Holomorphic Q Classes

po(z, z2)

= inf

{1

~~:112 :

1

7 is an arc in D from z 1 to

z,}.

We note that a simple calculation shows pn(O, z) = log(1 + lzl)/(1-lzl), and so (1- lzl)- 1

:::;

exp(pn(O, z)) :::; 2(1 -lzl)- 1 .

(2.12)

This distance is invariant under Aut(D), and therefore transfers to a natural conformally invariant metric on any simply connected proper subset G of C. If f : D --+ G is any conformal map, the hyperbolic distance on G is given by pe(wt, w2) = pn(zt, z2), where Wj = f(zj) for j = 1, 2. Furthermore, the pe(·, ·) can also be computed by integrating a density function he over arcs in G. A change of variable argument shows that he(f(z)) = ((1-lzl 2)lf'(z)l)- 1 • A useful geometric estimate for he that follows from the Koebe distortion theorem is that (2.13) :::; he(w) :::;

8e~w)

8e~w)'

where be denotes the Euclidean distance of point w E G to C \ G.

Theorem 2.3.1. Let a,p E (0, oo) and let 4>: D--+ D be univalent. Then (i) C

1

sup wE
exp (- PP
< oo.

(ii) C

E QP and

.

hm sup

r~1 wE
1


exp ( __; PP
(iii) C

(O),w))
(iv) C

E Ba and 8(O),w))1.ar(w)=0. r~1 wE
2.3 Geometric Characterizations

21

Proof. Let G = ¢(D). Since ¢ : D --+ D is conformal, a E D is equivalent to w =¢(a) E G. If 'lj; stands for the inverse map of¢, then by (2.12)

1-laa('l/J(z))l 2 ~ exp(-po(O,aa('l/J(z))) = exp(-pa(cl>(a),z)), and hence { l4>'(u)l2(1 -laa(u)I2)P dm(u) = { (1 -laa('l/J(z))I2)P dm(z) (1- l¢(u)l 2)2a Jo (1- lzl 2)2a

Jo

,..., { exp (- PP¢(D)(w, z)) d ( ) ( 1 - I Z 12 ) 2a m z .

"' JG

This, together with Theorem 2.2.1 (i) and (ii), leads to Theorem 3.1.1 (i) and (ii). Because of (2.12) and (2.13), the above argument, now using Theorem 2.2.1 (iii) and (iv), yields the rest two results in Theorem 2.3.1. As an application of these theorems, we present an example of a composition operator that distinguishes QP from Ba. Example 2.3.1. Let a E (0, 1) and p E (0, oo). Then there exists a univalent self-map¢ of D such that Cq, : Ba --+ QP is bounded but llci>IIHoo = 1. Proof. Set Go = D(O, 1/2), the disk with radius 1/2 and center 0. For k E N, choose ak E D such that po(O, ak) ~ k and the disks Gk = O"ak (Go) have disjoint projections onto T. We note for use below that the diameter of Gk is comparable to 1-lakl· Let Sk be the convex hull of Gk u {0} and define G = G 0 u U~ 1 Sk. Then G is a simply connected subdomain of D, and we can take ¢ to be the Riemann map from D onto G with ¢(0) = 0 and ¢' (0) > 0. We demonstrate that ¢is the required map. Note that c5o(ak) --+ 0, since c5o(ak) ::; 1 -lakl and po(O, ak) --+ oo. Accordingly, llct>IIH = 1. Next, for a E (0, 1) and p E (0, oo) fixed, we use Theorem 2.3.1 to show that Cq, : Ba --+ QP is bounded. First, let w E Go. Since Gk C G, the comparison principle for the hyperbolic metric yields that pa ::; pak. Then, for z E G, exp (- ppa(w,z)) ~ (1-lwi)P, by (2.12), since po(O,w)::; log3. Hence 00

Since 1-lakl::; 2exp(-po(O,ak))::; 2exp(-k), it follows that { exp( -ppa(w,z)) ~ . (1 -lzl2)2a dm(z) ~ exp(-J(p- 2a + 2)) ~ 1

Jo

f;:o

22

2. Composite Embedding

for any w E Go. Next suppose w E Sk. Then, as above,

1

Uj-f.kSj

exp (- PPc(w,z)) ~ . ( 1 -I l2 ) 2 a dm(z) ~ L.J exp( -J(p- 2a + 2)) ~ 1, Z J:ftk

and { exp (- PPc(w, z)) dm(z) (1 -lzl 2 ) 2a

Jsk

-< flak! 1 -lakl dr-< 1, - Jo (1- r) 2a -

since a E (0, 1] and exp (- ppc(w, z)) ~ 1. Putting the previous estimates altogether, we get that sup

wEG

exp (- ppc(w,z)) ( 1 I !2 ) 2 dm(z) - Z a

1 G

< oo,

and so that Ccf>: sa--+ QP is a bounded operator, by Theorem 3.1.1.

Notes 2.1 Theorem 2.1.1 can be reformulated as: there is a holomorphic map F : D --7 C 2 such that (1-lzl 2 )aiF'(z)l ~ 1 for all zED (for more information, see also Gauthier-Xiao [69]). In case of the Bloch space (viz. a= 1), this theorem is due to Ramey and Ullrich [103]. In [143], Zhao gave some characterizations of sa after the fashion of QP. 2.2 See the books of Cowen and MacCluer [43] and Shapiro [110], as well as the conference proceeding edited by Jafari et. al [81] for the discussions of composition operators on the classical spaces of holomorphic functions. Theorem 2.2.1 generalizes some results of Smith-Zhao [115], Xiao [137], Madigan [88] and Madigan-Matheson [89]. Very recently, Lindstrom, Makhmutov and Taskinen [87] proved essentially that in case of a = 1, the left-hand sides of (2.5) and (2.6) are comparable with the norm and the essential norm of Ccf> : S --7 QP respectively, and their result improves Montes-Rodriguez's [93]. Meantime, it is worthwhile to remark that (2. 7) and (2.8) are independent of p E (0, oo), and thus they are the conditions for Ccf>: S--+ sa to be bounded and compact respectively (cf. [139]). It would be interesting to characterize the composition operators Ccf> sending the spaces QP to themselves in terms of the function theoretic properties of¢. In the connection with this topic, we would like to mention: Aulaskari-Zhao [28], Bourdon-Cima-Matheson [34], Danikas-Ruscheweyh-Siskakis [47], Lou [86], Smith [114], Tjani[124], Wirths-Xiao [129], Xiao [137] and Zorboska[145]. 2.3 For those facts related to the hyperbolic metric, see, for example, Ahlfors [4, Chapter 1] and Shapiro [110, p.157]. The analogs of both Theorem 2.3.1 (i) and Example 2.3.1 for a = 1 was established earlier in Smith-Zhao [115].

3. Series Expansions

We saw in the previous two chapters that the series with Hadamard gaps play an important role in studying QP. Accordingly, it is necessary to investigate the properties of the coefficients of the usual or random power series living on QP in detail.

3.1 Power Series Recall that if f(z) = En=O anzn and p E [0, oo), then 00

llflli>P ~ L

n

1

-Pianl 2.

(3.1)

n=l

However, regarding Qp, we have the following result which is quite complicated. Theorem 3.1.1. Let p E (O,oo) and f E 1l with f(z) = E~=oanzn. Then f E Qp if and only if

·

~ (1- lwi 2 )P I~ (m + l)am+lr(n- m + p) w-n-m1 sup L...J L...J

wED n=O

(n

+ 1)P+l

m=O

(n- m)!

2

< oo.

Proof. Under the assumption above, we have 00

f'(z)

00

r(n +

)

-,---..;~.....:._.,.- = L(n + 1)an lZn""" p wnzn (1- wz)P n=O + ~ n!F(p)

=·f (t n=O

m=O

(m+1)am+lr(n-m+p)wn-m) zn (n- m)!F(p)

This, together with Parseval's formula, implies Fp(f, w) = [ (1- lwi 2)P }0 =

[ }0

lf'(z)l2 (1 -lzi2)P dm(z) 11- wzi 2P

~~ (~ (m + 1)am+lr(n- m + p)) znl ~

~

(n- m)!F(p)wm-n

2

dm(z) (1- lzi 2)-P

24

3. Series Expansions

= 27r1

1

= 7r

f;:a

f= It 0

n=O k=O =

2 2 (m + 1)am+tF(n- m + p) 1 r ndr ~ (n- m)!F(p)wn-m (1- r2)-P

~ ~

(m + 1)ak+lr(n- m + p) wn-ml211 (1- r)Prndr (n- m)!F(p) o

~ B( n + 1,p + 1) I~ L....t

7r L..J n=O

(m + 1)am+tF(n- m + p) -n-ml2 (n- m)!F( ) w m=O p

Note that the Beta function B ( ·, ·) ensures B(

1 1) = F(n+ 1)F(p+ 1) ~ F(p+ 1) n+ ,p+ F(n+p+2) (n+1)P+l'

which comes from a simple application of Stirling's formula. So, the equivalence stated in Theorem 3.1.1 follows from Theorem 1.1.1 and the preceding calculation.

Corollary 3.1.1. Let p E (0, oo) and f E 1l with f(z) = L::~=O anzn. (i) The condition

oo (1- r 2)P (Ln (m + 1)lam+ll r n-m) L o~r
2 (3.2)


implies that f E QP. (ii) If an~ 0 for n EN U {0} and f E QP, then {3.2) holds. (iii) lfg(z) = L::~=obnzn andbn t !ani forn E NU{O} then II911Qv t

11/IIQv·

Proof. The argument for Theorem 3.1.1 implies actually that for wED, oo n (m + 1)am+tF(n- m + p) lwln-m 12 F: (f w)-<'"" (1-lwl 2 )P '"" P ' - L..J (n + 1)P+l L..J (n- m)! ' n=O m=O

which yields (i), due to Theorem 1.1.1. Conversely, if an ~ 0 then it is easy to get (3.2) from f E QP. Now, (iii) is a consequence of (i) and (ii). Remark that the above assertion (iii) is particularly useful when one wishes to determine the Qp-membership of a power series whose Taylor coefficients decay rapidly at the infinity. For example, any f E 1l with f(z) = E~=O anzn; !ani j n- 1 , must belong to QP, p > 0, in that so does log(l- z).

3.2 Partial Sums The partial sums of power series can be selected to characterize every individual function in QP. Let f(z) = E~o akzk be in 11.. For n EN U {0}, define

3.2 Partial Sums

25

n

sn(f)(z) =

L akzk k=O

and

O"n(f)(z) =

:t

:t (1-

Bk(f)(z) = k=O n + 1 k=O

It is known that

O"n(f)(z) =

_k_)akzk. n +1

2~ l Kn(()f(z()jd(j,

where

is the Fejer kernel with the identical property: 1=

2~ l Kn(()jd(j.

Theorem 3.2.1. Let p E (0, oo) and let f E 1i. Then f E Qp if and only if

sup{llun(f)IIQP : n EN U {0}} < oo. Proof. If f

E QP, then

(un(f))'(z) =

2~ l

Kn(()f'(z()(jd(j.

Putting fc;(z) = f((z) and using Minkowski's inequality, we see that for any wED, Fp(un(/),w):,;

2~ l

Fp(f<,w)Kn(()id(l:,;

11/IIQP·

Applying Theorem 1.1.1, we find that un(f) E QP with llun(f)IIQP ~ llfllgP. Conversely, let supnENU{O} llun(f)IIQP be finite. Noticing 00

fr(z) = f(rz) = (1- r) 2 L(n + 1)un(f)(z)rn, r E (0, 1), n=O we have that for w E D, 00

Fp(fr, w) ~ (1- r) 2 I:
3. Series Expansions

26

3.3 Nonnegative Coefficients Although the quintessential example -log(1- z) does not distinguish each QP, its special expansion 2:~= 1 n -l zn suggests us to consider the question of characterizing those f E QP for which f(z) = E~=O anzn with an 2::: 0. with f(z) = E~=O anzn and an 2::: 0 for n E

f E 1l

Theorem 3.3.1. Let Nu {O}.

(i) If p E (0, 1] then f E QP if and only if 2

L L(n + 1)1-p m=O 00

(min(n,k)

sup k-P n=O kEN

a2n-m+1 (m + 1)1-p

)

< oo.

(3.3)

(ii) lfp E (1,oo) then f E QP if and only ifsupnENn- 1 2:~= 1 kak < oo. Proof. (i) From the proof of Theorem 3.1.1 it follows that for wED, f'(z)

(1- Wz)P

= ~(n+ l)c,.+Iz oo

n

(~c,.z oo

=

n

'

) ,

where _ ~ (m + 1)am+1T(n- m + p) -n-m (n + 1) Cn+1 - f;:o . w (n- m)!T(p) Now invoking (3.1), we get

Fp(f, w) ,...., (1 - lwi2)P ,....,

-<

~(

f;:o n +

1)1-pl

Cn+1

12

~(n + 1)1-p (~

-L-

n=O

(m + 1)am+dwln-m ) 2 L-(n+1)(n-m+1) 1-P m=O

Let k be the positive integer satisfying: (k+ 1)- 1 < 1-lwl ~ k- 1. Using the above inequalities, we see that it suffices to verify sup k-P Ik < oo, kEN

(3.4)

~(n+ 1)1-p (~ (m+ 1)am+1(1-: (k+ 1)-l)n-m)2. (n + 1)(n- m + 1)1-p ~ f;:o

(3.5)

where, for each k E N, one sets

Ik =

We assume now that the sequence of coefficients an off obeys (3.3). Then

3.3 Nonnegative Coefficients

f(n + 1)1-p (t (m+ a2n-)il 1

n=k

-p

m=O

and hence

2k

2n

L (L

laml)

27

)2-< kP, kEN, -

2 ::5 k, kEN.

n=k m=n This estimation gives

2n Lam ::51, n EN, m=n

and consequently,

n

L

miami ::5 n, n E N. m=1 Using (3.6) we can simplify (3.4) by observing that for k E N,

~(n+ 1)1-p (

L....J n=O

1 " ' (m+ 1)am+1(1- (k+ 1)- )n-m) L....J (n + 1)(n- m + 1)1-P O$m$n/2

(3.6)

2

-< kP -

due to the binomial theorem. It now follows that we need only to prove that (3.3) implies

~(n+ 1 )1-p ( L....J

n=O

1 " ' an-m+1(1-k- )m)" -:!,k" L....J (m + 1) 1 -P O$m$n/2

or, more simply, Jk ::5 kP fork EN, where Jk =

~(n+ 1)1-p (~ a2n-m+1(1-1 k-l)m)2.

L....J n=O

L....J m=O

(m + 1)

(3.7)

-P

As our last reduction we first notice that for 0 :::; m :::; k we have (1- k- 1 ) m ~ 1. Fixing a large integer N, then splitting the sum in (3. 7) into two parts and applying (3.3), we obtain

oo

Jk < "'(n + 1)1-p 2 - L....J n=O

+

~

L....J n=kN

(min(n,kN) ) "' a2n-m+1 L....J (m + 1)1-p m=O

(n + l)1-p (

::$ (kN)P

+

~

2

a2n-m+1(1- k-l)m) L....J (m + 1) 1 -P m=kN l - k-1 )2kN JkN. ( 1 - (kN)-1

2

28

3. Series Expansions

Thus

Jk -< p sup -k _ N kEN p

JkN

+ kEN sup (kN) p

provided N is sufficiently large. This yields supkEN Jkk-P ::::5 NP provided f is a polynomial. A limit argument concludes the proof of sufficiency part of (i). The proof of necessity part of (i) is easy. By reversing the first step in the above proof, we obtain (1-lwi2)P

~(n + 1)1-p (~ (m + 1)am+11wln-m ) 2-< 11/112 .L...J .L...J (n + 1)(n- m + 1)1-p -

n=O

Qp

m=O

for all w E D. Now (3.3) is established by replacing 1-lwl with k- 1 and lwln-m with 1 provided n- m ~ k. The remaining terms can be ignored. (ii) In case of p E (1, oo), one has QP =B. So, iff E Qp, then for j EN, II/IlB 2::

2 sup (1 -lzl )1/'(z)l z=1-j- 1 j

> _ J·-1""" .L...J nan ( 1 - J·-1)n-1 n=1 j

> _ J·-1 (1 - J·-1)j-1""" .L...J nan n=1 j

>_ J·-1""" .L...J nan, n=1 . fi m"t e. and hence supjEN J..:. . 1 ""'j Lm= 1 nan IS On the other hand, if supjEN j- 1 2:~= 1 nan < oo, then

2k+l

L

an ::::51,

kEN,

n=2k

and hence for z E D, 00

lf'(z)l

2k+l_1.

=I L L

nanzn-!1 k=O n=2k 00 2k+ 1 -1 ~ 2k+1 anlzl2k-1

L

L

00

::::5

L 2klz12k-1

k=O ::::5 (1 - lzl)-1'

which implies

f

E Qp. The proof is finished.

3.3 Nonnegative Coefficients

29

As a direct consequence of Theorem 3.3.1, the following conclusion supplies us with a surprising reason why log(1- z) lies in each Qw Corollary 3.3.1. Let p E {0, oo) and let f(z) = E::o anzn with an nonnegative and nonincreasing. Then f E QP if and only if supnEN nan < oo. Proof. For convenience, let C = supnEN nan. Case 1: p E {0, 1]. Suppose that f E Qp and 2 00

S(k) = """'(n + 1)1-p ~ n=O

(min(n,k)

"""' ~ m=O

)

a2n-m+1

{m+1) 1 -P

'

for k E N. Using the assumption that {an} is a nonnegative and nonincreasing sequence, we obtain 2 k

S(k) ~ """'(n + 1)1-p

(min(n,k}

"""' ~

f='o

a2n-m+1

{m + 1) 1 -P 2

>~(n+ 1 )1-p(~ a2n+1 ~ (m + 1)

- ~ n=O

)

1 -P

m=O

)

k

~ a~k+1 l:(n + 1)1+p n=O kP+2a~k+1'

t

and soC< oo, by Theorem 3.3.1 (i). Conversely, under the condition that C is finite, we dominate the upper bound of S(k) as follows:

oo

k

S(k) =

(

I:+ I: n=O

)

(n + 1)

m=O an+1

~ (m +

- ~

n=O

m=O (n + 1)1-p

+ """' 00

n=k+1

(

~

-

n=O

c2 +

n~ 1 00

1) 1 -P

~

m=O (n + 1)1-p ( (2n- k + 1) 2

1

(

)

)2

k """' a2n-k+1 ~ (m+1) 1 -P m=O

k n -< c2 """'(n + 1)-1-p ( """'

::-~i~P

I:

-p

n=k+1

k ( n <"""'(n+ 1)1-p """'

~

1

2 (min(n,k)

)2

(m+ 1) 1 -P

)2 2

k

~

1 ) (m + 1)1-P

)

30

3. Series Expansions

(n

~ c2 (~(n + 1)p-1 + ~

+ 1)1-p ) Ll (2n - k + 1) 2 n=k+1

Ll n=O

-< c2 (kP +

-

~

~

1

n~ 1 (n + 1)1+P

)

C2kP.

This gives that k-PS(k) ~ C 2 and so that f E QP, by Theorem 3.3.1 (i). Case 2: p E (1, oo). At this point, we have Qp = B. Note that under the hypothesis of Corollary 3.3.1, one has n

nan ~ - - :::; n- 1 L; kak :::; C, 2

nEN.

k=1

Thus the desired result follows from Theorem 3.3.1 (ii). Another direct consequence of Theorem 3.3.1 is the construction of some special functions suggesting that Qp is a large subspace of Vp. Example 3.3.1. Let p E (0, 1). Then there exists

f

Proof. For aj = (2i( 1 -P)j 2 <1+.B))- 112 , j EN, and j

00

J,a(z)

=L

L

aj

j=1

Step 1: we first show

f.a

=

\ Qp

bnzn.

n2:1

Since

j

00

L lail L (m + 2i/-P ~ Lj-< +

1 2

2

m=O

j=1

f.a

n ( np
f3 E (O,p/2), let

=L

23

m=O

E Vp.

