# Riesz Transform Characterization of Weighted Hardy Spaces Associated with Schrödinger Operators

Authors: Hua Zhu
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

We characterize the weighted local Hardy spacesh
ρ
1
(
ω
) related to the critical radius function ρ and weights ω
∈
A
1
ρ
,
∞
(
R
n
) by localized Riesz transforms R
^
j; in addition, we give a characterization of weighted Hardy spaces H
L
1
(
ω
) via Riesz transforms associated with Schrödinger operator L, where L
=
-
Δ
+
V is a Schrödinger operator on R
n (n
≥
3) and V is a nonnegative function satisfying the reverse Hölder inequality.


---

## Body

## 1. Introduction

LetL
=
-
Δ
+
V be a Schrödinger operator on R
n, n
≥
3, where V
≢
0 is a fixed nonnegative potential. We assume that V belongs to the reverse Hölder class R
H
s
(
R
n
) for some s
≥
n
/
2; that is, there exists C
=
C
(
s
,
V
)
>
0 such that(1)
1
B
∫
B
V
x
s
d
x
1
/
s
≤
C
1
B
∫
B
V
x
d
x
,for every ball B
⊂
R
n. With regard to the Schrödinger operator L, we know that the operators derived from L behave “locally” quite similar to those corresponding to the Laplacian (see [1, 2]). The notion of locality is given by the critical radius function(2)
ρ
x
=
1
m
V
x
=
s
u
p
r
>
0
r
:
1
r
n
-
2
∫
B
x
,
r
V
y
d
y
≤
1
.Throughout the paper we assume that V
≢
0, so that 0
<
ρ
(
x
)
<
∞ (see [2]).The study of Schrödinger operatorL
=
-
Δ
+
V recently attracted much attention; see [2–14]. In particularly, Dziubański and Zienkiewicz [5, 6] studied Hardy space H
L
1 associated with Schrödinger operators L. In [8], we introduce the weighted Hardy space H
L
1
(
ω
) with ω
∈
A
1
(
R
n
) (for Muckenhoupt’s weights, see [15–19]), and in [14], we obtain some weighted estimates for bilinear operators on H
L
1
(
ω
).Recently, Bongioanni et al. [3] introduced new classes of weights, related to Schrödinger operators L, that is, A
p
ρ
,
∞
(
R
n
) weight, which are in general larger than Muckenhoupt’s (see Section 2 for notions of A
p
ρ
,
∞
(
R
n
) weight). In [9, 10, 20], Tang gave some results about weighted norm inequalities with weights ω
∈
A
p
ρ
,
∞
(
R
n
). In [21], we have obtained an atomic characterization for weighted Hardy space H
L
1
(
ω
) with ω
∈
A
1
ρ
,
∞
(
R
n
), by introducing the weighted local Hardy spaces h
ρ
p
(
ω
) with A
q
ρ
,
∞
(
R
n
) weights and establishing the atomic characterization of the weighted local Hardy spaces h
ρ
p
(
ω
) with ω
∈
A
q
ρ
,
∞
(
R
n
) weights.An interesting question is whether we can give a characterization of weighted Hardy spacesH
L
1
(
ω
) with ω
∈
A
1
ρ
,
∞
(
R
n
) via Riesz transforms associated with Schrödinger operator L. Since it is difficult to give the characterization by the same method in [5, 8], in this paper, we will establish the characterization by a new method, which will take advantage of weighted local Hardy spaces h
ρ
p
(
ω
) and localized Riesz transforms. In fact, by this method, we can give the Riesz transforms characterization of h
ρ
p
(
ω
) at the same time.The paper is organized as follows. In Section2, we review some notions and notations concerning the weight classes A
p
ρ
,
θ
(
R
n
) introduced in [3, 9, 10], and we also review some results about h
ρ
1
(
ω
) and H
L
1
(
ω
). In Section 3, we establish the Riesz transforms characterization of weighted Hardy spaces H
L
1
(
ω
).Throughout this paper, we letC denote constants that are independent of the main parameters involved but whose value may differ from line to line. By A
~
B, we mean that there exists a constant C
>
1 such that 1
/
C
≤
A
/
B
≤
C. The symbol A
≲
B means that A
≤
C
B. The symbol [
s
] for s
∈
R denotes the maximal integer not more than s. We also set N
≡
{
1,2
,
…
} and Z
+
≡
N
∪
0.


## 2. Preliminaries

In this section, we first review some notions and notations concerning the weight classesA
p
ρ
,
θ
(
R
n
) introduced in [3, 9, 10]. Given B
=
B
(
x
,
r
) and λ
>
0, we will write λ
B for the λ-dilate ball, which is the ball with the same center x and with radius λ
r. Similarly, Q
(
x
,
r
) denotes the cube centered at x with side length r (here and below only cubes with sides parallel to the axes are considered), and λ
Q
(
x
,
r
)
=
Q
(
x
,
λ
r
). Particularly, we will denote 2
B by B
∗ and 2
Q by Q
∗. The following lemma is a basic property of the critical radius function ρ
(
x
).Lemma 1 (see [2]).
There existC
0
≥
1 and k
0
≥
1 so that for all x
,
y
∈
R
n
(3)
C
0
-
1
ρ
x
1
+
x
-
y
ρ
x
-
k
0
≤
ρ
y
≤
C
0
ρ
x
1
+
x
-
y
ρ
x
k
0
/
k
0
+
1
.In particular, ρ
(
x
)
~
ρ
(
y
) when y
∈
B
(
x
,
r
) and r
≤
C
ρ
(
x
), where C is a positive constant.Now we recall the covering ofR
n and the partition of unity related to ρ from [5]. For m
∈
Z we defined the sets B
m by B
m
=
{
x
:
2
-
(
m
+
1
)
/
2
<
ρ
(
x
)
≤
2
-
m
/
2
}, where critical radii ρ
(
x
) have been defined in (3). Since 0
<
ρ
(
x
)
<
∞, we have R
n
=
⋃
m
∈
Z
B
m.Lemma 2 (see [5]).
