An Introductory Course in Lebesgue Spaces

463 Pages·2016·4.069 MB·English

by  Rene Erlin Castillo, Humberto Rafeiro (auth.)

CMS Books in Mathematics
Canadian
Mathematical Society
René Erlín Castillo
Société mathématique
Humberto Rafeiro du Canada
An Introductory
Course in
Lebesgue Spaces
CanadianMathematicalSociety
Socie´te´ mathe´matiqueduCanada
Editors-in-Chief
Re´dacteurs-en-chef
K.Dilcher
K.Taylor
AdvisoryBoard
Comite´ consultatif
M.Barlow
H.Bauschke
L.Edelstein-Keshet
N.Kamran
M.Kotchetov
Moreinformationaboutthisseriesathttp://www.springer.com/series/4318

Rene´ Erl´ın Castillo • Humberto Rafeiro
An Introductory Course
in Lebesgue Spaces
123
Rene´Erl´ınCastillo HumbertoRafeiro
UniversidadNacionaldeColombia PontificiaUniversidadJaveriana
Bogota´,Colombia Bogota´,Colombia
ISSN1613-5237 ISSN2197-4152 (electronic)
CMSBooksinMathematics
ISBN978-3-319-30032-0 ISBN978-3-319-30034-4 (eBook)
DOI10.1007/978-3-319-30034-4
LibraryofCongressControlNumber:2016935410
MathematicsSubjectClassification(2010):46E30,26A51,42B20,42B25
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AlamemoriadeJulietamimama´-abuela
AmiesposaHilciadelCarmen
Amishijos:Rene´ Jose´
ManuelAlejandro
IreneGabrielay
RenzoRafael
R.E.C.
a` Danielapeloseuamorepacieˆncia
H.R.

Preface
This book is not a treatise on Lebesgue spaces, since this would not be a feasible
work due to the extension of their usage, e.g., in physics, probability, statistics,
economy,engineering,amongothers.Theobjectiveismorerealistic,beingthein-
troductionofthereadertothestudyofdifferentvariantsofLebesguespacesandthe
commontechniquesusedinthisarea.SincetheLebesguespacesmeasureintegra-
bilityofafunction,theycanbeseenasthefatherofallintegrablefunctionspaces
wheremorefinepropertiesoffunctionsaresought.
WecanfindmanybookswherethesubjectofLebesguespacesistouchedupon,
for example, books dealing with measure theory and integration. In the literature,
we can also find some books dealing with Sobolev spaces and they dedicate, in
general, not more than one chapter to Lebesgue spaces. A book dedicated solely
to Lebesgue spaces is unknown to the authors. With this in mind, we decided to
writeabookdevotedexclusivelytoLebesguespacesandtheirdirectderivedspaces,
viz.,Marcinkiewiczspaces,Lorentzspaces,andthemorerecentvariableexponent
LebesguespacesandgrandLebesguespacesandalsotobasicharmonicanalysisin
those spaces. We think this will be a welcome to any serious student of analysis,
since it will give access to information that otherwise is spread among different
booksandarticles,aswellasmorethantwohundredproblems.
Forexample,oneoftheattractiveness ofLorentzandweakLebesgue spacesis
that the subject is sufficiently concrete and yet the spaces have fine structure and
importanceinapplications.Moreover,theareaisquiteaccessibleforyoungpeople,
leading them to gain sophistication in mathematical analysis in a relatively short
timeduringtheirgraduatestudies.Thesefeatures,amongothers,makethesubject
particularlyinteresting.
Wethinkitisappropriatetocommentonthechoiceofthewritingstyleandsome
peculiarities.Inthefirstpartdealingwithfunctionspaces,wetriedtobeasthorough
aspossibleintheproofs,althoughthiscouldsoundprolixforsomereaders.Another
aspect is the inclusion of proofs of classical results that deviate from the standard
ones,e.g.,intheproofoftheMinkowskiinequality,weusedtheclassicalapproach
vii
viii Preface
via Ho¨lder’s inequality for the Lebesgue sequence space and the less-known di-
rectapproachfortheLebesguespaces.Wealsodecidedtoincludeachapterbriefly
touchingupontheso-callednonstandardLebesguespaces,namely,onvariableex-
ponentLebesguespacesandongrandLebesguespacessincetheseareareaswhere
intenseresearchisbeingmadenowadays.Thetopicofvariableexponentspacesbe-
came very fashionable in recent years, not only due to mathematical curiosity, but
alsotothewidevarietyoftheirapplications,e.g.,inthemodelingofelectrorheolog-
icalfluidsaswellasthermorheologicalfluids,inthestudyofimageprocessing,and
indifferentialequationswithnonstandardgrowth.GrandLebesguespacesattracted
the attention of many researchers and turned out to be the right spaces in which
somenonlinearequationsinthetheoryofPDEshavetobeconsidered,amongother
applications.
Thistextisaddressedtoanyonethatknowsmeasuretheoryandintegration,func-
tionalanalysis,andrudimentsofcomplexanalysis.
Part of the content of this book has been tested with the students from Univer-
sidadNacionaldeColombiaandalsofromthePontificiaUniversidadJaverianain
theclassesofadvancedtopicsinanalysisandalsoinmeasuretheoryandintegration.
Since many of the results were collected in personal notebooks throughout the
years,aconsiderablenumberofexactreferenceswerelost.Wewanttoemphasize
that the content is NOT original of the authors, except maybe the rearrangement
of the topics and some (hopefully a small number) mistakes. If the reader finds
misprintsanderrors,pleaseletusknow.
H.R. was partially supported by the research project “Study of non-standard
Banach spaces”, ID-PPTA: 6326 in the Faculty of Sciences of the Pontificia Uni-
versidadJaveriana,Bogota´,Colombia.
Bogota´,Colombia Rene´ Erl´ınCastillo
Bogota´,Colombia HumbertoRafeiro
October2015
Contents
Preface............................................................ vii
1 ConvexFunctionsandInequalities ............................... 1
1.1 ConvexFunctions .......................................... 1
1.2 YoungInequality........................................... 7
1.3 Problems ................................................. 14
1.4 NotesandBibliographicReferences........................... 19
PartI FunctionSpaces
2 LebesgueSequenceSpaces ...................................... 23
2.1 Ho¨lderandMinkowskiInequalities............................ 23
2.2 LebesgueSequenceSpaces .................................. 26
2.3 SpaceofBoundedSequences................................. 31
2.4 HardyandHilbertInequalities................................ 35
2.5 Problems ................................................. 40
2.6 NotesandBibliographicReferences........................... 42
3 LebesgueSpaces ............................................... 43
3.1 EssentiallyBoundedFunctions ............................... 43
3.2 LebesgueSpaceswith p≥1 ................................. 45
3.3 Approximations............................................ 63
3.4 Duality ................................................... 70
3.5 Reflexivity ................................................ 88
3.6 WeakConvergence ......................................... 90
3.7 ContinuityoftheTranslation .................................101
3.8 WeightedLebesgueSpaces ..................................103
3.9 UniformConvexity .........................................108
3.10 Isometries.................................................114
3.11 LebesgueSpaceswith0<p<1..............................120
ix