JENAER SCHRIFTEN MATHEMATIK UND INFORMATIK

Transcription

1 Friedrich-Schiller- Universität Jena JENAER SCHRIFTEN ZUR MATHEMATIK UND INFORMATIK Eingang : November 30, 2006 Math/Inf/20/06 Als Manuskript gedruckt Local characterization of generalized 2-microlocal spaces Henning Kempka Abstract This paper deals with a generalization of 2-microlocal spaces in the sense of weighted Besov spaces. We define 2-microlocal Besov spaces Bpq (R n, w) and describe first properties. The key theorem is the local means characterization for these spaces. Using this characterization we prove some conclusions as a pointwise multiplier assertion and the invariance of the spaces Bpq (R n, w) under the action of diffeomorphisms. Keywords: Besov spaces, 2-microlocal spaces, local means 2000 MSC: 42B15, 42B25, 42B35 This work was supported by the DFG within the SFB/TR7 Gravitational Wave Astronomy

2

3 Contents 1 Introduction 3 2 The 2-microlocal Besov spaces B pq (R n, w) Preliminaries Definitions and basic properties Lift property and equivalent norms Local Means Preliminaries The Peetre maximal operator Helpful lemmas Comparison of different Peetre maximal operators Boundedness of the Peetre maximal operator Local means characterization Further properties Embedding theorems General embeddings Embeddings for 2-microlocal Besov spaces Pointwise multipliers Invariance under Diffeomorphisms References 37 1

4 2

5 1 Introduction The concept of 2-microlocal analysis or 2-microlocal function spaces is due to J.M. Bony (see [Bo84]). It is an appropriate instrument to describe the local regularity and the oscillatory behavior of functions near to 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 [Ja91], [JaMey96], [LVSeu04], [Mey97], [MeyXu97], [MoYa04] and [Xu96]. The main achievements are closely related to the use of wavelet methods and, as a consequence, wavelet characterizations of 2-microlocal spaces. Here, we intend to give a unified Fourier-analytical approach to generalize 2-microlocal Besov spaces and we are interested in local characterizations of the spaces under consideration. Therefore, let {ϕ j } j N0 be a smooth resolution of unity (see Subsection 2.2 for the precise definition) and let {w j } j N0 be a sequence of weight functions satisfying 0 < w j (x) Cw j (y)(1 + 2 j x y ) α (1.1) 2 α 1 w j (x) w j+1 (x) 2 α 2 w j (x), (1.2) for x, y R n, j N 0 and α, α 1, α 2 0. F and F 1 stand for the Fourier transform and its inverse in the space S (R n ) of tempered distributions, respectively. Let 0 < p, q and s R. Then we introduce Bpq (R n, w) as the space of all f S (R n ) such that f Bpq (R n, w) ( = 2 jsq w j F 1 (ϕ j Ff) L p (R n ) ) 1/q q < (1.3) for 0 < q < and j=0 f Bp (R n, w) = sup 2 js w j F 1 (ϕ j Ff) L p (R n ) <, (1.4) j N 0 for q =. As a special case, let w j (x) = (1 + 2 j x x 0 ) s for s R, x 0 R n and j N 0. If p = q = 2 we obtain the spaces Hx s,s 0 considered by Bony in [Bo84]. The case p = q = yields the 2-microlocal spaces Cx s,s 0 introduced by Jaffard in [Ja91] and extensively treated by Meyer, Jaffard and Lévy-Vehel ([JaMey96],[LVSeu04]). The more general case 1 p, q, and characterizations of chirp-like signals as well as relations to gravitational wave signals, has been studied by Xu, Meyer and Moritoh,Yamada ([Xu96],[MeyXu97],[MoYa04]). We can rewrite [F 1 (ϕ j Ff)](x) = [(F 1 ϕ j ) f](x). (1.5) 3

6 The functions F 1 ϕ j do not have compact support. In particular, to compute the building blocks (F 1 ϕ j Ff) in x R n we need f globally. Roughly speaking, we shall show that the functions F 1 ϕ j in (1.5) and (1.3),(1.4), respectively, can be replaced by smooth functions with compact support in a ball of radius c2 j (c is a constant). This leads to local characterizations of our spaces. Characterizations of such a type are well known for weighted and unweighted Besov spaces (see for instance [Tri92] and [Tri06]) and turned out to be very useful to solve some key problems as the behavior by pointwise multiplication and invariance under diffeomorphisms. Moreover, it paves the way to atomic and wavelet representations as well as to discretizations (see [Tri06] for classical Besov spaces) and isomorphisms to corresponding sequence spaces. The paper is organized as follows. Section 2 contains all definitions and some basic properties such as the independence of Bpq (R n, w) of the choice of the resolution of unity {ϕ j } j N0 and the lift property. Here we rely on Fourier multiplier theorems for weighted spaces of entire analytic functions which can be found in [SchmTri87]. The main part is Section 3, where we give the characterization by local means. We use maximal function and inequalities and follow ideas in [Tri92], [Ry99] and [Vyb06] in a different context. The final Section 4 deals with embedding theorems for different metrics based on weighted Nikols kij inequalities ([SchmTri87]). Moreover, we apply the results of Section 3 (local means) to prove a theorem on pointwise multiplication. Finally, we use the local means characterization to prove that the spaces Bpq (R n, w) are invariant under a special class of diffeomorphisms. 4

7 2 The 2-microlocal Besov spaces B pq (R n, w) 2.1 Preliminaries As usual R n symbolizes 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 R n are denoted by x = (x 1,..., x n ), y = (y 1,..., y 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 D β = β / β1 β n We put x β = x β 1 1 x β n n. have to be understood in the distributional sense. 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 ϕ k,l = sup (1 + x ) k D β ϕ(x),k, l N 0. (2.1) x R n β l A linear mapping f : S(R n ) C is called a tempered distribution, if there is a constant c > 0 and k, l N 0 such that f(ϕ) c ϕ k,l 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 (Ff)(ϕ) := f(fϕ), ϕ S(R n ),f S (R n ) Fϕ(ξ) := 1 e ix ξ ϕ(x)dx. (2π) n/2 R n 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 ˆϕ instead of Fϕ. Vector-valued sequence spaces As usual L p (R n ) for 0 < p stands for the Lebesgue spaces on R n normed by (quasi-normed for p < 1) f L p (R n ) = 1/p f(x) p dx for 0 < p < and R n f L (R n ) = ess-sup x R n f(x). 5

8 If w is a non-negative measurable function on R n, we denote the weighted Lebesgue spaces by L p (R n, w) and they are defined for 0 < p by f L p (R n, w) = wf L p (R n ) = R n f(x) p w p (x)dx with the usual modification if p =. For a complex valued sequence a = {a j } j=0 the sequence spaces l q for 0 < q are normed by (quasi-normed for q < 1) 1/p, ( ) 1/q a l q = a j q j=0 a l = sup j N 0 a j. for 0 < q < and Let {f k } k N0 be a sequence of complex valued measurable functions, 0 < p and 0 < q. Then we put f k (x) l q (L p ) = {f k } k N0 l q (L p ) = k=0 R n f k (x) p dx q/p 1/q ( ) 1/q = f k L p q k=0, also with the above modifications for p = or q =. The constant c adds up 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. 2.2 Definitions and basic properties In this section we present the Fourier analytical definition of generalized 2-microlocal Besov spaces Bpq (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 our admissible weight sequences w = {w j } j N0. Definition 2.1 (Admissible weight sequence): Let α, α 1, α 2 0. We say that a sequence of non-negative 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 0 < w j (x) Cw j (y) ( j x y ) α for all j N 0 and all x, y R n. (2.2) (ii) For all j N 0 we have 2 α 1 w j (x) w j+1 (x) 2 α 2 w j (x) for all x R n. (2.3) 6

