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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 583156, 23 pages
Carleson Measure and Tent Spaces on the Siegel Upper Half Space
College of Mathematics, Qingdao University, Qingdao 266071, China
Received 22 April 2012; Accepted 31 October 2012
Academic Editor: Natig Atakishiyev
Copyright © 2012 Kai Zhao. 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.
The Hausdorff capacity on the Heisenberg group is introduced. The Choquet integrals with respect to the Hausdorff capacity on the Heisenberg group are defined. Then the fractional Carleson measures on the Siegel upper half space are discussed. Some characterized results and the dual of the fractional Carleson measures on the Siegel upper half space are studied. Therefore, the tent spaces on the Siegel upper half space in terms of the Choquet integrals are introduced and investigated. The atomic decomposition and the dual spaces of the tent spaces are obtained at the last.
It is well known that harmonic analysis plays an important role in partial differential equations. The theory of function spaces constitutes an important part of harmonic analysis. Heisenberg group, just as its name coming from the physicists Heisenberg, is very useful in quantum mechanics. Therefore, to discuss new function or distribution spaces and some characterizations of them is very significant in modern harmonic analysis and partial differential equations. Especially, the function spaces related to the Heisenberg group will be used in partial differential equations and physics. It is precisely the reason in which we are interested.
The Hausdorff capacity on is introduced by Adams in . Some limiting form is the classical Hausdorff measure. Adams also discussed some boundedness of Hardy-Littlewood maximal functions related to it. The capacity and Choquet integrals, in some sense, are from and applied to partial differential equations (see [2, 3]). As we know, the tent spaces have been considered by many authors and play an important role in harmonic analysis on (cf. [4, 5]). By using the Choquet integrals with respect to Hausdorff capacity on , the new tent spaces and their applications on the duality results for fractional Carleson measures, spaces, and Hardy-Hausdorff spaces were discussed (see ). The theory of spaces can be found in [7–11]). Inspired by the literature , in this paper, we will discuss the fractional Carleson measures and the tent spaces on the Siegel upper half space. In order to get the results in Section 3, some boundedness of Hardy-Littlewood maximal functions on the Choquet integral space with Hausdorff capacity on the Heisenberg group are discussed in Section 2. In Section 3, by the Choquet integrals with respect to Hausdorff capacity on the Heisenberg group, we will introduce the fractional Carleson measures on the Siegel upper half space. The characterizations of the fractional Carleson measures on the Siegel upper half space are obtained. And the dual of the fractional Carleson measures on the Siegel upper half space is also obtained. In the last section, the tent spaces on the Siegel upper half space in terms of Choquet integrals with respect to the Hausdorff capacity on the Heisenberg group are introduced. Then, the atomic decomposition and the dual spaces of the tent spaces are obtained. The fractional Carleson measures and the tent spaces on the Siegel upper half space will be used for spaces and the Hardy-Hausdorff spaces on the Heisenberg group (which will be discussed in another paper by us). On the other hand, they may be used in partial differential equations and quantum mechanics.
Let , , and be the sets of all integers, real numbers, and complex numbers, and , , and be -dimensional , , and , respectively. The Siegel upper half space in is defined by where . The boundary of is
The Heisenberg group on , denoted by , is a noncommutative nilpotent Lie group with the underlying manifold . The group law is given by where is the Hermitean product on .
It is easy to check that the inverse of the element is , and the unitary element is . The Haar measure on coincides with the Lebesgue measure on .
For each element of , the following affine self-mapping of : is an action of the group on the space . Observe that the mapping (1.4) gives us a realization of as a group of affine holomorphic bijections of (see ).
The dilations on are defined by , , and the rotation on is defined by with a unitary map of . The conjugation of is . The norm function is given by which is homogeneous of degree 1 and satisfies and for some absolute constant . The distance function of point and in is defined by .
For (or for , ), we define where , and .
