Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 275791,18 pages doi:10.1155/2012/275791

Research Article

Characterizations of Multiparameter Besov and Triebel-Lizorkin Spaces Associated with Flag Singular Integrals

Xinfeng Wu and Zongguang Liu

Department of Mathematics, China University of Mining and Technology (Beijing), Beijing 100083, China

Correspondence should be addressed to Xinfeng Wu, wuxf@cumtb.edu.cn Received 25 February 2012; Accepted 9 April 2012 Academic Editor: Yongsheng S. Han

Copyright © 2012 X. Wu and Z. Liu. 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 inhomogeneous multiparameter Besov and Triebel-Lizorkin spaces associated with flag singular integrals via the Littlewood-Paley-Stein theory. We establish difference characterizations and Peetre's maximal function characterizations of these spaces.

1. Introduction and Main Results

The flag singular integral operators were first introduced by Muller, Ricci, and Stein when they studied the Marcinkiewicz multiplier on the Heisenberg groups in [1]. To study the □-complex on certain CR submanifolds of Cn, in 2001, Nagel et al. [2] studied a class of product singular integrals with flag kernel. They proved, among other things, the Lp boundedness of flag singular integrals. More recently, Nagel et al. in [3, 4] have generalized these results to a more general setting, namely, homogeneous group. For other related results, see [5, 6].

For 0 <p < 1, Han and Lu [7] developed Hardy spaces HF(Rn xRm) with respect to the flag multiparameter structure via the discrete Littlewood-Paley-Stein analysis and discrete Calderon's identity and proved the HPF(RnxRm) ^ HPF(R"xRm) and HPF(R"xRm) ^ LP(Rnx Rm) boundedness for flag singular integral operators. The duality of HF(Rn x Rm) was also established. More recently, Ding et al. studied the homogeneous Besov spaces and Triebel-Lizorkin spaces associated with flag singular integrals in [8] and proved the boundedness of flag singular integrals on these spaces. Similar results can also be found in [9].

The aim of this paper is to give the new difference characterization as well as Petree's maximal function characterization of multiparameter Besov and Triebel-Lizorkin spaces associated with flag singular integrals, which reflect that the Besov spaces and Triebel-Lizorkin spaces have flag multiparameter structure. These characterizations are established for

the inhomogeneous Besov and Triebel-Lizorkin spaces, but the argument goes through with only minor alterations in the homogeneous ones introduced in [8].

In order to describe more precisely questions and results studied in this paper, we begin with basic notations and notions. Let f(1) e S(Rn x Rm) with

supp </f(1) c j (In) e Rn x Rm : 2 < < 2

2~jS, 2-jn)\2 = 1 V(In) e Rn x Rm \{(0,0)},

and let f(2) e S(Rm) with

supp f(2) c j n e Rm : ^ < \n\ < 2

£p(2-kn) | = 1 V n e R™ \{0}.

Let f(1) (x, y) = 2jnf(1) (2jx, 2jy), f® (y) = 2kmf(2) (2ky), and

fi,k{x,y) = fj(1)*2fk2) (x,y) = J f<j1)(x,y - (1.3)

then, for / e L2(Rn x Rm), the following Calderon's reproducing formula holds:

f = X X j * fj,k * f

jeZ keZ

where the series converges in L2(Rn x Rm).

A Schwartz function f e S(Rn+2m) is said to be a product test function in (Rn+m x Rm) if it satisfies

f(x,y,z)xayPdxdy=\ f(x,y,z)zJdz = 0, (1.5)

Jr™x Rn JR™

for all multi-indices a e Nn,ß,j e Nm.

Definition 1.1. A function f (x,y) defined on Rn xRm is said tobe a test function in Sg(Rn xRm) if there exists a function f # e 5o(Rm+n x Rm) such that

f(x,y) = f#(x,y - z,z)dz, (1.6)

and the seminorm of f is defined by

= infj \\f \\ ß : for all representations of f in (1.7) f, (1.7)

where || ■ denote the seminorm in S(Rn+2m). Denote by SF'(Rn x Rm) the dual of SF(Rn x Rm). '

Choose a Schwartz function y on Rn x Rm such that

^ ^ 2 2 MM I2 = 1 -EE|v®(2^1,2-j&)| ■

j=1fc=i

Note that y eSg(Rn xRm) with Fourier transform is supported in : |(£, n)I < 1}. Define

the operator So by Sof = y * f, f e Ss'(R" x Rm).

