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Toeplitz Operators on Harmonic Bergman Spaces
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TOEPLITZ OPERATORS ON HARMONIC BERGMAN
SPACES

By

Jie Mz'ao

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1997

ABSTRACT

TOEPLITZ OPERATORS ON HARMONIC BERGMAN SPACES

By

Jz'e Miao

In this dissertation, we study Toeplitz operators on harmonic Bergman spaces
of the unit ball in R" for n _>_ 2. We give characterizations for Toeplitz operators
with positive symbols to be bounded, compact, and in Schatten classes. We ob-
tain compactness criteria for Toeplitz operators with continuous symbols and with
bounded radial symbols. Our results are analogous to well known results on ana-
lytic Bergman spaces. However in R" for n > 2, some methods that are effective
in dealing with analytic Bergman spaces, such as using Mobius transformations,
are not available. The reproducing kernels for harmonic Bergman spaces are also
more complicated than those for analytic Bergman spaces. Our study focuses on
reproducing kernels for harmonic Bergman spaces. We also give some applications

of these reproducing kernels.

To my parents and wife

iii

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to Professor Sheldon Axler, my

advisor, for his constant encouragement, help, and excellent guidance.

iv

TABLE OF CONTENTS

1 Preliminaries 3
1.1 Harmonic Bergman Spaces ....................... 3
1.2 The Reproducing Kernel ........................ 8
1.3 Toeplitz and Hankel Operators .................... 11
2 Toeplitz Operators 13
2.1 Introduction ............................... 13
2.2 Teoplitz Operators with Positive Symbols .............. 13
2.3 Toeplitz Operators with Continuous Symbols ............. 26
2.4 Toeplitz Operators with Bounded Radial Symbols .......... 31
3 Weighted Harmonic Bergman Spaces 37
3.1 Introduction ............................... 37
3.2 Reproducing Kernels on the Unit Disk ................ 38
3.3 Some Properties of the Reproducing Kernels ............. 41
3.4 Application to an Inequality for Harmonic Functions ........ 45
3.5 Application to Toeplitz operators on Harmonic Bergman Spaces . . 49
BIBLIOGRAPHY 52
Bibliography 52

iv

Introduction

Toeplitz operators on analytic Bergman spaces have been well studied. Mc-
Donald and Sundberg [11], Luecking [9], Zhu [23], Korenblum and Zhu [7], Axler
and Zheng [3] considered Toeplitz operators on analytic Bergman spaces and ob-
tained criteria for Toeplitz operators to be bounded, compact, or in Schatten classes
for different type of symbols such as positive, continuous, bounded, or bounded
radial symbols.

We study Toeplitz (as well as Hankel operators) on harmonic Bergman spaces
of the unit ball in R" for n 2 2. Compared to those on analytic Bergman spaces,
Toeplitz and Hankel operators on harmonic Bergman spaces have not been as well
studied and understood. Recently, Hankel operators on harmonic Bergman spaces
of the unit ball in R" for n 2 2 were studied by J ovovic’ [6], and Toeplitz and Hankel
operators on harmonic Bergman spaces of the unit disk were studied by Wu [22].
We obtain results for Toeplitz and Hankel operators on harmonic Bergman spaces
analogous to those for analytic Bergman spaces. Our results improve and extend
the results in [6], [7], [9], and [22]. This dissertation is organized as follows.

In the first chapter we introduce the definitions for harmonic Bergman spaces,

the reproducing kernel for harmonic Bergman spaces, and Toeplitz and Hankel

operators on harmonic Bergman spaces. We also introduce some results that we
will need for harmonic Bergman spaces such as duality results.

The second chapter is devoted to Toeplitz operators. We give characteriza-
tions for Toeplitz operators with positive symbols to be bounded, compact, and in
Schatten classes. We obtain compactness criteria for the Toeplitz operators with
continuous and bounded radial symbols.

In the third chapter we study reproducing kernels for weighted harmonic
Bergman spaces. We obtain new properties for these reproducing kernels and
give some applications of these properties. As one application, we extend the re-
sults for Toeplitz Operators with positive symbols on harmonic Bergman spaces to
weighted harmonic Bergman spaces.

Throughout this dissertation, all constants that depend only on n or other
parameters and do not depend on functions and variables will be denoted by a
single letter “C”. The symbol “El” will denote the end of a proof and “z” will
indicate that the quotient of two positive quantities is bounded above and below

by constants.

CHAPTER 1

Preliminaries

1.1 Harmonic Bergman Spaces

Let B denote the open unit ball in R” for n 2 2. Let V be Lebesgue volume
measure on R" and LP(B) = LP(B,dV) for 1 g p S 00. For 1 S p < 00, the

harmonic Bergman space bP(B) is the set of all complex-valued harmonic functions

1/17
Hump: (/ lulpdV) < 00.
B

As is well known, b”(B) is a closed subspace of LP(B). When p = 2, there is an

11. on B such that

orthogonal projection Q from the Hilbert space L2(B) onto b2(B).
For each a: E B, the map 11. 0—) u(:c) is a bounded linear functional on b2(B).

Thus there exists a unique function R(:I:, ) E b2(B) such that

no) = Lu(y)R(x,y)dV(y)

for every u 6 b2(B). The function R on B x B is called the reproducing kernel of
b2(B). For f E L2(B,dV) and at E B we have

mm = [B f(y)R(cv, y) My).

In this section, we provide some basic results for harmonic Bergman spaces.
These results are analogous to well known results for analytic Bergman spaces (see
[4]) and they can be proved in a similar manner. We refer to a recent paper [21]

by Stroethoff for Theorems 1.1-1.5 .

Theorem 1.1 Let 1 < p < 00. Then Q is a bounded projection of LP(B) onto
bp(B).

The following duality result for b”(B) for 1 < p < oo follows easily from

Theorem 1.1. For 1 < p < 00, we use p’ to denote the conjugate of p, i.e.,

l+_];—1
p p’ '

Theorem 1.2 Let 1 < p < 00. Then the dual of b”(B) can be identified with

I

b” (B). More precisely, every bounded linear functional on b”(B) is of the form
f H / fgdv
B

for some unique 9 E bp'(B). Furthermore, the norm of the linear functional on

bp(B) induced by g E bp'(B) is equivalent to [[ngI.

A harmonic function u on B is said to be a Bloch function if

HUIIB = sup{(1- |$|2)|V u(Hill = :v E B} < 00-

The harmonic Bloch space B is the set of all harmonic Bloch functions on B. If
u is a constant function, then [lull 3 = 0, so the Bloch norm || [[3 is not actually a
norm on the harmonic Bloch space. However, [[uHB + |u(0)| does define a norm on
B. Whenever we refer to properties that require a norm for B, it will be this norm

that we have in mind.
Theorem 1.3 Q maps L°°(B) boundedly onto the harmonic Bloch space 8.
The following duality result for b1(B) follows easily from Theorem 1.3.

Theorem 1.4 The dual of b1(B) can be identified with the harmonic Bloch

space B. More precisely, every bounded linear functional on bl(B) is of the form
f »—> / fg dV
8

for some unique 9 E 8. Furthermore, the norm of the linear functional on b1(B)

induced by g E B is equivalent to [Igllg + [9(0)].

The harmonic little Bloch space 80 is the set of functions u harmonic on B
such that

(1 - lxl2)l VU($)| -+ 0

as [2:] -—> 1. It is easy to see that 80 is a closed subspace of B and that all harmonic
polynomials belong to 80. The following result shows that the harmonic little

Bloch space is the pre-dual of the harmonic Bergman space b1(B).

Theorem 1.5 The dual of the harmonic little Bloch space can be identified

with b1(B). More precisely, every bounded linear functional on [30 is of the form
f H / fg av
B

for some unique 9 E bl(B), and the norm of the linear functional on Bo induced

by g E b1(B) is equivalent to ||g||1.
Now we give a few more results that we will need.

Theorem 1.6 Q maps C(B) boundedly onto the harmonic little Bloch space
80.

