Weighted Hardy Spaces Associated to Discrete Laplacians on Graphs and Applications

Potential AnalDOI 10.1007/s11118-014-9395-8

Weighted Hardy Spaces Associated to DiscreteLaplacians on Graphs and Applications

The Anh Bui

Received: 16 May 2013 / Accepted: 10 February 2014© Springer Science+Business Media Dordrecht 2014

Abstract Let � be a infinite graph with a weight μ and let d and m be the distance and themeasure associated with μ such that (�, d,m) is a space of homogeneous type. Let p(·, ·) bethe natural reversible Markov kernel on (�, d,m) and its associated operator be defined byPf (x) = ∑

y p(x, y)f (y). Then the discrete Laplacian on L2(�) is defined by L = I −P .

In this paper we investigate the theory of weighted Hardy spaces HpL (�,w) associated to

the discrete Laplacian L for 0 < p ≤ 1 and w ∈ A∞. Like the classical results, we provethat the weighted Hardy spaces H

pL(�,w) can be characterized in terms of discrete area

operators and atomic decompositions as well. As applications, we study the boundednessof singular integrals on (�, d,m) such as square functions, spectral multipliers and Riesztransforms on these weighted Hardy spaces Hp

L (�,w).

Keywords Graphs · Discrete Laplacian · Hardy spaces · Spectral multipliers · Squarefunctions · Riesz transforms

Mathematics Subject Classifications (2010) 60J10 · 42B20 · 42B25

1 Introduction

Recently, the studies on random walks on graphs have been played an important role inboth pure and applied mathematics. According to [36], the theory of on random walks ongraphs ranges from topics such as the type problem of Riemannian manifolds to modelingphenomena such as the spread of cancer, see [10, 27]. For further information about thetheory of random walks on graphs and applications, we refer the reader to [23, 33, 36] and

T. A. Bui (�)Department of Mathematics, Macquarie University, NSW 2109, Australiae-mail: [email protected]; bt [email protected]

T. A. BuiDepartment of Mathematics, University of Pedagogy, Ho Chi Minh City, Vietnam

mailto:[email protected]

mailto:[email protected]

T.A. Bui

the references therein. Before coming to details, we would like to recall some backgroundsof the graphs in [7, 15, 28, 36].

1.1 Backgrounds of Graphs

Let � be a countably infinite set and let μ(x, y) be a weight on � satisfying μ(x, y) =μ(y, x) ≥ 0 for all x, y ∈ �. The weight μ(x, y) induces a graph structure on �. Forx, y ∈ �, we say the vertices x and y neighbors and write x ∼ y if μ(x, y) > 0. Thediscrete measure m on � associated to μ is defined by

m(x) =∑

y∼x

μ(x, y), x ∈ �.

For a subset E ⊂ �, we define m(E) =∑

x∈Em(x). Then we set Lp(�) = Lp(�,m) for

0 < p < ∞.A path of length n joining the vertices x and y is a sequence of vertices x =

x0, x1, . . . , xn = y such that xi ∼ xi−1, i = 1, . . . , n. In this article, we also assume that �is connected, i.e. for any x, y ∈ �, there exists a path joining x and y. The distance d(x, y)is then defined as the infimum of the lengths of paths joining x and y.

Let B(x, r) = {y ∈ � : d(x, y) ≤ r} denote the ball of center x and radius r . Inthe sequel, we assume that � satisfies the locally uniformly finite property, i.e. there existsN ≥ 1 so that for any x ∈ �, �B(x, 1) ≤ N where �E denotes the cardinal of the subsetE ⊂ �.

We consider the Markov kernel p(x, y) defined by

p(x, y) = μ(x, y)

m(x), x, y ∈ �.

It is not difficult to see that p(x, y) = 0 if d(x, y) ≥ 2. Note that p(x, y) may not besymmetric. However, the following identity holds

p(x, y)m(x) = p(y, x)m(y)

for all x, y ∈ �. Moreover, for x ∈ �∑

y∈�p(x, y) = 1.

Associated to the kernel p(·, ·), the Markov operator P is defined by

Pf (x) =∑

y∈�p(x, y)f (y).

Then we set L = I − P to be the discrete Laplacian on �. Let us set

p0(x, y) = δx(y), p1(x, y) = p(x, y)

where δx is the Dirac mass at x. Let pn(x, y), n ∈ N, be the nth convolution power ofp(x, y) defined by

pn(x, y) =∑

z∈�p(x, z)pn−1(z, y).

Then we havePnf (x) =

∑

y∈�pn(x, y)f (y), x ∈ �, n ∈ N.

Weighted Hardy Spaces Associated to Discrete Laplacians on Graphs

Remark 1.1 It can be verified that if supp ϕ ⊂ B(x, k) then Pnϕ ⊂ B(x, k + n) for alln ∈ N.

The operator “length of the gradient” ∇ is defined by

∇f (x) =⎛

⎝1

2

∑

y∈�p(x, y)|f (x)− f (y)|2

⎞

⎠

1/2

for any function f on � and x ∈ �. It is easy to check that

〈(I − P )f, f 〉L2 = ‖∇f ‖2L2 .

See, for example [7, Section 1].Throughout this paper, as in [7], we assume the following conditions:

(D) Doubling property: The graph (�, d,m) satisfies the doubling property (D), i.e.,there exists a constant C > 0 such that

V (x, 2r) ≤ CV (x, r), x ∈ �, r > 0 (1)

where V (x, r) = m(B(x, r)).(Sα) Uniform lower bound condition for p(x, y): Given α > 0, we say that (�, d,m)

satisfies the condition (Sα) if for all x, y ∈ �

x ∼ y ⇒ μ(x, y) ≥ αm(x) and x ∼ x for all x ∈ �.

(UE) Upper estimate for pn(x, y): We say that (�,μ) satisfies the condition (UE) if thereexist C, c > 0 such that

pn(x, y) ≤ Cm(y)

V (x,√n)

exp

(

−cd(x, y)2

n

)

for all n ∈ N and x, y ∈ �.

We would like to make some relevant comments on the assumptions above.

Remark 1.2

(i) Note that the doubling condition (D) implies that there exist constants c,D > 0 suchthat

V (x, r) ≤ c( r

s

)DV (x, s), for r > s > 0. (2)

Moreover under the doubling assumption (D) that the graph � is a space ofhomogeneous type in the sense of Coifman and Weiss [17].

(ii) The condition (Sα) is just a technical assumption. This condition is a necessary con-dition to obtain the analyticity of the Markov operators P , see [15]. By the similararguments as in the proof of [19, Theorem 1.1], we can get the temporal regularityof the Markov kernel. More precisely, for n ∈ N+, k ∈ N let us denote by pn,k theassociated kernel to (I − P )kP n. Under the conditions (D), (Sα) and (UE), we have

|pn,k(x, y)| ≤ Cm(y)

nkV (x,√n)

exp

(

−cd(x, y)2

n

)

. (3)

T.A. Bui

Since the proof of (3) is completely the same as that of [19, Theorem 1.1]. We omitdetails here.

(iii) Moreover, it was proved in [7] that the condition (Sα) implies that 0 does not belongto the spectrum of L on L2(�). Hence, L is one-to-one on L2(�). Since L2(�) :=R(L) ⊕ N (L), where R(L) stands for the range R(L) := {Lu : u ∈ L2(�)} andN (L) stands for the nullspace of L. Hence, in this situation, R(L) = L2(�). Inaddition, since L is bounded on L2(�), R(L) = L2(�).

1.2 The Aims

Recently, the studies on some singular integrals on graphs have been paid a lot of atten-tion such as the study of spectral multipliers of the discrete Laplace operators, see forexample [28], the study of Riesz transforms and square functions associated to the discreteLaplace operators, see for example [7, 30, 31], the study on weighted norm inequalities ofthese singular integral operators on graphs, see [8], and the study of singular integral oper-ators on graphs in the scale of Hardy spaces associated to the discrete Laplace operators,see [4].

We would like to emphasize that due to the lack of regularity assumption on the Markovkernels pn(x, y), singular integral operators on graphs such as Riesz transforms, squarefunctions and spectral multipliers considered in this paper, see Section 4, may not fall withinthe scope of Calderon-Zygmund theory. Hence, the theory of Hardy spaces in the sense ofCoifman and Weiss (see [18]) may not be applicable in our situation. This shows the need ofthe new variant of Hardy spaces which are suitable to this kind of singular integral operators.

Inspiring from the works of [4, 6, 20, 22, 24–26, 32], the aim of this paper is tostudy the theory of weighted Hardy spaces associated to the discrete Laplace opera-tors and then investigate the boundedness of singular integral operators on graphs suchas Riesz transforms, square functions and spectral multipliers on these weighted Hardyspaces. More precisely, firstly we characterize the weighted Hardy spaces associated tooperators in terms of the discrete area functions and in terms of atomic decompositions.Secondly, we study the boundedness of singular integral operators on these weighted Hardyspaces.

The organization of this paper is as follows. In Section 2, we summarize some basic prop-erties of the class of Muckenhoupt weights on graphs. Then we also introduce the weighteddiscrete tent spaces and give the atomic decomposition for the weighted tent spaces. Theweighted Hardy spaces associated to the discrete Laplace operators are investigated in Sec-tion 3. Similar to the classical case, we also prove that these weighted Hardy spaces canbe characterized in terms of the discrete area functions and the atomic decompositions. Asapplications, Section 4 studies the boundedness of singular integral operators on graphs onthe scale of these weighted Hardy spaces.