For f e L2(Rn x Rm), by taking the Fourier transform,

f = y * y * f + ^Evk * j * f,

j=ik=i

where the series converges in L2(Rn x Rm) norm.

Now, we introduce the definition of inhomogeneous Besov spaces and Triebel-Lizorkin spaces associated with flag singular integrals.

Definition 1.2. Let a, 6 e (-to, to) and p, q e (1, to). The inhomogeneous Triebel-Lizorkin space

associated with flag singular integrals Fp,q (Rn x Rm) is defined to be the collection of all f e SF'(Rn x Rm) such that

Fpß (R"xRm)

:= HSof ||

LP(R"xRm)

/ \ 1/q / » » \

(EE 2jaq2kßqIvjk * f |q ) j k=1 /

< TO, (1.10)

LP (R"xRm)

and the inhomogeneous Besov space associated with flag singular integrals B^R" x Rm) is defined to be the collection of all f e Ss'(R" x Rm) such that

/ TO TO

HSof || ( EE2jaq2kßqufj,k *fiiLp(r ™xRm)

Bp,ß(R"xRm)

< TO. (1.11)

\j=1 k=1

Throughout this paper, we always work on R" x Rm for some fixed n, m and use Faq

a 6 a 6

to denote ?„ a(Rn x Rm), similarly for Ss, B^6, and so forth. We would like to point out that the multiparameter structures are involved in the definitions of B^'6 and Fp^. The following result shows that the definition of the Besov spaces Bp6 and Triebel-Lizorkin spaces Fa6 is independent of the choice of (^(1), ^(2), y); thus, the Besov spaces B^'6 and the Triebel-Lizorkin spaces Fa6 are well defined.

Theorem 1.3. If dj/k satisfies the same conditions as fj,k, and $ is defined similar to (1.8) with f replaced by 9, then for a,p e R and p,q e (1, to) and f e ,

aß ~ 11/11^ ~ I\fWU- (1.12)

Remark 1.4. As the classical case, it is not hard to show that || ■ H^ and || ■ ||f ^ are norms of

B^ and Fvj, respectively. Moreover, B^ and F^ are complete with respect to these norms and hence are Banach spaces. We omit the details.

Throughout this paper, we use the notations j A k = min{j,k} and j V k = max{j,k}. We introduce the following flag multi-parameter Peetre maximal functions (with respect to f). For j, k e Z+ and b = (b, bz) e R+ x R+, define

/ S/ s Wik * fix - u,y - v) 1 fl,(f)(x,y) = sup m'k f\-'-y-J-b. (1.13)

rhjkKJJK * (u,v) eRn xR™ (1 + 2j |u|)b1 (1 + 2jAk |v|)b2

For j = k = 0, define

W = suip ■ (1.14)

* f(x - M,y - p) I (1 + |M|)b1 (1 + \v\)b

An index b = (b1,b2) is said to be admissible if

bi > ——, b2 >——. (1.15)

p A q p A q v '

We point out again that the flag multiparameter structure is involved in the definition of Peetre's maximal functions. The maximal function characterizations of Besov spaces and Triebel-Lizorkin spaces are as follows.

Theorem 1.5. Let a, fi e R and p,q e (1, o).Ifb is admissible, then for f e , one has

(i) ||f H^^ := ||Sof ||p + ||{2a/2fikHj(f)||p}hk\\^ ~ ||f W^,

(ii) Hf hfo^ := ||Sof Hp + 11|{2aj2?kyZ . (f |kp ~ ||f |u.

Tp,q,(1) b,],k " Tp,q

Here and in what follows, one uses the following notation:

( ^ 1/q

I 00 00

IbjL := \ZZ\aj'k\q \ ■ (1.16)

j k=1 J

In order to state our result for flag singular integrals, we need to recall some definitions given in [2]. Following closely from [2], we begin with the definitions of a class of

distributions on an Euclidean space Rd. A k-normalized bump function on a space Rd is a Ck function supported on the unit ball with Ck norm bounded by 1. As pointed out in [2], the definitions given below are independent of the choices of k, and thus we will simply refer to "normalized bump function" without specifying k.

