Hölder and locally Hölder Continuous Functions, and Open Sets of Class C^k, C^{k,lambda}

160 Pages·2016·2.461 MB·English

by  Renato Fiorenza (auth.)

Frontiers in Mathematics
Renato Fiorenza
Hölder
and
locally Hölder
,
Continuous Functions
and
Open Sets

k k,l
of Class C , C

Frontiers in Mathematics
Advisory Editorial Board
Leonid Bunimovich (Georgia Institute of Technology, Atlanta)
William Y.C. Chen (Nankai University, Tianjin, China)
Benoît Perthame (Université Pierre et Marie Curie, Paris)
Laurent Saloff-Coste (Cornell University, Ithaca)
Igor Shparlinski (Macquarie University, New South Wales)
Wolfgang Sprößig (TU Bergakademie Freiberg)
Cédric Villani (Institut Henri Poincaré, Paris)
More information about this series at http://www.springer.com/series/5388
Renato Fiorenza
Hölder and locally Hölder
Continuous Functions, and
k k,λ
Open Sets of Class C , C
Renato Fiorenza
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
Università Federico II di Napoli
Napoli, Italy
ISSN 1660-8046 ISSN 1660-8054 (electronic)
Frontiers in Mathematics
ISBN 978-3-319-47939-2 ISBN 978-3-319-47940-8 (eBook)
DOI 10.1007/978-3-319-47940-8

Library of Congress Control Number: 2016963063

Mathematics Subject Classification (2010): 26-01, 26A15, 26A16, 26B35, 35A09, 35J25, 46E35

