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RSITY LIBRARIES

Illllllllllulwlull Will

 

 

 

 

 

 

 

 

This is to certify that the

dissertation entitled

Some Operators and Carleson Measures on
Weighted Norm Spaces

presented by

Dang-sheng Gu

has been accepted towards fulfillment
of the requirements for

 

 

PhoDo degree in Mathematics

quu

Major professor

 

Date May 10. 1991

MS U is an Affirmative Action/Equal Opportunity Institution 042771

 

LIERARY
Michigan State
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MSU Is An Affirmative Action/Equal Opportunity Institution
emomS-q

I III if VIA

SOME OPERATORS AND CARLESON MEASURES
ON
WEIGHTED NORM SPACES

By

Dangsheng Gu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1991

ABSTRACT

SOME OPERATORS AND CARLESON MEASURES
ON
WEIGHTED NORM SPACES

By

Dangsheng Gu

Suppose (X, V, d) is a homogeneous space. Harmander has constructed a max-
imal operator to study problems involving Carleson measures in this situation. In
particular examples of homogeneous spaces, for example, in R" and in the unit ball
of C", a maximal averaging operator has proved to be useful. The first goal of
this paper is to study the weighted norm inequalities for the Hormander maximal
operator and the generalization of the maximal averaging operator. Using the con-
cept of the “balayée” of a measure, we characterize those positive measures it on
X+ = X x R+ such that the inequality ”Hy f I] L"(#) S C M f H LPN), where q < p, holds
for the Hormander maximal operator Hu, and those positive measures p on X such
that the similar inequality “M”, f H L901) S C II f Hum), where q < 1), holds for the

maximal averaging operator MW. defined by

Mu,rf($) = sup ”7793—7),- BM |f(u)|dv(u),

tZr
where B($, t) is the ball centered at z with radius t.

The second goal of this paper is to study the analytic functions on the unit ball

of C”. Let U be the unit ball in C” and Q be a positive measure on U satisfying
Békollé’s B: condition for some a > -1. The first result of this part is a Carleson
measure theorem for weighted Bergman spaces. We characterize those positive mea-
sures p on U such that IIfIILqp) S CllfllAp(Q) (1 < p S q) for any function f in the
weighted Bergman space 119(0). The second result concerns the Bergman operator on
weighted mixed norm spaces. Using an interpolation theorem between the LP spaces
on U and the LP spaces on the boundary of U with different weights, we prove that
for some weights satisfying Békollé’s B; condition, the Bergman operator induces a
bounded projection on the weighted mixed norm space on U. Thus we are able to

identify the dual of those weighted mixed norm spaces of analytic functions.

To my parents and my wife.

iv

ACKNOWLEDGMENTS

I would like to thank Professor William T. Sledd, my dissertation advisor, for all

his help, encouragement and advise. His knowledge and enthusiasm were invaluable.

TABLE OF CONTENTS

Introduction 1
Chapter 1 Preliminary 10
§1.l Homogeneous Spaces ......................................... 10
§1.2 Analytic Function Spaces on the Unit Ball of CN ............ 15

Chapter 2 Harmander Maximal Operator

and Carleson Measures on X+ 21
§2.l a-Carleson Measures on X+ with a Z l ...................... 21
§2.2 Hormander Maximal Operator and Space W3 ................ 25
§2.3 Another Characterization of W3 ............................. 29
§2.4 Carleson Measure Theorem on Weighted Hardy Spaces ....... 32

Chapter 3 Maximal Averaging Operator

and Carleson Measures on X 39
§3.1 a-Carleson Measures on X with a 2 1 ........................ 39
§3.2 Characterization of W31, for 0 < a < 1 ....................... 46
§3.3 Two-Weight Norm Inequalities ............................... 49

Chapter 4 Carleson Measure Theorem
in Weighted Bergman Spaces 54
§4.1 Carleson Measure Theorem
in Weighted Bergman Spaces ................................ 54

§4.2 Multipliers on Weighted Bergman Spaces ..................... 61

vi

Chapter 5 Bergman Operator

in Weighted Mixed-normed Spaces 65

§5.1 Preliminaries ................................................ 65

§5.2 Interpolation Spaces ......................................... 69

§5.3 Bergman Operator ........................................... 73

§5.4 Duality Theorems ............................................ 81
Bibliography 86

vii

INTRODUCTION

The purpose of this work is to study several operators acting on the spaces of
functions in a homogeneous space (X, V, d).
A homogeneous space (X, V, d) can be defined as a quasi~metric space (X, d) with
a positive measure V on X satisfying the following condition:

There is a constant Cu, 0,, > 1, such that
0 < u(B(:c,2r)) S C..V(B(:r,r)) 5 00

for all r > 0 and any a: E X, where B(a:,r) = {y|d(:r,y) < r}.

We shall study the following operators.

1. Harmander maximal operator H... An operator defined on the space
of locally integrable functions on X which maps a function f on X to a

function Hy f on X x R1“:

(Huf)($, t) = SUP )|f(U)ldV(U),

__1__.
"(3(y, 8)) Bah:

where the supremum is taken over all balls B(y, s) I) B(:c, t).

2. maximal averaging operator M”. An operator defined on the space
of locally integrable functions on X which maps a function f on X to a
function MW. f on X.

Mani-(x) = SUP

1
‘2' 14—31575? B(:r,t) |f(u)ld"(u)'

3. The analytic embedding operator I. The restriction of the identity

operator to the Bergman spaces in the unit ball U of C":

If=f.

4. The Bergman operator T5. An operator defined on the space of inte-

grable functions on the unit ball of C":

N
Taf(2) = ( 1:3 ) jU Kn(z,w)f(w)dmfi(w), 2' E U,

where
K5(z,w) = (1— < z,w >)'N'l'a fl > —1.

The first half of this paper is devoted to the study of the maximal operators H”
and M”. The problem that we are concerned with is to characterize those measures
p defined on X+ in the HV case and on X in the My... case, respectively, such that
the corresponding operator is bounded from LP(Q) to L901), where Q is a “weighted
measure” on X defined by dfl = wdu with a positive weight function (.0. We shall
refer to these two problems as problem I and problem 11, respectively.

We first consider the Hormander operator.

For as = 1, the unweighted case, when l < p = q < 00, the solutions of problem I
are known as the “Carleson measures”. In [7], Carleson characterized those finite

positive measures p on the unit ball U in C1 such that

(/U IU(2)I"d/t)"’ s Cllfllm

for every function f in the Hardy space H10 (0 < p < 00), where U (z) is the Poisson
integral of f. He showed that the above inequality holds if and only if 11(5) S Ch for

every set of the form

S:{re‘9:l—hSr<1,00S0S00+h}.

Such a measure p is now often called a Carleson measure. In order to generalize
Carleson’s result, Harmander [11] introduced the operator H... Using the Marcinkiewicz
interpolation theorem and a simple covering argument, he proved that the Carleson
measures are the solutions to the problem I when l < p = q < 00.

In [9], Duren extended Carleson’s theorem to the indexes O < p S q < 00. He

proved that, for 0 < p S q < 00

(/U |U(z)l"du)"° s Cllfllm

for every f in HP, if and only if MS) S Ch“, where 1 S a = q/p. Such a measure is
called an a -Carleson measure.
In general, an a -Carleson measure on X+ with respect to a positive Borel measure

A on X is a measure ,u on X‘l' such that

|#|(T(B(~’v,t))) S ClA(B($, t))l"',
where
T(B(I,t)) = {(31,506 X+|B(y,3) C B($,t)}

is the “tent” over the ball B(:r, t). We shall see that, using H6rmander’s idea, it is
not hard to show that if 1 < p S q < co and a = q/ p, then the a -Carleson measures
are the solutions to the problem I.

For the weighted case, when p = q, X = R" and u = m, where m is the Lebesgue

measure, the problem I has been solved by Francisco J. Ruiz and José L. Torrea [21].

In the case to satisfies Muckenhoupt’s A, condition, it will be shown that, similar
to the unweighted case, the solutions to the range 1 < p S q < 00 are the
a -Carleson measures with respect to Q.

The difficult part is the case when 0 < q < p < 00. It is natural to guess that the
solution must be an extension of a -Carleson measure with respect to 0. Using the
concept of the “balayée” of a measure p as employed by E. Amar and A. Bonami [1],
we are able to prove the following theorem which is contained in Theorem 2.9:

Theorem 1 Let 0 < (1 <1, and let q > 0, p >1, q/p = a. Let p be a positive
measure on X+. Suppose w E A? and set dfl = wdu. Then there is a constant C such

that

IIHquILem S Cllfllmfl)

for every f E LP(Q) if and only if

 

Sll ”(TB($ar)) 173-;
..E 9(B($,r)) 6" (n). (1)

Note that if q = p, then a = l and the condition (1) shows that p is an a-Carleson
measure with respect to 0. Therefore we have an unified approach to the solutions
of problem I.

The above result enables us to extend Carleson’s theorem to the weighted Hardy
spaces HP(Q) with p, q in the range 1 < p S q < 00 and in the range p > 1,
0 < q < p. It turns out that the solutions to the Carleson measure problem on the
weighted Hardy spaces H ”(9) are the same as the solutions of problem I. The results
for unweighted Hardy spaces when 0 < q < p < 00 were obtained by Videnskii [26]

in the one dimension case and by Luecking [15] in higher dimension case.

Now we consider the maximal averaging operator.

In order to study problem II in a general homogeneous space, we first introduce

the following concept:
We shall call a measure p on X an a-Carleson measure with respect to a positive
measure A on X if there exits a fixed r > 0 and a constant C, such that for any ball

B(:r,r) centered at a: with radius r,
|#|(B(1‘a")) .<_ CrlA(B($,"))l°'-

The reason to call such a measure an a-Carleson measure is that V. L. Oleinik
and B. S. Pavlov [18] have proved the following theorem which is an analogue of the
Carleson’s theorem mentioned in the discussion of the Hbrmander maximal operator:

Suppose U is the unit ball of C1. Then for l < p S q < 00,

(/U We”? 5 C(fU Ifrdmr/P

if and only if

1413(2)) S C[m(E(Z))]°’

for every 2 E U and any function f in the Bergman space A? , where E(z) is a
“suitable” subset of U and a = q/ p Z 1.

Similar characterizations were studied by Hastings [10] for the polydics DN and
by Cima and Wogen [8] for the unit ball U of CN.

We shall refer to the problem of characterizing those measures it on X such that

the inequality

llfllmu) S Cllfllmm

holds for all functions in weighted Bergman spaces Ap(9) as the “Carleson measure

problem on X”.

The reason to study the operator MW. is that many functions, for example sub-
harmonic functions, are controlled by the operator M”, and that Carleson measures
can be applied in the study of the operator Mm...

Applying similar ideas used in the study of problem I, one can show that, in a
homogeneous space, if no satisfies the condition AP, then the solutions to problem II

when l < p S q < 00 are those measures it on X satisfying
|#|(B(-’c,r)) S Cr[Q(B(1=,r))l"

for any a: with fixed r.

When p > 1 and q < p, we prove the following characterization theorem which is
contained in Theorem 3.11:

Theorem 2 Suppose p is a positive measure on X and suppose w 6 A,. Let

q>0,p>1, q/p=a<1. Then

(/ ward/1)”? s C(/ mun)“: f e mm,
if and only if

#(B(-’B,r)) 1%; 00
W E L (Q) < . (2)

When p = q, then a = l and the condition (2) implies that p is an a-Carleson
measure on X. Therefore we have reached an unified approach to the solution of
problem II.

Using the method of the proof of Theorem 2, we are able to characterize those
measures on a general homogeneous space such that the Hardy-Littlewood maximal
operator is bounded from LP(Q) to Lq(p) (1 < p < oo, 0 < q < 00). When 1 < p S .

q < co and X = R”, such a characterization have been obtained by E. Sawyer [24].

6

In the second half of this paper, we shall restrict ourselves to a special homogeneous
space, the unit ball U of C". We shall always consider the measure defined by
dmp = (1 — r2)”dm, S > —1, as the “unweighted” measure in U, and shall refer to
the “weighted” measure as the form dfl = wdmg.

As in the previous discussion, we have seen that Carleson measures play an im-
portant role in the study of maximal operators. Our third problem, which will be
referred to as problem III, is to determine the sufficient and necessary conditions un-
der which the embedding operator is bounded from AP(Q) to L901). This is, in fact,
equivalent to solving the Carleson measure problem on U, or, to set up a Carleson
measure theorem in the weighted Bergman spaces.

As we have mentioned before, for the unweighted case, when 1 < p S q < co, the
problem III was solved by Oleinik and Pavlov [18] in the one dimension case. The
higher dimension case was solved by Cima and Wogen [8] for q = p = 2, and was
generalized by Luecking [14] to O < p S q < 00. When 1 < q < p < 00, it was solved
by Luecking [15].

In the weighted case, a general technique to find a sufficient condition such that

“fllAP(u) S Cllfllmn)

was obtained by Luecking [14].

In this paper, we solve the problem for those weights w satisfying Békollé’s BE(w)
conditions studied by Békollé in [3]. We prove the following theorem which will be
restated as Theorem 4.7 in chapter 4:

Theorem 3 Leta Z 1 and let 1 < p S q < 00 such that q/p = 0. Suppose w

satisfies the 85(0)) condition. Then

”fllAvos) S C]]f”AP(Q)

7

for any f in the weighted Bergman spaces 149(0) if and only if there is a r, 1 > r > 0,

such that
#(E(a,r)) S Crl9(E(a,r))l°‘

for any a E U, where E(a, r) is the psudohyperbolic ball centered at :r with radius r.

The last problem in this paper concerns the boundedness of the Bergman operator
on the weighted mixed norm spaces in the unit ball of CN . We shall refer this problem
to problem IV.

