Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 879073,26 pages doi:10.1155/2012/879073

Research Article

Generalized Carleson Measure Spaces and Their Applications

Chin-Cheng Lin1 and Kunchuan Wang2

1 Department of Mathematics, National Central University, Chung-Li 320, Taiwan

2 Department of Applied Mathematics, National Dong Hwa University, Hualien 970, Taiwan

Correspondence should be addressed to Chin-Cheng Lin, clin@math.ncu.edu.tw Received 10 October 2011; Revised 20 February 2012; Accepted 12 March 2012 Academic Editor: Stevo Stevic

Copyright © 2012 C.-C. Lin and K. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the generalized Carleson measure spaces CMOa'q that extend BMO. Using Frazier and Jawerth's ^-transform and sequence spaces, we show that, for a € R and 0 < p < 1, the duals

of homogeneous Triebel-Lizorkin spaces fp>'q for 1 < q < g and 0 < q < 1 are CMOq/pyq/q)

and CMO;a+(n/p)-n'~ (for any r € R), respectively. As applications, we give the necessary and sufficient conditions for the boundedness of wavelet multipliers and paraproduct operators acting on homogeneous Triebel-Lizorkin spaces.

1. Introduction

In 1972, Fefferman and Stein [1] proved that the dual of H1 is the BMO space. In 1990, Frazier and Jawerth [2, Theorem 5.13] generalized the above duality to homogeneous Triebel-Lizorkin spaces Fpq. More precisely, they showed that the dual of F1q is F^ for a € R and 0 < q < g, where q' is the conjugate index of q. Throughout the paper, q' is interpreted as q' = g whenever 0 < q < 1, and q' = q/(q - 1) for 1 < q < g.

BMO = FGg2. For a € R, 0 <p< 1, and 0 <q< g, it is known (cf. [2-4]) that the dual of F^ is

F<£+(l//P n'G. Here, we will give another characterization for the duals of FP'q in terms of the generalized Carleson measure spaces for a € R, 0 <p < 1, and 0 < q < g.

We say that a cube Q c Rn is dyadic if Q = Qjk = {x = (x\, x2xn) € Rn : 2-jki < Xi < 2-j (ki +1), i = 1,2,...,n} for some j € Z and k = (ki,k2,...,kn) € Zn. Denote by £(Q) = 2-j the side length of Q and by xq = 2-jk the "left lower corner" of Q when Q = Qjk. We use supp and Xp to express the supremum and summation taken over all dyadic cubes P, respectively. Also, denote the summation taken over all dyadic cubes Q contained in P by ^qcp . For any dyadic cubes P and Q, either P and Q are nonoverlapping or one contains the other. For any

function f defined on R", j e Z, and dyadic cube Q = Qjk, set

/q(x) = \Q\ ^/(XQ1) = 2in/2f(2ix - k),

fj (x) = 2jnf ( 2jx), f(x) = f-x).

It is clear that gj * f (xq) = | Q| X/1{f,gQ), where (J,g) denotes the paring in the usual sense for g in a Frechet space X and f in the dual of X.

Choose a fixed function y in Schwartz class S = S(R"), the collection of rapidly decreasing Cg functions on R", satisfying

supp(<^) c jl: 2 <\l\< 2}, 3 5

|£(l)| > 0 0 if 5 <\l\< 3.

For a e R and 0 < p,q < +<x>, we say that f belongs to the homogeneous Triebel-Lizorkin space tp,q if f eS'/P, the tempered distributions modulo polynomials, satisfies

P a,q Fp

Z(2kaWk * f I)

for 0 <p < GO,

if G V^

\\P\-1 E (2ka|^k * f (x)ßqdxV < GO for p = G.

J Pk=-log2^(P) j

When 0 < p < go and q = to, the above ¿"?-norm is modified to be the supremum norm as usual, and t^T is defined to be B, which is

to» := sup sup 2kaIfk * f (x)| « sup\Q\-(a/n)-(1/2)

keZ xeQ Q

e(Q)=2-k

Kf'fQ}1 < G

We now introduce a new space CMO^ as follows.

Definition 1.1. Let y e S satisfy (1.2). For a,r e R and 0 < q < g, the generalized Carleson measure spaces CMOp'q is the collection of all f e S'/P satisfying ||f ||cmom < g, where

su4\P\-f E( \ Q \ -(a/n)-(1/2) Kf'VQ)IlQ(x))qdx

P PQ P

sup sup Q

-(a/n)-(1/2)K f,yQ) I = sup \ Q \-(a/n)-(1/2)K f,yQ) |, Q

0 < q < go, q = g,

and xq denotes the characteristic function of Q.

Remark 1.2. By definition, we immediately have CMO"'œ = Fœœ for a,r e R, and it is easy to check CMOar'q = {0} for r < 0 and 0 < q < œ. Note that the zero element in CMO/ means the class of polynomials. Also note that CMO^ = F^ with equivalent norms for a e R and 0 < q < œ. It follows from Proposition 3.3 that CMO^ = for a e R and 0 < q < œ. In particular, CMO^'2 = BMO, and hence the spaces CMO^ generalize BMO.

Remark 1.3. For a dyadic cube P, denote by kP = -log2£(P); that is, kP is the integer so that i(P) = 2-kp. In [5, 6], Yang and Yuan introduced the so-called "unified and generalized" Triebel-Lizorkin-type spaces f7^ with four parameters by

:= sup IP

E {2ka\n * f (x)\)q

for a,T e R, p e (0, to), q e (0, to], and f e S/P. Note that in [5] the space was defined for t e [0, to), p e (1, to), and q e (1, to]. It follows from [6, Theorem 3.1] that

sup|P rjf E(lQI(-a/")-(1/2)Kf,^Q >\XQ(x))

P \J P .QcP

It is clear that CMO^ = F^"1 for 0 <q < to, and hence CMO^ "looks like" a special case of Fpq. In fact, it was proved in [7, 8] that the space F^ is the "same" as the space

CMO"'q1 , .

rq+1-q/p

The definition of CMOaq is independent of the choice of y e S satisfying (1.2). To show that, we need the following Plancherel-Polya inequalities.

