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MIXED NORM GENERALIZATIONS
OF WEIGHTED BERGMAN SPACES
IN THE UNIT BALL OF [IN
By

Steven Charles Gadbois

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1985

ABSTRACT
MIXED NORM GENERALIZATIONS
OF WEIGHTED BERGMAN SPACES
IN THE UNIT BALL OF It“
By

Steven Charles Gadbois

Let f be an analytic function in the unit ball B of it” for which the "mixed

norm"
(I01 (I33 |f(rt)|P dc(t))q/pw(r)r2N-1 dr)"q

is finite. Here 0 < p < 00, 0 < q < 0°, 0) is a suitable radial weight function, and o
is normalized Lebesgue measure on as. Note that when p = q, because of the
"polar coordinates” formula, the space of all such functions is just the Bergman
space with weight (0. General mixed norm spaces were studied extensively by
Benedek and Panzone.

We begin by generalizing a collection of results gotten by Luecking forthe
Bergman spaces or the Hardy spaces. Boundedness of certain Bergman
projections is proven first, using vector-valued integration and some facts due to
Forelli and Rudin. Representation of the dual space of our mixed norm Spaces
follows from this. Then a representation of functions in our mixed norm spaces is
obtained (by using duality) and several equivalent norms are produced (by
refining arguments of Luecking).

We also state a general "Carleson measure theorem" for our mixed norm

spaces whose proof depends largely on geometry and the connection between

Steven Charles Gadbois

the nonisotropic metric and the invariant Poisson kernel. Several consequences
are noted, including a theorem originally due to Cima and Wogen. Other related
methods and results are given, among them a Carleson measure theorem for

mixed norm spaces in the polydisc, a generalization of a result of Hastings.

To my wife, parents, and family
for their patience and love

ACKNOWLEDGMENTS

I am indebted to my advisor, Professor William Sledd, for his suggestions,
support, and patience during my work on this dissertation and the reading that
preceded it. I have benefited greatly from his knowledge of the literature and his
remarkable intuition.

There are many others at Michigan State University whom I must also
thank, among them Professors Sheldon Axler, Wade Ramey, and Joel Shapiro,
for many inspiring analysis courses and seminars, and Professors Habib Salehi

and Lee Sonneborn, for special guidance and advice.

TABLE OF CONTENTS

INTRODUCTION

CHAPTER 1
§1. Notation and Definitions
§2. Basic Properties of Ali:
§3. The Nonisotropic Metric and the Pseudohyperbolic "Metric"

CHAPTER 2
§1. Boundedness of the Bergman Projection on LE5
§2. Representation of the Dual Space of Ali?
§3. Two Norm-Representation Theorems
§4. Representation of Ali? Functions

CHAPTER 3
§1. Some History
§2. Some Estimates
§3. A Carleson Measure Theorem
§4. Some Corollarles
§5. Related Methods and Results

BIBLIOGRAPHY

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41

INTRODUCTION

Let f be an analytic function in the unit ball B of it” for which the "mixed

norm"
(I01 (I38 |f(rrc)lp do(‘c))q’pw(r)r2N'1 dr)1/q

is finite. Here 0 < p < 00, 0 < q < 00, a) is a suitable radial weight function (e.g.,
(o(r) = (1 - r2)“ for a > -1), and o is normalized Lebesgue measure on BB. The
space of all such functions is denoted by Afi’j. Note that when p = q, because of
the "polar coordinates" formula, this is just the weighted Bergman space A50.
General mixed norm spaces were studied extensively by Benedek and Panzone
in [2].

The preliminaries are taken care of in Chapter 1. All notation and
definitions are given, several metrics to be used later are discussed, and ABE is
shown to be a Banach space for certain to.

Chapter 2 generalizes results gotten by Luecking in [11] for the Bergman
spaces Aior the Hardy spaces HP. Boundedness of certain Bergman
projections is proven first, using vector-valued integration and some facts due to
Forelli and Rudin in [7]. Representation of the dual space of Afj for certain p, q,
and (0 follows from this. Then by refining arguments of Luecking a representation
of functions in AZ? is obtained (duality is also used) and several equivalent

norms are produced.

In Chapter 3 we state a general "Carleson measure theorem" for our
mixed norm spaces whose proof depends largely on geometry and the
connection between the nonisotropic metric and the invariant Poisson kernel.
Several consequences are noted, including a theorem originally due to Cima
and Wogen in [4]. Other related methods and results are given, among them a
Carleson measure theorem for mixed norm spaces in the polydisc which

generalizes a result of Hastings in [8].

CHAPTER 1

In section one of this chapter, a mixed norm generalization of the
weighted Bergman space in the unit ball of it“ is defined, and notation is set
forth. Basic properties of our spaces of functions are given in section two, and the

various metrics we will find useful are described in section three.

§1 Ngtation and Definitigns

The classical Bergman space A9 on the unit ball B = BN in [EN (0 < p < 00)

is the set of functions f e H(BN) satisfying
(IB |f(z)|F’ dm(z))1/p < oo.

Here N is a positive integer, [IN is equipped with the usual inner product defined
for z =(21, . . . ,zN) and w = (w,, . . . ,wN) in It” by (z, w) = .51 2in and with the
associated norm [2] = (z, z)1/2, H(BN) is the set of holomorphic functions on BN, and
m is Lebesgue measure on BN normalized so that m(BN) = 1. Using "polar

cooordinatesv. (see [15], 1.4.3), this integral may be written as
(2N I! (Is lf(r1:)|P d0(‘t)) rZI‘I'1 dr)1/p

where l = [0, 1), S = SN = aBN = {z e (EN | [2| = 1}, and o=oN is the rotation

3

invariant positive Borel measure on SN with C(SN) = 1.
We shall study the following weighted "mixed norm" generalizations of the
Bergman spaces. If 0 < p, q < 00 and it o) is a nonnegative weight function on

[0, 1) satisfying I, (o(r)r2N" dr < 00, define
A53 = {t e H(B) | [ltllpm = (II (IS |I(l"c)|p do(t))q/pu)(r)r2N'1 dr)1/q < ea},

Note that when to a 1 and q = p, this is precisely the Bergman space. Also note

that
ltlpq.=( (.l Hill We art-dove.

where, for O s r < 1, fr is the function defined on S by Ir(’t) = f(rt). Using this

notation, we also define
A? = {t e HIBll iitil..,,,.. = (l. ltiitoaemoN-t dot/o < co}.

Ap°°= HP: {fe H(B)| llfllpoo= sup Ilf “9(3) < 0°}.

00"(1

and A°°°° = H°° = {i e H(B )Illfllm— = 3,ng lli noelL < 0°}-
The Spaces HID and H°° are the classical Hardy spaces. In the special case that
w(r) = (1 - r2)“ with a > -1 [or co 5 1], we write A2? [or APO] instead of A53, and
II “pm [or II “qu instead of II “pm. We also write dma(z) = (1 - I212)“ dm(z).
The set of (equivalence classes of) measurablefunctions satisfying the

integrability condition defining AZ? is denoted by L23. It is easy to check that

it + 9|!ng —<. llfllgm + Melitta

where S = min{p, q, 1}; thus (A23, ll NEW») and (LEE, ll Ilgfq’w) are metric
spaces, and are normed linear spaces if 1 s p, q s 00. The basic reference for
mixed norm spaces (including, but not limited to, our spaces LIZ?) is Benedek
and Panzone, [2].

We will follow the usual practice of writing c and C for positive constants
("small" and "large" respectively) that may vary from line to line. Dependence on
some parameter(s), for example p and N, may be emphasized by writing
C = C(p, N). If two positive quantities A(x) and B(x) have ratio bounded above
and below as x ranges over some index set X, we say A(x) and B(x) are
equivalent, and we write A(x) ~ B(x) for every x e X. For example, (1 - r2) ~ (1 - r)
for every 0 s r s 1; this particular fact will be used repeatedly without mention.

