Abstract and Applied Analysis

Volume 2013 (2013), Article ID 193420, 22 pages

http://dx.doi.org/10.1155/2013/193420

Research Article

Wavelets, Sobolev Multipliers, and Application to Schrödinger Type Operators with Nonsmooth Potentials

Pengtao Li,¹ Qixiang Yang,² and Yueping Zhu³

¹Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China

²School of Mathematics and Statics, Wuhan University, Wuhan 430072, China

³School of Sciences, Nantong University, Nantong 226007, China

Received 31 July 2013; Revised 17 September 2013; Accepted 18 September 2013

Academic Editor: Natig M. Atakishiyev

Copyright © 2013 Pengtao Li et al. 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 employ Meyer wavelets to characterize multiplier space without using capacity. Further, we introduce logarithmic Morrey spaces to establish the inclusion relation between Morrey spaces and multiplier spaces. By fractal skills, we construct a counterexample to show that the scope of the index of is sharp. As an application, we consider a Schrödinger type operator with potentials in .

1. Introduction

As useful tools, multipliers on the spaces of differential functions are applied to the study of various problems in harmonic analysis and differential equations. A function is called a multiplier from a Sobolev space to another Sobolev space if for every function , the product . We denote by the class of all such functions . In the famous book [1], Maz'ya and Shaposhnikova gave many characterizations of different kinds of multiplier spaces. For different indices , , and , the multiplier spaces can adapt to the needs of different problems. Further, for different indices , , and , they need different skills to deal with different multiplier spaces. In this paper, we consider multiplier spaces defined as follows.

Definition 1 (see [1]). Given , and , the multiplier space is defined as the set of all the functions such that

For a compact set , the capacity is defined by

where denotes the Schwartz class of rapidly decreasing smooth functions on .

Lemma 2 (see [1]). Given and .

(i)For , if and only if

(ii)For and any cube with a length less than 1, the capacity is less than .

Our motivation is based on the following consideration. For complicated compact sets, it is very difficult to compute the capacity. The main aim of this paper is to establish a relation between and the Morrey spaces , , and by wavelets. See also Liang et al. [2, 3], Triebel [4], and Yuan et al. [5] for further information on wavelet characterization of Morrey spaces. The special case has been studied by Yang [6] and Yang and Zhou [7]. We point out that the result for is not a simple generalization. For , the Sobolev space becomes Lebesgue space . It is well known that

where is the Hardy-Littlewood maximal operator. Yang and Zhou [7] used this equivalence to characterize . See also Yang [6]. However, (4) does not hold for .

For the case , it is necessary to make some progress in technique. The difficulty is to deal with the impact of the maximal operator on the frequencies when we split a product of two functions. To overcome this difficulty, we introduce an almost local operator . See Definition 12. Let , with . In Theorem 21, we characterize by Meyer wavelets without using capacity. Also our method can be applied to study the relation between multiplier spaces and Morrey spaces.

Lemma 2 implies that . For the converse imbedding, Fefferman [8] established the following relation:

Let and . From the counterexample in Theorem 33, we can see that the product may produce a logarithmic type blowup on the fractal sets with Hausdorff dimension . To eliminate this defect, we introduce a logarithmic type Morrey space and prove that, for ,

where , , and . See Section 4.1. In (6), the scope of is , where . In Section 4.2, our counterexample implies that, for , there exists some function , but . See Section 4.2 for the details. Theorems 24 and 33 illustrate the difference between Morrey spaces and multiplier spaces.

Remark 3. For the case , some similar counterexamples have been obtained. Lemarié-Rieusset [9] gave a counterexample to show that , where is an integer. Recently, Lemarié-Rieusset [10] and Yang and Zhou [7] constructed some counterexamples for and . The counterexamples there are independent of the wavelet characterization of . Our counterexample depends on the wavelet characterization of multiplier space, Theorem 24, and fractal skills. See Theorem 33.

As an application, we apply our results obtained in Section 4 to the Schrödinger operator , where is the potential function. Maz'ya and Verbitsky [11] considered the multipliers from to . For a Schrödinger operator , they got many sufficient and necessary conditions such that is a multiplier from to . For more information, we refer the reader to Jiang et al. [12], Lemarié-Rieusset [9], Maz'ya and Shaposhnikova [1], Maz'ya and Verbitsky [11, 13], Yang and Yang [14], Yang et al. [15–17], Yang and Zhou [7], and the references therein.