2 Preliminaries

We now specify some notations which will often be used in the sequel.We will often write B for B(xB, rB) and V (E) for m(E) for any measurable subset

E ⊂ �. Also given λ > 0, we write λB for the λ-dilated ball, which is the ball with thesame center as B and with radius rλB = λrB . For each ball B ⊂ � we set

S0(B) = B and Sj (B) = 2jB\2j−1B for j ∈ N.

We also write the sumβ∑

k=α

with real values α ≥ 0, β > 0 for the sum over k = [α], · · · [β]where [α], [β] denote the integer parts of α, β and use the convention that the constants C, ccan change from each of their appearance.

We also denote by M the Hardy-Littlewood maximal function defined by

Mf (x) = supB�x

1

V (B)

∑

y∈B|f (y)|m(y).

2.1 Muckenhoupt Weights on Graphs

Since (�, d,m) is a space of homogeneous type, according to [35], we can consider theclass of Muckenhoupt weights Ap(�), shortly Ap , on �.

A weight w is a non-negative measurable and locally integrable function on �. We saythat w ∈ Ap , 1 < p < ∞, if there exists a constant C such that for every ball B ⊂ �,

(1

V (B)

∑

x∈Bw(x)m(x)

)(1

V (B)

∑

x∈Bw−1/(p−1)(x)m(x)

)p−1

≤ C.

For p = 1, we say that w ∈ A1 if there is a constant C such that for every ball B ⊂ �,

1

V (B)

∑

y∈Bw(y)m(y) ≤ Cw(x) for a.e. x ∈ B.

We set A∞ = ∪1≤p<∞Ap .For 1 < q < ∞, the reverse Holder classes are defined as the set of all weights w

satisfying the following condition for any balls B ⊂ �,(

1

V (B)

∑

x∈Bwq(x)m(x)

)1/q

≤ C

V (B)

∑

x∈Bw(x)m(x).

When q = ∞, we say that w ∈ RH∞ if there is a constant C such that for any ball B ⊂ �,

w(x) ≤ C

V (B)

∑

y∈Bw(y)m(y) for a.e. x ∈ B.

Let w ∈ A∞, for 0 < p < ∞, the weighted spaces Lp(�,w) can be defined by{f :

∑

x∈�|f (x)|pw(x)m(x) < ∞

}

with the norm

‖f ‖Lp(�,w) =(∑

x∈�|f (x)|pw(x)m(x)

)1/p

.

We sum up some of the properties of Ap classes in the following results, see [35].

Lemma 2.1 The following properties hold:

(i) A1 ⊂ Ap ⊂ Aq for 1 ≤ p ≤ q ≤ ∞.(ii) RH∞ ⊂ RHq ⊂ RHp for 1 < p ≤ q ≤ ∞.

(iii) If w ∈ Ap, 1 < p < ∞, then there exists 1 < q < p such that w ∈ Aq .

T.A. Bui

(iv) If w ∈ RHq, 1 < q < ∞, then there exists q 1 such that for any ball B and any measurable subset E of B ,

C−1(V (E)

V (B)

)q

≤ w(E)

w(B)≤ C

(V (E)

V (B)

) r−1r

.

In the rest of paper, by the weight w, we shall mean w ∈ A∞ and we denote

qw = inf{q : w ∈ Aq} and rw = sup{r : w ∈ RHr}.For a measurable set E ⊂ �, we denote

w(E) =∑

x∈Ew(x)m(x).

2.2 Weighted Tent Spaces

Recall that the concept of tent spaces were introduced in [16]. In this section, adapting someideas in [16], the weighted tent spaces on graphs are investigated. We also can consideredthe weighted tent spaces on graphs as the discrete version of those in [16].

For any x ∈ �, denote by ϒ(x) the cone with vertex x, namely:

ϒ(x) = {(y, k) ∈ � × N+ : d(y, x) < k}.For any subset F ⊂ �, we denote R(F ) = ∪x∈Fϒ(x).

Let O be a subset of � × N+, the tent over O is defined by O = {(x, k) ∈ � × N+ :dist(x,Oc) ≥ k}. It can be verified that O = [R(Oc)]c. In the particular case, the tent overB := B(xB, rB) is defined by

B := {(x, k) ∈ � × N+ : d(xB, x) ≤ rB − k}.Let F ⊂ � be a closed set and O = �\F . For any fixed γ ∈ (0, 1), the set of points with

global γ -density with respect to F is defined by

F ∗ ={x ∈ � : m(B(x, r) ∩ F)

|V (x, r)| ≥ γ for all r > 0}.

Lemma 2.3 There exists a constant γ ∈ (0, 1) so that for any closes set F ⊂ � withm(Fc) < ∞ and any nonnegative summable function H on � ×N+,

∑

(y,k)∈R(F ∗)H(y, k)V (y, k)m(y) ≤ C

∑

x∈F

⎛

⎝∑

(y,k)∈ϒ(x)

H(y, k)m(y)

⎞

⎠m(x).

The proof of this lemma is similar to that of [29, Lemma 2.1]. Hence, we omit detailshere.

For any function on f on � × N+ and any x ∈ �, define

(Af )(x) =⎛

⎝∑

d(y,x)<k

∞∑

k=1

|f (y, k)|2kV (x, k)

μ(y)

⎞

⎠

1/2

.

For 0 < p < ∞ and w ∈ A∞, we say that f ∈ T p(�,w), if

‖f ‖T p(�,w) := ‖Af ‖Lp(�,w) < +∞.

In the particular, when w ≡ 1, we write T p(�) instead of T p(�,w).For the unweighted tent spaces, we have the following result.

Proposition 2.4 Let 1 < p < ∞. For f ∈ T p and g ∈ T p′we have

∑

x∈�

∞∑

k=1

|f (x, k)g(x, k)|k

μ(x) ≤∑

x∈�Af (x)Ag(x)μ(x).

The proof of Proposition 2.4 is similar to that of [16, Theorem 1] and we omit the detailshere.

Let us describe the notion of (w, p,∞) atoms.

Definition 2.5 Let p ∈ (0, 1] and w ∈ A∞. A measurable function a on � × N+ is said tobe a (w, p,∞) atom if there exists a ball B ⊂ � such that

(i) a is supported in B;(ii) for all 1 ≤ q < ∞, ‖a‖T p(�) ≤ V (B)1/qw(B)−1/p.

One can check that a (w, p,∞) atom is also a T p(�,w) function. Conversely, in thefollowing proposition, we will claim that a T p(�,w) function with 0 < p ≤ 1 can becharacterized in a linear decomposition of (w, p,∞) atoms.

Proposition 2.6 Let 0 < p ≤ 1 and w ∈ A∞. Then for any f ∈ T p(�,w), there exist asequence (λn)n∈N ∈ lp and a sequence of (w, p,∞) atoms (an)n∈N such that

f =∞∑

n=1

λnan (4)

and∞∑

n=1

|λn|p ≤ C‖f ‖pT p(�,w). (5)

In addition, if f ∈ T p(�,w)∩T 2(�) then the series in (4) converges in both T p(�,w) andT 2(�).

Proof We exploit some ideas in [16] to our situation (see also [29]).For � ∈ Z, we set E� = {x : AF(x) > 2�} and �� = {x : M(χE�

)(x) > 1 − γ }where γ is the constant in Lemma 2.3 and M is the Hardy-Littlewood maximal function.Then E� ⊂ �� and m(��) ≤ Cm(E�) for all �. Moreover since w ∈ A∞ we also obtainw(��) ≤ C(w)w(E�) for sufficiently small γ . It can be verified that suppf ⊂ ∪��.

For each �, using the covering Lemma in [18], there exist a family of balls {Qj� }j of ��

and two constants κ > 0 and c0 > 1 such that:

1. �k = ∪jQj� ;

2.∑

j χQj�

≤ κ;

3. c0Qj� ∩ (��)

c �= ∅.

T.A. Bui

Setting Bj

� = 4c0Qj

� , it can be verified that ��\��+1 ⊂ ∪jAj

� , where Aj

� = Bj

� ∩ (Qj

� ×N+) ∩ (��\��+1).

We set aj� = 2−(�+1)w(Bj� )

−1/pf χAj�

and λj� = 2(�+1)w(B

j� )

1/p . Obviously,

f =∑

�,j

λj�a

j� .

Let 1 < q < ∞ and h ∈ T q ′(�) so that ‖h‖T q′ (�) = 1. Note that Aj

� ⊂ (��+1)c =

R(F ∗�+1) where F�+1 = �\E�+1. Hence, thanks to Lemma 2.3 and Holder’s inequality, we

have

∣∣∣∑

y∈�

∑

k∈N

aj

� (y, k)h(y, k)

km(y)

∣∣∣ ≤

∑

y∈�

∑

k∈N

∣∣∣(a

j�χAj

�

)(y, k)h(y, k)

k

∣∣∣m(y)

≤∑

(y,k)∈R(F ∗�+1)

∣∣∣aj� (y, k)h(y, k)

k

∣∣∣m(y)

≤∑

x∈F�+1

⎛

⎝∑

(y,k)∈�(x)|ajk (y, k)h(y, k)|

m(y)

kV (y, k)

⎞

⎠m(x)

≤∑

x∈F�+1

⎛

⎝∑

(y,k)∈�(x)|ajk (y, k)h(y, k)|

m(y)

kV (x, k)

⎞

⎠m(x)

≤ C∑

x∈F�+1

A(ajk )(x)A(h)(x)m(x)

≤ C2−(�+1)w(Bj� )

−1/p

(∫

F�+1∩Bj�

|A(F )(x)|qdx)1/q

≤ CV (Bj� )

1/qw(Bj� )

−1/p

Therefore, aj� is a multiple of a (w, p,∞) atom.Furthermore, we have

∑

�,j

|λj� |p =∑

�,j

2p(�+1)w(Bj� ) ≤ C(w)

∑

�,j

2p(�+1)w(Qj�).

