Mean Oscillation and Boundedness of Multilinear Integral Operators ...

Archivum Mathematicum

Liu Lanzhe Mean Oscillation and Boundedness of Multilinear Integral Operators with General Kernels Archivum Mathematicum, Vol. 50 (2014), No. 2, 77--96

Persistent URL: http://dml.cz/dmlcz/143781

Terms of use: © Masaryk University, 2014 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz

ARCHIVUM MATHEMATICUM (BRNO) Tomus 50 (2014), 77–96

MEAN OSCILLATION AND BOUNDEDNESS OF MULTILINEAR INTEGRAL OPERATORS WITH GENERAL KERNELS Liu Lanzhe Abstract. In this paper, the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces are proved. The integral operators include singular integral operator with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

1. Introduction and results As the development of singular integral operators, their commutators and multilinear operators have been well studied (see [3]–[7], [18]–[20]). Let T be the Calderón-Zygmund singular integral operator and b ∈ BMO(Rn ), a classical result of Coifman, Rochberg and Weiss (see [6]) stated that the commutator [b, T ](f ) = T (bf ) − bT (f ) is bounded on Lp (Rn ) for 1 < p < ∞. The purpose of this paper is to introduce some multilinear operator associated to certain integral operators with general kernels (see [1, 10, 15]) and prove the boundedness properties of the multilinear operators from Lebesgue spaces to Orlicz spaces. In this paper, we are going to consider some integral operators as following (see [1]). Let l and mj be the positive integers (j = 1, . . . , l), m1 + · · · + ml = m and bj be the functions on Rn (j = 1, . . . , l). Set, for 1 ≤ j ≤ l, X 1 Rmj +1 (bj ; x, y) = bj (x) − Dα bj (y)(x − y)α . α! |α|≤mj

Definition 1. Let T : S → S 0 be a linear operator such that T is bounded on L2 (Rn ) and has a kernel K, that is there exists a locally integrable function K(x, y) 2010 Mathematics Subject Classification: primary 42B20; secondary 42B25. Key words and phrases: multilinear operator, singular integral operator, BMO space, Orlicz space, Littlewood-Paley operator, Marcinkiewicz operator, Bochner-Riesz operator. Supported by the Scientific Research Fund of Hunan Provincial Education Departments (13K013). Received November 11, 2013, revised February 2014. Editor V. Müller. DOI: 10.5817/AM2014-2-77

78

L. LANZHE

on Rn × Rn \ {(x, y) ∈ Rn × Rn : x = y} such that Z T (f )(x) = K(x, y)f (y)dy Rn

for every bounded and compactly supported function f , where K satisfies: |K(x, y)| ≤ C|x − y|−n , Z (|K(x, y) − K(x, z)| + |K(y, x) − K(z, x)|) dx ≤ C , 2|y−z| 0. The Luxemburg norm is given by (see [21]) Z −1 kf kLρ = inf λ 1+ ρ(λ|f (x)|) dx . λ>0

