Boll. Un. Mat. Ital. B (7) 10 (1996), no. 4, 843-883.

1

Commutators of Singular Integrals

on Homogeneous SpacesMarco BRAMANTI and M. Cristina CERUTTI

Sunto. Sia uno spazio omogeneo, un operatore di integrale singolare, continuo da a Ð Ñ Ð Ñ Ð ÑX,d, K X X. _ _: :

per ogni e tale che il suo nucleo soddisfi opportune disuguaglianze puntuali. Sia ;p a BMO X− Ð"ß_Ñ − Ð Ñsi dimostra che il commutatore

C K,a f aKf K afÒ ÓÐ Ñ œ Ð Ñ

è continuo su per ogni e soddisfa la seguente stima:_: p − Ð"ß_Ñ

² C K,a f c p,K,X a fÒ ÓÐ Ñ ² Ÿ Ð Ñ ² ² ² ²: ‡ :

dove è la seminorma . Si mostra anche come applicare questo risultato ad alcuni esempi² † ² ‡ BMO

concreti di operatori di integrale singolare su spazi omogenei di misura finita o infinita, fornendo

condizioni sul nucleo di , sufficienti affinche l'operatore di integrale singolare sia ben definito e´O Ocontinuo su ._:

INTRODUCTION

A well-known result of 1976 by Coifman-Rochberg-Weiss (see [CRW]) is that, if is a Calderon-´K

Zygmund operator on and , then the _ ‘: 8Ð Ñ a BMO commutator−

C K,a f aKf K afÒ ÓÐ Ñ œ Ð Ñ

is a well defined linear continuous operator from into , for every and_ _: : p 1,− Ð _Ñ

² Ð ÑC K,a f c p,n,K a f .Ò ÓÐ Ñ ² Ÿ Ð Ñ ² ² ² ²: ‡ : 0.1

Over the years several extensions of this result have been proved. For instance, Chanillo [Cha] in '82

proved an analogous result for fractional integral operators; Bloom [Bl] in '85 proved a weighted estimate

for the commutator of the conjugate operator; Segovia and Torrea [ST1] in '89 generalized the above result

to vector valued functions and more recently (see [ST2]) they considered higher order commutators. In this

paper we consider singular integral operators of Calderon-Zygmund type, defined on a homogeneous space,´

and prove the Coifman-Rochberg-Weiss' theorem in this context. In another one [BC2] we deal with

commutators of integrals on homogeneous spaces.fractional

The motivation for this paper arose when, proving estimates for parabolic equations with _: VMO

coefficients, we needed to extend [CRW]'s result to parabolic Calderon-Zygmund operators (see [BC1]).

That of homogeneous spaces seems a natural setting for this problem since the basic real analysis tools, as

maximal function, sharp function and , can be defined in a natural way in this context; also, theBMO

related results, analogous to the Euclidean case, have been proved (see [CW1], [Bu]). Moreover, the present

extension of (0.1) to homogeneous spaces turns out to be the suitable tool to prove -estimates for some_:

ultraparabolic operators with coefficients. This problem is considered by Bramanti-Cerutti-ZQSManfredini in [BCM]. See also Remark 4.13 for other notes about applications of our results to problems

arising from P.D.E.'s.

In § 1-2 we discuss definitions of homogeneous space and of Calderon-Zygmund type operator, as´

well as recall known results. In § 2-3 we prove our main results, namely the estimate (0.1) for the_:

commutator, respectively in the case of an infinite measure space and in a finite measure one. It turns out

that, while the infinite measure case is fairly straightforward, the finite measure one is significantly

different. We want to stress that our definition of Calderon-Zygmund operator requires continuity: in_:

other words, our results say that, under suitable assumptions on the kernel and the space, the estimate on_:

the operator implies 0.1) for its commutator. So, if we are given a concrete singular kernel on aK Ðhomogeneous space, to apply this theory we need to know that an continuous operator is actually_:

defined. This problem is discussed in § 4. Namely we first discuss some general sufficient conditions on the

kernel under which -continuity of the singular integral can be assured, and then give specific examples_:

arising in the applications in which our theorem (both in the infinite and finite measure case) can be

applied.

Finally, we wish to point out that our results are obtained under quite general assumptions on the

space , namely does not need to be endowed with a group structure neither the measure of a ball in X X X

needs to behave like a fixed power of the radius.

2

Acknowledgements. We wish to thank Prof. E. Fabes for some helpful discussions.

1. SOME BASIC FACTS ABOUT HOMOGENEOUS SPACES

Several different definitions of homogeneous spaces can be found in the literature. Here we refer

particularly to [CW1,2], [MS], [Bu].

Definition 1.1. Let be a set. A function is called if:X d:X X 0, quasidistance ‚ Ä Ò _Ñi d x,y 0 x yÑ Ð Ñ œ Í œ ;

ii d x,y d y,xÑ Ð Ñ œ Ð Ñ;iii c 1 x, y, z XÑ   −there exists a constant such that for every .

d x,y c d x,z d z,y .Ð Ñ Ÿ Ò Ð Ñ � Ð ÑÓ. (1.1)

At the end of section 4.a we will show that, for the results contained in this paper, condition ) can be33weakened to require only "quasisymmetry", instead of symmetry. (See Remark 4.6)

If is a set endowed with a quasidistance, the : (forÐX,d balls B x B x,r y X d x,y rÑ Ð Ñ ´ Ð Ñ œ − Ð Ñ �< š ›x X r 0 X X − �and ) form a base for a complete system of neighborhoods of , so that is a Hausdorff space.

Note that the balls are in general open sets; if they are, then they form a base for the topology of .not X

Definition 1.2. We say that is a if:ÐX,d, homogeneous space.Ñi X dÑ is a set endowed with a quasidistance , such that the balls are open sets in the topology induced by

d;

ii X doubling conditionÑ is a positive Borel measure on , satisfying the :.

0 B x c B x � Ð Ð ÑÑ Ÿ † Ð Ð ÑÑ � _. .#< <. (1.2)

for every some constant x X, r 0, c 1.− � �.

The numbers in (1.1)-(1.2) will be called and we will write for a constantc , c constants of X c X. . Ð Ñdepending on the constants of X.

We note that every homogeneous space is separable. This follows, for instance, from Theorem (1.2) of

[CW].

A fundamental result about quasidistances is the following one:

Theorem 1.3. (Macias-Segovia, [MS]).

Let be a quasidistance on a set . Then there exists a quasidistance on such that:d X X$

Ði d c , c x, y XÑ − and are equivalent, that is there exist constants such that for every :$ " #

c x,y d x,y c x,y" #$ $Ð Ñ Ÿ Ð Ñ Ÿ Ð Ñ;

Ð − Ðii 0,1 c 0 x,Ñ Ó �$ is "locally Holder continuous"; more precisely there exist and such that for every ¨ !

y, z X− :

± Ð Ñ Ð Ñ ± Ÿ Ð Ñ Ò Ð Ñ � Ð ÑÓ$ $ $ $ $x,z y,z c x,y x,z y,z .! !"�

If is a distance, is Lipschitz, and the previous result is trivial.d d

We say that if has bounded support andf f− V(!

² ² ´ � _Á

f sup .x y

(± Ð Ñ� Рѱ

Ð Ñf x f y

d x,y (

If is a homogeneous space with a regular measure, then the space is dense in for everyÐ ÑX,d,. . V _(

!:

( − Ð0,1], . (This follows from Thm. 1.3).p 1,− Ò _Ñ We will use also the following space of test funcitons

W _ f X f œ − Ð Ñš _ ± the support of is bounded .›Note that is dense in for every W _: p 1, .− Ò _Ñ The definition of standard real analysis tools, such as the , the ,maximal function Mf sharp function f #

the seminorm and the space, naturally carries over to this context, namely:BMO f BMO ² ² ‡

Mf x sup f y d yx B

Ð Ñ œ ± Ð Ñ ± Ð Ñ−

'B

. ;

3

where the is taken over all balls containing and ;sup x ... ... 'B B

1B

œ.Ð Ñ

'f x sup f y f d y

x B

#B

Ð Ñ œ ± Ð Ñ ± Ð Ñ−

' F .

where ;f f x d xF œ Ð Ñ Ð Ñ'B

.

² ² œ Ð Ñ œ ± Ð Ñ ± Ð Ñf sup f x sup f y f d yx B

‡ F#

B' . ;

BMO f X f´ − Ð Ñ ² � _ : .š _"69- ‡² ›

We recall three results related to these concepts, which have been proved in the context of

homogeneous spaces in [CW1], [Bu].

Theorem 1.4. ( ).Maximal Inequality

For every there exists a constant such that for every :p 1, c X,p f− Ð _Ó Ð Ñ − _:

² ² Ÿ ² ²Mf c f .: :

Theorem 1.5. ÐJohn-Nirenberg Lemma .ÑFor every there exists a constant such that for every every ball :p 1, c X,p f BMO, B− Ò _Ñ Ð Ñ −

Š ‹' ± Ð Ñ ± Ð Ñ Ÿ ² ²B

f x f d x c f .F ‡:

"Î:

.

Theorem 1.6. ÐSharp Inequality .ÑFor every there exists a constant such that for every :p 1, c X,p f− Ò _Ñ Ð Ñ − _:

if :.Ð Ñ œ _ ² ² Ÿ ² ²X f c f: :#

X f f c f .if :.Ð Ñ � _ ² ² Ÿ ² ²'X

#: :

Moreover, the following two lemmas about functions hold:BMO

Lemma 1.7. Let and . Then for every ball and every positive integer :a BMO M 1 B x j− � Ð Ñ<

± ± Ÿ † ² ²a a c j aF F ‡Q <4 <

where depends on and on the dubling constant .c M c.

Proof. ± ± œ ± Ð Ñ Ð Ñ ± Ÿa a a x a d x F FQ< <' Š ‹

B< Q<F .

Ÿ ± Ð Ñ ± Ð Ñ Ÿ ² ² a x a d x c a .cB B *.Ð ÑQ< Q<

'FQ<

.

