Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 396418, 15 pages http://dx.doi.org/10.1155/2014/396418

Research Article

Smooth Decompositions of Triebel-Lizorkin and Besov Spaces on Product Spaces of Homogeneous Type

Fanghui Liao, Zongguang Liu, and Xiaojin Zhang

Department of Mathematics, China University of Mining & Technology (Beijing), Beijing 100083, China Correspondence should be addressed to Fanghui Liao; liaofanghui1028@163.com Received 1 March 2014; Revised 28 May 2014; Accepted 16 June 2014; Published 22 July 2014 Academic Editor: Josip E. Pecaric

Copyright © 2014 Fanghui Liao et al. 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 Triebel-Lizorkin and Besov spaces by Calderon's reproducing formula on product spaces of homogeneous type. We also obtain smooth atomic and molecular decompositions for these spaces.

1. Introduction and Main Results

Atomic and molecular decompositions are significant tools in studying function spaces and operators in harmonic analysis. The atomic decomposition for Hardy spaces on R was first introduced by Coifman in [1] and was extended to Rn by Latter in [2]. Molecules in Hardy spaces were also introduced by Coifman in [3]. Coifman and Weiss in [4] extended molecules to more general setting in Hardy spaces.

The atomic decomposition of product Hardy spaces HP(R x R) was established by Chang and Fefferman in [5, 6]. Recently, general atomic decomposition on product Hardy spaces was constructed by Han et al. in [7] and Han et al. in [8].

The smooth atomic decomposition for the Triebel-Lizorkin spaces on R" first is considered by Frazier and Jawerth in [9] and later extended to spaces of homogeneous type by Han and Sawyer in [10]; for similar results, see [11]. Smooth molecules of the Triebel-Lizorkin spaces on bidisc were given by Wang in [12].

In [13], Hardy spaces associated with different homogeneities were constructed. Wu in [14] introduced related Triebel-Lizorkin and Besov spaces similarly. In [7, 15], the authors studied the Hardy spaces on product spaces of homogeneous type and showed some properties of these spaces. A natural question arises: whether we introduce the Triebel-Lizorkin and Besov spaces on product spaces of homogeneous type and study some properties of these spaces.

The purpose of this paper is to answer the question. Namely, we introduce the Triebel-Lizorkin and Besov spaces on product spaces of homogeneous type. A theory of atomic and molecular decompositions for these spaces is also presented.

We begin with some necessary definitions and notation on spaces of homogeneous type.

A quasimetric p on a set X is a function p: X x X ^ [0, ot) satisfying that (i) p(x, y) = 0 if and only if x = y, (ii) p(x, y) = p(y, x) for all x,y e X, and (iii) there exists a constant A e [1, ot) such that for all x, y, and z e X

p(x,y)<A[p(x,z) + p(z,y)]. (1)

Any quasimetric defines a topology, for which the balls B(x, r) = [y e X : p(y, x) < r} with all x e X and all r > 0 form a basis.

We now state definitions of spaces of homogeneous type.

Definition 1. A space of homogeneous type (X, p, p) is a set X with a quasimetric p and a nonnegative Borel regular measure p on X such that 0 < p(B(x,r)) < ot, and there exists a positive constant C < ot such that

p(B(x,2r)) <Cp(B(x,r)) (2)

for all x e X and all r > 0,where^ is assumed to be defined on a u-algebra which contains all Borel sets and all balls B(x, r).

We suppose that p(X) = >x> and ft({x}) = 0 for all x e X. Further, suppose there exist constant C > 0 and regularity exponent 0 < d < 1 such that, for all 0 < r < >x> and all x,x,y e X, p(B(x,r)) ~ r and \p(x, y) - p(x , y)\ <

Cp(x,x ')e[p(x,y) + p(x',y)]1 .

Let us recall the definition of an approximation to the identity on spaces of homogeneous type.

Definition 2 (see [7]). Let X be a space of homogeneous type as in Definition 1 and constant A satisfying (1). A sequence {Sk}keZ of linear operators is said to be an approximation to the identity of order e e (0, d] if there exists C > 0 such that, for all x, x', y, and y e X, Sk(x, y), the kernel of Sk, is a function from XxX into C satisfying

(i) \Sk(x,y)\<C(2-k£/(2-k + p(x,y))1+£);

(ii) \Sk(x,y) - Sk(x',y)\ < C(p(x,x')/(2-k + p(x, y))Y(2kel(2k +p(x,y))1+£)

for p(x,x') < (1/2A)(2-k +p(x,y));

(iii) \Sk(x,y) - Sk(x,y')\ < C(p(y,y')/(2-k + p(x, y)))£(2-k£/(2-k +p(x,y))1+£)

for p(y,y')<(1/2A)(2-k + p(x,y));

(iv) \[Sk(x,y) - Sk(x,y')] - [Sk(x',y) - Sk(x',y')]\ < C(p(x,x')/(2-k+p(x,y)))£(p(y,y')/(2-k + p(x, y)))£(2-k£/(2-k +p(x,y))1+£)

for p(x,x) < (1/2A)(2-k + p(x,y)) and p(y,y') < (1/2A)(2-k + p(x,y));

(v) \x sk(x>y)dp(y) = Jx sk(x,y)dp(x) = 1.

Moreover, a sequence {Sk}keZ of linear operators is said to be an approximation to the identity of older e e (0, d] having compact support if there exists a constant C > 0 such that, for all k e Z and all x, x , y, and y e X, Sk(x, y), the kernel of Sk is a function from XxX into C satisfying (v) and

(vi) Sk(x, y) = 0 if p(x, y) > C2-k and ||Sfc||iM(XxX) < C2k;

(vii) \Sk(x,y) - Sk(x',y)\ < C2k(1^p(x,x'Y;

(viii) \Sk(x,y) - Sk(x,y')\ < C2k(1+e)p(y,y')e;

(ix) \[Sk(x,y) - Sk(x,y')] - [Sk(x',y) - Sk(x',y')]\ < C2k(1+2£)p(x,x')£p(y,y')£.

