ATOMIC, MOLECULAR AND WAVELET DECOMPOSITION OF GENERALIZED 2-MICROLOCAL BESOV SPACES

We introduce generalized 2-microlocal Besov spaces and give characterizations in decomposition spaces by atoms, molecules and wavelets. We apply the wavelet decomposition to prove that the 2-microlocal spaces are invariant under the action of pseudodifferential operators of order 0.


Introduction
The concept of 2-microlocal analysis or 2-microlocal function spaces is due to J.M. Bony (see [2]).It is an appropriate instrument to describe the local regularity and the oscillatory behavior of functions near singularities.The approach is Fourier-analytical using Littlewood-Paley-analysis of distributions.The theory has been elaborated and widely used in fractal analysis and signal processing by several authors.We refer to S. Jaffard ([8], [9]), Y. Meyer ([14]) and J. Lévy Véhel, S. Seuret ( [12]).These works have been generalized in different directions by P. Andersson ([1]), H. Xu([15], [23]), and S. Moritoh, T. Yamada ([16]).The main achievements are closely related to the use of wavelet methods and, as a consequence, to wavelet characterizations of 2-microlocal spaces.Here, we intend to give a unified Fourier-analytical approach to generalized 2-microlocal Besov spaces and we are interested in characterizations by various decompositions.The paper is organized as follows.After recalling some preliminaries and notation in Section 2 we define the concept of 2-microlocal Besov spaces and repeat some results from [10] in Section 3. In the main Sections 4 and 5 we present the characterizations of the 2-microlocal Besov spaces by decomposition in atoms, molecules and in wavelets.Only the proofs of the main theorems can be found in these sections, whereas the proofs of the more technical lemmas are shifted to the Appendix.Finally, in Section 6 we show that the 2-microlocal Besov spaces are invariant under the action of certain Calderón-Zygmund operators introduced in [13] and the action of pseudodifferential operators of order 0.

Preliminaries
As usual R n denotes the n−dimensional Euclidean space, N is the collection of all natural numbers and N 0 = N ∪ {0}.Z and C stand for the sets of integers and complex numbers, respectively.The points of the Euclidian space x ∈ R n are denoted by x = (x 1 , . . ., x n ).If β = (β 1 , . . ., β n ) ∈ N n 0 is a multi-index, then its length is denoted by |β| = n j=1 β j .The derivatives n have to be understood in the distributional sense.We put The Schwartz space S(R n ) is the space of all complex valued rapidly decreasing infinitely differentiable functions on R n .Its topology is generated by the norms A linear mapping f : S(R n ) → C is called a tempered distribution, if there exists a constant c > 0 and k, l ∈ N 0 such that holds for all ϕ ∈ S(R n ).The collection of all such mappings is denoted by S (R n ).The Fourier transform is defined on both spaces S(R n ) and S (R n ) and is given by where Here x • ξ = x 1 ξ 1 + • • • + x n ξ n stands for the inner product.The inverse Fourier transform is denoted by F −1 ϕ or ϕ ∨ and we often write φ instead of Fϕ.We say a vector space E is a quasi-Banach space, if it is complete and quasi-normed by •| E .That means •| E fulfills the norm conditions but the triangle inequality changes to for c ≥ 1.If c = 1 in (1) then •| E is a norm and E is a Banach space.As usual L p (R n ) for 0 < p ≤ ∞ stands for the Lebesgue spaces on R n normed by (quasi-normed if p < 1) Let w be a positive measurable function.We define the weighted Lebesgue space If E = C then we shortly denote q (C) by q .The constant c stands for all unimportant constants.So the value of the constant c may change from one occurrence to another.By a k ∼ b k we mean that there are two constants c 1 , c 2 > 0 such that c 1 a k ≤ b k ≤ c 2 a k for all admissible k.

