Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators

Dachun YANG, Dongyong YANG
1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics
and Complex Systems, Ministry of Education, Beijing 100875, China
2 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
c
Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract Let ϕ be a growth function, and let A := −(∇−ia)·(∇−ia)+V be a
magnetic Schr¨odinger operator on L2(Rn), n 2, where a := (a1, a2, . . . , an) ∈
L2
loc(Rn,Rn) and 0 V ∈ L1
loc(Rn). We establish the equivalent characterizations
of the Musielak-Orlicz-Hardy space HA,ϕ(Rn), defined by the Lusin area
function associated with {e−t2A}t>0, in terms of the Lusin area function
associated with {e−t
√
A}t>0, the radial maximal functions and the nontangential
maximal functions associated with {e−t2A}t>0 and {e−t
√
A}t>0,
respectively. The boundedness of the Riesz transforms LkA−1/2, k ∈ {1, 2, . . . ,
n}, from HA,ϕ(Rn) to Lϕ(Rn) is also presented, where Lk is the closure of
∂
∂xk
− iak in L2(Rn). These results are new even when ϕ(x, t) := ω(x)tp for
all x ∈ Rn and t ∈ (0,+∞) with p ∈ (0, 1] and ω ∈ A∞(Rn) (the class of
Muckenhoupt weights on Rn).
Keywords Magnetic Schr¨odinger operator, Musielak-Orlicz-Hardy space, Lusin
area function, growth function, maximal function, Riesz transform
MSC 42B25, 42B20, 42B30, 42B35
1 Introduction
The development of the theory of Hardy spaces Hp(Rn), p ∈ (0, 1], was initiated
by Stein and Weiss [37], and was originally tied to harmonic functions. In
1972, real variable methods were introduced into this subject by Fefferman and
Stein [17]. Later, the advent of their atomic or molecular characterizations
enabled the extension of Hp(Rn) to far more general settings such as spaces
Received July 23, 2014; accepted September 22, 2014
Corresponding author: Dongyong YANG, E-mail: dyyang@xmu.edu.cn
1204 Dachun YANG, Dongyong YANG
of homogeneous type in the sense of Coifman and Weiss [8]. Nowadays, the
theory of Hardy spaces has played an important role in analysis and partial
differential equations; see, for example, [19,36]. It is known that Hp(Rn) is
essentially related to the Laplacian Δ and there are many settings in which
these classical spaces are not applicable. For instance, the Riesz transforms
∇L−1/2 may not be bounded from H1(Rn) to L1(Rn) when L is a second order
divergence form elliptic operator with complex bounded measurable coefficients;
see [21].
Recently, the study of the theory of Hardy spaces associated with operators
has been paid a lot of attention; see, for example, [1,4,7,12,14,15,20,22,25,40]
and references therein. In particular, let
A :=
n
k=1
L
∗
kLk + V
be a magnetic Schr¨odinger operator, where Lk is the closure in L2(Rn)
of ∂
∂xk
− iak, L∗
k the adjoint operator of Lk in L2(Rn), k ∈ {1, 2, . . . , n},
a := (a1, a2, . . . , an): Rn → Rn the magnetic potential, and V : Rn → R the
electrical potential. Auscher et al. [1] first investigated the theory of Hardy
spaces Hp
L(Rn), defined by the Lusin area function associated with the semigroup
{e−tL}t>0, where the infinitesimal generator L satisfies that the kernels
of {e−tL}t>0 have a Gaussian upper bound, and includes A as a special case.
Duong and Yan [15] further showed that the dual space of H1A
(Rn) isBMOA(Rn)
associated with A. Duong et al. [13] established the boundedness of the Riesz
transforms LkA−1/2 with k ∈ {1, 2, . . . , n} from the Hardy space H1A
(Rn) to
L1(Rn). Let X be a metric space, and let L be a nonnegative self-adjoint
operator satisfying the so-called Davies-Gaffney estimate. Hofmann et al. [20]
introduced and characterized the space H1L(X ) in terms of atoms, molecules
and the Lusin area function associated with the semigroup {e−t
√
L}t>0. These
characterizations were, in [20], applied to the Schr¨odinger operator A on Rn
with a = 0 to establish the equivalent characterizations of H1A
(Rn) in terms
of the non-tangential maximal functions and the radial maximal functions
associated with {e−t2A}t>0 and {e−t
√
A}t>0, respectively. All these results were
further generalized to Orlicz-Hardy spaces in [6,25], which include the Hardy
spaces Hp
A(Rn) with p ∈ (0, 1] as special cases. Inspired by [20,25], for general
a, the equivalent characterizations of Hp
A(Rn) in terms of the non-tangential
maximal functions and the radial maximal functions associated with {e−t2A}t>0
and {e−t
√
A}t>0 were established in [26].
On the other hand, Ky [29] studied Hardy spaces of Musielak-Orlicz type,
which generalize the Orlicz-Hardy spaces introduced by Str¨omberg [38] and
Janson [24] and the weighted Hardy spaces by Garc´ıa-Cuerva [18] and Str¨omberg
and Torchinsky [39]. We point out that the motivation to study function spaces
of Musielak-Orlicz type comes from applications to many fields of mathematics
and physics; see [2,3,10,11,28,30]. Let L be a nonnegative self-adjoint operator
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1205
on a metric measure space X , whose heat kernels satisfy Davies-Gaffney
estimates, and let
ϕ: X × [0,+∞) → [0,+∞)
be a function such that ϕ ∈ A∞(X ), the class of uniformly Muckenhoupt
weights (see Definition 1.1 below), its critical uniformly upper type index I(ϕ) ∈
(0, 1] and ϕ(·, t) ∈ RH
2/[2−I(ϕ)](X ), namely, ϕ satisfies the uniformly reverse
H¨older inequality of order 2/[2 − I(ϕ)] (see Definition 2.2 below). In [42],
the Musielak-Orlicz-Hardy space Hϕ,L(X ) was introduced and characterized
in terms of atoms, molecules, and the Lusin-area function associated with
the Poisson semigroup {e−t
√
A}t>0, and applied to the Schr¨odinger operator
A on Rn with a = 0 to obtain the equivalent characterizations of Hϕ,A(Rn)
in terms of aforementioned four maximal functions. Recently, Bui et al. [5]
further investigated Hϕ,L(X ) when L is a one-to-one operator of type ω, has
a bounded H∞-functional calculus in L2(X ) satisfying the reinforced (pL, qL)
off-diagonal estimates on balls, and ϕ(·, t) ∈ RH(qL/I(ϕ)) (X ), where pL ∈ [1, 2),
qL ∈ (2,+∞], and (qL/I(ϕ)) denotes the conjugate exponent of qL/I(ϕ). Here
and hereafter, for any index q ∈ [1,+∞], q denotes its conjugate index, that is,
1
q
+
1
q = 1.
Let ak ∈ L2
loc(Rn) be real-valued, k ∈ {1, 2, . . . , n}, and 0 V ∈ L1
loc(Rn).
The aim of this article is to characterize the Musielak-Orlicz-Hardy space
Hϕ,A(Rn) in terms of the Lusin-area function associated with the Poisson semigroup
{e−t
√
A}t>0 and aforementioned four maximal functions, and obtain the
boundedness of the Riesz transforms LkA−1/2 for k ∈ {1, 2, . . . , n} from the
Musielak-Orlicz-Hardy space Hϕ,A(Rn) to the Musielak-Orlicz space Lϕ(Rn).
To state our main results, we first recall some necessary notions and
notation. In this article, for k ∈ {1, 2, . . . , n}, Lk denotes the closure in L2(Rn)
of ∂
∂xk
− iak with domain C∞
c (Rn) (the set of C∞(Rn) functions with compact
support). The corresponding sesquilinear form Q is defined by setting, for all
f, g ∈ D(Q),
Q(f, g) :=
n
k=1