In [3], Békollé found a necessary and sufficient condition for weight functions such
that the Bergman operator is bounded on the corresponding weighted L1D spaces in
the unit ball of C”. In [13], M. Jevtié proved that there are bounded projections from
general mixed norm spaces onto the weighted mixed norm spaces of analytic functions
with the normal-function weights. The projections he studied are very similar to
the Bergman operator. Here, we show that the Bergman operator is bounded on
weighted LP spaces on the boundary of the unit ball of C” with normal-function
weights. Then we determine the weighted mixed norm spaces on the unit ball of C"
as the interpolation spaces between weighted L" spaces on the unit ball of C" and the
weighted LP spaces on the boundary of the unit ball CN with different weights. These
facts enable us to prove that the Bergman operator is bounded on weighted mixed
norm spaces with radial weights satisfying Békollé’s conditions. The main result of
this part is the following theorem which is contained in Theorem 5.16:

Theorem 4 Suppose p S q S 00, l < p < co, and that cp is a normal function.If

a radial function w(r) on [0,1) satisfies condition B§(tpP(r)w(r));
1
1..
1 ' ,
X (/ hw-%(r)p”” (r)(1 — 1:2)”‘1-2N'4dr):Fr S Chla“)?
1—

8

forallO<h< 1, where b+fir=L

Then, for i + :7 = 1,

(1) To, is bounded on LP'9(<p9w(1 — r2)°');

(2) To, is bounded on LPI'9'(go’q'w'gi'(l — r2)°').

As an application, we show that Jevtié’s result is a special case of our result.
Using the Bergman operator, we have obtained several duality theorems of weighted
mixed norm spaces.

Our exposition is organized in the following way.

We start by introducing the homogeneous spaces and analytic function spaces
on the unit ball of CN and some of their basic properties, the A, and Ba? weights,
definitions and notations of operators and the concept of a “balayée” of a measure.
This is done in chapter 1, immediately after this introduction.

In chapter 2, we first collect some results concerning the a-Carleson measures on
X+ with a Z 1. Then we present the main result concerning the boundedness of
the Hbrmander operator from LP(Q) to L9(p) when q < p. The extension of the
Carleson measure theorem in weighted Hardy spaces is presented in the last section
of chapter 2.

Chapter 3 is devoted to study the maximal averaging operator.

A Carleson measure theorem in the weighted Bergman space is presented in
chapter 4. As its application, we discuss the multipliers between different weighted
Bergman spaces.

The last chapter is devoted to the study of the Bergman operator in weighted

mixed norm spaces.

Chapter 1

PRELIMINARY

We introduce the homogeneous space and some of its basic properties in the
first section. Some notations and basic facts concerning the analytic functions in the

unit ball of C” are presented in the second section.
§1.1 Homogeneous Space

Let X be a topological space with a positive measure V. Let d be a real-valued
function in X x X. We shall call the triple (X, 11, d) a homogeneous space if it satisfies

the following conditions:

1. d(zr,x) = 0;
2. d(x, y) = d(y,:r) > 0 if 1' 75 y;

3. there is a constant Cd such that d(x, 2) S Cd[d(a:,y) + d(y, 2)] for all :r, y

and z;

4. given a neighborhood N of a point at, there is a r, r > 0, such that the

sphere B(:r,r) = {y|d(x,y) < r} with center at a: is contained in N;

10

5. the spheres B(a:,r) = {y|d(a:,y) < r} are measurable and there is a con-

stant Cu, Cu > 1, such that
0 < V(B(:r,2r)) S CuV(B(x,r)) S 00
for all r and :r.

A measure satisfying condition 5 is called a doubling measure. The doubling
measure V has the following property:

For any K > 0, there is a constant CK > 0 such that
V(B(:c, Kr)) S CKV(B(:L', r))

for all :r and r.
The family of balls in a homogeneous space satisfies the following geometric prop-

erties:

Lemma 1.1 Leta > 0. Then there is a constant C > 0 such that ifr S ar' and

B(:c,r) Fl B(y,r') 75 96, then B(x,r) C B(y,Cr').

Lemma 1.2 Let F be a family of {E(a, r)} of balls with bounded radii. Then there is
a countable subfamily {B(a:,,r,)} consisting of pairwise disjoint balls such that each
ball in F is contained in one of the balls B(a:,-,br,-), where b = 3C} and Cd is the

constant in condition 3 .
For the proof of Lemma 1.1 and 1.2 , see A. P. Calderon [6].

Lemma 1.3 Let p be a positive measure in X. Let a 2 1. If there is a 1'0 > 0 such

that #(B($a 70)) _<_ C[V(B($,To))]°’ for any a: E X, then for any r 2 r0 ,

#(B(-"=,r)) S CCi’[V(B($,r))]“,

11

where Cb depends only on the constant b in Lemma 1.2 and the constant C; in con-

dition 3 of the definition of homogeneous space.

Proof: Let r > re and let
To
E = {3(y. T) = y E B(x.r)}.
where b is as in Lemma 1.2. Then
Bo, r) c mass. 5,9).

By Lemma 1.2 , there exists {yr} C B(:c,r) such that B(:r,r) C UB(y,-,ro) and

{E(yg, 53)}221 is a disjoint family. Note that
UB(y,-, Tb—o) C E(a, Cd(r + 7%)) C B(:c,2Cdr)
since we may assume b _>_ 1. By the doubling property of V, there is a C1 such that
u(B(y.~,ro)) s c.u(B(y.-, 1'51)).
Thus

#(B(=v. r)}

imam»

i=1

|/\

C§[u(8(y.,ro))r

i=1

|/\

wait/(Bot 55-)»:

i=1

COW/(BOT, 20%))1"

|/\ |/\

|/\

CCf'[1/(B(a:, r))]".

The proof is complete.

12

Suppose (0(3) 2 0 is a positive locally integrable function on X. We say that a
measure 9, defined by dfl = wdu, satisfies Muckenhoupt’s A, condition relative to u

if for any ball B,

‘fi' P-1 < P .
./deV[/Bw du] _ Cw[1/(B)] 1 < p < oo,

[8de S CwV(B)essinf,€Bw(x) p = 1.

Note that if no satisfies the condition A, for some p > 1, then it is a doubling measure.

In fact, by Hfilder’s inequality and the fact that u is a doubling measure, we have
fl(B(:c, 2r))

= / wdu
B(z,2r)
[V(B($,2r))l’
w [fB(z,2r) ‘0— '17"le ’1
0,0, [u(B(x,1r))r
lfB(:r,r) w-p—deylp—l
fB(.z',r) W1" ileum) w“ 55‘1le- 1

[fB(J:,r) w' F1I‘M"-1

: CwCy wdu
B(:r,r)

= CWCVQ(B(2:, r))

 

|/\

|/\

 

 

l/\

0,0,,

foranyxEXandr>0.
By Halder’s inequality, the condition A, implies the condition A, if q > p. In [6],

A. P. Calderon proved the following theorem:

Theorem 1.4 Suppose that all continuous functions with bounded support is dense

in L1(V), then the A, condition implies the A, condition for some 7 < p.

In this paper, we shall always assume that the class of compactly supported con-

tinuous functions is dense in the space of integrable functions L1(V).

13

Definition 1.5 Let Q be any positive measure on X. The Hardy-Littlewood maximal

operator is defined by

l
M ... __ / d9.
9“”) £133 f2(B(x,t)) 3m) lfl
Let X"’ = X x R+ with the product topology. Denote
T(B($,t)) = {(31.3) 6 X+|B(y.$) C B($13.15)}-

Let Q be a positive measure on X. Following the notation of E. Amar and A.
Bonami [1], for 0 S a < 00, we shall call a Borel measure u on X+ an a-Carleson

measure relative to (I if
l/1|(T(B(1‘,t))) S ClQ(B($,t))l°'-
Definition 1.6 Let it be a positive measure on X. For f 2 0, define

59(23, y, t) = WXBhdfly)’

(sans, t) = [x 5am, mommy).

Definition 1.7 The Ho'rmander maximal operator is defined by

(Hans, t) = sup m m...) If(u)ldfl(u).

where the supremum is taken over all balls B(y,s) D B(:t:,t).

Definition 1.8 The nontangential maximal operator on X+ is defined by
N (U)(x) = $11p{ lu(v.t)| = d(x,y) _<_ t} = sup{lu(y. t)l = (3M) 6 1155)}.
where u is a function in X+ and

11$) = {(v.t) = d(31,15) S t}-

14

Definition 1.9 The weighted Hardy space is defined by
H”(Q) = {u : u is harmonic in RE“, N(u)(:r) E L”(fl)}
with llullmm) = ||N(U)HLP(9)-

Definition 1.10 Let 0 S a < 00 and let p be a Borel measure on X1“. Define

56/431) = /x+ 5906,31. t)d#(x,t)-
Vt? = {it = |#|T(B(x.t)) S C[Q(B(x. t))]°‘}-

We = {u = Saul e Lit-um}.

We shall call Sal/1| the balayée of u with respect to 0. For 0 < a < 1, W3 is the

complex interpolation space (V8, V6), ( see [1]).

§1.2 Analytic Function Spaces
on the Unit Ball of G”

Let U denote the unit ball in C", N Z 1. Denote by m Lebesgue measure on
C” = R2” normalized so that m(U) = 1. For a _>_ —1, let dma = ca(l — |z|2)°'dm
with ca, chosen so that when a > —1, mo,(U) = 1. Denote by V0 the surface measure
on the boundary S of U normalized so that 110(5) = 1.

A positive continuous function 4,0(7') on [0,1) is normal if there exist a, b, 0 < a < b,

such that

u r e a o o r) _
(2) fi 23 non — mcreaszng, 11m,_,1_ (33%;? — 0.
(1.1)

(ii) 11%;)? is non — decreasing, lim,_.1— :1: = 00.

15

We shall denote b = inf{b: b satisfies (ii) of (1.1)}.
The functions {tp , t/z } will be called a normal pair if cp is normal and if for some

b satisfying (1.1), there exists A > b, such that

cp(r)1/)(r) = (1 — r2)” 0 S r <1. (1.2)

If cp is normal, then there exists 112 such that { (,0 , w } is a normal pair and then
7,!) is normal [23].
For 2 = (21,22,...,ZN) and w = (w1,w2,...,wN) in C”, let
N
< z, w >= 2 2.17);
i=1
so that |z|2 =< z, z >. Following [19], for a E U, a 74 0, let (P, denote the automor-
phism of U taking 0 to a defined by

a — Paz — (1 — |a|2)i‘Qaz

(D, =
(Z) 1— < z,a >

 

1

where P, is the projection of C” onto the one-dimensional subspace spanned by a
and Q, = I — Pa.
For a E U, let

K(a) 5 {45(2) : Re < z,a >S 0},

then [20]
mama» ~ (1— IaI2)°+”+‘.

Define the pseudohyperbolic metric p on U by

p(z.€) = I‘I’e(z)l-

16

For0<r<1, let
E(a,r) = {z 6 U : p(z,a) < r}.
Then we have
ma(E(a,")) ~ m«(K(GD ~ (1 -|a|)N+1+°’-

For basic properties of K(a) and E(a,r), see [16] and [20].
Let 1 < p < 00. For a positive function w E L1(U,dma), the B§(w) condition is
the following:

There is a constant C such that for every K = K (a), a E U,
/ wdma(/ ...-WW)? 3 cmgur),
K K

1 1 _
where 5 + 17 — 1.
In the case at is a radial function, that is, w(r) is a measurable function on [0, 1),

using the fact that K (a) is “nearly”

S((,h)={zeU:|1—<2,C>|<h}

for C = 1%, h = 1 — |a| (see [16, p.321]), the condition Bg(w) can be written in the

form

1
_ 2 a 2N-1
/1_hw(r)(l r ) r dr

1 I
x l/ ,w(r)-%(1— r2)°r2N-ldr]fr g Ch(°'+‘)”.
1-

Let 1 < p < 00. For a positive function as E L1(U, dma), the C, condition is the
following:

There is a constant C such that for any E = E(a,r), a E U,

17

_L’ 2r
p < p ,
Lwdm(/Ew rdm) _ Cm (E)

The condition C, is a consequence of B}; for any a > —1 [16].
Let A(z) be a non-negative measurable function on U and B(r), C(r) be non-

negative measurable functions on [0,1) such that
Mr E [0.1) = 0(1‘) = 0}| = 0.

where [E] denotes the Lebesgue measure of E in R‘. For a measurable function f on

U and 26 S, let

”frllftm = LIf(rz)lpA(rz)d1/o(z), 0 S r < 1, 1 S p S 00.

Since I f (rz)|PA(rz) is a measurable function on U, ||f,||f,,,, is a measurable function

on [0, 1) ( see [20, p.150] ).

Definition 1.11 Let

1
llfllimw) = [Miniswm-ldr 15q<oo.

llfllLr.°°(A.C) = SUP llfrllA.pC(’")-
rE[O,l)

The mixed norm spaces are defined by

LP"’(A,B) = {f = ||fl|A.B.p.q < 00}-

L”’°°(A,C') = {f = llfllA.C.p.oo < 00}-

We shall denote H (U) the space of analytic functions on U and

18

HP:9(A, B) = Lp'°(A, B) n H(U);

new, 0) = new, C) n H(U).

In the case A = A(r) is a radial function, and B(r) = w(r)(l — r2)“, C(r) E 1,

denote

Haw/mu — r2)°‘) = LM(A, B),

L”’°°(A) = L”’°°(A, C),
and

Harm/mu — r2)°‘) = HM(A, B),

HP’°°(A) = H”’°°(A,C).
In the case A(z) = w(z), B = (1 — r2)“, C(r) _=_ 1, and p = q, we have
Definition 1.12 The weighted Bergman spaces are defined by
A"’(wdma) = Hp’p(w(z), (1 - r2)°’)~
Let
Ka(z,w) = (1— < z,w >)—N—1'°'

with a > —1, z, w E U. The Bergman operator T, is defined by [19]

N+a

) / Ka(z,w)f(w)dma(w) z e U.
N U

19

Define
713(2) = ( N;a ) [u lKa(Z,w)|f(w)dma(w) 2 E U-

Note that T; is a linear operator.

In [3], B. Békollé proved the following:

Theorem 1.13 T; is bounded on LP(wdma) if and only ifw satisfies B§(w) condi-

tion.

20

Chapter 2

HoRMANDER MAXIMAL
OPERATOR
AND CARLESON MEASURES ON X+

In this chapter, we restrict ourselves to the space X+ = X x R+ where X is a
homogeneous space. We study the characterization of measures p on X+ such that
the inequality ”H, f I] 1M») S C [I f I] 0(9) holds for the maximal operator H, studied
by Harmander. The solution when q < p utilizes the concept of the “balayée” of the
measure p. Using this characterization we extend Duren’s Carleson measure theorem
to the weighted Hardy spaces.