00

llf.alli,p

zm+

E Vp

.8)

< oo.

j=1

Step 2: we next show f.a E np
L

23+1

L

bm:::;

m=n

bm ~

n-( 1 -p)/ 2 ,

m=23-l

and hence

lf~(z)l

00

=

2k+ 1 -1

IL L

nbnzn-

1

1

k=O n=2k 2k+ 1 -1

00

: :; L k=O

2k+1

L

n=2k

bnlzl2k-1

3.4 Random Series

31

L 2(1+p)k/21zl2k-1 00

j

k=O 00

j=O

which implies ff3 E B(l+p)/ 2 • Step 3: we finally show ff3 to show sup

kEN

tt

~

k -p

~

QP

as f3

n 1-p (

n=22k

< pl2. By Theorem 3.3.1 (i) it suffices

~

2

b2n-m+1

~

m=O

)

(m + 1)1-p

(3.8)

=oo.

Let j 2:: 2k and observe that the interval 2j + j I 4 ~ n ~ 2j + j 12 contains [j I 4] (the integer part of il4) integers each of which satisfies 2j+ 1 ~ 2n- m + 1 ~ 2j+ 1 + j + 1 and hence

Consequently

_!_

~

kP ~

n1-p

n=22k

(~ ~

m=O

2 b2n-m+1

(m+ 1)1-p

)

=

2

kP ~

~

1 n1-p

j =2k n=2i

>provided {3

~ J~

_!_

oo

~ ~

kP

'1+2(3

~ kp- 2 (3

(~ ~

m=O

2 b2n-m+1

(m+ 1) 1-P

)

-+ 00

j=2k J

< pl2. Thus (3.8) follows and the proof is complete.

3.4 Random Series Now let cn(w) be a Bernoulli sequence of random variables on a probability space. In other words, the random variables are independent and each En takes the values 1 and -1 with equal probability 112. If f(z) = E~=O anzn is in tl, then we let fw(z) = L:~=O Enanzn, and call fw the random series of f. Moreover, a.s. means "almost surely"; that is, "for almost every choice of signs" .

32

3. Series Expansions

Theorem 3.4.1. Let p E (0, 1) and f E 1l with f(z) = L::=o anzn. Iff E QP then L::=o n 1 -Pianl 2 < oo. Conversely, if L::=o n 1 -Pianl 2 < oo then fw E QP a.s .. Proof. The first part of the theorem is trivial since QP ~ Vp. However, the second part is surprising. When 2::= 1 n 1-Pianl 2 is finite, we have f E Vp. By Theorem 2 in (42] fw is a pointwise multiplier of Vp, that is, fwg E Vp for any g E VP. Upon taking

1-lwi2)P/2

g(z) = ( 1 -wz -

'

wED,

and applying Theorem 1.1(c) in [117], we get that and lg'(z)l 2 (1-lzi 2 )Pif~(z)l 2 dm(z)

L

IIYIIvp

~

1 (cf. Lemma 1.4.1)

~ I!Yiltp ~ 1,

which certainly implies fw E QP. We are done. Next we show that sense.

L::=o n 1-P!an! 2 < oo

is best possible in a very strong

Theorem 3.4.2. Let p E (0, 1) and f E 1l with f(z) = L::=o anzn. Given a sequence {Cn}, Cn ~ 0, one can choose coefficients an > 0 such that L::=o n 1-Pa;cn < oo but fw ¢. QP for any choice of w. Proof. Let {cn} be a sequence of positive constants decreasing monotonically to 0. Choose integers nk which satisfy:

(i) no= 1, (ii) nk > 2nk-b kEN, 00

(iii)

I: c~

2

<

oo.

k=O

oo

oo

nk-1

L n -Pa!en = L L 1

n=1

k=1

n=nk-1

oo

nk-1 ~

~ < _ ~

00

~

n 1-Pa!cn

-1 -1/2 nk cnk-1 Cnk-1

::; L c~~l < oo. k=l

Notes

33

On the other hand,

thus fw

~ VP

and hence fw

~ QP

for any choice of w. The theorem is proved.

Notes 3.1 Theorem 3.1.1 and its proof (due to Aulaskari, Girela and Wulan [18]), are motivated by Proposition of Aulaskari-Xiao-Zhao [27]. Meanwhile, Corollary 1.1.1 (i) can be extended to the Hadamard product. More precisely, for f(z) = E~=O anzn and g( z) = E~=O bnzn, we define the Hadamard product of f and g as f * g(z) = E~=O anbnzn (see, for example, Anderson-Clunie-Pommerenke [8]). If p, q E [0, oo) and f E 1l, then by (3.1) it follows that f E QP if and only if sup ll(f o aw- f(w)) * kp,qliVq < oo, wED

where

00

kp,q(z)

= L n(q-p)f2zn. n=O

See also Aulaskari-Girela-Wulan [17]. Moreover, Corollary 3.1.1 has been used by Aulaskari-Girela-Wulan [18] to prove that if the Taylor coefficients an of f E QP are nonnegative, and if g E B, then f * g E Qw Concerning the other types of algebraic properties of QP, we refer the reader to Aulaskari-Danikas-Zhao [16]. 3.2 For the versions of Band BMOA of Theorem 3.2.1, see Holland-Walsh [77]. 3.3 Theorem 3.3.1 (i) is taken from Aulaskari-Stegenga-Xiao [24]. For BMOA, this is a well-known unpublished result ofFefferman, see for example [113]. Theorem 3.3.1 (ii) is from Girela [70]. For more general topics on the Taylor coefficients of functions in B, see Anderson-Clunie-Pommerenke [8] and Bennett-StegengaTimoney [32). Corollary 3.3.1 has its root in [33) where Bergh dealt with those BMO and B functions with nonnegative Taylor coefficients in an approach based on the Fourier series. Of course, Corollary 3.3.1 corresponds nicely to Corollary 1.3.1 which displays a close relation among B, BMOA and QP, p E (0, 1) through the value distribution of holomorphic functions under consideration. Example 3.3.1 should be compared with a constructive example of HoilandTwomey [75). For the reader's convenience, we give the construction of the example in the sequel. Given kEN, let m(k) be the integer part of 2V!G- 2v'k-1, put Fk = {2k + j : j = 0, 1, · · ·, m(k)}, where Fo = {0}, and set

Ek = {2k + j : j E Uo::;z::;k-lFz},

34

3. Series Expansions

where Eo = {1}. Define an = 2-v'k resp. an Ur?:.oEj. If f is the function determined by 00

f(z)

= :2: anzn = :2: 2-v'k

= 0 whenever n

:2: zn,

E

Ek resp. n ~

zED,

n=O

then a further application of Theorem 3.3.1 (cf. [70, Theorem 9.13) for details) gives that

f

E

Bn ( no
where Hq stands for the Hardy space on T (see also Section 4.2 of next chapter for the definition of Hq). 3.4 Theorems 3.4.1 and 3.4.2 are from Aulaskari-Stegenga-Zhao [25). According to Theorems 3.4.1 and 1.2.1, random series are similar to lacunary series in case of QP, p E [0, 1): they are very well behaved if the coefficients are weightedly square-summable and very badly behaved if not. However, for p = 1 Theorem 3.4.1 fails, as shown by Sledd and Stegenga [113). In case of B, it is easy to figure out that Theorem 3.4.1 is false, see Duren [51) and Sledd [112]. The proof of Theorem 3.4.2 is analogous to that of Theorem 3(b) of Cohran-Shapiro-Ullrich [42).

4. Modified Carleson Measures

In this chapter, we show that QP can be equivalently characterized by means of a modified Carleson measure. In the subsequent three sections, this geometric characterization is used to compare QP with the class of mean Lipschitz functions as well as the Besov space (as one of representatives of the conformally invariant classes of holomorphic functions), and to discuss the mean growth of the derivatives of functions in QP.

4.1 An Integral Form For p E (0, oo) we say that a complex Borel measure J.L given on D is a p-Carleson measure provided

IIJ.LIIc

= sup

I~T

p

IJ.LI(S(I)) < oo, IIIP

where the supremum is taken over all arcs I ~ T. Here and elsewhere in the forthcoming chapters, III stands for the arclength of I, and that S(I) means the Carleson box based on I:

S(I) = {zED : 1 - III ::; lzl < 1,

z

l;j

E I}.

Note that 0 E S(I) if and only if III 2:: 1. So we will always take III < 1 for granted (unless a special remark is made). When p = 1, we get the standard definition of the original Carleson measure. As in [66, p. 239], any p-Carleson measure has an integral representation. Lemma 4.1.1. Let p E (0, oo) and let J..L be a complex Borel measure on D. Then J.L is a p- Carleson measure if and only if

IIIJ.LIIIcp =

sup

f (

wEDjD

~w\ 2 )p diJ.LI(z) < oo.

:1 - WZ

Proof. Suppose that IIIJ.LIIIcP < oo. Then, for the Carleson box S(I) = {z E D : 1- h::; lzl < 1, 10- arg zl ::; h} with h = III, we take w = (1- h)ei(O+h/ 2 ), and so have

36

4. Modified Carleson Measures

~

IIIJ-tlllc P

inf ( 1- ~wl

11 - wzl 2

zES(I)

)P IJ-ti(S(J)) t

IJ-ti(S(J))

IIIP

'

which implies IIJ-tllcp ::5 IIIJ-tlllcp < oo. Conversely, assume that J-t is a p-Carleson measure, that is, IIJ-tllcP If wE D(O, 3/4), then

< oo.

Lc:.: -~::2 r

dll'l(z) :; !I'I(D):; lll'llc,.

If wED\ D(O, 3/4), then we put En= {zED: lz- w/lwll < 2n(1-lwl)} and hence get IJ-ti(En) ::5 IIJ-tllcP2nP(1 -lwi)P for n EN. We also have

1 - lwl2

11 - wzl 2 and so for n ~ 1 and Eo=

1-lwl2

1_

_J

__

-

1-

'

ZE

lwl'

E t,

0, _J

I1 - wz 12 -'

1 22n ( 1 - lw I),

Z

E En\ En-l·

Consequently,

00

::5 IIJ-tllcP

L 2-np, n=l

that is to say, IIIJ-tlllcp

< oo.

This lemma provides us a geometric approach to study QP.

Theorem 4.1.1. Let p E ( 0, oo) and f E 1-l with dJ-tJ,p(z) = lf'(z)l 2 (1-lzi 2 )Pdm(z), zED.

Then f E Qp if and only if J-tJ,p is a p-Carleson measure. Proof. This is a direct by-product of Lemma 4.1.1 and Theorem 1.1.1.

4.2 Relating to Mean Lipschitz Spaces For p E (0, oo], the Hardy space HP consists of those functions

f

E 1-l for which

4.2 Relating to Mean Lipschitz Spaces

llfiiHP

=

sup Mp(j, r)

O
37

< oo,

where

p E (0, oo); and

Moo(!, r) =max lf(r()l, (ET

p = oo.

When p E (1, oo] and a E (0, 1], we say that f E 1i belongs to A(p, a) provided Mp(f', r) ~ (1- r)l-a, r E (0, 1).

Lemma 4.2.1. Let p E (1, oo], a E (0, 1) and let f(z) = Then f E A(p, a) if and only if supkEN !ak!nk < oo.

E%:0 akznk

lie in HG.

Proof. Assume that f E A(p, a) and r E (0, 1). We have nkak = (27ri)- 1

f'(z)z-nkdz.

{

Jlzl=r For p E (1, oo), HOlder's inequality implies that

nk!akl ~ rl-nk IIJ;IIHP ~ rl-nk (1- r)a- 1. For p = oo, we just estimate the integral. Choosing r = 1 - n;;\ we obtain laklnk ~ 1. Conversely, let lak Ink ~ 1. Since the number of the Taylor coefficients ak when nk E In= {j EN: 2n ~ j < 2n+l} is at most (logc2] + 1, we get 00

Moo(!', r) ~ r- 1

L L

n}-arnj ~ r- 1(1- r)a-l.

n=OnjEln

Since Mp(f',r) ~ Moo(f',r), f E A(p,a) and the lemma is proved. The spaces A(p, a) are called the mean Lipschitz spaces and discussed in (35] where it is proved that the spaces A(p, 1/p) increase with p and are all contained in BMOA. This inclusion suggests a comparison with the Qp-spaces.

Theorem 4.2.1. Let p E (2, oo) and q = 1- 2/p. Then (i) A(p, 1/p) c ne>O Qq+e . (ii) HG n Qq c A(p, 1/p) . (iii) There exists a function f E 1i satisfying

fEn q>O

Qq \

U A(p,1/p).

p
38

4. Modified Carleson Measures

Proof. (i) We suppose first that f E A(p, 1/p) so that Mp(f',s) :::5 (1- s) 11P- 1 , s E (0, 1). Then, for the Carleson box S(I) = {z E D : 1- h ::s; lzl < 1, 10arg zl ::s; h/2}; h = III, we get by Holder's inequality and the assumptions p > 2 and q = 1 - 2/p that 2 (!.e+h/ 1f'(seicf>)l 2d¢>) (1- s 2)q+esds 1-h 9-h/2 2 1 ( 9+h/2 ) 1P 1 2 ::s; /, lf'(seicf>)jPd h - 1P(1- s 2)q+eds l-h 9-h/2 1

/-tJ,q(S(I)) = /,

J.

:::5 h 1 - 21P {

1

(Mp(/ 1 , s))\1- s)q+eds

lt-h

:::5 hl-2/p/,1 (1- s)2(1/p-1)(1- s)q+eds l-h

~ lllq+e.

By Theorem 4.1.1, f E Qq+e for all € > 0 and thus the inclusion is proved. In order to prove the strict inclusion, we consider a function f(z) = E:.o akznk where ak = k 1122-kfp and nk = 2k. Then laklnk!P = n 112 , and by Lemma 4.2.1, f tJ. A(p, 1/p). On the other hand, 00

00

L

L 2n(l-(q+e)) ( lak12) = L n2-ne < oo, n=O nkEln n=O and, by Theorem 1.2.1 (i), f E Qq+e for all € E (0, 2/p]. (ii) Suppose that f(z) = E:.o akznk belongs to HG n Ql-2/p· By Lemma 4.2.1, it suffices to show lakl 2 n~IP :::5 1. But this is obvious since the Taylor coefficients ak off E HG n Q1 _ 2 /p satisfy (1.4). To verify the strict inclusion, we construct a Hadamard gap series 00

fp(z)

=L k=O

00

akznk

=L

2-nfpz2n.

n=O

Since laklnk!P = 1, it follows from Lemma 4.2.1 that /p E A(p, 1/p). On the other hand, 00

00

L nk-(l-2/p) ( L lak12) = L(2n)2fp2-2n/p = oo. k=O nkEln n=O By Theorem 1.2.1 (i), f ¢ Ql-2/p· (iii) Hereafter, we use ll·llp, p E (0, oo), to represent the usual LP-norm. Suppose that we can select a function f(z) = E~=O anzn satisfying two conditions below: (a) IILlnfll2 ~ 2-n, where (Llnf)(() = EkEln ak(k for n EN and ( E T;

4.3 Comparison with Besov Spaces

(b) there exists n

Then for q

= n(p, m), for p = 3, 4, · · · and m

E

39

N, such that

> 0, 2

L 2n(1-q) ( L lakl) ~ L 2n(1-q)2n L 00

n=O

00

n=O

kEln

lakl2

kEln

00

=

L 2n( -q)2niiL.\nfll~ 1

n=O 00

~ 2:2-nq

< 00,

n=O

and so f E Qq, thanks to the proof of Theorem 1.2.1 (i). Since the spaces A(p, 1/p) are monotonically increasing (see also Corollary A(p, 1/p) for p = 3, 4, · · ·. Fix such a 2.3 in [35]), it suffices to show that f p. By (b) there exists {nm} such that IIL.\n.,.,.fiiP 2:: m2-mjp for m E N. Thus, supn IIL.\nfllp2nfp = oo and hence f A(p, 1/p), by Theorem 3.1 in [35). The construction. Let r 1, r2, · · · , be an enumeration of the pairs { (p, m) : p = 3, 4, ···;mEN}. We need to find integers nj: n1 < n2 < · · ·, and polynomials fi obeying: (c) fi polynomials of degree ~ 2ni ; (d) IIJill2 ~ 2-ni ; (e) llhll7rl{rj) 2:: 7r2(rj)2-ni/7ri(ri), where 7rj, j = 1,2, are projections on first and second coordinates of the pairs r j. Assume that {h} have been constructed, then define

tt

tt

00

f(z) =

L fj(z)z

2 ni.

j=1 It is then easily seen that

f satisfies (a) and (b) so we are done once we construct

{fi}. Construction of the sequence {fj}: Given ni_ 1 , p = 3, 4, · · · , and m E N, we must find ni > ni -1 and polynomials fi of degree 2ni such that (d)' IIJill2 ~ 2-ni ; (e)' llhiiP 2:: m2-ni/P. But the existence of fi follows immediately from the density of polynomials in the Hardy space HP and HP ~ H 2, p > 2. The proof of the theorem is completed.

4.3 Comparison with Besov Spaces For p E ( 1, oo), let Bp be the space of all functions

f

E

1-l such that

40

4. Modified Carleson Measures

II/IlB, =

(L 1/'(z)IP{l- lzi

2

)P-

2

dm(z)) I/p < oo.

The spaces Bp are the so-called Besov spaces. It is well known that every Bp is conformally invariant according to II/ o ai!Bp = II!IIBp for all f E Bp and a E Aut(D). Of course, it becomes a natural topic to compare QP with Bp.

Lemma 4.3.1. Let p E (1, oo) and let f(z) = f E Bp if and only if E%:o nklakiP < oo.

E%:o akznk

belong to HG. Then

Proof. The argument is similar to that of Theorem 1.2.1 (i), so we give the key steps of the argument. In fact, if tn = En 3 Eln n]lail 2 and In = {j EN: 2n :::; j < 2n+l }, then one has 00

IIJII~P ~

L

2-n{p-l}t~2.

n=O

Since the number of the Taylor coefficients ai is at most [loge 2] + 1 when ni E In, t~/2 ~

2pn

L

laj IP.

n3Eln

The above two estimates lead to 00

llfii~P ~

L nklakiP. k=O

Theorem 4.3.1. Let p E [1, oo). Then B2p c nl-l/p 1 and 1- 1/p < q < 1. Iff E B2p, then, for the Carleson box S(I) = { z E D : 1 - h :::; lzl < 1, 10- arg zl :::; h/2}; h = III, we get by Holder's inequality that

By Theorem 4.1.1, f E Qq for all q E (1 - 1/p, 1) and thus the inclusion is proved. To prove the strict inclusion, we choose a function f(z) = E%: 0 akznk where ak = 2-k/{ 2P) and nk = 2k. Then E%: 1 laki 2Pnk = oo, and by Lemma 4.3.1, f tt. B2p· Nevertheless,

4.4 Mean Growth

L 00

2n(1-q)

n=O

L

00

lakl2 =

nkEln

f

and, by Theorem 1.2.1 (i),

L

41

E Qq

2n(1-q-1fp)

< oo,

n=O

for all q E (1 - 1/p, 1).

4.4 Mean Growth The discussions carried out in last two sections suggest a consideration of the mean growth of the derivatives of functions in QP. Theorem 4.4.1. Let p E (0, 1) and f E 1-l. If

1 1

{1- r)"(M00 {! 1 ,r)) 2 dr < oo,

then f E QP. Furthermore, the exponent p cannot be increased, i.e., given there exists an f E 1-l such that

€

>0

Proof. Let f E 1-l satisfy 2

[ {1- r)P(M00 {! 1 ,r)) dr < oo. For the Carleson box S(I) = {z E D : 1- h ~ lzl h =III, we have

< 1, 10- argzl

~

h/2};

Hence,

[1 ( !.8+h/21f'(rei¢)12d¢) l1-h

8-h/2

rdr - j IIIP [1 (Moo(!'' ~))2 dr. (1- r 2) P Jo (1- r) P

By Theorem 4.1.1, f E QP. In order to prove that the above exponent p cannot be increased, we take

L 00

f(z) =

2k(p-1)/2z2k.

k=O

Then Theorem 1.2.1 (i) shows

f ¢;

QP. Now, it is easy to see that

42

4. Modified Carleson Measures

Moo(!', r) ::S (1 - r)-(P+ 1)1 2,

0

< r < 1.

