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Hankel Operators on Harmonic Bergman Spaces

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Mirjana Jovovic

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MSU Is An Alfirmotivo Action/Equal Opportunity Institution
Wows-9.1

HANKEL OPERATORS ON HARMONIC
BERGMAN SPACES

By

Mirjana J ovovié

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1994

ABSTRACT

HANKEL OPERATORS ON HARMONIC BERGMAN SPACES

BY

Mirjana Jovovié

We study Toeplitz and Hankel operators on the harmonic Bergman space b2(B),
where B is the open unit ball in R", n _>_ 2. Gregory Adams (see [1]) considered the
case n = 2 and mostly studied the properties of the operator T2. We show that if
f is in C(B) then the Hankel operator with symbol f is compact. For the proof we
have to extend the definition of Hankel operators to the spaces b”(B), l < p < 00,
and use an interpolation theorem. We also use the explicit formula for the orthogonal

projection of L2(B,dV) onto (22(8). This result implies that the commutator and

semi-commutator of Toeplitz operators with symbols in C (B) are compact.

We also show that the space b2(B) decomposes as 62(8) 2 @fi=on(B), where
’Hm(B) denotes the space of all homogeneous harmonic polynomials on B of degree m.
We prove that ngsz(Hm(B)) C Hm(B) for all j = 1,...,n and all m. We have
partial results that give some of the eigenvalues and eigenvectors of ngij. Further,
motivated by some calculations obtained with the aid of Mathematica, we have a

conjecture concerning the eigenspace decomposition of the restriction of ngH31. to

Hm(B).

DEDICATION

To my parents

iii

ACKNOWLEDGEMENTS

I would like to express my deep gratitude to my advisor, Professor Sheldon Axler.
His guidance, encouragement and enthusiasm were invaluable. I would also like to
thank the members of my thesis committee, Professor Joel Shapiro, Professor Alexan-
der Volberg, Professor Wade Ramey and Professor Bernd Ulrich. The faculty and
staff of the Mathematics Department at Michigan State University has generously
supported me during my graduate career. This includes funding for conferences
and providing me with an opportunity to spend two semesters at the Mathemati-
cal Sciences Research Institute at Berkeley. Special thanks to the department chair,

Professor Richard Phillips.

Contents

1 Preliminary Results
1.1 Spherical Harmonics ...........................

1.2 Harmonic Bergman Spaces ........................

1.3 The Kelvin Transform ..........................
2 Toeplitz and Hankel Operators
3 Hankel Operators on Hm(B)

BIBLIOGRAPHY

1

4

10

29

42

Chapter 1

Preliminary Results

1 . 1 Spherical Harmonics

Let Hm(R”), m 2 0, denote the space of all homogeneous harmonic polynomials on
R” of degree m. A spherical harmonic of degree m is the restriction to the unit sphere
S of an element of Hm(R”). The collection of all spherical harmonics of degree m is

denoted by ’Hm(.5') Note that ’Hm(R"), and hence ’Hm(S), is a complex vector space.

Denote by ’Pm(R”) the complex vector space of all homogeneous polynomials on

R” of degree m. Define an inner product ( , )m on ’Pm(R”) in the following way:
For p<e> = 2...”. end q<e> = step... bnxfi set <p, q>m = pawn]. where pa?) is

the partial differential operator Zlalzm (101)".
Note: The proofs of almost all the statements in this chapter can be found in [3].

The next theorem shows that any polynomial on R", when restricted to S, is a

sum of spherical harmonics. (See [3], Theorem 5.5.)

Theorem 1.1 Every [3 E ’Pm(R”) can be uniquely written in the form
p(:v) = 22mm) + Irvlzpmem) + + leQkpm_2k(e),

where k = [%] and pm_2j E Hm-2j(R”) forj = 0,1,..., k.

1

Corollary 1.2 Up is a polynomial on R" of degree m, then the restriction ofp to

S is a sum of spherical harmonics of degrees at most m.

Let L2(S) be the usual Hilbert space of square-integrable functions on S with inner
product defined by ( f, g) = f3 f g do, where a is normalized surface-area measure on

the unit sphere 5'.

Let us now View Hm(S) as an inner product space with the L2(S) inner product.

Then we have the following two theorems. (See [3], Theorem 5.3 and Theorem 5.8.)
Theorem 1.3 Ifm 75 k, then Hm(5) is orthogonal to ’Hk(S) in L2(S).
Theorem 1.4 L2(S) = 692:0 ’Hm(S)

Fix a point 77 E S, and consider the map A : Hm(S) —> C defined by A(p) 2 12(7)).
The map A is clearly linear. By the self-duality of the finite dimensional Hilbert space

’Hm(S') there exists a unique Z,7 E ’Hm(S') such that

13(77): (nZn) =/Sp-Z_,,d0

for all p E Hm(S). The spherical harmonic Z,7 is called the zonal harmonic of degree

m with pole 77. At times it is convenient to write 2,,(0 : Z(77, C) = Zm(n,C).

It is easy to show that each Z,7 is real valued, Z,,((,‘) : ZC(77) for all 17,C E S and

2,,(77) = ||Z,,||§ for all 7] E S, where H - [[2 denotes the norm in L2(S',da).

The dimension of Hm(R") is given in the following lemma. (See [3], page 82.)

Lemma 1.5 Let hm denote the dimension (over C) of the vector space Hm(S). Then

h _ n+m—1 __ n+m—3
m_ n—1 n—1

formZZandhlzn, ho=1.

It is not hard to show that Zn(77) = hm for all 77 E S.

00

Our previous decomposition L2(S) = $171.20 Hm(S) has the following restatement

in terms of zonal harmonics. (See [3], Theorem 5.14.)
Theorem 1.6 [ff 6 L2(S), then f(n) = 32:“ f, Zm(n,~) ) in L2(S).

Every element of Hm(S) has a unique extension to an element of Hm(R"); given
1) E Hm(S) we will let p denote this extension as well. (Note that this implies that
the dimension of 'Hm(R") is hm.) In particular, the notation Zm( - , n) will often refer

to the extension of this zonal harmonic to an element of ’Hm(R”).

It is easy to show that for a: E R" and u E ’Hm(R”)

u(e) = Lawnmower (1.1)

An explicit formula for zonal harmonics is given in the following theorem. (See [3],

Theorem 5.24.)

Theorem 1.7 Let a: E R", C E S. Then

1%]

Zm(x.<) : (n+2m —2)Z(—-1)

k=0

kn(n + 2)...(n + 2m — 2k — 4)(
2kkl(m — 2k)!

 

€23 _ C)m—2klxl2k

for each m > 0.

Let us now extend the zonal harmonic Zm to a function on R” X R”. We do this
by making Zm homogeneous in the second variable as well as in the first; in other

words, we set
Zm(rv,y) = lmlmlylmZmW/lmlw/lyll-
If either :8 or y is 0, we define Zm(:r,y) to be 0 when m > 0; when m = 0, we define

Z0 to be identically 1. With this definition, Zm(:z:, - ) E Hm(R") for each a: E R”;

also, Zm(:c,y) = Zm(y,:1:) for all :r,y E R”.

By using polar coordinates we can obtain the analogue of (1.1) for integration

over B. For it E ’Hm(R"), we have

n + 2m
“(55) — ml? ”(y)Zm($ay)dV(y) (1-2)
for each :1: E R”. In other words, for every u E Hm(R"), u(:c) equals the inner product
of u with %Zm(x, - ).

1.2 Harmonic Bergman Spaces

Let B be the open unit ball in R” for n 2 2. Let V be Lebesgue volume measure on
R" and let 1 S p < 00. The harmonic Bergman space bp(B) is the set of harmonic

functions u on B such that
llullp = (/3 lul” am”? < eo.

We often View bp(B) as a subspace of LP(B, dV).

