# Vector-Valued Inequalities in the Morrey Type Spaces

Authors: Hua Wang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

---

## Abstract

We will obtain the strong type and weak type estimates for vector-valued analogues of classical Hardy-Littlewood maximal function, weighted maximal function, and singular integral operators in the weighted Morrey spacesL
p
,
κ
(
w
) when 1
≤
p
<
∞ and 0
<
κ
<
1, and in the generalized Morrey spaces L
p
,
Φ for 1
≤
p
<
∞, where Φ is a growth function on (
0
,
∞
) satisfying the doubling condition.


---

## Body

## 1. Introduction

The classical Hardy-Littlewood maximal functionM
f
(
x
) is defined for a locally integrable function f on R
n by
(1)
M
f
(
x
)
=
sup
⁡
x
∈
B
1
|
B
|
∫
B
‍
|
f
(
y
)
|
d
y
,
where the supremum is taken over all balls B containing x. It is well known that the maximal operator M maps L
p
(
R
n
) into L
p
(
R
n
) for all 1
<
p
≤
∞ and L
1
(
R
n
) into W
L
1
(
R
n
). Let f
→
=
(
f
1
,
f
2
,
…
) be a sequence of locally integrable functions on R
n. For any x
∈
R
n, we define M
(
f
→
)
=
(
M
(
f
1
)
,
M
(
f
2
)
,
…
) and
(2)
M
q
(
f
→
)
(
x
)
=
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
1
/
q
,
1
<
q
<
∞
.This nonlinear operator was introduced by Fefferman and Stein in [1], and since then it has played an important role in the development of modern harmonic analysis. In this remarkable paper [1], Fefferman and Stein extended the classical maximal theorem to the case of vector-valued functions.Theorem 1 (see [1]).
Let1
<
p
<
∞. Then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(3)
∥
(
∑
j
‍
|
M
(
f
j
)
|
q
)
1
/
q
∥
L
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
.Theorem 2 (see [1]).
Letp
=
1. Then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(4)
∥
(
∑
j
‍
|
M
(
f
j
)
|
q
)
1
/
q
∥
W
L
1
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
.A weightw is a nonnegative, locally integrable function on R
n; B
=
B
(
x
0
,
r
B
) denotes the ball with the center x
0 and radius r
B. For 1
<
p
<
∞, a weight function w is said to belong to A
p, if there is a constant C
>
0 such that, for every ball B
⊆
R
n (see [2, 3]),
(5)
(
1
|
B
|
∫
B
‍
w
(
x
)
d
x
)
(
1
|
B
|
∫
B
‍
w
(
x
)
-
1
/
(
p
-
1
)
d
x
)
p
-
1
≤
C
.For the casep
=
1,  w
∈
A
1, if there is a constant C
>
0 such that, for every ball B
⊆
R
n,
(6)
1
|
B
|
∫
B
‍
w
(
x
)
d
x
≤
C
·
essinf
⁡
x
∈
B
w
(
x
)
.A weight functionw
∈
A
∞ if it satisfies the A
p condition for some 1
≤
p
<
∞. We say that w
∈
Δ
2, if for any ball B there exists an absolute constant C
>
0 such that
(7)
w
(
2
B
)
≤
C
w
(
B
)
.It is well known that ifw
∈
A
p with 1
≤
p
<
∞, then w
∈
Δ
2. Moreover, if w
∈
A
∞, then, for all balls B and all measurable subsets E of B, there exists δ
>
0 such that
(8)
w
(
E
)
w
(
B
)
≤
C
(
|
E
|
|
B
|
)
δ
.Given a ballB and λ
>
0, λ
B denotes the ball with the same center as B whose radius is λ times that of B. For a given weight function w and a measurable set E, we also denote the Lebesgue measure of E by |
E
| and the weighted measure of E by w
(
E
), where w
(
E
)
=
∫
E
‍
w
(
x
)
d
x.Given a weight functionw on R
n, for 1
≤
p
<
∞, the weighted Lebesgue space L
w
p
(
R
n
) is defined as the set of all functions f such that
(9)
∥
f
∥
L
w
p
=
(
∫
R
n
‍
|
f
(
x
)
|
p
w
(
x
)
d
x
)
1
/
p
<
∞
.We also denote byW
L
w
1
(
R
n
) the weighted weak space consisting of all measurable functions f such that
(10)
∥
f
∥
W
L
w
1
=
sup
⁡
λ
>
0
λ
·
w
(
{
x
∈
R
n
:
|
f
(
x
)
|
>
λ
}
)
<
∞
.In particular, whenw equals to a constant function, we will denote L
w
p
(
R
n
) and W
L
w
1
(
R
n
) simply by L
p
(
R
n
) and W
L
1
(
R
n
).In [4], Andersen and John considered the weighted version of Fefferman-Stein maximal inequality and showed the following.Theorem 3 (see [4]).
Let1
<
p
<
∞ and w
∈
A
p. Then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(11)
∥
(
∑
j
‍
|
M
(
f
j
)
|
q
)
1
/
q
∥
L
w
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
w
p
.Theorem 4 (see [4]).
Letp
=
1 and w
∈
A
1. Then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(12)
∥
(
∑
j
‍
|
M
(
f
j
)
|
q
)
1
/
q
∥
W
L
w
1
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
w
1
.Given a weightw, the weighted maximal function is defined as
(13)
M
w
f
(
x
)
=
sup
⁡
x
∈
B
1
w
(
B
)
∫
B
‍
|
f
(
y
)
|
w
(
y
)
d
y
,
where the supremum is taken over all balls B containing x. One says that u
∈
A
p
(
w
) with 1
<
p
<
∞, if u
,
w
∈
Δ
2 and there is a constant C
>
0 such that, for every ball B
⊆
R
n,
(14)
u
(
B
)
w
(
B
)
(
1
w
(
B
)
∫
B
‍
(
w
(
x
)
u
(
x
)
)
1
/
(
p
-
1
)
w
(
x
)
d
x
)
p
-
1
≤
C
.Iff
→
=
(
f
1
,
f
2
,
…
) is a sequence of locally integrable functions on R
n, then, for any x
∈
R
n, we can also define M
w
(
f
→
)
=
(
M
w
(
f
1
)
,
M
w
(
f
2
)
,
…
) and
(15)
M
w
,
q
(
f
→
)
(
x
)
=
(
∑
j
‍
|
M
w
(
f
j
)
(
x
)
|
q
)
1
/
q
,
1
<
q
<
∞
.In [5], Wu proved the following theorem.Theorem 5 (see [5]).
Let1
<
p
<
∞ and u
∈
A
p
(
w
). Then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(16)
∥
(
∑
j
‍
|
M
w
(
f
j
)
|
q
)
1
/
q
∥
L
u
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
u
p
.Obviously, by the definition, we havew
∈
A
p
(
w
) with 1
<
p
<
∞, if w
∈
Δ
2. Thus, as a direct consequence of Theorem 5, we get the following theorem.Theorem  5′.Let
1
<
p
<
∞
and
w
∈
Δ
2
. Then, for every
1
<
q
<
∞
, there exists a constant
C
>
0
independent of
f
→
=
(
f
1
,
f
2
,
…
)
such that
(17)
∥
(
∑
j
‍
|
M
w
(
f
j
)
|
q
)
1
/
q
∥
L
w
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
w
p
.Let us now consider the vector-valued singular integral operators. Suppose thatS
n
-
1 is the unit sphere in R
n (n
≥
2) equipped with the normalized Lebesgue measure d
σ. Let Ω
∈
L
s
(
S
n
-
1
) with 1
<
s
≤
∞ be homogeneous of degree zero and satisfy the cancellation condition
(18)
∫
S
n
-
1
‍
Ω
(
x
′
)
d
σ
(
x
′
)
=
0
,
where x
′
=
x
/
|
x
| for any x
≠
0. Then, the homogeneous singular integral operator T
Ω is defined by
(19)
T
Ω
f
(
x
)
=
P
.
V
.
∫
R
n
‍
Ω
(
x
-
y
)
|
x
-
y
|
n
f
(
y
)
d
y
.For1
<
s
≤
∞, we say that Ω satisfies the L
s-Dini condition and write Ω
∈
(
D
s
), if Ω
∈
L
s
(
S
n
-
1
) and
(20)
∫
0
1
‍
ω
s
(
δ
)
δ
d
δ
<
∞
,
where ω
s
(
δ
) denotes the integral modulus of continuity of order s of Ω defined by
(21)
ω
s
(
δ
)
:
=
sup
⁡
|
ρ
|
<
δ
(
∫
S
n
-
1
‍
|
Ω
(
ρ
x
′
)
-
Ω
(
x
′
)
|
s
d
σ
(
x
′
)
)
1
/
s
,
ρ is a rotation in R
n, and |
ρ
|
=
sup
⁡
z
′
∈
S
n
-
1
|
ρ
z
′
-
z
′
|. When s
=
∞, ω
s
(
δ
) is defined by the following expression:
(22)
ω
∞
(
δ
)
=
sup
⁡
|
x
′
-
y
′
|
<
δ
{
|
Ω
(
x
′
)
-
Ω
(
y
′
)
|
:
x
′
,
y
′
∈
S
n
-
1
}
.Letf
→
=
(
f
1
,
f
2
,
…
) be a sequence of locally integrable functions on R
n. For any x
∈
R
n, as above, the vector-valued singular integral operators with kernels Ω
∈
(
D
s
) can be defined as T
Ω
(
f
→
)
=
(
T
Ω
(
f
1
)
,
T
Ω
(
f
2
)
,
…
) and
(23)
T
Ω
,
q
(
f
→
)
(
x
)
=
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
1
/
q
,
1
<
q
<
∞
.In [4], Andersen and John derived the weighted strong and weak type estimates for vector-valued singular integral operators with kernels Ω
∈
(
D
∞
) (see also [6, 7]).Theorem 6 (see [4]).
LetT
Ω be a singular integral operator with kernel Ω
∈
(
D
∞
). If 1
<
p
<
∞ and w
∈
A
p, then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(24)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
L
w
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
w
p
.Theorem 7 (see [4]).
LetT
Ω be a singular integral operator with kernel Ω
∈
(
D
∞
). If p
=
1 and w
∈
A
1, then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(25)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
W
L
w
1
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
w
1
.Theorem6 was later extended by Rubio de Francia et al. in [7].Theorem 8 (see [7]).
Let1
<
s
<
∞ and let T
Ω be a singular integral operator with kernel Ω
∈
(
D
s
). If s
′
≤
p
<
∞ and w
∈
A
p
/
s
′, then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(26)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
L
w
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
w
p
.In particular, if we takew to be a constant function, then we immediately get the following unweighted results (see also [8]).Theorem 9.
LetT
Ω be a singular integral operator with kernel Ω
∈
(
D
∞
) and 1
<
p
<
∞. Then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(27)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
L
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
.Theorem 10.
LetT
Ω be a singular integral operator with kernel Ω
∈
(
D
∞
) and p
=
1. Then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(28)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
W
L
1
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
.Theorem 11.
Let1
<
s
<
∞ and let T
Ω be a singular integral operator with kernel Ω
∈
(
D
s
). If s
′
≤
p
<
∞, then, for every 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(29)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
L
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
.


