Function Theory in the Unit Ball of Cn: Reprint of the 1st Ed Berlin Heidelberg New York 1980

Put v = \f\^'^. Then

5.6. The Spaces N(B) and H'iB)

87

Theorem 5.6.2, with (p(x) = e''^'^, shows therefore that v < P[/j] for some /i6LV),with||/»||2 = ||/||P'^Hence

and we conclude from Theorem 5.4.10 (applied to h, with exponent 2) that (\MJ\''dcj
h'd,J Js

= A(o{)\\fr,.

5.6.6. Theorem. Suppose 0 < p < oo, / e H'iB), and put (1)

/ * ( 0 = (X-lim/)(0

(a.e.onS).

Then (2)

Um f | / * - / , | ' ' d < T = 0. Note that / * ( 0 exists a.e., by Theorem 5.6.4.

Proof. Put F = M.^^f. Then F < M ^ / if a > 2, and Theorem 5.6.5 shows that F e L\(J). The inequalities

^ ^y>

-\^P

+ yP

1),

(0
valid for X > 0, j ; > 0, show that (3)

* ....[2^^^ \r-fr\'< [2F^

(P^l) (0
1).

Since X ^ / * a.e. as r / * 1, (2) follows from (3) and the dominated convergence theorem. 5.6.7. Definition. We let A(S) denote the class of all fe C(S) that are restrictions to S of members of the ball algebra A(B). [Note that ^4(5) is a closed subalgebra ofC(S), relative to the sup-norm, by the maximum modulus theorem.] For 0 < p < 00, we let H^iS) be the L^-closure of ^(5). 5.6.8. Theorem, (a) Iffe IF(B) thenf* e IP{S) and |[/'*||p = \\f\\p. (b) Ifp >\andge H\S\ then P[^] = C[^] e H\B\ and g = C|^]* a.e.

88

5. Boundary Behavior of Poisson Integrals

Proof. Part (a) is a corollary of Theorem 5.6.6, since f e A(S) for 0 < r < 1, and since the subharmonicity of | / | ^ impHes that

lim||/,||,= sup II/,||p= 11/11,. r-^l

0
Next, Pig'] = Clg] when g e A(SX by Theorem 3.3.2. Hence Pig] = Clg] for every g e H^iS). Since Clg] is holomorphic, the rest of (b) follows from Theorems 3.3.4(b) and 5.4.8. Corollary. The map f-^ f* is a linear isometry ofH\B)

onto H\S).

5.6.9. Theorem. The map f -^ C [ / ] * is the orthogonal projection of V-{o) onto H^(S), and (1)

[\M^Clf]\'dG
[m^da Js

for every f e L^{a). Proof. Every feV'{a) has an orthogonal decomposition f = g -\- h, where g eH\S) and h 1 H^S). Fix zeB and define t/,(0 = C(z,0. Then u^ e A(S\ and therefore

=1

Clh'](z)=

hu,d(7 = 0.

Js

Thus C [ / ] = Clg], and C [ / ] * = ^ by Theorem 5.6.8. This proves the first assertion. Next, (1) holds with g in place of/, by the Hardy-Littlewood theorem (5.6.5). Since C [ / ] = C[^] and Wgh < 11/II2, (1) holds f o r / a s well.

5.7. Appendix: Marcinkiew^icz Interpolation This appendix contains the special cases of the Marcinkiewicz interpolation theorem that occur in the present book. The theorem is usually stated for a wider variety of function spaces (see, for instance, Zygmund [2], [3; Chap. XII] or Stein and Weiss [1; Chap. 5], and the proofs are correspondingly more complicated. We consider two positive measures a and fi on sets 5 and X, respectively, and we let Tbe an operator that associates to every/GL^((T) a measurable function Tf.X ^ [0, 00], and that is sub-additive: r ( / + g) < Tf + Tg.

5.7. Appendix: Marcinkiewicz Interpolation

89

The various maximal functions that occur in Chapters 5 and 6 are examples of such operators. We let c^ and c^o be the smallest constants such that the inequahties li{Tf > t] < c^r' \ l/r^d, \\Tf\\^
{iff Js

da.

(b) / / r > 1, then f (Tf)dfi < KXM, Jx

Ml c) + K'Xci, c,) f \f\log^\f\dc7. Js

The constants Kp, K\ K" are finite whenever the indicated parameters are finite (and are + oo otherwise). Proof. The formula (1)

[pPdii^p Jx

f ii{F>t}tP-^ Jo

dt

holds for every measurable F : X - > [0, oo] and for every pe(0, oo). To prove (1), define (p on X x [0, oo) by (p(x, t) = pt^'^ if 0 < r < F(x\ 0 otherwise, and apply Fubini's theorem to the integral of cp over X x [0, oo) with respect to the product measure dju x dt. Fix feL^((j). For every te(0, oo) split / into f = gt + K, where gji£) = / ( O if 1/(01 > t, g,(0 = 0 if 1/(01 < t. Put

(2)

G(t) = Jrg, > | | ,

Hit) = JTK > ^ j ,

and (3)

GM = ^

f \f\dG, t J\f\>t

H,(t) = ^ t

f l / r da. J|yl<(

90

5. Boundary Behavior of Poisson Integrals

The definitions of Ci and c^ (for r < oo) show that (4)

G(t)<2c,t-'

(5)

H(t) < I'Crt-'

j\gM(^=G,(tl f \K\' da = H,(t). Js

Integration of (3) yields (6)

pG,it)dt

(7)

p rG,(tr-'dt Jo

(8)

p rH.ity-' Jo

=

2c,j\f\\og^\f\dG

= ^ ^ f l/l^ da P - ^ Js

(l
dt = ^ ^ \\f\'da r - p Js

^X

(0
Since Tf < Tg^ + Th,, (4) and (5) give (9)

ii{Tf >t}<

G{t) + H{t) < G,{t) + H,{t\

Hence (1) impHes: \x{Tfy dji < (7) + (8) if 1 < p < r. This proves (a) in the case r < oo. If r = 00 in (a), assume c^ <^, without loss of generality. (Replace T by Tjlc^ if necessary.) Since ||/iJloo ^ U we have ||T/i(||^ < t/2, hence //(O = 0, and (9) becomes ^{Tf > t) < G^(t). Now (a) follows from (1) and (7). Since fi{Tf > t} < \\iii\\,(9) yields /•oo

/t{r/ > t}cit < 11/^11 +

Jo

/•oo Ji

GiCOrft +

/•oo Jo

so that jx C^TMiU < lliull + (6) + (8) (with p = 1). Since \\f\da<e\\a\\+ Js (b) follows.

Js

Uf\\og^\f\da,

i/i(Orff,

Chapter 6

Boundary Behavior of Cauchy Integrals

One major difference between the Poisson kernel and the Cauchy kernel is. that the former is positive and the latter is not. The positivity of P(z, Q made it possible to be rather unsubtle in the proof of the basic maximal theorem 5.4.5: jj, was replaced by |/i|, and it was then just a matter of making size estimates without giving too much away. No cancellation effects came into play. No such approach can succeed with the Cauchy integral, for the simple reason that

b\C(z,0\dc7(0^oo as z approaches any boundary point of B. (This is the case c = 0 of 1.4.10.) Nevertheless, Cauchy integrals of measures do have finite X-Hmits at almost all points of S (Theorem 6.2.3), and a weak-type (1, 1) maximal theorem (6.2.2) does exist, even though M^Cljii] is not pointwise dominated by MJJ,. The proofs of these facts proceed along a route that has by now become standard when deahng with singular integrals, but they are quite selfcontained and rely on no previous acquaintance with singular integrals. (These methods were originated by Calderon and Zygmund [1]; there are now many good expositions, for instance by Riviere [1], Coifman-Weiss [1], Stein [1].) There is one feature that should perhaps be emphasized. Usually, singular integrals involve a kernel K(x, y) in which both x and y live on the same set; the singularity occurs when x = y, and one deals with it by truncating K. In the Cauchy kernel C(z, CX C hves on 5, z Hves in B, and no truncations will be made. In this respect, the present treatment differs from the one used by Koranyi-Vagi [1; Theorem 7.1]. They introduce kernels C^z, 0 (the notation is different) in which both z and C mc on 5, by

^ '

[0

otherwise, 91

92

6. Boundary Behavior of Cauchy Integrals

and show, for feUip), Sby

1 < p < oo, that the functions G C / ] defined on

Q[/](z)= fcXz, 0/(QMO converge in the norm topology of L^(o-), as s \ 0, thus obtaining a generahzation of the Marcel Riesz projection theorem. This will be obtained more easily in Theorem 6.3.1 below. Among the other topics taken up in this chapter, let us just mention that the complex tangent spaces play an essential role in Section 6.4.

6.1. An Inequality 6.1.1. Lemma. Suppose C, rj, coe S, d(co, rj) < S, d(co, Q > 2^» ^^d zeDJ^Q. Then \C(z,ri) - C(z,(o)\ < (16ay^'5\l

-
(Asin§5.1.1,rf(a),^) = | l - < a ; , ^ > r / 2 . ) Proof. The proof will be based on the identity (1)

C(z, n) - C(z, (o) = 2^ By 5.4.3(1),

(2)

|l-|-^<4a|l-r^

and (3)

|1 - {z,ri>\-' < 4a|l - < C , ^ > r ' < 16a|l - r^;

the last inequality amounts to d(C, (o) < 2 d(C, rj), which is a consequence of the triangle inequality and the hypotheses of the lemma. We claim that (4)

I - > ^ ^

In the proof of (4), take co = e^, without loss of generality, and let z', t]' be the components of z, rj orthogonal to e^. The left side of (4) is then \
6.1. An Inequality

93

Since 11 - ^i | < ^ ^ < :^| 1 - Ci | < a| 1 - z^ |, by (2) and the hypotheses, one obtains

Kz,r,} - iz,co}\ < 5(2 + u'")\l -

z,\'l\

which proves (4), since a > 1. Returning to (1), we first apply (4), then (2) and (3), and conclude that I C(z, ff) — C(z, (D) I is at most n-l

The lemma follows. 6.1.2. Definition. For C, (o e S,(x > 1,5 > 0, we define a maximal difference, (1)

A(C, CO, oi, S) = sup I C(z, rj) - C(z, co) \,

the supremum being taken over all rj e Q(co, 3) and over all z e D^OThus Lemma 6.1.1 says that (2)

A(C,co,a,^)<(16a)" + M|l - r«-^/^

ifd(C, CO) > 23, i.e., if C ^ Q(co, 23). If we integrate (2), we obtain an inequality that will be needed in the proof of Theorem 6.2.2: 6.1.3. Theorem. Ifco e S,a > 1,3 > 0, then (1)

A(C, CO, a, 3)d(j(0 < A(oc),

where R(co, 3) = S\Q(co, 23).

Here, as in the rest of this chapter, ^(a) denotes a finite constant, that may also depend on the dimension n, and that is not necessarily the same at each occurrence. The integral in (1) is clearly independent of co. The significant assertion of the theorem is that there is a bound that is independent of 3. Proof. By 6.1.2(2), it is enough to show that

(2)

f

|l-|-"-i/^J(T(0<4-

94

6. Boundary Behavior of Cauchy Integrals

When n > 1, formula 1.4.5(2) shows that this integral equals

where £(<5) = {A: |A| < 1, |1 - A| > 43^}. Replace 1 - |Ap by 2|1 - A|, then put 1 — A = re'*, to see that (3) is less than 2 ' ' - 2 ( n - 1)

fr-^'Ur

2"-\n - 1)

This proves (2). If n = 1, (2) still holds, since the integral is then

6.2. Cauchy Integrals of Measures We need a covering lemma that is more elaborate than 5.2.3, one that is specifically adjusted to the maximal function of a given measure. In place of the constant A^ that occurred in 5.2.3, we now use A^ = sup

^(46)
By Proposition 5.1.4, ^4 < oo. 6.2.1. Lemma. Let ja be a complex measure on S. If t > ||/i||, then there are balls Qi and pairwise disjoint Borel sets Vi c= Q^ such that (i)

(M/i > 0 c= U, Q, = (J, ^^,

(ii) <^(Qi) t}. Assume n>0, without loss of generahty. To each C e £, corresponds then a largest 5 such that the ball q = Q(C, d) satisfies (1)

Kq) > t(T(q).

6.2. Cauchy Integrals of Measures

95

Since t > ||iu||, we have q ¥" S; hence (2)

fi(4q) < tC7(4q).

Et is thus covered by a collection T^ of balls q that satisfy (1) and (2). Let ri be the supremum of the radii of the members of F j , and choose qieT^ with radius >3ri/4. Discard all members of r^ that intersect q^. Call the remaining collection r 2 , let r2 be the supremum of the radii of the members of r 2 , and choose q2^'^2 with radius > 3^2/4. Discard all members of r2 that intersect ^2 > let T^ be the remaining collection, etc. (The process stops if some F^ is empty, otherwise it continues through the natural numbers.) We thus get disjoint balls qi. Put Qi = 4^^. It is then easy to find disjoint Borel sets P^ such that q^ cz Vi cz Q. for each i, and [jiVi= [jiQi, If some qeT^ was discarded at the ith stage, then q intersects qi, and r(q) < 4r(^i)/3, where r{q) is the radius of q. Since 1 + f + f < 4, we have q c 4qi = Qi. Thus E^ cz | J . Q^., and (i) is proved. Next, (3)

(y(Qi) < AMqd

< A^r

'Kqtl

by (1). This proves (ii). If we add the inequahties (3), the disjointness of {qi} proves (iii). Finally, fi(Qi) < tcr(gf), by (2), so that (iv) follows from KVd < KQi) < t(T{Qd < A^taiqd. 6.2.2. Theorem. To every a > 1 corresponds a constant A((x) < 00 such that (1)

t}
for all complex measures fion S and for all t > 0. Let us recall that M , C M ( 0 = sup{|CM(z): zeZ)«(0}, in accordance with Definition 5.4.4. Proof. Fix IX and t. If t < \\p\\, then (1) holds provided that .4(a) > 1. So assume t > ||//||, put E^ = {Mfi > r}, and choose {QJ, {T^} in accordance with Lemma 6.2. L Let (2)

dn= fd(T -\- dii,

be the Lebesgue decomposition of ^. We construct another decomposition (3)

dfi = g d(T -\- dp

96

6. Boundary Behavior of Cauchy Integrals

in which ^ is a "good" function (one that is not too large) and P = Zj5j is the "bad" part of the measure fi, by defining (4)

c , = (^)(P.)

^^

^^^^ 1/(0

(6)

p,(E) =

if C^S\[j,V, (n-c,c7)mnE).

Since n^ is concentrated in £, <= (J,, t^., it is not hard to verify that (3) holds, i.e., that (7)

KE)= \gdc7 + p(E) -L'

for every Borel set £ ci 5, by considering separately the cases £ <= T^ and En\JVi = 0.(ln the latter case, //,(£) = 0.) We shall first deal with g. Since Mf < Mfi, the corollary to Theorem 5.3.1 implies that | / ( 0 | < t a.e. outside E^, hence \g\^ < f | / | , so that

f

\g\'da
\\f\da
Part (iv) of Lemma 6.2.1 shows that \Ci\ < At (where A = A^, and part (iii) imphes therefore that

i

JVi

i

i

Thus (8)

j\g\'da<(l+A')ML

Setting G = M^C^g^, Theorem 5.6.9 (the Hardy-Littlewood maximal theorem) gives {G^ da < Aioc)\\fi\\t. Js

6.2. Cauchy Integrals of Measures

97

Since a{G > t} < t~^ j G^rfcr,we conclude that (9)

(^{Mo,CW\>-\< 2J -

t

Our next objective is the analogous estimate

(10)

<x<(M.Cm>^j<'^^"^' 2J -

t

This will complete the proof, because M , C M < M,C|>] + M,C[fi], so that (1) follows from (9) and (10). By (4), cMVb = MK). Hence i|^;|| < 2\ii\{V^, by (6), and thus (11)

Z m\ < 2ii/'ii. i

Let Q be the set that occurs in (10). Put W = |J,- {IQ^. Then

i

By part (iii) of Lemma 6.2.1, (12)

a(W) < Z(7(2Q,) < AI^cTiQd < ^ ^ .

Note that pi is concentrated on P^ cz Q., and that Pi(Qi) = 0. This is a crucial feature of the proof, for it enables us to write the Cauchy integral of Pi in the form (13)

ClP^-Jiz) = r {C(z, ri) - C(z, cod}dPiifi),

where cOf is the center of Qi = Q(cOiy di). Using the notation introduced in §6.1.2, it follows that (14)

|C[iS,](z)|
for all z G DjiO' This inequality persists if its left side is replaced by its supremum as z ranges over DJ^CX i.e., by M^ClPJiO- If we then integrate,

98

6. Boundary Behavior of Cauchy Integrals

and appeal to Theorem 6.1.3, we see that (15)

f

M^Cmdd

< A(a)||AI|.

Js\2Qi

In (15), we can replace S\2Qi by its subset Q\l^, and then add, to obtain (16)

f

Jn\w

M^CmdG

< A(oi) X Ili?,|| < i

2A(a)M,

by (11). The integrand in (16) exceeds t/2 at every point of Q. Hence (16) shows that

(17)

a(«\P^)<'^^"^ll'^ll t

This completes the proof, since (12) and (17) imply (10). 6.2.3. Theorem. If fi is a complex Borel measure on S, then C[//] e IP{B) for all p < 1. CoroJlary. C[/i] has finite K-limits C\_iiY^ at almost all points ofS, Proof. Fix a > 2, put F = M^C\_ix\. Theorem 6.2.2 shows that G{F > t) < xjt, where x = A(a) ||ju||. Hence, for 0 < p < 1, [fPd(T = p r a{F > t}tP-^ dt Js Jo

Since |C[/x](rO| < F(0, the theorem is proved. The corollary follows from Theorem 5.6.4. 6.2.4. Remark. There exist bounded function w on 5* whose Cauchy integrals tend to 00 along some radius of B. For example, let u be the real part of i log(l — Zj). This shows that the precise analogue of Theorem 5.4.5 fails for Cauchy integrals, since (M/i)(0 may be finite at a point where M^C\ji], or even MradCM is infinite. The conclusion of Theorem 6.2.3 cannot be strengthened to C[//] e H^(B). Theorem 6.3.5 shows this.

6.3. Cauchy Integrals of L^-Functions

99

6.3. Cauchy Integrals of L^-Functions 6.3.1. Theorem (Koranyi-Vagi [1]). If \ < p < co and a > 1, there exists A(a, p) < GO such that (1)

f\M,Clfl\''da
Js for every f e L\G).

Corollary. / / 1 < p < oo and feL\G\ then C\_f^eH\B). f -^ C [ / ] * is a bounded linear projection ofL\G) onto H^(S).

The map

Proof. The case p = lis Theorem 5.6.9. Since

f^M,cu:i is subadditive, the Marcinkiewicz interpolation theorem, combined with Theorem 6.2.2, proves (1) if 1 < p < 2. A standard duality argument completes the proof: Fix p, 2 < p < oc, let q be the conjugate exponent, pick / G L^(o-), h e L\a). For 0 < r < 1, [KQCU^{rQd(T{Q= •Is

Js

Since 1 < ^ < 2, ||C[/j],L < Aiq)\\hl,

IIh-CU\da s

\c\K]{rr])W)dcj{n\

so that


This holds for every h e L^((J). Hence

IIC[/li|,<^(^) 11/11, forallrG(0, 1). This says that C [ / ] e ff^(B) and that ||C[/]||p < A{q)\\f\\p. Now (1) follows from the Hardy-Littlewood maximal theorem. For the corollary, see Theorem 5.6.8. 6.3.2. Definition. The class of all measurable functions/on S for which

Jl/|l0g^|/|d(7<^ is called L log L.

100

6. Boundary Behavior of Cauchy Integrals

6.3.3. Theorem. IffeL

log L then C [ / ] e H\B).

Proof, By Marcinkiewicz interpolation, this follows from Theorems 6.2.2 and 6.3.1. L log L is actually the largest class in which this conclusion holds. Theorem 6.3.5 will show this. But first we show that there is a close connection between the maximal function Mfi and the class L log L. 6.3.4. Theorem. If pi is a complex measure on S for which Mfi e L^ia), then there is anfeLlogL such that dfx = fda. Stein [4] proved this for functions rather than measures, but the proof is essentially the same. Proof. Since M/J, — M\fi\ we may assume that jU > 0, and that \\fi\\ = 1. Our first objective is the inequahty (1)

fi{Mfi > t} < At(p(t)

(t > 1)

where (p(t) = G{A^M^ > t}, and A is the constant A^ in Lemma 6.2.1. Choose Qi.Vi'm accordance with that lemma, and put E^ = {Mfi > t}. Pick some g, = g(C, S), If co e Q,-, it follows that Qi cz Q(a), 23). Hence ( M M ) M > ^ (6(0), 2^)) > ^ ^ ^ > - 2 , by6.2.1(ii). Hence (Jf ^ ^ {^/^ > t/A"^}. By 6.2.1(iv),

KEt) < I KVd -^J. This is (1). Observe next that (p(t) decreases when t increases, and that (2)

f (p(t)dt = A^ UMfi)da < 00, Jo Js by formula 5.7(1). It follows that t(p(t) ^ 0 as t -> oo. Hence //(£,) -• 0 as t -^ 00, by (1). Since fi^ is concentrated on nE^ (Theorem 5.2.7), we conclude that fi^ = 0, hence dfi = fda for some / e L^(cr), and Mf = Mfi. At every Lebesgue point of/, / ( O < (Mf)(Q. Thus / < t a.e. outside £,, or {/ > r} <= £f. Hence (1) shows that (3)

f

Jf>t

fdc7
(r>l)

6.4. Cauchy Integrals of Lipschitz Functions

101

SO that finally (by Fubini's theorem)

A r(p(t)dt>

f f

f fda=

\f\og^fda,

which proves the theorem, because of (2). 6.3.5. Theorem. Iff e H\B) and M = R e / > 0, then u* e L log L. Proof. By the Hardy-Littlewood maximal theorem, M^feL^{c\ hence M^ueL^{G) (for any a > 1). There is a measure ju > 0 such that u = P M Since M^ueL^(cTX it follows from Theorem 5.4.12 that MiieL^{G\ and Theorem 6.3.4 gives the desired conclusion. 6.3.6. When n = 1, Theorems 6.2.3, 6.3.1, 6.3.3, and 6.3.4 are due to Kolmogorov, M. Riesz, Zygmund, and M. Riesz, respectively. See Zygmund [3], Chap. VII, §2, and p. 381, for detailed references.

6.4. Cauchy Integrals of Lipschitz Functions 6.4.1. Definitions. If 0 < a < 1, and if/is a complex function with domain S, or B, or S, we say that / satisfies a Lipschitz condition of order a (briefly: / e Lip a) provided that (1)

|/(z)-/(w)|
for some c = Cj < oo and all z, w in the domain of/. Note that this definition of Lip a is based on the euclidean metric, even when the domain of/is S, rather than on the nonisotropic metric introduced in §5.1.1. A holomorphic Lipschitz function is one that Hes in A(B) n Lip a. A complex-tangential curve is a C^-map y:I ^ S, where / is some interval on the real fine, whose derivative yXt) lies in the complex tangent space to S at y(tX for all tel. (See §5.4.2.) Analytically, the requirement is that (2)


(teiy

Stein [3] showed that holomorphic Lipschitz functions of order a are, roughly speaking, twice as smooth on complex-tangential curves. The same conclusion holds if the Lipschitz condition is only imposed on the slices of a holomorphic function (Rudin [12]). Ahern and Schneider [5] went even further, and showed that it holds for Cauchy integrals of arbitrary L^functions (not necessarily boundary values of holomorphic functions) whose slices are in Lip a. Theorems 6.4.9 and 6.4.10 summarize these results.

102

6. Boundary Behavior of Cauchy Integrals

But first it seems advisable to describe some geometric features of the curves that are involved. 6.4.2. Complex-Tangential Curves. Let y: I ^ S be a C^-curve. Then (y, y> = 1, and if we differentiate this, we obtain Re = 0 which characterizes the complex-tangential curves is thus equivalent to (1)

Im = 0. Associate to each C e 5 the curve T^ defined by

(2)

r^(Q) = e'^C

i-n
Note that F^ traces a circle, namely the intersection of S with the complex line through 0 and C, and that r^(0) = I'C. If y is a C^-curve in S, and if C = y(tX it follows that (3)

yXt) • r^(0) = ,Re<7tO, iy(0> = lm;

the dot-product is as in §5.4.2. Comparison of (1) and (3) leads to the following characterization: A C^-curve y in S is complex-tangential if and only if y is perpendicular to every circle F^ that it intersects. For any n > 1 there is a large supply of such curves. To illustrate this, consider the case n = 2. Let r = r(0, ^ =- 0(0 be of class C^ on /, so that y^ = re'^ defines a C^curve 7o i^i the complex plane. Choose real functions w, t; on / such that (4)

t;' = j - ^ T ^ ,

u=

v-e,

and define

(5)

re" y='

Then 7 is a curve on S. Consider the map of S onto the Riemann sphere (more appropriately, in the present context, onto the complex projective space of dimension 1) that sends (2, w) to w/z. Its fibers are the circles F^. It carries y onto our prescribed curve yo, since t; — w = 6, by (4).

6.4. Cauchy Integrals of Lipschitz Functions

103

Differentiation of (5) leads to (6)

=

i(u' + r^v') 1 + r2

which is 0, by (4). Hence y is complex-tangential This construction can be used to connect any point (a, b)eS to any (z, w) G 5 by a complex-tangential curve: Perform a unitary change of variables, if needed, so that none of a,fc,z, w are 0. Let / = [0,1]. Find a positive function r e C\I) with r(0) = \b/a\, r(l) = |w/z|, then find 0 e C\I) so that e(0) = arg('^j,

0(1)

= a,gQ

and

j\l+r'r'Q'dt

= SiTg(^\.

Put t!(0) = arg b, define u and v by (4), and y by (5). Then y(0) = (a, b), y(l) = (z,w). 6.4.3. A theorem of Hardy and Littlewood (see, for example, Duren [1], p. 74) characterizes the holomorphic function in U that belong to Lip a (0 < a < 1) by the growth condition

i/'(z)i = o((i - izir-i). We shall reprove this in the n-variable context, but mention it here because it furnishes much of the motivation for the work in this section. 6.4.4. The Radial Derivative ^ / . L e t / e H{B) have the homogeneous expans i o n / = ZJF^. We define (1)

(^/)(z) = f/cF,(z)

(zeB).

k= 0

^f is related to the derivative of the slice functions /^ by

(2)

m)W) = ¥lW

(CeSAeUy

The advantage of ^f over f[ is that the former is a holomorphic function in 5.

104

6. Boundary Behavior of Cauchy Integrals

If / is a Cauchy integral, say / = C[JK] for some measure fj, on S, one can use (2) to derive a Cauchy formula for ^ / , namely

(3)

^""^^^'^-U-tor^''^^^-

6.4.5. Lemma. Iffe

H{B) and P is a multi-index, then

(1)

D^^f-diDPf=\p\D''f

and, for 0 < r < l,CeS, P ^ 0,

(2)

r^^Wf)(,rO= fV^/)(tOf"""'rft.

Proof. By linearity, it suffices to prove (1) and (2) for / = F^, a homogeneous polynomial of degree k. Then D^F^ has degree k — \p\ (negative degree indicates the zero-function), so that - ^D^JF, = [feD" - (/c - \P\)D':\F, =

\p\D'F„

which gives (1). Next,

so that the integral in (2) is r\D^F,)(0

=

r\^WF,)(rO.

6.4.6. Lemma. Ifg e H(B), c > 0, and (1)

|^(z)|<(l-|z|)-

(zeB)

thenjor /c = 1, 2, 3 , . . . , 0 < r < 1, (2)

mg)(re,)\

< AU^ -

rr^-'^\

Recall that D2 = d/dz2. The derivative on the left of (2) is thus the feth derivative in the direction of ^2, which is orthogonal to ei. The pair ei, 62 could be replaced in the lemma by any pair of orthogonal unit vectors inC".

6.4. Cauchy Integrals of Lipschitz Functions

105

The point of the lemma is the appearance of /c/2 in the exponent. The growth of (D\g)(re^) can be much more rapid. For example, if g(z) = (1 - z^y then iD\g)(re,) = a,,,(l - r ) — \ where a^ ,. = c(c + 1 ) . . . (c + /c — 1). Proof. Fix r and define (3)

KX) = g(re, + Ae^)

for a l U 6 C with | Ap < 1 - r^ Then, by (1), (4)

\h(l)\ < 2%l - r ' -

\X\Y\

since 2(1 - |z|) > 1 - |zp. Let F be the circle on which |Ap = ^(1 - r^). The Cauchy formula (5)

;,W(0) = ^ 2ni

(x-'-'h(X)dX JY

gives the estimate (6)

\h\0)\
Since (D^^Xrei) = /I'^'CO), (2) follows from (6). 6.4.7. Lemma. Suppose fe H(B), 0 < a < 1, and (1)

l(^/)(z)l<(l-Uir-'

(zeB).

Then, for 2 <j < n and 0 < r < 1, (2) (3)

mDjf)ire,)\

< ^(1 -

rf-"\

l(Z>?/)(rei)|<^„(l-rr-i,

and (4)

l(D,/)(r.,)| < ,^^

^.^^^^^_

106

6. Boundary Behavior of Cauchy Integrals

Here, and later, A^ denotes a finite constant, not necessarily the same at each occurrence. Proof. Lemma 6.4.6, with ^f in place of g, shows that (5)

\(Dj^f)(re,)\

(6)

< A,(l -

rT-''\

\(DJ^f)(re,)\
6.4.8. Lemma. If the gradient of a function u: B -^ C satisfies (1)

|(gradM)(z)|<(l-|z|r'

(zeB)

for some a G (0, 1), then u e Lip a. (It follows, trivially, that u has a Lip a extension to B.) Proof Choose aeB, beB, with 0 < |a| < \b\ < 1. Put \b - a\ = S, and define a = -—— a, \ci\

b = —-— b. \b\

Three cases occur: (i) When^ < 1 - |b|then Igradi/I < ( 1 - \b\f-'

<^"-^

on the line from a to b. Hence | u(a) — u(b) \ < (5". (ii) Whenl - |fo| < ^ < 1 - | a | , t h e n | a - b'\ < \a-b\,Sind \u(a) - u(b)\ < \u(a) - u(b')\ + \u{b') - u(b)\. The first term on the right is estimated as in (i); the second is at most r\b\

(1

-xr-Ux
(iii) When 1 - |a| < (5, then \a' - b'\ < \a - ft|, and \u(a) - u(b)\ is at most \u(a) - u(a')\ + (w(a') - u(b')\ + \u(b') - u(b)\. The first and third term is estimated as in (ii), the second by (i).

6.4. Cauchy Integrals of Lipschitz Functions

107

6.4.9. Theorem. Assume 0 < a < 1 , / G L^(a), and

(1)

\f{^^)-f{e'X)\<W'-e'r

for all C^S and all real 6, t. The radial derivative of the Cauchy integral F = C [ / ] must then satisfy the growth condition (2)

\i^F)iz)\
Proof If we combine formulas 6.4.4(3) and 1.4.7(1), we see that (^F){z) is equal to

Fix z, C, for the moment, and denote the inner integral by J. Write
' • ^ I -271 - JJ__JJ11 -r^^-(^-9)|"+i -"'-e''-^'^^-

Note that 11 - A | < 2 | 1 - r A | i f | A | = 1. Hence (5)

l-^'-Sr

|l-
If we insert this into (3), and use 1.4.7(1) once more, we see that (6)

|(^F)(z)| < 2 n | | l - < z , O r " - ' ^ " M C ) .

Now (2) follows from (6) and Proposition 1.4.10. Remark. The hypothesis of Theorem 6.4.9 holds for every / e Lip a, and in particular for every feA(B) n Lip a, in which case F = f The converse of this remark is part of the following theorem. 6.4.10. Theorem. Suppose 0 < a < 1 , / G H(B), and (1)

|(^/)(z)|<(l-|z|)«-i

(ZG5).

Then f has a continuous extension to B (still called / ) that lies in A(B) n Lip a.

108

6. Boundary Behavior of Cauchy Integrals

/ / } ' : I -^ S is a complex-tangential C^-curve and ifg =fo y, then (a) g e Lip(2a) on I when 0 < a < j , (b) g is differentiable and g' e Lip(2a — 1) on I when ^ < a < 1. Proof, Since j ^ (1 - 0''"' dt = a ' ^ l - rf, we infer from (1) that / ( r Q converges, uniformly on S, to a limit that we call f(Q. This extends / , so that / e ^(B). Note also that (2)

\m-f(rO\
for all Ce 5,0 < r < 1. Lemma 6.4.7 shows that the gradient of / satisfies the hypothesis of 6.4.8. T h u s / e Lip a. Assume now, without loss of generahty, that our complex-tangential y has ly'l = L (This amounts to parametrizing y by arc length.) Define (3)

gXt) = firyit))

(t6/,0
For any t e /, there is a unitary change of coordinates in C" such that 7(0 = ^1 ^iid y'(t) = ^2- The chain rule shows then that (4)

grit) =

r(D2f)(re,)

and (5)

g:(t) = r\Dlf){re,)

+ ^

r{DJ){re,)y]{t).

7=1

Fix ti, ^2 e / so that t2 = t^ -^ h,0 < h < I. Put r = 1 - /i^. If 0 < a < ^, it follows from (2) that \g — gA < A^h^^", and it follows from (4) and Lemma 6.4.7 that \g',\ < A^h^"'\ so that (6)

\gr(t2) - gr(tl)\ < (t2 " 1^^"''

=

A,h^\

Hence \g{t2) — g{ti)\< A^h^^", and (a) is proved. Finally, assume ^ < a < 1. By (5) and Lemma 6.4.7, Ig^l < A^^yh^""'^, so that (7)

\g'Xt2)-9'Xh)\
Furthermore, it follows from 6.4.7(2) that (8)

Writ) - g'lt)\ < A^l - rf-'"

=

A^h'^-'

6.4. Cauchy Integrals of Lipschitz Functions

109

if r < 5 < 1. Hence {g's} converges, uniformly on /, as s -^ 1, so that g' exists and (8) holds with g' in place of g's. Combined with (7), this proves that (9)

\gXt2)-gXh)\
which is what (b) asserts. Remark. In the omitted case a = i, it is still true that Ig^l < Ayh~^. From this and (2) it is easy to deduce that (10)

\g(t + /i) + g(t -h)-

2g(t)\ < A,K

i.e., that g Hes in the space that is often called Aj. (Chapter V of Stein [1] describes the spaces A^ and their relation to Lip a.) But g need not be in Lipl. To see an example (with n = 2), take (11)

/(^) = f ^ l o g ^ — . Zi

1 -

Zi

Then (^/)(z) = Z2/(l - z j , and since |z2p < 1 - |zi P,

(12) Thus feA{B) but

mm)\ ^ {(T4^}'• n Lip^. If y(t) = (cos t, sin r), then y is complex-tangential,

/(7(0) = tanr-log

1 — cos t

is not in Lip 1 and is not differentiable at t = 0. The combination of Theorems 6.4.9 and 6.4.10 shows that the Cauchy integral tends to preserve smoothness on slices, while introducing additional smoothness in the complex-tangential directions. An extreme example of this phenomenon is furnished by any feL^ that is constant on each shce. The conclusion—probably surprising at first glance—is that C [ / ] is then constant in B. Actually, the proof follows immediately from the ^-invariance of the Cauchy transform: If Uz = e^^z, then f°U = fby assumption, hence C [ / ] o [/ = C [ / ] , which shows that the holomorphic function C [ / ] is constant on each shce. Hence it is constant in B, since all sHces intersect at the center of B.

110

6. Boundary Behavior of Cauchy Integrals

6.5. Toeplitz Operators 6.5.1. Introduction. To every cp e L°°((j) is associated a linear operator T^—the Toeplitz operator with symbol (^—that is defined for / G L ^ ( ( T ) by (1)

(T^f)(z) = acpn(z)

(zeB).

The holomorphic functions T^f have K-limits at almost all points of S (Theorem 6.2.3). This extends the domain of T^/is a natural way to the closed ball B, with the possible exception of a set of measure zero on 5. The domain of 7^ is often restricted to certain subclasses of L^((T). For example, every T^ is a bounded linear operator on the Hilbert space if ^(5), and the relations between the symbol cp and operator-theoretic properties of 7^ provide an interesting topic. This, however, will not be pursued here. (See Coburn [1], Boutet de Monvel [1], Janas [1], McDonald [2], [3].) Theorems 6.4.9 and 6.4.10 imply that T^ preserves Lip a (0 < a < 1) if (p e Lip a (since Lip a is an algebra). Theorem 6.3.1 shows that every T^ carries L^(cr) into H^B) if 1 < p < oc. This is false for p = 1, even when (p = 1: iff > 0 on iS, Theorem 6.3.5 shows that / must be in L log L in order for T^f = C [ / ] to be in H\B). It is also not true that every T^ carries H"^ to /f*: Let g be an unbounded holomorphic function in B whose real part is bounded, with ^(0) = 0. If (P = RQ ^*, then T^(2) -=g$H'^. (See Theorem 3.2.5.) There is a rather weak smoothness assumption which, when imposed on (p, avoids this last difficulty. Theorem 6.5.4 shows this in detail. We define the modulus of continuity co^ of a function (p:S^Chy (2)

(o^(t) = sup{\(p(0 - (p(ri)\: \C - r]\ < t}.

We say that (pis a. Dini function if

(3)

r^

. ^dt

o)^(t)—< Jo t

00.