Definition 1.6. A flag kernel on R" x Rm is a distribution K on R"+m which coincides with a C™ function away from the coordinate subspaces (0, y), where (0, y) e R" x Rm and satisfies

(1) (Differential inequalities) for any multi-indices a and j5,

|d«ayK(x,y)| < Ca,p| X |1 a '( |X | + | y|)-m- 15 1 , (1.17)

for all (x, y) e R" x Rm with | x | = 0,

(2) (Cancellation condition)

daXK(x,y)$1(Sy)dy < Ca|x|-"-|a|, (1.18)

for all multi-index a and every normalized bump function on Rm and every 6> 0,

dyKix, y) (Sx) dx

< Cß|yr-|ß| (1.19)

for every multi-index ¡5 and every normalized bump function on R" and every 6> 0,

J R"+m

K^x, y)$3( S1x, S2y)dxdy

< C, (1.20)

for every normalized bump function on R+m and every 61 > 0 and 62 > 0.

The boundedness of flag singular integrals on these inhomogeneous Besov spaces and Triebel-Lizorkin spaces is given by the following theorem, whose proof is quite similar to that in homogeneous case in [8]. We omit the proof here.

Theorem 1.7. Suppose that T is a flag singular integral defined on R" x Rm with the flag kernel K,

a 5 a 5

then, for p,q > 1 and a, 5 e R, T is bounded on and on Bp';q.

As in the classical inhomogeneous Besov spaces and Triebel-Lizorkin spaces, we will

a 5 a 5

give the difference characterization for and . However, the new feature is that the differences of functions are associated with the "flag." More precisely, for (u,v) e R" x Rm and w e Rm, we define the first flag difference (associated the flag {(0,0)} c {(0, y)}) in R" x Rm by

^vwfixy) = [Au,v ◦ AW2^ f(x,y) (1 21)

= f(x + u,y + v + w) - f(x + u,y + ^ - f(x,y + w) + f(x,y),

where Auv is the difference operator on Rm+n, and Aw is the difference operator on Rm. For k e Z+ and k > 2, the kthflag difference operator (AFvw)k can be defined inductively by

(AF )k = AF o(Af )k-1

(1.22)

Theorem 1.8. if a, ^ > 0,1 < p < to, 1 < q < to, and M > [a v f5]+1, where [•] denotes the greatest integer function, one defines

■W(2)

rp,q,(2)

i(As )M

du dv dw

Rn+mxRm

JR"+mxRm

|(u,v)|a|w|ß j |(u,v)|n+m|w|m |(AFv;w)Mf|\q dudvdw ^ /q

(1.23)

|(u,v)|a|w|ß / |(u,v)|n+m|w|m

then \\f\\3a, ~|l/llB*o, \\f Hw ~Wf Wf*■

p,q,(2) P,q ' p,q,(2) ' f,q

As mentioned before, by slightly modifying the proof, we can prove difference characterizations and Peetre's maximal function characterizations of homogeneous Besov and Triebel-Lizorkin spaces, introduced in [8]. We leave the details to the interested reader.

The following of the paper is organized as follows. In Section 2, we give some lemmas. The proof of Theorems 1.3 and 1.5 is presented in Section 3. Section 4 is devoted to the proof of Theorem 1.8.

2. Some Lemmas

In this section, we present some lemmas, which will be used in the proofs of the theorems.

2.1. Inhomogeneous Calderon's Reproducing Formula in S^ Lemma 2.1. The inhomogeneous Calderon's reproducing formula holds

f = y * y * f + ^ ¿jfj,k * fj,k * f, j=1k=1

where the series converges in LP (Rn+m) (1 <p< to), Ss(R" x Rm) and S^ (Rn x Rm )■

We point out that in [8] the homogeneous Calderon's reproducing formula was provided. Note that the convergence of these two kind of producing formulas are different. See [8] for homogeneous case.

