Some endpoint inequalities for multilinear integral operators
In this paper, the endpoint estimates for some multilinear operators related to certain fractional singular integral operators are obtained. The operators include Calder´on–Zygmund singular integral operator and fractional integral operator.
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Liu, L. 2017-09-29T10:49:55Z 2017-09-29T10:49:55Z 2006 Some endpoint inequalities for multilinear integral operators / L. Liu // Український математичний вісник. — 2006. — Т. 3, № 4. — С. 504-519. — Бібліогр.: 13 назв. — англ. 1810-3200 2000 MSC. 42B20, 42B25. https://nasplib.isofts.kiev.ua/handle/123456789/124565 In this paper, the endpoint estimates for some multilinear operators related to certain fractional singular integral operators are obtained. The operators include Calder´on–Zygmund singular integral operator and fractional integral operator. The author would like to express his deep gratitude to the referee for his valuable comments and suggestions. en Інститут прикладної математики і механіки НАН України Український математичний вісник Some endpoint inequalities for multilinear integral operators Article published earlier |
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Some endpoint inequalities for multilinear integral operators |
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Some endpoint inequalities for multilinear integral operators Liu, L. |
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Some endpoint inequalities for multilinear integral operators |
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Some endpoint inequalities for multilinear integral operators |
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Some endpoint inequalities for multilinear integral operators |
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Some endpoint inequalities for multilinear integral operators |
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some endpoint inequalities for multilinear integral operators |
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Український математичний вісник |
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Інститут прикладної математики і механіки НАН України |
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In this paper, the endpoint estimates for some multilinear operators related to certain fractional singular integral operators are obtained. The operators include Calder´on–Zygmund singular integral operator and fractional integral operator.
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Some endpoint inequalities for multilinear integral operators / L. Liu // Український математичний вісник. — 2006. — Т. 3, № 4. — С. 504-519. — Бібліогр.: 13 назв. — англ. |
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AT liul someendpointinequalitiesformultilinearintegraloperators |
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2025-11-25T22:54:36Z |
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Український математичний вiсник
Том 3 (2006), № 4, 504 – 519
Some endpoint inequalities for multilinear
integral operators
Lanzhe Liu
(Presented by V. Ya. Gutlanskii)
Abstract. In this paper, the endpoint estimates for some multilinear
operators related to certain fractional singular integral operators are
obtained. The operators include Calderón–Zygmund singular integral
operator and fractional integral operator.
2000 MSC. 42B20, 42B25.
Key words and phrases. Multilinear operator, Fractional singular
integral operators, BMO space, Hardy space.
1. Introduction
Let T be the Calderón–Zygmund singular integral operator, the clas-
sical result by Coifman, Rochberg and Weiss (see [6]) states that the com-
mutator [b, T ](f) = T (bf) − bT (f) (where b ∈ BMO(Rn)) is bounded
on Lp(Rn) for 1 < p < ∞; Chanillo (see [1]) has proved a similar result
when T was replaced by the fractional integral operator; in [9], the end-
point boundedness of the commutators was obtained. The main purpose
of this paper is to establish the endpoint boundedness of some multilin-
ear operators related to certain non-convolution type fractional singular
integral operators. As an application, the endpoint boundedness of the
multilinear operators related to the Calderón–Zygmund singular integral
operator and fractional integral operator is obtained.
2. Notations and results
Throughout this paper, Q will denote a cube of Rn with sides par-
allel to the axes. For a cube Q and a locally integrable function f , let
fQ = |Q|−1
∫
Q f(x) dx and f#(x) = supx∈Q |Q|−1
∫
Q |f(y) − fQ| dy. For
Received 28.03.2005
ISSN 1810 – 3200. c© Iнститут математики НАН України
L. Liu 505
a weight function w, f is said to belong to BMO(w) if f# ∈ L∞(w). Set
‖f‖BMO(w) = ‖f#‖L∞(w). Note that BMO(w) = BMO(Rn) if w = 1.
A function a is called an H1 atom if there exists a cube Q such that a is
supported in Q, ‖a‖L∞(w) ≤ w(Q)−1 and
∫
a(x)dx = 0. It is well known
that the Hardy space H1(w) has the atomic decomposition characteriza-
tion (see [8, 12]).