We present test functions on spaces of homogeneous type X before we give test functions on product spaces of homogeneous type X1 x X2.

Definition 3 (see [15]). For fixed x0 e X,let 0 < p, y < d, and r > 0, where d is the regularity exponent on X. A function f(x,y) defined on XxX is said to be a test function of type (x0, r, ft, y), if, for all x,yeX, f satisfies the following conditions:

(i) \f(x)\<C(r/(r + p(x,x0)))1+V;

(ii) \f(x)-f(y)\<c(p(x,y)/(r + p(x,x0)))p(r/(r + p(x, Xo)))1+y for p(x, y) < (1/2A)(r + p(x, Xo)).

If f is a test function of type (x0, r, ft, y), we write f e G(x0, r, ft, y) and the norm of f is defined by

\\fhMy) = inf {C '■ (i)-(ii) hold}. (3)

We denote by y) the class of G(x0, r, ft, y) with r = 1 for fixed x0 e X. Set f(x) e G0(ft,y) if f(x) e G(ft,y) with jx f(x)dp(x) = 0. It is easy to see that G(x1,r, ft, y) = G(ft, y) with an equivalent norm for all x1 e X and r > 0. Furthermore, we can check that G(ft, y) is a Banach space with respect to the norm in G(ft,y). Let G°(ft,y) be the completion of the space G0(Q, 9) in G(ft, y) with 0 < ft, y < 0. If fe G(ft,y), we then define llflVg.{fi^ = H/ll^).

We define the distribution space functionals L from G°(ft, y) to C with the property that there exists C > 0 such that \L(f)\ < CHfHG,(fiY) for all f e G°(ft,y).

Let (Xt, p^ft) for i = 1,2 be two spaces of homogeneous type as in Definition 1 and pt satisfies (1) with A replaced by At for i = 1,2. We define spaces of test functions on product space X1 x X2 of spaces of homogeneous type.

Definition 4 (see [15]). Let (x0,y0) e X1 x X2, 0 < ftt,yt < 9t, and ri > 0, where di is the regularity exponent on Xj for i = 1,2. A function f(x, y) defined on X1 x X2 is said to be a test function of type (x0, y0; r1,r2; ft1,ft2; y1, y2) if, for any fixed y, y e X2, f(x, y), as a function of variable of x, is a test function of G(x0 ,r1,ft1 ,y1) on X1. Similarly, for any fixed x,x' e X1, f(x,y), as a function of variable of y, is a test function of G(y0,r2, ft2,y2) on X2. Moreover the following conditions are satisfied:

(i) Hf(;y)llG(Xo,rlA,yi) < C(r2/(r2 + P2(y,y0)))1+n;

(ii) Hf(;y)-f(;y')HGMl,yi) < c(p2(y\y)/(r2 +

P2(y,y0))f2(r2/(r2 + P2(y,y0)))1+n for all y,y' e X2 with P2(y,y') < (1/2A2W2 + P2(y,y0));

(iii) properties (i)-(ii) also hold with x and y interchanged.

If f is a test function of type (x0, y0; r1, r2; ft1, ft2; y1,y2), we write

f e G (X0,y0;rvr2 ;ftl,ft2;Yl,Y2), (4)

and the norm of f is defined by

\\f\\G(Xo^wM2;Y^2) = inf {C ■ « - (rn) hold}. (5)

Similarly, we denote by G(ft1, ft2;y1,y2) the class of G(x0,y0;r1,r2;ft1,ft2;y1,y2) with r1 = r2 = 1 for fixed (X0,y0) e X1 x X2. Set f(x,y) e &0(ft1,ft7;Y1,Y2) if f(x,y) e G(ft1,ft2;Y1,Y2) with J^ f(x,y)d^(x) =

\x f(x, y)^ft2(y) = 0. It is easy to see that

G (X0,y0;r1,r2;ft1,ft2;Y1,Y2) = G (ft1,ft2;;Y1 ,Y2) (6)

with an equivalent norm for all (x0, y0) e X1 xX2 and r1,r2 > 0. Furthermore, we cancheckthat G(ft1, ft2; y1,y2) is a Banach space with respect to the norm in G(ft1, ft2;y1,y2).

Let G°(ft1, ft2,y1,y2) be the completion of the space G0(9l,e2;d1,d2) in G(ft1,ft2,y1,y2) with 0 < fti,yi < dt for i = 1,2. lffe G(j$i, fa ji, y2),wethendefine H/llG°(A,ft,yi,y2) =

WI&tfi&.riM)-

Also, we define the distribution space ('"(ft]^, ft2; y1, y2)) by all linear functionals L from G°(ft1,ft2; y1, y2) to C with the property that there exists a constant C > 0 such that \L(f)\ < CI/Ig.,^1^ i^) for all fe G°(ftv

We give Calderon's reproducing formulas on product spaces of homogeneous type.

Lemma 5 (see [7,15]). Suppose that i = 1,2. Let ei e (0,9i], let |Sfc }k be an approximation to the identity of order ei on spaces of homogeneous type Xi, and let Dk = Sk - Sk -1 for

ki e Z. Then there are families of linear operators {Dk and {Dk.}keZ on Xi such that, for all f e (&°(ft1,ft2;y1,y2))' with fti,yi e (0, ei),

f=1 Bk2 Dki Dk2 (f)= 1 Dh Dk2 Dh (f),

k1,k2ez k1,k2ez

where the series converges in the norm of both the spaces (&°(ft[,ft2;y'i, Y*2)) with ft\ e (0,fti),y'i e (0,y,). Moreover, for xi,yi e Xi and all ki e Z, Dk (xi,yi), the kernel of Dt, satisfies conditions (i) and (ii) in Definition 2 with ei replaced by any e^ e (0, ei) and

\x Dk, (xyd d^i (yi) = J^ D^ (x^ yt) d^i (xt) = (8)

Dk(x,y), the kernel of Dk, satisfies conditions (i) and (iii) in Definition 2 with e replaced by e for 0 < e < e and (8).

We also need the following result, which gives an analogue of the grid Euclidean dyadic cubes on a space of homogeneous type.

