PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 95 (109) (2014), 201–214 DOI: 10.2298/PIM1409201L

ESTIMATES OF MULTILINEAR SINGULAR INTEGRAL OPERATORS AND MEAN OSCILLATION

Lanzhe Liu

Communicated by Stevan Pilipović

Abstract. We prove the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces. The operators include Calderón–Zygmund singular integral operator, Littlewood–Paley operator and Marcinkiewicz operator.

1. Introduction

We are going to consider some integral operators as follows. Let m be a positive integer and A be a function on R^{n}. We denote

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

Definition 1.1. Let T : S → S′ be a linear operator and there exists a locally integrable function K(x, y) on R^{n} × R^{n} ∖ {x = y} such that

T f(x) = ∫_{R^{n}} K(x, y)f(y) dy

for every bounded and compactly supported function f and x ∉ suppf, where K satisfies: |K(x, y)| ⩽ C|x − y|^{−n} and for fixed 0 < ε ⩽ 1,

|K(y, x) − K(z, x)| ⩽ C|y − z|^{ε}|x − z|^{−n−ε}

if 2|y − z| ⩽ |x − z|. The multilinear operator related to the integral operator T is defined by

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

(13K013).

2010 Mathematics Subject Classification: 42B20; 42B25.

Supported by the Scientific Research Fund of Hunan Provincial Education Departments

Definition 1.2. Let F_{t}(x, y) be defined on R^{n} × R^{n} × [0, +∞); we denote
F_{t}(f)(x) = ∫_{R^{n}} F_{t}(x, y)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_{t}(x, y)f(y) dy.

Let H = {h : ||h|| < ∞} be a Banach space. For each fixed x ∈ R^{n}, we consider F_{t}(f)(x) and F_{t}^{A}(f)(x) as mappings from [0, +∞) to H. Then, the multilinear operators related to F_{t} are defined by S^{A}(f)(x) = ∥F_{t}^{A}(f)(x)∥, where ∥F_{t}(x, y)∥ ⩽ C|x − y|^{−n} and for fixed ε > 0

∥F_{t}(y, x) − F_{t}(z, x)∥ ⩽ C|y − z|^{ε}|x − z|^{−n−ε}

if 2|y − z| ⩽ |x − z|. We also define S(f)(x) = ∥F_{t}(f)(x)∥.

Note that when m = 0, T^{A} and S^{A} are just the commutators of T and S with A (see [4, 7–9, 12]). While when m > 0, they are nontrivial generalizations of the commutators. Let T be the Calderón–Zygmund singular integral operator. A classical result of Coifman, Rochberg and Weiss [4] states that the commutator [b, T] = T (bf) − bT f (where b ∈ BMO(R^{n})) is bounded on L^{p}(R^{n}) for 1 < p < ∞. Chanillo [1] proves a similar result when T is replaced by a fractional integral operator. In [8], the boundedness properties for the commutators related to the Calderón–Zygmund singular integral operators from Lebesgue spaces to Orlicz ones are obtained. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [2,3,5]). Our main purpose is to prove the boundedness properties for the multilinear operators T^{A} and S^{A} from Lebesgue spaces to Orlicz ones.

Let us introduce some notations. Throughout the paper, Q will denote a cube of R^{n} with sides parallel to the axes. For any locally integrable function f, the sharp function of f is defined by

f^{#} (x) = sup_{Q∋x} \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 [6] that

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

Let M be the Hardy–Littlewood maximal operator defined by

M(f)(x) = sup_{Q∋x} |Q|^{−1} ∫_{Q} |f(y)| dy.

We write M_{p}f = (M(f^{p}))^{1/p}. For 1 ⩽ r < ∞ and 0 ⩽ β < n, let

M_{β,r} (f)(x) = sup_{Q∋x} ( \frac{1}{|Q|^{1−rβ/n}} ∫_{Q} |f(y)|^{r}dy) ^{1/r} .

We say that f belongs to BMO(R^{n}) if f^{#} belongs to L^{∞} (R^{n}) and ∥f∥_{BMO} = ∥f^{#} ∥_{L^{∞}} . More generally, let φ be a nondecreasing positive function and define BMO_{φ} (R^{n}) as the space of all the functions f satisfying

\frac{1}{|Q(x, r)|} ∫_{Q(x,r)} |f(y) − f_{Q}| dy ⩽ Cφ(r).