Rn
Lkf(x)Lk(x)g(x) dx +

Rn
V (x)f(x)g(x) dx,
where
D(Q) := {f ∈ L2(Rn) : Lkf ∈ L2(Rn), k ∈ {1, 2, . . . , n},
√
V f ∈ L2(Rn)}.
The form Q is symmetric and closed. It was showed by Simon [35] that this form
coincides with the minimal closure of the form given by the same expression
but defined on C∞
c (Rn). Let
D(A) :=

f ∈ D(Q): ∃ g ∈ L2(Rn) such that ∀ ϕ ∈ D(Q),
Q(f,ϕ) =

Rn
g(x)ϕ(x)dx

(1.1)
1206 Dachun YANG, Dongyong YANG
and let Af := g for all f ∈ D(A) and g ∈ L2(Rn) as in (1.1). Then the
magnetic Schr¨odinger operator A is a self-adjoint operator by the symmetry of
Q; see [33]. Formally, we write
Af =
n
k=1
L
∗
kLkf + V f (1.2)
or
A = −(∇−ia) · (∇−ia) + V.
On the other hand, a function Φ: [0,+∞) → [0,+∞) is called an Orlicz
function if it is nondecreasing, Φ(0) = 0, Φ(t) > 0 for t ∈ (0,+∞), and
limt→+∞ Φ(t) = +∞ (see, for example, [32,34]). Unlike the classical case,
an Orlicz function in this article may not be convex. The function Φ is said to
be of upper (resp. lower) type p for some p ∈ [0,+∞), if there exists a positive
constant C such that, for all s ∈ [1,+∞) (resp. s ∈ [0, 1]) and t ∈ [0,+∞),
Φ(st) CspΦ(t).
For a given function ϕ: Rn×[0,+∞) → [0,+∞) such that, for any given x ∈
Rn, ϕ(x, ·) is an Orlicz function, ϕ is said to be of uniformly upper (resp. lower)
type p for some p ∈ [0,+∞) if there exists a positive constant C such that, for
all x ∈ Rn, s ∈ [1,+∞) (resp. s ∈ [0, 1]) and t ∈ [0,+∞),
ϕ(x, st) Cspϕ(x, t).
Moreover, let I(ϕ) and i(ϕ) be, respectively, the critical uniformly upper type
index and the critical uniformly lower type index defined, respectively, by
I(ϕ) := inf{p ∈ (0,+∞): ϕ is of uniformly upper type p} (1.3)
and
i(ϕ) := sup{p ∈ (0,+∞): ϕ is of uniformly lower type p}. (1.4)
Observe that i(ϕ) and I(ϕ) may not be attainable, namely, ϕ may not be of
uniformly lower type i(ϕ) or of uniformly upper type I(ϕ) (see [23,31] for some
examples).
Definition 1.1 [29] A function ϕ: Rn × [0,+∞) → [0,+∞) is said to satisfy
the uniformly Muckenhoupt condition for some q ∈ [1,+∞), denoted by ϕ ∈
Aq(Rn), if, when q ∈ (1,+∞),
sup
t∈(0,+∞)
sup
B⊂Rn
1
|B|

B
ϕ(x, t)dx

1
|B|

B
[ϕ(x, t)]−q /qdx
q/q
< +∞,
or
sup
t∈(0,+∞)
sup
B⊂Rn
1
|B|

B
ϕ(x, t)dx

ess sup
y∈B
[ϕ(y, t)]−1
< +∞.
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1207
Here, the first suprema are taken over all t ∈ (0,+∞) and the second ones over
all balls B ⊂ Rn.
Observe that, in Definition 1.1, if ϕ is independent of t, then ϕ ∈ Aq(Rn) for
q ∈ [1,+∞) just means ϕ ∈ Aq(Rn), the classical class of Muckenhoupt weights
(see, for example, [19,39]).
Definition 1.2 [29] A function ϕ: Rn×[0,+∞) → [0,+∞) is called a growth
function if the following hold true:
(i) ϕ is a Musielak-Orlicz function, namely,
(a) the function ϕ(x, ·): [0,+∞) → [0,+∞) is an Orlicz function for any
given x ∈ Rn;
(b) the function ϕ(·, t) is a measurable function for any given t ∈ [0,+∞);
(ii)
ϕ ∈ A∞(Rn) :=

q∈[1,+∞)
Aq(Rn);
(iii) the function ϕ is of uniformly upper type 1 and of uniformly lower
type p for some p ∈ (0, 1].
Throughout the article, we always assume that ϕ is a growth function as in
Definition 1.2. Clearly, the functions
ϕ(x, t) := tp, ∀ x ∈ Rn, t ∈ (0,+∞), p ∈ (0, 1], (1.5)
and
ϕ(x, t) := ω(x)tp, ∀ x ∈ Rn, t ∈ (0,+∞), p ∈ (0, 1], (1.6)
are both growth functions in Definition 1.2, where
ω ∈ A∞(Rn) :=

q∈[1,+∞)
Aq(Rn).
Another typical growth function is
ϕ(x, t) := t
log(e + |x|) + log(e + t) , ∀ x ∈ Rn, t ∈ (0,+∞). (1.7)
If ϕ is as in (1.7), then it is easy to show that ϕ ∈ A1(Rn), I(ϕ) = i(ϕ) = 1,
i(ϕ) is not attainable, but I(ϕ) is attainable (see [5,29]). For more examples of
growth functions, see, for example, [5,23,29]. The Musielak-Orlicz space Lϕ(Rn)
is then defined as the set of all measurable functions f such that

Rn
ϕ(x, |f(x)|)dx < +∞
with Luxemburg norm

f

Lϕ(Rn) := inf

λ ∈ (0,+∞):

Rn
ϕ

x,
|f(x)|
λ


dx 1

.
1208 Dachun YANG, Dongyong YANG
We now recall the definition of Hϕ,A(Rn) in [5]; see also [1,20,26] for the
definition of Hp
A(Rn) with p ∈ (0, 1], which corresponds to ϕ as in (1.5). For all
functions f ∈ L2(Rn) and x ∈ Rn, define the Lusin-area function SAf by
SAf(x) :=

Γ(x)
|t2Ae−t2Af(y)|2 dydt
tn+1
1/2
,
where, for all x ∈ Rn,
Γ(x) := {(y, t) ∈ Rn × (0,+∞): |x − y| < t}. (1.8)
It is known that SA is bounded on L2(Rn); see, for example, [13,20].
Definition 1.3 Let ϕ and A be as in Definition 1.2 and (1.2), respectively.
A function f ∈ L2(Rn) is said to be in
Hϕ,A(Rn) if SAf ∈ Lϕ(Rn); moreover,
define

f

Hϕ,A(Rn) :=
SAf

Lϕ(Rn).
The Musielak-Orlicz-Hardy space Hϕ,A(Rn) is then defined as the completion
of
Hϕ,A(Rn) in the quasi-norm
 ·

Hϕ,A(Rn).
Remark 1.4 The space Hϕ,L(X ) was introduced in [42] (see also [5]) whenX
is a metric measure space, L is a nonnegative self-adjoint operator satisfying the
Davies-Gaffney estimates, and ϕ is a growth function satisfying the additional
assumption that ϕ ∈ RH2/[2−I(ϕ)](X ), namely, ϕ(·, t) satisfies the uniformly
reverse H¨older inequality of order 2/[2 − I(ϕ)] (see Definition 2.2 below).
We now recall the Lusin-area function SP f and the maximal functions Nhf,
NP f, Rhf, and RP f in [26], respectively, as follows. For all β ∈ (0,+∞),
f ∈ L2(Rn), and x ∈ Rn, let
SP f(x) :=

Γ(x)
|t
√
Ae−t
√
Af(y)|2 dydt
tn+1
1/2
,
N β
h f(x) := sup
y∈B(x,βt), t>0
|e−t2Af(y)|, N β
P f(x) := sup
y∈B(x,βt), t>0
|e−t
√
Af(y)|,
Rhf(x) := sup
t>0
|e−t2Af(x)|, RP f(x) := sup
t>0
|e−t
√
Af(x)|,
where Γ(x) for x ∈ Rn is as in (1.8). Denote N 1
h f and N 1
P f simply by Nhf
and NP f, respectively. It is known that all these operators are bounded on
L2(Rn); see the proofs of [25, Theorem 5.2] and [26, Theorem 1.4].
Definition 1.5 A function f ∈ L2(Rn) is said to be in
H ϕ,M
(Rn) if M f ∈
Lϕ(Rn), where M f is one of SP f, Nhf, NP f, Rhf, and RP f as above;
moreover, let

f

Hϕ,M
(Rn) :=
M f

Lϕ(Rn).
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1209
The Musielak-Orlicz-Hardy space Hϕ,M (Rn) is then defined as the completion
of
H ϕ,M (Rn) in
 ·