9 Such a system {w j } j=0 Wα α 1,α 2 is called admissible weight sequence. Remark 2.2: A non-negative measurable function ϱ is called an admissible weight function if there exist constants α ϱ 0 and C ϱ > 0, such that 0 < ϱ(x) C ϱ ϱ(y)(1 + x y ) α ϱ holds for every x, y R n. (2.4) 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 wj depends on j N 0. Remark 2.3: If we use w Wα α 1,α 2 arbitrary but fixed numbers. without any restrictions, then α, α 1, α 2 0 are Remark 2.4: If w W α α 1,α 2, w W β β 1,β 2 and λ > 0, it is easy to check: (a) The sequence w 1 = {w 1 j } j=0 belongs to the class W α α 2,α 1. (b) The sequence λw belongs to the class W α α 1,α 2. (c) The sequence w λ = {w λ j } j=0 belongs to the class W λα λα 1,λα 2. (d) The sequence w + w belongs to the class W max(α,β) max(α 1,β 1 ),max(α 2,β 2 ). (e) The sequence w w belongs to the class W α+β α 1 +β 1,α 2 +β 2. Example 2.5: Let U be a subset of R n. We denote by dist(x, U) = inf z U x z the distance of x R n from U. A typical admissible weight sequence is for fixed U R n and s R given by We have for s 0 w j (x) := ( j dist(x, U) ) s for j N 0. (2.5) w j (x) w j+1 (x) 2 s w j (x) and for s < 0 2 s w j (x) w j+1 (x) w j (x). Hence, for all j N 0 and all fixed s R 2 max(0, s ) w j (x) w j+1 (x) 2 max(0,s ) w j (x) for every x R n. (2.6) From the inequality dist(x, U) x y + dist(y, U) we derive for s 0 w j (x) = ( j dist(x, U) ) s Since a + b 2ab for a, b 1, we get ( j x y + 2 j dist(y, U) ) s. w j (x) [ 2 ( j dist(y, U) ) (1 + 2 j x y ) ] s = 2 s w j (y)(1 + 2 j x y ) s, 7

10 for all x, y R n and all j N 0. If s < 0 we can do the same calculation for the inverse weight sequence w 1 and according to Remark 2.4(a) we find w j (x) 2 s w j (y)(1 + 2 j x y ) s, for all x, y R n and all j N 0. Finally, we have for fixed s R and all j N 0 0 < w j (x) 2 s w j (y)(1 + 2 j x y ) s, (2.7) for all x, y R n. Together with (2.6) we get w W α α 1,α 2 if s α, max(0, s ) α 1 and max(0, s ) α 2. A special case is U = {x 0 } for x 0 R n. Then dist(u, x) = x x 0 and we get the well known two-microlocal weights [JaMey96]: w j (x) = (1 + 2 j x x 0 ) s for j N 0. (2.8) If U is an open subset of R n, we get the weight sequence Moritoh and Yamada used in [MoYa04]. Example 2.6: Let w : R n [0, ) be a measurable function with the properties: There are constants C 1, C 2 1 and β 1 such that for all x, y R n For fixed s R and all j N 0 we define 0 w(x) C 1 w(y) + C 2 x y β. (2.9) w j (x) = ( j w(x) ) s /β for all x R n. (2.10) By analogy to Example 2.5 above we get 0 < w j (x) (2 C 1 C 2 ) s w j (y) ( j x y ) s and (2.11) 2 max(0, s ) w j (x) w j+1 (x) 2 max(0,s ) w j (x) holds for all x, y R n and j N 0. (2.12) Hence, we have w = (w j ) j N0 W α α 1,α 2 for all α s and α 1 max(0, s ), α 2 max(0, s ). As a special case we choose w : R n [0, ) subadditiv, that is 0 w(x + y) c 1 (w(x) + w(y)) and in addition we need w(x) c 2 x β for all x R n and fixed c 1, c 2, β 1. Thus we have (2.9) with C 1 = c 1 und C 2 = c 1 c 2 and we can define the admissible weight sequence as in (2.10). Next we define the resolution of unity. Definition 2.7 (Resolution of unity): A system ϕ = {ϕ j } j=0 S(R n ) belongs to the class Φ(R n ) if and only if 8

11 (i) supp ϕ 0 {x R n : x 2} and supp ϕ j {x R n : 2 j 1 x 2 j+1 } (ii) For each β N n 0 there exist constants c β > 0 such that (iii) For all x R n we have 2 j β sup x R n D β ϕ j (x) c β holds for all j N 0. ϕ j (x) = 1. j=0 Remark 2.8: 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 ϕ j (x) = ϕ 0 (2 j x) ϕ 0 (2 j+1 x). Now it is obvious that ϕ = {ϕ j } j N0 Φ(R n ). Definition 2.9: Let (ϕ j ) j N0 Φ(R n ) be a resolution of unity and let w = (w j ) j N0 W α α 1,α 2. Further, let 0 < p, 0 < q and s R. Then we define { Bpq (R n, w) = f S : f Bpq (R n, w) } <, where (2.13) ϕ f Bpq (R n, w) ( ) ϕ = 1/q 2 jsq wj (ϕ j ˆf) L p (R n ) q, (2.14) with the usual modifications if p or q are equal to infinity. j=0 Remark 2.10: One recognizes immediately that for w j 1 one obtains the usual Besov spaces, see [Tri83]. If one defines the admissible weight sequence as w j (x) = ϱ(x) for each j N 0 and ϱ being an admissible weight, we obtain the usual weighted Besov spaces, see [EdTri96, Chapter 4]. Firstly, we have to prove that Definition 2.9 is independent of the chosen system (ϕ j ) j N0 Φ(R n ). We need a Fourier multiplier theorem for weighted Lebesgue spaces of entire analytic functions as in [SchmTri87]. We define the Sobolev spaces W k 2 (R n ) for k N 0. A function f L 2 (R n ) belongs to W k 2 (R n ) if f W k 2 (R n ) := γ k D γ f L 2 (R n ) 2 1/2 <. (2.15) 9

12 Theorem 2.11 ([SchmTri87]): Let ϱ : R n R be an admissible weight which satisfies (2.4) for some α ϱ 0. Furthermore, let B b = {y R n : y b} for b > 0 and 0 < p. Then for every κ N with ( 1 κ > n min(1, p) 1 ) + α ϱ (2.16) 2 there exists a constant c > 0 (depending on b) such that ϱf 1 MFf Lp (R n ) c M W κ 2 (R n ) ϱf L p (R n ) (2.17) holds for all f L p (R n, ϱ) S (R n ) with supp Ff B b and all M S(R n ). Remark 2.12: Additionally, we need a corollary of the Theorem Let 0 < p and let We assume that the weight satisfies B b = {y R n : y b} for b > 0. 0 < ϱ(x) C ϱ ϱ(y) (1 + ab x y ) α ϱ for fixed a > 0 and all x, y R n. If f L p (R n, ϱ) with supp Ff B b, then supp F[f(b 1 x)] B 1 and by the properties of the Fourier transform ( ϱf 1 MFf ) (x) = { ϱ(b 1 )F 1 [ M(b ) ( Ff(b 1 ) ) ( ) ]} (bx). (2.18) Therefore, we obtain ( ϱf 1 MFf ) (x) Lp (R n ) { = ϱ(b 1 )F [ 1 M(b ) ( Ff(b 1 ) ) ( ) ]} (bx) Lp (R n ) = b n p { ϱ(b 1 )F [ 1 M(b ) ( Ff(b 1 ) ) ( ) ]} (x) Lp (R n ). For the weight function r(x) = ϱ(b 1 x) we have α r = α ϱ and 0 < r(x) max(1, a)c ϱ r(y) (1 + x y ) α ϱ = C ϱ r(y) (1 + x y ) α ϱ. We can apply Theorem 2.11 and obtain ( ϱf 1 MFf ) (x) Lp (R n ) c C ϱ b n p M(b ) W κ 2 (R n ) ϱ(b 1 )f(b 1 ) Lp (R n ) ( for κ > n 1 min(1,p) 1 2 ) + α ϱ. = cc ϱ M(b ) W κ 2 (R n ) ϱf L p (R n ) (2.19) Now, we are ready to show that Definition 2.9 of the spaces Bpq (R n, w) is independent of the chosen resolution of unity ϕ Φ(R n ). 10

