Function spaces with dominating mixed smoothness [1. Auflage] 9783037191958, 3037191953

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Series of Lectures in Mathematics

Function Spaces with Dominating Mixed Smoothness

These notes are addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of Besov–Sobolev type. In particular, it will be of interest for researchers dealing with approximation theory, numerical integration and discrepancy.

ISBN 978-3-03719-195-8

www.ems-ph.org

Triebel_ELM(30) | Rotis Sans | Pantone 287, Pantone 116 | RB: 11 mm

Function Spaces with Dominating Mixed Smoothness

The first part of this book is devoted to function spaces in Euclidean n-space with dominating mixed smoothness. Some new properties are derived and applied in the second part where weighted spaces with dominating mixed smoothness in arbitrary bounded domains in Euclidean n-space are introduced and studied. This includes wavelet frames, numerical integration and discrepancy, measuring the deviation of sets of points from uniformity.

Hans Triebel

Hans Triebel

Hans Triebel

Function Spaces with Dominating Mixed Smoothness

EMS Series of Lectures in Mathematics Edited by Ari Laptev (Imperial College, London, UK) EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series (for a complete listing see our homepage at www.ems-ph.org): Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp and María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions Koichiro Harada, “Moonshine” of Finite Groups Yurii A. Neretin, Lectures on Gaussian Integral Operators and Classical Groups Damien Calaque and Carlo A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry Claudio Carmeli, Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry Hans Triebel, Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration Koen Thas, A Course on Elation Quadrangles Benoît Grébert and Thomas Kappeler, The Defocusing NLS Equation and Its Normal Form Armen Sergeev, Lectures on Universal Teichmüller Space Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, 3-Manifold Groups Hans Triebel, Tempered Homogeneous Function Spaces Kathrin Bringmann, Yann Bugeaud, Titus Hilberdink and Jürgen Sander, Four Faces of Number Theory Alberto Cavicchioli, Friedrich Hegenbarth and Dušan Repovš, Higher-Dimensional Generalized Manifolds: Surgery and Constructions Davide Barilari, Ugo Boscain and Mario Sigalotti, Geometry, Analysis and Dynamics on subRiemannian Manifolds, Volume I Davide Barilari, Ugo Boscain and Mario Sigalotti, Geometry, Analysis and Dynamics on subRiemannian Manifolds, Volume II Dynamics Done with Your Bare Hands, Françoise Dal’Bo, François Ledrappier and Amie Wilkinson (Eds.) Hans Triebel, PDE Models for Chemotaxis and Hydrodynamics in Supercritical Function Spaces Françoise Michel and Claude Weber, Higher-Dimensional Knots According to Michel Kervaire Local Representation Theory and Simple Groups, Radha Kessar, Gunter Malle and Donna Testerman(Eds.)

Hans Triebel

Function Spaces with Dominating Mixed Smoothness

Author: Hans Triebel Friedrich-Schiller-Universität Jena Fakultät für Mathematik und Informatik Institut für Mathematik 07737 Jena Germany E-mail: [email protected]

2010 Mathematics Subject Classification: 46-02, 46E35, 42C40, 42B35, 41A55 Key words: Function spaces, dominating mixed smoothness, spaces on domains, wavelets, Faber frames, Haar frames

ISBN 978-3-03719-195-8 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2019 Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 CH-8092 Zürich Switzerland

Email: [email protected] Homepage: www.ems-ph.org

Typeset using the author’s TEX files: Alison Durham, Manchester, UK Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface

r A(Rn ) of Besov–Sobolev type with dominating These notes deal with spaces Sp,q mixed smoothness, where A ∈ {B, F}, r ∈ R, 0 < p ≤ ∞, 0 < q ≤ ∞, and their r A[Ω, s,

L > σp(n) − s,

(1.58)

s (Rn ) if, and only if, it can be and d > 1 be fixed. Then f ∈ S 0(Rn ) belongs to Bp,q represented as ∞ Õ Õ f = λ j,m a j,m, (1.59) j=0 m∈Z n

where a j,m are (s, p)-atoms and λ ∈ b p,q . Furthermore, s (Rn )k ∼ inf kλ |b k f |Bp,q p,q k

(1.60)

are equivalent quasi-norms, where the infimum is taken over all admissible representations (1.59) (for fixed K, L, d). (ii) Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R. Let K ∈ N0 , L ∈ N0 , d ∈ R with K > s,

(n) L > σp,q − s,

(1.61)

1 Spaces on Rn

10

s (Rn ) if, and only if, it can be and d > 1 be fixed. Then f ∈ S 0(Rn ) belongs to Fp,q represented by (1.59) where a j,m are (s, p)-atoms and λ ∈ fp,q . Furthermore, s (Rn )k ∼ inf kλ | f k f |Fp,q p,q k

(1.62)

are equivalent quasi-norms where the infimum is taken over all admissible representations (1.59) (for fixed K, L, d). This coincides essentially with [T08, Thm. 1.7, p. 5] and [T10, Thm. 1.7, p. 5]. There one also finds comments, explanations and related references, in particular to [T06, Sect. 1.5.1]. We have incorporated the above material because we wish r A(Rn ) with dominating mixed smoothness to compare assertions for the spaces Sp,q with their counterparts for the isotropic spaces Asp,q (Rn ). r A(Rn ) in a more formal way. Next we deal with atomic representations for Sp,q We follow essentially [T10, Sects. 1.2.2, 1.2.5]. Let  (1.63) Rk,m = x = (x1, . . . , xn ) ∈ Rn : 2−k j m j < x j < 2−k j (m j + 1) , with k ∈ N0n and m ∈ Zn , be the rectangular counterparts of the above cubes Q k,m = 2−k m + 2−k (0, 1)n , k ∈ N0 , m ∈ Zn and let (p)

χk,m (x) = 2[k]/p χk,m (x),

k ∈ N0n, m ∈ Zn, x ∈ Rn

(1.64)

be the p-normalized modification of its characteristic function χk,m where 0 < p ≤ ∞. Í Recall [k] = nj=1 k j . Let d Rk,m with d > 0 be the rectangle concentric with Rk,m and with side length d 2−k j , j = 1, . . . , n. Definition 1.6. Let 0 < p ≤ ∞, 0 < q ≤ ∞. Then s p,q b is the collection of all sequences  λ = λk,m ∈ C : k ∈ N0n, m ∈ Zn (1.65) such that kλ |s p,q bk =

 Õ  Õ k ∈N0n

|λk,m | p

 q/p  1/q

< ∞,

(1.66)

m∈Z n

and s p,q f is the collection of all sequences according to (1.65) such that

 Õ

 1/q

(p) kλ |s p,q f k = |λk,m χk,m (·)| q |L p (Rn ) < ∞,

(1.67)

k,m

with the usual modifications if p = ∞ and/or q = ∞. Remark 1.7. This is the n-dimensional version of [T10, Def. 1.40, p. 26] as already indicated in [T10, Sect. 1.2.5]. One has s p, p b = s p, p f . Next we describe the counterpart of the above isotropic (s, p)-atoms.

11

1.1 Definitions and basic assertions

Definition 1.8. Let r ∈ R, 0 < p ≤ ∞, K ∈ N0 , L ∈ N0 and d > 1. Then the L∞ -functions ak,m : Rn 7→ C with k ∈ N0n , m ∈ Zn are called (r, p)-atoms if supp ak,m ⊂ dRk,m,

k ∈ N0n, m ∈ Zn,

(1.68)

there exist all (classical) derivatives Dα ak,m with α = (α1, . . . αn ) ∈ N0n , 0 ≤ α j ≤ K such that Ín 1 (1.69) |Dα ak,m (x)| ≤ 2−[k](r− p )+ j=1 k j α j , k ∈ N0n, m ∈ Zn, x ∈ Rn and ∫ β x j ak,m (x) dx j = 0,

j = 1, . . . , n; β < L, k j ∈ N, m ∈ Zn .

(1.70)

R

Remark 1.9. This is the n-dimensional version of [T10, Def. 1.42, pp. 26/27] which in turn coincides essentially with [Vyb06, Def. 2.3, p. 25] and [Schm07, Def. 4.4, p. 180] (with different normalizations and some generalizations). One may also consult 1 [Baz05]. If K = 0 then (1.69) means ak,m ∈ L∞ (Rn ) and |ak,m (x)| ≤ 2−[k](r− p ) . If k j = 0 or L = 0 then no moment conditions in (1.70) are required. r A(Rn ). Next we formulate the atomic representation theorem for the spaces Sp,q

Let σp = σp(1) = max

1
, 1 − 1, p

(1) σp,q = σp,q = max

1 1
, ,1 −1 p q

(1.71)

be the one-dimensional version of (1.57). Theorem 1.10. (i) Let 0 < p ≤ ∞, 0 < q ≤ ∞, r ∈ R. Let K ∈ N0 , L ∈ N0 , d ∈ R, with K > r, L > σp − r, (1.72) r B(Rn ) if, and only if, it can be and d > 1 be fixed. Then f ∈ S 0(Rn ) belongs to Sp,q represented as Õ Õ f = λk,m ak,m, (1.73) k ∈N0n m∈Z n

where ak,m are related (r, p)-atoms according to Definition 1.8 and λ ∈ s p,q b. Furthermore, r k f |Sp,q B(Rn )k ∼ inf kλ |s p,q bk (1.74) are equivalent quasi-norms where the infimum is taken over all admissible representations (1.73) (for fixed K, L, d).

1 Spaces on Rn

12

(ii) Let 0 < p < ∞, 0 < q ≤ ∞, r ∈ R. Let K ∈ N0 , L ∈ N0 , d ∈ R, with K > r,

L > σp,q − r,

(1.75)

r F(Rn ) if, and only if, it can be and d > 1 be fixed. Then f ∈ S 0(Rn ) belongs to Sp,q represented by (1.73) where ak,m are related (r, p)-atoms according to Definition 1.8 and λ ∈ s p,q f . Furthermore, r k f |Sp,q F(Rn )k ∼ inf kλ |s p,q f k

(1.76)

are equivalent quasi-norms where the infimum is taken over all admissible representations (1.73) (for fixed K, L, d). Remark 1.11. This is the n-dimensional version of [T10, Thm. 1.44, p. 27] as already indicated in [T10, p. 34]. Further comments may be found in [T10, Rem. 1.45, pp. 27– 28]. It is remarkable that the restrictions for L in (1.72), (1.75) are the same as in (1.58), (1.61) with n = 1. This will be of some service below and it was also a substantial point in [T10]. A detailed proof of the above theorem may be found in [Vyb06, Sect. 2.2]. Some modifications are indicated in [T10, Rem. 1.45, pp. 27–28]. In particular, if α j = K then one can replace the corresponding derivative in (1.69) by a Lipschitz condition. This will also be of some service to us later on. 1.1.4 Wavelets. In addition to atoms, we rely on wavelet representations for the r A(Rn ). Again we compare corresponding assertions for spaces with domspaces Sp,q inating mixed smoothness with their counterparts for the inhomogeneous isotropic spaces Asp,q (Rn ). Somewhat in contrast to atoms, wavelet expansions for Asp,q (Rn ), r A(Rn ). 2 ≤ n ∈ N, are technically more complicated than for Sp,q We follow [T10, Sect. 1.1.4, pp. 8–11] closely. We suppose that the reader is familiar with wavelets in Rn of Daubechies type and the related multiresolution analysis. The standard references are [Dau92, Mal99, Mey92, Woj97]. A short summary of what is needed may also be found in [T06, Sect. 1.7]. First we give a brief description of some basic notation. As usual, C u (R) with u ∈ N collects all complex-valued continuous functions on R having continuous bounded derivatives up to order u inclusively. Let ψF ∈ C u (R),

ψ M ∈ C u (R),

u∈N

be real compactly supported Daubechies wavelets with ∫ ψ M (x) x v dx = 0 for all v ∈ N0 with v < u.

(1.77)

(1.78)

R

Recall that ψF is called the scaling function (father wavelet) and ψ M the associated wavelet (mother wavelet). We extend these wavelets from R to Rn by the usual

13

1.1 Definitions and basic assertions

multiresolution procedure. Let G = (G1, . . . , G n ) ∈ G0 = {F, M } n,

(1.79)

which means that Gr is either F or M. Let G = (G1, . . . , G n ) ∈ G j = {F, M } n∗,

j ∈ N,

(1.80)

which means that Gr is either F or M, where ∗ indicates that at least one of the components of G must be an M. Hence G0 has 2n elements, whereas G j with j ∈ N has 2n − 1 elements. Let n Ö j ψG,m (x) = ψGl 2 j xl − ml ), G ∈ G j , m ∈ Zn, (1.81) l=1

where (now) j ∈ N0 . We always assume that ψF and ψ M in (1.77) have L2 -norm 1. Then n o j 2 jn/2 ψG,m : j ∈ N0, G ∈ G j , m ∈ Zn (1.82) is an orthonormal basis in L2 (Rn ) (for any u ∈ N) and f =

∞ Õ Õ Õ j=0

with j,G

j,G

λm = λm ( f ) = 2 jn

j,G

j

λm ψG,m,

(1.83)

G ∈G j m∈Z n

∫ Rn


j j f (x) ψG,m (x) dx = 2 jn f , ψG,m ,

(1.84) j

is the corresponding expansion with respect to the L∞ -normalized functions ψG,m . (p)

Let χj,m be the same p-normalized characteristic function of the cube Q j,m = 2−j m + 2−j (0, 1)n as in (1.50), 0 < p ≤ ∞. We need the (differently normalized) counterparts of the sequence spaces b p,q and fp,q according to (1.51)–(1.53). Let s ∈ R, 0 < p ≤ ∞, 0 < q ≤ ∞. Then bsp,q is the collection of all sequences  j,G λ = λm ∈ C : j ∈ N0, G ∈ G j , m ∈ Zn (1.85) such that kλ

|bsp,q k

=

∞ Õ

n

2 j(s− p )q

j=0

Õ  Õ G ∈G j

j,G

|λm | p

 q/p  1/q

max(s, σp(n) − s). (1.88) s (Rn ) if, and only if, it can be represented as Let f ∈ S 0(Rn ). Then f ∈ Bp,q

f =

Õ

j,G

j

λm ψG,m,

λ ∈ bsp,q .

(1.89)

j,G,m

The representation (1.89) is unique, j

j,G

λm = 2 jn f , ψG,m




(1.90)

and 
j I : f 7→ 2 jn f , ψG,m

(1.91)

s (Rn ) onto bs . is an isomorphic map of Bp,q p,q

(ii) Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R, and (n) u > max(s, σp,q − s).

(1.92)

s (Rn ) if, and only if, it can be represented as Let f ∈ S 0(Rn ). Then f ∈ Fp,q

f =

Õ

j,G

j

λm ψG,m,

s λ ∈ fp,q .

(1.93)

j,G,m

The representation is unique with (1.90). Furthermore, I in (1.91) is an isomorphic s (Rn ) onto f s . map of Fp,q p,q This is an adapted and shortened version of [T10, Thm. 1.18, pp. 10–11]. There one finds further explanation, including how the convergence of (1.89), (1.93) must be understood and references to where related proofs can be found. r A(Rn ) in a more formal way. Next we deal with wavelet representations for Sp,q We follow essentially [T10, Sects. 1.2.4, 1.2.5, pp. 30–35]. Let ψF ∈ C u (R) and ψ M ∈ C u (R) be the same compactly supported real Daubechies wavelets as in (1.77), (1.78) with L2 -norms 1. Let √ (1.94) ψ−1,m (t) = 2 ψF (t − m) and ψk,m (t) = ψ M (2k t − m)

15

1.1 Definitions and basic assertions

with k ∈ N0 and m ∈ Z. Let N−1 = N0 ∪ {−1}. Then  k/2 2 ψk,m (t) : k ∈ N−1, m ∈ Z

(1.95)

is an orthonormal basis in L2 (R). Let n N−1 = {k = (k 1, . . . , k n ), k j ∈ N−1 },

and ψk,m (x) =

n Ö

ψk j ,m j (x j ),

n∈N

(1.96)

n k ∈ N−1 , m ∈ Zn .

(1.97)

j=1

Then 

n 2[k]/2 ψk,m : k ∈ N−1 , m ∈ Zn



(1.98)

with (again) [k] = j=1 k j is an orthonormal basis in L2 (Rn ). In [T10, Sect. 1.2.4, pp. 31, 34] one finds some technical explanation. We adapt the sequence spaces s p,q b and s p,q f in (1.66), (1.67) to the above situation. Let 0 < p ≤ ∞, 0 < q ≤ ∞. Then s p,q b− is the collection of all sequences  n λ = λk,m ∈ C : k ∈ N−1 , m ∈ Zn (1.99) Ín

such that kλ |s p,q b− k =

 Õ  Õ n k ∈N−1

|λk,m | p

 q/p  1/q

max(r, σp − r).

(1.102)

r B(Rn ) if, and only if, it can be represented as Let f ∈ S 0(Rn ). Then f ∈ Sp,q

f =

Õ Õ n m∈Z n k ∈N−1

1

λk,m 2−[k](r− p ) ψk,m,

λ ∈ s p,q b− .

(1.103)

1 Spaces on Rn

16

The representation (1.103) is unique, λk,m = 2[k](r− p +1) ( f , ψk,m ), 1

and

n k ∈ N−1 , m ∈ Zn,

n 1
o J : f 7→ 2[k](r− p +1) f , ψk,m

(1.104)

(1.105)

− r B(Rn ) onto s is an isomorphic map of Sp,q p,q b .

(ii) Let 0 < p < ∞, 0 < q ≤ ∞, r ∈ R, and u > max(r, σp,q − r).

(1.106)

r F(Rn ) if, and only if, it can be represented as Let f ∈ S 0(Rn ). Then f ∈ Sp,q

f =

Õ Õ n k ∈N−1

1

λk,m 2−[k](r− p ) ψk,m,

λ ∈ s p,q f − .

(1.107)

m∈Z n

The representation (1.107) is unique with (1.104). Furthermore, J in (1.105) is an r F(Rn ) onto s − isomorphic map of Sp,q p,q f . Remark 1.13. This is essentially the extension of [T10, Thm. 1.54, p. 32] from n = 2 to n ∈ N, as already indicated in [T10, Sect. 1.2.5, pp. 33–35]. There one also finds some technicalities, in particular how the convergence in (1.103) and (1.107) must be understood. Assertions of this type go back to [Vyb06, Thm. 2.11, p. 41] covering also wavelet representations of some more general spaces with dominating mixed smoothness. There is also a discussion of some further technicalities. One may also consult [Schm07, Thm. 4.9, p. 184] and the additional references given there. Remark 1.14. We have the same situation as previously discussed in Remark 1.11 with respect to the atomic representation Theorem 1.10. The conditions for u in (1.102), (1.106) are the same as in (1.88), (1.92) with n = 1. This will be of some service to us later on. 1.1.5 Complements. Recall that these notes are not a comprehensive report of the theory of functions with dominating mixed smoothness. Quite the contrary: we wish to complement the already existing literature as mentioned in the preface by some new aspects of these spaces on Rn in this Chapter 1 and, in particular, on arbitrary bounded domains in Chapter 2. But we collect a few more or less known basic properties which will be of some service to us. Other more specific assertions will be described later on when they are needed (and compared with their counterparts in terms of the isotropic spaces Asp,q (Rn )). Recall that ,→ means continuous embedding.

1.1 Definitions and basic assertions

17

r A(Rn ) be the spaces according to Definition 1.4 and Proposition 1.15. Let Sp,q Remark 1.5 where 0 < p ≤ ∞ (p < ∞ for F-spaces), 0 < q ≤ ∞ and r ∈ R.

(i) Then r S(Rn ) ,→ Sp,q A(Rn ) ,→ S 0(Rn ).

(1.108)

(ii) If, in addition, p < ∞ then r r r Sp,min(p,q) B(Rn ) ,→ Sp,q F(Rn ) ,→ Sp,max(p,q) B(Rn ).

(1.109)

(iii) If, in addition, ε > 0 and 0 < q0, q1 ≤ ∞, then r+ε r r−ε Sp,q B(Rn ) ,→ Sp,q B(Rn ) ,→ Sp,q B(Rn ). 0 1

(1.110)

r A(Rn ). (iv) If, in addition, p < ∞, q < ∞, then S(Rn ) is dense in Sp,q r A(Rn ) are usually proved Remark 1.16. These basic properties of the spaces Sp,q directly after their Fourier-analytical introduction according to Definition 1.4 complemented by the comments in Remark 1.5. We refer the reader to [ST87, Prop. 2.2.3/2, pp. 88–89, Thm. 2.2.4, pp. 92–93]. But it might be illuminating to look at the above r A(Rn ) assertions from the point of view of the powerful wavelet isomorphism of Sp,q according to Theorem 1.12 onto the related sequence spaces. Then (1.109) is a matter of the monotonicity of the `q -spaces and the triangle inequality for Lt -spaces, t ≥ 1. Here u = min(p, q) as the largest index for the left-hand embedding and v = max(p, q) as the smallest index for the right-hand embedding are sharp; see [ScS04, Thm. 1, pp. 121–122]. If p < ∞, q < ∞ then it follows from Theorem 1.12 that finite linr A(Rn ). The ear combinations of sufficiently smooth wavelets ψk,m are dense in Sp,q product structure of ψk,m according to (1.97) reduces the question of whether S(Rn ) r A(Rn ) to the one-dimensional case, which means to Ar (R). But the is dense in Sp,q p,q density of S(R) in Arp,q (R), p < ∞, q < ∞ is a very classical assertion. r A(Rn ) if According to part (iv) of the above proposition, S(Rn ) is dense in Sp,q r A(Rn ) 0 r ∈ R and 0 < p, q < ∞. In particular, it makes sense to interpret its dual Sp,q
within the framework of the dual pairing S(Rn ), S 0(Rn ) . Based on Definition 1.4 the duality theory for the (inhomogeneous) isotropic spaces Asp,q (Rn ) as developed r A(Rn ) in [T83, Sect. 2.11] can be carried over. This applies in particular to Sp,q with 1 ≤ p, q < ∞. But quite recently these Fourier-analytical arguments have been extended to 0 < p, q ≤ ∞, including some limiting cases, in [NgS17a, Sect. 4.3] and [NgS17b, Sect. 4.3]. Duality plays only a marginal role in what follows. But we give a new short wavelet proof of those assertions which are of interest later on.

Proposition 1.17. (i) Let 1 < p, q < ∞ and 1 1 1 1 + 0 = + 0 = 1, p p q q

r ∈ R.

(1.111)

1 Spaces on Rn

18 Then

r Sp,q B(Rn )0 = Sp−r0,q0 B(Rn ) and

r Sp,q F(Rn )0 = Sp−r0,q0 F(Rn ).

(1.112)

(ii) Let 1 ≤ p, q < ∞ and p0, q 0, r as in (1.111). Then r Sp,q B(Rn )0 = Sp−r0,q0 B(Rn ).

(1.113)

◦

r B(Rn ). Then (iii) Let Srp,∞ B(Rn ) be the completion of S(Rn ) in Sp,∞ ◦

Srp,∞ B(Rn )0 = Sp−r0,1 B(Rn ),

1 1 + = 1, r ∈ R. p p0

1 ≤ p ≤ ∞,

(1.114)

Proof. Step 1. Part (i) avoids any limiting situations and the corresponding Fourieranalytical proofs for the related isotropic spaces Asp,q (Rn ) in [T83, Sect. 2.11] can be transferred. Step 2. Parts (ii) and (iii) (in addition to part (i)) also cover limiting situations which will be of some use to us later on. First we give a new direct proof of part (ii) based r B(Rn ) be expanded according to (1.103), (1.104) on Theorem 1.12(i). Let f ∈ Sp,q n and let ϕ ∈ S(R ). Then one has Õ ( f , ϕ) = λk,m µk,m (1.115) n ,m∈Z n k ∈N−1

with λk,m as in (1.104) and µ = {µk,m },

1

µk,m = 2−[k](r− p )

∫ Rn

Recall that `q (A)0 = `q0 (A0),

1 ≤ q < ∞,

ϕ(x) ψk,m (x) dx.

(1.116)

1 1 + =1 q q0

(1.117)

in the usual interpretation, where A is a Banach space and A0 its dual; see [T78, Lem. 1.11.1, p. 68]. One has in particular `q (`p )0 = `q0 (`p0 ) with 1 ≤ p, q < ∞. Then r B(Rn ) onto s − it follows from (1.115) and the isomorphic map (1.105) of Sp,q p,q b normed by (1.100),  r r B(Rn )k ≤ 1 ∼ k µ |`q0 (`p0 )k. (1.118) kϕ |Sp,q B(Rn )0 k = sup |( f , ϕ)| : k f |Sp,q On the other hand, µk,m coincides with the coefficients in (1.104) if one replaces r by −r and p1 by p10 there. Then (1.118) and the counterpart of (1.100) prove (1.113).

19

1.1 Definitions and basic assertions ◦

Step 3. We prove part (iii). Since (by definition) S(Rn ) is dense in Srp,∞ B(Rn ), its
dual can again be interpreted in the framework of the dual pairing S(Rn ), S 0(Rn ) . Then one has the wavelet representation (1.103), (1.104) now being an isomorphism onto c0 (`p ) (with c0 (c0 ) if p = ∞) where c0 collects all sequences {a j }∞ j=1 ⊂ C 0 0 with a j → 0 if j → ∞. If A is a Banach space then c0 (A) = `1 (A ) in the usual interpretation; see [T78, Lem. 1.11.1, p. 68]. Otherwise one can follow the above arguments resulting in (1.114) instead of (1.113).  Remark 1.18. The duality theory for the spaces Asp,q (Rn ) goes essentially back to [T83, Sect. 2.11]. Some limiting cases are proved later on and related references may be found in [RuS96, Sect. 2.1.5], all based on Fourier-analytical arguments. As already mentioned, this theory has been transferred in [NgS17a, Sect. 4.3] and r A(Rn ). The above proof of (1.113), (1.114) [NgS17b, Sect. 4.3] to the spaces Sp,q relies on wavelet expansions and related isomorphisms to distinguished sequence spaces. It may well be the case that this approach can be extended to other spaces r A(Rn ) or to other types of spaces. But this will not be done here. Sp,q In the following assertion we assume that the dyadic resolution of unity ϕ = {ϕk }k ∈N0n in Definition 1.4 is fixed. Proposition 1.19. Let 2 ≤ n ∈ N. Let 0 < p ≤ ∞ (p < ∞ for F-spaces), 0 < q ≤ ∞ and r ∈ R. If { f j } nj=1 ⊂ Brp,q (R) then f (x) =

n Ö

r r f j (x j ) ∈ Sp,q B(Rn ) and k f |Sp,q B(Rn )k =

j=1

n Ö

k f j |Brp,q (R)k. (1.119)

j=1

r (R) then If { f j } nj=1 ⊂ Fp,q

f (x) =

n Ö j=1

r r f j (x j ) ∈ Sp,q F(Rn ) and k f |Sp,q F(Rn )k =

n Ö

r k f j |Fp,q (R)k. (1.120)

j=1

Proof. This follows immediately from the product structure of all ingredients in (1.31) and (1.33) (and the above assumption that ϕ is fixed).  Interpolation plays only a marginal role in what follows. But occasionally we use the complex interpolation of the above spaces. The classical complex interpolation [A0, A1 ]θ , 0 < θ < 1 for Banach spaces A0 , A1 being an interpolation couple goes back to [Cal64]. A detailed discussion and also further references may be found in [T78, Sect. 1.9]. An extension of this theory to arbitrary quasi-Banach spaces is not possible. But there is a distinguished class of quasi-Banach spaces to which the basic constructions of the complex interpolation for Banach spaces still work.

1 Spaces on Rn

20


This applies, in particular, to spaces of type `q (`p ), `q L p (Rn ) and L p (Rn, `q ) with 0 < p, q < ∞ (and also some limiting cases with max(p, q) = ∞). It can be transferred to several types of function spaces. We refer the reader to [KMM07] and the literature mentioned there, especially [MeM00]. We fix the outcome for the r A(Rn ). The extended complex interpolation method will again be above spaces Sp,q denoted by [A0, A1 ]θ . Proposition 1.20. Let 0 < p0, p1, q0, q1 ≤ ∞ (p0 < ∞, p1 < ∞ for F-spaces) with min(q0, q1 ) < ∞ and r0 ∈ R, r1 ∈ R. Let 0 < θ < 1 and 1 1−θ θ = + , p p0 p1

1 1−θ θ = + , q q0 q1

r = (1 − θ)r0 + θr1 .

(1.121)

Then 

Spr00,q0 B(Rn ), Spr11,q1 B(Rn )

θ

r = Sp,q B(Rn )

(1.122)

θ

r = Sp,q F(Rn ).