In this paper, we consider a class of multilinear integral operators
defined in the following way.
First, given a fixed locally integrable function K(x, y) on Rn × Rn,
set
TK(f)(x) =
∫
Rn
K(x, y)f(y) dy
for every bounded and compactly supported function f . We write K ∈ Σδ
for δ ≥ 0 if
|K(x, y)| ≤ C|x − y|−n+δ
and
|K(y, x) − K(z, x)| + |K(x, y) − K(x, z)| ≤ C|y − z|ε|x − z|−n−ε+δ
and 2|y − z| ≤ |x− z| for a fixed ε > 0. TK is called a fractional singular
integral operator if K ∈ Σδ for some δ ≥ 0.
Now, let m be a positive integer and A be a function on Rn. Set
Rm+1(A; x, y) = A(x) −
∑
|α|≤m
1
α!
DαA(y)(x − y)α,
and
Qm+1(A; x, y) = Rm(A; x, y) −
∑
|α|=m
1
α!
DαA(x)(x − y)α.
The multilinear operator associated with the fractional singular integral
operator TK is defined by
TA
K(f)(x) =
∫
Rm+1(A; x, y)
|x − y|m K(x, y)f(y) dy.
We also consider the variant of TA
K , which is defined by
T̃A
K(f)(x) =
∫
Rn
Qm+1(A; x, y)
|x − y|m K(x, y)f(y) dy.
506 Some endpoint inequalities...
Note that T̃A
K is closely related to TA
K , for
Rm+1(A; x, y) − Qm+1(A; x, y) =
∑
|α|=m
1
α!
(x − y)α(DαA(x) − DαA(y)).
Note that when m = 0, TA
K is just the commutators of TK and
A(see [1, 6, 9]). It is well known that multilinear operator, as an ex-
tension of commutator, is of great interest in harmonic analysis and has
been widely studied by many authors(see, e.g. [2–5]). In [7] and [10], the
weighted Lp (p > 1) and Hp(0 < p ≤ 1) boundedness of the multilin-
ear operator related to the Calderón–Zygmund singular integral operator
was obtained; in [2], the weak (H1, L1) boundedness of the multilinear
operator related to some singular integral operator was obtained.
Now we state our results as following.
Theorem 2.1. Let 0 ≤ δ < n and DαA ∈ BMO(Rn) for all α with
|α| = m. Suppose TK is bounded from Lp(Rn) to Lq(Rn) for any p, q ∈
(1, +∞) and 1/q = 1/p − δ/n. If K ∈ Σδ, then
(a) TA
K is bounded from Ln/δ(Rn) to BMO(Rn);
(b) T̃A
K is bounded from H1(Rn) to Ln/(n−δ)(Rn);
(c) TA
K is bounded from H1(Rn) to weak Ln/(n−δ)(Rn).
Theorem 2.2. Let DαA ∈ BMO(Rn) for all α with |α| = m and w ∈
A1. Suppose TK is bounded on Lp(w) for all 1 < p ≤ ∞. If K ∈ Σ0,
then
(i) TA
K is bounded from L∞(w) to BMO(w);
(ii) T̃A
K is bounded from H1(w) to L1(w);
(iii) TA
K is bounded from H1(w) to weak L1(w).
Remark 2.1. The boundedness is uniform with respect to K ∈ Σδ
and K ∈ Σ0, respectively. In general, TA
K is not (H1, Ln/(n−δ)) or
(H1(w), L1(w)) bounded.
L. Liu 507
3. Proofs of the theorems
To prove these theorems, we need the following lemmas.
Lemma 3.1 (see [5, p. 448]). Let A be a function on Rn and DαA ∈
Lq(Rn) for |α| = m and some q > n. Then
|Rm(A; x, y)| ≤ C|x − y|m
∑
|α|=m
(
1
|Q̃(x, y)|
∫
Q̃(x,y)
|DαA(z)|q dz
)1/q
,
where Q̃(x, y) is the cube centered at x and having side length 5
√
n|x−y|.
Lemma 3.2 (see [1, p. 8]). Let b ∈ BMO(Rn) and Cb be the commu-
tator defined by
Cb(f)(x) =
∫
Rn
b(x) − b(y)
|x − y|n−δ
f(y) dy.
