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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CONTENTS
§1.
Introduction
1
§2.
Tri is a Calder6n-Zygmund operator
1
§3.
The Calder6n-type reproducing formula on spaces o£ homogeneous type
36
§4.
The Besov and Triebel-Lizorkin spaces on spaces o£ homogeneous type
56
§5.
The T1 theorems £or
§6.
Atomic decomposition o£
§7.
Duality and interpolation o£
:ea,q and ta,q p p
69
:e;,q and r;,q :e;,q and r;,q
88 116
125
References
v
ABSTRACT
We develop Littlewood-Paley theory on spacea of homogeneous type. A key tool iB the CODBtruction of a Calderon reproducing formula that permits the introduction of Besov and Triebel-Lisorld.n spaceB in this general setting. We then obtain Tl and Tb theorems, atomic deoompositions and duality and interpolation results.
vi
§1. Introduction The purpose of this paper is to develop Littlewood-Paley theory on spaces of homogeneous type. It has been known for some time that most of the function and
If,
distribution spaces on
namely the Besov and Triebel-Lizorkin spaces, share a
common underlying structure, and that by using Littlewood-Paley theory these spaces can be characterized through the action of appropriate families of convolution operators. Classically, the Fourier transform is the basic tool for studying Littlewood-Paley characterizations of these spaces, since this operator diagonalizes convolution. precise, let
(1.1)
For k
eI
f{J
To be
be a function with the following properties:
(i)
rp E ~(If),
(ii)
supp VJ t
U £If: ~ s Ie1
(iii)
l;,c{)l ~
c>O
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
<
p
<
m, namely that for 1
<
p
<
m there
Y.S. HAN AND E.T. SAWYER
4
(1.7)
Since an appropriate family of operators {Dk}kel can be constructed on a space of homogeneous type, the above result of David, Journe and Semmes holds also in this setting. In (HJTW] it was shown that the Besov and Triebel-Lizorkin spaces on ~ can
be characterized by such a family of operators {Dk}kel' namely
(1.8)
II~IJio.q • {~1 (ok"11Dk(IJ11p]qt/q p
and (1.9)
ll~lra,q = ll{k~1(2ka1Dk(f)l)q}1/qllp· p
The key step there was to show that at most
C 2-N 6 for some
6
~
is a Calder6n-Zygmund operator with norm
> 0 and then the T1 theorems for the Besov and
Triebel-Lizorkin spaces on ~ can be used to show that TN 1 is bounded on the Besov and Triebel-Lizorkin spaces on
all.
The essential ingredient used in (DJS) and [HJTW)
is the following reproducing-type formula
(1.10)
where
1
I = T TN N
N Dk =
depending on
E
lli~N
N.
=
N
-1
E DkDkT N
kel
Dk+l satisfies the same estimates as
Dk
but with constants
The difference between the formula in (1.10) and the Calder6n
reproducing formula in (1.5) is the presence of the operator
TN 1.
However the
~ 2 boundedness of TN on spaces other than L depends on knowing that the T1 theorem
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
5
holds for these spaces. In [DJS), David, Joume and Semmes established the Tl theorem for L2 on spaces of homogeneous type by proving the boundedness of
Tj:i,
and thus
obtained Littlewood-Paley characterizations of LP. But this approach does not work for developing Littlewood-Paley characterizations of the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type.
Using the family of operators
{Dkhel , one can
formally introduce the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. However the formula in (1.10) doesn't allow one to show that these spaces are independent of the choice of the family of operators {Dk}kel· Instead, one needs an analogue of the stronger Calderon reproducing formula. On the other hand, the method in [HJTW}
does not help either since the key results used in [HJTW)
were the Tl
theorems for the Besov and Triebel-Lizorkin spaces. In other words, to use the method in [HJTW) one first needs the Tl theorems for the Besov and Triebel-Lizorkin spaces which ultimately depend on the Calderon reproducing formula (1.5). In order to develop Littlewood-Paley characterizations of the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, it suffices to show that Ti1 is a Calderon-Zygmund operator (whose kernel has some additional second order smoothness, namely (2.14) (iii)) since then
TN 1 D~
will satisfy nice estimates
sufficiently similar to those satisfied by Dk itself. By a result of Lemarie ([M]) one can show that
Hi = RN• RN
is a Calderon- Zygmund operator but the kernel of
necessarily has less smoothness. This prevents us from showing that
TN 1
R~ is a
Calder6n-Zygmund operator by simply iterating RN (either the smoothness disappears or the constants blow up). In this paper, using Cotlar-Btein techniques, we obtain sharp estimates for
R;, m = 1,2, ... , which allow us to show that
Ti1
is a
Calder6n-Zygmund operator, and hence establish a Calderon reproducing formula analogous to (1.5) on spaces of homogeneous type.
From this we then develop
Littlewood-Paley characterizations of the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type just as in the case of
an.
Y.S. HAN AND E.T. SAWYER
6
A brief description of the contents of this paper follows.
In section 2 we give a
presentation of basic notions of Calder6n-Zygmund operator theory on spaces of homogeneous type, and prove the fundamental result that Tfi1 is a Calder6n-Zygmund operator (with the additional smoothness (2.14) (iii)). In section 3 we introduce a class of distributions on spaces of homogeneous type and then establish a Calder6n reproducing formula for this class. The Littlewood.,-Paley characterizations of the Besov and Triebel-Lizorkin spaces will be established in section 4, and some basic properties of these spaces will be studied there. In section 5 we give several versions of Tl theorems on the Besov and Triebel-Lizorkin spaces and as an application, show that for
a
~
0, the
Littlewood-Paley characterizations are valid with only half (depending on the sign of a) the usual smoothness and cancellation conditions on the Dk. The atomic decompositions of these spaces will be given in secton 6.
We also give an atomic decomposition into
atoms with vanishing b-moment, where b is a para-accretive function, which is new even for the case of b
If.
On the real line, we come close to characterizing the functions
for which such atomic decompositions are possible for compactly supported
B~• 2 (1R), a > 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 <
= 6(y,x)
for all x, y e X,
6(x,y) ~ A[6{x,z)
111
such that for all x, y, z e X,
+ 6{z,y)).
Any quasi-metric defines a topology for which the balls form a base. However, the balls themselves need not be open when 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'
<
<
111
such that for all
111
> 0,
I-'{B(x,2r)) ~ A' 1-'{B(x,r)).
Here 1-' is assumed to be defined on a u-algebra which contains all Borel sets and all balls B(x,r). We suppose that 1-'{X)
= 111
and 1-'{{x})
=0
for all x
f
X. Macias and
Segovia have shown ([MS) Theorems 2 and 3) that in this case, the functional
7
8
Y.S. HAN AND E.T. SAWYER
p'{x,y) = inf{p(B): B is a ball containing x and y}
is a quasi-metric on
X
6
yielding the same topology as
quasi-metric p equivalent to p', and C
and that there is 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)
<
r}.
From (2.4), we obtain (by decomposing into annuli for
example) the often used formulae
I
p{z,x)E-1dp(z)
1:1
rf and
B(x,r) Let
Cd, '1 > 0,
( I )c p{z,x)-1-Edp(z) B x,r
1:1
r-f for
Endow
=
Cd
> 0.
be the space of all continuous functions on X with compact
support such that
(2.6)
E
sup If(x) - f(y) I x#y p{x,y)'l
with the natural topology and let
(Cdl'
be its dual space.
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
Definition (2.7)
A continuous complex-valued function
K(x,y)
{(x,y) e X • X: x # y} is called a standard kernel if there exist 0
defined on
< f ~ 6,
and C
9
0 =
<m
such that for all x, y e X with x # y,
(i)
IK(x,y)j ~ C p{x,y)-1,
(ii)
jK(x,y)- K(x,y')l
+ jK(y,x)- K(y' ,x)l ~
C p{y,y')fp{x,y)-{l+f)
for all x, y and y' e X with p(y,y') ~
h p{x,y).
For future reference, we say that a standard kernel has second order smoothness if in addition,
(iii)
I{K(x,y) -
K(x' ,y)]- [K(x,y')- K(x' ,y')] I
~ C p{x,x')fp{y,y')fp(x,y)-{1+2f) for p{x,x') ~ ~ p{x,y) and p(y,y') ~ ~ p{x,y). 3A 3A We remark that the second difference condition (iii) is satisfied by virtually all
classical examples of Calder6n-Zygmund operators.
In any event, we only need this
condition for the kernel of the operator TN"1 (which is used to derive the Calder6n reproducing formula) and it plays no further role in the paper.
Definition (2.8) A continuous linear operator T: Cd
-t
(Cl/>'
is a singular integral
operator if there is a standard kernel K such that
for all f, g e
c[/ with
=
!.!
supp f n supp g
K(x,y)f(y)g(x)dp(y)dp(x)
= +·
We then write T e CZK(f).
Ilefinition (2.9) A singular integral operator T is a Calder6n-Zygmund operator if T can be extended to a bounded operator on t 2. For such operators T we write T e CZO.