(1.123)



and 

Spr00,q0 F(Rn ), Spr11,q1 F(Rn )



r A(Rn ) as described in TheoProof. The wavelet isomorphism for the spaces Sp,q rem 1.12 reduces the above complex interpolation to corresponding assertions for the related sequence spaces s p,q b− in (1.100) and s p,q f − in (1.101). But these sequence spaces are of the above-mentioned type and one can apply the theories developed in [MeM00, KMM07] (which differ in some limiting cases). This has been done in detail in [Vyb06, Sect. 4] (for spaces on bounded domains Ω in Rn instead of Rn , but this does not matter for the above purpose). In particular, [Vyb06, Thm. 4.6, p. 60] is the sequence version of (1.122), (1.123). 

Remark 1.21. The above proposition is also essentially covered by [NgS17a, Prop. 4.4, pp. 371–372] for the B-spaces and by [NgS17b, Prop. 4.9, p. 317] for the F-spaces. Of interest to us later on is the special limiting case  r0  r n S∞,∞ B(Rn ), Spr11, p1 B(Rn ) θ = Sp, 0 < θ < 1, (1.124) p B(R ), r0 ∈ R, r1 ∈ R, 0 < p1 < ∞ and r, p as in (1.121). One may also consult [T10, (3.139), p. 146]. Otherwise the above proposition is the counterpart of a corresponding complex interpolation theorem for the (inhomogeneous and homogeneous) isotropic s (Rn ) and F s (Rn ) as may be found in [KMM07, Thm. 9.1, p. 157] spaces Bp,q p,q (extending [MeM00]). One may also consult [T10, Thm. 1.22, Rem. 1.23, p. 12]. It is well known that there is an elaborated real interpolation theory for the spaces r A(Rn ) Asp,q (Rn ). The corresponding real interpolation theory for the spaces Sp,q requires some care. This will not be needed here. Some assertions may be found in [Schm07] and, in particular, in [ScS04].

1.2 Properties, I

21

1.2 Properties, I 1.2.1 Introduction. As already mentioned in the preface it is the main aim of this Chapter 1 to complement the already existing substantial theory of the spaces r A(Rn ). Preference is given to those new properties which we need in ChapSp,q r A[Ω, p1 − 1 then it follows from (1.134) that it makes sense to ask whether (1.140)

24

1 Spaces on Rn

is also an equivalent quasi-norm in the corresponding spaces Spr B(Rn ). This might be the case, but it is not covered by the literature as we have already mentioned in connection with (1.48). But like there, we refer the reader to [NUU17]. Let χu (a) = 1 if a ∈ u and χu (a) = 0 if a < u. Let x ∈ Q, h ∈ Q and χu h = ( χu (1)h1, . . . , χu (n)hn ). Then (
m f (x) if x + m χ h ∈ Q,
∆h,u u m (1.142) ∆h,u,Q f (x) = 0 otherwise, m f where all points involved are in Q. The Q-counterparts selects those differences ∆h,u of (1.140), (1.141) are now given by Õ ∫ p dh dx  1/p −r p m r k f |Sp B(Q)km = f (x) (h)u ∆h,u,Q , (1.143) (h)u Q×Q u

where again 1 ≤ p < ∞, 0 < r < m ∈ N, modified by Õ m k f |Sr C (Q)km = sup (h)−r u ∆h,u,Q f (x)

(1.144)

u h ∈Q, x ∈Q

if p = ∞, 0 < r < m ∈ N. Similarly, k f |Spr W(Q)k =

Õ

kDα f |L p (Q)k

(1.145)

α∈N0n, 0≤α j ≤r

with 1 < p < ∞ and r ∈ N0 is the Q-version of (1.36). As we have already stated, there is little hope of finding extension operators of type (1.136), (1.137) for arbitrary r A(Ω) in (maybe even smooth) domains Ω ⊂ Rn , 2 ≤ n ∈ N. The situation spaces Sp,q improves if Ω is a rectangular domain of type (1.63) with Q = (0, 1)n as the prototype r A(Q) according to Definition 1.22(i) are specified as the and if the related spaces Sp,q above distinguished spaces Spr W(Q), 1 < p < ∞, r ∈ N0 and Spr B(Q), 1 ≤ p ≤ ∞, r B(Q) = S r C (Q). r > 0. We formulate the outcome. Recall S∞ Proposition 1.25. Let Q = (0, 1)n where n ∈ N. (i) Let 1 ≤ p ≤ ∞, m ∈ N and 0 < r < m. Then there exists a common extension operator extm , extm : Spr B(Q) ,→ Spr B(Rn ). (1.146) Furthermore, Spr B(Q) is the collection of all f ∈ L1 (Q) (or likewise f ∈ L p (Q)) such that k f |Spr B(Q)km < ∞ (1.147) according to (1.143), modified by (1.144) if p = ∞, equivalent norms.

1.2 Properties, I

25

(ii) Let 1 < p < ∞, m ∈ N and r ∈ N0 with r ≤ m. Then the extension operator extm in (1.146) is also a common extension operator, extm : Spr W(Q) ,→ Spr W(Rn ).

(1.148)

Furthermore, Spr W(Q) is the collection of all f ∈ L1 (Q) (or likewise f ∈ L p (Q)) such that k f |Spr W(Q)k < ∞ (1.149) according to (1.145), equivalent norms. Proof. The rather specific proof in terms of the classical extension method by Hestenes in [T10, Thm. 1.67, pp. 46–47] can be extended from n = 2 to 1 ≤ n ∈ N. There one also finds related references. One may also consult Section 1.3.7 below.  Remark 1.26. A common extension operator for given m ∈ N means that one asks for a linear and bounded extension operator from L p (Q) to L p (Rn ), 1 ≤ p ≤ ∞, such that its restriction to Spr B(Q), 1 ≤ p ≤ ∞, 0 < r < m is an extension operator from Spr B(Q) to Spr B(Rn ) and its restriction to Spr W(Q), 1 < p < ∞, r ∈ N, r ≤ m is an extension operator from Spr W(Q) to Spr W(Rn ). One obtains, essentially as a by-product, that (1.147) and (1.149) are characterizing equivalent norms. Further information can be found in [T10, Sects. 1.2.6–1.2.8, pp. 35–50]. In particular, one can extend [T10, Thm. 1.70, p. 50] as follows. Let v ∈ N and σp be as in (1.71). Then there exists a common extension operator extv , extv : Spr B(Q) ,→ Spr B(Rn ),

(1.150)

  0 < p < ∞, σp < r < 1/p,    1 < p < ∞, r = 0,    0 < p ≤ ∞, r < 0, 

(1.151)

v > max(r, σp − r).

(1.152)

for all spaces Spr B(Q) with

and The proof relies on rather specific orthogonal wavelet expansions for f ∈ Brp (I) = Brp, p (I) with I = (0, 1) and their extensions to Spr B(Q) similar to (1.97). We refer the reader to [T10, p. 50]. Extensions as described in Proposition 1.25 and in (1.150)–(1.152) are excepr A(Ω) as introduced in Deftions. In general one cannot expect that the spaces Sp,q r A(Rn ) even if Ω is inition 1.22 admit (common) bounded linear extensions to Sp,q

1 Spaces on Rn

26

a bounded smooth domain in Rn , 2 ≤ n ∈ N. We return to this problem in Section 1.3.7 below. We take this (negative) observation as a starting point in Chapter 2 r A[Ω, 0, 1 ≤ p ≤ ∞ are special cases of (1.129), (1.132) based on (1.138), (1.36) and (1.139)– (1.141). We use the same notation as in (1.47), (1.48) and (1.140), (1.141), (1.156), Î n , and (h) = (h)1 if u = {1, . . . , n}. (1.157) with (h)d = na=1 had , d ∈ R, h ∈ R++ Then Proposition 1.25 can be complemented as follows.

27

1.2 Properties, I

Theorem 1.27. Let Q = (0, 1)n where n ∈ N. (i) Let 1 < p < ∞ and r ∈ N0 . Then k∂ r,n f |L p (Q)k ∼ k f |Spr W(Rn )k,

f ∈ Sepr W(Q)

(1.158)

(equivalent norms). (ii) Let 1 ≤ p ≤ ∞, m ∈ N and 0 < r < m. Then ∫  1/p ∫ m,n ∆ f (x) p dx dh ∼ k f |Spr B(Rn )km, (h)−r p h n (h) R++ Rn

f ∈ Sepr B(Q) (1.159)

if 1 ≤ p < ∞ and sup

n ,x ∈R n h ∈R++

(h)−r ∆hm,n f (x) ∼ k f |Sr C (Rn )km,

f ∈ Ser C (Q)

(1.160)

if p = ∞ (equivalent norms). Proof. Step 1. Part (i) follows essentially from [T10, Props. 1.68, p. 48, 3.22, p. 154] extended from r = 1 to r ∈ N. But we give a new independent proof which can also be used to justify part (ii). By density arguments (and completion afterwards) we assume that all functions involved are smooth. Let I = (0, 1). Then we justify first the very classical assertion that there is a positive constant c such that k f |L p (I)k ≤ c k f (r) |L p (I)k,

e pr (I). f ∈W

(1.161)

We assume that there is no positive number c with (1.161). Then there are functions er { f j }∞ j=1 ⊂ W p (I) such that 1 = k f j |L p (I)k > j k f j(r) |L p (I)k.

(1.162)

e pr (I) into L p (I) is compact we may assume that f j converges Since the embedding of W in L p (I) to f ∈ L p (I) with k f |L p (I)k = 1. By (1.162) this is also a convergence e pr (I) and f (r) = 0. Then f = 0, which contradicts k f |L p (I)k = 1. For fixed in W 0 x = (x2, . . . , xn ) one can apply (1.161) to f (x1, x 0). Integration over x 0 results in k f |L p (Q)k ≤ k∂1r f |L p (Q)k.

(1.163)

Iteration proves k f |L p (Q)k ≤ c k∂ r,n f |L p (Q)k,

f ∈ Sepr W(Q).

(1.164)

By (1.36) and corresponding estimates for lower derivatives one obtains (1.158).

1 Spaces on Rn

28

Step 2. We prove (ii). We may again assume that all functions involved are smooth (at least continuous in the case of p = ∞). Let ∆hm f (x) with x ∈ R and h ∈ R be as in (1.45) with n = 1. Then k f |L p (I)k ≤ c

∫

1

h

−r p

∫

0

|∆hm f (x)| p

R

dx dh  1/p , h

erp (I), f ∈B

(1.165)

0 < r < m ∈ N, 1 ≤ p ≤ ∞ (modification if p = ∞) for some c > 0 is the counterpart erp (I) in L p (I) (or likewise L p (R)) is compact. If of (1.161). The embedding of B there is no positive constant c with (1.165) then it follows by the same arguments as erp (I) with k f |L p (I)k = 1 and ∆m f (x) = 0 in Step 1 that there is a function f ∈ B h a.e. in I × R. Then f must be a polynomial of degree less than m. One may consult [T06, pp. 200-201] for this plausible but not obvious conclusion. Then f = 0, which contradicts k f |L p (I)k = 1. Furthermore, ∆hm f (x) in (1.165) is zero outside some finite square I × (a, b), −∞ < a < b < ∞, depending on m. Now one can argue as in Step 1. Then one obtains k f |L p (Q)k ≤ c

∫

1

0

−r p h1

∫ Rn

as the counterpart of (1.163) and ∫ ∫ −r p k f |L p (Q)k ≤ c (h) Q

Rn

m |∆h,1 f (x)| p

dx dh1  1/p h1

(1.166)

dx dh  1/p (h)

(1.167)

|∆hm,n f (x)| p

n and incorporate as the counterpart of (1.164). One can replace Q in (1.167) by R++ lower differences with h ∈ Q on the right-hand side. Then part (ii) follows from the n ). above discussion (replacement of h ∈ Q by h ∈ R++  n in (1.159), Remark 1.28. It follows by the above arguments that one can replace R++ (1.160) by Q = (0, 1)n (similarly to (1.156), (1.157) compared with (1.140), (1.141)). But the above version will be more convenient for us.

1.2.3 Homogeneity. Later on we deal with the problem of pointwise multipliers in r A(Rn ). The related theory for the (inhomogeneous) isotropic spaces some spaces Sp,q s n Ap,q (R ) relies, at least partly, on local homogeneity assertions. We wish to do the same for some distinguished spaces with dominating mixed smoothness. But first we recall what is known in the case of isotropic spaces Asp,q (Rn ) as far as it is of relevance for our later purposes. Again let σp(n) be as in (1.57), n ∈ N. Let 0 < p ≤ ∞,

0 < q ≤ ∞ and

s > σp(n)

(1.168)

1.2 Properties, I

29

(p < ∞ for the F-spaces). Then n

k f (λ·) | Asp,q (Rn )k ∼ λ s− p k f | Asp,q (Rn )k

(1.169)

for all f ∈ Asp,q (Rn ),

supp f ⊂ {y : |y| ≤ λ}, 0 < λ ≤ 1,

(1.170)

A ∈ {B, F}, where the equivalence constants are independent of λ. This assertion has some history. The B-case is covered by [T08, Thm. 2.11, p. 34], whereas there (n) we needed for the F-case the more severe restriction s > σp,q according to (1.57). The above improvement of (1.169) with (1.168), p < ∞ for A = F, came out only quite recently; see [T15, Corol. 3.55, p. 116]. Of interest are counterparts for some spaces with dominating mixed smoothness. Again let Spr W(Rn ), r ∈ N0 , 1 < p < ∞ be the Sobolev spaces according to r B(Rn ) with S r C (Rn ) = S r B(Rn ) as introduced in (1.35), (1.36) and Spr B(Rn ) = Sp, p ∞ (1.139). We complement Q = (0, 1)n by the rectangular domains  Qλ = x = (x1, . . . , xn ) : 0 < x j < λ j , λ = (λ1, . . . , λn ), 0 < λ j < 1. (1.171) n according to (1.155) and x ∈ Rn then we put λx = (λ x , . . . , λ x ). As If λ ∈ R++ 1 1 n n Î n and d ∈ R. Let σ be as in (1.71). before, let (λ)d = nj=1 λ dj , λ ∈ R++ p

Theorem 1.29. (i) Let 1 < p < ∞ and r ∈ N0 . Then 1

k f (λ·) |Spr W(Rn )k ∼ (λ)r− p k f |Spr W(Rn )k

(1.172)

for all λ = (λ1, . . . , λn ), 0 < λ j ≤ 1 and all f ∈ Spr W(Rn ) with supp f ⊂ Qλ .

(1.173)

(ii) Let k · |Spr B(Rn )k be a fixed quasi-norm in Spr B(Rn ) with 0 < p ≤ ∞ and r > σp . Then

1

k f (λ·) |Spr B(Rn )k ∼ (λ)r− p k f |Spr B(Rn )k

(1.174)

(1.175)

for all λ = (λ1, . . . , λn ), 0 < λ j ≤ 1 and all f ∈ Spr B(Rn ) with supp f ⊂ Qλ .

(1.176)

Proof. Step 1. Part (i) follows from (1.158) and

∂ r,n f (λ·) (x) = (λ)r ∂ r,n f (λx).

(1.177)

1 Spaces on Rn

30

Let 1 ≤ p ≤ ∞ and 0 < r < m ∈ N. Then (1.175) is a consequence of (1.159), (1.160) with f (λx) in place of f (x), and   m,n
n f (λx), x ∈ Rn, h ∈ R++ ∆hm,n f (λx) = ∆λh , (1.178) where again λx = (λ1 x1, . . . , λn xn ) and λh = (λ1 h1, . . . , λn hn ). Step 2. The above arguments cover (1.175) if p ≥ 1 and r > 0. Then we could rely on (1.159) which in turn was based on (1.140). But as mentioned there, it is not clear whether (1.140) can be extended to p < 1 with (1.174) although it follows from (1.134) that the question itself makes sense. We prove (1.175) for all p and r with (1.174) by wavelet arguments. Let x = (x1, x 0) with x 0 = (x2, . . . , xn ), n ≥ 2. We expand f ∈ Spr B(Rn ) according to (1.103), (1.104) and the indicated isomorphic map onto s p, p b− which is a related `p -space. Using the product structure (1.97) of ψk,m and the orthonormality of (1.98) in Rn and Rn−1 one obtains Õ Õ 1 0 f = fk 0,m0 (x1 ) 2−[k ](r− p ) ψk 0,m0 (x 0) (1.179) n−1 m0 ∈Z n−1 k 0 ∈N−1

with [k 0 ](r− p1 +1)

fk 0,m0 (x1 ) = 2 =

∫

f (x1, x 0) ψk 0,m0 (x 0) dx 0

R n−1

∞ Õ Õ

2[k](r− p +1) 1

∫

1

Rn

k1 =−1 m1 ∈Z

f (x) ψk,m (x) dx · 2−k1 (r− p ) ψk1,m1 (x1 ), (1.180)

where m = (m1, m 0) and k = (k 1, k 0). This is an expansion of fk 0,m0 by the onedimensional version of Theorem 1.12 with the coefficients ∫ 0 0 [k](r− p1 +1) λkk1,m = 2 f (x) ψk,m dx, k1 ∈ N−1, m1 ∈ Z. (1.181) ,m1 Rn

But these are the same coefficients as in the expansion (1.103), (1.104) with p = q in (1.100). In particular,  k 0,m0 λk1,m1 : k1 ∈ N−1, m1 ∈ Z ∈ `p (1.182) and fk 0,m0 ∈ Brp (R) with k fk 0,m0 |Brp (R)k ∼

∞  Õ Õ 0 0 p  1/p λ k ,m . k1 =−1 m1 ∈Z

k1,m1

(1.183)

31

1.2 Properties, I

Then one has by Theorem 1.12, k f |Spr B(Rn )k ∼

 Õ

Õ

k fk 0,m0 |Brp (R)k p

 1/p

.

(1.184)

n−1 m0 ∈Z n−1 k 0 ∈N−1

We apply this observation to f (λ1 x1, x 0) with f ∈ Spr B(Rn ),

supp f ⊂ (0, λ1 ) × Rn−1,

0 < λ1 ≤ 1.

(1.185)

The expansion of f (λ1 x1, x 0) is the same as in (1.179) with fk 0,m0 (λ1 x1 ) in place of fk 0,m0 (x1 ). Then one obtains from the one-dimensional case of (1.168)–(1.170) and (1.184) that r− p1

k f (λ1 x1, x 0) |Spr B(Rn )k ∼ λ1 Iterative application proves (1.175) for f ∈

k f |Spr (Rn )k.

Spr B(Rn )

(1.186)

satisfying (1.176). 

Remark 1.30. We proved (1.184) in [T10, Thm. 1.58, pp. 37–39] for n = 2 and 0 < p ≤ ∞, r ∈ R. We relied on wavelet expansion as above. But the step from n = 2 to n > 2 is not totally obvious. This is the main reason why we inserted a proof of (1.184) which again applies to all Spr B(Rn ) with 0 < p ≤ ∞, r ∈ R. Otherwise one finds in [T10, Sect. 1.27, pp. 36–44] some further discussion, modifications and applications. Our aim here is different. We rely on the above theorem in order to introduce non-smooth atoms in the related spaces which, in turn, are used to deal with pointwise multipliers and spaces with dominating mixed smoothness in arbitrary domains. Problem 1.31. For isotropic spaces with positive smoothness s > 0 one has the rather final homogeneity assertions (1.168)–(1.170). One may ask whether Theorem 1.29 can be extended from the above spaces Spr W(Rn ) and Spr B(Rn ) to other r A(Rn ) with dominating mixed smoothness. This would be of great use in spaces Sp,q r A[Ω, σp,

(1.187)

for some v with 1 < v ≤ ∞ as a consequence of well-known embeddings for the spaces Spr B(Rn ); see [ST87, Sect. 2.4.1, pp. 127–133]. A continuous function ak,m according to Definition 1.8 satisfying (1.68), (1.69) is called a smooth (r, p)K -atom (we do not need the moment conditions (1.70)). Then, for the spaces Spr B(Rn )

1 Spaces on Rn

32

in (1.187), one has the representation (1.73), (1.74) by smooth (r, p)K -atoms with K > r. We extend this theory to atomic expansions in terms of non-smooth atoms. Let σp(n) be as in (1.57). Then the isotropic counterpart, decompositions of the (inhomogeneous) isotropic spaces s n Bps (Rn ) = Bp, p (R ),

0 < p ≤ ∞, s > σp(n)

(1.188)

by non-smooth (isotropic) atoms, may be found in [T06, Sect. 2.2, pp. 131–136] based on [Tri03]. This extension of expansions by (isotropic) smooth atoms as described in (1.58)–(1.60) has proved very useful for several purposes, including in particular the study of pointwise multipliers in these spaces. It comes out that the arguments in [T06, Sect. 2.2] can be transferred to the spaces Spr B(Rn ) in (1.187) in a rather straightforward manner, including applications to pointwise multipliers. Let Rk,m with k ∈ N0n and m ∈ Zn be the same rectangles introduced in (1.63). There we explained also what is meant by CRk,m , C > 0. Again let Spr B(Rn ) be as in r B(Rn ) = S r C (Rn ). Recall [k] = Ín k where k ∈ Nn . Let σ be (1.139) with S∞ p j=1 j 0 as in (1.71). Definition 1.32. Let C > 1, 0 < p ≤ ∞,

σp < r < σ < ∞.

(1.189)

Then ak,m ∈ Spσ B(Rn ) is called an (r, p)σ -atom (more precisely an (r, p)σ -C-atom) if supp ak,m ⊂ C Rk,m, k ∈ N0n, m ∈ Zn (1.190) and

kak,m |Spσ B(Rn )k ≤ 2[k](σ−r) .

(1.191)

Remark 1.33. This is the direct counterpart of [T06, Def. 2.7, p. 131]. We always assume that d > 1 in (1.68) and C > 1 in (1.190) are fixed (but suitably chosen if compared) and that related quasi-norms in Spr B(Rn ) are selected in such a way that additional equivalence constants can be neglected. Proposition 1.34. Let p, r, σ be as in (1.189). (i) Let r < σ < K ∈ N. Then any (r, p)K -atom is an (r, p)σ -atom. (ii) Let ak,m be an (r, p)σ -atom. Then kak,m |Spr B(Rn )k ≤ 1

and

kak,m |L p (Rn )k ≤ 2−[k]r .

(1.192)

Proof. Step 1. This is the counterpart of [T06, Prop. 2.9, p. 131]. Part (i) follows from Theorem 1.10 applied to Spσ B(Rn ) and (1.68), (1.69).

33

1.2 Properties, I

Step 2. The proof of part (ii) is an adapted copy of the related arguments in [T06, p. 132]. But it seems to be reasonable to fix the details. It is sufficient to prove (1.192) for ak = ak,0 with  supp ak ⊂ C x : 0 < x j < 2−k j ,

k ∈ N0n .

(1.193)

With λ j = 2−k j in (1.171) one has by (1.175) and λx = (λ1 x1, . . . , λn xn ), 1

kak |Spr B(Rn )k ∼ 2[k](r− p ) kak (2−k ·) |Spr B(Rn )k 1

≤ 2[k](r− p ) kak (2−k ·) |Spσ B(Rn )k ≤ 2[k](r−σ) kak |Spσ B(Rn )k

(1.194)

≤ 1. This proves the first inequality in (1.192). The second assertion follows by the same arguments with L p (Rn ) in place of Spr (Rn ).  Remark 1.35. As already stated, the above proof, in particular (1.194), is the direct counterpart of related arguments in [T06, p. 132]. One has to replace the homogeneity s (Rn ) as covered by (1.168)–(1.170) by the for the isotropic spaces Bps (Rn ) = Bp, p homogeneity according to Theorem 1.29. For the spaces Spr B(Rn ) in (1.187) one has the atomic representation (1.73), (1.74) by smooth (r, p)K -atoms with K > r. If p = q then we shorten s p, p b in (1.66) to kλ |b p k = kλ |s p, p bk =



Õ

|λk,m | p

 1/p

,

(1.195)

k ∈N0n,m∈Z n

0 < p ≤ ∞ (usual modification if p = ∞). Otherwise we wish to extend Theorem 1.10 for the spaces Spr B(Rn ) to the related non-smooth atoms according to Definition 1.32. We are in a comfortable position. As explained in the proof of Proposition 1.34 one has to replace the homogeneity for suitable isotropic spaces Bps (Rn ) by the corresponding homogeneity of the above spaces Spr B(Rn ). This gives the possibility of transferring the expansion by non-smooth atoms of suitable isotropic spaces Bps (Rn ) according to [T06, Thm. 2.13, p. 133] to the spaces Spr B(Rn ) in (1.187) as follows. Let σp be as in (1.71). Theorem 1.36. Let n ∈ N, C > 1, 0 < p ≤ ∞ and σp < r < σ < ∞.

(1.196)

1 Spaces on Rn

34

Then Spr B(Rn ) is the collection of all f ∈ S 0(Rn ) which can be represented as Õ f = λk,m ak,m with λ ∈ b p, (1.197) k ∈N0n,m∈Z n

where ak,m are (r, p)σ -C-atoms according to Definition 1.32. Furthermore, k f |Spr B(Rn )k ∼ inf kλ |b p k

(1.198)

is an equivalent quasi-norm where the infimum is taken over all admissible representations (1.197). Proof. Step 1. If p ≤ 1 then one obtains from (1.192) that

Õ

p Õ

λk,m ak,m |Spr B(Rn ) ≤ c |λk,m | p kak,m |Spr B(Rn )k p < ∞.

k,m

(1.199)

k,m

In particular, the right-hand side of (1.197) converges in S 0(Rn ). If 1 < p ≤ ∞ then it follows from (1.190) and (1.192) that

Õ Õ  Õ  1/p Õ

λk,m ak,m |L p (Rn ) ≤ c 2−[k]r |λk,m | p < ∞. (1.200)

k ∈N0n m∈Z n

k ∈N0n

m∈Z n

This ensures again that the right-hand side of (1.197) converges in S 0(Rn ). By Proposition 1.34(ii) and Theorem 1.10 it remains to prove that k f |Spr B(Rn )k ≤ c kλ |b p k

(1.201)

for any expansion f according to (1.197). This follows immediately from (1.199) if p ≤ 1. Step 2. It remains to prove (1.201) for 1 < p ≤ ∞. But here we are in a comfortable position. Similarly to (1.194) being the mixed counterpart of [T06, (2.32), p. 132], one can transfer the arguments in [T06, Step 2, p. 134–136], line for line, from the related isotropic case Bps (Rn ) to Spr B(Rn ). This will not be repeated here.  Remark 1.37. If p ≤ 1 then one does not need (1.191) for some σ > r. It is sufficient to assume that the ak,m , localized by (1.190), are normed by 1 in Spr B(Rn ) as in the first part of (1.192) (the second part in (1.192) is again a consequence of the first part). Remark 1.38. Non-smooth atoms for the spaces Bps (Rn ) with (1.188) go back to [Tri03] where they are used to study pointwise multipliers in these spaces. Our

35

1.2 Properties, I

intention now is rather similar. Relying on non-smooth atoms as introduced in Definition 1.32 we deal in Section 1.2.6 with pointwise multipliers for the spaces covered by (1.187). However, the original intention to look for non-smooth building blocks was different. We wanted to contribute to the theory of function spaces, especially of Besov type, on non-smooth structures (instead of Rn ) such as quasimetric spaces furnished with a doubling Radon measure or, as a prototype, a compact fractal d-set in Rn with 0 < d < n. Then one has at best Lipschitz-continuity for corresponding intrinsic compact building blocks. But it came out that non-smooth atoms for the isotropic spaces in (1.188) are very helpful in this context. The corresponding theory may be found in [Tri05] and [T06, Chap. 8]. This approach has been extended in [CaL09] to other types of Besov spaces on quasi-metric spaces. On the other hand, the (originally second) application of the theory of non-smooth atomic decompositions to pointwise multipliers in diverse types of Besov spaces attracted some attention. We refer the reader to [MPP07, ScV12, ScV13, Scha13, GoK16, GoK17] where sometimes non-smooth is related to [TrW96], [ET96, Sect. 2.5, pp. 57–65], and also to [Tri05]. 1.2.5 Pointwise multipliers and localizations. Pointwise multipliers and multiplir A(Rn ) will play some role in these notes. cation properties for spaces of type Sp,q This Section 1.2.5 may be considered a first step in this direction. We deal with smooth pointwise multipliers belonging to  Cb∞ (Rn ) = w ∈ C ∞ (Rn ) : sup |Dα w(x)| < ∞ for all α ∈ N0n . (1.202) x ∈R n

Then, obviously, f 7→ w f maps

S(Rn ) into S(Rn ) and

S 0(Rn ) into S 0(Rn ).

