CONTINUITY OF MULTILINEAR OPERATORS ON TRIEBEL-LIZORKIN SPACES

CONTINUITY OF MULTILINEAR OPERATORS ON TRIEBEL-LIZORKIN SPACES

LANZHE LIU

Received 4 February 2006; Revised 20 September 2006; Accepted 28 September 2006

The continuity of some multilinear operators related to certain convolution operators on the Triebel-Lizorkin space is obtained. The operators include Littlewood-Paley operator and Marcinkiewicz operator.

Copyright © 2006 Lanzhe Liu. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetTbe the Calder ´on-Zygmund singular integral operator, a well-known result of Coif- man et al. (see [6]) states that the commutator [b,T](f)=T(b f)−bT(f) (whereb∈ BMO) is bounded onL^p(Rⁿ) (1< p <∞); Chanillo (see [1]) proves a similar result when T is replaced by the fractional integral operator; in [8,9], these results on the Triebel- Lizorkin spaces and the caseb∈Lipβ(where Lipβis the homogeneous Lipschitz space) are obtained. The main purpose of this paper is to study the continuity of some multi- linear operators related to certain convolution operators on the Triebel-Lizorkin spaces.

In fact, we will obtain the continuity on the Triebel-Lizorkin spaces for the multilinear operators only under certain conditions on the size of the operators. As the applications, the continuity of the multilinear operators related to the Littlewood-Paley operator and Marcinkiewicz operator on the Triebel-Lizorkin spaces are obtained.

2. Notations and results

Throughout this paper,Q will denote a cube ofRⁿwith side parallel to the axes, and for a cubeQ, let f_Q= |Q|⁻¹

Qf(x)dx and f^#(x)=sup_x∈Q|Q|⁻¹

Q|f(y)−f_Q|d y. For 1≤r <∞and 0≤δ < n, let

Mδ,r(f)(x)=sup

x∈Q

1

|Q|^1−δr/n

Q

f(y)^rd y 1/r

, (2.1)

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 58473, Pages1–11 DOI 10.1155/JIA/2006/58473

we denoteMδ,r(f)=Mr(f) ifδ=0, which is the Hardy-Littlewood maximal function whenr=1 (see [10]). Forβ >0 andp >1, let ˙Fp^β,∞be the homogeneous Triebel-Lizorkin space, and let the Lipschitz space ˙∧βbe the space of functions f such that

f_∧˙_β= sup

x,h∈Rⁿ,h=0

Δ^[β]+1_h f(x)

|h|^β<∞ , (2.2)

whereΔ^khdenotes thekth difference operator (see [9]).

We are going to study the multilinear operator as follows.

Letmbe a positive integer and letAbe a function onRⁿ. We denote Rm+1(A;x,y)=A(x)−

|α|≤m

1

α!D^αA(y)(x−y)^α. (2.3) Definition 2.1. LetF(x,t) define onRⁿ×[0, +∞), denote

Ft(f)(x)=

RⁿF(x−y,t)f(y)d y, F_t^A(f)(x)=

Rⁿ

Rm+1(A;x,y)

|x−y|^m F(x−y,t)f(y)d y.

(2.4)

LetHbe the Hilbert spaceH= {h:h<∞}such that, for each fixedx∈Rⁿ,F_t(f)(x) andF_t^A(f)(x) may be viewed as a mapping from [0, +∞) to H. Then, the multilinear operators related toFtis defined by

T^A(f)(x)=F_t^A(f)(x); (2.5) and also defineT(f)(x)= F_t(f)(x).

In particular, consider the following two sublinear operators.