Lemma 6 (see [15]). Let X be a space of homogeneous type as in Definition 1. Then there exists a collection {Q^ c X : k e Z,a e Ik} of open subsets, where Ik is some index set, and constants C1,C2 > 0 such that

(i) p(X \ UaQka) = 0 for each fixed k and Qka nQkp = 0, if a = ft;

(ii) for any a, ft, k, I with I > k, either Qlp c Qkk or Qlp n

oL = 0;

(iii) for each (k, a) and each I < k, there is a unique ft such that Qkk c Qlp;

(iv) diam(Qk) < C12-k;

(v) each Qka contains some ball B(yk,C22~k), where yk e X.

We think of as being a dyadic cube with diameter 2-k and center yk. When k e Z and r e Ik, we denote by Qlk'V,

where v = 1,2,..., N(t, k), the set of all cubes Qk+i c where j is a fixed large positive integer, and ykv a point in

Throughout the paper, we use C to denote positive constants, whose value may change from one occurrence to the next. We denote by f ~ g that there exists a constant C > 0 independent of the main parameters such that C-1g < f < Cg. \A is the characteristic function of A. Let a+ = max(0, a) for any a e R. We also denote M1, M2 are the Hardy-Littlewood maximal function on spaces of homogeneous type X1, X2, respectively.

We introduce Triebel-Lizorkin space Kf? and Besov space Bpq on product spaces of homogeneous type.

Definition 7. Suppose that i = 1,2, s = (s1,s2), si e (-ei,ei) and let {D^}k gZ be the same as in Lemma 5. Let 1 < p,q <

rn and 0 < fti,yi < e{ < Q„ f e (' (ft1, ft2;y1,y2))' .Then

Triebel-Lizorkin space Fpq(X1 x X2) is defined by

Y 2MI<?2fc^|DfciDk^ (f)\

vfcj ,k2ez

The Besov space B^(X1 x X2) is defined by

Y 2klhq2<

vkj,k2ez

KDki (f)\\l ) (10)

In order to check that the definitions for Fpq and Bpq are independent of the choice of approximations to the identity, we recall almost orthogonality estimate.

Lemma 8 (see [15]). Let i = 1,2. Suppose that {Sk.}k£Z and

are two approximations to the identity on spaces of

homogeneous type, and D^ = Sk. -SK._1, E^ = Pk -Pk-1. Then for ei e (0,9) there exists a positive constant C depending only on ei such that D^D^Ek^Ek^(x1,y1;x2,y2), the kernel of D^ D;2 Eki Ek2, satisfies the following estimate:

l°i1 DhEkiEh (x1,y1 -,X2,y2)\ < C2~lk'-ilIe!-l2l£2_

-(k1Al1)e1

(2-(k^i) + Pi (xi,yi))

-(k2A 12)e2

By Calderon's reproducing formula and almost orthogonality estimate, we can get the following product-type Plancherel-Pôlya inequalities. The proofs of the following theorems are similar to Proposition 4.1 in [10]. Here we omit the details.

Theorem 9. Suppose that Dk and Ek are the same as in Lemma 8 for k e Z. Let 1 < p,q < œ>, and < et for i = 1,2.

Then there exists a constant C > 0 such that for all f e ,72))' with 0 < P, y < e; < 0f for i = 1,2,

and then

X X(2fclSl2fc2S2!DfciDfc, (/))

fc,ezfc,ez

!Iez i2ez

X X(2fcisi2:,S2||DfciDfc2 COL)

I? 1/?

fciezfc,ez

xx(2'isi2^2^£,2 (/)||Lp)?

iiezi2ez

Remark 10. Let D^, D^, Dk, and Dfc be as in Lemma 5. Then the kernel of D^ D^ has compact support, but not of Dki. Since (11) holds for D^ and D^ satisfying only the smoothness condition for the second variable, we conclude that Theorem 9 holds with D^ D^ being replaced by Dk Dfc .

2. Smooth Atomic Decomposition

In this section, smooth atomic decomposition is presented. We first give definitions of the smooth atoms for and

Definition 11. Suppose collections of open subset {Q^^1 c X1 : fc1 e Z,r1 e Iki,v1 = 1,...,N(fc1,r1)| and {Q^2'V2 c X2 : ^2 e Z,T2 e 7k2, V2 = 1,...,N(fc2,T2)} satisfying the conditions in Lemma 6. Lete = (e1,e2) and R = Qk 'Vl xQ^2, and a function % defined on X1 x X2 is said to be a e-smooth atom for R if

supp c B1 (^, A 1C2-kl) x B2 (zk22,V2, A2C2-k2) , (13) where zk"Vi is the center of Qk"V' for i = 1,2;

(%1, ^2) (^2) = I % (^1,^2) (^1) = 0;

Jx, Jx,

(^^(q^ )-1/2F2(Qkf2)-1/2, (15)

<CF1 (Q^i )-1/2-Ei ^ - )-1/2^1(x,x')Ei,

1% (^y) y')l

<cF1 (g*- p^oir )~1/2-2 M^P

i y) - % - % p - % PI

<cpqp )"1/2-Ei pqp ) i ft^P'2.

-1/2-2

Now, we also define certain spaces of sequences of indexed by dyadic rectangles {Qp x Qp} in X1 x X2 which will character the coefficients in our decomposition of and Suppose that s = (sj,s2), e = (e1,e2) with -e; < < e; for i = 1,2, 1 < _p, ^ < œ>. Let R be as in Definition 11. Let and be the collection of sequences

s={sqixq2}q1xq2g{^| suchthat

^■m -si-1/2

XXX p(Qp )

fci>fc2ti>t2vlv2

X^2 (Qp)-S2-1/2

< œ>,

X(XX ( (pQp ri

^ Mr r■

< œ>.

Smooth atomic decompositions for and can be

stated as follows.

Theorem 12. Let R be as in Definition 11. Suppose that 1 < p, q < >x>, s = (s1, s2), e = (e1, e2) with -e; < < e;for« = 1,2.

