BOUNDEDNESS FOR MULTILINEAR MARCINKIEWICZ OPERATORS ON CERTAIN HARDY SPACES

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BOUNDEDNESS FOR MULTILINEAR MARCINKIEWICZ OPERATORS ON CERTAIN HARDY SPACES

LIU LANZHE Received 14 January 2002

The boundedness for the multilinear Marcinkiewicz operators on certain Hardy and Herz-Hardy spaces are obtained.

2000 Mathematics Subject Classification: 42B25, 42B20.

1. Introduction and definitions. Suppose thatSⁿ⁻¹is the unit sphere ofRⁿ (n≥2)equipped with normalized Lebesgue measuredσ=dσ (x). LetΩbe homogeneous of degree zero and satisfy the following two conditions:

(i) Ω(x)is continuous onSⁿ⁻¹and satisfies the Lipγ condition onSⁿ⁻¹ (0≤γ≤1), that is,

Ω x

−Ω

y≤Mx−y^γ, x,y∈Sⁿ⁻¹; (1.1) (ii)

Sⁿ⁻¹S(x)dx=0.

Let m be a positive integer and A be a function on Rⁿ. The multilinear Marcinkiewicz integral operator is defined by

µ_Ω^A(f )(x)= _∞

0

Ft^A(f )(x)²dt t³

1/2

, (1.2)

where

Ft^A(f )(x)=

|x−y|≤t

Ω(x−y)

|x−y|ⁿ⁻¹

Rm+1(A;x,y)

|x−y|^m f (y)dy, Rm+1(A;x,y)=A(x)−

|α|≤m

1

α!D^αA(y)(x−y)^β.

(1.3)

We denote that Ft(f )(x)=f|x−y|≤t(Ω(x−y)/|x−y|ⁿ⁻¹)f (y)dy. We also denote that

µΩ(f )(x)= ^∞

0

Ft(f )(x)²dt t³

1/2

, (1.4)

which is the Marcinkiewicz integral operator (see [5,6,12]).

Note that whenm=0,µ^A_Ωis just the commutator of Marcinkiewicz operator (see [5,12]). It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [1,2, 3,4,5]). The main purpose of this paper is to consider the continuity of the multilinear Marcinkiewicz operators on certain Hardy and Herz-Hardy spaces.

We first introduce some definitions (see [7,8,9,10,11]).

Definition1.1. LetAbe a function onRⁿ,ma positive integer, and 0<

p≤1. A bounded measurable functionaonRⁿis said to be a(p,D^mA)-atom if

(i) suppa⊂B=B(x0,r ), (ii) aL^∞≤ |B|^−1/p, (iii)

a(y)dy=

a(y)D^αA(y)dy=0,|α| =m.

A temperate distributionf is said to belong toH_D^pmA(Rⁿ), if, in the Schwartz distributional sense, it can be written as

f (x)= ∞ j=0

λjaj(x), (1.5)

where aj’s are (p,D^mA)-atoms, λj ∈ C, and _∞

j=0|λj|^p < ∞. Moreover, f_H^p

DmA(Rⁿ)∼(_∞

j=0|λj|^p)^1/p.

LetBk= {x∈Rⁿ:|x| ≤2^k}, Ck=Bk\Bk−1, k∈Z, and mk(λ,f )= |{x∈ Ck:|f (x)|> λ}|; fork∈N, let ˜mk(λ,f )=mk(λ,f )and ˜m0(λ,f )= |{x∈B0:

|f (x)|> λ}|.