## 2. Our Main Results

The classical Morrey spacesL
p
,
λ were originally introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations. For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator, and the Calderón-Zygmund singular integral operator on these spaces, we refer the reader to [10–12]. In [13], Mizuhara introduced the generalized Morrey spaces L
p
,
Φ which were later extended and studied by many authors (see [14–18]). In [19], Komori and Shirai defined the weighted Morrey spaces L
p
,
κ
(
w
) which could be viewed as an extension of weighted Lebesgue spaces and then studied the boundedness of the above classical operators in harmonic analysis on these weighted spaces.Definition A (see [19]).
Let1
≤
p
<
∞, let  0
<
κ
<
1, and let w be a weight function on R
n. Then, the weighted Morrey space is defined by
(30)
L
p
,
κ
(
w
)
=
{
f
∈
L
loc
⁡
p
(
w
)
:
∥
f
∥
L
p
,
κ
(
w
)
<
∞
}
,
where
(31)
∥
f
∥
L
p
,
κ
(
w
)
=
sup
⁡
B
(
1
w
(
B
)
κ
∫
B
‍
|
f
(
x
)
|
p
w
(
x
)
d
x
)
1
/
p
and the supremum is taken over all balls B in R
n.Forp
=
1 and 0
<
κ
<
1, we also denote by W
L
1
,
κ
(
w
) the weighted weak Morrey spaces of all measurable functions f satisfying
(32)
∥
f
∥
W
L
1
,
κ
(
w
)
=
sup
⁡
B
sup
⁡
λ
>
0
1
w
(
B
)
κ
λ
·
w
(
{
x
∈
B
:
|
f
(
x
)
|
>
λ
}
)
<
∞
.LetΦ
=
Φ
(
r
), r
>
0, be a growth function, that is, a positive increasing function in (
0
,
∞
), and satisfy the following doubling condition:
(33)
Φ
(
2
r
)
≤
D
·
Φ
(
r
)
,
∀
r
>
0
,
where D
=
D
(
Φ
)
≥
1 is a doubling constant independent of r.Definition B (see [13]).
Let1
≤
p
<
∞. One denotes by L
p
,
Φ
=
L
p
,
Φ
(
R
n
) the space of all locally integrable functions f defined on R
n, such that for every x
0
∈
R
n and all r
>
0
(34)
∫
B
(
x
0
,
r
)
‍
|
f
(
x
)
|
p
d
x
≤
C
p
·
Φ
(
r
)
,
where B
(
x
0
,
r
)
=
{
x
∈
R
n
:
|
x
-
x
0
|
<
r
} is the ball centered at x
0 and with radius r
>
0. Then, one lets ∥
f
∥
L
p
,
Φ be the smallest constant C
>
0 satisfying (34) and L
p
,
Φ
(
R
n
) becomes a Banach space with norm ∥
·
∥
L
p
,
Φ.Obviously, whenΦ
(
r
)
=
r
λ with 0
<
λ
<
n, L
p
,
Φ is just the classical Morrey spaces introduced in [9]. We also denote by W
L
1
,
Φ
=
W
L
1
,
Φ
(
R
n
) the generalized weak Morrey spaces of all measurable functions f for which
(35)
sup
⁡
λ
>
0
λ
·
|
{
x
∈
B
(
x
0
,
r
)
:
|
f
(
x
)
|
>
λ
}
|
≤
C
·
Φ
(
r
)
,
for every x
0
∈
R
n and all r
>
0. The smallest constant C
>
0 satisfying (35) is also denoted by ∥
f
∥
W
L
1
,
Φ.Letf
→
=
(
f
1
,
f
2
,
…
) be a sequence of locally integrable functions on R
n. Now, let us formulate our main results as follows. For the continuity properties of vector-valued Hardy-Littlewood maximal function, weighted maximal function, and singular integral operators in the weighted Morrey spaces L
p
,
κ
(
w
) for all 1
≤
p
<
∞ and 0
<
κ
<
1, we will prove the following.Theorem 12.
Let1
<
p
<
∞, 0
<
κ
<
1, and w
∈
A
p. Then, for all 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(36)
∥
(
∑
j
‍
|
M
(
f
j
)
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.Theorem 13.
Letp
=
1, 0
<
κ
<
1, and w
∈
A
1. Then, for all 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(37)
∥
(
∑
j
‍
|
M
(
f
j
)
|
q
)
1
/
q
∥
W
L
1
,
κ
(
w
)
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.Theorem 14.
Let1
<
p
<
∞, 0
<
κ
<
1, and w
∈
Δ
2. Then, for all 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(38)
∥
(
∑
j
‍
|
M
w
(
f
j
)
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.Theorem 15.
LetT
Ω be a singular integral operator with kernel Ω
∈
(
D
∞
). If 1
<
p
<
∞, 0
<
κ
<
1, and w
∈
A
p, then, for all 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(39)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.Theorem 16.
LetT
Ω be a singular integral operator with kernel Ω
∈
(
D
∞
). If p
=
1, 0
<
κ
<
1, and w
∈
A
1, then, for all 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(40)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
W
L
1
,
κ
(
w
)
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.Theorem 17.
Let1
<
s
<
∞ and let T
Ω be a singular integral operator with kernel Ω
∈
(
D
s
). If s
′
≤
p
<
∞, 0
<
κ
<
1, and w
∈
A
p
/
s
′, then, for all 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(41)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.For the vector-valued inequalities in the generalized Morrey spacesL
p
,
Φ for all 1
≤
p
<
∞, we will show the following.Theorem 18.
Assume thatΦ satisfies (33) and 1
≤
D
(
Φ
)
<
2
n. Then, for all 1
<
p
≤
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(42)
∥
(
∑
j
‍
|
M
(
f
j
)
|
q
)
1
/
q
∥
L
p
,
Φ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.Theorem 19.
Assume thatΦ satisfies (33) and 1
≤
D
(
Φ
)
<
2
n. Then, for p
=
1 and all 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(43)
∥
(
∑
j
‍
|
M
(
f
j
)
|
q
)
1
/
q
∥
W
L
1
,
Φ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.Theorem 20.
LetT
Ω be a singular integral operator with kernel Ω
∈
(
D
∞
). Assume that Φ satisfies (33) and 1
≤
D
(
Φ
)
<
2
n; then, for all 1
<
p
,
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(44)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
L
p
,
Φ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.Theorem 21.
LetT
Ω be a singular integral operator with kernel Ω
∈
(
D
∞
). Assume that Φ satisfies (33) and 1
≤
D
(
Φ
)
<
2
n; then, for p
=
1 and all 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(45)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
W
L
1
,
Φ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.Theorem 22.
Let1
<
s
<
∞ and let T
Ω be a singular integral operator with kernel Ω
∈
(
D
s
). Assume that Φ satisfies (33) and 1
≤
D
(
Φ
)
<
2
n; then, for s
′
≤
p
<
∞ and all 1
<
q
<
∞, there exists a constant C
>
0 independent of f
→
=
(
f
1
,
f
2
,
…
) such that
(46)
∥
(
∑
j
‍
|
T
Ω
(
f
j
)
|
q
)
1
/
q
∥
L
p
,
Φ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.Throughout this paper, the letterC always denotes a positive constant independent of the main parameters involved, but it may be different from line to line. By A
~
B, we mean that there exists a constant C
>
1 such that 1
/
C
≤
A
/
B
≤
C. Moreover, we will denote the conjugate exponent of p
>
1 by p
′
=
p
/
(
p
-
1
).