Rn

We shall prove the following theorems in Section 2. Theorem 1. Let 0 < β ≤ 1, q 0 < p < n/lβ and ϕ, ψ be two non-decreasing positive functions on [0, +∞) with (ψ l )−1 (t) = t1/p ϕl (t−1/n ). Suppose that ψ is convex, ψ(0) = 0, ψ(2t) ≤ Cψ(t). Let T be the same as in Definition 1 and the sequence {k l Ck } ∈ l1 . Then T b is bounded from Lp (Rn ) to Lψl (Rn ) if Dα bj ∈ BMO(Rn ) for all α with |α| = mj and j = 1, . . . , l. Theorem 2. Let 0 < β ≤ 1, q 0 < p < n/mβ and ϕ, ψ be two non-decreasing positive functions on [0, +∞) with (ψ l )−1 (t) = t1/p ϕl (t−1/n ). Suppose that ψ is convex, ψ(0) = 0, ψ(2t) ≤ Cψ(t). Let S be the same as in Definition 2 and the sequence {Ck } ∈ l1 . Then S b is bounded from Lp (Rn ) to Lψl (Rn ) if Dα bj ∈ BMO(Rn ) for all α with |α| = mj and j = 1, . . . , l. Remark. (a) If l = 1 and ψ −1 (t) = t1/p ϕ(t−1/n ), then T b and S b are all bounded on from Lp (Rn ) to Lψ (Rn ) under the conditions of Theorems 1 and 2. (b) If l = 1, ϕ(t) ≡ 1 and ψ(t) = tp for 1 < p < ∞, then T b and S b are all bounded on Lp (Rn ) if Dα b ∈ BMOϕ (Rn ) for all α with |α| = m. (c) If l = 1, ψ(t) = ts and ϕ(t) = tn(1/p−1/s) for 1 < p < s < ∞, then, by BMOtβ (Rn ) = Lipβ (Rn ) (see [9, Lemma 4]), T b and S b are all bounded from Lp (Rn ) to Ls (Rn ) if Dα b ∈ Lipn(1/p−1/s) (Rn ) for all α with |α| = m. 2. Proof of theorems We begin with the following preliminary lemmas. Lemma 1 (see [1]). Let T and S be the the same as Definitions 1 and 2, the sequence {Ck } ∈ l1 . Then T and S are bounded on Lp (Rn ) for 1 < p < ∞. Lemma 2 (see [9]). Let ρ be a non-decreasing positive function on [0, +∞) and n ηR be an infinitely differentiable function that R on R with compact support such t t η(x) dx = 1. Denote that b (x) = b(x − ty)η(y) dy. Then kb − b k ≤ BMO n n R R Cρ(t)kbkBMOρ . Lemma 3 (see [1]). Let 0 < β < 1 or β = 1 and ρ be a non-decreasing positive function on [0, +∞). Then kbt kLipβ ≤ Ct−β ρ(t)kbkBMOρ . Lemma 4 (see [1]). Suppose 1 ≤ p2 < p < p1 < ∞, ρ is a non-increasing function on R+ , B is a linear or sublinear operator such that mB(f ) (t1/p1 ρ(t)) ≤ Ct−1 if R∞ kf kLp1 ≤ 1 and mB(f ) (t1/p2 ρ(t)) ≤ Ct−1 if kf kLp2 ≤ 1. Then 0 mB(f ) (t1/p ρ(t)) dt ≤ C if kf kLp ≤ (p/p1 )1/p . Lemma 5 (see [2]). Suppose that 0 < β < n, 1 ≤ r < p < n/β and 1/s = 1/p−β/n. Then kMβ,r (f )kLs ≤ Ckf kLp .

MULTILINEAR INTEGRAL OPERATORS

81

Lemma 6 (see [5]). Let b be a function on Rn and Dα A ∈ Lq (Rn ) for all α with |α| = m and some q > n. Then |Rm (b; x, y)| ≤ C|x − y|

m

X |α|=m

1 ˜ |Q(x, y)|

Z

|Dα b(z)|q dz

1/q

,

˜ Q(x,y)

˜ is the cube centered at x and having side length 5√n|x − y|. where Q To prove the theorems of the paper, we need the following Key Lemma. Let T and S be the same as in Definitions 1 and 2. Suppose that Q = Q(x0 , d) is a cube with supp f ⊂ (2Q)c and x, x ˜ ∈ Q. (I) If the sequence {k l Ck } ∈ l1 and Dα bj ∈ BMO(Rn ) for all α with |α| = mj and j = 1, . . . , l, then l X Y

|T b (f )(x) − T b (f )(x0 )| ≤ C

j=1

kDαj bj kBMO Mr (f )(˜ x) for any r > q 0 ;

|αj |=mj

(II) If the sequence {Ck } ∈ l1 , 0 < β ≤ 1 and Dα bj ∈ Lipβ (Rn ) for all α with |α| = mj and j = 1, . . . , l, then