Definition1.2. Let 0< p,q <∞, andα∈R. (1) The homogeneous Herz space is defined by

K˙q^α,p= f∈L^qloc

Rⁿ\{0}

:fK˙_q^α,p(Rⁿ)<∞

, (1.6)

where

fK˙^α,p_q (Rⁿ)=



 ^∞

k=−∞

2^kαpf χk^p

L^q





1/p

. (1.7)

(2) The nonhomogeneous Herz space is defined by

K^α,pq Rⁿ

= f∈L^qloc

Rⁿ

:f_Kq^α,p_(Rn)<∞

, (1.8)

where

f_Kq^α,p_(Rn)=



^∞

k=1

2^kαpf χk^p

L^q+f χB0^p

L^q





1/p

, (1.9)

...

where

f_{W Kq}^α,p_(Rn)=sup

λ>0λ



^∞

k=0

2^kαpm˜k(λ,f )^p/q





1/p

. (1.10)

Definition1.3. Letmbe a positive integer andAa function onRⁿ,α∈R, and 1< q≤ ∞. A functiona(x)onRⁿis called a central(α,q,D^mA)-atom (or a central(α,q,D^mA)-atom of restrict type), if

(1) suppa⊂B(0,r )for somer >0 (or for somer≥1), (2) aL^q≤ |B(0,r )|^−α/n,

(3)

a(x)dx=

a(x)D^βA(x)dx=0,|β| =m.

A temperate distributionfis said to belong toHK˙^α,pq,D^mA(Rⁿ)(orHKq,D^α,pmA(Rⁿ)) if it can be written asf=_∞

j=−∞λjaj(orf=_∞

j=0λjaj) in theS(Rⁿ)sense, whereajis a central(α,q,D^mA)-atom (or a central(α,q,D^mA)-atom of restrict type) supported onB(0,2^j)and_∞

j=−∞|λj|^p<∞(or_∞

j=0|λj|^p<∞). Moreover, f_HK˙^α,p_q,DmA(orf_HK^α,p

q,DmA)∼(

j|λj|^p)^1/p.

2. Theorems and proofs. We begin with some preliminary lemmas.

Lemma2.1(see [2]). LetAbe a function onRⁿandD^αA∈L^q(Rⁿ)for|α| = mand someq > n. Then,

Rm(A:x,y)≤C|x−y|^m

|α|=m

Q(x,y)˜ 1

Q(x,y)˜

D^αA(z)^qdz 1/q

, (2.1)

whereQ˜is the cube centered atxand having side length5√

n|x−y|.

Lemma2.2. Let1< p <∞ andD^αA∈L^r(Rⁿ), |α| =m, 1< r ≤ ∞, and 1/q=1/p+1/r. Then,µ_Ω^Ais bound fromL^p(Rⁿ)toL^q(Rⁿ), that is,

µ_Ω^A(f )

L^q≤C

|α|=m

D^αA

L^rfL^p. (2.2)

Proof. By Minkowski inequality and the condition ofΩ, we have

µ^A_Ω(f )(x)≤

Rⁿ

f (y)Rm+1(A;x,y)

|x−y|^m

∞

|x−y|

dt t³

1/2

dy

≤C

Rⁿ

Rm+1(A;x,y)

|x−y|^m+n f (y)dy.

(2.3)

Thus, the lemma follows from [3,4].

Theorem2.3. Let1≥p > n/(n+γ), and letD^βA∈BMO(Rⁿ)for|β| =m. Then,µΩ^Ais bounded fromHD^pmA(Rⁿ)toL^p(Rⁿ).

Proof. It suffices to show that there exists a constantc >0 such that, for every(p,D^mA)-atoma,

µΩ^A(a)

L^p≤C. (2.4)

Letabe a(p,D^mA)-atom supported on a ballB=B(x0,r ). We write

Rⁿ

µ_Ω^A(a)(x)p

dx=

|x−x0|≤2r

µ^A_Ω(a)(x)p

dx +

|x−x0|>2r

µ^A_Ω(a)(x)p

dx

≡I+II.

(2.5)

ForI, takingq >1 and by Hölder’s inequality and theL^q-boundedness ofµ^AΩ

(seeLemma 2.2), we see that

I≤Cµ^A_Ω(a)^p_Lq·B

x0,2r^1−p/q

≤Ca^p_Lq|B|^1−p/q

≤C.

