Besovish Spaces Through Atomic Decomposition

BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION

DANIEL SMANIA

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 we obtain results on multipliers and left compositions in this setting.

Contents 1. Introduction2 2. Notation5

I. DIVIDE AND RULE. 6 3. Measure spaces and grids6 4. A bag of tricks.7 5. Atoms 10 6. Besov-ish spaces 10 7. Scales of spaces 16 8. Transmutation of atoms 18 9. Good grids 23 10. Induced spaces 23 11. Examples of classes of atoms. 23 11.1. Souza’s atoms 24 11.2. H¨olderatoms 24 11.3. Bounded variation atoms 24

II. SPACES DEFINED BY SOUZA’S ATOMS. 25 12. Besov spaces in a measure space with a good grid 25 13. Positive cone 25 14. Unbalanced Haar wavelets 25 arXiv:1903.06937v2 [math.CA] 1 Sep 2020 15. Alternative characterizations I: Messing with norms. 27 15.1. Haar representation 27 15.2. Standard atomic representation 28 15.3. Mean oscillation 29 15.4. These norms are equivalent 29 16. Alternative characterizations II: Messing with atoms. 33 16.1. Using Besov’s atoms 33

2010 Mathematics Subject Classiﬁcation. 37C30, 30H25, 42B35, 42C15, 42C40. Key words and phrases. atomic decomposition, Besov space, harmonic analysis, wavelets. 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 Tem´atico2017/06463-3. 1 2 D. SMANIA

16.2. Using H¨olderatoms 35 16.3. Using bounded variation atoms 36 17. Dirac’s approximations 37

III. APPLICATIONS. 40 18. Pointwise multipliers acting on s 40 Bp,q 18.1. Non-Archimedean behaviour in β 42 Bp,q,selfs˜ 18.2. Strongly regular domains 44 1/p 18.3. Functions on p, L∞ 45 s B ∞ ∩ 19. p,q L∞ is a quasi-algebra 48 19.1.B Regular∩ domains 49 s 20. A remarkable description of 1,1. 51 21. Left compositions. B 53 References 53

1. Introduction s n Besov spaces Bp,q(R ) were introduced by Besov [4]. This scale of spaces has been a favorite over the years, with thousands of references available. Perhaps two of its most interesting features is that many earlier classes of function spaces appear in this scale, as Sobolev spaces, and also that there are many equivalent ways to s n deﬁne Bp,q(R ), in 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 Rn. For a historical account on Besov spaces and related topics, see Triebel [47] and the shorter but useful Jaﬀard [30], Yuan, Sickel and Yang [50] and Besov and Kalyabin [3].

In the last decades there 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 Rn by something with a poorer structure. It turns out that for small s > 0 and p, q 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 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 ) construction of Besov spaces s , Bp,q with p, q 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. BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 3

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 class of examples.

The primary tool in this work is the concept of atomic decomposition. An atomic decomposition represents each function in a space of functions as a (in- ﬁnite) linear combination of fractions of the so-called atoms. The advantage of atomic decompositions is that the atoms are functions that are often far more regular than a typical element of the space. 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, 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 Hp(R) and Latter [35] found an atomic decomposition for Hp(Rn). The inﬂuential work of Frazier and Jawerth [22] gave us an atomic decomposition for the Besov spaces s n Bp,q(R ). In the context of homogeneous spaces, we have results by Han, Lu and Yang [27] on the atomic decomposition of Besov spaces by H¨olderatoms.

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

On the study of Besov spaces on Rn 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 faith- fully/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 dy- namical systems, transfer operators play a huge role. Those operators act on Lebesgue spaces L1(m), but they are more useful if one can show they have a (good) action on more regular function spaces. Unfortunately, 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 4 D. SMANIA as in certain dissipative PDEs, since the positive 1-eigenvectors of this operator, of utmost 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 smoothness s does not seem to be a very useful concept since we can deﬁne Bp,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 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 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 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 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, 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 the generality of the setting, its statement is 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 do not change the location of the support and also the ”mass” of the representation is 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 by the cost of the original atomic decomposition. We will use this result many times 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 it using combinations of Souza’s atoms, it is not clear a priori how rich are those spaces. So BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 5

We give a bunch of alternative characterisations of these Besov spaces. We show that using more ﬂexible classes of atoms (piecewise H¨olderatoms, p-bounded variation atoms and even Besov atoms with higher regularity), we obtain the same Besov s space. This often allows us to easily verify if a given function belongs to Bp,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 Calder´onreproducing 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, we study the boundedness of pointwise multipliers acting in the Besov space. Since it is eﬀort- less to multiply Souza’s atoms, the proofs of these results are concise and easy to understand, generalising some of the results for Besov spaces in Rn by Triebel[44] and Sickel [41]. We also study the boundedness of left composition in Besov spaces s n of measure spaces with a grid, similar to some results for Bp,q(R ) 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 C1,C2, ...... for positive constants and λ1, λ2,... for positive constants smaller than one. 6 D. SMANIA

Table 1. Most important notation/symbols/constants

Symbol Description I phase space m finite measure in the phase space I aP , bP an atom supported on P a class of atoms A(Q) set of atoms of class supported on Q A sz class of (s, p)-Souza’sA atoms As,p h class of (s, β, p)-Hölderatoms As,β,p bv class of (s, β, p)-bounded variation atoms As,β,p bs class of (s, β, p, q)-Besov’s atoms As,β,p,q grid of I Pk family subsets of I at the k-th level of P P λ1 λ2 describes geometry of the grid s ≤( ) (s,p,q)-Banach space defined by the classP of atoms Bp,q A A s or s ( sz ) (s,p,q)-Banach space defined by Souza’s atoms Bp,q Bp,q As,p P, Q, W elements of the grid P Lp or Lp(m) Lesbesgue spaces of (I, A, m) norm in Lp, p (0, ]. | · |p ∈ ∞ p0 1/p + 1/p0 = 1, with p [1, ]. ρ min 1, p, q . ∈ ∞ { } tˆ max t, 1 . { }

I. DIVIDE AND RULE.

In Part I. we are going to assume s > 0, p (0, ) and q (0, ]. ∈ ∞ ∈ ∞

3. Measure spaces and grids Let I be a measure space with a σ-algebra A and m be a measure on (I, A), m(I) < . Given a measurable set J I denote J = m(J). We denote the ∞ ⊂ | | Lebesgue spaes of (I, A, m) by Lp.A grid is a sequence of ﬁnite families of measur- k able sets with positive measure = ( )k N, so that at least one of these families is non empty and P P ∈ G . Given Q k, let 1 ∈ P Ωk = P k : P Q = . Q { ∈ P ∩ 6 ∅} Then k C1 = sup sup #ΩQ < . k Q k ∞ ∈P Deﬁne k = sup Q : Q k . To simplify the notation we also assume that P = Q for|P every| P {| |i and∈Q P } j satisfying i = j. We often abuse notation 6 k∈ P ∈k P 6 using for both ( )k and k . P P ∈N ∪ P BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 7

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 = ( k) given by D D k k = [i/2k, (i + 1)/2k), 0 i < 2k . D { ≤ } k n We can also consider the dyadic grid n = ( n)k of [0, 1) , endowed with the Lebesgue measure, given by D D k = J J , with J k . Dn { 1 × · · · × n i ∈ D } and also the corresponding d-adic grids replacing 2k by dk 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 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 precisely the same measure 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 result by Christ [9]. See [42]. Remark 3.4. One can constructs grids for smooth compact manifolds and bounded polyhedral domains in Rn 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 `q(`pP ) of all indexed sequences

x = (xP )P , ∈P with xP C, satisfying ∈ 1/q p q/p x ` (` ) = xP < , | | q pP | | ∞ k P k X X∈P with the usual modification when q = . Then (` (` ), ) is a complex q pP `q (`pP ) ∞ | · | ρ ρ-Banach space with ρ = min 1, p, q , that is, d(x, y) = x y ` (` ) is a complete { } | − | q pP metric in `q(`pP ). The following is a pair of arguments we will use across this paper to estimate norms in `p and `q(`pP ). 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 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 (0, ] we defined tˆ= max 1, t . ∈ ∞ { } 8 D. SMANIA

Proposition 4.1 (Hölder-like trick). Let t (0, ) and q (0, ]. Let a = ∈ ∞ ∈ ∞ (ak)k, b = (bk)k, c = (ck)k nonnegative sequences such that for every k ˆ ˆ ˆ a1/t C1/tˆb1/tc1/t. k ≤ k k Then if q < we have ∞ ˆ 1/q ( a1/t)t/tˆ C1/tC (t, q, b) cq/t , k ≤ 2 k Xk Xk and if q = ∞ 1/tˆ t/tˆ 1/t 1/t ( ak ) C C2(t, q, b) sup ck . ≤ k Xk where q0/t A. If t 1 and q 1 then C (t, q, b) = ( b )1/q0 if q > 1, ≥ ≥ 2 k k and C (t, 1, b) = sup b1/t if q = 1. 2 k k P B. If t 1 and q 1 then C (t, q, b) = sup b1/t. ≥ ≤ 2 k k (q/t)0 1/(t(q/t)0) C. If t < 1 and q/t 1 then C2(t, q, b) = ( k bk ) if q < t, ≥ 1/t and C2(t, q, b) = supk bk if q = t. P 1/t D. If t < 1 and q/t < 1 then C2(t, q, b) = supk bk . Proof. We have Case A. If t 1 and q 1, by the Hölderinequality for the pair (q, q0) ≥ ≥ ˆ 1/q a1/t = a1/t C1/t( bq0/t)1/q0 cq/t . k k ≤ k k k k k k X X X X Case B. If t 1 and q 1 then the triangular inequality for q implies ≥ ≤ | · | ˆ q ( a1/t)q = ( a1/t)q Cq/t b1/tc1/t k k ≤ k k k k k X X q X Cq/t b1/tc1/t ≤ k k k X q/t q/t q/t C (sup bk ) ck ≤ k Xk Case C. If t < 1 and q/t 1 then t/tˆ = 1/t and by the Hölderinequality for the ≥ pair (q/t, (q/t)0) ˆ t/q a1/t C( b(q/t)0 )1/(q/t)0 cq/t , k ≤ k k k k k X X X Case D. if t < 1 and q/t < 1 then using the triangular inequality for q/t | · | ˆ q/t ( a1/t)q/t Cq/t b c k ≤ k k Xk Xk q/t Cq/t b c ≤ k k k X q/t q/t q/t C (sup bk ) ck . ≤ k Xk BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 9

Proposition 4.2 (Convolution trick). Let p, q (0, ). Let a = (ak)k , b = ∈ ∞ ∈Z (bk)k , c = (ck)k 0 be such that for every k ∈Z ∈Z ≥ 1/pˆ 1/pˆ 1/pˆ 1/pˆ ak C bk ici . ≤ − i X∈Z Then 1/q ( aq/p)1/q C1/pC (p, q, b) cq/p , k ≤ 3 k Xk Xk where C 1 satisﬁes 3 ≥ 1/p A. If p 1 and q 1 then C3(p, q, b) = k bk . ≥ ≥ ∈Z B. If p 1 and q 1 then C (p, q, b) = ( bq/p)1/q. ≥ ≤ 3 P k Z k C. If p < 1 and q/p 1 then C (p, q, b) = ( ∈ b )1/p. 3 P k Z k ≥ ∈ q/p 1/q D. If p < 1 and q/p < 1 then C3(p, q, b) = ( k bk ) P ∈Z Proof. We have P Case A. If p 1 and q 1, by the Young’s inequality for the pair (1, q) ≥ ≥ 1/q ( aq/p)1/q C1/p b1/p cq/p . k ≤ k k k k k X X X Case B. If p 1 and q 1 then the triangular inequality for q and the Young’s inequality for≥ the pair (1≤, 1) imply | · |

q/p q/p 1/p 1/p q ak = C bk ici − k k i X X X∈Z q/p q/p q/pˆ C bk ici ≤ − k i X X∈Z Cq/p( bq/p) cq/p ≤ k k Xk Xk Case C. If p < 1 and q/p 1 then by the Young’s inequality for the pair (1, q/p) ≥ p/q ( aq/p)p/q C( b ) cq/p . k ≤ k k k k k X X X Case D. If p < 1 and q/p < 1 then using the triangular inequality for q/p and the Young’s inequality for the pair (1, 1) | · |

q/p q/p q/p a C bk ici k ≤ − k k i X X X∈Z q/p q/p q/p C bk ici ≤ − k i X X∈Z Cq/p( bq/p) cq/p . ≤ k k Xk Xk 10 D. SMANIA

5. Atoms Let be a grid. Let p [1, ), u [1, ], and s > 0. A family of atoms asso- P ∈ ∞ ∈ ∞ k ciated with of type (s, p, u) is an indexed family of pairs ( (Q), (Q))Q k where P A B A ∈∪ P

A . (Q) is a complex Banach space contained in Lpu. 1 B A2. If φ (Q) then φ(x) = 0 for every x Q. A . (Q∈)B is a convex subset of (Q) such6∈ that φ (Q) if and only if 3 A B ∈ A σ φ (Q) for every σ C satisfying σ = 1. ∈ A ∈ | | A4. We have s 1 φ Q − u0p | |pu ≤ | | for every φ (Q). ∈ A We will say that φ (Q) is an -atom of type (s, p, u) supported on Q and that (Q) is the local Banach∈ A space onA Q. Sometimes we also assume B

A5. For every Q we have that (Q) is a compact subset in the strong topology of L∈p. P A or

A6. We have p [1, ) and every Q the set (Q) is a sequentially compact subset in the∈ weak∞ topology of L∈p P. A or even

A7. For every Q we have that (Q) is ﬁnite dimensional and (Q) contains a neighborhood∈ P of 0 in (Q). B A B We provide examples of classes of atoms in Section 11.