To define the cube in , let , where denotes the largest integer less than or equal to , and . We define the function by and set . Then where (see ). Now a “cube" (in fact, called “tile" more precisely) with center and edge sidelength is defined by . Let (with the Lebesgue measure) be the volume of with length . It is easy to see that . Obviously, the diameter of denoted by is equal to , where is a constant depending only on . The “dyadic cubes" on can be defined by , , . We also call the “cube" and “dyadic cube" as cube and dyadic cube, respectively.
A ball in with center and radius is denoted as . The tent based on the set is defined by
The Schwartz class of rapidly decreasing smooth function on will be denoted by . The dual of is , the space of tempered distributions on .
2. Hausdorff Capacity on Heisenberg Group
In order to discuss the fractional Carleson measures on the Siegel upper half space, we introduce the -dimensional Hausdorff capacity and Choquet integral on the Heisenberg group in the following.
Definition 2.1. For and , the -dimensional Hausdorff capacity of is defined by where the infimum ranges only over covers of by dyadic cubes.
Remark 2.2. It is easy to check that satisfies the following properties.(i)If is nondecreasing, then .(ii)If is nonincreasing, then .(iii) has the strong subadditivity condition.
For a nonnegative function on , the Choquet integral with respect to the Hausdorff capacity on the Heisenberg group is defined by
Since is monotonous, the integrand on the right hand side in (2.3) is nonincreasing and then is Lebesgue measurable on . It is easy to see that the equality (2.3) with the Hausdorff measure is similar to the equality of the distribution function with Lebesgue measure.
In order to characterize the fractional Carleson measure in the Siegel upper half space in Section 3, we need to discuss the Hardy-Littlewood maximal operator on the Heisenberg group.
Let . , the dyadic Hardy-Littlewood maximal operator of , is defined by where the supremum is taken over all dyadic cubes containing .
The boundedness of the dyadic Hardy-Littlewood maximal operators on the Choquet integral spaces is as follows.
Theorem 2.3. Suppose . There exists a constant depending only on and such that
To prove Theorem 2.3, we need to prove the following lemmas first.
Lemma 2.4. For , let be a sequence of dyadic cubes in such that . Then there is a sequence of disjoint dyadic cubes so that and . Moreover, if , then the following tent inclusion holds, where is the cube with the same center as and times the sidelength ( is some constant depending only on ).
Proof. Since , , we have that . Thus the union of cannot form arbitrarily large dyadic cubes. Thus, each must be included in some maximal dyadic cube , where denotes the collection of all dyadic cubes . If we write these maximal cubes as a sequence , then is disjoint and . By the definition of in , we know that . Jensen’s inequality gives for . Consequently,
Suppose that and . Then and for some fixed . For this , we consider the parent dyadic cube , namely, the unique dyadic cube containing whose sidelength is double of that of . Since is maximal in , we have that is not a union of the ’s, that is, contains a point . Denoting the boundary of by , there is If is the cube with the same center as , and times the sidelength, then which means that . The proof is completed.
Lemma 2.5. Let be the characteristic function on the cube . Then
Proof. Let be the center of . By the definition of the maximal function , we obtain Since , there is The proof of Lemma 2.5 is complete.
Lemma 2.6. Let be a family of nonoverlapping dyadic cubes. Then there is a maximal subfamily such that for every dyadic cube ,
Proof. Similar to the proof in , if , then obviously satisfies (2.13). If have been chosen so that (2.13) holds, then we define as the first index such that the family satisfies (2.13). Continuing this proceeding, therefore, we have that is a maximal subfamily of satisfying (2.13). Hence (2.13) holds.
To prove (2.14), let be an index such that for some . Then by the proof of (2.13), there exists a dyadic cube such that Therefore, We can assume that . Otherwise (2.14) is obviously correct. Then the sequence is bounded. Thus, we can consider the family of maximal cubes of the family . Hence Consequently, by the definition of , we obtain The proof is complete.
Proof of Theorem 2.3. By the definition of , for any , there exist such that
where . If , we can choose . If , then
By Lemma 2.4, for each integer , there is a family of nonoverlapping dyadic cubes such that
Set , where is the characteristic function of . Then .