Definition 2.2. Fixε >0,n > δ≥0. Letψbe a fixed function which satisfies the following properties:

(1)ψ(x)dx=0;

(2)|ψ(x)| ≤C(1 +|x|)^{−(n+1−δ)};

(3)|ψ(x+y)−ψ(x)| ≤C|y|^ε(1 +|x|)^{−(n+1+ε−δ)}when 2|y|<|x|. The multilinear Littlewood-Paley operator is defined by

g_δ^A(f)(x)= _∞

0

F_t^A(f)(x)²dt t

1/2

, (2.6)

where

F_t^A(f)(x)=

Rⁿ

Rm+1(A;x,y)

|x−y|^m ψt(x−y)f(y)d y (2.7)

andψt(x)=t^−n+δψ(x/t) fort >0. Denote thatFt(f)=ψt∗f, and also define that gδ(f)(x)=

_∞

0

Ft(f)(x)²dt t

1/2

, (2.8)

which is the Littlewood-Paleygfunction whenδ=0 (see [11]).

LetHbe the spaceH= {h:h =(₀^∞|h(t)|²dt/t)^1/2<∞}, then, for each fixedx∈Rⁿ, F_t^A(f)(x) may be viewed as a mapping from [0, +∞) toH, and it is clear that

gδ(f)(x)=Ft(f)(x), g_δ^A(f)(x)=F_t^A(f)(x). (2.9) Definition 2.3. Let 0≤δ < n, 0< γ≤1 andΩbe homogeneous of degree zero onRⁿ such that_Sn−1Ω(x)dσ(x)=0. Assume thatΩ∈Lip_γ(Sⁿ⁻¹), that is, there exists a con- stantM >0 such that for anyx,y∈Sⁿ⁻¹,|Ω(x)−Ω(y)| ≤M|x−y|^γ. The multilinear Marcinkiewicz operator is defined by

μ^A_δ(f)(x)= _∞

0

F_t^A(f)(x)²dt t³

1/2

, (2.10)

where

F_t^A(f)(x)=

|x−y|≤t

Ω(x−y)

|x−y|^n−1−δ

R_m+1(A;x,y)

|x−y|^m f(y)d y; (2.11) denote

F_t(f)(x)=

|x−y|≤t

Ω(x−y)

|x−y|^n−1−δ f(y)d y, (2.12) and also define that

μδ(f)(x)= _∞

0

Ft(f)(x)²dt t³

1/2

, (2.13)

which is the Marcinkiewicz operator whenδ=0 (see [12]).

LetHbe the spaceH= {h:h =(₀^∞|h(t)|²dt/t³)^1/2<∞}. Then, it is clear that μδ(f)(x)=Ft(f)(x), μ^A_δ(f)(x)=F_t^A(f)(x). (2.14) It is clear that Definitions2.2and 2.3are the particular examples ofDefinition 2.1.

Note that whenm=0,T^Ais just the commutator ofFt andA, while whenm >0, it is nontrivial generalizations of the commutators. It is well known that multilinear oper- ators are of great interest in harmonic analysis and have been widely studied by many authors (see [2–5,7]). The main purpose of this paper is to study the continuity for the multilinear operators on the Triebel-Lizorkin spaces. We will prove the following theo- rems inSection 3.

Theorem 2.4. Letg_δ^A be the multilinear Littlewood-Paley operator as inDefinition 2.2. If 0< β <min(1,ε) andD^αA∈_∧˙_βfor|α| =m, then

(a)g_δ^AmapsL^p(Rⁿ) continuously into ˙Fq^β,∞(Rⁿ), for 1< p < n/δand 1/q=1/ p−δ/n;

(b)g_δ^A maps L^p(Rⁿ) continuously into L^q(Rⁿ) for 1< p < n/(δ+β) and 1/ p−1/q= (δ+β)/n.

Theorem 2.5. Letμ^A_δ be the multilinear Marcinkiewiz operator as inDefinition 2.3. If 0<

β <min(1/2,γ) andD^αA∈_∧˙_βfor|α| =m, then

(a)μ^A_δ mapsL^p(Rⁿ) continuously into ˙Fq^β,∞(Rⁿ) for 1< p < n/δand 1/q=1/ p−δ/n, (b)μ^A_δ mapsL^p(Rⁿ) continuously intoL^q(Rⁿ) for 1< p < n/(δ+β) and 1/ p−1/q=

(δ+β)/n.

