Linear and Quasilinear Parabolic Problems: Volume II: Function Spaces (Monographs in Mathematics)

476 Pages·2019·2.177 MB·English

by  Herbert Amann

Monographs in Mathematics
106
Herbert Amann
Linear and
Quasilinear
Parabolic
Problems
Volume II: Function Spaces

Monographs in Mathematics
Vol. 106
Series Editors:
Herbert Amann
Universität Zürich, Zürich, Switzerland
Jean-Pierre Bourguignon
IHES, Bures-sur-Yvette, France
William Y. C. Chen
Nankai University, Tianjin, China
Associate Editors:
Huzihiro Araki, Kyoto University, Kyoto, Japan
John Ball, Heriot-Watt University, Edinburgh, UK
Franco Brezzi, Università degli Studi di Pavia, Pavia, Italy
Kung Ching Chang, Peking University, Beijing, China
Nigel Hitchin, University of Oxford, Oxford, UK
Helmut Hofer, Courant Institute of Mathematical Sciences, New York, USA
Horst Knörrer, ETH Zürich, Zürich, Switzerland
Don Zagier, Max-Planck-Institute, Bonn, Germany
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Herbert Amann
Linear and Quasilinear
Parabolic Problems
Volume II: Function Spaces
Herbert Amann
Institut für Mathematik
UniversitätZürich
Zürich,Switzerland
ISSN 1017-0480 ISSN 2296-4886 (electronic)
Monographsin Mathematics
ISBN978-3-030-11762-7 ISBN978-3-030-11763-4 (eBook)
https://doi.org/10.1007/978-3-030-11763-4
LibraryofCongressControlNumber:2019932606
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Nil humani quin corrigi possit1
L. Naumann
Systematik der Kochkunst
Preface Dresden 1887
In this volume we present a systematic and detailed exposition of the theory of
function spaces in an Euclidean setting. Particular emphasis is put on Besov and
Besselpotentialspaceswhichformtheframeforthestudyofparabolicdifierential
equations in the next volume.
The presentation includes several new features which lets it stand out from
other accounts. First, it consistently develops anisotropic spaces. Second, it ex-
pounds the whole theory for functions and distributions taking their values in
Banach spaces on which we impose only the necessary restrictions. Thus none in
the case of Besov spaces, except for re(cid:176)exivity assumptions in duality theorems.
Third, the theory is set forth for spaces whose elements are deflned on rectangu-
lar corners of Euclidean spaces. By this we pave the way for the investigation of
function spaces on Riemannian manifolds, possibly possessing corners and other
singularities. This is also put on hold for the third volume.
Our approach builds basically on two cornerstones: on Fourier analysis and
multipliertheorems,andonextension-restrictiontechniques.Bythiswecangivea
unifled presentation incorporating, in particular, Sobolev{Slobodeckii and H˜older
space scales. The rather detailed study of these spaces, which are of great impor-
tancefortheinvestigationofdifierentialequations,isafurthercharacteristictrait
of our treatise.
This volume consists of three chapters and an appendix. The flrst chapter,
whichisofrathertechnicalnature,collectspreparatorymaterial.Itsuppliesaflrm
basis for the main text which covers Chapters VII and VIII. The flrst one thereof
contains a systematic treatment of anisotropic vector-valued function spaces on
corners. In the second one we give a detailed and unifled account of trace and
boundary operators.
Forthereader’sconvenience,intheappendixweincludeadownscaledversion
of L. Schwartz’ theory of vector-valued distributions by admitting only Banach
spaces as targets. Particular weight is given on tensor products and convolutions
since, in the main text, we make use of such results. It should be mentioned that,
already in 2003, I had put a preliminary, slightly more comprehensive version of
this appendix on my homepage.
1Thereisnothingonearththatcouldnotbeimproved.
v
vi Preface
In essence, this volume forms a profound expansion and amelioration of my
earlier lecture notes ‘Anisotropic Function Spaces and Maximal Regularity for
Parabolic Problems. Part 1: Function Spaces’ [Ama09]. Besides of adding much
more material, I have corrected numerous (cid:176)aws and imprecisions which observing
readers have brought to my attention.
Once more, I could rely on the help of Pavol Quittner, Gieri Simonett, and
Christoph Walker. They read critically and carefully large parts of the flrst draft,
pointed out plenty of mistakes and misprints, and suggested very advantageous
changes and improvements. Sincere thanks are given to all of them for their gen-
erous support.
Last but not least, I could again experience the immensely valuable support
of my wife Gisela who transformed countless barely readable preliminary versions
and revisions into TEX flles and provided this perfect layout on hand. I am more
than deeply grateful to her.
Zu˜rich, January 2019 Herbert Amann
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter VI Auxiliary Material
1 Restriction-Extension Pairs
1.1 Smooth Functions on Corners . . . . . . . . . . . . . . . . . . . . . . 4
Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Restriction-Extension Operators . . . . . . . . . . . . . . . . . . . 6
Approximation by Test Functions . . . . . . . . . . . . . . . . . . 9
1.2 Tempered Distributions on Corners . . . . . . . . . . . . . . . . . . 10
Duality Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Sequence Spaces
2.1 Duality of Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . 19
Deflnitions and Embeddings . . . . . . . . . . . . . . . . . . . . . 19
Duality Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Weighted Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . 25
Image Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Embeddings and Duality . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Unweighted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Weighted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Anisotropy
