Archivum Mathematicum
Liu Lanzhe Mean Oscillation and Boundedness of Multilinear Integral Operators with General Kernels Archivum Mathematicum, Vol. 50 (2014), No. 2, 77--96
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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
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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
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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
MULTILINEAR INTEGRAL OPERATORS
≤
∞ 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
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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.
MULTILINEAR INTEGRAL OPERATORS
95
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Department of Mathematics, Hunan University, Changsha 410082, P. R. of China E-mail:
[email protected]