Hence:

± ± Ÿ ± ± � ± ± � á � ± ± Ÿ † ² ²a a a a a a a a c j aF F F F F F F FQ < Q < Q < Q <4 < < # 4 4Q< Q< -1 * . �

Lemma 1.8. Let and let:f BMO, −

f x

f x if f x n

n if f x n

n if f x n8Ð Ñ œ

Ð Ñ ± Ð Ñ ± ŸÐ Ñ  

Ð Ñ Ÿ

ÚÛÜ

Then and:f BMO 8 −

² f c f8 ‡ ‡² Ÿ ² ²

with an absolute constant.c

Finally, we recall the following result about homogeneous spaces of finite measure. For sake of

completeness we include the proof, having been unable to find a written reference.

Lemma 1.9. Let be a homogeneous space. Then if and only if is bounded.Ð Ñ Ð Ñ � _X,d, X X. .

Proof. We have to prove that if then is bounded (the reverse follows from the definition of.Ð Ñ � _X X

doubling condition).

4

By contradiction. x X x r d x ,x . Let be a fixed point of and a sequence of points such that By! 5 5 ! 5´ Ð Ñ Å _the triangle inequality we can prove that:

B x , B x , B x , B x , HrÐ Ñ � Ð Ñ Ð Ñ © Ð Ñ! 5 ! 5 5r r r

2c 2c 2c5 5 5

. . .œ g Ð"Þ Ñ and 3

with H c .œ " �."%-Š ‹

.

Then so that, by (1.3), But then, by doubling and (1.3), B x , X B x , 0. .Š ‹Ð Ñ Ð Ñ Ä! 5r r

2c 2c5 5

. .Å Ð Ñ.

. . .Š ‹ Š ‹ Š ‹B x , B x , Hr c X B x , 0Ð Ñ Ÿ Ð Ñ Ÿ Ð Ñ † Ð Ñ Ä! 5 5 5r r

2c 2c5 5

. .

so that 0, contradiction. .Ð ÑX œ �

2. COMMUTATORS OF CALDERON-ZYGMUND OPERATORS ON HOMOGENEOUS SPACES

Definition 2.1. We say that is a Calderon-Zygmund operator ( operator) on the homogeneous space´K CZ

Ð ÑX,d,. if:

i K: X X p 1,Ñ Ð Ñ Ä Ð Ñ − Ð _Ñ is linear continuous for every ;_ _: :

ii k:X X f x sprt fÑ ‚ Ä − Â there exists a measurable function such that for every , for a.e. :‘ W

Kf x k x,y f y d yÐ Ñ œ Ð Ñ Ð Ñ Ð Ñ'X

. ;

iii k pointwise Hormander conditionÑ the kernel satisfies the following :

there exist constants such that for every ,c 0, 0, M 1 x X, r 0, x B x , y B xO ! < ! Q< !� � � − � − Ð Ñ Â Ð Ñ"

± Ð Ñ Ð Ñ ± Ÿ †k x ,y k x,y ! Ð Ð ÑÑ Ð ÑÐ Ñc

B x ;y d x ,y

d x ,xO

! !

!

.

"

" (H1)

where, is the ball with center and radius . We will write for a constant depending onB x;y x 2d x,y c p,KÐ Ñ Ð Ñ Ð Ñthe number , the norm of and the constants in (H1). The following holds:p 1, K− Ð _Ñ _:

Proposition 2.2. Condition (H1) implies the following : there exist constantsintegral Hormander condition

A 1, B 0 x , x X� � −such that for every :!'d x ,y A d x ,xÐ Ñ  Ð Ñ !

! !± Ð Ñ Ð Ñ ± Ð Ñ Ÿk x ,y k x,y d y B.. (H0)

(Condition (H0) is assumed in [CG], [CW1] to prove the fundamental result about continuity of an_:

integral operator which is continuous on . This result will be recalled and applied in § 4)._#

Proof. Fix and apply (H1) with :x , x X r 2d x ,x! !− œ Ð Ñ

'd x ,y 2M d x ,xÐ Ñ  Ð Ñ !

! !± Ð Ñ Ð Ñ ± Ð Ñ Ÿk x ,y k x,y d y .

Ÿ † Ð Ñ œ c d x ,x O ! Ð Ñ  Ð ÑÐ Ñ

Ð Ð Ñц Ð Ñ" .

.'d x ,y 2M d x ,x

d y

B x ;y d x ,y! ! ! !"

œ † Ð Ñ Ÿ c d x ,x O !4œ"

_

Ð ÑŸ Ð ÑŸ Ð ÑÐ Ñ

Ð Ð Ñц Ð Ñ" .

.!'

2 Md x ,x d x ,y 2 M d x ,x

d y

B x ;y d x ,y4 4�"! ! ! ! !

"

Ÿ † Ð Ñ œ † Ð Ñ c d x ,x c c M, .O ! O4œ"

_

Ò Ð ÑÓ" ! 1

2 Md x ,x4!

" " �

Examples of operators on homogeneous spaces will be discussed in detail in § 4. For a operatorCZ CZ

K, consider the commutator

C K,a f aKf K afÒ ÓÐ Ñ œ Ð Ñ

where (When is implicitly understood, we shall write more briefly: ). Our goala BMO. K T f C K,a f− œ Ò ÓÐ Ñ+

is to prove the following estimate:

² T f c p,K,X a f+ : ‡ :² Ÿ Ð Ñ ² ² ² ² (2.1)

for every . The above definition of is, up to now, purely formal: if and ,p 1, C K,a f f a BMO− Ð _Ñ Ò ÓÐ Ñ − −_:

K af a BMOÐ Ñ − may not be well defined. However, if , by the John-Nirenberg lemma (Theorem 1.5),

5

a p 1, . f a BMO T f − − Ò _Ñ − −_ W:69- +for every Therefore if and , is well defined and belongs to for_

:69-

every ; if, moreover, , then for every . The following proposition is ap 1, a T f p 1,− Ò _Ñ − − − Ð _Ñ_ __ :+

consequence of the above remark, the density of in and Lemma 1.8.W _:

Proposition 2.3. If (2.1) holds for every and every , then for every the operator a f a BMO T− − −_ W_+

can be continuously extended to for every , and (2.1) holds for every , _: p 1, a BMO f .− Ð _Ñ − − _:

Let us now state the first main result.

Theorem 2.4. Let be a operator; then for every there exists a constant suchK CZ p 1, c c p,K,X− Ð _Ñ œ Ð Ñthat for every every :a , f− −_ W_

² Ð Ñ ² Ÿ ² ² ² ²T f c a f .+ : ‡ :#

Proof. We shall prove the following inequality:

for every there exists a constant such thatr 1, c r,K,X − Ð _Ñ Ð Ñ

± Ð Ñ Ð Ñ ± Ÿ † ² ² Ð ± ± ÑÐ Ñ � Ð ± ± ÑÐ Ñ −T f x c a M Kf x M f x x X.– – – –+ ‡

< <"Î< "Î<

# ŸŠ ‹ Š ‹ for every (2.2)

From (2.2) the result follows by the continuity of and the maximal inequality (Theorem 1.4). To proveK

(2.2), let us fix a ball containing , and let be any point of Let us write:B B x x x B . –$ $ $œ Ð Ñ!

T f x K af x a x K f x + Ð Ñ œ Ð ÑÐ Ñ Ð Ñ Ð ÑÐ Ñ œ

œ Ð Ñ Ð Ñ � Ð Ñ Ð Ñ Ð Ð Ñ Ñ † Ð ÑÐ Ñ ´ K a a f x K a a f x a x a K f x Š ‹ Š ‹B B B X\B B$ $ $ $ $; ;

Q Q

´ Ð Ñ � Ð Ñ � Ð Ñ A x B x C x . (2.3)

Following [To], p. 418, we get, by Theorem 1.5, Lemma 1.7 and the estimate on :_: K

' Š Š ‹ ‹ ± Ð Ñ ± Ð Ñ Ÿ Ð Ñ ² ² † ± Ð Ñ ± Ð ÑB *$

C x C d x c X,r a M K f x–B$. <

"Î<

; (2.4)

' Š Š ‹ ‹ ± Ð Ñ ± Ð Ñ Ÿ Ð Ñ ² ² ± ± Ð ÑB *$

A x A d x c r,K,X a M f x .–B$

. <"Î<

(2.5)

To bound the middle term, observe that' ' ± Ð Ñ ± Ð Ñ Ÿ ± Ð Ñ Ð Ñ ± Ð ÑB B$ $$

B x B d x 2 B x B x d xF !. . (2.6)

and that, by H1Ð Ñ

± Ð Ñ Ð Ñ ± Ÿ ± Ð Ñ Ð Ñ ± ± Ð Ñ ± ± Ð Ñ ± Ð Ñ ŸB x B x k x,y k x ,y a y a f y d y ! ! FÐ Ñ'X B x\ Q !$ $

.

Ÿ † Ð Ñ Ð Ñ Ÿ c d x ,x d y O ! Ð Ñ

± Ð Ñ� ±± Рѱ

Ð Ð Ñц Ð Ñ"

.'X B x

a y a f y

B x ;y d x ,y\ Q ! !!

F

$

$" .

Ÿ † Ð Ñ † c d yO Ð Ñ

± Ð Ñ� ±

Ð Ð Ñц Ð Ñ

"Î<

$ .".� '

X B x

a y a

B x ;y d x ,y\ Q ! !!

F<w

w

$

$"

† Ð Ñ� 'X B x

f y

B x ;y d x ,y\ Q ! !!

<

$"Ð Ñ

± РѱРРÑц Ð Ñ

"Î<

d y ..

. (2.7)

Moreover:

'X B x

f y

B x ;y d x ,y\ Q ! !!

<

$"Ð Ñ

± РѱРРÑц Ð Ñ d y

..Ð Ñ œ

œ Ð Ñ Ÿ d y!'4œ"

_

Ÿ Ð Ñ�± Рѱ

Ð Ð Ñц Ð ÑM d x ,y M

f y

B x ;y d x ,y4 4�"!

<

! !$ $ . " .

6

Ÿ † ± Ð Ñ ± Ð Ñ Ÿ! '4œ"

_

Ò Ó Ð Ð † ÑÑ Ð Ñ< f y d y 1 1

M B x , 2 M B x ,M4 4! !

4�"$ . $ $" .

