Some Sharp Weighted Estimates

for Multilinear Operators^{1}

Liu Lanzhe

Received: August 8, 2004 Revised: November 25, 2004

Communicated by Heinz Siedentop

Abstract. In this paper, we establish a sharp inequality for some multilinear operators related to certain integral operators. The operators include Calderón-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.

2000 Mathematics Subject Classification: 42B20, 42B25. Keywords and Phrases: Multilinear Operator; Calderón-Zygmund operator; Littlewood-Paley operator; Marcinkiewicz operator; Bochner-Riesz operator; Sharp estimate; BMO.

1. 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

T^{A}(f)(x) = ∫_{R^{n}} \frac{R_{m+1}(A; x, y)}{|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, 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.

1Supported by the NNSF (Grant: 10271071)

Documenta Mathematica 9 (2004) 607–622

2. Notations and Results

First, let us introduce some notations(see[6][12-14]). 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

f^{#}(x) = sup_{x∈Q} \frac{1}{|Q|} ∫_{Q} |f(y) − f_{Q}|dy,

where, and in what follows, f_{Q} = |Q|^{−1} ∫_{Q} f(x)dx. It is well-known that(see[6])

f^{#}(x) = sup_{x∈Q} inf_{c∈C} \frac{1}{|Q|} ∫_{Q} |f(y) − c|dy.

We say that f belongs to BMO(R^{n}) if f^{#} belongs to L^{∞}(R^{n}) and ||f||_{BMO} = ||f^{#}||_{L^{∞}}. For 0 < r < ∞, we denote f_{r}^{#} by

f_{r}^{#}(x) = [(|f|^{r})^{#}(x)]^{1/r}.

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 B̃ be the complementary associated to B, we denote that, for a function f

||f||_{B, Q} = inf \{λ > 0 : \frac{1}{|Q|} ∫_{Q} B (\frac{|f(y)|}{λ}) dy ≤ 1\}

and the maximal function by

M_{B}(f)(x) = sup_{x∈Q} ||f||_{B, Q};

The main Young function to be using in this paper is B(t) = t(1 + log^{+}t) and its complementary B̃(t) = expt, the corresponding maximal denoted by M_{LlogL} and M_{expL}. We have the generalized Hölder’s inequality(see[12])

\frac{1}{|Q|} ∫_{Q} |f(y)g(y)|dy ≤ ||f||_{B, Q}||g||_{B, Q}

and the following inequality (in fact they are equivalent), for any x ∈ R^{n},

M_{LlogL}(f)(x) ≤ CM^{2}(f)(x)

and the following inequalities, for all cubes Q any b ∈ BMO(R^{n}),

||b − b_{Q}||_{exp L, Q} ≤ C||b||_{BMO}, |b_{2^{k+1}Q} − b_{2Q}| ≤ 2k||b||_{BMO}.

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

R_{m+1}(A; x, y) = A(x) − ∑_{|α|≤m} \frac{1}{α!}D^{α}A(y)(x − y)^{α}.

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 K(x, y) on R^{n} × R^{n} such that

T (f)(x) = ∫_{R^{n}} K(x, y)f(y)dy

for every bounded and compactly supported function f. The multilinear operator related to the integral operator T is defined by

T^{A}(f)(x) = ∫_{R^{n}} \frac{R_{m+1}(A; x, y)}{|x − y|^{m}}K(x, y)f(y)dy.

Definition 2. Let F (x, y, t) defined on R^{n} × R^{n} × [0, +∞). Set

F_{t}(f)(x) = ∫_{R^{n}} F (x, y, t)f(y)dy

for every bounded and compactly supported function f and

F_{t}^{A}(f)(x) = ∫_{R^{n}} \frac{R_{m+1}(A; x, y)}{|x − y|^{m}}F (x, y, t)f(y)dy.

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_{t}^{A}(f)(x) as a mapping from [0, +∞) to H. Then, the multilinear operators related to F_{t} is defined by

S^{A}(f)(x) = ||F_{t}^{A}(f)(x)||;

We also define that S(f)(x) = ||F_{t}(f)(x)||. 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 ∈ BMO(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:

|T^{A}(f)(x) − T^{A}(f)(x_{0})| ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x)

for any cube Q = Q(x_{0}, d) with suppf ⊂ (2Q)^{c}, x ∈ Q = Q(x_{0}, d). Then for any 0 < r < 1, there exists a constant C > 0 such that for any f ∈ C_{0}^{∞}(R^{n}) and any x ∈ R^{n},

(T^{A}(f))_{r}^{#}(x) ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x).

