209 LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF

italian journal of pure and applied mathematics – n.

27−2010 (209−225)

209

LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF LITTLEWOOD-PALEY OPERATOR

Ying Shen Lanzhe Liu Department of Mathematics Changsha University of Science and Technology Changsha, 410077 P.R. of China e-mail: [email protected]

Abstract. In this paper, we will study the continuity of multilinear commutator generated by Littlewood-Paley operator and the functions bj on Triebel-Lizorkin space, Hardy space and Herz-Hardy space, where the functions bj belong to Lipschiz space. 2000 Mathematics Subject Classification: 42B20, 42B25. Keywords and phrases: Littlewood-Paley operator; Multilinear commutator; TriebelLizorkin space; Herz-Hardy space; Herz space; Lipschitz space.

1. Introduction We know, the commutator [b, T ](f )(x) = b(x)T (f )(x) − T (bf )(x) is bounded on Lp (Rn ) for 1 < p < ∞ when T is the Calder´on-Zygmund operator and b ∈ BM O(Rn ). Janson and Paluszynski study the commutator for the TriebelLizorkin space and the case b ∈ Lipβ (Rn ), where Lipβ (Rn ) is the homogeneous Lipschitz space. Chanillo (see [2]) proves a similar result when T is replaced by the fractional operators. The main purpose of this paper is to discuss the boundedness of Littlewood-Paley multilinear commutator generated by Littlewood-Paley operator and Lipschiz functions on Triebel-Lizorkin space, Hardy space and HerzHardy space.

2. Preliminaries and Definitions Throughout this paper, M (f ) will denote the Hardy-Littlewood maximal function of f , and write Mp (f ) = (M (f p ))1/p for 0 < p < ∞. Q will denote a cube of Rn with side parallel toZthe axes. Z −1 # −1 |f (y) − fQ |dy denote the f (x)dx and f (x) = sup |Q| Let fQ = |Q| Q

x∈Q

Q

Hardy spaces by H p (Rn ). It is well known that H p (Rn ) (0 < p ≤ 1) has the

210

ying shen, lanzhe liu

atomic decomposition characterization (see [11], [16], [17]). For β > 0 and p > 1, let F˙ pβ,∞ (Rn ) be the homogeneous Tribel-Lizorkin space. The Lipschitz space Lipβ (Rn ) is the space of functions f such that ||f ||Lipβ = sup

x,y∈Rn x6=y

|f (x) − f (y)| < ∞. |x − y|β

Lemma 1. (see [15]) For 0 < β < 1, 1 < p < ∞, we have ||f ||F˙pβ,∞

¯¯ ¯¯ ¯¯ ¯¯ Z ¯¯ ¯¯ 1 ¯¯ |f (x) − f |dx ≈ ¯¯¯¯sup Q β ¯¯ ¯¯ p ¯¯ Q |Q|1+ n Q L ¯¯ ¯¯ ¯¯ ¯¯ Z ¯¯ ¯¯ 1 ¯ ¯ ≈ ¯¯sup inf |f (x) − c|dx¯¯¯¯ β ¯¯ ·∈Q c |Q|1+ n Q ¯¯

.

Lp

Lemma 2. (see [15]) For 0 < β < 1, 1 ≤ p ≤ ∞, we have ||f ||Lipβ ≈ sup Q

≈ sup Q

Z

1 β

|Q|1+ n

Ã

1 β

|Q| n

Q

|f (x) − fQ |dx

1 Z |f (x) − fQ |p dx |Q| Q

!1/p

.

Lemma 3. (see [2]) For 1 ≤ r < ∞ and β > 0, let 

Mβ,r (f )(x) = sup  x∈Q

1 |Q|

1− βr n

1/r

Z Q

|f (y)|r dy 

,

suppose that r < p < n/β, and 1/q = 1/p − β/n, then ||Mβ,r (f )||Lq ≤ C||f ||Lp . Lemma 4. (see [5]) Let Q1 ⊂ Q2 , then |fQ1 − fQ2 | ≤ C||f ||Λ˙ β |Q2 |β/n . Definition 1. Let 0 < p, q < ∞, α ∈ R, Bk = {x ∈ Rn , |x| ≤ 2k }, Ak = Bk \Bk−1 and χk = χAk for k ∈ Z. 1) The homogeneous Herz space is defined by K˙ qα,p (Rn ) = {f ∈ LqLoc (Rn \{0}) : ||f ||K˙ qα,p < ∞}, where



||f ||K˙ qα,p = 

∞ X

k=−∞

1/p

2kαp ||f χk ||pLq 

;

lipschitz estimates for multilinear commutator ...

