is the Fefferman duality theorem. We start by introducing the space BMO of functions
of bounded mean oscillation. Let f (T). For every interval if. We say that f EBMO(T) if we put fI
=h
IC T
The quantity above is not a norm (it is zero for the constant function) so we define the norm by the formula
It follows from the John-Nierenberg inequality (cf.[J-N],[Ner]) that the above norm is equivalent (for all p, 1 < p < - ) to
JJf/
T
SUP(^
ICT
1
In particular
J J f- € I
1
BMO(T
The fundamental Fefferman duality theorem asserts THEOREM 2.1. The dual of ReH1 (T) is BMO(T); more precisely, for every bounded linear functional x* [ReH, (T)] * there exists a unique function cp EBMO(T) such that for f tL2(T) we have
x*(f) =lf(t)q(t)dt T
and conversely there exists a constant BMO(T) and f EL2(T) we have
cp
C
such that for every
8
P: Wojtaszczyk
REMARK.
f EL2(T) i n t h i s statement i s t h a t
The r e a s o n f o r i n v o k i n g f ( t ) V. ( t )
for
f t R e H 1 ( T ) and
cp
E B M O ( T ) need n o t b e Lebesgue
integrable. The above Theorem 2 . 1 .
implies t h e following description of
H 1 (D)*.
THEOREM 2 . 2 .
H1 (D)*
BMOA = H 1 (D) n B M O ( T ) .
c a n be i d e n t i f i e d w i t h t h e s p a c e
Now l e t us e x p l a i n a n o t h e r e q u i v a l e n t form o f t h e D u a l i t y Theorem. W e w i l l call the function
either
a(t) E l
a ( t ) d e f i n e d on
a n atom i f
or
we h a v e
for some interval
Ic T
supp a ( t ) CI
l a ( t )1 2 - q
and
T
'
and
i a ( t ) d t =O. T
THEOREM 2 . 3 . A function
f ( t )belongs t o
R e H l ( T ) i f and o n l y i f
f =E?, . a
where
~j
over a l l t h e a ' s are a t o m s and C \ X j \ < m . Moreover i n f Z 1 A . I j 7 a l l a t o m i c r e p r e s e n t a t i o n s o f f i s a norm e q u i v a l e n t t o
I I I I ReHl
(T) '
PROOF. An e a s y c a l c u l a t i o n shows t h a t
a t o m s h a v e u n i f o r m l y bounded norm i n
I L,
(T)
f o r a l l a t o m s , so a l l
ReH.,(T). To show
the other
i n c l u s i o n it i s enough t o show t h a t t h e s e t o f a l l atoms norms BMO(T).This f o l l o w s i m m e d i a t e l y from t h e o b s e r v a t i o n t h a t mean z e r o functions i n
L_
o f norm
<1 -
norm mean z e r o f u n c t i o n s i n
L1.
Two comments a b o u t Theorem 2 . 3 are i n o r d e r . a ) T h i s i s p o t e n t i a l l y a v e r y u s e f u l t h e o r e m . I t g i v e s a k i n d of " e x t r e m e p o i n t " d e s c r i p t i o n of
ReH1(T).
Obviously i n o r d e r t o
e s t a b l i s h t h e c o n t i n u i t y o f a n o p e r a t o r d e f i n e d on enough t o c h e c k t h a t i t i s
ReH1(T) it i s
u n i f o r m l y bounded o n a l l a t o m s . S i n c e
atoms a r e r a t h e r s i m p l e f u n c t i o n s t h i s t a s k s h o u l d b e much e a s i e r t h a n t h e i n i t i a l problem.
9
The Banach space H 7
b ) The n o t i o n o f a n atom i s v e r y e a s y t o g e n e r a l i s e .
In particular
w e c a n c o n s i d e r atoms on t h e i n t e r v a l a n d w e c a n d e f i n e t h e s p a c e Ha(O,II) a s t h e space of a l l f u n c t i o n s f d e f i n e d on [ O , I I ] such 1 t h a t f =CA.a w h e r e C I A j [ < m and a a r e atoms d e f i n e d on i j j [ O , i l ] . The norm i n H;(O,Il) i s by d e f i n i t i o n i n f C I A . I o v e r a l l
a t o m i c r e p r e s e n t a t i o n s of
I
f.
Now w e e s t a b l i s h a n e a s y p r o p o s i t i o n which w i l l b e u s e d i n S e c t i o n 5 .
For a f u n c t i o n f d e f i n e d o n
we put
[O,II]
PROPOSITION 2 . 4 . For
f EH;(O,fl)
operator
T
we define
Tf = € R e f -EImf + i ( E R e f + & I m f ) . The
e s t a b l i s h e s a n isomorphism b e t w e e n
H;(O,II)
and
H1 ( D ) .
PROOF. I t i s e a s y t o check t h a t
T
i s an a n a l y t i c f u n c t i o n . For
i s a complex l i n e a r map and t h a t a
a n atom on
[O,n]
Ea
Tf
i s a sum of
'L" i s c o n t i n u o u s on ReH1 ( T ) , i s i n R e H 1 ( T ) t h u s T i s continuous. To check t h a t -1 T i s c o n t i n u o u s i t i s enough t o c o n s i d e r f u n c t i o n s o f t h e form h = a +i"a where a i s a r e a l atom o n t h e c i r c l e . W e h a v e h =Tf two atoms on t h e c i r c l e . Moreover
so
€Ref - E I m f
where
€Ref -EImf = a . The norm o f
remembers t h a t
&Ref
is even and
f
i s e a s y t o e s t i m a t e i f one
EImf
i s odd.
NOTES. T h e r e are many p r o o f s of Theorem 2 . 1 .
i n t h e l i t e r a t u r e . The
t h e o r e m w a s announced by C.Fefferman i n [ F e ] a n d t h e p r o o f i n t h e context of
Rn
appeared i n [F-S].
The p r o o f s f o r t h e u n i t d i s c c a n
b e found i n [Koo] a n d [ S a r ] . The J o h n - N i e r e n b e r g i n e q u a l i t y m e n t i o n e d b e f o r e Theorem 2 . 1 .
was p r o v e d i n [ J - N ] .
found i n [ N e r ] o r [ K o o ] .
The s i m p l e r p r o o f s c a n be
The f a c t t h a t Theorem 2 . 1 .
Theorem 2 . 3 w a s o b s e r v e d by C.Fefferman
is equivalent t o
(unpublished). D i r e c t (i.e.
w i t h o u t u s e o f t h e d u a l i t y t h e o r e m ) p r o o f s o f Theorem 2 . 3 . c a n b e found i n [ C o ] and [ W i ] .
A v e r y g e n e r a l e x t e n s i o n of t h e
u s i n g atoms i s p r e s e n t e d i n [C-W].
H -theory
P
P. Wojtaszczyk
10
i
SECTION 3; S t e i n ' s theorem and i t s consequences.
