# The Boundedness of Some Integral Operators on Weighted Hardy Spaces Associated with Schrödinger Operators

Authors: Hua Wang
Journal: Journal of Function Spaces
 (2015)

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## Abstract

LetL=-Δ+V be a Schrödinger operator acting on L2(Rn), n≥1, where V≢0 is a nonnegative locally integrable function on Rn. In this paper, we will first define molecules for weighted Hardy spaces HLp(w)(0<p≤1) associated with L and establish their molecular characterizations. Then, by using the atomic decomposition and molecular characterization of HLp(w), we will show that the imaginary power Liγ is bounded on HLp(w) for n/(n+1)<p≤1, and the fractional integral operator L-α/2 is bounded from HLp(w) to HLq(wq/p), where 0<α<min{n/2,1}, n/(n+1)<p≤n/(n+α), and 1/q=1/p-α/n.


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## Body

## 1. Introduction

Letn≥1 and V be a nonnegative locally integrable function defined on Rn, not identically zero. We define the form Q by(1)Qu,v≔∫Rn∇u·∇vdx+∫RnVuvdxwith domain D(Q)=V×V, where (2)
V
≔
u
∈
L
2
R
n
:
∂
u
∂
x
k
∈
L
2
R
n
for
k
=
1,2
,
…
,
n
,
V
u
∈
L
2
R
n
.It is well known that this symmetric form is closed. Note also that it was shown by Simon [1] that this form coincides with the minimal closure of the form given by the same expression but defined on C0∞(Rn) (the space of C∞ functions with compact supports). In other words, C0∞(Rn) is a core of the form Q.Let us denote byL the self-adjoint operator associated with Q. The domain of L is given by (3)
D
L
≔
u
∈
V
:
∃
v
∈
L
2
R
n
such  that
Q
u
,
φ
=
∫
R
n
v
·
φ
d
x
,
∀
φ
∈
V
.Formally, we write L=-Δ+V as a Schrödinger operator with potential V. Let {e-tL}t>0 be the semigroup of linear operators generated by -L and let pt(x,y) be their kernels. Since V is a locally integrable nonnegative function on Rn, then the Feynman-Kac formula implies that the semigroup kernels pt(x,y) associated with e-tL satisfy the estimates:(4)0≤ptx,y≤14πtn/2exp⁡-x-y24tfor all t>0 and all x,y∈Rn.Since the Schrödinger operatorL is a self-adjoint and positive definite operator acting on L2(Rn), then L admits the following spectral decomposition: (5)L=∫0∞λdELλ,where EL(λ) denotes its spectral resolution. For any γ∈R, we can define the imaginary power Liγ associated with L by the formula(6)Liγ≔∫0∞λiγdELλ.By the functional calculus for L, we can also define the operator Liγ as follows:(7)Liγfx≔1Γ-iγ∫0∞t-iγ-1e-tLfxdt.By the spectral theory, we know that LiγL2→L2=1 for all γ∈R. Moreover, it was proved by Shen [2] that Liγ is a Calderón-Zygmund operator provided that V∈RHn/2 (reverse Hölder class). We refer the reader to [2–4] for related results concerning the imaginary powers of self-adjoint operators.For any0<α<n, the fractional integrals L-α/2 associated with L are defined by(8)L-α/2fx≔1Γα/2∫0∞tα/2-1e-tLfxdt.Since the kernel pt(x,y) of {e-tL}t>0 satisfies the Gaussian upper bounds (4), then it is easy to check that L-α/2fx≤C·Iα(f)(x) for all x∈Rn, where Iα denotes the classical fractional integral operator (see [5]):(9)Iαfx≔Γn-α/22απn/2Γα/2∫Rnfyx-yn-αdy.Hence, by using the Lp-Lq boundedness of Iα (see [5]), we have(10)L-α/2fLq≤CIαfLq≤CfLp,where 1<p<n/α and 1/q=1/p-α/n. For more information about the fractional integrals L-α/2 associated with some classes of operators, one can see [6–9].In [10], Song and Yan introduced the weighted Hardy space HL1(w) associated with L in terms of the square function and established its atomic decomposition theory. Furthermore, they also showed that the Riesz transform ∇L-1/2 associated with L is bounded on Lp(w) for 1<p<2 and bounded from HL1(w) to the classical weighted Hardy space H1(w) (see [11, 12] for H1(w)).Recently, in [13], we defined the weighted Hardy spaces HLp(w) associated with L for 0<p<1 and gave their atomic decompositions. We also obtained that ∇L-1/2 is bounded from HLp(w) to the classical weighted Hardy space Hp(w) (see also [11, 12] for Hp(w)) when n/(n+1)<p<1. In this paper, we first define molecules for the weighted Hardy spaces HLp(w) associated with L and then establish their molecular characterizations. As applications of the molecular characterization combining with the atomic decomposition of HLp(w), we will obtain some estimates of Liγ and L-α/2 on HLp(w) for n/(n+1)<p≤1. Our main results are stated as follows.Theorem 1.
LetL=-Δ+V, n/(n+1)<p≤1, and w∈A1∩RH2/p′. Then, for any γ∈R, the imaginary power Liγ is bounded from HLp(w) to the weighted Lebesgue space Lp(w).Theorem 2.
LetL=-Δ+V, n/(n+1)<p≤1, and w∈A1∩RH2/p′. Then, for any γ∈R, the imaginary power Liγ is bounded on HLp(w).Theorem 3.
Suppose thatL=-Δ+V. Let 0<α<n/2, n/(n+1)<p≤1, 1/q=1/p-α/n, and w∈A1∩RH2/p′. Then, the fractional integral operator L-α/2 is bounded from HLp(w) to Lq(wq/p).Theorem 4.
Suppose thatL=-Δ+V. Let 0<α<min⁡{n/2,1}, n/(n+1)<p≤n/(n+α), 1/q=1/p-α/n, and w∈A1∩RH2/p′. Then, the fractional integral operator L-α/2 is bounded from HLp(w) to HLq(wq/p).


