Theory of Hp Spaces

If(4l I ( 1 - l~12)-1'pllfllp~ I4 < 1. Show that the inequality is sharp for each fixed z.

5. For fixed z, IzI < 1, and for each positive integer n, find the maximum of If(")(z)l over all H 2 functions f with llfl12 2 1. Determine the extremal functions. 6. ForfE H 2 and for (zl/< 1, Iz2(< 1, and z1 # z 2 , prove

and show that the constant is best possible. 7. For each fixed z, Iz( < 1, and for each odd integer n = 2m + 1, prove the sharp inequality

and identify the extremal functions. (Hint: Show that the natural kernel is extremal.) (SzBsz [2]; Macintyre and Rogosinski [2].) 8. Let M,, denote the maximum of la, + a, + ... + a,,j among all H" functions f(z) = C akzkwith I l f l I , I 1. Show that

Hence M ,

N

I/. log n as n --+ a.(Hint: Observe that

(Landau [1,21.)

9. Find the distance from H' to the function k(z) = (1 - z-"-l)(l - z)-';

that is, find the minimum of Jjk-fill 10. Show that forf(z)

=

for allfe H I .

akzkE H ' ,

max l a o + a l + . . . + a , l

IIf II 1 5 1

+

1 (n 1)n ==-sec-. 2 2n+3

(Hint: Observe that the eigenvalue problem X , - ~ + X , - ~ + ~~ -~* + X , , = ? ~ . X ~ ,

k = = Q , I.,. . . 17,

has the solution

1 2

+

(n l)n* 2n+3'

A = - sec ____

xk = sin-

(k 4- l ) n 2n -t3 '

I<- 0, 1 ,

I

.

.

,

..>

(Egervdry [I].) NOTES

The theory of extremal problems in H p spaces has evolved from a long history of scattered work on special examples. Landau [ 1,2] solved thecoefficient problem (i) in Section 8.5 in the case p = GO and c, = c1 = . . . = L, = 1. Szzisz [3, 41 discussed the more general problem in 15". CarathCodory and FejCr [2] proposed the "minimal interpolation " problem to minimize Ilfil among all f~ H" whose Taylor coefficients a,, a , , . . . , a, at the origin are prescribed. They proved the existence and uniqueness of a wlution, and they gave an algebraic method for computing the minimum. Gronwall [l ] simplified this work. F. Riesz [l] discussed the same problem in N',proved the existelice and uniqueness of a solution, and characterized it using a variational method. Fejtr [ l ] solved the coefficient problem (i) of Section 8.5 in N ' , as well as its dual, the Garathtodory-Fejer problem, essentially by the method described in (iv) of Section 8.5. Egervgry [I] obtained a completely explicit solution i i i the case co = c1 = = c, = 1. Kakeya [I, 21 considered the more general interpolation problem (ii) of Section 8.5 and found the form of the solution. In the H" case, this problem was also discussed by Pick [ I , 21, Sclur [l], and R. Nevanlinna [I]. Geronimus [ I , 2, 31 treated some problems similar to those of Caratheodory-FejCr and Riesz. Doob [1] considered the problem of approximating an arbitrary rational kernel by an HI fimction, proved the existence and uniqueness of a solution, and found its qiiditative form. Goluzin [I, 21 discussed a wide variety of extremal problems. The first systematic attempt to unify the theory and to extend it to H P (1 5 p 5 co) was the paper of Macintyre and Rogosinski [I]. They formulated the duality relation in full generality, proved the existence and uniqueness of extremal functions and extremal kernels, described their form in the case of a rational kernel, and applied the theory to obtain explicit solutions to a great many extremal problems. Nevertheless, the theory achieved complete unity and elegance only when Havinson [1, 21 and (independently) Rogosinski and Shapiro [I] introduced 3 .

146

8 EXTAEMAL PROBLEMS

the methods of functional analysis. A. L. Shields discusses this development in a historical note following his translation of Havinson’s paper [2]. Subsequent refinements by H. S. Shapiro [l], Havinson [3], Bonsall [I], and others have led to the main existence and uniqueness theorem as stated and proved in Sections 8.1 and 8.2. The three counterexamples given in Section 8.3 are essentially due to Rogosinski and Shapiro [l]; the third is given in a simplified form suggested by Caughran [l]. Extremal problems for H P spaces over multiply connected domains (see Chapter 10) have been studied by Ahlfors [l], Garabedian [l], Nehari [2,3,4], Rudin [I, 21, Havinson [3], Penez [I], Lax [l], Tumarkin and Havinson [8], and others. Akutowicz and Carleson [l] have studied the analytic continuation properties of extremal interpolatory functions.

INTERPOLATION THEORY

CHAPTER 9

Let zl, z2 , . . . be distinct points in the open unit disk, and let ivl, w 2 ,. . . be arbitrary complex numbers. The general interpolation problem is to characterize the pairs of sequences {zk} and { w k } for which there exists a function J’E H P with f (zk) = wk , k = 1,2, . . . , and to find all of the interpolating functions. But in such general form the problem is difficult, and has not yet been fully solved (see Notes). In this chapter we confine attention to the more modest problem of describing the “ universal interpolation sequences.” These are the sequences {zk} which admit an H“ interpolation for every bounded sequence ( w k ) . This restricted problem has a natural generalization to H P (0 < p < 00) which we also discuss. The chapter concludes with a theorem of Carleson which generalizes a special result used in proving the main interpolation theorem. Carleson’s theorem plays an important part in the proof of the corona theorem (Chapter 12), and it has other applications. 9.1. UNIVERSAL INTERPOLATION SEQUENCES

Consider the problem of finding the most general function f E H P such that = W k , k = 1, 2, . . . . If a particular solutionfcan be found, it is an easy

f(Zk)

148

9 INTERPOL4TION THEORY

matter to describe the general solution. Indeed, the interpolation f is unique if and only if 1 (1 - lzkl) = co,since the difference of two interpolations would vanish at the points z,. If (1 - lzkl)< co, then the general solution is f + B g , where B is the Blaschke product with zeros (2,) and g is an arbitrary HPfunction. In discussing the existence of an H P interpolation, it is helpful to adopt a more abstract point of view. Fixing p (0 < p 5 co) and a sequence {z,}, consider the linear operator T which assigns the sequence ( f ( z k ) } to each f~ H P . The range of Tis clearly the set of all sequences {w,}for which interpolation is possible. The problem is then to characterize the range T(HP) for each sequence (2,). It is obvious that T ( H “ ) c em for every sequence (2,); and it follows from Theorem 5.9 that if Iz,~ + 1, then lim W,(I - Izkl)’’P = 0 k+ m

for each {w,} E T(HP), 0 < p < 00. It is of interest to identify the sequences {z,} for which T(H“) = em. These are called universal interpolation sequences, since to every {w,} E em there corresponds a functionfE H“ withf(z,) = w,. It is intuitively clear that { z k }cannot be a universal interpolation sequence if the points are “ too close together,” because it would then be impossible to interpolate a “ highly oscillatory” sequence {w,} by a function with “reasonably small ” derivative. This can be made precise. A sequence {zk}is said to be uniformly separated if there is a number 6 > 0 such that

The convergence of the Blaschke product implies, in particular, that C (1 - lzkl) < co. It turns out that {z,} is a universal interpolation sequence if and only if it is uniformly separated. This surprisingly sharp result can be extended to an arbitrary H P space. For this purpose it is convenient to consider a kind of weighted interpolation, Given a sequence (zk}, let T, be the linear operator on H P (0 < p 5’00) defined by In particular, T, = T. We noted above that if p < co, T, maps HP into the space of sequences which tend to zero, provided only that ( z k } has no limit point in the open disk. On the other hand, T, need not map HP into P, even if (1 - lzkl)
9.2 PROOF OF THE MAIN THEOREM

149

For 0 < p i 00, T,(HP)= L p

THEOREM 9.1 ( M a i n interpolation theorem).

if and only if { z k }is uniformly separated. 9.2. PROOF OF THE MAIN THEOREM

Let us first assume 1 I p i co, deferring the case 0 < p < 1 to Section 9.3. It is obvious that for every sequence { z k } , T = T, is a bounded operator from H“ to L“. More generally, if { z k }is such that T,(HP)c P,then T, is a bounded operator from H P into P.This will follow from the closed graph theorem if it can be shown that T, is closed. But suppose f,EHP,

L+f,

and Tp(f,)

= {(l

- Izk12)”pfn(zk)}

-+

Lp*

’v

Thenf,(z,) + f ( z k ) for each k,so T,(f)= w.This proves T, is a closed operator. Suppose now that T,(HP)= P,and let

N P = (fE H” : f ( Z k ) = 0, k

=

1, 2,

. . .)

denote the kernel of T, . The quotient space H P / N Pis a Banach space under the usual norm. (See Section 7.1.) Since T,(HP) = L”, T, induces a one-one bounded linear operator 9,from H P / N p onto P.By the open mapping theorem, 9,’ is bounded. In other words, corresponding to each w = { w k } E P,there is a function f E H P with

k = 1, 2, .. .

(1 - I z k 1 2 ) ’ / ” f ( z k ) = wk,

(1)

and

llfll 5 M I l M 4 , where M is a constant depending only on p . In particular, for each positive integer k there is a functionf, E H P with llfkll I M and .fiCzj>

=

((1 -

,

j =k j#k.

For n > k, let

Then F , k E H P and IlFnklI = llfkll 5 M. On the other hand, for any F follows from Theorem 5.9 that IF(z>l i CIIFIl(1 - 142)-1/p,

E H P , it

150 9 INTERPOLATION THEORY

where C depends only on p . Consequently,

ICllFnkI/ 5 C M .

This shows that {zk} is uniformly separated if Tp(Hp)= eP. The converse is more difficult. The plan is to prove it first in the special case p = 2, then to apply this to establish the general result. Two lemmas will be needed. LEMMA I.If

{zk}is a uniformly separated sequence, then

where A is a constant independent of k. PROOF.

A simple calculation gives

Thus

This gives the desired result with A = 1 - 2 log 6. LEMMA 2.

Let ajk ( j , k = 1, 2,

...,n)

be complex numbers such that

akj = aiikand n

C lajkl I M , j=1 Then for any numbers xl,

...,xn,

k = 1,2,

... , n.

9.2 PROOF OF THE MAIN THEOREM

151

PROOF. Write lajk x j

%I = ( l a j k ( 1 / 2 1 x j ~ ” a j k 1 1 / ’ ~ ~ ~ ~ )

and apply the Cauchy-Schwarz inequality twice. Since the calculation is straightforward, the details will be omitted. We return now to the proof of the main interpolation theorem. Forfand g in H’, let

It will be convenient also to use the notation

bnk

= Bnk(2k)*

Given w = ( w k } E 8‘ and a uniformly separated sequence show that a functionfe H 2 exists such that Tz(f)= w.Let and let

Then and

{zk},

we have to

152

9 INTERPOLATION THEORY

it follows that I(gnj

7

gnk)l

5 2(1 - lZjl2)(l - lzk12)11 - z j % l - 2 *

By Lemma 1, this implies n

C

I(qnj3

j=1

grid/ 5 2 A .

Consequently, in view of (2) and Lemma 2,

llf,l12

5 2A 8-411Jvl127

where the fact that { z k } is uniformly separated ( l b n k l 2 8) has been used. It follows that the functions f n form a normal family, so that a subsequence tends uniformly in each disk JzI 5 R < 1 to a functionfE H 2 for which T2(f)= w

and

llfll 5 CII4I,

(3)

where C = (2AI1l2 It remains to show that T,(f) E L2 for every f E H 2 . Let i v = { w k } be an arbitrary t2sequence, and let /z,(z) be the H 2 function of minimal norm such that 2 -112 h n ( z k ) = (1 - lzkl ) 7 k = 1, 2, * . 12. 9

We have just proved the existence of a function f E H 2 satisfying the conditions (3). Clearly, then,

I l ~ f l l5l cll~4. Appealing to relation (15) of Chapter 8, we therefore see that each f E H 2 satisfies

Taking the supremum over all iv E L2 with )/ivJI= 1, we conclude that

1. In particular, T 2 ( f )E t2for every f~ H 2 . since Ib,,k/I It is now an easy step to show that T p ( f )E L P for everyfE H P , 0 < p < a. Simply write

f(4=~(z)cs(~)12’p> where B is a Blaschke product and g is a nonvanishing H 2 function. Then m

co

1( l - IZk12>/.f(zk)lp k= 1

5 h=l

(l - lzk1z)>1g(zh)12

5 C211gl12= c

~II~II~.

9.3 THE PROOF F O R p < 1 153

That is,

ll~p(f)ll5 C2’pllfll,

.fE

HP.

(4)

Finally, it has to be shown that Tp(HP)3 C p (I I p I co) if {zk) is uniformly separated. Given {wk}E Gp, let g,(z) be the N P function of minimal norm which satisfies k = 1, 2,. . . > N. g r t ( Z k ) = (1 - ] z k \ 2 ) - 1 ’ p l r k , Appealing again to relation (15) of Chapter 8, we have

for somefE H q with IlfIl

5 6-

=

1, where q is the coiljugate index. Thus, by (4),

* c2’qw / /

,

Trivial modifications are required in the cases p = I and p = co, but the final result is the same. The functions g,(z) therefore constitute a normal family, and a subsequence converges to a function g E HP for which Tp(g)= IV. Since w was arbitrary, this proves Tp(HP)3 P,1 I p 5 a. 9.3. T H E PROOF FOR p < 1

We have already shown, even for p < I , that Tp(HP)c l pif {zk} is uniformly separated. On the other hand, the proof that Tp(HP)2 C p for 1 I p 5 co was based on the duality relation, which has no counterpart for p < 1. Nevertheless, a direct and surprisingly simple construction is available in this case. Let ( z k )be uniformly separated, and let w = ( w k ) E CP, 0 < p < 1. we shall construct a function J E H P such that T,(f) = IY. This for arbitrary I V E C p will prove Tp(HP)3 Cp. Let

bk(Z) =

m

IZ.1

z.- z

JJ J J j=1 j #k

zj 1- z j z

and gk(Z)= (1

Since Ibk(zk)l 2 6 > 0, the series

-kz)-2’p.

154 9 INTERPOLATION THEORY

converges uniformly in each disk Iz( I R c 1. Hence f is analytic in IzI < 1. Furthermore, m

proving that f E f I P . It is easy to see that Tp(f) = w. This shows that Tp(Hp)= C p if (zh) is uniformly separated. The converse can be proved by virtually the same argument used in the case 1 5 p 5 co. Of course, H P and C p are no longer Banach spaces if p < 1, but they are F-spaces (see Section 6.4) under the translation invariant metrics

and m

d(x, y ) =

Ixh h= 1

- yhIp,

respectively. Hence the closed graph and open mapping theorems are still available. It has to be checked that the quotient space H P / N Pis again an F-space, but this presents no difficulty. In fact, if Xis any F-space and S is a closed subspace, X / S is an F-space under the obvious metric. 9.4. UNIFORMLY SEPARATED SEQUENCES

The condition for uniform separation is difficult both to understand intuitively and to verify in practice. In fact, the very existence of uniformly separated sequences is not obvious. It is important to find a more accessible characterization. This is a purely geometric problem which will be discussed independently of the interpolation theory. LEMMA.

PROOF,

It is enough to assume a = a > 0. Then

--

9.4 UNIFORMLY SEPARATED SEQUENCES

155

where

The problem is to find the minimum of A+x B+x

d x )=-

for - 1 Ix I1. But in this interval q’(x) =

B-A

( B + x)’ > O’ ~

since 1 IA
which proves the lemma. THEOREM 9.2. If there is a constant c

1 - 1z,+11 I c(1 -

IZ”l),

< 1 such that n

=

1, 2 , . ..)

(5 )

then {z,} is uniformly separated. The condition (5) is also necessary if 0 5 z 1 < z 2 < ..*. PROOF.

The condition ( 5 ) implies 1 - lZjl I C j P k ( l - l z k l ) , j > k ;

In particular,

k = 1, 2, .. .

.

2 (1 - Izjl) < 00. It follows from (6) that f0r.j > k lzjl

- lzkl

2 (1 - d-’)>(1 - lzkl)

and 1 - IZjZkl = 1 - ] Z j l

+ IZjl(l

- I Z J ) 5 (1 + cj-”>(r - IZfJ).

Hence, by the lemma,

For j < k this inequality takes the form 2-

j < k.

(6)

156

9 INTERPOLATION THEORY

Consequently,

which shows that {zk} is uniformly separated. Now suppose 0 < zI < z2< .* * and k = l , 2 ,....

26>0, Then

z,+~- z, 2 6(1 - Z,Z,+~),

n

=

1, 2 , . .. ,

so that

Thus {z,} satisfies ( 5 ) , as claimed. Sequences (z,} which satisfy the condition (5) are called exponentiul sequences. 9.5. A THEOREM OF CARLESON

Part of the proof of the main interpolation theorem consisted in showing that if {z,} is uniformly separated a n d f c H P (0 < y < a), then

In other words, if

1-1 is

the discrete measure on IzI < 1 defined by

p(z,)

=

1 - Iz,~

.

n = 1, 2, . . ,

then

I,

zl
If(z>lPM z ) < 0-3

for eachfc H P , and the injection mapping from H P to the space LPp(with the obvious definition) is bounded. I t is natural to ask what other measures p have this property. One trivial example is the ordinary Lebesgue measxe on IzJ< 1. As it turns out, the exact class of admissible measures can be characterized very neatly. A finite

9.5 A THEOREM OF CARLESON

157

measure p on IzI < 1 will be called a Carleson measure if there is a constant A such that p(S) 2 Ah for every set S of the form S={z=re“:l - h s r < l ; 8 , ~ ~ 1 6 ~ + h ) . The theorem may now be stated as follows.

