Documenta Math. 607 Some Sharp Weighted Estimates for Multilinear Operators 1

In this paper, we establish a sharp inequality for some multilinear operators related to certain integral operators. The operators include Calderon-Zygmund singular integral operator, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator. As application, we obtain the weighted norm inequalities and L log L type estimate for the multilinear operators.


Introduction
Let T be a singular integral operator.In [1][2] [3], Cohen and Gosselin studied the L p (p > 1) boundedness of the multilinear singular integral operator T A defined by |x − y| m K(x, y)f (y)dy.
In [6], Hu and Yang obtain a variant sharp estimate for the multilinear singular integral operators.The main purpose of this paper is to prove a sharp inequality for some multilinear operators related to certain non-convolution type integral operators.In fact, we shall establish the sharp inequality for the multilinear operators only under certain conditions on the size of the integral operators.
The integral operators include Calderón-Zygmund singular integral operator,

Liu Lanzhe
Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.As applications, we obtain weighted norm inequalities and L log L type estimates for these multilinear operators.

Notations and Results
First, let us introduce some notations(see [6][ [12][13][14]).Throughout this paper, Q will denote a cube of R n with side parallel to the axes.For any locally integrable function f , the sharp function of f is defined by where, and in what follows, It is well-known that(see [6]) We say that f belongs to Let M be the Hardy-Littlewood maximal operator defined by M (f )(x) = sup x∈Q |Q| −1 Q |f (y)|dy, we write M p (f ) = (M (f p )) 1/p for 0 < p < ∞; For k ∈ N , we denote by M k the operator M iterated k times, i.e., M 1 (f )(x) = M (f )(x) and M k (f )(x) = M (M k−1 (f ))(x) for k ≥ 2. Let B be a Young function and B be the complementary associated to B, we denote that, for a function and the maximal function by The main Young function to be using in this paper is B(t) = t(1 + log + t) and its complementary B(t) = expt, the corresponding maximal denoted by M LlogL and M expL .We have the generalized Hölder's inequality(see [12]) and the following inequality (in fact they are equivalent), for any and the following inequalities, for all cubes We denote the Muckenhoupt weights by A p for 1 ≤ p < ∞(see [6]).
We are going to consider some integral operators as following.
Let m be a positive integer and A be a function on R n .We denote that Definition 1.Let S and S ′ be Schwartz space and its dual and T : S → S ′ be a linear operator.Suppose there exists a locally integrable function for every bounded and compactly supported function f .The multilinear operator related to the integral operator T is defined by for every bounded and compactly supported function f and Let H be a Banach space of functions h : [0, +∞) → R. For each fixed x ∈ R n , we view F t (f )(x) and F A t (f )(x) as a mapping from [0, +∞) to H.Then, the multilinear operators related to F t is defined by We also define that S(f Note that when m = 0, T A and S A are just the commutators of T , S and A. While when m > 0, it is non-trivial generalizations of the commutators.It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [1-5][7]).The main purpose of this paper is to prove a sharp inequality for the multilinear operators T A and S A .We shall prove the following theorems in Section 3. Theorem 1.Let D α A ∈ BM O(R n ) for all α with |α| = m.Suppose that T is the same as in Definition 1 such that T is bounded on L p (w) for all w ∈ A p with 1 < p < ∞ and weak bounded of (L 1 (w), L 1 (w)) for all w ∈ A 1 .If T A satisfies the following size condition: for any cube Then for any 0 < r < 1, there exists a constant C > 0 such that for any ) for all α with |α| = m.Suppose that S is the same as in Definition 2 such that S is bounded on L p (w) for all w ∈ A p , 1 < p < ∞ and weak bounded of (L 1 (w), L 1 (w)) for all w ∈ A 1 .If S A satisfies the following size condition: Then for any 0 < r < 1, there exists a constant C > 0 such that for any From the theorems, we get the following Corollary.Let D α A ∈ BM O(R n ) for all α with |α| = m.Suppose that T A , T and S A , S satisfy the conditions of Theorem 1 and Theorem 2.
(a).If w ∈ A p for 1 < p < ∞.Then T A and S A are all bounded on L p (w), that is and Then there exists a constant C > 0 such that for each λ > 0,

