Continuity for Multilinear Integral Operators on Some Hardy and Herz Type Spaces

Continuity for Multilinear Integral Operators on Some Hardy and Herz Type Spaces

Chen Qiong and Liu Lanzhe

College of Mathematics, Changsha University of Science and Technology, Changsha 410077, P.R. of China

e-mail : [email protected] Abstract

The continuity for some multilinear operators generated by certain integral operators and Lipschitz functions on some Hardy and Herz-type spaces are obtained. The operators in- clude Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.

Keywords: Multilinear operator; Lipschitz function; Hardy space; Herz space; Herz type Hardy space; Littlewood-Paley operator; Marcinkiewicz oper- ator; Bochner-Riesz operator.

2000 Mathematics Subject: 42B20, 42B25.

1 Introduction and Preliminaries

As the development of singular integral operators T, their commutators and multilinear operators have been well studied (see [1],[4-7]). From [8] and [9], we know that the commutators and multilinear operators generated byT and the BM O functions are bounded on L^p(Rⁿ) for 1 < p < ∞. Chanillo (see [2]) proves a similar result when T is replaced by the fractional integral operator. However, it was observed that the commutators and multilinear operators are not bounded, in general, fromH^p(Rⁿ) to L^p(Rⁿ) for 0< p≤1.

But, the boundedness holds if the BM O functions are replaced by the the Lipschitz functions (see [3], [11], [16] and [19]). This show the difference of the BM Ofunctions and the Lipschitz functions. The purpose of this paper is to establish the continuity properties for some multilinear operators generated by certain non-convolution type integral operators and Lipschitz functions on some Hardy and Herz-type spaces. The operators include Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.

First, let us introduce some notations(see [10], [17-21]). Throughout this paper, Q will denote a cube of Rⁿ with sides parallel to the axes. For a cube Q and a locally integrable function f, let fQ = |Q|⁻¹R

Qf(x)dx. Denote the Hardy spaces by H^p(Rⁿ). It is well known that H^p(Rⁿ)(0 < p ≤ 1) has the atomic decomposition characterization (see [20],[21]). Forβ >0, the Lipschitz spaceLip_β(Rⁿ) is the space of functions f such that (see [19])

||f||_Lipβ = sup

x,h∈Rⁿ, h>0

|f(x+h)−f(x)|/|h|^β <∞.

Definition 1.1 Let 0 < p, q < ∞, α ∈ R. For k ∈ Z, define B_k = {x ∈ Rⁿ :|x| ≤ 2^k} and C_k = B_k\Bk−1. Denote by χ_k the characteristic function of C_k and χ₀ the characteristic function of B₀.

(1) The homogeneous Herz space is defined by

K˙_q^α,p(Rⁿ) = {f ∈L^q_loc(Rⁿ\ {0}) :||f||K˙q^α,p <∞}, where

||f||K˙q^α,p =

" _∞ X

k=−∞

2^kαp||f χ_k||^p_Lq

#1/p

; (2) The nonhomogeneous Herz space is defined by

K_q^α,p(Rⁿ) = {f ∈L^q_loc(Rⁿ) :||f||_Kq^α,p <∞}, where

||f||_Kq^α,p =

" _∞ X

k=1

2^kαp||f χ_k||^p_Lq+||f χ₀||^p_Lq

#1/p

. Definition 1.2 Letα∈R, 0< p, q < ∞.

(1) The homogeneous Herz type Hardy space is defined by HK˙_q^α,p(Rⁿ) ={f ∈S⁰(Rⁿ) :G(f)∈K˙_q^α,p(Rⁿ)}, and

||f||_HK˙q^α,p =||G(f)||K˙q^α,p;

(2) The nonhomogeneous Herz type Hardy space is defined by HK_q^α,p(Rⁿ) ={f ∈S⁰(Rⁿ) :G(f)∈K_q^α,p(Rⁿ)}, and

||f||_HKq^α,p =||G(f)||_Kq^α,p; whereG(f) is the grand maximal function of f.

The Herz type Hardy spaces have the atomic decomposition characteriza- tion.

Definition 1.3 Let α∈R, 1< q < ∞. A function a(x) on Rⁿ is called a central (α, q)-atom (or a central (a, q)-atom of restrict type), if

1) Suppa ⊂B(0, r) for somer >0 (or for some r≥1), 2)||a||_L^q ≤ |B(0, r)|^−α/n,

3)R

a(x)x^γdx= 0 for |γ| ≤[α−n(1−1/q)].

Lemma 1.1(see[10],[18]]) Let 0< p <∞, 1< q <∞and α≥n(1−1/q).

