Paraproduct in Besov--Morrey spaces
Yoshihiro Sawano

TL;DR
This paper extends the theory of paraproducts to Besov--Morrey spaces, which are important in analyzing highly singular partial differential equations, providing a self-contained mathematical framework.
Contribution
It introduces and develops the theory of paraproducts specifically for Besov--Morrey spaces, filling a gap in the mathematical analysis of singular PDEs.
Findings
Established paraproduct bounds in Besov--Morrey spaces
Provided a self-contained framework for these spaces
Facilitated analysis of highly singular PDEs
Abstract
Recently it turned out that the paraproduct plays the key role in some highly singular partial differential equations. In this note the counterparts for Besov--Morrey spaces are obtained. This note is organized in a self-contained manner.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Mathematical Physics Problems · Differential Equations and Boundary Problems · Advanced Harmonic Analysis Research
Paraproduct in Besov–Morrey spaces
Yoshihiro Sawano
Abstract.
Recently it turned out that the paraproduct plays the key role in some highly singular partial differential equations. In this note the counterparts for Besov–Morrey spaces are obtained. This note is organized in a self-contained manner.
Key words and phrases:
Besov–Morrey spaces, paraproduct
1. Introduction
In this note we investigate the boundedness property of the pointwise multiplier of the functions in Hölder–Zygmund spaces and Besov–Morrey spaces including the commutators. Starting from the seminal papers [2, 3, 4], we investigate these operators from the viewpoint of harmonic analysis.
To describe our first result, we recall some notation. First, we use the following convention on balls in here and below: We denote by the ball centered at of radius . Namely, we write
[TABLE]
when and . Given a ball , we denote by its center and by its radius. We write instead of , where . Keeping this definition of balls in mind, we define Morrey spaces. Let . Define the Morrey norm by
[TABLE]
for a measurable function . The Morrey space is the set of all the measurable functions for which is finite. We move on to the definition of Besov–Morrey spaces. Choose so that
[TABLE]
We write
[TABLE]
for and .
For , define the Fourier transform and the inverse Fourier transform by:
[TABLE]
Here and below we write for and . It is known that and it satisfies
[TABLE]
for all . We define
[TABLE]
for .
Let , and . The space , which we call the Besov–Morrey space, is the set of all for which the norm is finite. The parameter describes the differential property in terms of Morrey spaces as is indicated by the relations and for all and . It is also clear from the triangle inequality in that . The main results in this note are the following:
Theorem 1.1**.**
Let , , , , and . Assume that
[TABLE]
Then for and the product makes sense and satisfies
[TABLE]
Theorem 1.1 is an extension of the inequality
[TABLE]
for and . The proof of Theorem 1.1 hinges on the paraproduct introduced by Bony [1]. Let . The (right) paraproduct is defined to be
[TABLE]
while the (left) paraproduct is defined to be
[TABLE]
Furthermore, the resonant operator is defined by
[TABLE]
We need some assumptions on and to justify these definitions. These three linear operators are key linear operators used in the proof of Theorem 1.1.
Another aim of this paper is to extend the results used in [2, 4], which also use these operators, to the Morrey setting:
Theorem 1.2**.**
Assume that the parameters satisfy
[TABLE]
Then for , and
[TABLE]
This result is a counterpart to [2, Lemma 2.4].
Here we briefly recall how Besov–Morrey spaces emerged. See [12, 19] for an exhaustive account. The first paper dates back to 1984. In [9] Netrusov considered Besov–Morrey spaces. Later on Kozono and Yamazaki investigated Besov–Morrey spaces and applied them to the Navier–Stokes equations [6]. Mazzucato expanded this application more in [8]. Decompositions of Besov–Morrey spaces can be found in [7, 14, 16]. After that Yang and Yuan defined Besov-type spaces and Triebel–Lizorkin-type spaces in [17, 18]. A close relation between these spaces is pointed out in [15]. Recently more and more is investigated. For example, Haroske and Skrzypczak investigated embedding relation of Besov–Morrrey spaces [5]. One of the important consequence of definining the Besov–Morrey spaces is that we have the embedding
[TABLE]
for . See [13].
We organize this paper as follows: Section 2 is devoted to collecting some preliminary facts. In Section 3 we prove Theorem 1.1 and in Section 4 we prove Theorem 1.2.