211

2) The nonhomogeneous Herz space is defined by Kqα,p (Rn ) = {f ∈ LqLoc (Rn ) : ||f ||Kqα,p < ∞}, where ||f ||Kqα,p =

"∞ X

#1/p

2kαp ||f χk ||pLq

+

||f χB0 ||pLq

.

k=1

Definition 2. Let α ∈ R, 0 < p, q < ∞. (1) The homogeneous Herz type Hardy space is defined by H K˙ qα,p (Rn ) = {f ∈ S 0 (Rn ) : G(f ) ∈ K˙ qα,p (Rn )}, and ||f ||H K˙ qα,p = ||G(f )||K˙ qα,p ; (2) The nonhomogeneous Herz type Hardy space is defined by HKqα,p (Rn ) = {f ∈ S 0 (Rn ) : G(f ) ∈ Kqα,p (Rn )}, and ||f ||HKqα,p = ||G(f )||Kqα,p ; where G(f ) is the grand maximal function of f . The Herz type Hardy spaces have the atomic decomposition characterization. Definition 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 some r > 0 (or for some r ≥ 1), 2) ||a||Lq ≤ |B(0, r)|−α/n , Z

3)

Rn

a(x)xη dx = 0 for |η| ≤ [α − n(1 − 1/q)].

Lemma 5. (see [6], [14]) Let 0 < p < ∞, 1 < q < ∞ and α ≥ n(1 − 1/q). A temperate distribution f belongs to H K˙ qα,p (Rn ) (or HKqα,p (Rn )) if and only if there exist central (α, q)-atoms (or central (α, q)-atoms of restrict type) aj supported on Bj = B(0, 2j ) and constants λj , (or f =

∞ X

X

|λj |p < ∞ such that f =

j

∞ X

j=−∞

λj aj ) in the S 0 (Rn ) sense, and

j=0

1/p  X ||f ||H K˙ qα,p (or ||f ||HKqα,p ) ∼  |λj |p  . j

λj aj

212


Definition 4. Let 0 < δ < n, ε > 0 and ψ be a fixed function which satisfies the following properties: Z

1)

Rn

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|. Let m be a positive integer and bj (1 ≤ j ≤ m) be the locally integrable function, set ~b = (b1 , · · · , bm ). The multilinear commutator of Littlewood-Paley operator is defined by ~b gψ,δ (f )(x)

where ~

Z

Ftb (f )(x) =

ÃZ

=

m Y

∞

dt ~ |Ftb (x)|2

!1/2

,

t

0

(bj (x) − bj (y))ψt (x − y)f (y)dy,

Rn j=1

and ψt (x) = t−n+δ ψ(x/t) for t > 0. Set Ft (f ) = ψt ∗ f. We also define that gψ,δ (f )(x) =

ÃZ ∞ 0

|Ft (f )(x)|

2 dt

!1/2

t

,

which is the Littlewood-Paley g function (see [17]). Let H be the space H(Rn ) = {h : ||h|| =

µZ ∞ 0

¶1/2

|h(t)|2 dt/t

< ∞},

then, for each fixed x ∈ Rn Ft (f )(x) may be viewed as a mapping from [0, +∞) to H, and it is clear that ~

~

b gψ,δ (f )(x) = ||Ft (f )(x)|| and gψ,δ (f )(x) = ||Ftb (f )(x)||. ˜

b is just the m order commutator. It is well Note that when b1 = · · · = bm , gψ,δ known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [1-4], [7-10], [12], [15]). Our main purpose is to establish the boundedness of the multilinear commutator on Triebel-Lizorkin space, Hardy space and Herz-Hardy space. Given a positive integer m and 1 ≤ j ≤ m, we set