(2.6)

To obtain the estimate ofII, we need to estimateµ^A_Ω(a)(x)forx∈(2B)^c. Let B˜=5√

nB, and let ˜A(x)=A(x)−

|α|=m(1/α!)(D^αA)B˜·x^α. Then,Rm(A;x,y)=

Rm(A˜;x,y). By the vanishing moment ofa, we write

Ft^A(a)(x)=

|x−y|≤t

Ω(x−y)

|x−y|^m+n−1− Ω x−x0

x−x0^m+n−1

Rm

A˜;x,y

a(y)dy

+

|x−y|≤t

Ω x−x0

x−x0^m+n−1

Rm

A˜;x,y

−Rm

A˜;x,x0

a(y)dy

−

|α|=m

1 α!

|x−y|≤t

Ω(x−y)(x−y)^α

|x−y|^m+n−1

D^αA(y)− D^αA

B

a(y)dy, (2.7)

...

thus,

µ_Ω^A(a)(x)≤



_∞

0 |x−y|≤t

Ω(x−y)

|x−y|^m+n−1− Ω x−x0

x−x0^m+n−1

×RmA˜;x,ya(y)dy 2

dt t³





1/2

+



_∞

0 |x−y|≤t

Ω

x−x0 x−x0^m+n−1

×Rm

A˜;x,y

−Rm

A˜;x,x0a(y)dy 2

dt t³





1/2

+



_∞

0

|α|=m

1 α!

|x−y|≤t

Ω(x−y)(x−y)^α

|x−y|^m+n−1

×

D^αA(y)− D^αA

B

a(y)dy

2dt t³





1/2

≡II1+II2+II3.

(2.8)

ByLemma 2.1, fory∈Bandx∈2^k+1B\2^kB, we know RmA˜;x,y≤C|x−y|^m

|α|=m

D^αA(x)−

D^αA

2^kB. (2.9) By the condition ofΩand Minkowski’s inequality, and noting that|x−y| ∼

|x−x0|fory∈Bandx∈Rⁿ\B, we obtain

Ω(x−y)

|x−y|^m+n−1− Ω x−x0

x−x0^m+n−1

≤C



 r

x−x0^m+n+ r^γ x−x0^m+n+γ−1



. (2.10) Thus,

II1≤C

B

Rm

A˜;x,ya(y) ^∞

|x−y|

dt t³

1/2

×



 r

x−x0^m+n+ r^γ x−x0^m+n+γ−1



dy

≤C



 r

x−x0ⁿ⁺¹+ r^γ x−x0^n+γ



|B|^1−1/p

|α|=m

D^αA(x)−

D^αA

2kB. (2.11)

On the other hand, by the following formula (see [2]):

Rm

A˜;x,y

−Rm

A˜;x,x0

=

|β|<m

1 β!Rm−|β|

D^βA˜;y,x0

x−x0

β

(2.12)

andLemma 2.1, we get Rm

A˜;x,y

−Rm

A˜;x,x0

≤C

|β|<m

|α|=m

x0−y^m−|β|x−x0^|β|D^αA_BMO, (2.13)

so that

II²≤C

B

x−x^0−(n+m)

|β|<m

Rm−|β|

D^βA˜;y,x⁰x−x^0|β|a(y)dy

≤C

B

x−x0^−(n+m)

|β|<m

x0−y^m−|β|x−x0^|β|

×

|α|=m

D^αA_BMOa(y)dy

≤C

|α|=m

D^αA_BMO

B

x0−y

x−x0ⁿ⁺¹a(y)dy

≤C

|α|=m

D^αA_BMOx−x0⁻ⁿ⁻¹|B|^1/n−1/p+1.

(2.14)

ForII3, and by the vanishing moment ofa, we write, Ω(x−y)(x−y)^α

|x−y|^m

D^αA(y)−

D^αA

B

a(y)dy

= Ω(x−y)(x−y)^α

|x−y|^m −Ω x−x0

x−x0

α

x−x0^m+n−1

D^αA(y)− D^αA

B

a(y)dy.