(1.203)

It is well known that f 7→ w f is also a linear and bounded map in all (inhomogeneous) isotropic spaces Asp,q (Rn ) according to Definition 1.1; see [T92, Thm. 4.2.2, p. 203]. r A(Rn ) as introduced in We prove the counterpart of this assertion for the spaces Sp,q Definition 1.4 and Remark 1.5. Theorem 1.39. Let 0 < p ≤ ∞ (p < ∞ for F-spaces), 0 < q ≤ ∞ and r ∈ R. Let w ∈ Cb∞ (Rn ) according to (1.202). Then Tw : f 7→ w f

(1.204)

r A(Rn ) into itself. is a linear and bounded map from Sp,q

Proof. Step 1. In addition, let r > σp for B-spaces

and r > σp,q for F-spaces

(1.205)

1 Spaces on Rn

36

with σp , σp,q as in (1.71). Then one does not need moment conditions for atomic expansions in these spaces according to Theorem 1.10. In particular, if one multiplies f in (1.73) with w then one again has an atomic decomposition and |Dα w|

(1.206)

and r + 2l > σp,q for F-spaces.

(1.207)

kTw k ≤ c

sup

x ∈R n, |α | ≤K

with K as in (1.72), (1.75) for the operator norm. Step 2. Let r ∈ R and l ∈ N such that r + 2l > σp for B-spaces

r+2l A(Rn ). By (1.38), (1.39) we have the lifting Then we can apply Step 1 to Sp,q n Ö


l r+2l r id − ∂j2 Sp,q A(Rn ) = Sp,q A(Rn ).

(1.208)

j=1

Applied to f =

În

j=1 (id

r+2l A(Rn ) one has − ∂j2 )l g with g ∈ Sp,q

wf =

n Ö


l id − ∂j2 (wg) +

j=1

Õ

Dα (wα g),

(1.209)

α∈N0n, 0≤α j ≤2l

where wα ∈ Cb∞ (Rn ) are linear combinations of derivatives of w. By (1.208), a related assertion for Dα according to [ST87, Thms. 2.2.6/1, 2.2.6/2, pp. 98–99], and Step 1 it follows that Õ r r+2l r+2l kw f |Sp,q A(Rn )k ≤ c kwg |Sp,q A(Rn )k + c kwα g |Sp,q A(Rn )k α∈N0n, 0≤α j ≤2l r+2l ≤ c 0 kg |Sp,q A(Rn )k

(1.210)

r ∼ k f |Sp,q A(Rn )k. r A(Rn ) and completes the proof of the theorem. This applies to any f ∈ Sp,q

 We apply the above observation to discuss localization properties of the spaces r A(Rn ) and to introduce some uniform and self-similar versions of these spaces Sp,q which will be of some use later on for more sophisticated pointwise multiplier assertions.

37

1.2 Properties, I

First we describe what in [T92, Sect. 2.4.7, pp. 124–129] we called the localization principle for the (inhomogeneous) isotropic spaces Asp,q (Rn ). Let 1=

Õ

ψl (x),

x ∈ Rn,

(1.211)

l ∈Z n

where ψ is a compactly supported C ∞ function and ψl (x) = ψ(x − l). (i) Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R. Then s (Rn )k ∼ k f |Fp,q

Õ

s (Rn )k p kψl f |Fp,q

 1/p

,

s (Rn ) f ∈ Fp,q

(1.212)

l ∈Z n

(equivalent quasi-norms). (ii) Let 0 < p, q, v ≤ ∞ and s ∈ R. Then Õ

s (Rn )k v kψl f |Bp,q

 1/v

,

s (Rn ) f ∈ Bp,q

(1.213)

l ∈Z n s (Rn ) if, and only if, (usual modification if v = ∞) is an equivalent quasi-norm in Bp,q p = q = v.

The proof of this assertion in [T92, Sect. 2.4.7, pp. 124–128] is based on local means and explicit construction of counterexamples. Nowadays one would use wavelet expansions for the spaces Asp,q (Rn ) as described in Section 1.1.4 (similarly r A(Rn )). We are interested in the counterpart of this to below for the spaces Sp,q r assertion for the spaces Sp,q A(Rn ). Theorem 1.40. (i) Let 0 < p < ∞, 0 < q ≤ ∞, r ∈ R. Then r k f |Sp,q F(Rn )k ∼

Õ

r kψl f |Sp,q F(Rn )k p

 1/p

,

r f ∈ Sp,q F(Rn )

(1.214)

l ∈Z n

(equivalent quasi-norms). (ii) Let 0 < p, q, v ≤ ∞ and r ∈ R. Then Õ

r kψl f |Sp,q B(Rn )k v

 1/v

,

r f ∈ Sp,q B(Rn )

(1.215)

l ∈Z n r B(Rn ) if, and only (usual modification if v = ∞) is an equivalent quasi-norms in Sp,q if, p = q = v.

1 Spaces on Rn

38

r F(Rn ) by (1.107) with λ Proof. Step 1. We prove part (i). We expand f ∈ Sp,q k,m = Í λk,m ( f ) as in (1.104). Replacing f in (1.104), (1.101) by l ∈Zn ψl f then it follows from the locations of the supports of both of the wavelets ψk,m and ψl that the left-hand side of (1.212) can be estimated from above by the right-hand side. We prove the converse. It follows from Theorem 1.39 that ψl is a pointwise multiplier in r F(Rn ), uniformly in l ∈ Zn . Then one has by (1.101) that for some c > 0, C > 0 Sp,q and all l ∈ Zn , ∫ Õ  p/q (p) r n p kψl f |Sp,q F(R )k ≤ c |λk,m ( f ) χk,m (x)| q dx (1.216) {y:|y−l | ≤C }

k,m

(modification if q = ∞). Summation over l ∈ Zn shows that the right-hand side of (1.212) can be estimated from above by the left-hand side. r B(Rn ) = Step 2. We prove part (ii). The above proof covers (1.214) for Sp, p r n r n r n Sp, p F(R ), 0 < p < ∞, and can be extended to S C (R ) = S∞,∞ B(R ). We have to show that these are the only cases for which (1.215) is an equivalent quasi-norm in r B(Rn ). Let ψ ∈ C ∞ (R) with supp ψ ⊂ (−1, 1) and Sp,q Õ ψ(t − m) = 1, t ∈ R. (1.217) m∈Z

Then Õ

ψl (x) = 1,

ψl (x) =

n Ö

l ∈Z n

ψ(x j − l j ),

x ∈ Rn

(1.218)

j=1

is a special system of type (1.211). Let n ≥ 2 and f (x) =

n−1 Ö

ψ(x j ) g(xn ) with g ∈ Brp,q (R).

(1.219)

j=1

Then one has by Proposition 1.19, r k f |Sp,q B(Rn )k ∼ kg |Brp,q (R)k.

(1.220)

With ψl as above it follows again from Proposition 1.19, Õ l ∈Z n

r kψl f |Sp,q B(Rn )k v

 1/v

∼

Õ

kψ(· − m)g |Brp,q (R)k v

 1/v

(1.221)

m∈Z

(modification if v = ∞). If the left-hand sides (1.220) and (1.221) are equivalent r B(Rn ) then the related right-hand sides are equivalent quasiquasi-norms in Sp,q r norms in Bp,q (R). But as mentioned above this requires p = q = v. This proves

1.2 Properties, I

39

part (ii) for the above special system ψl . But the pointwise multiplier Theorem 1.39 shows that admitted resolutions of unity according to (1.211) can be reduced to the above special resolution of unity in (1.218). This completes the proof.  Remark 1.41. Theorem 1.39 shows also that (1.211) in the above theorem can be relaxed by Õ 0 < c1 ≤ ψl (x) ≤ c2 < ∞, x ∈ Rn . (1.222) l ∈Z n

So far, we have the multiplier Theorem 1.39 for smooth functions w ∈ Cb∞ (Rn ). We are interested in a more detailed description of pointwise multipliers in some r A(Rn ). This will be done in Section 1.2.6, again largely parallel to corspaces Sp,q responding assertions for the isotropic spaces Asp,q (Rn ) in [T06, Sect. 2.3], based on [Tri03]. For this purpose we introduced in [T06, Def. 2.19, p. 137] the corresponding uniform and self-similar spaces as follows. Let {ψl }l ∈Zn be the resolution of unity according to (1.211) or its modification (1.222) based on ψ. Let A(Rn ) = Asp,q (Rn ) with 0 < p ≤ ∞ (p < ∞ for F-spaces), 0 < q ≤ ∞, s ∈ R be the isotropic spaces as introduced in Definition 1.1, Remark 1.2. Then Aunif (Rn ) is the collection of all f ∈ S 0(Rn ) such that k f | Aunif (Rn )kψ = sup kψl f | A(Rn )k < ∞

(1.223)

l ∈Z n

and Aselfs (Rn ) is the collection of all f ∈ S 0(Rn ) such that k f | Aselfs (Rn )kψ =

sup

j ∈N0,l ∈Z n

kψl (·) f (2−j ·) | A(Rn )k < ∞.

(1.224)

Elementary pointwise multiplier assertions for Asp,q (Rn ) show that these spaces are independent of ψ. We used these spaces to have a closer look at (non-smooth) pointwise multipliers in Asp,q (Rn ). We return to this point in Section 1.2.6. For the same reason we introduce now the dominating mixed counterparts of the above uniform and self-similar versions of the isotropic spaces Asp,q (Rn ). In a slight abuse of notation we let λx = (λ1 x1, . . . , λn xn ) if λ = (λ1, . . . , λn ) ∈ Rn , x = (x1, . . . , xn ) ∈ Rn , as well as λx = (λx1, . . . , λxn ) if λ ∈ R, x = (x1, . . . , xn ) ∈ Rn . Which is meant will be clear from the context. Definition 1.42. Let ψ with (1.211) be as above or as indicated in Remark 1.41. Let r A(Rn ) with A ∈ {B, F} as introduced in Definition 1.4. Then A n A(Rn ) = Sp,q unif (R ) 0 n is the collection of all f ∈ S (R ) such that k f | Aunif (Rn )kψ = sup kψl f | A(Rn )k < ∞ l ∈Z n

(1.225)

1 Spaces on Rn

40

and Aselfs (Rn ) is the collection of all f ∈ S 0(Rn ) such that k f | Aselfs (Rn )kψ =

sup

k ∈N0n,l ∈Z n

kψl (·) f (2−k ·) | A(Rn )k < ∞.

(1.226)

Remark 1.43. This
is the direct counterpart of (1.223), (1.224). Recall 2−k x = 2−k1 x1, . . . , 2−kn xn . It follows from Theorem 1.39 and Remark 1.41 that both Aunif (Rn ) and Aselfs (Rn ) are independent of ψ (equivalent quasi-norms). This justifies our omission of ψ on the left-hand sides of (1.225), (1.226) in the sequel. Again by Theorem 1.39 one has A(Rn ) ,→ Aunif (Rn ) and

Aselfs (Rn ) ,→ Aunif (Rn ).

(1.227)

Furthermore, there is no preference for the origin in (1.226). This follows from
sup kψ(· − l) f 2−k · | A(Rn )k k ∈N0n,l ∈Z n

∼

sup

y,z ∈R n, λ=(λ1,...,λ n ),0 σp(n) according to (1.57). The basic assertions obtained there, especially s (Rn ), rely on the localization as described in (1.213) and the for Bps (Rn ) = Bp, p local homogeneity according to (1.168)–(1.170). A more detailed formulation will be given later on; see (1.367)–(1.369). Theorems 1.29 and 1.40 show that these instruments are available for Spr B(Rn ) with 0 < p ≤ ∞ and r > σp,

(1.230)

where we used the abbreviations introduced in (1.71) and (1.139). But we split the counterparts of [T06, Sect. 2.3, pp. 136–146] and shift some considerations to

1.2 Properties, I

41

Section 1.4.3 below. In the present Section 1.2.6 we rely mainly on expansions by non-smooth atoms according to Theorem 1.36 combined with the local homogeneity according to Theorem 1.29 for the spaces in (1.230). As already mentioned we have (1.134) which can be strengthened as in (1.187) by r Sp,q A(Rn ) ,→ Lv (Rn ),

0 < p ≤ ∞, r > σp

(1.231)

(p < ∞ for F-spaces), 0 < q ≤ ∞, for some v with 1 < v ≤ ∞. This gives the possibility of transferring [T06, Def. 2.15, p. 136] from the isotropic spaces Asp,q (Rn ) r A(Rn ) as follows. Again let σ be as in (1.71). to the spaces Sp,q p r A(Rn ) according to Definition 1.4, A ∈ {B, F}, Definition 1.44. (i) Let A(Rn ) be Sp,q with 0 < p ≤ ∞ (p < ∞ for F-spaces), 0 < q ≤ ∞, r > σp . A locally integrable function w on Rn is called a pointwise multiplier for A(Rn ) if

generates a bounded map in A(Rn ).
The collection of all pointwise multipliers is denoted by M A(Rn ) . Tw : f 7→ w f

(1.232)

(ii) A quasi-Banach space A(Rn ) on Rn with S(Rn ) ,→ A(Rn ) ,→ S 0(Rn ),

A(Rn ) ⊂ L1loc (Rn )

(1.233)

is said to be a multiplication algebra if w1 w2 ∈ A(Rn ) for w1 ∈ A(Rn ), w2 ∈ A(Rn ) and if there is a number c > 0 such that kw1 w2 | A(Rn )k ≤ c kw1 | A(Rn )k · kw2 | A(Rn )k

(1.234)

for all w1 ∈ A(Rn ), w2 ∈ A(Rn ). Remark 1.45. It follows from (1.231) that w f in (1.232) makes sense pointwise and one may ask under what circumstances w f can be interpreted in S 0(Rn ) and whether Tw generates a linear and bounded map in A(Rn ). Then Tw , or w for short, belongs to the quasi-Banach space of all linear and bounded maps of A(Rn ) into itself, 
kw |M A(Rn ) k = sup kw f | A(Rn )k : k f | A(Rn )k ≤ 1 . (1.235) If w ∈ Cb∞ (Rn ) then w f is well defined for any f ∈ S 0(Rn ) and we have Theorem 1.39 r A(Rn ) without the restriction r > σ . But otherwise, to say what for all spaces Sp,q p r A(Rn ) (without the assumption is meant by w f with f ∈ Asp,q (Rn ) or f ∈ Sp,q r > σp ) is slightly tricky. A detailed discussion based on the Fourier-analytical definition of Asp,q (Rn ) may be found in [RuS96, Chap. 4]. These considerations can r A(Rn ) as introduced in Definition 1.4. One may also consult [T83, be extended to Sp,q Sect. 2.8] and [T92, Sect. 4.2] where one finds further references. Our restriction to spaces A(Rn ) consisting of regular tempered distributions ensures that the above definition is adequate. Nevertheless it seems to be desirable to add a few (more or less well-known) comments.

1 Spaces on Rn

42


Proposition 1.46. The collection M A(Rn ) of all pointwise multipliers in A(Rn ) = r A(Rn ) according to Definition 1.44(i) and (1.235) is a translation-invariant Sp,q quasi-Banach space and a multiplication algebra with
M A(Rn ) ,→ L∞ (Rn ). (1.236) Proof. Step 1.
We prove (1.236). Let ϕ ∈ D(Rn ) with ϕ(x) = 1 if |x| ≤ N ∈ N and w ∈ M A(Rn ) . Then w K ϕ ∈ A(Rn ), K ∈ N, and one obtains from (1.231), ∫  1/K  ∫  1/K K |w(x)| dx ≤ |w(x)| K |ϕ(x)| dx |x | ≤ N

≤c

Rn 1/K

≤ c1/K

kw K ϕ | A(Rn )k 1/K
kw |M A(Rn ) k · kϕ | A(Rn )k 1/K

(1.237)

for some c > 0. Then K → ∞ proves
kw |L∞ (Rn )k ≤ kw |M A(Rn ) k

(1.238)

and (1.236).

Step 2. We prove that M A(Rn ) is a closed subspace of L A(Rn ) , being the quasiBanach space of all linear and bounded maps of A(Rn ) into itself. Let {Tw j }∞ j=1 be a
n corresponding Cauchy sequence and T ∈ L A(R ) its limit. Using (1.231) we may assume that for any f ∈ A(Rn ), (w j f )(x) = (Tw j f )(x) → (T f )(x) if j → ∞

(1.239)

a.e. in Rn . On the other hand, it follows from (1.238) that we may also assume w j (x) → w(x) if j → ∞ for some w(x) ∈ L∞ (Rn ). Together with (1.239) one

n obtains T = Tw . In particular, M A(R ) is a closed subspace of L A(Rn ) . All other assertions of the above proposition are obvious.  Remark 1.47. The elegant argument (1.237) resulting in (1.238) goes back to [MaS85, Prop. 4, p. 47] in the wider context of pointwise multipliers. Further comments and references may be found in [T06, Sect. 2.3.1, pp. 136–137]. We need the following complement of (1.227) and (1.229). Let Aselfs (Rn ) with r A(Rn ) be as in Definition 1.42 and (1.231), in particular r > σ ≥ 0. A(Rn ) = Sp,q p Then Aselfs (Rn ) ,→ L∞ (Rn ). (1.240) This follows for p = ∞ from Aselfs (Rn ) ,→ Aunif (Rn ) ,→ Sr Cunif (Rn ) = Sr C (Rn ) ,→ C(Rn ),

(1.241)

43

1.2 Properties, I

where we used (1.227), elementary embeddings according to (1.109) as far as q is concerned, Theorem 1.40(ii) with p = q = ∞ and (1.47). For p < ∞ one has (1.231) with some 1 < v < ∞. Then it follows from (1.228) with 0 < λ j = λ < 1 and λ

−n

∫

| f (z − x)| dx = v

|x | ≤Cλ

∫

| f (z − λx)| v dx ≤ c k f | Aselfs (Rn )k v (1.242) |x | ≤C

that f is uniformly bounded at its Lebesgue points. This proves (1.240). After this preparation one can carry over [T06, Prop. 2.22(ii), Thm. 2.25, pp. 140–142] from s (Rn ) to the spaces S r B(Rn ) = S r B(Rn ) with the isotropic spaces Bps (Rn ) = Bp, p, p p p r n r n S C (R ) = S∞ B(R ). A more detailed formulation will be given later on in (1.367)–
(1.369). Let M Spr B(Rn ) be the space of all pointwise multipliers as introduced in Definition 1.44 for the spaces A(Rn ) = Spr B(Rn ) according to (1.230). Let Aselfs (Rn ) be the related spaces according to Definition 1.42. Then we have both (1.236) and (1.240). Theorem 1.48. Let 0 < p ≤ ∞ and

σp < r < σ.

(1.243)

Then
Spσ Bselfs (Rn ) ,→ M Spr B(Rn ) ,→ Spr Bselfs (Rn ).

(1.244)

If, in addition, p ≤ 1 then
Spr Bselfs (Rn ) = M Spr B(Rn ) .

(1.245)

Proof. Step 1. We prove the right-hand side of (1.244). Let w ∈ M(A) with A = Spr B(Rn ) and let ψ be as in (1.211) and (1.226). Then one has by Theorem 1.29 Í applied to ψ(2k ·)w with 2k x = (2k1 x1, . . . , 2kn xn ) and [k] = nj=1 k j , 1

kψw(2−k ·) | Ak ∼ 2−[k](r− p ) kψ(2k ·)w | Ak 1

≤ kw |M(A)k · 2−[k](r− p ) kψ(2k ·) | Ak ≤ c kw |M(A)k.

(1.246)

The right-hand side of (1.244) follows from (1.226) and the translation invariance both of A and M(A). Step 2. We prove the left-hand side of (1.244). Let Õ f = µk,m ak,m, µ = {µk,m } ∈ b p k ∈N0n,m∈Z n

(1.247)

44

1 Spaces on Rn

be an optimal decomposition of f ∈ Spr B(Rn ) by normalized (r, p)-atoms according to Definition 1.8 and Theorem 1.10, where we used s p, p b = b p as previously in (1.195). We wish to prove that wak,m with w ∈ Spσ Bselfs (Rn ) in Õ wf = µk,m (w ak,m ) (1.248) k ∈N0n,m∈Z n

(after uniform normalization) are (r, p)σ -atoms according to Definition 1.32. The support condition (1.190) is obvious. Let ak = ak,m with m = 0. Then supp ak (2−k ·) ⊂ {y : |y| ≤ C},

k ∈ N0n,

for some C > 0. One has by Theorem 1.39 and (1.206) for some c > 0, 1
kak (2−k ·) |M Spσ B(Rn ) k ≤ c 2−[k](r− p ), k ∈ N0n .

(1.249)

(1.250)

With the help of a suitable function ψ one obtains from Theorem 1.29, 1

kwak |Spσ B(Rn )k ∼ 2[k](σ− p ) kψ w(2−k ·) ak (2−k ·) |Spσ B(Rn )k ≤ c 2[k](σ−r) kψ w(2−k ·) |Spσ B(Rn )k

(1.251)

≤ c 2[k](σ−r) kw |Spσ Bselfs (Rn )k. It follows from the translation invariance both of Spσ B(Rn ) and Spσ Bselfs (Rn ) that kw ak,m |Spσ B(Rn )k ≤ c 2[k](σ−r) kw |Spσ Bselfs (Rn )k,

k ∈ N0n, m ∈ Zn . (1.252)

Application of Theorem 1.36, based on Definition 1.32, shows that w f ∈ Spr B(Rn ) and kw f |Spr B(Rn )k ≤ c kw |Spσ Bselfs (Rn )k · k f |Spr B(Rn )k. (1.253) This proves the left-hand side of (1.244). Step 3. Let, in addition, p ≤ 1. Then it follows from (1.252) with σ = r, Remark 1.37 and (1.199) that kw f |Spr B(Rn )k ≤ c kw |Spr Bselfs (Rn )k · k f |Spr B(Rn )k.

(1.254)

This shows that one can replace σ on the left-hand side of (1.244) by r resulting in (1.245).  Remark 1.49. The above proof is essentially a copy of corresponding arguments in the context of some isotropic spaces Asp,q (Rn ) according to [T06, Sect. 2.3.3, pp. 140– 143]. We discussed in [T06, Sect. 2.3, pp. 136–146] how conditions ensuring that
A(Rn ) is a multiplication algebra are related to pointwise multiplier spaces M A(Rn ) , uniform spaces Aunif (Rn ) and self-similar spaces Aselfs (Rn ). We return to this point in Section 1.4.3 below.

45

1.2 Properties, I

1.2.7 Local embeddings and isomorphic structure. In Section 1.3.2 below we r A(Rn ) and compare them recall some known embeddings between the spaces Sp,q s s n with related assertions for Ap,q (R) and Ap,q (R ). We introduced and recalled in Definition 1.22 and Remark 1.23 corresponding spaces on domains, repeated also in (1.265) and (1.266) below. In contrast to the spaces Asp,q (Ω) it does not make r A(Ω) if Ω is an arbitrary (bounded smooth) much sense to deal with the spaces Sp,q domain. After further discussion in Section 1.3 we will summarize our point of view r A[Ω, 0 such that f ∈ Brp0,q0 (Rn ) and

f < Brp1,q1 (Rn ).

(1.261)

This proves the only-if part of (1.255). Inserting f in (1.87) one obtains ∞

 Õ

r k f |Fp,q (Rn )k ∼

Õ

2 jr q |λ j | q χj,m (·)

 1/q

|L p (Rn )

j=0 0≤ml ≤2 j ∞

 Õ

 1/q

∼ 2 jr q |λ j | q |L p (Q)

∼

j=0 ∞ Õ

2 jr q |λ j | q

 1/q

(1.262)

.

j=0

This is the same as in (1.260) and a suitable choice of λ j results in the F-version of (1.261). Step 3. We extend (1.259) to f =

∞ Õ Õ

j

λ j,m ψG,m,

G = {M, . . . , M }.

(1.263)

j=0 m∈Z n r A(Rn ) according to (1.103) with k = · · · = This is an admitted expansion in Sp,q 1 n k n = j. Then [k](r − p1 ) = j(nr − np ). With µ j,m = λ j,m 2 j(nr− p ) the quasi-norms in (1.100) and (1.86) on the one hand, and in (1.101) and (1.87) on the other hand, coincide. This shows that

 j span ψG,m : j ∈ N0, m ∈ Zn, G = {M, . . . , M }

(1.264)

n (Rn ) and S r A(Rn ) with respect spans the same complemented subspace of both Arp,q p,q to the sequence spaces {λ j,m }. If one assumes that one has (1.257) or (1.258) then enr one also obtains a corresponding embedding for the related subspaces of A p,q (Ω). But by Step 2 this requires q0 ≤ q1 . 

Recall that  Asp,q (Ω) = f ∈ D 0(Ω) : f = g|Ω for some g ∈ Asp,q (Rn ) , kf

| Asp,q (Ω)k

= inf kg

| Asp,q (Rn )k

(1.265) (1.266)

47

1.2 Properties, I

is the usual way to define the spaces Asp,q (Ω) on domains Ω in Rn by restriction, where the infimum is taken over all g ∈ Asp,q (Rn ) with g|Ω = f . It is the isotropic counterpart of (1.126)–(1.128). Let the restriction operator re, where re g = g|Ω : Asp,q (Rn ) ,→ Asp,q (Ω),

(1.267)

be the classical isotropic counterpart of (1.135). As there, one asks whether there is a linear and bounded extension operator ext, ext : Asp,q (Ω) ,→ Asp,q (Rn ),

ext f |Ω = f ,

(1.268)

rewritten as re ◦ ext = id


identity in Asp,q (Ω) .

(1.269)

Some comments and references may be found in Remark 1.24. We return to the extension problem in Section 1.3.7 below, preferably in the context of the spaces r A(Rn ). If Ω is a bounded Lipschitz domain in Rn according to [T06, Def. 1.103, Sp,q p. 64] then there is a universal extension operator for all spaces Asp,q (Ω); see [Ryc99b]. Related references will be given in Section 1.3.7. One can use this observation to reformulate (1.255), (1.256) in a more handsome way as follows. Corollary 1.51. Let Ω be a bounded Lipschitz domain in Rn (interval if n = 1). Let 0 < p1 ≤ p0 ≤ ∞ (p0 < ∞ for F-spaces), 0 < q1 ≤ ∞, 0 < q0 ≤ ∞ and s ∈ R. Then Bps 0,q0 (Ω) ,→ Bps 1,q1 (Ω) if, and only if, q0 ≤ q1,

(1.270)

Fps0,q0 (Ω) ,→ Fps1,q1 (Ω) if, and only if, q0 ≤ q1 .

(1.271)

Proof. This follows by standard arguments from (1.255), (1.256) and (1.268) applied to bounded Lipschitz domains.  Remark 1.52. At least the if parts of the above corollary are well known and mentioned occasionally in passing. They follow immediately from the characterization of Asp,q (Rn ) in terms of local means; see [T92, Thms. 2.4.6, 2.5.3, pp. 122, 138], [T06, Thm. 1.10, Rem. 4.41, pp. 10, 227] and Hölder’s inequality for L p -spaces. In the same way one can justify the if parts of (1.257), (1.258) using the characterizations r A(Rn ) in terms of local means according to [Vyb06, Thm. 1.25, p. 23]. The of Sp,q only-if parts of the above corollary are something like mathematical folklore: no doubt, no proof. But this question has attracted some attention recently. One may consult [MiS15] and the references within. There one finds the following assertion. Let Ω be a bounded domain in Rn and 0 < s < N, 1 ≤ p1 ≤ p0 ≤ ∞. Then Bps 0 (Ω) ,→ Bps 1 (Ω) if, and only if, p0 = p1,

(1.272)

s (Ω). There is an immediate counterpart for the spaces where again Bps (Ω) = Bp, p r r Sp B(Ω) = Sp, p B(Ω).

1 Spaces on Rn

48

Corollary 1.53. Let Ω be a bounded domain in Rn , n ∈ N. Let r ∈ R and 0 < p1 ≤ p0 ≤ ∞. Then Brp0 (Ω) ,→ Brp1 (Ω)

if, and only if, p0 = p1 ,

(1.273)

,→

if, and only if, p0 = p1 .

(1.274)

Spr 0 B(Ω)

Spr 1 B(Ω)

Proof. Let ω be a domain with ω ⊂ Ω and let w ∈ D(ω) = C0∞ (ω) with w(x) = 1 in some ball. We assume that the embedding in (1.274) holds for some p1 < p0 . Then we combine Sepr 0 B(ω) ,→ Spr 0 B(Ω) ,→ Spr 1 B(Ω) (1.275) with f 7→ w f : Spr 1 B(Ω) ,→ Sepr 1 B(ω)

(1.276)

according to Theorem 1.39. But this contradicts the constructions in Steps 2 and 3 of the proof of Theorem 1.50, and similarly for (1.273).  Let Ω be a bounded Lipschitz domain in Rn . Let r ∈ N and 1 < p < ∞. Then the classical Sobolev spaces W pr (Ω) defined in the usual way as a restriction of W pr (Rn ) to Ω can be equivalently normed by Õ k f |W pr (Ω)k = kDα f |L p (Ω)k; (1.277) |α | ≤r

see [T78, Thm. 4.2.4, p. 316], [T06, Thm. 1.122, p. 77]. Then one has by Hölder’s inequality in contrast to (1.273), W pr 0 (Ω) ,→ W pr 1 (Ω),

1 < p1 ≤ p0 < ∞, r ∈ N.