In the first section we collect the results for a-Carleson measures with a 2 1. We
shall prove the main result of this chapter in section 2 and section 3. In the last

section we shall prove a Carleson measure theorem on weighted Hardy space.
§2.1 a—Carleson Measures on X+ with a 2 1

In this section, we always assume u is a positive measure.
The method of the proof of following theorem is essentially due to Hbrmander

[11], which gives a relation between an a-Carleson measure and the Lq-norm of the

operator H n-

21

Theorem 2.1 Leta 2 1, p > 1. Suppose that Q is a positive doubling measure on

X. Then u 6 V5” if and only if

llanllmu) S Cllfllum), f 6 PM)

where q/p = a.

Proof: That ”Ha f H 1440‘) S C I] f II ”(0) implies u 6 V6” follows from the standard
argument by taking f = xgwy).

For each n > 0, we define

" — en —1———— u
(Hflfxx, t) "' sSn,B(y,s])DB(x,t)“(BU/13)) B(y,s) ]f(U)[dQ( )

and we shall show that the inequality above holds with H9 replaced by H3 with C
independent of n. Once this is established, the theorem will follow by letting n tend
to infinity.
It is clear that H5 is of type (00,00). If we can show that H3 is also of weak type
(1, a), then the conclusion will follow from Marcinkiewicz interpolation theorem.
Let A > 0 and let E = {(z,t) E X+ : H3f(a:, t) > A}. For each (x, t) E E, there

is a ball B(y,r) containing a: such that n 2 r 2 t and

1

“(BU/.70) B(w) If(u)ldn(u) > A'

Let B be the collection of all such balls and let {B (y,-, r,-)} be the countable subfamily
of pairwise disjoint balls of B as in Lemma 1.2 . Then U3 B(y, r) C UB(y,-, br,-) and

that each B E B is contained in one of B(y,, br,).

22

It is clear that E C U T(B(y,-, br,)). Therefore

|/\

#(U T(B(ya. bra)»

Z ”(T(B(yia brill)

C 2(9(B(ys ban)“
0 Z(fl((B(y,-, rill)“
f—a 21/3 Ifldnr

(time)

C .
3:12 [B Ifldfl)

ham)

5 gt] man)“.

”(15)

IA |/\ l/\ l/\

l/\

That is H3 is of weak type (1, a). The conclusion follows.
Next we give a similar estimate to the operator Hy.
Let 7 > 1 and dfl = wdv. If to E A,, by Halder’s inequality, it is easy to show

that
(Huf)($, t) S C[13’n(|f|”)]””.
where C only depends on the A, condition. Thus we have:

Theorem 2.2 Leta Z 1. Ifw E A, and let d0 = wdu, then p 6 V5" if and only if

IIHuflqup.) S Cllfllem), f 6 Him)
for any p >1, q > 0, such that q/p = a.

Proof: That ”H, f H mm S C I] f [I ”(9) implies p 6 V3 follows from the standard
argument by taking f = XB($.t)-

Now suppose p 6 V6.

Since p > 1, by Theorem 1.4, there is a 1 < 7 < p, such that w E A, . Note that

w E A, implies that Q is a doubling measure. Therefore

23

q
/x+lef| dfl
vi
5 0/X IHnlfl I du

s (Ii/x, ureter/P.

The last inequality follows from Theorem 2.1, since iv = q/ p = a and g > 1. The
proof is complete.

The next lemma is due to E. Amar and A. Bonami [1].
Lemma 2.3 Let p be a positive measure on X+. Let
as = [m Sam. odutx. t).
If we define
A(E) = j, Sn(l/g)($,t)dn($,t),

then

A 6 Vol.
Proof: We need. to show that for any ball B

By definition

I = [Tapas/meanness)
= / fawn] flso(x,y,t)Tl,dn(y)1du(x.t)

 

24

Since (1:,t) E T(B) and y E B(x, t) imply that B(z, t) C B and y E B, then

XT(B)($,t)Xs(x,t)(y) S XB(y)XB(z,t)(y)°

Thus
1 x ac,¢(v)
1 _<_ [mimmmmnmm
= [801mm
= (2(3).

The proof is complete.

The last theorem of this section is due to Calderon in [6].
Theorem 2.4 [fl < p < 00, d0 = wdu with w E A,, then
[/ lMufl’dfll‘/’ 3 Cl] Want/P

for f E L”(Q).

§2.2 Hfirmander Maximal Operator

and Space W5

The following theorem shows the relation between the Hfirmander maximal oper-

ator and the space of “balayées”.

Theorem 2.5 Let 0 < a < l, and let q > 0, p > 1, q/p = a. Let p be a positive
measure on X1“. Suppose w E A, and set d9 = wdu. If” 6 W3 then there is a
constant C such that

llHufllLuu) S Cllfllum)

25

for every f 6 0(0).
Conversely, let 0 < q < p < co and let a = q/p. Suppose that Q is a doubling

measure on X. If

"Sufllbflid S Cllfller)

for every f E L”(fl), then u 6 W3.
Proof: Suppose u 6 W3 and q/p = a, p > 1. Let

go) = /x, 50($,y,t)d#($a t).

Then it 6 W3 implies g E Lil—0(9). Note that by Hblder’s inequality

[~9‘t2(1/.9)(9=,i)l‘l S (Sag)($,t)-

If f E L"(fl), then

f,“ IHuflqd/t
= fx,lHuflql5r2(1/y)($,t)l“So(1/g)(a=.t)dp(:c,t)
s f,+ leflq(509)($,t)Sn(1/9)($,t)d/1(x,t)
If,“ leflp50(1/g)(:r, t)dp(:c, t)]q/P
xi]x+ Ksngxx.t)II1-«s..(1/g)(x,t)d,.(., tux—«m»
l/x+ leflp50(1/g)(x, t)dp(:r, t)]9/P
x[ [x I(Hog)(x, t)|Té35rz(1/g)(:c,t)dp(x,t)]1"9/P

= AxB.

M

|/\

By Lemma 2.3, Sn(1/g)(a:, t)u 6 V3. It follows from Theorem 2.2 that

A 5 Cl [x lflpdfllq/p

26

and from Theorem 2.1 that

B s 61],, viewer-W.

Therefore
<1
A+lHufldfl
< p 0/9 1,. l—Q/p
_ CI/xlfldfll {/XIgII—dSI]
S Cllfllipmy

For the converse, suppose that Q is a doubling measure on X, and that

IISuflqu(,.) S Cllfllem)

for every f E Lp(fl). From the definition of W3, we need to show 9 E Lil—«(9).
Let f be in Lp/9(Q) which is the dual of Lit-LNG). For any y E B(m, t), by Lemma

1.1 and the fact that Q is a doubling measure, we have

(50f)($,t) S CMnf(y)-

Hence
I(Sof)(x,t)|""
73%;, BM anf(y)|"°du(y)
... CSu(|Mofll/")($,t)-
Therefore

I fxgwowl
s A,I(Snf)(x,t)ld#(x.t)
= /x+|(Sof)(w,t)|“’°’°du(w,t)

27

s 0/“[Su(lMofl"")l"dfl(x.t)
g C[/x(Mnf)P/qdfl]q/P (by the hypothesis)

< Gil/it Iflp/qqu/p < 00.

Since p/ q > 1, the last inequality follows from a similar argument used in the proof
of Theorem 2.1, we leave the details to the reader. Therefore g E Lia—5(0). The proof

is complete.

Corollary 2.6 Let 0 < q < p, 1 < p < 00 such that a = q/p. Let f E LP(RN)
and let U(:c,t) denote the Poisson integral off. Let p be a positive measure and let m

denote the Lebesgue measure on R". Then it 6 W3, if and only if there is a constant

C such that
(/ IU(w.t)I°diu)"" _<_ C(/ Ifl’dm)‘/”.

Proof: It suffices to prove the theorem for positive functions f 2 0.

Let m denote the Lebesgue measure on R" and let

_ CNt
(lwl’ + was

 

P(a:, t)

be the Poisson kernel in Rf +1. Let U (1:, t) be the Poisson integral of f. Then there

exist C1, 02 such that
ClSmf(a:,t) S U(:c, t) S Cszf($,t)

for all (:r, t).
Therefore the conclusion follows immediately from Theorem 2.5 .

Remark:

1. Corollary 2.6 is still true when R": +1 is replaced by the unit ball of C1.

We leave the details to the reader.

28

2. In Corollary 2.6, the space (111,“, m) can be replaced by the homogeneous

space (Rf+1,wdm, d) under the assumptions of Theorem 2.5 .

§2.3 Another Characterization of W3

Let Q be a positive measure on X defined by d9 = wdu. Let

_ lul(TB(m.r))
“3)“??? mar» ‘

 

Theorem 2.7 Let 0 < a < 1. Suppose Q is a doubling measure on X. Then
W3 c {u : K, e La-l-a(o)}.

0n the other hand, suppose 0 < a < 1 and w e A, for some 7 Z 1. then
W3 3 {u = K. e Lea-(9)},

Proof: Suppose p 6 W3. Then Sfilul E Lil—0(0). We may assume that u is

positive. Then for any y E X and r > 0,

1
“(,BW , ————/(W supreme)

 

 

 

_ ' 1 XB(y,r)(3)XB(a-,t)(3) s a:
‘ away, _—r—)) x+ x Q(B(:r,t)) M M“ "l
_ XB(y,r)nB(z,t)(3)

“ Q(B(1y,r))/x+/x Q(B(m,t)) d“(3)d”("’")
_ 1 fl(B(y,r)flB(a:,t)) x

‘ Q(B(y,r)) x+ f2(B(a:,t)) 5’“ ’2

2 1 o(B(y, r) n B(x, t)) dp(:r, t).

 

fl(B(y, r)) TB(y,r) Q(B(a:, t))

29

Since if (x, t) 6 TB(y,r), then B(x,t) C B(y,r). Thus

1
”(BU/ii.» B(lh")
> _1_
‘ Q(B(y,r)) TB(W)
#(TB(y,r))
n(B(y,r)l °

Therefore Mn(S§|p|)(y) _>_ K,(y). By Theorem 2.4, if Sfilpl E LTEEUI), then

Sx"i|/I|(-9)dfl(8)

duh, t)

 

Mg(Sfi|u|) E Liéifll). Hence K,“ E LIST-(Q).

Conversely, suppose K ,, 6 L736!) and w E A,. We first prove the following:
Lemma 2.8 {Sg(%;)(x,t)}p 6 V3.

Proof: Given any B (y, r), we need to prove that

l
¢/TB(y,r) 50(E)(x’ t)dfl($’ t) S CQ(B(y, r))

with C independent of y and r.
Note that if s E B(z,t) and (1:, t) E TB(y,r), then s E B(y,r). By Lemma 1.1,

there are C1, C2 > 0 independent of s, y and r, such that
B(y,r) C B(Sfllr) C B(y.02r)-

Since it is a doubling measure, we have

_1_ < Q(B(s,C1r))
Ku(3) _ I‘(TB(3,CIT))
Q(B(y,C2r))
#(TB(y.r))
”(BU/’7‘»
#(TB(y.r))°

 

_<_

 

S

 

Therefore

30

limos) SUI—EX“ ”‘24“: t)

__1___ duo) .
-/TB(y,r)fl(B($,t)) B(x,t)K,(3)dp( ,t)

WHafl
frees) C#(TB(y, r”(Mi/I” t)

= 0903(va-

 

The proof of the lemma is complete.

Now, similar to the proof of the first part of Theorem 2.5 (with g replaced by K ,),
for any f 6 L762), take q < 7 such that g = a, we have ||H,f||Lq(,) S C||f||m(g).
Then since we may assume 7 > 1, the second part of Theorem 2.5 implies that
p 6 W3. The proof of Theorem 2.7 is complete.

Combining Theorem 2.5 and Theorem 2.7, we have proven the following:

Theorem 2.9 Let 0 < a < 1, and let q > 0, p > 1, q/p = a. Let u be a positive
measure on X+. Suppose w E A, and set dfl = wdu. If K, E Lil—«((2), then there is

a constant C such that
llHufllLva) S Cllfller)

for every f E LP(Q).
Conversely, let 0 < q < p < co and let a = q/ p. Suppose that Q is a doubling

measure on X. If
”SNMMSQMWM

for every f E LP(Q), then K, E Lrl—afll),

31

§2.4 Carleson Measure Theorem

on Weighted Hardy Spaces

On RN, let (I be a doubling measure such that d0 = wdm, where m denotes the

Lebesgue measure. Recall that the weighted Hardy space is defined by
H’(fl) = {u : u is harmonic in Rf”, N(u)(:c) E L”(Q)}
with “Human = ||N(U)||Lr(m-

Lemma 2.10 Let
Pa) = {(y,t) = d(fcw) s t}.

(1) If (y, t) e I‘(a:), for any function f defined on x, we have
(Hof)(y, t) s CMnf(x)-

(2) For any x, we have
N(H9f)(:c) g CMgf(x).

Proof: Without lost of generality, we may assume f 2 0. we have

1

(Hflle/a t) = B(z,ss)gg(y,t) W B(Z”) f(U)dil(u).

Since for any (y,t) E F(:c) and B(z,s) D B(y,t), we have x E B(y,t) C B(z,s).
Therefore, by Lemma 1.1 there are constants C1 > C2 > 0 independent of 2:, y, z, s
and t such that

B(z,s) C B($,C23) C B(z,Cls).
Since it is a doubling measure, there is constant A such that

Q(B(z,s)) 2 AQ(B(z, 013)) 2 Amen, (3,3)).

32

Therefore

l
we? 3» [B(m) f‘“)d“(“)

1
S CQ(B($,018)) B(x,C1s)f(u)dn(U)

S CMnf(:r).

 

The conclusion (1) follows from the above inequality.

The conclusion (2) follows from ( 1) and the definition of operator N.

Theorem 2.11 Leta 2 1. Let Q be a doubling measure on X. Then u 6 V3 if and
only if
IIU($,1)IIL«(u) S CIIN(U)||LP(0)

for any measurable function u satisfying N(u)(x) E LP(Q) with q/p = a.

In particular, if X = R” and dfl = wdm, then

(1) Suppose w E A,. pr > 1 and ||u(:r,t)||Lq(,) S C]]N(U)HLP(0) for any harmonic
function u(.r, t) satisfying N(u) E L”(Q), then u 6 V3;

(2) Suppose w E A,. for some r 2 1. pr S 1 and ||u(x,t)]|Lq(,) S C]|N(u)IILp(n)

for any subharmonic function satisfying N (u) E L"(fl), then u 6 V3.
Proof: Suppose p > 1 and p 6 V3. If y E B(x, t), then
Mac, 0] S N(UXylo
Thus

HQ(N(‘U))(.’B,t)
1
Z W B(w)

Z lu($, t)|-

N (v)(y)dfl(y)

33

Therefore

IIU($,t)||Lc(u) S C||H9(N(u))||L.(,) S C||N(U)||Lr(0)-

The last inequality follows from Theorem 2.1 .