Using (i) and (ii), we have∑

�,j

|λj� |p ≤ C∑

�

2p(�+1)w(��) ≤ C∑

�

2p(�+1)w(E�)

= C∑

�

2p(�+1)w{x : Af (x) > 2�}

≤ C‖Af ‖pLp(w)

= ‖f ‖pT p(�,w)

.

It remains to claim that the series in (4) converges in both T p(�,w) and T 2(�) providedf ∈ T p(�,w) ∩ T 2(�). The proof of this statement is standard (see for example [26]) andhence we omit details here.

This completes our proof.


2.3 Calderon Reproducing Formula on Graphs

In this section, we recall a Calderon reproducing formula in [4] which plays an importantrole in obtaining the atomic decomposition of the weighted Hardy spaces.

Lemma 2.7 Let M ≥ 0. Then for any f ∈ L2(�), we have

f =∞∑

k=0

ck,M(I − P )MP kf (6)

on L2(�), where the coefficients ck,M are defined as follows(i) ck,1 = 1 for all k = 0, 1, 2, . . .;(ii) ck,M+1 = ∑k

j=0 cj,M for all k = 0, 1, 2, . . .;

(iii) ck,M ≤ kM−1 for all k and M .

Proof The result of this lemma is taken from [4]. For the sake of completeness, we providethe proof here.

We will prove lemma 2.7 by induction for f ∈ L2(�).For M = 1, we have

∥∥∥∥∥

N∑

k=0

(I − P )P kf − f

∥∥∥∥∥L2

= ‖PNf ‖L2 .

Due to Remark 1.2, we can pick a L2-function g so that f = Lg = (I − P )g. UsingGaussian upper bound (3) condition for (I − P )PN we imply that

‖PNf ‖L2 = ‖(I − P )PNg‖L2 ≤ C

N‖g‖L2 .

Hence,

limN→∞

∥∥∥∥∥

N∑

k=0

(I − P )P kf − f

∥∥∥∥∥L2

= 0.

This tells us that (6) holds for M = 1.We now assume that (6) holds for M = M , that is,

limN→∞

∥∥∥∥∥

N∑

k=0

ck,M (I − P )MP kf − f

∥∥∥∥∥L2

= 0. (7)

We will claim that (6) holds for M = M + 1. Indeed, we have, for any N ∈ N+,

N∑

k=0

ck,M+1(I − P )M+1P kf − f =N∑

k=0

ck,M+1(I − P )M(P k − P k+1)f − f

T.A. Bui

HenceN∑

k=0

ck,M+1(I − P)M+1P kf − f

= (I − P)M +N∑

k=0

(ck+1,M+1 − ck,M + 1)(I − P)MP k+1f − cN,M+1(I − P)MPN+1f − f

=(

(I − P)M +N∑

k=0

ck+1,M (I − P)MP k+1f − f

)

− cN,M+1(I − P)MPN+1f

=(

N∑

k=0

ck,M (I − P)MP kf − f

)

+ cN,M+1

(N + 1)M+1(N + 1)M+1(I − P)M+1PN+1g

where f = (I − P )g.Due to (7),

limN→∞

∥∥∥∥∥

N∑

k=0

ck,M (I − P )MP kf − f

∥∥∥∥∥L2

= 0.

Using (3) and the fact that cN,M+1 ≤ NM , one obtains

limN→∞

∥∥∥

cN,M+1

(N + 1)M+1(N + 1)M+1(I − P )M+1PN+1g

∥∥∥L2

= 0.

This completes our proof.

3 Weighted Hardy Spaces on Graphs

For given integer N ≥ 1, we consider the following discrete square functions

GL,Nf (x) =( ∞∑

k=1

|kN(I − P )NP kf |2k

)1/2

and

Shf (x) =( ∞∑

k=1

k|(I − P )P [ k2 ]f (x)|2)1/2

.

Using the similar argument as in [7, p. 286], we get that GL,M is bounded on L2(�). More-over, it can be verified that Sh(|f |) ≤ C(|f | + GL,1(|f |)). Hence, Sh is also bounded onL2(�).

We next consider the discrete area function defined by

SLf (x) =⎛

⎝∑

d(y,x)<k

∞∑

k=1

|k(I − P )P [ k2 ]f (y)|2kV (y, k)

m(y)

⎞

⎠

1/2

.

It is not difficult to check that |SL(f )| ≤ Sh(|f |). This implies that

‖SLf ‖L2 ≤ C‖Shf ‖L2 ≤ C‖f ‖L2 .

Moreover, it is proved in Proposition 5. 1 (Appendix) that SL is bounded on Lp(�) for all1 < p < ∞.

For 0 < p ≤ 1 and w ∈ A∞, the Hardy space HpL(�,w) is defined as the completion of

{f ∈ L2(�) : SLf ∈ Lp(�,w)}in the norm ‖f ‖Hp

L (�,w) = ‖SLf ‖Lp(�,w).

Similar to the Hardy spaces, we will show that the Hardy space HpL (�,w) can be char-

acterized in terms of atomic decompositions. We now consider the notion of (M,p, q,w)

atoms associated to the operator L.

Definition 3.1 Let w ∈ A∞ and 0 < p ≤ 1 < q ≤ ∞. We say that a function a ∈ Lq(�) isan (M,p, q,w)-atom associated to the operator L, if there exist a function b which belongsto Lq , and a ball B ⊂ � with rB ≥ 1 such that

(i) a = LMb;(ii) supp Lkb ⊂ B, k = 0, 1, . . . ,M;

(iii) ‖(r2BL)

kb‖Lq ≤ rMB V (B)1/qw(B)−1/p, k = 0, 1, . . . ,M .

By Holder’s inequality, it is easy to claim that an (M,p, q1, w)-atom is also an(M,p, q2, w)-atom whenever q1 ≥ q2.

Let w ∈ A∞ and 0 < p ≤ 1 < q ≤ ∞. We say that the given function f has an atomic(M,p, q,w)-representation if

f =∑

j

λj aj in L2(�)

where {λj }∞j=0 ∈ lp , each aj is an (M,p, q,w) atom. Then we define

Hp,qL,M,at (�,w) := {f : f has an atomic (M,p, q,w)-representation}

with the norm

‖f ‖Hp,q

L,M,at(�,w)= inf

⎧⎪⎨

⎪⎩

⎛

⎝∞∑

j=0

|λj |p⎞

⎠

1/p

: f =∑

j

λj aj is an atomic (M,p, q,w)-representation

⎫⎪⎬

⎪⎭.

The space Hp,qL,M,at (�,w) is then defined as the completion of Hp,q

L,M,at (�,w) with respectto this norm.

We would like to emphasize that in definition of Hp,qL,M,at (�,w), we only require that

the identity f = ∑j λj aj converges in L2(�), not in Lq(�). This brings some advan-

tages in proving the boundedness of singular integrals on these weighted Hardy spaces, seeSection 4.

Like the classical Hardy spaces, our weighted Hardy spaces can be characterized in termsof the atomic decomposition.

Theorem 3.2 Let 0 Dqwp

and q ≥ max{2, pr ′w}.

Theorem 3.2 will be a direct consequence of the following two propositions.

T.A. Bui

Proposition 3.3 Let 0 < p ≤ 1 and w ∈ A∞. Then for any f ∈ HpL (�,w) ∩ L2(�), there

exist a sequence of numbers {λj }j ∈ lp and a sequence of (M,p, q,w) atoms {aj }j withsome q ≥ 2 so that f = ∑

j λj aj in both L2(�) and HpL (�,w).

Proposition 3.4 Let 0 Dqwp

and q ≥ max{2, pr ′w}.

We first give proof of Proposition 3.3.

Proof Let f ∈ HpL (�,w) ∩ L2(�). By definition, we have F(·, ·) ∈ T p(�,w) ∩ T 2(�)

where F(y, k) = k(I − P )P [ k2 ]f (y) for all (y, k) ∈ � × N+. Hence by Proposition 2.6,there exist a sequence (λn)n∈N ∈ lp and a sequence of (w, p,∞) atoms (αn)n∈N so that

k(I − P )P [ k2 ]f (x) =∑

j≥0

λjαj (x, k), (x, k) ∈ � × N+ (8)

and ∑

j≥0

|λj |p ≤ C‖F‖pT p(�,w)

.

Now for any f ∈ L2, by Theorem 2.7,

f =∞∑

k=0

ck,M+1(I − P)M+1P kf =∞∑

k=0

ck,M+1

k + 1(I − P)MP k−[ k+1

2 ] ((k + 1)(I − P)P [ k+12 ]f

)

on L2(�), where the coefficients ck,M+1 are defined as in Theorem 2.7.Assume that each (w, p,∞) atom αj is supported in Qj for some ball Qj , j ∈ N.

Hence, αj (·, k) = 0 whenever k ≥ rBj. This along with (8) allows us to write

f =∑

j≥0

λj

rBj −1∑

k=0

ck,M+1

k + 1(I − P )MP k−[ k+1

2 ]αj (·, k + 1).

Let aj = LMbj where

bj =rBj −1∑

k=0

ck,M+1

k + 1P k−[ k+1

2 ]αj (·, k + 1).