For example, (3) holds if co 0; every (peLip a is thus a Dini function. (The terminology comes from Dini's test for the convergence of the conjugate Fourier series; see Zygmund [3], vol. I., p. 52.) If (^ is a Dini function on S, let us extend cpto a continuous function on B (with the same sup-norm), in such a way that (p(z) = (p(z/\z\) for ^ < |z| < 1, and let us define

(4) iorzeB,C€S,z¥^C-

rM) = {kz)-(p(0}c(z,o

6.5. Toeplitz Operators

111

6.5.2. Lemma. Ifcp is a Dinifunction, then {T^: zeB) is uniformly integrable. More explicitly: To every £ > 0 corresponds a ^ > 0 such that j ^ \T^\dG < e for all z e S and for all £ c S with a(E) < 5. Proof. Iir^ll^ is bounded for |z| < i . It is thus sufficient to consider z of the form z = rrj, ^ < r < 1, rjeS. Since |1 - /i| < 2|1 — r/l| for \X\ < 1, we have (1)

|C(rf/,OI<21C(^,OI

and therefore (2)

Ir,(C)I < 2^co(I^ - CI)IC(rj, 01 = 2"F,(0,

say, where a> = a>^. Since {F^:^eS} is ^-invariant, in the sense that Fu^f = F^o (7~ \ the lemma will follow as soon as we know that (3)

(F,(OdcT(0 < 00. Js

When n = 1, (3) is elementary. When n > 1, we use formula 1.4.5(2). Since co is nondecreasing and \rj — l^\^ = 2Re(l —
^

£ ^^ ~ ' ^ ' ^ r ' « ( V 2 | r ^ y m 3 ( A ) .

Now use 1 — | / l p < 2 | l — / l | , replace U by the half-disc D with center atA = 1 and radius 2 that covers [/, write A = 1 - se'^(0 < s < 2,|9| < n/I), and then replace 25 by t^. It follows that (4) is less than (5)

(n-

1)2"-^ f co(t)-< 00. Jo ^

6.5.3. Lemma. If (p is a Dini function, then

(1)

z ^ r,

is a uniformly continuous map ofB into L^icr). Proof, The uniform integrability of {F^}, combined with Egorov's theorem, gives (2)

lim \\r,-rjd(7 = 0

z-*w Js

112

6. Boundary Behavior of Cauchy Integrals

for every weB. This shows that (1) is continuous; the compactness of B gives uniform continuity. 6.5.4. Theorem. Let cp be a Dini function. (a) / / V^ is defined by

VJ=cpf-TJ then V^p carries the closed unit ball of H"^ into a bounded equicontinuous subset of C(B). (b) T ; carries H"^ into H"^, and A(S) into A(B). (c) T^ carries H^ into HK Note. Since bounded equicontinuous subsets of C(B) have compact closure (Ascoh's theorem), (a) asserts that V^ is a compact operator from if °° into C(B). Proof. For z e B.feH^,

(1)

the Cauchy formula shows that

(VJKz) = [{(piz) -
i.e., that

(2)

(vjKz)=

{fr^dc. Js

I f / G / / ^ it follows that

(3)

\(vj)(z) - (vj)(w)\ < 11/11, fir, - rjdcT Js

for zeB, we B. By Lemma 6.5.3, (3) imphes that V^fis uniformly continuous in B, hence has a continuous extension to B. The equicontinuity asserted in (a) is then also a consequence of (3). Since T^f = (pf — V^f, (a) shows that the holomorphic function T^/ is bounded i f / e if °°, which proves (b), and (c) will follow from (4)

sup

0
\\{VJ){rn)\d<j{n) {\{VJ){r ^S

<'X,

6.5. Toeplitz Operators

113

By (2) and Fubini's theorem, the estimate (5)

sup

0 < r < l JSS

\r^iO\d<Jiri) < CO rti\

implies (4). The computation used in the proof of Lemma 6.5.2 estabhshes (5). Theorem 6.5.4 will be used in the solution of Gleason's problem (Section 6.6) and in the proof of Henkin's theorem (Chapter 9). Our first application, however, is to the space f/°° + C, the set of all sums g -\- h *with g e C(SX /IE if ^(5): 6.5.5. Theorem (Rudin [7]). H'^iS) -h C(S) is a closed sub-algebra ofL'^ia). Proof. I f / G H ° ° ( S ) and cp is a Dini function. Theorem 6.5.4 shows, after restricting to S, that the product

(pf=T4+VJ hes in H'^{S) + C(5). Since the Dini functions are dense in C (we omit S in the rest of this proof) it remains to be shown that if °° + C is closed with respect to convergence in the essential supremum norm. The following observation is the key: / / / sH"^ + C, then f = g + hfor some geQhe if*, that satisfy ll^lloo<2||/|L, 11/^11 00 < 3 | | / | L . Indeed, / = G + if, for some GeCHeH"^. The case ||/||^ = 0 being trivial, assume ||/||oo > 0. Then there exists r < 1 such that the Poisson integral P[G] satisfies

I|G-P[G1L<||/IL. Define g = G-

P[G], + P [ / ] „

h = H - PIH^,. ThQngeC,hGH-,f=g + K ||^|U < 2||/iU, and \\h\\^ = \\f - g\\^ < 3II/II00. Finally, if / is in the closure of H"^ + C, there exist /• G if °^ 4- C, f = 1,2,3,...^ such that il/ilL < 2"^ for i > 2, and f = f^f f. Consequently, f = gi + hi (i > 1), where giEC, hiEH"^, and (for / > 2)\\gi\\^ < 2'-\ m^ < 2'-\ Then g = YA 9i is in C, /i = ^T h^ is in H'^, and hence / = ^ + ft is in if °° + C.

114

6. Boundary Behavior of Cauchy Integrals

6.5.6. Remarks. The case n= 1 of Theorem 6.5.5 was proved by Sarason [1]. When n = 1, then //°° + C is the smallest closed subalgebra of L°° that contains //°^ and is larger than i/°°. (See Sarason [2].) This is false when n > 1. For example, let E be an arbitrary circular subset of 5 with 0 < (7(E) < 1; then £ is a union of circles F^ introduced in §6.4.2. Let XE be the subalgebra of H°°(S) + C(S) consisting of all / whose restriction to almost every F^ c= £ belongs to i/°°(T). It is clear that XE is closed, that H'^iS) ^ XE ^ H'^iS) + C(5), and that both inclusions are strict. Aytuna and Chollet [1] have extended Theorem 6.5.5 to strictly pseudoconvex domains. If B and S are, however, replaced by a polydisc W (n > 1) and its distinguished boundary T", the situation is different: H'^{T'') + CiT"") is still a closed subspace of L°°(T"), but it fails to be an algebra (Rudin [7]).

6.6. Gleason's Problem 6.6.1. Let X be some class of holomorphic functions in a region Q cz C". Gleason's problem for X is the following: IfaeQ and fe X, do there exist functions g^,.. .,g„eX such that (1)

/(^) - m

= E (z, - a,)g,(z) k=l

forallzeQl Gleason [1] originally asked this for Q = B, a = 0, and X = A(BX the ball algebra. (In other words, do the coordinates functions generate the maximal ideal consisting of 3\\fEA(B) with /(O) = 0?) The difficulty of the problem depends on Q. For instance, if B is replaced by a polydisc, Gleason's question becomes trivial: take n = 2, for simpHcity, and decompose a n y / e H(U^) that is continuous on U^ into a sum (2)

/(z, w) = /(O, 0) + zg,(z, w) + wg2(z, w)

where ,,, (3)

, , /(z,0)-/(Q,Q) ^i(z, w) = , z

^ ^ /(z,w)-/(z,0) g2(z, w) = . w

The point is that z # 0 and w 7^ 0 on the torus T^, so that the functions (3) belong to the prescribed class. Put in different terms, i f / G //°°(L/^) and ^2 is as in (3), then ^2 e H'^iU^). This is not true in B:w^/(1 — z) is bounded in B2, but w/(l — z) is not; to see the latter, take (z, w) = (cos t, sin t).

6.6. Gleason's Problem

115

Gleason's original question was solved by Leibenson (see Henkin [4]), who observed that (4)

/(z) - /(O) = C ^f(tz)dt JQ

at

= t z, f^=i

C(DJ)(tz)dt JQ

and then proved that the functions (5)

g,(z) = f (DJ)(tz)dt Jo

(l
lie in A(B) if feA(B). (This proof is similar to some of the work in Section 6.4; details may be found on pp. 151-153 of Stout [1].) Kerzman and Nagel [1] used sheaf-theoretic methods to solve Gleason's problem in smoothly bounded strictly pseudoconvex regions, for functions subject to Lipschitz conditions that could vary from point to point. The simplest and most far-reaching solution is probably the one found by Ahern and Schneider [4]. They worked in strictly pseudo-convex domains. Here is the special case of the ball: 6.6.2. The Ahern-Schneider Solution. If / e H\B) and a e B, the Cauchy formula gives (1)

/(z) - / ( a ) = f lC(z, 0 - C(a, OmOd^iO Js

(z 6 B)

where /(C) denotes the K-limit of / at C e 5. This suggests consideration of the functions

Js

iz - a, C>

since they clearly satisfy n

(3)

f{z) - f(a) = X (^fc - «/c)^fe(«» z)' The identity

= c(z, 0 L shows that every gffc(a, z) is (for fixed aeB) a finite sum of products, each product being a holomorphic monomial z^" (of degree |a| < n — 1) times a

116

6. Boundary Behavior of Cauchy Integrals

Toeplitz transform of/whose symbol is in C°°(5); in fact, the symbols are conjugate-holomorphic functions in a neighborhood of B. The results of Section 6.5 prove therefore cases (i), (ii), and (iii) (for 0 < a < 1) of the following conclusion: The formulas (2) and (3) solve Gleason's problem in any of the following spaces X: (i) X = A(B). (ii)

X = H\B\

1 < /? < 00.

(iii) X = A{B) n Lip a, 0 < a < 1. (iv) X = H^(BX the space of all feH'^(B) extensions to B u E, where E c: S. (v) X = A(B, EAoc}).

that have continuous

The spaces (v) are defined as follows: let £ c: 5 be arbitrary, associate to each C G £ a number a(C), 0 < a(0 < j , and let X consist of all fe A(B) for which

zeB

1^ ~ U

(These spaces occur in Kerzman-Nagel [1], with a(Q < 1/2.) Note, furthermore, that the g^s given by (2) depend linearly on f when a is fixed, and that they depend holomorphically on a when f is fixed. The proofs of the remaining cases (iv), (v), and Lip 1 depend on a more detailed look at the operators K^ of Theorem 6.5.4, where cp is any of the conjugate-holomorphic symbols that play a role in the definition of gf^, and that are defined on a neighborhood of B. Recall that (5)

VJ=cpf-TJ

and that (6)

iyj){z)

= f {cp{z) - 9(0}C(z, 0 / ( O M C ) . Js

IffeH"^ then V^fe C(B) (Theorem 6.5.4); by (5), this proves case (iv). Similarly, case (v) will follow from (7)

|(gradF^/)(z)|
since V^f is then in Lip ^, by Lemma 4.6.8, and the case a = 1 of (iii) will follow if we show that the above gradient is bounded in B, since then V^fe Lipl.

6.6. Gleason's Problem

117

The gradient of V^f is obtained by differentiating (6). Since it suffices to consider (8)

f {cp(z) - cpiO} Y,

/Vxxn^i

(peC^(B\

mdo{Q

for 1 < 7 < n. Since \(p(z) - (p(0\ < c|z - CI < 2c|l -
(9)

cii/Lii-
and now (7) follows, for / G //°°, by Proposition 1.4.10. Finally, suppose feA(B) n Lip 1, and extend / to a bounded Lip 1 — function on C", for instance by setting f{z) = f(z/\z\) when |z| > 1. Put z = reiin (8), without loss of generahty, and put (10)

.^(0 = •A,,;© = nCjMre,) -
Writing C = (Ci, C')> (8) is then the sum of (11)

f ,,

' ^ y , „ ^ i [/(Ci, O - fir, O]rf
and (by 1.4.7(2)) (12)

£

f(r,ndv„.,(n^f

HW, CO ^e.

(1 - r C i ^ - T

Since | / ( C ^ , 0 - / ( r , O I < c|Ci - r| <^c|l - rC,\ and |iA(C)l < k^i - CI < c| 1 — rCi r^^, (11) is bounded. Since cp is holomorphic, the inner integral in (12) is (/^(O, C'X so that the absolute value of (12) is at most 2n||/||oo II<)^HocThis completes the proof. 6.6.3. A Multiplier Theorem. We shall now consider the Ahern-Schneider solution in the special case a = 0. Differentiate the Cauchy formula 6.6.2(1) with respect to z^, then replace z by tz, and integrate over 0 < t < 1. Comparison with 6.6.2(2) shows that (1)

f{DJ)(tz)dt

= g,iO,z).

The Ahern-Schneider solution thus turns out to be identical with Leibenson's, when a = 0.

118

6. Boundary Behavior of Cauchy Integrals

If f(z) = ^ c(a)z'* (the summation extends over all multi-indices a), one computes easily that (2)

z, f W ) ( f z ) d t = Z

^c(a)z°'

where |a| = aj + • • • -h a„, as usual. In view of the properties of the functions ^^ that were estabhshed in §6.6.2, the following multiplier theorem emei:ges: For k = 1,..., n, the transformation (3)

Xc(a)z"-^E ^ c ( a ) z "

maps X into X, where X is any of the spaces A(B), H\B) A{B) n Lip a (0 < a < 1), H^{B\ or A(B, £, {a}). The same is true if the right side of (3) is divided by Zj,.

(1 < p < oo),

6.6.4. Homomorphisms of H'^(B). The fact that Gleason's problem can be solved in H°°(JB) has impHcations concerning the possible homomorphisms of the algebra H°^(B) (relative to pointwise multiplication) into any algebra of holomorphic functions. This is Theorem 6.6.5 below. Its proof also depends on the following two easy propositions: (a)

To each homomorphism A ofA(B) onto C corresponds a point weB such that A/ = f(w)for every feA(B). (b) / / Q is a region in C", F:Q^ C" is a holomorphic map with range in B, and F(wo) e S for some WQ e Q, then F is constant. Proof of (di). Let TC^ denote the ith coordinate function: ni(z) = Zf, 1 < i < n. Put Wf = ATCf, w = ( w i , . . . , w„). Thus ATC^ = TC^W). Since A is Hnear and multipHcative, it follows that A/ = f(w) for all polynomials / . Since A is a complex homomorphism of a Banach algebra, ||A|| = 1, and the fact that the polynomials are dense in A(B) implies now that A/ = /(w) for all / e A(B). Proof of (b). Put g(w) = . Then gcQ -> C is holomorphic, 1^1 < 1, giwo) = 1. By the maximum modulus theorem, g(w) = 1 for all w G Q. Since F(w) G B, it follows that F(w) = F(wo) for all weQ. 6.6.5. Theorem. Suppose B is the unit ball in C , Q is a region in C", and T is a multiplicative linear map ofH'^(B) into H(Q). If Tf is nonconstant for at least one f e A(BX then there is a holomorphic map ij/iQ^B such that Tf = fo^for every feH'^(B), Corollary. The range of T lies in ff °°(Q).

6.6. Gleason's Problem

119

Proof. The multiplicativity of T implies that Tl = 1. If z G Q, it follows that / - > (Tf)(z) is a homomorphisms of A(B) onto C. Hence there is a point il/(z) e B such that (Tf)(z) = /(Hz)) for ei\lfeA(B). In particular, n^oxj/ = Tn^.l < i < n. Thus i/^ is a holomorphic map, and since Tf ^ const, for some / e ^(B), i/^ cannot be constant. Hence i/^(Q) cz B. Now let feH'^{B), pick Z G Q , and apply the solution of Gleason's problem to / at the point a = ^{z)\ there are functions QI E H'^{B) such that

/-/(•/'(z))=

ti^i-UzM.

Apply T to this equation and evaluate at z: (TfXz) - m{z))

= t liTT^diz) i=l

U^mTgdiz).

Since Tui = j/^^, the right side is 0, and thus Tf = fo\j/. This proof is quite similar to one used by Ahern and Schneider [1]. Corollary. / / T is an automorphism of the algebra H'^(B), then Tf = foil/ for some ij/ G Aut(B). This follows if the theorem (with Q = B)is applied to both 7and T~^. The assumption that Tf is nonconstant for at least one / e A(B) is needed for the validity of the theorem. In fact, there exist homomorphisms of ff °°(C/) onto H'^iU) that send every member of the disc algebra to a constant. For a proof of this (which depends on the existence of analytic embeddings of U in "fibers" of the maximal ideal space of H'^(U)), we refer to pp. 166-169 Hoffman's book [1].

Chapter 7

Some L^-Topics

7.1. Projections of Bergman Type 7.1.1. A Class of Reproducing Kernels. Fix n > 1, write B„ for B, temporarily, let s be a nonnegative integer, and let P be the orthogonal projection of C""^^ onto C" (regarding C" as the subspace of C""^* defined by z„+ ^ = • • • = z^+s = 0). I f / e (L' n H){B„), the Bergman formula can be applied to fo PmB„+s'.

If zeBn then Pz = z, =
r (1 - iwpy /(z) = c(n, 5) J^ ( ^ _ ^ ^ ^ ^ ^ y . . . i /(wMv.(w).

The constants c(n, s) can be computed by taking / = l , z = Oin(l): (3)

l=c(n,s)

f

(l-\w\'ydv(w).

JBn

Reverting to our usual notation {B and v in place of 5„ and v„) we obtain the following analogues of the Bergman formula: (4)

''''-[«

\ ^KXz, w)/(w)iiv(w) w)/(H

(z G B)

where (5)

t ' (- uA _ ^sVZ, W) —

for s = 0, 1, 2,.... 120

(1 - iwpy - < z , w » " + ^+^

(z e J5, w e 5)

7.1. Projections of Bergman Type

121

Both (5) and the right side of (4) make perfectly good sense for arbitrary complex s = a -\- it (provided (T > — 1, for obvious integrability reasons), if the complex powers are understood to be the usual principal branches: both numerator and denominator in (5) are 1 when w = 0. The binomial coefficient in (4) is then r(n + s + 1) r ( n + l)r(s + 1)* This suggests the introduction of operators 7^ defined by (6)

(TJ)(z)=l

(°r)//=<^ j

jJCXz,w)f(w)dv(w)

for z e B , for Re s = (7 > —1, and for any fGL^(v) {not necessarily holomorphic) for which the integrands are in L^(v). One can now use a complex interpolation argument to prove that all of these operators have the reproducing property (4): 7.1.2. Proposition. IffeH'^(B)

and a > -1, then TJ = f and Tj = /(O).

(The restriction to /f * will be removed in Theorem 7.1.4.) Proof. The computation that yielded the value of c(n, s) in §7.1.1 was valid for all s with a > —1. Hence 7]1 = 1. To prove that T^f = f amounts therefore to showing that fU

r flHHf_?

/(W)-/(Z)

JB [1 - <2, vv>j (1 - <^z,wyy^^

,

for z e B, (J > —1. Regard z and / as fixed, and denote the integral (1) by H{s). Then H is holomorphic in the half-plane a > -1, and H(s) = 0 when s is a nonnegative integer, because of the reproducing formula 71.1(4). Since ^ "'""'' < 2 |l-|

and

|arg(l -
there is a constant y = y(f, z) < oo such that (2)

\H(s)\ < yl'^e''^'^'^ Now define

(s = a + it, a > 0).

122

7. Some L^-Topics

Since 21 sin(7rs/2)| > | e^^'^ — e~'^^'^ |, G is a bounded holomorphic function in the half-plane c > 0. Since H vanishes at every positive integer, G vanishes at every odd positive integer. This zero-set of G violates the Blaschke condition for bounded holomorphic functions in a half-plane. Hence G is identically 0, and so is H. Thus (1) holds, and T^f = f. To evaluate 7^/, it is enough to consider the case /(O) = 0. Insert / into 7.1.1(6), use the explicit formula for K^z, w), and integrate in polar coordinates (§1.4.3). It is then easy to see that the integrals over S are 0, by the mean-value property of conjugate-holomorphic functions. 7.1.3. The Case s = n -\- 1. This case is particularly simple, because

(1)

\K„^,{z, w)\ = {j^^

l_|w|2

T T ^ ^

>+i

=

(JR
(see Theorem 2.2.6), so that (2)

\\K„^,(z,w)\dv(z)=l. JB

Consequently,

(3)

lj'--^i'"^C"r)Jyi*

for every feL^(v). Thus 7;,+ i is a bounded linear operator on L^(v), with range in (L^ n H)(B), Since /f °^(5) is dense in (L^ n H)(B), the conclusions of Proposition 7.1.2 hold i f / G (L^ n H){B). Hence T„+^ is a bounded linear projection ofL^{v) onto (L^ n H)(B). This is a special case of the next theorem, which says precisely on which L^ a given 7^ is bounded, or, equivalently, which 7^ are bounded on a given L^. Since KQ is the Bergman kernel, the theorem shows that the Bergman transform is not bounded on L^(v), but that it is bounded on L^(v) for 1 < P < 00.

7.1.4. Theorem (ForeUi-Rudin [1]) (a) For 1 < p < oo, Tg is a bounded linear operator on L^(v) if and only if (1)

Re(l + 5) > - . P

7.1. Projections of Bergman Type

(b)

123

/ / (1) holds then T, projects L^(v) onto {U n H){B): in fact

(2)

TJ = f and Tj = W)

for every

fe{UnH){B).

Proof. By Proposition 7.1.2, part (b) is an immediate consequence of (a), since H°°(B) is dense in {U n H)(B). The estimate (3)

e-^''l/2|X,(z, w)| < |K,(z, w)| < ^^'^'/^|X,(z, w)|

(where s = a + it, as before) will be used in the proof of (a); it holds because 1 — has positive real part, so that its argument Hes between —n/2 and n/2. We first take p = 1, in which case (1) becomes a > 0. If (7 < 0 and zeB, then X/z, w) is not a bounded function in B, and therefore 7^/fails to exist for some feL^(B). So assume a > 0. The duahty between L^ and L** shows that 7^ is bounded on L^ if and only if (4)

sup

|X,(z, w)\dv(z) < 00.

By (3), we can replace s by a in (4), and now Proposition 1.4.10 implies that (4) fails when a = 0 but that (4) holds when a > 0. This proves (a) in the case p = I. If 1 < p < 00 and 1 + a < 1/p, then aq < —1, where 1/p + 1/q = 1. Hence (5)

\\KXz,w)\Uv(w)

= 00

for every zeB, and 7]/fails to exist for some / e L^(v). There remains the case l (6)

hiz) = (1 - \z\'r^

1/p. Put c = 1/pq, and (zeB).

Since a — gc = tr — 1/p > — 1, Proposition 1.4.10 implies that (7)

r

r (1 - ivvp)""'"^

JjK„(z, w)|;,(w)*^v(w) = J ^ | i _ ^ ; J ) | . ^ i . . ^ H w ) < [«A(z)]«

124

7. Some L^-Topics

for some a = a(n, a, q) < oo. Also, pc = 1/q < 1 and a + pc = a + 1/q > 0, so that

(8)

jjKXz,w)\Kzydviz) = (1 - |w|Y

l^i^l^,]'Syr\..Mz)

< [bh(w)y for some b = b(n, a, q) < oo, by another application of 1.4.10. Now fix some / e L''(v), and put (9)

Fiz)=

f|K,(z,w)||/(w)|dv(w)

(zeB).

JB

By Holder's inequality and (7), Fiz)=

(10)

{\K„\''->h-\KJ'"'

dv

JB

p


"ji/p

(w)dv(M')

If we raise (10) to the pth power and apply (8), we obtain (11)

{F''dv
{\{\(w)dv(w) «

JB

<(' aby

{\K,(z,w)\h(zydv(z) JB

fi/rdv.

JB

If we combine (3), (9), and (11), we see that 7] is bounded on L^(v) if 1 < p < 00 and 1 -h o" > 1/p. This completes the proof. One consequence of the theorem just proved is that the L^(v)-norm of any holomorphic function in B is controlled by the corresponding norm of its real part: 7.1.5. Theorem. To every p, 1 < p < oo, corresponds a constant Mp = M^ „ < 00 with the following property: IffeH(B)J(0) = 0, M = R e / , then (1)

{\f\'dv<Mp

JB

JB

{\ufdv.

Proof. Fix p and choose s so that 1 + cr > 1/p. Then there exists >1 < oo such that IIT,^ lip < AII ^11 p for every g e L^(v). Apply this tog =f and g = f^

7.1. Projections of Bergman Type

125

(where /^z) = f(rz), 0 < r < 1, as usual). By Theorem 7.1.4, X = TXfr) = Ufr + /.) = ITXUrX SO that { IfA-dv = 2" \ \ Uu.Wdv

JB

JB

< {lAf

f luj" dv

JB

or (2)

f \f\Pdv < (2Ay f \u\P dv JrB JrB

(0 < r < 1).

Now (1) follows as r / * 1 in (2). 7.1.6. The real part of the function /(z) = f l o g ( l - z O is bounded in B, but its imaginary part is not. Theorem 7.1.5 can therefore not be extended to the case /? = oo. As in the proof of 7.1.5 shows, Theorem 7.1.4 also fails for p = oo, no matter what s we choose. 7.1.7. The Koranyi-Vagi theorem shows that Theorem 7.1.5 has an analogue in which integration over B is replaced by integration over S: There are constants Mp^„ < co, for l < p < o o , n = l , 2, 3 , . . . such that

(1)

{\f\''da<M,^J\u\''da

Js

Js

for every f = u -\- iv in H^{S) with v(0) = 0. In contrast to Theorem 7.1.5 this fails when p = 1 and n = 1; the functions (2)

/Xz) = y ^ T ^

(0 < r < 1)

show this. However, it is not known whether there exist constants M^ „ < oo for n = 2, 3, 4 , . . . , such that (1) holds. In other words, if 5 is (for example) the unit sphere in C^, it is not known whether (3)

sup|4q^
126

7. Some L^-Topics

where the supremum is taken over all holomorphic polynomials / # 0, with/(0) real, and w = Re / It seems quite likely that (3) is true when n > 1. This conjecture is closely related to the inner function problem which will be discussed in Chapter 19.

7.2. Relations between H^ and L^ r\H I.IA, Plurisubharmonic Functions. Let Q be a region in C". An upper semicontinuous function M:Q -• [—oo, oo) is said to be plurisubharmonic if to each a G Q and b G C" corresponds a neighborhood F of 0 in C such that (1)

2 -> M(a + lb)

is subharmonic in V. Less precisely, the restrictions of u to all complex Hues should be subharmonic. The most obvious examples of plurisubharmonic functions are l o g | / | and l / r ( c > 0), for any / e //(Q). The slice-integration formula shows that every plurisubharmonic function M in Q is subharmonic in Q (see the proof of Proposition 1.5.4) and that therefore AM > 0, provided, of course, that u e C^. When Q. = B, the class of plurisubharmonic functions is Moebius invariant. It follows that every plurisubharmonic u e C^(B) satisfies (2)

AM

> 0 and

AM

> 0.

Contrary to what one might expect by analogy with Theorem 4.4.9, (the equivalence of (b) and (c)), this last statement does not have a converse (at least not when n > 2): The function (3)

M(Z) = ZjZi + Z2Z2 -

Z3Z3

has AM = 4 and Au = 4(1 - |z|^)(l - M) > 0 in B, but M(0, 0, W) = - 1 wl^ is not subharmonic. Plurisubharmonic functions will play an important role in Chapter 17. They are also closely related to the characterization of domains of holomorphy (see, for instance, §2.6 in Hormander [2]). At present, we only need the following inequality. Its statement uses the familiar notation z = (z', z„) for zeB„, where z' = (z^,..., Z „ _ I ) E B „ _ I . 7.2.2. Proposition. Ifu is plurisubharmonic in JB„ (n > 1) and M > 0, then (1)

f ^Bn-i

M(Z^OMV„_I(Z')<

sup 0 < r < l 'JS

\u(rOda(0

7.2. Relations between H^ and U r\ H

111

where S is the boundary ofB„. If , furthermore, u(z\ z„) is a harmonic function ofz„,for every z' eB„_i, then equality holds in (1). Notes, (i) The assumption w > 0 can be removed. It is made because its presence assures the existence of all integrals that occur in the proof and it allows us to apply Fubini's theorem without any argument. (ii) The integrals on the right of (1) are nondecreasing functions of r, since u is subharmonic. The supremum is therefore equal to the limit, as Proof. To each z'sB^-i 0 < r < 1, (2)

corresponds a

Z „ G C SO

that {z\z^eS.

For

u{rz\ 0) < ^ J " u{rz\ rz^e'^Q,

since u is plurisubharmonic. By formula 1.4.7(2), it follows that (3)

f

u(rz\ 0)dv„_ i(zO < f u(rOdcT(0.

JSn-i

JS

Setting rz' = w\ the left side of (3) is seen to be I

JrBn- 1

u(w\, <0)dv(w').

Hence (1) follows from (3) as r y 1. If M(Z', Z„) is harmonic in z„, then equality holds in (2), hence also in (3) and (1). 7.2.3. Restrictions and Extensions. With B„ and B„_i as above, let / and g be functions with domains B„ and B„_ i, respectively, and define a restriction operator p and an extension operator E by (1)

(p/)(z') = / ( z ' , 0)

(2)

(Eg)(z\z„) = g(z')

(Z'G5„_,), (zeB„).

Since {pEg){z') = (Eg)(z\ 0) = g(z'\ pE is the identity operator. 7.2.4. Theorem. Assume n > l , 0 < p < o o . (a) (b)

The extension E is a linear isometry of(L^ n H)(B„_ j) into H^(B„). The restriction p is a linear norm-decreasing map of H^(B„) onto (UnH)(B,.,).

Proof Apply Proposition 7.2.2 to the plurisubharmonic function u = | / | ^ where / 6 //(B). This gives (a) and most of (b). The fact that p(H%B„)) is all of (U n H)(B„-1) follows from (a) and the identity pEg = g.

128

7. Some L^-Topics

Note: An analogue of this result holds in polydiscs, for restrictions to the diagonal, but it is not quite so easy. See Rudin [1], pp. 53, 69, DurenShields [1], Horowitz-Oberlin [1], J. H. Shapiro [1], Moulin-Rosay [1], and Detraz [1]. 7.2.5. Theorem. Suppose n > 1, 0 < p < oo. (a)

lffeH\B,)then

(1)

1/(^)1 <2''/^ll/IWl-1^1)-"/^

and (2)

lim(l-|z|r/^|/(z)HO

{ZEB;).

|z|-l

(b) Iffe(L^

n H)(B„) then (1) and (2) hold with n replaced by n -\- 1.

Proof. Assume fEH^(B„). Theorems 5.6.2 and 5.6.3 show that | / | ^ has an ^-harmonic majorant u = P\_dfi], where /x is a positive measure on 5 whose total mass is ||/||^. The estimate n^

P^ n

(1 - \z\'T

^ [1 + \z\

therefore yields (1). If we apply (1) to / — X ^^ place o f / and recall that | | / — /r||p ^ 0 as r / 1 (Theorem 5.5.6), we obtain (2). Next, '\ife(U n //)(5„), (a) can be applied to the function EfeHP(B„+ J ; see Theorem 7.2.4. This gives (b). Notes: (i) When p > 1, Theorem 7.2.5 can be proved more directly by applying Holder's inequality to the Cauchy formula and the Bergman formula, respectively. (ii) If c < n/p and / ( z ) = (1 - z^y then feH\B„\ by Proposition 1.4.10. The exponents that occur in (1) and (2) can therefore not be replaced by smaller ones. 7.2.6. In several complex variables it is much harder to construct functions that exhibit interesting phenomena than it is in one. The required techniques are not sufficiently developed. We shall now present two rather modest examples (7.2.9 and 7.2.10) that illustrate the possible boundary behavior of functions in H^(B„) and in (L^ n H)(B„X for n = 2 and n = 3. Similar examples undoubtedly exist when n >3, but different constructions would be required for them.

7.2. Relations between H^ and L^ n H

129

The following "scrambling lemma" works, however, for all n. It is basically due to Calderon (see Zygmund [3], Vol. II, p. 165) and to Stein [5; p. 146]. 7.2.7. Lemma. Suppose {/J is a sequence of Borel functions on S,0 < f < 1, and t

(1)

\ftdcr=cx>.

1 = 1 Js

Then there exist unitary Ui such that O O Y.fi(UiO = oo a.e. on S.

(2)

i=l

Proof. Let ^ = ^(n) be the unitary group on C", and let ^°° be the cartesian product of countably many copies of ^ , with the Tychonoff topology, so that ^"^ is a compact group whose Haar measure is the usual product of the Haar measures of the copies of %. Consider the function (3)

giC,{Ui})=f[U-MU^o^ i=l

on S X %'^. The definition of the product measure dU on ^"^ shows, for any CeS, that (4)

f g(L {U,})du =f[

f [1 -

f^u.oidUi.

By Proposition 1.4.7(3), (4) is the same as

(5)

f 0(c, {t/,})dc/= n [ i - f/i^4 i=i L

Jm^

^s

J

By (1), this infinite product is 0. Since gf > 0, Fubini's theorem implies now that (6)

[gii. {^J)d^(0 = 0

Js

for almost all {[/J G^°^. For any such choice of {(7J, ^(C, {t/J) = 0 for almost all ^eS, and therefore (2) holds. 7.2.8. Proposition. For z = ( z j , . . . , z„) G C", define F,(z) = n''"(zi-.-2„)^''

(/c = l , 2 , 3 , . . . ) .

130

7. Some L'-Topics

Then (i) \F^(z)\ oo. Proof. I F^(z) I attains its maximum on B when

This gives (i). By 1.4.9, the integral in (ii) is n^%n- 1)1 (kiy (n - 1 + nk)\ Hence (ii) follows from Stirling's formula, and (iii) is just (ii) with 2k in place of k. Since F^ is homogeneous of degree Ink, an integration in polar coordinates shows that

b'^^'''=Thkb'^^''^Thus (iii) impHes (iv). 7.2.9. Example. There is an feH^(B) (when n = 2, 3) such that almost no slice function /^ has a Taylor series that converges absolutely on the closed unit disc: Whenn = 2 or 3, Proposition 7.2.8 shows that there exists {Cfc},0 < c^ < 1, such that 00 /• (1)

X

\CkFk\d(j = 00

but

(2)

£ { \c,F,\'da < c^. k=i

Js

(Take c^ = k' ^'^ if n = 2, c^ = l/log(/c + 2) if n = 3.)

7.2. Relations between H^ and L^ n H

131

Since Ic^Ffcl < 1, it follows from Lemma 7.2.7 that there exist JJ^e'^ such that (3)

I kfeF,([/,OI = cx) a.e. on 5. fc=i

If/(z) = YJ ^k^ki^k^X the orthogonahty of the functions F^ ° U^ in shows that feH^(BX by (2). Finally, note that (4)

L^((T)

f,W=t^kFk(U,0^'"\

If C satisfies (3) then the series (4) does not converge absolutely on the unit circle. 7.2.10. Example. There is an feiL^n

H){B) (when n = 2,3) such that

sup |/(rOI = 00

0
for almost all C^S: When w = 2 or 3, Proposition 7.2.8 shows that 00 /• (1)

X

i n i ' ^v < a,

whereas (2)

X

UF,\'da=<x.

Since [F^l < 1, it follows from Lemma 7.2.7 that there exist such that (3)

U^e^

I i n ( ^ . O I ' = cx) a.e. on 5. fe=i

Now let {(pk(t)} be the Rademacher functions (independent random variables that take the values + 1 with probabihty ^) and define (4)

gt(z)=Y.
(0
Since the functions Fj, o JJ^ are mutually orthogonal in L^(v), (1) shows that every gt belongs to (L^ n H)(B).

132

7. Some L^-Topics

If Ce5 satisfies (3), it follows from a well-known theorem about Rademacher series that (5)

sup|^,(rC) = sup I

Fk(UuOr''''(p,it) — 00

for almost all t e [0, 1]. (See Zygmund [3], vol. II, pp. 212, 205.) By Fubini's theorem, it is therefore true for almost every t that the supremum in (5) is oo for almost all C e S. Putting f = Qt for such a value of t gives our example. 7.2.11. Example, li feH^(B2\ function of/ satisfies (1)

Theorem 7.2.4(b) implies that every slice

fl/c(A)P^v(A)< 11/11

the norm on the right being the //^-norm of/ In particular, the left side of (1) is a bounded function of C on S. One may ask whether there is a converse to this, i.e., whether H^ can be characterized by the slice functions forming a bounded set in the Bergman space. The answer is negative: Put F^{z) = (2zlZ2)^ for /c = 1, 2, 3, As in Proposition 7.2.8, (2)

Js

whereas another simple computation gives (3)

f |F,(AOP^v(A) =

Ju

\F,{0\' ^ 1 < 2/c + 1 " 2/c + 1

for every C e S. If c^ G C is chosen so that

and if / = J^c^F^, then (2) shows that / is not in H^(B2\ although (5) for every C^S.

r |/,(A)p dv(X) < 1 Ju

7.3. Zero-Varieties

133

7.3. Zero-Varieties 7.3.1. Introduction. If Q is a region in C" and / G / / ( Q ) , we write Z ( / ) for the set of all z e Q at which /(z) = 0. We call a set E cz Q a zero-variety in Q if £ = Z ( / ) for some feH(QX / # 0. If X is a subspace of /f(Q) and E = Z(f) for some / G X, / # 0, then £ is said to be a zero-variety for X. At the other extreme, E is said to be a determining set for X if the assumptions feX,Ecz Z(f) force / = 0. The local structure of zero-varieties is described by the Weierstrass preparation theorem (Chapter 14). The present section is devoted to quantitative global properties of zero varieties of certain subspaces of H{BX with special emphasis on the differences that exist between the case n = 1 and the cases n > 1. When n = 1, the zero-varieties in U are precisely the discrete subsets of U. Such a set £ is a zero-variety for H'^(U) (hence also for all H^{U) and for the Nevanhnna class N{U)) ii E = {a J satisfies the Blaschke condition Y,{\-

la,.I) < 00.

Conversely, £ is a determining set for iV(t/) (hence also for all H\U\ including H'^(U)) if £ violates the Blaschke condition. (Duren [1], Hoffman [1], Rudin [3], etc.) We turn ton > LA recent very impressive theorem, proved independently by Henkin and by Skoda, characterizes the zero-varieties of N(B) by an analogue of the Blaschke condition. This forms the subject of Chapter 17. One result of the present section (Theorem 7.3.8) is that to distinct values of p correspond distinct classes of zero-varieties. But no characterizations of any of these classes are obtained. In view of some examples, it seems quite unlikely that any reasonably simple characterizations can exist. Theorem 7.3.3 gives a simple necessary condition, which is however very far from being sufficient. In one variable, zeros of holomorphic functions can occur with multipHcities greater than 1. The Blaschke condition is usually stated so as to take these multiphcities into account. When n > 1, one can also assign multiplicities (positive integers) to the irreducible branches (we shall not define this) of a zero-variety. This leads to the concept of a divisor. The introduction of divisors can be avoided by the following simple device which will accompHsh the same thing for us: Two functions fge H(B) are said to have the same zeros if both f/g and g/f are holomorphic in B, i.e., if / = gh^ and g = //I2, with /i^, h2eH(B). Having the same zeros is clearly an equivalence relation, and many properties involving zeros (such as the consequences of Jensen's formula discussed in §7.3.2) depend only on the equivalence class to which a function belongs. Some of our results will be stated for the spaces H^(B) defined in §5.6.1.

