Weighted norm inequalities for a class of rough singular integrals

WEIGHTED NORM INEQUALITIES FOR A CLASS OF

ROUGH SINGULAR INTEGRALS

H. M. Al-QASSEM

Received 11 November 2004 and in revised form 6 February 2005

Weighted norm inequalities are proved for a rough homogeneous singular integral oper-ator and its corresponding maximal truncated singular operoper-ator. Our results are essential improvements as well as extensions of some known results on the weighted boundedness of singular integrals.

1. Introduction

Throughout this paper, we letξ denoteξ/|ξ| forξ∈Rn_{\{0}and p denote the dual}

exponent to p defined by 1/ p+ 1/ p=1. Letn≥2 andSn−1 _{represent the unit sphere}

inRn_{equipped with the normalized Lebesgue measuredσ=dσ(·). LetKbe a kernel of} Calderon-Zygmund-type onRn_{given by}

K(x)=Ω(x)

|x|n, (1.1)

whereΩis a homogeneous function of degree 0, integrable overSn−1_{, and satisfies}

Sn−1Ω(u)dσ(u)=0. (1.2)

Let ωbe a measurable, almost everywhere positive function onRn_{. We call suchωa} weight function. We denote byLp_{(ω) theL}p_{(p >0) space of all measurable functions f} onRn_{such thatf}

Lp_(ω)=(_Rn|f(x)|pω(x)dx)1/ p<∞.

LetΓ(t) be aC1_{function on the intervalR}

+. We define the singular integral operator

TΓ,Ωand its maximal truncated singular integral operatorTΓ∗,Ωby

TΓ,Ωf(x)=p.v.

Rnf

x−Γ|y|yK(y)d y,

T_Γ∗,Ωf(x)=sup

ε>0

|y|>εf

x−Γ|y|yK(y)d y,

(1.3)

wherey=y/|y| ∈Sn−1_{, and f ∈᏿(R}n_{), the space of Schwartz functions.}

Copyright©2005 Hindawi Publishing Corporation

IfΓ(t)=t, we will denoteTΓ,ΩbyTΩ,T_Γ∗,Ωand byTΩ∗.

The investigation of theLp _{boundedness ofT}

ΩandTΩ∗was begun by Calder ´on and

Zygmund in [3] and then continued by many authors. In 1956, Calder ´on and Zygmund showed that theLp _{(1< p <∞) boundedness ofTΩ andT}∗

Ω holds ifΩ∈Llog+L(Sn−1)

and that this condition is essentially the weakest possible size condition onΩfor theLp (1< p <∞) boundedness ofTΩto hold (see [3]). Some years later, Connett (see [5]) and

Coifman and Weiss (see [4]) independently showed thatTΩis bounded onLp(1< p <∞)

forΩ∈H1_(Sn−1_{). Here,H}1_(Sn−1_{) is the Hardy space on the unit sphere which contains}

the spaceLlog+L(Sn−1_{) as a proper space. The study of theL}p _{boundedness of the more} general class of operatorsTΓ,ΩandTΓ∗,Ωwas initiated by Fan and Pan in [8] and continued

by many authors. For a sampling of past studies of these operators, see [1,2,7,8,9]. In 1998, Grafakos and Stefanov in [12] introduced the condition

sup ξ∈Sn−1

Sn−1

_{Ω(y)log|ξ·y|}−11+α

dσ(y)<∞ (1.4)

and showed that it implies theLp_{boundedness ofTΩandT}∗

Ωforpin a range dependent

on the positive exponentα. For anyα >0, letFα(Sn−

1_{) denote the family ofΩ’s which are}

integrable overSn−1_{and satisfy (1.4).}

Theorem1.1 (see [12]). LetΩ∈Fα(Sn−

1_{)for someα >0and satisfy (1.2). Then}

(a)ifα >0,TΩis bounded onLp_(Rn_{)forp∈((2 +α)/(1 +α), 2 +α);} (b)ifα >1,T_Ω∗is bounded onLp_(Rn_{)forp∈(1 + 3/(1 + 2α), 2(2 +α)/3).}