Since the atom αj is supported in Qj , suppαj ⊂ Qj . This in combination with Remark 1.1yields that supp bj ⊂ 2Qj . Hence, applying Remark 1.1 again, we have

suppLkbj ⊂ B(xQj,M + 2rQj

) ⊂ (M + 2)Qj := Bj , k = 0, . . . ,M.

We now check that for each j , aj is an (M,p, q,w) atom for q ≥ 2. To do this, we fix0 ≤ N ≤ M . For any h ∈ Lq ′(�) with ‖h‖

Lq′ = 1 and supp h ⊂ Bj . We have

∑

x

LNbj (x)h(x)μ(x) =⎛

⎝∑

x

rBj −1∑

k=0

ck,M+1

k + 1(I − P )NP k−[ k+1

2 ]αj (x, k + 1)h(x)m(x)

⎞

⎠

≤⎛

⎝∑

x

rBj −1∑

k=0

∣∣∣ck,M+1

(k + 1)Nαj (x, k + 1)(k + 1)N (I − P )NP k−[ k+1

2 ]h(x)k + 1

∣∣∣m(x)

⎞

⎠ .

By (iii) of Lemma 2.7 and the fact that k + 1 ≤ rBj, we get that

∣∣∣ck,M+1

(k + 1)N

∣∣∣ ≤ rM−N

Bj.

This together with Proposition 2.4 gives

∑

x

LNbj (x)h(x)μ(x) ≤ rM−NBj

⎛

⎝∑

x

rBj −1∑

k=0

∣∣∣αj (x, k + 1)(k + 1)N (I − P )NP k−[ k+1

2 ]h(x)k + 1

∣∣∣μ(x)

⎞

⎠

≤ CrM−NBj

∑

x∈�A(αj )(x)Sk,L(h)(x)m(x)

≤ CrM−NBj

‖A(αj )‖Lq ‖Sk,L(h)‖Lq′

≤ CrM−NBj

‖αj ‖T q (�)‖SN,L(h)‖Lq′

where SN,L is the function defined by

SN,L(g)(x) =⎛

⎝∑

d(x,y)<k

∞∑

k=1

|(k + 1)N(I − P )NP k−[ k+12 ]g(y)|2

kV (x, k)m(y)

⎞

⎠

1/2

.

Since aj is a (w, p,∞) atom, and SN,L is bounded on Lr for all 1 < r < ∞, seeProposition 5.1, we have

∑

x

LNbj (x)h(x)μ(x)≤ CrM−NBj

V (Bj )1/qw(Bj )

−1/p‖h‖Lq′ .

This implies that‖LNbj‖Lq(Bj ) ≤ CrM−N

BjV (Bj )

1/qw(Bj )−1/p

for all N = 0, 1, . . . ,M .Hence, aj ′s are, up to a harmless multiplicative constant, (M,p, q,w) atoms associated

to the balls Bj for all j . This completes our proof.

We are now ready to give the proof of Proposition 3.4.

Proof Let f ∈ Hp,qL,M,at (�,w). Then we can write f = ∑

j λj aj and the series converges

in L2(�), where aj is an (M,p, q,w) atom for all j and∑

j |λj |p ≈ ‖f ‖pHp,qL,M,at (�,w)

.

Since SL is bounded on L2(�), SLf (x) ≤ ∑j |λj |SL(aj )(x) for a.e. x ∈ �. Therefore, to

show that f ∈ HpL (�,w), it suffices to claim that there exists a constant C so that

‖SLa‖Lp(�,w) ≤ C

for all (M,p, q,w) atoms a.Indeed, for an (M,p, q,w) atom a associated to the ball B , by Holder’s inequality, we

write

‖SLa‖pLp(�,w)=

∑

j≥0

⎛

⎝∑

x∈Sj (B)|SLa(x)|pw(x)m(x)

⎞

⎠

≤∑

j≥0

⎛

⎝∑

x∈Sj (B)|SLa(x)|qw(x)m(x)

⎞

⎠

p/q⎛

⎝∑

x∈Sj (B)[w(x)](q/p)′m(x)

⎞

⎠

1(q/p)′

.

T.A. Bui

Since q ≥ max{2, pr ′w}, w ∈ RH(q/p)′ . Hence, for j ≥ 0, we have

⎛

⎝∑

x∈Sj (B)[w(x)](q/p)′m(x)

⎞

⎠

1(q/p)′

≤ CV (2jB)−p/qw(2jB).

Therefore,

‖SLa‖pLp(�,w)≤ C

∑

j≥0

‖SLa‖pLq(Sj (B))V (2jB)−p/qw(2jB)

≤∑

j≥0

Ij .

For j = 0, 1, 2, 3, the Lq -boundedness of SL gives

Ij ≤ C‖a‖pLqV (2jB)−p/qw(2jB) ≤ C.

For j ≥ 4, we write

‖SLa(x)‖qLq(Sj (B))=

∑

x∈Sj (B)

∣∣∣

∑

d(y,x)<k

∞∑

k=1

|k(I − P )P [ k2 ]a(y)|2kV (y, k)

m(y)

∣∣∣q/2

m(x)

≤∑

x∈Sj (B)

∣∣∣

∑

d(y,x)<1

|(I − P )a(y)|2V (y,1)

m(y)

∣∣∣q/2

m(x)

+∑

x∈Sj (B)

∣∣∣

∑

d(y,x)<k

d(xB ,x)/4∑

k=2

|k(I − P )P [ k2 ]a(y)|2kV (y, k)

m(y)

∣∣∣q/2

m(x)

+∑

x∈Sj (B)

∣∣∣

∑

d(y,x)<k

∞∑

k=d(xB ,x)/4

|k(I − P )M+1P [ k2 ]b(y)|2kV (y, k)

m(y)

∣∣∣q/2

m(x)

:= Ej,1 + Ej,2 + Ej,3

where a = LMb.For the term E1, note that {y : d(x, y) < 1} = {x}. Moreover, for x ∈ Sj (B) with j ≥ 4,

we have

(I − P )a(x) = Pa(x).

This together with the condition (UE) implies that

Ej,1 =∑

x∈Sj (B)

∣∣∣|(I − P )a(x)|2

V (x,1)m(x)

∣∣∣q/2

m(x) =∑

x∈Sj (B)

∣∣∣|Pa(x)|2V (x,1)

m(x)

∣∣∣q/2

m(x)

≤∑

x∈Sj (B)|Pa(x)|qm(x) ≤ Ce−2j rB‖a‖qLq

≤ Ce−2j V (2jB)w(2jB)−q/p (since rB ≥ 1).

Concerning the term Ej,2, due to k ≈ [k/2] for all k ≥ 2 and (3), we write

Ej,2 ≤ C∑

x∈Sj (B)

∣∣∣∣∣∣

∑

d(y,x)<k

d(xB ,x)/4∑

k=2

∣∣∣∣∣∣

∑

z∈B

1

V (z,√k)

exp

(

−cd(z, y)2

k

)

|a(z)|m(z)

∣∣∣2 m(y)

kV (y, k)

∣∣∣q/2

m(x).

Note that in this situation d(z, y) ≈ 2j rB and

1

V (z,√k)

≤ 1

V (z, 2j rB)

(2j rB√

k

)D

≈ 1

V (2jB)

(2j rB√

k

)D

.

This yields that, for any n0 > 0,

∑

z∈B

1

V (z,√k)

exp

(

−cd(z, y)2

k

)

|a(z)|m(z)

≤ C∑

z∈B

1

V (2jB)

(2j rB√

k

)D

exp

(

−c(2j rB)2

k

)

|a(z)|m(z)

≤ C

(k

(2j rB)2

)n0 1

V (2jB)‖a‖L1

≤ C

(k

(2j rB)2

)n0 1

V (2jB)V (B)w(B)−1/p.

Therefore,

Ej,2 ≤ C∑

x∈Sj (B)

∣∣∣∣∣∣

∑

d(y,x)<k

d(xB ,x)/4∑

k=2

(k

(2j rB )2

)2n0 1

V (2jB)2V (B)2w(B)−2/p m(y)

kV (y, k)

∣∣∣∣∣∣

q/2

m(x)

≤ C∑

x∈Sj (B)

∣∣∣

(2j rB

(2j rB )2

)2n0 1

V (2jB)2V (B)2w(B)−2/p

∣∣∣q/2

m(x)

(

sinced(x, xB)

4≈ 2j rB

)

≤ C(2j rB)−qn0V (2jB)

(V (B)

V (2jB)

)q

w(B)−q/p

≤ C2−jq(n0−Dqw/p)V (2jB)w(2jB)−q/p.

For the last term Ej,3, using (3), we have

Ej,3 ≤ C∑

x∈Sj (B)

∣∣∣∣∣∣∣

∑

d(y,x)<k

∞∑

k= d(xB,x)

4

|kM+1(I − P )M+1P [ k2 ]b(y)|2k2M+1V (x, k)

m(y)

∣∣∣∣∣∣∣

q/2

m(x)

≤ C∑

x∈Sj (B)

∣∣∣∣∣∣∣

∞∑

k= d(xB ,x)

4

∑

y∈�

|kM+1(I − P )M+1P [ k2 ]b(y)|2k2M+1V (2jB)

m(y)

∣∣∣∣∣∣∣

q/2

m(x)

≤ C∑

x∈Sj (B)

∣∣∣

‖b‖2L2

d(x, xB)2MV (2jB)

∣∣∣q/2

m(x)

≤ C2−jMqV (2jB)

(V (B)

V (2jB)

)q/2

w(B)−q/p.

Since rB ≥ 1, we conclude that

Ej,3 ≤ C2−jq(M−Dqw/p)V (2jB)w(2jB)−q/p.