134

7. Some L^-Topics

7.3.2. Counting Functions. \{ feH{U\ the customary notation for the number of zeros of/(counted with multipUcities) in the disc rC/is ny(0-If/(O) "^ 0> there is also the integrated counting function f

dt n/0Jo '

Nf(r)=

(1)

(0
Jensen's formula (see Titchmarsh [1], p. 126) expresses Nf(r) directly in terms of/: (2)

Nfir) = - J

l o g i / ( r O M e - log|/(0)|.

Assume now that feH{B\ and /(O) = 1 for simplicity. For each (,eS, let nj(C, 0 and A/^/C? 0 be associated to the shce function /^ in the above manner. Thus (3)

N/C, '•) = ^ f

log|/(/-e'''0|de

(0 < r < 1).

Note that both sides of (3) are unchanged iff is replaced by some g e H(B) which has the same zeros as f {provided that g(0) = 1 ) . If (p is any nonnegative increasing convex function on (— oo, oo), Jensen's inequality leads from (3) to (4)

(piNfiC, r)) < ^ J '

cp(\og\f(re''0\)dQ

and if (4) is integrated over S one obtains (5)

f cpiNfiC, r))dc7iO < f cpQog I / 1 )dG, Js Js

by 1.4.7. Since N/(C» r) increases as r increases, the monotone convergence theorem leads from (5) to the inequality (6)

[(^(AT/C, 1)^^7(0 < sup

0 < r < l Js

Js

L(\og\f\)da.

This provides a necessary condition which the zeros of certain holomorphic functions must satisfy (Weyland [1]): 7.3.3. Theorem. Suppose f e H(B\ f(0) = 1. The inequality

(1)

{cp(N^c,m
7.3. Zero-Varieties

135

holds then iff e H^(B), In particular,

(2)

jN/C,l)MO
iff is in the Nevanlinna class N{B), and (3) iffeH\E)for

Jexp{piV/C, l)}dG(0 < O) some p,0 < p < GO.

Part (1) was proved above. The case (p(x) = x'^ gives (2), and (p(x) = e^"^ gives (3). The Henkin-Skoda theorem that was mentioned in §7.3.1 gives a converse of (2): / / f satisfies (2), then f has the same zeros as some g e N(B). (Chapter 17.) On the other hand, (3) does not ensure that / has the same zeros as some geH\B). We shall see this in §7.3.5, after the following theorem, which (for n = \) occurs in Shapiro-Shields [1]. Points of B will be written in the form Z = ( Z i , Z'),

Z' =

(Z2,...,Z„).

7.3.4. Theorem. Suppose S(l — x^) = oo, where 0 < x,- < 1 for all i, and let E he the set of all ZEB that have z^ e {x,}. / / / G H{B) satisfies the growth condition (1)

|/(z)|<exp|^—^|,, =3 E, then / = 0.

for some c < oo and some oc < j , and ifZ(f)

Note that £ is a countable union of balls of complex dimension n — 1, and that E is the zero-variety of some holomorphic function in B that depends only on z^. The theorem asserts that £ is a determining set for the class of functions described by (1). Proof Let Q be the set of all w G 5 with (2)

|2wi-l|
and

|w'|<|l-Wi|.

Fix w G Q. Since Q is a nonempty open set, it is enough to prove that /(w) = 0. Define (3)

h(X)

/I + A

1-A

\

^. ^^^

136

7. Some ^''-Topics

Put 5 = 1 — | w ' p | l - Wi|"^.ThenO < ^ < LA simple computation shows that (4)

4(1 - \h(re'%\^) = 2(1 - r^) + 3(1 - 2r cos Q + r^) > d sin^ 0

for 0 < r < 1,161 < n:. It follows that h maps U into B, so that one can define (5)

g(X) = fiHX))

(XeU).

By (1) and (4), (6)

^og\g(re«>)\ < ^ "^ ,>eM2 1 - \h(re'^)\ <1^f=^^l^'"®l

for 0 < r < 1, |0| < TT. Since a < | , (6) implies that g e N{U). Next, note that (7)

9(2.

^"'•-^^=^(^"r^^') = «

for all U since Z(f) => E. The sequence {2xj — 1} violates the Blaschke condition. Since g e N(U\ it follows that g(X) = 0 for all A G C/. In particular, 2wi — 1 G (7, so that (8)

/(w) = f(w,, wO = g(2w, - 1) = 0.

7.3.5. Example. If n > 1, there is a function g G H(B) such that

(a)

J^exp{pN,(C,l)}MO < 00

for every p < oo, although (b) no product of the form f = gh (with h e H(B)) satisfies a growth condition of the form 7.3.4(1), unless h = 0. In particular, no feH^{B) has the same zeros as g. To construct g, put Xi,= \ — {k log k)'^ for fc = 3,4, 5 , . . . , let G be any holomorphic function in U with simple zeros at precisely the points x^, and put g{z) = g(z^, z') = G(z^). Since 1(1 - x^) = oo, Theorem 7.3.4 shows that g has property (b). To prove (a) we need an upper estimate for iV^(C, !)• For C^S, rigiC, t) is the number of A's in tU such that ACi G {XJ,}. This is the number of /c's such thatXj, < t\Ci\, i.e., such that /clog fc <

i-fKil

7.3. Zero-Varieties

137

If /clog /c = A, then A < k^, hence A log kr

k = z

<

2A log A

Taking ^ = (1 - t|Ci | ) " \ it follows that 2 "«(^'^)<(l_(|^J)10g{l/(l-f|(J)}Since Jo

^

it follows that NJC, 1) < c log log

1-ICil

for some c < oo and all |Ci I sufficiently close to 1. Hence exp{pJV,(C, 1)} < ^log This implies (a). 7.3.6. Example. Suppose V^ and V2 are zero-varieties in B, V^ a V2, and V2 is a zero-variety for some class X c H(B). In some sense this imposes a restriction on the size of V2, and one might expect that V^ should therefore also be a zero-variety for X. But it need not be: Take n = 2, write (z, w) in place of (z^, Z2). Fix p > 0, let m be an integer, m > 4/p. Put Z) = { ( z , 0 ) : | z | < l } , r, = l - 2 - ^

E, = E=

/c= 1,2,3,...,

{iz,w)eB:zeT^}, Ij E,.

138

7. Some L^-Topics

Each Efc is the union of m2^ discs "in the w-direction" and D is a disc orthogonal to those that make up E. This example has the following properties: (a) £ is a zero-variety, (b) EVJD = Z(h) for some h e A(B\ but (c) no ^ G HP(B) has Z(g) = E.

The proof of (a) and (b) uses the product

Since (rfc)^" > | , we have 00

CO

(2) \P{z)\ < n of{1I's+ in4'"|zr2'} - 1expansion + ^4'"^<^>|zr where c{t) is the number the binary of t. Thus c{t) < log t + 1/log 2, so that

(3)

^

c

ip(z)i> 1 + I (r + i)''"izr <(1-lzl)^--^^

Since P{z) = 0 if and only if z G T^ for some k, the function g(z, w) = P(z) estabUshes (a). Since |wp < 1 - |z|^ in B, the function (4)

/i(z, w) = w^'^+^PCz)

estabhshes (b), because of (3). To prove (c), suppose g e H{B\ Z(g) ZD £, but ^(0) = 1. (Note that 0 ^ E.) Putting /(z) = g(z, 0), Theorem 7.2.4(b) shows that it suffices to prove that (5)

(\ffdv=^. Ju As in §7.3.2,

(6)

f f dt 1 r J nf(t)dt < J n / 0 j = 2^ J log|/(re^'«)Me.

7.3. Zero-Varieties

139

Multiply (6) by p, exponentiate, apply the geometric vs. arithmetic mean inequahty on the right, multiply by r dr, and integrate: (7)

£ jexp j\nf(t)dt\r

dr < ^ jjf\^r

dr dQ.

If 7 < t < 1, then Tfc < t < rfc+1 for some k. Since / has m2^ zeros in 7^, it follows that (8)

prifit) > pm2' >

4 > 1 — r^

2 -. 1— r

The left side of (7) is thus oo, and (5) is proved. Note. Each of the m2^ discs in Ej, has radius (1 -riy/2

^2-^'-'^^^

so that the area of E^ is about 2nm, for every k. Thus (b) shows that E u D is a zero-variety for A(B\ of infinite area. No such example seems to be known when n > 2 (where "area" must of course be replaced by (2n — 2)-dimensional volume), even with H'^{B) in place of A{B). The following lemma will be used in the proof of Theorem 7.3.8. 7.3.7. Lemma. Suppose (a) n is a finite positive measure on a set Q; (b) V is a real measurable function on Q, with 0 < y < 1 a.e., whose essential supremum is 1; (c) 0 is a continuous nondecreasing real function on [0, oo), 0(0) = 0, 0(x) -^ CO as X -^ ao\ (d) 0 < ^ < 00, 0 < t < 00. Then there exist constants c^ e (0, oo\for k = 1, 2, 3 , . . . , such that (1)

{ ^(c,v')dfi = S.

//I a I < 1 and i/1^ = 1^(0 is the set of all coeQat which Ckv\aj) > t, then (2) lim Ck(x^ = 0 k->^oo and (3)

hm f ^(CkV^)dfi = S.

140

7. Some L^-Topics

Proof. The monotone convergence theorem shows that (4)

c-^

\^(cv^)dfi

is, for each k, a continuous mapping of [0, oo) onto [0, oo) which carries 0 to 0. Hence (1) holds for some c^ e (0, oo). Assume |a| < j5 < 1, and let E be the set of all co e Q at which v(co) > fi. Since the essential supremum of r is 1, ix{E) > 0. Also, (5)

{cJ^)ii{E) < [{c,v')dii
This shows that {cj,fi^}is a bounded sequence, and (2) holds because \OL\ < p. To prove (3), define 0 GkioS) - ^ _ , Ckv\o))

(6)

if if

ojsY^ co^Yj,.

Then 0 < g^ < t. Since 0 < r < 1 a.e., (2) implies that gk(co) -^ 0 a.e. on Q. Thus (7)

f

^(c,v')dfi=

{Q>(g,)dfi^O

as /c ^ 00, by the dominated convergence theorem. Now (3) follows from (l)and(7). 7.3.8. Theorem. Fix n > 1. Assume that (p and ij/ are nonconstant, nonnegative, nondecreasing convex functions defined on (—oo, oo), and that \l/(t) -^-^+00 (p(t)

(1)

as

r^+oo.

Suitably chosen points Ci^-S', positive integers /Cf, and positive numbers a^ produce then a function (2)

/(z)= n(l-a.
such that (i) feH^iBXbut (ii) ifb e H-(B), g e H(B), 3 # 0, and (3)

h = if + b)g

then some constant multiple of h fails to be in H^(B).

7.3. Zero-Varieties

141

The classes H^(B) were defined in §5.6.1. Some comments on the theorem will follow its proof, which, for convenience, is spUt into four steps. Step 1. We claim that there exist nonempty circular sets K and X^ (i = 1, 2, 3,...) on S such that (a) K is compact, a(K) > 0, (b) the sets X^ are pairwise disjoint open subsets of S\K, and (c) every circular open subset of S that intersects K contains infinitely many X,.. The standard projection TT of S onto the complex projective space P of dimension n — 1 provides an easy way to see this. By definition, n(0 = n(rj) if and only if rj = e'^C for some real 9; a set £ c P is open if and only if TT" ^(E) is open in 5; and a(n~^(E)) defines a Borel measure on P. Now let K' be a compact subset of P, of positive measure, with empty interior, let {pj be a sequence of distinct points in P\K' whose set of subsequential Umits is exactly K\ pick pairwise disjoint neighborhoods X- of p,- that do not intersect K\ and define

Step 2. We now construct / . It will be convenient to assume that (4)

(p(t) = 0 if

t
To see that this involves no loss of generaHty, put (Pi(t) = (p(t) — (p(l) if t > 1, and put (p^(t) = 0 if t < 1. Then cp - cp^ is bounded, hence H^ = H^^, and (pi satisfies (4). For t > 0, define (5)

0 ( 0 = (pil + log(l + 0),

^ ( 0 = ^(1 + log(l + 0).

By (1), there are numbers ti > i (i = 1, 2, 3,,..) such that (6)

^ ( 0 > m(t)

if

t>ti.

Let the sets Xj be as in Step 1. For each f, pick d^^iWe now apply Lemma 7.3.7, with (5, a) in place of (Q, //), with v(z) = I I, with Si = 2/f, and with a^ the maximum of | I on S\Xi (so that ai < 1). By the lemma, there exist positive numbers a,- = c^. (where ki is a sufficiently large positive integer) such that, setting (7)

Ft(z) = a,{z,Ciy'

(zeC")

142

7. Some L^-Topics

we have

(8) (9)

fo(|F,|y(7 = | ,

Js \Fi(z)\<2-'

^

if zeS\Xi

(10)

and if

|z| < 1 - ^, I

r 0(|f,|>i(r > 4,

where 7^ = {CeS: |Fi(OI > h}. We define (11)

f(z)=f[(l-F,{z))

(zeB).

This product has the form (2). Because of (9), the product converges uniformly on compact subsets of B (thus / e H{B)\ and it is 0 only where one of the factors is 0. Note also that Y^ <= Jf,-, by (9), and that (6) and (10) imply (12)

f 4^(1^,1)^.7 > / .

Step 3. To prove that / e i/^(B), put (13)

M^{z) = n ii - Fii^)i

^ ( ^ ) = n {1 + i^^(^)i}-

Note that Y\{\ + 2~') < e x p ( ^ 2 " 0 = e. Since the sets X, are disjoint, (9) imphes therefore that

By (4) and (5), it follows that (15)

[0 ^O°g^(0)^^^(l^^.|)

in S\UX.. .„ ^^

Hence (8) implies (16)

{ip{\ogM)dG<

f

|<4.

7.3. Zero-Varieties

143

Mjv is the absolute value of a holomorphic function. Hence log M^ is subharmonic, for each AT, and so is (p{\og M^vX since (p is convex and nondecreasing. For 0 < r < 1, it follows that (17)

f (p(\og MM))dc7(0

< f (p(log M^)dG < 4.

If we fix r and let N -> oo, M^{r(,) -> | / ( r O | uniformly on S. Hence (17) gives (18)

J'

\ cp{\og\fA)dG < A

(0
Thus/E//^(B). S^^p 4. We now prove part (ii). Suppose b e H°^iB), g e H{B\ h = (f -\- b)g, and ^ # 0. Then ^ = G^ + G^+i 4- • • •, where each G^ is a homogeneous polynomial of degree i, and G^ ^ 0. Referring to Step 1, a(K) > 0, hence G^(0 T*^ 0 for some C^K. Since |G^(^'®0| = |G^(OI for all real 9, |G^| is bounded from 0 on some circular open subset of S that intersects K. By Step 1, there is a ^ > 0 and there is an infinite set J of natural numbers, such that (19)

|GJOI>^

if CeX,

and

ieJ.

We shall see that ch is not in H^(B) if c = 16e/3, by factoring (20)

ch =

Se(f-\-b)-23-'g.

Since ti -> oo (see (6)) we may, after discarding at most finitely members of J, assume that (21)

t,>5

+ S\\b\\^

(ieJ),

We now fix7 G J and choose r = r(J) < 1 so that (22)

2r>l

and

(1 - r^O 11^^,11 oo < L

Fix C^Yj and put (23)

y(X) = 2S-n-"^g(X0

Then 7 e i / ( l / ) , |y(0)| = 2d-'\G^(0\ 25-'\g(re'%)\

(XeU)

> 2 > r"=

ny(re'%

144

7. Some L^-Topics

and the subharmonicity of log 171 implies therefore that (24)

^ j ^ log 12(5 - 'gire'%) \ dQ > log | r-y(O) | > 0.

Next we note that

(25)

n(i-2--)>i 1=1

If C e Y; and I A| = r, it follows from (9) and (11) that 4|/(A0I > \FjiXO\ - 1 = r*'|f/OI - 1 > \FjiO\ - 2, by (22). Since IF/OI > tj on Yj, (21) implies now that (26)

8|/(A0 + b(AOI > 2 | F / 0 I - 4 - 8||Z>|L > 1 + 1^/01

By (20), (24), and (26), (27)

^ J " log I ch(re'''01 ^6 > log{e(l + | F/QI )}•

Since ij/ is convex and nondecreasing, Jensen's inequality leads from (27) to (28)

^ I " ./.(log I cHre'^O I )rfe > W( \ F / 0 1 )•

This holds for every C^Yj. If we integrate (28) over Yj and recall the rotation-invariance of a, (12) shows that (29)

f il/(\og\cHrO\)d(T(0

>I

JYJ

This was done for fixed r = r(j). But (29) impHes obviously that (30)

sup

U(\og\cHrO\)d(T(0

>j

for every j e J. The left side of (30) is therefore infinite, and we have proved that chis not in H^. 7.3.9. Remarks, (a) The appearance of a "constant multiple" may be a bit puzzHng in the conclusion of Theorem 7.3.8. The phrase may be omitted

7.4. Pluriharmonic Majorants

145

when \l/ satisfies the growth condition Mt + 1) (1)

Hm s u p —77-T— < 00

since H^(B) is then closed under scalar multiplication. In that case, the conclusion is simply that h ^ H^(B), The case (p(t) = exp(eO, il/(t) = (p(t + 1), shows the need for the "constant multiple": For every / G i / ^ we have e'^feH^. Thus H^ and H^ have the same zero-varieties, although \j/{t)/q){t) ^ 00 as t -> 00. (b) Theorem 7.3.8 states that no "bounded perturbation" f -\- b oi f has the same zeros as any member of H^(B), Letting b range over the constants, this gives some information about the level sets of/, or, in other words, about the distribution of values of/. (c) Fixp,0 < p < 00, and define (2)

(p(t) = e'\

m

= (2 + p^t^K.

Then ij/ is convex and increasing, H^ = H^, and H^ c H^ for all q > p. Since ij/ satisfies (1), Theorem 73.S furnishes anfe H^(B) whose zero-variety is a determining set for every H\B) with q > p. (d) With a gap series in place of a product. Theorem 7.3.8 appears in Rudin [9]. An earlier result, for polydiscs, is due to Miles [1]; it strengthened Theorem 4.1.1 of Rudin [1]. Similar theorems have also been proved for Bergman spaces, by Horowitz [1] and J. H. Shapiro [2].

7.4. Pluriharmonic Majorants 7.4.1. Introduction. If Q is a region in C" and 0 < p < 00, Lumefs Hardy space (LHYiQ) is defined to consist of all feH(a) such that | / | ^ < w for some pluriharmonic u. (Lumer [1].) Whenn = 1, pluriharmonic is the same as harmonic, so that this definition coincides with the classical one that involves harmonic majorants. But when n > 1, then (Lif)^(Q) is a proper subclass of what is usually called H^(Q); see Stein [2], Stout [5], for instance. The use of pluriharmonic majorants leads to some appeahng function-theoretic properties of (LH)^(Q); from the standpoint of functional analysis, however, (LHy(B) has some unexpectedly pathological properties, as we shall see presently; for example, {LHy{B) is not even a Hilbert space. One appealing property is what may be called holomorphic invariance: IfQ>: Qi ^ Q2 is holomorphic and f e (LHy(Q2) then / o O 6 (LHf (Qj). This is trivial, for if u is pluriharmonic, so is w o O, simply because u is locally the real part of a holomorphic function (Theorem 4.4.9).

146

7. Some L^-Topics

As regards zeros, (LHy(Q) behaves much better than H%B) does: / / Q is simply connected, then every fe(LHy(Q) has the same zeros as some heH'^iO). This is a corollary of the following result: 7.4.2. Proposition. Suppose Q is a simply connected region in C" and f e H(Q). Then f has the same zeros as some h G H * ( Q ) if and only if there is age H(Q) such that (1)

log|/|
Proof If (1) holds, put h = f-Qxp(—g). Then |/z| < 1 and h has the same zeros as /.Conversely, if he//°°(Q) has the same zeros as / , then f/h is a zero-free holomorphic function which has a holomorphic logarithm g, since Q is simply connected. 7.4.3. For most of the remainder of this section we confine ourselves to the case Q = B, n > 1. In addition to the difference in zero-varieties that we just saw, there are at least two other ways of seeing that {LHy{B) is a rather small subclass of H\B). The first of these involves rates of growth. If w is a positive pluriharmonic function in J5, then each slice function u^ is a positive harmonic function in U, hence u^{X) < 2w(0)/(l - \X\). lffe(LHy(B), it follows that (1)

\f(z)\
(zeB)

for some c = c(f) < oo. This is much more restrictive than the corresponding H^ bound (Theorem 7.2.5). For instance, if 1/p < t < n/p, and g(z) = (1 - z j then g e H\B) but g i (LHy{B). The second difference involves the norm

(2)

\\\f\\\, = Mu(Oy"',

the infimum being taken over all pluriharmonic majorants of | / | ^ in B. As pointed out by Lumer [1], it is easy to prove that this norm turns {LHy(B) into a Banach space if p > 1. If WfWp and ll/Jp denote the norms of / and /^ in H^iB) and if^((7), respectively, it follows that (3)

11/11

|jjl/cll?M0j'

7.4. Pluriharmonic Majorants

147

by slice-integration of | / | ^ whereas (4)

lll/lllp>sup||/,||,

since u^ is a harmonic majorant of | / J ^ for every u that competes in (2). There is a variant of (4) in which equality holds, and in which the right side is replaced by an expression that involves all representing measures of the origin (not just those that come from slices). This will be taken up in Section 9.7. EquaHty need not hold in (4). In fact, the left side may be oo although the right side is finite: 7.4.4. Example. When n = 2, there is a n / e H^(B) which extends continuously to B except for one boundary point, such that

(1)

l/(rA)prfe
^f

for allC e 5, 0 < r < 1, although / is not in (LHyiB): Writing (z, w) in place of (z^, Z2), define (2) let c^>0 (3)

g^(z,w) =

(l-zr--'w'-^\

satisfy fc^-l

but

m=1

fm^/^4=oo, m=1

and put (4)

f{z.y^)=Y.cJ

m=l

\ ^ J

^.fevv).

To see an example of (3), put c^ = 0 unless m is a power of 16. If m = 16'', put c , = 2-'^ (/c = 1,2,3,...). The proof that (4) defines a function with the desired properties is as follows. First, we claim, for 0 < (5 < 1, that (5)

l^.fevv)|<(^)''\2-^)\ /

for all (z, W)eB that satisfy 11 — z| > ^. Indeed, since | wp < 1 — |z|^ in 5, (6)

\gJ,z,w)\<\\-z\--\\-\z\'r^'l\

148

7. Some L^-Topics

On the set defined by |z| < 1, 11 — z| > 5, the right side of (6) attains its maximum at the point z = 1 — 3. This gives (5). By Stirhng's formula, (7)

0

Hence (5) shows that the mth term in (4) is dominated by a constant times

if 11 — z I > <5. Thus / is continuous on B, except at the point (1, 0). Next, we claim that

for C e 5, 0 < r < 1. It is of course enough to do this when r = 1. Insert the binomial expansion (10)

(1 - z)-

into (2), and apply Parseval's theorem; if C = (z, H ) e S and r = 1, the left side of (9) equals

(11)

iwr^^f l^^"^] 2 izr. 2k

It is easily verified that (12)

/k + mV ^/2m\/k + 2m\ \ m / \m J \ 2m /

If (12) is inserted into (11), another appHcation of the binomial theorem shows that (11) is at most

(13)

(^jji'^r^'d - \^\T""-' = M .

This proves (9). Since Sc^ = 1, (9) and (4) show that every/^ is a convex combination of functions whose if^-norm is at most 1. Hence / satisfies (1). It follows of course that / G if ^(J5). Finally, assume that | / | ^ < M for some pluriharmonic u in B. This will lead to a contradiction:

7.4. Pluriharmonic Majorants

149

For 0 < X < 1, u(x, w) is a harmonic function of w in |w|^ < 1 — x^. For 0 < r < 1 it follows that

(14)

u(x, o) = y r u(x, (1 - x^ywye

Let r ^ 1 in the last integral, and apply Parseval's theorem to (4), regarded as a power series in w. The result is «>

(15)

(l-xMx,0)>X4 m=l

/2m\ ~ ^ \ ^ /

(1+x)2m+l

As X / * 1, the sum of this series tends to infinity, by (3) and (7). But M(Z, 0) is a positive harmonic function in (7, so that (1 — x)u(x, 0) must be bounded. This contradiction shows that | / | ^ has no pluriharmonic majorant in B. Hence / is not in {LH)\B). 7.4.5. Notation. As usual, /°° is the Banach space of all bounded complex sequences, with the sup-norm, and CQ is the subspace of/°° consisting of those sequences that converge to 0. The following theorem describes some pathological features of (LHy(B): 7.4.6. Theorem (Rudin [11]). Fix n > 1,1 < p < oo, s > 0. (a)

There exists a linear map of /°° into (LHy(B) which assigns to each yel"^ a function fy that satisfies

(1)

lly|loo
(b) / / y is not in CQ , then U ^ fyoJJ is a discontinuous map of ^ into (LHYiB). (c) Ify is not in CQ , then {fy\ does not converge to fy in the norm topology of (LH)P(Bl as r y 1, Before we turn to the proof, let us Hst some consequences of the theorem; recall that two Banach spaces are said to be isomorphic if there is a linear homeomorphism of one onto the other. (i) (LHy(B) contains a closed subspace that is isomorphic to /°° and lies in H'^iB), (ii) (LHy(B) is not separable. (iii) A(B) is not dense in (LHy(B). (iv) (LHy(B) is not isomorphic to any Hilbert space.

150

7. Some L^-Topics

Indeed, (i) follows immediately from (a), and (i) obviously implies (ii). Since A(B) is separable in the sup-norm topology, it is a fortiori separable in the norm topology of (LHy(B); thus (ii) implies (iii). Finally, (iv) follows from (i), since every closed subspace of a Hilbert space is a Hilbert space, but /* (not being reflexive) is not isomorphic to any Hilbert space. Proof. As in the proof of Theorem 7.3.8, let {XJ be a sequence of pairwise disjoint, nonempty, circular open subsets of S. Enlarge each Xi to a set V^ that is open in C", so that VinVj= 0 if i^ j . Pick Ct e X^, pick Ui e % so that {[/J converges to the identity element of ^ as i ^ oo, and so that I l < Ij and pick positive integers n^ such that (2)

l
if

z^B\V,

and (3)

i<^iC.-,or
The linear map mentioned in part (a) is the one that assigns to every 7 = {c J 6 r" the function (4)

/,(z)= l c , < z , 0 " ^

(zeB).

Since no two of the sets VJ intersect, the inequality (2) fails (for any given zeB^ for at most one term of the series (4). Thus l/.(z)l
kJ2-<(l+8)||yL

SO that ||/,|L < (1 + e)ll7lloc. That |||/J|^ < ||/,|U is trivial, and the first inequality in (1) is a consequence of the following, with /^ in place of ^: Ifg = ZGfc is the homogeneous expansion of some g e (LHy(B\ then (5)

|0,(OI < lll^lllp

(CG5,/C =

1,2,3,...).

To see this, note that GjXO is a coefficient in the Taylor series of the sHce function g^, so that |G,(OI < il^dlp ^ iH^IHp' by 7-4.3(4). The proof of (a) is now comj^lete. Observe next that 00

7.4. Pluriharmonic Majorants

151

When z = Cfc, the absolute value of the kth term of this series is |cJ|i-"M>ik.|, by (3). Another application of (5) shows therefore that i lim suplCfcl < lim sup|||/, - fy o U^WlpHence fy o JJ^ does not converge to fy in (LHy(B) if {cj fails to converge to 0. This proves (b). The proof of (c) is quite similar. If r^ is chosen so that (rf)"' = j , then /,(z) - f,ir,z)

= f q{l - (r,)"'}
dy.

When z = Ck^^ follows from (5) that lil/.-(/.)Jlp>ik,| for fc = 1, 2, 3,

Thus lim sup III/y - (fyXWlp > i h m supjcj,

which proves (c). We conclude this section with an observation which is just as easily proved for arbitrary regions: 7.4.7. Proposition. Suppose Q is a region in C", feH(Q), 0 < t < n/2. Then there is an heH(Q) such that

| I m / | < 1, and

Qxp{t\f\} e^" cos t. The same holds with —/in place of/ Since | / | < 1 + |w|, it follows that

cost Corollary. Under the same hypotheses, fe (LHy(Q) for every p < co.

152

7. Some L^-Topics

Proof. I/I^ < (p/teYe'^^l (This corollary occurs in Stout [2], for star-shaped domains, with a different proof.) Note: The Proposition fails when t = K/2. The function /(z) = - l o g | i ^ n \ — z^

(ZEB)

shows this, since

exp|||/(z)|j> and (1 + zi)/(l - zi) is not in

1 + Zi

H\U),

7.5. The Isometrics of H\B) 7.5.1. Introduction. The isometrics in question are the linear maps T of H\B) onto H\B) that satisfy

(1)

l|T/||,= 11/11,

for every feH^(B).

These will be completely described, for all M > 1 and

for all p 7^ 2, 0 < p < 00.

It should be clear why the case p = lis special: H^(B) is a Hilbert space, isometrically isomorphic to every other separable Hilbert space (of infinite dimension), and its isometrics are therefore the unitary operators; the fact that H^(B) is also a space of holomorphic functions plays no role here. For any p, the following construction yields isometrics: Choose xj/ e Aut(B), c e C, | c | = 1, put i^ ~ ^(0) = a, and define

(2)

(T/)(z) = c^^^_^^j^^^,„,^/Wz)). It is easily seen that these satisfy (1), since f IT/1" dcr = r P{a, Q \ m{Q) Js Js

I" do{0

= i'CI/r-i/'DCa) = PUfnma))

=

[ifl'da.

7.5. The Isometrics of HP{B)

153

This computation made use of the ^-invariance of Poisson integrals (Theorem 3.3.8) and the fact that \l/{a) = 0; we have identified / and Tf with their boundary values / * and {Tf)* and will continue to do so, unless there seems to be some danger of confusion. It is also easily seen that the range of any T given by (2) is all of H\B\ for if ^G/f ^(5) and /(^) = ^

(1 - \a\yip

^(^ ^^)^'

then/GH^(B)and7y = gf. The principal result of this section, Theorem 7.5.6, is that every linear isometry ofH^{B) onto H\E) has the form (2), when p ^ 2. The first case of this (n = 1, p = 1) was proved by deLeeuw-RudinWermer [1], using the fact that the extreme points of the closed unit ball of H^(U) were completely known. Forelli [2] devised a different method that handled H^(U) for all p, and which was then appHed by Schneider [1] to HP(U"X and by ForeUi [3] to HP(B) for p > 2, The proof of Theorem 7.5.6 (Rudin [8]) is based on these ideas, in combination with some very general facts about L^-isometries on arbitrary finite measure spaces, namely Theorems 7.5.2 and 7.5.3. As shown by Koranyi-Vagi [2], this proof extends without difficulty to arbitrary bounded symmetric domains in place of balls. Stephenson [1] has studied the isometrics of the Nevanlinna class. In the following two theorems, fi^ and JLI2 will be finite positive measures, on some sets that will play no role at all and that will not even be named. 7.5.2. Theorem. Assume 0 < p < 00, p 7^ 2, 4, 6, and

(1)

If f^ I-^il^iX i = 1> 2,

J l l + A/iTd^. = J l l + A/^rd/i^

for every XeC, then

(2)

jihof,)dfi, = jihof^)dti2

for every bounded Borel function h:C^C, function h on C.

and also for every positive Borel

Proof. It is enough to prove that (2) holds for every heCo = Co(C), the space of all continuous functions on C that vanish at infinity. Let X be the set of all heCo such that (1) implies (2). If/j, /2 satisfy (1), then so do c -h / i , c -h /2 and c/j, c/2, for any ceC. It follows that J!f is a uniformly closed subspace of CQ which is invariant under translations, dilations, and rotations. We have to prove that X = CQ.

154

7. Some L^-Topics

Define (3)

11 + ^e'^ r ^9

u(X) = -^ f

(^ e C).

By Fubini's theorem,

(4)

jiu o y;.)d^, = _L J " rfe J11 + e'«/;. i" rf/i,

for 7 = 1, 2, so that (1) impHes (5)

J (u o f^)dfii "" J (" ° -^2)^/^2 •

Equation (5) remains true if u is replaced by w,, where w^^) = w(tA), t > 0, XeC, hence also if w is replaced by any finite linear combination (6)

v{X)=

Y.^i
As we shall now see, a^ and t,- can be so chosen that VECQ, hence veX. Since p is not an even integer, none of the coefficients b^ in the binomial expansion (7)

(l+Xy''

(|A|<1)

= tb,X' k= 0

is zero. Replace A by Xe^^ in (7) and substitute into (3). It follows that (8)

(|A|<1).

ua) = f:bi\M"' k= 0

By (3), M(A) = IXfuil/X). Hence (9)

00 u(A) = Xfe*'|A|''-^''

(|A|>1).

k= 0

Now choose f^, 0 < t^ <•"< 0, 1, 2 , . . . , define

(10)

c, = Z X a,.t?', a.f?',

tj^ < 00, where N > 2 + p. For /c =

y, y. = X E aitf-^\

7.5. The Isometrics of H^iB)

155

Then yo = 7i = --- = yiv-2 = 0 i s a homogeneous linear system of N — 1 equations in N unknowns « ! , . . . , % • Pick some nontrivial solution, and use it to define v by (6). Since the ti are distinct, c^ can be 0 for at most finitely many k. For |>l| < l/r^y, 00 v{X)^Y^c^bl\l\^\

(11)

by (6), (8), (10). Thus V is not identically 0. For |A| > 1/^1, one sees similarly that (12)

r(A)=

Z k=

y,bi\Xr"'

=

0(\X\-'-''),

N-l

since p - 2(N - 1) < -2 - p. Thus VECQ, hence veX; in fact v is also in L \ relative to Lebesgue measure of the plane, by (12). Suppose now that /i is a measure on C = R^ such that j /i d/x = 0 for every heX. Since X contains v and all its dilates t;^, and since X is translation invariant, it follows that all convolutions v^* fi are 0. Taking Fourier transforms, we see that (13)

0 (-] fiix) = 0

(xeR\t>

0),

where i), fi are the Fourier transforms of f, //. Note that t; is a radial function. Hence so is f), and if x ^ 0, t{xlt) ^ 0 for some t > 0, since t) ^ 0. Thus /2(x) = 0 for all x ^ 0, hence also for X = 0, by continuity. Therefore fi — 0, X = CQ, and the proof is complete. 7.5*3. Theorem. Assume 0 < p < co, p ^ 2. Assume that M a L'^{n^\ that 1 G M, and that M is an algebra over C, relative to pointmse multiplication. Let Abe a linear map ofM into L\p.^, such that y41 = 1 and

(1) for every feM. (2) (3)

JM/rrf/i, = Ji/r^/ii Then a.e. [//J,

Aifg) = Af'Ag ^Af'Agdn2

=

Ugdfi^

156

7. Some L^-Topics

and (4)

M/lloc =

for all feM,ge

M. Moreover,

(5)

JM4/'I,

for all fi,.. .^fj^eM or positive.

ll/lloc

• • •, Af^)dli2 = jhif,,..., f^)dfi,

and for every Borel function /i: C" ^ C that is bounded

Proof If p is not an even integer, then Theorem 7.5.2, w i t h / a n d Af in place of/i and /2, shows that

(6)

j\Af\""dfi2 = j\f\""dn,

for m = 1, 2, 3, If p is an even integer, then (1) is the same as (6), for some m > 2. Thus we may assume (6), for every fsM, and for some integer m > 2 which is fixed from now on. P i c k / e M, geM, and put u = Af,v = Ag, w = A(fg). For any oc, P.ye C, the integral (7)

(1 + aw 4- jSi; + ywT(l + ow -h ^ + y^)'" diii2

is then equal to

(8)

J ( l 4- a/ -f iS^ + yfgni +^+?g

+ yTgr dfi,.

The coefficients of aj5aj5, aj^y, yajS, yy are the same in (7) and (8). (It is here that m > 2 is used.) Each of the 4 integrals (9)

UVUV,

UVW,

WUV,

WW

is thus equal to j \fg\^, hence they are equal to each other, and therefore (10)

{\w -uv\^dn2

= 0.

It follows that w = uv a.e. This proves (2). Comparison of the coefficients of ap in (7) and (8) leads to (3).

7.5. The Isometries of H'iB)

By (2), (Aff

(11)

= Aif)

157

for / e M, /c = 1, 2, 3 , . . . . Hence

J M / I " = j\A{f)\' = J i r p = Ji/p^

the second of these equalities follows from (3). If we raise (11) to the exponent 1/2/c and then let /c ^ oo, we obtain (4). We turn to the proof of (5). If /i is a monomial in z^,..., z^, Z j , . . . , z^, then (5) holds by repeated appHcation of (2) and (3). Hence (5) holds if h is any polynomial in these variables. The range of ( / i , . . • ,/iv) has compact closure in C^; the same is true for (4fi,.. •, 4/]vX by (4). The StoneWeirstrass theorem shows therefore that (5) holds for every continuous h. The final assertion of the theorem follows now by standard approximation arguments. 7.5.4. Lemma. Suppose / , g, heN(B), / * = g*h* a.e. on S. Then f = gh in B.

and their boundary values satisfy

Proof. Since gh e N{B) and (gh)* = g*h* a.e., we have ( / — gh)* = 0 a.e. Apply Theorem 5.6.4(b). For the next lemma we recall that H^(S) is the class of all boundary functions of members of H^{B\ as in §5.6.7. 7.5.5. Lemma (Schneider [1]). Suppose 0 < p < oo, ueL'^iSX g ^0 a.e., and u^g e H^iS) for k = 1,2,3,.... Then u e if °"(5).

geH\S\

Proof. Without loss of generahty, suppose |w| < 1 and \\g\\p < 1. Put K = u^Q' Denote the holomorphic extensions of /i^ and g into B by the same letters. Then (1)

h\ = (ugf = h,,g^-^ a.e. on S.