W e s t a r t w i t h t h e f o r m u l a t i o n o f t h e E.M.
S t e i n m u l t i p l i e r theorem.
THEOREM 3 . 1 .
Let
(u ( n ) ) n = O W
b e a s e q u e n c e of complex numbers s u c h t h a t
n=O H1 (D)
a zn
n
-
W
c a n u ( n ) zn n=O
i s a c o n t i n u o u s map from
i n t o i t s e l f a n d i t s norm d e p e n d s o n l y o n
Now w e d e f i n e m u l t i p l i e r s
An,"
2 2 n < k < 22n+1
or
<2Tn-i
-
=1,2,3,
...
C.
as follows
,22n+2
otherwise. W e also put
1
k =0,1,2
0
kL4
. W
I t i s e a s y t o check t h a t
every
E
n
=+1
C An(k) = 1 f o r a l l k and t h a t f o r n=O m m t h e sequence ( C E ~ A ~ ( ~ ) s a) t ~i s f= i e~ s t h e n=O
a s s u m p t i o n s o f Theorem 3 . 1 .
with t h e s a m e
C.
So w e h a v e
COROLLARY 3.2. m
For
f € H I (D)
l y convergent.
we have
C
n=O
An(f) = f
a n d t h e series i s u n c o n d i t i o n a l -
By a s t a n d a r d a p p l i c a t i o n o f t h e K h i n t c h i n e i n e q u a l i t y w e o b t a i n
11
The Banach space H 1
COROLLARY 3.3. For
f EH1(D) we have
Let us recall the following DEFINITION 3 . 4 .
A Banach space
X
has
an unconditional finite dimensional expansion
of identity if there exists a sequence 0: finite dimensional operators m x EX, x = c Tn(x) and the series is (Tn)n=O such that for every n=O unconditionally convergent. Obviously an unconditional basis is a special example of a finite dimensional expansion of identity (for the definition see Section 5). Now we have the following THEOREM 3.5. L1 (T) with an uncond tional fin te dimensional expansion of identity. Then X is isomorphic to a subspace of H1(D).
Let
X
be a subspace of
PROOF.
-
x
m
C Tn (x) and the n=O series is unconditionally convergent. This implies that ( ( \xi(1 =
We have
m
Tn : X
12)’
X I finite dimensional and
=
1
J ( C ITn(x) is an equivalent norm on X. By a standard T n=O kn perturbation argument we can assume that ImTnCspan{eiJt}. 1=pn We choose integers way that
pn
and
wnI vAs
[pn +pn,pn + k , ] c [22vn,22"n+1I.
Now it is obvious that the map
x
-
.
being all different in such
m
C
e
n=O
ipnt
Tn(x)
is an
a
P. Wojtaszczyk
12
i s o m o r p h i c embedding from
X
into
H1(D).
H (D) c o n t a i n s s u b s p a c e s i s o m o r p h i c t o L p ( T ) , 1 < p 52, 1 d o e s . S i n c e by a t h e o r e m o f R o s e n t h a l [ R o s ] e v e r y r e f l e x i -
I n particular
L1
since
L1 i s isomorphic t o a w e g e t t h a t e v e r y r e f l e x i v e subspace of
v e s u b s p a c e of
s u b s p a c e o f some L1
is
a
Lpl
p 71
s u b s p a c e of
T h i s l a s t f a c t was p r o v e d i n a much s i m p l e r way i n [K-PI.
H1(D).
A s a n e x t a p p l i c a t i o n of S t e i n ' s theorem w e w i l l p r o v e t h e f o l l o w i n g
r e s u l t due t o S.Kwapien a n d A . P e l c z y n s k i
[K-PI.
THEOREM 3 . 6 .
L e t (f,)
be a s e q u e n c e i n
H, (D) s u c h t h a t
11
Canfnll H,
(Dr
1
2 2 (Clan/ )
-
Then t h e r e e x i s t s a s u b s e q u e n c e (f ) s u c h t h a t s p a n ( f ) i s nk "k complemented i n H 1 ( D ) . PROOF.
B y a s t a n d a r d g l i d i n g hump a r g u m e n t w e c a n assume t h a t t h e r e e x i s t a
subsequence ( f a)
(f
"k
=f
nk
) and a sequence o f m u l t i p l i e r s
m
(rk(j))j=l
such t h a t
"k
c ) f o r a l l c h o i c e s of
Let
gk € H I ( D ) *
=*I
t h e sum
CEkrk(j) satisfies the k a s s u m p t i o n s o f Theorem 3 . 7 w i t h t h e c o n s t a n t i n d e p e n d e n t of ( E ~ ) . E~
be s u c h t h a t
ne
a
p r o j e c t i o n o n t o s p a n ( f n ) by t h e f o r m u l a
k
13
The Banach space H 1
1
er that
P
1
is a cont nuous projection onto span
f ). nk
REMARK. f = z kn where kn+l/kn LX > I for all n, n then the above argument works without passing to a subsequence, so we obtain the following classical inequality of Paley [Pa]. Let us observe that if
COROLLARY 3.7.
n f = Z anz m
Let
kn+,/kn A,
>1
and let
n =O
be in
H I ( D ) . We have
1
n
I akn
REMARK Some of the results of this section follow also from the existence of an unconditional basis in H, ( D ) (cf.Section 5 ) . NOTES. Theorem 3 . 1 and Corollaries 3 . 2 and 3 . 3 . were proved in [Stel. A nice proof using atoms can be found in [C-W], Theorem 1 . 2 0 . Theorem 3.5. is a routine observation which, to the best of our knowledge, does not appear in the literature. Theorem 3.6. was proved in [K-P1,Theorem 3 . 1 . The proof given there is different and rather more complicated. We may remark that Theorem 3 . 5 shows that Theorem 3 . 6 holds also for H I (Tn); the H I space on the polydisc. SECTION 4; Isometric questions. We already know several isomorphic representations of H 1 ( D ) and we will see many other later. In this situation the importance of isometric theory is rather diminished. On the other hand there are several interesting results and open problems pertaining to the (in this particular norm),so we decided isometric theory of H 1 ( D ) to present them in our exposition. The existing isometric results
P. Wojraszczyk
14
c o n c e n t r a t e on t w o q u e s t i o n s :
a ) d e s c r i p t i o n of i s o m e t r i e s b) extreme p o i n t s t r u c t u r e . H - s p a c e s on more g e n e r a l P t h a n t h e u n i t d i s c ( c f . [ Ru 1 ] and t h e b i b l i o g r a p h y
The i s o m e t r i e s h a v e b e e n d e s c r i b e d f o r
En
domains i n
g i v e n t h e r e ) . The t h e o r e m f o r
H,(D)
reads a s follows:
THEOREM 4 . 1 .