A temperate distributionf belongs toHK˙_q^α,p(Rⁿ)(or HK_q^α,p(Rⁿ)) if and only if there exist central (α, q)-atoms(or central (α, q) -atoms of restrict type)a_j supported on B_j = B(0,2^j) and constants λ_j, P

j|λ_j|^p < ∞ such that f = P∞

j=−∞λ_ja_j(or f =P∞

j=0λ_ja_j)in theS⁰(Rⁿ) sense, and

||f||_HK˙^α,pq ( or ||f||_HKq^α,p)≈ X

j

|λ_j|^p

!1/p

.

2 Theorems

In this paper, we will study a class of multilinear operators related to some integral operators, whose definitions are follows.

Fixed 0 ≤ δ < n and ε > 0. Let m_i be the positive integers(i = 1,· · ·, l), m₁+· · ·+m_l=m and A_i be the functions on Rⁿ (i= 1,· · ·, l). Set

R_mi+1(A_i;x, y) =A_i(x)− X

|γ|≤mi

1

γ!D^γA_i(y)(x−y)^γ and

Q_mi+1(A_i;x, y) =R_mi(A_i;x, y)− X

|γ|=mi

1

γ!D^γA_i(x)(x−y)^γ. LetFt(x, y) define onRⁿ×Rⁿ×[0,+∞). Set

Ft(f)(x) = Z

Rⁿ

Ft(x, y)f(y)dy and

F_t^A(f)(x) = Z

Rⁿ

Ql

j=1R_mj+1(A_j;x, y)

|x−y|^m F_t(x, y)f(y)dy

for every bounded and compactly supported function f. Let H be the Ba- nach space H = {h : ||h|| < ∞} such that, for each fixed x ∈ Rⁿ, F_t(f)(x) and F_t^A(f)(x) may be viewed as a mapping from [0,+∞) to H. Then, the multilinear operator related toF_t is defined by

T^A(f)(x) =||F_t^A(f)(x)||,

whereF_t satisfies:

||F_t(x, y)|| ≤C|x−y|^−n+δ and

||F_t(y, x)−F_t(z, x)|| ≤C|y−z|^ε|x−z|^−n−ε+δ

if 2|y−z| ≤ |x−z|. Let T(f)(x) =||F_t(f)(x)||. We also consider the variant of T^A, which is defined by

T˜^A(f)(x) =||F˜_t^A(f)(x)||, where

F˜_t^A(f)(x) = Z

Rⁿ

Ql

i=1Q_mi+1(A_i;x, y)

|x−y|^m F_t(x, y)f(y)dy.

Note that whenm= 0,T^Ais just higher order commutator of the operators T andA(see [1],[12-14],[19]), while whenm >0, it is non-trivial generalizations of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors when A has derivatives of order m in BM O(Rⁿ)(see [4-6],[9]). The purpose of this paper is to prove the continuity properties of the multilinear operators T^A and ˜T^A on Hardy and Herz-type spaces. In Section 4, some examples of Theorems in this paper are given.

We shall prove the following theorems in Section 3.

Theorem 2.1 Let 0 < β ≤ 1, 0 ≤δ < n−lβ and D^γA_i ∈ Lip_β(Rⁿ) for allγ with |γ|=m_i and i= 1,· · ·, l.

(a) Suppose that T^A maps L^s(Rⁿ) continuously into L^r(Rⁿ) for any 1 <

r < n/µ and 1/s = 1/r − µ/n. If max(n/(n + β), n/(n + ε)) < p ≤ 1, 1/p−1/q= (δ+lβ)/n, then T^A mapsH^p(Rⁿ) continuously into L^q(Rⁿ).

(b) Suppose that ˜T^A maps L^s(Rⁿ) continuously into L^r(Rⁿ) for any 1 <

r < n/µ and 1/s = 1/r− µ/n. If 0 < β < min(1/l, ε/l), then ˜T^A maps H^n/(n+lβ)(Rⁿ) continuously into L^n/(n−δ)(Rⁿ).

Theorem 2.2 Let 0< β ≤1, 0≤δ < n−lβ, 0< p <∞, 1< q₁, q₂ <∞, 1/q₁ −1/q₂ = (δ+lβ)/n and D^γA_i ∈ Lip_β(Rⁿ) for all γ with |γ| = m_i and i= 1,· · ·, l.

(i) Suppose that T^A maps L^s(Rⁿ) continuously into L^r(Rⁿ) for any 1 <

r < n/µand 1/s= 1/r−µ/n. Ifn(1−1/q₁)≤α <min(n(1−1/q₁) +lβ, n(1− 1/q₁) +ε), thenT^A maps HK˙_q^α,p₁ (Rⁿ) continuously into ˙K_q^α,p₂ (Rⁿ).