Let Q be the set of all z G B where g(z) # 0. If z G Q, then (1) and Lemma 7.5.4 give (2)

(^)H'H(^)-

Since \\h),\\p < 1, Theorem 7.2.5 shows that {hf,(z)} is a bounded sequence; the bound depends on z and p. If we take /cth roots in (2) and let /c -^ oo, we conclude that \hjg\ < 1 at all points of Q. The corollary to Theorem 4.4.7 shows therefore that there exists feH'^(B) such that hi = gf in B. Since hi = gu on S, we see that u = / * .

158

7. Some L^-Topics

We now come to the main result of this section. 7.5.6. Theorem. Suppose 0 < p < co,p ^ 2,n > 1, and T is a linear isometry of H\B) onto H\B). Then there is a ^ek\xi{B) and a ce£,\c\ = I, such that

for allfGHP(B), ZEB, where a = il/~\0). The word "linear" refers to complex scalars, naturally. The reason for mentioning this explicitly is that there are other isometrics. For instance, when « = 1, the map that associates /(z) to f(z) is an isometry of H^(U) onto H^(U) is is not C-Hnear. Proof The natural one-to-one correspondence between fsH^{B) and its boundary function / * e H^(S) is a linear isometry, and can be used to identify HP(B) with H^iS). We may accordingly regard T as acting on H^iS), and shall make no notational distinction between / and / * . Put g = Tl. Then ||^||p = 1, hence g ^ 0 a.e. on S, and (2)

Af=^ G

is well-defined a.e. on S, for every feH\S). becomes (3)

^\Af\^\gY

The isometry hypothesis on T

da = ^\f\^

da

for all feH^iS). Since >11 = 1, we can apply Theorem 7.5.3, with ju^ = a, dpi! = \g\^d(7, and M = /f°°(S). Since \g\PeL\(T) and \g\ ^ 0 a.e. [1 is a multiplicative linear map of i/°°(S) into- L°^(/X2) = ^°°(cr) that preserves L°^norms. The multiplicativity of A shows that (4)

g. (Aff

= g • A(f') = T(f')

G

H^S)

for /c = 1, 2, 3 , . . . a n d / e if °°(S). By Lemma 7.5.5, Afe //°"(5). Passing from H'^iS) to H'^{BX we have now proved that A is a homomorphism ofH'^iB) into H'^iB). Theorem 6.6.5 shows therefore that (Af)(z) = f(il/(z)) for some holomorphic ij/.B -^ B. Thus (5)

(Tf)iz) = giz)f(i^iz))

(zeB)

7.5. The Isometrics of HP(B)

159

for every/G if °°(B). Since H'^{B) is dense in H\B) and evaluations at points of B are continuous linear functionals on H\B) (Theorem 7.2.5), (5) holds actually for all/G/f^(B). We now use the assumption that the range of Tis all of H^(B). Everything proved so far for Tis thus equally true for T " ^ In particular, the analogue of (5) is (6)

(T-'f)(z)

= h(z)f(cp(z))

where T/i = 1 and (p:B ^ Bis holomorphic. By (5) and (6), (7)

f=T-'Tf=h'l(Tf)ocp-]

=

h^(gocp)^(foil;ocp)

for QVQvy f e HP(B). With / = 1, (7) becomes (8)

h-(gocp)=l

so that f = foil/ o cp for QVQry f G H%B). Hence ij/ o (pis the identity map on B. The same argument (interchange T and T~Mn (7)) shows that cp oij/ is the identity map. We conclude that ij/ e Aut(B). It remains to identify g. Put a = il/~\0). By (5), T ( / o xj/'^) = gf. Hence (9)

j\fg\'da=

j\foi^-^\Pda

=

PUfn(a)

or

(10)

jjf\''\9\''dcT= imgl" da = jji{/{"P^da

for all / e //""(S); here P„(0 = P(fl, C)Another application of Theorem 7.5.3 leads now from (10) to (11)

j/J2\9\'da^

jjj2Pada

for, say, all/j, /2 in the ball algebra. The linear span of these functions/1/2 is a self-adjoint subalgebra of C(5). It follows therefore from (11) and the StoneWeierstrass theorem that (12)

1^(01 = ^ ^ , 0 ' / ' '

a.e.

onS.

160

7. Some L^-Topics

Hence g e H'^iB), and (5) implies that T maps if °^(J5) into H'^iB). The same is true of T" ^; thus h e H'^iB), and (8) shows that 1/g e H'^iB). Finally, put (1 -
(13)

^(-)=

(i-i^iy.

(-^)-

By (12) and (13), |^/c| = 1 a.e. on 5. Also, both gk and 1/^/c are in /f'^(fi). Thus gk = c,2i constant of absolute value 1, and (1) follows from (5) and (13). 7.5.7. As regards the sup-norm isometrics of i/°°(5) onto if °°(J5), and of A{B) onto A{B\ their characterization depends on the following general theorem: Every linear isometry Tofa sup-norm algebra A onto A has the form Tf=oi-TJ, where oce A, 1/ae A,\a\ = 1, and T^ is an automorphism of A. A proof of this may be found on pp. 144-147 of Hoffman [1], and in dcLeeuw-Rudin-Wermer [1]. Since the automorphisms of H'^(B\ and likewise of A(BX have the form f ^ f°il/ for some ij/eAut(B) (Theorem 6.6.5), one obtains the following description of their surjective isometrics: If T is a linear isometry ofH°^(B) onto H°^(BX or ofA(B) onto A(B), then there is ail/ e A\xi(B) and a ceC,\c\ = 1, such that Tf^cfoij,. Note that formally this is a limiting case of 7.5.6(1), obtained by letting p tend to 00.

Chapter 8

Consequences of the Schwarz Lemma

8.1. The Schwarz Lemma in B 8.1.1. The famihar classical Schwarz lemma deals with functions defined in the open unit disc U cz C, and asserts the following: (a) / / / : U -^ U is holomorphic, then |/'(0)| < 1, except whenf(X) = ck for some CGC with \c\ = 1. (b) If also /(O) = 0, then \ f(X) \<\X\for every X E ( 7 \ { 0 } , except when f(X) = cX, as in (a). As we shall see, this imphes a variety of analogous results in several variables. Our first example concerns holomorphic maps of one balanced region into another; a set £ cz C" is said to be balanced if XZGE whenever z e £ and A e C, m < 1. This terminology is customary in functional analysis. Balanced open sets in C" are also known as star-shaped circular regions. Note that every balanced region is a neighborhood of the origin. 8.1.2. Theorem. Suppose that (i) Q^ and Q2 ^^^ balanced regions in C" and C" respectively, (ii) Q2 i^ convex and bounded, (iii) F: Qi -> Q2 is holomorphic. Then (a) F'(0) maps Q^ into 0.2^ ^^^ (b) F(ra,)czra2(0
= 0.

Recall that F'(0) is a Hnear operator carrying C" into C"; see §1.3.6. Proof. The assumptions made on Q2 show that C" may be regarded as a Banach space Y whose unit ball is Q2. The corresponding norm is (1)

||w|| =inf{c>0:c-^wGO2}161

162

8. Consequences of the Schwarz Lemma

Fix z G rQi, where 0 < r < 1. Since Qj is open, z e tQ^ for some t < r. Let L be a linear functional on 7, of norm 1. Then (2)

g{X) =

LFih-'z)

defines a holomorphic map g of U into U, By the chain rule, (3)

gXO) = LFXO)t-'z.

Since |^'(0)| < 1, by 8.1.1(a), and since this holds for every L of norm 1, the Hahn-Banach theorem imphes that (4)

||F(0)r^z||
Thus F'(0)z G ^Q2 c: rQ2 • This proves (a). If also F(0) = 0 and gf is given by (2), then ^(0) = 0, hence \g(X)\ < |A|, and (b) follows by the same argument that gave (a). Remark. If Q^ is also convex and bounded, then C" is a Banach space X with unit ball Q^, and (a) asserts that F'(0): Z -> 7 is a hnear operator of norm at most 1. By analogy with the classical Schwarz lemma, one may ask whether F must then be linear whenever ||F'(0)|| = 1. This is not so when n > 1, even in the case Qj = B„, Q2 = ^mj we shall see this in §8.1.5. But the linearity of F does follow if F^O) is assumed to be an isometry: 8.1.3. Theorem. IfF:B„-^B^ is holomorphic and F'(0) is an isometry o/C" into C", then F(z) == F'(0)zfor all zeB„. Proof. Put F'(0) = v4, F(0) = a,G = cp^oF, where cp^ e Aut(J5 J is as in §2.2.1. We claim that a = 0. If ZEB„ and w = Az, the chain rule gives (1)

G'(0)z = q>:{a)w.

By hypothesis, |w| = Mz| = |z|. By Theorem 8.1.2, |0'(0)z| < |z|. Hence Theorem 2.2.2 shows that (2)

\s~^Pw + s~^Qw\ < |w| = \Pw + Qw|,

where s = (1 - \a\^y'^ and Pw 1 Qw. This can only happen when s = 1, i.e., a = 0. Thus F(0) = 0, hence |F(z)| < |z|, by Theorem 8.1.2. Pick Ce CM CI = 1, and define (3)

/i(A) =

(AGI/).

8.1. The Schwarz Lemma in B

163

Then h is a. holomorphic map of U into U with h'(0) = \AC\^ = I, so that h(X) = X, or (4)

a-^F(A0,^O = l

(0<|A|-<1).

Since |F(/10| < |A|, the left side of (4) is the inner product of two vectors in B^. This can only be 1 when the two vectors are equal (and have norm 1). Hence F(XQ — XAC, which gives the desired conclusion, since A is Hnear. As in the case in one variable, part (b) of the Schwarz lemma can be generalized by applying automorphisms to both the domain and the range ofF: 8.1.4. Theorem. / / F : 5„ ^ B^ is holomorphic, aeB„, and F(d) = b, then (1)

\niF{z))\<\
(zeB„).

Equivalently, (2)

|l-p ^ \l-
It is of course understood that cp^ e A\xt(B„) and (pjj e Aut(B^); see §2.2.1. Assertion (1) can be stated in geometric terms: F maps each ellipsoid E(a, s) (see §2.2.7) into the elHpsoid E{F{a\ e). Proof. Since (pj,oFo (p^ maps B^ into B^ and takes 0 to 0, Theorem 8.1.2 shows that \niF{cPa(zm<\zU

which gives (1) if z is replaced by (pjiz)- If we square (1), subtract from 1, and apply the identity 2.2.2(iv), we obtain (2). Note: If m = n and F eAut(B„), then equality holds in (2). To see this, apply (2) to F~ ^ as well as to F. 8.1.5. Examples, (i) Suppose f:B„^U linear functional that takes zeB„ to

(1)

is holomorphic. Then /'(O) is the

Z (DJ)iO)z,

k=l

164

8. Consequences of the Schwarz Lemma

which Hes in U, by Theorem 8.1.2 with m = 1. It follows that

(2)

t K^kfim' < 1.

fc=i

(ii) Suppose F:IJ -^ B^ is holomorphic, F = (f^,., .,fn). Then F'(0) is the linear map that takes A G L/ to the vector (3)

(/'i(0)A,...,/;„(0)A)

in B„, by Theorem 8.1.2 with « = 1. Hence

(4)

Z |/KO)P < 1.

(iii) As regards the remark that precedes Theorem 8.1.3, we shall now see that the extremal functions related to the Schwarz lemma need not be unique, even in the simplest case Q^ = ^ 2 , Q2 = V. The power series

(5)

i _ y r ^ = fc^t/c

(1^1
has Cfc > 0 for all k. Let the functions QI, be arbitrary members of //*(52), subject only to the inequality H^fclloo ^ ^k^ ^^^ define (6)

/(z, w) = z + w^g^iz, w) -f w'^g2(z, w) + w^^3(z, w) + • • •.

If I z 1^ + I w |2 < 1, it follows that (7)

|/(z, w)| < |z| + 1 - V l - | w | ^ < 1.

Every f given by (6) is thus a holomorphic map 0/82 into U. lfheH'^(B2) and \\h\\^ < 1, Theorem 8.1.2(a) implies that \\hXO)\\ < 1. Equality holds for ei;er3;/of the form (6), since/XO)ei = 1 and/X0)e2 = 0. If / I G H * ( B 2 ) , ||/I|L < 1, and ;i(0, 0) = 0, Theorem 8.1.2(b) implies that I /i(z, 0) I < I z I; again, equaHty holds for every f given by (6) Simple examples of (6) are (8)

z + iw^

or

z + 1 - y/l - w^.

8.2. Fixed-Point Sets in B

165

8.2. Fixed-Point Sets in B In Section 2.4 we saw that the fixed-point sets of automorphisms of B are affine. Theorem 8.2.3 will show that this property is shared by all holomorphic maps of B into B. But we first consider a somewhat more general situation. 8.2.1. Definition. Let Q be a balanced, convex, bounded region in C . As pointed out in the proof of Theorem 8.1.2, C may then be regarded as a Banach space X whose unit ball is Q. We say that Q is strictly convex if to every linear functional L on X, with ||L|| = 1, corresponds just one zeQ (the closure of Q) such that Lz = 1. Evidently, B is strictly convex. 8.2.2. Theorem (Rudin [13]). Let Q be a balanced, bounded, strictly convex region in C". / / F : Q -> Q is holomorphic and F(0) = 0, then F and the linear operator F'{G)fix the same points ofQ. Proof. Let X be the Banach space whose unit ball is Q. We shall use || • || for the norm in X, for the corresponding norms of linear functionals on X, and for the norms of linear operators on X. Put F'iO) = A, By Theorem 8.1.2, (1)

MII<1

and

||F(z)|| < ||z||

(zeQ).

Fix z G Q, z = rw, where 0 < r < 1, ||M|| = 1. By the Hahn-Banach theorem, there is a linear functional L on X with (2)

||L|| = 1,

Lu= 1.

Put (3)

g(X) = LF(Xu)

(XeU).

Then g:U ^ U is holomorphic, ^(0) = 0, and ^'(0) = LAu, IfF(z) = z,then^(r) = L(ru) = r, hence gf(A) = A for all A, hence ^'(0) = 1. Thus LAu = 1. The strict convexity of Q, combined with (2), implies now that Au = u. Hence Az = z. Conversely, assume Az = z. Then gXO) = Lu = 1, hence g(r) = r, or (4)

L(r-'F(ru))=

1.

By (IX Ik ^F('^u)\\ ^ 1- The strict convexity of Q, combined with (2) and (4), implies now that r~^F(ru) = u, hence F{z) = z.

166

8. Consequences of the Schwarz Lemma

8.2.3. Theorem (Rudin [13]). If F.B^B point set EofF is ajfine.

is holomorphic, then the fixed-

Proof. Suppose aeE, and let E^ denote the fixed point set of cp^^ F o (p^. Then O G £ ^ , and Theorem 8.2.2 implies that E^ is affine. Since E = (paiEJ, it follows from Proposition 2.4.2 that E is affine. 8.2.4. Holomorphic Retracts. A map F:B -^ Bis said to be a retraction of B if F(F(z)) = F(z) for every ZEB. The range of F is then exactly its fixedpoint set. A holomorphic retract of B is, by definition, the range of some holomorphic retraction of B. Theorem 8.2.3 thus has the following corollary. Corollary (Suffridge [1]). The holomorphic retracts of B are exactly the affine subsets ofB. Indeed, if £ c= B is affine and a e £, then (Pa{E) = B n Y, where Y is a subspace of C". Let P be the orthogonal projection of C onto 7. Then (paP(Pa is a holomorphic retraction of B onto E. The converse follows from Theorem 8.2.3. Although the holomorphic retracts of B are thus very simple, there exist very complicated holomorphic retractions. For example, let feH'^{B2) be any one of the functions described by 8.1.5(6), and put F(z, w) = (/(z, w), 0). Since/(z, 0) = z, F retracts B onto the set {(z, 0): |z| < 1}.

8.3. An Extension Problem 8.3.1. Statement of the Problem. Suppose 1 < n < m, and let O: B„ ^ 5^ be holomorphic. Let us say that O has the norm-preserving H°^ extension property (or property (*), for brevity) if the following is true: (*) To every / G f/°°(B„) corresponds (a) ^ o O = / , and (b) Halloo =

SLQE H'^{B^

such that

The problem is: Which O have property (*)? The reason for caUing this an extension problem is quite simple. Clearly, (*) implies that O is one-to-one. E v e r y / e ff°°(5„) corresponds therefore to a function / on 0(B„) such that / o O = / , and any g that satisfies (a) is an extension of / . The requirement (b) is of course extremely strong, and one should expect that only very special CD's can satisfy it. Theorem 8.3.2 confirms this expectation. If O has property (*) one sees very easily that \jj o^ has property (*) for every ^ e Aut(JB^). Theorem 8.3.2 implies therefore that every O with property (*) has affine range.

8.3. An Extension Problem

167

8.3.2. Theorem. For a holomorphic map 0 : 5 „ -> B^ with 0(0) = 0, the following are equivalent: (i) O has property (*). (ii) O is a linear isometry. (iii) There is a multiplicative linear operator

such that (Ef) o^=.ffor

every fe W^iB^).

Proof. Assume (i). Pick CeC", |CI = 1, and put f(z) = (z,0' Then feH^(B„l li/ll^ = 1. Hence there is a geH'^iBJ, with H^filoo = 1, such that g(<^(z)) = . With z = AC, this becomes (1)

gmXO) = ^

(?ieU),

Since
gXOWm

= 1.

By Theorem 8.1.2, O ' ( 0 ) C G S ^ and ^'(0) is a hnear functional on C", of norm at most 1. Hence (2) implies that Q>XO)C is a unit vector in C" for every unit vector C in C". This says that (^'(O) is an isometry, hence Q>(z) = O'(0)z, by Theorem 8.1.3. Thus (i) implies (ii). If (ii) holds, then 0(z) = Az, where ^4 is a linear isometry of C" onto a subspace 7 of C". Let P be the orthogonal projection of C" onto 7, and define (3)

iEf)iw)=f{A-'Pw)

(weBJ

for all / G i/°°(B„). (Note that A~ Ms linear and well-defined on the range of P, and that P maps B^ onto Y n B^.) It is clear that E is linear and multiplicative; also, (Ef) o O = / , because A~^P(^(z) = z. Thus (ii) implies (iii). Finally, assume (iii). Since E is multiplicative, Ef = E(f • 1) = (Ef) • (£1), hence El = 1. (Note that Ef = 0 implies / = 0.) If fg = 1, it follows that (Ef)' (Eg) = E(fg) = 1. Thus Ef is invertible in ff°°(BJ whenever / is invertible in H'^(B„). It follows that the sets f(B„) and (Ef)(BJ have the same closures in C. In particular, ||E/||oo = ll/lloo- Thus (iii) implies (i). Note: The only fGH'^(B„) that were needed to prove the implication (i) -^ (ii) were the Hnear functions . 8.3.3. This problem has been treated by Stanton [1] with finite Riemann surfaces in place of B„. If one drops condition (b) in §8.3.1, one obtains what is usually called the H"^ extension problem. Henkin [5] and Adachi [1] have studied this in

168

8. Consequences of the Schwarz Lemma

Strictly pseudoconvex domains. With polydiscs in place of balls, extension problems of this type occur in Chapter 7 of Rudin [1]. For extension theorems in the context of Bergman spaces and Hardy spaces, we refer to Amar [1], [3], and to Cumenge [1]. Extension theorems with C^-data were investigated by Elgueta [1].

8.4. The Lindelof-Cirka Theorem The classical theorem of Lindelof which Cirka extended to several variables concerns the Umit of a function /e/f°°(C/) at a single boundary point. It is thus not a theorem of Fatou type. Although Lindelof's theorem is an elementary consequence of the maximum modulus principle, it does not seem to appear in the standard elementary texts. For this reason, a proof is included here. 8.4.1. Theorem (Lindelof [1]). Suppose f e H°"((7) and y: 10, 1)^U tinuous curve such that y(t) -^ 1 as t ^ 1. If (1)

is a con-

lim/(y(0) = L

exists, then f has nontangential limit L at the point 1. Note that there is no restriction on the manner in which ^(0 tends to 1, except that y(t) must lie in U for all t < 1. Proof. Without loss of generality, assume ||/||oo = 1 and L = 0. Let Z be the strip defined by |Re z| < 1. Let (/> be a conformal map of U onto S, with (p(0) = 0, such that, setting T = cpoy^ we have Im r(t) -> + oo as t -^ 1, Replace/by F =fo cp'K Then F e H'^^L), \F\ < 1, F(r(r)) ^ 0 as r ^ 1. Given d e (0, 1), we have to prove that F(x + iy) ^ 0 SiS y ^ -f oo, uniformly in |x| < 1 - d. Fix e, 0 < e < 1. Choose any >; > Im r(0), so large that |F(r(0)| < e whenever Im r(t) > y. We claim that then (2)

|F(x + i » | <8^/^ if

\x\
The theorem follows obviously from (2). To prove (2), assume j; = 0, without loss of generality (by a vertical translation of Z), choose to so that Im r(to) = 0 but Im r(t) > 0 if to < t < UIQXE = {r(t): to
8.4. The Lindelof-Cirka Theorem

169

Assume XQ < x < 1 — 3, Define

(3>

^"^'^^

l+,(l+z)

(^"^^

where rj is a, positive parameter. Then G ^ G / / ° ° ( Z ) . On E, \F(z)\ < e; on E, \F(z)\ < s; hence | G^| < s on £ u E. On the right edge of Z, the boundary values of \G^\ are <e. When | I m z | is sufficiently large, then |G^(z)| < e because of the denominator in (3). These facts imply that | G^(x) | < e, by the maximum modulus principle, appHed to G^ in the component of I\(E u E) that contains x. Letting ?; -^ 0, we obtain therefore

since 1 — x > S. I f — l + ( 5 < x < XQ, replace 1 + z by 1 — z in (3); this leads to the same conclusion. Thus (2) holds, and the proof is complete. 8.4.2. Remark. We stated in Lindelof's theorem in the disc U but proved it in the strip Z. Other conformal maps will of course transfer the theorem to other regions in C. For example, let U^ = {z = re'^: r > 0, | e | < a}. If / G H ° " ( n j , / - > L along some curve yo ^^ n^ that approaches 0, and p < CL, then / tends to L along every curve y that approaches 0 within 11^. 8.4.3. Approach Curves in B. A curve in B that approaches a point C e 5 will be called a C-cwn;^. More precisely, a C-curve is a continuous map F: [0, 1) -> 5 such that F(0 -> C as t ^ 1. Usually, however, it will not be necessary to refer to any parametrization. With each C-curve F we associate its orthogonal projection (1)

y =
into the complex line through 0 and C- Then (T — y) 1 y, so that (2)

\r - y\'+

\y\'=

Since | F | < 1,(2) implies f'X\

\^)

ir-yp i-|y|'

< 1.

\r\\

170

8. Consequences of the Schwarz Lemma

As i^-curve F is said to be special if

|r(t)-y(OP lim';^,..;;/; =o 7": i-\y(t)\'

(4)

and is said to be restricted if it satisfies both (4) and

for some A < oo. The restricted C-curves T are thus the special ones whose projection y is nontangential. There is a simple relation between restricted C-curves and the Koranyi regions D^CO- Recall that z e D^O precisely when (6)

|l-
Assume that F satisfies (5), and also (3), but with some c < 1 in place of 1. Then (2) leads to 1 - |F|^ == 1 - \y\' - IF - y\' > (1 - c)(l -

\y\')

and (5) shows that

|i -
1-iri^

( i - c ) ( i + i7i)

which tends to A/2(l — c) as t -> 1. We conclude: If (x> A/(l — c), then F lies in Z)^(0 eventually; that is to say, F(0 e ^a(C) for all t that are suflBciently close to 1. If (3) is replaced by (4), the above holds for arbitrarily small c. Thus: Every restricted (,-curve F satisfying (5) lies eventually in DJ^Q, for all a> A, Conversely, every C-curve F that lies in DJX) satisfies (5) with A = on. We shall say that a function f:B ^ C has restricted K-limit L at C if hm/(F(t)) = L as t ^ 1, for every restricted C-curve F. The preceding discussion shows that this happens whenever / has a Klimit at C- However, an / G / / ° ° ( B ) may have a restricted X-limit at a point C,

8.4. The Lindelof-Cirka Theorem

171

without having a K-limit at C. The simplest example of this is probably given by the function (8)

/(z, w) =

1 -z^

which is in H'^(B2), has restricted X-limit 0 at (1, 0), but fails to have a Xlimit there, since (9)

f(U c V l - t') = c'

(0
for every ceU. The expansion of (8), namely (10)

f(z, w) = f z^'^w' k= 0

is a very simple example of a power series that converges absolutely at every point of S although the convergence is not uniform. 8.4.4. Theorem (Cirka [1]). Suppose feH'^(B\ and (1)

I^GS^TQ is a special C-curve,

lim/(ro(0) = L.

Then f has restricted K-limit L at CProof. Let F be any special C-curve. Fix t e [0, 1) for the moment. Since (F — y) J_ 7, the point (1 — X)y(t) + AF(0 lies in B whenever |y|^ + m ^ | F - yp < 1, i.e., whenever \X\ < R = R(t\ where (2)

R^ = ^ - I ^ l ^

By 8.4.3(3), R > I. Since F is special, R(t) ^ oo as f ^ 1. If I A| < R we can define (3)

giX) = / ( ( I - X)y{t) + Ar(0).

The Schwarz lemma, applied to g(X) — g(0) in the disc {| A| < R}, shows that (4)

19(1)-gm<^^.

172

Since R(t) -^ (5)

8. Consequences of the Schwarz Lemma OO,WQ

conclude from (3) and (4) that l i m { / ( r ( / ) ) - / ( y ( 0 ) } = 0.

We now apply (5) to the given curve FQ and to an arbitrary restricted C-curve r . By (1) and (5), /(yo(0) ^ ^- Since y is nontangential, Lindelof's theorem (applied in the disc {ACIAGL/}) shows that f(y{t))^L. Hence f(T(t)) -^ L, by (5), and the proof is complete. 8.4.5. Asymptotic Values. If / is a function in 5, F is a (-curve, and /(z) tends to L as z tends to C along F, then L is said to be an asymptotic value

of/ate.

Lindelof's theorem implies that no feH°^(U) can have more than one asymptotic value at any boundary point. This is false if U is replaced by B; the function /mentioned at the end of §8.4.3 has every c with |c| < 1 as an asymptotic value at (1, 0), even though | / | < 1. But Cirka's theorem shows that we still have uniqueness if we restrict ourselves to special C-curves: If f G H'^(B\ C^-S", and f tends to L^ and L2 along special ^-curves F^ and F2, then L^ = L2. 8.4.6. Example (Nagel-Rudin [2]). Here is an example that is a bit more ambitious than the one given at the end of §8.4.3. It exhibits a function feH'^{B2) whose restricted X-limit is 0 at every point on the circle {(e'^, 0): — 71 < 6 < 7r} but which has no X-limit at any of these points. To do this, pick positive integers n^ and corresponding radii rj = 1 — l/n^, so that ni = 2, (1)

nj > 10(n, + . . . + nj_,)

(j = 2, 3, 4,...)

and (2)

n,expj-^Ur'

(l
Define (3)

f(z,w)

= w'g(z) = w'f^njZ^K 7=1

Since nj\z\"^ < Kjij — n j _ i ) | z p < 2 ^ Izl"*, where the sum extends over all m with nj.^ < m < nj, it follows that \g(z)\ < 2/(1 — |z|), hence I /(z, w) I < 4 in JB2. Thus / e if^^C^^). Since /(z, 0) = 0, Cirka's theorem implies that / has restricted X-Hmit 0 at all points (z, 0) with |z| = 1.

8.4. The Lindelof-Cirka Theorem

173

For /c > 2, i < (1 - llkf < lie. If |z| = r^, it follows from (1), (2), (3) that

(4,

|^„|>5_1_|,-,

Thus \g{z)\ > rip/lO = 1/(20(1 — r^)) as soon as Up is large enough, and therefore (5)

l/(rpe'",cVr^)|>^

if |c| < 1. The points at which / is evaluated in (5) he in DJ^e'^, 0) when a > 2/(1 - |c|2). Hence/has no K-limit at (e'^ 0). 8.4.7. Example. Fix a constant c > j and define / in B2 by (1)

f(z, w) = (1 -

zy^w.

Then / ^ H°^(52), but / e ^^(^2) for all p < 4/(2c - 1). If i < ^ < c and (2)

r ( 0 = (t, (1 - 0')

(0 < ^ < 1)

then r tends to (1, 0) restrictedly, and (3)

/ ( r ( 0 ) = (1 - 0'"^ -

^.

Since f(z, 0) = 0, we see that / has no restricted X-limit at (1, 0). Take a point (a,b)eS,a^l, and consider the rectilinear path (4)

r ( 0 = (t + (1 - t)a, (1 - 0^)

from (a, Z?) to (1, 0). On this path, (5)

f(r(t))

= ---^-(i-ty-\ (1 - a)

When c < 1, this tends to 0 as r -^ 1. Thus all "rectilinear limits" o f / at (1, 0) are 0. In fact, /(z, w) ^ 0 as (z, w) ^ (1, 0) within any cone in B whose vertex is at (1, 0), although (as we saw above) the restricted X-limit of/does not exist there. When c = 1, then /(z, w) = w/(l — z), and (5) shows that / is constant on each of the lines (4). All rectilinear limits of/exist therefore at (1, 0), but they are not equal. In fact, they cover C.

174

8. Consequences of the Schwarz Lemma

By Cirka's theorem, no / e if °°(JB) can behave in this way. The following version of Cirka's theorem will be used in Section 8.5. It will be clear from the proof that the hypotheses could be varied considerably, but it seems best to stick to a simple statement. 8.4.8. Theorem. Suppose feH{B\ ^eS, f is bounded in every region DJ^Q, and the radial limit off exists at C- Then the restricted K-limit off exists at CProof. Let yo(0 = tC, 0 < t < I, and let F be any restricted C-curve, with projection y as in §8.4.3. Then y is nontangential. Lindelof's theorem shows therefore that / has the same limit along y and yo. It is thus enough to prove that (1)

lim{/(r(0) - f(y(t))} = 0. (-•1

We saw in §8.4.3 that r(t) e D^{Q eventually, for some a. Choose J? > a. A slight modification of the proof of Theorem 8.4.4 shows that (1 — X)y + XTeDp whenever \X\ < R = R(0, where ...

^'^

„2

^ =

I - \y\'-

(2/m

ir^TP

- y\

•

If r(t) e D „ then 11 - y| < (a/2)(l - \y\^), so that (3)

R'>

P-oc

l-|yp

P

\r-y\

which tends to oo as t ^ 1, since F is special. Now define g(X) = / ( ( I - A)7 + XT) for \X\ < R, use the fact that / is bounded in Dp, and estimate g(l) — g(0) by the Schwarz lemma, as in the proof of Theorem 8.4.4. This leads to (1).

8.5. The Julia-Caratheodory Theorem 8.5.1. In the present section, the following one-variable facts will be generalized to holomorphic maps from one ball into another: Suppose f:U ^ U is holomorphic. If there is some sequence {zj in U, with Zi -> 1 and /(z,) -^ 1, along which

1 - i/fe)i 1 - k.-i is bounded, then / maps each circular disc in U that has 1 in its boundary into a disc of the same sort. This (in a more quantitative form) is Julia's

8.5. The Julia-Caratheodory Theorem

175

theorem. Caratheodory added that /'(z) then has a nontangential positive finite Hmit at z = 1. Full details of this may be found in vol. 2 of Caratheodory's book [3]. The generalizations to several variables will be proved directly, without any reference to the theorems just mentioned. In fact, if one takes n = 1 and m = 1, the proofs that follow are the classical ones. 8.5.2. The Setting. Throughout this section, m and n will be fixed, F will be a holomorphic map of B„ into B^, C will be a fixed boundary point of B„, and we define (1)

L = limi„f-^-|^(^>l^ 1 - |z| Z-; The basic assumption we make is that L < oo. There is then a sequence {aj in J5„ that converges to C, such that

and such that F{ai) converges to some boundary point of B^. By unitary transformations we may choose coordinates so that C = ^i and F(a;) converges to ^1. (The symbol e^ is here used with two meanings; it designates the first element in the standard basis of C" as well as C". It is unlikely that this will cause any confusion.) Let / i , . . . , /^ be the components of F. The Schwarz lemma (Theorem 8.1.4) states that ^^^

|l-P^l-|F(a,)P

l-|f(z)|^

-

l-|a..p

|l-p

l-|zp

for all zGfi„. As i -> 00,
One incidental consequence of (1) is that L > 0. The inequahty (1) has an appeahng geometric interpretation that involves ellipsoids: For 0 < c < 1, let E^ be the set of all zeB„ that satisfy (2)

^ ^ < 1-lzP

' 1-c

176

8. Consequences of the Schwarz Lemma

Writing z = (z^ z') in the usual way, a little computing shows that (2) is the same as (3)

|z.-(l-c)P_^|zf^^ c c

Thus E^ is an ellipsoid in B„ that has e^ as a boundary point, has its center at (1 — c)ei, has radius c in the ^j-plane, and has radius ^/c in the directions orthogonal to e^. If 7/(1 — y) = Lc/(1 — c) and if Ey denotes the corresponding ellipsoid in B^, then it follows from (1) and (2) that F maps E^ into E^, where (4)

1 + Lc - c

Let us now add an inessential assumption that will simplify the statements of some inequalities, namely: F(0) = 0. Then \F(z)\ < \z\ (Theorem 8.1.2), hence L > 1, and thus (4) implies the simpler statement y < Lc. This proves the first part of the following geometric version of JuHa's theorem: 8.5.4. Theorem. IfF is as in §8.5.2 and if also F(0) = 0, then (i) F(Ec) <= Ei^c when 0 < c < 1/L, and (ii) F(DJc^D,^foralU>L To prove the assertion about the Koranyi regions D^ = DJ^e^), simply multiply the inequalities 8.5.3(1) and 1 1 I - \F(z)\'- < 1 - \z\'' Then take square roots, to obtain

|i-/.(z)l^^;^.[i^<,^/Z

l-|F(z)p-^

l-|z

if zeD^. We shall need the following relation between D^ and Dp. 8.5.5. Lemma. Suppose 1 < a < j8, ^ = j ( l / a — 1/j^), and z = (z^, z') e D^. (i) If\M <<5|1 - z^\then(z^ 4 - / 1 , Z ' ) G D ^ . (ii) If\w'\ < 5 | 1 -Zi\^^^then(z,,z' + w')eDp.

8.5. The Julia-Caratheodory Theorem

177

Proof. The condition that zeDg, can be written in the form (1)

|/|2
in which Zj and z' are separated. Since |z, | < 1, |A| < 1, j8 > 1, and 53 + 2/j8 < 2/a, we have |zi + Ap + | | 1 - zi - A| < Izip + 5|A| + | | 1 - zil <|zip+^|l-z,|
8.5.6. Theorem. Suppose F = (f^,..., f^) is a holomorphic map of B„ into B^,F(0) = 0, (1)

L = lim inf^ ~_|^^^^' < 00,

and F(ai) -* e^ for some sequence {flj} in B„ such that a,- -»• e^ and (2)

,._ 1 - \F(a,)\' Hm ^ ' , _ '3' = L.

1 - Ia,l^

Suppose 2 <j < m and 2 < k < n.

178

8. Consequences of the Schwarz Lemma

The following functions are then bounded in every region DX^i): (i) ( l - / i ( z ) ) / ( l - z i ) (ii) (Di/i)(z) (iii) fj(z)/ii-z,y" (iv) (I 2,y'\D,fj)(z) (V) (o,/i)(z)/(i - z,y" (vi) (D,fj)(z). Moreover, the functions (i), (ii) have restricted K-limit L at e^ and the functions (iii), (iv), (v) have restricted K-limit 0 at e^. Corollary. In the case m = n, the Jacobian JF ofF is bounded in every region Because of its length, the proof will be divided into several steps. Step 1. Radial Behavior. We shall first prove that

(3)

limi^4^ =L

and (4)

l i m - - ^ ^ = 0

(2<;<m),

where it is understood that 0 < x < 1. Suppose that actually \ — x < 1/L. Put \ — x = 2c. Then xe^ is a boundary point of E^. By Theorem 8.5.4, F{xe^) Hes in the closure of Ej^. Since ILc < 1, it follows that \F{xei)\ ^ 1 - '^I-c, which is the same as (5)

\-\F{xe,)\
Since F(0) = 0, 1 + \F{x)\ < 1 + x. Hence (5) implies

(^^

1 - \F{xe,)\' ^ 1 - \H^e,)\ 1-x^ 1-x

-^'

By the definition of L as the lower limit (1), it follows from (6) that

(7)

l i m i ^ l l ^ ^ = L.

8.5. The Julia-Caratheodory Theorem

179

To simplify the notation, we now write w = w(x) for fiixei). By (6) and Theorem 8.5.3,

(1 — xY

1 — x"^

Since 1 — \F(xei)\ < 1 — | w | < | l - w | , we conclude from (6), (7), and (8) that

(9)

liml^M^=UmV^ = ^-

jc-,1

y.^1

1 — X

1 — X

The ratio of the two numerators in (9) converges therefore to 1 as x ^ 1. This impHes that also

(10)

^

1

^^1 1 -

-

^ = ,,

|W(X)|

and (3) is thus a consequence of (9). Since w(x) -^ 1 as x -> 1, (3) is the same as (11)

lim^-y'<-->I^L.

x^l

1 — X

Now (4) follows from (7) and (11), because (12)

\F\' = \A\' + --- + \fJ'.

Step 2. The Functions (i) and (iii). Fix a > 1, and assume z G D^ei) is so close to ei that Lc < 1 if c = (a/2)| 1 — z^\. Then \1 - z,\^ = (2c/a)|l - z j < c(l - |z|^). Since c < c/(l - c), it follows that zeE^ (see 8.5.3(2)), hence F(z) G E^^^, and therefore (13)

|l-/i(z)|<2Lc = aL|l-Zi|.