(1) If 0 ≤ δ < n, 1 < p < ∞ and 1/q = 1/p − δ/n, then Cb is bounded
from Lp(Rn) to Lq(Rn) and from H1(Rn) to weak Ln/(n−δ)(Rn).
(2) If δ = 0, 1 < p < ∞ and w ∈ A1, then Cb is bounded on Lp(w) and
from H1(w) to weak L1(w).
Lemma 3.3 (see [5, p. 454(28)] and [12, p. 222]). Let Q be a
cube and Ã(x) = A(x) −∑|α|=m
1
α!(D
αA)Q̃xα. Then Rm+1(A; x, y) =
Rm+1(Ã; x, y).
Lemma 3.4 (see [3, p. 695, Lemma 2.2]). Let Q1 and Q2 be the
cubes with Q1 ⊂ Q2. Then
|bQ1 − bQ2 | ≤ C (1 + |log(|Q1|/|Q2|)|) ‖b‖BMO.
Proof of Theorem 2.1. (a) It suffices to prove that there exists a constant
CQ such that
1
|Q|
∫
Q
|TA
K(f)(x) − CQ| dx ≤ C‖f‖Ln/δ
holds for any cube Q. Fix a cube Q = Q(x0, d). Let Q̃ = 5
√
nQ and
Ã(x) = A(x)−∑|α|=m
1
α!(D
αA)Q̃xα, then Rm+1(A; x, y) = Rm+1(Ã; x, y)
by induction and Dαà = DαA − (DαA)Q̃ for all α with |α| = m. We
write, for f1 = fχQ̃ and f2 = fχRn\Q̃,
508 Some endpoint inequalities...
TA
K(f)(x) =
∫
Rn
Rm+1(Ã; x, y)
|x − y|m K(x, y)f(y) dy
=
∫
Rn
Rm(Ã; x, y)
|x − y|m K(x, y)f1(y) dy
−
∑
|α|=m
1
α!
∫
Rn
K(x, y)(x − y)α
|x − y|m DαÃ(y)f1(y) dy
+
∫
Rn
Rm+1(Ã; x, y)
|x − y|m K(x, y)f2(y) dy,
then
∣
∣TA
K(f)(x) − T Ã
K(f2)(x0)
∣
∣ ≤
∣
∣
∣
∣
TK
(
Rm(Ã; x, ·)
|x − ·|m f1
)
(x)
∣
∣
∣
∣
+
∑
|α|=m
1
α!
∣
∣
∣
∣
TK
(
(x − ·)α
|x − ·|m DαÃf1
)
(x)
∣
∣
∣
∣
+
∣
∣T Ã
K(f2)(x) − T Ã
K(f2)(x0)
∣
∣
:= I(x) + II(x) + III(x),
and, thus,
1
|Q|
∫
Q
∣
∣TA
K(f)(x) − T Ã
K(f2)(x0)
∣
∣ dx
≤ 1
|Q|
∫
Q
I(x) dx +
1
|Q|
∫
Q
II(x) dx +
1
|Q|
∫
Q
III(x) dx
:= I + II + III.