Theorem 2. Let D^{α}A ∈ BMO(R^{n}) 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:

||F_{t}^{A}(f)(x) − F_{t}^{A}(f)(x_{0})|| ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x)

for any cube Q = Q(x_{0}, d) with suppf ⊂ (2Q)^{c}, x ∈ Q = Q(x_{0}, d). Then for any 0 < r < 1, there exists a constant C > 0 such that for any f ∈ C_{0}^{∞}(R^{n}) and any x ∈ R^{n},

(S^{A}(f))_{r}^{#}(x) ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x).

From the theorems, we get the following Corollary. Let D^{α}A ∈ BMO(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

||T^{A}(f)||_{L^{p}(w)} ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}||f||_{L^{p}(w)}

and

||S^{A}(f)||_{L^{p}(w)} ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}||f||_{L^{p}(w)}.

(b). If w ∈ A_{1}. Then there exists a constant C > 0 such that for each λ > 0,

w({x ∈ R^{n} : |T^{A}(f)(x)| > λ})

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO} ∫_{R^{n}} \frac{|f(x)|}{λ} (1 + log^{+} (\frac{|f(x)|}{λ})) w(x)dx

and

w({x ∈ R^{n} : |S^{A}(f)(x)| > λ})

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO} ∫_{R^{n}} \frac{|f(x)|}{λ} (1 + log^{+} (\frac{|f(x)|}{λ})) w(x)dx.

3. 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

||f||_{W L^{q}} = sup_{λ>0} λ|{x ∈ R^{n} : f(x) > λ}|^{1/q}, N_{p,q}(f) = sup_{E} ||fχ_{E}||_{L^{p}}/||χ_{E}||_{L^{r}},

where the sup is taken for all measurable sets E with 0 < |E| < ∞. Then

||f||_{W L^{q}} ≤ N_{p,q}(f) ≤ (q/(q − p))^{1/p}||f||_{W L^{q}}.

Lemma 2([12, p.165]) Let w ∈ A_{1}. Then there exists a constant C > 0 such that for any function f and for all λ > 0,

w({y ∈ R^{n} : M^{2}f(y) > λ}) ≤ Cλ^{−1} ∫_{R^{n}} |f(y)|(1 + log^{+}(λ^{−1}|f(y)|))w(y)dy.

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

|R_{m}(A; x, y)| ≤ C|x − y|^{m} ∑_{|α|=m} (\frac{1}{|Q̃(x, y)|} ∫_{Q̃(x,y)} |D^{α}A(z)|^{q}dz)^{1/q} ,

where Q̃ is the cube centered at x and having side length 5\sqrt{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:

(\frac{1}{|Q|} ∫_{Q} |T^{A}(f)(x) − C_{0}|^{r}dx)^{1/r} ≤ CM^{2}(f).

Fix a cube Q = Q(x_{0}, d) and x̃ ∈ Q. Let Q̃ = 5\sqrt{n}Q and Ã(x) =

A(x) − ∑_{|α|=m} \frac{1}{α!}(D^{α}A)Q̃x^{α}, then R_{m}(A; x, y) = R_{m}(Ã; x, y) and D^{α}Ã =

D^{α}A − (D^{α}A)Q̃ for |α| = m. We write, for f_{1} = fχQ̃ and f_{2} = fχ_{R^{n}\Q̃},

T^{A}(f)(x) = ∫_{R^{n}} \frac{R_{m+1}(A; x, y)}{|x − y|^{m}}K(x, y)f(y)dy
= ∫_{R^{n}} \frac{R_{m+1}(A; x, y)}{|x − y|^{m}}K(x, y)f_{2}(y)dy

Documenta Mathematica 9 (2004) 607–622

+ ∫_{R^{n}} \frac{R_{m}(Ã; x, y)}{|x − y|^{m}}K(x, y)f_{1}(y)dy

− ∑_{|α|=m} \frac{1}{α!} ∫_{R^{n}} \frac{K(x, y)(x − y)^{α}}{|x − y|^{m}} D^{α}Ã(y)f_{1}(y)dy

then

|T^{A}(f)(x) − T^{A}(f_{2})(x_{0})|
≤ |T (\frac{R_{m}(Ã; x, ·)}{|x − ·|^{m}}f_{1}) (x)| + ∑_{|α|=m} \frac{1}{α!} |T (\frac{(x − ·)^{α}}{|x − ·|^{m}} D^{α}Ãf_{1}) (x)|
+ |T^{A}(f_{2})(x) − T^{A}(f_{2})(x_{0})|
:= I(x) + II(x) + III(x),

thus,

(\frac{1}{|Q|} ∫_{Q} |T^{A}(f)(x) − T^{A}(f_{2})(x_{0})|^{r} dx)^{1/r}

≤ (\frac{C}{|Q|} ∫_{Q} I(x)^{r}dx)^{1/r} + (\frac{C}{|Q|} ∫_{Q} II(x)^{r}dx)^{1/r} + (\frac{C}{|Q|} ∫_{Q} III(x)^{r}dx)^{1/r}
:= I + II + III.

