Table Of ContentMonographs in Mathematics
107
Hans Triebel
Theory
of Function
Spaces IV
Monographs in Mathematics
Volume 107
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Hans Triebel
Theory of Function
Spaces IV
Hans Triebel
Mathematisches Institut
Friedrich-Schiller-Universität
Jena, Germany
ISSN 1017-0480 ISSN 2296-4886 (electronic)
Monographsin Mathematics
ISBN978-3-030-35890-7 ISBN978-3-030-35891-4 (eBook)
https://doi.org/10.1007/978-3-030-35891-4
MathematicsSubjectClassification(2010): 46–02,46E35,42C40,42B35
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Contents
Preface ix
1 Fundamental principles 1
1.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Global spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Hybrid spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Wavelet characterizations . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Global spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Hybrid spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Basic assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Relations between global and hybrid spaces . . . . . . . . . 12
1.3.2 Lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.3 Distinguished representations . . . . . . . . . . . . . . . . . 17
1.3.4 Fatou property . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 The essentials, key theorems 21
2.1 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Embeddings in L∞(Rn) and C(Rn). . . . . . . . . . . . . . 21
2.1.2 Embeddings in the space of locally integrable functions . . 23
2.1.3 Sharp embeddings: Constant differential dimensions . . . . 25
2.1.4 Sharp embeddings: Constant smoothness . . . . . . . . . . 26
2.1.5 Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.6 Gagliardo-Nirenberg inequalities . . . . . . . . . . . . . . . 28
2.2 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 The trace theorem . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Dichotomy: Traces versus density . . . . . . . . . . . . . . . 35
2.3 Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Multipliers, localizations and multiplication algebras . . . . . . . . 40
2.4.1 Smooth multipliers . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.2 Localizations . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.3 Multiplication algebras. . . . . . . . . . . . . . . . . . . . . 44
v
vi Contents
2.4.4 Characteristic functions as multipliers . . . . . . . . . . . . 46
2.4.5 Rough multipliers . . . . . . . . . . . . . . . . . . . . . . . 53
2.5 Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5.1 Introduction and distinguished lifts . . . . . . . . . . . . . . 57
2.5.2 Main assertions . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.6 Spaces on domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.6.1 The essentials: Why and how . . . . . . . . . . . . . . . . . 62
2.6.2 Preliminaries and definitions . . . . . . . . . . . . . . . . . 63
2.6.3 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.6.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.6.5 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.6.6 Characteristics of distributions . . . . . . . . . . . . . . . . 72
2.7 Multipliers: Further properties . . . . . . . . . . . . . . . . . . . . 74
2.7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.7.2 Main assertions . . . . . . . . . . . . . . . . . . . . . . . . . 77
3 Further topics 83
3.1 Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3 Local homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4 Refined localization spaces . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.1 Multipliers, revisited . . . . . . . . . . . . . . . . . . . . . . 93
3.4.2 Main assertions . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5 Haar wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.6 Fubini property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.7 Characterizations in terms of Lusin functions and heat kernels . . 105
3.7.1 Lusin functions . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.7.2 Heat kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.7.3 Caloric smoothing . . . . . . . . . . . . . . . . . . . . . . . 109
4 Complements 115
4.1 Tempered homogeneous spaces . . . . . . . . . . . . . . . . . . . . 115
4.1.1 Introduction and motivation . . . . . . . . . . . . . . . . . 115
4.1.2 Spaces with negative smoothness . . . . . . . . . . . . . . . 118
4.1.3 Spaces in the distinguished strip . . . . . . . . . . . . . . . 120
4.1.4 Some properties . . . . . . . . . . . . . . . . . . . . . . . . 125
4.1.5 Caloric smoothing . . . . . . . . . . . . . . . . . . . . . . . 126
4.2 Natural habitats and the homogeneity rule . . . . . . . . . . . . . 128
4.2.1 Natural habitats . . . . . . . . . . . . . . . . . . . . . . . . 128
4.2.2 The homogeneity rule . . . . . . . . . . . . . . . . . . . . . 129
4.3 Spaces and tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.3.1 Classical Sobolev spaces . . . . . . . . . . . . . . . . . . . . 132
4.3.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.3.3 Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Contents vii
4.3.4 Fourier-analytical decompositions and paramultiplication . 135
4.3.5 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.3.6 Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.3.7 Wavelets and paramultiplication . . . . . . . . . . . . . . . 142
4.3.8 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Bibliography 147
Symbols 157
Index 159
Preface
Themainmotivationtowritethisbookoriginatesfromtheattempttoincorporate
the nowadays fashionable spaces Fs (Rn), s ∈ R, 0 < q ≤ ∞, into the existing
∞,q
elaborated theory of the spaces As (Rn), A ∈ {B,F}, s ∈ R, 0 < p ≤ ∞,
p,q
0 < q ≤ ∞ with p < ∞ for the F-spaces. But we decided to extend this task by
offering a guided tour through the jungle of these spaces of Sobolev-Besov type.