Given , , , and , we consider the following equation:

where and . If is a function of Hölder class, one usual method to deal with (7) is the boundedness of Calderón-Zygmund operators. As a function in , may not be a function. In Section 5, by Theorem 32, we prove that if , (7) has an unique solution in the Sobolev space .

The rest of this paper is organized as follows. In Section 2, we state some notations and known results which will be used throughout this paper. In Section 3, we give a wavelet characterization of . In Section 4, we introduce a class of logarithmic Morrey spaces such that . Further, we construct a counterexample to prove the sharpness of the scope of the index . In the last section, we consider an application to PDE problem.

Notations. represents that there is a constant such that whose right inequality is also written as . Similarly, if , we denote .

2. Some Preliminaries

In this section, we state some notations, knowledge, and preliminary lemmas which will be used in the sequel. Firstly, we recall some background knowledge of wavelets and multiresolution analysis.

For any and , let and denote by the set of all dyadic cubes . For arbitrary set , we denote by the -multiple of . Finally, let , be the characteristic functions of the unit cube and , respectively.

We will adopt real-valued tensor product wavelets to study the multiplier spaces in this paper. Let be an orthogonal multiresolution in with the scaling function . Denote by the orthogonal complement space of in ; that is, . Let be an orthogonal basis in . For , denote .

In the proof, we use only Meyer wavelets and regular Daubechies wavelets. We say a Daubechies wavelet is regular if it has sufficient vanishing moment until order and , where the regularity exponent is large enough and is determined by ; see [18, 19] for more details. For any , , and , we denote . For and , let and be the characteristic functions on and , respectively. For simplicity, we denote by and for short.

In addition we define

For fixed tempered distribution , if we use wavelets which are sufficiently regular, then we can define . And the wavelet representation holds in the sense of distribution.

Let be the orthogonal multiresolution in with the scaling function . Denote by the orthogonal complement space of in ; that is, . Denote by and the projection operators from to and , respectively. Dobynski got a decomposition of the product of two functions and , which is similar to Bony's paraproduct (see [20]). Denote

By the projection operators and , we divide the product into the following terms:

To facilitate our use, we make a modification to (10) and use special wavelets for different cases. Let be a positive integer. We decompose the product as

and the term can be decomposed as

In 1970s, Triebel introduced Triebel-Lizorkin spaces ([21]). Many function spaces can be seen as the special cases for . For example, is the fractional Hardy space. For , are the Sobolev spaces . For , is the space defined as

where and denote the unit operator and the Laplace operator, respectively. Here denotes the set of all measurable functions with

where denotes the mean value of on . See also Section 3.1 of [1].

For and , it is well known that . The following lemma gives a characterization of via Meyer wavelets and regular Daubechies wavelets. For the proof, we refer the reader to Chapters 5 and 6 of Meyer [18]. See also Yuan et al. [22] and Yang [23].

Lemma 4. Given , let be Meyer wavelets or -regular Daubechies wavelets. Then,

(i)for ,

(ii) if and only if there exists such that for ,

The wavelet characterizations of function spaces have been studied by many authors. In [18, Chapters 5 and 6], Meyer established wavelet characterizations for many function spaces, for example, Hardy space, BMO spaces, Besov spaces, and Bloch space. Yang et al. [24] used wavelets to characterize Lorentz type Triebel-Lizorkin spaces and Lorentz type Besov spaces. For the wavelet characterization of Besov type Morrey spaces and Triebel-Lizorkin type Morrey spaces, We refer to Yang and Yuan [25, 26], Yang [23], and Yuan et al. [22].

Morrey spaces were introduced by Morrey in 1938 and played an important role in the research of partial differential equations. Xiao [27] established a relation between the homogeneous Morrey space and -type space by heat semigroup and the fractional integrals. In the recent 20 years, -type spaces are studied extensively. See Essèn et al. [28], Dafni and Xiao [29, 30], Peng and Yang [31], Wu and Xie [32], Yang [23], and Yuan et al. [22].

Define as

The Morrey spaces are defined as follows.

Definition 5. Given and , the Morrey space is defined as the set of all measurable functions such that and

where is any cube in with .