For β > 0, the Lipschitz space Lip_{β}(R^{n}) is the space of functions f satisfying

∥f∥_{Lip_{β}} = sup_{x≠y}|f(x) − f(y)|/|x − y|^{β} < ∞.

For an f, let m_{f} (t) = |{x ∈ R^{n} : |f(x)| > t}| denote the distribution function of f.

Let ψ be a nondecreasing convex function on R^{+} with ψ(0) = 0 and let ψ^{−1} be its inverse function. The Orlicz space L_{ψ} (R^{n}) is defined by the set of functions f such that ∫ ψ(λ|f(x)|) dx < ∞ for some λ > 0. The norm is given by ∥f∥_{L_{ψ}} = inf_{λ>0} λ^{−1} (1 + ∫ ψ(λ|f(x)|) dx).

We shall prove the following theorems in Section 2.

Theorem 1.1. Let 1 < p < ∞ and φ, ψ be two nondecreasing positive functions on R^{+} with φ(t) = t^{n/p} ψ^{−1} (t^{−n} ) (or equivalently ψ^{−1} (t) = t^{1/p}φ(t^{−1/n} ). Suppose that ψ is convex, ψ(0) = 0, ψ(2t) ⩽ Cψ(t). Let T be the same as in Definition 1.1 and bounded on L^{p}(R^{n}) for all 1 < p < ∞. Then T^{A} : L^{p}(R^{n}) → L_{ψ} (R^{n}) is bounded if D^{α} A ∈ BMO_{φ} (R^{n}) for all α with |α| = m.

Theorem 1.2. Let 1 < p < ∞ and φ, ψ be two non-decreasing positive functions on R^{+} with φ(t) = t^{n/p} ψ^{−1} (t^{−n} ) (or equivalently ψ^{−1} (t) = t^{1/p}φ(t^{−1/n} ). Suppose that ψ is convex, ψ(0) = 0, ψ(2t) ⩽ Cψ(t). Let S be the same as in Definition 1.2 and bounded on L^{p}(R^{n}) for all 1 < p < ∞. Then S^{A} : L^{p}(R^{n}) → L_{ψ} (R^{n}) is bounded if D^{α} A ∈ BMO_{φ} (R^{n}) for all α with |α| = m.

2. Proofs of Theorems

We begin with the following preliminary lemmas.

Lemma 2.1. [7] Let φ be a nondecreasing positive function on R^{+} and η be an infinitely differentiable function on R^{n} with compact support such that ∫ η(x) dx = 1 . Denote b_{t}(x) = ∫_{R^{n}} b(x − ty)η(y) dy. Then ∥b − b_{t}∥_{BMO} ⩽ Cφ(t)∥b∥_{BMO _{φ}} .

Lemma 2.2. [7] Let 0 < β < 1 and φ be a nondecreasing positive function on R^{+} or β = 1. Then ∥b_{t}∥_{Lip_{β}} ⩽ Ct^{−β} φ(t)∥b∥_{BMO _{φ}} .

Lemma 2.3. [7] Suppose 1 ⩽ p_{2} < p < p_{1} < ∞, ρ is a nonincreasing function on R^{+} , B is a linear operator such that m_{B(f)} (t^{1/p_{1}} ρ(t)) ⩽ Ct^{−1} if ∥f∥_{L^{p_{1}}} ⩽ 1 and m_{B(f)} (t^{1/p_{2}} ρ(t)) ⩽ Ct^{−1} if ∥f∥_{L^{p_{2}}} ⩽ 1. Then ∫_{0}^{∞} m_{B(f)} (t^{1/p}ρ(t)) dt ⩽ C if ∥f∥_{L^{p}} ⩽ (p/p_{1})^{1/p}.

Lemma 2.4. [1] Suppose that 1 ⩽ r < p < n/β and 1/q = 1/p − β/n. Then ∥M_{β,r} (f)∥_{L^{q}} ⩽ C∥f∥_{L^{p}} .

