Besov-ish spaces through atomic decomposition

TL;DR

This paper introduces new function spaces akin to Besov spaces using atomic decomposition in measure spaces with grids, providing insights into multipliers and compositions for low-regularity cases.

Contribution

It develops a novel approach to constructing Besov-like spaces via atomic decomposition in measure spaces with minimal assumptions.

Findings

New Besov-like spaces constructed using atomic decomposition.

Results on multipliers for low-regularity spaces.

Analysis of left compositions in the new spaces.

Abstract

We use the method of atomic decomposition to build new families of function spaces, similar to Besov spaces, in measure spaces with grids, a very mild assumption. Besov spaces with low regularity are considered in measure spaces with good grids, and results on multipliers and left compositions are obtained.

Figures

Tables

Table 1. Table 1. Most important notation/symbols/constants

Symbol	Description
$I$	phase space
$m$	finite measure in the phase space $I$
$a_P,b_P$	an atom supported on P
$𝒜$	a class of atoms
$𝒜\left(Q\right)$	set of atoms of class $𝒜$ supported on $Q$
${𝒜}_{s,p}^{sz}$	class of $(s,p)$-Souza’s atoms
${𝒜}_{s,\beta ,p}^h$	class of $(s,\beta ,p)$-Hölder atoms
${𝒜}_{s,\beta ,p}^{bv}$	class of $(s,\beta ,p)$-bounded variation atoms
${𝒜}_{s,\beta ,p,q}^{bs}$	class of $(s,\beta ,p,q)$-Besov’s atoms
$𝒫$	grid of $I$
${𝒫}^k$	family subsets of $I$ at the $k$-th level of $𝒫$
$\text{Cr}menor\le \text{Cr}maior$	describes geometry of the grid $𝒫$
${ℬ}_{p,q}^s\left(𝒜\right)$	(s,p,q)-Banach space defined by the class of atoms $𝒜$
${ℬ}_{p,q}^s$ or ${ℬ}_{p,q}^s\left({𝒜}_{s,p}^{sz}\right)$	(s,p,q)-Banach space defined by Souza’s atoms
$P,Q,W$	elements of the grid $𝒫$
$L^p$ or $L^p\left(m\right)$	Lesbesgue spaces of $(I,𝔸,m)$
$|\cdot {|}_p$	norm in $L^p$, $p\in (0,\infty ].$
$p^{\prime }$	$1/p+1/p^{\prime }=1$, with $p\in [1,\infty ]$.
$\rho$	$\min \{1,p,q\}$.
$\hat{t}$	$\max \{t,1\}.$

Equations

$${\Omega }_Q^k=\{P\in {\mathcal{P}}^k:P\cap Q\ne \varnothing \}.$$

$$\text{Cl}[normal]mult1=\underset{k}{\sup }\underset{Q\in {\mathcal{P}}^k}{\sup }\#{\Omega }_Q^k<\infty .$$

$${\mathcal{D}}^k=\{[i/2^k,(i+1)/2^k),0\le i<2^k\}.$$

$${\mathcal{D}}_n^k=\{J_1\times \cdots \times J_n,with{J}_i\in {\mathcal{D}}^k\}.$$

$$x=(x_P{)}_{P\in \mathcal{P}},$$

$$|x|_{\ell_{q}(\ell_{p}^{\mathcal{P}})}=\Big{(}\sum_{k}\big{(}\sum_{P\in\mathcal{P}^{k}}|x_{P}|^{p}\big{)}^{q/p}\Big{)}^{1/q}<\infty,$$

$$a_k^{1/\hat{t}}\le C^{1/\hat{t}}{b}_k^{1/\hat{t}}{c}_k^{1/\hat{t}}.$$

$$(\sum_{k}a_{k}^{1/\hat{t}})^{\hat{t}/t}\leq C^{1/t}\Cl[normal]{co}(t,q,b)\Big{(}\sum_{k}c_{k}^{q/t}\Big{)}^{1/q},$$

$$(\sum _k{a}_k^{1/\hat{t}}{)}^{\hat{t}/t}\le C^{1/t}\text{Cr}co(t,q,b)\underset{k}{\sup }{c}_k^{1/t}.$$

$$\sum_{k}a_{k}^{1/\hat{t}}=\sum_{k}a_{k}^{1/t}\leq C^{1/t}(\sum_{k}b_{k}^{q^{\prime}/t})^{1/q^{\prime}}\big{(}\sum_{k}c_{k}^{q/t}\big{)}^{1/q}.$$

$$(\sum _k{a}_k^{1/\hat{t}}{)}^q$$

$$\sum_{k}a_{k}^{1/\hat{t}}\leq C(\sum_{k}b_{k}^{(q/t)^{\prime}})^{1/(q/t)^{\prime}}\big{(}\sum_{k}c_{k}^{q/t}\big{)}^{t/q},$$

$$(\sum _k{a}_k^{1/\hat{t}}{)}^{q/t}$$

$$\displaystyle\leq C^{q/t}\sum_{k}\big{(}b_{k}c_{k}\big{)}^{q/t}$$

$$\displaystyle\leq C^{q/t}(\sup_{k}b_{k}^{q/t})\Big{(}\sum_{k}c_{k}^{q/t}\Big{)}.$$

$$a_k^{1/\hat{p}}\le C^{1/\hat{p}}\sum _{i\in \mathbb{Z}}{b}_{k-i}^{1/\hat{p}}{c}_i^{1/\hat{p}}.$$

$$(\sum_{k}a_{k}^{q/p})^{1/q}\leq C^{1/p}\Cr{co2}(p,q,b)\Big{(}\sum_{k}c_{k}^{q/p}\Big{)}^{1/q},$$

$$(\sum_{k}a_{k}^{q/p})^{1/q}\leq C^{1/p}\big{(}\sum_{k}b_{k}^{1/p}\big{)}\Big{(}\sum_{k}c_{k}^{q/p}\Big{)}^{1/q}.$$

$$\sum _k{a}_k^{q/p}$$

$$(\sum_{k}a_{k}^{q/p})^{p/q}\leq C(\sum_{k}b_{k})\big{(}\sum_{k}c_{k}^{q/p}\big{)}^{p/q}.$$

$$\sum _k{a}_k^{q/p}$$

$$|\varphi {|}_{pu}\le |Q{|}^{s-\frac{1}{u^{\prime }p}}$$

$$\text{Cl}[normal]decaypart=\text{Cr}co(p,q,(|{\mathcal{P}}^k{|}^s{)}_k)<\infty .$$

$$\underset{k}{\lim }|{\mathcal{P}}^k|=0.$$

$$g=\sum _{k=0}^{\infty }\sum _{Q\in {\mathcal{P}}^k}{s}_Q{a}_Q$$

$$\big{(}\sum_{k=0}^{\infty}(\sum_{Q\in\mathcal{P}^{k}}|s_{Q}|^{p})^{q/p}\big{)}^{1/q}<\infty.$$

$$\sum_{k=0}^{\infty}\big{|}\sum_{Q\in\mathcal{P}^{k}}s_{Q}a_{Q}\big{|}_{p}^{p/\hat{p}}<\infty.$$

$$|g|_{\mathcal{B}^{s}_{p,q}(I,\mathcal{P},\mathcal{A})}=\inf\big{(}\sum_{k=0}^{\infty}(\sum_{Q\in\mathcal{P}^{k}}|s_{Q}|^{p})^{q/p}\big{)}^{1/q},$$

$$s-\frac{1}{p}+\frac{1}{t}\ge 0,p\le t\le pu$$

$$\text{Cl}[normal]kt=\text{Cr}{mult1}^{1+1/t}\text{Cr}co(t,q,(|{\mathcal{P}}^k{|}^{t(s-\frac{1}{p}+\frac{1}{t})}{)}_k)<\infty ,$$

$$|\sum _{Q\in {\mathcal{P}}^k}{s}_Q{a}_Q{|}_t$$

Full text

We would like the thank the referee for the useful suggestions. D.S. was partially supported by CNPq 306622/2019-0, CNPq 307617/2016-5, CNPq Universal 430351/2018-6 and FAPESP Projeto Temático 2017/06463-3.

1. 1 Introduction

2. 2 Notation

3. 3 Measure spaces and grids

4. 4 A bag of tricks.

5. 5 Atoms

6. 6 Besov-ish spaces

7. 7 Scales of spaces

8. 8 Transmutation of atoms

9. 9 Good grids

10. 10 Induced spaces

11. 11 Examples of classes of atoms.

12. 11.1 Souza’s atoms

13. 11.2 Hölder atoms

14. 11.3 Bounded variation atoms

15. 12 Besov spaces in a measure space with a good grid

16. 13 Positive cone

17. 14 Unbalanced Haar wavelets

18. 15 Alternative characterizations I: Messing with norms.

19. 15.1 Haar representation

20. 15.2 Standard atomic representation

21. 15.3 Mean oscillation

22. 15.4 These norms are equivalent

23. 16 Alternative characterizations II: Messing with atoms.

24. 16.1 Using Besov’s atoms

25. 16.2 Using Hölder atoms

26. 16.3 Using bounded variation atoms

27. 17 Dirac’s approximations

28. 18 **Pointwise multipliers acting on **$\mathcal{B}^{s}_{p,q}$

29. 18.1 **Non-Archimedean behaviour in **$\mathcal{B}^{\beta}_{p,\tilde{q},selfs}$

30. 18.2 **Strongly regular domains **

31. 18.3 **Functions on $\mathcal{B}^{1/p}_{p,\infty}\cap L^{\infty}$ **

32. 19 ** $\mathcal{B}^{s}_{p,q}\cap L^{\infty}$ is a quasi-algebra**

33. 19.1 Regular domains

34. 20 A remarkable description of $\mathcal{B}^{s}_{1,1}$.

35. 21 Left compositions.

1. Introduction

inline,linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 1- We changed the abstract a little bit. Besov spaces $B^{s}_{p,q}(\mathbb{R}^{n})$ were introduced by Besov [4]. This scale of spaces has been a favorite over linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 2- we replaced ”thought” by ”over” the years, with thousands of references available. Perhaps two of its most interesting features is that many earlier classes of function spaces appear linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 3- we replaced ”appears” by ”appear” in this scale, as Sobolev spaces, and also that there are many equivalent ways to define $B^{s}_{p,q}(\mathbb{R}^{n})$, in such way that you linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 4- we replaced ”in such way you” by ïn such way that you” can pick the more suitable one for your purpose. The reader may see Stein [43], Peetre [39], Triebel [47] for an introduction of Besov spaces on $\mathbb{R}^{n}$. For a historical account on Besov spaces and related topics, see Triebel [47] and the shorter linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 5- Replace ”more short” by ”shorter” but useful Jaffard [30], Yuan, Sickel and Yang [50] and Besov and Kalyabin [3].

In the last decades there was linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 6- we replaced ”were” by ”was” a huge amount of activity on the generalisation of harmonic analysis (see Deng and Han [17]), and Besov spaces in particular, to less regular phase spaces, replacing $\mathbb{R}^{n}$ by something with a poorer structure. It turns out that for small $s>0$ and $p,q\geq 1$, a proper definition demands strikingly weak assumptions. There is a large body of literature that provides a definition and properties of Besov spaces on homogeneous spaces, as defined by Coifman and Weiss [12]. Those are quasi-metric spaces with a doubling measure, which includes in particular Ahlfors regular metric spaces. We refer to the linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 7- we replaced ”pionner”by ”pioneer”pioneer work of Han and Sawyer [28] and Han, Lu and Yang [27] for Besov spaces on homogeneous spaces and more recently Alvarado and Mitrea [1] and Koskela, Yang, and Zhou [33] (in this case for metric measure spaces) and Triebel [48][46] for Besov spaces on fractals. There is a long list of examples of homogeneous spaces in Coifman and Weiss [13]. The d-sets as defined in Triebel [48] are also examples of homogeneous spaces.

*We give a very elementary (and yet practical linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 8- we replaced ”pratical” by ”practical”) construction of Besov spaces $\mathcal{B}^{s}_{p,q}$, with $p,q\geq 1$ and $0<s<1/p$, for measure spaces endowed with a grid, that is, a sequence of finite partitions of measurable sets satisfying certain mild properties. *

This construction is close to the martingale Besov spaces as defined by Gu and Taibleson [25], however we deal with ”nonisotropic splitting” in our grid without applying the Gu-Taibleson recalibration procedure to our grid, which simplifies the definition and it allows a broader linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 9- we replaced ”wider” by ”broader” class of examples.

The linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 10- we replaced ”main” by ”primary” primary tool in this work is the concept of atomic decomposition. An atomic decomposition represents linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 11- we replaced ”is the representation of” by ”represents” each function in a space of functions as a (infinite) linear combination of fractions of the so-called atoms. The advantage of atomic decompositions is linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 12- we replaced ”in” by ”is” that the atoms are functions that are often far more regular than a typical element of the space. linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 13- we replaced ”But” by ”However” However a distinctive feature (compared with Fourier series with either Hilbert basis or unconditional basis) is that such atomic decomposition is in general not unique. Nevertheless, in a successful atomic decomposition of a normed space of functions, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 14- comma added we can attribute a ”cost” to each possible representation, and the norm of the space is equivalent to the minimal cost (or infimum) among all representations. A function represented by a single atom has norm at most one, so the term ”atom” seems to be appropriated.

Coifman [11] introduced the atomic decomposition of the real Hardy space $H^{p}(\mathbb{R})$ and Latter [35] found an atomic decomposition for $H^{p}(\mathbb{R}^{n}).$ The influential work of Frazier and Jawerth [22] gave us an atomic decomposition for the Besov spaces $B^{s}_{p,q}(\mathbb{R}^{n})$. In the context of homogeneous spaces, we have results by Han, Lu and Yang [27] on the linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 15- we included ”the” atomic decomposition of Besov spaces by Hölder atoms.