Proof of Theorem
To prove the theorems, we need the following lemmas.Lemma 1 (Kolmogorov, [6, p.485]).Let 0 < p < q < ∞ and for any function f ≥ 0. We define that, for 1/r = 1/p − 1/q where the sup is taken for all measurable sets E with 0 < |E| < ∞.Then Then there exists a constant C > 0 such that for any function f and for all λ > 0, Lemma 3.([3, p.448]) Let A be a function on R n and D α A ∈ L q (R n ) for all α with |α| = m and some q > n.Then , where Q is the cube centered at x and having side length 5 √ n|x − y|.
Proof of Theorem 1.
It suffices to prove for f ∈ C ∞ 0 (R n ) and some constant C 0 , the following inequality holds: Now, let us estimate I, II and III, respectively.First, for x ∈ Q and y ∈ Q, using Lemma 3, we get thus, by Lemma 1 and the weak type (1,1) of T , we get For II, similar to the proof of I, we get For III, using Hölder' inequality and the size condition of T , we have This completes the proof of Theorem 1.
Proof of Theorem 2.
It is only to prove for f ∈ C ∞ 0 (R n ) and some constant C 0 , the following inequality holds: Fix a cube Q = Q(x 0 , d) and x ∈ Q.Let Q and Ã(x) be the same as the proof of Theorem 1.We write, for Now, similar to the proof of Theorem 1, we have For JJJ, using the size condition of S, we have This completes the proof of Theorem 2.
From Theorem 1, 2 and the weighted boundedness of T and S, we may obtain the conclusion of Corollary(a).
From Theorem 1, 2 and Lemma 2, we may obtain the conclusion of Corollary(b).

Applications
In this section we shall apply the Theorem 1, 2 and Corollary of the paper to some particular operators such as the Calderón-Zygmund singular integral operator, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.Application 1. Calderón-Zygmund singular integral operator.
Let T be the Calderón-Zygmund operator(see [6][14] [15]), the multilinear operator related to T is defined by Then it is easily to see that T satisfies the conditions in Theorem 1 and Corollary.In fact, it is only to verify that T A satisfies the size condition in Theorem 1, which has done in [6](see also [12][13]).Thus the conclusions of Theorem 1 and Corollary hold for T A .Application 2. Littlewood-Paley operator.Let ε > 0 and ψ be a fixed function which satisfies the following properties: (1) The multilinear Littlewood-Paley operator is defined by(see [8]) , where and ψ t (x) = t −n ψ(x/t) for t > 0. We write F t (f ) = ψ t * f .We also define that , which is the Littlewood-Paley operator(see [15]); Let H be a space of functions h : Then, for each fixed x ∈ R n , F A t (f )(x) may be viewed as a mapping from [0, +∞) to H, and it is clear that It is known that g ψ is bounded on L p (w) for all w ∈ A p , 1 < p < ∞ and weak (L 1 (w), L 1 (w)) bounded for all w ∈ A 1 .Thus it is only to verify that g A ψ satisfies the size condition in Theorem 2. In fact, we write, for a cube By Lemma 3 and the following inequality(see [14]) By the condition on ψ and Minkowski' inequality , we obtain For I 2 , by the formula (see [3]): and Lemma 3, we have similar to the estimates of I 1 , we get For I 3 , similar to the proof of I 1 , we obtain Documenta Mathematica 9 (2004) 607-622 From the above estimates, we know that Theorem 2 and Corollary hold for g A ψ .Application 3. Marcinkiewicz operator.
Let Ω be homogeneous of degree zero on R n and S n−1 Ω(x ′ )dσ(x ′ ) = 0. Assume that Ω ∈ Lip γ (S n−1 ) for 0 < γ ≤ 1, that is there exists a constant M > 0 such that for any x, y ∈ S n−1 , |Ω(x) − Ω(y)| ≤ M |x − y| γ .The multilinear Marcinkiewicz operator is defined by(see [9]) , where we write that We also define that , which is the Marcinkiewicz operator(see [16]); Let H be a space of functions h : Now, we will verify that µ A Ω satisfies the size condition in Theorem 2. In fact, for a cube and For J 3 , by the following inequality (see [16]): we gain Documenta Mathematica 9 (2004) 607-622 For J 4 , similar to the proof of J 1 , J 2 and J 3 , we obtain