204 LIU

Lemma 2.5. [9] Suppose that 1 ⩽ r < ∞ and b ∈ Lip_{β}. Then

∥(b − b_{Q})fχ_{2Q}∥_{L^{r}} ⩽ C|Q|^{1/r} ∥b∥_{Lip_{β}} M_{β,r} (f)(x)

where χ_{2Q} is the characteristic function of 2Q.

Lemma 2.6. [3] 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 on x and having side length 5\sqrt{n}|x − y|. To prove the theorems of the paper, we need the following Key Lemma 1. Let 0 < β < n, T and S be the same as in Definitions 1.1 and

1.2. Suppose that Q = Q(x_{0}, d) is a cube with suppf ⊂ (2Q)^{c} and x, x̃ ∈ Q. (a) If D^{α} A ∈ BMO(R^{n}) for all α with |α| = m, then

|T^{A}(f)(x) − T^{A}(f)(x_{0})| ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} M_{r}(f)(x̃) for any r > 1;

(b) If D^{α} A ∈ Lip_{β}(R^{n}) for all α with |α| = m, then

|T^{A}(f)(x) − T^{A}(f)(x_{0})| ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} M_{β,1} (f)(x̃);

(c) If D^{α} A ∈ BMO(R^{n}) for all α with |α| = m, then

∥F_{t}^{A}(f)(x) − F_{t}^{A}(f)(x_{0})∥ ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} M_{r}(f)(x̃) for any r > 1;

(d) If D^{α} A ∈ Lip_{β}(R^{n}) for all α with |α| = m, then

∥F_{t}^{A}(f)(x) − F_{t}^{A}(f)(x_{0})∥ ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} M_{β,1} (f)(x̃).

Proof. Let Ã(x) = A(x) − ∑ _{|α|=m} \frac{1}{α!} (D^{α} A)_{Q}x^{α}, then R_{m+1} (A; x, y) = R_{m+1} (Ã; x, y) and D^{α} Ã = D^{α} A − (D^{α} A)_{Q} for |α| = m.

Suppose that suppf ⊂ (2Q)^{c} and x, x̃ ∈ Q = Q(x_{0}, d). Note that |x_{0} − y| ≈ |x − y| for y ∈ (2Q)^{c}. We write

T^{A}(f)(x) − T^{A}(f)(x_{0}) = ∫_{R^{n}} [ \frac{K(x, y)}{|x − y|^{m}} − \frac{K(x_{0}, y)}{|x_{0} − y|^{m}} ] R_{m}(Ã; x, y)f(y) dy

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

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

:= I + II + III.

ESTIMATES OF MULTILINEAR SINGULAR INTEGRAL OPERATORS 205

(a) By Lemma 2.6 and the following inequality (see [10]), for b ∈ BMO(R^{n}), |b_{Q_{1}} − b_{Q_{2}} | ⩽ C log(|Q_{2}|/|Q_{1}|)∥b∥_{BMO} for Q_{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)_{Q} − (D^{α} A)_{Q(x,y)} |)

⩽ Ck|x − y|^{m} ∑_{|α|=m} ∥D^{α} A∥_{BMO} ,

thus

|I| ⩽ C ∫_{R^{n} ∖2Q} ( \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^{−kε} )\frac{1}{|2^{k+1} Q|} ∫_{2^{k+1} Q} |f(y)| dy

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} M(f)(x̃).

For II, by the formula (see [3]):

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

and Lemma 2.6, we get

|II| ⩽ C ∫_{R^{n} ∖2Q} \frac{|R_{m}(Ã; x, y) − R_{m}(Ã; x_{0}, y)|}{|x_{0} − y|^{m+n}} |f(y)| dy

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

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} M(f)(x̃).