6. Besov-ish spaces k Let p (0, ), u [1, ], q (0, ], s > 0, = ( )k 0 be a grid and let be a family∈ of∞ atoms∈ of type∞ (s,∈ p, u).∞ We will alsoP assumeP that≥ A

G2. We have C = C (p, q, ( k s) ) < . 4 2 |P | k ∞ and lim k = 0. k |P | s p The Besov-ish space p,q(I, , ) is the set of all complex valued functions g L that can be representedB by anP absolutelyA convergent series on Lp ∈

∞ (6.1) g = sQaQ k=0 Q k X X∈P where aQ is in (Q), sQ C and with ﬁnite cost A ∈ ∞ 1/q (6.2) ( s p)q/p < . | Q| ∞ k=0 Q k X X∈P BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 11

Note that the inner sum in (6.1) is ﬁnite. By absolutely convergence in Lp we mean that

∞ p/pˆ s a < . Q Q p ∞ k=0 Q k X X∈P

The series in (6.1) is called a s (I, , )-representation of the function g. Deﬁne Bp,q P A

∞ p q/p 1/q g s (I, , ) = inf ( sQ ) , | |Bp,q P A | | k=0 Q k X X∈P 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 s s I and/or the grid we will write either p,q( , ) or even p,q( ) instead of s P s B P A B A p,q(I, , ). Whenever we write just p,q it means that we choose the Souza’s B PszA B atoms s,p, with parameters s and p, a class of atoms we properly deﬁne in Section 11.1. A

Proposition 6.1. Assume G -G and A -A . Let t (0, ) be such that 1 2 1 4 ∈ ∞ 1 1 (6.3) s + 0, p t pu − p t ≥ ≤ ≤ and suppose

1+1/t k t(s 1 + 1 ) (6.4) C = C C (t, q, ( − p t ) ) < , 5 1 2 |P | k ∞

Then for every coeﬃcients sQ satisfying (6.2) and every -atoms aQ on Q the series (6.1) converges absolutely on Lt. Indeed A

1/p 1+1/t k s 1 + 1 p (6.5) s a C − p t s | Q Q|t ≤ 1 |P | | Q| Q k Q k X∈P X∈P and

(6.6) g t C5 φ s ( ). | | ≤ | |Bp,q A Proof. Firstly note that if p t pu ≤ ≤

a t = a (x) t1 dm(x) | P |t | P | P Z t a pu 1 pu P t P pu t ≤ || | | | | − pu t t − a P pu ≤ | P |pu| | t pu t 1 1 ts − t(s + ) P − u0p P pu = P − p t . ≤ | | | | | | 12 D. SMANIA

Consequently

s a t s a t dm | Q Q|t ≤ | Q Q| Q k Z Q k X∈P X∈P t sP aP dm ≤ Q | | Q k Z P k X∈P X∈P t sP aP dm ≤ Q | | Q k Z P Ωk X∈P X∈ Q t t C1 sP aP dm ≤ Q | | Q k Z P Ωk X∈P X∈ Q Ct s a t dm ≤ 1 | P P | Q k P Ωk Z X∈P X∈ Q Ct s t a t dm ≤ 1 | P | | P |t Q k P Ωk Z X∈P X∈ Q t t(s 1 + 1 ) t C P − p t s ≤ 1 | | | P | Q k P Ωk X∈P X∈ Q t+1 t(s 1 + 1 ) t C P − p t s ≤ 1 | | | P | P k X∈P t+1 k t(s 1 + 1 ) t C − p t s ≤ 1 |P | | P | P k X∈P By Proposition 4.1 (H¨older-like trick) and p t we have ≤ ˆ 1/tˆ t/t s a s a t | Q Q|t ≤ | Q Q|t k Q k k Q k X X∈P X X∈P 1 1/q 1+ k t(s 1 + 1 ) t q/t C t C (t, q, ( − p t ) ) s . ≤ 1 2 |P | k | P | k P k X X∈P 1 1/q 1+ k t(s 1 + 1 ) p q/p C t C (t, q, ( − p t ) ) s . ≤ 1 2 |P | k | P | k P k X X∈P

Remark 6.2. Note that due to G2 if t = p then C5 < . Sometimes it is convenient to use sharper estimates than (6.5) and (6.6) replacing∞ k by the sequence |P | Ck = max Q : Q k, s = 0 . {| | ∈ P Q 6 } k For instance, if sQ = 0 for every Q with k N, then we can replace k t(s 1 + 1 ) k∈t(s P 1 + 1 ) ≤ C2(t, q, ( − p t )k) by C2(t, q, ( − p t 1(N, )(k))k) in (6.4). Here 1(N, ) denotes the|P | indicator function of the|P set| (N, ). ∞ ∞ ∞ Proposition 6.3. Assume G -G and A -A . Then s ( ) is a complex linear 1 2 1 4 Bp,q A space and s ( ) is a ρ-norm, with ρ = min 1, p, q . Moreover the linear inclusion |·|Bp,q A { } s p ı:( p,q( ), s ( )) (L , p) B A | · |Bp,q A → | · | BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 13 is continuous. Proof. Le f, g s ( ). Then there are s ( )-representations ∈ Bp,q A Bp,q A ∞ ∞ f = sQ0 aQ0 and g = sQaQ. k=0 Q k k=0 Q k X X∈P X X∈P Let sgn 0 = 0 and sgn z = z/ z otherwise. Of course | | (6.7) cQbQ = sQ0 aQ0 + sQaQ, Q k Q k Q k X∈P X∈P X∈P where1 sQ0 sQ b = | | sgn(s0 )a0 + | | sgn(s0 )a , Q s + s Q Q s + s Q Q | Q| | Q0 | | Q| | Q0 | and c = s0 + s . Q | Q| | Q| Note that sgn(sQ0 )aQ0 , sgn(sQ)aQ are atoms due to A3. So by A3 we have that bQ is also an atom, since it is a convex combination of atoms. In particular

cQbQ k Q k X X∈P p s converges absolutely in L to f + g. It remains to prove that this is a p,q( )- representation of f + g. Indeed B A

∞ p q/p ρ/q ∞ p q/p ρ/q ∞ p q/p ρ/q ( c ) ( s0 ) + ( s ) . | Q| ≤ | Q| | Q| k=0 Q k k=0 Q k k=0 Q k X X∈P X X∈P X X∈P Taking the inﬁmum over all possible s ( )-representations of f and g we obtain Bp,q A ρ ρ ρ f + g s ( ) f s ( ) + g s ( ). | |Bp,q A ≤ | |Bp,q A | |Bp,q A The identity γf s ( ) = γ f s ( ) is obvious. By Proposition 6.1 we have that | |Bp,q A | || |Bp,q A if f s ( ) = 0 then f p = 0, so f = 0, so s ( ) is a ρ-norm, moreover (6.6) | |Bp,q A | | | · |Bp,q A tell us that ı is continuous.

Proposition 6.4. Assume G1-G2 and A1-A4. Suppose that gn are functions in s ( ) with s ( )-representations Bp,q A Bp,q A ∞ n n gn = sQaQ, k=0 Q k X X∈P where an is a -atom supported on Q, satisfying Q A i. There is C such that for every n

∞ (6.8) ( ( sn p)q/p)1/q C. | Q| ≤ k=0 Q k X X∈P ii. For every Q we have that s = lim sn exists. ∈ P Q n Q iii. For every Q there is a (Q) such that ∈ P Q ∈ A 1 0 We don’t need to worry so much if |sQ| + |sQ| = 0, since in this case cQ = 0 and we can choose bQ to be an arbitrary atom (for instance aQ) 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. 14 D. SMANIA

n p (1) either the sequence aQ converges to aQ in the strong topology of L , or (2) we have p [1, ) and an weakly converges to a . ∈ ∞ Q Q then g either strongly or weakly converges in Lp, respectively, to g s ( ), n ∈ Bp,q A where g has the s ( )-representation Bp,q A ∞ (6.9) g = sQaQ k=0 Q k X X∈P that satisﬁes

∞ (6.10) ( ( sn p)q/p)1/q C | Q| ≤ k=0 Q k X X∈P s Proof. By (6.8) it follows that (6.10) holds and that (6.9) is indeed a p,q( )- B pA representation of a function g. It remains to prove that gn converges to g in L in the topology under consideration. Given > 0, ﬁx N large enough such that 1+1/p k ps ρ 1/ρ p/pˆ C1 C2(p, q, ( 1[N, )(k))k)(2C + 1) < (/2) . |P | ∞ We can write g g = (sn an s a ) + cn bn , n − Q Q − Q Q Q Q k N Q k k>N Q k X≤ X∈P X X∈P where sn s bn = | Q| sgn(sn )an + | Q| sgn( s )a , Q sn + s Q Q sn + s − Q Q | Q| | Q| | Q| | Q| is an atom in (Q), and A cn = sn + s . Q | Q| | Q| Note that the series in the r.h.s. converges absolutely in Lp. Of course

∞ ( ( cn p)q/p)1/q (2Cρ + 1)1/ρ, | Q| ≤ k=0 Q k X X∈P So by (6.6) in Proposition 6.1 (see also Remark 6.2) we have n n 1+1/p k ps ρ 1/ρ p/pˆ cQbQ p C1 C2(p, q, ( 1[N, )(k))k)(2C + 1) < (/2) . | | ≤ |P | ∞ k>N Q k X X∈P In the case ii.1, note that if n is large enough then (sn an s a ) p/pˆ < /2, | Q Q − Q Q |p k N Q k X≤ X∈P p/pˆ and consequently g gn p < . So gn strongly converges to g. In the case ii.2, given| −φ | (Lp)?, with p 1 we have that for n large enough ∈ ≥ n n φ( (s a s a )) φ p ? /2, | Q Q − Q Q | ≤ | |(L ) k N Q k X≤ X∈P and of course n n φ( c b ) φ p ? /2, | Q Q|p | ≤ | |(L ) k>N Q k X X∈P p so gn weakly converges to g in L . BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 15

Corollary 6.5. Assume G1-G2 and A1-A4, and

(1) either A5 or (2) we have p 1 and A . ≥ 6 Then s i. Let gn p,q( ) be such that gn s ( ) C for every n. Then there is a ∈ B A | |Bp,q A ≤ subsequence that converges either strongly or weakly in Lp, respectively, to s some g ( ) with g s C. p,q p,q ( ) ∈ B As | |B A ≤ ii. In both cases ( p,q( ), s ( )) is a complex ρ-Banach space, with ρ = B A | · |Bp,q A min 1, p, q , { } iii If A5 holds then the inclusion s p ı:( p,q( ), s ( )) (L , p) B A | · |Bp,q A → | · | is a compact linear inclusion. Proof of i. There are s ( )-representations Bp,q A ∞ n n gn = sQaQ, k=0 Q k X X∈P where an is a -atom supported on Q and Q A ∞ (6.11) ( ( sn p)q/p)1/q C + ε , | Q| ≤ n k=0 Q k X X∈P n k and 1 εn 0. In particular, sQ C+1. Since the set k is countable, by the Cantor≥ diagonal→ argument, taking| | a ≤ subsequence we can assume∪ P that sn s for Q →n Q some sQ C. Due to A5 (A6) and the Cantor diagonal argument, we can suppose that an strongly∈ (weakly) converges in Lp to some a (Q). We set Q Q ∈ A ∞ g = sQaQ. k=0 Q k X X∈P s By Proposition 6.4 we conclude that g ( ) with g s C, and that g p,q p,q ( ) n p ∈ B A | |B A ≤ converges to g in L . s Proof of ii. Let gn be a Cauchy sequence on p,q( ). By Proposition 6.1 we have p B A p that gn is also a Cauchy sequence in L . Let g be its limit in L . By Corollary 6.5.i have that g s ( ). Note that for large m and n ∈ Bp,q A gn gm s ( ) , | − |Bp,q A ≤ and g g converges to g g in Lp, so again by Corollary 6.5.i we have that n − m n − gn g s ( ) , | − |Bp,q A ≤ so g converges to g in s ( ). n Bp,q A Proof of iii. It follows from i. The proof of the following result is quite similar.