10
Y.S. HAN AND E.T. SAWYER
It was shown in [CW] that any Calder6n-Zygmund operator is bounded on LP for
all p e (1,m) and of weak type (1,1), and also bounded from Lm to BMO. A basic problem is to decide when a singular integral operator Calder6n-Zygmund operator.
T
The first significant characterization on
theorem of David and Journe [DJ].
is actually a
If
is the T1
The T1 theorem was generalized to spaces of
homogeneous type by Coifman, and the proof of this result was given in [DJS]. To state this result we need the notion of weak boundedness.
Definition (2.10) A singular integral operator T is said to have the weak boundedness property if there exist 0 < f'J
(2.11)
I I
where ~. ¢
t
11¢11'1 ~
~
~
1 and C <
m
such that for all x e X and all r > 0,
C r,
Cd with supp ~. supp ,Pt. B(x,r), ll~llm ~ 1,
llt/JIIm
~ 1, and 11~11'1 ~ r-'1,
r-TI. We write T e WBP.
Theorem (2.12) (The T1 theorem [DJS]) A singular integral operator T on a space of homogeneous type is bounded on L2 if and only if (a) Tl e BMO, (b) T*1 e BMO, where T* is the transpose of T, and (c) T e WBP.
The hypotheses that ~X)
=m
and ~{x})
=0
for all x e X allow us to
construct an approximation to the identity.
Definition (2.13) A sequence (Sk)kel of operators is said to be an approximation to the identity if there exist 0 < y,y' e X,
~(x,y),
t ~
0 and C < m such that for all k e l and all x, x' and
the kernel of Sk, is a function from X " X into ( satisfying ,
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
=0
if p{x,y)
11
~ C 2-k and IISkllm 5 C2k,
(i)
Sk(x,y)
(ii)
ISk(x,y)- Sk(x' ,y)l 5 C 2k(l+E)p{x,x')f,
(iii)
ISk(x,y)- Sk(x,y')l 5 C 2k(l+E)p{y,y')f,
(iv)
I[Sk(x,y) - Sk(x' ,y)]- [Sk(x,y')- Sk(x' ,y')]l
5 C 2k(1+2E)p{x,x')fp{y,y')f, (v)
LSk(x,y)dp(y)
=
1,
(vi)
LSk(x,y)dp(x)
=
1.
The second difference property (iv) is usually omitted in this definition.
However, the
usual construction of an approximation to the identity satisfies (iv) anyway. Indeed, in [DJS] it was pointed out that one can construct a sequence (Sk)kel satisfying (i),(ii), (iii) and (v), (vi) as follows. Take a smooth function h: 1+ - 1+ equal to 1 on [0,
!1
and
to 0 on [2, m]. Let Tk be the operator with kernel 2kh(2kp{x,y)). The properties of p imply that
t 5 Tk1 5 C
of multiplication by
for some 0 < C < m. Let Mk and Wk be the operators
T~l
and
[Tk[T~rJr1
respectively, and let
Sk =
MkTkWkTkMk. It is clear that Sk(x,y), the kernel of Sk, satisfies (2.13) (i),(ii),(iii) and (v),(vi). The second difference property (iv) also follows easily once we write
Furthermore, lim Sk k-tm
=I
and
lim Sk k-t-m
=0
in the strong operator topology on L2.
Y.S. HAN AND E.T. SAWYER
12
AB mentioned in the introduction, it wa8 shown in [DJS] that TN 1 (where TN = E D~Dk., Dk = Sk+l- ~~ is as in (1.6)) is bounded on 1 2 and hence TN1 has the kel weak boundedness property. However, for our purposes we need a stronger version of the -1
weak boundedness property for TN .
Definition Let K be the kernel of an operator T. We say that T has the strong weak boundedness property, written T e SWBP, if there exists a constant C > 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' <
(ii)
E
there exists C > 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),
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
(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 <
< E there exist constants C > 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
Y.S. HAN AND E.T. SAWYER
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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
)"
Y.S. HAN AND E.T. SAWYER
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'
<
E,
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
where a,
<
P> 0
and
a + P= 1,
and this proves (2.25) with 'Y
= at > 0
and
Similarly, (2.20) and (2.28) yield (2.26). To see (2.27), we first consider k
t.
19
f'
= Pt
~
L In
that case, using the smoothness of Dk, we have
=
IJ[Dk(x,z)- Dk(x,y)) [Dt
which proves (2.27) for the case where k
~
L The proofs of (2.27) for
since D* t<1) = 0
k~
l and of (2.28)
are similar. Note that in the above estimates there are two cases: when k, the index of the left operator, is less than
l, the index of the right operator, we subtract the constant
Dk(x,y) from the left operator; while in the case where k
~
l, we do not need to subtract
anything. This feature will recur repeatedly.
We now turn to the proof of inequality (2.16). often write x
(2.30)
= z0, y = zm
and y'
= zm+l"
For convenience in notation, we
In particular we then have)
Y.S. HAN AND E.T. SAWYER
20
where Ej
= Ej[l,k] = Dl.+k· Dk.' 1 ~ j ~ m. Let J
J
sj
J
= kj A (lj +
kj) , 1 ~ j ~ m,
80
that
-II·
the kernel Ej(zj-l, zj) is supported within C 2 J of the diagonal, i.e., Ej(zj-l' zj) = 0 -II·
-II
for p(zj-l' zj) > c2 J. Define sm+l e I by 2 m+l choose successively
(2.31)
h < ~ < ... <.it = m+l
80
-1
that
s. and s. > S· for j > h• lt J
s.
=
s.
=
min s. and sj > s~ for j > ~· h<j~m+l J
s. Ja
=
min s. and s. > s. for j > j3, Ja J ~<j~m+l J
lt ~
min
-II
< p(y,y') ~ 2 m+l. Now
lt = IJm = IJm[l,k])
(s.
l~Hm+l J
etc., and let J = {jl' ~····.it-1 }. It is for precisely these j's that we must consider EJ{zj-l' zj)Ej(zj-l' zj+ 1) in place of Ej(zj-1' zj) and
(2.32)
-
Ej(zj-l' zj, zj+l)
=
80
we set
{Ej(zj-l' zj)- Ej(zj-l' zj+l), Ej(zj-l' zj)
We now claim that
To prove (2.33) we use descending induction to establish
,
j
eJ
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
-
21
-
Ep(zp-1,zp,zp+1) ... Em(zm-1'zm,zm+1)dp(z1) ... dp(zm-1),
for
1
~
p
~
m.
Em(zm_1,zm,zm+ 1)
The ease
p = m
follows immediately from the definition
= Em(zm_1,zm)- Em(zm_1,zm+ 1).
Now suppose (2.34) holds for
some fixed p with 1 < p < m. We show (2.34) then holds with p replaced by p- 1. Indeed, if p - 1 e J, then
-
-
-
-
Ep(zp-1,zp,zp+ 1)... Em(zm_1,zm,zm+ 1)dJ.&(z1)... dJ.&(zm-l),
Ep(zp-1'zp,zp+l) ... Em(zm-1'zm,zm+1)dp(z1) ... dp(zm-1),
JEp(zp-1,zp,zp+1)dJ.&(zp-1) = 0
since
and the variable zp-} doesn't occur in the
~em&ining fa.ctors E 1(z0,z1) ... Ep-2(zp-3,zp-2)Ep-1(zp-2,zp) Epj-l(zp,zp+ 1,zp+2) ...
Em(zm_1,zm,zm+l). On the other hand, if p - 1
t
J, then Ep-1(zp-2,zp-1,zp)
Ep-1(zp-2,zp-1) and so in eit~er ca.se we obtain (2.34) with p- 1 in place of p. If j
(2.35)
t J,
-
we estimate Ej in (2.32) by
IEj(zj-1,zj,zj+l) I
=
Y.S. HAN AND E.T. SAWYER
22
-
which follows from (2.20). To estimate Ej in (2.32) if j e J, we first observe that if
- -
the integrand E1" .. Em in (2.33) is non-zero, then
for jeJ,
(2.36)
where C depends on the pseudometric p. Indeed, for M large enough, d(z,w) = 1
p(z,w)1i/I is (equivalent to) a metric on X ((MS]). Fix j e J and let k be the least integer (if it exists) such that j
~
m +1 and k
t
J. Then the integrand in (2.33)
includes the factor
which clearly vanishes unless
(i) (ii)
(iii)
But since j, j + 1, ... k- 1 e J, (2.31) shows that sj < sj+ 1 <...sk_1 < sk and so
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
23
which yields (2.36). In the case j, j+1,... m+1 are all in J, the above argument still -fi
applies since p(zm, zm+ 1) ~ 2 m+l. Combining (2.36) with the definition (2.32) of
- -
-
Ej, we conclude that when Er .. Em # 0,
1~
(2.37)
H m+l. -s.
Indeed, if Ej(z}-1, zj) # 0, then p(z}-1, zj) ~ C2 J while if Ej(z}-1, zj+ 1) # 0 in case j e J, then
by (2.36) which yields (2.37) since sj < sj+ 1 for j the matter.