For fixed :1: E B, the map u I——> u(:c) is a linear functional on bP(B). We refer to this
map as point evaluation at :13. The following proposition shows that point evaluation

is continuous on bp(B). (See [3], Proposition 8.1.)

Proposition 1.8 Suppose :1: E B. Then

1

 

for every u E bp(B).

The next result shows that bP(B) is a Banach space. (See [3], Proposition 8.3.)

Proposition 1.9 The space bp(B) is a closed subspace of LP(B,dV).

5

Taking p = ‘2, we see that Proposition 1.9 shows that b2(B) is a Hilbert space

with inner product

(u,v) 2/81“) dV. (1.3)

Because the map u r——> u(a:) is a bounded linear functional on b2(B) for each a: E B,

there exists a unique function RB(a:, - ) E b2(B) such that

WU) = Luz/mafia) dV(y)

for every u E b2(B). The function 33, which can be viewed as a function on B X B,

is called the reproducing kernel of B. We list some basic properties of 123.

(i) Each R3 is real valued.
(ii) RB(a:,y) = R3(y,:c) for all :c,y E B.

(iii) [[RB(:I:, -)||2 = (123(33, o), RB(:L', -)) = RB(:I:,:1:) for every :1: E B.

The following lemma is useful in finding an explicit formula for the reproducing

kernel of the ball. (See [3], Lemma 8.8.)
Lemma 1.10 The set of harmonic polynomials is dense in bp(B), for 1 S p < 00.

The reproducing kernel of the ball is given by the following theorem. (See [3],
Theorem 8.9.)

Theorem 1.11 Ifx,y E B then

 

RB(:I:, y) = 71V1(B) if: (n + 2m)Zm(;r, y).

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

Let Q denote the orthogonal projection of L2(B, dV) onto b2(B). Then for every

f E L2(B,dV) and a: E B we have

(and): <62}: Rae. -)> = (f, Rue. -)> = panama/(y) (1.4)

Hence we can show the following theorem. (See [3], Theorem 8.14.)

Theorem 1.12 Let p be a polynomial on R" of degree m. Then Qp is a polynomial

of degree at most m. Moreover

1 m
MB) ,sz + 2k) [812(31)Zk($,y) dV(y)

=0

(Qp)($) =

 

for every :1: E B.

Now we can get a formula in closed form for RB(.v, y). (See [3], Theorem 8.13.)

Theorem 1.13 Let any E B. Then

(n — 4)|~’v|“ly|“ + (8x - y — 2n - 4)|$l2|y|2 + n

RB(~Tay) = nV(B)(1 _ 2:1: . y + I$I2ly|2)1+n/2

 

An easy calculation shows that

n(1--|~’v|2|3/|2)2 — 4lwl2lyl"’((1—liL‘l2)(1—|y|"’)+|3c - ylz)
nV(B)((1—|a:|2)(1—|y|2)+|x — .y|2)1+’“°/2 '

 

R3033?!) =

Remark: It is interesting to notice that R3 is not positive everywhere on B x B;
in fact it is not even bounded from below on B X B. To see this, we can choose :1: and
y from B such that [:c — yl2 =1/j,1—|:r|2 =1/j2, 1— |y|2 =1/j2, j 2 2. Then

R( ) n(1_(1—ji2)2)2—4(1*fi)2(317+%)
B x’y : 1 1 n
nV(B)(34- + ;)1+ /2

 

8 4 n—l _4 —4 n—2 71—4
—4+;2-+—le1+7.—+—1,?—1+—,7—
nl/(B)(%)7(1+ ,is)1+"/2

 

If we letj —> 00, then RB(a:,y) -—> —00.

Since R3 is a real valued function it follows that for every u E b2(B)
abs) = [B U(y)RB(ar,y) dvo).

Note that for fixed :1: E B, the function R3(x, - ) is bounded on B. Thus it makes
sense to ask whether the above equation holds not only for u E b2(B), but also for

u E b”(B), where 1 S p < 00. The following lemma answers that question.

Lemma 1.14 For all u E b”(B) and for all p E [1,00)
u<e> = Luz/mam) dV(y)- (1.5)

Proof: Note that the left-hand side and the right-hand side of (1.5) are bounded
linear functionals on b”(B) that agree on harmonic polynomials. Since, by Lemma

1.10, the set of harmonic polynomials is dense in bp(B) it follows that

use) = [B u(y)RB(rv,y) dV(y)
for all u E bP(B). [:1
Let 'Hm(B) denote the space of all homogeneous harmonic polynomials on B of

degree m. Next we prove that the Hilbert space b2(B), with inner product defined

by (1.3) is the direct sum of the spaces Hm(B).
Theorem 1.15 b2(B) = 693:0 Hm(B)

Proof: The finite dimensionality of Hm (B) implies that ’Hm (B) is a closed subspace

of b2(B). Note that dimension of ’Hm(B) is hm.

Let m gé k, and p E Hm(B), q E ’Hk(B). Since p and q can be extended uniquely

to R", using polar coordinates and the homogeneity of p and q we have

[3 p(e)q(e> dV<e) = MB) [01me dr [5 pmqm dam = 0

by Theorem 1.3. Hence ’Hm(B) is orthogonal to ’Hk(B) in b2(B).

To finish the proof it is enough to show that the linear span of U:=o ’Hm(B) is
dense in b2(B). Since, by Lemma 1.10, the set of harmonic polynomials is dense in
b2(B), it is enough to show that every harmonic polynomial belongs to the linear span
of Ufn°=0 Hm(B). Let p be a harmonic polynomial of degree k. Then p = Zi=0 p],
where p,- E ’Pj(B), j = 0,...,k. Also, 0 = Ap = 2;“:0 Apj, where Apj E ’Pj_2(B),
j = 2, ..., h, and Ape = Apl = 0. This implies that Apj = 0 for j = 0, ..., k; in other

words, 12,- E Hj(B), and the lemma is proved. D

1.3 The Kelvin Transform

The map a: H a:*, where

*

x: 0 ifxzoo

oo ifa:=0

{cc/[3:]2 ifz;£0,oo

is called the inversion of R" U {00} relative to the unit sphere. For any subset E of
R” U {00}, we define E“ = {:r“ : a: E E}.

Given a function u defined on a set E C R”\{O}, we define the function K[u] on
E'“ by

KEN-1‘): |$|2'"u($*)-
The function K [u] is called the Kelvin transform of u.

We easily see that K[K[u]] = u for all functions u as above. The transform K

is also linear: If u, v are functions on E and b, c are constants, then [{[bu + co] =

bK [u] + cK[v] on E“.

The crucial property of the Kelvin transform is given in the following theorem.

(See [3], Theorem 4.4.)

Theorem 1.16 [fit C R”\{0}, then u is harmonic on Q if and only if K[u] is

harmonic on (2“.

For p a homogeneous harmonic polynomial we have the following identity. (See

[3] , Theorem 5.32.)

Theorem 1.17 Let n > 2 and let p E Hm(R”). Then
1? = cm1([p(D)|$|2’"l,

where cm = ;«"=1(4 — n — 2j)‘1 for m > 0 and c0 = 1.
Now we have the following theorem. (See [3], Theorem 5.33.)

Theorem 1.18 Let n > 2. The space Hm(R") is the linear span of

{1"lD°|$|2_”lila|= m}
and ’Hm(S) is the linear span of

{(Dalxlz'n) ls: lal = m}-

The next theorem gives a basis of Hm(R") and HMS). (See [3], Theorem 5.34.)
Theorem 1.19 Let n > 2. The set
{K[D°'|:v|2‘"] : [al 2 m, an = 0 or 1}
is a vector space basis for ’Hm(R") and the set
{Dalrlz’n : [a] = m, on = 0 or 1}

is a vector space basis for Hm(S).

Note: We have concetrated here on the case n > 2. Analogous results hold when

n = 2 if lez‘" is replaced by log |:1:|.