Since (13) holds for every zeDJ^e^ that is sufficiently close to e^, we conclude that the function (1 — /i)/(l — z^) is bounded in every Dj^e})\ by (3) and Theorem 8.4.8, its restricted X-hmit at e^ is L. If 2 < j < m, the inclusion ¥(z) e E^^ shows that (14)

|//z)|2
Hence fj{z)l{\ — z^Y'^ is bounded in every DJ^e^), and its restricted K-limit at e^ is 0, because of (4) and Theorem 8.4.8.

180

8. Consequences of the Schwarz Lemma

Step 3. The Functions (ii) and (iv). These involve differentiation with respect to Zj. Suppose 1 < oc < p, choose S as in Lemma 8.5.5, let ZED^, and put (15)

r = r(z) =

S\l-z,\.

Then (z^ + A, z') e Dp for all A with |A| < r. By the Cauchy formula, (16)

iDJ,)(z)

= -^ f

Z7CI J|;t|=r

/i(z, + A, z')A-^ t i l

The integral is unchanged if/i is replaced by/^ — 1. Do this, then multiply and divide the integrand by z^ + A — 1, and put A = re'®, to obtain

(17)

(D,/,)(z) = - J__^ ——^-—-^

. |1 - -^;^|de.

The first factor in the integrand is bounded, by Step 2, since (zj + re'^, z') e Dp{ei). The second factor is at most 1 + 1/3, by (15). We conclude that D^f^ is bounded in D^ei). When z = xe^ in (17), then the second factor in the integrand is 1 — d~^e~'^, and the first factor converges boundedly to L as x ^ 1, since X + r(x)e'^ -^ 1 nontangentially, for every 0, by (15). Hence (Difi)(xei) -^ L as X -> 1, by the dominated convergence theorem. Another application of Theorem 8.4.8 shows now that D^f^ has restricted X-Hmit L at e^. If 2 < 7 < m, a similar application of the Cauchy formula gives

(18)

fj(z, + re^\ z') (1 - z, - re^y ^^ie (^ifO(^)-2nj_^(l-z,-re^r 72

.. ^^'

from which it follows exactly as above (using Step 2 and Theorem 8.4.8) that (1 — z^y^{Pifj){z) is bounded in D^ei) and that its restricted X-limit at e^ is 0. Step 4. The Functions (v) and (vi). These involve differentiation with respect to Zfc for 2 < /c < n. Without loss of generahty, take k = 2. Suppose 1 < a < j5, choose S as in Lemma 8.5.5, let z e DJ^ei), and put (19)

p = p(z) =

d\l-z,\ 1/2

Then (z^, z' + w')EDp(ei) for all w' with |w'| < p. If we apply the Cauchy formula as in Step 3, we obtain (20)

(g2/i)(z) ^ _ (1 - ^lY" 1 r l-Aiz„z, (1-Zi)^/^ p(z) '271 J _ .

+ pe'',...) 1-z,

_,3

^e

8.5. The Julia-Caratheodory Theorem

181

and, for j > 2,

(21) ^^^m^^ = —p(~^•Yn]_;—i^^^-^t^—'

^®-

The integrands are bounded, by the bounds of (i) and (iii) in D^{e^. In view of (19), the left sides of (20) and (21) are therefore bounded in DJ^e^. To finish, we have to prove that the left side of (20) has restricted iC-Hmit 0 at e^. By Theorem 8.4.8 it is enough to prove this for the radial limit. Moreover, it involves now no loss of generality to assume n = 2, m = 1, in which case / i = F. Writing (z, w) in place of (z^, Z2), we can expand F in the form (22)

F(z, w) = /(z) + 2w(l - zy^^Q{2) + £ QJ^z)v^K

Then (D2F)(z, 0)/(l — zY'^ = 2g(z). It is therefore enough to show that (23)

^(x)^0

as

x/l.

We know that (1 — /(z))/(l - z) -^ Las z ^ 1 nontangentially, and that g is nontangentially bounded at 1, and we make one further reduction: //I YJO ^k^^\ < 1 i^ ^ certain disc with center at 0, then also \CQ + ^C^W | < 1 in this same disc. This is so because CQ + ^C^W is the arithmetic mean of the first two partial sums of the power series. If we apply this to (22), we see that (23) is a consequence of the following proposition (in which there is some redundancy in the hypotheses): 8.5.7. Proposition. Suppose h: B2 -^ U has the form (1)

h(z, w) = f(z) + w(l -

zy^'g(z)

where / , g e H(U\ (1 —/(z))/(l — z) has finite nontangential limit L at z = 1, and g is nontangentially bounded at 1. Then (2)

^(x)-^ 0 as X y 1.

Proof. Choose e > 0, put c = L^/s^, let z tend to 1 along the Hne z = x 4ic(l — x). Then 1 — z = (1 — ic)(l — x\ hence (3)

|l-z|>c(l-x),

182

8. Consequences of the Schwarz Lemma

and also 1 — \z\^ > 1 — x if c^/(l + c^) < x < 1, an assumption that will be made in the rest of this proof. Note that (4)

/(z) = 1 - (L + o(l))(l - ic)(l - X)

SO that (5)

R e / ( z ) = l - ( L + o(l))(l-x).

Associate with every z under consideration a w e C with |wp = l — | z p > 1 — X, whose argument is so chosen that (6)

w(l - zY^'giz) = |w(l - zy/'g{z)\ > c'^\l

- x)\g(z)U

by (3). Hence, by (5) and (6), (7)

1 > Re h(z, w) > 1 + {c'^^\g(z)\ - L - o(l)}(l - x). Consequently,

(8)

lim sup \g(x + ic(l - x))\ < Lc~^'^ = G.

The same estimate holds on the Hne z = x — ic{\ — x). Since g{z) is bounded as z -^ 1 between these two lines, it follows that (9)

limsup|6f(x)| < e ,

which proves (2), since £ was arbitrary. 8.5.8. Examples. We shall now show that the conclusions of Theorem 8.5.6 are optimal. The numbers (i) through (vi) will refer to Theorem 8.5.6. The first two examples will use the function (1)

g{z) = exp^ - ^ - / log(l -z)\

Note that \g\ < \mU, (2)

{ze U).

and that g'{z) =

1- z

As z -• 1, g(z) spirals around the origin without approaching it.

8.5. The Julia-Caratheodory Theorem

183

First Example. Take n = m = 2, define F : B2 ^ ^2 by (3)

F(z, w) = (z, wg(z)).

The hypotheses of Theorem 8.5.6 hold, with L = 1. Since D^/i = 1 and D2/1 = 0, we have (4)

(JFKz, w) = (D,f,)(z,

w) = g(z).

Therefore the radial limit 0/1)2/2 cind ofJF does not exist at e^. This dealt with (vi). As regards (iv),

(5)

IW

(1 - zyi\D, f,)iz, w) = ^^^r^2 • 9(z).

This has no K-limit at e^, although its restricted X-Umit is 0. We see also that the boundedness assertion made about (iv) becomes false if the exponent ^ is replaced by any smaller one. Second Example. Take n == 2,m = 1, put (6)

F(z, w) = z +

Wg(z)'

Example 8.1.5 shows that F maps B2 into U. The hypotheses of Theorem 8.5.6 hold again with L = 1. Since F = f^ ,we now have

1 -z (8) (9)

(DiA)iz,^ w) = (D2A)(Z , w) (1_

2)1/2

'

^V-h

2 ( 1 - z)

iw^ 1 1 •QiA ^ ' 2(1 - z) w -(l_z)l/2

.

f

,(r\

Hence (i), (ii), and (v) n^^rf /tare no K-limit at e^, and the boundedness assertion made about (v) becomes false if\ is replaced by any larger exponent. Third Example. This will show that the exponent | is best possible in (iii). Take n = 1, m = 2. Pick a > 0, put

(10)

''(^)=^rj-^i«i"^^«'

and note that h hes in the disc algebra, that h{\) = 0, and that Re h(z) > 0 for all other zeU.

184

8. Consequences of the Schwarz Lemma

Put c = lIlTi^^' and define F = (/i,/2) by (11)

f,{z) = ze-^'^'^K

Mz) = c'^\l

-

zr^-^^'z.

Let u = Re[ch]. Since |1 -^^"^l < | e | if |e| < TT, we have I/2P < w on the unit circle, hence I/21^ < w in t/, because I/2 P is subharmonic. Therefore (12)

l/iP + l / 2 p < " + e - ^ " < l

in U; the last inequahty holds because 0 < u <^by our choice of c. Thus F maps U into B2 and F(0) = 0. To show that F satisfies the other hypothesis of Theorem 8.5.6, it is enough to show that //(x) has a finite limit as x / ^ l , since then (1 - \F(x)\^)/(l - x^) is bounded. By (10),

(13)

'''W = ^£(?^l^'^^'de.

Since \e'^ — x\> sin|0/2| if 0 < x < 1, |0| < TT, the dominated convergence theorem leads from (13) to (14) ^ ^

lim h\x) = - - ^ de ^_i ^ 271 J_^ 1 - cos 0

which is finite because £ > 0. By (14) and (11), Um f\(x) is also finite. (Ahern and Clark [1] have proved much more general theorems about derivatives of functions of the form (10).) By (11), fzM/il — xY^^^^'^^ is unbounded as x/" 1. The boundedness assertion concerning (iii) becomes therefore false if^ is replaced by any larger exponent. Finally, we note that the map F defined by (3) furnishes an example in which the function (iii) has no X-Hmit at e^.

Chapter 9

Measures Related to the Ball Algebra

9.1. Introduction This chapter deals with two types of topics. The material of Sections 9.2 and 9.3 is function-theoretic. The measures that are discussed there are intimately related to the holomorphic functions in B. On the other hand, Sections 9.4 and 9.5 describe some measure-theoretic aspects of the theory of function algebras in general. These would not become any simpler by specializing to the ball algebra. Both aspects are used in the proof of the Cole-Range theorem (Section 9.6), which is one of several modern generalizations of the classical theorem of F. and M. Riesz. The results of this chapter will be used in Chapters 10 and 11. Here are some of the relevant definitions. 9.1.1. Function Algebras. Let X be a compact Hausdorff space. A function algebra on X is a. subalgebra A of C(X) which is closed in the sup-norm topology, which contains the constants, and which separates points on X. In the case of greatest interest to us, X will be S, and A will be A(SX the restriction of the ball algebra A(B) to the sphere S. Let us note that A(S) and A{B) are isometrically isomorphic Banach algebras. This is an immediate consequence of the maximum modulus theorem. 9.1.2. Representing Measures. If X is a compact Hausdorff space, the set of all regular complex Borel measures on X will be denoted by M{X). With respect to the total variation norm, M{X) is a Banach space which can be identified with C(X)*, the dual space of C(X). Hence M(X) also has the corresponding weak*-topology. Suppose ^ is a function algebra on X, and /i is a multipHcative linear functional on A,h ^ 0. (I.e., /i is a homomorphism of A onto C.) Since h(l) = 1 and \\h\\ = 1, there is at least one probability measure p e M(X) which represents h, in the sense that (1)

Kf)={

fdp

(fBA).

Jx 185

186

9. Measures Related to the Ball Algebra

The set M^ of all such p is clearly a convex subset of M(X) which is also weak*-compact. In the special case A = A{SX we associate to every z e B the set M^ of all probability measures p e M(S) that "represent z" in the sense that

(2)

f(z)=

[fdp Jss

for every fe A(B) = A(S). For example, if P^O = P(z^ 0 (the invariant Poisson kernel), then P^a e M^.ln particular, a e MQ. There is an interesting difference here between the cases n = 1 and n > l.Whenn = 1, every point of the open unit disc (7 has a wm^we representing measure on the circle T. But when n > 1, every z e B has many representing measures on S. For example, for every C e S we have

(3)

/(O) = ^ J ' fie^'OdQ

(fe A(B)).

Thus there is a p e MQ that is concentrated on the circle {e'^C- —7i 0 is chosen so that P^ > cP^ on S, and p^ e M^. Define

Then p^ e M^ and p^ <^ p^,. We conclude: (a) If p(E) = 0 for every p e M^ then also p(E) = 0 for every p e M^. (b) If p 1 p for every peM^ then also p ± pfor every p e M^. 9.1.4. Annihilating Measures. If ^ c: C(X), and if v e M{X) satisfies j fdv = 0 for every fe A, we write v ± ^ or v G A-^. The members of A-^ are the annihilating measures for A. As regards notation, the letter v will denote annihilating measures, although we used it earlier for Lebesgue measure on C . However, the letter a will continue to stand for the usual rotation-invariant probability measure on S. 9.1.5. Henkin Measures. Suppose/^ G A(B) for / = 1, 2, 3 , . . . , the sequence {fi} is uniformly bounded on 5, and /(z) ^ 0 as i ^ oo, for every z e B. (No

9.2. Valskii's Decomposition

187

convergence is assumed at points of S.) Under these circumstances, {/J is said to be a Mont el sequence, following Pelczynski [1]. By Montel's theorem on normal families, every Montel sequence {fi} converges uniformly to 0 on every compact subset of 5, and the same is true of the derivatives D"/f, for every multi-index a. A measure fi e M(S) is a Henkin measure if lim fi dfi i-^oo Js for every Montel sequence {fi}. These measures were introduced in Henkin's paper [1]. They have also been called L-measures, ^-measures, and analytic measures. Examples of Henkin measures are (i) every v G A(S)^, (ii) every p that represents a point of B, (iii) every fi <^ a. Of these, the first two are quite obvious. To prove (iii), we have to show that ga is a Henkin measure for every g e L^ia). Since the class of all Henkin measures is a norm-closed subspace of M(iS), it suffices to prove (iii) for fi = go, where ^ is a monomial, say g{z) = z'^z^. If {/J is a Montel sequence, then so is {z^^y^z)}, hence D^{z^f^ -»> 0 as z -^ oo. Since

(iii) follows.

9.2. Valskii's Decomposition We shall now prove that every Henkin measure is a sum of the types that we just looked at. 9.2.1. Theorem (Valskii [1]). Iffi is a Henkin measure than there exist v e A(S)^ and g E L^((7) such that fi = v -\- gc. Proof. Let us write A for A{S\ and /I* for the dual of A. Thus A* is (isometrically isomorphic to) the quotient space M{S)IA^. For fi e MiS\ let \\fi\\A* denote its norm as a linear functional on A, and let \\fi\\ be its total variation, as usual. We break the proof into two steps.

188

9. Measures Related to the Ball Algebra

Step 1. If X is a Henkin measure and £ > 0, then there exists h G L^((T) such that ||/i||i < ||/l|| and ||A - h(7\\^* < s. To prove this, put u = P[X], the Poisson integral of >l, and let uXO = u(rC\ as usual, where CeS, 0 < r < 1. We claim that (1)

lim

U-UMA*

= 0.

Since ||M^||I < ||>l||, (1) implies that h = u, has the desired properties if r is close enough to 1. Assume, to reach a contradiction, that (1) fails. Then there exist 5 > 0, ri / 1, and fe A with || yjlU < 1, such that (2)

\\fidXI Js

Js

\f,u,Aa > ^

0 = 1,2,3...).

Since u^. = P[/l]^, Fubini's theorem gives

J\fUrdG= fu, do= \ f.\fdk

(3)

for / e ^ , 0 < r < 1. Thus (2) becomes

(4)

I

[uio-firimdm

> d.

Js But if ^i(^) = fiiz) — fiiriz) for z e 5, then {gi} is a Montel sequence (since {/J is equicontinuous on every compact subset of B\ and the assumption that A is a Henkin measure shows therefore that ^sQidX-^Oasi-^ oo, contradicting (4). Step 2, We now complete the proof of the theorem. Choose Si > 0, so that ^1 > Ili^lL* and Y,T ^i < 00. Put fii = fi and make the induction hypothesis that fc > 1 and that /^^ is a Henkin measure with WU^WA* < ^k-^Y the HahnBanach theorem, this means that Wfi^ - v^H < a^ for some V^GX"^. Apply Step 1 to /ifc — Vfci there is an hj, e L^((7), such that ||^fe|| ^ < Sj, and (5)

ll/^fe - Vfc - hk(T\\A* < ek + i-

Define f^k+i = f^k — ^k — K^^^ and proceed. It follows, for /c = 1, 2, 3 , . . . , that k

(6)

k

l^ = lik+1 + Z ^ i + Z^i^-

9.3. Henkin's Theorem

189

Put 0 = X r h. Then g e L^a), H^IU < ^ f ^i, and k (7)

fi-

00

gcT = //fe+i + XI V,. -

X ^i^-

Since v^ e A^,

(8)

ll/i-^^L*
twhih,

fc+1

The right side of (8) tends to 0 as /c -• oc. Thus fx —go e A^, and the proof is complete. 9.2.2. Remarks, (a) if £ > 0, the £f can be so chosen in the preceding proof that Y,T ^i < ll/^L* + £• The conclusion of the theorem can therefore be strengthened: g G L^ia) can be so chosen that ||^||i < ||)u||^* + £ and fi — ga e A^. (b) The Valskii decomposition is far from unique, since A^ n L^(a) ^ {0}. In fact, i f / e H\B) and /(O) = 0, then /*(T e A-^ n L\a). (c) When n = 1, the F. and M. Riesz theorem asserts that A-^ <= L^(a). In that case, the Henkin measures are thus exactly those that are absolutely continuous with respect to a, the Lebesgue measure on the unit circle.

9.3. Henkin's Theorem At the end of §9.1.5 we saw that the relation fi <4 a impHes that ^ is a Henkin measure. This can be significantly generalized: 9.3.1. Theorem (Henkin [1]). IfX is a Henkin measure and ii < X, then ji is a Henkin measure. Recall that /x <^ A is the same, by definition, as // <^ |A|. By the RadonNikodym theorem, the hypothesis implies therefore that fi = cpX, for some Proof. Since the set of Henkin measures is norm-closed in M(5), it suffices to prove the theorem under the assumption /x = (pX, cpeC^. Valskii's decomposition shows therefore that it is enough to prove the following proposition: IfH e A-^ and cp e C^(B\ then (1)

lim \ft(pdfi i-^oo Js

for every Montel sequence {f}.

=0

190

9. Measures Related to the Ball Algebra

To do this, we use the ToepUtz operator T^ and the related operator P^, defined as in Theorem 6.5.4 by (2)

(7;/)(z)=

\f(CMOC(z,Od<j(0

and

(3)

(v^ fxz) = J noMz) - cp(oic(z, od^icy

By Theorem 6.5.4, V^ maps the unit ball of//°°(5) into a uniformly bounded equicontinuous subset of C(B), Since (2) and (3) imply that (4)

Mz)cp(z) = (T^f,)(z) + (V^Mz)

(z e B)

it follows that T^fi extends continuously to 5. Thus T^fiEA(B). fi e A-^, (4) gives

(5)

Since

jjicpdti= jiV^fddfi.

Return to (3) and appeal once again to the fact that every measure absolutely continuous with respect to o- is a Henkin measure, to conclude that (y - o o . (Theorem 5.6.4.) Hence (j(E) = 0. The theorem asserts much more, namely that p(E) = 0 for every p e MQ. (See §9.1.3.) Proof. Put g = ^(l -\- / ) . Then | ^* | = 1 if and only if ^* = 1, which happens if and only if/* = 1. Pick a representing measure p eMQ. Let A = pl^, the restriction of p to E; i.e., X{X) = p{E n X). Then X
2 = V + /i(T

(v G yl-L, h e

L\G)).

9.4. A General Lebesgue Decomposition

191

If r < 1 then g^ e A, so that (2)

fgrd^= Js

forhda Js

(fe= 1,2,3,...).

The integral on the left of (2) extends only over £, and g*(0 = 1 for C e £. Letting r / * 1, it follows that (3)

p(E) = A(£) =

j(g*fh (g^fh dG

(k = 1, 2, 3 , . . .>

Since \g*\ < 1 a.e. [c], the last integral converges to 0 as /c ^ oo. Thus p(E) = 0. Corollary. Suppose fe H{B\ Re / > 0, and E is the set of all (,eS at which Re f{rQ -> + co asr /" \. Then E is totally null. Proof. Apply the theorem to ^(1 + / ) .

9.4. A General Lebesgue Decomposition 9.4.1. In the present Section, X is a compact Hausdorff space, M{X) = C(X)* is the space of all regular Borel measures on X, and X is a nonempty convex weak*-compact subset of M(X), consisting of probabihty measures. The object is to obtain a "Lebesgue decomposition" of an arbitrary p e M(X) relative to K. When X is a singleton, say K = {p}, then every peM(X) decomposes into p = Pa -^ ^s^ where p^ < p, and p^ is concentrated on a set E with p{E) = 0. This is the ordinary Lebesgue decomposition of p relative to p. There are at least two ways in which one can try to extend this to larger sets K. We shall describe these before proving anything. Glicksberg [1] used the following approach: Among all Borel sets £ <= X that are K-null (this means, by definition, that p(E) = 0 for every pe K) find one, say if, that maximizes \p\(E\ put Ps = P\H ^^^ t^a = f^ ~ f^sThen p^ is concentrated on a set that is K-null, and \pa\(E) = 0 for every E that is K-nuW. But it is not at all clear whether p^ is absolutely continuous with respect to any p e K. Konig and Seever [1] attacked the problem from the opposite direction: For every peK, let p = p^ -\- p'^ be the Lebesgue decomposition of p relative to p, with Pp <^ p, p^ 1 p. Find PQE K for which \\pp\\ is maximal. Let Pa be this p^^ and put p^ = p — p^. Then p^ < Pp^ and Ps -^ p for every p G K, but it is not clear whether p^ is K-singular in the strong sense of being concentrated on a set that is K-nuU.

192

9. Measures Related to the Ball Algebra

Rainwater [1] proved that this is in fact the case, thereby showing that the two decompositions are actually the same. Theorem 9.4.4 summarizes this line of development. Rainwater's lemma 9.4.3 depends on Glickberg's version of von Neumann's minimax theorem. The main point of the simple proof given by Glicksberg [1] is that it requires only one of the two convex sets to be compact. This is Theorem 9.4.2. When F is a function of two variables, we define F^ and F^ in the customary way by FM

= Fix,y) = Fy(x).

The vector spaces that occur in the minimax theorem are understood to have real scalars. 9.4.2. The Minimax Theorem. Suppose (i) G is a convex subset of some vector space, (ii) K is a compact convex subset of some topological vector space, and (iii) F:G x K ^ R satisfies (a) F^ is convex on Gfor every ye K, (b) F^ is concave and continuous on Kfor every xeG. Then sup inf F(x, y) = inf sup F(x, y). yeK

xeG

xeG

yeK

Proof Let a = sup inf, jS = inf sup. Since inf/(x, yo) ^ / ( ^ o , ^o) ^ sup/(xo, y) y

for all (XQ , yo) e G X K, we have OL < ^Ai P = — oo there is nothing left to prove. Since K is compact, jS < oo. So we have to consider the case — oo < j5 < 00. Replacement of F by F — ^ shows that the assumption p = 0 involves no loss of generaHty. Let H be the convex hull of the set {F^: xeG} a CR(K\ the space of all real-valued continuous functions on K. We claim that sup{h(y): yeK}>0 for every heH. Let h = ^A^i^xi^ where x,- e G, ti > 0, ^ ^f = 1- Put x' = Y, hx^. Then x' eG and F{x\ y')> ^ = 0 for some y' e K. The convexity of F^' shows therefore that

o
= h(yy

We conclude from this that the convex set H is disjoint from the (open) negative cone in CR(K). By the Hahn-Banach separation theorem and the

9.4. A General Lebesgue Decomposition

193

Riesz representation theorem there is a probabihty measure fi on K such that ^Khdfi>0 for every he H. Put yo = jx 3^ d^(y)' Then y^ G K, since K is compact and convex. (The simple facts about vector-valued integrals that are needed here may be found in Theorem 3.28 of Rudin [2]; the point yo is often called the barycenter of fi.) For every xeG.F^ is concave and F^ e H. It follows that

F(x,yo)>

JK

{F(x,y)dfi{y)>0,

Hence a > 0 = )S, and the proof is complete. Note: The function F(x, y) = y/(x + y) on (0, oo) x (0, oo) shows that the compactness of K cannot be omitted from the hypotheses. 9.4.3. Rainwater's Lemma. Let X and K be as in §9.4.1. Suppose v e M(X), and V J_ pfor every pe K. Then v is concentrated on a set E a X, of type F„, such that p(E) = Ofor every pe K. (Recall that a set is of type F^ if it is a union of countably many closed sets.) Proof. Replacing v by | v |, we see that we may assume v > 0, without loss of generality. Let G be the set of all continuous functions taking X into [0, 1]. Define F on G x K by (1)

F(K p)=

[hdv+ Jx

f (1 - h)dp. Jx

By the definition of the weak*-topology in M(X), F is a continuous function of p, for every fixed h. Since F is affine in each variable, F satisfies the hypotheses of the minimax theorem. Now fix pe K and recall that v _L p. This implies that there are disjoint compact sets in X, one of which carries most of the mass of p, whereas the other carries most of the mass of v. Urysohn's lemma provides therefore an heOfor which both integrals in (1) are very small. In other words, (2)

inf F(h, p) = 0 heG

for every p e K,so that (3)

inf sup FQi, p) = 0 heG

peG

194

9. Measures Related to the Ball Algebra

by the minimax theorem. By (3), there are functions hie G (i = 1, 2, 3,...) such that (4)

{ hidv-\- f (1

-hi)dp<2-'

for every p e K. Put Qo = Yj^i^Oi ~ Z(^ — ^i)' ^^^ ^^^ ^ ^^ ^^^ ^^^ where QQ < oo. Since ^0 is lower semi-continuous, £ is of type F„. By (4) QQ e L^(v), so that v is concentrated on E. But (4) shows also that g^e L^{p) for every p E K; since ^1 = 00 on £, p{E) = 0. This proves the lemma. The initials that name the following Theorem refer to Glicksberg, Konig, and Seever; see §9.4.1. 9.4.4. The GKS Decomposition Theorem. IfK is a weak*-compact convex nonempty set of regular Borel probability measures on a compact Hausdorff space X, then every p e M(X) has a unique decomposition

in which p^ ^ Po fa^ some pQ e X, and p^ ^^ concentrated on a set of type F„ that is K-null. Proof The uniqueness of such a decomposition is trivial. To prove existence, we begin by associating to every p G K the Lebesgue decomposition of p with respect to p. Thus p = p^ + p'^, where p^ ^ p, p^ 1. p. Put (1)

t=

sup{\\pJ:peK}.

Choose ti < t2 < "',so that t^ -^ t as i -^ oo. There exist ptE K (i = 1, 2, 3, ...) with \\pp.\\ > ti, and there are Borel sets A^ (^ X such that pp. = /x|^.. Put A = \Ji Ai, and define Pa = I^IASetting PQ = Yj ^'Pt^ the convexity and compactness of K imply that Po G K. Suppose E cz X satisfies Po{E) = 0. Then pf(E) = 0 for all i, hence also \pp.\(E) — 0, or \p\(E n Ai) = 0. Taking the union over all i, it follows that \p\(E n A) = 0, or |jU„|(£) = 0. This proves that p^ <^ p^. Note also that A ^ A^, so that Wp^W > \\Pp.\\ > t^ for all I Consequently, Finally, put p^ = p — p^ = p |^y^. Pick p EK. Then (2)

Pa + {lisX < i(Po + P) e X,

so that \\pa -\- (ps)p\\ < t. Since p^ and (ps)p are concentrated on A and X\A, respectively, the norm of their sum is the sum of their norms, and since

9.5. A General F. and M. Riesz Theorem

195

ll/^flll = t, we conclude that ||(iuj^|| = 0. Thus fi^ J- p, for every p e K, and an appeal to Rainwater's lemma shows that fi^ is concentrated on a set of type F„ that is X-null.

9.5. A General F. and M. Riesz Theorem 9.5.1. In this Section, X is a compact Hausdorff space, as before, A is a function algebra on X, O is a multipHcative linear functional on A, and M^ is the set of all probability ipeasures p e M(X) that represent O. As was pointed out in §9.1.2, M^ is nonempty, convex, and weak*-compact. Hence M^ can play the role of K in Theorem 9.4.4. The main result (Theorem 9.5.6) states that if v e A^ and if v = v^ + v^ is its GKS decomposition relative to M^, then both v^ and v^ He in A-^. (In the original version of the F. and M. Riesz theorem, A was the disc algebra on the unit circle, and the conclusion was that v^ = 0, i.e. that V <^ (7.)

9.5.2. Definition. I f / G ^ and w = Re /, we define Ow = Re Q>f. Then (1)

Q>u = \ udp Jx

for every p e M^, simply because every such p is a real measure. The set of all real parts of members of ^ will be denoted by Re A. Clearly, Re A is a subspace of

CR(X).

Observe that if p G M(X) is a probability measure that satisfies (1) for every w G Re ^4, then p e M^. For iffeA, then (1) holds for the real parts of / and of // 9.5.3. If ^ G Cj^(X), ueRe A, u > g, and p e M^, then obviously (1)

Jx

dp < Ow.

Hence the supremum of the left side, over all p, is at most equal to the infimum of the right side, over all ueRe A such that u > g. The following lemma asserts that equahty actually holds. 9.5.4. Lemma. If g e CR(X) then there exists po G M^ such that (1)

g dpo = infjOw: w > ^, w G Re A}. Jx

196

9. Measures Related to the Ball Algebra

Proof. For h e (2)

CR(XI

define p(h) = inf{OM: M > fc, w e Re A}.

Then pQi^ 4- /12) < p(hi) -\- K/iiX and p(th) = tpQi) for scalars t >0. In particular, 0 = p(0) < p(h) + pi — h), hence — K^) < p{ — h\ so that fp(/i) < p(th) for all real scalars t and all /i e C^{X), If we define A(fgf) = tp{g\ it follows that A is a linear functional on the one-dimensional space spanned by g, and that A < p there. One of the standard versions of the HahnBanach theorem (see, for example, Theorem 3.2 in Rudin [2]) asserts now that A extends to a linear functional on CR{X) that satisfies (3)

-p(-h)
for every h e CR(X). Since \p(h)\ < ||/i||oo, A is continuous, ||A|| < 1. When ^ G Re .4, then (2) gives p(h) = ^h, so that (3) gives Ah = ^h. Thus A is a norm-preserving linear extension of . Consequently, there is a po ^ ^o such that (4)

A/i= ihdpo Jx

(heC^iX)).

Since Ag = p(g), the proof is complete. 9.5.5. Lemma (ForelH [1]). Suppose E a X is a set of type F^, and p(E) = 0 for every p e M^. Then there is a sequence {/„} in A, with \\f^\\ < I for all m, such that (i) Um„_oo /mW = ^fo^ every xe E, but (ii) lim^-.oo /mW = 1 a.e. Ip^for every peM^. Proof By assumption, £ = U £^, where each £^ is compact, and £^ ci £;^+j. Fix m, for the moment. By the minimax theorem, (1)

inf sup I hdp = sup inf h

p

Jx

p

h Jx

h dp,

where p ranges over M^ and h ranges over all h G CR(X) that satisfy h > m on E^,h>0 on X. Since p(Ef„) = 0 for every p, the right side of (1) is 0. Hence (1) shows that for some ft = /z^ in our class, (2)

\h„dp<^

9.5. A General F. and M. Riesz Theorem

197

for every p e M^. Lemma 9.5.4 (with h^ in place of g) implies therefore that there exists M^ e Re ^ such that u^>monE^,u^>()on X, and Ow^ < 1/m^. Now choose g^e A so that u^ = Re g^ and O^^ is real, and put f^ = exp(-0. Then \fj = e x p ( - M j < 1 on X. On £^, l/^l < e"'", which proves (i). If p G M^ then (3)

\ fmdp = ^fm = e x p ( - % J = e x p ( - 0 i i j , Jx

and since Ow^ < 1/m^, (4)

£(l-/J.p^l-exp(-i,)
Consequently, ^ ^ Re(l — fm) < oo a.e. [p]. Thus Re / ^ ^ 1 a.e. [p], and since | / ^ | < 1, it follows that/^ -^ 1 a.e. [p]. This proves (ii). We are now ready for the Glicksberg-Konig-Seever generalization of the F. and M. Riesz theorem. (Part 3 of GHcksberg [2], Chapter II.7 of Gamelin [1], and Section 23 of Stout [1] contain more information on this topic.) 9.5.6. Theorem. Let (^ be a multiplicative linear functional on a function algebra A on X, let M^ be the set of representing measures for O, let v G A ^ , and let V = v« + V,

be the GKS-decomposition ofv relative to M^. Then v^ G A ^ and v^ G A-^. Proof By Theorem 9.4.4, v^ is concentrated on a set £ c: X, of type F„, such that p(E) = 0 for every p ^ M ^ , and v^ is the restriction of v to X\E. Associate {/^} to £, as in Lemma 9.5.5. For any fe A, we have ff^ e A, hence ^xffm dv = 0, or

(ffmdvs + fff.ndv„ = 0.

Jx

J)i

The first of these integrals tends to 0 as m -^ oo, since/^ -^ 0 at every point of E. By Theorem 9.4.4, v^ <^ Po for some po G M ^ . Since/„ ->• 1 a.e. [po], the second integral converges to jx fdva as m ^ oo. Thus j /^v^ = 0, v^E A-^, and the same is then true of v^ = v — v^.

198

9. Measures Related to the Ball Algebra

9.6. The Cole-Range Theorem We return now to the ball algebra A = A(B) which, as noted in §9.1.1, can be regarded as a function algebra on S. As in §9.1.2, we let MQ denote the set of all probabihty measures p e M(S) that represent the evaluation functional at the origin of B, in the sense that /(O) =

L'

fdp

for every/e A(B). 9.6.1. Theorem (Cole-Range [1]). Every Henkin measure fi e M(S) is absolutely continuous with respect to some p e MQ . Proof. Let p = p^ + fi^ he the GKS-decomposition of p with respect to MQ (Theorem 9.4.4): p^ is concentrated on a Borel set that is totally null (see §9.1.3) and Pa< P for some p e MQ. We shall prove that p^ = 0. Let h be any bounded Borel function on S. Since hp^ < p, Theorem 9.3.1 asserts that hp^ is a Henkin measure, so that hp^ has a Valskii decomposition hps = v-^ go where \ e A-^ and g e L^ia). Since hp^ is concentrated on a set that is totally null, the uniqueness part of Theorem 9.4.4 shows that V = -g<j

+

hp,

is the GKS decomposition of v with respect to MQ . Thus hp^ = v^ E A ^ , by Theorem 9.5.6. In particular, ^ hdp^ = 0. The arbitrariness of h implies now that p^ = 0, and the proof is complete. 9.6.2. Remark. It is a corollary of Theorem 9.6.1 that every pe A^ satisfies p <^ piov some p G MQ . This makes it clear that the Cole-Range theorem contains the original F. and M. Riesz theorem, since in the original setting MQ had only one member, namely Lebesgue measure on the unit circle.

9.7. Pluriharmonic Majorants 9.7.1. In this section we return to Lumer's Hardy spaces (LH)^(J5) that were discussed in Section 7.4. The reason is that Lemma 9.5.4 makes it possible to express the norms defined in §7.4.3 in terms of representing measures (Theorem 9.7.4), and this leads to further information about these norms.

9.7. Pluriharmonic Majorants

199

As regards notation, recall that the real pluriharmonic functions in B are exactly the real parts of holomorphic functions (Theorem 4.4.9). We therefore denote them by RP(5), and we define (1)

a(^) = inf{w(0): u>g,uE

RP(B)}

if ^ is a real function with domain B, and (2)

Pig) = inf(M(0): M > ^, M G Re A(B)}

if g is bounded above on B, In terms of this notation, the norm |||/|||p of a function/G(LH)^(J5), as defined in §7.4.3, satisfies (3)

lll/lll? = a(|/|'').

Here 0 < p < oo. W h e n / e A(BX then a can be replaced by P in (3): 9.7.2. Lemma. Iffe

A(B) then |||/|||? =

P(\fn

Proof, Pick 8 > 0. Since | / 1 ' ' is uniformly continuous on 5, there exists r < 1 such that ii/r-i/.n<£ on B. The definition of |||/|||p shows that there is a w G R P ( B ) , such that W(0) < III/111^ + 8. Put M = e + w,. Then u € Re A{BX u>s

+

\fA''>\ff

on B, and u(0) = e + w(0) < |||/|||^ + 2e. Hence i ? ( i / n < Ill/Ill?. Since a < j5 is trivial, the lemma follows from 9.7.1(3). 9.7.3. Lemma. Iffe

(LH)^(B) then

(1)

lim IIIXIII, = Ill/It = sup |||/,|||,.

w>\ff,

200

9. Measures Related to the Ball Algebra

Note that the first of these equaUties holds although |||/ —Xlllp need not tend to 0 as r -^ 1 (Theorem 7.4.6). The second equahty in (1) impHes that III X III p is a nondecreasing function of r. Proof. If w > l/l^ then u, > |/,|^ hence

(2)

\\\fr\K ^ uxo) = u(oy

Taking the infimum over all such u e RP(B) gives (3)

|||/,|||p < lll/lllp

(0 < r < 1).

Next, associate to each r < 1 a function i/^''^ e RP(B) such that (4)

u^''>\fr\'

and

u^'KO)<\\\fr\K^Sr,

where 8 ^ \ 0 as r / ^ 1. Since u^"^ > 0 and {w^''^(0)} is bounded, {u^''^} is equicontinuous on every compact subset of B. Some subsequence of {w^''^} converges therefore, uniformly on compact subsets of B, to a function u E RF(B) which satisfies u > \ff and

(5)

lll/IIU
by (4). Now (1) follows from (3) and (5). 9.7.4. Theorem. IfO
oo andfe(LUy(B\

Ill/Ill? = s u p | j j / , r dp:0
then MX

Here Mo is the set of all representing measure corresponding to the origin of B, as in Section 9.6. The fact that |||/|||p is the supremum of a large family of L^-norms may account for some of the pathology of the spaces (LH)^(B). Proof. For g e (2)

CR(S),

a special case of Lemma 9.5.4 asserts that

supj \gdp:pEMQV

= inf {w(0): u>g

If r < 1 then/, e A(B), so that

(3)

\\\fr¥p = P(\frn

on S, w e Re A(B)}.