- -
e J. Inequality (2.37) is the crux of
-
For j e J and E1... Em # 0, we estimate Ej by
-
jE.(z. 1,z.,zj+. 1)1 JJ- J
(2.39)
~
[
p(z ., z.+ 1)] f'
~-il/
= IE-(z. ,z.)- E.(z. 1 ,zj+. 1)1 JJ-1 J JJ-
s. -fi, -s. --riL-l C 2 lx({p(z}-1,zj) ~ C2 J} U {p(z}-l'zj+1) ~ C2 J}) 2 J by (2.25),
by (2.36) and (2.37). UBiDg estimates (2.35) for j obtain
t
J and (2.39} for j
e J in (2.33} we
Y.S. BAN AND E.T. SAWYER
(2.40}
since
i
= 1,...t. To estimate the integral in (2.40), begin by integrating in succession the variables
zl' z2,...zjC1 and using
I
~.
dp(zj) ~. ~ C 2 J for j
< j1
and then integrating
p(zj-1 ,zj)~c2 J the variables zj1,... zm_1 and using
I
~.
<_C2J+
dp( zj) ~.+1
1
for.>. J- h
to
p(zj'z j+ 1 )~c2 J conclude that the integral in (2.40) is at moat
(2.41}
~·
(Note that we skip over the weakest inequality: p(zj1_ 1,zj1) (2.38), the integral
J...J Er··Em diA{z1) ...dp(zm_1) s.
~ C 2 lt.)
Finally, by
will vanish unless d(x,y) = s. s. J1 -i,( -y- .
m m -1( d(z0,zm) ~ E d(zj-1' zy ~ C E 2 ~ Cm max 2 j=O j=O 1 ~ j~m
= Cm 2
, 1.e., p(x,y)
~.
~ CmM 2 h.
Thus combining (2.40) and (2.41) with this observation, we obtain
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
(2.42)
25
IRW[l,k]{x,y)- RW[l,k](x,y')l
for p(y,y') ~ J-p{x,y).
Now apply (2.23) to both RW[l,k](x,y) and RW[l,k](x,y')
and then take the geometric mean of the resulting inequality with (2.42) to obtain (recall
that s.
h
= 8m )
(2.43)
IRW[l,k](x,y) - RW[l,k](x,y') I
8 (1+Pe) Pe -8 2m p{y,y') x({p{x,y) S CmMA~ m})
where a, Therefore,
P> 0
and
a+ p = 1 (with P close to 1).
Y.S. HAN AND E.T. SAWYER
26
m+1
E
...
-{~l+ ... +llml).87
E 2 I~I>N llmi>N
E
oez
l
2-I I,+lm-km-11 +... +I I, +J<,-1,1 )<><
m [ E E j=1 all k such that u= 8 [l,k)=(l-+k.)Ak. m J J J
2o( 1+.8f')
p(y,y'lf'
x({p(x,y)
~
CmMAm2-u})
This shows (2.16) with 6 = .87 > 0 and ,8f' in place of f' (.8 close to 1). The proof of (2.16)' is similar.
The proof of (2.17) is similar in spirit to that of (2.16). Roughly speaking, in the worst case, we apply (2.13) (iv) in an appropriate way to a second difference of the kernel Ej1 (to get the factor p(x,y)-{ 1+ 2 f')), then argue with first differences to the right of j 1 as in the proof of (2.16) (to get the factor p(y,y')f'), and finally argue to the left of j 1 in the same way (to get the factor p( x,x' ) f'). We first establish the following estimate: For all 0
< f' <
t:,
there are constants 7 > 0 and C > 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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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)}
Y.S. HAN AND E.T. SAWYER
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 ~ <
h
and j0
= j1
are somewhat different. In particular, we only need the second difference condition {2.44) in the latter case. In the case j0 < j1, we can readily extend (2.33) to
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
29
[RW[l,k)(x' ,y)- RN[l,k)(x,y))- [RN[l,k)(x' ,y')- RN[l,k)(x,y')]
-
where Ej is given as before by (2.32) if j
~
h• but now by
E.(z. I' z.)- E.(z. 2, z.), J .r J J .r J { E.(z. 2, z. I' z.) = l r r l Ej(z~I· zj) , -
(2.46)
if j ~ ~- In the case ~
j
eJ
j ¢J
= ji, the analogue of (2.33) is
[RN[l,k)(x' ,y) - RN[l,k)(x,y)) - [RN[l,k)(x' ,y') - RN(l,k)(x,y' )]
-
J...J EI(z_l'zO,zi) ... E~-I(zj0_2 ,z~-I·z~) E~(zj0_2 ,zj0-l'z~,z~+I)
=
-
x
-
E~+l(z~,z~+l·z~+2) ... Em(zm-I'zm,~m+I)d~zi) ... diJ{zm-I),
--
where E. (z. 2,z. l'z. ,z. +I) Jo
Jo- Jo- lo
Jo
is defined by (2.32) for j ~ ~ =
- (z. I,z. ,z. +I) = E. lo Jo- Jo Jo h
-
-
lo
-
Jo
Jo- Jo
Jo
J
and by (2.46) for j < j0. We also have the analogue
of (2.37): When EIE 2 ... E. IE. E. +I ... E Jo-
-
E. (z. 2,z. ,z. +I) and E.
lo
-
IE
m-
m
is non-zero, then
Y.S. HAN AND E.T. SAWYER
30
0 5 j 5 m+l.
-
- -
Indeed, if E~+l ... Em_1Em # 0, then just as in the proof of (2.16), we have p(zj-l'zj) 5
-s.
C 2 J for ~+1 5 j 5 m+l.
-
On the other hand, if E 1E2 ... E~_ 1 # 0, then arguing
-
-s.
analogously from left to right with Ej as in (2.46), we obtain p(zj-1,zj) 5 C 2 J for 05H~·
-
Now estimate Ej as in the proof of (2.16), and in the case
~
= j 1, estimate
E~
by (2.44), to obtain the following analogue of (2.40) - (2.42) (just as in (2.41), we skip -6·
over the weakest inequality: p(zh_1,zh) 5 C 2
I[R~[l,k](x' ,y) -
(2.47)
h).
R~[l,k](x,y)] - [R~[l,k](x' ,y') - R~(l,k](x,y' )] I
Now apply (2.23) to each of the four terms on the first line of (2.47) and then take the geometric mean of the resulting inequality with (2.47) to get (recall again that
8
m
= s.) h
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
31
I(RN[l,k](x' ,y) - RN[l,k](x,y)) - [RN[l,k](x' ,y') - RN[l,k)(x,y' )] I
for a, fJ
> 0 and a + fJ =
1. With
fJ cloae to 1, we now obtain (2.17) by
adding
these estimates over k and l as in the proof of (2.16) (see the argument following (2.43)). Before turning to the strong weak boundedneas property, we show that (2.14) (iii) holds with
f
in place of
£'
even when Sk fails to satisfy the second difference
condition (2.13) (iv). To see this, use (2.43) for both x and x' to obtain
I[RN[l,k){x' ,y) - RN[l,k)(x,y)] - [RN[l,k)(x' ,y') - RN[l,k](x,y' )] I
Similarly, arguing as above, but to the left of obtain
io
rather than to the right of j 1, we also
32
Y.S. HAN AND E.T. SAWYER
I [R~[l,k)(x' ,y) -
R~[l,k)(x,y)) - [R~[l,k)(x' ,y') - R~[l,k)(x,y' )] I
Now take the product of the square roots of these two estimates and then sum over k and l as before to obtain (2.17) with
f-
in place of f'. Our assertion follows from
this. We now establish the strong weak boundedness property for the operator TN1. To -1 e sWBP, we write show that TN
(2.48)
Using the same notation as above and the estimates in (2.35) we obtain
(2.49)
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
On the other hand, we also have
Combining the estimates
JFm(zm, y)f(x,y)dJ.I(y) I =
I
and
IJ Fm(zm, y)f(x,y)dJ.I(y)l with the estimates for Fi, 0
and
~
i
~
~
(k +l ) C2 m
m-1, yields
m r,
33
Y.S. HAN AND E.T. SAWYER
34
-k Set r=2
°. Then we have
(2.50)
and (2.51)
By taking the geometric average of (2.49) and (2.50), and then (2.49) and (2.51), we
obtain
(2.52)
~
and
(2.53)
-{l~l+ ... +llml)af -{lk1+'--k l+... +lkm-1 +lm-1-km I)PE J. 2 2
0m 2
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
35
Substituting (2.52) and (2.53) into (2.48) yields
(2.54)
which proves
R;
e SWBP with constant em 2-N 6m. For N so large that
C 2-N 6 < 1, we obtain TN1 E SWBP.
§3. The C!ldgrOn - tyPe repmdudnr tormula on spaa:s of homnaeneoUI type To establish a Calderon reproducing formula on spaces of homogeneous type, we
need to introduce a suitable class of distributions on these spaces.