Closer to the spirit of this work we have the atomic decomposition of Besov space $B^{s}_{1,1}([0,1])$, with $s\in(0,1)$, by de Souza [14] using special atoms, that we call Souza’s atoms (see also De Souza [15] and de Souza, O’Neil and Sampson [16]). A Souza’s atom $a_{J}$ on an interval $J$ is quite simple, consisting of a function whose support is $J$ and $a_{J}$ is constant on $J$.

We also refer to the results on the B-spline atomic decomposition of the Besov space of the unit cube of $\mathbb{R}^{n}$ in DeVore and Popov [18] (with Ciesielski [10] as a precursor), that in the case $0<s<1/p$ reduces to an atomic decomposition by Souza’s atoms, and the work of Oswald [37] [38] on finite element approximation in bounded polyhedral domains on $\mathbb{R}^{n}$.

On the study of Besov spaces on $\mathbb{R}^{n}$ and smooth manifolds, Souza’s atoms may seem to have setbacks that restrict its usefulness. They are not smooth, so it is fair to doubt the effectiveness of atomic decomposition by Souza’s atoms to obtain a better understanding of a partial differential equation or to represent data faithfully/without artifacts, a constant concern in applications of smooth wavelets (see Jaffard, Meyer and Ryan [31]).

On the other hand, in the study of ergodic properties of piecewise smooth dynamical systems, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 16- comma inserted linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 17- we deleted ”the” transfer operators play linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 18- we replaced ”plays” by ”play” a huge role. Those operators act linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 19- we replaced ”acts” by ”act” on Lebesgue spaces $L^{1}(m)$, but they are more useful if one can show linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 20- we replaced ”it has a” by ”they have a” they have a (good) action on more regular function spaces. Unfortunately, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 21- we included a comma in many cases the transfer operator does not preserve neither smoothness nor even continuity of a function. Discontinuities are a fact of life in this setting, and they do not go away as in certain dissipative PDEs, since the positive $1$-eigenvectors of this operator, of utmost linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 22- repeated word ”importance” erased importance in its study, often have discontinuities themselves. The works of Lasota and Yorke [34] and Hofbauer and Keller [29] are landmark results in this direction. See also Baladi [2] and Broise [8] for more details. Atomic decomposition with Souza’s atoms, being the simplest possible kind of atom with discontinuities, might come in handy in these cases. That was a major motivation for this work.

Besides this fact, in an abstract homogeneous space higher-order linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 23- we replaced ”higher order” by ”higher-order” smoothness does not seem to be a very useful concept linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 24- comma removed since we can define $B^{s}_{p,q}$ just for small values of $s$, so atomic decompositions by Souza’s atoms sounds far more attractive.

Indeed, the simplicity of Souza’s atoms allows linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 25- we replaced ”allow” by ”allows” us to throw away the necessity of a metric/topological structure on the phase space. We define Besov spaces on every measure space with a non-atomic linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 26- we replaced ”non atomic” by ”non-atomic” finite measure, provided we endowed it with a good grid. A good grid is just a sequence of finite partitions of measurable sets satisfying certain mild properties. We give the definition of Besov space on measure space with a (good) grid in Part II.

In linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 27- we replaced ”On” by ”In” the literature we usually see a known space of functions be described using atomic decomposition. This typically comes after a careful study of such space, and it is often a challenging task. More rare is the path we follow here. We start by *defining * the Besov spaces through atomic decomposition by Souza’s atoms. This construction of the spaces and the study of its basic properties, as its completeness and linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 28- we removed the extra ”its” compact inclusion in Lebesgue spaces, uses fairly general and simple arguments, and it does not depend on the particular nature of the atoms used, except for very mild conditions on its regularity. In Part I we describe this construction in full generality.

We construct Besov-ish spaces. There are far more general than Besov spaces. In particular, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 29- we added a comma the nature of the atoms may change with its location and scale in the phase space and the grid itself can be very irregular.

The main result in Part I is Proposition 8.1. Due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 30- we added ”to” the generality of the setting, its statement is linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 31- we replaced ”hopeless” by ” hopelessly” hopelessly technical in nature, however this very powerful result has a simple meaning. Suppose we have an atomic decomposition of a function. If we replace each of those atoms by a combination of atoms (possibly of a different class) in such way that we linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 32- we replaced ”in such way we” by ”in such way that we” do not change the location of the support and also the ”mass” of the representation is concentrated linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 33- we replaced ”concentrate” by ”concentrated” pretty much on the same original scale, then we obtain a new atomic decomposition, and the cost of this atomic decomposition can be estimated linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 34- we replaced ”estimate” by ”estimated” by the cost of the original atomic decomposition. We will use this result many times throughout linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 35- we replaced ”along” by ”throughout” this work. This is obviously connected with almost diagonal operators as in Frazier and Jawerth [22][23].

In Part II we also offer a detailed analysis of the Besov spaces defined there. Since we define linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 36- we replaced ”defined” by ”define” it using combinations of Souza’s atoms, it is not clear a priori how rich are those spaces. So

We give a bunch of alternative characterisations of these Besov spaces. We show that using more flexible classes of atoms (piecewise Hölder atoms, $p$-bounded variation atoms and even Besov atoms with higher regularity), we obtain the same Besov space. This often allows us to easily verify if a given function belongs to $B^{s}_{p,q}$. We also have a mean oscillation characterisation in the spirit of Dorronsoro [19] and Gu and Taibleson [25], and we also obtained another one using Haar wavelets.

Haar wavelets were introduced by Haar [26] in the real line. The elegant construction of unbalanced Haar wavelets in general measure spaces with a grid by Girardi and Sweldens [24] will play an essential role here. If in general homogeneous spaces the Calderón reproducing formula appears to be the bit of harmonic analysis that survives in it and it allows to carry out the theory, in finite measure spaces with a good grid (and in particular *compact * homogenous spaces) full-blown Haar systems are alive and well. Recently a Haar system was used by Kairema, Li, Pereyra and Ward [32] to study dyadic versions of the Hardy and BMO spaces in homogeneous spaces.

We also provided a few applications in part III. In particular, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 37- we added a comma we study the boundedness of pointwise multipliers acting in the Besov space. Since it is effortless linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 38- we replaced ”very easy” by ”effortless” to multiply Souza’s atoms, the proofs of these results are concise linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 39- we replaced ”very short” by ”concise” and easy to understand, generalising some of the results for Besov spaces in $\mathbb{R}^{n}$ by Triebel[44] and Sickel [41]. We also study the boundedness of left composition in Besov spaces of measure spaces with a grid, similar to some results for $B^{s}_{p,q}(\mathbb{R}^{n})$ in Bourdaud and Kateb [6] (see also Bourdaud and Kateb [5][7]).

It may come as a surprise to the reader that Besov spaces on compact homogeneous spaces as defined by Han, Lu and Yang [27] (and in particular Gu-Taibleson recalibrated martingale Besov spaces [25]) are indeed particular cases of Besov spaces defined here, provided $0<s<1/p$ and $s$ is small. We show this in a forthcoming work [42].

2. Notation

We will use $C_{1},C_{2},\dots...$ for positive constants and $\lambda_{1},\lambda_{2},\dots$ for positive constants smaller than one.

I. DIVIDE AND RULE.

**

In Part I. we are going to assume $s>0$, $p\in(0,\infty)$ and $q\in(0,\infty].$

---

 

**

3. Measure spaces and grids

Let $I$ be a measure space with a $\sigma$-algebra $\mathbb{A}$ and $m$ be a measure on $(I,\mathbb{A})$, $m(I)<\infty$. Given a measurable set $J\subset I$ denote $|J|=m(J)$. We denote the Lebesgue spaes of $(I,\mathbb{A},m)$ by $L^{p}$. A grid is a sequence of finite families of measurable sets with positive measure $\mathcal{P}=(\mathcal{P}^{k})_{k\in\mathbb{N}}$, so that at least one of these families is non empty and

• ${\Cl[G]{fin}}$.

Given $Q\in\mathcal{P}^{k}$, let

[TABLE]

Then

[TABLE]

Define $|\mathcal{P}^{k}|=\sup\{|Q|\colon Q\in\mathcal{P}^{k}\}$. To simplify the notation we also assume that $P\neq Q$ for every $P\in\mathcal{P}^{i}$ and $Q\in\mathcal{P}^{j}$ satisfying $i\neq j$. We often abuse notation using $\mathcal{P}$ for both $(\mathcal{P}^{k})_{k\in\mathbb{N}}$ and $\cup_{k}\mathcal{P}^{k}$.

Remark 3.1**.**

There are plenty of examples of spaces with grids. Perhaps the simplest one is obtained considering $[0,1)$ with the Lebesgue measure and the dyadic grid $\mathcal{D}=(\mathcal{D}^{k})_{k}$ given by

[TABLE]

We can also consider the dyadic grid $\mathcal{D}_{n}=(\mathcal{D}^{k}_{n})_{k}$ of $[0,1)^{n}$, endowed with the Lebesgue measure, given by

[TABLE]

and also the corresponding $d$-adic grids replacing $2^{k}$ by $d^{k}$ in the above definitions. The above grids are somehow special since they are nested sequence of partitions of the phase space $I$ and all elements on the same level have linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 40- we replaced ”has” by ”have” exactly the same measure.

Remark 3.2**.**

Indeed, any measure space with a finite non-atomic measure can be endowed with a grid made of a nested sequence of partitions and such that all elements on the same level have linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 41- we replaced ”has” by ”have” linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 42- we replaced ”exactly” by ”precisely” precisely the same measure linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 43- we removed a comma since any such measure space is measure-theoretically the same that a finite interval with the Lebesgue measure.

Remark 3.3**.**

If we consider a (quasi)-metric space $I$ with a finite measure $m$, we would like to construct ”nice” grids on $(I,m)$. It turns out that if $(I,m)$ is a homogeneous space one can construct a nested sequence of partitions of the phase space $I$ and all elements on the same level are open subsets and have ”essentially” the same measure. This is an easy consequence of a remarkable linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 44- we replaced ’famous”by ”remarkable” result by Christ [9]. See [42].

Remark 3.4**.**

One can constructs grids for smooth compact manifolds and bounded polyhedral domains in $\mathbb{R}^{n}$ using successive subdivisions of an initial triangulation of the domain (see for instance Oswald [37][38] ).

4. A bag of tricks.

Following closely the notation of Triebel ****[48]****, consider the set $\ell_{q}(\ell_{p}^{\mathcal{P}})$ of all indexed sequences

[TABLE]

with $x_{P}\in\mathbb{C}$, satisfying

[TABLE]

with the usual modification when $q=\infty$. Then $(\ell_{q}(\ell_{p}^{\mathcal{P}}),|\cdot|_{\ell_{q}(\ell_{p}^{\mathcal{P}})})$ is a complex $\rho$-Banach space with $\rho=\min\{1,p,q\}$, that is, $d(x,y)=|x-y|_{\ell_{q}(\ell_{p}^{\mathcal{P}})}^{\rho}$ is a complete metric in $\ell_{q}(\ell_{p}^{\mathcal{P}}).$

The following is a pair of arguments we will use across linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 45- we replaced ”along” by ”across” this paper to estimate norms in $\ell_{p}$ and $\ell_{q}(\ell_{p}^{\mathcal{P}})$. Those are very elementary, and we do not claim any originality. We collect them here to simplify the exposition. The reader can skip this for the cases $p,q\geq 1$, when the results bellow reduce to the familiar Hölder’s and Young’s inequalities. Their proofs were mostly based on ****[22, Proof of Theorem 3.1]****. Recall that for $t\in(0,\infty]$ we defined $\hat{t}=\max\{1,t\}$.

Proposition 4.1** (Hölder-like trick).**

Let $t\in(0,\infty)$ and $q\in(0,\infty]$. Let $a=(a_{k})_{k},b=(b_{k})_{k},c=(c_{k})_{k}$ nonnegative sequences such that for every $k$

[TABLE]

Then if $q<\infty$ we have

[TABLE]

and if $q=\infty$

[TABLE]

where

• A.

*If $t\geq 1$ and $q\geq 1$ then $\Cr{co}(t,q,b)=(\sum_{k}b_{k}^{q^{\prime}/t})^{1/q^{\prime}}$ if $q>1$, *

and $\Cr{co}(t,1,b)=\sup_{k}b_{k}^{1/t}$ if $q=1$.

• B.

If $t\geq 1$ and $q\leq 1$ then $\Cr{co}(t,q,b)=\sup_{k}b_{k}^{1/t}$.

• C.

*If $t<1$ and $q/t\geq 1$ then $\Cr{co}(t,q,b)=(\sum_{k}b_{k}^{(q/t)^{\prime}})^{1/(t(q/t)^{\prime})}$ if $q<t$, *

and $\Cr{co}(t,q,b)=\sup_{k}b_{k}^{1/t}$ if $q=t$.

• D.

If $t<1$ and $q/t<1$ then $\Cr{co}(t,q,b)=\sup_{k}b_{k}^{1/t}$.

Proof.