Now, let us estimate I, II and III, respectively. First, for x ∈ Q and y ∈ Q̃, using Lemma 3, we get

R_{m}(Ã; x, y) ≤ C|x − y|^{m} ∑_{|α|=m} ||D^{α}A||_{BMO},

thus, by Lemma 1 and the weak type (1,1) of T, we get

I ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}|Q|^{−1} \frac{||T (f_{1})χ_{Q}||_{L^{r}}}{||χ_{Q}||_{L^{r/(1−r)}}}
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}|Q|^{−1}||T (f_{1})||_{W L^{1}}

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}|Q̃|^{−1} ∫Q̃ |f(y)|dy ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x̃);

For II, similar to the proof of I, we get

II ≤ C ∑_{|α|=m} |Q|^{−1} \frac{||T (D^{α}Ãf_{1})χ_{Q}||_{L^{r}}}{||χ_{Q}||_{L^{r/(1−r)}}} ≤ C ∑_{|α|=m} |Q|^{−1}||T (D^{α}Ãf_{1})||_{W L^{1}}

≤ C ∑_{|α|=m} |Q̃|^{−1} ∫Q̃ |D^{α}Ã(y)||f(y)|dy ≤ C ∑_{|α|=m} ||D^{α}A||_{exp L,Q̃}||f||_{LlogL,Q̃}
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M_{L log L}(f)(x̃) ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x̃);

Documenta Mathematica 9 (2004) 607–622

For III, using Hölder’ inequality and the size condition of T, we have

III ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x̃).

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:

(\frac{1}{|Q|} ∫_{Q} |S^{A}(f)(x) − C_{0}|^{r}dx)^{1/r} ≤ CM^{2}(f).

Fix a cube Q = Q(x_{0}, d) and x̃ ∈ Q. Let Q̃ and Ã(x) be the same as the proof of Theorem 1. We write, for f_{1} = fχQ̃ and f_{2} = fχ_{R^{n}\Q̃},

F_{t}^{A}(f)(x) = ∫_{R^{n}} \frac{R_{m}(Ã; x, y)}{|x − y|^{m}}F (x, y, t)f_{1}(y)dy

− ∑_{|α|=m} \frac{1}{α!} ∫_{R^{n}} \frac{F (x, y, t)(x − y)^{α}}{|x − y|^{m}} D^{α}Ã(y)f_{1}(y)dy

+ ∫_{R^{n}} \frac{R_{m+1}(A; x, y)}{|x − y|^{m}}F (x, y, t)f_{2}(y)dy,

then

|S^{A}(f)(x) − S^{A}(f_{2})(x_{0})| = |||F_{t}^{A}(f)(x)|| − ||F_{t}^{A}(f_{2})(x_{0})|||
≤ ||F_{t}^{A}(f)(x) − F_{t}^{A}(f_{2})(x_{0})||
≤ ||F_{t} (\frac{R_{m}(Ã; x, ·)}{|x − ·|^{m}}f_{1}) (x)|| + ∑_{|α|=m} \frac{1}{α!} ||F_{t} (\frac{(x − ·)^{α}}{|x − ·|^{m}} D^{α}Ãf_{1}) (x)||
+||F_{t}^{A}(f_{2})(x) − F_{t}^{A}(f_{2})(x_{0})||
:= J(x) + JJ(x) + JJJ(x),

thus,

(\frac{1}{|Q|} ∫_{Q} |S^{A}(f)(x) − S^{A}(f_{2})(x_{0})|^{r}dx)^{1/r}

≤ (\frac{C}{|Q|} ∫_{Q} J(x)^{r}dx)^{1/r} + (\frac{C}{|Q|} ∫_{Q} JJ(x)^{r}dx)^{1/r} + (\frac{C}{|Q|} ∫_{Q} JJJ(x)^{r}dx)^{1/r}
:= J + JJ + JJJ.