Definition (3.1) Fix two exponents 0
< {J ~
1 and 7
> 0. A function f defined on X
is said to be a smooth molecule of type ({J,7) centred at
Xo e X
with width d
> o, if
f
satisfies the following conditions:
(i)
(3.2)
(ii)
(iii)
JXf(x)d!J(x)
=
0.
This definition was first introduced in [M] for the case X = Rn. A function f is said to be a strong smooth molecule of type ({J,7) centred at
Xo e X with width
d
> 0 iff satisfies
the above conditions but with (ii) replaced by]
(ii)'
lf(x) -f(x')l
~ C (~~~(~'l, 0)]p(d + pt'Y
x,~
~ ~ (d+p(x~)).
36
))l+'Y for p(x,x')
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
The collection of all smooth molecules of type (fJ,'Y) centred at will be denoted by
vK (,8,-y)(~,d).
H
f E vK
~
37
e X with width d
(,8,-y)(~,d),
the norm of
f
>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
Y.S. HAN AND E.T. SAWYER
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 <
E'
<
E
there exists C > 0 such that
(i)
-
2-kf'
1}\(x~y)l ~ C (2=-t + p{x 1 y))l+f'
1
(ii)
for p(x,x')
~ h
It is easy to see that if {Dk}kel and {Dk}kel are as in Theorem (3.4), then
D:(f) and Dk*(f) belong to A (11,7) whenever f e A
(11,1)
with 0 <
11,
7 <
£,
This
allows one to define Dk(g) and D:(g) as elements of (A (11, 7))' for g e ( .6 (11, 7))' with 0 <
11.1 < f
by
for all f e A(/l,7). We now have
Theorem (3.§) Suppose that
{Dk}kel is as in Theorem (3.4). Then there exists a
family of operators {Dk}kel whose kernels satisfy the estimates {i) - (iii) in Theorem
(3.4), such that for all f e ( .6(11,1)),
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
39
(3.7)
where the series converges in the sense tha.t for all g e .-K(/3' •7') with /3' >
(3.8)
-
I im < E
M~m
1k1~M
=
DkDk(f), g>
P, 7' > 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)
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
To estimate p, let ..\u(x,y)
(3.14)
p =
= (J [-p{x,y)] u - . Then
+
..\u(x,y)x0(y)[IP(y) -IP{x)]t/J(x)>
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))
Y.S. HAN AND E.T. SAWYER
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
43
Assume (3.18) and (3.19) for the moment. Then since T satisfies the strong weak boundedness property, we have]
I
.E
~u(x,y)x0 (y)[cp(y)- cp(x)]tf.{x)>
I
JEr
I
~u(x,y)fij(y)x0 (y)[cp(y)- cp(x)],p(x)>
~ 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
Y.S. HAN AND E.T. SAWYER
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:
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
< 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<
E,
T maps 6-smooth atoms centred at
Xo
e X with width
> 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
d6 p(x,xo)I+b
Y.S. HAN AND E.T. SAWYER
46
and this proves the claim. Now note that when p(x,x0) ~ 3A2d and p(x 1y) ~ ~ (d+ p(x,x0)) 1 then x and
4A
y are not in the support of a. Thus
ITa(x)-Ta(y)l
=
=
1/[K(x,z)- K(y,z))a(z)diJ(z)l
I /{[K(x,z)- K(y,z))- [K(x,Xo)- K(y,x0)]}a(z)dl-'(z) I since Ja(z)diJ(z)
by (2. 7) (iii) since
p(x,y)
4~2
~
(d+
p(x~xo)) ~ 4~2
=0
(3:2 + 1) p(x,xo)
~
~ p(x1Xo) and p(z,Xo) ~ d ~ ~ p(x,Xo)· The final integral above is at most 3A
3A
x ·)£ C (PM!)
d£ l+
AAi()J p(x~xo)
which shows that Ta satisfies the smoothness condition for a £
strong smooth molecule and hence proves Lemma (3.20).
Remark For the case X
= of
1
it was shown in [M) that if T satisfies the hypotheses of
Lemma (3.20) 1 then T maps smooth molecules of type (P1 7) centred at Xo with width d>O to smooth molecules of type
(P' 7') 0
1
1
width d > 0. Since the proof is based on Lemma (3.12) this result holds also on spaces of homogeneous type. ~
Proof of Theorems (3.4). (3.6) and (3.9) Let Dk
N = TN1(Dk).
By Theorem (2.14) with
N large enough, T'N1 satisfies the hypothesis of Lemma (3.20) with t: replaced by t:' 1 0 < f' < £, and it is easy to see that
D~( · ,y) is an £-Smooth atom centred at y with
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
47
width C 2-k > 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 <
£
since then TN and T'N1 are bounded from
.,K(fJ,'Y) to .-K(fJ' ,'Y') by the remark following Lemma (3.20). Thus to show Theorem (3.4), we need only prove that for all f e .-K(fJ,'Y) with 'Y > 'Y' and fJ > fJ',
(3.22)
lim IITN 1 E DNkDk(f)ll (fJ' ') M-110 lki>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' <
'Y' < 'Y·
=
0.
Y.S. HAN AND E.T. SAWYER
48
We now claim that for 7' < 7 and
11' < {J' < fJ there exist constants C > 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 <
.,., < .,.,
and 0< IJ' < {J' <
fl.
Now (3.22} follows from (3.23}, (3.24} and (3.26}, since these estimates imply that
lkf>M D~Dk(f) e .6(11',7') for IJ' < fJ and 7' < 7 and moreover, N
II lki>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 <
f'
<
f
and with constants
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
49
E D~Dk(l) = E Dk(D~)''(l) = 0. We now lki>M lki>M D~Dk has the strong weak boundedness property with constants
independent of M. Moreover, show that
E lki>M independent of M. For all f e Cti(X
x
X) with supp f .t. B(Xo, r)
x
B(y0, r), llfll 111 ~ 1,
llf(·,y)II 11 Sr-f1 and llf(x, ·)llflsr-fl, wehave
< E lki>M
D~Dk, f>
=
E lki>M
JJJ D~(x,z)Dk(z,y)f(x,y)diJ(z)diJ(y)diJ(x).
Since
and
JJJ D~(x,z)Dk(z,y)f(x,y)diJ(z)diJ(y)diJ(x) I
I
= IJJJ D~(x,z)Dk(z,y)[f(x,y)- f(x,z))diJ(z)diJ(y)diJ(x) I
we have
~
E I lki>M
III D~(x,z)Dk(z,y)f(x,y)diJ(z)diJ(y)diJ(x)l
Y.S. HAN AND E.T. SAWYER
50
Thia shows that
E lki>M independent of M. Thua,
D~Dk
satiafiea the hypotheaea of the Lemma (3.20) with norm
E D~Dk maps smooth molecules of type (fl,7) to smooth lki>M molecules of type (fl' ,7') with fl' < fl < f and 7' < 7 < f, and this yields (3.25). To show (3.24), suppose f
(3.21)
I
E lki>M
e
.6(fl,7). Then if p(x.XO)
~ 2,
D~ Dk f(x)l ~ ·E I/ D~ Dk(x,y)[f(y}-{(x)]dp(y)l k>M
since D~ Dk(x,y) aatiafiea the conditions (i) (ii) and (iii) in (2.13}
where rr > 0 and 0
<
7'
< 7, and this proves (3.24} for p(x,Xo) ~
2.
On the other hand, if p(x,Xo) > 2 then
I
E DNk DL f(x)l lki>M ..
+ I E
k<-M
Now p(x,y)
~ I E I k>M
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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)
Y.S. HAN AND E.T. SAWYER
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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)
Y.S. HAN AND E.T. SAWYER
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)
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
where the series converges in the sense that for all g
(3..W)
1 im <
M-tm
E
lki~M
-
Dk Dk(f), g
>
=
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 <
£
and that {Sk}kel and {Pk}kel are two
approximations to the identity and let Dk = Sk- Sk_1 and Ek = Pk- Pk_1. Then there exists a constant C > 0 such that for all f e ( .Jt(fJ,'Y)), with 0 < {J, 'Y <
£,
(4:.2}
for 1 ~ p, q
~ m
(4:.3}
for 1 < p, q < m.
Proof We may assume that the right hand sides of (4.2) and (4.3) are finite. Theorem (3.6), there exists
{Dk}kel'
(iii) in Theorem (3.4) with 0 < (fJ.'Y)), with 0 < {J, 'Y <
£ '
£'
Ek(f)
By
a family of operators satisfying the properties (i) <
£
and -£' < a < £' such that for all f e ( .Jt
=
Ek>
56
=< E
l£1l
Dj!J/.f). Ek>
= EkE
l£11
Dj!J/.f)
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
57
since Ek(x,·) e .Jt'(t,t). Therefore,
(4.4) Applying the proof of Lemma (2.18), one can show that
(4.5)
-
-
where ~Dj.x,y) is the kernel of ~Dl and a V b denotes the maximum of a and b. It then follows from (4.5) that
Substituting the above estimate into (4.4) yields
by Minkowski's inequality (or Young's inequality for a discrete convolution) since
E 2-lll t'+ia < 1£"8.
m
for
-f'
< a<
E.
