16 APPUNTI LECTURENOTES AlessandraLunardi DipartimentodiScienzeMatematiche, FisicheeInformatiche Universita`diParma ParcoAreadelleScienze,53/A 43124Parma,Italia InterpolationTheory Alessandra Lunardi Interpolation Theory (cid:2)c 2018ScuolaNormaleSuperiorePisa Terzaedizione Secondaedizione:2009 Primaedizione1999 isbn 978-88-7642-638-4 (eBook) DOI 10.1007/978-88-7642-638-4 issn2532-991X(print) issn2611-2248(online) Contents Foreword vii Introduction ix Nomenclature xiii 1 Realinterpolation 1 1.1 The K-method . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Examples . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . 18 1.2 Thetracemethod . . . . . . . . . . . . . . . . . . . . . 20 1.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . 26 1.3 Dualspacesofrealinterpolationspaces . . . . . . . . . 28 1.4 Intermediatespacesandreiteration . . . . . . . . . . . . 30 1.4.1 Examples . . . . . . . . . . . . . . . . . . . . . 37 1.4.2 Applications: thetheoremsofMarcinkiewicz andStampacchia;regularityinellipticPDE’s . . 39 1.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . 42 2 Complexinterpolation 45 2.1 Deﬁnitionsandproperties . . . . . . . . . . . . . . . . . 47 2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . 55 2.1.2 ThetheoremsofHausdorff-YoungandStein . . 58 2.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . 61 3 Interpolationanddomainsofoperators 63 3.1 Operatorswithraysofminimalgrowth . . . . . . . . . . 63 3.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . 69 3.2 Twoormoreoperators . . . . . . . . . . . . . . . . . . 70 3.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . 75 vi AlessandraLunardi 3.2.2 Thesumoftwocommutingoperators . . . . . . 75 3.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . 83 4 Powersofpositiveoperators 85 4.1 Deﬁnitionsandgeneralproperties . . . . . . . . . . . . 85 4.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . 94 4.1.2 Powersofnonnegativeoperators . . . . . . . . . 95 4.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . 98 4.2 Operatorswithboundedimaginarypowers . . . . . . . . 98 4.2.1 Thesumoftwocommutingoperators withboundedimaginarypowers . . . . . . . . . 105 4.2.2 Maximal Lp regularityforequations inUMDspaces . . . . . . . . . . . . . . . . . . 110 4.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . 114 4.3 M-accretiveoperatorsinHilbertspaces . . . . . . . . . 115 4.3.1 Self-adjointoperatorsinHilbertspaces . . . . . 122 5 Interpolationandsemigroups 129 5.1 RealinterpolationbetweenBanachspacesanddomains ofgenerators . . . . . . . . . . . . . . . . . . . . . . . 135 5.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . 140 5.2 Examplesandapplications . . . . . . . . . . . . . . . . 141 5.3 RegularityinellipticPDE’sbyinterpolation . . . . . . . 148 5.4 Regularityinevolutionequationsbyinterpolation . . . . 156 6 Analyticsemigroupsandinterpolation 161 6.1 Characterizationofrealinterpolationspaces . . . . . . . 162 6.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . 165 6.2 Generationofanalyticsemigroupsbyinterpolation . . . 165 6.3 Regularityinabstractparabolicequations . . . . . . . . 167 6.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . 176 6.4 Anapplication: space–timeHo¨lderregularity inparabolicPDE’s . . . . . . . . . . . . . . . . . . . . 177 A TheBochnerintegral 181 A.1 Integralsovermeasurablerealsets . . . . . . . . . . . . 181 A.2 Lp andSobolevspaces . . . . . . . . . . . . . . . . . . 