- Research Article
- Open Access

- Jaesung Lee
^{1}Email author and - KyungSoo Rim
^{1}

**2010**:435450

https://doi.org/10.1155/2010/435450

© J. Lee and K. S. Rim. 2010

**Received:**30 November 2009**Accepted:**17 March 2010**Published:**30 March 2010

## Abstract

## Keywords

- Positive Constant
- Integral Operator
- Maximal Operator
- Maximal Function
- Nonnegative Function

## 1. Introduction

Let be the unit ball of with norm where is the Hermitian inner product, let be the unit sphere, and, be the rotation-invariant probability measure on .

In [1], for , we defined the kernel by

where is the Cauchy kernel and is the invariant Poisson kernel. Thus for each , the kernel is -harmonic. And for all , the ball algebra, such that is real, the reproducing property of ( of [2]) gives

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

For , , the -harmonic conjugate function of , on is defined by

since the limit exists almost everywhere. For , the definition of is the same as the classical harmonic conjugate function [3, 4]. Many properties of -harmonic conjugate function come from those of Cauchy integral and invariant Poisson integral. Indeed the following properties of follow directly from Chapters 5 and 6 of [2].

()As an operator, is of weak type (1.5) and bounded on for .

()If , then for all and if , then .

()If is in the Euclidean Lipschitz space of order for , then so is .

Also, in [1], it is shown that is bounded on the Euclidean Lipschitz space of order for , and bounded on .

In this paper, we focus on the weighted norm inequality for -harmonic conjugate functions. In the past, there have been many results on weighted norm inequalities and related subjects, for which the two books [3, 4] provide good references. Some classical results include those of M. Riesz in 1924 about the boundedness of harmonic conjugate functions on the unit circle for [3, Theorem of Chapter 3] and [3, Theorems and of Chapter 6] about the close relation between -condition of the weight and the boundedness of the Hardy-Littlewood maximal operator and Hilbert transform on . In 1973, Hunt et al. [5] proved that, for , conjugate functions are bounded on weighted measured Lebesgue space if and only if the weight satisfies -condition. It should be noted that in 1986 the boundedness of the Cauchy transform on the Siegel upper half-plane in was proved by Dorronsoro [6]. Here in this paper, we provide an analogue of that of [5] and [3, Theorems and of Chapter 6].

To define the -condition on , we let be a nonnegative integrable function on . For , we say that satisfies the -condition if

where is a nonisotropic ball of .

Here is the first and the main theorem.

Theorem 1.1.

if and only if satisfies the -condition.

In succession of classical weighted norm inequalities, starting from Muckenhoupt's result in 1975 [7], there have been extensive studies on two-weighted norm inequalities. Here, we define the -condition for two weights. For a pair of two nonnegative integrable functions, we say that satisfies the -condition if

where is a nonisotropic ball of . As mentioned above, in [7], Muckenhoupt derives a necessary and sufficient condition on two-weighted norm inequalities for the Poisson integral operator, and then in [8], Muckenhoupt and Wheeden provided two-weighted norm 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.

In this paper we provide a two-weighted norm inequality for -harmonic conjugate operator as our next theorem, by the method somewhat similar to the proof of the main theorem. For a pair , the generalization of the necessity in Theorem (1.5) is as follows.

Theorem 1.2.

then the pair satisfies the -condition.

The proofs of Theorems 1.1 and 1.2 will be given in Section 2. We start Section 2 by 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 and then, as an application, we attach some properties of weights on in relation with , which are similar to those on the Euclidean space.

## 2. Proofs

Definition 2.1.

where the supremum is taken over all the nonisotropic balls containing and stands for the average of over .

The sharp maximal operator is an analogue of the Hardy-Littlewood maximal operator , which satisfies . The proof of the following lemma is essentially the same as that of the Theorem of [4]; so we omit its proof.

Lemma 2.2.

Now we will prove Theorem 1.1.

Proof of Theorem 1.1.

First, we prove that (1.5) implies that satisfies the -condition.

If , then by a direct calculation we get

Therefore, the integrals of over and are equivalent.

Now for a given constant , put in (2.6) and integrate over . We have

where the constant is independent of . Consequently, we have the desired -condition. And this proves the necessity of the -condition for (1.5).

Conversely, we suppose that
and
satisfies the
-condition and then we will prove that (1.5) holds. To do this we will first prove the following. Claim (i). *Let*
*. Then for*
*, there is a constant*
*such that*
*for almost all*
*.*

To prove Claim (i), for a fixed , it suffices to show that for each there are constants and depending only on such that

Define

where is an absolute constant.

Write Then the integral of (2.23) is equal to

as we complete the proof of the claim.

Next, we fix and let . Then by Lemma maximal inequality there is a constant such that

where two constants and depend on and , which proves (1.5) and this completes the proof of Theorem 1.1.

Now, we will prove Theorem 1.2 by taking slightly a roundabout way from the proof of Theorem 1.1.

Proof of Theorem 1.2.

Assume the inequality (1.7). Let and be nonintersecting nonisotropic balls with positive distance, and with radius sufficiently small .

Let be supported in . Then from (2.4), there is a positive constant such that for all ,

For a constant which will be chosen later, put in (2.30), multiply on both sides, and integrate over . We have

for all balls , with radius less than or equal to and the distance between two balls greater then at any point of .

Here, unlike the proof of Theorem 1.1, we can not derive the equivalence between and in a straightforward method, for . For this reason, it is not allowed to replace by directly in (2.36). However, such difficulty can be overcome using the following method. By the symmetric process of the proof, we can interchange with in (2.36). Thus, for all such balls,

Now multiply two equations (2.36) and (2.37) by side. Since , we have

and the proof of Theorem 1.2 is complete.

## Declarations

### 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.

## Authors’ Affiliations

## References

- Lee J, Rim KS: Properties of the -harmonic conjugate operator.
*Canadian Mathematical Bulletin*2003, 46(1):113–121. 10.4153/CMB-2003-011-xMATHMathSciNetView ArticleGoogle Scholar - Rudin W:
*Function Theory in the Unit Ball of ℂn*. Springer, New York, NY, USA; 1980.View ArticleGoogle Scholar - Garnett JB:
*Bounded Analytic Functions, Pure and Applied Mathematics*.*Volume 96*. Academic Press, New York, NY, USA; 1981:xvi+467.Google Scholar - García-Cuerva J, Rubio de Francia JL:
*Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies*.*Volume 116*. North-Holland, Amsterdam, The Netherlands; 1985:x+604.Google Scholar - Hunt R, Muckenhoupt B, Wheeden R: Weighted norm inequalities for the conjugate function and Hilbert transform.
*Transactions of the American Mathematical Society*1973, 176: 227–251.MATHMathSciNetView ArticleGoogle Scholar - Dorronsoro JR: Weighted Hardy spaces on Siegel's half planes.
*Mathematische Nachrichten*1986, 125: 103–119.MATHMathSciNetGoogle Scholar - Muckenhoupt B: Two weight function norm inequalities for the Poisson integral.
*Transactions of the American Mathematical Society*1975, 210: 225–231.MATHMathSciNetView ArticleGoogle Scholar - Muckenhoupt B, Wheeden RL: Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform.
*Studia Mathematica*1976, 55(3):279–294.MATHMathSciNetGoogle Scholar

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