For p S 1, take r > 0 such that p/r > 1. Let C(x, t) = |u(:c,t)|’, then
N06”) = |N(u)(-’”)lr E [JP/Tn)-

The conclusion follows from the case p > 1.
The “only if” part follows by letting u(y, s) = XT(B(,,,))(y, s).
We now prove the particular case.
(1) Let XB(v.o) be the characteristic function of B (y, 3). Let U (re, t) be the Poisson

integral of x3(,,,). Then there are 0;, C2 > 0 such that
CIHm($, t) _>_. U($at) Z 025m(Xs(y,.))($, t)
for all (x, t). Thus if (:r,t) E TB(y,s), then
U(:c, t) 2 C2Sm(x3(,,,))(a:,t) Z Cg.

Hence

(smear/q s CHI/”Lum-

By Lemma 2.10,

N(HmXB(y,s))($) S CMm(XB(y,s))-

Therefore

(#(TB(31,3)))‘/°

C H U ”mm

|/\

l/\

C IIN (U )IIer)

|/\

C]|N(HmXB(v.8))llLP(9)

34

S Clle(XB(y’.))”Lp(Q) (Lemma 2.10)

S CllXB(u.a)“LP(0)

= C(“(B(y,s)))”‘°-

The last inequality follows from Theorem 2.4 .

(2) Suppose p S 1, w E A,. for some 1‘ Z 1 and suppose
”14$, t)llL°(u) S C||N(U)||LP(0)

for all subharmonic functions with N (u) E LP(Q). Let I > r. For any harmonic

function u E L'(Q), take k 2 1 such that U]: = p. Then
613,1) = lu($,t)|k
is subharmonic and N(G) = |N(u)|" E L”(fl). Thus

llullL‘°(u)

1 I:
= “CHILI/k0,)

I:
= ”Gilli-p0.)
k
s Uncut“,

= Cllullmoy

The conclusion follows from the case p > 1.

We now turn to the main result of this section:
Theorem 2.12 Let 0 < a < l and let q/p = 0. Then
||u($,t)||u(u) S C||N(U)llmm
for all u(x, t) satisfying N(u) E LP(Q) if and only ifu 6 W3.

35

In particular, if X 2 RN and d0 = wdm, where m denotes the Lebesgue measure,
then

(I) u e We implies ||u(x,t)llu(u) s C||N(U)llmn)i

(2) Suppose w E A,. pr > 1 and ||u(a:,t)|ILq(,) S CIIN(u)llLr(n) for all harmonic
functions u(z, t) satisfying N (u) E LP(Q), then u 6 W3;

(3) Suppose w 6 Ar for some 1‘ Z 1- IfP S 1 and ||u($,t)||m(u) S CHNWWLPW)

for all subharmonic functions satisfying N (u) E LP(Q), then [1 6 W3 .

Proof: We only prove the special case. The proof for the general case is similar.
(1) Suppose p 6 W3. Let g be the balayée of p w.r.t. Q as in Lemma 2.3. Note

that by Hfilder’s inequality

[30(1/9)(€l=,t)l'1 S (Sag)($,t)-

Then

j,+ Inc». are

/x+ [u(‘c,illqlSRU/QX‘”,ill-130(1/9)($,t)du(:r,t)
/x+ Mx’ t)[q(Sgg)(x, t)~5'0(1/s)(9'l', 04142:, t)

{/x, Mm, t)l"’Sa(1/g)(x, t)dp(a:, 019/»

x l A, l(Sng)($,t)|11—°Sn(1/g)(x,t)dp(x,ml-q/p
(fx, Mm, tllpSn(1/g)(:r, t)dp)9/P

”(in lHfl(9)($,t)lié;Sn(1/g)(x, t)dp)1‘9/P

_<. 0(/x |N(u)|"dfl)"”’-

l/\

|/\

|/\

The last inequality follows from Theorem 2.11 and Theorem 2.2 since by Lemma 2.3,

50(1/g)($,t)p 6 V3.

36

(2) Suppose for all harmonic functions u(x, t) with N (u) E L”(Q), we have
”H(iv, t)lllfl(u) S C||N(u)||Lr(o)-

Suppose p > 1 and that g is as above. Note that similar to the proof of Theorem
2.5, for any y E B(sr, t), by Lemma 1.1 and the fact that Q is a doubling measure, we

have

(50f)($,t) S CMof(y)-

Hence

[(SOf)($,t)l1/q
C 1 q
S W B(z’tHMnfU/ll / 619(9)

= 050(IM0f|1/°)($,t)-
Let f E LID/9(9). Then

I fxg(v)f(y)dfl(y)l
[XJSaIfKaoldp
/x+[(5rzlfl)”"]"du
C A,l50((Mnlfl)"")lqdn

c /M |U((Ma|f|)‘/°)|"du,

|/\ l/\ |/\

|/\

where U ((Mnl f [)1/ ‘1) denotes the Poisson integral of (MRI f [)1/9. Then by the hypoth-

esis,

| [x g(v)f(y)d9(y)l
s 0(/x |N[U((Mn|fl)"‘_’)]IPdW’”

s (“A INle((Mnlfl)"°)]|”d9)"/P

37

< (:(/x |Mm[(Mn|f|)‘/q]|’dfl)"/P (by Lemma 2.10)

C(/X(Molfl)”"'d0)"”

_<_ C(j)‘: Iflp/qdn)q/p S 00.

|/\

The last two inequalities follow from Theorem 2.4 since p > 1, p/ q > 1 and w E A,.
Therefore g E Lil—«(0), that is, p 6 W3.

(3) Similar to the proof of particular case (2) of Theorem 2.11 .

38

Chapter 3

MAXIMAL AVERAGING OPERATOR
AND CARLESON MEASURES ON X

In this chapter, we characterize those measures p such that the maximal averaging
operator defined on a homogeneous space (X, u,d) is bounded from L”(Q) to Lq(/1)
with 0 < q < p, where Q is a measure on X satisfying Muckenhoupt’s A, condition. In
the proof, we use the “balayée” of measure p with respect to it which is an analogue
of the balayée defined on X1“.

We shall collect some results for a-Carleson measures on X with a Z 1 in the
first section. In the second section we shall discuss some properties of the space of
“balayées” on X. The ideas there follow directly from the paper of E. Amar and A.

Bonami [1]. The main result of this chapter will be presented in the last section.

§3.1 a-Carleson Measures on X with a _>_ 1

In this and the next chapter, we shall state our results in the following generality.
The role of the family {E(x) : a: E X} below will vary in different situations that we
will subsequently study.

Let (X, V) be a measurable space satisfying the following condition:

39

For any a: 6 X, there is a V-measurable subset E (2:) containing at and a V- mea-
surable subset E2(z) D E (2:) with the following properties:
(1) ”(E(xll > 0;

(2) “Doubling pr0perty”:
”(E2($)) S Cut/(EM);

(3) “Covering property”:
If B C X and if A C U,eBE(:c), then there exists {xdfi-ii, C B such that {E,'}
(where E,- = E(a,) ) is a disjoint family and A C U§1E2($,).

We now give the definition of oz-Carleson measure on X:

Definition 3.1 Let (X, V) be as above. Let p be a measure on X and let it be a

positive measure on X. If
|#|(E2($)) S C[9(Ez($))l“
for any a: E X, where 00 > a Z 0, then we call u an a-Carleson measure w. r. t. 0.

Let
Vs? = {I1 = I#I(E2($)) S C[9(E2(3))l°}
with
IIPIIVg = inffc' = |#I(E2($)) S Cl9(E2($))l°‘}-

It is not hard to see that V3 becomes a linear normed space.

For f E L1 (V), define

loc

1
mm) ’ xiii?» V(15121)) Eu) ”Id”

40

Lemma 3.2 Let a Z 1 and p be a positive measure on X. Suppose that (X, V)
satisfies the assumptions made in the beginning of this section. If p is an a-Carleson

measure w. r. t. V, then
(/ ImflquW" s Cllulltffl/ lflpdVW” f e PM
for any 1 < p S q < 00 such that q/p= a.

Proof: Suppose p is an a-Carleson measure w. r. t. V.

For f E L°°(V), it is clear that

llmf||L°°(u) S “filmma-

If we can show that m is of weak type (1, a), the conclusion will follow from
Marcinkiewicz interpolation theorem.
Let A > 0, A = {z E X]mf(:r) > A}. Then for any a: 6 A, there exists y such

that a: E E(y) and
l
A < —— dV
u(Ea» a.) 'f'

The covering property implies that there is {yahfil C X, such that {E(y,)} is a
disjoint family, A C U§1E2(y,) and A < Wham) |f|dV.
Let E,- = E(yg), E? = E2(y,-). Then

u(A)
M(U:':1 E?»

few?)

i=1

llullvaiMEifl"

i=1

l/\

l/\

|/\

C.i"||#|lv.,a Ell/(1301“ (doubling property)

i=1

l/\

41

s CfllitllvslidEul“ (eel)

s alumni} / ‘ under

i=1

1
3 annual, / lfldvl".

Hence m is of weak type (1,0). By the Marcinkiewicz interpolation theorem, if

1/p=0,1/q=£—,0<0<1,then

Mafia“, 3 CIIuIWIIfIIW),

where C only depends on doubling constant C,. The proof is complete.
Remark: The doubling property and covering property can be replaced by the
assumption: if A C U353E(x), then there is {23;} C B such that A C U‘fglE(x,-) and

the sequence E (x,) can be distributed in N families of disjoint subfamilies.

Definition 3.3 Let 0 < oz < co and let 9 be any positive measure on X. Fix r > 0,

define

Pn,r($,y) = WXsaafly);
Po,rf($) = / Pa..(x.y)f(y)dn;

1
M9,, f x = su _—
( ) B(y,t))g(x,r) Q(B(y, t)) B(th)

P6,.#(y) = / Pn.r(w,y)dfl($);

V6,, = {l1 = |#|(B($,20ar)) S C[9(B(x,20ar))l°’};

If Idfl;

W3,. = {It = P5,.litl 6 LT1_°(0)};

V° = {/1 = |#|(X) < 00-}

Amar and Bonami have used the term “balayée” in a different but similar context

to describe the function P3,}; ( see Definition 1.10 in Chapter 1 ). We shall adopt

42

their usage and call P3,,Ipl the balayée of u w. r. t. (D, r). Under the norm

lli‘llwg, = llPfl,r”llL1-l;(n)a

W3., becomes a linear normed space.

Let (X, V,d) be a homogeneous space. Let w(x) Z 0, w E L}OC(V) be such that the
measure (I, defined by dfl = w(x)dV, is a doubling measure. Note that (X, fl,d) is
also a homogeneous space.

Fix r > 0. Let E(x) = B(x,r) and E2(x) = B(x,2Cdr), then
Ux€E(y)E(y) C 307,201")-

It follows from Lemma 1.1 and Lemma 1.2 that the assumptions of Lemma 3.2 are
satisfied by the space (X, 0, d). We shall call a measure p on X an a-Carleson measure

with respect to (fl, r) if there is a constant C, > 0 such that
#(Ez($)) S Crl9(E($))]°'
for every x E X.

Theorem 3.4 Let 1 S a < co, and let 1 < p S (1 such that q/p = a. Let p be a

positive measure and It be a positive doubling measure. Then

(/ [Mayflqdpr/‘l S C(flflrdfl)1/r
if and only if”(B(‘7”20‘"')) 5 C'[Q(B(3i20dr))la for every x 6 X.

Proof: Note that (X, (I, d) is a homogeneous space.

We only prove the “if” part. Fix R > r and define

1

MR, z = su _—
n, f ( ) “magmas-List!“(B(y’t» B(y,t)

If ldfl

43

The conclusion will follow by taking R —» 00, if we can prove that u is an a-Carleson

measure w. r. t. (0,1') implies

(/ |M&.f|"du)"q s C(/ Iflpdfl)"”

with C independent of R.
But this is a consequence of the proof of Lemma 3.2 with the applications of
Lemma 1.2 and Lemma 1.3. We leave the details to the reader. The proof is complete.
From the above proof, it is clear that if pB(a:,204r) S C'1[QB(2:,2C'dr)]°’ with C1
independent of r, then IIMn,rf||Lq(u) S C II f II ”(9) with C independent of r. Letting

r —+ 0, we have the following:

Corollary 3.5 Let 1 S a < co, and let 1 < p S q such that q/p = a. Let u be a

positive measure and Q be a positive doubling measure. Then

(/ IMnflqduW" s C(/ mun)”:

if and only iffor any r > 0, p(B(:c,2C'dr)) S C'1[fl(B(:r,2Cdr))]" for any a: E X with

C1 independent of 1'.

Now we turn to two-weight norm problem.

Let p be a positive measure in (X, V, d).

Theorem 3.6 Leta Z 1 and p > 1. Ifw E A,, and dfl = wdu, then for any q 2 p

such that q/ p = a,
(/ lMu,rf|"du)‘/° s C(/ mun)”: f e Um)
if and only ifp(B(:r,2C'dr)) _<_ C,[Q(B(x,2Cdr))]°‘ for any a: 6 X.

Proof: The “only if” part follows from taking f = XB(Z,2cd,).

44

Conversely, suppose p 6 V6”. Since w E A,,, by Theorem 1.4, there is a 7, '7 < p

such that w E A,. By Holder’s inequality, we have
M.,.f(x) s C[M9.r(|fl")(a=)]i,

where C only depends on A, constant.
Note that by Holder’s inequality, w 6 A,, implies that Q is a doubling measure.

Thus

/ |Mu,rf|"d/l
6' [warmth
Cl] Iflpdfl]°”’

l/\

|/\

with 0 depends on A, constant and the constant in the conclusion of Theorem 3.4.

The last inequality follows from Theorem 3.4 , since :- > 1, and i = a Z 1. The

7

proof is complete.
Similar to Corollary 3.5, if pB(z, 20,11“) S C1 [03(3, 2Cdr)]°' with Cl independent

of r, then we have

Corollary 3.7 Let p Z 1 and a Z 1. Ifw 6 A,,, then for any q 2 p such that
qm=a,
(/ IMuflqd/IW" s C(/ mum”: f e um)

if and only iffor any 1' > 0, p(B(:e,2Cdr)) S Cl[Q(B(:c,2Cdr))]°‘ for any a: E X with

01 independent of r.