Thus, if E > 0 then

This concludes the proof. From Theorem 4.1.1 it turns out that for 0

f E QP ==>

{1

Jo

< p::; 1, 2

(1- r)P(M2(f', r)) dr < oo.

In fact, we can obtain a better result.

Theorem 4.4.2. Let p E (0, 1), q E (0, 2] and f E 1-l. Iff E Qp then

{1

Jo

2

(1- r)P(Mq(J', r)) dr < oo.

Moreover this mean growth is sharp in the following sense: let be a nonnegative nondecreasing function defined on (0, 1) such that

1'

(1- r)P¢2 (r)dr < oo,

(4.1)

then there exists a function f E QP such that Mq(f', r) ~ (r),

0

< r < 1.

(4.2)

Proof. Since Mq(J', r) is a nondecreasing function of q, it suffices to show the "sharp" part of the theorem. Let p and be as above. We start with considering the case q = 2. Set rk = 1- 2-k, k E N. Since¢ is nondecreasing, we apply (4.1) to get

00

~

2::: (rk+1 - rk) (1- rk+1)P¢ (rk) 2

k=l 00

~

2::: 2-k(1+p)¢2(rk)· k=1

Thus Theorem 1.2.1 (i) shows that 00

f(z) = (ri)z + e 4

2::: 2-k(rk)z2

k

k=1

Notes

43

is a member of QP. However,

2 (M2(!', r) ) 2:: ¢ 2(rt)

00

+ e8 L: ¢ 2(rk)r 2k+l,

r E (0, 1).

(4.3)

k=l

This estimate, together with the fact that 4> is nondecreasing, implies

(4.4) Also, using the elementary in~quality (1- n- 1 ) n 2:: e- 2 which is valid for all integers n 2:: 2, and keeping in mind that 4> is nondecreasing, we find that (4.3) implies that if j 2:: 1 and Tj ~ r ~ ri+l then 2

"+2

(M2(!',r)) 2:: es¢2(ri+t)r2J

"+2

2:: es¢2(r)rr

2:: 4>2(r).

This, together with (4.4), gives

and hence the proof is done for q = 2. Take now q, 0 < q < 2. Let f be the function constructed in the previous case. Since f is given by a power series with Hadamard gaps, Theorem 8.20 in [146, p. 215] shows

Mq(f', r)

t: M2(!', r) t: ¢(r),

0 < r < 1.

Therefore, the proof is complete.

Notes 4.1 When p = 1, Lemma 4.1.1 and Theorem 4.1.1 (taken from AulaskariStegenga-Xiao [24]) are well known and are contained in works of Fefferman, Garcia and Pommerenke, see Baernstein [30] for an exposition on these works. For p = 2, see also Xiao [133]. Lemma 4.1.1 can be further applied to study the meromorphic Q classes; see e.g. Aulaskari-Wulan-Zhao [26], Essen-Wulan [61] and Wulan [131]. Theorem 4.1.1 has been extended to the higher dimensions, see Andersson-Carlsson [9] and Yang [141] [142]. Classically, a nonnegative Borel measure J-L on D is a Carleson measure for Vp, p E (0, oo), provided

For p = 1, Carleson characterized these measures and applied them to solve the corona theorem [38]. These measures were also important in Fefferman-Stein's duality for H 1 [64]. It is known that for p 2:: 1, /-lis a Carleson measure for Vp

44

4. Modified Carleson Measures

if and only if ll~tllcP < oo; and, if p E (0, 1), then the condition ll~tllcp < oo is necessary, but not sufficient. In fact, such a Carleson measure is characterized in terms of the Bessel capacity. When p = 0, the classical logarithmic capacity must be used. See Stegenga [117] for details. On the other hand, the Carleson measures for Vp may be described by single Carleson box; see Kerman-Sawyer [84] and Arcozzi-Sawyer-Rochberg [7]. Here, it is also worth mentioning that Ahern and Jevtic [3] used the strong Hausdorff capacity estimates of Adams [1] to obtain that a nonnegative Borel measure IL on D is a p-Carleson measure if and only if

L

lfldp

where a function provided

f

~ lf(O)I + llfiiBl-p'

f E BLP'

E 1-l is called to belong to the Besov space

IIJIIBLP =

B{_P, p

E (0, 1),

L

lf'(z)l(1-lzi 2 )P-ldm(z) < oo.

4.2 Lemma 4.2.1 should be compared with Theorem 1.2.1. And Theorem 4.2.1 improves actually the result that A(p, 1/p) increases with p and is contained in BMOA; see [35]. For a further discussion, consult Essen-Xiao [62]. 4.3 An elementary argument for Theorem 4.3.1 can be found in AulaskariCsordas [15]. It is clear that QP and Bp behave similarly in the sense of the limit space: limp-?oo Bp = B and QP = B for p > 1. Nevertheless, Bp is much smaller than Qp. 4.4 Theorems 4.4.1 and 4.4.2 come from Aulaskari-Girela-Wulan [18], but also have a close relation with the main results in the paper [71] by Girela-Marquez.

5. Inner-Outer Structure

Based on the classical factorization of the Hardy space HP, this chapter focuses on: characterizing the inner and outer functions in Qp; giving the canonical factorization of Qp; and representing each Qp-function as the quotient of two functions in Hoc n QP.

5.1 Singular Factors Since Hoc c BMOA, it is a natural question to ask about: is Hoc a subset of QP for 0 < p < 1? Unfortunately, this question has a negative answer. To this end, we give some examples. Example 5.1.1. There are functions in Hoc\ (Uo
Proof. For

q E (1, oo),

we consider oc

oc

fi(z) = L(klogqk)- 1 z 2 k; h(z) = Lk-qz 2 k. k=2

k=l

For lzl ~ 1, both these power series are absolutely convergent and it is clear that fi and hare in Hoc. Furthermore, if aj = (j logq j)- 1, we see that for p E (0, 1), oc

L2k(1-p)

k=2

L

oc

lai12 = L2k(1-p)a% = 00.

2k$2i<2k+l

It follows from Theorem 1.2.1 (i) that ment will show !2 ffi Uo
k=2

!I ffi

Uo
Example 5.1.1 suggests us to work with inner functions of Hoc. Recall that a function BE Hoc is called an inner function provided IIBIIHoo ~ 1 and its radial limit limr-+ 1 IB(r()l = 1 for almost all ( E T. It is well known that any inner function lies in BMOA, and yet, this is no longer true in case of QP, p E (0, 1). In order to see this, we need an estimation exchanging the derivative and the difference quotient.

46

5. Inner-Outer Structure

Theorem 5.1.1. Let p, o E (0, 1), ( E T and B be an inner function. Then

J.'

IB'(r()l 2 (1- r 2 )"dr "='

l

(1-IB(r(}l 2 ) 2 (1- r 2 )P- 2 dr.

(5.1)

Consequently, B E QP if and only if (1 - IB(z)l 2) 2(1 - lzi 2)P- 2dm(z) is a pCarleson measure. Proof. Since B is inner, one always has (1 - lzi 2)IB'(z)l ~ 1 - IB(z)l 2 and it suffices to prove the left-hand side inequality of (5.1), i.e.,

J.

1

1B'(r(}l 2 (1- r 2 )Pdr :::5

J.'

(1-IB(r(WJ 2 (1- r 2 )P- 2 dr.

Note that for almost all ( E T,

. 1-IB(r()l :0: IB(()- B(r()i

=I[

B'(((1- t)r + t)(}(1- r)dtl.

Applying Minkowski's inequality, we obtain

f. ··· 1

(

) 1/2

=

(

f.

1 22 2 ) 1/2 (1 -IB(r()l ) (1- r 2)P- dr

21 f. =21 [ :0:

1(

1

1(

1

IB'(s()12(11-s)P1ds - t

lt+(1-t)8

0

1

1 (

~2 o

[1

)8 IB'(s()l2(1- s)Pds

4

= - -

IB'((t + (1- t}r}()l 2 (1- r)Pdr

1-p (

f )8

1

1 2 /

)

IB'(s()l 2(1- s)Pds

-

) 1/2

) 1/2

t

dt

dt

dt (1- t)
) 1/2

The proof is complete.

!1;=

Corollary 5.1.1. Let p E (0, 1), n EN and B = 1 Bi, where each Bi is an inner function. Then BE QP if and only if Bj E QP for j = 1, 2, ... n.

Proof. If Bj E QP for j = 1, 2, ... n, we use induction to show BE QP. On the other hand, let BE QP. Since we know that IBil ~ IBI for all j and that for the Carleson box S (I), [

j S(I)

(1-IBj(z)l 2) 2(1 -lzi 2)P- 2dm(z)

~

[

j S(I)

(1-IB(z)l 2) 2(1-lzi 2)P- 2dm(z).

Theorem 5.1.1, together with the assumption B E QP, proves Bj E QP, j 1,2, .. . n.

=

5.1 Singular Factors

47

The forthcoming examples show that all singular inner functions are outside all Qp, p E (0, 1). Recall that a singular inner function is a function of the form Ss(z) = exp (

where the measure "singular" .

L: ~ ~dvs(()),

is nonnegative and singular to

V8

Example 5.1.2. Let p E (0, 1). Then 8 8

ld(l, and the index s means

~ QP.

Proof. First of all, we deal with the simplest case where explicitly, for 1 E (0, 1) and rJ E T let

V8

is atomic. More

z+rJ) •

s'"'f,7J(z) = exp ('Y z- 'f}

Then S'"'f,7J ~ QP. In fact, if otherwise, S'"'f,7J E QP, then for the Carleson box S(I), we get by Theorems 4.1.1 and 5.1.1 that

Consequently, f'"Y/(4111)

Jo

(1- e-s)2sp-2ds j

III1-p'Yp-1

which is impossible as III~ 0. Hence s'"'/,'f/ ~ Qp. Next, let us prove Ss ~ QP. If Ss contains a factor S'"'f,(' from Corollary 5.1.1 and the above simple case it follows that Ss ~ QP. It remains to consider the case when V 8 is nonatomic. Let

w(8)

= sup{vs(I): I subarc

of T,

III= 8}.

We know that w(8) is continuous and that lim8-+ow(8)/8 = oo (see Theorem 8.11 in (108]). If 8 8 E QP, then Theorem 5.1.1 holds with B replaced by 8 8 • Consequently, for the Carleson box S(I) with III= 8 < 1/2 one has

48

5. Inner-Outer Structure

which deduces immediately that

as 8 ---t 0. This is a contradiction, and so our assumption that 8 8 E QP is wrong. The proof is complete.

5.2 Blaschke Products Corresponding to those singular inner functions are the Blaschke products. By definition, a Blaschke product on D is a function of the form

B( { Zn}, z) =

IT nE N

lznl Zn--= z ' Zn 1- ZnZ

where {zn} CD is a sequence satisfying the condition E:'=l (1-lznl 2 ) < oo. Note that if Zn = 0 then lznl/ Zn is replaced by 1. Of course, every Blaschke product is an inner function, but not all Blaschke products are in QP because [107, Theorem 1] tells us that there exists a Blaschke product B satisfying J0 IB' (z) Idm( z) = oo. The existence of such a Blaschke product B leads to that oo =

L

IB'(z)ldm(z):::; Fp(B,O)

(

L

1/2

(1-lzi 2 )-Pdm(z) )

holds for p E (0, 1). This implies B ¢. QP. A careful analysis reveals that every Blaschke product in QP is closely related to its zeros. The precise relation can be presented by a full description of all inner functions in QP.

Theorem 5.2.1. Let p E (0, 1) and B be inner. Then BE QP if and only if B is a Blaschke product whose zero set {zn} is such that J.l{zn},p is a p-Carleson measure, where

5.2 Blaschke Products

49

00

d~-t{zn},p = L(l-lzni 2 )Pozn n=l

and oc, denotes a Dirac measure at (. Proof. Necessity: assuming that B E QP which, by Theorems 4.1.1 and 5.1.1, leads to (1- IB(z)l 2 ) 2 (1- lzi 2 )P- 2 dm(z) ::S IIBII~ IIIP (5.2) ls(I) v

r

for the Carleson box S(I), we claim that B must be a Blaschke product. In fact, we will prove that if B is not a Blaschke product, then it follows that B (j. QP. By the classical factorization theorem (cf. (66, p.74, Theorem 5.5]), every inner function B can be represented as a product of a complex number rJ E T, a Blaschke product and a singular inner function generated by a singular measure V 8 on T:

Ss(z) = exp (

l: ~ ~dv8 (()).

By Example 5.1.2 we know that 8 8 is not a member of QP. Furthermore, it turns out from Corollary 5.1.1 that each inner function B E QP contains its Blaschke product only. It remains to prove that J.t{zn},p is a p-Carleson measure. In the following argument, we assume without loss of generality that III :::; 1/2. To shorten our formulas, we introduce 00

(1-IO"zn (z)l 2 )

T( {zn}, z) = L n=l

R({zn},I)

=

L

(1-lznl 2 ).

ZnES(I)

Since

we have also 1 -IB(z)l 2 ~ 1- exp{ -2T( {zn}, z)}. Combining (5.2) and (5.3), we see

(5.3)

50

5. Inner-Outer Structure

This implies

so that L:~=l (1 -lznl 2 )8zn is a 1-Carleson measure, namely,

M = supT({zn},z) < oo, zED

owing to Lemma 4.1.1. Because 1 - e-2t 1 - e-2M 2t 2::: 2M = Mt, 0 ~ t ~ M, it follows from (5.3) that 1-IB(z)l 2 2::: 2M1T({zn},z), so that

1 -IB o O"w(z)l 2 !::: M1T( {O"w(zn)}, z). Using the assumption BE Qp, Theorems 1.1.1 and 5.1.1, and Lemma 1.4.1, we get that for wED,

(Fp(B, w)) 2 = fo!(B o aw)'(z)l 2(1 -lzi 2)Pdm(z)

!::: !::::

l

(1 -IB o aw(z)l 2 ) 2 (1

~ lzi 2)P- 2dm(z)

L

2 (T( {aw(zn)}, z)) (1-lzi 2)P- 2dm(z)

!:::: f(l-law(zn)l2)2 { n=l

(1 -lzi2)P 4 dm(z)

Jn 11- ZO"w(zn)l

00

!:::: L(l-law(zn)I 2)P. n=l

These inequalities imply IIIJi.{zn},plllcv < oo. In other words, Jl.{zn},p is a p-Carleson measure, due to Lemma 4.1.1. Sufficiency: if the inner function B is a Blaschke product whose zeros {Zn} are such that Jl.{zn},p is a p-Carleson measure, then we show that B belongs to QP. Writing B(z) = B( {zn}, z), we obtain

5.3 Outer Functions

and thus

51

1- ~znl2 . L..J 11zn zl2 n=l Applying the conformal invariance of ll·ll.a, as well as Lemma 1.4.1, we see that for wED, IB'(z)l

(Fp(B, w)) 2 ~

~

~~

IIBII~ fo1(B o aw)'(z)l(1 -lzi 2)P-ldm(z) 2 1 f(l- law(zn)l 2) Jn{ 11( -lzi )P-\ zaw(zn)l

dm(z)

n=l 00

n=l

This, together with Theorem 1.1.1, implies BE QP. The proof is complete.

5.3 Outer Functions An outer function for H 2 is the function of the form

O,p(z)

= 17exp

(,L ~ ~:

log,P(() ~~;1), 17 E T,

where '1/J > 0 a.e. on T, log'lj; E £ 1 (T) and 'ljJ E £ 2 (T). In what follows, for zED and ( E T let dm(z) d,.\(z) = (1- lzl2)2

1- lzl 2 ld(l

resp.

d!-£z(() = I(~ zl2 27r

be the hyperbolic measure on D resp. the Poisson measure on T. Before giving a description of the outer functions in Qp, we present a new characterization of QP. Theorem 5.3.1. Let p E (0, 1) and f E H 2 . Then the following conditions holds: (i)

f E QP if and only if one of

sup { ( { lf'(u)l 2g(u, z)dm(u)) (1-law(z)I 2)Pd,.\(z)

wenln (ii)

(iii)

ln

< oo.

52

5. Inner-Outer Structure

Proof. Since limz~T(1 -lzl 2 ) = 0, we apply Green's theorem [66, p. 236] to get that for zED, and p E (0, 1),

(1-lzi 2 )P = =

~ 2

L( :;u(1-lui JP)

p

7r

2

8

iogl<7z(u)idm(u)

r (1- plul )(1-lui )Pg(u, z)d.A(u). 2

Jo

2

Furthermore, from Fubini's theorem, Hardy-Littlewood's identity (cf. [66, p. 238, (3.3)]), it follows that

L(LIf'( 1 ~L

1T(Fp;~· O)) = 2

z) 2 g( u, z )dm(z)) (1 -

plul 2 ) ( 1 - lui 2 )P d>.( u)

(1-lui 2 )P(fr1f(()- f(u)l 2 d~tz(())d.A(u)

=

Since

d~tz

L(1-lui )P(fr1f(()l d~tu(() 2

2

-lf(u)l 2 )d.A(u).

and d.A are conformally invariant, when ( E T, w, zED one has d~tz(()

= d~tuw(z)(uw(()) and d.A(z) = d.A(uw(z)).

Theorem 1.1.1, with the help of some elementary calculations, implies the desired equivalence. Concerning the outer functions in QP, p E (0, 1), we can establish the following result.

Theorem 5.3.2. Let p E (0, 1) and let 'lj; > 0 a.e. on T, log¢ E L 1 (T) and 'lj; E L 2 (T). Then 0'1/J E QP if and only if

!~!',

L(l 1/l

2

2

2

d!'z- exp ( frlog,P dpz)) (1 -IC7w(z)l )"dA(z) < oo.

Proof. This follows immediately from both Theorem 5.3.1 (iii) with f = 0'1/J and the fact that IO'!fJI = 'lj; a.e. on T and IO'I/J(z)l = exp

(frlog'l/Jd~tz ),

zED.

5.4 Canonical Factorization One of the most essential properties on H 2 is that H 2 has an inner-outer factorization. That is to say, every nonzero H 2 -function can be factored in the form f = BO, where B is inner and 0 E H 2 is outer. Conversely, such a function BO belongs to H 2. Then, it is natural to ask: how about QP?. Now, Theorems 5.1.2 and 5.3.2 enable us to derive a solution to this question.

5.4 Canonical Factorization

53

Theorem 5.4.1. Let p E (0, 1) and let f E H 2 with f ¢= 0. Then f E QP if and only iff= BO, where B is an inner function and 0 E QP is an outer function for which

Proof. Because f is a member of H 2 with f ¢. 0, f must be of the form BO, where B is an inner function and 0 is an outer function for H 2 • A simple computation produces a formula related to pointwise multiplication below:

llBOl 2 dJLz- IB(z)O(z)l 2 = ll0l 2 dJL,- IO(z)l 2 + IO(z)l 2 (1- IB(z)l 2 ) Thanks to the fact that IB(z)l :::; 1 for all z ED, f E QP is equivalent to that 0 lies in QP and 101 2 (1 - IBI 2 ) meets the requirement of (5.4). The proof is complete. Theorem 5.4.1 has indeed illustrated that every nonzero Qv-function f has a unique representation BO, but also the outer factor 0 inherits the smoothness of f.

Corollary 5.4.1. Let p E (0, 1) and let f E H 2 be such that fIB E H 2 for an inner function B. Iff E QP then fIB E QP. Proof. Since f = (f I B)B, the corollary follows from Theorem 5.4.1.

Theorems 5.3.1 and 5.3.2 will be used to construct the cut-off outer functions in QP and hence to represent every Qv-function as the ratio of two H 00 -functions in QP. To see this, we cite Aleman's lemma as follows.