The range forpwas later improved (even for the more general operatorsTΓ,ΩandTΓ∗,Ω

andΓis a polynomial) to ((2 + 2α)/(1 + 2α), 2 + 2α) (in part (a)) and ((1 + 2α)/2α, 1 + 2α) withα >1/2 (in part (b)) (see [7]). However, it is still unknown whether the latter ranges of indices are sharp. It should be noted that Grafakos and Stefanov in [12] showed that

q>1

Lq_Sn−1_F α(Sn−

1_{) for anyα >0,}

α>0

Fα

Sn−1_H1_Sn−1

α>0

Fα

Sn−1_. (1.5)

In the meantime, the study of the weightedLp_{boundedness ofT}

Ωhas also attracted

the attention of many authors. For relevant results, one may consult [6,10,13,14,16, 17,20], among others. We will content ourselves here with recalling only the following pertinent results.

In 1993, Duoandikoetxea [6] proved the following two results.

Theorem1.2 (see [6]). Suppose thatΩ∈Lq_(Sn−1_{)for someq >1. ThenT}

ΩandT_Ω∗are bounded onLp_{(ω)ifq≤p <∞, p=1andω∈A}

p/q, whereA_p(Rn)denotes the Muck-enhoupt class of weights for which the classical Hardy-Littlewood maximal functionM f is bounded inLp_{(ω). (For the definition and properties ofA}

pweights, see[11].)

For a special class of radial weights Ap(R+) (see the definition in Section 2),

Theorem1.3 (see [6]). Ifω∈Ap(R+)for1< p <∞, thenTΩis bounded onLp(ω)

pro-vided thatΩ∈Llog+L(Sn−1_).

Later on,Theorem 1.3was improved and extended by Fan, Pan and Yang as described in the following theorem.

Theorem1.4 (see [10]). Ifω∈AI

p(R+)for1< p <∞, thenTΓ,Ω andTΓ∗,Ω are bounded

onLp_{(ω)provided thatΩ∈H}1_(Sn−1_{)andΓsatisfies either hypothesis I or hypothesis D}

defined below. Here,AI

p(R+)is a subclass ofAp(R+)and its definition will be reviewed in

Section 2.

In this paper, we will investigate the weightedLp_{(ω) boundedness of the operatorsT}

Γ,Ω

andT_Γ∗,Ωforω∈AIp(R+) andΩ∈Fα(Sn−1) for someα >0. To state our main results, we

need some definitions.

We will need the following definitions from [10]. Definition 1.5. A functionΓsatisfies “hypothesis I” if

(a)Γis a nonnegativeC1_{function on (0,∞);}

(b)Γis strictly increasing, Γ(2t)≥λΓ(t) for some fixedλ >1 andΓ(2t)≤cΓ(t) for some constantc≥λ >1;

(c)Γ(t)≥C1Γ(t)/ton (0,∞) for some fixedC1∈(0, log2c] andΓ(t) is monotone on

(0,∞).

Definition 1.6. Γsatisfies “hypothesis D” if (a’)Γis a nonnegativeC1_{function on (0,∞);}

(b’)Γis strictly decreasing,Γ(t)≥λΓ(2t) for some fixed λ >1 andΓ(t)≤cΓ(2t) for some constantc≥λ >1;

(c’)|Γ_{(t)| ≥C1Γ(t)/ton (0,∞) for some fixedC1∈(0, log}

2c] andΓ(t) is monotone

on (0,∞).

Model functions for theΓsatisfying hypothesis I areΓ(t)=td_{withd >0, and their} lin-ear combinations with positive coefficients. Model functions for theΓsatisfying hypoth-esis D areΓ(t)=tr_{withr <0, and their linear combinations with positive coefficients.} Theorem 1.7. Suppose thatΩsatisfies (1.2) andΩ∈Fα(Sn−

1_{)for someα >0. Assume}

thatΓsatisfies either hypothesis I or hypothesis D. If p∈((2 + 2α)/(1 + 2α), 2 + 2α)and ω∈AI

p(R+), then the operatorTΓ,Ωis bounded onLp(ω).

