báo cáo hóa học:" Research Article Estimates of M-Harmonic Conjugate Operator" pptx

For f ∈ L1S, Kf, the M-harmonic conjugate function of f, on S is defined by Kf ζ lim r→ 1 S function come from those of Cauchy integral and invariant Poisson integral.. In the past, th

Volume 2010, Article ID 435450, 13 pages

doi:10.1155/2010/435450

Research Article

Jaesung Lee and Kyung Soo Rim

Department of Mathematics, Sogang University, 1 Sinsu-dong, Mapo-gu, Seoul 121-742, South Korea

Correspondence should be addressed to Jaesung Lee,jalee@sogang.ac.kr

Received 30 November 2009; Revised 23 February 2010; Accepted 17 March 2010

Academic Editor: Shusen Ding

Copyrightq 2010 J Lee and K S Rim 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

We define theM-harmonic conjugate operator K and prove that for 1 < p < ∞, there is a constant

C psuch that

S |Kf| p ωdσ ≤ C p



S |f| p ωdσ for all f ∈ L p ω if and only if the nonnegative weight

ω satisfies the A p -condition Also, we prove that if there is a constant C psuch that

S |Kf| p vdσ ≤

C p



S |f| p wdσ for all f ∈ L p w, then the pair of weights v, w satisfies the A p-condition

1 Introduction

product, let S be the unit sphere, and, σ be the rotation-invariant probability measure on S.

In1, for z ∈ B, ξ ∈ S, we defined the kernel Kz, ξ by

where Cz, ξ  1 − z, ξ −n is the Cauchy kernel and P z, ξ  1 − |z|2n /|1 − z, ξ|2n is

3.2.5 of 2 gives



S

For that reason, Kz, ξ is called the M-harmonic conjugate kernel.

For f ∈ L1S, Kf, the M-harmonic conjugate function of f, on S is defined by



Kf

ζ  lim

r→ 1



S

function come from those of Cauchy integral and invariant Poisson integral Indeed the

1 As an operator, K is of weak type 1.5 and bounded on L p S for 1 < p < ∞.

2 If f ∈ L1S, then Kf ∈ L p S for all 0 < p < 1 and if f ∈ L log L, then Kf ∈ L1S.

3 If f is in the Euclidean Lipschitz space of order α for 0 < α < 1, then so is Kf.

0 < α < 1/2, and bounded on BMO.

functions In the past, there have been many results on weighted norm inequalities and related subjects, for which the two books3,4 provide good references Some classical results

3, Theorems 6.1 and 6.2 of Chapter 6.

For p > 1, we say that ω satisfies the A p-condition if

sup

Q

1

σQ



Q

ωdσ

 1

σQ



Q

ω −1/p−1 dσ

p−1

where Q  Qξ, δ  {η ∈ S : dξ, η  |1 − ξ, η| 1/2 < δ} is a nonisotropic ball of S.

Here is the first and the main theorem

Theorem 1.1 Let ω be a nonnegative integrable function on S Then for 1 < p < ∞, there is a

constant C p such that



S

Kfp

ωdσ ≤ C p



S

fp

if and only if ω satisfies the A p -condition.

In succession of classical weighted norm inequalities, starting from Muckenhoupt’s

we define the A p-condition for two weights For a pairv, w of two nonnegative integrable

functions, we say thatv, w satisfies the A p-condition if

sup

Q

1

σQ



Q

vdσ

 1

σQ



Q

w −1/p−1 dσ

p−1

a necessary and sufficient condition on two-weighted norm inequalities for the Poisson

inequalities for the Hardy-Littlewood maximal operator and the Hilbert transform We admit that there are, henceforth, numerous splendid results on two-weighted norm inequalities but left unmentioned here

operator as our next theorem, by the method somewhat similar to the proof of the main

Theorem 1.2 Let v, w be a pair of nonnegative integrable functions on S If for 1 < p < ∞, there

is a constant C p such that



S

Kfp

vdσ ≤ C p



S

fp

then the pair v, w satisfies the A p -condition.