Definitions and basic properties
In this section we present the Fourier analytical definition of generalized 2microlocal Besov spaces B s,mloc pq (R n , w) and we prove the basic properties in analogy to the classical Besov spaces.To this end we need smooth resolutions of unity and we introduce admissible weight sequences w = {w j } j∈N 0 .Definition 1 (Admissible weight sequence): Let α, α 1 , α 2 ≥ 0. A sequence of nonnegative measurable functions w = {w j } ∞ j=0 belongs to the class W α α 1 ,α 2 if and only if (i) There exists a constant C > 0 such that (ii) For all j ∈ N 0 we have is called admissible weight sequence.A non-negative measurable function is called an admissible weight function if there exist constants α ≥ 0 and C > 0, such that If w = {w j } ∞ j=0 is an admissible weight sequence, each w j is an admissible weight function, but in general the constant C w j depends on j ∈ N 0 .If we use w ∈ W α α 1 ,α 2 without any restrictions, then α, α 1 , α 2 ≥ 0 are arbitrary but fixed numbers.A fundamental example of an admissible weight sequence is given by the 2-microlocal weights.For a fixed nonempty set U ⊂ R n and s ∈ R they are given by where dist(x, and we get the well known 2-microlocal weights [9] treated by many authors If U is an open subset of R n , we get the weight sequence Moritoh and Yamada used in [16].Another example of an admissible weight sequence is given for fixed s ∈ R by where w : R n → [0, ∞) is a measurable function with the following properties: There are constants As a special case we choose w : ) and in addition we need w(x) ≤ c2 |x| β for all x, y ∈ R n and fixed c1 , c2 , β ≥ 1.
Thus we have (6) with C 1 = c1 and C 2 = c1 c2 and we can define the admissible weight sequence as in (5).For all examples of admissible weight sequences, considered above, it is easy to show, that w Next we define the resolution of unity.

Definition 2 (Resolution of unity):
(ii) For each β ∈ N n 0 there exist constants c β > 0 such that Remark 1: Such a resolution of unity can easily be constructed.Consider the following example.Let ϕ 0 ∈ S(R n ) with ϕ 0 (x) = 1 for |x| ≤ 1 and supp ϕ 0 ⊆ {x ∈ R n : |x| ≤ 2}.For j ≥ 1 we define , with the usual modifications if q is equal to infinity.One recognizes immediately that for w j (x) = 1 for all x ∈ R n and all j ∈ N 0 we get the usual Besov spaces, see [18].If one defines the admissible weight sequence as w j (x) = (x) for each j ∈ N 0 and being an admissible weight satisfying (2), we obtain weighted Besov spaces, see [4,Chapter 4].Applying a Fourier multiplier theorem for weighted Lebesgue spaces of entire analytic functions as in Subsection 1.7.5 in [17] we can prove that Definition 3 is independent of the chosen resolution of unity {ϕ j } j∈N 0 ∈ Φ(R n ), in the sense of equivalent quasi-norms.To this end, we can suppress the index ϕ in the notation of the norm.Moreover, as in [18,Subsection 2.3.3]we can prove that B s,mloc pq (R n , w) is a quasi-Banach space for all s ∈ R and 0 < p, q ≤ ∞ and even a Banach space in the case p, q ≥ 1.Also, we have for all admissible weight sequences and all s, p and q where S(R n ) is dense in B s,mloc pq (R n , w) for 0 < p, q < ∞.See [10] for details.Now, let us recall some results from [10] we need later on.
The spaces B s,mloc pq (R n , w) have the lift property.To be precise, let us define the lift operator I σ by The next result is the characterization of B s,mloc pq (R n , w) by local means.Therefore, we define for and for some ε > 0. Further, let If R = 0, then we do not need any moment conditions (9) on ψ 1 .Now, we present an elementary embedding in the scale of B s,mloc pq (R n , w).
For further embeddings in the general scale of B s,mloc pq (R n , w) and also in the special scale B s,s pq (R n , U ) with the 2-microlocal weights w j (x) = (1+2 j dist(x, U )) s we refer to [10].