Morrey spaces can be also characterized by wavelets. We state it as the following theorem and refer to Yuan et al. [22] for the proof.

Theorem 6. Given , and ,

if and only if for any with

The following lemma can be obtained by Lemmas 2 and 4 immediately.

Lemma 7. Given , and , then .

Now we give two lemmas about .

Lemma 8. Given , and , then .

Proof. For any dyadic cube , we have

For any , and , we denote . We can get the following result.

Lemma 9. Suppose that and . The wavelet coefficients of satisfy

Proof. Take and . We consider two cases and separately.

(i) For , by Lemma 4, we get

 It is easy to see that . (ii) For ,

 Because , we have

Let and be two functions such that

Write . The following lemma can be found in Chapter 8 of Meyer [18] or Chapter 6 of Yang [23].

Lemma 10 (see [18, Chapter 8, Lemma 1]). Let . For , the coefficients satisfy the following condition:

By Lemma 4, the boundedness of Calderón-Zygmund operators on is equivalent to the following lemma. We refer the reader to [18, 23, 33] for the proof.

Lemma 11. Suppose that and . Let . If the coefficients satisfy (27), then

We say that is a local operator if there exists some constant such that, for all and , maps a distribution with the support to another distribution supported on the ball . If is not a nonnegative integer, the operator is not a local operator. Now we use wavelets to construct some special fractional differential operators , which are almost local operators and will be used in the proof of our main result.

Definition 12. For and , we call that is an operator associated to the kernel if

It is easy to prove that is the identity operator and for . Furthermore, we have the following.

Lemma 13. Suppose . For any and ,

where is the Hardy-Littlewood maximal operator.

Proof. If , the proof was given by Meyer [18]. Now we consider the case . It is easy to verify that

By the fact that , we have

Hence, we can get

This completes the proof of Lemma 13.

In the rest of this section, we give a decomposition of Sobolev spaces associated with combination atoms. For and , denote

and, for , denote also .

Definition 14. Let and . For arbitrary measurable set with finite measure, we say that is a -combination atom if and . If is a dyadic cube, then we say is a -atom.

In the next theorem, we give a combination atom decomposition of Sobolev spaces. This result is a generalization of that of Yang [34].

Theorem 15. If , and , there exists a series of -combination atoms such that .

Proof. Denote

For , let . By wavelet characterization of Sobolev spaces, we have . Let , where are disjoint maximal dyadic cubes with . Let be the set of dyadic cubes contained in but not in , and . Let , and we can write also , where are disjoint maximal dyadic cubes in . The related set is defined as .

For any , we write

Then is a desired combination atom. This completes the proof.

3. Wavelet Characterization of the Multiplier Spaces

In this section, we use Meyer wavelets to characterize . Let be a function in satisfying and . For any , define

The function space is defined as follows.

Definition 16. Given , and , we say if and

where and .

Now we give a wavelet characterization of . Let and , be the scaling function and wavelet functions, respectively. For , , and , let

Furthermore, for , , , and , let

By the same method of [18, Chapter 8, Lemma 1], we could prove the following lemma.

Lemma 17. There exist sufficient big integers , , and such that and the following estimates hold.

(i)If , , , and , then

(ii)If , , , , , and , then

(iii)If , , , and , then

(iv)If , , , , , and , then

3.1. Decomposition of Products via Multilinear Analysis

At first, we give a wavelet decomposition of the product of . Let and be the scaling function and wavelet functions of Meyer wavelets, respectively. There exists an integer such that

Denote

For , , , and , we denote

For , or , and , denote

By (11) and (12), we can decompose as follows:

If , write . For , and , we define

Let

It is easy to see that

3.2. Several Technical Lemmas

Now we estimate the quantities , , , and separately. Let

For fixed , there is only one such that and the number of is finite. Then

Let be the Hardy-Littlewood maximal function. Then, if and , we have

Lemma 18. Given , , and . If , then

(i)for and ,

(ii)for and ,

Proof. (1) For , we have

By Lemma 17, we know

By Lemma 9, implies . Now we can get

Because ,

Now we estimate the term

Because , by Lemma 9, we have . By Lemma 17,

We can get, similarly,

(2) If , the estimates of and are easier than those of and . For example, we estimate the term

Because , by Lemma 9, . By Lemma 17,

Because , we can obtain

The estimate for can be obtained similarly. By the same methods used in (1) and (2), we can get the estimate of the term . We omit the details.