||~b||Lipβ =

m Y j=1

||bj ||Lipβ


213

and denote by Cjm the family of all finite subsets σ = {σ(1), ···, σ(j)} of {1, ···, m} of j different elements. For σ ∈ Cjm , set σ c = {1, · · ·, m} \ σ. For ~b = (b1 , · · ·, bm ) and σ = {σ(1), · · ·, σ(j)} ∈ Cjm , set ~bσ = (bσ(1) , · · ·, bσ(j) ), bσ = bσ(1) · · · bσ(j) and ||~bσ ||Lipβ = ||bσ(1) ||Lipβ · · · ||bσ(j) ||Lipβ . Lemma 6. (see [10]) Let 0 < β ≤ 1,0 < δ < n, 1 < p < n/β, 1/q = 1/p − β/n b and b ∈ Lipβ (Rn ). Then gψ,δ is bounded from Lp (Rn ) to Lq (Rn ).

3. Theorems and proofs Theorem 1. Let 0 < δ < n, 0 < β < min(1, ε/m), 1 < p < ∞, ~b = (b1 , · · · , bm ) ~b with bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m and gψ,δ be the multilinear commutator of Littlewood-Paley operator as in Definition 4. Then ~b a) gψ,δ is bounded from Lp (Rn ) to F˙ pmβ,∞ (Rn ) for 1 0 such that for every H p -atom a, ~

b ||gψ,δ (a)||Lq ≤ C.

Let a be a H p -atom, that is that a supported on a cube Q = Q(x0 , r), Z ||a||L∞ ≤ |Q|−1/p and

Rn

a(x)xγ dx = 0 for |γ| ≤ [n(1/p − 1)].

When m = 1, see [10]. Now consider the case m ≥ 2. Write ~b ||gψ,δ (a)(x)||Lq

ÃZ

≤

|x−x0 |≤2r

~b |gψ,δ (a)(x)|q dx

!1/q

ÃZ

+

|x−x0 |>2r

~b |gψ,δ (a)(x)|q dx

!1/q

= I + II. For I, choose 1 < p1 < n/(mβ + δ) and q1 such that 1/q1 = 1/p1 − mβ + δ/n. ~b By the boundednss of gψ,δ from Lp1 (Rn ) to Lq1 (Rn ) (see Theorem 1), we get ~b I ≤ C||gψ,δ (a)||qLq1 |Q(x0 , 2r)|1−q/q1 ≤ C||a||qLp1 ||~b||Lipβ |Q|1−q/q1

≤ C||~b||Lipβ |Q|−q/p+q/p1 +1−q/q1 ≤ C||~b||Lipβ . For II, let τ, τ 0 ∈ N such that τ + τ 0 = m, and τ 0 6= 0. We get ~ |Ftb (a)(x)|

+

Z

≤ |(b1 (x)−b1 (x0 )) · · · (bm (x)−bm (x0 ))

m X X

≤ C||~b||Lipβ |x − x0 |mβ · X

+C||~b||Lipβ

τ +τ 0 =m

≤ C||~b||Lipβ

B

(b(y) − b(x0 ))σ ψt (x − y)a(y)dy|

Z B

|x − x0 |

|ψt (x − y) − ψt (x − x0 )||a(y)|dy Z

0

τβ B

|y − x0 |τ β |ψt (x − y)||a(y)|dy

Z |x − x0 | t |x0 − y|ε |a(y)|dy (t + |x − x0 |)n+1+ε−δ B mβ

Z t 0 |x − x0 | |y − x0 |τ β |a(y)|dy n+1−δ (t + |x − x0 |) B τ +τ 0 =m X

≤ C||~b||Lipβ +C||~b||Lipβ

(ψt (x−y)−ψt (x−x0 ))a(y)dy|

Z

|(b(x) − b(x0 ))σc

j=1 σ∈Cjm

+C||~b||Lipβ

B

τβ

1 t · rmβ+ε+n(1− p ) n+1+ε−δ (t + |x − x0 |)

1 t · rmβ+n(1− p ) , n+1−δ (t + |x − x0 |)

218


thus ~b |gψ,δ (a)(x)|

≤

 Z ∞Ã C||~b||Lipβ 

+

 Z ∞Ã C||~b||Lipβ 

0

0

t (t + |x − x0 |)n+1+ε−δ t (t + |x − x0 |)n+1−δ

!2

!2

1/2

dt  t

1

· rmβ+ε+n(1− p )

1/2

dt  t

1

· rmβ+n(1− p )

1 ≤ C||~b||Lipβ |x − x0 |−n+δ · rmβ+n(1− p ) ,

so, II ≤ C||~b||Lipβ · r

mβ+n(1− p1 )

!1/q

ÃZ |x−x0 |>2r

|x − x0 |−nq+qδ dx

≤ C||~b||Lipβ .