(7)

THEOREM 9.3 (Carleson). Let p be a finite measure on Iz( < 1, and suppose 0 < p < 03. Then in order that these exist a constant C such that

it is necessary and sufficient that p be a Carleson measure. PROOF OF NECESSITY. It is easy to see that the validity of (8) for any given p is equivalent to its validity for p = 2. (Simply factor out Blaschke products and use the standard argument.) Suppose, then, that (8) holds with p = 2. Let zo = pein # 0 be an arbitrary point in the open disk, and consider the H 2 function

g(2) = (1 - z,z)-l,

whose norm is

llsllz = (1 - P 21-112

*

Let h = 1 - p, and let S be the set of points z = reiesuch that p

If (8) holds with p

sr<1 = 2,

and

CI

- h/2 5 0s ct + h/2.

then

1

lg(z)12 d p ( z ) I C2(1 - p

y

s C2h-I

But a simple geometric argument shows that for z E S, I(l/p)eia- z~ I I(l/p)e’“ - pei(a+h12) I

1

= - [l

P

- 2p2 cos(h/2) + ~ ~ 1 ’ ’ ~ ;

(9)

158

9 INTERPOLATION THEORY

Combining this with (8), we find

which was to be shown. PROOF OF SUFFICIENCY. This is much more difficult. We first give an outline of the argument, which is essentially based on the Marcinkiewicz interpolation theorem (see Zygmund [4], Chap. XII), although the special case actually needed is relatively easy, and will be proved independently. The first step is to show that (8) will follow if it can be shown that a certain sublinear operator T is of “ type ” (2,2). The second and most difficult step is to show that T is of “weak type ” (1, I). Here the argument relies upon a certain covering lemma, and it is essential that ,u be a Carleson measure. The third step, which could be avoided by direct appeal to the Marcinkiewicz theorem, is to deduce that Tis of type (2,2); it is trivially of type (GO,a).

First Step. For eachf E H 2 we have

If(z)l

1 5

1 P(r, t - e)l.f(e’‘)l 2n

dt.

0

Therefore, to show that (8) holds for a given measure p, it will be sufficient to prove that

if u(z) is the Poisson integral of an L2 function q(t) 2 0. With each point z = reie # 0 in the open disk we associate the boundary arc

I, = { e i t : e - +(I - r ) I tI e + +(I - r ) } . Choose 0 = 0 if z = 0. (See Figure 3.) It will be convenient to identify the arc I, with the corresponding segment on the real line (choosing 0 I 8 t271). Given an integrable function q(t)2 0, periodic with period 271, define

where the supremum is taken over all intervals I containing I,, of length 111 < 1. It is evident that @(z)is continuous in 0 < IzI < 1. We shall now prove that

44 I 1 6 n Z C W+ llqll11,

I4 < 1,

(11)

9.5

A THEOREM

OF CARLESON

159

Figure 3

where

is the Poisson integral of cp. Later, in the third step of the proof, we will show that (10) holds with u replaced by 4; this together with (11) will prove (10). To prove (Il), fix the point z = re", let t, = 2"-'(1 - r), and define the intervals 0, =

[--t,) t,],

n = 0, 1, .. . ) N ,

where N is the largest integer n such that t, < 3.Now define the sets Go=wo;

G,=w,-w,-l,

n=l,

GN+1 = [-n, n] - w N . Then 1

cJ c

N+1

P(r, t ) q ( t + 6 ) d t 2nn=0 cn 1 N+1 _< P(r, t,-l 2n ,=O

u(z)=-

..., N ;

160

9 INTERPOLATION THEORY

where tLl

= 0.

But from the inequality

one finds

p(r, t,-

1)

1671' 4"(1 - r)'

+ir
I

It follows that for r 2 +,

c

8n N + l u(z) I - 4-" 1 - r ,,=o

J

cp(t

+ 0) d t

G,

N

by the definitions of i j and N . If r < 3,we have trivially 2 " u(.) i q(t 71

J

+ 0) d t = 4/lcpl/,.

--K

This proves (1 1).

Second Step. The operator T : cp + i j given by the relation

assigns to every periodic q E 15' a function @ continuous in 0 < IzI < 1. We are going to show that if ,u is a Carleson measure, then T is of iveak type (1, 1):

A'%) cs-'llcpll,,

(12)

where E,

= {Z

: @(z) > S } ,

s

> 0.

For the proof, we define for each E > 0 the sets A," = [ z :

I'

Iq(t)l dt > S(E

+ lIzI))

1,

and B," = ( z : I, c I,,, for some Observe that the sets B," expand as

E -+

E, =

0 and

u B,". EZO

iv E

A,").

9.5 A THEOREM

OF

CARLESON

161

Hence p ( E J = lim p ( B i ) . &+O

If z,

E A,"

and the arcs Iznare disjoint, then s

c

(6

II

+ II,,,I>
1

Izn

icp(t)l d t 5 2nllpll1.

(14)

In particular, there can be at most a j n i t e number of points z, in A," whose associated arcs IZnare disjoint. The following lemma is now needed. Covering lemma. Let A be a nonempty set in (zI< 1 which contains no infinite sequence of points whose associated arcs 1," are pairwise disjoint. Then there exist a finite number of points zl, . . . ,zm in A such that the arcs I,, are disjoint and m

where J, is the arc of length 511,l whose center coincides with that of I,. Before proving the lemma, let us use it to complete the proof of (12). If E, is empty, there is nothing to prove. Otherwise, A," is nonempty for sufficiently small E , so that

u {z m

A," c

: I , c J,,>,

n=l

where z,

E A,"

and the arcs I,, are disjoint. It follows easily that m

B," = (j { z : I,

= J,J.

,= 1

But since p is a Carleson measure, we have

p ( { z : I , c J,,}) 5 5AlIZnl, Hence, by (15) and (14),

~1

= 1, . .

. , m.

m

In view of (13), this completes the proof of (12). Proof of covering lemma. Let p1 = inf{lzj : z

and choose z1 E A with Iz, 1 < f(l zl, . . . , z, - let pn = inf{lzl : z

EA

EA},

+ pl). If 0 E A , let z1 = 0. Having chosen

and I, n Izj = 0, j = 1, . . . , n - 1).

162 9 INTERPOLATION THEORY

Then choose z, E A with I,, n Z,, = 0 for j = 1,. .., n - 1 and lz,l < +(I p,). By hypothesis, this process must terminate after the choice of a finite number m of the points z,. Now for each z E A we have I, n I," # 0 for some n (n = 1, . .., m) ; suppose n is the smallest such index. Then IzI 2 p, , which gives 1 - [ z [I1 - p , < 2(1 - [z,l).

+

But this together with the fact that 1, meets Zznimplies I, c Jz, ,which proves the lemma.

Third Step. It now remains to make use of (12) to show that T is of type (2,2) :

4

14(41'

442) 5

cp EL2.

Kllcpl12Z,

(17)

21<1

For s > 0, let cp = $,

+ x,, where

*'

wherever ]cp[ > s/2C; otherwise.

icp = 10

Here C is the constant in (12); we assume C 2 1. Then

4(z) 5 3,(4 + l?s(z> 5 3,w +

2c S

9

from which it follows that

In particular, p(E,) 5 p(F,). Letting

-1

~ ( s )= p(E,),

1/41' d p =

we therefore have by (12)

lo m

00

S'

da(s) 5 2

0

SN(S)

ds

But with the notation 0s

=

it : Idt)l > s/2C),

we have

2cldr)I

= 1;711cp(t)lJ

0

27I

ds d t = 2c

jb I 4 w d 4

which proves (17). This completes the proof of the theorem.

9.5 A THEOREM OF CARLESON

163

REMARK. For later reference (in Chapter 12), we note that if 11 is a Carleson measure with constant A 2 1, then the constant C in (8) may be taken to be 4(80)4A2. This can be verified by a careful examination of the proof. Finally, it should be observed that the argument, with appropriate modifications, actually proves the following more general theorem. THEOREM 9.4. Let p be a finite measure on 121 < 1, and suppose 0 < p I q < 00. Then in order that there exist a constant C such that

for allfE H P , it is necessary and sufficient. that p(S) IAhq/Pfor every set S of the form (7). PROOF. The general theorem can be reduced as before to the case p = 2 I q < co. The necessity can then be proved as it was in the case q = p . To prove the sufficiency, one again uses the Marcinkiewicz interpolation theorem (in more general form). What now has to be shown is that under the hypothesis p(S) I Ahq, the operator T : cp -P 4 is of weak type (1, q) for q 2 1: PL(Q 5

c~-qll~l114,4 2 1,

(18)

for all sets E,defined as before. (Here we have replaced q/2 by q, for notational convenience.) The proof of (18) is the same as for q = 1, except that (16) must be replaced by

s c(27C)4s-q

cpIl14.

Theorem 9.4 can be applied to prove the Hardy-Littlewood theorem (Theorem 5.1 1) that f~ H P implies

Another application is a generalization of the FejCr-Riesz theorem (aside from the value of the constant):

-=

for allfE HP, 0

00.

164 9 INTERPOLATION THEORY

EXERCISES

1. Show that a necessary and sufficient condition for T, to map H P into OG) is that the measure ki defined by ~ ( ( z ,= ) 1 - 1zkI2be a Carleson measure. 2. Show that if zk = 1 then

L'J' (0 < p 5

rn

for somefE H P (0 < p < co), even though (1 - lzkl) < 00. 3. Show that if { z k }is an exponential sequence, then its associated measure p (as defined in Exercise 1) is a Carleson measure.

-

4. Construct a sequence { z k } with 0 < z1 < z2 < * . which is not exponential, but whose associated measure p is a Carleson measure. 5. Construct a sequence ( z k } such that T, maps H P into Cp, but not onto. 6. Provide the details for the two applications of Theorem 9.4 mentioned at the end of Section 9.5. 7. Use Theorem 9.4 to prove thatf' E H112impliesf€ H1, a special case of Theorem 5.12. NOTES

The problem of describing the universal interpolation sequences for H" was proposed by R. C. Buck. After partial solutions by Hayman [2] and by Newman [l], Carleson [l] proved Theorem 9.1 in the case p = co. Shortly thereafter, Shapiro and Shields [l] gave a simpler proof (which we have essentially followed) and generalized the theorem to 1 5 p 5 03. The extension to 0 < p < 1 is due to K a b a h [2, 51. Hayman actually constructed an interpolating function under hypotheses a little stronger than uniform separation. P. Beurling later proved the existence of an absolutely convergent interpolation series whenever { z k } is uniformly separated; see Carleson [3]. Theorem 9.2 was found independently by Hayman [2], Kab& [I], and Newman [l]. Newman's paper contains a further result of this type. Newman also showed, in an unpublished 1961 manuscript, that {z,} is uniformly separated if and only if the Blaschke product B(z) with zeros z, has the property that for sufficiently small E > 0, the level set IB(z)[= E decomposes into a system of disjoint curves, each surrounding exactly one point z,. This justifies the term " uniformly separated ". The result recently reappeared in a paper of Hoffman [2].

NOTES

165

Nevanlinna and Pick solved the general problem of characterizing the pairs of sequences { z k }and { w k } for which there exists f~H" with f ( z k )= w k , k = 1,2, . . . . An account of this work is in the paper of R. Nevanlinna [2]. Krein and Rechtman [I] gave another solution based on a certain moment theorem. More recently, Sz-Nagy and Koranyi [I, 21 based a proof on the spectral theorem for a unitary operator in Hilbert space. Carleson [I, 21 proved Theorem 9.3 en route to the corona theorem (see Chapter 12). The idea of basing the proof on the Marcinkiewicz interpolation theorem is due to Hormander [I], although the proof that Tis of weak type (1, 1) is similar to Carleson's original argument. The generalization to Theorem 9.4 is in Duren [2]. For a statement and proof of the Marcinkiewicz interpolation theorem, see Zygmund [4].

This Page Intentionally Left Blank

Hp SPACES OVER GENERAL DOMAINS

CHAPTER 10

Up to this point we havedealt almost exclusivelywith functions analytic in the unit disk. Our objective is now to extend the H P theory to other domains. Even in the case of a simply connected domain, there are two “natural” generalizations of the space H P which need not coincide. We begin by presenting these two definitions and exploring the connection between them. Specializing to the case of a Jordan domain with rectifiable boundary, we then discuss boundary behavior and introduce the concept of a Smirnov domain, which we investigate in some detail. We conclude by indicating some extensions of the theory to multiply connected domains. The discussion of H P spaces over a half plane is reserved for the next chapter. 10.1. SIMPLY CONNECTED DOMAINS

Let D be an arbitrary simply connected domain with at least two boundary points. There is no difficulty in defining the space H“(D) of bounded analytic functions in D ; it is a Banach space under the norm

llfll = SUP If(z)l* Z S D

168 10 HP SPACES OVER GENERAL DOMAINS

If 0 < p < co,there are at least two alternatives. One can require the boundedness of the integrals of I f / ” over certain curves tending to the boundary, or one can demand that Ifl” have a harmonic majorant (see Section 2.6). The latter definition is simpler, and will be considered first. A functionfanalytic in D is said to belong to the class HP(D) if the subharmonic function If(z)lp has a harmonic majorant in D. The norm can then be defined as I l f l I = [U(Z,)]~’~,where zo is some fixed point in D and u is the least harmonic majorant of I f l ” . It is easy to see that the space H P ( D )is conformally invariant. That is, i f f € H P ( D ) and if z = q(w) is a conformal mapping of a domain D* onto D, thenf(q(w)) E HP(D*).Furthermore, if the norm in HP(D*) is defined in terms of the point w o = q-’(z0), this correspondence f -f 0 q~ is an isometric isomorphism. A function f analytic in D is said to be of class EP(D) if there exists a sequence of rectifiable Jordan curves C,, C, , . .. in D, tending to the boundary inthesensethat C,eventuallysurroundseachcompactsubdomainof D,suchthat

It is not immediately clear that E P ( D )reduces to the usual H P space when D is the unit disk. As it turns out, however, it suffices to consider curves C, which are level curves of an arbitrary conformal mapping of the unit disk onto D. THEOREM 10.1. Let q(w) map JwI< 1 conformally onto D, and let r, be the image under q of the circle IwI = r. Then for each functionfF Ep(D),

PROOF. Let zo = q(O), and suppose ~ ’ ( 0> ) 0. LetfE EP(D), and let {C,} be a corresponding sequence of curves. Let D, denote the interior of C,. Without loss of generality, suppose zo belongs to all of the domains D, . Let qn(w) be the conformal mapping of [wI< 1 onto D,, , normalized by Pn’(0) > 0. ~ n ( 0= ) zo > Then 9,‘E H I , and f ( q n ( w ) )is continuous in ( w [i 1. By the CarathCodory convergence theorem, q, tends to q uniformly in each disk IwI I R < 1. Thus

10.2 J O R D A N DOMAINS WITH RECTIFIABLE BOUNDARY

COROLLARY.

169

The following are equivalent:

( 0 f ( Z ) E E P ( D ); (ii) F(w)=f(cp(w)) [cp'(~)]''~ E H P for some conformal mapping ~ ( wof) Iwl < 1 onto D ; (iii) F(w) E H P for all such mappings cp. This last result points out the difference between HP(B)and EP(D).In fact, E H P , while f E E P ( D )if and only if

f E H P ( D )if and only iff(cp(w))

f(cp("~Ccp'(~~l''" E HP. The two spaces therefore coincide if Icp'(w)l is bounded away from 0 and a. This will be the case, for example, if D is the interior of an analytic Jordan curve. (See, however, Exercise 3.) It is a surprising fact that the obvious sufficient condition for the equivalence of H P ( D )and E P ( D )is also necessary. THEOREM 10.2. Let cp(w) be an arbitrary conformal mapping of IwI < 1 onto the domain D.Then HP(D)= E P ( D )if and only if there are positive constants a and b such that

a I\cp'(w)\ I b,

w E D.

(1)

PROOF. It should be observed that (1) is really a property of D, not of q. We have already shown that (1) implies the equivalence of the two spaces. Conversely, suppose H P ( D )= EP(D).This means that for functionsf analytic in D, f ( c p ( W ) ) E H P*f(cp"~cp'(~~l''"E H P . The choice f ( z ) = 1 therefore shows that cp' E H I . On the other hand, iff is chosen so thatf"p(~))[q'(w)]''~ = 1, it follows that l/cp' E H'. Furthermore, since the modulus of an H P function can be prescribed arbitrarily on the boundary, subject only to the conditions f ( e " ) E Lp and loglf(e")l E L', the hypothesis implies that g E L1* cp'g E L'.

+

(Write g = g1 g 2 , where log)g,l E L' and g 2 EL"'.) But the operator T : g + c p ' g from L' to L1 is closed, hence bounded, by the closed graph E L". Since cp' E H', it follows from Theorem theorem. In other words, cp'"") , is half of (1). A similar argument shows that 2.11 that ~ ' E H " which l/q' E H". 10.2. JORDAN DOMAINS WITH RECTIFIABLE BOUNDARY

Suppose now that D is a domain bounded by a rectifiable Jordan curve C. Let ~ ( wmap ) IwI < 1 conformally onto D, and let $(z) be the inverse mapping.

170 10

H P

SPACES OVER GENERAL DOMAINS

Since C is rectifiable, it makes sense to speak of a tangential direction almost everywhere. THEOREM 10.3. Each function f of class E P ( D ) or H P ( D ) has a nontangential limit almost everywhere on C, which cannot vanish on a set of positive measure unless f(z) G 0. Furthermore,

If(z)lp Idzl < 03

if f

E E”(D),

JC

and Jc

If(z>IPl$’(z)l Idzl < co

if f E Hp(D).

PROOF. I f f € E P ( D ) ,then

F(w) = f(cp(”cp’(w)l’’P E HP. As observed in Section 3.5, cp preserves sets of measure zero on the boundary, and it preserves angles at almost every boundary point. But F(w) has a nontangential limit, and so does [cp’(w)]’’”,since cp‘ E H’. Hence f(q(bv)), and therefore f ( z ) , has a nontangential limit almost everywhere. If f ( z ) = 0 on a boundary set of positive measure, the same is true for F(w); thus F(1c) = 0, andf(z) = 0. The integrability of If(z)lpfollows from the fact that (F(w)lP is integrable over I wI = 1. The discussion is similar for f~ HP(D).

One may now ask under what conditions an analytic function f’ can be recovered from its boundary values by the Cauchy integral formula. In the case of the unit disk, we know (Theorem 3.6) this is true if and only i f f € H’. It will be convenient to use the notation L p ( C )for the class of measurable functions g on C such that Ig(z)Ip is integrable with respect to arclength. THEOREM 10.4.

EachfE E’(D) has a Cauchy representation

and the integral vanishes for all z outside C. Conversely, if g E L’(C) and /$“g(z)

dz = 0,

n = 0, 1,2, . . . ,

then

and g coincides almost everywhere on C with the nontangential limit off:

(3)

10.2 JORDAN DOMAINS WITH RECTIFIABLE BOUNDARY

171

REMARK. It should be noted that the condition (3) is equivalent to the identical vanishing of the integral (4) outside C, as a series expansion of the Cauchy kernel shows. PROOF OF THEOREM.

Iff’€ E’(D), then

so that 1 F(w)=-,j 2nz

F ( w ) do > o-w ~

/wl=l

IWI

< 1.