13 Theorem 2.13 (Independence of the resolution of unity): Let ϕ = (ϕ j ) j N0, φ = (φ j ) j N0 Φ(R n ) be two resolutions of unity and let w = {w j } j N0 Wα α 1,α 2 be an admissible weight sequence. If 0 < p, 0 < q and s R, then we have f B pq (R n, w) ϕ f B pq (R n, w) for all f S φ (R n ). Proof: It is sufficient to show that there is a c > 0 such that for all f S (R n ) we have f B pq (R n, w) c φ f B pq (R n, w). Interchanging ϕ and φ we derive ϕ the result we aim at. Putting ϕ 1 = 0 we see 1 φ j (x) = φ j (x) ϕ j+k (x) for all j N 0. k= 1 By the properties of the Fourier transform 1 { ( [ w j {F 1 φ j (Ff)} = w j F 1 φ j F F 1 ϕ j+k (Ff) ])}. k= 1 Now, we apply (2.19) with b = 2 j+2, M = φ j und f = F 1 ϕ j+k (Ff) for k { 1, 0, 1}. We get for every j N 0 wj { F 1 φ j ( F [ F 1 ϕ j+k (Ff) ])} Lp (R n ) ( with κ > n 1 min(1,p) 1 2 c φj (2 j+2 ) W κ 2 (R n ) { wj F 1 ϕ j+k (Ff) } Lp (R n ), (2.20) ) + α. By (2.2) and formula (2.19) the constant c does not depend on j N. Since supp φ j (2 j+2 ) B 1 and using the properties of the resolution of unity, we have We conclude that sup φ l (2 l+2 ) W2 κ (R n ) c sup l N 0 wj {F 1 φ j (Ff)} Lp (R n ) c c 1 k= 1 1 k= 1 sup l N 0 β κ sup 2 l β (D β φ l )(x) < c κ. x R n wj { F 1 φ j ( F [ F 1 ϕ j+k (Ff) ])} Lp (R n ) w j { F 1 ϕ j+k (Ff) } L p (R n ). Finally, multiplying by 2 js, using the property (2.3) of the admissible weight sequence and taking the l q quasi-norm with respect to j, we see that ( 2 jsq wj {F 1 φ j (Ff)} Lp (R n ) ) 1/q q c (2 s+α s+α 1 ) f B pq (R n, w) ϕ. j=0 This completes the proof. Remark 2.14: As in Theorem in [Tri83] we can prove that Bpq (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. 11

14 2.3 Lift property and equivalent norms We introduce the lift operator as in the classical case of Besov spaces, [Tri83]. If σ R, the operator I σ is defined by ( I σ : f ξ ˆf) σ (2.21) where ξ = (1 + ξ 2 ) 1/2. Theorem 2.15: Let s, σ R and w = (w j ) j N0 Wα α 1,α 2. Moreover, let 0 < p and 0 < q. Then I σ maps Bpq (R n, w) isomorphically onto Bpq s σ,mloc (R n, w) and Iσ f Bpq s σ,mloc (R n, w) is an equivalent quasi-norm on B pq (R n, w). Proof: To prove the theorem we show that Iσ f Bpq s σ,mloc (R n, w) ( ) 1/q = 2 j(s σ)q wj (ϕ j ξ σ ˆf) L p (R n ) q j=0 f B pq (R n, w). Let φ S(R n ) with 1 φ(x) = 1 if x 2 and 2 supp φ {ξ R n : 14 } ξ 4. Then we have for j 1 ( ) ( ϕ j ξ σ ˆf = ξ σ φ(2 j ξ)ϕ j ˆf), and we define M j (ξ) := 2 σj ξ σ φ(2 j ξ), whereas, supp ϕ j ˆf {ξ R n : ξ 2 j+1 }. ( ) Now, we can apply (2.19) with b = 2 j+2 1 and κ N with κ > n 1 + α and min(1,p) 2 we obtain ( w j 2 σj ξ σ φ(2 j ξ)ϕ j ˆf) L p (R n ) c sup Ml (2 l+2 ) W κ 2 (R n ) wj (ϕ j ˆf) L p (R n ) l N 0 for all j N 0 and 0 < p. If β N n 0 is a multi-index with β κ, we have D β ( M l (2 l+2 ) ) (x) = D β ( 2 σl 2 l+2 σ φ(4 ) ) (x) 2 2σ γ β c β,γ D γ ( 2 2(l+2) + x 2) σ/2 ( D β γ φ ) (4x) 4 β γ (2.22) ( 2 2(σ+κ) sup sup D δ φ ) (y) D c ( γ β,γ 2 2(l+2) + x 2) σ/2. δ κ y R n γ β (2.23) 12

15 Since φ S(R n ) and supp φ {ξ R n : 1 ξ 4} we obtain 4 ( sup sup D δ φ ) (y) c. (2.24) y R n δ κ Furthermore, we have D ( γ 2 2(l+2) + x 2) σ/2 ( cσ,γ 2 2(l+2) + x 2) σ/2 γ /2 and (2.25) supp M l (2 l+2 ) { } x R n 1 : 16 x 1. (2.26) Finally, we get from (2.23)-(2.27) for 0 < σ < κ ( D β M l (2 l+2 ) ) (x) ( ) β ( c c σ,γ 2 2(l+2) + x 2) σ/2 γ /2 γ γ β c ( ) β ( c σ,γ 2 2(l+2) + 1 ) σ/2 γ /2 + γ γ σ c. σ<γ κ ( ( β )c σ,γ 2 2(l+2) + 1 ) σ/2 γ /2 γ 16 This implies for all l N 0 M l (2 l+2 ) W2 κ (R n ) = For j = 0 we have to define φ 0 as β κ D ( β M l (2 l+2 ) ) L 2 (R n ) 1/2 2 <. φ 0 (x) = 1 if x 2 and supp φ 0 {ξ R n : ξ 4}. By a similar calculation as above (j = 0) we can show for all j N 0 ( w j 2 σj ξ σ ϕ j ˆf) L p (R n ) c wj (ϕ j ˆf) L p (R n ), (2.27) where c is independent of j N 0. Now, taking the l q quasi-norm in (2.22) leads to This proves the theorem. Iσ f Bpq s σ,mloc (R n, w) c f B pq (R n, w). The next theorem is a characterization of Bpq (R n, w) by derivatives. We follow closely Theorem in [Tri83] and use the weighted Fourier multiplier theorem

16 Theorem 2.16: Let s R, w = (w j ) j N0 Wα α 1,α 2, 0 < p, q and let m N 0. Then D β f B s m,mloc pq (R n, w) and f B s m,mloc pq (R n, w) n + m f x m Bs m,mloc pq (R n, w) i β m are equivalent quasi-norms on Bpq (R n, w). Proof: First Step: We define φ S(R n ) as φ(x) = 1 for 1/2 x 2 and supp φ {x R n : 1/4 x 4}. (2.28) Then for all j N F 1 ϕ j FD β f = cf 1 φ(2 j )x β ϕ j Ff x β = cf 1 φ(2 j ) (1 + x 2 ) m/2 FF 1 ϕ j (1 + x 2 ) m/2 Ff. Now, using (2.19) with b = 2 j+2 and M = φ(2 j ) we get for ( ) (1+ x 2 ) m/2 1 κ > n 1 + α min(1,p) 2 w j F 1 ϕ j FD β f L p (R n ) = c w jf 1 φ(2 j ) x β (1 + x 2 ) m/2 FF 1 ϕ j (1 + x 2 ) m/2 Ff L p(r n ) φ(4 ) (2 j+2 x) β (1 + 2 j+2 x 2 ) m/2 W 2 κ (R n ) wj F 1 ϕ j (1 + x 2 ) m/2 Ff Lp (R n ), (2.29) (2 for all 0 < p and j N. Since j+2 x) β < c for β m and φ S(R n ), we (1+ 2 j+2 x 2 ) m/2 get φ(4 ) (2 j+2 x) β (1 + 2 j+2 x 2 ) m/2 W 2 κ (R n ) c κ,m independently of j N. For j = 0 we obtain (2.29) by similar arguments and φ 0 S(R n ) with φ 0 (x) = 1 for x 2 and supp φ 0 {x R n : x 4}. Hence, we have for all j N 0 wj F 1 ϕ j FD β f Lp (R n ) cκ,m wj F 1 ϕ j (1 + x 2 ) m/2 Ff Lp (R n ), where the constant c κ,m is independent of j N 0 and β m. Finally, multiplying by 2 j(s m) and applying the l q quasi-norm in respect to j, we get for all β m D β f B s m,mloc pq (R n, w) c Im f Bpq s m,mloc (R n, w) c f B pq (R n, w). (2.30) Second Step: Now, we assume that f Bpq s m,mloc (R n, w) and m f B x m pq s m,mloc (R n, w) i for i = 1,..., n. We want to show that f belongs to Bpq (R n, w). Theorem 2.15 shows f B pq (R n, w) c Im f Bpq s m,mloc (R n, w) ( = c 2 j(s m)q wj F 1 (1 + x 2 ) m/2 ϕ j Ff Lp (R n ) ) q. (2.31) j=0 x β i=1 14