Proof. First we show that Q maps C (B) boundedly into 80. By the Stone-
Weierstrass Theorem, C(B) = L°°-closure { polynomials on R"}. Thus we only
need to show Q(p) 6 80 for any polynomial p, since 80 is closed in B. By Theorem
8.14 of [2], Q(p) is a polynomial of degree no more than that of p. Hence Q(p) 6 Bo.
To show that Q maps C(B) onto 80, we can use the same argument as for the

proof of Theorem 2.11 in [4]. The details are omitted here. D

Let 1 S p < 00 and let u be a positive Borel measure on B. The Closed Graph
Theorem shows that bP(B) is contained in LP(B, du) if and only if the inclusion
map from b” (B) to L"(B, du) is a bounded linear operator. Furthermore we can
ask when the inclusion map from b" (B) to LP(B, du) is a compact linear operator.
The following theorem gives a necessary and sufficient condition on u for this to
happen. First we introduce a covering lemma.

Fix r 6 (0,1). For a: E B, let K,(:r) = {y E B : [y — ml < r(1— |:1:|)}.

The following covering lemma says that we can cover B with K,(.7:)’s that do not

intersect too often. The proof of the following lemma is essentially the same as

that for Lemma on Coverings of [13].
Lemma 1.7 There exists a sequence {mi} in B such that
(1) 21K§($i): Bi

(2) There is a positive integer N such that each K,(:r,-) intersects at most N

spheres of {K,.(a:J-)}.

The number N depends on r for this lemma. We omit the details of the proof
here.
We always assume that {23,-} is a sequence given by Lemma 1.7 in this disser-

tation. If {33,-} is such a sequence, then it is clear that [23,-] —> 1 as i —> 00.

Theorem 1.8 Let 0 < r < 1. Let 1 g p < 00 and u be a positive Borel

measure on B.

(i) The inclusion map from b”(B) to LP(B,du) is bounded if and only if

_(__uKrCL‘ .)) -_
Vi(Kr($ ——)—-) is bounded for z — 1,2, ,
)

(ii The inclusion map from b”(B) to LP(B,du) is compact if and only if

#( Kr($i))
V(K,.(:z:,-))

The same argument as in [23] (see pages 338, 342, and 343) can be used to

—>0asi—>oo.

prove Theorem 1.8, so we will not give the details. Note that the subharmonicity
of |u|p for a harmonic function u on B and the decomposition from Lemma 1.7 are

needed for the proof.

1.2 The Reproducing Kernel

In this section we give an introduction of the reproducing kernel for b2(B). We
need to introduce zonal harmonics first.

Let ’Hm(R") denote the space of all homogeneous harmonic polynomials on
R" of degree m. A spherical harmonic of degree m is the restriction to S, the
unit sphere, of an element of Hm (Rn). The collection of all spherical harmonics of
degree m is denoted by ’Hm(S). For every 17 E S, there exists a unique Zm(n, ) E
’Hm(S) such that

pa) = [S p(<)zm(n,o dam

for all p E ’Hm(S), where a is the normalized surface-area measure on S. The
spherical harmonic Zm(77, ) is called the zonal harmonic of degree m. One can
extend the zonal harmonic to a function on R" X R" by making Zm homogeneous
of degree 777. in the second variable as well as in the first. Let hm denote the
dimension (over C) of the vector space ’Hm(S). One can compute hm explicitly

(see Exercise 5.5 of [1]):
(1.1) hm = + ,

for m > 0. Also, ho = 1.
The following lemma states some properties of zonal harmonics that we will

need. For more information on zonal harmonics, see Chapter 5 of [2].

Lemma 1.9 Let m be a non—negative integer.

(i) If (.17 E S, then Zm(C.C) = Zm(n.n) = hm;
(ii) IfC e S. then maxnes lZm(C, n>| = Zm<c C) = I
We have the following representation for R (Theorem 8.9 of [2]).

Theorem 1.10 If 2:, y E B, then

1 (X)

R(a:, y) = Z (n + 2m)Zm(a:, y).

 

The series converges absolutely and uniformly on K x B for every compact K C B.

Since Zm is real valued for each m (see Theorem 5.24 of [2]), we see that R is
real valued.
For 2:,y E B, let P(2:, y) be the “extended Poisson kernel” for B. Then ( see
pages 156 and 157 of [2])
°° 1 - livlglt/l2

(1-2) P(:r.y) = Z Zm($,y) = (1_ 235.1” l$|2|y|2)i’

m=0

 

:r,y€B.

From the equation above and Theorem 1.10, we have the following beautiful equa-
tion

(um, y) + 31m, tats).

ROE, y) = dt

 

nV(B)

This simple representation gives us a formula in closed form for R(.r, y) (The-

orem 8.13 of [2]).

Theorem 1.11 Let 1:,y E B. Then

(n — 4)|.r|4|yl4 + (8x - y — 2n — 4)]:rl2IyI2 + n
nV(B)(1 - 2:13-21 + |117|2l1u|2)‘+"/2

 

1303.11) =

10

It follows easily from Theorem 1.11 that

(13) Rm :1) = 30/, It), 13(17. 7‘31) = RW, y)

for 2:, y E B,r 6 (0,1). It is also clear that R(2:, y) is bounded for [2:] g r < 1, [y] <
1 for fixed r < 1.

A simple computation gives

"(1‘ ]-’E]2]Z/|2)2 — 4]$]2]y]2((1 — |:r]2)(1 — [31]?) + [3; _ 31]?)
nV(B)((1-—]:1:|2)(1—[y]2)+]x — 31]?)1+n/2

 

(1-4) R($, y) =

It follows easily from Theorem 1.11 that

4

1.5 R < .
( ) I (x,y)l_ (Flt-31+|~'L‘|2|y|2)”/2

 

Note that the reproducing kernel is much more complicated than that for analytic
Bergman spaces (see Chapter 3 of [16]). One of our tasks is to establish properties
for R analogous to those for the analytic Bergman kernel.

For f E L1(B) and 2: E B, we now see that Qf(2:) = [Bf(y)R(:r,y)dV(y) is

well defined. We end this section with the following lemma (see Lemma 2.1 of [6]).

Lemma 1.12 For all u E b1(B) we have

um = /B mam/we).

11
1.3 Toeplitz and Hankel Operators

Let 1 S p < 00. For a function f E L1(B,dV), the Toeplitz and the Hankel

operators with symbol f are densely defined on b” (B) by

T101) = Q(fU), H101): (1 - Q)(fU)

for u E b” (B) (1 L°°(B) (noting that harmonic polynomials are dense in b” (B) and
Q(g) is well defined for g e L1(B)). If T, is a bounded operator (when we put the
LP norm on b” (B) H L°°(B)), then T!” extends to a bounded operator from b”(B)
to bID (B), also denoted by T]. We do the same for H f.

In this section we give some preliminary results for Toeplitz and Hankel oper-
ators.

For f E L°°(B), it is easy to see that T, and H I are bounded linear operators
on bp(B) with

llell S ||f||oo. llell S Ilflloo.

In contrast to Hardy space Toeplitz operators, Toeplitz operators with unbounded
symbols can be bounded on Bergman spaces. For 1 < p < 00, f E LP(B), g E
LP'(B), define

(£9) = [B fgdv.

Since bp'(B) is the dual of bp(B) with respect to the pairing (-, -), define the adjoint

12

of T]: bP(B) H bP(B) to be the operator T; : bP'(B) t—) bP'(B) such that

(T;u, v) = (u, va)

for u E bp'(B) and v 6 b"(B).
Similarly, define the adjoint of H I : bP(B) I—) LP(B) to be the operator H }
mapping LP'(B) to bp'(B) such that

(H;u, v) = (u, va)

for u E LP'(B) and v E b"(B).

The following lemmas are standard (see Lemma 3.1 and 3.2 of [6])
Lemma 1.13 Let p 6 (1,00), a,b scalars, and f,g E L1(B). Then
(i) TOM, = an + ng on bP(B).

(ii) T; = T,— on bP'(B).

The connection between Toeplitz and Hankel operators is provided by the fol-

lowing lemma.

Lemma 1.14 Let p 6 (1,00) and either f or g E L°°(B). Then ng — Tng =
H}'.Hg on bP(B).

We end this section with the following result that was proved in [6].

Theorem 1.15 Let p 6 (1,00) and let f be a continuous function on B. Then

the Hankel operator Hf: bP(B) —> LP(B) is compact.