Chapter 2

Toeplitz and Hankel Operators

Since b2(B) is a closed subspace of the Hilbert space L2(B, dV), there exists a unique
orthogonal projection Q of L2(B, dV) onto b2(B). Then for f E L2(B, dV) and :1: E B

(1.4) gives

(Qf)(rr) = b rammed) dV(y)-

Since the above integral makes sense whenever f belongs to L1(B,dV) we can

extend the definition of Q to L1(B,dV). For f E L1(B,dV) and :1: E B define

(and) = [B f(y)RB($,y) dV(y)- (2.1)

The symmetry of RB and the harmonicity of RB(:1:, - ) imply that RB( - ,y) is a
harmonic function on B for every fixed y E B. Differentiation with respect to a: under
the integral sign shows that Qf is a harmonic function on B for every f E L1(B, dV).
By Lemma 1.14, Q f equals f for all f E b1(B). We already know that Q is a bounded
projection of L2(B, dV) onto b2(B). In [7] Ligocka shows that Q, defined by (2.1), is
a bounded projection from L”(B,dV) onto bP(B), and that the dual of bP(B) is

bp'(B), where 1/p+1/p’ :1 and 1 < p < 00.

Remark: In [8] Ligocka shows that Q maps L°°(B,clV) continuously onto the

space of Bloch harmonic functions on B, denoted by BlHarm(B). BlHarm(B) is
10

11

defined to be the set of harmonic functions u on B such that

:30 - |$|)| v u(:13)! < 00-
Ligocka also shows that the space of Bloch harmonic functions on B is the dual of
the space b1(B).
For f E L°°(B, dV), define the Toeplitz operator T; : b2(B) ——> b2(B) with symbol

f by
T1" = QUU)

for u E b2(B). It is clear that Tf is a bounded operator and that [leH S [[flIoo.

Lemma 2.1 Let f, f1,f2 E L°°(B,dV) and a, b scalars. Then

(i) Taf1+bf2 = aTh + 5Tb

{ii} T; : Tf.
Proof: (i) is obvious. To prove (ii), let u,v E b2(B). Then

<T;u.v> = arm = have» = mm

= u(Z)f(2')v(z)dV(Z)= (Ufw) = (62010.0)

We used that Q is the orthogonal projection from L2(B, dV) onto b2(B). D
For f E L°°(B, dV), define the Hankel operator Hf : b2(B) —> b2(B)i with symbol

f by
Hi” = (I - Q)(fU)

for u E b2(B).

Lemma 2.2 Let f E L°°(B,dV). Then Hf is a bounded operator and “HI” S [[fHoo.

12

Proof: Let u E b2(B). Then
”Hialli = IIfUI|§ - ||Q(fU)l|§ S llfUIlg S ||fIIEOIIUI|§
Hence ”Hf“ S ||f||oo- '3
The next lemma gives a formula for the adjoint of a Hankel operator.
Lemma 2.3 Let f E L°°(B,dV). Then If; : b2(B)i —+ b2(B) is given by
11;}. = can

for h e b2(B)i.

Proof: Let u E b2(B), h E b2(B)*. Then

(H;h,u) = (thful : (hv(1"Q)(fu)>
: (hvful _ (haQ(fu)> : (hvful
= (Th/u) = (QUhlaul-

Thus H3212 = Q(fh). 1:1

The connection between Hankel and Toeplitz operators is provided by the formula

given in the following lemma.

Lemma 2.4 Let f,g E L°°(B,dV). Then

1“,, — Tfrg = H}:Hg.
Proof: Let u E b2(B). Then

(H}-Hg)u = H}(1 — Q)(gU) = H}(gu - Q(gU)) = H}(gu) - H}(Q(9U))
= QUE“) _ Q(fQ(gu)) = ngu — Tf(Q(gu)) = ngu — (Tng)u
= (ng _ Tng)u.

13
Therefore T19 — Tng = H}Hg. B

Let f E L°°(B,dV). Define Sf : b2(B)J' —> b2(B)i by

51h = (I - came)

for h E b2(B)i.

Lemma 2.5 Let f,g E L°°(B,dV). Then

Hfg = 5ng + Hng.

Proof: Let u E b2(B). Then

(Sng + HIE)” = (51119)u + (H/Tglu = SIU - QM“) + Hf(Q(9u))
= 51(9u — Q(gU)) + (I - Q)(fQ(9U))
= (1 — Q)(fg'u — fQ(gzt)) + fQ(.cIU) — Q(fQ(QU))
= fun f fQ(gu) - Q0921) + Q(fQ(9'U)) + fQ(.W)
— Q(fQ(91t)) = (1 — Q)(ng) = [1ng-

Thus Hfg=Sng+Hng. D

Now, we have the following corollary.

Corollary 2.6 The set of all functions f E C(B) such that H; is a compact operator
is a closed subalgebra of C(B).

Proof: The only nontrivial part of the corollary is the assertion that the set in

question is closed under multiplication. This follows from Lemma 2.5. C]

In order to prove our main result we need to extend the definition of Toeplitz and

Hankel operators to the spaces bP(B) for p E (1,00).

14
Let f E L°°(B,dV) and let 1 < p < 00. The Toeplitz operator with symbol f is

the operator Tf : bP(B) —-+ bP(B) defined by

T!” = Q(fU)
for u E bp(B). The Hankel operator Hf : bP(B) —+ LP(B,clV) with symbol f is defined
by
HI“ =(1 - Q)(fU)
for u E bP(B).

Note that T; and H I depend upon p, although p does not appear in the notation.

The domain of T; and H f will always be clear from the context.

Remark: We know that Q : LP(B, dV) —> bp(B), defined by

(ogre) = /Bg(y)RB($a:l/) dV(y)
for g E Lp(B,dV), is a bounded operator for every p E (1, 00). Also,
T1 = QM!
and
Hf = (I - Q)Mf,

where M f is a bounded multiplication operator since f E L°°(B, dV). Thus it follows

that Ty and H f are bounded operators for every p E (1, 00) and every f E L°°(B, dV).

We can extend the domain of the inner product given by (1.3) to include all pairs
of functions f,g measurable on B such that fg E L1(B,dV). For such a pair of

functions define (f,g) by
(f,g) = foedv.

From now on, let p’ denote the number such that 1/p+1/p’ : 1, where p E (1, oo).

15

Since bP'(B) is the dual of bP(B) with respect to the pairing ( , ), define the
Banach space adjoint of Tf : bP(B) —> b”(B) to be the operator T; : bp'(B) —-> bp'(B)
such that

<T;u.e> = <u.:rre>

for u E bp'(B) and v E bP(B).

Proposition 2.7 Let p E (1,00), a,b scalars, and f, f1,f2 E L°°(B,dV). Let Tf,

Tfl, Ty, and Taf1+bf2 be operators on bP(B). Then

(i) Tafi-t-bfz = ani + be2

ii T“ = T-, where T- is the Toeplitz operator with symbol _ defined on bp' B .
f f f

Proof: (i) is obvious. To prove (ii), first we show that

((1 —— Q)g, Qh) = 0 (2.2)

for every g e LP(B,dV) and every h e LP'(B,dV). We know that Q, defined by
(2.1), is the bounded projection from Lq(B, dV) onto b‘I(B) for every q 6 (1,00). For
q = 2, Q is the orthogonal projection from L2(B,dV) onto b2(B), and hence (2.2)
holds. Since L2(B,dV)flLP(B, dV) is dense in LP(B, dV) and L2(B, dV)flL7"(B, dV)
is dense in LP'(B, dV), (2.2) holds for every g e LP(B, dV) and every h e LP'(B,dV).

Now, let u E bp'(B) and v E b”(B). Then

( fu, v) = (u, va)
= (u, Q(fv))
= (u, (I -(1 - Q))(fv)>
-_— (fu, v)
= (((I — Q) + Q)(TU)r v)
= (Ti—“r v)-

16

We used that Qw = w for w E bq(B), 1 < q < 00, and (2.2). D

Let p E (1,00) and Hf : bp(B) —+ LP(B,dV). The Banach space adjoint of Hf is
the operator H} : L”'(B,dV) ——> bp'(B) such that

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

where u E L”’(B,dV) and v E b”(B).