Theorem1.8. Suppose thatΩsatisfies (1.2) andΩ∈Fα(Sn−1)for someα >1/2. Assume

that Γsatisfies either hypothesis I or hypothesis D. If p∈((1 + 2α)/2α, 1 + 2α)andω∈

AI

p(R+), thenT_Γ∗,Ωis bounded onLp(ω).

Remark 1.9. Obviously, Theorems1.7 and1.8 represent an extension of Theorem 1.1 and an improvement and extension overTheorem 1.2in the caseω∈AI

p(R+) becauseΩ

is allowed to be in the spaceFα(Sn−

1_{) for someα >0; and bearing in mind the relation,}

Ld_(Sn−1_)⊂F α(Sn−

1_{) for alld >1.}

2. Some definitions and lemmas

We start this section by reviewing the definition of some special classes of weights and some of their important properties which are relevant to our current study.

Definition 2.1. Letω(t)≥0 andω∈L1

loc(R+). For 1< p <∞,ω∈Ap(R+) if there is a

positive constantCsuch that for any intervalI⊂R+,

|I|−1

Iω(t)dt |I|

−1

Iω(t)

−1/(p−1)_dtp−1_{≤C <∞. (2.1)}

A1(R+) is the class of weightsωfor which the Hardy-Littlewood maximal operatorM

satisfies a weak-type estimate inL1_(ω).

It is well known that the classA1(R+) is also characterized by all weightsωfor which

Mω(t)≤Cω(t) for a.e.t∈R+and for some positive constantC.

Definition 2.2. Let 1≤p <∞.ω∈Ap(R+) ifω(x)=ν1(|x|)ν2(|x|)1−p, where eitherνi∈

A1(R+) is decreasing orν2i ∈A1(R+),i=1, 2.

LetAI

p(Rn) be the weight class defined by exchanging the cubes in the definitions of Ap for alln-dimensional intervals with sides parallel to coordinate axes (see [15]). Let

AI

p=Ap∩AIp. Ifω∈Ap, it follows from [6] that the Hardy-Littlewood maximal function M f is bounded onLp_(Rn_{,ω(|x|)dx). Therefore, ifω(t)∈A}

p(R+), thenω(|x|)∈Ap(Rn). By following the same argument as in the proof of the elementary properties ofAp weight class (see, e.g., [11]), we get the following lemma.

Lemma2.3. If1≤p <∞, then the weight classAI

p(R+)has the following properties:

(i)AI p1⊂A

I

p2, if1≤p1< p2<∞; (ii)for anyω∈AI

p, there exists anε >0such thatω1+ε∈AIp; (iii)for anyω∈AI

pandp >1, there exists anε >0such thatp−ε >1andω∈AIp−ε. Definition 2.4. LetΓ(t) be aC1_{function on the intervalR}

+. Define the sequence of

mea-sures{σk,Ω:k∈Z}and its corresponding maximal operatorσΩ∗onRnby

Rn f d σk,Ω=

2k_≤|y|<2k+1 f

Γ|y|yΩ(y

₎ |y|n d y, σ_Ω∗f(x)=sup

k∈Z

_{σk,Ω∗f(x),} (2.2)

where|σk,Ω|is defined in the same way asσk,Ω, but withΩreplaced by|Ω|.

Lemma 2.5. Suppose thatΩsatisfies (1.2) and Ω∈Fα(Sn−

1_{)for someα >0. Then ifΓ}

satisfies hypothesis I,

_{σk,Ω≤C, (2.3)}

σk,Ω(ξ)≤Cakξ, (2.4)

σk,Ω(ξ)≤Clogakξ−α−1 forakξ> e, (2.5) and ifΓsatisfies hypothesis D,

_σk,Ω≤C,

σk,Ω(ξ)≤Ca−_k1ξ, (2.6)

σk,Ω(ξ)≤Cloga−k1ξ

−α−1

fora−1

k ξ> e,

for some positive constantCindependent ofkandξ, whereσk,Ωstands for the total

vari-ation ofσk,Ω.