183 A.3 Weighted Lp spaces . . . . . . . . . . . . . . . . . . . . 185 References 187 Index 195 Foreword This book is the second edition of my 1999 lecture notes of the courses on interpolation theory that I delivered at the Scuola Normale in 1998 and 1999. Later I held other lectureson interpolation theory in different universities,andtheoriginallecturenoteswerepolishedandenriched. In the mathematical literature there are many good books on the sub- ject, but none of them is very elementary, and in many cases the basic principlesarehiddenbelowgreatgenerality. IntheselecturesItriedtoillustratetheprinciplesofinterpolationthe- oryaimingatsimpliﬁcationratherthangenerality. Ireducedtheabstract theoryasfaraspossible,andgavemanyexamplesandapplications,espe- ciallytooperatortheoryandtoregularityinpartialdifferentialequations. Moreoverthetreatmentisself-contained,theonlyprerequisitebeingthe knowledgeofbasicfunctionalanalysis. I would like to thank colleagues and friends who followed parts of mylectures,readolderversionsofthesenotesandgavesuggestionsand help for improvements: P. Celada, S. Cerrai, Ph. Cle´ment, G. Da Prato, C.Villani,J. Zabczyk,A.Zaccagnini. Foreword to the third edition. The third edition is just a revised ver- sionofthesecondone: someproofsweremodiﬁedoradded, newrefer- ences were given, and a number of mistakes and misprints were correc- ted. Many of them were detected by colleagues who used this book for theirlectures, andtoldme. Inparticular, IwouldliketothankH.Abels, R.FarwigandJ.Voigtfortheirpreciousremarks. Introduction Let X, Y be two real or complex Banach spaces. By X = Y we mean that X and Y have the same elements and equivalent norms. By Y ⊂ X we mean that Y is continuously embedded in X. As usual, we denote by L(X,Y) (in short, L(X) if X = Y) the space of all linear bounded operatorsfrom X toY,endowedbythenorm(cid:4)T(cid:4)L(X,Y) := sup{(cid:4)Tx(cid:4)Y : (cid:4)x(cid:4) = 1}. X The couple of Banach spaces (X,Y) is said to be an interpolation couple if both X and Y are continuously embedded in a Hausdorff to- pological vector space V. In this case the intersection X ∩Y is a linear subspaceofV,anditisaBanachspaceunderthenorm (cid:4)v(cid:4)X∩Y := max{(cid:4)v(cid:4)X, (cid:4)v(cid:4)Y}. Alsothesum X +Y := {x + y : x ∈ X, y ∈ Y}isalinearsubspaceof V. Itisendowedwiththenorm (cid:4)v(cid:4)X+Y := inf (cid:4)x(cid:4)X +(cid:4)y(cid:4)Y. x∈X,y∈Y,x+y=v As easily seen, X + Y is isometric to the quotient space (X × Y)/D, where D = {(x,−x) : x ∈ X ∩Y}. SinceV isaHausdorffspace, then D is closed, and X +Y is a Banach space. Moreover, (cid:4)x(cid:4)X ≥ (cid:4)x(cid:4)X+Y and (cid:4)y(cid:4)Y ≥ (cid:4)y(cid:4)X+Y for all x ∈ X, y ∈ Y, so that both X and Y are continuouslyembeddedin X +Y. ThespaceV isusedonlytoguaranteethat X+Y isaBanachspace. It willdisappearfromthegeneraltheory. If(X,Y)isaninterpolationcouple,anintermediatespaceisanyBanach space E suchthat X ∩Y ⊂ E ⊂ X +Y. Aninterpolationspacebetween X andY isanyintermediatespacesuch that for every T ∈ L(X) ∩ L(Y) (that is, for every linear operator T : x AlessandraLunardi X + Y (cid:8)→ X + Y whose restriction to X belongs to L(X) and whose restriction to Y belongs to L(Y)), the restriction of T to E belongs to L(E). This implies that there is a constant independent of T such that (cid:4)T(cid:4)L(E) ≤ Cmax{(cid:4)T(cid:4)L(X),(cid:4)T(cid:4)L(Y)), as next lemma (taken from [63]) shows. Lemma0.1. Let (X,Y) be an interpolation couple, and let E be an in- terpolation space between X