-
This proves the first inequality in (4.2).
starting with the family of operators {~hez• this shows the other side in (4.2).
By
Y.S. HAN AND E.T. SAWYER
58
To prove the first inequality in (4.3), note tha.t the estimate in (4.5) implies
where M is the Hardy-Littlewood maximal function. Thus,
by Minkowski's or Young's inequality again. It then follows from the Fefferman-stein vector-valued maximal inequality ([FS]) that
which proves the first inequality in (4.3). The second inequality in (4.3) follows in the same way.
Remark Since (4.5) holds for Ek satisfying only the smoothness condition for the second variable, we conclude that Proposition (4.1) holds with
aa ia Theorem (3.36).
~replaced
-
-
by Dk , where Dk is
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
59
We now introduce the following norms which will define the Besov and TriebelLizorkin spaces on spaces of homogeneous type.
Definition (4.7) Suppose that {Sk}kel is an approximation to the identity, let Dk Sk- Sk_1, and let 0 < {J,7 <
(4.9)
E,
-E
< a<
E
where
f
=
is as in (2.14). For all f e ( .(((P,7)), with
define
II~IFa,q = ll{k~l (2kal Dk(f) Dq} 1/qllp
for 1 < p, q <
Ill.
p
It then follows from Proposition (4.1) that these norms are independent of the choice of the families of operators {Dk}kel which satisfy the properties in Theorem (2.14). To introduce the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, we need the following results:
Suppose that f e (.(((fJ,7)),
Prooosition (4.10) max{O,-a} < 7 <
where
E,
f
is as in (4. 7), and
with max{O, a}< {J < -f
< a <
E.
Then
11~1
1 ~ p, q < m, if and only if
= 0
for all g e .(((fJ' IY') with
Similarly, 11~1- a q= 0, 1 < p, q <
F' p
=
0
111,
P' > 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
Y.S. HAN AND E.T. SAWYER
60
Proof
n
is easy to see that if f e (.6 (/1,7))'
max(O,-a) < 7 <
E,
with
max(O, a) <
p<
E and
and = 0 for all g e .6 (11' ,7') with /1' > P and 7' > 7,
then = 0 for all k e I, since Dk(x,·) e .6(E,E), and hence
11~1· a,q =
0.
Bp
Conversely, iff e (.6 (11,7))' with max (0, a) <
P < E and
max(O,-a) < 7 < E, and
11~1· a,q= 0, 1 ~ p, q < ..,, then, by Theorem (3.6), we have that for all g e .6(/1' ,7') Bp
with /1' > 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'
<
m,
for l
and this then shows = 0 for all g e ,.(11' ,7') with /1' > p and 7' > 7·
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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.
<7<
f
It is easy to
Fp
see that
(4.12) respectively (4.13)
1 imll E DkDk(f) -~I· a q M-t111 JkJ~M Fp'
Let gM (x)
=
=
0.
J
E Dk(x,y)Dk(f)(y)dp(y), where x 0 is a fixed point in X as JkJSM p(y,xo)SM
in the definition of ..6(11,7). Then gM E ..K(f',f') for all M
is show that for any given u
II
I
E
> 0 there exists M1 > 0 such that for all M > M1,
JkJ~M p(y~)~M
respectively
> 0. All we need do now
Dk(x,y)Dk(f)(y)dp(y)-
E
JkJ~M
DkDk(f)ll ·a q Bp'
I
< u,
II E Dk(x,y)Dk(f)(y)dp(y) - E DkDk(f)IJ· a,q < u. Jkj~M p(y~)~M jkJ~M Fp
To do this, we write
I
E Dk(x,y)Dk(f)(y)dJ(y) - E DkDk(f)(x) jkJSM P{yoXcJ)SM jkJ~M
62
Y.S. HAN AND E.T. SAWYER
Then, by using the idea in the proof of Proposition (4.1),
Since f
t:
B:•q, there exists M0 > 0 such that
and it is easy to see that there exists M1 > M0 such that for all M > M1
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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
Y.S. HAN AND E.T. SAWYER
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'
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
65
This completes the proof of proposition (4.11).
Pmposition (4.14) Suppose
-f
and f e ( ~ (flo,'Yo)), with
<
o
<
max(O,o)
f,
11~1· o,q < •·
1
~
< flo <
p, q
< •·
f
and max(O,-a) respectively
f.
11~1· o,q < •·
Bp 1
< 10 < Fp
< p,q < 111. Then there exists g E ( ~(fl,'Y)), for all pairs (fl,'Y) satisfyiJ18 max(O,o)
< fl <
and
f
llg-fiiFo,q p
= 0,
<
max(O,-a)
1
<
f,
such that
llg-fii:Bo,q p
=
0,
respectively
and
I< g, h >I ~
C llgii:Bo,qllhll ~(fl,'Y) , p
respectively
I< g, h >I ~ c
llgii:Fo,qllhll ~(fJ, 1). p
f!QQf Suppose first that f e ( .,.
f.
~ e .,.(f' ,e), 0 < the sequence
f
and
By Proposition (4.11), there exists a sequence {~}n!l such that
f'
{~}n! 1
< f,
for n = 1,2, ... and lim
n.....
converges in ( ~ (fl,'Y)),.
.,.(fl,'Y) with max(O,o) < fl <
f' and max(O,-o)
II~- ~I· 0 q = 0.
We claim that
Bp'
To see this, note that for all h E
< 1 < f',
~ [k~l[2ko11Dk(~~)llp)j1/q (k~1(2-koii(Dk)•hllp't'Jl/q' by H6lder•s inequality
Y.S. HAN AND E.T. SAWYER
66
~ C IISn- ~IIBa,q llhll A({J, 7) p
by the remark following Theorem (6.20)
which shows lim gn exists in ( A(P,7)),. Moreover, if we set 1im Sn = g in ~~( A(P,7)),, then the above proof also shows that {gn}n:, 1 converges to g in the norm of Ba,q and hence p ll ~ lim I<Sn,h>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<
E,
where
E
is as in (2.14). The Besov space B:'q, 1 ~ p, q < m, is the
collection of all f e ( A(P,7)), with max{O,a) < that 11~1 . a,q < • and Bp
p<
E
and max(O,-a) < 7 <
E
such
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
67
(4.16} for all h
e .,K(fJ,7)_
H we interchange the order of summation and integration in the above definition we
obtain the Triebel-Lizorlrin space :F;•q, 1 < p, q < .... More precisly this space is the collection of all f that
llfll. aq < 111 F • p
E(
.,K({J,7))' with max(O,a) < {J <
f
and max(O,-a) < {J <
such
f
and
(4.17} for all h E
.,K ({J, 7).
In view of Proposition (4.14), the requirements (4.16) and (4.17) simply choose a
"good" canonical representative from each equivalence class.
By Propositions (4.1),
(4.10) and (4.14), these spaces are well defined and independent of the choice of the families of operators and the pair ({J,7) with max(O,a) < {J <
Proposition {4.18) Suppose that
-f
< a<
f.
f
and max(O,-a) < 1 <
f.
Then :Biq• 1 ~ p, q < ..., and :F;•q,
1 < p, q < ... , are Banach spaces.
~ Let {fn}:=1 e Bpa,q with exists f such that max(O,-a) < 1 <
Ill
E llfnll· a q < .... It suffices to show that there n=1 Bp '
E
f = f converges in ( ..N (fJ,1))' with max(O,a) < {J < n=1n f,
f
and
and also converges in the norm of :B;•q. To see this, take h E
({J,7) with max {O,a} < {J <
f
and max {0,-a} < 1 <
f.
Then
.,K
68
Y.S. HAN AND E.T. SAWYER
The last term goes to zero as n - t . , and this shows f
= •E fk k=l
of
sa.q p . The proof for :F;•q is similar.
converges in the norm
t¥nmn• m
§5. The TI
:Biq
ad
:Fiq Biq
In this sec:Uon we establish the T1 theorems for the Besov spaces
Triebel-Lizorkin spaces
F:•q.
and
Suppose that T is a singular in~ operator as in
Definiuon (2.8). Then T can be easily enended to a conunuous linear operator from
11, 7 ~
.,«(11,7), 0 < with 0 <
11,
7
~
(J
fJ (where fJ is as in (2.5)) to (Cd(x))'. To see this, let f e .,«(11,7) and g e
cg.
Choosing
e
E
cg(x)
with ((x)
=1
for
X
e supp g,
we then define
{5.1)
=
Now f( e
cg(x),
+
.
and so is well defined. Since 7 > 0, we have both f(l-e)
and g in L1(X), and because supp f(1-() n supp g =
!.!
+,
it follows that
K(x,y)f(y)(I-e(y))g(x)dp(y)dp(x)
converges absolutely, and hence that is well defined. that (5.1) is independent of the choice of
n is easy
to check
e.
We now state the main result in this sec:Uon.
