Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces

ME~fu~IRS American Mathematical Society Number 530 Littlewood-Paley Theory on Spaces of Homogeneous Type and the Class...

Author: Yongsheng Han | Eric T. Sawyer

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ME~fu~IRS American Mathematical Society

Number 530

Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces Y. S. Han

E. T. Sawyer

July 1994 • Volume llO • Number 530 (fifth of6 numbers) • ISSN 0065-9266

American Mathematical Society Providence, Rhode Island

1991 Mathematics Subject Classification. Primary 42B30, 42Bl5. Library of Congress Cataloging-in-Publication Data Han, Yongsheng. Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces I Y. S. Han, E. T. Sawyer. p. em. - (Memoirs of the American Mathematical Society, ISSN 0065-9266; no. 530) Includes bibliographical references. ISBN 0-8218-2592-5 1. Littlewood-Paley theory. 2. Multipliers (Mathematical analysis) 3. Hardy spaces. 4. Function spaces. I. Sawyer, E. T. (Eric T.), 1951-. II. Title. III. Series. QA3.A57 no. 530 [QA403.5} 510s-dc20 94-13336 [515 1 .2433] CIP

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A

we set V\(x)

is the collection of all f

1

= ,;n.vJ...~x).

£ ~'

f P,

if

s 2},

is 1{1 s£.

The Besov space

B:''l(lf),

a

e I,

1 S p, q S 111,

tempered distributions modulo polynomials, such that

Y.S. Han was partially supported by NSERC grant #OGP0105729. E.T. Sawyer was partially supported by NSERC grant #A5149. Received by the editor March 17, 1992, and in revised form April 20, 1993.

1

Y.S. HAN AND E.T. SAWYER

2

H we interchange the order of summation and integration in the above definition we

obtain the Triebel-Lizorkin space F~·q(of), a is the collection of all f

E

E

IR, 1 ~ p < m and 1 ~

q~

m. This space

rt/' / P such that

It is well known that the Calder6n reproducing formula ([C]) plays an important role in studying these spaces. For instance, this formula allows us to show that the definitions of the Besov and Triebel-Lizorkin spaces are independent of the choice of

tp

satisfying

(1.1) (i), (ii) and (iii), and to obtain atomic decompositions for these spaces. Moreover, it permits the identification of most of the classical function spaces - such as Hardy spaces, Sobolev spaces, 1P spaces, Lipschitz spaces and BMO, together with their traces on subspaces - as special cases of B~·~of) and

Fiq(of).

For these and other facts

about these spaces the reader is referred to (FJW], (P] and [Tr]. The Calder6n reproducing formula can be stated as follows:

Theorem (1.4) (The Calder6n reproducing formula)

Suppose that the function

tp

satisfies the properties (1.1) (i), (ii) and (iii). Then there exists a function '1/J satisfying the properties (1.1) (i) and (ii) such that for each f

E

rt/' /P,

(1.5)

where the series converges in the distribution sense. More precisely, this means that there exist polynomials {PN}N=-m such that

LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE

3

converges to f in #' as N ---t m.

See [FJW) and [P] for the proof and some applications of this formula. Further applications can be found in [C], [CF], [CDMS], [FJ], [FS], [GS] and [U] for example. We now replace Rn with the more general space of homogeneous type (introduced by Coifman and Weiss in [CW]). There are in general no translations or dilations on such spaces, and no analogue of the Fourier transform or convolution operation.

To

develop LiUlewood-Paley charaderizations of Besov and Triebel-Lizorkin spaces in this context we use an idea of Coifman. Let {Dk}kel be a family of operators whose kernels satisfy certain size; smoothness and moment conditions and the nondegeneracy condition,

(1.6)

I

=

E D kel k

2 on L.

Coifman's idea is to rewrite {1.6) as

I

where

[ E D ] [ E D1 = E E D n kel k I.El lJ lli>N kel k+rlt

-

+ E [ E

ltel lliSN

ln

D

k+lJ It

~=

and where

E E Dk+l Dk and TN = E D~Dk with D~ = E Dk+ ., lli>N ltel kel UISN J N is a fixed large integer. In [DJS] it was shown, using CoUar-Btein

techniques, that .for N sufficiently large,

Ti1

exists and is bounded on L2, thus

providing a key step in obtaining the Tb theorem. This in turn permitted them to establish LiUlewood-Paley theory for Lp, 1 exists a positive constant CP, such that

0. spaces.

f E

Finally, in section 7 we discuss duality and interpolation for these

§2. Til it a Calderon-Zygmund operaw We begin by brie:O.y reviewing spaces of homogeneous type and Calder6n-Zygmund operator theory on these spaces. A quasi-metric 6 on a set X is a function 6: X

-

x

X

[0,111] satisfying:

(2.1)

(i)

6(x,y) = 0 if and only if x = y,

(ii)

6(x,y)

(iii)

There exists a constant A
1.

Definition (2.2) ([CW]) A space of homogeneous type (X, 6, ~-&) is a set

X together

with a quasi-metric 6 and a nonnegative measure 1-' on X such that 1-'{B(x,r)) for all x x

EX

(2.3)

f

X and all r

and all r

> 0,

and such that there exists A'

0 and 0 < 8 < 1 satisfying

(2.4)

t

{2.5)

jp{x,y)- p{x' ,y) I S C p(x,x') 8 (p(x,y)

for all x E X, r

r S p(B(x,r)) S C r

> 0,

+ p(x' ,y)] 1- 8

for all x, x' and y

E X.

1 Moreover, there is a positive constant M such that d{x,y) = p{x,y)J/I. is equivalent to a

metric on X

x

X. In (2.4), and for the remainder of the paper, all balls are p-balls

B(x,r) = {yeX: p{y,x)

0 such that

Il

~ Cr

cg(x " X) with supp f c B(x0,r) " B(y0,r),

for all f e

~·

Yo e X,

11:£11 10

~

1,

llf(·,y)ll,~r-fl, and llf(x,·)ll"'~r-fl forall x,yeX.

We can now state and prove the main result of this section.

Theorem (2.14) Suppose that (Sk)kel is an approximation to the identity and set Dk. = Sk+l -

~·

Then

TN1

Moreover, smaller

E,

E D~Dk, where D~ = E Dk+. and N is a positive integer. kel UI~N J exists and is a Calder6n-Zygmund operator if N is large enough.