3. Main theorem and proof We first prove a general theorem.

Theorem 3.1 (main theorem). Let 0≤δ < n, 0< β <1, andD^αA∈_∧˙_βfor|α| =m. Sup- poseFt,T, andT^Aare the same as inDefinition 2.1, ifTis bounded fromL^p(Rⁿ) toL^q(Rⁿ) for 1< p < n/δand 1/q=1/ p−δ/n, andTsatisfies the following size condition:

F_t^A(f)(x)−F_t^A(f) x0≤C

|α|=m

D^αA_∧˙β|Q|^β/nMδ,1f(x) (3.1)

for any cubeQwith suppf ⊂(2Q)^candx∈Q, then

(a)T^Ais bounded fromL^p(Rⁿ) to ˙Fq^β,∞(Rⁿ) for 1< p < n/δand 1/q=1/ p−δ/n, (b)T^A is bounded fromL^p(Rⁿ) toL^q(Rⁿ) for 1< p < n/(δ+β) and 1/q=1/ p−(δ+

β)/n.

To prove the theorem, we need the following lemmas.

Lemma 3.2 (see [9]). For 0< β <1, 1< p <∞, fF˙^β,p^∞ ≈

sup

Q

1

|Q|^1+β/n

Q

f(x)−fQdx

L^p

≈ sup

·∈Q

inf

c

1

|Q|^1+β/n

Q

f(x)−cdx

L^p

.

(3.2)

Lemma 3.3 (see [9]). For 0< β <1, 1≤p≤ ∞, f_∧˙_β≈sup

Q

1

|Q|^1+β/n

Q

f(x)−fQdx

≈sup

Q

1

|Q|^β/n 1

|Q|

Q

f(x)₋f_Q^pdx 1/ p

.

(3.3)

Lemma 3.4 (see [1,2]). Suppose that 1≤r < p < n/δand 1/q=1/ p−δ/n. Then

M_δ,r(f)_Lq≤CfL^p. (3.4)

Lemma 3.5 (see [5]). LetAbe a function onRⁿandD^αA∈L^q(Rⁿ) for|α| =mand some q > n. Then

Rm(A;x,y)≤C|x−y|^m

|α|=m

1 Q(x,y)

Q(x,y)

D^αA(z)^qdz 1/q

, (3.5)

whereQ(x, y) is the cube centered atxand has side length 5^√n|x−y|.

Proof ofTheorem 3.1(main theorem). Fix a cubeQ=Q(x0,l) andx∈Q. LetQ=5^√nQ and A(x) =A(x)−

|α|=m(1/α!)(D^αA)_Qx^α, then Rm(A;x,y)=Rm(A;x, y) and D^αA= D^αA−(D^αA)_Qfor|α| =m. We write, forf1=f χ_Qand f2= f χ_Rn\Q,

F_t^A(f)(x)=

Rⁿ

Rm+1 A;x,y

|x−y|^m F(x−y,t)f(y)d y

=

Rⁿ

Rm+1 A;x,y

|x−y|^m F(x−y,t)f2(y)d y +

Rⁿ

Rm A;x,y

|x−y|^m F(x−y,t)f1(y)d y

−

|α|=m

1 α!

Rⁿ

F(x−y,t)(x−y)^α

|x−y|^m D^αA(y) f1(y)d y,

(3.6)

then

T^A(f)(x)−T^A f2 x0=F_t^A(f)(x)−F_t^A f2 x0

≤ Ft

Rm A;x,·

|x− ·|^m f1

(x)

+

|α|=m

1 α!

Ft

(x− ·)^α

|x− ·|^mD^αA f 1

(x) +F_t^A f2

(x)−F_t^A f2 x0=A(x) +B(x) +C(x), (3.7)

thus,

1

|Q|^1+β/n

Q

T^A(f)(x)−T^A(f) x0dx

≤ 1

|Q|^1+β/n

QA(x)dx+ 1

|Q|^1+β/n

QB(x)dx

+ 1

|Q_|^1+β/n

QC(x)dx:=I+II+III.