introducing the sharp maximal function and a lemma on the sharp maximal function, which plays an important role in the proof of the main theorem In the final section, as an appendix,

we introduce John-Nirenberg’s inequality on S and then, as an application, we attach some

Euclidean space

2 Proofs

Definition 2.1 For f ∈ L1S and 0 < p < ∞, the sharp maximal function f#p

on S is defined

by

f#p

ξ  sup

Q

 1

σQ



Q

f − f Qp

dσ

1/p

the average of f over Q.

is an analogue of the Hardy-Littlewood maximal

operator M, which satisfies f#p

ξ ≤ 2Mfξ The proof of the following lemma is essentially

Lemma 2.2 Let 0 < p < ∞ and ω satisfy A p -condition Then there is a constant C p such that



S



Mfp

ωdσ ≤ C p



S

for all f ∈ L p ω.

Proof of Theorem 1.1 First, we prove that1.5 implies that ω satisfies the A p-condition

If ξ, η ∈ S, then by a direct calculation we get

K

ξ, η





1−η, ξn

2−1−ξ, ηn

1−

If ξ /  − η and 1 − η, ξ n 2 − 1 − ξ, η n   0, then we get ξ  η Thus if ξ / η, then for ξ ≈ η,









0<dξ,η<δ K

ξ, η

f

η

dσ

η



 ≥



0<dξ,η<δ

C

1−

ξ, η2n f

η

dσ

η

2.4

C depends only on the distance between ξ and η.

radius sufficiently small δ, and that they are contained in another small nonisotropic ball, for

Kfξ 

Q1

K

ξ, η

f

η

dσ

η



 ≥



Q1

C

1−

ξ, η2n f

η

dσ

η

Since σQ1 ≈ δ 2n , there is a constant C > 0 such that I ≥ C1/σQ1Q1 fdσ Thus for almost all ξ ∈ Q2, we get

Kfξp ≥ C p C p

 1

σQ1



Q1

fdσ

p



Q2

C p C p



Q2

Kfξp



Q2

Kfp ω dσ≤



S

Kfp ω dσ ≤ C p



S

fp ω dσ  C p



Q1

Thus we get



Q2

C p C p



Q1

using1.5, we also have



Q1

C p C p



Q2

Now for a given constant α, put f  ω α χ Q1in2.6 and integrate over Q2 We have



Q2

Kfξp

ω dσ ≥ C p C p

 1

σQ1



Q1

ω α dσ

p

Q2

Thus we get

 1

σQ1



Q1

ω α dσ

p

Q2

C p C p



Q1

1

σQ1



Q1

ω dσ

 1

σQ1



Q1

ω −1/p−1 dσ

p−1

≤

p

C p C p

2

1

σQ



Q

ω dσ

 1

σQ



Q

ω −1/p−1 dσ

p−1

And this proves the necessity of the A p-condition for1.5

f ∈ L1S Then for q > 1, there is a constant C q > 0 such that Kf# 1

ξ ≤ C q f#q

ξ, for almost all ξ.

To prove Claimi, for a fixed Q  Qξ Q , δ, it suffices to show that for each q > 1 there

1

σQ



Q

Kf

η

− λdσ ≤ C q f#q

ξ Q

Now, we write

f

η

f

η

− f Q



χ 2Q



η

f

η

− f Q



χ S \2Q

η

 f Q  f1



η

 f2



η

Define

gz 



S

integral in2.15 is estimated as



Q

Kf

η

 igξ Qdσ

η

≤



Q

Kf1dσ

Q

Kf2 ig

Estimate of I1 By H ¨older’s inequality we get

1

σQ



Q

Kf1dσ≤ 1

σQ



Q

Kf1q

dσ

1/q

≤

 1

σQ



S

Kf1q

dσ

1/q

σQ 1/qf1

q ,

2.19

C alone will denote a positive constant, independent of δ, whose value may change from line

to line. Now by replacing f1by f − f Q, we get

f1

q



2Q

f − f Qq

dσ

1/q

≤



2Q

f − f 2Qq

dσ

1/q

 σ2Q 1/qf 2Q − f Q. 2.20

Thus by applying H ¨older’s inequality in the last term of the above, we see that there is a