Let TN =

Tri<x,y), the kernel of Ti1, satisfies the estimates (2.7) (i) - (iii) for namely: for 0 < E'
0 such that

IT'N1(x,y)- T'N1(x,y')l + IT'N1(y,x}- TN1(y' ,x)l

~ C l(y,y')E' p(x,y)-{l+E')

for p(y,y') < !xp(x,y),


(iii)

13

I[Ti1(x,y)- Ti1(x' ,y)] -[TN1(x,y') -TN1(x' ,y')] I

~ C p(x,x')E' p(y,y')f' p(x,y)-{1+2t') for p(x,x') ~ ~ p(x,y) and p(y,y') ~ ~ p(x,y). 3A 3A Moreover, TNl E SWBP. Finally, if the sequence (Sk)kEI satisfies only (i), (ii), (iii), (v) and (vi) of definition (2.13), i.e. the second difference condition is not assumed, then the conclusions ~f Theorem (2.14) still hold, but with

f

in place of f' in (2.14) (iii).

fiQQf First, we show (i) - (iii) in Theorem (2.14). To do this, we will show that RN(x,y), the kernel of RN, m f'

= 1,2,... as in §1,

satisfies the following estimates: for 0
0, and 6 > 0 such that

(2.16)

IRN(x,y)- RN(x,y')l ~ em 2-N6mp(y,y')E' p(x,y)-{l+E') for p(y,y') ~

(2.16)'

IRN(x,y)- RN(x' J)l ~ em 2-N6mp(x,x')E' p(x,y)-{1+e) for p(x,x') ~

(2.17)

h p(x,y), h p(x,y),

I[RN(x,y) - RN(x' ,y)] - [RN(x,y') - RN(x' ,y' )] I

~

em 2-N6m p(x,x')f' p(y,y')f' p(x,y)-{1+2E')

for p(x,x') ~ ~ p(x,;y) and p(;y,;y') ~ ~ p(x,;y). 3A 3A Estimates (2.15), (2.16) and (2.16)', to~her with the T1 theorem for L2 on a space of homogeneous type show that the operator norm of

~

is less than one for N large

14


since RN1

= RN• 1 = 0 in BMO (see (DJSJ).

-1 Thus TN

= I+ RN

2 + RN + ... ,and

the estimates (i)- (iii) for the kernel of TN1 now follow.

To show (2.15) we need the following fact:

Mmma

(2.18) Suppose that {Dk}kEI is a family of operators whose kernelB satisfy the

properties (i), (ii), (iii) in (2.13) and

for all k E I.

(2.19)

Then DkD ~x,y), the kernel of DkDt satisfies the following estimate:

where C is a constant, a A b denotes the minimum of a and b, and x(E) denotes the characteristic function of E.

fiQQf It is enough to consider k

=

~

L In that case, we have

J

I Dk(x,z)[D~z,y)- Djx,y))dp(z)l by (ii) in (2.13)

by (i) in (2.13) and since k ~ L


15

This proves (2.20) for the case where k ~ l and hence Lemma (2.18).

We now turn to the estimate of Dl.+k_Dk.' 1 ~ j ~ m, where J J J

RW(x,y)

Let z0

and kl' k2,... ,km e l. Then

=

= x and zm = y.

It now follows that

(2.22)

f:l. ~····~

RW, m = 12, ... , in (2.15).

IRW[l,k](x,y) I

By Lemma (2.18), we have that for 1 ~ j ~ m,

Set Ej[l,k) =

16

Y.S. BAN AND E.T. SAWYER

where 8m= 8m[t,k] = k1A(k1+ ~)~A(k2 +~)A ...kmA(km+ lm)· Now set Fj[t,k] = D k Dl +k , 1 ~ j ~ m -1. Then j j+1 j+1 RN[t,k](x,y) =

By Lemma (2.18), we have that for 1 ~ j ~ m -1,

~

-llj+1+kj+Ckjl E [kl(lj+l+kj+l)] { -lkl(lj+l+kj+l)]} C2 2 x( p{zj,zj+l) ~ CA2 )

It now follows that


17

By taking the geometric average between (2.22) and (2.23), we obtain that for a, {J

>0

and

ll+/J = 1,

(} 2 m x({p(x,y)

-0 m})

~ CmA~

and since for each u and j, the sum over k in square brackets is bounded by a constant, the above is dominated by

<em -

" z.

I~I>N

...

E

llmi>N

-{l~+ ... +llml)af m-1

2

Thus,

where 6 = Of

> 0, and this gives (2.15).

C

m

CmAm

l

p x,y

)"


18

To show (2.16), we need further estimates on DkDi

Lemma (2.24) Suppose {Dk}ke1l is as in Lemma (2.18). there are constants 7

Then for all 0

0 and C > 0 such that

Proof It suffices to show that

Indeed, by taking the geometric average between (2.20) and (2.27), it follows that

f'

0

and

a+ p = 1 (with P close to 1).


26

N llmi>N

E

oez

l

2-I I,+lm-km-11 +... +I I, +J 0 such that

X{min(p(x,y),p(x,y'),p(x' ,y),p(x' ,y')] 5 CA 2-{kAl)} To show (2.44) it suffices to prove


27

X{min[p(x,y),p{x,y'),p{x' ,y),p(x' ,y')) ~ CA 2-{kAl)}. Indeed, if a, {J

> 0 with a+

{J = 1, then by taking the geometric average of {2.45) and

(2.20), we obtain

X{min[p(x,y),p{x,y'),p(x' ,y),p{x' ,y')) ~ CA 2-{kAl)} which yields the desired estimate if we choose {J close to 1. To see (2.45), suppose first that k ~ l. Using the second order difference condition (2.13) (iv) for Dk, we have

= I 1 (Dk(x,z) =

Dk(x' ,z)] (DJz,y)- DJzJ')] dp(z)

I 1 {[Dk(x,z)- Dk(x' ,z))- (Dk(xJ)- Dk(x' ,y)]} [DJz,y)- DJz,y')] dp{z) I since D* !1)

~

I

= 0,

C ~(1+ 2 t) p{x,x')fp{y,y')f X{min[p(x,y),p(x,y'),p{x' ,y),p{x' ,y')]

~ CA 2-{kAl)}


28

which proves the case k 5 l of (2.45). The proof for the case k

> l is similar..