(5)

On the other hand, for fixed w,the function R(w) = is analytic in

cp‘(w)

1

-wcp(w>- rp(w)

w

101 < 1 and continuous in IwI I 1. Thus F(w)R(w)E H ’ ,

Adding this to

and

(9,we obtain F(w) dw q(w> - cp(w)’

which is equivalent to (2). To prove that the integral vanishes if z is outside C, we need only note that

since F E H’. Conversely, given g E L’(C) satisfying (3), the relation (5) allows us to write the function (4) in the form

where G(w) =g(cp(w))cp’(w). But I?($([)) is analytic in D and continuous in B, so by Walsh’s theorem it can be approximated uniformly in D by a polynomial. It now follows from (3) that the second integral in (6) is equal to zero. Hence F(w) is represented as a Cauchy integral which vanishes outside the unit circle (as another application of Walsh’s theorem shows). But this implies F E H’, or f~ E’(D). The uniqueness of the Cauchy representation

172 10

H p SPACES OVER GENERAL DOMAINS

(with the integral vanishing identically outside C ) is obvious. This completes the proof. We now recall that in the unit disk the space H' coincides not only with the class of Cauchy integrals, but also with the class of analytic Poisson integrals (Theorem 3.1). For a general Jordan domain with rectifiable boundary, the Poisson formula would generalize to Green's f u r m i h

where G ( [ ;z ) is Green's function of D with pole at z and ajan indicates the exterior normal derivative. We recall that Green's function is the unique function of the form G(i; 2)

= log15 -

ZI + A([),

zE

D,

where h([) is harmonic in D and continuous in the closure B, and G ( i ;z ) = 0 for on C. By a Green integral over C we mean a function of the form (7), withf(4') replaced by an arbitrary function k ( i ) for which the integral exists. Just as E 1 ( D ) coincides with the set of Cauchy integrals, it turns out that H ' ( D ) is the class of analytic Green integrals over the boundary.

c

THEOREM 10.5. EachfE H 1 ( D ) satisfies Green's formula (7). Conversely, every Green integral over C which is analytic in D is of class H 1 ( D ) . PROOF. Let i= q ( w ) map IwI < 1 onto = cp(0). The Green's function of D is

let z

D, let w then

= $([)

be its inverse, and

G ( i ; z ) = hMOl, and a straightforward calculation shows that

ButfE H 1 ( D ) impliesf(q(w))

E

H', and it follows that

which is equivalent to (7). Conversely, iff is the Green integral of k , then

will serve as a harmonic majorant of If(z)l.

10.3 S M I R N O V DOMAINS

173

10.3. SMIRNOV DOMAINS

Again let D be a Jordan domain with rectifiable boundary C, let z = q ( w ) map D onto Iwl < 1, and let w = $(z) be the inverse mapping. We have seen that everyfE EP(D)has a boundary function of class Lp(C). By analogy with the result for the unit disk, one might expect this set of boundary functions to coincide with the L pclosure of the polynomials in z. However, this turns out to be true only for a certain subclass of domains which we are about to describe. Since q‘ is an H’ fiinction with no zeros, it has a canonical factorization of the form q ’ ( 4= S ( W ) W W ) , where S is a singular inner function and CD is an outer function. D is called a Smirnov domain if S(w) 3 1 ; that is, if q’ is purely outer. It is easy to check that this is a property only of the domain D, and is independent of the choice of mapping function. Indeed, any other mapping q, has the form ql(w) = q(A(w)), where ilis some linear fractional mapping of the unit disk onto itself. Thus q l y W ) = qylb(w))ayw) = s(a(w))~(a(w))a’(w). But if S(w) is a nontrivial inner function, so is S(A(w)). (See Exercise 7, Chapter 2.) A function g e L P ( C )will be said to belong to the L p ( C ) closure of the polynomials if there is a sequence {qn(z)) of polynomials such that lim

JJdZ)

n+m

- 4n(z)IP Id4 = 0.

It is convenient to identify E p ( D ) with its set of boundary functions. Thus EP(D)is a closed subspace of Lp(C)which contains all polynomials, hence also their closure. THEOREM 10.6. Let D be a Jordan domain with rectifiable boundary C, and let 1 p < co. Then Ep(D)coincides with the L p ( C )closure of the polynomials if and only if D is a Smirnov domain.

PROOF.

Iff€ EP(D),then

F(w) =f(~(”~’(w)l’’p

E

HP.

If D is a Smirnov domain, then [q‘]’’pis outer, and by Beurling’s theorem (Theorem 7.4), there is a polynomial Q(w) such that IlP -

Q[~‘l’’pll <4 2 .

(8)

174 10

HP

SPACES OVER GENERAL DOMAINS

On the other hand, Q($(z)) is analytic in D and continuous in D, so by Walsh’s theorem (see Notes) there exists a polynomial q(z) such that

I Q($(z>)- 4 Z ) I < (@)L-’/‘

on C,

where L is the length of C. This implies

II[Q

-4

0

cplC~’l’~pll <~ / 2 .

(9)

Combination of (8) and (9) gives

which was to be shown. Conversely, if every f E EP(D)can be approximated by polynomials, a similar argument shows that the polynomial multiples of [ y ‘ ] l ‘ pare dense in H P ,so [4”]‘’p is outer, by Beurling’s theorem. Hence cp‘ is outer, and D is a Smirnov domain. (See Exercise 1.) If D is not a Smirnov domain, the problem arises to characterize the L p ( C )closure of the polynomials. This can be done in terms of the class N which was discussed in Section 2.5. +

THEOREM 10.7. For 1 < p < co, a function f E E P ( D ) is in the L p ( C ) closure of the polynomials if and only iff (cp(w)) E N + for some mapping cp of Iw[< 1 onto D (and therefore for all such mappings).

PROOF. LetfE E P ( D ) ,so that

F(w) =f (cpO(w>)ccp‘(w)ll’p E H P . Thusf(y(w)) E N in any case, since it is the quotient of two H P functions. Let S(w) be the singular factor of cp’(w). By appeal to Beurling’s theorem in its full strength, the proof of Theorem 10.3 may be adapted to show that f belongs to the L p ( C ) closure of the polynomials if and only if [S(w)]”” divides the inner factor of F(w). This is clearly equivalent to saying that f ( V ( 4 ) EN+. A simple sufficient condition for D to be a Smirnov domain is that log y’(w) E Hi.

This follows from Corollary 3 to Theorem 3.1. In fact, it is enough that arg ~ ’ ( wE) h’,

10.3 SMIRNOV DOMAINS

175

since log(qo'(w)l E h' whenever D has rectifiable boundary. This follows from the mean value theorem for harmonic functions and the fact that

In particular, D is a Smirnov domain if arg cp'(w) is bounded either from above or from below. Geometrically, this means that the local rotation of the mapping is bounded ; loosely speaking, the boundary curve cannot spiral too much. These considerations show that a domain is of Smirnov type if it is starlike (or even " close-to-convex "), or if it has analytic boundary. Smirnov domains also arise in the study of polynomial expansions of analytic functions. Our next aim is to generalize the simple fact that H 2 of the unit disk is the class of power series with square-summable coefficients. Associated with the rectifiable Jordan curve C is a unique sequence of polynomials po(z),p l ( z ) , . . . such that p n ( z ) = C,,Zn

+ C n , n - l Z " - l + * . . + C"05

C",

> 0,

and

where L is the length of C. These are called the Szego polynomials of C ; they can be constructed by orthonormalization of the sequence { z " } . A full discussion of these orthogonal polynomials and their remarkable connection with conformal mapping is beyond the scope of this book. If C is the unit circle, the nth Szego polynomial reduces to z", n = 0, 1, . . . . THEOREM 10.8.

If D is a Smirnov domain, every function f E E 2 ( D )has a

unique expansion

Furthermore, every series of the form (10) converges uniformly in each closed subdomain of D t o a function f E E 2 ( D ) . PROOF. First let (a,} be an arbitrary square-summable complex sequence. In the space L2(C),the functions n sn(z)

=

k=O

ak P k ( Z )

form a Cauchy sequence, so there is a functionfE L 2 ( C )such that

176 10 H p SPACES OVER GENERAL DOMAINS

Since f is in the L 2 ( C )closure of the polynomials, it is the boundary function of somefE E2(D). By Theorem 10.4, the Cauchy formula

is valid. Using the Schwarz inequality, we now see that s,(z) + f ( z ) uniformly in each closed subdomain of D. It should be observed that, for this half of the theorem, D need not be a Smirnov domain. (See Exercise 4, however.) Conversely, supposefE E 2 ( D ) , and let

be the “ Fourier coefficients ” of J: Among all polynomials of degree n, the “ Fourier polynomial ” sn(z) approximates f most closely in the L 2 ( C )sense. Thus it follows from Theorem 10.6 that (1 1) holds if D is a Smirnov domain. Applying the Schwarz inequality to the formula (12), we conclude as before that sn(z) + f ( z ) in D. By Parseval’s relation,

In particular, this proves the uniqueness of the coefficients. 10.4. DOMAINS NOT OF SMIRNOV TYPE

The notion of a Smirnov domain has been seen to play a decisive role in the theory of approximation and polynomial expansion. The question arises whether there actually exist Jordan domains with rectifiable boundary which are not of Smirnov type. It turns out that non-Smirnov domains do exist, but they are extremely pathological. They can be constructed by an elaborate geometric process due to Keldysh and Lavrentiev. In this section we shall outline a different approach to the problem which shows its close relation to a certain “ real-variables ” question. The discussion is based on a remarkably simple criterion for univalence, involving the Schwarzian derivative

THEOREM (Nehari; Ahlfors-Weill).

Let f ( z ) be analytic in Iz[ < 1, and

suppose I{f(z), z>I I k(1 - r2)-’,

r

= tzt,

(13)

10.4 DOMAINS NOT OF SMIRNOV TYPE

177

where k < 2. Thenf(z) maps IzI < 1 conformally onto a Jordan domain (on the Riemann sphere). The proof is beyond the scope of this book. (See Notes.) For a function p ( t ) of bounded variation over [0, 2n], it is convenient to adopt the normalization 0 < t 5 2n. p(t) = p(t-), For t E [0, 2n], define

and let v(t) be extended periodically to - co < t < co. Thus v(t) is continuous on (- co, co) if and only if p(t) is continuous on [O, 2n]. Let us say that ,u E A, if v E A,, in the sense of Chapter 5 . THEOREM 10.9,

Let p(t) be a normalized real-valued function of bounded

variation, and let J

A t ) = P,(t>

+ J0 w ( 7 ) dz

be its canonical decomposition into singular and absolutely continuous parts. Let

Then there exists a constant a > 0 such that exp{ -aF(z)} is the derivative of a fuiictionf(z) which maps Iz1 < 1 conformally onto a Jordan domain, if and only if p E A,. The boundary of this domain is rectifiable if and only if pL,(t)is nondecreasing and exp{ - 2naw(t)}is integrable. REMARK. This theorem shows, in particular, that the construction of a Jordan domain with rectifiable boundary, the derivative of whose mapping function is a singular inner function alone (as in the Keldysh-Lavrentiev example), is equivalent to the problem of constructing a singular, nondecreasing, bounded function p ( t ) of class A,. This latter construction can be carried out directly. (See Notes.) PROOF OF THEOREM.

Set

178 10

H p SPACES OVER GENERAL DOMAINS

so that ”2

- {f(z), z } = aF”(2)

+ 7[F’(z)I2. U

Let us first observe that there exists a number a > 0 such thatf(z) maps ( z [< 1 conformally onto a Jordan domain, if and only if

:r).

F’(z) = 0 (1

Indeed, if (1 5) holds, then F”(z) = O((1 - r)-’), and the inequality (1 3) can be achieved by a suitably small choice of a. Conversely, iff(.) is univalent in ( z (< 1, it must satisfy the elementary inequality

(See, e.g., Hayman [l], p. 5 or Nehari [ 5 ] , p. 216.) This implies (15). The next step is t o show that (15) holds if and only if p E A,. Let

where v is defined by (14). Integrating by parts, we find

Thus F’(z) = ig’(z)+ izg”(z). This shows that (15) is equivalent to the condition

But according to Theorem 5.3, (16) holds if and only if

gEA

and g(eiB) E A,.

(17)

We claim that g has these properties if and only if v E A,. Indeed, suppose g(z) = u(z) + iv(z) satisfies (17). Since u(z) is the Poisson integral of v,

u(eie)=v(O) wherever v is continuous. Hence, by the normalization of p, v is continuous everywhere, and is of class A*. Conversely, if v E A,, then u(z) is continuous in [z[s 1 and u(e“) E A,. Since A+ is preserved under conjugation (Theorem 5.8), u(z) has the same properties, so g(z) satisfies (17).

10.5 MULTIPLY CONNECTED DOMAINS

179

It remains to discuss the rectifiability of the boundary. Here we suppose f ( z ) maps IzI < 1 conformally onto a Jordan domain and has a derivative of the form

where a > 0, p,(t) is a singular function of bounded variation, and w(t) is integrable. The canonical factorization theorem (Theorem 2.9) makes it clear that f‘E N and that loglf’(eit)( = - 2naw(t)

a.e.

It now follows from the factorization theorem for H’ functions (Theorem 2.8) that f’E H’ if and only if ~ 1 is, nondecreasing ~ and If’(e’‘)l = exp{ -2naw(t)} E L ‘ .

Sincef‘ E H’ is equivalent to the rectifiability of the boundary (Theorem 3.12), the proof is complete. 10.5. MULTIPLY CONNECTED DOMAINS

The definitions of the classes HP(D)and Ep(D),given in Section 10.1 for a simply connected domain, are easily extended to the case in which D is multiply connected. Suppose, then, that D is an arbitrary domain in the complex plane. For 0 < p < 00, let HP(D) again be the space of analytic functionsfsuch that If(z)[” has a harmonic majorant in D. Fixing an arbitrary point zo E D, let

llfll

= [u(zo)l‘~”,

f~ HP(D),

(18)

where u is the least harmonic majorant of Ifl”. As we shall see shortly, this is a genuine norm if p 2 1. It is not difficult t o show that different points of reference zo induce equivalent norms (see Exercise 2). If D is simply connected (and has at least two boundary points), we know that f E HP(D)if and only if f(z)[$’(~)]l/~ E EP(D),where $ is any conformal mapping of D onto the unit disk. The following theorem may be viewed as a generalization of this result to multiply connected domains. THEOREM 10.10. Fix a point zo E D, letf(z) be analytic in D , and let A be any subdomain of D, containing z o , whose boundary r consists of a finite

180 10 H P SPACES OVER GENERAL DOMAINS

number of continuously differentiable Jordan curves in D. ThenfE H P ( D )if and only if there exists a constant M , independent of A, such that

where G(5; zo) is the Green’s function of A with pole at zo . PROOF. If Iflp has a harmonic majorant u in D, the integral (19) is always less than or equal to u(zo). Conversely, suppose (19) holds and let ( A k } be an expanding sequence of domains of the type described in the theorem, whose union is D.Let rkdenote the boundary of A , , and let Gk(c;z ) be the Green’s function with pole at z E Ak . Then the function

is harmonic in A , . Since I f l p is subharmonic, U k ( z ) 5 uk+l(z) for all z E A k ; while (19) implies that vk(zo)<M for all k . By Harnack’s principle (see Ahlfors [2], p. 236), the sequence { U k } therefore converges to a function u harmonic in D which is clearly a majorant (in fact, the least harmonic majorant) of Iflp. As a corollary to the proof, we note that iff E HP(D),then

where uk is defined by (20) and [[f[i by (18). In particular, the limit is independent of the sequence ( A k } .From (21) and Minkowski’s inequality, it follows that I( [( is a genuine norm if p 2 1. The space HP(D)can be identified in a natural way with a certain subspace of HP over the unit disk. To develop this correspondence, we first discuss a basic result which is essentially the uniformization theorem for planar domains. (See Goluzin [3], Chap. VI, Section 1 ; and Ahlfors and Sario [l].) If D has at least three boundary points, there exists a function q ( w ) analytic and locally univalent in IwI < 1, whose range is precisely D and which is invariant under a group 9 of linear fractional mappings of the unit disk onto itself: 9 E 9. c p M W ) ) = cp(w), Furthermore, if zo is an arbitrary point in D, cp may be chosen so that q(0)= z o and p’(0) > 0; and these conditions determine cp uniquely. In other words, the pair (Iwl < 1, p) is the universal covering surface of D,and 9 is the group of cover trartsformations, or the azrtomorphic group of D. If q(w,) = q ( w 2 ) , there is some g E 9 such that g(w,) = w 2 .

10.5 MULTIPLY CONNECTED DOMAINS

THEOREM 10.11.

181

The mapping

f ( z > F(w) =f ( q ( 4 ) is an isometric isomorphism of HP(D)onto the subspace of HP invariant under 9. +

REMARK. If D is simply connected, then 9 is trivial, and the theorem reduces to an observation already made in Section 10.1. PROOF OF THEOREM. The function F associated with a givenfis clearly invariant under 9.If 1f (z)\” has a harmonic majorant u(z), then u(q(w)) is a harmonic majorant of F(w). Conversely, let F be an arbitrary H P function invariant under 9,and let U(w) be the least harmonic majorant of IF(w)lp.In fact, U is simply the Poisson integral of (FIPover the unit circle, so (IF((= [U(O)]’/”. Since F is invariant under 9,U(g(w)) is also the least harmonic majorant of JF(w)IP, and we have U(g(w)) = U(w) for all g E 9.In other words, U is invariant under 9.This means that if q(w,) = q(w,), then U(w,)= U(w,),since there is some g E 9 for which g(w,) = w, . Hence the function u(z) = U(q-’(z)) is well defined and is the least harmonic majorant of If(z)lP, wheref(z) = F ( q - ’ ( z ) ) . In particular,

llfll = Cu(z0)l”P = CU(0>l1’P = IlFll, so the correspondence between f and F is an isometry. If D is,finitely connected, certain questions about functions in HP(D)can be reduced to the simply connected case by means of a decomposition theorem. For simplicity, we assume that no component of the complement of D reduces to a point, so that (by successive applications of the Riemann mapping theorem) D is conformally equivalent to a domain bounded by a finite number of analytic Jordan curves. THEOREM 10.12. Let D be a finitely connected domain whose boundary consists of disjoint Jordan curves C,, C, , . . . , C,,. Let Dk be the domain with boundary Ck which contains D ( k = 1, 2, . . . ,n). Then every f E HP(D)can be represented in the form

f ( z >=fi(z) + fZ(z) + * * * where fk E HP(Dk),k = 1, 2, . . . , n.