Now, similar to the proof of Theorem 1, we have

J ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO} \frac{1}{|Q̃|} ∫Q̃ |f(x)|dx ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x̃)

and

JJ ≤ C ∑_{|α|=m} |Q|^{−1} \frac{||S(D^{α}Ãf_{1})χ_{Q}||_{L^{r}}}{||χ_{Q}||_{L^{r/(1−r)}}} ≤ C ∑_{|α|=m} |Q|^{−1}||S(D^{α}Ãf_{1})||_{W L^{1}}

≤ C ∑_{|α|=m} |Q̃|^{−1} ∫Q̃ |D^{α}Ã(y)||f(y)|dy ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x̃);

For JJJ, using the size condition of S, we have

JJJ ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x̃).

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).

4. 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

T^{A}(f)(x) = ∫ \frac{R_{m+1}(A; x, y)}{|x − y|^{m}}K(x, y)f(y)dy.

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) ∫_{R^{n}} ψ(x)dx = 0,

(2) |ψ(x)| ≤ C(1 + |x|)^{−(n+1)},

(3) |ψ(x + y) − ψ(x)| ≤ C|y|^{ε}(1 + |x|)^{−(n+1+ε)} when 2|y| < |x|; The multilinear Littlewood-Paley operator is defined by(see[8])

g_{ψ}^{A}(f)(x) = (∫_{0}^{∞} |F_{t}^{A}(f)(x)|^{2} \frac{dt}{t})^{1/2} ,

where

F_{t}^{A}(f)(x) = ∫_{R^{n}} \frac{R_{m+1}(A; x, y)}{|x − y|^{m}}ψ_{t}(x − y)f(y)dy

Documenta Mathematica 9 (2004) 607–622

and ψ_{t}(x) = t^{−n}ψ(x/t) for t > 0. We write F_{t}(f) = ψ_{t} ∗ f. We also define that

g_{ψ}(f)(x) = (∫_{0}^{∞} |F_{t}(f)(x)|^{2} \frac{dt}{t})^{1/2} ,

which is the Littlewood-Paley operator(see [15]); Let H be a space of functions h : [0, +∞) → R, normed by ||h|| = (∫_{0}^{∞} |h(t)|^{2}dt/t)^{1/2} < ∞. Then, for each fixed x ∈ R^{n}, F_{t}^{A}(f)(x) may be viewed as a mapping from [0, +∞) to H, and it is clear that

g_{ψ}(f)(x) = ||F_{t}(f)(x)|| and g_{ψ}^{A}(f)(x) = ||F_{t}^{A}(f)(x)||.

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_{ψ}^{A} satisfies the size condition in Theorem 2. In fact, we write, for a cube Q = Q(x_{0}, d) with suppf ⊂ (Q̃)^{c}, x ∈ Q = Q(x_{0}, d),

F_{t}^{A}(f)(x) − F_{t}^{A}(f)(x_{0})

= ∫_{R^{n}} (\frac{ψ_{t}(x − y)}{|x − y|^{m}} − \frac{ψ_{t}(x_{0} − y)}{|x_{0} − y|^{m}}) R_{m}(Ã; x, y)f(y)dy

+ ∫_{R^{n}} \frac{ψ_{t}(x_{0} − y)}{|x_{0} − y|^{m}} (R_{m}(Ã; x, y) − R_{m}(Ã; x_{0}, y))f(y)dy

− ∑_{|α|=m} \frac{1}{α!} ∫_{R^{n}} (\frac{(x − y)^{α}ψ_{t}(x − y)}{|x − y|^{m}} − \frac{(x_{0} − y)^{α}ψ_{t}(x_{0} − y)}{|x_{0} − y|^{m}}) D^{α}Ã(y)f(y)dy

:= I_{1} + I_{2} + I_{3}.

By Lemma 3 and the following inequality(see[14])

|b_{Q_{1}} − b_{Q_{2}}| ≤ C log(|Q_{2}|/|Q_{1}|)||b||_{BMO}, forQ_{1} ⊂ Q_{2},

we know that, for x ∈ Q and y ∈ 2^{k+1}Q \ 2^{k}Q with k ≥ 1,

|R_{m}(Ã; x, y)| ≤ C|x − y|^{m} ∑_{|α|=m}(||D^{α}A||_{BMO} + |(D^{α}A)_{Q̃(x,y)} − (D^{α}A)Q̃|)
≤ Ck|x − y|^{m} ∑_{|α|=m} ||D^{α}A||_{BMO}.