As in the case p = 2, we have the following connection between Toeplitz and

Hankel operators.

Lemma 2.8 Letp E (1,00) and let f,g E L°°(B,dV). Then
ng _ Tng : er’Hgv

where

Tf,Tg,ng : bP(B) -—+ bP(B), Hg : bp(B) —+ LP(B,dV), H;- : LP(B,dV) —> bP(B).
Proof: Let u E b”(B), v E bp'(B). Then

<H}ngr v) = (ngv Hrvl = ((1— QXW), (1— Q)(fv))
= (an, iv) - (an, Q(fv)) - (QM), f'v) + (QM), Q(fv)>
= ((Q + (I - Q))(ng)v v) - ((Q +(1 - Q))(9u)r Q(fv))

- (42(9’“), (Q + (1 - Q))(fv)> + (QWU), 62080))

= (nguv 0) - (Tngu, v) = ((ng — Tng)u, ’0)-

We used that Qv = v for v E bp'(B), (2.2), and Proposition 2.7. CI

Let f E L°°(B,dV) and p E (1,00). The operator Sf : LP(B,dV) ——> LP(B,dV) is
defined by
51h =(1 — Q)(fh)-

17

It is clear that S f is a bounded linear operator for every f E L°°(B, dV) and every

p E (1, 00). Straightforward. calculations, as in Lemma 2.5, give the following lemma.

Lemma 2.9 Letp E (1,00) and let f,g E L°°(B,dV). Then
Hfg = 5'ng + LIng,
where
Hf,Hg,Hfg : bp(B) -—> LP(B,dV), T9 : bP(B) —> bP(B), Sf : LP(B,dV) ——> LP(B,dV).

As in the case p = 2 we have the next corollary.

Corollary 2.10 Let p E (1, 00). The set offunctions f E C(B) such that the Hankel

operator Hf : bP(B) —) L”(B,dV) is compact is a closed subalgebra of C(B).
Proof: This follows from Lemma 2.9 and the relation Hf = (I — Q)Mf. D

An operator T that maps a subspace X of LP(B, dV) into LP(B, dV) is called an
integral operator if there exists a complex-valued, measurable function I: defined on

B X B such that
(mm = [B uedvu) My)

for all f E X and almost all :r E B. The function h is called the kernel of T.

Lemma 2.11 Let f E L°°(B,dV) and p E (1,00). Then Hf : bP(B) -—+ LP(B,dV) is

an integral operator.

Proof: Let u E bp(B). Then

(HruXe) = (I—Q)(fu)(rv) =f<e>u<e> —Q(fu>(e>
he) /3 unseen) dV(y) — /B flour/mom) dvo)
= /B(f(e)—f(v))Re<e,y>u(y)dV(r/) = /B kr(:v,y)U(y)dV(y).

18

where kf(:1:,y) = (f(a:) — f(y))RB(x,y) is the kernel of Hf. C]

For 1 S p,q < 00 the mixed norm space Lp’q(B X B) is defined to be the space of

functions It on B X B such that
/B(/B |k(:v.y)lp dV(y))q/P dV(:1:) < oo.

In [4] Benedek and Panzone prove that Lp'q(B X B) is a Banach space with the

norm
”kn... = (id/B leer/)1? dV(y))"/” more
They also show that the dual of Lp'q(B X B) can be identified with Lpl'q'(B X B).

More precisely, every bounded linear functional (,0 on L1M (B X B) is of the form

We) = /B [B k(x,y)v(x,y)dV(y)dV($)
for some unique v E L”"9'(B X B). Furthermore, “90” = [I’UHPI’qu

The next lemma shows that each h E LP'*P(B X B) defines a bounded integral

operator on Lp( B , dV).

Lemma 2.12 Let p E (1,00) and let I: E Lp"p(B X B). Then the integral operator

T: LP(B,dV) —+ LP(B,dV), defined by
(Tf)(:v) = [B k(e.v>f<y) due),

is bounded.

Proof: Let f E LP(B,dV). Then we have

lleIIZ = [Barret/(e): [BI/Humandvuwdvo)
s /B(/B Ik(e.y>r' dV(y))”/’"(/B mew dV(y)) dV<e> = “rituals...

Hence IITH S llkllp’m' D

19

For g E LP(B,dV) and h E Lp'(B, dV), the tensor product of g and h is a function
on B X B defined by

(9 (29 We, 3/) = 9($)h(y)-
Lemma 2.13 Let p E (1,00). Then the linear span of the set
{g s h : g e L”(B,dV), h e LP'(B,dV)}
is dense in L”"P(B X B).

Proof: First we show that for g E LP(B,dV) and h E Lp'(B,dV), g (8) h belongs
to LP"P(B X B). We have

/B(/B |(g o z.)(e.y>r’ dqu Me) = /B(/B rumor dV(y))”/’" Me)
= [B Ivonne“: Me)
= thlilllgllg < 00-
Therefore g (8) h E LP"P(B X B).
Suppose go is a bounded linear functional on LP"p(B X B) such that cp(g (X) h) = 0
for every g E L”(B,dV) and every h E LP'(B,dV). Since 99 E (Lp"P(B X B))* S
Lp'p'(B X B), there exists v E Lp'p'(B X B) such that

viz/B /Blc(:1: (o y )de( )dV(:1:)

for all k E LP"P(B X B).

Therefore for every g E LP(B, dV) and every h E LP'(B, dV)

O:p(g®h) : /B/Bg($ (33 yldvft/ fill/(33)
= [B an [B 11(y)v(w,y)dV(y) due). (23)

 

20

We know that
[B h(y)v(:v,y) (Mg) 6 Nady)
because
</B l [B anew) de’ cit/(envy s Ilhllplllvllpr' < oo.
Then (2.8) implies that
[B how, r) dvu) = 0
for almost every :13, where the set of measure zero can depend on h E LP’(B, dV). Be.

cause LP’(B, dV) is separable, there exists a countable dense set {h 33:, in LP’(B, dV)

and for every j, there exists a set of measure zero Ej such that

[B br(y)v(ar,y) CMy) = 0
for x E Ej.
Let E’ = $11 Ej. Then E’ is a set of measure zero, and for every :1: E E’ and
everyj we have f8 hj(y)v(a:,y) dV(y) = 0.
Since v E Lp'p’(B X B), which means that fB(fB |v($,y)|p dV(y))P’/p dV(:1:) < 00,
it follows that (fB [v(:1:,y)["dV(y))1/7’ < 00 for almost every :1:. Hence there exists a

set of measure zero E” C B such that (f3 |v(:1:,y)|7’ dV(y))1/7’ < 00 for :1: E E”, i.e.,

vx(-) = v(:c, -) E LP(B,dV) for all x E E”.

Let E = E’UE”. The measure of the set E is zero, and for every :1: E E, v(:c, - )

defines a bounded linear functional 90,, on LP'(B, dV) by

v.02) = [8 under) dV(y)-

For every j, gox(hj) = 0, and the density of {hjljiir in LP'(B, dV) implies that 90,, E 0
as a linear functional on LP’(B,dV). Hence v$(-) = 0 in Lp(B,dV), which means
(f3 |v(a:,y)[pdV(y))l/p : 0 for all a: E E. Then it follows that v E 0 in Lp’p’(B X B)

21

since (fB(fB |v(a:,y)|” dl/(y))7”/’D dV(:1:))1/P’ = 0. Therefore 99 E 0 on LP”P(B X B),

which proves the lemma. CJ

The following proposition shows that the integral operator T with kernel I: in

L”"P(B X B) is compact.

Proposition 2.14 Let p E (1,00) and let 1:: E Lp”p(B X B). Then the integral

operator T : L”(B, dV) —> LP(B, dV), defined by

(Tf)(x) = [B k(e.y>f(y) My),

is compact.