There is a positive constantC and a collection of balls B
m
,
k
=
B
x
m
,
k
,
2
2
-
m
/
2, m
∈
Z, k
=
1,2
,
…, such that x
(
m
,
k
)
∈
B
m, B
m
⊂
⋃
k
B
(
x
(
m
,
k
)
,
2
-
m
/
2
), and for every (
m
,
k
) and R
≥
2
(4)
#
m
′
,
k
′
:
B
x
m
,
k
,
R
2
-
m
/
2
∩
B
x
m
′
,
k
′
,
R
2
-
m
′
/
2
≠
∅
≤
R
C
; here and in what follows, for any set E, #
E denotes its cardinality.Lemma 3 (see [5]).
There are nonnegative functionsψ
(
m
,
k
) such that (i)
for allm
∈
Z and k
∈
N, ψ
(
m
,
k
)
∈
C
c
∞
(
B
(
x
(
m
,
k
)
,
2
1
-
m
/
2
)
);
(ii)
∑
(
m
,
k
)
ψ
(
m
,
k
)
(
x
)
=
1 for all x
∈
R
n;
(iii)
there exists a positive constantC s.t. for all m
∈
Z, k
∈
N, and ∇
ψ
m
,
k
L
∞
(
R
n
)
≤
C
2
m
.In this paper, we writeΨ
θ
(
B
)
=
(
1
+
r
/
ρ
(
x
0
)
)
θ, where θ
≥
0 and x
0 and r denote the center and radius of B, respectively. As in [3], we say that a weight ω belongs to the class A
p
ρ
,
θ
(
R
n
) for 1
<
p
<
∞, if there is a constant C such that for all balls B
(5)
1
Ψ
θ
B
B
∫
B
ω
y
d
y
1
Ψ
θ
B
B
∫
B
ω
-
1
/
p
-
1
y
d
y
p
-
1
≤
C
. We also say that a nonnegative function ω satisfies the A
1
ρ
,
θ
(
R
n
) condition if there exists a constant C such that M
V
,
θ
(
ω
)
(
x
)
≤
C
ω
(
x
), a.e. x
∈
R
n, where (6)
M
V
,
θ
f
x
≡
s
u
p
x
∈
B
1
Ψ
θ
B
B
∫
B
f
y
d
y
.When V
=
0, we denote M
0
f
(
x
) by M
f
(
x
) (the standard Hardy-Littlewood maximal function). It is easy to see that |
f
(
x
)
|
≤
M
V
,
θ
f
(
x
)
≤
M
f
(
x
) for a.e. x
∈
R
n and any θ
≥
0. Clearly, the classes A
p
ρ
,
θ are increasing with θ, and we denote A
p
ρ
,
∞
=
⋃
θ
≥
0
A
p
ρ
,
θ and A
∞
ρ
,
∞
=
⋃
p
≥
1
A
p
ρ
,
∞. In addition, for 1
≤
p
≤
∞, denote by p
′ the adjoint number of p; that is, 1
/
p
+
1
/
p
′
=
1.We remark that balls can be replaced by cubes in definition ofA
p
ρ
,
θ and M
V
,
θ, since Ψ
(
B
)
≤
Ψ
(
2
B
)
≤
2
θ
Ψ
(
B
)
. In fact, for the cube Q
=
Q
(
x
0
,
r
), we can also define Ψ
θ
(
Q
)
=
(
1
+
r
/
ρ
(
x
0
)
)
θ. Now, we give some properties of weights class A
p
ρ
,
θ for p
≥
1.Lemma 4 (see [3, 10, 21]).
Letω
∈
A
p
ρ
,
∞
=
⋃
θ
≥
0
A
p
ρ
,
θ for p
≥
1. Then (i)
if1
≤
p
1
<
p
2
<
∞, then A
p
1
ρ
,
θ
⊂
A
p
2
ρ
,
θ;
(ii)
ifω
∈
A
p
ρ
,
∞, 1
<
p
<
∞, then there exists ϵ
>
0 such that ω
∈
A
p
-
ϵ
ρ
,
∞;
(iii)
forω
∈
A
p
ρ
,
θ, Q
=
Q
(
x
,
r
), and λ
>
1, there exists a positive constant C such that ω
(
λ
Q
)
≤
C
(
Ψ
θ
(
λ
Q
)
)
p
λ
n
p
ω
(
Q
)
.For anyω
∈
A
∞
ρ
,
∞
(
R
n
), define the critical index of ω by(7)
q
ω
≡
i
n
f
p
∈
1
,
∞
:
ω
∈
A
p
ρ
,
∞
R
n
.For anyφ
∈
D
(
R
n
), let φ
t
(
x
)
=
t
-
n
φ
(
x
/
t
) for t
>
0 and φ
j
(
x
)
=
2
j
n
φ
(
2
j
x
) for j
∈
Z. Now let us introduce the space H
L
1
(
ω
). Let {
T
t
}
t
>
0 be the semigroup of linear operators generated by L and let T
t
(
x
,
y
) be their kernels; that is, T
t
f
(
x
)
=
e
-
t
L
f
(
x
)
=
∫
R
n
T
t
(
x
,
y
)
f
(
y
)
d
y, for t
>
0 and f
∈
L
2
(
R
n
)
. A weighted Hardy-type space related to L with A
1
ρ
,
θ
(
R
n
) weights is naturally defined by H
L
1
ω
≡
f
∈
L
ω
1
R
n
:
T
∗
f
x
∈
L
ω
1
R
n, with f
H
L
1
(
ω
)
≡
T
∗
f
L
ω
1
(
R
n
). H
L
1
(
ω
) with ω
∈
A
1
(
R
n
) has been studied in [8, 14], and we have the following theorem.Theorem A (see [8]).