CHAPTER 2

Toeplitz Operators

2.1 Introduction

In this chapter we study Toeplitz operators on harmonic Bergman spaces b"(B) for
1 < p < 00. We will look at three special classes of symbols. For positive symbols,
we give characterizations for Toeplitz operators to be bounded, compact, and in
Schatten classes. For continuous symbols, a compactness criteria is obtained. In
fact, the essential spectrum of a Toeplitz operator with a continuous symbol is
found. We also obtain a compactness criteria for Toeplitz operators with bounded
radial symbols. A sufficient condition for Hankel operators to be compact, which

improves Theorem 1.15, is given along the way.

2.2 Teoplitz Operators with Positive Symbols

First we give three lemmas for the reproducing kernel R.

13

14

Lemma 2.1 Let :r E B. Then R($,y) z n for y E Kr(27).

_1__
(1 - IIL‘I)
Proof. We use the following formula (1.4) for R(2:, y):

R(:z: y) = n(1— lefty?)2 - 4]1’3]2|’t/]2((1—I.’r]2)(1—lylr")+].’16 — m2) = Il($,y)
’ nV(B>(<1— |x|2)(1— Iy|2>+lx — male/2 12(22, yr

 

 

If y E Kr(2:), then (1 - r)(1— ]2:[) < 1— [y] < (1+ r)(1— [27]). It is clear that
12(2:,y) z (1 — ]a:])2+" when y E Kr(2:). It is also clear that 11(22, y) _<_ C'(1—]2:])2

if y E K,(2:). Next we will try to find a lower bound for 11(23, y). Since n 2 2,

11(17, y) 2 2(1+|117||y|)2(1- livllyl)2 - 4|$|2|y|2(1- l$|2)(1- lylg) - 4|$|2ly|2l$ - ylg-

Let

1303,31) = 2(1+|$||y|)2(1- latllyl)2 = 2(1+|:17|lyl)2(1-|y|+lyl(1- Infill)?

So we have

13(17, y) = 2(1+l$||y|)2(1- lyl)2 + 4|y|(1+|-'Irl|y|)2(1- [900- III)

+2|y|2(1+|17||y|)2(1-lxl)?

Since

2(1+|~’L‘llyl)2(1-lyl)2 > 2(1- 7‘)2(1-|-77l)2

15

and

4lyl(1+ll‘|ly|)2(1-[31])(1-lxl) > 4|$|2ly|2(1- [3:[2)(1_ lyl2),

we will have 11 (2:, y) > 2(1 — r)2(1 — [27])2 if we can show the following inequality:
2l:t/|2(1+l13||y|)2(1 - ISIII)2 Z 4|$|2|yl2|$ — :I/l2
for y E K, (2:) This can be reduced to showing that
(1+ [itllyll2 2 2ll]2

for y E K,(2:). If [2:] < 0.7, then [2:]2 < 0.49 and the above inequality is trivial. If

[2:] 2 0.7, then [y — 2:] < 1— [2:] g 0.3. So [y] > [2:] — 0.3. So we have
(1+lirllyl)2 Z 4|$lly| > 4ll‘l(|$| - 0-3) > 2le2,

for y E Kr(2:). This finishes the proof of Lemma 2.1. C]

Now we estimate the LP-norm of the reproducing kernel.

Lemma 2.2 If1< p < 00, then [[R(2:, .)||,, z (1 — [2:[)_1('%u

Proof. [[R(2:,-)[[f, S C(l — [2])-"(VD follows from Lemma 3.2 (c) in [5]. On

the other hand, by Lemma 2.1 we have

lime-MI; 2 [Kr(z)|R(I,y)lpdV(1/)

1

C(T—W lea W)

IV

16

(1- IIEI)"
(1 - IIEI)"”'

This proves the lemma. El

Lemma 2.2 was known when p = 2 (see [2], Exercise 8.15).

R($ )
||R(IL‘ )llp
Proof. Let v E b”,(B). By Lemma 2.2

Lemma 2.3 [fl < p < 00, then —> 0 weakly in bp(B) as [2:] —-> 1.

NEE-)—
”RC” )[lp

Using Exercise 8.2 of [2], we see that the quantity above has limit 0 as [2:] —> 1. El

v)|~ ~ (1 - lid)? |v(:v)|-

Now we extend the notation of Toeplitz operators to the case where we allow
measures as symbols. Let u be a finite complex Borel measure on B. We densely

define the Toeplitz operator with symbol u on b”(B) by

amn=LRanmmww

for u E b"(B) fl L°°(B,dV). If du(y) = f(y) dV(y), then Tu = T,.

Let u be a finite positive Borel measure. For our purpose we will consider the
following two cases:

(1) Suppose u, v are both harmonic on B and continuous on B. If Tuu E b1 (B),

then we have

(T,,u,v) = lim BTpuvdV

=fim/B/fl McW()uwmm

r—>1

17

= lim Br"?7(rzy)U(z/)du(y)

T—tl

= f m? do,
8

where we used Fubini’s Theorem in the second step (R(2:, y) is bounded for [2:] S

r < 1) and the following properties of R (see (1.3)):

ROE. y) = R(y,-’r), R(y, r2) = Rey, Z)

for 2:, y, z E B.
(2) Suppose u satisfies (1) of Theorem 1.8 and u E b”(B) for 1 < p < 00.
Then Theorem 1.8 shows that T “u is well defined. By the proof of Lemma 3.2

of [5], one can show that |[R(2:, )[[1 < Cln—-— + C for any 2: E B. Thus

1
—17| I
|[R(2:, )[[1 S —q— for each oz > 0. So we have

(1 - IIEI)“

”11.th s [B/BIR($,y)IIU(y)|du(y)dV(:v)
= f/1R<zyndwehuwmh)
S 0/8( My ———):;du(y)

for some a > 0 to be decided later. The same argument mentioned after Theorem

1.8 and Holder’s inequality give

“7",.th s c/ lu(y dV(y)

“(ll-[ylla

C[[u[[p [[3 W dV(y)

1
p

l/\

18

if a is small enough.
Hence if u satisfies (1) of Theorem 1.7, u E bp(B) for 1 < p < 00, and v is a

bounded harmonic function on B, then Fubini’s Theorem gives

(Tuu,v) 2/ ut') du.
B

Now we can characterize the boundedness and compactness of positive Toeplitz

operators.

Theorem 2.4. Let 1 < p < 00 and u be a finite positive Borel measure on B.

Then the following conditions are equivalent:

(i) T,u is bounded on bp(B);
”(Kr($))

ii —— is bounded for 2: E B;
( ) V(K.(:v))

V(Kr (3%))

Proof. (i)=>(ii). By Lemma 2.1 and 2.2, and (I) mentioned just before the

is bounded for i = 1,2, - - ._

theorem we have

R($1) RC”, )

1
m “Re, on; “Re, 11h)

z ||R(x, outline, out I<T2R(m, -)1 Rev ->>|

(1 - |$|)"/B IR($,y)I2du(y)

<1— 1x1)" [W laterals)

u(Kr($))
V(Kr(1‘))

 

 

 

IV 22

22

19

for 2: E B. This shows that (ii) follows from (i).
(ii)=>(iii). This direction is trivial.
(iii)=>(i). Let u E b”(B) and v be a bounded harmonic function on B. Then

by Hblder’s inequality,

 

I<Ttu.v>l = [/ wade
B
1 a
s (/ lurdu)” (/ Ivr’de)”
B B
s Cllullpllvllp'

using (i) of Theorem 1.8 in the last inequality. Since the set of harmonic polyno~
mials is dense in bp'(B), the duality argument shows that T,, is bounded on bP(B).

This completes the proof of the theorem. C]
The following theorem is the little 0 version of Theorem 2.4.

Theorem 2.5 Let 1 < p < 00 and u be a finite Borel measure on B. Then

the following conditions are equivalent:

(1) Tu is compact on b”(B);

H(Kr($))
V(K rIB( )
(111)” ——(————VK :1“ )

Proof. (i):>(ii). It follows from Lemma 2.3 and the proof of (i):>(ii) of the

—>0 as [2:] —+ 1;

(ii)
—>0 asi—>oo.