(2.15)

Similar to the estimate ofII1, we obtain

II3≤C

|α|=m



 r

x−x⁰ⁿ⁺¹+ r^γ x−x0^n+γ





×

B

x0−yD^αA(y)− D^αA

Ba(y)dy

≤C

|α|=m

D^αA_BMO|B|^1−1/p



 r

x−x0ⁿ⁺¹+ r^γ x−x0^n+γ



.

(2.16)

...

Recalling thatp > n/(n+γ), therefore,

II≤ ∞ k=1

2^k+1B\2^kB

µ^A_Ω(a)(x)p

dx

≤C ∞ k=1

2k+1B\2kB



 r

x−x0ⁿ⁺¹+ r^γ x−x^0n+γ





p

|B|^p−1

×





|α|=m

D^αA(x)−

D^αA

2^k+1B





p

dx

+C





|α|=m

D^αA_BMO



p∞ k=1

2k+1B\2kB

x−x0^−p(n+1)|B|^{p(1+1/n−1/p)}dx

≤C





|α|=m

D^αA_BMO



p∞ k=1

2^{k(n−p−pn)}+2^{k(n−pn−pγ)}

≤C





|α|=m

D^αA_BMO



p

,

(2.17)

which, together with the estimate forI, yields the desired result. This finishes the proof ofTheorem 2.3.

Theorem2.4. Let0< p <∞,1< q <∞,n(1−1/q)≤α < n(1−1/q)+γ, andD^βA∈BMO(Rⁿ)for|β| =m. Then,µ^A_Ωis bounded fromHK˙q,D^α,pmA(Rⁿ)to K˙^α,pq (Rⁿ).

Proof. Let f ∈HK˙_q,D^α,pmA(Rⁿ) and f (x)=_∞

j=−∞λjaj(x) be the atomic decomposition forfas inDefinition 1.3. We write

µΩ^A(f )_˙

K^α,p_q (Rⁿ)≤C



 ^∞

k=−∞

2^kαp



 ^k−3

j=−∞

λjµΩ^A aj

χk

L^q





p



1/p

+C



 ^∞

k=−∞

2^kαp



 ^∞

j=k−2

λjµ_Ω^A aj

χk

L^q





p



1/p

≡I+II.

(2.18)

ForII, and by the boundedness ofµ^A_ΩonL^q(Rⁿ)(seeLemma 2.2), we have

II≤C



 ^∞

k=−∞

2^kαp



 ^∞

j=k−2

λjaj

L^q





p



1/p

≤C



 ^∞

k=−∞

2^kαp



 ^∞

j=k−2

λj2^−jα





p



1/p

≤C





















 ^∞

k=−∞

2^kαp ∞ j=k−2

λj^p2^−jαp





1/p

, 0< p≤1



 ^∞

k=−∞

2^kαp



 ^∞

j=k−2

λj^p2^−jαp/2







 ^∞

j=k−2

2^−jαp/2





p/p



1/p

, p >1

≤C





















 ^∞

j=−∞

λj^p

 ^j+2

k=−∞

2^(k−j)αp



λj^p2^−jαp





1/p

, 0< p≤1



 ^∞

j=−∞

λj^p

 ^j+2

k=−∞

2^(k−j)αp/2









1/p

, p >1

≤C



 ^∞

j=−∞

λj^p



1/p

≤Cf_HK˙_q,DmA^α,p (Rⁿ).

