Dissertationes mathematicae. 484: Weighted diffeomorphism groups of Banach spaces and weighted mapping groups (Warszawa, 2012). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаDissertationes mathematicae. 484: Weighted diffeomorphism groups of Banach spaces and weighted mapping groups / B.Walter; Institute of Mathematics, Polish Academy of Sciences. - Warszawa: Instytut matematyczny PAN, 2012. - 128 p. - Ref.: p.124-125. - Ind.: p.128. - ISSN 0012-3862
 

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Оглавление / Contents
 
1  Introduction ................................................. 6
2  Preliminaries and notation ................................... 8
   2.1  Notation ................................................ 8
   2.2  Differential calculus of maps between locally convex
        spaces .................................................. 8
   2.3  Frechet differentiability ............................... 9
3  Weighted function spaces .................................... 10
   3.1  Definition and examples ................................ 10
   3.2  Topological and uniform structure ...................... 12
        3.2.1  Reduction to lower order ........................ 12
        3.2.2  Projective limits and the topology of Cw(U,Y) ... 13
        3.2.3  A completeness criterion ........................ 14
   3.3  Composition on weighted functions and superposition
        operators .............................................. 17
        3.3.1  Composition with a multilinear map .............. 17
        3.3.2  Composition of weighted functions with bounded
               functions ....................................... 21
        3.3.3  Composition of weighted functions with an
               analytic map .................................... 26
   3.4  Weighted maps into locally convex spaces ............... 30
        3.4.1  Definition and topological structure ............ 30
        3.4.2  Weighted decreasing maps ........................ 33
        3.4.3  Composition and superposition ................... 35
4  Lie groups of weighted diffeomorphisms ...................... 41
   4.1  Weighted diffeomorphisms and endomorphisms ............. 41
        4.1.1  Composition of weighted endomorphisms in
               charts .......................................... 42
        4.1.2  Smooth monoids of weighted endomorphisms ........ 47
   4.2  Lie group structures on weighted diffeomorphisms ....... 48
        4.2.1  The Lie group structure of Diffw(X) ............. 48
        4.2.2  On decreasing weighted diffeomorphisms and
               dense subgroups ................................. 54
        4.2.3  On diffeomorphisms that are weighted
               endomorphisms ................................... 55
   4.3  Regularity ............................................. 57
        4.3.1  The tangent group and the regularity
               differential equation of Diffw(X) ............... 57
        4.3.2  Conclusion and calculation of one-parameter
               groups .......................................... 61
5  Integration of certain Lie algebras of vector fields ........ 62
   5.1  On the smoothness of the conjugation action on
        Diffw(X)0 .............................................. 63
        5.1.1  Contravariant composition on weighted
               functions ....................................... 64
   5.2  Conclusion and examples ................................ 67
6  Lie group structures on weighted mapping groups ............. 69
   6.1. Weighted maps into Banach Lie groups ................... 70
        6.1.1  Construction of the Lie group ................... 70
        6.1.2  Regularity ...................................... 74
        6.1.3  Semidirect products with weighted
               diffeomorphisms ................................. 77
   6.2  Weighted maps into locally convex Lie groups ........... 78
        6.2.1  Construction of the Lie group ................... 78
        6.2.2  A larger Lie group of weighted mappings ......... 81
A  Differential calculus ....................................... 95
   A.l  Differential calculus of maps between locally convex
        spaces ................................................. 95
        A.1.1  Curves and integrals ............................ 95
        A.1.2  Differentiable maps ............................. 98
   A.2  Frechet differentiability ............................. 104
   A.3  Relation between the differential calculi ............. 108
   A.4  Some facts concerning ordinary differential
        equations ............................................. 1ll
        A.4.1  Maximal solutions of ODEs ...................... 1ll
        A.4.2  Flows and dependence on parameters and
               initial values ................................. 113
B  Locally convex Lie groups .................................. 116
   B.l  Locally convex manifolds .............................. 116
   B.2. Lie groups ............................................ 117
        B.2.1  Generation of Lie groups ....................... 118
        B.2.2  Regularity ..................................... 118
        B.2.3  Group actions .................................. 119
C  Quasi-inversion in algebras ................................ 120
   C.l  Definition ............................................ 121
   C.2  Topological monoids and algebras with continuous
        quasi-inversion ....................................... 121

References .................................................... 124

Notation ...................................................... 126

Index ......................................................... 128


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