Proof. For any f e SF(R" x Rm), then by definition, there exists f# e S(R"+m x Rm) such that f (x, y) = JRm f #(x, y - z, z)dz. We need to show that for all N e Z+,

* / * /# + E j * j * f#,

;,keZ+: (2.2)

j<N or k<N

tends to f in the topology of S(R"+2m) as N ^ +<x>. We only consider the case when k < N in the summation in (2.2) since the other case can be dealt with in the same way. Denote, in this case, the expression (2.2) by f#. By Fourier inversion,

f # (x, y) - f#N (x, y) = (J# - f*N)\x, y), for every (x, y) e R"+m x Rm. (2.3) Let hN(ln,x,y) = e^+^f *^) £»n+! (2-kn) |2 so that

f # (x, y) - fN (x, y) = cn\ hN{l n x, y) dn dl

J Rn+mxRm

Since |hN(¿„ц,x,y)| < |f#(£,n)| e L1(R"+mxRm),Lebesgue's dominated convergence theorem yields

lim fN (x, y) = f# (x, y), for every (x, y) e R"+m x Rm. (2.5)

N ^ +<x>

On the other hand, using the cancellation conditions of ^(1) * * ^k^, and the

smoothness of f #, we can get

|j * j *f#(x,y)|< 2-j2-k(1 + |x| + |y|)-L, Vj,k,L e Z+. (2.6)

Now, (2.5) together with the estimate (2.6) implies that, for f e Ss(R" x Rm),

|f#(x) - fN(x,y)| = E E |j * j * f#(x,y)|< 2-N(1 + |x| + |y|)-L. (2.7)

;eZ+k>N+1

Applying this to duf# (here u denotes any multi-index in N"+2m) and noting that dufN = (duf #)N, we obtain

lim sup (1 + |X| + |y|)L|du(f# - fN) (X,y)| = 0. (2.8)

^ OT(x,y)eR"xRm

This proves the convergence of series in (2.1) in Sg(R" xRm). The convergence in S^(R" xRm) follows by a standard duality argument. The convergence in Lp(R"+m) can be proved similar to the product case, see [10, Theorem 1.1]. □

2.2. Almost Orthogonality Estimates

The following lemma is the almost orthogonality estimates, which will be frequently used. See [7] for a proof.

Lemma 2.2. Let x e Rn,y e Rm. Given any positive integers L and M, there exists a constant C = C(L,M) > 0 suchthat

• -,,r „ ,nr 2(iAi')M 2(iAi'AkAk')M

j * wr k (x,y) I < C2-|/-/|L2-|k-k|L----, (2.9)

^ ,yJ\~ (2jA;, + !x |)n+M (2jAj'AkAk' + \y\, ^ 9

where f, y are defined as in Section 1.

2.3. Maximal Function Estimates

The maximal function estimates are given as follows.

Lemma 2.3. For j, k, j', k' e Z+, and for any L> 0 and b = (b\, b2) e R+ x R+, there exists a constant C = C(L, b) depending only on L and b, but independent on j, k, j', kk, such that

\ (j * fj'k * fj'k * f )(x, y) \ < C2-LI j-j'I 2-LI k-k' I j (f)(x, y). (2.10)

Proof. By the almost orthogonality estimate in Lemma 2.2, for any L> 0 and M > b1 V b2, we have the pointwise estimate

\ (fj ,k * fj',k' * fj'k * / (x , y) \

< 2- j-j' 1L' 2-1 k-k' 1L'

2(jAj')n2(jAkAj'Ak')m \ frk * f(x - u,y - v) \

Jvnx Rm

(1 + 2jAj' I u I )M+^ 1 + 2j AkAj'Ak' | v | )M+m

< 2- 1 H 1 L2- 1 k-k 1 ^ kX/ (x)

R"x Rm

2(jAf)n2(jAkAfAk')m(1 + 2 | u | )b1 (1 + 2j'Ak' | v |) , , (2.11)

-TT-7-7-du dv

(1 + 2jAj' | u I )M+n(1 + 2jAkAj'Ak' | v | )M+m

< 2- 1 j-f 1 (L-b1)2- 1 k-k 1 (L-b1-bl)flr v (f) (x,y)