Ifω
∈
A
1
(
R
n
) and V
∈
R
H
n
(
R
n
) is a nonnegative potential, then, for all f
∈
H
L
1
(
ω
), one has f
H
L
1
(
ω
)
~
f
L
ω
1
(
R
n
)
+
∑
j
=
1
n
R
j
f
L
ω
1
(
R
n
)
.Then let us recall some results about weighted local Hardy spacesh
ρ
p
(
ω
). We first introduce some local maximal functions. For N
∈
Z
+ and R
∈
(
0
,
∞
), let (8)
D
N
,
R
R
n
≡
φ
∈
D
R
n
:
s
u
p
p
φ
⊂
B
0
,
R
,
φ
D
N
R
n
≡
s
u
p
x
∈
R
n
,
α
∈
Z
+
n
,
α
≤
N
∂
α
φ
x
≤
1
.For anyf
∈
D
′
R
n, the local nontangential grand maximal function M
~
N
,
R
(
f
) of f is defined as(9)
M
~
N
,
R
f
x
≡
s
u
p
φ
l
∗
f
z
:
x
-
z
<
2
-
l
<
ρ
x
,
φ
∈
D
N
,
R
R
n
,and the local vertical grand maximal function M
N
,
R
(
f
) of f is defined as(10)
M
N
,
R
f
x
≡
s
u
p
φ
l
∗
f
x
:
0
<
2
-
l
<
ρ
x
,
φ
∈
D
N
,
R
R
n
.For the sake of convenience, we denoteD
N
,
R
(
R
n
), M
~
N
,
R
(
f
), and M
N
,
R
(
f
) as D
N
0
(
R
n
), M
~
N
0
(
f
), and M
N
0
(
f
), when R
=
1, and as D
N
(
R
n
), M
~
N
(
f
), and M
N
(
f
), when R
=
m
a
x
{
R
1
,
R
2
,
R
3
}
>
1 (where R
1, R
2, and R
3 are defined as in Lemmas 4.2, 4.4, and  4.8 in [21]). For any N
∈
Z
+ and x
∈
R
n, obviously, M
N
0
(
f
)
(
x
)
≤
M
N
(
f
)
(
x
)
≤
M
~
N
(
f
)
(
x
)
.Definition 5.
Letω
∈
A
∞
ρ
,
∞
(
R
n
) and let q
ω be as in (7), p
∈
(
0,1
], and N
~
p
,
ω
≡
n
q
ω
/
p
-
1
+
2
. For each N
∈
N with N
≥
N
~
p
,
ω, the weighted local Hardy space is defined by h
ρ
,
N
p
ω
≡
f
∈
D
′
R
n
:
M
N
f
∈
L
ω
p
R
n
. Moreover, let f
h
ρ
,
N
p
(
ω
)
≡
M
N
(
f
)
L
ω
p
(
R
n
).Here and in what follows, we define(11)
N
p
,
ω
≡
m
a
x
N
~
p
,
ω
,
N
0
,where N
~
p
,
ω and N
0 are, respectively, as in Definition 2.2 and Theorem 3.1 of [21].Theorem B (see [21]).
Letω
∈
A
∞
ρ
,
∞
(
R
n
) and let N
p
,
ω be as in (11). Then for any integer N
≥
N
p
,
ω and f
∈
h
ρ
,
N
p
(
ω
), one has(12)
f
h
ρ
,
N
p
ω
~
M
~
N
f
L
ω
p
R
n
~
M
~
N
0
f
L
ω
p
R
n
~
M
N
0
f
L
ω
p
R
n
,where the implicit constants are independent of f.Next, we introduce the weighted local atoms and weighted atomic local Hardy space.Definition 6.
Letω
∈
A
∞
ρ
,
∞
(
R
n
) and let q
ω be as in (7). A triplet (
p
,
q
,
s
)
ω is called admissible, if p
∈
(
0,1
], q
∈
(
q
ω
,
∞
], and s
∈
N with s
≥
[
n
(
q
ω
/
p
-
1
)
]. A function a on R
n is said to be a (
p
,
q
,
s
)
ω-atom if (i)
s
u
p
p
a
⊂
Q
(
x
,
r
) and r
≤
L
1
ρ
(
x
),
(ii)
a
L
ω
q
(
R
n
)
≤
[
ω
(
Q
)
]
1
/
q
-
1
/
p,
(iii)
∫
R
n
a
(
x
)
x
α
d
x
=
0 for all α
∈
Z
+
n with |
α
|
≤
s, when Q
=
Q
x
,
r, r
<
L
2
ρ
(
x
), where L
1
=
4
C
0
(
3
n
)
k
0, L
2
=
1
/
C
0
2
(
3
n
)
k
0
+
1, and C
0 and k
0 are constants given in Lemma 1. Moreover, a function a
(
x
) on R
n is called a (
p
,
q
)
ω-single-atom with q
∈
(
q
ω
,
∞
], if a
L
ω
q
(
R
n
)
≤
[
ω
(
R
n
)
]
1
/
q
-
1
/
p
.Lemma 7 (see [21]).
Letω
∈
A
q
ρ
,
θ
(
R
n
) and a be a (
p
,
q
,
s
)
ω-atom, which satisfies s
u
p
p
a
⊂
Q
(
x
0
,
r
); then there exists a constant C such that a
L
1
(
R
n
)
≤
C
|
Q
|
ω
(
Q
)
-
1
/
p
Ψ
θ
(
Q
)
.Definition 8.
