Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 593521, 10 pages

http://dx.doi.org/10.1155/2013/593521

Research Article

The Fractional Carleson Measures on the Unit Ball of

Dongfang Wang1,2 and Bolin Ma²

¹College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

²College of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China

Received 7 March 2013; Accepted 9 May 2013

Academic Editor: Kehe Zhu

Copyright © 2013 Dongfang Wang and Bolin Ma. 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.

Abstract.

We construct a quantity in terms of integral of the Jacobian of a conformal self-map on the unit ball of . Then, we characterize the fractional Carleson measures on the unit ball by the quantity.

1. Introduction

Let be the unit disc of the complex plane and let be the boundary of . For any arc , let be the normalized length of . The Carleson square based on an arc is the set

We set when . Then, for , a nonnegative measure on is called an -Carleson measure if there exists a constant such that

is said to be a compact -Carleson measure if is an -Carleson measure and

The -Carleson measure is the classical Carleson measure (see [1, 2]).

Carleson measures are related to certain holomorphic function spaces, such as BMO, Morrey spaces, and spaces in a natural way (see [2–5]). Besides these, much research has been done about the characterizations of Carleson measures (see [6–10]). A well-known result is that a nonnegative measure on is a -Carleson measure if and only if

(see [2]). The -Carleson measures can be characterized by modifying (4) (see [4, 7, 10–12]). For instance, in [4], for , it is shown that a nonnegative measure on is an -Carleson measure or a compact -Carleson measure if and only if

Recently, Wu [13] provided a different way for the characterization of -Carleson measures in terms of the estimates instead of in (5). Suppose that , , and is a nonnegative measure on . For , , write

where is the cone in with the vertex . In [13], it is shown that

(1) if and only if is an -Carleson measure;(2) as if and only if is a compact -Carleson measure.

The relations among the Carleson measures, quantities , and some function spaces defined on are also displayed, which are applied to characterizing the boundedness and compactness of Volterra-type operators from Hardy spaces to some holomorphic spaces. One can refer to [13] for more details.

In order to show that the Jacobian of a conformal self-map of the unit ball in obeys the weak Harnack inequality, Kotilainen et al. introduced an integral form of the fractional Carleson measures on the unit ball (see [14]). For , set

and set . For , a nonnegative measure on is called a -Carleson measure or a compact -Carleson measure if and only if

(see [14, 15]). For , a conformal self-map of is defined by

Let stand for the Jacobian matrix of at . Then the Jacobian of is

For more details about this conformal self-map, one can refer to [16–19]. Then, it is shown in [14] that is a -Carleson measure or a compact -Carleson measure on if and only if

which is the analogue of (4) and (5).

Pursuing the above, in this paper, analogically to (6), we will construct a quantity on the unit ball by using the Jacobian of and establish the connections between the fractional Carleson measures on and the quantity. In Section 2, we give some preliminaries, which contain the fractional Carleson measure defined in terms of tents or Carleson boxes. In Section 3, we state our main results and their proof. The results are the extension of the ones in [13], and the real analysis techniques used in this paper should have an application in studying the operators on the function spaces defined on the unit sphere in future.

2. Preliminaries

Throughout this paper, denotes a positive constant that may change from one step to the next. For fixed , we call the set

a spherical cap centered on with radius . It is easy to see that is the projection of on the unit sphere. We always write a spherical cap as without pointing out its center and radius, if there are no confusing cases. For a spherical cap , we also denote the radius of by and the Lebesgue measure of by . Clearly, there is the estimate

where we say that and are equivalent, denoted by , if there are two positive constants and such that . The Carleson box based on a spherical cap is defined by

The tent based on is defined by

The cone in with the vertex is defined by

For any fixed , set

Clearly, if , is just the spherical cap in with center and radius .

For and a measurable function defined on , we denote by

the Poisson extension of onto . The nontangential maximal function of is the function

defined on .

For and any , we write

which is also an extension of onto .

For , we call the function

the centered Hardy-Littlewood maximal function of defined on .

The following two lemmas give a lower estimate and an upper estimate for the Poisson integral on . In the case of , one can refer to [20, Theorems 2.4 and  2.5].

Lemma 1. There exists a constant , such that

holds for all and , where is the Poisson extension of .

Proof. It is sufficient to prove that the estimate

holds for any . Since as , by the estimate in (13), it is easy to obtain the conclusion.

Lemma 2. There exists a constant , such that

holds for all and .