)
____)_
)

previous theorem.

(ii)=>(iii). This direction is trivial.

20

(iii)=>(i). Let u,- —+ 0 weakly in b”(B) as i ——> 00. For any bounded harmonic

function u on B, we have

1

I(Thut,v)I s C ([8 lard/t)” ana

It follows that

llTauz-llp s C (h Intlpdu)

So [[Tpuillp ——> 0 by (ii) of Theorem 1.8. This shows that T” is compact on bP(B)

and completes the proof of the theorem. [:1

When p = 2, the equivalence of (i) and (ii) for both Theorem 2.4 and 2.5 can
be deduced from Theorem 1 of [14].

In the rest of this section, we will prove a trace ideal criteria for the positive
Toeplitz operators on b2(B). The techniques used here were developed in [9] as
well as in [23], however the approach in [23] will be used for our purpose.

If T is a compact operator on a separable Hilbert space H, then there exist
numbers 30(T) Z 31(T) 2 ---, called the singular numbers of T, and orthonormal

vectors {e,} and {f,-} such that

Ta: = f: s.-(T)($,61>ft

for 2: E H. For 1 S p < 00, the Schatten ideal Sp(H) is defined to be the set of all
compact Operators T for which [[TIISP 2: (2,20 3,-(T)”)i < 00. As is well known,
Sp(H) is a Banach space with the norm [I ' [[3, and is a two-sided ideal in the space

of bounded linear operators on H. If T E 51(H) and {e,-} is an orthonormal basis

21

for H, then

LT(T) = Z<T8i,8i),
i=0
where the series is convergent and independent of {(2,}. If T E Sp(H) and T 2 0,
then [[Tllgp = [tr(T”)]i for 1 S p < 00.

We need the following three lemmas.

Lemma 2.6 IfT is either in Sl(b2(B)) or positive, then
W) = / (TR($. .), R(:r, .), dvm.
B

The proof of Lemma 2.6 is similar to Proposition 3.5 of [1] or Lemma 13 of

[23]. We omit the details here.

Lemma 2.7 If T1 and T2 are compact and 0 S T1 S T2, then 3,-(T1) S s,(T2)
f0ri=0,1,--~.

This is Lemma 14 of [23].

Lemma 2.8 Ifu Z 0, then there exists a constant C depending only on r such

that
C

H(Kr($)) S W

[W u(Kr(y)) dV(y)

for all 2: E B.

Proof. For 2: E B, we have

/K.(a)”(K'(y))dV(3/) fKr(x)dV(t/)AXK.(t/)(Z)d/L(Z)

: /8 (111(2) /1(r(x)X"’r(y)(zldV(?/)-

 

22

If [y — z] < 1+7" —(1 — [2]), then it is easy to see that [31 - z] < r(1—[y]). This

gives that XKr(y)(z z) 2 XKfi-_(z)(y). So we have
K dV / a I z / av ? .
[W m (y )1 (y )> W ( ) Krwrw (a)
If 2 E Kr(2:), then 1— [z] > (1 — r)(1—[2:]). It is a clear geometric fact that

CV(K,(2:)

     

,1, (2)) Z Witt-(33))

for all z E K,(2:). Combining the above two inequalities we prove the lemma. [:1

Now we can characterize the positive Toeplitz operators that lie in the Schatten

p—class.

Theorem 2.9 Let 1 S p < 00 and u be a finite positive Borel measure on B.

Then the following conditions are equivalent:

(i) T e S p(b'~’(B));
u u__(___K (117))
( ’ V(K(a: >1

0° #(Kr($i)) p
(HQZIWI <00.

Proof. (i)=>(ii). Suppose T“ E Sp(b2(B)). We have “Tun; = tr(T,f) since

6 L”(B, (1 — [2:[)“"dV(2:));

i=1

T], Z 0. Lemma 2.6 and Lemma 2.2 together with 6.4 of [1] give

Inns. ——- / aim, me, ->> dV(x)

_ (a: > RU: >
‘ l ”R H2<T"||1£1(:2=,-art,)||2’||R( held”)

 

23

 

x Rll‘l') Re,» P ,5
> CA” " "(l T”IIR(x,->II2’IIRIx,->II2>l W”

By Lemma 2.1 we get

llTull’E. 2 Ct. "<1—IxI>"/BIRIx,y)I2du<y>]”(1—Inn-"dV(x)
C/B (1—III)"/Krm|R(:r,y)|2du(y)] (l—lzvl)‘"dV(:v)
(7/8 W] (1—|a:|)‘"dV(:1:).

IV

IV

 

(ii)=>(iii). Suppose [B [W] (1 — |:z:|)‘" dV(fE) < 00. Then we have

 

 

°° iH/t (Kr (13)) ))]P
d .
g K( )(1_|x|)np+n V(:c)<oo
It follows that
0° 1
Z:1 (1— |:z:,-|)"P+n [KT(Ii)[u(Kr(x))]P all/(:12) < 00'

By Lemma 2.8 and Holder’s inequality we get

illfi‘IKi‘ . lll ill—-—Iifilllp<oo~

i=1

(iii)=>(i). We prove this direction by complex interpolation.

First consider the case where p = 1. We have

llTullsl = tr(Tu)

24

= Lama, .), R(a:, -)> dV(sr)
= [B/B|R(a:,y)|2dlt(y)dV(fI?)
= [B/B |R(:r,y)|2dV(-T)dll-(?/)
[8(1— IyI)”"d/t(y)

:flamquram>

0° ”(Kr($i))
(:CgVMWMY

22

|/\

Now consider the case where 1 < p < 00. We will show ||T,.||"_§p 3

oo Kr i P
C: [m] . For a complex number C with 0 g ReC g 1, we can define a
£21

V(K,.(x.-))
finite Borel measure on B by

and the Toeplitz operator on 122(3) by

mefl=LMaWMMMM-

It is easy to see that both T“ and T“1 are compact and T ,1 l 2 T" 2 0. Thus
P P

complex interpolation and Lemma 2.7 give
1_l .1.
mmanQaSMinn,

where M0 = {HTWII : Rec 2 0} and M1={llT,i<|lsl :ReC =1}.

25

Let Rec: 2 0. Then we have

°° (51%)) ‘1
lucl(K mczlglff— W) loam) n mm»
for k :2 1,2,---. Suppose Kr(:rk) fl Kr(a:,-) is not empty, it is easy to see that
1+ r

 

(1 — lfcil) <
CV(Kr(:z:k)) for k = 1, 2, - ~ -. Holder’s inequality and Theorem 1.8 give

% %
I<T...u,v>I s UBlulzdlucl) (f3 IvI2dIu4I)

Cllul|2llvll2

_ r(1 — lxkl). Thus by Lemma 1. 7 one can show |;i(|(K (2:0) 5

l/\

for all u, v E b2(B). This shows that IITpCII S C for all Re( 2 0. So Mo 3 C.
Let ReC = 1 and {21,-},{15} be two orthonormal bases for b2(B). It can be

shown in the exactly same way as in [23] (see page 351) that

00 ”(r-K xi P
E: [(Tpcuiwi) |< 0:1[W—KL(—)j-))-] ,

i=1

which implies that

M1 <CZ[—((I{(il)]p.

Hence we have
00

llTplls. s c{: _—

i=1

 

This finishes the proof of the theorem. [:1

26
2.3 Toeplitz Operators with Continuous Sym-

bols

In this section we will find the essential spectra of Toeplitz operators with contin-
uous symbols.

The first theorem of this section gives a sufficient condition for Hankel operators
to be bounded or compact on I)” (B) for 1 < p < 00. The techniques used to prove
this theorem were established in [10] and when n = 2 this condition was proved in
[22]. Since the same method in [22] can be applied to the case n 2 2, we will only

outline the proof.

Let 1 < p < 00 and (bp'(B))L = {u E LP(B,dV) : (21,22) 2 0 V2) 6 bp'(B)}.
Lemma 2.10 (l)l"'(B))i = Lp-closure {Ah : h E C§°(B)}.