Corollary 6.6. Assume G1-G2 and A1-A4, and

(1) either A5 or (2) we have p 1 and A . ≥ 6 16 D. SMANIA

Then for every f s ( ) there is a s ( )-representation ∈ Bp,q A Bp,q A f = cP aP k P k X X∈P such that 1/q p q/p f s ( ) = cP . | |Bp,q A | | k P k X X∈P We refer to Edmunds and Triebel [20] for more information on compact linear transformations between quasi-Banach spaces.

Corollary 6.7. Assume G1-G2 and A1-A4. If for every Q we have that (Q) ∈ P s B is finite-dimensional and (Q) is a closed subset of (Q) then ( p,q( ), s ( )) A B B A | · |Bp,q A is a ρ-Banach space, with ρ = min 1, p, q . { } Proof. Since all norms are equivalent in (Q) we have that A4 implies that (Q) is a closed and bounded subset of (Q),B so it is compact. By Corollary 6.5A.ii it s B follows that ( p,q( ), s ( )) is a ρ-Banach space. B A | · |Bp,q A 7. Scales of spaces Note that a family of atoms of type (s, p, u) induces an one-parameter scale A s˜ , → As,p˜ where is the family of atoms of type (˜s, p, u) defined by As,p˜ s˜ s (Q) = Q − a : a . As,p˜ {| | Q Q ∈ A} Moreover a family of atoms of type (s, p, ) induces a two-parameter scale A ∞ (˜s, p˜) , → As,˜ p˜ where is the family of atoms of type (˜s, p,˜ ) defined by As,˜ p˜ ∞ s˜ s+1/p 1/p˜ (Q) = Q − − a : a . Ap,˜ s˜ {| | Q Q ∈ A} Proposition 7.1. Assume G1-G2. Suppose that the (s, p, )-atoms satisfy A1- A . Let 0 s < s˜ and q, q˜ [1, ]. Suppose ∞ A 4 ≤ ∈ ∞ 1/q k q(˜s s) − < . |P | ∞ Xk Then s˜ s A. We have p,q˜( s,p˜ ) p,q( s,p) and the inclusion is a continuous linear map. B A ⊂ B A s˜ B. Suppose that also satisfies A5. Let gn ( s,p˜ ) be such that gn s˜ p,q˜ p,q˜( s,p˜ ) ∈ B A | s |B A ≤ C for every n. Then there is a subsequence that converges in p,q( s,p) to s˜ B A some g p,q˜( s,p) with g s˜ ( ) C. ∈ B A | |Bp,q˜ As,p˜ ≤ s˜ s C. Suppose that also satisfies A7. The inclusion ı: p,q˜( s,p˜ ) p,q( s,p) is a compact linear map. B A 7→ B A Proof. Consider a s˜ ( )-representation Bp,q˜ As,p˜ ∞ f = sQaQ, k=0 Q k X X∈P BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 17

s s˜ Since aQ is an s,p˜ -atom, we have that bQ = aQ Q − is an s,p-atom. In particular, we can writeA | | A

∞ s˜ s f = s Q − b . Q| | Q k=0 Q k X X∈P If k k then ≥ 0 1/p p p(˜s s) k s˜ s p 1/p ( s Q − − ( s ) | Q| | | ≤ |P | | Q| Q k Q k X∈P X∈P 1/q˜ k s˜ s p q/p˜ − ( s ) , ≤ |P | | Q| k k0 Q k X≥ X∈P so q/p 1/q p p(˜s s) ( s Q − | Q| | | k k0 Q k X≥ X∈P 1/q 1/q˜ k q(˜s s) p q/p˜ (7.12) − ( s ) . ≤ |P | | Q| k k0 k k0 Q k X≥ X≥ X∈P s˜ s Proof of A. In particular, taking k0 = 0 we conclude that p,q˜( s,p˜ ) p,q( s,p) and B A ⊂ B A 1/q k q(˜s s) s s˜ f ( p,s) − f ( ). | |Bp,q A ≤ |P | | |Bp,q˜ Ap,s˜ Xk Proof of B. By deﬁnition, there exist sn C, such that Q ∈

∞ n n gn = sQaQ, k=0 Q k X X∈P where an is a -atom supported on Q and Q As,p˜ ∞ (7.13) ( ( sn p)q/p˜ )1/q˜ C + ε , | Q| ≤ n k=0 Q k X X∈P n k where εn 0. In particular, sQ C + εn. Since the set k is countable, by → | | ≤ ∪ P n the Cantor diagonal argument, taking a subsequence we can assume that sQ sQ n p → and (due to A5) that aQ converges in (Q) and L to some aQ s,p˜ . By Lemma p B ∈ A 6.4 the sequence gn converge in L to a function g such that g s˜ ( ) C and | |Bp,q˜ As,p˜ ≤ with s˜ ( )-representation Bp,q˜ As,p ∞ g = sQaQ. k=0 Q k X X∈P s It remains to show that the convergence indeed occurs in the topology of p,q( s,p). For every k 0 and δ > 0 we can write B A 0 ≥ n s˜ s n n g g = δd + Q − c b , n − Q | | Q Q k 0, choose k such that bn . Given > 0, Q ∈ As,p 0 Q ∈ As,p choose k0 such that 1/q k q(˜s s) 1/ρ − (2C + 1) (/2) . |P | ≤ k k X≥ 0 By (7.12) and (7.13) for each n large enough we have

q/p 1/q 1/q n p p(˜s s) k q(˜s s) 1/ρ ( c Q − − (2C + 1) < (/2) . | Q| | | ≤ |P | k k0 Q k k k0 X≥ X∈P X≥ In particular s˜ s n n ρ Q − cQbQ s ( ) < /2. | | Bp,q As,p k k0 Q k X≥ X∈P Choose δ > such that q/p ρ/q ( δp < /2.

k 0 such that for every Q , with k < k0, if h (Q) ∈ P n n ∈ B satisﬁes h (Q) η then h p,s(Q). Since limn sQaQ = sQaQ in (Q) we conclude| that|B for≤n large enough∈ A we have B 1 dn = (sn an s a ) (Q) Q δ Q Q − Q Q ∈ As,p for every Q k, with k < k . In particular ∈ P 0 n ρ δdQ s ( ) < /2. Bp,q As,p k

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 2 be a class of (s, p, u2)-atoms for a grid , satisfying A1-A4 and G -GA . Let be also a grid satisfying G -G . W 1 2 G 1 2 BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 19

Figure 1. In the transmutation of atoms we replace atoms whose supports are represented in orange (supports with distinct scales are drawn in distinct lines for a better understanding) by a linear combination of atoms (whose supports are represented in green), in such way that we do not change too much the location of the support and also most of the ”mass” of the representation concen- trates on the same original scale.

II. Let ki N for i 0 be a sequence such that there is α > 0 and A, B R satisfying∈ ≥ ∈ αi + A k αi + B ≤ i ≤ for every i. III. There is λ (0, 1) such that the following holds. For every Q and P satisfying∈ P Q there are atoms b (P ) and corresponding∈ G ∈ W ⊂ P,Q ∈ A2 sP,Q C such that ∈ hQ = sP,QbP,Q. k P k,P Q X ∈WX ⊂ is a s ( )-representation of a function h , with s = 0 for every Bp,q A2 Q P,Q Q i, P k with k < k and moreover ∈ G ∈ W i p k k (8.14) s C λ − i . | P,Q| ≤ 6 P k,P Q ∈WX ⊂ for every k k . ≥ i

Let k = P Q: P k and s = 0 . H { ⊂ ∈ W P,Q 6 } Q [∈G Then

A. For every coeﬃcients (cQ)Q such that ∈G q/p 1/q c p < | Q| ∞ i Q i X X∈G 20 D. SMANIA

we have that the sequence (8.15) N c h 7→ Q Q i N Q i X≤ X∈G converges in Lp to a function in s ( ) that has a s ( )-representation Bp,q A2 Bp,q A2 (8.16) mP dP k P k X X∈H where m 0 for every P and P ≥ q/p 1/q m p | P | k P k X PX∈W ∈H 1/p B 1/q p q/p 1/q (8.17) C C λ− p C (p, q, b)C c ≤ 1 6 3 7 | Q| i Q i X X∈G Here C7 = max ` N, ` < α + 1 and b = (bn)n is defined by { ∈ } ∈Z αn A λ if n > α 1, bn = − 0 if n A 1, ( ≤ α − B. Suppose that the assumptions of A. hold and that sP,Q are non negative real numbers and bP,Q > 0 on P for every P,Q. Then mP = 0 and dP = 0 k 6 6 on P imply that P supp hQ for some Q satisfying cQ = 0 and s > 0. If we additionally⊂ assume that c ∈0 Wfor every Q then6 m = 0 P,Q Q ≥ P 6 also implies dP > 0 on P . C. Let 1 be a class of (s, p, u1)-atoms for the grid satisfying A1-A4. Suppose thatA there is λ < 1 such that for every atom a G (Q) we can find s Q ∈ A1 P,Q and bP,Q in III. such that hQ = aQ. Then s ( ) s ( ) Bp,q A1 ⊂ Bp,q A2 and this inclusion is continuous. Indeed 1/p B 1/q s p s φ ( 2) C1C6 λ− C3(p, q, b)C7 φ ( 1) | |Bp,q A ≤ | |Bp,q A for every φ s ( ). ∈ Bp,q A1 Proof. For every P k, with k N, and N N define ∈ H ∈ ∈ ∪ {∞} m = c s = c s P,N | Q P,Q| | Q P,Q| i N i k k i ≤ Q i≤ Q X PX∈GQ iXN PX∈GQ ⊂ ≤ ⊂ Due to G2 this sum has a finite number of terms. If this sum has zero terms define mP,N = 0 and let dP,N be the zero function. Otherwise define 1 (8.18) dP,N = cQsP,QbP,Q. mP,N i N i ≤ Q X PX∈GQ ⊂ We have that d is an (P )-atom. P,N A2

Claim I. We claim that for N N ∈ mP,N dP,N = cQhQ. k P k i N Q i X X∈H X≤ X∈G BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 21

Note that if Q i then due to (6.5), with t = p and (8.14) we have ∈ G p/pˆ s b < . P,Q P,Q p ∞ k P k X PX∈HQ ⊂ Consequently we can do the following manipulation in Lp

cQhQ = sP,QbP,Q i N Q i i N Q i k P k X≤ X∈G X≤ X∈G X PX∈HQ ⊂ = sP,QbP,Q k P k i N Q i P Q X X∈H X≤ X∈G X⊂ = mP,N dP,N . k P k X X∈H This concludes the proof of Claim I.

Claim II. For every N N we claim that ∈ ∪ {∞}

(8.19) mP,N dP,N k P k X X∈H is a s ( )-representation and Bp,q A2 q/p 1/q m p | P,N | k P k X X∈H 1/p B 1/q p q/p 1/q (8.20) C C λ− p C (p, q, b)C c . ≤ 1 6 3 7 | Q| i Q i X X∈G Indeed 1/pˆ 1/pˆ p p mP,N = cQsP,Q | | i | | P k P k ki k ≤ Q X∈H X∈W iXN PX∈GQ ≤ ⊂ p 1/pˆ cQsP,Q ≤ i | | ki k P k ≤ Q iXN X∈W PX∈GQ ≤ ⊂ p/pˆ p p 1/pˆ C1 cQ sP,Q ≤ i | | | | ki k P k ≤ Q iXN X∈W PX∈GQ ≤ ⊂ p/pˆ p p 1/pˆ C1 cQ sP,Q ≤ i | | | | ki k Q P k iX≤N X∈G PX∈WQ ≤ ⊂ p/pˆ 1/pˆ (k ki)/pˆ p 1/pˆ C1 C6 λ − cQ ≤ i | | ki k Q iX≤N X∈G ≤ p/pˆ 1/pˆ (k αi B)/pˆ p 1/pˆ (8.21) C C λ − − c . ≤ 1 6 | Q| αi+A k Q i X≤ X∈G 22 D. SMANIA