Theorem (5.2} Suppose that T is a singular integral operator with the kernel sausf:rins (2.7)(i) and the following half of (2.7)(ii) (i.e., smoothness in the x variable only)
69
Y.S. HAN AND E.T. SAWYER
70
IK(x,y)- K(x' ,y) I S C p{x,x')Ep(x,y)-{l+E) for all y, x and x' e X with p{x,x') S
and also T(1)
=0
1
can be extended to a bounded linear
and T e WBP. Then T
operator on
0
< a < E, 1 S
p,q
h- p(x,y),
<
m,
and on
Fa,q for
0
p
< a<
E,
< p,q
~ By Proposition (4.11), .,.(e ,e), 0 < 1S
p, q <
if f e
m and in
I
p
I
0
< a<
E',
m. Thus it suffices to show thai
-
I
I
,
I
and if f e ...K(f' ,f') n Fa,q with 0 p
(5.4)
B:•q for
a< e 1 < p q < 111 then
with o <
IIT£11 B. a q S C 11£11 B· a q p
< E, is dense in
F:·q for 0 < a < E', 1 < p, q <
.,.<e ,f') n :Ba,q p
(5.3)
f'
< a< f', 1 < p, q < m, then
IIT£11. a q S C 11£11. a q , F• F• p p
where C is a constant independent of f. Now
T
can be extended to a continuous linear operator from
(cg(x))', and thus, for f
e .,.(e ,e) n :B:·q,
...K (/1,7)
to
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
=
E DkDk(f), where the series converges in the norm of kel 0 < {J, 7 < f'. Hence,
since f
71
A ({J,7) for all
IIT~I· aq
B• p
since Dj.x, .) e C~
To estimate the last term above, we need the following lemma:
Lemm.a (5.5)
Suppose that
T
satisfies the hypotheses of Theorem (5.2), and that
(Dk)kel is as in the definitions of B~,q and
r;•q,
and 0 < a <
f'
<
f.
Then there
exists a constant C independent of f such that
(5.6)
_1\
( IDl l'Dk(x,y) I ~ c (2 k-..,e
" 1)
-k -l)f' 2 v 2 (2=1[ V 2=1. + p(x,y))l+f' . (
The proof of Lemma (5.5) is similar to the proof of Lemma (4.3} in [HS) (see p.37}, and we leave the details to the reader. Note that it is in this estimate where the one-sided smoothness and moment assumptions on the kernel of T are relevant. Using Lemma (5.5), we then have
Y.S. HAN AND E.T. SAWYER
72
=I+ II.
Term I satisfies
by Young's inequality since E 2(6--k)a <
111
for a > 0
k~L
by the remark following Proposition (4.1).
Similarly,
by Young's inequality since E 2(L-k)a <
111
for a > 0
k~L
by the remark following Proposition (4.1)
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
73
which proves (5.3).
The proof of (5.4) is similar:
by (5.6) and where M is
the Hardy-
LiUlewood maximal
operator
For term I, we have I
~ c 11{ E
{ E 2(l-k)o .jt.o M(D
~~ k~l
(f))]q}l/qll
k
p
since E 2(l-«.)o < • for a> 0 k~l
by the Feffermu-stein vectorvalued maximal inequality
by the remark following Proposition.(4.1).
Y.S. HAN AND E.T. SAWYER
Similarly,
which together with the above estimates proves (5.4).
This completes the proof of
Theorem (5.2).
As an application of the T1 theorem (5.1) (which is new even in the case X = Rn), the Besov spaces B;•q, for 0 < spaces F;•q, for 0 <
Ial <
Ial <
f
and 1 ~ p,q ~
and 1 < p, q <
f
111,
111,
and the Triebel-Lizorkin
can be characterized by more
general operators whose kernels satisfy only half (depending on the sign of a) of the usual smoothness and cancellation conditions (in particular, no second difference condition is needed). More precisely, let (Sk)kel be a sequence of functions from X such that for all k e l and all x, x', y e X,
(5.7)
=0
(i)
Sk(x,y)
(ii)
ISk(x,y) -
(iii)
!
if p(x,y)
~ C2-k, and 11Skll 111 ~ C2k;
Sk(x' ,y) I ~ C2k(l+E) p(x,x' )E;
Sk(x,y)d/J(y) = 1.
Theorem (5.8} For 0 < a < E, we have
IC
X into IR
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
~
75
p, q <
(5.9)
for 1
(5.10)
for 1 < p, q <
while for
111,
111,
< a < 0, we have
-f
(5.9)'
for 1 ~ p, q <
(5.10)'
for 1
•
Note that
'\:
satisfies the same hypotheses as
Ek
< p,
q
111,
< 111,
but with the variables
interchanged.
Proof We consider only the case 0
< a<
f,
the proof for the other case being the same
(or use the duality results in the next section). We first prove
(5.11)
{~[2ka11Ek(f)llp] q} 1/q ~ C II~IBa,q
for 0
< a < f and 1 ~ p, q < 111,
p
(5.12)
11{~[2ka1Ek(f)l]q} 1 /qllp ~ C ll~l:fa,q
for 0
f
and 1
< p,
q
< 111.
p
Suppose that f
(5.13)
e B~,q for
'\(f)
0
<
a
<
f
and 1 ~ p,q
< 111. Then, by Theorem (3.6),
Y.S. HAN AND E.T. SAWYER
76
To estimate the last term above we need the following lemma:
!.mnma(5.14) Suppose that
(5.15)
F-El )I< I-k t:-x,y -
(~)kel is
as in Theorem (5.8). Then
C(2(l-k)E'A1)
,....k -l E' (2 V2 ) (2JiV2-Z +p(x,y))l+e ,
This lemma can be proved in the same manner as Lemma (5.5). Using Lemma (5.14) we obtain
(see the proof of Theorem (5.2))
= c llfJI·Ba,q p
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
77
which shows (5.11). Similarly,
by (5.15) and where M is the HardyLitUewood maximal function
(see the proof of Theorem (5.2))
by the Fefferman and Stein vector-valued maximal function inequality
which shows (5.12).
The proofs of the other halves of (5.9) and (5.10) will use the T1 theorem (5.1) and an idea used in section 2. More precisely, as in section 2, I= RN +TN, where TN=
E E:~. E: = E Ek+J"· We will show that I im TN = I in the norm of Bpa,q and k UI~N N-t .. F;•q for the range of a, p, q indicated in Theorem (5.8). Assuming, for the moment, that this has been done, we then have
78
Y.S. HAN AND E.T. SAWYER
Using Lemma (5.14) we obtain
(5.17)
IITN~IBa,q ~
C
{~[~ 2ka (2(l-k)E' A 1)IIEJf)llp] q} 1/q
p
by Young's or Minkowski 's inequality since
~ [2-jf'
A 1)) ~a <
ID
for 0 < a <
E' . By
our assumption, TN 1 exists and is bounded on Biq for large Nand this together with the above estimates yields
which proves the other direction in (5.9).
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
79
The proof for F;•q is similar:
To estimate
For k
~
DkE~(g), we need the following estimate,
l, this follows easily from Lemma (5.14) while for k < l we simply note that
IDkE~(x,y)l ~ eN 2kx{p(x,y) ~
c2-k}.
We now have
by the Fefferman-8tein vectorvalued ma.ximal inequality
which shows the other direction in (5.10).
80
Y.S. HAN AND E.T. SAWYER
n
remains to show that 1 im TN ~~
=I
in the norm of Ba,q and Fa,q for the p p
range of a, p and q indicated in Theorem (5.8). EE E !t-ki>N
A
and consider the sums for k - i
To do this, we write
I -TN
> N and i - k > N separately.
Lemma (5.21) Suppose 0 < a< e. Then there exists C > 0 such that
< ~.
(5.22)
for 1
~
(5.23)
for 1
< p, q < ~-
p, q
Proof We first prove (5.22). By the definition of B;•q,
II
EE Ei k-i>N
~
C
=
C
~(£)11·B aq•
{~ [ EE
J k-i>N
{1: [ EE j
p
=
1 {Ej [~ann.J k-i>N EE Ei ~II ]q} /q P
[2(l-j)f' A 1)
~all~(£)11
] q} 1/q
by (5.14)
p
[2(i-j)f' A 1]2(j--l)a 2ia liE (£)11 ]q} 1/q k-i>N k p
=
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
81
E (2(l-j)E' A 1] 2(j-l)a 2(l--k)a k-l>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
Y.S. HAN AND E.T. SAWYER
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 <
£rnQ{. We will prove that the operators 2N
m.
'rN = 2N 6
EE Ef'k satisfy both the l-k>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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
(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 <
f'
<
E,
~ C 2-N 6 r
C~ with supp rp, supp 1/J t. B(x,r), IIIPII .. ~ 1, 111/111.. ~ 1, IIIPII, ~ r-fl and
111/111, ~ r -'1.
With this done, (5.25) and (5.26) will follow from the Tl theorem (5.2) with a < f'
<
E.
