This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
: D--+ D be univalent. Then (i) C
_ 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 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
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 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 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)
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
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. 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 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
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,
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)
-