Note that by two applications of Holder's inequality, Li‘s“ C LE}?

ifehher

p2 5. p1 and q2 < q1 and (0:1 + 1)/q1 < (a2 + 1)/q‘2
or

p2 5 p1 and q2 =q1 and 0:13 a2.

In either case, the containment is proper (unless p1 = p2, q1 = q2, and a1 = (12),
since then there is some 3 satisfying N/p1 + (or1 + 1)/q1 < s < N/p2 + (ct2 + 1)/q2,
and f(2) = (1 - (2, C))'5 (with Ce 8 fixed) defines a function in A5322 not in

Alas“:

Ilill ‘4 = I. (IS I1 -<r§. orsp actor/pit -r2)°‘r2N" dr

PIEII‘X

~ I, ((1 - r)N‘SP)q/P(1 - r)°‘r2""1 dr
by Proposition 1.4.10 of [15], and this integral is finite if and only if
s < N/p + (or +1)/q.
For 1 s p _<_°°, p' is the conjugate of p defined by 1/p + 1/p' = 1. The dual
space of a Banach space X is denoted by X*. We will use 2 and w for typical
elements of BN, and we may write 2 = r1 and w = pn, where r, p e l and 'c, n 6 SN.

When N = 1, B1 and S1 will also be written as [D and TI respectively, and ll

will often be identified with [0, 21:) without warning.
§2 Basic Properties of A23

The completeness of Ali? is a consequence of the following growth

condition. Our statements and proofs here will resemble those in [16].
Proposition 1.2.1 : If f e A? (0 < p, q 500, -1 < a), then

|f(z)| s Cllfllp,q,a(1 - |2|)'(N/P+(°‘+1)/q> for every 2 e B
for some C independent of f.

Prggf : First suppose 0 < p < 00. Since |f|P is subharmonic, for 0 < r < p < 1 and

13 e S we have

Ittrolip 5 Is |f(pnll" P((r/p)i:. n) dam) s 2”(p-r)‘NIs |f(pnllpd6m).

 

(2.1) Iiirrliip- W” s Cllfpll Ins,
Here P denotes the invariant Poisson kernel defined for z e B and t; e S by

Ptz. i.) = ((1 'lZ|2)/|1 -<z. or)”.

For basic facts concerning the invariant Poisson kernel, see [15], section 3.3. If

q = 00, the result follows from (2.1) immediately. If 0 < q < 00, we then have

tin no In 1 -r)N°/P<1 -p>op2~-i up s anti...

Letting x = (p - r)/(1 - r), for 1/2 3 r <1 it follows that

> |f( row: (1 - r)(N’P+ (Mil/shit, 1 qu/p (1 - x)“ [(1 - r)x + r12N1dx

~ "(mud (1- r)(N/p+ (a+1)q)q J01 qu/p(1_x)a CIX

Cllfllpqa

so the result follows. If 0 s r < 1/2, the result follows from the maximum modulus
theorem.
Now suppose p = 00. Then |f(r1:)| s llfpll L°°(S) for 0 s r s p < 1, and the

result for 0 < q s 00 is proven by the same procedures. 0
This growth estimate will also be a consequence of Theorem 3.3.1.

r ll 1.2.2 : A? is a closed subspace of Li? (0 < p, q s 00, -1 < a), and is

hence complete. 80 A3)? is a Banach space if 1 ._<.. p, q s 00.

 

Prggf : Suppose fn —> f in LES with tn 6 A35. By Proposition 1.2.1 fn is uniformly

Cauchy on compact subsets of B, so fn is uniformly convergent on compact

subsets of B to some 9 since L3? is complete (see [2], p. 304). But g is analytic
_ - p

([15], 1.1.4) and f _ g a.e., so g 6 Ag? and fn —> g in A03. El

Prgpgsitign 1.2.3 : If f e Afj (o < p, q < oo), then [iii] llfr-fllp’w = o.

This follows immediately from the dominated convergence theorem. (For
details, see [16], Proposition 3.3.) So the functions analytic in a neighborhood of

B form a dense subset of A23.
Th N ni r i M ri n h P h Ii "Metri"

There are several notions of "distance" in BN (or B1,, or SN) and each has
its own advantages. We will have occasion to use three.

The isotropic metric will refer to the usual metric in if“, and B(z, r) will
denote a corresponding ball, i.e., B(z, r) = {w 6 (EN | )z - W| < r}.

The nonisotropic metric d is defined on B]; by d(z, w) = (1 - (z, w)(1/2. It
satisfies the triangle inequality on B7] and is a metric on SN; see [15], 5.1.2.
Define 5(0, 5) = SN, and for 5 > o, p e (o, 1), and n e SN, define
§(pn, 5) = {1: e SN | (on, n))2 < 5(1 - p)}. Note that oénm, 5) ~ 5N(t - p)”; see
[15], 5.1.4.

The pseudohyperbolic "metric" p is defined on BN by p(z, w) = |<I>w(z)|

where (Dw is the automorphism of BN given for w #5 0 by

<Dw(z) = (w-(z, w)w/(w,w)- (1 - )W(2)1/2(z- (z, w)w/(w, w)))/ (1 -(z, w))

 

and for w = 0 by (Do(z) = -z. The corresponding "balls" are
E(w, 5) = (Dw'1(88N) = {z e BNI p(z, w) < 5} for w 6 EN and o < 5 < 1. Note that
mE(w, 8) ~ 82N(1 - Iwi)N+1; see [15], 2.2.7.

We will have need of the following.
Lemma 1.3.1 : Fix 0 < r <1 and 0 < 8 small. Then
(r - 8)/(1 - r8) < )2] < (r + 8)/(1 + r8) for every 2 e E(r, 8).
(Here E(r, 8) means E(w, 8), with w = (r, 0, . . . ,0) e BN.)
flog : Write z = (21, 22, . . . ,zN) = (21, z') and suppose z e E(r, 8). Then

8% - r21)2 > (r - 21)2 + (1 - r2))z'I?-,

i.e., 2r(1 - 82)Re 21 > (r2 - 82) + (1 - 82r2)|z1|2 + (1 - r2)|z').
Now, 4rRe 21 + r212]2 < r2 + 4r|z1| - r2|z1i2 3 4r < 4, so

(r2 - 82) + (1 - 82r2))zi2
= ((r2 - 82) + (1 - 82r2)|z1)2 + (1 - r2)lz’(2) + r2(1 - 82)Iz')2
< 2r(1 - 82)(Re 21 + riz')2/2) s 2r(1 - 82)()z1)2 + rlz'IZRe z, + r2lZ'I4/4)“2

< 2r(1 - 82)()z1|2 + )z')2)1/2 = 2r(1 - 82)|zl.

Hence ([2], 0') e E(r, 8), and (r - 8)/(1 - r8) < [2) < (r + 8)/(1 + r8). 0

 

 

 

CHAPTER 2

Most of the results of this chapter are generalizations of work of
Luecking in [11]. Section one deals with boundedness of the Bergman
projection, and this result is used to identify the dual Space of our mixed norm
spaces in section two. Sections three and four are concerned with

representations of the mixed norms and of functions in the mixed norm spaces.
§1 Boundedness of the Bergman Projection on L3?
Suppose s > -1. The Bergman kernel Ks is defined by
Ks(z. W) = (1 - lWlZIS/(l - <2. W>)N+1+s

forz, we BN. Note that:
(a) for fixed w e B, Ks( - ,w) 5 Ali?
and (b) for fixed 2 e B, Ks(z, - ) 6 L53 if II (1 - r2)$‘lco(r)r2"“'1 dr < 00, 9.9., if
co(r) ~ (1 - r2)“ with sq > -(a + 1). (But Ks(z, - ) is not conjugate holomorphic
unless s = 0.)
Both observations follow because the respective denominators are bounded
above and below in B.