Now, let us estimate I, II and III, respectively. First, we have known
(see [12, p. 144]), for b ∈ BMO(Rn),
‖b‖BMO ≈ sup
Q
(
1
|Q|
∫
Q
|b(y) − bQ|p dy
)1/p
,
then, for x ∈ Q and y ∈ Q̃, using Lemma 3.1 and Lemma 3.4, we get
Rm(Ã; x, y) ≤ C|x−y|m
∑
|α|=m
[
1
|Q̃(x, y)|
∫
Q̃(x,y)
(
|DαA(z)−(DαA)Q̃(x,y)|
+ |(DαA)Q̃(x,y) − (DαA)Q̃|
)q
dz
]1/q
L. Liu 509
≤ C|x − y|m
∑
|α|=m
(
‖DαA‖BMO + 1 +
∣
∣log |Q(x, y)|/|Q̃|
∣
∣
)
≤ C|x − y|m
∑
|α|=m
‖DαA‖BMO,
thus, by the (Ln/δ, L∞)-boundedness of TK , we have
I ≤ C
|Q|
∫
Q
∣
∣
∣
∣
Tδ
(
∑
|α|=m
‖DαA‖BMOf1
)
(x)
∣
∣
∣
∣
dx
≤ C
∑
|α|=m
‖DαA‖BMO‖Tδ(f1)‖L∞
≤ C
∑
|α|=m
‖DαA‖BMO‖f‖Ln/δ ;
Secondly, by the (Lp, Lq)-boundedness of TK for 1/q = 1/p− δ/n, p > 1
and Hölder’s inequality, we gain
II ≤ C
|Q|
∫
Q
|Tδ
(
∑
|α|=m
(DαA − (DαA)Q̃)f1
)
(x)| dx
≤ C
∑
|α|=m
(
1
|Q|
∫
Q
|Tδ((D
αA − (DαA)Q̃)f1)(x)|q dx
)1/q
≤ C|Q|−1/q
∑
|α|=m
‖(DαA − (DαA)Q̃)f1‖Lp
≤ C
∑
|α|=m
(
1
|Q|
∫
Q̃
|DαA(y) − (DαA)Q̃|q dy
)1/q
‖f‖Ln/δ
≤ C
∑
|α|=m
‖DαA‖BMO‖f‖Ln/δ .
To estimate III, we write
T Ã
K(f2)(x) − T Ã
K(f2)(x0)
=
∫
Rn
[
K(x, y)
|x − y|m − K(x0, y)
|x0 − y|m
]
Rm(Ã; x, y)f2(y) dy
+
∫
Rn
K(x0, y)f2(y)
|x0 − y|m [Rm(Ã; x, y) − Rm(Ã; x0, y)] dy
510 Some endpoint inequalities...
−
∑
|α|=m
1
α!
∫
Rn
(
K(x, y)(x − y)α
|x − y|m − K(x0, y)(x0 − y)α
|x0 − y|m
)
DαÃ(y)f2(y) dy
:= III1 + III2 + III3;
By Lemma 3.1 and Lemma 3.4, we know that, for x ∈ Q and y ∈
2k+1Q̃ \ 2kQ̃,
|Rm(Ã; x, y)| ≤ C|x−y|m
∑
|α|=m
(‖DαA‖BMO+|(DαA)Q̃(x,y)−(DαA)Q̃|)
≤ Ck|x − y|m
∑
|α|=m
‖DαA‖BMO.
Note that |x− y| ∼ |x0 − y| for x ∈ Q and y ∈ Rn \ Q̃, we obtain, by the
condition on K,
|III1| ≤ C
∫
Rn
( |x − x0|
|x0 − y|m+n+1−δ
+
|x − x0|ε
|x0 − y|m+n+ε−δ
)
|Rm(Ã; x, y)||f2(y)| dy
≤ C
∑
|α|=m
‖DαA‖BMO
∞
∑
k=0
∫
2k+1Q̃\2kQ̃
k
( |x − x0|
|x0 − y|n+1−δ
+
|x − x0|ε
|x0 − y|n+ε−δ
)
|f(y)| dy
≤ C
∑
|α|=m
‖DαA‖BMO‖f‖Ln/δ
∞
∑
k=1
k(2−k + 2−εk)
≤ C
∑
|α|=m
‖DαA‖BMO‖f‖Ln/δ ;
For III2, by the formula (see (39) in [5]):
Rm(Ã; x, y) − Rm(Ã; x0, y) =
∑
|β|<m
1
β!
Rm−|β|(D
βÃ; x, x0)(x − y)β
and Lemma 3.1, we have
|Rm(Ã; x, y) − Rm(Ã; x0, y)|
≤ C
∑
|β|<m
∑
|α|=m
|x − x0|m−|β||x − y||β|‖DαA‖BMO,
L. Liu 511
similar to the estimates of III1, we get
|III2| ≤ C
∑
|α|=m
‖DαA‖BMO
∞
∑
k=0
∫
2k+1Q̃\2kQ̃
|x − x0|
|x0 − y|n+1−δ
|f(y)| dy
≤ C‖DαA‖BMO‖f‖Ln/δ ;
For III3, by taking r > 1 such that 1/r+δ/n = 1, similar to the estimates
of III1, we get
|III3| ≤ C
∑
|α|=m
∞
∑
k=0
∫
2k+1Q̃\2kQ̃
( |x − x0|
|x0 − y|n+1−δ
+
|x − x0|ε
|x0 − y|n+ε−δ
)
|DαÃ(y)||f(y)| dy
≤C
∑
|α|=m
∞
∑
k=1
(2−k+2−εk)
(
|2kQ̃|−1
∫
2kQ̃
|DαA(y)−(DαA)Q̃|r dy
)1/r
‖f‖Ln/δ
≤ C
∑
|α|=m
‖DαA‖BMO‖f‖Ln/δ .