For III, similar to the estimates of I, we obtain, for any r > 1 with 1/r + 1/r′ = 1,

|III| ⩽ C ∫_{R^{n} ∖2Q} ( \frac{|x − x_{0}|}{|x_{0} − y|^{n+1}} + \frac{|x − x_{0}|^{ε}}{|x_{0} − y|^{n+ε}} ) |D^{α} A(y) − (D^{α} A)_{Q}∥f(y)| dy

⩽ C ∑_{|α|=m} ∑_{k=1}^{∞} (2^{−k} + 2^{−kε} ) ( \frac{1}{|2^{k+1} Q|} ∫_{2^{k+1} Q} |f(y)|^{r}dy) ^{1/r}

× ( \frac{1}{|2^{k+1} Q|} ∫_{2^{k+1} Q} |D^{α} A(x) − (D^{α} A)_{Q}|^{r′}dx) ^{1/r′}

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} M_{r}(f)(x̃).

Thus

|T^{A}(f)(x) − T^{A}(f)(x_{0})| ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} M_{r}(f)(x̃).

(b) By Lemma 2.6 and the following inequality, for b ∈ Lip_{β},

|b(x) − b_{Q}| ⩽ \frac{1}{|Q|} ∫_{Q} ∥b∥_{Lip_{β}} |x − y|^{β}dy ⩽ ∥b∥_{Lip_{β}} (|x − x_{0}| + d)^{β},

we get

|R_{m}(Ã; x, y)| ⩽ ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} (|x − y| + d)^{m+β} ,

then

|I| ⩽ C ∫_{R^{n} ∖2Q} ( \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∥_{Lip_{β}} ∑_{k=1}^{∞} ∫_{2^{k+1} Q∖2^{k} Q}( \frac{|x − x_{0}|}{|x_{0}−y|^{n+1−β}} + \frac{|x − x_{0}|^{ε}}{|x_{0}−y|^{n+ε−β}} ) |f(y)| dy

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} ∑_{k=1}^{∞} (2^{−k} + 2^{−kε} )\frac{1}{|2^{k+1} Q|^{1−β/n}} ∫_{2^{k+1} Q} |f(y)| dy

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} M_{β,1} (f)(x̃),

|II| ⩽ C ∫_{R^{n} ∖2Q} \frac{|R_{m}(Ã; x, y) − R_{m}(Ã; x_{0}, y)|}{|x_{0} − y|^{m+n}} |f(y)| dy

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} ∑_{k=1}^{∞} ∫_{2^{k+1} Q∖2^{k} Q} \frac{|x − x_{0}|}{|x_{0} − y|^{n+1−β}} |f(y)| dy

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} M_{β,1} (f)(x̃),

|III| ⩽ C ∫_{R^{n} ∖2Q} ( \frac{|x − x_{0}|}{|x_{0} − y|^{n+1−β}} + \frac{|x − x_{0}|^{ε}}{|x_{0} − y|^{n+ε−β}} ) |D^{α} A(y) − (D^{α} A)_{Q}∥f(y)| dy

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} ∑_{k=1}^{∞} (2^{−k} + 2^{−kε} )\frac{1}{|2^{k+1} Q|^{1−β/n}} ∫_{2^{k+1} Q} |f(y)| dy

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} M_{β,1} (f)(x̃).

Thus

|T^{A}(f)(x) − T^{A}(f)(x_{0})| ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} M_{β,1} (f)(x̃).

The same argument as in the proof of (a) and (b) will give the proof of (c) and (d), we omit the details. □

Now we are in the position to prove our theorems.

Proof of Theorem 1.1. We prove it in several steps. First, we prove

(1)  (T^{A}(f))^{#} ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} M_{r}(f)

for any 1 < r < ∞. Fix a cube Q = Q(x_{0}, d) and x̃ ∈ Q. Let Ã(x) = A(x) − ∑ _{|α|=m} \frac{1}{α!} (D^{α} A)_{Q}x^{α}. We write, for f_{1} = fχ_{2Q} and f_{2} = fχ_{R^{n} ∖2Q} ,

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 + ∫_{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

\frac{1}{|Q|} ∫_{Q} |T^{A}(f)(x) − T^{A}(f_{2})(x_{0})| dx ⩽ \frac{1}{|Q|} ∫_{Q} |T ( \frac{R_{m}(Ã; x, ·)}{|x − ·|^{m}}f_{1}) (x)|dx

+\frac{1}{|Q|} ∫_{Q} ∑_{|α|=m} \frac{1}{α!} |T ( \frac{(x − ·)^{α}}{|x − ·|^{m}} D^{α} Ãf_{1}) (x)|dx

+\frac{1}{|Q|} ∫_{Q} |T^{A}(f_{2})(x) − T^{A}(f_{2})(x_{0})| dx := I_{1} + I_{2} + I_{3}.