To prove (5.27), we write EE l-k>N
EA =
~ ~+A· We then have
. by (5. 7) (i)
This shows that
-=
E E ~+A and let T l l:>N k
Y.S. HAN AND E.T. SAWYER
For the smoothness condition, we have that if p(x,x')
~ ~~
~ 2-1, p{x,y), then
J[Ek+l(x,z) -Ek+Jx',z)] [~ (z,y) -Ek(x,y)]dp(z)l since ~+Jl)
=0
by (5.7) (i) and (ii) since p{x,x') < 2-lp{x,y) ~ C2-{k+l)
On the other hand, if 2-lp{x,y)
-
-
< p{x,x') ~
..
J p{x,y), then by (5.28), ..
ITJx,y)- TJx',y)j ~ ITJx,y)l + ITJx',y)l
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
85
Taking the geometric average of this inequality and (5.29), we obtain that for p(x,x') ~
h- p(x,y) IT~x,y)- T~x' ,y) I
(5.30)
~ C 2-1.6 p(x,x')f' p(x,y)-(l+f')
with 0 < f' < f and 6 = f- f'. This shows that for p(x,x') ~ ~ p(x,y), we have
for 0 <
f'
< f and 6 = f - f' > 0.
Finally, we show {5.27) {iii), i.e., that TN satisfies the weak boundedness property with constant
xa EX (5.31)
~ C 2-N 6. Fix two functions
tp and '1/J with supp tp, supp '1/J
t. B(xa,r) for
and r > 0, IIIPIIID ~ 1, 11'1/J!IID ~ 1, II'PIIfl ~ r-'~, and 11'1/JIIfl ~ r-'~. We have
I
I =
J: Ek+l ~( tp){x)¢(x)dl£{x)
I
I
86
Y.S. HAN AND E.T. SAWYER
+ C
E 2-(k+l)£ 2k£ 2-k"IIIPII 111/111 r k '1 m 2-
by (5.7) (i) and (ii)
Thus
which proves (5.27) with 6 = £.
Now using Lemma (5.21) and (5.24), we obtain
by Lemma (5.21) and (5.24).
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
87
Similarly,
(5.33)
11(1- TN)fll· a q ~ Fp '
This shows that
E
N~
TN
=I
C (2-Na + 2-N6J 11£11·
J
F~q
by Lemma (5.21) and (5.24).
p
in the norm of Bpa,q and Fa,q for the range of a, p p
and q indicated in Theorem (5.8), and hence, for N large, TN1 is bounded on B:•q and F:•q for the same rail.ge of a, p and q. This completes the proof of Theorem (5.8).
We remark that Theorem (5.8) generalizes a result of [HJTW] for the case X
= ~-
§6. Momic decom])Oilition of
s:q and F:q
The Calder6n reproducing formula is the key to obtaining atomic decompositions for the Besov and Triebel-Lizorkin spaces on Rn (see [FJ]). In this section, our Calder6n type reproducing formula will provide an atomic decomposition for the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type.
First we recall a construction of
Christ ([Ch]), which provides an analogue of the grid of Euclidean dyadic cubes on a space of homogeneous type.
Theorem(6.1) Given 6 > 0 sufficiently small, there exists a collection of open subsets
{Q~ c X: k e l,
r
e Ik},
where Ik denotes some ( possibly finite ) index set depending
on k, and positive constants a0 , fl, and C such that
=0
(i)
Jj{X\U Qk)
(ii)
If l
(iii)
For each (k,r) and each l
(iv)
diameter
(v)
Each
T
~
T
Vk,
k then either Q:, c Q~ or Q:, n Q~
Q~
(Q~) ~
C
= +.
< k there is a unique
r' such that
Q~ c Q:,
j-,
contains some ball
B(z~,
a0 j-).
We can now introduce (dyadic) smooth atoms on a space of homogeneous type in order to obtain atomic decompositions as in the case X= Rn.
88
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
Definition (6.2) Fix 6 > 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<
f,
1 ~ p,
biq
be the
[~.: (JJI..Qk)(-a-1/ 2+1/P} Is 1]p]qfp}1/q kEI .,. Qk
= { E
T
< a<
E,
1 < p,
.,.
q< ..,
let
fiq
(6.()
xQA'.,.
we let
such that
such that
is finite, where
q < m,
is the characteristic function of Qk. T
be the space of such sequences
Y.S. HAN AND E.T. SAWYER
90
We now have the following atomic decomposition for
Theorem (6.5) Suppose
(A) {s k}
Qr
e
I! 1 ~ p,
b:·q
convergence in
(6.6)
and
-E
q<
and
F:•q.
E.
and
ID
B:•q
f
E
B:•q,
then there exist a sequence
s =
E-tlmooth atoms
Ba,q and p
nan.b
q
Q
p
I
Similarly, if 1 < p,
s c nfll B. a q . p
q<
ID
I
and f
E
F:•q,
then there exist a sequence s = {s k}
E
£iq
Qr and E-tlmooth atoms {a k} such that f = E s k a k with convergence in Qr k,r Qr Qr
F:•q and
(6.7)
(B) Conversely, suppose f =
E s k a k' where a k are E-tlmooth atoms. k,r Qr QT Qr
Then
(6.8)
nfll· a,q
S C
Bp
(6.9)
nfll·F a q ~ p
I
nsn· a q for
1 S p,
nsn·f a q for
1<
q<
ID 1
p, q <
ID.
bp'
C
p
I
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
Proof The proof of this theorem is similar to the case of that 6 =
+
(see [FJ]). Assume first
in Theorem (6.1).
(A) Suppose f
=
If
91
e B:•q.
By Theorem (3.36), we have
E E s k a k•
kElT QT QT
where
and the above series converges in the norm of
P> max(O,a),
B:•q
and in
( .K (P.'Y))',
with
7 > max(O,-a). It is easy to see that a k satisfies conditions (i) (ii) QT
and (iii) in (6.2) for C large enough. To see (6.6), we have
92
(6.10)
Y.S. HAN AND E.T. SAWYER
llslll,a,q p
~ c ~~~ [zkop ~ I'( Q~-p/p. [J kIi\
The last inequality follows from the remark following Proposition (4.1). The proof in the case f
e F;•q
.
is similar, but using the vector-valued maximal
.
inequality. Indeed, suppose that f e F;•q. As in the proof for Biq• we have
f
=
E E s k a k' kel ,.. Q,. Q,.
where s k and a k are as above, and the aeries converges in the norm of F;•q and Q,. Q,. in ( .Jt(P,7))',
~th P> max(O,a), 7 > max(O,-a). To show (6. 7), we compute
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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
Y.S. HAN AND E.T. SAWYER
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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
Y.S. HAN AND E.T. SAWYER
~ 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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
(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
Y.S. HAN AND E.T. SAWYER
98
It follows from (4.5) that for 0
<
E'
<
f
and -f'
< a<
E',
-
-
where QkDj.x,y) is the kernel of QkDt This estimate implies
by Young's inequality since E ~~-UI E' < j
111
= c llfll· cr,q· Bp
To prove the other inequality in (6.18), note that Theorem (3.6) holds also with (Dk)kel replaced by (Qk)kel· Then repeating the above proof yields
which proves the other inequality in (6.18). The proof of (6.19} is similar (using (4.6)) and this completes the proof of Theorem (6.5).
R.emark (6.20) For 0
<
a
<
E,
the vanishing moment condition (6.2}(ii) is not needed
for part (B) of Theorem (6.5). This is because (6.15) holds with
E
=0
even without
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
99
(6.2)(ii). For future reference, we will refer to functions a k satisfying merely (6.2)(i) Qr and (iii) as
Q~-atoms.
As in the case of Rn, there is a decomposition involving "smooth molecules".
Definition (6.21) We say that m k is a (11,7)- smooth molecule for a dyadic cube Qk if
Q
1"
1"
(i) (ii)
(iii)
Our molecular decomposition is the following.
Theorem (6.22)
{Q~: k e I,
Suppose that
r
e Ik}
are dyadic cubes in X as in
Theorem (6.1) and that m k is a (11,7)- smooth molecule for Qk with max (O,a) <
Q
1"
1"
11 <
E
and max (0,-a) < 7 <
E.
Then for s = {s Q~} Q~ dyadic
for
{2)
II
E s km
krQ I 1"
kll· a q ~ cllsll· a q
Q1" F• p
f• p
for
-E
-E
< a < E and 1 ~ p,q < 111 1
< a < E and 1 < p,q < 111.
Y.S. HAN AND E.T. SAWYER
100
The proof of Theorem (6.22) is similar to the proof of Theorem (6.5). We leave the details to the reader.
Remark We remark that if h e .,K(f:J,'Y) with max (0, a) the above theorem implies h e B:·q with with
-£
<
a
<
E
and 1
-£
<
a
<
E
< {:J and
max (0, -a)
and 1 S p,q
<
111 ,
<
7, then
and h e F:·q
< p,q < 111.
Using Theorem (5.8) we can obtain a more general atomic decomposition for B:•q
> 0, p and q indicated in Theorem (5.8), involving
and F:•q for the range of a
atoms with vanishing b- moment which is new even in the case X= Rn. Before we state this result, we need the following definitions.