(i) Then there exist sequence S = {sb} e and e-smooth atoms {ab} such that

N(k1,T1) N(k2>T2)

/= I I I I I (20)

k1>k2ezTi6itl Vi = 1 T26it2 V2=1

wiife convergence in B^, and

< cjj/jj^

Similarly, there exist sequence S = |sR} e fSpq and e-smooth atoms |aR} such that

N(k1,r1) N(k2,r2)

f=l 1 1 1 1 sRaR (22)

k1,k2£Zr1iIkl v1 = 1 r2ilk2 v2=1

with convergence in F^ and

IISII^ <C\\f\\p;q. (23)

(ii) Conversely, if aR are e-smooth atoms and

N(k1 ,r1) N(k2,r2)

f= 1 1 1 1 1 sRaR' (24)

k1,k2eZr1iIk1 v1 = i r2iIk2 v2=1

\\% <£11%«, Mi? <dISIIf«■ (25)

Proof. We first prove part (i). Let R be as in Definition 11. Suppose f e B^; by Lemma 5, we have

N(k1,r) N(k2,r2) -

f(X'y)= 111111 Dki (x,s)

k1,k2eZr1eIk1 v1 = i r2eIk2 v2=1 jR

x Dk2 (t, y) DkiDki (f) (s, t) d^i (s) (t) (26)

N(k1,r1) N(k2,r2)

:= 1 1 1 1 1 sRaR'

k1,k2eZr1eIk1 V1 = 1 ^eIk2 V2=1

Sr = C,i(Qk^1 )-1'2to(<% V2 )-1/2

f____(27)

x fjD^ Dki (f) (s,t)\dpi (s)d^2 (t),

aR = c-1,i(Qk:>V1 f^Q^ )1/2

x(|R|Dk1 Dh (f)(s,t)\dft (s)d^ (t))

x f Dk1 (x, s) Dk2 (t, y) Dki Dk2 (f) (s, t)

x d^1 (s) dp2 (t).

Similar to [7,10], we can prove the convergence of the above series converges in Bpq and (&°(p[,fi2,y[,y'2)) with < ^ < ej and yj < y' < ej for i = 1,2. We now show that aR satisfies conditions in Definition 11. Here we only show that aR satisfies condition (iii) and we first prove that aR satisfies

(15). By the size condition of Dk. and Q?k"Vi) ~ 2 k for i = 1,2,we have

\aR (x' y)\

f^Q^ )1/2

x Qr |Dk1 Dh (f) (s, i)| d^i (s) d^2 (t))

x I -2-;--2-(29)

Jr (2—k1 +Pi (x,s))1+£1 (2—k2 +P2 (y,t))1+£2 x\Dk1 Dki (f) (s,t)\dpi (s)d^2 (t) <C,i(Qkf1 f^Q^ )1/22k12k2

<0^1 )-l%2{Qr?2 J1'2.

We now verify (16); here, we only consider the case p1(x,x') < (1/2A)(2—Ik1 + pi(x,s)) and similarly we estimate the case p1(x,x') > (1/2A)(2—k1 + p^x, s));weget

\aR fr y) - aR (x',y)\

r^Q^ )1/2

x (|r |Dk1 Dh (f) (s, i)| d^i (s) d^2 (t)) x|R \Dk1 (x,s)-Dki (x',s)\\Dk2 (t,y)\

x\DkiDk2 (f) (s,t)\dpi (s)d^2 (t) f'2^^ )1/2

x Qr |Dk1 Dki (f) (s, i)| d^i (s) d^2 (t)) (30)

x f ( ? (X,X') ) Jr \ 2—k1 + Pi (x, s) J

2—k1e1

x-;—

(2—k1 +pi (x,s))i+e1

x-i—

(2—k2 + P2(y,t))i+e2

x\DkiDk2 (f) (s,t)\dpi (s)d^2 (t)

<0^1 )—i'2—£1,2(Qk2V2 )—U2Ri(Xy)£1,

which is a desired estimate. The proof of (17) is similar to that of (16) by symmetry. Here we omit the details.

We now estimate (18). If ft (%,%') < (1/2A)(2 fcl +ft(%,s)) and ft(y,/) < (1/2A)(2-fc2 + ft(y, i)),then

<CFl (Qp P^Qp )1/2 x(Jj5tiDfc2 (/) (s,f)|dft (s)d№ (i)

x Jpfci (x,s)-Dti (*',s) |

x|Ofc2 (i,y)-Ofc2 (f,/) | x| Dti Dfc2 (/) (s,i)|d^i (s)d№ (i)

<cFi (Qpp^Qp )1/2

x (£ |DfciDfc2 (/) (s, i)| (s) (i)) (31)

A YY ^ )

2-fc! + ft (%, s) / \ 2-t + ^ (y, i)

(2-fci + ft (X,S))1+E1

(2-t + ft(y,i))1+ %

x|Dfci Dfc2 (/) (s,i)|d^i (s)d№ (i)

(Qp)-1/2-E1^ - )-1/2-E2

which is the desired result. Similarly we can consider another three cases.

Then, by Holder's inequality and Remark 10, we obtain

llSllj,«

( N(fci.Ti) Nfe^)

I ( I I I I

fci^eZ \Tieiti Vi = 1 T2ÊÎt2 V2=1

Fl^i ,V i )-Si-1/2+1/P ^Q^ 2 p-1/2+1/p X^* )-1/2^(qM )-1/2

X £ |Dfci Dfc2 (/) (5, i)| (5) (i))P)

( N(fci.Ti) Nfe^)

I ( I I I I 2klSlp2k2S2p

fci^ez \Tieiti vi=1 T2ÊÎt2 v2=1

x([ Äti Ôfc2 (/)(5,i)

(s) (i)

I (2tiSi2^2||DfciDfc2 (/)||p

fci ,fc2ez

<C||/|U

The proof of the case / e F^ goes by the analogous argument to the case that / e but using the vector-valued maximal inequality. Indeed, suppose that / e F^; as in the proof of ¿I4, we have

N(fci,Ti ) N(fc2 ,T2)