Every i s o m e t r y
where
I
of
H1(D)
is a non-constant
@
I b ( t ) d t = J ( b o @ )( t )1 g ( t ) I d t T T
i s g i v e n by
H,(D)
inner function i n
D,
q EH,(D)
and
f o r a l l bounded, Bore1 f u n c t i o n s
b(t)
C o n v e r s e l y e v e r y map d e s c r i b e d above i s a n i s o m e t r y
d e f i n e d on
T.
from
into itself.
H.,(D)
into
T h i s theorem was p r o v e d by F . F o r e l l i
[Fo]. For e a r l i e r r e s u l t s t h e
r e a d e r may c o n s u l t [ F o ] , [ R u 3 ] , [ H o ] . I t i s known t h a t a s u b s p a c e o f
i s norm one complemented. A s f a r as w e P know t h e c o m p l e m e n t a t i o n o f s u b s p a c e s of H I (D) i s o m e t r i c t o H , ( D ) L p , p 21
isometric t o
L
wa.s n o t i n v e s t i g a t e d i n g e n e r a l . Some p a r t i a l r e s u l t s are c o n t a i n e d i n [Ba]
. The
o t h e r open problem c o n n e c t e d w i t h i s o m e t r i e s i s t h e
H (D) contain a s u b s p a c e isometric t o H p ( D ) , 1 1 < p < 2 ? I t i s known ( c f . Theorem 3 . 5 . a n d f o l l o w i n g r e m a r k s ) t h a t
following: does H
P
(D),1 < p '2
i s i s o m o r p h i c t o a s u b s p a c e of
HI (D).
The e x i s t i n g i n f o r m a t i o n a b o u t e x t r e m e p o i n t s t r u c t u r e of i s presented i n
[Ho],
Chapter 9 .
H1(D)
L e t u s quote t h e d e s c r i p t i o n of
extreme p o i n t s . THEOREM 4 . 2 .
Let
f
be a f u n c t i o n i n
the unit
b a l l of
H,(D)
H1(D).
Then
i s a n e x t r e m e p o i n t of
f
i f and o n l y i f
o f norm 1 . W e do n o t know a b o u t any f u r t h e r r e s u l t s .
f
is an o u t e r f u n c t i o n
In particular strongly
exposed p o i n t s o r d e n t i n g p o i n t s of t h e u n i t
ball in
H,(D)
are
15
The Banach space H I
n o t d e s c r i b e d . F o r d e f i n i t i o n s of t h o s e n o t i o n s and t h e i r i m p o r t a n c e The problem o f d e s c r i p t i o n of s t r o n g l y
t h e r e a d e r may c o n s u l t [ D - U ] . exposed p o i n t s i s d i s c u s s e d i n
[ G a J ] , IV.5.
general theory (cf.[D-U]) t h a t
H , ( D ) h a s many p o i n t s o f b o t h k i n d s .
I t f o l l o w s from t h e
The l a s t i s o m e t r i c r e s u l t w e want t o m e n t i o n i s t h e f o l l o w i n g t h e o -
r e m o f Newman's [ N e w ] . THEOREM 4 . 3 . Let
m
(fn)n=,
and
11
weakly and
f
fn/l
-
be i n
Assume t h a t
H1(D).
11 f II .
Then
fn
tends t o
fn
tends t o
f
f
i n norm.
Newman named t h i s p r o p e r t y " p s e u d o - u n i f o r m c o n v e x i t y " . I n t h e Banach s p a c e t h e o r y t h i s i s e x p r e s s e d a s " t h e Kadec-Klee norm". Kadec a n d K l e e h a v e shown t h a t e v e r y Banach s p a c e c a n b e renormed t o have t h e For a d i s c u s s i o n o f t h i s n o t i o n
p r o p e r t y d e s c r i b e d i n Theorem 4 . 3 .
i n t h e framework o f g e n e r a l Eanach s p a c e s see
[Di].
SECTION 5 ; Eases a n d v a r i o u s i s o m o r p h i c r e p r e s e n t a t i o n s .
F o r a n a r b i t r a r y Banach s p a c e
t h e s y s t e m of v e c t o r s
X
i s c a l l e d a Schauder b a s i s i f f o r e v e r y element m
a u n i q u e s e q u e n c e o f s c a l a r s ( an ) n=O converges t o
x
i n t h e norm o f
X
x
m
C x:(x) n=O I' system
x-. I1
x;(xm) = O
,
x* € X * n
, so
a
i .e. a system such t h a t
m
o n l y i f t h e c l o s e d l i n e a r span o f {xnln=O e q u a l s
I f t h e b a s i s (x,)
is actually
an
t h e series h a s t h e form
f o r n # m , t h e n such a system is a basis f o r
of p a r t i a l sum o p e r a t o r s
C anxn n=O
X.
It i s a l s o known t h a t i f w e a r e q i v e n
( X ~ , X : ) ; =X~xC X*
there Zxists
s u c h t h a t t h e series
I t i s w e l l known t h a t i n t h i s case e a c h c o e f f i c i e n t
g i v e n by a l i n e a r f u n c t i o n a l
m
( ~ ~X ) ~ = ~
N * P (x) = C x (x)xn n=O n N
X
biorthoqonal x* ( x ) = 1 n n X
and
i f and
and t h e f a m i l y
i s u n i f o r m l y bounded.
has t h e a d d i t i o n a l property t h a t f o r every
x EX
m
t h e series
C x:(x) xn c o n v e r g e s u n c o n d i t i o n a l l y , w e c a l l s u c h a n=O b a s i s u n c o n d i t i o n a l . The b a s i s i s c a l l e d monotone i f IIPNII = I for
P. Wojtaszczyk
16
N =011,2, m
...
and u n c o n d i t i o n a l l y monotone i f f o r e v e r y
x = C x*(x)x n n=O n m
11
w e have
and f o r e v e r y s e q u e n c e of numbers
C cnx:(x)
n=O
x n l l 511x11
E~
long t i m e .
a
for quite
a
1
~
~
=I
1
-
The problem o f t h e e x i s t e n c e o f a S c h a u d e r b a s i s i n constructed
with
H 1 ( D ) was a r o u n d
I t w a s s o l v e d by P . B i l l a r d i n [ B i ] who
basis for
H1(D).
H i s p r o o f was r e a l v a r i a b l e i n
s p i r i t a n d u s e d t h e Haar f u n c t i o n s . The g e n e r a l scheme i s as f o l l o w s . Let
m
(fnInzo
be a n o r t h o n o r m a l s y s t e m o n
L e t us d e f i n e
[o,III w i t h
-.
fo(t) = 1
fi
( a s i n Proposition 2.4.)
n1
fn(t)
tE[O,
fn(-t)
tr[-nlol.
& f n ( t )=
n,
C l e a r l y ( & f n ) n = O i s a n o r t h o g o n a l s y s t e m on
T.