Now we consider the term . We have the following claim.

Lemma 19. Given , , and , if , then

Proof. In fact, for , and , let

We have

Because , by Lemmas 9 and 13, we have

Because

we can get

Hence, we have

Finally, we obtain, by Hölder's inequality,

This completes the proof of Lemma 19.

In order to deal with the term

we need the following estimate.

Lemma 20. Given , with , if and only if

Proof. In fact, for ,

By Lemma 4, using Hölder's inequality, we have

Because , we can see that .

Conversely, let

Denote . For , we write as

It is easy to see that is equivalent to .

By the wavelet characterization of , we get

Further, we can deduce that

Hence and . This completes the proof of this lemma.

3.3. Wavelet Characterization of Sobolev Multipliers

Theorem 21. Let , with . Then,

Proof. We first assume . Let and . By the decomposition given in Section 3.1, we have

Notice that implies that . By Lemmas 18–20, we obtain that, for , and

which gives

Hence, we can get .

Conversely, assume that . By Lemma 7, we have . We apply Lemmas 18 and 19 again to deduce

Then, Lemma 20 implies that . This completes the proof of this theorem.

Now we characterize by the fractional integration. Let , with on . It is well known that if and only if there exists such that

For , let

Definition 22. Given , and , then if and only if

where and .

Let and . Define

Similar to Lemma 20, we can get the following.

Lemma 23. Given , and , then if and only if

where and .

Now we give another characterization of .

Theorem 24. Given , and , then

Proof. By modifying the coefficients such that , we could suppose that . By Theorem 21 (ii) and Lemma 23, we know that is equivalent to

By a simple calculation, we get

So (95) is equivalent to the following inequality:

That is to say

Let

We have . Then, the inequality (98) is equivalent to

The above inequality is equivalent to .

4. Logarithmic Morrey Spaces and Multipliers

4.1. A Logarithmic Condition

Lemma 7 implies that . In this section, we consider the reverse inclusion relation. At first we introduce a logarithmic Morrey spaces.

Definition 25. Fix and . We say if and

for any cube with .

Similar to Theorem 6, we have the following wavelet characterization of .

Theorem 26. Given , , , belongs to if and only if

where with .

Proof. Similar to that of Theorem 6, the proof of this theorem can be obtained by the characterization of Triebel-Lizorkin spaces. See Lemma 4. We omit the detail.

In [8], Fefferman established the following relation:

where . In this section, we use wavelet characterization to give a logarithmic type inclusion. Let , , , and . In Theorem 32, we prove that is a subspace of . Hence,

Lemma 27. Given , , , and , if , then

Proof. Because , then for ,

We have

For , , and , take . By Lemma 4, we get

which gives

When ,

Since

we have

Let , , where denotes the set of all positive integers. Denote by the derivative . We have the following two lemmas.

Lemma 28. Given , , , and , if , the derivative , where .

Proof. If and , by Theorem 26, we have

Denote by the wavelet coefficients of . We can get and

This implies that .

To get the sufficient condition for multiplier spaces, we need to consider carefully the relationship of different dyadic cubes relative to combination atoms. Because of this reason, we use always the same Daubechies wavelets in the rest of this section. For a cube and a measurable set , if , we say that is a cube in the set .

For -combination atoms defined in Definition 14, the measure of is finite. Hence, the number of biggest dyadic cubes in is finite. We denote the number of biggest dyadic cubes in by . Denote by the set . If , we denote such cube by . The volume of is denoted by ; that is, . Denote .

The measure of is finite. We denote the number of biggest dyadic cubes in by . Denote by the set . If , we denote such cube by . The volume of is denoted by ; that is, . Denote .

We continue this process until there exists some such that is empty. For , we denote , and and are empty sets. Otherwise we continue until infinity. Then and , where .

To compute the norm of , we need to find out a special set of dyadic cubes denoted by such that . is nearly function on and satisfies the estimate of Lemma 29. We divide such process into the following three steps.

Step 1. For all , if   and with , we denote . Denote . For , we denote . For , we denote . In the next step, we choose a special subcover to the support of .

Step 2. We define now , and .

