1. Introduction and Main Results
The classical extrapolation theorem is due to Rubio de Francia [1], who showed that if sublinear operator
T
is bounded on
L
p
0
(
w
)
for some
p
0
,
1
≤
p
0
<
∞
, and every
w
∈
A
p
0
, then
T
is bounded on
L
p
(
w
)
for every
p
,
1
<
p
<
∞
and every
w
∈
A
p
. CruzUribe et al. [2] and Curbera et al. [3] generalized the extrapolation theorem of Rubio de Francia and got a lot of new extrapolation theorems associated with
A
∞
weights. These theorems have proved to be the key to solving many problems in harmonic analysis.
CruzUribe and Pérez [4] gave several other extrapolation theorems for pairs of weights of the forms
(
w
,
M
k
w
)
and
(
w
,
(
M
w
/
w
)
r
w
)
, where
w
is any nonnegative function,
r
>
1
, and
M
k
is the
k
th iteration of the HardyLittlewood maximal operator. The purpose of this paper is to extend the extrapolation theorems in [4] and derive some extrapolation results on Lorentz spaces.
Let
w
(
x
)
,
x
∈
R
n
, be a nonnegative, locally integrable weight function. For a measurable function
f
,
(1)
λ
f
w
(
t
)
=
w
x
∈
R
n
:
f
x
>
t
,
t
≥
0
,
is the distribution function of
f
with respect to the measure
w
(
x
)
d
x
. For
0
<
p
,
q
≤
∞
, the weighted Lorentz spaces
L
p
,
q
(
w
)
are the collection of all functions
f
, such that
f
L
p
,
q
(
w
)
<
∞
, where
(2)
f
L
p
,
q
(
w
)
=
q
∫
0
∞
λ
f
w
t
q
/
p
t
q