Proof. We will only present the proof of the lemma ifΓsatisfies hypothesis I, since the proof for the case thatΓsatisfies hypothesis D will be essentially the same. It is easy to verify that (2.3) holds for some positive constantC. By a change of variable, we have

σk,Ω(ξ)=

Sn−1 2

1e

−iΓ(2k_t)ξ·x

Ω(x)dt

t dσ(x). (2.7)

By (2.7) and the mean zero property (1.2) ofΩ, we get σk,Ω(ξ)≤

Sn−1 2

1

_e−iΓ(2k_t)ξ·x

−1Ω(x)dt

t dσ(x). (2.8)

which easily yields the estimate in (2.4).

Now, we turn to the proof of (2.5). Again, by (2.7), we have σk,Ω(ξ)≤

Sn−1

_{Ik(t,x,ξ)Ω(x)dσ(x), (2.9)}

where

Ik(t,x,ξ)= 2

1e

−iΓ(2k_t)ξ·xdt

t . (2.10)

We notice that

Ik(t,x,ξ)= 2

1H _(t)dt

t , H(t)=

t

1e

−iΓ(2k_s)ξ·x

ds, 1≤s≤2. (2.11)

Now, using the assumptions onΓ, we obtain

d ds

Γ2k_s=2k_Γ2k_s≥C

1Γ(2

k_s)

s ≥C1 Γ(2k₎

By applying van der Corput’s lemma (see [18]), |H(t)| ≤C−1

1 |Γ(2k)ξ|−1|ξ·x|−1t, for

1≤t≤2. Thus by integrating by parts, we have _{Ik(t,x,ξ)≤CΓ}

2kξ−1|ξ·x|−1_{. (2.13)}

Now combining this bound with the trivial bound|Ik(t,x,ξ)| ≤(log 2) and using the fact thattα+1_{/logtis increasing on (e,∞), we get}

_{Ik(t,x,ξ)≤C}log(3/2)|ξ·x|−1α+1

logΓ2k_)ξα+1 if _Γ

2kξ> e. (2.14)

Therefore, by (2.7), (2.9), and the assumption onΩ, we obtain (2.5). This completes the

proof of the lemma.

Lemma2.6. Letω∈Ap(R+)andΩ∈L1(Sn−1). Assume thatΓsatisfies either hypothesis I

or hypothesis D. Then _σ∗

Ω(f)Lp_(ω)≤CpΩL1_(Sn−1₎f_Lp_(ω) (2.15)

for1< p <∞, whereCpis independent of f. Proof. By definition ofσk,Ω, we have

_{σk,Ω∗f(x)≤}2k+1

2k

Sn−1

_{Ω(y)fx−Γ(t)ydσ(y)}dt

t

≤C

Sn−1

_Ω(y)ᏹ

Γ,y|f|(x)dσ(y)

,

(2.16)

where

ᏹΓ,yf(x)=sup k∈Z

2k+1

2k f

x−Γ(t)ydt t

. (2.17)

Lets=Γ(t). Assume first thatΓsatisfies hypothesis I. By the assumptions onΓ, we have dt/t≤ds/C1s. So, by a change of variable, we have

ᏹΓ,yf(x)≤ 1 C1

sup k∈Z

_Γ(2k+1₎

Γ(2k₎

_f(x−sy)ds

s

≤_C1 1supr>0

_Γ(r)

Γ(r/2)

_f(x−sy)ds

s

.

(2.18)

Now, by condition (a) or condition (b), we have

ᏹΓ,yf(x)≤ 1 C1supr>0

cΓ(r/2)

Γ(r/2)

_f(x−sy)ds

s

≤CMyf(x),

where

Myf(x)=sup R>0

R−1

R

0

_{f(x−sy)ds (2.20)}

is the Hardy-Littlewood maximal function of f in the direction ofy. On the other hand, ifΓsatisfies hypothesis D, we use a similar argument as above to getdt/t≤ −ds/C1sand

ᏹΓ,yf(x)≤ 1 C1

sup r>0

_Γ(r/2)

Γ(r)

_f(x−sy)ds

s

≤ 1

C1supr>0

_Γ(r/2)

(1/c)Γ(r/2)

_f(x−sy)ds

s

≤CMyf(x).