For , denote and . For , denote , if there exists , such that and . We know that . Denote and .

For , denote , if there exists , such that and . We know that . Denote and .

We continue this process until infinity. For , maybe, a part of , , and is empty set.

Step 3. Let . It is easy to see that the support of is contained in . For a -combination atom and , we have the following estimate.

Lemma 29. Given , , and , for and , we have .

Proof. By the definition of , we have

Because is a -combination atom, . Hence, for every , . We can obtain

Theorem 30. Suppose that , , , and . If and is a -combination atom, then, for , , and , we have

Proof. First, for , we prove . Let and . Denote by the set

For any dyadic cube , by the formulas (11) and (12), we decompose as

The rest of the proof is divided into three steps.

Step 1. For , we estimate . By Lemma 4, we obtain

Let

We have .

(1) Because , by Lemma 27,

By Lemma 29, we have . Hence we can get

Because , we have

(2) Now we estimate

Because , we have

Let

Then, we can get

Because is a -combination atom,

(3) Since is a -combination atom, for , , , and ,

Let

We have, by ,

By the fact that and , we get

(4) Now we estimate the term . Let

Because , we have

Then we can get, by the fact that is a -combination atom,

(5) Now we estimate the term

Because the function plays the role as that of , we have

(6) To estimate the term , we take . Let

By the orthogonality of the wavelet function, we have

Then,

By Hölder's inequality and , it can be deduced that

Finally, we can get

Because takes over all functions in , it is obvious that

Step 2. Assume that . Let and

We need to prove that

The index sets , are the same as Step 1.

(1) For the term , we have

Then, we have

(2) For the term

we have

Because is a -combination atom, we have

(3) Because is a -combination atom, for , , and , we can get

From the fact that and , we deduce that

(4) For the term , because and is a -combination atom, we have and

Then, we can get

(5) Now we estimate the term

Because the function plays the role as that of , we have

(6) For the term , we take . By the orthogonality of the wavelet functions, we have

We can get

By Hölder's inequality, we have

Because implies that , we can get

Because takes over all functions in , we can get

This completes the proof for the case .

Step 3. Now we consider the case . In this case, there exists an integer such that . For any with , the derivative of the product can be represented as

where . Denote and by and , respectively. Applying the conclusion in Step 1, we only need to prove

that is, .

If is a -combination atom, then

Hence,

On the other hand, if , then

and . We have

that is, .

For any cube , the function , and any dyadic cube , we divide the product into the following parts:

Similar to the method used in the case , we can complete the proof of the case . This completes the proof of this theorem.

By Theorem 30, we can get the following lemma.

Lemma 31. Given , , , and , if and is a -combination atom, then

Theorem 32. Given , , , and , then .

Proof. By Lemma 31, we have

4.2. The Sharpness for

In this section, applying our wavelet characterization of multiplier spaces and fractal theory, we prove that the scope of the index of obtained in Theorem 32 is sharp for . Precisely, by Meyer wavelets, we construct a counterexample to show that Theorem 32 is not true for the case .

Our key idea is to construct a group of sets composed by special dyadic cubes and fractal set with Hausdorff dimension . Denote by the union and . By the above dyadic cubes , we construct a special function , which is bounded on for all . The fractional integration bumps on the fractal set . Then we construct a multiplier such that its wavelet coefficients are based on these special dyadic cubes for all . Applying our wavelet characterization of multiplier spaces, we prove that the product of the above multiplier and the function will go out to the desired space .

Theorem 33. If , there exists such that .

Proof. We use Meyer wavelets and suppose that and . First of all, we construct a group of self-similar cubes such that the limitation is a set with Hausdorff measure .

We construct two series of integers and such that

Denote and . We take such that there exists satisfying that

For and , denote if , and for , we have

Denote , and, for , denote , if there exists and such that .

We divide the unit dyadic cube into dyadic cubes. Then, we reserve the dyadic cubes which are near the vertex points; that is, we reserve and denote , if there exists such that .

For the dyadic cube , we divide it into dyadic cubes, and we reserve the dyadic cubes which are near the vertex points; that is, we reserve

For , denote

and denote if there exists such that .

We continue this process until infinity, and we get a series of dyadic cubes and sets . We know that and the limitation of is a fractal set with Hausdorff dimension .