(ii) Suppose that ˜T^A maps L^s(Rⁿ) continuously into L^r(Rⁿ) for any 1 <

r < n/µ and 1/s= 1/r−µ/n. If 0 < p≤ 1 and 0 < β < min(1/l, ε/l), then T˜^A maps HK˙q^n(1−1/q1 ^1)+lβ,p(Rⁿ) continuously into ˙Kq^n(1−1/q2 ^1)+lβ,p(Rⁿ).

Remark. Theorem 2 also hold for the nonhomogeneous Herz and Herz type Hardy space.

3 Proofs of Theorems

We begin with a preliminary lemma.

Lemma 3.1(see [6]) LetA be a function onRⁿ such that D^γA∈L^q_loc(Rⁿ) for|γ|=m and some q > n. Then

|R_m(A;x, y)| ≤C|x−y|^m X

|γ|=m

1

|Q(x, y)|˜ Z

Q(x,y)˜

|D^γA(z)|^qdz 1/q

,

where ˜Q(x, y) is the cube centered at x and having side length 5√

n|x−y|.

Proof of Theorem 2.1(a).It suffices to show that there exists a constant C >0 such that for everyH^p-atoma, there is

||T^A(a)||_L^q ≤C.

Without loss of generality, we may assumel = 2. Leta be a H^p-atom, that is thatasupported on a cubeQ=Q(x0, d),||a||L^∞ ≤ |Q|^−1/pandR

Rⁿa(x)x^ηdx= 0 for|η| ≤[n(1/p−1)]. We write

Z

Rⁿ

|T^A(a)(x)|^qdx= Z

|x−x₀|≤2d

+ Z

|x−x₀|>2d

|T^A(a)(x)|^qdx=I₁+I₂. For I₁, taking q₁ > q and 1 < p₁ < n/(δ + 2β) such that 1/p₁ − 1/q₁ = (δ+ 2β)/n, by H¨older’s inequality and the (L^p1, L^q1)-boundedness of T^A, we get

I₁ ≤C||T^A(a)||^q_Lq1|2Q|^1−q/q1 ≤C||a||^q_Lp1|Q|^1−q/q1 ≤C.

To estimate I₂, we need to estimate T^A(a)(x) for x ∈ (2Q)^c. Let ˜A_i(x) = A_i(x)−P

|γ|=m_i 1

γ!(D^γA_i)_Qx^γ. ThenR_mi(A_i;x, y) = R_mi( ˜A_i;x, y) andD^γA˜_i(y) = D^γA_i(y)−(D^γA_i)_Q. We write, by the vanishing moment of a,

F_t^A(a)(x)

= Z

Rⁿ

F_t(x, y)

|x−y|^m − F_t(x, x₀)

|x−x₀|^m

R_m1( ˜A₁;x, y)R_m2( ˜A₂;x, y)a(y)dy +

Z

Rⁿ

F_t(x, x₀)

|x−x₀|^m[R_m1( ˜A₁;x, y)−R_m1( ˜A₁;x, x₀)]R_m2( ˜A₂;x, y)a(y)dy +

Z

Rⁿ

F_t(x, x₀)

|x−x₀|^m[R_m2( ˜A₂;x, y)−R_m2( ˜A₂;x, x₀)]R_m1( ˜A₁;x, x₀)a(y)dy

− X

|γ2|=m2

1 γ₂!

Z

Rⁿ

Rm1( ˜A1;x, y)D^γ2A˜2(y)(x−y)^γ2

|x−y|^m F_t(x, y)a(y)dy

− X

|γ₁|=m₁

1 γ₁!

Z

Rⁿ

R_m2( ˜A₂;x, y)D^γ1A˜₁(y)(x−y)^γ1

|x−y|^m F_t(x, y)a(y)dy

+ X

|γ₁|=m₁,|γ₂|=m₂

1 γ₁!

1 γ₂!