In order t o obtain
a (complex) o r t h o g o n a l s y s t e m o f a n a l y t i c f u n c t i o n s on
T
we define
T h e theorem o f B i l l a r d [ B i l s a y s t h a t i f w e s t a r t w i t h t h e Haar
s y s t e m o n [ O , I I I a s (fn);=o H1(D).
follows
Let
if
F n ( z ) i s a Schauder b a s i s i n
u s r e c a l l t h a t t h e Haar s y s t e m o n
[O,n] i s d e f i n e d a s
I
h ( t )=- 1 0
then
di?
I
n =2k + j , 0 < j <2k, k =0,1,2
,...
then
I
0
otherwise.
I t c a n be s e e n from P r o p o s i t i o n 2.4.
t h a t t h e theorem o f B i l l a r d can
17
The Banach space H 7
be f o r m u l a t e d as f o l l o w s . THEOREM 5 . 1 .
.
(Billard)
The Haar s y s t e m (hn);=o
is a b a s i s i n
H:(O,ll).
PROOF. Obviously t h e l i n e a r span o f t h e Haar s y s t e m e q u a l s p a r t i a l sum o p e r a t o r N P f = C N j=o
where
1
lIJi
I!s 3
PN
PNa
i s a c o n s t a n t m u l t i p l e of a n atom. I n t h e c a s e
.I
7
4
diam ( s u p p P N a )
so
+4/N
w e see t h a t f o r
f o r a t most two
#O
1 1 PNal I co ( 2 N .
1 1 P N a /I mz1 a /1
a ( t ) , supp a c I . S i n c e /PNa = 0 ,
d i a m ( s u p p P N a ) (111 flI
1
a r e d i s j o i n t i n t e r v a l s c o v e r i n g [ O , l l I and
and
1I
The
XIj
L e t us t a k e a n atom
have
HY(O,II).
i s given as an averaging o p e r a t o r
Obviously
111
Lm1
the function
<%1
111
we
j ' s . T h i s means t h a t
fPNa =O.
Moreover
I
II7J IJ,al 5TT-J-
T h i s shows t h a t a l s o i n t h i s c a s e
c o n s t a n t m u l t i p l e o f an atom.
1
1
7
PNa
is a
I t i s r e l a t i v e l y e a s y t o check ( c f . t h e l a s t P r o p o s i t i o n of [ K - P I )
t h a t t h e Haar system i s n o t an u n c o n d i t i o n a l b a s i s i n q u e s t i o n o f t h e e x i s t e n c e o f an u n c o n d i t i o n a l b a s i s i n
The
H;(O,II). H1(D)
was
r a i s e d by s e v e r a l m a t h e m a t i c i a n s ( c f . [ E I , [ K - P ] , [ P e ] ) . B e f o r e w e p r e s e n t t h e s o l u t i o n t o t h i s problem w e need t o d e f i n e H -space, 1
namely
H.,(d), t h e
new
H1-space c o n n e c t e d w i t h t h e c a n o n i c a l
The g e n e r a l t h e o r y of t h i s s p a c e c a n be found
dyadic martingale. in
a
[ G a r ] . From o u r p o i n t of view t h e most c o n v e n i e n t d e f i n i t i o n i s
t h e following.
For a f u n c t i o n
f
on
[O,II]
w e d e f i n e i t s norm i n
Hl(d) as
The s p a c e of a l l
f
such t h a t
H l ( d ) . I t i s known t h a t
I I f l I H1
(dl
i s f i n i t e i s denoted by
L [ O , I I ] c H 1 (d)CL1[O,II]
for a l l p > I . P Moreover i t i s o b v i o u s t h a t t h e Haar system is an u n c o n d i t i o n a l
P. Wojtaszczyk
18
basis i n
H , ( d ) . The q u e s t i o n a b o u t t h e e x i s t e n c e o f a n u n c o n d i t i o n a l
basis i n
H l ( D ) was a n s w e r e d b y t h e f o l l o w i n g t h e o r e m o f Maurey's
[ Mau
1.
THEOREM 5 . 2 .
The s p a c e s H 1 ( D ) a n d
H ( d ) are isomorphic.
1
Maurey's p r o o f o f Theorem 5 . 2 . h a s
one drawback, i t i s n o t c o n s t r u c -
t i v e . T h i s w a s remedied by C a r l e s o n [ C a ] and t h e a u t h o r [WO~].I n o r d e r t o d e s c r i b e t h e r e s u l t from [ W O ~ w]e need o n e more d e f i n i t i o n .
W e define points
to = O f i f
tn a s f o l l o w s
n =2k + j , k =0,1,2,...,
The F r a n k l i n s y s t e m
( t )=-
f O
I
/
1
i
fn(t)
O s j <2k;
t
n
=(j+1)2-kII.
i s d e f i n e d by t h e f o l l o w i n g c o n d i t i o n s :
,
i
f ( t )is a piecewise l i n e a r f u n c t i o n on [ O , l I ] w i t h nodes a t p o i n t s n t o f tl , . . . , t which i s o r t h o g o n a l t o a l l f j ' s , j < n and I I f n / 1 2 = 1 . n W e have t h e f o l l o w i n g t h e o r e m p r o v e d i n [WO~]. THEOREM 5 . 3 .
-
d e f i n e d by
The o p e r a t o r
T
from
onto
H:(O,II)
map F n ( z )
hn
Hl(d). If
T ( f ) = h n e x t e n d s t o a n isomorphism n F n ( z ) i s d e f i n e d as i n ( 1 ) t h e n t h e
i n d u c e s t h e isomorphism between
H1(D)
and
Hl(d).
I n p a r t i c u l a r t h e F r a n k l i n system i s an u n co n d itio n al b a s i s i n
Ha ( 0 , JI) and t h e c o r r e s p o n d i n g s y s t e m
1 basis i n
Fn ( z ) i s a n u n c o n d i t i o n a l
H1(D).
I t i s c l e a r t h a t o n l y t h e f i r s t a s s e r t i o n n e e d s a p r o o f . T h i s c a n be
found i n [ W O ~ ] and [ C i e ] o r i n a g r e a t e r g e n e r a l i t y i n [ W O ~ ] o r
[s-SI. The r e a d e r may be i n t e r e s t e d t o n o t e t h a t t h e e x i s t e n c e o f a n u n c o n d i t i o n a l b a s i s f o r H (D) i s a phenomenon from t h e i s o m o r p h i c t h e o r y . 1 The c o r r e s p o n d i n g i s o m e t r i c f a c t i s f a l s e as i s g i v e n i n t h e following
19
The Banach space H I
PROPOSITION 5 . 4 . H1(D) is not isometric to a subspace of a Banach space with an unconditionally monotone basis. The outline of the proof can be found in [ W O ~ ] . The following isometric problem concerning
H q (D) is open.