Lemma 5.4.1. Let (X, f..L) be a probability space and let 'lj; E L 1 (J.t) with 'lj; > 0 J.t-a.e. on X and log'lj; E L 1 (J.t). Let E(,P)

=

L

'1/JdJL- exp

(L

log'I/JdJL) .

Then

max { E(min{1, 'ljJ} ), E(max{l, 'ljJ})} :::; E 1 ('¢J).

(5.5)

Proof. Without loss of generality, assume that A = {x E X : 'l/;(x) ;;::: 1} and a= J.L(A) E (0, 1). The inequality E(min{l,'l/J}):::; E 1 ('¢J) is equivalent to

Note that the right-hand side of the la:;;t inequality is less than or equal to

54

5. Inner-Outer Structure

due to Jensen's inequality. Now it is easy to show by differentiation that t - {3 ";?: (t/ (3)!3- 1 holds for t ";?: 0 and f3 E (0, 1). Thus the inequality E(min{1, 'l/J}) ~ E ('l/J) follows from t = '1/JdJ-L and f3 =a. The proof for the other inequality in (5.5) can be given similarly with A replaced by X \ A and a by 1 - a.

JA

1

Theorem 5.4.2. Let p E (0, 1). Then every function in QP is the quotient of two functions in H(X) n QP. Proof. The proof relies upon the constructions of two cut-off outer functions attached to a given outer function. To this end, suppose that f E QP is such that f ¢= 0 (otherwise there is nothing to argue). Let BO be the inner-outer factorization off as in Theorem 5.4.1. In particular,

O(z)

= 17exp

(l ~ ~: Iog(IO{()I)I~~I),

where 'fJ E T and zED. This outer factor 0 is equipped with two cut-off outer functions below:

O+(z) = Jfiexp

(l ~ ~; log(max{IO(()I, 1}) 1~1)

= Jfiexp

(l ~ ~: log{min{IO{()I, 1}) ~~~~).

and

O_(z)

It is clear that 0_ and 1/0+ lie in H(X). A key observation is that 0 = 0+0_, IO-(z)l ~ IO(z)l and IO+(z)l ~ 1 for all zED. For convenience, we put

E(,P, z) = l 1/Jd!-'z- exp (l!og,Pdl-'z) , where 'l/J > 0 a.e on T and '1/J,log'l/J E L 1 (T). Since J-lz is a probability measure on T, Lemma 5.4.1 shows

Notice that f can be rewritten as BO+O- = (BO_)j(1f0+)· Accordingly, it suffices to verify that both g = BO_ and h = 1/0+ are members of QP. On the one hand, hE QP is obvious. As a matter of fact, owing to h' = -O+JO~, IO+I ~ 1 and

Notes

fr10+I 2 d~tz -IO+(z)l 2 =

2

55

2

E(IO+I ,z)::; E(IOI ,z)

::;

fr!OI 2 d~tz -IO(z)l 2 + IO(z)j 2 (1 -IB(z)l 2 )

::;

fr!fl 2 d~tz -lf(z)l 2 ,

it follows from Theorems 5.3.1 and 5.3.2 that 0+ E QP and hence hE QP. On the other hand, g E QP comes from Theorems 5.3.1 and 5.3.2 as well as the estimates below:

fr!BO_j 2 d~tz -IB(z)O_(z)l 2 =

E(IO-I 2 ,z) + jO_(z)I 2 (1-IB(z)l 2 )

=

+ jO(z)j 2 (1 - IB(z)j 2 ) fr10I 2 d~tz - IO(z)l 2 + IO(z)l 2 (1- IB(z)l 2 )

::;

fr1fl 2 d~tz -lf(z)l 2 •

::; E(IOI 2 , z)

This concludes the proof.

Notes 5.1 The first section of Chapter 5 is from Essen-Xiao (63]. For a result analogous to Theorem 5.1.1, we refer to Verbitskii [125, Lemma 2.2] which was proved by using HOlder's inequality. For an inner function B, the quotient (1-IB'(z)j 2 )/(1lzl2) has an operator-theoretic explanation. Let kw(z) = (1- lwl 2 ) 112 /(1- wz) be the normalized reproducing kernel of the Hardy space H 2 with respect to w E D. If C8 denotes the adjoint of the composition operator CB with the symbol B, then IIC8kwll~2 = (1-lwl 2 )/(1 -IB'(w)l 2 ) holds for every inner function B. For a proof, see also Shapiro [110, p.43-44].

5.2 Theorem 5.2.1 is one of the main results in Essen-Xiao [63]. As one of its consequences, it was proved by Nicolau-Xiao [95] that any Blaschke product in QP has small mean variation on many subarcs ofT. That is to say, if BE QP is a Blaschke product with {zn} and Lr ={zED: infnlcrz(zn)l2:: r}, r E (0,1), then lim sup 111-p { IB'(z)l 2 (1-lzi 2 )P1L)z)dm(z) = 0. r-+l /~T j S(I) Moreover, Resendis and Tovar [104] showed that if subordinated, i.e.,

E:=l (1 -

lzni 2 )P is a p-

56

5. Inner-Outer Structure 00

L

(1-lzni 2)P ::5 (1-lzki 2)P,

kEN,

n=k+1

then JL{zn},p is a p-Carleson measure. So, the criterion in Theorem 5.2.1 may be checked by using some appropriate requirements only on the distribution of {zn}· For more information, see Resendis-Tovar [104], Danikas-Mouratides [46] and Aulaskari-Wulan-Zhao [26]. The characterization of the inner functions in Vp, p E [0, 1) (in terms of the zero distribution of such functions) can be found in Ahern [2] and Carleson [37]. For f E 1-l with f(z) = L::~=O anzn; zED, the Cesaro operator Cis defined by oo

(Cf)(z) = L n=O

n

((n+ 1)-1 Lan)zn. k=O

Although H is not a subspace of QP, p E (0, 1), the Cesaro operator maps H 00 into QP thanks to 00

z(Cf)'(z) = f(z) - {1 f(tz) dt. } 0 1- tz 1- z See Essen-Xiao [63, Theorem 5.4], as well as Danikas-Siskakis [48] for BMOA. For more information about the Cesaro operators acting on different holomorphic function spaces, we refer to Siskakis [111] and Benke-Chang [31]. 5.3 The third section of Chapter 5 is from Xiao's paper [134] (see also Xiao [138] for meromorphic case). Notice that f E BMOA if and only iff = BO, where B is an inner function and () is an outer function in BMOA for which IO(z)l 2 (1 -IB(z)l 2 ) is bounded on D. This result is due to Dyakonov [55]. Theorem 5.3.1 is important since it gives a way to recognize QP, p E (0, 1) via BMOA. This theorem can be used to study some isoperimetric inequalities involved in Qp; see the paper [23] of Aulaskari, Perez-Gonzalez and Wulan. In particular, iff EBMOA has a hyperbolic image region: fl = f(D) for which the Green function is denoted by gn(·, ·), then the condition

sup { ( { gn(u, f(z))dm(u)) (1-luw(z)I 2 )Pd.A(z) wEDjD

ln

< oo

implies f E QP, p E (0, 1). Its converse is valid for every universal covering map. In case of BMOA, we refer to Metzger's paper (91] as well as Gotoh's preprint (72]. Regarding the Q classes on Riemann surfaces, we refer the interested reader to Aulaskari-He-Ristioja-Zhao (19], Aulaskari-Chen [14] and references therein. 5.4 With regard to the last section of Chapter 5, we mention that all Vp, p E [0, 1), have analogous results, see Aleman [5] and Dyakonov (54]. Theorem 5.4.2 depends on Lemma 5.4.1. But, the known fact that any BMOA-function equals the ratio of two functions in H 00 can be also worked out from the corresponding decomposition of the Nevanlinna functions {50]).

6. Pseudo-holomorphic Extension

In this chapter we first give a full boundary value description that f is in QP, p E (0, 1), and secondly provide a characterization of QP via the pseudo-holomorphic extension and, as a corollary, we prove that QP has the K-property. The latter means that, for any 'ljJ E H 00 , the Toeplitz operator T-~[J maps QP into itself.

6.1 Boundary Value Behavior A good way to know much more information about QP is to find out how a Qpfunction behaves on T. To understand this view-point, we, from now on, assume that for p > 0 and an arc I ~ T, pi denotes the subarc of T with the same center as I and with the arclength pill, but also we need a description of the boundary value functions of elements in Vp. · Lemma 6.1.1. Let p E (0, 1) and f E H 2. Then f E Vp if and only if

ntntp .• =

LL~~~~-11(.~; ·1d(lldql 11 1

< 00.

Proof. Since f E H 2, we may assume f(z) = L~=oanzn. A simple calculation involving Parseval's formula implies that for each ( E T,

Lif(z()- f(z)l ldzl "'~ lanl l(n- lj 2

2

2

.

This estimation leads to

llfll2 'Dv•*

=

{

{ lf(z)- f(w)j2ldzlldwl

}T }T

""L (L

lz - wl2-p

if(z()- f(z)l

"'~ Ia, I" 00

:=::::

2

Li(n -li"K -w-"ld(l

L lanl 2nl-p:::::: llfll~v' n=O

ldzl) I( -W- 2 Id(l

58

6. Pseudo-holomorphic Extension

so that Lemma 6.1.1 follows. By Lemma 6.1.1 and Theorem 1.1.1 we obtain a characterization of the boundary function off in QP. Theorem 6.1.1. Let p E (0, 1) and let f E H 2 • Then f E QP if and only if

where the supremum is taken over all arcs I

~

T.

Proof. By a change of variables: z = o-w(u), wED, we can easily establish

where

is the Poisson kernel. Necessity: iff E QP, then llfiiQv,* < oo. Arbitrarily pick a subarc I ofT. If I-/= T, then we choose a point wED\ {0} such that w/lwl and 211"(1-lwl) are the center and arclength of I, respectively. If I= T, then we take w = 0. With such a point w, as well as the inequality cost ;:::: 1 - 2- 1 t 2 for t E ( -oo, oo), we get that for u E I,

1

Pw(u)

1

2:: 1 -lwl ~TIT'

(6.2)

Corollary 1.1.1, Lemma 6.1.1 and an application of (6.2) to (6.1) show

Sufficiency: if llfiiQv•* < oo then f E QP. To each point w E T \ {0} we associate the subarc Iw with center wflwl and arclength 211"(1-lwl). For w = 0, we set Iw = T. Also, we set

In = 2n I~,

n = 0, 1, ... , N- 1,

where N is the smallest integer such that 2NIIwl ;:::: 211". And then, we put IN= T. Through the help of the elementary inequality cost ~ 1 - 21r- 2 t 2 for t E [-1r, 1r], we know that for every point u E T, 1

Pw(u)

Furthermore, for u E T \ In we have

~ 1-lwl'

(6.3)

6.1 Boundary Value Behavior

59

1

Pw(u) ~ 22nlwi1Iwl' In the sequel, we may assume lwl 2:: 1/2, otherwise, the result is obviously true. Therefore, if u E In+ 1 \ In, then

(6.4) With the above notations, we break

I

~

For convenience, we recall that T, namely,

!I=

!I

llf o uwll 2v

P•

* of (6.1) into two parts. .

stands for the average of

f on the arc

1~1~ f(z}ldzl.

By (6.3), (6.4), the definition of BMO (see Chapter 1), the triangle inequality and the following identity

we have

XI= {

{ (- ..

}lw }lw

)+I:' {• • • { (--.) n=O J1 + \1 }lw

When handling X 2 , we omit the integrated functions (for simplicity), and use the same manner as dominating X 1 to obtain

60

6. Pseudo-holomorphic Extension

2

X =

~{L+~v• + ~~L+~v•lm+Ivm N-1

:::5

N-lN-1

llfii~P•* + "f.Lvw L+'\Im + ~ "f.L+>v• L+>vm N-1

:::5

11/II~P•* + ~ ( ~ + l~) L+l\[• L+l\fm 00

00

:::5

IIJII~P•* + ~ ;n + ~ 2Pn IIJIItMo

j

llfii~P•*'

1 )

2

(

Combining the estimates of X 1 and X 2 , as well as using Corollary 1.1.1 and Lemma 6.1.1, we reach

which concludes the proof.

6.2 Weight Condition By a careful checking with Theorem 1.1.1 and its proof, we find that the weight (1-law(z) I2 )P can be replaced by its reflection (law(z)l- 2 -1)P in case p E (0, 1). More precisely, we have the following theorem.

f

Theorem 6.2.1. Let p E (0, 1) and SUp

E 1l. Then

r lf'(z)l (law(z)l2

wEDJo

2

f

E QP if and only if

-l)Pdm(z) <

00.

Proof. It suffices to prove that for w E D, the left-hand side integral and Fp(f, w)) 2 are controlled by each other, namely,

fo1f'(z)l 2 (law(z)l- 2

-

1)Pdm(z)

~ (Fp(f, w)) 2 •

By a change of variables, it is enough to check the last equivalence for w = 0. In other words,

1'

2

(M2(f', r)) (r- 2

-

l)Pr dr""

1'

2

(M2(f', r)) (1- r 2 )P r dr

which is equivalent to

1'

2

(M2(f', r)) (1- r 2 )P r 1 -

2

P

dr

""1'

2

(M2(f', r)) (1- r 2 )P r dr.

This, however, follows without difficulty, since p E (0, 1) and M2(f, r) is a nondecreasing function of r.

6.2 Weight Condition

61

For p E (0, 1), w E D and z E C, let 1 IP (1-lwi2)P llzl2 -liP Uw(z) = Uw,p(z) = 1- luw(z)l2 = lz- wl2p I This notation, together with Theorem 6.2.1, leads to a consideration of the weighted norm inequalities. Given a nonnegative weight function w E Lioc(C). For p > 1 and any Euclidean disk L1 in C, let

Av(w,Ll) =

(m~Ll)

i

i

m~Ll) (~ f(p-1) dmr-1

wdm) (

We say that w is an Ap-weight provided

Ap(w) = sup Ap(w, Ll) < oo, .ace

where the supremum ranges over all Euclidean disks L1 inC with the area m(Ll). Theorem 6.2.2. Let p E (0, 1). Then there exists a positive constant C such that Uw,p is an A2-weight with A2(Uw,p) ::; C for every w E D.

Proof. Fix w E D and put z E C.

It is clear that

A2(Uw) = A2(Uw,p) = A2(Vw,p) = A2(Vw), So, we can write Vw(z) = W(z)Yw(z), where W(z) = Wp(z) = llzl 2 -liP, Yw(z)

= Yw,p(z) =

w E D.

lz- wi- 2P.

It is well known (see e.g. [119, p.218]) that, since p E (0, 1), the weight Y0 (z) = lzi- 2P satisfies the Aq-condition for all q > 1. Since the Yw are translates of Yo, it follows that for every q > 1 there exists a constant Cq > 0 (depending on q) such that (6.5) holds for all wE D. Take and fix r E (1, 1/p), and let L1 be any disk in C. Then, for wED and r' = r/(r- 1), we have

r ) (m(Ll) 1 r dm(z)) } Vw(z) 1 r ) ( 1 r dm(z) ) = ( m(Ll) } W(z)Yw(z) dm(z) m(Ll) } W(z)Yw(z) 1

A 2(Vw, Ll) = ( m(Ll) } .a Vw(z) dm(z)

.a

.a

:::;

(~~~ W(z)) 1

X

Ll

(mtLl)

r dm(z) )

( m(Ll) } .a (W(z)Y

i 1

Yw(z)dm(z))