Hϕ,M
(Rn).
The main result of this article is as follows.
Theorem 1.6 Let ϕ and A be as in Definition 1.2 and (1.2), respectively.
Then the spaces Hϕ,A(Rn), Hϕ,SP (Rn), Hϕ,Rh(Rn), Hϕ,RP (Rn), Hϕ,Nh(Rn), and
Hϕ,NP (Rn) coincide with equivalent norms.
Remark 1.7 To the best of our knowledge, Theorem 1.6 is known only when
ϕ is as in (1.5), and new for other cases. To be precise, when ϕ is as in (1.5)
with p ∈ (0, 1] therein, we see that
Hϕ,A(Rn) = Hp
A(Rn),
which was introduced in [25]. Theorem 1.6 in this case was established in
[25, Theorem 5.2] and [26, Theorem 1.4]. Otherwise, Theorem 1.6 is new, even
when ϕ is as in (1.6) or in (1.7).
We also obtain the following boundedness of the Riesz transforms LkA−1/2,
k ∈ {1, 2, . . . , n}, from the space Hϕ,A(Rn) to Lϕ(Rn).
Theorem 1.8 Let ϕ be as in Definition 1.2, and let I(ϕ) and r(ϕ) be as in
(1.3) and (2.3), respectively. Assume that r(ϕ) ∈ (2/[2 − I(ϕ)],+∞]. Then
the Riesz transforms LkA−1/2, k ∈ {1, 2, . . . , n}, are bounded from Hϕ,A(Rn) to
Lϕ(Rn).
Remark 1.9 (i) To the best of our knowledge, Theorem 1.8 is known only
when ϕ is as in (1.5), and new for other cases. To be precise, when ϕ is as in
(1.5) with p ∈ (0, 1] therein, we see that, in this case, r(ϕ) = +∞ and I(ϕ) = p,
and hence, the assumption r(ϕ) ∈ (2/[2 − I(ϕ)],+∞] of Theorem 1.8 holds
true automatically. In this case, Theorem 1.8 is [26, Theorem 1.5]. Otherwise,
Theorem 1.8 is new, even when ϕ is as in (1.6) or in (1.7).
(ii) We mention that the range of r(ϕ) ∈ (2/[2 − I(ϕ)],+∞] is determined
by the atomic characterization of Hϕ,A(Rn) and the Lq(Rn)-boundedness of the
Riesz transforms LkA−1/2. To be precise, Bui et al. [5, Theorem 5.4] showed
that Hϕ,A(Rn) and HM,q
ϕ,A,at(Rn) coincide for all M ∈ N with M > n
2
q(ϕ)
i(ϕ) , and
q > r(ϕ)I(ϕ)/(r(ϕ) − 1) (this is equivalent to that r(ϕ) > q/(q − I(ϕ))). On
the other hand, Duong et al. [13] showed that the Riesz transforms LkA−1/2,
k ∈ {1, 2, . . . , n}, are bounded on Lq(Rn) for all q ∈ (1, 2]. In the proof of
Theorem 1.8, we need to use the aforementioned two facts at the same time,
which induce that the best choice for r(ϕ) is r(ϕ) ∈ (2/[2 − I(ϕ)],+∞].
We also notice that, when A is the Schr¨odinger operator L := −Δ + V,
it was shown in [5, Theorem 8.5(i)] that, if the Riesz transforms ∇L−1/2 are
bounded on Lq(Rn) for all q ∈ (1, p0) with p0 ∈ (2,+∞), then ∇L−1/2 are
also bounded from the Musielak-Orlicz-Hardy space Hϕ,L(Rn) to Lϕ(Rn) with
r(ϕ) ∈ (p0/(p0 − I(ϕ)),+∞]. Thus, the range of r(ϕ) in Theorem 1.8 coincides
with this range.
1210 Dachun YANG, Dongyong YANG
The organization of the article is as follows.
Section 2 is devoted to some basic lemmas needed in Sections 3 and 4. We
recall some known basic properties of Musielak-Orlicz functions established in
[23,29] and a bounded criterion of linear operators from Hϕ,A(Rn) to Lϕ(Rn)
in [5].
The proof of Theorem 1.6 is presented in Section 3. As in [20], we show
Theorem 1.6 by proving the following inclusion link:
Hϕ,A(Rn) ∩ L2(Rn) ⊂ Hϕ,Nh(Rn) ∩ L2(Rn)
⊂ Hϕ,Rh(Rn) ∩ L2(Rn)
⊂ Hϕ,RP (Rn) ∩ L2(Rn)
⊂ Hϕ,NP (Rn) ∩ L2(Rn)
⊂ Hϕ,SP (Rn) ∩ L2(Rn)
⊂ Hϕ,A(Rn) ∩ L2(Rn).
We point out that important tools used in the proof of Theorem 1.6 include
the properties of ϕ (see Lemmas 2.1 and 2.3 below), the Besicovitch covering
lemma, the Whitney decomposition, the semigroup properties of {e−tA}t>0,
the Caccioppoli inequality associated with A (this was established in [26]; see
also Lemma 3.1 below), and the fact that the kernels of {e−tA}t>0 satisfy the
Gaussian upper bound (see (3.3) below). Besides these key tools, by the atomic
characterization of Hϕ,A(Rn) and the bounded criterion, from Hϕ,A(Rn) to
Lϕ(Rn), of linear operators from [5], we show the inclusion
Hϕ,A(Rn) ∩ L2(Rn) ⊂ Hϕ,Nh(Rn) ∩ L2(Rn).
On the other hand, as in [42], we show the inclusion
Hϕ,SP (Rn) ∩ L2(Rn) ⊂ Hϕ,A(Rn) ∩ L2(Rn)
by establishing a pointwise estimate concerning a truncated Lusin-area
function
Sε,R,τ
P f and the non-tangential maximal function NP f (see (3.1) for
the definition of
Sε,R,τ
P f), and a ‘good-λ inequality’ between these two
operators. Also, it is worth to point out that, in the proof of this inclusion, the
special differential structure of the operator A itself plays an essential role.
In Section 4, by using the properties of ϕ, the semigroup properties of
{e−tA}t>0, the Davies-Gaffney estimates of {tLke−t2A}t>0 from [26], the atomic
characterization of Hϕ,A(Rn), and the bounded criterion, from Hϕ,A(Rn) to
Lϕ(Rn), of linear operators from [5], we show Theorem 1.8. We mention that
this result is different from some known results, for example, [5, Theorem
8.5 (ii)], where the Riesz transform ∇L−1/2, associated with the Schr¨odinger
operator L := −Δ + V, is bounded from the Musielak-Orlicz-Hardy space
Hϕ,L(Rn) to Hϕ(Rn). The reason for this difference is that, because of the
existence of a, it is unclear whether

Rn
LkA
−1/2α(x)dx = 0
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1211
for k ∈ {1, 2, . . . , n} and any molecule α associated with A, is true or not.
We now make some conventions on notation. Throughout this article, we
always use C to denote a positive constant that is independent of the main
parameters involved, but it may differ from line to line. The symbol f g
means that f Cg for some positive number C independent of f and g, and
f ∼ g means f g f. For any complex number z, its real part is denoted by
Re z. For any x ∈ Rn and λ, r > 0, let Q := Q(x, r) be the cube centered at x
with side length r and λQ := Q(x, λr); similarly, B := B(x, r) denotes the ball
centered at x with radius r and λB := B(x, λr). Moreover, for any ball B ⊂ Rn,
let
S0(B) := B, Sj(B) := (2jB) \ (2j−1B), j∈ N := {1, 2, . . . }. (1.9)
Let Z+ := N ∪ {0}. Also, for any set E ⊂ Rn, χE denotes its characteristic
function. Moreover, for all sets E,F ⊂ Rn and z ∈ Rn,
dist(E,F) := inf
x∈E, y∈F
|x − y|, dist(z,E) := inf
x∈E
|x − z|.
Finally, for any growth function ϕ, measurable subset E of Rn, and t ∈ [0,+∞),
let
ϕ(E, t) :=

E
ϕ(x, t)dx.
2 Basic lemmas
In this section, we recall some basic lemmas used in Sections 3 and 4. We begin
with the following lemma on the estimates of ϕ, which was first established by
Ky in [29].
Lemma 2.1 Let ϕ be as in Definition 1.2. Then the following statements hold
true.
(i) There exists a positive constant C such that, for all (x, tj) ∈ Rn×[0,+∞)
with j ∈ N,
ϕ

x,
+∞
j=1
tj

C
+∞
j=1
ϕ(x, tj ).
(ii) For all (x, t) ∈ Rn × [0,+∞), let

ϕ(x, t) :=
t
0
ϕ(x, s)
s
ds.
Then
ϕ is a growth function equivalent to ϕ; moreover,
ϕ is continuous and
strictly increasing.
(iii) For all f ∈ Lϕ(Rn) \ {0},

Rn
ϕ

x,
|f(x)|

f

Lϕ(Rn)


dx = 1.
1212 Dachun YANG, Dongyong YANG
Similar to the class A∞(Rn) of Muckenhoupt weights, it turns out that
uniformly weights in A∞(Rn) also have some very useful properties, including
the following uniformly reverse H¨older condition introduced in [23].
Definition 2.2 A function ϕ: Rn × [0,+∞) → [0,+∞) is said to satisfy
the uniformly reverse H¨older condition for some q ∈ (1,+∞], denoted by ϕ ∈
RHq(Rn), if, when q ∈ (1,+∞),
sup
t∈(0,+∞)
sup
B⊂Rn

1
|B|

B
[ϕ(x, t)]qdx
1/q
1
|B|

B
ϕ(x, t)dx
−1
< +∞,
or
sup
t∈(0,+∞)
sup
B⊂Rn

ess sup
y∈B
ϕ(y, t)

