PSEUDO-DIFFERENTIAL OPERATORS AND RELATED TOPICS

By ROBERT BROWN

CONTENTS INTRODUCTION SETS, COLLECTIONS, FAMILIES RELATIONS, FUNCTIONS, EQUIVALENCES AND ORDERS DOMINOES TUMBLING AND TRANS?NITE INDUCTION AXIOM OF CHOICE: EQUIVALENT FORMULATIONS WELL-ORDERING PRINCIPLE REVISITED METRIC SPACES TOPOLOGICAL SPACES KURATOWSKI’S CLOSURE COMPLETE METRIC SPACES CONTINUITY AND HOMEOMORPHISMS COMPACT TOPOLOGICAL SPACES COMPACT HAUSDOR?SPACES SEQUENTIAL COMPACTNESS STONE–WEIERSTRASS THEOREM MANIFOLDS CONNECTEDNESS AND PATH-CONNECTEDNESS CO-INDUCTION AND QUOTIENT SPACES INDUCTION AND PRODUCT SPACES METRISABLE TOPOLOGIES TOPOLOGY VIA GENERALISED SEQUENCES VECTOR SPACES TENSOR PRODUCTS TOPOLOGICAL VECTOR SPACES LOCALLY CONVEX SPACES TOPOLOGICAL TENSOR PRODUCTS BANACH SPACES BANACH SPACE ADJOINT HILBERT SPACES TRACE CLASS, HILBERT–SCHMIDT, AND SCHATTEN CLASSES MEASURES AND OUTER MEASURES MEASURING SETS BOREL REGULARITY ON LEBESGUE MEASURE LEBESGUE NON-MEASURABLE SETS MEASURABLE FUNCTIONS WELL-BEHAVING FUNCTIONS SEQUENCES OF MEASURABLE FUNCTIONS APPROXIMATING MEASURABLE FUNCTIONS INTEGRATION INTEGRATING SIMPLE NON-NEGATIVE FUNCTIONS INTEGRATING NON-NEGATIVE FUNCTIONS INTEGRATION IN GENERAL INTEGRAL AS A FUNCTIONAL LEBESGUE SPACES LP(?) SIGNED MEASURES DERIVATIVES OF SIGNED MEASURES INTEGRATION AS FUNCTIONAL ON FUNCTION SPACES INTEGRATION AS FUNCTIONAL ON LP(?) INTEGRATION AS FUNCTIONAL ON C(X) PRODUCT MEASURE AND INTEGRAL ALGEBRAS TOPOLOGICAL ALGEBRAS BANACH ALGEBRAS COMMUTATIVE BANACH ALGEBRAS C?-ALGEBRAS APPENDIX: LIOUVILLE’S THEOREM CONTENTS PART II COMMUTATIVE SYMMETRIES BASIC PROPERTIES OF THE FOURIER TRANSFORM USEFUL INEQUALITIES TEMPERED DISTRIBUTIONS FOURIER TRANSFORM OF TEMPERED DISTRIBUTIONS OPERATIONS WITH DISTRIBUTIONS APPROXIMATING BY SMOOTH FUNCTIONS DISTRIBUTIONS LOCALISATION OF LP-SPACES AND DISTRIBUTIONS CONVOLUTION OF DISTRIBUTIONS SOBOLEV SPACES WEAK DERIVATIVES AND SOBOLEV SPACES SOME PROPERTIES OF SOBOLEV SPACES MOLLI?ERS APPROXIMATION OF SOBOLEV SPACE FUNCTIONS INTERPOLATION MOTIVATION AND DE?NITION AMPLITUDE REPRESENTATION OF PSEUDO-DIFFERENTIAL OPERATORS KERNEL REPRESENTATION OF PSEUDO-DI?ERENTIAL OPERATORS BOUNDEDNESS ON L2(RN) CALCULUS OF PSEUDO-DI?ERENTIAL OPERATORS COMPOSITION FORMULAE CHANGES OF VARIABLES PRINCIPAL SYMBOL AND CLASSICAL SYMBOLS CALCULUS PROOF OF L2-BOUNDEDNESS ASYMPTOTIC SUMS APPLICATIONS TO PARTIAL DI?ERENTIAL EQUATIONS FREEZING PRINCIPLE FOR PDES ELLIPTIC OPERATORS SOBOLEV SPACES REVISITED DISTRIBUTIONS AND FOURIER TRANSFORMS ON TN AND ZN SOBOLEV SPACES HS(TN) DISCRETE ANALYSIS TOOLKIT CALCULUS OF ?NITE DI?ERENCES DISCRETE TAYLOR EXPANSION AND POLYNOMIALS ON ZN SEVERAL DISCRETE INEQUALITIES LINKING DI?ERENCES TO DERIVATIVES PERIODIC TAYLOR EXPANSION APPENDIX: ON OPERATORS IN BANACH SPACES TOROIDAL SYMBOLS QUANTIZATION OF OPERATORS ON TN TOROIDAL AMPLITUDES PSEUDO-DI?ERENTIAL OPERATORS ON SOBOLEV SPACES KERNELS OF PERIODIC