/r (

r

1

dm(z) ) /r' m(Ll) } .a (Yw(z)Y' 1

62

6. Pseudo-holomorphic Extension

Now it can be easily proved by a direct calculation that

(

1(:Cz~)r

d ( ) ) lfr 1 ( m(Ll) .a :::5 1.

~~~ W(z) )

Consequently, we have 1

1

A2(Vw, Ll) :::5 ( m(Ll) .a Yw(z) dm(z) Next, we set q = 1 + 1/r', and q' the last inequality as 1

)( 1

1 dm( z) ) lfr' m(Ll) .a (Yw(z))r'

= qf(q -1) so that q/q' = 1/r' and rewrite

1

A 2(Vw, Ll) :::5 ( m(Ll) .a Yw(z)dm(z)

)( 1

1 dm(z) )qfq' m(Ll) .a (Yw(z))q' fq

This, together with (6.5), implies A2 (Vw, Ll) :::5 Cq for every disk Ll every wE D. We are done.

c

C and

Each A2-weight induces an L 2-bounded Calder6n-Zygmund operator. To be precise, we say that a kernel k of the type k(z) = Q(z)lzl- 2 (z E C) is Calder6nZygmund kernel if Q is homogeneous of degree zero: Q(cz) = Q(z) for z E C and c > 0, but also has mean value zero on the unit circle: il(z)ldzl = 0. For such a kernel we consider the singular integral

JT

(Kf)(z) = p.v. =lim e-+O

L

f(z- w)D(w)lwl- 2dm(w)

1

f(z- w)D(w)lwl- 2dm(w),

lz-wl>e

and the operator K : f -+ K f is called a Calder6n-Zygmund operator. The well known result on the boundedness (cf. (44] for example) of Calder6nZygmund operator is to say that if w is an A 2-weight with A 2 (w) :::5 C and if Q satisfies also a "Dini-type" condition, i.e.,

1'

sup{j!J(zt)- !J(z2)l: Zt,Z2 E T, lzt- z2l < p}p- 1dp < oo,

then one has

L

2

I(Kg)(z)l w(z) dm(z) :::5

L

lg(z)l 2 w(z) dm(z),

g E L 2 (w).

(6.6)

Here it is worth pointing out that the constant before the right-hand side integral of (6.6) depends only on both C (the A2-bound of w above) and IIKII£2-t£2 (the norm of K in the unweighted L 2-space).

6.3 Pseudo-holomorphic Continuation

63

6.3 Pseudo-holomorphic Continuation In this section, f> denotes the closed unit disk and Dc the region C \ z* = 1/z for z E C \ {0}. For z = x + iy, let

f>. Put

be the Cauchy-Riemann operators.

Theorem 6.3.1. Let p E (0, 1) and f E nqE(O,oo)Hq. Then f E QP if and only if there exists a function F E C 1 (Dc) satisfying: (i) F(z) = 0(1) as z-+ oo, (ii) limr~l+ F(r() = f(() a.e. on T and in Lq(T) for all q E [l,oo),

(iii) 8 1 F8Z~z) wED}Dc sup {

2 1

(l¢w(z)l 2 - l)P dm(z) < oo.

Proof. Necessity: let f E QP, then we show that the above F exists. Set F(z) = f(z*) for z E Dc. It is clear that F is C 1 on Dc and satisfies (i) and (ii). Now letting w E D, making the change of variables z = u* in the integral which appears in (iii) and noticing that I8F(z)j8zl = lf'(z*)llz*l 2 , we obtain

l" ~a~;z)

2 1

(lw(z)l 2 - 1)" dm(z)

=

2

fn1J'(u}l (lw(uW 2 - 1)" dm(u).

Then (iii) follows from Theorem 6.2.1. Sufficiency: suppose that there is an F E C 1 (Dc) such that the above conditions (i), (ii) and (iii) hold, we verify f E QP. Fix z E D and R > 1. In view of (ii), the Cauchy-Green formula applied to the function that equals f in D and Fin Dc gives

f(z) = _1 { F(() d( _ .!:_ f 8F~~) dm(~). 27ri jlt;I=R (- z 7r jl
f'(z) = _.!:_ { 7r

Jnc

8F~~) dm(~) . 8~

Put

4>(z) = { 8F(z)j8z, 0,

(~- z) 2

if z E Dc, if zED.

Let now T be the Beurling transformation as follows:

f

g(~)

(Tg)(z) = p.v. Jc (~ _ z) 2 dm(~).

64

6. Pseudo-holomorphic Extension

It is well known that such a transformation is a Calderon-Zygmund operator. Using the previous formulas, we see

J'(z) = -7r- 1 (TcP)( z),

zED.

By Theorems 6.2.1 and 6.2.2 and (6.6)- the boundedness of Calder6n-Zygmund operator, we get that for w E D,

(Fp(J, w))P

~ =

L

lf'(z)l 2 (law(z)l- 2

-

1)Pdm(z)

L

lf'(z)I 2 Uw,p(z) dm(z)

L ~L

~

I(TcP)(z)I 2 Uw,p(z) dm(z)

l(TcP)(z)I 2 Uw,p(z) dm(z)

~

L

lcP(z)I 2 Uw,p(z) dm(z)

:S { :S

l ~n uw,p(z)dm(z) 8

L. l ~~z) 8

2

2 1

(luw(z)l 2 -l)P dm(z).

Therefore, the condition (iii) and Theorem 1.1.1 complete the proof.

6.4 JC-property To present some consequences of Theorem 6.3.1, we introduce further terminology. We recall first that, given a function v E L 00 (T), the associated Toeplitz operator Tv is defined by

(Tvf)(z) =

~

{ v(()f(() d(,

27rz }T (- z

f E Hl, zED.

A subspace X of H 1 is said to have the .IC-property provided T;p(X) ~X for any 'if; E H 00 • Moreover, iffI B E X whenever f E X and B is an inner function with fIB E H 1 , then we say that X has the F-property. It is known that the .IC-property implies the F-property: indeed, iff E H 1 , B is inner and fIB E H 1 then JIB= Tsf·

Theorem 6.4.1. Let p E (0, 1). Then QP has the JC-property. Proof. Suppose f E QP and 'if; E H 00 • We have to show that g = T;pf is neces-

sarily in QP.

Notes

65

Since g is the orthonormal projection of j{; onto H 2 , one has j{; = g + h for some hE H6, the space off E H 2 with f(O) = 0. Thus, g = f{;

-li a.e. on T.

(6.7)

Now, since f E QP, Theorem 6.3.1 tells us that there is a function FE C 1 (Dc) obeying (i), (ii) and (iii). Further, we set, for z E De,

lli(z)

= '1/J(z*);

H(z)

= h(z*)

and finally

G(z) = F(z)lli(z)- H(z). In what follows, we claim that GIT = g (the boundary values are taken in the sense of radial convergence a. e. on T and in each Lq with q < oo) and

I

la;~z) ~ II.PIIHoo la~~z)

I,

z

nc.

E

(6.8)

In fact, the equation GIT = g follows from (6.7) and the facts:

nc, and so

while (6.8) holds because lli and Hare holomorphic in

8G(z)

az

= lli(

z

)8F(z)

az '

zE

nc

·

Since G is obviously C 1 on nc and bounded at oo, we now conclude from GIT = g and (6.8) that the analogies of (i), (ii) and (iii) hold true with G and gin place ofF and f. Another application of Theorem 6.3.1 yields g E QP, as desired. Of course, this theorem in turn implies that each QP also enjoys the :Fproperty.

Notes 6.1 Most of Section 6.1 is from the papers by Nicolau-Xiao [95] and Xiao [136]. For an alternate proof of Theorem 6.1.1, see Essen-Xiao [63].

6.2 Holding the properties of the boundary value functions of Qp, we may consider the pseudo-holomorphic extension of the Qp-functions across the unit circle T. This idea is closely related to Dyakonov-Girela's work [56] whose main contents are adapted as the above Sections 6.2, 6.3 and 6.4. We refer to Dyn'kin's paper (57] for similar descriptions of the classical smoothness spaces, as well as to. Dyakonov's work [53] for other important applications of the pseudo-holomorphic extension method. In addition, Theorem 6.2.1 has a sort of higher dimension version; see Gurlebeck-Kahler-Shapiro-Tovar [73] for details.

66

6. Pseudo-holomorphic Extension

6.3 Though all the results in this chapter live on the condition p E (0, 1), it is readily seen from Ravin [74] that the endpoint spaces V and BMOA enjoy the K:-property. Here it is also worth pointing out that the Bloch space B fails at this property; see Anderson [6]. For the setting of the weighted Dirichlet spaces, see also Dyakonov [52], [54] and Rabindranathan [102].

7. Representation via 8-equation

The Fefferman-Stein decomposition theorem for BMOA is to say that every BMOA-function f can be written as the sum !1 +ih where !1, f2 E 1l and Refi E L 00 (T). The main aim of this chapter is to extend this result to QP, p E (0, 1). This aim will be realized via: introducing Qp(T), the non-holomorphic version of Qp; finding the Qp(T) n L 00 (T)-solutions to the a-equation; and presenting the Fefferman-Stein type decomposition for QP. As certain applications of the a-equation, we give the corona theorems for both Qp n H 00 and Qp, and then show the interpolation theorem for QP n H 00 •

7.1 Harmonic Extension Theorem 6.1.1 induces the following concept.

Definition 7.1.1. Let p E (0, 1) and f E L 2 (T). Then we say f E Qp(T) provided llfll 2 * = sup III-p /, /, If(()- f~)l QP, I~T I I I( -?JI 2 P

where the supremum is taken over all arcs I For

f

E L 1 (T), let

~

2

ld(lld1JI < oo,

T.

j be the harmonic extension off to D, i.e., J(z) =

L

f(()df-lz((), zED,

where f-lz is still the Poisson measure defined in Chapter 5. Below is the Stegenga's lemma.

Lemma 7.1.1. Let I and J be two arcs on T centered at (o = eiso with IJI ~ 3II!. Iff E L 2 (T). Then

{

Js(I)

IVJ(z)l 2 (1-lzi 2 )Pdm(z)

~

{ { }J}J

+ III2+p

lf(e~t)- ((e~)l le~t-ezsl2

( {

llti~21Jif3

2

dtds

P

lf(ei(t+s:))- fJ!dt)2

t

68

7. Representation via 8-equation

Proof. Assume without loss of generality that (o is 1 and

In the integral with the gradient square, !I contributes nothing since it is constant. Accordingly, IV/1 2 is dominated by IV/31 2 and IV/2! 2 • For z = rei 9 in the Carleson box S(I),

i9 l\7!3(re )I~ A

1271" (1 o

l!3(eit)l + (0 t)2dt ~ - . r)2

1 lti~2IJI/3

dt it lf(e ) - !Jit2

and therefore by the elementary estimates,

{ j S(I)

IV/3(z)l (1 -lzi 2)Pdm(z) 2

~ III

2 2

+P (

lf(eit)- JJI d:) t jlt1~21Jif3 {

Now for the integral over S(I) of IV/2! 2 we replace S(I) with D, and using the Fourier series of !2 (see also the proof of Lemma 6.1.1) we obtain

For T1, we observe that for (, 'T] E T,

Thus, we need only to estimate

which gives the estimate for T1. The T2 and T3 terms are handled similarly as the last estimate, using h(() = 0 for ( rt J. Combining the above inequalities implies the lemma.

7.1 Harmonic Extension

69

Motivated by those characterizations of QP in the previous chapters, we can easily establish the following result.

Theorem 7.1.1. Let p E (0, 1) and let f E BMO(T). Then the following conditions are equivalent:

(i) f E Qp(T). (ii) sup

( f lf(ei(9+t))- f(ei9)12do) dt < oo. {III~ 2

I<;Tlo

t

P

}I

where the supremum is taken over all arcs I ~ T. (iii) IV' /(z)l 2 (1 -lzi 2 )Pdm(z) is a p-Carleson measure. (iv)

(v)

(vi)

(vii)

Proof. It suffices to verify that (i), (ii) and (iii) are equivalent. Let (iii) hold. Assume without loss of generality that I is an interval (0, III) with III S ~· If t E (0, III) then, Minkowski's inequality is applied to imply

(!.I lf(ei(v+t))- f(e'")l 2dv) ~ $ 2 L, (L I~ (ue'•)l ds) •du 1

2

+

1

21' (L l:~((l-t)e'•)l ds) •du

= Int1

2

+ Int2,

where af 1an and af 1ao are the directional derivative of f relative to the radius and the argument, respectively. Making use of Hardy's inequality (cf. [118, p.272]), we obtain

7. Representation via 8-equation

70

tl (~~~~ ~ tl (L I:/,.((1- t)e'•JI ds) 2

2

tP

dt

:5

r

j S(4I)

dt

IV /(z)l 2 (1- lzi)Pdm(z).

Meanwhile, I nt2 obeys

tl (~~~~ ~ tl (L I~~ (re'•JI ds) 2

2

tP

dt

:5

r

J8(41)

dt

IV](z)l 2 (1 -lzi)Pdm(z).

Putting these inequalities in order, we see that (iii) implies (ii). Suppose that (ii) is valid, to prove (i), it suffices to consider small subarc I ofT (say III :::; 1/4). Of course, I may be assumed to equal a subinterval (a, b) of [0, 211'), and hence some elementary calculations give

which implies (i). Finally, let us prove the implication (i) => (iii). Note that leis- eitl :::; III holds for eis, eit E I. So, if (i) holds, then

Consequently,

llf(eis)- !1lds :5 llfiiQp,*III. For simplicity, let J = 3I, III < 1/3, and (o = 0 in Lemma 7.1.1. Since flies in BMO(T), this implies that for any subarcs I and J ofT satisfying I C J and IJI:::; 3III, one has

IJI- !JI 2 Thus

:::;

l~l11f(eis)- !JI

2

ds:::; 3llf111Mo·

7.1 Harmonic Extension

71

This, together with Lemma 7.1.1, produces (iii). From the argument of the above implication (iii) ==> (i) it can be readily seen that every Qp(T)-function admits different extension than the harmonic extension. More precisely, we have Corollary 7.1.1. Let p E {0, 1) and F be a function defined on D such that FE C 1 (D) and FIT= f. If IVF(z)l 2 {1-lzi 2 )Pdm{z) is a p-Carleson measure, then f E Qp(T). For

f

E L 1 (T) let

j denote its harmonic conjugate, namely,

r (' z)

f(z) = 17r }TIm ( + _ z f(()id(i, zED. 2 And for ( E T put ](() = limr~l ](r(), the radial limit function of]. Corollary 7.1.2. Let p E {0, 1). Iff E Qp(T) then Proof. Since every

f

E

J E Qp(T).

BMO(T) enjoys the following identity:

the corollary follows directly from Theorem 7.1.1. Once P stands for the Szego projection from L 2 (T) onto H 2 : Pf(z) =

~

{ !((2 ld(l, 27r }T 1- (z

f E L2 (T), zED,

Corollary 7.1.2 has an interesting consequence. Corollary 7.1.3. Let p E {0, 1). Then P: Qp(T) tive.

--7

QP is bounded and surjec-

72

7. Representation via 8-equation

Proof. A simple calculation shows ](z)

+ i/(z) =

1 { 27r JT

~ + z f(()ld(l = 2(P f)(z)- ](0), ':.Z

zED,

which gives the boundedness of P by Corollary 7.1.2. Since P f = f E QP, Pis onto and hence the proof is complete.

f whenever

7.2 a-estimates Invoking a basic theorem due to Carleson [39], Hormander [78] proved that if gdm is a 1-Carleson measure, then there exists a function f defined on f> such that 8f j8z = g on D, and such that the boundary value function f is in L 00 (T).

Here and hereafter we use the same letter for a function on f> and its boundary value function on T. Later, Jones [83] gave two constructive methods to solve the 8-equation. Observe that L 00 (T) is not a subclass of Qp(T), p E (0, 1) (see the examples at the beginning of Chapter 5), so it is of interest to solve the 8-equation with boundary value function in Qv(T) n L 00 (T).

It K (-It()

Lemma 7.2.1. Let

llttllc1 'z,

be a 1-Carleson measure. If for z E

f>

and ( E T,

1- 1(12

=

1r(1 - (z)(z- () x exp

(1 ( + w( lwl~l(l

+ wz) dlttl(w) ) , -1 -;-- -1 -_1 - W~:, 1 - WZ

then (7.1)

satisfies So(tt) E L 1 (D) and 8So(tt)/8z = tt on D in the sense of distribution. Moreover, if z E T, then the integral in {7.1} converges absolutely and

In particular, So(tt) E L 00 (T). Proof. This is one of Jones' 8-solutions. For completeness, we give a proof. On the one hand, if his coo and has compact support contained in D, then

La(So(f'~~)h(z)) and hence

dm(z) =-

;i l

So({')(z)h(z)dz = 0,

7.2 a-estimates

L

So({t)(z) a~~) dm(z) =-

73

L

h(z) aso~~)(z) dm(z).

However, by (7.1) and Fubini's theorem, it is easy to see that

{ ( { 8h(z) lc{ So(~-t)(z) 8h(z) 8z dm(z) = Jo Jc 8z K

=-

Lh(z)d~-t(z),

(

1.£

)

)

ll~-tllc1 'z, ( dm(z) d~-t(()

so that 8S0 (~-t)(z)j8z = 1.£ follows by letting h run through the translates of an approximate identity (see also [109, p. 31]). In fact, the most important is to prove the last claim of the theorem; the other two claims follow easily from the proof given below. By the form of So(!.£) it is enough to prove the last claim for the case 1.£ ~ 0 and ll~-tllc 1 =, 1. We first note that if w, (ED and lwl ~ 1(1, then

2 w() -< 2(1-1(1 ). l1-w(l2

Re (1 + 1-w(

We also observe that the normalized reproducing kernel

obeys

llkdH2

~

2. Consequently,

Fix a point w E T. Since

w()

Re (- 1 + 1-w(

2 l1-w(l 2 '

= _ 2(1- 1(1 )

the proof of Lemma 7.2.1 will follow immediately from

(7.2) However, this follows from the integral formula J0 e-tdt = 1. Suppose for example that 1.£ = Ef=l ajbt;,i is a finite weighted sum of Dirac measures. Let 1(11 ~ 1(21 ~ .. · ~ I(NI and put 00

. - aj(l-l(jl2) T b1 , wE . 11- (jwl2

74

7. Representation via 8-equation

Then since la£;(w)l = 1 for (ED and wET,

II"::; ~b;exp t,b;) < (-

1,

because the last sum is a lower Riemann sum for J0 e-tdt. Standard measure theoretic arguments now complete the proof of (7.2). 00

Before reaching the main result of this section, we need another lemma which says that some p-Carleson measures are stable under a special integral operator.

Lemma 7.2.2. Let p E (0, 1) and define

(Tf)(z) =

f(w)

{

Jn 11- zwl2dm(w).

If dJ-1-(z) = lf(z)l 2(1 - lzl 2)Pdm(z) is a p-Carleson measure, then dv(z) = l(Tf)(z)l 2(1-lzl 2)Pdm(z) is also a p-Carleson measure. Proof. For the Carleson box S (I), we have v(S(I)) =

f

l(Tf)(z)l 2 (l-lzl 2)Pdm(z)

ls(I)

dm(w)) 2 dm(z) r )lllf(UJlll2 ~ jfS(l) (l- lzl2)p (( jrS(2l) + jD\S(2l) WZ

~ J{S(I) (1-lzi

2 )P (

lf(~)l wzl

{

JS(2I) 11 -

(1-lzi 2)P ( { jD\S(2I) j S(I)

+ {

= Intg

2 2

dm(w))

dm(z)

lf(~)l dm(w)) 11 - wzl 2

2

dm(z)

+ Int4.

For Intg, we use Schur's lemma [144, p.42]. Indeed, we consider

and its induced integral operator

(Lf)(z~ =

L

f(w)k(z, w) dm(w).

Taking a E ( -1, -p/2) and applying Lemma 1.4.1, we get

L

k(z, w)(1-

lwl 2 Y~ dm(w) ~ (1- lzi 2 Y~

7.2 8-estimates

and

L

k(z, w)(1- lzl 2 ) 0 dm(z) ::S (1 -lwl 2 ) 0

75

•

Therefore the operator L is bounded from £ 2 (D) to £ 2 (D). Once the function fin Lf is replaced by g(w) = (1 -lwi 2 )PI 2 If(w)l1s(2l)(w), we have

Intg j

::S

L(L L

g(w)k(z,w)dm(w))" dm(z) 2

lg(z)l dm(z)

= [

J8(21)

lf(z)l 2 (1 -lzi 2 )P dm(z)

::S ll~tllcpiJIP. Since d~t(z) = lf(z)l 2 (1 - lzi 2 )Pdm(z) is a p-Carleson measure, lf(z)ldm(z) is a 1-Carleson measure. In fact, the Cauchy-Schwarz inequality gives that for the Carleson box S(I),

This deduces

These estimates on I nt 3 and I nt 4 imply that v is a p-Carleson measure. Theorem 7.2.1. Let p E (0, 1). If lg(z)l 2 (1-lzi 2 )Pdm(z) is a p-Carleson measure, then there is a function f defined on D such that

8f(z) = g(z), 82 and such that the boundary value function

z E D,

f belongs to Qp(T) n L (T). 00

Proof. By the hypothesis of Theorem 7.2.1 and the Cauchy-Schwarz inequality, gdm is a 1-Carleson measure. Thus, by Lemma 7.2.1, the function f = So(Jt) (where d~t = gdm) is defined on D. More importantly, the function f = So(Jt) satisfies the equation 8j /8z = g on D. Furthermore, the boundary value function

7. Representation via 8-equation

76

f is in L (T). However, our aim is to verify that the boundary value function flies in Qp(T), so we must show llfiiQP•* < oo. For this purpose, let 00

F

i

{

(z) = ; lo

x exp

1-1(12 11- (zl 2

(1 ( + w( lwl~l
wz) lg(w)ldm(w) ) g(()dm((). -_-1 -;-- -1 + 1 - w":. 1 - wz

Observe that F(z) has the same boundary values as zf(z) on T. So, from Corollary 7.1.1 we find it to be sufficient to check that l\7 F(z)l 2 (1- lzi 2 )Pdm(z) is a p-Carleson measure. Since gdm is a 1-Carleson measure, by the proof of Lemma 7.2.1 one has

Re ( {

Jlwl~l
~~lg(w)ldm(w)) ~ 1

1 + 1 - w":.

and

thus I\7F(z)l

~

{

lg(~)l

Jo 11- wzl2

dm(w).

(7.3)

Since lg(z)l 2 (1 - lzi 2 )Pdm(z) is a p-Carleson measure, an application of Lemma 7.2.2 to (7.3) produces that l\7 F(z)l 2 (1 - lzi 2 )Pdm(z) is a p-Carleson measure. The proof is complete.

7.3 Fefferman-Stein Type Decomposition As is well known, there is a close relation between the a-equation and the Fefferman-Stein decomposition asserting that any f E BMO(T) can be decomposed into f = u + v, where u, v E L 00 (T). So, it is not surprising that solving the a-equation with appropriate estimates leads to the following assertion.

Theorem 7.3.1. Let p E (0, 1) and f E L2 (T). Then f E Qp(T) if and only if f = u + v, where u, v E Qp(T) n L 00 (T).

Proof. Iff= u + v, u, v E Qp(T) n L00 (T), then it follows from Corollary 7.1.2 that v E Qp(T) and hence f E Qp(T). On the other hand, it is enough to consider the case that f E Qp(T) is realvalued. We find immediately that F = f +if E Qp(T) and its harmonic extension F E QP. It turns out from Theorem 7.1.1 that l\7 F(z) 12 (1 - lzi 2 )P dm(z), and then l8f(z)/8zl 2 (1 - lzi 2 )P dm(z) is a p-Carleson measure. Let dp,(z) =

7.4 Corona Data and Solutions

77

(8f(z)/8z)dm(z) and let f,_,(z) be the function given by Theorem 7.2.1; then 8f,_,(z)/8z = f.L and f,_, E Qp(T) n L 00 (T). Hence g = f- f,_, is holomorphic and g E QP. Put u =Ref,_,, then f- u =-!mg. So f = u + v, where u =Ref,_, and v = -Img belong to Qp(T) n L 00 (T).

Theorem 7.3.2. Let p E (0, 1) and f E H 2 • Then f E QP if and only iff= !I+ h where !I, hE 1-l and Re!l, Ref2 E Qp(T) n L00 (T). Proof. This follows immediately from Theorem 7.3.1.

The following result improves Corollary 7.1.3.

Theorem 7.3.3. Let p E (0, 1). Then the Szego projection P maps Qp(T) n L 00 (T) onto QP. Proof. It suffices to show that Pis onto. By Theorem 7.3.1, we see that iff E QP then there are g, hE Qp(T) n L 00 (T) such that f = g +h. This gives f = Pf = Pg +Ph= Pg

+ 2- 1 (ih + h- h(O)) =

P(g- ih)

+ h(O)- h(O),

concluding the proof.

7.4 Corona Data and Solutions Carleson's corona theorem [38] asserts that if g~, 92, · · ·, 9n (corona data) in H 00 satisfy infzED E;= 1 lgj(z)l 2:: 1 > 0, then there exist f~, h, · · ·, fn (corona solutions) in H 00 such that Lj=l fi9i = 1. More is true.

Theorem 7.4.1. Let p E (0, 1) and n EN. If g~, · · · ,gn E QP n H 00 satisfy n

1 = inf

zED

then there exist !1, f2, .. ·, fn E QP

L

k=l

nH

l9k(z)l > 0,

00

such that

(7.4)

E;=l /jgj =

1.

Proof. The argument will use Theorem 7.2.1 and Wolff's 8-approach (of proving Carleson's corona theorem). Suppose (gl, ... ,gn) E Qp n H 00 X ... X Qp n H 00 obeys (7.4). To find (g~, .. ·, 9n) E QP n H 00 x .. · x QP n H 00 such that E;=l /jgj = 1, let hk = ""'n

!Jk

L.Jj=l

IgJ·! 2 ,

k = 1, · · ·, n.

(7.5)

Then (h 1 , · • ·, hn) is a solution to the equation L~=l gkhk = 1. But this solution is not holomorphic. So, like in the H 00 setting (see [66, pp. 324-325]), we must modify (7.5). Without loss of generality, by the normal family principle we may assume that each 9k is holomorphic on some neighborhood off>. Suppose that we can find functions bj,k, 1 ~ j, k ~ n, defined on f> such that

78

7. Representation via 8-equation

8bj,k(z) = h ·(z) 8hk(z)

a-z

a-z ,

J

z

E

D

,

and such that the boundary value functions bj,k are in Qp(T) n L 00 (T). Then n

fk = hk + 'L)bk,j - bj,k)gj j=l

belongs to QP n H 00 and satisfies E~=l fkgk = 1. Thus we have only to show that these 8-equations admit Qp(T)nL 00 (T) solutions. It is enough to deal with an equation 8b/8z = h, where b = bj,k and h = hj8hk/8z. Because each gk is in QP n H 00 , lg~(z)l 2 (1-lzi 2 )Pdm(z) is a p-Carleson measure. Also because of

lh(z)l 2 (1 - lzi 2 )Pdm(z) is a p-Carleson measure. Therefore, with the help of Theorem 7.2.1, we get a function b defined on f> such that b satisfies 8bj8z = h on D, and such that the boundary value function b lies in Qp(T) n L 00 (T), as desired. Theorem 7.4.1 can be extended to QP via its multiplier space. To see this, denote by M(Qp) the set of pointwise multipliers of QP, i.e.,

