Báo cáo hóa học: "ON WEIGHTED INEQUALITIES FOR PARAMETRIC MARCINKIEWICZ INTEGRALS" potx

AL-QASSEM Received 25 February 2005; Revised 30 May 2005; Accepted 3 July 2005 We establish a weightedL pboundedness of a parametric Marcinkiewicz integral operator ᏹρ Ω,hifΩ is allowed

MARCINKIEWICZ INTEGRALS

H M AL-QASSEM

Received 25 February 2005; Revised 30 May 2005; Accepted 3 July 2005

We establish a weightedL pboundedness of a parametric Marcinkiewicz integral operator

ᏹρ

Ω,hifΩ is allowed to be in the block space B(0,−1/2)

q (Sn−1) for someq > 1 and h

satis-fies a mild integrability condition We apply this conclusion to obtain the weightedL p

boundedness for a class of the parametric Marcinkiewicz integral operatorsᏹ∗,ρ

Ω,h,λand

ᏹρ

Ω,h,Srelated to the Littlewood-Paleyg λ ∗-function and the area integralS, respectively.

It is known that the conditionΩ∈ B(0,q −1/2)(Sn−1) is optimal for theL2boundedness of

ᏹ1

Ω,1

Copyright © 2006 H M Al-Qassem This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

Suppose that Sn−1is the unit sphere of Rn(n ≥2) equipped with the normalized Lebesgue measuredσ = dσ( ·) LetΩ be a function defined on Sn−1withΩ∈ L1(Sn−1) and satisfies the vanishing condition



Forγ > 1, letΔγ(R+) denote the set of all measurable functionsh on R+such that

sup

R>0

1

R

R

0

h(t)γ

It is easy to see that the following inclusions hold and are proper:

L ∞

R+ 

⊂Δβ



R+ 

⊂Δα



R+ 

Throughout this paper, we letx denotex/ | x |forx ∈Rn \{0}and p denote the con-jugate index ofp; that is, 1/ p + 1/ p  =1

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 91541, Pages 1 17

DOI 10.1155/JIA/2006/91541

Suppose thatΓ(t) is a strictly monotonic C1function on R+andh : R+ →C is a

mea-surable function Define the parametric Marcinkiewicz integral operatorᏹρ

Ω,Γ,hby

ᏹρ

Ω,Γ,h f (x) =

0

F ρ

Ω,Γ,h f (t, x) 2dt

t

 1/2

where

F Ω,Γ,h ρ f (t, x) = t1ρ



|u|≤t f

x −Γ| u |)u Ωu 

| u | n−ρ h

| u |du, (1.5)

ρ = σ + iτ (σ, τ ∈R withσ > 0), f ∈᏿(Rn), the space of Schwartz functions

For the sake of simplicity, we denoteᏹρ

Ω,Γ,h =ᏹρ

Ω,hifΓ(t) ≡ t.

It is well-known thatᏹ1

Ω,1is the classical Marcinkiewicz integral operator of higher di-mension, corresponding to the Littlewood-Paleyg-function, introduced by Stein in [17] Stein showed thatᏹ1

Ω,1is bounded onL p(Rn) forp ∈(1, 2] ifΩ∈Lipα(Sn−1) (0< α ≤1) Subsequently, Benedek et al proved thatᏹ1

Ω,1is bounded on L p(Rn) for p ∈(1,∞) if

Ω∈ C1(Sn−1) (see [3]) Later on, the case of rough kernels (Ω satisfies only size and cance-lation conditions but no regularity is assumed) became the interest of many authors For

a sample of past studies, see ([1,2,4,5]) In [2], Al-Qassem and Al-Salman showed that

ᏹ1

Ω,1is bounded onL p(Rn) forp ∈(1,∞) ifΩ belongs to the block space B(0,−1/2)

q (Sn−1) and that the conditionΩ∈ B q(0,−1/2)(Sn−1) is optimal in the sense that there exists anΩ which lies inB q(0,υ)(Sn−1) for all−1< υ < −1/2 such thatᏹ1

