Estimates of -Harmonic Conjugate Operator

Abstract

We define the -harmonic conjugate operator and prove that for , there is a constant such that for all if and only if the nonnegative weight satisfies the -condition. Also, we prove that if there is a constant such that for all , then the pair of weights satisfies the -condition.

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

(1.1)

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

(1.2)

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

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

(1.3)

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

(1.4)

where is a nonisotropic ball of .

Here is the first and the main theorem.

Theorem 1.1.

Let be a nonnegative integrable function on . Then for , there is a constant such that

(1.5)

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

(1.6)

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.

Let be a pair of nonnegative integrable functions on . If for , there is a constant such that

(1.7)

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.

For and , the sharp maximal function on is defined by

(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.

Let and satisfy -condition. Then there is a constant such that

(2.2)

for all .

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

(2.3)

If and , then we get . Thus if , then for , we have . Hence there exist positive constants and such that

(2.4)

for any nonnegative function , where depends only on the distance between and . Suppose that and are nonintersecting with positive distance nonisotropic balls with radius sufficiently small , and that they are contained in another small nonisotropic ball, for example, with radius . Choose a nonnegative function supported in . Then from (2.4), for almost all we have

(2.5)

Since , there is a constant such that . Thus for almost all , we get

(2.6)

Putting and integrating (2.6) over after being multiplied by , we get

(2.7)

However by (1.5) there exists a number such that

(2.8)

Thus we get

(2.9)

Similarly, putting and integrating (2.6) over after being multiplied by and then using (1.5), we also have

(2.10)

Therefore, the integrals of over and are equivalent.

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

(2.11)

Thus we get

(2.12)

Finally take and apply (2.10) to (2.12), then we have the inequality

(2.13)

for every ball with radius less than or equal to at any point of . (Here, note that the right hand side of the above is independent of and particularly because depends only on the distance between and .) Therefore,

(2.14)

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 constantsuch thatfor 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

(2.15)

Now, we write

(2.16)

Since , we have

Define

(2.17)

Then is continuous on . By setting in (2.15), we shall prove the Claim. The integral in (2.15) is estimated as

(2.18)

Estimate of . By Hölder's inequality we get

(2.19)

since is bounded on . (Here, throughout the proof for notational simplicity, the letter alone will denote a positive constant, independent of , whose value may change from line to line.) Now by replacing by , we get

(2.20)

Thus by applying Hölder's inequality in the last term of the above, we see that there is a constant such that

(2.21)

Now we estimate . Since on , we have

(2.22)

By Lemma of [2], we get an upper bound such that

(2.23)

where is an absolute constant.

Write Then the integral of (2.23) is equal to

(2.24)

Thus there exist and such that

(2.25)

as we complete the proof of the claim.

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

(2.26)

Take such that . By the above Claim (i), the last term of the above inequalities is bounded by some constant (depending on and ) times

(2.27)

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 ,

(2.28)

where depends only on the distance between and . Also from the fact that , for some constant depending only on , the integral of (2.28) has the lower bound such as

(2.29)

Thus for almost all , we get

(2.30)

Putting and integrating (2.30) over after being multiplied by , we get

(2.31)

However, by (1.7) there exists a number such that

(2.32)

Thus,

(2.33)

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

(2.34)

By (1.7), we arrive at

(2.35)

Taking in (2.35), we have the inequality

(2.36)

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,

(2.37)

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

(2.38)

Let us note that depends on the distance between and . Taking supremum over all -balls, we get

(2.39)

and the proof of Theorem 1.2 is complete.

Appendix

-Condition and BMO

Let be a nonisotropic ball of . The space consists of all satisfying

(A.1)

where is the average of over . becomes a Banach space provided that we identify functions which differ by a constant. Since both definitions of -condition and are concerned about the local average of a function, it is natural for us to mention the relation between these concepts. In this section, we show that an weight on is indeed closely related to the . Proposition A.4 and Lemma A.3 tell about it. The proof 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 and be not intersecting the north pole. Then there exist positive constants and , independent of and , such that

(A.2)

for every .

The proof of Lemma A.3 is parallel to the proof of the classical John-Nirenberg's inequality on [3, Theorem of Chapter 6]. However, it is somewhat more complicated, and moreover, the details of the proof run off our aim of the paper. So we decide to omit the proof of Lemma A.3.

The next proposition is about the weight and on . Likewise, on the Euclidean space, by Jensen's inequality and the classical John-Nirenberg's inequality, we can see that the Euclidean analogue of Proposition A.4 is also true.

Proposition A.4.

Let be a nonnegative integrable function on . Then if and only if satisfies the -condition for some .

Proof.

We prove the necessity first. Suppose . Let denote a nonisotropic ball, and . Now consider integral

(A.3)

which is less than or equal to

(A.4)

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

(A.5)

John-Nirenberg's inequality implies that there exist positive constants and , independent of , such that

(A.6)

Now we take , and then we define

(A.7)

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

(A.8)

Therefore we have

(A.9)

which means that satisfies the -condition.

Conversely, suppose that there is such that satisfies the -condition. Then by Jensen's inequality it suffices to show that

(A.10)

Let us note that

(A.11)

Since both integrals and are bounded in essentially the same way, we only do . From Jensen's inequality once more, we have

(A.12)

Since satisfies the -condition, we finish the sufficiency and this completes the proof of the proposition.

Let satisfy the -condition and . Then, since , Hölder's inequality implies that

(A.13)

This means that satisfies the -condition. Also we can easily see that satisfies the -condition for . From this and Proposition A.4, we get the following corollary.

Corollary A.5.

Let and let be a nonnegative integrable function on such that satisfies the -condition for some . Then .

Proof.

If , then satisfies the -condition. Thus Proposition A.4 implies . If , then satisfies the -condition for , which implies that satisfies the -condition. Thus by Proposition A.4, we get consequently .