1
|B|

B
ϕ(x, t)dx
−1
< +∞,
where the first suprema are taken over all t ∈ (0,+∞) and the second ones over
all balls B ⊂ Rn.
The following lemma was obtained in [23]; see also [29,31].
Lemma 2.3 The following statements hold true:
(i) A1(Rn) ⊂ Ap(Rn) ⊂ Aq(Rn) for 1 p q < +∞;
(ii) RH∞(Rn) ⊂ RHp(Rn) ⊂ RHq(Rn) for 1 < q p +∞;
(iii) if ϕ ∈ Ap(Rn) with p ∈ (1,+∞), then there exists q ∈ (1, p) such that
ϕ ∈ Aq(Rn);
(iv) if ϕ ∈ RHq(Rn) with q ∈ [1,+∞), then there exists p ∈ (q,+∞) such
that ϕ ∈ RHp(Rn);
(v) A∞(Rn) = ∪
p∈[1,+∞)
Ap(Rn) = ∪
q∈(1,+∞]
RHq(Rn);
(vi) if p ∈ (1,+∞) and ϕ ∈ Ap(Rn), then there exists a positive constant
C such that, for all measurable functions f on Rn and t ∈ [0,+∞),

Rn
[M(f)(x)]pϕ(x, t)dx C

Rn
|f(x)|pϕ(x, t)dx,
where M denotes the Hardy-Littlewood maximal function on Rn, defined by
setting, for all x ∈ Rn,
Mf(x) := sup
B x
1
|B|

B
|f(y)|dy (2.1)
and the supremum is taken over all balls B containing x;
(vii) if ϕ ∈ Ap(Rn) with p ∈ [1,+∞), then there exists a positive constant
C such that, for all balls B1,B2 ⊂ Rn with B1 ⊂ B2 and t ∈ (0,+∞),
ϕ(B2, t)
ϕ(B1, t) C
|B2|
|B1|
p
;
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1213
(viii) if ϕ ∈ RHq(Rn) with q ∈ [1,+∞), then there exists a positive constant
C such that, for all balls B1,B2 ⊂ Rn with B1 ⊂ B2 and t ∈ (0,+∞),
ϕ(B2, t)
ϕ(B1, t) C
|B2|
|B1|
1−1
q .
For ϕ ∈ A∞(Rn), its critical indices of ϕ, q(ϕ) and r(ϕ), are defined,
respectively, as follows:
q(ϕ) := inf{q ∈ [1,+∞): ϕ ∈ Aq(Rn)} (2.2)
r(ϕ) := sup{q ∈ [1,+∞): ϕ ∈ RHq(Rn)}. (2.3)
Recall that, if q(ϕ) ∈ (1,+∞), then, by Lemma 2.3 (iii), we see that ϕ ∈
Aq(ϕ)(Rn), and there exists ϕ ∈ A1(Rn) such that q(ϕ) = 1 (see, for example,
[27]). Similarly, if r(ϕ) ∈ (1,+∞), then, by Lemma 2.3 (iv), we find that
ϕ ∈ RH
r(ϕ)(Rn), and there exists ϕ ∈ RH∞(Rn) such that r(ϕ) = +∞ (see, for
example, [9]).
We now recall the atomic characterization of Hϕ,A(Rn) from [5]. First, we
recall the following notion of (ϕ, q,M)A-atoms.
Definition 2.4 Let ϕ be a growth function as in Definition 1.2, M ∈ N and
q ∈ (1,+∞). A function α ∈ Lq(Rn) is called a (ϕ, q,M)A-atom associated to
A if there exists a function b ∈ D(AM) and a ball B := B(xB, rB) for xB ∈ Rn
and rB ∈ (0,+∞) such that
(A)i α = AMb;
(A)ii supp(Akb) ⊂ B, k ∈ {0, 1, . . . ,M};
(A)iii
(r2B
A)kb

Lq(Rn) r2M
B
|B|1/q
χB
−1
Lϕ(Rn), k ∈ {0, 1, . . .,M}.
A function f ∈ L2(Rn) is said to have an atomic (ϕ, q,M)A-representation,
f =

j λjαj , if, for each j, αj is a (ϕ, q,M)A-atom associated to a ball Bj ⊂ Rn,
the summation converges in L2(Rn) and {λj}j ⊂ C satisfies

j
ϕ(Bj , |λj|
χBj

−1
Lϕ(Rn)) < +∞.
Let

H
M,q
ϕ,A,at(Rn) := {f : f has an atomic (ϕ, q,M)A-representation}
with the quasi-norm
 ·

H M,q
ϕ,A,at(Rn) given by setting, for all f ∈
H M,q
ϕ,A,at(Rn),

f

HM,q
ϕ,A,at(Rn) := inf

Λ({λjαj}j): f =

j
λjαj is an atomic
(ϕ, q,M)A-representation

,
1214 Dachun YANG, Dongyong YANG
where the infimum is taken over all the atomic (ϕ, q,M)A-representation of f
as above and
Λ({λjαj}j) := inf

λ ∈ (0,+∞):

j
ϕ

Bj ,
|λj |
λ
χBj


Lϕ(Rn)


1

.
The atomic Musielak-Orlicz-Hardy space HM,q
ϕ,A,at(Rn) is then defined as the
completion of
HM,q
ϕ,A,at(Rn) with respect to the quasi-norm
 ·

HM,q
ϕ,A,at(Rn).
The following atomic characterization of Hϕ,A(Rn) was established in
[5, Theorem 5.4].
Lemma 2.5 Let ϕ be as in Definition 1.2, A as in (1.2),
q ∈
r(ϕ)
r(ϕ) − 1 I(ϕ),+∞


,
and M ∈ N with M > n
2
q(ϕ)
i(ϕ) , where i(ϕ) and q(ϕ) are as in (1.4) and (2.2),
respectively. Then the spaces Hϕ,A(Rn) and HM,q
ϕ,A,at(Rn) coincide with
equivalent quasi-norms.
From Lemma 2.5 and the boundedness criterion for nonnegative (sub)linear
operators in [5, Lemma 5.7], we deduce the following conclusion. Recall that
a sublinear operator T is said to be nonnegative if, for any function f in its
domain, Tf 0.
Lemma 2.6 Let ϕ be as in Definition 1.2, A as in (1.2),
q ∈
r(ϕ)
r(ϕ) − 1 I(ϕ),+∞


,
and M ∈ N with M > n
2
q(ϕ)
i(ϕ) , where i(ϕ) and q(ϕ) are as in (1.4) and (2.2),
respectively. Assume that T is a linear (resp. nonnegative sublinear) operator
which maps L2(Rn) continuously into L2,∞(Rn). If there exists a positive
constant C such that, for any λ ∈ C and (ϕ, q,M)-atom α associated with
the ball B,
Rn
ϕ(x, |T(λα)(x)|)dx Cϕ

B,
|λ|

χB

Lϕ(Rn)


,
then T extends to a bounded linear (resp. sublinear) operator from Hϕ,A(Rn) to
Lϕ(Rn).
3 Proof of Theorem 1.6
In this section, we show Theorem 1.6. To this end, we first recall some notation.
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1215
For the moment, we denote by Ln+1 the closure in L2(Rn+1) of ∂
∂t and write
∂
∂t as ∂
∂xn+1
. To prove the inclusion
Hϕ,NP (Rn) ∩ L2(Rn) ⊂ Hϕ,SP (Rn) ∩ L2(Rn)
in Theorem 1.6, we need to establish a pointwise estimate for the following
truncated operator
Sε,R,τ
P f, defined by setting, for all f ∈ L2(Rn) and x ∈ Rn,

Sε,R,τ
P f(x) :=

Γε,R
τ (x)
n +1
k=1
|tLke−t
√
Af(y)|2 dydt
tn+1
1/2
, (3.1)
where τ ∈ (0,+∞), ε,R ∈ (0,+∞) with ε < R and
Γε,R
τ (x) := {(y, t) ∈ Rn × (ε,R): |x − y| < τt}.
To this end, we recall a Caccioppoli inequality for weak solutions of the
equation
−∂2u
∂t2 + Au = 0 (3.2)
in an open ball
 B of Rn+1 in [26]. Define
W1,2
a,V (
 B) := {u ∈ L2(
 B): Lku ∈ L2(
 B), k ∈ {1, 2, . . . , n + 1},
√
V u ∈ L2(
 B)},
and let W1,2
a,V,0(
 B) be the subspace of W1,2
a,V (
 B) with trace 0 on ∂
 B. Here and
hereafter, for k ∈ {1, 2, . . . , n} and all u ∈ L2(
 B),
Lku(x, t) := Lk(u(·, t))(x), ∀ (x, t) ∈
 B.
The function u ∈ W1,2
a,V (
 B) is called a weak solution of (3.2) in
 B if
n +1
k=1


B
LkuLkϕdydt +


B
uV ϕdydt = 0, ∀ ϕ ∈ W1,2
a,V,0(
 B).
The following is the Caccioppoli inequality.
Lemma 3.1 [26, Lemma 2.1] Let (x0, t0) ∈ Rn+1, let R ∈ (0,+∞), and let u
be a weak solution of (3.2) in the ball B((x0, t0), 2R) ⊂ Rn+1. Then there exists
a positive constant C, independent of (x0, t0) ∈ Rn+1, R, and u, such that
n +1
k=1