The Bergman projection Ts is defined by

10

11

Tsf(z) = (N a S) L, Ks(z, w)f(w) dm(w)

for z e BN and f for which the integrands are in L‘(dm). In general, the binomial
coefficient (N NS) is I‘(N + S + 1)/I‘(N + 1)I‘(s + 1). It is clear that, for fixed S, Tsf is
holomorphic when defined.

In this section, a condition on s, p, q, and on will be found which ensures
that Ts is bounded on LEE; there will be no dependence on p other than p 2 1. In
[7], Forelli and Rudin Showed that Ts is bounded on LP(dm) (1 s p < 00) if and only
if (s + 1)p > 1. Then Békollé ([1]) showed that Ts is bounded on LP(dma)

(1 < p < 00, -1 < a) if and only if (s + 1)p > a + 1. (He actually showed this for
more general weights satisfying a ”8,, condition”, a condition analogous to

Muckenhoupt's Ap condition introduced in [13].) An important tool will be the

following pair of facts due to Forelli and Rudin in [7], Proposition 2.7:

(1.1a) I8 |Ks(z, w)|(1 - |w12)‘° dm(w) .<. C(1 - (2)2)°°for every 2 e B if 0 < c < 3 +1;

(1 .1b) I8 |Ks(z, w)|(1 - (2|2)'° dm(z) 5 C(1 -|w12)cfor every w e B if 0 < c + s < S +1.

Theorem 2.1.1 : Ts maps LEE (1 s p s 00, 1 < q < 00, -1 < or) boundedly into
AZ? if (s + 1)q > on + 1. Furthermore, Tsf = f and Tsf a f(0) for every f e Afifl.
Proof : As noted in [15], Proposition 7.1.2, Tsf = f and TSTEITE) is true for
is H°°(B), hence for f e A? by density of H°°(B) in Af’xq, once continuity is
vefified.

If 1 < p < 00, vector-valued integration and HCIder's inequality yield that

H (1.30,“ Lp(S) = CHI] IS Ks(r ' t mepm) dG(TI)p2N-1 deILP(S)

12

s CI. II-Is Ks(r- .Pnlfpm) domilltpapm‘ dp
=c I. (IS IIS Kart. Pnlfpm) domll" dott))1’9p2N"dp
s C II [IS (IS le(r’c. pnlllfptan dcml)
><(IS IKSUI. on)! d0(n))p’p'd0(t)]“°pz”" do-

But 3 IKs(l"l:, pn)| dam) is independent of 1:, so by Fubini's theorem, the

expression above is less than

cI. [(ISIK.Irr. pn)l d0(n))p""+ 1(IS prmllp do(n))]"ppz”" dp
= cl . I lesm. pn)l doth) Iliplltp.s.p2N-‘ do

This estimate can also be verified in a similar way if p = 1 or 00.

Using this estimate and Ho'lder's inequality, we have

||Tsf|| 3,... = oi. ||(Tsf).|lEp(s)(1 - Wit

3 CI . [I . IS )Ksm, pm) dom) Ilfplle(s)p2N-1 dp]q(. - ,2)a,2N-t d,
s cI. (IB IKs(rt. pniiii 12)-.. dm(pn))°/q'
><(IB le(r'r. pn)l(1 -p2>tqlli.,l|Ep.s) dm(pn))(1 - recto“-1 dr.

where 8 will be Chosen later. But
Is leirr. pnllii - pzl‘i’q' dm(pn) S C(1 - r2)“
by (1.1a), as long as 0 < 8q' < s + 1. Using this and Fubini's theorem, we have

llTsill 3...,

13

< cI.( ((1 - r2>-11')1/1'(IBIK.Irt. pn)l(1 -p2>11II1.,IIEp.., dm(pn))(1 - r2)11r2N-1 dr
= cI. Ilip ||Ep<s)( (1 - 1221111. IS IK.(r1:. pnii dam) (1 - r2111-11r1'1-1 dr 112”1 do.

But

II J’s IKsIVT» PTlII C1001) (1 " Izla'f’qrm" dr
=IBIK.<r1. pnilti -r2)-<2+11>dm(n)
5 C(1 - p2)'('a+5QI = C(1 - p2)a-8q

by (1.1b), as long asO<-a+8q+s<s+1. 80

11.111 .01 11.11121...“ >111p2-=~1dp Cllfllq

PQ‘X pq,'o<

To choose suitable 3, note that there exists 8 satisfying
O<8q'<s+1andO<-ot+8q+s<s+1
if and only if

(8+1)q>0t+1. [I

As in [7], p. 594, we immediately get the following.

Qorollam 2.1.2 : For1_<_p<°°,1<q<°°,and-1<or,

nt||mm 5 cl] Re Illpm for every f e H(B) with 1(0) = o.

14

Proof: Choose s>(or+1)/q - 1. Let u = Refand fix 0 < r, p <1. Then
fp = Ts(fp) = Ts(fp + fp) = 2Ts(up), so

Is |fp( rt)lpdo() )=2PIST |Ts (r)t|pdo 1:).
Thus

“I”:q o. =I. (SI Ifp (rt) [9 do“ ))q/p(1 _r2)ar2N-1 dr
= 2“ I ( s [Tsup (r1: )|9 do( ))q/p( 1 _ r2) 01er 1 dr

=2qllT.U pllq <2‘W‘llup llq

PJI‘X- FUN

and the result follows upon letting p —> 1. Cl
The "inner norm" LP(S) was not critical in Theorem 2.1.1. If N = 1, the

Bergman kernel satisfies Ks(re‘°, pelt) = Ks(rei(e ' (P), p) and a change of variables

is possible, so LP(S.) may be replaced by any Banach space X on 81 satisfying

|| FeIIX s CH F||X for all 6 e [0, 21:) and for all F e X. (Here, and only here, we

write Fe forthe function defined on S1 by Fe(ei‘P) = F(ei(‘p ' 9)). No confusion with fr

should result.) The beginning of the proof is then

”In K. Ire" pe‘111.I e19) dellx= llIn Ks(r. pate-111N211) dellx
=||InK.( Krpei1)I1...)I (e') d<I>HX< <0lli.llxI..IK.Ir.pe11)Id<p

and the rest of the proof goes through.

If N > 1,this can be imitated to some extent. Write e1 = (1 , 0, . . . ,0) 6 SN.
For 16 8, let III be a unitary transformation on S (i.e., (11111., [11112) = (n1, n2) for
every 711,112 15 S) with [1181 = 1:. So

15

I S K.(ri. ptlfphil don) = Is K.(rl1."1:. pu."vlf.(v) don)
= IS K.Ire.. pnli.Iu.n> doInl

where n = 11.47. If X is a Banach space on SN satisfying

(1.2) II F(u,n)II X s CII F(-)IIX for every n 6 SN and every F e x,

then III. K.Ir- . pvl1.Ivl doIvlllx s C "1.1). I. lK.Ire.. pnll dam). The
inequality (1.2) is trivial when X = C(SN) for any N. When N = 2, we may take

‘1 “‘2
“1: = ..
T2 ”‘1

A computation then shows that It. - 1:2] = “‘11“ - [.1121] I for every 1:1, 12, n e 82, so
that (1.2) holds when x = LipaSZ, o < a s 1. (Recall that
Lipa82= {fe C(82) I "(1:1) -f(1:2)I/II1- tzla < K(f) < 00 for all 11, 1:26 82 with 11¢12}.)