1
d
t
1
/
q
,
0
<
q
<
∞
,
sup
t
>
0
t
λ
f
w
t
1
/
p
,
q
=
∞
.
For
1
≤
p
<
∞
,
p
′
denote the conjugate exponent of
p
:
1
/
p
+
1
/
p
′
=
1
.
In the case when either
1
<
p
<
∞
and
1
≤
q
≤
∞
, or
p
=
q
=
1
, or
p
=
q
=
∞
,
L
p
,
q
(
w
)
is a Banach space. Furthermore, we have the relationship
(3)
C
1
f
L
p
,
q
(
w
)
≤
sup
‖
g
‖
L
p
′
,
q
′
w
≤
1
∫
R
n
f
x
g
x
w
x
d
x
≤
C
2
f
L
p
,
q
(
w
)
.
Our main results are the following.
Theorem 1.
Let
S
and
T
be operators (not necessarily linear), and let
f
be a function in a suitable test class for both
S
and
T
. Suppose that there exist positive constants
p
0
and
C
0
and a positive integer
k
such that for all weights
w
(4)
T
f
L
p
0
(
w
)
≤
C
0
S
f
L
p
0
(
M
k
w
)
.
Then, for all
p
,
p
0
<
p
<
∞
,
p
0
≤
q
≤
∞
, there exists a constant
C
p
depending only on
C
0
,
p
,
p
0
,
k
, and
n
, such that, for all weights
w
,
(5)
T
f
L
p
,
q
(
w
)
≤
C
p
S
f
L
p
,
q
(
M
[
k
p
/
p
0
]
+
1
w
)
,
where
[
k
p
/
p
0
]
is the largest integer less than or equal to
k
p
/
p
0
.
Theorem 2.
Let
S
and
T
be operators (not necessarily linear), and let
f
be a function in a suitable test class for both
S
and
T
. Suppose that there exist positive constants
p
0
and
C
0
and a positive integer
k
such that for all weights
w
(6)
T
f
L
p
0
(
w
)
≤
C
0
S
f
L
p
0
(
M
w
)
.
Then, for all
p
,
p
0
<
p
<
∞
,
p
0
≤
q
≤
∞
, there exists a constant
C
p
depending only on
C
0
,
p
,
p
0
,
k
, and
n
, such that, for all weights
w
,
(7)
T
f
L
p
,
q
(
w
)
≤
C
p
S
f
L
p
,
q
(
(
M
w
/
w
)
[
p
/
p
0
]
M
w
)
.
2. Applications
Let
M
be the HardyLittlewood maximal operator; Fefferman and Stein [5] showed that, for every
p
,
1
<
p
<
∞
, every nonnegative function
w
, and every function
f
,
(8)
∫
R
n
M
f
x
p
w
x
d
x
≤
C
∫
R
n
f
x
p
M
w
x
d
x
.
Using Theorem 1, we can get the following result.
Proposition 3.
For
p
,
1
<
p
<
∞
,
1
≤
q
≤
∞
, there exists a constant
C
>
0
, such that, for all weights
w
and function
f
,
(9)
M
f
L
p
,
q
(
w
)
≤
C
f
L
p
,
q
(
M
w
)
.
Let
T
be a CalderónZygmund operator. Wilson [6] and Pérez [7] showed that if
1
<
p
<
∞
, then, for any weight
w
and every
f
,
(10)
∫
R
n
T
f
x
p
w
x
d
x
≤
C
∫
R
n
f
x
p
M
[
p
]
+
1
w
(
x
)
d
x
,
where the exponent
[
p
]
+
1
is sharp. Using Theorem 1 we can get the following result.
Proposition 4.
Let
T
be a CalderónZygmund operator,
1
<
p
<
∞
,
1
≤
q
≤
∞
; then, for any weight
w
and every
f
,
(11)
T
f
L
p
,
q
(
w
)
≤
C
f
L
p
,
q
(
M
[
p
]
+
1
w
)
.
For the commutators of singular integral operators with a BMO function
[
T
,
b
]
, Pérez [8] showed the analogous weighted inequalities as in (10). By Theorem 1, we can give similar results as in Proposition 4; details are omitted.
Lerner [9] establishes
(12)
∫
R
n
T
f
x
w
x
d
x
≤
C
∫
R
n
M
f
x
M
w
x
d
x
,
for any CalderónZygmund operator
T
and any arbitrary weight
w
. Using Theorems 1 and 2, we can get the following results.
Proposition 5.
Let
T
be a CalderónZygmund operator,
1
<
p
<
∞
,
1
≤
q
≤
∞
; then, for any weight
w
and all
f
,
(13)
T
f
L
p
,
q
(
w
)
≤
C
M
f
L
p
,
q
(
M
[
p
]
w
)
,
T
f
L
p
,
q
(
w
)
≤
C
M
f
L
p
,
q
(
(
M
w
/
w
)
[
p
]
M
w
)
.
Now, we consider the potential type operators and commutators. For a nonnegative, locally integrable function
Φ
on
R
n
, assume that
Φ
satisfies the following weak growth condition: there are constants
δ
,
c
>
0
,
0
≤
ɛ
<
1
with the property that, for all
k
∈
Z
,
(14)
sup
2
k
<

x

≤
2
k
+
1
Φ
x
≤
c
2
k
n
∫
δ
(
1

ɛ
)
2
k
<

y

≤
2
δ
(
1
+
ɛ
)
2
k
Φ
(
y
)
d
y
.
The basic examples are provided by the Riesz potential of order
α
,
I
α
, defined by the kernel
Φ
(
x
)
=

x

α

n
,
0
<
α
<
n
.
Define the potential type operator
T
Φ
and the maximal operator
M
Φ
~
by
(15)
T
Φ
f
(
x
)
=
∫
R
n
Φ
x

y
f
y
d
y
,
M
Φ
~
f
x
=
sup
x
∈
Q
Φ
~
l
Q
Q
∫
Q
f
x
d
x
,
where
(16)
Φ
~
(
t
)
=
∫

