The Structure of Functions

436 Pages·2001·3.699 MB·English

by  Hans Triebel (auth.)

Modern Birkha¨user Classics
Many of the original research and survey monographs in pure and
applied mathematics published by Birkhäuser in recent decades have
been groundbreaking and have come to be regarded as foundational to
the subject. Through the MBC Series, a select number of these modern
classics, entirely uncorrected, are being re-released in paperback (and as
eBooks) to ensure that these treasures remain accessible to new
generations of students, scholars, and researchers.
Hans Triebel
The Structure of Functions
Reprint of the 2001 Edition
Hans Triebel
Mathematical Institute
Friedrich-Schiller-University Jena
(cid:74)(cid:101)(cid:110)(cid:97)
(cid:71)(cid:101)(cid:114)(cid:109)(cid:97)(cid:110)(cid:121)
ISBN 978-3-0348-0568-1 ISBN 978-3-0348-0569-8 (eBook)
DOI10.1007/978-3-0348-0569-8
Springer Basel Heidelberg New York Dordrecht London
Library of Congress Control Number: 2012953657
Mathematics Subject Classification (2010): 46-02, 46E35, 28A80, 42C40, 35Pxx, 47B06, 42Bxx, 35J60,
41A17

© Springer Basel 2001
Reprint of the 1st edition 2001 by Birkhäuser Verlag, Switzerland
Originally published as Volume 97 in the Monographs in Mathematics series
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with
reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed
on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts
thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current
version, and permission for use must always be obtained from Springer. Permissions for use may be obtained
through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the
respective Copyright Law.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
While the advice and information in this book are believed to be true and accurate at the date of publication,
neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or
omissions that may be made. The publisher makes no warranty, express or implied, with respect to the
material contained herein.
Cover design: deblik, Berlin
Printedonacid-freepaper
Springer Basel is part of Springer Science+Business Media (www.springer.com)
Contents
Preface ............................................................... xi
I Decompositions of Functions
1 Introduction, heuristics, and preliminaries ....................... 1
Classical function spaces; quarkonial representations;
Weierstrassian approach
2 Spaces on Rn: the regular case ................................... 10
Sequence spaces b and f ; β-quarks; Bs (Rn) with s>σ and
pq pq pq p
Fs (Rn) with s>σ ; unconditional convergence and optimal
pq pq
coefficients; Weierstrassian approach and Cauchy formula;
generalized quarks
3 Spaces on Rn: the general case ................................... 27
β-quarks; Bs (Rn) and Fs (Rn) with s∈R; optimal coefficients
pq pq
and Weierstrassian approach
4 An application: the Fubini property .............................. 34
Definition; theorem
5 Spaces on domains: localization and Hardy inequalities ........... 41
(cid:2) ◦
Definition of As (Ω), As (Ω), As (Ω) with A=B or A=F;
pq pq pq
relations between these spaces; density assertions; Hardy
inequalities for Fs -spaces; Hardy inequalities for Bs -spaces;
pq pq
refined localization for Fs (Ω) spaces with s>σ ; equivalent
pq pq
quasi-norms for Fs -spaces; pointwise multipliers; approximate
pq
resolutions of unity; refined localization for other cases
6 Spaces on domains: decompositions .............................. 71
Approximate resolutions of unity; β-quarks; sequence spaces;
quarkonial decompositions for Fs -spaces on domains;
pq
unconditional convergence and optimal coefficients; alternative
approach; references
v
vi Contents
7 Spaces on manifolds ............................................. 81
Riemannian manifolds (M,g) with bounded geometry and positive
injectivity radius; d-domains; resolution of unity; the spaces
Fs (M,gκ); refined localizations; Laplace-Beltrami operator and
pq
related lifts; embeddings; families of approximate resolutions of
unity; β-quarks; quarkonial decompositions of Fs (M,gκ); optimal
pq
coefficients
8 Taylor expansions of distributions ................................ 101
Weighted spaces and tempered distributions in Rn; related
β-quarks; quarkonial decompositions of weighted Bs-spaces on Rn;
p
quarkonial decompositions and Taylor expansions of tempered
distributions on Rn; tempered distributions in domains; related
β-quarks; quarkonial decompositions and Taylor expansions of
tempered distributions on domains
9 Traces on sets, related function spaces and
their decompositions ............................................. 120
Finite compactly supported Radon measures in Rn as
distributions; trace operator and identification operator; local
means; criteria for the existence of traces of Fs (Rn) on compact
pq
sets; related potentials and maximal functions; trace spaces;
related comments and references; ball condition; criterion for sets
satisfying the ball condition; traces on sets satisfying the ball
condition; approximate lattices and subordinated resolutions of
unity on compact sets; β-quarks on sets; Bs -spaces on sets and
pq
related quarkonial decompositions; trace spaces on sets; comments
and references
II Sharp Inequalities
10 Introduction: Outline of methods and results .................... 161
Preliminary versions of: rearrangements; three cases: sub-critical,
critical, super-critical; growth and continuity envelopes and
envelope functions; related Borel measures on the real line; related
Hardy inequalities
11 Classical inequalities ............................................. 167