This complete the proof of Theorem 2. Theorem 3. Let 0 < β ≤ 1, 0 < δ < n, 0 < p < ∞, 1 < q1 , q2 < ∞, 1/q1 − 1/q2 = mβ + δ/n, n(1 − 1/q1 ) ≤ α < n(1 − 1/q1 ) + β + δ/m, ~b = (b1 , · · · , bm ) ~b with bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m. Then gψ,δ is bounded from H K˙ qα,p (Rn ) to 1 (Rn ). K˙ qα,p 2 Proof. By Lemma 5, let f ∈ H K˙ qα,p (Rn ) and f = 1 ∞ X

B(0, 2j ), aj be a central (α, q)−atom, and

∞ X

λj aj , supp aj ⊂ Bj =

j=−∞

|λj |p < ∞. We have

j=−∞ ~

b ||gψ,δ (f )||pK˙ α,p ≤ C q2

+ C

∞ X k=−∞ ∞ X k=−∞



2kαp  

2kαp 

k−2 X j=−∞ ∞ X

p ~

b |λj |||gψ,δ (aj )χk ||Lq2 

p ~

b |λj |||gψ,δ (aj )χk ||Lq2 

j=k−1

= I + II. ~

b For II, by the boundedness of gψ,δ on (Lq1 , Lq2 ), we have

II ≤ C||~b||pLipβ ≤ C||~b||pLipβ

∞ X



2kαp 

k=−∞ ∞ X

∞ X

p

|λj |||aj ||Lq1 

j=k−1



2kαp 

k=−∞  ∞ X

∞ X

p

|λj | · 2−jα 

j=k−1 ∞ X

   |λj |p · 2(k−j)αp , 0 < p ≤ 1     k=−∞ j=k−1 p/p0   ≤ C||~b||pLipβ  ∞ ∞ ∞  X X X 0    2−jαp /2  , 1 < p < ∞ |λj |p · 2−jαp/2  2kαp   k=−∞

j=k−1

j=k−1


≤ C||~b||pLipβ

∞ X

219

|λj |p .

j=−∞

For I, when m = 1, we have |Ftb1 (aj )(x)|

≤ +

¯ ¯ Z ¯ ¯ ¯ ¯ ¯(b1 (x) − b1 (0)) (ψ t (x − y) − ψt (x))aj (y)dy ¯ ¯ ¯ Bj ¯Z ¯ ¯ ¯ ¯ ¯ ¯ ψ t (b1 (y) − b1 (0))aj (y)dy ¯ ¯ Bj ¯ "Z β ε

≤ C||b1 ||Lipβ

|x| |y| t · |aj (y)|dy (t + |x|)n+1+ε−δ

Bj

#

Z

+ ≤ + ≤ + thus b1 gψ,δ (aj )(x) ≤

t|y|β · |aj (y)|dy Bj (t + |x − y|)n+1−δ " Z |x|β t C||b1 ||Lipβ |y|ε |aj (y)|dy (t + |x|)n+1+ε−δ Bj # Z t ε |y| |aj (y)|dy (t + |x|)n+1−δ Bj " |x|β t j(ε+n(1− q1 )−α) 1 ·2 C||b1 ||Lipβ n+1+ε−δ (t + |x|) # t j(β+n(1− q1 )−α) 1 ·2 , (t + |x|)n+1−δ

 Ã Z  ∞ C||b1 ||Lipβ  0



+ 

Z ∞Ã 0

t (t + |x|)n+1+ε−δ

t (t + |x|)n+1−δ

!2

≤ C||b1 ||Lipβ |x|



1/2

dt  t

· −(n+ε−δ)

!2 1/2 1  · |x|β · 2j(ε+n(1− q1 )−α)

j(β+n(1− q1 )−α) 

·2

j(ε+n(1− q1 )−α)

β

· |x| · 2

1

j(β+n(1− q1 )−α)