and Y. Then there is C > 0 such that for each linear operator T : X + Y (cid:8)→ X + Y such that T|X ∈ L(X), T|Y ∈ L(Y)wehave (cid:4)T(cid:4)L(E) ≤Cmax{(cid:4)T(cid:4)L(X),(cid:4)T(cid:4)L(Y)}. Proof. The space B of the linear operators T : X +Y (cid:8)→ X +Y such that T|X ∈ L(X), T|Y ∈ L(Y) is easily seen to be a Banach space with the norm (cid:4)T(cid:4) := max{(cid:4)T(cid:4)L(X),(cid:4)T(cid:4)L(Y)}. Indeed, if {Tn} is a Cauchy sequence,Tn|X convergestosomeTX inL(X),Tn|Y convergestosomeTY inL(Y),thelimitoperatorT : X+Y (cid:8)→ X+Y,T(x+y) = T|Xx+T|Yy iswelldeﬁned(andobviouslylinear)in X +Y,andT → T inB. n Deﬁne a linear operator (cid:2) : B (cid:8)→ L(E) by (cid:2)(T) := T|E. We shall provethatthegraphof(cid:2)isclosed,sothat(cid:2)isboundedandthestatement follows. Let T → T inB besuchthat(cid:2)(T ) → S inL(E)asn → ∞. Then n n foreachw ∈ X +Y andforeachdecompositionw = x +y,withx ∈ X and y ∈ Y,wehave (cid:4)Tnw−Tw(cid:4)X+Y≤(cid:4)(Tn−T)x(cid:4)X+(cid:4)(Tn−T)y(cid:4)Y≤(cid:4)Tn−T(cid:4)((cid:4)x(cid:4)X+(cid:4)y(cid:4)Y) andtakingtheinﬁmumoverallthedecompositionsofw weget (cid:4)Tnw−Tw(cid:4)X+Y ≤ (cid:4)Tn −T(cid:4)(cid:4)w(cid:4)X+Y sothatlimn→∞Tnw = Tw in X +Y. Ontheotherhand,ifw ∈ E then limn→∞Tnw = Sw in E andhencein X +Y. Therefore, Tw = Sw for eachw ∈ E,i.e. S = (cid:2)(T). The general interpolation theory is not devoted to characterize all the interpolation spaces between X and Y but rather to construct suitable families of interpolation spaces and to study their properties. The most knownandusefulfamiliesofinterpolationspacesaretherealinterpola- tionspaceswhichwillbetreatedinChapter1,andthecomplexinterpol- ationspaceswhichwillbetreatedinChapter2.

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16 APPUNTI LECTURENOTES AlessandraLunardi DipartimentodiScienzeMatematiche, FisicheeInformatiche Universita`diParma ParcoAreadelleScienze,53/A 43124Parma,Italia InterpolationTheory Alessandra Lunardi Interpolation Theory (cid:2)c 2018ScuolaNormaleSuperiorePisa Terzaedizione Secondaedizione:2009 Primaedizione1999 isbn 978-88-7642-638-4 (eBook) DOI 10.1007/978-88-7642-638-4 issn2532-991X(print) issn2611-2248(online) Contents Foreword vii Introduction ix Nomenclature xiii 1 Realinterpolation 1 1.1 The K-method . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Examples . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . 18 1.2 Thetracemethod . . . . . . . . . . . . . . . . . . . . . 20 1.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . 26 1.3 Dualspacesofrealinterpolationspaces . . . . . . . . . 28 1.4 Intermediatespacesandreiteration . . . . . . . . . . . . 30 1.4.1 Examples . . . . . . . . . . . . . . . . . . . . . 37 1.4.2 Applications: thetheoremsofMarcinkiewicz andStampacchia;regularityinellipticPDE’s . . 39 1.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . 42 2 Complexinterpolation 45 2.1 Deﬁnitionsandproperties . . . . . . . . . . . . . . . . . 47 2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . 55 2.1.2 ThetheoremsofHausdorff-YoungandStein . . 58 2.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . 61 3 Interpolationanddomainsofoperators 63 3.1 Operatorswithraysofminimalgrowth . . . . . . . . . . 63 3.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . 69 3.2 Twoormoreoperators . . . . . . . . . . . . . . . . . . 70 3.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . 75 vi AlessandraLunardi 3.2.2 Thesumoftwocommutingoperators . . . . . . 75 3.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . 83 4 Powersofpositiveoperators 85 4.1 Deﬁnitionsandgeneralproperties . . . . . . . . . . . . 85 4.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . 94 4.1.2 Powersofnonnegativeoperators . . . . . . . . . 95 4.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . 98 4.2 Operatorswithboundedimaginarypowers . . . . . . . . 98 4.2.1 Thesumoftwocommutingoperators withboundedimaginarypowers . . . . . . . . . 105 4.2.2 Maximal Lp regularityforequations inUMDspaces . . . . . . . . . . . . . . . . . . 110 4.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . 114 4.3 M-accretiveoperatorsinHilbertspaces . . . . . . . . . 115 4.3.1 Self-adjointoperatorsinHilbertspaces . . . . . 122 5 Interpolationandsemigroups 129 5.1 RealinterpolationbetweenBanachspacesanddomains ofgenerators . . . . . . . . . . . . . . . . . . . . . . . 135 5.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . 140 5.2 Examplesandapplications . . . . . . . . . . . . . . . . 141 5.3 RegularityinellipticPDE’sbyinterpolation . . . . . . . 148 5.4 Regularityinevolutionequationsbyinterpolation . . . . 156 6 Analyticsemigroupsandinterpolation 161 6.1 Characterizationofrealinterpolationspaces . . . . . . . 162 6.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . 165 6.2 Generationofanalyticsemigroupsbyinterpolation . . . 165 6.3 Regularityinabstractparabolicequations . . . . . . . . 167 6.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . 176 6.4 Anapplication: space–timeHo¨lderregularity inparabolicPDE’s . . . . . . . . . . . . . . . . . . . . 177 A TheBochnerintegral 181 A.1 Integralsovermeasurablerealsets . . . . . . . . . . . . 181 A.2 Lp andSobolevspaces . . . . . . . . . . . . . . . . . . 183 A.3 Weighted Lp spaces . . . . . . . . . . . . . . . . . . . . 185 References 187 Index 195 Foreword This book is the second edition of my 1999 lecture notes of the courses on interpolation theory that I delivered at the Scuola Normale in 1998 and 1999. Later I held other lectureson interpolation theory in different universities,andtheoriginallecturenoteswerepolishedandenriched. In the mathematical literature there are many good books on the sub- ject, but none of them is very elementary, and in many cases the basic principlesarehiddenbelowgreatgenerality. IntheselecturesItriedtoillustratetheprinciplesofinterpolationthe- oryaimingatsimpliﬁcationratherthangenerality. Ireducedtheabstract theoryasfaraspossible,andgavemanyexamplesandapplications,espe- ciallytooperatortheoryandtoregularityinpartialdifferentialequations. Moreoverthetreatmentisself-contained,theonlyprerequisitebeingthe knowledgeofbasicfunctionalanalysis. I would like to thank colleagues and friends who followed parts of mylectures,readolderversionsofthesenotesandgavesuggestionsand help for improvements: P. Celada, S. Cerrai, Ph. Cle´ment, G. Da Prato, C.Villani,J. Zabczyk,A.Zaccagnini. Foreword to the third edition. The third edition is just a revised ver- sionofthesecondone: someproofsweremodiﬁedoradded, newrefer- ences were given, and a number of mistakes and misprints were correc- ted. Many of them were detected by colleagues who used this book for theirlectures, andtoldme. Inparticular, IwouldliketothankH.Abels, R.FarwigandJ.Voigtfortheirpreciousremarks. Introduction Let X, Y be two real or complex Banach spaces. By X = Y we mean that X and Y have the same elements and equivalent norms. By Y ⊂ X we mean that Y is continuously embedded in X. As usual, we denote by