(2.21)

So in either case, by (2.19)–(2.21) and Minkowski’s inequality for integrals, we get _σ∗

ΩfLp_(ω)≤C

Sn−1

_Ω(y)M

y|f|_Lp_(ω)dσ(y)

. (2.22)

By [6, (8)] and sinceω∈Ap(R+), we have

_Myf_Lp_(ω)≤CfLp_(ω) (2.23)

withCindependent ofy. By (2.22) and (2.23), we get (2.15) which finishes the proof of

the lemma.

Lemma2.7. Let1< p <∞andω∈Ap(R+). Then there exists a positive constantCpsuch that the inequality

k∈Z

_σk,Ω∗gk2

1/2

Lp_(ω)

≤Cp

k∈Z _gk2

1/2

Lp_(ω)

(2.24)

holds for any sequence of functions{gk}k∈ZonRn. Proof. ByLemma 2.6, we get

sup

k∈Z

_σk,Ω∗gk Lp_(ω)≤

σ_Ω∗ sup

k∈Z _gk

Lp_(ω)≤C

sup

k∈Z _gk

Lp_(ω). (2.25)

On the other hand, there exists a nonnegative function f in Lp

(ω) withfLp_(ω)≤1

such that

k∈Z

_σk,Ω∗gk

Lp_(ω)

=

Rn

k∈Z

Thus, by Fubini’s theorem and H¨older’s inequality, we get

k∈Z

_σk,Ω∗gk

Lp_(ω)

≤

Rn

k∈Z

_gk(x)σ∗

Ω(ω f)(−x)dx

≤

k∈Z _gk

Lp_(ω)

σ_Ω∗(ω f)

Lp_(ω1−p₎,

(2.27)

whereu(x) =u(−x). Sinceω∈Ap(R+) if and only ifω1−p

∈Ap(R₊), byLemma 2.6, we get

k∈Z

_σk,Ω∗gk

Lp_(ω)

≤Cp

k∈Z _gk

Lp_(ω)

. (2.28)

Therefore, we can interpolate (2.25) and (2.28) (see [11, page 481] for the vector-valued

interpolation) to get (2.24). The lemma is proved.

3. Proof of Theorems1.7and1.8

Proof ofTheorem 1.7. We will present the proof ofTheorem 1.7only for the case where Γsatisfies hypothesis I, since the proof for the case whereΓsatisfies hypothesis D is es-sentially the same. Let{ψk}∞−∞ be a smooth partition of unity in (0,∞) adapted to the

intervalsIk=[(ak+1)−1, (ak−1)−1]. To be precise, we require the following:

ψk∈C∞, 0≤ψk≤1,

k

ψk(t) 2

=1,

suppψk⊆Ik, dsψk(t)

dts ≤Cs

ts,

(3.1)

whereC∞ denotes the class of all infinitely differentiable functions on (0,∞) andCs is independent of the lacunary sequence{Γ(2k_{) :k∈Z}. (We remark at this point that ifΓ} satisfies hypothesis D, the partition of unity needed in our proof should have the same properties as above except that we need it to be adapted to the intervals [ak−1,ak+1]).

Define the multiplier operatorsSkinRnby

(Skf)(ξ)=ψk

|ξ| f(ξ). (3.2)

Define

Qj(f)=

k∈Z Sk+j

σk,Ω∗Sk+jf

. (3.3)

Now

TΓ,Ω(f)=

k∈Z

σk,Ω∗f =

k∈Z σk,Ω∗

j∈Z

Sk+jSk+jf

=

j∈Z

holds at least for f ∈᏿(Rn_{). By Plancherel’s theorem, we have Qj(f)}2

L2≤

k∈Z

Rn f(ξ)

2

χ∆_k+j(ξ)σk,Ω(ξ)

2

dξ, (3.5)

where

∆k=

ξ∈Rn_:a k+1

−1

≤ |ξ|<ak−1

−1

. (3.6)