Letω
∈
A
∞
ρ
,
∞
(
R
n
) and let q
ω be as in (7), and let (
p
,
q
,
s
)
ω be admissible. The weighted atomic local Hardy space h
ρ
p
,
q
,
s
(
ω
) is defined as the set of all f
∈
D
′
R
n satisfying that f
=
∑
i
=
0
∞
λ
i
a
i in D
′
R
n, where {
a
i
}
i
∈
N are (
p
,
q
,
s
)
ω-atoms, a
0 is a (
p
,
q
)
ω-single-atom, and {
λ
i
}
i
∈
Z
+
⊂
C. Moreover, the quasi-norm of f
∈
h
ρ
p
,
q
,
s
(
ω
) is defined by f
h
ρ
p
,
q
,
s
(
ω
)
≡
inf
⁡
∑
i
=
0
∞
λ
i
p
1
/
p, where the infimum is taken over all the decompositions of f as above.Theorem C (see [21]).
Letω
∈
A
∞
ρ
,
∞
(
R
n
), and let q
ω and N
p
,
ω be, respectively, as in (7) and (11). If q
∈
q
ω
,
∞, p
∈
(
0,1
] and integers s and N satisfy N
≥
N
p
,
ω and N
>
s
≥
[
n
(
q
ω
/
p
-
1
)
], then h
ρ
p
,
q
,
s
(
ω
)
=
h
ρ
,
N
p
(
ω
)
=
h
ρ
,
N
p
,
ω
p
(
ω
) with equivalent norms.In virtue of the above theorem, for notational simplicity, we denote byh
ρ
p
(
ω
) the weighted local Hardy space h
ρ
,
N
p
(
ω
) when N
≥
N
p
,
ω.Definition 9.
Letω
∈
A
∞
ρ
,
∞
(
R
n
) and let (
p
,
q
,
s
)
ω be admissible; then h
ρ
,
f
i
n
p
,
q
,
s
(
ω
) is defined as the vector space of all finite linear combinations of (
p
,
q
,
s
)
ω-atoms and a (
p
,
q
)
ω-single-atom, and the norm of f in h
ρ
,
f
i
n
p
,
q
,
s
(
ω
) is defined by f
h
ρ
,
f
i
n
p
,
q
,
s
ω
≡
inf
⁡
∑
i
=
0
k
λ
i
p
1
/
p, where f
=
∑
i
=
0
k
λ
i
a
i, k
∈
Z
+, {
λ
i
}
i
=
0
k
⊂
C, {
a
i
}
i
=
1
k are (
p
,
q
,
s
)
ω-atoms, and a
0 is a (
p
,
q
)
ω-single-atom.Letβ
∈
(
0,1
]. A quasi-Banach space B
β with the quasi-norm ·
B
β is called a β-quasi-Banach space if f
+
g
B
β
β
≤
f
B
β
β
+
g
B
β
β for all f
,
g
∈
B
β. For any given β-quasi-Banach space B
β with β
∈
(
0,1
] and a linear space Y, an operator T from Y to B
β is said to be B
β-sublinear if, for any f
,
g
∈
B
β and λ
,
ν
∈
C, T
(
λ
f
+
ν
g
)
B
β
≤
(
|
λ
|
β
T
(
f
)
B
β
β
+
|
ν
|
β
T
(
g
)
B
β
β
)
1
/
β and T
(
f
)
-
T
(
g
)
B
β
≤
T
(
f
-
g
)
B
β.Theorem D (see [21]).
Letω
∈
A
∞
ρ
,
∞
R
n, 0
<
p
≤
β
≤
1, and B
β be a β-quasi-Banach space. Suppose q
∈
(
q
ω
,
∞
) and T
:
h
ρ
,
f
i
n
p
,
q
,
s
(
ω
)
→
B
β is a B
β-sublinear operator such that S
≡
sup
⁡
{
T
a
B
β
:
a
i
s
a
(
p
,
q
,
s
)
ω
-
a
t
o
m
o
r
(
p
,
q
)
ω
s
i
n
g
l
e
-
a
t
o
m
}
<
∞
. Then there exists a unique bounded B
β-sublinear operator T
~ from h
ρ
p
(
ω
) to B
β which extends T.Theorem E (see [21]).
Let0
≢
V
∈
R
H
n
/
2 and ω
∈
A
1
ρ
,
∞
(
R
n
); then h
ρ
1
(
ω
)
=
H
L
1
(
ω
) with equivalent norms; that is, f
h
ρ
1
(
ω
)
~
f
H
L
1
(
ω
).


## 3. Riesz Transform Characterization of Weighted Hardy Spaces

The main purpose of this section is to give a characterization of weighted Hardy spacesH
L
1
(
ω
) with ω
∈
A
1
ρ
,
∞
(
R
n
) via Riesz transforms associated with Schrödinger operator L. We begin with some useful lemmas.Lemma 10 (see [8]).
Ifω
∈
A
1
(
R
n
), then there is a constant C
>
0 such that (13)
∑
m
,
k
R
j
ψ
m
,
k
f
-
ψ
m
,
k
R
j
f
L
1
ω
≤
C
f
L
1
ω
.Lemma 11 (see [5]).
There exists a sequence of pointsx
j
∈
R
n, j
≥
1, such that the family B
j
=
B
(
x
j
,
ρ
(
x
j
)
), j
≥
1, satisfies the following:(a)
⋃
j
B
j
=
R
n.
