Spaces on the Unit Circle

Abstract

We introduce a new space of Lebesgue measurable functions on the unit circle connecting closely with the Sobolev space. We obtain a necessary and sufficient condition on such that , as well as a general criterion on weight functions and , , such that . We also prove that a measurable function belongs to if and only if it is Möbius bounded in the Sobolev space . Finally, we obtain a dyadic characterization of functions in spaces in terms of dyadic arcs on the unit circle.

1. Introduction

In recent years a new class of Möbius invariant function spaces, called spaces, has attracted a lot of attention. These spaces were originally defined in [1] as spaces of analytic functions in the unit disc in the complex plane . Later on, some further generalizations such as and appeared; see [2, 3], for example. Let be the boundary of . For , Xiao studied the space in paper [4], consisting of all Lebesgue measurable functions with where the supremum is taken over all subarcs and is the arc length of . A series of results of can be found in [4–6]. Note that if , then coincides with , the space of measurable functions of bounded mean oscillation on introduced by John and Nirenberg in [7]. For any given arc and function , the square mean oscillation of on is defined by where Then a function is said to belong to the space if and only if .

In paper [2], Essén and Wulan studied spaces of holomorphic functions on the unit disc and developed their general theory. Later on, Wulan and Zhou gave a decomposition theorem on spaces and built a relationship between spaces of analytic functions and the Morrey type space; see [8, 9], for example. Our aim in this paper will be to extend these ideas to the real spaces so that we may obtain related results on the “real spaces” by using known results on real Hardy spaces. Historically, the “real variable” theory of Hardy spaces has proved to be important in the development of harmonic analysis. We feel that these spaces are intrinsically interesting and that understanding them better will help inform our study of spaces of holomorphic functions.

As a continuation of [2], Essén et al. described the boundary values behavior of analytic functions in spaces [10] as follows.

Theorem EWX. Let be nondecreasing and satisfy the conditions where Then belongs to the space if and only if

The above theorem suggests the following definition of spaces on the unit circle. Let be a nondecreasing function. The space consists of all Lebesgue measurable functions on for which (7) holds. If , , coincides with . The space first appeared in [11], where Pau gave that the Szegö projection from to is bounded and surjective. By [10] and [11] we know that if the weight function satisfies conditions (4) and (5).

In addition, (for two functions and ) means that there is a constant (independent of and ) such that . We say that (i.e., is comparable with ) whenever . In the whole paper we assume that is doubling; that is, .

2. BMO and Spaces

In this section, we investigate the relationship between spaces and and study how depends on the weight function .

The following identity is easily verified:

Proposition 1. is a subset of for all .

Proof. For , it is easy to see that Note that the area measure of is zero. For , we have Suppose that . For and integer , denote by the subarcs of with arc length . Then We have by (8).

Corollary 2. The space is Banach with the norm of , where is the supremum of (7).

Proof. Let be a Cauchy sequence in . By Proposition 1 we know that is subset of . Hence is a Cauchy sequence in as well and in for some . It follows from Fatou’s lemma that, for every integer , This gives in .

Theorem 3. if and only if

Proof. Assume that and (13) holds. We use for the arc in which has the same center as and length for a nonnegative integer . For any given and , then
By the inequality for and the above estimate, we have The above estimate shows that . This and Proposition 1 imply .
Conversely, suppose that . If we can choose an integer sequence such that Define a function as follows: Then ([12], page 178). By assumption we have . It is easy to see that We give the following estimate which will be proved later: By (17), (19), and (20), we have
We now prove (20). Note that is valid for all and . Then The proof is complete.

It is reasonable to assume that for otherwise weight function basically dose not play any role. Moreover, the function must be at least locally on the boundary when belongs to the spaces. Therefore the weight function plays a role only if is small. Then the following result is obvious.

Theorem 4. Let such that , and set . Then .

Proof. Since and is nondecreasing, it is easy to see that . We now prove . Note that there exists an integer such that . If , then by Proposition 1. For any , divide into the subarcs of length . For , denote the th subarcs, arranged in the natural order. Let be the smallest subarcs containing and . Then we have The above estimate gives Hence . So we have . The proof is complete.