We have

Case A. If $t\geq 1$ and $q\geq 1$, by the Hölder inequality for the pair $(q,q^{\prime})$

[TABLE]

Case B. If $t\geq 1$ and $q\leq 1$ then the triangular inequality for $|\cdot|^{q}$ implies

[TABLE]

Case C. If $t<1$ and $q/t\geq 1$ then $\hat{t}/t=1/t$ and by the Hölder inequality for the pair $(q/t,(q/t)^{\prime})$

[TABLE]

Case D. if $t<1$ and $q/t<1$ then using the triangular inequality for $|\cdot|^{q/t}$

[TABLE]

∎

Proposition 4.2** (Convolution trick).**

Let $p,q\in(0,\infty)$. Let $a=(a_{k})_{k\in\mathbb{Z}},b=(b_{k})_{k\in\mathbb{Z}},c=(c_{k})_{k\in\mathbb{Z}}\geq 0$ be such that for every $k$

[TABLE]

Then

[TABLE]

where $\Cl[normal]{co2}\geq 1$ satisfies

• A.

If $p\geq 1$ and $q\geq 1$ then $\Cr{co2}(p,q,b)=\sum_{k\in\mathbb{Z}}b_{k}^{1/p}$.

• B.

If $p\geq 1$ and $q\leq 1$ then $\Cr{co2}(p,q,b)=(\sum_{k\in\mathbb{Z}}b_{k}^{q/p})^{1/q}$.

• C.

If $p<1$ and $q/p\geq 1$ then $\Cr{co2}(p,q,b)=(\sum_{k\in\mathbb{Z}}b_{k})^{1/p}$.

• D.

If $p<1$ and $q/p<1$ then $\Cr{co2}(p,q,b)=(\sum_{k\in\mathbb{Z}}b_{k}^{q/p})^{1/q}$

Proof.

We have

Case A. If $p\geq 1$ and $q\geq 1$, by the Young’s inequality for the pair $(1,q)$

[TABLE]

Case B. If $p\geq 1$ and $q\leq 1$ then the triangular inequality for $|\cdot|^{q}$ and the Young’s inequality for the pair $(1,1)$ imply

[TABLE]

Case C. If $p<1$ and $q/p\geq 1$ then by the Young’s inequality for the pair $(1,q/p)$

[TABLE]

Case D. If $p<1$ and $q/p<1$ then using the triangular inequality for $|\cdot|^{q/p}$ and the Young’s inequality for the pair $(1,1)$

[TABLE]

∎

5. Atoms

**Let $\mathcal{P}$ be a grid. Let $p\in[1,\infty),u\in[1,\infty]$, and $s>0$. A family of atoms associated with $\mathcal{P}$ of type $(s,p,u)$ is an indexed family $\mathcal{A}$ of pairs $(\mathcal{B}(Q),\mathcal{A}(Q))_{Q\in\cup_{k}\mathcal{P}^{k}}$ where **

• ${\Cl[A]{banach}}$.

$\mathcal{B}(Q)$** is a complex Banach space contained in $L^{pu}$.**

• ${\Cl[A]{suporte}}$.

If $\phi\in\mathcal{B}(Q)$ then $\phi(x)=0$ for every $x\not\in Q$.

• ${\Cl[A]{convex}}$.

$\mathcal{A}(Q)$** is a convex subset of $\mathcal{B}(Q)$ such that $\phi\in\mathcal{A}(Q)$ if and only if $\sigma~{}\phi~{}\in~{}\mathcal{A}(Q)$ for every $\sigma\in\mathbb{C}$ satisfying $|\sigma|=1$.**

• ${\Cl[A]{atom}}$.

We have

[TABLE]

for every $\phi\in\mathcal{A}(Q)$.

**We will say that $\phi\in\mathcal{A}(Q)$ is an $\mathcal{A}$-atom of type $(s,p,u)$ supported on $Q$ and that $\mathcal{B}(Q)$ is the local Banach space on $Q$. Sometimes we also assume **

• ${\Cl[A]{compact}}$.

For every $Q\in\mathcal{P}$ we have that $\mathcal{A}(Q)$ is a compact subset in the strong topology of $L^{p}$.

or

• ${\Cl[A]{compactw}}$.

**We have $p\in[1,\infty)$ and every $Q\in\mathcal{P}$ the set $\mathcal{A}(Q)$ is a sequentially compact subset in the *weak *** topology of $L^{p}$.

**or even **

• ${\Cl[A]{finite}}$.

For every $Q\in\mathcal{P}$ we have that $\mathcal{B}(Q)$ is finite dimensional and $\mathcal{A}(Q)$ contains a neighborhood of [math] in $\mathcal{B}(Q)$.

We provide examples of classes of atoms in Section 11.

6. Besov-ish spaces

Let $p\in(0,\infty)$, $u\in[1,\infty]$, $q\in(0,\infty]$, $s>0$, $\mathcal{P}=(\mathcal{P}^{k})_{k\geq 0}$ be a grid and let $\mathcal{A}$ be a family of atoms of type $(s,p,u)$. We will also assume that

• ${\Cl[G]{sum}}.$

We have

[TABLE]

and

[TABLE]

The Besov-ish space $\mathcal{B}^{s}_{p,q}(I,\mathcal{P},\mathcal{A})$ is the set of all complex valued functions $g\in L^{p}$ that can be represented by an absolutely convergent series on $L^{p}$

[TABLE]

where $a_{Q}$ is in $\mathcal{A}(Q)$, $s_{Q}\in\mathbb{C}$ and with finite cost

[TABLE]

Note that the inner sum in (6.1) is finite. By absolutely convergence in $L^{p}$ we mean that

[TABLE]

The series in (6.1) is called a $\mathcal{B}^{s}_{p,q}(I,\mathcal{P},\mathcal{A})$-representation of the function $g$. Define

[TABLE]

where the infimum runs over all possible representations of $g$ as in (6.1).

Quite often along this work, when it is obvious the choice of the measure space $I$ and/or the grid $\mathcal{P}$ we will write either $\mathcal{B}^{s}_{p,q}(\mathcal{P},\mathcal{A})$ or even $\mathcal{B}^{s}_{p,q}(\mathcal{A})$ instead of $\mathcal{B}^{s}_{p,q}(I,\mathcal{P},\mathcal{A})$. Whenever we write just $\mathcal{B}^{s}_{p,q}$ it means that we choose the Souza’s atoms $\mathcal{A}^{sz}_{s,p}$, with parameters $s$ and $p$, a class of atoms we properly define in Section 11.1.

Proposition 6.1**.**

Assume ${\Cr{fin}}$-${\Cr{sum}}$ and ${\Cr{banach}}$-${\Cr{atom}}$. Let $t\in(0,\infty)$ be such that

[TABLE]

and suppose

[TABLE]

Then for every coefficients $s_{Q}$ satisfying (6.2) and every $\mathcal{A}$-atoms $a_{Q}$ on $Q$ the series (6.1) converges absolutely on $L^{t}$. Indeed

[TABLE]

and

[TABLE]

Proof.

Firstly note that if $p\leq t\leq pu$

[TABLE]

Consequently

[TABLE]

By Proposition 4.1 (Hölder-like trick) and $p\leq t$ we have

[TABLE]

∎

Remark 6.2**.**

Note that due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 46- we added ”to” ${\Cr{sum}}$ if $t=p$ then $\Cr{kt}<\infty$. Sometimes it is convenient to use sharper estimates than (6.5) and (6.6) replacing $|\mathcal{P}^{k}|$ by the sequence

[TABLE]

For instance, if $s_{Q}=0$ for every $Q\in\mathcal{P}^{k}$ with $k\leq N$, then we can replace $\Cr{co}(t,q,(|\mathcal{P}^{k}|^{t(s-\frac{1}{p}+\frac{1}{t})})_{k})$ by $\Cr{co}(t,q,(|\mathcal{P}^{k}|^{t(s-\frac{1}{p}+\frac{1}{t})}1_{(N,\infty)}(k))_{k})$ in (6.4). Here $1_{(N,\infty)}$ denotes the indicator function of the set $(N,\infty)$.

Proposition 6.3**.**

Assume ${\Cr{fin}}$-${\Cr{sum}}$ and ${\Cr{banach}}$-${\Cr{atom}}$. Then $\mathcal{B}^{s}_{p,q}(\mathcal{A})$ is a complex linear space and $|\cdot|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})}$ is a $\rho$-norm, with $\rho=\min\{1,p,q\}$. Moreover the linear inclusion

[TABLE]

is continuous.

Proof.

Le $f,g\in\mathcal{B}^{s}_{p,q}(\mathcal{A})$. Then there are $\mathcal{B}^{s}_{p,q}(\mathcal{A})$-representations

[TABLE]

Let $sgn\ 0=0$ and $sgn\ z=z/|z|$ otherwise. Of course

[TABLE]

where111We don’t need to worry so much if $|s_{Q}|+|s_{Q}^{\prime}|=0$, since in this case $c_{Q}=0$ and we can choose $b_{Q}$ to be an arbitrary atom (for instance $a_{Q}$) in such way that (6.7) holds. For this reason we are not going to explicitly deal with similar situations ( that is going to appear quite often) along the paper.

[TABLE]

and

[TABLE]

Note that $sgn(s_{Q}^{\prime})a_{Q}^{\prime},sgn(s_{Q})a_{Q}$ are atoms due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 47- we added ”to” ${\Cr{convex}}$. So by ${\Cr{convex}}$ we have that $b_{Q}$ is also an atom, since it is a convex combination of atoms. In particular

[TABLE]

converges absolutely in $L^{p}$ to $f+g$. It remains to prove that this is a $\mathcal{B}^{s}_{p,q}(\mathcal{A})$-representation of $f+g$. Indeed

[TABLE]

Taking the infimum over all possible $\mathcal{B}^{s}_{p,q}(\mathcal{A})$-representations of $f$ and $g$ we obtain

[TABLE]

The identity $|\gamma f|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})}=|\gamma||f|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})}$ is obvious. By Proposition 6.1 we have that if $|f|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})}=0$ then $|f|_{p}=0$, so $f=0$, so $|\cdot|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})}$ is a $\rho$-norm, moreover (6.6) tell us that $\imath$ is continuous. ∎

Proposition 6.4**.**

Assume ${\Cr{fin}}$-${\Cr{sum}}$ and ${\Cr{banach}}$-${\Cr{atom}}$. Suppose that $g_{n}$ are functions in $\mathcal{B}^{s}_{p,q}(\mathcal{A})$ with $\mathcal{B}^{s}_{p,q}(\mathcal{A})$-representations

[TABLE]

where $a_{Q}^{n}$ is a $\mathcal{A}$-atom supported on $Q$, satisfying

• i.

There is $C$ such that for every $n$

[TABLE]

• ii.

For every $Q\in\mathcal{P}$ we have that $s_{Q}=\lim_{n}s_{Q}^{n}$ exists.

• iii.

For every $Q\in\mathcal{P}$ there is $a_{Q}\in\mathcal{A}(Q)$ such that

1. (1)

either the sequence $a_{Q}^{n}$ converges to $a_{Q}$ in the strong topology of $L^{p}$, or 2. (2)

we have $p\in[1,\infty)$ and $a_{Q}^{n}$ weakly converges to $a_{Q}$.

then $g_{n}$ either strongly or weakly converges in $L^{p}$, respectively, to $g\in\mathcal{B}^{s}_{p,q}(\mathcal{A})$, where $g$ has the $\mathcal{B}^{s}_{p,q}(\mathcal{A})$-representation

[TABLE]

that satisfies

[TABLE]

Proof.

By (6.8) it follows that (6.10) holds and that (6.9) is indeed a $\mathcal{B}^{s}_{p,q}(\mathcal{A})$-representation of a function $g$. It remains to prove that $g_{n}$ converges to $g$ in $L^{p}$ in the topology under consideration. Given $\epsilon>0$, fix $N$ large enough such that

[TABLE]

We can write

[TABLE]

where

[TABLE]

is an atom in $\mathcal{A}(Q)$, and

[TABLE]

Note that the series in the r.h.s. converges absolutely in $L^{p}$. Of course

[TABLE]

So by (6.6) in Proposition 6.1 (see also Remark 6.2) we have

[TABLE]

In the case $ii.1$, note that if $n$ is large enough then

[TABLE]

and consequently $|g-g_{n}|_{p}^{p/\hat{p}}<\epsilon$. So $g_{n}$ strongly converges to $g$.

In the case $ii.2$, given $\phi\in(L^{p})^{\star}$, with $p\geq 1$ we have that for $n$ large enough

[TABLE]

and of course

[TABLE]

so $g_{n}$ weakly converges to $g$ in $L^{p}$. ∎

Corollary 6.5**.**

Assume ${\Cr{fin}}$-${\Cr{sum}}$ and ${\Cr{banach}}$-${\Cr{atom}}$, and

1. (1)

either ${\Cr{compact}}$ or 2. (2)

we have $p\geq 1$ and ${\Cr{compactw}}$.

Then

• i.

Let $g_{n}\in\mathcal{B}^{s}_{p,q}(\mathcal{A})$ be such that $|g_{n}|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})}\leq C$ for every $n$. Then there is a subsequence that converges either strongly or weakly in $L^{p}$, respectively, to some $g\in\mathcal{B}^{s}_{p,q}(\mathcal{A})$ with $|g|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})}\leq~{}C$.

• ii.

In both cases $(\mathcal{B}^{s}_{p,q}(\mathcal{A}),|\cdot|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})})$ is a complex $\rho$-Banach space, with $\rho=\min\{1,p,q\}$,

• iii

If ${\Cr{compact}}$ holds then the inclusion

[TABLE]

is a compact linear inclusion.

Proof of i..