|T b (f )(x)−T b (f )(x0 )| ≤ C

l X Y j=1

kDαj bj kLipβ Mlβ,r (f )(˜ x) for any r > q 0 ;

|αj |=mj

(III) If the sequence {k l Ck } ∈ l1 and Dα bj ∈ BMO(Rn ) for all α with |α| = mj and j = 1, . . . , l, then

kFtb (f )(x)−Ftb (f )(x0 )k ≤ C

l X Y j=1

kDαj bj kBMO Mr (f )(˜ x) for any r > q 0 ;

|αj |=mj

(IV) If the sequence {k l Ck } ∈ l1 , 0 < β ≤ 1 and Dα bj ∈ Lipβ (Rn ) for all α with |α| = mj and j = 1, . . . , l, then

kFtb (f )(x)−Ftb (f )(x0 )k ≤ C

l X Y j=1

kDαj bj kLipβ Mlβ,r (f )(˜ x) for any r > q 0 .

|αj |=mj

˜ = 5√nQ and Proof. Without P loss of generality, we may assume l = 2. Let Q 1 α α α˜ ˜bj (x) = bj (x) − ˜ ˜ x , then Rm (bj ; x, y) = Rm (bj ; x, y) and D bj = α! (D bj )Q |α|=m

82

L. LANZHE

Dα bj − (Dα bj )Q˜ for |α| = mj . We write, for supp f ⊂ (2Q)c and x, x ˜ ∈ Q, T b (f )(x) − T b (f )(x0 ) =

2 K(x, y) K(x0 , y) Y − Rmj (˜bj ; x, y)f (y) dy m |x0 − y|m j=1 Rn |x − y|

Z

Rm2 (˜b2 ; x, y) K(x0 , y)f (y) dy Rm1 (˜b1 ; x, y) − Rm1 (˜b1 ; x0 , y) |x0 − y|m Rn Z Rm1 (˜b1 ; x0 , y) + Rm2 (˜b2 ; x, y) − Rm2 (˜b2 ; x0 , y) K(x0 , y)f (y) dy |x0 − y|m Rn Z i h R (˜b ; x, y)(x−y)α1 X 1 Rm2 (˜b2 ; x0 , y)(x0−y)α1 m2 2 − K(x, y) − K(x , y) 0 α1 ! R n |x−y|m |x0−y|m Z

+

|α1 |=m1

× Dα1 ˜b1 (y)f (y) dy i X 1 Z h Rm (˜b1 ; x, y)(x−y)α2 Rm1 (˜b1 ; x0 , y)(x0−y)α2 1 − K(x, y) − K(x , y) 0 α2 ! R n |x−y|m |x0−y|m |α2 |=m2

× Dα2 ˜b2 (y)f (y) dy Z h i X (x − y)α1 +α2 (x0 − y)α1 +α2 1 K(x, y) − K(x0 , y) + m m α1 !α2 ! Rn |x − y| |x0 − y| |α1 |=m1 , |α2 |=m2

× Dα1 ˜b1 (y)Dα2 ˜b2 (y)f (y) dy = I1 + I2 + I3 + I4 + I5 + I6 . (I). By Lemma 6 and the following inequality (see [10]), for b ∈ BMO(Rn ), |bQ1 − bQ2 | ≤ C log(|Q2 |/|Q1 |)kbkBMO for Q1 ⊂ Q2 , we know that, for x ∈ Q and y ∈ 2k+1 Q \ 2k Q with k ≥ 1, X

|Rm (˜b; x, y)| ≤ C|x − y|m

(kDα bkBMO + |(Dα b)Q(x,y) − (Dα b)Q˜ |) ˜

|α|=m m

≤ Ck|x − y|

X

kDα bkBMO .

