Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 365463, 6 pages http://dx.doi.org/10.1155/2013/365463

Research Article

Relationship between Hardy Spaces Associated with Different Homogeneities and One-Parameter Hardy Spaces

Xinfeng Wu

Department of Mathematics, China University of Mining & Technology, Beijing 100083, China Correspondence should be addressed to Xinfeng Wu; wuxf@cumtb.edu.cn Received 30 May 2013; Accepted 25 July 2013 Academic Editor: Yongsheng S. Han

Copyright © 2013 Xinfeng Wu. 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 prove that the Hardy spaces associated with different homogeneities H^m, 0 < p < m, are continuously embedded into the intersection of the isotropic Hardy spaces H^so and the nonisotropic Hardy spaces H%on. As a consequence, we obtain that any operator bounded from either H?o or H£on into LP must be bounded from H^m to Lp.

1. Introduction

For x = (x , xn) € RM x R and § > 0, we consider two kinds of homogeneities on R":

s°e {x,xn) = (Sx',Sxn), / / 2

s°h (x',xn) = {Sx',s2xn)-

The first are the classical isotropic dilations occurring in the classical Calderon-Zygmund singular integrals, while the second are nonisotropic and related to the heat equations (also the Heisenberg groups). Let e(%) and h(%) be functions on R" homogeneous of degree 0 in the isotropic sense and in the nonisotropic sense, respectively, and both smooth away from the origin. Then, it is well known that the Fourier multipliers T1 defined by T1(f)(^) = e($)f(%) and T2 given by T2(f)(%) = h(^)f(^) are both bounded on Lp for 1 < p < m, of weak-type (1,1), and bounded on the classical isotropic Hardy spaces H^o and, nonisotropic Hardy spaces H^, respectively. Riviere in [1] asked the question is the composition T1 ° T2 still of weak-type (1,1)? Phong and Stein in [2] answered this question and gave a necessary and sufficient condition for which T1 ° T2 is of weak-type (1,1). The operators Phong and Stein studied are in fact a composition of operators with different kinds

of homogeneities which arise naturally in the 3-Neumann problem. Recently, Han et al. [3] developed a theory of the Hardy spaces H?om, 0 < p < m, associated with the different homogeneities and proved that the composition of the two Calderoon-Zygmund convolution operators with different homogeneities is bounded on H?om(Rn). Weighted function spaces associated with different homogeneities and boundedness of composition of operators on them were recently investigated in [4-6].

The Hardy spaces HCom introduced in [3] have surprising multiparameter structures which reflect the mixed homogeneities arising from the two operators under consideration. A natural question arises: Is there any relationship between H?om and the two classical one-parameter Hardy spaces H^o and H^? The main purpose of this paper is to answer this question. We shall prove that HCom are continuously embedded into the intersection of H^o and H?on. As an application, we show that any operator boundedness from either H^o or into Lp must be

bounded from H?om into Lp. Our methods are to use the partially discrete Calderon-type formula and the Littlewood-Paley theory in this context, which are appropriately developed.

Before stating the results more precisely, we first recall some notions and notations. For x = (x',xn) € R"-1 x R,

we denote |*|e = (|x'|2 + |*J2)1/2 and |*|h = (|x'|2 + K|)1/2. Let e S(R") satisfy

supp ^ <2},

^^ (2j°e?) = 1 V^e R" \{0}.

The isotropic discrete square function ^o(/) is defined by

0?so (/)(*) :=

i i k */(*q)| xq (*)

jeZQeQs o

where ^j1)(x) = 2 j"^(1)(2 j°ex), Qjso denotes the set of all dyadic cubes with sidelength 2j, and denotes the left-lower corner of Q. TheisotropicHardyspacesH^o(R"), 0 < p < to, are defined by

p iDn\

f 6 P : ll^fso c^iLr») < TO

LP(R")

Similarly, let ^(2) e S(R") satisfy

supp V(2) (Ç) Ç : 2 < 14 - 2) >

(2V) = 1 R" \{0}.

The nonisotropic discrete square function #fon(/) is defined by

(/) m :=

I I K2) •/(*

fceZ Qgg^

where = 2-fc("+1)^(2)(2-fc<>hx), Qj;on is the set of dyadic

nonisotropic "cubes" Q = Q1 xQ2 c R"-1 x R with sidelength l(Q1) = 2k and |Q21 = 22fc, and xq is the left-lower corner of Q. The nonisotropic Hardy spaces H^, 0 < p < to, are defined by

f 6 SP : |#non (/)|

For e Z, let = ^j1) * y®. The discrete

square function ^om associated with different homogeneities is given by

&om (/) M :=

I I xr (x)

j.fceZ

1/2 ^ , (8)

where denote the set of dyadic rectangles R = Q1 x Q2 in R"-1 x R with sidelength l(Q1) = 2-/Vfc and |Q2| = 2jv2fc and

is the left-lower corner of .R. The Hardy spaces H^, 0 < p < to, associated with different homogeneities are defined by

f 6 p : l^com (/)|

The main result of this paper is as follows. Theorem 1. Let 0 < p < to. One has

HLn (R")-HfSo (R>Hin (R").