(3.8)

Now, let us estimateI,II, andIII, respectively. First, forx∈Qandy∈Q, using Lemmas 3.3and3.5, we get

R_m A;x,y≤C|x−y|^m

|α|=m

sup

x∈Q

D^αA(x)− D^αA_Q

≤C|x−y|^m|Q|^β/n

|α|=m

D^αA_∧˙

β,

(3.9)

thus, takingr,ssuch that 1≤r < pand 1/s=1/r−δ/n, by the (L^r,L^s) boundedness ofT and Holder’ inequality, we obtain

I≤C

|α|=m

D^αA_∧˙

β

1

|Q|

Q

T f1

(x)dx≤C

|α|=m

D^αA_∧˙

βT f1

L^s|Q|^−1/s

≤C

|α|=m

D^αA_∧˙βf1

L^r|Q|^−1/s≤C

|α|=m

D^αA_∧˙βM_δ,r(f)(x).

(3.10) Secondly, using the following inequality (see [9]):

D^αA− D^αA_Qf χ_QLr≤C|Q|^1/s+β/nD^αA_∧˙βMδ,r(f)(x), (3.11) and similar to the proof ofI, we gain

II≤C

|α|=m

D^αA_∧˙βM_δ,r(f)(x). (3.12)

ForIII, using the size condition ofT, we have III≤C

|α|=m

D^αA_∧˙βMδ,1(f)(x). (3.13)

We now put these estimates together; and taking the supremum over allQ such that

x∈Q, and using Lemmas3.2and3.4, we obtain T^A(f)_F˙^β,∞

q ≤C

|α|=m

D^αA_∧˙βfL^p. (3.14)

This completes the proof of (a).

(b) By same argument as in proof of (a), we have 1

|Q|

Q

T^A(f)(x)−T^A f2 x0dx

≤C

|α|=m

D^αA_∧˙_β Mδ+β,r(f) +Mδ+β,1(f), (3.15)

thus,

T^A(f)^#≤C

|α|=m

D^αA_∧˙β Mδ+β,r(f) +Mδ+β,1(f). (3.16)

Now, usingLemma 3.4, we gain T^A(f)_Lq≤C T^A(f)^#_Lq

≤C

|α|=m

D^αA_∧˙β Mδ+β,r(f)_Lq+Mδ+β,1(f)_Lq

≤CfL^p. (3.17)

This completes the proof of (b) and the theorem.

To prove Theorems2.4and2.5, sinceg_δandμ_δare all bounded fromL^p(Rⁿ) toL^q(Rⁿ) for 1< p < n/δ and 1/q=1/ p−δ/n(see [11,12]), it suffices to verify thatg_δ^A andμ^A_δ satisfy the size condition inTheorem 3.1(main theorem).

Suppose suppf ⊂(2Q)^c and x∈Q=Q(x0,l). Note that |x0−y| ≈ |x−y|for y∈ (2Q)^c.

Forg_δ^A, we write F_t^A(f)(x)−F_t^A(f) x0

=

Rⁿ\Q

ψ_t(x−y)

|x−y|^m −ψ_t x0−y x0−y^m

Rm A;x,yf(y)d y +

Rⁿ\Q

ψt x0−yf(y) x0−y^m

R_m A;x,y−R_m A;x0,yd y

−

|α|=m

1 α!

Rⁿ\Q

ψt(x−y)(x−y)^α

|x−y|^m −ψt x0−y x0−y^α x0−y^m

D^αA(y) f(y)d y

=I1+I2+I3.

(3.18) By the condition onψ, we obtain

I1≤C

Rⁿ\Q

x−x0 x0−y^m+1

Rm A;x,yf(y) _∞

0

tdt

t+x0−y^2(n+1−δ) 1/2

d y

+C

Rⁿ\Q

x−x0^ε

x0−y^mR_m A;x,yf(y) _∞

0

tdt

t+x0−y^{2(n+1+ε−δ)} 1/2

d y

≤C

|α|=m

D^αA_∧˙_β|Q|^β/n ∞ k=0

2^k+1Q\2^k+1Q

x−x0

x0−y^n+1−δ+ x−x0^ε x0−y^n+ε−δ

f(y)d y

≤C

|α|=m

D^αA_∧˙β|Q|^β/n ∞ k=1

2^−k+ 2^−kε 1

2^kQ^1−δ/n

2^kQ

f(y)d y

≤C

|α|=m

D^αA_∧˙β|Q|^β/nMδ,1(f)(x).