§2 Representation of the Dual eeeee (21A;q

Representation of the dual space of AI”: will follow from boundedness of
the Bergman projection (Theorem 2.1.1). The case N = 1, ct = 0 was handled by
Shapiro in [16], Corollary 3.6. In [11], Theorem 2.1, Luecking identified the dual
space of Ag using the boundedness of the Bergman projection on A: and

P
ANN”.

- p’cl’
leen 9 e Autioq’f

Lgf = (f, g) = I315 dm for f e AE’Xq. By two applications of Hb'lder's inequality,

I

define the linear functional Lg on A2? by

16

KI: g>I S IIfIIp,q,aIIgIIp',q',a(1-q°)'

Theorem 2.2.1 : Suppose that -1 < or, 1 < p < 00, and max{1, a + 1} < q < 00.,
Then the map taking 9 to L9 is a linear homeomorphism of Agg:q,)onto the

m
dual space of A,x .

Emof : As noted above, any 9 e Azgjqqdefines a bounded linear functional
Lg on Afij, with I LgII .<. “9 up...“1 _q.,.

Now take any L e (Agjf. Extend to L e (Eff by the Hahn-Banach
theorem. Write 03(r) = (1 - r2)“, and note thatj e Leo if and only if j/oa”q 6 LEE. So
define A e (qu). by A] = L(j/m"q); then there exists some k e U” such that
A] = I B jk dm. (See [2], Theorem 3.1 forthe generalized representation theorem.)
Let h = REST/'5. We have h e Lg’gfq,)and Lf = (f, h) for all f e LEE. Since
K0(z, w) = Ko(w, z), Fubini's theorem implies that

(2.1) (101., 12) = (f., Tofz) for 111 6 LE:1 0 L2 and 12 e Lg’gfq,)n L2.

(To justify the application of Fubini's theorem, note that Tot1 and Tot2 are in L2 since
f1 and 12 are, either by Theorem 2.1.1 or by Békollé’s result.) Now, q > at + 1 and

q' > a(1 - q') + 1, so by Theorem 2.1.1, To is bounded on La? and Liam By

continuity of To and density of the respective spaces ([2], p. 308), (2.1) is also true

for f1 6 LIZ? and (2 E [$ng Let g = Toh. 80 g e Agar” and forfe Ali?

we have Lf = (f, h) = (Tot, h) = (f, Toh) = (f, 9), Le, L = Lg.
If g e Ai’giqqdetines the zero functional, then since Ko(z, -) e A? for
any fixed 2 e B, we have 0 = (Ko (2, -), g) = To g(z) = 97?), Le, g a 0. So the map

 

 

taking 9 to L9 is a one-to-one, continuous, linear transformation of Agfiiqqonto

 

17

(Af’xqf. By the open mapping theorem, the map is actually a linear

homeomorphism. Cl

One can use other 0.9., weighted) duality pairings (and other kernels) to

- PR
get other representations of (Au)

§3 Two Norm-Reoreeentetion Theoreme

In [11], Theorem 5.1, Luecking shows that
km
II‘IIIHp ~ (sup kZ:1|f(amk)IP(1 - rm)N)1/P for all f e H9

where 0 s r0 < r1 < . . . —> 1 and {amk} satisfies
(1)1amk|= rmforeach m=0, 1,2,. . .and each k= 1, 2, . . . ,km,
(2) rmSN C LkJE(amk, 8) for each m for some 8 = 8(p) sufficiently small,
and (3) E(amk, e) n E(amkv, e) = o for each m and each k¢ k' for some 0 < e < 8.
Such a set {amk} will be called an 15-8 lattice; notice that this differs slightly from
Luecking's use of the term, for he requires the condition BN = knJLkJE(amk, 8)
instead of (2).

A close analysis of Luecking's proof yields the following.

Theorem 2.3.1 : Fix 0 < p < 00, 0 < q < 00, and -1< on. Let rm =1 - 2"“ for
m = 0, 1, 2, . . . and suppose {amk} is an c-8 lattice for 8 = 8(p, q, or) sufficiently

small. Then

 

18
00
Ilfllp’q’a~ ( Z (k )i(amk)lp2-mN)q/22-m<2+1l)1/1i for everyte AEE.

Proof : Write Emk = E(amk,s 6,) Am = UE(amk,e e),r me- =(rm + c)/(1 + r me) and
m. = [(rm - e)/(1 - rme), (rm + e)/(1 + rme)). Then by subharmonicity of IfIP, the

separation property (3), and Lemma 1.3.1,

§|f(rna k-)|22 111“ <2 mN(z C Em k 1112 dm /m(E Emk))

s 02 mN2m1N+1l gIEmk IfIP dm = C2111IAm IfIF’ dm
= C2mIlm8 (IS |f(r1c)||° d<s(1:))r2N'1 dr

5 CZmIIlme '2'“ dr) II frmeII 52(3) 5 CIIfl'mcII 52(5)-

Since we may assume that e < 1/3 (at the cost of increasing the constant C), we

have rms < (rm + 1)/2 = r and thus

m+1'

§(§Ia |f(am ll12 m”)1’1’2 "1°11" < c: IIIrm+1IIEP(S)2-m(a+1)

<cl. 1111...,(1 -ror2~1dr=cltl...

In the other direction, Luecking uses a change of variables, Fubini's theorem, and

the ”denseness" property (2) and actually shows that
CIIfrmIIp LP(S)"‘ < CSDIIfrmEII EP (8) + ; |f(amk)|p2-mN
so

Cllfllq . q .. < § (Cfipllfrmh II [13.8, +2 |f(amk)IP2'mN)q/P2'm(°1+1)
.<. Kg (C8<1|Ifrm+1 "5%) + (E; If(amk)lpg-mN)q/p)2-m(a+1)

19
where K=1ifq$pandK=2q’F*1ifp<q. Thus

Cllfllgmx < CSQZ IIfrm+1IILP(3)2 mm“) + Cg (I; |f(amk)lp2'mN)q/p2-m(a+1)
s C8q||f|I 3 q o, + cg (g |f(amk)lp2'mN)q/92'm(a+1)

and the result follows for 8 sufficiently small. D

We wish to note here that the same proof works if (1-r)°‘ is replaced by any
radial weight a) satisfying co(2r - 1) s Cw(r) for every r 6 [1/2, 1).

The fact that IIfII dominates the weighted sum will also be a

11.12.01
consequence of Theorem 3.3.1.

As a consequence of this theorem, no such e-8 lattice can be a subset of
the zero set of an A? function not identically zero.

The second theorem of this section concerns subsets G of B that are
"large enough" so that “III“... and "fo II one are comparable for every
f 6 A33. (The constant of comparability may depend on G, as well as p, q, and 0).)

We make the following assumptions:

(1) 0=ro<r1<...—>1,

(2) {am} is an e-8 lattice for some a and 8 sufficiently small (8 = 8(p, q, K)
where K is as in (6) below),

(3
(4
(

(rm +1)/2 < r "HM for each m = 0, 1, 2, . . .for some integer M,
|(rm -rm -r

m rm)| >Syforeach m¢m' forsomeO<y<s,

m’ rm+1]’

(6
and (7

(rm+1)(rm+1rm) < K0.)( rm+1-1M)(rm+1+M rm+M) for eaCh m

l

)

5) co( r)< Cw()(rm+1) for each m and for each r e (r

) (o

) (co((r,,,+1)(rm+1 - rm))F’/q s Cco(r)(1- rm.) for each m and for each r

 

20

satisfying(|(r-m+1r/)(1 - rm+.r)| < yfor some p and q.
If p = q, these conditions are satisfied by rm =1 - 2"“ and co(r) = (1 - r2)0L with a > -1.
Conditions (3) and (4) say that rm —> 1 "not too slowly", while condition (6) says

that rrn —-> 1 "not too quickly", at least for weights 0) with suitably controlled growth.