By a straightforward computation and (2.3)–(2.5), we get _Qj(f)

L2≤C|j|−α−1f_L2 ifj≤ −1, _Qj(f)

L2≤Cλ−jfL2 ifj≥0,

(3.7)

and hence,

_Qj(f) L2≤C

1 +|j−α−1fL2 ∀j∈Z. (3.8)

On the other hand, for everyp∈(1,∞) andω∈AI p(R+),

_Qj(f)

Lp_(ω)≤Cp

k∈Z

_{σk,Ω∗Sk+jf}2

1/2

Lp_(ω)

≤Cp

k∈Z

_Sk+jf2

1/2

Lp_(ω)

≤CpfLp_(ω),

(3.9)

where the first and the last inequalities follow by the weighted Littlewood-Paley the-ory sinceω∈AI

p(R+)⊂Ap(R+)⊂Ap(R+), whereas the second inequality follows from Lemma 2.7. Thus, byLemma 2.3, there is anε >0 such that

_Qj(f)

Lp_(ω1+ε₎≤CpfLp_(ω1+ε₎ (3.10)

for everyω∈AI

p(R+) and 1< p <∞. By interpolating between (3.8) and (3.9) withω=1,

for everyp∈((2 + 2α)/(1 + 2α), 2 + 2α), there is aθp>1 such that _Qj(f)

Lp≤C

1 +|j|−θp

fLp (3.11)

holds for j∈Z. Using Stein and Weiss’ interpolation theorem with change of measure, we interpolate (3.10) with (3.11) to get that, for everyp∈((2 + 2α)/(1 + 2α), 2 + 2α) and ω∈AI

p(R+), there is anηp>1 such that _Qj(f)

Lp_(ω)≤C

1 +|j|−ηp

holds forj∈Z. Therefore, by (3.4) and (3.12), we get _TΓ,Ω(f)

Lp_(ω)≤

j∈Z

_Qj(f)

Lp_(ω)≤CfLp_(ω) (3.13)

for every p∈((2 + 2α)/(1 + 2α), 2 + 2α) andω∈AI

p(R+). This completes the proof of

Theorem 1.7.

Proof ofTheorem 1.8. Assume thatp∈((1 + 2α)/2α, 1 + 2α) andω∈AI

p(R+). For anyε >

0, there is an integerksuch thatak≤ε < ak+1. So we have

T_Γ∗,Ω(f)≤σΩ∗(f) +H(f), (3.14)

whereH(f)=sup_k∈Z|Tk(f)|andTk(f)= _∞

j=kσj,Ω∗f. By (3.14) andLemma 2.6, we

notice that the proof ofTheorem 1.8is completed if we can show that _H(f)

Lp_(ω)≤CpfLp_(ω) (3.15)

for p∈((1 + 2α)/2α, 1 + 2α) and ω∈AI

p(R+). So, we turn to the proof of (3.15). Let

ϕ∈᏿(Rn_{) be such thatϕ(ξ)=1 for|ξ|<1/λandϕ(ξ)=0 for|ξ|> λ. Defineφ,φ} kby (φ) =ϕandφk(ξ)=(1/(ak)n)φk(ξ/ak). DecomposeTk(f) as

Tk(f)=δ−φk∗

∞

j=k

σj,Ω∗f +φk∗TΓ,Ω(f)−φk∗ k−1

j=−∞

σj,Ω∗f

=T_k(1)f +T_k(2)f +T_k(3)f,

(3.16)

whereδis the Dirac delta function. ByTheorem 1.7and sinceω∈Ap(R+)⊂Ap(R+), we

immediately get sup

k∈Z _T(2)

k f

Lp_(ω)≤C

_M

TΓ,Ω_Lp_(ω)≤CpfLp_(ω) (3.17)

forp∈((1 + 2α)/2α, 1 + 2α) andω∈AI

p(R+). By definition ofT_k(3)f, we get

sup k∈Z

_T(3)

k f≤

∞

j=1

sup k∈Z

_{σk−j,Ω∗φk∗f(x)=}∞

j=1

Uj(f). (3.18)