L(X,Y) (in short, L(X) if X = Y) the space of all linear bounded operatorsfrom X toY,endowedbythenorm(cid:4)T(cid:4)L(X,Y) := sup{(cid:4)Tx(cid:4)Y : (cid:4)x(cid:4) = 1}. X The couple of Banach spaces (X,Y) is said to be an interpolation couple if both X and Y are continuously embedded in a Hausdorff to- pological vector space V. In this case the intersection X ∩Y is a linear subspaceofV,anditisaBanachspaceunderthenorm (cid:4)v(cid:4)X∩Y := max{(cid:4)v(cid:4)X, (cid:4)v(cid:4)Y}. Alsothesum X +Y := {x + y : x ∈ X, y ∈ Y}isalinearsubspaceof V. Itisendowedwiththenorm (cid:4)v(cid:4)X+Y := inf (cid:4)x(cid:4)X +(cid:4)y(cid:4)Y. x∈X,y∈Y,x+y=v As easily seen, X + Y is isometric to the quotient space (X × Y)/D, where D = {(x,−x) : x ∈ X ∩Y}. SinceV isaHausdorffspace, then D is closed, and X +Y is a Banach space. Moreover, (cid:4)x(cid:4)X ≥ (cid:4)x(cid:4)X+Y and (cid:4)y(cid:4)Y ≥ (cid:4)y(cid:4)X+Y for all x ∈ X, y ∈ Y, so that both X and Y are continuouslyembeddedin X +Y. ThespaceV isusedonlytoguaranteethat X+Y isaBanachspace. It willdisappearfromthegeneraltheory. If(X,Y)isaninterpolationcouple,anintermediatespaceisanyBanach space E suchthat X ∩Y ⊂ E ⊂ X +Y. Aninterpolationspacebetween X andY isanyintermediatespacesuch that for every T ∈ L(X) ∩ L(Y) (that is, for every linear operator T : x AlessandraLunardi X + Y (cid:8)→ X + Y whose restriction to X belongs to L(X) and whose restriction to Y belongs to L(Y)), the restriction of T to E belongs to L(E). This implies that there is a constant independent of T such that (cid:4)T(cid:4)L(E) ≤ Cmax{(cid:4)T(cid:4)L(X),(cid:4)T(cid:4)L(Y)), as next lemma (taken from [63]) shows. Lemma0.1. Let (X,Y) be an interpolation couple, and let E be an in- terpolation space between X and Y. Then there is C > 0 such that for each linear operator T : X + Y (cid:8)→ X + Y such that T|X ∈ L(X), T|Y ∈ L(Y)wehave (cid:4)T(cid:4)L(E) ≤Cmax{(cid:4)T(cid:4)L(X),(cid:4)T(cid:4)L(Y)}. Proof. The space B of the linear operators T : X +Y (cid:8)→ X +Y such that T|X ∈ L(X), T|Y ∈ L(Y) is easily seen to be a Banach space with the norm (cid:4)T(cid:4) := max{(cid:4)T(cid:4)L(X),(cid:4)T(cid:4)L(Y)}. Indeed, if {Tn} is a Cauchy sequence,Tn|X convergestosomeTX inL(X),Tn|Y convergestosomeTY inL(Y),thelimitoperatorT : X+Y (cid:8)→ X+Y,T(x+y) = T|Xx+T|Yy iswelldeﬁned(andobviouslylinear)in X +Y,andT → T inB. n Deﬁne a linear operator (cid:2) : B (cid:8)→ L(E) by (cid:2)(T) := T|E. We shall provethatthegraphof(cid:2)isclosed,sothat(cid:2)isboundedandthestatement follows. Let T → T inB besuchthat(cid:2)(T ) → S inL(E)asn → ∞. Then n n foreachw ∈ X +Y andforeachdecompositionw = x +y,withx ∈ X and y ∈ Y,wehave (cid:4)Tnw−Tw(cid:4)X+Y≤(cid:4)(Tn−T)x(cid:4)X+(cid:4)(Tn−T)y(cid:4)Y≤(cid:4)Tn−T(cid:4)((cid:4)x(cid:4)X+(cid:4)y(cid:4)Y) andtakingtheinﬁmumoverallthedecompositionsofw weget (cid:4)Tnw−Tw(cid:4)X+Y ≤ (cid:4)Tn −T(cid:4)(cid:4)w(cid:4)X+Y sothatlimn→∞Tnw = Tw in X +Y. Ontheotherhand,ifw ∈ E then limn→∞Tnw = Sw in E andhencein X +Y. Therefore, Tw = Sw for eachw ∈ E,i.e. S = (cid:2)(T). The general interpolation theory is not devoted to characterize all the interpolation spaces between X and Y but rather to construct suitable families of interpolation spaces and to study their properties. The most knownandusefulfamiliesofinterpolationspacesaretherealinterpola- tionspaceswhichwillbetreatedinChapter1,andthecomplexinterpol- ationspaceswhichwillbetreatedinChapter2.

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