B((x0,t0),R)
|Lku(y, t)|2dydt C
R2

B((x0,t0),2R)
|u(y, t)|2dydt.
Using Lemma 3.1, we now prove the following conclusion.
1216 Dachun YANG, Dongyong YANG
Lemma 3.2 Let α ∈ (0, 1), and let ε,R ∈ (0,+∞) such that ε < R. Then
there exists a positive constant C such that, for all f ∈ L2(Rn) and x ∈ Rn,

Sε,R,α
P f(x) C

1 +log R
ε

1/2
NP f(x).
Proof Let
u(y, t) := e−t
√
Af(y), ∀ (y, t) ∈ Rn × (0,+∞).
Observe that the kernel pt(y, z) of e−tA satisfies the Gaussian upper bound that,
for all t ∈ (0,+∞) and almost all y, z ∈ Rn,
|pt(y, z)| (4πt)−n/2 exp

−
|y − z|2
4t


; (3.3)
see [16]. This fact, together with the well-known subordination formula that,
for all f ∈ L2(Rn),
e−t
√
Af =
1 √
π
+∞
0
e−u
√
u
e− t2
4uAfdu, (3.4)
implies that e−t
√
A is bounded on L2(Rn) for each t ∈ (0,+∞). Let α ∈ (0, 1),
ε,R ∈ (0,+∞) with ε < R, and x ∈ Rn. Moreover, for any (z, τ) ∈ Γε,R
α (x), let

 B(z, τ) := B((z, τ), r)
with r := δτ, where δ ∈ (0, 1) is small enough. By the Besicovitch covering
lemma, we know that there exists a subsequence
{
 Bj}j := {B((zj, τj), rj)}j
of balls covering Γε,R
α (x) with bounded overlap. Observe that, for any (y, t) ∈

 Bj, t ∼ dj , where dj denotes the distance between
 Bj and the bottom boundary
Rn×{0}. Also, we see that, if (y, t) ∈ 2
 Bj , then (y, t) ∈ Γ(x) for δ small enough.
Hence,
|e−t
√
Af(y)| NP f(x).
On the other hand, by the semigroup property, we find that, for fixed t ∈
(0,+∞),
Au(·, t) − ∂2
∂t2 u(·, t) = 0
in L2(Rn), which implies that u is a weak solution of (3.2) for each 2
 Bj . By the
bounded overlap of {
 Bj}j and Lemma 3.1, we conclude that, for all x ∈ Rn,
[
Sε,R,α
P f(x)]2

j


B
j
n +1
k=1
|tLke−t
√
Af(y)|2 dydt
tn+1
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1217


j
r
−(n−1)
j


B
j
n +1
k=1
|Lke−t
√
Af(y)|2dydt


j
r
−(n+1)
j

2
Bj
n +1
k=1
|e−t
√
Af(y)|2dydt
[NP f(x)]2

j
r
−(n+1)
j
|
 Bj |
∼ [NP f(x)]2

j


B
j
dydt
tn+1
∼ [NP f(x)]2

Γε,R
α (x)
dydt
tn+1


1 + log R
ε


[NP f(x)]2,
which implies the desired conclusion. This finishes the proof.
Proof of Theorem 1.6 Step 1 Show
Hϕ,A(Rn) ∩ L2(Rn) ⊂ Hϕ,Nh(Rn) ∩ L2(Rn).
To this end, by Lemma 2.6, it suffices to show that, for any λ ∈ C and
(ϕ, q,M)A-atom α associated with a ball B := B(xB, rB) for some xB ∈ Rn
and rB ∈ (0,+∞),

Rn
ϕ(x,Nh(λα)(x))dx ϕ

B,
|λ|

χB

Lϕ(Rn)


, (3.5)
where M ∈ N with M > n
2
q(ϕ)
i(ϕ) . Indeed, we first observe that Nh is bounded
on Lq(Rn) for all q ∈ (1,+∞) (see, for example, the proof of [26, Theorem
1.4]). Since ϕ ∈ A∞(Rn), from (1.3), (1.4), (2.2), and (iii)–(v) of Lemma
2.3, it follows that there exist q0 ∈ (q(ϕ),+∞), p2 ∈ (0, i(ϕ)], p1 ∈ [I(ϕ), 1],
and q ∈ (I(ϕ)/[r(ϕ)] ,+∞) such that ϕ is of uniformly upper type p1 and of
uniformly lower type p2, ϕ ∈ Aq0(Rn), and ϕ ∈ RH
(q/p1) (Rn). We now write

Rn
ϕ(x,Nh(λα)(x))dx =
+∞
j=0

Sj (B)
ϕ(x,Nh(λα)(x))dx =:
+∞
j=0
Ij ,
where {Sj(B)}j∈Z+ are as in (1.9).
For j ∈ {0, 1, 2, 3, 4}, by the fact that ϕ is of uniformly upper type p1 and
of uniformly lower type p2, the H¨older inequality, the Lq(Rn)-boundedness of
Nh, Definition 2.4 (A)iii, ϕ ∈ Aq0(Rn) and ϕ ∈ RH(q/p1) (Rn), p2 p1, (vi) and
(vii) of Lemma 2.3, we have
1218 Dachun YANG, Dongyong YANG
Ij
2
i=1

χB
pi
Lϕ(Rn)

Sj (B)
ϕ(x, |λ|
χB
−1
Lϕ(Rn))[Nh(α)(x)]pidx

2
i=1

χB
pi
Lϕ(Rn)

Sj (B)
[Nh(α)(x)]qdx
pi/q
×

Sj (B)
[ϕ(x, |λ|
χB
−1
Lϕ(Rn))](q/pi) dx
1/(q/pi)

2
i=1

χB
pi
Lϕ(Rn)

α
pi
Lq (Rn)
|2jB|−pi/qϕ(2jB, |λ|
χB
−1
Lϕ(Rn))

2
i=1

χB
pi
Lϕ(Rn)
|B|pi/q
χB
−pi
Lϕ(Rn)
|2jB|−pi/q2jnq0
×ϕ(B, |λ|
χB
−1
Lϕ(Rn))
2−jnp2( 1
q
−q0
p2
)ϕ(B, |λ|
χB
−1
Lϕ(Rn)). (3.6)
Now, we turn to the case when j 5. From the fact that ϕ is of uniformly
upper type p1 and of uniformly lower type p2, we deduce that
Ij
2
i=1

χB
pi
Lϕ(Rn)

Sj (B)
ϕ(x, |λ|
χB
−1
Lϕ(Rn))[Nh(α)(x)]pidx.
By the H¨older inequality, the facts that ϕ ∈ RH
(q/p1) (Rn) ∩ Aq0(Rn), and
(vii) and (viii) of Lemma 2.3, we further conclude that, for all j ∈ N∩ [5,+∞),
Ij
2
i=1

χB
pi
Lϕ(Rn)

Nh(α)χSj (B)

pi
Lq(Rn)

ϕ

·,
|λ|

χB

Lϕ(Rn)


χSj (B)

L(q/pi)

(Rn)

2
i=1

χB
pi
Lϕ(Rn)

Nh(α)χSj (B)

pi
Lq(Rn)
|2jB|−pi/qϕ

2jB,
|λ|

χB

Lϕ(Rn)



2
i=1

χB
pi
Lϕ(Rn)

Nh(α)χSj (B)

pi
Lq(Rn)
|2jB|−pi/q2jq0nϕ

B,
|λ|

χB

Lϕ(Rn)


.
Let a ∈ (0, 1) such that ap2(2M + n) > n. We see that, for all j ∈ N ∩ [5,+∞)
and x ∈ Sj(B),
Nh(α)(x) sup
y∈B(x,t), t 2aj−2rB
|e−t2Aα(y)|+ sup
y∈B(x,t), t>2aj−2rB
· · · =: Ij,1 + Ij,2.
On one hand, by (3.3), the definition of α, and the H¨older inequality, we
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1219
know that, for all j ∈ N ∩ [5,+∞),
Ij,1 sup
y∈B(x,t), t 2aj−2rB
t
−n

B
e−|y−z|2/(4t2)|α(z)|dz
sup
t 2aj−2rB
t
−n
t
2jrB

N+n

α

L1(Rn)
(2aj−2rB)N(2jrB)−(N+n)|B|1/q |B|1/q
χB
−1
Lϕ(Rn)
2−j[n+(1−a)N]
χB
−1
Lϕ(Rn),
where N satisfies p2[n+(1−a)N] > q0n. On the other hand, recall that, for all
k ∈ N, there exist positive constants C(k) and
 C(k), depending on k, such that,
for almost all y, z ∈ Rn,

∂k
∂tk pt(y, z)