Z

Rⁿ

D^γ1A˜₁(y)D^γ2A˜₂(y)(x−y)^γ1+γ2

|x−y|^m F_t(x, y)a(y)dy;

By Lemma 3.1 and the following inequality

|b(x)−b_Q| ≤ 1

|Q|

Z

Q

||b||_Lipβ|x−y|^βdy≤ ||b||_Lipβ(|x−x₀|+d)^β, we get

|R_mi( ˜A_i;x, y)| ≤ X

|γ|=m_i

||D^γA_i||_Lipβ(|x−y|+d)^mi+β;

By the formula (see [6]):

R_mi( ˜A_i;x, y)−R_mi( ˜A_i;x, x₀) = X

|η|<m

1

η!R_mi−|η|(D^ηA˜_i;x₀, y)(x−x₀)^η, and note that |x−y| ∼ |x−x₀|for y ∈Q and x∈Rⁿ\2Q, we obtain

|T^A(a)(x)|=||F_t^A(a)(x)|| ≤C

2

Y

i=1



 X

|γ_i|=m_i

||D^γiA_i||_Lipβ



 Z

Q

|y−x₀|

|x−x₀|^{n+1−δ−2β} + |y−x₀|^ε

|x−x₀|^{n+ε−δ−2β} + |y−x₀|^β

|x−x₀|^n−δ−β + |y−x₀|^2β

|x−x₀|^n−δ

|a(y)|dy

≤ C

2

Y

i=1



 X

|γ_i|=m_i

||D^γiA_i||_Lipβ





|Q|^1/n+1−1/p

|x−x₀|^{n+1−δ−2β} + |Q|^ε/n+1−1/p

|x−x₀|^{n+ε−δ−2β} + |Q|^β/n+1−1/p

|x−x₀|^n−δ−β + |Q|^{2β/n+1−1/p}

|x−x₀|^n−δ

; Thus

I₂ ≤

∞

X

k=1

Z

2^k+1Q\2^kQ

|T^A(a)(x)|^qdx

≤ C





2

Y

i=1



 X

|γ_i|=m_i

||D^γiA_i||_Lipβ









q

×

∞

X

k=1

2kqn(1/p−(n+1)/n)

+ 2kqn(1/p−(n+ε)/n)

+ 2kqn(1/p−(n+β)/n)

≤ C





2

Y

i=1



 X

|γi|=mi

||D^γiA_i||_Lipβ









q

≤C,

which together with the estimate forI₁ yields the desired result.

(b). Without loss of generality, we may assume l = 2. It is only to prove that there exists a constant C > 0 such that for every H^n/(n+2β)-atom a sup- ported onQ=Q(x0, d), there is

||T˜^A(a)||_L^n/(n−δ) ≤C.

We write Z

Rⁿ

|T˜^A(a)(x)|^n/(n−δ)dx= Z

|x−x₀|≤2d

+ Z

|x−x₀|>2d

|T˜^A(a)(x)|^n/(n−δ)dx:=J₁+J₂.

For J₁, by the (L^p, L^q)-boundedness of ˜T^A for 1 < p < n/(δ + 2β), q >

n/(n−δ)and 1/q= 1/p−(δ+ 2β)/n, we get

J₁ ≤C||T˜^A(a)||^n/(n−δ)_Lq |2Q|1−n/((n−δ)q)≤C||a||^n/(n−δ)_Lp |Q|1−n/((n−δ)q) ≤C.

To obtain the estimate ofJ₂, we denote ˜A_i(x) = A_i(x)−P

|γ|=mi

1

γ!(D^γA_i)_2Qx^γ. Then Qmi(Ai;x, y) =Qmi( ˜Ai;x, y). We write, by the vanishing moment of a andQ_mi+1(A_i;x, y) =R_mi(A_i;x, y)−P

|γ|=mi

1

γ!D^γA_i(x)(x−y)^γ, forx∈(2Q)^c, F˜_t^A(a)(x)

= Z

Rⁿ

F_t(x, y)

|x−y|^m − F_t(x, x₀)

|x−x₀|^m

R_m1( ˜A₁;x, y)R_m2( ˜A₂;x, y)a(y)dy +

Z

Rⁿ

F_t(x, x₀)

|x−x₀|^m[R_m1( ˜A₁;x, y)−R_m1( ˜A₁;x, x₀)]R_m2( ˜A₂;x, y)a(y)dy +

Z

Rⁿ

F_t(x, x₀)

|x−x₀|^m[R_m2( ˜A₂;x, y)−R_m2( ˜A₂;x, x₀)]R_m1( ˜A₁;x, x₀)a(y)dy

− X

|γ₂|=m₂

Z

Rⁿ

F_t(x, y)(x−y)^γ2

|x−y|^m − F_t(x, x₀)(x−x₀)^γ2

|x−x₀|^m

×R_m1( ˜A₁;x, y)D^γ2A˜₂(x)a(y)dy

− X

|γ₂|=m₂

Z

Rⁿ

F_t(x, x₀)(x−x₀)^γ2

|x−x₀|^m [R_m1( ˜A₁;x, y)−R_m1( ˜A₁;x, x₀)]