JRnx Rm

2(jAj')n2(jAkAj'Ak')m

-du dv

(1 + 2jAj' | u I )M+n-b1 (1 + 2j AkAj'Ak' | v | )M+m-b2

< 2-1/-j|(L)2-|k-k| (L)fbj7,kXf)(x,y), where L = L' - bi - b2. This proves (2.10). □

Remark 2.4. Since the almost orthogonality estimates hold with ^0/0 or 00/0 is replaced by y, repeating the same argument as (2.11), we see that the estimate (2.10) is still valid if y0/0 or 60/0 is replaced by y.

Denote by Ms the strong maximal operator defined by

Ms/ (x) = sup f \/(y)\dy, (2.12)

R3x J R

where the supremum is taken over all open rectangles R in Rn x Rm that contain the point x.

Lemma 2.5. Let 0 < ci,c2 < to, and 0 < r < to, then for all j,k e Z and for all C1 functions u on Rn x Rm whose Fourier transform is supported in the rectangle {£ : |£'| < c12/,\£,n\ < c22jAk}, one has the estimate

,u(x - u,y - v)| . 1/r

sup-' \/r y . !l m/r < CMs|u|0(x,y) ■ (2.13)

(1 + 2/ |u|)n/r (1 + 2/Ak\v\)m/

In particular, ifb = (b1, b2) with b1 > n/r and b2 > m/r, then for all / e Lp(Rn x Rm) (1 <p < to),

V/WXy) < Ms( W/,k * /\r) (x,y)x/r. (2.14)

Lemma 2.5 can be proved as in the classical one-parameter case. We refer the reader to

2.4. An Embedding Result

The following lemma is an embedding result.

Lemma 2.6. For a,p> 0 and p,q e (1, to), one has the following continuous embedding:

B^ Lp, f^ LP ■ (2.15)

Proof. For / e Sg(Rn x Rm), by inhomogeneous Calderon's reproducing formula (2.1),

/ \ 1/q

/ to TO \

I V V?/aq?¥qwm;,.* twq \ nJ

(2.16)

< 11^0/wp + EEHm * /wp < 11^0/wp +1 EE2/aq2k*||/*/wp i ~ i/11

p /=1 k=1 p p \/=1 k=1 v p,q

where we have used Holder's inequality in the last inequality. This proves Lemma 2.6 for Besov spaces.

For the Triebel-Lizorkin spaces, by (2.1), the pointwise inequality f/rk * /(x,y)| < f* fc(/)(x,y), Holder's inequality, and Fefferman-Stein's vector-valued inequality, we have

p < lly * (Sof)llp

EE| j * j *f | j=1k=1

< llSof H

< ll Sof l

/ oo oo

(EE2jaq2kßqjf)q \

/ \ 1/q

(EE 2jai>2k^nk * f ^ ) k=1 /

(2.17)

This ends the proof of Lemma 2.6.

3. Proof of Theorems 1.3 and 1.5

We first prove Theorems 1.3 and 1.5 for Triebel-Lizorkin spaces by showing

„aß < lf ^a/ <

Tpq Tp,,q,(1)

ia-P < lf PW-

Tp,,q,(1) Tp'i

The first inequality in (3.1) follows from the pointwise inequality

I j * f(x,v) I < y*brj f (x , v), Vj, k e Z+ -

Next, for any admissible b, fix b. Since b is admissible, we can choose r < p A q such that b1 > n/r, b2 > m/r, and the thus inequality (2.14) holds. We apply Lemma 2.5 and the Lp/r (£q/r) boundedness of Ms to deduce

Za.P Tp,q, (1)

2ja2kß0* (f)

L?(£q)

j^Msfljk * f ^ )1/r}

lp(£q)

■¡■aß ,

which gives the third inequality in (3.1).