Proof. Let u E LP'(B, dV). Then u E b”,(B) if and only if
(Au, h) = (u, Ah) 2 O

for all h 6 C§°(B). This implies the conclusion of the lemma. D

By Lemma 2.10 and the boundedness of Q in LP norm, we have LP (B, dV) 2
b”(B) EB (I)"”(B))i = b”(B) EB Lp-closure {Ah : h E C§°(B)}.

Lemma 2.11. Let 1 < p < 00. Then

|h(rc)|” x IvhmIP ,
[Bu—Issued” )SC 3(1—Ixnvdvl‘)

27

and

lvh(:r)l” p
Ema/(3:) 5 CL | AIM)! dV(sv)

for all h E C8°(B).

One can use the same argument as for Lemma 3 of [10] or Lemma 5.2 of [22]

to prove the above two inequalities. Note that we need

Iv how 3 mung“, 1} 2": Ih..I:L-)IP

i=1

and Proposition 3 of [21] (page 59) for the second inequality. The details are

omitted here.

Theorem 2.12 Let 1 < p < 00 and f E LP(B,dV). Suppose f = f1+ f2 with
f1 6 01(8).

1
llbth 1— 21—] Pdv 12 dd B,
(l f 0 lVf1($)l( lxl) an V(K,(:I:)) Kr(x)|f2l are mm 6 foer
then Hf is bounded on bP(B);

(2) If both Ivf1(a:)|(1—|:1:|) and m [mm Ifglp dV approach 0 as |:1:| —+ 1,
then Hf is compact on b”(B).

Sketch of the proof. We have Hf = Hf, + Hf, and Hf, = (I — Q)Mf2, where
M f, is the multiplication by f2. If f2 satisfies the condition of (1) or (2), then M f2
is bounded or compact on bp(B), respectively, according to Theorem 1.8. Thus
H ,2 is bounded or compact on b” (B), respectively.

So we only need to deal with H f,. By Lemma 2.10 we only need to show

|(H;1(U). Ah)! = |(f1u, Ah)! S CllUllpll A hllp'

28

for any u E bp(B) fl L°°(B, dV), h E C§°(B) in order to prove the boundedness of
H f:- The above inequality will follow from the following identity (which is from

integration by parts) and Lemma 2.11

(f1U,Ah> : "(U Vflavh’) + (Vfl ' Vuah>a

where (u v f1, vh) = Z / u(f1)x,hx,dV. Similarly if {u,-} is a sequence tending
i=1 3
to O weakly in b”(B), one can show that ||Hf1ui||p —> 0 as i —> 00. This gives the

compactness of H h- C]
Now we show that Theorem 1.15 follows from Theorem 2.12.
Corollary 2.13 Let 1 < p < 00 and f E C(B). Then H; is compact on bp(B).

Proof. We have

f = P(fls) + (f - P(f|s) = f1+f2.

where P(f|5) is the Poisson integral of f|3,f1 = P(f|5), and f2 = f — P(f|S).
Since fig 6 0(5), we have f2 —> 0 as |x| —+ 1. On the other hand, fl is harmonic
on B and f1 6 C(B). By Theorem 1.6, fl = Q(fl) 6 80. So f1 and f2 satisfy the

conditions in (2) of Theorem 2.12. So H f is compact. [:1

The proof above only requires that f be continuous on 5'. Hence this corollary
and the remaining results of this section are valid for a larger class of symbols than
the continuous functions on B.

We need two more lemmas.

29

Lemma 2.14 Let 1 < p < 00. If f,g E C(B), then both ng — Tng and

Tng —— Tng are compact on b”(B).
This is a consequence of Corollary 4.5 of [6]

Lemma 2.15 Let 1 S p < 00. Iff E C(B) and f = 0 on S, then T, is

compact on bp(B).

Proof. It is easy to see that there exists f,- E C (B) such that each f,- = 0 on a
neighborhood of S and Hf,- - f “00 —+ 0 as i —> 00. Theorem 1.8 shows that each

M f, is compact on b”(B); thus so is each T f,. Since T f, —> Tf, T, is compact. E]

For 1 < p < 00, let B(bp(B)) be the set of bounded linear Operators on b” (B),

and let 08(T) denote the essential spectrum of T E B (1)” (8)).
Theorem 2.16 Iff E C(B), then (78(Tf) =2 f(S).

Proof. First we show f (S) C 08(Tf). Without loss of generality we assume
f (17) 2 0 for some 17 E S. We need to show T; is not a Fredholm operator.
We prove this by contradiction. Suppose T, is a Fredholm operator. Then by
Atkinson’s Theorem, there exists P E B(bp(B)) such that PI} — I is compact on
b”(B). By Lemma 2.2,
ROE, -) Rb, -)

PT; HR

(2:, Mt: ‘ IIR<x,->II. * 0

as |$| —+ 1.

On the other hand, we have
PT P < C M
ll fllR llp —- ll fllR

(00")llp alt)”;

3O

_ )pl_____R(rv yll” ,
‘ C/B ”(9 'IIRIa: WW

: C([l + 12),

where 11 is the integral over A = {y E B : ly —- 17] < 25} and 12 is the integral over

B\A for6>0. Given c>0, since f(x) —>Oas:r:-—>r), we have

|R($, yll”

I < e
1 _ B||R($. '3)“

pdV(y )2:

 

if 6 is small enough. It is easy to get that |R(:c, y)| g for any 3:, y 6 B by

C
III? — yl"

(1.5). So we have

dV(y)

I2 S C(1—l17l)"(p_1)
Iy—nIza la? - yl"

<Ce

 

if la: —— 77| is small enough. This gives a contradiction.

Now we show 08(Tf) C f (S) Without loss of generality we assume 0 is not in
f (S) We need to show that T; is a Fredholm operator. Let g E C(B) be such
that g = %~ on S. By Lemma 2.15, ng - I = T fg_1 is compact. Thus by Lemma
2.14, T ng — I and T 9T, — I are both compact. Again by Atkinson’s Theorem we

conclude that TI is a Fredholm operator. [:1
Theorem 2.16 implies the following corollary.

Corollary 2.17 Let 1 < p < oo. Iff E C(B), then T, is compact on bp(B) if
and only iff = O on S.

31
2.4 Toeplitz Operators with Bounded Radial
Symbols

Let D denote the Open unit disk in the complex plane C and let A be the normalized
area measure on D. The analytic Bergman space on D, denoted L§(D), consists

of the analytic functions f on D with

|lf||§ = [D Iflsz < oo.

Let P denote the orthogonal projection from L2(D) onto its closed subspace
L§(D). For f E L°°(D), the Toeplitz operator T, is defined on L§(D) by ng =
P(fg). It is easy to see that T, is bounded with “T,“ S ||f||oo.

Since point evaluation is a bounded linear functional on L§(D), for each 2: E D

there exists a unique K; E L§(D) such that

f(Z) = (f, K.)

for all f E L§(D). The functions K, (2: E D) are called reproducing kernels for

L§(D); they have the explicit form

1
Kz(w):(T:——2w—)2, 11261).

For every z E D, let kz(w) = Kz(w)/||Kz||2. Then kZ (z E D) are called

normalized reproducing kernels for LZ(D). For f E L°°(D), the function f, called

32

the Berezin transform of f, is defined on D by

222) = (TI/cum = fo(w)|kz(w)I2dA(w)-

Recently, the following compactness characterization for Toeplitz operators

with bounded radial symbols on the Bergman space of the unit disk was proved in
WI
Theorem 2.18 Let f be a bounded radial function on D. Then the following

conditions are equivalent:
(i) Tf : L,2,(D) ——) LZ(D) is compact;

(ii) f(z) —> 0 as |z| —) 1‘;

 

(iii) 1:r/rlf(t)dt—>0 asr—> 1".

More recently, Axler and Zheng [3] showed that (i) and (ii) above are equivalent
even for nonradial bounded functions on the disk. The purpose of this section is
to extend Theorem 2.18 to spaces of harmonic functions in higher dimensions.
We consider Toeplitz operators on the harmonic Bergman space of the unit ball
in R" for n 2 2. We use the same basic approach as in [7], but our context
of harmonic functions and higher dimensions requires new estimates. Although
there appears tolbe no canonical choice for an orthonormal basis for the harmonic
Bergman space, and reproducing kernels for the harmonic Bergman space appear

to be quite different from analytic Bergman kernels when n > 2, it turns out that

a similar approach can be used.