≤ C||b1 ||Lipβ |x|−n+δ · 2

1



1

|x|

−n+δ

·2

j(β+n(1− q1 )−α) 1

,

from that we have b1 ||gψ,δ (aj )χk ||Lq2

j(β+n(1− q1 )−α)

≤ C||b1 ||Lipβ · 2

µZ

Bk |x|

1

j(β+n(1− q1 )−α)

≤ C||b1 ||Lipβ · 2

1

−nq2 +q2 δ

−kn(1− q1 )+kδ

·2

2

[j(β+n(1− q1 )−α)−k(β+n(1− q1 ))]

≤ C||b1 ||Lipβ · 2

1

dx

1

,

so, I ≤ C||b1 ||pLipβ

∞ X k=−∞



2kαp 

∞ X

j=−∞

p [j(β+n(1− q1 )−α)−k(β+n(1− q1 ))] 

|λj | · 2

1

1

¶1/q2

¸

220

ying shen, lanzhe liu  ∞ k−2  X X  (j−k)(β+n(1− q1 )−α)p   1 |λj |p · 2 , 00

2l(n(1−1/q1 )+β+δ/m)p (2l )np/q2  1/p 

(2l )(n−(m−1)β−δ) 

l=−∞



≤ sup λ2lλ (n−(m−1)β−δ) ≤ C||~b||Lipβ  λ>0

1/p

∞ X

|λk |p 

.

k=−∞

Now, combining the above estimates for G1 and G2 , we obtain  ~

b (f )||W K˙ n(1−1/q1 )+β+δ/m,p ≤ C||~b||Lipβ  ||gψ,δ q2

∞ X

1/p

|λk |p 

.

k=−∞

Theorem 4 follows by taking the infimum over all central atomic decompositions. References ´rez, C., Weighted [1] J. Alvarez, R. J. Babgy, D. S. Kurtz and C. Pe estimates for commutators of linear operators, Studia Math., 104 (1993), 195-209. [2] Chanillo, S., A not on commutators, Indiana Univ. Math. J., 31 (1982), 7-16. [3] Chen, W,G., Besov estimates for a class of multilinear singular integrals, Acta Math. Sinica, 16 (2000), 613-626. [4] R. Coifman, R. Rochberg and G. Weiss, G., Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635. [5] Devore, R.A. and Sharply, R.C., Maximal functions measuring smoothness, Mem. Amer. Math. Soc., 47(1984). [6] J. Garcia-Cuerva and M. J. L. Herrero, J.L., A theory of Hardy spaces associated to Herz spaces, Proc. London Math. Soc., 69 (1994), 605-628. [7] Janson, S., Mean Oscillation and commutators of singular integral operators, Ark. Math., 16 (1978), 263-270. [8] Liu, L.Z., Boundedness of multilinear operator on Triebel-Lizorkin spaces, Inter. J. of Math. and Math. Sci., 5 (2004), 259-271. [9] Liu, L.Z., The continuity of commutators on Triebel-Lizorkin spaces, Integral Equations and Operator Theory, 49 (2004), 65-76.

224


[10] Liu, L.Z., Boundedness for multilinear Littlewood-Paley operators on Hardy and Herz-Hardy spaces, Extracta Math., 19 (2) (2004), 243-255. [11] Lu, S.Z., Four lectures on real H p spaces, World Scientific, River Edge, NI, 1995. [12] Lu, S.Z., Q. Wu, Q. and Yang, D.C., Boundedness of commutators on Hardy type spaces, Sci. in China (Ser. A), 45 (2002), 984-997. [13] Lu, S.Z. and Yang, D.C., The decomposition of the weighted Herz spaces and its applications, Sci. in China (Ser. A), 38 (1995), 147-158. [14] Lu, S.Z. and Yang, D.C., The weighted Herz type Hardy spaces and its applications, Sci. in China (Ser. A), 38 (1995), 662-673. [15] Paluszynski, M., Characterization of the Besov spaces via the commutator operator of Coifman, Rochbeg and Weiss, Indiana Univ. Math. J., 44 (1995), 1-17. [16] Stein, E.M., Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals, Princeton, Princeton Univ. Press, 1993. [17] Torchinsky, A., Real variable methods in harmonic analysis, Pure and Applied Math., 123, Academic Press, New York, 1986. Accepted: 16.02.2009