Thus, to finish the proof of (3.1), it remains to verify the second inequality. For j, k e Z+, $ * 9jrk is nonzero only when j = 1 and k = 1. Thus, applying Calderon's identity (2.1), Minkowski's inequality, Remark 2.4, and Lemma 2.5, we deduce that

iiy * / iiP < iiy * 0H1 ■ ii0 * / iL+ik

p II b,1,1

ii0 * f iip

2ja2kßdl (f)

LP(£q)

Fp,q, (1)

To finish the proof, it remains to show

LP (£q)

rp,q, (1)

By the inhomogeneous Calderon's identity (2.1), we have

\fj,k * f(x,y) \ < (\fj,k * * * f\) (x,y) + E (\fjk * efk \ * |%k'* f \) (x,y). (3.6)

j',k'eZ+

It follows that

jX* (f)

LP(£1)

2ja2kß

(Ifj,k * 0|*|0 * f |)('- u, --v)

(u,v)eRn xRm (1 + 2j IuI)b1 (1 + 2jAk I vI)

LP(£q)

2ja2kß Y> sup

j',k'>0 (u,v) eRnxRm

(Ifj,k * Oj'k'I * j' * f I)( - u, ■- v)

(1 + 2j IuI)b1 (1 + 2jAk IvI)b2

LP(£q)

:= I1 +12.

We first estimate Ji. By the support properties of fjrk and y and Young's inequality,

I1 <iifU * Ml ■ ii0 * f iip < ii0 * f iip

Next, we give the estimate for I2. For any (u,v) € R" x Rm,

( | fjrk * erk | * | Qj',v * f 0 (x - u,y - v)

(1 + 2j | u | )bl(1 + 2jAk | v | )

jRnx Rm

1 fj,k * dj',k (u', v' )11 Qf k * f(x - u - u',y - v - v')1

(1 + 2j | u | )&1(1 + 2jAk | v | )

du'dv'

< Qb r v (f)(xy) f | j * ej'k (^ v')|

,k ./«"xR™

(1 + 2j'|u + u'|)b1 (1 + 2j'Ak'|v + v'|)

(1 + 2j |u|)b1 (1 + 2jAk |v|)

< 2-m-r\2-Lk-k,\eif kX f)(x,y),

where we have used Lemma 2.2 in the last inequality. Applying Minkowski's inequality and Holder's inequality yields

\2ja2kp J ( 1 j * Qj'k 1 * 1 j( * f D (x - u, y - v)

\ (u,v)ER"xR" (1 + 2j| u|)b1 (1 + 2jAk| v |)b2

I J / J 2- |j-j' ||a |)2- 1 k-k' | q(L-e-| ^ | )

I j',k' eZ+ \j,keZ+ /

(3.10)

x2j'aq2k'^qe**.jk k, (f )(x,y)q

f, k'( №,y)

where we have chosen e as a small positive constant less than L - (jaj V j/3j). Therefore,

2j'a2k'Pe^*jj, kx f

(3.11)

Lp(gq )

Combining the estimates (3.8) and (3.11), we obtain (3.5). This finishes the proof of (3.1), and hence, Theorems 1.3 and 1.5 for Triebel-Lizorkin spaces follow.

The proofs of Theorems 1.3 and 1.5 for Besov spaces are similar. By (3.2) and the maximal function estimate (2.14), \\fj,k * f ||p ~ . (f )||p. The conclusion of Theorem 1.5

for Besov spaces follows. By Calderon's reproducing formula (2.1), Young's inequality,

the almost orthogonality estimate in Lemma 2.2, Holder's inequality, and Minkowski's inequality, we have

ZZ2jaq2kßqllfj,k * f ll j=1 k=1

< 2aq2ßqll„1,1 * y||?||y * f Up

oo oo / oo oo

+ ^ ^WEElljk * Ork ye,,** f ll

j=1 k=1

\ j'=1 k'=1

ll* * f Up + XZ2jaq2kßq(ZZ2-(L^q(|jj|+Ik-k' ' ) llOj'k' * f Up

j=1k=1

\j'=1k'=1

lly * f up + x Z2j'aq2kH0f,k'* f Up,

j'=1 k'=1

(3.12)

as desired. This ends the proof of Theorem 1.3 for Besov case. Hence, the proofs of Theorems 1.3 and 1.5 are complete.