We now turn to the proof of (2.17). As in the proof of (2.16), we write y

= zm

and y' = zm+ 1 - but now let x = z_1 and x' = z0. Define s0 and sm+ 1 in Z -s -1 -s -s -1 -s by 2 < p(x,x') 5 2 and 2 m+ 1 < p(y,y') 5 2 m+ 1. As before, choose j1

°

°

=

min s. and s. > s. for j > h• and this time choose also j0 so 05Hm+1 J J h that s. = min s. (= sJ. ) and sJ. > si... for j < ~- Of course j0 may equal Jo 05j5m+1 J 1 "'U h· Now choose 0 = j-u < Lu+l < ... < j_1 < ~ 5 j1 < j2 < ... < .it,_1 < .it, = m+1 so that sJ.1

so that etc.

for j < j_l' sj and s. > s. J L1 0

s. J_1

= 05 min j <j

s.

=

min s. and s. > s. for j < ~· J Jo 05j5m+1 J

s.

=

min s. and s. > s. for j > j1, J h 05j5m+1 J

8·

min s. = h<j5m+1 J

Jo h ~

and s. > s. for j > j2, J J2

etc,

and let J

= {j-u,Lu+1,... ,j_l'~'j 1 ,~, ... ..it,_l'.it,}.

The two cases ~
0 in

vK(,8,-y)(x0 ,d) is defined by

(3.3)

- llfll

vK

(P,'Y)(

) xo,d

=

inf {C ~ 0: (3.1) (i), (ii) and (iii) hold}.

We now fix a point x0 E X and denote the class of all f e vK(,8,-y)(x0,1) by vK(,B,-y)_ It is easy to see that vK(P,'Y) is a Banach space under the norm llfll vK(,B,-y)·

Just as the space of distributions

rt/'

is defined on ~. we may introduce the

dual space ( vK (,8, 'Y))' consisting of all linear functional& l from

vK (,8, 'Y) to ( with

the property that there exists a finite constant C such that for all f E vK (,8, 'Y), ll( f) I

~

C llfllj.P.'Y)" We denote by the natural pairing of elements h E ( .J((P,'Y)),

and f e .Jt(P,'Y)_ It is also easy to see that .Jt(,8,-y) (x1,d) d

= .Jt(,B,-y)

for x 1 E X and

> 0 with equivalent norms (but with constants depending on x0, xl' and d). Thus

is well defined for all h E ( .Jt(,8,-y)), and for all f E .Jt(P,'Y) (xpd) with x 1 e X and d

> 0.

We can now state our Calderon - type reproducing formulas:

Theorem (3.4) Suppose that {Sk}ke7l is an approximation to the identity and let Dk = Sk+l - Sk. f E .J((,8,-y)

(3.5)

Then there exists a family of operators

{Dkhe7l

such that for all

38


where the aeries converges to f in the norm of .6 (11' 1 7') if Moreover

Dk(x y) 1

1

the kernel of

Dk

1

11' < 11

and 7' < 1·

satisfies the following estimates: for 0
7,

.

-

Theorem (3.9} Suppose that {Dk}kel and {Dk}kel are as in Theorem (3.6). Then for all f e LP, 1 < p < m,

(3.10)

where the series converges in the LP norm.

-

We will construct Dk so that Dk( ·,y) is actually a strong smooth molecule of type ( e, e) centred at y with width 2-k. For this we begin by noting tha.t D~ ( · ,y) (the kernel of D~ is defined in §2) is a smooth atom. Here a function a(x) defined on X is said to be a 6-smooth atom centred at x0 with width d > 0 if a(x) satisfies the following properties:

(3.11)

c

B(~,d),

(i)

supp a

(ii)

1a(x) 1 ~ d-1 and 1 a(x) - a(y) 1 ~ d-1- 6 p(x,y) 5,

I

(iii)

a(x)dJJ(x)

X -

We then take Dk

N = TN-1 (Dk),

= 0. 1

-

where TN is defined in section 2. To see that Dk

satisfies the estimates (i) - (iii) in (3.4), it suffices to show tha.t if an operator

T

satisfies T e CZK(E) with a standard kernel having second order smoothness, i.e. (i), (ii)


To estimate p, let ..\u(x,y)

(3.14)

p =

= (J [-p{x,y)] u - . Then

+

Since K is loe&lly integrable on {}

= {(x,y)

f

X • X: x

=

P 1,u

+ P2,u.

# y}, the first term on the right

hand side of (3.14) satisfies

IP1,ul = ~

C

~

C

ll

K(x,y)(1- ..\u(x,y))x0(y)[IP(y)

II I I It/J(x)l~x)

-IP(x)]t/J(x)d~y)d~x)l

K(x,y)XQ(Y)[IP(y) -IP(x)]t/J(x)l ~y)d~x)

= C

llt/1111'

Thus it remains to show that 1 im p 2 = 0, i.e., u-tO ,u

(3.15)

1 im = 0, U-tO

and it is here that we use the strong weak boundedness property of T:

(3.16)

41

I I

~ C ~B(Xo,r))


42

for all f e

c[J(x

x

X) satisfying supp f

c B(x0,r)

x

B(y0,r), II f 11 111 S 1, II f( · ,y) 11,., S r-n

and II f(x, ·) 11,., S r _,for all x,y E X. Now let {yj}jel e X be a maximal collection of points satisfying

for all k.

(3.17)

By the maximality of {yj} jel' we have that for each x E X there exists a point Yj such that p(x,yj) S o-. Let '7j(y)

p(y ,y.>,

= 0[~J

and "ijj(y)

=

[

~ fli(y)

]-1

flj(y). To see that "ijj

is well defined, it suffices to show that for any y e X, there are only finite many '7j with '7j(y) # 0. This follows from the following fact:

"'j(y) # 0 if and only if p(y,yj) S 2o-

and hence this implies that B(yj,o) ~ B(y, 4Ao). Inequality (3.17) shows B(yj, B(yk, ~

h> = +.

B(y, 4Ao-).

lsupp x0 I

Rl

h> n

and thus there are at most CA points yj e X such that B(yj, u/4A)

r = {j: lsupp "ijj I Rl u.

Now let

r and

"ijj(Y)XQ(Y) # 0}.

Note that

# r

S Crfu

since

Thus we can write

and we obtain

It is then easy to check that

supp Au(x,y)"ijj(y)x0(y)[cp(y) - cp(x))¢(x) ~ B(yj,3Au)

x

B(yj,2u) and IIAJx,y)"ijj(y)x0(y)[cp(y) - cp(x)]¢(x)ll 111 S C u'~, where C is a constant depending only on 0, rp, t/J,

'1>• and r

but not on u and j. We claim that


43

Assume (3.18) and (3.19) for the moment. Then since T satisfies the strong weak boundedness property, we have]

I

I

~ E C JJ(B(yJ.,3Au)) u'l ~ C !. CAu u'l )r u

I

= CAru'l

which yields (3.15). U remains to show (3.18) and (3.19).