More precisely, there is a constant C depending on p and n such that, for all f e H{om (R"),

||Hfso (R») +

l|HP„0n (R") - C||/||HLm (R")•

In [3], it was proved that the composition of the two Calderon-Zygmund convolution operators with different homogeneities is bounded on HCom (see [3, Theorem 1.9]). This result can be improved by the following.

Corollary 2. Let 0 < p < to. One has that

(1) any operator T bounded from either H^o (R") or H^on (R") into If (R") must be bounded from H^m(R") to L^(R");

(2) if r1 is bounded from H{so (R") to L^(R") and T2 is bounded from H^on (R") to i/(R"), then T1 ° T2 is bounded from H{om (R") to i/(R").

2. New Square Function Characterizations of

Let 0(1) e S(R") satisfy supp 0(1) ç B(0,1),

0(1) (x)x"dx = 0 V|a|<Mp,

(2V) = 1 V^e R" \ {0},

where M^ is a sufficiently large constant depending on p and n. Similarly let 0(2) e S(R") satisfy supp 0(2) c £(0,1),

0(2) (%) x^dx = 0 V | ß| - MP,

I^ (2V) = 1 V^6 R" \ {0}.

Let 0j.1)(x) = 2-j"0(2-joex), 0<2)(x) = 2-fc("+1)0(2-\x), and 0j,fc = 0j1) * 0(2). Then, Hfom can be characterized via the continuous square function ^com defined by

#com (/) M =

I |0;Jt */(^)f

j,fceZ

The following Calderon-type identities are well known (cf. [7-9]).

Lemma 3. For f € L2(R"), there is a sufficiently large integer N such that

Lemma 5 (see [10]). Let Q e @J.soN and M > 0. Then,forany u, 9q e Q, 9qi e Q', and 8 e (n/(n + M), 1], one has that

2M0v/)

ieZQe@'-N

= 1 1 № (-*q)W

keZQe@*no

where the two series converge in L2(R") and 9q denotes any fixed point of Q.

The Calderoon-type identities in Lemma 3 lead to the following square functions:

Ä (f)M :=

¿om (dm ■=

1 1 *f(xQ)\xQ (x)

¿keZQeCN

1 1 \$j,k *f(xQ)\XQ (x)

j,fc6ZQ6@n-nN

(2^? + \m-0q,\):

n+M \9j' (dQ'

< cN2nij-f |((i/5)-i)

1 \df (9q'fc

where M denotes the classical isotropic Hardy-Littlewood maximal operator and avb = max (a, b).

We first assume that f € L2nHfom(R"). Set fk = ^f* f. Applying the discrete Calderon-type identity in (15), the classical almost orthogonality estimate, and Lemma 5, we deduce that for any u,9q € Q, 9qi € Q', and 8 € (n/(n + M), min (p, 1)),

k\k * f(u)\

Q'e@LN

where N is the sufficiently large integer in Lemma 3. The purpose of this section is to prove the following.

Theorem 4. For p € (0, m), one has that

a(1) (f)|| _ 11 „(2) ( r)

3 com (J )|Lp _ № com (J )

Proof. We only prove that

¿om (/)

com \J / WlP

as llfll^p

ll^com(f)Wie can be obtained in the same

manner. To prove (18), it suffices to show that

1 1 supKfc *f(u)\2XQ

J* q6@;;on "eQ

1 1 inQ\$f,k *f(u)\2XQ

To verify (19), we need the following lemma.

1 1 ^ ) (U-9q/ W.1) *fk (9q )

j'^Q'^

< 1 2-li-/li°M

2M(jvf)

$) .fk (V

(2W + \m-0q'\)

n+M \<f

< 1 2-lj-flM j'eZ

1 ) .fk (8q )\

.Q'eQ^

Since u and 9q are arbitrary points in Q and Q', respectively,

sup \$hk * f(u)\

X infU;.:' */fc(v) I

исПЧ J

which, by the Cauchy-Schwarz inequality, implies that I I2

sup|0j>fc * /(и)|

-ij-j' |M

X infk1' */fc (v)i

^^ исПЧ J

where we have used Xj'eZ 2 |j j < C. Multiplying ^q from

both sides and summing over j, k e Z, Q e QjsoN yield that, for any 9q e Q,

X X sup|fj,fc */(m)Uq m

ij-j'iM

J'fceZ Q^ "6Q

< X X 2-

j.fceZ j'gZ

X infUJ1' */fc (v)|

^^ ксПМ J

< X Xх

j'eZfceZ

X inf^1' */fc (V)|

It follows that

X X supkj,fc */(m)|2^q M

j'sZfceZ

X inf>J' */fc (V)|

Finally, taking the L^ norm on both sides and applying the Fefferman-Stein vector-valued inequality yield (19). Since L2 П Hfom is dense in HCom (see [3]), a limiting argument concludes the proof of Theorem 4.