+ fn(z),

PROOF. Since HP(D) is conformally invariant, we may suppose that C, is a circle which surrounds C , , . . . , C,,. Let r,, T, , . . . , r,,be disjoint rectifiable Jordan curves in D which are homologous to C , , Cz ,. . . ,C,,,respectively.

182 10

HP

SPACES OVER GENERAL DOMAINS

+ + ... +f,, where

Then f =f, f i

is analytic in Dk (after the obvious analytic continuation). To show that f k E HP(Dk),we need only deal with the case k = 1, since any of the curves C, can be made to play the role of C, by a suitable conformal mapping. Let R c D be an annulus with outer boundary C, and inner boundary r, a circle in D. By hypothesis, If I p has a harmonic majorant u in D. Thus, since f , , . . . ,f , are bounded in R, there is a constant a such that

+

z E R. If,(z)l” I u(z) a, But u = u, + u2 , where u1 is harmonic inside C, and u2 is harmonic outside and on r. In particular, u2 is bounded in R, so Ifl(Z>l” I u,(z) + b, z E R, (22) for some constant b. But (ul + b) is harmonic and If,l” is subharmonic throughout D,, so the inequality (22) holds throughout D,. Hencef, E HP(D,), and the proof is complete.

If the boundary curves C, C2 , . . ., C,, are rectifiable, it follows at once ) a nonfrom the decomposition theorem that a function f E H H P ( D has tangential limit at almost every boundary point, and the boundary function cannot vanish on a set of positive measure unless f = 0. The classes Ep(D)can also be considered in multiply connected domains. A function f analytic in D is said to belong to Ep(D)if there is a sequence {Av} of domains whose boundaries {r,} consist of a finite number of rectifiable Jordan curves, such that A” eventually contains each compact subset of D, the lengths of the I’,are bounded, and

1 f(z)l” ldzl < a.

lim sup v’m

y

Suppose now that D is finitely connected, and that its boundary C consists of rectifiable Jordan curves C,, C, , . . . , C, . If y1 = 1, we know by Theorem 10.1 that the boundedness of the lengths of the rvis a superfluous requirement in the definition of Ep(D);but whether this condition can be dispensed with in the multiply connected case is still unknown. For domains D of this type, it is easy to prove a decomposition theorem analogous to Theorem 10.12 for functions f E Ep(D). It then follows that every such f has a nontangential limit almost everywhere on C, and that (if p 2 1)f can be recovered from its boundary function by a Cauchy integral over C. Also, EP(D)= HP(D)if all the boundary curves are analytic.

EXERCISES

183

If D is a finitely connected domain, none of whose boundary components consists of a single point, there is a characterization of Ep(D)which generalizes a result proved earlier for simply connected domains. Let w = $(z) be any conformal mapping of D onto a domain bounded by analytic Jordan curves, and let z = cp(w) be the inverse mapping. Iff(z) is analytic in D , thenfE EP(D) if and only if If(cp(w))lp Icp’(w)l has a harmonic majorant in G = $(D). In particular, if arg(cp’(w)) is single-valued in G, one can say that f E Ep(D)if and only iff(cp(~)>[cp’(w)]’~~ E HP(G).The proof is omitted, since it is similar to the proof of Theorem 10.10. EXERCISES

1. Let D be a Jordan domain with rectifiable boundary and let cp map

IwI < 1 conformally onto D . For 0 < p < co, show that every F E H P has the form F(w) =f(40(w))[cp’(w)11’” for some f~ EP(D). 2. Let D be an arbitrary domain, let z1 and z2 be any points in D , and let

llflll = cu(zl>ll/p?

l l f l l z = cu(z2>1’/p

be the corresponding norms on HP(D).Prove the existence of positive constants A and B such that AIIfIIl 5

I I f I I z 5 B Ilflli,

fE

HP(D).

(Hint: Choose a simply connected subdomain containing z1 and z 2 , map it onto the unit disk, and apply the Poisson formula.) 3. Let D be the interior of a Jordan curve which is analytic except at one point, where it has a corner with interior angle a. For 0 < p < co, show that EP(D)5 HP(D)if 0 < a < n;while HP(D)5 Ep(D)if n < a < 27c. 4. Let D be a Jordan domain with rectifiable boundary C, not a Smirnov domain. Show that f E E 2(D)has an expansion f(z> = C anPn(z),

C Ian12 < 00,

in terms of the Szego polynomials p n , if and only iffcan be approximated in the L2(C) sense by polynomials. Hence show that the first statement in Theorem 10.8 is false if D is not a Smirnov domain. 5. Let D be a Jordan domain with rectifiable boundary C, and let w = $(z) map D onto IwI < 1. Show that $’ E E 1 ( D ) .Show, however, that $’ $ E P ( D ) for a nyp > 1 if D is not a Smirnov domain.

184 10 HP

SPACES OVER GENERAL DOMAINS

6. Let D be a Jordan domain with rectifiable boundary C. Show that if f E E P ( D )and its boundary function is in L4(C)for some q > p, it need not follow that f E E4(D). Show, however, that the conclusion is true if D is a Smirnov domain. (Hint: See Exercise 5 and Theorem 2.1 1.) 7. Let D be a Jordan domain with rectifiable boundary C. According to Theorem 10.4, the set of boundary functions o f f € E 1 ( D )is precisely the class of functions f E L 1 ( C )such that JCznf(z)d z

= 0,

n = 0, 1, . . . .

Show that the analogous statement for 1 < p 5 domain, but need not be true otherwise.

GO

is true if D is a Smirnov

8. Show that if q maps IwI < 1 onto a Jordan domain D with rectifiable boundary and log(cp’(e”)(E L log’L, then D is a Smirnov domain. Thus D is a Smirnov domain if lq’1 is not “too small” on the boundary. NOTES

For a Jordan domain with rectifiable boundary, the spaces E p ( D )were first considered by Smirnov [3], who defined them in terms of level curves. The equivalence with the apparently more general definition (Theorem 10.1) was proved by Keldysh and Lavrentiev [ I ] ; see Privalov [4]. Smirnov also introduced the definition of H P ( D ) and proved Theorems 10.4 and 10.5. The recent book of Smirnov and Lebedev [I] discusses these matters. Theorem 10.2 is due to Tumarkin and Havinson [3]. Privalov [4] gives further information on integrals of Cauchy-Stieltjes type and generalizations of the F. and M. Riesz theorem. A discussion of the Carathtodory convergence theorem and Walsh’s theorem can be found in Goluzin [3]. Smirnov gave a “Hilbert space ” proof of Theorem 10.6 in the case p = 2 (see also Goluzin [3]). Since our proof is based on Beurling’s theorem, it is valid even for p < I (see Gamelin [I]). The general result (for 0 < y < a)was stated by Keldysh [l]. Theorem 10.7 is due to Tumarkin [2], with a different proof; it is also valid f o r p < 1 . Theorem 10.8 goes back to Smirnov [I, 31: For further information on the Szego polynomials, the reader is referred to Szego [2, 31. Smirnov apparently tried without success to prove that every Jordan domain with rectifiable boundary is a Smirnov domain (see Smirnov [3], p. 353), but several years later Keldysh and Lavrentiev [l] produced a counterexample. Their construction was simplified in the book of Privalov [4], but the technical details remain formidable. The relatively simple approach described in Section 10.4, is due to Duren, Shapiro, and Shields [l]. Nehari [I] obtained

NOTES

185

a Schwarzian derivative criterion for univalence in the open disk, the condition (13) with k replaced by 2 . Ahlfors and Weill [l] sharpened it to the form given. See also Duren [5]. Constructions of a singular nondecreasing function of class A, have been carried out by Piranian [I], Kahane [I], and H. S. Shapiro [4]. For sufficient conditions that a domain be of Smirnov type, see Privalov [4], Tumarkin [3], and H. S. Shapiro [3]. Rudin [I, 21 developed the theory of H P ( D )in the multiply connected case and proved Theorems 10.10, 10.11, and 10.12. For the decomposition theorem in EP(D),see Tumarkin and Havinson [4]. They also found [2] the characterization of Ep(D)mentioned at the end of Section 10.5. Their survey paper [7] gives a clear account of the theory and contains further references. For a discussion of H P spaces over an annulus, see Kas’yanyuk [l, 21, Sarason [l], and Coifman and Weiss [I]. There is a large and rapidly growing literature on H P spaces over Riemann surfaces. Some of the relevant papers are Parreau [I 1, Royden [I], Voichick [l, 2, 31, Voichick and Zalcman [I], Gamelin and Voichick [l], Forelli [3,4], Fisher [l], Heins [l], and Earle and Marden [ I , 21. The notes of Heins [2] survey certain aspects of this theory. Rudin [9] describes some generalizations of the H P theory to several complex variables, where the domain is a polydisk. Some of the one-variable theory extends t o higher dimensions, but there are many counterexamples. Recently, various aspects of the H P theory have been generalized to the abstract setting of function algebras. See, for example, the surveys of Gamelin and Lumer [I], Lumer [l], and Gamelin [2], which give further references.

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H p SPACES OVER A HALF-PLANE

CHAPTER 11

This chapter deals with functions analytic in the upper half-plane D

= {Z = x

+ iy : y > O}.

It turns out that the spaces H P ( D )and E P ( D ) ,as defined in Chapter 10, do not coincide in this case, and in fact E P ( D )is properly contained in H P ( D ) .It is natural also to consider the space 5 j p (0 < p < GO) of functionsfanalytic in D, such that If(x + iy)lp is integrable for each y > 0 and

is bounded, 0 < y < GO. fjmwill denote the space of bounded analytic functions in D. Eventually it will turn out that 5 j p = EP(D),but the general theory of Chapter 10 is not entirely applicable because the boundary of D is not rectifiable. It is possible to develop the theory of bPby mapping the half-plane onto the unit disk, but this approach runs into difficulties because the lines y = y o are mapped onto circles tangent to the unit circle. Mainly to deal with this problem, or rather to avoid it, we begin with some lemmas on subharmonic

188 11

Hp

SPACES OVER A HALF-PLANE

functions in D. We then discuss boundary behavior, factorization, and integral representations of s j p functions, basing most of the proofs on known properties of H P functions in the disk. The chapter concludes with the Paley-Wiener theorem, in which a Fourier transform plays the role of the Taylor coefficients. 11.1. SUBHARMONIC FUNCTIONS L E M M A 1.

If g ( z ) 2 0 is subharmonic in the upper half-plane D and

+ iy) dx 5 M ,

/:mg(x

y > 0,

then

PROOF. Fix zo = xo

+ iyo ( y o > 0), and map D onto the unit disk by w

- zo z--f;; 2

=; -,

Then

G(w)= g ( ' O - z o 1-w

")

is subharmonic in lwI < 1, so by the mean value theorem, PC

4

where w = li + iu. Letting p

-+

1, we therefore have

L E M M A 2. If a subharmonic function g ( z ) satisfies the hypotheses of Lemma I , then it has a harmonic majorant in D.

11 2 BOUNDARY BEHAVIOR

PROOF.

189

Map D onto lwl < 1 by

w

i(1

z-i

+ w)

z = q(w) = ~. 1-w

= -.

= I)(.)

z+i'

The line y = b then corresponds to the circle c,, with center b(l + b)-' and radius R = (1 + b)-'. By Lemma 1, C(w)= g((p(w)) is bounded inside cb, so it has a least harmonic majorant U,(w) there. If a < b, it is clear that U,(w) 2 Ub(w) for each w inside Cb. Thus by Harnack's principle (see Ahlfors [2], p. 236) and a diagonalization argument, lim U,(w)

=

( w I < 1;

U(w),

a+O

and U(w) is a harmonic majorant of G(w) unless U(w) = GO. Hence u(z) = U($(z)) is a harmonic majorant of g(z) if U(w) GO. To show that U(w) f co,let rb be a circle concentric with C, and having radius p < R. Let V,(w) be the Poisson integral of C over rb. Then, in particular,

+

But as p -+ R, vb(w) -+ ub(w) inside convergence theorem that

5

cb;

l+b 7 /-,g(x

so it follows from (2) and the bounded

+ ib) dx 5 (1 +7cb)M

As b + 0, this shows U(0) 5 M / x . Thus U(w)

+

00, and

the proof is complete.

11.2. BOUNDARY BEHAVIOR

The following theorem is an immediate consequence of Lemma 2. THEOREM 11.1. COROLLARY.

If 0 < p < GO and f E g p , thenfE H P ( D ) .

I f , f E !$', then the boundary function

f ( x ) = limf(x Y+O

+ iy)

190 11

HP

SPACES OVER A HALF-PLANE

exists almost everywhere, f

E Lp,and

PROOF OF COROLLARY. As in the proof of Theorem 10.3, the existence (more generally) of a nontangential limit f (x) = limz+xf ( z ) follows from the fact that F(w) = f ( q ( w ) ) is in H P , where q is the mapping (1). Fatou’s lemma shows If(x)lp is integrable over ( - co, co). Finally, (3) follows from the fact (Theorem 2.2) that

It is also true thatf(x + iy) tends to f ( x ) in the Lp mean. Before showing this, it is convenient to prove a Poisson integral representation for functions and a factorization theorem analogous to that of F. Riesz (Theorem 2.5). THEOREM 11.2.

I f f € sjp, 1 I p I co, then

Conversely, if h E L P (1 5 p I a) and f(z)

is analytic in D,then f

1

=n1 -“m ( X -

E 5”

t y + y h ( t ) dt

and its boundary functionf(x)

= h(x)

a.e.

PROOF. Since F(w) =f ( q ( w ) ) is in H P (1 i p I co), it has a Poisson representation (Theorem 3.1)

1 F(w) = 27c

s

2~ 0

+w

eie

Re(---]F(eiB) e” - w

do.

This gives (4) after a change of variable and a straightforward calculation. The converse is obtained from Jensen’s inequality. COROLLARY.

Iff E sjp, 1 I p < co,then W

lim y-0

SO0

-m

If(x

+ iy)Ip dx = j- m lf(x)lp dx.

(5)

11.2 BOUNDARY BEHAVIOR

191

PROOF. Applying Jensen’s inequality to (4), we have

Jy

+ iy)Ip dx

If(x

m

m

If(x>lp dx,

Y > 0.

(6)

This together with Fatou’s lemma gives the result. THEOREM 11.3. I f f € 5jP(0 < p I co) and f ( z ) $0, then f ( z ) = b(z)g(z), where g is a nonvanishing !+’ function with Ig(x)l = If(x)l a.e., and

is a Blaschke product for the upper half-plane. Here m is a nonnegative integer and z, are the zeros (z, # i ) off in D,finite or infinite in number. Furthermore,

PROOF. According to Theorem 2.5,

. f ( d w ) ) = B(w)G(w),

where B is a Blaschke product in the disk and G E H P has no zeros. If we define b(z) = B($(z)j, the expression (7) follows from the corresponding formula for B ; and (8) is equivalent to (1 - I$(z,)l) < co. It remains to show that the nonvanishing function g(z) = G($(z)) is in JjP. But by Theorem 11.2, or rather by its proof, [g(z)lPis the Poisson integral of its boundary function; hence g p E &’. COROLLARY 1.

Iff€

aP,0 < p < co, then (5) holds. \

PROOF. Since

J;mls(x

g p E &I,

+ iY)lP dx 4

1; I& m

+ iy)IPdx

+

1; Is(x>l”dx 1;

If(x)l” dx

=

m

m

.

COROLLARY 2. If f e & p , 0 < p < co, then f ( z ) -+0 as z -+ co within each half-plane y 2 6 > 0.

PROOF. Since If(z)lpI Ig(z)lp,it is enough to prove this for functions. But each f~ 5$’ has a Poisson representation of the form (4). Given E > 0, choose T large enough so that

192 11

H P SPACES

OVER A HALF-PLANE

Then (4) gives

It is now a short step to the theorem on mean convergence. THEOREM 11.4.

Iff€

SSP (0 < p

< a),then

PROOF. Apply Corollary 1 above and Lemma 1 in Section 2.3.

A further application of the factorization theorem shows that 9Xp(y,f ) is a nonincreasing function of y i f f € S j p . This is expressed by the following theorem. THEOREM 11.5.

I f f € S j p (0 < p < a)and 0 < y1 < y 2 , then

~ ~ , 0 1f1> >2 q J ( Y 2 f 1. 2

+

fl

PROOF. Let f i ( z ) = f ( z iyl). Then fi E S j p , so it has the factorization = b,g, as in Theorem 11.3. But since glPE $ j l , an application of (6) gives mpoi2- Y l , . f l ) I

5qY2

- y,, Sl)

‘JJZ,(O, Sl>=

qo>f,),

which proves the theorem. The hypothesis f

E

$ j p is essential, as the example

f ( z ) = e-iz(i

+ z)-’/P

shows. Here

rnn,P(Y,f )

= neY(y

+ 1)

- >

which increases to infinity with y. 11.3. CANONICAL FACTORIZATION

The factorizationf= bg given jn Theorem 11.3 can be refined as it was in the case of the disk (Theorem 2.8) to produce a canonical factorization for $ j p functions. The space H P ( D )will be considered first.

11.3 CANONICAL FACTORIZATION

THEOREM 11.6.

Each function f

E H P ( D ) ,0

193

< p < co,has a unique factor-

ization of the form f (z) = e'"'b(z)~(z)G(z),

where

CI

(9)

2 0, b(z) is a Blaschke product of the form (7),

1+tz

s ( z ) = exp(i

t - z dV(t)\

for some nondecreasing function v(t) of bounded variation with v'(t) = 0 a.e., and a!

~ ( z=) eiY exp

+

(1 t z ) log w ( t ) ( t - z)(l t 2 )

+

for some real number y and some measurable function w ( t ) 2 0 with

Conversely, iff has the form (9), where CI 2 0, b(z) is an arbitrary Blaschke product, and the functions v(t) and w(t) have the properties indicated, then j-€ HP(D). PROOF.

Iff

E H P ( D ) ,then

by Theorem 2.8,

F(w) =f(cp(wN

=

B(w)S(w)@(w),

where B(w) is a Blaschke product for the disk,

p(e) being a bounded nondecreasing singular function, and 2 R eie

+

loglF(eie)I d e ) . As in the proof of Theorem 11.3, b(z) = B(rc/(z)) is a Blaschke product of the form (7). With w = $(z) and eie = $(t), a calculation gives

eie + w 1 + tz cis - w i(t - z) and 2dt do= 1 t2'

+

194 11 H p SPACES OVER A HALF-PLANE

Thus G(z) = a($(.)) has the form (1 l), with w(t) = If(t)l. The properties (12) of o follow from the properties loglF(eie)IE L1 and F(eie)E Lp.Finally, taking into account the possible jumps of p at 0 and at 2n, we have log S($(z))

=

i

s

1+tz

-d v ( t ) + iaz,

a t - z

where v ( t ) = p(arg{$(t)}) and a = p(O+) - p(0)

+ p(271) - p(2n-)

2 0.