Theorem 1.4 (Plancherel-Polya inequality for 0 < q < to). Let e S satisfy (1.2). For a,r e R and 0 <q < to, if f eS' / P satisfies

k=-log2^(P) QCP

e(Q)=2-k

E E 2kasup\^k * f (u)\ IQI

k=-log2^(P) QCP

e(Q)=2r<

IP r Ë E (2kasup\^k * f (u)\) IQI

IP Ir E E [2kaUnQ \(pk * f (u)\) IQI

k=-log2£(P) QCP \ ueQ

e(Q)=i-k

Theorem 1.5 (Plancherel-Polya inequality for q = to). Let eS satisfy (1.2). For a,r e R, if f eS'/P satisfies

sup IQI

<a/n)-rsup\ykQ * f («)\ ) < TO,

(1.10)

sW |Q|-(a/n)-r sup \(^kQ * f (M)\j « sup( IQ|-(a/n)-rMnQ* f (M)\) . (1.11)

Remark 1.6. Let e S satisfy (1.2). Denote by CMO^(y) the collection of all f e S'/P satisfying ||f ||CMO«-q(^) < to defined in Definition 1.1 with respect to y. Then, by Theorem 1.4,

CMor < ^F1'

IPI-r E E (2kasup\^k * f (m)\ ) IQI

k=-log/(P) QçP \ «eQ /

e(Q)=2-k

< C sup

IPI-r E E 2kainf* f (u)\) IQI

k=-log2^(P) QçP V "eQ

£(Q)=2-k

(1.12)

< Cllf llcMo^-q w for0 <q< TO.

Similarly, ||f ||CMO«-qy < C||f |CMO«-q(^ by interchanging the roles of y and Hence, the definition of CMO^y) is independent of the choice of y and, for short, denoted by CMO^. Also, Theorem 1.5 shows that CMOa,TO is independent of the choice of y satisfying (1.2) in the same argument.

Remark 1.7. The classical Plancherel-Polya inequality [9] concludes that if {xk} is an appropriate set of points in R", for example, lattice points, where the length of the mesh

is sufficiently small, then

E\f(*k)\M «Hfl. (1.13)

for all 0 <p <to with a modification if p = to.

Using the CalderOn reproducing formula (either continuous or discrete version), several authors obtain the variant Plancherel-Polya inequalities [10-13]. These inequalities give characterizations of the Besov spaces and the Triebel-Lizorkin spaces. Moreover, using these inequalities, one can show that the Littlewood-Paley ^-function and Lusin area S-function are equivalent in Lp-norm.

Define a linear map Sv from S'/P into the family of complex sequences by

Sf = { if'V^Q- (1-14)

Let So denote the family of f e S satisfying Jxkf (x)dx = 0 for all k e (N U {0})". For g e CMO-a'q, define a linear functional Lg by

Lg (f) = S(g), S,(f )> = Z(g'fQXf'VQ> for f e So. (i 15)

We now state our first main result as follows. Theorem 1.8 (duality for f^)- Suppose that a e R, 0 <p < 1, and 0 < q < to.

(a) For 1 <q < to, the dual of Fpq is CMO^/,^^) in the following sense.

(i) For g e CMOj;pHq,/q), the linear functional Lg given by (1.15), defined initially on S0, extends to a continuous linear functional on tO1 with ||Lg || < C\\g\\ O-a,q .

CMO(q'/p)-(q'/q)

(ii) Conversely, every continuous linear functional L on Fp'q satisfies L = Lg for some g e CMO-;'i)-(q'/q) With llgllCMO-' ( '/ ) < C"L".

(q' /p)-(q/q)

(b) For 0 <q < 1,the dual of ta'q is CMO-a+("/p)-"'TO (any r e R) in the following sense.

(i) For g e CMO-a+("/p)-"'TO, the linear functional Lg given by (1.15), defined initially on S0, extends to a continuous linear functional on F^ with ||Lg|| <

C|lgllCMO-a+("/p)-"'TO.

(ii) Conversely, every continuous linear functional L on f^ satisfies L = Lg for some

g e CMO-a+("/p)-"'TO with 11g11CMO-a+("/p)-",TO < CllLll.

Remark 1.9. For 0 < p < 1 and 0 < q < 1, it follows immediately from [2, 3] (Verbitsky [4] corrected a gap of the proof) and definition that (t^)' = = CMO-a+("/p)-"'TO

(any r e R). Theorem 1.8 (b) shows a different approach to the duality and includes the case of p = 1.

For p = 1 < q < to, we have CMO-"^/^ = (fa'q)' = F^TOa'q. For 0 < p < 1 < q < to,

CMO-a'q = (fa'q)' = f-a+("/p)-"'TO and hence CMO-a'q = CMO-a+("/p)-"'TO That CMO(q'/p)-(q'/q) (Fp ) Fto , and hence CMO(q'/p)-(q'/q) CMOr . That

is, each CMOaq,/p)-{q/ql) coincides with CMOa+("/p)-"'TO for a,r e R and 0 <p < 1 <q< to.

Remark 1.10. In Remark 1.2 we are aware that CMOa'q generalize BMO by the viewpoint of spaces directly. Choosing a = 0 and q = 2 in Theorem 1.8, we immediately have (Hp)' = (tp-2)' = CMO02/p)-1 for 0 < p < 1. In particular, BMO = CMO0'2. Once again, we obtain

that CMOa'q generalize BMO by the viewpoint of duality. It was also proved in [14] that the dual of the multiparameter product Hardy space is the generalized multiparameter Carleson measure space (cf. [14] for more details).

Remark 1.11. For a,r e R, in order to make each index works, we defined CMOa/<x> to be supP\P\-rsupQrp\Q\-(a/n)-(i/2)\(J,fQ)\ in our earlier 7

for 0 < p,q < 1, the dual of F^ would be CMO-*/^. In this paper, however, we follow the referee's suggestion and adopt a more "natural" definition of CMO^ in Definition 1.1, that is, the limit of CMO^ as q ^ to. The sequence space c^ given in Definition 2.1 has a similar story as well.

As applications, we first recall the Haar multipliers introduced in [15, 16]. Given a sequence t = [tI}j, where the I's are dyadic intervals in R, a Haar multiplier on L2(R) is a linear operator of the form

Htf (x) := £ ti{f,hi)hi(x), f e L2(R), (1.16)

where hI are the Haar functions corresponding to I.

Using Meyer's wavelets, we may generalize the above Haar multiplier to Rn and obtain a necessary and sufficient condition for the boundedness on Triebel-Lizorkin spaces. Let [yv} for i e E := [1,2,...2n - 1} be Meyer's wavelets (cf. [17], [18, pages 71-109]). Then, [yQ}, where i e E and Q's are dyadic cubes in Rn, is a frame for Fpq for a e R and 0 <p,q <to; that is, \\f \\paq a £ieE \\[(f,yQ)}QII aq for f e Fpq. For t = [tQ}Q, define a wavelet

p Q Jp

multiplier Xt on Rn by

Ttf) = EE \Q\-1/2tQ f,yQ )yQ (1.17)

for f e S/P such that the above summation is well defined. Theorem 1.12. Suppose that a,p e R, 0 <p < 1 and 0 < q < to. Then,

(a) for 1 < q < to, Tt is bounded from Fpq into Fa^1 if and only if t e ^/y^/y

(b) for 0 < q < 1 and r e R, Tt is bounded from F(p'q into Fp^1 if and only if t e cf+(n/p)-n,TO, where c^ is given in Definition 2.1.