First, assume that . Then the Hölder inequality tells us Therefore, by Lemma 2.5, we have Now, assume that . Since , there is By using Jensen’s inequality, we obtain Thus, also by Lemma 2.5, there is That means that (2.5) holds.
For a given , let be the family of maximal dyadic cubes such that Note that is dyadic and . Since , there is Hence where the last inequality is due to the fact that Therefore, by (2.27) and (2.29), we obtain For the above , by Lemma 2.6 and (2.31), there exists a subfamily such that The proof of Theorem 2.3 is complete.
3. -Carleson Measure on Siegel Upper Half Space
Let be a cube in with center . The Carleson box based on is defined by
For , a positive Borel measure on the Siegel upper half space is called a -Carleson measure if there exists a constant such that
For , let be the cone at . Suppose that is a measurable function on . The nontangential maximal function of is defined by
Since the nontangential maximal function and Hausdorff capacity are defined, we are paying attention on the characterizations of the -Carleson measures.
Theorem 3.1. For , let be a positive Borel measure on . Then the following three conclusions are equivalent.(i) is a -Carleson measure.(ii) There exists a constant such that
holds for all Borel measurable functions on .(iii) For every , there exists a constant such that
holds for all Borel measurable functions on .
Proof. (i)(ii). Assume that is a -Carleson measure, and is a Borel measurable function on . For , let . If the integral on the right hand side of (3.4) is finite, we may assume that . Let be any of the dyadic cubes covering of with . Then Lemma 2.4 tells us that there is a sequence of dyadic cubes with mutually disjointed so that , and , where is the cube with the same center and times sidelength of .
If satisfies , then for all . Thus , and hence By Lemma 2.4, we obtain Taking an infimum over all such dyadic cube coverings, we have Therefore,
(ii)(i). Suppose that (3.4) is valid for all Borel measurable functions on . For a cube , let . Note that if and only if , there is . Then, by (3.4), we have It means that is a -Carleson measure.
(ii)(iii). Replacing with in (3.4) for , we immediately obtain
(iii)(ii). If we set in (3.5), then (3.4) holds.
The proof of Theorem 3.1 is complete.
To continue the characterization of the -Carleson measures on the Siegel upper half space, we need to prove the following Lemma 3.2. It is said that the nontangential maximal functions are dominated by the dyadic Hardy-Littlewood maximal operators on the Heisenberg group.
Lemma 3.2. Let be a locally integrable function on , and be a nonnegative radial and decreasing function with . Then where and is real.
Proof. Since , we can suppose that for some . Then Hence Lemma 3.2 is proved.
Theorem 3.3. For , , let be a positive Borel measure on . Then is a -Carleson measure if and only if holds for all functions on which can be written as , where is a locally integrable function on , and is the function in Lemma 3.2.
Proof. Suppose that is a -Carleson measure. By the conclusions of Theorems 3.1 and 2.3 and Lemma 3.2, there is
Conversely, for any cube , if we set , then for , there is Thus, by using the inequality (3.15), we obtain This ends the proof of Theorem 3.3.
Let be the space of all Borel measurable functions on satisfying Then gives a (quasi) norm and is complete.
The following result says that is the dual of -Carleson measure.
Theorem 3.4. Let . Then there exists a duality between the space of -Carleson measures and in the following sense.(i) Every -Carleson measure on defines a bounded linear functional on via the pairing (ii) Let be the closure in of the continuous functions with compact support in . Then every bounded linear functional on given via the pairing (3.20) by a Borel measure on is a -Carleson measure.
Proof. Assume that is a -Carleson measure and . By (3.19) and in Theorem 3.1, we have
Thus, is well defined. Hence, every -Carleson measure defines a bounded linear functional on . The part is proved.
For part , let be a bounded linear functional on . By the continuity and the closure of , applying the Riesz representation theorem, we obtain a Borel measure on having If for any cube , then Hence, is a -Carleson measure with . This ends the proof of Theorem 3.4.
4. Tent Spaces with Hausdorff Capacity
With Hausdorff capacity on the Heisenberg group discussed above, in this section, we introduce the tent spaces on the Siegel upper half space, an analogy of the Coifman-Meyer-Stein tent space on (cf. [4, 6]). Then the atomic decomposition of the tent spaces and the duality of the tent space are discussed.