Note that |x − y| ∼ |x_{0} − y| for x ∈ Q and y ∈ R^{n} \ Q. By the condition on ψ and Minkowski’ inequality , we obtain

||I_{1}|| ≤ C ∫_{R^{n}} \frac{|R_{m}(Ã; x, y)||f(y)|}{|x_{0} − y|^{m}}

[∫_{0}^{∞} (\frac{t|x − x_{0}|}{|x_{0} − y|(t + |x_{0} − y|)^{n+1}} + \frac{t|x − x_{0}|^{ε}}{(t + |x_{0} − y|)^{n+1+ε}} )^{2} \frac{dt}{t}]^{1/2} dy

Documenta Mathematica 9 (2004) 607–622

616 Liu Lanzhe

≤ C ∫_{(2Q)^{c}} (\frac{|x − x_{0}|}{|x_{0} − y|^{m+n+1}} + \frac{|x − x_{0}|^{ε}}{|x_{0} − y|^{m+n+ε}} ) |R_{m}(Ã; x, y)||f(y)|dy
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}

∑_{k=1}^{∞} ∫_{2^{k+1}Q\2^{k}Q} k (\frac{|x − x_{0}|}{|x_{0} − y|^{n+1}} + \frac{|x − x_{0}|^{ε}}{|x_{0} − y|^{n+ε}} ) |f(y)|dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO} ∑_{k=1}^{∞} k(2^{−k} + 2^{−εk})|2^{k+1}Q|^{−1} ∫_{2^{k+1}Q} |f(y)|dy
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x);

For I_{2}, by the formula (see [3]):

R_{m}(Ã; x, y) − R_{m}(Ã; x_{0}, y) = ∑_{|β|<m} \frac{1}{β!}R_{m−|β|}(D^{β}Ã; x, x_{0})(x − y)^{β}

and Lemma 3, we have

|R_{m}(Ã; x, y)−R_{m}(Ã; x_{0}, y)| ≤ C ∑_{|β|<m} ∑_{|α|=m} |x−x_{0}|^{m−|β|}|x−y|^{|β|}||D^{α}A||_{BMO},

similar to the estimates of I_{1}, we get

||I_{2}|| ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO} ∑_{k=1}^{∞} ∫_{2^{k+1}\2^{k}Q} \frac{k|x − x_{0}|}{|x_{0} − y|^{n+1}} |f(y)|dy
≤ C||D^{α}A||_{BMO} ∑_{k=1}^{∞} k2^{−k}|2^{k+1}Q|^{−1} ∫_{2^{k+1}Q} |f(y)|dy
≤ C||D^{α}A||_{BMO}M(f)(x);

For I_{3}, similar to the proof of I_{1}, we obtain

||I_{3}|| ≤ C ∑_{|α|=m} ∑_{k=1}^{∞} ∫_{2^{k+1}\2^{k}Q} (\frac{|x − x_{0}|}{|x_{0} − y|^{n+1}} + \frac{|x − x_{0}|^{ε}}{|x_{0} − y|^{n+ε}} ) |D^{α}Ã(y)||f(y)|dy
≤ C ∑_{|α|=m} ∑_{k=1}^{∞} k(2^{−k} + 2^{−εk}) \frac{1}{|2^{k+1}Q|} ∫_{2^{k+1}Q} |D^{α}Ã(y)||f(y)|dy
≤ C ∑_{|α|=m} ∑_{k=1}^{∞} k(2^{−k} + 2^{−εk})

(||D^{α}A||_{exp L,2^{k+1}Q}||f||_{LlogL,2^{k+1}Q} + ||D^{α}A||_{BMO}M(f)(x))

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}(M_{L log L}(f)(x) + M(f)(x))
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x).

From the above estimates, we know that Theorem 2 and Corollary hold for g_{ψ}^{A}. 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])

μ_{Ω}^{A}(f)(x) = (∫_{0}^{∞} |F_{t}^{A}(f)(x)|^{2} \frac{dt}{t^{3}})^{1/2} ,

where

F_{t}^{A}(f)(x) = ∫_{|x−y|≤t} \frac{Ω(x − y)}{|x − y|^{n−1}} \frac{R_{m+1}(A; x, y)}{|x − y|^{m}}f(y)dy,

we write that

F_{t}(f)(x) = ∫_{|x−y|≤t} \frac{Ω(x − y)}{|x − y|^{n−1}} f(y)dy.

We also define that

μ_{Ω}(f)(x) = (∫_{0}^{∞} |F_{t}(f)(x)|^{2} \frac{dt}{t^{3}})^{1/2} ,

which is the Marcinkiewicz operator(see [16]); Let H be a space of functions h : [0, +∞) → R, normed by ||h|| = (∫_{0}^{∞} |h(t)|^{2}dt/t^{3})^{1/2} < ∞. Then, it is clear that

μ_{Ω}(f)(x) = ||F_{t}(f)(x)|| and μ_{Ω}^{A}(f)(x) = ||F_{t}^{A}(f)(x)||.