Proof: Let h E Lp"p(B x B). By Lemma 2.13, for every 6 > 0 there exist functions
gj E ”(B, dV), hj E L”’(B,dV) and complex numbers aj, j = 1, ...,m(e), such that
”k _ 227:? aigi 8’ hillr’m < 6-

Define an operator TC : LP(B,dV) ——> L”(B,dV) by

m(e)

(Tm )=/( 23699312?) y))f(y)dV(:y)

for f E Lp(B, dV). TC is a finite rank operator since

m(c)

Tr: Zed (fy hd y)My ))gy
We also have

”(T — Tdfllt = [B |(T — Tour Me)

m(c)

= [I [an (ey) Zen-(e) >Iy(y »f(y)dV<y)rMe>

m(6)

s (/ |f(y )(yrdv m/ (/ |k(:vy)— Econ-(e) y)’r My >)P/P’dV<e>)

m(6)

||f||£ llk - Z (1,-g, ® brllp ,p _ <€pllf|l§-

[\D
(0

Hence [[T — Tell S 6, which implies that T is a norm limit of finite rank operators,

and therefore compact. E]

In order to prove our main result we will need the following interpolation theorem

given in [5]. (See Theorem 2.9, Chapter IV)

Theorem 2.15 Let (X,u) and (Y, V) be finite measure spaces. Suppose that 1 S

(Ijr'rj S 00, (j = 0,1), and let T be a linear operator that satisfies
T : Lq°(u) —> LT°(1/) boundedly

and

T : L‘“(p) ——> L1”1 (u) compactly.

IfO < 0 <1 and q,r are defined by

 

 

then

T : Lq(u) ——> LT(V) compactly.

Now we can prove our main result.

Theorem 2.16 Let p E (1,00) and let f be a continuous function on the closure of

B. Then the Hankel operator Hf : b”(B) —> LP(B,dV) is compact.

Proof : By Corollary 2.10 A = { f E C(B) : Hf is compact } is a closed

subalgebra of C(B) . We want to show that A = C(B).

By the Stone—Weierstrass Theorem it will be enough to show that Hat) is com-
pact for every j = 1, ...,n. Note that because of the symmetry with respect to the

coordinates, it is enough to consider only Hm. Hence it is enough to show that

Hm,1 : bp(B) —-> LP(B,dV), defined by Hx,(u) = (I — Q)(:1:1u), is compact.

By Lemma 2.11

(H..y)(e) (I — WWW

f3 yr(e,y)y(y)dV(y),

where k1(:1:,y)=(:1:1 — y1)RB(:c,y).

 

Let
I = M. W or «wow/r Me).
where
k.(e.y) = ($1 — y1><n<1—Ier)yr)2— elyrlyrul—Ier)<1—)yr)+1y — yr)

nV(B)((1— |:r|2)(1 — lyl2)+|:1: _ y[2)1+n/2 '

If we can show that I is finite for some q, that would imply that k1 E Lq”q(B X B)
and hence, by Proposition 2.14, it would follow that Hr] : bq(B) -—) Lq(B,dV) is
compact. Then, using Theorem 2.15, we could prove that H$1 : bP(B) —> LP(B,dV)

is compact.

In order to show that I is finite, we divide the region of integration into three

parts.

I s C(q){/ry(/..-..o-... Ikr(e,y)r’ dV<y))q/q’ Me)

J
v

I1

+ f (/ |k1(:v,y)|"'dV(y))"""dV(fv)
\3 l —y|>1—|rl.Ix—y|>1-|y|

J

12

+ L/‘3(/i-le<lx-ylSI—Iy| [b71(:lr,y)[q’ dl/(l/llcl/ql (fl/(30}

J
v

[3

We used the following inequality : If oz > 1, then for arbitrary complex numbers a

24

and b

|a + bl“ S 2""‘(Ialor + lbl"). (2-4)

To show the finiteness of [1,12 and 13 we will need the following relationships as

well:
1— |x|2|y|2 = (1 - lrlz) + liv|2(1 - |y|2) (2-5)
and
(1— lxl2)(1— |y|2)+|rv — yr 2 [(0 — [:1:|2)2+(1 — lyrr’) (2.6)
for :1:,y E B.

In 11, [:13 — y] S l — [3:], which implies 1 -— [y] S 2(1 —|:1:|), and using (2.5) and
(2.6)

|k1(:c,y)| S Cr(n)lfcr - y1|(]1_—l,[:[)lo)+n = C1(n)(-|i$—::l-§)in-

 

Hence

[331 " yllq’ q/q'
1. s 0101.61)fB(fI$_yl_<_1$le W My)) Me).

By the change of variables formula with z = :1: — y we get

1 q’ q q’ :1:
Ir_<.Cr(n,q) jB (1_W</lz|51_lx| lzrl dV(z))’ M ).

Using the polar coordinates formula for integration on R" we have

1

1—|:1:|
n-1+q’ 9/9’
(1_ lxl)”q(/o r dr) dV(:1:)

|/\

 

H _ (n+9’M/q'
/ (1 lxl) (mm)

C] (n,Q) B (1—]$])nq
s C{'(n,q)f3(1-lxl)q‘" Me).

The last integral is finite if and only if q > n — 1, and hence 11 is finite for

q>n—1.

25

Similarly for 12, since [:1: — yl > 1 — [:1:| and la: — y| > 1—[y|, using (2.5)

 

2
|k1($ry)l S 02(nll501— 3/1ng = 2(“lHfii-
Hence
_ q,
12 S 02mg)/B(fly—yl>1-IeI.Ie-y|>1—1ylHi— dWyDQ/q’ all/(xi
The change of variables 2 = a: — y gives
1. 3 021m) / (/ 'Zl'q', dV(z))""" My).
8 1—1e1<)z|52 lzln"

Using the polar coordinates formula we have

2
I. s Cé(nrq) /B(/1_le WW der/‘1' My).

We consider two cases:

(i) n—1+q’—nq’#—1,i.e., qaén
Then

2W 0— lrvl)”+""”"'

I< ’ , f — q/q’dv .
2-0.(ny)3(n+q,_nq, n+q,_nq, ) (e)

 

 

Using (2.4) we get
I. s Cé'(n,q) + 05"(nyq)f8(1— Ie))<”+q’-W’>q/q’ Me)

3 cam) + 03"(nrq)f8(1— |sv|)"‘” Me).

The last integral is finite if and only if q > n — 1, and hence 12 is finite for

q>n—1andq#n.

(ii) q= n

Then q/q’ = n — 1 and we have

[2 S C;(n)/B(ln2 —ln(1—|a:|))"_1dV(:r).

By (2.4) it follows
123 03m) + 0;"(n> [B |1n(1 — |x|)l"“ dvm,

and the last integral is finite.

Hence for every q > n — 1, 12 is finite.

To show the finiteness of 13, we proceed in the similar way. Using 1 — |a:| <

Ia: — yl S 1 — [y|, (2.5), and (2.6) we have

_ 2 _ _
(1 lyl) <03(n)l1‘1 y1l<03(n)l$1 yll

k :13, SC n a: — _ — °
l 1( yll 3( )l 1 y1|(1_ly|)2+n (l—lyl)” lit-Ell"

Now, as in the case of [2,

l$1 — yllq’

1 < —-—dV q/q’dv .
3"03(n’q)/B(/1—I:rl<Ix—yISI—Iyllib—ylnq' (y)) (w)

Using the change of variables formula with z = a: — y we have

 

IZIIq’ / I
< q q .
13 _ C3(”a‘1)/ (/l—lzl<lzlsl lzlnq' dl (2)) dl (3")

Using the polar coordinates formula we have

13 S Cé(n7q)/

1
B(/1_lxlrn-1+q’-W’ dry/9’ dV(a:).

We consider the following two cases:

(i) n—1+q’—nq’7é—l,i.e.,q;£n
Then

, 1 (1 — la: "”"W’
13 S 03(n,Q)/( I I — n + qll)_ nql

 

 

)q/q’ dV(a:).
By (2.4) it follows that
13 S Célmql + CQHUW) /B(1 - |$|)("+""""')"/"' (ll/(9:)-

The last integral is finite if and only if q > n — 1, and hence 13 is finite for

q>n—1andq#n.