|α|=m

˜ by the conditions on K and Note that |x − y| ∼ |x0 − y| for x ∈ Q and y ∈ Rn \ Q, recalling r > q 0 , we obtain Z |I1 | ≤ Rn \2Q

2 Y 1 1 − y)| |Rmj (˜bj ; x, y)| |f (y)| dy |K(x, |x − y|m |x0 − y|m j=1

Z + Rn \2Q

|K(x, y) − K(x0 , y)| |x0 − y|−m

2 Y j=1

|Rmj (˜bj ; x, y)| |f (y)| dy


≤

∞ Z X 2k+1 Q\2k Q

k=1

+

|K(x, y) − K(x0 , y)| |x0 − y|−m

2k+1 Q\2k Q

2 X Y j=1

+C

kDαj bj kBMO

∞ X

kDαj bj kBMO

2k+1 Q\2k Q

∞ X

|αj |=mj

Z

k2

Z

k=1

X

2 Y

|Rmj (˜bj ; x, y)| |f (y)| dy

j=1

|αj |=mj

2 Y j=1

×

2 Y 1 1 |K(x, y)| − |Rmj (˜bj ; x, y)||f (y)| dy |x − y|m |x0 − y|m j=1

∞ Z X k=1

≤C

83

k2

|x − x0 | |f (y)| dy |x0 − y|n+1

Z

0

|f (y)|q dy

1/q0

2k+1 Q\2k Q

k=1

1/q

|K(x, y) − K(x0 , y)|q dy

2k+1 Q\2k Q

≤C

2 Y j=1

≤C

X

kD bj kBMO

|αj |=mj

2 X Y j=1

αj

∞ X

2

k (2

−k

+Ck )

Z

1

|2k+1 Q|

k=1

|f (y)|r dy

1/r

2k+1 Q

kDαj bj kBMO Mr (f )(˜ x) .

|αj |=mj

For I2 , by the formula (see [5]): Rm (˜b; x, y) − Rm (˜b; x0 , y) =

X 1 Rm−|γ| (Dγ ˜b; x, x0 )(x − y)γ γ!

|γ| 0 such that

94

L. LANZHE

for any x, y ∈ S n−1 , |Ω(x) − Ω(y)| ≤ M |x − y|γ . The multilinear Marcinkiewicz operator is defined by Z ∞ dt 1/2 µbΩ (f )(x) = |Ftb (f )(x)|2 3 , t 0 where Ftb (f )(x)

Z = |x−y|≤t

Ω(x − y) |x − y|n−1

Ql

j=1

Rmj +1 (bj ; x, y) |x − y|m

f (y) dy ,

we write that Z Ft (f )(x) = |x−y|≤t

Ω(x − y) f (y) dy . |x − y|n−1

We also define that µΩ (f )(x) =

∞

Z

|Ft (f )(x)|2

0

dt 1/2 , t3

which is the Marcinkiewicz operator (see [24]). 1/2 R∞ Let H be the space H = h : khk = 0 |h(t)|2 dt/t3 < ∞ . Then, it is clear that µΩ (f )(x) = kFt (f )(x)k

and µbΩ (f )(x) = kFtb (f )(x)k .

It is easily to see that µbΩ satisfies the conditions of Theorem 2 (see [12]–[16, 24]), thus Theorem 2 holds for µbΩ . Application 4. Bochner-Riesz operator. ˆ = (1 − t2 |ξ|2 )δ fˆ(ξ) and B δ (z) = t−n B δ (z/t) for Let δ > (n − 1)/2, Ftδ (f )(ξ) + t t > 0. The maximal Bochner-Riesz operator is defined by (see [17]) Bδ,∗ (f )(x) = sup |Ftδ (f )(x)|. t>0

Set H be the space H = {h : khk = sup |h(t)| < ∞}. The multilinear operator t>0

related to the maximal Bochner-Riesz operator is defined by b b Bδ,∗ (f )(x) = sup |Fδ,t (f )(x)| , t>0

where b Fδ,t (f )(x)

Ql

Z

j=1

= Rn

Rmj +1 (bj ; x, y) |x − y|m

Btδ (x − y)f (y) dy ,

We know b b Bδ,∗ (f )(x) = kBδ,t (f )(x)k . b It is easily to see that Bδ,∗ satisfies the conditions of Theorem 2 (see [12]–[13, 25]), b thus Theorem 2 holds for Bδ,∗ .