17 From [Tri83] we adopt the construction of functions ϱ 1,..., ϱ n C (R n ) from the third step of the proof of Theorem If m is even, then ϱ i (x) = 1 has the desired properties but for odd m the situation is a bit more complicated. These functions fulfill 1 + n ϱ i (x)x m i c(1 + x 2 ) m/2 for all x R n. i=1 Thus we have M(x) := (1 + x 2 ) m/2 [1 + n ϱ i (x)x m i i=1 ] 1 c for all x R n. With the function φ S(R n ) as in (2.28) and (2.19) with b = 2 j+2 and κ > 0 large enough we get for all j N wj F 1 (1 + x 2 ) m/2 ϕ j Ff Lp (R n ) [ ] = w jf 1 M(x)φ(2 j )FF n 1 ϕ j 1 + ϱ i (x)x m i Ff L p(r n ) i=1 c M(2 j+2 )φ(4 ) W κ 2 (R n ) [ ] n w jf 1 ϕ j 1 + ϱ i (x)x m i Ff L p(r n ). From the properties of M and φ S(R n ) we get that the Sobolev space norm is bounded, independent of j N. For j = 0 we can do the same calculation with φ 0 instead of Φ(2 j ). By an analogous procedure as above we can use (2.19) again and obtain wj F 1 (1 + x 2 ) m/2 ϕ j Ff Lp (R n ) c w j F 1 ϕ j [1 + c w j F 1 ϕ j Ff L p (R n ) + c c wj F 1 ϕ j Ff Lp (R n ) + c ( for κ > n 1 min(1,p) 1 2 i=1 i=1 ] n ϱ i (x)x m i Ff L p(r n ) n w j F 1 ϱ i (x)φ(2 j )FF 1 x m i ϕ j Ff L p (R n ) i=1 n ϱi (2 j+2 x)φ(4 ) W κ 2 (R n ) wj F 1 x m i ϕ j Ff Lp (R n ), i=1 ) + α. Since ϱ i C (R n ) and φ S(R n ) we get that the Sobolev space norm is bounded by a constant independently of j N. If j = 0, then we use the usual replacement by φ 0. Finally, we have wj F 1 (1 + x 2 ) m/2 ϕ j Ff Lp (R n ) c wj F 1 ϕ j Ff Lp (R n ) + c wj F 1 x m i ϕ j Ff Lp (R n ) = c w j F 1 ϕ j Ff L p (R n ) + c w jf 1 ϕ j F m f L p(r n ) x m i 15

18 for all j N 0. Using (2.31) we get f B pq (R n, w) c f B s m,mloc pq (R n, w) n + c m f x m i Finally, this and (2.30) prove the theorem. i=1 pq (R n, w). Bs m,mloc Now, we present a characterization of the 2-microlocal spaces with the special weight sequence w j (x) = (1 + 2 j dist(x, U)) s for U R n. Definition 2.17: Let {ϕ j } j N0 Φ(R n ) be a resolution of unity. Let U R n and s R be fixed. Further, let 0 < p, 0 < q and s R. Then we define { } Bpq s,s (R n, U) = f S : f Bpq s,s (R n, U) <, where ( ) f Bpq s,s (R n 1/q, U) = 2 jsq (1 + 2 j dist(x, U)) s (ϕ j ˆf) L p (R n ) q, with the usual modifications if p or q are equal to infinity. j=0 Remark 2.18: In slight abuse of notation we write Bpq s,s (R n, x 0 ) if U = {x 0 } R n. If U = {x 0 } R n then B (R s,s n, x 0 ) = Cx s,s 0, see [JaMey96, Definition 1.1]. For p = q = 2 we get B s,s 22 (R n, x 0 ) = Hx s,s 0. Both types are the 2-microlocal spaces introduced by Bony [Bo84] and Jaffard [Ja91]. Corollary 2.19: Let s, s R and let U R n. Further, let 0 < p, q and m N 0, then the following statements are equivalent (i) f B s,s pq (R n, U) (ii) D β f Bpq s m,s (R n, U) for all 0 β m (iii) f Bpq s m,s (R n, U) and m f x m i B s m,s pq (R n, U) for each i = 1,..., n. Remark 2.20: This corollary coincides essentially with Corollary 3.1 in [Mey97] for the special case p = q = and U = {x 0 } R n. 16

19 3 Local Means 3.1 Preliminaries In this part we present the main technical tool. We characterize the spaces Bpq (R n, w) by so called local means. We follow closely the method presented by Rychkov [Ry99] and by Vybiral [Vyb06]. Recall the specific system {ϕ j } j N0 S(R n ) which we fix now for the rest of our work: Let ϕ 0 S(R n ) with { 1, if x 1 ϕ 0 (x) =. 0, if x 2 We put ϕ(x) = ϕ 0 (x) ϕ 0 (2x) and ϕ j (x) = ϕ(2 j x) for all x R n and all j N The Peetre maximal operator The Peetre maximal operator was introduced by Jaak Peetre in [Pe75]. The operator assigns to each system {ψ j } j N0 S(R n ), to each distribution f S (R n ) and to each number a > 0 the following quantities sup y R n (ψ k ˆf) (y) k (y x) a, x Rn,k N 0. (3.1) Since ψ k S(R n ) for all k N 0 the operator is well-defined because (ψ k ˆf) = c(ψ k f) is well-defined for every distribution f S (R n ). Given a system {ψ k } k N0 S(R n ), we set Ψ k = ˆψ k S(R n ) and reformulate the Peetre maximal operator (3.1) for every f S (R n ) and a > 0 as Helpful lemmas (Ψ (Ψ k f)(y) kf) a (x) = sup y R n k (y x), x a Rn and k N 0. (3.2) Before proving the local means characterization we present some technical lemmas without proof, which appeared in the papers of Rychkov [Ry99] and Vybiral [Vyb06]. The first lemma describes the use of the so called moment conditions. Lemma 3.1: Let g, h S(R n ) and let M 1 be an integer. Suppose that (D β ĝ)(0) = 0 for 0 β M. (3.3) 17

20 Then for each N N 0 there is a constant C N such that where g t (x) = t n g(x/t). sup (g t h)(z) (1 + z N ) C N t M+1, for 0 < t < 1, (3.4) z R n Remark 3.2: If M = 1, the condition (3.3) is empty. The next lemma is a discrete convolution inequality which we will need later on. Lemma 3.3: Let 0 < p, q and δ > 0. Let {g k } k N0 be a sequence of non-negative measurable functions on R n and let G ν (x) = 2 ν k δ g k (x), x R n, ν N 0. (3.5) k=0 Then there is some constant c = c(p, q, δ) such that G k l q (L p ) c g k l q (L p ). (3.6) Lemma 3.4: Let 0 < r 1 and let {γ ν } ν N0, {β ν } ν N0 in (0, ). Assume that for some N 0 N 0, be two sequences taking values γ ν = O(2 νn 0 ), for ν. Furthermore, we assume that for any N N γ ν C N k=0 2 kn β k+ν γ 1 r k+ν, ν N 0, C N < holds, then for any N N γ r ν C N k=0 2 kn β k+ν, ν N 0 (3.7) holds with the same constants C N. The proofs of the lemmas can be found in [Ry99] and [Vyb06] Comparison of different Peetre maximal operators In this subsection we present an inequality between different Peetre maximal operators. We start with two given functions ψ 0, ψ 1 S(R n ). We define ψ j (x) = ψ 1 (2 j+1 x), for x R n and j N. (3.8) Furthermore, for all j N 0 we write Ψ j = ˆψ j and in an analogous manner we define Φ j from two starting functions φ 0, φ 1 S(R n ). Using this notation we are ready to formulate the theorem. 18