We consider another application. Let y and y in S satisfy (1.2) and (3.1). Choose a function O e S supported on [0,1]n and J® = 1. For a e R and g e FTOtoTO, define the paraproduct operator ng by

ng (f) = E{g,yQ )\Q\-1/2{f, Oq )yQ. (1 18)

Thus, the adjoint operator ng is

ng(f) = Z{g,yQ)\Q\-m(f,yQ) Oq. (1.19)

Then, ng 1 = g and ng 1 = 0 since (1,Oq) = \Q\1/2 and (1,yQ) = 0. Also, if g e F°TOTO, then

both ng and ng are singular integral operators satisfying the weak boundedness property. Moreover, ng is a Calderon-Zygmund operator (i.e., ng is bounded on L2(R")) if and only if g e fTO2 by David-Journe's T1 theorem [19] (also see [12, Theorems 5.4 and 5.8]). The authors showed a more general type of paraproduct operators in [12, page 688], which were derived from the discrete Calderon reproducing formula.

Theorem 1.13. Suppose that ¡5 e R, 0 <r < 1 and 0 <p < r < q <r/(1 - r).

(i) For a < 0, ng is bounded from tp,''1 into F;+5'r if and only if g e CMO5''q-/;()q/pr()q-r).

(ii) If a e R with a + ¡5 > 0 and g e CMO¡;rP/pr()q-r), then ng is bounded from tp' into

■a+fi/r

Remark 1.14. When r = 1,0 <p < 1 < q < to, and ¡5 e R, Theorem 1.13 says that ng is bounded from p;" into fa+5'1 if and only if g e CMO;/p)-(q'/q) for a < 0, and ng is bounded from p;" into Fa,+5'1 for a > -5 provided g e CMO^p)-(qVq). In 1995, Youssfi [20] showed that, for ¡5 e R, 1 < p < to, 1 < q < 2, and g e fTO'TO, ng is bounded from tp' into tp,'v if and only if g e fTOp. The special case of Theorem 1.13(i), p = r, generalizes Youssfi's result to 0 < p < 1. More precisely, for a < 0, 5 e R, 0 <p < 1, and p < q <p/(1 - p), ng is bounded from P,'' to

f;+5'p if and only if g e CMOfq/(q-p) = fTOpq/(q-p).

The paper is organized as follows. In Section 2, we introduce the discrete version of the generalized Carleson measure spaces c;'q and show that the duals of sequence Triebel-Lizorkin spaces fP'q for 1 < q < to and 0 < q < 1 are c;/y/) and cr;+("/p) "'TO (for any r e R), respectively. In Section 3, we prove the duals of homogeneous Triebel-Lizorkin spaces Fpq for 1 < q < to and 0 < q < 1 to be the generalized Carleson measure spaces CMO(;'/p)-(q,/q)

and CMO-;+("/p) "'TO (for any r e R), respectively. In Section 4, we prove the Plancherel-Polya inequalities that give us the independence of the choice of y for the definition of the generalized Carleson measure spaces. In the last section, we show the boundedness of wavelet multipliers and paraproduct operators. Throughout, we use C to denote a universal constant that does not depend on the main variables but may differ from line to line. Also, Q and P always mean the dyadic cubes in R", and, for r > 0, we denote by rQ the cube concentric with Q whose each edge is r times as long.

2. Sequence Spaces

In this section, we introduce sequence spaces c;'q and then characterize the duals of fp'q by means of c;'q. Let us recall the definition of these sequence spaces fp'q defined in [2]. For a e R and 0 <p; <to, the space fp'q consists all such sequences s = {sq }q satisfying

|s|f;'q :=

X(lQ|-(a/")-(1/2)|sQ|XQ)

< to if 0 <p < to,

Lp , 1/q (^

sup I IP Z (|Q|(-a/")-(1/2) | Sq | XQ (x) )qdx\ < TO if p = TO.

As before, the previous £q-norm is modified to the supremum norm for 0 <p < to and q = to. For p = q = to, we adopt the norm

J := supiQr^-^lsQl. (2.2)

Note that \\s\\qaq is equivalent to the Carleson norm of the measure

E( i Q i-(a/n)-(1/2)|sd)q i Q i <W(Q)), (2.3)

where 6(x t) is the point mass at (x,t) e R++1. See [2] for the details

To study the duals of fp^,q, we introduce a discrete version of the generalized Carleson

measure spaces ca'q.

Definition 2.1. For a,r e R and 0 < q < to, the space cr is the collection of all sequences t = [tQ}q satisfying WtW^q < to, where

'sup \ P \ ~r[ E ( \ Q \-(a/n)-(1/2)|tQ|XQ(x))qdx for 0 <q< TO,

P PQ P

lltllc-q :=

supsup iQ r(a/n)-(1/2)ltQl = sup iQ i-(a/n)-(1/2)|fQ| for q = to.

P QCP Q

It is obvious that

\P\-r E (\ Q \-(r/n)-(1/2)+(1/q)|tQ|)^ for 0 <q< TO (2.5)

and M^to = \\t\\fTOTO for a,r e R. Using embedding theorem, Frazier and Jawerth [2, equation (5.14) and Theorem 5.9] obtained that, for a e R and 0 < q < to, the dual of

fP,q is fTO-^+(n/p)-n,TO when 0 <p < 1, and the dual of fpq is f^ . Note that c^ = [0} for r < 0

T^TorC» TA70 rri^70 fVl Q H 1 1 Q 1 f01 d \ O V* 1 t~\ CDHIIDnrD cncipoc -f

and 0 < q < to. Here we give the dual relationship between sequence spaces fp,q and c^.

Theorem 2.2 (duality for fp,q). Suppose that a e R, 0 <p < 1, and 0 < q < to.

(a) For 1 < q < to, the dual of fp,q is c-^)^, /) in the following sense.

(i) For t = [tQ }q e cq*)W the linear functional ¿t on fp" given by £t(s) = Xq SQtQ is continuous with \\£t\\ < C\t\for s = [sq}q e fp,q.

(q'/p)-(q'/q)

(ii) Conversely, every continuous linear functional i on f p^ satisfies i = £t for some

t e %/v)-(q'/q) with m^ < C\\i\\.

(q'/p)-(q'/q)

(b) For 0 <q < 1,the dual of fp,q is c-r+(n/p)-n,TO (any r e R) in the following sense.

(i) For t = {tQ }q e c/+("/p) "'TO, the linear functional 4t on fP'q given by 4t(s) =

Xq SQtQ is continuous with ||4t|| < C|t|c-a+(»/p)-»'TO for s = {sq}q e fp'q. (ii) Conversely, every continuous linear functional 4 on fP'q satisfies 4 = 4t for some t e c-a+("/p)-"'TO with ||t||c-a+(»/p)-»'» < C||4||.