For , let

where . Let . Applying the wavelet characterization of Morrey spaces, . In fact, for and any cube with , we have

Hence, we get

Let be a sufficient small positive real number. For , then . For and , we take

Then, we have . In fact,

Now we estimate the coefficients . We divide the estimate into two cases.

(1) For ,

(2) For , there exists such that . It is easy to see that is equivalent to .

Finally, we have

This completes the proof.

5. An Application to Schrödinger Type Operators

Let be a Schrödinger operator. Maz'ya and Verbitsky [11] established many sufficient and necessary conditions such that is a multiplier from to . Their results can be used to study the Schrödinger operators . In Section 4, Theorem 32 gives a relation between Morrey spaces and Sobolev multipliers. In this section, we give an application of the wavelet characterization of to the Schrödinger type operator .

For , , and , we want to find a solution to the equation

Remark 34. Fixed , , and .

(i)If there exists a such that is sufficiently small, according to the continuity of Calderón-Zygmund operator , (185) can be solved easily. But if we consider a nonsmooth potential , applying the same proof in Lemmas 8 and 9, it is possible that is not a function.(ii)The condition cannot be weaken to . In fact, according to our counterexample in Section 5, if , there exists some such that the operator is not continuous from to .

Now, we use our sufficient condition of multiplier spaces to get the solution of (185). We need the following two operators. For , let

Lemma 35. Given , , , and , if , the operator is bounded from to with the operator norm less than , where denotes a constant associated with , , , and .

Proof. By Theorem 32, for any , we have

Then, is a bounded operator on with the norm less than .

In the following lemma, we prove that is invertible in and the inverse can be written formally as .

Lemma 36. Given , , , and , if , the operator is invertible in .

Proof. By Lemma 35, is bounded on . Hence, for any ,

If , the above series is convergent in . Further,

Similarly, we can also get

that is, the operator is invertible in .

Theorem 37. Given , , , and , if , then, for , there exists a unique solution for (185).

Proof. Because , we have . By Lemma 36, the operator is invertible in . Hence, we can get that there exists a unique solution to the following equation in :

where . Hence, for the above ,

is a solution to (191). Write . Then, is equivalent to . It is easy to verify that is a solution to (185). This completes the proof.

Conflict of Interests

The authors declare that they have no conflict of interests in this submitted paper.

Acknowledgments

The research is supported by NSFC nos. 11171203, 11201280, and 11271209; New Teachers’ Fund for Doctor Stations, Ministry of Education 20114402120003; Guangdong Natural Science Foundation S2011040004131; MYRG116(Y1-L3)-FST13-QT; MYRG115(Y1-L4)-FST13-QT; and FDCT 098/2012/A3.

References

V. Maz’ya and T. O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, vol. 23 of Monographs and Studies in Mathematics, Pitman, 1985.

Y. Liang, Y. Sawano, T. Ullrich, D. Yang, and W. Yuan, “New characterizations of Besov-Triebel-Lizorkin-Hausdorff spaces including coorbits and wavelets,” The Journal of Fourier Analysis and Applications, vol. 18, no. 5, pp. 1067–1111, 2012.

Y. Liang, Y. Sawano, T. Ullrich, D. Yang, and W. Yuan, “A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces,” Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 489, 114 pages, 2013.

H. Triebel, Local Function Spaces, Heat and Naiver-Stokes Equations, vol. 20 of EMS Tracts in Mathematics, European Mathematical Society, Zürich, Switzerland, 2013.

W. Yuan, W. Sickel, and D. Yang, “On the coincidence of certain approaches to smoothness spaces related to Morrey spaces,” Mathematische Nachrichten, 2013.

Q. Yang, “Characterization of multiplier spaces with Daubechies wavelets,” Acta Mathematica Scientia B, vol. 32, no. 6, pp. 2315–2321, 2012.

D. Yang and Y. Zhou, “Localized Hardy spaces related to admissible functions on RD-spaces and applications to Schrödinger operators,” Transactions of the American Mathematical Society, vol. 363, no. 3, pp. 1197–1239, 2011.

C. L. Fefferman, “The uncertainty principle,” Bulletin of the American Mathematical Society, vol. 9, no. 2, pp. 129–206, 1983.

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, vol. 431, Chapman & Hall, Boca Raton, Fla, USA, 2002.