Thus
III ≤ C
∑
|α|=m
‖DαA‖BMO‖f‖Ln/δ .
(b) It is only to show that there exists a constant C > 0 such that
for every H1-atom a(that is that a satisfies: supp a ⊂ Q = Q(x0, d),
‖a‖L∞ ≤ |Q|−1 and
∫
a(y) dy = 0 (see [8])), the following holds:
‖T̃A
K(a)‖Ln/(n−δ) ≤ C.
We write
∫
Rn
[
T̃A
K(a)(x)
]n/n−δ
dx
=
[
∫
|x−x0|≤2r
+
∫
|x−x0|>2r
]
[
T̃A
K(a)(x)
]n/(n−δ)
dx := J + JJ.
For J , by the following equality
Qm+1(A; x, y) = Rm+1(A; x, y) +
∑
|α|=m
1
α!
(x − y)α(DαA(x) − DαA(y)),
512 Some endpoint inequalities...
we have,
|T̃A
K(a)(x)| ≤ |TA
K(a)(x)| + C
∑
|α|=m
∫
Rn
|DαA(x) − DαA(y)|
|x − y|n−δ
|a(y)| dy,
thus, T̃A
K is (Lp, Lq)-bounded by Lemma 3.2 and (a), where 1/q = 1/p−
δ/n. We see that
J ≤ C‖T̃A
K(a)‖n/((n−δ)q)
Lq |2Q|1−n/((n−δ)q)
≤ C‖a‖n/(n−δ)
Lp |Q|1−n/((n−δ)q) ≤ C.
To obtain the estimate of JJ , we denote Ã(x) = A(x) −∑|α|=m
1
α! ×
(DαA)2Qxα. Then Qm(A; x, y) = Qm(Ã; x, y). We write, by the vanish-
ing moment of a,
T̃A
K(a)(x) =
∫
Rn
K(x, y)Rm(A; x, y)
|x − y|m a(y)dy
−
∑
|α|=m
1
α!
∫
Rn
K(x, y)DαÃ(x)(x − y)α
|x − y|m a(y) dy
=
∫
Rn
[
K(x, y)
|x − y|m − K(x, x0)
|x − x0|m
]
Rm(Ã; x, y)a(y) dy
+
∫
Rn
K(x, x0)
|x − x0|m
[Rm(Ã; x, y) − Rm(Ã; x, x0)]a(y) dy
−
∑
|α|=m
1
α!
∫
Rn
[
K(x, y)(x − y)α
|x − y|m − K(x, x0)(x − x0)
α
|x − x0|m
]
DαÃ(x)a(y) dy,
:= JJ1 + JJ2 + JJ3.
Now, similar to the proof of III, we obtain, for x ∈ (2Q)c
|JJ1| ≤ C
∫
Rn
[ |y − x0|
|x − y|n+m+1−δ
+
|y − x0|ε
|x − y|n+m+ε−δ
]
|Rm(Ã; x, y)||a(y)| dy
≤ C
∑
|α|=m
‖DαA‖BMO(|Q|1/n|x − x0|−n−1+δ + |Q|ε/n|x − x0|−n−ε+δ),
|JJ2| ≤ C
∫
Rn
|Rm(Ã; x, y) − Rm(Ã; x, x0)||a(y)|
|x − y|m+n−δ
dy
L. Liu 513
≤ C
∑
|α|=m
‖DαA‖BMO
∫
Rn
|x0 − y‖a(y)|
|x − x0|n+1−δ
dy
≤ C
∑
|α|=m
‖DαA‖BMO|Q|1/n|x − x0|−n−1+δ
and
|JJ3| ≤ C
∫
Rn
|x0 − y|
|x − y|n+1−δ
∑
|α|=m
|DαÃ(x)||a(y)| dy
≤ C
∑
|α|=m
|DαÃ(x)|(|Q|1/n|x − x0|−n−1+δ + |Q|ε/n|x − x0|−n−ε+δ).