Remark 2.3. For a e R and 0 < q < to, sequence spaces cp' = fTO'q and c;'TO = fTO'TO (for any

r e R) by definitions. Theorem 2.2 shows that (fP'q)' = f-;'q, which gives a different but simpler proof of Frazier-Jawerth's result for the duality of f,'q (cf. [2, Theorem 5.9]).

Proof of Theorem 2.2. For s = {sq }q e fp'q and t = {tQ}Q e c-a';, set s = {Sq}q and t = {?q}q to

be Q Q Q Q

Sq = ¿q = IQIa/"tQ. (2.6)

Then, £f(s) = 4(s). Also,

llf = fp^' |i||c0'q, = 1111 . (2.7)

Without loss of generality, we may assume that a = 0.

We first consider the case 1 < q < to. Let t e c;'/) and define a linear functional

4 on f°'q by

4(S) = X sQtQ for s e fpq. (2.8)

For S = {sq}q e f^,let

vq{x) :=(|] (|Q|-1/2|sq|xq(x^ ) . (2.9)

For k e Z, let

Qk : = {x e Rn :2k < Vq(x) < 2k+^,

Qk : = { x e R" : Mxak (x) > 1J' (2.10)

Bk := {dyadic Q : |Q n > Q' |Q n j1 < Q for some j > k

where M is the Hardy-Littlewood maximal function. Then, for each dyadic cube Q, there exists exactly a k e Z such that Q e Bk. For every Q e Bk, let Q denote the maximal

dyadic cube in Bk containing Q. Then all of such Q's are pairwise disjoint. Thus, by Holder's inequality for q and the inequality (a + b)p < ap + bp for 0 <p < 1,

ZsQtQ Q

ee e( iQ i-(1/2)+(1/?)|sqI)( i Q i (1/2)-(1/q)m)

k(=-Z QB QÇQ

E E I E ( iQ i-(1/2)+(1/q)M)q I I E (i Q r(1/2)+(1/q,)M)q

kG~Z QeBk\ QÇQ I \ QCQ

Wfp)-W fq)

EE|E( i Q i -(1/2)+(1/q)|sqi)qI |Q|

teZ QpP„. V qcQ I

p/q 1fP

q 1 ^|i-(p/q)

QeBk \ QÇQ QB

(2.11)

Since Q e Bk implies Q c k, the disjointness of Q's and Holder's inequality yield

p/q1/p

E sq^q Q

< lltll

(q'/p)-(q'/q) ^kez

Îe|ôkr/q)(E (i Q i -(1/2)+(1/q)|SQ0'

I kez \Q(=-Bk

(2.12)

We claim that £QeBk ( \Q\-(1/2)+(1/q) \sq\)q < C2kq\£k\ for k e Z and 0 < q < to. Assume the claim for the moment. The weak (1,1) boundedness of M gives \££k\ < C\£k \, and hence

E SQ^Q Q

< C||t|| oq

(q'/p)-(q' fq)

(e|ö k

1-(p/q)

E 2kp i ik i

< cpy^ _

(q'fp)-(q'fq) \kez

< C|t|c0^ \\Vq\\Lp

(qlfp)-(ql fq)

= C|t|c0,ql ||s|/0,q.

(q'fp)-(q' fq) Jp

To prove the claim, we note that, for k e Z and 0 < q < to, 2q(k+1)|£k 1 >

(2.13)

k > (Vq(x))qdx

}Q k\U f=k+1 ï

E (|QI-1/2|sqIiq(x))qdx

J ikM]^ ï Q v 7

(|QI-1/2|sq|)q| (ïk \ j n q| for some j > k + 1,

Abstract and Applied Analysis which implies

2q(fc+1)|6fc|> (|Q|

-(1/2)+(1/q)

(2.15)

For 0 < q < 1, with a modification, we have

ZSQtQ Q

(i/p)-i

<XX X(lQl1/2|sQO(lQI-(1/p)+(1/2)|tQl)v M ,

keZ Q eB^Q-Q \|Q|/

<l|t||

,(n/p) -n,c

XXIX IQI1/2|SQ|

QeBk \ Q-Q

< C||t||c(n/p)-n. ( X|"fc|1-^2fc|ñJ)p

< c|t|c(n/P)-n,~ y vqy^

(1/p)-1

(2.16)

On the other hand, suppose that £ is a continuous linear functional on fP'q. For each dyadic cube P, write eP = {(ePto be the sequence defined by

1 if Q = P, Q 10 if Q/ P•

(2.17)

Let tP = £(eP) and t = {tP}P. Then, for s = {sq}q e ,

¿(s) = X sQtQ = 4(s)

(2.18)

Fix a dyadic cube P. For 1 < q < to, let X be the sequence space consisting of s = {sq}qcp, and define a counting measure on dyadic cubes Q c P by da(Q) = |Q|/|P|(q /p)-(q /q). Then,

(q'/p)-(q'/q)

X (QI-(1/2)+(1/q')|tQ|)q

\\s\\eq (X'da)

IP|(q/p)-(q/q) qU IQI SQ|Q|-1/2|tQ|

<\\£\\ sup

\\eq(X'do)

(q'/p)-(q'/q)

Note that

{| pPS-/'^} QÇP - I PI (q'/l)-(q'/q)^EE Q ^^ ^ ^ ^ }

f0,q I I \QÇP / I (2.20)

< C||s||m?(X,dc).

1 q \ 1/q

__L_ , Q j(-l/2)+(l/q') |tQ|)^ < Cm. (2.21)

and hence t e c0qq/p)-(qyq). For 0 < q < 1, consider eP defined before. Then, \\ePW^ =

\ p\-(i/2)+(i/p) and

( \ P \ (1/2)-(1/p)\ tp\)||ep|| = \ tp\ = |^(ep) | < \m\||ep|| (2.22)

Hence, HtH^/p)-^ = supP \ P\(1/2)-(1/p)\ tP\ < \\e\\. This completes the proof. □

3. Proof of the Main Theorem

Let us recall the y-transform identity given by Frazier and Jawerth [2]. Choose a function y eS satisfying (1.2). Then there exists a function y eS satisfying the same conditions as y such that Xy(2-j£)y(2-j£) = 1 for = 0. The y-transform identity is given by

f = E{f,yQ)yQ, (3.1)

where the identity holds in the sense of S' / P, So, and Fpq -norm.

Define a linear map Sy from S' /P into the family of complex sequences by

SyJ) = {{J,yQ)}Q, (3.2)

and another linear map Ty from the family of complex sequences into S'/P by

^(Mq) =x SQyQ. (3.3)

Then, Ty o Sy\paq is the identity on Fp,q by [2, Theorem 2.2].