(b)
For everyσ
≥
1 there exist constants C and N
1 such that Σ
j
χ
σ
B
j
≤
C
σ
N
1.Forj
=
1,2
,
…
,
n, let us define the Riesz transforms R
j associated with Schrödinger operator L by R
j
=
∂
/
∂
x
j
L
-
1
/
2. In addition, as in [22], for all j
∈
{
1,2
,
…
,
n
} and x
∈
R
n, we define localized Riesz transforms as(14)
R
j
l
f
x
≡
p
.
v
.
c
n
∫
R
n
x
j
-
y
j
x
-
y
n
+
1
η
x
-
y
ρ
x
f
y
d
y
,where and in what follows c
n
≡
Γ
(
(
n
+
1
)
/
2
)
/
[
π
(
n
+
1
)
/
2
], η
∈
C
1
(
R
n
) supported in (
-
1,1
), and η
(
t
)
=
1 if |
t
|
≤
1
/
2. For all j
∈
{
1,2
,
…
,
n
} and x
∈
R
n, we define (15)
R
j
,
∗
f
x
≡
c
n
s
u
p
ε
>
0
∫
x
-
y
≥
ε
x
j
-
y
j
x
-
y
n
+
1
f
y
d
y
.With these operators, we can get the following boundedness.Lemma 12.
LetR
j
l be as in (14); then (i)
R
j
l
(
f
)
L
ω
p
(
R
n
)
≤
C
p
,
ω
f
L
ω
p
(
R
n
), for 1
<
p
<
∞ and ω
∈
A
p
(
R
n
),
(ii)
R
j
l
(
f
)
L
ω
1
,
∞
(
R
n
)
≤
C
ω
f
L
ω
1
(
R
n
), for ω
∈
A
1
(
R
n
).Proof.
Letj
∈
{
1,2
,
…
,
n
}; then by [15, 18], we know that R
j
,
∗ is bounded on L
ω
p
(
R
n
) for 1
<
p
<
∞ and bounded from L
ω
1
(
R
n
) to L
ω
1
,
∞
(
R
n
). Let {
R
~
j
}
j
=
1
n be the classical Riesz transforms; then for all x
∈
R
n, we have (16)
R
j
l
f
x
-
R
~
j
f
x
≲
∫
1
/
2
ρ
x
≤
x
-
y
<
ρ
x
x
j
-
y
j
x
-
y
n
+
1
1
-
η
x
-
y
ρ
x
f
y
d
y
+
∫
x
-
y
≥
ρ
x
x
j
-
y
j
x
-
y
n
+
1
f
y
d
y
≲
M
f
x
+
R
j
,
∗
f
x
,where and in what follows M denotes the Hardy-Littlewood maximal operator. By the above estimate, combining with the boundedness on L
ω
p
(
R
n
) for 1
<
p
<
∞ and boundedness from L
ω
1
(
R
n
) to L
ω
1
,
∞
(
R
n
) of M, R
~
j, and R
j
,
∗, we can obtain the conclusions of the lemma.Then forω
∈
A
∞
ρ
,
∞
(
R
n
), we can obtain the following lemma.Lemma 13.
LetR
j
l be as in (14); then (i)
R
j
l
(
f
)
L
ω
p
(
R
n
)
≤
C
p
,
ω
f
L
ω
p
(
R
n
), for 1
<
p
<
∞ and ω
∈
A
p
ρ
,
∞
(
R
n
),
(ii)
R
j
l
(
f
)
L
ω
1
,
∞
(
R
n
)
≤
C
ω
f
L
ω
1
(
R
n
), for ω
∈
A
1
ρ
,
∞
(
R
n
).Proof.
By Lemma11, there exists a sequence of points x
j
∈
R
n, j
≥
1, such that ⋃
j
B
j
=
⋃
j
B
(
x
j
,
ρ
(
x
j
)
)
=
R
n. Let σ
1
=
C
0
2
k
0
/
(
k
0
+
1
)
+
1, with C
0
,
k
0 as in Lemma 1, and set B
~
j
=
σ
1
B
j. Then by Lemma 1 of [3], for ω
∈
A
p
ρ
,
∞
(
R
n
) and any B
j, there exists ω
¯
∈
A
p
(
R
n
) such that ω
¯
=
ω on B
~
j. Thus, by Lemma 12, we have (17)
R
j
l
f
L
ω
p
B
j
=
R
j
l
χ
B
~
j
f
L
ω
p
B
j
≲
R
j
l
χ
B
~
j
f
L
ω
¯
p
B
j
≲
χ
B
~
j
f
L
ω
¯
p
R
n
≲
f
L
ω
p
B
~
j
.Summing up all balls of {
B
j
}
j gives (i).
For (ii), as in (i), forω
∈
A
1
ρ
,
∞
(
R
n
) and any B
j, there exists ω
¯
∈
A
1
(
R
n
) such that ω
¯
=
ω on B
~
j. Then, by Lemma 12, we have (18)
ω
x
∈
B
j
:
R
j
l
f
x
>
λ
≤
ω
x
∈
B
j
:
R
j
l
χ
B
~
j
f
x
>
λ
=
ω
¯
x
∈
B
j
:
R
j
l
χ
B
~
j
f
x
>
λ
≲
λ
-
1
χ
B
~
j
f
L
ω
¯
1
R
n
≲
λ
-
1
f
L
ω
1
B
~
j
.Summing up all balls of {
B
j
}
j gives (ii).
The proof is complete.Now we can establish the Riesz transform characterization as follows.Theorem 14.