Definition (6.23) A function b
e L111(X) is
such that for all x e X and r
>
said to be para-accretive if there exists 6 > 0
0, there exist x' e B(x,r) and r'
>
0 such that
B(x' ,r') .C. B(x,r) and
II'(B(!,r))
(6.24)
b,
there exists
b(y)dl'{y)l
~
6.
p(B(x,r)) ~:~ p(B(x' ,r')),
Note that if (6.24) holds, then para-accretive
r
~(x' ,r')
c
>
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,
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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 _. (
Y.S. HAN AND E.T. SAWYER
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.,.
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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
,
Y.S. HAN AND E.T. SAWYER
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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
Y.S. HAN AND E.T. SAWYER
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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,
Y.S. HAN AND E.T. SAWYER
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
(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
Y.S. HAN AND E.T. SAWYER
(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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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
Y.S. HAN AND E.T. SAWYER
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
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
Y.S. HAN AND E.T. SAWYER
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
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
115
(6.49)
Finally, we can write
tk ,rr =
d
k rr -
_1_ E sO 1k rr k k+l .,. ~ •.,. ~.r ' r. 2 rr<~2 rr
where dk rr is a (fixed multiple of a) [2krr, 2k+lrr~tom with vanishing b-moment and
'
by {6.49},
by (6.48},
~
c,
This completes the proof of Theorem 6.28.
by (6.49) again.
§7.
Duality and interpolation of :Biq and Fiq In this section we study duality and interpolation of the spaces B~,q and F~'q·
Theorem (7.1} Suppose -E < a<
E.
Then
(A) (B:·q>* = :B-:!q' for 1 < p, q < · preo"sely,. gtven g
m
with
~+
!, = ~ + !, =
1. More
· nal on e B.-a,q' p' , t hen -. g(f) = defines a linear funct1o d
.J((e ,e) n :Ba,q with 0 < p
f'
<
E
such that
and this linear functional can be extended to Bpa,q with norm at most C
llgll.B-a,q , . P'
Conversely, if
$
is a linear functional on B:•q, then there exists a unique
g e :B-a,q' such that
P'
116
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
117
defines a linear functional on .6(f' ,f') n Ba,q p , and ~ is the extension of ~g with llgll·-a,q' ~ B P'
c 11-2'11·
'Q
(B) (Fp'q)
* = F--;!q . '
precisely, given
g e :F--;:q',
,.,.ce,e) n :Fo,q p
with
o<
f'
for 1 < p, q < then <
f
111
with
1 p1 + P' = q1 + q'1 =
1. More
~ gCt) = defines a linear functional on
such that
and this linear functional can be extended to Fiq with norm at moat C llglljrll,q' .
P' Conversely, if
~ is a linear functional on F:•q, then there exists a unique
g e F--;:q' such that
defines a linear functional on .6(e ,e) n :Fa,q p , and ~ is the extension of ~g with llgll·-a,q' ~
F P'
c 11.2'11·
!EQf Since the proof of (B) is similar to the proof of (A), we prove only (A). Suppose that g e B-::q' and f
e .6(e ,e) n B:•q.
By Theorem (3.6), we have
Y.S. HAN AND E.T. SAWYER
118
by the remark following Proposition (4.1).
By Proposition (4.11), ..K(E' ,E') n a;•q is dense in Biq• and thus .it ( E' 'E')
a linear functional defined on
n Bpo,q
I
~g(!)
= is
and can be extended to Bpo,q with
norm at most C llgll. -a q•·
B
p;
To show the converse, we need the following lemma. Lemma {7.2)
Suppose
{gk}kel
[k:1 (2-k 0 llgkllp,]q'] 1fq' < g(x) = E Dkgk(x) kel
E
111,
with
is a sequence of functions that satisfies -E
< o <
E
and
1
<
p', q'
<
111.
Then
. o,q where {Dk}kel is as in the definition of BP .
B-o,q' P'
Moreover,
llgll
·-aq' B P;
<_
C { E [2-kallg II ,)q'}1/q', kel kp
where C is a constant.
Assuming this lemma for the moment, suppose that
f
e :Bo.q p
I
$
E [a;•qr. Then for all
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
I .t'(f) I
If f e B:•q,
then the sequence
{(fk)kei ll(fk)llfl,a(LP)
i
functional
=[
~
11~1
II .&'II
.
Q
B• p
q·
{Dkf}kel is in the sequence space
f (tc llfkllp]q)l/q < a} 0
on a subset of the sequence space
tl• 0 (LP)
We
-
=
11-t"(f)ll
~ 11-t'IIII~IBa,q =
tl• 0 (LP) =
now define a linear
as follows:
It is clear that .t' is bounded on this subset of the sequence space
li(cnkf))l
111
tl• 0 (LP):
11-t"IIII(Dkf)llti,a(LPf
p
-
By the Hahn-Banach theorem .t' can be extended to functional.
Since (fl• 0 (Lp))*
= ti' ,-a(LP')
for 1
tl• 0 (LP) ~ p,q <
sequence (gk)kel E ti' ,-a(LP') such that ll(gk)llti' ,-a(LP')
m,
as a continuous linear there exists a unique
~ C 11-t"ll and
(7.3)
for all (fk) E fl• 0 (LP). It then follows that if f E -f'
< a<
f,
~ ( E' ,E') n B:•q with 0 <
then
.t'(f)
Let g =
f D:gk.
Then by Lemma (7.2), g E a;~·q'. Thus,
t'
<
t,
120
Y.S. HAN AND E.T. SAWYER
~(f)
=
~g
(f)
= < f,g >
for all f e .A'(E',E')n Bpa,q. Since .A'(E',E')n :Ba,q p is dense in Ba,q p , we have that ~ is the extension of ~g· Moreover, by Lemma (7.2),
(7.5)
We now return to the proof of Lemma (7.2). By the definition of :Biq•
(7.6)
by Young's inequality since
E 2-UI E+ja < jell
111
for
-f
<
a
<
f.
This completes the
proof of part (A).
We turn now to the interpolation of :B;•q and :F;•q by the complex method.
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
121
Theorem (7.7} Let [A,B]B' 0 ~ 9 ~ 1, denote the intermediate space between A and B obtained by complex interpolation. Suppose - f < flo• a 1 <
t,
and a = (1 - 8)flo
+
Oar
Then
(7.8)
(7.9)
where 1/p
= (1- 6)/Po + 9/p1
and 1/q = (1- 8)/~
+ 6/q1.
The idea of the proof of Theorem (7.7) is to reduce interpolation of :B;•q and Fa,q to interpolation of Apa,q and Gpa,qI where Apa,q is the space of measurable p functions F(x,k) defined on X x Z such that
(7.10)
and o;•q is the space of measurable functions F(x,k) defined on X x Z such that
(7.11)
For complex interpolation of A;•q and o;•q, we have
Prooosition (7.12} Suppose
-f
< flo, a 1 <
t
and a= (1- O)flo
+ Oar
Then
122
Y.S. HAN AND E.T. SAWYER
(7.13)
(7.1()
where 1/p
= (1 -
O)/p0 + O/p1 and 1/q
= (1 -
0)/~ +
Ofqr
This proposition can be proved using an idea in [DJS] (see p.53 in [DJS]). We now return to the proof of Theorem (7.7). Define a mapping
:T from A a,q p
to Ba,q by p
F(x,k)
---+
E Dk(F(.,k))(x). k
We claim that this mapping takes A:•q onto Biq for
-E
<
a<
f
and 1 < p,q <
Indeed, we have
by Young's inequality since E2-UIE+ja < j
= C
IIFII A• aq· p
111
for-E<
a<
E
111.
LITTLEWOOD-PALEY THEORY ON SPACES OF HOMOGENEOUS TYPE
1.23
To see that :J' is onto, define a mapping :F from Ba,q (to Aa,q) by p
f
-----t
p
-
F(x,k) = Dk(f),
-
where Dk and Dk are as in Theorem (2.14). We claim that :J' composed with :F is the identity. Indeed,
C
llfll·B• aq p
by the remark following Proposition (4.1). This shows that F(x,k)
E Dk(F(.,k)) = E DkDk(f) = f k
k
e A;•q and thus
in Ba,q
p '
which shows that :J' takes Aa,q onto Ba,q By interpolation, the image of [Ailo•
Conversely, the mapping f
-----t
F(x,k) = Dk(f)(x) takes :B;•q into A;•q. To
see this, we have
llfll.B• aq· p
Y.S. HAN AND E.T. SAWYER
124
Hence, it sends
[B;g·'lo, B:rql]o
into
[A;g•'lo, A:rql]o
f e [B;g•'lo, B:fql]O' then
{~(2ka11Dk(f)llp)q}l/q <
[B£ro•'lo, Bal,ql] c Ba,q_
This proves (7.8).
Po
Pt
o
P
completes the proof of Theorem (7.7).
111,
=
A:·q.
Thus, if
and so f e B:•q. Therefore,
As the proof of (7.9) is similar, this
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G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, Math. Notes 28, Princeton University press, 1982.
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Y.S. Han Department of Mathematics Auburn University Auburn, Alabama 36849 U.S.A. E.T. Sawyer Department of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1 Canada