There are $\mathcal{B}^{s}_{p,q}(\mathcal{A})$-representations

[TABLE]

where $a_{Q}^{n}$ is a $\mathcal{A}$-atom supported on $Q$ and

[TABLE]

and $1\geq\varepsilon_{n}\to 0$. In particular, $|s_{Q}^{n}|\leq C+1$. Since the set $\cup_{k}\mathcal{P}^{k}$ is countable, by the Cantor diagonal argument, taking a subsequence we can assume that $s_{Q}^{n}\to_{n}s_{Q}$ for some $s_{Q}\in\mathbb{C}$. Due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 48- we added ”to” ${\Cr{compact}}$ (${\Cr{compactw}}$) and the Cantor diagonal argument, we can suppose that $a_{Q}^{n}$ strongly (weakly) converges in $L^{p}$ to some $a_{Q}\in\mathcal{A}(Q)$. We set

[TABLE]

By Proposition 6.4 we conclude that $g\in\mathcal{B}^{s}_{p,q}(\mathcal{A})$ with $|g|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})}\leq C$, and that $g_{n}$ converges to $g$ in $L^{p}$. ∎

Proof of ii..

Let $g_{n}$ be a Cauchy sequence on $\mathcal{B}^{s}_{p,q}(\mathcal{A})$. By Proposition 6.1 we have that $g_{n}$ is also a Cauchy sequence in $L^{p}$. Let $g$ be its limit in $L^{p}$. By Corollary 6.5.i have that $g\in\mathcal{B}^{s}_{p,q}(\mathcal{A})$. Note that for large $m$ and $n$

[TABLE]

and $g_{n}-g_{m}$ converges to $g_{n}-g$ in $L^{p}$, so again by Corollary 6.5.i we have that

[TABLE]

so $g_{n}$ converges to $g$ in $\mathcal{B}^{s}_{p,q}(\mathcal{A})$. ∎

Proof of iii..

It follows from i. ∎

The proof of the following result linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 49- we replaced ”resukt” by ”result” is quite similar.

Corollary 6.6**.**

Assume ${\Cr{fin}}$-${\Cr{sum}}$ and ${\Cr{banach}}$-${\Cr{atom}}$, and

1. (1)

either ${\Cr{compact}}$ or 2. (2)

we have $p\geq 1$ and ${\Cr{compactw}}$.

Then for every $f\in\mathcal{B}^{s}_{p,q}(\mathcal{A})$ there is a $\mathcal{B}^{s}_{p,q}(\mathcal{A})$-representation

[TABLE]

such that

[TABLE]

We refer to Edmunds and Triebel ****[20]**** for more information on compact linear transformations between quasi-Banach spaces.

Corollary 6.7**.**

Assume ${\Cr{fin}}$-${\Cr{sum}}$ and ${\Cr{banach}}$-${\Cr{atom}}$. If for every $Q\in\mathcal{P}$ we have that $\mathcal{B}(Q)$ is finite-dimensional and $\mathcal{A}(Q)$ is a closed subset of $\mathcal{B}(Q)$ then $(\mathcal{B}^{s}_{p,q}(\mathcal{A}),|\cdot|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})})$ is a $\rho$-Banach space, with $\rho=\min\{1,p,q\}$.

Proof.

Since all norms are equivalent in $\mathcal{B}(Q)$ we have that ${\Cr{atom}}$ implies that $\mathcal{A}(Q)$ is a closed and bounded subset of $\mathcal{B}(Q)$, so it is compact. By Corollary 6.5.ii it follows that $(\mathcal{B}^{s}_{p,q}(\mathcal{A}),|\cdot|_{\mathcal{B}^{s}_{p,q}(\mathcal{A})})$ is a $\rho$-Banach space. ∎

7. Scales of spaces

Note that a family of atoms $\mathcal{A}$ of type $(s,p,u)$ induces an one-parameter scale

[TABLE]

where $\mathcal{A}_{\tilde{s},p}$ is the family of atoms of type $(\tilde{s},p,u)$ defined by

[TABLE]

Moreover a family of atoms $\mathcal{A}$ of type $(s,p,\infty)$ induces a two-parameter scale

[TABLE]

where $\mathcal{A}_{\tilde{s},\tilde{p}}$ is the family of atoms of type $(\tilde{s},\tilde{p},\infty)$ defined by

[TABLE]

Proposition 7.1**.**

Assume ${\Cr{fin}}$-${\Cr{sum}}$. Suppose that the $(s,p,\infty)$-atoms $\mathcal{A}$ satisfy ${\Cr{banach}}$-${\Cr{atom}}$. Let $0\leq s<\tilde{s}$ and $q,\tilde{q}\in[1,\infty]$. Suppose

[TABLE]

Then

• A.

We have $\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{\tilde{s},p})\subset{\mathcal{B}}^{s}_{p,q}(\mathcal{A}_{s,p})$ and the inclusion is a continuous linear map.

• B.

Suppose that also satisfies ${\Cr{compact}}$. Let $g_{n}\in\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{\tilde{s},p})$ be such that $|g_{n}|_{\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{\tilde{s},p})}\leq C$ for every $n$. Then there is a subsequence that converges in ${\mathcal{B}}^{s}_{p,q}(\mathcal{A}_{s,p})$ to some $g\in\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{s,p})$ with $|g|_{\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{\tilde{s},p})}\leq C$.

• C.

Suppose that also satisfies ${\Cr{finite}}$. The inclusion $\imath\colon\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{\tilde{s},p})\mapsto{\mathcal{B}}^{s}_{p,q}(\mathcal{A}_{s,p})$ is a compact linear map.

Proof.

Consider a $\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{\tilde{s},p})$-representation

[TABLE]

Since $a_{Q}$ is an $\mathcal{A}_{\tilde{s},p}$-atom, we have that $b_{Q}=a_{Q}|Q|^{s-\tilde{s}}$ is an $\mathcal{A}_{s,p}$-atom. In particular, we can write

[TABLE]

If $k\geq k_{0}$ then

[TABLE]

so

[TABLE]

Proof of A. In particular, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 50- we added a comma taking $k_{0}=0$ we conclude that $\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{\tilde{s},p})\subset{\mathcal{B}}^{s}_{p,q}(\mathcal{A}_{s,p})$ and

[TABLE]

Proof of B. By definition, there exist $s^{n}_{Q}\in\mathbb{C}$, such that

[TABLE]

where $a_{Q}^{n}$ is a $\mathcal{A}_{\tilde{s},p}$-atom supported on $Q$ and

[TABLE]

where $\varepsilon_{n}\to 0$. In particular, $|s_{Q}^{n}|\leq C+\varepsilon_{n}$. Since the set $\cup_{k}\mathcal{P}^{k}$ is countable, by the Cantor diagonal argument, taking a subsequence we can assume that $s_{Q}^{n}\to s_{Q}$ and (due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 51- we added ”to” ${\Cr{compact}}$) that $a_{Q}^{n}$ converges in $\mathcal{B}(Q)$ and $L^{p}$ to some $a_{Q}\in\mathcal{A}_{\tilde{s},p}$. By Lemma 6.4 the sequence $g_{n}$ converge in $L^{p}$ to a function $g$ such that $|g|_{\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{\tilde{s},p})}\leq C$ and with $\mathcal{B}^{\tilde{s}}_{p,\tilde{q}}(\mathcal{A}_{s,p})$-representation

[TABLE]

It remains to show that the convergence indeed occurs in the topology of ${\mathcal{B}}^{s}_{p,q}(\mathcal{A}_{s,p})$. For every $k_{0}\geq 0$ and $\delta>0$ we can write

[TABLE]

where

[TABLE]

and with $b_{Q}^{n}\in\mathcal{A}_{s,p}$ given by

[TABLE]

and

[TABLE]

Note that $b_{Q}^{n}\in\mathcal{A}_{s,p}$. Given $\epsilon>0$, choose $k_{0}$ such that $b_{Q}^{n}\in\mathcal{A}_{s,p}$. Given $\epsilon>0$, choose $k_{0}$ such that

[TABLE]

By (7.12) and (7.13) for each $n$ large enough we have

[TABLE]

In particular

[TABLE]

Choose $\delta>$ such that

[TABLE]

Due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 52- we added ”to” ${\Cr{finite}}$ there is $\eta>0$ such that for every $Q\in\mathcal{P}^{k}$, with $k<k_{0}$, if $h\in\mathcal{B}(Q)$ satisfies $|h|_{\mathcal{B}(Q)}\leq\eta$ then $h\in\mathcal{A}_{p,s}(Q)$. Since $\lim_{n}s_{Q}^{n}a_{Q}^{n}=s_{Q}a_{Q}$ in $\mathcal{B}(Q)$ we conclude that for $n$ large enough we have

[TABLE]

for every $Q\in\mathcal{P}^{k}$, with $k<k_{0}$. In particular

[TABLE]

We conclude that

[TABLE]

for $n$ large enough, so the sequence $g_{n}$ converges (due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 53- we added ”to” Corollary 6.5) to $g$ in the topology of ${\mathcal{B}}^{s}_{p,q}(\mathcal{A}_{s,p})$.

∎

8. Transmutation of atoms

It turns out that sometimes a Besov-ish space can be obtained using different classes of atoms. The key result in Part I is the following

Proposition 8.1** (Transmutation of atoms).**

*Assume

• I.

Let $\mathcal{A}_{2}$ be a class of $(s,p,u_{2})$-atoms for a grid $\mathcal{W}$, satisfying ${\Cr{banach}}$-${\Cr{atom}}$* and ${\Cr{fin}}$-${\Cr{sum}}$. Let $\mathcal{G}$ be also a grid satisfying ${\Cr{fin}}$-${\Cr{sum}}$.*

• II.

Let $k_{i}\in\mathbb{N}$ for $i\geq 0$ be a sequence such that there is $\alpha>0$ and $A,B\in\mathbb{R}$ satisfying

[TABLE]

for every $i$.

• III.

There is $\lambda\in(0,1)$ such that the following holds. For every $Q\in\mathcal{G}$ and $P\in\mathcal{W}$ satisfying $P\subset Q$ there are atoms $b_{P,Q}\in\mathcal{A}_{2}(P)$ and corresponding $s_{P,Q}\in\mathbb{C}$ such that

[TABLE]

is a $\mathcal{B}^{s}_{p,q}(\mathcal{A}_{2})$-representation of a function $h_{Q}$, with $s_{P,Q}=0$ for every $Q\in\mathcal{G}^{i}$, $P\in\mathcal{W}^{k}$ with $k<k_{i}$ and moreover

[TABLE]

*for every $k\geq k_{i}$. *

Let

[TABLE]

*Then

• A.

For every coefficients $(c_{Q})_{Q\in\mathcal{G}}$ such that

[TABLE]

we have that the sequence

[TABLE]

converges in $L^{p}$ to a function in $\mathcal{B}^{s}_{p,q}(\mathcal{A}_{2})$ that has a $\mathcal{B}^{s}_{p,q}(\mathcal{A}_{2})$-representation

[TABLE]

where $m_{P}\geq 0$ for every $P$ and

[TABLE]

Here $\Cl[normal]{m1}=\max\{\ell\in\mathbb{N},\ell<\alpha\}+1$ and $b=(b_{n})_{n\in\mathbb{Z}}$ is defined by

[TABLE]

• B.

Suppose that the assumptions of A. hold and that $s_{P,Q}$ are non negative real numbers and $b_{P,Q}>0$ on $P$ for every $P,Q$. Then $m_{P}\neq 0$ and $d_{P}\neq 0$ on $P$ imply that $P\subset supp\ h_{Q}$ for some $Q\in\mathcal{W}^{k}$ satisfying $c_{Q}\neq 0$ and $s_{P,Q}>0$. If we additionally assume that $c_{Q}\geq 0$ for every $Q$ then $m_{P}\neq 0$ also implies $d_{P}>0$ on $P$.

• C.

Let $\mathcal{A}_{1}$ be a class of $(s,p,u_{1})$-atoms for the grid $\mathcal{G}$ satisfying ${\Cr{banach}}$-${\Cr{atom}}$. Suppose that there is $\lambda<1$ such that for every atom $a_{Q}\in\mathcal{A}_{1}(Q)$ we can find $s_{P,Q}$ and $b_{P,Q}$ in III. such that $h_{Q}=a_{Q}$. Then

[TABLE]

and this inclusion is continuous. Indeed

[TABLE]

for every $\phi\in\mathcal{B}^{s}_{p,q}(\mathcal{A}_{1}).$

inline,linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 54- In the caption of the figure 1 we replaced ”are concentrated” by ”concentrates”

Proof.

For every $P\in\mathcal{H}^{k}$, with $k\in\mathbb{N}$, and $N\in\mathbb{N}\cup\{\infty\}$ define

[TABLE]

Due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 55- we added ”to” $\Cr{sum}$ this sum has a finite number of terms. If this sum has zero terms define $m_{P,N}=0$ and let $d_{P,N}$ be the zero function. Otherwise define

[TABLE]

We have that $d_{P,N}$ is an $\mathcal{A}_{2}(P)$-atom.

Claim I. We claim that for $N\in\mathbb{N}$

[TABLE]

Note that if $Q\in\mathcal{G}^{i}$ then due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 56- we added ”to” (6.5), with $t=p$ and (8.14) we have

[TABLE]

Consequently we can do the following manipulation in $L^{p}$

[TABLE]

This concludes the proof of Claim I.