/= I I I I I

fci,fc2eZTieiti Vi = 1 T2ÊÎt2 V2=1

where and % are as in (27) and (28), respectively. From [7, 10], we can also similarly obtain the series converges in

the norm of F*,<? and in , fa, , ?2))' with fa < fa < e;

and y < y' < e; for i = 1,2. To show the conclusion, using Fefferman-Stein vector-valued maximal inequality and Remark 10, we have

N(fci.Ti) Nfe^) ,

I I I I I (^1(Qti'Vi P

fci>fc2eZTieiti Vi=1 t2eit2 v2=1

fc2,V^ -S2-1/^ /nfci,V^-1/^nfc2 M-1/2

X^2 (Qp P"'CPQP ) >2^* )

X li|DfciDfc2 (S, ^ ^^ ^(Oxq^i.vi XQ^

I (2fciSi2^2M2 (M1 (DtiD*. (/))))'

fci>fc2ez

I (2fciSi2*^D*. (/))'

fci>fc2ez

< C||/|U.

To prove part (ii), we need the discrete version of the Hardy-Littlewood maximal function estimate on one single factor X, which is an analogue to a result of [9]. Also see [10].

Lemma 13. Suppose that X is a space of homogeneous type as in Definition 1 Let 0 < a < r < >x> and X > r/a. Fix ^, q with e Z and for any x e X

2-(k1-p1)£i X -:-

(1 + 2^ Pl (x,z%-V' ))1-1

2-(k2-^2)£2

X -;-.

(1 + 2^2p2 (X,Z?2'V2))1-2

N(?,t)

Y Y I (1 + min{2\2»}p(x,zïV)y

<c2(^(m( y y k-iv

where z^'v is the center of dyadic cube and C depends only on X, r, a.

We now turn to proof of part (ii). In this part, we denote R = x Q^'V2. Suppose that {Dk,}k gZ for i = 1,2 is as in Lemma 5 and aR is an e-smooth atom. For D^Dk (aR)(x, y), we consider it by four cases. We first estimate the case kt > Pi,k2 > In this case, applying the cancellation conditions of Dki and Dk2,

\Dk, Dk2 (aR)(x,y)\

= lf Dki (x,s)Dk2 (y,t)

x [aR (s, t) - aR (x, t) - aR (s, y) + aR (x, y)] x (s) dp2 (t)

y1/2-1 ^(Q??2)

-1/2-62

p1 (s,x)<2C2 P2

{iyj<2CT*2 (2-kj +P1 (x,s)) £l

1+e2 P1(X'S)ei

2 k262

(2-k2 +P2 (t,y)) Xp2(t,yï2d^1 (s)dfo (t)

iC^Q?*1 y1/2-1 )

X 1 ^(s,X)<2C2-^ (s)d^2 (f)

Pi(t,y)<2C2-k2

-1/2-62

(1+2* P1 (x,z?'Vj )r£j

(1+2*2 p2 (y,4f2 )f+2

To estimate the case k1 > ^1,k2 < p2, we only estimate p2(t,zuxf2) < (l/2A)(2-k2 + p2(y,t)), and afterwards we can consider the case p2(t, zU2'V2) > (1/2A)(2-k + p2(y,t)) similarly. In this case, we have

\Dk,Dk2 M (x,y)\

= |f Dh (x,s)[Dk2 (y,t)-Dk2 (y,zUf2)] \ JX, xX 1 2

x [aR (s, t) - aR (x, i)] d^1 (s) d^2 (t)

<CV1(Q^J )-1/2-1 ^(Q'f2 )

X 1 pj(s,x)<2c2-tj (2-k + . ,^+ej P^ZrP )<2C2-f2 (2 1 + P1 (X, S))

P2 (t,<f2 ) V 2-k2

2-k2 +p2 (t,y)J (2-k2 + p2 (t,y))U x p1 (x, s)£j d^1 (s) dp2 (t)

iC^f1 )-1/2-61 fc(Q?'V2)

p1(s,x)<2C2 '

p2(t^2 )<2C2->2 (1+2Ï1 P1 (x,Zrl'Vl ))+£l

2 2k2^2

(1+2k2 p2(y,z>?f2 ))1-2

d^1 (s) dp2 (t)

<CV1(Q^ y1/2-1 ^(Q^2)

X| P1(s,X)<2C2-k1 (s)d^2 (t)

P2(t,zU2V2 )<2C2-^2

2k12-k1e1

(1+2*1 P1 (X,z?-V1 ))1+61

2 2 ^2^2

(1 + 2k2 P2(y,z>i^'V2 ))1+£2

tel.. v=1

<cfi(q^ )-1/2 Pq^2 )

(1+2* ft (*,<V ))1+£1

2-(^2-fc2)(1+e2)

(1+2*2^ (y,<2'V2))

1+e, '

The estimate of the case ^ < ^, fc2 > is similar to the case fcj > fc2 < by symmetry and we obtain

K Öfc2 (%)(*> >0|

<CF1 (Q^1 )-1/2^2(Q^2'V2) 2-(^i-fci)(1+ei)

(*2 ^2)e2

(1 + 2*1 (^ (1 + 2*ft (*,<2'V2))1+£2

For fc1 < < when ft1(s,z"i1,Vi) < (1/2A)(2 * + ft1(s,x)) and ft2(f,z"22'V2) < (1/2A)(2-*2 + ft2(y, i)),then

KÖfc2 (%)(*> >0|

= ||XixX2 [^ (*.*)-(*.*r)]

x[d,2 (y,i)-Dfc2 (^ 'V2 )]

x (s, i) (s) (i)

<cF1 (Q^ii'Vi )-1/2№(q£'v2)

X |pi(s>Z?i1'V1 )<2C2-^1 I ft(t.4f2 )<2C2-^2

/ ft (^r) )1

\2-fci + ft1 (*,s); _)

(2-*i +ft1 (x,S))1+eA 2-*2 + ft2 (i,^)/

(2-*2 +ft2 (f,>0)

(s) (i)

sC^Q*'1)- U\2 (Qif2)

2-^iei

)<2C2-^1 (2-fci +ft1 (x)s))^1

(1+2fci ft1 (x,S))1+£i

(2-fc2 +ft2(i,y))£

(1+2*2 ft2(i,y)) <0^)-1/2^2 (O^)-1/2

(1 + 2*i ft1 (^

2-(^2-fc2)(1+e2)

(s) (i)

(1 + 2*2ft2 (y,<2'V2))

1+e, '

We can similarly get the same result for another three cases; here we omit the details.