T.A. Bui

From the estimates of Ej,1, Ej,2 and Ej,3, we obtain that

‖SLa‖pLp(�,w)≤ C + C

∑

j≥4

[e−2jp/q + 2−jp(n0−Dqw/p) + 2−jp(M−Dqw/p)

]

≤ C

as long as n0 >Dqwp

and M >Dqwp

. This completes our proof.

We will end this section by considering the notion of (M,p, q,w, ε)-molecules whichwill be useful in the sequel.

Suppose ε > 0, w ∈ A∞ and 0 < p ≤ 1 < q ≤ ∞. We say that a function α ∈ Lq(�) isan (M,p, q,w, ε)-molecule associated to the operator L, if there exists a function b whichbelongs to Lq , and a ball B ⊂ � with rB ≥ 1 such that

(i) α = LMb;(ii) ‖(r2

BL)kb‖Lq(Sj (B)) ≤ 2−jεrMB V (B)1/qw(B)−1/p, k = 0, 1, . . . ,M .

The following result give the sufficient condition so that an (M,p, q,w, ε) moleculebelongs to the weighted Hardy spaces.

Proposition 3.5 Let w ∈ A∞ and 0 Dqwp

, q ≥ max{2, pr ′w} and

ε > D(qwp

− 1q). Then there exists a constant C > 0 so that

‖α‖HpL (�,w) ≤ C

for all (M,p, q,w, ε) molecules α.

Proof By definition, we need only to prove that there exists C > 0 such that

‖SLα‖Lp(�,w) ≤ C

for all (M,p, q,w, ε) molecules α.Assume that α is an (M,p, q,w, ε) molecule associated to some ball B with rB ≥ 1.

We set αj = αχSj (B) for all j ≥ 0. Then we can write

‖SLα‖pLp(�,w)≤

∑

j

‖SLαj‖pLp(�,w):=

∑

j

Ij .

For each j , by Holder’s inequality and the fact that w ∈ RH(q/p)′ , we write

Ij =∑

k≥0

‖SLαj‖pLp(Sk(2j B),w)

≤ C‖SLαj‖pLq(Sk(2j B))V (2j+kB)−p/qw(2j+kB)

:=∑

k≥0

Ijk.

For k = 0, 1, 2, 3, using the Lq -boundedness of SL to give

Ijk ≤ C‖αj‖pLqV (2j+kB)−p/qw(2j+kB)

≤ C2−εjpV (B)p/qw(B)−1V (2j+kB)−p/qw(2j+kB).

Due to Lemma 2.2, we get

V (B)

V (2j+kB)≤ C

(w(B)

w(2j+kB)

)1/qw.

Therefore,

Ijk ≤ C2−εjp

(w(2j+kB)

w(B)

)1− pqqw

≤ C2−jp(ε−D(qwp − 1

q )).

Hence,∑

j≥0

∑

k=0,1,2,3

Ijk ≤ C

as long as ε > D(qwp

− 1q).

It remains to show that∑

j≥0

∑

k≥4

Ijk ≤ C.

To do this, we can adapt the argument in Proposition 3.4 to our situation with the minormodification and hence we leave to the interested reader. The proof is completed.

4 Boundedness of Singular Integrals on Hardy Spaces HpL(�,w)

The weighted Lp norm inequalities for some singular integrals on graphs such as the squarefunctions, Riesz transforms and spectral multipliers were studied in [5, 8] for p ≥ 1. Theaim of this section is to study the weighted norm inequalities for these integral operators onthe scale of weighted Hardy spaces Hp

L (�,w) with 0 < p ≤ 1.

4.1 Square Functions

Recall that for N ≥ 1, the square function GL,N is defined by

GL,Nf (x) =( ∞∑

k=1

|kN(I − P )NP kf (x)|2k

)1/2

Note that when N = 1, the operator GL,1 is bounded on Lp(�), see [7]. The weightedestimates for GL,1 were investigated in [8]. In the general case N ≥ 1, it will be proved inProposition 5.1 that GL,N is bounded on Lp(�) for all 1 < p < ∞. In the following, wewill investigate the boundedness of GL,N on Hardy spaces Hp

L (�,w) with 0 < p ≤ 1 andw ∈ A∞.

Theorem 4.1 For M ≥ 1, the square functionGL,N is bounded fromHpL (�,w) to Lp(�,w)

for all 0 < p ≤ 1 and w ∈ A∞.

Proof Fix p ∈ (0, 1] and w ∈ A∞. Since w ∈ A∞, we can pick q ≥ 2 so that w ∈RH(q/p)′ . Let f ∈ H

pL (�,w)∩L2(�). Then by Proposition 3.3, we can write f = ∑

j λj aj

in L2(�) so that ‖f ‖pH

pL (�,w)

≈ ∑j |λj |p, where aj is an (M,p, q,w) atom associated to

T.A. Bui

the ball Bj with rBj≥ 1 for each j . From the fact that GL,N is bounded on L2(�), we get

that

GL,Nf (x) ≤∑

j

|λj |GL,Naj (x), for a.e. x ∈ �.

Hence, to complete the proof, we need only to show that there exists C > 0 so that

‖GL,Na‖Lp(�,w) ≤ C

for all (M,p, q,w) atoms a = LMb with M >Dqwp

.Indeed, we write, by Holder’s inequality,

‖GL,Na‖pLp(�,w)=

∑

j≥0

⎛

⎝∑

x∈Sj (B)|GL,Na(x)|pw(x)m(x)

⎞

⎠

≤∑

j≥0

⎛

⎝∑

x∈Sj (B)|GL,Na(x)|qm(x)

⎞

⎠

p/q ⎛

⎝∑

x∈Sj (B)w(x)(q/p)

′m(x)

⎞

⎠

1(q/p)′

.

The fact that w ∈ RH(q/p)′ implies

⎛

⎝∑

x∈Sj (B)w(x)(q/p)

′m(x)

⎞

⎠

1(q/p)′

≤ CV (2jB)−p/qw(2jB).

Hence,

‖GL,Na‖pLp(�,w) ≤ C∑

j≥0

‖GL,Na‖pLq(Sj (B))V (2jB)−p/qw(2jB)

≤ C∑

j≥0

Ij .

Using Lq -boundedness of GL,N , we arrive at

Ij ≤ C‖a‖pLqV (2jB)−p/qw(2jB) ≤ CV (B)w(B)−1V (2jB)−p/qw(2jB) ≤ C

for j = 0, 1, 2, 3, 4.For j ≥ 4, we write

‖GL,Na‖qLq(Sj (B))=

∑

x∈Sj (B)

( ∞∑

k=1

|kN(I − P )NP ka(x)|2k

)q/2

m(x)

≤ C∑

x∈Sj (B)

⎛

⎝r2B∑

k=1

|kN(I − P )NP ka(x)|2k

⎞

⎠

q/2

m(x)

+ C∑

x∈Sj (B)

⎛

⎜⎝

∞∑

k=r2B

|kN(I − P )M+NP kb(x)|2k

⎞

⎟⎠

q/2

m(x)

:= Ij,1 + Ij,2.


Using Minkowski’s inequality, we obtain that

Ij,1 ≤ C

⎛

⎝r2B∑

k=1

‖kN(I − P )NP ka(x)‖2Lq(Sj (B))

k

⎞

⎠

q/2

.

Note that since the associated kernel of kM(I − P )MP k satisfies (3), we have, for anyn0 > 0,

‖kN(I − P )NP ka(x)‖Lq(Sj (B)) ≤ C‖a‖Lq exp

(

−c(2j rB)2

k

)

≤ C

( √k

2j rB

)n0

‖a‖Lq .

This implies that

Ij,1 ≤ C‖a‖qLq

⎛

⎝r2B∑

k=1

kn0

k(2j rB)2n0

⎞

⎠

q/2

≤ C2−jqn0V (B)w(B)−q/p.

We now take care of Ij,2. We first rewrite, by using Minkowski’s inequality,

Ij,2 =∑

x∈Sj (B)

⎛

⎜⎝

∞∑

k=r2B

|kM+N(I − P )M+NP kb(x)|2kM+1

⎞

⎟⎠

q/2

m(x)

≤⎛

⎜⎝

∞∑

k=r2B

‖kM+N(I − P )M+NP kb(x)‖2Lq(Sj (B))

kM+1

⎞

⎟⎠

q/2

.

(9)

Using (3) again, we have

‖kM+N(I − P )M+NP kb(x)‖Lq(Sj (B)) ≤ C‖b‖Lq exp

(

−c(2j rB)2

k

)

≤ ‖b‖Lq

( √k

2j rB

)M/2

.

This together with (9) yields

Ij,2 ≤ C‖b‖qLq

⎛

⎜⎝

∞∑

k=r2B

1

kM/2+1(2j rB)M

⎞

⎟⎠

q/2

≤ C2−jMq/2V (B)w(B)−q/p.