3. Proofs of Theorem 1 and Corollary 2

We first give the following.

Proof of Theorem 1. For 1 < p < то, Theorem 1 is trivial since яшш = = Hnon = We now assume that 0 < p < 1. We only prove ll/llH£on < ll/llH&m as the inequality ||/||hp <

||/||hp can be proved similarly. Since L2 n HCom is dense in H£m, we may assume that / e L2 n -H^. For i e Z, set

Q , = {xe R" (/)(*) >2 '}.

Denote that

B , = {(j,Q):Qe Sj-oN,|QnQ ,|

> -—, |Q n О ;+11 < 2 |Q| ,+1I 2 |Q|

where N is the sufficiently large integer in Lemma 3.

Applying the discrete Calderon-type reproducing formula in Lemma 3,

a (*) = X X |Q|^(1) *a (*q)0? (*-*q).

ieZ (j,Q)eBj (28)

к e Z,

where := 0(2) * / and the series converges in the L2(R") norm. We claim that

X |Q|^j1} *Л (Xq) ^ (• - *Q)

(j.Q)eBj

C2 ip|n,|.

Assume the claim for the moment. Then, by the continuous square function characterization of H^on,

Г" j z

J ft ь

Z wf *fk ЫФ

x(x- Xq)

Z Z ^Ф? ЫФ'

>eZ (j,Q)6Bj

x|(x-^Q)|2

N1-P/2Z к

Z wf */к (^q) ф

(j.Q)eBj

X ( - ^

ieZ keZ

(j.Q)eBj

Z wj *fkЫФ)

(j.Q)eBj

X(X - xq)

^Z2'P N<\k(om w\\p -u

We now estimate the last L2 norm by duality argument. For ( with ||£||2 < 1, applying the Cauchy-Schwarz inequality yields

Z № ф] *fk (xq)фу (--xq)'^

,(j.q)eBf /

Z \q\ ф? * fk (xq) j фу (x - xq) Ck (x) dx

■""■л f- (la J

where in the second inequality, we have used Minkowski's inequality, for I2 and in the third inequality used the inequality (I ,\at\)p <1 ,\at\P for 0<p<1.

Thus, to finish the proof of Theorem 1, it suffices to verify

(j,Q)eB,

<( z iQikr */k м:

V (j.Q)eBi

x( Z IQllW *<(xq)

claim (29). Since is supported in unit ball of R, for where ф^)(х) = ( x) Since

(j, Q) € B{, )(- - xq)| are supported in

ai = {x: M (xnt)(x)> 2}-

Z IQI^(1)

(j.Q)eBj

Thus, by Holder's inequality and \Q ¡\ < C\Q;\ (which follows from L boundedness of M),

Z k(1) *<M*Q (X)

Z IQI^j1) *fk (xq) ^j (■ - xq)

(j.Q)eBj

<f kfso (0(x)l2dx<\\t;\\2L2 <1,

(j.Q)eBj

(1/2)-2

we now have that

Z W? */kЫФ)

(j.Q)eBj

x(x- Xq)

Z Z *fk Ы ^j1) (-Xq)

k (j,Q)eBj

<Z Z IQik?*/kы|2>

k (j,Q)eBj

which is, in turn, bounded by

2X X k? *A M2 |Qn(n,-n 1+1)|

k (j,Q)6Bi

Z Z */(*q)Uq (x)

fc (j,Q)eBj

Jnj\ni4

2f [Ä (/)(*)fd*<22i |Q,

Jni\ni+1

(1/2)-2

[10] M. Frazier and B. Jawerth, "A discrete transform and decompositions of distribution spaces," Journal of Functional Analysis, vol. 93, no. 1, pp. 34-170,1990.

where in the first inequality, we have used |Q| < 2|Q n (Q; \ Qi+1)| when (j, Q) e B ;. Substituting this estimate back to (32) verifies the claim (29), and hence, Theorem 1 follows. □

Finally, we give the following.

Proof of Corollary 2. (1) follows directly from Theorem 1, while (2) follows from (1). □

Acknowledgments

This research was supported by NNSF, China (Grant no.

11101423), and supported in part by NNSF, China (Grant no.

11171345). The author would like to express his deep gratitude

to the referee for his/her valuable comments and suggestions.

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