1

σQ



Q

Kf1dσ ≤ C q f#q

ξ Q

Now we estimate I2 Since f2≡ 0 on 2Q, we have

I2



Q

f2 iKf2− g

ξ Qdσ≤

S \2Q2f2

η

Q

C

ξ, η

− Cξ Q , ηdσξdσ

η



S \2Q

f2

η

1−

η, ξ Qn 1/2 dσ

η

where C is an absolute constant.

Write S \ 2Q ∞k12k1Q\ 2k Q Then the integral of2.23 is equal to

∞



k1



2k1 Q\2k Q

f

η

− f Q

1−

η, ξ Qn 1/2 dσ

η

≤∞

k1

1

22n1k δ 2n1



2k1 Q\2k Q

f − f Qdσ

≤∞

k1

1

22n1k δ 2n1

⎛

2k1 Q

f − f2k1 Qdσk

j0



2k1 Q

f2j1 Q − f2j Qdσ⎞⎠.

2.24

1

σQ



Q

Kf2 ig

ξ Qdσ ≤ C∞

k1

k

2k f#1

ξ Q



≤ C q f#q

ξ Q



as we complete the proof of the claim



S

Kfp

ω dσ≤



S

M

Kfp

ω dσ ≤ C p



S



Kf#1p

Take q > 0 such that p/q > 1 By the above Claimi, the last term of the above inequalities is



S



f#qp

ω dσ ≤ C



S



Mfqp/q

ω dσ ≤ C

S

fp

ofTheorem 1.1

Proof of Theorem 1.2 Assume the inequality 1.7 Let Q1 and Q2 be nonintersecting non-isotropic balls with positive distance, and with radius sufficiently small δ

all ξ ∈ Q2,

Q1

1

1−

ξ, η2n f

η

dσ

η

C depends only on the distance between ξ and η Also from the fact that σQ1 ≈ δ 2n,

as

C

 1

σQ1



Q1

fdσ



Thus for almost all ξ ∈ Q2, we get

Kfξp ≥ C p C p

 1

σQ1



Q1

fdσ

p



Q2

C p C p



Q2

Kfξp



Q2

Kfp

v dσ≤



S

Kfp

v dσ ≤ C p



S

fp

w dσ  C p



Q1

Thus,



Q2

v dσ≤ C p

C p C p



Q1



Q2

Kfξp

v dσ ≥ C p C p

 1

σQ1



Q1

w α dσ

p

Q2

By1.7, we arrive at

 1

σQ1



Q1

w α dσ

p

Q2

v dσ≤ C p

C p C p



Q1

1

σQ1



Q2

v dσ

 1

σQ1



Q1

w −1/p−1 dσ

p−1

≤

p

C p C p

2

greater then δ at any point of S.



Qi v dσ and

Qj w dσ in a straightforward method, for i /  j i, j  1, 2 For this reason, it is

with Q2in2.36 Thus, for all such balls,

1

σQ2



Q1

v dσ

 1

σQ2



Q2

w −1/p−1 dσ

p−1

≤

p

C p C p

2

1

σQ1



Q1

v dσ

 1

σQ2



Q2

w −1/p−1 dσ

p−1

×σQ1

2



Q2

v dσ

 1

σQ1



Q1

w −1/p−1 dσ

p−1

≤

p

C p C p

4

.