T.A. Bui

From the estimates of Ij,2 and Ij,2, we conclude that

‖GL,Na‖pLp(�,w)≤ C + C

∑

j≥4

[2−jpn0V (B)p/qw(B)−1V (2jB)−p/qw(2jB)

+ 2−jMp/2V (B)p/qw(B)−1V (2jB)−p/qw(2jB)]

≤ C + C∑

j≥4

[2−jp(n0−Dqw/p) + 2−jp(M/2−Dqw/p)

]

≤ C

provided n0 >Dqwp

and M >Dqwp


4.2 Riesz Transforms

Recall that the operator “length of the gradient” ∇ is defined by

∇f (x) =⎛

⎝1

2

∑

y∈�p(x, y)|f (x)− f (y)|2

⎞

⎠

1/2

for any function f on � and x ∈ �. In this section, we consider the boundedness of Riesztransforms ∇(I − P )−1/2 on Hardy spaces Hp

L (�,w). Note that in this situation, the Riesztransform is a nonnegative sublinear operator, not a linear operator. It was proved in [30] thatunder the assumptions (D), (Sα) and (UE), the Riesz transform ∇(I − P )−1/2 is boundedon Lp(�) for 1 < p ≤ 2. Moreover, the Riesz transform is of weak-type (1, 1). We set

q = sup{q : ∇(I − P )−1/2 is bounded on Lq(�)}.

Obviously, in our setting, q ≥ 2.The main result in this section is formulated by the following theorem.

Theorem 4.2 The Riesz transform ∇(I − P )−1/2 is bounded from HpL (�,w) to Lp(�,w)

for all 0 0 so that

(∑

x∈�|∇xpn,k(x, y)|re

γ d(x,y)2

k μ(x)

)1/r

≤ Cμ(y)

kn+1/2V (y,√k)1−1/r

for all k ∈ N+ and y ∈ �.

In the case n = 0, the estimate in Lemma 4.3 was obtained in [7, Lemma 3.1] whoseproof may be adapted to this situation with minor modifications. Hence, we omit detailshere.

We are ready to give the proof of Theorem 4.2.

Proof of Theorem 4.2 Fix p ∈ (0,1] and w ∈ RH(q/p)′ . Then we can pick 2 ≤ q < q

if q > 2, or q = 2 if q = 2 so that w ∈ RH(q/p)′ . By the argument used in the proof ofTheorem 4.1, we need to show that there exists a constant C > 0 so that

‖∇(I − P )−1/2a‖Lp(�,w) ≤ C

for any (M,p, q,w) atoms a = LMb associated to some ball B with M >Dqwp

.

Denote T = ∇(I −P )−1/2. Like the first step in the proof of Theorem 4.1, we can write

‖T a‖pLp(�,w)

≤ C∑

j≥0

‖T a‖pLq(Sj (B))

V (2jB)−p/qw(2jB)

≤ C∑

j≥0

Ij .

The Lq -boundedness of T implies

Ij ≤ C‖a‖pLqV (2jB)−p/qw(2jB) ≤ CV (B)p/qw(B)−1V (2jB)−p/qw(2jB) ≤ C

for j = 0, 1, 2, 3, 4.For j ≥ 4, as in [7], the following identity holds

∇(I − P )−1/2f = ∇⎛

⎝∑

k≥0

akPk

⎞

⎠ f in L2(�)

for all f ∈ E := {f ∈ L2 : f = L1/2g for some g ∈ L2}, where 0 ≤ ak ≤ 1√k, k ≥ 1.

Therefore, if a is an (M,p, q,w) atom with q ≥ 2, then b ∈ L2. Hence we can write,a = L1/2g with g = LM−1/2b. Since the spectrum of L is contained in [0, 2] and L isnon-negative self adjoint, by the spectral theory, g = LM−1/2a ∈ L2. This concludes thata ∈ E. For this reason, we can write, for x ∈ Sj (B), j ≥ 4

T a(x) =∑

k≥0

ak∇P ka(x).

Hence, for j ≥ 4,

Ij =∥∥∥∑

k≥3

ak∇P ka

∥∥∥Lq(Sj (B))

≤∥∥∥

r2B∑

k=3

ak∇P ka

∥∥∥Lq (Sj (B))

+∥∥∥∑

k≥r2B

ak∇(I − P )MP ka

∥∥∥Lq (Sj (B))

≤∥∥∥∥∥∥

r2B∑

k=3

ak

⎛

⎝∑

y∈B∇xpk(x, y)a(y)

⎞

⎠

∥∥∥∥∥∥Lq(Sj (B),m(x))

+

∥∥∥∥∥∥∥

∑

k≥r2B

ak

⎛

⎝∑

y∈B∇x pM,k(x, y)a(y)

⎞

⎠

∥∥∥∥∥∥∥Lq(Sj (B),m(x))

= Ij,1 + Ij,2.

where pM,k(x, y) is a associated kernel to (I − P )MP k .

T.A. Bui

For the term Ij,1, we use the estimate in Lemma 4.3 and Minkowski’s inequality to obtainthat

Ij,1 ≤r2B∑

k=3

ak

∥∥∥

⎛

⎝∑

y∈B∇xpk(x, y)a(y)

⎞

⎠∥∥∥Lq(Sj (B),m(x))

≤r2B∑

k=3

ak∑

y∈B‖∇xpk(x, y)‖Lq(Sj (B),m(x))a(y)

≤r2B∑

k=3

ak supy∈B

C√kV (y,

√k)1−1/q

exp

(

−cγ22j r2

B

k

)

‖a‖L1 .

We have, for any n0 > 0,

supy∈B

C√kV (y,

√k)1−1/q

exp

(

−cγ22j r2

B

k

)

≤ C√kV (2jB)1−1/q

( √k

2j rB

)n0

.

This together with the fact that 0 ≤ ak ≤ 1/√k gives, for any n0 > 0,

Ij,1 ≤ C

r2B∑

k=3

C

k

( √k

2j rB

)n0

V (2jB)1/qw(B)−1/p

≤ C2−j (n0−Dqw/p)V (2jB)1/qw(2jB)−1/p.

Similarly, we obtain

Ij,2 ≤∑

k>r2B

ak

∥∥∥∑

y∈B∇x pM,k(x, y)b(y)

∥∥∥L2(Sj (B),m(x))

≤∑

k>r2B

akC

kM+1/2V (y,√k)1−1/q

exp

(

−cγ22j r2

B

k

)

‖b‖L1

≤∑

k>r2B

C

kM+1V (2jB)1−1/q

( √k

2j rB

)M

rMB V (B)w(B)−1/p

≤ C2−j (M−Dqw/p)V (2jB)w(2jB)−1/p.

These two estimates Ij,1 and Ij,2 tell us that

‖T a‖Lp(�,w) ≤ C + C∑

j≥4

2−jp(n0−Dqw/p) + C∑

j≥4

2−jp(M−Dqw/p) ≤ C

as long as M >Dqwp

and n0 >Dqwp


4.3 Spectral Multipliers of the Discrete Laplacian

Since ‖L‖L2→L2 = ‖I − P ‖L2→L2 ≤ 2, it admits the spectral resolution L = ∫ 20 λdEL(λ).

Let F : [0, 2] → C be a bounded Borel measurable function, we define the operator

F(L) =∫ 2

0F(λ)dEL(λ)

which is bounded on L2(�).Let s > 0. For each function f , we define

‖f ‖Cs =[s]∑

k=0

‖f (k)‖∞ +Ms(f )

where

Ms(f ) = sup{ |f ([s])(x + t)− f ([s])(x)|

t s−[s] : t > 0, x ∈ R

}.

We setCs(R) := {f : ‖f ‖Cs < ∞}.

It turn out that each function in Cs can be approximated by a polynomial. More precisely,we have we the following approximation result. See for example [1].

Lemma 4.4 Let s > 0 and f ∈ Cs(R) with supp f ⊂ [−4, 4]. Then there exists c > 0such that for all k ∈ N+, there is a polynomial Q with deg(Q) ≤ k such that

‖f −Q‖L∞([−4,4]) ≤ cMs(f )

ks.

Note that when p > 1, it was proved in [4] that F(L) is bounded on Lp for all 1 D/2. When 0 D(

1p− 1

2

)and

Poincare inequality. In this section, without the Poincare inequality assumption, we establishboundedness of F(L) on Hardy spaces Hp

L (�,w) for 0 D(

1p− 1

2

). If F be a bounded Borel measurable

function on [0, 2] satisfying the following condition

supt>0

‖η(λ)F (tλ)‖Cs < ∞, (10)

where η ∈ C∞c (0,∞) is a fixed function not identically zero, then F(L) is bounded on

HpL (�,w) for all w ∈ RH(2/p)′ ∩ Aq with 1 ≤ q < p( s

D+ 1

2 ).

Proof Fix w ∈ Aq ∩RH(2/p)′ . Since 1 ≤ q < p(

sD+ 1

2

), s > D

(qp− 1

2

). Since F(L) is

bounded on L2, due to Proposition 3.5, it suffices to claim that for each (2M,p, 2, w) atoma = L2Mb associated to the ball B , F(L)a is a multiple of an (M,p, 2, w, ε) molecule

with ε = s ′ where s is a number satisfying s > s ′ > D(qp− 1

2

)and M >

Dqp

. If we write

F(L)a = LM(F(L)LMb) then we need to claim that, for all k = 0, . . . ,M ,

‖Lk(F (L)LMb)‖L2(Sj (B))≤ C2−s′j rM−k

B V (B)1/2w(B)−1/p for all k = 0, 1, 2, . . ..(11)

T.A. Bui

Using the L2-boundedness of F(L), it is easy to check that (11) holds for j = 0, 1, 2, 3. Itremains to claim (11) for j ≥ 4. Following the decomposition as in [1, 28], we write

F(λ) =∑

�≥1

F(λ)θ(λ)ϕ(2�λ)+ (1 − θ(λ))F (λ) :=∑

�≥1

F�(λ)+ F0(λ)

where 0 ≤ θ ∈ C∞c (R) and ϕ ∈ C∞

c satisfy

θ(λ) = 1 for λ ∈ [−1/4, 1/4] and θ(λ) = 0 for λ /∈ [−1/2, 1/2]and

suppϕ ⊂ (1/4, 3/4),∑

�≥1

ϕ(2�−1λ) = 1, λ ∈ (0,1/2].