C(k)
tk+n
2
exp

−
|y − z|2

 C(k)t


;
see [33, Theorem 6.16]. From this, the semigroup property, the definition of α,
and the H¨older inequality, we deduce that, for all j ∈ N ∩ [5,+∞),
Ij,2= sup
y∈B(x,t), t>2aj−2rB
|AMe−t2Ab(y)|
= sup
y∈B(x,t), t>2aj−2rB

∂
∂s

M
s=t2e−sAb(y)

sup
y∈B(x,t), t>2aj−2rB
t
−(2M+n)

Rn
exp

−
|y − z|2

 CMt2


|b(z)|dz
(2ajrB)−(2M+n)
b

L1(Rn)
(2ajrB)−(2M+n)r2M
B
|B|1/q |B|1/q
χB
−1
Lϕ(Rn)
2−aj(2M+n)
χB
−1
Lϕ(Rn).
Combining these two inequalities, we find that, for all j ∈ N ∩ [5,+∞) and
x ∈ Rn,
Nh(α)(x) {2−j[n+(1−a)N] + 2−aj(2M+n)}
χB
−1
Lϕ(Rn).
By this, we conclude that, for all j ∈ N ∩ [5,+∞),
Ij
2
i=1

χB
pi
Lϕ(Rn)

χB
−pi
Lϕ(Rn)
|2jB|pi/q |2jB|−pi/q2jq0nϕ

B,
|λ|

χB

Lϕ(Rn)


× {2−pij[n+(1−a)N] + 2−ajpi(2M+n)}

2
i=1
{2j(q0n−pi[n+(1−a)N]) + 2j[q0n−api(2M+n)]}ϕ

B,
|λ|

χB

Lϕ(Rn)


,
1220 Dachun YANG, Dongyong YANG
which, together with the facts that
p2 p1, p2[n + (1 − a)N] > q0n, ap2(2M + n) > q0n,
implies that
+∞
j=5
Ij ϕ

B,
|λ|

χB

Lϕ(Rn)


.
From this and (3.6), we deduce (3.5), which further implies the desired inclusion
relation that
Hϕ,A(Rn) ∩ L2(Rn) ⊂ Hϕ,Nh(Rn) ∩ L2(Rn).
Step 2 Prove
Hϕ,Nh(Rn) ∩ L2(Rn) ⊂ Hϕ,Rh(Rn) ∩ L2(Rn).
Observe that, for all f ∈ L2(Rn),
Rhf Nhf.
By this fact, we conclude that, for all f ∈ Hϕ,Nh(Rn) ∩ L2(Rn),
f ∈ Hϕ,Rh(Rn),
f

Hϕ,Rh(Rn)
f

Hϕ,Nh(Rn),
which further implies the desired conclusion.
Step 3 Show
Hϕ,Rh(Rn) ∩ L2(Rn) ⊂ Hϕ,RP (Rn) ∩ L2(Rn).
From the subordination formula (3.4), it follows that, for all f ∈ L2(Rn)
and x ∈ Rn,
RP f(x) sup
t>0
+∞
0
e−u
√
u
|e−t2A/(4u)f(x)|du Rhf(x)
+∞
0
e−u
√
u
du Rhf(x),
which further implies that, for all f ∈ Hϕ,Rh(Rn) ∩ L2(Rn),
f ∈ Hϕ,RP (Rn),
f

Hϕ,RP (Rn)
f

Hϕ,Rh
(Rn).
From this, we deduce the desired inclusion relation.
Step 4 Prove
Hϕ,RP (Rn) ∩ L2(Rn) ⊂ Hϕ,NP (Rn) ∩ L2(Rn).
By [26, (2.12)], we know that, for any q ∈ (0, 1) and x ∈ Rn,
N 1/4
P f(x) [M ([RP f]q) (x)]1/q , (3.7)
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1221
where Mf denotes the Hardy-Littlewood maximal function of f as in (2.1).
Using this fact, Lemma 2.1 (ii), Lemma 2.3 (vi), Definition 1.2, and arguing as
the proof of [42, (7.17)], we conclude that, for all f ∈ L2(Rn),
N 1/4
P f ∈ Lϕ(Rn), NP f ∈ Lϕ(Rn),
NP f

Lϕ(Rn)
∼
N 1/4
P f

Lϕ(Rn).
This, together with (3.7), Definition 1.2, Lemma 2.1 (ii), Lemma 2.3 (vi), and
an argument similar to [42, (7.16)], further implies that, for all f ∈ Hϕ,RP (Rn)∩
L2(Rn),

NP f

Lϕ(Rn)
∼
N 1/4
P f

Lϕ(Rn)
RP f

Lϕ(Rn).
By this, we obtain the desired inclusion relation.
Step 5 Show
Hϕ,NP (Rn) ∩ L2(Rn) ⊂ Hϕ,SP (Rn) ∩ L2(Rn).
By an argument similar to that used in the proof of [42, Proposition 7.6], we
see that, to prove the desired inclusion relation, it suffices to show that there
exist positive constants C and ε0 ∈ (0, 1) such that, for all γ ∈ (0, 1], λ, ε,R ∈
(0,+∞) with ε < R, f ∈ Hϕ,NP (Rn) ∩ L2(Rn), and t ∈ (0,+∞),

{x∈Rn :
Sε,R,1/20
P f(x)>2λ,NP f(x) γλ}
ϕ(x, t)dx
Cγε0

{x∈Rn :
Sε,R,1/2
P f(x)>λ}
ϕ(x, t)dx. (3.8)
To this end, fix 0 < ε < R < +∞, γ ∈ (0, 1], and λ ∈ (0,+∞). Let
f ∈ Hϕ,NP (Rn) ∩ L2(Rn), O:= {x ∈ Rn :
Sε,R,1/2
P f(x) > λ}.
Then O is an open subset of Rn. Let O = ∪kQk be a Whitney decomposition
of O such that {Qk}k has disjoint interiors,
2Qk ⊂ O, 4Qk ∩ (Rn \ O) = ∅.
By O = ∪kQk and {Qk}k is disjoint mutually, to show (3.8), it suffices to prove
that, for each k,

{x∈Qk :
Sε,R,1/20
P f(x)>2λ,NP f(x) γλ}
ϕ(x, t)dx γε0

Qk
ϕ(x, t)dx. (3.9)
Denote by k the side length of Qk. Observe that, if x ∈ Qk, then

Smax{10
k,ε},R,1/20
P f(x) λ; (3.10)
see [42, (7.8)]. It follows, from (3.10), that, if ε 10 k, then
{x ∈ Qk :
Sε,R,1/20
P f(x) > 2λ, NP f(x) γλ} = ∅,
1222 Dachun YANG, Dongyong YANG
and hence, (3.9) holds true. When ε < 10 k, by (3.10) and the fact that

Sε,R,1/20
P f
Sε,10
k,1/20
P f +
S10
k,R,1/20
P f,
it remains to show that there exists ε0 ∈ (0, 1) such that, for all t ∈ (0,+∞),

{x∈Qk :
S
ε,10 k,1/20
P f(x)>λ,NP f(x) γλ}
ϕ(x, t)dx γε0

Qk
ϕ(x, t)dx. (3.11)
Let
F := {x ∈ Rn : NP f(x) γλ} .
Then we claim that (3.11) can be deduced from the following inequality that

Qk∩F
[
Sε,10
k,1/20
P f(x)]2dx (γλ)2|Qk|. (3.12)
Indeed, if (3.12) holds true, we first deduce, from the Tchebychev inequality,
that
|{x ∈ Qk ∩ F :
Sε,10
k,1/20
P f(x) > λ}| γ2|Qk|. (3.13)
On the other hand, by the fact that ϕ ∈ A∞(Rn) and Lemma 2.3 (v), we
conclude that there exists r ∈ (1,+∞) such that ϕ ∈ RHr(Rn), which, together
with (3.13) and Lemma 2.3 (vii), implies that, for all t ∈ (0,+∞),
1
ϕ(Qk, t)

{x∈Qk∩F :
S
ε,10 k,1/20
P f(x)>λ}
ϕ(x, t)dx

|{x ∈ Qk ∩ F :
Sε,10
k,1/20
P f(x) > λ}|
|Qk|
(r−1)/r
γ2(r−1)/r.
Let
ε0 :=
2(r − 1)
r
.
Then we have

{x∈Qk∩F :
S
ε,10 k,1/20
P f(x)>λ}
ϕ(x, t)dx γε0ϕ(Qk, t),
which implies (3.11). Thus, the claim holds true.
Now, we show (3.12). If ε 5 k, then, by the definitions of
Sε,10
k,1/20
P f
and F, together with Lemma 3.2, we conclude that

Qk∩F
[
Sε,10
k,1/20
P f(x)]2dx

Qk∩F
[NP f(x)]2dx (γλ)2|Qk|.
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1223
Assume that ε < 5 k. Let
Gk :=