21 Theorem 3.5: Let w = {w j } j N0 W α α 1,α 2, 0 < p, q and s, a R with a > 0. Moreover, let R + 1 N 0 with R + 1 > s + α 2, and for some ε > 0 D β ψ 1 (0) = 0, 0 β R (3.9) φ 0 (x) > 0 on {x R n : x < ε} (3.10) φ 1 (x) > 0 on {x R n : ε/2 < x < 2ε} (3.11) then 2 ks (Ψ kf) a w k lq (L p ) c 2 ks (Φ kf) a w k lq (L p ) (3.12) holds for every f S (R n ). Proof: We define the functions {λ j } j N0 by λ j (x) = ϕ ( 2x ) j ε φ j (x). It follows from the Tauberian conditions (3.10) and (3.11) that they satisfy λ j (x)φ j (x) = 1, x R n (3.13) j=0 λ j (x) = λ 1 (2 j+1 x), x R n, j N (3.14) supp λ 0 {x R n : x ε} and supp λ 1 {x R n : ε/2 x 2ε}. (3.15) Furthermore, we denote Λ k = ˆλ k for k N 0 and obtain together with (3.13) the following identities (convergence in S (R n )) f = Λ k Φ k f, Ψ ν f = k=0 We have (Ψ ν Λ k Φ k f)(y) R n Ψ ν Λ k Φ k f. (3.16) k=0 (Ψ ν Λ k )(z) (Φ k f)(y z) dz (Φ kf) a (y) (Ψ ν Λ k )(z) (1 + 2 k z a )dz (3.17) R n =: (Φ kf) a (y)i ν,k, where I ν,k := R n (Ψ ν Λ k )(z) (1 + 2 k z a )dz. 19

22 According to Lemma 3.1 we get { 2 (k ν)(r+1),k ν I ν,k c 2 (ν k)(a+ s +1+α 1),ν k. (3.18) Namely, we have for 1 k < ν with the change of variables 2 k z z I ν,k = 2 n (Ψ ν k Λ 1 ( /2))(z) (1 + z a )dz R n c sup z R n (Ψ ν k Λ 1 ( /2))(z) (1 + z ) a+n+1 c2 (k ν)(r+1). Similarly, we get for 1 ν < k with the substitution 2 ν z z I ν,k = 2 n (Ψ 1 ( /2) Λ k ν)(z) (1 + 2 k ν z a )dz R n c2 (ν k)(m+1 a). M can be taken arbitrarily large because Λ has infinite vanishing moments. Taking M > 2a + s + α 1 we derive (3.18) for the cases k, ν 1 with k ν. The missing cases can be treated separably in an analogous manner. The needed moment conditions are always satisfied by (3.9) and (3.15). The case k = ν = 0 is covered by the constant c in (3.18). Furthermore, we have (Φ kf) a (y) (Φ kf) a (x)(1 + 2 k (x y) a ) (Φ kf) a (x)(1 + 2 ν (x y) a ) max(1, 2 (k ν)a ). We put this into (3.17) and get { (Ψ ν Λ k Φ k f)(y) sup c(φ 2 (k ν)(r+1),k ν y R n ν (x y) kf) a a (x) 2 (ν k)( s +1+α 1),k ν. Multiplying both sides with w ν (x) and using { 2 (k ν)( α 2),k ν w ν (x) w k (x) 2 (ν k)( α 1),k ν, (3.19) leads us to { (Ψ ν Λ k Φ k f)(y) sup w y R n ν (x y) a ν (x) c(φ 2 (k ν)(r+1 α 2),k ν kf) a (x)w k (x) 2 (ν k)( s +1),k ν. This inequality together with (3.16) gives for δ := min(1, R + 1 α 2 s) > 0 2 νs (Ψ νf) a (x)w ν (x) c 2 k ν δ 2 ks (Φ kf) a (x)w k (x), x R n. k=0 Then, Lemma 3.3 yields immediately the desired result. 20

23 Remark 3.6: The conditions (3.9) are usually called moment conditions while (3.10) and (3.11) are the so called Tauberian conditions. If R = 1 in Theorem 3.5, then there are no moment conditions on ψ Boundedness of the Peetre maximal operator We will present a theorem which describes the boundedness of the Peetre maximal operator. We use the same notation introduced in the beginning of the last subsection. Especially, we have the functions ψ k S(R n ) and Ψ k = ˆψ k S(R n ) for all k N 0. Theorem 3.7: Let {w k } k N0 W α α 1,α 2, a, s R and 0 < p, q. For some ε > 0 we assume ψ 0, ψ 1 S(R n ) with ψ 0 > 0 on {x R n : x < ε} (3.20) ψ 1 > 0 on {x R n : ε/2 < x < 2ε}. (3.21) If a > n p + α, then 2 ks (Ψ kf) a w k lq (L p ) c 2 ks (Ψ k f)w k lq (L p ) (3.22) holds for all f S (R n ). Proof: As in the last proof we find the functions {λ j } j N0 with the properties (3.14)- (3.15) and λ k (2 ν x)ψ k (2 ν x) = 1 for all ν N 0. (3.23) k=0 Instead of (3.16) we get the identity Ψ ν f = Λ k,ν Ψ k,ν Ψ ν f, (3.24) k=0 where Λ k,ν (ξ) = [λ k (2 ν )] (ξ) = 2 νn Λ k (2 ν ξ) for all ν, k N 0. The Ψ k,ν are defined similarly. For k 1 and ν N 0 we have Ψ k,ν = Ψ k+ν and with the notation { ψ 0 (2 ν x), if k = 0 σ k,ν (x) = ψ ν (x), otherwise we get ψ k (2 ν x)ψ ν (x) = σ k,ν (x)ψ k+ν (x). Hence, we can rewrite (3.24) as Ψ ν f = Λ k,ν ˆσ k,ν Ψ k+ν f. (3.25) k=0 21

24 For k 1 we get from Lemma km (Λ k,ν ˆσ k,ν )(z) = 2 νn (Λ k Ψ)(2 ν z) C M 2 νn (1 + 2 ν z a ) (3.26) for all k, ν N 0 and arbitrarily large M N. For k = 0 we get the estimate (3.26) by using Lemma 3.1 with M = 1. This together with (3.25) gives us (Ψ ν f)(y) C M 2 νn k=0 R n 2 km (1 + 2 ν (y z) a ) (Ψ k+ν f)(z) dz. (3.27) For fixed r (0, 1] we divide both sides of (3.27) by (1 + 2 ν (x y) a ) and we take the supremum in respect to y R n. Using the inequalities (1 + 2 ν (y z) a )(1 + 2 ν (x y) a ) c(1 + 2 ν (x z) a ), and we get (Ψ k+ν f)(z) (Ψ k+ν f)(z) r (Ψ k+νf) a (x) 1 r (1 + 2 k+ν (x y) a ) 1 r (1 + 2 k+ν (x z) a ) 1 r (1 + 2 ν (x y) a ) 2 ka (1 + 2 k+ν (x y) a ) r, (Ψ νf) a (x) C M 2 k(m+n a) (Ψ k+νf) a (x) 1 r k=0 R n 2 (k+ν)n (Ψ k+ν f)(z) r dz. (3.28) (1 + 2 k+ν (x y) a ) r Now, we apply Lemma 3.4 with γ ν = (Ψ νf) a (x), β ν = R n 2 νn (Ψ ν f)(z) r (1 + 2 ν (x y) a ) r dz, ν N 0 N = M + n a, C N = C M + n a and N 0 giving the order of the distribution f. By Lemma 3.4 we obtain for every N N, x R n and ν N 0 (Ψ νf) a (x) r C N 2 knr 2 (k+ν)n (Ψ k+ν f)(z) r dz. (3.29) (1 + 2 k+ν (x y) a ) r k=0 R n We point out that (3.29) holds also for r > 1, where the proof is much simpler. We only have to take (3.27) with a + n instead of a, divide both sides by (1 + 2 ν (x y) a ) and apply Hölder s inequality with respect to k and then z. Multiplying (3.29) by w ν (x) r we derive with the properties of our weight sequence (Ψ νf) a (x) r w ν (x) r C N k=0 2 k(n α 1)r R n 2 (k+ν)n (Ψ k+ν f)(z) r w k+ν (z) r (1 + 2 k+ν (x y) a α ) r dz, (3.30) 22