P. G. Lemarié-Rieusset, “Multipliers and Morrey spaces,” Potential Analysis, vol. 38, no. 3, pp. 741–752, 2013.

V. G. Maz'ya and I. E. Verbitsky, “The Schrödinger operator on the energy space: boundedness and compactness criteria,” Acta Mathematica, vol. 188, no. 2, pp. 263–302, 2002.

R. Jiang, D. Yang, and Y. Zhou, “Localized Hardy spaces associated with operators,” Applicable Analysis, vol. 88, no. 9, pp. 1409–1427, 2009.

V. G. Maz’ya and I. E. Verbitsky, “Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers,” Arkiv för Matematik, vol. 33, no. 1, pp. 81–115, 1995.

D. Yang and D. Yang, “Characterizations of localized via commutators of localized Riesz transforms and fractional integrals associated to Schrödinger operators,” Collectanea Mathematica, vol. 61, no. 1, pp. 65–79, 2010.

D. Yang, D. Yang, and Y. Zhou, “Endpoint properties of localized Riesz transforms and fractional integrals associated to Schrödinger operators,” Potential Analysis, vol. 30, no. 3, pp. 271–300, 2009.

D. Yang, D. Yang, and Y. Zhou, “Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators,” Communications on Pure and Applied Analysis, vol. 9, no. 3, pp. 779–812, 2010.

D. Yang, D. Yang, and Y. Zhou, “Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators,” Nagoya Mathematical Journal, vol. 198, pp. 77–119, 2010.

Y. Meyer, Ondelettes et Opérateurs, I et II, Hermann, Paris, France, 1992.

P. Wojtaszczyk, A Mathematical Introduction to Wavelets, vol. 37 of London Mathematical Society Student Texts, Cambridge University Press, 1997.

R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, “Compensated compactness and Hardy spaces,” Journal de Mathématiques Pures et Appliquées. Neuvième Série, vol. 72, no. 3, pp. 247–286, 1993.

H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1983.

W. Yuan, W. Sickel, and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, vol. 2005 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.

Q. Yang, Wavelet and Distribution, Beijing Science and Technology Press, 2002.

Q. X. Yang, Z. X. Chen, and L. Z. Peng, “Uniform characterization of function spaces by wavelets,” Acta Mathematica Scientia A, vol. 25, no. 1, pp. 130–144, 2005.

D. Yang and W. Yuan, “A new class of function spaces connecting Triebel-Lizorkin spaces and spaces,” Journal of Functional Analysis, vol. 255, no. 10, pp. 2760–2809, 2008.

D. Yang and W. Yuan, “New Besov-type spaces and Triebel-Lizorkin-type spaces including spaces,” Mathematische Zeitschrift, vol. 265, no. 2, pp. 451–480, 2010.

J. Xiao, “Homothetic variant of fractional Sobolev space with application to Navier-Stokes system,” Dynamics of Partial Differential Equations, vol. 4, no. 3, pp. 227–245, 2007.

M. Essén, S. Janson, L. Peng, and J. Xiao, “ spaces of several real variables,” Indiana University Mathematics Journal, vol. 49, no. 2, pp. 575–615, 2000.

G. Dafni and J. Xiao, “Some new tent spaces and duality theorems for fractional Carleson measures and ,” Journal of Functional Analysis, vol. 208, no. 2, pp. 377–422, 2004.

G. Dafni and J. Xiao, “The dyadic structure and atomic decomposition of spaces in several real variables,” The Tohoku Mathematical Journal, vol. 57, no. 1, pp. 119–145, 2005.

L. Peng and Q. Yang, “Predual spaces for spaces,” Acta Mathematica Scientia B, vol. 29, no. 2, pp. 243–250, 2009.

Z. Wu and C. Xie, “ spaces and Morrey spaces,” Journal of Functional Analysis, vol. 201, no. 1, pp. 282–297, 2003.

Y. Meyer and Q. X. Yang, “Continuity of Calderón-Zygmund operators on Besov or Triebel-Lizorkin spaces,” Analysis and Applications, vol. 6, no. 1, pp. 51–81, 2008.

Q. Yang, “Atomic decomposition in ,” Indian Journal of Pure and Applied Mathematics, vol. 31, no. 9, pp. 1081–1087, 2000.