45

§3.2 Characterization of W31, for 0 < a < 1

In [1], E. Amar and A. Bonami worked on X+ and showed that the space of
“balayées” is the interpolation space between the space of bounded measures and the
space of Carleson measures on X+. We shall prove that in our situation, the parallel
result still holds. We shall show that the space W3,, with 0 < a < 1, is the complex
interpolation space between V0 and V6,," The idea of the proof follows from E. Amar

and A. Bonami.
In this section we always assume that Q is a doubling measure on (X, V, d). Note

that (X, 9, d) is also a homogeneous space.

Lemma 3.8 Up is a positive measure, for any 1' > 0, let

943/) = Pam).

Then there is a constant C > 0 independent ofr such that if we define

ME) = LPn,r(£-:;)d#,

then

A,.(BCB, 20d?» S 09(B($, 2041‘».

Proof: Fix 1', let E(x) = E(a,r) and E2(a:) = B(x,2Cdr). It suffices to show that
for any a: E X,

l 2
[W Po.r(;)(y)dn(y) s cm (on)

with C independent of 1'. Note that

XE2(r)(y)XE(y)(t) S XB(x,303r)(t)XE(y)(t)-

46

We have

[M )ngixwdule)
[13ml] Pnr(y,t)—— —(—g)dfl(t)]dfl(y)

= / were / ”(35%“ (”dew/1e)

X 3 z (31))“: v (t )
= l 9.0)] E (Kimmy)? ”(3’)de

1 Xs(z,3cgr)(t)XE(y)(t)
/ 9.0)] may» WW“)

 

 

 

 

 

= Q(B(:c, 3037‘»

g CQ(E2(a:)).

The last inequality follows from the doubling property of Q and hence C depends

only on doubling constant . The proof is complete.

Lemma 3.9 If p 6 W3,, then there exists positive yo 6 V6,, and h 6 L”(po) such
that
fl = hl‘Oi

where l/p =1—- 0.

Proof: We may assume that p is positive.

Take p0 = 130.45%)!" h = [Pg,,.(;—r)]‘1 with g, defined in the previous lemma. By
the assumption, 9, 6 LP(Q).

By Schwarz inequality, h S P9,,gp It suffices to show P9,,g, E LP(po).

From the previous lemma, #0 6 Vfiw. Since g, E LP(Q), P9,,g, < M9,,gr, and

p > 1, Lemma 3.2 implies that

[lpflwgrlpdflo S 0/ Igrlpdfl < 00.

The proof is complete.

47

We now prove the main result of this section.

Theorem 3.10
W3,, = (V0: Vfl,r)0“

Proof: W3,, '—-> (V0, V‘},,)a follows from Lemma 3.9 .
In fact, suppose p 6 W3,,” By Lemma 3.9, there exist po 6 Ki, and h E L”(po),

l/p =1— 0, such that
u = hflo-

Since h E (Ll(uo),L°°(po))a and L1(po), L°°(po) can be identified as a subspace of
V0 and V6”, respectively, the conclusion follows.
Next show (V0, V&,,)a ¢—i W3”.

Define a multilinear map by

my, h. u) = A(Po,rf)(Pn.r9)hd/‘-

Then on L2(Q) x L2(fl) x L°°(p) x V6,, by Schwarz inequality and Lemma 3.2,

we have

|T(f,g, h, u)!
S llhllL°°(u)llPfl,rfllL’(u)llP0.rgllL’(u)

S Clll‘llVg,llhllL°°(u)llfllL’(9)ll9llL2(9)°
Similarly, on L°°(Q) x L°°(Q) x L°°(p) x V0, we have
|T(f,g, h,#)l S Clll‘llWllhllL°°(#)”fllL°°(9)llgllL°°(9)°
By multilinear interpolation theorem [5, p.96], on
L2"(9) X 112%”) X L°°(#) X (V0, Vfl,r)03

48

where 1 / q = a, we have

|T(f, g, h, fill S Cllflllwoygm). llhlle(u)llf||L2v(n)llgllmmr

Fix h such that Ihl = l and hdp = dlpl. Let f = g E L2°(Q). Then

[x an,.-f|’d|/t| s Cllullwmvin.llfllizqmy

Since 9 is a doubling measure, for any y E B(x, r), IPQ,,f(:c)| S CM9,,f(y), it follows
that

1 2 C 1 2
an,rf($)l / S W B(I’r)an,rf(3/)l / €190!)

CPQ.’[(M0,rf)1/2l($)‘

Thus, if f e L9(Q) = [LF17(Q)]', then

AIP5,.IuI(z/)]|f(y)ldfl(y)
_<_. APn,.-If|(x)dlul(x)
s C[K(Pm[(Mn,.f)"’](x))2dlul(w)

Cllt‘ll(V°.V;§,,)a ll(M9.rf)1/2llL29(fl)

IA

S Clll‘llwoyg'ga”filmm-

Therefore (V0, V(§’,)a c_, War. The proof is complete.
§3.3 Two-Weight Norm Inequalities

Let (X, 11, d) be a homogeneous space. Let a be a positive measure on X and let

(I be a positive measure on X defined by d0 = wdu. Let r > 0 be fixed. Define

, _ ”(B(x,r))
W) " W

49

Theorem 3.11 Fix r > 0. Suppose p >1 and w E A,,. Let q > 0 and q/p = a <1.

If
Kr 6 Lil—0(0) < (X),

then there is a C, > 0 such that
(/x lMu,.-f|"dn)"" s or]x lfl’dfl)‘/” f 6 m0).

Conversely, let 0 < q < p and a = q/p. If!) is a doubling measure on X, and if

(/x IPu.rf|"d#)‘/” s cur/x lflpdfl)"’ f e um).
then

K. e Ltéam) < 00.

Proof: Note that w 6 A,, implies that Q is a doubling measure.
Suppose ”K'IILTEWO) < 00, and p, q, a as in the assumption. Let g, be the
balayée of p w. r. t. (Q,r) as in Lemma 3.8 . By Lemma 1.1, it is clear that there

are constants A and B independent of r such that
AK.- S gr S BKr.
Then g, E LT-J'WQ). By Schwarz inequality, [Pg,,(;1;)]‘l(x) S Pn,,g,.(x). We have

/ IMu,rf|°d#
= / IMv.rfIqIPn..-(i)(x>1-1Pn.r(;—r)(x)dy(x)
S fIMu,rflqlP0,rgr($)]Pfl,i-(51:)(1')dfl($)
[/ |wa IPIPn.r(gl—r)(x)1dp<z)r/P
xl/ Warm)!ri—«Pn.r(g17)(x)dy(x)ll-°/P
If IMv.rfIPIPo.r<ixxnmmm

xl/ wise):fiPn..(;—r)(x)du(x)r-q/P.

|/\

|/\

50

By Lemma 3.8 , there is a constant C independent of r such that

[B(mcdr) P9,,( ixwdmy) S C”(B(it, 20,110),

It follows from Theorem 3.4 and Theorem 3.6 that

/ War/Imp

S C[/|f|Pdn]Q/P[/|grlil—adQ]l-q/p

|/\

Cllgrlngflmllfllipm)

|/\

CllKrlngzmfllfllbm)

with C independent of r.
Conversely, suppose “MW f || (My) S C,” f || ”(9) with p, q and a as in the as-

sumption. By the discussion at the beginning of above proof, we need to show that

For r > 0 fixed and :c E X, since 0 is a doubling measure, there is a constant C

only depending on the doubling constant of 0 such that for any 31 E B(x, r),

P0.r|f|(i€) S CMo,rf(y) S CMnf(y)-

Thus

C

[Pn,.-|fl(x)l”" S m B(z,r)(Mnf(y))1/"dl/(y)

= CP.,.[(Mof)‘/°](x).
Now if f belongs to LP/9(Q), the dual space of Lia—0(9), we have
I / grfdm

s / Pn,.|fl(x)du(x)
= /[Po.r|fl(x)l“"”"d#(rv)

51

l/\

C [(Pm[(Mnf)""](x))°dp(x)
S CC,[/(Mnf)p/qdfl]q/p (by assumption)

3 004/ mummy? < 00.

The last inequality follows from Corollary 3.5 with p = 9. Thus the constant C in the
last inequality is independent of 1'. Therefore g, E Lid—«(0) and ”shun-33(9) S C C...
The proof is complete.

Next we turn to discuss Hardy-Littlewood maximal operator.

Note that under the assumption that continuous compact supported functions are

dense in L‘(u), Calderon showed [6] that if w E A,,, then
1133(1) Pfl,rf(x) : f($)

almost everywhere on X. In particular, My f (2:) Z | f (2:)I almost everywhere. Then
“mum s Cllfllmn) implies "qu) s Cllfllum). Therefore em = gen for some
g. Now it not hard to prove that IIMyfIILqM S Cllfllum) if and only ifg 6 LII—0(5)).

In a general homogeneous space, applying the method used in the proof of Theo-
rem 3.11, we can obtain the following two-weight norm inequality for Hardy-Littlewood

maximal operator My.

Theorem 3.12 Let X be a general homogeneous space. Suppose p > 7 Z 1 and

cue/4,. Letq>0 andq/p=a<1. If

33g IIKrIILrgm) S C < 00,

then

(/x IMuflqdu)"" s C(jx liven)”: f e um).

52

Conversely, let 0 < q < p and o: = q/p. If!) is a doubling measure on X, and if

for any r > 0

(/x IPu.rf|°du)"" _<_ C(jx mum”? f e mm),

with C independent ofr, then
K, 1 < C < oo.
Sigh)” ”L "(9)

Proof: Suppose supr>0||KrllL,—_1_;m) < C < 00. From the proof of Theorem 3.11,

we have
/ IMy.ef|"d/z s ngp IlKrllLrgzmfllfllipm)

with C independent of r.

Now let r —+ 0, since 1W”, f increases, it follows from Fatou’s lemma that

“Myfllmm S C Ilf ”mm-

The converse part is a direct consequence of the proof of Theorem 3.11. The proof

is complete.

53

Chapter 4

CARLESON MEASURE THEOREM
IN WEIGHTED BERGMAN SPACES

Let U be the unit ball in C” and Q be a positive measure on U satisfying Békollé’s
BE condition. We characterize those positive measures p on U such that the inequality
[I f I] L'U‘) S C N f H ”(9) (1 < p S q) holds for any function f in the weighted Bergman

space AP(Q). As an application, we characterize the multipliers from 119(0) to Aq(fl)

(q 2 p)-

§4.1 Carleson Measure Theorem in Weighted

Bergman Spaces

In [8], Cima and Wogen proved the following Carleson measure theorem for

A2(dmg) in the unit ball U of C”:
Theorem 4.1 Let ,8 > —1. Then
2 < 2
[Um em _ C/U lfl dms
for any f E L2(dmg) if and only iffor some fixed r, 0 < r < 1,
p(E(a,r)) S Cm5(E(a,r)) a E U.

54

In [14], Luecking developed a general technique to find a sufficient condition for

llfllLP(u) S Cllflle(n)-

The following lemma is a generalization of Luecking’s work in a homogeneous

space.

Lemma 4.2 Let 6 > 0 and let (X,V), E(z), a Z 1 as in Lemma 3.2 . Let p be a

positive measure such that
1452(3)) S Co[V(E2($))l°~

Let C1 > 0. Then for p > 6, q/p = a, and any f satisfying

01

|f(a:)|5 S m 3(3) ”|st a: E X,

there is a C > 0 such that
llfllwu) S CllfllLP(u) f 6 ”(V)-

Proof: We may assume f E L”(V). Since q 2 p > 6 and i = 0:, Lemma 3.2

implies that

[/ lire/11":
-—— l / (WW/11W
Cl/(m(|f|6))*dnl”°
Cl/(Ifl‘)‘dv]"”
= Cl] Iflpdvl‘”.

l/\

IA

The proof is complete.
In this section, we shall work with the homogeneous space (U,wdmp, p).

We shall refer all definitions and notations in this chapter to §1.2.

55

Recall that p(z,§) = |<I>£(z)| is a metric onU and E(z,r) = {w E U : p(z,w) < 7‘}.

As the second application of the generality stated in the beginning of section 3.1,
chapter 3, we shall take E(z) = E(z,r/3) for some fixed r, 0 < r < 1, E2(z) =
E(z,r). Then the A, (7 > 1) condition in the space (U,wdm5, p) is equivalent to
the C, condition defined in §1.2. By Holder’s inequality and the fact that m5 is a
doubling measure, d9 = wdmp is a doubling measure. Therefore (U, I), p) becomes a
homogeneous space. From Lemma 1.2 , the assumptions of Lemma 3.2 are satisfied
by (11.9.11)-

In [16, Lemma 3.1], D. Luecking proved the following:

Lemma 4.3 Ifw satisfies the C, condition for some 7 > 1 and cl!) = wdmg, then

for any f analytic in U, any q > 0, and any 2 E U,

flap) ”|qu

|f(z)|q S cm

with C depends only on 6, 7, r, and C, constant.

From Lemma 4.3 and Lemma 4.2 , we have the following generalization of Theo-

rem 4.1:

Theorem 4.4 Let a _>_ 1. Let p,q > 0 such that q/p = a. Ifw satisfies the C,
condition for some 7 > 1 and p is a positive a-Carleson measure w. r. t. (O, r), then
for any f E A”(Q)

I] lflqd/zl‘“ S Ct] lfl’dfll"”-

We next prove that being an a-Carleson measure is also a necessary condition for
llfllA'(u) S CllfHAp(fl) if an satisfies Bg(w) condition.
We shall use the following well known facts in the proof of next two lemmas.

(1) For every a E B, <I>,,(0) = a and <1),,(a) = 0.

56

(2) The identity

(1— < a,a >)(l— < z,w >)
(1— < z,a >)(1— < a,w >)

 

1— < <I>a(z),<I>a(w) >=

holds for all z E E, w E B.

(3) The identity

1- |a|2)(1- l2?)
1— (Pa 2 = (
I (2” |1— < z,a > [2

holds for every 2 E B.

(4) The real Jacobian of <I>a at z E B is

(JRo.)(z) = ( fing )

For the proof of these facts, see [19, p.26].

N+l

Lemma 4.5 Let a E U and 0 < r <1. Then
sup{|1— < a,z > I :z E E(a,r)} =(1— |a|2)(1— r|a|)'l
Proof:

sup{|1— < a,z > I : z E E(a,r)}

= sup{|1— < <I>a(0),<I>a(/\) > | : A E rU}

1- lal2

— < a,/\ >
= (1-|a|2)(1-r|a|)"-

|:/\ErU}

 

= sup{|1

The proof is complete.