## 2. Notations and Preliminaries

Let us first recall some standard definitions and notations. The classicalAp weight theory was first introduced by Muckenhoupt in the study of weighted Lp boundedness of Hardy-Littlewood maximal functions in [14]. A weight w is a nonnegative, locally integrable function defined on Rn, and B=B(x0,rB) denotes the ball with the center x0 and radius rB. We say that w∈A1, if (11)1B∫Bwxdx≤C·essinfx∈Bwx,for  every  ballB⊆Rn,where C is a positive constant which is independent of B. A weight function w is said to belong to the reverse Hölder class RHs, if there exist two constants s>1 and C>0 such that the following reverse Hölder inequality holds: (12)1B∫Bwxsdx1/s≤C1B∫Bwxdx,for  every  ballB⊆Rn.Given a ballB and λ>0, λB 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 denote the Lebesgue measure of E by E and the weighted measure of E by w(E), where w(E)=∫Ew(x)dx.We give the following results which will be often used in the sequel.Lemma 5 (see [15]).
Letw∈A1. Then, for any ball B, there exists an absolute constant C>0 such that (13)w2B≤CwB.In general, for any λ>1, we have (14)wλB≤C·λnwB,where C does not depend on B nor on λ.Lemma 6 (see [15]).
Letw∈A1. Then, there exists a constant C>0 such that (15)C·EB≤wEwBfor any measurable subset E of a ball B.Given a weight functionw on Rn, for 0<p<∞, we denote by Lp(w) the weighted space of all functions f satisfying (16)fLpw≔∫Rnfxpwxdx1/p<∞.In particular, when w equals a constant function, we will denote Lp(w) simply by Lp(Rn) and define (17)fLp≔∫Rnfxpdx1/p<∞.Throughout this paper, we will useC to denote a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence. By A~B, we mean that there exists a constant C>1 such that 1/C≤A/B≤C. Moreover, we denote the conjugate exponent of s>1 by s′=s/(s-1).