PROBLEM. a) Does
H (D) have a monotone Schauder basis? 1 b) Describe the norm one, finite dimensional projections in
H1(D).
It is quite likely that the answer to a) is negative. I do not
know about any norm one, finite dimensional projection in H1(D) whose rank has dimension greater than 1. It was shown in [ W O ~ ] that any norm one finite dimensional projection in L1/H1 is actually one dimensional.
As the reader may have noticed we have dealt with three different H I spaces, and all of them turned out to be isomorphic. It is only a tip of an iceberg. There is a vast proliferation of HI-spaces important in analysis, cf. [C-W] or [ F o - S ] . T o decide exactly which are isomorphic to H ( D ) and which are not may be 1 of interest. Maurey in [Maul considers a wide class of martingale H -spaces associated with different sequences of a-fields and 1 shows them to be isomorphic to HI (d), He also shows that the Fefferman-Stein spaces H I (Rn) (for definitions see [ F - S ] ) are isomorphic to H,(d). If we adhere to the spirit of complex function theory the picture becomes more complicated. It is possible to consider HI-spaces of functions analytic on relatively wild sets in (I: (cf. [Dur],Chapter 10) but instead we prefer to discuss the situation in several complex variables. There are at least two extremely nice subsets of En, the ball Bn and the polydisc Dn. Both admit natural HI spaces, which can be defined as a closure of analytic polynomials in a certain norm. To get HI (D") we take the norm
20 where
P. Wojtaszczyk
i s a n o r m a l i s e d i n v a r i a n t measure o n a t o r u s
v
s p a c e H1 (B,)
where sphere
o
Tn.
The
i s o b t a i n e d i f w e t a k e a s a norm
i s a n o r m a l i s e d , r o t a t i o n - i n v a r i a n t measure on t h e u n i t S.
About t h o s e s p a c e s w e have t h e f o l l o w i n g THEOREM 5 . 5 .
( a ) H1 ( B n ) (b) I f
is isomorphic t o
H1(Dn)
H1 ( D ) , n =1,2,3,.
i s isomorphic t o
H1 (Dm)
then
..
[WO~].
n =m [Boul],[BouZ].
Both t h o s e f a c t s a r e q u i t e c o m p l i c a t e d t o p r o v e . L e t m e remark o n l y t h a t i n b o t h c a s e s t h e isomorphism between
H,(D)
and
H l ( d ) is a
v i t a l p a r t of t h e p r o o f . W e feel t h a t a l o t r e m a i n s t o b e done i n c o n n e c t i o n w i t h t h i s Theorem.
I n p a r t i c u l a r w e do n o t know
what i s t h e Banach-Mazur d i s t a n c e
H1(B,) and H , ( D ) . A l s o w e do n o t know w h a t i s t h e s i t u a t i o n f o r o t h e r n i c e domains i n En. W e feel t h a t it s h o u l d b e p o s s i b l e
between
t o extend ( a )
t o a l l s t r i c t l y pseudoconvex domains. The methods o f
[ W O ~ ]g i v e many domains which are n o t s t r i c t l y pseudoconvex f o r which ( a ) h o l d s . The o t h e r class o f domains which a d m i t n i c e bounded homogeneous
domains
.
H -spaces a r e 1 Those are f u l l y c l a s s i f i e d a n d r a t h e r
w e l l u n d e r s t o o d , c f . [Hua] ,[H-MI i n v e s t i g a t e and c l a s s i f y
.
I t i s a v e r y i n t e r e s t i n g problem t o
H 1 - s p a c e s o n t h o s e d o m a i n s . Some i n t e r e s -
t i n g p a r t i a l r e s u l t s i n t h i s d i r e c t i o n h a v e b e e n o b t a i n e d by T . Wolniewicz [wall. L e t m e d i s c u s s b r i e f l y t h e method o f p r o o f o f Theorem 5 . 5 .
(b)
( c f . [Bou 1 1 ) s i n c e i t g i v e s a v e r y i n t e r e s t i n g i n f o r m a t i o n a b o u t
H1(D). W e w i l l f o r m u l a t e i t i n t e r m s o f t h e Haar s y s t e m i n H . , ( d ) . N L e t H ( d ) d e n o t e t h e s p a n of t h e f i r s t N H a a r f u n c t i o n s . W e s a y 1 ( f o l l o w i n g J . B o u r g a i n [ B o u l ] ) t h a t a map 5 :H 1N ( d ) H1 ( d ) i s order inversing i f c ( h , ) = C akhk and t h e A n ' s a r e f i n i t e , p a i r w i s e
-
kEAn
d i s j o i n t sets w i t h min
Ak >max
As
whenever
k < s . I t is a n easy
e x e r c i s e ( c f - T h e o r e m 3 . 5 . ) t h a t t h e r e e x i s t o r d e r i n v e r s i n g isomorHNl ( d ) i n t o H l ( d ) w i t h u n i f o r m c o n s t a n t s . The
p h i c embeddings o f
21
The Banach space H 1
-
main r e s u l t o f [ B o u l ] a s s e r t s however t h a t e v e r y p r o j e c t i o n o n t o t h e r a n g e o f s u c h o r d e r i n v e r s i n g i s o m o r p h i s m s h a s b i g norm. More p r e c i s e l y : f o r every
C >1
there e x i s t s a function
f o r e v e r y o r d e r i n v e r s i n g isomorphism
I I f l l 'IIcfl/ HI (d) onto
N
cp(N)
6 :Hy(d)
m’N
such t h a t
m
Hl(d)
with
l C l l f l [ t f t H l ( d ) , t h e n o r m o f e v e r y p r o j e c t i o n from N < ( H I ( a ) ) is g r e a t e r than cp(N).
w e wish t o conclude t h i s s e c t i o n w i t h t h e d e s c r i p t i o n of a r e c e n t b e a u t i f u l and d e e p r e s u l t o f P . J o n e s [ J o n ] . L e t u s s t a r t w i t h t h e d e f i n i t i o n of t h e uniform approximation p r o p e r t y . A Banach s p a c e
X
i s s a i d t o have t h e uniform approximation p r o p e r t y
(cf.[P-R]) i f there e x i s t
a constant
C
t h a t f o r e v e r y f i n i t e system of v e c t o r s e x i s t s a f i n i t e dimensional operator
T
and a f u n c t i o n
-
XI,
:X
... , x N X
in
cp(N)
X
such
there
such t h a t
T h i s i s a p r o p e r t y s t r o n g e r t h a n t h e bounded a p p r o x i m a t i o n p r o p e r t y s i n c e t h e bound
on t h e d i m e n s i o n of t h e r a n g e of a n o p e r a t o r
T
is
imposed. P . J o n e s proved i n [ J o n ] t h a t
BMO
h a s t h e uniform approximation
p r o p e r t y . I t was p r e v i o u s l y unknown ( c f . [ P e l ] ) w h e t h e r
BMO
has the
approximation p r o p e r t y . S i n c e uniform approximation p r o p e r t y i s a self-dual property (i.e. X [ H e i ] , w e see t h a t
has i t i f and o n l y i f
X*
h a s it),c f .