Claim II. For every $N\in\mathbb{N}\cup\{\infty\}$ we claim that

[TABLE]

*is a $\mathcal{B}^{s}_{p,q}(\mathcal{A}_{2})$-representation and *

[TABLE]

Indeed

[TABLE]

If $\alpha=1$ and $A=B=0$ then this is a convolution, so we can use Proposition 4.2 (the convolution trick) and it easily follows (8.17). In the general case, consider

[TABLE]

Every $k\in\mathbb{N}$ can be written in an unique way as $k=\alpha j_{k}+\ell_{k}+r_{k}$, with $j_{k}\in\mathbb{N}$, $\ell_{k}\in\mathbb{N}$, $\ell_{k}+r_{k}<\alpha$ and $r_{k}\in[0,1)$. Fix $\ell\in[0,\alpha)\cap\mathbb{N}$ and $j\in\mathbb{N}$. Then there is at most one $k^{\prime}\in\mathbb{N}$ such that $\ell_{k}=\ell$ and $j_{k}=j$. Indeed, if $k^{\prime}=\alpha j+\ell+r^{\prime}$ and $k^{\prime\prime}=\alpha j+\ell+r^{\prime\prime}$, with $r^{\prime},r^{\prime\prime}\in[0,1)$ and $\ell+r^{\prime}$ and $\ell+r^{\prime\prime}$ smaller than $\alpha$, then $k^{\prime}-k^{\prime\prime}=r^{\prime}-r^{\prime\prime}\in(-1,1)$, so $k^{\prime}=k^{\prime\prime}$ and $r^{\prime}=r^{\prime\prime}$. If such $k^{\prime}$ exists, denote $k(\ell,j)=k^{\prime}$ and $r(\ell,j)=k(\ell,j)-\alpha j-\ell$ and $a_{\ell,j}=u_{k(\ell,j)}$. Otherwise let $a_{\ell,j}=0$. Then (8.21) implies

[TABLE]

Here $b_{n}=\lambda^{\alpha n}$ if $n>A/\alpha-1$, and $b_{n}=0$ otherwise. Fixing $\ell\in\mathbb{N},\ \ell<\alpha$, Proposition 4.2 (the convolution trick) gives us

[TABLE]

and

[TABLE]

This implies in particular that the sum in (8.19) is a $\mathcal{B}^{s}_{p,q}(\mathcal{A}_{2})$-representation. This proves Claim II.

Claim III. We have that in the strong topology of $L^{p}$

[TABLE]

For each $P\in\mathcal{H}$ the sequence

[TABLE]

is eventually constant, therefore convergent. The same happens with

[TABLE]

Estimate (8.20) and Proposition 6.4 imply that that (8.15) converges in $L^{p}$ to a function with $\mathcal{B}^{s}_{p,q}(\mathcal{A}_{2})$-representation (8.19) with $N=\infty$. This concludes the proof of Claim III.

Then Claim I, II, and III imply A. taking $m_{p}=m_{P,\infty}$ and $d_{P}=d_{P,\infty}.$ We have that C. is an immediate consequence of A. Note that (8.18) and A. give B. ∎

9. Good grids

A $(\Cl[c]{menor},\Cl[c]{maior})$-good grid , with $0<\Cr{menor}<\Cr{maior}<1$, is a grid $\mathcal{P}=(\mathcal{P}^{k})_{k\in\mathbb{N}}$ with the following properties:

• ${\Cl[G]{g2}}.$

We have $\mathcal{P}^{0}=\{I\}$.

• ${\Cl[G]{g3}}.$

We have $I=\cup_{Q\in\mathcal{P}^{k}}Q$ (up to a set of zero $m$-measure).

• ${\Cl[G]{g4}}.$

The elements of the family $\{Q\}_{Q\in\mathcal{P}^{k}}$ are pairwise disjoint.

• ${\Cl[G]{g5}}.$

For every $Q\in\mathcal{P}^{k}$ and $k>0$ there exists $P\in\mathcal{P}^{k-1}$ such that $Q\subset P$.

• ${\Cl[G]{g6}}.$

We have

[TABLE]

for every $Q\subset P$ satisfying $Q\in\mathcal{P}^{k+1}$ and $P\in\mathcal{P}^{k}$ for some $k\geq 0$.

• ${\Cl[G]{g7}}.$

The family $\cup_{k}\mathcal{P}^{k}$ generate the $\sigma$-algebra $\mathbb{A}$.

10. Induced spaces

Consider a Besov-ish space $\mathcal{B}^{s}_{p,q}(I,\mathcal{P},\mathcal{A})$, where $\mathcal{P}$ is a good grid. Given $Q\in\mathcal{P}^{k_{0}}$, we can consider the sequence of finite families of subsets $\mathcal{P}_{Q}=(\mathcal{P}_{Q}^{i})_{i\geq 0}$ of $Q$ given by

[TABLE]

Let $\mathcal{A}_{Q}$ be the restriction of the indexed family $\mathcal{A}$ of pairs $(\mathcal{B}(P),\mathcal{A}(P))_{P\in\mathcal{P}}$ to indices belonging to $\mathcal{P}_{Q}$. Then we can consider the induced Besov-ish space $\mathcal{B}^{s}_{p,q}(Q,\mathcal{P}_{Q},\mathcal{A}_{Q})$. Of course the inclusion

[TABLE]

is well-defined linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 57- we replaced ”well defined”by ”well-defined” and it is a weak contraction, that is

[TABLE]

Under the degree of generality we are considering here, the restriction transformation

[TABLE]

given by $r(f)=f\cdot 1_{Q}$, is a bounded linear transformation, however it is easy to find examples of Besov-ish spaces where $f1_{Q}\not\in\mathcal{B}^{s}_{p,q}(Q,\mathcal{P}_{Q},\mathcal{A}_{Q}).$

11. Examples of classes of atoms.

There are many classes of atoms one may consider. We list here just a few of them.

11.1. Souza’s atoms

Let $Q\in\mathcal{P}$. A $(s,p)$-Souza’s atom supported on $Q$ is a function $a\colon I\rightarrow\mathbb{C}$ such that $a(x)=0$ for every $x\not\in Q$ and $a$ is constant on $Q$, with

[TABLE]

The set of Souza’s atoms supported on $Q$ will be denoted by $\mathcal{A}^{sz}_{s,p}(Q).$ A canonical Souza’s atom on $Q$ is the Souza’s atom such that $a(x)=|Q|^{s-1/p}$ for every $x\in Q$. Souza’s atoms are $(s,p,\infty)$-type atoms.

11.2. Hölder atoms

Suppose that $I$ is a quasi-metric space with a quasi-distance $d(\cdot,\cdot)$, such that every $Q\in\mathcal{P}$ is a bounded set and there is $\Cl[c]{hh},\Cl[c]{hhh}\in(0,1)$ such that

[TABLE]

for every $P\subset Q$ with $P\in\mathcal{P}^{k+1}$ and $Q\in\mathcal{P}^{k}$. Additionally assume that there is $\Cl[normal]{kki}\geq 0$ and $D\geq 0$ such that

[TABLE]

Let

[TABLE]

For every $Q\in\mathcal{P}$, Let $C^{\alpha}(Q)$ be the Banach space of all functions $\phi$ such that $\phi(x)=0$ for $x\not\in Q$, and

[TABLE]

Let $\mathcal{A}^{h}_{s,\beta,p}(Q)\subset C^{\beta D}(Q)$ be the convex subset of all functions $\phi$ satisfying

[TABLE]

We say that $\mathcal{A}^{h}_{s,\beta,p}(Q)$ is the set of $(s,\beta,p)$-Hölder atoms supported on $Q$. Of course $\mathcal{A}^{h}_{s,\beta,p}$-atoms are $(s,p,\infty)$-type atoms and $\mathcal{A}^{sz}_{s,p}(Q)\subset\mathcal{A}^{h}_{s,\beta,p}(Q)$.

11.3. Bounded variation atoms

Now suppose that $I$ is an interval of $\mathbb{R}$ with length $1$, $m$ is the Lebesgue measure on it and the partitions in the grid $\mathcal{P}$ are partitions by intervals. Let $Q$ be an interval and $s\leq\beta$, $p\in[1,\infty)$. A $(s,\beta,p)$-bounded variation atom on $Q$ is a function $a\colon\mathbb{R}\rightarrow\mathbb{C}$ such that $a(x)=0$ for every $x\not\in Q$,

[TABLE]

and

[TABLE]

Here $var_{1/\beta}(\cdot,Q)$ is the pseudo-norm

[TABLE]

where the sup runs over all possible sequences $x_{1}<x_{1}<\dots<x_{n}$, with $x_{i}$ in the interior of $Q$. We will denote the set of bounded variation atoms on $Q$ as $\mathcal{A}^{bv}_{s,p,\beta}(Q)$. Bounded variation atoms are also $(s,p,\infty)$-type atoms.

II. SPACES DEFINED BY SOUZA’S ATOMS.

**

In Part II. we suppose that $s>0$, $p\in[1,\infty)$ and $q\in[1,\infty]$.

---

 **

12. Besov spaces in a measure space with a good grid

We will study the Besov-ish spaces $\mathcal{B}^{s}_{p,q}(\mathcal{P},\mathcal{A}^{sz}_{s,p})$ associated with the measure space with a good grid $(I,\mathcal{P},m)$. We denote $\mathcal{B}^{s}_{p,q}=\mathcal{B}^{s}_{p,q}(\mathcal{P},\mathcal{A}^{sz}_{s,p})$. Note that $\mathcal{A}^{sz}_{s,p}$ satisfies ${\Cr{banach}}$-${\Cr{finite}}$. Note that by Proposition 6.1 there is $\beta>1$ such that $\mathcal{B}^{s}_{p,q}\subset L^{\beta}$.

If $p\in[1,\infty)$, $q\in[1,\infty]$ and $0<s<\frac{1}{p}$ we wil say that $\mathcal{B}^{s}_{p,q}$ is a Besov space.

13. Positive cone

We say that $f$ is $\mathcal{B}^{s}_{p,q}$-positive if there is a $\mathcal{B}^{s}_{p,q}$-representation

[TABLE]

where $c_{P}\geq 0$ and $a_{P}$ is the standard $(s,p)$-Souza’s atom supported on $P$. The set of all $\mathcal{B}^{s}_{p,q}$-positive functions is a convex cone in $\mathcal{B}^{s}_{p,q}$, denoted $\mathcal{B}^{s+}_{p,q}$. We can define a “norm” on $\mathcal{B}^{s+}_{p,q}$ as

[TABLE]

where the infimum runs over all possible $\mathcal{B}^{s}_{p,q}$-positive representations of $f$. Of course for every $f,g\in\mathcal{B}^{s+}_{p,q}$ and $\alpha\geq 0$ we have

[TABLE]

Moreover if $f\in\mathcal{B}^{s}_{p,q}$ is a real-valued function then one can find $f_{+},f_{-}\in\mathcal{B}^{s+}_{p,q}$ such that $f=f_{+}-f_{-}$ and

[TABLE]

An obvious but important observation is

Proposition 13.1**.**

if $f\in\mathcal{B}^{s+}_{p,q}$ then its support

[TABLE]

is (up to a set of zero measure) is a countable union of elements of $\mathcal{P}$.

14. Unbalanced Haar wavelets

Let $\mathcal{P}=(\mathcal{P}^{k})_{k}$ be a good grid. For every $Q\in\mathcal{P}^{k}$ let $\Omega_{Q}=\{P_{1}^{Q},\dots,P_{n_{Q}}^{Q}\}$, $n_{Q}\geq 2$, be the family of elements $\mathcal{P}^{k+1}$ such that $P_{i}^{Q}\subset Q$ for every $i$, and ordered in some arbitrary way. The elements of $\Omega_{Q}$ will be called children of $Q$. Note that every $Q\in\mathcal{P}$ has at least two children. We will use one of the method described (Type I tree with the logarithmic subtrees construction) in Girardi and Sweldens ****[24]**** to construct an unconditional basis of $L^{\beta}$, for every $1<\beta<\infty$.

Let $\mathcal{H}_{Q}$ be the family of pairs $(S_{1},S_{2})$, with $S_{i}\subset\Omega_{Q}$ and $S_{1}\cap S_{2}=\emptyset$, defined as

[TABLE]

where $\mathcal{H}_{Q,j}$ are constructed recursively in the following way. Let $\mathcal{H}_{Q,0}=\{(A,B)\}$, where $A=\{P_{1}^{Q},\dots,P_{[n_{Q}/2]}^{Q}\}$ and $B=\{P_{[n_{Q}/2]+1}^{Q},\dots,P_{n_{Q}}^{Q}\}$. Here $[x]$ denotes the integer part of $x\geq 0$. Suppose that we have defined $\mathcal{H}_{Q,j}$. For each element $(S_{1},S_{2})\in\mathcal{H}_{Q,j}$, fix an ordering $S_{1}=\{R_{1}^{1},\dots,R_{n_{1}}^{1}\}$ and $S_{2}=\{R_{1}^{2},\dots,R_{n_{2}}^{2}\}$. For each $i=1,2$ such that $n_{i}\geq 2$, define $T_{i}^{1}=\{R_{1}^{i},\dots,R_{[n_{i}/2]}^{i}\}$ and $T_{i}^{2}=\{R_{[n_{i}/2]+1}^{i},\dots,R_{n_{i}}^{i}\}$ and add $(T_{i}^{1},T_{i}^{2})$ to $\mathcal{H}_{Q,j+1}$. This defines $\mathcal{H}_{Q,j+1}.$

Note that since $\mathcal{P}$ is a good grid we have $\mathcal{H}_{Q,j}=\emptyset$ for large $j$ and indeed

[TABLE]

Define $\mathcal{H}=\cup_{Q\in\mathcal{P}}\mathcal{H}_{Q}$. For every $S=(S_{1},S_{2})\in\mathcal{H}_{Q}$ define

[TABLE]

where

[TABLE]

Note that

[TABLE]

Since $1\leq\#S_{i}\leq 1/\Cr{menor}$ we have

[TABLE]

so

[TABLE]

Consequently

[TABLE]

for every $x\in\cup_{P\in S_{1}\cup S_{2}}P.$ Here

[TABLE]

Let

[TABLE]

and define

[TABLE]

Then by Girardi and Sweldens ****[24]**** we have that

[TABLE]

is an unconditional basis of $L^{\beta}$ for every $\beta>1$.