Recall that T3 is the Bergman operator and

N
Tare) = ( 1:3 ] [U lKa(z.w)|f(w)dma(w) z e v.

57

Lemma 4.6 Let p > 1 and q/p = a. Suppose (.0 satisfies the BE(w) condition and

d9 = wdmg. Then ||f||Aq(,,) S C”f”Ap(n) implies that for fixed r > 0
#(E(a,")) S Cr[9(E(ae7‘))l°'
for any a E U.

Proof: Suppose for any f E AP(Q), ||f||Aq(,,) S C”f”Ap(Q)o For any a E U, take

xg(a’,)(w)(1- < a,w >)‘6

u (1— < z,w >)~+1+fi dm(w).

 

f(2) =

Then f (z) is analytic and

 

_ XE(,,,,.)(w)(1- < a,w >)‘6
Inna...) — [U I [U (,_ < w >),,,,., etm<w>|vdn(z)

XE(0',.)(w)[1— < a,w > '3 d pd”
/U| U |1—- < z,w > |N+1+fi(1—|w|2)3 "m(w“ (z)
— < a,w >
1— [w]2

 

 

‘_ I
= llTfi(XE(a.r)(w)l lalellirm)’

Since no satisfies 850.0) condition, Theorem 1.13 implies that T5 is bounded on

LP(Q). Hence

Il— < a,U}>|fi
1—|w|2

l- < a,w >
= c [U xE(.,,.)(w)I 1—|w|2 lppdmw)

 

llfllfipm) S CIIXE(a.r)(w) “3(a)

 

fip
supE(a,r) l1_ < a,w > l a
CAI XE(a,r)(w) [1 _ lwlzlfip d (w)

(1— |a|"’)"”(1 — wan-fir
C/U XE(a,r)(w) '1 _ lwlglgp dint")

S Cr / dew),

E(a.r) ll - lwlzl‘”

 

|/\

 

|/\

Since on E(a,r), (1 — |w|2) ~ (1 — Ialz), we have
P _.
llfllmn) s 0.- lg...) dmw) -— anew».

58

On the other hand

Ilfllq — j | XE(a.r)(w)(l— <a,w>)a

(1— < a,w >)3
d 9d .
/E(aer) I[E(a,r) (l— < z,w >)N+1+fl m(w)| ”(2)

 

d""t(w)l"alle(2)

 

Let w = <I>,,(/\) in the second integral, then 2 = (1)607) for some 17 and A, 17 E rU. Thus

IV

IV

IV

llflliqu)

(1- < <I>a(0).<1>.(A) >)‘3 1— M2 1 q
v/E(a.r) I er (1- < <I>a(q),<1>a(A) >)N+1+fi(|1_ < 3,0 > l2)N+ dm(/\)I d#(2)

/ I ( 1-|a|2 )5((1—<n,a>)(1—<a,A>))N+1+fi
E(a,r) rUl

 

 

 

 

 

 

 

 

— < a,). > (1 -|a|2)(1— < n,A >)
><(|,_1<‘,',‘:[2> l2)"“dm(»\)|"du(z)
law) “1'. < "’0 >)NHW /rU (1— < 1,1; >)Nflrzgl‘l— < a,A >)N+1 “Cl/1(2)
2N
[E(a,r) K1- < 17,a >)N+l+fi [U (1— < rt,n >);+1:lg(lt: < a,rt >)N+1|qdp(z)
C m...) |(1— < ma >)~+1+a,~2~[T,( (1_ < m: >)~+1+ fi)(ra)]|9dp(z)
C E(a,.) |(1— < 77,a >)N+1+/3r2~(1_ < r20?” >)~+1+a |°dp(z)

l
C 30...)“ “ ”'N+‘+fi"2"“27vm)°d”(zl

C.#(E(a, r))-

Since “ill/N(u) S CIIfIIAPm), it follows that

#(E(ae7')) S Cr[fl(E(a,T))l°'

with C, only depends on r. The proof is complete.

Since the BEG») condition implies the Cp condition, combining Theorem 4.4 and

Lemma 4.6 , we have proved the following:

Theorem 4.7 Let (1 Z 1 and q _>_ p > 1 such that q/p = 0. Suppose w satisfies

B50») condition. Then

llfllAeoe) S C'HfllAvm) f 6 AW),

59

if and only ifp(E(a,r)) S C,[Q(E(a,r))]°‘ for any a E U.

From Theorem 1.13, w E B§(w) implies that IngfllApm) S C“f”Lp(Q). Note that

T3 f is analytic, we have

Corollary 4.8 Under the assumption of Theorem 4.7 , for any f E L"(Q),
llTfifllAc(u) _<_ Cllfllem)
if and only ifp E V00}-
We close this section by considering the case q < p.

Theorem 4.9 Leta = q/p < 1 and 1 > r > 0. Then

1. IfO < q, p > mas:{1,q} andw 6 B50»), then " B :1 6 LTEF(Q) for some
r implies

llTsfllAuu) S Cllfller) f 6 mm.

In particular, W E LTl—a'(0) for some r implies
llfllA°(u) S Cllfllewm) f 6 ”(Q)-
2. [[0 < q < p and Q is a doubling measure, then
llTifllLuu) S Cllfllmfl) f 6 L”(9),
implies W E Lia—«19) for any r > 0.
Proof: 1. Since T5 f is analytic, w E B§(w) implies (by Lemma 4.3 )

|T3f(2)| S CMn,r[Tpf(Z)l-

60

Since B5(w) implies C,p which is equivalent to the A,, condition in (U,wdmp, p). By

Theorem 3.11 and Theorem 1.13 , if p 6 W3,, and w 6 85(0)) then

llTfifllAq(u) S C||M9,.[Tgf]||u(,.)
S CllTfifllApm)

S Cllfllerr
2. For any 1 > r > 0, by the fact that
ma(E(aaT)) ~ (1 - |a|2)N+1+s
and Lemma 4.5, there is a constant C, > 0 such that Pm,” f S C,T5| fl Therefore

IImefllLem S CrllTEIflllLuu) S CrllfllLr(0)-

By Theorem 3.11, p 6 W3”.

§4.2 Multipliers on Weighted Bergman Spaces

Let M (p, B, 7) denote the collection of all functions f which multiply A’(wdmg)
into Ap(wdm,), that is, f g 6 A"(wdm,) for any 9 E AP(deg).

Let N (p, q, 3) denote the collection of all functions f which multiply Ap(wdmg)
into A"(wdmp).

In [25], G. D. Taylor proved that

(1) iffl > 7. M(2,fi.7) = {0};

(2) iffl S 7. M(2.fl,7) = {f = f is analytic. |f(2)l = 0(1-|2|)951}-

In [2], K. R. M. Attele proved that

(1) if? < q, N(ntzfl) = {0};

61

(2) ifp = q, N(p,p,fl) : H°°;
(3) ifp > q. N(p.q,fl) = {f E A”(017715): l;=1/q -1/p}-
The (3) of Attele’s result has been generalized by Luecking ( see [17] ).

Applying Theorem 4.7 , we have the following results for the weighted Bergman

spaces.

Theorem 4.10 Let 1 < p < 00. Suppose (.0 satisfies the B5(w) condition. Then
(1)iffl > 7, M(afln) = {0};
(2) tffl S 7. M(p,/3,7) = {f = f 18 analytic, |f(z)| = 0(1-IZI) P }-

Proof: Since f E M(p,fl, 7) if and only if for any g 6 Ap(wdmg)

/ lgfl’wqu s c / Igl’wdma.

from Theorem 4.7, we have that f E M (p, ,6, 7) if and only if for any 0 < r < 1, there

is a C > 0 depending only on r, such that for any 2 E U,

f pwdm S C/ wdm .
/E(2,r)| I 1 E(z,r) fl

Let dfl = wdm,. Since ma(E(z,r)) ~ (1 — |z|2)° for any a > 1, the above inequality

is equivalent to
1

— PdQ<C1—zzfi-7.
“(E(Zfl'» E(z,r) [fl — ( l l )

Then Lemma 4.3 implies that
|f(2)l’ S 00- |z|)B'"-

Conversely, it is clear that |f(z)|p S C(l — [25])3"y for any 2 E U implies that
f E M(p, fl, 7). Therefore , if 3 > 7, letting |z[ —+ 1, it follows that f E 0; if 5 S 7,
then [f(z)| = 0(1 -— |z|)£;_1. The proof is complete.

Let H °° = {f : f is a bounded analytic function in U}.

62

Theorem 4.11 Let p > 1 and q > 0. Ifw satisfies the condition B5(w), then

N(P.p,fl) = H°° and N(nmfl) = {0} ifp< q.

Proof: Let p S q. By Hblder’s inequality B5(w) implies B5(w). Let d9 = wdmg.
Similar to the proof of Theorem 4.10 , f E N (p,q, 3) if and only if

W)“ — W a 'f '°wdmfiSCI9(E(e,r))1q/P-l

with C depending only on r.

pr = q, it is clear that f E H°°; ifp < q letting |z| —-> 1, it follows that f(2) ;-= 0
on U. The proof is complete.

We close this section by giving an example of Theorem 4.10 and Theorem 4.11.

Let {d(r), zp(r)} be the normal pair defined in (1.1), Chapter 1. Let /\ > 0 be the
real number in (1.2), Chapter 1.

For a normal function ¢(r), if p > 1, there exists a nonnegative number t Z 0
such that (¢(r)(1 — r)“)"%, is integrable in L1(dmg+t). We may assume that t is big
enough.

We now prove that W = ¢(r)(1 — r)“ satisfies 85+,(W).

In fact, fix 20 E U, denote K = K (20). Since ($37 is non—increasing and if

z E K(zo) . lzl > Izolo

AWdr;(g+)t(z)
" C K(l-

¢(|20l)
S C(1_I_z_—OI I)“ “Admfi+t+a(z)

S C(—————¢(IT:DD,(1—|20|)”+‘+°+”“

= C¢(IZOI)(1 - [Zelzlmwmle

r“) ——dmfi+t+a(2)

The third inequality follows from mfi+t+a(K) ~ (1 — Izol2)fl+‘+“+N +1.

63

Similarly, note that
[¢(r)(1 — r)-*]-% = c(¢(,~))%(1 _ rye-A);
Hence

/K W’%dmp+,(z)
= c/K(¢(r))"edm,,,+(,_,,é(z)

I I

S C(¢(|zo|))%(1 _ |z0|2)3+‘+(t-A)%+N+1.

Now it is clear that B5 +,(W) is satisfied.
Since AP(¢(r)dmg) = Ap(deg+¢) and Aq(¢(r)dm,) = A"(de,+t), Theorem

4.10 and Theorem 4.11 imply the following:

Theorem 4.12 Consider the spaces AP(¢(r)dmg) and Av(¢(r)dm,), where ¢(r) is a
normal function. Then

(1)1ffl > 7, M(nfln) = {0};

(221m 5 7. M(nfln) = {f : f .3 analytic. |f(z)| = 0(1-Izl)"—?"};

(3) N(nnfl) = H°° and N(Mfl) = {0} ifp < q.

64

Chapter 5

BERGMAN OPERATOR
IN WEIGHTED MIXED-NORMED
SPACES

In this chapter, we use an interpolation theorem between weighted norm spaces
to determine the weighted mixed norm spaces on U, the unit ball of C”, as the
interpolation spaces between the LP spaces on U and the LP spaces on the boundary
3 of U with different weights. Using these facts, we prove that for some appropriate
weights, the Bergman operator induces a bounded projection on the weighted mixed
norm space. Thus we are able to identify the dual of those weighted mixed norm
spaces of analytic functions.

In section 1 we give some preliminaries. In section 2 we prove an interpolation
theorem of mixed norm spaces. We shall present the main result of this chapter in

section 3. Several duality theorems will be presented in the last section.
§5.1 Preliminaries

We shall refer all definitions and notations in this chapter to §1.2.
Let {<p,z/)} be the normal pair as in (1.2), Chapter 1. Suppose that A = <p”(r),

B = w(r)(1 — r2)“, C E 1. We shall need the following lemmas.

65

Lemma 5.1
1
/ (1— pr)-*(1— r)-‘ee(r)dr s oer-1o) o s p < 1.
o
For the proof, see [23, p.291].

Lemma 5.2 [ft > 0, w E U, then

ls duo(z) = 0( 1

|1- < w > IN“ (1— ler)‘

 

For the proof, see [19, p.17].
Lemma 5.3 For7 > —1, and m > 1+ 7,
/01(1— pr)“m(1 — r)"dr S C(l — p)1+7_"‘ 0 S p <1.
For the proof, see [23, p.291].
Lemma 5.4 For any f E H°°, Ta(f) = f.

For the proof, see [19, p.121].

Let L” and H“ as in Definition 1.11, §1.2.
Lemma 5.5 Forl _<_ p < oo, 1 s q < oo,IIfr — fllHP'9(<pqw(1_,-2)a) —» 0 as r -—> 1-.

This follows immediately from the dominated convergence theorem. (For details,
see [22, Proposition 3.3 D.

We shall use the following pairing between functions in Lp’q(<pqw(1 — r2)°’) and

I

functions in LPI'9’(cp‘q'w_g<1—(1 — 13)"):
< 1.9 >= [U f(z)§(z)dme(z)- (5.3)
In [4, p.304 ], A. Benedek and R. Panzone showed that the dual space of the mixed
norm space LM (cpqw(1 — r2)°') can be identified with LP"°’(<p9w(1 — r2)°’) under the
pairing

< f, g >= fU f(z)§(z)<p"wdma(z).

66

Lemma 5.6 Forl S p < oo, 1 S q < 00, under the pairing (5.3), the dual of

L”*°(<p"w(1 — r2)°’) can be identified with L”""'(gp'q'w_gi(1 -— r2)°’).