(by the doubling condition) (2.8) c c ,M, M f x .–Ÿ Ð Ñ † † Ð ± ± ÑÐ Ñ. $" 1

"<

Similarly:

'X B x

a y a

B x ;y d x ,y\ Q ! !!

F<w

$

$"Ð Ñ

± Ð Ñ� ±

Ð Ð Ñц Ð Ñ d y .

.Ð Ñ Ÿ

Ÿ Ð Ñ † † ± Ð Ñ ± Ð Ñ Ÿc c a y a d y . $ $! '4œ"

_

Ò Ó Ð Ñ F<1

M B x ,M4!

4�"

w

" $.

(by Theorem 1.5 and Lemma 1.7)

Ÿ Ð Ñ ² ² œ Ð Ñ † † ² ² c r,K,X a c r,K,X a .1 11 j

M$ $" " "! Š ‹4œ"

_< <‡ ‡

w w<w

4 (2.9)

Finally (2.2) follows from (2.3) (2.9), and the theorem is proved. �

As an immediate consequence of Theorems 2.4, 2.3 and 1.6, we obtain the -estimate (2.1) in the_:

case :.Ð Ñ œ _X

Theorem 2.5. Let be a homogeneous space with , a operator, . Then for everyX X K CZ a BMO.Ð Ñ œ _ −p 1, T − Ð _Ñ the operator can be continuously extended to , and+ _:

² T f c p,K,X a f .+ : ‡ :² Ÿ Ð Ñ ² ² ² ²

Let us note that in the case , applying Theorem 1.4 to Theorem 2.4 we get:.Ð Ñ � _X

² T f T f c p,K,X a f+ + : ‡ : ² Ÿ Ð Ñ ² ² ² ²'X

(2.10)

for every In the next section we will show how to get the desired estimate froma , f , p 1, .− − − Ð _Ñ_ W_

this result.

3. ESTIMATE ON THE COMMUTATOR OF CALDERON-ZYGMUND OPERATORS

ON FINITE MEASURE HOMOGENEOUS SPACES

Throughout this section will be a finite measure homogeneous space, unless otherwise stated. Let usX

state the main result we are going to prove:

Theorem 3.1. Let be a homogeneous space and a operator on such that:Ð ÑX,d, K CZ X.

3Ñ Ð\Ñ � _ and is regular;. .

ii k k k x,y k y,xÑ Ð Ñ œ Ð ÑÑ the kernels and satisfy condition (H1) (where ;‡ ‡

iii kÑ satisfies the growth condition:

± Ð Ñ ± Ÿ −k x,y x, y X ccB x;y.Ð Ð ÑÑ for every , some constant ; (3.1)

Then for every , , the operator can be continuously extended to anda BMO p 1, T − − Ð _Ñ + _:

² T f c a f+ : ‡ :² Ÿ ² ² ² ²

with depending on , the norm of and the constants of and appearing in conditions (H1),c p, X K k k_: ‡

(3.1).

Let us sketch the line of the proof of Theorem 3.1. By (2.10), to get the result we need to estimate

' Ð ÑÐ Ñ Ð ÑX

T f x d x .+ .

Since the function cannot be written explicitely using our "abstract" definition of operator, we shallT f CZ+

consider suitable "approximating" integral operators , and estimate the mean value of the commutators K C%

Ò ÓÐ ÑK ,a f K K ~

% %. Finally, we shall prove that converges, in an appropriate weak sense, to some operator which

is not necessarily , but has the same commutator.K

The first step is the following:

Theorem 3.2. Suppose that and satisfy condition (H1), is a bounded function and k k k x,y d y‡ 'X± Ð Ñ ± Ð Ñ.

the same is true for Set, for :k . f‡ − W

7

Kf x k x,y f y d yÐ Ñ œ Ð Ñ Ð Ñ Ð Ñ'X

.

K f x k x,y f y d y .‡ ‡Ð Ñ œ Ð Ñ Ð Ñ Ð Ñ'X

.

Let , and be the commutators of and , respectively. Then, for every therea T T K K p 1,− − Ð _Ñ__ ‡ ‡+ +

exists a positive constant , depending on , , the norms of and , and the constants appearing inc p X K K_: ‡

condition (H1) for and , such thatk k‡

² T f c p,K,X a f+ : ‡ :² Ÿ Ð Ñ ² ² ² ² .

Proof. By (2.10) we have that for every :a , f− −_ W_

² ² Ÿ ² ² ² ² � Ð Ñ ± Ð ÑÐ Ñ Ð Ñ ±T f c a f X T f x d x . + : ‡ : +"Î:�". .'

X(3.2)

Therefore, let us consider:

' ' 'X X XÐ ÑÐ Ñ Ð Ñ œ Ð Ñ Ð Ñ Ò Ð Ñ Ð ÑÓ Ð Ñ Ð ÑT f x d x d x k x,y a x a y f y d y .+ . . .

Since are bounded functions and , the integral on the right hand side of the last equationa, f X.Ð Ñ � _converges absolutely and equals

Ð Ñ Ð Ñ Ð Ñ Ò Ð Ñ Ð ÑÓ Ð Ñ œ Ð Ñ Ð Ñ Ð Ñ' ' 'X X Xf y d y k x,y a y a x d x f y T 1 y d y .. . .+

‡

Hence:

± Ð ÑÐ Ñ Ð Ñ ± Ÿ ² ² ² ²'X

T f x d x f T 1+ : :+‡. w. (3.3)

Since also satisfies (H1), (2.10) holds for , too, andk T‡ ‡+

² ² Ÿ ² ² Ð Ñ � Ð Ñ ± Ð ÑÐ Ñ Ð Ñ ±T 1 c a X X T 1 x d x .+ +‡ "Î: �"Î: ‡

: ‡ww

. . .'X

(3.4)

Again by Fubini's theorem, we get:

' ' 'X X XÐ ÑÐ Ñ Ð Ñ œ Ð ÑÐ Ñ Ð Ñ Ð ÑÐ Ñ Ð ÑT 1 x d x Ka y d y K a x d x .+‡ ‡. . . (3.5)

By Holder and the estimate on :¨ _: K

± 'XÐ ÑÐ Ñ Ð Ñ ± Ÿ ² ² † Ð Ñ Ÿ ² ² † Ð ÑKa y d y Ka X c a X .. . .: :

"Î: "Î:w w(3.6)

Without loss of generality, we can assume

'Xa x d x 0 Ð Ñ Ð Ñ œ.

since the commutator we are estimating is not affected by adding a constant to . We claim that a

² ² Ÿ ² ² Ð Ña c a X .: : ‡"Î:.

To see this, recall that, by Lemma 1.9, is bounded, so it coincides with some ball . Then, by TheoremX B

1.5,

' 'X B± Ð Ñ ± Ð Ñ œ Ð Ñ † ± Ð Ñ ± Ð Ñ Ÿ Ð Ñ ² ²a x d x X a x a d x c X a: :

F:‡. . . . .

Therefore from (3.6) we get:

Ka y d y c a X± Ð ÑÐ Ñ Ð Ñ ± Ÿ ² ² Ð Ñ'X

. .‡ (3.7)

An analogous estimate holds for the second term in (3.5) and from (3.2) (3.7) the result follows. �

The second step is the construction of suitable "truncated operators" to which we will apply the

previous theorem. Since we will need this construction also in § 4, while considering general homogeneous

spaces (not necessarily of finite measure) we drop for a moment the assumption By Theorem.Ð Ñ � _X .

1.3, we can endow with a quasidistance equivalent to and locally Holder of exponent . Consider the¨X d$ !

functions : ( ) defined by:<3 Ò _Ñ Ä Ò Ó œ0, 0,1 i 1, 2

<"Ð Ñ œ! Ÿ " " " � �"  

t

t

t t 2

t 2.

ÚÛÜ

for

for

for

<#Ð Ñ œ" Ÿ � �!  

t

t 2

3 t 2 t 3

t 3.

ÚÛÜ

for

for

for

8

For any set% − Ð Ñ0,1

<%Ðt 0 t 2

t t 2Ñ œ

Ð Ñ Ÿ Ÿ

Ð Ñ  œ<

< %

"

#

t%

for

for

k x,y k x,y x,y .% %Ð Ñ œ Ð Ñ † Ð Ð ÑÑ< $ (3.8)

Lemma 3.3. Suppose 0,1 is a number for which Theorem 1.3 ( ) holds for , and satisfies! − Ð Ó ii X K

condition (H1) with exponent ; moreover, suppose satisfies the growth condition (3.1). Then defined" k k%above satisfies condition (H1), with constants independent of . More precisely, the exponent appearing in%

(H1) is , .# ! "œ Ð Ñmin

Remark 3.4. Observe that if satisfies (H1) and (3.1) with respect to and , then it satisfies similark d .

conditions with respect to and (for any quasidistance equivalent to ); in particular, (H1) holds with$ . $ d

the same exponent . Therefore to prove Lemma 3.3, we don't need to distinguish and : henceforth " $d d

will be the quasidistance of that satisfies a Holder condition of exponent (recall Theorem 1.3).¨X !

Proof of Lemma 3.3. Let us start by noting that the triangle inequality implies that if andx B x− Ð Ñ< !

y B x c c c MÂ Ð ÑQ< ! " # ., there exist two positive constants , depending on , such that

c d x,y d x ,y c d x,y" ! #Ð Ñ Ÿ Ð Ñ Ÿ Ð Ñ Ð Ñ3.9

(assuming ).M c� .

Now, let .x X, r 0, x B x , y B x! < ! Q< !− � − Ð Ñ Â Ð Ñ

± Ð Ñ Ð Ñ ± Ÿk x ,y k x,y % %!

Ÿ ± Ð Ð ÑÑÒ Ð Ñ Ð ÑÓ ± � ± Ð ÑÒ Ð Ð ÑÑ Ð Ð ÑÑÓ ± œ � d x ,y k x ,y k x,y k x,y d x ,y d x,y I II.< < <% % %! ! !

By definition of and (H1)<%

I .Ÿ †c

B x ;y d x ,y

d x ,xO

! !

!