(1.278)

There is a counterpart at least for the Sobolev spaces Spr W(Q) with Q = (0, 1)n , 1 < p < ∞ and r ∈ N. They can be equivalently normed by Õ kDα f |L p (Q)k; (1.279) k f |Spr W(Q)k = α∈N0n, 0≤α j ≤r

see (1.145), Proposition 1.25. Then by Hölder’s inequality again one has Spr 0 W(Q) ,→ Spr 1 W(Q),

1 < p1 ≤ p0 < ∞, r ∈ N,

(1.280)

in contrast to (1.274). One must be aware that the isomorphic structure of the Sobolev spaces of type W pr , Spr W with 1 < p < ∞, r ∈ N on the one hand, and of the Besov spaces Brp , Spr B r B with r ∈ R, 0 < p ≤ ∞, 0 < q ≤ ∞ on the other and more generally Brp,q , Sp,q

49

1.2 Properties, I

hand, is totally different. For this reason it seems to be reasonable to abbreviate Brp, p as Brp and not as W pr , 0 < r < N, 1 < p < ∞, squeezing the classical Sobolev spaces W pr , 1 < p < ∞, r ∈ N0 , and the Slobodeckij spaces W pr , 1 < p < ∞, 0 < r < N, notationally into the same scale. The notation W pr , 0 < r < N, 1 < p < ∞ goes back to Slobodeckij in the mid-1950s, when nothing was known about the isomorphic structure of these spaces. Although we will not use it later on it might be of some interest to add a few comments about the isomorphic structure of the above spaces. If A and B are two quasi-Banach spaces then A  B means that there is a linear isomorphic map of A onto B. First we remark that Spr W(Rn )  W pr (Rn )  L p (I),

r ∈ N0, 1 < p < ∞,

(1.281)

with I = (0, 1). This is covered by the lifts (1.16), (1.18) for W pr (Rn ) and (1.38)–(1.42) for Spr W(Rn ) combined with L p (Ω)  L p (I),

1 < p < ∞, Ω a domain in Rn ;

(1.282)

see [T78, Thm. 2.11.1/1, p. 236] and the references given there. Second, we clarify r B(Rn ) as introduced in the isomorphic structure of the spaces Brp,q (Rn ) and Sp,q Definitions 1.1 and 1.4. Let `q (`p ) with 0 < p, q ≤ ∞ be the collection of all sequences  λ = λ j,k ∈ C : j ∈ N, k ∈ N (1.283) such that kλ |`q (`p )k =

Õ ∞ ∞ Õ j=1

|λ j,k |

p

 q/p  1/q

(1.284)

k=1

is finite (usual modification if max(p, q) = ∞). Of course, `p (`p )  `p , where `p has the usual meaning. Then r Brp,q (Rn )  Sp,q B(Rn )  `q (`p ),

r ∈ R, 0 < p ≤ ∞, 0 < q ≤ ∞

(1.285)

follows from the wavelet representation (1.89)–(1.91) and Theorem 1.12(i). The isomorphic structure of the spaces L p (I) and `q (`p ) is rather different. We collect some related assertions. (i) Let 1 < p0, p1 < ∞. Then L p0 (I)  L p1 (I) if, and only if, p0 = p1, L p0 (I)  `p1 if, and only if, p0 = p1 = 2.

(1.286) (1.287)

This is a very classical assertion going back to [Ban32, Chap. 12]. Further information and references may be found in [T78, Sect. 2.11.1, pp. 236–237].

50

1 Spaces on Rn

(ii) Let 0 < q0, q1, p0, p1 ≤ ∞. Then `q0 (`p0 )  `q1 (`p1 ) if, and only if, q0 = q1 and p0 = p1 ;

(1.288)

see [AlA15]. This generalizes an earlier assertion in [CeM11] where all parameters involved are assumed to be between 1 and infinity (Banach spaces). This shows that the spaces in (1.281) and (1.285) belong (in general) to different isomorphic classes. Below we wish to look at isomorphic classes of these spaces on domains. Then one needs the following modification of the spaces `q (`p ). Again let K 0 < p, q ≤ ∞ and K = {K j }∞ j=1 with K j ∈ N. Then `q (`p ) collects all sequences  λ = λ j,k ∈ C : j ∈ N, k = 1, . . . , K j such that kλ |`q (`p )K k =

Õ Kj ∞ Õ j=1

|λ j,k | p

 q/p  1/q

(1.289)

(1.290)

k=1

is finite (usual modification if max(p, q) = ∞). Let `q (`p )◦ = `q (`p )K if K j = j. According to [AlA17] one has `q (`p )K  `q (`p )◦

if K = {K j } with K j → ∞.

(1.291)

Furthermore, one finds in [AlA17] a final answer under which conditions `q0 (`p0 )◦ and `q1 (`p1 ) are isomorphic. In addition to the obvious case p0 = q0 = p1 = q1 there are a few (rather specific) cases. But with these exceptions the spaces `q0 (`p0 )◦ and `q1 (`p1 ) are not isomorphic. A decisive tool for the step from Rn in (1.281), (1.285) to corresponding spaces on domains is the so-called Pełczyński decomposition technique. A subspace B of a quasi-Banach space A is called complemented if there is a linear and bounded operator P, called projection, with P : A ,→ A,

P A = B, P2 = P.

(1.292)

If b ∈ B and b = Pa for some a ∈ A then Pb = P2 a = Pa = b,

b ∈ B.

(1.293)

In particular, P, restricted to B, is the identity and B = {a ∈ A : Pa = a}. Proposition 1.54. Let A and B be two quasi-Banach spaces and let `p (A)  A for some 0 < p ≤ ∞. If A is isomorphic to a complemented subspace of B and B is isomorphic to a complemented subspace of A then A  B.

1.2 Properties, I

51

Remark 1.55. We refer the reader to [T78, Lem. 3.7, pp. 289–290] or [AlK06, Thm. 2.2.3, p. 34] for a short elegant proof (extended from Banach spaces to quasiBanach spaces) going back to [Pel60]. Further comments and (historical) references may be found in [Pie07, Sect. 4.9.3, pp. 134–137]. In our context (1.268), (1.269) is a typical example with the projection P = ext ◦ re

of Asp,q (Rn ) onto P Asp,q (Rn )

(1.294)

and ext as the isomorphic map of Asp,q (Ω) onto P Asp,q (Rn ), where Ω is a bounded Lipschitz domain in Rn according to [T06, Def. 1.103, p. 64]. Recall I = (0, 1). Theorem 1.56. (i) Let Ω be a bounded Lipschitz domain in Rn , n ∈ N (bounded interval if n = 1). Then W pr (Ω)  L p (I),

1 < p < ∞, r ∈ N0

(1.295)

and Brp,q (Ω)  `q (`p )◦,

1 < p < ∞, 0 < q ≤ ∞, r ∈ R.

(1.296)

1 < p < ∞, r ∈ N0

(1.297)

(ii) Let Q = (0, 1)n . Then Spr W(Q)  L p (I), and r Sp,q B(Q)  `q (`p )◦,

0 < p, q ≤ ∞,

1
1 − 1 < r < min , 1 p p

(1.298)

0 < p, q < ∞,

1 1
< r < 1 + min , 1 . p p

(1.299)

(1 < q ≤ ∞ if p = ∞), r Sp,q B(Q)  `q (`p )◦,

Proof. Step 1. We prove (1.295). If Ω is a bounded C ∞ domain in Rn then the desired assertion follows from W pr (Ω)  L p (Ω) according to [T78, Thms. 4.9.2, 4.9.3, pp. 335, 337], combined with (1.282). If Ω is a bounded Lipschitz domain and if Ω1 , Ω2 are bounded C ∞ domains with Ω1 ⊂ Ω and Ω ⊂ Ω2 then one has (1.268), (1.269) and (1.294) with Ω1 , Ω and Ω, Ω2 in place of Ω, Rn . We apply Proposition 1.54 with A = L p (I)  W pr (Ωl ), l = 1, 2 and B = W pr (Ω) combined with some translation and dilation. This proves (1.295).

1 Spaces on Rn

52

Step 2. We prove (1.296). Using (1.291),   1
1 1
if 0 < p, q ≤ ∞, max n − 1 , − 1 < r < min , 1 p p p (1.300) follows from the isomorphic representation of these spaces in terms of Haar bases according to [T10, Thm. 2.26, p. 96]. We wish to apply Proposition 1.54 with A = `q (`p )◦ and q in place of p, Brp,q (Q)  `q (`p )◦

  `q `q (`p )◦  `q (`p )◦ . We distribute N such that N =

Ð∞

l=1

(1.301)

K l with K l = {K jl }∞ j=1 in a one-to-one way. Then

 l `q `q (`p )K = `q (`p )◦

(1.302)

and (1.301) follows from (1.291) (possible equivalence constants do not matter). If Ω is a bounded Lipschitz domain then one can use again (1.268), (1.269) and (1.294) with Brp,q in place of Asp,q and Brp,q (Q) as the reference space. Then one obtains  1 
1 1
if 0 < p, q ≤ ∞, max n − 1 , − 1 < r < min , 1 , p p p (1.303) where Ω is now a bounded Lipschitz domain. If Ω is a bounded C ∞ domain and r ∈ R, 1 < p < ∞, 0 < q ≤ ∞, then one can use again the lifts according to [T78, Thm. 4.9.2, p. 335] extended from 1 ≤ q ≤ ∞ to 0 < q ≤ ∞ by real interpolation. This proves (1.296) for bounded C ∞ domains. Afterwards one can argue as in Step 1 and extend this assertion to bounded Lipschitz domains. Brp,q (Ω)  `q (`p )◦

Step 3. We prove (1.297). By the above method based on the extension (1.148) and (1.281) it follows that Spr W(Q) is isomorphic to a complemented subspace of L p (I). Let Q1 ⊂ Rn be a cube and let f ∈ L p (Rn ) = Sp0 F(Rn ), 1 < p < ∞ with supp f ⊂ Q1 be expanded according to Theorem 1.12. Let Q2 be a second cube such that all supp ψk,m ⊂ Q2 for all wavelets involved. Then L p (Q1 ) is isomorphic to a complemented subspace of Spr W(Q2 ). In particular, L p (I) is isomorphic to a complemented subspace of Spr W(Q2 ). Then one can argue as above based on Proposition 1.54 with A = L p (I). This proves (1.297). The isomorphic structure r B(Q) in (1.298) and (1.299) follows from the expansions in these spaces in of Sp,q terms of Haar tensor bases and Faber bases according to [T10, Thms. 2.41, 3.16, pp. 108, 145] and their indicated extension from n = 2 to n ∈ N in [T10, Sects. 2.4.5, 3.2.5, pp. 109–111, 152–154]. 

1.3 Intermezzo: Key problems

53

Remark 1.57. Let 0 < p, q ≤ ∞ and r ∈ R. Then there are good reasons to conjecture that r Brp,q (Ω)  Sp,q B(Q)  `q (`p )◦, (1.304) where Ω is a bounded Lipschitz domain in Rn and Q = (0, 1)n . This is supported by the above theorem and (1.303) where the restrictions for p, q, r are caused by our method. Further affirmative assertions may be found in [AlA17, Thm. 4.3 and the comments afterwards] dealing with Besov spaces denoted there as Brp,q [Q] with a reference to [DeP88], where they are defined intrinsically in terms of differences. But according to [DeS93] these spaces coincide with Brp,q (Q) introduced by restriction as in (1.265), (1.266). Further references and discussion may be found in [T06, Thm. 1.118, Rem. 1.119, pp. 74–76] and [T08, Sect. 4.3.3, pp. 117–122].

1.3 Intermezzo: Key problems Until now we have described (old and new) aspects of the theory of the spaces r A(Rn ) with dominating mixed smoothness parallel to corresponding properties Sp,q of the isotropic spaces Asp,q (Rn ). It has come out so far, and will also be supported by what follows, that there are three types of problem and related assertions. r A(Rn ), n ∈ N are strikingly similar to reType I. A few properties for the spaces Sp,q s lated ones for the spaces Ap,q (R) in one dimension. This is considered a strong r A(Rn ) compared with As (Rn ) in motivation and justification to prefer Sp,q p,q adequate problems when switching from one to several dimensions. r A(Rn ) and As (R) Type II. In some cases one has again parallel assertions for Sp,q p,q r A(Rn ) having in one dimension complemented by some new properties for Sp,q no counterpart for Asp,q (R) or Asp,q (Rn ). r A(Rn ) Type III. One may take it as a further justification to study the spaces Sp,q that there are some aspects which are totally different from corresponding comparable properties for the isotropic spaces Asp,q (Rn ).

It is one aim of Chapter 1 of these notes to indicate to which of the above three r A(Rn ) might belong (provided that it types of assertions a specific problem for Sp,q fits in this scheme). For this purpose it seems to be reasonable to touch upon key properties and technicalities for the isotropic spaces Asp,q (Rn ) and to ask how their r A(Rn ) look or may look. On the one hand, we deal in counterparts for the spaces Sp,q these notes, preferably, with those topics where we have something new to say. But on the other hand it is reasonable to list further crucial properties on which we rely and to give some references. But in addition to the above three types it may happen

54

1 Spaces on Rn

that the dimension n simply does not matter. This applies for example to properties discussed in Section 1.1.5 including duality and complex interpolation. r A(Rn ) with 0 < 1.3.1 Fourier multipliers. Let A(Rn ) be either Asp,q (Rn ) or Sp,q p ≤ ∞ (p < ∞ for F-spaces), 0 < q ≤ ∞ and s ∈ R, r ∈ R. Let M be a C ∞ function in Rn such that all derivatives Dα M, α ∈ N0n have at most polynomial growth. Then M is said to be a (smooth) Fourier multiplier for A(Rn ) if there is a constant c > 0 such that

(M b f )∨ | A(Rn )k ≤ c k f | A(Rn )k, f ∈ A(Rn ). (1.305)

This coincides with [T83, Def. 2.3.7, p. 57] and [ST87, Def. 2.2.5, p. 95]. In particular, (M b f )∨ ∈ S 0(Rn ) is well defined for any f ∈ S 0(Rn ) and the request (1.305) (n) (1) makes sense. Let σp(n) , σp,q and σp = σp(1) , σp,q = σp,q be as in (1.57), (1.71). Let ϕ0 , ϕ1 be as in (1.4), (1.5) and kM |h2 1/p. Then r r tr Sp,q B(Rn ) = Sp,q B(Rn−1 ),

r r tr Sp,q F(Rn ) = Sp,q F(Rn−1 ).

(1.330)

Furthermore, there exists a linear and bounded extension operator ext with r r ext : Sp,q B(Rn−1 ) ,→ Sp,q B(Rn ),

r tr ◦ ext = id identity in Sp,q B(Rn−1 ) (1.331)

and r r ext : Sp,q F(Rn−1 ) ,→ Sp,q F(Rn ),

r tr ◦ ext = id identity in Sp,q F(Rn−1 ). (1.332)

r A(R) = Ar (R) may A proof of this assertion for n = 2 and the trace spaces Sp,q p,q be found in [ST87, Thm. 2.4.2, p. 133] based on the Fourier-analytical Definition 1.4. There is little doubt that the arguments given there can be extended from n = 2 to 2 ≤ n ∈ N. But it seems reasonable to give a short wavelet proof as follows. r A(Rn ) be given by (1.103) or (1.107). Then Let x = (x 0, 0) and let f ∈ Sp,q

f (x 0, 0) =

Õ

Õ

n−1 k 0 ∈N−1

m0 ∈Z n−1

with µk 0,m0 =

∞ Õ

0

1

µk 0,m0 2−[k ](r− p ) ψk 0,m0 + + +

1

λ(k 0,kn ),(m0,0) 2−kn (r− p ) ψkn,0 (0)

(1.333)

(1.334)

k n =−1

where + + + indicates terms (k 0, k n ), (m 0, mn ) with |mn | ≤ M for some M. Inserted in (1.100), (1.101) it follows from r > 1/p and the min(p, q)-triangle inequality, r r tr : Sp,q A(Rn ) ,→ Sp,q A(Rn−1 ).

(1.335)

Let ϕ be as in (1.25). Then it follows from Proposition 1.19 that g(x 0) 7→ ϕ(xn )g(x 0) r A(Rn−1 ) into S r A(Rn ). This proves (1.330) and is an extension operator from Sp,q p,q (1.331), (1.332).

59

1.3 Intermezzo: Key problems

We glance at limiting cases both for isotropic spaces and spaces with dominating mixed smoothness. First we remark that 1/p

tr Bp,1 (Rn ) = L p (Rn−1 ),

1 ≤ p < ∞, n ≥ 2.

(1.336)

But there is no linear and bounded extension operator with a counterpart of (1.328) as mentioned in [T83, Rem. 2.7.2/5, p. 139] where one also finds the related references. 1/p Later on, this remarkable observation was extended in several directions, Bp,q (Rn ) 1/p with 0 < p < ∞, q ≤ min(1, p), and Fp,q (Rn ), 0 < p ≤ 1, 0 < q ≤ ∞ (see [T92, n Sect. 4.4.3, p. 220]) and on d-sets in R , 0 < d < n (see [T97, Thm. 18.6, p. 139], [T06, Prop. 1.172, p. 105], [T08, Prop. 6.64, Rem. 6.65, p. 220]). There one also finds the necessary explanations of how these modifications must be understood. From the same arguments as in (1.333)–(1.335) it follows for 0 < p ≤ ∞, q ≤ min(1, p), 1/p

1/p

tr Sp,q B(Rn ) = Sp,q B(Rn−1 ).

(1.337)

The product property according to Proposition 1.19 ensures similarly to (1.331) that there is a linear and bounded extension operator (in sharp contrast to (1.336)). Although not obvious, one can assume that one has for 0 < p ≤ 1 and 0 < q ≤ ∞, 1/p

1/p

tr Sp,q F(Rn ) = Sp,q F(Rn−1 )

(1.338)

again with a counterpart of (1.332). But this must be checked. In all cases covered by (1.330) and (1.337), (1.338) one has r id : Sp,q A(Rn ) ,→ C(Rn ).

(1.339)

If r > 1/p then (1.339) follows from 1

r r id : Sp,q A(Rn ) ,→ Sp,∞ B(Rn ) ,→ Sr− p C (Rn ) ,→ C(Rn ),

(1.340)

where we used (1.109), (1.312), the notation (1.139) and (1.144). Furthermore, 1/p

0 Sp,q B(Rn ) ,→ S∞,1 B(Rn ) ,→ C(Rn )

(1.341)

if 0 < p ≤ ∞ and 0 < q ≤ 1 where the first embedding is covered by (1.312) and the second one by [ST87, Rem. 2.4.1/2, p. 132] (but both assertions follow easily from the wavelet representation Theorem 1.12). Using (1.323) with p ≤ 1 and (1.341) one has 1/p 0 Sp,q F(Rn ) ,→ S∞,1 B(Rn ) ,→ C(Rn ). (1.342) In the case of the isotropic spaces Asp,q (Rn ) the trace assertions (1.327) and (1.336) require that one has to clarify what this means. For the spaces with dominating

1 Spaces on Rn

60

r A(Rn ) considered so far, one has always (1.339) and the trace mixed smoothness Sp,q in (1.324), (1.325) can simply be taken pointwise. But the situation is different when one asks for traces on hyperplanes which are not parallel to the axes of coordinates. The simplest case is the diagonal

 Γ = x = (x1, x2 ) ∈ R2 : x1 = x2 in the plane R2 . Let trΓ be as in (1.324). Let 1 ≤ p ≤ ∞ and r > from [Tri89, Thm., p. 479] that min(2r− p1 ,r)

r trΓ Sp,1 B(R2 ) = Bp,1

(R).

(1.343) 1 2p .

Then it follows

(1.344)

Furthermore, there exists a linear and bounded extension operator similar to (1.331). 1 If 2p < r < p1 then this diagonal embedding is different from (1.330) and the embedding in C(Rn ) as in (1.340), (1.341) is not ensured any longer. Then one has to rely on the explanations given above for what is meant by (1.324) with Γ as in (1.343). The proof in [Tri89] relies on the Fourier-analytical definition of the related spaces. But nowadays one can argue more efficiently as follows. Of interest are the rectangles Rk,m according to (1.63) in R2 with Rk,m ∩ Γ , ∅. The restriction of f , expanded according to (1.103), to Γ is given by f |Γ =

∞ Õ Õ

1

Õ

1

µk,m 2−k1 (2r− p − p ) ψk1,m1 (x1 ) ψk2,m2 (x1 ) + + + (1.345)

k1 =−1 m1 ∈Z −1≤k2 ≤k1, R k, m ∩Γ,∅

with 1

µk,m = 2−(k1 −k2 )( p −r) λk,m,

(1.346)

where + + + indicates similar terms with x1 , caused by the controlled overlapping of m1 -terms ψk1,m1 for fixed k 1 , and related terms with x2 . Let 1 ≤ p < ∞ and 2r− 1

< r ≤ p1 . Then (1.345) is an atomic expansion according to (1.59) in Bp,q p (R) 1 (no moment conditions are required). If either 2p < r < p1 , 0 < q ≤ ∞ or r = p1 , 0 < q ≤ 1 then one has by (1.345), (1.346), 1 2p

2r− 1

r trΓ : Sp,q B(R2 ) ,→ Bp,q p (R).

(1.347)

Furthermore, there exists a linear and bounded extension operator based on the well-known counterpart for isotropic spaces and 2r− 1

2 r 2 ext : Bp,q p (R) ,→ B2r p,q (R ) ,→ Sp,q B(R ),

(1.348)

1.3 Intermezzo: Key problems

61

where the second embedding is a special case of (1.431) below. Together with (1.347) one has 2r− 1 r trΓ Sp,q B(R2 ) = Bp,q p (R). (1.349) 1 This covers in particular (1.344) if 2p < r < p1 . The reason for this slightly surprising effect becomes clearer if one compares (1.345), (1.346) with its counterparts (1.333), (1.334) where one needs the summation over all k 2 ∈ N−1 . These assertions have r B(R2 ) and S r F(R2 ) in been substantially extended in [Vyb05] to further spaces Sp,q p,q the plane R2 , again using atomic arguments. It is remarkable that the study of traces r A(Rn ) on oblique hyperplanes in Rn , n ≥ 3 apparently causes a of the spaces Sp,q lot of trouble. We refer the reader to [VyS07] dealing with traces of some spaces r A(R3 ) on the hyperplane {x ∈ R3 : x + x + x = 0}. Further assertions and Sp,q 1 2 3 references may be found in [Schm07, pp. 155, 184–187]. r A(Rn ) according to If one wishes to classify trace problems for the spaces Sp,q the above scheme at the beginning of Section 1.3 then Type II might be a reasonable choice. Some restrictions for the parameters in connection with (1.330), (1.339) are similar to the isotropic Asp,q (Rn ), especially if n = 1. But there are also significant differences. This applies in particular to trace assertions of type (1.344), (1.349).

Remark 1.58. It is one aim of this Section 1.3 to compare properties of the spaces r A(Rn ) with dominating mixed smoothness with related ones for the isotropic Sp,q spaces Asp,q (R) and Asp,q (Rn ). But somewhat between isotropic spaces and spaces with dominating mixed smoothness are the anisotropic spaces n As,α p,q (R ),

0 < p ≤ ∞, 0 < q ≤ ∞, s ∈ R

(1.350)

(p < ∞ for F-spaces) with the anisotropy n α = (α1, . . . , αn ) ∈ R++ ,

n Õ

α j = n,

(1.351)

j=1

where R++ has the same meaning as in (1.155). The theory of these spaces (and related references in particular to the Russian literature according to [Nik77, BIN75]) may be found in [ST87, Chap. 4] and [T06, Chap. 5]. Anisotropic spaces might be of interest in the above context for two reasons. First, spaces with dominating mixed smoothness in Rn can be considered as the union of suitable anisotropic spaces at least as long as no moment conditions for related atoms are needed. This follows from corresponding atomic and wavelet characterizations as treated in [T06, Chap. 5] where one has to adapt the related anisotropic rectangles in [T06, p. 244] to Rk,m in (1.63). Second, we studied in [ST87, Chap. 4], and in particular in [MaT91], traces of anisotropic spaces (in R2 and R3 ) on non-rectangular domains (for example the unit circle). This might be considered the anisotropic counterpart of (1.344), (1.349) and,

1 Spaces on Rn

62

in particular, to the assertions in [VyS07]. One may ask whether these considerations are related to each other. 1.3.4 Dichotomy. Let 0 < d < n ∈ N and let Γ = supp µ be a compact d-set, where µ is the corresponding Hausdorff measure. Let L p (Γ) with 0 < p < ∞ be quasi-normed by ∫  1/p k f |L p (Γ)k = | f (γ)| p µ(dγ) . (1.352) Γ

Let  Ap (Rn ) = Asp,q (Rn ) : 0 < q < ∞, s ∈ R ,

0 < p < ∞,

(1.353)

A ∈ {B, F}. In particular, both C0∞ (Rn ) = D(Rn ) and S(Rn ) are dense in the admitted spaces. Let DΓ = D(Rn \ Γ) = C0∞ (Rn \ Γ) be the collection of all complex-valued infinitely differentiable functions in Rn with compact support in Rn \ Γ. Let σ ∈ R. Then

  D Ap (Rn ), L p (Γ) = (σ, u) with 0 < u < ∞  is called the dichotomy of Ap (Rn ), L p (Γ) if ( s > σ, 0 < q < ∞, trΓ : Asp,q (Rn ) ,→ L p (Γ) exists for s = σ, 0 < q ≤ u, and

( DΓ is dense in

Asp,q (Rn )

for

s = σ, s < σ,

u < q < ∞, 0 < q < ∞.

(1.354)

(1.355)

(1.356)

Furthermore, means that

  D Ap (Rn ), L p (Γ) = (σ, 0)

(1.357)

trΓ exists for s > σ, 0 < q < ∞, DΓ is dense in Asp,q (Rn ) for s ≤ σ, 0 < q < ∞,

(1.358)

  D Ap (Rn ), L p (Γ) = (σ, ∞)

(1.359)

trΓ exists for s ≥ σ, 0 < q < ∞, DΓ is dense in Asp,q (Rn ) for s < σ, 0 < q < ∞.

(1.360)

(

and means that (

We dealt in [T08, Sect. 6.4, pp. 215–236], based on [Tri08], with this natural dichotomy traces versus density. The above definitions coincide essentially with [T08,

63

1.3 Intermezzo: Key problems

Def. 6.66, p. 221]. There one also finds further explanations. According to [T08, Thm. 6.68, p. 222] one has the following rather final assertion:  



   

n−d p ,1



 D Bp (Rn ), L p (Γ) =    n−d , p  p

if p > 1, (1.361) if p ≤ 1,

and  

   



n−d p ,0



 D Fp (Rn ), L p (Γ) =    n−d , ∞  p

if p > 1, (1.362) if p ≤ 1.

There is a natural counterpart with L p (Rd ), n > d ∈ N in place of L p (Γ),  



   

n−d p ,1



 D Bp (Rn ), L p (Rd ) =    n−d , p  p

if p > 1, (1.363) if p ≤ 1,

and  



   

n−d p ,0



 D Fp (Rn ), L p (Rd ) =    n−d , ∞  p

if p > 1, (1.364) if p ≤ 1.