The following result is natural in view of Proposition 1 and Theorem 3.

Theorem 5. Let and assume that as . If then .

Proof. Obviously, we have . We assume that . The open mapping theorem tells us that the identity map from one of those spaces into the other one is continuous. Therefore there exists a constant such that . By the assumption, there exists an integer such that for . For any , divide into the subarcs of length . For , denote by the th subarcs, arranged in the natural order. Applying the same manner in handing A and B in the proof of Theorem 4, we can deduce that if , then where is a constant which is dependent on . Consequently, for any and , we have
A simple computation shows that for . So all polynomials belong to spaces. For any given , denote by the truncation of the function . Then and . Applying Fatou’s lemma, we deduce that Equation (28) and Proposition 1 show . It follows from Theorem 3 that the integral (13) with must be convergent, which contradicts our assumption. We conclude that we must have .

3. Möbius Invariant Spaces

Let be a nondecreasing function. The Sobolev type space consists of those Lebesgue measurable functions satisfying If , then are sobolev spaces and are introduced in [4]. See [13] about the theory of Sobolev spaces. If , then is a subspace of . From Section 2 it turns out that is closely related to the Sobolev type space on the unit circle. By (7) and (29) it follows that is a subspace of . As a matter of fact, we have the following result.

Theorem 6. Let satisfy condition (4). Then , if and only if where is a Möbius transformation of the unit disk for .

Proof. We acknowledge that this proof is suggested by the technique of [4]. Firstly, we give the following equality for and :
Sufficiency. Suppose that (30) holds. Choose an arc . Without loss of generality, we assume . We choose a point of such that and are the center and arc length of , respectively. We have the following estimate: Then By (31) we complete the proof of sufficiency.
Necessity. We assume that . For any , let be the arc in with the midpoint of and the arc length of . If , we set . Also, define where is the smallest integer such that ; that is, . Then where For any given , we have By (32) and (37), we obtain that
On the other hand, by Lemma 2.1 of [10], condition (4) implies that for small enough . Then Here we apply the following estimate: The above estimate gives that
Applying the same manner in handing , we have . Therefore, we obtain The proof is complete.

Corollary 7. is a Möbius invariant space in the sense that for any and .

Proof. Corollary 7 is obvious by Theorem 6.

4. Dyadic Characterization

For given arc , denote by the set of the arcs of length obtained by successive bipartition of . The discrete characterization of space is given in [5]. We will prove a discrete characterization of spaces. The following is the principle result of this section.

Theorem 8. Let satisfy condition (4). Then belongs to the space , if and only if

We first acknowledge that this proof is suggested by the technique of [5]. To prove Theorem 8, we need the following lemmas.

Lemma 9. Let be an arc. If , then where

Proof. The following result can be found in [5]: Note that and . By (46), we have The proof is complete.

Lemma 10. Let satisfy condition (4). Let be three arcs of equal length: , such that and are adjacent and . Then for any , we have

Proof. See [5] about the proof of the following inequality: Without loss of generality, we assume that and . For each integer , let be the set of the dyadic arcs of length contained in , arranged in the natural order. If , then for some ; by (48) we have The different choices of yield different . Summing over all and , we have It is easy to see that The following estimate about the final double sum first appeared in Lemma 1 of [5]. Consider If satisfies condition (4), Lemma 2.1 in [10] implies that there exists some small enough such that is nondecreasing. Substituting and summing over , we finally obtain Thus we have proved (49) and hence the proof is complete.

Lemma 11. If satisfies condition (4), then there exists a such that is nonincreasing. Furthermore, for any .

Proof. Lemma 11 can be found in [14].

Proof of Theorem 8. We now prove the necessity. It is easy to see that By (55), we have where and , for , and , for .
Note that because of . Since satisfies condition (4), by Lemma 11 we may assume that is nonincreasing. In fact, if , we can replace with by Theorem 3. Then This gives
For sufficiency, we claim that where for .
In fact, by (56) and Fubini’s theorem, we have This and (56) show that it suffices to verify First, suppose that with and let be such that . Noting that and thus