Conversely, iff is an arbitrary function of the form (9), then

by the geometric-arithmetic mean inequality. Thus If(z)lp has a harmonic majorant in D. THEOREM 11.7. Each functionye Jjp, 0 < p < co, has a unique factorization of the form (9), with the factors defined as in Theorem 11.6 except that the second condition in (12) is replaced by w E Lp. Conversely, each product of such factors belongs to !$'. PROOF. Since , f !$ ~ ' implies f E H P ( D ) and f ( t ) E Lp, the first statement follows from Theorem 11.6. The converse is proved by integrating (13) with respect to x.

11.4. CAUCHY INTEGRALS

We now wish to show that every function in Jjp (1 Ip < 00) can be recovered from its boundary function by a Cauchy integral. It would be possible to prove this by mapping the half-plane onto the disk, as we did in Section 10.2 for the case of a domain bounded by a rectifiable Jordan curve. However, it is much easier to base the discussion on the identity

and essentially to follow the argument used for the disk in Section 3.3.

11.5 FOURIER TRANSFORMS

THEOREM 11.8.

Iff

E

195

S j p (1 I p < co), then

and the integral vanishes for all y < 0. Conversely, if h E Lp (1 Ip < co) and dt

eo,

y

then for y > 0 this integral represents a function f tion f ( x ) = h(x) a.e. PROOF.

Iff

E


S j p whose boundary func-

5', the Cauchy integral

is analytic in both of the half-planes y > 0 and y < 0. According to (14), it is related to the Poisson integral by the identity

Therefore, in view of Theorem 11.2, y > 0. F(Z) = F(z) - f (z), In particular, F(?) is analytic for y > 0, so F(z) must be identically constant in the lower half-plane. But since F(z) -+ 0 as z -+ 00, the constant is zero. Thus F(z) =f ( z ) in y > 0 and F(z) = 0 in y < 0, which was to be shown. The

converse follows immediately from Theorem 11.2. 11.5. FOURIER TRANSFORMS

We come now to the Paley-Wiener theorem, a half-plane analogue of the fact that H 2 is the class of powcr series a, Z" with la,J2< 00. For functions analytic in the upper half-plane, the Fourier integral

c

c

W

f(z> =

joeiZ'F(t) d t

(15)

plays the role of a power series. Before stating the Paley-Wiener theorem, we recall a few facts about Fourier transforms of L2 functions. I f f € L2,its Fourier transform is defined as R

f ( x ) = 1.i.m. R-tw

j e-'"'f(t) d t , 2n -R

196 1 1

H P SPACES OVER A HALF-PLANE

where “1.i.m.” stands for “limit in mean” in the L2 sense. It is a theorem of Plancherel that ?exists, lifi122= 2n I l f 1Iz2, and R

f(t) = 1.i.m. R+m

1- eixtf^(x)

dx.

R

If g is another L2 function with Fourier transform @, the PIunchereI formula is (16)

-m

THEOREM 11.9 (Paley-Wiener). A functionf(z) belongs to Sj2 if and only if it has the form (15) for some F E L2.

PROOF. Iff has the form (15) with F E L2, it is analytic in the upper halfplane, as an application of Morera’s theorem shows. For fixed y > 0, the functionf,(x) = f ( x + iy) is the inverse Fourier transform of

Hence

Jymlf(x + i y ) I 2 d x = 271

lorn

e-2y‘lF(t)12 d t

.m

S 2n

j,

IF(t)I2 dt < co,

s2.

showing that f E Conversely, each f in 5j2 is the Cauchy integral of its boundary function, by Theorem 11.8 : f(z)

1 2ni

= -Jm

-m

f(t> dt, t-z

y >O.

But

where o(5)= eiz6 for 5 2 0 and a(() formula combined with (17) gives

=0

which proves the Paley-Wiener theorem.

for

< < 0. Thus

the Plancherel

EXERCISES

COROLLARY.

tion, thenf((5)

197

I f f € Bz and f is the Fourier transform of its boundary funcfor almost all < 0.

<

=0

PROOF. By Theorem 11.8, the Cauchy integral (17) vanishes for all y < 0. But if y < 0,

1 27ci(z - t )

1 =-J’ 271

0

e-it< i z y

e

dS,

-00

and we find as before that 0

J’-

e i z y ( ( )d< E 0,

y < 0.

m

In particular,

which provesf((5)

=0

for almost all

5 < 0.

The argument can be generalized to give a similar representation for 5jp functions, 1 < p < 2. We shall content ourselves with a discussion of B‘. If f~L1, its Fourier transform

.f(x)

1 m eCix‘f(t)d t 27c - 0 0

=-

J’

is continuous on - co < x < co, andf(x) -,0 as x -+ rt co. If also g E L1,the formula (16) is a simple consequence of Fubini’s theorem. Thus the proof of the Paley-Wiener theorem can be adapted to obtain the following result. THEOREM 11.10. Ifj’E &’ a ndpis the = 0 for all 5 0 and

function, then?(<)

Fourier transform of the boundary

5

EXERCISES

1. Show that , f ~ if and only if f(q(w))[q‘(~>]’’~ E H P , 0 < p < 03. Hence show that 3jp = E P ( D ) ,where D is the upper half-plane. [Suggestion: Use the canonical factorization theorems (Theorems 11.6 and 11.7).] 2. Show that E P ( D )is properly contained in H p ( D )if D is the upper halfplane.

198 11

H P

SPACES OVER A HALF-PLANE

3. Give an example of a function f(z) which is analytic in a half-plane y > -6 (6 > 0), with f ( x ) E I.’, but which is not the Cauchy integral off(x). 4. Let p(t) be a complex-valued function of bounded variation over (- co, a), such that /:meixt

dp(t) = o

for all x > 0.

Show that dp is absolutely continuous with respect to Lebesgue measure.

5. Prove the half-plane analogue of Hardy’s inequality (Section 3.6): If f~ 6’and .f is the Fourier transform of its boundary function, then

(Hille and Tamarkin [3]. See Exercise 4 of Chapter 3.)

6. For f E S j p , 0 < p < co, prove

(This analogue of the FejCr-Riesz theorem is due to M. Riesz [l].) 7. Show that if b(z) is a Blaschke product in the upper half-plane, then

lim y’o

+ 1- loglb(x 1+ x 2

iy)J

dx

= 0.

m

Conversely, show that if f ( z ) is analytic in y > 0, If(z)I < 1, and lim y’o

/-

m

loglf(x + iY)l dx 1+X2

= 0,

thenf(z) = ei(Y+az)b(z), where y is a real number, a 2 0, and b(z) is a Blaschke product (Akutowicz [l]). NOTES

Most of the results in Section 11.2 are due to Hille and Tamarkin [3], who considered only 1 I p < co. They proved the key result &” c H P ( D )using a lemma of Gabriel [ 2 ] on subharmonic functions. But in order to apply Gabriel’s lemma, it must first be shown that a functionfe BP tends to a limit as z -+ co within each half-plane y 2 6 > 0. Hille and Tamarkin were able to show this only after a difficult argument proving the Poisson representation

NOTES

199

(Theorem 11.2) from first principles. Kawata [l] extended the Hille-Tamarkin results to 0 < p < 1. The relatively simple approach via harmonic majorants, as presented in the text, is due to Krylov [l]. Krylov also obtained the canonical factorization theorems of Section 11.3. Theorem 11.9 is in the book of Paley and Wiener [11. For proofs of the Plancherel theorems and other information about Fourier transforms, see Goldberg [l]. Theorem 11.10 is due to Hille and Tamarkin [l]; see their papers [2, 31 for further results. Kawata [l] proved theorems on the growth of 9Jlp(y,f) analogous to those of Hardy and Littlewood for the disk.

This Page Intentionally Left Blank

THE CORONA THEOREM

CHAPTER 1 2

The purpose of this final chapter is to give a self-contained proof of the “ corona theorem,” which concerns the maximal ideal space of the Banach algebra H “ . After describing the result in its abstract form, we show how it reduces to a certain “concrete theorem. Here the discussion must presuppose an elementary acquaintance with the theory of Banach algebras. However, the proof of the reduced theorem (which occupies most of the chapter) uses purely classical methods, and makes no further reference to Banach algebras. ”

12.1. MAXIMAL IDEALS

Let A be a commutative Banach algebra with unit, and let A2 be its maximal ideal space, endowed with the Gelfand topology. In other words, the basic neighborhoods of a point M* E 4 have the form % ! = (hf E &:if l f k ( M ) - Ak(hf*)l < &, k = 1, . . .,/?>, where E > 0, the xk are arbitrary elements of A , and A, is the Gelfand transform of xk. That is, A ( M ) = CpM(x),where (pM is the multiplicative linear functional with kernel M . Now let

yk = xk - Ak(M*)e,

202

1 2 THE

CORONA THEOREM

where e is the unit element of A . Then y , the equivalent form

E

M" (since j,(M*) = 0), and

% = { M E 4 : ljk(M)I < E ,

k

=

takes

1, . . . , ti}.

It is well known that A is a compact Hausdorff space under the Gelfand topology. Associated with each fixed point [, < 1, the Banach algebra H" has the maximal ideal

[cl

M,

=

{YE H" :f([)=O}.

The problem arises to describe the closure of these ideals M, in the maximal ideal space 4 of H " , under the Gelfand topology. Are there points in &' which are outside this closure? To put the question in more picturesque language, does the unit disk have a "corona"? As it turns out, the answer is negative. CORONA THEOREM. The maximal ideals M , , maximal ideal space of H".

< 1, are dense in the

The corona theorem is a direct consequence of the following purely function-theoretic result. THEOREM 12.1 (Reduced corona theorem).

Let,f,,f2, . . . , f, be functions

in H" such that

+ I f Z ( Z ) I + ... + Ifn(4l 2 6, Izl < 1, for some 6 > 0. Then there are functions gl, g 2 , . . . , g, in H" fi(z)g,(Z) +fz(z)gZ(z)+ . . . +fn(z)gn(z) 1. Ifl(4I

such that

To derive the corona theorem from Theorem 12.1, suppose the maximal ideals M , are not dense in A. Then some M* E A has a neighborhood of the form % = { M E Jt' : Ifk(M)I < E,

k

=

1, . . . , n } ,

fk E M",

which contains no ideal M , . In other words, to each point [ (/[I < 1) there corresponds an integer k ( k = 1, . . . , 17) such that

Ifk(0I = Ifk(M<)I2 8 . In particular, If,([)l + ... +\)[(,fI 2 &forall [, < 1. But by Theorem 12.1, this implies the existence of gl, . . . , g, in H" such that figl + * - . + f , g , = e .

12.2 INTERPOLATION AND THE CORONA THEOREM

203

Sincef, E M Y ( k = 1, . . . , n ) , it then follows that e E M * , which is impossible. Thus the ideals M , are dense in A?,as the corona theorem asserts. The rest of the chapter will be devoted t o the proof of Theorem 12.1. 12.2. INTERPOLATION AND THE CORONA THEOREM

The most difficult step in the proof of the corona theorem is to show that the zeros of an arbitrary finite Blaschke product can be surrounded by a contour which is not too long,” and on which the Blaschke product i s neither “too large” nor “too small”. That the contour is not too long will mean that the arclength measure it induces on the unit disk is a Carleson measure, in the sense of Section 9.5. The precise statement is as follows. “

L E M M A 1 (Carleson’s lemma). There exist absolute constants (0 < K < 1) and C > 0 for which the following is true. Corresponding to each E (0 < E 5 $) and to each finite Blaschke product IC

there is a rectifiable contour r such that (i) r has winding number I about each point a,; (ii) E I IB(z)I I cK for all z E r; (iii) the measure p defined on (zI < 1 by letting p ( E ) be the arclength of that part of which lies in the set E, has the property p ( S ) 5 CE-2h

for each set S o f the form

S = { z = r e i 0 :1 - h < r <

1; O , ~ O I ~ , + / I } .

Since the proof of this lemma is long and technical, we shall defer it to the end of the chapter (Sections 12.3-12.5). It seems advantageous first to motivate the lemma by showing how it leads to a proof of the (reduced) corona theorem, by way of the following result on interpolation. L E M M A 2. Let B(z) be a finite Blaschke product of the form (1) with distinct zeros a, (v = 1, . . . , s). For 6 < 3,let E(z) be analytic in the (possibly disconnected) set { z : lB(z)l < S} and satisfy IF(z)I < 1 there. Then there existsf€ H“ withf(a,,) = F(a,), v = 1, . . . , s; and i l f l l _< 6-“, where a is an absolute constant.

204

12 THE CORONA T H E O R E M

PROOF. Obviously, there are many functions f E H“ with f ( a v )= F(av), v = 1, . .., s. Choose such an interpolating function f with minimal norm. Then by the duality relation [see Section 8.5, example (ii)],

for some g E H’ of norm llgl!l= 1. With K as in Carleson’s lemma, choose such that E~ = 612, and let T’be the corresponding contour. Then we have

E

But the arclength measure ,LL induced by r is a Carleson measure [Lemma 1, (iii)], so by Carleson’s theorem (Theorem 9.3 and the remark following the proof), there is an absolute constant c such that

!lfll, where

CI

CE-’(Igj!1

= CE-’

I 6-“,

is an absolute constant. This proves Lemma 2.

In using Lemma 2 to prove the corona theorem, one must approximate certain functions by finite Blaschke products. The following lemma is slightly stronger than what is actually needed. Letf(z) be analytic in the open unit disk D and continuous in I 1 on Iz] = 1 and let E be the subset of / z J= 1 on which If(z)l < 1. Suppose E is nonempty. Then there exists a sequence (B,(z)) of finite Blaschke products with simple zeros, such that IB,(z)I + If(z)[ uniformly in each closed subset of ( D - E ) , and B,(z) + f ( z ) uniformly in each closed subset of D. L E M M A 3.

D. Suppose 0 < If(z)l

PROOF. Let S be an arbitrary closed subset of (6 - E). For 0 < p < 1, let fJz) = f ( p z ) , and let ED be the subset of Iz[ = 1 for which If,(z)[ < 1. Then fp(z)+f(z) uniformly in 6 as p -+ 1, and for p sufficiently near 1, S c (D-Ep). Hence it will suffice to prove the lemma under the assumption thatfis analytic in 6, and therefore has at most a finite number of zeros in D.Then, since it is clear that a finite Blaschke product can be approximated by one with simple zeros, uniformly in D, it is enough to suppose that f does not vanish in Assuming for convenience that f (0) > 0, we then have

a.

12.2 INTERPOLATION A N D THE CORONA THEOREM

205

Now let wk =eZnik/"be the nth roots of unity (k = 1, . . ., n), and let

Elementary considerations show thatf,(z) +f(z) uniformly in S. Let &k

= -

n1 loglf(ok)l

>

so that 1 0 5 &k 5 - -log n

= an,

say, where p is the minimum of If(z)l on IzI 6, < 5, let 1 - p: = 2~~

=

1. Choosing n so large that

and ak = p k w,;

and define

Note that (ak(= I if ck = 0, so that the corresponding factor in B,(z) is trivial. A calculation gives n

2 log(B,(z)l

=

-2(1 - 1212)

1 F k ( l - akz1-2 + O(S,2), k= 1

uniformly in S. From this it follows that loglBn(z)l = loglf,(z)I

+ 0(6l>>

uniformly in S. Hence log(B,(z)I -log(f(z)(, which implies (Bn(z)l-,l f ( z ) [ , uniformly in S. Since B,(O) > 0, it also follows (by analytic completion of the Poisson formula) that &(z) + f ( z ) uniformly in each disk (zI 5 ro < 1. Using Lemmas 2 and 3, we can now carry out the proof of the corona theorem. In fact, the argument will give Theorem 12.1 in the following sharper form. THEOREM 12.2.

Letfi,. . . ,f,be H" functions with

l i f k l l I 1 ( k = 1 , . . . ,n)

and Ifl(Z)I

+ ... + If"(Z)I

2 6,

IZI

< 1,

(2)

where 0 < 6 < +.Then there exist functions gl, . .. , gnin H" such that

f,(z)g,(z) + ... +f,(z)gn(z) = 1, and /(gkl/ 5 f i - p n , where

I4 < 1,

Pn is a constant depending only on n.

(3)

206

1 2 T H E CORONA THEOREM

PROOF. The argument will proceed by induction on n. The theorem is trivial for n = 1. Suppose it has been proved for all collections of (n - 1) functions, and letf,, ..., f , satisfy the given hypotheses. Suppose first thatf, is a finite Blaschke product:

with distinct a,, 0 5 la,( < 1. Then

where the Djare the simply connected components of S, . Since Ifl(Z>l

+ . * . + lfn-I(Z)l

2 8/27

z E s,>

and since the statement of the theorem is conformally invariant, it follows from the inductive hypothesis that in each domain Dj there exist (n - 1) bounded analytic functions Gjk with

and n- 1

c

fk(Z)Gjk(Z)

=

k=l

for all z in Dj.By Lemma 2,there exist functionsg,, .. . ,g , - l in H" such that gk(%)

= Gjk(av),

Dj

3

j =

*.

* 3

m,

and

Now define

This function is analytic in [ z (< 1, since

The relation (3) is automatically satisfied, and for all z $ S,,

12.3 HARMONIC MEASURES

207

where y, depends only on n. By the maximum modulus theorem, the same estimate holds throughout IzJ < 1. This concludes the proof for the case in whichf, is a finite Blaschke product with simple zeros. Continuing the inductive argument, suppose next that, more generally, f,(z) is analytic in (zI 5 1 and f , ( z ) # 0 on IzI = 1. Let Fn(z) be analytic and nonvanishing in IzJ < 1, continuous in IzI i 1, and satisfy

(Fn(z)l= min{lf,(z)]-', 3/6}

on

( z J= 1.