## 3. Boundedness in the Weighted Morrey Spaces

### 3.1. Proofs of Theorems12, 13, and 14

Proof of Theorem12.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
κ
(
w
) with 1
<
p
,
q
<
∞, and 0
<
κ
<
1. Fix a ball B
=
B
(
x
0
,
r
B
)
⊆
R
n and decompose f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B and χ
2
B denotes the characteristic function of 2
B
=
B
(
x
0
,
2
r
B
), j
=
1,2
,
…. Then, we write
(47)
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
(
f
j
0
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
+
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
=
I
1
+
I
2
.
Using Theorem3 and inequality (7), we get
(48)
I
1
≤
1
w
(
B
)
κ
/
p
∥
(
∑
j
‍
|
M
(
f
j
0
)
|
q
)
1
/
q
∥
L
w
p
≤
C
·
1
w
(
B
)
κ
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
w
(
2
B
)
κ
/
p
w
(
B
)
κ
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
Let us now turn to estimate the other termI
2. For any x
∈
B and any fixed ball B
¯
=
B
¯
(
x
,
r
¯
), by definition (1), one can write
(49)
M
(
f
j
∞
)
(
x
)
=
sup
⁡
r
¯
>
0
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
d
y
.
A simple geometric observation shows that whenx
∈
B
(
x
0
,
r
B
) and y
∈
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c, then we have B
(
x
0
,
r
B
)
⊆
2
B
¯
(
x
,
r
¯
)
=
B
¯
(
x
,
2
r
¯
). This fact together with the A
p condition and Hölder’s inequality implies that
(50)
(
∑
j
‍
|
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
d
y
|
q
)
1
/
q
≤
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
1
|
B
¯
(
x
,
r
¯
)
|
(
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
×
(
∫
B
¯
(
x
,
r
¯
)
‍
w
(
y
)
-
p
′
/
p
d
y
)
1
/
p
′
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
¯
(
x
,
r
¯
)
)
(
κ
-
1
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
/
p
.
Therefore, for anyx
∈
B,
(51)
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
/
p
.
So we have(52)
I
2
≤
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
w
(
B
(
x
0
,
r
B
)
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
Combining the above estimates forI
1 and I
2 and then taking the supremum over all balls B
⊆
R
n, we complete the proof of Theorem 12.Proof of Theorem13.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
1
,
κ
(
w
) with 1
<
q
<
∞ and 0
<
κ
<
1. Fix a ball B
=
B
(
x
0
,
r
B
)
⊆
R
n; then, we set f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, for any given λ
>
0, one writes
(53)
w
(
{
x
∈
B
:
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
)
≤
w
(
{
x
∈
B
:
(
∑
j
‍
|
M
(
f
j
0
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
)
+
w
(
{
x
∈
B
:
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
)
=
I
1
′
+
I
2
′
.
Theorem4 and inequality (7) yield
(54)
I
1
′
≤
2
λ
·
∥
(
∑
j
‍
|
M
(
f
j
0
)
|
q
)
1
/
q
∥
W
L
w
1
≤
C
λ
·
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
1
/
q
w
(
x
)
d
x
)
≤
C
·
w
(
2
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
Next, let us turn to deal with the termI
2
′. We apply Chebyshev’s inequality to obtain
(55)
I
2
′
=
∫
{
x
∈
B
:
(
∑
j
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
/
2
}
‍
w
(
x
)
d
x
≤
2
λ
×
∫
{
x
∈
B
:
(
∑
j
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
/
2
}
‍
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
×
w
(
x
)
d
x
.
For anyx
∈
B and any fixed ball B
¯
=
B
¯
(
x
,
r
¯
), we still have B
(
x
0
,
r
B
)
⊆
2
B
¯
(
x
,
r
¯
)
=
B
¯
(
x
,
2
r
¯
) if x
∈
B
(
x
0
,
r
B
) and y
∈
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c. It then follows from (49) and the A
1 condition that
(56)
(
∑
j
‍
|
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
d
y
|
q
)
1
/
q
≤
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
essinf
⁡
y
∈
B
¯
(
x
,
r
¯
)
⁡
w
(
y
)
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
1
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
w
(
y
)
d
y
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
w
(
B
¯
(
x
,
r
¯
)
)
(
κ
-
1
)
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
.
Hence, for anyx
∈
B,
(57)
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
.
Therefore,(58)
I
2
′
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
w
(
B
)
(
κ
-
1
)
·
2
λ
∫
B
‍
w
(
x
)
d
x
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
Summing up the above estimates forI
1
′ and I
2
′ and then taking the supremum over all balls B
⊆
R
n and all λ
>
0, we finish the proof of Theorem 13.Proof of Theorem14.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
κ
(
w
) with 1
<
p
,
q
<
∞, and 0
<
κ
<
1. For any ball B
=
B
(
x
0
,
r
B
)
⊆
R
n and let f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, we have
(59)
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
w
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
w
(
f
j
0
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
+
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
w
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
=
I
1
′′
+
I
2
′′
.
By using Theorem 5′ and inequality (7), we thus obtain
(60)
I
1
′′
≤
1
w
(
B
)
κ
/
p
∥
(
∑
j
‍
|
M
w
(
f
j
0
)
|
q
)
1
/
q
∥
L
w
p
≤
C
·
1
w
(
B
)
κ
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
w
(
2
B
)
κ
/
p
w
(
B
)
κ
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
To estimate the other termI
2
′′, it follows from definition (13) that, for any x
∈
B and any fixed ball B
¯
=
B
¯
(
x
,
r
¯
),
(61)
M
w
(
f
j
∞
)
(
x
)
=
sup
⁡
r
¯
>
0
1
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
w
(
y
)
d
y
.
Moreover, as before, we know thatB
(
x
0
,
r
B
)
⊆
2
B
¯
(
x
,
r
¯
)
=
B
¯
(
x
,
2
r
¯
) if x
∈
B
(
x
0
,
r
B
) and y
∈
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c. Applying the Δ
2 condition and Hölder’s inequality, we can deduce that
(62)
(
∑
j
‍
|
1
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
w
(
y
)
d
y
|
q
)
1
/
q
≤
1
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
w
(
y
)
d
y
≤
1
w
(
B
¯
(
x
,
r
¯
)
)
(
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
×
(
∫
B
¯
(
x
,
r
¯
)
‍
w
(
y
)
d
y
)
1
/
p
′
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
¯
(
x
,
r
¯
)
)
(
κ
-
1
)
/
p
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
/
p
.
Therefore, for anyx
∈
B,
(63)
(
∑
j
‍
|
M
w
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
/
p
.
Consequently,(64)
I
2
′′
≤
(
∑
j
‍
|
M
w
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
w
(
B
(
x
0
,
r
B
)
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
Collecting all the above estimates forI
1
′′ and I
2
′′ and then taking the supremum over all balls B
⊆
R
n, we complete the proof of Theorem 14.