33

For f 6 L°°(B), the Berezin transform f is defined on B by

~

f(rv) = «are, was. .)> = [B f(y)|r(x.y)|2dV(y),

where r(a‘, ) = R(:1:, )/||R(:r, )[[2. Although a formula for R(;1:, y) in closed form
is available, we will not use it. We will use Theorem 1.10 instead.

We need two lemmas from [7].

Lemma 2.19 Let A Z 1. Suppose |am+1—am| S C(m+l)’\'2 for some positive

constant C and all m 2 0. Then limmsoo am/(m + 1)A'1 = 0 if and only if

t—H’

lim (1 — t)’\ Z amtm = 0.
m=0

Proof. This can be proved using Lemma 1 of [7] and the same proof as for

Theorem 2 of [7]. Cl

Lemma 2.20 Let k be a nonnegative integer. Suppose f E L°°[0, 1). Then

 

, 1
11m
r—+1‘ l— 7‘

frlf(t)dt=0

if and only if

1
lim m f(t)t2m+'° dt = 0.

m—mo 0

Proof. Note that the boundedness of f implies lim,,Hoe f01 f (t)t2m+" dt = 0,
and the condition lim,,Hoe m fol f (t)t2’"+" dt = 0 implies limHoo 3 [J f (t)ts dt = 0.

Thus the lemma follows from Theorem 4 of [7]. E]

For a radial function f on B, we define a function f * on [0, 1) to be the function

34
such that f*(|:c|) = f (:23) Now we can prove an analogue of Theorem 2.18 for the
harmonic Bergman space.

Theorem 2.21 Let f be a bounded radial function on B. Then the following

conditions are equivalent:
(i) Tf : b2(B) —> b2(B) is compact;

(ii) f(x) —> 0 as [ml —> 1‘;

 

1
/f*(t)dt—> 0 as r -—> 1‘.

(1”)1 —r .

Proof. For at E B, by Theorem 1.10 and using the fact that spherical harmonics

of different degrees are mutually orthogonal to each other, we have

f(x) = ||R(a: )I|§/ f(y) )|R(:v y)|2dV(y )
z ()1—|x|"2()n+2m hm|z|2mf f“(t(t)t2"‘+" ldt
____ (1_ [TD )1: Z am(f )lx,|2m
m=0

where am(f) = (n + 2m)2hm fol f"‘(t)tr""‘+"‘1 dt. In order to apply Lemma 2.19, we

need to estimate |am+1(f) — am(f)| for m 2 0. We have

am+1( f) — am( f) = (n+ 2m + 2)'~’h,,,+1 [01 f*(t)t2m+”+1dt
—(n + 2m)2hm [01 f*(t)t2m+"‘1dt
= [(n + 2m + 2)2hm+1 — (n + 2m)2hm]/01f*(t)t2m+"+1dt
+(n + 2m)2hm /01 f*(t)(t2m+”+1 — t2m+"“) dt

= I1(f) +12”),

35

where 11(f) denotes the first term and 12( f ) the second term.
Since hm g C(m +1)""2 for all m 2 0, we have [12(f)| S C(m +1)"‘2 for all

m 2 0. Thus if we can Show that
(2.1) |(n + 2m + 2)2hm+1 — (n + 2m)2hml s C(m +1)“,

we will have |Il(f)| _<_ C(m+1)"‘2, and consequently |am+1(f) —am(f)| g C(m+

1)"‘2 for all m 2 0. Clearly
(n+2m+2)2hm+1—(n+2m)2hm = (n+2m)2(hm+1—hm)+4(n+2m)hm+1+4hm+1.

It follows easily from (1.1) that

n + m — 2 n + m — 3
hm+l — hm : +
n — 3 n — 3

Combining the identities above we have the desired inequality (2.1). By Lemma
2.19, the condition (ii) holds if and only if lim,,Hoo am(f)/(m + 1)"‘1 = 0. So (ii)
holds if and only if lim"HOG m [01 f*(t)t2m+"‘1dt = 0. From Lemma 2.20, we see
conditions (ii) and (iii) are equivalent.

On the other hand, every function in b2(B) is a sum of homogeneous harmonic

polynomials. For m 2 0, let pm,1,- - °,Pm,h,,, be an orthonormal basis for ’Hm(S).

Then

00
U {CmpmJa ' ' ' a Cmpmfim}
m=0

36

 

is an orthonormal basis for b2(B), where cm 2 \/(n + 2m) / nV(B). It is easy to
see that T} is a diagonal operator with respect to this basis since f is a radial

function. For each j E {1, - - - , hm} we have

1
(TmePm,j,Cum,j) = (n+2m)/0 f'(t)t2m+"‘1dt

am(f)
(n + 2m)hm'

 

Thus T, is compact on b2(B) if and only 1im,,Hoe am(f)/(n + 2m)hm = 0. It is
clear that (n + 2m)hm z (m + 1)”‘1. Again by Lemma 2.19, the condition (ii)
holds if and only if lim,,Moo am( f ) / (n + 2m)hm = 0. This finishes the proof of the

equivalence of (i) and (ii) and the proof of the theorem. C]

The equivalence of (i) and (ii) in Theorem 2.18 was extended to higher dimen-

sions in [20] for the analytic Bergman spaces of the unit ball in C".

CHAPTER 3

Weighted Harmonic Bergman

Spaces

3. 1 Introduction

The harmonic Bergman space bf,(B), with a > —1, is the set of all complex-valued

harmonic functions u on B with

IIuIIl. = (l. lu(:c)|2(1- lwl2)"dV(:v))2 < oo.

Point evaluation is a bounded linear functional on bf,(B). Hence for every

2: E B, there exists a unique Ra(:z:, ) E b§(B) such that

qu> = [B u(y)Ra(:v.y)(1 — IyI2rdv<y>

37

38

for all u 6 bf,(B). The functions Ra(x, ) are called reproducing kernels for 11,2,(8).
Obviously R0 = R. We will see that each R0 is real valued for a > —1 in Section
3.3.

The purpose of this chapter is to study these reproducing kernels. These re-
producing kernels have been studied by different authors in [2], [5], [8], and [15].
While reproducing kernels for (analytic) Bergman spaces of the unit ball in C"
have simple formulas in closed form, those for harmonic Bergman spaces are much
more complicated, and it appears to be impossible to find formulas in closed form
for R0 in general, except when n = 2. In Section 2.2, we point out how harmonic
reproducing kernels behave differently from analytic ones on the unit disk. In Sec-
tion 3.3, we give a representation for R0 in terms of zonal harmonics in higher
dimensions and establish some properties for R0. We use an estimate on R, given
recently in [15] to prove one property for R0. In the last two sections, we give

some applications of these properties.

3.2 Reproducing Kernels on the Unit Disk

We consider Ra when n = 2 in this section. Let D denote the open unit disk in the
complex plane C and A be Lebesgue area measure on D. For a > —1, the analytic
Bergman space Lfi’a(D) is the set of all analytic functions in L2(D, (1 — lzl2)°dA(z)).

Let K a be the reproducing kernel for A§(D), i.e.,

2(2) = [Df(w)1?a(z,w)(1—lwl2)”dA(w). z e D.

39
for all f E Ag(D). We know that

_ _a+1 1

Ka(z,w) — 7r (1_ 2117?”, z,w E D.

 

The reproducing kernels for bZ(D) are closely related to K0,(z, w). We have (see

page 357 of [22])

0+1 1
Ra(z,w)= 7r (2Re(1—2217)2+0—1)’ z,w€D.

 

 

For 2: E D,r 6 (0,1), let D,(z) = {w E C: |w — z] < r(1—— |z|)}. An important

property for Ka(z, w) is that

[Ka(z,w)| z 1/(1— |z|)2+a, w E D,(z).

For the unit disk, one usually uses the pesudo—hyperbolic disk instead of D,(z)
because of its connection with Mobius transformations; see [4] for example. How-
ever we will use the obvious extension of D,.(z) for higher dimensions in the next
section.

We find that Ra(z,w) behaves quite differently from Ka(z,w), which never

vanishes.

Proposition 3.1 For each r 6 (0,1), there exist 2 E D and a > —1 such that

Ra(z,w) = O for some w E Dr(z).