Theorem 2.3.2 Szuppose 0 < p < q < 00, conditions (1 )- (7 ) hold, and
UUE(a 5)c G c 8. Then IIfII

amk, ~ IIfoIID'q’m for every fe ASE.

moo

fleet : Since we may assume that y < 1/3, condition (3) yields
rmM_2rm -1 <(rm -y)/ (1 -r mm1y)<(r +y)/(1+rmy)S(rm+1)/2.<_rm+M.
Then, as noted in the proof of Theorem 2.3.1, Luecking actually shows that
cllfrmll to... s 061111....” It... +5 |f(amkllp(1 -r...lN
so

CR ”Inn“ EPIs)w(rmlrm2”“(rm - in.-.)
‘- 05? IIfrm+MIIEP(S)m(rm)rm2N'1(rm - n.-.)

+ 2 (§( |f( amkllpU - r...)N)“’F’co(rm)rmZN'l(rrn - rm)
< K0511: ”frmllq M90) m), m2N-1(rm _ rm)

“1% (E213 |f( amkllpU ' r...)"’)“’pcn(rm)rm2""1(rm - rm_1 )

by condition (6). Thus for 8 small,

21

Z IIfrmIILP(3)(°(rm Irmz NIIm " r"1.1)
s 05 (.2111 2....ll1I1 - r...) '1rl11wtmirmZN-1Irm - r...)

(The reverse inequality is also true but is not needed here.)
Write lnn = [rm, rm“), Emk = E(amk, 7), Am = 9513111“): and

my = [(rrn - y)/(1 - rmy), (rm + y)/(1 + rm'y)). Then using condition (5) and the above

estimate,

lili..= .2. 1.11 l1 .npeo 21-1
S 0:50 IIffm+1IILP(S)w(rm+1)rm+12N-1(rm+1 - Irm)

s cg (g |f(amk)|P(1 - rm))N)2/po(rm)rm2N-1(rm - rm).

By subharmonicity of mi), disjointness of the Emk's, and condition (7), the above

sum is less than

cg (g (1 - rm NI Em, 11111 dm /m(Emk))q/Pco(rm)rm2N'1(rm - rm_1)
s cg(( (1-r M)-1I rnmr|i|r>dm)11/i>ro(r) rmZN-1Irm-rm_,)

$02 (I IS II(f’t) Ip xAm(rt) do(1: ) (It)(r)lrr2""1 dr)q/P.

Finally, using Jensen's inequality (since q/p 2 1) and disjointness of the les

(condition (4)). this is dominated by

0% I I (IS |f(r1c)|p )(Am(r1:) do(i:))‘i’pco(r)r2”1 dr
5 cg I lm.(Is |f(rt)|p xenon) do(t))q/pro(r)r2N-1 dr
5 CII (IS II(l"t)Ip xG(r1:) do(1r))‘i’po)(r)r‘ZN'1 dr

22

Luecking shows in [10] that IIIIIW ~ II foII p,a for every f 6 Ag if and
only if

(3.1) ma([D n B) ~ ma(G n B) for every isotropic ball B centered on li

when N = 1, p = q, and (00') = (1 - r2)01 with or > -1. (He also gives a more general
result of the same nature in [11].) In this case we can also show directly that

conditions (1) - (7) along with LIA) E(a 8) C G imply (3.1). The key is the

mk1

geometric estimate
km 2 C(rm2 - 82)/(1 - rm?) (m it 0)
where c = C(8, r1).

§4 Reoreeentetion of A125 Fonotione

The duality result (Theorem 2.2.1) and the equivalence of norms result
(Theorem 2.3.1) can be used to obtain a representation of Ag? functions as
sums of kernel functions. This generalizes Luecking's Corollary 4.4 in [11].

If p is a weight function on {0, 1, 2, . . .}, we write c = {ka}m,k e Ill;q if

0° K“ / 1/
(mgo (Kzfl 'ka'p)q pum) q = ”C” 9.0.1) < 0°"
Jheorem 2.4.] : Suppose that 1 < p < 00,1 < q < 00, and -1 < a. Let rm =1 - 2m
and cm = (1 - rm)1 + Nq'lp't 01(1‘11'lrmZN'1 for m = 0, 1,2, . . . , and suppose {amk} is an

2-8 lattice for 8 = 8(p, q, to) sufficiently small. Then every f 6 A? is of the form

23

K

0° '“ N 1
“2) = mgo k2“ ka‘UmU-(Z, amk»- -
for some C e tiff, and any f of this form is in Nix”.
Note that no claim of uniqueness of c e If? is being made.

Proof : AS in the proof of Theorem 2.3.1,

IIgIIp',q',a(1-q°)" (m( Z(§ |g(a mk)Ip) )Q/P (1 - rm)1+NqH/p +oI(1-'mq)r 2N- -'1)1/q

i.e., IIgII p',q',oi(1-q') N IIg( amk) II p',q',t)'

Thus the map R. A111 -> 119:" defined by (R9)...k = g(amk) is a linear

0([1- J
isomorphism. Hence :1 is one-to-one with closed range, and R‘: 113:1 —-> All? is
onto. (Since max{1, a(1 - q') + 1} < q' < 00, we have(AEq‘I :1?”qu by
Theorem 2.2.1, where the duality pairing does not involve a weight, and we have
(tips) ~ 111:? by Theorem 3.1 of [2], where the duality pairing does involve
the weight 1).)

To identify R*, take g 6 A5“? q )and c e ll‘fj, supposing first that C has
only finitely many nonzero terms. Then

 

I B IR‘cig dm = IR‘c. g) = Ic. Rg) = g (; c..,..dIa1.,.l)1>m
= g go...» ml}. gIz )(1 -Ia.....z>l-N-1 dm(Z)

=IBm( (Z i: cmkomfi - (z, amk))'N’1)g z) dm(z),

 

so R C(Z =2 g cmkom( (-z, amk))‘N'1. To get the result for general c e 113:1,

use finite approximations to c (for which the result was just verified), the continuity

24

of R*, and the fact that convergence in AZ? implies pointwise convergence. 0

 

CHAPTER 3

This chapter is concerned with "Carleson measure theorems". Some of
the classical results are described in the first section, some estimates (mostly
geometric) are derived in the second section, and the third section is devoted to
the statement and proof of the main theorem. Corollaries and related methods

and results are given in sections four and five.

§1 §Qme History

In the course of proving the corona theorem in [3], Carleson characterized
those finite positive measures u on B1 such that (J31 If]p du)”p .<. C||f|| Hp for every
fe Hp (O < p < 00). He showed that this holds if and only if [IS 5 C(1 - p) for every
set S of the form 8 = Sp60 = {rei9 | p .<. r<1 and 90 - 1t(1- p) 59 < 00 + 1t(1- p)}; such
a measure It is now often called a Carleson measure, and such sets 8 are called
Carleson sets. The necessity of this geometric condition is easily shown by the
proper choice of a function f e Hp which is suitably large on 8; this procedure will be
demonstrated in the proof of our Theorem 3.3.1. Carleson‘s proof of the sufficiency
of the geometric condition used a complicated covering argument. Hormander ([9])
derived a version for more general regions in it“ using a maximal function,
Marcinkiewicz interpolation, and a simpler covering argument. Using some of

Hormander’s ideas, Duren ([5]) generalized the theorem to the following:

25

 

26
(L3 Iflp2 du)1’92 .<. Cllflal1 for every f e H91 (0 < p1 s p2 < 00)
if and only if
us 3 C(1 - p)|°2’PI for every Carleson set S = Spoo-