### 3.2. Proofs of Theorems15, 16, and 17

Proof of Theorem15.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
κ
(
w
) with 1
<
p
,
q
<
∞, and 0
<
κ
<
1. Fix a ball B
=
B
(
x
0
,
r
B
)
⊆
R
n and decompose f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, we write
(65)
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
+
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
=
J
1
+
J
2
.
It follows from Theorem6 and inequality (7) that
(66)
J
1
≤
1
w
(
B
)
κ
/
p
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
L
w
p
≤
C
·
1
w
(
B
)
κ
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
w
(
2
B
)
κ
/
p
w
(
B
)
κ
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
To estimate the termJ
2, we first observe that when x
∈
B and y
∈
(
2
B
)
c, then |
y
-
x
|
~
|
y
-
x
0
|. This fact together with the condition Ω
∈
L
∞
(
S
n
-
1
) yields
(67)
|
T
Ω
(
f
j
∞
)
(
x
)
|
=
|
∫
(
2
B
)
c
‍
Ω
(
x
-
y
)
|
x
-
y
|
n
f
j
(
y
)
d
y
|
≤
C
∑
l
=
1
∞
‍
∫
2
l
+
1
B
∖
2
l
B
‍
1
|
x
-
y
|
n
·
|
f
j
(
y
)
|
d
y
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
∖
2
l
B
‍
|
f
j
(
y
)
|
d
y
.
Then, by duality and Hölder’s inequality with exponentq
>
1, we get
(68)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
(
∑
j
‍
|
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
∖
2
l
B
‍
|
f
j
(
y
)
|
d
y
|
q
)
1
/
q
≤
C
sup
⁡
(
∑
j
|
ζ
j
|
q
′
)
1
/
q
′
≤
1
∑
j
‍
(
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
|
f
j
(
y
)
|
d
y
·
|
ζ
j
|
)
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
sup
⁡
(
∑
j
|
ζ
j
|
q
′
)
1
/
q
′
≤
1
(
∑
j
‍
|
f
j
(
y
)
|
·
|
ζ
j
|
)
d
y
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
.
Furthermore, it follows from Hölder’s inequality, (68), and the A
p condition that
(69)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
×
(
∫
2
l
+
1
B
‍
w
(
y
)
-
p
′
/
p
d
y
)
1
/
p
′
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
w
(
2
l
+
1
B
)
(
κ
-
1
)
/
p
.
Sincew
∈
A
p
⊂
A
∞, thus, by inequality (8), we find that
(70)
J
2
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
w
(
B
)
(
1
-
κ
)
/
p
w
(
2
l
+
1
B
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
(
|
B
|
|
2
l
+
1
B
|
)
δ
·
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
,
where the last series is convergent since 0
<
κ
<
1 and δ
>
0. Combining the above estimates for J
1 and J
2 and then taking the supremum over all balls B
⊆
R
n, we complete the proof of Theorem 15.Proof of Theorem16.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
1
,
κ
(
w
) with 1
<
q
<
∞ and 0
<
κ
<
1. For any ball B
=
B
(
x
0
,
r
B
)
⊆
R
n and decompose f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B,  j
=
1,2
,
…. Then, for any given λ
>
0, we can write
(71)
w
(
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
)
≤
w
(
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
)
+
w
(
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
)
=
J
1
′
+
J
2
′
.
Theorem7 and inequality (7) imply that
(72)
J
1
′
≤
2
λ
·
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
W
L
w
1
≤
C
λ
·
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
1
/
q
w
(
x
)
d
x
)
≤
C
·
w
(
2
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
In the proof of Theorem15, we have already showed that, for any x
∈
B,
(73)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
.
On the other hand, it follows directly from theA
1 condition that
(74)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
essinf
⁡
y
∈
2
l
+
1
B
⁡
w
(
y
)
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
C
∑
l
=
1
∞
‍
1
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
w
(
y
)
d
y
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
∑
l
=
1
∞
‍
1
w
(
2
l
+
1
B
)
1
-
κ
.
In addition, sincew
∈
A
1
⊂
A
∞, then by inequality (8), we can see that for all x
∈
B
(75)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
·
1
w
(
B
)
1
-
κ
∑
l
=
1
∞
‍
w
(
B
)
1
-
κ
w
(
2
l
+
1
B
)
1
-
κ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
·
1
w
(
B
)
1
-
κ
∑
l
=
1
∞
‍
(
|
B
|
|
2
l
+
1
B
|
)
δ
*
·
(
1
-
κ
)
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
·
1
w
(
B
)
1
-
κ
,
where in the last inequality we have used the fact that δ
*
·
(
1
-
κ
)
>
0. If
(76)
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
=
∅
,
then the inequality
(77)
J
2
′
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
holds trivially. Now, we may suppose that
(78)
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
≠
∅
.
Then, by the pointwise inequality (75), we can see that
(79)
λ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
·
1
w
(
B
)
1
-
κ
,
which in turn is equivalent to
(80)
w
(
B
)
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
Therefore,(81)
J
2
′
≤
w
(
B
)
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
Summing up the above estimates forJ
1
′ and J
2
′ and then taking the supremum over all balls B
=
B
(
x
0
,
r
B
)
⊆
R
n and all λ
>
0, we conclude the proof of Theorem 16.Proof of Theorem17.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
κ
(
w
) with 1
<
p,  q
<
∞, and 0
<
κ
<
1. For each fixed ball B
=
B
(
x
0
,
r
B
)
⊆
R
n, we set f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, we have
(82)
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
+
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
=
J
1
′′
+
J
2
′′
.
Applying Theorem8 and inequality (7), we can get
(83)
J
1
′′
≤
1
w
(
B
)
κ
/
p
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
L
w
p
≤
C
·
1
w
(
B
)
κ
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
w
(
2
B
)
κ
/
p
w
(
B
)
κ
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
We now turn to deal with the termJ
2
′′. An application of Hölder’s inequality gives us that
(84)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
|
∫
(
2
B
)
c
‍
Ω
(
x
-
y
)
|
x
-
y
|
n
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
|
≤
∑
l
=
1
∞
‍
(
∫
2
l
+
1
B
∖
2
l
B
‍
|
Ω
(
x
-
y
)
|
s
d
y
)
1
/
s
×
(
∫
2
l
+
1
B
∖
2
l
B
‍
1
|
x
-
y
|
n
s
′
(
∑
j
‍
|
f
j
(
y
)
|
q
)
s
′
/
q
d
y
)
1
/
s
′
.
Whenx
∈
B and y
∈
2
l
+
1
B
∖
2
l
B, then we can easily see that 2
l
-
1
r
B
≤
|
y
-
x
|
<
2
l
+
2
r
B. Since Ω
∈
L
s
(
S
n
-
1
) with 1
<
s
<
∞, then
(85)
(
∫
2
l
+
1
B
∖
2
l
B
‍
|
Ω
(
x
-
y
)
|
s
d
y
)
1
/
s
≤
(
∫
2
l
-
1
r
B
∞
‍
∫
S
n
-
1
‍
|
Ω
(
z
′
)
|
s
d
σ
(
z
′
)
ρ
n
-
1
d
ρ
)
1
/
s
≤
C
∥
Ω
∥
L
s
(
S
n
-
1
)
|
2
l
+
1
B
|
1
/
s
.
Consequently,(86)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
Ω
∥
L
s
(
S
n
-
1
)
∑
l
=
1
∞
‍
|
2
l
+
1
B
|
1
/
s
·
1
|
2
l
+
1
B
|
×
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
s
′
/
q
d
y
)
1
/
s
′
=
C
∥
Ω
∥
L
s
(
S
n
-
1
)
×
∑
l
=
1
∞
‍
(
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
s
′
/
q
d
y
)
1
/
s
′
.
We will consider two cases. Whenp
=
s
′, then it follows directly from the A
1 condition that
(87)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
(
1
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
.
For the case ofp
>
s
′, we set t
=
p
/
s
′
>
1. Then, by using Hölder’s inequality with exponent t
>
1 and the fact that w
∈
A
t
⊂
A
∞, we deduce that
(88)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
1
/
s
′
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
×
(
∫
2
l
+
1
B
‍
w
(
y
)
-
t
′
/
t
d
y
)
1
/
t
′
s
′
≤
C
∑
l
=
1
∞
‍
(
1
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
.
Hence, for everys
′
≤
p
<
∞, by the pointwise estimates (87) and (88) together with inequality (8), we finally obtain
(89)
J
2
′′
≤
C
∑
l
=
1
∞
‍
(
1
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
·
w
(
B
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
w
(
B
)
(
1
-
κ
)
/
p
w
(
2
l
+
1
B
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
(
|
B
|
|
2
l
+
1
B
|
)
δ
*
*
·
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
Combining the above estimates forJ
1
′′ and J
2
′′ and then taking the supremum over all balls B
⊆
R
n, we are done.