It is easy to check that supp F0 ⊂ [1/2, 2] and supp F� ⊂ [2−(�+1), 2−�] for � ≥ 1. Then,we have

‖Lk(F (L)LMb‖L2(Sj (B))≤

∑

�≥0

‖LM+kF�(L)b‖L2(Sj (B)).

For � = 0 and j ≥ 4, we have

‖LM+k(F0(L)b)‖L2(Sj (B))= ‖F0(L)(L

M+kb)‖L2(Sj (B))

=∥∥∥∥∥∥

∑

y∈BKF0(L)(·, y)LM+kb(y)

∥∥∥∥∥∥L2(Sj (B))

≤ ‖LM+kb(y)‖L1 supy∈B

‖KF0(L)(·, y)‖L2(Sj (B))

≤ rM−kB V (B)w(B)−1/p sup

y∈B‖KF0(L)(·, y)‖L2(Sj (B))

.

(12)

Note that KF0(

√L)(·, y) = (F0(L)p0(·, y))(x). Applying Lemma 4.4, there exists a

polynomial Q1 so that

deg(Q1) ≤ [2j−2rB ] and ‖F0 −Q1‖L∞([1/2,2]) ≤ cMs(F0)[2j−2rB ]−s ≈ 2−sj r−sB .

Since x ∈ Sj (B), y ∈ B , by Remark 1.1,

‖KF0(L)(·, y)‖L2(Sj (B))= ‖[(F0(L)−Q1(L))p0(·, y)](x)‖L2(Sj (B))

≤ C‖F0 −Q1‖L∞([1/2,2])‖p0(·, y)‖L2

≤ C2−sj r−s

B

V (y, 1)1/2≤ c

2−sj r−(s−D/2)B

V (B)1/2.

This together with (12) gives

‖LM+kF0(L)b‖L2(Sj (B))≤ CrM−k

B 2−sj r−(s−D/2)B V (B)1/2w(B)−1/p

≤ CrM−kB 2−s′jV (B)1/2w(B)−1/p.

Hence, for k = 0, 1, . . . ,M and j ≥ 4,

‖LM+kF0(L)b‖L2(Sj (B))≤ CrM−k

B 2−s′jV (B)1/2w(B)−1/p. (13)

For � ≥ 1, set

F�(λ) = (1 − λ)−2�F�(λ).

Then we have‖F�‖∞ ≤ c‖F�‖∞ and F�(L) = F�(L)P

2� .

We write, for k = 0, 1, . . . ,M and j ≥ 4,∑

�≥1

‖LM+kF�(L)b‖L2(Sj (B))≤

∑

�:2�≥2j rB

‖F�(L)(LM+kP 2�b)‖L2(Sj (B))

+∑

�:2�<2j rB

‖F�(L)LM+kb‖L2(Sj (B))

:= I1 + I2.

Thanks to (3), we have

I1 ≤ C∑

�:2�≥2j rB

‖LM+kP 2�b‖L2 ≤∑

�:2�≥2j rB

c

2�(M+k)‖b‖L2

≤ C∑

�:2�≥2j rB

(2j rB

2l

)M+k

2−j (M+k)rM−kB V (B)1/2w(B)−1/p

≤ C2−js′rM−kB V (B)1/2w(B)−1/p.

To estimate I2, for each � with 2� < 2j rB , we can apply Lemma 4.4 to pick a polynomialQ2 of degree at most 2j rB − rB − 2 so that

‖F� −Q2‖L∞([2−(�+1),2−�]) ≤ [2j rB − rB − 2]−s ≈ (2j rB)−s .

This in combination with Remark 1.1 implies

I2 ≤ c∑

�:2�<2j rB

‖(F� −Q2)LM+kb‖L2 ≤ c

∑

�:2�<2j rB

‖F� −Q2‖L∞([2−(�+1),2−�])‖LM+kb‖L2

≤ C∑

�:2�<2j rB

(2j rB )−srM−k

B V (B)1/2w(B)−1/p

≤ C(2j rB)−s+s′ log2(2

j rB)× (2j rB)−s′rM−k

B V (B)1/2w(B)−1/p

≤ C2−js′ rM−kB V (B)1/2w(B)−1/p

where in the last inequality we use the fact that x−α log2 x ≤ C for all x ≥ 1 and α > 0.From these three estimates of I1, I2 and (13), we conclude that F(L)a is a multiple of

an (M,p, 2, w, s ′) molecule. Hence our proof is complete.

Acknowledgments The author was supported by ARC (Australian Research Council). He would like tothank the referee for his/her useful comments to improve the paper.

Appendix

In this section, for N ∈ N, we consider the following the square function and discrete areafunctions defined by

GL,Nf (x) =( ∞∑

k=1

|kN(I − P )NP kf (x)|2k

)1/2

,

SL,N(f )(x) =⎛

⎝∑

d(x,y)<k

∞∑

k=1

|kN(I − P )NP [k]f (y)|2kV (x, k)

m(y)

⎞

⎠

1/2

.

T.A. Bui

and

SL,N(f )(x) =⎛

⎝∑

d(x,y)<k

∞∑

k=1

|(k + 1)N(I − P )NP k−[ k+12 ]f (y)|2

kV (x, k)m(y)

⎞

⎠

1/2

,

By using the similar arguments to those in [7, p. 286] and Section 3, it can be verified thatthe operators GL,N , SL,N and SL,N are bounded on L2(�). In the following proposition, wewill show that these operators are bounded on Lp(�) for all 1 < p < ∞.

Proposition 4.6 Let N ∈ N. Then

(i) GL,N is bounded on Lp(�) for all 1 < p < ∞;(ii) SL,N and SL,N (g) are bounded on Lp(�) for all 1 2, respectively.

Theorem 4.7 Let p0 ∈ [1, 2). Suppose that T is a sublinear operator of strong type (2, 2),and let At, t > 0, be a family of linear operators acting on L2(�). Assume for j ≥ 3

⎛

⎝1

V (2jB)

∑

x∈Sj (B)|T (I − ArB )f (x)|2m(x)

⎞

⎠

1/2

≤ α(j)

(1

V (B)

∑

x∈B|f (x)|p0m(x)

)1/p0

(14)and for j ≥ 3

⎛

⎝1

V (2jB)

∑

x∈Sj (B)|ArBf (x)|2m(x)

⎞

⎠

1/2

≤ α(j)

(1

V (B)

∑

x∈B|f (x)|p0m(x)

)1/p0

(15)

for all balls B and all f supported in B . If∑

j α(j)2Dj < ∞, then T is of weak type

(p0, p0).

Theorem 4.8 Let p0 ∈ (2,∞]. Suppose that T is a sublinear operator acting on L2, andlet At, t > 0, be a family of linear operators acting on L2(�). Assume

(1

V (B)

∑

x∈B|T (I −ArB )f (x)|2m(x)

)1/2

≤ C(M(|f |2))1/2(y) (16)

and(

1

V (B)

∑

x∈B|T ArB f (x)|p0m(x)

)1/p0

≤ C(M(|Tf |2))1/2(y), (17)

for all f ∈ L2(�), all balls B and all y ∈ B . Then T is of strong type (p0, p0).

We are ready to give the proof for Proposition 5.1.

Proof of Proposition 5.1 (i) We adapt some ideas in [2] to our present situation to considertwo cases.

Case 1: 1 D/2.Since ArB satisfies Gaussian upper bound as in (UE), it is easy to verify that the condition(15) holds. Hence, it remains to check that for j ≥ 3

⎛

⎝ 1

V (2jB)

∑

x∈Sj (B)

|GL,N(I − P [r2B ])nf (x)|2m(x)

⎞

⎠

1/2

≤ 2−2jn

(1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

(18)

for all balls B and all f supported in B .We can assume that rB ≥ 1. Then we can write

(1

V (2j B)

∑

x∈Sj (B)|GL,N (I − P [r2

B ])nf (x)|2m(x)

⎞

⎠

1/2

=⎛

⎝1

V (2j B)

∑

x∈Sj (B)

( ∞∑

k=1

|kN(I − P )N+nP k(I + P + . . .+ P [r2B ]−1)nf (x)|2

k

)

m(x)

⎞

⎠

1/2

≤ C sup0≤�≤n([r2

B ]−1)

⎛

⎝ 1

V (2j B)

∑

x∈Sj (B)

⎛

⎝(2j rB )2∑

k=1

|kN (I − P )N+nP k+�f (x)|2k

⎞

⎠m(x)

⎞

⎠

1/2

+ C sup0≤�≤n([r2

B ]−1)

⎛

⎝ 1

V (2j B)

∑

x∈Sj (B)

⎛

⎝∞∑

k=(2j rB )2

|kN(I − P )N+nP k+�f (x)|2k

⎞

⎠m(x)

⎞

⎠

1/2

:= I + I I.

For the first term I , we have

I ≤ C sup0≤�≤n([r2

B ]−1)

⎛

⎝ 1

V (2jB)

(2j rB )2∑

k=1

‖kN(k + �)N+n(I − P )N+nP k+�f ‖2L2(Sj (B))

k(k + �)2N+2n

⎞

⎠

1/2

.

Since the kernel of (I − P )N+nP k+� satisfies (3), it can be verified that⎛

⎝1

V (2jB)

∑

x∈Sj (B)|(k + �)N+n(I − P )N+nP kf (x)|2m(x)

)1/2

≤ C

(k + �

(2j rB)2

)N+n(

1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

Therefore,

I ≤ C

⎛

⎝(2j rB )2∑

k=1

k2N

k(2j rB)4(N+n)

⎞

⎠

1/2 (1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

≤ C2−2jn

(1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

.