Thus, if /=z^zz^ zPPz^ zsr2)

by Lemma 13 and above estimates, then

X (2fciSi2*2*2|D*1 D*2 (/)|J

fc1>fc2ez

X ( 2*1%2*2%

fc1>fc2ez

N(ii,Ti) N(i2>T2)

D*i d*2 ( X X X X X

>i2Ti6i„1 Vi = 1 T26i V2 = 1

X ( X X 2*1s12*2s2 2^1/22^2/2 2(ii-*1 )£12(i2-*2)£2

*1>*2ez \ ii =-toi2=-ra

N(i2>T2)

N(ii ,Ti)

m2 ( X X m1 ( X X m^r )xq?f2

v^lo v2=1 vi^m v1 = 1 /

2(ft-fc2)M,

Nfe,^) / N(^i.Ti) \

X X Mi( X X M^r Uq^

X ( X X 2fcisi 2fc2s22^i/22^2/22-(^i-fci)(i+ei)2(^2-fc2)£2

fci>fc2ez \ ^i=fci + 1^2=-to

/ Nfe^) / N(^i.Ti) \

2(^i-fci)M2 ( X X M1 ( X X talXQT JXQe--V2

v^lo v2=1 vi^ vi=1 /

X ( X X 2^i5i 2^2522^i/22^2/22"(^i"fci)(1+ei)2"(^2"fc2)(1+e2)

( Nfe^) ( N(^i.Ti) \

2(ft-fc2)M2 ( X X ( X X WXQST W2

V^ft V2 = 1 Vi^ Vi = 1 /

/1 + /2 + /3 + ^

Let A, = (^ ^ i^^i'xr)17'- We

now consider 71. By Holder's inequality and noting that ki > e; > for « = 1,2, we have

/1 < C

( h k2 \q X ( X X 2fcisi 2fc2s22^/22^2/22(^i-fci)£i 2(^2-^2)^2^

fc^eZ\ft=-TO ^2=-TO

fci fc2

X X X 2-(fci-^i)(£i-si)2-(fc2-^2)(£2-s2)(2^i(1/2+si-1/p)2^2(1/2+s2-1/p)^ )q

fci>fc2ez ^i=-to ^2=-to

X (2^i(1/2+si-1/i)2^2(1/2+S2-1/p)A )q

Similar to the proof of we get desired results for 72, i3, and

To show the second inequality of (25), let fl(x, y) = ^^z With

«1 '«2

Nfe/T,)

= M2 ( X X )

-1/2-S2

N(ft'Ti)

( X X ^(Qir)-1/2-Sl

\Ti Vi = 1

X M )(7)-

Applying Lemma 13 and Holder's inequality, we get

X (2fclSl2^ |D,iDfc2 (/)|)*

fc1 ,fc2ez

/ fci ¿2

X ( X X 2-(fci-^i)(£i-si)2-(fc2-^2)(£2-s2)

fci'fc26z \ft=-toft=-to

TO «2

X X X 2

^'^2ezfci=^i ¿2=-to

(fci-^i)(ei-Si)2(fc2-^2)(e2+s2)

«1 TO

X X X 2

(fci )(£, +S, ) 2 (¿2 ^2x^2 ^2)

«1 «2

XXX 2(fci -^i)(£i+5i)2(fc2-^2)(e2 +S,)

«1 ¿i=-TO ¿2 =-TO

(fci TO

^i=-TO^2=fc2+1

-(fci-^i)(ei -si )2(fc2-ft)(e2+s2)

Hence, by Fefferman-Stein vector-valued maximal inequality, we obtain

X a«,'«,

TO ¿2

X ( X X 2(fci

fc1'fc26z \«1=fc1 + 1«2=-to

(fci-ft)(ei+%)2-(fc2-ft)fe-S2)

X ( X X 2(fc1-«l)(e1+5l)2(fc2-«2)(e2+52) \«1=^1 + 1 «,=¿2 + 1

X X X 2

«1 '«2 € z k 1 =«1 ¿2 =«2

(¿1 -«1 )(e, -S,) 2-(k2-«2)(e2-s2)

X (2^2^ ^¿2(/)|)?

'¿2 ez

(N(«i'Ti) N(«2'T,)

XXX X* (QT)-

t2eim v1=1 v2=1

1/2-s,

X^Q^ )-1/2-S2

< C||S|U,

l/^ which completes the proof of Theorem 12.

3. Smooth Molecular Decomposition

In this section, we show smooth molecular decomposition of Triebel-Lizorkin and Besov spaces. We first give definitions of the smooth molecules as follows.

v2 = 1

Definition 14. Suppose that collections of open subset {Qp cXj : kx e Z.Tj e Iki,v1 = 1,...,N(fc1,r1)} and {Qp c X2 : fc2 e Z,r2 e 42, v2 = 1)...,N(fc2)r2)} satisfy the conditions in Lemma 5. Let ft = (fa, fa), y = (y1,y2)

and R = Qk1,Vl x Qk2 ,V2, and zk1,Vl ,zk1,Vl are the centers of ri r2 ri ri

Qp, Q^^2, respectively. A function defined on X1 x X2 is said to be a (ft, y)-smooth molecule for R if

\(-i+r2)

(xj, x2) (x2) = I (x^ x2) (xj = 0; Jx, Jx,

K (^)|<cpQpp№(Qp)

x(l + 2 1 ft (x,zTi1 1))

-(1+V2)

x(l + 2-k2ft (y,p2))

< CpQp )-1/2-^1 ^*)-1/2 Xft(*yf (l+2-k2 ft ))(-1+r2) x {(i+2-k1 ft (^p ))-(1+r1) +(l + 2-k1,1 )p+p,

\-1/2-ft

< cpQp ) P(qP)

x^2(y,y')'2 (l + 2-k1 ft (*,zp ))-(1+r1) X {(l + 2-k2ft (7,-k22" ))-(1+r2) + (l+2-k2 ,2 ))-(1+r2)},

% j) - ^ - / ) - P P| <CpQp )-1/2 pQp)-1/2

xft(xpft ft^p^

fft o-k1 ( k1>vftN-(1+r1) x {(l + 2 1 ft (^ 1))

+ (l+2 1 ft 1)) }

x{(l + 2-k2ft (y,p2))(

+ (l+2-k2 ,2 ))-(1+r2 )}.