3.1 Anisotropic Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Weight Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
vii
viii Contents
3.2 Quasinorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Parametric Augmentations . . . . . . . . . . . . . . . . . . . . . . . 42
Augmented Quasinorms . . . . . . . . . . . . . . . . . . . . . . . . 43
Positive Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . 45
Difierentiating Inverses . . . . . . . . . . . . . . . . . . . . . . . . 47
Slowly Increasing Functions . . . . . . . . . . . . . . . . . . . . . 49
3.4 Fourier Multipliers and Multiplier Spaces . . . . . . . . . . . . . . . 52
Elementary Fourier Multiplier Theorems . . . . . . . . . . . . . . 52
Fourier Multiplier Spaces . . . . . . . . . . . . . . . . . . . . . . . 54
Resolvent Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5 Multiplier Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Resolvent Estimates for Homogeneous Symbols. . . . . . . . . . . 58
Functions of Homogeneous Symbols . . . . . . . . . . . . . . . . . 59
Dunford Integral Representations . . . . . . . . . . . . . . . . . . 61
Powers and Exponentials . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 Dyadic Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . 67
Preliminary Fourier Multiplier Theorems . . . . . . . . . . . . . . 68
3.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Chapter VII Function Spaces
1 Classical Spaces
1.1 Bounded Continuous Functions . . . . . . . . . . . . . . . . . . . . . 78
Banach Spaces of Bounded Continuous Functions . . . . . . . . . 78
Vector Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
1.2 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Regular Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 84
Basic Deflnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
1.3 Restrictions and Extensions . . . . . . . . . . . . . . . . . . . . . . . 86
1.4 Distributional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 90
1.5 Re(cid:176)exivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
1.6 Embeddings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
1.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2 Besov Spaces
2.1 The Deflnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Preliminary Estimates. . . . . . . . . . . . . . . . . . . . . . . . . 100
A Retraction-Coretraction Pair . . . . . . . . . . . . . . . . . . . 102
The Final Deflnition. . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.2 Embedding Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Little Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Embeddings With Varying Target Spaces . . . . . . . . . . . . . . 109
2.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.4 Fourier Multiplier Theorems . . . . . . . . . . . . . . . . . . . . . . 112
Contents ix
2.5 Operators of Positive Type . . . . . . . . . . . . . . . . . . . . . . . 115
Resolvent Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A Representation Theorem . . . . . . . . . . . . . . . . . . . . . . 117
Bounded Imaginary Powers . . . . . . . . . . . . . . . . . . . . . . 118
Interpolation-Extrapolation Scales . . . . . . . . . . . . . . . . . . 119
2.6 Renorming by Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 120
Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Sandwich Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Sobolev Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.7 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Real and Complex Interpolation . . . . . . . . . . . . . . . . . . . 125
Interpolation with Difierent Target Spaces . . . . . . . . . . . . . 126
Embeddings of Intersection Spaces . . . . . . . . . . . . . . . . . . 127
Interpolation of Classical Spaces . . . . . . . . . . . . . . . . . . . 128
2.8 Besov Spaces on Corners . . . . . . . . . . . . . . . . . . . . . . . . 128
2.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3 Intrinsic Norms, Slobodeckii and H˜older Spaces
3.1 Commuting Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.2 Semigroups and Interpolation . . . . . . . . . . . . . . . . . . . . . . 137
Preliminary Estimates. . . . . . . . . . . . . . . . . . . . . . . . . 138
Renorming Intersections of Interpolation Spaces . . . . . . . . . . 141
3.3 Translation Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.4 Renorming Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . 149
3.5 Intersection-Space Characterizations . . . . . . . . . . . . . . . . . . 151
Intersection Space Representations. . . . . . . . . . . . . . . . . . 152
Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Nikol0ski‚‡ Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
3.6 Besov{Slobodeckii and Besov{H˜older Spaces . . . . . . . . . . . . . 155
Mixed Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Slobodeckii, H˜older, and Little H˜older Spaces . . . . . . . . . . . 157
3.7 Little H˜older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Very Little H˜older Spaces . . . . . . . . . . . . . . . . . . . . . . . 164
3.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4 Bessel Potential Spaces
4.1 Basic Facts, Embeddings, and Real Interpolation . . . . . . . . . . . 168
4.2 A Marcinkiewicz Multiplier Theorem. . . . . . . . . . . . . . . . . . 171
4.3 Renorming by Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 173
4.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.5 Complex Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 176
A Holomorphic Semigroup . . . . . . . . . . . . . . . . . . . . . . 176
Interpolation with Difierent Target Spaces . . . . . . . . . . . . . 178