z

<
t
Φ
(
z
)
d
z
,
t
≥
0
.
For the Riesz potential of order
α
,
Φ
~
(
t
)
=
t
α
. Pérez [10] studies the twoweight strong type
(
p
,
q
)
inequalities for
T
Φ
,
1
<
p
≤
q
<
∞
.
Lemma 6 (see [<xref reftype="bibr" rid="B10">10</xref>]).
Let
T
Φ
be the potential type operator with
Φ
satisfying (14), let
f
and
g
be bounded functions with compact support, and let
a
>
2
n
; then, there exist a family of cubes
Q
k
,
j
and a family of pairwise disjoint subsets
E
k
,
j
,
E
k
,
j
⊂
Q
k
,
j
, with
Q
k
,
j
<
(
1

2
n
/
a
)

1
E
k
,
j
for all
k
,
j
, such that
(17)
∫
R
n
T
Φ
f
x
g
x
d
x
≤
C
∑
k
,
j
1
3
Q
k
,
j
∫
3
Q
k
,
j
f
x
d
x
Φ
~
l
ρ
Q
k
,
j
ρ
Q
k
,
j
×
∫
ρ
Q
k
,
j
g
x
d
x
E
k
,
j
,
where
ρ
=
δ
(
1
+
ɛ
)
.
By Lemma 6, we can easily prove the following result.
Lemma 7.
Let
T
Φ
be the potential type operator with
Φ
satisfying (14); then, there is a constant
C
>
0
such that, for any weight
w
and all
f
,
(18)
∫
R
n
T
Φ
f
x
w
x
d
x
≤
C
∫
R
n
M
Φ
~
f
x
M
w
x
d
x
.
Using (18) and Theorems 1 and 2, we can get the following results.
Proposition 8.
Let
T
Φ
be the potential type operator with
Φ
satisfying (14),
1
<
p
<
∞
,
1
≤
q
≤
∞
; then, for any weight
w
and every
f
,
(19)
T
Φ
f
L
p
,
q
(
w
)
≤
C
M
Φ
~
f
L
p
,
q
(
M
[
p
]
w
)
,
T
Φ
f
L
p
,
q
(
w
)
≤
C
M
Φ
~
f
L
p
,
q
(
(
M
w
/
w
)
[
p
]
M
w
)
.
For
k
≥
1
and
b
∈
BMO
, the commutators of potential type integral operator with a BMO function are defined by
(20)
T
Φ
,
b
k
f
(
x
)
=
∫
R
n
b
x

b
y
k
Φ
x

y
f
y
d
y
.
Li [11] gave the twoweight strong type
(
p
,
q
)
inequalities for
T
Φ
,
b
k
,
1
<
p
≤
q
<
∞
. We can easily get similar results as in Lemmas 6 and 7 from [11]; by Theorem 1, we can obtain some weighted inequalities for the commutators of potential type integral operators; details are omitted.
3. Proofs of Theorems
We need some notations and facts. Let
B
(
t
)
:
[
0
,
∞
)
→
[
0
,
∞
)
be a Young function, that is, a continuous, convex, and increasing function with
B
(
0
)
=
0
such that
B
(
t
)
→
∞
as
t
→
∞
. Given a Young function
B
, we define the
B
average of a function
f
over a cube
Q
by
(21)
f
B
,
Q
=
inf
λ
>
0
:
1
Q
∫
Q
B
f
x
λ
d
x
≤
1
.
In particular, for the Young function
B
k
(
t
)
=
t
(
1
+
log
+
t
)
k
,
k
=
1,2
,
…
, the
B
k
average of a function
f
over a cube
Q
is denoted by
f
L
(
log
L
)
k
,
Q
. Define the maximal function associated with the Young function
B
k
as
(22)
M
L
(
log
L
)
k
f
x
=
sup
Q
∋
x
f
L
(
log
L
)
k
,
Q
.
Pérez [8] obtained that
M
L
(
log
L
)
k
f
(
y
)
≈
M
k
+
1
f
(
y
)
.
Proof of Theorem <xref reftype="statement" rid="thm1.1">1</xref>.
Fix
p
,
p
0
<
p
<
∞
,
p
0
≤
q
≤
∞
, and let
r
=
p
/
p
0
,
s
=
q
/
p
0
. Then, by duality,
(23)
T
f
L
p
,
q
(
w
)
p
0
=
T
f
p
0
L
r
,
s
(
w
)
≤
sup
g
∈
L
r
′
,
s
′
(
w
)
,
‖
g
‖
L
r
′
,
s
′
(
w
)
≤
1
∫
R
n
T
f
x
p
0
g
x
w
x
d
x
.
The last inequality follows since
L
r
′
,
s
′
(
w
)
and
L
r
,
s
(
w
)
are associate spaces. Fixing one of these
g
’s, we use (4) to continue with
(24)
∫
R
n
T
f
x
p
0
g
x
w
x
d
x
≤
C
∫
R
n
S
f
x
p
0
M
k
g
w
x
d
x
≤
C
∫
R
n
S
f
x
p
0
M
L
(
log
L
)
k