M(Qp) = {f E QP: Mig= fg E QP whenever g E Qp}· The following conclusion gives a description of M(Qp)·

Theorem 7.4.2. Let p E (0, oo). Iff E M(Qp) then f E H 00 and log2 2

llfiiL(Q ) = sup P

I~T

III TIT p

1

S(I)

2

lf'Cz)l (1 -lzi)Pdm(z)

< oo,

(7.6)

where the supremum ranges over all subarcs I ofT. Conversely, iff E H 00 and lf'(z)l 2 (1-lzi)Plog2 (1-lzl)dm(z) is a p-Carleson measure, then f E M(Qp)· Proof. Let f E M(Qp)· Observe that for a fixed w E D, the function gw(z) = log(2/(1- wz)) belongs to Qp with SUPwED llgwiiQp ::5 1 (cf. Corollary 3.1.1 (iii)). Then fgw E QP with lllfgwlll ::51 for all wE D. Since any function g E Qp has the following growth (cf. ( 1. 7)): 2 lg(z)l ::5 lllglllQP log 1 -lzl, zED, this, together with lllfgwlll ::5 1, gives that 2

if(z)gw(z)l ::5 lllfgwiiiQP log 1 -lzl' zED,

(7.7)

7.4 Corona Data and Solutions

79

so that f E H 00 • Concerning (7.6), we argue as follows. Because off E M(Qp), it follows from Theorem 4.1.1 that for the Carleson box S(I),

and so that 2

(lll/9wlll~ + 11/llhoo)IIIP .

2

{ lf'(z)l l9w(z)l (1-lzi)Pdm(z) :::5 ./s(I) 9

p

9

Note that if w = (1 - III)ei and ei is taken as the center of I then for all z E S(I), log2/III ~ l9w(z)l. Whence (7.6) is forced to come out. On the other hand, assume that f E H 00 and l/'(z)l 2 (1 - lzi)P log2 (1 lzl)dm(z) is a p-Carleson measure. With the help of (7.7) we deduce that if 9 E QP then for the Carleson box S (I),

f (·. ·) = f ./S(l)

l(f9)'(z)l 2 (1-lzi)Pdm(z)

./S(I)

:::5

1119111~

{ P ./S(I)

+ llfllhoo

If' (z)l 2 (1 -

lzi)P log 2 (1 - lzl)dm(z)

l9'(z)l 2 (1 -lzi)Pdm(z),

{ ./S(I)

and hence !9 E QP. In other words, f E M(Qp)· The proof is complete. The QP, p E ( 0, 1), corona theorem is formulated below.

Theorem 7.4.3. Let p E (0, 1) and (91, · · ·, 9n) E 1l x 1-l· · · x 1-l. Also for (f1 , · · · , f n) E 1l X 1l · · · X 1l let n

M(gt,-··,gn)(fl, · · · 'fn) =

L fk9k· k=l

Then M(g 1 ,92 , .• ·,gn) : QP X QP X • • • X QP --+ QP is surjective if and only if M( Qp) x M( Qp) X • • • x M( Qp) sat~sfies (7.4).

(91, 92, · · ·, 9n) E

Proof. Suppose that M(g 1 , •• ·,gn) : QP x QP x · · · x QP --+ QP is surjective. Evidently, it is enough to check (7.4). For this, we use the open map theorem to get that to f E QP there correspond !1, !2, · · ·, fn E QP with 111/kiiiQp :::5 IIIJIIIQP and f = I:~=l fk9k· In particular, by taking f(z) = log(1 - ze-i 9 )/2 we obtain ·

log

I which implies (7.4).

1

-

;e

-i9

I

2

::>log l-lzl

f; lak(z)l, n

80

7. Representation via 8-equation

On the other hand, let (gt,g 2 , • • • ,gn) E M(Qp) x M(Qp) x · · · x M(Qp) and (7.4) hold. In order to show that Mc 91 , .• ·,gn) : QP X QP x · · · x QP-+ QP is surjective, we must verify that for every f E QP, there are ft, h, · · ·, fn E QP to ensure the equation: 2:::~= 1 fk9k =f. By the proof of Theorem 7.4.1, we see that hk in (7.5) are non-holomorphic functions satisfying 2:::~= 1 gkhk = 1. However, if we can find functions bj,k (j, k = 1, 2, · · ·, n) defined on D to guarantee bj,k E Qp(T) and

on D, then

n

1i =

+ 2~/bj,k- bk,j)gk

fhj

k=1

just meet the requirements: 2:::~= 1 !k9k = f and fj E QP. Note that fhj E Qp(T) can be figured out from the following argument. Obviously, we are required only to prove that 8bj8z = fh (where b = bj,k and h = hj8hk/8z) admits Qp(T)solution. To this end, we choose a standard solution to 8bj8z = jh, that is,

r f(()h(() dm((). z- (

b(z) = ~

(7.8)

1o

1r

It is easy to see that this solution is C 2 on D, but also continuous on C. Certainly, we cannot help checking whether or not such a solution belongs to Qp(T). From the conditions f E QP and 9k E M(Qp) as well as Theorems 7.4.2 it turns out that for the Carleson box S(I),

1

(···)a =

S(l)

1

S(I)

=

8b(z) 12 8z

- _ (1 -lzi)Pdm(z) 1

r

lf(z)h(z)j 2 (l -lzl)Pdm(z)

1s(I)

::5

+

t 1r

lf'(z)gk(z)l 2 (l-lzl)Pdm(z)

k=1

S(I)

k=1

S(I)

t 1r

l(fgk)'(z)l 2 (1-lzl)Pdm(z).

For convenience, we reformulate the Beurling transform of a function Lloc(C) as (T(~))(z) = p.v. If~=

1

(

~(w)

c z-w

~ E

)2 dm(w), z E C.

fh on D and~= 0 on De, then 8bj8z = Carleson box S (I),

(T(~))(z)

and hence for the

7.4 Corona Data and Solutions

{

j 8(1)

81

18b(z) 12 (1 -lzi)Pdm(z)

(·.·)a= { } 8(1)

OZ

I(T(18(2/)4>))(z)l 2(1- lzi)Pdm(z)

:::; 2 { 18(1)

+2

I(T((1-18(2/)4>))(z)l 2(1-lzi)Pdm(z)

{ }8(1)

:::; 4

L

I(T(18(21)4>)(z)l 211- lziiPdm(z) 2

+4 f

J8(1)

=

(f

lf(w)h(w)l dm(w)) (1 -lzi)Pdm(z)

JD\8(21)

lw - zl 2

Int1 + Int2.

Since 11 - lziiP is an A2-weight for p E (0, 1) (cf. [44]) and the Beurling transform is a Calder6n-Zygmund operator, it follows that

Int1

::::5 ::::5 ::::5

L L

IT(18(2J)cP)(z)l 211-lziiPdm(z)

2 l(18(2l)fh)(z)l 11 -lzi!Pdm(z)

{

J8(21)

lf(z)h(z)i 2 (1-lzi)Pdm(z)

IIIP,

::::5

where the constants involved above and below may depend on the norms of the Beurling transform and the given functions f, h and 9k. Due to 9k E M(Qp) once again, Theorem 7.4.2 implies

{

J8(1)

lg~(z)l 2 (1-lzi)Pdm(z) ::::5 I~IP 2

•

log T1T

Accordingly, by the Cauchy-Schwarz inequality one has

{

j 8(1)

lf(z)h(z)ldm(z)

::::5

t j{ k=l

(lf'(z)gk(z)l

+ l(gfk)

1

z)l)dm(z)

::::5

III,

8(1)

that is to say, fhdm is a 1-Carleson measure. This fact is applied to deduce

Int2

::::5

{ }8(1)

::::5 ::::5

{

j 8(1) IIIP.

(£: f (£:

i=l}8(2i+IJ)\8(2il)

j=l

l~(w)h(p)i dm(w)) w-z

2

(1 -lzl}"dm(z) 2

1 22J-1112 j{8(2i+ll) lf(w)h(w)ldm(w)) (1-lzl}"dm(z)

82

7. Representation via 8-equation

The above estimates on Inti, j = 1, 2 tell us that

{ ls(I)

laba(z) 12(1-lzi)Pdm(z) :j IJIP, z

and so that

{

j S(I)

lVb(z)l 2 (1-lzi)Pdm(z) .

:j

IJIP.

By Corollary 7.1.1 we see that b lies in Qp(T). This completes the proof.

7.5 Interpolating Sequences In order to solve the interpolation problem for QP n H 00 , we pause briefly to work with Khinchin's inequality. Given finitely many complex numbers a 1 , ···,an, consider the 2n possible sums LJ=l ±aj obtained as the plus-minus signs vary in the 2n possible ways. For q > 0 let

denote the average value of I LJ=l ±ajlq over the 2n choices of sign. The following lemma is a special case of the so-called Khinchin's inequality. Lemma 7.5.1. Let q E (0, 2]. Then

(7.9) Proof. The proof is from Garnett's book [66, p.302], but we include the proof for completeness. Let n EN and [l be the set of 2n points w = (wb w2, · · ·, wn), where Wj = ±1. Define the probability f.J, on [l so that each point w has probability 2n. Also define X(w) = Ej= 1 ajWj· Then X(w) is a more rigorous expression for E ±aj, and by definition

Meanwhile, let Xj(w) = Wj, j = 1, 2, · · ·, n. Then IXj(w)l 2 = 1 and for j =F k, £(XjXk) = 0 since XjXk takes each value ±1 with probability 1/2. This means that {Xb X2 , • · ·, Xn} are orthonormal in L 2 (f.J,). Because X= Ej= 1 ajXj and because q E (0, 2], Holder's inequality implies ·

7.5 Interpolating Sequences

83

A sequence {Zn} C D is called an interpolating sequence for QP n H 00 if for each bounded sequence {Wn} C C there exists a function f E QP n H 00 such that f(zn) = Wn for all n E N. With Lemma 7.5.1, we can establish the following theorem.

Theorem 7.5.1. Let p E (0, 1). Then a sequence {zn} CD is an interpolating sequence for Qp n H 00 if and only if {Zn} is separated, i.e.

·rlzn-Zml

0 m >, m#n 1- ZnZm

and at the same time dJl-{zn},p = En(l-lzni 2 )P8zn is a p-Carleson measure. Proof. The part of necessity combines Khinchin's inequality and a reproducing formula for Vp, p > 0. The reproducing formula of Rochberg and Wu (105] asserts that for f E Vp, one has f(z) = f(O)

+fo

j'(w)K(z, w)(l-lwi 2 )P dm(w), zED,

where

K(z, w)

=

(7.10)

(1 - zw) 1+P- 1

w(1- zw)l+P ·

Now assume that { zn} is an interpolating sequence for QP n H 00 • Then for €~) = ±1, j = 1, · · ·, 2n, k = 1, · · ·, n, there are fi E QP n H 00 such that fi(zk) = f~), k = 1, ···,nand llhiiH= + lllhiiiQP ~ 1. Applying (7.10) to fi oaw at aw(zk) we get

Since n

n

L(l-law(zk)I )P =

L €~) fi(zk)(1- law(zk)I )P

k=l

k=l

2

2

n

=

fi(w) LE~)(1-Iaw(zk)I 2 )P k=l {

+ Jn =

(

fi

0

T1 +T2,

I

aw)

~ €~) K(aw(zk), e)

dm(e)

(e)~ (1- law(zk)I 2 )-P (1 -lei 2 )-P

84

7. Representation via 8-equation

we may compute the expectation of both sides of this equality. Observe that by (7.9) with q = 1 we find

In the meantime, applying the Cauchy-Schwarz inequality, Lemma 1.4.1 and (7.9) with q = 2, we get

£(T2) ::; sup II! o crwllvv j

So, the estimates involving £(Tl) and £(T2 ) indicate that the second condition of Theorem 7.5.1 holds. Since {zn} is also an interpolating sequence for H 00 , the first condition holds as well. To demonstrate the part of sufficiency we suppose that { zn} satisfies the above assumption. By the Cauchy-Schwarz inequality we see that Ln (1 2 lznl )8zn is a 1-Carleson measure and then by the argument in [66, p. 287] that { Zn} is uniformly separated, namely,

. IT I1 _ _

Zn- Zm

'f/ = mf

Now, for any {wn} E

zoo

I > 0.

ZnZm

n mi=n

let

Here and afterwards, Bn { Z)

=

IT mi=n

in which

lzm

I/Zm

lzm Zm

I(

Zm

=- Z

1- ZmZ

)

is replaced by 1 if Zm = 0. Besides,

,

Notes

85

for 'Y = 1/(2log(e/7J2 )). It is clear that f E H 00 and f(zn) = Wn for n EN. However, what we want is: f E QP. As in the proof of Theorem 7.2.1, we consider

then f(() = B(()F(() for ( E T, where

B (z) =

I ( Zm =- Z ) IJ lzm Zm 1- ZmZ

,

m

in which lzml/zm is replaced by 1 if Zm = 0. By Theorem 5.2.1 we know that BE QP. Therefore, in order to prove f E QP we only need to show that IV'F(z)l 2 (1lzi2)Pdm(z) is a p-Carleson measure, due to Corollary 7.1.1. The same argument as that leading to (7.3) gives that {En} E zoo, and so that

{

IV'F(z)l ~ ll{wn}lloo ln

G(w)

11- wzl2dm(w),

(7.11)

where m

Thus for the Carleson box S (I) one has that

and so that IG(z)l 2 (1- lzi 2)Pdm(z) is a p-Carleson measure. Employing (7.11) and Lemma 7.2.2 we finally obtain that IV'F(z)l 2(1-lzi 2)Pdm(z) is a p-Carleson measure. Therefore, the proof is complete.

Notes 7.1 Lemma 7.1.1 is from Stegenga (117]. Theorem 7.1.1 gives another proof for Theorem 6.1.1. The equivalences among (i), (ii), (iii) and (iv) of Theorem 7.1.1 are from the papers of Nicolau-Xiao [95] and Xiao (136], respectively. In fact,

86

7. Representation via 8-equation

these equivalences show that Qp(T) consists of all Mobius bounded functions in the Sobolev space .C~(T) on T, namely, f E Qp(T) if and only if sup

wED

llf o CTw- f(w)ll.c2(T) < oo; P

see also (136]. Here we say that a measurable function fonT belongs to .C~(T) provided

llfll.c2(T) P

= (/, /, T T

) ! lf(w)- f(z)l 2 Iw-z 12 _P JdzlldwJ < oo.

For some relations between BMO(T) and the Sobolev spaces, we refer to Strichartz (120].

7.2 Lemma 7.2.2 and Theorem 7.2.1 are also from [95]. Lemma 7.2.2 has been employed by Suarez to study meromorphic functions [123]. 7.3 The results in Sections 7.3 and their proofs can be found in (95]. Note that the argument for Theorem 7.3.3 does not require the predual of QP' At this point, QP is different from BMOA. Nevertheless, it would be interesting to characterize the predual of QP for each p E (0, 1).

7.4 Theorems 7.4.2 and 7.4.3 are in Xiao [135]. Observe that only necessity of Theorem 7.4.1 is useful in the proof of Theorem 7.4.2. So, these theorems have been reasonably generalized by Andersson and Carlsson to the Q spaces over strongly pseudoconvex domains of en; see [9]. However, it would be interesting to give a full description of M(Qp), p E (0, 1), since the cases p ~ 1 have been figured out by Stegenga [116], Ortega-Fabrega [96] and Brown-Shields [36], respectively. Theorem 7.4.3 is available for the cases p = 0 and p ~ 1; see Nicolau (94] and Ortega-Fabrega (96],[97]. 7.5 Concerning Theorem 7.5.1 (cf. [95]), we would like to point out that Earl's constructive solution (58] for H 00 -interpolation may be modified to prove the sufficiency part of Theorem 7.5.1. In fact, Earl's construction indicates that when { Zn} is an interpolating sequence for QP n H 00 there exist interpolating functions of the form KB(z), where K is a constant and B(z) is a Blaschke product. The Blaschke product B(z) has simple zeros {(n} which are hyperbolically very close to the {Zn}. It follows that {(n} is also an interpolating sequence for QP n H 00 • Another proof involved in 8-techniques is presented in (95].

8. Dyadic Localization

This chapter contains a local analysis of QP (T) based on the dyadic portions. First of all, we give an alternate characterization of QP in terms of the square mean oscillations over successive bipartitions of arcs in T. Next, we consider the dyadic counterpart Q~(T) of Qp(T), in particular, we show that f E Qp(T) if and only if (almost) all its translates belong to Q~(T); conversely, functions in Qp(T) may be obtained by averaging translates of functions in Qp(T). Finally, as a natural application of the dyadic model of QP (T), we present a wavelet expansion theorem of QP (T).

8.1 Square Mean Oscillation From now on, using the map: t---+ e21rit, we identify T with the unit interval [0, 1), where subintervals may wrap around 0. Meanwhile, a subarc ofT corresponds to a subinterval of (0, 1). A dyadic interval is an interval of the type: [m2-n, (m + 1)2-n), n E N U {0}, k = 0, 1, · · ·, 2n - 1. Denote by I the set of all dyadic subintervals ofT (including T itself), and let In, n EN U {0} be the set of the 2n dyadic intervals of length 2-n. Similarly, if I ~ T is any interval, dyadic or not, we let In(I), n E N U {0}, denote the set of the 2n subintervals of length 2-niii obtained by n successive bipartition of I. Of course, III still denotes the length of interval I ~ T. For the sake of simplicity, we rewrite, for any interval I ~ T and an £ 2 (I) function j,

f(I) =!I=

l~l ~ f(x)dx,

the mean of f on I, and define

ifJJ(I) =

l~l ~ lf(x)- f(I)I 2 dx,

the square mean oscillation of f on I. Obviously, ifJt(I) < oo if and only if f E L2 (I); we may extend the definition to all measurable functions f on I by letting ifJt(I) = oo when f tt L 2 (I). Recall that f E BMO(T) if and only if sup 1 ifJt(I) < oo, where the supremum is taken over all intervals in T. Moreover, the forthcoming two identities are easily verified.

88

8. Dyadic Localization

1~111f(x)- al 2 dx = !Pt(I) +If(!)- al 2 , a E C;

(8.1)

~~ 2 111f(x)- f(y)l 2 dxdy = 2~J(/).

(8.2)

and

Furthermore, if

Is; J, then by (8.1), (8.3)

Similarly,

If(!)- f(Jll 2

:,;

1

1~/!Pt(J).

(8.4)

2-pk~~(J).

(8.5)

Given an interval I ~ T and an £ 2 (I) function

f, set

00

lffJ,p
L: I: k=O JEik(l)

The goal of introducing (8.5) is to investigate mean oscillation of each Qp(T) function. To achieve this we need two preparatory lemmas. Lemma 8.1.1. Let p E (0, 1) and I~ T be an interval. Iff E L2 (I) then

~j(I) = 2- 1

L

~J(J)

J EI1 (/)

and lffJ,p(I) ~

L

L

+ 2- 2

lf(J)- f(I)I 2

tYj,p(J)

L

+

JEI1(/)

lf(J)- f(I)l 2 •

JEI1(I)

Proof. (8.6) follows from (8.1) and

~J(I) = III- 1

L 1lf(x)- f(I)l dx L (~J(J) + lf(J)- f(I)l 2

JEI1(/)

= 2-

1

J

2

).

JEI1(I)

Next, this and (8.5) yield, since Ik(I) = 00

tYj,p(I) = ~J(I)

+L

'E

UJEI1(1)

L

Ik-1 (J) for kEN, 2-pk~J(K)

k=1 JEI1(/) KEik-l(J)

= ~J{I)

+

L

2-PtlfJ,p(J)

JEI1(I)

~

L

(tPJ,p(J)

+ ~J(J) + lf(J)- f(I)I 2 )

(tPJ,p(J)

+ lf(J)- f(I)I 2 ),

JEI1(J)

~

L JEI1(/)

(8.6)

J EI1 (I)

(8.7)

8.1 Square Mean Oscillation

89

which implies (8.7).

Lemma 8.1.2. Let p E (0, 1) and I~ T be an interval. Iff E L 2 (I), then t/1 f,p

(I) -< _1 { { lf(x)- f(y)l2 dxd . - IIIP JI JI lx- Yl 2-p y

(8.8)

Proof. By (8.5) and (8.2), l[tf,p(I) = 2- 1

f

I;

k=O

=

ll

2-pk(2-klll)-

JEik(/)

2

111f(x)- /(y)/ 2 dxdy J J

ai(x, y)lf(x)- f(y)l 2 dxdy.

(8.9)

Here and afterwards, 00

ai(x, y) = 2- 1

L L

2( 2-p)kiii- 21J(x)1J(y).

(8.10)

k=O JEik(I)

Since x, y E J E Ik(I) implies lx- Yl ~ 2-kiii, one has

ai(x, y)

~ 2k~1IIflx-yl

furthermore ai(x, y) = 0 unless x, y E I. Consequently, (8.8) follows. Although we will show the converse inequality of (8.8), before doing so we define I + t = { x + t; x E J} for I (an interval in T) and t E R, and then give a slightly weaker but more general form.

Lemma 8.1.3. Let p E (0, 1) and I~ T be an interval. Iff E £ 2 (1), then 1 { { lf(x) - f(y)l 2 1 /_III IIIP }I }I lx- Yl2-p dxdy ~ Tff -III WJ,p(I + t) dt

+ WJ,p(I).

Proof. By (8.9) and Fubini's theorem, { ( { 1 /_III ) Tff1 /_III -III WJ,p(I + t) dt = JT JT Tff -III ai+t(x, y) dt

2

lf(x)- f(y)l dx dy.

This and (8.9) show that it suffices to verify 1 /_III Tff -III ai+t(x, y) dt + ai(x, y)

t

lx - yiP- 2 IIIP ,

x,y E I.

(8.11)

First, suppose that x, y E I with lx- Yl ~ III/2 and let lEN U {0} be such that 2-l- 2III < lx- Yl ~ 2-l-liii. Then, by (8.10), and noting that x r;fi I+ t and thus ai+t(x, y) = 0 when ltl > III,

90

8. Dyadic Localization

It is easily seen that the final integral, for each J, equals IJI - lx- Yl 2:: III/2, and thus the sum over J is at least IJI/2, and (8.11) holds for lx- Yl :::; III/2. Finally, if x, y E I with lx- Yl > III/2, then, by (8.10),

aq(x, y) 2:: 2-1111-2 ~ III-Pix- Ylp-2 and (8.11) holds in this case too. To produce a full converse to the inequality in Lemma 8.1.2, we still need two more lemmas, which may also have independent interest.

Lemma 8.1.4. Let p E (0, 1). Let I, I', I" ~ T be three intervals of equal size: III = II' I = II" I, such that I' and I" are adjacent and I~ I' U I''. Then, for any

f

E

£ 2 (1' U J"), (8.12)

and

WJ,p(I) :j Wj,p(I')

+ tPj,p(I") +If(!')- f(/")1 2.

(8.13)

Proof. It follows from (8.3) and (8.6) that


The 2i different choices of J E Ij(I) yield different k, and summing over all j and J we thus obtain,

8.1 Square Mean Oscillation

91

00

WJ,p(/) =

L L:

2-Piq)J(J)

j=O JEii(I) 00

2i+l

< 2 """""'"' q>J(I~,k)

-

2PJ

L_- L_j=O k=l

+"' """""' 00

2i+l_l

L_-

j=O

L_k=l

2

IJ(lj,k)- J.(Ij,k+dl . 2PJ

(8 .14)

The first double sum on the right hand side of the last inequality is just WJ,p(I') + WJ,p(I"). In order to dominate the final double sum, consider a pair (j, k) with j 2 0 and 1 ::; k < 2i+l - 1. Let I* be the smallest dyadic interval that contains Ij,k U Ij,k+b and let the length of I* be 2-j+m, where m E N. Moreover, for 0 ::; l ::; m, let Jl and Kt be the dyadic intervals of length 2-i+l that contain Ij,k and Ij,k+I, respectively; thus Ij,k = Jo C J1 C · · · C Jm =I* and Ij,k+l = Ko C · · · C Km =I*. Using the Cauchy-Schwarz inequality and (8.4), we get m

lf(Ij,k)- J(Ij,k+dl

2

l=l m

~

Ll

2

IJ(Kl)- J(Kt-1)1)

l=l m

lf(Jt)- f(Jt-1)1 2

l=l m

~

2

m

L lf(Jt-d- f(Jl)l + L

::; (

+I: Z lf(Kt)- f(Kt-1)1 2

2

l=l

L:t (q>J(Jt) + q>J(Kt)). 2

(8.15)

l=l

If k f= 2i, then Ij,k U Ij,k+I ~I' or I", and thus II* I::; 1 and m::; j. If k = 2i however, then I* = [0, 2) and m = j + 1; in this case we modify (8.15) by observing that Jj = I' and Kj = I" and thus m

lf(Ij,k)- f(lj,k+I)I ::;

m

L lf(Jt-I)- f(Jt)l + L lf(Kt)- f(Kt-dl l=l

l=l

+ IJ(I')-

f(I")I

which by the same argument implies m

lf(li,k) - J(Ij,k+l)l ~ 2

Ll

2

( q>J(Jl)

+ q>J(Kt)) + lf(I') -

j(I")I 2 • (8.16)

l=1

We now keep j 2 0 fixed and sum (8.15) or (8.16) (when k = 2i) for 1 ::; k ::; 2i+I - 1. We observe that the intervals Jl and Kt that appear belong to Ij-t(I') U Ij-t(I"), with 1 ::; l ::; j. Moreover, each dyadic interval J in Ij-t(I') U Ij-t(I") appears at most four times as a Jl or a Kt (viz. when, in It(J), Ij,k is the rightmost interval, Ij,k+I is the leftmost interval or Ij,k and Ij,k+I are the two middle intervals). By noting that (8.16) is used only once, we then have

92

8. Dyadic Localization

2i+ 1 -1

L

j

lf(Ij,k)- f(lj,k+dl 2 ~

k=l

L

L

l=l

JEij-t(l')Uij-I(/")

l2 if!J(J) +If(!')- f(J")I 2 •

Substituting j = k + l and summing over j, we finally obtain 00

2i+ 1 -1

I: I: j=O

00

2-pjlf(lj,k)- f(lj,k+l)l 2 ~

00

:z::

I: I:

2-pk-pll 2 if!1 (J)

l=l k=O JEik(/')Uik(/")

k=l

00

+ L 2-pjlf(I')- f(J")I 2 j=O

~ Pj,p(I')

+ Pj,p(I") + If(!') -

f(I") 12 '

which completes the proof of (8.13) via (8.14).

Theorem 8.1.1. Let p E (0, 1) and f E £ 2 (T). Then f E Qp(T) if and only if sup ICT tfJf,p (I) is finite, where I ranges over all intervals in T. Moreover, for any interval I~ T and f E £ 2 (/), l[J

(I)~ _1 { { lf(x)- f(y)l dxdy.

J,p

2

IIIP J1 J1 lx- YI 2 -P

Proof. Necessity follows immediately from Lemma 8.1.2. For sufficiency, we may assume that f is defined on the real line R with constant f (I) outside I. Let I_ and I+ be the two intervals of the same length as I that are adjacent to I on the left and the right, respectively. Note that then PJ,p(l_) = PJ,p(I+) = 0 and that f(J_) = f(l+) = f(l). For each t with ltl < III, either I+ t C /_ U I or I+ t C I+ U I, and in both cases Lemma 8.1.4 shows PJ,p(I + t) ~ tfJJ,p(I). The result follows by Lemma 8.1.3. Corollary 8.1.1. Let p E (0, 1). Then Qp(T) equals the space of all f E £ 2 (T) such that sup 1 PJ,p(I) is finite, where I ranges over all intervals in T with dyadic length 2-n, n E N U {0}

Proof. Since every interval I is contained in an interval J with dyadic length it is obvious that it suffices to consider intervals of dyadic lengths in the definition of Qp(T). The proof is completed by Theorem 8.1.1.

IJI < 2111,

8.2 Dyadic Model Motivated by Corollary 8.1.1, we define a dyadic analogue of Qp(T) and give some results involving the two spaces. First, the dyadic distance d(x, y) between two points in Tis determined by

8.2 Dyadic Model

93

d(x, y) = inf{III : x, y E IE I}. For p E (0, 1), the space Q~(T) is defined as the class of all functions such that 2 2 1 J,J,If(x)- f(y)l _ II!IIQdP =sup -III (d( )) dxdy < oo. I EI p I I X, Y 2 p

f E L2 (T)

Meanwhile, BMOd(T) stands for the dyadic version of BMO(T), namely, BMOd(T) provided f is an L 2 (T) function satisfying

llfii1Mo< =

f

E