(y, t) ∈ Rn × (ε, 10 k): Ψk(y) := dist(y,Qk ∩ F) <
t
20

.
By the definition of
Sε,10
k,1/20
P f, we see that

Qk∩F
[
Sε,10
k,1/20
P f(x)]2dx

Gk
n +1
k=1
|tLke−t
√
Af(y)|2 dydt
t
∼

Gk
n +1
k=1
t|Lke−t
√
Af(y)|2dydt.
Let
Ek :=

y ∈ Rn: there exists t ∈ (ε, 10 k) such that Ψk(y) <
t
20

.
Then we claim that Ek ⊂ 2Qk. Indeed, if y ∈ Ek, then there exists t ∈ (ε, 10 k)
such that (y, t) ∈ Gk. Furthermore, we see that there exists x ∈ Qk ∩ F such
that |x − y| < t/20. By t < 10 k, we know that |x − y| < k/2, which implies
that Ek ⊂ 2Qk, and hence, the claim holds true.
Let

G
k :=

(y, t) ∈ Rn ×
ε
5, 40 k


: Ψk(y) <
t
10

.
Then, for any (y, t) ∈
Gk,
|u(y, t)| := |e−t
√
Af(y)| γλ.
Indeed, for any (y, t) ∈
Gk, there exists x ∈ Qk ∩ F such that |x − y| < t and
t ∈ (ε/5, 40 k). This implies that (y, t) ∈ Γ(x), where Γ(x) is as in (1.8). Thus,
from the definitions of F and NP (f), it follows that, for all (y, t) ∈ Γ(x),
|e−t
√
Af(y)| NP (f)(x) γλ.
Let
Gk,1 :=

(y, t) ∈ Rn ×
ε
2, 20 k


: Ψk(y) <
t
10

.
Then, by [41, Lemma 3.6], we see that there exists a function
ξ ∈ C
∞

Rn ×
ε
2, 20 k



∩ C

Rn ×
ε
2, 20 k


such that supp(ξ) ⊂ Gk,1, 0 ξ 1, ξ ≡ 1 on Gk, and |
∇ ξ(y, t)| t−1 for any
1224 Dachun YANG, Dongyong YANG
(y, t) ∈ Rn × (ε/2, 20 k). By 0 ξ 1 and ξ ≡ 1 on Gk, we conclude that

Gk
t
n +1
k=1
|Lke−t
√
Af(y)|2dydt


Rn×(0,+∞)
t
n +1
k=1
|Lke−t
√
Af(y)|2ξ(y, t)dydt
=
n +1
k=1

Gk,1
tLku(y, t)Lku(y, t) ξ(y, t)dydt
∼ Re
n +1
k=1

Gk,1
tLku(y, t)

Lk(uξ)(y, t) − u(y, t) ∂ξ(y, t)
∂xk

dydt

.
From integration by parts, the fact that, for fixed t,
Au(·, t) − ∂2
∂t2 u(·, t) = 0
in L2(Rn) and the definition of A, it follows that
Re

Gk,1
tLn+1u(y, t)Ln+1(uξ)(y, t) dydt

= − Re

Gk,1

t
∂2
∂t2 u(y, t)u(y, t) ξ(y, t) + ∂u(y, t)
∂t
u(y, t) ξ(y, t)

dydt

= − Re

Gk,1

tAu(y, t)u(y, t) ξ(y, t) + ∂u(y, t)
∂t
u(y, t) ξ(y, t)

dydt

= − Re
n
k=1

Gk,1
tLku(y, t)Lk(uξ)(y, t) dydt

−

Gk,1

t|u(y, t)|2V (y)ξ(y, t) +
1
2
∂|u(y, t)|2
∂t
ξ(y, t)

dydt.
Since ξ, V 0, from this fact, integration by parts, the choice of ξ, and the
Cauchy-Schwarz inequality, we further deduce that

Gk
t
n +1
k=1
|Lke−t
√
Af(y)|2dydt


Gk,1

− Re
n +1
k=1
tLku(y, t)u(y, t) ∂ξ(y, t)
∂xk

+
1
2
|u(y, t)|2 ∂ξ(y, t)
∂t

dydt


Gk,1\Gk
n +1
k=1
|Lku(y, t)u(y, t)| + t
−1|u(y, t)|2

dydt
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1225


Gk,1\Gk
n +1
k=1
t|Lku(y, t)|2dydt +

Gk,1\Gk
t
−1|u(y, t)|2dydt
=: J1 + J2.
We first estimate J2. By the fact that Gk,1 ⊂
Gk, we conclude that
|u(y, t)| γλ, ∀ (y, t) ∈ Gk,1 \ Gk.
Moreover, we write
Gk,1 \ Gk ⊂

(y, t) ∈ Rn ×
ε
2, 20 k


: t
20 Ψk(y) <
t
10

∪

(y, t) ∈ Rn ×
ε
2, 20 k


: Ψk(y) <
t
10,
ε
2 < t ε

∪

(y, t) ∈ Rn ×
ε
2, 20 k


: Ψk(y) <
t
10, 10 k t < 20 k

.
From these facts, we deduce that
J2

Gk,1\Gk
(γλ)2 dydt
t
(γλ)2

Hk,1
ε
ε/2
dt
t
+
20
k
10
k
dt
t
+
20Ψk(y)
10Ψk(y)
dt
t

dy
(γλ)2|Hk,1|,
where
Hk,1 :=

y ∈ Rn: there exists t ∈
ε
2, 20 k


such that (y, t) ∈ Gk,1

.
Moreover, we claim that Hk,1 ⊂ 5Qk. Indeed, for any y ∈ Hk,1, there exists
t ∈ (ε/2, 20 k ) such that (y, t) ∈ Gk,1. From this and the definition of Gk,1, it
follows that there exists x ∈ Qk ∩F such that |x−y| < t/10 and t ∈ (ε/2, 20 k).
This implies that |x−y| < 2 k, and hence, y ∈ 5Qk. Thus, the claim holds true,
from which it follows that
|J2| (γλ)2|Qk|.
To estimate J1, for any (y, t) ∈ (Gk,1 \ Gk) and δ ∈ (0, 1), let
E(y,t) := B((y, t), r)
with r := δt, and
 E(y,t) := B((y, t), 2r). Take δ small enough such that, for any
(y, t) ∈ (Gk,1 \ Gk),

 E(y,t)
⊂

(z, s) ∈ Rn ×
ε
5, 30 k


: s
40 < Ψk(z) <
s
10

∪

(z, s) ∈ Rn ×
ε
5, 30 k


: Ψk(z) <
s
2,
ε
5 s <
2
ε

∪

(z, s) ∈ Rn ×
ε
5, 30 k


: Ψk(z) <
s
5, 5 k s 30 k

=: Gk,2.
1226 Dachun YANG, Dongyong YANG
By the Besicovitch covering lemma, we know that there exists a subsequence
{E(yj ,tj )
}j of balls such that
(Gk,1 \ Gk) ⊂

j
E(yj ,tj)
and {E(yj ,tj )
}j has bounded overlap. Observe that, for any j and (y, t) ∈ Ej ,
t ∼ tj ∼ rj .
From this, the fact that Gk,2 ⊂
Gk, and Lemma 3.1, we deduce that
J1

j

Ej
n +1
k=1
t|Lku(y, t)|2dydt


j
1
r2
j


E
j
t|u(y, t)|2dydt


j
1
rj
(γλ)2


E
j
dydt


j
1
rj
(γλ)2|Ej |


j
(γλ)2

Ej
dydt
t
(γλ)2

Gk,2
dydt
t
(γλ)2

Hk,2
2ε
ε/5
dt
t
+
30
k
5
k
dt
t
+
40Ψk(y)
10Ψk(y)
dt
t

dy
(γλ)2|Hk,2|,
where
Hk,2 :=

y ∈ Rn: there exists t ∈
ε
5, 30 k


such that (y, t) ∈ Gk,2

.
By an argument similar to that used in the estimate of Hk,1, we find that
|Hk,2| |Qk|,
which implies that
J1 (γλ)2|Qk|.
This shows (3.12) when ε < 5 k.
Step 6 Prove
Hϕ,SP (Rn) ∩ L2(Rn) ⊂ Hϕ,A(Rn) ∩ L2(Rn).
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1227
To this end, for all f ∈ L2(Rn+1
+ ) with compact support and x ∈ Rn, define
ΠΨ,A(f)(x) := C(M)
+∞
0
(t2A)M+1e−t2A(f(·, t))(x)
dt
t
,
where C(M) is a positive constant, depending on M, such that
C(M)
+∞
0
t2(M+2)e−2t2 dt
t
= 1.
It was shown that ΠΨ,A is bounded from T2
2 (Rn+1
+ ) to L2(Rn) in [25, Proposition
4.2 (i)] and from Tϕ(Rn+1
+ ) to Hϕ,A(Rn) in [5, Proposition 4.5 (ii)], where, for
any measurable function g on Rn+1
+ and x ∈ Rn,
A (g)(x) :=