These assertions have been generalized in [Har12, CaH15] to larger classes of isotropic spaces in Rn and more general fractals. It is quite natural to ask for counterparts of (1.353) and (1.354) for spaces with dominating mixed smoothness,  r Sp A(Rn ) = Sp,q A(Rn ) : 0 < q < ∞, r ∈ R ,

0 σp(n) for B-spaces

and

(n) s > σp,q for F-spaces,

(1.377)

(n) where σp(n) and σp,q have the same meaning as in (1.57). Appropriately chosen numbers 0 < λ0 < · · · < λ L and al ensure that ext L , ( f (x), xn > 0, ext L f (x) = Í L (1.378) 0 l=0 al f (x , −λl xn ), xn ≤ 0,

is a linear and bounded extension operator from Asp,q (R+n ) to Asp,q (Rn ); see [T92, Thm. 4.5.2, pp. 223–224]. An illustrated version of these procedures may also be found in [HaT08, Sect. 3.4, pp. 72–79]. As for the history of this so-called Hestenes extension (see [Hes41]), we refer the reader to [HaT08, Sect. 4.6.1, p. 112]. There is also an extension to all s ∈ R (see [T92, Sect. 4.5.5, pp. 228–235]), including related references. In any case, the construction (1.378) respects the axes of coordinates and it makes sense to try to use this method to construct extensions of type (1.375). But this has not yet been done. Extension operators exist so far for the rather special spaces covered by Proposition 1.25 and Remark 1.26. 1.3.8 Diffeomorphisms. A continuous one-to-one map of Rn onto itself,
y = ψ(x) = ψ1 (x), . . . , ψn (x) , x ∈ Rn, (1.379)
−1 −1 −1 n x = ψ (y) = ψ1 (y), . . . , ψn (y) , y ∈ R , (1.380) ∞ is called a diffeomorphism if all components ψ j (x) and ψ −1 j (y) are real C functions n on R and for j = 1, . . . , n,
n sup |Dα ψ j (x)| + |Dα ψ −1 (1.381) j (x)| < ∞ for all α ∈ N0 with |α| > 0. x ∈R n

67

1.3 Intermezzo: Key problems


Then ϕ 7→ ϕ ◦ ψ, given by (ϕ ◦ ψ)(x) = ϕ ψ(x) , is a one-to-one map of S(Rn ) onto itself. This can be extended to a one-to-one map of S 0(Rn ) onto itself by
( f ◦ ψ)(ϕ) = f , |det ψ∗−1 | ϕ ◦ ψ −1 , f ∈ S 0(Rn ), ϕ ∈ S(Rn ), (1.382) as the distributional version of ∫ ∫
f ◦ ψ (x) ϕ(x) dx = Rn

Rn


f (y) ϕ ◦ ψ −1 (y) det ψ∗−1 (y) dy

(1.383)

(change of variables), where ψ∗−1 is the Jacobian and det ψ∗−1 its determinant. It is well known that f 7→ f ◦ ψ is an isomorphic map of Asp,q (Rn ) onto itself for all isotropic spaces Asp,q (Rn ) according to Definition 1.1.

(1.384)

We refer the reader to [T92, Thm. 4.3.2, p. 209]. The proof given there relies on local means. This requires a lot of effort and can be simplified as follows. It is sufficient to prove that there is a number c > 0 such that k f ◦ ψ | Asp,q (Rn )k ≤ c k f | Asp,q (Rn )k,

f ∈ Asp,q (Rn ).

(1.385)

We rely on the atomic representation as described in (1.58)–(1.62). If s > σp(n) for the (n) B-spaces and s > σp,q for the F-spaces then no moment conditions for the related atoms a j,m are required. This shows that a j,m ◦ ψ are essentially again admitted atoms with (1.385) as a consequence. Let s ∈ R and l ∈ N such that s + 2l > σp(n) (n) for B-spaces and s + 2l > σp,q for F-spaces. Let I2l : f 7→ (id − ∆)l f ,

l∈N

(1.386)

n be the lift according to (1.14), (1.15). Let f ∈ Asp,q (Rn ) and g ∈ As+2l p,q (R ) with I2l g = f . We have Õ


cα (x) Dα (g ◦ ψ) ψ −1 (x) (1.387) f (x) = (id − ∆)l g ψ −1 ◦ ψ(x) = |α | ≤2l

for some cα ∈ Cb∞ (Rn ) according to (1.202) (being pointwise multipliers in all spaces Asp,q (Rn )). Replacing x in (1.387) by x = ψ(y) it follows that Õ kDα (g ◦ ψ) | Asp,q (Rn )k k f ◦ ψ | Asp,q (Rn )k ≤ c1 |α | ≤2l n ≤ c2 kg ◦ ψ | As+2l p,q (R )k

≤ ∼ This completes the proof of (1.385).

n c3 kg | As+2l p,q (R )k k f | Asp,q (Rn )k.

(1.388)

1 Spaces on Rn

68

One may ask to what extent these arguments can be transferred to the spaces r A(Rn ). On the one hand, there is the promising lift (1.208) as a substitute for Sp,q (1.386). On the other hand, for general diffeomorphic maps ψ according to (1.379), (1.380) the properties (1.68), (1.69) are not preserved if one switches from ak,m to ak,m ◦ ψ. In any case, the situation improves if one assumes that the diffeomorphism ψ is fibre preserving, which means
y = ψ(x) = ψ1 (x1 ), . . . , ψn (xn ) , x ∈ Rn, (1.389)
−1 n −1 −1 (1.390) x = ψ (y) = ψ1 (y1 ), . . . , ψn (yn ) , y ∈ R , where ψ j (t), j = 1, . . . , n are strictly monotonically decreasing or increasing C ∞ functions on R, satisfying (1.381). Proposition 1.60. Let ψ be a fibre-preserving diffeomorphism. Then f 7→ f ◦ ψ

r is an isomorphic map of Sp,q A(Rn ) onto itself

(1.391)

r A(Rn ), A ∈ {B, F}, according to Definition 1.4. for all spaces Sp,q

Proof. We modify the above arguments for the isotropic spaces. It is sufficient to prove that there is a number c > 0 such that r r k f ◦ ψ |Sp,q A(Rn )k ≤ c k f |Sp,q A(Rn )k,

r f ∈ Sp,q A(Rn ).

(1.392)

If r > σp for the B-spaces and r > σp,q for the F-spaces then no moment conditions for the related atoms ak,m in Theorem 1.10 are required. Since ψ in (1.389), (1.390) is fibre preserving it follows from (1.68), (1.69) that ak,m ◦ ψ are essentially again admitted atoms with (1.392) as a consequence. Let r ∈ R and l ∈ N such that r + 2l > σp for B-spaces and r + 2l > σp,q for F-spaces. We rely on the lifting r A(Rn ) and g ∈ S r+2l A(Rn ) with f = În (id − ∂ 2 )l g. Then (1.208). Let f ∈ Sp,q p,q j=1 j the counterpart of (1.387) is given by n Ö
f (x) = (id − ∂j2 )l g ψ −1 ◦ ψ(x) = j=1

Õ


cα (x) Dα (g ◦ ψ) (ψ −1 (x)) (1.393)

α∈N0n, 0≤α j ≤2l

for some cα ∈ Cb∞ (Rn ), being pointwise multipliers according to Theorem 1.39. Now one can argue as in (1.388) and as in connection with (1.209) (as far as derivatives are concerned). This completes the proof.  The above considerations show that mappings of type f 7→ f ◦ ψ for the spaces are at best of Type II in the classification at the beginning of Section 1.3: the same as for Asp,q (R) in one dimension, but too restrictive in higher dimensions. But the change of variables plays a role in the recent theory of numerical integration r A(Q), Q = (0, 1)n ; see [NUU17]. in the context of the spaces Sp,q r A(Rn ) Sp,q

1.4 Properties, II

69

1.3.9 Résumé. The Sobolev spaces Spr W(Rn ), 1 < p < ∞, r ∈ N, with dominatr B(Rn ), ing mixed smoothness, normed by (1.36), and their Besov counterparts Sp,∞ r > 0, 1 ≤ p < ∞, normed by (1.48), (1.49) (with q = ∞), were introduced by S. M. Nikol’skij in [Nik62, Nik63a, Nik63b]. Further detailed references may be found in [ST87, Sect. 2.1, pp. 80–81]. Otherwise we refer the reader to the more recent surveys and books already mentioned in the preface and the literature within. r B(Rn ) and their counterparts S r W(Q), S r B(Q) on the The spaces Spr W(Rn ), Sp,q p,q p n cubes Q = (0, 1) proved to be very effective especially in connection with approximation, sampling, numerical integration, etc. One may ask whether these advantages (compared with their isotropic counterparts) can be preserved when switching from Q to arbitrary (bounded) domains Ω in Rn and how related spaces may look. The r A(Ω) as introduced in Definition 1.22. Similarly to first candidates are the spaces Sp,q the case of their isotropic counterparts Asp,q (Ω) one would try to reduce these spaces r A(Rn ). As already mentioned above, one needs for to the corresponding spaces Sp,q this purpose smooth pointwise multipliers (covered by Theorem 1.39 in a satisfactory way), extensions and diffeomorphisms (as discussed in Sections 1.3.7, 1.3.8). But the considerations make clear that the outcome is now totally different compared with the spaces Asp,q (Ω). The theory of reasonable spaces with dominating mixed smoothness on domains seems to be of Type III in the classification at the beginning of Section 1.3. Then it might be reasonable (in contrast to their isotropic counterr A[Ω, σp . As already stated in Remark 1.45, it is slightly tricky to say what is meant by a pointwise multiplier for spaces with r ≤ σp . For our purpose it is sufficient to argue as follows. If 1 < p ≤ ∞ and r < 0 then



M Spr B(Rn ) = M Sp−r0 B(Rn ) , M Brp (R) = M B−r (1.394) p0 (R)

1 Spaces on Rn

70

with p1 + p10 = 1 is justified by (1.113) and its counterpart for Brp (R), interpreting pointwise multipliers as symmetric linear operators in Sp−r0 B(Rn ) or B−r p0 (R). Further
rj n more, if w ∈ M Sp j B(R ) , j = 1, 2, then it follows from the complex interpolation (1.122) for 0 < θ < 1 that
w ∈ M Spr B(Rn ) ,

θ 1 1−θ = + , p p1 p2

r = (1 − θ)r1 + θr2

(1.395)


r makes sense, and similarly for w ∈ M Bpjj (R) . This is sufficient for our purposes, although it can be extended to other cases. Proposition 1.61. Let 2 ≤ n ∈ N, 0 < p ≤ ∞, r ∈ R and
w j ∈ M Brp (R) , j = 1, . . . , n. Then w(x) =

n Ö


w j (x j ) ∈ M Spr B(Rn ) .

(1.396)

(1.397)

j=1

Proof. As mentioned in Remark 1.30 the expansion (1.179), (1.180) with (1.184) applies to all f ∈ Spr B(Rn ). Then one has the unique expansion Õ

w1 f =

0

1

w1 (x1 ) fk 0,m0 (x1 ) 2−[k ](r− p ) ψk 0,m0 (x 0).

(1.398)

n−1 k 0 ∈N−1

Inserted in (1.184) one obtains kw1 f |Spr B(Rn )k ≤ kw1 |M Brp (R)k · k f |Spr B(Rn )k. Iteration proves (1.397).

(1.399) 

We apply Proposition 1.61 to the characteristic functions of the (bounded and unbounded) rectangles  Ra,b = x = (x1, . . . , xn ) ∈ Rn : a j < x j < b j , j = 1, . . . , n , (1.400) where − ∞ ≤ a j < b j ≤ ∞,

j = 1, . . . , n.

(1.401)

Proposition 1.62. Let n ∈ N, 0 < p ≤ ∞ and r ∈ R. Then the characteristic function χa,b of the rectangle Ra,b in (1.400) is a pointwise multiplier in Spr B(Rn ) if, and only if, 1 1 −1 σp . (i) If, in addition, r < 1/2p then
χΩ ∈ M Spr B(R2 ) .

(1.419)

(ii) If, in addition, 1 ≤ p < ∞ and r > 1/2p then χΩ < Spr B(R2 ) and


χΩ < M Spr B(R2 ) .

(1.420)

Proof. Step 1. We prove part (i). We modify the arguments in [Tri03, pp. 475– 476] where we dealt with characteristic functions as pointwise multipliers in some

75

1.4 Properties, II

isotropic spaces Bps (Rn ). We may assume that sup{|x − y| : x ∈ Ω, y ∈ Ω} < 1. Let Γ = ∂Ω and  Ωk = x ∈ Ω : 2−k−2 ≤ dist(x, Γ) ≤ 2−k , k ∈ N0, (1.421) where dist(x, Γ) = min{|x − y| : y ∈ Γ},

x ∈ Ω.

(1.422)

Let 

ϕlk : k ∈ N0 ; l = 1, . . . , Mk ⊂ D(Ω) = C0∞ (Ω)

(1.423)

be a resolution of unity, Mk ∞ Õ Õ

ϕlk (x) = 1

if x ∈ Ω

(1.424)

k=0 l=1

with  supp ϕlk ⊂ x : |x − xlk | ≤ c 2−k ⊂ Ωk and

|Dα ϕlk (x)| ≤ cα 2 |α |k ,

α ∈ N0n,

(1.425) (1.426)

for suitably chosen positive constants c and cα . One may assume that χΩ (x) =

Mk ∞ Õ Õ

1

1

22k(r− p ) 2−2k(r− p ) ϕlk (x),

x ∈ R2,

(1.427)

k=0 l=1

where Mk ∼ 2k an atomic decomposition of χΩ in Spr B(R2 ) according to Theorem 1.10 (no moment conditions are required). By (1.74) one has k χΩ |Spr B(R2 )k ≤ c

∞ Õ

2k(2r p−1)

 1/p

< ∞,

(1.428)

k=0

where we used r < 1/2p. We wish to apply Theorem 1.48 based on Definition 1.42. Of course χΩ ∈ Spσ B(R2 ) if r < σ < 1/2p. Next we show that χΩ belongs to Spσ Bselfs (R2 ) based on (1.226) with A(Rn ) = Spσ (R2 ) and f = χΩ . One needs χΩ 2−m x) = χ2m Ω (x)  with 2m Ω = (x1, x2 ) ∈ R2 : x1 = 2m1 y1, x2 = 2m2 y2, y ∈ Ω ,

(1.429)

m ∈ N20 . This anisotropic blow-up also improves the local properties of 2m Γ. In
particular, one can apply the above covering with 2m Ω ∩ l + (0, C)2 , C > 0 in place

1 Spaces on Rn

76

of Ω, where m ∈ N20 and l ∈ Z2 , with Mk ∼ 2k uniformly in the counterparts of (1.427). This applies also to ψl χ2m Ω as needed in (1.226) resulting in χΩ ∈ Spσ Bselfs (R2 ),

σp < r < σ < 1/2p.

(1.430)

Then (1.419) follows from Theorem 1.48.
Step 2. If χΩ ∈ M Spr B(R2 ) then χΩ ∈ Spr B(R2 ). In other words, it is sufficient to prove the first assertion in (1.420). Let χΩ ∈ Spr B(R2 ) for 1 ≤ p < ∞ and some r > 0. The boundary Γ of the bounded C ∞ domain Ω in R2 has oblique parts where the outer normal has two non-vanishing components. Application of a suitable fibrepreserving diffeomorphism according to Proposition 1.60 shows that we may assume that 0 < x1 = x2 < ε is part of the boundary Γ. Then it follows from χΩ ∈ Spr B(R2 ) that Spr B(R2 ) has no trace on this part of the boundary. But this contradicts (1.347) if r > 1/2p.  Remark 1.70. Let Ω be a rectangle according to (1.400) (or a finite union of such rectangles). Then it follows from (1.330) and Proposition 1.62 that r = 1/p is the breaking point for traces of functions f ∈ Spr B(R2 ). But these are exceptions and r = 1/2p is the breaking point for more general domains in R2 . This applies in particular to triangles in R2 , but also to polygons having at least one oblique line-segment as part of the boundary. One can try to rephrase and strengthen this observation in terms of the dichotomy as described in Section 1.3.4. Remark 1.71. Let 2 ≤ n ∈ N, 0 < p ≤ ∞, 0 < q ≤ ∞ and r > σp , (1.57), (1.71). nr (Rn ) in (1.59), (1.60) (no moment conditions Then the atomic expansion of f ∈ Bp,q r B(Rn ) according to Theorem 1.10 are needed) is also an atomic expansion in Sp,q (with the correct normalizations in (1.55), s = nr and in (1.69)). Then one has nr r Bp,q (Rn ) ,→ Sp,q B(Rn ),

0 < p ≤ ∞, 0 < q ≤ ∞, r > σp .

(1.431)

Similarly, for the F-spaces, based on (1.61), (1.62) and again Theorem 1.10, nr r Fp,q (Rn ) ,→ Sp,q F(Rn ),

0 < p < ∞, 0 < q ≤ ∞, r > σp,q .

(1.432)

A detailed study (mainly based on Fourier-analytical arguments) of how isotropic r A(Rn ) with dominating mixed smoothness are related spaces Asp,q (Rn ) and spaces Sp,q to each other may be found in [NgS17a, NgS17b]. In our context we interpreted the representation (1.427) of χΩ in terms of isotropic atoms as an expansion in Spr B(R2 ) with the outcome (1.428). Usually one would not expect that optimal expansions n r n in Anr p,q (R ) are also optimal expansions in Sp,q A(R ). But Theorem 1.69 suggests that the representation (1.427) is an exception. One may ask what happens in higher

1.4 Properties, II

77

dimensions. The breaking point for traces of Asp,q (Rn ) on, say, (n − 1)-dimensional hyperplanes is s = 1/p. One even has the substantial refinements (1.361)–(1.364). r A(Rn ) one has the traces in (1.330) on hyperplanes In the case of the spaces Sp,q parallel to the axes of coordinates. This suggests that r = 1/p is the related breaking r A(Rn ) on oblique (n − 1)-dimensional point. But the determination of traces of Sp,q hyperplanes or boundaries of bounded smooth domains is a rather tricky task. We discussed this point at the end of Section 1.3.5. The first candidate for a general breaking point might be r = 1/np, n ≥ 2. This is supported by Theorem 1.69, n = 2 and the embeddings (1.431), (1.432). But if n ≥ 3 then the situation might be much more complicated. Problem 1.72. According to Proposition 1.62 one has a final answer to the question of under what circumstances χa,b is a pointwise multiplier in Spr B(Rn ). It would be desirable to clarify the situation if r = 1/2p in Theorem 1.69. 1.4.2 Multiplication algebras. According to Definition 1.44(ii) a quasi-Banach space A(Rn ) on Rn with (1.233) is said to be a multiplication algebra if w1 ·w2 ∈ A(Rn ) for w1 ∈ A(Rn ), w2 ∈ A(Rn ) and if there is a number c > 0 such that kw1 · w2 | A(Rn )k ≤ c kw1 | A(Rn )k · kw2 | A(Rn )k

(1.433)

for all w1 ∈ A(Rn ), w2 ∈ A(Rn ). One has A(Rn ) ,→ L∞ (Rn )

(1.434)

by the same arguments as in the proof of Proposition 1.46. First we recall what is known in the case of the isotropic spaces Asp,q (Rn ) as introduced in Definition 1.1. We copy the formulation given in [T13, Thm. 1.16, Rem. 1.17, p. 12]. Let 0 < p, q ≤ ∞ (p < ∞ for the F-spaces) and s ∈ R. Then the following assertions are pairwise equivalent: (i) Asp,q (Rn ) is a multiplication algebra. (ii) s > 0 and Asp,q (Rn ) ,→ L∞ (Rn ). (iii) Either ( s (Rn ) with Asp,q (Rn ) = Bp,q

s > n/p where 0 < p, q ≤ ∞, s = n/p where 0 < p < ∞, 0 < q ≤ 1,

(1.435)

s > n/p where 0 < p < ∞, 0 < q ≤ ∞, s = n/p where 0 < p ≤ 1, 0 < q ≤ ∞.

(1.436)

or ( Asp,q (Rn )

=

s (Rn ) Fp,q

with

1 Spaces on Rn

78

This assertion has a little history. It begins with [Tri77b], [Tri78, Sect. 2.6.2] and [T83, Sect. 2.8.3]. This has been complemented and corrected since. The final version and related references may be found in [SiT95]. One may also consult [RuS96, pp. 221–222, 258] for a more detailed description of the history. Let Aunif (Rn ) and Aselfs (Rn ) with A(Rn ) = Asp,q (Rn ) be as in (1.223), (1.224). We dealt in [T06, Sects. 2.3.1, 2.3.2, pp. 136–140] with the close connection of multiplication algebras A(Rn ), Aunif (Rn ) and Aselfs (Rn ). It is quite natural to ask for counterparts r A(Rn ) with dominating mixed smoothness. First results have been for the spaces Sp,q obtained quite recently for some spaces Spr B(Rn ) and Spr W(Rn ) in [NgS17d], closely related to corresponding pointwise multipliers. This has been extended in [NgS17c] r B(Rn ) based on related characterizations in terms of derivatives to some spaces Sp,q (Sobolev spaces) and differences (Besov spaces). We look at this problem from the point of view of wavelets. Let 0 < p ≤ ∞ and r > σp with σp as in (1.71). Then Brp (R) = Brp, p (R) is a multiplication algebra if, and only if, ( either 1 < p ≤ ∞, r > 1/p, (1.437) or 0 < p ≤ 1, r ≥ 1/p. This is a special case of (1.435). As already mentioned above, being a multiplication algebra is also equivalent to Brp (R) ,→ L∞ (R),

r > σp .

(1.438)

We wish to give a new proof of these assertions in terms of wavelets, which can be extended afterwards to corresponding spaces Spr B(Rn ). For this purpose we rely on the wavelet representation of Spr B(R) = Brp (R) according to Theorem 1.12. In particular, f ∈ Brp (R) can be expanded as f =

1

Õ

λk,m 2−k(r− p ) ψk,m

(1.439)

k ∈N−1,m∈Z

with λk,m = λk,m ( f ) = 2

k(r− p1 +1)

∫

f (y) ψk,m (y) dy

(1.440)

R

and k f |Brp (R)k ∼ kλ |`p k,

f ∈ Brp (R), λ = {λk,m }.

(1.441)

Let f1, f2 ∈ Brp (R) with 0 < p ≤ ∞, r > σp,

(1.442)

again with σp as in (1.71). The question of whether f = f1 f2 belongs to Brp (R) can be reduced to (1.439)–(1.441). In particular, inserting the related expansions for f1

79

1.4 Properties, II

and f2 in (1.440) one obtains ∫ k(r− p1 +1) λk,m ( f ) = 2 f1 (y) f2 (y)ψk,m (y) dy R Õ 1 1 1 2 = 2k(r− p +1) λk 1,m1 ( f1 )λk 2,m2 ( f2 ) 2−(k +k )(r− p ) Ikk,m 1,k 2,m1,m2 , k 1,k 2 ∈N−1, m1,m2 ∈Z

(1.443) where Ikk,m 1,k 2,m1,m2

=

∫

ψk,m (y) ψk 1,m1 (y) ψk 2,m2 (y) dy.

(1.444)

R

Recall that (1.441) is an isomorphic map of Brp (R) onto `p . This shows that the question of whether (1.445) {µ1k 1,m1 } × {µ2k 2,m2 } ,→ {µk,m } with µk,m = 2k(r− p +1) 1

Õ

µ1k 1,m1 µ2k 2,m2 2−(k

1 +k 2 )(r− 1 ) p

Ikk,m 1,k 2,m1,m2

(1.446)

k 1,k 2 ∈N−1, m1,m2 ∈Z

is a bounded bilinear map Mp : `p × `p ,→ `p,

0 < p ≤ ∞,

(1.447)

is equivalent to the problem of whether Brp (R) is a multiplication algebra. Of course, Mp = MpΨ depends on the chosen sufficiently smooth wavelet basis Ψ = {ψk,m }. But this will not be indicated. We concentrate on the sufficient part of the above equivalence, showing that Mp with (1.445)–(1.447) is a bounded bilinear map if p and r are related as in (1.437). As far as the necessary part is concerned we rely on the above equivalence: if Mp is a bounded bilinear map, then Brp (R) is a multiplication algebra and (1.437) follows from (1.435) and the equivalence mentioned there. But we return to this point later on in Remark 1.74 below. Proposition 1.73. Let 0 < p ≤ ∞ and ( r > p1 if 1 < p ≤ ∞, r≥

1 p

if 0 < p ≤ 1.

(1.448)

Then Mp according to (1.445)–(1.447) is a bounded bilinear map. Proof. Let k ∈ N−1 and m ∈ Z. Then there is ∼ 1 relevant term in (1.444) with 1 2 k 2 ≤ k 1 ≤ k and 2−k m ∼ 2−k m1 ∼ 2−k m2 . Using the support and cancellation properties of ψk,m and expanding ψk 1,m1 (y) ψk 2,m2 (y) at y = 2−k m one obtains k,m I 1 2 1 2 ≤ c 2−k 2−(k−k 1 )l, k 2 ≤ k 1 ≤ k, (1.449) k ,k ,m ,m

1 Spaces on Rn

80

1

where l ∈ N is at our disposal. Let k < k 1 and again k 2 ≤ k 1 . Then there are ∼ 2k −k 1 1 2 relevant terms with 2−k m1 ∼ 2−k m and ∼ 1 term with 2−k m1 ∼ 2−k m2 . Instead of (1.449) one now has 1 1 k,m (1.450) Ik 1,k 2,m1,m2 ≤ c 2−k 2−(k −k)l, k < k 1, k 2 ≤ k 1, 2

1

where again l ∈ N is at our disposal. Since r ≥ 1/p we always have 2−k (r− p ) ≤ c. Then it follows from the above estimates (including the number of relevant terms) and its counterpart with k 1 ≤ k 2 , using an `1 -convolution argument, Õ Õ | µk,m | ≤ c | µ1k 1,m1 | · | µ2k 2,m2 |. (1.451) k ∈N−1,m∈Z

k 1,k 2 ∈N−1, m1,m2 ∈Z


p Í∞ This proves (1.447) with p = 1. If p < 1 then we use in addition ≤ j=1 a j Í∞ p j=1 a j , a j ≥ 0, with the same outcome, that is, (1.447) with 0 < p < 1. Let p > 1. Then we have r > 1/p. The estimates (1.449) and (1.450) are independent of k 2 and we can estimate the related k 2 -terms in (1.446) by Õ  Õ  1/p 1 2 | µ2k 2,m2 | p µ2k 2,m2 2−k (r− p ) ≤ c . k 2 ≤k 1

(1.452)

k 2 ≤k1

A corresponding estimate with respect to k 1 can be based again on (1.449), (1.450). Afterwards one can argue as above.  Remark 1.74. As already stated, (1.448) is not only sufficient to ensure that Mp according to (1.445)–(1.447) is a bounded bilinear map, but also necessary. This follows from the equivalence of this assertion with the question of whether Brp (R) is a multiplication algebra according to (1.437) based on (1.435) with n = 1 and p = q. But one may ask for a direct proof in terms of sequences. One may consider (1.443) as a one-to-one vehicle to transfer multiplication properties between function spaces to corresponding assertions in terms of sequence spaces based on (1.444)–(1.446). In some sense, Brp (R) with (1.437), (1.441) on the one hand, and Mp with (1.447) on the other hand, is the simplest case. It might be of interest to extend these considerations to Arp,q (R), in particular to those cases where Arp,q (R) is a multiplication algebra satisfying (1.435), (1.436) with n = 1 and r = s. Furthermore, it seems to be possible to step from n = 1 to 2 ≤ n ∈ N and to discuss multiplication properties for the spaces Arp,q (Rn ) in this context, especially if they are multiplication algebras according to (1.435) with r = s. The n-dimensional counterpart of the wavelet expansion (1.439)–(1.441) is given by (1.89), (1.90) with a different normalization j according to (1.86), (1.87) where the wavelets ψG,m in Rn are products of wavelets

81

1.4 Properties, II

in R. Then the counterpart of (1.444) (with an integration over Rn ) is the product of corresponding one-dimensional integrals as in (1.444) and one can still rely on estimates of type (1.449), (1.450). But this will not be done here. The original proofs of multiplication properties of function spaces Arp,q (Rn ), including the question of under what circumstances Arp,q (Rn ) is a multiplication algebra, as described at the beginning of this Section 1.4.2, are based on paramultiplication on the Fourier (or frequency) side. The above approach shifts paramultiplication from the Fourier side to the space side and relies on the examination of the integrals in (1.444). Similar techniques have also been used in the context of some bilinear mappings in function spaces. We refer the reader to the comments in [T13, Rem. 1.18, pp. 12–13] and [MaN09, BMNT10]. We are interested in the n-dimensional generalization of Proposition 1.73 and its one-to-one relation to the question of whether Spr B(Rn ) is a multiplication algebra if, and only if, ( either 1 < p ≤ ∞, r > 1/p, (1.453) or 0 < p ≤ 1, r ≥ 1/p, as in (1.437) referring to the one-dimensional case Spr B(R) = Brp (R) = Brp, p (R). For this purpose we use the wavelet expansion of f ∈ Spr B(Rn ) according to Theorem 1.12, Õ 1 (1.454) f (x) = λk,m 2−[k](r− p ) ψk,m (x) n ,m∈Z n k ∈N−1

with [k] =

Ín

j=1

kj, [k](r− p1 +1)

λk,m = λk,m ( f ) = 2

∫ Rn

f (x) ψk,m (x) dx

(1.455)

and k f |Spr B(Rn )k ∼ kλ |`p k,

f ∈ Spr B(Rn ), λ = {λk,m }.

(1.456)

Recall the product structure of ψk,m (x) according to (1.97), ψk,m (x) =

n Ö

ψk j ,m j (x j ),

n k ∈ N−1 , m ∈ Zn .