We prove only (3.18), the proof of (3.19)

being similar. To show (3.18) it suffices to show that for x, x 1 E X and p(x, x 1) ~ u,

since if p(x,x1) > u, then t.he expression on the left above is clearly bounded by

~ C p(x,x1)'1. By the construction of fij, it. followa that. lfij(Y)x0(y) I ~ C for all y e X. Thus


Recall that p(x,x1)

~

u. If p(x,y) > Cu (where C is a constant depending on A but

not on u) then >.u(x,y) = >.u(xl'y) = 0 a.nd so I p(x,y)

~

= 0.

Thus we may assume that

Cu and so, with 0 as in (2.5),

since we may assume '7

~

0. Terms II and

since we can assume that u of Lemma (3.12).

m are easy to estimate:

< 1. This completes the proof of (3.18) and hence the proof

We note in passing that the calculations above, together with the

dominated convergence theorem and T1 = 0, yield the following integral representation:


< Trp,'I/J >

= p +q =

45

lim p 1 u + q 0'-tO I

= AK(x,y) bo(Y)[rp(y)- rp(x})- xl(y)rp(x}} '1/J(x) dp(y)dp(x). Next we prove the following lemma which has a somewhat stronger conclusion than one of the results in [FHJW].

Lemm.a (3.20) Suppose that T: C~(x) ... (C~(x))' is a singular integral operator satisfying T e CZK(E) with a standard kernel having second order smoothness, i.e. (i), (ii) and (iii} of definition (2.7) hold, T(l) = T*(l) = 0 and that T has the strong weak boundedness property. Then for 0 d

< 6
0 to strong smooth molecules of type ( 6, 6) centred at Xo with width d > 0.

£mQf We claim that by the argument in

[FHJW) (see the proof of Theorem (1.15)

there), we need only check that if a is a 6-tmooth atom centred at x 0 with width d > 0 then

when p(x,xa) (d+ p(x.xo))

~

3A 2d and p(x,y)

~ ~ (d+

p(x.xo)). Indeed, If p(x,Xo)

4A

< p(x,y) ~

~

3A 2d

and~ 4A

h- (d+ p(x,x0)), then by the size estimate for Ta in [FHJW},

ITa(x)-Ta(y)l

~

ITa(x)l

+

ITa(y)l

0 for some constant C depending on N but not on k. It then follows from Lemma (3.20) that Dk( · ,y) is a strong smooth molecule of type ( £', f') centred at y with width C 2-k > 0 and this shows that Dk(x,y) satisfies properties (i) -(iii) in Theorem (3.4). We now have

-

(3.21)

DD(f)= lkiSM k k

whenever f

E

E

.,K(fJ,'Y) with 0 < fJ,'Y
M vi ·'Y

Note that

T'N1 satisfies the hypotheses of Lemma (3.20) and again, by the remark

following Lemma (3.20),

where fJ' < {!' < fJ and 'Y'
0 and

.,. > 0 such that

(3.24)

(3.25)

Assumiq (3.24} and (3.25) for the moment, take their geometric mean to obtain

(3.26)

where 0
M E Dk~(f)ll (IJ' 7,} .6 ,

-T'M

~ C2

11~1 u(IJ,'Y) ,..

for some r' > 0. Let M go to • and then (3.23) together with the above estimates yields (3.22).

It remaiDB to show (3.24} and (3.25}. Note first that it is easy to check that the operator

E

lki>M

D~Dk is in CZK( f') with 0
2 then

I

E DNk DL f(x)l lki>M ..

+ I E

kM

nNk D,_(x,y)(f(y)-l(x)]d#J(y)l · ..

J D~ Dk(x,y)f(y)dp(y) I = I + n.

~ C2-k < C2-M for k > M and hence p(x,y) < 1 if M is larger

than 1Df..JC. Thua term I is bounded by a constant times


For term II, we first estimate

I D~ Dk(x,y)f(y)d~y). Choose C so large and c so

small that p(x,x0 ) > C2-k and where

p(x,x0) $ C2-k,

51

D~Dk(x,y) # 0 imply p(y,xa) > cp(x,x0).

we use the facts that

In the case

I f(y)d~y)=O and that D~Dk(x,y)

satisfies the same conditions as Dk(x,y), to obtain

s p(y,xo) 1$ CA2-k(

e k

p(y,xa)

-i

2

)

2 (

1 (

l+p y,xo

))l+r d~y)

llfll

.-K

(P.r)


52

where u

==

"f- "f'

> 0. On 'he o'her hand, in the case where p(x,x0) > C2-k, then

p(y,Xo) > cp(x,x0), and so also

(3.30)

(3.31)

IJ D~ Dk(x,y)f(y)dp(y) I

Combining (3.29) and (3.31) we obtain

(3.32)

which together with (3.27) and (3.28) implies (3.24). Theorem (3.4).

To prove Theorem (3.6), it is enough to show

(3.33)

This completes the proof of


for all g E .,K (fJ' '1') with /J' > fJ and 7' > 1· Now N *(TN -1 )*(g) (Dk)

E

lki~M

D~ D~(g)

=

53

E

lki~M

D:

-1 )* = (TN) * -1 . Thus the proof of Theorem (2.14) can be and (TN

. wtth . Dk replaced by Dk* and I = TN-1 TN by I = TN* (TN) * -1 , and (3.33) applied follows from the proof of Theorem (3.4). We leave the details to the reader.

Finally, to prove Theorem (3.9), note first that f- E f - TN1[TN

E

lki~M

D~ Dk(f)) = TN 1( E

lki>M

lki~M

-

Dk Dk(f)

=

D~(f)) on LP. We now need only

show

(3.M)

By a result of David, Joume and Semmes ([DJS]),

To see that the last term above goes to zero as M -+ m, we again use a result of David, Journe and Semmes ([DJS]):

(3.35)


The last term above goes to zero as M ---+

111

since

is finite. Applying the proof of Theorem (3.4) with I=TN TN"1 and Dk=D~(TN)-1 ,

we

have

Theorem (3.36) Suppose that {Dk}kell is as in Theorem (3.4). Then there exists a family of operators {Dk}kell such that for all f e .,K(fJ,'Y),

{3.37)

where the series converges in the norm of

-

-

vK (fJ' ''Y') with

/1' < fJ

and 7' < 'Y·

Moreover, Dk(x,y), the kernel of Dk, satisfies the conditions in Theorem (3.4), but with

(ii) replaced by

-

-

IDk(x,y)-Dk(x,y')l

~

c [ ___£ ( y,y

-kt'

I )

2=-k+p(x,y)

Theorem(3.38) Suppose that {Dk}kell

]

t' ---,,........=2-----.--,----,-

(2-k +p(x,y))l+f'

is as in Theorem (3.4). Then there exists a

family of operators {Dk}kell whose kernels satisfy the estimates in Theorem (3.36) such that for all f E ( vK(fJ,'Y)),

(3.39)


where the series converges in the sense that for all g

(3..W)

1 im

=

e

A

55

(fJ' ,7') with fJ' > fJ and

< f, g >.