Proposition 3.1. Suppose that a e R and, 0 <p,q < +œ, and y,y in S satisfy (1.2) and (3.1). The

7y • Fp " Jp and Ty • Jp

linear operators Sy : FJq i—> jp'^ and Ty : fa i—> Fa* defined by (3.2) and (3.3), respectively, are

F aq Fp

Dual relation by Theorem 2.2 (a)

(q '/p)-(q/q)

Dual relation by Theorem 1.8(a)_ cmo-"'17'

CMO(q'/p)-(q'/q)

Figure 1: Diagram for spaces and maps for 1 < q < to.

faq jP

Dual relation by Theorem 2.2 (b)

(n/p)-U'!X

a+(n/p)-

Figure 2: Diagram for spaces and maps for 0 < q < 1.

bounded. Furthermore, Tf o Sf is the identity on p;". In particular, ||f Hp;; « ||Sy(f)Wfa; and t;'q can be identified with a complemented subspace of fP'q.

Figures 1 and 2 illustrate the relationship among P,'', fP'q, CMO;'q, and c;'q.

One recalls the almost diagonality given by Frazier and Jawerth [2]. For a e R and 0 <p'q < to, let J = n/(min{1,p,q}). One says that a matrix A = {aQP}qp is (a,p^)-almost diagonal if there exists e> 0 such that

sup |aQP I < +TO' (3.4)

Q'P wQP (e)

() (e(Q)YA |xQ - xp | VJ-£ . f/4(Q)\("+£)/2 / 4(P) x(("+-)/2)+J-"| WQP(e)n^j V + max(*(P),e(Q))J im) I

Lemma 3.2. For a,r e R and 0 < q < to, an (a + nr,q,q)-almost diagonal matrix is bounded on c;'q. Furthermore, when r > 0,an (a + nr' <tov TO)-almost diagonal matrix is bounded on c;'TO.

We postpone the proof of Lemma 3.2 until the end of Section 4.

Let r,r e R. For q — œ, we have car'™ — fT and CMOr'œ = FT. Thus, Sy • CMOr'œ ^ ca,co and Ty • cr,œ ^ CMOr-œ are bounded by Proposition 3.1. For 0 <q< œ and f e CMO^, let s — (sq}q — Sy(f). Then, the y-transform identity (3.1) shows that f — ^QsQyQ and Wfiicmo^ — l^y^W^q — Mcy.In particular, \\f \\CMOr'q — l^yWW^q — WSy(f)Wp « \\fh%<. Furthermore, for s e c°r'q,

\\Tf (s)\\cMOaq " Zspyp =

P cMoa,q

Yl^p'Vq P I JQ

= ||As||cM, (3.6)

where A := [(yP,yQ)}qp is (a + nr,q,q)-almost diagonal (cf. [2, Lemma 3.6]) and hence A is bounded on cp'q by Lemma 3.2. Therefore, Sy is bounded from CMOp'q to cp'q and Ty is bounded from cp'q to CMOp'q.

We summarize that Ty o Sy\CMOpq is also the identity on CMOp'q.

Proposition 3.3. For (a,r,q) e R x R x (0, to) or (a,r,q) e R x R x[to}, the linear operators Sy : CMOp'q — c^ and Ty : cp'q — CMOp'q are bounded. Furthermore, Ty o Sy is the identity on CMOp'q and \\f hcmo^ = \\Syf H^q. In particular, \\f hcmo^ = \\Sy(f = \\Sy(f )\\p a \\f H^ for a e R and 0 < q < to, and \\f \\CMOa-TO = \\Sy(f )\\caTO = \\Sy(f )\\j^,to a \\f \\p«,to for a,r e R.

Theorem 1.8 can be proved as a consequence of Propositions 3.1-3.3 and a duality result between two sequence spaces.

>-a,q'

Proof of Theorem 1.8. First let us consider the case for 1 < q < to. Let g e CMO(q,/p)-(q,/q) by Proposition 3.3, \\g\\CMO-pq' = \\Sy(g)\\ -r,q . It follows from Theorem 2.2 that Cs

(q'/p)-(q'/q) _ fy /p)-(q'/q)

is a continuous linear functional on fp'q and \\£Sy(g) \\ a \\Sy (g) \\ -aq . Hence, for f e S0,

c(^/p)-(q'/q)

lLgf l < C\\Sy (g )\\

c V , II 9

(q fp)-(q fq)

sy(f) \U < C\\g\\

cmo a,q

Wfp)-Wfq)

Fa,q. Fp

Since J>0 is dense in Fp'q, the functional Lg can be extended to a continuous linear functional

on F^ satisfying №gN < CHgN

cmo:y, )

(q fp)-(q fq)

Conversely, let L e (FP'q)', and set ê — L ◦ Ty on fp'q. By Proposition 3.1, £ e (fp'q)'.

ÇJpMqi/q) sUchthat

Thus, by Theorem 2.2, there exists t — (îq}q e c^,

— X sQlQ for {qq e f

and HtH^ « UH < CHLH. For f e Fp'q, we have

(q'fp)-(q'fq)

m ◦ Sy(f ) = L ◦ Ty ◦ Sy(f ) = L(f ). (3.9)

So, for f eSo and letting g = T9 (t) = £Q ïq^q,

Lf = e ◦ Sf = £ {f,yQ)tQ = <t,S,(f)>. (3.10)

It follows from [2, equations (2.7)-(2.8)] that (g,f> = (Sv(g),S(p(f)> and (t,S9(f)> = (Tv (t),f > for f e So and g e S'/P. This shows that L(f ) = (Ty(t),f > = Lg(f) for f e So. Proposition 3.3 and Theorem 2.2 give

||g||CMO-«q < CHtnc^ < C||L||. (3.11)

(qVp)-(q' /q) (q'/p)-(q'/q)

A similar argument gives the desired result for 0 < q < 1 with a slight modification, and hence the proof is finished. □

Remark 3.4. As pointed out by one of the referees, Yang and Yuan [8, Theorem 1] show that if t > 1/p and 0 < p,q < œ, then f7^ = Fœ+nT (n/p)/x>, where the definition of Fpq is given in Remark 1.3. Thus, for0 <p < 1 and 1 < q < œ,

(FT)' = F^"^ = fqJ(1/p)-(1/q) = CMO^/p )-(q'/q), (3.12)

which demonstrates a different approach to the duality.

4. Proofs of the Plancherel-Polya Inequalities

In this section we demonstrate the Plancherel-Polya inequalities.