27

(ii) q= n

Then q/q’ = n — 1 and we have
13 S Cé(n)/B(ln1—ln(1—|:I:|))"—ldV(:I:),

13 s 03(n)/B(—1n<1—le))"-‘ dvm,

and the above integral is finite.

Hence for every q > n — 1, 13 is finite.

Therefore k1 e LQ’v4(B x B) for q > n — 1. By Proposition 2.14 it follows that the
operator T1 : L°(B,dV) —+ Lq(B,dV), defined by (Tlf)(:z:) 2 IB k1(:c,y)f(y) dV(y),
is compact whenever q > n — 1.

In the Interpolation Theorem 2.15 let X = Y = B, p = V = V, q,- = r,, j = 0,1,
and T = [~1le : L9(B,dV) —> Lq(B,(lV) for q 6 (1,00). Then T is bounded on
L"(B,dV), for q 6 (1,00) since HI, and Q are bounded for 1 < q < 00, and T is
compact for q > n — 1 since T = TlQ. Fix qo such that 1 < qo < p and q1
such that q] > max{n — 1,p}. By Theorem 2.15 T maps Lq(B,dV) compactly into

Lq(B,dV) for every q such that qo < q < ql, and hence for q = p.

Since Hag1 : bP(B) —+ LP(B,dV) is the restriction of T : L”(B,dV) —> LP(B,dV)

onto bp(B), it follows that H1:l is compact, completing the proof. D

We can now prove the following corollary.

Corollary 2.17 Let f,g E L°°(B,dV). If eitherf org is in C(B), then the operators

ng — Tng and Tng — Tng on b7’(B), where 1 < p < 00, are compact.

Proof: This follows from Theorem 2.16 and Lemma 2.8. E]

28

Let H be a separable Hilbert space, {e,n}$,f=1 an orthonormal basis for H. A
bounded operator A on. H is called. trace class if and only if ZfidflAlem, em) < 00.

We define the trace of A to be

trA = Z (Aem, em).

m=1
This sum is finite and independent of the orthonormal basis {em}fi=1. (See [9],

Theorem 6.24.)

For any bounded linear operator T on b2(B) define
||T||,. : {tr(T"'T)r/2}1/T, 1 S r < 00.

The Schatten r-class S, is the set of operators T with ||T||T < 00.

Proposition 2.18 If r > max{n — 1,2}, the Hankel operators H3]. mapping b2(B)

to b2(B)J*, j = 1, ...,n, belong to Schatten r-class 5,.

Proof: By the Hausdorff-Young Theorem for integral operators [11], (see also [6],

Theorem A), and Proposition 2.14 it follows that for r > max{n — 1, 2}

lle,llr S (llkjllr’wllkfllr’wlln < 00,

where mm) = low) and mm) = (so - y.)RB(x,y),j=1,...n.
Thus HI). 6 5,. for r > max{n — 1,2} and j = 1,...,n. Cl

Remark: If we want to show Theorem 2.16 only for p = 2 we could use Proposition

2.18 to show that ij :l)2(B)—+122(B)i is compact for j = 1, ...,n.

Chapter 3

Hankel Operators on Hm(B)

The Hankel operator Hf : b2(B) ——> b2(B)i is defined by Hf(U) = (I — Q)(fu). We
consider the operators 11;].ij : b2(B) —> b2(B), j : 1, ..., n and we show that ngij

maps Hm(B) into Hm(B) for every m 2 0 and j = 1, ...,n.

We will need the following lemmas. (See [3], Exercise 1.20 and Corollary 5.23.)
Lemma 3.1 A polynomialp is homogeneous of degree m if and only ipr - :1: 2 mp.

Lemma 3.2 [fu is a harmonic function on B, then there exist hm E Hm(R") such

that

for all a: E B, the series converging absolutely and uniformly on compact subsets ofB.

Now we can show the following proposition.
Proposition 3.3 Let u be harmonic. Then Ak+1(|:c|2ku) = 0 for every k 2 0.

Proof: Lemma 3.2 implies that u(:c) 2 £3sz hm(:c) for hm E Hm(R“). Hence it
is enough to show that
Ak+1(l$l2khm) : 0
29

30
for hm E Hm(R"), m 2 0 and k _>_ 0.

We will use the following product rule. (See [3], page 13.)
A(uv) = uAv + 2Vu - Vv + vAu. (3.1)
Let hm E Hm(R"). Then
A(Icclzkhm) = A(|:1:|2l“)hm + 2V|x|2k . th.

We have
Alfv 12k=2 W + 2(’C- 1))|;L~|2(k-1)

and

Vla'IZk = 2k|ar|2(k"1)x.

By Lemma 3.1 it follows that
A(|x|2khm) = 2k(n + 2(lc — 1) + 2m)|:c|2(k'1)hm.
After applying the Laplacian k-times to |x|2khm we have
k
N(|x|2khm) = kklfl( (n + 2(k )+ 2m)hm .
Therefore Ak+1(|x|2khm) = 0. D

We also have the next proposition.

Proposition 3.4 Let u E Hm(R”). Then

9311433) = hm+1(~’15)+ l$l2hm—1($) (3'2)

and
mine) = hm+2<w> + lemme) + mum—2e) (3.3)

where h,- E H,(R"), i = m — 2, m — 1, m, m +1, m + 2, j = 1,...,n.

 

31

Proof: Theorem 1.1 implies that
W03) = hm+1($) + |:c|2hm_1(:c) + + |$|2khm+1—2k($),

where k = [$31] and hm+1_2j E Hm+1_2j(R”),j = 0, ...,k.

For m g 2 (3.2) is obvious. Suppose now that m _>_ 3. We first show that xju is
orthogonal to |x|2k7Dm+1_2k(R") with respect to the inner product ( , )m+1 for k 2 2

and j = 1,...,n.

Let q E Pm+1_2k(R”) and let I: _>_ 2. Then
( larleqfiJ-u >m+1 = (1(D)[Ak(xjfl)l-

Using (3.1) we have

A(xja) = ija + 2Vx, - Va + xjAa = 2—3:
3'
and hence
on 8
2 ‘— _ —— = ‘ — - = .
A (:cJu) __ A(20xj) 282:,(Au) 0

Thus q(D)[Ak(:c,-u)] = 0 for k 2 2; i.e., a'J-u is orthogonal to |x|2k’Pm+1_2k(R") with

respect to ( , )m+1.

Also hm+1 and |x|2hm_1 are orthogonal to lezkpm+1_2k(R”) with respect to the

inner product ( , )m+1 for k 2 2. To see this let q E 77m+1_2k(R”). Then

<l$l2kqahm+1>m+1 : <l$l2k_2quhm+1>m—l :0

since hm+1 is harmonic. Proposition 3.3 implies that A2(|a'|2hm_1) = 0 and hence

(val2kqalrvl2hm—llmn = (lxlzk'4q,A2(I$lzhm—1)>m—3 =0-

Therefore :cju—hm+1—|:c|2lim_1 is orthogonal to |$|4?m_3(R”) with respect to ( , )m+1
and :cJ-u — hm+1 — |zc|2hm_1 = |a:|4hm_3 + + |as|2"‘hm+1_2;c E |$|47)m_3(R”), and (3.2)

is proved.

32

To show (3.3) we use (3.2) in the following way:

wine) = o<hm+1(x)+lxrhm-l(x))
= xjhm+1($) + |$12$jhm—1(~Tl
= amen) + lezama) + lemme) + lwlzbm—2(x))

= hm+2($) + l$|2hm($) + l$|4hm—2($)-

Hence (3.3) holds. [:1

Now we can prove the following theorem.