15. Alternative characterizations I: Messing with norms.

We are going to describe three norms that are equivalent to $|\cdot|_{\mathcal{B}^{s}_{p,q}}$. Their advantage is that they are far more concrete, in the sense that we do not need to consider arbitrary atomic decompositions to define them.

15.1. Haar representation

For every $f\in L^{\beta}$, $\beta>1$, the series

[TABLE]

is converges unconditionally in $L^{\beta}$, where $d_{S}^{f}=\int f\phi_{S}\ dm.$ We will call the r.h.s. of (15.26) the Haar representation of $f$. Define

[TABLE]

15.2. Standard atomic representation

Note that

[TABLE]

where $k_{I}=|I|^{1/p-s-1/2}d_{I}^{f}$ and $a_{I}$ is the canonical Souza’s atom on $I$. Let $S\in\mathcal{H}$. Then $S\in\mathcal{H}_{Q}$, with $S=(S_{1},S_{2})$ and some $Q\in\mathcal{P}^{k}$, with $k\geq 0$. It is easy to see that for every $P\in S_{1}\cup S_{2}$ the function

[TABLE]

is a Souza’s atom on $P$. Choose

[TABLE]

Note that

[TABLE]

For every child $P$ of $Q\in\mathcal{P}^{k}$, $k\geq 0$, define

[TABLE]

where

[TABLE]

The (finite) number of terms on this sum depends only on the geometry of $\mathcal{P}$. Then $\tilde{a}_{P}^{f}$ is a Souza’s atom on $P$ and

[TABLE]

Let $a_{P}$ be the canonical $(s,p)$-Souza’s atom on $P$ and choose $x_{P}\in P$. Denote

[TABLE]

In particular, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 58- we added a comma for every $P$

[TABLE]

extends to a bounded linear functional in $L^{1}$. We have $|k_{P}^{f}|\leq\tilde{k}_{P}^{f}$ and

[TABLE]

where this series converges unconditionally in $L^{\beta}$. Here $a_{P}$ is the canonical Souza’s atom. We will call the r.h.s. of (15.29) the standard atomic representation of $f$. Let

[TABLE]

15.3. Mean oscillation

Define for $p\in[1,\infty)$

[TABLE]

and

[TABLE]

Denote for every $p\in[1,\infty)$ and $q\in[1,\infty]$

[TABLE]

with the obvious adaptation for $q=\infty$. Let

[TABLE]

15.4. These norms are equivalent

We have

Theorem 15.1**.**

Suppose $s>0$, $p\in[1,\infty)$ and $q\in[1,\infty]$. Each one of the norms $|f|_{\mathcal{B}^{s}_{p,q}}$, $N_{st}(f)$, $N_{haar}(f)$, $N_{osc}(f)$ is finite if and only if $f\in\mathcal{B}^{s}_{p,q}$. Furthermore these norms are equivalent on $\mathcal{B}^{s}_{p,q}$. Indeed

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

Proof.

The inequality (15.31) is obvious. To simplify the notation we write $d_{S},k_{P}$ instead of $d_{S}^{f},k_{P}^{f}$. 

Proof of (15.32). The number of terms in the r.h.s. of (15.28) depends only on the geometry of $\mathcal{P}$. Indeed

[TABLE]

Consider the standard atomic representation of $f$ given by (15.29). Note that by (15.27)

[TABLE]

for every $k$. Consequently

[TABLE]

This completes the proof of (15.32).

Proof of (15.33). Note that

[TABLE]

Given $\epsilon>0$ and $Q\in\mathcal{P}$, choose $c_{Q}\in Q$ such that

[TABLE]

Since $\phi_{S}$ has zero mean on $Q$ for every $S\in\mathcal{H}_{Q}$ we have

[TABLE]

Since $\epsilon$ is arbitrary, this concludes the proof of (15.33).

Proof of (15.34). Finally note that if $f\in\mathcal{B}^{s}_{p,q}$ and $\epsilon>0$ then there is a $\mathcal{B}^{s}_{p,q}$-representation of $f$

[TABLE]

such that

[TABLE]

For each $J\in\mathcal{P}^{k_{0}}$, choose $x_{J}\in J$. Then

[TABLE]

This is a convolution, so

[TABLE]

and since $\epsilon>0$ is arbitrary, by Proposition 6.1 we obtain

[TABLE]

This proves (15.34). ∎

The following is an important consequence of this section.

Corollary 15.2**.**

For each $P\in\mathcal{P}$ there exists a linear functional in $L^{1}$

[TABLE]

with the following property. The so-called standard $\mathcal{B}^{s}_{p,q}$-representation of $f\in\mathcal{B}^{s}_{p,q}$ given by

[TABLE]

satisfies

[TABLE]

16. Alternative characterizations II: Messing with atoms.

Here we move to alternative descriptions of $\mathcal{B}^{s}_{p,q}$, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 59- we added a comma which are quite different from linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 60- we replaced preposition ”of” by ”from” those in Section 15. Instead of choosing a definitive representation of elements of $\mathcal{B}^{s}_{p,q}$, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 61- we added a comma we indeed give atomic decompositions of $\mathcal{B}^{s}_{p,q}$ using far more general classes of atoms.

16.1. Using Besov’s atoms

The advantage of Besov’s atoms is that it is a wide and general class of atoms, that includes even unbounded functions. They can be considered in every measure space endowed with a good grid as in Section 3. Moreover in appropriate settings it contains Hölder and bounded variations atoms, which will be quite useful in the get other characterizations of $\mathcal{B}^{s}_{p,q}(\mathcal{P},\mathcal{A}^{sz}_{s,p})$. The atomic decompositions of $B^{s}_{p,q}(\mathbb{R}^{n})$ by Besov’s atoms were considered by Triebel ****[45]**** in the case $s>0$, $p=q\in[1,\infty]$ and by Schneider and Vybíral ****[40]**** in the case $s>0$, $p,q\in[1,\infty]$. They are refered there as ”non-smooth atomic decompositions”.

Let $s<\beta$ and $\tilde{q}\in[1,\infty]$. A $(s,\beta,p,\tilde{q})$-Besov atom on the interval $Q$ is a function $a\in\mathcal{B}^{\beta}_{p,\tilde{q}}(Q,\mathcal{P}_{Q},\mathcal{A}^{sz}_{\beta,p})$ such that $a(x)=0$ for $x\not\in Q$, and

[TABLE]

where

[TABLE]

The family of $(s,\beta,p,\tilde{q})$-Besov atoms supported on $Q$ will be denoted by $\mathcal{A}^{bv}_{s,\beta,p,\tilde{q}}(Q)$. Naturally $\mathcal{A}^{sz}_{s,p}(Q)\subset\mathcal{A}^{bs}_{s,\beta,p,\tilde{q}}(Q)$. By Proposition 6.1 we have

[TABLE]

**so a $(s,\beta,p,\tilde{q})$-Besov atom is an atom of type $(s,p,1)$. **

The following result says there are many ways to define $\mathcal{B}^{s}_{p,q}$ using various classes of atoms.

Proposition 16.1** (Souza’s atoms and Besov’s atoms).**

Let $\mathcal{P}$ be a good grid. Let $\mathcal{A}$ be a class of $(s,p,u)$-atoms , with $u\geq 1$, such that for some $s<\beta$, $\tilde{q}\in[1,\infty]$, and $\Cr{56}$, $\Cr{566}\geq 0$ we have that for every $Q\in\mathcal{P}$

[TABLE]

Then

[TABLE]

Moreover

[TABLE]

Proof.

The first inequality is obvious. To prove the second inequality, recall that due to linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 62- we added ”to” Proposition 8.1 it is enough to show the following claim

Claim. Let $b_{J}$ be a $(s,\beta,p,\tilde{q})$-Besov atom on $J\in\mathcal{P}^{j}$. Then for every $P\subset J$ with $P\in\mathcal{P}$ there is $m_{P}\in\mathbb{C}$ such that

[TABLE]

where $a_{P}$ is the canonical $(s,p)$-Souza’s atom on $P$ and

[TABLE]

Indeed, since

[TABLE]

there exists a $\mathcal{B}^{\beta}_{p,\tilde{q}}(J,\mathcal{P},\mathcal{A}^{s}_{\beta,p})$-representation

[TABLE]

where $d_{P}$ is the canonical $(\beta,p)$-Souza’s atom on $P$ and

[TABLE]

Then

[TABLE]

is a $(s,p)$-Souza’s atom and

[TABLE]

with $m_{P}=c_{P}|P|^{\beta-s}$ and

[TABLE]

∎

16.2. Using Hölder atoms

Suppose that $I$ is a quasi-metric space with a quasi-distance $d(\cdot,\cdot)$ and a good grid satisfying the assumptions in Section 11.2.

Proposition 16.2**.**

Suppose

[TABLE]

$p\in[1,\infty)$* and $\tilde{q}\in[1,\infty].$ For every $Q\in\mathcal{P}$ we have*

[TABLE]

for some $\Cr{at3}>0$. Moreover

[TABLE]

for some $\Cr{at4}>0.$ In particular

[TABLE]

and the corresponding norms are equivalent.

Proof.

Let $\phi\in\mathcal{A}^{h}_{s,\tilde{\beta},p}(Q)$. Then $\phi$ has a continuous extension to $\overline{Q}$. So firstly we assume that $\phi\geq 0$ has a continuous extension to $\overline{Q}$. Define

[TABLE]

and for every $P\subset W\subset Q\in\mathcal{P}^{j}$ with $P\in\mathcal{P}^{k+1}$ and $W\in\mathcal{P}^{k}$ define

[TABLE]

Of course in this case $c_{P}\geq 0$ and

[TABLE]

Here $\Cr{super}=\sup_{Q\in\mathcal{P}}|Q|^{\tilde{\beta}-\beta}.$ Consequently

[TABLE]

so

[TABLE]

This implies that

[TABLE]

where $a_{P}$ is the canonical $(\beta,p)$-atom on $P$, is a $\mathcal{B}^{\beta}_{p,\tilde{q}}(Q,\mathcal{P}_{Q}\mathcal{A}^{sz}_{\beta,p})$-representation of a function $\tilde{\phi}$. From (16.41) it follows that

[TABLE]

for every $x\in W\in\mathcal{P}^{N}.$ In particular

[TABLE]

for almost every $x$, so $\tilde{\phi}=\phi$. So

[TABLE]

where

[TABLE]

In the general case, note that $\phi_{+}(x)=\max\{\phi,0\},\phi_{-}(x)=\min\{-\phi,0\}\in\mathcal{A}^{h}_{s,\tilde{\beta},p}(Q)$ and $\phi=\phi_{+}-\phi_{-}$. Applying (16.43) to $\phi_{+}$ and $\phi_{-}$ we obtain (16.38).

The second inclusion (16.39) in the proposition can be obtained taking $\tilde{\beta}=\beta$ in (16.2) and a few modifications in the above argument. By Proposition 16.1 we have (16.40). ∎

16.3. Using bounded variation atoms

Now suppose that $I$ is an interval of $\mathbb{R}$ of length $1$, $m$ is the Lebesgue measure on it and the partitions in $\mathcal{P}$ are partitions by intervals.

Proposition 16.3**.**

If

[TABLE]

then

[TABLE]

for every $Q\in\mathcal{P}.$ If

[TABLE]

then

[TABLE]

for every $q\in[1,\infty]$. In particular

[TABLE]

and the corresponding norms are equivalents.

Proof.

Suppose

[TABLE]

Let $Q\in\mathcal{P}^{j}$ and $a_{Q}\in\mathcal{A}^{bv}_{s,\tilde{\beta},p}(Q)$. We have

[TABLE]

where $\Cr{supe}=\sup_{Q\in\mathcal{P}}|Q|^{\beta}.$ Note that for every $k\geq j$

[TABLE]

Note that in the case $\tilde{\beta}p=1$ the argument above needs a simple modification. For $k<j$ let $W_{Q}\in\mathcal{P}^{k}$ be such that $Q\subset W_{Q}$. Then

[TABLE]

By Theorem 15.1 we have that if $\beta=\tilde{\beta}$ then $\Cl[normal]{at1}\mathcal{A}^{bv}_{s,\beta,p}(Q)\subset\mathcal{A}^{bs}_{s,\beta,p,\infty}(Q)$ for some $\Cr{at1}>0$, and if $\beta<\tilde{\beta}$ we have that for every $q\in[1,\infty)$

[TABLE]

so $\Cl[normal]{at2}\mathcal{A}^{bv}_{s,\tilde{\beta},p}(Q)\subset\mathcal{A}^{bs}_{s,\beta,p,q}(Q)$ for some $\Cr{at2}>0$. ∎

17. Dirac’s approximations

We will use the Haar basis and notation defined by Section 14. For every $x_{0}\in I$ and $k_{0}\in\mathbb{N}$ define the finite family

[TABLE]

Let $N(x_{0},k_{0})=\#\mathcal{S}_{x_{0}}^{k_{0}}$. Then we can enumerate the elements

[TABLE]

of $\mathcal{S}_{x_{0}}^{k_{0}}$ such that $S^{i}=(S^{i}_{1},S^{i}_{2})$ satisfies

[TABLE]

for some $a_{i}\in\{1,2\}$ and

[TABLE]

for every $i$. Let

[TABLE]

and define for $i>0$

[TABLE]

One can prove by induction on $j$ that for $j>0$

[TABLE]

and in particular

[TABLE]

where $x_{0}\in Q_{k_{0}}\in\mathcal{P}^{k_{0}}.$ In other words

[TABLE]

Note that

[TABLE]

Multiplying (17.44) by $f$ and integrating it term by term, and using (17.45) we obtain

[TABLE]

If $f\in L^{\beta}$, with $\beta>1$, it can be written as

[TABLE]

with $d_{S}=\int f\phi_{S}\ dm$, where this series converges unconditionally on $L^{\beta}$. Let

[TABLE]

Then

[TABLE]

Let

[TABLE]

be the series given by (15.29). Note that

[TABLE]

Consequently

Proposition 17.1** (Dirac’s Approximations).**

Let $f\in L^{\beta}$, with $\beta>1$. Let

[TABLE]

• A.