If α = 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 u = m p and c = c p k | P,N | i | Q| P k Q i X∈H X∈G Every k N can be written in an unique way as k = αjk + `k + rk, with jk N, ∈ ∈ `k N, `k + rk < α and rk [0, 1). Fix ` [0, α) N and j N. Then there ∈ ∈ ∈ ∩ ∈ is at most one k0 N such that `k = ` and jk = j. Indeed, if k0 = αj + ` + r0 ∈ and k00 = αj + ` + r00, with r0, r00 [0, 1) and ` + r0 and ` + r00 smaller than α, ∈ then k0 k00 = r0 r00 ( 1, 1), so k0 = k00 and r0 = r00. If such k0 exists, denote − − ∈ − k(`, j) = k0 and r(`, j) = k(`, j) αj ` and a`,j = uk(`,j). Otherwise let a`,j = 0. Then (8.21) implies − − 1/pˆ p/pˆ 1/pˆ (αj+`+r(`,j) αi B)/pˆ 1/pˆ a C C λ − − c `,j ≤ 1 6 i αi+A αj+`+r(`,j) ≤ X p/pˆ 1/pˆ B/pˆ α(j i)/pˆ 1/pˆ C C λ− λ − c ≤ 1 6 i A+`+r(`,j) i j+ − ≤ Xα p/pˆ 1/pˆ B/pˆ α(j i)/pˆ 1/pˆ C1 C6 λ− λ − ci ≤ A i A/α 1, and bn = 0 otherwise. Fixing ` N, ` < α, Proposition 4.2 (the convolution− trick) gives us ∈ q/p 1/q q/p 1/q K` = uk = a`,j k N j `Xk∈=` X 1/p B p q/p 1/q C C λ− p C (p, q, b) c . ≤ 1 6 3 | Q| i Q i X X∈G and q/p 1/q 1/q m p = uq/p | P,N | k k P k k X X∈H X q/p 1/q q 1/q = uk = K` 0 `<α k N 0 `<α ≤X `Xk∈=` ≤X 1/p B 1/q p q/p 1/q (8.22) C C λ− p C (p, q, b)C c . ≤ 1 6 3 7 | Q| i Q i X X∈G s This implies in particular that the sum in (8.19) is a p,q( 2)-representation. This proves Claim II. B A

Claim III. We have that in the strong topology of Lp

lim mP,N dP,N = mP, dP, . N ∞ ∞ →∞ k P k k P k X X∈H X X∈H For each P the sequence ∈ H (8.23) N m 7→ P,N BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 23 is eventually constant, therefore convergent. The same happens with (8.24) N d . 7→ P,N Estimate (8.20) and Proposition 6.4 imply that that (8.15) converges in Lp to a s function with p,q( 2)-representation (8.19) with N = . This concludes the proof of Claim III. B A ∞ Then Claim I, II, and III imply A. taking mp = mP, and dP = dP, . We have ∞ ∞ that C. is an immediate consequence of A. Note that (8.18) and A. give B.

9. Good grids k A(λ1, λ2)-good grid , with 0 < λ1 < λ2 < 1, is a grid = ( )k N with the following properties: P P ∈ G . We have 0 = I . 3 P { } G4. We have I = Q k Q (up to a set of zero m-measure). ∪ ∈P G5. The elements of the family Q Q k are pairwise disjoint. k { } ∈P k 1 G . For every Q and k > 0 there exists P − such that Q P . 6 ∈ P ∈ P ⊂ G7. We have Q λ | | λ 1 ≤ P ≤ 2 | | for every Q P satisfying Q k+1 and P k for some k 0. ⊂ k ∈ P ∈ P ≥ G8. The family k generate the σ-algebra A. ∪ P 10. Induced spaces

s k0 Consider a Besov-ish space p,q(I, , ), where is a good grid. Given Q , B P A P i ∈ P we can consider the sequence of ﬁnite families of subsets Q = ( Q)i 0 of Q given by P P ≥ i = P k0+i,P Q . PQ { ∈ P ⊂ } Let Q be the restriction of the indexed family of pairs ( (P ), (P ))P to A A B A ∈P indices belonging to Q. Then we can consider the induced Besov-ish space s (Q, , ). Of courseP the inclusion Bp,q PQ AQ i: s (Q, , ) s (I, , ) Bp,q PQ AQ → Bp,q P A is well-deﬁned and it is a weak contraction, that is

f s (I, , ) f s (Q, , ). | |Bp,q P A ≤ | |Bp,q PQ AQ Under the degree of generality we are considering here, the restriction transformation r : s (I, , ) Lp Bp,q P A → given by r(f) = f 1Q, is a bounded linear transformation, however it is easy to ﬁnd examples of Besov-ish· spaces where f1 s (Q, , ). Q 6∈ Bp,q PQ AQ 11. Examples of classes of atoms. There are many classes of atoms one may consider. We list here just a few of them. 24 D. SMANIA

11.1. Souza’s atoms. Let Q .A(s, p)-Souza’s atom supported on Q is a ∈ P function a: I C such that a(x) = 0 for every x Q and a is constant on Q, with → 6∈ s 1/p a Q − . | |∞ ≤ | | sz The set of Souza’s atoms supported on Q will be denoted by s,p(Q). A canon- A s 1/p ical Souza’s atom on Q is the Souza’s atom such that a(x) = Q − for every x Q. Souza’s atoms are (s, p, )-type atoms. | | ∈ ∞ 11.2. H¨olderatoms. Suppose that I is a quasi-metric space with a quasi-distance d( , ), such that every Q is a bounded set and there is λ , λ (0, 1) such that · · ∈ P 3 4 ∈ diam P λ λ 3 ≤ diam Q ≤ 4 for every P Q with P k+1 and Q k. Additionally assume that there is C 0 and D⊂ 0 such that∈ P ∈ P 8 ≥ ≥ 1 D D D Q (diam Q) C8 Q . C8 | | ≤ ≤ | | Let 1 0 < s < , s < β. p For every Q , Let Cα(Q) be the Banach space of all functions φ such that φ(x) = 0 for x∈ PQ, and 6∈ φ(x) φ(y) α φ C (Q) = φ + sup | − α | < . | | | |∞ x,y Q d(x, y) ∞ x=∈y 6 Let h (Q) CβD(Q) be the convex subset of all functions φ satisfying As,β,p ⊂ φ(x) φ(y) s 1/p β s 1/p sup | − Dβ | Q − − and φ Q − . x,y Q d(x, y) ≤ | | | |∞ ≤ | | x=∈y 6 We say that h (Q) is the set of (s, β, p)-H¨olderatoms supported on Q. Of As,β,p course h -atoms are (s, p, )-type atoms and sz (Q) h (Q). As,β,p ∞ As,p ⊂ As,β,p 11.3. Bounded variation atoms. Now suppose that I is an interval of R with length 1, m is the Lebesgue measure on it and the partitions in the grid are partitions by intervals. Let Q be an interval and s β, p [1, ). A (s,P β, p)- ≤ ∈ ∞ bounded variation atom on Q is a function a: R C such that a(x) = 0 for every x Q, → 6∈ s 1/p a Q − . | |∞ ≤ | | and s 1/p var (a, Q) Q − . 1/β ≤ | | Here var ( ,Q) is the pseudo-norm 1/β · var (a, Q) = sup( a(x ) a(x ) 1/β)β, 1/β | i+1 − i | i X where the sup runs over all possible sequences x1 < x1 < < xn, with xi in the interior of Q. We will denote the set of bounded variation atoms··· on Q as bv (Q). As,p,β Bounded variation atoms are also (s, p, )-type atoms. ∞ BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 25

II. SPACES DEFINED BY SOUZA’S ATOMS.

In Part II. we suppose that s > 0, p [1, ) and q [1, ]. ∈ ∞ ∈ ∞

12. Besov spaces in a measure space with a good grid We will study the Besov-ish spaces s ( , sz ) associated with the measure Bp,q P As,p space with a good grid (I, , m). We denote s = s ( , sz ). Note that sz P Bp,q Bp,q P As,p As,p satisfies A -A . Note that by Proposition 6.1 there is β > 1 such that s Lβ. 1 7 Bp,q ⊂ If p [1, ), q [1, ] and 0 < s < 1 we wil say that s is a Besov space. ∈ ∞ ∈ ∞ p Bp,q 13. Positive cone We say that f is s -positive if there is a s -representation Bp,q Bp,q f = cP aP k P k X X∈P where cP 0 and aP is the standard (s, p)-Souza’s atom supported on P . The set of all s ≥-positive functions is a convex cone in s , denoted s+. We can define Bp,q Bp,q Bp,q a “norm” on s+ as Bp,q 1/q p q/p f s+ = inf cP , | |Bp,q k P k X X∈P where the infimum runs over all possible s -positive representations of f. Of Bp,q course for every f, g s+ and α 0 we have ∈ Bp,q ≥ s+ s+ s+ s+ s+ s s+ αf = α f , f + g f + g , f p,q f . | |Bp,q | |Bp,q | |Bp,q ≤ | |Bp,q | |Bp,q | |B ≤ | |Bp,q s s+ Moreover if f p,q is a real-valued function then one can find f+, f p,q such ∈ B − ∈ B that f = f+ f and − − s+ s s+ s f+ f p,q and f f p,q . | |Bp,q ≤ | |B | −|Bp,q ≤ | |B An obvious but important observation is Proposition 13.1. if f s+ then its support ∈ Bp,q supp f = x I : f(x) = 0 . { ∈ 6 } is (up to a set of zero measure) is a countable union of elements of . P 14. Unbalanced Haar wavelets Let = ( k) be a good grid. For every Q k let Ω = P Q,...,P Q , P P k ∈ P Q { 1 nQ } n 2, be the family of elements k+1 such that P Q Q for every i, and ordered Q ≥ P i ⊂ in some arbitrary way. The elements of ΩQ will be called children of Q. Note that every Q has at least two children. We will use one of the method described (Type I tree∈ P with the logarithmic subtrees construction) in Girardi and Sweldens [24] to construct an unconditional basis of Lβ, for every 1 < β < . ∞ 26 D. SMANIA

Let Q be the family of pairs (S1,S2), with Si ΩQ and S1 S2 = , defined as H ⊂ ∩ ∅ Q = j Q,j, H ∪ ∈NH where are constructed recursively in the following way. Let = (A, B) , HQ,j HQ,0 { } where A = P Q,...,P Q and B = P Q ,...,P Q . Here [x] denotes { 1 [nQ/2]} { [nQ/2]+1 nQ } the integer part of x 0. Suppose that we have defined Q,j. For each element (S ,S ) , fix ≥an ordering S = R1,...,R1 andH S = R2,...,R2 . 1 2 ∈ HQ,j 1 { 1 n1 } 2 { 1 n2 } For each i = 1, 2 such that n 2, define T 1 = Ri ,...,Ri and T 2 = i i 1 [ni/2] i i i 1 ≥ 2 { } R ,...,R and add (T ,T ) to Q,j+1. This defines Q,j+1. { [ni/2]+1 ni } i i H H Note that since is a good grid we have = for large j and indeed P HQ,j ∅ sup # Q < . Q H ∞ ∈P Define = Q Q. For every S = (S1,S2) Q define H ∪ ∈P H ∈ H 1 1 1 P S1 P R S2 R φS = ∈ ∈ m(S1,S2) P S P − R S R P ∈ 1 | | P ∈ 2 | | where P P 1 1 1/2 m = + . (S1,S2) P R P S1 | | R S2 | | ∈ ∈ Note that P P φS dm = 0. ZQ Since 1 #S 1/λ we have ≤ i ≤ 1 λ2 λ1 Q P Q , | | ≤ | | ≤ λ1 | | P S X∈ i so 1/2 1/2 2λ1 1 2 1 1/2 m(S1,S2) 1/2 λ2 Q ≤ ≤ λ1 Q | | | | Consequently

C9 1 1 1 1/2 min , Q ≤ m(S1,S2) { P S P R S R } | | ∈ 1 | | ∈ 2 | | 1 1 1 φS(x) P max P , ≤ | | ≤ m(S1,S2) { P S P R S R } ∈ 1 | | ∈ 2 | | C (14.25) 10 . P P ≤ Q 1/2 | | for every x P S S P. Here ∈ ∪ ∈ 1∪ 2 λ3/2 λ1/2 C = 1 and C = 2 + 1. 9 √ 10 3/2 2λ2 √2λ1 Let ˆ = I H { } ∪ H and deﬁne 1 φ = I . I I 1/2 | | BESOV-ISHBESOV-ISH SPACES SPACES THROUGH ATOMIC ATOMIC DECOMPOSITION DECOMPOSITION 27 27

Q Q Q P P 2∪ 3 P2 P3 ! P3 P4 P5 ∪ S1 ! ∪ ! ∪ S2 S2 S1 ! P1 P2 P3 P3 P4 P5

P 1 S1 ! P2 S1 P2 ! ! S2 P1

S2′ ! P4 P5 S2 ∪ ! P3 P2 ! S2 S1 ! P2 S1′ ! Figure 2. P3

Construction of the unbalanced Haar basis, following Girardi and Sweldens, in the case when is a grid of intervals. Every Q gives originP to a family of bounded functions∈ P indexed by the subtree . We give examples HQ of the (logarithmic) subtree construction when P4 ! Q has two, tree and ﬁve children. Then S1 HQ have one, two and four elements, respectively. S2 ! P5

Figure 2. Then by Girardi and Sweldens [24] we have that Then by Girardi and Sweldens [24]wehavethat φS S ˆ { } ∈H is an unconditional basis of Lβ for everyφS βS >ˆ1. { } ∈H is an unconditional basis of Lβ for every β > 1. 15. Alternative characterizations I: Messing with norms.