PSEUDO-DI?ERENTIAL OPERATORS ASYMPTOTIC SUMS AND AMPLITUDE OPERATORS EXTENSION OF TOROIDAL SYMBOLS PERIODISATION OF PSEUDO-DI?ERENTIAL OPERATORS SYMBOLIC CALCULUS OPERATORS ON L2(TN) AND SOBOLEV SPACES ELLIPTIC PSEUDO-DI?ERENTIAL OPERATORS ON TN SMOOTHNESS PROPERTIES AN APPLICATION TO PERIODIC INTEGRAL OPERATORS TOROIDAL WAVE FRONT SETS FOURIER SERIES OPERATORS BOUNDEDNESS OF FOURIER SERIES OPERATORS ON L2(TN) AN APPLICATION TO HYPERBOLIC EQUATIONS EUCLIDEAN COMMUTATOR CHARACTERISATION PSEUDO-DI?ERENTIAL OPERATORS ON MANIFOLDS COMMUTATOR CHARACTERISATION ON CLOSED MANIFOLDS TOROIDAL COMMUTATOR CHARACTERISATION PART III REPRESENTATION THEORY OF COMPACT GROUPS INTRODUCTION GROUPS WITHOUT TOPOLOGY GROUP ACTIONS AND REPRESENTATIONS TOPOLOGICAL GROUPS REPRESENTATIONS OF TOPOLOGICAL GROUPS COMPACT GROUPS HAAR MEASURE AND INTEGRAL INTEGRATION ON QUOTIENT SPACES PETER–WEYL DECOMPOSITION OF REPRESENTATIONS FOURIER SERIES AND TRIGONOMETRIC POLYNOMIALS CONVOLUTIONS CHARACTERS INDUCED REPRESENTATIONS EXPONENTIAL MAP NO SMALL SUBGROUPS FOR LIE, PLEASE LIE GROUPS AND LIE ALGEBRAS UNIVERSAL ENVELOPING ALGEBRA CASIMIR ELEMENT AND LAPLACE OPERATOR COMMUTATIVE C?-ALGEBRAS HOPF ALGEBRAS FOURIER SERIES ON COMPACT LIE GROUPS FUNCTION SPACES ON THE UNITARY DUAL 1031 SPACES ON THE GROUP G 1032 SPACES ON THE DUAL G 1033 SPACES LP( G) SYMBOLS OF OPERATORS 1041 FULL SYMBOLS 1042 CONJUGATION PROPERTIES OF SYMBOLS BOUNDEDNESS OF OPERATORS ON L2(G) TAYLOR EXPANSION ON LIE GROUPS 1071 DI?ERENCE OPERATORS 1072 COMMUTATOR CHARACTERISATION 1073 CALCULUS 1074 LEIBNIZ FORMULA BOUNDEDNESS ON SOBOLEV SPACES HS(G) SYMBOL CLASSES ON COMPACT LIE GROUPS 1091 SOME PROPERTIES OF SYMBOLS OF ?M(G) CONTENTS 1092 SYMBOL CLASSES ?M(G) 1010 FULL SYMBOLS ON COMPACT MANIFOLDS 1011 OPERATOR-VALUED SYMBOLS 10111EXAMPLE ON THE TORUS TN 1012 APPENDIX: INTEGRAL KERNELS PRELIMINARIES: GROUPS U(1), SO(2), AND SO(3) 1111 EULER ANGLES ON SO(3) 1112 PARTIAL DERIVATIVES ON SO(3) 1113 INVARIANT INTEGRATION ON SO(3) GENERAL PROPERTIES OF SU(2) EULER ANGLE PARAMETRISATION OF SU(2) QUATERNIONS 1141 QUATERNIONS AND SU(2) 1142 QUATERNIONS AND SO(3) 1143 INVARIANT INTEGRATION ON SU(2) 1144 SYMPLECTIC GROUPS LIE ALGEBRA AND DI?ERENTIAL OPERATORS ON SU(2) IRREDUCIBLE UNITARY REPRESENTATIONS OF SU(2) 1161 REPRESENTATIONS OF SO(3) MATRIX ELEMENTS OF REPRESENTATIONS OF SU(2) MULTIPLICATION FORMULAE FOR REPRESENTATIONS OF SU(2) LAPLACIAN AND DERIVATIVES OF REPRESENTATIONS ON SU(2) 1110 FOURIER SERIES ON SU(2) AND ON SO(3) SYMBOLS OF OPERATORS ON SU(2) SYMBOLS OF ?+, ??, ?0 AND LAPLACIAN L DI?ERENCE OPERATORS FOR SYMBOLS 1231 DI?ERENCE OPERATORS ON SU(2) 1232 DI?ERENCES FOR SYMBOLS OF ?+, ??, ?0 AND LAPLACIAN L 1233 DI?ERENCES FOR A??0 SYMBOL CLASSES ON SU(2) PSEUDO-DI?ERENTIAL OPERATORS ON S3 APPENDIX: IN?NITE MATRICES ANALYSIS ON CLOSED MANIFOLDS ANALYSIS ON COMPACT HOMOGENEOUS SPACES ANALYSIS ON K\G, K A TORUS LIFTING OF OPERATORS