25 for all x R n, ν N 0 and all N N. Now, choosing r > 0 with < r < p the function n a α 1 (1 + z ) r(a α) L 1(R n ) and by the majorant property of the Hardy-Littlewood maximal operator (see [StWe71], Chapter 2) it follows (Ψ νf) a (x) r w ν (x) r C N 2 k(n α1)r M( Ψ k+ν f r wk+ν)(x) r. (3.31) k=0 We choose N > 0 such that N > s + α 1 and denote From (3.31) we derive g k (x) = 2 krs M( Ψ k f r w r k)(x). G ν (x) = (Ψ νf) a (x) r w ν (x) r C 2 k(n α1)r g k (x). So, for 0 < δ < N + s α 1, we can apply Lemma 3.3 with the l q/r (L p/r ) norm. This gives us 2 krs (Ψ kf) a (x) r w k (x) r l q/r (L p/r ) c 2 krs M( Ψ k f r w r k)(x) l q/r (L p/r ) (3.32) Rewriting the left hand side of (3.32) and using the scalar Hardy-Littlewood Theorem [FeS71] (we recall r < p) on the right hand side, we finally get 2 ks (Ψ kf) a w k lq (L p ) c 2 ks (Ψ k f)w k lq (L p ), and the proof is complete. 3.2 Local means characterization In this section we only combine the two previous subsections to derive the usual local means characterization as in [Tri92] and [Ry99]. The Peetre maximal operator was defined in section and the functions ψ 0, ψ 1 belong to S(R n ). Theorem 3.8: Let w = {w k } k N0 W α α 1,α 2, 0 < p, q and let s, a R, R + 1 N 0 with a > n p + α and R + 1 > s + α 2. If and k ν D β ψ 1 (0) = 0, for 0 β R, (3.33) ψ 0 (x) > 0 on {x R n : x < ε} (3.34) ψ 1 (x) > 0 on {x R n : ε/2 < x < 2ε} (3.35) for some ε > 0, then f B pq (R n, w) 2 ks (Ψ k f)w k lq (L p ) 2 ks (Ψ kf) a w k lq (L p ) holds for all f S (R n ). 23

26 Remark 3.9: (a) The proof of Theorem 3.8 is is just a reformulation of Theorem 3.5 and Theorem 3.7. (b) If R = 1, then there are no moment conditions (3.33) on ψ 1. (c) In [Vyb06] the proof for the local means characterization was made for the dominating mixed smoothness case. It is not hard to see that we can also generalize our weight functions in the following sense: We can use tensor products of weights, i.e. w k (x) = n wk(x i i ) i=1 where the one-dimensional measurable functions wk i (t) have to satisfy the weight conditions 0 < w i k(t) C i w i k(r)(1 + 2 k t r ) α i, 2 αi 1 w i k (t) w i k+1(t) 2 αi 2 w i k (t). Finally, we get the weight class W α α 1,α 2 with α = (α 1,..., α n ) and α 1 = (α 1 1,..., α n 1), α 2 = (α 1 2,..., α n 2). The local means characterization with this weights can be seen directly if one compares the above Theorem with Theorem 1.23 in [Vyb06]. Next we reformulate the Theorem 3.8 in the sense of [Tri92]. Let B = {x R n : x < 1} be the unit ball and k S(R n ) a function with support in B. For a distribution f S (R n ) the corresponding local means are defined by (at least formally) ( ) y x k(t, f)(x) = k(y)f(x + ty)dy = t n k f(y)dy, x R n, t > 0. (3.36) t R n Let k 0, k 0 S(R n ) be two functions with and R n supp k 0 B, supp k 0 B, (3.37) For N N 0 we define the iterated Laplacian ˆk 0 (0) 0, ˆk0 (0) 0. (3.38) k(y) := N k 0 (y) = ( n 2 y 2 j=1 j ) N k 0 (y), y R n. (3.39) 24

27 It follows easily that ǩ(x) = x 2N ǩ 0 (x) and that implies (3.40) D β ǩ(0) = 0 for 0 β < 2N. (3.41) Using this notation we come to the usual local means characterization. Theorem 3.10: Let w = {w k } k N0 Wα α 1,α 2, 0 < p, q, s R. Furthermore, let N N 0 with 2N > s + α 2 and let k 0, k 0 S(R n ) and the function k be defined as above. Then ( k 0 (1, f)w 0 L p (R n ) + 2 jsq k(2 j, f)w j Lp (R n ) ) 1/q q j=1 f B pq (R n, w) (3.42) holds for all f S (R n ). Proof: We put ψ 0 = k 0, ψ 1 = k ( /2). Then the Tauberian conditions (3.34) and (3.35) are satisfied and due to (3.41) also the moment conditions (3.33) are fulfilled. If we define ψ j for j N 0 as in (3.8), then we get (ψ j ˆf) (x) = c(ψj f)(x) = c For j = 0 we get Fψ 0 = k 0 and for j 1 we obtain R n (Fψ j )(y)f(x + y)dy. (3.43) (Fψ j )(y) = (Fψ 1 (2 j+1 ))(y) = 2 (j 1)n (Fψ 1 )(2 j 1 y) = 2 jn k(2 j y). This and the equation (3.43) lead to (ψ j ˆf) (x) = c2 jn R n k(2 j y)f(x + y)dy = ck(2 j, f)(x), j N 0, x R n. Together with Theorem 3.8 the proof is complete. Remark 3.11: If we take w j 1 for all j N 0, we obtain the usual Besov spaces. If we now compare our result with section in [Tri92], we get an improvement with respect to N N 0. The condition in [Tri92] is 2N > max(s, σ p ) where σ p = max(0, n(1/p 1)). We derived 2N > s in Theorem 3.10 (α 2 = 0 for w j 1) wich seems to be more natural. Furthermore, we proved the equivalence of the (quasi-)norms for all f S (R n ) by this method where in [Tri92] the equivalence does only hold for f B s pq(r n ). 25

28 For the last modification of the local means representation we introduce some necessary notation. For ν N 0, m Z n we denote by Q νm the cube centred at the point 2 ν m = (2 ν m 1,..., 2 ν m n ) with sides parallel to coordinate axes and of length 2 ν. Hence Q νm = {x R n : x i 2 ν m i 2 ν 1, i = 1,..., n}, ν N 0, m Z n. (3.44) If γ > 0, then γq νm denotes a cube concentric with Q νm with sides also parallel to coordinate axes and of length γ2 ν. Defining the Peetre maximal operator by (3.2), we get (Ψ νf) a (x) c sup x y γq νm (Ψ ν f)(y), ν N 0, x R n, where the constant c only depends on a > 0, γ > 0 and does not depend on x R n and ν N 0. With this simple observation we get immediately the following conclusion of Theorem 3.8. Theorem 3.12: Let w = {w k } k N0 Wα α 1,α 2, 0 < p, q, s R. For N N 0 with 2N > s + α 2 let k 0, k 0, k be as in Theorem Then for every γ > 0 f B pq (R n, w) sup k 0 (1, f)(y) (x y) γq 0,0 L p(r n, w 0 ) ( q) 1/q + 2 jsq sup k(2 j, f)(y) (x y) γq j,0 L p(r n, w j ), holds for all f S (R n ). j=1 (3.45) 26