Now, we will verify that μ_{Ω}^{A} satisfies the size condition in Theorem 2. In fact, for a cube Q = Q(x_{0}, d) with suppf ⊂ (2Q)^{c}, x ∈ Q = Q(x_{0}, d), we have

||F_{t}^{A}(f)(x) − F_{t}^{A}(f)(x_{0})||

≤ (∫_{0}^{∞} |∫_{|x−y|≤t} \frac{Ω(x − y)R_{m}(Ã; x, y)}{|x − y|^{m+n−1}}f(y)dy

− ∫_{|x_{0}−y|≤t} \frac{Ω(x_{0} − y)R_{m}(Ã; x_{0}, y)}{|x_{0} − y|^{m+n−1}}f(y)dy|^{2} \frac{dt}{t^{3}} )^{1/2}

+ ∑_{|α|=m} (∫_{0}^{∞} |∫_{|x−y|≤t} (\frac{Ω(x − y)(x − y)^{α}}{|x − y|^{m+n−1}}

− ∫_{|x_{0}−y|≤t} \frac{Ω(x_{0} − y)(x_{0} − y)^{α}}{|x_{0} − y|^{m+n−1}} ) D^{α}Ã(y)f(y)dy|^{2} \frac{dt}{t^{3}} )^{1/2}

Documenta Mathematica 9 (2004) 607–622

≤ (∫_{0}^{∞} [∫_{|x−y|≤t, |x_{0}−y|>t} \frac{|Ω(x − y)||R_{m}(Ã; x, y)|}{|x − y|^{m+n−1}}|f(y)|dy]^{2} \frac{dt}{t^{3}} )^{1/2}

+ (∫_{0}^{∞} [∫_{|x−y|>t, |x_{0}−y|≤t} \frac{|Ω(x_{0} − y)||R_{m}(Ã; x_{0}, y)|}{|x_{0} − y|^{m+n−1}}|f(y)|dy]^{2} \frac{dt}{t^{3}} )^{1/2}

+ (∫_{0}^{∞} [∫_{|x−y|≤t,|x_{0}−y|≤t} |\frac{Ω(x − y)R_{m}(Ã; x, y)}{|x − y|^{m+n−1}}

−\frac{Ω(x_{0} − y)R_{m}(Ã; x_{0}, y)}{|x_{0} − y|^{m+n−1}} | |f(y)|dy]^{2} \frac{dt}{t^{3}} )^{1/2}

+ ∑_{|α|=m} (∫_{0}^{∞} |∫_{|x−y|≤t} (\frac{Ω(x − y)(x − y)^{α}}{|x − y|^{m+n−1}}

− ∫_{|x_{0}−y|≤t} \frac{Ω(x_{0} − y)(x_{0} − y)^{α}}{|x_{0} − y|^{m+n−1}} ) D^{α}Ã(y)f(y)dy|^{2} \frac{dt}{t^{3}} )^{1/2}
:= J_{1} + J_{2} + J_{3} + J_{4}

and

J_{1} ≤ C ∫_{R^{n}} \frac{|f(y)||R_{m}(Ã; x, y)|}{|x − y|^{m+n−1}} (∫_{|x−y|≤t<|x_{0}−y|} \frac{dt}{t^{3}})^{1/2} dy
≤ C ∫_{R^{n}} \frac{|f(y)||R_{m}(Ã; x, y)|}{|x − y|^{m+n−1}} (\frac{1}{|x − y|^{2}} − \frac{1}{|x_{0} − y|^{2}} )^{1/2} dy
≤ C ∫_{(2Q)^{c}} \frac{|f(y)||R_{m}(Ã; x, y)|}{|x − y|^{m+n−1}} \frac{|x_{0} − x|^{1/2}}{|x − y|^{3/2}} dy
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO} ∑_{k=1}^{∞} k2^{−k/2}|2^{k+1}Q|^{−1} ∫_{2^{k+1}Q} |f(y)|dy
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x),

similarly, we have J_{2} ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x); For J_{3}, by the following inequality (see [16]):

|\frac{Ω(x − y)}{|x − y|^{n−1}} − \frac{Ω(x_{0} − y)}{|x_{0} − y|^{n−1}} | ≤ C (\frac{|x − x_{0}|}{|x_{0} − y|^{n}} + \frac{|x − x_{0}|^{γ}}{|x_{0} − y|^{n−1+γ}} ) ,

we gain

Documenta Mathematica 9 (2004) 607–622

J_{3} ≤ C ∑_{|α|=m} ||D^{α}A||_{BMO} ∫_{(2Q)^{c}} (\frac{|x − x_{0}|}{|x_{0} − y|^{n}} + \frac{|x − x_{0}|^{γ}}{|x_{0} − y|^{n−1+γ}} )