Ω,1is not bounded onL2(Rn)

In H¨ormander [10] defined the parametric Marcinkiewicz operatorᏹρ

Ω,1forρ > 0 and

proved that ᏹ1

Ω,1 is bounded onL p(Rn) for p ∈(1,∞) ifΩ∈Lipα(Sn−1) (0< α ≤1) Sakamoto and Yabuta [15] studied theL p-boundedness of the more general parametric Marcinkiewicz integral operatorᏹρ

Ω,1ifρ is complex and proved thatᏹρ

Ω,1is bounded

onL p(Rn) for p ∈(1,∞) if Re(ρ) = σ > 0 andΩ∈Lipα(Sn−1) (0< α ≤1) Recently, in [1] the author of this paper gave that the more general operatorᏹρ

Ω,Γ,h is bounded on

L p(Rn) forp satisfying |1/ p −1/2 | ≤min{1/2, 1/γ  }if Re(ρ) = σ > 0,Γ satisfies a convex-ity condition,Ω∈ B(0,q −1/2)(Sn−1) andh ∈Δγ(R+) for someq, γ > 1 This is an essential

improvement and extension of the results mentioned above

On the other hand, the weightedL pboundedness ofᏹ1

Ω,hhas also attracted the atten-tion of many authors in the recent years Indeed, Torchinsky and Wang in [19] proved that

ifΩ∈Lipα(Sn−1), (0< α ≤1), thenᏹ1

Ω,1is bounded onL p(ω) for p ∈(1,∞) andω ∈ A p

(The Muckenhoupt’s weight class, see [9] for the definition) In Sato in [16] improved the weightedL pboundedness of Torchinsky-Wang by proving thatᏹ1

Ω,his bounded onL p(ω)

for p ∈(1,∞) provided thath ∈ L ∞(R+),Ω∈ L ∞(Sn−1) andω ∈ A p(Rn) Subsequently,

in Ding et al in [5] were able to show thatᏹ1

Ω,h is bounded onL p(ω) for p ∈(1,∞) provided thath ∈ L ∞(R+),Ω∈ L q(Sn−1),q > 1 and ω q 

∈ A p(Rn) In a recent paper, Lee and Lin in [13] showed thatᏹ1

Ω,h is bounded onL p(ω) for p ∈(1,∞) if h ∈ L ∞(R+),

Ω∈ H1(Sn−1) andω ∈  A I

p(Rn), where H1(Sn−1) is the Hardy space on the unit sphere andAI

p(Rn) is a special class of radial weights introduced by Duoandikoetxea [6] whose definition will recalled inSection 2

In this paper, we will investigate the weighted L p(ω) boundedness of the

paramet-ric Marcinkiewicz operatorᏹρ

Ω,Γ,hforω ∈  A I

p(Rn) and under the natural conditionΩ∈

B(0,q −1/2)(Sn−1) To state our results, we will need the following definitions from [8]

Definition 1.1 We say that a functionΓ satisfies “hypothesis I” if

(a)Γ is a nonnegative C1function on (0,∞),

(b)Γ is strictly increasing, Γ(2t) ≥ η Γ(t) for some fixed η > 1 and Γ(2t) ≤ c Γ(t) for

some constantc ≥ η > 1.

(c)Γ(t) ≥ α Γ(t)/t on (0, ∞) for some fixedα ∈(0, log2c] andΓ(t) is monotone on

(0,∞)

Definition 1.2 We say thatΓ satisfies “hypothesis D” if

(a)Γ is a nonnegative C1function on (0,∞),

(b)Γ is strictly decreasing, Γ(t) ≥ η Γ(2t) for some fixed η > 1 and Γ(t) ≤ c Γ(2t) for

some constantc ≥ η > 1.