Acknowledgement. The author would like to express his gratitude to the referee for his comments and suggestions.


95

References [1] Chang, D.C., Li, J.F., Xiao, J., Weighted scale estimates for Calderón-Zygmund type operators, Contemp. Math. 446 (2007), 61–70. [2] Chanillo, S., A note on commutators, Indiana Univ. Math. J. 31 (1982), 7–16. [3] Cohen, J., A sharp estimate for a multilinear singular integral on Rn , Indiana Univ. Math. J. 30 (1981), 693–702. [4] Cohen, J., Gosselin, J., On multilinear singular integral operators on Rn , Studia Math. 72 (1982), 199–223. [5] Cohen, J., Gosselin, J., A BMO estimate for multilinear singular integral operators, Illinois J. Math. 30 (1986), 445–465. [6] Coifman, R., Rochberg, R., Weiss, G., Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611–635. [7] Ding, Y., Lu, S.Z., Weighted boundedness for a class rough multilinear operators, Acta Math. Sinica 17 (2001), 517–526. [8] Garcia-Cuerva, J., Rubio de Francia, J.L., Weighted norm inequalities and related topics, North-Holland Math., vol. 116, Amsterdam, 1985. [9] Janson, S., Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), 263–270. [10] Lin, Y., Sharp maximal function estimates for Calderón-Zygmund type operators and commutators, Acta Math. Scientia 31(A) (2011), 206–215. [11] Liu, L.Z., Continuity for commutators of Littlewood-Paley operator on certain Hardy spaces, J. Korean Math. Soc. 40 (2003), 41–60. [12] Liu, L.Z., The continuity of commutators on Triebel-Lizorkin spaces, Integral Equations Operator Theory 49 (2004), 65–76. [13] Liu, L.Z., Endpoint estimates for multilinear integral operators, J. Korean Math. Soc. 44 (2007), 541–564. [14] Liu, L.Z., Sharp and weighted inequalities for multilinear integral operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 101 (2007), 99–111. [15] Liu, L.Z., Sharp maximal function estimates and boundedness for commutators associated with general integral operator, Filomat 25 (2011), 137–151. [16] Liu, L.Z., Weighted boundedness for multilinear Littlewood-Paley and Marcinkiewicz operators on Morrey spaces, J. Contemp. Math. Anal. 46 (2011), 49–66. [17] Lu, S.Z., Four lectures on real H p spaces, World Scientific, River Edge, NI, 1995. [18] Paluszynski, M., Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J. 44 (1995), 1–17. [19] Pérez, C., Pradolini, G., Sharp weighted endpoint estimates for commutators of singular integral operators, Michigan Math. J. 49 (2001), 23–37. [20] Pérez, C., Trujillo-Gonzalez, R., Sharp weighted estimates for multilinear commutators, J. London Math. Soc. 65 (2002), 672–692. [21] Rao, M.M., Ren, Z.D., Theory of Orlicz spaces, Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1991. [22] Stein, E.M., Harmonic analysis: real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton NJ, 1993. [23] Torchinsky, A., Real variable methods in harmonic analysis, Pure and Applied Math., Academic Press, New York 123 (1986).

96

L. LANZHE

[24] Torchinsky, A., Wang, S., A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), 235–243. [25] Wu, B.S., Liu, L.Z., A sharp estimate for multilinear Bochner-Riesz operator, Studia Sci. Math. Hungar. 42 (1) (2005), 47–59.

Department of Mathematics, Hunan University, Changsha 410082, P. R. of China E-mail: [email protected]