Research Article | Open Access
Wavelets, Sobolev Multipliers, and Application to Schrödinger Type Operators with Nonsmooth Potentials
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 .
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 , 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 ). 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 ). 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 , and Yuan et al.  for further information on wavelet characterization of Morrey spaces. The special case has been studied by Yang  and Yang and Zhou . 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  used this equivalence to characterize . See also Yang . 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.
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  gave a counterexample to show that , where is an integer. Recently, Lemarié-Rieusset  and Yang and Zhou  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  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. , Lemarié-Rieusset , Maz'ya and Shaposhnikova , Maz'ya and Verbitsky [11, 13], Yang and Yang , Yang et al. [15–17], Yang and Zhou , 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 ). 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 (). 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 .
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 . See also Yuan et al.  and Yang .
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.  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 , and Yuan et al. .
Morrey spaces were introduced by Morrey in 1938 and played an important role in the research of partial differential equations. Xiao  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. , Dafni and Xiao [29, 30], Peng and Yang , Wu and Xie , Yang , and Yuan et al. .
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.  for the proof.
Theorem 6. Given , and , if and only if for any with
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
Lemma 10 (see [18, Chapter 8, Lemma 1]). Let . For , the coefficients satisfy the following condition:
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.
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 .
Theorem 15. If , and , there exists a series of -combination atoms such that .
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 ,