## 3.1. Proofs of Theorems12, 13, and 14

Proof of Theorem12.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
κ
(
w
) with 1
<
p
,
q
<
∞, and 0
<
κ
<
1. Fix a ball B
=
B
(
x
0
,
r
B
)
⊆
R
n and decompose f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B and χ
2
B denotes the characteristic function of 2
B
=
B
(
x
0
,
2
r
B
), j
=
1,2
,
…. Then, we write
(47)
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
(
f
j
0
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
+
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
=
I
1
+
I
2
.
Using Theorem3 and inequality (7), we get
(48)
I
1
≤
1
w
(
B
)
κ
/
p
∥
(
∑
j
‍
|
M
(
f
j
0
)
|
q
)
1
/
q
∥
L
w
p
≤
C
·
1
w
(
B
)
κ
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
w
(
2
B
)
κ
/
p
w
(
B
)
κ
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
Let us now turn to estimate the other termI
2. For any x
∈
B and any fixed ball B
¯
=
B
¯
(
x
,
r
¯
), by definition (1), one can write
(49)
M
(
f
j
∞
)
(
x
)
=
sup
⁡
r
¯
>
0
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
d
y
.
A simple geometric observation shows that whenx
∈
B
(
x
0
,
r
B
) and y
∈
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c, then we have B
(
x
0
,
r
B
)
⊆
2
B
¯
(
x
,
r
¯
)
=
B
¯
(
x
,
2
r
¯
). This fact together with the A
p condition and Hölder’s inequality implies that
(50)
(
∑
j
‍
|
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
d
y
|
q
)
1
/
q
≤
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
1
|
B
¯
(
x
,
r
¯
)
|
(
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
×
(
∫
B
¯
(
x
,
r
¯
)
‍
w
(
y
)
-
p
′
/
p
d
y
)
1
/
p
′
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
¯
(
x
,
r
¯
)
)
(
κ
-
1
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
/
p
.
Therefore, for anyx
∈
B,
(51)
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
/
p
.
So we have(52)
I
2
≤
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
w
(
B
(
x
0
,
r
B
)
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
Combining the above estimates forI
1 and I
2 and then taking the supremum over all balls B
⊆
R
n, we complete the proof of Theorem 12.Proof of Theorem13.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
1
,
κ
(
w
) with 1
<
q
<
∞ and 0
<
κ
<
1. Fix a ball B
=
B
(
x
0
,
r
B
)
⊆
R
n; then, we set f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, for any given λ
>
0, one writes
(53)
w
(
{
x
∈
B
:
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
)
≤
w
(
{
x
∈
B
:
(
∑
j
‍
|
M
(
f
j
0
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
)
+
w
(
{
x
∈
B
:
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
)
=
I
1
′
+
I
2
′
.
Theorem4 and inequality (7) yield
(54)
I
1
′
≤
2
λ
·
∥
(
∑
j
‍
|
M
(
f
j
0
)
|
q
)
1
/
q
∥
W
L
w
1
≤
C
λ
·
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
1
/
q
w
(
x
)
d
x
)
≤
C
·
w
(
2
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
Next, let us turn to deal with the termI
2
′. We apply Chebyshev’s inequality to obtain
(55)
I
2
′
=
∫
{
x
∈
B
:
(
∑
j
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
/
2
}
‍
w
(
x
)
d
x
≤
2
λ
×
∫
{
x
∈
B
:
(
∑
j
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
/
2
}
‍
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
×
w
(
x
)
d
x
.
For anyx
∈
B and any fixed ball B
¯
=
B
¯
(
x
,
r
¯
), we still have B
(
x
0
,
r
B
)
⊆
2
B
¯
(
x
,
r
¯
)
=
B
¯
(
x
,
2
r
¯
) if x
∈
B
(
x
0
,
r
B
) and y
∈
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c. It then follows from (49) and the A
1 condition that
(56)
(
∑
j
‍
|
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
d
y
|
q
)
1
/
q
≤
1
|
B
¯
(
x
,
r
¯
)
|
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
essinf
⁡
y
∈
B
¯
(
x
,
r
¯
)
⁡
w
(
y
)
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
1
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
w
(
y
)
d
y
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
w
(
B
¯
(
x
,
r
¯
)
)
(
κ
-
1
)
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
.
Hence, for anyx
∈
B,
(57)
(
∑
j
‍
|
M
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
.
Therefore,(58)
I
2
′
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
w
(
B
)
(
κ
-
1
)
·
2
λ
∫
B
‍
w
(
x
)
d
x
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
Summing up the above estimates forI
1
′ and I
2
′ and then taking the supremum over all balls B
⊆
R
n and all λ
>
0, we finish the proof of Theorem 13.Proof of Theorem14.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
κ
(
w
) with 1
<
p
,
q
<
∞, and 0
<
κ
<
1. For any ball B
=
B
(
x
0
,
r
B
)
⊆
R
n and let f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, we have
(59)
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
w
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
w
(
f
j
0
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
+
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
M
w
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
=
I
1
′′
+
I
2
′′
.
By using Theorem 5′ and inequality (7), we thus obtain
(60)
I
1
′′
≤
1
w
(
B
)
κ
/
p
∥
(
∑
j
‍
|
M
w
(
f
j
0
)
|
q
)
1
/
q
∥
L
w
p
≤
C
·
1
w
(
B
)
κ
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
w
(
2
B
)
κ
/
p
w
(
B
)
κ
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
To estimate the other termI
2
′′, it follows from definition (13) that, for any x
∈
B and any fixed ball B
¯
=
B
¯
(
x
,
r
¯
),
(61)
M
w
(
f
j
∞
)
(
x
)
=
sup
⁡
r
¯
>
0
1
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
w
(
y
)
d
y
.
Moreover, as before, we know thatB
(
x
0
,
r
B
)
⊆
2
B
¯
(
x
,
r
¯
)
=
B
¯
(
x
,
2
r
¯
) if x
∈
B
(
x
0
,
r
B
) and y
∈
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c. Applying the Δ
2 condition and Hölder’s inequality, we can deduce that
(62)
(
∑
j
‍
|
1
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
∩
(
2
B
(
x
0
,
r
B
)
)
c
≠
∅
‍
|
f
j
(
y
)
|
w
(
y
)
d
y
|
q
)
1
/
q
≤
1
w
(
B
¯
(
x
,
r
¯
)
)
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
w
(
y
)
d
y
≤
1
w
(
B
¯
(
x
,
r
¯
)
)
(
∫
B
¯
(
x
,
r
¯
)
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
×
(
∫
B
¯
(
x
,
r
¯
)
‍
w
(
y
)
d
y
)
1
/
p
′
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
¯
(
x
,
r
¯
)
)
(
κ
-
1
)
/
p
≤
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
/
p
.
Therefore, for anyx
∈
B,
(63)
(
∑
j
‍
|
M
w
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
w
(
B
(
x
0
,
r
B
)
)
(
κ
-
1
)
/
p
.
Consequently,(64)
I
2
′′
≤
(
∑
j
‍
|
M
w
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
w
(
B
(
x
0
,
r
B
)
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
Collecting all the above estimates forI
1
′′ and I
2
′′ and then taking the supremum over all balls B
⊆
R
n, we complete the proof of Theorem 14.