(c)|Γ(t) | ≥ α Γ(t)/t on (0, ∞) for some fixedα ∈(0, log2c] andΓ(t) is monotone on

(0,∞)

Model functions for theΓ satisfy hypothesis I are Γ(t) = t d withd > 0, and their linear

combinations with positive coefficients Model functions for the Γ satisfy hypothesis D areΓ(t) = t rwithr < 0, and their linear combinations with positive coefficients

Theorem 1.3 Let h ∈Δγ(R+) for some γ > 1 Assume that Γ satisfies either hypothesis I or hypothesis D andΩ∈ B q(0,−1/2)(Sn−1) for some q > 1 Then

ᏹρ

Ω,Ψ,h(f )

L p(Rn)≤ C p Ω

B(0,q −1/2)(Sn −1 ) f

is bounded on L p(Rn ) for |1/ p −1/2 | < min {1/γ , 1/2 }

Theorem 1.4 Let h ∈Δγ(R+) for some γ ≥ 2, 1 < p < ∞ Assume that Γ satisfies either

hypothesis I or hypothesis D andΩ∈ B(0,q −1/2)(Sn−1) for some q > 1 Then there exists C p > 0 such that the following inequality holds:

ᏹρ

Ω,Γ,h(f )

L p(ω) ≤ C p Ω

B(0,q −1/2)(Sn −1 ) f

for γ  < p < ∞ and ω ∈  A I

p/γ (R+).

Remark 1.5 (a) In order to make a comparison among the above mentioned results, we

remark that on Sn−1, for anyq > 1, 0 < α ≤1 and−1< υ, the following inclusions hold

and are proper:

C1 

Sn−1 

⊂Lipα

Sn−1 

⊂ L q

Sn−1 

⊂ L

log+L

Sn−1 

⊂ H1 

Sn−1 

,

r>1

L r

Sn−1 

⊂ B(0,υ) q



Sn−1 

With regard to the relationship betweenB q(0,υ)(Sn−1) andH1(Sn−1) (forυ > −1) remains open

(b) We point out that the result inTheorem 1.3extends the result of Al-Qassem and Al-Salman [2] who obtained Theorem 1.3 in the special case h ≡1 and Γ(t) ≡ t and

also improves substantially the result of Sakamoto and Yabuta [15] We remark also that

Theorem 1.4 represents an improvement and extension of [5, Theorem 1] in the case

ω ∈  A I

p(R+)

(c) The method employed in this paper is based in part on ideas from [1,2,7,8,16], among others

The paper is organized as follows InSection 2we give some definitions and we estab-lish the main estimates needed in the proofs of our main results The proofs of Theorems

1.3and1.4will be given inSection 3 Additional results can be found inSection 4 Throughout the rest of the paper the letterC will denote a positive constant whose

value may change at each occurrence

2 Definitions and lemmas

Let us begin by recalling the definition of some special classes of weights and some of their important properties

Definition 2.1 Let ω(t) ≥0 andω ∈ L1

loc(R+) For 1< p < ∞, we say thatω ∈ A p(R+) if there is a positive constantC such that for any interval I ⊂R+,



| I | −1



I ω(t)dt



| I | −1



I ω(t) −1/(p−1)dt

p−1

A1(R+) is the class of weightsω for which M satisfies a weak-type estimate in L1(ω), where M( f ) is the Hardy-Littlewood maximal function of f

It is well-known that the classA1(R+) is also characterized by all weightsω for which Mω(t) ≤ Cω(t) for a.e t ∈R+and for some positive constantC.