.Ð Ð ÑÑ Ð ÑÐ Ñ"

"

Also, observe that the quantity

± < <% %Ð Ð ÑÑ Ð Ð ÑÑ ± ´d x ,y d x,y A!

can be nonzero only if one of the following cases occurs:

1 case: one of the numbers and is in => d x ,y d x,y 0, 2 .Ð Ñ Ð Ñ Ñ! Ð %

d x ,y d x,y 2 , 2 .2 case: one of the numbers and is in , the other is 8. Ð Ñ Ð Ñ Ð Ñ! % %3%

�

1 case.=>

A Ÿ Ÿ±d x ,y d x,yÐ Ñ� Рѱ!

%(by Theorem 1.3)

Ÿ (by (3.9))c d x ,x † Ð Ñ † Ÿ!Ò Ð Ñ Ð ÑÓ!

%

d x ,y d x,y!"!

Ÿ † c .d x ,x

d x ,y

Ð ÑÐ Ñ

!

!

!

!

2 case.8.

A c d x ,x d x ,y d x,x Ÿ † Ð Ñ Ò Ð Ñ � Ð Ñ Ó Ÿ% ! ! !"� "�! ! ! (by (3.9))

Ÿ † c .d x ,x

d x ,y

Ð ÑÐ Ñ

!

!

!

!

Therefore

II Ÿ †cB x ;y d x ,y

d x ,x

.Ð Ð ÑÑ Ð ÑÐ Ñ

! !

!!

! ,

and the theorem is proved. �

Let's now consider the operators:

K f x k x,y f y d y .% %Ð Ñ œ Ð Ñ Ð Ñ Ð Ñ'X

. (3.10)

Note that condition 3.1 and the doubling condition implyÐ Ñ

' 'X d x,y

d y

B x;y± Ð Ñ ± Ð Ñ Ÿ Ÿ Ð Ñk x,y d y c x% %

.

.. %� Р�

Ð ÑÐ Ð ÑÑ3

% uniformly in

9

and the same is true for In particular, for and are well defined and bounded.k . f , K f K f ‡ ‡− W % %

The following continuity estimate on with constants independent of , may be obtained_ %: K%

adapting to our context the proof of a known result by M. Cotlar, regarding Calderon-Zygmund operators´

on (see for instance [Me], p. 241).‘8

Theorem 3.5. Let be a operator; let the kernel satisfy (3.1); let be defined as in (3.8), (3.10).K CZ k K %Then for every there exists a constant independent of such that:q 1, c− Ð _Ñ %

± Ð Ñ ± Ÿ Ð ÑÐ Ñ � Ð ± ± ÑÐ Ñ � Ð Ñ − −K f x c M Kf x M f x Mf x x X, f% ŸŠ ‹; ;"Î;

for every ._

Corollary 3.6. Under the same assumptions of Theorem 3.5, let

K f x sup K f x .æ Ð Ñ œ ± Ð Ñ ±� !%

%

Then is continuous from to , for every . K p 1,æ _ _: : − Ð _Ñ

The following proposition is now a consequence of Theorem 3.2, Lemma 3.3 and Corollary 3.6.

Proposition 3.7. Let be a operator; let the kernels and satisfy condition (H1); moreover, supposeK CZ k k‡

k satisfies (3.1). Then the following estimate holds

² Ò ÓÐ Ñ ² Ÿ ² ² ² ²C K , a f c a f% : ‡ :

for every , with depending on and the constants involved in the assumptions,p 1, a , f , c p− Ð _Ñ − −_ W_

but not on .%

Next, we need to specify in what sense the operators approximate .K K%

Theorem 3.8. Let and be as in Proposition 3.7, and assume that the measure is regular. Then thereK K% .

exist a sequence and a function such that for every , every we have:% _ W8_Ä − − Ð _Ñ −0 b p 1, f, g

' ' 'X X Xf K g f Kg bfg n%8 Ä � Ä_for .

A similar result has been proved for Calderon-Zygmund operators in (see for instance [M], p.´ ‘8

227). The proof of this result, thought, uses tools charachteristic of (Fourier transform, dyadic‘8

decompositions etc.) and therefore it cannot be repeated in our context. Our proof, on the other hand, relies

entirely on topological and measure theorethical properties.

Proof. We know that:

² ² Ÿ ² ²K f c f% : : : (3.11)

for every , and so for every with independent of .p 1, f , f , c− Ð _Ñ − −W _: : %

Observe that the space is separable. To see this, note that since is separable (as we noted after_: X

Definition 1.2) and the balls form a base for the topology, has a countable base. Also, the measure isX

regular, hence Borel sets can be approximated, in measure, with open sets, so that the Borel -algebra is5

separable, and is separable for every ._: p − Ò"ß_Ñ From (3.11), by compactness, we can construct a sequence such that converges weakly in%8 Ä 0 K f %8

_ _: :, for , for every belonging to some dense subset of . But then, again by (3.11),n fÄ_

' 'X X g K f g Kf

~%8 Ä (3.12)

for in particular for every with linear continuous.every f , g , f, g , K:~

− − − Ä_ _ W _ _: : : :w

In order to study the relationship between and , let such that .K K f, g dist sprt f, sprt g 0~

− Ð Ñ   �W %!Then:

' ' 'X sprt f sprt g f K g x d x f x d x k x,y g y d y Ð ÑÐ Ñ Ð Ñ œ Ð Ñ Ð Ñ Ð Ñ Ð Ñ Ð Ñ œ% %8 8. . .

(for every %8 #� %! Ñ

œ Ð ÑÐ Ñ Ð Ñ'X f Kg x d x ..

Taking the limit for , we havenÄ_

' 'X X f Kg f Kg.~

œ (3.13)

10

Now fix , let be its support. Set ; then \ . Ifg C C x X: dist x,C C X C− œ − Ð Ñ   œW 8 88

š › -1n

E C E f© œ8 I, a Borel set, taking in (3.13) we get;

' 'E EKg Kg~

œ ,

i.e. Kg Kg X C~œ a.e. in \ . Therefore the operator

K K K~

" ´ ,

is , that islocal

for every for a.e. g , K g x 0 x sprt g.− Ð Ñ œ ÂW "

Let us now show that is the multiplication by a bounded function. Let be a Borel set, anK E C " 8

increasing sequence of closed sets and a decreasing sequence of open sets such thatA8.Ð Ñ Ä © ©A C 0, C E A .8 8 8 8\ Set:

b x K 1 x K x K x K xÐ Ñ ´ Ð ÑÐ Ñ œ Ð ÑÐ Ñ � Ð ÑÐ Ñ � Ð ÑÐ Ñ" " G " "E G E; ; ;8 8 8 8-\ (3.14)

and take limits in (3.14) for Since \ in , and a subsequencen . A C 0, K 0 Ä_ Ð Ñ Ä Ä. ;8 8 " E G8 8\ _:

converges a.e. Since in , in and a subsequence converges a.e. Moreover; ; ; ;G I " G " I8 8 K K Ä Ä_ _: :

ÐK x 0 x E K x E" "E; 8- ÑÐ Ñ œ − −for a.e. , since is local. Then for a.e. we have:

Ð ÑÐ Ñ œ Ð ÑK x b x ." I;

On the other hand for a.e. , and therefore for a.e. . This impliesÐ ÑÐ Ñ œ  РÑÐ Ñ œ ÂK x 0 x E K x 0 x E" G " I; ;8

that for every Borel set E

Ð ÑÐ Ñ œ Ð Ñ † Ð ÑK x b x x" I I; ;

and, by linearity, an anologous identity holds for every simple function and so, by density, for every function

in :W

Ð ÑÐ Ñ œ Ð ÑÐ Ñ œ Ð Ñ † Ð Ñ −Kf Kf x K f x b x f x f .~

" for every (3.15)W

Moreover since for every b f, g− __ − W

± Ð ÑÐ Ñ Ð Ñ ± œ ± Ð ÑÐ Ñ Ð Ñ ± Ÿ ² ² ² ²' 'X Xbfg x d x g K f x d x c f g . ." : :w ,

and, taking and letting we have:h , h 0, f h , g h −   œ œW "Î: "Î:w

± Ð ÑÐ Ñ Ð Ñ ± Ÿ ² ²'Xbh x d x c h. " (3.16)

From (3.12), (3.15), (3.16) the theorem follows. �

The previous theorem can also be read as a "representation theorem for operators": if is a CZ K CZ

operator, there exist a sequence and a function such that for every we have:%8 Ä 0 b f, g − −_ W_

' ' ' 'X X X XÐ ÑÐ Ñ Ð Ñ œ Ð Ñ Ð Ñ Ð Ñ Ð Ñ Ð Ñ � Ð ÑÐ Ñ Ð Ñ

8f Kg x d x lim f x d x k x,y g y d y bfg x d x .. . . .%8

In other words, the "abstract" operator allows the above "concrete" representation; note, however, that ifK,

we only know that satisfies definition 2.1 with kernel , the function remains unknown and is K k b K not

really determined; on the other hand, the of is uniquely determined by the kernel.commutator K

Proof of Theorem 3.1. Let By Theorem 3.8 we have:a , f, g . − −_ W_

'XgC K , a f Ò ÓÐ Ñ œ%8

œ Ð Ñ Ä � Ð Ñ � œ g a K f gK af g a Kf bf g K af baf g T f.' ' 'Š ‹ š Š ‹ Š ‹›X X X% %8 8 +

By Proposition 3.7 we have:

± ' 'X Xg T f lim g C K , a f c g a f .+ : ‡ :8± œ ± Ò ÓÐ Ñ ± Ÿ ² ² ² ² ² ²

8 % w

Hence, by the density of in :W _:

² T f c a f+ : ‡ :² Ÿ ² ² ² ²

for every By Proposition 2.3, the theorem follows. a , f . − −_ W_�

4. CONTINUITY OF SINGULAR INTEGRAL OPERATORS._:

EXAMPLES AND APPLICATIONS

11

In this section we want to show how our results about the commutator of a operator we haveCZ

proved so far (i.e. Theorem 2.5 in the case Theorem 3.1 in the case ) can be applied.Ð Ñ œ _ Ð Ñ � _X , X.

to given concrete singular integral operators. The problem here is to assure that the kernel actually definesk

a singular integral operator, continuous on . Our goal is to prove this fact under some natural_:

assumptions on the kernel. This is possible basically through two known theorems, which will be recalled

later. Our main results are contained in Theorems 4.1, 4.2, 4.3, 4.5.