4. Proof of Theorem 1.8

4.1. Proof of Theorem 1.8 for Besov Space.

By the moment conditions of f(1)'s and f® 's, we may write

fj,k * f{x,y) =

Rn+mxRm

q,(1)(u,v)„k2)(W)

/ F \M (A?u,-v;-w) f(x,y)

du dv dw- (4.1)

By Minkowski's inequality and Holder's inequality and noting that

(AVv;-w )Mf|| =||(AF,v;w )Mf||

we have

Urn * f ll q =(ff Uf)(u,v)|Lk2)(w)|||(Af/v;w )Mf|| dudvdwX

' \J J Rn+mxRm I N III llp /

Rn+mxRm

| j(u,v) | |q£2)(w)| U (AFvw)Mfllqdudvdw-

Hence,

YZ2m2kiSqhj,k * f U j=1k=1

1 Rn+m xRm < ll ll q

^ 2jaq|qj1 (u,v)|

2 2kßq|„k2)(w)| )||(Af/v;w )MflHdu dvdw (4.4)

,a,f> '

where the last inequality follows from

J2kßq|qk2)(w)| < J2kßq-

1 (2-k + | w | )m+L ~ | w | m+ßq

(1)(u v)| <_1_

j K , ~ |(u,v)|n+m+aq-

J^2jaq|q(1) (u,v)| < j=1

This inequality together with the trivial inequality \S0/\p < \\/\\p yields \\/\Ba,p To prove the converse, by Lemma 2.6, it suffices to show that

Rn+mxRm

|(Alv;w)Mf Up \ du dvdw |(u,v)|a|w|ß J |(u,v)|n+m|w|m

< UfUq

p,q,(2)

By the Calderon's identity (2.1), we write

(A|v;w)M/ =(jKASUvw)M^j * (S0/ + EE((A|v;^Mf/^) * f/k * /, (4.7)

/=1 k=1

where the series on the right hand side converges in inequality and Young's inequality, we conclude that

1). Thus, by Minkowski's

MM MM o o mm

(A|v;w) f < (A|v;w) y US0f Up + XX (Alvw) j llj * f Up- (4.8)

V ' P K ' 1 P /=1 k=1 ^ ' 1 P

It follows that the left hand side in (4.6) is dominated, up to a constant, by the sum of

III1 Z2faq2k'ßq (A|vw) y llSof UP,

j'eZ k'eZ 1

III2 - S E 2j'aq2k'ßq(z £ U(A|v;w)Mj ||1 llq/k * f |L

j'eZ k'eZ \j=1 k=1M 111 f

Journal of Function Spaces and Applications For III1, we have

IIIi < <

^2-fq(M-a)2-k'q(M-P) + ^ 2-i'q(M-a)2k'qP

j>0 k'>0

j '>0 k'<0

^ 2j/q«2-k/q(M-^) ^ 2j'qa2k''

j'<0 k'>0

j'<0 k'< 0

(4.10)

«50/y; < 11S0/ «;,

where we have used the estimate

\L-iu/v;w J

< 2M-min(0/-j'/-k'/-j'-k'}

(4.11)

To estimate III2, by the estimate

(a® ^

\L-iu/v;w J

< 2M-min(0/j-j'/k-k'/j-j'+k-k'}

(4.12)

and Holder's inequality, we see that III2 is majorized by

^ | ^ 2q[M-e-(aV^)](j-j'+k-k') + ^ 2q[(M-a-e)(j-/) + (k'-k)(p-e)]

j',k'eZ I 0<j<j' 0<j<j'

\0<k<k' k>(k'v0)

+ ^ 2q(M-?-e)(k-k,)2q(j'-j)(a-e) + ^ 2q'"-j')(a-e)2q(k'-k)(P-e) | (4.13)

j>(j'v0) j>(j'V0) j

0<k<k' k>(k'v0) /

x 2jaq2k^q||^j,k */y;

<2 * / y;,

j>0 k>0

proving (4.6), where e is a positive number such that 0 < e < (a A p) < (a V p)+ e < M. This concludes the proof of Theorem 1.8 for Besov spaces.

