Abstract
In this paper, we consider some properties of homogeneous Besov–Lorentz spaces. First, we get some relationship between and Besov–Lorentz spaces, and then, we obtain the scaling property of and .
1. Introduction
In [1], Yang-Cheng-Peng introduced Besov–Lorentz spaces and Triebel–Lizorkin–Lorentz spaces by Littlewood–Paley decomposition and proved that the real interpolation spaces fall into the Triebel–Lizorkin–Lorentz spaces .
For the real interpolation of homogeneous Besov spaces , it is well known that when the index is fixed, are still Besov spaces. If , generally speaking, will fall outside of the scale of Besov spaces. There are many works which considered the real interpolation, see [1–12]. But does be the Besov–Lorentz space which is given in [1]? In this paper, we partly answer this question and get some relationship between and . Since the properties of function spaces are significant for PDE, furthermore, we consider the scaling property of and .
For homogeneous Besov spaces and Triebel–Lizorkin spaces, we use the characterization based on the homogeneous Littlewood–Paley decomposition, see [11]. Given a nonnegative function such that and if . Define
Then be a family of function satisfying
Define , the is called the th dyadic block of the Littlewood–Paley decomposition of . Let be the space of all Schwartz functions on . The space of all tempered distributions on equipped with the weak- topology is denoted by . Let be the space of all polynomials on , and let denote by the space of all tempered distributions modulated polynomials equipped with the weak- topology. For any , we recollect the definition of and .
Definition 1. Let , , and . Then,(i)for , , if ,(ii)for , if .When , it should be replaced by the supremum.
Based on the homogeneous Littlewood–Paley decomposition , we recall the definition of Triebel–Lizorkin–Lorentz spaces and Besov–Lorentz spaces which have been studied in [1]. Let , we denote by the Lebesgue measure of .
Definition 2. We assume that , , , and . Then,(i), if (ii), if As or , it should be modified by supremum.
The definition of the above spaces are independent of the choice of , see [1, 4, 11, 13, 14].
First, we consider some relationship between and . Then, we get the scaling property of and . These properties are important in Cauchy problem for nonlinear PDE, such as Navier–Stokes equations.
This paper is organized as follows. In Section 2, we list some background that shall be used in this paper. In Section 3, we give some relationship between and . In Section 4, we prove that and have scaling property.
In this paper, we denote by an estimate of the form with some constant C which is independent of the main parameters, but it may vary from line to line, means and .
2. Preliminaries
2.1. Real Interpolation
We recall that if is a pair of quasi-normed spaces which are continuously embedded in a Hausdorff space , then the is defined for all with and .
Definition 3. Let and , thenIf , then
In this subsection, we shall replace the continuous by a discrete variable . The relationship between and is . This discretization turns out to be a very helpful technical tool. Assume that . Let us denote by the sequences such that
The following result implies a discrete representation of the spaces . It was proved in [4].
Lemma 1. Let . If , then if and only if belong to . Moreover, we have
By Lemma 1, it is easy to see the following remark:
Remark 1.
Moreover, in this subsection we give some notations. We assume that is an arbitrary real number. We denote
Let . Observe the form of Definition 1, it is easy to see that is a retract of and is a retract of . It is
2.2. Lorentz Spaces
In the following part, we review the definition of Lorentz spaces which are a generalization of Lebesgue spaces. For , the distribution and rearrangement function given by the following formulas:
Then, for and , Lorentz spaces are defined in the following way
For ,
It is not difficult to see that . When , corresponds to the weak spaces. However, the previous formula is not very useful because it depends on the rearrangement function and we will use an equivalent characterization which has been studied in [1].
Definition 4. Supposed that , , and . Then, , ifas , it should be replaced by the supremum.
Remark 2. Observe that (14) is a form of the value of a function multiplied by the measure, then in comparison with Definition 4, we can discover the space is a retract of and is a retract of . It can be written as
So, we can obtain that Triebel–Lizorkin–Lorentz spaces and Besov–Lorentz spaces are the generalized Besov spaces and Triebel–Lizorkin spaces based on the Lorentz spaces . The following lemma is a classical result of real interpolation of Lebesgue spaces, see [4].
Lemma 2. Given , , and . We have
Furthermore, we recall the vector valued version of Minkowski’s inequality.