If this representation is either as in (15.29) or $k_{P}\geq 0$ for every $P$ then we have For every $Q\in\mathcal{P}$

[TABLE]

• B.

In the case of the representation (15.29) we also have

[TABLE]

Proof.

We have that $A.$ is obvious if $k_{P}\geq 0$ for every $P$. In the other case note that for every $x_{0}\in Q\in\mathcal{P}^{k_{0}}$ we have

[TABLE]

so A. and B. follows. ∎

III. APPLICATIONS.

**

In Part III. we suppose that $0<s<1/p$, $p\in[1,\infty)$ and $q\in[1,\infty]$.

---

 **

18. Pointwise multipliers acting on $\mathcal{B}^{s}_{p,q}$

Here we will apply the previous sections to study pointwise multipliers of $\mathcal{B}^{s}_{p,q}$. To be more precise, Let $g\colon I\rightarrow\mathbb{C}$ be a measurable function. We say that $g$ is a pointwise multiplier acting on $\mathcal{B}^{s}_{p,q}$ if the transformation

[TABLE]

defines a bounded operator in $\mathcal{B}^{s}_{p,q}$. We denote the set of pointwise multipliers by $M(\mathcal{B}^{s}_{p,q})$. We can consider the norm on $M(\mathcal{B}^{s}_{p,q})$ given by

[TABLE]

Of course a necessary condition for a function to be a multiplier is that

[TABLE]

Denote

[TABLE]

The linear space $\mathcal{B}^{s}_{p,q,selfs}$ endowed with $|\cdot|_{\mathcal{B}^{s}_{p,q,selfs}}$is a normed space introduced by Triebel ****[45]****. We have

[TABLE]

In the following three propositions we see that many results of Triebel ****[45]**** and Schneider and Vybíral ****[40]**** for Besov spaces in $\mathbb{R}^{n}$ can be easily moved to our setting. The simplest case occurs when $p=q=1$.

Proposition 18.1**.**

We have that $M(\mathcal{B}^{s}_{1,1})=\mathcal{B}^{s}_{1,1,selfs}$.

Proof.

Let $g\in\mathcal{B}^{s}_{1,1,selfs}.$ Given $f\in\mathcal{B}^{s}_{1,1}$ and $\epsilon>0$ one can find a $\mathcal{B}^{s}_{1,1}$-representation

[TABLE]

where $a_{Q}$ is a $(s,1)$-Souza’s atom and

[TABLE]

so

[TABLE]

and consequently

[TABLE]

Since $\epsilon$ is arbitrary we get

[TABLE]

∎

Lemma 18.2**.**

Let $W\in\mathcal{P}$. The restriction application

[TABLE]

given by $r(f)=1_{W}f$ is continuous. Indeed there is $\Cl[normal]{incint}\geq 1$, that does not depend on $W$, such that

1. A.

For every $f\in\mathcal{B}^{s}_{p,q}$ we have

[TABLE]

In particular $1_{W}\in M(\mathcal{B}^{s}_{p,q})$. 2. B.

For every $\mathcal{B}^{s+}_{p,q}$-representation

[TABLE]

one can find a $\mathcal{B}^{s+}_{p,q}(W,\mathcal{P}_{W},\mathcal{A}^{sz}_{s,p})$-representation

[TABLE]

such that

[TABLE]

Moreover $d_{Q}\neq 0$ implies $Q\subset\ supp\ f$.

Proof.

Let $Q\in\mathcal{P}.$ Denote by $a_{Q}$ the canonical $(s,p)$-Souza’s atom supported on $Q$. If $W\cap Q=\emptyset$ we can write $1_{W}a_{Q}=0a_{Q}.$ If $Q\subset W$ then $1_{W}a_{Q}=1a_{Q}.$ If $W\subset Q$ then

[TABLE]

where

[TABLE]

In every case we can write

[TABLE]

with

[TABLE]

By Proposition 8.1.A and 8.1.B there is $\Cr{incint}$ such that A. and B. hold. ∎

Proposition 18.3**.**

We have that $\mathcal{B}^{s}_{p,q,selfs}\subset L^{\infty}$ and this inclusion is continuous.

Proof.

Let $g\in\mathcal{B}^{s}_{p,q,selfs}$. Then $ga_{Q}\in\mathcal{B}^{s}_{p,q}$ and by Lemma 18.2 we have

[TABLE]

By Proposition 6.1 (taking t = $p$) we have $(s,p)$-Souza’s atom $a_{Q}$ we have

[TABLE]

for some constant $\Cr{nova}$. In other words

[TABLE]

so

[TABLE]

for every $Q\in\mathcal{P}$. Due to the fact that $\cup_{k}\mathcal{P}^{k}$ generates the $\sigma$-algebra $\mathbb{A}$, by Lévy’s Upward Theorem (see Williams [49]) for almost every $x\in I$ the following holds. If $x\in Q_{k}\in\mathcal{P}^{k}$ then

[TABLE]

So

[TABLE]

∎

inline,linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 63- In the caption of Figure 2, we replaced ”. the” by ”. The” (missing capitalization) and we replaced ”contributing for” by ”contribution to”

18.1. Non-Archimedean behaviour in $\mathcal{B}^{\beta}_{p,\tilde{q},selfs}$

If we have a sequence $g_{i}\in M(\mathcal{B}^{s}_{p,q})$ we can get the naive estimate

[TABLE]

Nevertheless linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 64- we replaced ”But it is remarkable that” by ”Nevertheless, remarkably,”, remarkably, sometimes one can get a far better estimate. To state the result we need to define

[TABLE]

It is easy to see that this norm is equivalent to $|\cdot|_{\mathcal{B}^{s}_{p,q,selfs}}$.

Proposition 18.4**.**

Let $\beta>s$. There is $\Cl[normal]{gen}$ with the following property. Let $g_{i}\in\mathcal{B}^{\beta,t_{i}}_{p,\tilde{q},selfs}$, with $i\in\Lambda\subset\mathbb{N}$, and $t_{i}\in\mathbb{N}$.

Consider a function $f$ with a $\mathcal{B}^{s}_{p,q}$-representation

[TABLE]

satisfying

• A.

We have

[TABLE]

• B.

If $Q\in\mathcal{P}^{k}$ satisfies $c_{Q}\neq 0$ and $Q\cap supp\ g_{i}\neq\emptyset$ then $k\geq t_{i}$.

Then we can find a $\mathcal{B}^{s}_{p,q}$-representation

[TABLE]

such that

[TABLE]

Proof.

It is enough to prove the result for the case when $\Lambda$ is finite. Let $Q\in\mathcal{P}^{k_{0}}$ with $c_{Q}\neq 0$. There is $\{i_{1},\dots,i_{j}\}\subset\Lambda$, such that

[TABLE]

and $Q\cap supp\ g_{i_{\ell}}\neq\emptyset$ for every $\ell$. In particular $k_{0}\geq\max_{\ell}t_{i_{\ell}}.$ By Lemma 18.2 for each $\ell\leq j$ we can find a $\mathcal{B}^{\beta}_{p,q}$-representation

[TABLE]

such that $b_{P}$ is the canonical $(\beta,p)$-Souza’s atom supported on $P$ and

[TABLE]

Since $b_{P}=|P|^{\beta-s}a_{P}$, where $a_{P}$ is the canonical $(s,p)$-Souza’s atoms supported on $P$, we can write

[TABLE]

with $s_{P,Q}^{\ell}=\tilde{s}_{P,Q}^{\ell}|P|^{\beta-s}$ satisfying

[TABLE]

so we can write

[TABLE]

with

[TABLE]

satisfying

[TABLE]

By Proposition 8.1.A we can find a $\mathcal{B}^{s}_{p,q}$-representation (18.51) satisfying (18.4). ∎

Remark 18.5**.**

If $g$ is $\mathcal{B}^{\beta}_{p,q}$-positive we can define

[TABLE]

If we assume additionally that $g_{i}$ are $\mathcal{B}^{\beta,t_{i}}_{p,q}$-positive, Proposition 18.4 remains true if we replace all the instances of $|\cdot|_{\mathcal{B}^{s,t}_{p,q,selfs}}$ by $|\cdot|_{\mathcal{B}^{s+,t}_{p,q,selfs}}$ in its statement. Moreover by Proposition 8.1.B and Lemma 18.2. B we can conclude that

• i.

if $c_{Q}\geq 0$ for every $Q$ then $d_{Q}\geq 0$ for every $Q$,

• ii.

If $Q$ is such that $d_{Q}\neq 0$ then $Q\subset supp\ g_{i}$, for some $i\in\Lambda$.

Corollary 18.6**.**

For every $\beta>s$ and $\tilde{q}\in[1,\infty]$ we have $\mathcal{B}^{\beta}_{p,\tilde{q},selfs}\subset M(\mathcal{B}^{s}_{p,q})$. Moreover this inclusion is continuous.

18.2. Strongly regular domains

We may wonder on which conditions the characteristic function of a set $\Omega$ is a pointwise multiplier in $\mathcal{B}^{s}_{p,q}$.

Definition 18.7**.**

A measurable set $\Omega\subset I$ is a $(\alpha,\Cl[normal]{rp2},\Cl[normal]{k1})$-strongly regular domain if for every $Q\in\mathcal{P}^{j}$, with $j\geq\Cr{k1}$, there is family $\mathcal{F}^{k}(Q\cap\Omega)\subset\mathcal{P}^{k}$ such that

• i.

We have $Q\cap\Omega=\cup_{k}\cup_{P\in\mathcal{F}^{k}(Q\cap\Omega)}P$.

• ii.

If $P,W\in\cup_{k}\mathcal{F}^{k}(Q\cap\Omega)$ and $P\neq W$ then $P\cap W=\emptyset$.

• iii.

We have

[TABLE]

The following result can be associated with results in Triebel ****[45]**** for $B^{s}_{p,p}(\mathbb{R}^{n})$, especially linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 65- we replaced ”specially” by ”especially” when we consider the setting of Besov spaces in compact homogenous spaces. See Section 18.2 for details.See also Schneider and Vybíral ****[40]****.

Proposition 18.8**.**

If $\Omega$ is a $(1-\beta p,\Cr{rp2},\Cr{k1})$-strongly regular domain then

[TABLE]

Proof.

Given $Q\in\mathcal{P}^{j}$, with $j\geq\Cr{k1}$ we can write

[TABLE]

where $a_{P}$ is a $(\beta,p)$-atom. Note that

[TABLE]

so (18.56) holds. ∎

Proposition 18.9** (Pointwise Multipliers I).**

There is $\Cl[normal]{gen2}$ with the following property. Suppose that $\Omega_{i}$ are $(1-\beta p,K_{i},t_{i})$-strongly regular domains, $i\in\Lambda\subset\mathbb{N}$, and $\Theta_{i}>0$ for every $i\in\Lambda_{i}$. Consider a function $f$ with a $\mathcal{B}^{s}_{p,q}$-representation

[TABLE]

satisfying

• A.

We have

[TABLE]

• B.

If $Q\in\mathcal{P}^{k}$ satisfies $c_{Q}\neq 0$ and $Q\cap\Omega_{i}\neq\emptyset$ then $k\geq t_{i}$.

Then we can find a $\mathcal{B}^{s}_{p,q}$-representation

[TABLE]

such that

[TABLE]

Moreover

• i.

If $Q$ satisfies $d_{Q}\neq 0$ then $Q\subset\Omega_{i}$ for some $i\in\Lambda$.

• ii.

If $c_{Q}\geq 0$ for every $Q$ then $d_{Q}\geq 0$ for every $Q$.

Proof.

It follows from Proposition 18.4, Proposition 18.8 and Remark 18.5. ∎

18.3. Functions on $\mathcal{B}^{1/p}_{p,\infty}\cap L^{\infty}$

We want linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 66- ”we replaced ”We would like to give” by ”We want to give” to give explicit examples of multipliers in $\mathcal{B}^{s}_{p,q}$. One should compare the following result with the study by Triebel**[44]**** of the regularity of the multiplication on Besov spaces. See also Maz’ya and Shaposhnikova [36] for more information on multipliers in classical Besov spaces.**

Proposition 18.10** (Pointwise multipliers II).**

Let $g\in\mathcal{B}^{1/p}_{p,\infty}\cap L^{\infty}$. Then the multiplier operator

[TABLE]

defined by $G(f)=gf$ is a well-defined linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 67- we replaced ”well defined” by ”well-defined” and bounded operator acting on $(\mathcal{B}^{s}_{p,q},|~{}\cdot~{}|_{\mathcal{B}^{s}_{p,q}})$. Indeed

[TABLE]

where $\Cr{e}=\Cr{e}(1/p,p,\infty)$ and $\Cr{no}=\Cr{no}(1/p,p,\infty)$ are as in Corollary 15.2.

Remark 18.11**.**

We can get a similar result replacing $\mathcal{B}^{a}_{p,b}$ by $\mathcal{B}^{a+}_{p,b}$ everywhere.

Proof.