Thus
JJ ≤
∫
(2Q)c
(|JJ1 + JJ2 + JJ3|)n/(n−δ) dx
≤ C
(
∑
|α|=m
‖DαA‖BMO
)n/(n−δ) ∞
∑
k=1
k[2−kn/(n−δ) + 2−knε/(n−δ)] ≤ C.
(c) By the following equality
Rm+1(A; x, y) = Qm+1(A; x, y) +
∑
|α|=m
1
α!
(x − y)α(DαA(x) − DαA(y)),
we have
|TA
K(f)(x)| ≤ |T̃A
K(f)(x)| + C
∑
|α|=m
∫
Rn
|DαA(x) − DαA(y)|
|x − y|n−δ
|f(y)| dy,
thus, by Lemma 3.2 and (b), we obtain
|{x ∈ Rn : |TA
K(f)(x)| > λ}| ≤ |{x ∈ Rn : |T̃A
K(f)(x)| > λ/2}|
+
∣
∣
∣
∣
∣
{
x ∈ Rn :
∑
|α|=m
∫
Rn
|DαA(x) − DαA(y)|
|x − y|n−δ
|f(y)| dy > Cλ
}∣
∣
∣
∣
∣
≤ C(‖f‖H1/λ)n/(n−δ).
This completes the proof of Theorem 2.1.
514 Some endpoint inequalities...
Proof of Theorem 2.2. (i) It is only to prove that there exists a constant
CQ such that
1
w(Q)
∫
Q
|TA
K(f)(x) − CQ|w(x) dx ≤ C‖f‖L∞(w)
holds for any cube Q. Fix a cube Q = Q(x0, d). Let Q̃ and Ã(x) be
the same as the proof of Theorem 2.1. We have, similar to the proof of
Theorem 2.1, for f1 = fχQ̃ and f2 = fχRn\Q̃,
∣
∣TA
K(f)(x) − T Ã
K(f2)(x0)
∣
∣ ≤
∣
∣
∣
∣
TK
(
Rm(Ã; x, ·)
|x − ·|m f1
)
(x)
∣
∣
∣
∣
+
∑
|α|=m
1
α!
∣
∣
∣
∣
TK
(
(x − ·)α
|x − ·|m DαÃf1
)
(x)
∣
∣
∣
∣
+
∣
∣T Ã
K(f2)(x) − T Ã
K(f2)(x0)
∣
∣
:= I(x) + II(x) + III(x),
and, thus,
1
w(Q)
∫
Q
∣
∣TA
K(f)(x) − T Ã
K(f2)(x0)
∣
∣w(x) dx
≤ 1
w(Q)
∫
Q
I(x)w(x) dx +
1
w(Q)
∫
Q
II(x)w(x) dx
+
1
w(Q)
∫
Q
III(x)w(x) dx := I + II + III.