Lemma 3. Let , , and , then
2.3. Triebel–Lizorkin–Lorentz Spaces
Fixed the indices , the real interpolation spaces of Triebel–Lizorkin spaces are Triebel–Lizorkin–Lorentz spaces. The following theorem has been proved in [1].
Theorem 1. Let . Then,(i)(ii)
By (10) and (15), we can rewrite Theorem 1 as given in the following:
Remark 3.
Especially, for , we have
Corollary 1. Let . If , thenor equivalently,
3. Relationship between and
In this section, we give the relationship between the real interpolation spaces and Besov–Lorentz spaces .
Theorem 2. Let , , , and . Then,(i), if (ii), if (iii), if
Before the proof of Theorem 2, we first list several useful lemmas. Define the functional by
First, we recall an important lemma about , see [8].
Lemma 4. Let be a couple of quasi-normed spaces. For any , we have
We do not prove Theorem 2 directly. We prove the following equivalent theorem:
Theorem 3. Let , , and . Then,(i), if (ii), if (iii), if
It is easy to see that Theorem 3 implies Theorem 2.
Remark 4. Denote . For function and , denote . We haveHence, Theorem 3 is equivalent to Theorem 2.
Since , by Theorem 3, we can see that a part of real interpolation spaces are still Besov spaces. In fact, we have
Remark 5. Let . If , thenConsequently,
Now, we come to prove Theorem 3.
Proof. Let denote the sequence , andFor and , we deduceFor , we haveAs , by (28), Lemma 4 and using a discrete representation of the space which is described in Lemma 1, we deduce thatIf , by (29), thenWhen ,As , applying a discrete representation of the space , we haveApplying (33), we know thatWhen ,For ,(i)If , then . From (30), (34) and Minkowski’s inequality (17), it follows that As , by (32) and (35), we obtain So(ii)Conversely, if , then . In a similar way, by (30), (34) with Minkowski’s inequality (17), we obtain When , by (31) and (36) and the same as we did in (i), we omit the details. Hence,(iii)For , from (i) and (ii), it is easy to see that We finish the proof of Theorem 3.
By Remark 3, as contrasted with the real interpolation of Triebel–Lizorkin spaces, we conclude the following result:
Remark 6. Fixed the indices and , the real interpolation of Triebel–Lizorkin spaces satisfyThe real interpolation of Triebel–Lizorkin spaces for index can be directly interpolated as , but the real interpolation of Besov spaces for index can not be directly interpolated as , the relationship between depends on whether or .
Moreover, we need to point out that for the inhomogeneous spaces and , all the results of real interpolation are also true.
4. Scaling Property
In this section, we get the scaling property of and . Homogeneous spaces can be the critical spaces of many nonlinear partial differential equations. Critical spaces hold an important status in nonlinear partial differential equations. First, we recall the scaling property of Lorentz spaces, see [15].
Lemma 5. The Lorentz space is homogeneous, for any strictly positive number and any belonging to :
Suppose that is a tempered distribution, and consider the tempered distribution defined by , then, we have the following proposition.
Proposition 1. We assumed that , , , , , an integer , and a distribution of , then, we have
Proof. Let us recall:and then by definition of and the change of variable , we obtainBy Lemma 5, we haveThen,for the homogeneous Triebel–Lizorkin–Lorentz spaces, the proof is similar, we omit the details, see [16].
More generally, there exists a constant , depending only on , such that for all positive , we haveand the similar for . Then we have the following corollary.
Corollary 2. Let , , , and a distribution of , for any strictly positive number , we have
Remark 7. There have been a lot of results about the properties of these two types spaces, such as the wavelet decomposition characterization of and is already obtained, see [1], the boundedness of operators in the generalized Besov-type was considered in [17], and so on. These properties are important in Cauchy problem for nonlinear PDE. For instance, based on these properties, we can consider the well-posedness of the Navier–Stokes equations in and .
Data Availability
The data used to support the findings of this study are cited at relevant places within the text as references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank Dr. Fan Geng for her invaluable discussions and suggestions. This work was partially supported by the National Natural Science Foundation of China (Grant nos. 12071229 and 12261075), Scientific and Technological Innovation Team of Nonlinear Analysis and Algebra with Their Applications in Universities of Yunnan Province, China (Grant no. 2020CXTD25), and the Project of Yunnan Education Department Scientific Research Fund (Grant no. 2022J0809).