Proof: For any linear functional L of LP’°(<p°w(1 — r2)°'), there is a unique function

h E LPI'q’(cp°w(1 — r2)°‘) such that for any f E Lp'q((pqw(1 — r2)°’),
LU) = [U f(2)7z(z)s0q(r)w(r)dme(z)

and ”L” = llhlle’,q’(,pqw(1_,.2)a) [4]

Let
g = h90"(7‘)w(7‘)-

Then

I
”all" ,

Lp'n' ((p-q' w" %'(1_,-2)a)
l ' -’ a; 4': 2N-1 2
= A(nglpe Pee/ow .. (l-rrdr

1 ' ' ' —' 1’7 —9: 2N—1 2
= /(/S|hl‘°sp"”w”<p 1”dl/ofiw qr (1-r)“dr
0

1 ' "fr '— ' '—1’— 2N-1 2
/(/SlhlpdV0)’ 9qu qwq qr (l—r)°’dr
0
”hlle’,qI (¢qw(1_r2)a)'

I

Thus g E LPI'9'(<,o'q'w-%(l — r2)°') and

L(f) = /Uf(z)e(z)dm.(z).

I

Conversely for any g E L”"q'(go‘9'w-ga"(1 — 73)"), by Hélder’s inequality

[U f(z)§(2)dme(z) = Lgm

is a bounded linear functional on Lp’q(cpqw(l — r2)°‘). The proof is complete.

Let U” denote the unit ball of LM(B(r)).

67

Lemma 5.7

llfllLP-¢(B)= 811p /Uf§B(r)dm.
geUP'ec'

For the proof see [4, p.303].

Lemma 5.8 Let 0 < p < oo, 0 < q < 00. Suppose w1(r),w2(r) E L1(dr) are two
positive functions on [0,1). If there exists a re > 0 such that for r0 < r < 1, w} ~ (.02,
then

Hp’q(w1(7‘)) N Hp'q(w2(7'))-

Proof: There are C1, C2 > 0 such that if r0 < r < 1, C1w2(r) < w1(r) < C2w2(r).

Let 2 E S and f E Hp'q(w1(r)). Let

_ r0 p % 2N—1
I—jo (lslflduo)w1r dr.

Since if f is analytic, then fs I f (roz)|"duo is an increasing function of r. Thus

I s [0” w1(r)r2N'ldr(/S Wessex/0e»?

Let
C(ro) = (/ mow-Mm

Then

I = C(ro) / 1w,(,,),,2~—1d,,
S C v/r:(/S |f("oz)lpdV0(Z))%w2(P)P2N'1dP
s 015/8 ”(snide/0(2))%we(p)p2N-ldp

S Cllfllimw,

where C depends only on re.

68

Therefore

”fut...”
= folds Ifl’dVo(2))iae(r)r2N"dr
= (f + 1:)(Llflpduo<e))%wf”-ldr
s C(ro)l|fllle...(.,,, + 0: [:(jslflpdyo(z))%w§1v_ldr

S (C(ro) + Czlllfllin..(w,)-

Similarly
llfllimn» S Cllfllimmr

The proof is complete.
§5.2 Interpolation Spaces

Throughout this section, we will follow the notations of [5].

We first list some basic definitions of real interpolation method.

Definition 5.9 Let X0,X1 be two topological vector spaces. X0,X1 are said to be
compatible if there is a Hausdorfl' topological vector space U such that X0,X1 are

sub-spaces of U.

Let X = (X0,X1) denote a compatible couple of two quasi-normed spaces X0

anXm.
Definition 5.10 Let a E 2L0 X,-. Define

K(t,a) = K(t,a;Xo,X1)
= inf{||ao[|x0 +t|[a1|[x, : a = a0 + a1,ao E Xo,a1 6 X1}.

69

°° dt
“anal = [0 [t‘K(t.a)]"7 o<o<1,o<qseo.

1
(X0,X1)9,q,x = {a G in 1 llall9,q,X < Cole

i=0
Theorem 5.11 Let)? = (X0,X1),I’ = (Y0,Y1). Suppose T is a linear map from

23:0 X, ——+ Zioni such that for any a,- E X,-, i = 0,1,
[[TaillY, S Killaillxp

Then
T: (X0, X1)”; —» (Y0,Y1)9,q,Y

with

llTall0,q.Y S Ki’oKfllalloes-

For the proof, see [5].

Throughout this section, we will assume that C (r) is a non-negative function on
[0,1) such that [{r 6 [0,1) : C(r) = 0}] = 0, where IEI is the Lebesgue measure of
E on [0,1). Suppose B(r) is a non-negative function on [0,1) and suppose that the
measure [1, defined by dp = B(r)C(r)""r2N"1dr, where 0 < 7 < 00, is a o-finite

measure on [0,1). Let
m(af) = #{TE [0.1) = llfrllA.pC(r) >p}-

f‘(t) inf{p = m(p. f) S t}-

°° dt
||f||$,, [0 strung 0<r30e,0<qsee.

The vector valued Lorentz space L( p, 1', q) is defined by

L(pmq) = {ft llfllm S 00}.

For the properties of m(p, f), f“ and L(p, T, q), see [12] and [5].

70

Lemma 5.12
L(p, (M) = L”"'(A, 30"")-
Proof:

llfllZ,, = /O°°Ir(t)lth
= [umnnmrnwe

1
= f0Ilfr||3.,,.B(r)0(r)"‘”r2”“dr

‘— llfllLP.q(A,BCv-‘1)°

The proof is complete.
Assuming L = (LP"'(A, B), LP’°°(A, C )) is a compatible couple, we have the fol-

lowing vector valued version of Theorem 5.2.1 of [5].

Theorem 5.13 Suppose f E LP'7(A,B)+LP'°°(A,C), 1 S p < 00,0 < 7 < 00. Then
(1) K(t,f; Len/4,3), Lem, 0)) ~ 0;" warden,-

1_1-0
(2) F0r7<qSoo,;——,—

(LPN/1, 3), L”’°°(A, C))0,q,L = L(P, W1)-

Proof: (1) The proof will follow from the argument in the proof of Theorem 5.2.1
in [5] once we make a decomposition of f.

(i) “ S ” part. For 2 E 5, let

fem) = I f(rz) — 153315,), if u “mom > f*(t")

0 otherwise

and let f1 = f— f0.
Let

E = {7‘ 6 [0, 1) = ||(fo)r||A.pC(r) 75 0}-

71

Then since f“ is non-increasing, we have
/1(E)= m(f'(t”),f) = |{3 = f"(8) > f"(t”)}| S t”
and f*(s) is constant on [u(E), t7]. Thus

K(t, f; LM(A, B), LP'°°(A, C))

|/\

”fella-ms) + tllfllleoMp)

_ r2 _ f*(t7)f("2) 1 r r2N—1 r 1; .. ,

_ {/E llf( ) ”fr“A,pC(r)“A’pB( ) d l + if (t )

= I], nuruwm — f“(t)ll1,pB(r)C(r)'"r’"“dr]i + we")
u(E) ; ‘7 L

= {/0 we) - murder + I /, (murder

= (furs) — mmrdsfi +I/0”(f*(t~)>~de1‘e

3[/oh(f‘(t”))"ds]i-

 

|/\

(ii) “ Z ” part. Assume f = f0 + f1, f0 6 LP'7(A,B), and f1 6 Lp’°°(A, C). Since

||(fo + f1).||,4,,,C(r) S ||(fo)r||A.pC(7‘) + ||(f1).||A,,,C(r),

we have
m(Pl + Pzef) S m(p12f0)+ m(Pz: f1)-
Hence
{m + p. =m<p1+ p2.f) s s} 2 {e +22 :m(p1.fo) s (1 — 6)8;m(p2,f1) 3 es}
for 0 < 6 <1. Since
inf{Pl + P2 3m(P1,fo) S (1 - €)3;m(P2, f1) S 63}
=infiP1 3 m(p13f0) S (1 - €)3} + {P2 1 m(P2ef1) S 63}-

It follows that
f’(3) S f6(1- 6) + files)-

72

Thus

[figurines

{/n(f6((1 — omens + l /( f. an)" 2.1%

Ilium — £)s))7d3]% + We)
{/01(HUG)’HA"C(r))qB(T)C(")"’r2N“dr]i(1 — e)'% +t||f1l|Lp.ee(A,c)

|/\ |/\ |/\

-2
(1 — é) ”llfOIILr-~(A.B) +t|lf1llLPv°°(A.C)-

Let c ——+ 0, we have proved < 1 >.
(2) See the proof of Theorem 5.2.1 of [5].

Let 7 = p,0 = 1 — 5,C(r) E 1, we have
Corollary 5.14

mm, B) = (WA. 8), L”'°°(Ae 1)).-5...

for q > p.

In particular
L”"’(<p"w(1 - r2)"') = (L””°(<P”w(1 - r1')"'),L’"'°°(<p”))1—5,z.e-

Proof: It follows from Lemma 5.12 and (2) of Theorem 5.13.

§5.3 Bergman Operator

In this section, we first prove that the Bergman operator is bounded on the
weighted Hardy type spaces LP'°°(cpP), 1 S p S 00. Then our main result of this

chapter will follow from this fact and Corollary 5.14.

Theorem 5.15 Let cp be a normal function. Let b > 0 be as defined after (1.1),

Chapter I. Ifa — b > —1, then T; is bounded on LP'°°(cpp).

73

Proof: Let a > 0 as in (1.1). (,£ 6 S and z =

p(,w=r£,0Sp<1,0Sr<1.

Let k + l = N + a + 1, where lc and I will be determined later.

For1<p<oo

[3 IT;f(2)|’cp”(p)dz/o(<)

 

 

 

 

 

 

_<_ [SI U I,_ < 'fffl' I“...emergent/om)
[SI U |1- <”$15,511,190”,dm.(w)rsop(p)duo(<)
/s[ e |1_E(:llp:([3:e(.)dma(w>1
XI/U 11— < :f‘;<:>l,,,(r)sense/0(4).
The second factor of the integrand is
I = [u |1— <:me:(:)|le’ emf,
= {/0 ( s |1—:C:of)> W ')(l 90(5) rm—ldrlfr'

Since a — b > —1,
from Lemma 5.2 that
1 _ 2 a
I 0/ (1 r ) dr ]£,
0

e<r><1 — my» -~
(1 — r) (1

—r)°"“dr

 

— o e(r)(1— rp)”""

+ A! (1 -r)"(1

there is a b > 0 such that a — b > —1. If lp’ — N > 0, it follows

 

— r)°"bdr]f,
s0(r)(1 - rp)"’"”

If [p — N > a — a + 1, by lemma 5.3, since (17-3-1 is non-decreasing, 9&3: is non-

increasing, we have

 

 

l dma(w) 1:,
U |1— < z, w > [‘P'w(r)
(1— p)a —lp’+N+a-a+l+
_ Cl— <P(P) (1- P)
(1 _ p)N+1+a—Ip' f’
C[ 90(9) 1 '

74

(l -p)”
90(1)) —(1

p)—zp’+N+e—b+1 l fr

Then

[S IT;f(z)IPeP(p)duo(o

 

 

 

 

 

 

 

 

 

 

s “(Fitz/yes], SI U|,_ {(25:52:55)elm.(w)1ep(p)duo(<)
S C[(1— eras-21%, s |1- 375): ,..>'f(w)'p::ff)“’p(’) m(w)
S GIG-223“"? fr u|1—p1r|kv-N |f(w)|”:(”:)/I)r”(r) dma(w)

s a“ - P;“(’;;*“"”'sums-..) j; (,<1_;,31:f;<;g,,ee

s Cllflan°°(w)l(l " ”Wm-”'15 (1 " PlNilia’kpeoup)

90(p) c2(2)

= CllfllLrewwr
The third inequality holds if kp — N > O, and the fifth inequality holds if
kp — N > a — a + 1.

Therefore if we choose k, I such that

kp—N>0
lp'—N>0
kp—N>a-a+1
lp'—N>a-a+1

01'
{ 1+ 2r > N+1+a
P P
g Nilia
k+p> p

then ||Tgf||u.ee(,pp) S CIIfIILMeWp). Since lc + l = N + 1+ 0, we can let I = N—firfl,
lc = 3%. The proof for l < p < 00 is complete.

For p = 1 and p = 00, the arguments are similar. The proof is complete.

75

Remark:

1. T; is not bounded on “unweighted Hardy type” spaces LP'°°(1). In fact,

1 E Lp’°°(1) but T;(1) is not in LP'°°(1).

2. The condition a —b > —1 can not be omitted. In fact, take 30(r) = (1 — r)c
for some 6 > 0, then b: c. Suppose a — c = —1. Let f = (1 — r)‘°. Then

f E Lp’°°(gop), but T;( f) is undefined.
We now prove the main result of this chapter.

Theorem 5.16 Suppose p S q S 00, 1 < p < oo, cp is a normal function, and
a — b > —1, where b is as defined after (1.1), Chapter 1. Ifa radial function w(r) on

[0,1) satisfies condition B§(<pp(r)w(r)):

fl; w(r)gop(r)(1 — r2)°’r2N"ldr

l ' I
x (/ hw'%(r)<p"’ (r)(1 — r2)°’r2N'1dr)fr S Ch(°’+1)” (5.4)
1-

forallO<h< 1, where %+fir=1.
1 1 _
Then, for 7'- ;r — 1,
(1) T; is bounded on LP*9(go°w(1 — r2)");

(2) T; is bounded on Lpl'q'(cp‘q'w'%(1 — r2)c').
Proof: (1) From Corollary 5.14,
L'r”"(¢>"cv(1 - r2)"') = (Lp’”(sa’w(1 - r0“). L”'°°(<p”))1_g,q-

By Theorem 1.13 , (5.4) implies that T; is bounded on LP'P(cp”w(1 — r2)°’)). Then
it follows from Theorem 5.15 and Corollary 5.14 that T; is bounded on the space

Lp’q(<pqw(1 — r2)°’).

76

I

(2) For f e Lp'.e'(,.-e'w-i(1 - r20), 9 e L”"’(<P"w(1 — r20),

l fuy(2)—T;f(z)dme(2)l
s /U|g(z)llT;f(z)ldme(z)
s [UT;|g(z)I|f(w)ldme(w)

_<_ C||T3|g(2)l“uneven-er)”f“, '

p’.q’ (.p-q'w- ger(1_r2)a)

|/\

CllgllLe~(ew(1—r2)a) ||f|| ,

L» «’(e-«w’ta-rvof
The last inequality holds because of part < l >. Lemma 5.6 and Lemma 5.7 then
implies that T; is bounded on LP”°'(<p"q'w'g<T(1 — 73)“). The proof is complete.