2.38

C depends on the distance between Q1 and Q2 Taking supremum over all

δ-balls, we get

⎛

⎝sup

Q

1

σQ



Q

v dσ

 1

σQ



Q

w −1/p−1 dσ

p−1⎞

⎠

2

≤

p

C p C p

4

Ap-Condition and BMO

sup

Q

1

σQ



Q

f − f Qdσf

BMO are concerned about the local average of a function, it is natural for us to mention

of Proposition A.4 comes from John-Nirenberg’s inequality Lemma A.3 which states as follows

Lemma A.3 John-Nirenberg’s inequality Let f ∈ BMO and E ⊂ S be not intersecting the north

pole Then there exist positive constants C1and C2, independent of f and E, such that

σ

η ∈ E :f

η

− f E> λ

for every λ > 0.

and moreover, the details of the proof run off our aim of the paper So we decide to omit the

Euclidean space, by Jensen’s inequality and the classical John-Nirenberg’s inequality, we can

Proposition A.4 Let ω be a nonnegative integrable function on S Then log ω ∈ BMO if and only

if ω α satisfies the A2-condition for some α > 0.

Proof We prove the necessity first Suppose log ω ∈ BMO Let Q denote a nonisotropic ball, and α > 0 Now consider integral

1

σQ



Q

e α | log ω−log ω Q|dσ, A.3

which is less than or equal to

σQ

1

σ 

By change of variables, the integral term of the above is equal to

α σQ

∞

0

σ 

η ∈ Q :log ω

η

−log ω

indepen-dent of Q, such that

σ 

η ∈ Q :log ω

η

−log ω

Q > λ ≤ C

1e −C2λ/ log ω BMO σQ. A.6

C2− αlog ω

BMO

By the above choice of α and M, for each nonisotropic ball Q, we have the inequality

1

σQ



Q

e ±αlog ω−log ω Qdσ ≤ M  1. A.8

Therefore we have

sup

Q

1

σQ



Q

e α log ω dσ

 1

σQ



Q

e −α log ω dσ



Conversely, suppose that there is α > 0 such that ω α satisfies the A2-condition Then

by Jensen’s inequality it suffices to show that

sup

Q

1

σQ



Q

e α | log ω−log ω Q|dσ < ∞. A.10

Let us note that

1

σQ



Q

e α | log ω−log ω Q|dσ≤ σQ1



Q

e α log ω dσ e −αlog ω Q σQ1



Q

e −α log ω dσ e α log ω Q

 I  II.

A.11

Since both integrals I and II are bounded in essentially the same way, we only do I From

Jensen’s inequality once more, we have

I 



1

σQ



Q

e α log ω dσ



e σ Q−1



Q log ω −α dσ ≤

 1

σQ



Q

ω α dσ



1

σQ



Q

ω −α dσ



.

A.12

the proposition

Let ω satisfy the A p -condition and r > p Then, since 1/r − 1 < 1/p − 1, H¨older’s

inequality implies that



1

σQ



Q

ω −1/r−1 dσ

1/r−1

≤

 1

σQ



Q

ω −1/p−1 dσ

1/p−1

This means that ω satisfies the A r -condition Also we can easily see that ω −1/p−1satisfies the

Corollary A.5 Let p > 1 and let ω be a nonnegative integrable function on S such that ω α satisfies the A p -condition for some α > 0 Then log ω ∈ BMO.

Proof If p ≤ 2, then ω α satisfies the A2-condition Thus Proposition A.4 implies log ω ∈

BMO If p > 2, then ω −α/p−1 satisfies the A q -condition for q  p/p − 1 < 2, which implies

Acknowledgments

The authors want to express their heartfelt gratitude to the anonymous referee and the editor for their important comments which are significantly helpful for the authors’ further research The authors were partially supported by Grant no 200811014.01 and no 200911051, Sogang University

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fp

ofTheorem 1.1

Proof of Theorem 1.2 Assume the inequality... λ

for every λ > 0.

and moreover, the details of the proof run off our aim of the paper So we decide to omit the

Euclidean space, by Jensen’s inequality... “Properties of the M-harmonic conjugate operator,” Canadian Mathematical Bulletin, vol 46, no 1, pp 113–121, 2003.

2 W Rudin, Function Theory in the Unit Ball of< /i>Cn,