## 3.2. Proofs of Theorems15, 16, and 17

Proof of Theorem15.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
κ
(
w
) with 1
<
p
,
q
<
∞, and 0
<
κ
<
1. Fix a ball B
=
B
(
x
0
,
r
B
)
⊆
R
n and decompose f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, we write
(65)
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
+
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
=
J
1
+
J
2
.
It follows from Theorem6 and inequality (7) that
(66)
J
1
≤
1
w
(
B
)
κ
/
p
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
L
w
p
≤
C
·
1
w
(
B
)
κ
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
w
(
2
B
)
κ
/
p
w
(
B
)
κ
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
To estimate the termJ
2, we first observe that when x
∈
B and y
∈
(
2
B
)
c, then |
y
-
x
|
~
|
y
-
x
0
|. This fact together with the condition Ω
∈
L
∞
(
S
n
-
1
) yields
(67)
|
T
Ω
(
f
j
∞
)
(
x
)
|
=
|
∫
(
2
B
)
c
‍
Ω
(
x
-
y
)
|
x
-
y
|
n
f
j
(
y
)
d
y
|
≤
C
∑
l
=
1
∞
‍
∫
2
l
+
1
B
∖
2
l
B
‍
1
|
x
-
y
|
n
·
|
f
j
(
y
)
|
d
y
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
∖
2
l
B
‍
|
f
j
(
y
)
|
d
y
.
Then, by duality and Hölder’s inequality with exponentq
>
1, we get
(68)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
(
∑
j
‍
|
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
∖
2
l
B
‍
|
f
j
(
y
)
|
d
y
|
q
)
1
/
q
≤
C
sup
⁡
(
∑
j
|
ζ
j
|
q
′
)
1
/
q
′
≤
1
∑
j
‍
(
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
|
f
j
(
y
)
|
d
y
·
|
ζ
j
|
)
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
sup
⁡
(
∑
j
|
ζ
j
|
q
′
)
1
/
q
′
≤
1
(
∑
j
‍
|
f
j
(
y
)
|
·
|
ζ
j
|
)
d
y
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
.
Furthermore, it follows from Hölder’s inequality, (68), and the A
p condition that
(69)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
×
(
∫
2
l
+
1
B
‍
w
(
y
)
-
p
′
/
p
d
y
)
1
/
p
′
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
w
(
2
l
+
1
B
)
(
κ
-
1
)
/
p
.
Sincew
∈
A
p
⊂
A
∞, thus, by inequality (8), we find that
(70)
J
2
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
w
(
B
)
(
1
-
κ
)
/
p
w
(
2
l
+
1
B
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
(
|
B
|
|
2
l
+
1
B
|
)
δ
·
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
,
where the last series is convergent since 0
<
κ
<
1 and δ
>
0. Combining the above estimates for J
1 and J
2 and then taking the supremum over all balls B
⊆
R
n, we complete the proof of Theorem 15.Proof of Theorem16.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
1
,
κ
(
w
) with 1
<
q
<
∞ and 0
<
κ
<
1. For any ball B
=
B
(
x
0
,
r
B
)
⊆
R
n and decompose f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B,  j
=
1,2
,
…. Then, for any given λ
>
0, we can write
(71)
w
(
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
)
≤
w
(
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
)
+
w
(
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
)
=
J
1
′
+
J
2
′
.
Theorem7 and inequality (7) imply that
(72)
J
1
′
≤
2
λ
·
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
W
L
w
1
≤
C
λ
·
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
1
/
q
w
(
x
)
d
x
)
≤
C
·
w
(
2
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
In the proof of Theorem15, we have already showed that, for any x
∈
B,
(73)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
.
On the other hand, it follows directly from theA
1 condition that
(74)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
essinf
⁡
y
∈
2
l
+
1
B
⁡
w
(
y
)
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
C
∑
l
=
1
∞
‍
1
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
w
(
y
)
d
y
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
∑
l
=
1
∞
‍
1
w
(
2
l
+
1
B
)
1
-
κ
.
In addition, sincew
∈
A
1
⊂
A
∞, then by inequality (8), we can see that for all x
∈
B
(75)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
·
1
w
(
B
)
1
-
κ
∑
l
=
1
∞
‍
w
(
B
)
1
-
κ
w
(
2
l
+
1
B
)
1
-
κ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
·
1
w
(
B
)
1
-
κ
∑
l
=
1
∞
‍
(
|
B
|
|
2
l
+
1
B
|
)
δ
*
·
(
1
-
κ
)
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
·
1
w
(
B
)
1
-
κ
,
where in the last inequality we have used the fact that δ
*
·
(
1
-
κ
)
>
0. If
(76)
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
=
∅
,
then the inequality
(77)
J
2
′
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
holds trivially. Now, we may suppose that
(78)
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
≠
∅
.
Then, by the pointwise inequality (75), we can see that
(79)
λ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
·
1
w
(
B
)
1
-
κ
,
which in turn is equivalent to
(80)
w
(
B
)
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
Therefore,(81)
J
2
′
≤
w
(
B
)
≤
C
·
w
(
B
)
κ
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
κ
(
w
)
.
Summing up the above estimates forJ
1
′ and J
2
′ and then taking the supremum over all balls B
=
B
(
x
0
,
r
B
)
⊆
R
n and all λ
>
0, we conclude the proof of Theorem 16.Proof of Theorem17.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
κ
(
w
) with 1
<
p,  q
<
∞, and 0
<
κ
<
1. For each fixed ball B
=
B
(
x
0
,
r
B
)
⊆
R
n, we set f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, we have
(82)
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
+
1
w
(
B
)
κ
/
p
(
∫
B
‍
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
=
J
1
′′
+
J
2
′′
.
Applying Theorem8 and inequality (7), we can get
(83)
J
1
′′
≤
1
w
(
B
)
κ
/
p
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
L
w
p
≤
C
·
1
w
(
B
)
κ
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
w
(
2
B
)
κ
/
p
w
(
B
)
κ
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
We now turn to deal with the termJ
2
′′. An application of Hölder’s inequality gives us that
(84)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
|
∫
(
2
B
)
c
‍
Ω
(
x
-
y
)
|
x
-
y
|
n
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
|
≤
∑
l
=
1
∞
‍
(
∫
2
l
+
1
B
∖
2
l
B
‍
|
Ω
(
x
-
y
)
|
s
d
y
)
1
/
s
×
(
∫
2
l
+
1
B
∖
2
l
B
‍
1
|
x
-
y
|
n
s
′
(
∑
j
‍
|
f
j
(
y
)
|
q
)
s
′
/
q
d
y
)
1
/
s
′
.
Whenx
∈
B and y
∈
2
l
+
1
B
∖
2
l
B, then we can easily see that 2
l
-
1
r
B
≤
|
y
-
x
|
<
2
l
+
2
r
B. Since Ω
∈
L
s
(
S
n
-
1
) with 1
<
s
<
∞, then
(85)
(
∫
2
l
+
1
B
∖
2
l
B
‍
|
Ω
(
x
-
y
)
|
s
d
y
)
1
/
s
≤
(
∫
2
l
-
1
r
B
∞
‍
∫
S
n
-
1
‍
|
Ω
(
z
′
)
|
s
d
σ
(
z
′
)
ρ
n
-
1
d
ρ
)
1
/
s
≤
C
∥
Ω
∥
L
s
(
S
n
-
1
)
|
2
l
+
1
B
|
1
/
s
.
Consequently,(86)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
Ω
∥
L
s
(
S
n
-
1
)
∑
l
=
1
∞
‍
|
2
l
+
1
B
|
1
/
s
·
1
|
2
l
+
1
B
|
×
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
s
′
/
q
d
y
)
1
/
s
′
=
C
∥
Ω
∥
L
s
(
S
n
-
1
)
×
∑
l
=
1
∞
‍
(
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
s
′
/
q
d
y
)
1
/
s
′
.
We will consider two cases. Whenp
=
s
′, then it follows directly from the A
1 condition that
(87)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
(
1
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
.
For the case ofp
>
s
′, we set t
=
p
/
s
′
>
1. Then, by using Hölder’s inequality with exponent t
>
1 and the fact that w
∈
A
t
⊂
A
∞, we deduce that
(88)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
1
/
s
′
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
×
(
∫
2
l
+
1
B
‍
w
(
y
)
-
t
′
/
t
d
y
)
1
/
t
′
s
′
≤
C
∑
l
=
1
∞
‍
(
1
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
.
Hence, for everys
′
≤
p
<
∞, by the pointwise estimates (87) and (88) together with inequality (8), we finally obtain
(89)
J
2
′′
≤
C
∑
l
=
1
∞
‍
(
1
w
(
2
l
+
1
B
)
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
w
(
y
)
d
y
)
1
/
p
·
w
(
B
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
w
(
B
)
(
1
-
κ
)
/
p
w
(
2
l
+
1
B
)
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
·
∑
l
=
1
∞
‍
(
|
B
|
|
2
l
+
1
B
|
)
δ
*
*
·
(
1
-
κ
)
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
κ
(
w
)
.
Combining the above estimates forJ
1
′′ and J
2
′′ and then taking the supremum over all balls B
⊆
R
n, we are done.


## 4. Boundedness in the Generalized Morrey Spaces

### 4.1. Proofs of Theorems18 and 19

In order to prove the main theorems of this section, we need to establish the following technical lemma.Lemma 23.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
Φ with 1
<
q
<
∞ and the doubling constant D
(
Φ
) satisfy 1
≤
D
(
Φ
)
<
2
n. Then, for any 1
≤
p
<
∞ and any ball B
=
B
(
x
0
,
r
), there exists a constant C
>
0 depending only on p and q such that
(90)
(
∫
R
n
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
≤
C
·
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.Proof.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
Φ with 1
<
q
<
∞ and 1
≤
p
<
∞. Then,
(91)
(
∫
R
n
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
≤
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
+
(
∫
(
2
B
)
c
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
=
I
1
+
I
2
.
By a standard estimate of maximal functionsM
(
χ
B
), we get
(92)
I
1
≤
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
Φ
(
2
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
≤
C
·
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
,
I
2
≤
∑
l
=
1
∞
‍
(
∫
2
l
+
1
B
∖
2
l
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
≤
C
∑
l
=
1
∞
‍
(
1
2
(
l
+
1
)
n
∫
2
l
+
1
B
∖
2
l
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
C
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
1
/
p
[
2
(
l
+
1
)
n
]
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
≤
C
·
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
,
where the last inequality is due to 1
≤
D
(
Φ
)
<
2
n. We are done.Proof of Theorem18.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
Φ with 1
<
p
≤
q
<
∞. We first recall the following vector-valued maximal inequality established by Fefferman and Stein (see [1]); there exists a constant C
>
0 depending on p,  q, and n such that
(93)
∫
R
n
‍
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
w
(
x
)
d
x
≤
∫
R
n
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
M
(
w
)
(
y
)
d
y
holds for any positive functions f
1
,
f
2
,
… and any weight function w. Moreover, it can be easily shown using vector-valued interpolation that, for all 1
<
p
≤
q (see [1, 20]),
(94)
∫
R
n
‍
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
≤
∫
R
n
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
M
(
w
)
(
y
)
d
y
.
It is worth pointing out that the above estimate (94) is false in general in the range p
>
q (see [21]). For any ball B
=
B
(
x
0
,
r
)
⊆
R
n with x
0
∈
R
n and r
>
0, we take w
(
x
) as the characteristic function (
χ
B
)
(
x
) of the ball B
=
B
(
x
0
,
r
); then, by Lemma 23 and (94), we get
(95)
1
Φ
(
r
)
1
/
p
(
∫
B
(
x
0
,
r
)
‍
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
1
Φ
(
r
)
1
/
p
(
∫
R
n
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
≤
C
·
1
Φ
(
r
)
1
/
p
·
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.
Therefore, taking the supremum over all ballsB
=
B
(
x
0
,
r
)
⊆
R
n, we complete the proof of Theorem 18.Proof of Theorem19.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
1
,
Φ with 1
<
q
<
∞. In [21], Pérez proved that, for each weight w and any positive functions f
1
,
f
2
,
…, there exists a constant C
>
0 depending only on q and n such that
(96)
w
(
{
x
∈
R
n
:
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
)
≤
C
λ
∫
R
n
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
M
w
(
y
)
d
y
holds for all λ
>
0. For each fixed ball B
=
B
(
x
0
,
r
)
⊆
R
n, we again take w
(
x
) to be the characteristic function (
χ
B
)
(
x
) of the ball B
=
B
(
x
0
,
r
); then, by using Lemma 23 and (96), we obtain
(97)
|
{
x
∈
B
:
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
|
=
|
{
x
∈
R
n
:
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
∩
B
|
≤
C
λ
∫
R
n
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
M
(
χ
B
)
(
y
)
d
y
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.
Taking the supremum over all ballsB
=
B
(
x
0
,
r
)
⊆
R
n and all λ
>
0, we finish the proof of Theorem 19.