Now we can state the smooth molecular decomposition for Triebel-Lizorkin and Besov spaces.

Theorem 15. Let R be as in Definition 11. Suppose that for i = 1,2, s = (s^), e = (£1,62), ft = (ft^), 7 = (71,72), -£i < Sj < e;, 1 < ft ^ < ra, is a (ft, y)-smooth molecule with max(0, s;) < ft; < e; and max(0, -s;) < y < e;for« = 1,2.If

N(k1,r1) W(k2>r2)

/=1111! ^^ (47)

k1,k2 ezr^i^ V1=1 T26it2 V2=1

ifeen ||/||£m < C||S||jw and y/y^ < C||S||

Proof. Suppose that R = Q£,V1 x Q^2'V2. From the proof of Theorem 12, we only need to claim the following estimates. If fc1 < fc1 < then

P Ok2 K)(x,y)|

<cpQ^v1 ^P^f2)

-(^1-k1)(1+y1)

-(^2-k2 )(1+T2)

(l + 2k1 ft (*,<1,v1 ))1+r1 (l + 2k2ft ))

A>v2\\ 1+^2 '

If fc1 > fc1 < then |ök1 AC2 K) (X,y)|

sc^q*'1 )-1/2^2(Qi2,v2 )-1/2

-(k1-^1)^1

-(^2-k2)(1+?2)

(l + 2^1 ft (*,<1,v1 ))1+r1 (l +2k2^2 (y,<2'v2))

If fc1 < > then

|ök1 Ök2 K)

<cF1(Qi'11'v1 )-1/2^2(Qi2,v2)

2-(ft-k1)(1+r1)

-(k2-^2)^2

(l + 2k1 ft (*,<1,v1 ))1+r1 (l +2^2(y,Zi*f2))

1+?2 ' (50)

If k1 > fa ,k1 > fa ,then \Dk, Dk2 (mR)(x,y)]^ <Cfa(Q™ yinfa(Q>;.f2)

-(JC2-fi2)ß2

(1+2*1 Pi(x,zM))1+» (1+2*2Pl(y,z*f2))

*2'V2\\1+^2 '

We now prove the above estimates. Let W1 = {x : p1(x,z^l'Vl) < 4Cl2-k^}, W2 = {y : p^z^V) < 4C22-k2}, W3 = {x : p1 (x,s) < 4C12-k}, and W4 = {y : p2(y,t) < 4C22-k}, where 2Ct is the smallest constant which satisfies supp Dk.(x,y) c {(x,y) e Xt x Xt : p(x, y) < 2C{2-ki} for

i = 1,2, and Wf, W2C, W3C, and W4C are their complements, respectively We first give the proof of (48). Note that k1 < fa, k2 < fa, and then

\Dk1 Dk2 (mR) (x,y)\

<[ \Dh (x,s)-Dki )\

J^xX2 1

x\Dk2 (y,t)-Dk2 (y,z^V2 )\ x \mR (s,t)\dfa (s)fa (t)

<\Xw1 Xw2 \ + Xw? Xw2 \

\ 1 2 JW3xW4 1 2 JW3xW4

+ Xw1 Xw2r \ +Xw1 Xw2 I r 1 2 Sw3xw, 1 2 Jw^xw.

1 Xw2 \

1 2 Jw

+ Xw, + Xw,c Xw2

, + Xwf Xw?

+ Xw,c Xw2

Jw,xw4

+ Xw1 Xw? I +Xw1 Xw? I

1 2 Jw^xw4 1 2 Jw3xw^

+ Xwl Xw2 \ r ^ + Xw? Xw^ \ „ 1 2 Jwfxw 1 2 Jw3rxw4

+ XwP Xw? \ ^ + Xw<r Xw2 \ ^ r 2 Jw3xwf 1 ¡w^xw^

+ Xw, Xw2 \ r „ + Xw1c Xw2c \ r „ 2 Jwfxw 1 2 Jwfxw

x\Dki (x,s)-Dki (x,z%-V1 )|

x\Dk2 (y,t)-°k2 (y,z?2'V2 )|

x \mR (s, i)| dfa (s) fa2 (t) := (48)i +(48)2 + --- + (48)i6.

By the support of Dk and the definition of Wj for i = 1,2 and j = 1,2,3,4, we can easily get (48)7 = (48)10 = (48)12 = (48)13 = (48) 14 = (48)15 = (48)16 = 0. Here we only estimate the first five terms, and we can similarly estimate other terms. Now, we first prove (48)1 :

\(48)1\ <C2k1(1+e1)2k2(1+e2)

xfa1 y1/2fa2(Q^2 )-1/2Xw1 Xw2

w,xw4 (1 + 2*1 Pl (s^tf1 )f+y1)

(t,Z*f2 )

dfa (s) dfa (t)

(1+2*2 P2 (t,z*f2 )f+*> <Cfai(Q*:V1 )-1/2fa{Q*f2 )-1/2

x 2k1(1+£1)2k2(1+£2)2-*1 (1+y1)2*2(1+y2)

Pi(s,z*!'V y r1 1dfai (s)

-i-kt v 1 '

xw^2 ^1(i,-*22'v2W

<Cfai(Q**:V1 y 1/2fa(Q*f2)

0-(*1-k1)(1+Y1) 0-(*2-k2)(1+Y2) ... ...