The results mentioned so far have concerned the Hardy spaces; i.e.,
measures on B are compared to ”measures on 8". Other results have been
gotten concerning the (weighted) Bergman spaces; i.e., measures on B are
compared to measures on B. For example, there is a theorem due to Hastings

([8]) for the polydisc lDN (the product of N copies of [D = 8,):

(IBM [flpz dp)1/pz _<_ C(Jmulflpt dm)"PI for every analytic function f on [DN

(0 < 91 -<- 92 < GO)
if and only if

us 3 C'[(1 - p1) - .(1 - pN)]292’PI for every set S ofthe form 8 = Splemx- - «8390".
The result of Luecking in [11], already discussed at the end of Section 3 of
Chapter 2, can be viewed as a Carleson measure theorem for the measures
demainduced by "large enough" sets G in B1. Cima and Wogen proved a
Carleson measure theorem for weighted Bergman spaces in BN ([4]); the
Cima-Wogen theorem is a consequence of Luecking's general technique in [12],

and also of the proof of the main result here (see Theorem 3.4.3). The main

27

result here, Theorem 3.3.1, is for weighted mixed norm generaliZations of

Bergman spaces in I”, and is thus of the second type mentioned.

§2 ngg Estimates

Recall from section 3 of Chapter 1 that for p e- (0, 1), n 6 SN, and 8> 0 we
defined B( (apt), )-.-{t e le |(d(1:, 11))2 < 5(1 - p)}, where d is the nonisotropic
metric on sN given by d(1?, n) = H - (1:,11)|1/2;We also defined Rm, 6) = 3N. Also
recall that oB(pr), 8) ~ (1 - p)”. (Now the constants of comparability involve 8.)

Our first lemma is the construction of a "nice" covering of S by sets B.

Lemma 3.2.1 : Suppose p e (O, 1) and 8 > 0 are fixed. Then
é(pn1, 5), . . . , §(pnkp, 8) can be chosen inductively so that

(1) S C 91911108).
(2) B(pm, 8/2), . . . , B(pnkp, 8/2) are pain/vise disjoint,
and (3) no point of S is in more than L of the sets B(pnk, 8), where L = L(N, 8) < 00

isindependentofp.

M : In the induction scheme, if there exists some n e S \ LkJB(pnk, 8), then
add B(pn, 8) to the collection; condition (2) will continue to hold because d is a
metric. The scheme will terminate in finitely many steps because

oB(pnk, 8/2) >c(1 - p.)N oS =1, and (2) holds. To demonstrate condition (3),
take any point I] e S; then B(pr), 28) contains all of the sets B(pnk, 8) which
contain n because d is a metric; but (2) holds, oB(pnk, 8/2) 2 c(1 - p)N , and
C(1 - p)N 2 cam 25). El

28

The geometric facts above do not depend strongly on the particular metric
being used. However, the nonisotropic metric is being used here because of its
close connection with (the denominator of) the invariant Poisson kernel
P(z, c) = ((1 - Iziz)/|1 - (2, QB)". in particular, it allows the following argument.

Fix f e H(BN) and r0 6 (O, 1). Choose a positive integer M so large that
8 <1 - 2"M and then let R = 1 - (1 - ro)2'M. Note that O < r0 < R < 1, so there is no

problem with the existence of fR on S. Define u on RB = {R2 | z e B} by

Ulr’r) =13 lfn(C)Ip((R2 - rzi/IR - <rt. or)“ claim.

i.e., u(rt) = Is |fR(i;)|PP((r/R)t, C) do(i;) = P[|fR|P] ((r/R)t). Then if]p s u on RB (see
the proof of Theorem 5.6.2 in [15]) and Is ur do = u(O) =13 IfRIp do for 0 s r < R
(use the definition of u, Fubini's theorem, and the fact that Is P((l’/R)‘E, C) do(c) = 1).

Most importantly, we have the following.

Lemma 3.2.2 : With the setup above, if 20 = r01), 2' = r015, and z = r1, where

r s [r0, (r0 + 1)/2] and ‘t e B(zo, 8), then u(zo) ~ u(z') ~ u(z).

Prggf : Write r0' = ro/R and r' = r/R. Fix Ce 8. Then by the triangle inequality,

the fact that ’t e B(zo, 8), and the choice of M, we have

I1 - (ro'r. OI"2
S I1 -<ro't.'t>l"2+l1-<T.n>|"2+l1-<n.ro'rl>l"2 +l1-<ro'n.i;>l“2
S (1 - ro-)1l2 + 51/2“ - ro)1/2 + (1 , rov)1/2 +11 _ (ro'Tl. 011/2

5 (1 - To')“2 + (1 - 2"“)1’20 - r0)”2 + (1 ~ [dim + I1 -<ro'11. OI”2

29
S 4|1 - (ro'n. DI"?
and I1 - (ro'n, QIVZ s 4|1 -(ro"c,C)|1’2 similarly, so P(ro'r), Q ~ P(ro't, C), and
u(zo ) ~ u(z') follows by integrating over S with respect to do(i;). To show that
u(z') ~ u(z), first note that
1 - r2 31 -, ro'2 < 2((2M -1)/(2'V'-1 -1))(1 - r') 3 C(1 - r'2).

Secondly, for C e S,

I1 - (1’1, QUIZ 5 I1 - (r'T, ro'1)|1/2 + [1 -(l’o'1:, 911/2

5 C(1 - ro')“2 + |1 - (TO'T, 9|”2 s CH - (TO'T, QIVZ.
Finally, for C e S,

I1 - (ro't. C>I2 = (1 - ro'l<t, Oi)2 + 2ro‘(l<'t. C>l - Rea. Z2))
S((2M - 1)/(2M'1 - 1))2(1 - r'l<r. C»)2 + 2r'(l<r. €>l - Rah. Q)
S C|1 - (r'T, QIZ.

So u(z‘) ~ u(z) follows as before by integrating.
The claims made above also hold if r0 = 0; some of the estimates become

simpler. D

We will also need the following.

 

30

Lemma 3.2.3: Fix a, b, c e lR satisfying 0 > -1, b < -1 - c, and a < -1 - b - c. Then
for 0 s p s 1,I1°°((1-p)x + p)a xb (x - 1)° dx is finite and bounded above by a

constant C = C(a, b, c) independent of p.

Prgof : Use the fact that 1 s (1-p)x + p s x, consider the cases a < O and a 2 O

separately, and write I 1°° =11? + .12”. D

§3 A Qarlesgn Measgre Thegrem

With the preliminaries disposed of, we proceed to the statement and the
proof of a Carleson measure theorem for mixed norm generalizations of Bergman

spaces in the unit ball of it”.

Thegrem 3.1: Suppose O<p<°°, O <q1 Sq2<°°, -1 <a, and O <8< 1, and
suppose [I and v are finite positive measures on S and [0, 1) respectively. Then

the following are equivalent:
(3.1) (l. (is ltmlp dulrlhz’p dv<r>)l’qz s cllillpn... for every t 6 Ali“.

(3.2) (p(é‘xpn, 5)))1/p(v([p, 1)))1/Q2 3 C(1 - p)N/p + (wt/qt for every n e s and
every p e [0, 1).