Decomposition by Atoms and Molecules
In this chapter we present two decomposition theorems.We characterize the spaces B s,mloc pq (R n , w) via decompositions by atoms and molecules.First we introduce the basic notation.4.1.Sequence Spaces.First of all, we define for ν ∈ N 0 and m ∈ Z n the closed cube Q νm with center in 2 −ν m and with sides parallel to the axes and length 2 −ν .By χ νm we denote the characteristic function of the cube Q νm , defined by with the usual modifications if p or q are equal to infinity.Remark 2: Observe, that 4.2.Atoms and Molecules.Atoms are the building blocks for atomic decompositions.
and if If an atom a is centered at Q νm , that means if it fulfills (13), then we denote it by a νm .We recall the definition n and point out that for ν = 0 or L = 0 there are no moment conditions (15) required. and Remark 3: (a) For L = 0 or ν = 0 there are no moment conditions (17) required.If a molecule is concentrated in Q νm , that means it satisfies (16), then it is denoted by First, we show the convergence of the molecular decomposition.To this end, we introduce the number σ p , defined by σ p = max(0, n(1/p − 1)).
We also need a representation formula of Calderón type.The proof can be found in [5,Theorem 2.6].Lemma 2: Let {ϕ j } j∈N 0 ∈ Φ(R n ) be a resolution of unity and let M ∈ N. Then there exist functions θ 0 , θ ∈ S(R n ) with: where the functions ψ 0 , ψ ∈ S(R n ) are defined by We have already seen that the sum in (18) converges in S (R n ) under the conditions of Lemma 1.Now we come to the atomic decomposition theorem. holds.Moreover, where the constant c is universal for all f ∈ B s,mloc pq (R n , w).
Proof.Since [K, L]-atoms are [K, L, M ]-molecules for every M > 0 the convergence in S (R n ) has already been stated in Lemma 1.
The proof relies on the method used in the proof of [7,Theorem 5.11].We use Lemma 2 with M = L − 1, the functions θ 0 , θ ∈ S(R n ) with the properties ( 19)-( 23) and the functions ψ 0 , ψ ∈ S(R n ) with (24).Let f ∈ B s,mloc pq (R n , w), then we get from the Lemma 2 where ψ ν (•) = 2 νn ψ(2 ν •).Now, splitting the integration with respect to the cubes Q νm we derive We define for each ν ∈ N and all m ∈ Z n where otherwise we set a νm (x) = 0.The a 0m atoms and λ 0m are defined similarly using θ 0 and ψ 0 .Clearly, (25) and the properties of θ 0 , ψ 0 , θ and ψ ν ensure that a νm are [K, L]-atoms.It remains to prove, that there exists a constant c such that λ| b s,mloc pq (w) ≤ c f | B s,mloc pq (R n , w) .We have for fixed ν ∈ N 0 and a > n p + α Here, (ψ * ν f ) a denotes the Peetre maximal operator, defined in (8).Therefore, we have using ( 12) Since ψ 0 ∈ S(R n ) and ψ ∈ S(R n ) are two kernels which fulfill the moment conditions (9) and the Tauberian conditions (10) and (11), we can use Proposition 3 with a > n p + α and derive from (28) To prove the converse direction of the atomic decomposition, we take the more general molecules in the above sense.To this end, we need two technical lemmas as used in [5].We want to estimate the size of ϕ ∨ j * µ νm , where {ϕ j } ∈ Φ(R n ) and µ νm is a [K, L, M ]-molecule concentrated in Q νm .We use the resolution of unity introduced in Remark 1.We recall Lemma 3.3 in [5].Lemma 3: Let {ϕ j } j∈N 0 ∈ Φ(R n ) be a resolution of unity and {µ νm } ν∈N 0 ,m∈Z n are [K, L, M ]-molecules.Then we have for all and The next lemma is analogous to Lemma 3.4 in [5] and the proof is in the Appendix. where Finally, we can state the converse direction of the decomposition theorem by molecules.
is an element of B s,mloc pq (R n , w) and Proof.We have the representation of f ∈ S (R n ) by (31) and we know by Lemma 1 that this representation converges.Now, we estimate the norm of f Let p ≥ 1.We estimate the first L p Norm in (32).We use (30) and we derive from Lemma 4 with R = M > n + α For the second term in (32) we use (29) and Lemma 4 with R = M − L − n > n + α and get . Now, we consider the case 0 < p < 1.We use the embedding p → 1 and obtain for the first term in (32) By applying (30) and using the properties of the weight sequence, we estimate So, we get for the first L p norm . By a similar calculation we obtain the second estimate (M > n p + L + n + α) We denote p := min(1, p) and t := min(1, p, q).We can rewrite our results for the first term as p , for all 0 < p, q ≤ ∞.Finally, we conclude with that notation and t/ p → 1 , and Young's inequality gives us With the same notation a similar estimate can be achieved for 2 js II j q t .Here one has to use ζ := L − σ p + s − α 1 > 0 and this finishes the proof.
For every M > 0 every [K, L] atom is a [K, L, M ] molecule.So we get an easy corollary for the atomic decomposition.
belongs to the space B s,mloc pq (R n , w) and there exists a constant c > 0 with The constant c is universal for all λ and a νm .
(ii) For each f ∈ B s,mloc pq (R n , w) there exist λ ∈ b s,mloc pq (w) and [K, L]-atoms centered at Q νm such that f can be represented as in (33) (convergence being in S (R n )), where and the constant c is universal for all f ∈ B s,mloc pq (R n , w).