Proof of Theorem 1.4. Without loss of generality, we may assume that a = 0. By (3.1), we rewrite <pj * f (u) as

< * f(u) = ^(f^Q) <(u - x)fQ(x)dx

E X iQi(f,Vk('-xQ)) (u - x)Wk(x - xQ)d'

^ z Q ^

~xQ)) I < (u - x)wk(x - xQ)dx.

e(Q)=2-k

Using the inequality [2, page 151, equation (B.5)]

^(^j (u - x)fk (x - xq)dx

< C2-Klj-fci-

2-(jAfc)

(2-(jAfc) + |u - xq i)

where j A k = min{ j, k} and K > 1 + nr, we obtain

i _ i 9-(jAk)

|*j * f (u)|< ^ E rK|j-k||Q| TTjk I-^ ^ * I- (4.3)

kez Q (2-(jAk) + |u - Xq|) v ;

e(Q)=2-k

Thus, for e(Q) = 2-j,

sup | d>j * f (u)|) < C[ V V 2-K|j-k|| Q |---|Sk * f (xq)|

uuQ % nV ~ ^ Q ^ (2-(jAk) + Xq- - XQ|)n+1 m fQn

e(Q)=2-k / (4.4)

2-(jAk)

< ^ E 2-K|j-k|Q|7j-2-^^*f(Xq)|

kez Q (2-(jAk) + | Xq- Xq ^

e(Q)=2-k

where the last inequality is followed by Holder's inequality and

2-(jAk)

£ -^ < c (4.5)

Q (2-(jAk) + | Xq- Xq |)

e(Q)=2-k

Denote Tq by

TQ := S|<^k * f (u)|q- (4.6)

Since xq can be replaced by any point in Q in the last inequality,

2-(jAk)

/ i~ iV 2-(jAk)

(sup|fc * f (u)|) < ^ E 2-K|j-k||Q|72-"Ak)—|-^TQ- (4.7)

\ueQ- '/ kez Q (2-(jAk) + | Xq- Xq |)

Given a dyadic cube P with e(P) = 2 ko, the above estimates yield

■x, / \ q

E E sup|<^j * f (u)| |Q'|

j=ko QQP \ueQ- /

e(Q-)=2-j

2-(jAk) (4.8)

_T1 i i v '

x - jAk

< E 2-K|j-k||Q'|--—-2-^tq|q|

j=ko Q'CP kez Q (2-(jAk) + |xQ, - XQ|)

e(Q-)=2-j e(Q)=2-k

:= CAi + CA2,

& O-(j'Ak)

* = E E E E -Tq 1 Q 1 -

j=k° Q'CP k>k° Q (2-(jAk) + \XQ'- Xq |)

e(Q')=2-J rn)=2-k , .

& 2-(jAk) v '

* E 2-K|j-k||Q'|-2--r-2-^Tq | Q | .

j=k° Q'CP k<ko Q (2-(jAk> + | XQ' Xq|)

)=2-J e(Q)=2-k

Then, A1 can be further decomposed as

& O-dAk)

*i = E E E E 2-K 1 d-k llQ'l72d^+2-^ TQ i Q i

j=k° Q'CP k>k° QC3P (2 (jAk) + | XQ' XQ |)

e(Q')=2-i e(Q)=2-k

K1 d-k iQ'i • n+1

j=ko QCP k>ko Qr3P=0 (2-(jAk) + \XQ - XQ^"+

e(o' )=2-J e(Q)=2-k

:= *ii + *12.

2-(dAk) (4.10)

EEH 2-K|d-k||Q'|-—--+tTq | Q|

There are 3n dyadic cubes in 3P with the same side length as P, so

E TqIQI < 3n sup E TqIQI. (411)

Qc3P P'C3P QCP' (4.t1)

e(Q)<e(P) e(P)=e(P) e(Q)<e(P')

& J-(jAk)

Ipi-rAn <cipiE E E 2-Kli-klQ'|72--r2-^ttqiQI

j=ko Q'CP k>k° QC3P (2 (jAk> + ^Q- XP |)

e(Q'),2-l t(Q)=2-k (4 12)

< Csup|P'|-r E E inQ№k * f (u)^IQI.

P' k=-log2£(P') QCP' ueQ

e(Q)=2-k

Next we decompose the set of dyadic cubes {Q : Q n 3P = C(Q) = C(P)} into {B{} according to the distance between each Q and P. Namely, for each i e N,

Bi := {P' : P'n 3P = 0, £(P) = £(P'), 2i-ko < ^p - yP| < 2i-k°+T},

(4.13)

where yQ denotes the center of Q. Then, we obtain

\p\-ra12 < \pVZ E E E 2-кjj-kj |q'\

2"(jAfc)

i=1 P'eBi j=ko Q'CP k>ko QCP'

e(Q')=2-i ¿(Q)=2-k

(4.14)

(2-(jAk) + \xp, - xpj)

Tq\Q\.

Since X Q-cP |Q'| = |P| for each j > k0 and | xp - xp| « 2i-k° for P' e Bj, the right-hand side of

e(o' )=2-j

(4.14) is dominated by

^E EjP\ 2(i-ko)(n+i) E ( E20

¿=1 P'eBi 2 k>ko\j=ko

-(jAk)-jk-jj

PT E Tq\Q\

¿(Q)=2-k

(4.15)

There are at most 2(t+2)n cubes in B¿, and hence

\P\-rA12 < C\

sup|P'rs E Tq\Q\

P' k>ko QCP'

e(Q)=2-k

oo 2-ko

e\P L.......2"

2(i-ko)(n+1) '

= Csup|P'|-r E E inQI£k */(«)T\Q\

k=-log2^(P') QCP'

i(Q)=2-;

(4.16)

To estimate A2, for i e N and k < ko, set

E,k := {Q : ¿(Q) = 2-k, xq e 2iP \ 2i-1p}.

(4.17)

Then, \xq - xp\« 2i-ko for Q e Ei,k and

a2 = eee e

2-K\j-k\\p \ 2-(jAk)

j=ko k<ko ¿=1 QeEi,k \QP (2-(jAk) + |xq - xp|)'

-\Q\-rTQ\Q\. (4.18)

Since, for Q e Ei/k,

\ Q \-rTQ \ Q \ < sup|P'|-r E E Tq'IQ'I (4.19)

pl m=-log2^(P') Q'CP'

¿(Q')=2-m

and the number of dyadic cubes contained in Ei/k is at most 2(i+k-k0)n,

| PI r A2 < C

sup|p'|-r E E tqQ

2-k(n+1)

P' m=-log2£(P) Q'CP'

e(o' )=2-m

X V V V2(ko-k)nr2K(k-j) _2_2(i+k-ko)n (4.20)

Z-i Zj Zj 2(i-k0)(n+1)

j=ko k<ko i=1

= C sup|p'r E E fm * / (M)I^IQ'I,

p' m=-log/(P') Q'CP' "e<Q

e(Ql)=2-m

where the condition K > 1 + nr is used in the last equality. Combining the estimates of A1 and A2, we prove Theorem 1.4. □

By modifying the proof above, we may easily show Theorem 1.5. Detailed verifications are left to the reader.

We now return to show Lemma 3.2.

Proof of Lemma 3.2. For r < 0, c^ = {0}, and hence the result holds. For r = 0, c^ = /a'q, and so the matrix is bounded by [2, Theorem 3.3]. To complete the proof, it suffices to show the boundedness of (a + nr, q, q)-almost diagonal matrices for the case r > 0.