Letω
∈
A
1
ρ
,
∞
(
R
n
) and V
∈
R
H
n
(
R
n
); then one has (19)
f
H
L
1
ω
~
f
h
ρ
1
ω
~
f
L
ω
1
R
n
+
∑
j
=
1
n
R
j
l
f
L
ω
1
R
n
~
f
L
ω
1
R
n
+
∑
j
=
1
n
R
j
f
L
ω
1
R
n
.Proof.
f
H
L
1
(
ω
)
~
f
h
ρ
1
(
ω
) is obvious by Theorem E. We now prove (20)
f
h
ρ
1
ω
~
f
L
ω
1
R
n
+
∑
j
=
1
n
R
j
l
f
L
ω
1
R
n
. We first assume that (21)
f
L
ω
1
R
n
+
∑
j
=
1
n
R
j
l
f
L
ω
1
R
n
<
∞
, and here we will borrow some idea from [23]. Let B
(
m
,
k
) be a ball as in Lemma 2, and let ζ
a
1
B
(
m
,
k
) be a C
0
∞ nonnegative function supported in 2
a
1
B
(
m
,
k
) and ζ
a
1
B
(
m
,
k
)
=
1 on a
1
B
(
m
,
k
), where a
1
=
3
C
0
+
1. By properties of weights, we can set ω
¯
∈
A
1
(
R
n
) so that ω
¯
=
ω on a
5
B
(
m
,
k
), where a
5 is a constant independent of B
(
m
,
k
) and will be given in the following proof. Taking φ
∈
D
N
0
(
R
n
), by Theorem A, we have(22)
s
u
p
0
<
2
-
l
<
ρ
x
φ
l
∗
f
L
ω
1
B
m
,
k
≲
s
u
p
0
<
2
-
l
<
ρ
x
φ
l
∗
f
ζ
a
1
B
m
,
k
L
ω
¯
1
R
n
≲
f
ζ
a
1
B
m
,
k
H
L
1
ω
¯
≲
f
ζ
a
1
B
m
,
k
L
ω
¯
1
R
n
+
∑
j
=
1
n
R
j
f
ζ
a
1
B
m
,
k
L
ω
¯
1
R
n
.Let us denote the integral kernel of the operator (
∂
/
∂
x
j
)
L
-
1
/
2 by R
j
(
x
,
y
), and let R
~
j
(
x
,
y
) be the kernel of the classical Riesz transform R
~
j
f
=
(
∂
/
∂
x
j
)
Δ
-
1
/
2
f. Then we can define (23)
R
¯
j
f
x
≡
∫
R
n
R
j
x
,
y
η
x
-
y
ρ
x
f
y
d
y
,and we also have (24)
R
j
l
f
x
=
∫
R
n
R
~
j
x
,
y
η
x
-
y
ρ
x
f
y
d
y
.Let a
2
=
C
0
(
1
+
16
a
1
)
+
2
a
1
<
a
5; then we have (25)
R
j
f
ζ
a
1
B
m
,
k
-
ζ
a
1
B
m
,
k
R
j
l
f
L
ω
¯
1
R
n
≲
R
j
f
ζ
a
1
B
m
,
k
χ
a
2
B
m
,
k
-
ζ
a
1
B
m
,
k
R
j
f
χ
a
2
B
m
,
k
L
ω
¯
1
R
n
+
ζ
a
1
B
m
,
k
R
j
f
χ
a
2
B
m
,
k
-
ζ
a
1
B
m
,
k
R
¯
j
f
χ
a
2
B
m
,
k
L
ω
¯
1
R
n
+
ζ
a
1
B
m
,
k
R
¯
j
f
χ
a
2
B
m
,
k
-
ζ
a
1
B
m
,
k
R
j
l
f
χ
a
2
B
m
,
k
L
ω
¯
1
R
n
≡
I
1
+
I
2
+
I
3
. For I
1, by the same method of the proof of Lemma 10 (Lemma 3.14 of [8]), we get(26)
I
1
≲
f
L
1
a
2
B
m
,
k
,
ω
¯
≲
f
L
ω
1
a
2
B
m
,
k
. For I
2, by Lemma 3 of [24], we have(27)
I
2
≲
ζ
a
1
B
m
,
k
R
j
f
χ
a
2
B
m
,
k
-
ζ
a
1
B
m
,
k
R
¯
j
f
χ
a
2
B
m
,
k
L
ω
¯
1
R
n
=
R
j
f
χ
a
2
B
m
,
k
-
R
¯
j
f
χ
a
2
B
m
,
k
L
1
2
a
1
B
m
,
k
,
ω
¯
≲
∫
2
a
1
B
m
,
k
∫
x
-
y
>
ρ
x
/
2
R
j
x
,
y
1
-
η
x
-
y
ρ
x
f
χ
a
2
B
m
,
k
y
d
y
ω
¯
x
d
x
≲
∫
2
a
1
B
m
,
k
∫
x
-
y
>
ρ
x
/
2
f
χ
a
2
B
m
,
k
y
x
-
y
n
d
y
ω
¯
x
d
x
≲
∫
2
a
1
B
m
,
k
∫
x
-
y
>
ρ
x
/
2
f
χ
a
2
B
m
,
k
y
ρ
x
n
d
y
ω
¯
x
d
x
.Since x
∈
2
a
1
B
(
m
,
k
) and y
∈
a
2
B
(
m
,
k
), we have ρ
(
x
)
~
ρ
(
x
(
m
,
k
)
)
~
ρ
(
y
), and there is a constant a
3
>
1 independent of B
(
m
,
k
) such that (28)
I
2
≲
∫
a
2
B
m
,
k
∫
x
-
y
<
a
3
ρ
y
ω
¯
x
ρ
y
n
d
x
f
χ
a
2
B
m
,
k
y
d
y
≲
∫
a
2
B
m
,
k
f
y
ω
¯
y
d
y
≲
f
L
ω
1
a
2
B
m
,
k
.We now estimate I
3. For x
∈
2
a
1
B
(
m
,
k