Then On= F,f, has the property l(Dn(z)l5 1 in JzI I 1, and l@,,(z)J = 1 on JzJ= 1 except on the set where Ifn(z)I < 6/3. Thus by Lemma 3, there is a sequence {B,(z)} of finite Blaschke products with simple zeros, such that B,,(z) + @,(z)in Iz( < 1 as m a3 ; and --f

lBrn(z>l2 I@n(z)l - 6/25

m 2 mo

9

(4)

uniformly on the set where Ifn(z)\ 2 Sj2. Since IF,,(z)l 2 1 in Jzl i 1, it now follows from (2) and (4) that Ifi(z>I + ... + Ifn-1(z)l

+ IBrn(2)l 2 6/2,

IzI < 1, m 2 mo.

Thus by what has already been proved, there are H" functions glm, . . . , z m,) such that

g,lrn(m

fi(z)g,rn(z)

+ .. + f n -

,(z)gn- l,m(z) -t Bm(z)gnm(z) = 1

in (z(< 1 and

such that for a sufficiently large choice of /I,. Now choose a sequence {mi} gkrni(z) -gk(z) uniformly in each disk Iz] 5 Y,, < 1. Then& E H", llgklli 6-'., and (3) holds. Finally, iff, is an arbitrary H" function, we may choose p < 1 such that ,f,(z) # 0 on IzI = p. Then by what has just been proved, there are functions 9:"' E H" with lig?)Il I and f,(pz)g'P)(z)

+

*

. . + f,(pz)g$)(z)

E

1.

Now let p -+ 1 through a suitable sequence, so that gp) tends pointwise to k = 1, . . . , n. This concludes the proof.

g k E H",

12.3. HARMONIC MEASURES

In order to complete the proof of the corona theorem, it now remains only to verify Carleson's lemma. This we shall do in several stages. In the present section, we digress to establish a general result ("Hall's lemma") on the

208 12 THE

CORONA THEOREM

harmonic measures of certain sets, and to apply it to obtain a special estimate which will enter into the argument at a later stage (Section 12.5). Hall’s lemma also has more direct applications to function theory, but these lie outside the scope of the present discussion. Let E be a closed subset of the right half-plane

H

=

{z : Re{z} > O}.

Suppose E does not divide the plane, and that the Dirichlet problem is solvable for D = H - E. Let dE denote the boundary of E, and let E*

=

{ i ( z (: z

E}

E

be the circular projection of E onto the positive imaginary axis. Let w(z) be the harmonic measure of E with respect to D. In other words, w(z) is the bounded harmonic function in D for which w(iy) = 0 (- GO < y < GO) and w(z) = 1 for z E aE. Finally, let

1

w”(z) = -

j xz +x( dty - t)” ’

z = x + iy,

E*

be the harmonic measure of E* with respect to H . L E M M A 4 (Hall’s Lemma).

w(x

For (x+iy)

E

D,

+ iy) 2 fo*(x - ilyl).

(5)

PROOF. Suppose first that E consists of a finite number of radial segments

k = 1, 2, . . . , n ; /8,1 < 7112, I I’ 5 b k } , with the intervals (ak, bk) disjoint. Let {reiek

ak

denote the Green’s function of H , and consider the function

where ds is the element of arclength on E. We claim that

o*(x) I U(x),

x > 0;

and

U(z) < +,

Re{z} > 0.

12.3 HARMONIC MEASlJRES

209

To prove (6), observe that on the semicircle = p, Re{<} 2 0, the function t-'C(x, [) attains its minimum for 5 = 0; hence 1

- G(x, [)

5

2x

l i l = P.

2 ___ x2 P2'

+

But this gives 1 U(x)Z-j" 7L E*

x

___

x2

+p

d p = W*(X).

The proof of (7) is somewhat more difficult. Fix z = x + iy, and let M ( p ) be the maximum of <-'G(z, <) over the part of the circle I[ - zI = p where Re{<) 2 0. Since on this circle

is a decreasing function of value o f t . Thus

4 , the maximum occurs for the smallest possible

Now let $ ( p ) denote the total length of the part of E which lies in the disk - zI < p . Since $(p) I2 p and M ( p ) is a decreasing function, integration by parts gives

From (6) and (7), it is easy to deduce (5). Indeed, it follows from (7) that

3w(z) - U(2) 2 0 on the boundary of D,so by the maximum principle, the same is true in D. This and (6) show that the function q ( z ) = 3w(z) - w*(z)

is non-negative on the positive real axis; while cp(iy) = 0 for y < 0, and q(z) 2 5 for z E E. By the maximum principle, then, o ( x + iy) 2 +w*(x+ iy),

x > 0, y < 0.

210 1 2 THE

CORONA THEOREM

Hence by symmetry, w(x

+ iy) 2 +w*(x - iy),

x > 0, y > 0.

This proves the lemma for the special case in which E is a union of radial segments with nonoverlapping projections. For a general compact set E, choose E > 0 and consider the set

s, = { z : W(.)

> 1-E}.

Clearly, aE c S,. Choose a set E" c S, which consists of a finite number of radial segments with nonoverlapping projections, for which E"* = E*. (To see that this is possible, cover E by open disks in S, and apply the Heine-Bore1 theorem.) Let &(z) be the harmonic measure of E". By what has just been proved, &(x iy) 2 +*(x - i lyl),

+

since E"* = E*. But the function [w(z) - &(z)]vanishes on the imaginary axis, is 2 0 on aE, and is 2 - E wherever it is defined on E". Thus by the maximum principle, w(x

+ iy) + E 2 6(x + iy) 2 +w*(x - I iyl)

+

for (x iy) E D.Now let E -+ 0, and the lemma is proved for compact sets E. Finally, suppose E is closed but unbounded. Let E, be the intersection of E with the disk Iz1 5 r, let E," be its circular projection, and let w,(z) and w,*(z) denote the respective harmonic measures. Then w(x

+ iy) 2 w,(x + iy) 2 $w,*(x - ilyl)

for each point (x + iy) E D.But it is clear from the integral representations that w,*(z) + w*(z) pointwise as r co. This completes the proof. --f

Hall's lemma will now be applied to obtain a special result needed in the proof of Carleson's lemma. Let R be the annulus p < IzI < 1, and let El be a closed subset of R which does not divide the plane. Let wl(z) be the harmonic measure of El with respect to ( R - El), and let El*

=

{eie : reie E E l }

be the radial projection of El onto the outer boundary of R. For fixed B < n/ llog P I , let Fl* be the part of El* such that 161 5 p llog P I . Then the total length IFl*/ of Fl* can be estimated as follows. LEMMA 5.

1 f p 3 6 E,,

12.4 CONSTRUCTION O F THE CONTOUR

PROOF.

r

211

The multiple-valued function

maps R onto the right half-plane Re{(} > 0. Let z valued) inverse, and let

E

=

{i: cp(i)

= q([) denote

the (single-

E El}.

Then E is a closed (unbounded) subset of H , and Hall's lemma may be = applied. The harmonic measure of E with respect to D = H - E is o([) wl((p(5)). Since cp maps the circular projection E* of E onto El*, the harmonic measure of E* with respect to H is w * ( i ) = ol*(cp(i)), where wl*(z) is the harmonic measure of El* with respect to R. Thus Hall's lemma gives Wl(cpP(t

+ i d ) 2 3Wl*(cp(5

- ilrll)),

5 + iy E D.

Choosing y 2 0, we obtain in particular o ~ ( z )2 +01*(~/2),

zE

(R - E l ) ; IzI

2 p"'.

(8)

On the other hand, if F* is the image of F,* under the restriction of cp-l to 161 < rc, we have

2 2110g pl-'[~osh(rc~/2>1-~ IF1*[ I

This together with (8) proves Lemma 5. 12.4. CONSTRUCTION OF THE CONTOUR

r

We are now ready to prove Carleson's lemma (Lemma l), the key to the proof of the corona theorem. Let B(z) be a finite Blaschke product, suppose 0 < E < 1, and consider the sets d(E) =

{ z : lB(z)l 4 E l ,

9?(E) = ( 2

For n

= 0,

1,

: IZI 4 1, lB(z)l 2 E } .

... and k = 1, 2, . . ., 2"+', let

Rnk= {reie : 1 - 2-" I r 2 1 - 2-"-l,

(k - 1 ) 2 3 I 6 4 k2-"rc}.

212

1 2 THE CORONA T H E O R E M

Let N be a positive integer (to be chosen later), and subdivide Rnkinto 22N parts by means of the radial lines

6 = ( k - 1 +j2-N)2-"n,

j = 1, . .., 2N- 1 ,

and the circular arcs = 1 - 2-n

+j 2 - N - " - 1

,

j=1,

...) 2 N - l .

Denote these parts (boundaries included) by Rnk(i),i = 1 , . . . , 22N.The actual numbering scheme is not important. The sets Rnk(i)will be called the blocks of the rectangle R,, . Now fix E 5 4 and let d obe the union of all the blocks Rnk(i)which meet &(E). Thus d ( ~ c) do. LEMMA 6.

If 2 - N < 618, t h e n d o c d ( 2 ~ ) .

PROOF. If zo €.do, then zo is in some block Rnk(i)which contains a point z1 E ~ ( E ) Then .

lB(z,)I 5 I&,>

- %)I

+ 8.

But it follows easily from the Cauchy formula that IB'(z)l I(1 - - 1 ~ 1 ~ )so- ~ ,

5 (1

+ 2 ~ ) 2 - ~ - " 2<"8.2-N 5 E.

Thus IB(zo)l< 28, which shows that zo E d ( 2 ~ )This . proves Lemma 6. From now on, let N be the smallest integer such that 2-N I 4 8 . (In partiSince E I 4 and K < 5, cular, N 2 5.) For fixed IC (0 < IC < +), let go= it is easily seen that 8 > 2 ~Hence . by Lemma 6 , d 0 n a0= @. For later use (in Section 12.51, we now record a lemma on the harmonic measure o ( z ; d o ) o f d o with respect to the unit disk. Roughly speaking, it is an estimate on the distance from d oto go,which increases as IC decreases. LEMMA 7. PROOF.

But for z

For

ZE

go,

@(z;d o )2 2K.

By Lemma 6,

E

go, loglB(z)l 2 K log E 2 2 K log ( 2 4 ,

since E 5 $. This gives the desired result.

12.4 CONSTRUCTION OF THE CONTOUR

r

213

Before turning directly to the proof of Carleson's lemma, it will be helpful to introduce some further notation. If n 2 1 and Rnk(i)[resp., Rnk]has the form

{re'@: a I r I b, c I 0 I d } , let vnk(i) [resp., vnk]consist of the circular arc {r = a, c 5 0 5 d } plus the two radial segments {a 5 r < 1, 0 = c, d } . Let Snk(i) [resp., S,k] be the set

{re'@: a I r < 1, c I 0 5 d). Finally, let Gnk(i)be the union of v n k ( i ) and the boundaries of all the blocks RmI(jc ) Snk(i) such that m < n + N . This set Gnk(i)will be known as the grating of Rnk(i).(See Figure 4,where N = 2 for the sake of the illustration.)

Figure 4 The grating GIz(i) of a block R l z ( i ) (shaded).

In terms of the Set do,we shall define a set Erik associated with Rnk,as follows. A block Rml(j ) c S n k n d owill be called a leading block of Snkif it is not contained in Smr1,(j') for any other block RmrIr(j') c Snk n do.Then Enkis defined as the union of v,k and the gratings G m l ( j )of all the leading blocks Rml(j)of S n k . There can be at most a finite number of gratings in Enk, since Bo contains some annular region ro < JzII1. Conceivably, E,,kmay consist of Vnkalone.

214 1 2 THE CORONA THEOREM

Within each set s , k we now construct a set Fnkcalled itsjinal set. A rectangle RmIc s n k will be called a leading rectangle of s,k if R,, meets W,but is not contained in S,,,, for any other rectangle RrnrlI c snkwhich meets 93,.Thus the leading rectangles of snk are defined in terms of W,just as the leading blocks are defined in terms of d o .Although s,k may have no leading blocks, it does have finitely many leading rectangles, since W,contains the annulus ro < IzI I 1. The jinal set Fnkof s,k is now defined as the union of the sets Em,associated with the leading rectangles R,, of snk. We are now equipped to describe a set A composed of certain radial segments and circular arcs in the open unit disk. The set required for Carleson's lemma will be an appropriate subset of A. It will be shown in Section 12.5 that the arclength measure induced by A (hence that induced by r) satisfies the metrical condition (iii) of Carleson's lemma. Let A, be the union of the boundaries of the blocks ROk(i),k = 1, 2; i = 1, . .., 2',. The sets S l k (k = 1, 2, 3, 4) will be known as residual sets of the zeroth generation. Let At be the union of the final sets F l k , k = 1, 2, 3, 4. Adjacent to each grating Gnk(i)in Al, and within the region snk(i) is a well determined set S,,,, I ,called a residual set of thejirst generation. Specifically, if S,k(i)

=

{reie: a I r < 1, c I eI c + 2-n-N7c},

then the corresponding residual set is S,,,,,

=

{reie:1 - 2-"-" I r < 1; c I eI c + 2-n-N71}.

Now let A2 be the union of the final sets Fn+N,I of all the residual sets S,,, I of the first generation. Adjacent to each grating in A2 is a residual set of the second generation; let A3 be the union of the final sets of these residual sets. The process continues until, after finitely many steps, a set AM is constructed which contains no gratings. Let

A

= A0

v A1 v

. * *

v AM.

It follows from the construction that s , k n d o= @ for each leading rectangle Rnkof a residual set of ( M - 1)st generation. Thus A surrounds d oand separates it from the boundary of the disk. We claim that A separates d ofrom W,, in the sense that each continuous curve in IzI < 1 joining a point z1 E d o to a point z2 e g o must meet A (perhaps at z1 or z2). It is clear that z1 and z 2 cannot belong to the same block Rnk(i),since d on W,= @. If 1z21 I+ then A,, separates z2 from zl. If 1z21 > +, let S be the residual set of highest generation which contains z 2 . Then z 2 E snk for some leading rectangle Rnkof S. If z16 Sn,, then vnkc A separates z1 from z 2 . If z1 E s n k , then z1 E S m l ( j )for some leading block

12.5 ARCLENGTH OF

r

215

R J j ) of s n k . But then the grating G m l ( j belongs ) to A and separates z1from z 2 , since z2 belongs to no residual set of generation higher than that of S. Thus A separates d ofrom go.Now let A be the set formed from A by deleting all interior points of d oand of go.It is clear that A still separates d ofrom go.Let R be the union of all the components of the complement of A which meet d o and , let r be the boundary of 0.Then &(E) c d oc R; and gon R = /a, since A separates d ofrom go.Consequently, E

I ~ B ( z )I( E~

if

ZE

r.

Finally, let r have the orientation compatible with the counter-clockwise orientation of the blocks &k(i). Then I? is a contour having winding number 1 about each point in R, and in particular, about each zero of the Blaschke product B(z). Hence r has properties (i) and (ii) of Carleson's lemma. We shall show in Section 12.5 that it also has property (iii). 12.5. ARCLENGTH OF

r

To complete the proof of Carleson's lemma, it remains to show that the arclength measure p induced by r is a Carleson measure. More specifically, it has to be shown that there is an absolute constant C such that p ( S ) ICE-2h for each set S of the form S = {reio: 1 - h I r < 1;

eo I e I eo + h } .

We shall prove this (if K is sufficiently small) for the arclength measure corresponding to the larger set A. In fact, the reader may find it convenient to identify p with this larger measure. It is clearly sufficient to show that p(&)

I cE-22-n

(9)

for all the sets Snk.Observe that p(RoK)I 2 N <~1 6 n ~ - ' < 1 6 ~ ~ - ~k ,= 1,2,

since ~ / < 8 2-N+1.(Recall that N is the smallest integer with 2-N Hence we need only prove (9) for n 2 1 .

Let Gnk(i) be an arbitrary grating, subtending an angle a = at the origin. Then

L E M M A 8. 2 - n - N 7~

48.)

p(G,k(i)) IbE-2a,

where b is an absolute constant.

216 1 2 T H E CORONA THEOREM

PROOF. Directly from the definition, one sees that the radial segments in G,,(i) have total length less than ( N + 4)2-"-', while the circular arcs have total length less than

( N i - 1)2-"n.

Since N 2 5, these two estimates give p(Gnk(i))< 5N2-" < bs-'a,

where b = 256. Next we consider rectangles which contain points of the set .go.The following lemma is the crucial step in the proof of (9). Recall that the parameter ti (0 < K <+) is still at our disposal. LEMMA 9. Suppose ti I K~ = Q[cosh (n2/2)]-', let R,, (n 2 1) be a rectangle which meets go,and let the leading blocks of S,,, subtend angles cq,a2,. . . , a, at the origin. Then c('

+

c(2

f

'

. . + G!"

I 42,

where a = 2-"n is the angle which Rnksubtends at the origin. PROOF. We shall apply Lemma 5. Choose

z o = p 1 i 3 e i e ~ E R,, n go. Let L,, be the union of the leading blocks of S,,,, and let L:k be the radial projection of L,, onto the unit circle. Note that

L,,cR= ( z : p < ( z I < l } , since (1 - 2-"-1)3 < 1 - 2-". Note alse that L:k has total length

IL,*,l

= 011

+ + * . . + a, a2

*

Let ol(z) be the harmonic measure of L,, with respect to R,and let ol*(z) be the harmonic measure of L:, with respect to R. Let w ( z ; L n k )denote the harmonic measure of L,, with respect to the unit'disk. Since L,, c d o ,it follows from the maximum principle and Lemma 7 that o1(zo) 5 o ( z 0 ; L,,)

Io(z0 ;d o ) I 2lC.

(10)

On the other hand, the sector 10 - OoI I p [log p ( contains S,,, (and hence Lnk)if p llog pI 2 2-%. Since 2-" 5 IIOg(1 - 2-")( I /log p ( I3(10g(l - 2-")[ _< 2'-"7~,

12.5 ARCLENGTH OF I?

this will be the case if and (10) give

p = 71.

217

L,k, since d‘, n W,= 0, so Lemma 5

But z,

I G l 5 l h PI c o s h Z ( ~ p / 2 > ~ , ( z o > < 2’-”71ti cosh2(n2/2)I a12 if

ti

5 tco. This proves Lemma 9.

The inequality (9) can now be proved for an arbitrary residual set. As above, E* will denote the radial projection of a set E onto the boundary of the unit disk. We assume henceforth that ti I t c 0 . L E M M A 10. Let S be a residual set of the mth generation, and let T be the union of the (m + 1)st generation residual sets contained in S. Then

p(S n Am+1)I Cc-’IS*I

(1 1)

and

IT*( I+ (SYI.