First, using Lemma 3.1 and the L∞(w)-boundedness of TK , we have
I ≤ C
w(Q)
∫
Q
∣
∣
∣
∣
TK
(
∑
|α|=m
‖DαA‖BMOf1
)
(x)
∣
∣
∣
∣
w(x) dx
≤ C
∑
|α|=m
‖DαA‖BMO‖Tf1‖L∞(w) ≤ C
∑
|α|=m
‖DαA‖BMO‖f‖L∞(w);
Secondly, since w ∈ A1, w satisfies the reverse of Hölder’s inequality:
(
1
|Q|
∫
Q
w(x)q dx
)1/q
≤ C
|Q|
∫
Q
w(x) dx
L. Liu 515
for all cube Q and some 1 < q < ∞ (see [12]), thus, taking p > 1 and
1/p+1/p′ = 1, by the Lp(w)-boundedness of TK and Hölder’s inequality,
we gain
II ≤ C
w(Q)
∫
Q
∣
∣
∣
∣
T
(
∑
|α|=m
(DαA − (DαA)Q̃)f1
)
(x)
∣
∣
∣
∣
w(x) dx
≤ C
∑
|α|=m
(
1
w(Q)
∫
Q
|T ((DαA − (DαA)Q̃)f1)(x)|pw(x)dx
)1/p
≤ C
∑
|α|=m
(
1
w(Q)
∫
Q
|(DαA(x) − (DαA)Q̃)f1(x)|pw(x) dx
)1/p
≤ C
∑
|α|=m
w(Q)−1/p
(
∫
Q̃
|DαA(x) − (DαA)Q̃|pq′ dx
)1/pq′
×
(
∫
Q̃
w(x)q dx
)1/pq
‖f‖L∞(w)
≤ C
∑
|α|=m
(
1
|Q|
∫
Q̃
|DαA(x) − (DαA)Q̃|pq′ dx
)1/pq′
×
(
1
|Q|
∫
Q̃
w(x)q dx
)1/pq
( |Q|
w(Q)
)1/p
‖f‖L∞(w)
≤ C
∑
|α|=m
‖DαA‖BMO
(
1
|Q|
∫
Q̃
w(x) dx
)1/p
( |Q|
w(Q)
)1/p
‖f‖L∞(w)
≤ C
∑
|α|=m
‖DαA‖BMO‖f‖L∞(w);
For III, similar to the proof of Theorem 2.1, we obtain
III ≤ C
∑
|α|=m
‖DαA‖BMO
1
w(Q)
×
∫
Q
∞
∑
k=0
∫
2k+1Q̃\2kQ̃
k
( |x − x0|
|x0 − y|n+1
+
|x − x0|ε
|x0 − y|n+ε
)
|f(y)| dy w(x) dx
516 Some endpoint inequalities...
+ C
∑
|α|=m
1
w(Q)
∫
Q
∞
∑
k=0
∫
2k+1Q̃\2kQ̃
( |x − x0|
|x0 − y|n+1
+
|x − x0|ε
|x0 − y|n+ε
)
× |DαÃ(y)||f(y)| dy w(x) dx
≤ C
∑
|α|=m
‖DαA‖BMO‖f‖L∞(w)
∞
∑
k=1
k(2−k + 2−kε)
≤ C
∑
|α|=m
‖DαA‖BMO‖f‖L∞(w).
(ii) It suffices to show that there exists a constant C > 0 such that
for every H1(w)-atom a (that is that a satisfy: supp a ⊂ Q = Q(x0, r),
‖a‖L∞(w) ≤ w(Q)−1 and
∫
a(y)dy = 0 (see [8])), we have
‖T̃A
K(a)‖L1(w) ≤ C.
We write
∫
Rn
T̃A
K(a)(x)w(x) dx =
[
∫
2Q
+
∫
(2Q)c
]
T̃A
K(a)(x)w(x) dx := J + JJ.
For J , similar to the proof of Theorem 2.1, we get
|T̃A
K(a)(x)| ≤ |TA(a)(x)| + C
∑
|α|=m
∫
Rn
|DαA(x) − DαA(y)|
|x − y|n |a(y)| dy,
thus, T̃A
K is Lp(w)-bounded by Lemma 3.2 and (i). We see that
J ≤ C‖T̃A
K(a)‖L∞(w)w(2Q) ≤ C‖a‖L∞(w)w(Q) ≤ C;
For JJ , notice that if w ∈ A1, then w(Q2)
|Q2|
|Q1|
w(Q1) ≤ C for all cubes Q1, Q2
with Q1 ⊂ Q2. Thus, by Hölder’s inequality and the reverse of Hölder’s
inequality for w ∈ A1 and some 1 < q < ∞, taking p > 1 and 1/p+1/p′ =
1, similarly, we obtain
JJ ≤ C
∑
|α|=m
‖DαA‖BMO
∞
∑
k=1
(2−k + 2−εk)
( |Q|
w(Q)
w(2k+1Q)
|2k+1Q|
)
+ C
∑
|α|=m
∞
∑
k=1
(2−k + 2−εk)
|Q|
w(Q)
(
1
|2k+1Q|
∫
2k+1Q
|DαÃ(x)|p dx
)1/p
L. Liu 517
×
(
1
|2k+1Q|
∫
2k+1Q
w(x)p′ dx
)1/p′
≤ C
∑
|α|=m
‖DαA‖BMO
∞
∑
k=1
k(2−k + 2−εk)
(
w(2k+1Q)
|2k+1Q|
|Q|
w(Q)
)
≤ C.