Wavelet decomposition
In this section we describe the characterization of B s,mloc pq (R n , w) by a decomposition in wavelets.We follow closely the ideas expressed in [19], [20] and [11].
5.1.Preliminaries.First of all, we recall some results from wavelet theory.

Proposition 5:
(i) There are a real scaling function ψ F ∈ S(R) and a real associated wavelet ψ M ∈ S(R) such that their Fourier transforms have compact supports, ψ F (0) = (2π) −1/2 and (ii) For any k ∈ N there are a real compactly supported scaling function ψ F ∈ C k (R) and a real compactly supported associated wavelet In both cases we have, that {ψ νm : ν ∈ N 0 , m ∈ Z} is an orthonormal basis in L 2 (R), where and the functions ψ M , ψ F are according to (i) or (ii).This Proposition is taken over from [21,Theorem 1.61].The wavelets in the first part are called Meyer wavelets.They do not have a compact support but they are rapidly decaying functions (ψ F , ψ M ∈ S(R)) and ψ M has infinitely many moment conditions.The wavelets from the second part are called Daubechies wavelets.
Here the functions ψ M , ψ F do have compact support, but they have only limited smoothness.Both types of wavelets are well described in [22], chapters 3 and 4.This orthonormal basis can be generalized to the n-dimensional case by a tensor product procedure.We take over the notation from [21, Subsection 4.2.1] with l = 0. Let ψ M , ψ F be the Meyer or Daubechies wavelets described above.Now, we define where the * indicates, that at least one where To get the wavelet characterization we use local means with kernels which only have limited smoothness and we use the molecular decomposition described in the previous section.This idea goes back to [20], [11] and [7].First, we recall the local means with kernel k With t = 2 −j , x = 2 −j l where j ∈ N 0 and l ∈ Z n , one gets First, assume that the expression (35) makes sense, at least formally.Later on we show that (35) is a dual pairing.Now, the usual properties for k are transferred to the kernels k jl .
It is clear from the definition, that {2 −jn k jl } are [A, B, C] molecules.