## 3. Atomic Decomposition and Molecular Characterization of Weighted Hardy Spaces

LetL=-Δ+V. For any t>0, we define Pt=e-tL and (18)Qt,k≔-tkdkPsdsks=t=tLke-tL,k=1,2,….We denote Qt,k simply by Qt when k=1. First note that Gaussian upper bounds carry over from heat kernels to their time derivatives.Lemma 7 (see [16, 17]).
For everyk=1,2,…, there exist two positive constants Ck and ck such that the kernel pt,k(x,y) of the operator Qt,k satisfies (19)pt,kx,y≤Ck4πtn/2exp⁡-x-y2cktfor all t>0 and almost all x,y∈Rn.LetR+n+1=Rn×(0,∞). For any x∈Rn, we set (20)Γx=y,t∈R+n+1:x-y<t.For a given function f∈L2(Rn), we consider the square function associated with Schrödinger operator L, which is defined by (see [18, 19])(21)SLfx≔∬ΓxQt2fy2dydttn+11/2,x∈Rn.Set (22)HL2Rn=RL¯=Lu∈L2Rn:u∈L2Rn¯,where R(L) stands for the range of L in L2(Rn). Given a weight function w on Rn, in [10, 13], the authors defined the weighted Hardy spaces HLp(w) for 0<p≤1 as the completion of HL2(Rn) in the norm given by the Lp(w)-quasinorm of square functions; that is, (23)fHLpw≔SLfLpw.For the Schrödinger operator L=-Δ+V, it can be shown that HL2(Rn)=L2(Rn) (see, e.g., [19]). In [10], Song and Yan characterized weighted Hardy spaces HL1(w) in terms of atoms in the following way and obtained their atomic characterizations.Definition 8 (see [10]).
LetM∈N and p=1. A function a(x)∈L2(Rn) is called a (1,2,M)-atom with respect to w (or a w-(1,2,M)-atom) if there exist a ball B=B(x0,rB) and a function b∈D(LM) such that(a)
a=LMb;(b)
supp⁡Lkb⊆B, k=0,1,…,M;(c)
rB2LkbL2(B)≤rB2MB1/2[w(B)]-1, k=0,1,…,M.Theorem 9 (see [10]).
LetM∈N and w∈A1∩RH2. If f∈HL1(w)∩L2(Rn), then there exist a family of w-(1,2,M)-atoms {aj} and a sequence of real numbers {λj} with ∑j‍|λj|≤CfHL1(w) such that f can be represented in the form f(x)=∑j‍λjaj(x), and the sum converges in the sense of both L2(Rn)-norm and HL1(w)-norm.Similarly, in [13], we introduced the notion of weighted atoms for HLp(w) (0<p<1) and proved their atomic characterizations.Definition 10 (see [13]).
LetM∈N and 0<p<1. A function a(x)∈L2(Rn) is called a (p,M;L)-atom with respect to w (or a w-(p,M;L)-atom) if there exist a ball B=B(x0,rB) and a function b∈D(LM) such that(a′)
a=LMb;(b′)
supp⁡Lkb⊆B, k=0,1,…,M;c′
(rB2L)kbL2(B)≤rB2M|B|1/2[w(B)]-1/p, k=0,1,…,M.Theorem 11 (see [13]).
LetM∈N and 0<p<1.(i)
Ifw∈A1 and f∈HLp(w)∩L2(Rn), then there exist a family of w-(p,M;L)-atoms {aj} and a sequence of real numbers {λj} with ∑j‍|λj|p≤CfHLp(w)p such that f can be represented in the form f(x)=∑j‍λjaj(x), and the sum converges in the sense of both L2(Rn)-norm and HLp(w)-norm.(ii)
Ifw∈A1∩RH2/p′ and n/(n+1)<p<1, then every w-(p,M;L)-atom a is in HLp(w) and its HLp(w)-norm is uniformly bounded; that is, there exists a constant C>0 independent of a such that aHLp(w)≤C.For every bounded Borel functionF:[0,∞)→C, we define the operator F(L):L2(Rn)→L2(Rn) by the following formula: (24)FL≔∫0∞FλdELλ,where EL(λ) is the spectral decomposition of L. Therefore, the operator cos⁡(tL) is well defined on L2(Rn). Moreover, it follows from [20] that there exists a constant c0 such that the Schwartz kernel Kcos(tL)(x,y) of cos⁡(tL) has support contained in {(x,y)∈Rn×Rn:x-y≤c0t}. By the functional calculus for L and Fourier inversion formula, whenever F is an even bounded Borel function with F^∈L1(R), we can write F(L) in terms of cos⁡(tL). More precisely (25)FL=2π-1∫-∞∞F^tcos⁡tLdt,which gives (26)KFLx,y=2π-1∫t≥c0-1x-yF^tKcos⁡tLx,ydt.Lemma 12 (see [10, 19]).