Let $a_{Q}=|Q|^{s-1/p}1_{Q}$ be the canonical $(s,p)$-Souza’s atom on $Q$ and $b_{J}=1_{J}$ be the canonical $(1/p,p)$-Souza’s atom on $J$. Given $\epsilon>0$, let

[TABLE]

be a $\mathcal{B}^{s}_{p,q}$-representation of $f$ such that

[TABLE]

and

[TABLE]

be a $\mathcal{B}^{1/p}_{p,\infty}$-representation of $g$ given by Corollary 15.2 (in the case of Remark 18.11 we can consider an optimal $\mathcal{B}^{1/p}_{p,\infty}$-positive representation of $g$). We claim that

[TABLE]

[TABLE]

are $\mathcal{B}^{s}_{p,q}$-representations of functions $u_{i}\in\mathcal{B}^{s}_{p,q}$. Firstly note that the inner sums are finite. Moreover if $J\in\mathcal{P}^{j}$ we denote by $Q_{k}(J)$ the unique element of $\mathcal{P}^{k}$, with $k\leq j$ that satisfies $J\subset Q_{k}(J)$ then

[TABLE]

The right hand side is a convolution, so we can easily get

[TABLE]

Moreover by Proposition 17.1.B, with $s=1/p$, we obtain

[TABLE]

So

[TABLE]

We claim that $gf=u_{1}+u_{2}$. Indeed let

[TABLE]

and

[TABLE]

By Proposition 6.1 we have

[TABLE]

and

[TABLE]

So

[TABLE]

Note that

• A.

If $Q\subset J$ then $a_{Q}b_{J}=a_{Q}$,

• B.

If $J\subset Q$ then

[TABLE]

So

[TABLE]

Note that

[TABLE]

Now we can use Corollary 6.5.i to conclude the proof. ∎

19. $\mathcal{B}^{s}_{p,q}\cap L^{\infty}$ is a quasi-algebra

Multipliers in $\mathcal{B}^{s}_{p,q}\cap L^{\infty}$ are indeed much easier to come by.

Proposition 19.1** (Pointwise multipliers III).**

Let $g,f\in\mathcal{B}^{s}_{p,q}\cap L^{\infty}$. Then $g\cdot f\in\mathcal{B}^{s}_{p,q}\cap L^{\infty}$ and

[TABLE]

So $\mathcal{B}^{s}_{p,q}\cap L^{\infty}$ is a quasi-Banach linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 68- we replaced ”quasi Banach” by ”quasi-Banach” algebra. Here $\Cr{e}=\Cr{e}(s,p,q)$ and $\Cr{no}=\Cr{no}(s,p,q)$ are as in Corollary 15.2.

Proof.

Of course $|f\cdot g|_{\infty}\leq|f|_{\infty}|g|_{\infty}$. Let $a_{Q}=|Q|^{s-1/p}1_{Q}$ be the canonical $(s,p)$-Souza’s atom on $Q$. Let

[TABLE]

and

[TABLE]

be $\mathcal{B}^{s}_{p,q}$-representations of $f$ and $g$ given by Corollary 15.2. We claim that

[TABLE]

[TABLE]

are $\mathcal{B}^{s}_{p,q}$-representations of functions $u_{i}\in\mathcal{B}^{s}_{p,q}$. Moreover by Proposition 17.1.A we have

[TABLE]

So

[TABLE]

and by an analogous argument

[TABLE]

Define $f_{k_{0}}$ and $g_{k_{0}}$ as in the proof of Proposition 18.10. By Proposition 17.1 we have $|f_{k_{0}}|\leq|f|_{\infty}$ and $|g_{k_{0}}|\leq|g|_{\infty}$. Since $\lim_{k_{0}}f_{k_{0}}=f$ and $\lim_{k_{0}}g_{k_{0}}=g$ in $L^{p}$, we can assume, taking a subsequence if necessary, that $f_{k_{0}}g_{k_{0}}$ converges pointwise to $fg$. So by the Theorem of Dominated Convergence we have $\lim_{k_{0}}f_{k_{0}}g_{k_{0}}=fg$ in $L^{1}$. Finally note that if $J\subset Q$ then

[TABLE]

Now we can use the same argument as in the proof of Proposition 18.10 to conclude that $gf=u_{1}+u_{2}$. This concludes the proof. ∎

19.1. Regular domains

Here we will give sufficient conditions for the characteristic function of a set to define a bounded pointwise multiplier either on $\mathcal{B}^{s}_{p,q}\cap L^{\infty}$. For every set $\Omega$, let

[TABLE]

Definition 19.2**.**

We say that a countable family of pairwise disjoint measurable sets $\{\Omega_{r}\}_{r\in\Lambda}$ is $(\alpha,\Cl[normal]{domain},\Cl[c]{domainc})$-regular family if one can find families $\mathcal{F}^{k}(\Omega_{r})\subset\mathcal{P}^{k}$, $k\geq k_{0}(\Omega_{r})$, such that

• A.

We have $\Omega_{r}=\cup_{k\geq k_{0}(\Omega_{r})}\cup_{Q\in\mathcal{F}^{k}(\Omega_{r})}Q$.

• B.

If $P,Q\in\cup_{k\geq k_{0}(\Omega_{r})}\mathcal{F}^{k}(\Omega_{r})$ and $P\neq Q$ then $P\cap Q=\emptyset$.

• C.

We have

[TABLE]

We say that a measurable set $\Omega$ is a $(\alpha,\Cr{domain},\Cr{domainc})$-regular domain if $\{\Omega\}$ is a $(\alpha,\Cr{domain},\Cr{domainc})$-regular family.

Proposition 19.3**.**

Let $\beta>s$. Every $(1-\beta p,C,0)$-strongly regular domain is a $(1-sp,C^{\prime},\Cr{maior}^{(\beta-s)p})$-regular domain, for some $C^{\prime}$.

Proof.

Consider a $(1-\beta p,\Cr{rp2},0)$-strongly regular domain $\Omega$. There are at most $\Cr{menor}^{-k_{0}(\Omega)}$ elements in $\mathcal{P}^{k_{0}(\Omega)}$ and

[TABLE]

for every $Q,W\in\mathcal{P}^{k_{0}(\Omega)}$. Consequently

[TABLE]

for each $Q\in\mathcal{P}^{k_{0}(\Omega)}$. For every $Q\in\mathcal{P}^{k_{0}(\Omega)}$ there is a family $\mathcal{F}^{k}(Q\cap\Omega)$ such that

[TABLE]

and

[TABLE]

Let

[TABLE]

We have

[TABLE]

This concludes the proof. ∎

Remark 19.4**.**

Suppose that there is $\Cl[normal]{compara}$ such that for every $k$ and every $Q,W\in\mathcal{P}^{k}$ we have

[TABLE]

and

[TABLE]

Then it is easy to see that one can choose $C^{\prime}=\Cr{numero}\Cr{compara}C$.

The following result is similar to results for Sobolev spaces by Faraco and Rogers ****[21]****. See also Sickel ****[41]****.

Corollary 19.5**.**

If $\{\Omega_{r}\}_{r\in\Lambda}$ is a $(1-ps,\Cr{domain},\Cr{domainc})$-regular family then there is $\Cl[normal]{rel}$ such that for every $g\in\mathcal{B}^{s}_{p,q}\cap L^{\infty}$ and $r\in\Lambda$ we can find a $\mathcal{B}^{s}_{p,q}$-representation

[TABLE]

such that

[TABLE]

Note that

[TABLE]

is a $(1-ps,\Cr{domain},\Cr{domainc})$-regular domain and $F(g)=g1_{\Omega}$ is a bounded operator in $\mathcal{B}^{s}_{p,q}\cap L^{\infty}$ satisfying

[TABLE]

Moreover

[TABLE]

Proof.

Notice that

[TABLE]

where $c_{Q}=|Q|^{1/p-s}$ for every $Q\in\cup_{k}\cup_{r}\mathcal{F}^{k}(\Omega_{r})$ and $c_{Q}=0$ otherwise. Let

[TABLE]

be $\mathcal{B}^{s}_{p,q}$-representations $g$ given by Corollary 15.2. Consider $u_{1},u_{2}$ as in the proof of Proposition 19.1. By Proposition 17.1 we can get exactly the same estimate as in the proof of Proposition 19.1.

Note that those $Q\in\mathcal{P}^{k}$ for which the corresponding atom $a_{Q}$ has a non-vanishing linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 69- we replaced ”no vanishing” by ”non-vanishing” coefficient in the definition of $u_{1}$ belongs to $\cup_{r}\cup_{j}\mathcal{F}^{j}(\Omega_{r})$, and moreover every $J\in\mathcal{P}^{k}$ for which the corresponding atom linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 70- we replaced ”atoms” by ”atom” $a_{J}$ has non-vanishing linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 71- we replaced ”no vanishing” by ”non-vanishing” coefficients in the definition of $u_{2}$ is contained in some $Q\in\mathcal{F}^{j}(\Omega_{r})$, for some $j$ and $r$. In particular $J\subset\Omega_{r}$. So (19.61) holds, with

[TABLE]

for every $Q\subset\Omega_{r}$.

Note also that

[TABLE]

so (19.64) and consequently (19.63) hold. ∎

Remark 19.6**.**

Using the methods in Faraco and Rogers [21] one can show that quasiballs in $[0,1]^{n}$ (and in particular quasidisks in $[0,1]^{2}$, that is, domains delimited by quasicircles) give examples of regular domains in $[0,1]^{n}$ endowed with the good grid of dyadic $n$-cubes and the Lebesgue measure $m$.

20. A remarkable description of $\mathcal{B}^{s}_{1,1}$.

When $p=q=1$ (and $s>0$ small), something curious happens. We can skip the good grid and characterise the Besov space $\mathcal{B}^{s}_{1,1}$ of a homogeneous space using regular domains. Fix $\Cr{domain}\geq 1$ and $\Cr{domainc}\in(0,1)$. Let $\mathcal{W}$ be the family of all $(1-s,\Cr{domain},\Cr{domainc})$-regular domains. Of course $\mathcal{P}\subset\mathcal{W}$. Let $\hat{\mathcal{W}}$ be a family of sets satisfying

[TABLE]

Define $B^{1-s}$ as the set of all functions $f\in L^{1/(1-s)}$ that can be written as

[TABLE]

where $A_{i}\in\hat{\mathcal{W}}$ for every $i\in\mathbb{N}$ and

[TABLE]

It is easy to see that

[TABLE]

Define

[TABLE]

where the infimum runs over all possible representations (20.65). One can see that $(B^{1-s},|\cdot|_{B^{1-s}})$ is a normed vector space.

Proposition 20.1**.**

We have that $B^{1-s}=\mathcal{B}^{s}_{1,1}(\mathcal{P})$ and the corresponding norms are equivalent.

Proof.

Note that (19.64) says that there is $C$ such that if $A\in\mathcal{W}$ then $1_{A}\in\mathcal{B}^{s}_{1,1}(\mathcal{P})$ and

[TABLE]

In particular, linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 72- we added a comma if $f$ has a representation (20.65) we conclude that

[TABLE]

In particular $B^{1-s}\subset\mathcal{B}^{s}_{1,1}(\mathcal{P}).$ On the other hand if $g\in\mathcal{B}^{s}_{1,1}(\mathcal{P}).$ then we can write

[TABLE]

[TABLE]

and $|g|_{\mathcal{B}^{s}_{1,1}(\mathcal{P})}$ is the infimum of $\sum_{P\in\mathcal{P}}|s_{Q}|$ over all possible representations. In particular $g\in B^{1-s}$ and

[TABLE]

∎

Remark 20.2**.**

Let $I=[0,1]$ with the dyadic grid $\mathcal{D}$ and the Lebesgue measure $m$. We prove in Part IV that $\mathcal{B}^{s}_{1,1}(\mathcal{D})$, with $0<s<1$, is the Besov space $B^{s}_{1,1}([0,1])$, and its norms are equivalent. Note that every interval $[a,b]\subset[0,1]$ is a $(1-s,2,2^{s-1})$-regular domain. So we can apply Proposition 20.1 with $\hat{\mathcal{W}}=\{[a,b],\ 0\leq a<b\leq 1\}.$ That is, $f$ belongs to $B^{s}_{1,1}([0,1])$ if and only if it can be written as in (20.65), where every $A_{i}$ is an interval and $\sum_{i}|c_{i}|<\infty$, and the norm in $B^{s}_{1,1}([0,1])$ is equivalent to the infimum of $\sum_{i}|c_{i}|$ over all possible such representations. This characterisation of the Besov space $B^{s}_{1,1}([0,1])$ was first obtained by Souza [14].

21. Left compositions.

The following linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 73- we replaced ”folllowing” by ”following” result generalizes a well-known result on left composition operators acting on Besov spaces of $\mathbb{R}^{n}$. See Bourdaud and Kateb ****[6]**** ****[5]****[7]**** for recent linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 74- we replaced ”recents” by ”recent” developments on the study of left compositions on Besov spaces of $\mathbb{R}^{n}$.

Proposition 21.1**.**

Let

[TABLE]

be a Lipchitz function such that $g(0)=0$. Then the left composition

[TABLE]

defined by $L_{g}(f)=g\circ f$ is well-defined linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 75- we replaced ”well defined”by ”well-defined” and

[TABLE]

where $K$ is the Lipchitz constant of $g$. Consequently there exists $C$ such that

[TABLE]

for every $f\in\mathcal{B}^{s}_{p,q}$.

Proof.

Note that

[TABLE]

So it easily follows that $osc^{s}_{p,q}(g\circ f)\leq Kosc^{s}_{p,q}(f)$. Of course $|g\circ f|_{p}\leq K|f|_{p}$. In particular $g\circ f\in\mathcal{B}^{s}_{p,q}.$ ∎

inline,linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,size=tiny,]Revision 76- Reference Smania, D. ”Classic and Exotic Besov spaces defined by good grids” was published in The Journal of Geometric Analysis. We updated the reference.