Theorem 4.1. Let be a homogeneous space, a regular measure. Let \ aÐ Ñ ‚ œ × ÄX,d, k: X X x y . ‘. Ökernel satisfying the following conditions:

the growth condition (3.1); GÐ Ñthe Hormander condition with respect to both variables, i. e. both and satisfy (H1); H¨ k k‡ Ð Ñthe cancellation property with respect to both variables: CÐ Ñ there exists such that for every , , a.e. C 0 r, R 0 r R x� � � � _

± Ð Ñ Ð Ñ ± Ÿ'r d x,y R� Ð Ñ� k x,y d y C,.

k .and the same is true for ‡

Let be the truncated kernel defined as in § 3, and for any , setk f%(− V!

K f x k x,y f y d y .% %Ð Ñ œ Ð Ñ Ð Ñ Ð Ñ'X

.

Then can be extended to a linear continuous operator from into for every , andK p 1,% _ _: : − Ð _Ñ

² ² Ÿ ² ² −K f c f f% : :: for every ,_

where the constant depends on and all the constants involved in the assumptions, but not on .c X, p %

Theorem 4.2. Suppose all the assumptions of Theorem 4.1 hold. Then there exist a sequence and a%8 Ä 0

CZ K f, goperator such that for every − W

' 'X Xf K g f K g 0† Ä †%8 for .%8 Ä

Moreover, if are two operators obtained in this way by two different sequences , there exists aK , K " # %8function such thatb − __

K f K f b f" # œ †

for every and therefore and have the same commutator (for which the results of sections 2-3f , K K− _: " #

hold).

In the particular case of antisymmetric kernels, the previous result can be improved:

Theorem 4.3. Suppose all the assumptions of Thm. 4.1 hold. Moreover, let

k x,y k y,x x, y X x y.Ð Ñ œ Ð Ñ − Á for every ,

Then for any f, g ,− V(

!

Ø Ù Ä Ø Ù ´ Ð ÑÒ Ð Ñ Ð Ñ Ð Ñ Ð ÑÓ Ð Ñ Ð Ñ ÐK f,g Kf,g k x,y f y g x f x g y d x d y%"#' 'X X

. . 4.1Ñ

for In particular, the limit operator is unique.% Ä 0. CZ

For a kernel which is not necessarily antisymmetric, we are interested in getting a better result about

the convergence of , that is uniqueness of the limit and strong convergence in . These facts can beK% _:

proved under stronger assumptions on . Before stating the result, we need one morek%

Definition 4.4. Let be an unbounded homogeneous space. For every , defineÐ Ñ −X,d, x X.

r sup r 0: B x x B <œ � Ð Ñ œ Ö ×š ›(with the convention ). We say that satisfies a if there existsup 0 X,d, reverse doubling condition g œ Ð Ñ.c 1, M 1 x X, r rw

B. � � − �such that for every :

. .Ð Ð ÑÑ   † Ð Ð ÑÑB x c B xQ< <w. . 4.2Ð Ñ

It can be proved that, under some general additional geometric assumptions on the space , (4.2) isÐ ÑX,d

actually a consequence of the doubling condition on . (See [BC2])..

12

Theorem 4.5. Let be a homogeneous space such that is regular and, if is unbounded, theÐ ÑX,d, X. .

reverse doubling condition (4.2) holds. Let be a kernel satisfying all the assumptions of Theorem 4.1.k

Moreover assume that

for a.e. there existsx X−

lim k x,y d y b x .0%

.Ä

Ð Ñ Ð Ñ ´ Ð Ñ Ð'd x,y 1Ð Ñ� % LÑ

Then for every , converges (strongly) in for .f p 1, , K f 0− − Ð _Ñ_: % _ %: Ä

Remark 4.6. (Homogeneous spaces with a quasisymmetric quasidistance).

Although, in the definition of homogeneous space, we have required the symmetry of , the above results.still hold if is only , that is satisfies:. quasisymmetric

.ÐCß BÑ Ÿ - † .ÐBß CÑ Ð Ñ4.3

for some positive constant , every . (See Examples B and C in section 4.b for applications in- Bß C − \which the natural quasidistance is quasisymmetric).

Note that, if satisfies of Definition 1.1 and (4.3), setting. Ð3Ñß Ð333Ñ

. Bß C œ . Bß C � .ÐCß BÑw( ) ( )

we get a symmetric quasidistance , equivalent to . Therefore if ) is a kernel satisfying condition. . 5ÐBß Cw

(G) or (H) with respect to ( , it satisfies the same condition (possibly with other constants) with\ß .ß Ñ.respect to ( , which a homogeneous space. Moreover, if it satisfies (C) (G) with respect to\ß . ß Ñw . is and

( , let, for :\ß .ß Ñ ! � < � V � _.

( ) ; ( ) /( ;G œ CÀ < � . Bß C � V G œ CÀ < � . Bß C � V " � -Ñ" #wš › š ›

/( ) ( ) ; /( ) ( ) ( ,G œ CÀ < " � - Ÿ . Bß C Ÿ < G œ CÀ V " � - Ÿ . Bß C Ÿ " � -ÑV$ %š › š ›with as in (4.3). Then by (C), (G) and the doubling condition,-

¸ ¸ ¸ ¸' ' 'G G G G" # " #5ÐBß CÑ . ÐCÑ œ 5ÐBß CÑ . ÐCÑ � 5ÐBß CÑ . ÐCÑ Ÿ. . .

\

Ÿ - � ± 5ÐBß CÑ ± . ÐCÑ � ± 5ÐBß CÑ ± . ÐCÑ Ÿ' 'G G$ %

. . const.

So satisfies (C) with respect to ( .5 \ß . ß Ñw . �

Analogously, we can see that if satisfies (L) (G) with respect to ( , then it satisfies (L)5 \ß .ß Ñand .

with respect to ( .\ß . ß Ñw .

As a consequence, Theorems 4.1, 4.2, 4.3, 4.5 still hold if in the definition of homogeneous space we

weaken the assumptions, asking to be quasisymmetric, instead of symmetric; moreover, the same is true.for the commutator estimate (Theorems 2.5-3.1).

All the theorems we have stated so far depend on the following two known results.

Theorem 4.7, by Coifman-de Guzmàn. (See [CG]. A detailed proof is exposed in [CW1]).

Let be a homogeneous space, : a measurable function satisfying an integralÐ ÄX,d, k X X.Ñ ‚ ‘

Hormander condition (H0) (see Proposition 2.2). Assume that the operator¨

Kf x k x,y f y d yÐ Ñ œ Ð Ñ Ð Ñ Ð Ñ'X

.

is continuous from to . Then for every there exists a constant depending on , the _ _ _# # #p 1,2 c p− Ò Ó :

norm of and the constants appearing in (H0 , such that for every :K f Ñ − �_ _# :

² ² Ÿ ² ² � ŸKf c f 1 p 2: : : if ;

.š ›x X: Kf x t c t 0 p 1.− ± Ð Ñ ± � Ÿ � œ"² ²ft" for every , if

If also satisfies condition (H0), then, by duality, we get that is also continuous for k K p 2, .‡ _: − Ð _Ñ

Note that the theorem cannot be directly applied to a kernel, that is with the growth allowedsingular

by condition (G), because the integral of such a kernel with an function does not converge._#

13

The second result we need is an extension to homogeneous spaces of the so called " theorem" byT 1)ÐDavid-Journe, [DJ]. The extension we use is due to Christ, [Chr]. (A less general extension was given by´

David-Journe-Semmes in [DJS]). To state the theorem we need some more definitions.´

Let such that is dense in for every , and let be the dual space of( V V− Ð Ð Ð Ñ0,1 X X p 1,Ó Ñ Ð Ñ − Ò _Ñ( (

! !w_:

V(

!.

Definition 4.8. We say that a linear continuous operator , continuous in the naturalT: V V( (

! !wÄ Ð Ñ

topologies, satisfies the (WBP) if there exists a constant such that for anyweak boundedness property c 0�x X, r 0, f, g B x f g f g r! < ! _ _!

�− � − Ð Ñ ² ² ² ²V( ( with support contained in , , 1, , :Ÿ ² ² ² ² Ÿ( (

± Ø Ù ± Ÿ † Ð Ð ÑÑTf,g c B x .. < ! Ð ÑWBP

Theorem 4.9, by Christ (see [Chr]).

Let be a homogeneous space, a linear continuous operator satisfying (WBP).Ð Ñ Ä Ð ÑX,d, T: . V V( (

! !w

Suppose there exists a kernel \ , satisfying (G) and (H) (see Thm. 4.1) such that fork: X X x y‚ œ × ÄÖ ‘

any with disjoint supportsf, g − V(

!

Ø Ù œ Ð Ñ Ð Ñ Ð Ñ Ð Ñ Ð ÑTf,g k x,y f y g x d y d x .' 'X X

. .

Moreover suppose that and belong to . Then can be extended to a continuous operatorT 1 T 1 BMO TÐ Ñ Ð Ñ‡

from into ._ _# #

Remark 4.10. The definition of and are not in general trivial. However we will apply thisT 1 T 1Ð Ñ Ð Ñ‡

theorem to the truncated operators , so that the definition of and will be clear.K K 1 K 1% % %‡

4.a. PROOF OF THEOREMS 4.1, 4.2, 4.3, 4.5

Proof of Thm. 4.1. Let us consider the truncated kernels . We are going to check that all assumptions ofk%Thm. 4.9 are fulfilled, with constants independent of . Conditions C) and (G) imply that and are% Ð K 1 K 1 % %

‡

bounded functions, uniformly in , and therefore have norms bounded uniformly in . By (G) and the% %BMO

doubling condition,

'X± Ð Ñ ± Ð Ñ Ÿ Ð Ñk x,y d y c x% . % uniformly in ,

so that is well defined for any ; if also we can writeK f x f g %( (Ð Ñ − −V V! !