(∫_{|x_{0}−y|≤t,|x−y|≤t} \frac{dt}{t^{3}})^{1/2} |f(y)|dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO} ∑_{k=1}^{∞} k(2^{−k} + 2^{−γk})M(f)(x)
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x);

For J_{4}, similar to the proof of J_{1}, J_{2} and J_{3}, we obtain

||J_{4}|| ≤ C ∑_{|α|=m} ∑_{k=1}^{∞} ∫_{2^{k+1}Q\2^{k}Q} (\frac{|x − x_{0}|}{|x_{0} − y|^{n+1}}

+\frac{|x − x_{0}|^{1/2}}{|x_{0} − y|^{n+1/2}} + \frac{|x − x_{0}|^{γ}}{|x_{0} − y|^{n+γ}} ) |D^{α}Ã(y)||f(y)|dy

≤ C ∑_{|α|=m} ∑_{k=1}^{∞} k(2^{−k} + 2^{−k/2} + 2^{−γk})\frac{1}{|2^{k+1}Q|} ∫_{2^{k+1}Q} |D^{α}Ã(y)||f(y)|dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x).

Thus, Theorem 2 and Corollary hold for μ_{Ω}^{A}. Application 4. Bochner-Riesz operator. Let B_{t}^{δ}(f ˆ)(ξ) = (1 − t^{2}|ξ|^{2})_{+}^{δ}f̂(ξ). Denote

B_{δ, t}^{A}(f)(x) = ∫_{R^{n}} \frac{R_{m+1}(A; x, y)}{|x − y|^{m}}B_{t}^{δ}(x − y)f(y)dy,

where B_{t}^{δ}(z) = t^{−n}B^{δ}(z/t) for t > 0. The maximal multilinear Bochner-Riesz operator is defined by(see[9])

B_{δ,∗}^{A}(f)(x) = sup_{t>0} |B_{δ, t}^{A}(f)(x)|.

We also define

B_{∗}^{δ}(f)(x) = sup_{t>0} |B_{t}^{δ}(f)(x)|,

which is the maximal Bochner-Riesz operator (see [10][11]). Let H be the space of functions h(t) such that ||h|| = sup_{t>0} |h(t)| < ∞, where

h(t) maps [0, +∞) to H. Then it is clear that

B_{∗}^{δ}(f)(x) = ||B_{t}^{δ}(f)(x)|| and B_{δ,∗}^{A}(f)(x) = ||B_{δ, t}^{A}(f)(x)||.

620 Liu Lanzhe

Now, we will verify that B_{δ,∗}^{A} satisfies the size condition in Theorem 2. In fact, for a cube Q = Q(x_{0}, d) with suppf ⊂ (2Q)^{c}, x ∈ Q = Q(x_{0}, d), we have

BÃ ; t,δ(f)(x) − BÃ ; t,δ(f)(x_{0}) = ∫_{R^{n}} [\frac{B_{t}^{δ}(x−y)}{|x−y|^{m}} − \frac{B_{t}^{δ}(x_{0}−y)}{|x_{0}−y|^{m}}] R_{m}(Ã; x, y)f(y)dy

+ ∫_{R^{n}} \frac{B_{t}^{δ}(x_{0}−y)}{|x_{0}−y|^{m}}[R_{m}(Ã; x, y) − R_{m}(Ã; x_{0}, y)]f(y)dy

− ∑_{|α|=m} \frac{1}{α!} ∫_{R^{n}} (\frac{B_{t}^{δ}(x−y)(x−y)^{α}}{|x−y|^{m}} − \frac{B_{t}^{δ}(x_{0}−y)(x_{0}−y)^{α}}{|x_{0}−y|^{m}} ) D^{α}Ã(y)f(y)dy

= L_{1} + L_{2} + L_{3}.