Proof :To showthat (3.1) implies (3.2), fix r) e S and p 6 [1/2, 1). (For p near 0,
(3.2) is true with large enough constant 0' since It and v are finite measures.) Let

w = pr) and define f(2) = (1 - (z, M)5 where s > N/p + (a+1)/q1. Take 2 = r1 where

 

31
p < r <1 and ‘t e B(w, 5) and let 2' = p‘t. Then
|1 - (z, w)|“2 .<. H -(z, z')|1’2 + |1 - (2', MI"2
s I1 -<2, 2W2 + (l1 -<n. t>l + Kn. t) - (ml. pt>l)"2
s <1 - rpl12 + (5(1 -pl + <1 432))“2 s C(1 -p)"2.

i.e., |f(z)| > C(1 - p)'5 for such 2. So the left-hand side of (3.1) is greater than

(I91 (Iéipn. 5) C(1 ' Pi's" dm(w))‘h’p dv(r))1/qz
= C“ ' Pi's(u(B(pn. 8)))"P(v([p, 1 )))1/<tz,

On the other hand,

”filmy; = (II (Is I1 - (r1, w)|'sp d6(‘t))qllp(1 - r2)ar2N-1 dr)1/q1
~ (II ((1 - rp)N-sp)ql/p(1 - r)<1r2N-1 dr)1/q1

by Proposition 1.4.10 of [15], and underthe change of variables

x = (1/r - p)/(1 - 9). this becomes
C(1 _ p)N/p +(a+1)/q1- $0100 ((1-p)X + p)-(2N +1 + on + NqI/p - sq1)qu1/p - sq] (X _1)a dX)1/ql,

which is bounded above by C(1 - p)N/p +(0I+1)/q1-s’ with C independent of p, by

Lemma 3.2.3. Combining these two estimates, (3.1) yields

(u(glpn. 5)))“"(V(lp. 1)))"“2

 

32

50(1 - p)S(1 - p)N/p+(a+1)/qt 'S=C(1 - p)N/p+ (a+1)/q1’

which is (3.2).
The hard part of the theorem is showing that (3.2) implies (3.1). Fix
fe Aff‘. As in the previous section, choose a positive integer M so large that
8 < 1- 2"“; let rm = 1 - 2"”, Rm = 1 - 2““, and um = P[)me|P] on RmB for
m- - 0, 1, 2,. ' and by Lemma 3.2.1 for m = 1, 2, 3, . . . we can choose
B(r rmn1, 8),. B( rmnkm, 8) so that
(1) S C L(JBUmnk, 8),
(2) B(rmm, 8/2), . . . , B(rmnkm, 8/2) are painlvise disjoint,

and (3) no point of S is in more than L of the sets B(rmnk, 8), where L is
independent of m.

For convenience, write Im = [rm, rm“), zrnk = rmnk, Bmk = B(rmnk, 8), and

30k = Bo = SN. Then

i. (is lttrcllp du( ))°2"’ dvirl< <m°§ l... (:2: lg... Umlr‘t)dl1(1))q2’p dvir)

s 0% I; umlzmtlh u(B no i)‘ (v (I ill/aziplqz'p

by condition (1) and since Um(l"t) s Cum(zmk) for r 2 lm and 1: e Bmk (Lemma

3.2.2). By assumption (3.2) and because q1 s qz, this is less than

< c{z |f( um )(2- th/p+ (“+1)/Ql))P]QI/P}QZ/QI
< C{Z J|m( (g J‘Bmkz'jfll um (m) dO'(’t ))q,/p(1 ' r1ar2N-1 dr}qZ/q‘

where we have used the facts that I 1m (1 - r)°‘r2""1 dr ~ 2'm(°‘+‘), oBmk ~ 2"“, and

33

um(zmk) s Cum(rt:) for r 6 lm and ’C e Bmk (Lemma 3.2.2). Since condition (3) says

the sets Bmk have ”controlled overlap", we have

§- Iémk Um(r'r)d6(1) 5 Gig um(rT) dam
= Cum(0) = Cl. liRmtrllp dv(r) s C S If,'(i:)|° do(1:)

where r' = (r + 2M - 1)2'M, since Rm 3 r' for r 6 lm. (Note that this is the only place
where analyticity of f is used, and the full strength of analyticity is not needed.)

Inserting this estimate and using a change of variables, we have .’

Jl (Is ”(MID dl1(T))q2/p dv(r)
S OLE“: Ilm (Is Ifrirllp d0(’€))ql’p(1 - r)°‘r2N°1 dr}q2/qt
5 GUI (Is WWI" d0(‘t))ql’p(1 - r)°‘r2N'1 dr}Qz/Ql
S GUI Us I‘r'ltil" dv(r))‘il’w1 - r')“r'2”" dr‘}°2"‘i = cilill Sitte- El

The same proof actually yields the following.

Thegrem 3.3,1': Suppose O < p< 00, 0 <q15q2 < 00,-1< a, and 0 < 8< 1,
suppose v is a finite positive measure on [0, 1), and suppose for each r e [0, 1),
u, is a finite positive measure on S satisfying p,(B(pn, 8)) ~ pp(B(pn, 8)) for

r e [p, (p + 1)/2), p e [0, 1), and n e S. Then the following are equivalent:

(3-1') (II as Ifrmlp dll,(‘5))q2’p dV(r))"q2 s Cllfllp,q,,a for every f 5 Ag“,

(32') (It, (Eaten. 8)))"P(v(lp. (p + 1)/2)))"°2 s C(1 -plN/p+ (War for every

34
n e S and every p 6 [0,1).

For example, du,(T) could be g(r, T) du(t) for a suitable nonnegative
function g on [0, 1) x SN.

In light of the classical theorems (e.g., Duren's and Hastings'), it is natural
to wonder whether Theorem 3.3.1 is true if p is replaced by p2 on the left and by
p1 on the right, with O < p1 5 p2 < 00. Our method of proof sheds no light on this

quesfion.
4 m rllrie

Condition (3.2) is easily verified for certain choices of u and v, and thus

we get the following corollaries.

r II .4.1 : A331 CA‘ijzif0<pzsp1<°°,O<q23q1<°<>,-1 <ot1,

-1 < a2, and (0L1 + 1)/q1 s (on2 + 1)/q2.

Prggf : Take )J. = c and dv(r) = (1 - r)°‘2 dr, so that “f“ 3 out” by

pirQ2r°iz plIQl-ai
Theorem 3.3.1, hence Ali]? C Ag]? The slightly more general statement
given then follows from H'o'lder's inequality as noted in Section 1 of Chapter 1.

(However, Ho'lder's inequality alone is not enough to give this result.) I]

r ll .42 ; (L |f(rTO)IQZ(1 - r)Q2(N/p+(a+1)/qt) -1 dr)"qz 5 c||t||when

(0<p<°°,0<q1Sq2<°°,-1<a)foreveryfe A551 andeverytoe S.

 

35
Prggf : Take u = 8To ( = point mass at 1:0) and dv(r) = (1 - r)q2(N/P+(a+1)/qt)-1 dr. U

Theorem 3.3.1 also yields Proposition 1.2.1, which was the growth
condition used to show that AZ? is a Banach space: fix 2 e B, take
p. = (1 - 'Z|)N52/I2l and v = (1 - (21)““8'2'.

We can also get the easier direction of Theorem 2.3.1, namely that

(m2: (:2; (r(amk)|92-mN)q/Pz-mla+li)“q .<. clltllpm for every 1 e Ag?

(where {amk} is an e-8 lattice): for r e [1 - 2"“, 1 - 2"“), take )1, = :ZZSamk/lamkl
and dv(r) = 2"“U‘W9’r 0‘) dr; Theorem 3.3.1' applies since
oBmk ~ 2"“N ~ o{1: 5 SN | p(amk, lamkl’t) < 8} (recall that p is the pseudohyperbolic
metric) so uerk = number of amk's in (1 - 2"“)Bmk ~ "C.

One final consequence of Theorem 3.3.1 is actually a corollary of its proof.
First, we need some terminology. For 11 e S and 0 < t < 1, let
A(n, t) = {z e B | [1 - (2,11)|1/2 < t1’2}. It is easy to check that the sets A are
comparable to our Carleson sets (which were also defined in terms of the

nonisotropic metric) in the following sense:
(4.1) {rte B|(1-t/4)sr<1and1e 5((1 -t/4)n. 5)}
C A(n. 1)

C {rte B |(1 - 4t/52) S r <1 and TE g((i - 4V52)n, 5)}.