T.A. Bui

Likewise, for the second term II , we have⎛

⎝ 1

V (2j B)

∑

x∈Sj (B)|(k + �)N+n(I − P )N+nP kf (x)|2m(x)

⎞

⎠

1/2

≤ C

(1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

.

Hence, using the fact that k + � ≈ k, we have

II ≤ C

⎛

⎝∞∑

k=(2j rB )2

1

k2n+1

⎞

⎠

1/2 (1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

≤ C2−2jn

(1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

.

So, (18) holds and hence GL,N is bounded on Lp(�) for all 1 D/2. By similar argument as in the proof of case 1, we can show that

(16) holds. Hence, to complete the proof, we need only to show that

(1

V (B)

∑

x∈B|GL,N(I − (I − P [r2

B ])n)f (x)|pm(x)

)1/p

≤ C(M(|Tf |2))1/2(y), (19)

for all f ∈ L2, all balls B and all y ∈ B .Since I − (I − P [r2

B ])n = ∑n�=1 c�P

�[r2B ], we need to check that, for all � = 1, . . . , n,

there holds(

1

V (B)

∑

x∈B|GL,NP

�[r2B ]f (x)|pm(x)

)1/p

≤ C(M(|Tf |2))1/2(y), (20)

for all f ∈ L2, all balls B and all y ∈ B .Indeed, we write, by Minkowski’s inequality,

(1

V (B)

∑

x∈B|GL,NP

�[r2B ]f (x)|pm(x)

)1/p

≤⎛

⎝ 1

V (B)

∑

x∈B

( ∞∑

k=1

|kN(I − P )NP k+�[r2B ]f (x)|2

k

)p/2

m(x)

⎞

⎠

1/p

≤∑

j≥0

⎛

⎝ 1

V (B)

∞∑

k=1

‖kNP �[r2B ]((I − P )NP kf χSj (B))‖2

Lp(B)

k

⎞

⎠

1/2

Since P �[r2B ] satisfies (UE), we have, for any ε > 0,

1

V (B)‖P �[r2

B ]((I − P )NP kfχSj (B))‖2Lp(B)V (2j B) ≤ C2−2jε

V (2j B)

⎛

⎝∑

x∈Sj (B)|(I − P )NP kf (x)|2m(x)

⎞

⎠

Hence,(

1

V (B)

∑

x∈B|GL,NP

�[r2B ]f (x)|pm(x)

)1/p

≤ C∑

j≥0

2−jε

(1

V (2jB)

∑

x∈B|GL,Nf (x)|2m(x)

)1/2

≤ CM(GL,Nf )1/2(y)

for all y ∈ B .Hence, this completes the proof of (i).(ii) Since the proofs for SL,N and SL,N are the same. We need only to give the proof for

SL,N . To prove (ii), we also consider two cases.Case 1: 1 D2 + D

2 (1p− 1

2 ). Similar to the argument used in the proof of (i), we need only to claim thatfor j ≥ 3⎛

⎝ 1

V (2jB)

∑

x∈Sj (B)

|SL,N (I − P [r2B ])nf (x)|2m(x)

⎞

⎠

1/2

≤ C2−2jn

(1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

(21)

for all balls B and all f supported in B .Obviously, (21) holds for rB < 1. For rB ≥ 1, we can write

(1

V (2j B)

∑

x∈Sj (B)|SL,N (I − P [r2

B ])nf (x)|2m(x)

⎞

⎠

1/2

=⎛

⎝1

V (2j B)

∑

x∈Sj (B)

⎛

⎝∑

d(x,y)<k

∞∑

k=1

|kN(I − P )NP [k](I − P [r2B ])nf (y)|2

kV (x, k)m(y)

⎞

⎠m(x)

⎞

⎠

1/2

≤ C

⎛

⎜⎝

1

V (2j B)

∑

x∈Sj (B)

⎛

⎜⎝

∑

d(x,y)<k

d(x,xB )

4∑

k=1

. . .m(y)

⎞

⎟⎠m(x)

⎞

⎟⎠

1/2

+ C

⎛

⎜⎝

1

V (2j B)

∑

x∈Sj (B)

⎛

⎜⎝

∑

d(x,y)<k

∞∑

k= d(x,xB )

4

. . .m(y)

⎞

⎟⎠m(x)

⎞

⎟⎠

1/2

:= I + I I.

Let us take care of the term I first. We write

I ≤ C sup1≤�≤n[r2

B ]−1

⎛

⎝1

V (2j B)

∑

x∈Sj (B)

⎛

⎝∑

d(x,y)<k

d(x,xB )/4∑

k=1

|kN(I − P )N+nP [k]+�f (y)|2kV (x, k)

m(y)

⎞

⎠m(x)

⎞

⎠

1/2

.

Set F(B) := {z : d(z, Sj (B)) ≤ 2j−2rB}. Then d(F (B),B) ≈ 2j rB and m(F(B)) ≈V (2jB). Moreover,

I ≤ C sup1≤�≤n([r2

B ]−1)

⎛

⎜⎝

1

V (2j B)

∑

d(x,y)<k

⎛

⎜⎝

∑

y∈F(B)

d(x,xB )

4∑

k=1

|kN(I − P )N+nP [k]+�f (y)|2kV (x, k)

m(y)

⎞

⎟⎠m(x)

⎞

⎟⎠

1/2

≤ C sup1≤�≤n([r2

B ]−1)

⎛

⎝ 1

V (2j B)

⎛

⎝∑

y∈F(B)

2j rB∑

k=1

|kN(I − P )N+nP [k]+�f (y)|2k

m(y)

⎞

⎠

⎞

⎠

1/2

.

T.A. Bui

Since the kernel of (I − P )N+nP [k]+� satisfies (3), we have

supy∈F(B)

sup1≤�≤n[r2

B ]−1

|(I − P )N+nP [k]+�f (y)|

≤ sup1≤�≤n[r2

B ]−1

C

(k + �)N+nV (2jB)exp

(

−c(2j rB)2

k + �

)

‖f ‖L1

≤ C

(2j rB)2(N+n)

(1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

.

Therefore,

I ≤⎛

⎝ 1

V (2jB)

∑

y∈F(B)

2j rB∑

k=1

k2N

k(2j rB)4(N+n)m(y)

⎞

⎠

1/2 (1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

≤ C2−2jn

(1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

.

Let us take care of the second term II . We can write

I I ≤ C sup1≤�≤n([r2

B ]−1)

⎛

⎜⎝

1

V (2j B)

∑

x∈Sj (B)

⎛

⎜⎝

∑

d(x,y)<k

∞∑

k= d(x,xB )

4

|kN(I − P )N+nP [k]+�f (y)|2kV (x, k)

m(y)

⎞

⎟⎠m(x)

⎞

⎟⎠

1/2

≤ C sup1≤�≤n([r2

B ]−1)

⎛

⎝ 1

V (2j B)

∑

y∈�

∞∑

k=2j−3rB

∑

d(x,y)<k

|kN(I − P )N+nP [k]+�f (y)|2kV (x, k)

m(x)m(y)

⎞

⎠

1/2

≤ C sup1≤�≤n([r2

B ]−1)

⎛

⎝ 1

V (2j B)

∞∑

k=2j−3rB

∑

y∈�

|kN(I − P )N+nP [k]+�f (y)|2k

m(y)

⎞

⎠

1/2

.

(22)

Using (3) again, we can write

‖(I − P )N+nP [k]+�f (y)‖L2 ≤ supz∈B

C

(k + �)N+nV (z,√k + �)1/p−1/2

‖f ‖Lp

≤ C

(k + �)N+nV (2j B)1/p−1/2max

(

1,

(2j rB√k + �

)D(1/p−1/2))

‖f ‖Lp .

Substituting this information into (22), by a simple calculation we can dominate II by

≤ C2−j (2n−D( 1p− 1

2 ))

(1

V (B)

∑

x∈B|f (x)|pm(x)

)1/p

.

Hence, SL,N is bounded on Lp for all 1 < p < 2.Case 2: 2 < p < ∞Let h ∈ L(p/2)′(�). We have

∑

x∈�S2L,Nf (x)h(x)m(x)=

∑

x∈�

∑

d(x,y)<k

∞∑

k=1

|kN(I − P )NP [k]f (y)|2kV (x, k)

m(y)h(x)m(x)

=∑

y∈�

∞∑

k=1

∑

d(x,y)<k

|kN(I − P )NP [k]f (y)|2kV (x, k)

h(x)m(x)m(y)

≤ C∑

y∈�G2L,Nf (y)M(h)(y)m(y)

where

G2L,Ng(y) =

( ∞∑

k=1

|kN(I − P )NP [k]g(x)|2k

)1/2

.

By the similar argument as in (i), we also get that GL,N is bounded onLp for all 1 < p < ∞.This together with Holder’s inequality gives

∑

x∈�S2L,Nf (x)h(x)m(x)≤ ‖GL,Nf ‖2

Lp‖M(h)‖L(p/2)′

≤ C‖f ‖2Lp‖h‖L(p/2)′ .

This implies‖S2

L,Nf ‖Lp/2 ≤ C‖f ‖2Lp

or equivalently,‖SL,Nf ‖Lp ≤ C‖f ‖Lp

for all 2 < p < ∞. This completes our proof.�

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Weighted Hardy Spaces Associated to Discrete Laplacians on Graphs and Applications

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