Now, for I_{1}, if x ∈ Q and y ∈ 2Q, using Lemma 2.6, we get

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

thus, by the L^{r} boundedness of T for any 1 < r < ∞ and Holder’ inequality, we obtain

I_{1} ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} \frac{1}{|Q|} ∫_{Q}|T (f_{1})(x)| dx ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} ∥T (f_{1})∥_{L^{r}} |Q|^{−1/r}

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} ∥f_{1}∥_{L^{r}} |Q|^{−1/r} ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} M_{r}(f)(x̃).

For I_{2}, for any q > 1, l > 1 and denoting r = ql, by L^{q}-boundedness of T, we gain

I_{2} ⩽ \frac{C}{|Q|} ∫_{Q} |T ( ∑_{|α|=m} (D^{α} A − (D^{α} A)_{Q})f_{1}) (x)|dx

⩽ C ∑_{|α|=m} ( \frac{1}{|Q|} ∫_{Q} |T ((D^{α} A − (D^{α} A)_{Q})f_{1})(x)|^{q}dx) ^{1/q}

⩽ C|Q|^{−1/q} ∑_{|α|=m} ∥(D^{α} A − (D^{α} A)_{Q})f_{1}∥_{L^{q}}

⩽ C ∑_{|α|=m} ( \frac{1}{|Q|} ∫_{Q} |D^{α} A(y) − (D^{α} A)_{Q}|^{ql′}dy) ^{1/(ql′)} ( \frac{1}{|Q|} ∫_{Q} |f(y)|^{ql}dy) ^{1/(ql)}

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} M_{r}(f)(x̃).

For I_{3}, by using Key Lemma, we have I_{3} ⩽ C ∑ _{|α|=m} ∥D^{α} A∥_{BMO} M_{r}(f)(x̃). We now put these estimates together, and taking the supremum over all Q such that x̃ ∈ Q, we obtain (T^{A}(f))^{#} (x̃) ⩽ C ∑ _{|α|=m} ∥D^{α} A∥_{BMO} M_{r}(f)(x̃); Thus, taking r such that 1 < r < p, we obtain

(2)  ∥T^{A}(f)∥_{L^{p}} ⩽ C∥(T^{A}(f))^{#} ∥_{L^{p}} ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} ∥M_{r}(f)∥_{L^{p}}

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} ∥f∥_{L^{p}} .

Secondly, we prove that, for D^{α} A ∈ Lip_{β}(R^{n}) with |α| = m

(3)  (T^{A}(f))^{#} ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} (M_{β,r} ((f)) + M_{β,1} (f))

for any 1 ⩽ r < n/β. In fact, by Lemma 2.6, we have, for x ∈ Q and y ∈ 2Q

|R_{m}(Ã; x, y)| ⩽ C|x − y|^{m} ∑_{|α|=m} sup_{z∈2Q} |D^{α} A(z) − (D^{α} A)_{Q}|

⩽ C|x − y|^{m}|Q|^{β/n} ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}}

and by Lemma 2.5, we have

∥(D^{α} A − (D^{α} A)_{2Q})fχ_{2Q}∥_{L^{r}} ⩽ C|Q|^{1/r} ∥D^{α} A∥_{Lip_{β}} M_{β,r} (f)(x).

Similarly to the proof of (1), we obtain

\frac{1}{|Q|} ∫_{Q} |T^{A}(f)(x) − T^{A}(f)(x_{0})| dx ⩽ \frac{1}{|Q|} ∫_{Q} |T ( \frac{R_{m}(Ã; x, ·)}{|x − ·|^{m}}f_{1}) (x)| dx

+ \frac{1}{|Q|} ∫_{Q} ∑_{|α|=m} \frac{1}{α!} |T ( \frac{(x − ·)^{α}}{|x − ·|^{m}} D^{α} Ãf_{1}) (x)|dx