9.7. Pluriharmonic Majorants

201

by Lemma 9.7.2. If u e Re A(B) satisfies M > | /^^ on S, then the same inequahty holds in B, since | X T is subharmonic. Hence it follows from (2), (3), and the definition of j^ in 9.7.1(2) that (4)

;illj = ssuupp|jjJjj//,,|r ^ p : p G M o lil/.IIIJ

If we now take the supremum over r in (4), the theorem follows from Lemma 9.7.3. 9.7.5. The "obvious" members of MQ are the circular probabihty measures p e M(S). By definition, these satisfy (1)

\vie''Odp(0=

\vdp

for every v e C(S) and for every real 6. By Fubini's theorem, (1) imphes that (2)

jv dp = [dpiO • ~ f v(e">OdQ.

W i t h / 6 A(B) in place of f, the fact that the slice functions/^ are in A(U) shows therefore that j fdp — /(O). Thus p e MQ, as asserted. If fe H(B) and v is replaced by | / J ^ in (2), consideration of the slice functions shows that (3)

Js

flfrl'dp

is a nondecreasing function of r. But if p G MQ is not circular, then (3) may fail to be a monotonic function of r, for certain/, in spite of the monotonicity of |||/|||p (Lemma 9.7.3) and Theorem 9.7.4. Theorem 9.7.6 will show this. To construct some noncircular p e MQ , take n = 2, for simplicity, and let T be any probability measure on [7 c= C that satisfies (4)

f9dT = g(0)

for every g e A(Uy For example, T might be concentrated on a simple closed curve F in U that surrounds the origin, in such a way that T solves the Dirichlet problem at 0 relative to the domain bounded by F. The measure p that satisfies

(5)

jvdp = jdTiz). ^ I" viz, e ^ ' ^ y r ^)dd

202

9. Measures Related to the Ball Algebra

for every v e C(S) belongs then to MQ . To see this, simply note that the inner integral on the right side of (5), with v replaced b y / e A(B\ equals/(z, 0). The support of this p is the set of all (z, w)e S for which z lies in the support of T. 9.7.6. Theorem. Put f(z, w) = (1 — z)w in C^. Then there exists p e MQ such that Js | / . T ^P i^ ^ot a monotonic function ofr in [0, l],/or any p e (0, oo). Proof. Let / be the interval [|, ^] on the real axis in C, let Q = U\I, and let T be the probability measure on ^Q = / u T that satisfies (1)

hdT = h(0) Jdi Jen

for every h e C(Q) that is harmonic in Q. Use this T to define p as in 9.7.5(5). If we replace y by | X1^ in that formula, the inner integral is (2)

|r-r^z|^(l -

\z\y^\

This vanishes on T, where \z\ = 1. Hence (3)

J \fr\' dp = j(r - r'xni

- xy^' dz(x).

The integral on the right is 0 when r = 0. It is positive for every r e (0, 1]. It decreases as r increases from f to 1, since the integrand is then a decreasing function of r, for every xe I.

9.8. The Dual Space of ^(5) 9.8.1. The dual X* of a given Banach space X can often be described in several ways. For instance, A(B) is isometrically isomorphic to A(S\ a closed subspace of C(S). Since C(5)* = M(S), the space of complex Borel measures on 5, standard duality theory gives one description of A{B)'^ as a quotient space, namely (1)

A(B)* = M(S)/A\

(In this context, equality is understood to mean isometric isomorphism.) Here is another description: Let HM be the space of all Henkin measures on S, and let TS be the space of all totally singular ones. If we combine the theorems of Henkin and ColeRange, we see that HM consists of precisely those measures that are abso-

9.8. The Dual Space of A(B)

203

lutely continuous with respect to some representing measure p e MQ . The GKS decomposition theorem (with respect to MQ) says therefore that there is a direct sum decomposition (2)

M(S) = HM@ TS.

Observe next that A^ c= HM. Thus (1) can be replaced by (3)

A(B)* = (HM/A^) @ TS.

By Valskii's theorem, HM = L^(a) + A^. (Note that this sum is not direct, since L^(o-) n A-^ is far from being {0}.) The first summand on the right of (3) is thus the image of L^((T) under the quotient map with null space A-^. In particular, HM/A^ is separable. If/i G r S and A <^ /i then (trivially) A e TS. Thus TS is what Kakutani [1] has called an (L)-space. (Further bibliographic information on this topic may be found in Dunford-Schwartz [1], pp. 394-395.) Let us summarize these observations: 9.8.2. Theorem. A(B)* is the direct sum of an (Lyspace and a separable Banach space. Pelczynski [1; Theorem 11.5] used this information about .4(B)* to prove the following: Ifk > 2 and A(U^) is the polydisc algebra in k variables, then A(U^)'^ is not isomorphic to any closed subspace ofA(BJ*. Here "isomorphic" means: linearly homeomorphic. The proof uses more Banach space theory than can be presented here. We therefore omit it. But let us state the dual formulation of the result: 9.8.3. Theorem. If n> \ and /c > 2, then A{U^) is not isomorphic to any quotient space of A(B^. In other words, there is no continuous linear mapping of any A{B^ onioA{U^)'\{k> 2. Henkin [1] proved earher that A{Bj) and A{U^) are not isomorphic as Banach spaces. In fact, it was apparently for this purpose that he introduced his class of measures.

Chapter 10

Interpolation Sets for the Ball Algebra

10.1. Some Equivalences 10.1.1. Definitions. We shall be concerned with compact sets K c= S. For convenience, we introduce six labels, (Z), (P), (/), PI), (AT), and (TN), to denote certain properties that K may or may not have in relation to the ball algebra AiB). X is a (Z)-set (zero set) if there is a n / e A(B) such that/(C) = 0 for every C e X and/(z) ^ 0 for every z e B\K. X is a (P)-set (peak set) if there is a n / e A(B) such that/(C) = 1 for every Ce K and | f(z) | < 1 for every z e B\K. (Such a n / i s said to peak on K,) K is an (/)-set (interpolation set) if every complex continuous function on K extends to a member of A(B). X is a (P/)-set (peaknnterpolation set) if the following is true: to every g E C(K) (g ^ 0) corresponds a n / e A(B) such that / ( O = g(Q for every CG iCand|/(z)| < ||^|||j^ for every z e S\iC. (Here ||gf|||5: denotes the maximum

of 1^(01 on X.)

K is an (iV)-set if X is a null set for every v e A^. More exphcitly, the requirement is that |v|(X) = 0 whenever v e M(S) satisfies j fdv = 0 for every/G A(B). X is a (TN)-SQt if K is totally null, i.e., if p(K) = 0 for every representing measure p G MQ. (See §9.1.2.) These six properties turn out to be equivalent: 10.1.2. Theorem. / / a compact set K (^ S has one of the six properties (Z), (P), (/), (P/), (N), (TN), then it has the other five. This will be proved in accordance with the following diagram, in which single arrows indicate easy implications, whereas double arrows indicate substantial theorems.

204

10.1. Some Equivalences

205

It is trivial that (PI) implies (/) and (Z). Suppose/e A(B) has K as its zero set in B, and | / | < 1. Since B\K is simply connected, / = exp g for some g that is continuous on B\K and holomorphic in B. Clearly, Re ^ < 0 and Re g(z) -> — oo as z approaches any point of K. Hence h = g/(g — 1) is in A(B) and peaks on K. This shows that (Z) -^ (P). If/6yl(B) peaks on X, and p e M o , then/'"(O) = j / ' " ^p for m = 1, 2, 3, The integral converges to p(K) as m ^ oo, by the dominated convergence theorem, and/'^CO) ^ 0, since | / ( 0 ) | < 1. Thus p(K) = 0. This shows that (P) -> (TAT). That (TAT) =^ (N) follows from the Cole-Range theorem: Suppose K is totally null, and v e A^. Then v is a Henkin measure, hence v <^ p for some PEMQ. Since p(K) = 0, it follows that |v|(X) = 0. Finally, the implications (/) =^ (N) => (PI) are special cases of the theorems of Varopoulos and Bishop which will be proved in Sections 10.2 and 10.3. 10.1.3. Some Background. The case n = 1 of Theorem 10.1.2 (dealing with the disc algebra and with compact sets K on the unit circle) represents theorems that were proved over a rather long span of time. [When n — 1, (TN) is simply the property of having Lebesgue measure 0. We shall nevertheless continue to write (TN\ for the sake of uniformity and brevity.] In his thesis, Fatou [1] showed that (TN) imphes (Z). The brothers F. and M. Riesz [1], in their only joint paper, proved that (TN) implies (N). That (TN) also imphes (P) is implicit in their proof. About 40 years later, Carleson [2] and Rudin [4] showed, independently, that (TN) implies (/). Since it is easy to see that no set of positive Lebesgue measure has any of the properties (Z), (N), (I), these three were thus known to be equivalent in the disc algebra context, but only because each of them was known to be equivalent to (TN). Bishop [1] threw an entirely new light on this subject by proving that (N) implies (PI) in very general situations that have nothing to do with holomorphic functions or with (TN). This theorem occupies Section 10.3. In the polydisc setting (where K a T") it is known that (Z), (P), (/), (PI), and (N) are equivalent. Proofs may be found in Chap. 6 of Rudin (1) and in §21 of Stout [1]. Valskii [1] proved part of Theorem 10.1.2, namely the equivalence of (PI), (Z), (P), and (N), for smoothly bounded, star-shaped, strictly pseudoconvex domains in C . Chollet [1] showed that Theorem 10.1.2 holds in arbitrary strictly pseudoconvex domains (not necessarily simply connected) by giving a different proof of (Z) -^ (P). 10.1.4. Among the implications marked in the diagram that follows Theorem 10.1.2 there is only one, namely (TN) => (N), whose proof involves holomorphic functions in a significant way. The others (except for (Z) -> (P), where holomorphic functions do play a small role) are function algebra

206

10. Interpolation Sets for the Ball Algebra

theorems, and it is reasonable to ask questions such as the following: Does (P) imply (N) in every function algebra? Do (P) and (/) always imply each other? Here is a simple example that answers these negatively. LetX = K^U 1^2 be the following compact subset of C x i^:Ki consists of all points (z, 0) with \z\ < 1, K2 consists of all points (0, t) with 0 < t < 1. Thus X looks hke a thumbtack, and K^ can be identified with the closed unit disc in C. Let A consist of all/G C(X) whose restriction to K^ is holomorphic in the interior ofK^. Then A is a. function algebra on X (and, in fact, X is the maximal ideal space of A). Clearly, K^ is a (P)-set of A which is neither (/) nor (A^). (Note that X^ is a (P)-set which contains compact sets that are not (P)-sets!) On the other hand, K2 is an (/)-set for A which is neither (P) nor (AT). In view of Varopoulos' theorem (Section 10.2), it should be mentioned that only one point of K2, namely (0, 0), fails to be a peak point for A. That K2 is not (P) is due to the maximum modulus theorem. To see that K2 is not (AT), let V be the sum of normalized Lebesgue measure on the boundary of K^ and the point measure of mass — 1 at (0, 0) G K^ n X2 • Both Ki and K2 are (Z)-sets for A. 10.1.5. Consequences of Theorem 10.1.2. Properties (AT) and (TAT) are obviously hereditary (if K has one of them, so does every compact K^ cz K\ and they are preserved under the formation of countable unions. Since this is not so obvious for the other properties under consideration. Theorem 10.1.2 leads to some nontrivial conclusions, such as the following. (a) If a compact K cz S is the union of countahly many (Piysets, then K is a (Piyset. (b) If K c: Sis a (Z)-set, then every compact K^ a K is a (Z)-set. (c) For explicitly described sets K, properties (P) or (Z) are often the ones that are most easily verified. For example, let K be the set of all z E S = dB„ all of whose coordinates are real. Thus K is an (n — l)-sphere. Define

g(z) = zl + --' + zl Then |(1 -j- g) peaks on K. Conclusion: K is totally null, X is a (P/)-set, etc. (d) Here is an elaboration of (c): Let £ be a compact subset of the unit circle, of Lebesgue measure 0, let g be as in (c), and put K =

Sng-\E).

The case E = {1} occurred in (c). Since E has measure 0, £ is a peak-set for A(U). Let he A(U) peak on E. Then ho g e A{B) and hog peaks on K. If E is a Cantor set of measure 0, we conclude that there exist "Cantor sets of (n — l)-spheres" in S that are (P/)-sets.

10.2. A Theorem of Varopoulos

207

(e) Let E be as in (d), but this time define g(z) =

n"^^z,'"Z„.

The set of all z G 5 where \g(z)\ = 1 consists of the n-torus defined by

For any a 6 £, g~^((x) n S is thus an (n — l)-torus in S. Setting K = S n g~^(E\ it follows, as in (d), that there exist "Cantor sets of (n - l)-tori" in S that are (P/)-sets.

10.2. A Theorem of Varopoulos 10.2.1. The original proof of this theorem, as given by Varopoulos [1], relied on the fact that every compact Hausdorff space can be homeomorphically embedded in some (large) compact abelian group, and used some harmonic analysis on such groups. Glicksberg [2; p. 9] simplified the proof a great deal: the only group that appears is a finite cychc one, and no harmonic analysis is needed beyond the ability to sum a finite geometric progression. 10.2.2. Theorem. Suppose (i) (ii) (Hi) (iv)

A is a function algebra on a compact Hausdorff space X, K cz X is a compact interpolation set for A, every point ofK is a peak-point for A, veM{X)andveA'-.

Then |v|(K) = 0. Proof Fix e > 0. There is an open set Q c= X such that K cz Q and IVI {Q\K) < s. By (iii) there corresponds to every point x e K a, function fx^ A such that f^(x) = 1 but | f^iy) | < 1 for every y ^ x,y e X, We can replaceXc by/^, for some large integer m, so that | /^ | < s outside Q. The sets where \1 —fj < s form an open cover of K. Hence (writing f in place of fx) there are finitely many functions/i,... ,/v 6 A, with WfW = 1, such that I /• I < e outside Q, and such that the sets (1) cover K.

E, = { | 1 - / , ! < £ }

(i=l,...,iV)

208

10. Interpolation Sets for the Ball Algebra

Next, there are pairwise disjoint compact sets Ki (^ E^n K such that

("\y K}J |v|(Q\yxJ<2£.

(2)

To say that K is an interpolation set for A is the same as to say that the bounded hnear operator that assigns to every/e A its restriction to K has all of C(K) as its range. The open mapping theorem for Banach spaces impHes therefore that there is a constant a < oo with the following property: every g e C{K) has an extensionfe A such that \\f\\x < a||^||xBriefly: K is an (/)-set with constant a. Since X^ u • • • u K^y cz K, it follows from Tietze's extension theorem that Ki u • • • u K^ is also an (/)-set with constant a. Now put (o = Qxp(2ni/N). Since the sets K^ are pair wise disjoint, the preceding remarks show that there are functions Qr^ A, 1 < r < N, with ll^.llx < a, such that (3)

gXx) = (D^' if

xeK,.

Put (4)

h,ix) = N-'Y.

o^^'gXx)

(XEX,1
and

(5)

H = £ hlf,.

Then, obviously, H e A.lfxe

(6)

K^, (3) and (4) give

V^) = ^"'i«"^"''* = <5p. r=l

where (5^^ = 1 if p = s, ^^^ = 0 if p 7^ s. Hence H(x) =fsix) on K, a E^, so that (7)

\1-H(x)\<s

on

K.W'uKj,.

Next, for any xe X, X \h,(x)\' = N-'Y.9r(x)gs(x)T.^~''oj^' P

r,s

p

(The indices p, r, s run from 1 to N) Hence (8)

\\H\\x < a^

= N-' I \gXx)\' < a\ r

10.3. A Theorem of Bishop

209

and (9)

|if(x)|
if

xeX\Q.

Let us write H^ for H, and let e -^ 0 through the sequence 1/n^, for example. By (7), (8), (9), we obtain a uniformly bounded sequence {H^} in A which converges pointwise, a.e. [|v|], to the characteristic function of K, by (2). Since v e A-^ and H^ e A, it follows, by the dominated convergence theorem, that v(K) = 0. Finally, the hypotheses (ii) and (iii) hold for every compact K^ cz K. Thus v(Xi) = 0 for all such K^, and therefore |v|(X) = 0. 10.2.3. Every ( e 5 is a peak point for A(B). The function

shows this. Consequently, Theorem 10.2.2 gives the impHcation (/) => (N) in Theorem 10.1.2.

10.3. A Theorem of Bishop This will complete the proof of Theorem 10.2.1, by estabHshing the implication (N) => (PI), 10.3.1. Theorem (Bishop [1]). Suppose (a) X is a compact Hausdorff space, (b) A is a closed linear subspace ofC(X), (c) K is a compact subset ofX, of type G^, such that \ v | (K) — Ofor every V 6 M(X) that annihilates A. Then K is a peak-interpolation set for A. Note that A does not have to be an algebra. The assumption that K is of type Gs is of course satisfied by every compact X if X is metric; it will only play a role in the last of the three steps into which we spHt the proof. Step l.Ifge \f\
C(K\ \g\ < 1, then there exists fe A such thatf=

g on K and

Proof Let 11^ and Uc be the open unit balls in A and C(X), respectively. Let R be the restriction map that takes A into C(K). Then R(UA) is a convex balanced subset of (7cAssume, to reach a contradiction, that R(U^) is not dense in Uc- By the Hahn-Banach separation theorem, there is then a measure /i e M(X), concentrated on K, and a constant rj < 1, such that \^K^ dn\ < rj for every G e U^, whereas \j^F d^ = 1 for some F e UQ.

210

10. Interpolation Sets for the Ball Algebra

Another application of the Hahn-Banach theorem shows that the hnear functional G -^ j ^ ^ ^n, whose norm is
\.

JK

This contraction shows that R(UA) is dense in 17^ If how g e C(K) and |gf |||^ < r < 1, it follows that there is anf^e A with WfiWx < r and \\g -f^W^ < Kl - r). Next, there is an /^ G ^ ||/2||x < i ( l - r), and \\g - f, -MK < i ( l - r\ etc. I f / = I/-, t h e n / = g on K, / G A, and

||/||;, OonX,and \g\ < bonK. Then there exists fe A such thatf = g on K and \f\ < b on X. Proof. Let A^ = {f/b:feA}, and apply Step 1 to g/b, with A^ in place of A. (Note that ^ ± A if and only ifbfi 1 A^, and that |//| and \bfi\ vanish on the same sets.) Step 3. We now prove the theorem. There are open sets V^ :D V2 ^ V^ -> -" so that K= nVi. Put ri = l'''^. Choose g e C{K\ \\g\\j^ - 1. We will construct functions/^ G A such that (1) (2) (3) (4)

/, = (1 - 8r,)^ on X, |/,|
Put/i = 0. Then (1) and (2) hold when i = 1. Assume i > 1 and/- G A satisfies (1) and (2). By (1), there is an open set W,KczW cuVi, such that |/,| < 1 - 7r, in W. Choose b e C(X) so that b = Sri on K, b = r^ off W, and 0 < ^ < 6rj on X. By Step 2 there is an hiE A such that (a) hi = Ar^g = (8r^ - Sr^^ i)g on K, ip) \h,\<6r,onX, (y) \hi\
10.4. The Davie-0ksendal Theorem

211

P u t / = limy;-. By ( 3 ) , / G A By ( 1 ) , / = g on K,lf x^K such that x^ Vm.so that \f(x)\

< \Ux)\

+ f |/z,(x)| < 1 - 2r, + tr, m

there exists m

= 1,

m

since \hi\ < r^ outside Vi <^ V^. Thus | / ( x ) | < 1 outside K, and peakinterpolation is estabhshed. 10.3.2. To prove the implication (N) =^ (PI) in Theorem 10.1.2 we apply Theorem 10.3.1 with X = S. If X satisfies (N) and g e C(K\ g ^ 0, Theorem 10.3.1 yields anfeA(B) such t h a t / = ^ on X and |/(C) < ||^||x for every C G S\K. T h u s / i s not constant, and now | / ( z ) | < ||^|||^ follows for every z G 5, by the maximum modulus theorem.

10.4. The Davie-0ksendal Theorem 10.4.1. The shortest as well as the most elementary proof of the F. and M. Riesz theorem is undoubtedly the one found by 0ksendal [1]. Its idea was used by Davie and 0ksendal [1] to formulate a covering condition that implies peak-interpolation, in the context of strictly pseudoconvex domains. Specialized to the ball, this is Theorem 10.4.3 below. 0ksendars proof is the case M = 1 of the proof of Theorem 10.4.3. 10.4.2. Definition. For C G S and (5 > 0, put F(C,(5)={ZG5:|1-
Each F(C, 3) is a "nonisotropic ball" Q defined in §5.1.1; however, g(C, S) = F(C, S^). For our present purpose, it seems desirable to make this change of notation. When w = 1, then F(C, S) is an arc on the unit circle, of length 23, having C as midpoint. 10.4.3. Theorem. Suppose K is a compact subset of S, with the following property: To every s > 0 correspond finitely many sets V(Ci, 3iX 1 < i < m, with Ci^ K and Z^^ < s, such that

Then K is an (Nyset. Note: When n = 1, the hypothesis says simply that K has Lebesgue measure 0.

212

10. Interpolation Sets for the Ball Algebra

Proof, For each e > 0, choose Ci and Si as in the hypothesis, put c^ = and define X G A{B) by

SJ-^E,

We claim that (a) |/,(z)| < 2 for all ZG 5, (b) 11 - / / z ) I < ^ 6 for every z G X, and (c) Hm, _, 0 X(z) = 0 for every z G 5 \ X . Once these are proved, pick v e A{BY. Then ^ fdv = 0. By the dominated convergence theorem, v{K) = 0. The same applies to every compact Ki a K. Hence |v|(X) = 0, which is what the theorem asserts. Since Re(l — 0, each factor in (1) has absolute value < 1, for every z e B, This proves (a). If z G X, then at least one factor in (1) has absolute value <Si/Ci = y/s. This proves (b). Fix z G B\K. Then there exists t > 0 such that 11 — I > ^ for every CeK. Put (2)

u

'

1 + c, -
Then |MJ < c^/f, X l"il < ^~^ Z ^f ^ ^~^v^» and m

(3)

/,(z) = i - n ( i - « . x

so that

\fxz)\ = 1-1 +0(1 - ud\ < -1 + n a + i".-i) < - 1 + exp^lMjl < - 1 + e x p ( r ^ y e ) . This proves (c). 10.4.4. An Application. Let y be a complex-tangential curve, as defined in §6.4.1. Then y (or, more precisely, the range of y) is a (P/)-set. This follows from Theorems 10.4.3 and 10.1.2, since it can be verified that the hypothesis of 10.4.3 holds. Since we shall find a different proof in Section 10.5, we omit the details. 10.4.5. There are variations of the Davie-0ksendal technique that allow one to prove that certain uncountable unions of (P/)-sets are (PI).

10.4. The Davie-0ksendal Theorem

213

For instance, let H c: S be the set where a certain h e A(B) peaks, and assume also that h e Lip 1, i.e. that \h(z) — h(w)\ < c|z — w| for some fixed c and for all z,w e B. Let £ be a compact set of real numbers, of Lebesgue measure 0, and associate with each t e £ an automorphism i/^, e Aut(B), in such a way that \il/,(z) - il/Xz)\ < \t - s\ior 3LI\ z e B, t e E, s e E. Put ht = ho \j/^. Then hf peaks on H^ = \j/^ ^(H). Define K = IJ^g^ //^. Then K is compact. 10.4.6. Proposition. The set K described above is a {PI)-set. Proof. Since m{E) = 0, there are intervals L^ (i = 1, 2, 3,...) with centers tjG£, of length di, so that E^^ < oo and so that every point of E lies in infinitely many Lj. Define 1 -

/if.

^

where c is the Lipschitz constant of h. Note that Re(l - /i) > 0 on S. Hence |^,| < 1. Fix z e K. Then z e H^ for some t e E. Hence there are infinitely many / with t E Li, so that \t — ti\ < djl. For any such f, |1 - h,i^z)\ = \hiz) - Ki^z)\ < c\t -U\<

ic'(5,,

which imphes that |^,(^)| < j . This happens for infinitely many i. Thus 00,/or every z e K. ff^(z) -^0,asN-^ Next, let KQ be a compact subset of B\K. Note that (z, 0 - ^ 1 — Kiz) is a continuous zero-free function on KQ X E. Hence there is an f/ > 0 (depending on KQ) such that |1 — hf(z)\ > rj for all ZEKQ^I e E. This implies

so that X 11 "~ Gi(^)\ converges uniformly on KQ. It follows that {/^} converges, uniformly on KQ, to a continuous limit which is # 0 at every point ofXo. Thus/(z) = lim^_oo /iv(^) exists for all z e B,/(z) = 0 if and only if z e K, a n d / i s continuous on B\K. To prove tha,ifeA(B\ it is now enough to show that / is continuous at every point of K. Since K = {f= 0}, this amounts to showing that | / 1 is continuous at every point where it vanishes. But I / 1 is upper-semicontinuous (since | /v +11 < I /v I) and > 0, and all such functions are continuous wherever they are 0. We have now proved that K is a (Z)-set, hence it is a (P/)-set, by Theorem 10.1.2.

214

10. Interpolation Sets for the Ball Algebra

10.4.7. Example. Take n = 2, and let K be the set of all points (cos e, e'^ sin 9)

(-n

< 9<

TT,

r e £)

where E is compact, m(E) = 0. Since {(cos 9, sin 9)} is the peak set of (1 H- z^ + w^)/2, which is in Lip 1, Proposition 10.4.6 shows that K is a (P/)-set. K is a union of circles, one for each t e E; they all pass through the points (1,0) and ( - 1 , 0 ) .

10.5. Smooth Interpolation Sets 10.5.1. The following simple example illustrates the theme of this section. Fix C e -S, and put r(t) = e'X,

7(0 = (cos t, sin t, 0,..., 0),

where —n 5 is a map of class C \ The first-order partial derivatives of the components of O are thus assumed to be continuous functions in Q. For each xeQ, Q> has then a Frechet derivative 0'(x); this is the reallinear map of JR"* into C" that satisfies (1)

lim

mx + h)-o(x)-nx)h\ . —

= 0.

We say that 0 is complex-tangential if the orthogonality relation (2)

{(^\x)h, 0(x)> = 0

holds for every xeQ and for every h e R"".

10.5. Smooth Interpolation Sets

215

Let M = 0(Q) and assume (to simplify this discussion) that O is one-toone. Put C = ^(x). The tangent space 7J(M) consists then of all vectors 0'(x)/z, as h ranges over R"", and (2) is equivalent to the inclusion (3)

T^(M) c= Tf (S).

The latter is the complex tangent space of S at C- (See §5.4.2.) Here is yet another interpretation of (2): Let y: [0, 1] -• Q be any C^curve, and define F: [0, 1] ^ 5 by F = O o y. The chain rule shows then that is complex-tangential if and only if every F obtained in this way is complextangential. We shall say that O is nonsingular if the rank of (^'(x) is m for every x G Q. In that case, O is locally one-to-one, and there is a continuous positive function c on Q such that (4)

I c(x)\h\

(x 6 Q, /i e R"").

The precise manner in which the crucial hypothesis (2) enters the proof of Theorem 10.5.4 is contained in the following lemma. 10.5.3. Lemma. / / O : Q^ S is complex-tangential, then the inner products (1)

(!>{x + 3v) -
converge, as 3 \ 0, to

(2)

W(x)(v - u)\'

for all xeQ and all u,ve R"". Note that the denominator in (1) is 3^, not 3. Proof. Fix X, u, v. If 3 is small enough then (since Q is open) all points (3)

y(t) = X + (1 - t)3u -^ t3v

(0 < t < 1)

he in Q. Fix such a 3, for the moment, and define 7: [0, 1] ^ 5 by y(t) = ^(y(t)). Then (4)

y'(0 = 30mmv

- u).

216

10. Interpolation Sets for the Ball Algebra

Since O is complex tangential, it follows that (5)

= 0

(0 < f < 1).

Note that (1) equals ^~^<7(1) - y(0), 7(1)>. Using (5), one obtains <7(1) - 7(0), y(l)> = (^ \y(t)dt,

yil)j

= f
= j dt j\y{tiY(s)yds. Hence (4) shows that (1) equals

(6)

jdt j iny(t)(v - u), ny(s))(v - u)yds.

As ^ \ 0 , y(t) -> X, uniformly for 0 < t < 1. Since ^' is continuous, the integrand in (6) converges uniformly to \0'(x)(v — w)p. Since the double integral extends over one half of the unit square in the (s, 0-plane, (6) converges to (2) as 5 \ 0. 10.5.4. Theorem. IfQ is open in R^ and

is a C^-map that is nonsingular and complex-tangential, then ^(K) is a (PI)set for every compact K cz Q. Under stronger smoothness assumptions on O this was proved (with strictly pseudoconvex domains in place of balls) by Henkin (see CirkaHenkin [1]), by Nagel [1], and by Burns and Stout [1]. The proof that follows is basically the one in Rlidin [14], but the details are considerably simpler since we now deal only with the unit ball. Proof, By Bishop's theorem (Section 10.3) it suffices to show that every p E Q is the center of a compact ball K such that | v | (0(X)) = 0 for every

veA\

Since O is nonsingular, we can choose K so that there is an a > 0 such that (1)

|(^(^)_o(>;)| >a\x-y\

(x,yeK).

10.5. Smooth Interpolation Sets

217

For y eQ, define (2)

g(y)=

f {1 + hny)v\'r"^ dv.

Since |'(}^)t;| > c(3;)|i;|, by 10.5.2(4), g is continuous, and of course >0. Now pick a continuous/: R"" -^ C, with support in K, and define

for 5 > 0, z 6 B. (The integrand is understood to be 0 outside X.) Since I (x)y I < 1, it is clear that h^ e A(B). We claim that {h^} has the following properties: (4)

sup{\hs(z)\: z eB,3>0}

(5)

lim/z/z) = 0 if z e B\Q>(K).

(6)

lim h(
< oo,

if

>'€K.

Of these, (5) is obvious, since 1 — Re(x)> has a positive lower bound as X ranges over K. To prove (4) and (6), associate to each z e B a point y e K such that

(7)

|a)(^)_^l
for all xe K.By the triangle inequahty, (7) gives (8)

|a)();)-a>(x)|<2|z-(D(x)|. The change of variables x = y + dv converts (3) to (f/g)(y + Sv)dv

(9) By (8) and (1),

8 Re(l - 4| z - 0(x)P > 10)0;) - 0(x)|^ > a^lx - y\^ = a^3^\v\^

218

10. Interpolation Sets for the Ball Algebra

The integrand in (9) is thus dominated by the L^-function

1 + " l„|2(

(10)

This establishes (4), and shows that (6) follows from (9), (2), and the dominated convergence theorem, since (11)

1 - <^0;). 0(y + bvyy _ 1 ^ ,, lim ^^.^y^^^>. ^ ^ l^,(^)^p^ ^2

2'

" "

by Lemma 10.5.3 (with u = 0). To complete the proof, note that O is one-to-one on X, so that K is well defined and continuous. Since each h^ e MB\ it follows from (4), (5) and (6) that (12)

f

(/oa)-^)Jv = 0

for every v G A ^ , and for every/e C(K) that vanishes on the boundary of the ball K. Hence | v|(0(Xi)) = 0 for every compact K^ in the interior of K. As noted at the start of the proof, this gives the desired conclusion. 10.5.5. Totally Real Vector Spaces. A real vector space V a C" is said to be totally real if F n (iV) = {0}, i.e., if V contains no complex subspace of positive dimension. Suppose V is totally real, and V c= Tf, the complex tangent space of S at C, whose real dimension is 2n — 2. Then iV also lies in Tf. Consequently dim^j V < n — 1. This leads to the following upper bound on m in Theorem 10.5.4.

10.5.6. Theorem. Suppose that the hypotheses of Theorem 10.5.4 hold. Associate to each x eQthe real vector space

Then (a) V,-(iVJ = 0, (b) V^ is totally real, and (c) m < n — 1.

10.5. Smooth Interpolation Sets

219

Assertion (a) means, more explicitly, that z and iw are perpendicular to each other, with respect to the real dot-product on R^"" = C , for all z eV^, w eV^. (See §5.4.2.) This amounts to proving that (a') is real for all z,w e V^, Proof. Take x = 0, for simplicity, and write V in place of V^. Obviously, Vr\ (iV) = {0} of V • (iV) = 0. Thus (a) impHes (b). Since O is assumed to be nonsingular, dim^ V = m. Thus (b) implies (c). So we have to prove (a'). Let z = ^'(0)w, w = €>'(0)t;, where u,ve R"". Define Qs(u, v) = S-\^(Sv)

- ^(dul 0((5t;)>.

Since CD'(0)M = lim —^——^—^-^

it is easy to check that = lim{Qs(u, 0) + e,(0, v) - Qs(u, v)}, which is real, by Lemma 10.5.3. Note. If 0 is assumed to be of class C^ rather than just C \ there is a more appealing proof of Theorem 10.5.6: The hypothesis <0'(x)t;, ^(x)> = 0, when applied to basis vectors i;, shows, for 1 <j<m, that = 0, where Dj = d/dXj. Hence (1)

+ = 0

for all;, k. Since e C^, the first inner product in (1) is unchanged if j and k are switched. Hence so is the second. But switching j and k replaces , DfcO) by its complex conjugate. It follows that {DjO), 0^0} is real, for all 7, k. This impUes the theorem, as before. 10.5.7. We shall see later (as a Corollary to Theorem 1L2.5) that a C^-curve in S = dB„ cannot be a (P/)-set unless it is complex-tangential. All smooth (P/)-sets in S are thus described (at least locally) by Theorem 10.5.4, and we conclude from Theorem 10.5.6 that their dimension is at most n — 1. This top dimension can be attained: in §10.1.5 we met an (n — l)-sphere and an (n — l)-torus in 5 that are (P/)-sets. Stout [7] noticed that these are the only known examples of (P/)-sets of dimension n — 1 that are compact connected manifolds, and asked whether there were any others. The case

220

10. Interpolation Sets for the Ball Algebra

n = 3 is the first one that is of interest in this connection. Here the question may be asked as follows: Which compact connected l-manifolds admit complex-tangential embeddings in the 5-sphere dB^ ? Changing the subject slightly, it seems reasonable to conjecture that the topological dimension of no (P/)-set K in dB„ exceeds n — 1, even if no smoothness condition is imposed on K. In this connection, Tumanov [1] has constructed a (P/)-set in 882 which does not directly deal with this conjecture (it is totally disconnected, thus has topological dimension 0) but whose Hausdorff dimension is surprisingly large, namely f. The construction is quite intricate, and we shall not include it here. 10.5.8. Although it is difficult to find complex-tangential embeddings of compact manifolds of dimension w — 1, it is quite easy to do this with R"~^: Let a = (ai, ...,a„) be a nonsingular C^-map of R"~^ onto a hypersurface in R" whose normal has all components positive. This impHes that there are positive functions Fj on R"~^, of class C \ such that "

doi-

Moreover, one can adjust them so that

(2)

Y.FJ(x)=L

Now put O = ((^j,...,
(pjix) = Fjix)Qxp{iQCj(x)}

(1 < 7 < n).

Then 0 is a nonsingular C^-map of R"~^ into dB„ that satisfies

(4)

t ^(^M^)

j=l

OXj^

=0

(l
by (1) and (2). If e^ denotes the /cth unit vector in the standard basis of R"~^, (4) can be rewritten as (5)

<0'(xK, Q>(x)y = 0

Thus O is complex-tangential.

(1 < fc < n - 1).

10.5. Smooth Interpolation Sets

221

10.5.9. Curves in T^. Take n = 2, take r > 0, s > 0 so that r^ -{- s^ = 1. The boundary of B2 contains the torus consisting of the points (re'^, se^^), — TT < 6, (p < 71. Let 7 be a C^-curve in this torus; thus (1)

y(t) = (re'^^'K se'"'^'^)

(a < t < b).

Then {y, y> = 0 if and only if r^Q' + s^(p' = 0. In other words, the range of 7 is a (P/)-set for ^(^2) if and only if (2)

r^e(0 + s^(p(t) = const.

In the (G, (p)-plane, this represents a line with slope —r^/s^. Consequently, 7 cannot be a circle unless r^/s^ is rational. The torus contains many other smooth curves that are (P/)-sets for the polydisc algebra A(rU x sU) (see Rudin [1] or Stout [1]), but only those that satisfy (2) work in B2 which is of course much larger than rll x sU. 10.5.10. An Annulus in B2. Fix a,0 < a < 1, and consider the map ^ that takes A G C\{0} to (z, w) e C^, by

Assume 1 # ju. If A/z / a^ then z(X) ^ z(fi). If Aju # — 1 then w(X) ^ w(/i). Thus (Q), and, in particular, on F«. This suggests a problem that seems quite hard: Suppose X is a compact set in S, and K is not a (Z)-set. Every/e A(B) with / Ix = 0 must then have some zeros somewhere on B\K, by definition of (Z)-set. But let E^ be the set of all z G B\K such that /(z) = 0 for every f e A(B) that vanishes at every point of K. The question is: Must E^^ be nonempty! In the above example of the annulus, the answer was obvious. It is equally obvious whenever K contains any arc that forms part of the boundary of (say) an analytic disc embedded in B. But when there is no such analytic structure in evidence, the question remains open.

222

10. Interpolation Sets for the Ball Algebra

Here is a related (probably easier) question: IfE^ intersects <S, must Ej^ intersect B? 10.5.11. Remark. Suppose 0 < a < | , y is a complex-tangential curve in S, a n d / e A(B) n Lip a. Theorems 6.4.9 and 6.4.10 imply then that the restriction o f / t o y lies in Lip(2a). (For the moment, we ignore the distinction between the mapping y and its range.) One might therefore expect (replacing a by 0) that all restrictions of members of A(B) to y ought to be just a little smoother than merely continuous. However, y is an interpolation set, and thus exactly the opposite is true: every continuous function on y has an ^(B)-extension. Since restriction to y improves the Lipschitz behavior, some smoothness is lost in the extension process. For example, if some function on y hes in Lip(j — e) for every e > 0, but is not in Lip(|^), then none of its ^(B)-extensions can He in Lip(^).