2. Preliminaries
2.1. Schwartz distributions and the Fourier transform
Let us recall the notation of multi-indexes to define the Schwartz space . By “a multi-index”, we mean an element in . In this paper a tacit understanding is that all functions assume their value in . For a multi-index , we define . For a multi-index and , we set
[TABLE]
Definition 2.1** (Schwartz function space ).**
For multi-indexes and a function , write , temporarily. The Schwartz function space is the set of all the functions satisfying
[TABLE]
The elements in are called the test functions.
Denote by the set of all continuous linear mappings from to . Denote by the value of evaluated at ; .
Note that is embedded into and that mapsto isomorphically to itself. Thus by duality mapsto isomorphically to itself.
A function is said to have at most polynomial growth at infinity, if for all , there exist and such that:
[TABLE]
Here we are interested in the inclusion:
[TABLE]
for having at most polynomial growth at infinity. Usually we assume that is compactly supported.
Let be a bounded set in . Denote by the set of all distributions whose Fourier transform is contained in the closure . Define .
Lemma 2.2**.**
- (1)
For all , ,
[TABLE] 2. (2)
Let be a compact set. Then for all , , holds.
Proof.
- (1)
The proof of (2.3) is standard: Simply write out the convolution in full in terms of the integral to have
[TABLE]
Since is compact and is closed, is a closed set. Thus, taking the closure of the above inclusion, we conclude that (2.3) holds. 2. (2)
Inclusion is a consequence of and the fact that maps isomorphically.
∎
Define the convolution
by as long as the integral makes sense.
A band-limited distribution is a distribution whose Fourier transform is compactly supported.
Lemma 2.3**.**
For all band-limited distributions and all functions , holds.
Proof.
Let be such that . We need to show that
[TABLE]
By the definition of the Fourier transform this amounts to showing:
[TABLE]
Since , we have
[TABLE]
from the definition of the pointwise multiplication for and . We note that
[TABLE]
Thus, by the definition of the Fourier transform acting on
[TABLE]
From the definition of the Fourier transform if and only if . Since , we have
[TABLE]
thanks to Lemma 2.2. Thus, and holds. ∎
Corollary 2.4**.**
For all band-limited , holds.
Proof.
Let be such that . We need to show that . Let be such that and that . Then
[TABLE]
since
[TABLE]
in . If is chosen so that , then we have
[TABLE]
thanks to Lemma 2.3, since and are both band-limited due to Lemma 2.2. Thus, . ∎
2.2. Lipschitz spaces and Hölder–Zygmund spaces
Let . We let be the set of all bounded continuous functions for which the quantity is finite. Let satisfy (1.1). We write
[TABLE]
for and as before. Then the (Besov)–Hölder–Zygmund space with . is defined to be the set of all for which
[TABLE]
is finite. Noteworthy is the fact that and are isomorphic for all but that is a proper subset of .
Usually we replace (1.1) by . However, if we pose a stronger condition (1.1) on , we can quantify what we are doing. The following is an example of such an attempt.
Example 2.5**.**
Let satisfy .
- (1)
We note that only if . In this case, we have
[TABLE] 2. (2)
Assume . Then since
[TABLE]
[TABLE]
Consequently,
[TABLE]
only if or , or equivalently .
2.3. Some estimates in Besov–Morrey spaces
For the paraproducts, we use the following observation:
Lemma 2.6**.**
Let , and . Suppose that we are given a collection satisfying , , and
[TABLE]
Then
[TABLE]
with
[TABLE]
Let and . Then define .
Proof.
Let be as before for each . . Then
[TABLE]
Thus,
[TABLE]
As a consequence
[TABLE]
as required. ∎
3. Paraproduct
3.1. Paraproduct
For the paraproducts, we use the following observation:
Lemma 3.1**.**
Let , , , , , , and . Assume that
[TABLE]
Suppose that we are given collections satisfying , , and
[TABLE]
Then we have
[TABLE]
and satisfies
[TABLE]
Proof.