Proof. Clearly, it is the consequence of the definitions of and . Noticing that as , we have

for any . Using (13), it implies the required conclusion.

The following two lemmas are well known. One is the generalized maximal theorem; the other is the fact that the nontangential maximal function can be pointwise controlled by maximal function (see [21]).

Lemma 3. Let be a measurable function defined on .

(a)If and , is finite almost everywhere. (b)If , then for any ,

(c)If and , then and

where depends only on and .

Lemma 4. If , , then holds for almost every .

For the convenience, we define Carleson measures on in terms of Carleson boxes or tents.

Definition 5. Let and a spherical cap in . A nonnegative measure on is called an -Carleson measure if there exits a constant such that

is called a compact -Carleson measure if is an -Carleson measure and

Remark 6. (1) Comparing Definition 5 with the definition in (8), one can see that the in Definition 5 is equal to in (8).

(2) For any , we can replace with in Definition 5. (3) For , it goes back to the one in [2].

For , a well-known result which is due to Carleson in [1] for and Duren in [22] for says that a nonnegative measure on the unit disc of the complex plane is a bounded -Carleson measure if and only if

By Definition 5 and using the real analysis techniques, we obtain the following extension of this result on .

Theorem 7. Let be a nonnegative measure on .

(a)For , let be a -measurable function defined on and let be the nontangential maximal function of . If is a -Carleson measure on , then

(b)For , is a -Carleson measure on if and only if

for any , and here is the Poisson integral of .

Proof. It is sufficient to (31) if

Indeed, write , and above if the inequality holds, then

which is what we need.

Suppose that is a -Carleson measure on . Write . Recalling the Whitney decomposition (see [21]), we know that there exists a disjoint collection of spherical caps such that , and here is the spherical cap with the same center as but radius times. Now, we claim that

Indeed, let so that . By the definition of , we have that for all satisfying . Thus, , and clearly .

Clearly, we have

Thus, we have

By Minkowski’s inequality acting on the last inequality, we have

Then, (31) follows.

Now, turn to the proof of the “only if” part of (b). If is a -Carleson measure, we can obtain

as a direct sequence of (a). Noting Lemmas 3 and 4, we complete the proof of the “only if” part of (b).

The proof of the “if” part of (b) is easy. For any spherical cap in , if , we have . Let . Since the simple fact as and the estimate in (13), we obtain the useful estimate

Now, it is easy to see

which is to say that is a -Carleson measure. The proof of (b) is completed.

Let be a spherical cap in with center and radius . Let be a nonnegative integer and, be the greatest integer less than . Denote the spherical cap with the same center as but radius by . Then, and . Moreover, for any spherical cap , we have

Lemma 8. For fixed , let be a spherical cap in with center and radius . Then, one has the following estimates:

(i), if , (ii), if , .

Proof. If , then we have the estimates

Part (i) is yielded by the above.

For part (ii), if , we have

The proof is completed.

Combining together Definition 5, the decomposition of in (42), and Lemma 8, we can obtain the following characterization for -Carleson measures on , which is similar to [14, Theorems 2.3 and 2.4].

Theorem 9. Let and . A nonnegative measure on is an -Carleson measure or a compact -Carleson measure if and only if

or

In particular, when , is an -Carleson measure or a compact -Carleson measure on if and only if

3. The Carleson Measures Characterized by Behaviors

By the estimate in (13) and Theorem 9, we observe that

For , , and , write

By Theorem 9, it is clear that is an -Carleson measure on if and only if .

For a spherical cap and , define

It is also clear that is an -Carleson measure on if and only if for all .

Now, we are going to state our main arguments in this section, which are devoted to establishing the connections between -Carleson measures on and

Here, we emphasiz that the following results have been achieved when (see [13]).

Theorem 10. Let , , and a nonnegative measure on .

(i) if and only if for all spherical cap . (ii) if and only if for all spherical cap .

Remark 11. (1) In Theorem 10, the Carleson box can be replaced by the tent .

(2) The results in the above theorem hold for on (see [13, Theorem 1]), but they do only for here.

Proof. For any spherical cap , there must exist such that . By Lemma 8, we have

Thus,

The “only if” parts of (i) and (ii) are concluded from the above inequality.

To prove the “if” part of (i), let and . By the decomposition of in (42), it is clear that

By Lemma 8, we have

Taking norm on both sides of the above inequality, if , we have

If , then

The proof of the “if” part of (i) is completed.