10.6. Determining Sets 10.6.1. Definition. A set K c S is said to be a determining set, or simply a (D)-set, if only o n e / e A(B) (namely/ = 0) h a s / ( 0 = 0 for all C e K, In other words, every/e A(B) is determined by its restriction to K. To be a (D)-set, K thus has to be "sufficiently large" in some sense. However, as is the case for (Z)-sets, the important thing in this context is not just the size of a set, but its positioning. Before we develop the machinery that is needed for the main result of this section (Theorem 10.6.8), let us look at some simple examples. 10.6.2. Examples, (a) If (T(K) > 0 then X is a (D)-set, by Theorem 5.6.4. (b) If K is the set of all z e S with z„ = 0, then K is obviously not a (D)set. Note that K is a sphere of dimension 2n — 3. This should be compared to Theorem 10.6.9. (c) When n>2 then 1 < 2n — 2. Nevertheless, there exist smooth arcs in S that are (D)-sets: Let a^,..., a„ be positive numbers that are Hnearly independent over the rationals. Define O: C ^ C* by 0(A) = n - i / V ^ ^ . . . , e ' ^ ' ' ^ ) . Let / be an interval (or any set of positive measure) on the real axis R, and put K = a)(/). We claim that X is a (D)-set. Suppose/G A(B),f\j^ = 0. Then/o O is continuous on the closed upper half-plane, holomorphic in its interior, 0 on /. Hence / o O = 0. Thus / ( O = 0 for all C^(^(R). The arithmetic assumptions about {ai,...,a„} imply that 0(i?) is dense in the torus defined by |Ci | = ••• = ICnl = '^'^^^

10.6. Determining Sets

223

(Kronecker's theorem). By continuity,/vanishes at every point of this torus, hence (see §1.2.1) at every point of the polydisc defined by |Cil < n"^^^, 1 < i < n. It follows t h a t / = 0. 10.6.3. Totally Real Manifolds. A smooth manifold M in C is said to be totally real if the tangent spaces Tp(M) are totally real for every p e M, as in Definition 10.5.5. We saw in Section 10.5 that there exist smooth (P/)-sets of dimension n — 1 in S, and that these are all totally real. By way of contrast, Theorem 10.6.8 will show that every totally real M c S (of class C^), whose dimension is n, is a (D)-set. The torus that occurred in 10.6.2(c) is an example of such a set. The proof of Theorem 10.6.8 uses carefully controlled holomorphic maps of U into C" that carry part of the unit circle T into M. The existence of such maps will be proved by applying the Banach contraction theorem in an appropriate function space H: 10.6.4. Definition. For a fixed n > 1, let H be the space of all absolutely continuous maps u:T-^R" whose derivative is in L^, with the norm (1)

ll«ll = ll"||2 + Il"'ll2Here u' = du/dQ and

(2)

llt^ll2 = | ^ J " l " ( e ^ ' ) l ' ^ e j ' ' ' ,

where | M(e*®) | is the eucHdean norm of the vector u(e'^) e R". With every ue H one can associate its "harmonic conjugate" w*. One way to do this is to use the Fourier series (3)

jkB

«(e'«)= X«*e''

(whose coefficients lie in C" = R" + jR").Put£o = 0,6^= liffe > 0,6^ = - 1 if fc < 0, and define

(4)

M V ' ) = -iZe*%e"'*. — 00

so that (5)

(u + iu*)(e'^) = ao + 2 ^ a^e'JkQ 1

224

10. Interpolation Sets for the Ball Algebra

Setting/ = u -\- iu*, we see that/can be extended to a continuous map of U into C" which is holomorphic in U, such that/(0) = QQ. By Parseval's theorem and (4), u* e H whenever ue H;m fact (6)

||w*|| < \\ul

Equahty holds in (6) if and only if UQ = 0. Finally, note that the sup-norm of w on T satisfies (7)

||u|U<

00 ^|aj<2||u||, — 00

by the Schwarz inequality. 10.6.5. Proposition (Bishop [2]). Let Q be a convex neighborhood ofO in R", with compact closure Q. Let h: Q-^ R" be a C^-map, with h(0) = 0 and /z'(0) = 0. Then there is a constant K < co such that (1)

iih o u - h o v\\ < Ki\\u\\ + Wvmu - v\\

for all U,VEH (2)

whose range lies in Q. In particular, \\hou\\


Proof. For each xeQ, h'(x) is a hnear operator on R" whose norm we denote by I hXx) I. Since h is of class C^ on the compact set Q, there is a c < oo such that (3)

Wix) - hXy)\ < c\x - y\

(x,yeQ).

Thus \h'(x)\ < c|x|, and therefore (4)

ih(x) - h(y)\ < c(\x\ + \y\)\x -

yl

Now fix M, f e H, fix 0, put x = w(9), y = v{d) (writing 6 in place of e'% By (4) and 10.6.4(7), (5)

\(h o u)(Q) - (h o vm\

< 2c(\\u\\ + ||i;||)|w(0) - i;(9)|

so that (6)

| | / i o t / - / i o t ; | | 2 < 2 c ( | | i / | | + ||t;||)||ii - i;||.

10.6. Determining Sets

225

Next,

\(h o u)XQ) - (h o vym = \hXx)uXQ) -

hXyym

< \hXx) - hXy)\ \u'm

+ \hXy)\ \uXe) - vXQ)U

which, by (3) and 10.6.4(7), is at most 2c||w-t;|||u'(e)| + 2ci|i;|||MXe)-t;'(e)|. Minkowski's inequahty shows therefore that (7)

\\(h o uy - (h o vy\\2 < 2c(\\u\\ + ||t;||)||u - t;||.

Now (1) follows from (6) and (7), with K = 4c, and (2) is the special case of (1) in which v = 0. 10.6.6. Generic Manifolds. Let M be a C^-manifold in C". Following Pincuk [1], we say that M is generic if the C-span of the tangent space Tp(M) is all of C , for every p G M. In other words, it is required that (1)

T^(M) + iT,(M) = C"

(psMy

It is clear that if (1) holds for some PQ G M , then it holds also for all p G M that are sufficiently close to PQ . It is also clear that a totally real manifold is generic if and only if its dimension is n. Here are some other simple properties: (a) Every generic vector space V a C contains a totally real generic subspace X. Proof, By assumption, F is a real vector space such that V -h iV = C". Let P be an i^-basis of V. Since the C-span of P is C", j8 contains some n vectors that are Unearly independent over C. Their i^-span X has the desired properties. (b) IfX is as in (a), then there is an invertible C-linear map A o / C onto C" such that AX = jR", the R-span of the basis vectors Ci, ...,e„. Proof Let L be an K-linear map of X onto i^", and define A by A(x + iy) = Lx -\- iLy for

x,yeX.

226

10. Interpolation Sets for the Ball Algebra

Note that it may not be possible to find a unitary A such that AX = K". For example, take n = 2, let X be the set of all (a + iP)ei + ^62 with a, p e R. It is easily verified that X is totally real. Since X contains two vectors whose inner product is not real (namely e^ and ie^ + ej), no unitary map carries X onto R^. (c) Suppose V and X are as in (a), M is a manifold in C", peM, and V = Tp{M). Then M has a submanifold M^, with p 6 M^, such that Tp{Mi) = X. Proof. Take p = 0, without loss of generality. Any K-linear projection P of C" onto V is then a 1-1 map of some neighborhood N^ of 0 in M onto a neighborhood ^ 2 of 0 in K Let M^= N^n P~\N2 n X). (d) Suppose Q is a connected open set in C , M is a generic manifold in Q, fe H(Q), andf(z) = Qfor every z E M. Thenf = 0 in Q. Proof Fix p e M.By (b) and (c) we may assume, without loss of generahty, that Tp(M) = R". Thus f(p + tej)/t tends to 0 as t tends to 0 through real values. It follows that (Djf)(p) = 0 for 1 <j ^ e C} is totally real: If Q is a connected neighborhood of 0 in C ^ , / e H{Q), and/(A, I) = 0 whenever (A, I) e Q, t h e n / = 0 in Q. 10.6.7. We are now ready for the main result. Since it may not be possible to find a unitary transformation in the change of coordinates 10.6.6(b), we may as well state the theorem for arbitrary convex Q, rather than just for B. In fact, convexity of Q is not needed either, but then the proof (as given by Pincuk [1]) requires some further background. By analogy with A(B), we let ^(Q) denote the class of all/G C(Q) that are holomorphic in Q. 10.6.8. Theorem (Pincuk [1]). Suppose Q is a convex open set in C", M is a C^-manifold in 5Q, and there is a point p e M such that (1)

Tp(M) + iTp(M) = C". Then M is a {D)-setfor ^(Q).

10.6. Determining Sets

227

Proof. If (1) holds for one p, then it also holds for all points of M that are sufficiently close to p. By §10.6.6, we may therefore assume the following situation, without loss of generality: p = 0^ To(M) = /^", and there is a ball Q cz i^", with center 0, and a C^map h:Q-^ R" with h(0) = 0, h'iO) = 0, such that (2)

M =

{x-\-ih(x):xeQ}.

Moreover, M is generic, and there is a vector y e R", y ^^ 0, such that the translates M + ity of M lie in Q whenever 0 < t < 1. Now choose a constant K, so large that the radius of Q is at least 1/K, and so that the conclusion of Proposition 10.6.5 holds. Put d = 1/(32 K). Next, choose ue H (see §10.6.4) with the following properties: u(e'^) = t{e^^)y, where tis a real-valued function, 0 < t < l,t vanishes at every point of some arc 7 c: T, r(l) > 0, and ||w|| < S. Moreover, let u be an even function of 0( —7r < 6 < Tc). Then w* is an odd function of 0. In particular, w*(l) = 0. Our objective is now to solve the functional equation (3)

^ = c - w* - (/z o ^)*

where c e R", \c\ < 23, and g e H. To do this, let X = {g e H: \\g\\ < 43}. Every g e X maps T into Q, so that we can define ^ : X ^ H by (4)

^ ^ = c - w* - (/i o g)*.

Then \\(hog)*\\ < \\h o g\\ < K\\gf < 3, by Proposition 10.6.5. Since \c\ < 23 and ||w*|| < ||M|| < 3, we see that ^ maps X into X. If ^1, ^2 ^ ^» another application of Proposition 10.6.5 shows that (5)

||^^l-^^2ll
Thus ^ is a contraction of X into X, and since X is a complete metric space, ^ has a unique fixed point in X. This solves (3). More precisely, we have proved: To every c e R" with |c| < 2^ corresponds a unique gc^ H such that Wgdl < 43 and (6)

g, = c-u*-(ho

g^y.

We need one further property of {g^}: To every xeR" corresponds some c € R" with \c\ < 23, such that (7)

gXD = X.

with \x\ < 3

228

10. Interpolation Sets for the Ball Algebra

To prove (7), consider the map (8)

c^gXl)

=

c-(hog^r(l).

(Recall that w*(l) = 0.) A computation similar to (5) shows that (9)

\\gc,-9cJ<2\c,-C2l

Also, if |c| = 25, then (10)

\c-gXl)\<2\\hog^\\<2K\\g,f<S.

The continuous map (8) thus moves no point of the sphere {|c| = 23} by more than S. Therefore the image of the ball {| c | < 25} covers the ball {\x\<S}, This establishes (7). Now define/,: T ^ C " by (11)

fc = Qc + ihog^ -h iu

which, by (6) is the same as (12)

fc = c + ilh o g^ + i(h o g^Y + M + iu*l

By (12), each / has a continuous extension to U (which we still call X) whose restriction to 1/ is a holomorphic map into C . By (2), g^ + i(h o g^ maps T into M c 5Q. Hence (11) and our choice of u show t h a t / ( r ) <= Q and thatX(l) e Q, so that X(l/) c: Q, by the maximum modulus principle and the convexity of Q. MOVQOVQV JXi) <= M, where y is the arc in T on which M=

0.

Assume now that F e ^(Q) and F|j^f = 0. Then F of^sA{lJ) 7, hence on U. In particular (13)

F{fX\)) = 0

vanishes on

(|c| < IS),

If |x| < 5 and (7) holds, then (14)

X(l) = X + ih{x) + iu{\).

Thus (13) impHes that F vanishes on (15)

Ml = {x + ih{x) + iuiX): |x| < 5} c Q.

Since M^ is a translate of a portion of M, M^ is a generic manifold. By 10.6.6(d), F = 0 in n. This proves that M is a (D)-set for y4(Q). Here is an application, also due to Pincuk [1]:

10.7. Peak Sets for Smooth Functions

229

10.6.9. Theorem. If n> \ and M is a C^-manifold of dimension In — 2 in S = dB„, then M is a (Oysetfor A(B). Proof Since 2n — 2 > n — 1, it follows from Theorem 10.5.6 that M is not complex-tangential. Hence there is a p G M at which the complex vector space

is strictly larger than Tp{M). The real dimension of Xp is thus In. In other words, Xp = C". Now refer to Theorem 10.6.8.

10.7. Peak Sets for Smooth Functions 10.7.1. When A(B) is replaced by any of the algebras (1)

A'^iB) = A(B) n C^CS)

(m = 1, 2, 3 , . . . , oo)

then the peak sets, zero sets, and interpolation sets are no longer the same. This happens already when n = 1: Every peak set for A^(U) is finite, every finite subset of the unit circle is a peak set for A'^iU) (even for a rational function), but there exist perfect sets that are zero sets and interpolation sets for A'^(U\ even in the strong sense of the possibihty of interpolating all derivatives. These results are due to Carleson [1], Taylor-WilUams [1], [2], and Alexander-Taylor-WilHams [1]. When n > 1, complex-tangential conditions are again important, as in Theorem 10.5.4. 10.7.2. Definitions. For 1 < m < oo, a compact K <= 5 is a peak set for A'^iB) if there exists/e A'^iB) such t h a t / - 1 on X, | / 1 < 1 on B\K. We say that K is locally a peak set for ^'"(JB) if every point of K has a neighborhood V such that iC n F is a peak set for ^""(JB). If to every g e ^"(S) corresponds a n / e A'^iB) such t h a t / = g on K, then K is said to be an interpolation set for A"^(B). A C^-manifold M c S is said to be complex-tangential at a point C e M if

This is a pointwise version of the condition that was discussed in §10.5.2. As regards peak sets for A'^{B\ the following results are known.

230

10. Interpolation Sets for the Ball Algebra

10.7.3. Theorem. Let n> \. The following three properties of a compact K (^ S are equivalent: (a) K is locally a peak set for A°^(B). (b) Every point of K has a neighborhood V such that K n F lies in a totally real C^-manifold M c: S, of dimension n — 1, that is complextangential at every point ofKn V. (c) Same as (b), except that M is to be complex-tangential at every point ofM. The implications (c) => (a) => (b) were proved by Hakim-Sibony [2]. That (b) => (c) was added by Chaumat-Chollet [2], [3]; in the same paper, they prove that all compact subsets of peak sets for A°°(J5) are peak sets as well as interpolation sets for A°^(B), In an earlier paper (Chaumat-Chollet [1]) they obtained a global version of (c) => (a): 10.7.4. Theorem. If n > 1, K is compact, M is a complex-tangential C°°manifold in S, and K cz M, then K is a peak set for A'^(B). Nagel [1] showed earlier that every complex-tangential closed C^manifold M c: S is the zero set of some/ E A'^(B). It is not known whether all sets that are locally peak sets for .4** (5) are in fact peak sets for A'^(B), The difficulty is that the class of all peak sets for A'^(B) is not closed under the formation of finite unions; see §10.7.7. We shall prove the implications (a) => (b) => (c) of Theorem 10.7.3, but only in the case w = 2, where the manifold M reduces to a curve. For the rest, we refer to the above-cited papers. That (b) imphes (c) is quite easy in the case of curves: 10.7.5. Proposition. Suppose that n = 2, 1 < m < oc, £ c [— 1, 1], and y: \^— I, l^ -^ S is a nonsingular O^-curve that satisfies (1)

= 0

(fe£).

Then there is a S > 0 and a nonsingular C"'^-curve F : [ —5, <5] ^ 5 such that (2)

nt) = y(t)

(tE£n[-^,5])

and such that F(0> = Ofor every t E [ —^, 5]. (The term "nonsingular" means that y{t) ^ 0 for all t.) Proof Let y^, 72 be the components of y. By a unitary change of variables we can arrange it so that none of y^, 72 ? 17i T have a zero on [ — ^, (5], for some

10.7. Peak Sets for Smooth Functions

231

^ > 0. This means that there are real C^-functions x, u, p on [ — (5,3'\, with X > 0, x' > 0, such that (3)

^-(T^)"^'"

on I-6,3'].

y^ =(1 + xY'^

Define a by

(4)

jc'a = xu' + v'

and define F = (r^, r2) by (5)

ri=yie'^

r2 = y2e-'^\

By (4), a G C"" \ hence F e C"" ^ Also, F is nonsingular, since the real part of Fi exp{ — i(u + a)} is positive. A simple computation shows that == 0 amounts to showing that (6)

x(u + a)' + (v — xa)' — 0.

But (6) is an immediate consequence of (4). 10.7.6. Theorem. Suppose n = 2, 1 < m < oo, / G A'*'{B\ f peaks on K cz S, Co G K, and W is a neighborhood ofCoThen there is a neighborhood Vof[,Q,Vcz W, and there is a nonsingular C^'^'Curve y: [— 1, 1] ^ 5 n F, such that (i) K n V lies in the range ofy, and (ii) (y\t\ 7(0> = 0 whenever y(t) e K n V. Proof. Extend / to a ^"-function with domain C^. Assume that ^^ = ^2 = (0, 1), without loss of generahty. The Hopf lemma (proved in §15.3.7), applied to the function w ^ / ( O , w) in the unit disc, shows that (D2 /)(^2) > 0By the implicit function theorem, there is a small polydisc P cz W, centered at ^2 J whose projection into the z-plane we call Q, and there is a function a G C'"(Q), such that/(z, w) = 1 in P if and only if w = a(z). In particular, (1)

/(z,a(z))=l

(ZGQ).

Let KQ be the set of all z G Q such that (z, a(z)) G K. Thus KQ is the projection of K n P into the z-plane. We claim that doc

d^oc

,

_

-—r-r doc

-zz, ^—rz and z + a(z) — oz dz oz ocz are 0 at every point O/KQ .

232

10. Interpolation Sets for the Ball Algebra

If didz is applied to (1), the result is (2)

(^2/)§ + (52/)^ + 5i/=0

in Q; the derivatives D2 /, D2 /, D i / a r e evaluated at (z, a(z)). Since/e ^^(5), D2 / a n d B j / , as well as any of their first-order derivatives, are 0 on S. Hence (2) gives da/dz = 0 on KQ . If dIdz is applied to (2), one obtains that d^aldz dz = OonK^. Observe next that u = R e / i s maximized (relative to 5) on K, so that its directional derivatives along tangents to S are 0. Thus (3)

ZD2U — wD^u = 0

onK.

Since D ^ / = D2 / = 0 on S, (3) is the same as (4)

zD2f -wDJ

=0

onK.

=0

onKo.

Application of d/dz to (1) shows that (5)

DJ+(D2f)^

If we take conjugates in (4) and use (5), we obtain (6)

z + a(z)—= 0 cz

onKo,

as claimed. Next, define h e ^"(Q) by (7)

/z(z) = |z|^ + | a ( z ) | ^ - l .

Since | / 1 < 1 in 5, we have /i > 0 in Q, and h(z) = 0 for all z e KQ. Since da/dz = d^oc/dz dz = 0 on KQ, we have 2

dz dz

> 1

on KQ , hence, in particular, at z = 0. One can therefore rotate coordinates in C so that (9)

h = ax^ + by^ -\-o(\z\^l

10.7. Peak Sets for Smooth Functions

233

with b > 0, Then dh/dy = Iby + o(|z|), and another appUcation of the impHcit function theorem shows that there is a nonsingular C""^-curve y^ through 0 which is the zero set of dh/dy in a (perhaps smaller) neighborhood QQ of 0 in C. (For the moment, we ignore the distinction between a curve and its range.) Since every point of XQ is a local minimum of/i, it follows that Let (10)

PQ

= {(z, W)G P:zG Qo} and put ro(0 = 7o(0^i + a(7o(0)^2

where t ranges over some parameter interval. This To runs through all points of K n PQ. Since doc/dz = Oon KQ, (11)

n{t) = Yoit)L+^iyoit))e2

ifyo(0eXo.By(10),(ll),and(6),

(12)

= 0 if ro(t)eKnPo.

Finally, put y = TQ/ITQ]. Then 7 is a curve in S, y(t) = ro(0 when ro(0 e K, and for these values of t we also have y'(t) = ro(0? since | FQ | has a local minimum there. Thus (12) holds with y in place of FQ . This proves the theorem. 10.7.7. Example (Hakim-Sibony [2]). Take n = 2, let K^ and K2 consist of all points (cos 0, sin 9) and (cos 0, i sin 0), respectively, where — TT < 0 < TT. Then j(z^ + w^ + 1) peaks on K^, ^(z^ - w^ + 1) peaks on K2, but K^ u K2 violates the conclusion of Theorem 10.7.6, with m = 2, at the points (1,0) and ( - 1 , 0 ) . Thus, although K^ and K2 are peak sets of polynomials, their union is not contained in the peak set of a n y / e A^(B).

Chapter U

Boundary Behavior of/f °^-Functions

11.1. A Fatou Theorem in One Variable The objective of this preliminary section is Theorem 11.1.2, a one-variable Fatou-type theorem for nonholomorphic functions that will be needed in the proof of Theorem 11.2.4. We begin with a simple lemma about functions on the real Hne. 11.1.1. Lemma. Le^/i, ...,/^ be nonnegative even functions on (—oc, oo) that are nonincreasing on (0, oo), and define /»00

(1)

J(t v — CX)

for real numbers t^, ...,tj^. Then (2)

J(t,,...,t^)<J(0,...,0).

Proof It is enough to prove the lemma under the additional assumption that each/^- is continuously differentiable and has compact support. The case N = 1 is trivial. Assume N > 1, and assume that the lemma is true with A^ — 1 in place of N. Then (2) holds whenever some two of the tj are equal, since we can then replace the corresponding two functions/^ by their product, thus reducing the number of factors from AT to AT — 1. In the proof of (2) we may thus assume, without loss of generality, that ti < ••' < tj^^i < t^. Differentiate (1) with respect to tj^, then replace xhy x -\- tj^. Since/Jv is odd, we obtain (3)

^ =

[j9(-x)-g(x)-]f^ix)dx

where g(x) = f^{x -h t^ - fi) •••/N-I(:>C + r^, - r^v-i). For x > 0, g(x) < g( — x), since t^ — tj > 0. Hence dJ/dtj^ < 0. This implies that (4)

234

J{ti,...,

tj^-1, t^) < Jyti,...,

t^_ 1, t^_ i)

11.1. A Fatou Theorem in One Variable

235

whenever t^ < • • • < tjy-1 < ^jv- The right side of (4) is at most J ( 0 , . . . , 0), by our induction hypothesis. The lemma follows. 11.1.2. Theorem. Let Q = (a, b) x (0, c) be an open rectangle in the upper half of C. Suppose that (a) F:Q -^ C is a bounded C^-function, and (b) dFldz 6 U{Q) for some p > \. Then Hm F{x + iy) exists for almost every x e (a, b\ as y \ 0. Note that (b) represents a considerable weakening of the classical hypothesis that F G H'^iQX i.e., that dF/dz = 0. It seems to be unknown whether the theorem fails when p = 1. The original proof of the theorem (Nagel-Rudin [2]) involved an appeal to the theory of singular integrals. The more elementary proof that follows is patterned after pp. 60-61 of Carleson [3]. That such an elementary proof might exist was suggested by Ahern. Proof The hypothesis is preserved when p is replaced by any smaller value (>1). We may thus assume, without loss of generahty, that the conjugate exponent q is an integer. By shrinking (a, b) a Uttle and making c somewhat smaller, we may also assume, without loss of generahty, that F is defined and' C^ on all of Q, except, of course, on its lower edge [a, b~\. Since F is bounded, there is a sequence e^ \ 0 such that the functions X -> F{x + ie.^ converge, in the weak*-topology of L'^ilja, ft]), to some (p G L°^([a, fc]). Extend F to g by setting F{x) = (p{x\ a < x < b. Let Qj = (a, b) x (sj, c). If z G Q and j is large enough, then zeQ^. Since F e C^(Qj), a standard appHcation of Green's theorem (see, for example, p. 3 of Hormander [2]) gives then ... (1)

J,,. I f F(z) = 1^.

dQj

FJOdC 7
1 f " ' JQj

h(w yv —

-du dv z

where w = u + iv, h = dF/dz. The above-mentioned weak*-convergence, combined with the fact that h G LP(Q) c L\Q), shows that we can let; -^ oo in (1), to obtain F = G ~ H, where (2)

G(z) = —: 2711 JsQ C - Z

and

(3)

Hiz) = - f

^^^dudv.

236

11. Boundary Behavior of //"^-Functions

Since G is the Cauchy integral of a bounded function, it is classical (see, for instance, Lemma 2.6 in Chap. V of Stein-Weiss [1] that lim G(x + iy) exists, as ); \ 0, for almost every x e [a, fe]. Define the oscillation of if at x to be (4)

osc(H, x) = Hm [sup|/f(x, / ) - H(x, / ) | : 0 < / , / < SI

The theorem will be proved as soon as we show that (5)

osc(if, jc) = 0 a.e.

The maximal function (6)

H(x) = sup{\H(x + iy\:0 < y < c}

can be used for this purpose. Let (p be any continuous function on [a, fe], 0 < (p < H, that satisfies (p(x) < H(x) wherever H(x) > 0. Since H is lower semi-continuous, ft is the pointwise Hmit of an increasing sequence of such (p's. If XQ is such that H(xo) > 0, then (p(xo) < \ H{XQ -f- iyo) \ for some Jo» and therefore cp{x) < \H{x + /j^o)! for all x sufficiently close to XQ. It follows that there is a Borel function y: [a, b~\ -^ (0, c) such that (7)

^<(p(x)<\H{x

+ iy{x))\

(a < x < b).

By (3), Fubini's theorem, and Holder's inequaUty, (7) impHes (8)

J cp(x)dx < U \hn

'[jdudvU

j ^ ^

X — iy(x)

The ^th power of the second factor in (8) is /•b

(9)

fb

f>h

\ " \ ^-^^ -"dxq

f»c

du \ ij/dv

where (10)

^ = |w _ xi + i(v - y(x,))\-''

• • |w - X, + i(v -

y(xq))\~'.

To symmetrize, let (9') be (9) with ( — c,c) in place of (0, c). By Lemma ILLl, the y-integral over ( — c,c) is maximized when y{x) = 0. There are absolute constants, A^, A2 < 00 such that /.^x

(11)

C^

1

dx

,

. ^

b-a

r-7< ^ 1 + ^2 o g — — - .

11.2. Boundary Values on Curves in S

237

Hence (90 is less than some AQ = Ao(a, b, q). Since (8) holds for every eligible cp, we conclude that (12)

H{x)dx < Al o l K t I l p H(x)dx

To finish, let Qj be as in the beginning of this proof, let hj = 0 in Qj, hj = h in Q\Qj, and define Hj as in (3), with hj in place of h. Then H — Hj is continuous outside Qj, so that (13)

osc(//, x) = osc(Hj, x) < 2Hj(x)

(a < x < b).

Hence (12) impHes (14)

I osc(if, x)dx < 2Ao\\hj\\p,

Asj -^ 00, \\hj\\p -> 0, and (5) follows from (14).

11.2. Boundary Values on Curves in S 11.2.1. So far we have encountered two types of results concerning the boundary behavior of //°°-functions: Koranyi's generalization of Fatou's theorem, which asserts that every fe //°^(B) has jfC-Hmits at almost all points of S, and the Lindelof-Cirka theorem that deals with Hmits at a single point of S. This leaves many questions. For instance, if y is a smooth curve in 5, does every/e H'^(B) have some sort of limit at almost every point of y, relative to its arc-length measure? The answer turns out to be no or yes, depending on whether y is or is not complex-tangential. Since complextangential curves are peak-sets, the first case is contained in the following simple fact: 11.2.2. Proposition. If K a S is a peak-set for A(B\ then there exists an fe i/°°(B) which has no limit along any curve in B that ends at a point ofK. Proof Let g e A(B) peak on K. Then Re(l — ^) > 0 on B\K, so that there is a function h = log(l — g\ holomorphic i n ^ , with \lmh\ < 7i/2, such that Re h(z) = logl 1 — g(z)\ -> — oo SiS z -^ K. Put / = exp(z7z). Then exp(-7c/2) < | / ( z ) | < exp(7r/2) for all z e B. When z tends to K along any curve r , / ( z ) spirals around the origin infinitely many times.

238

11. Boundary Behavior of /f°°-Functions

11.2.3. We now turn our attention to C^-curves cp: I -^ S that are nowhere complex-tangential. Since Re<
(1)

i8W= - /

rW(tX(p(t)>dt

•'a

for xe I. Then PXx) > (5 > 0 for some S and every x e /, so that p has an inverse a G C \ on J = [0, j8(b)]. By the chain rule, <((p o ay, cp o oC) = oc\(p' o a, (^ o a> = i(P' o a)a' = i at every point of J. Setting i/^ = cp o a, i/^ is a reparametrization of cp that satisfies (2)

<(A', ^> = I

We now come to the main result of this section. It was first proved by Nagel and Rudin [2], under the assumption that cp' satisfies a Lipschitz condition of some positive order. Nagel and Wainger [1] modified the proof so as to ehminate the need for this Lipschitz condition. They introduced the reparametrization (2), and the sphtting into radial and tangential components that occurs in Step 3 of the proof of Theorem 11.2.4. The resulting proof appHes then to any absolutely continuous (p, provided that lm{(p', cp} is positive a.e.; in fact, they weaken the hypotheses even further, putting rectifiabihty of cp in place of absolute continuity. To avoid technicaUties, we confine ourselves here to the case of continuous (p'. Both of the above-mentioned papers deal with arbitrary smoothly bounded domains, not just with B. 11.2.4. Theorem. Suppose that cp: [a, b] -^ S is a C^-curve that satisfies

(1)

<(PVI

cp(t)y ^ 0

(a
Let f E H'^iB). The restricted K-limit of f exists then at cp{t\ for almost every t e [a, fc]. "Restricted K^-limits" are defined in §8.4.3. The word "restricted" cannot be omitted from the conclusion. Example 8.4.6 shows this, since the circle (p{t) = (e'\ 0) satisfies (1).

11.2. Boundary Values on Curves in S

239

Proof. The idea of the proof is to construct a C^-map 0 that carries a rectangle Q (as in Theorem 11.1.2) into B, in such a way that (i) 0, (iii) for every x e [a, fc], the curve F^ defined by r^(y) = 1. Once we have this, it follows from Theorem 11.1.2 that the limit of/along F^ exists for almost every x. The Lindelof-Cirka theorem 8.4.4 implies then that the restricted X-limits of/exist at the corresponding points q>{x). For convenience, we break the proof into three steps. Step L The Map O. The remarks made in §11.2.3 show that it involves no loss of generality to assume that (2)

W{x\

(p(x)y = i

(a<x

< b).

(The reparametrization in question carries sets of measure 0 to sets of measure 0.) Extend cp to a C^-map of (—oo, oo) into 5, with WW^c < oo. Choose a positive function i/^ on (— 1, 1), such that

f n)dt = i,

(3)

\

\m)\dt<j\.

Define (4)

u(x, y) =

(p(x + ty)il/(t)dt.

(5)

0(x + iy) = (p(x) - yu(x, y\

and put Q = (a, b) x (0, i). We claim that in Q (writing O for Q>(x + iy)X (6)

1 - |(Dp > y

and (7)


for some constants c < oo, Ci < oo.

< ciy

240

11. Boundary Behavior of //^-Functions

Differentiation of (4) gives \du/dx\ < Wcp'W^o^ \Su/dy\ < 1, so that, setting C = (p(x), we have

I r du\ \uix,y)-C\ = \\ -^\
(8)

\jo ^y\

Thus 1- y< Re. By (5), 1 - |Op = 2y Re
> 2y(l - y) -

y\

which gives (6) if y <^. Next, differentiation of (5) leads to d(S> 1

du

1

by another application of (8), since du/dz is bounded. This proves (7), since (2) impHes that (cp' - icp, cp} = 0. Step 2. The Curves F^. Fix x e [a, /?], put C = (^), and define r^(y) = Q>(x + iy\ 0 < y < i Write F in place of F^, and let y =
|r-y| |r-y| i-lyp-i-irp-'Hence (compare 8.4.3(4)) every T^ is a special C-curve, as y \ 0. Step 3. Estimation of(d/dzXfo O). Assume/e H'^(B), and define f: g -> C byF=/o4). To each z e g, let w = 0(z) = rC with C e S, r > 0. By (6) (9)

1 - r ^ = 1 - |wp >>;. Consider the vectors

(10)

a = ^_

^ (.),...,-^(z)j

11.2. Boundary Values on Curves in 5

241

and (11)

iS = ((Di/)(w),...,(D„/)(w)).

Since/is holomorphic, the chain rule gives (12)

dF aF^<'^'^>-

The vectors a, j8 have decompositions (13)

a =
jS = iP, OC + Pt

in which the "tangential" components a, and j8, are orthogonal to C- Hence (12) becomes (14)

| ^ = <«,0 + <«„)?,>.

Let us look at the three inner products in (14). First, (10) shows that (15)

1 /8^ \
by (7). Next, || < |j8| < ||/IL/(1 - |w|), by the Schwarz lemma. Hence (9) shows that (16)

|

-(;)•

The first summand on the right of (14) is thus bounded in Q. To estimate , note that |aj < |a| < c, where c is as in (7), and that <w, a,> = 0. If A 6 C and c|A| < y^'^, it follows from (9) that (17)

|w + M P = |wp + | A p | a , p < l .

The function (18)

/z(A)=/(w + Aa,)

is thus holomorphic in the disc \M < c~^y^'^. By the Schwarz lemma, (19)

|/>'(0)|<||/IL-c>;-i'^

242

11. Boundary Behavior of /r°°-Functions

By (10), (11), and the chain rule, (20)

/i'(0) = = ,

and comparison of (19) and (20) gives finally (21)

g

= 00'-1/2).

Consequently, dF/dz e L\Q) for every p < 2, This proves statement (iv) made at the start of this proof. As explained there, this imphes the theorem. The following consequence of Theorem 11.2.4 appears in Nagel [2], with a different proof. 11.2.5. Theorem. IfK is a (Piyset in S and (p:\^a,b2^S that (1)


is a C^-curve such

(a
then (p~ ^(K) has Lebesgue measure 0. Corollary. 7/(1) holds then (p(la, bj) is not a (Pl)-set, Proof. Let E = (p(la, bj). Then E n Kisa. peak-set. By Theorem 11.2.2 there is a n / e H'^(B) whose radial Hmit exists at no point of £ n K. By Theorem 11.2.4, (p~\E n K) = (p~\K) has measure 0. It is an open question whether the converse is true: If K ci S is compact and (p~ ^(K) has measure 0 for every cp that satisfies (1), is ^ a (P/)-set? In view of Theorems 10.5.4 and 10.1.2, the following is a consequence of 11.2.5: 11.2.6. Theorem. iSw/7/70S6 0 : 0 ^ 5 is a nonsingular C^-map of an open set Q c: R^ into S, Then ^(Q) is totally null if and only if O is complextangential 11.2.7. While we are on the subject of curves in S that are nowhere complextangential, let us (following Nagel [2]) look at Cauchy integrals of certain measures concentrated on such curves. To be specific, let
(1)

ydii

= ^'f{cp{t)mt)dt

11.2. Boundary Values on Curves in 5

243

for every/e C(S). Let F = C[^], the Cauchy integral of n. Then

Since <(j9, (/)'> # 0, we have 0 such that (3)

|S

izeVnB,a
When z G F n B, we can therefore rewrite (2) in the form

...

^^

p.. _ f

<2,ffl'(0>

_jKO_ .

^'-' J. {i-r"

and integrate by parts. Since ij/ and ij/' vanish at a and b,

(5)

^(^) = ;r^n: J, {i-(o>r-^ * it

{
Now let z = rCC^ S. For C e S\F, F(rC) is bounded, since ^([a, ft]) c= V, For C e 1^ n 5 and r sufficiently close to 1, the derivative in (5) is bounded, by (3). Hence there are constants c^, C2 < 00 such that

The last integral stays bounded as r y 1 (Proposition 1.4.10). We have thus proved the following: The Cauchy integral (2) of the measure fx defined by (1) is in H^(B). In particular, we see, when n > 1, that there exist measures on S that are singular with respect to a but whose Cauchy integrals are in H^(B). One can go much further if more differentiability is imposed on (p and ij/. For example, let cp G C°°, i/^ 6 C°^, in addition to the preceding requirements. If a is any multi-index and F is given by (2), then (D°'F)(z) is given by an integral like (2), with n + |a| in place of n, and with some C°°-numerator in place of il/(t). One can then integrate this by parts any number of times, each time multiplying and dividing by
244

11. Boundary Behavior of //"^-Functions

/ / (p:la,b']-^S is a C^-curve such that {(p', (p} is nowhere 0, if ijj is a complex C^-function with support in (a,b), and if fieM(S) is the measure concentrated on (p([_a, bj) that is defined by (1), then the Cauchy integral of fi lies in A'^iB) = A(B) n C°°(5). This topic was developed in more detail by Nagel [2] and Stout [5], [8]. 11.2.8. Example. When (p is complex-tangential, the situation is entirely different. For example, take n = 2, (p(t) = (cos t, sin t) on [ —TT, TC], i/^(r) = l/2n. The Cauchy integral under consideration is now

27r J_^

(1-zcost-wsmO

By contour integration, one sees that (2)

F(z,w) = (l - z ^ - w ^ ) - ^ / ^

[In fact, it is enough to prove (2) when z and w are real, and this case reduces (by a translation) to the case w = 0, 0 < z < 1.] There is a unitary transformation U such that, setting G = F o C/, we have (3)

G(z,w) = (l -2zw)-3/2

If we expand G^^^ by the binomial theorem and apply Parseval's theorem, we find that G^^^ is not in H^, hence F is not in H^, in spite of the extreme smoothness of cp and il/.

11.3. Weak*-Convergence 11.3.1. If 0 < r < 1, C e 5, a n d / e if °°(5), we use the famiHar notation/,(0 = /(rQ. In the present discussion, the symbol Hm will always refer to r / ^ 1. Consider the following four properties that a measure ju £ M(S) may or may not have: (a) (b)

For every/G H'^iB), lim fXO exists pointwise a.e. [ | /i | ]. For every/e H'^iB), lim/^ exists in the weak*-topology of L°°(\fi\X regarded as the dual of L^|)U|).

(c)

For every/G H'^{B),

hm js f df^ exists.