We may assume that a = 0 since the case implies the general case. The proof is similar to the proof of Theorem 1.4. Here, we only outline the proof. First let us consider the case for q > 1. Let A = {aQP}q p be an (nr,q,q)-almost diagonal matrix. Then, for £(Q) = 2-k,

| (As)q| < C£ E 2(j-k)(nr+((n+^)/2))(1 + 2j|xq - xp|)~n-£ispi' jez e(p)=2-i

(I Q r1/2|(As)o|)q < C E E 2(j-k)(nr+(^2))(1 + 2j|xq - xp|p( I P|-1/2 I sp I )q

jez i(P)=2-j

(4.21)

due to Holder's inequality. Given a dyadic cube R with £(R) = 2 s,

EE (I Q I-1/2|(As)q|) q I Q I < CI + CII' (4.22)

k>5 QCR

e(Q)=2-'

I = E E E E 2(j-k)(nr+n+(^/2))(1 + 2j|xq - xp|)-n-^IPI-1/2 I sp i)qIPl

k>S QCR j>s £(P)=2-i

e(Q)=2-k

II =E E E E 2(j-k)(nr+n+(£/2)^1 + 2j | xq - xp|)-n-^IPI-1/2 I sp i)qIPI.

k>s QCR j<s e(p)=2-l e(Q)=2-k

Then, I can be further decomposed as

I = E E E E ^H™^))/! + 2j|xq -„|)"~(|P|-1/2 |Sp|V| P|

k>6 QCR j>6 PC3R

e(0)=2-k e(P)=2-i

+ E E E E 2(j-k)(nr+n+(e/2))(1 + 2j 1 Xq - xp|)-"-£(|p|-1/2|sp|)?|p| (4-24)

k>6 QCR j>6 Pn3R=0

l(Q)=2-k i(P)=2-J

:= I11 + Ill-

The same argument showed in the proof of Theorem 1.4 for the term A1 gives us

|R|-rI < C||s|| V (4.25)

To estimate II, for i e N and j < 6, let

Ehj := {Q : £(Q)= 2-j, xq e 2iR \ 2i-1^. (4.26)

Then, using the same argument as Theorem 1.4 for A2, we have

|R|-rII < C||Sy V (4.27)

Both estimates for I and II show the desired result for q > 1.

When q < 1, we modify the previous proof by replacing Holder's inequality with q-triangle inequality to get the result.

When q = to and r > 0, the space c^ = f^, and hence an (a + nr, to, w)-almost diagonal matrix is bounded on c^ by Proposition 5.3. □

Remark 4.1. Note that caq = /TO'q. By a duality argument and [2, Theorem 3.3 and page 81], one can show that the (a + n, q, q)-almost diagonal matrix is bounded on /TO,q. When q > 1 and r > 1, we can prove Lemma 3.2 by duality in Theorem 2.2. Let A = {aQP}qp be an (nr,q, q)-almost diagonal matrix. Also define the transpose of A by A' = {aPQ}Q P. For q > 1 and r > 1, let p = (q + q')/(q'r + q). Then, p < 1. Since A is (nr,q,q)-almost diagonal, A' is (0,p, q')-almost diagonal by a calculation for a different value of e. Thus, by Theorem 2.2 (a) and Proposition 5.3, A' is bounded on c^.

5. Applications

We define another wavelet multiplier on Rn by using ^-transform identity as follows. Let (p and f in S satisfy (1.2) and (3.1). For a sequence t = {tQ}q, where the Q's are dyadic cubes in Rn, define the wavelet multiplier Tt by

t.(/) = e \Q\-1/2tQ(f,yQWQ Q

for f e S/P such that the above summation is well defined. Thus, we have the following characterization.

Theorem 5.1. Suppose that a, ft e R, 0 <p < 1, and 0 <q < <x>. Then,

(a) for 1 < q< œ, Tt is bounded from Fpp,q into f^'1 if t g /y/y

(b) for 0 < q < 1 and r e R, Tt is bounded from Fp,'q into Fa+5,1 if t e c5+(n/p) n,œ.

Proof. We show the case a = 0 only, which implies the general case by (2.7). For ¡5 e R, 0 < p < 1, and 1 < q < œ, let f e FP'q and t e CqVpHqVq). It follows from Theorem 2.2 and Proposition 3.1 that

Tt(/)||ffi < C {|Q|-1/2fQ(f,yq)}q

CZ(\Qrß/n\tQ\)\{f,VQ )\ c||{(/,^q)}q||,o„ {\Q\-ß/ntQ}c

'(qVpHqVq)

< C||f Htf<

q ll'IL M

(q'/p)-(q'/q)

This shows that Tt is bounded from F^ into Ff1 and \\Tt\\ < Cyty^q yields the boundedness of Tt for the case 0 < q < 1.

(qVpHq'/q)

. A similar argument □

In order to prove Theorem 1.12, we demonstrate a similar result in sequence spaces first. For a sequence t = [tQjg, define Dt by

Dt(s) = j|Q| 1/2iQSQ} for s = ^sJwith finitely many nonzero terms. (5.3)

Theorem 5.2. Suppose that a, 5 e R, 0 <p < 1, and 0 <q < œ. Then,

(a) for 1 < q < <x>, Dt is extendible to be bounded from fp,q into f^^ if and only if t e rM

C(q'/p)-(q'/q),

(b) for 0 < q < 1 and r e R, Dt is extendible to be bounded from fp,q into f^^"1 if and only if

t e cP+(n/p)-n"x'

Proof. We still assume that a = 0. For p e R, 0 <p < 1, and 1 < q < to, let s = {sq}q e fpq and

t = {tQ }q e C

M (q'/pMq'/q)

. It follows from Theorem 2.2 that

IIA(s% = £(\Qrß/n\tQ\)\sQ\

< C||s|U,

{\Q\-ß/ntQ}i

W /o)~W/q)

= C||s|U ||t|U,'

(qVpHqVq)

Conversely, suppose that Dt maps from f^ into /f'1 boundedly. For t = {tQ}Q, let T = {|Q|-f/ntQ}q. Define a linear functional by

^t(s) = E sQtQ for s = ^sJwith finitely many nonzero terms.

(s)\ < E(\Q\-ß/n\tQ\)\sQ\ = ||Dt(s)|/f!. (5.6)

The assumption shows that is a continuous linear functional on fpq. Using Theorem 2.2,

we have t e clq'/ph(ql/q),and hence t e 4ipHqVq).

For 0 < q < 1, a similar argument gives the desired result of (b). □

Proof of Theorem 1.12. The "if" part follows from Theorem 5.1. To show the "only if" part, define Xi by

Tif ) = E IQr1/2fQfQ>Q. (5.7)

The boundedness of Tt says that T is bounded from pp,q into P'a+p' . Clearly,

Sfi ◦ T ◦ Tfi (s) = Dt(s) for s e f^. (5.8)

It follows from Proposition 3.1 that Dt is bounded from f^ into f^'1, and hence t e c|qq/p)_(q,/q) for 1 < q < to and t e C+(n/p)-n'TO for 0 < q < 1 and r e R by Theorem 5.2. □

In order to study the boundedness of the paraproduct operators acting on Triebel-Lizorkin spaces, we need more results described as follows.