4.2. Proof of Theorem 1.8 for Triebel-Lizorkin Space.

Using the moment conditions on fjrk, we have

X 2jaq2kßq\fjrk * f (x,y)\q

j,keZ+

< E 2jaq2kßq\\ \Ä(1)(M/ü)\\yk2)(^)\\(Af/ü;ro)Mf{x,y)\qdudvdw

II TO) n+m ^TO> m I ^

j,keZ+

oo ■ \ / o \ m

^2jaq\\ff\u,v)\ )(E2kßq\yk2)(w)\) (A|v;w) f(x,y)

du dv dw

du dv dw

Rn+mxRm

|(u,v)|a|w|ß I |(u,v)| n+m\w\m'

where g(x) := g (-x). Therefore,

(4.14)

/ oo oo

(EE 2jaq2kßq\yhk * f\q

\J=1 k=1

Fp,q,(2)

(4.15)

which, together with the obvious inequality ||50/||; < ||/1|;, yields ||/|La,p < ||/|| a,p .

' ;,q 'p,q,(2)

To show the converse, write

(AFrv;w)Mf * (Sof) + EE((A|v;w)M j) * j *f (4.16)

^ ' j=1 k=1 ^ '

where the series converges in S^(Rn x Rm). It follows that

|(A|v;w)Mf | \ q du dv dw R-mxR^ |(u/v)|a|w|V |(u,v)|n+m|w|m

< ll(E ^^((Alvw)MV) * Sof |q}

, 1/q.

j'eZ k'eZ

¡EE2/aq2k^qŒE|((Af,v;w )Mtyj,k) * j * f|) }

j'eZ k'eZ

:= JV1 + JV2,

j=1 k=1

(4.17)

where |(u,v)|~ 2 j and |w| ~ 2 k'.

Journal of Function Spaces and Applications For IV2, by Lemma 2.2,

{Alv;w) j(u,v)

< 2M-min(0/j-j//k-k//j-j/+k-k/} _

2jn2(jAk)m

(1 + 2j |u| + 2jAk |v|)

(4.18)

Consequently,

J fj,k * fj,k *

< 2M-min(0/j-j'/k-k'j-j'+k-k'}

Vb,j/kf (x,y)-

(4.19)

Therefore,

ZZ2j'aq2k'ßq(ZZ( (Af,vw) j) * j * f^) )

j'eZk'eZ \j=1 k=1 ^ ' /

^ I ^ 2q[M-«-(aVß)](j-j'+k-k') + ^ 2q(M-a-e)(j-j')2q(k'-k)(ß-e)

j,keZ+\ j'>j \k>k

+ ^ 2q(M-ß-e)(k-k') 2q(f-j)(a-e) + ^ 2qj-j)(a-£) 2q(k'-k)(ß-£)

r<; k'>k

1<1 k'<k

x 2jaq2kßYUk(f)(x,y)q

< £ 2jaq2kßqfbjk(f)(x,y)

j,keZ+ '

(4.20)

which together with the maximal function characterization of fv'q implies

E 2jaq2kßqqb f

j,keZ+ '

„aß-

(4.21)

As for IV1, similar estimate to (4.19) yields

(Af,v;^ My * S0f (x, y) < 2-M maxf0'j''k''j'+k'}qb,0,0 (f )(x, y) - (4.22)

Therefore,

j'>0 k'>0

j'>0 k'<0

k'q(M-ß)

ypfqa2

j'<0 k'>0

j'<0 k'< 0

(4.23)

IIS0/ I

This estimate together with (4.21) and Lemma 2.6 yields \\f ¡La,?, < ||f ||. This ends the proof of Theorem 1.8.

Acknowledgments

The research was supported by NNSF of China (Grant nos. 11101423, 11171345) and the Fundamental Research Funds for the Central Universities of China (Grant no. 2009QS12). The authors would like to express their deep gratitude to the referee for his/her valuable comments and suggestions.

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[8] Y. Ding, G. Z. Lu, and B. L. Ma, "Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals," Acta Mathematica Sinica (English Series), vol. 26, no. 4, pp. 603-620, 2010.

[9] D. Yang, "Besov and Triebel-Lizorkin spaces related to singular integrals with flag kernels," Revista Matematica Complutense, vol. 22, no. 1, pp. 253-302, 2009.

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