1
g
w
x
M
~
w
(
x
)
M
~
(
w
)
(
x
)
d
x
≤
C
S
f
p
0
L
r
,
s
(
M
~
(
w
)
)
M
L
(
log
L
)
k

1
g
w
x
M
~
w
x
L
r
′
,
s
′
(
M
~
(
w
)
)
=
C
S
f
L
p
,
q
(
M
~
(
w
)
)
p
0
M
L
(
log
L
)
k

1
g
w
x
M
~
w
x
L
r
′
,
s
′
(
M
~
(
w
)
)
,
where
M
~
is an appropriate maximal operator to be chosen soon. To conclude, we just need to show that
(25)
M
L
log
L
k

1
g
w
x
M
~
w
x
L
r
′
,
s
′
(
M
~
(
w
)
)
≤
C
g
L
r
′
,
s
′
(
w
)
,
or equivalently
(26)
F
:
L
r
′
,
s
′
w
⟶
L
r
′
,
s
′
M
~
w
,
where
(27)
F
f
=
M
L
(
log
L
)
k

1
f
w
M
~
w
.
To do this, we choose
M
~
pointwise bigger than
M
L
(
log
L
)
k

1
; then, we trivially have
(28)
F
:
L
∞
w
⟶
L
∞
M
~
w
.
Therefore, by Marcinkiewicz’s interpolation theorem for Lorentz spaces in [12], it will be enough to show that for some
ɛ
>
0
(29)
F
:
L
r
+
ɛ
′
w
⟶
L
r
+
ɛ
′
M
~
w
,
which amounts to proving
(30)
∫
R
n
M
L
log
L
k

1
(
f
w
)
x
r
+
ɛ
′
M
~
w
x
1

r
+
ɛ
′
d
x
≤
C
∫
R
n
f
x
r
+
ɛ
′
w
(
x
)
d
x
.
But this result follows from [13]: it is shown there that, for
ν
>
1
and
η
>
0
,
(31)
∫
R
n
M
L
log
L
k

1
f
x
ν
′
M
L
log
L
k
ν

1
+
η
w
x
1

ν
′
d
x
≤
C
∫
R
n
f
x
ν
′
w
x
1

ν
′
d
x
.
We finally choose the appropriate parameters and weight. Let
ν
=
r
+
ɛ
,
η
=
ɛ
=
[
k
r
]
+
1