We15. areAlternative going to describe characterizations three norms that are I: Messing equivalent with to norms.s . Their | · |Bp,q advantage is that they are far more concrete, in the sense that we do not need to We are going to describe three norms that are equivalent to s . Their consider arbitrary atomic decompositions to define them. |·|Bp,q advantage is that they are far more concrete, in the sense that wedonotneedto consider15.1. Haar arbitrary representation. atomic decompositionsFor every f toL defineβ, β > them.1, the series ∈ f β 15.1.(15.26)Haar representation. Forf every= f dSφLS , β > 1, the series ∈ S ˆ X∈H f (15.26) f = dS φS is converges unconditionally in Lβ, where df = fφ dm. We will call the r.h.s. of S ˆS S (15.26) the Haar representation of f!.∈ DefineH β f R is converges unconditionally in L ,whered = fφS dm. We will call the r.h.s. of S p 1/q 1/p s 1/2 f 1 sp f p q/p 2 (15.26)theNhaarHaar(f) = representationI − − dI +of f.DefineQ − − dS , | | | | k "| | | | k Q S Q ∈P p ∈H 1/q 1/p s 1/2 f X X 1 sp X f p q/p N (f)= I − − d + Q − − 2 d , haar | | | I | | | | S| k Q k S # ! $ !∈P !∈HQ % & 28 D. SMANIA

15.2. Standard atomic representation. Note that f f kI aI = dI φI , 1/p s 1/2 f where kI = I − − dI and aI is the canonical Souza’s atom on I. Let S . | | k ∈ H Then S Q, with S = (S1,S2) and some Q , with k 0. It is easy to see that for∈ every H P S S the function ∈ P ≥ ∈ 1 ∪ 2 1/2 Q s 1/p aS,P = | | P − φS1P C10 | | is a Souza’s atom on P . Choose f 1/2 1/p s f c = C Q − P − d . S,P 10| | | | S Note that f 1/p s 1/p s 1/p s 1/2 f (15.27) c C max λ − , λ − Q − − d . | S,P | ≤ 10 { 2 1 }| | | S| For every child P of Q k, k 0, deﬁne ∈ P ≥ f 1 f a˜ = c aS,P , P k˜f S,P P S=(S ,S ) P S1 S2 1X2 ∈HQ ∈X∪ where (15.28) k˜f = cf . P | S,P | S=(S ,S ) P S1 S2 1X2 ∈HQ ∈X∪ The (ﬁnite) number of terms on this sum depends only on the geometry of . Then f P a˜P is a Souza’s atom on P and f ˜f f dSφS = kP a˜P . S Q P k+1 ∈H ∈PP Q X X⊂ Let a be the canonical (s, p)-Souza’s atom on P and choose x P . Denote P P ∈ a˜f (x ) 1 1 kf = P P k˜f = df φ (x ) = f φ (x )φ dm. P P s 1/p P P s 1/p S S P P s 1/p S P S − − S − S | | | | X∈HQ | | Z X∈HQ In particular, for every P f kf 7→ P extends to a bounded linear functional in L1. We have kf k˜f and | P | ≤ P f f f = kI aI + kP aP k Q k P k+1 ∈PP Q X X∈P X⊂ f (15.29) = kQaQ. i Q i X X∈P β where this series converges unconditionally in L . Here aP is the canonical Souza’s atom. We will call the r.h.s. of (15.29) the standard atomic representation of f. Let q/p 1/q N (f) = kf + kf p . st | I | | Q| k 1 Q k X≥ X∈P BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 29

15.3. Mean oscillation. Deﬁne for p [1, ) ∈ ∞

p 1/p oscp(f, Q) = inf ( f(x) c dm(x)) , c | − | ∈C ZQ and

osc (f, Q) = inf f c L (Q). ∞ c | − | ∞ ∈C Denote for every p [1, ) and q [1, ] ∈ ∞ ∈ ∞ 1/q s sp p q/p (15.30) osc (f) = Q − osc (f, Q) , p,q | | p k Q k X X∈P with the obvious adaptation for q = . Let ∞ s s N (f) = I − f + osc (f). osc | | | |p p,q

15.4. These norms are equivalent. We have

Theorem 15.1. Suppose s > 0, p [1, ) and q [1, ]. Each one of the norms ∈ ∞ ∈ ∞ s s , , , is ﬁnite if and only if . Furthermore f p,q Nst(f) Nhaar(f) Nosc(f) f p,q these| |B norms are equivalent on s . Indeed ∈ B Bp,q

(15.31) f s Nst(f), | |Bp,q ≤

(15.32) N (f) C N (f), st ≤ 11 haar

(15.33) N (f) C N (f), haar ≤ 10 osc

(15.34) Nosc(f) C12 f s . ≤ | |Bp,q where 1/p s 1/p s 2 1/p C = 1 + C max λ − , λ − λ− − , 11 10 { 2 1 } 1

1+1/p k sp s 1 C = C C (t, q, ( ) ) I − + . 12 1 2 |P | k | | 1 λs − 2

Proof. The inequality (15.31) is obvious. To simplify the notation we write dS, kP f f instead of dS, kP .

Proof of (15.32). The number of terms in the r.h.s. of (15.28) depends only on the geometry of . Indeed P 1 (15.35) sup #(S1 S2) 2 . Q ∪ ≤ λ1 ∈P S=(S1,S2) Q X ∈H 30 D. SMANIA

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

p p kP cS,P k k+1 | | ≤ k P S S | | Q P Q S Q ∈ 1∪ 2 ∈PP Q ∈H S=(S ,S ) X∈P X⊂ X∈P X X1 2 p p p 1 sp 1 sp 1 sp C10 max λ2− , λ1− Q − − 2 dS ≤ { } k | | P S S | | Q S Q ∈ 1∪ 2 X∈P X∈H S=(XS1,S2) p p 1 sp 1 sp 2p 1 sp p C10 max λ2− , λ1− λ1− Q − − 2 dS ≤ { } k | | P S S | | Q S Q ∈ 1∪ 2 X∈P X∈H S=(XS1,S2) p p 1 sp 1 sp 2p 1 1 sp p C max λ − , λ − λ− − Q − − 2 d . ≤ 10 { 2 1 } 1 | | | S| Q k S X∈P X∈HQ

for every k. Consequently

q/p 1/q k + k p | I | | P | ≤ k 1 P k X≥ X∈P 1 1 1 1 s s 2 p 1/q s 1/2 p − p − − − p 1 sp p q/p I p − − d + C max λ , λ λ Q − − 2 d . ≤ | | | I | 10 { 2 1 } 1 | | | S| k Q k S X X∈P X∈HQ

This completes the proof of (15.32).

Proof of (15.33). Note that

1/2 1/p d f φ dm f I − . | I | ≤ | || I | ≤ | |p| | ZI

Given > 0 and Q , choose c Q such that ∈ P Q ∈

1/p p f cQ dm (1 + )oscp(f, Q). Q | − | ≤ Z BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 31

Since φ has zero mean on Q for every S we have S ∈ HQ

1 sp p p 1/p Q − − 2 d | | | S| Q k S Q X∈P X∈H p 1 sp p 1/p Q − − 2 fφ dm ≤ | | | S | Q k S Q Z X∈P X∈H p 1 sp p 1/p Q − − 2 fφS cQφS dm ≤ | | | Q − | Q k S Q Z X∈P X∈H p 1 sp p 1/p Q − − 2 ( f cQ φS dm) ≤ | | Q | − || | Q k S Q Z X∈P X∈H p 1 sp 1/p0 1/2 p 1/p C Q − − 2 ((1 + )osc (f, Q) Q − ) ≤ 10 | | p | | Q k S Q Q∈PJ ∈H X⊂ X 1/p 1 sp+p/p p p C (1 + ) Q − 0− osc (f, Q) ≤ 10 | | p Q k S Q X∈P X∈H sp p 1/p C (1 + ) Q − osc (f, Q) . ≤ 10 | | p Q k S Q X∈P X∈H

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

s s Proof of (15.34). Finally note that if f p,q and > 0 then there is a p,q- representation of f ∈ B B

f = kP aP . P X∈P

such that

∞ p q/p 1/q ( kQ ) (1 + ) f s . | | ≤ | |Bp,q k=0 Q k X X∈P 32 D. SMANIA

For each J k0 , choose x J. Then ∈ P J ∈ 1/p sp p J − osc (f, J) | | p J k0 ∈PX 1/p sp p J − f(x) kQaQ(xJ ) dm ≤ | | J | − | J k0 Q Z J ∈PQ ∈PX X⊂ 1/p sp p J − k a dm ≤ | | | R R| J k0 k>k0 R k Z R∈PJ ∈PX X X⊂ 1/p s p J − k a dm ≤ | | | R R| k>k0 J k0 R k Z R∈PJ X ∈PX X⊂ 1/p s p J − k a dm ≤ | | | R R| k>k0 J k0 R k Z R∈PJ X ∈PX X⊂ 1/p sp p J − k a dm ≤ | | | R R| k>k0 J k0 R k Z R∈PJ X ∈PX X⊂ sup R : R k,R J sp 1/p {| | ∈ P ⊂ } k p ≤ J | R| k>k0 J k0 R k | | R∈PJ X ∈PX X⊂ 1/p sp(k k0) p λ − k ≤ 2 | R| k>k0 J k0 R k R∈PJ X ∈PX X⊂ s(k k0) p 1/p λ − k . ≤ 2 | R| k>k0 R k X X∈P This is a convolution, so q/p 1/q sp p 1 + s J − oscp(f, J) s f p,q . | | ≤ 1 λ2 | |B k0 J k0 X ∈PX − and since > 0 is arbitrary, by Proposition 6.1 we obtain q/p 1/q s sp p I − f + J − osc (f, J) | | | |p | | p k0 J k0 X ∈PX 1+1/p k sp s 1 s C1 C2(t, q, ( )k) I − + s f p,q . ≤ |P | | | 1 λ2 | |B − This proves (15.34). The following is an important consequence of this section. Corollary 15.2. For each P there exists a linear functional in L1 ∈ P f kf 7→ P s s with the following property. The so-called standard p,q-representation of f p,q given by B ∈ B f f = kP aP , k P k X X∈P BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 33 satisfies 1/p f p q/p k C10C11C12 f s . | P | ≤ | |Bp,q k P k X X∈P 16. Alternative characterizations II: Messing with atoms. s Here we move to alternative descriptions of p,q, which are quite different from those in Section 15. Instead of choosing a definitiveB representation of elements of s s p,q, we indeed give atomic decompositions of p,q using far more general classes ofB atoms. B

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 Section3. Moreover in appropriate settings it contains H¨olderand bounded variations atoms, s sz which will be quite useful in the get other characterizations of p,q( , s,p). The s n B P A atomic decompositions of Bp,q(R ) by Besov’s atoms were considered by Triebel [45] in the case s > 0, p = q [1, ] and by Schneider and Vyb´ıral[40] in the case s > 0, p, q [1, ]. They are∈ refered∞ there as ”non-smooth atomic decompositions”. Let s <∈ β and∞ q ˜ [1, ]. A (s, β, p, q˜)-Besov atom on the interval Q is a ∈ ∞ function a β (Q, , sz ) such that a(x) = 0 for x Q, and ∈ Bp,q˜ PQ Aβ,p 6∈ 1 s β a β (Q, , sz ) Q − . | |Bp,q˜ PQ Aβ,p ≤ C13 | | where 1/q˜ C = C1+1/p λkβq˜ 1. 13 1 2 ≥ Xk The family of (s, β, p, q˜)-Besov atoms supported on Q will be denoted by bv (Q). As,β,p,q˜ Naturally sz (Q) bs (Q). By Proposition 6.1 we have As,p ⊂ As,β,p,q˜ 1+1/p 1/q˜ C1 k βq˜ (16.36) a p Q a β (Q, , sz ) | | ≤ C13 |P | | |Bp,q˜ PQ Aβ,p Xk s β β s s 1 (16.37) Q − Q = Q = Q − . ≤ | | | | | | | | ∞ so a (s, β, p, q˜)-Besov atom is an atom of type (s, p, 1).

s The following result says there are many ways to deﬁne p,q using various classes of atoms. B Proposition 16.1 (Souza’s atoms and Besov’s atoms). Let be a good grid. Let be a class of (s, p, u)-atoms , with u 1, such that for someP s < β, q˜ [1, ], Aand C , C 0 we have that for every≥Q ∈ ∞ 14 15 ≥ ∈ P 1 sz bs s,p(Q) (Q) C15 s,β,p,q˜(Q). C14 A ⊂ A ⊂ A Then s ( , sz ) = s ( , ) = s ( , bs ). Bp,q P As,p Bp,q P A Bp,q P As,β,p,q˜ 34 D. SMANIA

Moreover

C15 f s C f s sz and f s sz f s . p,q ( ) 14 p,q ( s,p) p,q ( s,p) β s p,q ( ) | |B A ≤ | |B A | |B A ≤ 1 λ − | |B A − 2 Proof. The ﬁrst inequality is obvious. To prove the second inequality, recall that due to Proposition 8.1 it is enough to show the following claim