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

⩽ ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} \frac{C}{|Q|^{1/r−β/n}} ( ∫_{Q} |T (f_{1})(x)|^{r}dx) ^{1/r}

+ ∑_{|α|=m} ( \frac{C}{|Q|} ∫_{Q} |T (D^{α} Ãfχ_{2Q})(x)|^{r}dx) ^{1/r}

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

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} \frac{1}{|Q|^{1/r−β/n}} ∥f_{1}∥_{L^{r}}

+ C ∑_{|α|=m} ( \frac{1}{|Q|} ∫_{R^{n}} |(D^{α} A(x) − (D^{α} A)_{Q})f(x)χ_{2Q}(x)|^{r}dx) ^{1/r}

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

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} (M_{β,r} (f)(x̃) + M_{β,1} (f)(x̃));

Thus, taking 1 ⩽ r < p < n/β, 1/q = 1/p − β/n and by Lemma 2.4, we obtain

(4)  ∥T^{A}(f)∥_{L^{q}} ⩽ C∥(T^{A}(f))^{#} ∥_{L^{q}}

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} (∥M_{β,r} (f)∥_{L^{q}} + ∥M_{β,1} (f)∥_{L^{q}} )

⩽ C ∑_{|α|=m} ∥D^{α} A∥_{Lip_{β}} ∥f∥_{L^{p}} .

Now we verify that T^{A} satisfies the conditions of Lemma 2.3. In fact, for any 1 < p_{i} < n/β, 1/q_{i} = 1/p_{i} − β/n (i = 1, 2) and ∥f∥_{L^{p_{i}}} ⩽ 1, note that T^{A}(f)(x) = T^{A−A_{s}} (f)(x)+T^{A_{s}} (f)(x) and D^{α} (A_{s}) = (D^{α} A)_{s}, by (2) and Lemma 2.1, we obtain

∥T^{A−A_{s}} (f)∥_{L^{p_{i}}} ⩽ C ∑_{|α|=m} ∥D^{α} (A − A_{s})∥_{BMO} ∥f∥_{L^{p_{i}}}

⩽ C ∑_{|α|=m} ∥D^{α} A − (D^{α} A)_{s}∥_{BMO} ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO _{φ}} φ(s),

and by (4) and Lemma 2.2, we obtain

∥T^{A_{s}} (f)∥_{L^{q_{i}}} ⩽ C ∑_{|α|=m} ∥(D^{α} A)_{s}∥_{Lip_{β}} ∥f∥_{L^{p_{i}}} ⩽ Cs^{−β} φ(s) ∑_{|α|=m} ∥D^{α} A∥_{BMO _{φ}} .

Thus, for s = t^{−1/n} ,

m_{T^{A} (f)} (t^{1/p_{i}} φ(t^{−1/n} )) ⩽ m_{T^{A−As} (f)} (t^{1/p_{i}} φ(t^{−1/n} )/2) + m_{T^{As} (f)} (t^{1/p_{i}} φ(t^{−1/n} )/2)

⩽ C [( \frac{φ(s)}{t^{1/p_{i}} φ(s)} ) ^{p_{i}} + ( \frac{s^{−β} φ(s)}{t^{1/p_{i}} φ(s)} ) ^{q_{i}} ] = Ct^{−1} .

Taking 1 < p_{2} < p < p_{1} < n/β and by Lemma 2.3, we get, for ∥f∥_{L^{p}} ⩽ (p/p_{1})^{1/p} ,

∫_{R^{n}} ψ(|T^{A}(f)(x)|) dx = ∫_{0}^{∞} m_{T^{A} (f)} (ψ^{−1} (t)) dt ⩽ C,

and thus, ∥T^{A}(f)∥_{L_{ψ}} ⩽ C. This completes the proof of Theorem 1.1. □

Proof of Theorem 1.2. Let Q, Ã(x), f_{1} and f_{2} be the same as the proof of Theorem 1.1, we write

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

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

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

then

\frac{1}{|Q|} ∫_{Q} |S^{A}(f)(x) − S^{A}(f_{2})(x_{0})| dx = \frac{1}{|Q|} ∫_{Q} ∥|F_{t}^{A}(f)(x)∥ − ∥F_{t}^{A}(f_{2})(x_{0})∥| dx