Consider the following two cases: Case 1. 0 < t ≤ d. In this case, notice that (see [11])

|B^{δ}(z)| ≤ c(1 + |z|)^{−(δ+(n+1)/2)},

we obtain

|L_{1}| ≤ Ct^{−n} ∫_{R^{n}\Q̃} \frac{|f(y)||R_{m}(Ã; x, y)|}{|x_{0} − y|^{m}} (1 + |x − y|/t)^{−(δ+(n+1)/2)}dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}t^{−n} ∑_{k=0}^{∞} k ∫_{2^{k+1}Q̃\2^{k}Q̃} |f(y)||(1 + |x − y|/t)^{−(δ+(n+1)/2)}dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}(t/d)^{δ−(n−1)/2} ∑_{k=1}^{∞} k2^{k((n−1)/2−δ)}M(f)(x)
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x),

|L_{2}| ≤ Ct^{−n} ∫_{R^{n}\Q̃} \frac{|f(y)||R_{m}(Ã; x, y) − R_{m}(Ã; x_{0}, y)|}{|x_{0} − y|^{m}} (1 + |x − y|/t)^{−(δ+(n+1)/2)}dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}t^{−n}

∑_{k=0}^{∞} ∫_{2^{k+1}Q̃\2^{k}Q̃} \frac{|x − x_{0}||f(y)|}{|x_{0} − y|} (1 + |x − y|/t)^{−(δ+(n+1)/2)}dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x),

|L_{3}| ≤ C ∑_{|α|=m} t^{−n} ∑_{k=0}^{∞} ∫_{2^{k+1}Q̃\2^{k}Q̃} |f(y)||D^{α}Ã(y)|(1 + |x_{0} − y|/t)^{−(δ+(n+1)/2)}dy

≤ C ∑_{|α|=m}(t/d)^{δ− \frac{n−1}{2}} ∑_{k=0}^{∞} 2^{k( \frac{n−1}{2} −δ)} \frac{1}{|2^{k+1}Q̃|} ∫_{2^{k+1}Q̃} |f(y)||D^{α}A(y) − (D^{α}A)Q̃|dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x).

Documenta Mathematica 9 (2004) 607–622

Case 2. t > d. In this case, we choose δ_{0} such that (n − 1)/2 < δ_{0} < min(δ, (n + 1)/2), notice that (see [11])

|B^{δ}(x − y) − B^{δ}(x_{0} − y)| ≤ C|x − x_{0}|(1 + |x − y|)^{−(δ+(n+1)/2)},

similar to the proof of Case 1, we obtain

|L_{1}| ≤ Ct^{−n} ∫_{R^{n}\Q̃} \frac{|f(y)||R_{m}(Ã; x, y)|}{|x_{0} − y|^{m+1}} |x_{0} − x|(1 + |x_{0} − y|/t)^{−(δ_{0}+(n+1)/2)}dy

+Ct^{−n−1} ∫_{R^{n}\Q̃} \frac{|f(y)||R_{m}(Ã; x, y)|}{|x_{0} − y|^{m}} |x_{0} − x|(1 + |x_{0} − y|/t)^{−(δ_{0}+(n+1)/2)}dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}(d/t)^{(n+1)/2−δ_{0}} ∑_{k=1}^{∞} k2^{k((n−1)/2−δ_{0})}M(f)(x)
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x),

|L_{2}| ≤ Ct^{−n} ∫_{R^{n}\Q̃}\frac{|f(y)||R_{m}(Ã; x, y) − R_{m}(Ã; x_{0}, y)|}{|x_{0} − y|^{m}} (1 + |x_{0} − y|/t)^{−(δ_{0}+(n+1)/2)}dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}(d/t)^{(n+1)/2−δ_{0}} ∑_{k=1}^{∞} 2^{k((n−1)/2−δ_{0})}M(f)(x)
≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M(f)(x),

|L_{3}| ≤ C ∑_{|α|=m}(d/t)^{(n+1)/2−δ_{0}} ∑_{k=0}^{∞} 2^{k((n−1)/2−δ_{0})} \frac{1}{|2^{k+1}Q̃|} ∫_{2^{k+1}Q̃} |f(y)||D^{α}Ã(y)|dy

≤ C ∑_{|α|=m} ∑_{k=1}^{∞} k2^{k((n−1)/2−δ_{0})} \frac{1}{|2^{k+1}Q̃|} ∫_{2^{k+1}Q̃} |f(y)||D^{α}A(y) − (D^{α}A)Q̃|dy

≤ C ∑_{|α|=m} ||D^{α}A||_{BMO}M^{2}(f)(x).

Thus, Theorem 2 and Corollary hold for B_{δ,∗}^{A}. Acknowledgement. The author would like to express his deep gratitude to the referee for his valuable comments and suggestions.

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Liu Lanzhe

College of Mathematics and Computer Changsha University of Science and Technology Changsha 410077 P.R. of China lanzheliu@263.net