Claim. Let b be a (s, β, p, q˜)-Besov atom on J j. Then for every P J with J ∈ P ⊂ P there is mP C such that ∈ P ∈

bJ = mP aP , P J X⊂ where aP is the canonical (s, p)-Souza’s atom on P and

1/p p 2 (k j)(β s) mP λ2 − − . | | ≤ C13 P k,P J ∈PX ⊂

Indeed, since

1 s β bJ β (J, , s ) J − , | |Bp,q˜ P Aβ,p ≤ C13 | | there exists a β (J, , s )-representation Bp,q˜ P Aβ,p

bJ = cP dP , P ,P J ∈PX⊂ where dP is the canonical (β, p)-Souza’s atom on P and

bJ = mP aP , P ,P J ∈PX⊂ β s with m = c P − and P P | | p(β s) p 1/p P − p(β s) p 1/p m = | | J − c | P | J | | | P | P k,P J P k,P J | | ∈PX ⊂ ∈PX ⊂ (k j)(β s) β s p 1/p λ − − J − c ≤ 2 | | | P | P k,P J ∈PX ⊂ 2 (k j)(β s) λ2 − − . ≤ C13 BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 35

16.2. Using Hölderatoms. Suppose that I is a quasi-metric space with a quasi- distance d( , ) and a good grid satisfying the assumptions in Section 11.2. · · Proposition 16.2. Suppose 0 < s < β < β,˜ p [1, ) and q˜ [1, ]. For every Q we have ∈ ∞ ∈ ∞ ∈ P (16.38) C h (Q) bs (Q), 16As,β,p˜ ⊂ As,β,p,q˜ for some C16 > 0. Moreover h bs (16.39) C17 s,β,p(Q) s,β,p, (Q), A ⊂ A ∞ for some C17 > 0. In particular (16.40) s = s ( h ) = s ( h (Q)) Bp,q Bp,q As,β,p˜ Bp,q As,β,p and the corresponding norms are equivalent. Proof. Let φ h (Q). Then φ has a continuous extension to Q. So firstly we ∈ As,β,p˜ assume that φ 0 has a continuous extension to Q. Define ≥ 1/p β c = min φ(Q) Q − . Q | | and for every P W Q j with P k+1 and W k define ⊂ ⊂ ∈ P ∈ P ∈ P 1/p β (16.41) c = (inf φ(P ) inf φ(W )) P − P − | | Of course in this case c 0 and P ≥ 1/p β c (inf φ(P ) inf φ(W )) P − | P | ≤ − | | s 1/p β˜ βD˜ 1/p β Q − − (diam W ) P − ≤ | | | | ˜ Dβ 1/p C8 P s β˜ β˜ β ˜ | | Q − P − ≤ λβ Q | | | | 3 | | ˜ Dβ 1/p β˜ β C8 P s β P − ˜ C18 | | Q − | | . ≤ λβ Q | | Q 3 | | | | β˜ β Here C18 = supQ Q − . Consequently ∈P | | p Dβp˜ p p(s β) (k+1 j)p(β˜ β) C18C8 P cP Q − λ2 − − | | | | ≤ | | λpβ Q P k+1 P k+1 3 | | P∈PXQ P∈PXQ ⊂ ⊂ p Dβp˜ p(s β) C18C8 (k+1 j)p(β˜ β) − − (16.42) Q − βp λ2 . ≤ | | λ3 so ˜ 1/q˜ Dβ 1/q˜ p q/p˜ C18C8 1 s β cP Q − . | | ≤ λβ q˜(β˜ β) | | k P k+1 3 1 λ2 − X P∈PXQ − ⊂ This implies that

φ˜ = cP aP , P PX∈PQ ⊂ 36 D. SMANIA where a is the canonical (β, p)-atom on P , is a β (Q, sz )-representation of P Bp,q˜ PQAβ,p a function φ˜. From (16.41) it follows that

cP aP (x) = min φ(W ) k N P k X≤ X∈P for every x W N . In particular ∈ ∈ P lim cP aP (x) = φ(x). N →∞ k N P k X≤ X∈P for almost every x, so φ˜ = φ. So

1 s β (16.43) C16φ β (Q, , sz ) Q − | |Bp,q˜ PQ Aβ,p ≤ 2| | where ˜ Dβ 1/q˜ 1 C18C8 1 − C16 = 2 . β q˜(β˜ β) λ3 1 λ − − 2 In the general case, note that φ+(x) = max φ, 0 , φ (x) = min φ, 0 h { } − {− } ∈ ˜ (Q) and φ = φ+ φ . Applying (16.43) to φ+ and φ we obtain (16.38). As,β,p − − − The second inclusion (16.39) in the proposition can be obtained taking β˜ = β in (16.42) 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 R of length 1, m is the Lebesgue measure on it and the partitions in are partitions by intervals. P Proposition 16.3. If 1 0 < s β ≤ ≤ p then bv bs C20 s,β,p(Q) s,β,p, (Q). A ⊂ A ∞ for every Q . If ∈ P 1 0 < s β < β˜ ≤ ≤ p then C bv (Q) bs (Q) 21As,β,p˜ ⊂ As,β,p,q for every q [1, ]. In particular ∈ ∞ s = s ( bv ) = s ( bv ). Bp,q Bp,q As,β,p Bp,q As,β,p˜ and the corresponding norms are equivalents. Proof. Suppose 0 < s β β˜ 1 ≤ ≤ ≤ Let Q j and a bv (Q). We have ∈ P Q ∈ As,β,p˜ sp 1 1/p s s β a ( Q Q − ) = Q C Q − , | Q|p ≤ | || | | | ≤ 19| | BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 37

β where C19 = supQ Q . Note that for every k j ∈P | | ≥ βp p W − osc (a ,W ) | | p Q W k X∈P 1 βp p W − osc (aQ,W ) ≤ | | ∞ W Q,W k ⊂ X∈P 1 βp − 1 βp˜ ˜ βp˜ 1 βp˜ 1/β W − − osc (aQ,W ) ≤ | | ∞ W Q,W k W Q,W k ⊂ X∈P ⊂ X∈P (β˜ β)p 1 βp˜ − 1 βp˜ 1 βp˜ − − p max W − W (var1/β˜(aQ,Q)) ≤ W Q,W k | | | | ⊂ ∈P W Q,W k ⊂ X∈P (k j)(β˜ β)p (β˜ β)p 1 βp˜ sp 1 λ − − Q − Q − Q − ≤ 2 | | | | | | (k j)(β˜ β)p (s β)p λ − − Q − . ≤ 2 | | Note that in the case βp˜ = 1 the argument above needs a simple modification. For k < j let W k be such that Q W . Then Q ∈ P ⊂ Q βp p βp sp 1 W − osc (a ,W ) W − Q Q − | | p Q ≤ | Q| | || | W k X∈P βp βp sp Q (s β)p (j k)βp (s β)p WQ − Q = | | Q − λ2 − Q − . ≤ | | | | WQ | | ≤ | | | | ˜ bv bs By Theorem 15.1 we have that if β = β then C20 s,β,p(Q) s,β,p, (Q) for some A ⊂ A ∞ C > 0, and if β < β˜ we have that for every q [1, ) 20 ∈ ∞ 1/q s sp p q/p I − a + W − osc (a ,W ) | | | Q|p | | p Q k W k X X∈P s 1 1 1/q s β C19 I − + + Q − . q(β˜ β) qβ ≤ | | 1 λ − 1 λ2 | | − − so C bv (Q) bs (Q) for some C > 0. 21As,β,p˜ ⊂ As,β,p,q 21 17. Dirac’s approximations We will use the Haar basis and notation defined by Section 14. For every x I 0 ∈ and k0 N define the finite family ∈ k0 k x = S : S = (S1,S2) (Q), with Q , k < k0, x0 a=1,2 P Sa P S 0 { ∈ H ∈ P ∈ ∪ ∪ ∈ } Let N(x , k ) = # k0 . Then we can enumerate the elements 0 0 Sx0 S1,S2,...,SN(x0,k0) of k0 such that Si = (Si ,Si ) satisfies Sx0 1 2 x0 P Si P ∈ ∪ ∈ ai for some a 1, 2 and i ∈ { } P Si+1 Si+1 P Q Si Si Q ∪ ∈ 1 ∪ 2 ⊂ ∪ ∈ 1∪ 2 38 D. SMANIA for every i. Let 1 ψ = I 0 I | | and define for i > 0

j P Sj 1P ∈ aj ψi = . j P i=0 PP Sa X ∈ j | | and in particular P

R Si R R Si R i R + i P 1/2 ∈ 3 ai | | ∈ 3 ai | | R S2 P S1 − mSi = − ∈ | | ∈ | | P i 1 Q P i 1 Q ( i R )( i P ) Q Sa−i 1 Q Sa−i 1 P R S2 PP S1 ∈ − | | ∈ − | | ∈ | | ∈ | | P P i R 1/2P P R S3 a 1 ∈ − i | | (17.45) = 1/2 i i 1 P P S P ( Q S − Q ) ai ai 1 ∈ | | ∈ − | | Multiplying (17.44) by f and integratingP it termP by term, and using (17.45) we obtain

f = dSφS S ˆ( ) ∈XH P β with dS = fφS dm, where this series converges unconditionally on L . Let R (17.46) fk0 = dI φI + dSφS.

Then

N(x0,k0)

i i fk0 (x0) = dI φI + dS φS (x0). i=1 X N(x0,k0) dI ai+1 1 1 i = 1/2 + ( 1) dS I − mSi i P i=1 P Sa | | X ∈ i | | N(x0,k0) ( P i R )( i P ) 1/2 dI ai+1 R S2 P S1 1 i ∈ | | ∈ | | = 1/2 + ( 1) dS I − i R + i P i P i=1 PR S2 P P S1 P Sa | | X ∈ | | ∈ | | ∈ i | | N(x0,k0) PR Si R P1/2 P dI ai+1 3 ai 1 i ∈ − | | = 1/2 + ( 1) dS 1/2 i i 1 I − P P S P ( Q S − Q ) i=1 ai ai 1 | | X ∈ | | ∈ − | | 1Qk P P = f 0 dm. · Q Z | k0 | Let

(17.47) f = kP aP , k P k X X∈P be the series given by (15.29). Note that

(17.48) fk0 = kP aP ,

k k0 P k X≤ X∈P Consequently Proposition 17.1 (Dirac’s Approximations). Let f Lβ, with β > 1. Let ∈ f = kP aP . k P k X X∈P A. If this representation is either as in (15.29) or kP 0 for every P then we have For every Q ≥ ∈ P s 1/p kJ J − f1Q . | | ≤ | |∞ J ,Q J ∈PX⊂ B. In the case of the representation (15.29 ) we also have

s 1/p 1Q k J − = f dm. J | | · Q J ,Q J ∈PX⊂ Z | | Proof. We have that A. is obvious if k 0 for every P . In the other case note P ≥ that for every x Q k0 we have 0 ∈ ∈ P s 1/p 1Q f (x ) = k J − = f dm. k0 0 J | | · Q J ,Q J ∈PX⊂ Z | | so A. and B. follows. 40 D. SMANIA

III. APPLICATIONS.

In Part III. we suppose that 0 < s < 1/p, p [1, ) and q [1, ]. ∈ ∞ ∈ ∞

18. Pointwise multipliers acting on s Bp,q Here we will apply the previous sections to study pointwise multipliers of s . Bp,q To be more precise, Let g : I C be a measurable function. We say that g is a pointwise multiplier acting on→s if the transformation Bp,q G(f) = gf deﬁnes a bounded operator in s . We denote the set of pointwise multipliers by Bp,q M( s ). We can consider the norm on M( s ) given by Bp,q Bp,q g M( s ) = sup gf s s.t. f s 1 . | | Bp,q {| |Bp,q | |Bp,q ≤ } Of course a necessary condition for a function to be a multiplier is that

s s s p,q,selfs = g p,q : sup gaQ p,q < B { ∈ B a sz | |B ∞} Q∈As,p Denote g s = sup ga s . p,q,selfs Q p,q | |B a sz | |B Q∈As,p s The linear space p,q,selfs endowed with s is a normed space introduced B | · |Bp,q,selfs by Triebel [45]. We have g s g M( s ) | |Bp,q,selfs ≤ | | Bp,q In the following three propositions we see that many results of Triebel [45] and Schneider and Vyb´ıral [40] for Besov spaces in Rn can be easily moved to our setting. The simplest case occurs when p = q = 1. Proposition 18.1. We have that M( s ) = s . B1,1 B1,1,selfs Proof. Let g s . Given f s and > 0 one can ﬁnd a s -representation ∈ B1,1,selfs ∈ B1,1 B1,1

f = cQaQ, k Q k X X∈P where aQ is a (s, 1)-Souza’s atom and

Since is arbitrary we get

gf s g s f s . | |B1,1 ≤ | |B1,1,selfs | |B1,1 Lemma 18.2. Let W . The restriction application ∈ P r : s (I, , sz ) s (W, , sz ) Bp,q P As,p → Bp,q PW As,p given by r(f) = 1W f is continuous. Indeed there is C22 1, that does not depend on W , such that ≥ A. For every f s we have ∈ Bp,q (18.49) 1W f s (W, , sz ) C22 f s . | |Bp,q PW As,p ≤ | |Bp,q In particular 1 M( s ). W ∈ Bp,q B. For every s+-representation Bp,q

f = cQaQ k Q k X X∈P one can ﬁnd a s+(W, , sz )-representation Bp,q PW As,p

1W f = dQaQ k Q k X X∈P such that

q/p 1/q q/p 1/q (18.50) (d )p C (c )p . Q ≤ 22 Q k Q k k Q k X X∈P X X∈P Moreover d = 0 implies Q supp f. Q 6 ⊂ Proof. Let Q . Denote by a the canonical (s, p)-Souza’s atom supported on Q. ∈ P Q If W Q = we can write 1W aQ = 0aQ. If Q W then 1W aQ = 1aQ. If W Q then ∩ ∅ ⊂ ⊂ W 1/p s 1 a = | | − a , W Q Q W | | where 1/p s W − (k0(W ) k0(Q))(1/p s) | | λ − − Q ≤ 2 | | In every case we can write

hQ = 1W aQ = sP,QaP , k P k X PX∈PQ ⊂ with p (k k0(Q))(1 sp) s λ − − | P,Q| ≤ 2 P k PX∈PQ ⊂ By Proposition 8.1.A and 8.1.B there is C22 such that A. and B. hold.