~~~ ~~ 2 ~ ~ lf(x)- f(y)l 2 dxdy < oo.

Since d(x, y) ~ lx- yl, it follows immediately that Qp(T) ~ Q~(T). The inclusion is strict; for example, it is easily seen that if f(x) = logx, x E (0, 1), then f E Q~(T), but f ~ BMO(T) because of the infinite jump at 0, and thus f ~ Qp(T).

Theorem 8.2.1. Let p E (0, 1) and f E L2 (T). Then f E Q~(T) if and only if SUPiEI Wj,p(l) < 00. Proof. Necessity follows immediately from the definition. For sufficiency, we observe that if IE I, x, y E I and kEN U {0}, then x, y E J for some J E Ik(I) if and only if d(x, y) :::; 2-kiii, and thus, by (8.10),

2k$IJI/d(x,y)

The desired result follows from (8.9). For n E N U {0} let :Fn be the a-field generated by the partition In; and for f E L 2 (T) let E(fi:Fn) stand for the function that is constant f(I) on each dyadic interval I E In.

Theorem 8.2.2. Let p E (0, 1). Iff E L2 (T) and fn following conditions are equivalent:

=

E(fi:Fn), then the

(i) f E Q~(T). (ii) For n EN U {0}, 00

L 2(

1

-p)k E(lf-

fn+ki 2 I:Fn) ~ 1, a.s ..

k=O

(iii) For n EN U {0}, 00

L 2( k=O

1

-p)k E(lfn+k+1

- fn+ki 2 I:Fn) ~ 1, a.s ..

94

8. Dyadic Localization

Proof. If IE In and J E Ik(I) ?f>J(J) = and

IJI =

In+k, then

IJI- 1

i

lf(x)- fn+kl 2 dx,

2-kiii. Hence, by definition, 00

lftJ,p(I)

~

2(1-p)k

={:;-III- j

r

00

2

lf(x)- fn+kl dx

1

= {:; 2< 1-p)k E(lf- fn+ki 2 1Fn),

which together with Theorem 8.2.1 shows the equivalence {i) Furthermore,

<==?

{ii).

00

2

E(lf- fn+ki 1Fn)

= LE(Ifn+j+1- fn+ii 2 1Fn), j=k

and thus, interchanging the order of summation, 00

L 2( 1 -p)k E(lf- fn+ki 2 1Fn) k=O

00

j

=L

L 2( 1 -p)k E(lfn+j+1- fn+ji 2 1Fn) j=Ok=O 00

~ L 2(l-p)j E(lfn+j+1- fn+ii 2 1Fn),

j=O which implies that (ii) is equivalence to (iii). The next theorem gives a surprising relation linking Qp(T), BMO{T) and Q;(T). .

Theorem 8.2.3. Let p E {0, 1). Then Qp(T) = Q;(T)nBMO(T).

Proof. The inclusion Qp(T) ~ Q~{T)nBMO(T) follows directly from the related definitions. Conversely, suppose that f E Q;{T)n BMO(T). Let I~ T be an interval of dyadic length. Then there exist two adjacent dyadic intervals I' and I" of the same size as I, such that I C I' U I". Lemma 8.1.4, Theorem 8.2.1 and {8.4) produce

Hence WJ,p(I) is bounded uniformly for all intervals I of dyadic length, and the result follows from Corollary 8.1.1. It is clear that one reason for the discrepancy between Qp(T) and Q;(T) is that Qp(T) is translation (i.e. rotation) invariant while Q;(T) is not. In fact, the forthcoming theorem shows that a function belongs to Qp(T) if and only if all its translates lie in Q~(T). To be more precise, we denote by Tt the translation operator: (rtf)(x) = f(x- t). With this notation, we have the following conclusion.

8.2 Dyadic Model

95

Theorem 8.2.4. Let p E (0, 1) and f E L 2 (T). Then f E Q~(T) if and only if SUPtETIITt!IIQdp < 00. Proof. Since every interval of dyadic length is the translate of a dyadic interval, Corollary 8.1.1 and Theorem 8.2.1 prove that

and so that the result follows. The condition that Ttf E Q~(T) for all t E T may be relaxed considerably. To see this, we need two auxiliary lemmas. Lemma 8.2.1. Let I~ T be an interval of length 2-n, n EN, and let m(t), for t E T, be the smallest integer such that the translated interval I + t is contained in a dyadic interval of length 2-n+m(t). Then

l{t: m(t) > M}l::; 2-M, MEN

(8.17)

Especially, for r E (0, oo),

1

(m(t)Ydt

T

~

f>r2l-k <

00.

k=l

IJI = 2-n+m(t) ::; 1 by the definition of m(t), m(t) ::; n, and hence (8.17) is trivial for M ~ n. For the integer M : 0 ::; M < n, let I = [a, a+ 2-n), a E [0, 1 - 2-n). Note that n > m(t) > M is equivalent to I + t ~ J for any dyadic interval J with IJI = 2-n+M. While the latter is just to say t E [k2-n+M - 2-n -a, k2-n+M - a) for k = 1, 2, ... , 2n-M. So, {t : m(t) > M} consists of 2n-M intervals of length 2-n, and thus there is equality in (8.17). In particular, l{t: m(t) = M}l::; 21-M, and Proof. Since

1

(m(t)Ydt =

T

t

M=O

Mr/{t: m(t) = M}l

~

f>r2l-k <

oo.

k=l

We are done. Lemma 8.2.2. Let E ~ T be a set with positive measure. If Ttf E BMOd(T) fortE E, then f E BMO(T). Proof. Let M E N be such that 2-M < lEI. Since llrtfiiBMOd is a measurable function in t, there exists a positive constant C < oo and a subset Eo~ E with IEol >2-M such that llrtfllnMOd ::; C fortE Eo. Suppose that I ~ T is an interval of dyadic length 2-n with n ~ M, and let m(t) be in Lemma 8.2.1. By (8.17) and our assumptions,

l{t: m(t) > M}l ::; 2-M < IEol,

96

8. Dyadic Localization

and thus there is atE Eo such that m(t):::; M. Then llrtfiiBMOd :::; C and I +t is contained in a dyadic interval J with IJI/III = 2m(t) :::; 2M. Hence it follows from (8.3) that J(I) = ,.J(I +t):::;

:~II,.J(J):::; 2Mihfli1Mo•:::; 2MC2 •

Consequently, if! f (I) is uniformly bounded for all intervals I ~ T of dyadic length :::; 2-M; this easily implies, by (8.3) and (8.6), that if! 1 (1) is uniformly bounded for all intervals I~ T, i.e., f E BMO(T).

Theorem 8.2.5. Let p E (0, I) and f E L 2 (T) . Then the following conditions are equivalent: (i) f E Qp(T) . (ii) Ttf E Q~(T) for all t E T . (iii) rtf E Q~(T) for almost all t E T. (iv) rtf E Q~(T) fort E E ~ T with lEI

> 0.

Proof. By Theorem 8.2.4, it remains only to show that (iv) implies (i)~ If (iv) holds, then Ttf E BMOd(T) for t E E ~ T with lEI > 0 since Q~(T) ~ BMOd(T), and hence Lemma 8.2.2 gives f E BMO(T). Now, choose some t E E. Since BMO(T) is translation invariant, Ttf is in BMO(T); furthermore, Ttf belongs to Q~(T) by assumption. Hence Ttf E Qp(T) by Theorem 8.2.3, and finally f E Qp(T), namely, (i) holds, due to the translation invariance of QP (T).

Corollary 8.2.1. Let p E (0, 1) and f E L 2 (T). Iff E Qp(T) then rtf E Q~(T) for all t E T, while iff f:. Qp(T) then Ttf f:. Q~(T) for a.e. t E T. For q E (0, oo], define Lq(Q~) as the class of all measurable functions F on TxT, such that F(t, ·)is in Q~(T) for a.e. t E T and IIF(t, ·)IIQ~ (as a function oft) belongs to Lq(T). As the second immediate consequence of Theorem 8.2.5, we have

Corollary 8.2.2. Let p E (0, 1) and q E (0, oo]. Then f E Qp(T) if and only if rof(·) E Lq(Q~). On the other hand, if q > 1, beginning with any function in Lq(Q~), we may construct a function in Qp(T) as a suitable average. That is to say,

Theorem 8.2.6. Let p E (0, 1) and q E (1, oo]. If F(t, ·) E Lq(Q~) and g(x) =

JT F(t, x + t)dt,

then g E Qp(T).