Γ(x)
|g(y, t)|2 dydt
tn+1
1/2
,
Tϕ(Rn+1
+ ) := {g :
g

Tϕ(Rn+1
+ ) :=
A (g)

Lϕ(Rn) < +∞},
T2
2 (Rn+1
+ ) := {g :
g

Tp
2 (Rn+1
+ ) :=
A (g)

Lp(Rn) < +∞}.
Let f ∈ Hϕ,SP (Rn) ∩ L2(Rn). Then we have
SP f ∈ Lϕ(Rn) ∩ L2(Rn),
which implies that
t
√
Ae−t
√
A f ∈ Tϕ(Rn+1
+ ) ∩ T2
2 (Rn+1
+ ).
On one hand, by the H∞-functional calculus, we see that
f = CΠΨ,A(t
√
Ae−t
√
A f)
in L2(Rn). This, together with the boundedness of ΠΨ,A from Tϕ(Rn+1
+ ) to
Hϕ,A(Rn), further implies that f ∈ Hϕ,A(Rn), which, combined with the
inclusion relations in Steps 1–5, completes the proof of Theorem 1.6.
4 Proof of Theorem 1.8
This section is devoted to the proof of Theorem 1.8. To this end, we first recall
that the semigroup {tLke−t2A}t>0 for k ∈ {1, 2, . . . , n} satisfies the following
Davies-Gaffney estimates, which was established in [26, Lemma 3.1].
Lemma 4.1 There exist positive constants C and
 C such that, for all t ∈
(0,+∞), disjoint closed sets E,F ⊂ Rn, and f ∈ L2(Rn) with supp(f) ⊂ E,
n
k=1

χF tLke−t2Af

L2(Rn)
 C exp

− [dist(E,F)]2
Ct2



fχE

L2(Rn).
1228 Dachun YANG, Dongyong YANG
Proof of Theorem 1.8 Since r(ϕ) > 2/[2 − I(ϕ)], we see that [r(ϕ)] I(ϕ) < 2
and, by Lemma 2.5,
Hϕ,A(Rn) = HM,2
ϕ,A,at(Rn).
By Lemma 2.6, to prove Theorem 1.8, it suffices to show that, for any k ∈
{1, 2, . . . , n}, λ ∈ C, and (ϕ, 2,M)A-atom α associated with a ball
B := B(xB, rB)
for some xB ∈ Rn and rB ∈ (0,+∞),

Rn
ϕ(x, |LkA
−1/2(λα)(x)|)dx ϕ

B,
|λ|

χB

Lϕ(Rn)


. (4.1)
We first write

Rn
ϕ(x, |LkA
−1/2(λα)(x)|)dx =
+∞
j=0

Sj(B)
ϕ(x, |LkA
−1/2(λα)(x)|)dx
=:
+∞
j=0
Lj .
Recall that LkA−1/2 is bounded on Lp(Rn) for any p ∈ (1, 2]; see [13]. Using
this fact and an argument similar to (3.6), we find that, for j ∈ {0, 1, 2, 3, 4},
Lj ϕ

B,
|λ|

χB

Lϕ(Rn)


.
Now, we turn to the case when j 5. As in the proof of Theorem 1.6, since
ϕ ∈ A∞(Rn), from (1.3), (1.4), (2.2), and Lemma 2.3 (iii)–(v), it follows that
there exist q0 ∈ (q(ϕ),+∞), p2 ∈ (0, i(ϕ)], and p1 ∈ [I(ϕ), 1] such that ϕ is
of uniformly upper type p1 and of uniformly lower type p2, ϕ ∈ Aq0(Rn), and
ϕ ∈ RH(2/p1) (Rn). Recall that
A
−1/2 =
1 √
π
+∞
0
e−tA √dt
t
;
see [16,33]. From this, together with the fact that ϕ is of uniformly upper type
p1 and of uniformly lower type p2, the H¨older inequality, and the fact that
ϕ ∈ RH(2/p1) (Rn) and ϕ ∈ Aq0(Rn), we deduce that, for all j ∈ N ∩ [5,+∞),
Lj
2
i=1

χB
pi
Lϕ(Rn)

Sj (B)
ϕ(x, |λ|
χB
−1
Lϕ(Rn))|LkA
−1/2(α)(x)|pidx

2
i=1

χB
pi
Lϕ(Rn)

Sj(B)
|LkA
−1/2(α)(x)|2dx
pi/2
Musielak-Orlicz-Hardy spaces associated with magnetic Schr¨odinger operators 1229
×

ϕ

·,
|λ|

χB

Lϕ(Rn)


L(2/pi)

(Sj (B))

2
i=1

χB
pi
Lϕ(Rn)

Sj(B)

Lk
+∞
0
e−t2A(α)(x)dt

2
dx
pi/2
× |2jB|−pi/2

Sj(B)
ϕ(x, |λ|
χB
−1
Lϕ(Rn))dx

2
i=1

χB
pi
Lϕ(Rn)
+∞
0

Sj(B)
|Lke−t2A(α)(x)|2dx
1/2
dt
pi
× |2jB|−pi/22jq0nϕ

B,
|λ|

χB

Lϕ(Rn)


=
2
i=1

χB
pi
Lϕ(Rn)
rB
0

Sj(B)
|Lke−t2A(α)(x)|2dx
1/2
dt
pi
× |2jB|−pi/22jq0nϕ

B,
|λ|

χB

Lϕ(Rn)


+
2
i=1

χB
pi
Lϕ(Rn)
+∞
rB

Sj(B)
|Lke−t2A(α)(x)|2dx
1/2
dt
pi
× |2jB|−pi/22jq0nϕ

B,
|λ|

χB

Lϕ(Rn)


=:
2
i=1
Hj,i.
For Hj,1, by Lemma 4.1, the definition of α, and p2 p1, we see that
Hj,1
2
i=1

χB
pi
Lϕ(Rn)
rB
0
exp

− (2jrB)2
ct2



α

L2(Rn)
dt
t
pi
× |2jB|−pi/22jq0nϕ

B,
|λ|

χB

Lϕ(Rn)



2
i=1

χB
pi
Lϕ(Rn)
|B|pi/2
χB
−pi
Lϕ(Rn)
rB
0

− t
2jrB

2M dt
t
pi
× |2jB|−pi/22jq0nϕ

B,
|λ|

χB

Lϕ(Rn)


2−j(p2(2M+n
2 )−q0n)ϕ

B,
|λ|

χB

Lϕ(Rn)


,
where M is large enough such that p2(2M + n
2 ) > q0n.
To estimate Hj,2, we write
tLk(t2A)Me−t2A = tLke−t2A/2(t2A)Me−t2A/2.
1230 Dachun YANG, Dongyong YANG
Recall that the semigroup {(t2A)Me−t2A}t>0 satisfies the Davies-Gaffney
estimates (see [20, Proposition 3.1]). Then, using [21, Lemma 2.3] and Lemma
4.1, we find that {tLk(t2A)Me−t2A}t>0 also satisfies the Davies-Gaffney
estimates. By this fact, the H¨older inequality, and the Minkowski inequality,
we obtain
Hj,2 ∼
2
i=1

χB
pi
Lϕ(Rn)
+∞
rB

Sj (B)
|Lke−t2AAMb(x)|2dx
1/2
dt
pi
× |2jB|−pi/22jq0nϕ

B,
|λ|

χB

Lϕ(Rn)



2
i=1

χB
pi
Lϕ(Rn)
+∞
rB
exp

− (2jrB)2
ct2



b

L2(Rn)
dt
t2M+1
pi
× |2jB|−pi/22jq0nϕ

B,
|λ|

χB

Lϕ(Rn)



2
i=1

χB
pi
Lϕ(Rn)
+∞
rB
t
2jrB

2M−1 dt
t2M+1
pi
× (r2M
B
|B|1/2
χB
−1
Lϕ(Rn))pi |2jB|−pi/22jq0nϕ

B,
|λ|

χB

Lϕ(Rn)


2−j[p2(2M−1+n
2 )−q0n]ϕ

B,
|λ|

χB

Lϕ(Rn)


,
where M is large enough such that p2(2M − 1 + n
2 ) > q0n. Thus, if we choose
M satisfying p2(2M −1+ n
2 ) > q0n, then we have (4.1). This finishes the proof
of Theorem 1.8.
Acknowledgements Dachun Yang was supported by the National Natural Science
Foundation of China (Grant Nos. 11171027, 11361020), the Specialized Research Fund for
the Doctoral Program of Higher Education of China (Grant No. 20120003110003), and the
Fundamental Research Funds for Central Universities of China (Grant Nos. 2012LYB26,
2012CXQT09, 2013YB60, 2014kJJCA10). Dongyong Yang was supported by the National
Natural Science Foundation of China (Grant No. 11101339) and Fundamental Research Funds
for Central Universities of China (Grant No. 2013121004).
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