) and y
∈
a
2
B
(
m
,
k
), we have ρ
(
x
)
~
ρ
(
x
(
m
,
k
)
)
~
ρ
(
y
), and there exists a constant a
4
>
1 independent of B
(
m
,
k
) such that ρ
(
x
)
≤
a
4
ρ
(
y
). On the other hand, if V
∈
R
H
n
(
R
n
), then there exists ε
>
0, which depends only on n and the constant C in (3), such that V
∈
R
H
n
+
ε
(
R
n
) (see [25]). Thus, according to Lemma 3 of [24], we get (29)
I
3
≲
∫
2
a
1
B
m
,
k
∫
R
n
R
j
x
,
y
-
R
~
j
x
,
y
η
x
-
y
ρ
x
f
χ
a
2
B
m
,
k
y
d
y
ω
¯
x
d
x
≲
∫
2
a
1
B
m
,
k
∫
x
-
y
<
ρ
x
R
j
x
,
y
-
R
~
j
x
,
y
f
χ
a
2
B
m
,
k
y
d
y
ω
¯
x
d
x
≲
∫
2
a
1
B
m
,
k
∫
x
-
y
<
ρ
x
f
χ
a
2
B
m
,
k
y
x
-
y
n
x
-
y
ρ
x
2
-
n
/
n
+
ε
d
y
ω
¯
x
d
x
≲
∫
a
2
B
m
,
k
∫
x
-
y
<
a
4
ρ
y
1
x
-
y
n
x
-
y
ρ
y
2
-
n
/
n
+
ε
ω
¯
x
d
x
f
y
d
y
.For the inner integral, there exists a constant a
5 independent of B
(
m
,
k
) such that a
5
>
a
2 and B
(
y
,
m
a
x
{
a
3
,
2
a
4
}
ρ
(
y
)
)
⊂
a
5
B
(
m
,
k
) for all y
∈
a
2
B
(
m
,
k
); then by the properties of A
1 weights, we have (30)
∫
x
-
y
<
a
4
ρ
y
1
x
-
y
n
x
-
y
ρ
y
2
-
n
/
n
+
ε
ω
¯
x
d
x
≲
∑
k
=
k
0
∞
∫
x
-
y
~
2
-
k
ρ
y
1
x
-
y
n
x
-
y
ρ
y
2
-
n
/
n
+
ε
ω
¯
x
d
x
≲
∑
k
=
k
0
∞
∫
x
-
y
~
2
-
k
ρ
y
2
-
k
2
-
n
/
n
+
ε
2
-
k
ρ
y
n
ω
¯
x
d
x
≲
∑
k
=
k
0
∞
2
-
k
2
-
n
/
n
+
ε
∫
x
-
y
≤
2
-
k
ρ
y
ω
¯
x
2
-
k
ρ
y
n
d
x
≲
ω
¯
y
, where k
0 satisfies 2
-
k
0
-
1
<
a
5
≤
2
-
k
0. Hence, we obtain(31)
I
3
≲
∫
a
2
B
m
,
k
f
y
ω
¯
y
d
y
≲
f
L
ω
1
a
2
B
m
,
k
. Combining (22), (26), (28), and (31), we get (32)
s
u
p
0
<
2
-
l
<
ρ
x
φ
l
∗
f
L
ω
1
B
m
,
k
≲
f
L
ω
1
a
2
B
m
,
k
+
∑
j
=
1
n
R
j
l
f
L
ω
1
2
a
1
B
m
,
k
. Therefore, by Theorem B and Lemma 2, we obtain(33)
f
h
ρ
1
ω
≲
∑
m
,
k
s
u
p
0
<
2
-
l
<
ρ
x
φ
l
∗
f
L
ω
1
B
m
,
k
≲
∑
m
,
k
f
L
ω
1
a
2
B
m
,
k
+
∑
j
=
1
n
R
j
l
f
L
ω
1
2
a
1
B
m
,
k
≲
f
L
ω
1
R
n
+
∑
j
=
1
n
R
j
l
f
L
ω
1
R
n
.
In order to prove the converse inequality, by Theorem D, it suffices to show that, for allj and any (
1,2
,
s
)
ω-atom or (
1,2
)
ω-single-atom a, (34)
R
j
l
a
L
ω
1
R
n
≲
1
. If a is a single atom, by Hölder inequality and L
ω
2
(
R
n
) boundedness of R
j
l, we have (35)
R
j
l
a
L
ω
1
R
n
≲
R
j
l
a
L
ω
2
R
n
ω
R
n
1
/
2
≲
a
L
ω
2
R
n
ω
R
n
1
/
2
≲
1
.Next we assume that a is a (
1,2
,
s
)
ω-atom and s
u
p
p
a
⊂
Q
(
x
0
,
r
) with r
≤
L
1
ρ
(
x
0
); then ρ
(
x
)
≤
C
0
2
2
k
0
(
1
+
L
1
)
ρ
(
x
0
) for any x satisfying |
x
-
y
|
<
ρ
(
x
) and y
∈
Q
(
x
0
,
r
).
We first consider the atoma with L
2
ρ
(
x
0
)
≤
r
≤
L
1
ρ
(
x
0
). Taking b
1
=
1
+
C
0
2
2
k
0
(
1
+
L
1
)
/
L
2, then s
u
p
p
(
R
j
l
(
a
)
)
⊂
Q
(
x
0
,
b
1
r
); thus, by Hölder inequality, Lemma 4, and L
ω
2
(
R
n
) boundedness of R
j
l, we get (36)
R
j
l
a
L
ω
1
R
n
=
R
j
l
a
L
ω
1
Q
x
0
,
b
1
r
≲
a
L
ω
2
R
n
ω
Q
x
0
,
b
1
r
1
/
2
≲
a
L
ω
2
R
n
ω
Q
x
0
,
r
1
/
2
≲
1
.