(1.457)

j=1

This is the n-dimensional version of (1.439)–(1.441) for Brp (R) = Spr B(R). We proceed similarly to (1.442), (1.443) and assume f1, f2 ∈ Spr B(Rn ) with 0 < p ≤ ∞, r > σp,

(1.458)

with σp as in (1.71). By (1.231) and Definition 1.44(ii) it makes sense to ask for conditions ensuring that Spr B(Rn ) is a multiplication algebra. The question of whether

1 Spaces on Rn

82

f = f1 f2 also belongs to Spr B(Rn ) can be reduced to (1.454)–(1.456). Inserting the related expansions for f1 and f2 in (1.455) one now obtains λk,m ( f ) =

(1.459)

= 2kl (r− p +1) 2−(kl +kl )(r− p ) I k1l ,m2l

(1.460)

λk 1,m1 ( f1 ) λk 2,m2 ( f2 )

n , k 1,k 2 ∈N−1 1 2 m ,m ∈Z n

with

J k1l ,m2l

kl ,kl ,m1l ,m2l

l=1

1

kl ,kl ,m1l ,m2l

n Ö

J k1l ,m2l

Õ

1

1

2

kl ,kl ,m1l ,m2l

where I k1l ,m2l

is given by (1.444). Here we used that all factors, and also the
wavelets in (1.457), have a product structure. Recall σp = max p1 , 1 − 1. kl ,kl ,m1l ,m2l

Theorem 1.75. Let 2 ≤ n ∈ N, 0 < p ≤ ∞ and r > σp .

(1.461)

Then Spr B(Rn ) is a multiplication algebra if, and only if, (

either 1 < p ≤ ∞, r > 1/p, or 0 < p ≤ 1, r ≥ 1/p.

(1.462)

Proof. Step 1. Recall that the wavelet expansion (1.454)–(1.456) is an isomorphism. In other words, Spr B(Rn ) is a multiplication algebra if, and only if, Mp in (1.447), now with respect to µk,m =

Õ n , k 1,k 2 ∈N−1 1 2 m ,m ∈Z n

µk 1,m1 µk 2,m2

n Ö l=1

J k1l ,m2l

kl ,kl ,m1l ,m2l

,

(1.463)

n , m ∈ Zn , with (1.460), is a bounded bilinear map. This problem can k ∈ N−1 be reduced to the one-dimensional case. Let k = (k1, k 0) and m = (m1, m 0) with n−1 and m 0 ∈ Zn−1 . Similarly k 1 = (k 1, k 1 0 ), k 2 = (k 2, k 2 0 ), m1 = (m1, m1 0 ), k 0 ∈ N−1 1 1 1 0 0 0 0 0 m2 = (m12, m2 ). For frozen k 0, k 1 , k 2 and m 0, m1 , m2 one has the one-dimensional situation as considered in Proposition 1.73. Let p = 1. Then one has the counterpart of (1.451) with respect to k 1 , m1 and k11 , k12 , m11 , m12 . Now one can iterate this procedure. This proves (1.447) with p = 1. If p < 1 one again uses the p-triangle inequality as after (1.451). If p > 1 then one can argue as in connection with (1.452). In other words, one always has the bounded bilinear map Mp according to (1.447) (n-dimensional version). This proves that Spr B(Rn ) is a multiplication algebra if p and r are restricted according to (1.462).

1.4 Properties, II

83

Step 2. We prove that (1.462) is also necessary. Let A(Rn ) = Spr B(Rn ) be a multiplication algebra. Then one has (1.434). Let f (x) = g(x1 )ψ(x 0) where ψ ∈ S(Rn−1 ) is a fixed function, say, with ψ(0) = 1 and g ∈ Brp (R). Then one has by Proposition 1.19 that f ∈ Spr B(Rn ) and k f |Spr B(Rn )k ∼ kg |Brp (R)k. (1.464) This shows that Brp (R) ,→ L∞ (R)

(1.465)

and (1.462) follows from (1.437), (1.438), based on (1.435).  Corollary 1.76. Let 0 < p ≤ ∞,

r > σp .

(1.466)

Then Spr B(Rn ) is a multiplication algebra if, and only if, Spr B(Rn ) ,→ L∞ (Rn ).

(1.467)

Proof. If Spr B(Rn ) is a multiplication algebra then (1.467) follows from (1.434). Conversely, (1.467) ensures (1.464), (1.465) and, consequently (1.462), based on (1.435). This proves that Spr B(Rn ) is a multiplication algebra.  Remark 1.77. By Theorem 1.75 and Corollary 1.76 one now has the same equivar B(Rn ) in place of B s (Rn ) in (1.435) lences (i)–(iii) as above with Spr B(Rn ) = Sp, p p,q with n = 1. 1.4.3 Pointwise multipliers, revisited. Again let   1
1
(1.468) σp(n) = n max , 1 − 1 and σp = σp(1) = max , 1 − 1, p p
0 < p ≤ ∞. Let M A(Rn ) be the collection of all pointwise multipliers as introduced r A(Rn ), in particular for S r B(Rn ) = S r B(Rn ), in Definition 1.44 for the spaces Sp,q p p, p s (Rn ). Let A n and in Section 1.3.5 for the isotropic spaces Bps (Rn ) = Bp, unif (R ), p n Aselfs (R ) be as in (1.223), (1.224) for the isotropic spaces and as in Definition 1.42 r A(Rn ). As already mentioned in (1.367)–(1.369) we have for the spaces Sp,q
σ s Bp,selfs (Rn ) ,→ M Bps (Rn ) ,→ Bp,selfs (Rn ) for 0 < p ≤ ∞, σp(n) < s < σ. If, in addition, p ≤ 1 then
s M Bps (Rn ) = Bp,selfs (Rn ).

(1.469)

(1.470)

1 Spaces on Rn

84

The counterpart for the spaces Spr B(Rn ) is covered by Theorem 1.48. Then
Spσ Bselfs (Rn ) ,→ M Spr B(Rn ) ,→ Spr Bselfs (Rn ) for 0 < p ≤ ∞, σp < r < σ. If, in addition, p ≤ 1 then
M Spr B(Rn ) = Spr Bselfs (Rn ).

(1.471)

(1.472)

We wish to complement these assertions and recall first what is known for the isotropic spaces A(Rn ) = Asp,q (Rn ). This will be done in terms of multiplication algebras as introduced in Definition 1.44(ii). According to [T06, Thm. 2.21, pp. 138– 139] one has Aunif (Rn ) = Aselfs (Rn ) (1.473) if, and only if, A(Rn ) is a multiplication algebra, which is equivalent to (1.435), s (Rn ) is a multiplication algebra, which can be characterized (1.436). If A(Rn ) = Fp,q s n s (Rn ) = B s n by (1.436), or C (R ) = B∞ ∞,∞ (R ) with s > 0 then
M A(Rn ) = Aunif (Rn ) = Aselfs (Rn ); (1.474) see [T06, Prop. 2.22, pp. 140–141]. One may ask whether there are counterparts r A(Rn ). At least for the spaces S r B(Rn ) = of these observations for the spaces Sp,q p r n r n Sp, p B(R ) = Sp, p F(R ) one has the following assertions. Proposition 1.78. Let 0 < p ≤ ∞ and r > σp .

(1.475)

Let Spr B(Rn ) be a multiplication algebra. Then
M Spr B(Rn ) = Spr Bunif (Rn ) = Spr Bselfs (Rn ). Proof. According to (1.471) one has
M Spr B(Rn ) ,→ Spr Bselfs (Rn ) ,→ Spr Bunif (Rn ).

(1.476)

(1.477)

Let w ∈ Spr Bunif (Rn ). We apply (1.214), (1.215) with ψ 2 in place of ψ and obtain kw f |Spr B(Rn )k p Õ ∼ kψ 2 (· − l)w f |Spr B(Rn )k p l ∈Z n

≤c

Õ l ∈Z n

kψ(· − l) f |Spr B(Rn )k p sup kψ(· − m)w |Spr B(Rn )k p m∈Z n

≤ c 0 kw |Spr Bunif (Rn )k p · k f |Spr B(Rn )k p

(1.478)

1.4 Properties, II

85


(usual modification if p = ∞). This shows that w ∈ M Spr B(Rn ) and
Spr Bunif (Rn ) ,→ M Spr B(Rn ) . Then (1.476) follows from (1.477) and (1.479).

(1.479) 

Remark 1.79. The first equality in (1.476) restricted to p ≥ 1 is covered by [NgS17d, Thm. 3.12, p. 72]. There one also finds related assertions for the Sobolev spaces Spr W(Rn ), 1 < p < ∞, r ∈ N, as introduced in (1.35), (1.36). According to [NgS17c, r B(Rn ) Thm. 3.7, p. 2247] the first equality in (1.476) remains valid for the spaces Sp,q with 1 ≤ p ≤ q ≤ ∞ and r > 1/p. The arguments are based on characterizations of these spaces in terms of differences as in (1.48), (1.49). On the other hand, s (Rn ) one may ask whether (1.474) for the multiplication algebras A(Rn ) = Fp,q r n according to (1.436) has a suitable counterpart for related spaces Sp,q F(R ). For the r B(Rn ) one has the above-mentioned affirmative assertion with the decisive spaces Sp,q s (Rn ). restriction p ≤ q. The case q < p is different, even for the isotropic spaces Bp,q One may consult [T06, Rem. 2.29, pp. 144–145], the references given there and the recent paper [NgS18]. Let n ∈ N, 0 < p, q ≤ ∞,

s > σp(n)

(1.480)

(p < ∞ for F-spaces). Then k f1 f2 | Asp,q (Rn )k ≤ c k f1 | Asp,q (Rn )k · k f2 |L∞ (Rn )k + c k f1 |L∞ (Rn )k · k f2 | Asp,q (Rn )k

(1.481)

for some c > 0 and all fl ∈ Asp,q (Rn ) ∩ L∞ (Rn ), l = 1, 2; see [RuS96, Thm. 2, p. 222]. One may ask whether there is a counterpart of this useful inequality for r A(Rn ), n ≥ 2. But according to [NgS17c, NgS17d] the answer is the spaces Sp,q negative for those spaces of Besov–Sobolev type treated there using Fourier-analytical arguments, derivatives and differences. We comment on this question in terms of wavelet expansions. Proposition 1.80. Let n ≥ 2, 0 < p ≤ ∞ (p < ∞ for F-spaces), 0 < q ≤ ∞ and r > 0. Then there is no number c > 0 such that r k f1 f2 |Sp,q A(Rn )k r r ≤ c k f1 |Sp,q A(Rn )k · k f2 |L∞ (Rn )k + c k f1 |L∞ (Rn )k · k f2 |Sp,q A(Rn )k r A(Rn ) ∩ L (Rn ), l = 1, 2. for all fl ∈ Sp,q ∞

(1.482)

1 Spaces on Rn

86

Proof. There is no need to bother about what is meant by f1 f2 . We disprove (1.482) in terms of finite wavelet expansions according to Theorem 1.12. Then all functions in (1.482) are well defined. We may assume n = 2. Let f1 (x1, x2 ) = ψ0 (x2 )

2j Õ

ψ j,m (x1 ),

f2 (x1, x2 ) = ψ0 (x1 )

2j Õ

ψ j,m (x2 ),

(1.483)

m=1

m=1

where ψ j,m (t) = ψ M (2 j t − m), j ∈ N0 are the one-dimensional wavelets according to (1.94) and ψ0 ∈ D(R), ψ0 (t) = 1 if |t| ≤ C with C > 0 sufficiently large such that ( f1 f2 )(x) =

2j Õ

ψ j,m1 (x1 ) · ψ j,m2 (x2 ),

x = (x1, x2 ) ∈ R2 .

(1.484)

m1,m2 =1

Then one has by Theorem 1.12 and (1.100), (1.101) (no summation with respect to the q-index), 1

2j

r k f1 f2 |Sp,q A(R2 )k ∼ 22j(r− p ) 2 p = 22jr .

(1.485)

By Proposition 1.19 one has k

r fl |Sp,q A(R2 )k

2j

Õ

j 1

r ∼ ψ j,m |Bp (R) ∼ 2 j(r− p ) 2 p = 2 jr ,

(1.486)

m=1

l = 1, 2. Furthermore, k fl |L∞ (R2 )k ∼ 1. From (1.485) and (1.486) it follows that there is no constant c > 0 with (1.482).  Remark 1.81. Let Sr C(Rn ), n ≥ 2, r ∈ N be the collection of all f ∈ L∞ (Rn ) such that k f |Sr C(Rn )k = sup kDα f |L∞ (Rn )k (1.487) α∈N0n, 0≤α j ≤r

is finite. This extends (1.36) to p = ∞. According to [NgS17d, Thm. 3.5, p. 70] there is no number c > 0 such that k f1 f2 |Sr C(Rn )k (1.488) ≤ c k f1 |Sr C(Rn )k · k f2 |L∞ (Rn )k + c k f1 |L∞ (Rn )k · k f2 |Sr C(Rn )k r A(Rn ). But with for all fl ∈ Sr C(Rn ). These spaces do not fit in the scales Sp,q σ n σ n σ n S C (R ) = S∞ B(R ) = S∞,∞ B(R ) one has the embeddings

Sr1 C (Rn ) ,→ Sr C(Rn ) ,→ Sr2 C (Rn ),

0 < r2 < r < r1 .

(1.489)

87

1.4 Properties, II

This follows from [ST87, Prop. 4, p. 92]. One can disprove (1.488) by the same wavelet arguments as above. Again we may assume n = 2. Then one has by (1.485), (1.486), k f1 f2 |Sr2 C (R2 )k ∼ 22jr2 , k fl |Sr1 C (R2 )k ∼ 2 jr1 (1.490) and k fl |L∞ (R2 )k ∼ 1. Now it follows from (1.489) with 2r2 > r1 that there is no c > 0 with (1.488). 1.4.4 Hölder inequalities. We dealt in Section 1.4.2 with multiplication algebras r B(Rn ) based on the expansions (1.454)–(1.456) and the for the spaces Spr B(Rn ) = Sp, p multiplications (1.458)–(1.460) with (1.444). One may ask for further applications r A(Rn ). of this method in the context of pointwise multiplications of the spaces Sp,q This seems to be possible. We concentrate on so-called Hölder inequalities. First we recall what is known in the case of the isotropic inhomogeneous spaces Asp,q (Rn ). Let n ∈ N, 1 < p1, p2 < ∞, 1 1 1 = + < 1, p p1 p2 and

1 1 s = + , s p1 p1 n

1 1 s = + , s p2 p2 n

s>0

1 s 1 = + . s p p n

(1.491)

(1.492)

Let 0 < q1, q2, q ≤ ∞. Then Bps s ,q1 (Rn ) · Bps s ,q2 (Rn ) ,→ Bps s ,q (Rn ) 1

(1.493)

2

if, and only if, 0 < q1 ≤ p1,

0 < q2 ≤ p2

and

max(q1, q2 ) ≤ q ≤ ∞.

(1.494)

Similarly, Fps s ,q1 (Rn ) · Fps s ,q2 (Rn ) ,→ Fps s ,q (Rn )

(1.495)

max(q1, q2 ) ≤ q ≤ ∞.

(1.496)

1

2

if, and only if,

By (1.491) one has the Hölder inequality L p1 (Rn ) · L p2 (Rn ) ,→ L p (Rn ).

(1.497)

One can interpret (1.493), (1.495) as Hölder inequalities of type (1.497) shifted along lines of slope n in a ( p1 , s)-diagram up to the level of smoothness s > 0. This assertion goes back to [SiT95, Thm. 4.2.1, pp. 117–118] where one finds a complete proof based

1 Spaces on Rn

88

on paramultiplication (on the Fourier side). The if part has been repeated in [ET96, Thm. 2.4.3, pp. 51–54]. Furthermore, one may consult [RuS96, Chap. 4] where one finds numerous assertions about pointwise multiplications including the above Hölder inequalities, always preferably based on paramultiplication (on the Fourier r A(Rn ) with dominating side). One may ask for counterparts for the spaces Sp,q mixed smoothness. This might be a challenging task, but it seems to be possible to say at least something elaborating the technique as previously used in Section 1.4.2 (paramultiplication on the space side). First we modify (1.454)–(1.456) again based on Theorem 1.12. Any f ∈ r B(Rn ) with 0 < p, q ≤ ∞ and r ∈ R can be expanded by Sp,q 1

Õ

f (x) =

λk,m 2−[k](r− p ) ψk,m (x)

(1.498)

n ,m∈Z n k ∈N−1

with [k] =

Ín

j=1

kj, [k](r− p1 +1)

λk,m = λk,m ( f ) = 2

∫ Rn

f (x) ψk,m (x) dx

(1.499)

and r f ∈ Sp,q B(Rn ), λ = {λk,m }.

r k f |Sp,q B(Rn )k ∼ kλ |`q (`p )k,

(1.500)

Here kλ |`q (`p )k =

 Õ  Õ n k ∈N−1

|λk,m |

p

 q/p  1/q

(1.501)

m∈Z n

(usual modification if max(p, q) = ∞). First we wish to outline a new proof of (1.493) with n = 1 in the above context. One needs in addition some elementary embeddings of mixed `p -spaces. Let 0 < u ≤ v ≤ ∞. Then `u ,→ `v which means

∞ Õ

|al | v

and `u (`v ) ,→ `v (`u )

 1/v

≤

l=1

∞ Õ

|al | u

 1/u

,

(1.502)

al ∈ C

(1.503)

l=1

and Õ ∞ Õ ∞ l=1

k=1

|al,k |

u

 v/u  1/v

≤

Õ ∞ Õ ∞ k=1

l=1

|al,k |

v

 u/v  1/u

,

al,k ∈ C.

(1.504)

89

1.4 Properties, II

To justify (1.504) we apply the triangle inequality for `v/u to the left-hand side and obtain Õ ∞ ∞ Õ l=1

|al,k |

u

 v/u  uv · u1

Õ ∞ ∞ Õ

≤

k=1

=

k=1 Õ ∞

l=1 ∞ Õ

k=1

l=1

|al,k | |al,k |

u · uv

v

 u/v  1/u

 u/v  1/u

(1.505) .

Proposition 1.82. Let 1 < p1, p2 < ∞, 1 1 1 = + < 1, p p1 p2 1 1 = + r, r p1 p1

r > 0,

1 1 = + r, r p2 p2

(1.506)

1 1 = +r r p p

(1.507)

max(q1, q2 ) ≤ q ≤ ∞.

(1.508)

and 0 < q1 ≤ p1,

0 < q2 ≤ p2,

Then Brp r ,q1 (R) · Brp r ,q2 (R) ,→ Brp r ,q (R). 2

1

(1.509)

Proof. We remark that (1.318), (1.319) based on (1.507), (1.508) ensure Brp r ,q j (R) ,→ Fp0j ,2 (R) = L p j (R). j

(1.510)

In particular, (1.509) makes sense as a refinement of (1.497) at least in the context of L1loc (R)∩ S 0(R). We rely again on (1.443), (1.444) reformulated in terms of sequences according to (1.446) which means in our case, using (1.507), 1

µk,m = 2k(1− p )

Õ

k1

µ1k 1,m1 µ2k 2,m2 2 p1

2

+ kp

2

Ikk,m 1,k 2,m1,m2 ,

(1.511)

k 1,k 2 ∈N−1, m1,m2 ∈Z

k ∈ N−1 , m ∈ Z. One again has the estimates (1.449), (1.450). Similarly to there, we distinguish between two cases, max(k 1, k 2 ) ≤ k and max(k 1, k 2 ) > k. Let 0 k 2 ≤ k 1 ≤ k and let µk,m be the corresponding partial sum in (1.511). Then we have the same situation as at the beginning of the proof of Proposition 1.73, in particular m1 = m1 (m), m2 = m2 (m) almost uniquely. Using (1.449) and (1.506) one obtains Õ −(k−k 1 )(l+ p1 )−(k−k 2 ) p1 0 1 2, | µk,m |≤c | µ1k 1,m1 µ2k 2,m2 |2 (1.512) k 2 ≤k 1 ≤k, m1 (m),m2 (m)

1 Spaces on Rn

90

where l ∈ N is at our disposal. This is again an almost-diagonal situation with rapid decay outside the diagonal. By (1.506), (1.507) one has 1 1 1 = r + r, 2r p1 p2 p

p2r < pr .

(1.513)

Then it follows from Hölder’s inequality and the monotonicity (1.503),  Õ Õ k ∈N−1

≤

0 | µk,m |p

r

 q/p r  1/q

m∈Z

 Õ Õ

2r

 q/p2r  1/q

(1.514)

m∈Z

k ∈N−1

≤c

0 | µk,m |p

 Õ  Õ k 1 ∈N−1

r | µ1k 1,m1 | p1

 q/p2r  1/q  q/p1r  1/q  Õ  Õ p2r 2 . · | µk 2,m2 |

m1 ∈Z

k 2 ∈N−1

m2 ∈Z

By (1.508) and again the monotonicity (1.503) with v = q and u = q1, q2 one has  Õ Õ k ∈N−1

r 0 | µk,m |p

 q/p r  1/q

≤ c k f1 |Brp r ,q1 (R)k · k f2 |Brp r ,q2 (R)k,

(1.515)

2

1

m∈Z

where we used (1.500) specified by Spr r ,q j B(R) = Brp r ,q j (R), j = 1, 2. Let k < k 1 , j

j

00 be the corresponding partial sum in (1.511). Using (1.450) and k 2 ≤ k 1 and let µk,m (1.506) one obtains 00 | µk,m | ≤ c 2−(k

1 −k)

Õ

| µ1k 1,m1 µ2k 2,m2 | 2

k 1 −k p1

2

+ k p−k −(k 1 −k)l 2

,

(1.516)

k 1 then the left-hand side of (1.519) can be estimated by the right-hand (k 1 −k)

1 0

q 1 . But this factor is well compensated by the factor side multiplied by c 2 1 −k) −(k 2 in (1.516). For fixed k ∈ N−1 we take in (1.519) the `p1 -norm with respect to m ∈ Z. Since q1 ≤ p1 we can apply (1.504) with u = q1 and v = p1 . Then one has the `p1 -summation with respect to m1 inside and the `q1 -summation with respect to k 1 , k 2 outside. Using the monotonicity `p1r ,→ `p1 one can estimate the (now inside) `p1 -norm from above by the `p1r -quasi-norm. The related factors µ2k 2,m2 can first be estimated from above by sup | µ2k 2,m2 | where the supremum is taken over 2

2

2−k m2 ∼ 2−k m which is at most ∼ max(1, 2k −k ) terms well compensated by (1.517). But then one has essentially the same situation as in the case k 1 ≤ k. Using again the rapid decay with respect to k off the diagonal according to (1.517) one obtains 00 in place of µ 0 , and similarly for the cases with the counterpart of (1.515) with µk,m k,m 1 2 k ≤ k . Then one needs q2 ≤ p2 . This proves (1.509).  Remark 1.83. One can compare the above proof, paramultiplication on the space side, with the corresponding proof in [ET96, Thm. 2.4.3, pp. 52–54], based on [SiT95], paramultiplication on the Fourier side. Both proofs use Hölder inequalities based on (1.513), (1.518). Instead of (1.519) and `q1 (`p1 ) ,→ `p1 (`q1 ) we used in [ET96] the equivalent embedding (1.510), whereas the counterpart of the above case k 1 ≤ k in [ET96] uses (1.513) and sophisticated estimates in terms of paramultiplication on the Fourier side. Otherwise the method in [SiT95, ET96] can be applied, without any additional effort, to Rn for all n ∈ N, which means (1.491)–(1.493) for the B-spaces. We did not try to extend the above method from n = 1 to n ∈ N in the context of the isotropic spaces Brp,q (Rn ). This might be possible, but it may require additional effort. On the other hand, the above method allows us to step from Brp,q (R) r B(Rn ). to Sp,q Theorem 1.84. Let n ∈ N, 1 < p1 < ∞, 1 < p2 < ∞, 1 1 1 = + < 1, p p1 p2 and

1 1 = + r, pr1 p1

1 1 = + r, pr2 p2

r>0

1 1 = + r. pr p

(1.520)

(1.521)

1 Spaces on Rn

92 Let 0 < q1, q2, q ≤ ∞. Then

Spr r ,q1 B(Rn ) · Spr r ,q2 B(Rn ) ,→ Spr r ,q B(Rn ) 1

2

(1.522)

if, and only if, 0 < q1 ≤ p1,

0 < q2 ≤ p2,

max(q1, q2 ) ≤ q ≤ ∞.

(1.523)

Proof. Step 1. As far as the if part is concerned we are in a rather similar situation to the proof of Theorem 1.75 where we used the product structure of all factors in (1.463), based on (1.460). There we reduced the arguments to the one-dimensional case according to Proposition 1.73. Its role is now taken over by Proposition 1.82. In the related proof we distinguished between two cases, characterized by (1.513) and (1.518), where the latter case is divided into two subcases according to (1.517). Now one has the same situation with respect to any of the n coordinates x1, . . . , xn . Each of the 2n main cases can be treated separately using the above-mentioned product structure of all factors. First one deals with the cases characterized by (1.518) and (1.517), where one ends up with `p1r and `p2r as far as the Zn -parts are concerned. Afterwards one can incorporate the cases originating from (1.513). As far as the k-part according to (1.501) is concerned one relies again on the almost diagonal situation with exponential decay off the diagonal. Step 2. The only-if part relies on the multiplication property according to Proposition 1.19. We assume that we have (1.522) for given r > 0 and p’s as in (1.520), (1.521) and some q1 , q2 , q. Then we apply (1.119) in the same way as in Step 2 of the proof of Theorem 1.75. Let f1 (x) = g1 (x1 ) ψ(x 0) and

f2 (x) = g2 (x1 ) ψ(x 0),

(1.524)

where ψ ∈ S(Rn−1 ) is a fixed function, say, with ψ(0) = 1 and g1 ∈ Brp r ,q1 (R), 1 g2 ∈ Brp r ,q2 (R). This reduces (1.522) to 2

Brp r ,q1 (R) · Brp r ,q2 (R) ,→ Brp r ,q (R). 1

2

(1.525)

Then (1.523) follows from (1.494).  Remark 1.85. There should be an F-counterpart extending (1.495) to related spaces with dominating mixed smoothness. In other words, let n ∈ N, 1 < p1 < ∞, 1 < p2 < ∞ and p, r, pr1 , pr2 , pr be as in (1.520), (1.521). Let 0 < q1, q2, q ≤ ∞. Then (1.526) Spr r ,q1 F(Rn ) · Spr r ,q2 F(Rn ) ,→ Spr r ,q F(Rn ) 1

2

93

1.4 Properties, II

if, and only if, max(q1, q2 ) ≤ q ≤ ∞.

(1.527)

The one-dimensional case n = 1 is covered by (1.495), (1.496). But the n-dimensional case must be checked. Of special interest would be the (fractional) Sobolev spaces Spr H(Rn ) with dominating mixed smoothness according to (1.40), (1.41), r F(Rn ), Spr H(Rn ) = Sp,2

1 < p < ∞, r ∈ R.

(1.528)

Then Spr r H(Rn ) · Spr r H(Rn ) ,→ Spr r H(Rn ) 1

2

(1.529)

is a special case of (1.526) for the above parameters and pr > 1. Remark 1.86. If 0 < p < ∞, 0 < q < ∞ and 1 1
− 1 < r < min , 1 p p

(1.530)

r B(Rn ); see [T10, then the respective Haar systems are bases in Brp,q (R) and Sp,q Thm. 2.9, p. 80, Thm. 2.38, p. 104]. One may ask whether one can replace in these cases the Daubechies wavelets in the proofs of Proposition 1.82 and Theorem 1.84 by the simpler Haar wavelets. The integrals in (1.444) can now be calculated explicitly and many of them are zero. But one no longer has the rapid decay in the offdiagonal cases as in, for example, (1.517) choosing 2 ≤ l ∈ N. Furthermore, the multiplication of three Haar functions in (1.444) gives in the non-vanishing cases positive and negative terms and it is not clear how to cope with them as a substitute for the indicated rapid decay off the diagonal.