Definition 4.1. Let . A Lebesgue measurable function on is said to belong to if where runs over all balls in , and is a tent based on .
Definition 4.2. Let . The tent space consists of all measurable functions on for which
where the infimum is taken over all nonnegative Borel measurable functions on satisfying
and is allowed to vanish only where vanishes.
We will identify with a dual space of . In order to do this, we first introduce the -atom as follows.
Definition 4.3. A function on is said to be a -atom, if there exists a ball such that is supported in the tent and satisfies
For the tent space on and , we have the triangle inequality with a constant in the following lemma.
Lemma 4.4. Let . If , then and
Proof. Without loss of generality, we assume that for all . Set . Then and . Suppose that for all such that and According to the definition of , the above inequality holds obviously. By using the Cauchy-Schwarz inequality, we have Let . Notice that the vanishing of implies the vanishing of all , which can only happen whenever all the vanish, that is, , then Thus, by the inequality (4.7), we obtain Taking the infimum on the left above inequality, we have The proof of Lemma 4.4 is complete.
Remark 4.5. By Lemma 4.4, one can show that is a quasinorm and the tent space is complete.
The main result of this section is the atomic decomposition of the tent space as follows.
Theorem 4.6. Let . Then a function on belongs to if and only if there exist a sequence of -atoms and an -sequence such that Moreover, where the infimum is taken over all possible atomic decompositions of .
Note that the right hand side of (4.12) in fact defines a norm and then becomes a Banach space.
Proof. Suppose that is a -atom on . Then there exists a ball such that supp and
Fix , and let
where denotes the distance between and in , and is a suitable constant which will be chosen later. Obviously, is identically equal to on the upper half ball of radius (center is , and ) and decays radially outside the ball (outside the ball , and ). For , the distance in from the cone to is . Then
Thus, by choosing . On the other hand, let on . We have
Therefore, and .
Now, taking a sum , where and every is -atom on , by Lemma 4.4, the sum converges in the quasinorm to with That means .
Conversely, let . We choose a Borel measurable function on such that (4.3) holds and
For each , let . By Lemma 2.4 and the definition of , it follows that there exists a disjoint dyadic cubes sequence such that where . Define Then is disjoint in for different . Therefore, Note that , implies . And each is contained in a cube of sidelength in . By using the subadditivity of the -Hausdorff capacity and (4.3), there is as . Thus, by (4.21), where is a set of zero -Hausdorff capacity, that is, Since is allowed to vanish only where vanishes, we have a.e. on . Let where the is a cube as Lemma 2.4, that is, .
Thus Note that, by (4.21), , where is the ball with the same center as and radius ( is a constant). Thus, is supported in , and This means that every is a -atom. The remaining is to estimate .
Notice the fact that By the Cauchy-Schwarz inequality, (4.19) and (4.20), we have If and every is a -atom, then, by Lemma 4.4, we obtain Thus where the infimum is taken over all atomic decompositions of . The proof is complete.
The dual result is as follows.
Theorem 4.7. Let . Then the dual of can be identified with under the pairing
Proof. We first show that
holds for all and .
In fact, assume that is a nonnegative Borel measurable function on satisfying inequality (4.3) in Definition 4.2 and . Then there exists a constant for any ball satisfying It means that is a -Carleson measure, and . Hence, by Theorem 3.1, we obtain Thus, for , by using the Cauchy-Schwarz inequality, there is This gives (4.34). Thus, every induces a bounded linear functional on via the pairing (4.33). It suffices to prove the converse.
Let be a bounded linear functional on and fix a ball . If is supported in with , then Therefore, is a multiple of a -atom with Hence, induces a bounded linear functional on . Thus, there exists a function which is locally in such that whenever with support in some finite tent . By the atomic decomposition of tent function in Theorem 4.6, obviously, the subspace of such is dense in . Therefore, the rest of the proof is to show that and .
Now, again fix a ball and for every , let where is the truncated of .