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Contents
Introduction ix
1 Ho(cid:127)lder and locally Ho(cid:127)lder continuousfunctions.
The linear spacesCk((cid:10)), Ck;(cid:21)((cid:10)), and Ck;(cid:21)((cid:10)) 1
loc
1.1 The H(cid:127)older condition . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Sum, product, quotient, and composition of H(cid:127)older functions . . . 13
1.3 Inverse of a function with H(cid:127)olderian derivatives . . . . . . . . . . . 16
1.4 Locally H(cid:127)older continuous functions . . . . . . . . . . . . . . . . . 26
1.5 The linear spaces Ck((cid:10)), Ck;(cid:21)((cid:10)), and Ck;(cid:21)((cid:10)) . . . . . . . . . . . 30
loc
2 Coordinate changes in Rn. Rotations. Cones in Rn 37
2.1 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 Constructions of two coordinate systems . . . . . . . . . . . . . . . 47
2.4 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Open sets with boundary of class Ck and of class Ck;(cid:21).
The cone property 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2 Open sets with boundary . . . . . . . . . . . . . . . . . . . . . . . 80
3.3 The cone property . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Open sets of class Ck and of class Ck;(cid:21) 97
4.1 Open sets of class Ck;(cid:21) . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2 Admissible pairs of real numbers for an open set (cid:10) . . . . . . . . . 108
4.3 On open sets in R . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4 Functions in open sets with admissible pairs . . . . . . . . . . . . . 119
5 Majorization formulas for functions in Cm;(cid:21)((cid:10)), Cm;(cid:21)((cid:10)),
loc
and Cm((cid:10)) 125
5.1 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 Majorization formulas . . . . . . . . . . . . . . . . . . . . . . . . . 130
v
vi Contents
Bibliography 145
Index of Symbols 149
Subject Index 151
In Memory of My Beloved Enza
Introduction
Theaimofthisbookis,amongotherthings,topresentadetailedtreatmentofthe
classic(cid:21)-H(cid:127)oldercondition,andtointroducethenotionoflocallyH(cid:127)oldercontinuous
functioninanopenset(cid:10)inRn (Miranda[43]);thelinearspaceCk;(cid:21)((cid:10))consisting
loc
of those functions strictly contains the classic H(cid:127)older space Ck;(cid:21)((cid:10)) (Chapter 1),
and coincides with it under certain hypotheses of (cid:10) (see (4.66), p. 120).
For n (cid:21) 2 the study of the functions belonging to these spaces cannot be
separated from that of the boundary of (cid:10), since the properties of @(cid:10) interfere
with those of the functions de(cid:12)ned in (cid:10). Thus we will perform here an in-depth
examinationofthenotionofopensetofclassCk [Ck;(cid:21)],whichisimportantinthe
study of the functions, not only of the H(cid:127)older spaces Ck;(cid:21)((cid:10)) and their variable-
exponentversionCk;(cid:11)((cid:1))((cid:10))(see,e.g.,[6,7,18,27,28,49,11,14]),butalsoofthe
Sobolev spaces Wm;p((cid:10)) (see, e.g., [31, 42, 43, 1, 2, 35, 36, 37, 22]), as well as,
e.g., in the Sizing and Shape Optimization theory (SSO; see, e.g., [26]).
The properties that characterize an open set (cid:10) of class Ck [Ck;(cid:21)] can be
substantially reduced to two: one which concerns exclusively the points of @(cid:10),
and the other which involves also the points of (cid:10). In Chapter 3 we deal with the
opensetsthathaveonlythe(cid:12)rstproperty,andsoweintroducethenotionofopen
set with boundary of class Ck [Ck;(cid:21)], indicating for k (cid:21) 1 a further equivalent
formulation, which is sometimes preferable (see Proposition 3.2.2 on p. 89).
Chapter 4 is mainly devoted to the open sets of class Ck [Ck;(cid:21)], i.e., the
sets having both of the aforementioned properties; as a consequence of a result
established there (see Theorem 4.1.2 on p. 100), the notion can be expressed in a
formthate(cid:11)ectivelyhighlightswhatwejustobserved:anopenset(cid:10)isofclassCk
[Ck;(cid:21)] when it has a boundary of class Ck [Ck;(cid:21)] and coincides with the interior
of its closure.
Our approach, di(cid:11)erent from that one usually adopted (see, e.g., [1, 30, 9,
16, 24, 31, 36, 42, 45, 20]), has the advantage that some well-known results are
obtained under less restrictive hypotheses: for instance, the classic theorem for an
open set (cid:10) with a bounded boundary (see, e.g., [43, (53.IV) p. 311], where (cid:10) is
assumed bounded):
(cid:10) of class C0;1 =) (cid:10) has the cone property
will be established under the assumption that the boundary of (cid:10) is only of class
ix
x Introduction
C0;1(seeCorollary3.3.2 onp.96;note thatthe assertion is obtained throughthe
existenceofadmissibleconesfor(cid:10)).Thistheorem,evenwiththenewassumption,
does not admit a converse: at the end of Chapter 3 (see Proposition 3.3.3 on
p. 96) we give a very simple example of an open set having the cone property, but
not a boundary of class C0;1; the example also shows that an open set with the
conepropertydoesnotnecessarilysatisfyaconditionappearinginworkofAdams
and Fournier [2, 1.35, p. 13], which generalizes the notion of convex open set and
therefore here, being among the hypotheses of some theorems, is introduced as
subconvex open set (see Condition (S) on p. 9). Note that this condition, although
in a more restrictive form, was introduced some years before by Miranda [43, p.
313], who denoted it by L).
Moreover, in Chapter 4, we prove that for every open set of class C0;1 with
bounded boundary there exists a pair of real numbers, here called admissible for
(cid:10), which has an important property formulated on p. 108: they provide among
other things two criteria, respectively for the H(cid:127)older condition and the Lipschitz
condition.
Chapters 3 and 4 require, for the sake of completeness and clarity of expo-
sition, the background material given in Chapter 2, consisting of de(cid:12)nitions and
elementaryresultsonorthonormalmatrices,aswellasontheassociatedlinearop-
erators in Rn: we have thought it useful also to give the proofs for completeness,
but also because sometimes they are omitted or not detailed, being of undergrad-
uate level.
In order to establish a lemma on the cones in Rn (Lemma 2.5.10 on p. 68)
essentialforourpurposes,inChapter2wedwellextensivelyonthenotionofcone,
and we construct some linear operators in Rn that are of independent interest:
namely, a linear operator that performs a prescribed rotation around the vertex
(seeProposition2.5.2onp.58),orarounditsaxis(seeProposition2.5.3onp.59),
orthatreduces(orenlarges)theapertureofacone(seeProposition2.5.7onp.62).
Consequently, we thought it appropriate to formalize the notion of orthogonal
Cartesian axes of origin !, determined by an orthonormal basis in Rn. For an
openset(cid:10)whoseboundaryisofclassC1,wethencarryouttheconstructionofa
coordinate system whose origin is in a point of @(cid:10): the so-called tangent-normal
(to @(cid:10)) coordinate system. The way to proceed is natural, however, the detailed
exposition (see Theorem 3.2.1 on p. 82) is not so simple and it seems missing in
the literature (Miranda in [43, p. 314] assumes that the system of axes is the one
in question, and limits himself to asserting that with this hypothesis there is no
loss of generality).
ForcertainseminormsinthespaceCm;(cid:21)((cid:10)),with(cid:10)havingtheconeproperty,
loc
in the last chapter we prove two majorization formulas by Miranda [43, (54.XIII)
p.326,formulas(54.15),(54.16)]whosevaliditywaslimitedtothefunctionsinthe
space Cm;(cid:21)((cid:10)). We establish them for all the functions in Cm;(cid:21)((cid:10)) and combine
loc
them in a single formula (5.34), which also includes the case of (cid:10) open set in R:
this latter case is not explicitly considered in [43], nor is it excluded (as it should