Thanks to Corollary 2.4 we have for all . Thus by the equivalent expression (see Lemma 2.6) and the Hölder inequality, we have
[TABLE]
∎
3.2. Resonant part
To handle the resonant part, we use the following lemma. When we prove this type of estimates, we can use the atomic decomposition taking advanatage of the assumption and . Here we estimate the distributions directly. This corresponds to [2, Lemma A3].
Lemma 3.2**.**
Let , and . Suppose that we are given a collection satisfying , and
[TABLE]
Then
[TABLE]
Proof.
Let be as before for each . We have
[TABLE]
As a consequence, by the translation invariance of and the equality for all
[TABLE]
Since , by the Hölder inequality
[TABLE]
Thus, if we take the -norm, then we obtain
[TABLE]
∎
Corollary 3.3**.**
Let , , , , , , and . Assume that
[TABLE]
Suppose that we are given collections satisfying , and
[TABLE]
Then
[TABLE]
and
[TABLE]
Proof.
In fact, by Corollary 2.4, we see that . Thus, invoking Lemma 3.2 and using the Hölder inequality twice, we have
[TABLE]
∎
3.3. Conclusion of the proof of Theorem 1.1
We prove Theorem 1.1 as follows: If we use Lemma 3.1, then we have
[TABLE]
Since and , we have
[TABLE]
Thus,
[TABLE]
Recall that . Since , we have
[TABLE]
Likewise
[TABLE]
Meanwhile, we have
[TABLE]
by Corollary 3.3.
Putting together these observations, we obtain the desired result.
4. Commutator estimate
We recall the following lemma obtained in [2, Lemma 2.2]:
Lemma 4.1**.**
Let , , and let , . Then
[TABLE]
This is a slight extension of [2, Lemma 2.2] to the case where . Here for the sake of convenience for readers, we recall the whole proof.
Proof.
Since for all which grows polynomially at infinity,
[TABLE]
As a result, letting
[TABLE]
we have
[TABLE]
as required. ∎
Lemma 4.2**.**
Let , , and let , . Then we have
[TABLE]
This is also a slight extension of [2, Lemma 2.3] to the case where . Here for the sake of convenience for the readers we supply the proof.
Proof.
We assume ; otherwise we can mimic the argument below and we can readily incorporate the case where is not so large. We decompose
[TABLE]
Let be fixed. We use Lemma 4.1 to have
[TABLE]
Meanwhile, using
[TABLE]
for , we have
[TABLE]
∎
We prove Theorem 1.2 to conclude this note.
Proof.
We decompose
[TABLE]
We handle the first term; other two terms are dealt with similarly. We decompose
[TABLE]
Since
[TABLE]
for all , we have
[TABLE]
Using Example 2.5, we estimate the second term:
[TABLE]
Next, we note that
[TABLE]
Adding this estimate over , we have
[TABLE]
∎
5. Acknowledgement
The author thankful to Professor Alexey Karapetyants for his inviting me to the conference OTHA 2018. The author is also thankful to Professors Yuzuru Inahama and Masato Hoshino for their encouragement to write this note.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. (French), [Symbolic calculus and propagation of singularities Quantitative analysis in Sobolev imbedding theorems for and applications to spectral theory, nonlinear partial differential equations] Ann. Sci. École Norm. Sup. (4) 14 , no. 2, 209–246 (1981)
- 2[2] Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PD Es. Forum Math. Pi, 3:e 6, 75, 2015.
- 3[3] Gubinelli, M., Imkeller, P., Perkowski, N.: A Fourier approach to pathwise stochastic integration, Electronic J. of Probability 21 , Number 2016, paper no. 2, 37 pp. (2016)
- 4[4] Hairer, M.; A theory of regularity structures. Invent. Math., 198(2):269–504, (2014)
- 5[5] Haroske, D.D., Skrzypczak, L.: On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces, Rev. Mat. Complut. 27 , no. 2, 541–573 (2014)
- 6[6] Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data. Comm. PDE 19 , 959–1014 (1994)
- 7[7] Mazzucato, A. L.: Decomposition of Besov–Morrey spaces. Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), 279–294, Contemp. Math. 320 , Am. Math. Soc. Providence, RI, 2003.
- 8[8] Mazzucato, A. L.: Besov–Morrey spaces: Function space theory and applications to non-linear PDE, Trans. Am. Math. Soc. 355 , no. 4, 1297–1364 (2003)