### 4.2. Proofs of Theorems20, 21, and 22

Proof of Theorems20 and 22.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
Φ with 1
<
p,  q
<
∞. For any ball B
=
B
(
x
0
,
r
)
⊆
R
n with x
0
∈
R
n and r
>
0, we write f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, we have
(98)
1
Φ
(
r
)
1
/
p
(
∫
B
(
x
0
,
r
)
‍
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
1
Φ
(
r
)
1
/
p
(
∫
B
(
x
0
,
r
)
‍
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
+
1
Φ
(
r
)
1
/
p
(
∫
B
(
x
0
,
r
)
‍
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
=
J
1
+
J
2
.
Applying Theorems9 and 11 and the doubling condition (33), we obtain
(99)
J
1
≤
1
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
L
p
≤
C
·
1
Φ
(
r
)
1
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
Φ
(
2
r
)
1
/
p
Φ
(
r
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.
Let us analyze the second termJ
2. To this end, we will consider the following two cases. (i) When Ω
∈
L
∞
(
S
n
-
1
), we use the estimate (68) along with Hölder’s inequality to obtain
(100)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
×
(
∫
2
l
+
1
B
‍
1
d
x
)
1
/
p
′
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
.
(ii) The second case is whenΩ
∈
L
s
(
S
n
-
1
) with 1
<
s
<
∞. In the proof of Theorem 17, we have already proved that, for any x
∈
B (see (86)),
(101)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
(
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
s
′
/
q
d
y
)
1
/
s
′
.
For the case ofp
=
s
′, we have
(102)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
(
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
d
y
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
.
We now proceed to the case wherep
>
s
′; in the present situation, we also set t
=
p
/
s
′
>
1. Then, by using Hölder’s inequality with exponent t
>
1, we deduce that
(103)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
1
/
s
′
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
d
y
)
1
/
p
×
(
∫
2
l
+
1
B
‍
1
d
y
)
1
/
t
′
s
′
≤
C
∑
l
=
1
∞
‍
(
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
d
y
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
.
Summarizing the estimates (100)–(103) derived above, we conclude that
(104)
J
2
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
∑
l
=
1
∞
‍
|
B
(
x
0
,
r
)
|
1
/
p
Φ
(
r
)
1
/
p
·
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
.
Since1
≤
D
(
Φ
)
<
2
n, then by using the doubling condition (33) of Φ, we know that the above series is bounded by an absolute constant:
(105)
∑
l
=
1
∞
‍
|
B
(
x
0
,
r
)
|
1
/
p
Φ
(
r
)
1
/
p
·
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
≤
C
∑
l
=
1
∞
‍
(
D
(
Φ
)
2
n
)
(
l
+
1
)
/
p
≤
C
.
Therefore,(106)
J
2
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.
Combining the above estimates forJ
1 and J
2 and then taking the supremum over all balls B
=
B
(
x
0
,
r
)
⊆
R
n, we complete the proof of Theorems 20 and 22.Proof of Theorem21.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
1
,
Φ with 1
<
q
<
∞. For each fixed ball B
=
B
(
x
0
,
r
)
⊆
R
n, we again decompose f
j as f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. For any given λ
>
0, we write
(107)
|
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
|
≤
|
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
|
+
|
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
|
=
J
1
′
+
J
2
′
.
Let us start with the termJ
1
′. Theorem 10 and the doubling condition (33) imply that
(108)
J
1
′
≤
2
λ
·
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
W
L
1
≤
C
λ
·
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
1
/
q
d
x
)
≤
C
·
Φ
(
2
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.
We turn our attention to the estimate ofJ
2
′. It follows directly from the estimate (68) that, for any x
∈
B,
(109)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
·
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
|
B
(
x
0
,
2
l
+
1
r
)
|
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
·
Φ
(
r
)
|
B
(
x
0
,
r
)
|
×
∑
l
=
1
∞
‍
|
B
(
x
0
,
r
)
|
Φ
(
r
)
·
Φ
(
2
l
+
1
r
)
|
B
(
x
0
,
2
l
+
1
r
)
|
.
Notice that1
≤
D
(
Φ
)
<
2
n. Arguing as in the proof of (105), we can get
(110)
∑
l
=
1
∞
‍
|
B
(
x
0
,
r
)
|
Φ
(
r
)
·
Φ
(
2
l
+
1
r
)
|
B
(
x
0
,
2
l
+
1
r
)
|
≤
∑
l
=
1
∞
‍
(
D
(
Φ
)
2
n
)
l
+
1
≤
C
.
Hence, for anyx
∈
B
(
x
0
,
r
),
(111)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
·
Φ
(
r
)
|
B
(
x
0
,
r
)
|
.
If{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
/
2
}
=
∅, then the inequality
(112)
J
2
′
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
holds trivially. Now if instead we suppose that
(113)
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
≠
∅
,
then, by the pointwise inequality (111), we can see that
(114)
λ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
·
Φ
(
r
)
|
B
(
x
0
,
r
)
|
,
which in turn gives us that
(115)
|
B
(
x
0
,
r
)
|
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.
Therefore,(116)
J
2
′
≤
|
B
(
x
0
,
r
)
|
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.
Summing up the above estimates forJ
1
′ and J
2
′ and then taking the supremum over all balls B
=
B
(
x
0
,
r
)
⊆
R
n and all λ
>
0, we conclude the proof of Theorem 21.