⩽ \frac{1}{|Q|} ∫_{Q} ∥F_{t}( \frac{R_{m}(Ã; x, ·)}{|x − ·|^{m}}f_{1}) (x)∥ dx

+ ∑_{|α|=m} \frac{1}{α!} \frac{1}{|Q|} ∫_{Q} ∥F_{t} ( \frac{(x − ·)^{α}}{|x − ·|^{m}} D^{α} Ãf_{1}) (x)∥ dx

+ \frac{1}{|Q|} ∫_{Q} ∥F_{t}^{A}(f_{2})(x) − F_{t}^{A}(f_{2})(x_{0})∥ dx.

By using the same argument as in the proof of Theorem 1.1 will give the proof of Theorem 1.2, so we omit the details. □

3. Applications

In this section we shall apply Theorems 1.1 and 1.2 to some particular operators such as the Calderón–Zygmund singular integral operator, Littlewood–Paley operator, and Marcinkiewicz operator.

3.1. Calderón–Zygmund singular integral operator. Let T be the Calderón–Zygmund operator [4, 6, 10], i.e., 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 verify that Key Lemma holds for T^{A}, thus T satisfies the conditions in Theorem 1.1. So, the conclusion of Theorem 1.1 holds for T^{A}.

3.2. Littlewood–Paley operator. Let ε > 0 and ψ be a fixed functions satisfying

(1) |ψ(x)| ⩽ C(1 + |x|)^{−(n+1)} ,

(2) |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

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

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

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

which is the Littlewood–Paley g function [11];

Let H = {h : ∥h∥ = (∫_{0}^{∞} |h(t)|^{2}dt/t)^{1/2} < ∞} be space. Then, for each fixed x ∈ R^{n}, F_{t}^{A}(f)(x) may be regarded 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 has been known that g_{ψ} is bounded on L^{p}(R^{n}) for all 1 < p < ∞. Thus it is only to verify that Key Lemma holds for g_{ψ}^{A}. In fact, we write, for a cube Q = Q(x_{0}, d) with suppf ⊂ (2Q)^{c}, x, x̃ ∈ 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

:= J_{1} + J_{2} + J_{3};

By the condition of ψ and Minkowski’s inequality, we obtain, for any r > 1,

∥J_{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

⩽ 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} M(f)(x̃),

∥J_{2}∥ ⩽ C ∑_{|α|=m} ∥D^{α} A∥_{BMO} ∑_{k=1}^{∞} ∫_{2^{k+1} ∖2^{k} Q} \frac{|x − x_{0}|}{|x_{0} − y|^{n+1}} |f(y)| dy

⩽ C∥D^{α} A∥_{BMO} M(f)(x̃),

∥J_{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} ∥D^{α} A∥_{BMO} M_{r}(f)(x̃).

From the above estimates, we see that Theorem 1.2 holds for g_{ψ}^{A}.

3.3. Marcinkiewicz integral 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, i.e., 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 integral operator is defined by

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,

and we write

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

We also define

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

which is the Marcinkiewicz integral [12].

Let H = {h : ∥h∥ = (∫_{0}^{∞} |h(t)|^{2}dt/t^{3})^{1/2} < ∞} be a space. 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, it is only to verify that Key Lemma holds for μ_{Ω}^{A}. In fact, for a cube Q = Q(x_{0}, d) with suppf ⊂ (2Q)^{c}, x, x̃ ∈ Q = Q(x_{0}, d) and r > 1, 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}

⩽ ( ∫_{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}

:= L_{1} + L_{2} + L_{3} + L_{4} and

L_{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} M(f)(x̃).

Similarly, we have L_{2} ⩽ C ∑ _{|α|=m} ∥D^{α} A∥_{BMO} M(f)(x̃).

For L_{3}, by the following inequality [12]:

| \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

L_{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)(x̃);

Similarly to the cases of L_{1}, L_{2} and L_{3}, we obtain for L_{4}

L_{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_{r}(f)(x̃).

Thus, Theorem 1.2 holds for μ_{Ω}^{A}.

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Department of Mathematics (Received 11 06 2011) Hunan University Changsha 410082 P. R. of China

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