By Lemma 5.5 we have the following corollary:

Corollary 5.17 Under the same assumption of Theorem 5.16 , To, is a bounded
projection of LP’°(<p9w(l — r2)°') onto H P'9(cpqw(1 — r2)°') and a bounded projection of

I

L”'q’(<p"9'w'gc‘(1 — r2)°') onto HP"9'(w'°'w'%(l - r2)°') for p S q S 00, 1 < p < 00.
Remark:

1. The example in remark 2 after the proof of Theorem 5.15 shows that, in
general ( We assume I; w(1—r)°’dr < oo ), in order to make T; well defined

in Lp'q(cp°w(1 — r)°‘), we must have a — b > —1.

2. In order to make T; a bounded operator, it is not necessary that w and
(p satisfy the condition (5.4). In fact, for N = 1, fix q and p with q > p,
choosec> 0 and a> —1 such that a—c> —1 and a—c(q—p) < —1.

Let cp(r) = (1 — r)° and w(r) = (1 — r)'°". Then

1
/ wcpp(1- r2)°’dr = 00,
l—h

so that cp and to do not satisfy the condition B§(gopw).

77

However, since Lp’q(cpqw(1 — r2)"‘) = LM((1 — r2)"'), choose 7 > 0 very
small and let 95(1‘) = (1 — r)", (E(r) = (1 — r)‘°”, then it is not hard to see
that the condition 8505122)) is satisfied. Theorem 5.16 then implies that
T; is bounded on LP'°(<,3"<IJ(1 — r2)°') = LW((1 — 73)").

3. Suppose To, is bounded on Lp'q(<pqw(1—r2)°‘). Following the method Békollé
used in [3, p.311], if we put f(2) = w-gi(r)go‘°'(r)XK(a)(z), it can be shown

that (p and w satisfy the condition Bg(cp°w).

We next give an application of Theorem 5.16.

In [13] M. Jevtié showed the following:
Theorem 5.18 Forl S p S 00, l S q S 00, the transformation P defined by
Pf(w) = [U f(z)K,\_1(z,w)z/)(r)(1 — r2)"3’dm(z),
where w E U, is bounded from Lp’°(r1‘2N) onto Hp'q(cpqr1"2N(1 — r)‘1).

We now show that for 1 < p < oo, 1 < q < 00, this Theorem is a special case of
Theorem 5.16 .

Let w(r) = (1 — r2)'*r1'2N. For any f E L?’q(r1‘2N), define
PM = f(z)¢(r)(1 - r2)“
Then F E LP'°(<p°w(1 — r2)"“) and
llFlle.q(eqw(1—r2)A—1) = llflle.q(r1—2N)
Thus Pf(w) = T,\_1F(w). Since
H”’°(r°rl‘2N(1 - 7‘2)“) = H”"’(so"w(1 - 73)“).

Theorem 5.18 is now equivalent to the statement :

78

Th1 is a bounded operator mapping from the space LP'q(<pqw(l — r2)"‘1) onto the
space Hp'q(cpqw(l — r2)"'1).

It suffices to prove the boundedness of the operator T,\_1, since then it will follow
from Lemma 5.4 and Lemma 5.5 that T34 is an onto map.

By Theorem 5.16 , for q 2 p, it suffices to verify (5.4). We have

I I I

‘P-p'w'i? = $0_pe(1_r2)igr(2N—1)%
= cp‘P'(1 — r2)*(p'-1),.(2N—1)(p’-1)

= wp'rON-INH-llu _ r2)”.
Thus (5.4) is equivalent to

l
/ ep(r)(1-r2)-ldr
l-h
1 I I
x ( 1 his)» r<2N-1>P(1 —r2)-ldr)f' g cw. (5.5)

Condition (5.5) will be verified by the following

Lemma 5.19 For any normal pair {90,1/2} and t a non-negative real number,
1 1 I g, A
h Lp”(r)(l —— r)"1dr(/ h we)? (1 — r)t"1dr)9 g 0M +29 (5.6)
1— 1-
for all 0 < h < 1, as long as each factor makes sense.

Proof: Let 0 < h < 1. Since "’ r is non-increasing, where a > 0 is as in (1.1),

l-r ‘3

f1; cp”(r)(1 — r)t‘1dr

1 901,0.) t—l ap
C‘A—hW(I—T) + d?”

‘Pp(1-h) 1 _ t-l+aP
C h“? 1-_h(1 7‘) dr

|/\

S C(Pp(1 _ h) hap+t
h“?

= Ce"(1 — h)h‘.

79

Similarly
([1; ¢P'(1— r)‘-1dr)f' _<_ C¢P(1— h)h“’f'.
Since gpp(1 — h)z/fl’(l — h) = hip, (5.6) follows. The proof of the lemma is complete.
Since (5.6) implies (5.5), for p S q, Theorem 5.18 follows from Lemma 5.19 and
Theorem 5.16 . For q < p, we have qI > p'. Since in (5.6), the position of p, p',
(,0, :12 are symmetric, (2) of Theorem 5.16 implies that T;_, is bounded on the space

Lp'q(w‘9w—3$(1 — r2)*‘1). Since

<p(7‘)¢(r) = (1 - 7")",

and
w(r) = (1 — rm.)
we have
¢_qw-fr(1 — 7‘2)’\'1 = (pqwr(2N-l)§+l(1 — r2)A_1.
Therefore

Line-W (1 — 13>“) = Liq<<pwr”"“’e"“(1 — 13>“).
On the other hand, since
mew - ea“) c Lp’q(<p°wr(2N-l)i'+l(1 — H2“)
and TA_1f is analytic for f E L”"’(<p°w(1 — r2)*‘1), we have

llTA—lfllLPquwu—ri’y-l)

= [ITA-lfllHMMl-vU-T’V'll

-<- CllTX-lf” (2N—1).%.+1 ( by Lemma 5.8 )
Hp.q(¢qwr q (1_,.2).\—1)

< C _

_ IlfllLM(¢qw1-(2N 1)§+1(1_r2)x_1)

-<— CllfllLP.q(cpcw(1-r2)A-l)-

80

Thus Th1 is bounded on LP"1(<p"w(1 - r2)"‘1). Therefore Theorem 5.18 is also true

for q < p. The proof is complete.
§5.4 Duality Theorems

The following theorem is the main result of this section.

Theorem 5.20 Suppose p S q S 00, 1 < p < 00, and { cp, w } is a normal pair,
a + A — b > —1, where b is as defined after {1.1) and A is as in (1.2). Suppose

w(r) Z 0 satisfies

[1.14, w(1")gop(r)(1 _ r2)ar2N_1dr

x (A; w'é(r)ip'p'(r)(1 — r2)°’r2N‘ldr)fr g Ch(a+A+l)p (5.7)
for all 0 < h < 1, where 5+ fir = 1. Then the dual of Hp'q(cpqw(1 — r2)°') can be
identified with HP""'(z/fl'w'ii‘(1 — 73)“) under the pairing

< fig >= [U f(2)§(z)dme+x(z)- (5.8)
More precisely, ifg E HP”9’(w9'w'gi‘(1 — r2)°’) and if we define

L9(f) =< fag>

for all functions f E Hp'9(<p°w(1 -— r2)°’), then Lg G [Hp'q(<pqw(1 — r2)°’)]"' and
”L,” S CH9” , , , _g; -
Hm (in w 9(1—r2)°)

Conversely, given a linear functional L E [H 1M(<,o"(.o(1 — r2)°’)]“, there is a unique

I

g e. Hp’vq'w’w-Hl — eta)
such that L9 = L and
”all , S CHLII-

HP'.9'(¢Q'w‘g<T(1—r2)a)

81

Proof: Take (32(r) = w(r)(1 — r2)”. Then

I I I
I

go" iii—i? = cp'p’w'rie'fl — r2)“;T = ¢p'w—%;
L”"’(<pqw(1 — r2)°') = Lp’°(cpq&2(1 — r2)°’+A) (equal norm);
L”""'(¢°'w-gv'(1 — r2)°') = Lpl’q’(<p"q'di'gi(l — r2)°‘+’\) (equal norm).

Thus a: satisfies (5.7) implies that (I) satisfies (5.4). Theorem 5.16 then implies that

I

T‘H is a bounded map on both Lp'q(cp‘1(b(l —r2)°'+") and LPI'q'(<p‘q’iIJ—gi(l —r2)°’+t).

0:

Now, it suffices to prove that the dual of Lp'q(<pq52(l — r2)"+’\) can be identified
with LPl'9’(cp‘9’LD'gi’(l — r2)°‘+") under the pairing (5.8) for p S q.

I

Let g E LP"°’(go‘q'LD'gc—(l — r2)°'+*). It follows from Halder’s inequality that

l < fig > i < CllfllLPv‘l(¢9w(l- r2)°+’\)llglle, '

”"09"" w ‘1 (l-r’)°+")

So g defines a bounded linear functional L on H 1M(<,o"c.72(1 — r2)°‘+*) and

“L“ S 0Hall , _,_
LP 1 (¢-q Q q (1—r2)°+*).

Conversely, let L be a bounded linear functional on H P'9(cp"ib(1 — r2)°+*). Then

L can be extended to be a linear functional on LP'9(cp"cIi(l — r2)°"”). By Lemma 5.6,

I

there exists a h E L’l'9’(<p’° (D'%(1 — r2)°'+*) such that

L(f) =< M >
and

“LII ~ llhllL, ,

”(e-e w ‘m— «ac-+1)

Let

g = Ta+,\h.
Theorem 5.16 implies that g E H” “'(cp‘ 9&2 557(1 — r2)“+"). Now for f 6 H°°(U), by
Lemma 5.4 and Fubini’s theorem, we have
L(f) =< f, h >=< Ta+Afa h >=< f,Ta+x(h) >=< M >.

82

By Lemma 5.5 , H°° is dense in Hp'q(goqdi(1 — r2)°'+’\). Then the continuity of L

implies L(f) =< f,g > for all f E Hp’q(cpq£b(l — r2)°'+’\). We also have

LP .q'(¢p-q'Q—%’(1—r2)a+k) Lp'.q'(¢-s'o' q (1—r2)a+A)

llgll , , < Cllhll L. S CHLII-

I

If g E Lpl'q’ («p-9’42? %(1 — r2)°'+’\) defines a zero functional, then since the function

K0+A(z, -) E H°°(U), for any fixed 2 E U, we have, for some C > 0,

0 =< Ka+x(z, -).9 >= 09(2).

Hence g E 0. So there is a one-to—one, continuous, linear transformation from
Li’""'(cp‘q'&)’ i (1 — r2)°‘+’\) onto the dual space of Lp'q(cp‘1(b(1 — r2)°’+’\). The proof is
complete.

We next give some applications of Theorem 5.20.

In [16] D. Luecking used Theorem 1.13 to identify the dual of weighted Bergman

spaces. He proved the following:

Theorem 5.21 Suppose w(z) satisfies

_e’ ~ # ,,
Aweumrex/Kw .(.)dm.,(z)). 30mm) (5.9)

forl < p < oo, wheren > —1, 7 > -1, a = 5+5; andK is the region in the condition

I

A200). Then the dual of HP’P(w(z)dm,,) can be identified with HP""'(w(z)—i?dm,)

under the pairing {5.3).
In [13] M. Jevtié showed the following:

Theorem 5.22 Let 1 S p S 00, 1 S q < 00. Then the dual space of the space
HP'9(<pqr1"2N(1 - r2)'1) can be identified with HP”9'(w9'rl‘2N(1 — r2)‘1) under the
pairing

< f,g>= /Uf(z)g(z)dm,\_1.

83

In Theorem 5.20, taking cp = (1 — r2)‘, 1,!) = (1 — r2)j, a = —1 in Theorem 5.20,

where i, j > 0, we have i +j = A. Then (5.7) becomes

1 .
/ h w(r)(1— r2)""lr2N"ldr

1-
l ' . I
x ( hw'PF(r)(1 — r2)” "ler"1dr)fr S ChA”. (5.10)
1-
It is not hard to see that
H”’°(so"w(1 — r71) = Hqu — err-1)

and
Hp'eq'w'w-“iu — r71) = Hp'i’w-Hl — sit-1).

By Theorem 5.20, for q 2 p, if to satisfies the condition (5.10), then the dual space of

I

H M (w(1 — r2)“1‘1) can be identified with H 1"”, (w’gv'(1 -— r2)59"1) under the pairing
(5.8). Leti= 1?,j = 31:51. Thenn =iq—1,7=jq—-1 and A = i+j= t+1. These

observations give the following:

Theorem 5.23 Forl < p S q < oo, 17 > —1, 7 > —1, and t > —1 satisfy

ifw(r) satisfies
1 2
/ w(r)(1 — r2)("+1)q-1r2N’ldr
l-h

1 ’ '_
x (/ ,w-%(r)(1—r2)“’+"? 1r2N_1dr)fr _<_ Ch(‘+1)P, (5.11)
1-

then under the pairing
< f,g >= [U fadmi (5.12)

the dual of HM(w(1 — r2)") can be identified with HP"‘1'(w—7¢gr(1 — r2)'7).

84

Proof: (5.11) is now equivalent to (5.10). From the above discussion, the proof is
complete.

It is not hart to see that, for a radial weight function w(r), Theorem 5.20 gives
a generalization of Theorem 5.21 in mixed-norm spaces. In fact, in Theorem 5.23,
taking q = p, we immediately get Theorem 5.21.

Next we show that Theorem 5.22 is a special case of Theorem 5.20 if 1 < p < oo,
1 < q < 00.

In fact, by Lemma 5.8 , it suffices to show that the dual of Hp"’(<pq(1 — r2)‘1) can

be identified with H P"9'(¢9’(1 — r2)‘1) under the pairing

< f, 9 >= [U f(z)g(z)dm,\_1.

Taking w = 1, a = —1 in Theorem 5.20, (5.7) becomes
1 N 1 ' r,
l ,. 90“er — We? ~‘dr(/ ,ie (1 _ mam-1a) s can».
1" 1-

By Lemma 5.19 and the discussion after Lemma 5.19, any normal pair satisfies this
inequality. Therefore for q 2 p, Theorem 5.22 follows from Theorem 5.20 immediately.

For q < p, then q' > p'. Using the duality argument, it cam be shown that the dual
space of HP"9'(¢9’(1—r2)‘1) is HP'°(<p‘1(l—r2)’1). This implies that HP"1(<p"(1—r2)‘1)
is a closed subspace of Lp'q(cp9(1—r2)‘l). Since LP'q(cpq(1—r2)"1) is reflexive [4, p.306],
it follows that the dual of HP'9(go‘1(1 — r2)‘1) is Hpi'9’(i,b°’(1 — r2)‘1). We also get

Theorem 5.22.

85

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