k
r
/
k
+
1
>
0
, and pick the weight
(32)
M
~
w
=
M
L
(
log
L
)
[
k
r
]
w
.
This shows that
(33)
T
f
L
p
,
q
(
w
)
≤
C
S
f
L
p
,
q
(
M
L
(
log
L
)
[
k
p
/
p
0
]
(
w
)
)
.
We conclude the proof of (5).
Proof of Theorem <xref reftype="statement" rid="thm1.2">2</xref>.
The proof of this theorem proceeds exactly as that of (5) with minor changes. At inequality (23), fixing one of these
g
’s, we use (6) to continue with
(34)
∫
R
n
T
f
x
p
0
g
x
w
x
d
x
≤
C
∫
R
n
S
f
x
p
0
M
g
w
x
d
x
.
For
g
and
w
, we have
M
(
g
w
)
≤
C
M
c
(
g
w
)
, where
C
is a constant depending only on the dimension
n
and
M
c
is the unweighted centered HardyLittlewood maximal operator:
(35)
M
c
(
f
)
(
x
)
=
sup
r
>
0
1
B
r
x
∫
B
r
(
x
)
f
x
d
x
,
in which
B
r
(
x
)
is the ball of radius
r
centered at
x
. Furthermore,
(36)
M
c
(
g
w
)
(
x
)
=
sup
r
>
0
1
B
r
x
∫
B
r
(
x
)
g
y
w
y
d
y
=
sup
r
>
0
w
B
r
x
B
r
x
1
w
B
r
x
∫
B
r
(
x
)
g
y
w
y
d
y
≤
M
w
,
c
g
x
M
w
x
,
where
M
w
,
c
is the weighted centered maximal operator. Then, for
δ
>
0
,
(37)
∫
R
n
S
f
p
0
M
g
w
d
x
≤
C
∫
R
n
S
f
p
0
M
w
,
c
g
w
M
w
r
+
δ

1
M
w
w
r
+
δ
w
d
x
≤
C
S
f
p
0
L
r
,
s
(
(
M
w
/
w
)
r
+
δ
w
)
×
M
w
,
c
g
w
M
w
r
+
δ

1
L
r
′
,
s
′
(
(
M
w
/
w
)
r
+
δ
w
)
.
To conclude, we just need to show that
(38)
M
w
,
c
g
w
M
w
r
+
δ

1
L
r
′
,
s
′
(
(
M
w
/
w
)
r
+
δ
w
)
≤
C
g
L
r
′
,
s
′
(
w
)
,
or equivalently
(39)
F
:
L
r
′
,
s
′
w
⟶
L
r
′
,
s
′
M
w
w
r
+
δ
w
,
where
(40)
F
f
=
M
w
,
c
f
w
M
w
r
+
δ

1
.
We notice that
M
w
≤
(
M
w
/
w
)
r
+
δ
w
for each
w
. With this, we trivially have
(41)
F
:
L
∞
(
w
)
⟶
L
∞
M
w
w
r
+
δ
w
.
Therefore, by Marcinkiewicz’s interpolation theorem for Lorentz spaces, it will be enough to show that for some
ɛ
>
0
(42)
F
:
L
r
+
ɛ
′
w
⟶
L
r
+
ɛ
′
M
w
w
r
+
δ
w
,
which amounts to proving
(43)
∫
R
n
M
w
,
c
g
w
M
w
r
+
δ

1
r
+
ɛ
′
M
w
w
r
+
δ
w
d
x
≤
C
∫
R
n
g
r
+
ɛ
′
w
d
x
.
Taking
ɛ
=
δ
and using the wellknown fact that
M
w
,
c
is bounded on
L
p
(
w
)
,
1
<
p
<
∞
, with a constant that depends only on
p
and
n
, we get (43).
This shows that
(44)
T
f
L
p
,
q
(
w
)
p
0
≤
C
S
f
L
p
,
q
(
(
M
w
/
w
)
p
/
p
0
+
δ
w
)
p
0
.
Taking
δ
=
[
p
/
p
0
]
+
1

p
/
p
0
>
0
, we get (7). This ends the proof of Theorem 2.