X2 2 Xw1 Xw2

<Cfai(Q**:V1 )-1/2fa(Q*f2)

2-(*1-k1)(1+Y1)

(l + 2k1 pi (x,^1 ))1+V1

2-(*2-k2)(1+Y2)

x -:-,

(l+2k2 P2 (y,z*f2 )f+2

which is the desired estimate. We now estimate (48) 2 by

1(48)2

< ^2kl 2k2(1^fai(Q£1-V1 y mfa(Q*22V2 yL'\wc Xw2

xjp1(*,s) (lx((l + 2*1

<2C12-1

P1(s4r )

>P1 (x^1 )/2

xpi (s,z*r;>V1 ))(1+Y1)y]dfa (S)

Jw4 (1+2*ft (¿,4^2 < C2fcl 2fc2(1+£2)Fl(Q*i.Vi )-1/2

X ^Q^2 )-1/2XwCXW2

X _2_2-"2(1+r2)

2*i(1+ri)p1(x;Zi,ii'Vi )(1+ri)

.V,N 1/2

2~(*i -fci)(1+ri)

(1 + 2fci ft (*,<'Vi ))1+yi

2-(*2-fc2)(1+r2)

X -:-,

(1 + 2*2(^)p2

which verifies (48)2.

The verification of (48)3 is similar to that of (48)2 by symmetry.

Similarly we have

1(48)4 < C2fci2fc2(1+£2)F1(Q?ii,Vi )-1/2^2(Qi^'V2XW2 X I -71—rd^i (s)

W™-)>2Ci 2-fci (2*i p1 (S)Z£'Vi ))(1+7i)

ft Mf2 )£

W4 (1 + 2*p2 (i^f2))(1+r2)

< CpQ^1'1'1 )-1/2F2(Q?f2r'VXw2

X 2-(?i-fci)(1+^i)2-(?2-fc2)(1+r2)

2-(?i-fci)(1+yi)

(l + 2fci ft ))1+ri

2-(?2-fc2)(1+r2 )

X -:-,

(1+2*2(^f2))1+r2

which is also a desired estimate.

By an analogou argument to (48)4, we can verify (48)5. We now prove (49) j

= II öfc, (*, s) öfc2 (y, i) (s, i) (s) (0

|JxixX2 i 2

Lix*2 idfci mr2))|

X |mfi (s, i) - (%, i)| (s) (i)

_ I Xrn, I I +Xw2 I

Jx3xw4 2 Jx3xw4c 2 Jx3xw4

X |-Dfci (y,i)-Dfc2 MP )|

|mfi (s, i) - (%, i)| (s) (i) := (49)1 + (49)2 + (49)3.

For (49) 1, we have | (49)11

<Xw2 2fci 2fc2(1+£2)F1(Q?ii'Vi )-1/2-ft ^f2)

X I p (x S)A (_1_

kxw/1 ' V(1 + 2*ft (*,z*'Vi))1+ri

(1 + 2? ft (s,^1 ))1+ri

(1 + 2*2(i,^2))1+r2 <Xw2 2fci 2fc2(1+£2)

XF1 (QT ^Q^ )-1/2

ft(^)fi

(1 + 2?ft (*,<1,Vi )p ft^1

(s) (i)

(1 + 2? (s,<1,Vi ))1+ri

Jp2(t>Z?22,V2 )<8C22-fc2

-^2(1+72)

Xft(i,^2f-1-72^2 (i)

2-(?2-*2)(1+r2)

(1 + 2*2(y^f2))1+72

X I 2ttft+fci

Pi(x,s)<4Ci 2"*i (1 + 2? px (s,^1 ))1+ri

2-(k1-*1)ß1

2~(k1-pi)ß 1 x -:-

(l + 2*ipi (x>z%*))l+n

2-(^2-k2)(l+Y2)

(l + 2«2P2 (y,4f2))1+*2'

since 1 + 2* ftfazp*) < 2(2C1 + 1)(1 + 2* ftis.zfi*)) by kt > fa, which is the proof of (48) 1.

The verification of terms (49)2 and (49)3 is similar to that of (49) 1 and proof of (50) is similar to that of (49) by symmetry.

We now prove (51). In this case, we have

\Dk, Dk2 (mR) (x' y)\

<f K (x,s)\\Dk2 (y>t)

JX,XX, 1 i i 2

x \mR (s, t) - mR (x, t)

-mR (s, y) - mR (x, y)\ dfa (s) dfa (t) 2k 2k2 fa^fi )-1/2-ß fa2(Q^f2 )-1/2-ß2

Pi(x,s)ß

f P2(y,t)ß

(l + 2*iPi (x,z*V))1+yi l

(l + 2*iP1 (s,z*lV ))1+^

(1+2*2 p2 (y,Z*2*

+ (l+2*2P2 (t,z*f2))^2

<Cfa1(Q^ )-mfa2(Q*22 V )-m

2-(k\-*\)ß\

dfa1 (s)

dfa (t)

(l + 2*iP1 (x,z*'Vl ))1+^ 2^ß+k1 f P1(x,s)ßl

w, (l + 2*p1 (s,z*l'Vl))1m

dfa1 (s)

-(k2-*2)ß2

(l+2*2P2 (y,z*f2 ))1+V2 ■*2ß2+k2

iw4 (l+2*2P2 (t,z*f2 ))1+V2

dfa2 (*)

<Cfa1(Q^ )-1/2fa2(Q^^'V2)

2-(ki-*i)ßi

(l + 2*iP1 (x,z*'Vl ))1+yi

2-(k2-*2)ß2

(l + 2*2P2 (y,z*2'V2 )f+2'

(58) □

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for carefully reading the paper and for making valuable comments. The first and second authors are supported by NNSF, China (Grant no. 11171345) and the Doctoral Fund of Ministry of Education of China (Grant no. 20120023110003). The second author is also supported by the NNSF, China (Grant no. 51234005). The third author is supported by the Fundamental Research Funds for the Central Universities (Grant no. JCB2013B06).

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