## 4.1. Proofs of Theorems18 and 19

In order to prove the main theorems of this section, we need to establish the following technical lemma.Lemma 23.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
Φ with 1
<
q
<
∞ and the doubling constant D
(
Φ
) satisfy 1
≤
D
(
Φ
)
<
2
n. Then, for any 1
≤
p
<
∞ and any ball B
=
B
(
x
0
,
r
), there exists a constant C
>
0 depending only on p and q such that
(90)
(
∫
R
n
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
≤
C
·
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.Proof.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
Φ with 1
<
q
<
∞ and 1
≤
p
<
∞. Then,
(91)
(
∫
R
n
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
≤
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
+
(
∫
(
2
B
)
c
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
=
I
1
+
I
2
.
By a standard estimate of maximal functionsM
(
χ
B
), we get
(92)
I
1
≤
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
Φ
(
2
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
≤
C
·
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
,
I
2
≤
∑
l
=
1
∞
‍
(
∫
2
l
+
1
B
∖
2
l
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
≤
C
∑
l
=
1
∞
‍
(
1
2
(
l
+
1
)
n
∫
2
l
+
1
B
∖
2
l
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
C
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
1
/
p
[
2
(
l
+
1
)
n
]
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
≤
C
·
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
,
where the last inequality is due to 1
≤
D
(
Φ
)
<
2
n. We are done.Proof of Theorem18.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
Φ with 1
<
p
≤
q
<
∞. We first recall the following vector-valued maximal inequality established by Fefferman and Stein (see [1]); there exists a constant C
>
0 depending on p,  q, and n such that
(93)
∫
R
n
‍
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
w
(
x
)
d
x
≤
∫
R
n
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
M
(
w
)
(
y
)
d
y
holds for any positive functions f
1
,
f
2
,
… and any weight function w. Moreover, it can be easily shown using vector-valued interpolation that, for all 1
<
p
≤
q (see [1, 20]),
(94)
∫
R
n
‍
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
p
/
q
w
(
x
)
d
x
≤
∫
R
n
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
M
(
w
)
(
y
)
d
y
.
It is worth pointing out that the above estimate (94) is false in general in the range p
>
q (see [21]). For any ball B
=
B
(
x
0
,
r
)
⊆
R
n with x
0
∈
R
n and r
>
0, we take w
(
x
) as the characteristic function (
χ
B
)
(
x
) of the ball B
=
B
(
x
0
,
r
); then, by Lemma 23 and (94), we get
(95)
1
Φ
(
r
)
1
/
p
(
∫
B
(
x
0
,
r
)
‍
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
1
Φ
(
r
)
1
/
p
(
∫
R
n
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
M
(
χ
B
)
(
x
)
d
x
)
1
/
p
≤
C
·
1
Φ
(
r
)
1
/
p
·
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.
Therefore, taking the supremum over all ballsB
=
B
(
x
0
,
r
)
⊆
R
n, we complete the proof of Theorem 18.Proof of Theorem19.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
1
,
Φ with 1
<
q
<
∞. In [21], Pérez proved that, for each weight w and any positive functions f
1
,
f
2
,
…, there exists a constant C
>
0 depending only on q and n such that
(96)
w
(
{
x
∈
R
n
:
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
)
≤
C
λ
∫
R
n
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
M
w
(
y
)
d
y
holds for all λ
>
0. For each fixed ball B
=
B
(
x
0
,
r
)
⊆
R
n, we again take w
(
x
) to be the characteristic function (
χ
B
)
(
x
) of the ball B
=
B
(
x
0
,
r
); then, by using Lemma 23 and (96), we obtain
(97)
|
{
x
∈
B
:
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
|
=
|
{
x
∈
R
n
:
(
∑
j
‍
|
M
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
∩
B
|
≤
C
λ
∫
R
n
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
M
(
χ
B
)
(
y
)
d
y
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.
Taking the supremum over all ballsB
=
B
(
x
0
,
r
)
⊆
R
n and all λ
>
0, we finish the proof of Theorem 19.


## 4.2. Proofs of Theorems20, 21, and 22

Proof of Theorems20 and 22.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
p
,
Φ with 1
<
p,  q
<
∞. For any ball B
=
B
(
x
0
,
r
)
⊆
R
n with x
0
∈
R
n and r
>
0, we write f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. Then, we have
(98)
1
Φ
(
r
)
1
/
p
(
∫
B
(
x
0
,
r
)
‍
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
1
Φ
(
r
)
1
/
p
(
∫
B
(
x
0
,
r
)
‍
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
+
1
Φ
(
r
)
1
/
p
(
∫
B
(
x
0
,
r
)
‍
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
=
J
1
+
J
2
.
Applying Theorems9 and 11 and the doubling condition (33), we obtain
(99)
J
1
≤
1
Φ
(
r
)
1
/
p
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
L
p
≤
C
·
1
Φ
(
r
)
1
/
p
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
Φ
(
2
r
)
1
/
p
Φ
(
r
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.
Let us analyze the second termJ
2. To this end, we will consider the following two cases. (i) When Ω
∈
L
∞
(
S
n
-
1
), we use the estimate (68) along with Hölder’s inequality to obtain
(100)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
p
/
q
d
x
)
1
/
p
×
(
∫
2
l
+
1
B
‍
1
d
x
)
1
/
p
′
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
.
(ii) The second case is whenΩ
∈
L
s
(
S
n
-
1
) with 1
<
s
<
∞. In the proof of Theorem 17, we have already proved that, for any x
∈
B (see (86)),
(101)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
(
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
s
′
/
q
d
y
)
1
/
s
′
.
For the case ofp
=
s
′, we have
(102)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
(
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
d
y
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
.
We now proceed to the case wherep
>
s
′; in the present situation, we also set t
=
p
/
s
′
>
1. Then, by using Hölder’s inequality with exponent t
>
1, we deduce that
(103)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
1
/
s
′
(
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
d
y
)
1
/
p
×
(
∫
2
l
+
1
B
‍
1
d
y
)
1
/
t
′
s
′
≤
C
∑
l
=
1
∞
‍
(
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
p
/
q
d
y
)
1
/
p
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
.
Summarizing the estimates (100)–(103) derived above, we conclude that
(104)
J
2
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
·
∑
l
=
1
∞
‍
|
B
(
x
0
,
r
)
|
1
/
p
Φ
(
r
)
1
/
p
·
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
.
Since1
≤
D
(
Φ
)
<
2
n, then by using the doubling condition (33) of Φ, we know that the above series is bounded by an absolute constant:
(105)
∑
l
=
1
∞
‍
|
B
(
x
0
,
r
)
|
1
/
p
Φ
(
r
)
1
/
p
·
Φ
(
2
l
+
1
r
)
1
/
p
|
B
(
x
0
,
2
l
+
1
r
)
|
1
/
p
≤
C
∑
l
=
1
∞
‍
(
D
(
Φ
)
2
n
)
(
l
+
1
)
/
p
≤
C
.
Therefore,(106)
J
2
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
p
,
Φ
.
Combining the above estimates forJ
1 and J
2 and then taking the supremum over all balls B
=
B
(
x
0
,
r
)
⊆
R
n, we complete the proof of Theorems 20 and 22.Proof of Theorem21.
Let(
∑
j
‍
|
f
j
|
q
)
1
/
q
∈
L
1
,
Φ with 1
<
q
<
∞. For each fixed ball B
=
B
(
x
0
,
r
)
⊆
R
n, we again decompose f
j as f
j
=
f
j
0
+
f
j
∞, where f
j
0
=
f
j
·
χ
2
B, j
=
1,2
,
…. For any given λ
>
0, we write
(107)
|
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
)
(
x
)
|
q
)
1
/
q
>
λ
}
|
≤
|
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
0
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
|
+
|
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
|
=
J
1
′
+
J
2
′
.
Let us start with the termJ
1
′. Theorem 10 and the doubling condition (33) imply that
(108)
J
1
′
≤
2
λ
·
∥
(
∑
j
‍
|
T
Ω
(
f
j
0
)
|
q
)
1
/
q
∥
W
L
1
≤
C
λ
·
(
∫
2
B
‍
(
∑
j
‍
|
f
j
(
x
)
|
q
)
1
/
q
d
x
)
≤
C
·
Φ
(
2
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.
We turn our attention to the estimate ofJ
2
′. It follows directly from the estimate (68) that, for any x
∈
B,
(109)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∑
l
=
1
∞
‍
1
|
2
l
+
1
B
|
∫
2
l
+
1
B
‍
(
∑
j
‍
|
f
j
(
y
)
|
q
)
1
/
q
d
y
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
·
∑
l
=
1
∞
‍
Φ
(
2
l
+
1
r
)
|
B
(
x
0
,
2
l
+
1
r
)
|
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
·
Φ
(
r
)
|
B
(
x
0
,
r
)
|
×
∑
l
=
1
∞
‍
|
B
(
x
0
,
r
)
|
Φ
(
r
)
·
Φ
(
2
l
+
1
r
)
|
B
(
x
0
,
2
l
+
1
r
)
|
.
Notice that1
≤
D
(
Φ
)
<
2
n. Arguing as in the proof of (105), we can get
(110)
∑
l
=
1
∞
‍
|
B
(
x
0
,
r
)
|
Φ
(
r
)
·
Φ
(
2
l
+
1
r
)
|
B
(
x
0
,
2
l
+
1
r
)
|
≤
∑
l
=
1
∞
‍
(
D
(
Φ
)
2
n
)
l
+
1
≤
C
.
Hence, for anyx
∈
B
(
x
0
,
r
),
(111)
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
·
Φ
(
r
)
|
B
(
x
0
,
r
)
|
.
If{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
/
2
}
=
∅, then the inequality
(112)
J
2
′
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
holds trivially. Now if instead we suppose that
(113)
{
x
∈
B
:
(
∑
j
‍
|
T
Ω
(
f
j
∞
)
(
x
)
|
q
)
1
/
q
>
λ
2
}
≠
∅
,
then, by the pointwise inequality (111), we can see that
(114)
λ
≤
C
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
·
Φ
(
r
)
|
B
(
x
0
,
r
)
|
,
which in turn gives us that
(115)
|
B
(
x
0
,
r
)
|
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.
Therefore,(116)
J
2
′
≤
|
B
(
x
0
,
r
)
|
≤
C
·
Φ
(
r
)
λ
∥
(
∑
j
‍
|
f
j
|
q
)
1
/
q
∥
L
1
,
Φ
.
Summing up the above estimates forJ
1
′ and J
2
′ and then taking the supremum over all balls B
=
B
(
x
0
,
r
)
⊆
R
n and all λ
>
0, we conclude the proof of Theorem 21.



---

*Source: 738125-2014-05-15.xml*