Sharp embeddings of Bs (Rn) and Fs (Rn) in: Lloc(Rn), L (Rn),
pq pq 1 r
C(Rn), C1(Rn); Lorentz-Zygmund spaces; sharp refined classical
embeddings in limiting situations of Bs (Rn) and Hs(Rn) in
pq p
Lorentz spaces and Zygmund spaces and related inequalities;
historical comments and references
Contents vii
12 Envelopes ....................................................... 181
Rearrangement and growth functions; related Borel measures on
the real line; examples of admitted Borel measures; growth
envelope functions; equivalence classes of growth envelope
functions; growth envelopes; related inequalities; moduli of
continuity; continuity envelope functions; equivalence classes of
continuity envelope functions; continuity envelopes; inequalities
between moduli of continuity and rearrangements
13 The critical case ................................................. 202
n n
Calculation of the growth envelopes E Bp and E F p; related
G pq G pq
inequalities; extremal functions; historical comments and
references; spaces on domains and related growth envelopes; bmo
and its growth envelope
14 The super-critical case ........................................... 218
Calculation of the continuity envelopes E B1+np and E F1+np;
C pq C pq
extremal functions; related inequalities; borderline cases; envelope
functions and non-compactness;historical comments and references
15 The sub-critical case ............................................. 229
Calculation of the growth envelopes E Bs and E Fs in the
G pq G pq
sub-critical case; related inequalities; optimal embeddings in
Lorentz spaces and Zygmund spaces; references
16 Hardy inequalities ............................................... 235
Hardy inequalities as consequences of sharp embeddings: critical
case, sub-critical case; references; Hardy inequalities related to
half-space, domains, d-sets, Borel measures
17 Complements .................................................... 243
Green’s functions as envelope functions; further limiting
embeddings; moduli of continuity; generalized B-spaces and
Lip-spaces; logarithmic spaces; compact embeddings; references
III Fractal Elliptic Operators
18 Introduction ..................................................... 251
Preliminary versions of: d-sets and (d,Ψ)-sets, types of fractal
elliptic operators, related spectral problems
19 Spectral theory for the fractal Laplacian ......................... 253
Physical and mathematical background; the Dirichlet Laplacian
and its mapping properties; the fractal elliptic operators
B =(−Δ)−1◦trΓ and (−Δ)−1◦μ; density of D(Rn\Γ) in some
viii Contents
function spaces: spectral synthesis; related references; spectral
theory of the operator B related to d-sets: eigenvalues and
eigenfunctions; the case n=1; Weyl measures; strongly diffuse
measures; entropy numbers and approximation numbers; fractal
drums in the plane; the music of the ferns: Weylian or alien;
degenerate case
20 The fractal Dirichlet problem .................................... 294
Smooth case, single layer potentials; duality of H1-spaces in
arbitrary domains; Hs-spaces on d-sets; the operator B and single
layer potentials on d-sets; mapping properties; the fractal Dirichlet
problem; references; Dirichlet problem in domains with fractal
boundary; the smooth case; L -L -theory; classical solutions,
2 p
Wiener criterion
21 Spectral theory on manifolds .................................... 310
Continuation of Sect. 7: Riemannian manifolds M, function spaces
Fs (M,gκ), compact embeddings, related entropy numbers; the
pq
operator H =−Δ +(cid:4)id−βg−κ; calculation of the negative
β g
spectrum of H ; local singularities and hydrogen-like operators,
β
related entropy numbers and negative spectra; the opinion of the
physicists in such hyperbolic worlds
22 Isotropic fractals and related function spaces .................... 329
Analysis versus (fractal) geometry; (d,Ψ)-sets and properties of
admissible functions Ψ; related measures are Weyl measures;
pseudo self-similar sets and ψIFS; function spaces of type
B(s,Ψ)(Rn) and F(s,Ψ)(Rn); related atoms, β-quarks, and
pq pq
quarkonial decompositions; tailored spaces for (d,Ψ)-sets; spaces
on (d,Ψ)-sets and related entropy numbers
23 Isotropic fractal drums .......................................... 348
Extension of the spectral theory in Sect. 19 from d-sets to
(d,Ψ)-sets; Weyl measures
IV Truncations and Semi-linear Equations
24 Introduction ..................................................... 355
Preliminary versions: truncation property, truncation couple,
Q-operator, semi-linear equations
25 Truncations ...................................................... 357
Real spaces Bs (Rn), Fs (Rn), Hs(Rn); truncation operators T,
pq pq p
T+; examples; Fatou property; truncation property, uniform
Contents ix
continuity; truncation couples; calculation of all truncation
couples; truncation property for Fs (Rn); non-linear interpolation;
pq
references; truncation property for Hs(Rn) and bmo(Rn);
p
Lipschitz-continuity; H¨older-continuity
26 The Q-operator .................................................. 385
Definition of the Q-operator; properties: boundedness and
Lipschitz-continuity
27 Semi-linear equations; the Q-method ............................ 389
Semi-linear integral equations and related regularity theory: the
Q-method; semi-linear differential equations and related regularity
theory; local singularities and related regularity theory; the
Q-method revisited
References ............................................................ 403
Symbols .............................................................. 419
Index ................................................................. 423