s Proposition 18.3. We have that L∞ and this inclusion is continuous. Bp,q,selfs ⊂ 42 D. SMANIA

Figure 2. Non-Archimedean behaviour. Illustration for Propo- sition 18.4. The ﬁlled regions are the supports of the functions gi. The squares are the elements of the grid on which there are atoms contributing to the representation of f. Every square intercepts at most one support, so each atom “sees” only the function whose support is nearby.

Proof. Let g s . Then ga s and by Lemma 18.2 we have ∈ Bp,q,selfs Q ∈ Bp,q g1Q s (Q, , sz ) C22 g1Q s C22 g s . | |Bp,q PQ As,p ≤ | |Bp,q ≤ | |Bp,q,selfs By Proposition 6.1 (taking t = p) we have (s, p)-Souza’s atom aQ we have

1+1/p k sp s gaQ p C1 C2(p, q, ( Q )k)C22 g s < C23 Q g s . | | ≤ |P | | |Bp,q,selfs | | | |Bp,q,selfs for some constant C23. In other words 1/p sp 1 p s Q − g dm C Q g s , 23 p,q,selfs | | Q | | ≤ | | | |B Z so 1 p p p g dm C23 g s , Q | | ≤ | |Bp,q,selfs | | ZQ k for every Q . Due to the fact that k generates the σ-algebra A, by L´evy’s Upward Theorem∈ P (see Williams [49]) for∪ almostP every x I the following holds. If x Q k then ∈ ∈ k ∈ P 1 lim g p dm = g(x) p. k Q | | | | | k| ZQk So

g C23 g s . | |∞ ≤ | |Bp,q,selfs

18.1. Non-Archimedean behaviour in β . If we have a sequence g Bp,q,selfs˜ i ∈ M( s ) we can get the naive estimate Bp,q

( gi)f s gi M( s ) f s . | |Bp,q ≤ | | Bp,q | |Bp,q i i X X Nevertheless , remarkably, sometimes one can get a far better estimate. To state the result we need to deﬁne BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 43

s,t s g = sup gaQ p,q . | |Bp,q,selfs a sz | |B Q∈As,p Q j j∈Pt ≥ It is easy to see that this norm is equivalent to s . | · |Bp,q,selfs Proposition 18.4. Let β > s. There is C24 with the following property. Let β,ti gi , with i Λ N, and ti N. ∈ Bp,q,selfs˜ ∈ ⊂ ∈ Consider a function f with a s -representation Bp,q f = cQaQ k Q k X X∈P satisfying A. We have

sup gi β,ti N. Q | |Bp,q,selfs˜ ≤ ∈P Q supp g = c =0 ∩ i6 ∅ Q6 X B. If Q k satisfies c = 0 and Q supp g = then k t . ∈ P Q 6 ∩ i 6 ∅ ≥ i Then we can find a s -representation Bp,q (18.51) ( gi)f = dQaQ i k P k X X X∈P such that q/p 1/q d p | Q| k P k X X∈P q/p 1/q (18.52) C N c p ≤ 24 | Q| k P k X X∈P Proof. It is enough to prove the result for the case when Λ is finite. Let Q k0 with c = 0. There is i , . . . , i Λ, such that ∈ P Q 6 { 1 j} ⊂

(18.53) ( gi)aQ = gi` aQ. i ` j X X≤ and Q supp gi` = for every `. In particular k0 max` ti` . By Lemma 18.2 for each ` ∩ j we can6 ﬁnd∅ a β -representation ≥ ≤ Bp,q ` (18.54) gi` aQ = s˜P,QbP k P k X PX∈PQ ⊂ such that bP is the canonical (β, p)-Souza’s atom supported on P and 1/q˜ ` p q/p˜ β,t s˜P,Q 2C22 gi` i` . | | ≤ | | p,q,selfs˜ k P k B X PX∈PQ ⊂ β s Since bP = P − aP , where aP is the canonical (s, p)-Souza’s atoms supported on P , we can write| | ` gi` aQ = sP,QaP , k P k X PX∈PQ ⊂ 44 D. SMANIA

` ` β s with s =s ˜ P − satisfying P,Q P,Q| | 1/p s` p | P,Q| P k PX∈PQ ⊂ (β s)(k k0) β s β,t − − 2C22 gi` i` λ2 sup Q − , ≤ | | p,q,selfs˜ Q | | B ∈P so we can write ( gi)aQ = sP,QaP , i k P k X X PX∈PQ ⊂ with ` sP,Q = sP,Q X` satisfying p 1/p (β s)(k k0) β s sP,Q 2NC22λ2 − − sup Q − . | | ≤ Q | | P k ∈P PX∈PQ ⊂ By Proposition 8.1.A we can ﬁnd a s -representation (18.51) satisfying (18.52). Bp,q Remark 18.5. If g is β -positive we can deﬁne Bp,q g s+,t = sup ga s+ . Q p,q | |Bp,q,selfs a sz | |B Q∈As,p Q j j∈Pt ≥ If we assume additionally that g are β,ti -positive, Proposition 18.4 remains true if i Bp,q we replace all the instances of s,t by s+,t in its statement. Moreover | · |Bp,q,selfs | · |Bp,q,selfs by Proposition 8.1.B and Lemma 18.2. B we can conclude that

i. if cQ 0 for every Q then dQ 0 for every Q, ii. If Q is≥ such that d = 0 then Q≥ supp g , for some i Λ. Q 6 ⊂ i ∈ β s Corollary 18.6. For every β > s and q˜ [1, ] we have p,q,selfs˜ M( p,q). Moreover this inclusion is continuous. ∈ ∞ B ⊂ B 18.2. Strongly regular domains. We may wonder on which conditions the characteristic function of a set Ω is a pointwise multiplier in s . Bp,q Deﬁnition 18.7. A measurable set Ω I is a (α, C25,C26)-strongly regular j ⊂ k k domain if for every Q , with j C26, there is family (Q Ω) such that ∈ P ≥ F ∩ ⊂ P

i. We have Q Ω = k P k(Q Ω) P . ∩ k ∪ ∪ ∈F ∩ ii. If P,W k (Q Ω) and P = W then P W = . iii. We have∈ ∪ F ∩ 6 ∩ ∅ (18.55) P α C Q α. | | ≤ 25| | P k(Q Ω) ∈FX ∩ s n The following result can be associated with results in Triebel [45] for Bp,p(R ), especially when we consider the setting of Besov spaces in compact homogenous spaces. See Section 18.2 for details.See also Schneider and Vyb´ıral[40]. BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 45

Proposition 18.8. If Ω is a (1 βp, C ,C )-strongly regular domain then − 25 26 1/p (18.56) 1Ω β+,C26 C25 . p, ,selfs | |B ∞ ≤ Proof. Given Q j, with j C we can write ∈ P ≥ 26 P 1/p β (18.57) 1 a = | | − a . Ω Q Q P k P k(Q Ω) X ∈FX ∩ | | where aP is a (β, p)-atom. Note that

P 1 βp 1/p | | − C1/p, Q ≤ 25 P k(Q Ω) | | ∈FX ∩ so (18.56) holds.

Proposition 18.9 (Pointwise Multipliers I). There is C27 with the following property. Suppose that Ωi are (1 βp, Ki, ti)-strongly regular domains, i Λ N, and Θ > 0 for every i Λ . Consider− a function f with a s -representation∈ ⊂ i ∈ i Bp,q

f = cQaQ k Q k X X∈P satisfying A. We have 1/p sup ΘiKi N. Q ≤ ∈P Q Ω = c =0 ∩ i6 ∅ Q6 X B. If Q k satisﬁes c = 0 and Q Ω = then k t . ∈ P Q 6 ∩ i 6 ∅ ≥ i Then we can ﬁnd a s -representation Bp,q

(18.58) ( Θi1Ωi )f = dQaQ i k P k X X X∈P such that

q/p 1/q d p | Q| k P k X X∈P q/p 1/q (18.59) C N c p . ≤ 27 | Q| k P k X X∈P Moreover

i. If Q satisﬁes dQ = 0 then Q Ωi for some i Λ. ii. If c 0 for every6 Q then d ⊂ 0 for every Q∈. Q ≥ Q ≥ Proof. It follows from Proposition 18.4, Proposition 18.8 and Remark 18.5.

1/p 18.3. Functions on p, L∞. We want to give explicit examples of multipliers s B ∞ ∩ in p,q. One should compare the following result with the study by Triebel[44] of theB regularity of the multiplication on Besov spaces. See also Maz’ya and Shaposh- nikova [36] for more information on multipliers in classical Besov spaces. 46 D. SMANIA

1/p Proposition 18.10 (Pointwise multipliers II). Let g p, L∞. Then the multiplier operator ∈ B ∞ ∩ G: s s Bp,q → Bp,q s deﬁned by is a well-deﬁned and bounded operator acting on s . G(f) = gf ( p,q, p,q ) Indeed B | · |B g 1/p p, G s C C | |B ∞ + g , p,q 11 12 1/p s | |B ≤ 1 λ − | |∞ − 2 where C = C (1/p, p, ) and C = C (1/p, p, ) are as in Corollary 15.2. 11 11 ∞ 12 12 ∞ Remark 18.11. We can get a similar result replacing a by a+ everywhere. Bp,b Bp,b s 1/p Proof. Let aQ = Q − 1Q be the canonical (s, p)-Souza’s atom on Q and bJ = 1J be the canonical| (1/p,| p)-Souza’s atom on J. Given > 0, let

f = cQaQ k Q k X X∈P be a s -representation of f such that Bp,q 1/q p q/p cQ (1 + ) f s | | ≤ | |Bp,q k Q k X X∈P and

g = eJ bJ k J k X X∈P 1/p be a p, -representation of g given by Corollary 15.2 (in the case of Remark 18.11 B ∞ 1/p we can consider an optimal p, -positive representation of g). We claim that B ∞ J 1/p s u = | | − c e a , and 1 Q Q J J j J j J Q,Q=J,Q | | X X∈P ⊂ X6 ∈P

u2 = cQeJ aQ, k Q k Q J,J X X∈P ⊂X∈P s s are p,q-representations of functions ui p,q. Firstly note that the inner sums B j ∈ B k are ﬁnite. Moreover if J we denote by Qk(J) the unique element of , with k j that satisﬁes J Q∈ P(J) then P ≤ ⊂ k 1/p s J − p 1/p | | cQk(J)eJ Qk(J) J j k j | | X∈P X≤ (j k)(1/p s) p 1/p λ − − c e ≤ 2 | Qk(J) J | k j J j X≤ X∈P p 1/p (j k)(1/p s) eJ λ2 − − max cQ ≤ | | Q k | | J j k j ∈P X∈P X≤ p 1/p (j k)(1/p s) p 1/p max eJ λ2 − − ( cQ ) ≤ j | | | | J j k j Q k X∈P X≤ X∈P (j k)(1/p s) p 1/p C11C12 g 1/p λ − − ( cQ ) p, 2 ≤ | |B ∞ | | k j Q k X≤ X∈P BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 47