Definition 2.2 Let 1 ≤ p < ∞ We say thatω ∈  A p(R+) if

ω(x) = ν1| x |ν2| x |1−p, (2.2) where eitherν i ∈ A1(R+) is decreasing orν2

i ∈ A1(R+),i =1, 2

LetA I

p(Rn) be the weight class defined by exchanging the cubes in the definitions of

A p for alln-dimensional intervals with sides parallel to coordinate axes (see [12]) Let



A I

p =  A p ∩ A I

p Ifω ∈  A p, it follows from [6] that the classical Hardy-Littlewood maximal functionM f is bounded on L p(Rn,ω( | x |)dx) Therefore, if ω(t) ∈  A p(R+), thenω( | x |)∈

A p(Rn)

By following the same argument as in the proof of the elementary properties ofA p

weight class (see, e.g., [9]) we get the following lemma

Lemma 2.3 If 1 ≤ p < ∞ , then the weight class AI

p(R+) has the following properties:

(i)AI

p1⊂  A I

p1, if 1 ≤ p1 < p2 < ∞ ;

(ii) For any ω ∈  A I

p , there exists an ε > 0 such that ω1+ε ∈  A I

p ; (iii) For any ω ∈  A I

p and p > 1, there exists an ε > 0 such that p − ε > 1 and ω ∈  A I

p−ε

The block spaces originated in the work of Taibleson and Weiss on the convergence of the Fourier series in connection with the developments of the real Hardy spaces Below

we will recall the definition of block spaces on Sn−1 For further background information about the theory of spaces generated by blocks and its applications to harmonic analysis, see the book [14]

Definition 2.4 A q-block on S n−1is anL q(1< q ≤ ∞) functionb(x) that satisfies

(i) supp(b) ⊂ I;

(ii) b L q ≤ | I | −1/q 

where| I | = σ(I), and I = B(x0,θ0)= { x  ∈Sn−1:| x  − x0 | < θ0 }is a cap on Sn−1for some

x0 ∈Sn−1andθ0 ∈(0, 1]

Jiang and Lu introduced (see [14]) the class of block spacesB(0,q υ)(Sn−1) (forυ > −1) with respect to the study of homogeneous singular integral operators

Definition 2.5 The block space B q(0,υ)(Sn−1) is defined by

B(0,q υ)

Sn−1

=

Ω∈ L1

Sn−1

:Ω=

∞

μ=1

η μ b μ,M(0,q υ) η μ



< ∞



where eachη μis a complex number; eachb μ is aq-block supported on a cap I μon Sn−1,

υ > −1 and

M(0,υ)

=∞ μ=1

η μ1 + log(υ+1)

I μ−1 

LetΩ B(0,υ)

q (Sn −1 )=inf{ M q(0,υ)({ η μ }) :Ω=∞ μ=1η μ b μand eachb μis aq-block function

supported on a capI μon Sn−1} Then · B(0,υ)

q (Sn −1 ) is a norm on the spaceB(0,q υ)(Sn−1) and (B q(0,υ)(Sn−1), · B(0,υ)

q (Sn −1 )) is a Banach space

In their investigations of block spaces, Keitoku and Sato in [11] showed that these spaces enjoy the following properties:

B(0,υ2 )

q



Sn−1

⊂ B(0,υ1 )

q



Sn−1

ifυ2 > υ1 > −1;

B(0,υ)

q2



Sn−1 

⊂ B(0,υ)

q1



Sn−1 

if 1< q1 < q2, for anyυ > −1;

q>1

B(0,q υ)

Sn−1



q>1

L q

Sn−1

for anyυ > −1.

(2.6)

Definition 2.6 For a suitable C1functionΓ on R+, a measurable functionh : R+ →C and

a suitable functionbμ on Sn−1 we define the family of measures{ σb μ,t:t ∈R+}and the maximal operatorσb ∗



Rn f dσb

μ,t = 1

t ρ



(1/2)t<|y|≤t f

Γ| y |y 

h

| y | b μ(y )

| y | n−ρ d y,

σb ∗

μ f (x) =sup

t∈R



 σ



b μ,t



where| σ b

μ,t |is defined in the same way asσb

μ,t, but withb μreplaced by| b μ |andh replaced

by| h |

Fork ∈Z,μ ∈N∪ {0}, and a capI μ on Sn−1 with| I μ | < e −2, we letθ μ =[log| I μ | −1] andω μ =2θ μ, where [·] denotes the greatest integer function Now seta k,μ = Γ(ω k) ifΓ

satisfies hypothesis I anda k,μ =(Γ(ωk))−1ifΓ satisfies hypothesis D Then by the

con-ditions ofΓ, it is easy to see that{ a k,μ }is a lacunary sequence of positive numbers with infk∈Z(a k+1,μ /a k,μ)≥ η θ μ > 1.