(2.19)

ForI, and similar to the proof ofTheorem 2.3, we have, forx∈Ck,j≤k−3,

µ^A_Ω aj

(x)≤C

|x|^−n−m−1Bj^1/n+|x|^{−n−m−γ}Bj^γ/n

× B_j

aj(y)Rm

A˜;x,ydy

+C

|β|=m

D^βA_BMO|x|⁻ⁿ⁻¹Bj^1/n

B_j

a(y)dy +C

|x|⁻ⁿ⁻¹Bj^1+1/n+|x|^−n−γBj^1+γ/n

×

|β|=m

B_j

D^βA(y)−

D^βA

B_ja(y)dy

≤C

2^−k(n+1)2^{j(1+n(1−1/q)−α)}+2^−k(n+γ)2^{j(γ+n(1−1/q)−α)}

×





|β|=m

D^βA(x)−

D^βA

B_k





+C

|α|=m

D^αA_BMO(k−j)

×

2^−k(n+1)2j(1+n(1−1/q)−α)+2^−k(n+γ)2j(γ+n(1−1/q)−α) (2.20)

...

thus,

I≤C



 ^∞

k=−∞

2^kαp



 ^k−3

j=−∞

λj2−k(n+1)+j(1+n(1−1/q)−α)+2−k(n+γ)+j(γ+n(1−1/q)−α)

×

|α|=m B_k

D^αA(x)−

D^αA

B_k^qdx 1/q



p



1/p

+C



 ^∞

k=−∞

2^kαp



 ^k−3

j=−∞

λj(k−j)

2^{−k(n+1)+j(1+n(1−1/q)−α)}

+2−k(n+γ)+j(γ+n(1−1/q)−α)

2^kn/q

|α|=m

D^αA_BMO



p



1/p

≡I1+I2.

(2.21)

To estimateI1andI2, we consider two cases.

Case1(0< p≤1). We have

I1≤C



 ^∞

k=−∞

2^kαp

k−3 j=−∞

|λj|^p

2^{[−k(n+1)+j(1+n(1−1/q)−α)]p}

+2[−k(n+γ)+j(γ+n(1−1/q)−α)]p 2^knp/q





|β|=m

D^βA_BMO



p



1/p

=C

|β|=m

D^βA_BMO

 ^∞

j=−∞

λj^{p ∞}

k=j+3

2(j−k)(1+n(1−1/q)−α)p

+2(j−k)(γ+n(1−1/q)−α)p



1/p

≤C

|β|=m

D^βA_BMO

 ^∞

j=−∞

λj^p



1/p

≤Cf_HK˙_q,DmA^α,p (Rⁿ).

(2.22)

Similarly,

I2≤Cf_HK˙_q,DmA^α,p (Rⁿ). (2.23)

Case2(p >1). By Hölder’s inequality, we deduce that

I1≤C

|β|=m

D^βA_BMO

 ^∞

j=−∞



 ^k−3

j=−∞

λj^p2(j−k)p(γ+n(1−1/q)−α)/2





×



 ^k−3

j=−∞

2^{(j−k)p(γ+n(1−1/q)−α)/2}





p/p



1/p

≤C



 ^∞

j=−∞

λj^p



1/p

≤Cf_HK˙_q,DmA^α,p (Rⁿ), I2≤Cf_HK˙_q,DmA^α,p (Rⁿ).

(2.24)

This finishes the proof of Theorem 2.

Remark2.5. Theorem 2.4also holds for nonhomogeneous Herz-type space.

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Liu Lanzhe: Department of Applied Mathematics, Hunan University, Changsha 410082, China

E-mail address:[email protected]

Mathematical Problems in Engineering

Special Issue on

Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios

Call for Papers

Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.

Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Di

ff

erential Equations,”

allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.

This proposed special edition of the Mathematical Prob-

lems in Engineering aims to provide a picture of the impor-

tance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.

Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophis- ticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.

Authors should follow the Mathematical Problems in Engineering manuscript format described at

http://www .hindawi.com/journals/mpe/. Prospective authors should

submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at

http://

mts.hindawi.com/

according to the following timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

José Roberto Castilho Piqueira,

Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, 05508-970 São Paulo, Brazil;

[email protected]

Elbert E. Neher Macau,

Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 12227-010 São Paulo, Brazil ; [email protected]

Celso Grebogi,

Center for Applied Dynamics Research, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK; [email protected]

Hindawi Publishing Corporation http://www.hindawi.com