Ø Ù œ Ð Ñ Ð Ñ Ð Ñ Ð Ñ Ð ÑK f,g k x,y f y g x d y d x% %' 'X X

. .

where the integral is absolutely convergent. To estimate , let forØ Ù − § Ð ÑK f,g f, g , sprt f, sprt g B x%(V! < !

some Thenx X, r 0. ! − �

K f x k x,y f y f x d y f x k x,y d y % % %Ð Ñ œ Ð Ñ Ò Ð Ñ Ð ÑÓ Ð Ñ � Ð Ñ † Ð Ñ Ð Ñ �' 'd x,y r d x,y rÐ Ñ� Ð Ñ�. .

� Ð Ñ Ð Ñ Ð Ñ œ � � k x,y f y d y I II III.'d x,y rÐ Ñ  % .

± ± Ÿ ² ² Ð Ñ ŸI c f d y (by G Ð ÑÑ ( .'0 d x,y r

d x,y

B x;y� Ð Ñ�Ð ÑÐ Ð ÑÑ

(

.

(expanding the integral as and using the doubling property) ! 'n=1

d x,y

_

� Р�r r

2 28�" 8á

Ÿ c f r .² ² †((

± ± Ÿ ± Ð Ñ ± ± Ð Ñ Ð Ñ ± Ÿ ± Ð Ñ ±II f x k x,y d y c f x 'd x,y rÐ Ñ� % . since:

'd x,y rÐ Ñ� k x,y d y %Ð Ñ Ð Ñ œ.

œ Ð Ñ Ð Ñ � Ð Ñ Ð Ñ ´ � k x,y d y k x,y d y A B.' '2 d x,y r d x,y 2% % % %� Ð Ñ� � Ð ÑŸ. .

± ÐA c ± Ÿ by C).

Finally, by (G) and the doubling condition.± ŸB k x,y d y c ± Ÿ ± Ð Ñ ± Ð Ñ'% %� Ð ÑŸd x,y 2

.

± III c d y .± Ÿ Ð Ñ'd x,y r

f y

B x;yÐ Ñ ± РѱРРÑÑ.

.

14

Therefore:

± 'X g x K f x d x Ð Ñ Ð Ñ Ð Ñ ± Ÿ% .

Ÿ ± Ð Ñ ± ² ² † � ± Ð Ñ ± � Ð Ñ Ð Ñ Ÿc g x f r f x d y d x ' 'Š ‹B x d x,y r

f y

B x;y< !Ð Ñ Ð Ñ ± РѱРРÑÑ(

(.

. .

Ÿ ² ² † Ð Ð ÑÑ ² ² † � ² ² � ² ² † c g B x f r f f sup .x

_ < ! _ _ Ÿ Ð ÑŸ †Ð Ñ

Ð Ð ÑÑ. Š ‹'(

( .

.r d x,y c r

d y

B x;y

where we used the assumption on the supports of and . Since the last integral in brackets is boundedf g

uniformly in , by the doubling, we get:r

± Ø Ù ± Ÿ † Ð Ð ÑÑ ² ² ² ² † � ² ²K f,g c B x g f r f% ((. < ! _ _Š ‹

with an absolute constant. This proves that is continuous, and satisfies (WBC), withc K : %( (V V! !

wÄ Ð Ñ

constants independent of . Then by Thm. 4.9, can be extended to a continuous operator from into% _K %#

_ %#, with norm uniformly bounded in . From this fact and condition (H), by Thm. 4.7, we also get

² ² Ÿ ² ²K f c f% : : : Ð Ñ4.4

for every . Since the same reasoning can be applied to , by duality we get that (4.4) actuallyp 1,2 K− Ð Ó ‡%

holds for every . This proves Thm. 4.1. p 1,− Ð _Ñ �

For seak of brevity, we skip the proof of Thm. 4.2, since it is similar to that of Thm. 3.8, with minor

changes in the case .Ð Ñ œ _X .

To prove Thms. 4.3 and 4.5 we need the following technical lemma, which replaces in homogeneous

spaces the use of Young inequality to estimate some "convolution" integrals.

Lemma 4.11. Let , boundedly supported, and set, for every f f r 0− �_"

g x d y .< Ð Ñ-± РѱРРÑÑÐ Ñ œ Ð Ñ'

d x,y r

f y

B x;y..

Then, for every and , ; moreover in for p 1, r 0 g g 0 r .− Ð _Ñ � Ä Ä �_< <− _ _: :

Proof. Suppose first that is unbounded. Note that, by the doubling condition, is comparableX B x;y .Ð Ð ÑÑwith :.Ð Ð ÑÑB y;x

. .Ð Ð ÑÑ Ÿ Ð Ñ † Ð Ð ÑÑB x;y c c , c B y;x .. . (4.5)

Let .h − _:w

± Ð ÑÐ Ñ Ð Ñ ± Ÿ ± Ð Ñ ± Ð Ñ Ð Ñ Ÿ' ' 'X X d x,y r

h x

B x;yh g x d x f y d y d x < Ð Ñ-

± РѱРРÑÑ. . .

.

(by (4.5) and Holder)¨

Ÿ ² ² † ± Ð Ñ ± Ð Ñ h f y d y .: Ð Ñ-Ð Ñ

Ð Ð ÑÑ

"Î:

w :' '� X d x,y r

d x

B y;x

.

..

Now

' ! 'd x,y r 2 r d x,y 2 r

d x d x

B y;x B y;xÐ Ñ- � Ð ÑŸÐ Ñ Ð Ñ

Ð Ð ÑÑ Ð Ð ÑÑ8œ!

_

. .

. .: :8 8�"œ Ÿ

Ÿ Ÿ !n n

1B y,2 r

œ

_

Ð Ð ÑÑ!

8�" :".

(by (4.2), with : if , otherwise)n max n 2 r r r r n 0! C C !8œ Ö � × � œ

Ÿ † .Š ‹!8œ!

_

Ð Ñ Ð Ð ÑÑ1 1

c B y,rw Ð:"ÑÐ8�"Ñ :". .

15

Hence:

± Ð ÑÐ Ñ Ð Ñ ± Ÿ Ð Ñ ² ² Ð Ñ' 'X X

f y

B y;rh g x d x c p,X h d y .< :

± Рѱ

Ð Ð ÑÑ. .w

"Î:w.(4.6)

Recall that is boundedly supported, let . Let us show that for any fixed f sprt f B x r 0© Ð Ñ �V !

inf B y,r c 0.y B x− Ð Ñ

Ð Ð ÑÑ œ �V !

. (4.7)

To see this, observe that, for every , , so that, by the doubling condition,y B x B x B y− Ð Ñ Ð Ñ © Ð ÑV ! V ! #- V.

0 B x c X, r, R B y .� Ð Ð ÑÑ Ÿ Ð Ñ † Ð Ð ÑÑ. .V ! <

From (4.6)-(4.7) we have

± Ð ÑÐ Ñ Ð Ñ ± Ÿ Ð Ñ ² ² † Ð Ñ ² ² œ Ð Ñ ² ²'Xh g x d x c p,X h c X, r, R f c p,X,r,f h< : " :. w w

hence To show that in for observe that for every ,g . g 0 r r 1< <:− Ä Ä _  _ _:

± Ð Ñ ± Ÿ ± Ð Ñ ± − Ð Ñ Ä Ä _g x g x . g x 0 r< " <:_ Moreover a.e. for , as we can see applying Lebesgue's

theorem to

g x y d y .< Рѱ РѱРРÑÑÐ Ñ œ Ð Ñ Ð Ñ'

X X B xf y

B x;y; .\ < .

Therefore, again by Lebesgue's theorem, in for g 0 r .< Ä Ä__:

X g r r g 0.If is bounded, by (4.7) is bounded for every , and for sufficiently large So the lemma is< < ´proved. �

Proof of Theorem 4.3. Since for the 's Theorem 4.1 holds, it is enough to check the weak convergenceK%

(4.1). By antisymmetry and with a change of variables, for any fixed with support contained inf, g − V(

!

some ball :B

Ø Ù œ Ð ÑÒ Ð Ñ Ð Ñ Ð Ñ Ð ÑÓ Ð Ñ Ð ÑK f,g k x,y f y g x f x g y d x d y .% %"#' 'X X

. .

Now:

± Ø Ù Ø Ù ± Ÿ † Ð ÑK f,g Kf,g c d x %(

% %

.

.% .' '

B d x,y 2

d y

B x;y� Ð Ñ�Ð Ñ

Ð Ð ÑÑ �

� Ð Ñ Ð Ñ œ �' 'B d x,y

f y g x f x g y

B x;yd x d y I II.. .2 3

% %� Ð Ñ�± Ð Ñ Ð Ñ± ± Ð Ñ Ð Ñ±

Ð Ð ÑÑ.

Now

± I c B 0± Ÿ † Ð Ñ Ä% .(

while by Lemma 4.11.± ± ÄII 0 �

Proof of Theorem 4.5. For let us writef ,− V(

!

K f x k x,y f y f x d y f x k x,y d y % % %% %Ð Ñ œ Ð Ñ Ò Ð Ñ Ð ÑÓ Ð Ñ � Ð Ñ † Ð Ñ Ð Ñ �' '

� Р� � Р�d x,y 1 d x,y 1. .

� Ð Ñ Ð Ñ Ð Ñ œ � � k x,y f y d y I II III .'1 d x,yŸ Ð Ñ�3

%% % % %.

I k x,y f y f x d y k x,y f y f x d y % %% % %œ Ð Ñ Ò Ð Ñ Ð ÑÓ Ð Ñ � Ð Ñ Ò Ð Ñ Ð ÑÓ Ð Ñ œ' '

� Р� � Р�d x,y 2 2 d x,y 1. .

œ �A B .% %

With an estimate similar to that in the proof of Thm. 4.1, for The± ÄA c f 0 0.% ((± Ÿ ² ² † Ä% %

function is boundedly supported and converges uniformly, as , toB x 0%Ð Ñ % Ä

'0 d x,y 1� Ð Ñ� k x,y f y f x d yÐ Ñ Ò Ð Ñ Ð ÑÓ Ð Ñ. Ð Ñ4.8

which is bounded by , as a similar estimate shows. So converges to (4.8) in . c f B ² ² ( % _:

Since is compactly supported, our assumption (L) implies that uniformly, andf II b x f x % Ä Ð Ñ Ð Ñtherefore in . Here we used the fact that_:

'% %� Р�d x,y 1

k x,y d yÐ Ñ Ð Ñ.