H (D) h a s t h e u n i f o r m a p p r o x i m a t i o n p r o p e r t y . 1
SECTION 6 ; Complemented s u b s p a c e s . I n t h i s s e c t i o n w e a r e i n t e r e s t e d i n complemented s u b s p a c e s o f
H1(D).
i s complemented i f t h e r e e x i s t s a p r o j e c t i o n ( a n i d e m p o t e n t map) w i t h a r a n g e e q u a l t o X. L e t us
remind t h a t a s u b s p a c e
XCH1(D)
G e n e r a l l y , b e i n g a complemented s u b s p a c e o f a g i v e n s p a c e i s a much s t r o n g e r r e s t r i c t i o n t h e n merely b e i n g a s u b s p a c e . To i l l u s t r a t e
P. Wojtaszczyk
22
t h i s l e t m e remark t h a t f o r e v e r y
a subspace of and
H1(D)
p, 1 ( p 5 2 1 i s isomorphic t o P ( c f . Remarks a f t e r Theorem 3 . 5 . ) b u t o n l y 1,
l 2 a r e isomorphic t o
complemented s u b s p a c e s o f
i s immediate from C o r o l l a r y 2 . 1 .
of
r e f l e x i v e complemented s u b s p a c e o f
[K-PI which s a y s t h a t t h e o n l y H1(D)
is
12.
W e s t a r t however w i t h a more h a r m o n i c - a n a l y t i c
projections i n
H
1
problem: what a r e t h e
-
( D ) which commute w i t h r o t a t i o n s ? I t i s e a s y t o
see t h a t such a p r o j e c t i o n i s d e t e r m i n e d by a s u b s e t numbers, and i s g i v e n by t h e formula
terise
t h e idempotent sets, i . e .
o u s map on
H
This
H1(D).
c" anzn n=O
sets
A
of n a t u r a l n C anz To c h a r a c nEA A
.
f o r which it i s a c o n t i n u -
1 ( D ) i s a n i n t e r e s t i n g u n s o l v e d problem. The f o l l o w i n g
remarks a r e obvious: (1)
each f i n i t e set i s idempotent
(2)
if
A
and
are idempotent t h e n
B
A U B,
A nB
and
N\B
are
idempotent (3)
if
A
i s idempotent t h e n f o r e v e r y
(A-k) n N
k EN
the translate
i s idempotent.
(4) a p e r i o d i c sequence is i d e m p o t e n t . L e t us remark t h a t t h e above f a c t s ( e x c e p t ( 3 ) ) a r e v a l i d a l s o
for
L1(T).
I n t h i s case t h e i d e m p o t e n t s e t s have been d e s c r i b e d by
Helson [ H e l l as p e r i o d i c s e q u e n c e s mod a f i n i t e s e t . For
H1(D)
lary 3.7.)
t h e s i t u a t i o n i s d i f f e r e n t . By t h e P a l e y theorem ( C o r o l t h e r e is an i n v a r i a n t p r o j e c t i o n onto a H i l b e r t space.
The theorem o f Rudin [Ru4] d e s c r i b e s t h e i n v a r i a n t p r o j e c t i o n s o n t o H i l b e r t spaces i n
H ( D ) a s g i v e n by a f i n i t e sum of l a c u n a r y sets. 1
The above mentioned theorem of Kwapien and p e l c z y n s k i ([K-P],Coroll a r y 2.1.) s a y t h a t e v e r y complemented r e f l e x i v e subspace of
H1(D)
i s isomorphic t o a H i l b e r t s p a c e . The above mentioned f a c t s l e a d If
A
t o t h e following conjecture.
i s an i d e m p o t e n t s e t t h e n t h e r e e x i s t a p e r i o d i c s e t
B
and
l a c u n a r y s e t s ( i . e . s e t s s a t i s f y i n g t h e a s s u m p t i o n s of C o r o l l a r y 3 . 7 . ) ( C j ) j=1
and ( D . ) S 3 1=1
such t h a t
23
The Banach space H 1
A =B U(
k
S
( U D.).
U C.)
j=1 1
j=1
7
L e t us remark t h a t i n v a r i a n t p r o j e c t i o n s f o r
Hp(D),
p <1
have been
d e s c r i b e d i n [K-TI a s c o r r e s p o n d i n g t o a p e r i o d i c s e t mod a f i n i t e
s e t . F o r t h e i n v a r i a n t , norm one p r o j e c t i o n s on
H1(D)
the situation
i s c l e a r . W e have PROPOSITION 6.1. There is a c o n s t a n t
c >I
t i o n on
[lPli
form
HI (D) with
P is an i n v a r i a n t p r o j e c i s g i v e n by a s e t A of t h e
such t h a t i f
A ={a +bk, k = 0 , 1 , 2 ,
then
- .. I
P
f o r some
a
PROOF. I f the set p z q
p,q,
i s n o t of t h i s form t h e n t h e r e e x i s t i n t e g e r s
A
such t h a t
p €A
and e x a c t l y one o f
p+q
and
p -q
is i n
So w e i n f e r t h a t t h e r e e x i s t s a n i n v a r i a n t p r o j e c t i o n o f norm
from
span {zP-q,zP,zP+q~ onto
span {zP-q,zP~ o r onto
A.
(c
span
Both t h o s e cases r e d u c e t o t h e n a t u r a l p r o j e c t i o n from span{z,l,z} onto
s p a n {!,I}.
I t i s a n e l e m e n t a r y c a l c u l a t i o n t o show
t h a t t h e norm o f t h i s p r o j e c t i o n i s s t r i c t l y g r e a t e r t h a n 1 . Now w e t u r n t o o t h e r c l a s s o f d i s t i n g u i s h e d s u b s p a c e s o f
namely t o s u b s p a c e s i n v a r i a n t f o r m u l t i p l i c a t i o n by specified to
t e d B e u r l i n g theorem, [Koo], 1 V . E . )
reads a s follows: L e t
H (D)
Then t h e r e e x i s t s a n i n n e r f u n c t i o n p r o d u c t and
a
1
XCH I
1
HI (D),
z. The celebra-
(cf.[Dur],Sect.7.3.
( D ) be s u c h t h a t
or
z-XCX.
( i . e . a p r o d u c t of a Blaschke
s i n g u l a r i n n e r f u n c t i o n ) such t h a t
X =I*H, (D).
The f o l l o w i n g t h e o r e m h o l d s THEOREM 6 . 2 .
The m u l t i p l i c a t i o n i n v a r i a n t s u b s p a c e H1(D)
i f and o n l y i f
I
X =I-H,(D)
i s complemented i n
i s a B l a s c h k e p r o d u c t whose z e r o s form a
Carleson sequence. L e t m e r e c a l l t h a t a s e q u e n c e ( z )c D measure g i v i n g e a c h o f t h e measure.
