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Preface
ThetaskofthisbookistopresentthetheoryofthescalesofBanach
spacesandtheroletheyplayinthemoderntheoryofPartialDifferential
Equations.Somepartsofthetheoryofinterpolationareanalysedhere
too.Thebookgatherstheresultsofpreviousinvestigationsonthissubject,
completedwiththenewones.
Thepresentstudyisdirectedtothemathematicsstudentsfinishingal-
readytheiruniversitycareer,mathematiciansandotherpeopleinterestedin
mathematicalscience.Tounderstandit,thebasicknowledgefromthefields
suchasacourseonfunctionalanalysiscontainingthebasicsofSobolev
spacesandintegralcalculusinBanachspacesarerequired.Everyreader
shouldalsobefamiliarwiththetheoryofdistributionsandtheFourier
transform,buttheelementarytheoremsrelatedtobothofthemcanbe
foundinAppendixA.
Thebookisdividedintothreeparts.Thefirstoneintroducesthereader
intothetheoryoftheinterpolationspaces,givesitsbriefdescriptionand
presentsthebasicpropertiesofinterpolationspaces.
AstheprecursorsofthetheoryofinterpolationwecanconsiderM.Riesz
andO.Thorin,whointhethirtiesprovedthetheoremoftheinterpolationof
thespacesLp(Ω).In1939thegeneralizationoftheseresultswaspublished
byJ.Marcinkiewicz.AftertheSecondWorldWarthetheorywasinves-
tigatedbyi.a.A.Zygmund,A.P.Calderón,E.GagliardoandJ.L.Lions
alongwithJ.Petree,whoperceivedtheinterpolationspacesastracesfor
variantsofSobolevspaces.Themorecompletedescriptionofthetheory
ofinterpolationspacescanbefoundi.a.inthemonographsofH.Triebel,
L.TartarandA.Lunardi.
Theaimofthesecond,mainpartofthebook,istopresentthecon-
structionofthescaleoftheBanachspaces,generatedbysuchanoperator
A:D(A)⊆X→Xofthetype(ωjM)that0∈ρ(A).Inthispartwe