16

has limit for This follows from (L), as can be seen for instance approximating the cutoff function % <Ä 0. %

, which enters the definition of , by piecewise constant functions.k%

To prove that converges in , let us show that for , Suppose forIII III III 0 0.% % (_ % (::² ² Ä Ä

instance 0 .� �% (

III III k x,y f y d y k x,y f y d y % ( % œ Ð Ñ Ð Ñ Ð Ñ � Ð Ñ Ð Ñ Ð Ñ �' '2 2 2 3

( % % %� Рџ � Р�d x,y d x,y. .

Ð Ñ Ð Ñ Ð Ñ k x,y f y d y'2 3

( (� Ð Ñ�d x,y ( . . 4.9Ð Ñ

Each term in (4.9) is majorized by

c d y'd x,y

f y

B x;yÐ Ñ ± РѱРРÑÑ2

( ..Ð Ñ

which converges to zero in , as , by Lemma 4.11. This completes the proof of Thm. 4.5._ (: Ä 0 �

4. b. EXAMPLES OF SINGULAR INTEGRAL OPERATORS

ON HOMOGENEOUS SPACES

We start with the following

Remark 4.12. Whenever, in a homogeneous space, the following holds

c r B x c r" < #R RŸ Ð Ð ÑÑ Ÿ. (4.10)

for every some positive constants , condition (H1) can be rewritten as:r 0, x X, c , c , N� − " #

there exist such that for every :c 0, 0, M 1 x X, r 0, x B x , y B xO ! < ! Q< !� � � − � − Ð Ñ Â Ð Ñ"

± Ð Ñ Ð Ñ ± Ÿ †k x ,y k x,y c .! OÐ ÑÐ Ñd x ,x

d x ,y!

!�R

"

" (H2)

Example A. Singular integrals with mixed homogeneities, studied by Fabes and Rivière in [FR] and

applications to parabolic equations.

Let the Lebesgue measure, the distance defined as follows. Let , , be realX , d N œ ‘ .R ! !" Rá

numbers, with ; the equation1 Ÿ Ÿ Ÿ á Ÿ! ! !" # R

!4œ"

R

1x4#

# 43! œ

allows for every \{ } a unique positive solution Set and define Itx 0 x . 0 0 d x,y x y . − Ð Ñ Ð Ñ œ Ð Ñ œ Ð Ñ‘ 3 3R 3

can be proved (see [FR]) that is actually a distance. Introducing the polar type change of variablesd

ÚÝÝÛÝÝÜx cos cos cos

x cos sin

x sin

" " R�# R�"

# " R�"

R "

œ ᜠá

áœ

3 : : :

3 : :

3 :

!

!

!

"

#

R

we find , with and the surface meaure on . Thereforedx d d d x 1œ ± ± œ ×3 3 5 5!�" ! !œ œ Ö! !4œ"

R

4 R�"

we can compute:

.Ð Ð ÑÑ œ Ð Ñ † �B x c N r r 0<! for every . 4.11Ð Ñ

In particular (4.10) holds, and ( ) is a homogeneous space. The unit ball with respect to is the‘R ,d,dx d

euclidean unit ball. Let

k x,y Ð Ñ œH

3

Ð � ÑÐ � Ñx y

x y ! , 4.12Ð Ñ

where the function has the following mixed homogeneity of degree zeroH

H H ‘Ð á Ñ œ Ð á Ñ � −t x , t x , , t x x , ,x , t 0, x! ! !" # R" # R " R

Rfor every 4.13Ð Ñ

and satisfies the Holder condition:¨

there exist such that for every :c 0, 0,1 x, y� − Ð Ó −" !R�"

17

± Ð Ñ Ð Ñ ± Ÿ ± ±H Hx y c x y ." (4.14)

It is easy to check that the following condition (H2) is fulfilled:

if , are as above, there exists a constant such that for every :! " c x , x B x , y B x! < ! #< !R− − Ð Ñ Â Ð Ñ‘

± Ð Ñ Ð Ñ ± Ÿ †k x ,y k x,y c .!Ð ÑÐ Ñd x ,x

d x ,y!

!�

"

" ! (4.15)

The growth condition (G) follows from (4.11) and the definition of Moreover, if we assume that k. H

satisfies the vanishing property

'!

R�"H 5Ð Ñ Ð Ñ œx d x 0, (4.16)

then conditions (C) and (L) are trivial. Therefore, by Thms. 4.1, 4.5, for every it is well definedp 1,− Ð _Ñ

Kf x lim k x,y f y dy0

Ð Ñ œ Ð Ñ Ð ÑÄ%

'% ,

where the limit exists in sense, and is continuous. (This result was proved with a direct approach in_: K

[FR]). For this operator the commutator estimate of Thm. 2.5 holds.

Kernels of the above kind arise for instance considering parabolic operators with constant coefficients:

P œ + ` `!3ß4œ"

8

34 >#0 03 4

,

where is a symmetric and positive matrix. Here (we set ). Let be+ B ´ Ð ß >Ñ − ´ R œ 8� "348 " R !0 ‘ ‘ >

the fundamental solution for , with pole at the origin, and letP

5ÐBß CÑ œ Ð` Ñ B CÑ 3ß 4 œ "ßá ß 80 03 4

# !> ( , for any .

Then is a kernel as in (4.12)-(4.13), with 1, 2. For any test function we can5 œ á œ œ œ ?! ! !" 8 8 "

write:

?ÐBÑ œ ÐB CÑP?ÐCÑ .C'‘8�">

!

and, differentiating twice the above formula with respect to we find0

` ?ÐBÑ œ T ÞZ Þ ` ÐB CÑP?ÐCÑ .C � -Ð8Ñ † P?ÐBÑ0 0 ‘ 0 03 4 3 4

8�"# # !' > .

Therefore, the -continuity of the singular integral implies -estimates for the (spacial) second_ _: :

derivatives of , in terms of and . ? P? ? �

Example B. Kolmogorov-type operators, studied by Lanconelli-Polidoro in [LP].

Let us consider the following linear partial differential operator:

P ´ + ` � , B ` ` ´ ! !;

3ß4œ" 3ß4œ"34 B B 34 3 B >

R

3 4 4

´ + ` � ØBßFHÙ ` H œ Ð` ` á ` Ñ , , , ,!;3ß4œ"

34 B B > B B B3 4 " # R

where , is a constant, symmetric, positive matrix ( the principal part isÐBß >Ñ − Ö+ × ; ‚ ; 3Þ/Þ‘R " 34

uniformly elliptic on , ), and is an constant matrix with the following structure:‘; ; � R F R ‚R

F ´

F ! á !F á !

ã ã ã ä ãá Fá

Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÕ Ø

0

0 0

0 0 0

0 0 0 0

"

#

<

where for every , is a matrix with rank and ,5 œ "ßá ß < F : ‚ : : ; œ :   :   á   :5 5�" 5 5 ! " <

: � : �á � : œ R P F! " < . Under these assumptions, the operator is hypoelliptic, and the matrix induces

in the space a (noncommutative) ‘R " group of translations

ÐBß >Ñ ‰ ÐCß œ C � IÐ ÑBß > � IÐ Ñ œ F7 7 7 7 7) ( ), where exp( ),X

18

such that is invariant for left-translations. Moreover, there exists a on , which weP group of dilations ‘R "

denote by , where every is a group homomorphism, acting as follows:š ›HÐ Ñ HÐ Ñ- ---!

HÐ ÑÐBß >Ñ œ B á ß B > Ð Ñ- - - -( , , ) 4.17! !" R" R

#

where are suitable integers 1. The operator is homogeneous of degree 2 with respect to the group of!3   Pdilations:

P ?ÐHÐ ÑDÑ œ † HÐ Ñ P? ÐDÑŠ ‹ Š ‹- - -# ( ) ,

with . For and we have the simplest example:0

0 0D œ ÐBß >Ñ R œ # F œ

"” •P œ ` � B ` `B

#" B >" #

.

In this case the groups of translations and dilations are, respectively:

ÐB ß B ß >Ñ ‰ ÐC ß C ß Ñ œ ÐB � C B � C B ß > � Ñ" # " # " " # # "7 7 7, ;

HÐ Ñ ÐB ß B ß >Ñ œ Ð B ß B ß >Ñ- - - - ." # " #$ #

We can define a , setting, for any , ,homogeneous norm D − D Á !‘R "

² D ² œ Í ± HÐ ÑD ± œ3 1,1

3

with euclidean norm; also, let 0. Then:± † ± œ ² ! ² œ

3Ñ ² HÐ ÑD ² œ ² D ² D − � for every , 0;- - ‘ -R "

33Ñ ÖD − ² D ² × The set : =1 is the euclidean sphere ;‘R " R "!

333Ñ D − For every , ' ‘R "

² D ‰ ² Ÿ -Ð ² D ² � ² ² Ñ - œ -ÐFÑ  ' ' for some constant 1.

Defining:

.ÐDß Ñ œ ² ‰ D ²' '�" ,

we have that is a quasisymmetric quasidistance (see Remark 4.6). Since the Lebesgue's measure is left and.right-invariant under the group law, and

./>HÐ Ñ œ- -U #,

with , as in (4.17), we have that 1 , forU œ �á � ± ÐDß <Ñ ± œ ± Ð!ß <Ñ ± œ ± Ð!ß Ñ ± <! ! ! U U U" R 3U #

every and 0. Therefore is a with respect to the quasidistance , andD − < � .D .‘R " doubling measure

( , is a homogeneous space.‘R " .Dß .Ñ

Let be the fundamental solution for , with pole at the origin, which is homogeneous of degree>! PU HÐ Ñ with respect to , and let-

5ÐDß Ñ œ Ð` Ñ ‰ DÑ 3ß 4 œ "ßá ß ;' > 'B B# ! �"3 4

( , for any .