ByLemma 2.6, we have _Uj(f)

Lp_(ω)≤M

σ_Ω∗(f)_Lp_(ω)≤CpfLp_(ω) (3.19)

for everyω∈AI

p(R+) and 1< p <∞. Thus, byLemma 2.3, there is anε >0 such that

_Uj(f)

for everyω∈AI

p(R+) and 1< p <∞. On the other hand,

Uj(f)≤

k∈Z

_{σk−j,Ω∗φk∗f}2

1/2

. (3.21)

Thus, by Plancherel’s theorem andLemma 2.5, we have _Uj(f)2

2≤

Rn

k∈Z

σk−j,Ω(ξ)2ϕk(ξ)2 f(ξ)2dξ

≤

Rn

|akξ|≤λ

_ak−jξ2

f(ξ)2dξ

≤Cλ−2j_f2 2.

(3.22)

Thus, by interpolating between (3.22) and (3.20) (withω=1), we get _Uj(f)

p≤Cλ−jβ(p)fp (3.23)

for 1< p <∞and for someβ(p)>0. Now, using Stein and Weiss’ interpolation theorem with change of measures [19], we may interpolate between (3.20) and (3.23) to obtain a positive numberυpsuch that

_Uj(f)

Lp_(ω)≤Cλ−jυpfLp_(ω), (3.24)

which implies

sup

k∈Z _T(3)

k f

Lp_(ω)≤CpfL

p_(ω) (3.25)

for everyω∈AI

p(R+) and 1< p <∞.

Finally, we will prove that sup

k∈Z _T(1)

k f

Lp_(ω)≤CpfL

p_(ω) (3.26)

for p∈((1 + 2α)/2α, 1 + 2α) andω∈AI

p(R+). The proof of this inequality is similar to

the proof of (3.25). In fact, we have

sup k∈Z

_T(1)

k f≤

∞

j=1

sup k∈Z

_σk+j,Ω∗ δ−φk

∗f(x)

= ∞

j=1

Υj(f). (3.27)

ByLemma 2.6and theLp_{(ω) boundedness ofM, we get Υj(f)}

for everyω∈AI

p(R+) and 1< p <∞. Also, by Plancherel’s theorem,

_Υj(f)2 2≤

Rn

k∈Z

σk+j,Ω(ξ) 2

1−ϕk(ξ)2 f(ξ)2dξ

=

k∈Z

1/λ≤|akξ|

σk+j,Ω(ξ) 2

f(ξ)2dξ

≤ ∞

k=−∞ ∞

s=0

λs_{≤|λakξ|<λ}s+1

_logak+jξ−2α−2

f(ξ)2dξ

≤ ∞

k=−∞ ∞

s=0

1 j+s−1

2α+2

λs_{≤|λakξ|<λ}s+1 f(ξ) 2

dξ

≤ ∞

s=0

1 j+s−1

2α+2 f2

2

≤C j−1−2α_f12 2.

(3.29)

ThusTheorem 1.8is proved.

We end this section with the following result concerning power weights|x|γ

. One of the important special classes of radial weights is the power weights|x|γ

,γ∈R. It is known that|x|γ

∈Ap(Rn) if and only if−n < γ < n(p−1).

Our result regarding this class of weights is the following.

Theorem3.1. Let1< p <∞. Assume thatΓsatisfies either hypothesis I or hypothesis D. If

Ω∈Fα(Sn−1)for someα >0and satisfies (1.2), then forω(x)= |x| γ

withγ∈(−1,p−1), TΓ,Ωis bounded onLp(ω)forp∈((2 + 2α)/(1 + 2α), 2 + 2α)andTΓ∗,Ωis bounded onLp(ω)

forp∈((1 + 2α)/2α, 1 + 2α).

A proof of this theorem can be obtained by applying the results in Theorems1.7and 1.8and noticing that|x|γ

∈AI

p(R+) forγ∈(−1,p−1).

Acknowledgment

The author would like to thank the referees for their valuable comments and suggestions.

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H. M. Al-Qassem: Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, Jordan