Letφ∈C0∞(R) be even and supp⁡φ⊆[-c0-1,c0-1]. Let Φ denote the Fourier transform of φ. Then, for each j=0,1,… and for all t>0, the Schwartz kernel K(t2L)jΦ(tL)(x,y) of (t2L)jΦ(tL) satisfies (27)supp⁡Kt2LjΦtL⊆x,y∈Rn×Rn:x-y≤t.For a given real numbers>0, we define (28)
F
s
≔
ψ
:
C
⟶
C
measurable
,
ψ
z
≤
C
·
z
s
1
+
z
2
s
.Then, for any nonzero function ψ∈F(s), we can prove the following estimate (see [10, 19]):(29)∫0∞ψtLfL22dtt1/2=CfL2,where C=(∫0∞‍|ψ(t)|2dt/t)1/2. Inspired by [19, 21, 22], we are now going to define the weighted molecules corresponding to the weighted atoms mentioned above.Definition 13.
Letε>0, M∈N, and 0<p≤1. A function m(x)∈L2(Rn) is called a w-(p,M,ε;L)-molecule associated with L, if there exist a ball B=B(x0,rB) and a function b∈D(LM) such that(A)
m=LMb;(B)
(rB2L)kbL2(2B)≤rB2MB1/2[w(B)]-1/p, k=0,1,…,M;(C)
(rB2L)kbL2(2j+1B∖2jB)≤2-jεrB2M2jB1/2[w(2jB)]-1/p, k=0,1,…,M, j=1,2,….Clearly, for everyw-(p,M;L)-atom a, it is also a w-(p,M,ε;L)-molecule for all ε>0. Then, we are able to establish the following molecular characterizations for the weighted Hardy spaces HLp(w)  (0<p≤1) associated with L.Theorem 14.
Letε>0, M∈N, and 0<p≤1.(i)
Iff∈HLp(w)∩L2(Rn) and w∈A1, then there exist a family of w-(p,M,ε;L)-molecules {mj} and a sequence of real numbers {λj} with ∑j‍|λj|p≤CfHLp(w)p such that f(x)=∑j‍λjmj(x), and the sum converges in the sense of both L2(Rn)-norm and HLp(w)-norm.(ii)
Assume thatM>n/2(1/p-1/2) and w∈A1∩RH2/p′. Then, every w-(p,M,ε;L)-molecule m is in HLp(w). Moreover, there exists a constant C>0 independent of m such that mHLp(w)≤C.Proof.
(i) This is a straightforward consequence of Theorems9 and 11. (ii) We will use some ideas from [19, 22]. Suppose that m is a w-(p,M,ε;L)-molecule associated with a ball B=B(x0,rB). Let φ∈C0∞(R) be even with supp⁡φ⊆[-(2c0)-1,(2c0)-1] and let Φ denote the Fourier transform of φ. We set Ψ(x)=x2Φ(x), x∈R. By the L2-functional calculus of L, for every m∈L2(Rn), we can establish the following version of the Calderón reproducing formula:(30)mx=cΨ∫0∞t2LMΨ2tLmxdtt,where the above equality holds in the sense of L2(Rn)-norm. Set (31)U0B=2B,UjB=2j+1B∖2jB,j=1,2,….Then, we can decompose (32)Rn×0,∞=⋃j=0∞UjB×0,2jrB∪⋃j=1∞2jB×2j-1rB,2jrB.For any measurable set E in Rn, we denote χE the characteristic function of the set E. Hence, by formula (30), we are able to write (33)mx=cΨ∑j=0∞∫02jrBt2LMΨ2tLmχUjBxdtt+cΨ∑j=1∞∫2j-1rB2jrBt2LMΨ2tLmχ2jBxdtt≔∑j=0∞mj1x+∑j=1∞mj2x.Let us first consider the terms {mj(1)}j=0∞. We will show that each mj(1) is a multiple of a w-(p,M;L)-atom with a sequence of coefficients in lp. Indeed, for every j=0,1,2,…, one can write (34)mj1x=LMbjx,where (35)bjx=cΨ∫02jrBt2MΨ2tLmχUjBxdtt.By using Lemma 12, we can easily conclude that, for every k=0,1,…,M, supp⁡(Lkbj)⊆2j+1B. In addition, by the duality argument, we get (36)2j+1rB2LkbjL22j+1B=suphL22j+1B≤1∫2j+1B2j+1rB2Lkbjxhxdx.Then, it follows from Hölder’s inequality and estimate (29) that (37)