40

Proof. For z, w E D, we have

 

 

_ a +1 (2Re(1— Zw)2+°‘ )

Ra(z,w) — 7, [1_ zwl4+2a

1 .
Let 2 = t 6 (0,1) and f — w 2 3e“), where s > 0. It is easy to see that for

 

 

 

 

 

w E Dr(t),
rt r 1 — t r 1 — t rt
~— 2 — (1 ) < sin6 < 1 l = ,
1+t z—t f—t 1+t
_ _ rt , rt ,
and the range of 6 IS (— arcsm —, arcsm ) when w ranges over D,(t). Slnce

l-l-t 1+t
|1—t2Dl g (1 — t)(1 + t +rt) for w E D,.(t), we can choose t close enough to 1 such
S

that [1 — t2D|2+0 If we choose Oz large enough, then the range of cos(2 + (1)0

.1.
2.

is [—1, 1] when w ranges over D,(t). Hence the conclusion follows from

 

_ a +1 (2cos(2 + (1)0 —|1—tzv|2+a)

Ra(t,w) _ 7r [l-tw|2+°

It is not difficult to see from the proof above that we still have
Ra(z,w)%1/(1—|z|)2+°‘, w E D,(z),

provided that r is small enough (depending only on a). In the next section we will

prove this property for Ra(:r, y) in higher dimensions.

41
3.3 Some Properties of the Reproducing Kernels

We can describe the reproducing kernels in terms of zonal harmonics.

We have the following representation for R0.

Proposition 3.2 Leta > —1. If x, y E B, then

°° I‘(m+- 2+oz+1)
nV(B)()oz+1)_m0 (m+’2‘)

 

 

th'v, y)-

The series converges absolutely and uniformly on K x B for every compact K C B.

Proof. This can be proved using the same argument as for the proof of Theorem

8.9 of [2]. [:1

Since Zm is real valued for each m, we see that R0 is real valued.

Now we can give an estimate for Ra.
Proposition 3.3 Let a > —1. Then

(i) Ra(:r,:r) z 1/(1— |.1:|)"+" for :1: E B;

(ii) “Ram ')||3,.. % 1/(1- |$l)”+" for a: E B;
(iii) lRa($,y)| S C/(1 - lxllyllmm for may 6 B.

Proof. First we prove (1). By Proposition 3.2 and (i) of Lemma 1.9, we have

 

 

R(a::r)— iNij-E 2+oz-I-1)
a, —nV(B)2()=a+1m01—‘(m+3)

h,,,|:1:[2m.

Since hm % (m + 1)"’2, by Stirling’s formula we see the coefficients in the series

above are of order m"—1 as m -—> 00. This proves (i).

42

(ii) follows from ||Ra(ar, )||§a = Ra(:r,a:).

To Show (iii), for 12:,y E B, let .2: = IzrlC, y = |yln. Then by (ii) of Lemma 1.9

 

 

 

 

 

2 °° I‘( (m+- 2+oz+1) m
lRa(m.y)| S nV(B)1“()a+1 "A; “771+ n5) (Irvllyl) IZm(C.27)|
2 °° l"( (m+- 2+oz+ 1)
S ”V(B )1“(a+1),:4:0 I‘(m+%) (lmllyllm hm
C
(1 - lxllyl)"+“'

This finishes the proof. [I]

The following simple fact will be used:
1- lyl % 1- livl, y 6 KM)-

We have the following lower bound estimate for the reproducing kernels.

Proposition 3.4 Let a > -—1 and :1: 6 B. Then there exists r = r(a) 6 (0,1)

depending only on a such that Ra(:r, y) a: 1/(1 — |:L'|)"+" for y E K..(:1:).

Proof. It follows from Proposition 3.3 (iii) that Ra(:r,y) S C/(l — |a:|)"+" for
y E K,(a:).

To show the other direction, for y E K ,(r), by the mean value theorem we have

Ra(:1:,y) 2 R0, (1:, :r)— max IV” Ra(a:,u)||y—:z:|

“EKrfxl
C
— a u :1:,u —:1:.
WW“. ugIfo)lvR ( my I

IV

 

If u E K%(ar 13,) then 1 — [u] > —(1 — le). Thus for u E K1(:L ), Cauchy’ s estimates

43
(2.4 of [2]) gives

C

C
< —— < .
qu Ra($1u)l — (1_ [U])v61?{:)({u)|Ra($ ’U)[ — (1__ [$])n+a+1

 

Thus if r is chosen small enough, for y E K,(:r), we get

Cr C
I1 — |xl)"+" I1 — Iz|)"+° - I1 — IxIIn+a

 

 

 

120017.11) 2

This proves the proposition. C]

When a = 0, the proposition above was proved in Lemma 2.1 for any r E (0, 1)
using the explicit formula for Ro(:1:, y).
For :13, y E B, let P(z, y) be the “extended Poisson kernel” for B given by (1.2).

If a is a non—negative integer, then

 

Ra(:r,y) = (m+— n+-a) (m+

 

ll
Q
+
S
Q 3
\l/
H-
N13
+
Q
“U
C?
H
S

For a: E B, :13 ¢ 0, let :1: = :1: / [11:]2 be the inversion of .22. Notice that our reproducing
kernels are slightly different from those in [5] and [15] because we choose (1 — lxl2)“

as weights. We have the following lemma.
Lemma 3.5 Leta > —1.

(i) [Ra(a:,y)| S Clii: — y|‘"‘°‘ for :1:,y E B with [:17] >%

44
(ii) Ifa > n(% — 1) — i, then
fisa(C.y)l”do(C) s C(1 — |yl)"’l‘("+")”, y e B.

Proof. The same proof as for Lemma 2.3 of [15] yields (i) (although only the

case when a > 0 was considered in [15]). Now (ii) follows from (i) by the proof for
Lemma 3.2 of [5]. C]

In order to prove our next result, we need the following simple estimate (see
page 291 of [17]).

Lemma 3.6 Iffi > —1andm > 1+6, then forO g t < 1,
1
f (1 — tr)_’"(1 — near 3 C(1—t)1+/3‘"‘.
o

The following is the last property for Ra in this section.
. . n + B
Propomtlon 3.7 pr > —+—, B > —1, and a > —1, then
11 a

1

_ 5 ~
[8 lRa(:z:.y)|”(1 lyl) dV(y)~(1_|$|)(,,,a,p_(,,+,,,. x63.

Proof. For a: E B, by Proposition 3.4, we have

[8 lRa(1:.2/)|”(1-lyl)"dV(1/) 2 [Kr(z)lRa(1:,y)l”(1-lyl)"dV(y)

> C .
— (1 _ ]$])(n+a)p—(n+/3)

To show the other direction, using Lemma 3.5 (ii) and the fact that R0(r:1:, y) =

45
Ra(:1:, ry) for :13, y E B, O < r < 1 (which follows from Proposition 3.2), we have
1
f8 IR.(1r.y)l”(1 - lyl)"dV(.1/) = nV(B) / (1— nun-1 (f5 |R0(1:, rordan) dr

-_- 71V(B)/01(1—r)fir"_1(fSIRaULOIPdOKO dr

1
C/ (1 — r)B(l — r]x|)"‘l_("+°)” dr
0

|/\

|/\

C(l _ |$])n+B—(n+a)p,

where we used Lemma 3.6 in the last step. [I]

3.4 Application to an Inequality for Harmonic
Functions

The following result was proved in [8] and [18].

Theorem 3.8 Let G be a measurable subset ofB and p > 0,fl > —1. Then

the following conditions are equivalent:

(i) There is a constant C > 0 such that

/B|f(y)|"(1— lyllfidVQ/l s c l. |f(y)|”(1-lyl)"dV(y)

for each harmonic function f on B for which the left-hand side of the inequality

is finite;

(ii) There is a constant 6 > 0 such that V(C D K) 2 (ll/(B 0 K) for every ball

K whose center lies on S.