Using the duality according to Proposition 1.17 one can extend Theorem 1.84 to some spaces with r < 0. Corollary 1.87. Let n ∈ N, 1 < p1 < ∞, 1 < p2 < ∞, 1 1 1 = + < 1, p p1 p2 and 0
0,

(1.539) x ∈ Rn is the well-known Gauss–Weierstrass semigroup which can be written on the Fourier side as −t |ξ | 2 d W w b(ξ), ξ ∈ Rn, t > 0. (1.540) t w(ξ) = e The Fourier transform according to (1.3) and its S 0(Rn )-generalization is taken with respect to the space variables x ∈ Rn . Of course, (1.539), (1.540) must be interpreted as distributions. But we recall that (1.539) makes sense pointwise: it is the convo2 lution of w ∈ S 0(Rn ) and gt (y) = (4πt)−n/2 e−|y | /4t ∈ S(Rn ) with the consequence that w ∗ gt ∈ C ∞ (Rn ),

|(w ∗ gt )(x)| ≤ ct (1 + |x| 2 ) N /2,

x ∈ Rn,

(1.541)

for some ct > 0 and, say, N ∈ N. Basic information and references may be found j in [T14, Sect. 4.1, pp. 112–114]. Let ψG,m with j ∈ N0 and m ∈ Zn be the L∞ normalized n-dimensional isotropic Daubechies wavelets according to (1.81). Then ∫ 2 1 j j j bG,m (x, t) = Wt ψG,m (x) = e−|x−y | /4t ψG,m (y) dy, (1.542) n/2 (4πt) Rn

95

1.4 Properties, II

x ∈ Rn , t > 0 are L∞ -normalized molecules, called caloric wavelets. Details, explanations and proofs may be found in [T13, Sects. 2.4.1, 2.4.2, pp. 81–88]. We used this observation in [T13, Thm. 5.12, p. 171] and [T14, Thm. 4.1, p. 114] to study the smoothing effect of Wt in the context of local and hybrid function spaces LrAsp,q (Rn ) and L rAsp,q (Rn ). This covers in particular the related (inhomogeneous) isotropic spaces Asp,q (Rn ) according to Definition 1.1, Remark 1.2 with the following outcome, explicitly formulated in [T13, Thm. 5.30, p. 187]. Let 1 ≤ p, q ≤ ∞ (p < ∞ for F-spaces), s ∈ R and d ≥ 0. Then there is a constant c > 0 such that for all t with 0 < t ≤ 1 and all w ∈ Asp,q (Rn ),

n s n t d/2 Wt w | As+d p,q (R ) ≤ c kw | A p,q (R )k.

(1.543)

One may ask for a counterpart of this observation in the context of the spaces r A(Rn ) according to Definition 1.4, Remark 1.5. First one has to replace the Sp,q n-dimensional (isotropic) Gauss–Weierstrass semigroup Wt w according to (1.539) by a related product of Gauss–Weierstrass semigroups in R. Let n ≥ 2 and n t = (t1, . . . , tn ) ∈ R++

(1.544)

n as in (1.155). Instead of t ∈ Rn we also write 0 < t ∈ Rn . Let with R++ ++ În d d (t) = j=1 t j as previously used in Theorem 1.27. Then

W(t) w(x) = (4πt)

−1/2

= (4πt)

−1/2

∫



n Ö

R n j=1 n Ö

w,

e− |x j −y j |

e

2 /4t

j

w(y) dy (1.545)

− |x j −· | 2 /4t j



,

0 0 such that for all t = (t1, . . . , tn ) ∈ Rn , 0 < t j ≤ 1 and r A(Rn ), all w ∈ Sp,q

r+d r A(Rn ) ≤ c kw |Sp,q A(Rn )k. (1.549) (t)d/2 W(t) w |Sp,q Proof. Let w(x) =

Õ

1

λk,m 2−[k](r− p ) ψk,m (x)

(1.550)

n ,m∈Z n k ∈N−1

be the expansion according to Theorem 1.12. Then one can argue as in the proof of [T13, Thm. 5.12, pp. 171–173]. The needed lifting is covered by (1.38), (1.39). Otherwise one can modify the splitting in [T13, Steps 2, 3, pp. 172/173] now applied to W(t) w. This is the point where one needs the above comments about caloric wavelet r A(Rn ). Otherwise one can follow the arguments given there, decompositions of Sp,q appropriately modified.  Remark 1.90. If t > 0 and (t) = (t, . . . , t) then it follows from (1.539), (1.545) or (1.540), (1.546) that Wt w(x) = W(t) w(x),

t > 0, x ∈ Rn .

(1.551)

97

1.4 Properties, II

0 (Rn ) = S 0 F(Rn ), 1 < p < ∞, (1.11), (1.34). Then one has by Recall L p (Rn ) = Fp,2 p,2 (1.543) and (1.549) with 1 < p < ∞, d ≥ 0, d t d/2 kWt w |Fp,2 (Rn )k ≤ c kw |L p (Rn )k,

t

nd/2

kWt w

d |Sp,2 F(Rn )k

n

≤ c kw |L p (R )k,

0 < t ≤ 1,

(1.552)

0 < t ≤ 1.

(1.553)

d (Rn ) = H d (Rn ) and S d F(Rn ) = S d H(Rn ) are related (fractional) Recall that Fp,2 p p p,2 Sobolev spaces which can be equivalently normed according to (1.16), (1.17) and (1.40), (1.41). Corresponding multiplier assertions for L p (Rn ), 1 < p < ∞ show that

Spd H(Rn ) ,→ Hpd (Rn ) ,→ Spd/n H(Rn ),

d ≥ 0, 1 < p < ∞,

(1.554)

where the second embedding is a special case of (1.432). In any case, (1.554) is in good agreement with (1.552), (1.553) Remark 1.91. The smoothing (1.543) is one of the crucial instruments in our approach to non-linear heat equations and Navier–Stokes equations in [T13, T14] and even more in connection with PDE models for chemotaxis in [T17]. It would be of interest to find applications for its mixed counterpart (1.549). We do not know whether there are any relations between PDEs and spaces with dominating mixed smoothness. On the other hand, these spaces play a crucial role in diverse approximations, sampling and numerical integration. One may ask whether the controlled smoothing (1.549) can be used in this context. One can shift the numerical integration of compactly supported functions via ∫ ∫ f (x) dx = W(t) f (x) dx (1.555) Rn

Rn

from rough to smooth where one always has a precise control about decay in space and time as described, for example, in Remark 1.90. Instead of W(t) one can rely on other distinguished suitable functions as constructed in [UlU16]. 1.4.6 Thermic characterizations. We use the preceding considerations to deal r A(Rn ) based on W w according to with thermic characterizations of the spaces Sp,q (t) (1.545), (1.546). But first we recall how related assertions for the (inhomogeneous) isotropic spaces Asp,q (Rn ) as introduced in Definition 1.1, Remark 1.2 look. Again let Wt f be the Gauss–Weierstrass semigroup in Rn according to (1.539), (1.540) (with f in place of w). We follow [T15, pp. 14, 58], which is a commented reformulation of corresponding assertions in [T92, Sect. 2.6.4, pp. 151–155]. There one also finds detailed further references. Let s σp(n) =
n max( p1 , 1) − 1 then Asp,q (Rn ) ⊂ L1loc (Rn ) (see [T01, Thm. 11.2, pp. 168–169]), and the above assertions can be complemented as follows. Let 0 < p, q ≤ ∞ (p < ∞ for F-spaces),

s > σp(n), s/2 < m ∈ N.

(1.563)

99

1.4 Properties, II s (Rn ) is the collection of all f ∈ S 0 (Rn ) ∩ L loc (Rn ) such that Then Bp,q 1 s (Rn )k ∗ = k f |L (Rn )k k f |Bp,q p m ∫ 1 s dt  1/q + t (m− 2 )q k∂tmWt f |L p (Rn )k q t 0

(1.564)

s (Rn ) is the collection of all f ∈ S 0 (Rn ) ∩ is finite (equivalent quasi-norms) and Fp,q loc n L1 (R ) such that s ∗ k f |Fp,q (Rn )km = k f |L p (Rn )k

 ∫ 1 q dt  1/q s

|L p (Rn ) + t (m− 2 )q ∂tmWt f (·) t 0

(1.565)

is finite (equivalent quasi-norms), with the usual modification if q = ∞. We refer again to [T92, Thm. 2.6.4, p. 152]. We use f ∈ S 0(Rn ) ∩ L1loc (Rn ) always with the understanding f ∈ L1loc (Rn ) and | f | ∈ S 0(Rn ). We refer to the discussion in [T15, (3.2), (3.3), pp. 45, 58]. r A(Rn ) We are interested in corresponding characterizations of the spaces Sp,q according to Definition 1.4, Remark 1.5. Again let W(t) f be given by (1.545), (1.546) (with f in place of w) where 0 < t ∈ Rn abbreviates (1.544). Let m ∈ N and m = ∂(t)

n Ö j=1

∂tmj =

n Ö

∂ m /∂t jm,

t = (t1, . . . , tn ) ∈ Rn .

(1.566)

j=1

Applied to (1.545) one has on the Fourier side (1.546), m ∂(t) W(t) f


∧

m š (ξ) = ∂(t) W(t) f (ξ) = (−1)nm

n Ö

ξ j2m e−t j ξ j b f (ξ), 2

ξ ∈ Rn,

(1.567)

j=1

0 < t ∈ Rn . We reduced in [T92, Sect. 2.6.4, pp. 151–155] descriptions of type (1.561), (1.562) for the spaces Asp,q (Rn ) to more general Fourier-analytical decompositions according to [T92, Thm. 2.4.1, pp. 100/101, Thm. 2.5.1, p. 132] covering in particular (isotropic n-dimensional) Gauss–Weierstrass semigroups. We outlined in [T15, Sect. 2.6.3, pp. 31–36] how a counterpart of this reduction in the context of tempered homogeneous spaces with dominating mixed smoothness may look. This m W f according to (1.567) will play a will not be repeated here. Quite obviously, ∂(t) (t) decisive role. We first give a related formulation and afterwards add some comments. Because of the inhomogeneity, we need a refinement of (1.566), (1.567) that is the counterpart of (1.46)–(1.48). As there, let u ⊂ {1, . . . , n}, 2 ≤ n ∈ N, where u = ∅ is admitted, |u| = card u and u¯ = {1, . . . , n} \ u. Then Ö Ö
∧ 2 2 m ∂(t),u,1 W(t) f (ξ) = (−1) |u |m ξa2m e−ta ξa e−ξb b f (ξ), ξ ∈ Rn (1.568) a ∈u

b ∈u¯

1 Spaces on Rn

100 and m ∂(t),u,0 W(t) f


∧

(ξ) = (−1) |u |m

Ö

ξa2m e−ta ξa b f (ξ), 2

ξ ∈ Rn .

(1.569)

a ∈u

Temporarily let Wτ(b) , 0 ≤ τ < ∞ be the one-dimensional Gauss–Weierstrass semigroup with respect to the coordinate xb in Rn . Then one replaces ∂τmWτ(b) in (1.567) by W1(b) in (1.568) and by the identity id = W0(b) in (1.569). Again let Q = (0, 1)n and Î (t)ud = a ∈u tad with d ∈ R and t ∈ Rn . Put (t)u = (t)1u . If u = {1, . . . , n} then we and Theorem 1.27. Furwrite (t)ud = (t)d and (t)u = (t) in agreement with (1.545)
thermore, (t)ud = 1 if u = ∅. Recall σp = max p1 , 1 − 1, 0 < p ≤ ∞. Furthermore, according to (1.134) one has r Sp,q A(Rn ) ⊂ L1loc (Rn ) if r > σp .

(1.570)

We use S 0(Rn ) ∩ L1loc (Rn ) as indicated after (1.565). Theorem 1.92. Let 0 < p, q ≤ ∞ (p < ∞ for F-spaces). r B(Rn ) is the collection of all f ∈ S 0 (Rn ) such that (i) Let r < 0. Then Sp,q ∫ rq dt  1/q r k f |Sp,q B(Rn )kW = (t)− 2 kW(t) f |L p (Rn )k q (t) Q

(1.571)

r F(Rn ) is the collection of all f ∈ S 0 (Rn ) is finite (equivalent quasi-norm) and Sp,q such that

 ∫ q dt  1/q rq

r n |L p (Rn ) (1.572) k f |Sp,q F(R )kW = (t)− 2 W(t) f (·) (t) Q

is finite (equivalent quasi-norm), with the usual modification if q = ∞. r B(Rn ) is the collection of all f ∈ S 0 (Rn ) (ii) Let r ∈ R and r/2 < m ∈ N0 . Then Sp,q such that Õ ∫ dt  1/q (m− r )q r n m k f |Sp,q B(R )km = (t)u 2 k∂(t),u,1 W(t) f |L p (Rn )k q (1.573) (t)u Q u r F(Rn ) is the collection of all f ∈ S 0 (Rn ) is finite (equivalent quasi-norms) and Sp,q such that

Õ

 ∫ q dt  1/q (m− r2 )q m r n k f |Sp,q F(Rn )km = (t) |L (R ) ∂ W f (·)

(1.574) p u (t),u,1 (t) (t)u Q u

is finite (equivalent quasi-norms), with the usual modification if q = ∞.

1.4 Properties, II

101

r B(Rn ) is the collection of all (iii) Let r > σp and r/2 < m ∈ N. Then Sp,q 0 n loc n f ∈ S (R ) ∩ L1 (R ) such that Õ ∫ dt  1/q (m− r )q m r ∗ (1.575) (t)u 2 k∂(t),u,0 W(t) f |L p (Rn )k q k f |Sp,q B(Rn )km = (t)u Q u r F(Rn ) is the collection of all f ∈ S 0 (Rn ) ∩ is finite (equivalent quasi-norms) and Sp,q L1loc (Rn ) such that

Õ

 ∫ q dt  1/q (m− r )q m

r ∗ k f |Sp,q F(Rn )km = (t)u 2 ∂(t),u,0 W(t) f (·) |L p (Rn ) (1.576)

(t) u Q u

is finite (equivalent quasi-norms), with the usual modification if q = ∞. Î Proof (comments). Step 1. If f (x) = nj=1 f j (x j ) then one has by Proposition 1.19, r k f |Sp,q A(Rn )k =

n Ö

k f j | Arp,q (R)k.

(1.577)

j=1

If one introduces new quasi-norms then it is a good test to confirm this product property. This is the case for all quasi-norms in the above theorem. It also makes clear why one needs the sum over u. If u = ∅ then one has the term kW1 f |L p (Rn )k in (1.573), (1.574) with W1 f as in (1.559) and k f |L p (Rn )k in (1.575), (1.576). Step 2. There are two possibilities to prove the theorem. One can imitate the corresponding arguments for the isotropic spaces Asp,q (Rn ) resulting in (1.557), (1.558), as well as (1.561), (1.562) and (1.564), (1.565). This means that one begins with [T92, Sect. 2.6.4, pp. 151–155] which covers some specific aspects of thermic characterizations including the step from (1.561), (1.562) to (1.564), (1.565), based on (1.570). Otherwise the thermic characterizations of Asp,q (Rn ) are reduced to the rather general Fourier-analytical decomposition theorems [T92, Thm. 2.4.1, pp. 100–101, Thm. 2.5.1, p. 132]. There is little doubt that the related arguments in the context of r A(Rn ). the inhomogeneous isotropic spaces Asp,q (Rn ) can be shifted to the spaces Sp,q However, the details may require some effort. But there is a second, maybe more transparent and more efficient, way. The related isotropic version goes back to [Ryc99a]. Its dominating mixed counterpart may be found in [Vyb06, Thm. 1.23, p. 22]. One may also consult [Baz03, Baz05]. Combined with the already indicated peculiarities of thermic characterizations one has direct access to the assertions of the theorem. Another interesting special case and also a discussion about some weak points in the papers [Ryc99a, Vyb06] and related further references may be found in [UlU16]. 

1 Spaces on Rn

102

Remark 1.93. In [T15, Sect. 2.6.3, pp. 31–36] we discussed tempered homogeneous function spaces with dominating mixed smoothness (as a proposal). There one finds m W f from general Fourier-analytical charhow to recover expressions of type ∂(t) (t) acterizations of spaces with dominating mixed smoothness, the dominating mixed counterpart of [T92, Thms. 2.4.1, 2.5.1] or of [Vyb06]. 1.4.7 Tempered homogeneous spaces with negative smoothness. In [T15] we de∗

veloped the theory of the tempered homogeneous isotropic spaces Asp,q (Rn ), preferably in the distinguished strip
n 1 −1 < s < , (1.578) p p
within the framework of the dual pairing S(Rn ), S 0(Rn ) where again A ∈ {B, F}. The restriction (1.578) ensures 0 < p, q ≤ ∞,

n

∗

S(Rn ) ,→ Asp,q (Rn ) ,→ S 0(Rn ).

(1.579)

We complemented this theory in [Tri17a, Tri17b] by further properties, examples and limiting cases. It was our main aim to offer an alternative way to deal with ∗

the homogeneous spaces Asp,q (Rn ) in comparison with the homogeneous spaces
Û n ), SÛ 0(Rn ) and the AÛ sp,q (Rn ) considered in the framework of the dual pairing S(R unavoidable bitter fighting modulo polynomials. The interest in these homogeneous spaces comes from scaling properties of some distinguished PDE models in natural sciences (physics and biology), above all of the Navier–Stokes equations. One may ask for a corresponding theory of tempered homogeneous spaces with dominating ∗

r A(Rn ), n ≥ 2, although we have no applications so far. In mixed smoothness Sp,q [T15, Sect. 2.6.3, pp. 31–36] we outlined how a corresponding theory may look. Now we return to this topic restricting ourselves to some main ideas. Otherwise we follow [T15], dealing first in this Section 1.4.7 with spaces of negative smoothness r < 0, complemented afterwards in Sections 1.4.9 and 1.4.10 by spaces with positive and general smoothness. We use the same notation as in the preceding sections. This applies in particular to (1.544)–(1.546) and the abbreviations explained ahead of Theorem 1.92. ∗

r B(Rn ) Definition 1.94. Let 0 < p, q ≤ ∞ (p < ∞ for F-spaces) and r < 0. Then Sp,q 0 n is the collection of all f ∈ S (R ) such that ∫ ∗ rq dt  1/q n r (1.580) k f |Sp,q B(R )kW = (t)− 2 kW(t) f |L p (Rn )k q n (t) R++

103

1.4 Properties, II ∗

r F(Rn ) is the collection of all f ∈ S 0 (Rn ) such that is finite and Sp,q

 ∫ ∗

r k f |Sp,q F(Rn )kW =

(t)−

n R++

rq 2

 1/q

W(t) f (·) q dt |L p (Rn ) (t)

(1.581)

is finite, with the usual modification if q = ∞. Remark 1.95. This is the homogeneous version of the quasi-norms in Theorem 1.92(i). On the other hand, the above definition is the dominating mixed counterpart of [T15, Def. 3.1, (i), (ii), p. 46] where we introduced the tempered homogeneous ∗

isotropic spaces Asp,q (Rn ) with s < 0. In [T15, Thm. 3.3, pp. 48–49] we collected basic properties of these spaces. This can be carried over to the above spaces ∗

r A(Rn ) with A ∈ {B, F}, and r < 0 by more or less obvious modifications of the Sp,q related proofs. But it seems to be reasonable (also for our later purposes) to give explicit formulations and to comment on the underlying arguments.

According to the terminology introduced in [T15, Sect. 1.3, pp. 5–6] the quasi∗

r A(Rn ) are admissible: it makes norms (1.580), (1.581) in the spaces A(Rn ) = Sp,q 0 n sense to test any f ∈ S (R ) for whether it belongs to A(Rn ) or not. Within a given fixed quasi-Banach space A(Rn ), equivalent quasi-norms are called domestic. They are not necessarily reasonable for any S 0(Rn ) or admissible. Maybe the most dis∗

r A(Rn ) are their Fourier-analytical tinguished domestic quasi-norms in A(Rn ) = Sp,q descriptions. These will be needed later on. First we describe the homogeneous r A(Rn ) as introduced counterpart of the Fourier-analytical definition of the spaces Sp,q in Section 1.1.2. Let ϕ ∈ S(R) with

ϕ(y) = 1 if |y| ≤ 1

and

ϕ(y) = 0 if |y| ≥ 3/2,

ϕ j (y) = ϕ(2−j y) − ϕ(2−j+1 y),

y ∈ R, j ∈ Z

(1.582) (1.583)

and ϕk (x) =

n Ö

ϕkl (xl ),

k = (k 1, . . . , k n ) ∈ Zn, x = (x1, . . . , xn ) ∈ Rn .

(1.584)

l=1

Then Õ k ∈Z n

ϕ (x) = 1 k

n

if x ∈ R with

n Ö

x j , 0.

(1.585)

j=1

The function ϕk refers to  Rk = x ∈ Rn : 2kl −1 < |xl | < 2kl , l = 1, . . . , n

(1.586)

1 Spaces on Rn

104

(2n rectangles). This is the homogeneous counterpart of (1.25)–(1.29). Let A(Rn ) be a quasi-normed space in S 0(Rn ) with A(Rn ) ,→ S 0(Rn ). Then A(Rn ) is said to have the Fatou property if there is a positive constant c such that from sup kg j | A(Rn )k < ∞ and

g j → g in S 0(Rn )

(1.587)

j ∈N

it follows that g ∈ A(Rn ) and kg | A(Rn )k ≤ c sup kg j | A(Rn )k.

(1.588)

j ∈N

We brought this formulation over from Section 1.3.6, where one finds comments Í and references about this useful property. Again let [k] = nj=1 k j if k ∈ Zn and as before λx = (λ1 x1, . . . , λn xn ) if λ = (λ1, . . . , λn ) ∈ Rn and x = (x1, . . . , xn ) ∈ Rn . Î Furthermore, 0 < λ ∈ Rn means λ ∈ Rn with λ j > 0 and (λ)d = nj=1 λ dj , d ∈ R. For 0 < p, q ≤ ∞ and r ∈ R let ∗

r k f |Sp,q B(Rn )kϕ =

 Õ

q  1/q
∨ 2r[k]q ϕk b f |L p (Rn )

(1.589)

k ∈Z n

and

 Õ

∗
∨ q  1/q

r k f |Sp,q F(Rn )kϕ = 2r[k]q ϕk b f (·) |L p (Rn ) .

(1.590)

k ∈Z n

Recall A = {B, F}. ∗

r A(Rn ) with r < 0 and 0 < p, q ≤ ∞ (p < ∞ for Theorem 1.96. The spaces Sp,q F-spaces) according to Definition 1.94 are quasi-Banach spaces (Banach spaces if p ≥ 1, q ≥ 1). They have the Fatou property. In addition, ∗

r r Sp,q A(Rn ) ,→ Sp,q A(Rn ) ,→ S 0(Rn )

(1.591)

and ∗

1

∗

r r k f (λ·) |Sp,q A(Rn )k = (λ)r− p k f |Sp,q A(Rn )k,

Furthermore,

∗

r k f |Sp,q A(Rn )kϕ,

0 < λ ∈ Rn .

(1.592)

∗

r f ∈ Sp,q A(Rn ) ∗

(1.593)

r A(Rn ), with the usual modifications if are equivalent domestic quasi-norms in Sp,q q = ∞.

105

1.4 Properties, II

Proof. This is the direct counterpart of [T15, Thm. 3.3, pp. 48–51]. This applies also to its proof. But we indicate the necessary modifications. The embedding (1.591) follows from Definition 1.94 compared with Theorem 1.92(i) and (1.108). We prove ∗

n n r n the Fatou property. Let {g j }∞ j=1 ⊂ A(R ) with A(R ) = Sp,q A(R ) as in (1.587), (1.588). Then by (1.545) one has

W(t) g j (x) → W(t) g(x),

x ∈ Rn, 0 < t ∈ Rn .

(1.594)

But this reduces the Fatou property for A(Rn ) to the usual Fatou lemma for integrable functions. One may consult [Fra86] for some technical details. In addition to the n and above abbreviations, let (tλ2 ) = (t1 λ12, . . . , tn λn2 ) where t = (t1, . . . , tn ) ∈ R++ n λ = (λ1, . . . , λn ) ∈ R++ . Then by (1.545) one has

n W(tλ2 ) f (λx) = W(t) f (λ·) (x), x ∈ Rn, λ ∈ R++ . (1.595) Inserted into (1.580), (1.581) one obtains (1.592). In Step 2 of the proof of Theorem 1.92 we mentioned that (1.571), (1.572) follow from the mixed counterpart of the general Fourier-analytical decomposition theorem for inhomogeneous isotropic spaces according to [T92, Thms. 2.4.1, 2.5.1]. The (technically simpler) general Fourier-analytical decomposition theorem for homogeneous isotropic spaces may be found in [T15, Prop. 2.10, p. 18–19] based on [Tri88, Corols. 2, 8, pp. 172–173, 183]. We take for granted that there is a related counterpart for the above homogeneous spaces with dominating mixed smoothness using that ϕk and the kernels of W(t) have the desired product structure, (1.584) and (1.545), (1.546). Then (1.593) based on (1.589), (1.590) follows from (1.580), (1.581). We refer the reader to [T15, p. 50] as far as some technicalities are concerned.  Remark 1.97. Recall that (1.593) based on (1.589), (1.590) are equivalent domestic ∗

r A(Rn ) according to Definition 1.94. In quasi-norms in the already defined spaces Sp,q particular, they cannot be used to check whether f ∈ S 0(R n ) belongs to this space or  Î 0 n not. If f ∈ S (R ) with supp b f ⊂ y ∈ Rn : nj=1 y j = 0 then (1.589), (1.590) are zero (even worse than the fighting modulo polynomial as for homogeneous isotropic spaces).

One may ask for a counterpart of (1.579), ensured by (1.578), for the spaces ∗ r A(Rn ) Sp,q

so far with r < 0. This is indispensable for a duality theory within
the dual pairing S(Rn ), S 0(Rn ) . We complement the standard spaces S(Rn ) and D(Rn ) = C0∞ (Rn ) by ∫  D(Rn )◦ = ϕ ∈ D(Rn ) : Rn ϕ(x) dx = 0 . (1.596) Recall A ∈ {B, F}.

1 Spaces on Rn

106 ∗

r A(Rn ) be the spaces as introduced in Definition 1.94. If, Proposition 1.98. Let Sp,q in addition,

1 < p ≤ ∞ (p < ∞ for F-spaces), then

0 < q ≤ ∞ and

1 − 1 < r < 0, (1.597) p

∗

r r S(Rn ) ,→ Sp,q A(Rn ) ,→ Sp,q A(Rn ) ,→ S 0(Rn ).

(1.598)

If, in addition, 1 < p < ∞,

0 0, kW(t) ψ |L p (Rn )k ≥ c (t)− 2 + 2p , 1

1

0 < p < ∞, t j ≥ 1,

(1.608)

Î where again t = (t1, . . . , tn ) and (t)a = nj=1 t ja with (t) = (t)1 . Let 0 < q < ∞. Inserted in the right-hand side of (1.580) one has ∫ ∫ ∞ q  1 dτ n − r2q n q dt (t) kW(t) ψ |L p (R )k ≥c τ − 2 (r+1− p ) =∞ (1.609) n (t) τ R++ 1

1 Spaces on Rn

108 if r ≤

1 p

∗

r B(Rn ) if 0 < p, q < ∞ − 1. This shows that S(Rn ) is not a subset of Sp,q ∗

r F(Rn ). This makes it and r ≤ p1 − 1. By (1.600) one can extend this assertion to Sp,q clear that the counterpart of (1.578) is now the distinguished strip

0 < p, q ≤ ∞,

1 1 −1 σp(n) . In [T15] we took these descriptions as ∗

starting points to introduce corresponding homogeneous isotropic spaces Asp,q (Rn )
in the framework of the dual pairing S(Rn ), S 0(Rn ) . As far as related counterparts for spaces with dominating mixed smoothness are concerned, so far we have ∗

r A(Rn ) with r < 0 in introduced corresponding tempered homogeneous spaces Sp,q n n in (1.580), (1.581). Definition 1.94, replacing Q = (0, 1) in (1.571), (1.572) by R++ One may ask for an extension of this method to, say, r > σp again taking related ∗

assertions for Asp,q (Rn ) and Asp,q (Rn ) as a guide. This causes some problems and will be discussed in the following Sections 1.4.9, 1.4.10. In this Section 1.4.8 we adapt (1.575), (1.576) to this task which might be of self-contained interest. First we recall the isotropic counterpart following [T15, Rem. 3.10, p. 58], which in turn is based on [T92, Thm. 2.6.4, p. 152, Rem. 2.6.4, p. 155]. Again we use S 0(Rn ) ∩ L1loc (Rn ) as discussed after (1.565). Let s 0 < p, q ≤ ∞ (p < ∞ for F-spaces), s > σp(n), < m ∈ N. (1.612) 2 s (Rn ) is the collection of all f ∈ S 0 (Rn ) ∩ L loc (Rn ) such that Then Bp,q 1 ∫ ∞  1/q m (m− s2 )q n q dt s (Rn )k + = k f |L (Rn )k + (1.613) k f |Bp,q t k∂ W f |L (R )k p t p m t t 0 s (Rn ) is the collection of all f ∈ S 0 (Rn ) ∩ is finite (equivalent quasi-norms) and Fp,q loc n L1 (R ) such that s + k f |Fp,q (Rn )km = k f |L p (Rn )k

 ∫ ∞

q dt  1/q s

+ t (m− 2 )q ∂tmWt f (·) |L p (Rn ) t 0

(1.614)

109

1.4 Properties, II

is finite (equivalent quasi-norms), with the usual modification if q = ∞. In other words, one can replace the integration over t ∈ (0, 1) in (1.564), (1.565) by t ∈ (0, ∞). We ask for a related replacement of Q in (1.575), (1.576). But this requires some effort. We proved in [T92, Thm. 2.3.3, p. 98] that (1.613), (1.614) are respective equivalent quasi-norms by reduction to corresponding Fourier-analytical assertions. Now we are doing the same in the context of spaces with dominating mixed smoothness. Let ϕ ∈ S(R) be as in (1.25) = (1.582). Let {ϕk }k ∈N0n and {ϕk }k ∈Zn be the related inhomogeneous and homogeneous resolutions of unity according to (1.27), (1.28) and (1.584), (1.585). Let r k f |Sp,q A(Rn )kϕ

∗

r k f |Sp,q A(Rn )kϕ

and

(1.615)

be the related quasi-norms as introduced in Definition 1.4 and (1.589), (1.590), where 0 < p, q ≤ ∞ (p < ∞ for F-spaces), r ∈ R and A ∈ {B, F}. Let σp and σp,q be as in (1.611). Proposition 1.101. (i) Let 0 < p, q ≤ ∞ and r > σp . Then there is a constant c > 0 such that ∗

r r k f |Sp,q B(Rn )kϕ ≤ c k f |Sp,q B(Rn )kϕ,

r f ∈ Sp,q B(Rn ).

