© J. Lee and K. S. Rim. 2010
Received: 30 November 2009
Accepted: 17 March 2010
Published: 30 March 2010
In , 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 ) gives
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 .
Also, in , 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.  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 . Here in this paper, we provide an analogue of that of  and [3, Theorems and of Chapter 6].
Here is the first and the main theorem.
In succession of classical weighted norm inequalities, starting from Muckenhoupt's result in 1975 , 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 , Muckenhoupt derives a necessary and sufficient condition on two-weighted norm inequalities for the Poisson integral operator, and then in , 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.
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.
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 ; so we omit its proof.
Now we will prove Theorem 1.1.
Proof of Theorem 1.1.
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 .
as we complete the proof of the claim.
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.
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.
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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