Also observe that

36
(4.2) m,.(A(n. 1)) ~ We“ ~ m..(§(<1 -ctln. 5)).
Now, we offer another proof of this result of Cima and Wogen ([4]):

Thegrem 3.4.3 (Qima-ngen) : )3 |f|2 div .<. CJB |f|2 dma for every f e H(BN) if and
only if A(Am. t)) s C'ma(A(n. t)) for every n 6 SN and 0 < t < 1.

gm: Take p = q1 = q2 = 2 in the proof of Theorem 3.3.1. Then having a
"product measure" was not important there. The necessity of the condition

A(Am, t)) s C'ma(A(11, t)) follows from the second containment in (4.1 ), condition
(3.2), and (4.2). lts sufficiency follows from the first containment in (4.1), (4.2), and

condition (3.1). D
R late M h n R I

For 8 > O, p e (O, 1), and n 6 SN, define B(pn, 8) = {’C 6 SN) I‘C - n| < 8(1- p)}.
Note that o(B(pn, 8)) ~ 82N"(1 - p)2N". Considerthe two conditions

(5.1) (I. (is If,(r)lp du(‘t))q2/p dv(r))1/Q2 s Clllllp,q,,a for every t e Aliflt,

(5.2) (u(étpn. 8)))""(V(lp. t)))“qz s C'(1 - p)‘2”'1”°*(“*”’qi for every n e s
and every p e [0, 1).

Asusual,wesuppose0<p<°°,O<q1sq2<°°,-1<a,0<8<1,anduandv

are finite positive measures on S and [0, 1) respectively. Then (5.2) implies (5.1)

37

(but any attempt at a proof of (5.2) seems to require the exponent involving N, not
2N-1). Since sets B(prh, 8), . . . ,B(pnkp, 8) can be chosen to cover S with
”controlled overlap” as in Lemma 3.2.1, the key to the proof of this implication is to
show that u(z) ~ u(zo) for 2 near 20 where u is some suitable majorant of |f|p on
some large enough ball in BN. This can be done by a purely geometric argument
which we describe here.

Fix ye (1, 2) and choose a positive integer M so large that 7““ > 1/(2 - y).
For m = 0, 1, 2, . . . let rrn =1 - yr" and Rm =1 - W1“, and let um be the least
harmonic majorant of |f|p on RmB. Fix zm = rmn and z = r1 with rm .<. r < rm+1 and

t e g(zm, 8). Note that
[z - zml .<. [1'12 - l’m‘tl + lrmn - rm'rl s (1 + 8hrm - Ym‘1 - 872'“.

Let am = y’" - yr” and bm = am - |z - zml. Recall that B(z, r) is an isotropic ball; i.e.,

B(z, r) = {w 6 [EN | (z - wl < r}. Then, since we may assume 8 < (y- 1)/y2, we have
1> bm/am 2(7-1- 572 + (”ZN/(72 - 1) > (7-1 - 8y2)/(‘y2 - 1) > 0,
so m(B(z, bm)) ~ m(B(zm, am)), independently of m, and

um(z) = 08(2. bm) um dm)/m(B(z, bm))
(m(B(zm, am))/m(B(z, bm)))()B(zm, am) urn dm )/m(B(zm, am)) 3 Cum(zm)

IA

by the ordinary mean value property (which is not true for M-harmonic functions,

e.g., those gotten by integrating with the invariant Poisson kernel as in the proof of

 

38

Theorem 3.3.1). To get the other direction, let cm = 7”” - WW and dm = 0m - |z - zml.
Then, since )M" > 1/(2 - y) and since we may assume 8 < (W'1(2 - y) - 1)/yM, we

have

1 > dm/cm 2 (2)NH - yM - 1 - 5W + 5y'l/l-rii)/(y~'-1 - 1)
> (W"(2 -1()- 1 - 51M)/(1(M" - 1) > 0.

so m(B(zm, dm)) ~ m(B(z, cm)) and um(zm) s Cum(z) as above.

We remark that if 1 s p < 00 and 1 s q1 .<. q2 < 00, "f e A?” could be
replaced by "f positive subharmonic in BN" in the argument above.

This method doesn't seem to work with nonisotropic balls B because the
nonisotropic metric fails to give adequate control in all directions. This manifests
itself in the argument above in ”radii" bm and dm that may fail to be positive.

We close with a generalization of Hastings‘ result in [8] for the polydisc.

Thegrem 3.3.1 : Suppose O < p < 00, 0 < q1s q2 < 00, and -1 < a1, . . . , aN,
suppose 111, . . . , W are finite positive measures on ii = S1, and suppose
v1, . . . ,vN are finite positive measures on [0, 1). Then the following two

conditions are equivalent:

(5.3) (Jr "J1 Utr- - in |f(r1e‘91. . . . . rNeieimp du1(61)~-de(eN))Qz/p
dv1(r1)- . .va(rN))1/q2

S C((]: - 'L (In - a)“ |f( r1ei91, . . . , rNeieN)|Pde1- - ~deN)qr/p
(1 ' r1)<11 r1dr1n-(1 - rN)°lN rN drN)1/qt

for every function f analytic on ll)”,

39

(5.4) (111([901 - 1:51, 901 + 1:81)))“P x- - -x (ttho0N - 1:5,], SON + tr5N)))l/P
x (v1([1 - 81,1)))1’q2 x. - ex (vN([1 - 8N,1)))1/q2
g (:5 1(1/p + (a1+1)/qr) x. . .x 5N(1/p + (aN+1)/q1)
for every 901, . . . , 90M 6 Ti and every 81, . . . , 8N e (O,1].
grog: First fix 901,. . . ,OON a ii and 51,. . . ,5N e (0.1]. Let wj = (1 - 8j)ei90j for
j=1, . . .,Nanddefine

t(z,, . . . , 2N) = (1 - z,W,')~S. - -(1 - szfiys

where s > 1/p + (ctJ-+1)/q1 for each j. Then |f(z1, . . . , 2N)| > 081'5- - -8N's if zj = rjeiei
satisfies 6j e [901-- 1181-, 901+ “51) and rj e [1 - 8]., 1) for each j. (See [6], p. 157.) So
(5.3) implies (5.4) as before.

Forthe proof that (5.4) implies (5.3), fix ye (1, 2), choose M so large that
W" >1/(2 - 7), write rmi =1 - yr") and Rmj =1 - yM'mj forj = 1, . . . , N, and write
in = (m, . . . , mN) e {0, 1, 2, . . . }N. For each N-tuple m, let uan be the least
n-harmonic (i.e., harmonic in each variable separately) majorant of |f|p on
Rm1lD x- - -x RleD (see [14], p. 52). If ej e [90j - u(1 - rmj), 90]. + n(1 - rmj))

and rj e [rmr 'mj +1) for each j, then
um(r1ei91, . . . , rNeieN) ~ um(rm1ei901, . . . , rmNeieou)
by applying the geometric argument with isotropic balls given at the beginning of

this section. (The argument is applied N times, one time on each "slice”.) One

other needed observation (see [14], p. 52) is that

4O
um(0) = )n- - J11 |f(Rm1el91, . . . , RmNeieNMp d631- - -d9N.
Then the proof is completed in the manner of the proof of Theorem 3.3.1. [J

When 1 s p < 00 and 1 s q1 s q2 < 00 the theorem is also true with

"f analytic on 0”" replaced by "f positive N-subharmonic in [[DN".

BIBLIOGRAPHY

10.

11.

12.

13.

14

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42

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