With the same technique used in the proof of “if” part of (i), one can obtain the “if” part of (ii). Suppose that . We observe first that for , there must exist an integer such that

Then, we have

By the assumption of the “if” part of (ii), there must exist such that if

If , then we have

which is the desired result. With the same process, we can obtain the results when . The proof of Theorem 10 is completed.

Theorem 12. Suppose that , , and is a nonnegative measure on .

(i) or for all spherical cap if and only if   is an -Carleson measure for all . (ii) or for all spherical cap if and only if   is a compact -Carleson measure for all .

Proof. In the situation of , the results are deduced by Theorems 9 and 10 and the estimate

For , it is well known that if . By Lemma 2, we have

Now, by the estimate in (13) and Hölder’s inequality, we have

Combining with Theorem 10, we complete the proof of “only if” parts.

Turn to the proof of “if” parts. For and , suppose that is an -Carleson measure or a compact -Carleson measure. By Lemma 1, we know that

which implies that is an -Carleson measure or a compact -Carleson measure. By the estimate in (13), we have

Using the above estimate, the duality, and also Theorem 10, we deduce the desired results.

Now, we are going to consider the case of .

Theorem 13. Let and a nonnegative measure on . Then, the following are equivalent.

(a1) or for all spherical cap .(b1) is an -Carleson measure for all . Moreover, the following are equivalent.(a2) or for all spherical cap .(b2) is an -Carleson measure for all .(c2) is an -Carleson measure.

The equivalence above holds for the compact case also.

Proof. For and , we have

which implies the equivalence of (a1) and (b1) by duality theorem and Theorem 10.

Invoking Lemma 2, we observe that “(a2)⇒(b2)” is a direct consequence of the equivalence of (a1) and (b1).

To prove “(b2)(c2),” let be any fixed point in and let

be a function defined on . For , one should observe the fact and

by Lemma 8 and the estimate (13). Then, the assumption that is an -Carleson measure implies that

holds for any , which is to say that is an -Carleson measure.

To prove “(c2)⇒(a2),” assume that is an -Carleson measure. For any , let be the sphere cap . Let be a nonnegative integer. For some , write . Let be the spherical cap with the same center as but radius . We choose a number small enough such that

Here is the top of . We observe the fact that

holds for any . Now, it is easy to see that for any

which is to say that . The proof is completed.

Theorem 14. Let , , and a nonnegative measure on .

(i) or for all spherical cap if and only if is an -Carleson measure. (ii) or for all spherical cap if and only if is a compact -Carleson measure.

Proof. To prove the “if” parts, suppose that is an -Carleson measure or a compact -Carleson measure. By Theorem 10, it is sufficient to prove that

holds for all .

For , the results are trivial because of the fact of .

For , let and (here is the conjugate of , that is, ). By Lemma 1, we have

Let . It is easy to see that and (here is the conjugate of ). Now, by the previous assumption, we have that is a -Carleson measure. Using Hölder’s inequality and Theorem 7, we have

By the duality theorem, we conclude that

which completes the proof of “if” parts.

We now turn to the “only if” parts. Assume that or for all spherical cap . For a spherical cap , we have

Then, by the estimate in (13) and Fubini’s theorem, we obtain that

Since , Jensen’s inequality and the above inequality imply that

Moreover, by the assumption and the remarks below Definition 5 and Theorem 10, we have

Then, we conclude that is an -Carleson measure or a compact -Carleson measure. The proof of Theorem 14 is completed.

Theorems 10 and 14 imply the following characterizations for -Carleson measures on as .

Corollary 15. Let , and a nonnegative measure on .

(i) is an -Carleson measure on if and only if

(ii) is a compact -Carleson measure on if and only if

Remark 16. By Theorem 13, the above corollary holds if and .

For , we have the following results.

Theorem 17. Let and a nonnegative measure on .

(a)If   and is an -Carleson measure on , then

(b)If   and or for all , then is an -Carleson measure.(c)If , then for all if and only if

Proof. To prove (a), it is sufficient to prove

for any spherical cap . For and any spherical cap , by Hölder’s inequality and Fubini’s theorem, it is clear that

Then, by the assumption that is an -Carleson measure, it is easy to see that .

For (b), let , and then (here is also the conjugate of ). Noticing the fact that for any , we obtain

By Theorem 10, we complete the proof of (b).

For (c), since condition for all is equivalent to , it is the direct consequence of the estimate for

and the duality theorem with appropriate choices of .

Acknowledgments

The authors are partially supported by grants from the NSF of China (11271162) and the NSF of Zhejiang province (Y6100810).

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