Theorem 3.5 For m _>_ 0 andj = 1,...,n the operator ngHIJ. maps Hm(B) into

Hm(B).

Proof: Let u E Hm(B). Then

HQHI,“ = H;,((1 — Q)(~’L‘ju)) = Q($j($ju — Q(a:,-u))).

By Proposition 3.4
one) = hale) + mum—1e),

where h,- E H;(B) for i = m — 1, m +1. Then by Theorem 1.12

 

 

m+l
oo.-axe) = 7271(3) gm + 2k) [B Zions/my) dV(y)
1 m+1 2
= MB, :02 + 2k) /B zk(a:,y><hm+1<y)+ lyl hm_1(y)) dV(y)
= hm+1(a:) + n + 20'” _1)hm_1(:c).

n+2m

We used the polar coordinates formula, Theorem 1.3 and (1.2).

Hence
it + 2(m — 1)
n + 2m

 

(:cju _ Q(xju))($) = (1:312 — film—1(3)

 

33

 

and
_ _ 2 n+2(m—1)
$j($jU-Q($W))(l‘) — 331(lxl - n+2m )hm-1(1')
= (W — "’ + 2“” ‘1))(hm(x>+ Ixrhmeo»,

 

n+2m

where h.- E H;(B) for i = m, m — 2. In the same way as above

Haifa-”(33) = Q($j(93ju - Q($ju)))(-’L‘)

 

 

m+2
= n_V1(—B_) Z (n + 210/8 Zk($.y)yj(ij(y) - Q(ij)(y)) dV(y)
k=0
1 "1+2 . 2 n + 2(m — 1)
= Wt?) g(n+2i)/sz(e,y)(|y| — n+2m )hm(y) dV(y)
1 m” 2 n '1' 2(m — 1) 2
WW??? gown/B Zk(rv,y)(ly| — n +2”, )Iyl hm—2<y>dV<z/>

 

1 n + 2(m — 1)
_ n—1+2+2m _
— (n + 2m)(/O r dr n + 2771

1
/ rn—1+2m dT)hm($)
O

 

 

 

 

 

+ (n + 2(m — 2))(/01rn-1+2m dr — n 22:27; 1) [01er dr)hm_2(a:)
+ (n + 2(m — 2))hm'-2($)(n +12m _ n +n2-I(-n2r; 1) n + 2in — 2)
= (n + 2m)(:+ 2m + 2) hm”).
Therefore 11’;ij (Hm(B)) C Hm(B). [I]

Since b2(B) = 692:0 Hm(B) and II;IH$,(Hm(B)) C Hm(B) it is enough to study

the restriction of the operators ngHxJ to Hm(B). Note that 11;}.ij is self-adjoint

so its restriction to Hm( B) has an orthonormal basis of eigenvectors.

Motivated by some calculations obtained with the aid of Mathematica, we have

a conjecture concerning this orthonormal basis of eigenvectors. It will be stated for

34

H;1H,,,, but it holds for all ngij, j = 1, ...,n, with an appropriate choice of basis
for Hm(B).

Conjecture 3.6 Let n > 2. Then the eigenvalues of the restriction of Hngx, to

Hm(B) are:
4(m—k)(m+n+k—3)
(2m + n — 4)(2m + n — 2)(2m + n)(2m + n + 2)’

 

for Is: 2 0,1, ..., m, with multiplicities
n + k — 2 n + k — 4
n — 2 _ n — 2 ’

where we define ( i > 2 0 forl < r. The corresponding eigenvectors are obtained
from the basis {K[D°’|:c|2‘"] : Ial = m, a1 = m — 1:, an 2 0 or 1, k = 0,...,m},

using the Gram-Schmidt Orthogonalization Process.
We can prove this conjecture for k = 0, 1.

Lemma 3.7 Let n > 2 and m 2 1. Then for :1: aé 0

|:1:|2D{"+1 |:c|2_" : —(2m + n — 2)::31Din Iarlz'" — m(m + n — 3)D’{’"1|:c|2‘". (3.4)

Proof: Induction on m. It is easy to see that (3.4) holds for m = 1. Suppose it is

true for m = k. Then we have
|:1:|2Df+1 lez‘" = —(2k + n -— 2):1:1Df|a:|2‘" — k(k + n — 3)Df‘1|a‘|2_n.
Differentiating with respect to 3:1 we get

23:10?“ IwI“ + lele‘Hlez’” = —<2x.~ + n _ 2)th?”

—(2lc + n — 2):L‘1Dic+1 |a:|2“" — kUC + n — 3)Df|:c|2""’.

35

Hence
M2191c+1 lez‘” = —(2k + n)$lDl°|$|2_” — (’8 +1)(k + n — 2)Df|x|2—",
and the lemma is proved. [:1
Corollary 3.8 Let n > 2 and m 2 1. Then
K[D;"+1|a:|2‘”] = —(2m + n — 2):c1K[D;n|:c|2_”] — m(m + n — 3)|:c|2K[D;’"1|:c|2"”].
Proof: Using the definition of Kelvin transform and Lemma 3.7 we get

KlDi"+‘l:v|2'”] ——— lxl2‘"Di"“l:v*|2‘"

 

=lxl2‘" (l$*|201”+1|$*|2'")

lfl="'|2
= |513|2“"+2(-(2m + n - ‘ZlfBin‘llez—n - m(m + n - 3)Dl”‘1|$*|2'”)

= ——(2m + n _ 2)e11<[ogn|x|?-n] _ m(m + n — 3)|:c|’~’1<[D;"*1|a:I2‘“J,

and the corollary is proved. D
Remark: Corollary 3.8 implies that

1

[(Dm 2—n:_____
“31 [1133' l 2m+n—2

(K[D;"+1|a;|2—"]+ m(m + n — 3)|1F|2K[Di”’1 |$|2""l)-
Now we can prove the following theorem.

Theorem 3.9 Let n > 2. Then K[D;nl:r|2‘”] is an eigenvector ofI-Ig‘,‘1 H1,1 with eigen-

value
4m(m + n — 3)
(2m + n — 4)(2m + n — 2)(2m + n)(2m + n + 2).

 

Proof: From Proposition 3.4 using (3.2) it follows that for u E Hm(B) we have

$11433) = hm+1(x) + l‘vlzhm—IC’B)

36

and
:clhm_1(:r) = hm(a:) + lxlzhm_2(:c), (3.5)
where h,- E Hm(B), i = m — 2, m — 1, m, m +1.

The proof of Theorem 3.5 shows that

* 4
H1" Hx,u(:c) — (2m + n)(2m + n + 2) hm($)-

 

Hence it is enough to show that in the case u(a:) = K[D;”|:c|2‘”] the corresponding
hm from (3.5) is given by

m(m+n—3)
(2m+n—4)(2m+n—2

 

)U.

From the remark after Corollary 3.8 it follows that

mm+n—3) , m_ _n
hm_.(a:)=— 257.“.-2 MD. liar 1.

 

Similarly, the remark after Corollary 3.8 implies that

 

 

 

m(m + n — 3) 1
hm- :: — — I' Dm 2’" 2hm_ . .
Therefore
m(m + n — 3) 1
[m = i 7
I (at) 2m+n—2 2m+n—4u($)
which proves the theorem. D

We can also show that 1([DT—IDJ-lrcl2‘fl, j = 2, ..., n are eigenvectors of H;1H1‘1

with the eigenvalue

4(m —1)(m + n —- 2)
(2m + n — 4)(2m + n — 2)(2m + n)(2m + n + 2).

 

To show this we will need the following lemmas.

Lemma 3.10 Let n > 2 and m 2 2. Then forj = 2, ...,n and a: 74 0

x101”“0jlm|2‘" = 4(IxIZDrDjlxI2-n + omelet-n

2m + n —
+(m —-1)(m + n — 4)D’1”—2D,-|a:|2"").