Ifr
<
L
2
ρ
(
x
0
), then s
u
p
p
(
R
j
l
(
a
)
)
⊂
Q
(
x
0
,
b
2
ρ
(
x
0
)
), where b
2
=
C
0
2
2
k
0
(
1
+
L
1
)
+
L
2. Applying the Hölder inequality, Lemma 4 and L
ω
2
(
R
n
) boundedness of R
j
l give us that (37)
R
j
l
a
L
ω
1
Q
x
0
,
2
r
≲
a
L
ω
2
R
n
ω
Q
x
0
,
r
1
/
2
≲
1
.Moreover, for x
∈
Q
(
x
0
,
b
2
ρ
(
x
0
)
)
∖
Q
(
x
0
,
2
r
), there is a constant b
3
>
1 such that b
3
-
1
ρ
(
x
0
)
≤
ρ
(
x
)
≤
b
3
ρ
(
x
0
), and by the vanishing moment of a, we get (38)
R
j
l
a
x
≲
∫
R
n
x
j
-
y
j
x
-
y
n
+
1
η
x
-
y
ρ
x
-
1
a
y
d
y
+
∫
R
n
x
j
-
y
j
x
-
y
n
+
1
-
x
j
-
x
0
j
x
-
x
0
n
+
1
a
y
d
y
≲
∫
x
-
y
≥
ρ
x
0
/
2
b
3
a
y
x
-
y
n
d
y
+
∫
R
n
x
0
-
y
x
-
x
0
n
+
1
a
y
d
y
≲
1
ρ
x
0
n
+
r
x
-
x
0
n
+
1
a
L
1
R
n
,where, in the penultimate inequality, we use the fact that if |
x
-
y
|
<
ρ
(
x
0
)
/
2
b
2, then η
(
|
x
-
y
|
/
ρ
(
x
)
)
=
1. This implies that (39)
∫
Q
x
0
,
b
2
ρ
x
0
∖
Q
x
0
,
2
r
R
j
l
a
x
ω
x
d
x
≲
a
L
1
R
n
∫
Q
x
0
,
b
2
ρ
x
0
∖
Q
x
0
,
2
r
ω
x
ρ
x
0
n
d
x
+
a
L
1
R
n
∫
Q
x
0
,
b
2
ρ
x
0
∖
Q
x
0
,
2
r
r
ω
x
x
-
x
0
n
+
1
d
x
≡
I
+
I
I
. For I, since s
u
p
p
a
⊂
Q
0
≡
Q
(
x
0
,
r
)
⊂
Q
(
x
0
,
ρ
(
x
0
)
)
≡
Q, by Lemmas 4 and 7, we have (40)
I
≲
a
L
1
R
n
ω
Q
x
0
,
b
2
ρ
x
0
ρ
x
0
n
≲
Q
ω
Q
ω
Q
x
0
,
b
2
ρ
x
0
ρ
x
0
n
≲
1
. For I
I, by Lemmas 4 and 7, we get (41)
I
I
≲
a
L
1
R
n
∫
2
r
≤
x
-
x
0
≤
b
2
ρ
x
0
r
ω
x
x
-
x
0
n
+
1
d
x
≲
a
L
1
R
n
∑
j
=
2
j
0
∫
x
-
x
0
~
2
j
r
r
ω
x
x
-
x
0
n
+
1
d
x
≲
Q
0
ω
Q
0
∑
j
=
2
j
0
2
-
k
ω
2
j
Q
0
2
k
Q
0
≲
1
, where j
0 is an integer such that 2
j
0
-
1
r
<
b
2
ρ
(
x
0
)
≤
2
j
0
r. Thus, for r
<
L
2
ρ
(
x
0
) we obtain (42)
R
j
l
a
L
ω
1
R
n
≲
1
.
Finally, we provef
L
ω
1
(
R
n
)
+
∑
j
=
1
n
R
j
(
f
)
L
ω
1
(
R
n
)
~
f
L
ω
1
(
R
n
)
+
∑
j
=
1
n
R
j
l
(
f
)
L
ω
1
(
R
n
)
. Indeed, we only need to show that (43)
R
j
l
f
-
R
j
f
L
ω
1
R
n
≲
f
L
ω
1
R
n
. In fact, note that |
x
-
y
|
≤
ρ
(
x
); then there exists a constant β
>
0 such that ρ
(
y
)
/
β
≤
ρ
(
x
)
≤
β
ρ
(
y
) by Lemma 1. Set ω
∈
A
1
ρ
,
θ
(
R
n
) with θ
>
0. Hence, by the properties of A
1
ρ
,
θ
(
R
n
), we get (44)
R
j
l
f
-
R
j
f
L
ω
1
R
n
≤
∬
R
n
R
j
x
,
y
-
R
~
j
x
,
y
η
x
-
y
ρ
x
f
y
d
y
ω
x
d
x
≤
∫
R
n
∫
x
-
y
<
ρ
y
/
β
R
j
x
,
y
-
R
~
j
x
,
y
ω
x
d
x
f
y
d
y
+
∫
R
n
∫
ρ
y
/
β
≤
x
-
y
<
β
ρ
y
R
~
j
x
,
y
ω
x
d
x
f
y
d
y
+
∫
R
n
∫
ρ
y
/
β
≤
x
-
y
R
j
x
,
y
ω
x
d
x
f
y
d
y
≲
f
L
ω
1
R
n
+
∫
R
n
ω
Q
y
,
ρ
y
ρ
y
n
f
y
d
y
+
∫
R
n
∑
k
=
1
∞
2
-
k
N
2
k
ρ
y
-
n
∫
x
-
y
≤
2
k
ρ
y
/
β
ω
x
d
x
f
y
d
y
≲
f
L
ω
1
R
n
+
∫
R
n
∑
k
=
1
∞
2
-
k
N
-
k
0
+
1
θ
ω
Q
y
,
ρ
y
ρ
y
n
f
y
d
y
≲
f
L
ω
1
R
n
, if taking N
>
(
k
0
+
1
)
θ
+
1.
Therefore, the proof is complete.



---

*Source: 2856103-2016-07-26.xml*