(iii) Similarly, we know
|TA
K(f)(x)| ≤ |T̃A(f)(x)| + C
∑
|α|=m
∫
Rn
|DαA(x) − DαA(y)|
|x − y|n |f(y)| dy,
by Lemma 3.2 and (ii), we obtain
w({x ∈ Rn : |TA
K(f)(x)| > λ}) ≤ w({x ∈ Rn : |T̃A
K(f)(x)| > λ/2})
+ w
({
x ∈ Rn :
∑
|α|=m
∫
Rn
|DαA(x) − DαA(y)|
|x − y|n |f(y)| dy > Cλ
})
≤ C‖f‖H1(w)/λ.
This completes the proof of Theorem 2.2.
4. Applications
In this section we shall apply the Theorem 2.1 and 2.2 to some partic-
ular operators such as the Calderón–Zygmund singular integral operator
and fractional integral operator.
Aplication 1 (Calderón–Zygmund singular integral operator).
Let T be the Calderón–Zygmund operator defined by (see [8, 12])
T (f)(x) =
∫
K(x, y)f(y) dy,
the multilinear operator related to T is defined by
TA(f)(x) =
∫
Rm+1(A; x, y)
|x − y|m K(x, y)f(y) dy.
Then it is easily to see that TK satisfies the conditions in Theorem 2.2,
thus that TA is bounded from L∞(w) to BMO(w) and from H1(w) to
weak L1(w) and that T̃A is bounded from H1(w) to L1(w) for w ∈ A1
and DαA ∈ BMO(Rn) with |α| = m.
518 Some endpoint inequalities...
Aplication 2 (Fractional integral operator with rough kernel).
For 0 ≤ δ < n, let Tδ be the fractional integral operator with rough
kernel defined by (see [7, 9, 10])
Tδf(x) =
∫
Rn
Ω(x − y)
|x − y|n−δ
f(y) dy,
the multilinear operator related to Tδ is defined by
TA
δ f(x) =
∫
Rn
Rm+1(A; x, y)
|x − y|m+n−δ
Ω(x − y)f(y) dy,
where Ω is homogeneous of degree zero on Rn,
∫
Sn−1 Ω(x′)dσ(x′) = 0 and
Ω ∈ Lipγ(Sn−1) for 0 < γ ≤ 1, that is there exists a constant M > 0
such that for any x, y ∈ Sn−1, |Ω(x) − Ω(y)| ≤ M |x − y|γ . Then Tδ
satisfies the conditions in Theorem 3.1. In fact, for supp f ⊂ (2Q)c and
x ∈ Q = Q(x0, d), by the condition of Ω, we have (see [12])
∣
∣
∣
∣
Ω(x − y)
|x − y|n−δ
− Ω(x0 − y)
|x0 − y|n−δ
∣
∣
∣
∣
≤ C
( |x − x0|γ
|x0 − y|n+γ−δ
+
|x − x0|
|x0 − y|n+1−δ
)
,
thus, similar to the proof of Theorem 2.1,
|TA
δ (f)(x) − TA
δ (f)(x0)|
≤ C
∞
∑
k=1
k(2−γk + 2−k)‖DαA‖BMO‖f‖Ln/δ
≤ C‖DαA‖BMO‖f‖Ln/δ .
Therefore that TA
δ is bounded from Ln/δ(Rn) to BMO(Rn) and from
H1(Rn) to weak Ln/(n−δ)(Rn) and T̃A
δ is bounded from H1(Rn) to
Ln/(n−δ)(Rn) for all DαA ∈ BMO(Rn) with |α| = m.
Acknowledgement. The author would like to express his deep gratitude
to the referee for his valuable comments and suggestions.
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Contact information
Lanzhe Liu College of Mathematics
Changsha University of
Science and Technology
Changsha 410077,
P.R. of China
E-Mail: lanzheliu@163.com
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