Then is a kernel on the above homogeneous space, for which the -estimate of Thms. 4.1, 4.5 and5 G^ _:

the commutator estimate of Thm. 2.5 hold. �

Remark 4.13. Applications of the commutator theorem to get a priori estimates for partial

differential operators. In the above examples, -continuity of singular integral operators implies -_ _: :

estimates for (suitable) second derivatives of , in terms of and . The differential operator which is? P? ? Pconsidered has constant coefficients or, more generally, is translation invariant, with respect to a suitable

group. The estimate can also be used to get -estimates, for suitable classes of P.D.O.'s withcommutator _:

variable coefficients. This technique was used first by Chiarenza-Frasca-Longo [CFL1,2], who applied

Coifman-Rochberg-Weiss' Theorem to get estimates for uniformly elliptic operators in nondivergence_:

form, with coefficients. The class ("vanishing mean oscillation", see [Sarason]) is containedZQS ZQSin , and properly contains the space of uniformly continuous functions, so the theory for ellipticFQS _:

operators with coefficients is an interesting extension of the classical one. This kind of result hasZQSbeen extended to the (uniformly) parabolic case by Bramanti-Cerutti [BC1], who proved a parabolic version

of CRW Theorem. In another paper, Bramanti-Cerutti-Manfredini [BCM], using the commutator theorem

in homogeneous spaces, prove estimates for a special class of ultraparabolic equations, of Kolmogorov-_:

Fokker-Planck type (see Example B for the translation invariant case). It should be noted that, although

19

CRW-type Theorem provides a very powerful tool, the -estimates for second derivatives for operators_:

with variable coefficients are not a direct consequence of it. In fact, in these cases, the representation

formula for the derivatives of is obtained by "unfreezing" the coefficients of after having written the? Prepresentation formula for the operator with constant coefficients. Therefore the kernels one obtains are of

the kind and are themselves kernels; instead, one has to expand as a series in terms of5ÐBß B CÑ G^ 5not

kernels that turn out to be kernels. This technique requires a deep analysis of the properties of theG^singular kernel .5ÐBß B CÑ

Example C. The Kohn-Laplacian on the Heisenberg group. (See [St3] for the proofs of the facts listed

below).

Let us consider endowed with the group law:‚ ‘8 ‚

Ð >Ñ ‰ Ð =Ñ œ Ð � > � = � # Ð † ÑÑ0 ( 0 ( 0 (, , , Im ;–

Ð >Ñ œ Ð ß >Ñ0 0, .�"

Ð ‚ ß ‰ Ñ Ð ß ‰ Ñ‚ ‘ ‘ ‘8 #8 " #8 ", or, equivalently, , is a Lie group, for which the Lebesgue measure in is the

Haar measure. Moreover the dilations

HÐ ÑÐ >Ñ œ Ð >Ñ �- 0 -0 - -, , ( 0)#

are group automorphisms. We can define in a ‘#8 " homogeneous norm

± Ð >Ñ ± œ Ð ± ± � > Ñ0 0, % # "%

and a quasidistance

. Ð >Ñß Ð ß =Ñ œ ± Ð ß =Ñ ‰ Ð >Ñ ±Š ‹0 ( ( 0, , .�"

Then , with this quasidistance and the Lebesgue's measure, is a homogeneous space, such that the‘#8 "

measure of a ball of radius equals , with . (Therefore is called the< † < U œ #8 � # Uconstant U

homogeneous dimension of ).‘#8 "

Consider now the Lie algebra canonically associated to the Lie group. To describe it, let us write, from

now on, , , with ; let , , , respectively, be the left-invariant vector fields onÐ >Ñ œ ÐBß Cß >Ñ Bß C − \ ] X0 ‘8 4 4

‘#8 " 4 4 that equals , , at the origin. Then one can compute:`Î`B `Î`C `Î`>

\ œ � #C4 4` ``B `>4

;

] œ #B4 4` ``C `>4

;

X œ ``> .

Moreover,

[ ] for \ ß] œ %X 4 œ "ßá ß 84 4

and all the other commutators vanish. The operator

_ œ \ � ]!Š ‹4œ"

8# #4 4

is the Kohn-Laplacian on the Heisenberg group; it is a hypoelliptic operator, homogeneous of degree two

with respect to the dilations , and it has a fundamental solution , with pole at the origin, -HÐ Ñ HÐ Ñ- > -!

homogeneous of degree ; moreover, is explicitely known: ÐU #Ñ >!

> 0 0! % # 8Î#Ð >Ñ œ -Ð8ÑÎÐ ± ± � > Ñ, .

For any test function we can write:?

?ÐDÑ œ Ð ‰ DÑ ?Ð Ñ .'‘#8�"> ' _ ' '! �"

and, differentiating twice the above formula with respect to the vector fields or we find, for example,\ ]4 4

\ \ ?ÐDÑ œ T ÞZ Þ \ \ Ð ‰ DÑ ?Ð Ñ . � -Ð8Ñ † ?ÐDÑ3 4 3 4! �"'

‘#8�" > ' _ ' ' _ .

The kernel ) turns out to be a kernel on the homogeneous space described5ÐDß Ñ œ \ \ Ð ‰ D G^' > '3 4! �"

above; for the corresponding operator the estimates and the commutator estimate hold._:

20

We expect that this fact could be a starting point to get estimates for the "second derivatives" _: 3 4\ \? for a class of operators with variable coefficients, modeled on the Kohn-Laplacian. (See Remark 4.13). �

The next two examples are taken from [CW], pp.76-77.

Example D. Singular kernels on eucledean spaces with weighted measures. (This example is related to

some weighted singular integrals in studied by Stein in [St1]).‘8

Let with the eucledean distance. It is well known (see [GR],X , d x dx 0 n, dœ œ ± ± Ÿ �‘ . #8 �#

p. 407) that the function is an weight for hence with ± ± � � Ð Ñ ± ± Ÿ �x A - n n p 1 , x 0 n! #:

�! #

is an -weight for every In particular is doubling, so that is a homogeneous space; oneA p 1, . d X : − Ò _Ñ .

can prove that satisfies also a reverse doubling condition. Note that is translation invariant. Now,d d not. .

let

(4.18)k x,y x yÐ Ñ œ ± ± ± ±H # #Ð � ѱ � ±

x y

x y 8 Š ‹where is a homogeneous function of degree zero (in the usual sense) satisfying (4.14), (4.15). In this caseH

(4.10) does not hold, but the following inequalities can be easily verified:

.Ð Ð ÑÑ ŸB x;y c x y x x y x /2

c x y x y x /2.

± � ± ± ± ± � ±�± ±

± � ± ± � ± ± ±

8 �

8�

#

#

if

if

(4.19)

By (4.18) and (4.19), we can see that satisfies condition (G). Moreover, one can prove that satisfies (H)k k

with exponent . ( is the same number appearing in (4.15)). Finally, also (C) and (L) can be" "w œ min ,Ð Ñ" #

proved, the last one following from the analogous result for classical integrals on . So also in thisCZ ‘8

case, by Thms. 4.1, 4.5, the kernel defines a unique operator, for which the commutator estimatek CZ

holds.

Example E. Singular integrals on the sphere. (The following operators, discussed in [CW1], are related to

the Riesz transform on the sphere, studied in [KV2]).

Let the surface of the unit ball in , the euclidean distance (in ) and the surfaceX d œ D8�" ‘ ‘ .8 8

measure. We know that

c r B x c r" < #8�" 8�"Ÿ Ð Ð ÑÑ Ÿ.

for every , with x X, 0 c c .− ! � < � # � �" #

Let us consider the kernels:

k x,y 34�

± � ±Ð Ñ œx y x y

x y

3 4 4 3

8

where are two fixed indexes between and . (In the following we will suppose these indexesi, j i j 1 nÐ Á Ñfixed once and for all, and we will drop them from ). One can check that , so thatk k x,y c x yÐ Ñ Ÿ † ± ± 8�"

(G) holds for Moreover, a condition (H2) is satisfied:k.

there exists such that for every :c 0 x , r 0, x B x , y B x� − � − Ð Ñ Â Ð Ñ! < ! #< !8�"!

± Ð Ñ Ð Ñ ± Ÿ †k x ,y k x,y c .!± � ±± � ±x x

x y!

!8

From this condition (H) follows. Note that, for every r, R 0,�

'r x y R�± � ±� k x,y d 0Ð Ñ œ5

so that C) and (L) hold. Again, a unique operator is well defined, for which the commutator estimateÐ CZ

holds, by Thm. 3.1.

Example F. This example exhibits a kernel which defines infinitely many operators, all of them havingCZ

the same commutator. In other words, Thm. 4.1 and 4.2 apply, but Thm. 4.5 does not.

In , with the usual distance and measure, define the convolution kernel:‘8

k x,y k x y k x cos log x .Ð Ñ œ Ð Ñ Ð Ñ œ ± ±with 1

± ±x 8

Then:

± k x Ð Ñ ± Ÿ 1

± ±x 8 that is (G) holds;

21

± f Ð Ñk x ± Ÿ , which implies (H);1

± ±x 8�"

'r x R�± ±� 8 k x dx c sin log R sin log rÐ Ñ œ † Ò Ó,

so that (C) holds, but

lim k x dx 0% Ä

Ð Ñ'%�± ±�x 1

does not exists.

By Thm. 4.1 we know that

² ² Ÿ ² ² − Ð _ÑK f c f p 1,% : : : for every

with independent of . By Thm. 4.2 we also know that the commutator converges weakly toc C K ,a : % Ò Ó%

C K,a , K CZ Ò Ó where is any operator of an equivalence class. For this commutator the estimate of Thm. 2.5

holds. Nevertheless, does not converge in any sense to a operator. We can actually compute, for anyK f CZ %

sequence ,%5 Ä 0

lim K f x k x y f y f x dy k x y f y dy 0%5

± � ±� ± � ±-ÄÐ Ñ œ Ð Ñ Ò Ð Ñ Ð ÑÓ � Ð Ñ Ð Ñ �%5

' 'x y 1 x y 1

Ð Ñ † Ò ÓÄ

c f x lim sin log .0

8 55%

%

Hence in general does not converge, and there are sequences for which , where isK f K f Kf b f K % %5 5Ä � †

a fixed operator and is any number in the interval b c , c .Ò Ó8 8 �

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