∫
2
j
+
1
B
2
j
+
1
r
B
2
L
k
b
j
x
h
x
d
x
=
c
Ψ
2
j
+
1
r
B
2
k
·
∫
0
2
j
r
B
∫
2
j
+
1
B
t
2
M
L
k
Ψ
t
L
m
χ
U
j
B
y
·
Ψ
t
L
h
y
d
y
d
t
t
≤
c
Ψ
2
j
+
1
r
B
2
k
·
2
j
r
B
2
M
-
2
k
∫
0
2
j
r
B
∫
2
j
+
1
B
t
2
L
k
Ψ
t
L
·
m
χ
U
j
B
y
·
Ψ
t
L
h
y
d
y
d
t
t
≤
c
Ψ
2
j
+
1
r
B
2
M
∫
0
∞
t
2
L
k
Ψ
t
L
·
m
χ
U
j
B
L
2
2
d
t
t
1
/
2
∫
0
∞
Ψ
t
L
·
h
χ
2
j
+
1
B
L
2
2
d
t
t
1
/
2
≤
c
Ψ
2
j
+
1
r
B
2
M
·
m
χ
U
j
B
L
2
·
h
χ
2
j
+
1
B
L
2
≤
C
·
2
-
j
ε
2
j
+
1
r
B
2
M
·
2
j
+
1
B
1
/
2
w
2
j
+
1
B
-
1
/
p
.Hence, (38)2j+1rB2LkbjL22j+1B≤C·2-jε2j+1rB2M2j+1B1/2w2j+1B-1/p,which implies our desired result. Next we consider the terms {mj(2)}j=1∞. For every j=1,2,…, we write (39)mj2x=cΨ∫2j-1rB2jrBt2LMΨ2tLmxdtt-cΨ∫2j-1rB2jrBt2LMΨ2tLmχ2jBcxdtt≔mj21x-mj22x.To deal with the term mj(21), we recall that m=LMb for some b∈D(LM), and then (40)mj21x=cΨ∫2j-1rB2jrBt2LMΨ2tLLMbxdtt=LMbj21x,where (41)bj21x=cΨ∫2j-1rB2jrBt2LMΨ2tLbxdtt.Since b(x)=b(x)χ2jB(x)+∑l=j∞‍b(x)χUl(B)(x), then we can further write (42)bj21x=b1,j21x+∑l=j∞blj21x,where (43)b1,j21x=cΨ∫2j-1rB2jrBt2LMΨ2tLbχ2jBxdtt,blj21x=cΨ∫2j-1rB2jrBt2LMΨ2tLbχUlBxdtt.By using Lemma 12 again, we have supp⁡(Lkb1,j(21))⊆2jB and supp⁡(Lkblj(21))⊆2l+1B for every k=0,1,…,M. Moreover, it follows from Minkowski’s integral inequality that (44)2jrB2Lkb1,j21L22jB=cΨ2jrB2k·∫2j-1rB2jrBt2MLM+kΨ2tLbχ2jBdttL22jB≤cΨ2jrB2k·∫2j-1rB2jrBt2LM+kΨ2tLbχ2jBL22jBdtt2k+1≤Cbχ2jBL22jB≤C∑l=0j-1bχUlBL22jB≤C∑l=0j-12-lεrB2M2lB1/2w2lB-1/p.When 0≤l≤j-1, then 2lB⊆2jB. Since w∈A1, then, by using Lemma 6, we can get (45)w2lBw2jB≥C·2lB2jB.Consequently, (46)2jrB2Lkb1,j21L22jB≤C·2-j2M-n1/p-1/2·2jrB2M2jB1/2w2jB-1/p∑l=0∞12lε·12ln/p-n/2≤C·2-j2M-n1/p-1/2·2jrB2M2jB1/2·w2jB-1/p.On the other hand, (47)
2
l
+
1
r
B
2
L
k
b
l
j
21
L
2
2
l
+
1
B
=
c
Ψ
2
l
+
1
r
B
2
k
·
∫
2
j
-
1
r
B
2
j
r
B
t
2
M
L
M
+
k
Ψ
2
t
L
b
χ
U
l
B
d
t
t
L
2
2
l
+
1
B
≤
c
Ψ
2
l
+
1
r
B
2
k
∫
2
j
-
1
r
B
2
j
r
B
t
2
L
M
+
k
Ψ
2
t
L
·
b
χ
U
l
B
L
2
2
l
+
1
B
d
t
t
2
k
+
1
≤
C
2
l
+
1
r
B
2
k
·
b
χ
U
l
B
L
2
2
l
+
1
B
·
1
2
j
r
B
2
k
≤
C
·
2
-
l
ε
2
l
+
1
r
B
2
M
·
2
l
+
1
B
1
/
2
w
2
l
+
1
B
-
1
/
p
.Observe that 2M>n(1/p-1/2). Thus, from the above discussions, we have already proved that each mj(21) is a multiple of a w-(p,M;L)-atom with a sequence of coefficients in lp. Finally, we estimate the terms {mj(22)}j=1∞. For every j=1,2,…, we decompose mj(22) as follows: (48)mj22x=cΨ∑l=j∞∫2j-1rB2jrBt2LMΨ2tLmχUlBxdtt=∑l=j∞LMblj22x,where (49)blj22x=cΨ∫2j-1rB2jrBt2MΨ2tLmχUlBxdtt.It follows immediately from Lemma 12 that supp⁡(Lkblj(22))⊆2l+1B for every k=1,2,…,M and l≥j. Moreover, (50)2l+1rB2Lkblj22L22l+1B=cΨ2l+1rB2k·∫2j-1rB2jrBt2MLkΨ2tLmχUlBdttL22l+1B≤cΨ2l+1rB2k2lrB2M-2k·∫2j-1rB2jrBt2LkΨ2tLmχUlBL22l+1Bdtt≤cΨ2l+1rB2MmχUlBL22l+1B≤C·2-lε2l+1rB2M2l+1B1/2w2l+1B-1/p.Therefore, we have showed that each mj(22) is also a multiple of a w-(p,M;L)-atom with a sequence of coefficients in lp. This completes the proof of Theorem 14.