37

Proof: Let n > 2. From Lemma 3.7 we have
|x|2DT|m|2’” = —(2m + n — 4)::31D1n'1 Icclz‘" — (m —1)(m + n — 4)D§”‘2|a:|2‘".
Differentiating with respect to :13,- gives
2eDrIxI2-n + lamina-lei” = —(2m + n — 4)xlDT‘lDJ-lxl2‘"
—(m —1)(m + n — 4)D;”'2Dj|a:|2—n.

Therefore

—1
2m+n—4

+(m -1)(m + n - 4)Dl”_2Djlwl2'").

wIDT‘lesvt‘" = (IxIZDrDjlxr-n + 2ij{”|$|""”

and the lemma is proved.

Corollary 3.11 Let n > 2. Then forj = 2, ...,n,
——1

2m + n — 4

+(m —1)<m + n — 4>Ier<iDr-2DjIver-”1).

x11{[D;n-1Dj|x|2-n] : (I{[D;nDj|1L’l2—n] + 2$j1([D;n|fE|2—n]

Proof: From Lemma 3.10 we have

*m—' *—n _1 *m air—n *mx-n
$101 102193 12 = m(liv 1201 Djlx 12 +237le 15'? l2

+(m —1><m + n — 4)D{’“2Djlrv*|2’")-

The Kelvin transform gives

 

$1 -1 l :1:-
—K Dm‘lD- ' 2'” = K DmD‘ 2'" 2—-J—K Dm 2"”
lez i 1 2111 l 2m+n—4(|:z;|2 [ 1 Jlivl ]+ 19312 [ 1|:c| ]
+(m —1)(m + n — 4)K[D{”‘QDJ-|x|2‘"]).
Therefore
——1
:ciK[D’1"'lDJ-|:c|2'"] = ———————(K[D{”Dj|a‘|2_”] + 2a:,-K[D§”|:c|2‘”]

2m + n — 4
+(m — 1)(m + n — 4)|:I:|2K[D{”‘2Djleg—nD,

which finishes the proof.

38

Lemma 3.12 Let n > 2, m 2 2 andj = 2,...,n. Then fora: ¢ 0

m —n —1 m —n m— —n
ijl 13312 = m(lxl201 191151312 —m(m - 1)D1 20213312 )- (3-6)

Proof: Induction on m. It is easy to see that the statement holds for m = 2.

Suppose it holds for m = 1:. Hence we have

—n —1 ‘ —n — -n
ijflez = 'Qk—_i_';—_—2(l~"3|2DiDjli'3|2 — kfk ‘1)Di 20313312 )-

Differentiating with respect to $1 gives

—n —1 —n —n
$jDi+ll$|2 = 27_m(23310i0jl$l2 +l$lsz+1Djl$|2

—k(k —1)Df‘1D,-|x|2‘”).

Lemma 3.10 implies

—1 ( —2
2k+n—2 2k+n-—2
+k<k + n — 3>Df-‘D.IxI2-n> + IerrlDM-n

(|$|20i‘+le|$|2"" + 2x30?“ Ill/12’"

 

ijf+1|x|2-" =

—k(k —1)Df‘lDJ-l:v|2‘")-

Thus we have

 

 

 

4 .. —1 2k+n—4
1 __ _Dk+1 ‘ 2—n : , 2 k+1 , 2-n
2k(k + n — 3) + k(k —1)(2k + n — 2) k_, 2_n
- 2k+n_2 D1 DJlxl )'
In other words,
—n —1 ' —n ‘— —n
517J'Dic+ll~"3|2 = gig—+71%1513121)?”Dim2 - kfk +1lDi 10319312 )3

and the lemma is proved.

39

Corollary 3.13 Let n > 2 andj = 2, ...,n. Then

71 1 ’ m —'n
ij[D{"|:c|2‘] = *ml‘wi 19119712 1

m(m ‘ ” |x|2K[Di"‘2DJ-|w|2‘"]-

+2m+n—2

Proof: By Lemma 3.12

n- m :- —n —1 a: m an —n m— a: —n
33101 1‘” 12 = mm? 1201 Dill? 12 — m(m _1)Dl 203133 12 ).
The Kelvin transform gives

(L‘j m 2—n _1 1

|x|2
—m(m —1)1<[D{”‘ZDJ'|$|2‘”])-

KID?“ Di |$|2’"l

Therefore

—1

$jKlDinl$|2_nl = m(K[D{”D,-|x|2‘"] — m(m —1)|x|2K[D§"‘2D,-|:c|2‘”]),

which proves the corollary. [:1

Now we can prove the following theorem.

Theorem 3.14 Let n > 2 andj = 2, ...,n. Then 1([Din—1Dj|a:|2‘"] is an eigenvector

of H_,',‘;1H,:1 with eigenvalue
4(m —1)(m + n —2)
(2m + n — 4)(2m + n —— 2)(2m + n)(2m + n + 2).

Proof: As in the proof of Theorem 3.9, we know that for u E Hm(B)

:clu(:c) = hm+1(:c) + |:c|2hm_1(:c)

and

atlhm_1(a:) = hm(:c) + |$|2hm_2(:c),

40

where h.- E H:(B), i = m — 2, m — 1, m, m + 1. Also, the proof of Theorem 3.5

implies that
4

In 1: : hm
H1.) H ,u(:1:) (2m + n)(2m + n + 2)

 

(93)-
Hence it is enough to show that in the case u = [{[DT‘IDjIxIZ‘n] the corresponding
hm from (3.7) is given by

(m—1)(m+n—2) u
(2m+n—4)(2m+n—2)'

 

Using Corollary 3.11 and Corollary 3.13 we have

 

m— —n 1 m —n
1711(101 1Dj|ac|2 l = _——2m+n—4K[D1 D31342 l
2 1
— — K DmD- 2'"
2m+n—4( 2m+n—2 [1 3le ]
m(m “ 1) 2 m—2 , 2—n
—2m+n—2l$| K101 DJlxl l)

— (m ‘ ”(m + ” ‘ 4) lezKiDr-szlxr-"i.

 

 

 

2m + n — 4
Thus the corresponding hm_1 is
2m(m — 1) (m —1)(m + n — 4) _2 2_
hm_ = — — K Dm D- ”
1(a) ( (2m+n—4)(2m+n—2) 2m+n—4 ) [ 1 3137' ]

__2m(m—1)+(m—1)(m+n—4)(2m+n—2) m_2 -:c2‘"
_ (2m+n—4)(2m+n—2) KlDI DJ" ]

 

__(m—1)(m+n—2)

1' Dm_2D~. 2’".
(2m+n—2) ‘ll Jlrl l

 

Using Corollary 3.11 and Corollary 3.13 we get

_ (m-1)(m+7l—2) , m—2 , 2-7.
:clhm_1(a:) _ — 2m+n—2 :1:1[\[D1 0,le ]

 

(m-l)(m+n—2) 1

— 2m+n—2 (”2m+n—6

 

I&’[Di’“lDJ-I:c|2‘”]

M

 

 

 

2 1
— __—__—__I( Dm-lD, 2—n
2m+n—6( 2m+n—4 [1 Jlxl l
(Tn—1)(m—2) 2 —3 2-
KDm D- n
+ 2m+n—4 le [ 1 Jlxl l)
2m+n—6
Therefore
_. (m-1)(m+n—2) 1
2 m—l 2—n
+ )KlDi DleL‘I l

 

(2m+n—6)(2m+n—4)

(m —1)(m + n — 2)
(2m + n — 4)(2m + n -— 2)

Ix’iDr-lelxr-"L

 

which finishes the proof.

BIBLIOGRAPHY

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293-305

 

 

 

 

 

 
  

n.”

 

 

  

NICHIGQN STATE UN IV

312II III O[I II] IIIIII III II IlII