×D^γ2A˜₂(x)a(y)dy

− X

|γ1|=m1

Z

Rⁿ

Ft(x, y)(x−y)^γ1

|x−y|^m − Ft(x, x0)(x−x0)^γ1

|x−x₀|^m

×R_m2( ˜A₂;x, y)D^γ1A˜₁(x)a(y)dy

− X

|γ₁|=m₁

Z

Rⁿ

F_t(x, x₀)(x−x₀)^γ1

|x−x₀|^m [R_m2( ˜A₂;x, y)−R_m2( ˜A₂;x, x₀)]

×D^γ1A˜₁(x)a(y)dy

+ X

|γ1|=m1,|γ2|=m2

Z

Rⁿ

F_t(x, y)(x−y)^γ1+γ2

|x−y|^m − F_t(x, x₀)(x−x₀)^γ1+γ2

|x−x0|^m

×D^γ1A˜₁(x)D^γ2A˜₂(x)a(y)dy,

then, similar to the proof of (a), we obtain

|T˜^A(a)(x)|

≤ C

2

Y

i=1



 X

|γi|=mi

||D^γiA_i||_Lipβ



 Z

Q

|y−x₀|

|x−x0|^{n+1−δ−2β} + |y−x₀|^ε

|x−x0|^{n+ε−δ−2β}

|a(y)|dy

≤ C

2

Y

i=1



 X

|γ_i|=m_i

||D^γiA_i||_Lipβ





|Q|^(1−2β)/n

|x−x₀|^{n+1−δ−2β} + |Q|^(ε−2β)/n

|x−x₀|^{n+ε−δ−2β}

,

thus

J2 ≤C





2

Y

i=1



 X

|γi|=mi

||D^γiAi||Lip_β









n/(n−δ)

∞

X

k=1

[2kn(2β−1)/(n−δ)

+2kn(2β−ε)/(n−δ)

]≤C,

which together with the estimate for J₁ yields the desired result. This com- pletes the proof of Theorem 2.1.

Proof of Theorem 2.2(i). Without loss of generality, we may assume l = 2. Let f ∈ HK˙_q^α,p₁ (Rⁿ) and f(x) = P∞

j=−∞λ_ja_j(x) be the atomic decom- position forf as in Lemma 1.1. We write

||T^A(f)||^p_˙

Kq^α,p2

≤

∞

X

k=−∞

2^kαp

k−3

X

j=−∞

|λ_j|||T^A(a_j)χ_k||_L^q2

!^p

+

∞

X

k=−∞

2^kαp

∞

X

j=k−2

|λ_j|||T^A(a_j)χ_k||_L^q2

!p

=K₁+K₂.

ForK₂, by the (L^q1, L^q2) boundedness of T^A, we have

K₂ ≤ C

∞

X

k=−∞

2^kαp

∞

X

j=k−2

|λ_j|||a_j||_L^q1

!p

≤







CP∞

j=−∞|λ_j|^p Pj+2

k=−∞2^(k−j)αp

, 0< p≤1 CP∞

j=−∞|λ_j|^p Pj+2

k=−∞2^(k−j)αp/2 Pj+2

k=−∞2^{(k−j)αp0/2}p/p⁰

, p >1

≤ C

∞

X

j=−∞

|λ_j|^p ≤C||f||^p

HK˙q^α,p1

.

ForK₁, similar to the proof of Theorem 2.1(a), we get, forx∈C_k, j ≤k−3,

|T^A(a_j)(x)|

≤ C

|Bj|^1/n

|x|^{n+1−δ−2β} + |Bj|^ε/n

|x|^{n+ε−δ−2β} + |Bj|^β/n

|x|^n−δ−β + |Bj|^2β/n

|x|^n−δ Z

Rⁿ

|a_j(y)|dy

≤ C

2j(1+n(1−1/q₁)−α)

|x|^{n+1−δ−2β} + 2j(ε+n(1−1/q₁)−α)

|x|^{n+ε−δ−2β} +2j(β+n(1−1/q₁)−α)

|x|^{n−δ−β−n}

,

thus

||T^A(a_j)χ_k||_L^q2

≤ C2^−kα 2(j−k)(1+n(1−1/q1)−α)

+ 2(j−k)(ε+n(1−1/q1)−α)

+ 2(j−k)(β+n(1−1/q1)−α)

; To be simply, denote W(j, k) = 2(j−k)(1+n(1−1/q1)−α) + 2(j−k)(ε+n(1−1/q1)−α) + 2(j−k)(β+n(1−1/q₁)−α) and recall that α <min(n(1−1/q₁) +β, n(1−1/q₁) +ε), then

K₁ ≤ C

∞

X

k=−∞

k−3

X

j=−∞

|λ_j|W(j, k)

!^p

≤

( CP∞

j=−∞|λ_j|^pP∞

k=j+3W(j, k)^p, 0< p≤1 CP∞

j=−∞|λ_j|^ph P∞

k=j+3W(j, k)^p/2i h P∞

k=j+3W(j, k)^p0/2ip/p⁰

, p >1

≤ C

∞

X

j=−∞

|λ_j|^p ≤C||f||^p

HK˙q^α,p1

.