We leave details of the proOfs of Theorems (3.36) and (3.37) to the reader.

§4:. The Besoy and Trlebel-Lizorkin spaces on apaces of homoseneous type In this section we introduce Littlewood-Paley characterizations of the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. For this we need

Proposition (4.1) Suppose

-£

< a
0 such that for all f e ( .Jt(fJ,'Y)), with 0 < {J, 'Y
P and 7' > 7·

if and only if

for all g e .(((P' ,7') with fJ' > {J and 7' > 7·

f

and

. a q= 0, B' p


60

Proof

n

is easy to see that if f e (.6 (/1,7))'

max(O,-a) < 7
p and 7' > 7.

I I

=

-

I<E DkDk(f), g > I

In Theorem (6.20) below we will show that if g e and 7'

,.(11' ,7') with /1' > p > max(O,a)

> 7 >max (0,-a), then {k:1 [2-kall:i)k*(g)llp,]q'} 1/q' < •· This shows that the

last term above vanishes and hence = 0. Similarly, if 11~1· a,q = O, then Fp

In Theorem (6.20) below we will show that if g e

and 7' > 7 > max(O,-a), then

,.(11' ,7') with /1' > p > max(O,a)

ll{k!l[2-kal~*(g)l]q'} 1 /q'llp'

7·


Proposition (4.11) For all f' with 0

< f' < f, ..K(f' ,f')

61

is dense with respect to the

norms II II . ·a,q' 1 ~ p, q < 111 and II II:Fa,q' 1 < p, q < .... Bp p

fi2Qf Suppose that f E ( ..6(11,7))' with max(O, a) < 11 < and 11~1· a,q Bp

< 111, 1 ~ p, q ~ ... , respectively

11~1· a,q

f

and max(O,-a)

< ..., 1 < p, q < 111.

0 such that

and it is easy to see that there exists M1 > M0 such that for all M > M1


Thus, if M

> M1

~

c 2U1 (]' + c 2U1 (]' =

u.

Similarly, if xM is the characteristic function of the set {y e X: p{y,Xo)

> M}, then

63


Since

llfll . a q < • , F• p

there exists M0 > 0 such that

and there exists M 1 > M 0 such that for all M

Thus, for all M

~

Ml'

1

~ C 2U

CT

+

1 Cm

CT

=

CT.

~

Ml'


65

This completes the proof of proposition (4.11).

Pmposition (4.14) Suppose

-f

and f e ( ~ (flo,'Yo)), with

l n-+m

~ lim I< n-+m

-

E DkDk(gn), h> I

kel

by Theorem (3.6)

~ C lim IISnll· aqllhll. -aq' n-+m Bp ' BP' ' by the remark following Theorem (6.20).

The proof for 11~1· a,q < • is similar. Fp

We can now introduce the Besov and Triebel-Lizorkin spaces on a space of homogeneous type.

DefinitiOn (4.15) Suppose that {Dk}kel is a family of operators as in Theorem (2.14)

and

-E

< a
N ~ C2-Na, ~ (2(l-j)f' A 1] 2(j-l)a 2(l-k)a ~ C2(l-k)a, and the first half of Theorem

The last inequality follows from the facts that

J

(5.8). The proof of (5.23) is similar. By the definition of F;•q, we have

~

C II{~[ EE J k-l>N

~a [2(l-j)e A 1] M(~(f))q}l/qll

=

C 11{1:[ EE (2(l-j)f' A 1] 2(j-l)a 2(l-k)a 2ka M(E (f))]q}l/qll k · P j k-l>N

p

by (5.20)

82


since EE [2(l-j)E' A 1] 2(j-l)a 2(l-k)a ~ c2-Na and~ [2(l-j)E' A 1] 2(j-l)a ~ c k-l>N J

by the Fefferman-5tein vectorvalued maximal inequality

by the first half of Theorem (5.8).

Lemma (5.24) Suppose 0

< a < £. Then there exist C > 0 and 6 > 0 such that

(5.25)

for 1 ~ p, q < 111

(5.26)

for 1 < p, q
N size and smoothness conditions with E', where 0 < £' < £, the weak boundedness property uniformly in N, and also TN(1) = 0. More precisely, let

=

EE Ef'k· We claim that for 0 < E' < l-k>N such that TN(x,y) satisfies the following estimates, kernel of TN

£,

TN(x,y) be the

there is a constant C


(5.27)

(i)

ITN(x,y)l ~ C 2-N 6 p{x,y)-1,

(ii)

ITN(x,y) -TN(x',y)l

~ c 2-N 6 ~~~Jlr' p(x,y)-1

for p{x,x') ~

(iii)

for all rp, 1/J

E

Il

83

ix p (x,y), 0
0 and a collection of open subsets

{Q~ c X:

k

f

89

I, TEik}

satisfying the conditions in Theorem (6.1}. A function a k defined on X is said to be

Q.,. a -y-smooth atom for

Q~

if

(i}

(ii}

(iii)

As in the case X = indexed by "dyadic cubes"

B:•q

decomposition of

rt

(see (FJ]},

{Q~} in X, which

and

:Fiq· For

collection of sequences s = { sq}

(6.3}

llsll·

ba,q p

is finite, and, for

-f

we also define certain spaces of sequences

k

Qe{Q.,.}

-f

will characterize the coefficients in our

< a
max(O,a), 7 > max(O,-a). To show (6. 7), we compute


93

(6.11)

~ C 11{k~l[2ka IJlkf(x)l]q}l/qllp

by the Fefferman--.stein vedor-valued inequality

~ C 11~1· aq •

F• p

where the last inequality follows from the remark following Proposition (4.1). To prove part (B), we need the following lemma (see [FJ)).