Duality.
In this subsection we show that the expression makes sense as a dual pairing.Here, w ∈ W α α1,α2 , f ∈ B s,mloc pq (R n , w) and the function k : R n → C belongs to some weighted space of continuously differentiable functions Because of (37) and (39) the equation (36) makes sense at least for u > σ p − s + α 1 .Further, we mention that all functions {k jl } from Definition 8 with A ≥ u, B ∈ N 0 arbitrary and C ≥ κ belong to the space C u (R n , κ).So we see that (35) is well defined for A > σ p − s + α 1 and C ≥ α + n, but these conditions will always be fulfilled in the following theorems.5.3.Wavelet isomorphism.We want to use the molecular decomposition obtained in the last section.We assume, that {µ νm } are [K, L, M ] molecules and that the {k jl } are the above given functions from Definition 8. Before stating the theorem we have to prove some fundamental lemmas.where the last inequality comes from the atomic decomposition theorem.We recall, that atoms have compact support and that they are [K, L, M ] molecules for every M > 0. We split (42) dependent on j ∈ N 0 into To shorten notation we define A(m, l) Let ν ≤ j and 1 ≤ p ≤ ∞, then we get by Minkowski's inequality and Lemma 5 where in the last step we used Lemma 6 with R = C − α > n.
In the case 0 < p < 1 we have also the estimate (44) and get by p → 1 A direct calculation shows, that l∈Z n A(m, l) −p(C−α) ≤ c2 (j−ν)n for C > n p + α.Therefore, we get from (45) and (46) where Let us consider the case ν > j and 1 ≤ p ≤ ∞.Then we derive similar to the first case with Minkowski's inequality and Lemma 5   , where we used Lemma 6 with R = C − A − n − α > n in the last step.We obtain now with ρ For 0 < p < 1 we have again (48), use p → 1 and obtain   The sum over l ∈ Z n is bounded by a constant for C > n p + n + A + α.Hence, we get where ρ = A + s − α 1 − (n/p − n) > 0. Now the result (43) can be obtained from (47), ( 49) and (50) by standard arguments.
Remark 5: As shown in the proof it is enough to assume weight sequence w defined in (65).The corresponding sequence spaces are defined by the norm . Now, we can adapt everything from Theorem 4 and a wavelet decomposition for the 2-microlocal Besov spaces with respect to the Daubechies wavelets follows.
with unconditional convergence in S (R n ) and in any space B t,t pq (R n , U ) with t < s and t < s .The representation (67) is unique, Gm is an isomorphic map from B s,s pq (R n , U ) onto bs,s pq (U ).Moreover, if in addition max(p, q) < ∞ then {Ψ ν Gm } is in unconditional basis in B s,s pq (R n , U ).Let us say a few words about the convergence of (67).As in Theorem 4 we have unconditional convergence in B s,s pq (R n , U ) for max(p, q) < ∞ and arbitrary U ∈ R n , not necessarily bounded.For 0 < p < ∞ and 0 < q ≤ ∞ we have unconditional convergence in B t,s pq (R n , U ) with s > t and also arbitrary U .Only in the case of p = ∞ we need as an additional assumption that U ⊂ R n has to be bounded to get the unconditional convergence in B t,t pq (R n , U ) for all s > t and s > t .Remark 7: For p ≥ 1 we have σ p = 0 in condition (66).Now we can rewrite this condition as This is almost the same condition k > max(max(s, s + s ), max(−(s + n), −(s + s ))) used in [14, p.64, p.67] in connection with the orthonormal Daubechies wavelets.
According to Remark 6 we can also give a characterization of B s,s pq (R n , U ) by other wavelet bases {Ψ ν Gm } (for example the Meyer wavelets), where {Ψ ν Gm } are [K, K, M ] molecules with the condition (64) on K and M .
. Moreover, since every classical pseudodifferential operator belonging to S 0 1,0 is a sum of an operator in Op(M γ ) and a regularizing operator ([13, Chapter 7]), we conclude that the 2-microlocal spaces B s,mloc pq (R n , w) are invariant under the action of pseudodifferential operators out of S 0 1,0 .We can state now regularity result for solutions of elliptic partial differential equations.From Calderón and Zygmund [3] we know, that the inverse of an elliptic operator is a product of a fractional integration and a pseudodifferential operator of order 0. The latter one is a sum of an operator in Op(M γ ) and a regularizing operator and the first one is a lift operator (Proposition 2).Using this fact together with the last Theorem, we obtain the following assertion.
For ϕ ∈ S(R n ) we get from the moment conditions (17) for fixed ν ∈ N 0 where κ > 0 will be specified later on.We use Taylor expansion of ϕ up to the order L and get with ξ on the line segment joining y and 2 Using the properties of the weight sequence and y κ ≤ y − 2 −ν m κ ξ κ , we can estimate (71) by Hence, we derive from (71) Now, let us suppose that p ≥ 1 then we get applying Hölder's inequality on the integral in (72) with κ > By choosing M large enough (M > L + 2n + 2α) and using Lemma 4 with j = ν we get Putting this into (73) and taking the 1/p power, the first part of the lemma is proved.
Second Step: Now, we have j ≥ ν and we can use The same splitting of the integral in dyadic cubes as in the first step together with the above inequalities lead to the second case.
Proof of Lemma 5. To prove the case ν ≥ j one can modify the steps in [5,Lemma 3.3].We only need the moment conditions on µ νm and a sufficiently strong decay of the derivatives of k j,l .The proof for ν ≥ j follows easily by interchanging the roles of k jl and µ νm .
Proof of Lemma 6.Here we only give the proof for ν ≥ j because the other case follows by similar arguments.For k ∈ Z n we introduce the quantity Then we get for m ∈ j ν (k) , where the last inequality is due to card{l ∈ Z n : Q νm ⊆ Q jl + 2 −j u} ∼ 1 and R > n.

Theorem 7 :
Let w ∈ W α α 1 ,α 2 , 0 < p, q ≤ ∞ and s ∈ R. Let Λ be an elliptic partial differential operator of order m, with smooth coefficients.If Λf = g and g belongs to B s,mloc pq (R n , w), then f belongs to B s+m,mloc pq (R n , w).AppendixHere we present the more technical proofs of the Lemmas.Proof of Lemma 1.We have to prove that the limes lim r→∞ r ν=0 m∈Z n λ νm µ νm (x) exists in S (R n ).