(12)

PROOF. Let R,, be a leading rectangle of S, and let the leading blocks (if any) of S,, subtend angles a l , a’, . . . , a, at the origin. The set Enkassociated with R,, is the union of v,k and the gratings of these leading blocks. Thus by Lemmas 8 and 9,

+

+

p(E,,,) 5 p(V,,,) be-’(x1 .. . < 3 ~ ~ l E z k+I $ b f 2 IE,*kl.

+ a,)

Summing over all the leading rectangles of S, we obtain (11). A similar application of Lemma 9 gives (12). COROLLARY. There is an absolute constant C such that p(S) I C&-’1S*I for all residual sets S.

PROOF. Let S be a residual set of mth generation. Then successive applications of Lemma 10 show that P(S n A m +

j )

< 2-jt1C&-’ IS*/,

j = 1 , 2 , . ..,M

- m.

The corollary then follows by addition over j . It is now a simple matter to prove (9) for an arbitrary set S,,, . Indeed, let c1, a 2 , . . . , a,,be the maximal residual sets contained in s,k. That is, aj is a

218

12 THE CORONA THEOREM

residual set in S,, , not contained in any lower-generation residual set in S,, By the construction of A, s , k fl

A

C

c,, U 0 1 U

' * '

.

U 0,,

where c,k is the union of the gratings G,,(z), i = 1, . .. ,22N.Thus by Lemma 8 and the corollary to Lemma 10,

+ P(Ol> -I-* * ' + P(>,. 5 2Nb~-22-"-Nn C E - ' ( ~ O , * (

P(!jn!f) 5 P(G,L)

+

+

+ I.,*/)

I c , E - 2 IS,*,l. This establishes the required inequality (9), and the proof of Carleson's lemma is complete. This also finishes the proof of the corona theorem. EXERCISES

1. Let A be the Banach algebra of functionsf(z) analytic in IzI < 1 and continuous in IzI I 1, with the uniform norm. Show that iffi, . .. ,A, are functions in A with no common zero in IzI I 1, then there exist g l , ...,g, in A such that f1(z)g,(z) + . * +f,(z)g,(z) = 1, 121 5 1. 2. Prove the following " zero-one " interpolation theorem. Let {a,} and (b,} be sequences of complex numbers in IzI < 1 such that (1 - la,[) < co and (1 - lb,l) < co, and let A(z) and B(z) denote the respective Blaschke products. Then there existsf€ H" such thatf(a,) = 0 andf(b,) = 1 (n = 1,2, . ..) if and only if there is a number 6 > 0 such that

1441+ lWl 2 6,

14 < 1.

Under this condition, f c a n be chosen with llfll I C, a constant depending only on 6 (Carleson [2]). 3. Use the zero-one interpolation theorem (Exercise 2) to prove the main interpolation theorem (Theorem 9.1) for H" : if {z,} is uniformly separated and (w,} E d", then there exists f~ H" such that f(z,) = w,, n = 1, 2, . . . . (Carleson [a].) NOTES

The corona theorem was conjectured by S. Kakutani as early as 1941. Carleson [2]gave a proof in 1961, basing part of the argument on unpublished work of D. J. Newman. In particular, the deduction of Theorem 12.1 from

NOTES

219

Carleson's lemma (as presented in Section 12.2) is essentially due to Newman. Hormander 12, 31 recently found a somewhat different approach which uses techniques borrowed from the theory of partial differential equations. Ha!l's lemma and some of its function-theoretic applications may be found in Hall [l]. The book of Gelfand, Raikov, and Shilov [I] is a good source for the basic theory of Banach algebras. The works of Wermer [l], Browder [l], and Gamelin [2] survey the theory of function algebras. Further information about the maximal ideal space of H" is in the papers of Kakutani 111, Newman [2], Schark [l], Kerr-Lawson [I], and Hoffman [2].

This Page Intentionally Left Blank

RADEMACHER FUNCTIONS

The Rademacher functions cpl(t),

cp2(t),

APPENDIX A

. .. are

cp,(t) = sgn{sin (2"nt)},

0 s t 5 1.

Thus, for example, 1, o < t < *

0, t

= 0, f, 1.

In general, cp,(t) vanishes at all multiples of 2-" and takes the values + I elsewhere. Let R denote the set of all dyadic rationals in the interval [0, 11; that is, numbers of the form m2-" (rn = 0 , 1, . . . ,2"; n = 1,2, . . .). The set R is countable, and so has measure zero. Each number t E [0, 11 - R has a unique binary expansion t = O.b,b, b3 . ..,

b,

=0

or

1.

(1)

It is easy to see from the definition of the Rademacher functions that cp,(t) = 1 if 6, = 0, while cp,(t) = - 1 if b, = 1. The number t then determines a sequence of " signs " +_ 1 : (2) cpl(0, %(O, cp3(t), . * * ;

222 APPENDIX A, RADEMACHER FUNCTIONS

and different t’s generate different sign sequences. Since t .$ R,the sequence (2) cannot be eventually constant; it must assume each of the values + 1 and - 1 infinitely often. Furthermore, every sequence of signs { E , } (E, = k 1) not eventually constant is representable in the form (2) by a unique t E [O, 1) - R. We have only to set b, = (1 - cn)/2 and let the expansion (1) determine t. In short, the collection of all sign sequences { c n } not eventually constant is in one-one correspondence with the set [0, I] - R.It is now natural to define the measure of a given collection of sequences { E , } as the Lebesgue measure of the corresponding subset of [0, I], provided this set is measurable. The set of all eventually constant sequences ( E , } is assigned measure zero. It becomes meaningful now to speak of “almost every sequence of signs.” The Rademacher functions form an orthonormal system over the interval co, 11:

More generally, if n, < n2 <

< nk ,

fdqfll(t)qII,(t) ’ ’ ’ q , k ( t > d t = *’

(3)

To see this, observe that on each interval ( j / 2 n k - 1(, j + 1)/2”k-1), the product q,,(t) qnk-,(t)is constant, while qnk(t) takes the values + 1 and - 1 equally often. e . 0

Let a,, a 2 , ... be complex numbers such

THEOREM A . l (Rademacher).

that

c

la,J2 < co. Then the series m

converges almost everywhere. Equivalently, the theorem says that for any square-summable sequence +a, converges for almost every choice of signs.

{a,,}, the series

PROOF O F THEOREM. Let s,(t) = CE=, akqk(t). By the Riesz-Fischer theorem, there is an L’ function @(t) such that

lo I@(t)- s,(t)I2 d t 1

lim n4 m

= 0.

In particular, @(t)is integrable and (by the Schwarz inequality)

1

B

dt = n-r w

a

@(t)d t ,

0 I a < /3 5 1.

APPENDIX A

223

fi

For almost every t E [0, I], the indefinite integral @(u) du has a derivative equal to @(t); let to 4 R be such a point. For each integer m, to is contained in a unique interval (a,, p,) of the form (j/2,, ( j 1)/2"). ,On the interval (a,, p,), pk(t)is constant if k I m, but takes the values f1 equally often if k > m. Thus

+

/ar[sn(t)

for n > m.

- s,(t)] dt = 0

In view of (5), we conclude that

s"

m

0 = lim n-tw

dt

um

=

1 ~

Pm

[ @ ( t ) - s,(t)]

dt.

Urn

Hence, because s,(t) is constant on (a,, s,(to)

=I

Bm

- s,(t)]

[s,(t)

- am

fm@(t)

pm),

dt -+

as m -+ co.

@(to)

em

Thus ( s n ( t ) } converges almost everywhere. COROLLARY.

If ($,(x)} is an orthonormal system in L2[u,b], and if

< co,then for almost every choice of signs { e n ] , the series

JU,,~~

converges almost everywhere. PROOF. By the Lebesgue monotone convergence theorem, the series lan everywhere. Thus, for almost every

c,"= $,,(x)]' converges almost x [a, b], the series E

converges for almost every % E[0, I]. Let E be the set of all points ( t , x) at which (6) converges, and let r(t,x) be the characteristic function of E. We have just observed that for almost every x , r(t,x ) = 1 for almost all t. Fubini's theorem therefore gives

la b

(b-U) =

1

r(t,X) dt dx = /

1

b

O

a

r(t,x) dx dt.

If the corollary were false, however, the right-hand side would be less than (b - a).

224 APPENDIX A, RADEMACHER FUNCTIONS

THEOREM A.2 (Khinchin‘s inequality). Suppose as in Theorem A.1 that 00, and let @(t)be the sum (4). Then @ €LPIO,11 for everyp < 00,

candlunI2<

I f p = 2m is an even integer, the constant ( p / 2 PROOF.First let p

= 2m,

+

can be replaced by m1I2.

and let the coefficients aj be real. Then

where the sum is extended over all systems of nonnegative integers vj such that v1 * * * + v, = 2m. In view of the orthogonality property (3), however, all the terms vanish except those for which all vj are even integers (zero included). Thus the sum is equal to

+

On the other hand,

But the ratio of the respective coefficients is ( 2 m ) !k , ! k 2 ! * . - k,! <-( 2 m ) ! < mrn. m ! ( 2 k l ) ! ( 2 k 2 )...( ! 2k,)! - m ! 2 ” I 2 - k ; the inequality involving mm may be [If k > 0, k ! / ( 2 k ) !s (k + verified by induction.] Therefore,

if the u j are real. In the complex case, let uj = aj + iPj and let s, = u,

+ iu, = C aj ‘ p j + i C

Pj

yj.

By Minkowski’s inequality,

I m j=l

which is equivalent to (8).

aj2

+ m f1 j=

c laj12, n

pj2

=m

j= 1

APPENDIX A

225

If 2m - 2 < p < 2m, it follows that

Letting n -+ 00 here and in (8), we obtain (7) and its sharpened form for the case p = 2m. The argument also shows that s, + (D in the L p mean for every p < co. THEOREM A.3 (Khinchin-Kolmogorov).

If

= co, then the series

a, qn(t)diverges almost everywhere. PROOF, Suppose, in fact, that the partial sums sn(t) are bounded on a set E of measure [El > 0. Then

Is,(t) - sm(t)lI C,

1 I m < n ; t E E,

where C is a constant. It follows that

= IEI

k

lakI2

k=m+ 1

+

f

j , k=m+ 1 j
a j G

/

E

(9) qj(t)qk(t)

dt.

On the other hand, according to (3), the doubly indexed system of functions q j ( t ) ( P k ( t ) (1 - < j < k < 00) is orthonormal over [0, 11. We may regard J E q j ( t ) q k ( t ) dt as the “Fourier coefficient ” of r(t), the characteristic function of the set E. By Bessel’s inequality, then,

Therefore, for m sufficiently large,

It follows from (10) and the Schwarz inequality that the absolute value of the last term in (9) is no greater than

226 APPENDIX A. RADEMACHER FUNCTIONS

Introducing this estimate into (9), we find

The next theorem serves a special purpose in Section 4.6 and has other applications. THEOREM A.4. Let g,(z), g2(z), . . . be complex-valued functions each of which is continuous in IzI 5 I except perhaps at a finite number of points on IzI = 1. Suppose

,

Jg,(z)l < co in 1zI < 1, the convergence being uniform in each disk IzI I p < 1 ; and (ii) for each N , CFZN 1gn(reie)>12 co uniformly in 0 as r -+ 1. (i)

--f

Then, for almost every choice of signs

{gn},

the function

has a radial limit almost nowhere. (That is, the 0’s for which limr+l G(reie) exists constitute a set of measure zero.) PROOF. Because of hypothesis (i), the function

m,0, t> = c cpn(t)gn(rei8) m

n= 1

is well defined for all r E [0, I), 0 E [0,271], and t E [0, 11. It is continuous in r and 0. Let E = ((8, t) :lim F(r, 8, t ) does not exist.} r+l

We are going to show that for every 8 E [0,271], the “0-section” be = {t : (0, t ) E S}

has measure Idel = 1. It will then follow from Fubini’s theorem, as in the proof of the Corollary to Theorem A.1, that for almost every t E [0, 11, the set E‘

=

{e : (0, t ) E 6)

has measure 271, as the theorem asserts.

APPENDIX A

227

Suppose, then, that lSsl< 1 for some fixed 8. Then the complementary set has measure

80

a = I&,]

I

=1

- 1601 > 0,

and F(r, 8, t) has a (finite) radial limit for all t E 8 0 . Because of the continuity of the functions gn(z),we may conclude that m

FN(r,8, t ) =

C qn(t)gn(re") n=N

( N = 1,2, . . .)

has a radial limit for all t in a set A of measure a, obtained from Reby the deletion of a countable set. Now, for K = 1, 2, ...,let BK = { t E A : IF(r, 8, t)l I K for all r < I}.

Then B, c B, c . - -and &=, BK = A , so that lBKl -+ IAl = a. Choose K = K , so large that IBK,I 2 $a, and let A , = BK1.Then IF(r, 8, t)l IK , for all t E A , and r < 1. Similarly, there are a set Az c Al and a constant K, such that ] A z ]2 +a and IF,(r, 8, t)l IK, if t E Az and r < 1. Proceeding inductively, we may construct a sequence of sets

AxAl xAzx.-*xAN3*-. and a sequence of constants { K N }such that

IAN1 2 (a/2)(1

+2-N)

and

lFN(Y,8,t)l
tEAN; r < l .

Finally, let a,

B= n A N . N=l

Then IB( 2 a/2 > 0,and

We are now in a position to imitate the proof of Theorem A.3. First fix N so large that

m
228

APPENDIX A. RADEMACHER FUNCTIONS

In view of (I 1), an argument entirely similar to the proof of Theorem A.3 now gives

or m

But tAs contradicts hypothesis (ii). Hence Idel = 1 for every ”, and the proof is complete. THEOREM A.5.

Let a, , a,,

... be complex numbers such that

’ ~1. lim sup ~ q l l = n+

(i) If

C

m

< co, then for almost every choice of signs {q,}, m

f(z)=

C

E,u,z“EHP

for all p < 00.

n=O

1

(ii) If la,I2 = 00, then for almost every choice of signs { E , } , ~ ( z )has a radial limit almost nowhere. PROOF.

Part (ii) is an immediate corollary of Theorem A.4, with g,(z)

=

anz”.We have only to observe that m

02

m

n=N

n=N

n=N

c 1anznl2= C lan12r2n-+ c lanlZ= co

as r -+ 1. (i) If

C lanlZ< co,then W

g ( z , t> =

C qn(t)anzn E H Z

n= 1

for every t, so (by Theorem 2.1 1) it need only be shown that for almost every t E 10, I], the radial limit G(8, t ) = lim g(re”, t ) r -*1

belongs to LP, as a function of 0, for all p < co. By Khinchin’s inequality (Theorem A.2),

APPENDIX

A 229

On the other hand, Theorem A . l implies that for each fixed 8, the limit (12) exists for almost every t. By Fatou's lemma, then,

Thus IG(8, t)lP is integrable over the rectangle, and by Fubini's theorem, G(8, t ) E Lp for almost every t. To complete the proof, choose a sequence p1,p2,...-)a, andlet E, = (t :G(0, t ) 4 Ek};

E

=

{ t : G(8, t) $ Lp for some p < co}.

Then lEkl = 0 for each k , so that E = UF'

Ek is also of measure zero.

Even if log n < co,it can happen that E, a, Z" $ H of signs. However, the slightly stronger condition

for every choice

C [a,(2(logn)lfd< co

(6 > 0 ) forces 1E, a,, z" to be continuous in JzII 1 for almost every choice of signs. For the proofs we must refer the reader to the literature (see Notes). NOTES

More information on Rademacher functions and related questions can be found in the books of Alexits [l], Kaczmarz and Steinhaus [I], and Zygmund [4]. Paley and Zygmund [l, 21 used Rademacher functions to prove various theorems in function theory. Proofs of the assertions in the last paragraph (above) may be found in Paley and Zygmund [l]. Theorem A.5 is due to Littlewood [4].

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MAXIMAL THEOREMS

APPENDIX B

The Hardy-Littlewood “ maximal theorem ” becomes clearer if it is stated first in discrete form. Let al, u 2 , . . . , a, be given nonnegative numbers, and let b,, b,, . . . , b, be the same numbers rearranged in nonincreasing order. For each fixed k (k = 1, . . . , n), let u k = max lsjck

1 k-j

+ 1 2ai i=j

be the optimal average of successive ui terminating with a,; and let

be the corresponding quantities for the bi. Let s(x) be any nondecreasing function defined for all x 2 0. Then, as Hardy and Littlewood showed, k= 1

k= 1

Hardy and Littlewood interpreted the a, as cricket scores and s as the batsman’s “ satisfaction function.” The theorem then says that the batsman’s

232

APPENDIX 6, MAXIMAL THEOREMS

total satisfaction is maximized if he plays a given collection of innings in decreasing order. In order to state a continuous version of this theorem, it is necessary to define the “ rearrangement ” of a function. Let f(x) be nonnegative and integrable over a finite interval [0, a ] , and let p ( y ) be the measure of the set in whichf(x) > y . Note that p(y) is nonincreasing. Two functionsf,(x) and f2(x) are said to be equimeasurable if they give rise to the same function p(y). It is then clear from the definition of Lebesgue integral that “

”

J)dX)

dx

dx.

= f0k x )

If p ( y ) is associated as above with f(x), its inverse function f*(x) = p-’(x), normalized so that f*(x) =f*(x +), is called the decreasing rearrangement of f(x). It is easy to see that f*(x) and f(x) are equimeasurable. Finally, let A(x,

1

”

5 ) = A ( x , 5 ; f)= 2Jt f(t)

dt,

0

5: < X ;

and let O(x) = @(x; f) = sup A(x,

5:; f).

0564x

The maximal theorem may now be stated as follows. THEOREM B . l (Hardy-Littlewood maximal theorem).

If s(y) is any non-

decreasing function defined for y 2 0, then Jis(O(x;

f))d x

Hardy and Littlewood used a limiting process to base a proof on the discrete form of the theorem. Shortly afterwards, F. Riesz showed that in fact O(x;f*), O*(x;f) I

0 < x Ia,

(1)

which immediately implies the Hardy-Littlewood result. Riesz’s proof of (1) makes use of his “rising sun lemma,” which may be stated as follows. LEMMA. Let g(x) be continuous on the interval [0, a], and let E be the set of all x in (0, a ) for which there exists 5 in [0, x) with g(5) < g(x). Then E is an open set: E = U(ak, bk), where the intervals (ak, bk) are disjoint; and &k) dbk).