(3.19)

ForI2, by the formula (see [5]):

Rm A;x,y−Rm A;x0,y=

|η|<m

1

η!Rm−|η| D^ηA;x,x 0

(x−y)^η (3.20)

andLemma 3.5, we get

Rm A;x,y−Rm A;x0,y≤C

|α|=m

D^αA_∧˙_β|Q|^β/nx−x0x0−y^m−1, (3.21)

thus, similar to the proof ofI1,

I2≤C

Rⁿ\Q

R_m A;x,y−R_m A;x0,y x0−y^m+n−δ

f(y)d y

≤C

|α|=m

D^αA_∧˙

β|Q|^β/n∞

k=0

2^k+1Q\2^kQ

x−x0 x0−y^n+1−δ

f(y)d y

≤C

|α|=m

D^αA_∧˙β|Q|^β/nM_δ,1(f)(x).

(3.22)

ForI3, similar to the proof ofI1, we obtain

I3≤C

|α|=m

Rⁿ\Q

x−x0

x0−y^n+1−δ + x−x0^ε x0−y^n+ε−δ

f(y)D^αA(y) d y

≤C

|α|=m

D^αA_∧˙β|Q|^β/n ∞ k=1

2^k(β−1)+ 2^k(β−ε)Mδ,1(f)(x)

≤C

|α|=m

D^αA_∧˙β|Q|^β/nMδ,1(f)(x)

(3.23)

so that

F_t^A(f)(x)−F_t^A(f) x0≤C

|α|=m

D^αA_∧˙β|Q|^β/nMδ,1(f)(x). (3.24)

Forμ^A_δ, we write

F_t^A(f)(x)−F_t^A(f) x0

≤ _∞

0

|x−y|≤t,|x0−y|>t

Ω(x−y)Rm A;x,y

|x−y|^m+n−1−δ f(y)d y 2dt

t³ 1/2

+ _∞

0

|x−y|>t,|x0−y|≤t

Ω x0−yRm A;x0,y x0−y^m+n−1−δ

f(y)d y²dt t³

1/2

+ _∞

0

|x−y|≤t,|x0−y|≤t

Ω(x−y)Rm(A;x, y)

|x−y|^m+n−1−δ

−Ω x0−yRm A;x0,y x0−y^m+n−1−δ

f(y)d y 2dt

t³ 1/2

+C

|α|=m

_∞

0

|x−y|≤t

Ω(x−y)(x−y)^α

|x−y|^m+n−1−δ −

|x0−y|≤t

Ω x0−y x0−y^α x0−y^m+n−1−δ

×D^αA(y) f(y)d y

2dt t³

1/2

:=J1+J2+J3+J4.

(3.25) Then

J1≤C

Rⁿ\Q

f(y)R_m A;x,y

|x−y|^m+n−1−δ

|x−y|≤t<|x0−y|

dt t³

1/2

d y

≤C

Rⁿ\Q

f(y)R_m A;x,y

|x−y|^m+n−1−δ

x0−x^1/2

|x−y|^3/2 d y

≤C

|α|=m

D^αA_∧˙

β|Q|^β/n∞

k=1

2^−k/2 1 2^kQ^1−δ/n

2^kQ

f(y)d y

≤C

|α|=m

D^αA_∧˙β|Q|^β/nM_δ,1(f)(x),

(3.26)

similarly, we haveJ2≤C_|α|=mD^αA_∧˙_β|Q|^β/nMδ,1(f)(x).

ForJ3, by the following inequality (see [12]):

Ω(x−y)

|x−y|^m+n−1−δ − Ω x0−y x0−y^m+n−1−δ

≤C

x−x0

x0−y^m+n−δ+ x−x0^γ x0−y^{m+n−1−δ+γ}

, (3.27)