## 4. Proof of Theorems1 and 2

Proof of Theorem1.
By the known result, we have that, for anyγ∈R, the operator Liγ is linear and bounded on L2(Rn) (see, e.g., [2, 23]). Since HLp(w)∩L2(Rn) is dense in HLp(w), then, by Theorems 9 and 11 and a standard density argument, it is enough for us to show that, for any w-(p,M;L)-atom a, M∈N, there exists a constant C>0 independent of a such that Liγ(a)Lp(w)≤C. Let a be a w-(p,M;L)-atom associated with a ball B=B(x0,rB), aL2(B)≤B1/2wB-1/p. We write (51)LiγaLpwp=∫2BLiγaxpwxdx+∑k=1∞∫2k+1B∖2kBLiγaxpwxdx≔I1+I2.We set s=2/p>1. Note that w∈RHs′, and then it follows from Hölder’s inequality, the L2 boundedness of Liγ, and Lemma 5 that(52)I1≤∫2BLiγax2dxp/2∫2Bwxs′dx1/s′≤CaL2Bp·w2B2B1/s≤C.On the other hand, for any x∈2k+1B∖2kB, k=1,2,…, by expression (7), we can write (53)Liγax≤C∫0∞e-tLaxdtt≤C∫0rB2e-tLaxdtt+C∫rB2∞e-tLaxdtt≔I+II.For the term I, we observe that when x∈2k+1B∖2kB, y∈B, then x-y≥2k-1rB. Hence, by using Hölder’s inequality and estimate (4), we deduce(54)e-tLax≤C·t2k-1rBn+1∫Baydy≤C·t2krBn+1·aL2BB1/2≤C·wB-1/p·t2kn+1·rB.So we have (55)I≤C·12kn+1wB1/p·1rB∫0rB2dtt≤C·12kn+1wB1/p.We now turn to estimate the other term II. In this case, since there exists a function b∈D(LM) such that a=LMb and bL2(B)≤rB2M|B|1/2[w(B)]-1/p, then it follows from Hölder’s inequality and Lemma 7 that(56)e-tLax=tLMe-tLbx·1tM≤C·12k-1rBn+1∫Bbydy·1tM-1/2≤C·12krBn+1bL2BB1/2·1tM-1/2≤C·rB2M-12kn+1wB1/p·1tM-1/2.Consequently, (57)II≤C·12kn+1wB1/p·rB2M-1∫rB2∞dttM+1/2≤C·12kn+1wB1/p,where in the last inequality we have used the fact that M≥1. Therefore, by combining the above estimates for I and II, we obtain the following pointwise inequality:(58)Liγax≤C·12kn+1wB1/p,whenx∈2k+1B∖2kB.Notice that w∈A1. Substituting inequality (58) into the term I2 and using Lemma 5, then we have(59)I2≤C∑k=1∞12kpn+1wB·w2k+1B≤C∑k=1∞12kpn+1-n≤C,where the last series is convergent since p>n/(n+1). Summarizing estimates (52) and (59) derived above, we complete the proof of Theorem 1.Proof of Theorem2.
SinceHLp(w)∩L2(Rn) is dense in HLp(w) and the operator Liγ is linear and bounded on L2(Rn), then, in view of Theorems 9, 11, and 14, it suffices to verify that, for every w-(p,2M;L)-atom a, the function m=Liγ(a) is a multiple of a w-(p,M,ε;L)-molecule for some ε>0, and the multiplicative constant is independent of a. Let a be a w-(p,2M;L)-atom with supp⁡a⊆B=B(x0,rB). By the definition, there exists a function b∈D(L2M) such that (60)a=L2MbrB2LkbL2B≤rB4MB1/2wB-1/p,k=0,1,…,2M.We set b~=Liγ(LMb), and then m=LM(b~). Obviously, we have m(x)∈L2(Rn). Moreover, for k=0,1,…,M, we can deduce (61)rB2Lkb~L22B=1rB2MLiγrB2LM+kbL22B≤C·1rB2MrB2LM+kbL2B≤C·rB2MB1/2wB-1/p.It remains to estimate (rB2L)kb~L2(2j+1B∖2jB) for k=0,1,…,M, j=1,2,…. We write (62)rB2Lkb~x=LiγrB2LkLMbx≤C∫0rB2e-tLrB2kLM+kbxdtt+C∫rB2∞e-tLrB2kLM+kbxdtt≔I′+II′.As mentioned in the proof of Theorem 1, we know that when x∈2j+1B∖2jB, y∈B, then x-y≥2j-1rB, j=1,2,…. It follows from Hölder’s inequality and estimate (4) that(63)I′≤C∫0rB2t2j-1rBn+1rB2kLM+kbL2B1/2dtt≤C·12jrBn+11rB2M·rB4MB1/2wB-1/pB1/2·∫0rB2dtt≤C·12jn+1·rB2MwB-1/p.Since w∈A1 and B⊆2jB, j=1,2,…, then, by using Lemma 6, we can get(64)wBw2jB≥C·B2jB.Hence, (65)I′≤C·12jn+1-n/p·rB2Mw2jB-1/p,whenx∈2j+1B∖2jB.Applying Hölder’s inequality and Lemma 7, we obtain(66)II′≤C·rB2k∫rB2∞tLM+ke-tLbxdttM+k+1≤C·rB2k∫rB2∞12j-1rBn+1bL2BB1/2dttM+k+1/2≤C·12jrBn+1rB4M+2kBwB-1/p∫rB2∞dttM+k+1/2≤C·12jn+1·rB2MwB-1/p.It follows immediately from inequality (64) that (67)II′≤C·12jn+1-n/p·rB2Mw2jB-1/p,whenx∈2j+1B∖2jB.Combining the above estimates for I′ and II′, we thus obtain (68)rB2Lkb~L22j+1B∖2jB≤C·12jn+1-n/p·rB2M2jB1/2w2jB-1/p.Observe that p>n/(n+1). If we set ε=(n+1)-n/p, then we have ε>0. Therefore, we have proved that the function m=Liγ(a) is a multiple of a w-(p,M,ε;L)-molecule. This completes the proof of Theorem 2.