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

p 1/p cQeJ Q k J ,Q J X∈P ∈PX⊂ p 1/p c p e ≤ | Q| J Q k J ,Q J X∈P ∈PX⊂ 1/p p cQ g . ≤ | | | |∞ Q k X∈P So u1 s (1 + ) f s g . | |Bp,q ≤ | |Bp,q | |∞ We claim that gf = u1 + u2. Indeed let

fk0 = cQaQ

gk0 = eJ bJ

lim gk0 g p0 = 0 k0 | − | and

lim fk0 f p = 0. k0 | − | So

lim fk0 gk0 fg 1 = 0. k0 | − | Note that

A. If Q J then aQbJ = aQ, B. If J ⊂ Q then ⊂ J 1/p s a b = | | − a . Q J Q J | | So

fk0 gk0 = eJ cQaQbJ i k

Note that

s s lim ur,k0 ur 1 = 0 and ur,k0 Bp,q ur Bp,q for r = 1, 2. k0 | − | | | ≤ | | Now we can use Corollary 6.5.i to conclude the proof.

s 19. L∞ is a quasi-algebra Bp,q ∩ s Multipliers in L∞ are indeed much easier to come by. Bp,q ∩ s Proposition 19.1 (Pointwise multipliers III). Let g, f p,q L∞. Then g f s ∈ B ∩ · ∈ L∞ and Bp,q ∩ f g s + f g C11C12( f s + f )( g s + g ). | · |Bp,q | · |∞ ≤ | |Bp,q | |∞ | |Bp,q | |∞ s So L∞ is a quasi-Banach algebra. Here C = C (s, p, q) and C = Bp,q ∩ 11 11 12 C12(s, p, q) are as in Corollary 15.2. s 1/p Proof. Of course f g f g . Let aQ = Q − 1Q be the canonical (s, p)-Souza’s atom| on· Q|∞. Let≤ | |∞| |∞ | |

f = cQaQ k Q k X X∈P and g = eJ aJ k J k X X∈P be s -representations of f and g given by Corollary 15.2. We claim that Bp,q s 1/p u = J − c e a , 1 | | Q J Q k Q k Q J,J X X∈P ⊂X∈P s 1/p u = Q − c e a . 2 | | Q J J k J k J Q,Q=J,Q X X∈P ⊂ X6 ∈P s s are p,q-representations of functions ui p,q. Moreover by Proposition 17.1.A we haveB ∈ B 1/p s 1/p p J − c e | | Q J Q k J ,Q J X∈P ∈PX⊂ 1/p p s 1/p p c J − e ≤ | Q| | | J Q k J ,Q J X∈P ∈PX⊂ 1/p p cQ g . ≤ | | | |∞ Q k X∈P So u1 s C11C12 g f s , | |Bp,q ≤ | |∞| |Bp,q and by an analogous argument

u2 s C11C12 f g s . | |Bp,q ≤ | |∞| |Bp,q

Deﬁne fk0 and gk0 as in the proof of Proposition 18.10. By Proposition 17.1 we p have fk f and gk g . Since limk fk = f and limk gk = g in L , | 0 | ≤ | |∞ | 0 | ≤ | |∞ 0 0 0 0 we can assume, taking a subsequence if necessary, that fk0 gk0 converges pointwise BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 49

to fg. So by the Theorem of Dominated Convergence we have limk0 fk0 gk0 = fg in L1. Finally note that if J Q then ⊂ s 1/p a a = Q − a , Q J | | J Now we can use the same argument as in the proof of Proposition 18.10 to conclude that gf = u1 + u2. This concludes the proof.

19.1. Regular domains. Here we will give suﬃcient conditions for the character- s istic function of a set to deﬁne a bounded pointwise multiplier either on p,q L∞. For every set Ω, let B ∩

k (Ω) = min k 0: P k s.t. P Ω 0 { ≥ ∃ ∈ P ⊂ } Deﬁnition 19.2. We say that a countable family of pairwise disjoint measurable k k sets Ωr r Λ is (α, C28, λ5)-regular family if one can ﬁnd families (Ωr) , k k{ (Ω} ),∈ such that F ⊂ P ≥ 0 r k A. We have Ωr = k k0(Ωr ) Q (Ωr ) Q. ∪ ≥ k ∪ ∈F B. If P,Q k k0(Ωr ) (Ωr) and P = Q then P Q = . C. We have∈ ∪ ≥ F 6 ∩ ∅

α k k0( r Ωr ) α (19.60) Q C λ − ∪ Ω . | | ≤ 28 5 | ∪r r| r Λ Q k(Ω ) X∈ ∈FX r

We say that a measurable set Ω is a (α, C28, λ5)-regular domain if Ω is a (α, C28, λ5)- regular family. { }

Proposition 19.3. Let β > s. Every (1 βp, C, 0)-strongly regular domain is a (β s)p − (1 sp, C0, λ − )-regular domain, for some C0. − 2 Proof. Consider a (1 βp, C25, 0)-strongly regular domain Ω. There are at most k (Ω) − λ− 0 elements in k0(Ω) and 1 P λ k0(Ω) Q λ k0(Ω) 2 − | | 2 λ1 ≤ W ≤ λ1 | | for every Q, W k0(Ω). Consequently ∈ P k0(Ω) k0(Ω) λ2 − Ω k0(Ω) λ2 | | λ1− λ1 ≤ Q ≤ λ1 | | for each Q k0(Ω). For every Q k0(Ω) there is a family k(Q Ω) such that ∈ P ∈ P F ∩ P = Q Ω ∩ k P k(Q Ω) X ∈FX ∩ and 1 βp 1 βp P − C Q − . | | ≤ | | P k(Q Ω) ∈FX ∩ Let k k (Ω) = Q k0(Ω) (Q Ω). F ∪ ∈P F ∩ 50 D. SMANIA

We have 1 sp 1 βp (β s)p P − = P − P − | | | | | | Q k0(Ω) P k(Q Ω) P k(Q Ω) ∈PX ∈FX ∩ ∈FX ∩ (β s)p 1 βp max P − P − ≤ P k(Q Ω) | | | | Q k0(Ω) ∈F ∩ P k(Q Ω) ∈PX ∈FX ∩ (k k0(Ω))(β s)p (β s)p 1 βp λ − − Q − P − ≤ 2 | | | | Q k0(Ω) P k(Q Ω) ∈PX ∈FX ∩ (k k0(Ω))(β s)p (β s)p 1 βp Cλ − − Q − Q − ≤ 2 | | | | Q k0(Ω) ∈PX (k k0(Ω))(β s)p 1 sp Cλ − − Q − ≤ 2 | | Q k0(Ω) ∈PX k0(Ω)(1 sp) k0(Ω) λ2 − (k k0(Ω))(β s)p 1 sp Cλ1− λ2 − − Ω − ≤ λ1 | | This concludes the proof.

Remark 19.4. Suppose that there is C29 such that for every k and every Q, W k we have ∈ P 1 Q | | C , C ≤ W ≤ 29 29 | | and # P k0(Ω) : P Ω = C , { ∈ P ∩ 6 ∅} ≤ 30 Then it is easy to see that one can choose C0 = C30C29C. The following result is similar to results for Sobolev spaces by Faraco and Rogers [21]. See also Sickel [41].

Corollary 19.5. If Ωr r Λ is a (1 ps, C28, λ5)-regular family then there is C31 { }s ∈ − s such that for every g L∞ and r Λ we can ﬁnd a -representation ∈ Bp,q ∩ ∈ Bp,q (19.61) g 1 = dr a , · Ωr Q Q k Q k,Q Ω X ∈PX⊂ r such that 1/q r p q/p (19.62) dQ ) C31 g s . | | ≤ | |Bp,q j r Q j X X QX∈PΩ ⊂ r Note that Ω = Ω ∪r r is a (1 ps, C28, λ5)-regular domain and F (g) = g1Ω is a bounded operator in s − L∞ satisfying Bp,q ∩ 1/p C28 1/p s s (19.63) F L∞ C11C12 1 + Ω − . | |Bp,q ∩ ≤ (1 λq/p)1/q | | − 5 Moreover 1/p C28 1/p s (19.64) 1Ω s Ω − . | |Bp,q ≤ (1 λq/p)1/q | | − 5 BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 51

Proof. Notice that

f = 1 Ω = cQaQ, ∪r r k Q k X X∈P 1/p s k where c = Q − for every Q (Ω ) and c = 0 otherwise. Let Q | | ∈ ∪k ∪r F r Q

g = eJ aJ k J k X X∈P s be p,q-representations g given by Corollary 15.2. Consider u1, u2 as in the proof of PropositionB 19.1. By Proposition 17.1 we can get exactly the same estimate as in the proof of Proposition 19.1. k Note that those Q for which the corresponding atom aQ has a non-vanishing ∈ P j coefficient in the definition of u1 belongs to r j (Ωr), and moreover every k ∪ ∪ F J for which the corresponding atom aJ has non-vanishing coefficients in the ∈ P j definition of u2 is contained in some Q (Ωr), for some j and r. In particular J Ω . So (19.61) holds, with ∈ F ⊂ r r s 1/p s 1/p d = J − c e + J − c e Q | | Q J | | J Q Q J,J Q J,J=Q,J ⊂X∈P ⊂ X6 ∈P for every Q Ωr. Note also⊂ that 1/q 1 sp q/p Q − | | k r Q k(Ω ) X X ∈FX r 1/q 1/p (k k0(Ω))q/p 1/p s C λ − Ω − ≤ 28 5 | | k k ( Ω ) ≥ X0 ∪r r 1/p C28 1/p s Ω − . ≤ (1 λq/p)1/q | | − 5 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 s . B1,1 When p = q = 1 (and s > 0 small), something curious happens. We can skip the good grid and characterise the Besov space s of a homogeneous space B1,1 using regular domains. Fix C 1 and λ (0, 1). Let be the family of all 28 ≥ 5 ∈ W (1 s, C28, λ5)-regular domains. Of course . Let ˆ be a family of sets satisfying− P ⊂ W W ˆ P ⊂ W ⊂ W 1 s 1/(1 s) Deﬁne B − as the set of all functions f L − that can be written as ∈ ∞ 1 (20.65) f = c Ai , i A 1 s i=0 i − X | | 52 D. SMANIA

ˆ where Ai for every i N and ∈ W ∈ c < . | i| ∞ i X It is easy to see that

f 1/(1 s) ci . | | − ≤ | | i X Deﬁne

f 1 s = inf c , | |B − | i| i X where the inﬁmum runs over all possible representations (20.65). One can see that 1 s (B − , 1 s ) is a normed vector space. | · |B − 1 s s Proposition 20.1. We have that B − = 1,1( ) and the corresponding norms are equivalent. B P

s Proof. Note that (19.64) says that there is C such that if A then 1A 1,1( ) and ∈ W ∈ B P 1 s 1A s ( ) C A − . | |B1,1 P ≤ | | In particular, if f has a representation (20.65) we conclude that

f s ( ) C f B1 s . | |B1,1 P ≤ | | − 1 s s s In particular B − ( ). On the other hand if g ( ). then we can write ⊂ B1,1 P ∈ B1,1 P

∞ 1Q g = sQ 1 s Q − k=0 Q k X X∈P | |

∞ s = s < . | Q| | Q| ∞ P k=0 Q k X∈P X X∈P and g s ( ) is the inﬁmum of P sQ over all possible representations. In B1,1 P | | 1 s ∈P | | particular g B − and ∈ P g B1 s g s ( ). | | − ≤ | |B1,1 P

Remark 20.2. Let I = [0, 1] with the dyadic grid and the Lebesgue measure m. We prove in Part IV that s ( ), with 0 < s < 1,D is the Besov space Bs ([0, 1]), B1,1 D 1,1 and its norms are equivalent. Note that every interval [a, b] [0, 1] is a (1 s 1 ⊂ − s, 2, 2 − )-regular domain. So we can apply Proposition 20.1 with ˆ = [a, b], 0 a < b 1 . That is, f belongs to Bs ([0, 1]) if and only if it canW be written{ as in≤ ≤ } 1,1 (20.65), where every A is an interval and c < , and the norm in Bs ([0, 1]) i i | i| ∞ 1,1 is equivalent to the inﬁmum of i ci over all possible such representations. This characterisation of the Besov space|B|s ([0P, 1]) was ﬁrst obtained by Souza [14]. P 1,1 BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION 53

21. Left compositions. The following result generalizes a well-known result on left composition operators acting on Besov spaces of Rn. See Bourdaud and Kateb [6][5][7] for recent developments on the study of left compositions on Besov spaces of Rn. Proposition 21.1. Let g : I C → be a Lipchitz function such that g(0) = 0. Then the left composition L : s s g Bp,q → Bp,q deﬁned by L (f) = g f is well-deﬁned and g ◦ g f + oscs (g f) K( f + oscs (f)), | ◦ |p p,q ◦ ≤ | |p p,q where K is the Lipchitz constant of g. Consequently there exists C such that

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Departamento de Matematica,´ Instituto de Cienciasˆ Matematicas´ e de Computac¸ao-˜ Universidade de Sao˜ Paulo (ICMC/USP) - Sao˜ Carlos, Caixa Postal 668, Sao˜ Carlos-SP, CEP 13560-970, Brazil. E-mail address: [email protected] URL: https://sites.icmc.usp.br/smania/