(1.616)

(ii) Let 0 < p < ∞, 0 < q ≤ ∞ and r > σp,q . Then there is a constant c > 0 such that ∗ r r r k f |Sp,q F(Rn )kϕ ≤ c k f |Sp,q F(Rn )kϕ, f ∈ Sp,q F(Rn ). (1.617) Proof. Step 1. We prove (i). According to (1.589) one has to estimate the terms (ϕk b f )∨ with k ∈ Zn \ Nn by the right-hand side of (1.31). This can be reduced to the one-dimensional case. Let ϕ and ϕ j with j = −1, −2, . . . be as in (1.582), (1.583). Then ϕϕ j = ϕ j . Temporarily let F1 be the Fourier transform in R and F1−1 its inverse. Then one has by [T92, (6), p. 98], kF1−1 (ϕ j F1 h) |L p (R)k ≤ c 2−jσp kF1−1 (ϕF1 h) |L p (R)k,

− j ∈ N,

(1.618)

0 < p ≤ ∞. Let ϕk be as in (1.584), 0

ϕk (x) = ϕk1 (x1 ) ϕk (x 0),

x 0 = (x2, . . . , xn ), k 0 ∈ Zn−1, k 1 ≤ −1.

(1.619)

−1 its inverse, n ≥ 2. Then the Let Fn−1 be the Fourier transform in Rn−1 and Fn−1 crucial term in (1.589) with k 1 ≤ −1 can be written as 0

−1 k (ϕk b f )∨ = F1−1 ϕk1 F1 Fn−1 ϕ Fn−1 f = F1−1 ϕk1 F1 g k

0

(1.620)

1 Spaces on Rn

110


0 −1 ϕk 0 F 0 0 n−1 one can apply (1.618) with g k (x1, x 0) = Fn−1 n−1 f (x1, x ). For fixed x ∈ R n−1 to (1.620). Then afterwards taking the L p (R )-quasi-norm, by the Fubini theorem one obtains 0 k(ϕk b f )∨ |L p (Rn )k ≤ c 2−k1 σp k(ϕϕk b f )∨ |L p (Rn )k (1.621) 0

0

with ϕϕk = ϕ(ξ1 )ϕk (ξ 0). Using r > σp it follows that −1  Õ

q  1/q 0 2k1 r q (ϕk b f )∨ |L p (Rn ) ≤ c k(ϕϕk b f )∨ |L p (Rn )k.

(1.622)

k1 =−∞

If k 0 ∈ N0n−1 then the right-hand side fits in (1.31). Otherwise one eliminates the terms with −k2 ∈ N and continues by iteration. This proves (1.616). Step 2. We prove part (ii). We need a vector-valued version of (1.618). Let ϕ and 0 ϕk1 be as above. Then one has again ϕϕk1 = ϕk1 with −k 1 ∈ N. One can replace g k in (1.620) by
∨ 0 e g k (x1, x 0) = ϕϕk b f (x1, x 0), e k = (0, k 0), (1.623) similarly to (1.619) and used in (1.622). Let u < min(1, p, q) and let x 0 ∈ Rn−1 be fixed. Then it follows from the one-dimensional version of [T83, Sect. 1.5.1, (4), p. 25] applied to (1.620), (1.623), ∫ k
∨ u −1 k ϕ b F ϕ 1 (y1 ) u |gek (x1 − y1, x 0)| u dy1 . f (x1, x 0) ≤ c (1.624) 1 R


-norm to (1.624) with respect to x1 ∈ R and We apply the L p/u R, `q/u 0 e 0 n−1 k ∈Z and incorporate the weights 2r[k ] = 2r[k] . Together with F1−1 ϕk1 (y1 ) =
k −1 k e (2 1 y1 ) one obtains 2 1 F1 ϕ

 Õ

q  1/q
∨ 0 0

L p (R)

2r[k ]q ϕk1 ϕk b f (·, x 0)

(Zn−1 )

k 0 ∈Z n−1

 Õ

q  1/q 1 e e
∨ L p (R)

. ≤ c 2k1 (1− u ) 2r[k]q ϕk b f (·, x 0)

(1.625)

k 0 ∈Z n−1

Afterwards one can apply L p (Rn−1 ) with respect to x 0 ∈ Rn−1 . Finally one has to incorporate the weighted `q -sum with respect to −k 1 ∈ N and the weights 2rk1 . The related left-hand side of the Rn -version of (1.625) can be enlarged if one replaces q by qe < min (q, p) with respect to k1 (not to k 0 ∈ Zn−1 ). Then one can apply the n n triangle inequality with respect to L p/e q (R ) and one obtains the R -version of the right-hand side multiplied by the factor −1 Õ k1 =−∞

1

q

1 u

> max (1, p1 , q1 ). Iteration proves (1.617). 

After this preparation one can now complement Theorem 1.92 by asking for m counterparts of (1.612)–(1.614). We use ∂(t),u,0 W(t) f as in (1.569) with m ∂(t),u,0 W(t) f = f

if u = ∅.

(1.627)

It is now reasonable to isolate the corresponding term k f |L p (Rn )k in (1.574), (1.575) Í and to indicate by u,∅ the sum over all subsets u ⊂ {1, . . . , n} with 1 ≤ |u| = card u. Recall u = {1, . . . , n} \ u. Let (t)ud and (t)u be as used in Theorem 1.92 and  Ru++ = t = (t1, . . . , tn ) ∈ Rn : ta > 0 if a ∈ u and ta = 0 if a ∈ u .

(1.628)

Î The integration over Ru++ will be indicated by (dt)u = a ∈u dta . We use S 0(Rn ) ∩ L1loc (Rn ) as described after (1.565) based on (1.570). Let σp and σp,q be as in (1.611). Theorem 1.102. (i) Let 0 < p, q ≤ ∞, r > σp

and r/2 < m ∈ N.

(1.629)

r B(Rn ) is the collection of all f ∈ S 0 (Rn ) ∩ L loc (Rn ) such that Then Sp,q 1 r + k f |Sp,q B(Rn )km

= k f |L p (R )k + n

Õ ∫ u R++

u,∅

(m− r2 )q

∂ m W(t) (t),u,0

(t)u

q (dt)u  1/q (1.630) f |L p (Rn ) (t)u

is finite (equivalent quasi-norms), with the usual modification if q = ∞. (ii) Let 0 < p < ∞, 0 < q ≤ ∞, r > σp,q

and r/2 < m ∈ N.

(1.631)

r F(Rn ) is the collection of all f ∈ S 0 (Rn ) ∩ L loc (Rn ) such that Then Sp,q 1 r + k f |Sp,q F(Rn )km

Õ

 ∫ = k f |L p (R )k +

n

u,∅

u R++

(m− r2 )q m ∂(t),u,0W(t)

(t)u

q (dt)u  1/q

(1.632) f (·) |L p (Rn ) (t)u

is finite (equivalent quasi-norms), with the usual modification if q = ∞.

112

1 Spaces on Rn

Proof. One can replace Q = (0, 1)n in (1.575), (1.576) by Q ∩ Ru++ . The integration over the remaining ta with a ∈ u = {1, . . . , n} \ u is immaterial. But the n . Then (1.630), (1.632) are the situation is different when switching from Q to R++ adequate counterparts of (1.575), (1.576). Otherwise we take for granted that the Fourier-analytical arguments in the proof of Theorem 1.92(iii) complemented now by Proposition 1.101 justify the above theorem. As far as the incorporation of the additional Fourier-analytical and thermic terms are concerned one also may consult the proof of Theorem 1.96 and the references within where we argued in the opposite direction, from thermic to (domestic) Fourier-analytical characterizations.  Remark 1.103. In the isotropic case we have (1.613), (1.614) with s > σp(n) both for the B-spaces and the F-spaces. This means s > σp if n = 1. In the above theorem we needed r > σp for the B-spaces and r > σp,q for the F-spaces, going back to Proposition 1.101. The corresponding Fourier-analytical reduction in the case of the isotropic spaces is covered by [T92, Thm. 2.3.3, p. 98], where the related assertion is proved in a few lines. If n = 1 then also the proof of Proposition 1.101 can be simplified in this way. But if n ≥ 2 then it is not clear whether r > σp,q in Proposition 1.101 and as a consequence in Theorem 1.102 is a natural restriction for the F-spaces. 1.4.9 Tempered homogeneous spaces with positive smoothness. Let σp , σp,q and σp(n) = nσp be as at the beginning of Section 1.4.8. Let n ∈ N, 0 < p < ∞, 0 < q ≤ ∞, n n n σp(n) < s < , − = s − , (1.633) p w p and s/2 < m ∈ N. Then 1 < w < ∞. Recall ∂tm = ∂ m /∂t m . We introduced in ∗

[T15, Def. 3.9, p. 58] the tempered homogeneous isotropic spaces F sp,q (Rn ) as the collection of all f ∈ S 0(Rn ) ∩ L1loc (Rn ) such that

 ∫ ∞

q dt  1/q s

L p (Rn )

+ k f |Lw (Rn )k t (m− 2 )q ∂tmWt f (·) (1.634)

t 0 ∗

is finite and, under the additional restriction 0 < q ≤ w, the spaces B sp,q (Rn ) as the collection of all f ∈ S 0(Rn ) ∩ L1loc (Rn ) such that ∫ ∞

q dt  1/q s t (m− 2 )q ∂tmWt f |L p (Rn ) + k f |Lw (Rn )k (1.635) t 0 is finite. These constructions are based on the sharp embeddings for the inhomogeneous isotropic spaces s (Rn ) ,→ L (Rn ), Fp,q w

0 σp,q for the F-spaces. The first embedding in (1.644) is covered by ∗

r A(Rn ) with q < ∞ follows as in the proof of (1.108). The density of D(Rn )◦ in Sp,q Proposition 1.98 with a reference to [T15, p. 54]. 

Remark 1.107. All assertions of the theorem apply to the F-spaces where 0 < p < ∞, 0 < q ≤ ∞ and r, w as in (1.641) with the exception of the second embedding in (1.644) where the additional restriction r > σp,q comes from Theorem 1.102(ii). We discussed this point previously in Remark 1.103. Again it is not clear whether r > σp,q can be replaced by r > σp . Problem 1.108. According to Theorem 1.29 the inhomogeneous spaces Spr W(Rn ) and Spr B(Rn ) with r, p as there are homogeneous at the small, which means for λ = (λ1, . . . , λn ), 0 < λ j ≤ 1. Such an observation is very useful for spaces with dominating mixed smoothness on domains, as discussed in Chapter 2. We described in (1.168)–(1.170) the more general, rather satisfactory counterpart for related inhomogeneous spaces Asp,q (Rn ). Its final proof (as far as the F-spaces are concerned) goes back to [T15, Corol. 3.55, p. 116] based on the following observation.

1 Spaces on Rn

116 Let 0 < p < ∞, 0 < q ≤ ∞ and

σp(n) = nσp < s < n/p. Then

(1.651)

∗

k f | Asp,q (Rn )k ∼ k f | Asp,q (Rn )k

(1.652)

∗

for all f ∈ Asp,q (Rn ) with supp f ⊂ Q = (0, 1)n ; see [T15, (3.232), p. 85]. Combined with the global homogeneity ∗

n

∗

k f (λ·) | Asp,q (Rn )k ∼ λ s− p k f | Asp,q (Rn )k,

λ>0

(1.653)

(see [T15, Thm. 3.11, p. 60]), one can derive (1.169) for functions satisfying (1.170). ∗

r A(Rn ) according to Definition 1.104 A corresponding assertion for the spaces Sp,q would be of great use for related spaces on domains as treated in Chapter 2. We have so far the global homogeneity (1.645). But the counterpart of (1.652), ∗

r r k f |Sp,q A(Rn )k ∼ k f |Sp,q A(Rn )k,

(1.654)

∗

r A(Rn ) with supp f ⊂ Q is not so clear. One has to compare (as for, say, f ∈ Sp,q in the isotropic case) the quasi-norms in Theorem 1.102 on the one hand, with their homogeneous counterparts in Definition 1.104 and Theorem 1.106 on the other hand (even at the expense of modifying the latter ones). s (Rn ) with (1.633) and their domiProblem 1.109. Both for the isotropic spaces Bp,q r n nating mixed counterparts Sp,q B(R ) with (1.638) we have the respective embeddings (1.637), (1.640) where q is restricted by q ≤ w (there are no restrictions of this type for the F-spaces, (1.636), (1.639)). In the case of the isotropic spaces one can compensate this drawback by replacing the Lebesgue spaces Lw (Rn ) by the more general Lorentz spaces Lw,v (Rn ). This is based on the following well-known embeddings. Again let s, p, w be as in (1.633). Then n s (Rn ) ,→ L Fp,q w,v (R ) if, and only if, p ≤ v ≤ ∞

(1.655)

n s (Rn ) ,→ L Bp,q w,v (R ) if, and only if, q ≤ v ≤ ∞.

(1.656)

and

This means

∫ 0

∞

t 1/w f ∗ (t)


v dt  1/v s (Rn )k ≤ c k f |Fp,q t

(1.657)

117

1.4 Properties, II

s (Rn ) if, and only if, p ≤ v ≤ ∞ (usual modification for some c > 0 and all f ∈ Fp,q if v = ∞). Similarly, ∫ ∞
v dt  1/v s (Rn )k ≤ c k f |Bp,q (1.658) t 1/w f ∗ (t) t 0

if, and only if, q ≤ v ≤ ∞ (usual modification if v = ∞). Here f ∗ (t) is the well-known decreasing (which means non-increasing) rearrangement of f . We dealt in [T15, Sects. 3.6, 3.7, pp. 68–75] with Lorentz spaces and their relations to the tempered ∗

homogeneous spaces Asp,q (Rn ). There one also finds the necessary explanations and ∫ε ∫∞ references. In particular, (1.657), (1.658) with 0 , 0 < ε < 1 in place of 0 may be found in [T01, Sect. 15.3, pp. 232–233], repeated in [T06, Thm. 1.84, p. 51]. Otherwise the embeddings (1.655)–(1.658) may be found in [Har07, (8.6), (8.7), p. 120]. We justified (1.656) in [T15, pp. 72–73] by real interpolation as follows: let σp(n) < s0 < s1
0 t µh (t)1/w where  µh (t) = x ∈ Rn : |h(x)| > t ,

t>0

(1.669)

(see [T15, Thm. 3.15, p. 69], where one also finds some references). Then (1.667) follows from (1.668), (1.669) with h(x) = χ 0(x 0)g(x1 ). On the other hand, one has by [T10, Prop. 6.3, p. 250] that r χ 0 ∈ Sp,q A(Rn−1 ) if 0 < p < ∞, 0 < q ≤ ∞, r < 1/p.

(1.670)

Let 0 < p < ∞, 0 < q ≤ ∞ and r, w be as in (1.641). Then (1.667), (1.670) and Proposition 1.19 reduce the only-if part of (1.661), (1.662) to the only-if parts of (1.655), (1.656) with n = 1. This means again that p ≤ v ≤ ∞ in (1.665) and q ≤ v ≤ ∞ in (1.666). 1.4.10 Tempered homogeneous spaces with general smoothness. We described r A(Rn ) in Theorem 1.92 thermic characterizations for all inhomogeneous spaces Sp,q with 0 < p, q ≤ ∞ (p < ∞ for F-spaces), r ∈ R. The quasi-norms in (1.571)– (1.574) are admissible, which means that any f ∈ S 0(Rn ) can be tested for whether it belongs to the corresponding space or not. In the case of (1.575), (1.576) these tests must be restricted to regular f ∈ S 0(Rn ), which means f ∈ S 0(Rn ) ∩ L1loc (Rn ). As ∗

r A(Rn ) are concerned we far as the related tempered homogeneous counterparts Sp,q have Definition 1.94, Theorem 1.96 and, in particular, Proposition 1.98 if r < 0, and Definition 1.104, Theorem 1.106 if r > 0. Both Proposition 1.98 and Definition 1.104 suggest restricting the considerations to the distinguished strip

1 1 −1 0 the quasi-norms in (1.642), (1.643) require that f ∈ S 0(Rn ) is regular, which means f ∈ S 0(Rn ) ∩ L1loc (Rn ). They are not admissible quasi-norms for which it makes sense to test any f ∈ S 0(Rn ) for whether it belongs to the related space or not. The situation is very much the same as in [T15, ∗

Sect. 3.8] where we dealt with tempered isotropic homogeneous spaces Asp,q (Rn ) in the distinguished strip (1.578) ensuring (1.579). This can be carried over to related homogeneous spaces with dominating mixed smoothness.

1 Spaces on Rn

120

We begin with some preparation and rewrite (1.545) as ∫ √
2 −n/2 W(t) f (x) = (4π) e−|z | /4 f x + tz dz,

(1.672)

Rn

√ √ √ where again x + tz = (x1 + t1 z1, . . . , xn + tn zn ) with 0 < t ∈ Rn . Let 1 < p < ∞. Using W(2t) f = W(t)W(t) f one obtains by (1.672), 1

|W(2t) f (x)| ≤ c (t)− 2p kW(t) f |L p (Rn )k,

x ∈ Rn,

(1.673)

∗ ∗ Î σ B(Rn ) with where again we used (t)a = nj=1 t ja with a ∈ R. Let S σ C (Rn ) = S∞,∞ σ < 0 be normed by (1.580), ∗ σ k f |S σ C (Rn )k = sup (t)− 2 W(t) f (x) .

(1.674)

0 0 then the spaces introduced in Definition 1.104 coincide with their counterparts in the above definition. This follows from (1.641), ∗

Lw (Rn ) ,→ S −1/w C (Rn ) as a consequence of (1.673), as well as (1.676) and its Fcounterpart. In other words we extended Definitions 1.94 and 1.104 to r = 0 and ∗

r B(Rn ) with (1.641) and q > w. But otherwise the above arguments remain to Sp,q unchanged. We formulate the outcome. ∗

r A(Rn ) according to Definition 1.111 are quasiTheorem 1.113. The spaces Sp,q Banach spaces (Banach spaces if p ≥ 1, q ≥ 1), where (1.677) are equivalent ∗

∗

r B(Rn ) and (1.678) are equivalent quasi-norms in S r F(Rn ). quasi-norms in Sp,q p,q They have the Fatou property. Furthermore, ∗

r S(Rn ) ,→ Sp,q A(Rn ) ,→ S 0(Rn )

(1.679)

with the specification (1.598) for the spaces covered by Proposition 1.98 and (1.644) for the spaces covered by Theorem 1.106. All spaces have the homogeneity property ∗

∗

1

r r k f (λ·) |Sp,q A(Rn )k = (λ)r− p k f |Sp,q A(Rn )k,

0 < λ ∈ Rn .

(1.680)

In addition, ∫

m

q dt  1/q r W(t) f |L p (Rn ) , (t)(m− 2 )q ∂(t) n (t) R++  Õ

q  1/q
∨ 2r[k]q ϕk b f L p (Rn )

(1.681) (1.682)

k ∈Z n ∗

r B(Rn ) and are equivalent domestic quasi-norms in all spaces Sp,q

 ∫



m q dt  1/q r L p (Rn )

, (t)(m− 2 )q ∂(t) W(t) f (·) n (t) R++

(1.683)

 Õ


∨ q  1/q

L p (Rn )

2r[k]q ϕk b f (·)

(1.684)

k ∈Z n

∗

r F(Rn ). If p < ∞, q < ∞ then are equivalent domestic quasi-norms in all spaces Sp,q ∗

r A(Rn ). S(Rn ), D(Rn ) and D(Rn )◦ are dense in Sp,q

122

1 Spaces on Rn

Proof. This theorem extends Theorem 1.106 and the relevant parts of Theorem 1.96, Proposition 1.98. This applies also to the formulations. We explained above how the corresponding arguments must be modified. Otherwise one can follow the previous considerations. 

2 Spaces on domains

2.1 Introduction The inhomogeneous isotropic spaces Asp,q (Rn ) with A ∈ {B, F}, s ∈ R and 0 < p, q ≤ ∞ (p < ∞ for the F-spaces) are the natural extension of the corresponding spaces Asp,q (R) on the real line R to higher dimensions 2 ≤ n ∈ N. But there is an alternative way to step from one dimension to higher dimensions. These are the r A(Rn ) with dominating mixed smoothness, 2 ≤ n ∈ N, compared with spaces Sp,q r Ap,q (R), with the remarkable outcome that the restrictions for the parameters r, p, r A(Rn ) are the same as for Ar (R). It q for some distinguished properties of Sp,q p,q was one of the main aims of Chapter 1 to discuss this observation in detail. In r A(Rn ) and their Section 1.3 we suggested looking at assertions for the spaces Sp,q r r n relations to Ap,q (R) (and Ap,q (R )) in terms of the three types described there. Type I refers to properties which are very near to those of Arp,q (R), whereas Type III r A(Rn ), n ≥ 2 on the one hints at topics which are significantly different for Sp,q hand, and for Arp,q (R), Arp,q (Rn ) on the other hand. Maybe the most prominent and very interesting example of a Type III problem is apparently the question about spaces with dominating mixed smoothness on domains Ω in Rn . Definition 1.22 r A(Ω) imitates the well-known natural procedure for isotropic spaces resulting in Sp,q and the indicated modifications. But already the first discussion in Remark 1.24 and even more the summarizing résumé of Section 1.3.9 make clear that such an approach is rather inadequate for spaces with dominating mixed smoothness on arbitrary (smooth) domains. As indicated in the preface, the only exceptions seem to be cubes and rectangles with sides parallel to already fixed coordinate axes and near-by modifications. In the case of Ω = Q = (0, 1)n one has for some distinguished spaces Spr W(Q) and Spr B(Q) of Sobolev–Besov type satisfactory assertions as discussed in Section 1.2.2. But these seem to be exceptions which cannot be extended from Q to, say, bounded smooth domains Ω in Rn , 2 ≤ n ∈ N (and also not to other types of spaces). Of course, one can replace Q = (0, 1)n by rectangles (with sides parallel to the coordinate axes). There are also proposals based on finite unions of such rectangles and some diffeomorphic distortions, related tensor products of wavelets and domain decomposition procedures. We refer the reader to [CDFS13] where one finds adequate discussion and respective references. But we are looking for different devices. Guided by suitable constructions for isotropic spaces Asp,q (Rn ) and based r A(Rn ) as established in Chapter 1 we try to on related properties for the spaces Sp,q

124

2 Spaces on domains

introduce corresponding spaces on domains Ω in Rn intrinsically. The first promising isotropic candidates to be checked are the refined localization spaces s,rloc Fp,q (Ω),

(n) 0 < p < ∞, 0 < q ≤ ∞, s > σp,q ,

(2.1)

(n) where Ω is an arbitrary bounded domain in Rn and σp,q has the same meaning as in (1.57). The definition of these spaces relies on Whitney decompositions of Ω and related resolutions of unity % = { %J, L } described below in Section 2.2.1, resulting in s,rloc k f |Fp,q (Ω)k % =

Õ

s (Rn )k p k %J, L f |Fp,q

 1/p

;

(2.2)

J, L

see [T08, Def. 2.14, p. 36]. The justification of this approach relies on two ingredients. First, a global localization property as described in (1.212) for the F-spaces (and a related negative assertion for the B-spaces according to (1.213)). This explains the preference of the F-spaces, compared with their B-counterparts. Second, %J, L refers to cubes of side length ∼ 2−J and %J, L f should be reduced by
gJ, L (x) = (%J, L f ) 2−J (x + L) , supp gJ, L ⊂ (0, 2)n, (2.3) to a standard situation. The desirable retransformation to %J, L f requires a local homogeneity property as described in (1.168)–(1.170). Furthermore, one needs representations in terms of atoms according to (1.61)–(1.62) without moment conditions. (n) This explains the crucial restriction s > σp,q in (2.1). According to [T08, Thm. 3.28, p. 97] one has for so-called E-thick domains (covering in particular bounded Lipschitz domains; see [T08, Prop. 3.8, p. 75]) s,rloc s (Ω), ep,q Fp,q (Ω) = F

(n) 0 < p < ∞, 0 < q ≤ ∞, s > σp,q

(2.4)

with  s (Ω) = f ∈ F s (Rn ) : supp f ⊂ Ω ep,q F p,q

(2.5)

as previously used in Section 1.2.7. One may ask whether these constructions can be transferred to suitable spaces with dominating mixed smoothness. The satisfactory r F(Rn ) according to Theorem 1.40 global localization property for the spaces Sp,q paves the way for dealing with corresponding localization spaces at least in terms of (2.3). This will be done in Section 2.2. The desirable local homogeneity reducing gJ, L in (2.3) to %J, L f in (2.2) is not (yet) available in this generality, but according to Theorem 1.29 for the distinguished spaces Spr W(Rn ) and Spr B(Rn ). For these cases we develop in Section 2.3 a (weighted) counterpart of (2.2). Nothing like s (Ω) by S r F(Ω) as introduced in ep,q ep,q (2.4) can be expected if there one replaces F Definition 1.22. In Section 2.4 we ask for spaces with dominating mixed smoothness

125

2.2 Localization spaces

in bounded C ∞ domains having boundary values and related trace theorems. Here we take Section 1.3.3 and in particular (1.347)–(1.349) as a guide. Finally we illustrate in Section 2.5 the theory of these spaces by some specific properties and examples, including Haar frames and Faber frames, as well as some first assertions about numerical integration and discrepancy.

2.2 Localization spaces 2.2.1 Definitions and basic properties. Let Ω be a bounded domain (= open set) in Rn , n ∈ N, and let d(x) = inf{|x − y| : y ∈ Γ},

x ∈ Ω, Γ = ∂Ω.

(2.6)

Similarly to [T08, Sect. 2.1.2] we rely on the well-known Whitney decomposition as may be found in [Ste70, Thm. 3, p. 16, Thm. 1, p. 167] or [KiK13, Sect. 3.1.4, pp. 71–77, Sect. 8.1], adapted to our needs. Let Q0J, L = 2−J L+2−J (0, 1)n ⊂ Q1J, L = 2−J L+2−J (0, 2)n ⊂ Ω,

J ∈ N, L ∈ Zn (2.7)

be open cubes with sides parallel to the coordinate axes, 2−J L as the lower-left corner both for Q0J, L and Q1J, L and the respective side lengths 2−J and 2−J+1 such that the cubes Q0J, L are pairwise disjoint, Ø Ω= Q0J, L and inf d(y) ∼ 2−J . (2.8) J, L

y ∈Q1J, L

For adjacent cubes Q0J, L and Q0J 0, L0 one may assume |J − J 0 | ≤ 1. Let ZΩ be the collection of all lattice points 2−J L contributing to (2.8) (lower-left corners of Q1J, L ). Let  % = %J, L : 2−J L ∈ ZΩ , %J, L ∈ D(Ω), supp %J, L ⊂ Q1J, L (2.9) be an adapted resolution of unity, Õ %J, L (x) = 1, 0 ≤ %J, L (x) ≤ 1, |Dγ %J, L (x)| ≤ cγ 2J |γ |, x ∈ Ω, (2.10) 2−J L ∈ZΩ

γ ∈ N0n . One may again consult [Ste70, KiK13]. Here D(Ω) = C0∞ (Ω) and also D 0(Ω) have the same meaning as at the beginning of Section 1.2.2. If f ∈ D 0(Ω) then %J, L f ∈ S 0(Rn ) (usual abuse of notation, extending %J, L f outside Ω by zero) and
gJ, L (x) = (%J, L f ) 2−J (x + L) ∈ S 0(Rn ), supp gJ, L ⊂ Q1 = (0, 2)n (2.11) r F(Rn ) be the in the standard interpretation (extension by zero outside Q1 ). Let Sp,q spaces with dominating mixed smoothness according to Definition 1.4.

126

2 Spaces on domains

r F[Ω,