29 4 Further properties 4.1 Embedding theorems General embeddings For the spaces Bpq (R n, w) introduced above we want to show some general embedding theorems. We follow closely [Tri83], see Proposition 2.3.2/2 and Theorem We say a Banach space A 1 is continuously embedded in another Banach space A 2, A 1 A 2, if A 1 A 2 and there is a c > 0 such that a A 2 c a A 1 for all a A 1. First, we present an embedding theorem which connects the two-microlocal Besov spaces with the usual weighted Besov spaces [EdTri96]. We denote by Bp,q(R s n, α) the weighted Besov spaces, with respect to the weight x α = (1 + x 2 ) α/2 for α R. Theorem 4.1: Let w W α α 1,α 2, s R and 0 < p, q, then B s+α 2 pq (R n, α) Bpq (R n, w) B s α 1 pq (R n, α). Proof: Using the properties (2.2) and (2.3) we obtain w j (x) 2 jα 2 w 0 (x) C2 jα 2 w 0 (0)(1 + x 2 ) α/2 w j (x) 2 jα 1 w 0 (x) 1 C 2 jα 1 w 0 (0)(1 + x 2 ) α/2 for all x R n and every j N 0. It follows immediately c 1 2 jα 1 (1 + x 2 ) α/2 (ϕ j ˆf) L p (R n ) w j (ϕ j ˆf) L p (R n ) c 2 2 jα 2 (1 + x 2 ) α/2 (ϕ j ˆf) L p (R n ) and therefrom the theorem. The following is an easy consequence of the above theorem and B s pq(r n ) L max(1,p) (R n ) for s > σ p = n(1/p 1) +. Corollary 4.2: Let and w W α α 1,α 2 and let 0 < p, q, then for s > σ p + α 1 B pq (R n, w) L max(1,p) (R n, x α ). We need a special weighted version of Nikol skij s inequality. Proposition 4.3: Let ϱ be an admissible weight satisfying (a, b > 0) 0 < ϱ(x) C ϱ ϱ(y)(1 + ab x y ) α ϱ for all x, y R n. Further let 0 < p q and B b = {x R n : x b}. If β N n 0 is a multi-index, then there exists a positive constant c such that ϱd β ϕ L q (R n ) cb β +n ( 1 p 1 q) ϱϕ Lp (R n ) (4.1) holds for all ϕ L p (R n, ϱ) with supp ˆϕ B b where the c is independent of b > 0. 27

30 Proof: We substitute ϱ(x) := ϱ(b 1 x) ϕ(x) := ϕ(b 1 x). and Now the weight ϱ satisfies 0 < ϱ(x) C ϱ ϱ(y)(1 + x y ) α ϱ for all x, y R n. (4.2) Further, the function ϕ L p (R n, ϱ) with supp ϕ B 1. Now, we can apply Proposition in [SchmTri87]. After a resubstitution we derive the above statement (4.1). From Remark 2 in [SchmTri87, 1.4.2] we get that the constant c in (4.1) is independent of b and of the choice of the weight function (it depends only on C ϱ and α ϱ ). Theorem 4.4: Let s R and w, ϱ W α α 1,α 2 with w j(x) ϱ j (x) c for all j N 0 and x R n. (i) For 0 < p and 0 < q 1 q 2 we have Bpq 1 (R n, ϱ) Bpq 2 (R n, w). (4.3) (ii) If 0 < p, 0 < q 1, 0 < q 2 and ε > 0, then Bpq 1 (R n, ϱ) Bpq s ε,mloc 2 (R n, w). (4.4) (iii) For 0 < p 1 p 2, 0 < q and < s 2 s 1 < with s 1 n s 2 n we have B s 1,mloc p p 1 p 1 q (R n, ϱ) B s 2,mloc p 2 q (R n, w). (4.5) 2 Proof: The proof of 4.3 and 4.4 is the same as in Proposition 2.3.2/2 in [Tri83] one only has to plug in the weight sequence. To prove 4.5 we use Proposition 4.3 with b = 2 j+1, ϱ = w j and ϕ = (ϕ j ˆf) for each j N 0. Now, the substituted weight functions w j satisfy a condition as in (4.2), where the constants C wj and α wj do not depend on j N 0. Hence, Proposition 4.3 gives ( w j (ϕ j ˆf) L p2 (R n ) c2 jn 1 p 1 1 ) p wj 2 (ϕ j ˆf) L p1 (R n ), for all j N 0, where the constant c is independent of j N 0. After multiplying the inequality by 2 j(s 2 n/p 2 ) and using the conditions on s 1, s 2, p 1, p 2 and the weight sequences, we get 2 js 2 w j (ϕ j ˆf) L p2 c 2 js 1 ϱ j (ϕ j ˆf) L p1. Finally we apply the l q quasi-norm to find the desired result. With minor modifications we have an analogous theorem to Theorem in [Tri83]. 28

31 Theorem 4.5: Let w W α α 1,α 2, s R and 0 < p, q, then S(R n ) B pq (R n, w) S (R n ) holds. (4.6) If s R and 0 < p, q <, then S(R n ) is dense in Bpq (R n, w). Proof: The proof is essentially the same as in [Tri83, 2.3.3]. One only has to bring in the weight sequence and use its properties (2.2) and (2.3). Also the weighted Nikol skij inequality (Proposition 4.3) and section 1.5 in [SchmTri87] has to be used as a replacement for the unweighted ones in the proof in [Tri83] Embeddings for 2-microlocal Besov spaces In this subsection we present some special embedding theorems for the weight sequence of 2-microlocal weights, w j (x) = (1 + 2 j dist(x, U)) s for fixed U R n and s R. The spaces Bpq s,s (R n, U) were defined in Definition As shown in Example 2.5, the weight sequence belongs to W s max(0, s ),max(0,s ). We recall the spaces Bs pq(r n, α), with respect to the weight x α = (1 + x 2 ) α/2 for α R. An easy consequence of Theorem 4.1 and Theorem 4.4 is the following. Theorem 4.6: Let s R and let 0 < p, q. (i) For s R and U = {x 0 } R n we have (ii) For s 0 and U V R n we have B s,s pq (R n, x 0 ) C s n p,s x 0. Bpq s+s (R n, s ) Bpq s,s (R n, U) Bpq s,s (R n, V ) Bpq(R s n, s ). (iii) For s 0 and U V R n we have B s,s pq (R n, U) B s,s (iv) For s t and U R n we have pq (R n, V ) Bpq(R s n ) Bpq s, s B s,s pq (R n, U) B s,t pq (R n, U). (R n, V ) B s, s pq (R n, U). Corollary 4.7: Let s s 0 and let 0 < p, q. Further, if U R n, then B s,s pq (R n, U) B s pq(r n ) B s, s pq (R n, U). Remark 4.8: Corollary 4.7 coincides partially with Proposition 1.3 (1) and (2) in [JaMey96] for p = q = and U = {x 0 } and with Theorem 3.2 in [MoYa04] with p = q 1 and U be an open subset or U = {x 0 } R n. In the mentioned papers local versions of Bpq s,s (R n, U) have been used to treat further kinds of embeddings in the scale of Bpq s,s (R n, U). 29

32 4.2 Pointwise multipliers Let g be a bounded function on R n. We ask, under which conditions the mapping f gf makes sense and generates a bounded operator in a given space Bpq (R n, w). We follow closely [Tri92, 4.2.2] and adapt the proofs to our situation. First, we prove a lemma which is important for pointwise multipliers. Lemma 4.9: Let w Wα α 1,α 2 and let 0 < p, q. Then for s > n + α + α p 1 and all γ > 0 there is a constant c γ > 0 such that w 0( ) sup y γ f(y) L p(r n ) c γ f B pq (R n, w) holds for all f S (R n ). Proof: Let {ϕ j } j N0 Φ(R n ) be the chosen resolution of unity from the beginning of the chapter. Then we get for arbitrary ε > 0 w 0( ) sup f(y) y γ L p(r n ) c 2 jε w 0 ( ) sup (ϕ j ˆf) (y) y γ L p(r n ). For all a > 0 we have j=0 sup (ϕ j ˆf) (y) c2 ja (ϕ j ˆf) (x z) sup x y γ z R n j z a where the constant only depends on γ > 0. Using the property (2.3) of the weight sequence and Theorem 3.8, we obtain for arbitrary a > n/p + α and ε > 0 w 0( ) sup f(y) y γ L p(r n (ϕ ) c j f) a B a+α 1 +ε,mloc p1 (R n, w) c f B a+α 1 +ε,mloc p1 (R n, w) for s > n p + α + α 1 and f S (R n ). c f B pq (R n, w), Let k 0, k S(R n ) and k(t, f) be the same functions as in (3.36)-(3.39). For g C m (R n ) we study gf where f Bpq (R n, w). First, we prove the theorem and after that we discuss, how gf has to be understood. Theorem 4.10: Let w W α α 1,α 2, s R and let 0 < p, q. If m N is sufficiently large, then there exists a positive number c m such that gf Bpq (R n, w) c m β m for all g C m (R n ) and all f S (R n ). D β g L (R n ) f B pq (R n, w) (4.7) 30