Proposition 5.3 ([2, pages 54 and 81]). For a e R and 0 <p,q <to, an (a,p,q)-almost diagonal matrix is bounded on fp^'q.

Lemma 5.4. Define a matrix by G = {(fp,Oq)}q,p. Then, for a < 0 and 0 < p,q < +<x>, G is (a,p,q)-almost diagonal and hence is bounded on fp'.

Proof. For J(p) < J(Q), since J xYfP(x)dx = 0 for all j, by [2, page 150, Lemma B.1], we have

f O W< c(e(Q) \Y 1 + |XQ - xpI

^OQ)| <c{m) (1 + QQ)

J(P) N ((n+e)/2)+J-n

for s > 0 and a < J - n + (s/2), where J = n/min{1,p,q} and C is independent of P and Q. For £(Q) < £(P), by [2, page 152, Lemma B.2], we obtain

Kfp, Oq> | < c 1 +

1xq — xp1

-J—e

J(Q)\n/2 J(P)/

= cffl) X1

1xq —xp1

J (P )

-J—e

J(Q)\ J(P )/

(n—2a)/2

(5.10)

Choosing s = -2a, we obtain the result. We now can prove Theorem 1.13.

Proof of Theorem 1.13. To simplify notations, let q0 = qr / (q - r) and (1/p0) = (1/p) - (1/q) + (1/q0) . The requirement p < r < q < r/(1 - r) guarantees that p0 < 1 < q0. Now assume that g e CMO'

ß'0 (q0/p0) — (q0/q'0)

and f e Fp . To prove part (i), by (3.1) we rewrite ng(f) as

n(f) = X<^,^q>IQI—1l/2(Xf,yp)fp,®Q/fQ

= Z(g,fQ )IQ|—1/2(Gs)QfQ, Q

(5.11)

where s = {(f,yp )}p. Proposition 3.1 and Theorem 2.2 give

||n(f)11Fa+ß- < c {IQI—1/2 <g,fQ> (Gs)Q}

¿<x.a+,r

= C|( | Q |—(ß/n)—(1/2)+(1/2r)KgtVQ)|)r.( | Q |

—(a/n)—(1/2) + (1/2r)

|(gs)q|) r

{(|QI—(ß/n)—(1/2)+(1/2r)K 0r}

(a/n)—(1/2)+(1/2r)|(Gs)Q|)r}

,<Wr)'

~(('/r)'/(p/r))—(('/r)'/(q/r))

It is clear that

(ß/n)-(1/2) + (1/2r)

\(g><PQ)\) 1

o,(q/r)'

'((q/r)'/(0/r))-((q/r)'/(q/r))

= SuH \P\-r(q/r)'((i/p)-(i/q)) p l jpqcp

\ E (\Q\-(ß/n)-(1/2)\(g,PQ)\XQ(x))r{q/rYdx

Jpqcp x 7

S'Pq)}

qwj-Io

(qo/po) -(qo/p'0)

1/(q/r)'

(5.13)

-(a/n)-(1/2)+(1/2r)

(Gs)^r}

(a/n)-(1/2)

j o,(q/r)

\(gs)q\) XQ(x)

Hence, by Propositions 3.1 and 3.3, and Lemma 5.4,

(5.14)

lln (f ^ < c||i (Z'Pq)}q

IC10 ■■ "fp

(qo/oo)-(qo/q0)

||Gs||f;

< c||gllcMo^-qo , ||s||f;,q

(qo/po)-(qo/qo' )

< c||gwcMoßqo Wf WF;q.

(qofoo)-(qofqo ) 0

(5.15)

Next suppose that ng is bounded from Pp^ into P<a+fS,r. Without lost of generality, we may assume that a = 0. A computation yields

-qo((1/po) + (1/qo) -1)

IPQÇP = \p\-(1/p) + (1/q)

|pE(\Q\-(ß/n)-(1/2)\(g,PQ)\XQ (x))qo dx

|pE(\Q\-(ß/n)-(1/2)\(g,PQ)\XQ (x)) qr/iq~r)dx

f E ( \ Q \ -

(q-r) / qr

< c \ P\ -(1/^p E ( \ Q \ -{ß/nh(1/2) Kg,pq) \xq(x))rdx

Fix an integer N > (n/p) - n. Choose a function 0 e S(Rn) satisfying d(x) = 1 on [0,1]n, 0(x) = 0 if x / 3[0,1]n and JxY0(x)dx = 0 for all multi-indices y with |y| < N. By the

molecular theory [2, page 56], it follows that 0 e pp,q. For each dyadic cube P, define 0P by

0P(x) = 0

x - xP

(5.17)

Then, (0P,Oq) = j®Q(x)dx = |Q|1/2 for all dyadic cubes Q C P and ||0PHp* = C|P|1/p by the

translation invariance and the dilation properties of FP'q. By Proposition 3.1,

k (0P )W * ß,

{(g, Pq )\Q\-1/2( 0P, ®q)}

> ' ( \ Q \ -(ß/n)-(1/2)Kg,PQ)\XQ(x))rdx

(5.18)

and hence, by the boundedness of ng,

\ P \ -qo((1/po)+(1/qo)-1^ £ ^ \ Q \ -(ß/n)-(1/2)\(g,pQ )\XQ(x^ qo dx

(5.19)

Taking the supremum on P, we show that g e CMO'

(qo/po)-(qo/qo ).

To prove part (ii), assume that g e CMO' and s = {( f, fQ)}q. By Proposition 3.1,

(qo/po)-(qo/qo )

and f e F;,q. let t = {(g/ Pq >}q

E\ P \-1/2(g/PP )(®P/PQ )(f/fP )

= WGDts

>a+ß,r

(5.2o)

where (5 := {(®p,^q)}qp is the transpose of {(yP,®q)}q,p. Since a + ¡¡> 0, by Lemma 5.4, (5

is (a + ¡¡, r, r)-almost diagonal and hence is bounded on fa+5'r. Following the same argument as the proof of part (i), we get

|ng(f)||F«+p, < C||Dts||f;+p,

= C£(\ Q \ -(ß/n)-(1/2)+(1/2r)Kg,pQ)\)r.( \ Q \ -

< Cllg|lCMoß'

(;/n)-(1/2) + (1/2r)

\(f/fQ )\) '

(5.21)

(qo/po)-(qo/qo)

which completes the proof.

Acknowledgments

The authors are grateful to the referees for many invaluable suggestions. Research by both authors was supported by NSC of Taiwan under Grant nos. NSC 100-2115-M-008-002-MY3 and NSC 100-2115-M-259-001, respectively.

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