Lemma 2.7 Let μ ∈N∪ {0} and h ∈Δγ(R+) for some γ with 1 < γ ≤ 2 Letb μ be a function

on S n−1 satisfying (i)

Sn −1 b μ(y)dσ(y) = 0; (ii)  b μ  q ≤ | I μ | −1/q 

for some q > 1 and for some cap I μ on S n−1with | I μ | < e −2; and (iii)  b μ 1≤ 1 Then there exist constants C and

0< υ < 1/q  such that if Γ satisfies hypothesis I,

σ

ω k+1 μ

ω k σb

μ,t(ξ) 2dt

t ≤ Cθ μ

a k,μ−2υ/γ  θ μ | ξ | −2υ/γ  θ μ; (2.9)

ω k+1 μ

ω k σb μ,t(ξ) 2dt

t ≤ Cθ μ



a k,μ

 2υ/γ  θ μ

| ξ |2υ/γ  θ μ, (2.10)

and if Γ satisfies hypothesis D,

σ

b μ,t ≤ C;

ω k+1 μ

ω k



σb

μ,t(ξ)2dt

t ≤ Cθ μ

a k,μ−2υ/γ  θ μξ 2υ/γ  θ μ

;

ω k+1 μ

ω k



σb

μ,t(ξ)2dt

t ≤ Cθ μ

a k,μ 2υ/γ  θ μ

| ξ | −2υ/γ  θ μ,

(2.11)

where  σ b

μ,t stands for the total variation of σb

μ,t The constant C is independent of k, μ, ξ andΓ(· ).

Proof We will only present the proof of the lemma ifΓ satisfies hypothesis I, since the

proof for the case thatΓ satisfies hypothesis D will be essentially the same By (iii) and the

definition ofσb μ,t, one can easily see that (2.8) holds with a constantC independent of t

andμ Next we prove (2.9) By definition,



σ b

μ,t(ξ) = t1ρ

t

(1/2)t



Sn −1e −i Ψ(s)ξ ·xb μ(x) h(s)

By H¨older’s inequality, a change of variable and since|Sn −1e −i Ψ(s)ξ ·xb μ(x)dσ(x) | ≤1, we

obtain

σb

μ,t(ξ)  ≤t

(1/2)t

h(s)γ ds

s

 1/γt

(1/2)t





Sn −1e −i Ψ(s)ξ ·xb μ(x)dσ(x)

γ

 ds s

 1/γ 

≤ C

t

(1/2)t





Sn −1e −i Ψ(s)ξ ·x bμ(x)dσ(x)

2ds s 1/γ 

= C



Sn −1×Sn −1



b μ(x)b μ(y)I μ,t(ξ, x, y)dσ(x)dσ(y)1/γ



,

(2.13)

where

I μ,t(ξ, x, y) =

 1

1/2 e −i Γ(ts)ξ ·(x−y) ds

WriteI μ,t(ξ, x, y) as

I μ,t(ξ, x, y) =

 1

1/2 Y t (s) ds

where

Y t(s) =

s

1/2 e −i Γ(tw)ξ ·(x−y) dw, 1/2 ≤ s ≤1. (2.16) Now, using the assumptions onΓ, we obtain

d

dw



Γ(tw)= tΓ(tw) ≥ α Γ(tw)

w ≥ α Γ(t/2)

c

Γ(t)