These yield the desired result.

(ii). Without loss of generality, assumel = 2. Letf ∈HK˙q^n(1−1/q1 ^1)+2β,p(Rⁿ) and f(x) = P∞

j=−∞λ_ja_j(x) be the atomic decomposition for f as in Lemma

1.1. Write

||T˜^A(f)||^p_˙

Kq^n(1−1/q2 ^1)+2β,p

≤

∞

X

k=−∞

2^{kp(n(1−1/q1)+2β)}

k−3

X

j=−∞

|λ_j|||T˜^A(a_j)χ_k||_L^q2

!^p

+

∞

X

k=−∞

2^{kp(n(1−1/q1)+2β)}

∞

X

j=k−2

|λ_j|||T˜^A(a_j)χ_k||_L^q2

!p

= L₁ +L₂.

ForL₂, by the (L^q1, L^q2) boundedness of ˜T^A, we get L2 ≤ C

∞

X

k=−∞

2^{kp(n(1−1/q1)+2β)}

∞

X

j=k−2

|λj|||aj||L^q1

!p

≤ C

∞

X

j=−∞

|λj|^p

j+2

X

k=−∞

2(k−j)p(n(1−1/q1)+2β)

!

≤ C

∞

X

j=−∞

|λ_j|^p ≤C||f||^p

HK˙q^n(1−1/q1 ^1)+2β,p

.

ForL₁, similar to the proof of Theorem 2.1(b), we get, for x∈C_k,j ≤k−3,

|T˜^A(a)(x)| ≤ C

|B_j|^1/n

|x|^{n+1−δ−2β} + |B_j|^ε/n

|x|^{n+ε−δ−2β} Z

Rⁿ

|aj(y)|dy

≤ C

2^j(1−2β)

|x|^{n+1−δ−2β} + 2^j(ε−2β)

|x|^{n+ε−δ−2β}

, thus

L₁ ≤ C

∞

X

k=−∞

2^{kp(n(1−1/q1)+2β)}

k−3

X

j=−∞

|λ_j|^p 2^j(1−2β)

2k(n+1−δ−2β) + 2^j(ε−2β) 2k(n+ε−δ−2β)

!^p

2^knp/q2

≤ C

∞

X

j=−∞

|λ_j|^p

∞

X

k=j+3

2p(1−2β)(j−k)

+ 2p(ε−2β)(j−k)

≤ C

∞

X

j=−∞

|λj|^p ≤C||f||^p

HK˙q^n(1−1/q1 ^1)+2β,p

.

These yield the desired result and finish the proof of Theorem 2.2.

4 Examples

Now we give some examples including Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.

Example 1 Littlewood-Paley operator.

Fixed ε >0 and µ >(3n+ 2)/n. Let ψ be a fixed function which satisfies:

(1) R

Rⁿψ(x)dx= 0,

(2) |ψ(x)| ≤C(1 +|x|)⁻⁽ⁿ⁺¹⁾,

(3) |ψ(x+y)−ψ(x)| ≤C|y|^ε(1 +|x|)^−(n+1+ε) when 2|y|<|x|;

We denote that Γ(x) ={(y, t)∈R₊ⁿ⁺¹ :|x−y|< t} and the characteristic function of Γ(x) by χ_Γ(x). The Littlewood-Paley multilinear operators are defined by

g_ψ^A(f)(x) = Z ∞

0

|F_t^A(f)(x)|²dt t

1/2

,

S_ψ^A(f)(x) = Z Z

Γ(x)

|F_t^A(f)(x, y)|²dydt tⁿ⁺¹

1/2

and

g_µ^A(f)(x) =

"

Z Z

Rⁿ⁺¹₊

t t+|x−y|

nµ

|F_t^A(f)(x, y)|²dydt tⁿ⁺¹

#1/2

,

where

F_t^A(f)(x) = Z

Rⁿ

Ql

j=1R_mj+1(A_j;x, y)

|x−y|^m ψ_t(x−y)f(y)dy, F_t^A(f)(x, y) =

Z

Rⁿ

Ql

j=1R_mj+1(A_j;x, z)

|x−z|^m f(z)ψ_t(y−z)dz

and ψ_t(x) = t⁻ⁿψ(x/t) for t > 0. The variants of g^A_ψ, S_ψ^A and g^A_µ are defined by