Proof. Write ft(x) = F(t, x) and ht(x) = F(t, x + t). Assume that I ~ T is an interval of dyadic length 2-n, n E N. Fix t E T and let (ignoring the case when I+ t is dyadic) I' and I" be the two dyadic intervals of length 2-n that intersect I. Then I~ I' U I" and Lemma 8.1.4 gives

8.2 Dyadic Model

tfJht,p(I)

97

= tfJft,p(I + t) :5 tfJft,p(I') + tfJft,p{I") + lft(I') - ft(I")I 2 :5 llftll~dp + lft(I')- ftU")I 2.

Let m(t) be as in Lemma 8.2.1, and let, for l = 0, ... , m(t), Jl and Kl be the dyadic intervals of length 2-n+l that contain I' and I", respectively. Then Jm(t) = Km(t) and, using (8.4) we get m(t)

lft(I')- ft(I")I ~

L (lft(Jl-1)- f(Jl)l + lft(Kl-1)- f(Kz)l) l=1

m(t)

:5

L

((q>Jt(Jz))1f2

+ (q)Jt(Kl))1f2)

l=1

:5 2m(t)llftiiQ~· As a consequence, we moreover obtain

JT

Because g = htdt and (tfJJ,p{I)) 112 may be regarded as an £ 2 -norm, we may use Minkowski's inequality to obtain

So, it follows from Holder's inequality and choosing r E (1, oo) with 1/r+1/q = 1 that

(tJ!g,p(I)) 112 ::$ llm(t) ll£r(T) IIFIILq(Q~) · By Lemma 8.2.1, this shows that tfJ9 ,p(I) is uniformly bounded when I~ Tis an interval of dyadic length :::; 1/2. The case I= T (which may be written as [0, 2- 1 ) U [2- 1 , 1)) follows easily, and thus g E Qp(T) by Corollary 8.1.1. Once we introduce the linear operator: (Tf)(t,x) = f(x- t) (mapping functions on T to functions on T x T), we have the following result. Corollary 8.2.3. Let p E (0, 1) and q E (1, oo). Then (i) Qp(T) is isomorphic to the complemented subspace T(Qp(T)) of Lq(Q~). (ii) The adjoint operator T* ofT maps Lq(Q~) onto Qp(T).

Proof. Since T* is given by

T* F(x) =

l

F(t,x + t)dt,

T*T is the identity and TT* is a projection. This, together with Theorem 8.2.6 implies Corollary 8.2.3.

98

8. Dyadic Localization

8.3 Wavelets The purpose of this section is to show that the well known characterization of BMO(T) by means of a periodic wavelet basis can be extended to Qp(T). We start with recalling the Haar system on T. In this section, let H denote the Haar function:

1, t E (0, 1/2), -1, t E [1/2, 1), { 0, otherwise.

H(t) =

For j E N U {0} and k = 0, 1, · · ·, 2i -1 and define hj,k(t) = 2i/ 2 H(2it- k) I[O,l). Set also ho,o(t) = 1. The system {hj,k} is called the Haar system on T, and forms a complete orthonormal basis in L 2 (T). More precisely (cf. [65]), if Ij,k = [k2-j, (k + 1)2-i) is a dyadic interval in [0, 1), and if 2j-1

L

fj(x) =

f(lj,k)1I 3,k (x),

k=O

is an approximation off at the resolution 2-j, then it follows immediately from the Lebesgue differentiation theorem that limj-+oo fi(x) = f(x) a.e. on [0, 1). Thus, for almost every x E [0, 1), 00

f(x) = fo(x)

+

L (fi+l (x) -fi(x)). j=O

However a simple argument shows 2j-1

fi+t(x) -Ji(x) =

L (f, hj,k)hj,k(x), k=O

where (f, g) means the usual inner product 00

f = f(T)

JT f(x)g(x)dx. Therefore,

2j-1

+ 2: L (J, hj,k)hj,k, j=l k=O

which shows that the Haar system represents f a. e. on T, as well as in £ 2 (T). In what follows, as to A= (j, k) we write the shorter notation hj,k ash>. and denote by!(>..) the dyadic interval {t: 2it- k E [0, 1)}. Moreover, for IE I, a sequence a= {a(>..)}, and q E (0, 1], let

8.3 Wavelets

99

Lemma 8.3.1. Let p E (0, 1). Then for each IE I and sequence a= {a(A)}, 00

Ta,p(I) ~

L 2-pk L

Ta,l(J).

JEik(l)

k=O

Proof. The right hand side equals

L L (l:fl)P la(A)I2 = III-p L JEil (I)

/(>.)~J

I(>.)~/

Ill

III

L

~ III-p

L

la(A)I2

IJIP-1

JEil (/),J2_I(>.)

la(A)I 2II(A)IP-l.

/(>.)~/

Note that every function in BMOd(T) can be described by the Haar system, that is, an L 2 (T)-function f belongs to BMOd(T) if and only if its Haar coefficients a = {a (A)}: a(A)

= (f, h>.) =

L

f(t)h>.(t)dt

satisfy supTa,l(I) < oo.

(8.18)

lEI

See also (39]. Similarly, for Q~(T) we have Theorem 8.3.1. Let p E (0, 1). Iff E Q~(T), then the sequence of its Haar coefficients a = {a( A)} satisfies

supTa,p(I) < oo.

(8.19)

lEI

Conversely, every sequence a= {a(A)} satisfying {8.19) is the sequence of Haar coefficients of a unique f E Q~(T). Proof. If a( A) = (f, h>.) and I E I, then (f- f(I))1I =

L

a(A)h>.

/(>.)~1

and thus

~J(I)

= III- 1

L

la(A)I 2

= Ta,l(I).

/(>.)~/

It follows by the definition of WJ,p(I) and Lemma 8.3.1 that tPJ,p(I) ~ Ta,p(I), and the result follows by Theorem 8.2.1. This simple theorem suggests us to consider the wavelet bases. Recall that a wavelet is a function l]! E L 2 (R) such that the family of functions Wj,k(x) = 2il 2 w(2ix- k) where j and k range over Z (the set of all integers), is an orthonormal basis in L 2 (R). For such a family, let

100

8. Dyadic Localization

'!f;j,k(x) = l:wj,k(x + l). lEZ

Then each '!f;j,k is a function on T (i.e., a 1-periodic function on R). Moreover, '!f;j,k(x) = '!f;j,k+ 2i (x) and '!f;j,k+l (x) = '!f;j,k(x + 2-i). In particular, there exists a l/1 so that {1} U {'!f;j,k} (j = 0, 1, 2, · · ·; k = 0, 1, 2, · · ·, 2i- 1) is a complete orthonormal basis in L 2 (T), viz., the 1-periodic wavelet basis. For convenience, we will write the shorter notation '!f;j,k as 'lj;>.., where A= (j, k). And for simplicity we consider only "good" wavelets, and thus suppose that each l/1 satisfies max{ll/l(x)l, ll/l'(x)l} :j (1 + lxl)- 2, x E R; but also l/1 has a compact support so that the support set of each '!f;>. obeys: supp'!f;>. ~ ml(.X), where m is a constant (fixed throughout the rest part of this section). For these, we refer to Meyer [92, Section 11 in Chapter 3] and Wojtaszczyk [130, Section 2.5]. Observe that the wavelet coefficients b = {b(.X)} of a BMO(T)-function are entirely controlled by sup lEI Tb,l (I) < oo [92, p.162]. This can be extended to Qp(T) as follows. Theorem 8.3.2. Let p E (0, 1). Iff E Qp(T), then the sequence b = {b(.X)} of its wavelet coefficients:

b(.X)

= (f, '!f;>.) =

l

f(x)'!f;>.(x) dx,

satisfies

sup Tb,p(I)

< oo.

(8.20)

lEI

Conversely, every sequence b = {b(.X)} satisfying {8.20} is the sequence of wavelet coefficients of a unique f E Qp(T). Proof. First, let

f

f

E

Qp(T) and I E I. For J E Ik(I), k E N U {0}, put

= fmJ + (J-

fmJ)1mJ

+ (J-

fmJ)1T\mJ

= fl + f2 + j3.

Since supp'!f;>. ~ ml(.X), (h, '!f;>.) = 0 if J(.X) ~ J. On the other hand, the integral of each wavelet 'l/J>. is zero. So (j, '!f;>.) = (h, '!f;>.), and, by (8.2) one has

2::

I(J,'!f;>.)l ~ 2

2:: 1(!2,'l/J>.)I 2 = llhlli2 = lmJiq)J(mJ) )..

I(>.)~J

:::>

I~JJmJ L)f(x)- /(Y)I

2

d:I:dy.

This gives that for J E Ik (I), Th,l(J) =

l~l

2 L lb(A)I 2 :o; IJII~JI { { lf(x)- /(Y)I d:I:dy. I(>.)~J JmJ JmJ

8.3 Wavelets

101

Using Lemma 8.3.1, we obtain in the same manner as for Lemma 8.1.2 00

Tb,p(I) ~ 2:2-pk

~

{

E

Tb,l(J)

JEik(I)

k=O

{

Jml lml

~ III-p {

lf(x)- J(y)l2

f

lmJ2~27:(y) dxdy

L

II

k=O JEik(I)

{

lml lml

lf(x)- f(y)l2 dxdy lx- YI 2 -P

~ llfii~P•*' Thus (8.20) follows. Conversely, suppose that (8.20) holds; multiplying f by a constant, we may assume that Tb,p(I) ::; 1 for every I E I. In particular, Tb,l (I) ::; Tb,p(I) ::; 1 for every IE I, and so f = b(A)'l/J>. E BMO(T),

L )..

with the sum converging e.g. in the weak* topology on BMO(T). We will verify

f

E

Qp(T).

Fix a (not necessarily dyadic) interval I of dyadic length and consider an interval J E I 1 (I). Let A0 ( J) = {A : mi (A) n J =I= 0} and partition this set into

A1 = A1(J) ={A E Ao(J): II(A)I::; IJI}, A2 = A2(J) ={A E Ao(J): IJI < II(A)I::; III}, A3 = A3(J) ={A E Ao(J) : III < II(A)l}. Since 1/J>. = 0 on J unless A E Ao we have, on J, fj =

L

f

=

!I + h + j3,

where

b(A)'l/J>., j = 1, 2, 3.

>.EAi

Hence, using the Cauchy-Schwarz inequality we get (8.21) In what follows, we treat the three terms separately. First of all,

tPft (J) ::; IJI- 1 II!I lli2 = IJI-l

L

lb(A)I 2.

(8.22)

AEA1

Secondly, I\71/J>..I ~ ll(A)I- 3 12 , and thus lh(x)- h(y)l ~

L

lb(A)III(A)I- 312 Ix- Yl·

>.EA2

As a consequence, we have by letting € = (1+p)/2 and using the Cauchy-Schwarz inequality

102

8. Dyadic Localization

~J,(J) :5 !J!•( I: ,)t~~~~.)" .XEA3

:5\J,.

I: :~i~~~: c~~)'r I: (,)~)r

.XEA2

AEA2

If A E A2 , then I (A) is a dyadic interval contained in an interval with the same center as 1 and length (m + l)IJ(A)I +Ill~ (m + 2)111. Hence, for each kEN, there are at most m + 2 such intervals I(A) with II(A)I = 2kiJI. Moreover, there is a constant number of different A for each such interval, and so the number of elements of {A E A2 : II (A) I = 2k I11} is finite for each k E N. Consequently,

L (J!L )e-< £:2-ke-< 1 II(A)I - k=1 -

AEA2

and """

gjh (1) ::5 LJ lb(A)I

2(

2

Ill ) -e -1 II(A)I II( A) I .

(8.23)

.XEA2

Thirdly, we similarly have

by

lb(A)III(A)I- 1/ 2 ~ r~;:(I(A)) ~ 1. Again, there is a bounded number of terms for each I(A), and now II(A)I = 2kiJI, kEN; hence lf3(x)- !3(Y)I ::5 lx- YIIII- 1 and gj13(l) ::5 lli 2III- 2· (8.24)

Consequently, by the above estimates: (8.21) through (8.24), 2 2 2 1 """ 2 """ lb(A)I ( 111 ) - € (Ill ) gjj(l) ::5 LJ lb(~)l + LJ II(A)I II(A)I + TIT .

PT AEA1

AEA2

Summing over 1 E I 1 (I) we obtain

llif,p(I)

=

I: (',~:r~,(J) JEI1{I)

-<

2: 2:

lb(A)I2 (0)p - JEil(I) AEAl(J) Ill III 2 + lb(A)I 1JIP+2-EIJ(A)Ie- 3III-p

L

L

JEI1(/) AEA2(J)

+

('~l)p+2. JEil{I) I I

2:

(8.25)

8.3 Wavelets

103

The final sum equals

L 2j (2-j)P+2 = L 2-(1+p)j ::5 1. 00

00

j=O

j=O

In the two double sums, we interchange the order of summation. If .A occurs there, then II(.A)I ~ III and mi(.A) n I =f. 0; thus, if we let :J(I) be the set of dyadic intervals J of the same size as I with mJ n I =f. 0, it follows that I(.A) E I1(J) for some J E :J(I). Fix such a .A, with II(.A)I = 2-kiii. For each j ~ k, there are at most finite many intervals J E Ij(I) with .A E A1(J), each contributing 2(1-p)JIII- 1Ib(.A)I 2 to the first double sum in (8.25). Similarly, for each integer j > k, there are at most C2(j-k) (where C is an absolute constant) intervals J E Ij(I) with .A E A2(J), each contributing

to the second double sum in (8.25). These get together to yield at most

CIII-1Ib(.A)I2

~

(~ 2(1-p)j + ~

2(1-p)k lb(.A)I2. 2(1-p)k ) III L....J 2< 1+p-€) j=k+1

L....J j=O

As a result, (8.25) gives 00

tf!J,p(I) ::5

L L L

2(1-p)kiii-1Ib(.A)I2 + 1

JE.1(I) k=O I(>.)Eik(J)

::5

L

Tb,p(I')

+ 1.

JE.1(I) We have proved that tf!J,p(I) ::5 1 for every interval I of dyadic length. Since the same estimate applies to every translate I + t, Lemma 8.1.3 shows that

Jj I

I

lf(x)- f(y)l2 dxdy

lx- Yl 2 -p

::5 IIIP

is valid for every interval I of dyadic length. Therefore, Corollary 8.1.1 implies f E Qp(T). Uniqueness off follows from the uniqueness in BMO(T); if j, g E Qp(T) ~ BMO(T) have the same wavelet coefficients, then they define the same linear functional on the predual space of BMO(T) and thus f = g as elements of BMO(T) (i.e. modulo constants), see [92, Section 5.6] once again. Corollary 8.3.1. Let p E (0, 1). Then U : E b(.A)'l/J>. ~ E a(.A)h>. sets up an isomorphism between Qp(T) and Q~(T) with the sums interpreted formally or as converging in suitable weak topologies.

104

8. Dyadic Localization

Notes 8.1 Section 8.1 is one of the main topics of Janson's paper (80]. Corollary 8.1.1 tells us that restriction of Qp(T) to dyadic intervals would give Q~(T). 8.2 Section 8.2 is also from [80]. Theorem 8.2.2 is similar to the one for BMO(T) [64]. Of course, Theorem 8.2.3 reveals a close relation between Qp(T) and BMO(T). Note that Theorem 8.2.6 is an extension to Qp(T) of a result by Garnett and Jones (67] for BMO(T). Here, it is worth pointing out that the space Lq ( Q~) is not the usual Lebesgue space of Banach space valued functions, defined as the closure of simple functions in the obvious norm. The problem is that Q~(T) is not separable, and it is easily seen that if e.g. F(t, x) = 1{(t,x):O<x 0) and defined a family of functions {
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Index

Adjoint 55, 97 Aleman's lemma 53 Almost surely (a.s.) 31-32,93 Average 59, 82,96 Banach space 3, 13, 16, 104 Bergman spaces 3 Bernoulli sequence 31 Besov space(s) 35, 39-40,44 Bessel capacity 44 Beta function 8, 24 Beurling transform 64, 80-81 BiBloch-type mappings 13 Blaschke product(s) 48-50, 55,86 Bloch(-type) space(s) 1, 2, 11, 13, 22,66 Borel measure 35,43 Boundary value behavior 57 Bounded mean oscillation 2 Boundedness 16, 18-19,62,64, 72 Calder6n-Zygmund operator 62, 64,81 Canonical factorization 45, 52 Carleson's corona theorem 77 Cauchy(-Green formula, -Riemann operators, -Schwarz inequality, sequence) 3,63,75,81,84,91,101-102 Cesaro operator(s) 55-56 Compactness 16 Composite embedding 13 Composition operator(s) 13, 16,21-22, 55 Conformal (automorphisms, invariance, map) 1,19,21,51 Conformally invariant 1, 6, 15, 20, 35,52 Cut-off 53-54 Decomposition via 8-equation Dirac measure(s) 49,73 Dirichlet space(s) 1, 3, 66 Distribution 72 Dyadic localization 87 Euclidean (distance, disk(s))

67

F-property 64-65 Fatou's lemma 3, 25 Fefferman-Stein type decomposition 76 Fejer kernel 25 Fourier series 33, 68 Fubini 's theorem 6, 52, 73, 89 Fundamental material 1 Gamma function 7 Geometric characterization Green function 1, 2, 56 Green's theorem 52

20

Haar (coefficients, function, system) 98-99,104 Hadamard (gap(s), product, series) 3, 23,33,38,43 Hardy space(s) 3, 34, 36, 39, 45,55 Hardy-Littlewood's identity 52 Hardy's inequality 69 Harmonic (conjugate, extension) 67, 71, 76 Hausdorff capacity 44 Higher derivative(s) 1, 7 Holder's inequality 4, 37-38,40, 55, 82, 97 Hyperbolic (distance, measure, metric) 20-22,51,56 Image area 1, 6, 12 Inclusion 3, 38-40, 94 Inner-outer structure 45 Integral form 35 Interpolating sequence(s) 82-84, 86 Isomorphism 103 Jensen's inequality 54 Jones' 8-solutions 72

20,61

67,

/C-property 57, 64, 66 Khinchin's inequality 82-83

112

Index

Koebe distortion theorem

20

Lacunary series 34 Lebesgue differentiation theorem 98 Lipschitz (class, function, spaces) 13, 35-37 Logarithmic capacity 44 Maximal space 12 Mean (growth, oscillation, value, variation) 35,41-42, 55, 62, 87 Minkowski's inequality 25, 46, 69,97 Mobius (bounded functions, invariant spaces, transformations) 1, 3, 12, 86 Modified Carleson measures 35 Nevanlinna functions

56,

Orthonormal (basis, projection, property) 65,82,98-100,104 Parseval's formula 8, 23, 57 Partial sums 25 Pointwise (multiplication, multipliers) 32,53,78 Poisson (kernel, measure) 51, 58, 67 Predual 86, 103 Probability 31, 53-54, 82 Pseudo (-holomorphic, -hyperbolic) 1, 57,63,65

Q classes

1, 43, 56, 86, 104

Radial (convergence, limit) 45, 65, 71 Random (series, variables) 31, 34 Reproducing (formula, kernel) 55, 73, 83 Riemann (map, sum, surfaces) 21, 56, 74 Schur's lemma 74 Series expansions 23 Singular (factors, inner function(s), integral, measure) 45,47-49,62 Stegenga's lemma 67 Stirling's formula 7-8,24 Subharmonic 3, 9 Support(s) 72, 100, 104 Szego projection 71,77 Toeplitz operator 57, 64 Translation (invariance, invariant, operator) 94,96 Univalent (function(s), self-map) 20-21

6, 12,

Wavelet (s, bases, coefficients, expansion) 87,98-100,103-104 Weight (condition, function), 60-62,81 Weighted norm inequalities 61 Wolff's 8-approach 77