Therefore,thisbookcanalsoserveasanadvancedtextbookintroducingthistopic
ofSobolev-Besov-typespacesinamodernandconciseway.Weconcentrateonthe
essentials, complemented by a kaleidoscope of highlights exploring the grounds
of the fascinating mathematical landscape called function spaces. The required
hiking equipment is: Fourier analysis, measure theory, functional analysis, and
enthusiasm.Newproofsrelymostlyonwaveletrepresentations.Inallothercases,
we take over already existing assertions (quite often from our own books), which
are carefully commented and adapted. Detailed references are also provided.
Thetableofcontentsshowswhichtopicsaretreatedinthisbook,butletus
addafewcommentsatthispoint.Nowadays,aplethoraoffunctionspacesofmany
types prevails. In this book, however, the aim is not to provide an encyclopaedic
survey. Much rather, we deal almost exclusively with the above-mentioned inho-
mogeneous spaces As (Rn) and their counterparts As (Ω) on domains Ω in Rn.
p,q p,q
There are two exceptions. Firstly, we will occasionally rely on some assertions for
the so-called hybrid spaces LrAs (Rn), which have the remarkable property
p,q
L0Fs (Rn)=Fs (Rn), 0<q≤∞, s∈R, (0.1)
p,q ∞,q
for all 0 < p < ∞. Secondly, we will deal in the Sections 4.1 and 4.2 with the
∗
tempered homogeneous spaces As (Rn) within the dual pairing (cid:2)S(Rn),S(cid:4)(Rn)(cid:3)
p,q
mostly restricted to their natural habitat
0<p≤∞, n(cid:2)1 −1(cid:3)≤s≤ n. (0.2)
p p
This will be based on heat kernels as discussed in Section 3.7 for the spaces
As (Rn).Apartfromthat,wewillimmediatelyjumpfromtheFourier-analytical
p,q
definition of the spaces As (Rn) in Section 1.1 to their wavelet characterization
p,q
inSection1.2withoutdiscussinganyothermeansorspecialcases.Thisapproach
ix
x Preface
will be compensated to some extent in the final Section 4.3, where we discuss
Sobolev and Besov spaces in their more traditional versions based on derivatives
anddifferencesoffunctions,includingtherelatedreferences.Furthermore,wewill
glance at other means such as atoms and quarks. The fact that this discussion
is located at the very end of the book gives us the possibility to make clear how
all these special spaces, means and building blocks (including wavelets and heat
kernels) are interwoven and how they are connected with the preceding text.
Note that this book may be considered both as a supplement to the mono-
graphs [T83, T92, T06] with the same title and as an advanced companion of the
textbook [HT08].
Throughout the book, we shall use the symbol ∼ (equivalence) as follows:
Let I be an arbitrary index set. Then
a ∼b for i∈I (equivalence) (0.3)
i i
for two sets of positive numbers {a :i∈I} and {b :i∈I} means that there are
i i
two positive numbers c and c such that
1 2
c a ≤b ≤c a for all i∈I. (0.4)
1 i i 2 i
Lastbutnotleast,IwouldliketothankDorotheeD.Haroskeforproducing
the figures.
Hans Triebel, September 2019