Lemma(6.12) Suppose 1 ~ p ~ ., #'. 'I e I with 'I~ p and for " dyadic cubes "

where

(6.13)

z': is the "centre" of Q~ in {6.1) and

u

Q!;,

> 0 (recall that I'(~ 111 2--1'). Then


We now tum to the proof of (6.8). Suppose (Dk)kEI is aa in (4.15) and a 11 is Q., an Hlmooth atom. H k

~

p., then

(6.15)

H k

~

p., then

(6.16)

Thus, if f = J: s,.,.ua 11, then II,T '"f,- Q,-

~C

+

11

k J:

~~=-

Cll

#( 1 /2+~1 /P) A 2-{k"""II)(E-a)llfl

E

i~=k+1

by (6.16) and (6.13) with f1

= 11

II

~ 1 1 2 +~1 /P) A 2-{Jr-k)(a+E)IIfl by (6.15) and (6.14) with f1 = k II


95

Let a ={a#'} pel and b = {b#'}pel be defined by a#'= 2/J{ 1/ 2+a-1/P) A#' and

b#' = 2-IJ{t-a)

X{#':#'~ 0}. Then since f > a, k

II

E

F-

21J{ 112+a-1/P) A 2-{k-#')(t-a) lltt = lla #'

* bll

fl

Similarly, with a as above but with b = {b#'} where b#' = 2-IJ{a+t)

II E Fk+l

x{~£=

#' > 0},

2/J{1/2+a-lfp) A 2-{,Ht)(a+t)ll #' fl

These estimates yield (6.8). To show (6.9), let a(x) = {a#'(x)} pel with

~

0},

b = {bP}#'fl

with

2/J{ t+a)X{#': #'

< 0}. Applying Lemma (6.12) and the estimates in (6.15) and (6.16), we

have

b#' = 2-IJ{t-a) X{#': #'

and

c = {c#'}utl

with

c#' =

96


~ C ( I(x)

where I(x) =

II

+ II(x)

a* b llti

~ c II

),

a llti and II(x) =

II

a* c llti

~ c II

a llti since b, c

e

r.

Hence, integrating yields

valued

maximal

function

inequality = Cllsll·aq· f , p

To complete the proof of Theorem (6.5), we need to remove the assumption that

6 = 1/2. For convenience, we assume 6 = 2-i for some i ~ 1 in Theorem (6.1) (but the same proof carries over to the general case). Thus all we need do is show that if a family of operators (Pk:)kel satisfies the usual conditions, but scaled to 2-i, namely


(6.17)

=0

~ C2-ik, and IIPkll .. S ctk

(i)

Pk(x,y)

(ii)

IPk(x,y)- Pk(x' ,y)l ~ c:zik(1+E) p(x,x')E

(iii)

IPk(x,y)- Pk(x,y')l ~ ctk(1+E) p(y,y')E

(i•)

I[Pk(x,y)- Pk(x,y')]-{Pk(x' ,y)- Pk(x' ,y')] I

i i

(•) (vi)

if p(x,y)

97

S C tk(1+2E)p{x,x')E p{y,y')E Pk(x,y)dJ.(y)

=1

Pk(x,y)dJ.(x)

=1

(6.18)

(6.19)

-

Indeed, with this established, we simply repeat the above proof with f = E QkQk(f) in place of f

=

kel

-

E DkDk(f).

kel

To see (6.18), suppose f e

.-K {{J,7)

max (0,-a) < 7 < E. Then by Theorem (3.6),

Thus,

n Bp0 •q

with max (O,a)

< {J < E and


98

It follows from (4.5) that for 0

0

such that

Ib I

~ c

and also that for any

>

0 almost everywhere

because of the validity of Lebesgue's theorem on differentiation of integrals on spaces of homogeneous type. Our more general atomic decomposition is the following.

Theorem (6.25) Suppose that a complex-valued bounded function b is para-accretive. Then for 0 < a < t,


101

. (A) f

E

Biq iff there exist sequences of numbers {sQ}Q"dyadic" and unooth

atoms {aQ}Q"dyadic" which satisfy

supp aQ .c. CQ, where CQ =

(i)

B(z~, CAl') if Q = Q~ for

some .,. as in Theorem (6.1)

(6.26)

(B) f

(ii)

J

(iii)

laQ(x)l

E

aQ(x)b(x)dp(x) = 0

~ p(Q)-1/ 2

and laQ(x)- aQ(Y)I

~ p(Q)-1/ 2-E p(x,y)E,

Fiq iff there exist sequences of numbers {sQ}Q"dyadic" and smooth

atoms {aQ}Q"dyadic" which satisfy (6.26) (i), (ii) and (iii), such that f = E sQaQ and ll{sQ}II £a,q • II~IFa,q· p p Proof: The "if" halves of (A) and (B) follow from the atomic decomposition in Theorem The "only if" halves of (A) and (B) follow from

(6.5) together with Remark (6.20).

Theorem (5.8) and the following result in [DJS] concerning para-accretive functions:

PropoBition (6.27) ((DJS]) Suppose that b is a bounded complex-valued function on X and that i is a positive integer. The following are equivalent:

(a)

b is para-accretive

(b)

There exist C, u,

such that

E

>

o,

and for each k e I, a function Pk: X

x

X _. (


102

~ C 2ik; Pk(x,y)

(i)

IPk(x,y)l

(ii)

IPk(x,y)- Pk(x,y')l

I I

(iii)

(iv)

Pk(x,y)dp(y)

This proposition was proved for i

=1

=

I

if p(x,y) ~ C 2-ik

~ C ik(1+t) p(y,y')f

~

I Pk(x,y)b{y)dp(y)l

=0

f1

for p(y,y')

~

2-ik

for all x e X

Pk(x,y)dp(x)

=1

for all x, y e X.

in [DJS) and the proof for i

Suppose now that b is para-accretive, let ~

* = PkM

~

2 is similar.

1 PkMb, where Pk (Pkb)-

*

is as in (6.27)(b), Pk denotes the transpose of Pk and Mg denotes the operator of multiplication by g. It is easy to check that the Sk, k e I, satisfy the conditions in (5.7}, but relative to the scale {2ik}, namely

.

. -ik if p(x,y) ~ c2 , and IISkll.., ~

Sk(x,y)

(ii)

l~(x,y)- Sk(x' ,y)l ~

!

(iii)

Set Dk

=0

(i)

= Sk -

Sk(x,y)dp(y)

d 'k;

ctk(l+t) p(x,x')£;

= 1.

Sk-1' From Theorem (5.8) we know that 1 im TN = I in the norm of

N-t..,

.