˜

g_ψ^A(f)(x) = Z ∞

0

|F˜_t^A(f)(x)|²dt t

1/2

,

S˜_ψ^A(f)(x) = Z Z

Γ(x)

|F˜_t^A(f)(x, y)|²dydt tⁿ⁺¹

1/2

and

˜

g_µ^A(f)(x) =

"

Z Z

Rⁿ⁺¹₊

t t+|x−y|

nµ

|F˜_t^A(f)(x, y)|²dydt tⁿ⁺¹

#1/2

,

where

F˜_t^A(f)(x) = Z

Rⁿ

Ql

j=1Q_mj+1(A_j;x, y)

|x−y|^m ψ_t(x−y)f(y)dy and

F˜_t^A(f)(x, y) = Z

Rⁿ

Ql

j=1Q_mj+1(A_j;x, z)

|x−z|^m ψ_t(y−z)f(z)dz.

Set F_t(f)(y) =f ∗ψ_t(y). We also define that g_ψ(f)(x) =

Z ∞ 0

|F_t(f)(x)|²dt t

1/2

,

S_ψ(f)(x) = Z Z

Γ(x)

|F_t(f)(y)|²dydt tⁿ⁺¹

1/2

and

g_µ(f)(x) = Z Z

Rⁿ⁺¹₊

t t+|x−y|

nµ

|F_t(f)(y)|²dydt tⁿ⁺¹

!1/2

, which are the Littlewood-Paley operators (see [21]). LetH be the space

H = (

h:||h||= Z ∞

0

|h(t)|²dt/t 1/2

<∞ )

or

H=







h:||h||= Z Z

Rⁿ⁺¹₊

|h(y, t)|²dydt/tⁿ⁺¹

!1/2

<∞





 ,

then, for each fixedx ∈Rⁿ, F_t^A(f)(x) and F_t^A(f)(x, y) may be viewed as the mapping from [0,+∞) to H, and it is clear that

g_ψ^A(f)(x) = ||F_t^A(f)(x)||, gψ(f)(x) =||Ft(f)(x)||, S_ψ^A(f)(x) =

χ_Γ(x)F_t^A(f)(x, y)

, S_ψ(f)(x) =

χ_Γ(x)F_t(f)(y) and

g_µ^A(f)(x) =

t t+|x−y|

nµ/2

F_t^A(f)(x, y) ,

g_µ(f)(x) =

t t+|x−y|

nµ/2

F_t(f)(y) .

It is easily to see thatg_ψ,S_ψ andg_µsatisfy the conditions of Theorem 2.1 and 2.2, thus Theorem 2.1 and 2.2 hold forg_ψ^A and ˜g^A_ψ,S_ψ^A and ˜S_ψ^A,g^A_µ and ˜g_µ^A.

Example 2 Marcinkiewicz operator.

Fixed Fixλ >max(1,2n/(n+ 2)) and 0< γ ≤1. Let Ω be homogeneous of degree zero on Rⁿ with R

Sⁿ⁻¹Ω(x⁰)dσ(x⁰) = 0. Assume that Ω∈ Lip_γ(Sⁿ⁻¹).

The Marcinkiewicz multilinear operators are defined by µ^A_Ω(f)(x) =

Z ∞ 0

|F_t^A(f)(x)|²dt t³

1/2

,

µ^A_S(f)(x) = Z Z

Γ(x)

|F_t^A(f)(x, y)|²dydt tⁿ⁺³

1/2

and

µ^A_λ(f)(x) =

"

Z Z

Rⁿ⁺¹₊

t t+|x−y|

nλ

|F_t^A(f)(x, y)|²dydt tⁿ⁺³

#1/2

,

where

F_t^A(f)(x) = Z

|x−y|≤t

Ql

j=1R_mj+1(A_j;x, y)

|x−y|^m

Ω(x−y)

|x−y|ⁿ⁻¹f(y)dy and

F_t^A(f)(x, y) = Z

|y−z|≤t

Ql

j=1Rmj+1(Aj;y, z)

|y−z|^m

Ω(y−z)

|y−z|ⁿ⁻¹f(z)dz;