33 Proof: First Step: Firstly, we prove the theorem under the additional assumption s > n p + α + α 1. We use the Taylor expansion of g C m (R n ) g(x) = β m 1 D β g(y) (x y) β + β! β =m for θ (0, 1). By (3.36) we have k(2 j, f)(x) = k(y)f(x + 2 j y)g(x + 2 j y)dy = β m 1 R n D β g(x) 2 j β β! R n y β k(y)f(x + 2 j y)dy + 2 jm D β g(y + θ(x y)) (x y) β, (4.8) β! R n k(y)r m (x, 2 j, y)f(x + 2 j y)dy, where the remainder term in Taylor s expansion, r m (x, 2 j, y), is in any case uniformly bounded. If we choose N N 0 in (3.39) sufficiently large, for each β m 1 the function k β (y) = y β k(y) is a new kernel for which Theorem 3.10 holds. Thus, choosing m > s + α 2 and using Theorem 3.10 for every β m 1 we obtain ( 2 jsq wj k(2 j, f) Lp (R n ) ) 1/q q c D β g L (R n ) f B pq (R n, w) j=1 + c β m β m 1 D β g L (R n ) w 0( ) sup y 1 f(y) L p(r n ) Now, Lemma 4.9 with γ = 1 proves the theorem provided s > n + α + α p 1. Second Step: Let < s n +α+α p 1 and let l N with s+2l > n +α+α p 1. From the lift property (see Section 2.3) we get, that any f Bpq (R n, w) can be represented as f = (id +( ) l )h, with h Bpq s+2l,mloc (R n, w) and h B s+2l,mloc pq (R n, w) f B pq (R n, w). (4.9) We have gf = (id +( ) l )gh + β <2l D β (g β h), where each g β is a sum of terms of the type D β g with β 2l. Now, Theorem 2.16 shows gf B pq (R n, w) c gβ h Bpq s+2l,mloc (R n, w). β 2l If l N is sufficiently large, that is m 2l > s + 2l + α 2, we can apply the first step and and obtain gf Bpq (R n, w) c D β g L (R n ) h Bpq s+2l,mloc (R n, w). β m Finally, (4.9) proves the theorem. 31

34 Remark 4.11: The interpretation of g f is a bit sophisticated. We approximate f and g by smooth functions, f j and g j. The limit of g j f j exists in Bpq (R n, w), see [Tri92, Remark 1/4.2.2], and we define g f = lim j g j f j, where g j f j has to be understood in the usual pointwise sense, as limit element. For a more detailed discussion of this procedure we refer also to [RuSi96, Chapter 4]. 4.3 Invariance under Diffeomorphisms In this section we show that the spaces Bpq (R n, w) are invariant under diffeomorphisms. The result and the proof are closely related to Section 4.3 in[tri92]. Let m N, then we call an isomorphism ψ : R n R n an m-diffeomorphism if the components ψ j (x) of ψ(x) = (ψ 1 (x),..., ψ n (x)) have classical derivatives up to the order k and the functions D β ψ j (x) are bounded for all 0 < β m, 1 j n and all x R n. Furthermore, the Jacobian matrix ψ has to fulfill det ψ (x) d > 0 for all x R n. If y = ψ(x) is a m-diffeomorphism for every m N, then it is called diffeomorphism. We want to prove that f f ψ is a linear and bounded operator in all spaces B pq (R n, w). If ψ is a diffeomorphism, then f ψ(x) = f(ψ(x)) (4.10) makes sense for all f S (R n ). If ψ is only an m-diffeomorphism, then (4.10) has to be understood as an approximation procedure with smooth functions (see also Remark 4.11). In the proof we use the local means characterization in the form of Theorem First of all, we have to prove two lemmas which will be useful later on. We need a modification of Theorem Therefore, let k 0 and k 0 be kernels in the sense of (3.37)-(3.39) with N N 0 large enough and a(x) be an n n matrix with real-valued continuous entries a ij (x), where x R n and i, j {1,..., n}. Further, there exist two numbers d, d > 0 with a ij (x) d for all x R n, i, j {1,..., n} and (4.11) det a(x) d > 0 for all x R n. (4.12) Since, y ya(x) is an isomorphic mapping for fixed x R n we can generalize (3.36) by k(a, t, f)(x) = k(y)f(x + ta(x)y)dy. (4.13) R n Lemma 4.12: Let w Wα α 1,α 2, s R and let0 < p, q. Further, let a(x) be the above matrix with (4.11), (4.12) and let k 0 and k be the functions from (3.37)-(3.39). Then there exists a constant c such that ( k 0 (a, 1, f)w 0 L p (R n ) + 2 jsq k(a, 2 j, f)w j L p (R n ) ) 1/q q holds for all f S (R n ). j=1 c f Bpq (R n, w) (4.14) 32

35 Proof: Let B be the collection of all matrices b = {b ij } n i,j=1 satisfying (4.11) and (4.12). For fixed b B we derive by this properties k(b, t, f)(x) = k b (t, f)(x) whereas k b (y) = ck(b 1 y) (4.15) is a modified kernel in the sense of (3.37)-(3.39). The same holds for k b 0, so that we can apply now Theorem 3.10 with the new kernels, and get k b 0 (1, f)w 0 Lp (R n ) ( + 2 jsq k b (2 j, f)w j Lp (R n ) ) 1/q q j=1 c f B pq (R n, w) for all f S (R n ). Now, we obtain (4.14) from this formula in going over to the supremum over all b B inside the L p quasi-norms. The second Lemma is necessary for our weighted spaces. To get the invariance under diffeomorphisms of our spaces we also need a special restriction on the diffeomorphisms. From now on we consider only diffeomorphisms ψ which satisfy ψ(x) = x for x near to infinity ( x > R for some R > 0). With that restriction we are ready to formulate the next lemma. Lemma 4.13: Let w 0 be an admissible weight function. Let R > 0 and ψ be an m- diffeomorphism with ψ(x) = x for x > R, then there exists a constant c > 0 such that (w 0 ψ 1 )(x) cw 0 (x) holds for all x R n. Proof: If ψ is an m-diffeomorphism with the restriction above, then also ψ 1 is an m-diffeomorphism with ψ 1 (x) = x for x > R. We define a := max sup ψ 1 i 1 i,j n (x) x R n x j. (4.16) Using the properties of the weight function w 0 and Taylor expansion of ψ 1 we obtain w 0 (ψ 1 (x)) Cw 0 (x)(1 + x ψ 1 (x) ) α Cw 0 (x)(1 + x ψ 1 (0) ψ 1 (..) x ) α Cw 0 (x)2 α (1 + ψ 1 (0) ) α (1 + x ψ 1 (..) x ) α C w 0 (x)(1 + x ψ 1 (..) x ) α. Here ψ 1 (..) is the Jacobian where in every line different arguments from the line segment between 0 and x are possible. In every case, the absolute values from all entries of ψ 1 (..) are bounded by a. We can estimate from this property x ψ 1 (..) x x (1 + a n) for all x R n. (4.17) Finally, we get from ψ 1 (x) = x for x > R and the preceding calculation { w 0 ψ 1 (x) = w 0 (ψ 1 C R,α,ψ,n w 0 (x) for x R (x)) w 0 (x) for x > R, and this finishes the proof. 33