PROOF OF LEMMA. Since the inequalities 5 < x and g(5)
APPENDIX B

233

choosex, E ( a k ,bk)and consider the point x 1 where g(x) attains its minimum in [O, xO]. Then x, $ ( a k ,x,], since these points are in E. Thus x1 E [0, a k ] . But since uk $ E, g(ak)5 g(x) for all x E [0, ak). Hence x, = a,, and in particular, g(u,) 5 g(xo). Now let xo -+ b, to conclude that g(a,) 5 g(bk). PROOF OF (I). For fixed y o 2 0, apply the lemma to the function

dx) =

Jkr) d t - Yo x. 0

E is then the set of all x for which there exists 5 in [O, x) with A(x, 5 ; f ) > y o . In other words, E = {x : O(x;f) >yo). (2) On the other hand, the condition g(ak)I g(bk) says that A(bk, a k ; f )2 y o . Thus

where m ( E ) denotes the measure of E. Now let

All this is for arbitrary y o 2 0. Given xo E (0, a], choose y o = O * ( x o ; f ) . Then by (2) and the definitioii of O*, m ( E ) = xo. Hence (4) gives @(xo;f*) 2 Y o = @*(xo;f), which proves (1). The following application of the maximal theorem will serve to illustrate its usefulness. THEOREM B.2. If f ( x ) belongs to Lp (1 c p < co) over some interval [O, a ] , then the maximal function O(x) = O(x; I f / ) is also in Lp,and

234 APPENDIX 6, MAXIMAL THEOREMS

PROOF. Apply the maximal theorem with s(y) = y p , note that O ( x ; f * )= A(x, O ; f * ) , and use the following inequality. L E M M A (Hardy's inequality).

If 1 < p < co,g(x) is in Lp over (0, co),and

then G E Lp and

PROOF OF LEMMA.

Fix a > 0. Since

the continuous form of Minkowski's inequality gives

Now let a -+ co to obtain Hardy's inequality. The next theorem is essentially a restatement of Theorem B.2 in a form convenient for certain applications (see Section 1.6). THEOREM B.3. Letf(x) be periodic with period 2n, and suppose f E Lp = Lp(O,2n), I < p < co.Then

If jo T

F ( x ) = sup O<(T(sn

is also in Lp,and [IFlip PROOF.

f(x

+ t ) dt

I

C p ~ ~ fwhere l ~ p ,Cpis a constant depending only on p .

It is clearly enough to assumef(x) 2 0 and to consider

A 1- T f ( x + t ) d t O<Tin 0

F l ( x ) = sup

APPENDIX B

235

This function is easily compared with the maximal function O(x) = O ( x ; f ) associated withf. For x 2 rc, it follows from the definitions that Fl(x) 2 O(x). For 0 I x < rc, the periodicity offmay be used to show that F,(x)5 O ( x + 271). Hence by Theorem B.2,

as desired. The constant Cpcan be taken as 4””p(p - I)-’. NOTES

Maximal theorems are the invention of Hardy and Littlewood [4], who proved Theorem B.l and pointed out its applications to function theory. The simple proof given above is due to F. Riesz [8]. See also Hardy, Littlewood, and P6lya [I]. Riesz had introduced the “rising sun lemma” as a tool in differentiation theory (see, for example, Boas [2]).Hardy’s inequality and related results are in Hardy, Littlewood, and P6lya [l].

This Page Intentionally Left Blank

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This Page Intentionally Left Blank

AUTHOR INDEX

Numbers in italics show the page on which the complete reference is listed

A Ahlfors, L. V., 6 , 16740,146, 180,1859 189, 237 Akutowicz, E. J., 146, 198, 237 Alexits, G . , 229, 237

C Calderbn, A. P., 68, 238 Cantor, D. G., 107, 238 CarathCodory, C., 145, 238 Cargo, G. T., 31, 238 Carleson, L., 146,164,165,218,237,238 Caughran, J. G., 92, 146, 238 Caveny, J., 107, 239 Coifman, R., 185, 239 Collingwood, E. F., 239

B D Banach, S., 99, 237 Bary, N. K., 63, 238 Beurling, A., 31, 126, 238 Blaschke, W., 30, 238 Boas, R. P., 235, 238 Bonsall, F. F., 146, 238 Browder, A., 219, 238

Day, M. M., 127, 239 Doob, J. L., 145, 239 Douglas, R. G., 239 Dunford, N., 99, 111, 115,239 Duren, P. L., 90-92, 106, 107, 118, 127, 165, 184, 185 239

254

AUTHOR INDEX

E Earle, C. J., 185, 239 Egerviry, E., 145,239 Epstein, B., 240 Evgrafov, M. A,, 106, 145, 240 F Fatou, P., 14, 240 Fejer, L., 52, 145, 238, 240 Fichtenholz, G., 52, 240 Fisher, S. D., 185, 240 Flett, T. M., 92, 106, 240 Forelli, F., 127, 185, 240 Frostman, O., 31, 240

K Kabaila, V., 164,243 Kaczmarz, S., 107, 229, 243 Kahane, J.-P., 185, 243 Kakeya, S., 145, 244 Kakutani, S., 219, 244 Kas’yanyuk, S. A., 185, 244 Kawata, T., 199, 244 Keldysh, M., 184, 244 Keldysh, M. V., 184, 244 Kerr-Lawson, A., 219, 244 Kolmogorov, A., 68, 244 Koranyi, A., 165, 249 Kothe, G., 127, 244 Krein, M., 165, 244 Krylov, V. I., 199, 244

G 1

Gabriel, R. M., 14, 198, 240 Gamelin, T. W., 127, 184, 185, 219, 240,241 Garabedian, P. R., 146, 241 Gaudry, G. I., 241 Gelfand, 1. M., 219, 241 Geronimus, Ja. L., 145, 241 Gleason, A. M., 241 Goffrnan, C., 241 Goldberg, R. R., 199, 241 Golubev, V., 52, 241 Goluzin, G. M., 52, 145, 180, 184, 241 Gronwall, T. H., 145, 241 Gwilliam, A. E., 69, 241 H Hall, T., 219, 241 Hardy, G. H., 14, 52, 68, 91, 92, 106, 107, 235, 242 Havin, V. P., 242 Havinson, S. Ja., 145,146,184, 185,242,250 Hayrnan, W. K., 49,92, 164, 178, 242 Hedlund, J. H., 107,242 Heins, M., 185, 243 Helson, H., 52, 127, 243 Herglotz, G., 14, 243 Hille, E., 198, 199, 243 Hoffman, K., 164, 219, 243 Hormander, L., 165, 219, 243

Landau, E., 30,145,244 Landsberg, M., 127, 245 Lavrentiev, M. A., 184, 244 Lax, P. D., 146, 245 Lebedev, N. A., 184, 249 deleeuw, K., 127, 245 Littlewood, J. E., 14, 30, 52, 68, 91, 92, 106, 107,229,235,242,245 Livingston, A. E., 127,245 Lohwater, A. J., 52,92,239,245 Lumer, G., 185,240,245 M Macintyre, A. J., 144, 145,245 Marden, A., 185, 239 Marden, M., 141, 245 Minker, J., 240 N

Nagasawa, M., 127,245 Natanson, I. P., 2, 44, 245 Nehari, Z., 137, 146, 178, 184, 185, 245 Neuwirth, J., 246 Nevanlinna, F., 30, 246 Nevanlinna, R., 30, 145, 165, 246

AUTHOR INDEX

Newman, D. J., 107, 127, 164, 219, 246 Noshiro, K., 246 0

Ostrowski, A., 30,246

P Paley, R. E. A. C., 31,68,107,199,229,246 Parreau, M., 185, 246 Pedrick, G., 241 Penez, J., 146, 246 Pick, G., 145, 246 Piranian, G., 92, 185, 245, 247 Plessner, A., 14, 247 Pblya, G., 52, 235, 242 Porcelli, P., 247 Prawitz, H., 52, 247 Privalov, I. I., 31, 52, 92, 107, 184, 185, 247

R Rado, T., 14,247 Raikov, D., 219, 241 Rechtrnan, P., 165, 244 Riesz, F., 14, 29, 30, 52, 145, 235, 240, 247 Riesz, M., 30, 52, 68, 198,247 Rogosinski, W., 144-146,247 Rogosinski, W. W., 144-146, 245, 247 Romberg, B. W., 118,127,239,247 Royden, H. L., 185, 245 Rudin, W., 52, 73, 92, 107, 127, 146, 185, 239,245,248 Ryan, F. B., 52, 245 Ryff, J. V., 14, 248

S Sarason, D., 185, 248 Sario, L., 180, 237 Schark, 1. J., 219, 248 Schur, I., 145, 248 Schwartz, J. T., 99, 111, 115, 239 Shapiro, H. S., 90, 91, 107, 127, 145, 146, 164, 184, 185, 239, 246-248

255

Shapiro, J., 127, 248 Shields,A.L.,90,91, 107,118, 127, 164,184, 239,248 Shilov, G., 219,241 Smirnov, V. I., 30,31, 52,184,248,249 Somadasa, H., 31, 249 Stein, E. M., 107, 249 Stein, P., 68, 249 Steinhaus, H., 229, 243 Sunouchi, G., 249 ~

~

~F’268, 249 ~

Szego, G., 30, 184, 249 Sz.-Nagy, B., 165, 249

T Tamarkin, J. D., 198, 199, 243 Tanaka, C . , 30, 249 Taylor, A. E.3 14, 126, 250 Taylor, G. D., 92, 106, 107, 239, 250 Titchmarsh, E. C., 9, 72, 250 Turnarkin, G. C., 146, 184, 185, 250

V

Voichick, M., 127, 185, 241, 250, 251

W Walters, S. S., 127, 251 Weill, G., 185, 237 Weiss, G., 30, 185, 239, 249, 251 Weiss, M., 30, 251 Wells, J. H., 107, 251 Wermer, J., 127, 219, 245, 251 Whitney, H., 241 Widder, D. V., 2, 251 Wiener, N., 199, 246

Z Zalcman, L., 185, 251 Zygrnund, A., 29, 31, 52, 68, 92, 106, 107, 158, 165, 229, 246, 249, 251

~

SUBJECT INDEX

A

Abel summability, 62 Absolute continuity, 41, 42 Annihilator, 110 Arithmetic-geometric mean inequality, 29 Automorphic group, 180

B Banach algebra, 201 Banach space, 37 Beurling’s theorem, 114, 173 Blaschke product, 20, 191, 204 Block, 212 Boundary function, 6, 17, 35, 74, 170, 189 Bounded characteristic, 16 Bounded variation, 2, 39, 72

C

Carathkodory convergence theorem, 168, 184 Carathkodory’s theorem, 44 Carleson measure, 157 Carleson’s lemma, 203 Carleson’s theorem, 157 Cauchy integral, 39, 170, 194 Cauchy-Stieltjes integral, 39 Class A,, 72 Class A,, 72 Class N , 16 Class N + , 26 Coefficients, 93, 139, 142 Conformal mapping, 43 Conjugate function, 33, 53, 62 Conjugate harmonic function, 33, 53, 83 Conjugate index, 54 Conjugate Poisson kernel, 35 Conjugate trigonometric series, 61 Canonical factorization, 24, 25, 192

SUBJECT INDEX

Corona theorem, 202 Cover transformations, 180 Covering lemma, 161

D Divisor, 113 Dual extremal problem, 130 Dual space, 112, 115 Duality relation, 130

E Equimeasurable functions, 232 Exponential sequence, 156 Extremal function, 132 Extremal kernel, 132 Extremal problem, 129 Extreme point, 123

F F-space, 99, 115, 154 Fatou’s theorem, 14 Fejer-Riesz inequality, 46 Final set, 214 Fourier coefficients, 37 Fourier series, 61 Fourier-Stieltjes series, 61 Fourier transform, 195

G Gelfand topology, 201 Gelfand transform, 201 Grating, 213 Green integral, 172 Green’s function, 172 Green’s theorem, 55

257

on coefficients of H p functions, 95, 98 on comparative mean growth, 84, 87 on integrals of H p functions, 88 on multipliers, 103, 107 on rearrangements, 232 Hardy’s convexity theorem, 9 Hardy’s inequality, 48, 234 Harmonic conjugate, 33, 53, 83 Harmonic majorant, 28, 168, 179, 188 Harmonic measure, 208 Hausdorff-Young inequalities, 93 HeIIy selection theorem, 3 Herglotz representation, 3 Hilbert’s inequality, 48

1

Inner factor, 113 Inner function, 24 Integral modulus of continuity, 72 Interpolation, 140, 141, 147, 203 Invariant subspace, 126

J

Jensen’s inequality, 29 Jensen’s theorem, 16 Jordan curve, 43 Jordan domain. 169

K Keldysh-Lavrentiev example, 176, 184 Khinchin-Kolmogorov theorem, 225 Khinchin’s inequality, 224 Kite-shaped region, 5 Kolmogorov’s theorem, 57

L H Hall’s lemma, 208 Hardy-Littlewood theorem on boundary smoothness, 74, 78

Lacunary sequence, 104 Leading block, 213 Leading rectangle, 214 Least harmonic majorant, 28, 168, 179, 189

258 SUBJECT INDEX

Linear functionals on H p 1 c p < co, 112 0 < p < 1, 115 Lipschitz classes, 72 Littlewood‘s subordination theorem, 10

M Marcinkiewicz interpolation theorem, 95, 158, 163, 165 Maximal ideal, 201 Maximal theorems, 11, 23, 89, 231 Mean convergence, 21, 192 Mean growth, 74 Modulus of continuity, 71 Multipliers, 99 Multiply connected domains, 179 N

Natural kernel, 142 Nevanlinna class, 16 Nevanlinna theorem, 16 Nontangential limit, 6, 17 Normal family, 6 Normalized extremal function, 132

0 Orthogonal polynomials, 175, 184 Orthonormal system, 175, 222, 223 Outer function, 24, 25

P Paley-Wiener theorem, 196 Paley’s theorem, 104 Plancherel formula, 196 Poisson kernel, 2 Poisson integral, 2, 41, 190, 195 Poisson-Jensen formula, 16 Poisson-Stieltjes integral, 2, 40

Q Quotient space, 110

R Rademacher functions, 67, 87, 94, 221 Rademacher’s theorem, 222 Rational kernels, 136 Rearrangement, 232 Rectangle, 212 Rectifiable curve, 44 Residual set, 214 Riesz factorization theorem, 20 Riesz (F. and M.) theorem, 41 Riesz (M.) theorem, 54 Rising sun lemma, 232

S Schlicht functions, 49 Schwarzian derivative, 176 Singular inner function, 24 Smirnov domain, 173 Smoothness classes, 71 Subharmonic function, 7, 188 Subordinate function, 10 Symmetric derivative, 4 Szego polynomials, 175, 184 T Taylor coefficients, 93, 139, 142 Trigonometric series, 61

U Uniformly convex space, 123 Uniformly separated sequence, 148, 154 Universal covering surface, 180 Universal interpolation sequence, 148

w Walsh’s theorem, 174, 184 Weak type, 158, 160, 163, 165

Z Zeros, 18, 191 Zygmund’s theorem on boundary smoothness, 76 on conjugate functions, 58

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Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Paul A. Smith and Samuel Eilenberg

Columbia University, New York

1: ARNOLD SOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume V I )

2 : REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 3 : HERBERTBUSEMANN A N D PAULKELLY.Projective Geometry and Projective Metrics. 1953 4: STEFAN BERCMAN A N D M. SCHIFFER. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. 1953 5: RALPHPHILIP BOAS,JR. Entire Functions. 1954 6: HERBERT BUSEMANN. The Geometry of Geodesics. 1955 7 : CLAUDECHEVALLEY. Fundamental Concepts of Algebra. 1956 8: SZE-TSEN Hu. Homotopy Theory. 1959 9: A. M. OSTROWSKI. Solution of Equations and Systems of Equations. Second Edition. 1966 ~. on Analysis. Volume I, Foundations of Modern Analy10: J. D I E U W N NTreatise sis, enlarged and corrected printing, 1969. Volume 11, 1970. 11 : S. I. GOLDBERC. Curvature and Homology. 1962. 12 : SICURDUR HELGASON. Differential Geometry and Symmetric Spaces. 1962 13 : T. H. HILDEBRANDT. Introduction to the Theory of Integration. 1963. 14 : SHREERAM ABIIYANKAR. Local Analytic Geometry. 1964 15: RICHARD L. BISHOPA N D RICHARD J. CRITTENDEN. Geometry of Manifolds. 1964 16: STEVEN A. GAAL.Point Set Topology. 1964 17: BARRYMITCHELL.Theory of Categories. 1965 18: ANTHONY P. MORSE.A Theory of Sets. 1965

Pure and Applied Mathematics A Series of Monographs and Textbooks

19 : GUSTAVECHOQUET. Topology. 1966 20: 2. I. BOREVICH A K D I. R. SHAFAREVICH. Number Theory. 1966 21 : Josh LUIS MASSERA A N D J U A NJORGE SCHAFFER. Linear Differential Equations and Function Spaces. 1966 22 : RICHARD D. SCHAFER. An Introduction to Nonassociative Algebras. 1966 23: MARTINEICHLER. Introduction to the Theory of Algebraic Numbers and Functions. 1966 24 : SHREERAM ABHYANKAR. Resolution of Singularities of Embedded Algebraic Surfaces. 1966 25 : FRANCOIS TREVES. Topological Vector Spaces, Distributions, and Kernels. 1967 26: PETER D. LAXand RALPHS. PHILLIPS. Scattering Theory. 1967 27: OYSTEINORE.The Four Color Problem. 1967 28 : MAURICE HEINS.Complex Function Theory. 1968 29 : R. M. BLUMENTHAL A N D R. K. GETOOR. Markov Processes and Potential Theory. 1968 30 : L. J. MORDELL. Diophantine Equations. 1969 31 : J. BARKLEYROSSER.Simplified Independence Proofs : Boolean Valued Models of Set Theory. 1969 32: WILLIAMF. DONOGHUE, JR. Distributions and Fourier Transforms. 1969 33: MARSTONMORSEA N D STEWART S. CAIRNS.Critical Point Theory in Global Analysis and Differential Topology. 1969 34: EDWIN WEISS.Cohomology of Groups. 1969 35: HANSFREUDENTHAL AND H. DE VRIES. Linear Lie Groups. 1969 36 : LASZLO FUCHS. Infinite Abelian Groups : Volume I. 1970 37: KEIO NAGAMI.Dimension Theory. 1970 38: PETER L. DUREN.Theory of HP Spaces. 1970

In firebaration EDUARD PRUGOVECKI. Quantum Mechanics in Hilbert Space. BODOPAREICIS. Categories and Functors : An Introduction

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