The variants ofµ^A_Ω, µ^A_S and µ^A_λ are defined by

˜

µ^A_Ω(f)(x) = Z ∞

0

|F˜_t^A(f)(x)|²dt t³

1/2

,

˜

µ^A_S(f)(x) = Z Z

Γ(x)

|F˜_t^A(f)(x, y)|²dydt tⁿ⁺³

1/2

and

˜

µ^A_λ(f)(x) =

"

Z Z

Rⁿ⁺¹₊

t t+|x−y|

nλ

|F˜_t^A(f)(x, y)|²dydt tⁿ⁺³

#1/2

,

where

F˜_t^A(f)(x) = Z

|x−y|≤t

Ql

j=1Q_mj+1(A_j;x, y)

|x−y|^m

Ω(x−y)

|x−y|ⁿ⁻¹f(y)dy and

F˜_t^A(f)(x, y) = Z

|y−z|≤t

Ql

j=1Q_mj+1(A_j;y, z)

|y−z|^m

Ω(y−z)

|y−z|ⁿ⁻¹f(z)dz.

Set

Ft(f)(x) = Z

|x−y|≤t

Ω(x−y)

|x−y|ⁿ⁻¹f(y)dy;

We also define that

µΩ(f)(x) = Z ∞

0

|Ft(f)(x)|²dt t³

1/2

,

µ_S(f)(x) = Z Z

Γ(x)

|F_t(f)(y)|²dydt tⁿ⁺³

1/2

and

µ_λ(f)(x) = Z Z

Rⁿ⁺¹₊

t t+|x−y|

nλ

|F_t(f)(y)|²dydt tⁿ⁺³

!1/2

, which are the Marcinkiewicz operators(see [22]). Let H be the space

H = (

h:||h||= Z ∞

0

|h(t)|²dt/t³ 1/2

<∞ )

or

H=







h:||h||= Z Z

Rⁿ⁺¹₊

|h(y, t)|²dydt/tⁿ⁺³

!1/2

<∞





 . Then, it is clear that

µ^A_Ω(f)(x) =||F_t^A(f)(x)||, µ_Ω(f)(x) =||F_t(f)(x)||, µ^A_S(f)(x) =

χ_Γ(x)F_t^A(f)(x, y)

, µ_S(f)(x) =

χ_Γ(x)F_t(f)(y) and

µ^A_λ(f)(x) =

t t+|x−y|

nλ/2

F_t^A(f)(x, y) ,

µ_λ(f)(x) =

t t+|x−y|

nλ/2

F_t(f)(y) .

It is easily to see thatµ_Ω,µ_S andµ_λ satisfy the conditions of Theorem 2.1 and 2.2, thus Theorem 2.1 and 2.2 hold forµ^A_Ω and ˜µ^A_Ω,µ^A_S and ˜µ^A_S, µ^A_λ and ˜µ^A_λ.

Example 3 Bochner-Riesz operator .

Let δ > (n−1)/2, B_t^δ(f)(ξ) = (1ˆ −t²|ξ|²)^δ₊fˆ(ξ) and B_t^δ(z) = t⁻ⁿB^δ(z/t) fort >0. Set

F_δ,t^A(f)(x) = Z

Rⁿ

Ql

j=1R_mj+1(A_j;x, y)

|x−y|^m B_t^δ(x−y)f(y)dy and

F˜_δ,t^A(f)(x) = Z

Rⁿ

Ql

j=1Q_mj+1(A_j;x, y)

|x−y|^m B_t^δ(x−y)f(y)dy.

The maximal Bochner-Riesz multilinear operator and its the variants are de- fined by

B^A_δ,∗(f)(x) = sup

t>0

|B_δ,t^A(f)(x)| and ˜B_δ,∗^A(f)(x) = sup

t>0

|B˜_δ,t^A(f)(x)|.

We also define that

Bδ,∗(f)(x) = sup

t>0

|B_t^δ(f)(x)|,

which is the maximal Bochner-Riesz operator(see [15]). Let H be the space H={h:||h||= sup

t>0

|h(t)|<∞}, then

B_δ,∗^A(f)(x) =||B_δ,t^A(f)(x)||, B_∗^δ(f)(x) =||B_t^δ(f)(x)||.

It is easily to see thatB_δ,∗ satisfies the conditions of Theorem 2.1 and 2.2, thus Theorem 2.1 and 2.2 hold forB^A_δ,∗ and ˜B_δ,∗^A.

4 Open problem

In this paper, the boundedness properties of the multilinear operators gen- erated by certain non-convolution type integral operators and Lipschitz func- tions on some Hardy and Herz-type spaces are obtained. The operators include Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz oper- ator.

The open problem is to study the boundedness of the multilinear op- erators generated by the non-convolution type integral operators and others locally integrable functions on others spaces.

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