B;•q and F:•q for a certain range of a, p and q, and thus, if N is large enough, we have that f = TN(T'fi)f in the norm of Biq and Fiq· Therefore, for f e Biq•

=

EE s .a . j .,.

QJ.,. QJ.,.


103

where

It is easy to check that a . satisfies conditions (6.24) (i), (ii) and (iii). To see that QJ

r

ll{s }II. Q

p

ba,q

we have

S C

11£11. aq B• p

=

{E[E[p(Qj)-a-1/ 2+1/Pis ·I j r r QJ r

Jp]qfp} 1/q

~ C 11£11.

p

Ba,q

,


104

The proof for f E F;•q is similar using the Fefferman-Stein vector-valued inequality.

Theorem 6.25 does not hold in general for bounded functions with bounded inverse. For example, there is no atomic decomposition (as in Theorem (6.25)(A)) of

11;•2

into

atoms with vanishing b-moment if b(x) = eix on the real line. Indeed, we show below that for every compactly supported f e

11;•2(11) , 0 < a < 1/2, satisfying part (A) of

., 111

Theorem (6.25) with b = brJ.x) = e1 x ,

P>

1

a+ 2"• we have Jf(x)brJ.x)dx

= 0.

In

fact, for functions b that are locally para-accretive on I (i.e. (6.24) is required to hold only for intervals of length at most one - b/1' 0 ~ fJ ~ 1, are examples), we can come close to characterizing when the above atomic decomposition holds for compactly supported f, in terms of the mean oscillation,

MeanOsc(B,Q)

where BQ

1 = TQT

6B(x) dx,

of any antiderivative B of b on large intervals Q. More precisely we have,

Definition (6.28) A function b

e L111(1) is said to be locally para-accretive if there exists

6 > 0 such that for all x e IR and 0 < r

~

1, there exist x' E (x-r, x+r} and r'

>0

such that (x'-r', x'+r') C (x-r, x+r) and,

(6.29)

I~

x'+r'

I

b(y)dyl

~

6.

x'-r' Theorem (6.30) Suppose b is a bounded complex-valued function on I denote any a.ntideriva.tive of b. Let 0

< a < 1/2.

and let B


105

1

(A) Suppose that 1 im R R-1111

a-2"

sup MeanOsc(B,Q) = 0. Then /f(x)b(x)dx Qc[-R,R]

= 0 for every compactly supported f Theorem (6.25) (with X= I and

£

e B~· 2

with f = EsQaQ satisfying part (A) of

= 1). In particular, there is no atomic decomposition

into a sum of atoms with vanishing b-moment (as in Theorem (6.25)(A) with X= It and

f

= 1) for any compactly supported function f e B~· 2 with /f{x)b(x)dx # 0. a-1

~ in f

(B) Conversely, let A(R) = R

MeanOsc(B,Q). If b is locally

IQI=R

E (~ )2 < 111 , then for every compactly supported

para-accretive and

k=1 A(2 )

there are sequences of numbers

{sQ}Q"dyadic"

and smooth atoms

satisfying the conclusions of Theorem (6.25)(A) (with X= It and

Remark (6.31) Let brJ.x) = eilxlp{i/1 Then

+ (1-P)Ixi-IIJ

BrJ.x) = sgn(x) lxll-,8 eilxi,B

MeanOsc(BP'[s,t])

111

for lxl

E

f e

Ba• 2 2

I

{aQ}Q"dyadic"

= 1).

~ 1, and

is an antiderivative for

1 otherwise.

lxl~

1

and

min {t 1-P, t-il} for 0 < s < t. Theorem 6.30 now shows that all

compactly supported f e B~· 2 (0 < with vanishing b-moment if 0 ~

a< 1/2) have atomic decompositions into atoms

P< a +

1/2, but not if

P > a + 1/2.

The same

result is easily obtained for bp..x) = ei IxI 11•

R.emark (6.32) Theorem 6.30 can be generalized to ~ with MeanOsc(B,Q) replaced by 1

IQI~

sup 1/aQbl for any cube Q Q-atoms aQ

in~ (for n = 1 this is equivalent to the

previous definition- see the lemma below). However, our proof of this result requires the atoms in part (A) of the theorem to have vanishing b-moments up to order [j], wlrile our use of Theorem 6.5 in part (B) yields a decomposition into atoms having vanishing bmoment of order 0 only. U is for this reason that we restrict attention to the case n = 1 here.

106


To prove Theorem 6.30, we will use the following estimate on the size of IJaQbl where aQ is a Q-atom (as in Remark (6.20), we say that aQ is a Q-atom if it satisfies merely the usual size and smoothness conditions (6.26)(i) and (iii) with

E

= 1).

Lemma 6.33 Suppose B is an antiderivative of b. For all intervals Q,

(6.34) !IQI-112 Mean0sc(B,3Q)

~

sup IJaQbl Q-atoms aQ

Proof Suppose aQ is a Q-atom. Then llaQ'II 111

~ Conversely, fix

Q

~

~

3 IQI-112 Mean0sc(B,3Q)

IQI-3/ 2 and supp aQ c 3Q, so

3 IQI-112 Mean0sc(B,3Q).

and set

6(x) = sgn[B(x) - B3Q).

=

Let

03Q

=

o

j3QI-1

/6(x)dx and define aQ(x) liQI-3/2 I [6(t)- 3Q)dt. Then supp ~ ~n~~ aQ C 3Q, llaQ'II 111 = liQI-3/ 2 110- 83QIIm ~ !IQI-3/2, and so llaQIIm ~ I3QI llaQ'IIm

~

IQI-112. Thus aQ is a Q-atom and

= liQI-3/2

/6(x)[B(x)-BQ]dx 3Q

=

= liQI-3/ 2

!1 Qj-112 Mean0sc(B,3Q)

/IB(x)-B 3Qidx 3Q


107

This completes the proof of Lemma 6.33.