2:s
n
is a C a r l e s o n sequence i f t h e
t h e m a s s ( 1 -lzn
I 2)
i s a Carleson
P. Wojtaszczyk
24
The p r o o f o f t h i s Theorem i s a l m o s t t h e same b u t s i m p l e r t h a n t h e one g i v e n f o r t h e a n a l o g o u s r e s u l t i n t h e d i s c a l g e b r a c a s e i n [ c-P-s
I.
Now w e t u r n t o t h e ( p o s s i b l y open e n d e d ) p r o b l e m o f c l a s s i f y i n g i n f i n i t e d i m e n s i o n a l complemented s u b s p a c e s o f
H1(D).
To do i t
f o r a g i v e n Banach s p a c e i s one o f t h e f a v o r i t e o c c u p a t i o n s i n t h e geometry o f Banach s p a c e s . The case l i t e r a t u r e , t h e r e s u l t s o f [B -R -S ]
p '1
Lp'
has
generated a v a s t
i n d i c a t e t h a t such a c l a s s i f i c a -
t i o n i n t h i s c a s e h a s t o be very complicated or i s i m p o s s i b l e a t
a l l . I n contrast, t h e case
L1
seems t o be much s i m p l e r ; o n l y two
i s o m o r p h i c t y p e s o f complemented s u b s p a c e s o f namely
1,
and
L1.
L1
a r e known,
I t i s a w e l l known open p r o b l e m i f
that is all.
Our aim now i s t o w r i t e down a l l i s o m o r p h i c t y p e s of complemented H 1 ( D ) w e know o f a n d t o i n d i c a t e t h a t t h e y a r e i n d e e d
subspaces of different
.
W e have t h r e e b u i l d i n g b l o c k s ;
( a ) by P a l e y ' s t h e o r e m ( C o r o l 1 a r y 3 . 7 . ) complemented s u b s p a c e o f
l2
i s isomorphic t o a
H1(D)
( b ) l 1 i s i s o m o r p h i c t o a complemented s u b s p a c e of
H1(D)
(this
i s very easy) (c) H , ( D ) HY
h a s an u n c o n d i t i o n a l b a s i s ;
denote
t h e s p a n of t h e f i r s t
f i x a F r a n k l i n b a s i s and l e t n
elements of t h i s b a s i s .
W e w i l l put those blocks together with t h e a i d of t h e following
THEOREM 6.3.
H 1 ( D ) i s isomorphic t o i t s
The p r o o f c a n be found i n
THEOREM 6 . 4 .
(CH,(D))l.
[ W O ~ ] . A l s o t h e isomorphism
provides the a l t e r n a t i v e proof, since f o r easy.
I.,-sum
H
1
H1 (D)-'H,
( d ) t h e theorem i s
The f o l l o w i n g s p a c e s a r e complemented i n
H1(D):
(d)
25
The Banach space H I
The arrow
X
-
Y
means t h a t
X
embeds i n t o
Y
a s a complemen-
t e d s u b s p a c e . All t h o s e t e n s p a c e s a r e p a i r w i s e n o n - i s o m o r p h i c . PROOF.
E v e r y t h i n g e x c e p t t h e l a s t s t a t e m e n t f o l l o w s from p r e v i o u s comments. The p r o o f o f t h e l a s t c l a i m u s e s a l o t o f Banach s p a c e t h e o r y b u t o t h e r w i s e i s q u i t e b o r i n g . N e v e r t h e l e s s l e t u s g i v e some h i n t s . W e s t a r t from t h e t o p .
i s t h e o n l y r e f l e x i v e s p a c e on t h e l i s t
l2
l1
i s t h e o n l y one which
1; s
does n o t c o n t a i n
u n i f o r m l y complemented
l 1+ 1 2
h a s a unique up t o p e r m u t a t i o n u n c o n d i t i o n a l b a s i s
[E-W],which i s c l e a r l y
d i f f e r e n t from a l l o t h e r s p a c e s
cm
(
c 1:)1
n=l
a l s o h a s a u n i q u e up t o p e r m u t a t i o n u n c o n d i t i o n a l eJ
b a s i s ( c f . [B-C-L-T
)
so i s not isomorphic t o (
m
( C H:)l
n=l
does n o t c o n t a i n
n
C HI),
n=l
P. Wojtaszczyk m
n
( C 1 2 )+ 11 2
m
complemented copy o f m
( C
n=l
l;)l
(C12)
does n o t c o n t a i n
n=l
n
(Assume it d o e s , t h e n by [ W o l ]
C H,)l
(
n=l
would c o n t a i n
(
and does n o t c o n t a i n a
C HY)l n= 1
complemented, so b o t h would
be i s o m o r p h i c a n d i t i s n o t s o . ) m
l2
i n t o ( C HY) i s compact, by n= 1 m [E-W] w e o b t a i n t h a t e v e r y u n c o n d i t i o n a l b a s i s i n ( C HY) + l2 n=l
s i n c e e v e r y o p e r a t o r from
m
s p l i t s i n t o a p a r t spanning
l2 m
but the natural basis i n
( C H:)l
n= 1
a n d a p a r t s p a n n i n g ( Z H n1) 1 ' n= 1
+
does n o t have t h i s
(C12)1
property (C12)1
h a s a u n i q u e up t o p e r m u t a t i o n u n c o n d i t i o n a l b a s i s ,
see [B-C-L-TI m
( C HY)
+
1 < p (2,
but
n= 1
RGMARK
(El2)
does n o t c o n t a i n a subspace isomorphic t o
H, ( 0 )
1 P'
h a s such subspaces.
.
The i s o m o r p h i c t y p e of t h e s p a c e ( C H?) n= 1
d o e s n o t depend o n a
choice of b a s i s . This i s a s t a n d a r d decomposition argument. The v e r y n a t u r a l q u e s t i o n r a i s e d by t h e above d i s c u s s i o n i s t o c o n s t r u c t more complemented s u b s p a c e s of
HI (D)
.
I n view o f t h e mul-
t i t u d e of a p p a r e n t l y d i f f e r e n t i s o m o r p h i c r e p r e s e n t a t i o n s f o r
H1(D)
(see S e c t i o n 5 ) w e b e l i e v e t h a t such a c o n s t r u c t i o n i s p o s s i b l e . I t i s s t i l l an open problem r a i s e d i n [ C a s l i f
u s r e c a l l t h a t a Banach s p a c e tion X.
X =X 1 + X 2
X
H1(D)
is primary. L e t
i s p r i m a r y i f f o r e v e r y decomposi-
w e h a v e a t l e a s t o n e of
X1
or
X2
isomorphic t o
I t i s o u r b e l i e f t h a t t h e answer t o t h i s p r o b l e m i s p o s i t i v e .