## 5. Proof of Theorems3 and 4

Proof of Theorem3.
As in the proof of Theorem1, assuming first that f∈HLp(w)∩L2(Rn), we are going to prove that, for every w-(p,M;L)-atom a associated with a ball B=B(x0,rB), M>(3n)/4, there exists a constant C>0 independent of a such that L-α/2(a)Lq(wq/p)≤C. We write (69)L-α/2aLqwq/pq=∫2BL-α/2axqwxq/pdx+∑k=1∞∫2k+1B∖2kBL-α/2axqwxq/pdx≔J1+J2.By our assumption 0<α<n/2, 1/q=1/p-α/n, then we are able to choose a number μ>q such that 1/μ=1/2-α/n. Set s=2/p. By a simple calculation, we can see that (70)qp·μq′=1p·qμμ-q=1p·1p-12-1=s′,1-qμ=q1q-1μ=q1p-12=qps′.Applying Hölder’s inequality with exponent ν=μ/q>1, the L2-Lμ boundedness of L-α/2 (see (10)), Lemma 5, and w∈RHs′, we get(71)J1≤∫2BL-α/2axq·μ/qdxq/μ·∫2Bwxq/p·μ/q′dx1-q/μ=∫2BL-α/2axμdxq/μ·∫2Bwxs′dxq/ps′≤CaL2Bqw2B2B1/sq/p≤C.We now turn to deal with J2. For any x∈2k+1B∖2kB, k=1,2,…, by expression (8), we can write (72)L-α/2ax≤C∫0∞e-tLaxdtt1-α/2≤C∫0rB2e-tLaxdtt1-α/2+C∫rB2∞e-tLaxdtt1-α/2≔III+IV.For the term III, it follows immediately from (54) that (73)III≤C·12kn+1wB1/p·1rB∫0rB2dtt1/2-α/2≤C·rBα2kn+1wB1/p.For the other term IV, by the previous estimate (56), we thus have (74)IV≤C·12kn+1wB1/p·rB2M-1∫rB2∞dttM+1/2-α/2≤C·rBα2kn+1wB1/p,where the last inequality holds since M>(3n)/4>1/2+α/2. Combining the above estimates for III and IV, we obtain the following pointwise inequality:(75)L-α/2ax≤C·rBα2kn+1wB1/p,whenx∈2k+1B∖2kB.Note that w∈A1. Substituting inequality (75) into the term J2, we get (76)J2≤C∑k=1∞rBα2kn+1wB1/pq∫2k+1Bwxq/pdx.Using Hölder’s inequality with exponent ν=μ/q>1, Lemma 5, and the condition w∈A1∩RHs′, we can deduce that(77)J2≤C∑k=1∞rBα2kn+1wB1/pq·2k+1Bq/μ·∫2k+1Bwxq/p·μ/q′dx1-q/μ≤C∑k=1∞rBα2kn+1wB1/pq·2k+1Bq/μ·w2k+1B2k+1B1/sq/p≤C∑k=1∞12kqn+1-n≤C,where in the last inequality we have used the fact that q>p>n/(n+1). By combining inequality (77) with (71), we obtain the desired result. Therefore, by a standard density argument, we can show that the same conclusion holds for all f∈HLp(w). This concludes the proof of Theorem 3.Proof of Theorem4.