(d) fi is a, Henkin measure. For example, (a) holds when fi = a (Theorem 5.6.4), and when /i is arclength measure on a C^-curve in S that is nowhere complex-tangential (Theorem 11.2.4).

245

11.3. Weak* -Convergence

The following implications are known:

Of these, (a) -> (b) follows from the dominated convergence theorem, (b) -^ (c) is trivial, and (d) -^ (b) is quite easy: Suppose /z is a Henkin measure,/e if*(B), geL^(\fi\). Let n y 1, ti/^ 1. Then {/^. —/,.} is a Montel sequence. By Theorem 9.3.1, g\ix\ is a Henkin measure. Thus jfr,gd\fi\-

j

f,^gd\fi\^0

as i ^ 00. The arbitrariness of {rj and {ti} implies therefore that hm ^ frQ d\iLi\ exists. Thus (b) holds. The implication (c) => (d) will be established in Theorem 11.3.4. 11.3.2. Lemma. If K a S is a peak-set for A(BX then there is an F e Ef^iB) and a sequence r^ / ^ 1 such that \\F\\^ = 1 and

lim|f(rpO-(-l)''| = 0

(1)

p-^ao

uniformly on K.

Proof Fix some g e A(B) that peaks on K. Choose e^ > 0 so that SSp < oo. Let Ap be the open triangle in C whose vertices are at 0,1, is p. Put TQ = 0 and let /ii be a homeomorphism of U onto Aj that is holomorphic in U (a Riemann map), with /ii(l) = 1, and define/^ = h^ o g. Make the induction hypothesis, for some p > 1, that r^-i a n d / i , . . . , fpEA(B) are chosen, peaking on K. Then there is an r^, r ^ . j < r^ < 1, 1 — rp < Sp, such that (2)

Z|l-//r,0l<8p

i=i

(CeK\

and there is a Riemann map hp+i of U onto Ap+i, with hp+i(l) = 1, such that \hp+i\ < 8p+1 on thQ compact set g(rpB). This follows from a normal family argument, since hp+1(0) can be moved arbitrarily close to 0 in applying the Riemann mapping theorem. Put/^+i = /i^+i o g^ to complete the induction.

246

11. Boundary Behavior of/f°°-Functions

This defines {r^} and {fp} so that fp peaks on K, Im/p > 0 in 5, and l/p+il < £ p + i O n r p B . Put

(3)

F = Qxp]niY, fp\-

Then F e H(B) and | F | < 1 in B. For each p, (4)

P - 1

fjirpO

^

ll^-UrpOl+llfjirpCl p+i

j = i

When C^ K, the first sum on the right is <ep, by (2). In the second sum, \fj\ < Sj on Vj-iB 3 VpE. Thus (5)

P-

E fjirpO <1BJ

(CEK).

p

Now (1) follows from (3) and (5). 11.3.3. Lemma.///x has property (c) (§11.3.1) and K is a peak-set, then KK) = 0. Proof. Put fi(K) = a. Choose £ > 0. Associate F and {rp} to K, as in Lemma 11.3.2. If/i is a sufficiently high power of some member of A(B) that peaks on K, then (1)

f

\h\d\fi\ < £.

JS\K

The same inequahty holds then with h^. in place of h, if r is sufficiently close to 1. For such r, (1) implies (2)

\ FXdfi I Js

-

FrKdn < 8. JK

Since Fh e H'^iB) and /x satisfies (c), the integrals over S converge to some limit L as r / " 1. Lemma 11.3.2 shows that the integrals over K converge to a when r = rip and p ^ co, whereas they converge to — a when r = rip-iThus \L— a| < £, |L -f- a| < e, and therefore |a| < e. 11.3.4. Theorem. If fie M(5) and lim

r-*l Js

/ , dfi

exists for every fe H°°(B), then fi is a Henkin measure.

11.4. A Problem on Extreme Values

247

Proof, If K is any compact set in S that is totally null (i.e., p(K) = 0 for every representing measure p e MQ) then X is a peak-set (Theorem 10.1.2), hence fi(K) = 0, by Lemma 11.3.3. The totally singular part of ft in its GKSdecomposition relative to MQ (see Theorem 9.4.4) is therefore 0. Consequently, fi <^ pfoT some p e MQ, hence )U is a Henkin measure, by Theorem 9.3.1. 11.3.5. It is not known whether the missing arrow can be inserted into the diagram in §11.3.1, i.e., whether every/G H°° has radial limits a.e. with respect to every Henkin measure. Here is another way of asking the same question: / / / G if * ( B ) and Ef is the exceptional set of points ^eS at which lim / ( r Q fails to exist, does it follow that Ef is totally null?

11.4. A Problem on Extreme Values 11.4.1. A Conjecture. Suppose/e A{B) and/is not constant. Let E^{f) be the set where | / 1 attains its maximum. (Related sets E2 and E3 will be defined presently.) Thus E^{f) is the set of all C e 5 at which | / ( 0 I = ||/||oo. It is probably true that (*)

(7(£i) = 0 when

n > 1.

This has some (perhaps only superficial) resemblance to the inner function problem which will be discussed in Chapter 19. The following definitions will make it easier to describe the progress that has been made so far toward proving (*). 11.4.2. Definitions. Again,/G ^(5),/is not constant, and, from now on, n > 1. We associate three subsets of 5 t o / : E'l = £ ! ( / ) is as in §11.4.1. E2 = E2{f) is the set of all ^" e iS that have a neighborhood N^a S such t h a t / ( 0 is a boundary point of/(Ar^). In other words, C e £2 if and only if the restriction of/to S fails to be an open mapping at C. E3 = E^if) is the set of all C^ 5 that are not Hmit points of the associated variety K;={ZGB:/(Z)=/(0}.

Note that it is not required, in the definition of £3, that C be an isolated point of the set of all z in the closed ball B at which/(z) = /(O- For example, if / peaks on K, then K a E^{f), simply because V^ is empty for every

CeK.

248

11. Boundary Behavior of //°°-Functions

Since/"H5(/(B))) cz E2, it is clear that E^ c £2 and that, in general, El ¥= E2. We shall see presently that £2 ^ ^3 • For /c = 1, 2, 3 , . . . , let A\B) = A{B) n C\B). Sibony [1] has obtained the following results (always assuming n > 1): (i) Iffe A(B) n Lip 1, then (7(£i) = 0. (ii)

Iffe

^2«-2(B), then G{E2) = 0.

(iii) If fe A^(B), then the n-dimensional Hausdorff measure of E^ is finite. The conclusion of (iii) is of course much stronger than o-(£i) = 0, since n <2n - 1. The w-torus that occurs in 10.1.5(e) is an example that is relevant to (iii). We shall not deal with (iii) here, but will weaken the hypotheses and strengthen the conclusions of (i) and (ii): (iv) Iffe A(B) n Lip a/or some oc> j , then (riE^) = 0. Theorem 11.4.7 will actually give a more precise conclusion. We first demonstrate some other properties of £3. 11.4.3. Proposition. Suppose n > l,fe A(B\fix Q = {z e B:xi > t}, Ifp e Q and Vp= {z eQ:f(z)

t, - 1 < r < 1, and define

w = {z e S: x^ > t}.

= /(p)}, then the closure of Vp intersects a>.

(As usual, Xi = Re Zi.) This proposition follows from a general maximum modulus theorem on varieties (Theorem 14.1.5) but a simple ad hoc proof can be given: Proof Write V for Vp and assume/(p) = 0, without loss of generaUty. Since V is compact, there is a point ae V such that Re ai > Re z^ for all z eV.lf a G CO, we are done. Assume that aeV. Then there is an r > 0 such that the points a -\- sci H- Xe2 lie in B whenever 0 < £ < r, | A| < r. Define

for 0 < 8 < r, I A| < r. When e > 0, our choice of a shows that g^ has no zero in the disc |/i| < s. Since goiO) = 0 and g^ -> go uniformly as 8 \ 0 , we conclude that goW = 0 for all X with |A| < r. It follows that V^ B n L, where L is the complex line {a + 2^2: >1 e C}. The closure ofBnL intersects a; in a circle. Corollary 1. E^if) has empty interior.

11.4. A Problem on Extreme Values

249

Indeed, the proposition asserts that co contains points that are not in £ 3 , no matter how small 1 — tis. Corollary 2. /(Q) cif{co). In particular,/(B) cz/(S) for every/e A{B). Corollary 3. JE^aC/) ^ E^(f) iff is not constant. Proof Suppose ^i e £2 • Then t can be chosen in the proposition so that/(^i) is a boundary point of/(co). Since/(Q) is open, it follows from Corollary 2 that/(^i) is not a point of/(Q). Hence e^e E^. The function (1 — z^)^ shows that E2 can be a proper subset of £3. 11.4.4. Proposition. E^if) is a set of type F„, Proof Fix /, define V^ as in §11.4.2, and, for fc = 1, 2, 3 , . . . , let X^ be the set of all C e 5 whose distance from P^ is < l//c. Since s\£3 = n

^k.

it is enough to prove that each X^ is open. Fixfc,fixC e X|t. Then there is a z e T^ and an 8 > 0 such that |z-CI + 2 e < | . k Let W be the open ball with center z, radius e. Then/(Ti^) is a neighborhood of/(C) = /(z) in C. Hence there is a ^, 0 < ^ < e, such that/(//) € f{W) for all ;7 e 5 with | ?/ — CI < ^- Since V^ then intersects W, the distance from r] to V^ is less than

k so that rj e Xj,. Thus X^ is open and the proof is complete. 11.4.5. Lemma. Ifh e H{IJ\ 0 < |;i(A)| < 1/or all XEU, and h(0) = c, then |/i'(0)|<2|c|log

© •

Proo/ If ^ = (log h — log c)/(log h + log c) then g maps C/ into [/ and g'(0) = h\0)/2c log c. Now apply the Schwarz lemma.

250

11. Boundary Behavior of //"^-Functions

11.4.6. Carleson Sets. Given a compact X on a circle T, let J, (/ = 1, 2, 3,...) be the components of T\K (the collection {JJ is finite or countable), and let di be the length of the arc J^. Then K is said to be a Carleson set if (a) (b)

K has Lebesgue measure 0, and 2:(5,log(l/5,)
It is easy to find sets K (even countable ones with only one limit point) that satisfy (a) but violate (b). Suppose/G A(U) n Lip 8 for some s > 0 , / # 0, and K is the set of all boundary points of U w h e r e / = 0. The fact that

r

\og\f(e'^)\de>

-00

leads then fairly directly to the conclusion that X is a Carleson set. This was proved by Carleson [1]; he also obtained a converse, which will not concern us here: Every Carleson set is the set of zeros of some/e A'^iU). We are finally ready for the result stated at the end of §11.4.2. 11.4.7. Theorem. Suppose n> 1,^ < a < 1, and (i) (ii)

/ G A(B) n Lip a, the origin is not a critical point off.

Then there is at most one complex line L through the origin such that the closure ofEj^^f) n L fails to he a Carleson set. The meaning of (ii) is that the gradient of/is not 0 at the origin: at least one of the number (Di/)(0),..., (Z)„/)(0) is ^^0. Proof Assume | | / | | ^ = 1. Put a - ^ = a. Put (1)

^(A) = (/)2/)(A^i)

{XeU).

As soon as we show that (I) g extends to a function on U that belongs to A(U) n Lip e, and (II) gW = OifXe,eE,(f\ the proof is essentially done. To see this, let K be the closure of the set of all Xe T such that Xci e E^if)Then g(X) = 0 for all /I G A^, by (II) and the continuity of g on U, and if K is not a Carleson set, then (I) impUes that g = O.ln particular, (D2 /)(0) = 0. Of course, one proves in the same way that (Djf)(0) = 0 for ; = 2, 3 , . . . , «. Now let L be any complex line through the origin, not necessarily through ^1. By means of a unitary transformation, the preceding paragraph implies

11.4. A Problem on Extreme Values

251

the following: If the closure ofLn E^if) is not a Carleson set, then all firstorder directional derivatives of fat 0, in directions orthogonal to L, are 0. This can obviously not happen for two distinct L's, unless the gradient of /vanishes at the origin. The proof of (I) uses the radial derivative ^/defined in §6.4.4. The constants Ml, M2 below depend only on a and the Lipschitz constant of/ Theorem 6.4.9 shows that (2)

mf\z)\<M,{l-\z\r-'

(zeB)

and therefore Lemma 6.4.7 gives (3)

mD2f)(^e,)\

(Ae t/).

< M2(l -\My-'^'

By (1), (3) is the same as (4)

(A e l / ) .

WW\<M2(l-\Mr-'

The case n = 1 of Theorem 6.4.10 (this is the Hardy-Littlewood result mentioned in §6.4.3) shows that (4) impHes statement (I). To prove (II), suppose C = e'^e^ e E^if), for some fixed 0. There is then a neighborhood N of C in C" such that / — / ( Q has no zero in B n AT. If 0 < r < l , l — ris sufficiently small, and (5)

h(X) =m

+ ^2) - / ( O

(|A| < (1 -

r^)

it follows that 0 < |/i(A)| < 2. Hence (1) and Lemma 11.4.5 imply (6)

|3(re'»)| = |(D,/)(rO| = \h'm

<

^^^l^J^^^^

where c = h(0). Since/e Lip(i + e),

(7)

I c I = I h(0) I = I f(rO - /(OI < M(l - ry-" 'l\

By (6) and (7), g{r^^) ^ 0 as r ^ 1. This proves (II). Corollary. Ifn > 1, a > ^ andf e A{B) n Lip a, then (y(E^(f)) = 0. If/is constant, there is nothing to prove. If not, then some point of B is not a critical point of/ hence 0 is not a critical point of F = / o i/^ for some ij/ e Aut(B). The theorem applies to F and shows (by L4.7(l)) that (T(E^(F)) = 0. Finally, E^if) = il/iE^iF)), and ij/ preserves sets of measure 0.

252

11. Boundary Behavior of i/°°-Functions

11.4.8. Remark. The conclusion of Theorem 11.4.7 may fail if hypothesis (ii) is dropped: When n = 2 and /(z, w) = zw, then E^(f) is the 2-torus |z| = |w| = 1/^2. (In this example, E^ = E2 = E^.) 11.4.9. Remark. I f / e A\BXfis not constant, and n > 1, let E^(f) be the set of a\\C^ S at which the equations

hold for 7,fe= 1,..., n. (These are the conjugates of the tangential CauchyRiemann equations; see Chapter 18.) The proof of Lemma 2 in Sibony [1] shows that if one of these equations fails at a point C, i.e., if C ^ E^if), then C is a Hmit point of F^, hence C ^ E^if). Since E4,(f) is closed, it follows that the closure ofE^if) lies in EJ^f),

Chapter 12

Unitarily Invariant Function Spaces

This chapter deals with a subject that is basically a topic in harmonic analysis and which, at first glance, may seem to have little to do with our principal concern, namely with holomorphic functions. Nevertheless, one of its main results (Theorem 12.3.6) will be essential later in the classification of Moebius-invariant spaces, an obviously function-theoretic topic. An interesting aspect of the second half of the chapter—a description of the ^-invariant subalgebras of C(S)—is that the structure of these algebras is more compHcated in dimension 2 than in any other dimension. Almost everything in this chapter (except for Section 12.1) comes from Nagel-Rudin [1] and Rudin [15].

12.1. Spherical Harmonics This preparatory section contains proofs of two basic facts about spherical harmonics: their Hnear span is dense in C(5), and harmonics of different degrees are orthogonal. Since the complex structure of C" is irrelevant in this context, we shall temporarily use a eucHdean space R^ as our setting. 12.1.1. Definitions. For /c = 0, 1, 2 , . . . , ^^ denotes the space of all homogeneous complex-valued polynomials on R^ of degree k, and Jif^ is the space of a l l / 6 ^,^ that satisfy A/ = 0, where

dxj

dxjf'

Naturally, the term "homogeneous" refers here to real scalars: if/G ^j, then

f(tx) = t'^fix), for

xGR^,teR.

Begin harmonic, each fe Jfj^ is uniquely determined by its restriction to the unit sphere S. These restrictions are the so-called spherical harmonics of degree fe. We shall freely identify Jf^ with its restriction to S. 253

254

12. Unitarily Invariant Function Spaces

12.1.2. Theorem. Iffe

M'j,, g e Jf^, and k ^ m, then {fgdc7 = 0. Js

Proof. The homogeneity of / and g shows that df/dr = kf and dg/dr = mg on 5, where d/dr denotes the radial derivative. Since A/ = 0 = Ag, one of Green's identities gives

(. - k) J/, da = l[ff - 4 ) . . = 1 jw - m)dr = 0. 12.1.3. Theorem. Each ^^ is a direct sum (1)

^k = ^k®

M^^k-2 e \xf^k- 40 • • •

Here | x p = xf H- • • • + x^, and the sum stops when the subscript reaches lorO. Proof. Each fe^j, h^s the form f(x) = S/^x", where /« e C and a ranges over the multi-indices with | a | = /c. The inner product (2)

. = Za!/a^«

turns ^fc into a finite-dimensional Hilbert space. Suppose |a| = \p\ = k. Then D V = a! if a = JS, and is 0 if a T^ j5. Hence (2) is the same as (3)


where/(D) = Z/;D« = l.Wldx,r If h{x) = |xpgf(x) and ge^,,-2 fe^ii, (3) implies (4)

< | x p ^ , / X = MD)f=

(f,ge^,), •••{dldx^r. then ;i(£)) = ^g{p), so that, for any g(D)Af=


since A • g(D) = g(D) • A and A / = A/! This shows t h a t / I \x\^^k-2 (in the sense of the inner product (2)) if and only if A / ± ^ ^ . 2 , i.e., if and only if A / = 0 (since A/e ^k-i)- In other words (5)

^k =

^k®M'^k-2'

Repeat the same argument with ^fc_2 in place of ^fc, and proceed. Finally, note that ^ i = J f i and ^Q = J^Q.

12.2. The Spaces H{p, q)

255

Corollary. The linear span of {Jf^: /c = 0,1, 2,...} is dense in C(S). Proof. On 5, |x| = 1, so that every/e ^^ is a sum of spherical harmonics. The Stone-Weierstrass theorem implies that the Hnear span of {^k'-k = 0, 1, 2,...} is dense in C(5). 12.1.4. Although we shall not need it later, let us note one other consequence of 12.1.3(4): If ^6^fc_2 and g 1 A^^, then \x\^g 1 ^j,. Since \x\^ge^k, it follows that \x\^g = 0, hence g = 0. Conclusion: ^^_2 = A^^. Every polynomial is therefore the Laplacian of a polynomial. Chapter IV of Stein and Weiss [1] contains more details about spherical harmonics, and their relation to Fourier analysis. The inner product 12.1.3(2) occurs there.

12.2. The Spaces H(p, q) 12.2.1. Definition. For nonnegative integers p and q (and fixed dimension n), H(p, q) is the vector space of all harmonic homogeneous polynomials on C" that have total degree p in the variables Z j , . . . , z„ and total degree q in the variable z^,..., z„. Thus every/e H(p, q) has bidegree (p, q). Note that H(p,0) consists of holomorphic polynomials, and H(0,q) consists of polynomials whose complex conjugates are holomorphic. The case n = 1 is somewhat special. In that case, dim H(p, 0) = dim if(0, q) = I, but H(p, q) = {0} if both p > 0 and ^ > 0. These zero-dimensional spaces are henceforth excluded from consideration. If we identify C with 2^^", it is clear that H(p, q) a 3^^ whenever p + q = k. Actually, more is true: 12.2.2. Proposition. J f ^ is the sum of the pairwise orthogonal spaces H(p, q), where p -\- q = k. Proof. Suppose (p, q) ¥" (r, s), p -\- q = r + s = k, fe H(p, q), and g e H(r, s). Then p — q ^ r — s,so that

f (fg)(e''Ode = (fg)(0 f e''^-'^^^-^''de = o

J —n

J —n

for every C e 5. This gives the orthogonality

jfgdd^O.

256

12. Unitarily Invariant Function Spaces

Next, fix fej^j^. Then / = /o + / i + ••• +/k, where /^ has bidegree (i, k — i\ so that A/j has bidegree (/ — 1, A; — i — 1) (or is 0, when f = 0 or i = /c). No cancellation can therefore account for the vanishing of the sum k

XA/; 1 = A/=0.

i= 0

It follows that each /• is harmonic. Thus fi e H(U k — i). This completes the proof. 12.2.3. Theroem. L^((T) is the direct sum of the pairwise orthogonal spaces H(p, q\0 < p < co,0
\fQ do,

Js

12.2.5. Theorem. Fix (p, q). To every z E S corresponds then a unique K^ E H(p, q) that satisfies (1)

('ip,/)(z)=[/,XJ

(/€LV))-

12.2. The Spaces H{p, q)

257

These kernel functions K^ have the following additional properties: (2) (3)

KXw) = K^iz) ^pj=

(z.weSl

j fiOK, da(0

(4)

Ku, = K,oU-'

(5)

K, =

( / e L'(a)) {JJe^UX

K,oV

for all V e ^ that fix z, and (6)

K,(z) = K^(w) > 0

(z,we S).

Note: Since K^ e C(5), the vector-valued integral (3) allows us to extend the domain of Up^ to L^((T).

Proof Let us write n for Upq. Since / -

(nf)(z)

is a bounded Hnear functional on L^, there is a unique K^ e L^ that satisfies (1). Since nf=0 for e v e r y / I H(p, q\ it follows that K^ e H(p, q). When / e /f(/7, ^), (1) becomes

In particular, K^(z) = [K^, X J . This proves (2), and (2) shows that (3) is just another way of writing (1). Since n commutes with ^i/, [/, K^J = (nf)iUz) = n(fo [/)(z) = [/o [/, K J = [/, K^ o t / - i ] for every feL^ia). (The last equality depends on the -^-invariance of G.) This proves (4), hence also its special case (5). Finally, KaXUz) = (K,oU-')iUz)

= KXz)

is another consequence of (4). It proves (6), because K^z) = IK^, K J > 0. The reason for mentioning (5) in the theorem just proved is that this property plays a role in the following proposition, which may be regarded as another characterization of K^: 12.2.6. Proposition. For each zeS, H(p, q) contains a unique f such that f{z) = 1 and f = f° Vfor every Ve^ that fixes z.

258

12. Unitarily Invariant Function Spaces

Proof. The existence of/follows from 12.2.5(5). To prove the uniqueness, assume z = ^1, without loss of generahty, and write points w 6 C" in the form w = (wi, W). The invariance/ = / ^ F shows then, for each Wj, that/(wi, w') is a radial polynomial in w', hence is a polynomial in | W p. Having bidegree (p, ^),/therefore has the form (1)

/ ( w ) = Zc,|w'pV?--(wO''-' i=0

where r = min(p, g) and CQ, ..., c^ are constants, CQ = 1. Differentiation of (1) gives

k=i

dWk dWk

i=o

where (3)

bi = (p-

i)(q - i)Ci + (i + l)(n + i - l)Cf+i

(0 < i < r).

Since/is harmonic, (2) vanishes, so that bi = 0 for all i, and (3) successively determines Ci,..., c^. 12.2.7. Theorem. Suppose T: H(p, q) -^ H(r, s) is linear and commutes with %. When (r, s) # (p, q\ then T = 0. When (r, s) = (p, q\ then T = cl, where c is a constant and I is the identity operator. Proof. Let K^ e H(p, q) be as in Theorem 12.2.5 and let L^ be the corresponding function in H(r, s). \{Ve% fixes z, then (1)

( T X J o V = T(K, o V) = TK,

by 12.2.5(5), since T commutes with ^ . By Proposition 12.2.6, (1) shows that there corresponds to each z e 5 a c(z) e C such that TK^ = c(z)L^. Hence (2)

(TK^Xz) = c(z)LXz). By 12.2.5(6), L^z) is independent of z. If w = Uz, then, by 12.2.5(4),

(3)

(TXJ(w) = (TK, o u- ')(Uz) = (TK,)(z).

We conclude from this that c(z) = c, the same for all z G S.

12.3. ^-Invariant Spaces on S

259

If/G H(p, q), 12.2.5(3) shows that (4)

/=

{fiOK.dcTiO,

Apply T to (4), and use TK^ = cL^: (5)

Tf=

IfiOTK,

for every/G H(p, q). If (r, 5) ^ (p, qX n,J= theorem.

dcT(0 = c {f(OL,

da(0 = cn,J,

0. If (r, 5) = (p, q\ n,J = f. This proves the

12.2.8. Theorem. Each H(p, q) is '%-minimal. More explicitly, the assertion is that H(p, q) has no proper ^-invariant subspace. Proof. Since H{p, q) is finite-dimensional, it is closed in L^{o). Suppose Y is a ^-subspace of H{p, q). Then so is 7-^, the orthogonal complement of Y relative to if (p, q). The projection of H{p, q) onto 7, with null-space 7"^, commutes with '^, and would violate Theorem 12.2.7 (the case (p, ^) = (r, s)) unless 7 = H(p, ^) or 7 = {0}; for if cl is a projection, then c = 1 or c = 0.

12.3. ^-Invariant Spaces on S 12.3.1. Notation. In the rest of this chapter, Q will stand for the first quadrant of lattice points in the plane. Thus Q consists of all ordered pairs (p, q) in which p and q are nonnegative integers. For Q c: Q, the algebraic sum of all ff (p, q) with (p, g) e Q will be denoted by EQ. We adopt the convention that E^^ = {0} when Q is empty. The letter X will stand for any of the Banach spaces C(S) or L^((7), 1 < p < 00. The X-closure of E^ will be denoted by X^. Trivially, every X^ is a closed ^-space in X. The proof of the converse is the main topic of this section. We begin with the simplest case, X = L^ = L^ip). 12.3.2. Theorem. If 7 is a <%-invariant closed subspace of L^{G\ and if Q is the set of all (p,q)^Q such that Up^ 7 ^ {0}, then 7 = (L^)^. Proof Pick (p, q) e Q. Recall that Ttp^ is the orthogonal projection of L^ onto H(p, q). Since 7 is '^-invariant and itpq commutes with %, Kp^ 7 is a

260

12. Unitarily Invariant Function Spaces

nontrivial ^-space in H(p, q). The ^-minimality of if (p, q) (Theorem 12.2.8) shows therefore that Up^ Y = H(p, q). Let YQ = {/e Y:7ipqf= 0}, and let Y^ be the orthogonal complement of YQ in Y. Then YQ is ^-invariant (being the null-space of a map that commutes with '^X hence so is Y^. Moreover, TT^^: YJ -^ /f(p, g) is an isomorphism, whose inverse we denote by A. Choose (r, 5) e Q, (r, s) # (p, ^), and consider the linear map T = n^s" ^• It is clear that T commutes with ^ and that T carries H(p, q) into H{r, s). By Theorem 12.2.7, T = 0. Hence n,, Y^ = {0} for every (r, s) 7^ (p, ^). By Theorem 12.2.3, Y^ = if(p, ^). Thus H{p, q) c: Y, for every (p, ^) G Q. In other words, (L^)Q CZ y. Since TT^^ 7 = {0} for every (r, 5) not in Q, another look at Theorem 12.2.3 completes the proof of the theorem. The following lemmas will make it easy to pass from L^ to X. 12.3.3. Lemma. If feX

then U -^ fo U is a continuous map of % into X.

Proof If e > 0, then || / — ^|| < £ for some g e C(S). There is a neighborhood N of the identity in ^ such that ||^ - ^ o C/||^ < 8 for every U eN. Since I / - / 0 U\ <\f-g\

+ \g-goU\

+ \(g - / ) o u\,

we have || / — / ° U\\ < 38 for every U e N. 12.3.4. Lemma. / / Y is a closed %-space in X, then Y n C(S) is dense in Y. Proof. Pick/ e 7, choose N as in the proof of Lemma 12.3.3, let ^:^ be continuous, with support in N and ^ij/ dU = 1. Define

(1)

g=

^ [0, 00)

fHu)foUdu.

The integrand is a continuous 7-valued function. Hence geY. If and Vci — (,, the in variance of the Haar measure dU shows that (2)

^(0=

{^{VV-')f{UeddU.

Thus geY n C(S), Finally, the relation (3)

f-9=

U(UXf-foU)dU

gives 1 1 / - ^11 < 3£, since 1 1 / - / ° U\\ < 3si{U e N.

Ve^

12.3. ^-Invariant Spaces on 5

261

12.3.5. Lemma. If Y a C{S\ Y is a %-space, and some g e C(S) is not in the uniform closure of Y, then g is not in the L^-closure of Y. Proof There is a /x G M{S) such that J /^/z = 0 for a l l / e 7, but \ g dpi = I. There is a neighborhood N of the identity in '^ such that RQ \ g o JJ dfi > ^ for every U e N. Associate i/^ to N as in the proof of Lemma 12.3.4, and define A e C(S)* by (1)

Ah =

dfiiO

HU)h{UOdU.

The Schwarz inequality shows that the square of the absolute value of the inner integral in (1) is at most

(2)

r W' dU f \h(UO\' dU = ml

J<^ by 1.4.7(3), so that

J%

(3)

|A/i|
Js

(W da,

Thus A extends to a bounded Hnear functional on L^(cr). If we interchange the two integrals in (1), we see that A / = 0 for every feY, whereas Re Ag > | . This completes the proof. 12.3.6. Theorem (Nagel-Rudin [1]). / / Yis a ^-invariant closed subspace of X, and ifQ is the set of all {p, q) e Qfor which Up^ Y ^ {0}, then Y = X^. Proof Recall that the domain of Up^ has been extended to L^{o) :=> X, by 12.2.5(3). Define 7 to be the L^-closure of 7 n C, where C = C(S). Since Y is X-closed, 7 n C is uniformly closed, so that Lemma 12.3.5 gives (1)

YnC

= Y

nC.

Observe next that 7 n C is L^-dense in 7, by definition, and is X-dense in 7, by Lemma 12.3.4. Since each Tip^ is X-continuous as well as L^-continuous, it follows that TipqY = {0} if and only if np^Y = {0}. Theorem 12.3.2 shows therefore that 7 is the L^-closure of £Q. Since EQ C= C, another application of Lemma 12.3.5 gives (2)

Y r\ C = uniform closure of E^.

Since 7 n C is X-dense in 7 (Lemma 12.3.4), (1) and (2) imply that 7 is the X-closure of E^. This is the assertion of the theorem.

262

12. Unitarily Invariant Function Spaces

The following consequence will be used later: 12.3.7. Theorem. Let Ybe a ^-invariant closed subspace ofX. If there exists anfeY and age H(p, q) such that

(!)

J/S da

^0

then H(p, q) c= Y. Note that (1) is particularly easy to verify (without computation) if f=gil/ for some positive function il/. Proof. By 12.2.5(3) and Fubini's theorem,

J 9^pqfd(T = J fn^

da = \ fg da ^ 0,

since TT^^^ = g. Hence n^^f ^ 0. Now apply Theorem 12.3.6. Theorem 12.2.7 was used in the proof of Theorem 12.3.2. Here is another consequence: 12.3.8. Theorem. 5wppo5^ T.X^^X^ is a continuous linear map that commutes with %. Then there exist c(p, q) e C, for every (p, q) e Q, such that (1)

Tf = c(p, q)f

for all fe H(p, q).

The numbers c(p, q) determine T. Formally, i f / = I/^, with/^, e H(jp, q\ then T / = Sc(p, q)f,^. Thus T is a multipHer transformation. Proof. Ufs Tmaps H(p, q) into H{r, s) and commutes with ^ , hence n^sTf = 0 if feH(p, q) and (r, s) # (p, q). Consequently, TH{p, q) is a (finite-dimensional, hence closed) '^-invariant subspace of X2 that is annihilated by TC^^ whenever (r, s) ^ (p, q). By Theorem 12.3.6, TH(/7, q) = H(p/q) or {0}. Now (1) follows from Theorem 12.2.7. Here is an apphcation: 12.3.9. Theorem. There is no continuous linear projection of H\S).

L^((T)

onto

In other words, H^ is an uncomplemented subspace of L^ The proof is almost the same as in the case n = 1; see Hoffman [1], p. 154, or Rudin [5].

12.3. ^-Invariant Spaces on S

263

Proof, Assume, to reach a contradiction, that there is a continuous linear projection P of L^ onto HK Define T: L^ -^ H^ by (1)

Tf=

\lP(foU-')-]oUdU,

It is easy to verify (see Rudin [5], or Theorem 5.18 in Rudin [2]) that T is then a continuous Hnear projection of L^ onto H^ that commutes with ^ . Thus T is as in 12.3.8(1) (where X^= X2 = L^), with c(p, 0) = 1 for all p, and c(p, ^) = 0 whenever (^f > 0. This implies that

(2)

Tf= cur,

the boundary function of the Cauchy integral of/ But Theorems 6.3.3 and 6.3.5 show that the Cauchy integral of a positive/e L^ lies in H^ if and only if/e L log L. Hence T cannot map all of L^ into H^. 12.3.10. We conclude this section with a rather curious characterization of A(S), the restriction of the ball algebra A(B) to S, in terms of the behavior on certain circles on S. Fix r, 0 < r < 1. Associate to each orthogonal pair of vectors C, ^ in 5 (i.e.,
r^e")

= tC + (1 -

t'y"e'\

This is the intersection of S and a complex Une orthogonal to C> whose distance from the origin is t. I f / e A(B\ then

(2)

X^m

+

(l-t'y^nri)

is a function in the disc algebra whose boundary function i s / o r^ ,^. This obviously necessary condition turns out to be sufficient as well: 12.3.11. Theorem. Fix r, 0 < t < 1, and define T^,^ as in §12.3.10. Iffe C(S) and iffo F^^ extends to a member of the disc algebra for every orthogonal pair of vectors C, y\ in S, then f has a continuous extension to B that is holomorphic in B. Proof Let Y be the class of a l l / e C(S) that satisfy the hypothesis. It is clear that y is a closed subspace of C(S). If U e%, then

(1)

ur^_, = r.

Thus Y is ^-invariant. By Theorem 12.3.6, it is enough to show that Y contains no if (p, q) with ^ > 0.

264

12. Unitarily Invariant Function Spaces

Fix (p, q), q > 0, let h(z) = 2?(z2)^ take l^ = ei,rj = 62- Then h e H(p, q) and (2)

ih o r^J(e'') = t^il -

tyi^e-'^\

which has no extension to the disc algebra. Thus h$Y, and the proof is complete. With a stronger hypothesis (involving all complex lines that intersect B, not just those at distance t from the origin) this was first proved by Agranovskii-Valskii [1]. Stout [6] extended their result to arbitrary bounded regions in C" with C^-boundary. The theorem fails when t — 0, i.e., when the complex lines through the origin are the only ones that occur in the hypothesis. To see this, take/e C(jS) so that fie'^rj) = /(rj) for all f/ e 5, 161 < 7c, but which is otherwise arbitrary. For instance,/could be identically 0 on some circular open set in S without being 0 on all of S. Theorem 12.3.11 is most interesting when t is near 1, since the complex hues that are involved do not enter the ball tB,

12A. ^-Invariant Subalgebras of C(S) 12.4.1. If Q is any subset of the quadrant Q, as in Section 12.3, the uniform closure of the linear span of the spaces H{p, q) with (p,q)eQ will from now on be denoted by Y^. Theorem 12.3.6 shows that every closed ^-invariant subspace of C(S) is a 1^; the converse is trivial. The problem to which we now turn is to find those sets Q cz g for which the corresponding ^-space Y^ is an algebra, relative to pointwise multiplication. We call such sets algebra patterns. It will be helpful, throughout this section and the next, to think of Q as a geometric object, embedded in the plane, and as a semigroup relative to coordinatewise addition, rather than just as a collection of ordered pairs. When n > 3, the algebra patterns (hence the ^-invariant algebras) can be completely characterized by a simple combinatorial criterion (Theorem 12.4.5). Some consequences of this are summarized in Theorem 12.4.7. The case n = 2 is rather different and more complicated. It is taken up in the following section, 12.4.2. The case n = 1 is so simple that we dispose of it immediately: If Q is any additive semigroup of integers, then the set of all continuous functions on the unit circle whose Fourier coefficients vanish on the complement of Q is a closed ^-algebra, and there are no others. From now on we assume therefore that n > 1.

12.4. -^-Invariant Subalgebras of CiS)

265

12.4.3. The Spaces H(p, q) • H{r, s). If (p,q)eQ and (r, 5) e Q, then H(p, q) • H(r, s) is defined to be the vector space of finite sums l^fiQi with /^ e /f (p, ^), ^f G H(r, 5). It is clear that each H(p, q) • H(r, s) is a finite-dimensional '^space and that it is therefore a sum of finitely many H{a, b)'s, by Theorem 12.3.6. This theorem shows also that 1^ is an algebra if and only if f/(p, q) • H{r, s) c: y^ whenever (p, ^) e Q and (r, s) e Q. We associate with each pair of points (p, ^) e Q and (r, s)eQ the number (1)

^ = i^(p^q;r,s) = min(p, s) + min(r, (^f). It is easy to check that jj, is also given by ju = min(p + (^f, r -h s, p + r, ^ + s).

12.4.4. Theorem. Ifn>3, then (1)

(p, q) e Q, (r, 5) G g, anJ // = /i(p, q\r,s) is as above,

H(p,q)-H(r,s)=

f^Hip

j=o

+ r - j , q + s - j).

When n = 2, then H(p, q) • H(r, s) is a sub space of the sum in (1). As we shall see in the next section, equahty can actually fail in (1) when n = 2. When n > 3, the theorem has the following consequence: If Q is an algebra pattern that contains (p, q) and (r, 5), then, obviously, the left side of (1) lies in Y^, hence so does every summand on the right of (1); this says that Q contains the points (2)

(p + r -hq^

s-j)

(0 <j < fi).

These are ju + 1 adjacent lattice points on a line of slope 1 (parallel to the diagonal of Q) whose highest point is (p, q) + (r, s). Proof, We begin by proving the inclusion (3)

H(p,q)'H(r,s)<^

Y.H(p + r - j , q + s - j)

for any n > 2. Let/G H(p, q\ g e H(r, s). Then/^ is a polynomial that is homogeneous of degree p -\- q -\- r -\- s, with respect to real scalars. By Theorem 1213, fg

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12. Unitarily Invariant Function Spaces

coincides therefore on S with a finite sum of spherical harmonics of degree at most p -\- q -\- r -\- s. This yields an orthogonal decomposition (on S)

(4)

fg-lKt

in which h^t e H(a, b),a-\-b