The following r e s u l t s e e m s to j u s t i f y t h i s b e l i e f .
27
The Banach space H I
THEOREM 6 . 5 .
Let
b e a s u b s p a c e of
X
Than
H1(D). H
1
HI(D).
Assume t h a t
c o n t a i n s a s m a l l e r subspace
X
i s isomorphic t o
X
complemented i n
Y
.
H I (D)
( D ) and isomorphic t o
The p r o o f o f t h i s t h e o r e m i s a v e r b a t i m r e p e t i t i o n of t h e p r o o f of t h e analogous statement f o r
L
1
P'
g i v e n i n [J-M-S-T],pp.265-70.
Now w e w i l l d i s c u s s b r i e f l y t h e s p a c e of p o l y n o m i a l s . To be p r e c i s e ,
let
Pn
denote
spanil , z , .
the
.., z n }
in
H I (D). I t i s w e l l known
t h a t i n i t s n a t u r a l p o s i t i o n a s a s u b s p a c e of
H1(D)
the space
Pn
i s b a d l y complemented, more p r e c i s e l y t h e norm of t h e b e s t p r o j e c t i o n is of o r d e r
l o g ( n + l ) . T h i s r e s u l t however d e p e n d s o n t h e
p a r t i c u l a r p o s i t i o n of
Pn
in
It is a natural question
H1(D).
( c f . [Wo41 ) i f t h e P ' s a r e i s o m o r p h i c t o u n i f o r m l y complemented
n
subspaces of
H 1 ( D ) . A p o s i t i v e answer t o t h i s was g i v e n by
J.Bourgain and A.Pelczynski.
w e l l known
and
T h e i r argument r u n s a s f o l l o w s : A s i s
H 1 ( D ) is i s o m o r p h i c t o
qn:H1 ( D )
+
H I (D)
-
Pn
m
I I i n I\ 5 2
and
A c t u a l l y S.V.Bo&ariov
1 I qnl I
51
[Bo] has
i s uniformly isomorphic t o
Now w e d e f i n e
by
k m k n n-k q n ( C a k z , C b z ) = C - akzk k=O k=O k=O Since
H1(D)+ H 1 ( D ) .
Hn"
n
+ c
k=O
and
n-k
bkzn-k'
q n o in= i d
t h e c l a i m follows.
shown more. H e h a s p r o v e d t h a t
.
Pn
T h i s i s i n c o m p a r a b l y more c o m p l i c a -
ted. SECTION 7 ; C o n c l u d i n g Remarks. I n t h e previous sections t h e p a t i e n t reader has
f o u n d many o p e n
problems connected t o t h e r e s u l t s w e have been d i s c u s s i n g . I n t h i s s e c t i o n w e want t o p r e s e n t some d i r e c t i o n s f o r p o s s i b l e f u r t h e r
P. Wojtaszczyk
28
research. 1 ) M u l t i p l i e r s . I t i s a w e l l known open problem t o g i v e good m
c r i t e r i a f o r a s e q u e n c e ( p ( n ) )n=O
into
H I (D)
i s a f i n e example. Our knowledge
The Theorem 3.1.
H1(D).
t o b e a m u l t i p l i e r from
however i s s t i l l q u i t e s m a l l a s i s w i t n e s s e d by t h e p r o b l e m o f d e s c r i p t i o n of 0 , l v a l u e d m u l t i p l i e r s ( i . e . i n v a r i a n t p r o j e c t i o n s ) . There i s a v a s t l i t e r a t u r e o n m u l t i p l i e r s b o t h from i t s e l f and from
H1(D)
H (D) into 1 i n t o o t h e r n a t u r a l s p a c e s . W e do n o t want
t o d i s c u s s it h e r e . L e t u s m e n t i o n o n l y t h a t m u l t i p l i e r s from H1(D)
into
and from
H2(D)
H1(D)
into
l1
a r e f u l l y described
(cf.[Dur],Th.6.4 a n d [ S l - S t ] , r e s p ) . However, as f a r a s w e k n o w , a l l t h e e x i s t i n g t h e o r e m s d e a l w i t h b o u n d e d n e s s o r c o m p a c t n e s s of m u l t i p l i e r s . I t i s o u r b e l i e f t h a t i t i s w o r t h w h i l e t o i n v e s t i g a t e what m u l t i p -
l i e r s b e l o n g t o o t h e r n a t u r a l and i m p o r t a n t o p e r a t o r i d e a l s . The theory of operator i d e a l s ( c f . [ P i e ] ) i s a very important t o o l i n
its applications t o the d i s c algebra a r e d i s c u s s e d i n [ P e l ] . I n t h i s s i t u a t i o n w e b e l i e v e t h a t some
v a r i o u s p a r t s of a n a l y s i s ,
t h e o r e m s o f t h i s t y p e may t u r n o u t t o b e v e r y u s e f u l . 2 ) F i n i t e d i m e n s i o n a l s t r u c t u r e . The problem b a s i c a l l y i s t o d e v e l o p
t h e l o c a l theory of
H1(D).
I n p a r t i c u l a r to investigate various
p a r a m e t e r s a s s o c i a t e d w i t h i t s main f i n i t e d i m e n s i o n a l b u i l d i n g b l o c k s ,
H?. The p r e s e n t a t i o n o f some p a r t s of l o c a l t h e o r y
i.e. the spaces
o f Banach s p a c e ( i n t h e c o n t e x t o f c l a s s i c a l s p a c e s ) c a n b e found i n [ P e l l ] . The o n l y p a p e r which c o n t a i n s some r e s u l t s of t h i s H1(D) i s [G-R].
n a t u r e connected w i t h
3 ) Convergent T a y l o r s e r i e s . I t i s w e l l known t h a t t h e T a y l o r s e r i e s
n C anz n=O
of an
H1(D)
function
f
need n o t c o n v e r g e t o
f
in
norm. W e f e e l t h a t t h e s p a c e of c o n v e r g e n t T a y l o r s e r i e s , i . e . s p a c e c o n s i s t i n g of a l l
f =
m
C an?€ n=O
H1(D)
the
such t h a t
n=o
N
may be a n i n t e r e s t i n g o b j e c t t o s t u d y . I do n o t know o f a s i n g l e work d e a l i n g w i t h t h i s s p a c e . The a n a l o g o u s s p a c e f o r
p
=m,
i.e.
t h e s p a c e o f u n i f o r m l y c o n v e r g e n t T a y l o r series r e c e i v e d much a t t e n t i o n i n r e c e n t y e a r s . The b a s i c r e f e r e n c e f o r t h i s i s [ V i n ] . W e f e e l however t h a t t h e r e i s a v e r y s m a l l c o n n e c t i o n between p =1
and
p
=m
i n t h i s case.
29
The Banach space H I
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33