As in the proof of Theorem2, it is enough to show that, for every w-(p,2M;L)-atom a, function m=L-α/2(a) is a multiple of a w-(p,M,ε;L)-molecule for some ε>0 and the multiplicative constant is independent of a. Let a be a w-(p,2M;L)-atom with supp⁡a⊆B=B(x0,rB) and a=L2M(b), where M>(3n)/4>max⁡{n/2(1/p-1/2),1/2+α/2}, b∈D(L2M). Setting b~=L-α/2(LMb), then we have m=LM(b~). It is easy to check that m(x)∈L2(Rn). As before, since 0<α<n/2, then we may choose a number μ>2 such that 1/μ=1/2-α/n. For k=0,1,…,M, by using Hölder’s inequality and the L2-Lμ boundedness of L-α/2, we obtain(78)rB2Lkb~L22B≤1rB2ML-α/2rB2LM+kbLμ2B2B1/2-1/μ≤C·1rB2MrB2LM+kbL2BB1/2-1/μ≤C·rB2MB1/2+α/nwB-1/p.Noting that 1/q=1/p-α/n, then a straightforward computation yields that q/p<2/p′ whenever 0<α<n/2. By our assumption w∈RH2/p′, then we have w∈RHq/p. Consequently, (79)wq/pBp/q≤C·wBB1-p/q,which implies(80)wB-1/p≤C·B1/q-1/pwq/pB-1/q.Substituting inequality (80) into (78), we can get(81)rB2Lkb~L22B≤C·rB2MB1/2wq/pB-1/q.It remains to estimate (rB2L)kb~L2(2j+1B∖2jB) for k=0,1,…,M, j=1,2,…. We write (82)rB2Lkb~x=L-α/2rB2LkLMbx≤C∫0rB2e-tLrB2kLM+kbxdtt1-α/2+C∫rB2∞e-tLrB2kLM+kbxdtt1-α/2≔III′+IV′.For the term III′, by using the same arguments as in the proof of (63), we have (83)III′≤C·12jn+1·rB2MwB-1/p1rB∫0rB2dtt1/2-α/2≤C·12jn+1·rB2M+αwB-1/p.For the term IV′, we proceed as that of (66) and then obtain (84)IV′≤C·12jn+1·wB-1/prB4M+2k-1∫rB2∞dttM+k+1/2-α/2≤C·12jn+1·rB2M+αwB-1/p.Combining the above estimates for III′ and IV′, we can get (85)rB2Lkb~x≤C·12jn+1·rB2M+αwB-1/p,whenx∈2j+1B∖2jB.Since w∈A1, then it follows from inequality (64) that (86)rB2Lkb~x≤C·12jn+1-n/p·rB2M+αw2jB-1/p,whenx∈2j+1B∖2jB.Similar to the proof of (80), we can also show that (87)w2jB-1/p≤C·2jB1/q-1/pwq/p2jB-1/q.Hence, we finally obtain (88)rB2Lkb~x≤C·12jn+1-n/q·rB2Mwq/p2jB-1/q,whenx∈2j+1B∖2jB.Therefore,(89)rB2Lkb~L22j+1B∖2jB≤C·12jn+1-n/q·rB2M2jB1/2wq/p2jB-1/q.Observe that 1≥q>p>n/(n+1). If we set ε=(n+1)-n/q, then ε>0. Summarizing estimates (81) and (89) derived above, we conclude the proof of Theorem 4.



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*Source: 823862-2015-11-25.xml*