46

Luecking [8] proved (ii)=>(i), and (i):$(ii) only when p = 2, fl = 0. Later Sledd
proved (i)=>(ii) for all p > 0,3 > ——1 in [18] (I thank Professor William T. Sledd
for this reference). To prove (i):>(ii) in the case when p = 2,13 = 0, Luecking [8]
used R0(:1:, y) and suggested the use of Rg(:1:, y) for the case when p = 2, B > —1.
Sledd [18] developed a different approach by constructing harmonic functions using
the Poisson kernel.

We here provide another proof of (i)=>(ii) for Theorem 3.9. Our method is
similar to that in [8]. For 6 2 0, our proof is even shorter than that in [8], where
the explicit formula for R0012, y) was used. For —1 < B < 0, our proof uses a
careful argument. We believe the reproducing kernels are natural candidates for

this type of inequality.

Proof of (i)=>(ii). By the argument in the proof of Lemma 3 of [8], we only need
to show that given 2: > 0, there is a constant Cc (depending on e) such that for
every ball K with its center on S, there exists a harmonic function f (depending

on e and K) on B such that
(1) fB|f(y)|P(1—|y|)’6dV(y) 2 C, where C does not depend on K,e, and f;
(2) fB\Klf(l/)lp(1— Isl)” dV(y) < e;

(3) fGnKlf(l/)lp(1— lyllfi dV(y) S CAI/(G 0 K)/V(K 0 Bl)“ for some a > 0,

where a depends only on 6.

Without loss of generality let K have radius h < 1 and center 21 = (1,0, - - - ,0).
n + H

n+0:

. Let

 

Choose or large enough so that p >

n+B

f(y) = Ra($k1y)(1— |$k|ln+0_ P ,

 

47

where at], = ru, r > 0, and 1 — r 2 sh for small 3 > 0 to be chosen.
Condition (1) follows from Proposition 3.7.
The case B Z 0 is easier to deal with in order to show (2) and (3). Let B 2 0.

If y E K, then 1 — |y| < h. By Proposition 3.4, for y E K, we have

 

 

(1 - lyllfi < '1" 1

(1— IxIIW - Ishw __. Cs

2 P —2 B -
|f(J)| (1 M) :0 W )

This implies (3) for a = 1.
By Lemma 3.5, we have |R0(:I:k,y)| S C/lik — y|"+°‘ ifs < %. Notice that

(1 —— lyl) < lit), — y], y E B. We have

 

/ If(y)|”(1 — |y|)5 dV(y) 3 C(1_ [$k[)P(n+a)—(n+3)
B\K

w Til—l

S C(Sh)p(n+0)-(n+6)/ ———dr

h TP(n+al—B

< C(8)p(n+a)-(n+fil,

where we used the fact that B \ K C {y 6 R” : |y — Kirk] > h} in the second step.
If s is chosen small, then we have condition (2).
The case when —1 < 6 < 0 requires more work. First we choose q > 1 such

that qfl > —1. Let q’ denote the conjugate of q. Holder’s inequality gives

(B\K '1 (1)1”(1 — IyIIBdVIy) s (1 — lzkl)”("+°"‘"+m

ap—

(I. Wag/1% — IyIIfiqu/In)
\K

(f mantel"? dV(y)) .
B\K

48

If (n + a)p > 2(3 + S), then by Proposition 3.7

l

‘1 l
1’19 _ 50 /
(f,\KIR.Ia:I.,I/)I I1 lyl) 11(1)) 30(1—kaII<’*+“>‘%-<%+I’>'

 

If (n + a)p > 22,, then we have
9

.94...

 

|/\
Q
A
ax
8
‘1.
:5
+ ~3::
3 l
53..
‘3
V

(be lam. at? dVIyI)

Combining the inequalities above, we get

/ |f(y)lp(1— IyIII’dVII/I s Garret—1",
B\K

provided that a is large enough. This gives (2) if s is small enough.

We now show (3). We have

AnKlf(y)lp(1—lyl)"dV(1/) = (1—|11|)P<"+“)—<"+"> [GM |R0(:rk,y)|p(1—|y|)fidV(y).

By Holder’s inequality and Proposition 3.7, we get

1/0
fanKlRa(ark,y)lp(1-lyllfith/l s (I. lRa(rrk,y)|"”(1-Iyl)"”dV(y))

-(V(G r1 KIIW

C 1/'
S (1— kal)”‘"+"’“(%+5> MGM” , '

 

49

Hence we obtain that

V(G n 12))”1'< C (V(G n [0) W

LnKlf(y)lp(1-lyl)”dV(1/) s c (m m, r, K,

Thus the condition (3) is satisfied with a = 1 /q’. D

3.5 Application to Toeplitz operators on Har-
monic Bergman Spaces

Let u be a finite complex Borel measure on B. We densely define the Toeplitz

operator on b§(B) with symbol u by

T111417) = [a Ra($,y)U(y)du(1/)

for u 6 13(3) n L°°(B, (1 — (moat/(3)). If d;1(y) = f(y)(1 — (main/(y), then
we write T” = Tf. Let (-, -)0 denote the inner product for L2(B, (1 — |x|2)°‘dV(a‘)).

For bounded u, v E b§(B), it follows from Fubini’s Theorem that

(Tuu,v)a 2/ uvdu.
8

Suppose u _>_ 0 and let I denote the inclusion map from bf,(B) to L2(B,du).
It is clear that T“ is bounded (compact) on b§(B) if and only if I is bounded
(compact).

The characterization of boundedness and compactness for the inclusion map

50

was given in [14], where more general domains in R" and more general spaces were
considered, except for —1 < a < 0. We can extend the characterization to all
or > —1 in our case. From here on we always assume r is the number given in

Proposition 3.4 (which depends only on 01).

Proposition 3.9 Let a > —1 and u be a finite positive Borel measure on B.
Then the following conditions are equivalent:

(i) I is bounded (compact);

(ii) u(Kr(a:))/V(K,.(:r))1+9rf is bounded for :r E B (—> 0 as |:r| —> 1).

Proof. Oleinik and Pavlov [14] proved that (ii)=>(i). To prove the implication

in the other direction, suppose I is bounded. Then

2 < 2 _ 2 a
[B lul d1 _ c [B lu(y)| I1 lyl I dV(y)
for all u E bflB). For x E B, let u(y) = Ra(:r,y) E b§(B). Then

(ll:(l1:l)(2$(21)+~a) S C [K439 IROCE, y) I2 d#(y)

C/B lRa(:v, yll2 (111(9)

3 c [B IRa(1:.y)l2(1-|yl2)“dV(1/)
C
(1 — |12|)"+"’

 

l /\

 

where we used Proposition 3.3 (ii) in the last step. A modification of this argument
shows that compactness of I implies the little 0 condition; we omit the details. This

proves (ii). Cl

51

Now we can state Proposition 3.9 in terms of Toeplitz operators. Although [14]
only gives the continuous version, a discrete version can be easily obtained (see,

for example, Theorem 1.8).

Proposition 3.10 Let 02 > -1 and u be a finite positive Borel measure on B.
Then the following conditions are equivalent:

(i) T,, is bounded (compact) on bz(B);

(ii) p(K,(a:))/V(K,(:r))1+% is bounded for :1: E B (—> 0 as [:13] ——> 1);

(iii) u(K,(:1:,~))/V(K,(a:,-))1+% is bounded fori = 1,2, - -- (—> O as i —-> 00).

Now we can establish a trace ideal criteria for positive Toeplitz operators on

bf,(B). The case a = 0 was proved by Theorem 2.9 using ideas from [9] and [23].

That result can be extended to all a > —1.

Theorem 3.11 Let 1 S p < oo,a > —1, and u be a finite positive Borel

measure on B. Then the following conditions are equivalent:

(i) T11 E 5120133));

(ii) u(Kr($))/V(Kr(rr))1+% E If"(BI(1-- |$|2)‘"dV(-T));

(iii) 22:. (IIK.Ix.II/VIK.I1:.II‘+%)” < oo.

The proof of the theorem above is entirely analogous to that for Theorem
2.9, so we will not give a proof for it. We remark that the two properties for
the reproducing kernels needed for the proof are supplied by Proposition 3.3 and
3.4, and the Sp-norm of T” is related to the reproducing kernels by the following
identity:

“Tull”, = A<T5Ra(1r.-),Ra(xv,-)>a(1- |m|2)“dV(1‘)-

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