s for 1/2 ≤ w ≤ s ≤1. (2.17) Thus by van der Corput’s lemma,| Y t(s) | ≤(c/α) | Γ(t)ξ/s | −1| ξ  ·(x − y) | −1 By integration

by parts, we get

I μ,t(ξ, x, y)  ≤ CΓ(t)ξ−1 ξ  ·(x − y)−1

which when combined with the trivial estimate| I μ,t(ξ, x, y) | ≤log 2 and choosingτ such

that 0< τ < 1/q yields

I μ,t(ξ, x, y)  ≤  Γ(t)ξ−τξ  ·(x − y)−τ

By H¨older’s inequality and (ii) we get



σb

μ,t(ξ) ≤ CΓ(t)ξ−τ/γ  b

μ 2/γ  q

×



Sn −1×Sn −1

ξ  ·(x − y)−τq 

dσ(x)dσ(y)

 1/(q  γ )

≤ CΓ(t)ξ−τ/γ I μ−2/(q  γ )

.

(2.20)

ω k+1

μ

ω k σb

μ,t(ξ) 2dt

t ≤ C min

logI μ−1 

, logI μ−1 Γ

ω k μ

ξ−2τ/γ I μ−2/(q  γ ) 

≤ C logI μ−1 Γ

ω k

ξ−2τ/γ log(|I μ | −1 )

,

(2.21) which proves (2.9) To prove (2.10), we use the cancellation condition ofbμto get



σb

μ,t(ξ) ≤

Sn −1

t

t/2

e −i Γ(s)ξ ·x −1h(s)b μ(x)ds

Hence, by (iii) and sinceΓ is increasing we get



σb

μ,t(ξ) ≤ CΓ(t)ξ. (2.23)

By using the same argument as above we get (2.10) The lemma is proved 

Lemma 2.8 Let μ ∈N∪ {0} , h ∈Δγ(R+) for some γ > 1, γ  < p < ∞ and ω ∈  A p/γ (R+) Assume thatb μ ∈ L1(Sn−1) and Γ satisfies either hypothesis I or hypothesis D Then there

exists a positive constant C p such that

σ∗

b μ(f )

L p(ω) ≤ C p b

μ

L1 

Sn −1 f L p(ω) (2.24)

Proof By H¨older’s inequality, we have

σ

b μ,t ∗ f (x)

≤t

(1/2)t

h(s)γ ds

s

 1/γt

(1/2)t





Sn −1



b μ(y )f

x − Γ(s)y 

dσ(y )

γ  ds s 1/γ 

≤ C b μ 1/γ

L1(Sn −1 )

t

(1/2)t



Sn −1



b μ(y )f

x − Γ(s)y γ 

dσ(y )ds

s

 1/γ 

.

(2.25) Thus

σ ∗

b μ f (x) ≤ C b

μ 1/γ

L1(Sn −1 )



Sn −1



b μ(y )M

Γ,y 

| f | γ 

(x)dσ(y )

 1/γ 

where

M Γ,y  f (x) =sup

t∈R



t/2 t f

x − Γ(s)y ds

s



Letw = Γ(s) Assume first that Γ satisfies hypothesis I By the assumptions on Γ, we have

ds/s ≤ dw/αw So, by a change of variable we have

M Γ,y  f (x) ≤sup

t∈R+

Γ(t)

Γ(t/2)

f (x − w y )dw

w



≤sup

t∈R+

c Γ(t/2)

Γ(t/2)

f (x − w y )dw

w



≤ CM y  f (x),

(2.28)

where

M y  f (x) =sup

R>0

R −1

R

0

f (x − w y )dw (2.29)

is the Hardy-Littlewood maximal function of f in the direction of y  On the other hand,

ifΓ satisfies hypothesis D, as above we have ds/s ≤ − dw/αw and

M Γ,y  f (x) ≤1

α t∈supR+

Γ(t/2)

Γ(t)

f (x − w y )dw

w



≤1

α t∈supR+

Γ(t/2)

(1/c) Γ(t/2)

f

x − w y )dw

w



≤ CM y  f (x).