Proof of Theorem 6.30 (A)

Let

f E B~· 2

have compact support and the atomic decomposition

f = EsQaQ as in part (A) of Theorem (6.25) with X= I and

E

= 1. For R ~ 1, let

f/R(x) = 1/(x/R) where 11 is a smooth nonnegative function supported in [-3,3] and equal to one on [-1,1). Let g

= fiR(b

- bR) where the constant bR

chosen so that g has mean zero. Then g e

B2°•2

=

111 b _!!.__ is /7'/R

and so by duality (see §7),

(6.35)

From the definition of a Q-atom, we obtain and so

(6.36)

lfaQfiRI

~

C min {IQ1 112, RIQI-l/ 2},

108

Also,


since

cR-1/2'1n.

is a

[-R,R]-atom,

(6.32) yields

I/'7Rbl

~

c

Mean0sc(B,(-3R,3R]) and so

(6.37)

IIIRI ~ C R

a+l

1 lbRI ~ C R

0

-

2" Mean0sc(B,[-3R,3R])

-+ 0

as R

-t IIJ

1

by hypothesis. To show IR

-+

0 as R -+

111

we need only that R

Q-2"

sup Qc[-R,R]

MeanOsc(B,Q) is bounded. To see this, we write IR as

(6.38)

Now if xQ denotes the centre of the interval Q, then ['7R- '7R(xQ)]aQ is c ¥ times a Q-atom when I Q I ~ R and since aQ has vanishing b-moment, we have for Q c (-3R,3R],

{6.39)

~

C R-1 I Q 1112 Mean0sc(B,3Q)

by (6.34) and the hypothesis. Thus we have

~

1

C R-a- 2" I Q 11/ 2


(6.(())

S C R-a/ 2 ... 0 as R ...

111.

Similarly, using (6.39), we have

(6.41)

SC{

E

lsQ 12 IQl-2a } 1/2 ... 0 as R ... IIJ.

IQI>R1/ 2 Finally, to bound term VR in (6.38), we use the estimate

109

110


(6.42)

for IQI ~ R,

which follows from (6.34) and the hypothesis since '7RaQ is cR1/ 2 1QI-1/ 2 times a [-R,R)--atom for IQ I ~ R. Then

(6..f3)

From (6.38}, (6.40), (6.41) and (6.43), we obtain that IR ~ 0 as R ~ m, and together with (6.37), this shows that the right side of (6.35) tends to zero as R ~ m. Since f77R

=f

for R sufficiently large (supp f is compact) and bR ~ 0 as R ~ m (see (6.37)), we now conclude from (6.35) that /f,.,Rb ~ 0 as R ~CD. But f77Rb

= fb for R large and 80 /fb = 0.

(B) Since b is locally para-accretive, we can find Pk:IR

x

I ~ ( for each k ~ 0

(but no longer for k

< 0) satisfying (i)-{iv) of Proposition 6.27. As in the proof of

Theorem 6.25, set Sk

* = PkM

1 so that Dkb

* = 0. = Dkb

construction as in §2 we have

1 PkMb for k ~ 0, and Dk (Pkb)Then I

=

lim Sk

k~m

=

S0

+

= Sk -

Sk 1 for k > -

E Dk. Applying Coifman's

k~1


111

(6.«)

where

TN =

~ s0nt

t=1

u0 +

m N E DkDk, k=1

N Dk =

E

jk-lj~N

Dl and

u0

=

2

s0 +

l>1

As in Lemmas (5.21} and (5.24}, 1 ;m TN =I in the norm of N~m

N E DkSO k=1

:8~2 ,

for N sufficiently large, TN- 1 exists and is bounded on :8~ 2 . Thus for f

+

and so

e :8~· 2 ,

(6.45)

where

s~ and a~ are defined as in the proof of Theorem 6.23. Here also we have that

the atoms a~ satisfy (6.26} (i), (ii) and (iii) for k ~ 1, but only (6.26) (i) and (iii) for


112

k = 0.

k=1 {E r 11~1

ls~l 2 }112 ~

C

~

E E lskl 2 2-2ak }1/ 2

Moreover, {

11~1

T

a2 B2•

C

11~1·

B~·

T

= C (11~1 . a 2 + 11~1 2 ).

2 and for k = 0, we have

This completes the proof since

B2•

a 2 = 11~1 . a 2 for f compactly supported. Indeed, B2' B2'

A

llfllm ~ 11~1 1 ~ Cll~l 2 if f is compactly supported.

since

Now choose

f

sufficiently

small to obtain 11~1 2 ~ C f 11~1· a 2.

B2'

We now show, using the hypothesis in part (B) of Theorem 6.30, that each atom a 0 has an atomic decomposition a 0 = T

T

;; sk ek

k=O

IT

'T

into atoms ek

b-moment, and with uniform control on the constants involved. notation, we depart here from our usual convention and use k

with vanishing IT

For convenience in

> 0 (rather than k < 0)

to index an atom of width 2k. Without loss of generality we consider a

= a~

which

satisfies

(i) supp a

(6.46)

(ii) (iii)

c

k

[-2

°, 2k°]

IJa(x)b(x)dx I

(k 0 fixed)

~ 1

la(x)l 5 1 and la'(x)l ~ 1

For k

~

[-2,2]

and equal to one on [-1,1].

0 let '1k(x) = rl,..x/ak) where '7 is a smooth nonnegative function supported in Let

Qk

denote the interval

[-2k, 2k]

and for


k

~

k0 choose a Qk-atom

~

such that

/~b

Lemma 6.33 and the definition of A(R), we have

Now let tk =

f~b

for k

~

k0 and write

E d k>k k - 0

each

~

is a Qk-atom with vanishing b-moment and

For convenience, we take k0 = 0 and rewrite the above as

a0 T

= k=O E sk,Tek,T

where ~.,.. is a [r-2k, r+2k]-atom with vanishing b-moment and

'

113


114

{k=O E Isk,r 12 2-2ak } 1/ 2

(6.48)

It remains to show that if

f

~

C < m, for all

T

E Z.

is compactly supported, then these atomic

decompositions for the a~ can be combined into a single atomic decomposition for f. This will be accomplished by showing that the sequence {s~} is summable. Without loss of generality we may suppose that f is supported in [0,1] with

11~1

.a 2 ~

B2' if

Q~ = [r, r+1], we have for Irl ~ C, r+l =

c

I

ln~(Ti1 f](Y)I

dy

T

r+l = C

1

I III D~(y,z)(Ti1 (z,w)0

T

~

C

TN1(r,w)]f(w) dwdzl dy

r+1

1

T

0

I I I ID~(y,z)l [1~=;1]

f.'

lr:wl lf(w)l dwdzdy

since 11~1 1 ~ C a 11~1· a 2 for f supported in [0,1]. Thus we have L B 2•

1. Then


115

(6.49)

Finally, we can write

tk ,rr =

d

k rr -

_1_ E sO 1k rr k k+l .,. ~ •.,. ~.r ' r. 2 rr

Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces

Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces

Harmonic Analysis on Spaces of Homogeneous Type

Harmonic analysis on spaces of homogeneous type