(2.30)

By (2.26)–(2.30) and Minkowski’s inequality for integrals we get

σb ∗

μ(f )

L p(ω) ≤ C b

μ 1/γ

L1(Sn −1 )



Sn −1



b μ(y ) M

y 

| f | γ 

L p/γ (ω) dσ(y )

 1/γ 

. (2.31)

By [6, equation (8)] and sinceω ∈  A p/γ (R+) we have

M y  f

L p/γ (ω) ≤ C  f  L p/γ (ω) (2.32) withC independent of y  Thus, by (2.31)–(2.32) we get (2.24) This completes the proof

Lemma 2.9 Let μ ∈N∪{0} , h ∈Δγ(R+) for some γ ≥ 2, γ  < p < ∞ and ω ∈  A p/γ (R+) Assume thatb μ ∈ L1(Sn−1) and Γ satisfies either hypothesis I or hypothesis D Then there

exists a positive constant C p such that the inequality

k∈Z

ω k+1 μ

ω k



σb

μ,t ∗ g k2dt

t

 1/2

L p(ω)

≤ C p



logI μ−1  1/2 b

μ

L1(Sn −1 )

k∈Z

g k 2

 1/2

L p(ω)

,

(2.33)

holds for any sequence of functions { g k } k∈Zon R n

Proof Let γ  < p < ∞ By a change of variable, we have

k∈Z

ω k+1 μ

ω k



σb μ,t ∗ g k2dt

t

 1/2

≤

k∈Z

ω μ

1



σb μ,ω k t ∗ g k2dt

t

 1/2

By H¨older’s inequality and following a similar arguments as in the proof of (2.25) we get



σb

μ,t ∗ g k(x)γ 

≤ C b

μ

L1(Sn −1 )

t t/2



Sn −1



b μ(y )g

k



x − Γ(s)y γ 

dσ(y )ds

s



.

(2.35) Let d = p/γ  By duality, there is a nonnegative function f ∈ L d 

(ω1−d 

, Rn) satisfying

 f L d (ω1− d )≤1 such that

k∈Z

ω μ

1



σ b

μ,ω k t ∗ g kγ  dt

t

 1/γ 

γ 

L p(ω)

=



Rn

k∈Z

ω μ

1



σb

μ,ω k t ∗ g k(x)γ  dt

t f (x)dx.

(2.36) Therefore, by (2.35)–(2.36) and a change of variable we get

k∈Z

ω μ

1



σb

μ,ω k t ∗ g kγ  dt

t

 1/γ 

γ 

L p(ω)

≤ C

logI μ−1  b

μ

L1(Sn −1 )



Rn

k∈Z

g k(x)γ 

M ∗ μ f (x)dx,

(2.37)

where

M μ ∗ f (x) =sup

t∈R+



(1/2)t<|y|≤t f

x +Γ| y |y b

μ(y )| y | −n d y. (2.38)

By H¨older’s inequality, we obtain



k∈Z

ω μ

1



σb

μ,ω k t ∗ g kγ  dt

t

 1/γ 

γ 

L p(ω)

≤ C

logI μ−1  b

μ

L1(Sn −1 )



k∈Z

g kγ  1/γ 

γ 

L p(ω)

M μ ∗ f

L d (ω1− d ).

(2.39)

It is easy to verify thatω ∈  A d(R+) if and only ifω1−d 

∈  A d (R+) By the same argument

as in the proof ofLemma 2.8, we have

M ∗

μ f

L d (ω1− d )≤ C p b

μ

Sn−1

for anyυ > −1.

(2.6)

Definition 2.6 For a suitable C1functionΓ on R+,... | −1/q 

for some q > and for some cap I μ on S n−1with |...



Sn−1 

if 1< q1 < q2, for anyυ > −1;

q>1

B(0,q